1\. A. AKCeHOBHQ
H. H. PaKHHa
K. C. {J)apHHO


C!)M3MKa


'"
B CpeAHeM WKOne


TeopHSI. 3aAaHHSI. TeCTbl


y Qe6HOe noco6He AJISI yqpem.l{eHHH.
06eCneQHBalO
HX nOJIyqeHHe o6
ero Cpe.n;HerO 06pa30BaHHSI


I10.n; pe.n;aKIJ;HeH K. C. cl>apHHo


J!.onYl4 eHO
M UHUC mepc m80.M. 06pa308aHUR
Pecny6RuKu BeRapYCb


MHHCK
"A.AYKaItbISI i BbIxaBaHHe"
2004




YAK 53(075.3}
BBK 22.3S1 721
A42
P e  e H 3 e H T bI: Ka(}>e,lJ;pa MeTO,lJ;HKH npenO,lJ;aBaHH.H (}>H3HKH
BeJlOPYCCKoro rOCY,lJ;apCTBeHHoro ne,lJ;arOrHqeCKOrO YHHBepCHTeTa
HMeHH MaKCHMa TaHKa (A. A. JIYije6UIl, KaHHlIaT ne,lJ;arOrHqeCKHX
HaYK, ,AOn;eHT, 3aBe,n;YlOID;Hii Ka(}>e,Apoii); Ka(}>e,Apa 06ID;eii <pH3HKH
BeJlOPyccKoro rocY,AapCTBeHHoro YHHBepcHTeTa (JI. A. H CalleliK06a,
KaH,lJ;H,AaT <pH3HKo-MaTeMaTHqeCKHX HayK, ,lJ;On;eHT; B. H. Aliijy.n.e6UIl,
KaH,AH,lJ;aT <pH3HKo-MaTeMaTHqeCKHX HayK, ytIHTeJlb <pH3HKH rHMHa-
3HH N2 5 r. MHHcKa).
AKceHOBH'I JI. A.
A42 CPH3HK8 B cpe,AHeii IIIKOJIe: TeOpH.H. 3a.u;aHH.H. TecTbI: Yqe6.
noc06He ,AJI.H yqpem,lJ;eHHii, 06eCneqHBalOID;Hx nOJIyqeHHe 06ID;.
cpe,lJ;. 06p830B8HH.H / JI. A. AKceHOBHq, H. H. PaKHHa, K. C. CPa-
pHHO; IIo,A pe,A. K. C. CPapHHo. - MH.: MYKaD;bI.H i BbIxaBaH-
He, 2004. - 720 c.: HJI.
ISBN 985-471-123-4.
Coepm8HHe yqe6Horo noc06:H.H cooTBeTcTByeT nporpaMMe no
(}>H31:1Ke .nJl.H nocTynaroIUHx B BbICIIIHe yqe6HbIe 3aBe.neHH.H, a TaK-
me IIIKOJlbHbIM npOrpaMMaM :H3yqeH:H.H cp:H3:HK:H K8K H8 6830BOM,
TaK H Ha nOBbIIIIeHHOM H yrJly6J1eHHoM YPOBHSlX.
npHBe,lJ;eHbI 10 BapHaHTOB ne.narorHqeCKHX TeCTOB (c OTBeTa-
MH), KOTopbIe MOryT 6bITb HCnOJlb30BaHbI .nJl.H onpe.neJleHH.H ypOB-
H.H YCBoeHH.H yqHMHC.H Kypca <pH3HKR B YCJlOBH.HX .nec.HT:H6aJIJlb-
HOH CHCTeMbI 0eHHB8HH.H pe3YJlbTaTOB yqe6HoH .ne.HTeJlbHOCTH yqa-
:HXC.H.
llpeOlia3liaileli0 O.n.Sl YllalqUXCSl WKO.n.. eUM1ia3uu. .n.uee6. c.n.y-
wame.n.eu noozom06ume.n.blibtX omoe.n.eliuu 6Y306. Byoem no.n.e3libtM
npenooa6ame.n.SlM tjJU3UKU o.n..R opzaliU3auu caMocmOSlme.n.bHOU U
Uliou6uoya.n.bliou pa60mbt YllalqUXcSl. cocma6.n.eliUSl caMocmOSlme.n.b-
HbtX. KOlimpo.n.blibtX pa60m. noozom06Ku K umOZ060U ammecmauu
6btnYCKIiUK06.
YAK 53(075.3)
BBK 22.3J1 721
ISBN 985-471-123-4
@ AKceHOBHq JI. A.. PaKHHa H. H..
ct>8pHHO K. C.. 2004
@ O<popMneHHe. 113,n8TeJIbCTBO
"A.AYKabIJI i BbIx8B8HHe". 2004

BBEHHE
)];aHHoe yqe6HOe noc06He HMeeT eJlblO OKa3aTb yqaHMC.H nOMOb B
CHCTeMaTH3aHH, 0606eHHH H yrJly6JIeHHH 3HaHHH no H3HKe, OCBoeHHH
MeTO)J;OB H npHeMOB peIIIeHH.H 38.)J;aq npH nO)J;rOTOBKe K HTorOBOH aTTeCTa-
HH 3a KYpC cpelIHeH 06e06pa30BRTeJlbHOH IIIKOJlbI H BCTynHTeJlbHbIM HC-
nbITaHH.HM B BhICIIIHe yqe6HbIe 3aBe)J;eHH.H.
Co,nep.>KaHHe yqe6HOrO nOC06H.H COOTBeTcTByeT nporpaMMe no H3HKe
JI.H nocTynalOHx B BbICIIIHe yqe6Hble 3aBe)J;eHH.H, a TaKme IIIKOJIbHbIM
nporpaMMaM H3yqeHH.H H3HKH KaK Ha 6a30BOM, TaK H Ha nOBbIIIIeHHOM
H yrJly6J1eHHoM ypOBH.HX.
Yqe6HOe noc06He npe)J;Ha3HaqeHO )J;JI.H yqaHXC.H IIIKOJl, rHMHa3HH, JlH-
eeB, CJIYIIIaTeJIeH nO)J;rOTOBHTeJIbHbIX OT)J;eJIeHHH BbICIIIHX yqe6HhIx 3aBe-
eHHH, yqaHXC.H CHCTeMbI npoTex06pa30BaHH.H H CGY30B. IIoJIe3HhIM OHO
6Y)J;eT H npenO)J;aBaTeJl.HM H3HKH )J;JI.H opraHH3aHH caMOCTO.HTeJIbHOH H
HH)J;HBH)J;yaJlbHoH pa60TbI yqaHXC.H, COCTaBJIeHHs:I caMOCTO.HTeJIbHbIX, KOH-
TpOJlbHbI pa60T, nO)J;rOTOBKH K HTorOBOH aTTeCTaQ,HH BbInYCKHHKOB.
MaTepHaJl pacnpe)J;eJleH no IIIeCTH OCHOBHbIM rJlaBaM: "MexaHHKa",
"3J1eMeHTbI CTaTHCTHqeCKOH H3HKH H TepMOHHaMHKH", "3JIeKTpo,nHHaMHKa",
"KoJle6aHH.H H BOJlHbI", "OnTHKa", "KBRHTOBa.H <pH3HKR", BHYTPH rJIaB -
no pa3)J;eJlaM.
HanpHMep, rJIaBa "MexaHHKa" COCTOHT H3 CJIe)J;YIOHx pa3)J;eJlOB: "Oc-
HOBbI KHHeMaTHKH", "OCHOBbI )J;HHaMHKH", "3aKOHbI coxpaHeHH.H B MexaHH-
Ke", "3J1eMeHTbI CTaTHKH", "3JIeMeHTbI CTaTHKH H )J;HHaMHKH .>KH)J;KOCTeH
II ra30B".
Ka.>K)J;bIH pa3)J;eJl BKJIlOqaeT:
· nepeqeHb paCCMaTpHBaeMbIX B yqe6HOM noc06HH BonpOCOB;
· KpaTKoe H CHCTeMaTH3HpOBaHHoe H3JIOmeHHe OCHOBHoro TeOpeTHqeC-
Koro MaTepHaJla (opHeHTHpyeT Y'laHXC.H Ha YCBoeHHe nOH.HTHH, 3aKOHOB,
3aKOHOMepHOCTeH H T.)J;.);
· BonpOCbI H 3a,naHH.H I1.JI.H caMOKOHTpOJI.H (OHH nO)J;06paHbI H COPMY-
JIllpOBaHbI TaK, qT06bI yqaHeC.H MOrJlH npOBepHTb ypOBeHb CBOHX 3HaHHH
II YMeHHH no TeMe; BonpOChI H 38.)J;aHHs:I nOCTeneHHO YCJIO.>KH.HIOTC.H, qTO
Tpe6yeT OT yqaHXC.H ,nJl.H OTBeTa H peIIIeHH.H r JIy60Koro nOHHMaHH.H H-
3HQeCKHX 3aKOHOB, .HBJIeHHH H npoeCCOB, npHBJleqeHH.H 3HaHH:H H3 pa3-
JIHqHbIX pa3)J;eJlOB H3HKH);
· OCHOBHble <POPMYJlbI;
· MeTO)J;HqeCKHe peKOMeH)J;aQ,HH no peIIIeHHIO 3a)J;aq (nOCJle)J;OBaTeJlb-
HOCTb )J;e:HCTBH:H, KOTopbIe He06xo)J;HMO BbInOJlHHTb npH peIIIeHHH 38.)J;aq, -
OT aHaJlH3a YCJlOBH.H 3a,n;aqH (ero KpaTKOH 3anHCH, BbInOJlHeHH.H pHCYHKa,
3

cxeMbI, qepTema, nO.HCH.HIOHX YCJIOBHe 3a.u;aqH) )J;O aHaJlH3a H OI(eHKH no-
JlyqeHHoro OTBeTa);
· npHMepbI pemeHH.H 3a)J;aq (B )J;aHHOH qaCTH Kam,ltOrO pa3)J;eJIa Ha
npHMepe pemeHH.H HaH60JIee THnOBbIX 3a.naq lIeMOHcTpHpyeTC.H nponecc
nOCTpOeHH.H H HCnOJlb30BaHH.H aJlrOpHTMa pemeHHSI 3a)J;a'I Ha OCHOBe Me-
TO)J;HqeCKHX peKOMeH)J;aI(HH).
B npHJlOmeHHH npHBe)J;eHbI 10 BapHaHTOB ne)J;arOrHqeCKHX TeCTOB, KO-
TopbIe MorYT 6bITb HCnOJlb30BaHbI )J;JI.H onpe)J;eJIeHH.H YPOBH.H YCBoeHHH yqa-
HMHCH Kypca «pH3HKH B YCJIOBH.HX )J;eC.HTH6aJlJlbHOH CHCTeMbI OI(eHHBaHH.H
pe3YJlbTaTOB yqe6HoH )J;eHTeJIbHOCTH yqaHXCH.
TIpH nO)J;rOTOBKe nOC06H.H aBTopbI OnHpaJIHCb Ha CBOH MHOrOJIeTHHH
onbIT npenO)J;aBaHH.H «pH3HKH B nOJlHTeXHHqeCKOH rHMHa3HH N2 6
r. MHHCKa H JlHI(ee BeJloPyccKoro HanHOHaJlbHOrO TeXHHqeCKOrO YHH-
BepCHTeT8.
ABTOPCKHH KOJIJleKTHB BbIpamaeT HCKpeHHlO1O 6JIaro)J;apHocTb peI(eH3eH-
TaM 3a rJly60KHH aHaJlH3 cO)J;epmaHH.H yqe6Horo nOC06H.H, I(eHHbIe npe,nJlo-
meHH.H H nOmeJlaHH.H.
Bce 3aMeqaHHH, HMelOHe I(eJIblO YJlyqmHTb KaqeCTBO )J;aHHoro yqe6-
Horo nOC06H.H, 6Y)J;YT npHHHTbI C 6J1arO)J;apHOCTblO. npOCHM npHCbIJIaTb HX
no a)J;pecy: 220010, r. MHHCK, YJI. COBeTCKa.H, 9, MHHHcTepcTBo 06pa30Ba-
HH.H Pecny6JIHKH BeJIapycb, rJlaBHOe ynpaBJleHHe 06ero cpe)J;Hero 06pa-
30BaHH.H .

MEXAHKA
.<,;j a!,-t:JXf '.4\'iJ.C'1I.,5ti:'I!"t
.:'I':
.;""'
1. OCHOBM KHHEMATHKH
MexaHHeCKoe ABH>KeHHe. OTHOCHTenbHOCTb ABH>KeHHSI. CHCTeMa OTCe-
Ta. MaTepHanbHaSi TOKa. TpaeKTOpHSI. nYTb H nepeMeL4eHHe. CpeAHSISI CKO-
pOCTb. MrHOBeHHaSi CKOpOCTb. YCKopeHHe. PaBHOMepHoe H paBHOYCKopeHHoe
npSlMonHHeHoe ABH)I(eHHe. CnO)l(eHHe CKopOCTe.
rpaHKH 3aBHCHMOCTH KHHeMaTHeCKHx BenHHH OT BpeMeHH B paBHO-
MepHOM H paBHOYCKopeHHOM ABH>KeHHH.
CB060AHoe naAeHHe Ten. YCKopeHHe cB060AHoro naAeHHSI. ABH)I(eHHe
Tena, 6poweHHoro ropH30HTanbHO. ABH>KeHHe Tena, 6poweHHoro nOA yrnoM
K ropH30HTY.
ABH>KeHHe Tena no OKPY>KHOCTH C nOCTOSlHHOH no MOAyn CKOpOCTb.
YCKopeHHe npH ABH>KeHHH Tena no OKPY>KHOCTH C nOCTOSlHHOH no MOAyn
CKOpOCTb (....eHTpOCTpeMHTenbHoe YCKopeHHe).
1.1. OCHOBHLle nOHHTHH KHHeMaTHKH
KHHeMaTHKa - pa3AeJl MexaHHKH, H3yqalOHH ABHmeHHe TeJl 6e3 yqeTB
npHqHH, BbI3BaBillHX 3TO ABHmeHHe.
OCHOBHOH 3RRqeH KHHeMRTHKH HBJlHeTCH HaxO*,AeHHe nOJlOmeHH.H TeJlR B
JlI060H MOMeHT BpeMeHH. eCJlH H3BeCTHbI ero nOJlOmeHHe. CKOpOCTb H YCKope-
HHe B HRqRJIbHbIH MOMeHT BpeMeHH.
MeXaHHqeCKOe )UlHateHHe - 3TO H3MeHeHHe nOJlOmeHHH TeJl (HJIH qaCTeH
TeJIR) OTHOCHTeJlbHO Pyr pyra B npOCTpRHCTBe C TeqeHHeM BpeMeHH.
AJI.H onHCRHH.H MeXaHHqeCKOrO ,ABHmeHH.H HaAo BbI6pRTb CHcTeMY OT-
CqeTa.
y 0
0 x
z
Puc. 1.1
Teno OTC'lJeTa - TeJlO (HJlH rpynna TeJl) , npH-
HHMaeMoe B AaHHOM cJIyqae 3a HenOABHmHoe, OT-
HOCHTeJlbHO KOToporo paccMaTpHBaeTC.H ,l1BHmeHHe
pyrHx TeJl.
CHcTeMa OTCqeTa - 3TO CHCTeMa KOOpHHRT,
CB.H3aHHa.H C TeJlOM OTCqeTR, H BbI6paHHbIH cnoc06
H3MepeHHH BpeMeHH (pHC. l.I).
nOJIOmeHHe TeJlR MomHO onpeeJlHTb C nOMO-
blO paHyca-BeKTopa f HJlH C nOMoblO KOOpH-
HaT.
5

PHYC-BeKTOp r TOqKH M HanpaBJleHHbIH
OTpe30K np.HMOH, COeHH.HIOHH HaqaJIO OTCqeTa 0
C TOqKOH M (pHC. 1.2).
KOOp,lJ;HHaTa x TOqKH M - 3TO npOeKIJ;H.H
KOHIJ;a paHyca-BeKTopa TOqKH M Ha OCb Ox.
06blqHO nOJlb3YIOTC.H npSlMoyronbHoH CHCTeMOH KO-
op.zxHHaT. B 3TOM CJIyqae nOJIOmeHHe TOqKH M Ha
x JlHHHH, nJIOCKOCTH H B npOCTpaHCTBe onpeeJl.HIOT
COOTBeTCTBeHHO O.zxHHM (x). BYM.H (x, y) H TpeMSI
(x, y, z) qHCJlaMH - KoopHHaTaMH (pHC. 1.3).
B 3J1eMeHTapHOM Kypce cpH3HKH H3yqalOT KH-
HeMaTHKY .zxBHmeHH.H MaTepHaJlbHOH TOqKH.
MaTepHaJILHaJl TO'lKa - TeJlO, pa3MepaMH
KOToporo B naHHbIX YCJIOBH.HX MomHO npeHe6-
peqb.
3TOH MOeJlblO nonb3YIOTC.H B Tex cJlyqa.HX,
Kor.zxa JlHHeHHbIe pa3MepbI paCCMaTpHBaeMbIX TeJl
MHoro MeHbllIe Bcex npOqHX paCCTO.HHHH B .zxaHHoH
3a.zxaqe HJlH Kor.zxa TeJlO .zxBHmeTC.H nocTynaTeJlbHO.
IT ocmyname.n.blibl.M Ha3bIBaeTC.H .D;BHmeHHe TeJla,
npH KOTOpOM npSlMa.H, npOXOnSllIJ;a.H qepe3 JII06bIe nBe TOqKH TeJla. nepeMe-
aeTC.H, OCTaBa.HCb napaJlJIeJlbHOH caMoH ce6e. npH nocTynaTeJlbHOM BHme-
HHH Bce TOqKH TeJla onHCbIBalOT O,AHHaKOBble TpaeKTopHH H B JlI060H MOMeHT
BpeMeHH HMelOT OHHaKOBbIe CKOpOCTH H YCKopeHHSI. IT03TOMY .zxnSl onHCaHH.H
TaKoro BHmeHH.H TeJla OCTaTOqHO onHcaTb BHmeHHe ero OHOH rrpOH3BOJlb-
HOH TOqKH.
B .zxaJlbHeHllIeM no.zx CJlOBOM "Teno" 6yeM rrOHHMaTb "MaTepHanbHaSl
TOqKa" .
JIHHHSI, KOTOPYIO onHCbIBaeT BHmyeeC.H Teno B onpeneJIeHHOH CHCTeMe
OTCqeTa. Ha3bIBaeTC.H TpaeKTopHeH. Ha npaKTHKe cpOPMY TpaeKTopHH 3aalOT
C nOMOblO MaTeMaTHqeCKHX cpOPMYJl (y = f(x) - ypaBHeHHe TpaeKTopHH) HJIH
H306pamalOT Ha pHCYHKe. BH,A TpaeKTopHH 3aBHCHT OT BbI60pa CHCTeMbl OT-
CqeTa. HanpHMep, TpaeKTopHeH TeJla, CB060,AHO na.n;alOero B BarOHe, KOTOpbIii
BHmeTC.H paBHOMepHO H np.HMOJlHHeHHO, .HBJI.HeTC.H np.HMa.H BepTHKaJlbHa.H JlH-
HH.H B CHCTeMe OTCqeTa. CB.H3aHHOH C BarOHOM. H napa60Jla B CHCTeMe OTCqeTa,
CB.H3aHHOH C 3eMJleH.
B 3aBHCHMOCTH OT BHa TpaeKTopHH pa3J1HqalOT np.HMOnHHeHHoe H KPH-
BOJIHHeHHOe .zxBHmeHHe.
llYTL S - CKaJl.HpHaSl cpH3HqeCKa.H BeJlHqHHa, onpe.zxeJl.HeMa.H ,AJlHHOH Tpa-
eKTopHH. onHcaHHoH TeJlOM 3a HeKoTopbIH npOMemYTOK BpeMeHH. IIYTb Bce-
ra nOJlOmHTeJleH: s> O.
nepeMeeHHe /).r TeJla 3a onpeneJleHHbIH npOMemYTOK BpeMeHH - Ha-
npaBJIeHHbIH OTpe30K np.HMOH, COeHH.HIOlf(HH HaqaJIbHOe (TOqKa Mo) H KOHeq-
Hoe (TOqKa M) nOJlOmeHHe TeJla (CM. pHC. 1.2):
y
Xo
x
Puc. 1.2
y
------"'1
","", ",'" I
'" '''' I
(--- --"YM I
Y I :
I I
I IX I
: Z V', I /
I _____'"
Z
Puc. 1.3
M
x
!:if = f - "0 ,
re f H ;h - paHycLI-BeKTopbl TeJla B 3TH MOMeHTbl BpeMeHH.
6

llpOeKQ,H.H nepeMeeHH.H Ha OCb Ox: rx =!::ix = x - xo, r)J;e Xo H X - KO-
0PAHHaTbI TeJla B HaqaJlbHbIH H KOHeqHbIH MOMeHTbl BpeMeHH.
MOAYJlb nepeMeeHH.H He MomeT 6bITb 60JlbWe nYTH: Irl  s.
3HaK paBeHcTBa OTHOCHTC.H K cnyqalO np.HMOnHHeHHoro .l{BHmeHH.H. eCJlH
HanpaBneHHe .l{BHmeHH.H He H3MeH.HeTC.H.
3Ha.H nepeMeeHHe Ii HaqaJlbHOe nOJlOmeHHe TeJla, MomHO HaHTH ero no-
nomeHHe B MOMeHT BpeMeHH t:
r = ro + r;
{ X = Xo + rx;
Y = Yo + !::iry.
CKOpOCTL - Mepa MeXaHHqeCKOrO COCTO.HHH.H TeJla. OHa xapaKTepH3yeT
6bICTPOTY H3MeHeHH.H nOJlOmeHH.H TeJla OTHOCHTeJlbHO AaHHOH CHCTeMbI OTCqe-
T8 H .HBJI.HeTC.H BeKTopHOH H3HqeCKOH BenHqHHOH.
Cpe,lJ;HJUI CKOpOCTL < V > - BeKTOpHa.H H3HtIeC-
Ka.H BeJlHqHHa, qHCJleHHO paBHa.H OTHoweHHIO nepe-
MeeHH.H K npOMemYTKY BpeMeHH, 3a KOTOpbIH OHO
npOH30WJlO, H HanpaBneHHa.H B,nOJlb nepeMeeHH.H
(pHC. 1.4):
Puc. 1.4
_ r
< v> =-;
t
B CIf e,A"HHe" CKOpOCTH SlsnSleTCSI MeTp B ce-
< v > ii 6.r.
KYHAY ( : ).
Cpe.nH.H.H CKOpOCTb, HaHAeHHa.H no 3TOH cpopMYJle, xapaKTepH3yeT ,nBH-
meHHe TOJlbKO Ha TOM yqaCTKe TpaeKTopHH. .nn.H KOToporo OHa onpe.neJle-
Ha. Ha ,npyroM yqaCTKe TpaeKTopHH OHa MomeT 6bITb .npyrOH.
11 u S
HorAa nOJlb3YIOTC.H cpe,nHeH CKOpOCTblO nYTH: < v > = 6.t ' re s - nYTb,
npOH,AeHHbIH 3a npOMemYTOK BpeMeHH 6.t. Cpe.nH.H.H CKOpOCTb nYTH - 3TO
CKan.HpHa.H BeJlHqHHa.
MruoBeHHaH CKOpOCTL V TeJla - CKOpOCTb TeJla B ,AaHHbIH MOMeHT Bpe-
MeHH (HJlH B .naHHOH TOqKe TpaeKTopHH). OHa paBHa npe.neJlY, K KOTOPOMY
CTpeMHTC.H CpeAH.H.H CKOpOCTb 3a 6eCKOHeqHO MaJlbIH npOMemYTOK BpeMeHH
- 1 . !::ir -, 3 -,
v = 1m - = r. AeCb r - npOH3Bo,nHa.H OT pa.nHyca-BeKTopa no BpeMeHH.
6tO !::it
B npoeKHH Ha OCb OX:
1 . !::ix ,
v = 1m - = x .
x 6tO !::it
MrHOBeHHa.H CKOpOCTb TeJla HanpaBJleHa no KacaTenbHoH K TpaeKTopHH B
KamAoH ee TOqKe B CTOPOHY ABHmeHH.H (CM. pHC. 1.4).
7

Y CKopeHHe - BeKTOpHa.H cpH3HqeCKaH BeJlHqHHa, xapaKTepH3YIOa.H 6bIC-
TPOTY H3MeHeHH.H CKOpOCTH. OHO nOKa3bIBaeT, Ha KaKYIO BeJlHqHHY H3MeH.H-
eTC.H CKOpOCTb TeJla 3a eHHHY BpeMeHH.
Cpe.n;Hee YCKopeHlle - <pH3HqeCKa.H BenHqHHa, tIHCneHHO paBHaH OTHOIlIe-
HHIO H3MeHeHH.H CKOpOCTH KO BpeMeHH, 38 KOTopoe OHO npOH30IlIJlO:
_ l:1v V - Vo
<a>=-=-.
l:1t l:1t
BeKTOp < d > H8npaBJleH napaJlJleJlbHO BeKTOpy
H:3MeHeHH.H CKOpOCTH l:1v « ii > ii 6v) B CTOPOHY
BorHYTOCTH TpaeKTopHH (pHC. 1.5).
MrHoBeHHoe YCKopeHHe:
- 1 . l:1v -
a= lID-=V
6tO 6t
B CM e,AHHHQ.eH YCKopeHHSI SlBnSleTCSI MeTp Ha ce-
KYHAY B KB8AP8Te ( ;: 1.
B 06eM cnyqae MrHOBeHHoe YCKopeHHe Ha-
npaBJleHO no yrnoM K CKOpOCTH. 3Ha.H Tp8eKTO-
pHIO, MomHO onpe,AeJlHTb HanpaBJleHH:e CKOpOCTH,
HO He YCKopeHH.H. HanpaBneHHe YCKopeHH.H onpe-
eJlHeTCH HanpaBJleHHeM paBHoeHCTBYIOeH CHJI, eHCTBYIOHX Ha TeJlO.
TIpH np.HMOJlHHeHHOM BHmeHHH C B03p8CTalOeH no MOYJlIO CKOpOCTblO
(pHC. 1.6, a) BeKTopbI a H Vo COHanpaBJleHbI (a ii v o ) H npOeKH.H YCKope-
HHH Ha HanpaBJleHHe ,ABHmeHHH nOJlO>KHTenbHa.
TIpH npHMOJlHHeHHOM BHmeHHH C y6bIBalOeH no MoynlO CKOpOCTblO
(pHC. 1.6, 6) HanpaBJleHHH BeKTopOB a H Vo npOTHBOnOJlOmHbI (ii i J, v\» H
npOeKl.\H.H YCKopeHH.H Ha HanpaBJleHHe ,ABH>KeHH.H OTpHaTenbHa.
Puc. 1.5
a
o .
a Do
. .
M
v
8
6
a Do
.. x
0 .. .
" x
M
Puc. 1.6
BeKTOp ii npH KpHBonHHeHHoM ,ABH)KeHHH MO)K-
HO pa3J10mHTb H8 ltBe COCT8BJlHIOIUHe. HanpaBneH-
HbIe BOJlb CKOpOCTH aT H nepneHHKY JlHpHO CKO-
pOCTH an (pHC. 1.7), aT - TaHreHH8J1bHOe YCKO-
peHHe, xapaKTepH3YIOee 6bICTPOTY H3MeHeHHs:I MO-
yn.H CKOpOCTH npH KpHBOJlHHeHHOM BHmeHHH,
an - HopManbHoe YCKopeHHe, xap8KTepH3YIOlI.{ee
6bICTPOTY H:3MeHeHHH H8npaBneHHH BeKTopa CKOpO-
CTH npH Kp HBOJlHHe HHOM BHmeHHH MOYJlb YCKO-
peHHa a =  a; + a .

1.2. OTHOCHTeJILHOCTL ABHmeHHH. CJIOmeHHe CKopocTeit
KaK OTMeqaJlOCb BbIllie. .D:JI.H onHC8HH.H .D:BHmeHH.H TeJla He06xo.D:HMO BbI-
6paTb TeJlO OTCqeTa H CB.H3aTb C HHM CHCTeMY KOOp,lIHHaT. B KaqeCTBe TeJla
OTCqeTa MomeT BbIcTynaTb JlI060e TeJlO.
B pa3HbIX CHCTeMax OTCqeTa 6YYT pa3J1HqHbI BH TpaeKTopHH, 3HaqeHH.H
CKOpOCTH, nepeMeeHH.H H ,ApyrHx BenHqHH. B 3TOM H 3aKJIlOqaeTC.H OTHOCH-
TeJlbHOCTb BHmeHH.H.
x'
IT p H M e p. qeJIOBeK H.D:eT no naJIY-
6e napoxoa co CKOpOCTblO VI OTHOCH-
TeJlbHO napoxo)J;a. ITapoxo BHmeTC.H
nocTynaTeJlbHO co CKOpOCTblO V 2 OTHO-
CHTeJlbHO 6epera. HaHeM CKOpOCTb V
qeJlOBeKa OTHOCHTeJlbHO 6epera.
CB.HmeM HenOBHmHYIO CHCTeMY OT-
CqeTa (xOy) C 3eMJleH, a nOBHmHYIO
(x'O'y') - C napOXO.D:OM.
H3 PHCYHK8 1.8 BHHO, qTO nepe-
MeeHHe
 6.;: * 6.;:1' ( 1.1 )
t .
y'


r
o
x
Puc. 1.8
6.;: = 6.;:1 + 6.;:2
r,Ae 6. - nepeMeeHHe qeJlOBeKa OTHOCHTeJIbHO napoxoa, 6.;:2 - nepeMe-
eHHe napOXO.D:a OTHOCHTeJlbHO 6epera. 6.;: - nepeMemeHHe qeJlOBeKa OTHO-
CHTeJlbHO 6epera.
TaKHM 06pa30M, ec.n.u me.n.o Oali06peMenii0 Yllacm6yem 6 neCKo.n.bKUX
a6UJlCeliuSlx. mo pe3y.n.bmupYI014ee nepe.Me14enue m01lKU pa61i0 6eKmopiiOU
cY.MMe nepe.Me14eliuu. C06epwaeMblx elO 6 KaJICaOM U3 iJ6UJlCeIiUU. B 3TOM
COCTOHT YCTaHOBneHHbIH 3KcnepHMeHTaJlbHO npuliqun lie3a6UCU.MOCmu a6U'
JlCeliuu.
Pa3eJlHB ypaBHeHHe (1.1) Ha npOMemYTOK BpeMeHH, 3a KOTOpbIH npo-
H3011IJlH nepeMeeHH.H qeJlOBeKa H napOXO.D:a. nOJlyqHM 3aKOH CJlOeHHH
CKopocTeH:
V = VI + v2'
CKopocmb V meJla omliOCumeJLbno nenOa6U:HCIiOU CUCme.Mbl omcilema pa6-
lia zeoMempullecKou cYMMe CKopocmu VI me.n.a omliocume.n.bliO nOa6UJlCiiOU CU-
cmeMbl omc1lema U CKopocmu V 2 caMOU noiJ6UJlCiiOU cucmeMbl omc1lema om-
Hocume.n.bliO lienOa6U:HCnou.
3aKOH CJlOmeHH.H CKopOCTen cnpaBeJlHB H )J;JI.H HepaBHOMepHoro )J;BHme-
HH.H, TOJlbKO B 3TOM CJIyqae V, VI H V 2 - MrHOBeHHbIe CKOpOCTH.
3TOT 3aKOH 6bIJI YCTaHOBJleH r. raJlHJIeeM. OR cnpaBeJlHB TOJlbKO JI.H
,ABIUKeHHH co CKOpOCT.HMH, HaMHoro MeHhWHMH CKOpOCTH CBeT8 c = 3.10 8
M
C
(v « c). TaKHe CKOpOCTH B <pH3HKe Ha3bIBalOT liepe.n.Slmu6uCmCKU.MU
9

1.3. PaBHoMepHoe npJlMOJIHHeHHOe .lJ;BHmeHHe
PaBHOMepHoe npJlMOJIHHeHHOe ABHeHHe - 3TO BHmeHHe, npH KOTO-
pOM Teno 3a JlI06bIe paBHbIe npOMemYTKH BpeMeHH COBepmaeT paBHbIe ne-
peMeeHH.H, T. e. 3TO BHmeHHe C nOCTO.HHHOH no MOYJlIO 11 HanpaBJleHHIO
CKOpOCTblO:
a=O -
v = const - ypaBHeHHe CKOpOCTH,
v

 I1x
0 ,,- Xo
r71
./
Yx
Puc. 1.9
..
.x
ypaBHeHHe YCKopeHH.H.
IIYCTb B MOMeHT BpeMeHH t o = 0 KO-
opHHaTa TeJla xo, B MOMeHT t - x
(pHC. 1.9).
Tora 3a npOMemYTOK BpeMeHH
I1t = t - to = t KoopHHaTa TeJla H3Me-
HHJlaCb Ha BeJlHqHHY I1x = x - xo' CJleo-
BaTenbHO, npOeKII.H.H CKOpOCTH Tena
x-xo
v =- 
x t
x = xo + vxt - KHHeMaTHtIeCKOe ypaBHeHHe paB-
HOMepHoro BHmeHH.H (ypaBHeHHe 3aBHCHMOCTH Koop,nHHaTbI OT BpeMeHH).
IIpOeKII.HH nepeMeeHHH llr x = x - xo =>
llrx = vxt - ypaBHeHHe nepeMeeHH.H.
IIpH paBHOMepHOM npHMOJlHHeHHOM ,nBHmeHHH HanpaBneHHe CKOpOCTH He
H3MeH.HeTC.H, n03TOMY nYTb S = Illrxl. CJleOBaTeJlbHO, S = Ivxl t - ypaBHeHHe
nYTH.
3aBHcHMoCTb KHHeMaTHqeCKHX BeJlH-
vrn"'1 qHH OT BpeMeHH MomHO H306pa3HTb
  rpacpHqeCKH. M306pa3HM rpa<})HKH CKO-
x pOCTH. nepeMeI.I.teHH.H. nYTH H KOOp,AHHa-
TbI HJI.H Tpex TeJl: 1, 2, 3 (pHC. 1.10).
TeJla 1, 2 ,nBHmYTC.H B nOJlOmHTeJlbHOM
HanpaBJleHHH OCH OX, npHqeM v2 > vI;
TeJlO 3 ,ABHmeTCH B HanpaBneHHH, npo-
THBononomHOM OCH Ox; HX HaqaJlbHbIe
Koop,nHHaTbI COOTBeTCTBeHHO XOl' X02' X03' rpa<}>HKH CKOpOCTH npe,nCTaBJleHbI
Ha pHcYHKe 1.11. IIJlOrqa.u;b 3amTpHxoBaHHoro np.HMoyroJlbHHKa qHCJleHHO paB-
Ha nYTH S (MO,nYJlIO nepeMeeHH.H), npoH,neHHoMY TeJlOM 1 3a BpeMH tl' Ha
pHCYHKe 1.12 ,naHbI rpac}>HKH nepeMerqeHH.H I1rx = f(t). Ha pHCYHKe 1.13 - rpa-
<}>HKH nYTH S = f(t)
HaKJlOH rpa<}>HKa r x = f(t) KOCH BpeMeHH 3aBHCHT OT MO)J;YJl.H CKO-
pOCTH:
VI

O XOI
'-- y X 02
V 2
[])--+
./
1'X oa
./
Puc. 1.10
10
tg a. = v X '

VJr 6r" S
2
1 3
0 t 0
3 0 t
3
Puc. 1.11 Puc. 1.12 Puc. 1.13
x
o
Puc. 1.14
t
rpa<}>HKH llBHmeHH.H (38BHCHMOCTH KOOp,nH-
HaTbI OT BpeMeHH) H306p8meHbI Ha PHCYH-
Ke 1.14.
C nOMOblO rp8<}JHKa ,nBHmeHH.H MomHO on-
pe)J;eJlHTb: 1) Koop,nHHaTbI TeJl8 B Jl1060H: MO-
MeHT BpeMeHH; 2) nYTb, npOH:,neHHbIH: TeJlOM 3a
HeKoTopbIH npOMemYTOK BpeMeHH; 3) BpeM.H, 3a
KOTopoe npOH:)J;eH KaKOH:-TO nYTb; 4) KpaTqaH:-
wee p8CCTO.HHHe MemAY TeJl8MH B Jl1060H: MO-
MeHT BpeMeHH; 5) MOMeHT H MeCTO BCTpeqH TeJl
H )J;p.
1.4. PaBHoycKopeHHoe npJlMOJIHHeHHOe ,lJ;BH*eHHe
PaBHOYCKopeHHoe npJlMOJlHHeHHOe ABH3teHHe - 3TO BHmeHHe, npH KO-
TOpOM CKOpOCTb TeJl8 38 JlI06bIe p8BHbIe npOMemYTKH BpeMeHH H3MeH.HeTCJI
o,nHHaKOBO, T. e. 3TO ,nBHmeHHe C nOCTO.HHHbIM no MO)J;YJlIO Ii H8npaBJleHlllO
YCKopeHHeM.
a - const - yp8BHeHHe YCKopeHH.H.
II - v
o onpe)J;eJleHHIO YCKopeHH.R a = -.
t
IIYCTb B MOMeHT BpeMeHH t o - 0 CKOpOCTb TeJla paBH8 vo, B MOMeHT Bpe-
MeHH t - V. Tor,na 3a npOMemYTOK BpeMeHH t - t - to - t CKOpOCTb H3Me-
_ v - Vo
HHJlaCb Ha v - v - vo. CJleAOBaTeJlbHO, YCKopeHHe a.. - 
t
aa a l a 2 v - Vo + at - ypaBHeHHe CKOpOCTH.
  ....
V 03 VOl V 02 MJlH B npOeKUH.HX: V x - vox + axt.
...... .....  3TH
r3l rn 121 38BHCHMOCTH KHHeM8THleCKHX BeJlHqHH
. .. OT BpeMeHH H306pa3HM rpa<}JHqeCKH AJI.H Tpex
0 x TeJl (pHC. 1.15).
Puc. 1.15
11

rpaqJHKH YCKopeHH.H ax = f(t) npeCTaBJleHbI Ha pHCYHKe 1.16, a rpaqJHKH
CKOpOCTH V x = f(t) - Ha pHCYHKe 1.17.
AJI.H HaXOmeHH.H nepeMeeHH.H BOCnOJlb3yeMC.H rpaqJHKoM CKOpOCTH
(pHC. 1.18).
V x V x
3 
VOx
1 

0 t 
2 t 2 0 l1t t
Puc. 1.16 Puc. 1.17 Puc. 1.18
ax
o
.n;nH Manoro npoMemYTKa BpeMeHH l1t H3MeHeHHeM BeJlHqHHbI CKOpOCTH
MomHO npeHe6peqb H CKOpOCTb MomHO CqHTaTb nocToHHHOiL Tora nepeMe-
eHHe 3a npoMemYToK BpeMeHH l1t 6yeT paBHo nJlOaH Y3KOH rycTo 3a-
IIITpHxoBaHHoH nonOCKH. MbIcneHHo pa36HB Bce BpeM.H ,nBHmeHH.H TeJla Ha Ma-
nbIe npoMemYTKH BpeMeHH H Hai'i,n.H nepeMeeHHe 3a Kam,nbIH oT,neJlbHbIH
npoMemYToK BpeMeHH. CYMMHpyeM 3TH nepeMeeHHH. Mo,nYJlb npoeKllHH ne-
peMeeHH.H 3a npoMemYToK BpeMeHH I).t = t - to = t B npeeJle qHCJleHHO pa-
BeH nJlOaH 3aIIITpHXOBaHHOH Tpaneu,HH.
(Vox + vx)t
l1r x = .
2
V x = vox + axt B (1.2), nOJlyqHM:
Cne)J;OBaTeJlbHO,
(1.2)
IIocTaBHB 3HaqeHHe
a t 2
l1r x = voxt + T - ypaBHeHlle nepeMeIU;eHIIH B npoeKu,HHX;
- 2
A - - axt
U.r = vot + - - ypaBHeHHe nepeMeeHHH B BeKTopHoM BH,ne.
2
Y1.IHTbIBaH, qTO x = Xo + l1r, HMeeM:
a t 2
x = Xo + voxt +.....L- - KIiHeMaTH'leCKOe ypaBHeHlle
2
paBHOYCKopCHHoro ABHeIlIlH.
... _ _ _ iit 2
Ero BeKTopHbIH BH: r = ro + vot + - .
2
HCKnlOqa.H 1i3 ypaBHeHHH CKOpOCTH Ii nepeMeeHH.H BpeMH t, nonY1.IHM:
2 2
A r. = V x - VOx I 2 2 A
U. X => V x = "VOx + axu.r x '
2a x
CpaBHHBaH BbIpameHHe (1.2) C qJOPMY nOH l1r x = < V > x t. HaH,neM:
Vo + V
< V > = x x - npoeKIIH cpeAlieH CKOpOCTH
x 2
npH paBIIOYCKopeliliOM ABHeHHH.
12

rpa<pHKOM nepeMeeHH.H .HBJI.HeTC.H napa60JIa, nOJlOmeHHe BepllIHHbI KOTO-
pOH 3aBHCHT OT HanpaBJleHHH HaqaJIbHOH CKOpOCTH H YCKopeHHSl (pHC. 1.19).
6.r" 6.r" ax > 0 6.r"
ax> 0
0 t vox> 0 v o ,,< 0
a,,< 0
V o ,,= 0 0 t 0
t
6.r" 6.r" 6.r"
ax> 0
0 t V o ,,= 0
0 t
a,,< 0 a,,< 0
VOx < 0 0 V o " > 0
Puc. 1.19
1.5. CBo6oAHoe naeHHe Ten. YCKopeHHe cBo60Horo naeHHH
CBo6o,lJ.HOe na,lJ.eHHe - 3TO BHmeHHe TeJIa nop, p,eMcTBHeM TOJIbKO CHJIbI
T.HmeCTH.
Ha TeJlO, naalOee B B03p,yxe, KpoMe CHJlbI T.HmeCTH p,eHcTByeT CHJla CO-
npOTHBJleHH.H B03P,YX8, CJle,AOBaTeJIbHO, TaKoe ,ABHmeHHe He .HBn.HeTC.H CB060JJ;-
HbIM na,AeHHeM. CB060,AHOe n8,AeHHe - 3TO n8,AeHHe TeJl B B8KYYMe.
Y CKopeHHe g, KOTopoe co06aeT TeJlY CHJla T.HmeCTH, Ha3bIB81OT YCKope-
HlleM cB06o.o;Horo na.o;eHII.H. OHO nOK83bIB8eT, H8 KaKYIO BeJlHqHHY H3MeH.HeT-
C.H CKOpOCTb CB060JJ;HO n8p,alOero TeJI8 3a eJJ;HHHny BpeMeHH.
Y CKopeHHe cB060)J;Horo na)J;eHH.H g H8npaBJleHO BepTHK8J1bHO BHH3.
raJlHJleO raJIHJleH YCTaHOBHJI (3aKoH raJlHJle.H): 6ce me.n.a naiJalOm lia
n06epXliocmb 3eM.n.u noiJ iJeucm6Ue.M. 3eMiiOZO npUmSlJICeliUSl npu omcym.
cm6UU cu.n. conpomu6.n.eliUR C oiJuliaK06blM YCKopenueM. m. e. YCKopeliue C60-
60iJliozo naiJeliuSl lie 3a6ucum om .M.aCCbl me.n.a.
Y6eJJ;HTbC.H B 3TOM MomHO, HcnOJlb3Y.H Tpy6KY
HblOTOHa HJIH cTp060CKOnHqeCKHM MeTOJJ;.
Tpy6K8 HblOTOH8 npep,CT8BJI.HeT C060M CTeKJI.HHHYIO
TPy6KY p,JIHHOH OKOJlO 1 M, Op,HH KOHen KOTOpOH 3a-
na.HH, a p,pyroM cHa6meH KpaHOM (pHC. 1.20). IIoMe-
CTHM B Tpy6KY TPH pa3HbIX npe)J;MeTa, HanpHMep
P,P06HHKY, np06KY II nTHqbe nepo. 3aTeM 6bICTpO ne-
peBepHeM Tpy6KY. Bce TPH TeJla yn8p,YT H8 JJ;HO Tpy6-
KH. HO B pa3Hoe BpeM.H: CHaqaJIa ,AP06HHK8. 38TeM
np06Ka H, HaKOHeu, nepo. Ho TaK na,AalOT TeJl8 B TOM
cJlyqae, Korp,a B Tpy6Ke eCTb B03YX (pHC. 1.20, a).
CTOHT TOJlbKO B03p,yx OTKaqaTb HaCOCOM H CHOBa ne-
a
6

G
Puc. 1.20
13

peBepHYTb TPy6KY, MbI YBHHM, qTO Bce TPH TeJla ynaYT OHOBpeMeHHO
(pHC. 1.20, 6).
B 3eMHbIX YCJlOBH.HX g 3aBHCHT OT reorpaqJHqeCKOH lIIHpOTbI MeCTHOCTH.
H8H60nLwee 3H8'leHHe OHO HMeeT H8 nonroce (g  9,81 ;. ). H8HMeHLwee -
H8 3KB8Tope (g  9,75 ;. ). IIpH'lHHLI 3Toro: 1) C}'TO'lHOe Bp81I\eHHe 3eMnH
BOKpyr cBoeH OCH; 2) OTKJlOHeHHe qJOpMbI 3eMJlH OT ccpepHqeCKOH; 3) HeOHO-
pOHoe pacnpeeJIeHHe nJlOTHOCTH 3eMHbIX nopo.
YCKopeHHe cB060Horo naeHHJI 3aBHCHT OT BbICOTbI h TeJIS Ha nOBepx-
HOCTblO nJlaHeTbI (CM. HHme  2.13). Ero, eCJlH npeHe6peqb BpaeHHeM nJIa-
HeTbI, MomHO paCCqHTaTb no qJopMYJle:
GM
gh = ,
( R + h )2
nOCTO.HHHa.H, M
re G - rpaBHTaHOHHa.H
nnaHeTbI.
KaK CJleyeT H3 nOCJle)J;HeH qJOPMYJlbI, C YBeJlHqeHHeM BbICOTbI nO)J;'beMa
Tena Ha nOBepXHOCTblO nJlaHeTbI YCKopeHHe cB060)J;Horo naeHH.H YMeHblIIa-
eTC.H. ECJlH npeHe6peqb BpaeHHeM nJlaHeTbI, TO Ha nOBepXHOCTH nJlaHeTbI
paHYCOM R
Macca nJlaHeTbI, R - paHYC
GM
g = R 2 .
AJI.H He60JlblIIHX BbICOT (h« R) MomHO CqHTaTb g =const, )J;JlH TaKHX BbI-
COT cB060Hoe naeHHe .HBJI.HeTC.H paBHOYCKopeHHbIM )J;BHmeHHeM. ,LJ;JlJI ero
onHCaHHJI MomHO HCnOJlb30BaTb qJOPMYJlbI paBHOYCKopeHHoro BHmeHH.H:
ypaBHeHHe CKOpOCTH: V = Vo + gt;
KHHeMaTHqeCKOe ypaBHeHHe, onHCbIBalOee cB060Hoe na)J;eHHe TeJl:
_ _ _ gt 2 g t 2
r = ro + vot + -, HJlH B npoeKHH HB OCb Oy: y = Yo + vollt + -1!..-.
2 2
1.6. .n;BHeHHe TeJIa, 6pomeHHoro BepTHKaJILHO
CB060HO naalOee TeJlO MomeT BHraTbC.H np.HMOJlHHeHHO HJlH no KPH-
BOJIHHeHHOH TpaeKTopHH. :3TO 3aBHCHT OT HaqaJlbHbIX ycnoBHH. PaccMoTpHM
3TO nop06Hee.
CBo6oAHoe neHHe 6e3 HaqaJILHOH CKOpOCTH (vo=O) (pHC. 1.21)
IIpH BbI6paHHoH CHCTeMe Koop)J;HHaT )J;BHmeHHe TeJl8 onHCbIBaeTC.H ypaB-
gt 2
HeHHJIMH: VII = gt. Y =""2'
H3 nOCJleHeH qJOPMY JlbI MomHO HaHTH BpeM.H naeHH.H TeJla C BbICOTbI h:
t   . IIol\cT8Bn>1H H8HI\eHHOe BpeMH B <l>opMyny IIJIH CKOpOCTH, nonY"HM
MOYJlb CKOpOCTH TeJla B MOMeHT naeHH.H: v = 2hg .
14

.lI.BH3CeHHe TeJIa, 6pomeHHoro BepTHKaJILHO BBepx C HaqaJILHOH CKOpOC-
TLIO V O (pHC. 1.22)
0 --EJ v o = 0 Y
h i i
0
y
Puc. 1.21 Puc. 1.22
.n:BHmeHHe TeJla onHCbIB8eTC.H ypaBHeHH.HMH:
gt 2
v y = Vo - gt, y = vot - 2'
M3 ypaBHeHH.H CKOpOCTH BH)J;HO, qTO TeJlO BHmeTCSI paBH038MeJleHHO
BBepx, ,nOCTHraeT MaKCHManbHOH BbICOTbI, a 3aTeM ,nBHmeTC.H paBHOYCKopeHHO
BHH3. YqHTbIBa.H, qTO npH y = h max CKOpOCTb V y = 0 H B MOMeHT ,nOCTHmeHH.H
TeJlOM nepBOHaqaJlbHOro nonOmeHH.H y = 0, MomHO HaHTH:
V o
t 1 = - BpeMSI nO'beMa TeJla Ha MaKCHM8J1bHYIO BbICOTY;
g
v 2
hm =  - MaKCHMaJlbHa.H BbICOTa nO'beMa TeJla;
ax 2g
2vo
t 2 = 2t 1 = - - BpeM.H non eTa TeJla;
g
V2y = -vo - npOeKnH.H CKOpOCTH 8 MOMeHT ,nOCTHmeHH.H TenOM nepBOH8-
q8J1bHOro nOJlOmeHHSI.
1.7. ,l(BH)KeHHe TeJIa, 6pomeHHoro ropH30HTaJILHO
h
ECJlH CKOpOCTb V o HanpaBJleHB He BepTH-
K8J1bHO. TO ,nBHmeHHe TeJl8 6y,neT KpHBonHHeii-
HbIM.
PaCCMOTpHM BHmeHHe TeJla, 6poweHHoro
ropH30HTaJlbHO C BbICOTbI h CO CKOpOCTblO Vo
(pHC. 1.23). ConpOTHBJleHHeM B03yxa 6yeM
npeHe6peraTb. AJI.H onHCaHH.H ,nBH>KeHH.H He06-
XO,nHMO BbI6paTb )J;Be OCH Koop,nHHaT - Ox H
Oy. H8qaJlO OTCqeTa KoopHHaT COBMeCTHM C
HaqaJlbHbIM nOJlOmeHHeM TeJla. M3 PHCYH-
K8 1.23 BH,nHO. qTO vox = vo. vO y = O. gx = O. gll = g.
x
y
Puc. 1.23
15

TOr,lJ;8 ,lJ;BHmeHHe TeJla onHllleTCSI ypaBHeHHSlMH:
V x - Vo, x - vot;
gt 2
V g - gt, y.. -.
2
(1.3)
(1.4)
AHaJlH3 3THX c»OPMYJI nOKa3bIB8eT, qTO B ropH30HTaJlbHOM HanpaBJleHHH
CKOpOCTb TeJla OCTaeTCSI HeH3MeHHoH. T. e. TeJlO lIBHmeTC.H p8BHOMepHO. B Bep-
THK8.1IbHOM HanpaBJleHHH TeJlO ,lJ;BHmeTC.H paBHOYCKopeHHO C YCKopeHHeM g,
T. e. TaK me, KaK TeJlO, cB060HO na,lJ;alOee 6e3 HaqaJlbHOH CKOpOCTH. HaH-
,lJ;eM yp8BHeHHe TpaeKTopHH. AJI.H 3Toro H3 ypaBHeHH.H (1.3) HaHeM BpeM.H
x g 2
t - - H, nOCTaBHB ero 3HaqeHHe B c}>0PMYJlY (1.4), nOJlYQHM: y - x .
 2
STO ypaBHeHHe napa60JlbI. CJleOB8TeJlbHO, Teno. 6pOllleHHoe ropH30HTaJlb-
HO. BHmeTC.H no napa60Jle. CKOpOCTb TeJla B JlI060H MOMeHT BpeMeHH H8np8B-
JleH8 no KaCaTeJlbHOH K napa60Jle (CM. pHC. 1.23). MOn:YJlb CKOpOCTH MomHO
p8CC'lHTaTb no TeopeMe IIHc}>8ropa:
v ..  v; + v -  v5 + (gt )2 .
3Ha.H BbICOTY h, C KOTOpOH 6pOllieHO TeJlO, MomHO H8HTH BpeMSI t I' Qepe3
KOTopoe TeJlO yna,lJ;eT Ha 3eMJlIO. B 3TOT MOMeHT KoopHHaTa y paBHa BbICO-
gt 2
Te: YI = h. 113 ypaBHeHH.H (1.4) HaXOHM: h - -1... OTclOa
2
t 1 - f! .
(1.5)
<I>opMYJla (1.5) onpeeJl.HeT BpeM.H nOJleTa TeJla. 3a 3TO BpeM.H TeJlO npoH-
,lJ;eT B ropH30HT8.1IbHOM HanpaBJleHHH p8CCTO.HHHe l, KOTopoe Ha3bIBalOT ,lJ;aJlb-
HOCTblO nOJleT8 H KOTopoe MomHO HaHTH H8 OCHOB8HHH CPOPMYJlbI (1.3), yqH-
TblBDH. 'ITO I = x,. CJIe.z\OBDTeJILHO. I - Vo f! - I\ 8JILHOCTL nOJIeTD renD. Mo.
YJlb CKOpOCTH TeJla B 3TOT MOMeHT VI -  v5 + 2gh.
1.8. J];BH)KeBHe TeJIa, 6poweHHoro no,l); yrJIOM K ropH30HTY
y
o
Puc. 1.24
16
x
IIycTb TeJlO 6pollieHo no,lJ; yrJloM a K ro-
PH30HTY co CKOpOCTblO uo. K8K H B npebI-
,lJ;YHX cJlyqa.Hx, 6Y,lJ;eM npeHe6peraTb conpo-
THBJleHHeM B03yxa. AJI.H onHCaHH.H BHme-
HH.H He06xo,lJ;HMO BbI6paTb Be OCH KOOpH-
HaT - Ox H Oy (pHC. 1.24). Haq8.1Io OTCqe-
Ta COBMeCTHM C HaqaJlbHbIM nOJlOmeHHeM
TeJla. IIpoeKIJ.HH HaqaJlbHOH CKOpOCTH Ha
OCR Oy H Ox: Vo g ... V o sin a; vox'" Vo cos a.
IIpoeKIJ.HH YCKopeHH.H: gx= 0; gg= -g.

TOr,A8 ,ABHmeHHe TeJla 6Y)J;eT onHCbIBaTbC.H ypaBHeHH.HMH:
x = Vo cosa t; (1.6)
gt 2
Y = V o sin a t -2; (1.8)
V x = V o cosa; (1.7)
v y = V o sin a - gt. (1.9)
113 aTHX cpOPMYJI CJle,AyeT, qTO B ropH30HTaJIbHOM HanpaBJleHHH TeJlO ,ABH-
meTC.H p8BHOMepHO co CKOpOCTblO V x = Vo cosa, a B BepTHK8J1bHOM - paBHO-
YCKopeHHO (CM.  1.6).
TpaeKTopHeH ,ABHmeHHH TeJlB 6Y,AeT napa60JIa. YqHTbIBa.H, qTO B BepxHeH
TOqKe napa60JlbI v y = 0, MomHO HaHTH BpeM.H t l n011'heMa TeJla 110 BepxHeH TOq-
KH napa60JlbI:
V o sin a
o = Vo sin a - gtl => t l == .
g
(1.10)
IIO,ACT8BHB 3HaqeHHe t l B yp8BHeHHe (1.8), HaH11eM MaKCHMaJlbHYIO BbICO-
TY nO,A'heMa TeJla:
. Vo sin a _ ! v sin 2 a
h max = YI = Vo sIn a
g 2 g2
v 2 sin 2 a
h _ 0
'"'max - 2g
BpeM.H nOJleTa TeJla HaXO,AHM H3 YCJlOBH.H, qTO
2
. gt 2 0
Y2 == O. CJlelIOBaTeJlbHO, V o SIn a t 2 - 2 == .
MaKCHMaJIbHa.H BbICOTa nO)J;'heMa TeJla.
npH t = t 2 KOOp,AHHaTa
OrCIO.n8, t 2 == 2vo sin a. _ BpeMH
g
nOJleTa TeJl8. Cp8BHHBa.H aTY cpOPMYJlY C cpOPMYJlOM (1.10), BH11HM. qTO t 2 = 2t t .
BpeM.H ,ABHmeHH.H TeJl8 C M8KCHM8JIbHOH BbICOTbI t 3 = t 2 - t l == 2t l - t l = t l . CJle-
,AOBaTeJlbHO, CKOJlbKO BpeMeHH TeJlO nO,AHHMaeTC.H Ha MaKCHMaJlbHYIO BbICOTY,
CTOJlbKO BpeMeHH OHO onYCKaeTC.H C aTOM BbICOTbI. IIo,AcTaBJI.HH B ypaBHeHHe
Koop)J;HHaTbI x (1.6) 3HaqeHHe BpeMeHH t 2 , HaH,AeM:
l = 2v o cos a V o sin a = v sin 2a
-- )J;aJIbHOCTb nOJleTa TeJl8.
g g
MrHOBeHH8.H CKOpOCTb B JlI060H TOqKe Tp8eKTopHH H8np8BJleHa no KaCaTeJlb-
HOH K TpaeK Top HH (CM. pHC. 1.24). MOn;Y Jlb CKOpoCTH OnpelIeJl.HeTC.H no cpopMYJle
v =  v; + v; =  v cos 2 a + (vo s in a - gt)2 =
=  v - 2vogt sin a + g2t 2 .
TaKHM 06pa30M, )J;BHmeHHe TeJla, 6pomeHHoro no,A yrJloM K ropH30HTY HJlI1
B ropH30HTaJIbHOM HanpaBJleHHI1., MomHO paCCMaTpHBaTb KaK pe3YJlbTaT ,ABYX
He3aBHCHMbIX )J;BHmeHHH - ropH30HTaJlbHOrO paBHoMepHoro H BepTHKaJIbHOrO
paBHOYCKopeHHoro (cB060,AHOrO na,AeHHH 6e3 Ha-qaJlbHOH CKOpOCTH HJlH ,ABH-
meHH.H TeJla, 6pomeHHoro BepTHKaJIbHO BBepx).
17

.l(BH*eHHe TeJla no OKPY*HOCTH C nOCTOHHHOii no MOAYJlIO CKOpOCTLIO -
3TO ,n;BHmeHHe, npH KOTOpOM TeJlO 3a JlI06bIe paBHbIe npOMemYTKH BpeMeHH
onHCbIBaeT O)J;HHaKOBbIe ,n;yrH.
nOJlOmeHHe TeJla H8 oKpymHocTH onpe)J;eJl.HeTC.H
Pa,zJ;HYCOM-BeKTopOM r, npOBe)J;eHHbIM H3 IJ,eHTpa OK-
pymHoCTH. MO,n;YJlb pa.u;Hyca-BeKTopa paBeH pa.u;HYCY
oKpymHocTH R (pHC. 1.25).
3a BpeM.H At TeJlO, JJ.BIU8.HCb H3 TOqKH A B TOq-
KY B, COBepmaeT nepeMemeHHe tJ..;:, paBHoe xop,n;e
x
AB, H npoxo,n;HT nYTb, paBHbIH ,n;JlHHe )J;yrH t.
Pa,n;Hyc-BeKTop nOBOpaqHBaeTC.H Ha yroJl cp.
YroJl BbIp8m81OT B pa,n;HaHax.
CKOPOCTb v ,n;BHmeHH.H TeJla no TpaeKTopHH (OK-
pymHocTH) HanpaBJleH8 no KaCaTeJlbHOH Ie TpaeKTO-
pHH. OHa Ha3bIBaeTC.H J1HBeiiBOH CKOpOCTLIO. MO,n;YJlb JlHHeHHOH CKOpOCTH pa-
BeH OTHomeHHIO )J;JlHHbI )J;yrH oKpymHocTH l K npOMemYTKY BpeMeHH 6.t, 3a
KOTOpbIH 3Ta Ayra npOHlIeHa:
V B
Puc. 1.25
1.9. J];BH3teHHe TeJIa no OKPY3tHOCTH
C nOCTOHHHOH no MO.a;YJIIO CKOpOCThlO
l
v--.
At
CKaJI.HpHa.H cpH3HqeCKa.H BeJlHqHHa, qHCJleHHO paBHaSi OTHomeHHIO yrJla no-
BopoTa pa.u;Hyc8-BeKTopa K npoMemYTKY BpeMeHH, 3a 1e0TOpbIH 3TOT nOBopoT
npOH30meJl. Ha3bIBaeTC.H yrJloBoii CKOpOCTLIO:
Acp
w - -- -.
At
B CM e,ltHHHIIeH yr JlOBOH CKOpOCTH HBJIHeTCH pll,ltH8H B ceKYHl\Y (P).
IIpH paBHoMepHoM ,n;BHmeHHH no oKpymHocTH yrJlOBa.H CKOpOCTb H MO,n;YJlb
JlHHeHHOH CKOpOCTH - BeJlHqHHbI nOCTO.HHHbIe: w - const; v - const.
IIoJlomeHHe TeJla MomHO onpe,n;eJlHTb, eCJlH H3BeCTeH MO,n;YJlb pa)J;Hyca-
BeKTOpa r H yroJl q>, KOTOpbIH OH COCTaBJISieT C OCblO Ox (yrJIOB8..H Koop)J;HHaTa).
ECJlH B HaqaJIbHbIH MOMeHT BpeMeHH to - 0 yrJlOBa.H Koop)J;HHaTa p8.BHa CPo, a
B MOMeHT BpeMeHH t OHa paBH8 q>, TO yroJl nOBopOTa Acp pa,n;Hyca-BeKTopa 3a
BpeM.H tJ..t - t - to - t paBeH tJ..cp - cp - CPo' Tor)J;a H3 nOCJle)J;HeH <l>OPMYJlbI Mom-
HO nOJlyqHTb KHHeMaTHqeCKOe ypaBBeHHe ABH*eHHH MaTepHaJlLBOii TOqKH no
OKPY*HOCTH:
<p - CPo + wt.
OHO n03BOJlSleT onpe,n;eJlHTb nOJlOmeHHe TeJla B JlI060H MOMeHT BpeMeHH t.
l l v
YqHTbIBaH, qTO I1cp - R ' nOJlyqaeM: w - Rl1t - Ii 
v - wR - cpopMYJla CB.H3H Mem)J;y JlHH€HHOH
H yrJloBoH CKOpOCTblO.
18

llpOMemYTOK BpeMeHH T, B TeqeHHe KOToporo TeJlO COBepmaeT OHH nOJl-
HbIH 060POT. Ha3bIB8eTC.H nepHOOM BpaeHHJI:
T _ !!t
N'
r)J;e N - qHCJlO 060POTOB, COBepmeHHbIX TeJlOM 3a BpeM.H !!t.
3a BpeM.H !!t = T TeJlO npOXOHT nYTb l - 2nR. CJleOBaTeJlbHO,
2nR
v - . .
T'
21t
w-
T
BeJlH'iHH8 V, 06paTHa.H nepHOY, nOKa3bIBalO8.H, CKOJlbKO 060pOTOB COBep-
maeT TeJlO 3a e)J;HHHY BpeMeHH, Ha3bIBaeTC.H '1aCTOTOH BpaeHHJI:
CJle)J;OBaTeJlbHO,
1 N
v - - --.
T !!t
v - 2nvR; w - 2nv.
1.10. YCKopeHHe npH BHJKeHHH TeJIa no OKPYJKHOCTH C nOCTOJlHHOH
no MO.D;YJIIO CKOpOCTblO (eHTpocTpeMHTeJIbHOe YCKopeHHe)
llpH paBHOMepHOM Bp8Il1eHHH no oKpymHocTH MOYJlb CKOpOCTH v .l{BHme-
HHSI TeJla He H3MeH.HeTC.H, HO HanpaBJleHHe CKOpOCTH H3MeH.HeTC.H HenpepbIB-
HO. CJleOB8TeJlbHO, aHHoe BHmeHHe - )J;BHmeHHe C YCKopeHHeM. OHO xa-
paKTepH3yeT 6bICTPOTY H3MeHeHH.H CKOpOCTH no HanpaBJleHHIO.
II - u Tpe Q I A D
o onpeeJleHHIO cpeHero YCKopeHH.H < a > - -. yroJlbHHKH  H
At
BCD - paBH06e)J;peHHbIe (pHC. 1.26). YrJlbI npH BepmHHax - O)J;HHaKOBbIe
(K8K yrJlbI C COOTBeTCTBeHHO nepneH.l{HKYJI.HpHbIMH cTopoHaMH). OTclOa cJle-
)J;yeT, qTO !!OAB no06eH !!BCD.
I Au l l /:1r l v I Ar l
M3 nO)J;06H.H - - - => IAul- -.
v R R
A
V o
v IArl
TorlIa < a> - -'-.
R At
MrnOBeHHoe YCKopeHHe
a - lim < a > -
M-+O
v. I!!rl v 2
- -11m - --.
R t-+o!!t R
J3 - yroJl Memy Uo H Au ( < a > )
BHemHHH no OTHomeHHIO K !! BCD:
A A 180 - A<p 90 A<p
p - u<p + - + - .
2 2
TIpH At - 0 yrOJI !!cp - 0 H, CJleOBaTeJlbHO, f3 - 90°. TIepneHHKYJI.HpOM
K KaCaTeJlbHOH K oKpymHocTH .HBJI.HeTC.H pa)J;HYC. CJle)J;oBaTeJlbHO, a HanpaB-
Puc. 1.26
19

JleHO no paHYCY K eHTpy H n03TOMY Ha3bIBaeTCH eH-
TpOCTpeMHTeJILHLIM YCKopeHHeM:
v 2
a=-.
R
v
MO.qYJIb a = const, HanpaBJleHHe ii HenpepbIBHO H3-
MeH.HeTCH (pHC. 1.27). II03ToMY aHHoe BHmeHHe He
.HBJI.HeTC.H paBHOYCKopeHHbIM.
Puc. 1.27
1.11. BpaaTeJILHOe ]J;BHeHHe a6COJIIOTHO TBep]J;oro TeJIa
BOKpyr Heno]J;BHHOH OCH
A6COJllOTHO TBep,noe TeJIO - TeJlO, H3MeHeHHeM pa3MepOB H <}>OpMbJ KO-
Toporo MomHO npeHe6peqb.
AJI.H KHHeMaTHqeCKorO OllHCaHH.H BpaaTeJlbHOrO BHmeHH.H a6COJllOTHO
TBeporo TeJl8 BOKpyr HelloBHmHoH OCH HcnOJlb3YIOTC.H Te me BeJlHqHHbI, qTO
H JI.H OllHCaHH.H BH:>KeHH.H MaTepHaJlbHOH TOqKH no OKPY:>KHOCTH.
IIpoMemYToK BpeMeHH, B TeqeHHe KOToporo TeJIO COBepmaeT OHH nonHbIH
060POT BOKpyr OCH, - nepHO,n Bpam;eHHJI. BeJlHqHHa, 06paTHa.H nepHOy, -
qaCTOTa Bpam;eHHJI.
TIpH BpaaTeJlbHOM BHmeHHH a6COJllOTHO TBeporo TeJla BOKpyr Heno-
BHmHOH OCH YZ.n.bt nOBopoma paiJuyca-6eKmopa pa3.n.U'IllibtX mOlleK me.n.a oiJu-
liaK06bt.
Y rOJl nOBopOTa H yr JlOBa.H CKOpOCTb xapaKTepH3yIOT BHmeHHe Bcero a6co-
JlIOTHO TBeporo TeJla B neJIOM. J!UlieUIia.R CKopocmb KaKou-.n.u60 mOIlKU a6co.
.n.lOmliO m6epiJozo me.n.a nponOpijUOIiaJl.blia paccmOSlHUIO mOIlKU om OCU 6paU{eIiUSl:
21t
v = (J)R = 21tvR = -R.
T
TIpH paBHOMepHOM BpaeHHH 86cOJllOTHO TBeporo TeJla YZ.n.bt nOBopoma
me.n.a 3a .n.106bte pa6libte npO.MeJICymKu 6peMeliu oiJuliaK06bt. malizeliqua.n.blibte
YCKopeIiU.R y pa3.n.ulllibtX mOlleK me.n.a omcymcm6YlOm. a 1i0p.Ma.n.blioe YCKope-
liue mOIlKU me.n.a 3aBucum om ee pacCmO.RIiUSl iJo ocu 6paU{eIiUSl:
v 2 2 2 2 41t 2
an = - = (J) R = 41t V R = - R.
R T 2
BeKTOp iin HanpaBJIeH no paHYCY TpaeKTopHH TOqKH K OCH BpaeHHH.
TIpH HepaBHOMepHOM BpaeHHH a6COJllOTHO TBeporo TeJla 6bICTPOTY H3MeHe-
HH.H yrJloBoH CKOpOCTH co BpeMeHeM xapaKTepH3yeT yrnoBoe YCKopeHHe:
_ (J) _ (J)2 - (J)I
E--- .
t !J.t
ECJlH E = const, TO TeJlO COBepmaeT pa6liOYCKopeliHoe BpaeHHe H B MO-
MeHT BpeMeHH t yrJlOBa.H CKOpOCTb
(J) = (J)o + Et.
20

YroJl nOBOpOTa <p TeJla BOKpyr OCH 3a BpeM.H t npH paBHonepeMeHHOM BH-
meHHH onpe,neJl.HeTC.H no (}JopMYJle
Et 2
<p = root +-.
2
BOIIPOCbI If 3A]I;AHIf.H JJ;JUI CAMOKOHTPOJI.H
1. qmo nonUMalOm noiJ cucmeMOU omcllema? cucmeMOU KoopiJuliam? pa-
OU YCOM -6e K mopoM?
2. YKaJICUme, 6 KaKUX IiUJlCec.n.eoylOU{UX c.n.YllaSlX me.n.o MOJICIiO npUliSlmb
3a Mamepua.n.bliYIO mOIlKY. a) npu ycmalioBKe paKembt Ha cmapme; 6) npu
paclleme mpaeKmopuu paKembt,' 6) npu pacileme YZ.n.060U CKopocmu cymollHo
ZO 8paU{eliUSl 3eM.n.u 80KPYZ ocu.
3. KaKue cyU{ecm6YlOm cnoc06bl onucaliUSl iJ8UJlCeliUSl Mamepua.n.bliou
mOIlKu?
4. M Slll yna.n. c 8btCOmbt 5 M, omCKOIlU.n. om no.n.a U noiJn.RJlC.R Ha 2 M.
H auiJume nymb U nepeMeU{eliue MSllla.
5. KaKue KUlieMamUlleCKue 6e.n.UIlUlibt 3a6UCSlm om 6bt60pa cucmeMbt om-
cilema? oiJuliaK06bt 6 pa3.n.UlllibtX cucmeMax omcilema?
6. M oJICem .n.u lle.n.06eK. liaxooSlcb lia o6uJlCyU{e.MCSl 3cKa.n.amope Mempo.
6btmb 6 nOKoe 6 Cucme.Me omcilema. c6Sl3aliliOU c n06epXliocmblO 3eM.n.u?
7. C08naoaem .n.u lianpa6.n.eHue YCKopeliuSl C Hanpa6JleHUeM cKopocmu Ma-
mepua.n.bliou mOIlKU npu pa81i0ycKopelilioM iJ6UJlCeliuu? npu pa6li03aMeiJ.n.elilioM
iJ6UJlCe liUU?
8. KaKue KUlieMamUlleCKue xapaKmepucmUKU iJ6UJlCeliUR ocmalOmc.R nocmo-
SllilibtMU npu pa6liOMepiiOM npSl.Mo.n.ulieulioM 06UJlCeHuu? npu pa8lioycKopeliHoM
iJ6UJlCeliuu?
9. KaKue 8e.n.Ullulibt, xapaKmepu3YIOU{ue 06UJlCeIiUe, .MOJICliO onpeoe.n.umb no
zpacpuKY cKopocmu?
10. J(6a noe3iJa uiJym lia6cmpeilY iJpyz opyzy: OiJUIi YCKopeliliO lia ce6ep.
opyzou 3aMeiJ.n.elilio lia IOZ. KaK 6yiJym lianpa6.n.elibt 8eKmOpbt YCKopeliuu no
e3iJ06?
11. q eM om.n.UllalOmcSl iJ8UJlCeIiUSl, ypa6lieliUSl KomOpbtX npu6eiJelibt liUJlCe?
Xl = 3 - 5t - 2t 2 ; X 2 = -3 + 5t - 2t 2 .
12. Cpa6liUme 8peMSl naiJeliuSl me.n.a, 6poweliliozo zopU30nma.n.bH.O C H.eKO-
mopou 6btCOmbt U c60600liO naiJalOU{ezo c mou JlCe 6btCOmbt.
13. Tpu me.n.a 6powelibt maK: nep80e - 6liU3 6e3 Halla.n.bliOU CKopocmu,
6mopoe - 6liU3 C Halla.n.bliOU cKopocmblO. mpembe - 66epx. q mo MOJICIiO CKa.
3amb 06 YCKopeliuSlx 3mux meJl npu ux 06UJlCeliuu?
14. KaK 6yoem U3MeliSlmbCSl oa.n.bliocmb no.n.ema cliapSliJ06 npu Y6e.n.ulleHuu
yz.n.a liaK.n.Olia OpyOUSl K Zopu30limy?
15. KaK nanpa8.n.eliO YCKopeliue npu Kpu60.n.ulieuHoM iJ6U:HCeliuu?
16. KaK nanpa6JleHa .MZli08elilia.R CKopocmb Mamepua.n.bliou mOIlKU npu KpU-
80.n.UlieUliOM iJ8U:HCeliuu?
17. .H 8.n.SlemcSl .n.u iJ6UJlCeliUe no OKpYJICHocmu C nocmOSlliliOU no MOOY.n.1O
cKopocmblO pa8lioycKopelilibtM?
21

18. A6moMot5u.n.b iJ8UJlCemCSl lia n060pome. OiJuHaKo-
6ble .n.u paccmOSlHUSl npoxoiJSlm npu amOM npa8ble U .n.e-
6ble Ko.n.eca a6moMot5u.n.Sl?
19. Be.n.ocuneiJucm iJ8UJlCemCSl no npSlMo.n.uHeUIiO.MY
YltaCmKY iJopozu co cKopocmblO V (puc. 1.28). C Ka-
KOU cKopocmblO iJ8UJlCymcSl mOltKU A. B. C. D Ko.n.eca
6e.n.ocuneiJa omHocume.n.bHo OCU? omliocume.n.bliO iJo-
pozu?
20. Bblpa3ume 6 paiJuaHax YZo.n. n080poma llaC060U
cmpe.n.KU 3a 1 It. 3 It. 6 Il. 12 Il. 24 It.
21. Ko.n.eco 6pa14aemcSl 80KPYZ lienoiJ6UJlCiiOU OCU. npoxoiJSl14eU llepe3 ezo
ijelimp. Ot5.n.aiJaem .n.u .n.1Ot5aSl mOltKa lia ot5oiJe Ko.n.eca malizeliijua.n.bliblM U
liopMa.n.bliblM YCKopeliuSlMu, ec.n.u 6pa14eliUe npoucxoiJum c nocmOSlliliOU yz.n.o-
80U cKopocmblO? c nocmOSlHliblM YZ.n.06blM YCKopeHueM?
22. Kamep co cnopmCMeliOM lia 60iJHblX JlbtJICaX iJ6UJlCemCSl no o KPYJICIiOC-
mu. CnopmcMeli MOJICem c.n.eiJ06amb 3a IiUM no mou JlCe OKpYJlCIiOcmu. a maK-
JfCe 6He u 6Hympu amou OKpYJICliocmu. KaK060 coomHoweliue cKopocmeu
cnopmcMeHa u Kamepa 6 amux c.n.yltaJlx?
B
D
Puc. 1.28
Af - f - fo
_ Af
<v>--
At
OCHOBHbIE OPMYJIbI
- nepeMeeHlfle rel1a (MaTeplflal1bHOLii rOKIfI). rAe fo IfI f
PCiAlfIycbI-BeKTopbl B HaaI1bHbll7I to IfI KOHeHbll7I t MOMeHTbI Bpe-
MeHIfI.
- cpe.nH CKOpOCTb nepeMeeHIfI Tel1a, rAe l:i.r - nepeMee-
Hlfle Tel1a 3a npoMeoK BpeMeHIfI At.
Ax Ar x
< v x > - At - 6t- - npoeK cpeAHel7l CKOpOCTIfI Ha OCb Ox. rAe l:i.x - 1f13MeHeHlfle
KooPAIfIHaTbi Tel1a 3a npOMe)KyroK BpeMeHIfI 6. t, Arx - npoeK
nepeMeeHIfI Ha OCb Ox.
S
<v>--_.
6t
- 1 . 6r -
v - lID - - r'
61--0 6t
I . Ax ,
v - 1m - x
x 61--0 At
- cpeAH CKOpOCTb nyrlfl, rAe s - nyrb, npol7lAeHHbll7I rel10M 3a
npOMe)KyroK BpeMeHIfI At.
- MrHOBeHHa CKOPOCTb. rAe r' - npOIfl3BOAHa OT PaAlflyca-BeK-
Topa no BpeMeHIfI.
- npoeKIfI MrHoBeHHol7I CKOpOCTIfI Ha OCb Ox.
_ Au v - V o
< a > - -- - - - cpeAHee YCKopeHlfle Tel1a. rAe 6V - 1f13MeHeHlfle CKOpOCTIfI 3a
At At
npOMe)KyroK BpeMeHIfI At. V o IfI v - CKOpOCTIfI Tel1a B HaaI1b-
Hbll7I ." KOHe'iHbll7I MOMeHTbI BpeMeH...,.
22

- 1 . Av -I
a.. 1m - - v
At
v - VI + V2
Ar ,.. vt
Arx - vxt
ii-O
s -IArl -Iv xl t
x - xo + vxt
V - V o + iit
V x - VOx + axt
_ _ iit 2
I1r - vot + -
2
a t 2
Ar - V o t + .....L-
x x 2
(vox + v x )
Ar - t
x 2
2 2
V x - vOx
/).r x -
2a x
- MrHOBeHHoe YCKopeHio1e. rAe Vi - npoio13BoAHaSl OT CKOpOCHt
no BpeMeHio1.
3aKOH C110>KeHio1S1 CKopocTe B Kl1aCcw.ecKo MexaHio1Ke. rAe
V - CKOpOCTb Tel1a OTHOCio1Tel1bHO HenOABio1>KHO Cio1CTeMbl OTC'ie-
Ta, VI - CKOpOCTb Tel1a OTHOCio1Tel1bHO nOABio1>KHO Cio1CTeMbl OT-
C'ieTa. v2 - CKOpOCTb nOABio1>KHO Cio1CTeM OTC'ieTa OTHOCio1Tel1b-
HO HenOABio1>KHO.
PaBHOMepHoe npsMoHHHeHHoe ABHeHHe
- nepeMeeHio1e Tel1a 3a npoMe>t<yTOK BpeMeHio1 t, rAe V - const -
CKOpOCTb Tel1a. /).r x io1 V x - npOeKLtio1io1 flepeMeeHio1S1 io1 CKOpOCHt
Ha OCb Ox.
- YCKopeHio1e Tel1a.
- nyrb 3a npOMe>KyroK BpeMeMio1 t.
- Kio1HeMaTio1'ieCKOe ypaBHeHio1e paBHoMepHoro ABio1>KeHio1S1. rAe x -
KooPAio1HaTa Tel1a B MOMeHT BpeMeHio1 t. Xo - Ha'iaJ1bHaSl KOOPAio1-
HaTa Tel1a.
PaBHoYCKopeHHoe npsMoHHHeHHoe ABHeHHe
- CKOpOCTb Tel1a B MOMeHT BpeMeHa.t t. rAe Vo - CKOpOCTb Tel1a
B Ha'iaJ1bHbl MOMeHT BpeMeHio1 to - O. d - const - YCKopeHio1e Tel1a.
- nepeMeeHio1e Tel1a 3a npOMe>KyroK BpeMeHio1 t
- npOeKLtio1 nepeMeeHio1S1 Tel1a 3a npOMe>KYToK BpeMeHio1 t
Ha OCb Ox.
a t 2
X - Xo + voxt +.....L- - Kio1HeMaTio1'ieCKOe ypaBHeHio1e paBHoycKopeHHoro ABio1>KeHio1S1 Tel1a.
2
G M
gh - ( ) 2
R+h
M
g-G-
R 2
CBo6oAHoe D8AeHHe TeH
YCKopeHio1e cBo60AHoro naAeHio1S1 Ha BCOTe h (6e3 Y'ieTa cy-
TO'iHOrO BpaeHio1S1 nl1aHeTbl). r,ne G - rpaBio1Tau.io10HHa nOCTOSlH-
HaSl, M - Macca nl1aHeT, R - PaAio1YC nl1aHeT. h - BblCOTa Tel1a
HaA nOBepXHOCTblO nl1aHeTbl
- YCKopeHio1e cB060AHoro naAeHio1S1 Ha nOBepXHoCTio1 nl1aHeT (6e3
Y'ieTa ee cyrO'iHOrO BpaeHio1).
23

v., - Vo., + g.,t
- npOeKl.t&.1s:1 CKOpOCT&.1 CB060AHO n(tAalOw,ero Tena B MOMeHT Bpe-
MeH&.1 t. rAe vO y - npOeKl.t&.1s:1 Ha8nbHO CKOpOCT&.1. gJ/ - npOeKl.t&.1s:1
YCKopeH&.1s:1 cB060AHoro n(tAeH&.1s:1 Ha OCb Oy.
g t 2
Y - Yo + vol + -i- - K&.1HeMaT&.1eCKOe ypaBHeH&.1e. On&.1CblBalOw,ee cB06oAHoe n(tAeH&.1e
Ten. rAe Yo - Ha8nbHas:l KOOpAJ.'1HaTa Tella.
t h
t-
g
v - 2gh
t - Q
1 g
v 2
h.nax - 2
2vo
t 2 - -
g
f2h
I - Vo 'f g
Vo sin a
t 1 - - -
g
v sin 2 a
h.nax - 2g
2 . 2
I _ Vo sin a
g
- BpeMs:I n(tAeH&.1s:1 Tena Ha 3eMnio C BbiCOTbi h H(tA ee nOBepx-
HOCTblO 6e3 Ha8nbHO CKOpOCT&.1.
- CKOpOCTb rena B MOMeHT n(tAeH&.1s:1 (UpH Vo - 0) C BbiCOTbi h.
- BpeMs:I nO.D.beMa Tena. 6poweHHoro BepT&.1KanbHO BBepx. Ha MaK-
C&.1M8nbHYIO BblCOT)'. rAe Lio - CKOpOCTb B MOMeHT 6pOCaH&.1s:1
- MaKC&.1ManbHas:l BbiCOTa nOAbeMa rena. 6poweHHoro BepT&.1KanbHO
BBepx co CKOpOCTbtO Vo.
- BpeMs:I non eTa Tena. 6poweHHoro BepT&.1K8nbHO BBepx co CKO-
pOCTbtO Vo.
- AanbHOCTb noneTa Tena. 6poweHHoro rOp&.130HT8nbHO C BblCO-
Tbi h C Ha8nbHO CKOpOCTbtO Lio.
- BpeMs:I nOAbeMa Ha MaKC&.1M8nbHYtO BblCOT)' Tena. 6poweHHoro
nOA yrnoM a K rOp&.130HT)'. rAe Vo - Ha8nbHas:l CKOpOCTb Tena.
- MaKC&.1M8nbHas:l BbiCOTa nOAbeMa Tena. 6poweHHoro nOA yrnoM
K rOp&.130HT)'.
- ACU1bHOCTb nOlleTa rella. 6poweHHoro nOA yrnoM K rOp&.130HT)'
ABHeHHe TeJla no OKPYHOCTH C nOCTOSlHHOH no MO.IJ.YJlIO CKOpOCTLIO
I
u-
ht
hq>
w - - --
ht
v-wR
T _ ht
N
1
v----
T
24
- ll&.1HeHas:l CKOpOCTb Tena. rAe l - AJ1&.1Ha Ayr&.1 OKPY>KHOCT&.1.
npoiilAeHHoiil TenOM 3a npOMe>KyroK BpeMeH&.1 ht.
- yrnoBas:l CKOpOCTb Tella. rAe L\q> - yroll nOBopoTa PaA&.1yca-BeK-
Topa AB&.1>Kyw,erocs:I no OKPY>KHOCT&.1 Tella 3a npOMe>KyroK BpeMe-
H&.1 I1t.
- CBs:l3b n&.1HeiilHOiil CKOpOCT&.1 C yrnoBoli1. rAe R - PCiA&.1yc OK-
PY>KHOCT&.1
- nep&.10A Bpaw,eH&.1s:1. rAe N - &.1cno 060POTOB Tella 3a npoMe-
>KyrOK BpeMeH&.1 I1t
- aCTOTa Bpaw,eH&.1s:1.

21tR
v---21tRv
T
- CBS13b Me)t(J1Y lWIHeHO CKOpOCTblO, nepli10AOM Bpaw.eHIi1S1  a-
CTOTO.
v 2
a - -- - w 2 R -
U R
2 2 41t 2
- 41t V R - --- R - eHTpOCTpeMTenbHoe YCKopeHe.
T 2
METOJ(HQECKHE PEKOMEHAAHH
HeCMOTp.H Ha 60RbIlloe p83H006p83He 3a,u,aq no KHHeMaTHKe. MomHO npe,A-
JlOmHTb CJle)J;YIOHH aJIrOpHTM HX peIlleHH.H:
1. CAeJl8Tb CXeMaTHqeCKHH PHCYHOK, H306pa3HB HatIa.n:bHoe nOJlOmeHHe TeJl
H HX H8qaJIbHOe COCTO.HHHe, T. e. Vo H ii.
2. BbI6p8Tb CHCTeMY OTC'leT8 Ha OCHOB8HHH aHaJIH38 YCJlOBH.H 3a,u,8tIH. ,L(JI.H
3Toro !lymHo BbI6p8Tb TeJlO OTCqeTa H CB.H38Tb C HHM CHCTeMY KOOPAHH8T,
YK838B Haq8JIO OTCqeTa KOOPAHHaT, H8npaBJleHHe oceu KOOPAHHaT, MOMeHT H8-
tIaJla OTCtIeTa BpeMeHH. IIpH BbI60pe nOJlOmHTeJlbHbIX Hanp8BJleHHH PYKOBOA-
CTBYIOTC.H HanpaBJleHHeM ABHmeHH.H (CKOpOCTH) HJlH HanpaBJleHHeM YCKO-
peHH.H .
3. COCT8BHTb H8 OCHOB8HHH 38KOHOB )J;BHmeHH.H CHCTeMY YP8BHeHH" B BeK-
TOpHOM BHAe AJI.H Bcex TeJl, a 3aTeM B CKaJI.HpHOH c»opMe, CnpOeII.HpOB8B Ha
KOOPAHHaTHbIe OCH 3TH BeKTopHbIe ypaBHeHHSi ABHmeHH.H. IIpH 3anHCH 3THX
ypaBHeHHH cJleAyeT 06paTHTb BHHMaHHe H8 aHaK!i .. +" H .. -.. npOeKI1,HH BXO-
)J;.HHX B HHX BeKTopHbIX BeJlHqHH.
4. OTBeT He06xo,AHMO nOJlyqHTb B BHAe 8H8J1HTHqeCKOH <POPMYJlbI (B 06-
eM BHAe), a B KOHl1,e npOH3BeCTH qHCJlOBbIe p8C'leTbI.
nPHMEPbI PEIlIEHH.H 3A):(A q
1. CKonbKO BpeMeHH naCCWKHp, CH,ASIHH Y OKH8 noe3,Aa, KOTOpblH H,AeT co CKO-
KM
pOCTblO 54 -, 6Y,AeT BH,AeTb npOXO,ASIHH MHMO Hero BCTpe'lHblH nOe3,A, CKOpOCTb
'I
KM
KOToporo 36 -, a ,AnHH8 250 M?
'I
M M
)J,aHo: VI = 15 -, v2= 10 ,I = 250 M.
C C
HaHTH: t.
Pewe H He. HenO,ABH)KHYIO CHCTeMY OTCqeTa CBSI)KeM C 3eMneH, nO,ABHmHYIO -
C nOe3,AOM, B KOTOpOM H8XO,AHTC.H naCC8mHp. CornaCHO 38KOHY cnomeHHSI CKOpOC-
TeH V 2 ... V 2I + VI' r,Ae V21 - CKOpOCTb BCTpe'lHOrO nOe3,A8 OTHOCHTenbHO nepBoro.
B npOeKI1,HSlX Ha OCb Ox (pHC. 1.29):
v2 - v21 - VI => V21 - VI + V2.
TaK KaK nYTb, npOH,AeHHblH BCTpetIHblM nOe3,AOM
OTHOCHTenbHO nepBoro, p8BeH AnHHe noe3,Aa, TO BpeMSI
.
o
VI

V 21

V 2

,.
x
Puc. 1.29
t-
t -10 c.
V21
VI + v2
25

2. TIapoxo,A H,AeT OT HHmHero HOBropo.n;a ,AO ACTp8X8HH 5,0 CYTOK, a 06paT-
HO -- 7,0 CYTOK. KaK ,Aonro 6Y,AeT nJlbITb nnOT OT HHmHero HOBrOpO,Aa ,AO ACT-
paxaHH? CTOSlHKH If 3a.n;epmKH B ,ABHmeHHIf HCKnlO'llfTb.
AaHo: t l =5,O CYT., t 2 =7,O CYT.
HaHTH: t3'
Pew e H H e. HenO,ABHmHYIO cHcTeMY OTCqeTa CBSlmeM C 6eperoM, nO,ABHm-
HYIO -- C BO,AOH. By.n;eM C'lHT8Tb, qTO CKOpOCTb BO.n;bI Ha BceM nYTH O,AHHaKOBa I{
CKOpOCTb napoxo,Aa OTHOCHTenbHO BO.n;bI nOCTOSlHHa If paBHa Mo.n;ynlO MrHoBeHHoH
CKOpOCTH napoxo,Aa OTHOCHTenbHO BO,AbI.
T8K KaK nnOT ,ABHmeTCSI OTHOCHTenbHO 6epera co CKOpOCTblO Te'leHHSI peKH up'
8
TO BpeMSI ero ,ABHmeHHSI t 3 - -, r,Ae 8 p8CCTOSlHHe Mem,AY rOpO,A8MH. TIpH
V p
,ABHmeHHH napOXO,Aa no Te'leHHIO (pHC. 1.30) ero CKOpOCTb cornaCHO aaKOHY cno-
meHHSI CKopocTeH v] - VII + v p , HJIH B npoeKu.HH Ha
OCb Ox
Va V p VI
---.. --. .
. .
0 X
Puc. 1.30
VI - V n + Vp'
(1)
r,Ae VI -- CKOpOCTb napOXO,Aa OTHOCHTenbHO 6epera,
V n -- CKOpOCTb napOXO,Aa OTHOCHTeJlbHO peKH.
3HaSl BpeMSI ,ABHmeHHSI, MomHO HaHTH CKOpocTb:
8
VI - --.
t 1
(2)
8
H3 q>opMyn (1) If (2) HMeeM: - V n + Vp' (3)
t l
TIpR .n;B1:ImeHHH napoxo.n;a npOTHB TeqeHHSI
(pHC. 1.31) U2 - Un + Up, HnH B npoeKu.HH H8 OCb OX
V2 - V n - V p , r,Ae V2 - CKOpOCTb napOXO,Aa OTHOCHTenb-
HO 6eper8.
V 2 Va V p
..-- ......- --.
.. .
X 0
Puc. 1.31
8
C ,APyroH CTOpoHbI, V2 - ---. Tor,Aa
t 2
8
- VII - Vp'
t 2
PellIaSl CHCTeMY ypaBHeHHH (3) H (4) OTHOCHTenbHO VP' nonyqlfM:
(4)
\ 
t 2
- V n + V p
8(t 2 - td
=> V p - - .
2t l t 2
- V n - V p
 8 2t l t 2
HaH,lleM BpeMSI ,ABHmeHlfSl nnOTa: t 3 - . - --- -, t3 = 35 CYTOK.
V p t 2 - t l
3. llpH paBHOYCKopeHHOM ,ABHmeHHH Teno npOXO,AHT aa .n;Ba nepBbIX paBHbIX
nOCJIe,AOBaTenbHbIX npOMemYTKa BpeMeHH no 4,0 C Ka>K,AbIH nYTIf 81 = 24 M H
82 = 64 M COOTBeTCTBeHHO. Onpe,AenHTe Ha'l8nbHYIO CKOpOCTb If YCKopeHlfe TeJla.
AaHo: tl = t2= 4,0 C, 81 = 24 M, 82= 64 M.
H 8HTH: vo, a.
26

P e ill e H H e. 3anHweM yp8BHeHHH nYTH ,AnH 81 Ii (81 + 82) COOTBeTCTBeHHO
(pHC. 1.32). T8.K K8.K Ha'laJIbH8.H CKOpOCTb B 3TOM cnyqae O,AHHaKOBa, TO
V o VI
 
j. 8 1 .,. 8 2 .f
Puc. 1.32
! at2
81 - votl T-1-;
2
a (t 1 + t 2 )2
8 1 +  - Vo (t 1 + t 2 ) + 2
(1)
TaK KaK tl = t 2 , TO
Bt +  - 2v o t 1 + 2at; .
BbIp83HB H3 (1) Vo H nO,ACTaBHB ee B (2), nOJIyqHM:
82 -8 1 M
a - - 2 -, a - 2, 5 2'
t 1 C
(2)
Tor,Aa Ha'lanbH8S1 CKOpocTb
28 1 - at; M
Vo - , Vo - 1,0
2t 1 C
4. ABToM06Hnb, ,ASHraSlCb no npSlMonHHeHHoH TpaeKTopHH P8BHOYCKopeHHO C
M
Ha'laJIbHOH CKOpOCTblO 5,0 ,npowen 3a nepBYIO ceKYH,AY nYTb, paBHbIH 6,0 M.
C
HaH,AHTe YCKopeHHe 8BTOM06HJISI, MrHOBeHHYIO CKOpOCTb B KOHqe BTOpOH ceKYH,AbI
H nepeMelI\eHHe 3a 2.0 c.
M
y
,l'(8HO: vo-5,0 81-6,0 M, t 1 -1,0 c, t2==2,0 c.
C
HaHTH: a, V2, 8.
Pewe H H e. 3HaH nYTb, npOH,AeHHbIH TeJlOM 3a nepBYIO ceKYH,AY, MomHO HaH-
at; 2(8 - votd M
YCKopeHHe: B] - v o t 1 +  a - 2 . a - 2,0 2'
2  C
CKOPOCTb B KOHqe BTOpOH ceKYH,AbI HaH,AeM no <l>opMYJle
M
v2 - Vo + at 2 . V2 = 9.0 .
C
IIepeMelI\eHHe 38 2 C MomHO paCC'lHT8Tb no <l>opMyn8M:
2 2 2
at 2 v 2 - V o
6.r - V o t 2 T  HnH 6.r - -- -- r-14 M.
2 2a '
5. Teno 6poweHo BepTHKaJIbHO BBepx C Ha-
q8nbHOH CKOpOCTblO vo. Ior,Aa OHO ,AOCTHrno
MaKCHMaJlbHOH BblCOTbI, C Toro me MeCTa H C
A

I
I
I
I
Ji
TOH me Ha'lanbHOH CKOpOCTblO Vo 6poweHo BTO-
poe Teno. Ha KaKOH BbICOTe HAA nOBepXHOCTblO
3eMnH Tena BCTpeTSlTcSI?
,l'( a H 0: vo.
x H a H T H: YS'
P e ill e H H e. Ha'laJIo KooPAHHaTHOH OCH Oy CO-
BMeCTHM C nOJlOmeHHeM BTOporo TeJl8 (pHC 1.33).
TH
hmu
o
Puc. 1.33
27

38nHllleM KHHeM8THqeCKHe ypaBHeHHSI ,ABHmeHHSI .n;nSl nepBOrO H BToporo Ten
(OTC'leT BpeMeHH Be,AeM C MOMeHTa 6pocaHHSI BToporo Tena) B npOeK1I.HSlX Ha
OCb Oy:
g t 2 g t 2
Yl = YOI + V o1 i + -L..; Y2 = Y02 + V 02y t + -L..,
2 2
V 2
r.n;e YOl = h max = 2 ; v0 1y = 0; g y = -g; Y02 = 0; v 02y = v o .
v 2 gt 2 gt 2
Cne,AOB8TenbHo, Yl = 2 - 2; Y2 = Vo t - 2 .
B MOMeHT BCTpe'lH t B KOOp.n;HH8TbI Ten o.n;1iHaKOBbI: Yh = Y2B'
V5 gt; gt; V o
Tor,Aa - - - = vot - -  t = -.
2g 2 8 2 8 2g
TIO,lICT8BnSlSl 3H8'leHHe t B B <l>opMyny KOOp.n;1iH8TbI nepBoro Ten8, nonY'lHM:
v 2 g v 2 3 v 2
Y =__.----!L.=_
B 2g 2 4g 2 8 g .
6. C BepToneTa, HaXO,ASIerOCSI Ha BbICOTe 300 M, c6pollleH rpY3. HaH,A1iTe, 'le-
pe3 K8Koe BpeMSI ['J>Y3 ,AOCTHrHeT 3eMnH, ecnH BepToneT: 1) HenO,ABHmeH; 2) onyc-
M M
K8eTCSI co CKOpOCTblO 5.0
C
Y
Yo __!)_=:____2  ___lt v.
: Lio I
I I
i' i! ii
I I I
I I I
I I I
o
Puc. 1.34
3) nO,AHHMaeTCSI co CKOpOCTblO 5.0
C
M M
,ZJ;aHo: vo= 5,0 h = 300 M, g= 10 2'
C C
H8HTH: tl' t 2 , t3'
Pew e H He. COBMeCTHM Ha'l8nO KOOp,AHHaT C
nOBepXHOCTblO 3eMn1i (pHC. 1.34) H KOOp,AHHaTHYIO
OCb H8np8BHM BBepx. 38nHllleM KHHeM8TH'IeC-
Koe ypaBHeHHe ,ABHmeHH.H rpY3a B npOeK1I.HSlX Ha
OCb Oy:
x
g t 2
Y = Yo + vOyt + -1!-
2
B nepBOM cnY'l8e V Oy = 0: Yo = h: gy = -g.
gt 2
3H8'l1iT, Y = h - 2' B MOMeHT n8,AeHHSI rpY38 H8 3eMnIO t = t 1 KOOp.n;HH8T8 Yl = O.
2 f! h
T O = h _ gtl 77
or,Aa 2  t 1 = g' t 1 =, c.
gt 2
Bo BTOpOM cnyqae V Oy = -v o ; Yo = h; g y = -g. Cne,AOB8TenbHo. Y = h - vot - 2-
B MOMeHT n8,AeHHSI rpY3a t = t 2 KOOP,AHH8T8 Y2= O. TOr,A8
gt 2 -Vo +  v + 2gh
o = h - v o t 2 - -  gt 2 + 2v O t 2 - 2h = 0  t 2 = , t 2 = 7.1 c.
2 g
B TpeTbeM cnyq8e vo y = vo; Yo = h; gy = -g; np1i t = t3 KOOp,AHHRTR Y = O.
28

gt 2
o = h + v O t 3 - ---!. 
2
Vo -to  v5 -to 2gh
t 3 = , t 3 = 8.3 c.
g
M, npHKpenneH8 HHTb. 'tJepea 5 C nocne H8'l8n8
Hero H8MOTanOCb 10M HHTH 'tJeMY p8BH8 yr-
7. K B8nY, p8,AHYC KOToporo 5
p8SHOMepHoro Bp8eHHSI B8n8 Ha
nOBaH CKOpOCTb Bp8eHHSI?
JJ;aHo: R=5 M, t=5 C, l=10 M.
H8HTH: 00.
P e ill e H H e. 38 BpeMSI t npoH3BonbH8Sl
TOqKa H8 060,Ae B8na npOXO,AHT nYTb,
l
pa.BHbIH ,AnHHe HHTH. II03TOMY MO,Aynb JIHHeHHOH CKOpOCTH TOqKH V = -. Y'lHTbI-
t
BaSI CBH3b Mem,Ay nHHeHHOH H yrJIOBOH CKOpOCTSlMH, HaXO,AHM:
v l p
00=-=-, w=0,4 -
R Rt C
8. M8nb'lHK paBHoMepHo Bpa8eT K8MeHb, npHBSl38HHbIH K BepeBKe ,AnHHOH
06
50 CM, B BepTHK8nbHOH nnOCKOCTH, ,Aen8S1 3.0 -. H8 K8KYIO BbICOTY B3neTen K8-
C
MeHb, ecnH sepeBK8 060pBanacb B TOT MOMeHT. KOr,Aa nHHeHH8S1 CKOpOCTb 6bIn8 H8-
npaBneHa
BepTHK8nbHO BBepx?
,D;aHO: l=0,50 M. v=3,0 c- 1 .
H8HTH: h max .
P e ill e H He. M8KCHManbHYIO BbICOTY nOn'beMa KaMHSI
v 2
(pHC. 1.35) H8H,AeM H3 <l>opMynbI "-max =...JL. r,Ae Vo - H8'l8nb-
2g
HaSl CKOpOCTb K8MHSI, 6poweHHoro BepTHK8nbHO BBepx, p8BH8fl
nHHeHHOH CKOpOCTH K8MHSI npH ero ,ABHmeHHH no oKpymHocTH
Th mu
I
I
I
I
-
V o
p8,AHYCOM R = l. HaH,AeM 3TY CKOpOCTb no <l>opMyne V o = 21tVl.
Tor,Aa
Puc. 1.35
4n 2 v 2 l 2 2n 2 v 2 l 2
h = = hm" = 4,5 M.
max 2g g 
29

2.0CHOBMAHHAMHKH
3aKOHbi HbIOTOHa. npSlMaSl M 06paTHaSl 3aAa'lM AMHaMMKM. nepBblH 3a-
KOH HbIOTOHa. HepManbHaSl CMCTeMa OTC'IeTa. npMHMn OTHOCMTenbHOCTM
ranMneSl.
BTopoH 3aKOH HbIOTOHa. CMna. Macca.
TpeTMH 3aKOH HbIOTOHa.
CMI1b1 ynpyrOCTM. 3aKOH rYKa.
CMnbl TpeHMSI. K03cS>cS>MI,4MeHT TpeHMSI. ABM>KeHMe Ten C Y'leTOM CMn Tpe-
HMSI.
3aKOH BceMMpHoro TSlrOTeHMSI. rpaBMTaMOHHble CMnbl. CMna TSI>KeCTM.
ABM>KeHMe Tena nOA AeHCTBMeM CMnbl TSI>KeCTM. PaC'IeT nepBOH KOCMM'Ie-
CKOH CKOpOCTM. ABM>KeHMe MCKYCCTBeHHblX cnYTHMKOB.
Bec Tena. HeBeCOMOCTb.
HeMHepl,4ManbHble CMCTeMbl OTC'IeTa. SlBneHMSI, Ha6nlOAaeMbie B HeMHep-
ManbHblX CMCTeMax OTC'IeTa.
2.1. IIepBLIH 3aKOH HLIOTOHa
B HHaMHKe paCCM8TpHBaeTC.H BnH.HHHe B38HMo)J;eHcTBHSI Me.>KY TeJl8MH Ha
HX MeXaHHqeCKOe )J;BHmeHHe.
OCHOBHaJl 38)J;a1Ja AHHaMHKH COCTOHT B onpe)J;eJleHHH nOnO.>KeHH.H TeJla B
npOH3BOJlbHbIH MOMeHT BpeMeHH no H3BeCTHOMY HaqaJlbHOMY nOJlOmeHHIO, H8-
qaJlbHOH CKOpOCTH H CHJlaM, eHcTBYIOHM Ha TeJlO.
TIpli pemeHHH o6paTHoH 3a,zJ;a1JH HymHo YMeTb no 38KOHY BHmeHH.H Tena
onpe,AeJl.HTb eHcTBYIOHe H8 Hero HeH3BeCTHbIe CHnbI.
On bIT nOKa3bIBaeT, qTO H3MeHeHHe COCTO.HHHSI nOKO.H HJlH p8BHOMepHoro H
np.HMOJlHHeHHOro BHmeHH.H JlI060ro TeJla npOHCXO,AHT TOJlbKO npH B3aHMoeH-
CTBHH TeJl.
EcnH me B38HMO,AeHCTBHe OTCYTcTByeT HnH CKOMneHCHpOB8HO. TO TeJlO 6y-
,lIeT COXpaH.HTb COCTOSlHHe nOKO.H HJlH paBHOMepHoro H np.HMOnHHeHHoro .nBH-
meHH.H. TIepBblM K 3TOMY BbIBOY npHmeJl ranHJleO r8J1HJleH.
Ho BHmeHHe OTHOCHTenbHO. B OTHomeHHH nH JlI06bIX CHCTeM OTCqeT8 BbI-
nOnH.HeTCSI 3TO YTBep.>KeHHe?
OnbIT nOK83bIBaeT, qTO HeT. OTHOCHTeJlbHO YCKopeHHO BHmYHXC.H CHC-
TeM OHO He BbInOJlH.HeTC.H (B KaqeCTBe npHMepa paCCMOTpHTe nOBe)J;eHHe M.Hq8.
nem8ero H8 nOJlY B8rOHa, ,ABHmyerOC.H C YCKopeHHeM). 0606HB nO,A06HbIe
onbITbI, IIc8aK HblOTOH H ccpopMynHpOBaJl nepBbIH 3aKOH HH8MHKH.
llepBLIH 3aKOH HLlOToHa. Cyecm6YlOm maKue CUCme.Mbl omcilema. om.
liocume.n.bliO KomopblX nocmyname.n.bHo iJ6uJICyeecSl me.n.o coxpaliSlem CKO-
pocmb nocmOSlliliou. ec.n.u lia liezo lie iJeucm6YlOm iJpyzue me.n.a (u.n.u ux iJeu.
cm6ue KOMnelicupyemcSl):
n
L i; = 0, v = const, a = o.
1=1
TIepBbIH 3aKOH HblOTOHa BbInOJlH.HeTC.H B HHepl.\HaJlbHbIX CHCTeM8X OT-
CqeTa.
30

2.2. HHepHaJlbHaJl CHCTeMa OTC'IeTa
CHCTeMbI OTCqeTa, OTHOCHTeJlbHO KOTOpbIX TeJlO npH KOMneHcanHH BHelIIHHX
B03,neRcTBHR ,nBHmeTC.R paBHOMepHO H np.HMOJlHHeRHO, Ha3bIBaIOTC.R HHepHaJIL-
HLIMH CHCTeM8MH.
IIpH paCCMOTpeHHH HBJIeHHR, Ha KOTopbIe He BJIHHeT BpaeHHe 3eMJIH BOK-
pyr CBoeH OCH H ee lIBHmeHHe BOKpyr COJlHna, CHCTeMY, CB.H3aHHYIO C 3eMJleH
(reoneHTpHqeCKa.H CHCTeMa), MomHO CqHTaTb HHepnHaJlbHOR.
ECJlH me He06xo,AHMO yqHTbIBaTb ,nBHmeHHe 3eMJIH, TO HHepUHaJIbHOH CqH-
TalOT CHCTeMY, CB.H3aHHYIO C COJlHneM (reJlHoeHTpHqeCKa.H CHCTeMa).
ECJlH me H3BeCTHa XOT.H 6bI 0,nH8 HHepnHaJIbHa.R CHCTeMa, TO HHepnHaJIb-
HbIMH 6y,nYT Bce JJ;pyrHe CHCTeMbI OTCqeT8, JJ;BHmYHeC.H OTHOCHTeJlbHO ee np.H-
MOJIHHeRHO H paBHOMepHO.
2.3. rrpHHHn OTHOCHTeJlbHOCTH raJlHJleS
IIpoBeJJ;eM onbIT, HanpHMep, 6y,neM 6pocaTb M.Hq BepTHK8J1bHO BBepx B Ba-
rOHe, ,nBHmyeMC.H paBHOMepHO H npHMOJIHHeRHO, H Ha nJlaT«l>opMe. B 060HX
cJlyqa.Hx 6Y)J;eM BH,neTb O,nHO H TO me: CHaqaJla M.Hq ,nBHmeTCH 3aMeJJ;JleHHO
BBepx, 3aTeM YCKopeHHO BHH3 H na,naeT 06paTHO B PYKH.
3TO nOKa3bIBaeT, qTO HHKaKHMH MeXaHHqeCKHMH onbITaMH, npOBOJJ;HMbIMH
B JJ;aHHOR CHCTeMe OTCqeT8, HeJIb3.H YCTaHOBHTb, ,nBHmeTC.H JIH OHa paBHOMepHO
H np.HMOJlHHeHHO HJlH Haxo,nHTC.H B nOKoe (pHC. 2.1, a, 6).
a 6
Y 0 y' 0
t v t
0 0
0 x 0' x'
Puc. 2.1
3TO 06'bHCH.HeTC.H TeM, qTO 60 6cex uliepqua.n.bliblX cucmeMax omcllema 6ce
.MeXaIiUlleCKUe Sl6.n.eIiUSl npomeKalOm OOUliaK060 npu OOUliaK06blX lialla.n.bliblX
YC.n.06USlX (npHHHn OTHOCHTeJlLHOCTH raJIHJleH).
2.4. Macca
IIpH B03JJ;eRcTBHH O,nHHX TeJl Ha ,npyrHe TeJla H3MeH.HIOT CBOIO CKOpOCTb -
npH06peTalOT YCKopeHHe. IIpH 3TOM pa3HbIe TeJIa npH JJ;8HHOM B03,neRcTBHH
npH06peTalOT pa3Hoe YCKopeHHe (T. e. OKa3bIBalOT pa3Hoe conpOTHBJleHHe H3-
MeHeHHIO HX CKopocTeR).
CBOHCTBO Ten npH06peTaTb onpe,neJleHHOe YCKopeHHe npH ,AaHHOM B03,neH-
CTBHH Ha3bIBaeTC.R HHepTHOCTLIO. HHepTHOCTb COCTOHT B TOM, qTO .lInSl H3Me-
HeHH.H CKOpOCTH TeJla Ha 3aJJ;8HHYIO BeJlHqHHY HymHo, qT06bI Ha Hero ,neii-
CTBOBaJlO ,npyroe TeJlO H 3TO ,neRCTBHe ,nJlHJlOCb HeKOTopoe BpeM.H.
31

MHepTHOCTb - 3TO CBOHCTBO, npHcyee BceM TeJlaM. Macca TeJla - KO-
nHqeCTBeHHa.H Mepa ero HHepTHOCTH.
o Tene, KOTopoe B pe3YJlbTaTe B3aHMoeHCTBH.H MeHblIIe H3MeHHeT CBOIO
CKOpOCTb, rOBOp.HT, qTO OHO 60Jlee HHepTHO, Macca ero 60JlblIIe:
 =
lnt 
B Clf e,AHHHQ,eH MaCCbI Ten8 SlBnSleTCSI KHnorpaMM (Kr).
TaK K8K M8CC8 BXO,AHT B 3aKOH BceMHpHoro THI'OTeHH.H, TO OHa onpee-
JIHeT TaKme rpaBHTaHOHHoe B3aHMoeHCTBHe Ten.
Y CTaHOBJleHHaH TeopHeH OTHOCHTeJlbHOCTH B3aHMOCBH3b Memn;y MaCCOH H
3HepI'HeH W = mc 2 nOKa3bIBaeT, qTO Macca .HBn.HeTC.H MepOH nonHOH 3Hep-
I'HH Ten.
Macca Tena 06J1aaeT CBOHCTBOM a.nHTHBHOCTH (cnomeHHH): npH coeHHe-
HHH BYX TeJl B O,AHO MaCCbI 3THX TeJl CKnabIBaIOTC.H. CBOHCTBO aHTHBHOC-
TH M8CCbI OqeHb TOqHO BbInOJlHHeTC.H n.H MaKpOCKOnHqeCKHX Ten H HapY1II8-
eTCH nHlIIb TOI',Aa, KOI',Aa 3HeprH.H B3aHMoeHCTBH.H COCT8BHbIX qaCTeH TeJla Be-
nHKa, HanpHMep, npH coeHHeHHH npOTOHOB H HeHTpOHOB B aTOMHoe .Hpo.
CyeCTBYIOT ,ABa OCHOBHbIX cnoc06a onpeeneHHH MaCCbI Tena: 1) nYTeM
cpaBHeHH.H YCKopeHHH Tena HeH3BecTHoH M8CCbI Ii 3T8J10Ha M8CCbI npH HX B3a-
HMolleiicTBHH (m. =  ""1; 2) nyreM B3BelllHBRHHH HR PW{RJKHLIX BeCRX.
B Kn8CCHqeCKOH MexaHHKe HblOTOH8 C'tlHT8JOT, liTO:
1) M8CC8 TeJla He 3aBHCHT OT CKOpOCTH ero ,ABHmeHH.H;
2) M8CC8 Ten8 p8BH8 CYMMe M8CC Bcex q8CTH (HnH M8TepH8J1bHbIX TO-
qeK), H3 KOTOpbIX OHO COCTOHT;
3) .nn.H .naHHOH CHCTeMbI Ten BbInOnH.HeTCH 38KOH coxpaHeHHH MaCCbI: npH nlO-
6bIX npoueccax. npOHcxo.nHWHX B CHCTeMe Ten. ee Macca OCTaeTC.H HeH3MeHHOH.
2.5. CHJIa
lIpH eHcTBHH Ha TeJlO pyrHx Ten MomeT H3MeHHTbC.H nH60 cpopMa H pa3-
MepbI Ten8 (TeJlO ecpopMHpyeTC.H), nH60 CKOpOCTb Ten8 (Teno npH06peTaeT YC-
KopeHHe), JlH60 B03MomHO O,AHOBpeMeHHO H TO, H ,Apyroe.
<I>H3HqeCKa.H BeJlHqHHa, onHCbIBalOa.H H H3Mep.HlOaH B03eHCTBHe OHO-
ro TeJla Ha ,Apyroe, B pe3YJlbTaTe KOToporo TeJla npH06peTalOT YCKopeHHH HnH
ecpopMHpYIOTC.H, Ha3bIBaeTC.H CHJIOH.
CHna F - BeJlHqHHa BeKTOpHa.H. OH8 xapaKTepH3yeTcH MoyneM, TOq-
KOH npHJlOmeHHH H HanpaBJleHHeM B npOCTp8HCTBe.
TOqKY npHJlOmeHH.H CHJlbI B TBepOM TeJle Mom-
0' HO nepeHOCHTb TonbKO BOJlb JlHHHH ee eHCT-
BHfl 00', He H3MeHSI.H pe3YJlbTaTa ee eHCTBH.H Ha
Teno (pHC. 2.2).
B MexaHHKe H3yqalOT cHny ynpyrocTH, CHJlY Tpe-
HHH, CHJlY T.HmeCTH, cHny T.HrOTeHHH.
EHHHeH H3MepeHH.H CHJlbI B CM HBJlHeTCH HblO-
TOH (H). 06bIqHO JI.H H3MepeHH.H CHnbI nOJlb3YIOTC.H
HHaMOMeTpOM.
o
Puc. 2.2
32

2.6. BTOpOH 3aKOH HblOTOHa
Puc. 2.3
3aKOH YCTaHOBJleH Ha OCHOBaHHH 0606eHH.R MHO-
rOqHCneHHbIX onbITOB H Ha6nlOlIeHHH (pHC. 2.3). IIpo-
HJlJlIOCTpHpOBaTb ero MomHO C nOMOblO onbITa, cxe-
Ma KOToporo H306pameHa Ha pHCYHKe 2.3.
BTOpoii 3aKOH HLIOToHa. YCKopenue meJla npR.MO
nponopu,UOlia.n.bliO cu.n.e, iJeucmBYIOeu lia me.n.o. 06pam-
liO nponopuuolia.n.blio .Macce amozo me.n.a U lianpa6Jle-
no B cmopoliY oeucmBuR CUJlbL:
_ F
a =-.
m
ECJlH Ha Teno eHcTBYIOT HeCKonbKO CHJI, TO, KaK nOKa3bIBaeT onbIT, BbI-
nonHSIeTC.R npHHHn He3aBHCHMOCTH eHCTBH.R CHn. Kam,Aa.H CHna  coo6aeT
- F;
CBoe YCKopeHHe a i = -. Pe3YJlbTHPYIOee YCKopeHHe
m
ii = i iii ' ii = i i; =
i=l i=l m
1 n _ n
= - LPi, r,Ae F = Ii;
m i=l i=l
paBHoeHCTBYIOa.H CHna.
1 n _
ii = - L Pi - BTOpOH 3aKOH HbIOToHa.
m i=l
BOJlee nOnHa.H cpopMynHpoBKa BToporo 3aKOHa HbIOTOH8: YCKopeHue TeJIa
npHMO nponopUOHaJIbHO paBHOJJ;eiicTBYIOeii Bcex CUJI, JJ;eiicTBYIOUX Ha
TeJlO, o6paTHo nponopUOHaJILHO Macce TeJIa U HanpaBJIeHO B CTOPOHY paB-
HOJJ;eiicTByroeii CUJIbI.
BTOpOH 3aKOH HblOToHa BbInOnH.HeTC.H B HHepHanbHbIX CHCTeMax OT-
CqeTa.
BamHo nOH.HTb: H3 BToporo 3aKOHa HblOToHa cneyeT, qTO CHna onpee-
JI.HeT HanpaBneHHe YCKopeHH.H, a He HanpaBneHHe CKOpOCTH:
ii ii F.
TaK, HanpHMep, no eHCTBHeM CHnbI TSImeCTH Teno MomeT .D.BHraTbCSI Bep-
THKanbHO BHH3 (pHC. 2.4, a) H no napa60Jle (pHC. 2.4, 6).
a
6
-
'"
mg "
,
,
,
\
\
\
//////////////////)/////
4J _v o = 0
tmg
I
I
I
I
I
I
///////////>////////////
Puc. 2.4
33

2.7. TpeTHH 3aKOH HLIOToHa
On bIT nOKR3bIBReT, qTO ,neHCTBHe TeJl ,npyr Ha ,npyrR .HBJI.HeTC.H ,ABYCTOPOH-
HHM. Henb3.H 06HapymHTb TRKoro CJlyqR.H, qT06bI KaKoe-TO Teno ,AeHCTBOBRno
Ha ,Apyroe TeJlO H He HcnbITbIBRJlO 6bI npH 3TOM OTBeTHoro ,neHCTBH.H.
CU.n.bl. C KomopblMU 63aUMooeucm6Y1Om 06a me.n.a, OOliOU npupOObl. pa6libl
no MOOY.n.IO. nanpa6Jlelibl 600.n.b OOliOU npSlMOU 6 npomU60no.n.OJIClible cmopOlibl
U npu.n.OJICelibl K pa3liblM me.n.aM:
F 12 = - F 21 - TpeTHH 3aKOH HLIOTOHa
CHJlbI B3RHMo,neHCTBH.H B03HHKRIOT o,nHOBpeMeHHO H nonapHO. TaK KaK OHH
npHJIOmeHbI K pa3HbIM TeJlaM. TO He MorYT ypaBHoBellIHBaTb ,npyr ,Apyra.
CJIe,nyeT HMeTb B BH,ny, qTO TpeTHH 3aKOH cnpRBe,nJIHB He3aBHCHMO OT Toro,
nOKO.HTCSJ nH B3RHMO,AeHCTBYIOHe TeJIa HJlH me OHH ,ABHmYTC.H, H8XO,A.HTC.H
OHH B Henocpe,nCTBeHHOM KOHTaKTe ,Apyr C ,ApyroM HJlH pa3,neneHbI npOCTpaH-
CTBOM.
PRCCMOTPHM CJIe,nYIOHe npHMepbI:
1. KHHrR JIemHT Ha CTOJIe. Co CTOpOHbI CTOJIR ,neHcTByeT CHJla peRK-
lI.HH N p , co CTOpOHbI KHHrH - ee Bec P. P = -N p . Bec KHHrH npHJlOmeH K
CTOJIY, CHJlR peaKlI.HH onopbI - K KHHre (pHC. 2.5).
2. ABe MaTepHaJIbHble TOqKH MaccaMH m] H m2 npHT.HrHBRIOTC.H ,npyr K
,npyry, HRXO,n.HCb H8 pRCCTO.HHHH r (pHC. 2.6).
3. ABa MRJleHbKHX O,AHOHMeHHO 3ap.HmeHHbIX llIRpHKa B3aHMo,neikTBYIOT
,npyr C ,npyroM C CHJIRMH OTTRJlKHBRHH.H (pHC. 2.7).
m i Fi
. .
F 2 m 2
c .
F +qi +q2 F
.............. - - - . .
Puc. 2.5
Puc. 2.6
Puc. 2.7
2.8. CHJlLI ynpyrocTH
M3MeHeHH.H <}>OpMbI 11: pR3MepOB Tena Ha3bIBRIOT e4>opMaHHMH.
.D;e<}>opMRlI.H.H - pe3ynbTaT ,nBHmeHH.H qRCTeH TeJIa OTHOCHTeJIbHO ,npyr ,npy-
ra. .D;e<}>opMRu.HH MorYT 6bITb BbI3BRHbI ,neHcTBHeM Ha TeJlO BHellIHHX CHJI.
PR3J1HqaIOT ,ne<}>opMau.HH paCT.HmeHH.H (cmaTH.H), H3rH6a, c,nBHra, KpyqeHH.H.
.D;e<}>opMalI.HH, nOJIHOCTblO HCqe3alOllJ.He nocne npeKpallJ.eHH.H ,neHcTBHSJ Ha
TeJIO CHJI, Ha3bIBalOT ynpYZUMU, a ,ne<}>opMalI.HH, COXpaH.HIOHeC.H H nocJle Toro,
KaK BHellIHHe CHJIbI nepeCTaJIH ,neHcTBoBRTb Ha Teno, - n.n.acmUlleCKUMU.
Cnoc06HOCTb I( ynpyrHM H nJlaCTHqeCKHM ,Ae<}>opMalI.H.HM 38BHCHT OT npH-
pO,nbI BellJ.eCTBa, H3 KOToporo H3rOTOBJIeHO TeJIO, OT ycnoBHH, B KOTOpbIX OHO
HRXO,AHTC.H, OT cnoc06a ero H3rOTOBJleHH.H.
IIpH ,ne<}>opMalI.HH TBep,noro TeJla npOHcxo,nHT CMeeHHe qRCTHlI. (aTOMOB,
MOJIeKYJl, HOHOB) H3 nepBOHa'laJlbHbIX nOJlOmeHHH paBHOBeCH.H B HOBbIe nOJIO-
meHH.H. IIpH 3TOM H3MeH.HIOTC.H CHnOBbIe B3aHMO.lJ;eHCTBH.H Mem,Ay OT,AeJIbHbI-
34

MH qaCTHIJ;aMH TeJla. B pe3YJIbTaTe B eq)QpMHpOBaHHOM TeJle B03HHKalOT BHYT-
peHHHe CHJIbI, npen.HTCTBYIOHe ero ,Ae<l>opMaHH.
CHJlbI, B03HHKalOHe B TeJle npH ero ynpyroif e<l>opMaIJ;HH H HanpaBJleH-
HbIe B CTOPOHY, npOTHBonOJlomHYIO CMeeHHIO qaCTHIJ; npH e<l>opMaIJ;HH, Ha-
3bIBalOT CHJlaMH ynpyrOCTH.
CHJlbJ ynpyrocTH npen.HTCTBYIOT H3MeHeHHIO pa3MepOB 1'1 <l>OpMbI TeJla.
CHJIbJ ynpyrocTH eHCTBYIOT B JlI060M CeqeHHH .n;e<l>opMHpoBaHHoro TeJla, a
TaKme B MeCTe ero KOHTaKTa C TeJlOM, BbI3bIBalOHM e<l>opMaIJ;HH.
CHJlbI ynpyrocTH nepneHHKYJl.HpHbI nOBepXHOCTH conpHKOCHOBeHH.H B3aH-
MoeMCTBYIOHX TeJl, a eCJlH BO B3aHMoeMCTBHH yqaCTBYIOT TaKHe TeJla, Kal<
npymHHbI, HHTH, TO CHJlbI ynpyrocTH HanpaBneHbI BOJlb HX OCH.
CHJlY ynpyrocTH, eMCTBYIOW;YIO Ha TeJlO CO CTOpOHbI onopbI, qaCTO Ha3bI-
BalOT peaKHeH onopLI.
rrpHpoa ynpyrHx CHJI 3JIeKTpOMarHHTHaH.
JIHHeHHaJl Ae<l>opMaHJI (.ne<l>opMaUHJI pac-
T.HmeHH.H) - .ne<l>opMaUH.H. npH KOTOpOM npo-
.x HcxonHT H3MeHeHHe TOJIbKO onHoro nHHeMHoro
pa3Mepa TeJla. KOJlHqeCTBeHHO OHa xapaKTepH'
3yeTcJI a6COJllOTHbIM 6.l H OTHOCHTeJIbHbIM €
y.n;nHHeHHeM.
x = 6.l = l-1o a6COJllOTHOe yJlHHeHHe,
re l H lo - JlHHbI TeJla B e<l>opMHpoBaH-
HOM H He,AecpopMHpOBaHHOM COCTOJlHH.HX COOT-
.x BeTCTBeHHO (pHC. 2.8).
6l
€ = 1;; - OTHOCHTeJlbHOe yJlHHeHHe.
He60JIbIllHe H KpaTKOBpeMeHHbIe necpopMa-
HH C .nocTaTOqHOM CTeneHblO TOqHOCTH MorYT
paCCMaTpHBaTbC.H KaK ynpyrHe. ,IJ,JI.H TaKHX e<tJopMaHM P06epT rYK 3Kcne-
pHMeHTaJlbHO YCTaHOBHJI 3aKOH.
CUJla ynpyzocmu 8031iUKalO14aJl npu derpop.Maijuu meJla. np.Jl.MO nponopl{UO-
HaJlbHa a6COJllOmHO.MY yaJlUHeHUIO meJla U Hanpa8JleHa 8 cmopOHY. npomU80no-
1l0JICliYIO 1tanpa8.n.eliUIO C.Me14eliUJl llacmUl{ me.n.a'
F x= -kx - 3aKOH rYKa.
3,AeCb F;x - npOeKIJ;HJI CHJIbI Ha OCb Ox,
k - meCTKOCTb TeJIa, 3aBHC.Hw;aJl OT pa3MepOB
TMa H MaTepH8JIa, H3 KOTOporo OHO H3rOTOBJIeHO.
:H<:ecTKocTb qHCJIeHHO paBHa CHJle ynpyroc-
TH, B03HHKalOeM B TeJIe npH e,AHHHqHOM a6-
COJlIOTHOM ynHHeHHH.
B CM eHHHIJ;eii meCTKOCTH JlBJI.HeTC.H HblO-
TOH Ha MeTp [ : 1-
rpa<l>HqeCKH 3aBHCHMOCTb F x= f(x) npeCTaB-
JleHa Ha pHCYHKe 2.9.
lo
o I
I I
F==?
l 6l 0 1 . 6.l .!
. . -
O!F II
Puc. 2.8
Puc. 2.9
2.9. 3aKoH rYKa
x
35

Ha 3aKOHe rYKa OCHOBaHO )J;eHCTBHe HHaMOMeTpa - npH60pa, npeHa3-
HaqeHHOrO JlH H3MepeHH.H CHJI.
Ha npaKTHKe qaCTO TPYHO onpeeJlHTb BenHqHHY eopMaHH, TaK KaK
OHa Mana.
2.10. CHJILI TpeHHH. K034>4>HHeHT TpeHHH
CHJlOH TpeHHH Ha3bIBalOT cHny, B03HHKalOYIO npH conpHKOCHOBeHHH ,ABYX
Ten H npen.HTCTBYIOYIO HX OTHOCHTenbHOMY nepeMeeHHIO. OHa npHnomeHa
K TenaM BOJlb nOBepXHOCTH conpHKOCHOBeHH.H. TpeHHe, B03HHKalOee Mem-
Y nOBepXHOCTHMH pa3J1HqHbIX Ten, Ha3bIBalOT 61teWliUM TpeHHeM. ECJlH Tpe-
HHe npO.HBn.HeTCH Memy qaCT.HMH OHoro H Toro me Tena. TO OHO Ha3bIBaeT-
CH 6liympeli1tUM TpeHHeM.
TpeHHe Memy nOBepXHOCT.HMH ,ABYX conpHKacalOHXC.H TBep,AbIX TeJl npH
OTCYTCTBHH Memy HHMH mHKOH HJlH ra3006pa3HOH npocnOHKH Ha3bIBaeTC.H
CYXUM TpeHHeM.
TpeHHe Memy nOBepXHOCTblO TBeporo TeJla H oKPymalOeH ero mHKOH
HnH ra3006pa3HOH cpeOH, B KOTOpOH Teno BHmeTC.H. Ha3bIBaeTC.H 6Sl3KUM
TpeHHeM.
Pa3J1Hqa1OT mpeliue no KOSl, mpeliue CKo.n.bJICeIiUSl H mpeHue KalleHUJl.
CHJla TpeHH.H nOKO.H F Tp.n B03HHKaeT Memy HenOlIBHmHbIMH TBepbIMH Te-
naMH, Kora eCTb CHJlbI, eHCTBYIOHe B HanpaBJIeHHH B03MomHoro BHme-
HH.H TeJla.
CHJlR TpeHHH nOKO.H Bcera palma no MO,AYJlIO H HanpaBJleHa npOTHBono-
JIOmHO CHJle, napaJIJleJlbHOH nOBepXHOCTH conpHKOCHOBeHH.H H CTpeM.HeHC.H npH-
BeCTH 3TO TeJlO B BHmeHHe. YBeJlHqeHHe 3TOH npHJlOmeHHOH K TeJlY BHelllHeH
CHJlbI npHBOHT K B03paCTaHHIO H CHJlbJ TpeHH.H nOKO.H. CHJla TpeHH.H nOKO.H
HanpaBJleHa B CTOPOHY, npOTHBOnOJlOmHYIO B03MomHOMY nepeMeeHHIO TeJla
(pHC. 2.10 a, 6). FTp.n = -P. 0 < FTP.n < FTp.n.max' MaKCMMaJIbHa.H CMJIa TpeHH.H no-
KOH nponOpHOHaJIbHa MOYJIIO CHJIbI HopManbHoro aBneHH.H FA' npOH3BOHMO-
ro TeJlOM Ha onopy:
F Tp.n.max = IlOFn'
a
6
N p
mg
mg
Puc. 2.10
TaK KaK no TpeTbeMY 3aKOHY HblOToHa FA = N p (N p - CHJla HOpMaJlbHOH pe-
aKHH onopbI), TO FTp.n.max = lloN p ' 3eCb Jlo - K03Hu.HeHT TpeHHH nOKO.H.
3aBHCHHH OT MaTepHaJIR H COCTO.HHH.H TPYHXC.H nOBepXHOCTeH. CHJla Tpe-
36

HH.H nOKOH npen.HTCTByeT HaqaJlY .lI.BHmeHH.H. Ho 6bIBalOT cJlyqaH. KOr.lI.a CHJIa
TpeHH.H nOKO.H CJIymHT npHqHHOM B03HHKHOBeHHH JIBHJKeHH.H TeJIa. HanpHMep.
xO.lI.b6a qeJlOBeKa. TIpH xO.lI.b6e CHJIa TpeHH.H nOKO.H, .lI.eMCTBYIOaH Ha no.lI.O-
llIBY, co06aeT HaM YCKopeHHe. llO.lI.OllIBa He CKOJlb3HT Ha3a.lI., H, 3HaqHT, Tpe-
HHe MeJK.lI.Y HeM H .lI.OpOrOM - 3TO TpeHHe nOKOH.
PaCCMOTpHM 6PYCOK, JIemaHM Ha TenemKe
(pHC. 2.11). Ha Hero )leMcTByeT CHJIa F. CTpe-
M.Ha.HC.H C,ABHHYTb ero C MeCTa. B npOTHBono-
JlOmHOM HanpaBJIeHHH Ha 6pYCOK co CTOpOHbI
TeJleJKKH ,AeMcTByeT CHJIa TpeHH.H nOKO.H F -rp.n'
Ha TeJIemKY co CTOpOHbI 6pYCKa .lteHcTByeT Ta-
KaH JKe no MO.lI.YJlIO H npOTHBOnOJIOmHa.H no Ha-
npaBJIeHHIO CHna F-rp.n, npHBO.lI..Ha.H K .lI.BHme-
HHID TenemKH BnpaBO.
CHJIa TpeHH.H nOKO.H HrpaeT npHHHnHaJIbHYIO pOJIb B ,ABHJKeHHI1 MallIHH.
lIIHHbI Be.lI.YHX KOJleC aBTOM06HJIeH KaK 6bI OTTaJlKHBaIOTC.H OT .lI.OpOrH, H
npH OTCYTCTBHH np06YKCOBKH TonKalOa.H aBTOM06HJIb CHJIa - 3TO CHJIa Tpe-
HH.H nOKO.H.
CHJla TpeHI1H CKOJIbJKeHH.H F TP B03HHKaeT npH conpHKOCHOBeHHH .lI.BHmy-
HXC.H OTHOCHTeJlbHO .lI.pyr .n;pyra TeJl H 3aTpY.lI.HHeT HX .lI.BHJKeHHe. CHJla Tpe-
HH.H CKOJlbmeHH.H HanpaBJleHa B,AOJlb nOBepXHOCTH conpHKOCHOBeHH.H B CTOpO-
HY, npOTHBonOJIOmHYIO CKOpOCTH .lI.BHJKeHH.H. CHJIB TpeHH.H CKOJlbmeHHH F TP
np.HMO nponopHOHaJlbHa CHJle HOpMaJlbHOrO .lI.aBJIeHH.H:
F Tp = JlF n  F-rp = J.lN p '
r.lI.e J..l - K03<P<PHHeHT TpeHH.H CKOJIbmeHH.H. 3aBHCHHM OT KaqeCTBa 06pa-
60TKH nOBepXHOCTeM H HX MaTepHaJIa.
 F
 'fP.D
////// ///////
Puc. 2.11
F
Jl =  - const ,AJIH .n;aHHbIX TeJI.
N p
Jlo ;:::: Jl (J..lo HeCKOJlbKO 60Jlblile J.l - C.lI.BHHYTb TeJlO C MeCTa TpY.lI.Hee. qeM
npO,AOJlmaTb ero HaqaBlileeCH CKOJIbmeHHe).
CHJla TpeHH.H He 3aBHCHT OT nnOa.lI.H conpHKacalOHXC.H nOBepXHOCTeM TeJI
H HX nOJlOJKeHH.H OTHOCHTeJlbHO .lI.pyr ,Apyra. a TaKme OT M0.lI.YJI.H CKOpOCTH npH
He60JIbllIHX CKOpOCT.HX. HO 3aBHCHT OT HanpaBneHH.H CKOpOCTH: npH H3MeHeHHH
HBnpaBneHH.H CKOpOCTH H3MeHHeTCH H HanpaBJIeHHe F TP (pHC. 2.12). .l1:eMcTBHe
CHJI TpeHH.H CKOJlbmeHH.H COnpOBOJK,AaeTC.H npeBpaeHHeM MeXaHHqeCKOM 3Hep-
rHH BO BHYTpeHHIOIO.
F'fP"
v"
Puc. 2.12
CywecTBoBaHHe CHJI TpeHHH 06'bHCH.HeTC.H npo-
.HBJleHHeM CHJI 3J1eKTpOMarHHTHoro B3aHMOJIeM-
CTBH.H. CHJIbI TpeHH.H nOKOH BbI3bIBaIOTC.H B oc-
HOBHOM ynpyrHMH .lI.e<popMaH.HMH MHKpOBbICTY-
nOB Ha nOBepxHocTH TPYHXCH TeJI. CHJIbI Tpe-
HHH CKOJIbJKeHH.H B03HHKalOT B pe3ynbTaTe nnac-
THqeCKHX .lI.e<popMaHM MHKpoBblcTynoB H HX qa-
CTHqHOrO pa3pYlIleHH.H. a TaKJKe CHJI MemMoJleKY-
nHpHoro B3aHMO.lI.eMCTBH.H B 06nacTH KOHTaKTOB.
37

2.11. J];BH)KeHHe TeJIa C yqeToM CHJILI TpeHHH
F.
ii =  npOTHBOnOJlOmHO V (pHC. 2.13). CJle,AOBa-
m
TenbHO, ,AeHCTBHe CHJlbI TpeHH.H npHBO,AHT K YMeHb-
meHHIO MO,AYJlH CKOpOCTH. EcnH Ha Teno ,AeHcTByeT
F TP ' TO OHO 06H3aTeJlbHO OCTaHOBHTC.H, npOHJJ:.H HeKOTopoe paCCTO.H-
TOpM03HOH nYTb. PaCCqHTaeM ero.
ii
- 
/:/J//:/
Puc. 2.13
TOJlbKO
HHe -
CHJla TpeHH.H CKonbmeHH.H BCer,Aa HanpaBJleHa B
CTOPOHY, npoTHBononomHYIO CKOpOCTH ,ABHmeHH.H.
TI03TOMY YCKopeHHe, co06aeMoe CHJlOH TpeHHSI,
..
x
v 2 _ v 2
TIpH paBH03aMe,AJleHHOM ,ABHmeHHH npOeKH.H nepeMeeHH.H l:1r = 0
x -2a
F.. 2 2
TaR KaK a = -.2£.. H V = O. TO 6.r x = Vo = mvo .
m 2a 2FTp
mv 2
s=- 
2FTp
s - v! 06 3TOM HymHo nOMHHTb BO,AIITenHM MamHH H nemeXO,AaM, nepe-
ceKalOHM YJlHY.
TOpM03HOH nYTb
2.12. rpaBHTal.{HOHHble CHJIbI. 3aKoH BceMHpHoro THrOTeHHH
Bce TeJla B npHpO,Ae B3allMHO npHT.HrHBaIOTC.H ,Apyr K ,Apyry. BnepBbIe
HblOTOH ,AOKa3aJl, qTO npHqHHa, BbI3bIBalOa.H na,AeHHe KaMH.H Ha 3eMJlIO,
,ABHmeHHe JIYHbI BOKpyr 3eMnH H nJlaHeT BOKpyr ConHa, O,AHa H Ta me -
3TO CHna BceMHpHoro T.HrOTeHH.H (rpaBHTaHOHHa.H CHJla), ,AeHcTBYIOaH Mem-
,Ay JlI06bIMH TeJl8MH BCeJleHHOH.
rpaBHTaHoHHbIe CHJlbI - 3TO CHJlbI eHTpaJlbHbIe, T. e. OHH H8npaBJleHbJ
B,AOnb npHMoH, COe,AHHHIOeH B38HMO,AeHcTBYIOHe MaTepHanbHbIe TOqKH.
rpaBIITaHoHHoe B3aHMO,AeHCTBHe ocyecTBn.HeTC.H nOCpe,ACTBOM rpaBHTaH-
OHHoro nOJl.H. 3TO nOJle, Hap.H,AY C ,ApyrllMH nOn.HMII II BeeCTBOM, .HBn.HeTC.R
O,AHOH H3 cpO pM M8TepHH.
C K8m,AbIM TeJlOM Hepa3pbIBHO CB.H3aHO rpaBHTaHOHHoe nOJle, npO.HBJI.HIO-
eec.H B TOM, qTO H8 nOMeeHHYIO B nOJle MaTeplianbHYIO TOqKY ,AeHcTByeT
rpaBHT8HOHHa.H CHJla, nponopHOHaJlbHaH Macce 3TOH TOqKH.
Teno, rpaBIiTaIiOHHoe nOJle KOToporo HCCne,AyeTC.H. Ha3bIBaeTCH ucmOllliU
KOM no.n.Sl.
Ft
r r
F 2
.i
c
Puc. 2.14
38
rp8BHTaHOHHbIe CHJlbI 38BHC.HT OT nOJlOmeHH.H Ten
(KOOp,AIiHaT) (pHC. 2.14). Mx npOCTO paCCqliTaTb ,AJI.H
MaTepHaJlbHbIX TOqeK no 38KOHY BceMHpHoro T.HrOTe-
HIi.H: iJ6e .Mamepua.n.blibte mOIlKU npumSlzu6alOmcSl opyz
K iJpyzy c cu.n.oil, npSl.MO nponop4UOHaJlb1toil npou36eiJe-
liUIO ux .Macc U 06pam1t0 nponopl{UOlia.n.blioil K6aiJpa-
my paccmoSlliuSl .Me:JICiJy 1tU.MU.

J . r:'\
 nOCTO.HHHa.H.
rpaBHTaHOHHa.H nOCTO.HHHa.H onpe,AeneHa
onbITHbIM nYTeM. OHa QHCneHHO paBHa CHJle,
C KOTOpOH npHT.HrHBaIOTC.H ,ABa TeJla MaccaMH
1 Kr, HaXO,A.HHeC.H H8 paCCTO.HHHH 1 M APyr
OT ,Apyra.
IIo 3aKOHY BceMHpHoro T.HrOTeHH.H MO)KHO
paCCqHTaTb TaKme cHny npHT.HmeHH.H Mem,Ay
llBYM.H cc})epHqeCKHMH TenaMH (r - 3TO paCCTO.H-
HHe Mem,Ay Q,eHTpaMH cc})ep) HnH TeJlaMH, O,lIHO H3 KOTOpbIX - lIIap 60nblIIO-
ro pa,AHyca, a BTopoe - npoH3BonbHoH c})OPMbI, HO He60nblIIHx pa3MepOB
(pHC. 2.15, a, 6).
F = G ml
r 2
a

r
6
d- F 2
 _ ---r-- r-
Puc. 2.15
- 3aKOH BceMHpHoro TJIrOTeHHH.
H'M 2
G=6,67.10- 11
Kr 2
- rpaBHTaHOHHa.H
2.13. CHJIa TJDKeCTH
CHJIa TJla\:eCTH - 3TO CHna, ,AeHcTBYIOa.H Ha Teno co CTOpOHbI 3eMnH H
co06alOa.H TeJlY YCKopeHHe cBo6oAHoro naAeHH.H:
F T = mg .
JII060e TeJlO, HaXO,A.Heec.H Ha 3eMne (HnH
B6nH3H Hee), BMeCTe C 3eMneH BpaaeTC.H BOK-
Pyr ee OCH, T. e. TeJlO ,ABHmeTCSI no oKpym-
HOCTH pa,AHYCOM r C nOCTO.HHHOH no MOAYJlIO
CKOpOCTblO (pHC. 2.16).
Ha Teno Ha nOBepXHOCTH 3eMnH ,AeHCTBY-
lOT CHna T.HrOTeHH.H F H CHna co CTOpOHbI
3eMHOH nOBepXHOCTH N p'
Mx paBHoAeHcTBYIOa.H
Puc. 2.16
- - -
Fi=F+N p (2.1)
co06aeT TeJlY Q,eHTpOCTpeMHTenbHoe YCKopeHHe
v 2
au = - .
r
Pa3nO)KllM cHny T.HrOTeHH.H P Ha ABe co-
CTaBJI.HIOHe. OAHa H3 KOTOpbIX 6Y,AeT PI' T. e.
F = Fi + FT'
M3 ypaBHeHHH (2.1) H (2.2) Bll,AHM, qTO
P T = -N p .
(2.2)
39

TaKHM 06pa30M, CHJla T.HmeCTH F T - O,l(Ha H3 COCTaBJI.HIOHX CHnbI T.H-
rOTeHH.H F. BTOpa.H COCTaBJI.HIOa.H Fl co06aeT TeJIY IJ,eHTpOCTpeMHTenbHoe
YCKopeHHe.
B TO'llKe M Ha reorpa<pH'lIeCKOH IlIHpOTe q> CHJIa T.HmeCTH HanpaBJleHa He
no pa,l(HYCY 3eMnH, a no,l( HeKOTophlM yrnoM a. K HeMY. CHJI8 T.HmeCTH Ha-
npaBJIeHa no TaK Ha3h1BaeMOH OTBeCHOH np.HMOH (no BepTHKaJIH BHH3).
CHJIa T.HmeCTH paBH8 no MO,l(YJlIO H HanpaBJIeHHIO CHJIe T.HrOTeHH.H TOJIbKO
Ha nonlOcax. Ha 3KBaTope OHH COBna,l(aIOT no HanpaBJIeHHIO, a no MO,l(YJIIO
OTnH'lIHe HaH60JlbIlIee.
F T = F-F 1 . F T = F-mw 2 R,
r,l(e w - yrJIOBa.H CKOpOCTb BpaeHH.H 3eMJIH, R - pa,l(HYC 3eMJIH.
21t 2 . 3.14 PM -4 PM
w=-, w= - = 0,727.10 -.
T 24 . 3600 C C
TaK KaK W OqeHb MaJIa. TO F T ::::: F. CJIe,l(OBaTeJIbHO. CHna T.HmeCTH MaJIO
OTJlHqaeTC.H no MO,l(YJlIO OT CHJIbI T.HrOTeHH.H, n03TOMY ,l(aHHbIM pa3nHqHeM qa-
CTO MomHO npeHe6peqb.
GMm
Tor,l(a F ::::: F, mg =
1 (h + R)2
GM
=> g = .
(h + Rl
113 3TOH <POPMYJIbI BH,l(HO. qTO YCKopeHHe CBo60,AHOrO na,AeHH.H g He 3aBH-
CHT OT MaCCbI na,l(alOIIJ,ero TeJIa. HO 3aBHCHT OT BbICOTbI.
2.14. .L(BHmeHHe TeJIa no)]; J];eHcTBHeM CHJILI THmeCTH
ECJlH Ha TeJIO ,AeHcTByeT TOJlbKO CHna T.HmeCTH, TO TeJlO COBepIlIaeT CBo60,A-
Hoe na,l(eHHe. BH,A TpaeKTopHH ,ABHmeHH.H 3aBHCHT OT HanpaBJIeHH.H 11 MO,l(YJI.H
Ha'llaJlbHOH CKOpOCTH. TIpH 3TOM B03MomHbI CJle,l(YIOHe CJIyqaH ,l(BHmeHH.H TeJla:
1. ECJIH Ha'llaJIbHa.H CKOpOCTb Tena paBHa HynlO HnH napaJIneJIbHa CHJIe
T.HmeCTH. TeJlO COBepIlIaeT np.HMOJlHHeHHOe cB060,l(HOe na,l(eHHe (CM.  1.6).
2. ECJII1 HaqaJIbHa.H CKOpOCTb TeJla HanpaBneHa no,l( yrJloM K CHne T.HmeCTH,
TO TeJIO 6Y,AeT ,ABHraTbC.H no napa60Jle JlH60 no BeTBH napa60JIbI (CM.  1.7, L .8).
3. TeJIo MomeT ,l(BHraTbC.fI no KPyroBoH HJIH 3JInHnTHqeCKOH op6HTe BOK-
pyr nJIaHeTbI (CM.  2.17, 2.18).
Puc. 2.17
40
2.15. Bec TeJla
BeCOM TeJIa Ha3bIBaeTC.H CHJla, C KOTOpOH TeJlO ,l(eH-
cTByeT Ha HenO,l(BHmHYIO OTHOCHTeJIbHO Hero onopy
HJIH nO,l(Bec.
Bec TeJIa B03HHKaeT BCJle,l(CTBHe ero ,Ae<popMaIJ,HH.
BbI3BaHHoH ,l(eHCTBHeM CHnbI co CTOpOHbI onopbI (CHJIbl
peaKIJ,HH) HnH nO,l(BeCa (cHnbl HaT.H»<:eHH.H) Bec cYIIJ,e-
CTBeHHO OTJIHqaeTC.H OT CHJlbI T.H»<:eCTH:
1. 3TO CHJIbJ pa3HoH npHpO,AbI: CHna T.HmeCTH -
rpaBHTaIJ,HOHHa.H CHna, Bec - ynpyra.H CHna (3JIeKT-
pOMarHHTHoH npHpO,l(bI).

2. OHH npHJIOmeHbI K pa3HbIM TeJIaM: CHJIa TSDKeCTH - K TeJIY, Bec -
K onope (pHC. 2.17).
3. HanpaBJleHHe Beca TeJlB He 06H3aTeJIbHO COBna.n;aeT C OTBeCHbIM HanpaB-
neHHeM.
a 6 6
Y Y Y
N p1 N p2 Np3
&:O -ta h
mg
0 p. 0 P2 0 P 3
Puc. 2.18
4. CHna THmeCTH TeJIa B ,AaHHOM MeCTe 3eMnH nOCTO.HHHa.H H He 3aBHCHT OT
XapaKTepa .n;BHmeHH.H TeJIa; Bec 3aBHCHT OT YCKopeHH.H, C KOTOpbIM ,ABHmeTC.H Teno.
PaCCMOTpHM, KaK H3MeHHeTC.H Bec Tena, .n;BHmyerOC.H B BepTHKanbHOM Ha-
npaBJIeHHH BMeCTe C onopOH (pHC. 2.18 a, 6, 6). Ha TeJIO .n;eHCTBYIOT CHJIa
T.H:meCTH F T = mg H CHJla peaKHH onopbI 1\1 p'
OCHOBHoe ypaBHeHHe .n;HHaMHKH: mii = mg + N p . B npoeKHH Ha OCb Oy:
1) 0 = N pI - mg => N pI = mg. IIo TpeTbeMY 3aKOHY HblOTOHa MO.n;y JIH CHJI
N pI = Pl' CJle.n;OBaTeJlbHO, Bec TeJla PI = mg.
2) ma = N p2 - mg => N p2 = m (a + g). 3HaQHT. P 2 = m (a + g) => P 2 > mg
(Teno HcnbITbIBaeT neperPY3KH).
3) -ma = N p3 -mg => N p3 = m(g-a). CJIe.n;oBaTeJlbHO, Bec TeJIa
P 3 = m (g - a) =>  < mg. ECJlH a = g, TO P = O.
TaKHM 06pa30M, Bec TeJIa npH BepTHKaJIbHOM .n;BHmeHHH MO:meT 6bITb B 06-
eM cJlyqae BblpameH <POPMYJlOH
P=m(gta).
2.16. HeBeCOMOCTL
Puc. 2.19
MbICJIeHHO pa306beM Heno.n;BHmHoe Teno Ha ro-
pH30HTaJlbHbie CJIOH. Ha Kam.n;bIH H3 3THX CJIOeB
.n;eHcTByeT CHJla T.HmeCTH mg H Bec P BbIllIeJIe-
:maeH qaCTH TeJIa (pHC. 2.19). 3TOT Bec 6Y,AeT
CTaHOBHTbC.H TeM 60JlbllIe, qeM HHme JIemHT CJIOH.
II03ToMY no,A BnH.HHHeM Beca BbIllienemaHX qac-
TeH TeJla Kam.D;bIH CJlOH .D;e<popMHpyeTC.H H B HeM
B03HHKalOT ynpyrHe Hanp.HmeHH.H, KOTopble B03pa-
CTalOT no Mepe nepeXO,Aa OT BepXHeH qaCTH Tena
K HHmHeH.
41

EcnH TeJlO CB060HO na8eT (a = g), TO ero Bec paBeH HYJlIO. B TeJle HCqe-
3810T BCSiKHe ecpopM8HH H, HeCMOTpSi H8 COXp8HSiIOw;eeC.H eHCTBHe CHJIbI
T.HmeCTH, BepXHHe CJIOH He 6YYT 8BHTb Ha HHmHHe.
COCTO.HHHe, npH KOTOpOM B CB060,AHO BHmyw;eMC.H Tene HCqe3alOT ecpop-
M8HH H B38HMHbIe 8BJleHH.H, H83bIB8eTC.H HeBeCOMOCTLIO.
IIpHqHHa HeBeCOMOCTH 38KJIlOq8eTC.H B TOM, qTO CHn8 BceMHpHoro T.HrOTe-
HHSI co068eT TeJIY Hero onope 0HH8KOBoe YCKopeHHe.
2.17. IIepBas KOCMHqeCKaS CKOpOCTb
'"
./
/
I
I
I
,
\
\
\
'\
"
"-
'......--......
Puc. 2.20
llepBaH KOCMHQeCKaH CKOpOCTL JI.H aHHoH nJI8-
HeTbI - aTO CKOpOCTb, KOTOpylO Hao C006W;HTb Teny
npH 38nycKe, qT06bI OHO CTaJIO cnYTHHKOM nnaHeTbI
H npH aTOM BHranOCb no KPyroBoH op6HTe B6JIH3H
ee nOBepXHOCTH.
IIycTb HCKYCCTBeHHbIH cnYTHHK H8XO,AHTCSi H8 BbI-
COTe h H8 nOBepXHOCTblO nn8HeTbI. CHJla T.HrOTeHH.H
co06aeT cnYTHHKY neHTpOCTpeMHTeJlbHOe YCKopeHHe
(pHC. 2.20). 110 BTOPOMY 3aKOHY HblOToHa
ma = F  mv 2 = G mM 
( R + h ) ( R + h )2
v;  h - J1HHeiiHaH CKOpOCTb CIIYTHHKa,
.nBHmYIllerOC.H no KpyroBoH op6HTe (KpyroBa.H CKOpOCTb) Ha BbICOTe h Hall no-
BepXHOCTblO nJlaHeTbI. 3.necb G - rpaBHTaHOHHa.H nOCTO.HHHaSi, M - Macca
nn8HeTbI, R - ee paHYC.
B6J1H3H I1J1aHeTbI (h; 0) v; JC: - nepBaH KOCMHqeCKaH CKOpOCTb.
GM
TaK KaK B6J1H3H 3eMJlH g - =:::) GM = gR 2 , TO nepBaSJ KOCMHqeCKaSJ
- R 2
r:::;; KM
CKOpOCTb n.H 3eMJlH v = "gR = 7 , 9
C
2.18. .l(BHmeHHe HCKyccTBeHHbIx cnYTHHKOB
AJlSi Toro QT06bI TeJIO CTano HCKYCCTBeHHbIM cnYTHHKOM, ero Hao no-
H.HTb Ha HeKOTOpylO BbICOTY (BbIBecTIi Ha OP6HTY) H C006HTb eMY ropH30H-
TaJlbHYIO OTHOCHTeJlbHO nOBepXHOCTH 3eMJlH CKOpOCTb (pHC. 2.21).
AJI.H 3anYCKa Hcnonb3YIOT paKeTbI. IIocJIe BbIBoa Ha OP6HTY paKeTa pa3-
rOHSieT cnYTHHK o Tpe6yeMoH CKOpOCTH, nOCJIe aToro cnYTHHK OT,AeJISieT-
C.H OT paKeTbI-HOCHTeJI.H Ii npOOJlmaeT CBoe BHmeHHe no op6HTe TOJlbKO
no eHcTBHeM CHJlbI T.HrOTeHH.H. ECJlIi cnYTHHKY B6J1H3H 3eMnH co06-
KM
HTb CKOpOCTb V = 7,9 -, TO OH 6yeT BHraTbC.H no KpyroBoH op6HTe
C
42

ECJIH 7,9
KM KM
<v<11,2
C
KOH Op61iTe.
IT
.,
.,
\. IIapa60Jla
Puc 2.21
C
TO CnYTHIiR 6y,neT ,nBHraTbC.H no 3J1J1HnTliqeC-
CKOPOCTb, KOTOpylO Ha,no C006Il..tHTb
TeJIY. tIT06bI OHO. npeo,nOJIeB npliT.HmeHlie
nJIaHeTbI. CTano cnYTHIiKOM COJIHua. Ha-
3bIBalOT BTOpOH ROCMliqeCKOH CKOpOCTblO.
AnSI 3eMnli BTopaSl KOCMliqeCK8.H CKO-
pOCTb
v 2 = .J2gR == 11,2
KM
C
KM
Pa3BHB CKOpOCTb 11,2 Teno Ha-
c
qHHaeT ,nBHraTbC.H no napa60Jle. ECJlIi
KM
V > 11.2 -, TO TeJlO ,nBlimeTCSI no rli-
C
nep60JIe.
,Z:{BlijKeHlie IiCKYCCTBeHHbIX cnYTHIiKOB nJlaHeTbI npOlicxo,nIiT no,n ,neHCTBH-
eM TOJIbKO O,nHOH CIiJIbI - CHJIbI T.HrOTeHIi.H. 3T8 CHJla co06aeT cnYTHIiKY Ii
BceM HaXO.HIiMC.H H8 HeM TeJlaM OLIIiHaKOBoe YCKopeHHe - YCRopeHlie CBO-
60,nHoro na,neHIi.H. n03TOMY Bce TeJIa Ha cnYTHIiKax Haxo,n.HTC.H B COCTO.HHIiIi
HeBeCOMOCTIi .
2.19. CHcTeMa MaTepHaJlLHLIX TOqeK. YpaBHeHHe nocTynaTeJILHOrO
ABHJKeHHSI CHCTeMLI MaTepHaJILHLIX TOqeK. QeHTp HHepHH
COBoKynHocTb TeJl, paCCMaTpHB8eMbIX KaK e,nHHoe ueJIoe, Ha3bIB81OT Me-
xaHeCKOH CHCTeMOH.
CHJIbI, C KOTOpbIMH B3aHMo,neHcTBYIOT MaTepHaJIbHbIe TOqKH CIiCTeMbI Mem-
,ny C060H, Ha3bIBalOT BHyrpeHHHMH CHnaMIi. CIiJlbl, C KOTOpbIMIi Ha MaTepli-
anbHbIe TOqKIi CIiCTeMbI ,neHcTBYIOT TeJla, He Bxo,nSlIl..tlie B ,naHHYIO CIiCTeMY
(BHellIHHe TeJla), Ha3bIBalOT BHeWHHMH CIiJlaMH.
MeXaHHqeCKa.H CIiCTeMa TeJl. Ha KOTOpylO He LIeiicTBYIOT BHellIHlie CIiJIbl.
H83bIB8eTCSI 3aMKHYTOH, IiJIIi 1i30nlipOBaHHOH. CIiCTeMOH.
AJI.H npOCTOTbI paccym,neHIiH paCCMOTpliM BHaqaJIe CIiCTeMY, COCTO.HYIO
1i3 ,ABYX MaTeplianbHbIX TOqeK C MaccaMIi
ml H m2' KOTopbIe pacnOJlOmeHbI Ha OCH
a6culicc B TOqKaX C Koop)J;IiHaTaMH Xl H
X2 (pHC. 2.22). PaCCTO.HHHe Mem,ny 3TliMIi
TOqKaMH l = x 2 - XI' TOqKY C, KOTOpa.H ,ne-
JIIiT 3TO paCCTO.HHlie Ha OTpe3RIi. 06paTHo
X nponopUIiOHanbHbIe MaCC8M, Ha3bIBalOT eH-
TpOM Macc.
CJle,AOBaTeJIbHO, no onpe,n;eJIeHIiIO
y
z

Puc. 2.22
m 2
!1. = m2
 mI'
(2.3)
43

TIYCTb KoopHHaTa aTOM TOqKH xc, Tor8  = Xc - Xl;  = X 2 - Xc' TIocTa-
BHM B ypaBHeHHe (2.3), nOJlyqHM: n1-t (xc - Xl) = m2 (X2 - Xc). OTCIOa
n1-t Xl + m2 x 2
Xc = .
n1-t+
(2.4)
<l>opMYJla (2.4) n03BOJlHeT onpeeJIHTb KOOp)lHHaTY ueHTpa Macc CHCTeMbI,
COCTO.HeM H3 BYX M8TepHaJlbHbIX TOqeK.
BbIpameHHe (2.4) MomHO 0606II.\HTb Ha CHCTeMY n MaTepHaJIbHbIX TOqeK,
pacnOJlOmeHHbIX rrpOH3BOJIbHbIM 06pa30M. KoopHHaTa eHTpa Macc
n1-t Xl + m 2 x 2 + .. .mnX n
Xc = .
m l + m 2 + ... + m n
(2.5)
AHaJlOrHqHbIe BbIp8meHH.H nonyq8lOTC.H H n.H KOopHHaT Yc H Zc'
TIOJIOmeHHe eHTpa Macc MomHO TaKme OnpeeJlHTb C nOMOII.\blO pa,AHyca-
BeKTOpa:
n
m.r..
L.J rr
- i=l
rc = ,
ml + m2 + .. . + m n
re mi Ii  - MaCCa H P8HYC-BeKTOp i-Ii qaCTHbI.
TIpH BHmeHHH MaTepHanbHbIx TOqeK CHCTeMbI KoopHHaTbI HX H3MeHH-
IOTCH. 3anHcaB BbIpameHHe (2.5) JI.H BYX MOMeHTOB BpeMeHH t l H t 2 . BbI'l-
TeM OHO H3 Pyroro H nOJIyqHM:
ml1 + 2 +... + m n l1x n
6.x c = .
m l + m 2 +... + m n
Pa3eJlHB 06e qaCTH 3Toro BbIpameHH.H H8 npOMemYTOK BpeMeHH
6.t = t 2 - t l , HMeeM:
mlv lx + V2x +... + mnv nx
v cx = ,
m l + m2 + .. . + m n
re Vcx' Vlx ... V nx - npOeKHH Ha OCb Ox BeKTOpOB CKOpOCTH BHmeHHSI
eHTp8 MaCC H MaTepH8JIbHbIX TOqeK I, 2 ... n.
AHaJIOrHqHbIe BbIpameHH.H MomHO 3anHC8Tb H n.H npoeKHH CKOpOCTH H8
OCH Oy H Oz.
BbIpameHHe JI.H H8XOmeHH.H CKOpOCTll BHmeHH.H eHTpa Macc B BeKTOp-
HOM BH,Ae:
- fflt. VI ;- m 2 v 2 + ... ;- m n Un
V =
c
m l +  + ... + m n
(2.6)
TIpH BHmeHHH MaTepHaJlbHbIX TOqeK CHCTeMbI eHTp MaCC nepeMeaeT-
C.H. OnpeeJIHM, OT qerO 38BHCHT XapaKTep BHmeHH.H eHTpa MaCC.
An.H 3TOm paCCMOTpHM CHCTeMY ,ABYX MaTepHaJIbHbIX TOqeK. TIYCTb Ha 3TH Ma-
TepHaJIbHbIe TOqKH eMCTByroT BHelIIHHe CHJIbI PI H P 2 H 3TH TOqKH B3aHMOeH-
44

m l
m 2
CTByroT Mem.n;y C060M CHJlaMH F l2 H
F 21 (pHC. 2.23). 3anHllleM .n;JI.H Kam,AOM
MaTepHanbHoM TOqKH BTOpOH 3aKOH
HbIOTOHa:
PI '" _ _ / F 2
----
m1a l = F l2 + it;
a2 = F 21 + F 2 .
BbIpa3HM YCKopeHHe TOqeK qepe3 HX HaqanbHbIe VI H V2 H KOHeQHbIe
Puc 2.23
V H v CKOpOCTH:
_ _ v - vI .
a l --,
I:1t
_ v - v2
a 2 = I:1t .
IIo.n;cTaBHM nOJlyqeHHbIe BbIpameHH.H BO BTOpOM 3aKOH HblOToHa:
m l (v - vd = (F12 + Pdl:1t;
m 2 (v - v 2 ) = (F21 + F 2 ) I:1t.
CKJIa.n;bIBa.H 3TH paBeHCTBa, HMeeM:
(mlV +m2 v ;)-(m I V l +m2 V 2)=(F 12 +F 2 .)Llt+(ii +F 2 )Llt. (2.7)
IIoCKOJlbKY no TpeTbeMY 3aKOHY HblOToHa ii2 = -F 21 , TO F l2 + F 21 = O.
ll3 CPOPMYJlbI (2.6) nOJIyqaeM:
VI + v2 = v e ( + m 2 );
v; + m2iJ = v; ( + m 2 ),
r.n;e v e H v; - CKOpOCTH ,ABHmeHH.H eHTpa Macc B HaqaJlbHbIH MOMeHT Bpe-
MeHH H qepe3 npOMemYTOK BpeMeHH f1t. Tor.n;a paBeHCTBO (2.7) MomHO 3anH-
caTb TaK:
( + m 2 )( iJ; - v e ) = (ii + F 2 ) Llt.
(2.8)
BbIpameHHe (2.8) nOKa3bIBaeT, qTO H3MeHHTb CKOpOCTb ,ABHmeHH.H eHTpa
Macc CHCTeMbJ MaTepHaJlbHbIX TOqeK MorYT TOJlbKO BHelllHHe CHJlbl, BHYTpeH-
HHe CHJlbl H3MeH.HIOT CKOpOCTH .n;BHmeHHSI OT.n;eJIbHbIX MaTepHaJlbHbIX TOqeK.
0603HaqHM ml + m2 = M, r.n;e M - CYMMapHa.H Macca MaTepHaJlbHbIX TO-
qeK CHCTeMbl. CJle)J;oBaTeJlbHO,
M (v; - v c ) - - Llv - -
= FI + F 2 => M ---E.. = FI + F 2 =>
t Llt
n
Mile = PI + P 2 => Mile = Lit,
i=l
(2.9)
T. e. ijelimp .Macc CUCme.Mbt 06UJlCemCSl KaK .Mamepua.n.bliaSl mOIlKa 6 KOmO.
poil cocpeOOmOllelia .Macca BCeU CUCme.Mbt U lia KomopylO oeilcmByem cu.n.a.
45

pa6liaSl ZeO.MempUlleCKOU CYM.Me 6CeX 6lieWHUX CU.n.. BeUCm6YlOUJ,UX lia cucme-
.My. BbIpameHHe (2.9) npeCTaBJI.HeT C060H 3aKOH ABHjKeHHH eHTpa Macc.
n
ECJIH CHCTeMa MaTepHaJIbHbIX TOqeK 3aMKHYTas, T. e. L..F; = 0, TO Miic = O.
1=1
CJIe,nOBaTeJlbHO. ueHTp Macc 3aMKHYTOH CHCTeMbI MaTepHaJlbHbIX TOqeK
BHmeTC.H paBHOMepHO H np.HMOnHHeHHO JIH60 OCTaeTC.H HenOBHmHbIM He-
3aBHCHMO OT Toro, KaK BHmYTC.H OTeJlbHbIe MaTepHaJlbHbIe TOqKH CHC-
TeMbI.
TaK, eCJlH B HHepHaJlbHOH CHCTeMe OTCqeTa eHTp Macc 3aMKHYTOH CHC-
TeMbI 6bIJ1 HenOBHmeH B KaKOH-TO MOMeHT BpeMeHH, TO OH 6y,neT OCTaBaTb-
C.H B nOKoe, HeCMOTp.H Ha BHmeHHe OTeJlbHbIX MaTepHaJlbHbIX TOqeK CHCTe-
MbI B pe3YJlbTaTe HX B3aHMoeHCTBH.H.
TaKHM 06pa30M, u.eHTp Macc (u.eHTp HHepu.HH) CHCTeMbI. nOJlOmeHHe KO-
Toporo xapaKTepH3yeT pacnpe1l'.eneHHe Macc 3TOH CHCTeMbI H onpeeJl.HeTC.H
q,opMynoH (2.5), a CKOpOCTb KOToporo onpeeJI.HeTCH q,opMynoH (2.6), .HBJIS-
eTCH xapaKTepHOH TOqKOH JIS CHCTeMbI, nOBeeHHe KOTOpOH nO,AqHH.HeTC.H 3a-
KOHaM HblOTOHa H He 3aBHCHT OT npHpObI CHJI B3aHMoeHCTBH.H B CHCTeMe.
IIoH.HTHe u.eHTpa Macc OTHOCHTCS 11 K OHHOqHOMY TeJIY, KOTopoe MomHO
npeCTaBHTb KaK cOBoKynHocTb MaTepHaJlbHbIX TOqeK. l(eHTp Macc TeJIa Mom-
HO paCCMaTpHBaTb KaK TOqKY (HaXO.HYIOC.H BHYTPH TeJla HJIH BHe ero), B
KOTOpOH nepeceKalOTC.H JlHHHH eHCTBH.H CHJI, npHBo,n.HHX BaHlioe me.n.o 6
nocmyname.n.blioe B6UJlCeliue. l(eHTp Macc TeJIa COBna,AaeT C ero U.eHTpOM TS-
meCTH (CM.  4.5).
2.20. HeHHepn;HaJlLHLle CHCTeMLI OTC'IeTa
CHCTeMbI OTCqeTa, B KOTOpbIX cB060Has MaTepHaJlbHa.H TOqKa HJlH CBO-
60Hoe TeJIO He COXP8H.HIOT CKOpOCTb BHmeHH.H HeH3MeHHOH npH KOMneHca-
U.HH BHellIHHX B03eHCTBHH (HeHHepu.HaJlbHOe BHmeHHe), Ha3bIBaIOTC.H HeHHep-
lJ,HaJIbHLIMH CHCTeMaMH OTCqeTa (HCO).
HeHHepUHaJIbHOH .HBJlSeTC.H CHCTeMa OTCqeTa, BHmyaHC.H C YCKopeHHeM
OTHOCllTeJlbHO HHepu.HaJIbHOH CllCTeMbI OTCqeTa (RCO).
B 3JIeMeHTapHOM Kypce q,H3HKH paCCMaTpHBaIOTC.H npocTeHllIHe HeHHep-
u.HaJlbHbIe CHCTeMbI OTCqeTa, BHmYHeC.H nocTynaTenbHO C YCKopeHHeM. B
HeHHepu.HaJlbHbIX CHCTeMax OTCqeTa He BbInOJlHSIOTC.H 3aKOHbl HbIOTOHa. IIo-
KOH HJlH BHmeHHe MaTepHaJlbHbIX TOqeK H Ten B HeHHepu.llaJlbHbIX CHCTe-
Max OTCqeTa onHCbIBaIOTC.H ypaBHeHH.HMH, aHanOrHqHbIMH no q,opMe 3anHCH
ypaBHeHHIO BToporo 38KOHa HbIOTOHa. OHaKO B ypaBHeHHe BBO,A.HTC.H CHnbI
HHepUHH.
CHJIbI HHepu.HH He BbI3bIBalOTCS B03,AeHCTBH.HMll Ha ,AaHHYIO MaTepHaJlbHYIO
ToqKY HJlH TeJlO KaKHX-TO ,ApyrHx Ten HJIll nOJleH, OHH nHllIb OTpamalOT He-
HHepu.HanbHOCTb caMOH CHCTeMbI OTCqeTa.
C:HJIbI HHepu.HH, npHKJlabIBaeMbIe K KaKOH-JlH60 CHCTeMe MaTepHaJlbHbIX
TOqeK, HapYllialOT 3aMKHYTOCTb 3TOH CHCTeMbI H Bcera .HBJI.HIOTC.H ,nJl.H 3TOH
CHCTeMbI BHellIHHMH CHJlaMH.
46

2.21. .HBJIeHHH, Ha6JIIO,IJ;aeMLle B HeHHepHaJILHbIX CHCTeMax OTC'IeTa
x T.HHeTC.H Ha /)'l, nonomeHHe mapa B Haqa-
Jle ,ABHmeHH.H OTHOCHTenbHO BarOHeTKH H3-
MeHHTC.H no cpaBHeHHIO C ero nOJIOmeHH-
eM npH a... c = 0, a ,AaJlee OTHOCHTeJIbHO BarOHeTKH OH 6Y,AeT HenOlIBHmeH.
- - -
IIpH OTCYTCTBHH TpeHH.H mg + N p + F H + Fynp = 0, r,Ae mg - CHna T.H>Ke-
CTH, N p - CHJIa peaKHH nOJla, F ynp - CHJIa ynpyrocTH co CTOpOHbl paCT.H-
///!//J///
Puc. 2.24
Y
Y'!
a..C'.

N p
o
Puc. 2.25
IIpH onHcaHHH nocTynaTeJlbHoro ,nBH-
meHH.H Tena B HeHHepHaJlbHOR CHCTeMe
OTCqeTa (RCO), ,ABHmyeRC.H nocTynaTeJIb-
x'
HO, BBO,nHTC.H CHJIa HHepHH F i = -ma H . C '
r,ne a... c - YCKopeHHe caMoR RCO OTHO-
CHTeJlbHO KaKoH-To HHepllHaJlbHOR CHCTeMbl
OTCqeTa.
PaccMoTpHM npHMep C mapoM Mac-
COR m, Haxo,n.HHMC.H Ha rna,AKOM nony
BarOHeTKH H npHKpenJleHHbIM K ee nepe-
,AHeH CTeHKe npymHHoH (pHC. 2.24).
Ra pHCYHKe 2.25 CHCTeM8 xOy - He-
nO,ABHmHa.H HHepHaJlbHa.H CHCTeMa OTCqeTa,
a x'O'y' - HeHHepHaJlbHa.H CHCTeMa OTCqe-
Ta, CB.H3aHHa.H C BarOHeTKOR.
IIpH ,ABH>KeHHH BarOHeTKH C YCKopeHH-
eM a... c OTHOCHTenbHO xOy npY>KHHa pac-
HYTOH npymHHbI, F H - CHJla HHepHH.
B npOeKH.HX Ha OCH Oy H Ox HMeeM: mg = N p ; F H = Fynp' TaK KaK MO-
,nYJlb Fynp = kt1l, TO F}{ MomeT 6bITb H3MepeHa no paCT.HmeHHIO npymHHbI.
,ZJ;BHmeHHe mapa BMeCTe C BarOHeTKOR OTHOCHTeJlbHO HHepHaJlbHOH CHCTeMbl
xOy onHCblBaeTC.H BTOpbIM 3aKOHOM HbIOTOHa:
mg + N p + Fynp = ma... c ' HJlH B npoeKHH F ynp = ma u . c '
Tor,Aa F}{ = ma u c' a C yqeToM HanpaBJleHHR 1;1 = -ma H C'
BOIIPOCbI II 3A,lI;AHII.H ,lI;JUI CAMOKOHTPOJUI
1. llpu8eoume npUMepbl. Kozoa 3eM.n.1O MOJICliO Cilumamb UliepijUa.n.bliou cu-
cmeMOU omcilema.
2. 'C[mo nOliUMalOm noiJ "uHepijUeu" U "uliepmliocmblO"?
3. Kale deu:HCemc.1l meJW noiJ iJeucm6ueM nOCmO.1lHHOU CU.n.bl?
4. Kliuza .n.e:HCum lia cmo.n.e. YKaJICUme CU.n.bl. noiJlluliSlIOUJ,uecSl mpembeMY
3aKoliY H blOmOlia.
47

5. H auoume omlioweliue YCKopeliuu 06YX JlCe.n.e31ibLX wap06 npu ux cmo.n.-
Kli06eliUU. ec.n.u paouyc nep60zo wapa 6 2 pa3a .MelibWe paouyca 6mopozo.
6. llolle.MY me.n.a, liaXOO.RU{uecSl 6 KO.Mliame. lieC.MOmpSl Ha UX 63aU.MHOe
npum.RJlCeliue. lie npU6JlU:JfCalOmCSl opyz K opyzy?
7. KaK. no.n.b3Y.RCb 3aKOliO.M 6Ce.MUpHOZO m.RZOmeHUJl. .MOJICIiO liaumu .Maccy
3e.MJlu?
8. KaKue oeljJop.Ma4UU onUCbL6alOmcSl 3aKOliO.M rYKa?
9. q mo nOliU.MalOm noo a6co.n.lOmlibL.M yo.n.ulieliue.M? omliocume.n.blibL.M yo-
.n.ulieIiUe.M?
10. /(eucm6yem .n.u cu.n.Q mpeliuSl nOKOSl lia npeo.Mem. .n.eJICaU{uu IiQ zopu
30Iima.n.bIi0.M cmo.n.e? lia liaK.n.oliHoU nJlOCKocmu?
11. KaKaSl cu.n.a yoepJICu6aem me.n.o lia 6paU{alUl4e.MCSl oucKe? KaK OHa Ha
npa6Jlelia?
12. 3a6ucum .n.u mOp.M03liOU nymb a6mo.M06u.n.Sl om ezo .MaCCbt?
13. llpu KaKUX YC.n.06USlX .MaWUlia liQ 3aKpyz.n.eHuu oopozu lie 6yiJem 6bL'
6powelia lia 0601lUliY?
14. H a KaKOU cmaouu 06UJlCeliUSl KOC.MUlleCKOZO Kopa6.n.Sl KOCMOlia6m no.
IlY6cm6yem cocmOSlliue lie6eCO.MOCmu?
15. II olle.MY 6btZOOliO 3anycKamb paKembL-Hocume.n.u 6 n.n.OCKocmu 3K6a-
mopa?
_ F
a=-
m
Pi = -F 2
Fx = -kx
OCHOBHLIE «I>OPMY JILl
- BTOPO 3aKOH HblOToHa, rAe a - YCKopeHe Tella, F
paBHoAecTBYIOw.aSl Cll, AecTBYIOW.X Ha TellO, m - Macca
Tella.
- TpeT 3aKOH HblOTOHa, rAe FI  i2 - Cllbl B3aMOAe-
CTBSI ABYX Tell.
- 3aKOH rYKa, rAe F x - npoeKu.SI Cllbl ynpyrocT, k - >KeCT-
KOCTb Tella, x - a6COlllOTHoe YAJlHeHe
F - F - ,.1\1 - Clla TpeHSI CKOllb>KeHSI, rAe FT p .n.max - MaKCMallbHaSl Clla
-rp - -rp.n.max - f-U" p
TpeHSI nOKOSl, J.l - K034>4>u.eHT TpeHSI, N p - Clla peaKu.
OnOpbl, paBHaSl no MOAYlllO Clle HopMallbHoro AaBlleHSI Tella
Ha onopy
F = Gml
r 2
F T = mg
P=-N
p
48
- 3aKOH BceMpHoro TSlrOTeHSI, rAe F - Clla npTSI>KeHSI,
G - rpaBTau.OHHaSl nOCTOSlHHaSl, ml  m2 - MaCCbl Tell,
r - paCCTOSlHe Me>KAY HM.
- Clla TSI>KeCTVI, rAe g - YCKopeHVle cB060AHoro nCiAeHSI
- Bec Tell a , rAe N p - Clla peaKu. onopbl

v =  GM
R+h
- JU1HeHa CKOpOCTb cnyrHiIIKa, ABiII>Kyw.eroc no KpyroBO op-
6i11Te rAe G - rpaBiIITau.iII0HHa nOCTOHHa. M - Macca nl1a-
HeTbl, R - pa,n,iIIYc nl1aHeTbl, h - BblCOTa cnyrHiIIKa HaA nOBepx-
HOCTblO nl1aHeTbl.
v=JgR
- nepBa KOCMiIIeCKa CKOpOCTb Tel1a Afl nl1aHeTbl (h ::::: 0) rAe
g - YCKopeHille cBo6oAHoro naAeHiII Ha 3TO nl1aHeTe.
METOAHqECKHE PEKOMEH,n:AQHH
IIpH perneHHH 3aaq no TeMe "HHaMHKa MaTepHaJIbHOH TOqKH" CJIeyeT
npHepmHBaTbC.H cJleYIOll.\ero aJlrOpHTMa:
1. CeJlaTb cXeMaTHqeCKHH PHCYHOK, H306pa3HB Bce CHJlbI, eHcTBYIOHe
Ha Kamoe TeJlO paccMaTpHBaeMoH CHCTeMbI.
2. BbI6paTb CHCTeMY KoopHHaT xOy C TaKHM pacnOJlOmeHHeM oceH.
qT06bI HaH60JIbrnee KOnHqeCTBO BeKTopOB CHJl 6bIno HanpaBJIeHO B,AOJIb
oceH. B CJIyqae BHmeHH.H TeJla no oKpymHocTH OCb Ox OJl:>KHa 6bITb
HanpaBJleHa K ueHTpy oKpymHocTH (KaK H ueHTpOCTpeMHTeJlbHOe YCKO-
peHHe).
3. 3anHcaTb AJI.H Kamoro TeJIa B OT,AeJIbHOCTH BTOpOH 3aKOH HblOToHa B
n
BeKTopHoH cpopMe (HHaMHqeCKOe ypaBHeHHe BHmeHH.H); mli = L F;.
i=1
4. Cnpoen;HpOBaTb 3TH ypaBHeHH.H Ha BbI6paHHble OCH KoopHHaT.
5. OnOJlHHTb nOJlyqeHHYIO CHCTeMY B cnyqae He06xoHMOCTH KHHeMaTH-
qeCKHMH H ,AHHaMHqeCKHMH COOTHorneHHHMH H pernHTb ee OTHOCHTenbHO HC-
KOMOH BeJlHqHHbI.
nPIIMEPbI PEIDEHIUI 3A,z::(A q
1. ABTOM06HJlb MaCCOH 3000 Kr, pa3rOH.H.HCb H3 COCTO.HHHH nOKOH, o-
M
CTHraeT CKOpOCTH 8,0 -, a 3aTeM npO,AOJlmaeT BHmeHHe C BbIKJllOqeH-
C
HbIM BHraTeJleM o nOJlHOH OCTaHOBKH OnpeeJlHTe YCKopeHH.H aBToM06H-
JI.H npH pa3rOHe H npH TopMomeHHH, nOJlHOe BpeM.H BHmeHH.H Ii npoH-
eHHbIH nYTb. CHJIa THrH ABHraTeJI.H 3000 H, K03cpcpHHeHT conpOTHBne-
HH.H 0,020.
M
)J;aHO: m=3000 Kr, vo=O, vl=8,0
C '
v=O. F=3000 H, J,.l=0,020.
HaHTH: aI, a2, t, s.
P e III e H H e. Ha aBTOM06HJIb H8 pa3rOHHOM yqaCTKe eHCTBYIOT CHna TH-
:>KeCTH mg, CHJla HOpMaJIbHOH peaKHH nonOTHa oporH N p , CHJl8 T.HrH F,
49

CHna conpOTHBJleHH.H F:: (pHC. 2.26). 3anHweM OCHOBHoe ypaBHeHHe ,nHHaMH-
KH B BeKTopHoH cpopMe: mi4 = mg + F:: + N p + F.
B npOeKQ,H.HX Ha OCH OX H Oy:
mal = F - ; 0 = N p - mg.
TaK KaK Fc = JjN P' TO, pewa.H BMecre 3TH ypaBHe-
HH.H, HeM YCKopeHHe al:
F M
a l = - - J..lg, a l = 0,80 "2'
m c
IIyrb, npoH,neHHbIH aBTOM06HJleM npH pa3rOHe,
x HaH,neM no cpopMYJle
v 2
51 = -L, 51 = 40 M.
2a l
3anHlIIeM OCHOBHoe ypaBHeHHe ,nHHaMHKH ,nJl.H yqacTKa TopMOmeHH.H:
mii 2 = F:: + mg + N p .
B npOeKH.HX Ha OCH OX H Oy: ma 2 =; 0 = N p - mg.
Pewa.H BMeCTe 3TH ypaBHeHH.H, HaH,neM YCKopeHHe a2:
M
a 2 = J..lg. a 2 = 0, 20 2'
c
y
N p
a l

o
Fc
F
mg
Puc. 2.26
,lJ,nHHa yqacTKa TopMOmeHH.H
v 2
82 = _ 2 I , 8 2 = 1,6 . 10 2 M.
a2
llpOH,neHHbIH nYTb 8 = 81 + 82, 8 = 2,0 . 10 2 M.
llOJlHOe BpeM.H ,nBHmeHH.H
VI VI VI VI
t l = -. t 2 = -  t = - + -. t = 50 c.
al a2 a l a2
2. Ha r na,nKoH ropH30HTaJlbHOH nOBepXHOCTH Haxo,n.HTC.H TPH 6pycKa Mac-
caMH 1. 2, 4 Kr, CB.H3aHHbIe HeBeCOMbIMH HepaCT.HmHMbIMH HHT.HMH. K 6PYCKY
60nbweH MaCCbI npHnomeHa CHna 7 H. HaH,nHTe YCKopeHHe 6PYCKOB H CHnbI
HaT.HmeHH.H HHTeH.
,lJ,aHo: m3= 1 Kr. m2= 2 Kr. ml=4 Kr. F=7 H.
HaHTH: Tit T2' a.
Npa
a

N p2 N pi
PemeHHe. Ha pHCYHKe
2.27 nOKa3aHbI CHnbI, npHJlO-
meHHbIe K TeJlaM. Bce Tena
)J;BHmYTC.H C o,nHHaKOBbIMH
YCKopeHH.HMH ii.
3anHweM OCHOBHOe ypaB-
HeHHe ,AHHaMHKH ,nJl.H Kam,AO-
ro H3 Ten:
x
 ii = F + N p] + m l g + 7i ;
ii = N p2 + m 2 g+ T{ T T 2 ;
ii = N p3 + g + t;.
mal
m;g
mil
Puc. 2.27
50

YqHTbIBa.H, I:ITO MOYJlH CHn 7i = T{ H T 2 = T;, B npoeKHH Ha OCb Ox
nonYQHM:
m 1 a = F - 7i ;
a = 7] -T 2 ;
m 3 a = T 2 .
CKnabIBaSI 3TH ypaBHeHH.H. nonYQHM: (+  + rrz.a) a = F.
F M
OTKya a = , a = 1 2'
m} + m2 + ffl:J C
3aTeM H3 TpeTbero H BToporo ypaBHeHHH CHCTeMbI HaXOHM:
T 2 = In:3a, T 2 = 1 H.
1;. = 12 + m 2 a, 1;. = 3 H.
3. KaKoH yroJl C BepTHKaJlblO 06pa3yeT HHTb KOHHqeCKOrO MaSITHHKa (Teno,
nO,ABemeHHoe Ha HHTH, KOTOpa.H npH BHmeHHH Ma.HTHHKa onHCbIBaeT KOHH-
qeCKYIO nOBepxHocTb), JlHHO:H 1,2 M, eCJlH ero nepHO.n; 06paeHH.H 2,0 c?
,[(aHo: l=1,2 M, T=2,0 c.
HaHTH: a.
y P e III e H H e. Ha Ma.HTHHK .n;eHcTBYIOT CHJla
T.HmeCTH mg H CHna HaT.HmeHH.H HHTH F;t
(pHC. 2.28). YCKopeHHe MaSITHHKa HanpaBneHO
no paHYCY K eHTpy oKpymHoCTH. 3anHmeM
OCHOBHoe ypaBHeHHe .n;HHaMHKH: miiu = mg + };..
CnpoeHpyeM Ha OCH OX H Oy:
x
o
mg
Puc. 2.28
ma u = F;. sin a
ma u = 1;. sin a;
=>
o = 1;. cos a - mg mg = F;. cos a.
Pa3eJlHM OHO ypaBHeHHe Ha pyroe:
au = sin a
g cos a
Ho
v 2
a = -,
r
re r - paHYc oKpymHocTH, KOTOPYIO onHCbIBa-
eT Ma.HTHHK B ropH30HTaJlbHOH nJlOCKOCTH. M3 pHCYHKa 2.28 BHlIHO, qTO
. 2nr 4n 2 r 4n 2 lsina
r = l SIn ex. TaK KaK v = -. TO au = - =
T T 2 T 2
nO.n;CTaBHB 3TO 3HaqeHHe YCKopeHH.H, nOJlyqHM:
4n 2 l sin ex sin ex gT2
2 = => cosa =, cosa = 0,85, a = 34°.
T g cos a 4n l
4. HaHHTe nJlOTHOCTb BeeCTBa nJlaHeTbI, nepHO BpaeHH.H KOTOpOH
24 'I, ecnH Ha ee 3KBaTope TeJla HeBeCOMbI.
.n a H 0: T = 24 . 3600 c, P = O.
HaHTH: p.
P e III e H H e. Ha TeJlO, HaXO.Heec.H Ha 3KBaTope nJlaHeTbI, .n;eHcTByeT CH-
na T.HrOTeHHSI F H CHJla HopManbHoH peaKHH nOBepXHOCTH nnaHeTbI N p
51

Puc. 2.29
N p
(pHC. 2.29). Teno BMeCTe C nJlaHeTOM paBHOMepHO
BpaaeTC.H BOKpyr ee OCH CJleOBaTeJlbHO, YCKO-
peHHe 6Y,AeT H8npaBJleHO no paHYCY K eHTpy
oKpymHoCTH.
OCHOBHoe ypaBHeHHe ,AHHaMHKH 3anHllIeTC.H TaK:
mall. = F + N p ,
HnH B npoeKHSlX Ha OCb Ox: mall. = F - N P' I10
TpeTbeMY 3aKOHY HblOToHa P = N p . Ho no YCJlO-
BHIO P=O. CJle,AOBaTenbHo, Np=O. I103TOMY
GMm
ma = F = -, re G - rpaBHTaHOHHa.H no-
u. R 2
CTOSlHHa.H,
a R 2
f.-I - Macca nJlaHeTbI, R - ee paHYC. OTclOa M = . eH-
v 2 4n 2 R 4nR 3
TpOCTpeMHTenbHoe YCKopeHHe a = - = -. 3HaqHT, M = - I1cKoMa.H
u. R T 2 GT2 .
M 41tR 3
nnOTHOCTb p = V' re V = -----g- - 06'beM nJlaHeTbI. CneoBaTenbHo,
4n 2 R 3 .3 3n
p=
GT 2 . 4nR 3 - GT 2 '
p = 19
Kr
3 .
M
5. BbIqHCJlHTe nepBYIO KOCMH'ieCKYIO CKOpOCTb ,AnSI JIYHbI. PaHYc JIYHbI
npHHSlTb paBHbIM 1 760 KM. YCKopeHHe cB060JJ.Horo naJJ.eHH.H Ha nOBepXHOCTH
M
JIYHbI 1,60 2".
C
,l{aHO: R = 1.760 10 6 M. g= 1,60
M
2 .
C
HaMTH: V.
Pew e H H e. Ha cnYTHHK 6YAeT .neMCTBOBaTb TonbKO CHJla T.HmeCTH mg,
KOTOpa.H 6YAeT co06aTb eMY He06xOAHMoe eHTpOCTpeMHTenbHoe YCKopeHHe
mall. = T11:g, TaK KaK
v 2 v 2 JRi 3 M
a -TO - = g => v = R g , v = 1,68 .10 - .
U = R' R C
6. PaCCTOSlHHe MemAY eHTpaMH 3eMJlH H JIYHbI paBHO 60 3eMHbIM pa,AH-
ycaM. a Macca JIYHbI B 81 pa3 MeHbllIe MaCCbI 3eMnH. B KaKOM TOqKe npSl-
MOM, COeAHH.HIOeH HX neHTpbI, Teno 6Y,lIeT npHTSlrHBaTbC.H K 3eMJle H JIYHe C
oAHHaKoBbIMH cHnaMH?
.I(aHO: l=60, M3 =81M n ,  =6,4.10 6 M, F i =F 2 .
HaMTH: li'
52

P e ill e H H e. HaHeM CHJIbI THrOTeHIiH F 2 Me.my TeJIOM Ii 3eMJIeH H Fl
Mem,Ay JIYHOH Ii TeJIOM. ITo 3aKOHY BceMlipHoro T.HI'OTeHIiH (pIiC. 2.30)
3
F. - GM 3 m . F. _ GMnm
2 - (l _ )2' 1 - If
TaK Kal< no ycnoBli1O Fl = F2' TO
GM 3 m
(l _  )2
GMnm
if
=>
1 81
2 - (l _ 4)2 .
Puc. 2.30
IToCJIe H3BJIeqeHIi.H KBpaTHoro KOpH.H
1 9
--- =>
 l-
 =0,11=6, 4 =3,8.10 7 M.
7. Ha CKOJlbKO CMeCTliTC.H HenOBlimHa.H JIOKa MaccoH 20 Kr, ecnli qe-
nOBel< MaccoH 70 Kr nepeH,AeT C ee Hoca Ha KOPMY? PaccToHHlie OT Hoca ,AO
I<OpMbI 5 M, conpOTIiBJIeHlie BObI npeHe6pe.mIiMo MaJlO.
.D:aHo: M=280 Kr, m=70 KI', l=5 M.
HaHTIi: s.
P e ill e H H e. TaK KaK BHaqaJIe JlOKa Ii qe-
JlOBeK HenO.llBH.mHbI, a CHCTeMa "qenOBeK-JlO.ll-
Ka" - 3aMKHYTaH, TO eHTp Macc CIiCTeMbI
(TOqKa C) OTHOCHTenbHO IiHepliaJlbHOH CIiCTe-
MbI, CB.H3aHHOH C 6epeI'OM, 6Y,AeT HenOBlimeH.
BHaqane, O nepexoa qenOBeKa (pHC. 2.31),
KoopIiHaTa eHTpa Macc onpeeJl.HeTC.H no
cpopMYJIe
C. .0
S I
x
0
Puc. 2.31
Mi+ ml
x = 2 .
e M+m
B pe3ynbTaTe nepexo,Aa qeJlOBeKa C Hoca Ha KOPMY JlO,AKa CMeCTHTC.H Ha s
B CTOPOHY, npoTIiBOnOJIOmHYIO HanpaBJIeHlilO BlimeHIiH qeJIOBeKa. B aTOM cny-
qae KOOp,AHHaTa eHTpa Macc
x = M(  +s ) + ms
e M+m
Tal< KaK eHTp Macc HenOBH.meH, TO
l
M-+ml
2
M+m
M( +s )+ms
M+m
lIcKoMoe paCCTO.HHHe
ml
s= ,
M+m
s = 1 M.
53

3. 3AKOHbI COXP AHEHlI.H B MEXAHlIKE
Mnynbc Tena. Mnynbc cnbl. 3aKOH coxpaHeHSI Mnynbca. PeaKTB-
Hoe ABM)I(eHe. K. 3. LtMonKoBcKH - OCHOBOnOnO)l(HK yeHSI 0 peaKTBHOM
ABM)I(eH.
MexaHeCKaSl pa6oTa. Pa60Ta cnbl TSI)I(eCT, cnbl ynpyrocT, CMnbl
TpeHSI. MOU4HOCTb.
MexaHMecKaSl 3HeprSI. KHeTecKaSl  nOTeHu.anbHaSl 3HeprSI. nOTeH-
u.ManbHa 3Hepr Ten npM rpaBMTau.OHHOM B3aMOAeMCTB. nOTeH4Manb-
HaSl 3HeprMSI ynpyro AeopMMpOBaHHoro Tena. 3aKOH coxpaHeHSI 3Hepr B
MexaHMKe. 3MeHeHe 3HeprM B He3aMKHYTbiX CCTeMax. 3aKOH coxpaHeH
M npeBpaU4eHSI 3HeprM. Ynpyre  Heynpyre cOYAapeHSI Ten.
3.1. HMIIYJILC TeJIa
v
p
IIMnYJlLCOM TeJla (KoJlHqeCTBOM .D;BHmeHH.H) Ha3bI-
BaeTC.H BeKTOpHa.H <pH3HqeCKa.H BenHqHHa, paBHa.H npo-
H3Be.v;eHHIO MaCCbI TeJla m H8 ero CKOpOCTb V H H8-
npaBneHHa.H TaK me, KaK H CKOpOCTb (pHC. 3.1):
p=mv. (3.1)
E.nHHHueH HMnynbca Tena B CM SlBn.HeTCSI KHno-
Puc. 3.1
( Kr c ' M I .
rpaMM-MeTp H8 ceKYH,Ay J
IIycTb CKOpOCTb TeJl8 nOlI .D;eHCTBHeM CHJlbI F H3MeHSIeTC.H 3a BpeM.H t1t
OT VO .v;o V. COrJl8CHO OCHOBHOMY yp8BHeHHIO ,AHH8MHKH F = mil. YqHTbI-
_ v-v
BaSI, qTO a = --!!., nOJlYQHM
t1t
PD.t = mv - mvo  FD.t = P - Po = D.p =>
6.jJ = P D.t.
(3.2)
IIpoH3Be.v;eHHe CHJlbI H8 BpeMSI ee .v;eHcTBH.H Ha3bIBaeTC.H HMnYJlLCOM CHJlLI:
Fl!:.t. E.D;HHHl1eH HMnYJlbC8 CHnbI .HBJI.HeTC.H HbIOTOH-ceKYH.D;a (H. c).
<I>opMyn8 (3.2) BbIpamaeT BTOpOH 3aKOH HbIOTOH8, KOTOpbIH MomeT 6bITb
c<popMynHpoBaHo Cne,AYIOHM 06pa30M: U3Melieliue uMnYJlbCa me.n.a pa6liO UM
ny.n.bcy pa61i08eilcm6YIOeil CU.n., 8eilcm6YIOux lia 8alilioe me.n.o.
OrclO.v;a BH.D;HO, qTO UMny.n.bc me.n.a U3Me1i.RemCSl n08 oeilcm6ueM oalilioil CU.n.bl
oouliaK060 y me.n. Jl1060u MaCCbl. ec.n.u mo.n.bKO 6pe.M..R. 8eucm6uSl cu.n. oou lia K060.
MMnYJlbc Tena, KaK H CKOpOCTb, 3aBHCHT OT BbI60pa CHCTeMbI OTCqeTa. Y c-
KopeHHe .v;BHmeHH.H TeJla O,AHHaKOBO BO Bcex HHepl1HaJlbHbIX CHCTeMa:x OTCqe-
Ta. Cne.v;oBaTenbHo, CHna, a 3HaqHT, COrJlaCHO (3.2) H H3MeHeHHe HMnynbca
Tena He 3aBHCHT OT BbI60pa CHCTeMbI OTCqeTa. B Jl1060il UHepijUa.n.blioil CUC.
meMe omcilema U3M.elieHUe UM.ny.n.bca me.n.a 08UHaK060.
3.2. 3aKOH coxpaHeHHH HMIIYJILCa
PaCCMOTpHM CHCTeMY Tpex TeJl (pHC. 3.2). Ha TeJla .D;eHcTBYIOT BHelIIHHe
CHJlbI PI' P 2 , P 3 . CHnbJ ii.2' F21' l13' F31' F23' F 32 - BHYTpeHHHe CHJlbI. 3a-
nHlIIeM .v;n.H Kam.v;oro TeJla OCHOBHoe ypaBHeHHe .v;HHaMHKH:
54

Pl = iit + (F12 + F 13 )t.
b.P2 = F 2 b.t + (F21 + F 23 ) t,
P3 = F 3 b.t + (F31 + F 32 )t.
TIPOCYMMHpOBaB 3TH ypaBHeHH.H H yqHTbIBaH.
qTO cor JlaCHO TpeTbeMY aaKOHY HblOTOHa
112 = - F 21 , 113 = -F31' F 23 = -F32'
nOJlyqHM
b. (PI + P2 + P3) = (Fl + F 2 + F3 ) b.t :=)
:=) b.p = (Fi + F 2 T F3 )b.t,
r,Ae p = PI + P2 + P3 HMnynbc CHCTeMb] Ten.
H Mny.n.bC cucmeMbt me.n. pa6eli zeOMempUlleCKOU CYMMe UMnYJlbC06 me.n.
cucmeMbt. TaKHM 06pa30M. HMnynbc CHCTeMbI T'en MorYT H3MeHHTb TonbKO
BHellIHHe CHnbI. ECJlH CHCTeMa 3aMKHYTa, TO
Sp = 0 :=) p = const ;
Vl + V2 +... + mnV n = V + V; +... + mnv, (3.3)
PaBeHcTBo (3.3) BblpamaeT 3aKOH coxpaHeHHSI HMnynbca (3CI1): uMny.n.bC
3aMKliymou cucmeMbt me.n. coxpaliJlemCSl npu .n.106btX 63auMOaeucm6uSlx amux
meJl.
B cnyqae He38MKHYTOH CHCTeMbI 3CI1 Hcnonb3yeTcH, ecnH:
Fl
Puc. 3.2
n
a) reOMeTpHqeCKa.H cYMMa BHellIHHX CHn paBHa HynlO: L  = 0;
i=1
6) npOeKnH.H paBHO,AeHcTBYIOeH BHellIHHX CHn HB HeKoTopoe HanpaBJle-
n
HHe paBHa HYJlIO, T. e. eCJlH LFtx = 0, TO B,AOnb 3Toro HanpaBneHH.H HMnYJlbC
1=1
CHCTeMbI COXpaH.HeTC.H;
B) BpeM.H B3aHMO,AeHCTBH.H Mano (BbICTpeJl, B3PbIB. y,Aap H T.n.).
C nOMOblO 3CI1 MomHO BblqHCn.HTb CKOpOCTH TeJl, He 3Ha.H 3HaqeHH.H
CHn, ,AeHCTBYIOHX Ha HHX. 3CI1 .HBJlHeTC.H Bce06HM 3aKOHOM: OH npHMe-
HHM KaK K TeJlaM 06bI'lHbIX pa3MepOB, TaK H K KOCMHqeCKHM TeJlaM M 3ne-
MeHTapHbIM qaCTHnaM.
3.3. PeaKTHBHoe ABHmeHHe
IIo,A peaKTHBHLIM .lI.BHeHHeM nOHHMalOT ,nBHmeHHe Tena, B03HHKalOee
npli OT,AeJleHHH OT TeJla ero qaCTH C HeKoTopoH OTHOCHTenbHO TeJla CKOpOC-
TblO.
IIpH 3TOM nO.HBn.HeTC.H TaK Ha3bIBaeMa.H peaKTHBHaH CHJla, TonKalOa.H
Teno B CTOPOHY, npOTHBonOJlomHYIO HanpaBneHHIO ,nBHmeHHH oT,nenHIOIlJ;eHCH
OT Hero qaCTH TeJla.
55

Y
o
m
I  VI
Puc. 3.3
M H v 2
PeaKTHBHoe BHmeHHe cOBepllIaeT paKeTa (pHC. 3.3).
OCHOBHOH qaCTblO peaKTHBHoro )J;BHraTeJISI HBJI.HeTCH KaMe-
pa crOpaHH.H. B OHOH H3 ee CTeHOK HMeeTC.H OTBepCTHe -
peaKTHBHoe conJlO, npe,AHa3HaqeHHOe JlSI BbIxoa ra3a, 06-
pa3YIOerOC.H npFJ cropaHHH TOnJlHBa. BbICOKa.H TeMnepaTY-
pa H aBneHHe ra3a onpe,AeJl.HIOT 60JIbllIYIO CKOpOCTb HCTe-
qeHH.H ero H3 conJla.
.D:o pa60TbI ABHraTeJl.H HMny JlbC paKeTbJ H roplOqero 6bIJI
paBeH HYJlIO, CJle)J;OBaTeJIbHO, H nocJle BKJllOqeHH.H BHraTe-
JleH reOMeTpHqeCKa.H CYMMa HMnYJlbCOB paKeTbJ H HCTeKa-
IOHX ra30B paBHa HYJlIO:
mV 1 + MV 2 = 6,
r,Ae m H VI
Macca H CKOpOCTb BbI6paCbIBaeMbIX ra30B,
Macca H CKOpOCTb paKeTbI.
B npOeKl\lili Ha OCb Oy: M V2 - mV l = 0

m
V2 = - VI - CKOpOCTb paKeTbI.
M
He60JlbllIOrO li3MeHeHli.H MaCCbI
3Ta cI>opMYJla cnpaBeJlHBa npli YCJlOBHH
paKeTbI.
rJlaBHa.H oc06eHHOCTb peaKTHBHoro ABH)KeHli.H COCTOHT B TOM, qTO paKeTa
MomeT KaK YCKOp.HTbC.H, TaK H TOpM03HTbC.H H nOBOpaqHBaTbCH 6e3 KaKoro-
JlH60 B3aHMoeHCTBH.H C PyrHMH TeJlaMFJ B OT JIHqHe OT Bcex IJ;PyrHx TpaHC-
nopTHbIX cpenCTB.
IIo npliHl\liny peaKTliBHoro BHmeHli.H nepeBliraIOTC.H OCbMliHorli, KaJlb-
MapbI, KapaKaTHl\M, MeY3M.
BOJlbllIa.H 3acJlyra B pa3BHTHH TeopHH peaKTHBHoro )J;BlimeHH.H npHHaJIe-
mHT K. 3. HOJIKOBCKOMY. OH pa3pa60TaJl TeopHIO nOJleTa TeJla nepeMeHHoH
MaCCbI (paKeTbI) B O)J;HOpOHOM nOJle T.HrOTeHH.H H paCCqHTaJl 3anaCbI TOnJIH-
Ba, He06xoHMbIe JI.H npeOOneHH.H CHJlbI 3eMHoro npHT.HmeHHH, OCHOBbI Teo-
pHli mliKOCTHoro peaKTliBHoro BHraTeJI.H. a TaKme 3J1eMeHTbI ero KOHCTPYK-
l\HH. TeoplilO MHorocTyneHqaTbIX paKeT, npliqeM npeJlOmliJl Ba BapHaHTa:
napaJIJIeJIbHbIH (HeCKOJlbKO peaKTliBHbIX BHraTeJleH pa60TaeT oHoBpeMeHHo)
H nOCJleOBaTeJlbHbIH (peaKTHBHble )J;BHraTeJlH pa60TalOT pyr 3a pyroM).
K. 3. HOJIKOBCKHH cmpozo liaYiliiO ,AOKa3aJl B03MomHOCTb nOJIeTa B KOCMOC C
nOMOblO paKeT C )KHKOCTHbIM peaKTHBHbIM ,ABHraTeJleM. npeJlOmHJI cne-
l\HaJlbHbIe TpaeKTopHli nocaAKli KOCMliqeCKHX annapaTOB Ha 3eMJlIO, BbIBH-
HYJI lielO C03aHli.H MemnJlaHeTHbIX Op6HTaJlbHbIX CTaHl\liH, npeJIOmHJI lielO
aBTOMaTliqeCKOrO ynpaBJleHli.H paKeToH.
TpY,lIbI K. 3. HOJIKOBCKOro .HBHJIHCb TeOpeTliqeCKOH 6a30H JlSI pa3BHTliSl
cOBpeMeHHoH paKeTHoH TeXHHKH.
BOTIPOCbI II 3A}J;AHIIH }J;JIH CAMOKOHTPOJIH
1. Te.n.o aBU:HCemC.fL no OKpY:HCliOCmU c nocmOSlliliOU no Moay.n.1O CKO-
pocmblO. H 3MeliSlemcSl .n.u ezo uMny.n.bc?
56

2. Om Ilezo 3a6ucum U3.MelieliUe CKopocmu me.n.a? U3.MelieliUe U.Mny.n.bca
me.n.a?
3. OouliaKoBoe.n.u 3liailenUe U.Meem 6 pa3.n.UlllibtX Uliep4ua.n.bHblX cucme-
.Max omcilema U.Mny.n.bc me.n.a? U3.MelieliUe U.Mny.n.bca meJla?
4. KaK U.Mny.n.bc CUJlbl C6.R3an C U.Mny.n.bCO.M me.n.a?
5. A6mo.Mo6u.n.b npu mop.MOJICeliUU OBUJICemCR 3a.Meo.n.elilio. KaK lianpa6-
JleH 6eKmop U3.MelieliUSl ezo U.Mny.n.bca?
6. Te.n.o .MaCCOU m 6poweli0 6epmUKa.n.bli0 B6epx co CKopocmblO v O ' Hau-
oume U3.MelieliUe U.Mny.n.bca a) 3a 6pe.MSl noooe.Ma lia .MaKCU.Ma.n.bliYlO 6blcomy.
6) 3a 6ce 6pe.MSl 06 UJICeli USl.
7. cPym60.n.ucm npezpaJICoaem nymb .MSlIlY. Kozoa cu.n.bliee Oli nOIlYBcm6Y-
em yoap .MSllla: Kozoa 6eJICum liaBcmpellY .MSlIlY? Kozoa 6eJICum om .MSllla?
Kozoa cmoum lien006UJlC1i0?
8. Ycu.n.uu lieCKOJlbKUX lle.n.06eK oocmamoillio. Ilm06bl cOBuliymb C .Mecma
a6m06yc. llolle.MY 3mom JlCe a6m06yc lie COBuzaemcSl C .Mecma om nonaoa-
nUJl npomU60maliKOBOZO cliapSloa. np06u6alOU{ezo ezo lia6bl.n.em?
9. Mozym.n.u 8liympeliliue CU.n.bl U3.MeliUmb U.Mny.n.bC CUCme.Mbl me.n.? U.M-
ny.n.bCbl me.n. CUCme.Mbl?
10. B KaKUX c.n.YllaSlx 3aKOli coxpalieliUSl U.Mny.n.bca npU.MeliSllOm K lie3a.M-
Knymbl.M cucme.Ma.M?
11. M O:HCem .n.u lle.n.06eK. cmoSlU{uu lia uoea.n.blio z.n.aoKou .n.eoSlliou n06epx-
nocmu. c06uliymbcSl C .Mecma. lie ynupaSlcb OCmpbl.MU npeO.Mema.MU 6 .n.eo?
12. Mozym .n.u OCKO.n.KU 630p6a6weucSl zpaliambl .n.ememb 6 OOliO.M lianpa6-
JleHUU, ec.n.u 00 B3pblBa zpakama nOKOUJlacb?
13. CliapSlo B Bepxlieu mOIlKe mpaeKmopuu nOJlema pa30pBaJlC.R na 06a uc-
KOJlKa. ooun U3 KomopblX yna.n. lia 3e.M.n.1O mOllliO noo .Mecmo.M pa3pblBa. On.
peaeJlume 603.M0:HClibLe lianpaB.n.eliUSl nOJLema 8mop020 OCKOJLKa.
14. M ozym .n.u 06uzambcSl paKembl 6 6e36030YW1i0.M npocmpalicm6e? Ka-
Koe 6.n.USlliue lia 06UJlCeliUe paKembl OKa3bl8aem 6030YWliaSl cpeoa?
15. KaKU.M 06pa30.M .MO:HCliO U3.MeliSlmb 06UJlCeliUe KOCMUlleCKOZO Kopa6.n.Jl
noc.n.e 6bl6eoeliuSl lia op6umy?
16. JI.n.Sl Ilezo UCnOJlb3YlOmcSl .Mli02ocmynenllambte paKembt?
OCHOBHbIE OPMY JIbI
p=mv
- MnYllbC Tella (KollecTBo AB)!(eH), rAe m - Macca
Tella, V - ero CKOpOCTb.
F6.t = 6p
- BTOpO 3aKOH HblOToHa, rAe F 6.t - MnYllbC Cllbl F I
l:1t - BpeM ee AeLitcTB, 6.p - 3MeHeHe MnYllbca
Tella.
mlvl + m2 v 2 = mlv + m2v; - 3aKOH coxpaHeH MnYllbca, r.o.e mll\  V2 - M-
nYllbCbl Tell .0.0 B3aMOAeCTB. m 1 v  v; - MnYllb-
Cbl Tell noclle B3aMOAeCTB
57

METO.D:I1QECKI1E PEKOMEH.D:A:Q1111
TIpH pelIleHHH 3a,naq Ha 3aKOH coxpaHeHH.H HMnYJlbCa He06xOlIHMO:
1. YKa3aTb. KaKHe TeJla BXOIUIT B paCCMaTpHBaeMYIO CHCTeMY. Cn;eJlaTb PH-
CYHOK, H306pa3HB Ha HeM BeKTopbI HMnYJlbCOB TeJl Henocpe,nCTBeHHO nepe,n H
nocJle B3aHMo,neHCTBH.H. ECJlH HanpaBJleHHe HCKOMoro HMnYJlbCa 3apaHee He-
H3BeCTHO, TO ,nJI.H Hero npHHHMaeTC.R HeKOTopoe YCJlOBHOe HanpaBJleHHe
2. BbI.HCHHTb, .HBJI.HeTC.H CHCTeMa 3aMKHYToH HJlH HeT.
ECJlH CHCTeMa 3aMKHYTa HJlH BbInOJlH.HeTC.H O,nHO H3 CJle,nYIOHX YCJlOBHH:
a) BHelIlHHe CHJlbI ypaBHoBelIlHBaIOTC.H, 6) BpeM.H B3aHMo,neHcTBHH MaJlO,
B) npOeKH.H paBHO)J;eHCTBYIOeH BHelIlHHX CHJI Ha KaKoe-TO HanpaBJIeHHe
(OCb Ox) paBHa HYJlIO, TO 3anHcaTb 3aKOH coxpaHeHH.H HMnYJlbca:
n
L Pi = const
1=1
n
HJIH ,nJl.R nOCJle,nHero cJlyqa.R L Pix = const.
i=1
ECJIH CHCTeMa He3aMKHYTa, TO 3anHcaTb tiP = F I!1t, paCCMaTpHBa.H ,nBHme-
HHe Bcex TeJl CHCTeMbI B O,nHOH H TOH me HHepHaJlbHOH CHCTeMe OTCqeTa.
3. CnpoeHpoBaTb 3anHcaHHbIe ypaBHeHH.H Ha BbI6paHHbIe OCH Koop,nHHaT
Ox H Oy.
4. )J;onOJlHHTb npH He06xo,nHMOCTH CHCTeMY nOJlyqeHHbIX ypaBHeHHH KH-
HeMaTHqeCKHMH H ,nHHaMHqeCKHMH ypaBHeHH.HMH.
1. IIIapHK MaccoH m, JleT.HHH co CKOpOCTblO v,
y,nap.HeTC.H 0 CTeHKY no,n yrJlOM a K HeH H OTCKaKH-
BaeT no,n TeM me yrJIOM 6e3 nOTepH CKOpOCTH. On-
pe,neJlHTe HanpaBJleHHe H MO,nYJlb BeKTopa H3MeHeHH.H
HMnYJlbca lIlapHKa 3a BpeM.H y,napa.
)J;aHo: m, v, a.
HaHTH: l:1p.
P e III e H H e. TaK KaK HMnYJlbC - BeJlHqHHa BeK-
TOpHa.H, TO HCKOMoe H3MeHeHHe HMnYJlbca lIlapHKa
paBHO reOMeTpHqeCKOH pa3HOCTH KOHeqHOrO Ii Ha-
qaJlbHOro 3HaqeHHH HMny Jlbca:
l:1p = P2 - PI = mV 2 -mv 1 .
YqHTbIBa.H, qTO MO,nYJIb CKopOCTeH He H3MeH.HeT-
C.H, T.e. VI = V2= V, H3 pHCYHKa 3.4 Haxo,nHM MO,nYJlb H3MeHeHH.H HMnYJlbca:
l:1p = 2mvsin a.
2. Ky6HK H3 neHOnJlaCTa MaccoH 100 r JlemHT HB ropH30HTaJIbHOH no,nCTaB-
Ke. BbIcoTa Ky6HKa 10 CM. CHH3Y Ky6HK np06HBaeT BepTHKaJlbHO JleT.Ha.H
M
nYJl.H MaccoH 10 r. CKOpOCTb nYJlH npH Bxo,ne B Ky6HK 100
Puc. 3.4
npl1MEPbI PEIIIEHI1.H 3A.n:A Q
npH BbIJleTe -
95 M IIo,nnpbIrHeT JlH KYOHK?
C
C
58

AaHo: m 1 =0,100 Kr, h=O,lO M, m2=0,010 Kr,
M M
VI = 100 -, V2= 95 -
C C
HaHTH: F.
Pew e H He. Ky6HK nO,AnpbIrHeT, eCJlH MO,nYJlb
CHJlbI, ,neHCTBYIOeH H8 Ky61:1K co CTOpOHbI nYJlH, 6y-
,AeT 60JIbIlIe ero C1:IJlbI T.HmeCTH F > mlg (pHC. 3.5).
AJI.H Haxom,neHH.H 3TOH CHJlbI, paCCMOTpHM nYJIIO.
Ha Hee ,neHcTByeT C1:IJl8 T.HmeCTH H co CTOpOHbI
paBHa.H no MO,ny JlIO cor JlaCHO TpeTbeMY 3aKOHY HblOTOHa
o
'2g
+ F'
y
t V2

t v!
m 2
Puc. 3.5
F'
,
Ky61:1K8 CHJla
c1:IJle F.
3anHIlIeM BTOpOH 38KOH HblOTOH8 B HMnYJlbcHoH cpopMe: m2 v 2 - m2 v l =
= (F' + m 2 e) t. HJlH B npOeKlt1:l1:l H8 OCb Oy:
m2 v I - m2v2 = (F + m2g)t.
CqHTa.H, qTO nYJl.H BHYTPH Ky61:1K8 ,nBHmeTC.H paBH03aMe,AJleHHO, MomHO
HaHTH BpeM.H:
h = < V > l1t = VI + v2 l1t => l1t = 2h
2 VI + v2
IIo)J;cTaB1:IM BO BTOpOH 3aKOH HblOToHa:
2h
m2vI - m2v2 = (F;- g)
VI + V2
 (vi - vi )
OTKy,n8 F = m 2 g.
2h
IIOlICTaBHB qHCJleHHbIe 3HaqeH1:ISI. nOJlyq1:lM F = 50 H. CJle,noBaTeJIbHO, KY-
61:1K no,nnpbIrHeT, TaK KaK mlg= 1.0 H.
3. B pe3YJIbTaTe B3pbIB8 K8MeHb p83J1eTaeTC.H
Ha Tp1:l KYCKa. ABa KYCK8 JleT.HT nO)J; npSlMbIM yr-
JIOM ,npyr K ,npyry: KYCOK MaccoH 1, ° Kr co
M
CKOpOCTblO 12 , KYCOK M8CCOH 2,0 Kr co
P2 M
CKOpOCTblO 8,0 -. TpeTHH KYCOK OTJleT8eT co CKO-
C
Puc. 3.6
AaHo: ml = 1,0 Kr,
M
pOCTblO 40 -. KaKoBa MaCC8 TpeTbero KYCK8 H B
C
K8KOM HanpaBJleHHH OH JIeTHT?
M M M
vI=12 -, m2=2.0 Kr, V2= 8,0 , v3=40
C C C
HaHTH: m3, a.
P e ill e H H e CHcTeMa "K8MeHb- KYCKH" He3aMKHYTa BCJle,nCTBHe ,neHCTB1:ISI
C1:IJIbI T.HmeCTH. O,nHaKo 3a,naq8 MomeT 6bITI:. peIlIeHa Ha OCHOBaH1:I1:I 3CI1, Tal<
KaK BpeM.H pa3pbIBa M8J10. Ao pa3pbIBa KaMeHb nOK01:lJlC.H, n03TOMY HMnYJIbC
ero 6bIJI paBeH HYJlIO.
59

TIocJle pa3pbIBa HMnYJlbCbI KYCKOB COOTBeTCTBeHHO PIt P2' P3'
3anHllIeM 3CM: PI + P2 + P3 = O. feOMeTpHqeCKa.H CYMMa HMnYJlbCOB nep-
BbIX )lBYX KYCKOB Pl2 = PI + P2' M3 3CM P3 = - (PI + P2)'
CneOBaTeJlbHO, P3 = -PI2'
M3 pIICYHKa 3.6 HaXO,AIIM no TeopeMe TIII<paropa P12 =  P; + P: .
TI03TOMY
P3 =  P; + p  m3 v 3 =  (mlvd2 + (m2 v 2)2 
_  (mlvd2 + (m2 v 2)2 . _ PI _ m1v 1
m3 - , tga.----.
v3 P2 m2 v 2
TIocTaBIiM qIICJleHHbIe 3HaqeHII.H, nOJlyqIIM: m3=0,50 Kr; tga.=0,75;
a. = 53°.
M
4. MaJlbqHK, 6erYIIH co CKOpOCTblO 4,0 AOrOH.HeT TenemKY, BHmy-
C
M
YIOC.H co CKOpOCTblO 3.0 -, E BCKaKHBaeT Ha Hee. Macca MaJlbqIIKa 50 Kr,
C
TenemKH 80 Kr. HaH,AIITe CKOpOCTb TenemKII B TOT MOMeHT, Kora Ha Hee
BCKOqHJI MaJlbqIIK (pHC. 3.7).
)];aHO: v}=4,0
HaHTH: v.
M
C '
M
V2= 3,0
C
ml = 50 Kr, m2= 80 Kr.
. Vi
. v 2
.
0 X
Puc. 3.7
P e III e H H e. PaccMoTpHM cHcTeMY "MaJlbqIIK-Te-
nemKa" (plic. 3.7). fIOCKOJlbKY CIIJlbI B3aIIMOAeHCTBII.H
MaJlbqHKa Ii TeJlemKH C 3eMJleH He HMelOT COCTaBJI.HIO-
HX B rOpII30HTaJlbHOM HanpaBneHHH. TO )J;JI.H ropH30H-
TaJlbHbIX COCTaBn.HIOHX IIMnYJlbCOB MomHO 3anliCaTb
3CM.
)];0 B3aHMOAeHCTBII.H IIMnYJlbC MaJlbqHKa paBeH
Ul' HMnYJlbC TeJlemKH paBeH m2v2'
TIo 3aKOHY coxpaHeHH.H HMnYJlbCa
fflt VI +  V 2 = (fflt + m 2 ) V ,
HJlE B npOeKHII Ha OCb Ox:
( ) V =  v} -r  v 2 . ,
m} v} + m 2 v 2 =  + m 2 v :::) v = 3,3
mi + m2 C
M
3.4. MeXaHH'IeCKan pa60Ta
,ileHcTBHe CHJlbI, CB.H3aHHOe C nepeMemeHIIeM Tena. xapaKTepII3yeTC.H Me-
XaHIIqeCKOH pa60ToH.
MeXaHHqeCKaJl pa60Ta - 3TO CKaJl.HpHa.H <pII3HqeCKa.H BeJlHqIIHa. KOTOpa.H
xapaKTepII3yeT npoecc nepeMeeHH.H Tena no )J;ei1cTBlieM ClinbI II paBHa npo-
H3Be)J;eHHIO MOAYJI.H CHJlbI F Ha MOYJlb nepeMeeHH.H 6.r H Ha KOCHHYC yrJla a.
MemAY HHMH: A = F6.rcosa.
60

 
/
/
a r r-- a
/ / /) /> ; --
B CM e,AHHHu;eii pa60TbI SlSn.HeTC.H ,Amoynb (,1:(m).
3eCb F = const H a = const Ha BceM nepeMe-
eHHH (pHC. 3.8).
Pa60Ta - BeJlHqHHa CKaJl.HpHa.H H MomeT
6bITb Kal< nonmKHTeJlbHOii, TaK H oTpHnaTeJlbHoii.
1t 7t
A > 0, ecnH a < -; A < 0, ecnH a > -.
2 2
1t
IIpH .u.BHmeHHH TeJla pa60Ta CHJlbI paBHa HYJlIO. ecnH a = "2'
B 06eM cJlyqae CHJIa nepeMeHHa, nYTb KpHBonHHeeH. yroJI ex H3MeH.HeT-
C.H npOH3BOJIbHO. Tor,Aa ,AJI.H onpe,AeJIeHH.H pa60TbI HY:>KHO MbICJIeHHO pa36HTb
Bce nepeMeeHHe r Ha TaKHe MaJIbIe nepeMeeHH.H f1'i, Ha KOTOpbIX MO:>K-
HO CqHTaTb CHJlY H yron HeH3MeHHbIMH, H HaHTH 3JIeMeHTapHbIe pa60TbI no
<popMYJle
Puc. 3.8
A.4. = F:f1r: cos a..
 .. I
Pa60Ta Ha BceM nepeMeeHHH 6yeT paBHa aJIre6paHqeCKO:H CYMMe 3JIeMeH-
TapHbIX pa60T H TeM TOqHee, qeM MeHblIIe Kam.u.oe nepeMeeHHe H qeM 60JIb-
n 2
lIIe HX qHCJlO: A = L H B npeeJIe npH r -7 0 A = J F(r)cos a dr -
i=l I
pa60Ta CHJlbI F Ha BCe:H TpaeKTopHH BbIpamaeTC.H HHTerpaJlOM, BblqHCJI.HeMbIM
B,AOnb TpaeKTopHH, re 1 H 2 - paHYCbI-BeKTopbI HaqaJIbHO:H H KOHeqHO:H
TOqeK TpaeKTopHH.
,1:(JI.H BbIqHCJIeHHjI 3Toro HHTerpana Ha,AO 3HaTb a8BHCHMOCTb F(r) B,AOJlb
TpaeKTopHH. ,[{JljI onpe.u.eJleHH.H pa60TbI MomHO BOCnOJlb30BaTbC.H rpa<pHqecKHM
MeTo.u.OM (pHC. 3.9. a. 6. 8). Ha rpa<pHKe Fx= {(x) pa60Ta Ha nepeMell1eHHH
rx = 6x qHCneHHO paBHa nJlOa,AH 3alIITpHXOBaHHOH qJHrYPbI. Pa60TY MO:>K-
HO npeCTaBHTb KaK npoH3BeeHHe CpeHe:H CHJlbI Ha nepeMeeHHe:
A = < F> f1r. B qaCTHOCTH, eCJlH CHJIa H3MeH.HeTC.H nHHe:HHO OT Fl .u.o F 2 Ha
,l.(aHHOM nepeMeeHHH f1rx = X2 - xl = M., TO ee cpe,nHee 3HaqeHHe
< F > = FI + F 2 H Tor,na pa60TY MomHO paCCqHTaTb no qJopMYJle A = Fi + 12 .f:1x
22'
a
F"
6
8
F"
F
Xl
X 2
X 2
Xl

x
X
Xl

x
Puc. 3.9
ECJlH K .u.BHmyeMYC.H Teny npHJlOmeHO HeCKOJIbKO CHn, TO Kam,AaJI H3
HHX COBeplIIaeT pa60TY, a 06a.H pa60Ta paBHa a.n:re6paHqeCKO:H CYMMe pa-
60T, COBeplIIaeMbIX OT,neJlbHbIMH CHJlaMH.
61

3.5. Pa60Ta CHJILJ TJljKeCTH
IIycTb Teno nepeMeaeTC.H BepTHKaJlbHO BHH3 H3 nOJlOmeHH.H 1 B no-
nomeHHe 2, onpe,AeJl.HeMbIe COOTBeTCTBeHHO BbICOTaMH hI H h 2 Ha,zr; HYJle-
BbIM ypoBHeM (pHC. 3.10). Pa60T8 CHJlbI T.HmeCTH
A = mg/:1rcosO = mg( - h.;,).
IIpH nepeMeeHHH TeJla H3 nOJlOmeHH.H 1
B nonomeHHe 2 no TpaeKTopHH 1-3-2
pa60Ta CHJlbI T.HmeCTH A = A l3 + A 32 . Ho
A l3 = mg6.r l cosa. A 32 = mg6. r 2 cos90o = O.
M3 PHCYHK8 3.10 BH,AHO, qTO
1
hI
h 2
Puc. 3.10
/:1rl cos a = ht -   A = mg (ht -  ).
3TO 3HaqHT, qTO pa60Ta CHJlbI T.HmeCTH
He 3aBHCHT OT cpOpMbI TpaeKTopHH ,zr;BHme-
HH.H TeJla, a 38BHCHT TOJlbKO OT nepeMee-
BepTHKanH. Ha 38MKHYTOH TpaeKTopHH pa60Ta
HH.H u.eHTpa T.HmeCTH TeJla no
CHJlbI T.HmeCTH paBHa HynlO.
CHnbI, pa60Ta KOTOpbIX He 38BHCHT OT cpOpMbI TpaeKTopHH, a H8 38MKHY-
TO TpaeKTopHH paBHa HYJlIO, Ha3bIBaIOTC.H KOHcepBaTHBHLIMH. CJle,zr;OB8Tenb-
HO, CHna T.HmeCTH - KOHcepBaTHBH8.H CHn8.
3.6. Pa60Ta CHJILI ynpyrocTH
Lo
tOMfu I x

I- I
I
I X
10 -
I I FI 1

lOp 2
I 2
I
Puc. 3.11
..
x
nYCTb Teno. npHKpenJleHHOe K npymHHe
H H8xo,zr;.HlI.\eec.H Ha rJla,zr;KOM cTepmHe, nepe-
Me8eTC.H H3 nOJlOmeHH.H 1 B nOJlOmeHHe 2
(pHC. 3.11).
CHJla ynpyrocTH, ,zr;eHcTBYIOa.H Ha TeJlO
co CTOpOHbI )J;ecpopMHpoBaHHoH npymHHbI, He
OCTaeTC.H nOCTO.HHHOH, a H3MeH.HeTC.H corJlac-
HO 3aKOHY rYKa npOnOpI.\HOH8J1bHO a6COJlIOT-
HOMY YLIJlHHeHHIO:
FI = kX I H F 2 = kX2'
HaH)J;eM pa60TY CHJlbI ynpyrocTH Ha
3TOM nepeMeeHHH no CPOPMY Jle
_ ( ) _ FI + F 2 ( ) _ k (Xl + X 2 )( Xl - X 2 L
A - < F > Xl - x 2 - Xl - x 2 - ,
2 2
BOJlee cTporHH BblBOLI CPOPMYJlbI lIJI.H p8CqeT8 p860TbI
MomHO c,zr;eJl8Tb. HCnOJlb30BaB MeTOLI HHTerpHpOBaHH.H:
A = kx _ kx .
2 2
CHJlbl ynpyrocTH
X 2 X 2 x 2 2 ( 2 2 ) 2 2
kx X 2 kx kx kx kx
A= J Fxdx=J -kxdx=-kJ Xdx=-TL I =- -t--f =-f--f.
XI XI XI
62

MomHO nOKa3aTb, qTO pa60Ta CHJlbI ynpyrocTH He 3aBHCHT OT <pOpMbI
TpaeKTopHH H Ha 3aMKHYTOH TpaeKTopHH paBHa HYRIO. OHa 3aBHCHT TonbKO
OT B3aHMHoro nOJlOmeHH.H qaCTeH TeJla. CHJla ynpyrocTH Tome KOHcepBaTHB-
Ha.H CHJla.
3.7. Pa60Ta CHJILI TpeHHJI
  A-----'
.. , =:/ ///////: J';;/77JL
Puc. 3.12
2
Puc. 3.13
TaK KaK c1:IJIa TpeHH.H HanpwmeHa npo-
THBOnOJlOmHO nepeMeeHHIO (pHC. 3.12),
TO pa60Ta CHJlbI TpeHH.H 6y,neT
A", = FTprcos180o = -pr.
IIYCTh TeRO nepeMeIl\ae-rcSI H3 TOqKH 1
B TOqKY 2 no pa3HhIM TpaeKTOpH.HM. TaK
KaK MO,nYJlH nepeMeeHHH  rl H
(r2+ r3) Heo,nHHaKOBhI (pHC. 3.13). TO
CHJla TpeHH.H COBepIllaeT pa3HbIe pa60TbI.
TaKHM 06pa30M, B OTJlHlIHe OT CHRbI TSI-
meCTH H CHJlbI ynpyrocTH pa60Ta CHnbI
TpeHHSl 3B.BHCHT OT <pOpMbI TpaeKTopHH. no
KOTOpOH ,nBHmeTCH TeJlO. H Ha 3aMKHyrOH
TpaeKTopHH He pB.BHa HYJlIO. Pa60Ta CHJlbI
TpeHH.H He06paTHMO npeBpaI.I.taeT MexaHH-
qeCKOe ,nBHmeHHe TeJla B TennOBoe llBH>Ke-
HHe aTOMOB H MOJleKYJl.
3.8. MOHOCTL
Pa3J1HqHhIeMaIllHHbIHMexaHH3MbI.BbInORH.HIOIl\He o,nHHaKOBYIO pa60TY.
MorYT OTRHqaTbC.H MOHOCTblO. MOHOCTb xapaKTepH3yeT 6bICTPOTY COBepIlle-
HHSI pa60TbI. OqeBH,nHo, qTO qeM MeHhIllee BpeM.H Tpe6yeTC.H ,nR.H BblnORHe-
HH.H ,naHHoH pa60TbI, TeM 3<p<peKTHBHee pa60TaeT MaIllHHa, MexaHH3M H ,np.
IIpH ,nBHmeHHH JlI060ro TeJla Ha Hero B 06eM cJlyqae .n;eHcTByeT HeCKORh-
KO CHJI. Kam.n;a.H CHna cOBepIllaeT pa60TY, H, CJle.n;OBaTeRhHO, .n;R.H Kam.n;oi1
CHnhI MbI MomeM BblqHCRHTh MOI.I.tHOCTb.
Cpe.nHHH MOll.{HOCTL CHJlbl - CKanSlpHa.H <pH3HqeCKa.H BeRHqHHa N, paB-
Ha.H OTHOIlleHHIO pa60TbI A. cOBepIllaeMoH CHJlOH. K npoMemYTKY BpeMeHH  t,
B TeqeHHe KOToporo OHa COBepIllaeTC.H:
N = . B CM elIHHHueif MOIl1HOCTH SlBnSleTCSI BaTT (BT).
t
ECRH Teno ,nBHmeTC.H npSlMORHHeHHO H Ha Hero ,neHcTByeT nOCTOSlHHa.H
CHRa, TO OHa COBepIllaeT pa60TY A = F r cos a. II03TOMY MOI.I.tHOCTb 3TOH CHRbI
Frcosa
N = = Fv cosa = F . v,
t v
r,ne F v - npOeKnH.H CHRbI Ha HanpaBReHHe ,nBHmeHHSl.
IIo 3TOH <popMyne MO>KHO paCClIHTbIBaTb H Cp e.n;H 10 10 , H MrHOBeHHYIO MO-
HOCTH, no,nCTaBR.H.H 3HaqeHH.H cpe)J;HeH < v> HJlH MrHoBeHHoH v CKOpOCTH.
63

MmOBeHHaH MOOCTL - 3TO MOHOCTb CHJlbI B ,AaHHbIH MOMeHT BpeMeHH.
N Mni = lim M = A'.
M-+O 6.t
JII060H ,ABHraTeJIb HJlH MexaHH3M npe,AHa3HaqeHbI ,nJl.H BbInOJlHeHH.H onpe,Ae-
neHHoH MeXaHH'lleCKOll pa6oTbI. KOTOpylO Ha3bIBalOT nOJIe3HOH pa60ToH An-
Ho m060ll MallIHHe npHxo,nHTC.H cOBepllIaTb 6onbllIYIO pa6oTY. TaK KaK Bcne,n-
CTBl1e ,nellcTBH.H CHJI TpeHH.H qaCTb nO,ABO,AHMOll K MallIHHe 3HeprHH He MomeT
6bITb npe06pa30BaHa B MeXaHHqeCKYIO pa60TY. TI03TOMY 3<p<peKTHBHOCTb pa60-
TbI MallIHHbI xapaKTepH3YIOT K03<P<PHll;HeHTOM nOJIe3HOrO ,neHCTBH.H 11 (KTI.rO.
K03cJ>cJ>uu;neHT nOJIe3HOrO JJ:eHCTBU.H 11 - 3TO OTHOllIeHHe nOJIe3HOH pa60-
TbI An. COBepllIeHHOH MallIHHOll. KO Bcell 3aTpa't.!eHHOH pa60Te A3 (no,nBe,neH-
HOll 3HeprHH W):
11 =  =  = N n ,
A3 W N 3
r,Ae N n . N 3 - nOJle3Ha.H H 3aTpaqeHHa.H MOrn;HOCTH COOTBeTCTBeHHO.
KTI.n: 06bI'llHO BblpamalOT B npou;eHTax.
BOIIPOCbI JJ.JUI CAMOKOHTPOJUI
1. HaKue aBa rpaKmopa 06Jl3ameJlblibl aJlJl COBepWeHUJl MexaHUl£eCKoil pa-
60mbl?
2. COBepwaem.n.u lienOa6UJlCHO cmoSlU{uil lle.n.06eK M.eXaliUl£eCKYIO pa60my,
yaepJICUBaJl KaMelib Ha BblmSlliymoil pYKe?
3. C06epWaemCJl.n.u MexaliUlleCKaSl pa6oma. ec.n.u HG a6UJlCyU{eeCJl me/LO
aeilcmBYlOm 63aUMHO ypaBlio6eWU6alOU{uecSl CU.n.bl?
4. [[pu KaKUX YCJl06UJlX pa60ma aaHHOU CUJlbl, aeilcm6YIOU{eil lia aBUJlCY
U{eec.R me.n.o, .MaKCUM.a.n.bHa? pa6lia HYJlIO? ompuqame.n.blia?
5. 3a6ucum JlU pa60ma om 6bl60pa CUCme.Mbl omcilema?
6. HaK paCCllumamb pa60my nepe.MeliHoil CUJlbl?
7. HaK 3a6ucum M.OU{liocmb CU.n.bl om CKopocmu a6UJlCeHuSl?
8. q eJlOBeK onYCKaemcSl Ha napawlOme. KaKaSl CUJla C06epwaem nplJ.
:imOM nOJlOJICUmeJlbHYIO U KaKaR ompuqameJlbHYIO pa60my?
9. C06epwaem JlU pa60my CUJla npUmJlJICeHUR JIYHbt 3e.MJleil?
10. ECJLU u6mo.Mo6uJlb 6oe3J1Caem Ha zopy npu HeUj.MeHHOU .MOl«HOCmU a6U
zame.n.Jl, mo 60aUmeJlb Y.MeHbWaem cKopocmb OBUJlCeHUJl. [[olle.MY?
11. Om Ilezo 3a6ucum pa60ma CUJlbl mJlJICecmu?
12. l{ e.MY pa6lia pa60ma CUJlbl mJlJICecmu npu ZopU30limaJlbliOM a6UJlCeHUU
meJla? npu aBUJlCeHUU meJla no 3aMKHymou mpaeKmopuu?
13. TeJLO c8onooHO naiJaem C lielwmopou 6blcombl. OiJUHalWBYlO JlU pano
my c06epwaem CUJla mSlJICecmu 3a oouliaK06ble npO.MeJICymKu 6peM.eHu?
14. 3a6ucum JlU pa60ma CUJlbl ynpyzocmu om q;Op.J,tbl mpaeKmopuu?
15. B lle.M CXOacm60 pa60m, c06epWae.MblX CUJlOU ynpyzocmu U CUJLOU mJl
Jf{ecmu?
16. l{6e OaUHaKo6ble npYJICuHbl (JlCeJle3liaJl U .MeaHaJl) ynpyzo paCmJlHymbl
a) Ha OaUliaK06YlO aJIUliY. 6) OaUliaK06bl.MU CUJlaMU. Ha pacmJlJICellue Komo-
poil U.J liUX liaao 3ampamumb 60JlbWYlO pa60my 6 CJlYilaSlX a) U 6)?
17. Bcezaa JLU pa60ma CUJlbl mpeliU.R CKOJlbJICeHUSl ompuqameJlblia l
18. MOJICem JlU 6blmb pa60ma CUJlbl mpeHUSl CKOJlbJICeHUSl pa6Hoil HYJllO?
64

19. M oJICem JlU C06epwamb pa60my cu.n.a mpeliUSl nOKoSl?
20. Te.n.o .MaCCOU m noiJliU.MalOm lia 8blcomy h a) 8epmUKa.n.bHO 68epx,
6) no z.n.aiJKou HaK.n.OIiIiOU n.n.OCKocmu. 6) no wepOX06amou liaK.n.OliliOU n.n.oc-
Kocmu. Cpa6liUme pa60mbl 6lieWlieU CU.n.bl U CU.n.bl mJlJICecmu.
21. B lle.M om.n.ullue KOlicep6amU6liblX CUJl om lieKolicep6amU6liblx?
A = F6.rcosa
A = mg(h 1 -h 2 )
A = kx; _ kx
2 2
A = -FTp 6.r
A
N=-
6.t
N = Fu . v
11 = Au . 100 %
A
mv 2
w=-
K 2
6W K = A
W n = mgh
kx 2
w=-
n 2
W=WK+W n
W K + W n = const
6W = Ai + A 2
OCHOBHbIE <I>OPMY JIbI
- pa60Ta nocToHHO CL-1llbl F, rAe 11 r - MOAYllb nepe-
Mew.eHL-1, a - yroll Me)t(,D.y CL-1ll0 L-1 nepeMew.eHeM
- pa60Ta CL-1llbl T>KeCTL-1, rAe hi L-1 h 2 - HaallbHa L-1 KO-
HeHa BblCOTa Tell a OTHOCTellbHO HYlleBoro ypOBH.
- pa60Ta CL-1llbl ynpyroCTL-1, rAe k - >KeCTKOCTb npY>KL-1Hbl,
XI L-1 X2 - HaallbHoe  KOHeHoe 3HaeHL-1 llL-1HeVlHO Ae-
<popMau.L-1L-1.
- pa60Ta CL-1llbl TpeHL-1 FTp_
- MOW.HOCTb (cpeAH). rAe 6.t - npOMe>KyroK BpeMeHIi1
3a KOTOpbl COBepweHa pa60Ta A.
- MrHOBeHHa MOW.HOCTb CL-1llbl F. rAe F u - npOeK1I.HSI
CHnb] Ha HanpaBneHHe ABHmeHHSI, v - MrHOBeHHa CKO-
pOCTb.
- KnA MeXaHL-13Ma, rAe An - nOlle3Ha pa60Ta, A - BC
COBepweHHa pa60Ta.
- KIi1HeTIi1eCKa 3HeprL-1.
- TeopeMa 0 KIi1Hen1eCKOVI 3Hepn1L-1, rAe 6Wt< - 1i13Me-
HeHL-1e KIi1HeTIi1eCKOVI 3HeprL-1Ii1, A - pa60Ta paBHOAeCTBY-
lOw.eVl CL-1ll, AeCTBYIOW.L-1X Ha TellO.
- nOTeHu.li1allbHa 3Heprli1 rella, nOAHToro HaA 3eMJlelil,
rAe h - BblCOTa Tella OTHOCL-1TeJlbHO HYJleBOrO ypOBH.
- nOTeHu.L-1allbHa 3HeprL-1 Ae<popMli1pOBaHHoro Tella,
rAe x - llL-1HeVlHa Ae<popMau.li1
- nOllHa MeXaHL-1eCKa 3Heprli1 CL-1CTeMbl Tell.
- 3aKOH COXpaHeHIi1 MeXaHIi1eCKO 3Heprli1li1.
- L-13MeHeHli1e nOllHO 3Heprli1L-1 CL-1CTeMbl Tell, rAe Al - pa-
60Ta BHeWHL-1X CL-1ll, A 2 - pa60Ta CL-1llbl TpeHIi1.
65

METO,lUIqECKIIE PEKOMEHAA:QIIII
IIpE pemeHIUi 3a.n;aq Ha BbJl.IHCJIeHHe pa60TbI nOCTOSlHHOii CHJIbI He06xo,Z:(HMO
1. BbI.HCHHTb, pa60TY KaKoH CHJlbI Tpe6yeTC.H onpe,Z:(eJlHTb B 3a,Z:(aqe, H 3a-
nHcaTb HCXO,Z:(HYIO qJOPMYJlY A = F/)' r cos a.
2. C,Z:(eJlaTb PHCYHOK, YKa3aB Ha HeM CHJIbI, npHnomeHHble K Teny, H BeK-
TOp nepeMeeHH.H, a TaKme onpe,Z:(eJlHTb yroJl Mem,Ay ,Z:(aHHOH CHJlOH H nepe-
MeeHHeM.
3. ECJlH CHJla He 3a,Z:(aHa, HaHTH ee H3 OCHOBHoro ypaBHeHH.H ,Z:(HHaMHKH.
4. HaHTH nepeMeeHHe (eCJIH OHO He 3a,Z:(aHO) H3 KHHeMaTHqeCKHX ypaB-
HeHHH.
5. IIo,Z:(cTaBHTb 3HaqeHHSI CHJIbI H nepeMeeHH.H B CPOPMYJlY pa60TbI.
6. IIpH paCqeTe pa60TbI nepeMeHHoH CHnbI npOlue Bocnonb30BaTbC.H rpa-
qJH1.JeCKHM cnoc06oM.
IIpH pemeHHH 3a,Z:(aq Ha onpe,Z:(eJIeHHe MOHOCTH He06xo,Z:(HMO:
1. BbI.HCHHTb, KaKYIO MOHOCTb HymHo onpe,AeJlHTb - Cpe,AHIOIO HJIH
MrHOBeHHYIO.
2. 3anHcaTb qJopMynbI ,Z:(JI.H paCqeTa MOHOCTH:
N =  ; N = Fv cosa; N = Fv . v.
IIepBa.H H3 HHX n03BOJI.HeT paCC1.JHTaTb cpe,Z:(HIOIO MOHOCTb, a BTOpa.H H
TpeTbSl - KaK cpe,Z:(HIOIO, TaK H MrHOBeHHYIO B 3aBHCHMOCTH OT Toro, no,A-
CTaBJI.HIOT JlH 3HaqeHHe Cpe,AHeH CKOpOCTH HJlH MrHOBeHHOH.
3. 113 OCHOBHoro ypaBHeHH.H ,Z:(HH8MHKH onpe,AeJlHTb CHJIY T.HrH, c,Z:(eJIaB
npe,ABapHTenbHO PHCYHOK C YKa3aHHeM neHcTBYIOHX H8 TeJlO CHJI.
4. 11.3 38KOHOB KHHeMaTHKH onpe,Z:(eJIHTb cpe,Z:(HIOIO HJIH MrHOBeHHYIO CKO-
pOCTH) eCJlH OHH He 38)J;aHbI B ycnoBHH 38,Z:(aqH.
5. IIo,Z:(cTaBHTb H8H,Z:(eHHbIe 3HaqeHH.H CHJIbI T.HrH H CKOpOCTH B qJOPMYJlY
,Z:(JI.H p8CqeTa MOHOCTH.
nPIIMEPbI PElliEHIIH 3AAA q
1. BepTOJIeT, Macca KOToporo C rpY30M 6,0.10 3 Kr, 3a 2,5 MHH Ha6p8J1
BbICOTY 2250 M. Onpe.neJlHTe pa60TY ,Z:(BHraTeJl.H 3a 3TO BpeM.H, CqHTa.H nO,Z:('beM
BepTOJIeT8 paBHOYCKopeHHbIM.
A8HO: m= 6,0.10 3 Kr, t= 1,5.10 2 c, h=2250 M.
HaHTH: A.
P e III e H H e. I1cKOM8.H p860T8 A = Fh, r,Z:(e F -
CHna T.HrH ,Z:(BHraTeJI.H. AJI.H ee H8XO)K,AeHH.H paccMoT-
pHM, KaKHe CHJIbI ,Z:(eHcTBYIOT H8 BepTOJleT (pHC. 3.14):
y

t a
Puc. 3.14
CHJla T.HrH F H CHJla T.HmeCTH mg.
yp8BHeHHe lIHHaMHKH: ma = F + mg.
B npOeKI.\HH H8 OCb Oy:
ma = F - mg => F = ma + mg.
OCHOBHoe
o
66

at 2 2h
I1p1:l paBHOYCKOpeHHOM ,n;BlimeHli1:l npOH,n;eHHbIH rIYTb h = - ==> a = - li
2 t 2
F = m(  + g 1-
HCKoMaSI pa60Ta A = m(  + g )ho
l1o,n;cTaBJI.HH q1:lCJIeHHbIe 3HaqeHli.H. nonyqRM A = 1,4.10 2 M.D:m.
2. K nemaeMY Ha ropli30HTaJIbHOH nOBepXHOCT1:I 6PYCKY MaccoH 12 Kr
H
np1:lKpenJIeHa HeBeCOMa.H npym1:lHa meCTKOCTblO 300 -. K03qJqmlieHT Tpe-
M
H1:IH Mem,n;y 6PYCKOM 1:1 nOBepXHOCTblO 0,40. K cB060,n;HOMY KOHy npym1:lHbI
np1:lJIOmeHa BHeIIlH.HSI C1:IJIa no,n; yr JlOM 30° K ropli30HTY, no,n; ,n;eHCTB1:IeM KO-
TOPOH I'PY3 paBHOMepHO nepeMeaeTC.H Ha paCCTO.HHlie 4,0 M. Onpe,n;enliTe pa-
60TY, COBepIIleHHYIO BHeIIlHeH CliJIOH.
H
.D:aHo: m=12 IU, k=300 -, Jl=0,40, 0.=30°, r=4,0 M, a=O.
M
HaHT1:I: A.
P e III e H H e. IIo,n; ,n;eHCTB1:IeM C1:IJlbI, np1:lnOmeH-
HOH K npymliHe, npOlicxo,n;1:IT ,n;B1:ImeHlie TeJI8 li
ynpyra.H ,n;eqJopMau.liSl npymliHbI, n03TOMY 1:ICKO-
Ma.H pa60T8
y
o
mg
kx 2
A = Fl1rcosa. +-.
2
3a,Aaqa CBO,n;liTC.H K HaXOm,AeHlilO ClinbI, COBep-
IIlalOeH pa6oTY, li a6conlOTHoro y,n;JlliHeHli.H x.
Ha 6PYCOK (pliC. 3.15) ,n;eHcTByeT c1:IJla ynpy-
x
Puc. 3.15
- -
rOCT1:I npymliHbI F ynp' qliCJleHHO paBHa.H CliJle F.
nplinomeHHoH K npymliHe, CHna T.HmeCTli mg,
Clina TpeHliSl F TP 1:1 c1:IJla HOpMaJIbHOH peaKlili onopbI N P' OCHoBHoe ypaB-
HeHlie ,AliHaMliKli rIPH paBHoMepHoM ,n;BlimeHlili: F + F TP + mg + N p = 0,
eKlili Ha OCli Ox 1:1 Oy: F cos a. - FTp = 0; F sin a. + N p - mg = O.
Y'lliTbIBa.H, qTO FTp = JlN p ' li3 3T1:IX ypaBHeHliH Haxo,n;liM cliny:
Jlmg
B npo-
F=
cos a. + J.l sin a.
F J.lmg
110 3aKOHY rYKa F = kx. 1103TOMY X = - = .
k k ( cos a + J.l sin a. )
IIo,n;cTaBJI.HH 3HaqeH1:ISI F R x, nOJlyqaeM
J.lmg (2kl1r ( cos a. + Jl sin a. ) cos a. + Jlmg)
A= .
2k( coso. + Jlsin 0.)2
IIo,n;CTaBHB q1:lCJleHHble 3HaqeH1:I.H, nOJIyqHM A = 1,6.10 2 .D:m.
67

3. KaKYIO MHHHMaJlbHYIO pa60TY Ha,D;O COBeplIlHTb, qT06bI O,D;HOpO,D;HbIH Ky6.
HaXO,D;HHHC.H Ha ropH30HTanbHoH nnOCKOCTH, nepeBepHYTb C O,D;HOH rpaHH Ha
cOCe,AHIOIO? MaCC8 Ky6a 100 Kr, ,D;JlHHa ero pe6pa 50 CM.
,l]:aHo: m=100 Kr, a=0,50 M.
HaHTH: A.
a
, , / /
, /
a / ,
/ / , , hI
h'
Puc. 3.16
113 PHCYHKB BH,D;HO, qTO
Pew e H H e. BY,D;eM nepeBOpaqHBaTb Ky6
TaK, qT06bI OH He OTpbIBanCH OT ropH-
30HTanbHOH nnOCKOCTll H He CKonb3Hn no
HeH. eHTp T.HmeCTH Ky6a nepeMe8eT-
C.H npH 3TOM C BbICOTbI hi l1.0 BbICOTbI h 2
(pHC. 3.16, a. 6).
COBeplIlaeMa.H pa60Ta - 3TO pa60Ta npo-
THB CHJI T.HmeCTH Ky6a.
a a.J2
hi = -;  = -. Cne,D;OBaTeJlbHO,
2 2
A = m g ( - + a l = m:a (J2 -1).
ITocJle nO,D;CTaHOBKH qHCneHHbIX ,D;aHHbIX nonyqHM A = 98 ,r(m.
4. ABToM06Hnb MaccoH 2000 Kr TpOraeTC.H C MeCTa H ,D;BHmeTC.H B ropy,
YKnOH KOTOpOH paBeH 0,020. ITpOH,D;.H paCCTO.HHHe 100 M, OH pa3BHBaeT CKO-
KM
pOCTb 32,4 -. K03cpcpHHeHT TpeHH.H 0,050. Onpe,D;eJlHTe Cpe,AHIOIO MO-
q
HOCTb. pa3BHBaeMYlO MOTOpOM 8BToM06HnH npH 3TOM ,ABHmeHHll.
AaHo: m=2000 Kr. sina=0.020.L\r=100 M.
M
v = 9,0 -, Jl = 0,050, vo= O.
C
H a H T ll: < N> .
Pew e H H e. I1cKOMa.H MOHOCTb < N > =
= F < v >, r,D;e F - CHna T.Hrll aBTOM06llJlH,
v
< v > ="2 - Cpe,AH.HH CKOpOCTb l1.BHmeHll.H aB-
TOM06Hn.H. 3anHlIleM OCHOBHoe ypaBHeHHe ,D;llHa-
MHKH
mii = F + F -rp + N p + mg,
B npoeKHH Ha OCH OX H Oy (pllC. 3.17):
ma = F - F Tp - mg sin ex; 0 = N p - mg cos 0..
YqHTbIBa.H, qTO F Tp = JlN p . H3 3THX ypaBHeHHH HaH,D;eM
Puc. 3.17
F = m (a + gSlna + Jlgcosa), rlle cosa =  1- sin 2 a.
v 2
Y cKOpeHHf:' npH paBHoycKopeHHoM ,ABllmeHHH a = -, Cne,D;OB8TeJlbHO,
2L\r
<N>=m ( +gSina+Jl  1-Sin2a ) .!:.; <N> =10 KBT.
2L\r 2
68

5. OnpeeJlHTe MOHOCTb BHraTeJl.H, npHBO.Hero B BHmeHHe TpaHcnop-
Tep, KOTOpbIH 3a CYTKH nOHHMaeT 2,0.10 6 T meJle3HOH PYbI C nOBepxHocTH
3eMJlH Ha BbICOTY 5,0 M, ecnH Kn.n; TpaHcnopTepa paBeH 63 %.
.n; a H 0: t = 86400 c, m = 2,0. 10 9 Kr. h = 5,0 M, 11 = 0,63.
HaHTH: N.
N
PeweHHe. no onpe,AeneHHIO f} = ;; , re N - 3aTpaqeHHaSI MOHOCTb.
N n - nOJle3Ha.H MOHOCTb, T .e. MOHOCTb, nOTpe6J1.HeMa.H HenocpeCTBeHHO
JI.H nOH.HTHSI rpY3a:
A
Nil = - ,
t
re A = mgh - pa60Ta, COBepmeHHaSI npOTHB CHnbI T.HmeCTH rpY3a.
Cne,AOBaTeJlbHO, 11 mgh => N = mgh ; N = 1.8 KBT.
= tN 11t
3.9. MeXaHH'IeCKaB 3HeprHB
MeXaHHqeCKOe COCTOSIHHe Tena (cHcTeMbI TeJl) onpe.neJl.HeTC.H ero nOJlome-
HHeM OTHOCHTenbHO PyrHx Ten (KoopHHaTaMH) Hero CKOpOCTblO.
EcnH H3MeH.HeTC.H XOTSI 6bI OHa H3 3THX BenHqHH, TO rOBOp.HT 06 H3Me-
HeHHH MeXaHHqeCKOrO COCTOSIHH.H Tena.
COCTO.HHHe aHHoH CHCTeMbI TeJl 06.H3aTenbHo H3MeHSIeTC.H, ecnH BHelIlHHe
CHnbI COBepmalOT OTnHt{HYIO OT Hyn.H pa60TY.
KOnHqeCTBeHHo MeXaHHqeCKOe COCTO.HHHe CHCTeMbI Hero H3MeHeHHe xa-
paKTepH3yeTC.H MeXaHHqeCKOH 3HeprHeH W.
MexaHHtJ:eCKaH 3HeprHH - 3TO <pH3HqeCKaSI BeJlHt{HHa, .HBJlSIIOa.HCSI CPYHK-
HeH COCTO.HHH.H CHCTeMbI H xapaKTepH3YIOa.H cnoco6HOCTb CHCTeMbI COBep-
maTb pa60TY. M3MeHeHHe MexaHHt{ecKoH 3HeprHH W paBHo pa60Te npHno-
meHHbIX K CHCTeMe BHemHHX CHn:
w = A.
3HaqeHHe 3HeprHH CHCTeMbI B aHHoM COCTO.HHHH He 3aBHCHT OT nYTH ne-
pexoa ee B 3TO COCTO.HHHe.
3.10. KHHeTH'IeCKaB 3HeprHB
HaHeM, KaK 3HeprH.H TeJl 3aBHCHT OT
HX CKOpOCTH.
nYCTb Ha Teno MaCCOH m eHcTByeT
CHJIa F (3TO MomeT 6bITb OHa CHJIa HnH
paBHoeHcTBYIOa.H HeCKonbKHX CHn), Ha-
npaBJleHHa.H Bonb nepeMeeHH.H, H CKO-
pOCTb Tena H3MeH.HeTC.H OT VI O V2 (pHC. 3.18). Pa60Ta 3TOH CHnbI A=Fl1r.
no BTOPOMY 3aKOHY HblOToHa F = ma.
 F 6rr--, F
/ /: / / / / / / / / >;;;;J / /:/ /
Puc. 3.18
69

v 2 _ V 2
TIpll paBHOYCKOpeHHOM BIl)KeHIlIl a = 2 1 . CneOBaTeJ1bHO.
2r
2 2
A = mV 2 _ mV l .
2 2
<1J1l3H'ieCKa.H BeJ1HqllHa W K
mv 2
2
Ha3bIBaeTC.H KHHeTH'ieCKOH 3HeprlleH.
3Heprll.H, KOTOpOH 06J1a,AaeT TeJlO BCJleCTBlle CBoero BHmeHIl.H, Ha3bIBa-
eTC.H KHHeTH'IeCKOH 3l1eprHeH.
Tor,Aa A = W K2 - W K1 , T.e.
 W K = A TeopeMa 0 KHHeTH'IeCKOH 3HeprHH:
U3.MeHeHUe KUHemUlleCKOU 3Hepzuu meJla paBHO pa60me paBHo8eucm6Y1OUI,eU
Bcex cu.n.. 8eucmBY1OUI,UX Ha meJlo.
3Ta TeopeMa cnpaBeJlHBa He3aBHCHMO OT Toro, KaKHe CHJlbI eHCTBYIOT
Ha TeJlO: cllna ynpyroCTIl. CIlJIa TpeHIlSl Ilnll CIlJla T.HmeCTH.
mv 2
ECJ11l vI = 0 H v 2 = v, TO - = A.
2
TaKHM 06pa30M, KHHeTHqeCKa.H 3HeprH.H TeJ1a paBHa pa60Te, KOTOpylO He-
06XOHMO COBepIlIIlTb, qT06bI nOKO.HeMYC.H TeJlY C006IlTb CKOpOCTb v.
KHHeTllqeCKa.H 3HeprH.H 3aBHCIlT OT BbI60pa CIlCTeMbI OTCqeTa.
3.11. nOTeHHaJILHaH 3HeprHH TeJIa
rrpH rpaBHTaHOHHOM B3aHMoeHcTBHH
TIoTeHlJ,llaJ1bHa.H 3Heprll.H - 3TO 3Heprll.H CIlCTeMbI. onpeen.HeMa.H B3aIlM-
HbIM pacn0J10meHHeM TeJ1 (HJ11l qaCTeH TeJla n;Pyr OTHOCHTeJ1bHO n;pyra) H xa-
paKTepOM clln B3aHMoeHCTBIl.H Memy HHMII.
PaccMoTpHM BHaqaJle HeKOTopoe TeJ10, HaXO.Heec.H B rpaBHTalJ,HOHHOM
nOJ1e 3eMnH.
TIpH nepeMeeHHH TeJ1a
BbICOTbI h 2 (pIlC. 3.19) CHJla
r--,
I
L...
C BbICOTbI hI OTHOCHTeJlbHO HYJleBoro YPOBH.H o
T.HmeCTH cOBepllIaeT pa60TY
A = mg - mg = -(mg - mg).
W n = mgh - nOTeHHanLHaJl 3HeprHJI Tena B
none TJlrOTeHHSI.
mg
Tora A = -(W n2 - Wnd = -Wn'
ll3MeHeHHe nOTeHlJ,llaJlbHOH 3HeprHH TeJla, KOTO-
poe HaXOHTC.H B rpaBHTau,HOHHOM nOJle 3eMJlH, B3.H-
h = 0 Toe co 3HaKOM "-", paBHO pa60Te CHJlbI T.HmeCTIl.
TIoTeHlJ,HaJ1bHa.H 3HeprllSl onpeeJ1.HeTC.H C TOqHOCTblO
,AO HeKOTopOH nOCTO.HHHOH. Ee 3HaqeHIle 3aBHCIlT OT
BbI60pa HY JleBOrO ypOBH.H OTCqeTa BbICOTbI. TIoTeHlJ,Il-
aJ1bHYIO 3HeprHIO KaKorO-TO nOJIO)KeHH.H (h 2 = 0) C'iIlTaIOT paBHOH HYJ110
(W n2 = 0), a 3HeprlllO pyrllx [lOJlOmeHHH (hI = h) OTCqHTbIBalOT OTHOCIlTeJlbHO
HYJleBoro ypOBH.H: W n = mgh = A.
mW
h 2
Puc. 3.19
70

2D
h 2
TaKHM 06pa30M, nOTeHHanLHaH 3HeprHH
B none THI'OTeHHH - :3T0 3HeprHSI, 06YCJIOB-
JleHHa.H B3aHMoeHCTB1:IeM TeJla C 3eMJIeH; OHa
3aBHCHT OT HX B3aHMHoro nonomeHHSI H paB-
h = 0 Ha pa60Te, KOTOPYIO COBepmaeT CHJla T.HmeCTH
npH nepeMemeHHH TeJIa 1:13 .naHHoro nOJIome-
HH.H Ha HYJleBOH ypOBeHb.
TaK KaK HYJleBOH YPOBeHb Bw6HpaeTC.H npo-
H3BOJlbHO, TO nOTeHHanbHa.H 3HeprH.H MomeT
HMeTb H OTpHaTeJIbHOe 3HaqeHHe (pHC. 3.20):
W nl > 0; W n2 < O.
1
hI
Puc. 3.20
3.12. nOTeHl{HaJILHaH aHeprHH ynpyro .ne4>opMHpoBaHlloro TeJIa
TIpH 1:I3MeHeHHH BeJlH'tIHHbI ,Ae<t>opMaHH ynpyro ,Ae<t>opMHpoBaHHoro TeJIa
OT x] o X2 c1:IJla ynpyrocTH COBepmaeT pa60TY
A = kxf _ kx: = _ ( kX: _ kxf ) .
2 2 2 2
kx 2
BeJl1:lq1:lHa W n = - - nOTeHl(1:IaJlbHa.H 3HeprH.H ynpyro e<t>opMHpoBaH-
2
Horo TeJla. A = -(W n2 - W nl ) = -6W n . ECJlH Xl = 0, x 2 = X, TO
kx 2
W n =2=-A =II'
IIoTeHHanLHaH 3HeprHH ynpyro .ne4>opMHpoBaHHoro TeJla - 3HeprHSI,
o6ycnOBJleHHa.H B3aHMO,AeHCTBHeM qaCTeH TeJla Memy C060H. OHa paBHa pa-
60Te ABU' KOTOpylO COBepmalOT BHemHHe CHJlbI, qT06bI He,Ae<t>opMHpOBaHHYIO
npymHHY cmaTb (paCT.HHYTb) H8 BeJlH'tIHHY x.
3.13. 3aKoH coxpaHeHHH aHeprHH B MexaHHKe
Tena MorYT OHOBpeMeHHO 06J1aaTb H KHHeTHqeCKOH, H nOTeHHaJIbHOH
3HeprHeH. TIOJlHa.H MeXaH1:IqeCK8.H 3HeprH.H Tena (cHcTeMbI TeJI) W - 3TO CYM-
Ma KHHeTH'tIeCKOH Ii nOTeH[(HaJlbHOH 3HeprHH: W = W n + WK'
PaCCMOTpHM 3aMKHYTYIO CHCTeMY TeJl "TeJlo-3eMJI.H", Mem,Ay KOTOpbIMH
,AeHcTByeT TOJlbKO KOHcepBaTHBHa.H CHJla - CHJla T.HmeCTH (CM. pHC. 3.19). no
eHcTBHeM 3TOH CHJIbI H3MeH.HeTC.H H KHHeTHqeCKa.H. H nOTeH1:IaJIbHaSI 3Hepnf.H
TeJIa. TIpWIeM WK = A H -Wn = A, T.e. YBeJlHqeHHe KHHeTH'tIeCKOH 3Hep-
rHH CHCTeMbI paBHo y6bIJlH ee nOTeHIJ;HaJIbHOH 3HeprHH.
113 3THX ypaBHeHHH nOJlyqaeM
(WK + W n ) = 0 => W K + W n = W = const.
TaKOH me pe3YJlbTaT nOJIyqaeTC.H H npH eHCTBHH CHJIbI ynpyrocTH. Ta-
KHM 06pa30M, no.n.1iaSl MexaliUlleCKaSl 3liepZUSl 3aMKliymoil cucmeMbt me.n.
71

63aUMooeucmeYI014UX CU.n.aMU mSlZOmeliUSl U.n.U CU.n.aMU ynpyzocmu. oCmaemCSl
noc mOSlIiIiOU.
STO YTBepm.n;eHHe Ii eCTb 3aKOH COXpaHeHHH 3HeprHH B MeXaHHKe.
3.14. 1I3MeHeHHe MeXaHH'IeCKOH 3HeprHIt B He3aMKHYToH CHCTeMe
EcnH CHCTeMa He3aMKHYTa.H, T.e. .n;eHcTByeT BHeIlIH.H.H CH-
Jla F, TO MOmHO nOK83aTb, qTO nOJlHa.H MeXaHHqeCKa.H 3Hep-
rH.H He COXpaH.HeTC.H, a ee H3MeHeHHe .6.W = AF' r.n;e A F
pa60Ta BHeIlIHeH CHJlbI.
PaCCMOTpHM CHCTeMY Ten (TeJlo-3eMJI.H). Ha KOTOPYIO .u.eH-
cTByeT BHeIlIH.H.H CHJla F. IIycTh TeJlO no.n;HHMaIOT BBepx Ha BbI-
COTY h (pHC. 3.21). Pa60Ta BHeIlIHHX CHJI A F = Fh. Pa60Ta
BHYTpeHHeH CHJlbI (CHJlbI T.HmecTH)
Amg = -mgh = -/),. W n .
COrJlaCHO TeopeMe 0 KHHeTHqeCKOH 3HeprHH L\W K = A,
r.n;e A - CYMMapHa.H pa60Ta Bcex CHn, .n;eHcTBYIOHX H8 TeJlO:
A = A F + g'
Cne.n;oBaTenbHo, 6. W K = -/),. W n + A F =>
/),.W = A F .
r- -I
L_ _J
h

Puc. 3.21
F
mg
3.15. 3aKoH coxpaHeHHH H npeBpal.I..\eHHH 3HeprHH
ECJlH B 3aMKHYToH CHCTeMe Mem.n;y TenaMH .n;eHcTByeT CHJl8 TpeHH.H, TO
nOJlHa.H MeXaHHqeCKa.H 3HeprH.H y6bIBaeT H, Kal\ nOKa3hIB8eT onbIT, ee H3Me-
HeHHe paBHO pa60Te CHnbI TpeHH.H:
6.W = A,.P'
He COXpaH.HeTC.H nOJlHa.H MeXaHHqeCKa.H 3HeprH.H Ii B TOM cJlyqae, eCJlH B
CHCTeMe Ten npOHCXO,A.HT HeynpyrHe .n;ecpopM8nHH. Ho y6bIB8HHe MeX8HHqeC-
KOH 3HeprHH He 03HaqaeT, qTO 3HeprH.H HCqe3aeT 6eCCJle.n;HO. OHa npeBpaI.I..1;a-
eTC.H H3 MeXaHHqeCKOH B .n;pyrylO, B qaCTHOCTH 80 BHYTpeHHlO1O 3HeprHIO.
,n:JlH JlI060H 3aMKHYTOH CHCTeMbI Bcer.n;a BbInOnH.HeTC.H 3aKOH coxpaHeHHH
H npeBpalQ.eUHH 3HeprHH: ee.n.UIlUlia no.n.liou 3liepZUU (MexaliUllecKOU u iJpy-
ZOu) 3aMKHymou cucmeMbt ocmaemcSl nocmOSlIiHOU. IIpH 3TOM. 6yiJYllu lieco-
3uiJaeMou U H.eYliUllmOJICa(JMOU. 3liepZUSl .Mo}{C(Jm npe6pa14ambCSl U3 oaliOZO 6uoa
e iJpyzou.
3.16. YnpyrHe H HeynpyrHe cOY)J;apeHHH TeJI
IIpHMepoM npHMeHeHH.H 3aKOHOB coxpaHeHH.H HMnYJlbca 1-1 3HeprHH HBn.H-
eTC.H coy.n;apeHHe (y.n;ap) TeJl.
Y,lJ;ap - 3TO KpaTKOBpeMeHHoe B3aHMo.n;eHcTBHe conpHKacalOHXC.H Ten.
npHBo.n;Hee K 3HaqHTenbHOMY H3MeHeHHIO COCTO.HHHH HX .n;BHmeHH.H.
72

TeJla BO BpeM.H y,napa npeTepneBalOT ,necpOpMaUHIO. KllHeTHqeCKa.H 3Heprll.H
OTHOCllTenbHOrO ,nBllmeHll.H coy,nap.HlOIIJ;llXC.H TeJl Ha KOpOTKOe BpeM.H npe06pa3Y-
eTC.H B nOTeHUllanbHYIO 3HeprlllO ynpyro ,necpopMHpOBaHHbIX Ten. 3a BpeM.H y,na-
pa npOllCXO,AllT nepepacnpe,AeneHHe 3HeprllH Me:>K,ny cOY,Aap.HlOIIJ;llMllC.H TenaMll.
Y,Aap Ha3bIBaeTC.H eHTpaJIhHhIM, ecnll TeJla ,no y,napa ,nBllmYTC.H B,nonb
np.HMOH, npoxo,n.HeH Qepe3 llX ueHTpbI Macc.
Pa3J1llqalOT ,nBa npe,neJlbHbIX THna y,napa: a) y,nap a6COJllOTHO ynpyrllH,
6) y,nap a6COJllOTHO HeynpyrllH.
A6cOJIIOTHO ynpyrHM Ha3bIBaeTC.H y,nap, nocJle KOToporo B03HHKIlIlle B Te-
nax ,necpopMaUllll nonHOCTblO llCqe3alOT.
TIpll a6COJllOTHO ynpyroM y,nape BbInOJlH.HIOTC.H 3aKOH coxpaHeHll.H llMnynbca
H 3aKOH coxpaHeHH.H MeXaHHqeCKOH 3HeprHll.
A6cOJIIOTHO HeynpyrHH YAap - y,Aap. nocne KOToporo B03HHKIlIlle B Te-
nax ,necpopMaullll nOJlHOCTblO COXpaH.HIOTC.H. [JocJle a6COJllOTHO Heynpyroro y,na-
pa Tena ,nBllmYTC.H KaK e,nHHoe ueJloe. TaKOH y,nap Ha6nlO,naeTC.H npll CTonK-
HOBeHllll Ten ll3 M.HrKllX, nnaCTHqHbIX MaTepllanOB.
TIPll a6COJllOTHO HeynpyroM y,nape BblnOnH.HeTC.H TOJlbKO 3aKOH coxpaHeHll.R
HMnynbca, a KllHeTllqeCKa.H 3HeprH.H Ten He COXpaH.HeTC.H.
3.17. IIpocTeHlllHe MexaHH3MLI
n.H 06nerqeHH.H COBepIlIeHH.H MeXaHHqeCKOH pa60TbI H3,naBHa HcnOJlb3YIOT-
C.H pa3nHqHbIe npHcnoc06J1eHll.H - npOCTbIe MexaHll3MbI.
TIpocTble MexaHll3MbI - 3TO YCTpoHcTBa, B KOTOpbIX pa60Ta COBepIlIaeTC.H
TonbKO 3a CqeT MeXaHHqeCKOH 3HeprHH. TIpocTbIe MexaHH3MbI (pblqar, HaKnOH-
Ha.H nJlOCKOCTb, 6J10K II ,np.) cJlymaT ,nJl.H npe06pa30BaHll.H CllJlbI, llX npMMe-
H.HIOT npH COBepIlIeHllll pa60TbJ B Tex cJlyqa.HX, Kor,na Ha,no ,neHcTBlleM O,AHOH
CHnbI ypaBHoBeCHTb ,npyrylO cHny.
HaKJIOHHan nJIOCKOCTh. Ee llcnonb3YlOT B Tex cnyqa.HX, Kor,na Ha,no no,A-
H.HTb T.HmeJlbIH rpY3 Ha HeKOTOpylO BbICOTY.
PaccMoTpllM rJla,nKYIO HaKJlOHHYIO nJlOCKOCTb (pllC. 3.22). PaCCqllTaeM cllny
F, KOTOpyro Ha,nO npHnomllTb K TeJlY MaCCOH
m, tlTo6bI no,nH.HTb ero paBHOMepHO Ha BbICO-
TY h.
3anllIlIeM OCHOBHoe ypaBHeHlle ,AllHa-
MllKH: F + N + mg = O. CnpoeUllpyeM 3TO
paBeHCTBO Ha OCb Ox: F - mg sin a = O. OT-
h clO,na HCKOMa.H Cllna
F . h mg _
=mgslna=m g l  F--';'
Puc. 3.22 T.e ,nn.H paBHoMepHoro no,nH.HTll.H rpY3a c
nOMoblO HaKJlOHHOH nJlOCKOCTll He06xo,nH-
MO npllnomllTb CllJlY, BO CTOJlbKO pa3 MeHbIlIYIO CllnbI T.HmeCTll rpY3a, BO
CKOJlbKO pa3 ,AJlHHa HaKnoHHoH nJlOCKOCTH 60JlbIlIe ee BbICOTbI.
PhllIar. PbIqarOM Ha3bIBalOT llMelOee Heno,nBllmHYIO OCb BpaeHll.H TBep-
,noe TeJlO, Ha KOTopoe ,neHcTBYIOT CHnbI, CTpeM.HlleC.H nOBepHYTb ero BOKpyr
3TOH OCll. Pa3nllqalOT pbIqarll nepBoro H BToporo po,na.
73

PbPIarOM nep60Z0 poiJa Ha3bIBalOT pbPIar. OCb
BpaI.l.\eHH.H 0 KOTOpol'O paCnOJlOmeHa Mem,AY TOq-
KaMH A H B npHJlOmeHH.H CHJI. a caMH CHnbI
HanpaBneHbI B OHY CTOPOHY (pHC. 3.23. a). 3TO
KOpOMbICJlO paBHOnJleqHX BeCOB. :>KeJle3HO0-
pomHbIM IIIJlaI'6aYM. HO:>KHHl\bI H p.
PbIqar 6mopozo poiJa - pbIqal'. OCb Bpa-
I.l.\eHH.H 0 KOTOpOI'O pacnonomeHa no OHY
CTOPOHY OT TOqeK npHJlOmeHH.H CHn. a caMH
CHnbI HanpaBneHbI npOTHBononomHO Pyr PY-
ry (pHC. 3.23. 6). 3TO raeqHbIe KnlOqH. Hn-
l\bI n.H paCKanbIBaHHH opeXOB. BepH H p.
Y cnOBHe paBHOBeCH.H pbIq8ra BbITeKaeT "3
npaBHna MOMeHTOB M 1 = M 2 (CM.  4.3).
TaK KaK M 1 = F1 H M 2 = F 2 l 2 . re l1 H
l2 - nJleqH CHJI. 1(eHcTBYIOHX Ha pbrqar. TO
F 1 =
F 2 4 - YCJlOBHe paBHOBeCHH pbIqara.
IIpH paBHOBeCHH pbIqara no,A eMCTBHeM
BYX CHn MOYJlH 3THX CHJI 06paTHO nponop-
HOHanbHbI HX nJleqaM.
C nOMOblO pblqara MomHO nOJlHTb BbIHI'-
pbIIII B c1:IJIe. T.e. MeHblIIeM C1:IJI0M MomHO ypaB-
HOBecHTb 6onblIIyIO CHJlY.
EJIOK. BnOKH Hcnonb3YIOT ,AnH nO,lIH.HTH.H
rpY30B. BnOK npeCTaBJlHeT C060M Koneco C
men060M. YKpenJleHHOe B 060MMe. IIo meJlo-
6y 6nOKa nponYCKalOT BepeBKY. TpOC HnH
l\enb. HenO,lJ;BHHLIM Ha3bIBalOT TaKO:H 6nOK.
OCb KOToporo 3aKpenJleHa H npH nO,A'beMe rpy-
30B OHa He nOHHMaeTCH H He onYCKaeTCH
(pHC. 3.24. a. 6).
HenoBHmHbIM 6J10K MomHO paCCMaTpH-
BaTb KaK paBHOnneqHll pbrqal'. Y KOTOpOI'O
nneqH npHJlOmeHHbIX CHJI paBHbI paHYCY
Koneca. CJleOBaTenbHo. H3 npaBHJla MOMeH-
TOB mgr = Fr BbITeKaeT. qTO HenollBHmHbIM
6J10K BbIHl'pbIlIIa B CHne He aeT (F = mg).
OH n03BOJlHeT MeH.HTb HanpaBJleHHe eM-
Puc. 3.25 CTBHH CHnbI.
Ha pHCYHKe 3.25. a. 6 H306pameH no,lJ;-
BHHLIH 6JIOK (OCb 6J10Ka nOHHMaeTC.H H
onYCKaeTC.H BMeCTe C rpY30M). TaKoM 6noK nOBOpaqHBaeTC.H OKono MrHOBeH-
HOM OCH O. IIpaBHJlO MOMeHTOB JI.H Hero 6yeT HMeTb BH,A:
a
A
F!' I I t o l 2
,. ....
I
6 l 2
0 A
PI
Puc. 3.23
B
F '
a
mg
Puc. 3.24
a
o
74
6
H
mg P
F
6
P
OF

Puc. 3.26
mgr = F . 2r  F = mg .
2
TaKIiM 06pa30M. no,nBlljKHbIll 6noK ,naeT
BbIHrpbIlli B CIlJIe B ,nBa pa38.
06bIqHO Ha npaKTIlKe npllMeH.HIOT KOM61l-
HaU;HIO Heno,nBllmHoro 6JIOKa C no,nBlljKHbIM
(pIiC. 3.26). Heno,nBH>KHbIll 6nOK npIlMeH.HeT-
C.H TOJIbKO JJ,JIH YJJ,06cTBa. OH t 1l3MeH.HH Ha-
npaBJleHHe ,qellcTBH.H CHJlbI, n03BOn.HeT, Ha-
npliMep, nO,AHIlM8Tb rpY3. CTO.H Ha 3eMJIe.
F
3.18. PaBeHCTBO pa60T npH HCnOJIL30BaHHH
npOCTLIX MexaHH3MOB
MbI BHJJ,IlM, qTO C nOMo:r.u:blO npOCTbIX MexaHIl3MOB MomHO nOJIyt.IIITb BbJ-
IIrpbIlli B CIIJIe. A JJ,alOT nil npOCTbIe MexaHIl3MbI BbIHrpbIlli B pa60Te?
PaCCqllTaeM pa60TY, KOTOPYIO COBep1II8eT clina F npli nOJJ,'beMe rpY3a C no-
MOblO HaKJIOHHOll nJIOCKOCTH (CM. pHC. 3.22):
h
A F = Fi. lloJJ,cTaBIiM HallJJ,eHHbIe 3H8qeHIl.H CIlJlbI F = mg - II nOJlYt.IIlM
l
h
A F = mg-l = mgh.
l
TaKIiM 06pa30M, p860Ta A F paBH8 pa60Te, KOTOPYIO HymHo COBeplIIIlTb,
t.IT06bI paBHOMepHO nOJJ,H.HTb rpY3 Ha BbICOTY h, He Hcnonb3Y.H HaKJIOHHOll
nJIOCKOCTIi (CM.  3.6).
He JJ,aeT BbIHrpbIllia B pa60Te II pbIq8r. ,I:(eHcTBIiTenbHo, ecnll ypaBHo-
BelIIeHHbIll pbIqar (pHC. 3.27) npliBeCTH B JJ,BllmeHHe, TO TOqKIl npHJlOmeHIlH
CIlJl F I H F 2 3a O,qHO II TO me BpeM.H COBep-
lIIaT pa3HbIe nepeMelLteHIl.H  rl H  r2' llpH
3TOM (CqllTaeM yroJl a nOBopOT8 pbIq8ra He-
60JIblII1lM) rl = 4 a., 6. r 2 = L;,a. CJIeJJ,OBaTeJIb-
F HO, 3TIl CIlJlbI COBep1II8T p860TY
-
Al =fi6.r l =fia II A 2 =F2r2 =F20.a. TaK
K8K Fl4 = F 2 0., TO Al =.
llpll IlCnOJIb30BaHlili HenOJJ,BllmHoro 6noK8
MbI BIlJJ,IlM, qTO npHJIOmeHHbIe CIiJIbI F H mg
paBHbI H nYTH, npOllJJ,eHHbIe TOt.IKaMH npHJlOmeHHH CHJI npH nOJJ,'beMe rpY3a,
Tome O,AHHaKOBbI, a 3HaqHT, 0JJ,HH8KOBbI H pa60TbI.
't.JT06bI npH nOMOH nOJJ,BHmHoro 6JIOKa nOJJ,H.HTb rpY3 H8 BbICOTY h, He-
06XOJJ,HMO KOHeII, BepeBKH, K KOTOPOMY npHJIOmeH8 CHJI8 F. nepeMeCTHTb
mg
H8 2h. CJle,AOB8TeJlbHO, Al = mgh H  = F . 2h = 2' 2h = mgh.
T8KHM 06pa30M, nOJlyq8H BbIHrpbIlli B CHne B JJ,B8 pa3a, npoHrpbIBalOT B
JJ,Ba pa3a B nepeMeeHHH, CJle,AOB8TenbHO, H nOJJ,BHmHbIll 6J10K BbIHrpbIllia B
pa60Te He ,naeT.
FI
it
Puc. 3.27
75

MHorOBeKOBa.H npaKTHKa nOKa3aJIa, qTO HH O,nHH H3 npOCTbIX MeXaHH3-
MOB He ,naeT BbIHrpbIllia B pa60Te.
Eru;e ,npeBHHe }TLIeHbIe CCPOPMYJlHpOBaJIli npaBHJlO ("30JlOTOe npaBHJlO Mexa-
HHKH"), npHMeH.HeMOe KO BceM MexaHH3MaM: BO CKOJlbKO pa3 BbIHrpbIBaeM B
CHJIe, BO CTonbKO pa3 npOHrpbIBaeM B paCCTOSlHHH.
IIplJ paCCMOTpeHHH npOCTbIX MexaHH3MOB MbI He yqHTbIBaeM TpeHHe, a
TaKme Bec caMHX MexaHH3MOB. B peaJIbHbIX YCJIOBH.HX 3TO He06xo,nHMO }TLIH-
TbIBaTb. II03TOMY qaCTb pa60TbI COBepIllaeTC.H CHJlOH F Ha nepeMeeHHe OT-
)J;eJIbHbIX qaCTeH MexaHH3Ma H npOTHB CHJIbI TpeHH.H. Pa60Ta me no no,n'beMY
rpY3a An (nOJle3Ha.H pa60Ta) 6y,neT MeHbIlle nOJlHOH pa60TbI A (pa60TbI, KOTO-
PYIO COBepIllaeT CHJIa F).
3cp<peKTHBHOCTb pa60TbI MexaHH3Ma xapaKTepH3YlOT K03cpcpHu,HeHTOM nOJle3-
Horo .neHCTBHSI (KIIJJ: MexaHH3Ma):
K03cp<pHu,HeHT nOJle3HOrO ,AeHCTBH.H - cpH3HqeCKa.H BenHqHHa, paBHa.H OT-
HOIlleHHIO nOJIe3HOH pa60TbI An KO BceH 3aTpaqeHHOH pa60Te A:
11 =  -100 %.
A
BOITPOCbI 11 3A,[(AHll.H ,[(JI.H CAMOKOHTPOJI.H
1. B KaKUX c.n.YllaSlX U.Mny.n.bC me.n.a U3.MeHSlemcSl. a KUliemUlleCKaSl aliep
ZU.R ocmaemcSl neU3.Meliliou?
2. Kale U3.MeIiSlemCSl KUliemUlleClea.R aliepzu.R meJla. ec.n.u npu.n.OJICeliliaJl CUJla
cOBepwaem no.n.OJICUme.n.bIiYIO (ompuu,ame.n.bHYIO) pa60my?
3. KaleaSl 3liepZUSl SlB.n.SlemcSl .Mepou B3aU.MOOeUCm6uSl meJl?
4. MO:HCem .n.u nomeliu,ua.n.bliaSl aliepzuSl 6blmb ompUl{ame.n.bliou?
5. II olle.MY yp06elib lialla.n.a omcilema nOmelil{Ua.n.bliou aliepzuu 6 pa3liblX
3aoailax .MOJICliO 6bl6upamb npou360.n.bHO?
6. Om Ilezo 3a6ucum nOmelil{Ua.n.bliaSl aliepzuSl me.n.a. nooliSlmozo Hao 3e.M-
.n.eu? Kale U3.MenSlemCSl aHa npu 06U:HCenUU me.n.a 6Bepx? BliU3? KaK CBSl3alia
olia C pa60mou CUJlbl m.RJlCecmu?
7. Om Ilezo 3aBucum nomeliijua.n.bliaJl 31iepZUSl ynpyzo Be(jJop.MUpOBaliliozo
meJla? KaK C6Sl3aHa olia C pa60mou CU.n.bl ynpyzocmu?
8. Kale U3.MeIiSlemCSl nOmeHl{Ua.n.bliaSl aHepzuSl npY:HCUlibl npu ee pacmSlJICe-
IiUU? npu cJlCamuu?
9. KaKou aliepzueil 06.n.aoaem me.n.o. nooliSlmoe liao 3e.M.n.eu? c606001i0 na-
oalOw,ee me.n.o?
10. l{ mo n01iU.MalOm noo no.n.liou .MeXaIiUlleCleOU 3liepZueu? Bcezoa .n.u B
3a.Mleliymou cucme.Me no.n.liaSl .MeXaliUlleCKaSl aliepzuSl coxpaliSlemcSl?
11. Te.n.o 6poweli0 6epmuKa.n.blio 66epx. H 3.MeliSlemCSl .n.u no.n.liaSl Mexa-
liUlleCKaSl aliepzuSl? KaKue B3aU.Mlible npe6paw,eliuSl .MeXaliUllecKou aliepzuu
npoucxoo.Rm npu OBU:HCeliUU me.n.a. ec.n.u CUJlQ conpomuB.n.eliuSl lie oeu-
cm6yem?
12. Te.n.o c60600liO naoaem C 6blcombl h liao n06epxliocmblO 3e.M.n.u. IT o.
cmpoume zparjJuKu 3a6UCU.MOCmu ezo nomeliu,ua.n.b1iOu. KUliemUlleCKOU U nOJl-
Hoil aliepzuu 6 3a6UCU.MOCmu om 6blcombl me.n.a liao 3e.M.n.eu
76

13. Te.n.o COCKaJLb3bl.6aem 6e3 mpeliUR C IC.n.Ulia, .n.e:HCaw,ezo liQ ZopU30limaJlb-
"WU n.n.OCKocmu. o6a pa3a: nep6bl.U pa3 K.n.UIi 3aKpen.n.eli, 6mopou pa3 K.n.UIi
.MOJICem CKo.n.b3umb 6e3 mpeliUSl. Byoem .n.u CKopocmb me.n.a 6 ICOliije IC.n.Ulia
OOUliaK060U 8 060ux C.n.YllaJlX. ec.n.u me.n.o 06a pa3a COCKa.n.b3bL6a.n.0 C OaliOU U
mou JlCe 8bLcombL?
14. J( 6UJlCYUUCR 6aZOli cma.n.KU6aemcSl C npYJICUHlibL.M a.MOpmU3amOpO.M.
Bo Ilmo npe6paw,aemcSl ezo KUliemUlleClCaSl aliepzuSl?
15. KaIC U3.MeIiSlemCSl no.n.HaSl .MeXaIiUlleCKaJl aliepzuSl 3a.MICliymou cucme-
.Mbl me.n.. ec.n.u me.n.a 63aU.MOaeucm6YlOm .MeJICoy c060u CUJla.MU mpeHuSl?
16. t[ e.n.06eK cnYCKaemcSl lia napaWlOme. KaK U3.MeliSlemCSl ezo ICUliemu-
lleCICa.R aliepzuJl? nomeHquaJLbHaJl? nOJLHaJl .MeXaHUeCICaJl aHepZUJl?
17. t[e.M om.n.UllaemcSl a6COJL1OmliO lieynpyzuu yoap om a6co.n.lOmliO ynpy-
zozo? KaK U3MeIiSlemC.R KUliemUlleCKaSl U nomeliqua.n.bliaSl aliepzuu me.n. 6 npo-
ijecce coyaapeliUSl? CoxpaliSlemcSl .n.u npu amo.M no.n.liaSl .MeXaIiUlleCKa.R aliep-
ZUSl me.n.?
18. KaICu.Mu 3aKOlia.MU coxpalieliUSl .MOJICliO no.n.b308ambCSl npu paCC.Mompe
liUU ynpyzozo U Heynpyzozo yaap08?
19. TeJLo .MaCCOU mI, a6U:HCyeecSl co CICopocmblO v, ynpyzo yaapJlemCSl 0
nOICOJleeCJl me.n.o .Maccou m2' Om Ilezo 3a8ucum lianpa8JLeliUe a8UJlCeliUSl
me.n.Q Maccou ml noc.n.e yaapa? B KaICO.M c.n.Yllae OliO omCKOllum lia3aa?
ocmali06umcSl? 6yaem npoaO.n.:HCamb a6u:»celiue 6 mo.M JlCe Hanpa6.n.eliuu?
20. Kozoa noICoSluUCSl wap npu06pemaem 60.n.bWYIO cKopocmb om opyzo
zo maKOZO JlCe wapa: npu ynpyzo.M UJlU npu Heynpyzo.M coyoapeHuu?
21. II Olle.MY npu OaunaIC080.M yaape na.n.ICou 0 ICa.Melib U meliliUCHblU .MJlIl
noc.n.ealiuu .n.emum aa.n.bwe?
METO,rUlqECKIIE PEKOMEH,Z:(Al(1I11
IIpH pellIeHHH 38.u.aq H8 npHMeHeHHe 3aKOH8 npeBpaeHH.H H coxp8HeHH.H
3HeprHH Heo6xo.u.HMO:
1. C.u.eJl8Tb cXeMaTHqeCKHH PHCYHOK, OTMeTHB Ha HeM HaqaJIbHOe H KO-
HeqHOe nOJlOmeHH.H TeJl (1 H 2), YKa3aHHbIe B 3a.u.aqe.
2. BbI6p8Tb HyneBoH ypOBeHb OTCqeT8 nOTeHIl.H8JIbHOH 3HeprHH.
3. YKaaaTb CKOpOCTH 11 BbICOTbI (HJII1 KOOp.u.HHRTbI TeJla, .u.BHmyerOC.H no,lJ.
,AeHCTBHeM CHJIbI ynpyrocTH). X8paKTep113YIOIl111e COCTO.HHHe Ten8 B 060HX no-
JIOmeHH.HX.
4. 3anHcaTb «})OPMYJIY ,An.H paCqeT8 nOJIHOH MeXaHl1qeCKOM 3HeprHH B no-
JIOmeHH.HX 1 H 2.
5. BbI.HCHHTb, K8KHe CHJIbI B paccM8TpHB8eMoH MeX8HHqeCKOH CHCTeMe .HB-
JI.HIOTC.H BHeIlIHHMH, BHYTpeHHHMH, KOHcepBaTHBHbIMH, HeKOHcepBaTHBHbIMH.
6. AJI.H 38MKHYTOH CHCTeMbI, B KOTOPOH .u.eHcTBYIOT TOJlbKO KOHcepB8THB-
HbIe CHJIbI. 3anHC8Tb 3aKOH coxpaHeHH.H MeX8HHqeCKOH 3HeprHH WI = W 2'
ECJlI1 B 38MKHYTOH CHCTeMe .u.eHCTBYIOT CHJlbI TpeHH.H, 3anHcaTb OPMYJlY
W 2 - WI =. ECJII1 CHCTeMa He3aMKHYTa H BHellIHHe CHJlbI COBepllI81OT p8-
60TY A, TO W 2 - WI =A.
77

7. HanHcaTb CPOPMYJlbI n.H paCqeTa pa60TbI A HJUI A TP '
8. CocTaBHTb npH He06xoHMOCTH OnOJIHHTeJIbHbIe ypaBHeHH.H H3 HHa-
MHKH HJlH KHHeMRTHKH. B HeKOTopblX 3aaqaX Hao npHMeHHTb H 3aKOH co-
xpaHeHHH HMnYJlbca.
9. PeIllHTb nOJIY'leHHYIO CHCTeMY ypaBHeHHH.
IIPIIMEPbI PElliEHII.H 3A,L(A q
V
1- --2
L__-.J
1. TeJlo 6pomeHo BepTHKaJlbHO BBepx co
M
CKOpOCTblO 16 -. Ha KaKOH BblCOTe KHHeTH-
C
qeCKaH SHeprHH TeJla p8BHa ero nOTeHHaJIbHOH
sHeprHH? ConpOTHBJIeHHeM B03yxa npeHe6peqb.
M
.naHo: vo=16 - . Wi<=W n ,
C
HaHTH: h.
P e III e H He. CHCTeMa "TeJlo-3eMn.H" .HBJI.H-
eTC.H 3aMKHYTOH. CJIe,AOBaTeJIbHO, nOJlHa.H Mexa-
HHqeCKaH SHeprH.H COXpaH.HeTC.H.
3a HYJIeBOH YPOBeHb nOTeHHaJIbHOH SHeprHH npHHHMaeM ypOBeHb, C KO-
Toporo 6bIJIO 6pomeHO TeJIO (pHC. 3.28).
h
Va
1
h=O
Puc. 3.28
B COCTO.HHHH 1 nOJIHa.H MeXaHHqeCKa.H SHeprH.H
WI
mV5
=
2'
B COCTO.H-
HHH 2 OHa COCTaBJIHeT
mv 2
W 2 = - + mgh. Do 3aKOHY coxpaHeHHH 3HeprHH
2
mv 2 mv 2
WI = W 2 =>  = - + mgh. Do YCJIOBHIO
2 2
mv 2
- = mgh.
2
mv 2 u 2
CJIeOBaTeJIbHO,  = 2m g h => h = . h = 6,4 M.
2 4g'
2. qeJloBeK CTOHT Ha HeIIoBHmHoH TeJlemKe 11 6pocaeT ropH30HTaJlbHO
M
KaMeHb MaccoH 8 Kr co CKOpOCTblO 5 -. OIIpeeJIHTe, KaKYIO pa60TY COBep-
C
maeT qeJIOBeK. eCJIH ero Macca BMeCTe C TeJIemKOH 160 Kr.
M
,lI,aHo: mI=8 Kr, vl=5 -, m2=160 Kr.
C
HaHTH: A.
P e III e H H e. Pa60Ta. COBepmeHHa.H 'leJlOBeKOM B MOMeHT 6pocKa, paBHa H3-
MeHeHHIO MeXaHHqeCKOH 3HeprHH CHCTeMbI "qeJIOBeK C TenemKOH - KaMeHb":
A =  W. TaK KaK KaMeHb H qenOBeK B MOMeHT, HeIIocpeCTBeHHO npeme-
78

2 2
A = m1v 1 + m 2 v 2
CTBYIOHH 6POCKY, nOKOHJlHCb, TO WI = 0 H 2 2' r)J;e V2
CKOpOCTb TeJlemKH C qeJlOBeKOM B MOMeHT Henocpe)J;CTBeHHO nocJle 6pocKa.
mlv 1
no 3aKOHY coxpaHeHH.H HMnYJlbCa m1v] = v2 => v2 = -.
m2
m v 2 m2v  ( + m2)v
CJle,noBaTeJlbHO, A = -L!.. + --1- = A = 1 10 2 Am.
2 2 2
3. U;HpKOBOH apTHCT MaCCOH 60 Kr npblraeT C BbICOTbI 10 M Ha paCT.HHY-
TYIO ceTKY. C KaKoH cpe)J;HeH CHJlO:M OR ,naBHT Ha ceTKY, eCJlH OHa nporH6a-
eTC.H Ha 1,0 M?
A a H 0: m = 60 Kr, h = 10 M. X = 1,0 M.
H a H T H: < F > .
--,
Jl
h
x h=O
Puc. 3.29
P e 111 e H H e. CHcTeMa "apTHcT-3eMJI.H-
ceTKa" HBJI.HeTC.H 3aMKHYToii. CJle,noBaTeJlbHO,
BbInOJlH.HeTCH 3aKOH coxpaHeHH.H sHeprHH.
nOJIHa.H MeXaHHqeCKa.H SHeprH.H CHCTe-
MbI B COCTO.HHHH 1 (pHC. 3.29) COCTaBJI.SIeT
kx 2
WI = mg (h + x), B COCTO.HHHH 2 W 2 = - .
2
Ha OCHOBaHHH 3aKOHa coxpaHeHH.H SHeprHH
( ) kx2 2mg(h T x)
Wi = W 2 => mg h + x = - => k = 2 .
2 x
TaK KaK no 3aKOHY rYKa CHJIa ynpyrocTH nponopUHOHaJlbHa BenHqHHe
kx
Ae<l>opMaHH, TO < F ynp > = 2' no TpeTbeMY 3aKOHY HblOTOHa HCKOMa.H CHJla
mg(h + x)
< F > = < Fynp >. CJle,nOBaTeJIbHO, < F > = ; < F > = 6,6 KH.
x
4. KaMeHb MaccoH 50 r, 6pomeHHbIH no,n yrnoM K ropH30HTY C BbICOTbI
M
20 M Ha;J; nOBepXHOCTblO 3eMJlH co CKOpOCTblO 18 -, ynaJl H8 3eMnlO co CKO-
C
M
pOCTblO 24 Haii,nHTe pa60TY no npeo,nOJIeHHIO CHJI conpOTHBJleHH.H B03,nyxa.
C
AaHO: m=0,050 Kr, h=20 M, vo=18 v=24
C C
M
M
HaHTH A.
P e III e H H e. CHcTeMa "KaMeHb-3eMJI.H"
,neHcTByeT CHJla conpOTHBJleHH.H B03,nyxa.
SHeprH.H He COXpaH.HeTC.H Ii W = Ac'
3aMKHYTa.SI, HO Mem,ny TeJlaMH
nOSTOMY nOJIH8.H MeXaHHqeCKa.SI
79

h
Puc. 3.30
IIoJlHa.H MeXaHHqeCKa.H 3HeprH.H B
COCTO.HHHH 1 (pHC. 3.30) COCTaBJI.HeT
mv 2
WI = ---t + mgh, B COCTO.HHHH 2
mv 2
W 2 = -. CJleOBaTeJlbHO,
2
v
mv 2 mv 2
A = - ----.Q.. -m g h'
c 2 2 '
m (v 2 - v )
Ac = - mgh; Ac == -3,7 Am.
2
I1cKoMaSl pa60Ta A == - Ac; A == 3, 7 Am.
5. JleT.Ha.H ropH30HTaJlbHO co CKOpOCTblO v nynSl MaCCOH ml 3aCTpeBaeT
B 6pycKe MaCCOH m2' nOBeIlleHHOM Ha HeBeCOMOH HHTH. KaKoe KOJlHqeCTBO
TenJlOTbI BbIeJlHTCSI B 6pycKe? Ha KaKOH yroJl OTKJlOHHTC.H HHTb, eCJlH ee
)J;JlHHa l?
AaHo: mh m2' v, l.
HaHTH: Q, <x.
P e III e H H e. TaK KaK BpeM.H B3aHMoeHCTBH.H nYJlH H 6pycKa MaJIO, TO
HX CYMMapHbIH HMnYJlbC COXpaH.HeTC.H: 1'nJ. v == (1'nJ. +"'-2) VI' re VI - CKOpOCTb
miv
6pYCKa H nYJlH nocJle B3aHMOeHCTBH.H. OTCIOa VI =
+m2
BbIeJleHHOe KOJlHqeCTBO TenJlOTbI 6yeT paBHO pa3HOCTH KHHeTHqeCKOH
3HeprHH CHCTeMbI "nYJl.H-6pYCOK" O B3aHMOeHCTBH.H H nocJle Hero:
mIv 2 (mi + m 2 )v: mlm2v2
Q=Wi-W 2 =-- =
2 2 2(mi +m2)
PaccMoTpHM CHCTeMY "6pYCOK C nYJleH - 3eMJI.H". CHJla HaT.HmeHH.H HH-
TH - BHeIllH.H.H CHJla. Ho TaR KaR OHa nepneHHKYJI.HpHa CKOpOCTH, TO pa60Ta
3TOH CHJlbI p8.BHa HYJlIO. II03TOMY W == A == 0 Ii WI = W 2 , re WI H W 2 -
o
l
C
h=O
r
.
1
VI
Puc. 3.31
m 2 v 2
OTclOa h = I .
2(mi + m 2 )2 g
80
nOJlHaH 3HeprH.H CHCTeMbI B nOJlOmeHH-
SIX 1 H 2 (pHC. 3.31).
IIpHHHMa.H 3a HYJleBOH ypOBeHb OT-
CqeT8 BbICOTbI nOJIOmeHHe 1. HMeeM
(mi + )v:
= (mi + m2)gh.
2
IIocTaBHB 3HaqeHHe VI' nOJlyqHM
m:v 2
== gh.
2(mi + m2)2

l- h m 2 v 2
1I3 t:1 OBC cosa = -. Tora cosa = 1- 1 .
1 2lg(ml + )2
Puc. 3.32
Puc. 3.33
6. !{B8 nO,ABHmHbIX KnliH8 OHH8KO-
BOH M8CCbI M (pHC. 3.32) HMelOT nnaB-
Hble nepexoAbI H8 ropli30HTanbHYlO nnoc-
KOCTb. C JIeBOrO KnHHa C BblCOTbI H co-
CKanb3bIB8eT waH68 MacoH m. Ha K8KYlO
M8KCHM8nbHYlO BbICOTY h nO,AHHMeTCSI
waH6a Ha npaBOM KnliHe? TpeHlieM npe-
He6pe'lb.
)];8HO: M, m, H.
HaHTH: h.
P e 111 e H H e. T8K KaK TpeHHe OTCYT-
cTByeT, TO B rOpH30HTanbHOM HanpaBne-
HHH Ha CHCTeMY "KnIiH-W8H6a" BHeW-
Hlie ClinbI He ,AeHCTBYlOT. !{n.H onpe,Ae-
neHHSI CKOpOCTH v neBoro KnHH8 H CKO-
pOCTH VI maH6bI (pHC. 3.33) cpa3Y noc-
h ne cnYCK8 MO>KHO Bocnonb30saTbCfl 3aKO-
H8MIi coxpaHeHHSI HMnynbca H 3HeprHH:
M v 2 mv 2
-+=mgH.
2 2
Mv= mVl;
PemaSl CHCTeMY 3THX ypasHeHHH,
H8H.n;eM CKOpOCTb W8H6bI:
'" ; lMgH .
m+M
(1)
llIaH6a. liMeSl CKOpOCTb vI> H8'lHeT no.n;HHM8TbCSI Ha npaBbIH KnHH 38Me.n;neH-
HO. B MOMeHT H8HBbIcmero no.n;'beM8 maH6bI ee CKOpOCTb OTHOCHTenbHO npaBoro
KnHHa p8SH8 HynlO H CKOpOCTH KnHHa H waH6bI V2 6y.n;YT 0.n;HH8KOSbI. 3anHmeM
38KOHbI coxp8HeHHSI HMnynbc8 H 3HeprHIi:
mv; (m + M)v
mVl = (m + M)V2; - = + mgh.
2 2
113 ypaBHeHHH (1) Ii (2) HaH,AeM HCKOMYlO BbICOTY:
v 2 M M 2 H
h = --!... =
2g (m + M) (m + M)2 .
(2)
81

4. 3REMEHThI CTATMKM
C"'na. CnOJKeH",e c"'n. Ll.eHTp TSlJKeCT"'. PaBHOBec",e Ten. MOMeHT c"'nbl.
YcnOB",e paBHOBec",SI Tena, He "'MelOero oc'" BpaeH"'SI. YcnOB",e paB-
HOBec",SI Tena c 3aKpenneHHO OCblO BpaeH"'SI YcnOB"'SI paBHOBec",SI TBep-
Aoro Tena. YCTO"'BOCTb Ten. B"'Abl paBHOBec",SI.
CTaTHKa H3yqaeT ycnOBH.H paBHOBeCH.H MaTepHaJIbHOH TOqKH H a6conlOT-
Horo TBep,AOrO TeJIa.
A6COJIIOTHO TBep,noe Teno - TeJIO, pa3MepbI H cpOPMY KOToporo MomHO
CqHTaTb HeH3MeHHbIMH.
IIo,n YCJIOBHHMH paBHOBeCH.H nOHHMalOT YCJIOBH.H. npH KOTOpbIX Teno npH
HaJIHqHH BHeIllHero B03,neHCTBH.H MomeT Haxo,nHTbC.H B nOKoe OTHOCHTeJlbHO
HHepHaJIbHOH CHCTeMbI OTCqeTa; ,ABHraTbC.H nocTynaTeJIbHO, paBHOMepHO H
np.HMOJIHHeHHO; paBHOMepHO Bpall{aTbC.H OTHOCHTeJIbHO OCH. npOXO,A.Hll{eH qe-
pe3 l\eHTp Macc.
4.1. CHJla. CJlOmeHHe CHJI
OCHOBHbIe cpH3HqeCKMe BenHtIHHbI, Hcnonb3yeMbIe B CTaTHKe. - CHJIa 11
MOMeHT CHJlbI. CHJIa KaK BenHqHHa BeKTOpHa.H xapaKTepH3yeTC.H Mo,nYJIeM, Ha-
npaBneHHeM B npOCTpaHCTBe H TOqKOH npHJIOmeHHH.
Pe3YJlbTaT ,neHcTBHH CHJIbI Ha MaTepHanbHYIO TOqKY 3aBHCHT TOJIbKO OT
ee MO,nYJIH H HanpaBneHHH. TBep,AOe me TeJIO HMeeT onpe,neneHHbIe pa3Me-
PbI. I103TOMY OlIHHaKOBbIe no MOlIYJIIO H HanpaBJleHHIO CHnbI BbI3bIBa-
lOT pa3JIHtIHble ,nBHmeHHH TBep,noro TeJIa B 3aBHCHMOCTH OT TOqKH npHno-
meHHH.
TOqKY npHnOmeHH.H CHJIbI MomHO nepeHOCHTb TOJIbKO B,nonb npHMOH,
B,nOJIb KOTOPOH 3Ta CHna ,neHcTByeT. 06 3TOM Heo6xo,nHMO Bcer,na nOMHHTb npH
OCYll{eCTBneHHH pa3J1U'lHbIX Onepal\HH Ha,A CHJIaMH.
CHJIa R, KOTOpa.H npOH3Bo,nHT Ha TeJIO TaKoe me ,AeHCTBHe, KaK H He-
CKOJlbKO o,nHOBpeMeHHO ,neHcTBYIOHX Ha Hero CHJl, Ha3bIBaeTCS:I paBHo.a;eHCTBY-
n
lOm;eH. OHa paBHa reOMeTpHqeCKOH CYMMe 3THX CHn: if. = L F; .
;=1
CJIOmHTb CHJIbI - 3TO 3HatIHT HaHTH HX paB-
Ho,neHCTBYIOll{YIO.
EcnH K TeJIY npHJIOmeHO .lJ.Be CHJIbI B O.lJ.HOH
TOqKe. TO paBHO.lJ.eHCTBYIOll{YIO HaXO.lJ..HT no npaBH-
JlY napaJIJleJIOrpaMMa (pHC. 4.1). MO,nYJIb paBHO-
,AeHcTBYIOll{eH .lJ.BYX CHJI MomHO onpe,neJIHTb no Te-
opeMe KOCHHYCOB
R =  F12 + F 2 2 + 2F 1 F 2 cosa
HJlH npH a = 90°- no TeopeMe IIHcparopa.
-. Ii

F 2
Puc. 4.1
82

Puc. 4.2
ECJIH HenapaJIJIeJIbHbIe CHJIbI npHJIOmeHbI B
pa3HbIx TOqKaX TeJIa, TO JI.H HaXOmeHH.H HX
paBHOeRCTBYIOeH 3TH CHJIbI i1 H P 2 nepeHO-
C.HT B TOqKY 0 nepeCeqeHH.H np.HMbIX, BOJlb KO-
TOpbIX OHH ,AeHCTBYIOT (pHC. 4.2), a 3aTeM npo-
H3BOlI.HT HX BeKTopHoe cnomeHHe no npaBHJlY na-
paJIJIeJlOrpaMMa. TOqKOR npHJIOmeHH.H paBHO,AeH-
cTBYIOeH CHJIbI MomeT 6bITb JII06aSl TOqKa npSl-
MOH, BOJIb KOTOpOH OHa eHcTByeT.
4.2. Y CJIOBHe paBHOBeCHH TeJIa, He HMelOlI.(erO OCR BpalI.(eHHH
ECJIH no eHcTBHeM npHJIOmeHHbIx K TeJIY CHJI OHO BHmeTC.H nocTyna-
TenbHO, TO Bce ero TOqKH BHmYTCSI O,AHHaKOBO H OCTaTOqHO paCCMOTpeTb BH-
meHHe OHOH TOqKH. YCJIOBHe paBHOBeCHSI TaKoro TeJIa, KaK H YCJlOBHe paBHO-
BeCH.H MaTepHaJlbHOH TOqKH, BbITeKaeT H3 OCHOBHoro ypaBHeHHSI HHaMHKH:
n
ii = 0 H V = cons t, eCJlH L = O.
i=1
8TO BbIpameHHe MomHO 3anHcaTb B npOeKl{H.HX Ha OCH:
n n n
 F = 0;  F = 0;  F = O.
 IX  Ig  u
i=1 i=l i=l
TaKHM 06pa30M. Ilm06bl He6paalOeecSl me.n.o liaxoou.n.OCb 6 pa6li06eCUU,
lie06xoiJu.M0. Ilm06bl zeO.MempUlleCKa.R cY.M.Ma CU.n.. iJeucm6YlOUI,UX lia me.n.o. 6blJla
paBHa HYJlIO (UJlU aJlZe6paUlleCKa.R cY.M.Ma npoeKquu amux cu.n. lia OCU KOOp-
iJuliam 6bl.n.a pa6lia IiY.n.IO).
4.3. Y CJIOBRe paBHOBeCHlI TeJIa C 3aKpenJIeHHOH OCLIO BpalI.(eHHlI
BbI.HCHHM YCJlOBHe paBHOBeCH.H TeJIa. KOTopoe nOlI neHCTBHeM npHJIOmeH-
HbIX K HeMY CHJI nocTynaTeJIbHO He ,ABHmeTC.H, a nOBOpaqHBaeTC.H BOKpyr He-
KOTOpOH 3aKpenJleHHOH OCH. HanpHMep, pac-
CMOTpHM HeBeCOMbIH CTepmeHb BC, Ha KOTOpbIH
eHCTBYIOT CHJlbI PI Ii F 2 (pHC. 4.3).
I10BepHeM ero Ha He60JIbllIOH yron <p BOKpyr
OCR O. IIpH 3TOM TOqKH npRnOmeHH.H C1:IJI Fl Ii
F 2 COBepllIaT nepeMeeHlie BBl H CC 1 COOTBeT-
CTBeHHO. I1pH MBJIOM yrJIe nOBopOTa q> MomHO
CqHTaTb, qTO
Puc. 4.3
B = BO . q> =  cp; CC l = CO . cp = <P,
r,l1e 1 1 H 1 2 - nJleqH CHJI PI H i2 COOTBeTCTBeHHO, a yroJl q> BbIpamalOT B
paHaHax.
83

IIJle1Jo CHJlLI - 3TO KpaTqaRmee paCCTO.HHHe OT OCM Bparn;eHM.H ,n;o np.H-
MOR J B,n;OJlb KOTOpOH ,n;eRcTByeT CHJla.
3TH CHJlbI COBepmalOT pa60TY:
Al = Ficp;  = -F2<p.
IIpoM3Be,AeHHe CMJlbI Ha ee nJleqo Ha3bIBalOT MOMeHTOM CHJlLI OTHOCHTenb-
HO OCM BpaeHM.H:
M = =f::Fl.
IIpMH.HTO CqMTaTb MOMeHT CMJl, BpaalOHX TeJIO OTHOCMTenbHO ,AaHHOR
OCH no qaCOBOR CTpeJlKe, nOJlOmHTeJlbHbIM, a npOTHB qaCOBOH CTpeJlKM - OT-
pMaTenbHbIM.
MOMeHT CMJlbI xapaKTepM3yeT Bparn;aTeJIbHOe ,n;eHCTBHe 3TOH CHJlbI M BO
Bparn;aTeJIbHOM ,n;BHmeHHM HrpaeT TY me pOJlb, qTO H CHJIa B nocTynaTeJIbHOM
)J.BMmeHMM.
CYMMapHa.H pa60Ta, cOBepmaeMa.H CHnaMM FI M F2' COCTaBJI.HeT
A = (Fi - F2)<P = (MI + M 2 )<p.
(4.1)
Ha OCHOBaHHM TeopeMbI 0 KHHeTHqeCKOR 3HeprHH WK = A. IIpM paBHO-
BeCHH CKOpOCTb TeJIa He M3MeH.HeTC.H. 3HaqMT WK = 0  A = O. TaK KaK
<p:l= O. TO H3 BbIpameHM.H (4.1) cneyeT. qTO M 1 + M 2 = O. TaKMM 06pa30M. JI.H
Toro qT06bI me.n.o C 3aKpen.n.eliliou OCblO 6paU{eliuSl liaxoou.n.OCb 6 pa6liOBecuu.
ne06xoou.M0. Ilm06bl a.n.ze6paUlleCKaSl CY.MMa .M0.Melim06 cu.n.. npu.n.OJICeliliblX K
n
me.n.y, omliocume.n.bliO amou ocu 6blJla pa6lia ny.n. 10 , T.e. LM; = O.
i=1
4.4. Y CJIOBHH paBHOBeCHH TBep.u;oro TeJIa
B 06rn;eM cJlyqae, ecnH TeJIO no,n; ,n;eRcTBMeM npMJIOmeHHbIX K HeMY CMJI
MomeT M nOBOpaqMBaTbC.H, H ,ABHraTbC.H nocTynaTeJlbHO, TO ,n;n.H Toro, qT06bI
OHO H8xo,n;Mnocb B p8BHOBeCMM, Heo6xo,AMMO BbInonHeHMe ,n;BYX ycnoBMH:
1) zeO.MempUlleCKaSl cY.M.Ma npu.n.OJICeliliblX K meJlY cu.n. oo.n.JlCna 6blmb PU6-
n
L  = 0;
;=1
2) a.n.ze6paUlleCKaSl cY.M.Ma .M0.Melim06 amux cu.n. omnocume.n.bliO .n.1060u ocu
oo.n.JlClia 6 blmb pa6lia IiY.n.IO:
na liY.n.IO:
n
LM i = O.
i=1
4.5. CJIO)KeHHe napaJIJIeJIbHbIX CHJI. QeHTp TH)KeCTH
PaBHo,n;eHCTBYIOrn;a.H ,n;BYX napanJleJlbHbIX o,n;HHaKOBO H8npaBJleHHbIX CHJI
(pHC. 4.4, a) paBHa no Mo,n;ynlO CYMMe HX MOlIYJIeH. napaJlneJIbHa HM H Ha-
npaBJIeHa B TY me CTOPOHY, a JlMHM.H )J;eHcTBM.H paBHo,n;eRcTBYIOrn;eR ,n;eJIMT
84

OTpe30K, COe.v;HH.HIOHH TOqKH npHJIO-
meHH.H CJlaraeMbIX CHJI, Ha yqaCTKH,
06paTHO npOnOpHOHaJlbHbIe CHJIaM.
STO MomHO .v;OKa3aTb: eCJlH B npe.v;no-
JlaraeMOH TOqKe 0 npHJlOmeHH.H paB-
Ho.v;eHCTBYIOeH MbICJleHHO nOCTaBHTb
onopy. TO peaKH.H onopbI CKOMneHCH-
pyeT paBHOlleHCTBYIOYIO. CHCTeMa CHn
OKameTC.H ypaBHoBemeHHoii, H MomHO
BOCnOJlb30BaTbC.H nepBbIM H BTOpbIM
YCJlOBHeM paBHOBeCH.H (CM.  4.4):
OB = FI
OA 112'
PaBHo.v;eHcTBYIOa.H .v;BYX aHTHnapanJleJlbHbIX CHJI (pHC. 4.4, 6) paBH8 no
Mo.v;ynlO pa3HOCTH HX Mo.v;YJleH, napaJlJleJlbHa HM, HanpaBJleHa B CTOPOHY 60nb-
meii CHJlbI. a TOqKa npHJIOmeHHH paBHo.v;eiicTBYIOeH JIemHT H8 npOllOJlme-
HHH JlHHHH, coe.v;HH.HIOeH TOqKH npHJlOmeHH.H CJIaraeMbIX CHJI Ha paCCTOH-
HH.HX OT HHX, 06paTHo nponopnHOHaJlbHbIX CHJIaM:
)J;Be paBHbIe no MO)J;YJlIO H npOTHBOnOJlOmHbIe no HanpaBJIeHHIO CHJIbI, npH-
nomeHHbIe K TeJlY B pa3HbIX TOqKaX, 06pa3YlOT napy CHJI. IIo.v; )J;eiicTBHeM
napbI CHJI TeJlO He .v;BHmeTC.H nocTynaTeJlbHO, a TOJlbKO Bpa8eTCH BOKpyr
OCH, npoxo,A.HIl..teii qepe3 neHTp Macc lIaHHoro TeJla.
TaK KaK Macca TeJla pacnpe.v;eJIeHa no BceMY TeJlY. TO Ha KamylO qaCTb
TeJla (pHC. 4.5) .v;eHcTByeT co CTOpOHbI 3eMJlH CHJIa T.HmeCTH. Mx paBHo.v;eH-
CTBYIOYIO (CHJlY T.HmeCTH TeJla) MomHO HaRTH no
npaBHJlY CJlOmeHH.H napaJlnenbHbIX CHJI.
IJ;eHTp THeCTH 0 - TaKa.H TOqKa npHnOmeHH.H
paBHo.v;eiicTBYIOeii CHJI T.HmeCTH, .v;eHCTBYIOHX Ha
Bce qaCTH TeJla, KOTopaH He H3MeH.HeT CBoero nOJlo-
meHH.H npH JlI06bIX nOBopOTax TeJla.
OTHOCHTeJlbHO neHTpa T.HmeCTH anre6paHqeCKa.H
CYMMa MOMeHToB CHn T.HmeCTH Bcex qacTeii TeJla paB-
Ha HYJIIO (CM.  4.3). OnbITHbIM nYTeM nOJlOmeHHe
l.\eHTpa T.HmeCTH nJlOCKHX TeJl onpe.v;eJl.HIOT KaK TOq-
KY nepeCeqeHH.R. BepTHKaJIbHbIX JlHHHii, nonyqeHHbIX
npH no.v;BemHBaHHH TeJla B o.v;Hoii, a 3aTeM B .v;pyroii
TOqKax.
a
A 0 B
_ P -2
FI
R
Puc. 4.4
o
Puc. 4.5
6
FI
A
B
o
R
F 2
R = Fj + F2'
R=F 2 -F 1 ;
OB _ FI .
OA - F 2 '
R ii F 2 .
4.6. BH,lJ;LI paBHOBeCHSI
)J;JI.H Toro qT06bI Cy.v;HTb 0 nOBe.v;eHHH TeJIa B peaJlbHbIX YCJIOBH.HX, MaJlO
3HaTb, qTO OHO Haxo.v;HTCH B paBHOBeCHH. Ha.v;o ee OneHHTb 3TO paBHOBe-
CHe. Pa3J1Hqa1OT YCToiiqHBOe, HeycTOiiqHBOe H 6e3pa3J1HqHOe paBHOBeCHe.
85

PaBHOBeCHe TeJla Ha3bIBalOT yc-
TOHqHBbIM, eCnH npH OTKJlOHeHHH
OT Hero B03HHKalOT CHJlbI, B03Bpa-
alOHe TeJlO B nOJlOmeHHe paBHO-
BeCHSI (pHC. 4.6, a. nOJlOmeHHe 2).
B YCTOHqHBOM paBHOBeCHH eHTp
TSlmeCTH Tena 3aHHMaeT HaHHH3-
lIIee H3 Bce:x 6J1H3KHX nOJlOmeHHH.
IIOJlOmeHHe YCTOHqHBOrO paBHOBe-
CH.H CB.H3aHO C MHHHMYMOM nOTeH-
HaJlbHOH 3HeprHH no OTHOlIIeHHIO
KO BceM 6J1H3KHM COCeJI.HHM nono-
meHH.HM TeJla.
PaBHoBecHe TeJla Ha3bIBalOT He-
YCTOHqHBLIM, ecnH npH caMOM He-
3HaqHTeJlbHOM OTKJlOHeHHH OT Hero
paBHO)J.eHCTBYIOaSl )J.eHCTBYIOHX
Ha TeJlO CHJI BbI3bIBaeT )J.anbHeHlIIee
OTKJlOHeHHe TeJla 0'1' nOJlOmeHH.H
1 2 3 paBHOBeCH.H (pHC. 4.6, a, nOJlome-
HHe 1). B nOJlOmeHHH HeycTOHqH-
Boro paBHOBeCH.H BbICOTa I(eHTpa
T.HmeCTH MaKCHMaJlbHa ii nOTeHH-
aJlbHa.H 3HeprHSI MaKCHMaJlbHa no
OTHOlIIeHHIO K APyrHM 6nH3KHM
nonOmeHH.HM TeJla.
PaBHoBecHe. npH KOTOpOM CMe-
lqeHHe TeJla B nlO60M HanpaBJle-
HHH He BbI3bIBaeT ii3MeHeHH.H )J.eH-
1 2 3 CTBYIOHX Ha Hero CHJI H paBHO-
BeCHe TeJla COXpaH.HeTC.H, Ha3bIBa-
Puc. 4.6 10'1' 6e3pa3J1HqHLIM (pHC. 4.6, a,
nOJlOmeHHe 3).
Be3pa3J1HqHOe paBHOBeCHe CB.H3aHO C HeH3MeHHOH nOTeHI(HaJlbHOH 3HeprHeH
Bcex 6J1H3KHX COCTO.HHHH, H BbICOTa eHTpa T.HmeCTH O)J.HHaKOBa BO Bcex )J.OC-
TaTOqHO 6J1H3KH:X nonomeHHSlX.
TeJlo, HMelOee OCb BpaeHH.H (HanpHMep, O,AHOpO,AHa.H JlHHeHKa, KOTopaSl
MOme'I' BpalqaTbC.H BOKpyr OCH, npoXO)J;.HeH qepe3 TOqKY 0, H306pameHHa.H
Ha pHCYHKe 4.6, 6), HaXO,AHTCH B paBHOBeCHH, eCJlH BepTHKanbHaH npHMaSl,
npOXOJI..HmaSi qepe3 neHTp TSlmeCTH TeJla. npOXOJI.HT qepe3 OCb BpameHH.H. IIpH-
qeM eCJlH I(eHTp T.HmeCTH C BbIIIIe OCH BpaeHH.H (pHC. 4.6, 6; 1), TO npH
nlO60M OTKJlOHeHHH OT nOJlOmeHH.H paBHOBeCHH nOTeHI(HaJlbHa.H 3HeprH.H
YMeHblIIaeTC.H H MOMeHT CHJlbI T.HmeCTH OTHOCHTeJlbHO OCH 0 OTKJlOH.HeT Teno
)J.aJlbIIIe OT nOJlOmeHH.H paBHOBeCH.H. 3TO HeYCTOHqHBOe nOJlOmeHHe paBHoBe-
CH.H. ECJlH I(eHTp T.HmeCTH Haxo)J.HTcSI HHme OCH BpaeHHSl (pHC. 4.6, 6; 2),
TO paBHOBeCHe YCTOHqHBoe. ECJlH I(eHTp T.HmeCTH H OCb BpaeHH.H COBna)J.a-
lOT (pHC. 4.6, 6; 3), TO nOJlOmeHHe paBHOBeCH.H 6e3pa3nHqHoe.
a
6
86

TeJlo, HMelOIltee nJlOIltaAb OnOpbI. HaXOAHTC.H B paBHOBeCHH. eCJlI1 BepTH-
KanbHa.H np.HMa.H, npOXOA.HIIlaji qepe3 n;eHTp TjlmeCTH TeJla He BbIXOAHT 3a
npeAeJlbI nnoaAH onopbI 3Toro TeJla, T.e. 38 npeAenbI KOHTypa 06pa30BaHHO-
ro TOqKaMI1 conpHKOCHOBeHH.H Tena C onopOH PaBHOBeCl1e B 3TOM cJlyqae 3a-
BI1CI1T He TonbKO OT paCCTO.HHIi.H MemAY n;eHTpOM T.HmeCTIi Ii onopOH (T.e. OT
ero nOTeHn;l1anbHOH 3HeprHH B rpaBHTan;HOHHOM none 3eMJlH), HO 11 OT pac-
nonOmeHIi.H Ii pa3MepOB nnoa,A1i onopbI 3Toro TeJIa.
Ha pHCYHKe 4.6. B 1i306pameHo TeJlO. IiMelOee <t>OPMY n;IiJIIiH,Apa. ECJII1
ero HaKJIOHI1Tb Ha ManbIH yron, TO OHO B03BpaTIiTC.H B IiCXO,AHoe nonomeHlie 1
I1nli 2 ECJlH me ero OTKnOHHTb Ha yroJl p (nonomeHHe 3), TO Teno onpOKH-
HeTC.H. llpl1 3aAaHHOH Macce H nJlOaAH onopbI YCTOHqHBOCTb Tena TeM BbIllIe,
qeM HHme pacnonomeH ero n;eHTp T.HmeCTH, T.e. qeM MeHbllIe yrOJJ MemAY
npjlMOH, COeAHH.HIOeH n;eHTp T.HmeCTH TeJla 11 KpaHHlO1O TOqKY conpHKOCHO-
BeHI1.H nnoa,A11 onopbI C rOpl130HTaJlbHOH nJlOCKOCTblO.
BonpOCbI JJ;JUI CAMOKOHTPOJUl
1. Om Ilezo 3a6ucum pe3y.n.bmam oeucm6USl CUJlbl Ha Mamepua.n.bliYIO mOIlKY?
lia m6epooe me.n.o?
2. MOJIClio.n.u nepeliocumb cu.n.y napa.n..n.e.n.blio caMOU ce6e B .n.106ylO mOIl-
KY m6epoozo me.n.a?
3. KaK liaxooumcSl pa61i0oeucm6YIOU{aSl lieCKo.n.bKUX cu.n., npu.n.OJICeliliblX K
OOliOU mOIlKe? B KaKOM c.n.Yllae o6e CU.n.bl. npu.n.OJICelilible K pa3liblM mOIlKaM
me.n.a, .MOJICliO 3aMeliumb ux pa6Hooeucm6YIOU{eu?
4. roe liaxooumcSl mOIlKa npu.n.OJICeliUSl paBliooeucm6YIOU{eu 06YX napa.n.-
.n.e.n.bliblX U oouliaK060 lianpa6.n.eliliblx cu.n.? npomu60no.n.OJICli0 HanpaB.n.eliliblx
CUJl? f[ e.MY pa6libt .MooYJlU paBHooeucmBYIOU{eu 3mux cu.n.?
5. B KaKOM c.n.Yllae o6e napa.n..n.e.n.blible npomU60nOJlOJICliO lianpaB.n.elilible
CU.n.bt .MOJICliO 3a.MeliUmb OOliOU?
6. f[mo 3liailUm "me.n.o HaxooumcSl 6 pa6HoBecuu"?
7. KaKue 06UJfCeHUR .MOJICem c06epwamb me.n.o npu pa6li06eCUU cu.n.? f[mo
MOJfCliO CKU3amb 0 cKopocmu U yCKopeHuu 3mozo me.n.a?
8. M a.n.bIlUK. OBuzaSlCb pa6liO.MepliO. mSlHem 3a 6epeBKY caliKU. KaKa.R cu.n.a
ypaBiioBeWU6aem cu.n.y mpeliUSl caHOK 0 cliez?
9. llolleMY lie.n.b3Sl liamSlliymb Bepe6KY cmpozo ZopU30HmaJlbli0 2
10. f[mo .n.ezlle: yoepJICu6amb me.n.o lia liaK.n.OliliOU n.n.OCKocmu u.n.u mSlliymb
ezo paBliOMepliO no lieu?
11. KaKaSl cu.n.a .MOJICem 6bl36amb 6paU{eliue m6epoozo me.n.a 60KPYZ He
n006UJlCiiOU ocu?
12. n olle.MY U3.MeliSlemCSl pe3YJlbmam oeucmBuSl CU.n.bl Ha me.n.o. U.MelOU{ee
OCb 6paU{eIiUSl. ec.n.u ee nepeliocumb napa.n..n.e.n.blio ca.MOU ce6e lie BOO.n.b npR
.Mou. BOOJlb Komopou olia oeucm6yem?
13. nOl£e.MY OJlUlilibl.M. K.n.IOI£O.M zauKY omBepHymb .n.eZlle, lleM KOpOmKU.M?
14. ITp060.n.oKa, noo6eWelilia.R Ha Humu 3a cepeouliY. HaxooumcSl 6 pa6HO-
Becuu 6 ZopU30lima.n.bliOM no.n.OJICeliuu. Ocmaliemc.R .n.u olia 6 paBliOBecuu. eCJlU
OOUIi ee KOlieij cozliymb nono.n.a.M.?
15. rop liaxooumcSl 4elimp mSlJICecmu y K6aopama. coe.n.aHliozo U3 npoBo-
.n.OKU? y Ko.n.b4a? KaK onpeoe.n.umb no.n.OJICeliue ijelimpa m.RJlCecmu n.n.OCKUX
meJl lia onblme?
87

16. T{ eM om.n.U'laemc.R ycmOU'lU60e pa61i06eCUe om lieycmoullU60Z0? om 6e3-
pa3.n.U'lIiOZO?
17. llpu KaKOM YC.n.06UU me.n.o. UMelO14ee n.n.o14aob onOpbt. 6yoem liaxooumb-
CSl 8 pa8H.06ecuu?
18. T{ mo ycmoullu8ee: qu.n.uliiJp u.n.u KOIiYC. ec.n.u 8btCOma U n.n.014aiJb OCIiO-
6aliUSl y IiUX oouliaK06bt 2
R=ii+F2
OCHOBHbIE cI>OPMY JIbI
- paBHoAeillcTBYIOUJ,aSl ABYX C&Ml
R =  Fl2 + Fl + 2FJ.F2 cosa - MOAYJlb paBHoAecTBYIOUJ,e ABYX C&IIIl, rAe ex. - yroIl
Me)f(,[l,Y C&IIllaM&II PI &II F 2
R=Fl+F2
R=F2-FJ.
M = :t Fl
n n
L =0; LM j =0
;=1 '=1
F,! t R _
lL:1
A 0 B
Puc. 4.7
- MOAYllb paBHoAeillcTBYIOUJ,eill ABYX napallJlellbHblx OA&II-
HaKOBO HanpaBJleHHblX C&IIll P l &II P 2 (p&llc. 4.7),
(  = i. ).
- MOAYJlb paBHoAeillcTBYlOweill ABYX aHT&IInapallllellbHblx
CIIIJl (pIIIC. 4.8).
- MOMeHT C&IIllbl P. rAe 1 - nlleLJO C&IIllbl OTHOC&IITellbHO
OCIII, npoxoASlUJ,eill LJepe3 TOLJKY 0 (pIllC. 4.9).
- YCll0BIIISI paBHoBecIIISI TBepAoro TeJla, rAe it CIIIJlbl,
AeillcTBYIOUJ,lIIe Ha TellO, M, - MOMeHTbl 3TIIIX Clllll
Ft
A
o
R
B
F 2
Puc. 4.8
Puc. 4.9
METO)J;HqECKHE PEKOMEH)J;AIJ;HH
IIpH peweHHH 3aaq, OTHOC.HHXC.H K paBHoBecHIO TeJl, He06xoHMO:
1. CeJlaTb PHCYHOK, H306pa3HB Bce CHJlbI, eHcTBYIOHe Ha TeJlO, Haxo-
)J;.Heec.H B nOJlOmeHHH paBHOBeCH.H.
88

2. AJI.H Tena, He HMelOero OCH BpaeHH.H. .n;OCTaTOqHO HCnOJlb30BaTb nep-
n
Boe YCJlOBHe paBHOBeCHH: 2,  = O.
i=1
3. BbI6paTb OCH OX H Oy H 3anHcaTb 3TO ypaBHeHHe B npoeKHHX.
4. An.H TeJla C 3aKpenneHHoH OCblO BpaeHH.H ,AOCTaTOqHO HCnOJlb30BaTb
npaBHJlO MOMeHTOB. An.H 3Toro Ha.n;o HaHTH nneqH Bcex CHn OTHOCHTenbHO
.n;aHHoH 3aKpenJleHHOH OCH. COCTaBHTb aJlre6paHqeCKYlO CYMMY MOMeHTOB 3THX
n
CHn C yeToM 3HaKOB H npHpaBH.HTb ee K HYJlIO: 2, M i = O.
i=1
5. EcnH OCb BpaeHH.H He 3aKpenJleHa, Ha.n;o Hcnonb30B8Tb 06a ycnOBH.H
paBHOBeCHH. An.H 3anHCH npaBHna MOMeHTOB Heo6xo.n;HMO BbI6paTb OCb Bpa-
eHHH. JIyqme ee BbI6HpaTb TaK, 'IT06bI qepe3 Hee npOXO,AHJlO HaH60Jlbmee
qHCnO JlHHHH .n;eHCTBHH HeH3BeCTHbIX CHJI.
6. PemHTb nonyqeHHYro CHCTeMY ypaBHeHHH.
F
A
B
llPIIMEPbI PEllIEHII.H 3A,L(A q
F Y2
1. Y nHqHbIH cpoHapb MaccoH 20,0 Kr BHCHT Ha
,ABYX CTepmH.HX. npHKpenJleHHbIX K CTeHe H8 pac-
CTOHHHH 60.0 CM .n;pyr OT .n;pyra. ,L(JlHHa cTepmHeH:
AC = 90,0 CM, BC = 120 CM. Onpe.n;enHTe CHnbI, .n;ei1-
CTBYIOHe Ha CTepmHH B TOtIKe C no.n;Beca cpOHap.H.
LI:aHO: m=20.0 Kr, AB=O.600 M, AC=0,900 M,
BC = 1.20 M.
HaHTH: F I , F 2 , FYI' F y2 '
P e 1lI e H H e. Ha TOqKY C (pHC. 4.10) .n;eHcTByeT
CHna HaT.HmeHH.H HHTH F, paBHaH CHne T.HmeCTI1
no.n;BemeHHoro K HeH rpY3a: F = mg. Pa3nomHM 3TY
cHny Ha COCT8Bn.HIOHe FI H F 2 . Tenepb BH.n;HM,
qTO CHna ii paCT.HrHBaeT CTepmeHb AC, a CHna F 2 cmHMaeT CTepmeHb BC.
TpeyronbHHKH ACB H CFF I nO,A06HbI KaK TpeyroJlbHHKH C COOTBeTCTBeH-
HO napaJlJleJlbHbIMH CTopOHaMH. 113 HX no.n;06H.H cne.n;yeT:
AC Fl Fl AC
-=-=- ===> FI =mg-; FI =300 H;
AB F mg AB
Puc. 4.10
AB F mg BC
- - - - - => F 2 = mg -; F 2 = 400 H.
BC F 2 F 2 AB
B pe3Y nbTaTe .n;ecpopMaHH cTepmHeM B HHX B03HHKalOT CHJlbI ynpyrocTH
FYI H F y2 :
FyI =.Fi = 300 H; Fy2 = F 2 = 400 H.
PaBHo.n;eMcTBYIOa.H CHJI ynpyrocTH Fy = FYI + F y2 paBHa no MO.n;y JlIO F
H HanpaBJleHa npoTHBOnOJlOmHO eM.
89

2. BpYCOK M8CCOH 1,0 Kr HaXO,AHTC.H Ha HaKJlOHHOH nJlOCKOCTH C yrJloM
H8KJIOHa 30°. KaKYIO MHHHMaJIbHYIO ropH30HTaJlbHYIO CHJlY HymHo npHJlOmHTb
K 6PYCKY, qT06bI OH nOKOHnC.H? qT06bI OH ,ABHraJlC.H paBHOMepHO BBepx? Ko-
3<}Jcl>HHeHT TpeHH.H p8BeH 0,20.
A8HO: m=1.0 Kr. a=30°. Jl=0.20.
HaHTH: FI' F 2 .
PeweHHe. Ha nOKO.HeeC.H TeJlO (pHC. 4.11),
HaXO,A.Heec.H Ha HaKJlOHHOH nJlOCKOCTH, ,AeH-
CTBYIOT CHJla T.H.iKeCTH mg, CHJla HOpMaJlbHOH pe-
8KHH onopbl N p . CHJl8 TpeHH.H nOKO.H F TP' H8-
npaBJleHH8.H 8 CTOPOHY, npOTHBOnOJlOmHYIO B03-
MomHOMY BHmeHHIO TeJl8, H CHJla Fi.
Puc. 4.11 YCJlOBHe paBHOBeCHX: PI + PTp + N p + mg = O.
CnpoeHpyeM 3TH CHJlbI H8 OCH OX H Oy:
FI cosa. + FTp -mgsina. = 0; N p - mgcosa. - FI sin a. = 0_
OTCIO,Aa N p = mg cos a. + Fl sin a..
YqHTbIBa.H, qTO FTp = JlN p = J.l(mgcosa. + FI sin a.) . HMeeM
mg (sin ex - J.l cos a)
FI cos a + J.l (mg cos a + FI sin a) - mg sin a = 0 => FI = . ,
cosa + J.lslna
llO,ACT8BJI.H.H qHCJleHHbIe 3H8qeHH.H, nOJIyqHM Fl = 4,7 H.
T8K K8K npH pemeHHH 3a,AaqH ,AJI.H CHJlbI TpeHH.H nOKO.H MbI B3.HJlH ee
M8KCHM8J1bHOe 3HaqeHHe, TO HaH,AeHHOe 3HaqeHHe CHJIbI F I HBJI.HeTC.H ee MH-
HHMaJlbHbIM 3HaqeHHeM.
Ha TeJlO, ,ABHmYIQ;eeC.H p8BHOMepHO BBepx (pHC. 4.12). ,AeHcTBYIOT CHJla
TSimeCTH mg, CHJIa HOPMaJIbHOH peaKHH onopbI N p . CHJIB P 2 H CHna TpeHHSI
CKOJlb>KeHH.H P-rp, HanpaBJleHHa.H B CTOPOHY, npo-
THBOnOJlOmHYIO )J;BHmeHHIO TeJla. Y CJlOBHe paBHO-
BeCH.H:
Puc. 4.12
F 2 + FTp + N p + mg = O.
B npoeKHH H8 OCH OX H Oy:
F 2 cosa. - FTp - mg sin a. = 0;
- F 2 sin a. + N p - mg cos a. = O.
(J.lCosa + sina)mg
OTclO)J;a F 2 = . ; F 2 = 9,0 H.
cos a - J.lSln a
3. O,AHOPO,AHbIH MaccHBHbIH CTep.iKeHb C YKpenJIeHHbIMH Ha ero KOH8X rpy-
38MH 5,5 Kr H 1,0 Kr H8XO,AHTC.H B p8BHOBeCHH, eCJlH nO,AnepeTb ero Ha paCCTO-
1
.HHHH, paBHOM - ero )J;JlHHbI, OT 60Jlee T.HmeJlOro rpY3a. KaKOBa Macca CTepmH.H?
5
90

m,gi - m g (  -i)- m.+ - i) = O.
2
OTclO.n;a m = - (ml - 4m 2 ); m = 1 ,0 Kr.
3
4. no.n; KaKHM H8HMeHbWHM yrJloM K ro-
PH30HTY MomeT CTO.HTb npHCJlOHeHHaH K CTe-
He JleCTHHl.\a. ecnH H3BeCTHO, qTO K03cpcpHI.\H-
eHTbI TpeHH.H Memy JleCTHHUeH H nOJlOM H
Mem.n;y JleCTHHl.\eii H CTeHoii paBHbl J..l?
,l(aHO: J..l.
H8HTH: a.
P e ill e H H e. H8 JleCTHHl.\y (pHC. 4.14) .n;eiicTByeT CHJl8 THmeCTH mg, CHna
mIl
mg
Puc. 4.13
y
F'rpl
Puc. 4.14
l
,l(8HO: ml=5,5 Kr, m2=1,O Kr, ll=-'
5
HaiiTH: m.
P e ill e H H e. Ha CTepmeHb (pHC. 4.13) .n;eii-
cTByeT CHna T.HmeCTH mg, Bec rpY3a 1\, Bec
m 2 i rpY3a P2 H CHna HOpMaJlbHOH peaKI.\HH ono-
m 2
pbI N p . CTepmeHb HaXOlIHTC.H B p8BHOBeCHH.
n03TOMY CYMM8 MOMeHTOB Bcex .n;eiiCTBYIOHX
CHJI OTHOCHTeJlbHO OCH 0 6y.n;eT paBHa HYJlIO:
x
HOpMaJlbHOii peaKnHH CTeHbI N pI' CHJl8 TpeHH.H nOKO.H Mem)J;y CTeHoii H JleCT-
HHneii F Tpl' CHJla HOpMaJlbHOii peaKUHH nOJla N p2 H CHna TpeHH.H nOKO.H
Mem.n;y nonOM H JleCTHHneii F Tp2'
JIecTHHl.\a Haxo.n;HTC.H B nOKoe. Cne.n;OBaTeJlbHO,
FTpl + N p1 + FTp2 + N p2 + mg = O.
B npOeKI.\H.HX Ha OCH OX H Oy 3TO ypaBHeHHe 3anHWeTC.H:
N p1 -FTp2 = 0
F Tpl - mg + N p2 = 0

FTp2 = N p1
FTpl = mg - N p2 '
(1)
YqHTbIBa.H, qTO FTpl  J..lNpl H FTp2  J.lN p 2' nOJlyqHM
N p1  J.lN p 2; mg  (1 + Jl2)N p2 .
3anHweM npaBHJlO MOMeHTOB OTHOCHTeJlbHO OCH 0:
N p2l cos a - F TP2 l sin a - mg.!.. cos a = 0,
2
2N p2 -mg
r.n;e l - .n;JlHHa JleCTHHnbI. OTclO.n;a tg a = .
2F Tp2
(2)
91

1 - Jl2
tg amin =-.
2Jl
5. OnpenenHTe nOJlOmeHHe neHTpa T.HmeCTH o,n-
Hopo,lmoH KBa,npaTHoH nJl8CTHHhI co CTOpOHOH a B
KOTOpOH Bblpe3aHO KpyrJloe OTBepCTHe (,nHCK) pa,nHY-
a
COM "4 TaK, KaK YKa38HO H8 pHCYHKe 4.15.
a
.D:aHo: a, 4'
RaHTH: x.
[Jpe,nnOJlOmHM, qTO BbIpe3aHHhIH
Puc. 4.15
PeweHHe.
TIo,nCTaBHM BMeCTO FTp2 (1) H, yqHTbIBa.H Hepa-
BeHCTBO (2), nonyq1:lM
B
Kpyr nOMeeH Ha npemHee
MeCTO. Tor,n8 CHJla T.HmeCTH Bcero KBa,npaT8 mg = ffltg + m 2 g. r,ne m 1 g -
CHna T.HmeCTH ,nHCKa, npHJlOmeHHa.H B eHTpe 0 1 , "'-2g - CHJla T.HmeCTH nJla-
CT1:IHbI C OTBepCTHeM, npHJlOmeHHa.H B HCKOMOM eHTpe T.HmeCTH O 2 , OTHOC1:I-
TeJlbHO OCH, npoxo,n.H:r.qeH qepe3 061:Iii eHTp T.HmeCTH 0, CYMMa MOMeHTOB
CHn T.HmeCTH paBHa HYJlIO.
.D:JI.H Haxom,neH1:I.H nJleqa C1:IJlhI Tntg paCCMOTpHM TpeyroJlhHHI< OlAB. OH
np.HMoyroJlhHbIH (KacaTeJlbH8.H AB nepneH,nHKyn.HpH8 pa,nHYCY), H CTOpOHbI
a
°IA=AB= -.
4
TIo TeopeMe TIHcl>aropa 0 1 B = !!:..J2.
4
CJle,noBaTeJlbHo t nJleqo C1:IJlbI mlg OTHOC1:ITeJlbHO OC1:I 0 6y,neT
a.J2 a.J2 a.J2
---
2 4 4
aJ2
3anHllIeM npaBHJlO MOMeHTOB: m2gx - m1g - = O. r,ne x - p8CCTO.HHHe
4
TntaJ2
OT TOqKH 0 ,no TOqKH 02' OTCIO,na x - - - ITYCTb h - TOJl1:IHa nJlac-
- 4m2 .
THHK1:I, P
nJlOTHOCTb MaTepHaJla, H3 KOToporo OHa H3rOTOBJleHa.
( a ] 2 1tpha2
Tor,na ml = ph1t"4 = 16;
2 npha 2 2 16 -1t
m 2 =m-m 1 =pha -16=pha -16-'
1ta.J2
Tor,na HCKOMa.H BeJl1:lq1:lHa x = = 0,09a.
4(16-1t)
92

5. 3JIEMEHTbI CTATHKH H AIIHAMIIKH
jKHAKOCTE:R II rA30B
AaBneHMe. CMnbl AaBneHMSI. 3aKOH nacKanSi AnSI JKMAKOCTeM M ra30B.
npMHMn YCTpoMcTBa rMApaBnMecKoro npecca. rMAPoCTaTMeCKoe AaBneHMe.
Coo6alOMecSi COCYAbl. WnlO3bl. ATMoc<I>epHoe AaBneHMe. OnblT ToppMen-
nM. M3MeHeHMe aTMoc<I>epHoro AaBneHMSI C BbICOTOM. 5apoMeTp-aHepoMA.
ApxMMeAoBa CMna >KMAKOCTeM M ra30B. YcnoBMSI nnaBaHMSI Ten. BOAHblM
TpaHcnopT. B03AyxonnaBaHMe. 3aBMCMMOCTb AaBneHMSI >KMAKOCTM OT CKOpOC-
TM ee TeeHMSI. YpaBHeHMe 5epHynnM. nOA'beMHaSi CMna Kpblna caMoneTa.
rHpo-(a3po )CTaTHKa H3yqaeT YCJlOBH.H paBHOBeCH.H mHKOCTeH M r830B H
Ten, HaXO.HHXC.f.I B HHX.
B MeX8HHKe mHKOCTM H r83bI p8CCM8TpHB8IOTC.H K8K CnJlOllIHble, Henpe-
pbIBHO pacnpeneJleHHbIe B 38H.HTOH HMH q8CTH npOCTpaHCTBa. Bo MHorHX 381I8-
'lax cmHMaeMOCTblO mHKOCTH MomHO npeHe6peqb. B 3TOM cJlyqae nOJlb3YIOTC.H
nOH.HTHeM HecmHM8eMOH mHKOCTH - mHKOCTH, nJlOTHOCTb KOTOpOH BCIO.AY
OHHaKOBa M He H3MeH.HeTC.H co BpeMeHeM, T .e. aHHa.H Macca mHKOCTH HMe-
eT onpeeJleHHbIH 06'beM, 8 cpopMa MomeT 6bITb Jl106a.H (cpopMa COCY8).
IIpeHe6pera.H H3MeHeHH.HMH 06'beMOB mHKOCTH H r83a, He06xoHMO yqH-
TbiBaTb CHJlbI ynpyrocTH, )J;eHCTBYIOHe Memy coce)J;HHMH q8CTHI(aMH HJlH
CJlO.HMH 3THX BeeCTB. B OTJlHqHe OT TBep.AbIX TeJl, re CHJlbI ynpyrocTH B03-
HHKalOT npH H3MeHeHHH pa3MepOB H cpOpMbI TeJl. B mH,AKOCT.HX OHH B03HH-
K810T TOJIbKO npH paCT.HmeHHH HJlH cmaTHH, B ra3ax - TOJIbKO npH cm8-
THH. IIPH H3MeHeHHH CPOpMbI mHKOCTeH H r830B CHJlbI ynpyrocTH He B03HH-
K8IOT.
AHaJlOraMH MaTepHaJlbHbIX TOqeK B TBepbIX TeJlaX B rHpOCTaTHKe CJlY-
maT ,AOCTaTOqHO MaJlbIe no 06'beMY nOpI(HH ,)KHKOCTH (ra3a), BHYTpeHHeH
CTPYKTYPOH KOTOpbIX npeHe6peralOT.
5.1. )I;aBJIeHHe. CHJIa J(aBJIeHHH
KaK nOKa3bIBaeT onbIT, mH,AKOCTH eHcTBYIOT C HeKOTopbIMH CHJIaMH Ha JlIO-
6ylO nOBepXHOCTb TBeporo TeJla, rpaHHqaerO C HeH: Ha HO, Ha CTeHKH co-
CY,A8, B KOTOpOM HaXOHTC.H mHKOCTb, Ha nOBepXHOCTb TeJl8, nOMeeHHoro B
>KH,AKOCTb. co CTOpOHbI OHoro CJlO.f.l mHKOCTH H8 pyrOH. 8TH CHJlbI Ha3bIBa-
lOT CHJlaMH aBJleHH.H. OHH 06J1aaIOT p.H)J;OM oc06eHHocTeH: 1) no npHpoe 3TO
CHnbI ynpyrocTH cmaTblX mHKocTeH; 2) CHJlbI )J;aBJleHH.H 6J1aroap.H TeKyqecTH
Bcera nepneHHKYJl.HpHbI nOBepXHOCTH, Ha KOTOPYIO eHcTBYIOT; 3) CHJlbI ,AaB-
JIeHH.H pacnpeeJleHbI no BceH nOBepXHOCTH conpHKOCHOBeHH.H TBeporo TeJla C
>KHKOCTblO, n03TOMY CHJlbI aBJleHH.H 3aBHC.HT OT pa3MepOB 3TOH nOBepXHOCTH.
AJI.f.I Toro qT06bl oxapaKTepH30BaTb pacnpeeJleHHe CHJI aBJleHH.H BOJIb
nOBepXHOCTH conpHKOCHOBeHH.H, BBO.HT nOH.HTHe ,AaBJleHH.H.
J(aBJleHHe - cpH3HqeCKa.H BeJIHqHHa, H3Mep.HeMa.H OTHOllIeHHeM CHJlbI 8B-
neHH.H F. .AeHcTBYIOeH H8 nOBepXHOCTb, K nnO8.AH nOBepXHOCTH S:
F
p = S '
93

B CM e.n;HHHeH .n;aBJleHHH HBJI.HeTC.H naCKaJlb (lla).
] ITa - .n;aBJIeHHe, npOH3BO.n;HMOe CHJlOH B 1 H, paBHOMepHO pacnpe.n;e-
JleHHOH no nepneH)J;HKY JI.HpHOH K HeH nOBepXHOCTH nJlOI..IJ;a.n;blO 1 M 2 .
OnbIT nOKa3blB8eT, 'lTO .n;8BJleHHe B .n;8HHOM MeCTe He 3aBHCHT OT opHeH-
T8n;HH nJlO8.n;KH H BeJIHqHHbI ee nJIO8.n;H. OHO 3aBHCHT JlHWb OT CTeneHH
Cm8TH.H mH.n;KOCTH. jKH.n;KOCTb MomeT 6bITb cmaTa BCJle,ACTBHe Toro, qTO OH8
06J18.n;aeT BeCOM, HJlH nOTOMY, qTO Ha Hee .n;eHCTBYIOT BHeWHHe nOBepXHOCT-
HbIe CHJlbl.
3Ha.H .n;aBJleHHe B )J;aHHOM MeCTe, MomHO P8CCqHT8Tb CHJlY )J;aBJIeHH.H, ,AeH-
CTBYlOlltYIO H8 nOBepXHOCTb nJlO8.n;blO S: F = pS.
5.2. 3aKoH IIacKaJI.H AJI.H HKocTeH H ra30B
PaCCMOTpHM mH,AKOCTb, KOTOpa.H
Haxo.n;HTC.H B cocy.n;e no.n; nopWHeM
(pHC. 5.1») Kor.n;8 CHJlbI F, .n;eiicTBY-
IOHe Ha cB060.n;HYIO nOBepXHOCTb
:>KH.n;KOCTH, 3HaqHTeJlbHO 60JlbWe Beca
mH.n;KOCTH HJlH mH.n;KOCTb Haxo.n;HTC.H
B HeBeCOMOCTH, T. e. MomHO CqHT8Tb,
qTO Ha mH.n;KOCTb .n;eHcTBYIOT TOJIbKO
nOBepXHOCTHble CHJlbl, H BeCOM mH.n;'
KOCTH MomHO npeHe6peqb. BbI.n;eJlHM
MbICJleHHO K8KOH-TO M8J1bIH HJlHH,A'
pHqeCKHH npOH3BOJlbHO opHeHTHpo-
B8HHbIH 06'beM :>KH.n;KOCTH. Ha OCHOBa-
HHH SToro 06'beMa mH.n;KOCTH )J;eHcTBYIOT CHJlbI )J;aBJIeHH.H F I H F 2 OCTaJlbHOH
Puc. 5.1
mH.n;KOCTH, Ha 60KOBYIO nOBepxHocTb - CHJlbI )J;aBJleHH.H F 3 H F 4' Y CJlOBHe
paBHOBeCHH BbI.n;eJleHHOrO B :>KH.n;KOCTII MaJIOrO 06'beM8:
ii + F 2 + F3 + F4 = O.
B npoeKn;HH Ha OCb Ox:
FI - F 2 = 0 =:) PI S - P2S = 0 =:) PI = P2'
T.e. .n;8BJleHHe BO Bcex TOqK8X HeBeCOMOH HenO,ABH)KHOH mH.n;KOCTII O,AHH8KOBO.
I1pH H3MeHeHHH nOBepXHOCTHOH CHJlbI F 6y.n;YT H3MeH.HTbC.H BeJlHqHHbI PI II P2'
HO HX paBeHCTBO 6Y)J;eT COXpaH.HTbC.H. STO BnepBbIe YCTaHOBHJI B. ITaCKaJIb.
3aKOH fiaCKaJlH: JlCUOKOCmb (za3) nepeoaem npOU3600UMoe lia liee n06epx-
HocmliblMU cu.n.aMU 8lieWliee oa6.n.eliue no 6ceM lianpa8.n.eliU.R.M 6e3 U3Melie1iUSl.
)J;aBJleHHe, npOH3BO)J;HMoe H8 mH.n;KOCTb HJlH ra3, nepe.n;aeTC.H He TOJlbKO B
HanpaBJleHHH .n;eHCTBH.H CHJlbI, HO .Ii B Kam.n;ylO TOqKY mH.n;KOCTH (ra3a) 6J1a-
ro.n;ap.H nO,ABHmHOCTH MoneKYJI :>KH.n;KOCTH (ra3a).
5.3. rHpaBJIH'IeCKHH npecc
)J;eHCTBHe npecca OCHOBaHO Ha 38KOHe llaCKaJl.H. rH.n;paBJlHqeCKHH npecc
npe.n;CTaBJI.HeT c060ii .n;Ba C00681OHXC.H COCy.n;8, 3anOJIHeHHbIX mH.n;KOCTblO
94

F
(06bIqHO TeXHHqeCKHM MaCJlOM) H 3aKpbITbIX nop-
lIlHHMH pa3J1HqHOH nJlOa.n;H 8] H 8 2 (pHC. 5.2).
BHelIlH.H.H CHJla FI' ,ZJ;eHcTBYIOa.H Ha MaJlbIH nop-
F I
lIleHb. C031IaeT lIaBJleHHe PI = -. IIo 3aKoHY
8 1
IIaCKaJl.H OHO nepe,AaeTCH mH,ZJ;KOCTblO no BceM Ha-
npaBJleHH.HM 6e3 H3MeHeHH.H. IIOSTOMY co CTOpOHbI
mH,AKOCTH Ha BTOpOH noplIleHb ,ZJ;eHcTByeT CHJla
_ _ F 1 F 2 _ 52 .
F 2 -PI 8 2-- 8 2  ---,
8 1 }j 8.
FI h 2
hi
F 2 = 
F I 
rUOpa6.n.UlleCKUU npecc oaem 6btUZPbtW 6 cu.n.e
60 cmo.n.bKO paa, 60 CKO.n.bKO pa.J nJww,aOb 60Jlb'
wezo nopwH..R. npeBbtWaem n.n.ow,aOb .Ma.n.ozo nopWliSl.
CHJIa F 1 H3MeH.HeT TaKme nOTeHHaJlbHYIO 3HeprHIO mH,ZJ;KOCTH B npec-
ceo Ho TaK KaK CHJla T.HmeCTH STOH mHlIKOCTH HaMHoro MeHblIle CHJlbI F 1-
MbI C'lHTaJlH mH,ZJ;KOCTb HeBeCOMOH. B CB.H3H C 3THM He06xo,ZJ;HMO OTMeTHTb,
qTO B peaJIbHbIX YCJlOBH.HX ypaBHeHHe (5.1) BbInOJlHHeTC.H JlHlIlb npH6JIH-
meHHO.
BbIHrpbIlIla B pa60Te npecc He .n;aeT. ,l.(eHcTBHTeJlbHO, npH onYCKaHHH Ma-
(5.1)
Puc. 5.2
noro nOplIlH.H CHJla .F1 cOBeplIlaeT pa60TY Al = Fl , r,ZJ;e hI - XO,ZJ; MaJlOrO
nOplIlH.H. 'tfacTb mH,ZJ;KocTH H3 Y3Koro HJlHH,ZJ;pa BbITeCH.HeTC.H B lIlHPOKHH. H
6onblIloH nopIIIeHb nO,ZJ;HHMaeTC.H Ha h 2 . Pa60Ta CHJlbI F 2
Al FI hI
A2=F2-=-'-' (5.2)
A 2 F2
Ho mH,ZJ;KOCTb HecmHMaeMa. CJle,ZJ;oBaTeJlbHo. 06'beMbI mH.n;KocTeH, nepeme,A-
IIIHX 113 O,ZJ;Horo HJlHH,ZJ;pa B ,ZJ;pyroH, paBHbI, T.e.
  = 82  !!l = 8 2 .
 
IIo.n;CTaBHM STO ypaBHeHHe H ypaBHeHHe (5.1) B (5.2), nOJlyqHM Al =A 2 .
rH,ZJ;paBJlHqeCKHH npecc n03BOJl.HeT pa3BHBaTb KOJlOCCaJIbHbIe CHJlbI H HC-
nOJIb3yeTC.H ,ZJ;JI.H npecCOBaHH.H H3,ZJ;eJlHH (H3 MeTaJlJla. nJlaCTMaCcbI, H3 pa3J1H'tJ-
HbIX nopoIIIKOB), ,ZJ;JlH npo,ZJ;aBJIHBaHH.H oTBepCTHH B MeTaJIJIHqeCKHX JlHCTaX,
,ZJ;nH HcnbITaHH.H MaTepHaJlOB Ha npOqHOCTb, ,ZJ;JI.H nOlIH.HTH.H T.HmeCTeH, .ltJI.H BbI-
mHMaHH.R MaCJla H3 CeM.HH Ha MacJl060HHbIX 3aBo,ZJ;ax, ,ZJ;JI.H npecCOBaHH.H ClJa-
HepbI, KapToHa, ceHa. Ha MeTaJIJIyprHqeCKHX 3aBo.n;ax rH,ZJ;paBJIHqeCKHe npec-
CbI H3nOJlb3YlOT npH H3rOTOBJleHHH CTaJIbHbIX BaJIOB MaIIIHH, meJle3HO,ZJ;opom-
HbIX KOJleC H MHorHX ,ZJ;pyrHx H3.n;eJlHH.
5.4. rHpOCTaTHqeCKOe .n;aBJIeHHe
PaCCMOTpHM paBHOBeCHe O.n;HOPO.n;HOH mH,ZJ;KOCTH, Haxo.n;.HeHC.H B nOJle T.H-
rOTeHH.R 3eMnH.
95

Ha KamJJ:YIO qaCTHD;Y mHJJ:KOCTH, HaXO,n:.HeHC.H B nOJle T.HrOTeHH.H 3eMnH,
,n:eHcTByeT CHJla T.HmeCTH. ITOJJ: JJ:eHCTBHeM 3TOM CHnbI KaJKJJ:bIM cnOM mH,n:KO-
CTH onaBHT Ha pacnonomeHHbIe noon HHM cnOH. B pe3ynbTaTe JJ:aBneHHe BHYT-
pl1 JKHJJ:KOCTH Ha pa3HbIX ypOBH.HX He 6YJJ:eT OJJ:HHaKOBbIM. CJleJJ:OBaTeJlbHO, B
mHJJ:KOCT.HX cyecTByeT JJ:aBJleHHe, 06YCJlOBJleHHOe ee BeCOM.
,lIaBJleHHe. 06ycnoBneHHoe BeCOM JKHJJ:KOCTH, Ha3bIB81OT rH.IJ:pOCTaTHqeCKHM
AaBJIeHHeM.
)J.n.H KOnHqeCTBeHHoro paCqeTa MbIcneHHO BbI.qeJlHM B mM,AKOCTM ManbIH
06'beM D;HnHH,n:pMqeCKOH <}JOpMbI, p8CnOJlOmeHHbIH BepTHKMbHO, CeqeHHeM S H
BbICOTOH h (pHC. 5.3). B cnyqae He-
nO,ABMJKHOH JKM,AKOCTM Bec 3Toro
HnHHJJ:p8, a 3HaqMT, M CMJla JJ:aB-
neHHH Ha nnO8.AKY S B OCHOB8HHM
6YJJ:eT paBHa CHne THmecTH mg.
Tor,n:a ,n:aBneHHe Ha nnoaJJ:KY
Puc. 5.3
mg pVg phDg
p=-=-=-=pgh.
S S S
p = pgh
rH.IJ:pOCTaTeCKoe AaBJleHHe,
r,n:e p - nnOTHOCTb .)KMJJ:KOCTM. h - BbICOTa cTon6a .)KHJJ:KOCTM. T8KMM 06pa-
30M, rM,ApOCTaTHqeCKOe JJ:aBJleHMe paBHO Becy cTon6a *H,AKOCTH C eJJ:HHHqHblM
OCHOBaHMeM 11 BbICOTOH, paBHOH rJly6MHe norpymeHM.H TOqKH nOJJ: CB060,AHOH no-
BepXHOCTblO JKH,n:KOCTH.
rpa<}JHqeCKM 3aBHCMMOCTb ,AaBneHHH OT rJly6MHbI norpy.meHMH B *M,AKOCTb
npe,ACT8BneHa Ha pMCYHKe 5.4.
p
h
)J.aBneHHe *M,AKOCTM Ha ,AHO He
3aBHCMT OT <}JOpMbI cocY,Aa, a on-
pe,AenHeTCH TonbKO BbICOTOH ypOB-
HH JKMnKOCTH 11 ee nnOTHoCTblO.
Bo Bcex cJlyqa.HX, npHBeneHHbIX
Ha pHCYHKe 5.5, ,naBneHHe mMnKO-
CTM Ha nHO COCy,nOB onMHaKOBO.
JKM,AKOCTb JJ:aBMT H8 AaHHoH
rJly6HHe O,AHHaKOBO no BceM Ha-
npaBJleHM.HM - He TOJlbKO BHH3,
HO H BBepx, Ii B CTOpOHbI.
Cne,AOBaTeJlbHO, ,AaBneHMe Ha
CTeHKY Ha )J;aHHOH rJly6HHe 6YJJ:eT
T8KHM me, KaK H JJ:aBJleHHe Ha
ropH30HT8nbHYlO nnO8,AKY, pac.
nOJlomeHHYIO Ha TOH me rny6HHe.
ECJlH HaJJ: CB060,AHOH nOBepXHO-
CTblO mHJJ:KOCTH C03JJ:aeTC.H ,AaBJle-
HMe Po, TO ,AaBneHHt' B mH,AKOCTM
Ha rJly6MHe 6Y,AeT
p = Po + pgh.
Puc. 5.4
h
Puc. 5.5
96

06paTHTe BHHM8HHe Ha p83nHqHe BbIpameHHH: ",AaBJleHHe H.nKOCTH H8
rJly6HHe h" (p = pgh) H "aBneHHe B H.nKOCTH Ha I'ny6HHe h" (p = Po + pgh).
3TO Ha)J;O yqHTbIBaTb npH pemeHHH pa3J1HqHbIX 3aaq.
CHJlbI aBJleHHS Ha HO H Ha CTeHKH MomHO paCCqHTaTb no <popMynaM:
FJJ. = pghSJJ. - CHJla 8BJleHH.H mHKOCTH Ha ropH30HT8J1bHOe ,AHO, r,Ae
SA nJlOa,n;b Ha;
F = pgh S
CT 2 CT
- CHJl8 8BJleHH.H mH,AKOCTH H8 60KOBYlO npsMoyroJlbHYIO
BepTHKaJIbHYIO CTeHKY COCY8, re SCT - nJlOa,n;b CTeHKH.
B nOKO.HeHC.H »<HKOCTH cB060Ha.H nOBepXHOCTb »<HKOCTH BCer,A8 1'0pH-
30HTaJlbHa.
HepeKo BCTpeq810TCS CJlyqaH, Kora mHKOCTb, nOKOSCb OTHOCHTeJlbHO
cocY)J;a, BHmeTC.H BMeCTe C HHM. ECJlH npH 3TOM cocy BHmeTC.H paBHO-
MepHO H npSMOJlHHeHHO, TO cB060Ha.H nOBepXHOCTb mHKOCTH 6yeT ropH-
30HTaJlbHa. Ho eCJlH cocy BHmeTC.H C YCKopeHHeM, TO CHTY8U;H.R MeHSeTC.H
H B03HHKalOT BonpOCbI 0 <popMe CB060HOH nOBepXHOCTH mHKOCTH, 0 pac-
npeeJleHHH 8BJleHH.H B HeH.
TaK, B CJIyt{ae ropH30HTaJlbHOrO )J;BHmeHHS COCY8 C YCKopeHHeM ii B nOJle
T.HI'OTeHH.H 3eMJUf Jlr06aa qaCTb mOCTH MaCCOH m BHmeTCS C TeM me YCKO-
peHHeM ii no eHCTBHeM paBHOeHCTByroeH CHnbI aBJleHHS N JJ.' eHCTBYIO-
eH co CTOpOHbI OCT8J1bHOH »<HlIKOCTH H CUJlbI T.HmeCTH mg (puc. 5.6).
OCHOBHoe ypaBHeHue HH8MUKU:
a
.
NJJ. +mg = mii.
B pe3YJlbT8Te CB0601IH8S nOBepXHOCTb
mHKOCTH He 6yeT ropU30HTaJlbH8, a 06-
pa3yeT C ropH30HTOM YI'OJl a, KOTOpbIH
MomHO JlerKO H8HTH, ecnH cnpOelJ,UpOB8Tb
OCHOBHoe ypaBHeHHe HHaMHKH Ha ropH-
30HT8nbHYlO H BepTUKanbHYIO OCH:
N A sin a = ma; N A cosa = mg. OTCIO,A8
Nit
- mg - -
---
-
Puc. 5.6
a
tg a = -.
g
,l:(8BneHHe Ha I'OpH30HTanbHYlO nOBepx-
HOCTb (ropH30HT8J1bHOe HO) 6yeT B03pa-
CTaTb B Hanp8BneHHH, npoTHBOnOJlOmHOM
YCKopeHHIO.
5.5. Coo6alOHecJI COCYAbI
C006J.1..(alOJ.I..(HeCS COCYLI - 3TO COCY,AbI, coeHHeHHbIe Hume nOBepXHOCTH
)KHKOCTH, T8K qTO mHKOCTb MomeT nepeTeKaTb H3 OHoro COCY8 B PY-
I'OH. H3 YCJlOBH.H paBHOBeCH.H mH)J;KOCTH cneYeT 3aKOH C00681OHXC.H COCYOB:
97

oonopOOliaSl JlCUOKOCmb ycmaliaB.n.UBaemCSl 6 lien006U:J/CliblX coo6w,alOw,UXCR co
cyiJax maK. Ilmo iJaB.n.eliUe 60 6cex mOIlKaX. paCnOJlOJICeliliblX 6 OiJliOU ZOpU
30nmaJlbliOU n.n.OCKocmu. OOUliaK060.
T8K, B TOqK8X A Ii B (pIiC. 5.7)
P A = PB'
PIg + P2g = pggh g + P4 gh 4 3a-
KOB co06alOHXC.H COCYOB.
Ecnli B HenO,lJ.BHmHbIx C00681OHX-
C.H COCY,lJ.8X HaXO,lJ.HTC.H O,lJ.HOpO,lJ.HaH >KH,lJ.-
KOCTb (pHC. 5.8), TO ee CB060,lJ.H8.H no-
BepXHOCTb YCT8HaBJIIiB8eTC.H H8 OlIHOM
ypoBHe:
hI = h 2 .
EcnH HaJIHTbI ,lJ.Be HeCMeIllHBalOHe-
C.H O,lJ.HOpO,lJ.HbIe mH,lJ.KOCTH (pHC. 5.9), TO
BbICOTbI CTOJl60B mH,AKOCTeH H8,lJ. ypOB-
HeM pa3,lJ.eJl8 06paTHO nponopHOHaJlbHbI
nJlOTHOCTHM 3THX mH,lJ.KOCTeH:
hI P2
PIg = P2g =:) - = -
 PI
hi
P.;;
h.
h 2
PI
h3
P- 
- -- --- .-
A
--.=- B- -
Puc. 5.7
hi
Puc. 5.8
hi
././././././
././././././
./ ./ . Pi.... ./ ./
././././././ h 2
././././././
h 2
-=- - -_P 2..=. -- --=-=-
Puc. 5.9
IIpHMepbI co06alOHXC.H COCY,lJ.OB: IllJl 103 bI H8 KaHaJlaX H peK8X, BOlIOH8-
nopHbIe 6allIHH H BO,lJ.OHanOpHbIe ceTH, BO,lJ.OMepHbIe CTeKJI.HHHbIe Tpy6KH Ha
napOBbIX KOTJIaX, Q>OHTaHbI, qaHHHKH Ii ,lJ.p.
5.6. lliJi 103 bI
llIJII03bI peK H KaHaJIOB p86oTalOT no npliHIJ;liny coo6alOHXC.H COCY,lJ.OB.
B CMemHbIX IllJl1030BbIX K8Mepax, OT,AeJIeHHbIX ,lJ.pyr OT ,lJ.pyra IllJII030BbI-
Mli BopOTaMIi. BO,lJ.8 CTOIiT H8 p83HOM ypOBHe (pIiC. 5.10). IIo,lJ. BOpOTaMI1 npo-
XO,lJ.IiT nO,ABO,lJ.HbIH KaHaJI, COe,lJ.IiHHIOHH 06e KaMepbI; ero MomHO 3aKpbIBaTb
Ii OTKpbIBaTb.
IIpH OTKpbIBaHli1i nO,lJ.BO,lJ.HOrO K8HaJI8 06e K8MepbI npeBpa:r.q8IOTC.H B co06-
w;alOW;HeC.H COCY,lJ.bI, Ii BO,lJ.8 B HIiX YCTaH8BJIHB8eTCH Ha O,lJ.HOM ypOBHe. TOr,A8
98

! I 
- - --..- -... . - - ,.
-.--.- . . .--....
--- _. -_.
-- -- -.
-- -..--.- - ..--.-. -
 -.
- -
II
m
. __  T\

---::-.
'"
Puc. 5.10
MomHO OTKpbITb llIJlI030BbIe BopoTa H nepeBeCTIi CYHO H3 OHOH KaMepbI B
pyrylO .
B cJlyqae 60JlbllIOll pa3HHbI B ypOBH.HX BooeMoB YCTpaHBalOT eJIbIll pSl
llIJlI030BbIX KaMep, pa6oTalOHx oHa 3a Pyroll nOCJleOBaTeJlbHo.
5.7. ATMocepHoe ):.(aBJIeHue. OnblT TOppHlIeJIJIH.
H3MeHeHHe a TMocepHoro ):.(aBJIeHHH C BblCOTOH
3eMJI.H oKpymeHa aTMoc«l>epoll - B03n;YllIHoH 060JIOqKOR, COCTO.HeM H3
CMeCH pa3J1HqHbIX ra30B.
ATMoc«l>epHoe aBJleHHe 06ycJloBJleHo aBJleHHeM BepXHHX CJIOeB B03yxa
Ha HHmeJlemaHe CJlOIi BCJleCTBHe npHTSlmeHH.H .K 3eMJIe.
ATMoc«l>epa He HMeeT qeTKOR rpaHHbI, nJlOTHOCTb ee H3MeHSIeTC.H C BbICO-
TOll. ll03TOMY paCCqHTaTb BeJlHqHHY aTMoc«l>epHoro aBJIeHHSI no cpopMYJle JISI
BbIqHCJleHH.H aBJleHH.H cToJl6a mH,AKOCTH HeJlb3.H.
A TMoccpepHoe ,AaBJleHHe onbITHbIM
nYTeM BnepBble H3MepHJI 8. TOppH-
qeJlnH B 1643 r.
3ana.HHHYIO C O,AHOrO KOHlt.a CTeK-
n.HHHYIO Tpy6KY JIHHOM OKOJlO 1 M
HanOJlH.HJIH PTYTblO H, 3aKpbIB OT-
BepCTHe pyroro KOHa, nepeBOpaqH-
BaJIH H norpymaJlH B COCY)J; C PTYTblO
(pHC. 5.11).
3aTeM OTBepCTHe OTKpbIBaJlH.
qaCTb PTYTH BbIJlHBaJIaCb, a B Tpy6Ke
OCTaBaJlC.H CTOJl6 PTYTH onpeeneHHoM
BbICOTbI h. rHpOCTaTHqeCKOe aBJle-
HHe KOToporo ypaBHoBellIHBaeTC.H aT-
Moc«l>epHbIM aBJleHHeM. 1I3MepHB TaKHM 06pa30M BbICOTY cToJl6a PTYTH. Mom-
HO paCCqHTaTb aTMoc«l>epHoe aBJIeHHe:
Pa = pgh.
h
=
Puc. 5.11
re p - nJlOTHOCTb PTYTH.
Ha npaKTHKe aTMoc«l>epHoe aBJleHHe H3Mep.HIOT B MHJlJlHMeTpax pTYTHoro
cTon6a.
99

COrJlaCHO aTOll cpopMYJle
1 MM pT. CT. = 13,6 . 10 3 Kr . 9 81 !::.. 10- 3 M = 133 ITa.
M 3 ' c 2
ATMoccpepHoe ,naBneHHe Po, ypaBHoBemHBalOee npH ° °C ,naBJleHHe CTOJl-
6a PTYTH BbICOTOll 760 MM, CqHTaeTC.H HOpMaJlLHLIM H Ha3bIBaeTC.H 4>H3HlIe-
CKOH aTMoc4>epoH:
Po= 760 MM pT. CT.;
1 aTM = 760 MM pT. CT. = 1,013. 10 5 ITa.
p,aTM
Po
0,8
0,6
0,4
0,2
°
8 16 24 h,KM
Puc. 5.12
ATMoccpepHoe ,naBJleHHe YMeHbIIIaeT-
C.H c YBeJlHqeHHeM BbICOTbI no,n'beMa
H8,A 3eMneH.
3TO 06'b.HCH.HeTC.H TeM, qTO C YBe-
JlHqeHHeM BbICOTbI TonHHa cmHMalO-
ero CJlO.H aTMoccpepbI YMeHbIIIaeTC.H.
O,nHaKO B OTJlHqHe OT ,naBJleHH.H B
mH,nKoCT.HX H3MeHeHHe aTMoc<}JepHo-
ro ,AaBJleHH.H He nponopHOHaJlbHO h,
a npOHcxo,nHT rOpa3,A0 6bICTpee
(pHC. 5.12), qTO 06'b.HCH.HeTC.H YMeHb-
IIIeHHeM nJlOTHOCTH aTMoccpepbI C YBe-
nHqeHHeM BbICOTbI.
5.8. BapoMeTpbl. MaHoMeTpbl
BapOMeTpLI - npH60pbI, CJlymaHe ,nJl.H H3MepeHH.H aTMoc<}JepHoro .AaB-
JleHH.H. PTYTHbIH 6apoMeTp (pHC. 5.13) COCTOHT H3 3anOJlHeHHOll PTYTblO
U-06pa3HoH CTeKn.HHHOH TPy6KH, O,nHH
KOHe KOTOpOll 3ana.HH, a Ha ,nPyroM
HaXO.AHTC.H OTKpbITbIH pe3epByap C PTY-
TblO. BapoMeTp HMeeT IIIKaJlY C MHnJlH-
MeTpOBbIMH ,AeJleHH.HMH, no KOTOpOH He.
nocpe,nCTBeHHO H3Mep.HIOT aTMoccpepHoe
,naBJleHHe B MHJlJlHMeTpax pTYTHoro
cTon6a. OHO qHCneHHO paBHO BbICOTe
cTon6a PTYTH Mem,ny ee YPOBH.HMH B
3aKpbITOM H OTKpbITOM KOJleHaX 6apo-
MeTpa.
ITpeHMyw;ecTBo TaKHX 6apOMeTpOB -
60nbIlla.H TOqHOCTb nOKa3aHHH. He,AOCTaT-
KH - OHH rpOM03,AKH, xpynKH, napbI
PTYTH Bpe,nHbI ,nJlSi 3,nOpOBb.H JlIO,neH.
MeTaJlJlHqeCKHH 6apOMeTp - aHepo-
H;:J; (pHC. 5.14) COCTOHT H3 I1,HJlHH,npHqe-
CKOH KaMepbI K, H3 KOTOpOH OTKaqaH
B03,nyX. KaMepa repMeTHqHO 3aKpbITa
TOHKOH rocppHpoBaHHoH KpbIIIIKoH-MeM6-
paHoH M.
lliKa.n:a
h
Puc. 5.13
100

't.JT06bI aTMOCcpepHOe .naBJIeHHe He CnJIIOHJIO MeM-
6paHY, OHa coe.nHHeHa nocpe.nCTBOM T.HrM T C npy-
mHHOH ll, 3aKpenJleHHOH Ha Kopnyce npH60pa.
K npymMHe npHKpenJleHa CTpeJIKa C, KOHeu. KOTOpOH
nepeMeaeTC.f.I no IIIKaJle lll. npH H3MeHeHHH aTMOC-
cpepHoro .a;aBJIeHH.H MeM6paHa nporM6aeTC.H BHYTPb
MnH Hapymy M nepeMeaeT CTpeJlKY no IIIKaJIe.
TIpeMMyecTBa aHepOH.nOB B TOM, qTO OHH y.n06HbI
B pa6OTe, npOqHbI, Ma.TIora6apMTHbI. OCHOBHOH He.nOCTa-
TOK - OHH MeHee TOqHbI, qeM pTYTHbIe 6apOMeTpbI.
,l(JI.H H3MepeHH.H .naBneHH.H, 60JIbIIIerO HJlH MeHb-
IIIero aTMoccpepHoro, HCnOJIb3YlOT MaHOMeTpbI. MaHO-
MeTpbI 6bIBalOT mH.a;KoCTHbIe H MeTaJlJIH't.JeCKHe.
mH,ZJ;KOCTHbIH MaHOMeTp ,AeJlalOT B BM,Ae U -06pa3-
HOH Tpy6KM C mM,AKOCTblO (06bIqHO BO,AOH HJlM PTY-
TbIO), O.nHO KOJleHO KOTOpOH npMcoe.a;HH.HeTC.H K cocy.a;y, .a;aBJIeHHe B KOTOpOM
HymHo M3MepMTb (pMC. 5.15, a). YpoBeHb mM.nKOCTH B aTOM KOJleHe nOHM-
3HTC.H (eCJlM .a;aBJleHMe B cocy.a;e 60JlbIIIe aTMoccpepHoro) HJIM nOBbICHTC.H (eCJIM
OHO MeHbIIIe aTMoCqJepHoro) no cpaBHeHMIO C ypOBHeM mM,AKOCTM BO BTOpOM
KoneHe. 113Mep.HeMoe .naBJleHHe 6Y,AeT P = Pa :t pgh, r,Ae Pa - aTMocqJepHoe
,AaBJleHHe, pgh - rH.a;pOCTaTMqeCKOe ,naBJleHHe M36bITOqHOrO cToJl6a mHlIKO-
CTM B KOJleHe MaHOMeTpa.
K
Puc. 5.14
a
Puc. 5.15
8
,l(JI.H H3MepeHM.H TaKMM MaHoMeTpoM )J;aBJleHH.H BHYTPM mH.a;KOCTM K
O.nHOMY H3 ero KOJleH npMcoe.a;MH.HIOT C nOMOblO pe3HHoBoH Tpy6KH nJloc-
KYIO Kop060qKY, 0.a;H8 cTopoHa KOTOpOH 3aT.HHYTa pe3MHoBoH nJleHKOH
(pMC. 5.15, 6).
TIpocTeHIIIMH MeTaJlJlHqeCKMH M8HOMeTp YCTpoeH CJIe.a;YIOHM 06pa30M
(pHC. 5.15, 8). TOHKa.H ynpyra.H nJIaCTMHKa M - MeM6paH8 - repMeTMqeCKM
3aKpbIBaeT Kop06KY K, H3 KOTOpOH qaCTHqHO OTKaqaH B03.nyX. K MeM6paHe
npMcoenMHeH YKa3aTeJIb P, BpaaIOIl1HHC.H OKOJlO OCH O. TIpM norpymeHMM
npM60pa B mMnKOCTb MeM6paH8 nporM6aeTC.H non neHCTBMeM CMJI .naBJleHM.H.
Ii ee nporH6 nepe.a;aeTC.H YKa3aTeJlIO, nepe)J;BHralOeMYC.H no IIIKaJle.
101

5.9. ApXHMe,lJ;OBa CHJIa ,lJ;JIH mH,ZJ;KOCTeH H ra30B
CyruecTBoBaHHe rH,ApOCTaTHqeCKOrO ASBneHHH
npHBOHT K TOMY, qTO HS JlI060e Teno, HSXOlI.H-
eec.H B mH,AKOCTH HnH ra3e, AellcTByeT BblTan-
KHBalOa.H CHna. .D;JI.H ee paCqeTa pSCCMOTpHM
Teno Ky6HqeCKOH cpOPMbl, HaXOAHeeC.H B mHA-
KOCTH (pHC. 5.16).
Ha Bce rpaHH Tena, norpymeHHoro B mHA-
KOCTb, AellcTBYIOT CHnbI AaBneHHH.
PSBHoAellcTBYIOa.H 3THX CHJI aBneHH.H
BbITaJlKHBalOaH CHJIS:
FA = Fl + F 2 + F3 + F4 + F5 + F6'
CHJlbI ASBJleHH.H Ha 60KOBbIe (i Ha nepe,AHIOIO H 3SAHIOIO rpaHH ypaBHoBe-
Puc. 5.16
lllMBaIOTCH: F3 + F4 = 0, F5 + F6 = O.
Ha BepxHlO1O rpaHb ,AellcTByeT CHna
Fl = pghr S.
Ha HMmHlO1O rpaHb - CHJla
F 2 = pgh 2 S.
PaBHoAellcTBYIOa.H 3THX CHn HanpaBJIeHa BepTMKaJlbHO BBepx, npHJIOme-
HS K :u,eHTpy Macc BblTecHeHHoH TeJlOM mHAKOCTM - :u,eHTpy AaBJIeHMH 0
(pHC. 5.17) M npeACTaBJI.HeT C060ll BbITanKHBSIOYIO CHJIY FA:
FA = F 2 - Fl = P)f(g (h2 - hr)S = p)f(ghS = P)f(gV,
rAe V - 06'beM norpymeHHoH B mHAKOCTb 'lSC-
TH TeJla.
-- 0 - - -
- -
-- -
- -
--- - - - - -
-=..0 ___-
-- -
- -
Puc. 5.1 7
FA = P)f(gV - 3aKOH ApxHMe.n:a: lia me.n.o,
nozpYJlCeliliOe 6 JlCUOKOCmb (Za3). oeucmeyem 6bl-
ma.n.KUealOaSl cu.n.a, paeliaSl eecy JlCUa"ocmu
(Za3a) 6 06oe.Me nozpYJICeliliou Ilacmu me.n.a. lia-
npa6.n.eliliaSl eepmUKa.n.bliO 66epX U npUJlOJICeliliaR
6 qelimpe aae.n.eIi USl.
5.10. YCJIOBHH nJIaBaHHH TeJI
Ha TeJlO, HaXOA.HeeCH B mHAKOCTM (rs3e), AellcTBYIOT ABe npOTHBononom-
HO HanpaBJleHHbIe CHJlbI: CHJIa THmeCTM mg H spXMMeAOBa CHna FA' rIoBe,Ae-
HHe TeJIa 3aBHCMT OT COOTHOllleHH.H 3TMX CHJI. B03MomHbI TPH cJlyqS.H:
1) FA < mg (pHC. 5.18). PSBHoAellcTBYIOa.H 3TMX CHJI HSnpaBJleHa BHM3,
B pe3YJlbTaTe TeJIO TOHeT. EcnH TeJlO CnJlOlllHOe, TO 3TO 6YAeT npH YCJlOBHH,
KorAa nJlOTHOCTb TeJla 60Jlbllle nnOTHOCTH mHAKOCTH:
p)f(gV < PTg V ==> p)f( « PT)'
102

2) FA = mg (pliC. 5.19). TeJlo nJlaBaeT BHYTPH >KH,nKOCTIf H8 JlI060H rJly6H-
He (6e3pa3J1HqHOe paBHOBeCHe): Pm = (PT)'
3) mg < FA' P8.BHo,neHCTByroa.H 3TliX CliJl Hanp8.BJle-
Ha BBepx. H TeJlO H8qHH8eT BCnJlbIB8Tb. lIOK8 TeJlO He
.nOCTHrHeT CB060lIHOH nOBepXHOCTH mH,AKOCTli. BbITaJIKH-
Baroa.H CHJl8 He 6y,neT H3MeH.HTbC.H. lIpH ,naJIbHeHIlleM
no,n'beMe BbITaJIKHBaroa.H CHJl8 HaqHeT YMeHbIllaTbC.H H,
Kor)J;a OHa CT8.HeT paBHOH no MO,nYJlIO CHJle T.HmeCTH,
TeJlO OCT8HOBHTC.H 11 6y,neT nJlaBaTb Ha nOBepXHOCTH.
BCllJIbIBIllee TMO 6y,neT, T8KliM 06pa30M, qaCTHtJHO BbI-
CTYllaTb HB,D; nOBepXHOCTblO mH,ZJ;KOCTH (pHC. 5.20).
06'beM norpymeHHoH q8CTli V n TaKOB, qTO Bec BbI-
TeCHeHHOH mli,AKOCTH paBeH CHJle T.HmeCTH nJlaBalOe-
ro TeJla: FAI = mg. ECJlH cpe,nH.H.H nJlOTHOCTb TeJl8
MeHbIlle nJlOTHOCTH mH)J;KOCTH (pJK > (PT }). TO pJK I = (PT) V - YCJlOBHe nJl8-
B8HH.H CnJlOIllHOrO TeJla H8 nOBepXHOCTH >KH,nKOCTli.
1l0CKOJlbKY BbIT8J1KliB81Oa.H CliJla FA = pgV 3aBHCHT OT BHeIllHero 06'beM8
Ten8 H He 38BliCHT OT BemeCTB8 TeJIa Hero pacnpe,neJIeHli.H B TeJle (HaJIliqli.H
nonOCTeH, Heo,nHopo.nHOCTli nJlOTHOCTH H T..n.). TOqK8 npHJIOmeHH.H BbIT8J1KliB8-
lOeH CliJIbI FA He 06.H3aTeJlbHO ,nOJlmH8 COBnaTb C eHTpOM T.HmeCTH TeJla
(pliC. 5.21, a). TaK KaK B mli,AKOCTH HeT TpeHH.H nOKO.H, MOMeHTbI CHn mg H FA
Puc. 5.18
F Al
mg-_
Puc. 5.20
a
F A -
 - m -FA- -
- -
- -
--- -=-- 
- -
--= mg-_-
mg_ --=
Puc. 5.19
6
--
F -- -=-
A _-
r- -F --
_  A ___  
_ c _-
- -
- -
.... - -..--
=- =- mg  --
:- - --
mg -=-
Puc. 5.21
103

3aCTaB.HT TenO nOBepHYTbC.H TaK. qTO 06e YKa3aHHbIe CHnbI 6y,nYT pacnoJlome-
HbI B.nOJlb Ol1HOH BepTHKaJlH (pHC. 5.21, 6). AaJlee B 3aBHCHMOCTH OT COOTHO-
llIeHH.H Mo,nYJleH FA H mg TeJlO 6y,neT HJlH HaxO,nHTbC.H B paBHoBecHH, Hnli
BcnnbIBaTb, HnH onycKaTbC.H BHH3.
5.11. BOAHLIH TpaHcnopT
B03MomHOCTb nJlaBaHH.H Kopa6neH, napoxo,nOB H .lIp. OCHOBaHa Ha 3aKOHe
ApxHMe,na. TeJlo, HMelOlQee nOJlOCTH, Ky,na mH,nKOCTb He npOHHKaeT npH nJla-
BaHHH, BbITeCH.HeT TaKOH me 06'beM mli,nKOCTH.
KaK H cnnOllIHoe TeJlO. IT03TOMY H BbITanKH-
BalOa.H CHJla )J;JI.H TaKoro TeJla Ta me. qTO H
,AJlSI CnJlOllIHoro. Ho Macca Tena C nOJlOCT.HMH
MeHblIIe. BbITeCHeHHbIH 06'beM mli,AKOCTIi OKa-
3bIBaeTC.H 60nbllIe 06'beM8. 3aH.HTOrO BemeCTBOM
Tena nJlOTHOCTblO PT (pHC. 5.22). IT03TOMY OHO
MomeT nJlaBaTb .name B TOM cJlyqae, eCJlH
g - - PT > Pm' TaK, CTaJlbHOH Kopa6nb BbITeCH.HeT
-==- - m - -= OO'beM BO,nbI BO MHOrO pa3 60JlbllIHH, qeM
06'beM CTaJIH, H3 KOTOpOrO C,neJlaH Kopnyc cy,n-
Ha. Bec BO,nbI, BbITeCH.HeMOH nO,nBO,nHOH qaC-
TblO, paBeH CHJle T.HmeCTH, )J;eHCTBYIOeH Ha Kopa6J1b C rpY30M.
rJly6HHY, Ha KOTOPYIO Kopa6J1b HJlH ,Apyroe Cy,nHO norpymaeTC.H B BO,ny,
Ha3bIBalOT OC8,lJ:KOH.
HaH60JlbllIa.H ,nOnYCKaeMa.H OCa,nKa OTMeqeH8 Ha Kopnyce KpaCHOH nHHH-
eH, Ha3bIBaeMOH BaTepJlHHHeH. Bec BO.lIbI, BbITeCH.HeMOH CY,AHOM npH norpy-
meHHH 110 BaTepnHHIiH. Ha3bIBaeTC.H BOJl.OH3MemeHHeM CYJl.Ha.
Pa3HOCTb Mem,ny BO,nOH3MeeHHeM cy,nHa H BeCOM CaMOrO CY,AHa onpe,ne-
JI.HeT rpY30nO,lJ:'LeMHOCTL 3TOrO cy,nHa.
AJlSi Kopa6neH H nO,nBO,nHbIX nO,nOK BameH Bonpoc 06 YCTOHqHBOCTH HX
paBHOBeCH.H npH nJlaBaHHH, KOTOpa.H 3aBHCHT OT COOTHOllIeHIi.R CHJlbI T.HmeCTH
H apxHMe,nOBOH CHJlbI, a TaKme OT B3aHMHoro pacnonOmeHH.H n.eHTpa ,naBJle-
HH.H H n.eHTpa T.HmeCTH. AJI.H no,nBo,nHbIX JlO,nOK YCJlOBHeM YCToH"lIHBoro paB-
HOBeCH.H .HBJI.HeTC.H pacnonomeHlie n.eHTpa T.HmeCTH Hlime n.eHTpa ,AaBneHH.H.
ITpli HaKJlOHeHHIi Kopa6neH 1-13MeH.HeTC.H «popMa BbITeCH.HeMOrO 06'beMa, a
cne,nOBaTeJlbHO, H nOJlOmeHHe n.eHTpa ,naBJleHH.H OTHOCHTeJIbHO Kopa6J1.H. PaB-
HOBeCHe Kopa6J1.H 6y,neT YCTOHqHBbIM, ecnH TOqKa nepeCeqeHH.H BepTHKaJlH.
npoBe,neHHoH qepe3 n.eHTp ,naBneHH.H, co cpe,nHeH JlHHHeH cy,nHa (TaK Ha3bIBa-
eMbIH MeTan.eHTp) JIemHT BbIllIe n.eHTpa T.HmeCTH. ITpli 3TOM YCTOHqHBOCTb KO-
pa6JIeH B03paCTaeT npH YBeJlHqeHHH paCCTO.HHH.H Mem,ny n.eHTpOM T.HmeCTH H
MeTan.eHTpOM. [J03TOMY n.eHTp T.HmeCTH Kopa6n.H ,nonmeH 6bITb KaK MomHO
HHme: T.HmeJlbIe rpY3bI y6HpalOT C nany6bI. nopomHHe Kopa6J11i 3arpymalOT
6annaCTOM.
An.H 06eCneqeHH.H 6e30nacHocTH MopennaBaHH.H Bce BHYTpeHHee npOCTpaH-
CTBO Kopa6J1.H pa3,neJl.HIOT Ha Bo,nOHenpOHHn.aeMbIe OTceKH. B cJlyqae np06oH-
HbI BO.lIO:H 3anOJlH.HeTC.H TOJlbKO O,nHH OTceK, Cy,nHO npo,nonmaeT nJlaBaTh, XOT.H
H norpymaeTC.H rJly6me B BO,ny.
FA
Puc. 5.22
104

Ha 3aKOHe ApXHMe.n;a OCHOBaHO B03)J;yxOnJla-
BaHHe: nOJleT .n;HpHma6J1eH, aSpOCTaTOB, lIIapOB-
30H.n;OB. Mx nOJleT HanOMHHaeT nJlaBaHHe nO,nBO.n;-
HOH nO.n;KH no.n; BO,AOH.
ECJlH CHJla T.HmeCTH mg JleTaTeJlbHOrO anna-
paTa (pHC. 5.23) BMeCTe C CHJlOH T.H:1KeCTH ra3a,
3anOJlH.HIOerO 060JlOqKY. MeHblIle Beca B03,nyxa
B 06'beMe, BbITeCH.HeMOM annapaTOM, TO lIIap no.n;-
HHMaeTC.H BBepx; ecnH OHH paBHbI, TO lIIap He-
nO)J;BHmHO BHCHT B B03)J;yxe; eCJlH CHJla T.HmeC-
TH mg 60JlblIle Beca B03)J;yxa, lIIap onycKaeTC.H.
¥ CJlOBHe nJlaBaHH.H B03,nYIIIHoro lIIapa
FA  mg MomeT 6bITb BbInOJlHeHO. eCJlH nJlOT-
HOCTb ra3a, HanOJlH.HIOerO lIIap, MeHblIle nJlOT-
HOCTH B03)J;yxa. An.H SToro HcnoJlb3yeTC.H BO.n;O-
pO)J;, reJlHH HJlH HarpeTbIH B03)J;YX.
Pa3HocTb Mem.n;y BeCOM B03,nyxa B 06'beMe
060JlOqKH annapaTa Ii BeCOM JlerKOrO ra3a, 3a-
nOJlH.HIOerO STY 060JlOqKY, Ha3bIBalOT nO,lJ;'LeM-
HOB CHJIOB JleTaTeJlbHOrO annapaTa (aspocTaTa).
XOT.H 3aKOH ApxHMe.n;a .n;JI.H ra30B 06'b.HCH.HeT
nOJleT B03.n;YIIIHoro lIIapa, HO BbITaJlKHBalOa.H
CHJla B03HHKaeT 3,neCb He TaK, KaK B cJlyqae TBep-
.n;oro Tena. HHmHlO1O qaCTb 060JloqKH (pHC. 5.24)
OCTaBJI.HIOT OTKpbITOH; .n;aBJleHHe ra3a (HanpHMep,
Bo.n;opo.n;a) y HHmHero OTBepCTH.H paBHO .naBJleHHIO
B03)J;yxa. JI.aBJleHHe BO,AOpO,Aa H .n;aBJleHHe B03.n;y-
xa YMeHblIlaIOTC.H npH no,nH.HTHH BBepx. 3HaqHT,
.naBJleHHe KaK B03,nyxa. TaK H BOlIopO.na Ha pa3HbIX yqacTKax lIIapa 6y.n;eT MeHb-
me, qeM y HHmHero OTBepCTH.H. HO .naBJleHHe JlerKOrO Bo.nopo.na y6bIBaeT MelI-
JIeHHee, qeM B03.nyxa.
IIosToMY Ha 060JlOqKY BHYTPH 6y.n;eT )J;eHcTBoBaTb 60JlblIlee .naBJleHHe, npH-
qeM CaMa.H 3HaqHTeJlbHa.H pa3HHa .naBJleHHH Bonopona H B03nyxa nOJlyqHTC.H
B BepxHeH qaCTH 060JlOqKH. CJle.n;oBaTeJlbHo. CHJla, neHcTBYIOa.H Ha KynoJl
060JlOqKH H3HYTPH, 6Y)J;eT 60JlblIle, qeM cHapymH. Pa3HocTb Mem)J;y STHMH CH-
JIaMH H ypaBHoBelIlHBaeT CHJlY T.HmeCTH 060JlOqKH C rpY30M.
TaKHM 06pa30M, BbITaJlKHBalOa.H CHJla C03.n;aeTC.H He 6J1aro.n;ap.H pa3HOCTH
.n;aBneHHH Ha HHmHlO1O H BepXHlO1O qaCTH TeJla (KaK B cJlyqae TBep)J;oro TeJla) ,
a 6JIaro.n;ap.H pa3HOCTH ,naBneHHH H3HYTPH H CHapY:1KH Ha BepXHlO1O qaCTb 060-
JIOqKH.
B HaqaJle nOJleTa lIIap HanOJIHeH BO.n;OpO.n;OM HaCTOJlbKO, qTO BbITaJlKHBalO-
Il\a.H CHJla npeBocxo.n;HT ero CHJIY T.HmeCTH, H OH no.n;HHMaeTC.H BBepx. Kor.n;a
map .n;OCTHraeT CJlOeB B03.n;yxa C MeHblIlHM .n;aBJleHHeM, BO.n;OpO.n; paCIIIHp.HeTC.H
H qaCTb ero BbIXO.n;HT H3 HHmHero OTBepCTH.H. TaKHM 06pa30M, Ha BbICOTe
YMeHblIlaeTC.H H HapymHoe, H BHYTpeHHee .naBJleHHe, YMeHblIlaeTC.H H paBHO-
FA
Puc. 5.23
--
-..........
"
Puc. 5.24
5.12. B03.D;yxonJIaBaHUe
105

eHCTBYIOa.H 3THX CHJ1 aBJleHH.H, T.e. BbITaJ1KHBalOa.H CHJla. Ha HeKOTO-
pOH BbICOTe OHa CTaHeT paBHa Becy IIIapa C HaXOHHMCH B HeM ra30M, li
IIIap nOBliCHeT. ,n;J1.H Toro qT06bI onYCTHTbC.H Ha 3eMJ1m, CJ1eyeT BbInYCTliTb
li3 060JlOqKH qaCTb ra3a. ,n;JI.H 3Toro B BepXHeH qaCTH 6annOHa HMeeTC.H KJ1a-
naH, KOTOpb1H MomHO OTKpbITb npH nOMOli BepeBKH H3 KOp3HHKli llIapa.
IIpli OTKpbIBaHHH KnanaHa ra3, HMelOHH 60nbllIee aBJ1eHHe, 'tJeM oKpymalO-
liH B03YX, BbIXOHT Hapymy. 't.{epe3 HHmHee OTBepCTHe 3TO HeB03MomHO,
TaK KaK 3,AeCb aBneHHe BOOpO,Aa H B03yxa O,AliHaKOBO.
5.13. 3aBHCJlMOCTL J];aBJIeHHH mHJ];KOCTH OT CKOpOCTH ee TeqeHHH
B npebIyHX naparpacpax 6bIJ1H paCCMOTpeHbI 3aKOHbI paBHOBeCli.H mH-
KOCTen H ra30B. Tenepb paCCMOTpHM HeKOTopbIe .HBJ1eHli.H, CB.H3aHHble C liX
,ABlimeHlieM.
)IBlimeHHe mHKOCTH Ha3bIBalOT Te'leHHeM. a cOBoKynHocTb qaCTHI..1 Bli-
myeHC.H mHlIKOCTH nOTOKOM. IIpH onHcaHlili lIBHmeHH.H mli,AKOCTli onpelIe-
J1.HIOT CKOpOCTH, C KOTOpbIMli qaCTHIJ.bI mli,AKOCTli npOXO,AHT qepe3 aHHYIO
TOqKY npOCTpaHCTBa. ECJlli B KamoH TOqKe npOCTpaHCTBa, 3an0J1HeHHOro Bli-
myeHcH mHKOCTblO, CKOpOCTb He H3MeHHeTCH co BpeMeHeM, TO TaKoe BH-
meHlie Ha3bIBaeTCH YCTaHOBliBllIliMCH, HJ1H CTaIJ.HOHapHbIM. IIpli CTaIJ.liOHap-
HOM TeqeHHH J1106aH qaCTliIJ.a mHKOCTH npOXOHT qepe3 aHHYIO TOqKY npo-
CTpaHCTBa C OHHM H TeM me 3HaqeHHeM CKOpOCTH. MbI 6yeM paCCMaTpli-
BaTb TOJ1bKO CTaIJ.HOHapHoe TeqeHHe HeaJ1bHOH HecmHMaeMOH mli,AKOCTH. lI..a;e-
aJlLHOH Ha3bIBalOT mli,AKOCTb, B KOTOpOH OTCYTCTBYIOT CHJlb1 TpeHli.H.
KaK H3BeCTHO, HenOBHmHaH mHKOCTb B cocye, COrJlaCHO 3aKOHY IIac-
K8J1H, nepeaeT BHellIHee aBJ1eHHe KO BceM TOqKaM mH,AKOCTli 6e3 H3MeHe-
HliH. Ho Kora mHKOCTb TeqeT 6e3 TpeHHH no Tpy6e nepeMeHHoro nonepeq-
Horo CeqeHliH, ,ARBJ1eHlie B pa3Hb1X MeCTax Tpy6bI HeOliHaKOBO. OIJ.eHliTb pac-
npe,AeJ1eHHe aBJleHHH B Tpy6e, no KOTOpOH TeqeT mliKOCTb, MomHO C nOMO-
blO YCTaHOBKH. CXeMaTliqeCKH H306pameHHOH Ha pHCYHKe 5.25. BOJ1b TPY-
6b1 BnaliBalOT BepTliKaJIbHbIe OTKpbITbIe Tpy6Kli-MaHoMeTpb1. ECJIH mliKOCTb
B Tpy6e HaXOHTCH no ,AaBJle-
HHeM, TO B MaHOMeTPHqeCKOn
Tpy6Ke mliKOCTb nOHliMaeTCH
Ha HeKOTOpylO Bb1COTY, 3aBliC.H-
YIO OT aBneHli.H B aHHOM
MeCTe Tpy6b1. OnbIT nOKa3bIBa-
eT, qTO B Y3KHX MeCTax Tpy6bI
BbICOTa CToJ16liKa mliKOCTli
MeHbllIe, qeM B llIHpOKHX. 8TO
3HaqHT, QTO B 3TH X Y3KHX Me-
CTax aBJleHHe MeHbllIe. 't.{eM
3TO 06HCH.HeTC.H?
IIpenoJ10mHM, qTO Hecmli-
MaeMR.H mli,AKOCTb TeqeT no ro-
pli30HTanbHOH Tpy6e C nepeMeH-
Hb1M CeqeHHeM (pHC. 5.25). BbI-
eJ1liM MbICJleHHO HeCKOJ1bKO
Puc. 5.25
106

CeqeHlill B Tpy6e, nJIOa,nH KOTOpbIX 0603HaqliM 8 1 Ii 8 2 , TIpH cTaHoHap-
HOM TeqeHli1i qepe3 JlI060e nOnepeqHOe CeqeHHe Tpy6bI 3a paBHbIe npoMemYT-
KH BpeMeHIi nepeHOC.HTC.H o.n;IiHaKoBbIe 06'beMbI mli,nKOCTH.
TIYCTb VI - CKOpOCTb mli,nKOCTIi qepe3 CeqeHlie 8., V2 - CKOpOCTb mlilI-
KOCTIi qepe3 CeqeHlie 8 2 - 3a BpeM.H D.t 06'beMbI >K1i,AKOCTeH, npoTeKalOlix
qepe3 3TIi CeqeHIi.H. 6y.n;YT paBHbI:
L\Vj = 118} = v l L\t. 8 1 ;
L\ V 2 = 82 = v 2 f).t . 8 2 ,
TaK KaK mH,nKOCTb HeCmHMaeMa, TO D.1/i = L\V 2 . CJIe.n;OBaTeJlbHO, V1Sl = V2S2
HnH v8 = const .n;JI.H HeC>KIiMaeMOH mli.n;KOCTH. 8TO COOTHOIIIeHHe Ha3bIBaeTC.H
ypaBHeHlieM Hepa3pbIBHOCTH.
113 3TOrO ypaBHeHIi.H  = 8 2 , T.e. CKOpOCTIi >K1i,nKOCTIi B ,nBYX JII06bIX
v2 8 1
CeqeHH.HX 06paTHo nponopIiOHaJlbHbI nJlOa,n.HM CeqeHHH. 8TO 3HaqHT, QTO
QaCTIibI mH.n;KOCTIi npli nepexo.n;e 1i3 IIIlipOKOH qaCTIi Tpy6bI B Y3KYlO yc-
KOp.HIOTC.H. CJle,noBaTeJlbHO, Ha mH,nKocTb, nocTynalOYIO B 60JIee Y3KYlO qaCTb
Tpy6bI. ,AeHcTByeT co CTOpOHbI mH,nKOCTIi. eIUe Haxo,n.HIUeHC.H B IIIHpOKOH qa-
CTH Tpy6bI, HeKOTOpa.H CHJla. TaKa.H CIiJIa MomeT B03HHKHYTb TOJlbKO 3a cqeT
pa3HocTIi ,naBJleHIiH B pa3J1HqHbIX qaCT.HX mH,nKOCTH. TaK KaK CHJla HanpaB-
neHa B CTOPOHY Y3KOH qaCTH Tpy6bl, TO B IIIHpOKOM yqaCTKe Tpy6bI .n;aBJIe-
RHe ,nOJImHO 6bITb 6QJlbIIIe, QeM B Y3KOM. YqHTbIBa.H ypaBHeHlie Hepa3pbIB
HOCTH, MOmHO C,neJlaTb BbIBO,n: npu cmaquoliapliO.M melleH.UU JlCuihcocmu
Oa6.n.eliUe .MelibWe 6 mex .MeCmax. zoe 60.n.bWe CKOpOCmb melleliUSl, U. lia060-
porn. 60.n.bWe 6 mex .MeCmaX, zoe CKOpOCmb melleH.USl Melibwe.
K 3TOMY BbIBO.ny BnepBbIe npliIIIeJI ,il,. BepHYJIJIH, n03TOMY ,A8HHbIH 3aKOH
Ha3bIBalOT 3aKOHOM BepHYJlJlH.
llpHMeHeHHe 3aKOHa COXpaHeHH.H 3HeprHH K nOTOKY ,nBHmyeHC.H mli.n;KO-
CTH n03BOJl.HeT nOJlyqHTb ypaBHeHHe, BblpamalOee 3aKOH BepHY JInH (npHBO-
.nHM 6e3 BblBO.na):
PV 2 PV 2
p} + -1... = P2 + ---1.. - ypaBHeHHe BepHYJlJlH AJISI rOpH30HTaJlbHOii TPy6KH.
2 2
3,AeCb PI Ii P2 - CTaTIiQeCKlie ,naBJIeHIi.H, p - nnOTHOCTb >KH,nKOCTH. CTa-
TIit.IeCKOe .naBJIeHHe paBHo OTHOIIIeHHIO CHnbI ,naBJIeHH.H O,nHOH QaCTH >KH.nKO-
CTH Ha ,npyrylO K nJIOaH conpliKocHoBeHHSI, KOr,lIa CKOpOCTb HX OTHOCIiTeJIb-
Horo ,nBH>KeHH.H paBHa HYJIIO. TaKOe ,naBJIeHHe H3MepHJI 6bI MaHoMeTp. .n;BH-
>KYHHC.H BMeCTe C nOTOKOM. Heno,nBHmHa.H MOHOMeTpHqeCKa.H Tpy6Ka C OT-
PV 2
BepcTHeM, 06paeHHbIM HaBCTpeqy nOToKY, H3MepHT ,naBJIeHHe p = p} + -t.
pV; pV
CJIaraeMbIe Ii HMeIOT, C O.n;HOH CTOpOHbI, pa3MepHocTb ,AaBJIe-
2 2
HH.H, C ,ApyroH - pa3MepHOCTb 06'beMHOH nJlOTHOCTIi 3HeprHIi. T. e. 3HeprHIi.
mvl.
npHxo,n.HeHC.H Ha e)J;HHHy 06'beMa. .f(eHcTBHTeJIbHO, W = -, Macca
K 2
107

mH.n;KOCTH m = p V. ECJlH V = 1
M 3 ,
pv 2
TO W =-
K 2
Ha3bIBalOT
pv 2
II03ToMY -
2
AHHaMHQeCKHM AaBneHHeM. 3TO KHHeTHqeCKR.H 3HeprHH nOTOKa B e,AHHHqHOM
06'beMe mH.n;KOCTH (06'beMHRSI nJlOTHOCTb 3HeprHH).
ECJlH TPy6KR He ropH30HTaJlbHa.H, TO Ha.n;o yqHTbIBaTb H rH.n;pOCTaTHqeC-
Koe .n;RBneHHe mH.n;KOCTH. YpaBHeHHe BepHYJlnH 6y.n;eT HMeTb BH.n;:
pv 2 pv 2
PI + pght T -l = P2 + pg + ,
2 2
r.n;e hI H h 2 - BbICOTbI, HR KOTOpbIX HRXO.n;.HTC.H CeqeHH.H 8 1 H 8 2 ,
3aKoH BepHY JlJIH JlemHT B OCHOBe npHHHn8 .n;eHcTBH.H MHorHX TeXHHqeC-
KHX YCTPOHCTB It npH60poB: BO.n;OCTpYHHoro Hacoca, nYJlbBepH3aTopa, <}>OPCYH-
KH KRp6lOpaTopa. 3aKoH BepHYJlJlH n03BOJl.HeT 06'bHCHHTb B03HHKHOBeHHe
no.n;'beMHOH CHJlbI KpbIJla CRMOJleTa.
5.14. ABHmeHHe TeJI B mHItKOCTHX H ra3ax. CHJIa conpOTHBJIeHHH
Bo Bcex peanbHbIX mH.n;KOCTHX npH nepeMeeHHH O.n;HHX CJlOeB OTHOCH-
TenbHO .n;PyrHx B03HHKalOT 60Jlee HJlH MeHee 3HaqHTenbHbIe CHJlbI TpeHH.H.
Co CTOpOHbI cnOH, .n;BHmyerOC.H 6bIcTpee, HR CJlOH, .n;BHmYHHC.H Me.n;-
JleHHee, .n;eHcTByeT YCKopHIOa.H CHJla. Co CTOpOHbI me CJlO.H, .n;BHmye-
roc.H Me.n;JleHHee, HR CJlOH, .n;BHmYHHcH 6bIcTpee, .n;eHcTByeT TOpM03.HR.H
CHJla. 3TO BHYTpeHHee TpeHHe HR3bIBaeTCH BH3KOCThlO mH,AKOCTH IiJlH ra3R.
3TH CHJlbJ HRnpaBJleHbI no KacaTenbHOH K nOBepxHOCTH CJlOeB. IIycTb Mem,Ay
.n;BYM.H nJlOCKOCT.HMH HRXO.n;HTC.H CJlOH mH.n;KOCTH (pHC. 5.26); BepXHH.H nJloc-
KOCTb ,nBHmeTC.H OTHOCHTeJlbHO HHmHeH co CKOpOCTblO V. MbIcneHHo pa306'b-
eM mH,AKOCTb Ha OqeHb TOHKHe cnOH napaJlJlenbHbIMH nJlOCKOCT.HMH, OTCTO.H-
HMH Ha paCCTO.HHHH /::"x .n;pyr OT .n;pyra. CJlOH mH.n;KOCTH, KacalOHeC.H TBep-
.n;bIX Ten, npHnHnalOT K HHM. IIpOMemYTO'tJHbIe cnOH HMelOT pRCnpe,l:(eneHHe CKO-
pOCTeH, H306pRmeHHbIX HR pHC. 5.26. IIycTb
V pR3HOCTb CKopocTeii Mem,l:(y COCe,l:(HHMH CJlO-
V - B /::"v
HMH /::"v. enHqHHa - , KOTOpRH nOKR3bl-
/::"x
BaeT, K8K 6bICTpO MeH.HeTC.H CKOpOCTb npH
nepexo.n;e OT CJlOH K CJlOIO, HR3bIBaeTCSI rpa-
AHeHTOM CKOpOCTH.
PaCqeTbI nOKa3bIBRIOT. tITO CHna BHYT-
peHHero TpeHH.H Mem,l:(y COCe,l:(HHMH CJlO.H-
MH mH.n;KOCTH TeM 60JlbllIe, qeM 60JIbllIe
nnOR,l:(b nOBepXHOCTH conpHKOCHOBeHH.H
CJlOeB, H 3aBHCHT OT 6bICTpOTbI H3MeHeHH.H
K CJlOIO B HRnpRBneHHH OCH OX, nepneH,l:(H-
CJlOeB:
Puc. 5.26
CKOpOCTH npH nepexo,l:(e OT CJlO.H
KY nHpHoH CKOpOCTH )J;BHmeHH.H
108
F = 1l1  IS,

r.n;e S - nJlOa.n;b COnpHKOCHOBeHH.H CJlOeB, 11 - K03cpcpHHeHT BHYTpeHHerO
/),V
TpeHH.H, HJlH B.H3KOCTb mH.n;KOCTH, /),X - rpa.n;HeHT CKOpOCTH.
B.H3KOCTb 3aBHCHT OT TeMnepaTypbI. C pOCTOM TeMnepaTypbI B.H3KOCTb mH.n;-
KOCTH YMeHbWaeTC.H.
IIpH .n;BHmeHHH TBep,AOrO TeJIS B >KH,AKOCTH HJlH ra3e TaKme B03HHKseT
CHJla conpOTHBJleHH.H .n;BHmeHHIO, KOTOPYIO Ha3bIBalOT CHJlOH BH3Koro TpeHHH.
Ho B OTnHqHe OT cyxoro TpeHH.H B mH)J;KOCT.HX H ra3ax OTCYTCTByeT CHns
TpeHHH nOKO.H. HaJIHqHe CHJIbI conpOTHBJleHHH .n;BHme-
HHIO TeJla B cpe.n;e 06'bHCH.HeTCH cyecTBoBaHHeM BHYT-
peHHero TpeHH.H, 06YCJlOBJleHHoro OTHOCHTeJlbHbIM .n;BH-
meHHeM CJlOeB mH.n;KOCTH HnH ra3S.
YCTSHOBJIeHO. qTO CHJIa BH3Koro TpeHHH 3SBHCHT OT
CKOpOCTH .n;BHmeHHSI TeJlS. 3aBHCHMOCTb npoeKHH
CHJlbI B.H3KOrO TpeHH.H OT CKOpOCTH nOKa3SHS Ha PH-
CYHKe 5.27.
ECJlH CKOpOCTb .n;BHmeHH.H Tens HeBeJlHKa, TO
CHJla conpOTHBJleHHSI np.HMO nponopHOHSJlbHa MO-
.n;YJlIO CKOpOCTH: Fc=kv, r.n;e k - K03cpcpHHeHT npo-
nopHOHaJlbHOCTH. KOTOpbIH 3aBHCHT OT po.n;a B.H3KOH
cpe.n;bI, cpOpMbI H pa3MepOB TeJlS. ECJlH CKOpOCTb
.n;BHmeHHH TeJlS B03pSCTseT, TO B03paCTseT H CHna
conpOTHBJleHHH:
Fc s
Puc. 5.27
v
Fc =kv n , r.n;e n=2, 3.
IIpH YBenHqeHHH CKOpOCTH lIBHmeHH.H TeJIS B mHlIKOCTH HnH rS3e no-
HBnHIOTC.H BHXPH, TOpM03.HHe ,ABHmeHHe: BCJle.n;CTBHe B.H3KOCTH B 06J1SCTH.
npHJIeralOeH K nOBepXHOCTH TeJla, 06pa3yeTcH nOrpaHHqHbIH CJlOH qaCTH.
.n;BH>KYHXC.H C MeHbWHMH CKOpOCTHMH. B pe3YJlbTaTe TOpM03.HerO .n;eHcTBH.H
3Toro CJlOH B03HHKseT BpaeHHe qSCTH, H .n;BHmeHHe mH,AKOCTH B norps-
HHqHOM cJloe CTaHOBHTCH BHxpeBbIM. ECJlH TeJlO He HMeeT 06TeKaeMOH cpop-
MbI, TO nOrpaHHqHbIH CJlOH mH.n;KOCTH OTpbIBaeTC.H OT nOBepXHOCTH TeJlS. 3a
TenOM B03HHKaeT TeqeHHe >KH.n;KOCTH (ra3a), HanpSBJIeHIIOe npOTHBOnOnO>KHO
Hs6eralOeMY nOTOKY. OTopBaBWHHC.H nOrpaHHqHbIH CJlOH, CJlelIYH 3a 3THM
TeqeHHeM, 06pa3yeT BHXPH, BpaSIOHeC.H B npOTHBOnOJIOmHbIe CTOpOHbI.
(pHC. 5.28, 6). mH.n;KOCTb, BpaalOaHC.H B BHxpe, .n;BHmeTCH 6bIcTpee mH,A-
a
°lbo

6 6

01  j  0, _ Q Q""
J
  
Puc. 5.28
109

KOCTH B cTau,HOHapHOM nOTOKe (pHC. 5.28, a). II03ToMY C 3aHeH CTOpOHbI
06TeKaeMoro TeJIa, re o6pa30BaJIHCb BHXPH, AaBJIeHHe CTaHOBHTC.H MeHbIlIe,
qeM C nepeHeH. Pa3HOCTb AaBneHHH BnepeAH H n03aAH ABHmym;erocSI TeJIa
H C03,AaeT conpOTHBJIeHHe BHmeHHIO TeJla. B HTore C YBeJIHqeHHeM CKOpOC-
TH CHJIa conpOTHBJleHH.H paCTeT HeJlHHeHHO (CM. pHC. 5.27).
CHJIa conpOTHBJleHHSI 3aBHCHT OT <pOpMbI TeJIa. IIpHAaHHe Teny cneu,Hanb-
HO paCCqHTaHHOH o6TeKaeMoH cpOpMbI cym;ecTBeHHO YMeHbIlIaeT CHJIY con po-
THBneHH.H, TaK KaK B 3TOM CJIyqae mH)1KOCTb BCIOn;y npHJIeraeT K ero nOBep-
XHOCTH H n038H Hero He 38BHxpeH8 (pHC. 5.28. B).
5.15. TIO,lJ;'beMHaH CHJIa KpbIJIa CaMOJIeTa
KpbIJIbSl caMoneTOB HMelOT P83JIH'lHble
<l>OPMbI. HO Bce OHH HeCHMMeTpH'lHbI:
cBepxy KpbIJIO BbmYKnoe, CHH3Y cnerKa
BorHyroe HJIH nnOCKoe (pHC. 5.29).
 Ha6eraIOJ,.QHH: B03,AYWHbIH: nOTOK
 BbmYKnyIO nOBepXHOCTb KpbIna npoxo-
,lJ,HT C 60nbWeH: CKOPOCTbIO. qeM HHm-
HIOIO nJIOCKYIO. KpoMe Toro, npH o6Te-
K8HHH BepxHeii KpOMKH, CpbIB8S1Cb C
KpbIna. Q8CTHU,bI B03,Ayxa 06p83YlOT
BHXPb. B KOTOpOM B03AYX BpaJ,.Q8eTCfl
npOTHB 'l8COBOH: CTpenKH. BHXpH 3TH
P8CTYT. CpblB8IOTCSI. BHOBb 06p83YIOTCSI
H T.,A. BHXpb npOTHBononomHoro H8.
npaBneHHH B03HHKaeT BOKpyr KpbIn8.
8Ta onOnHHTeJlbHaSi U,HPKYJI.HU,HSI
B03yxa HaKJIallbIBaeTC.H Ha Ha6eralO-
HH Ha KpbIJIO B03YIlIHbIH nOTOK.
B pe3YJIbTaTe CKOpOCTb B03,AYlliHoro
nOTOKa Ha KpbIJlOM CTaHOBHTC.H
60JIbIlIe, QeM no KpbIJlOM (HaA KpbI-
JIOM CKOpOCTb u,HpKyn.Hu,HH H CKO'
pOCTb Ha6eralOero nOTOKa COBna)J;aIOT
no HanpaBJIeHHIO, a no KpbIJIOM OHH
npOTHBOnOnO:>KHbI). COrJIaCHO 3aKOHY
BepHYJIJIH no KpbIJIOM AaBJIeHHe 6yeT 60JIbIlIe, qeM HaA KpbIJIOM. llMeHHo
3Ta pa3HOCTb aBJleHHH Ha HH:>KHIOIO 11: BepXHlO1O nOBepXHOCTH KpbIJla C03aeT
nO'beMHYIO CHJIY F KpbIJIa. I1o'beMHaH CHJIa .HBJI.HeTC.H OHOH H3 COCTaBJI.HIO-
m;HX 06m;ero conpOTHBJIeHH.H R (pHC. 5.30).
COCTaBJI.HIOa.H Q npe)1CTaBJI.HeT C060H JI060BOe conpOTHBJIeHHe. MOn;YJlb
3THX CHJI 3aBHCHT OT CKOpOCTH Ha6eralOero nOTOKa, era nJlOTHOCTH, npocpH-
JI.H KpbIJIa H OT yrJIa a, no,A KOTOpbIM Ha6eraeT nOTOK (yroJl aTaKH).
TeOpH.H B03HHKHOBeHH.H nOA'beMHOH CHJIbI KpbIJIa npH 06TeK8HHH ero B03-
AYXOM 6bIJla BnepBble pa3pa60TaHa H. E. )KYKOBCKHM. KOToporo no npaBY Ha-
3bIBalOT OTU.OM PYCCKOH aBHau,HH.

c

c

Puc. 5.29
F
Puc. 5.30
110

BOllPOChI AJUI CAMOKOHTPOJI.H
1. If e.M Om.n.UllalOmCSl ynpyzue C60UCm6a JlCUdKOCmeu om ynpyzux c80ucm8
m8epdbtX me.n.?
2. II Olle.MY 3ICudKocmu U za3bt nepedalOm da8.n.eliue 80 t3ce cmopOlibL odu-
naKoeo?
3. /(eucm6yem .n.u 3aKOli llaCKaJlSl 6 CocmOJlliUU lieeeco.Mocmu?
4. Om Ilezo 3aeucum 8eJlUilUlia zUdpocmamullecKozo daeJleliuJl? 3a8ucum
.n.u olia om ljJOp.Mbt cmoJl6a JlCUdKocmu?
5. OduliaK060.n.u da6.n.eliue lia pa3libtX yp06nSlx e JlCUdKocmu. naxodSlw,eu.
CSl 6 no.n.e CU.n.bt mSlJICecmu? 6 YC.n.06USlX lie6eCO.MOCmu? na OdliO.M YP06ne JlCUO-
Kocmu 6 pa3libtX nanpa6.n.eliuSlx?
6. OduliaK06a .n.u cu.n.a da8JleliUJl na dno 6 cocyoax, 6 odun U3 KomOpbtX
na.n.uma eoda .MaCCOu m. a 6 dpyzou nocma6.n.elia ZUpSl mou JlCe .MaCCbt?
7. OduliaK06a.n.u CUJla da6.n.eliUJl na dlio U cmeliKU cocyda. eCJlU nJlow,adb
dna u cmenOK odunaK06a?
8. B KaKO.M c.n.Yllae CUJla daeJlenuJl na dno cocyoa 60JlbWe (.MelibWe UJlU
pa6na) CU.n.bt mSl3ICecmu 3ICudKocmu, lia.n.umou e cocyd?
9. OdUliaK08bt .n.u yp06liU JlCUdKOCmu 6 co06w,alOw,uxcSl cocydax. ec.n.u 6
nux lia.n.uma oi)liopodliaSl 3ICudKocmb? neCKOJlbKO JlCUOKOCmeu l
10. J(6a cmeK.n.SllilibtX cocyoa coeoulieHbt pe3uli060U mpy6Kou. OcmaliemcJl
JlU yp08elib 800bt 8 amux cocyoax oounaK06bt.M. ec.n.u npa6btU cocyo liaK.n.o-
numb? nOdliSlmb 66epx?
11. J(aem .n.u zudpa6.n.ullecKuu npecc 6btUZPbtW 6 cu.n.e? 6 pa60me? Bydem
JlU on pa60mamb e cocmOSlliUU lie6eCO.MOCmu?
12. KaKou Onbtm n0380JlU.n. onpede.n.umb da6JleliUe 3e.MIiOU am.MOcljJepbt?
13. KaK U3.MeIiSllOmCSl da6.n.eliue U n.n.omnocmb 803dyxa c 8btCOmou?
14. 3a6ucum JlU 8btCOma yp06liSl pmymu 6 Onbtme TOPPulleJlJlU om dua-
.M.empa mpy6KU? om ee liaK.n.ona K Zopu30limy?
15. II Olle.MY na me.n.o. nozpYJICennoe 6 .HCUOKocmb. deucmeyem 8btmaJlKU6a1O
aJl cu.n.a?
16. OouliaK08aSl .n.u 6btma.n.KU6alOw,aSl cu.n.a oeucm6yem lia me.n.o. no.n.lioc
mblO nozpYJICelilioe 6 OdliOPOOliYIO JlCUOKOCmb, ec.n.u OliO liaXOdumcSl Ha pa3HOu.
zJly6une?
17. llpu KaKO.M YC.n.06UU me.n.o 1l.n.a6aem 6 JlCUOKocmu? mOliem? 6CnJlbt6a
em?
18. Bymbt.n.Ka, nano.n.lielinaJl 60dOU, monem 8 Bode. Ymoliem JlU 6ymbtJlKa.
nano.n.lieliliaJl pmymblO, ec.n.u ee onycmumb 6 pmymb?
19. B cocyde c 60dOU n.n.a6aem KYCOK .n.bda. If mo 6ydem c yp06lie.M 60dbl
6 cocyde noc.n.e mozo. KaK .n.ed pacmaem? Ifmo 6yoem. ec.n.u 6 cocyd 6.MeCmO
600bt lia.n.umb JlCUOKOCmb c 60.n.bweu n.n.omliocmblO?
20. M 03IC1i0 .n.u ym6epJICdamb. llmo me.n.o. n.n.aealOw,ee 8liympu .HCUdKOCmu 6
6e3pa3.n.UIlIi0.M pa6li06ecuu. liaXOOumcJl 6 cocmOSlliUU lie6eCO.MOCmu?
21. KaKU.M Za30.M IiY:JfCliO 3ano.n.Humb 060.n.01lKY aapocmama. Ilm06bL c ezo
nO.M.ow,blO .MOJICliO 6bt.n.0 nodliSlmb 60.n.bWOU ZPY3?
22. II olle.MY me.n.o .MOJICem lie nOdliSlmbcSl c z.n.uliucmozo dlia 60dOe.Ma. XomJl
8 06btllliO.M cocmOSlliUU OliO n.n.a8aem lia n06epxnocmu 60dbt?
111

23. Ymoliem .n.u JlCe.n.e3libtU npeOMem 6 60iJe lia uCKyccm6eliliOM cnymliu-
Ke 3eM.n.u?
24. Cnpa6eiJ.n.u8 .n.u 3aKOH ApxuMeiJa lia JI Ylie?
25. KaK U3MeliSlemcSl CKopocmb JlCUOKOCmu npu npomeKaliUU no mpy6e. ce-
lleliUe Komopou YMelibwaemcSl? IIolleMY 8 Y3KUX llacm.RX mpy60np0800a CKO-
pocmb JlCUOKOCmu 60.n.bwe. lleM 6 WUpOKUX?
26. KaKaSl cu.n.a 8bt3bt6aem Y6e.n.Ulleliue CKopocmu JlCUOKOCmu npu nepexo-
oe ee U3 WUpOKOU 6 Y3KYlO v.acmb mpy60np06ooa?
27. KaK U3MeliSlemcSl cmamUlleCKOe oa6.n.eliue JlCUOKOCmu npu Y6e.n.UlleIiUU
CKopocmu nomoKa?
28. q mo nOIiUMalOm noo oUliaMullecKuM oa6.n.eliueM JlCUOKocmu? Om llezo
OliO 3a6ucum?
29. B mpy6Ke C c nepeMelilibtM celleliueM (8 2 > 8 1 ) mellem 60oa. B lieu
liaxooumcJl nY3btpele 603iJyxa. Kale U3MeliumcSl ezo iJuaMemp npu npoxoJICoe-
IiUU mpy6 KU?
30. II 01leMY cmpySl JlCUOKocmu. 6btmeKa.R U3 olia cocyoa. cYJICaemcSl KIiU3Y?
31. Omllezo nOJICaplibtU 6paliocnoum oe.n.alOm CYJICalOU{UMCSl lia KOliije?
32. J(6a .n.eZKUX wapUKa. noo6ewelilibte lia liumSlX lia 6.n.U3KOM paccmoSl-
IiUU. c6.n.uJlCalOmcSl. ec.n.u MeJICoy IiUMU npooymb 8030YX. II olleMY?
F
P=-
8
Fi = 8 1
F 2 S2
P = pgh
P = Po + pgh
F.a = p gh8 .a
F. = pgh S
CT 2 CT
FA = p)l(gV
n m
L Pi = L PII
i=1 11=1
112
OCHOBHbIE cI»OPMYJIbI
- AaBlleHli'le, rAe F - Cli'llla AaBlleHIi'ISI, 8 - nll0l1\.aAb.
- COOTHOWeHli'le Cli'lll B rIi'lApaBllli'1eCKOM npecce, rAe F 1 - Cli'llla,
AeCTBYIOl1\.aSl Ha Mallbl nOpWeHb, F 2 - Cli'llla AaBJ1eHIi'ISI >KIi'IAKO-
Cn., Ha 60J1bWO nOpWeHb npecca, SI Ii'I S2 - nJ10l1\.aAli'I Mall0rO Ii'I
60J1bWOrO nOpWHe COOTBeTCTBeHHO.
- rIi'lAPOCTaTIi'IeCKOe AaBlleHli'le. rAe p - nJ10THOCTb >KIi'IAKOCTIi'I. g -
YCKOpeHli'le cB060AHOrO naAeHIi'ISI. h - BbiCOTa cToJ16a >KIi'IAKOCTIi'I.
- nOllHOe AaBlleHli'le B 111060 TOKe >KIi'IAKOCTIi'I, rAe Po - AaBlle-
Hli'le Ha ee CB06oAHO nOBepXHOCTIi'I.
- cli'Illa A8BJ1eHIi'ISI >KII'I.D.KOCTIi'I Ha AHO COCYAa, rAe SA - nll0l1\.8Ab AHa
- Cli'llla AaBJ1eHIi'ISI >KIi'IAKOCTIi'I Ha npSlMoyrollbHYIO BepTIi'IKallbHYIO
CTeHKY COCYAa, rAe SCT - nJ10l1\.aAb CTeHKIi'I.
- 3aKOH Apxli'IMeAa. rAe FA - BbITaJ1KIi'IBalOl1\.aSl CIi'IJ1a. p)l(. - nJ10T-
HOCTb >KIi'IAKOCTIi'I, V - 06beM nOrpY>KeHHO B >KIi'IAKOCTb aCTIi'I TeJ1a.
n m
- 3aKOH co06l1\.alOl1\.li'IxcSI COCYAOB, rAe L Pi Ii'I L PII - CYMMbl
1=1 11=1
AaBlleHIi'I CT01160B >KIi'IAKOCTe, HaXOASll1\.Ii'IXCSI HaA HYlleBblM YPOB-
HeM HaA nOBepXHOCTblO, npOXOASll1\.e epe3 caMYIO HIi'I>KHIOIO
rpaHIi'ILJ.Y pa3AeJ1a >KIi'IAKOCTIi'I COOTBeTCTBeHHO B i-OM Ii'I k-OM co-
CYAax.

V 1 =V 2 ,
 = 82
vS = const
pv 2
p+pgh+-=
2
= const
- YCJ10Be HeC>KMaeMOCT >KAKOCT, rAe VI  V 2 - 06beMbi
nopu. >KAKOCT, nepeTeKalOw.e 3 OAHoro COCYAa B APyro
8 1  8 2 - nJ1ow.a.o. nonepeHoro ceeHSf COCYAOB, hi  h 2 -
BblCOTbl CT0J160B nepeTeKalOw.e >KAKOCT
- ypaBHeHe Hepa3pbIBHOCT. rAe v - CKOpOCTb HeC>KMaeMO
>KAKOCT. S - nJ1ow.a.o.b ceeHSf TPy6bl, no KOTOpO TeeT >KWJ.-
KOCTb.
pv 2
- ypaBHeHe 6epHYJ1J11 rAe P - CTaTeCKoe AaBJ1eHe, -
2
AHaMeCKoe AaBJ1eHe, pgh - rAPocTaTecKoe AaBJ1eHe
>KAKOCT .
METO,ll;WlECKHE PEKOMEH,lI;AQHH
rHpOCTaTHKa BKJllOqaeT 3alIaqH. CB.H3aHHbIe C HaxomlIeHHeM lIaBJleHH.H H
CHJI ,AaBJleHH.H B nOKO.HeHC.H mH,AKOCTH. 9TH 3a,AaqH pellialOTC.H Ha OCHOBe
3aKOHa IIacKaJl.H H BbITeKalOHX H3 Hero Cne,ACTB1:IH. ,lI;JI.H pelIIeHH.H He06xo-
,n1:lMO:
1. C,AeJlaTb qepTem H OTMeTHTb Bce paBHoBecHbIe ypOBHH mH,AKOCTH, KO-
Topble OHa 3aHHMaJla no YCJlOBHIO 3a)J;aqH, YKa3aTb rpaHHnbI pa3,AeJla mH,AKO-
CTeH, eCJlH HX HeCKOJlbKO, 1:1 BbICOTbI CTOJl60B mH)J;KOCTeH. B cJlyqae co06a-
IOHXC.H COCY)J;OB HymHo BbI6paTb ropH30HTaJlbHbIH ypOBeHb B O,AHOpO,AHOH
>K1:I,AKOCTH (06bIqHO caMYIO H1:ImHIOIO rpaHHUY pa3l1eJla cpel1)
2. CocTaBHTb ypaBHeHH.H paBHOBeCH.H mH)J;KOCTH ,nJl.H ,ABYX npOH3BOJlbHbIX
TOqeK, JlemaHX Ha BbI6paHHoM ropH30HTaJlbHOM YPOBHe:
n m
LPI = LPk.
1=1 k=1
3. ECJlH lJ.0 HacTynJleHH.H paBHoBecH.H mHlIKOCTb nepeJlHBanaCb H3 OlIHOH qa-
CTH cocY,Aa B ,npyryro, TO HymHo 3anHcaTb YCJlOBHe HecmHMaeMOCTH mH,nKOCTH:
Vi = V 2 HnH 8 1 h 1 = 82'
4. 3anHcaTb )J;onOJlHHTeJlbHbIe <l>OPMYJlbI, CB.H3bIBalOHe HCKOMbIe BeJlHq1:l-
HbI H ,naHHbIe no YCJlOBHIO 3a.D;aq1:l.
PelIIeHHe 3a,Aaq Ha ,nBHmeHHe TeJla B mH,AKOCTH 1:IJlH ra3e OCHOBaHO Ha
3aKOHax )J;HHaMHKH nocTynaTeJlbHoro ,ABHmeHH.H TBep,AOrO TeJla c yqeToM 3a-
KOHa ApXHMe,Aa. He06xo,AHMO:
1. ClJ.eJlaTb qepTe)K H YKa3aTb CHJlbI. l1eHcTBYIOHe Ha TeJlO. norpymeH-
Hoe B mH,AKOCTb, yqHTbIBa.H, qTO BbITaJlKHBalOa.H CHJla
FA = pgV.
n
2. CocTaBHTb OCHOBHoe ypaBHeHHe AHHaMHKH L  = mii HJlH
1=1
n
I, ii = 0,
1=1
n
a HHor,na H ypaBHeHHe MOMeHTOB LMt = O.
1=1
3. CocTaBHTb ,AOnOJlHHTeJlbHbIe ypaBHeHH.H COrJlaCHO YCJlOBHIO 3a)J;aqH.
113

IIPHMEPbI PEIlIEHHH 3A]J;A q
1. HecpTb XpaHHTC.R B 6aKe, HMelOeM cpOPMY npSlMoyrOJIbHOrO napanJIe-
JIenHne,Aa BbICOTOH 8,0 M, JlHHOi1 2.0 M, IIIHpHHOH 1,2 M. Onpe,AeJl1:lTe CHnbI.
C KOTOpbIMH HecpTb aBHT Ha HO H Ha 60KOBYIO CTeHKY 6aKa.
Kr
.II:aHo: h=8,O M, 1=2,0 M, d=I,2 M, p=760 M 3 '
HaHTH: F l , F 2 .
P e ill e H He. C1:IJIa, C KOTOpOH He<PTb LIaBHT Ha ,AHO, Fl = pSn' rLIe p = pgh -
rHpOCTaT1:IqeCKOe aBJIeHHe cToJl6a HecpTH, SD, = ld - nJlOab ,AHa. CJle,AO-
BaTeJlbHO, F] = pghld.
CHna, C KOTOpOH HecpTb aBHT na 60KOBYIO CTeHKY 6aKa. F 2 = < p > SCT'
re < p > = 0 + p =!!.. _ cpeHee aBJIeHHe Ha CTeHKY, SC1 = hd - nJIOa,Ab
2 2
CTeHKH.
Tor,Aa
F 2 = P . hd = pgh 2 d .
2 2
IIocTaBHB qHCJleHHbIe 3HaqeH1:I.H, nOJlyq1:lM Fl =0,15 MH, F 2 =0,29 MH.
2. CaMOJIeT COBepIIIaeT "MepTBYIO neTJlIO" paHY-
y
N p r
mg
0
Puc. 5.31
KM
COM 440 M co CKOpOCTblO 792 -. OnpelIeJIHTe lIaB-
q
JleH1:Ie 6eH3HHa Ha HO 6aKa, 3anOJIHeHHOro 6eH3HHOM
o BbICOTbI 1,0 M, B H1:ImHeH TOqKe "MepTBoH neTnH".
M Kr
.II: a H 0: h = 1,0 M, R = 440 M, v = 220 -, P = 700 3'
C M
HaHTH: p.
P e ill e H H e. Ha 6eH3HH B HHmHeH TOqKe neT nH
eHcTByeT CHJIa T.HmeCTH mg U CHJIa peaKnHU N p ,AHa
6aKa (pHC. 5.31). OCHoBHoe ypaBHeH1:Ie 1:IHaM1:IK1:I:
mii = N p + mg,
unu B npOeKII.IUI H8 OCb Oy:
ma u = N p -mg => N p = ma u + mg.
v 2
Yq1:lTbIBa.H, qTO neHTpOCTpeM1:ITeJIbHOe YCKopeH1:Ie au = 'Ii' a C1:IJIa peaK-
nUH LIHa no TpeTbeMY 3aKOHY HblOToHa paBHa no MO,AYJlIO CHJIe aBJleHH.H 6eH-
"HHa Ha I\HO, HMeeM: F. = m(  + g )-
114

FD.
HCKOMOe aBJleHHe P = S' BbIpa3HM Maccy 6eH3HHa qepe3 ero nJlOTHOCTb
H 06'beM: m = pV = pSh.
phS (V 2 + g J
Tori\a p = : = Ph[  + g} P =84 KlIa.
3. ,l];HaMeTp OlIHoro H3 co06alOHXC.H COCYOB, B KOTopbIe HaJlHTa PTYTb.
B Ba pa3a 60JlbIlle HaMeTpa Pyroro. B Y3KHH COcy HaJlHJIH CTOJl6 BObI
BMCOTOH 50 CM. OnpeeJIHTe, Ha CKOJlbKO H3MeHHTC.H ypOBeHb PTYTH B 060HX
cocyax.
,l];aHO: d 2 =2d lt h B =0,50 M.
HaHTH: hI' h 2 .
P e ill e H H e. ITYCTb nepBOHaqanbHbIH ypo-
BeHb PTYTH B cocyax AIA2 (pHC. 5.32). ITpH
!._ A 2 HaJIHBaHHH BObI B Y3KOM cocye PTYTb ony-
CTHTC.H Ha BeJlHqHHY hI' B IIIHpOKOM nO.D:HH-
MeTC.H Ha h 2 . mH.D:KOCTH HaXOH.HTC.H B COCTO.H-
HHH paBHOBeCH.H. BocnoJlb3yeMC.H 3aKOHOM co-
06alOHXC.H COCYOB.
,l];JI.H 3Toro paccMoTpHM TOqKH 1 H 2, Jle-
maHe Ha OHOM ropH30HTaJlbHOM ypOBHe,
Puc. 5.32 cooTBeTcTBYIOeM HHmHeH rpaHHlJ,e pa3,AeJla
>KHKOCTeH :
PI = P2; PBghB = PpTg( + ).
PTYTb HecmHMaeMa, CJle,AOBaTeJlbHO, nOplJ,H.H PTYTH, BbITecHeHHoH H3 Y3-
Koro B IllHpOKHH cocYJI., HMeeT onpeeJleHHbIll 06'beM, T.e. h1S 1 = S2' re
  2
8 1 = - - nJlOab CeqeHH.H Y3KOH TPy6KH, 8 2 = - = '1'tdl
4 4
CeqeHH.H IllHpOKOH TPy6KH.
TOr,A8
nJlOa,Ab
 1td; = ndf =) ht = 4.
4
ITocTaBHM B 3aKOH co06alOHXC.H COCY)J;OB PB = ppT5 =)  = PBh B ;
5 PPT
h] = 2,9 CM; h 2 = 0,70 CM.
4. MeHbIll Illap C BHYTpeHHeH nOJlOCTblO nOBeIlleH K .D:HHaMOMeTpy. ITo-
Ka3aHHe HHaMOMeTpa COCTaBJI.HeT 2.59 H. ECJlH Illap norpY3HTb B BOY, TO
nOKa3aHHe .D:HHaMOMeTpa CTaHeT paBHMM 2,17 H. Onpe,AeJlHTe 06'beM BHYT-
peHHell nOJlOCTH Illapa. BbITaJlKHBalOYIO CHJlY B03,Ayxa He yqHTbIBaTb.
Kr 3 Kr
,l];aHO: F 1 =2,59 H, F2=2,17 H, PM=8,9'103 3' PB=I,0.10 3'
M M
HaHTH: VI'
115

P e 111 e H H e. B B03.n;yxe Ha
map )J;eHCTBYeT CHJla T.HmecTH mg
H CHJla ynpyrOCTH it. npymHHbI
.n;HHaMOMeTpa (pHC. 5.33, a). Yc-
JlOBHe paBHOBeCH.H: FI + mg = 0,
HRH B npoeKHH Ha OCb Oy:
FI - mg = ° => FI = mg.
TIpH norpymeHIDf mapa B BO-
l1Y (pHC. 5.33, 6): FA + F 2 + mg = 0,
B npoeKHH Ha OCb Oy:
FA + F 2 - mg = ° => FA = mg - F 2 .
M3 3THX ypaBHeHHH
FA = FI - F 2 .
TIo 3aKoHY ApXHMe.n;a FA = PsgV, r.n;e V - HapymHbIH 06'beM mapa.
a
y
o
t5
y
m FI
V = VI + V 2 , r.n;e VI - 06'beM nOJlOCTH mapa, V 2 = - = - - 06'beM, 3aH.H-
PM PMg
- -
--
--
o _ -i
Puc. 5.33
TbIH Me.n;blO. CJle.n;oBaTeJlbHO,
( F.. ) F, -F. F,
Psg VI +-L =FI-F2 => VI = I 2 _-L; V l =13 CM 3 .
PMg Psg P...g
5. llIapHK .n;JI.H HrpbI B HaCTORbHbIH TeHHHC pa.n;HycoM 15 MM H MaCCOH
5,0 r norpymeH B BO.n;y Ha rJly6HHY 30 CM. Kor.n;a mapHK OTnYCTHJlH, OH
BbInpbIrHYJl H3 BO.n;bJ H8 BbICOTY 10 CM. KaKoe KORHqeCTBO TenROTbI BbI,AeRH-
JlOCh BCJle.n;cTBHe TpeHH mapHKa 0 BO,AY?
Kr
.D:aHo:R=0,015M, m=5,0.10- 3 Kr, h l =0,30M, h 2 =0,10M, p=I,0.10 3 3'
M
HaHTH: Q.
P e 111 e H H e. PaCCMOTpHM CHCTeMY "mapHK-3eMR.H". 8Ta CHCTeMa He3aM-
KHYTa.H. Ha Hee .n;eHCTBYIOT BHemHHe CHRbI:
BbITaJlKHBalOa.H CHna FA H CHR8 conpOTHB-
JleHH.H Fe (pHC. 5.34). llORHa.H MeXaHHqe-
CKa.H 3HeprH.H He COXpaH.HeTC.H, H ee H3Me-
HeHHe paBHO pa60Te 3THX CHJI, T .e.
L\W = AA +.
TIpHMeM 3a HYJleBOH ypoBeHb OTCqeTa
nOTeHHaJlhHOH 3HeprHH HHmHee nOJlome-
HHe 1 mapHKa. Tor.n;a WI =0. B BepXHeM
nOJlOmeHHH 2 mapHKa W 2 = mg ( + ) H
l\W = 'W2 - WI = mg( +).
2.
h 2
116
1- -
Fe
Puc. 5.34

Pa60Ta BbITanKHBalOeH CHJIbI AA = FA = pgV.
4 3 4 3
06'beM mapHKa V = S1tR . CJleOBaTeJlbHO, AA = pg S1tR .
B HTore nOJlyqHM:
4
mg( +) = _1tR3pg +Ac 
3
4
Ac = mg( + )_-1tR3pg.
3
TaK KaK CHJla conpOTHBJleHH.H COBepmaeT OTpHltaTeJlbHYIO pa60TY, TO Me-
XaHHqeCKaSl 3HeprH.H CHCTeMbI YMeHbmaeTC.H, OHa npeBpaaeTC.H BO BHYTpeH-
HIOIO 3HeprHIO. n03TOMY HCKOMoe KOJIHqeCTBO TenJlOTbI Q = -A c , T. e.
4
Q = S1tR3pg - mg( + ); Q = 22 M)J;m.
117

3nEMEHTbi
CTATMCTMECKO M3MKM
M TEPMOAMHAMMKM
6. OCHOBbI MOJIEKYJI.HPHO-KlIHETlIQECKOH
TEOPlIlI
OCHOBHLle nOJlOjKeHHH MOJleKYJlHpHo-KHHeTHqeCKOH TeopHH H HX onblTHoe
060CHOBaHHe. J];H4>4>Y3HH H 6POYHoBcKoe ABHjKeHHe. B3aHMOAeHCTBHe aTOMOB
H MOJleKYJI BeeCTBa. Macca H pa3MepLI MOJleKYJI. KOJlHqeCTBO BeeCTBa. IIo-
CTOHHHaH ABora.n:po. It1J:eaJIbHLIH ra3. CKOpOCTli MOJleKYJI ra3a. OCHOBHoe ypaB-
HeHHe MOJleKYJlHpHO-KHneTH'IeCKOH TeopHIl H,lJ.eaJlLHOrO ra3a. Te..lnepaTypa H
ee H3MepeHHe. TenJlOBO paBHOBeCHe CHCTeMLI. A6COJllOTHLIH HYJlb. H3Mepe-
HHe CKopocTeH ra30BLlX MOJleKYJI. OnLIT IDTepHa. Pacnpe,lJ.eJleHHe MaKCBeJl-
J1a. BepO.HTHOCTL H 3HTponHH. nOpHAOK " 6eCnOpHJ(OK B MaKpOCHCTeMax.
YpaBHeHHe COCTOHHHH HlIeaJlLHOrO ra3a Hero '1aCTHLIe cJlyqaH ,nJlH no-
CTOJlHHOrO 3Ha'leHHJI TeMllepaTypLI, 06'LeMa H ,lJ.aBJleHHH.
6.1. CTaTHCTHqeCKHH H TepMo,ZJ:HHaMHqeCKHH
MeTO,ZJ:bI onHCaHH.H MaKpOCKOnHqeCKHX CHCTeM
MOJIeKYJI.HpHa.H cpH3HKa H TepMQ.lI.HHaMHKa - pa31IeJIbI cpH3HKH. B KOTOpbIX
H3Y'laIOTC.H CBOHCTB8 TeJl H npOHCXOLI.HHe B HHX MaKpOCKOnHtIeCKHe npoeCCbI,
CB.H3aHHbIe C orpOMHbIM tIHCJIOM tIaCTH, co.n;epmaHXCH B TeJIax.
.n;JI.H HCCJle.n;OBaHHH 3THX [lpOeCCOB nonb3YIOTC.H .n;BYM.H MeTo.n;aMH: MOJleKY-
JI.HpHO- KHHeTHqeCKHM (cTa THCTHtIeCKHM) H TepMo.n;HHaMHtIeCKHM.
B OCHOBe MOJleKYJI.HpHOH «pH3HKH JlemHT MOJleKYJI.HpHO-KHHeTHqeCKa.H Teo-
pHH (MKT), KOTopaH 06'L.HCH.HeT CTpoeHHe H CBOHCTBa TeJl ,ABHmeHHeM H B3a-
HMo.n;eHcTBHeM qaCTH (MOJIeKYJI, aTOMOB, HOHOB), H3 KOTOpbIX COCTO.HT TeJla.
CBOHCTBa TeJl, KOTopbIe HenOCpe.lI.CTBeHHO Ha6J1IO.n;aIOTcH Ha onbITe (.n;aBJIeHHe,
TeMnepaTypa H .lI.p.), OHa HCTOJlKOBbIBaeT KaK CYMMaPHbIH pe3YJlbTaT .n;eHCTBHH
tIaCTH. npH 3TOM OHa nOJlb3yeTcH CTaTHCTHqeCKHM MeTo.n;OM, HHTepeCY.HCb He
HH.n;HBH.n;yaJlbHbIMH xapaKTepHCTHKaMH OT.n;eJlbHbIX 'tlaCTH, a JlHllIb cpe.n;HHMH
3HaqeHH.HMH «pH3H'tleCKHX BeJlH'tlHH, KOTopbIe xapaKTepH3YlOT .lI.BHmeHHe 'laCTH,
COCTaBJI.HIOWHX CHCTeMY.
B KJlaCCHtIeCKOH MexaHHKe npe.lI.nOJIaraeTC.H, tITO .n;BHmeHHe qaCTH npOHCXO-
.n;HT B COOTBeTCTBHH C 3aKOHaMH HbIOTOHa. ECJlH CHCTeMa COCTOHT H3 MflJlOrO
qHCJla qaCTH, TO, 3Ha.H HaqaJlbHbIe ycnOBHJI H pelllaJl ypaBHeHH.H MexaHHKH,
MomHO HaHTH Koop.n;HHaTbI H CKOpOCTH Bcex tIaCTH B Jl1060H MOMeHT BpeMeHH,
T.'3. onpe,AeJlHTb COCTO.HHHe CHCTeMbI. O.n;HaKO orpOMHoe tIHCJlO MOJIeKYJI (HanpH.
Mep, B 1 M 3 ra3a npH HOpMaJlbHbIX YCJIOBH.HX co.n;epmHTCH qHCJlO MOJleKYJI,
paBHoe N = 2,7. 10 25 M- 3 (nOCTO.HHHa.H JIolllMH.n;Ta), a B TBep.n;bIX H mH.n;KHX
Tenax - nOp.H.n;Ka 10 28 MOJIeKYJI) H XaOTHtIUOCTb HX .n;BHmeHHH C03.n;aeT Henpe-
118

o,nonHMbIe TPY)J;HOCTH HB nYTH HCnOJlb30BaHH.H )J;n.fl onHCaHH.H CHCTeMbI MeXaHH-
qeCKOrO nO)J;xo)J;a. MbI npaKTHqeCKH He MomeM peIlIaTb KOJlOCCaJIbHOe qHCJlO CO-
CTaBJleHHbIX ypaBHeHHH. KpOMe Toro, cornaCHO COOTHOIlIeHHIO Heonpe,neJleHHOCTeH,
OTKPbITOMY B 1927 r. B. reH3eH6eproM (CM.  21.2), onpe,neJlHTb TOqHO nono-
)KeHHe qaCTHW>I B npOCTpaHCTBe (ee Koop,nHHaTbI) H ee HMnYJlbC (a 3HaqHT, H
CKOpOCTb) o,nHOBpeMeHHO npHHI.{HnHaJIbHO HeB03MO)KHO. .n;BHmeHHe Kam,noH OT,nenb-
HO B3.HTOH qaCTHbI He TOJlbKO He MomeT 6bITb YCTaHOBJleHO, HO Ii He npe,n-
CTaBJI.HeT HHTepeca npH H3yqeHHH MaKpOCKOnHqecKHx .HBneHHH, paCCMaTpliBaeMbIX
B MOJleKYJI.HpHOH <pH3HKe. 3TH .HBJleHH.H 06YCJlOBJleHbJ He CTOJlbKO )J;eTaJl.HMH CTpO-
eHH.H aTOMOB H xapaKTepOM ynpaBnSIIOHX HMH 3aKOHOB, CKOJlbKO He06bIqaHHO
60nbIlIHM qHCJlOM caMHX aTOMOB B MaKpOCKOnHqeCKHX CHCTeMax. HanH'lHe 60Jlb-
IIIoro qHCJla qaCTHI.{ lIaeT B03MO)KHOCTb Sc}><peKTHBHO Hcnonb30BaTb CTaTHCTHqeC-
KHe MeTOLIbI. OKa3bIBaeTC.H. B xaoce LIBHmeHHH MHorHX qaCTlUI MO)KHO Bce-TaKH
HaHTH HeKOTopbIe 3aKOHOMepHOCTH B nOBe,neHHH cpe,nHHX napaMeTpOB (HnH TaK
Ha3bIBaeMble CTaTHCTHqeCKHe 3aKOHOMepHOCTH). HanpHMep, MbI He HMeeM B03Mom-
HOCTH onpe,neJlHTb CKOpOCTb Kam,nOH OT,nenbHOH qaCTHI.{bI ra3a, HO MomeM Y3-
HaTb cpe,nHIOIO CKOpOCTb, C KOTOpOH ,nBHmYTCH qaCTHI.{bI ,naHHOH rpynnbI MOJle-
KYJI npH ,naHHbIX YCJlOBHSIX. IIpH STOM, KaK nOKa3bIBWOT paCqeTbI, COCTOSIHHe
CHCTeMbI He 3aBHCHT OT HaqaJIbHbIX Koop,nHHaT H CKopOCTeH qaCTHI.{.
KHHeTHqeCKa.H 3HeprH.H ,nBHmeHHSI OT)J;enbHbIX MOJleKyn MomeT npHHHMaTb
pa3J1HqHbIe 3HaqeHH.H rrpH )J;aHHOM COCTO.HHHH BeeCTBa, cpe)J;H.H.H me 3HeprH.H
HMeeT npH STOM BnOJlHe onpe)J;eJleHHOe 3HaqeHHe. 3TO cpe,nHee 3HaqeHHe sHep-
rHH onpe,neJl.HeT TeMnepaTYPY TeJla.
O,AHaKO B03MomeH H ,nPyroH cnoc06 onHCRHH.fI .HBneHHH. MHorHe CBoHcTBa
BeecTB CB.fI3aHbI C npoHcxo,n.HHMH B HHX npOI.{eCCaMH rrpeBpalI..{eHH.fI sHep-
rHH H3 O,nHHX BH,nOB B ,nPyrHe. IIosToMY H H3yqaTb 3TH CBOHCTBa MomHO Ha
OCHOBe 3aKOHOB SHepreTHqeCKHX npeBpaeHHH.
TepMo,nHHaMHKa H3yqaeT 06He CBOHCTBa TeJl H pa3J1H'IHbIe npOI.{eCCbI B
HHX, conpOBom,nalOHeC.H npeBpaeHH.HMH 3HeprHH, Ha OCHOBe ,nBYX HaqaJl -
<PYH,naMeHTaJlbHbIX 3aKOHOB, YCTaHOBneHHbIX B pe3Y.JIbTaTe 0606eHH.H orpOM-
Horo qHCJla onbITHbIX <paKTOB, He HCnOJlb3YSI KaKYIO-JlH60 onpe,neJleHHYIO MO-
,nenb CTpOeHH.H BeeCTBB H He BbICKa3bIBa.H npe)J;nOJlOmeHH.H 0 3aKOHax B3aH-
MOAeHCTBH.H aTOMOB H MOJleKYJI.
B TepMo,nHHaMHKe H3yqa1OTC.H TenJlOBbIe npOI.{eCCbI - npOI.{eCCbI, CB.H3aH-
Hble C H3MeHeHHeM TeMnepaTypbI TeJla, a TaKme C H3MeHeHHeM ero arperaTHO-
ro COCTOSIHH.H.
TepMo,nHHaMHqeCKHH H MOJleKY JI.HpHO- KHHeTHqeCKHH MeTo,nbI, npHMeHSIeMbIe
K OAHHM H TeM me 06'beKTaM, ,nonOJlH.HIOT ,nPyr ,nPyra.
6.2. OCHoBHble llOJIOmeHHH MOJIeKYJIHpHO-KHHeTH'IeCKOH TeopHH
H HX OllLITHoe 060CHOBaHHe
OCHOBHbIe nOJlOmeHH.H MOJleKYJI.HpHO-KHHeTHqeCKOH TeopHH (MKT) BeeCTBa
3aKnlOqalOTC.fI B CJle,nYIOeM:
1) Bce 6eU{ecm6a cocmOSlm U3 Me.n.bllauwux Ilacmuq: Mo.n.eKY.n.. amOM06.
UOli06 U op.. pa30e.n.elilibtX MeJICoy c060u npoMeJICymKaMu.
MOJIeIYJIa - MeJlbqaHIlIa.H YCTOHqHBaSI qaCTHI.{a BeeCTBa, COXpaH.HIOa.H
ero OCHOBHbIe XHMHqeCKHe CBoHcTBa.
119

MOJleKYJlbI, 06pa3YIOrn;He aHHoe BeeCTBO, COBeplIIeHHO O,AHHaKOBbI; pa3-
nH'lHbIe BeeCTBa COCTO.HT H3 pa3nHqHbIX MOJleKYJI. B npHpO)J;e cyrn;eCTByeT
qpe3BbIqaHHO 60JlbliiOe KOJlHqeCTBO pa3J1HqHbIX MoneKY n.
MoneKYJlbI COCTO.HT H3 60Jlee MenRHX 'IaCTH - aTOMOB.
ATOMLI - MeJlbqaHIIIHe qaCTHbI XHMHqeCKOrO SJleMeHTa, COXpaH.HIOHe ero
XHMHqeCKHe CBOHCTBa.
't.JHCJlO pa3J1H'IHbIX aTOMOB cpaBHHTeJlbHO HeBeJlHKO H paBHO qHCJlY XHMH-
'leCRHX sneMeHTOB (116) H HX H30TonOB (OROJlO 1500).
A TOMbI npel1CTaBJI.HIOT C060H BeCbMa CJlOmHbIe 06pa30BaHH.H. HO KJlaCCHqe-
CRa.H MKT HCnOJlb3yeT MOeJlb aTOMOB B BHe TBepltbIX He,lIeJlHMbIX qaCTH'IeK
ccpepHqeCKOH <}:>OpMbI.
,Z:(oKa3aTeJlbCTBOM MOJleKYJI.HpHOrO CTpOeHH.H BeeCTBa HBJI.HIOTC.H HcpcpY3H.H,
3aROHbI coxpaHeHH.H MaCCbI, RpaTHblX oTHolIIeHHH H KpaTHbIX 06'beMOB npH
XHMHqeCKHX peaKH.HX. Ha6nlOaeMbIe .HBJleHHSI npOHHaeMOCTH, cmHMaeMOCTH
H paCTBopHMOCTH Bern;eCTB CBHeTeJlbCTBYIOT 0 TOM, qTO OHH He CnJlOIIIHbIe, a
COCTO.HT H3 OTeJlbHbIX, pa3eneHHbIX npOMemYTKaMH 'IaCTH. C nOMOblO CO-
BpeMeHHbIX MeTOOB HccneOBaHHSI (SJleKTpOHHbIH H 30HOBbIH MHKpOCKonbI)
Y,AaJIOCb nonY'IHTb H306pameHH.H MOJleKYJI.
2) q acmUijbl 6 6e14eCm6e c6Sl3aIi bl iJpyz c OPYZOM cu.n.aMU Mo.n.eKy.n.SlpIiOZO
63auMoiJeucm6uSl - npUmSlJlCeliUSl U omma.n.KU6aIiUSl.
3TH CHJlbI 3aBHC.HT OT paCCTO.HHH.H Memy qaCTHaMH (CM.  6.5).
OnbITHbIM l10Ka3aTeJlbCTBOM SToro nOJlOmeHH.H .HBn.HeTC.H HaJlHqHe npOqHOC-
TH H ynpyrocTH TeJl, nOBepXHOCTHoro HaT.HmeHH.H B mHl1ROCT.HX H T.l1.
3) M o.n.eKy.n.bl liaXOOSlmcSl 6 lienpepbl61i0M 6ecnopSliJoilHOM (men.n.060M) iJ6U-
JlCeHUU.
BH TennOBoro BHmeHH.H (nocTynaTenbHoe, KOJle6aTenbHoe, Bparn;aTenbHOe)
MoneKYJl 3aBHCHT OT xapaKTepa HX B3aHMoeHCTBH.H " H3MeH.HeTC.H npH nepe-
xoe BeeCTBa H3 OHoro arperaTHoro COCTO.HHH.H B pyroe.
I1HTeHcHBHoCTb TenJlOBOrO BHmeHH.H onpeeJl.HeT CTeneHb HarpeTOCTH TeJla,
xapaKTepH3yeMYlO TeMnepaTYPOH TeJla.
,Z:(OKa3aTeJlbCTBOM TenJlOBOrO BHmeHH.H. MOJleRYJl .HBJI.HeTC.H 6pOYHOBCKoe
BHmeHHe, ,AHcpCPY3H.H, HcnapeHHe Bern;eCTB H ,Ap.
6.3. )1.H«p«PY3HH
AHY3HJI - caMonpOH3BOJlbHOe B3aHMHoe npOHHRHOBeHHe MoneKYJI co-
npHKaCalOrn;HXC.H BemeCTB. npHBO.Hmee K BbIpaBHHB8HHIO ROHueHTpaUHH Be-
rn;eCTB8 no BceMY 06'beMY. IIpH HcpcpY3HH MoneKynbI rp8HHqarn;HX Memy co-
60ft TeJl, HaXO.HCb B HenpepbIBHOM BHmeHHH, npOHHK81OT B MemMoneKYJl.Hp-
HbIe npOMemYTKH pyr pyr8 H pacnpeen.HIOTC.H Mem,Ay HHMH.
,Z:(HCPCPY3H.H npO.HBJI.HeTC.H BO Bcex Tenax - B ra3fiX, mHKOCT.HX, TBepbIX
TeJl8X, HO B pa3HoH CTeneHH.
,Z:(HcpcpY3HIO B ra3ax MomHO 06H8pymHTb, eCJlH, HanpHMep, cocy C n8XY-
qHM ra30M OTKpbITb B nOMeeHHH. '-Iepe3 HeKOTopoe BpeM.H ra3 pacnpOCTpa-
HHTC.H no BceMY nOMern;eHHIO.
,Z:(HCPCPY3H.H B mHKOCT.HX npOHCXOHT 3HaqHTenbHO MeneHHee, qeM B ra-
3ax. HanpHMep, ecnH B CTaKaH HanHTb CHa'I8n8 cnOH paCTBopa MeHoro
Kynopoc8, 8 3aTeM O'IeHb OCTopomHO 06aBHTb cnOH BO,AbI H OCTaBHTb CT8-
120

KaH B nOMeeHHH C HeH3MeHHOH TeMnepaTYPOH, TO qepe3 HeKOTOpOe BpeM.H
HCqe3HeT pe3Ka.H rpaHHn;a Memy paCTBOpOM MeHOrO Kynopoca H BOOH, a
qepe3 HeCKOJlbKO HeH mH)J;KOCTH nepeMelIIaIOTC.H.
Aml>«PY3H.H. B TBep)J;bIX TeJlaX npOHCXOHT ee MeJleHHee, qeM B mHKO-
CT.HX (OT HeCKOJlbKHX qaCOB O HeCKOJlbKHX JleT). OHa MomeT Ha6J1IO,AaTbC.H
TonbKO B XOpOlIIO OTlIIJlH«poBaHHbIX TeJlaX, Kor)J;a paCCTO.HHH.H Memy nOBepx-
HOCT.HMH OTlIIJlH«poBaHHbIX Ten 6J1H3KH K MemMOJleKYJI.HpHOMY paCCTOSlHHIO
(10- 8 CM). TIpH 3TOM CKOpOCTb H«P«PY3HH YBeJlHqHBaeTC.H npH nOBbIlIIeHHH TeM-
nepaTypbI H aBJleHH.H.
Pa3HOBHHOCTblO H«P«PY3HH .HBJI.HeTC.H OCMOC - npOHHKHOBeHHe mHKOCTeH
H paCTBopOB Qepe3 nopHCTYIO nOJlynpOHHIJ;aeMYIO neperopOKY.
AH«P«PY3HjJ Ii OCMOC lirpalOT 60JlblIIYIO pOJlb B npHpoe II TeXHHKe. B npM-
pO)J;e 6J1aroap.H H<P«PY3HH, HanpHMep, ocyeCTBJI.HeTC.H nHTaHHe paCTeHHH H3
nOqBbI. OpraHH3M qeJlOBeKa M mHBOTHbIX BcaCbIBaeT 'lepe3 CTeHKH nHeBapH-
TeJlbHOrO TpaKTa nHTaTeJlbHbIe BeI.QeCTBa. B TeXHHKe C nOMOI.QblO )J;H«P«PY3HH,
HanpHMep, nOBepXHOCTHbIH CJlOH MeTaJIJlHqeCKHX H3eJlHH HaCbIaeTC.H yrJlepo-
OM (IJ;eMeHTaIJ;H.H) M T..
6.4. BpOYHoBcKoe ABH:lKeHHe
BpOYHOBCKoe ,ABHmeHHe OTKpbITO B 1827 r. aHrJlHHCKHM 60TaHHKoM P. Bpo-
YHOM, TeOpeTHqeCKOe 060cHoBaHHe C TO'lKH 3peHH.H MKT aHO B 1905 r.
A.8HHlIITeHHoM 11 M. CMOJlYXOBCKMM.
BpOYHoBcKoe ,l(BHJKeHHe - 3TO 6ecnOp.HOqHOe BHmeH1Ie Menb'l8HlIIHX
TBepbIX qaCTHI1;, "B3BelIIeHHbIx" B mHKOCT.HX (ra3ax).
"B3BelIIeHHble" qaCTHI1;bJ - 3TO qaCTHIJ;bI, nJlOTHOCTb BeeCTBa KOTOpblX
cpaBHMMa C nJlOTHOTblO cpebI, B KOTOpOH OHM HaXO.HTC.H.
An.H 6pOYHOBCKoro ,nBHmeHH.H xapaKTepHO cJleYlOrnee:
1) 6POYHoBcKHe qaCTHI1;bI COBeplIIalOT HenpepbIBHoe XaOTHqeCKOe BHmeHHe.
MHTeHCHBHOCTb KOToporo 38BHCHT OT TeMnepaTypbl Ii OT pa3MepOB 6POYHOB-
CKOH qaCTHltbI;
2) TpaeKTOpH.H lIBHmeHH.H 6pOYHOBCKOH qaCTHUbI OqeHb CJlOmHa.H. He 3aBM-
CMT OT npHpo,nbI BeeCTBa qaCTHIJ; H BHelIIHHX YCJlOBHH.
Ha pHCYHKe 6.1 nOKa3aHbi nOJlome-
HH.H 6pOYHOBCKOH qaCTHI1;bI qepe3 Ka>K-
,nbIe 30 c, a H8 Bpe3Ke - nOJlOmeHH.H
3TOH qaCTHIJ;bI Ha yqacTKe AB, 3acpHK-
CHpOBaHHbIe qepe3 KamylO ceKYHY .
8TH nOJlOmeHH.H coeHHeHbI np.HMbIMM
JlHHH.HMH. TpaeKTOpH.H me qaCTHI1;bI
ee 60Jlee CJlOmHa.H;
3) 6pOYHoBcKoe BHmeHHe Ha6J11O-
aeTC.H B mHKOCT.HX H ra3ax.
TIpHqHHaMH 6POYHoBcKoro BHme-
HMjJ .HBJI.HIOTC.H:
BHmeHHe MOJleKYJI cpebI, B KOTOpOH HaXOHT-
J ...-
 I D  JI
..... .... I\t
-.....-  l. 
I I · "'"  - .
I I  " .I
.... '....
:)<f 
.Jtt.f'_
I-I-'
B
Puc. 6.1
1) TenJlOBOe XaOTHqeCKOe
CH 6POYHoBCKa.H q8CTHIJ;a;
2) OTCYTCTBHe nOJlHOH KOMneHCaIJ;MH Y8POB MOJleKyn
C pa3nHqHbIX CTOpOH, TaK KaK )J;BHmeHHe MOJleKYJI HOCHT
cpebI 06 3TY 'laCTHIJ;Y
cnyqaHHbIH xapaKTep.
121

BHmYI.I..\HeCH MOJleKYJlbI >KHKOCTH npH CTOJIKHOBeHHli C KaKHMH-nH60 TBep-
bIMH qaCTHn;aMH nepealOT HM HeKOTOpOe KOJIHqeCTBO BHmeHH.H. ECJlli qac-
THn;a OCTaTOqHO BeJlHKa. TO qHCJIO MOJleKYJI, HaJleTalOl.I..\HX Ha Hee CO BCeX
CTOpOH, qpe3BbIqaHHO BeJIHKO, HX yapbI B KambIH ,AaHHbIH MOMeHT KOMneH-
ClipYIOTCH, H TaKa.H qaCTlin;a npaKTliqeCKH OCTaeTC.H HenOBHmHOH. Ho eCJIH
'-IaCTHn;a OCTaTOqHO MaJla, KOMneHCan;HH MomeT OKa3aTbC.H HenOJIHOH. CJIY-
qaHHO C O.lI.HOH CTOpOHbI 0 qaCTlin;y yapHT 3aMeTHO 60Jlblliee qHCJIO MOJIeKYJI.
qeM C ,npyrOH. H B pe3YJIbTaTe qaCTliUa npH.lI.eT B LlBlimeHlie.
BpOYHOBCKOe ,nBlimeHHe CBH,neTeJIbCTByeT, qTO MOJIeKYJIbI ,neHCTBHTeJIbHO cy-
l.I..\eCTBYIOT Ii qTO OHH HenpepblBHO II XaOTHqeCKH BHmYTC.H.
6.5. B3aHMoAeHcTBHe MeAY aTOMaMH H MOJIeKYJIaMH BeeCTBa.
CTpoeHHe TBepALIX, HAKHX H ra3006pa3HbIx TeJI
Memy MOJIeKYJIaMH BeeCTBa eHcTBYlOT OHOBpeMeHHO CliJlbI npHT.Hme-
HH.H H CliJIbI OTTaJlKHBaHH.H. 3TH CHnbI B 60JIblliOH CTeneHH 3aBHC.HT OT pac-
CTOHHliH Mem,ny MoneKYJIaMli.
Cor naCHO 3KcnepHMeHTa'1bHbIM li TeOpeTliqeCKliM liCCJIe,nOBaHHHM MemMOJIe-
KYJI.HpHbIe CliJlbI B3aHMo,neHCTBli.H 06paTHO nponopn;liOHaJIbHbI n-H CTeneHH pac-
CTOHHHH Mem,ny MCJIeKYJIaMH:
1
Fr - i-.
r n
r,ne ,AJI.H CliJI npHT.HmeHliH n = 7. a JIH CHn OTTanKHBaHHH n = 9 + 15.
B3aHMO.lI.eHCTBHe .lI.BYX MOJIeKYJI MomHO onlicaTb npH nOMOH rpacpliKa 3aBI1-
CHMOCTH npoeKn;HH paBHoeHcTBYlOl.I..\eH F r CH]] npliTHmeHH.H H OTTaJIKHBaHHH MO-
neKYJI OT paCCTOHHH.H r Mem.lI.Y HX n;eHTpaMH. HanpaBHM OCb r OT MOJIeKYJIbI 1.
n;eHTp KOTOpOH COBna,naeT C HaqaJIOM Koop,nliHaT, K HaXOLI.HIIleMYC.H OT Hero Ha
paCCTOHHHII rl n;eHTpy MOJleKYJIbl 2 (pHC. 6.2, a).
By,neM CqllTaTb. qTO MOJIeKYJla 1 HenO.lI.BlimHa, a MOJIeKYJIa 2 113MeH.HeT
CBoe nOJIOmeHlie OTHOCHTeJIbHO MOJIeKYJIbI 1.
Tor,na npOeKn;Il.H CHJlbl OTTaJIKllBaHHH MOJIeKYJIbI 2 OT MOJIeKYJIbI 1 Ha
OCb r 6Y,AeT nOJlOmllTeJlbHOH. llpOeKn;H.H ClinbI npllT.HmeHH.H MOJIeKYJlbI 2 K
MOJIeKYJIe 1 6y,neT OTplln;aTeJIbHoH.
a
6
CHn8 S38HMO,AeHCTBHSI
Fr 
r
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F:Ep '
.
r
r 1
Puc. 6.2
122

CHJlbI OTTaJlKHBaHH.H (pHC. 6.2, 6) ropa30 60JIbille CIIJI npHT.HmeHIIH Ha
MaJlbIX paCCTO.HHHHX (r < ro), HO ropa30 6blcTpee y6b1BalO'I' C YBeJlHqeHHeM r.
CHJIbI npIIT.HmeHHH Tome 6bICTpO y6bIBaIOT C YBeJlHqeHHeM r, TaK '1'1'0, Ha'lII-
Ha.H C HeKOToporo paCCTO.HHII.H r m' B3aHMO.lI.eHCTBHeM MOJIeKYJI MomHO npeHe6-
pe'lb. HaH60nbillee paCCTO.HHHe r m' Ha KOTOpOM MOJIeKYJIbI ern;e B3aHMO.lI.eH-
CTBYIOT, Ha3bIBaeTCH pa,n;HYCOM MOJleKyn.HpHOrO ,n;eHCTBH.H (r m - 1,57. 10- 9 M).
IIpH r= ro CHJIbI OTTanKIIBaHH.H no MO,AYJlIO paBHbJ CHJlaM npHT.HmeHH.H.
PaCCTO.HHHe ro cOOTBeTcTByeT YCTOHqHBOMY paBHoBecHoMY B3aHMHoMY nOJlo-
:>KeHHIO MOJleKYJI.
B pa3J1H'IHbIX arperaTHbIx COCTOHHIiHX Bern;ecTBa paccToHHHe Me:>K,n;y ero MO-
JIeKYJIaMH pa3J1HqHo. OTCIO,n;8 H pa3JIH'IHe B CHJlOBOM B3aHMO,AeHCTBIiH Mone-
KYJI H cyrn;ecTBeHHoe pa3JIH'IHe B xapaKTepe ,n;BIi:>KeHH.H MOJIeKYJI ra30B. :>KH,n;-
KocTeH H TBep,n;bIx TeJl.
B za3ax paCCTO.HHH.H Me:>K,AY MOJIeKYJlaMH B HeCKOJlbKO pa3 npeBbIIlIalOT pa3-
MepbI caMHX MOJleKYJI. BCJIe,n;CTBHe 3Toro CIiJIbl B3aHMo,n;eHCTBHH Mem,AY MOJle-
KynaMH ra3a MaJlbI II KHHeTHqeCKa.H 3HeprHH TenJIOBOrO ,ABIi:>KeHIi.H MOJleKYJI
HaMHoro npeBbIIlIaeT nOTeHII8JlbHYIO 3HeprHIO 1iJ( B3aHMO.lI.eHCTBHH. Ka:>K.lI.aSl MO-
neKYJla ,ABHmeTC.H CB060.lI.HO OT ,APyrHJ( MoneKYJI C orpoMHbIMH CKOpOCT.HMH (COT-
Hli MeTpoB B ceKYHY). MeH.H.H HanpaBJIeHHe H MO)J;YJIb CKOpOCTH npH CTOJIKHO-
BeHH.HX C ,APyrHMH MOJleKYJlaMH. )J;JIIiHa cB060,AHOrO np06era A MOJIeKYJI ra3a
3aBliCHT OT ,n;aBJleHIi.H H TeMnepaTypbI ra3a. IIpH HOpMaJlbHbIX YCJIOBHHX
A - 10- 7 M.
B :HCUOKOCmSlx paCCTO.HHHe Mem,AY MOJIeKYJlaMH 3Ha'lHTeJIbHO MeHbIlIe, qeM
B ra3ax. CHJlbJ B3aHMo,n;eHCTBH.H Mem,n;y MOJIeKYJlaMI1 BeJlHKH, H KHHeTH'IeC-
KaH 3HeprH.H ,ABlimeHH.H MOJIeKYJI cOli3MepHMa C nOTeHHaJIbHm1 3HeprIIeH HX
B3aHMO,AeHCTBH.H, BCJIe,n;CTBHe qero MOJleKYJlbI mH,n;KOCTH COBepilialOT KOJIe6a-
HIi.H OKOJlO HeKOToporo nOJlOmeHIi.H paBHOBeCH.H, 3aTeM CKR'IK006pa3HO nepe-
XO,A.HT B HOBbIe nOJlOmeHH.H paBHOBeCH.H qepe3 O'leHb MaJlbIe npOMemYTKH Bpe-
MeHH (10- 8 c). qTO npHBO.lI.HT K TeKyqeCTH :>KH.lI.KOCTH. TaKHM 06pa30M, B mli,n;-
KOCTH MOJleKYJIbI COBepilialOT B OCHOBHOM KOJle6aTeJlbHbIe Ii nocTynaTeJIbHbIe
,n;BIi:>KeHHH.
B m6epOblX me.n.ax CHJlbI B3aHMo,n;enCTBIi.H Mem,n;y MoneKYJlaMH HaCTOJlbKO
BenHKH, qTO KHHeTHqeCKa.H 3HeprH.H ,ABHmeHHH MOJleKYJI HaMHoro MeHbille no-
TeHIIaJlbHoi:f 3HeprHH HJ( B3aHMo,n;eHCTBH.H. MOJIeKYJIbI COBepilialOT JlHilib KO-
JIe6aHH.H C MaJlOH aMnJlHTy,n;OH OKOJlO HeKOToporo nOCTO.HHHOrO nOJlOmeHH.H
paBHOBeCH.H - Y3JIa KpHCTaJlJIH'IeCKOH peIlIeTKH.
6.6. Macca MOJIeKYJl
Onpe,n;eJlHTb Maccy MOJIeKYJlbI 06bI'IHbIM nYTeM, T.e. B3BeIlIHBaHHeM, KOHeq-
HO, HeB03MomHO. OHa ,AJlH 3Toro CJlHiliKOM MaJla. B HaCTO.Hee BpeM.H cyrn;e-
cTByeT MHoro MeTo,n;OB onpe,n;eJleHH.H Macc MOJIeKYJI, B '1aCTHOCTH, C nOMOrn;blO
Macc-cneKTporpacpa (CM.  11.14) onpe,n;eJleHbI MaCCbI mo Bcex aTOMOB Ta6J11i-
U;bI MeH,n;eneeBa.
TaK, ,n;JlH H30Tona yrJlepO,Aa lC mo= 1,995' 10- 26 Kr.
IIOCKOJIbKY MaCCbI aTOMOB H MOJIeKYJI qpe3BbI'IaHHO MaJIbl, TO npli pac'Ie-
Tax 06bI'IHO HCnOJIb3YIOT He a6COJlIOTHble, a OTHOCHTeJIbHbIe 3HaqeHIi.H Macc,
nonY'IaeMbIe nYTeM cpaBHeHH.H Macc aTOMOB H MOJleKYJI C aTOMHOH e,AHHHeH
123

1
MaCCbl, B KaqeCTBe KOTOpOH BbI6paHa qaCTb MaCCbI aTOMa H30Tona yrJle-
12
poa le:
1 _
1 a. e. M. = - moc= 1.660 . 10 27 Kr.
12
OTHOCHTeJlLHOH MOJleKYJI.HpHOH (HJlH aToMHoH) MaCCOH M r Ha3bIBalOT Be-
nHqHHY, nOKa3bIBalOYIO, BO CKOJlbKO pa3 Macca MOJleKYJlbI (HJlH aTOM a) 60nb-
me aTOMHOH eHHHbI MaCCbI:
OTHOCHTeJlbHa.H
BeJlHqHHOH.
OTHOCHTeJlbHbIe aTOMHbIe MaCCbI Bcex XHMHqeCKHX 3J1eMeHTOB YKa3aHbI B
Ta6nJne MeH,neJleeB8.. TaK, y Boopoa OHa paBHa 1,008, y reJlH.H - 4,0026.
ITpH paC'leTaX OTHOCHTenbHYIO aTOMHYIO Maccy oKpyrJlHIOT l1.0 6J1HmaHlIIero u.e-
norD qHCna. HanpHMep, Y Bo,nopo,na ,no 1, Y reJIH.H o 4
Omnocume.nbliaJl .MOJleKYJ/"RpliaR .Macca oalinozo 6e14eCmea pa6lia cY.M.Me om-
1i0cume.n.blibtX amO.MlibtX .Macc 3.ne.Melim06. 6XOOSl14UX 6 cocmae .Mo.n.eKy.n.bt oali-
1i0ZO 6e14ecm6a. Ee paCCqHTbIBaIOT, nonb3YHCb Ta6nHl.\eH MeHeJleeBa H XHMH-
qeCKOH «POPMYJlOH BeeCTBa.
TaK, ,nJl.H BObI H 2 0 OTHOCHTeJlbHaH MOJleKYJI.HpHa.H Macca paBHa M r = 1.2 +
+16=18.
M = mo
r 1
-'mo
12 c
MOJleKYJI.HpHaH (aToMHa.H)
(6.1)
Macca .HBJIHeTCH 6e3pa3MepHoH
6.7. KOJIH'IeCTBO BeeCTBa. nOCTO.HHHa.H ABOra,IJ;pO
KOJlH'leCTBO BeeCTBa, co,nepmaerOC.H B Tene, onpe,nen.HeTC.H qHCJlOM MO-
neKyn (HnH aToMoB) B 3TOM TeJle. ITOCKOJlbKY qHCnO MOJleKYJI B MaKpOCKonH-
qeCKHX TeJlaX OqeHb BeJlHKO, ,nn.H onpeeJleHH.H KOnHqeCTBa BeeCTBa B Tene
cpaBHHBalOT 'lHCnO MOJleKYJl B HeM C qHCnOM aTOMOB B 0,012 Kr H30Tona
yrnepoa le.
KOJlH'IeCTBO BeeCTBa v BenHqHHa, paBHa.H OTHOlIIeHHIO qHCJla MOJle-
KYJI (aToMoB) N B ,naHHOM TeJle K qHcny aTOMOB NAB 0,012 Kr H30Tona
yrJlepo,na le:
N
V=-.
N A
B ell e,nHHHueH KOJlHqeCTBa BemeCTBa .HBJI.HeTC.H MOJlb. 1 MOJIL - KOJlH-
qeCTBO BeeCTBa, B KOTOpOM CO)J;epmHTC.H CTOJlbKO me CTPYKTypHbIX 3neMeHTOB
(aToMoB, MOJleKYJI, HOHOB), CKOJlbKO aTOMOB B 0,012 Kr H30Tona yrnepoa IC.
'llHCJlO qaCTHl.\ B O,nHOM Mone BeeCTBa Ha3bIBaeTCH nOCTO.HHHOH ABorpo.
(6.2)
N _ 0,012 .
A - ,
ma c
N - 0,012 _ 6 02 . 10 23 -1
A - - , MOJlb .
1.995 . 10- 26
(6.3)
124

TaKHM 06pa30M, 1 Monb nlO60ro BeeCTBa CO,AepmHT O,AHO H TO me tIHC-
no tIaCTH - N A qaCTH. TaK KaK Macca mo tIaCTHbI Y pa3HbIX BeIneCTB
pa3nHtIHa, TO H MaCca N A tIaCTH Y pa3nHqHbIX BeeCTB pa3J1HtIHa
Maccy BeeCTBa, B3.HTOrO B KOnHtIeCTBe 1 Monb, Ha3bIBalOT MOJUlpHOH Mac-
COH M:
M=moN'A'
(6.4)
B CM e,AHHHeH MOn.HpHOH MaCCbI .HBn.HeTC.H KHnorpaMM Ha Monb ( )
M::b .
Mcnonb3YSl <}JopMynbI (6.1)-(6.4), MomHO YCTaHOBHTb CB.H3b Mem,Ay MOn.Hp-
HOH MaccoH M H OTHocHTenbHoH MoneKyn.HpHOH MaccoH M r :
M=M r .10- 3 .
TaK, MoneKYJlSlpHa.H Macca yrJleKHcnoro ra3a 44, MOJl.HpHa.H 44. 10- 3
Kr
MOJIb
3HaSl Maccy BeecTBa Hero MOJl.HpHYIO Maccy M, MomHO HaHTH qHCJlO
MOJIeH (KOJlHtIeCTBO BeeCTBa) B TeJle: v = !!!:..
M
Tor,Aa H3 <}JOPMYJlbJ (6.2) tIHCnO qaCTH B Tene
m
N=vNA=-N A .
M
3HaSl MOJISIPHYIO Maccy 1:1 nOCTO.HHHYIO ABOra,ApO, MomHO paCCtI1:ITaTb Maccy
O,AHOH MOJleKYJlbI:
M m
mo=-=-'
N A N
6.8. Pa3MepbI MOJIeKYJI
Pa3Mep MoneKynbI .HBJI.HeTC.H BeJlHtIHHOH YCJIOBHOH. Ero 0eH1:IBaIOT TaK.
Mem,Ay MOJleKYJlaMH Hap.H,Ay C C1:InaMH npHT.HmeHH.H ,AeHcTBYIOT H CHnbI OT-
TanKHBaHH.H, nOSTOMY MOJleKynbI MorYT C6n1:lmaTbC.H n1:l111b ,AO HeKOToporo pac-
CTO.HH1:I.H d (pHC. 6.3.).
PaCCTO.HHHe npe,AenbHOro C6J1HmeHH.H eHTpOB ,ABYX MOJleKYJl Ha3bIBalOT
sc}>c}>eKTHBHLIM AHaMeTpOM MOJleKYJlLI d (npH STOM CtIHTaIOT, qTO MoneKYJIbI
d 1:IMelOT c<}JepHtIeCKYIO CPOPMY).
B HaCTO.Hee BpeM.H cyecTByeT MHoro MeTO-
,AOB onpe,AeneHH.H pa3MepOB MoneKyn. CaMbIH npo-
CTOH. XOT.H H He caMbIH TOqHbIH. COCTOHT B CJIe-
,AYIOeM. B TBep,AbIX 1:1 mH,AK1:IX Tenax MOJIeKYJIbI
pacnonomeHbI OtIeHb 6JI1:I3KO O,AHa K ,APyrOH. notI-
TH BnJlOTHYIO. IIoSTOMY MomHO Cq1:lTaTb, 'lTO
06'beM V, 3aHHMaeMbIH TenOM HeKOTopOH MaCCbI m,
npH6n1:l31:1TenbHO p8BeH CYMMe 06'beMOB Bcex ero
MoneKyn.
,,-
/
, /
\/
' l:J ' . 
\ . \ /
, / , /
..... /..... /
....-J" ....--"
.,
/
/
/
....
.....
Puc. 6.3
125

V
TOr,lI8 06'beM OLIHOH MOJleKYJIbI 6YLIeT V o = - rLIe V - 06'beM TeJla.
N
m
N = M N A - qHCJlO MOJleKYJI B TeJle. CJle,lIOBaTeJlbHO,
VM
V o =-.
mN A
m
TaK KaK - = p, r,lIe p
V
CqHTa.H, qTO MOJleKYJla
r,lIe r - pa,lIHYC, HMeeM
M
nJlOTHOCTb serqecTBa, TO V O = -. (6.5)
pN A
MaJIeHbKHH wapHK, ,lIHaMeTp KO'l'OpOro d = 2r,
4 3 nd 3
V o = 3 nr = (5'
IIo,lIcTaBHB CIO,lIa 3HaqeHHe V o (6.5), nOnY'lHM
1!d 3 M OTClOl\a d =  6M .
6 pN A npN A
TaK, ,AJI.H BO,lIbI
d - 6 . 18 . 10- 3 - 3 8 . 1 0 -10
- 3 - , M.
3, 14 1. 6,02. 10 23
Pa3MepbI MOJleKYJI pa3J1HqHbIX BerqecTB HeO,lIHHaKOBbI, HO Bce OHH nop.H,lI-
Ka 10- 10 M, T.e. OqeHb MaJIbI-
6.9. eaJIbHLIiI ra3
OCHOBHMe CBOHCTB8 TeJla onpe)J;eJI.HeT COBMeCTHoe ,lIeHCTBHe ,lIBYX KOHKYPH-
PYlOrqHX cpaKTopoB: 1) B3aHMO,lIeHCTBHe MOJIeKYJI, KOTopoe CTpeMHTCSI YlIepmaTb
MOJIeKYJlbI Ha onpe.n;eJIeHHbIX paCCTO.HHH.HX ,lIPyr OT ,APyra; 2) XaOTHqeCKOe Ten-
JIOBOe ,lIBHmeHHe MOJIeKYJI, KOTopoe pa36pacbIBaeT HX no BceMY 06'beMY.
KaK nOKa3bIBaeT onbIT, MOJleKYJlbI ra3a pacnpe,lIeJl.HIOTCSI no BceMY npe,lIO-
CT8BneHHOMY ,AJISI Hero 06'beMY. CJIe,lIOBaTeJlbHO, rJIaBHYIO pOJIb B nOBep;eHHH ra3a
HrpaeT XaOTH'leCKOe ,lIBHmeHHe MOJleKYJI. a CHnbI B3aHMOlIeHCTBHSI MaJlbI. H HMH
MomHO npeHe6peqb. 8TO 03HaqaeT, qTO MOJIeKYJla ra3a LIBHmeTC.H paBHOMepHO H
np.HMOJlHHeHHO, nOKa He CTOJlKHeTC.H C ,lIpyroH MOJleKYJlOH. IIpH CTOJIKHOBeHHH
H3MeH.HeTC.H BeJlHqHHa H HanpaBJleHHe CKOpOCTH ,lIBHmeHHS:I MOJIeKYJIbI, H OHa CHO-
Ba .n;BHmeTCS:I paBHOMepHO H np.HMOJIHHeHHO ,lI0 CJle,lIYlOrqero CTOJIKHOBeHH.H. .D:JIH-
Ha CB060,lIHOrO np06era (paCCTO.HHHe Mem,lIY ,lIBYMS:I nOCJle,lIOBaTeJlbHbIMH CTOJIKHO-
BeHH.HMH MOJleKYJIbI) A - 10- 7 M. IIpH TaKoH ,lInHHe CB060,lIHOrO np06era TOJIbKO
0,04 % npOCTpaHCTBa, 3aH.HTOrO ra30M, npHXO,AHTCSI Ha c06cTBeHHbIH 06'beM
ero MoneKYJI. 8TO ,lIaeT npaBo BOCnOJlb30BaTbC.H MO,lIeJIblO H,lIeaJlbHOrO ra3a.
eaJILHLIH ra3 - 3TO ra3 C ,lIOCTaTOqHO npOCTbIMH CBoHcTBaMH:
1) c06cTBeHHbIM 06'beMOM MOJIeKYJl MomHO npeHe6peqb no cpaBHeHHIO C
06'beMOM cocY,lIa, B KOTOpOM HaXO,lIHTCS:I ra3;
126

2) MemAY MoneKYJlaMH HAeaJlbHOrO ra3a HeT CHn B3aHMoAeMCTBH.H; OHH
AeHcTBYIOT TonbKO npH CTOJlKHOBeHH.HX MoneKyn;
3) MoneKynbI HAeaJlbHOrO ra3a BeAYT ce6.H npH CTonKHOBeHH.HX KaK a6co-
nlOTHO ynpyrHe llIapHKH.
ITpH He60nbllIHX AaBneHH.HX H He OqeHb HH3KHX TeMnepaTypax peanbHbIe
ra3bI 6J1H3KH K HAeaJlbHOMY ra3Y. ITpH BbICOKHX AaBneHH.HX MoneKynbI ra3a
HaXOA.HTC.H TaK 6J1H3KO, 'lTO Me*AY HHMH B03HHK81OT 3aMeTHble cunhl B3au-
MOAeHCTBH.H. ITpeHe6peqb HX co6CTBeHHbIM 06'beMOM Henb3.H, H ra3 y*e He
.HBn.HeTC.H HlIeanbHbIM.
,l],n.H onHCaHH.H CBOHCTB ra30B MO*HO nonb30BaTbC.H: 1) MUKpocKonullecKu
MU napaMempaMu (CKOpOCTb, Macca MoneKynbI, ee 3HeprH.H HAP.), KOTopbIe
.HBn.HIOTC.H xapaKTepHCTHKaMH MoneKyn H cpeAHHe qHCneHHbIe 3HaqeHH.H KOTO-
pbIX HaXOA.HTC.H TonbKO paCqeTHbIM nYTeM; 2) MaKpOCKOnUl{,eCKUMU napaMem-
paMu (AaBneHHe. TeMnepaTypa, 06'beM ra3a), 3HaqeHHe KOTOpbIX onpeAen.HeTC.H co-
BMeCTHbIM AeHcTBHeM orpOMHoro 'lHCna MoneKyn. MaKponapaMeTpbl - 3TO napa-
MeTpbI, xapaKTepH3YIOHe COCTO.HHHe CHCTeMbI (ra3a) B enOM H He HMelOHe
cMbIcna B npHMeHeHHH K OTAenbHbIM qaCTHu.aM CHCTeMbI. 'llHcneHHbIe 3Ha'4e-
HH.H HX HaXOA.HT H3MepeHueM C nOMOlliblO npH60pOB u paCqeTHhlM nYTeM.
AaBneHue ra3a p - 3TO CpeAH.H.H cun8 YAapOB ero MoneKyn 0 Teno (Ha-
npHMep, 0 CTeHKH COCYAa), OTHeCeHHa.H K eAHHHu.e nOBepXHOCTH Tena.
A6conIOTHaH TeMnepaTYpa T - Mepa cpeAHeH KHHeTHqeCKOH 3HeprHH Ten-
nOBoro ABHJKeHH.H MoneKynbI (CM.  6.12).
IToA 06'beMOM ra3a V nOHHMalOT BHYTpeHHHH 06'beM COCYAa, B KOTOpOM
HaXOAHTC.H ra3.
6.10. CKOpOCTH MOJIeIYJI ra30B
.L(BHJKeHHe MoneKyn r83a nOA'4HH.HeTC.H 3aKOH8M CTaTHCTH'tleCKOM cpH3UKH.
B KaJKAbIH MOMeHT BpeMeHH CKOpOCTH OTAenbHhIX MoneKyn MorYT 3Ha'4HTenb-
HO OTnH'48TbC.H APyr OT APyr8, HO HX cpeAHHe 3H8'4eHH.H OAHHaKOBbI H npH
p8C'4eT8X Hcnonb3YIOTC.H He MrHOBeHHble CKOpOCTH OTAenhHbIX MoneKyn, 8 He-
KOTopble cpeAHHe 3HaqeHH.H. Pa3nHqalOT cpeAHlO1O apHcpMeTUqeCKYIO <v> H cpeA-
HIOIO KBaApaTHqHYIO <V KB > CKOpOCTH XaOTHqeCKOrO ABHmeHH.H MoneKyn.
ITYCTh UMeeTC.H N MoneKyn. CKOpOCTH KOTOpbIX COOTBeTCTBeHHO Vh V2' ..., VN'
CpeAH.H.H 8pHcpMeTHqeCK8.H CKOpOCTb XaOTHqeCKOrO ABHmeHU.H MoneKyn (npH
rpy60M npH6J1HmeHHH) no Mon;ynlO onpelIen.HeTC.H KaK CYMM8 MOlIyne:H: CKOpOC-
Te:H: MoneKyn ra3a, AeneHHR.H Ha HX 06ee qHCno:
( ) _ VI + V2 + .. . + v N
V - .
N
Bonee cTporoe onpelIeneHue cpenHeH CKOpOCTH n8HO B  6.15.
CpeAH.H.H KBaApaTHqH8.H CKOpOCTb XaOTHqeCKOrO ABHJKeHH.H MoneKyn
(V KB ) = JP) =
2 2 2
VI + v2 +... + V N
N
(6.6)
rAe <v 2 > - cpeAHuH KBaApaT CKOpOCTH ABHmeHH.H MoneKYJI Ero He cneAyeT
CMellIHBaTb C KBaApaTOM cpeAHeH CKOpOCTU: <v 2 >:f:. «V»2.
127

J8RT J3RT
KaK nOKa3bIBalOT paCqeTbI (CM.  6.15), (v) = -; (V KB ) = - M '
nM
YHllBepC8JIbHaH ra30BaH nOCTO.HHHa.H, M - MOJIHpHa.H Macca.
R
re
6.11. OCHoBHoe ypaBHeHHe MKT H,ZJ;eaJIbHOrO ra3a
3TO yp8BHeHHe CB.H3bIB8eT Me>KY C060H M8KponapaMeTp p COCTO.HHHH He-
aJIbHOrO ra38 H MHKponapaMeTp (V KB ) (HJlH (W K ), x8paKTepH3YIOHH BH>Ke-
HHe MOJleKYJlbI. .D:JIH BbIBO8 OCHOBHoro yp8BHeHHH MKT paCCMOTpHM OHO-
aTOMHbIH HeaJlbHbIH ra3, HaXO.HIlHC.H B TepMOHH8MHqeCKOM paBHOBeCHH,
T.e. B COCTOHHHH, B KOTOpOM Bce MaKpOCKOnHqeCKHe napaMeTpbI OCTaIOTC.H He-
H3MeHHbIMH BO BpeMeHH H no BceMY 06'beMY.
npeCTaBllM ce6e COCY,A B BHe Ky6a C
JlHHOH pe6pa l (pHC. 6.4), B KOTOpOM 6ecno.
p.HOqHO BH>KYTCH N MOJIeKYJI MaCCOH mo
K8>K,A8H. CTeHKH cocya nOBepralOTCH Henpe-
pBHOH 60M6apHpoBKe MOJIeKYJI8MH. Yap
MOJleKYJI 06YCJIOBJlHBalOT aBJleHHe r83a Ha
CTeHKll. BBll,AY 6eCnOpH,AOqHOCTH BllmeHH.H
MOJIeKYJI pe3YJIbTaT HX y8pa 0 CTeHKH TaKOB,
1
KaK eCJlH 6bI Bcex MOJleKyn BHr8n8Cb
3
np.HMOJIHHeHHO Me>KlIY
z
y
Puc. 6.4
x
npaBOH H JleBOH CTeH-
1
KaMH (BOJIb OCll Ox), MOJleKYJI - Me>KY
3
1
nepeHeH H 38HeH CTeHKaMH (BOJIb OCH Oy) H MOJIeKYJI - Memy Bepx-
3
HeH H HH>KHeH CTeHKaMH (BOJlb OCH Oz), npWIeM B nOJIO>KHTeJIbHOM H8npaB-
1
JIeHHH OCH BH>KeTCH '6 MOJIeKYJI.
OT,AeJIbH8.H MoneKyna, JIeTHl.I.ta.H nepneH,AHKYJlHPHO K O.nHOH 113 CTeHOK,
H8npHMep K JleBOH. co CKOpOCTblO V, B pe3ynbTaTe ynpyroro y.n8pa OTCKO-
'lHT H83a, H ee HMnYJlbC H3MeHHTCH H8 6.p = P - PO' HJlH B npOeKUH.HX:
l:1px = mov x - (-mov x ) = 2movx'
3TO H3MeHeHHe HMnYJlbC8 COrJl8CHO BTOPOMY 38KOHY HblOTOHa onpe,AeJlHT
HMnYJlbC CHJlbI ii., eHCTBYIOeH co CTOpOHbI CTeHKH H8 MoneKYJlY: FJ.6.t = lip,
HJIH B npOeKH.HX: F 1 l:1t = 2m o v x ' re l:1t - npOOJImHTeJlbHOCTb yapa. no
TpeTbeMY 38KOHY HblOTOH8 T8K8.H me no MOYJIIO, HO npOTHBOnOJIO>KH8.H no
HanpaBJleHHIO CHJl8 6yeT eHCTBOB8Tb H8 CTeHKY. CJleOB8TeJlbHO, MOYJIb
llMnYJlbC8 CHJlbl, eHCTBYIOeH H8 CTeHKY npH yape OHOH MOJleKYJIbI, paBeH
F 1 l:1t = 2m o v x '
3a npOMe>KYTOK BpeMeHH l:1t 0 CTeHKY nJIOablO S yapHTcH 'lHCJIO MO-
nV
neKYJI N = -, re n - KOHeHTpaH.H MOJIeKYJI ra38, V - 06'beM, paB-
2
HbIH V = Sl = Sv x 6.t. CJleOB8TeJIbHO, N = nSv x l:1t .
2
128

CpeAH.H.H c1:IJla YAapOB BCeX MoneKyn 0 CTeHKY
( F ) = N . 2movx = nSv x l1t . 2movx = nSv2mo.
l1t 2 l1t x
TaK KaK B COCTOHHH1:I TennOBoro paBHOBeCHH Bce HanpaBneHH.H JW.H BeK-
TOpOB CKopocTeR MoneKYJI paBHOnpaBHbI, TO CpeJIHHe 3HaqeHHH KBan;paTOB MO-
AyneH 1:IX npOeKl(1:IR Ha KOOPAHHaTHbIe OCH paBHbI MemAY C060R:
(v x )2 = (V y )2 = (V Z )2.
OTCIOAa cneAyeT, qTO
( V 2 ) = (V X) 2 + (V y / + (V Z ) 2 = 3 ( V x ) 2
H ( V ) 2 = (V 2 )
x 3'
rAe (v 2 ) -- cpeAHee 3HaqeHHe KBaApaTa CKOpOCTH MoneKyn ra3a.
CpeAHee ,AaBJleHHe (T.e. CpeAH.H.H CHna, npHXOA.Hl..QaHC.H Ha eAHHHl(Y nOBepx-
HOCTH) ra3a Ha CTeHKY COCYAa
p = () = nmo (V 2 ).
(6.7)
TaK KaK CpeAH.HH KHHeTHqeCKa.H 3HeprH.H nocTynaTenbHoro ABHmeHH.H MO-
neKY nbI
m ( V2 )
(W K ) = 0 2 . TO
2
P = -n(WI()'
3
(6.8)
YpaBHeHHe (6.7) H 3KB1:IBaJleHTHOe eMY ypaBHeHHe (6.8) Ha3bIBaeTC.H oc-
HOBHLIM ypaBHeHHeM MOJIeKYJIHpnO-KHHeTHqeCKOH TeopHl1 ra30B.
OHO CBH3bIBaeT MaKpOnapaMeTp p C MHKpOnapaMeTpOM (WI() {1:InH ( V 2 ) )
11 nOKa3bIBaeT, qTO nOHHTHe ,AaBneHH.H HMeeT CMbICn CpeAHeH BenHq1:lHbl 1:1 He-
npHMeHHMO K OTAenbHOH MoneKyne.
6.12. TeMIIepaTypa H TenJIOBOe pdBHOBeCHe CHCTeMbI
IToH.HTHe TeMnepaTypM TeCHO CB.H3aHO C nOHHTHeM TennOBoro paBHOBeCHH.
EcnH ABa Tena pa3HOH TeMnepaTypbI npHBeCTH B conpHKOCHOBeHHe. TO, KaK
nOKa3bIBaeT onbIT, MemAY HHMH 6y AeT npOHCXOAHTb TenJIo06MeH -- npOI:(eCC
nepeAaq1:l 3Hepr1:l1:l OT 60nee HarpeToro Tena K MeHee HarpeTOMY, conpOBomAa-
101..Q1:I1)C.H H3MeHeH1:I.HMH p.HAa (}>H3HqeCK1:IX napaMeTpOB. 1..Jepe3 HeKOTopoe BpeM.H
H3MeHeHHe MaKpOCKOnHQeCK1:IX napaMeTpOB Ten npeKpal..QaeTC.H, T.e. Tena np1:l-
XOA.HT B COCTO.HH1:Ie TennOBoro paBHOBeCH.H.
TenJIOBbIM paBHOBeCHeM Ha3bIBalOT TaKoe COCTO.HHHe CHCTeMbI, npH KOTOpOM
Bce MaKpOCKOnHQeCKHe napaMeTpbI CKonb yrOAHO Aonro OCTalOTC.fI HeH3MeHHbIMM.
Bo Bcex qaCT.HX CHCTeMbI Ten, HaXOA.HI..QHXC.H B COCTO.HHMM TenJlOBOrO paB-
HOBeCHH, TeMnepaTypa OAHHaKOBa.
JII06aH TepMOAHHaMMQeCKa.H CHCTeMa MomeT HaXOAMTbC.fI B pa3JIMtIHbIX CO-
CTO.HHM.HX TenJlOBOrO paBHOBeCMH. B KamAOM 1:13 3TMX COCTO.HHHR .n;aBneH1:Ie H
129

06'beM MOrYT 6bITb pa3J1H'tIHbI (HO nOCTO.HHHbI), a me.Mnepamypa U.Meem cmpo.
zo onpeiJe.n.elilioe 3lialleHue.
CJle,nOBaTenbHO, meMnepamypa xapaKmepu3yem cocmoSlliue men.n.060zo pa6
lioeecuSl mep.MoiJulia.MUlleCKOU CUCme.Mbl.
ECJlH npH KOHTaKTe ,nBYX TeJl HHKaKHe HX <pH3H'tIeCKHe napaMeTpbI, Ha-
npHMep 06'beM, ,naBJleHHe, He H3MeH.SIIOTC.H, TO Mem,ny TeJlaMH HeT Tenno06Me-
Ha 11 HX TeMnepaTypbl o,nHHaKOBbI. Cne,nOBaTenbHO TeMnepaTypa KaK MaKpo-
CKOnH'tIeCKHH tPH3HqeCKHH napaMeTp onpe,nen.SIeT B03MomHOCTb Tennonepe,naqH
OT o,nHoro Tena K ,nPyrOMY H HanpaBneHHe Tennonepe,na'tIH.
llOJlb3Y.HCb MOJleKYJl.HpHO-KHHeTHqeCKHMH npellCTaBJleHH.HMH, MomHO ,naTb
60Jlee Harn.H)1HOe TonKOBaHHe TennOBoro paBHOBeCH.H. EcnH npHBeCTH B CO-
npHKOCHOBeHHe ,nBa ra3a C pa3J1HqHbIMH 3HaqeHH.HMH cpe,nHeH KHHeTHqeCKOH
3HeprHH MoneKYJI, TO MOJleKYJlbI, .v;BHmYHeC.H C 60JlbIllHMH CKOpOCT.HMH, CTaJl-
KHBa.HCb C MOJleKYJlaMH ,npyroro ra3a, 6y,nYT HX YCKOp.HTb, caMH npH 3TOM
3aMe,nn.H.HCb. llPOHCXO,nHT nepe,naqa 3HeprHll, nOKa He HacTynllT MOMeHT, Kor-
,na cpe.v;HHe KllHeTHqeCKHe 3HeprHH MoneKyn 060HX ra30B BbIpaBH.HIOTC.H. 8TO
H eCTb COCTO.HHHe TenJlOBOrO paBHOBeCH.H, npH KOTOpOM nepexo,n BHYTpeHHeH
3HeprHH OT o,nHoro ra3a K .v;pyrOMY npeKpaaeTC.H, XOT.H CTonKHOBeHll.H 6ec-
nOp.H.v;O'tIHO ,nBHmYHXC.H MOJleKyn 060HX ra30B 6y,nYT npo,nonmaTbC.H.
TaKHM 06pa30M, npH conpHKOCHOBeHHH ,nBYX TeJl npOHCXO,AHT BbIpaBHHBa-
HHe H TeMnepaTYP, H cpe.v;HllX KllHeTllqeCKHX 3HeprHH MOJleKYJI. ECTeCTBeHHO
npe.v;nOJlOmHTb, 'tITO TeMnepaTypa MomeT CJlymHTb MepOH Cpe,AHeH KllHeTllqeC-
KOH 3HeprHH .v;BHmeHH.SI MOJleKYJI.
2
113 OCHOBHoro ypaBHeHH.H MKT (6.8) HMeeM p = -n(W K ). r,Ae
3
N
n =-.
V
Cne,nOBaTenbHO,
2N
P = "3 V (W K ). OTclO,na
pV =  ( W ) .
N 3 K
(6.9)
MomHO npOBeCTH cne,nYIOHH 3KcnepHMeHT. B3RTb COCy.v;bI C pa3HbIMH ra-
3aMH. Onpe.v;eJlHTb npe.v;BapHTeJlbHO HX 06'beMbI, MaCCbI H paCC'tIHTaTb 'tIHCnO
MoneKyn no <popMYJle N =.!.!2.N A' 3aTeM nOMeCTHTb cocy.v; B TalOHH Jle,n.
M
llocne HacTynneHH.H TenJlOBOrO paBHOBeCH.H onpe,AeJlHTb .v;aBJleHHe p H paCCqH-
pV
TaTb OTHOIlleHHe -. OnbIT nOKa3bIBaeT, 'tITO OHO o.v;HHaKoBo ,nJl.H Bcex ra-
N
30B 3aTeM 3TH COCy,nbI nOMealOT B KHn.HYIO BO.v;y. On.HTb 3TO OTHOIlleHHe
.v;JI.H Bcex ra30B onpe,neJleHHOe, HO 60JlblIIee, T.e. OHO - T.
BBe,n.H K03<P<PHHeHT nponopHOHaJlbHOCTH k, MomHO 3anHcaTb:
pV
- = kT.
N
CpaBHHBa.H STO paBeHCTBO C (6.9), HMeeM
2 3
kT = -(W1J. OTclO,na (W K ) = -kT,
3 2
(6.10)
(6.11)
K03<p(pHIJ;HeHT k = 1,38. 10- 23 Am Ha3bIBalOT nOCTO.SIHHOH BonbMaHa.
K
130

ITo cpopMYJle (6.11) MO.>KHO paCCt.IHTaTb cpeJJ.HIOIO KHHeTHt.IeCKYIO 3HeprHIO
MoneKYJlbl O}lHOaTOMHoro r83a. EcnH .>Ke ra3 MHOrOaTOMHbIH, TO
(W K ) = kT,
2
rJJ.e - t.IHCJlO CTeneHeH cB060JJ.bI, i = 3 - JI.H oJJ.HoaToMHoH MOJleKynbI ra3a.
TeMnepaTypa KOToporo He OqeHb BeJIHKa, i = 5 - AJI.H ,ABYX- H 60JIee aTOMHOH
nHHeHHOH MOJIeKYJIbl, l = 6 - AJI.H TpeX- H 60nee aTOMHOH HenHHeHHOH MOJIeKynbl.
Ha OCHOBaHHH (6.11) MO.>KHO YTBep.>K,AaTb, t.ITO a6co.n.lOmlia.R me.Mnepamypa T
ecmb .Mepa cpeolieu KUliemUlleCKOU 3liepZUU nocmyname.n.bliozo 06UJlCenUR .MO-
Jletey.n.bL.
Yt.IHTbIBa.H (6.11), OCHOBHoe ypaBHeHHe MKT (6.8) JI.H HeaJIbHOrO ra3a
MO.>KHO 3anHcaTb B BHe
p = nkT.
6.13. 113MepeliHe TeMnepaTypbI
"llT06bI H3MepHTb TeMnepaTYPY KaKOrO-JlH60 TeJIa, ero He06xo,AHMO npHBec-
TH B TenJIOBOH KOHTaKT C TepMOMeTpOM. TepMoMeTp - npH60p ,AJI.H H3Mepe-
HH.H TeMnepaTYP. OCHOBHOH t.IaCTblO TepMOMeTpa .HBJI.HeTC.H TepMOMeTpHt.IeCKOe
Teno, rrpHBOHMoe B TenJIOBOH KOHTaKT C 06'beKTOM, TeMnepaTYPY KOToporo
Hao H3MepHTb. B .>KHKOCTHbIX TepMOMeTpax TepMOMeTpHqeCKHM TenOM CJIY-
.>KHT JlH60 PTYTb, JlH60 nOKpaIlleHHbIH cnHpT. B TepMOMeTpax conpOTHBJIe-
HH.H TepMOMeTpHqeCKHM TenOM cnY.>KHT MeT8nJIH'leCKa.H npOBOJIOKa. a TeMnep8-
Typa onpeJJ.eJl.HeTC.H no ee 3neKTpHt.IeCKOMY conpOTHBJIeHHIO. TepMOMeTp He JJ.OJI-
meH HMeTb 60JIbIllOH MaCCbI: MaCCHBHbIH TepMOMeTp H3MeHHT TeMnepaTYPY Toro
Tena, C KOTOpbIM OH npHBe,AeH B TennOBOH KOHTaKT.
TepMOMeTp <pHKcHPyeT CBOIO c06CTBeHHYIO TeMnepaTYPY, paBHYIO TeMnepa-
Type TeJIa, C KOTOpbIM OH HaXOHTC.H B TepMOHHaMHqeCKOM paBHOBeCHH.
,n;JI.H H3MepeHH.H TeMnepaTypbI MO.>KHO BOCnOJlb30BaTbC.H 3aBHCHMOCTblO JIIO-
60H MaKpOCKOnHqeCKOH BenHt.IHHbI (06'beMa, aBJIeHH.H, 3neKTpHt.IeCKOrO conpo-
THBJIeHH.H H p.) OT TeMnepaTypbI. Ha npaKTHKe t.Iae Bcero HCnOJIb3YIOT 3a-
BHCHMOCTb 06'beMa mHJJ;KOCTH (pTYTH, cnHpTa) OT TeMnepaTypbl (mHJJ;KOCTHble
TepMOMeTpbI). He06xoHMO npe.>Ke Bcero C03JJ;aTb TeMnepaTypHYIO IllKaJIY. no-
3BOJI.HIOYIO npHnHCbIBaTb TeMnepaType onpeeJIeHHbIe t.IHCJIa. Y CTpOHCTBO 60JIb-
lIIHHCTBa TepMOMeTpOB OCHOBaHO Ha npe,AnOnOmeHHH, t.ITO nOJIO.>KeHHOe B OC-
HOBY H3MepeHH.H H3HqeCKOe CBOHCTBO TepMOMeTpHqeCKOrO TeJIa nHHeHHO He-
rrpepblBHO 3aBHCHT OT TeMnepaTypbl. ,n;JI.H nOCTpOeHH.H IllKanbI Bbl6HpalOT Be
TaK Ha3bIBaeMbIe penepHbIe TOqKH, KOTOpblM npHnHCbIBaIOTC.H npOH3BOJIbHbIe
3Hat.IeHH.H TeMnepaTYP, a IllKana Me.>KJJ.Y HHMH JJ.eJlHTC.H Ha paBHbIe qaCTH.
3THM YCTaHaBJIHBaeTC.H eHHH1.(a H3MepeHH.H TeMnepaTypbl. B MeTpH'leCKOH CH-
CTeMe JI.H npaKTH'leCKOrO ynoTpe6JIeHH.H npHH.HTa IllKana IJ;eJIbCHj) (Me.>KYHa-
pOHa.H npaKTHqeCKa.H IllKaJla TeMnepaTYP). ITpH rrOCTpoeHHH 3TOH III KaJI bI npH-
HHMaIOT, 'lTO npH HopManbHOM aTMoccpepHOM aBneHHH TeMnepaTypa ITJIaBne-
HII.H Jlba paBHa 0 °C, a TeMnepaTypa KHneHH.H BObI 100 °C (penepHbIe TOt.I-
KH). IllKaJIY Me.>KY TOt.IKaMII 0 II 100 eJI.HT Ha 100 paBHbIX qaCTeH, Ha3bl-
BaeMbIX rpaycaMH. rrepeMeeHHe YKa3aTeJI.H (B mHKOCTHbIX TepMOMeTpax -
KOH1.(a cToJl6HKa mHKocTH) Ha OHO eJIeHHe COOTBeTcTByeT H3MeHeHHIO TeM-
131

nepaTypbl Ha 1°C. 0603HaqeHHe TeMnepaTypbI no
lllKaJle r(eJlbCH.H - t °C.
TaKHe TepMOMeTpbI 06J1aaIOT cyecTBeHHbIMH
HeOCTaTKaMH: 1) Hana30H TeMnepaTYP OrpaHH'leH:
npH HH3KHX TeMnepaTypax mHKOCTH 3aTBepeBaIOT,
npH BbICOKHX HCnap.HIOTC.H; 2) nOKa3aHHSl pa3J1H'lHbIX
TepMOMeTpOB, HanpHMep pTYTHoro H cnHpTOBoro, co-
Bnaa.H npH 0 °C H 100°C, He COBnaaIOT npH
,APyrHx TeMnepaTypax B CHJIY Toro, qTO TeMnepaTYP-
HbIe KOSQ:>Q:>HII.HeHTbI 06'beMHOrO paClllHpeHH.H cnHpTa
H PTYTH no-pa3HOMY 3aBHCSlT OT TeMnepaTypbl.
B OTJlHqHe OT mHJJ:KOCTeH Bce pa3pemeHHbIe
(H.neaJlbHble) ra3bI npH HarpeBaHHH o.nHHaKOBO H3-
MeH.HIOT CBOH 06'beM H aBJIeHHe, npHtIeM ,AaBJIe-
Puc. 6.5 HHe ra3a nponopn;HOHaJlbHO TeMnepaType. CJIeOBa-
TeJlbHO, aBJleHHe ra3a (npH V = const) MomHO
npHHSlTb B KatIeCTBe KOJIHqeCTBeHHoH MepbI TeMnepaTypbI. CoeHHHB cocY,A,
B KOTOpOM HaXO,AHTC.H ra3 (qae BOOpO,A HJIH reJlHH), C MaHoMeTpoM H npo-
rpaYHpoBaB npH60p, MomHO 113Mep.HTb TeMnepaTYPy no nOKa3aHH.HM MaHO-
MeTpa. TaKoH npH60p Ha3bIBaeTC.H ra30BbIM TepMoMeTpoM (pHC. 6.5).
ra30BbIH TepMOMeTp HenpHroeH JlSl onpe,AeJIeHH.H TeMnepaTypbl B 06JIaCTH
BbICOKHX -reMnepaTYP, npH KOTOpbIX npOHCXO)J;HT TepMHqeCKaSl )J;HCCOII.HaII.H.H H
HOHH3aII.H.H, H OtIeHb HH3KHX TeMnepaTYP, npH KOTOpbIX Bce peaJlbHbIe ra3bI
KOH,AeHCHpYIOTC.H .
6.14. A6coJI]{)TnaJl TeMnepaTypnaJl WKaJIa.
A6cOJI]{)THbIH JIY JIb
Bbl60p B KaqeCTBe OCHOBHbIX TOqeK TeMnepaTYPHOH lllKaJlbI TOqeK Ta.H-
HH.H nba H KHneHH.H BO,AbI COBepllleHHO npOH3BOJIeH. IIoJlY"tJ:eHHa.H TaKHM 06-
pa30M TeMnepaTypHa.H lllKaJla OKa3aJlaCb Hey06HoH JI.H TeOpeTHqeCKHX HC-
CJIeOBaHHH .
OnHpa.HCb Ha 3aKOHbI TepMOHHaMHKH, KeJIbBHHY yaJlOCb nOCTpOHTh TaK
Ha3b1BaeMYIO a6coJllOTHYlO TeMnepaTypHYIO lllKaJIY (ee B HaCTO.Hee BpeM.H Ha-
3bIBalOT TepMoHHaMH'leCKOH lllKaJlOH TeMnepaTYP HJIH lllKaJlOH KeJlbBHHa), co-
BepllleHHO He 3aBHC.HYIO Hli OT npHpO,Abl TepMOMeTpHqeCKOrO Tena. Hli OT
H36paHHoro TepMOMeTpH'leCKOrO napaMeTpa. OHaKO npHHII.Hn nOCTpOeHH.H Ta-
KOH lllKaJlbI BbIXOHT 3a npeeJlbl lllKOJlbHOH nporpaMMbI.
Mbl paCCMOTpHM STOT Bonpoc, Hcnonb3Y.H pyrHe co06pameHH.H.
113 (}>OpMYJIbI (6.10) BbITeKalOT JJ:Ba B03MomHblX cnoc06a YCTaHOBneHH.H TeM-
nepaTYPHoH lllKaJlbI: HCnOJIb30BaHHe H3MeHeHHSl JJ:aBJIeHH.H onpeJJ:eJleHHOrO KOJIH.
qeCTBa ra3a npH nOCTOSlHHOM 06'beMe I1JIH H3MeHeHHe 06'beMa npH nOCTO.HHHOM
aBJIeHHH. TaKYIO lllKany Ha3bIBalOT H,ZJ.eaJIbHOH ra30BoH lI1KaJIOH TeMnepaTypbI.
TeMnepaTypa, onpe,AeJl.HeMa.H paBeHCTBOM (6.10), Ha3bIBaeTC.H a6COJI]{)THOH
TeMnepaTYPoH. A6coJlIOTHa.H TeMnepaTypa T He MomeT 6bITb OTpHII.aTeJlbHOH,
TaK KaK CJIeBa B paBeHCTBe (6.10) CTO.HT 3aBeOMO nOJIOmHTeJlbHbIe BeJlH'lH-
HbI (TOtIHee, OHa He MomeT 6bITb pa3HbIX 3HaKOB, OHa MomeT 6b1Tb JlH60
nOJlOmHTeJIbHOH, JIH60 OTpHII.aTeJlbHOR. 8TO 3aBliCHT OT BbI60pa 3HaKa nOCTO-
132

HHHOH k. TaK KaK YCJlOBHJlHCb TeMnepaTYPY TpOHHOH TOqKlI CqHTaTb nono-
mHTeJlbHOH, TO a6COJllOTHaH TeMnepaTypa MOmeT ObITb TOJlbKO nOJIOmHTeJIb-
HOH). CJle,nOBaTeJlbHO, HaHMeHblllee B03MOmHOe 3HaqeHHe TeMnepaTypbl T = °
eCTb TeMnepaTypa, KOr,na ,naBJleHHe HJlH 06'heM paBHbI HYJlIO.
IIpe,neJlbHa.H TeMnepaTypa, npH KOTOpOH ,naBneHlIe H,neaJlbHOrO ra3a oopa-
I..QaeTC.H B HY JIb npH cpHKCHpOBaHHOM 06'beMe HJlH 06'heM H,neaJlbHOrO ra3a CTpe-
MHTC.H K HYJlIO (T.e. ra3 KaK 6bI ,nOJlmeH CmaTbC.H B "TO'lKY") npll HeH3MeH-
HOM ,naBJIeHHH, Ha3bIBaeTC.H a6COJIIOTHbIM HYJIeM. 8TO CaMaSl HH3Ka.H TeMnepa-
Typa B npHpO,ne.
m ( V2 )
ll3 paBeHCTBa (6.11), YtJHTbIBa.H, tJTO (W K ) = 0 , BbITeKaeT (pH3HtJe-
2
CKHH CMbICJI a6COJIIOTHOrO HYJI.H: a6COJL1OmliblU tiYJLb - meMnepamljpa. npu
Komopou i)OJlJICltu npeKpUmUmbC.H menJW60e nocmynameJlbnOe i)6UJICeIiUe MOJle-
KYll. A6cOJlIOTHbIH HYJIb He,nOCTHmHM.
B Mem.n;YHapoJJ:HoH CHCTeMe e,nHHHU (Cll) HCnOJIb3YIOT a6COJIIOTHYIO TepMo.nH-
HaMH'leCKYIO lllKany TeMnepaTYp. 3a HYJIeBYIO TeMnepaTYPY no 3TOH lllKaJIe npH-
HHT a6COJllOTHbIH HYJIb. B Ka'leCTBe BTOpOH onopHoH TO'lKH npllH.HTa TeMnepaTY-
pa, npH KOTOpOH Haxo.n;HTC.H B AHHaMHtJeCKOM paBHOBeCHH Bo,na. Jle,n H HaCbI-
I..QeHHbIH nap, TaK Ha3bIBaeMa.H TpoHHaH TOqKa (no lllKaJle l(eJlbCH.H TeMnepaTY-
pa TpOHHOH TO'lKH paBHa 0,01 °C ). KamAa.H e,nHHHu.a a6COJIIOTHOH TeMnepaTY-
pbI, Ha3bIBaeMa.H KeJlbBHHOM (o603HatJaeTC.H 1 K), paBHa rpa,nycy l(eJlbCH.H.
IIorpyma.H KOJl6y ra30Boro TepMOMeTpa B TaIOHi1 JIe,n, a 3aTeM B KHn.H-
I..QYIO BO.n;y npH HOpMaJIbHOM aTMoccpepHoM ,naBJIeHHH, 06HapymHJIH, qTO .n;aB-
neHHe ra3a BO BTOpOM cJlY"t.!ae B 1,3661 pa3a 60nbllle. qeM B nepBOM. Y'lll-
TbIBa.H 3TO H nOJlb3Y.HCb CPOPMYJlOH (6.10), MomHO onpe.n;eJlHTb, tJTO TeMnepa-
Typa Ta.HHH.H Jlb,na To= 273,15 K.
,Z:(eHcTBHTeJIbHO, 3anHllleM ypaBHeHHe (6.10) .n;JI.H TeMnepdTypbI To TaRHlIH
nbAa H TeMnepaTYPbI KHneHH.H BOAbI ( To + 100):
PI V P2 V
- = kTo; - = k(T o + 100).
N N
Pa3,neJIHM BTopoe ypaBHeHHe Ha nepBoe, nOJIytJHM
P2 = To + 100
PI To
OTCIO,na
1: 0 =  = 100 = 273,15 K.
1, 3661-1
P2 -1
PI
Ha pHCYHKe 6.6 cxeMaTlI1.IeCKI1 nOKa3aHbI lllKana l(eJlbCH.H H TepMo,nllHaMHtJe-
CKa.H lllKana.
I
273,15 373,15
Puc. 6.6
T,K
TaK KaK 1 K = 1 uC, TO JlI060e 3HaqeHHe
TeMnepaTypbI T 6y,neT Ha 273,15 rpaAyca BbI-
llle cooTBeTcTBYlOlQeH TeMnepaTypbI t no l(eJIb-
CHIO: T= t + 273,15. Ho H3MeHeHHe a6COJIIOT-
HOH TeMnepaTypbI /j. T paBHO H3MeHeHHIO TeM-
nepaTypbI /j. t no lllKaJIe l(eJlbCH.H: /j. t = /j. T.
I
-273,15
I
I
I
°
I I
° 100
t, DC
133

6.15. H3MepeHHe CKopocTeH ra30BbIX MOJIeKYJI.
OnbIT llITeplla. PacnpeAeJIeHHe MaKCBeJIJIa
3 mo (V 2 )
113 cpOPMYJI (W K ) = 2. kT Ii (W K ) = 2 nonY'IaeM cpOPMYJIY .l\JlSI paC'Ie-
Ta Cpe,AHeH KBR.l\paTIi'IHOH CKOpOCTIi ,ABIi>KeHIiSl MOJIeKYJI O.l\HOaTOMHoro ra3a:
(V K .) = J(V 2 ) = l:: = l:t
(6.12)
M
TaK KaK mo = - H kN A = R, r.l\e R - YHIiBepCaJIbHa.H ra30BaSl nOCTOSlHHaSl.
N A
TaKIiM 06pa30M (V KB ) 3aBliCliT OT TeMnepaTypbI H nplipO.l\bl ra3a. TaK,
M M
npli 0 °C .fiJISI BO.fiOpO.l\a OHa paBHa 1800 -. .fiJISI a30Ta - 500
C C
BnepBble HR onblTe onpe,AenliJI CKOpOCTb
MOJleKYJI O. IllTepH. B KaMepe, H3 KOTOpOH
OTKa'laH B03.l\YX. HaXO,ASlTC.H .l\Ba KoaKClianb-
HbIX HJIHH.l\pa 1 H 2 (pHC. 6.7), KOTopbIe
MorYT Bpall\aTbC.H BOKpyr OCR 00 1 C nOCTO-
.HHHOH yrJlOBOH CKOpOCTblO 00.
B.l\Onb OCIi 00 1 HaT.HHYTa nnaTIiHOBaSl
nocepe6peHHa.H npOBOJIOKa, 'Iepe3 KOTOpylO
npOnYCKalOT 3JIeKTpliqeCKIii1 TOK. OHa Harpe-
BaeTC.H, H cepe6po IiCnap.HeTC.H. ATOMbl cepe6-
pa 'Iepe3 ll\eJIb 4 B CTeHKe HJIIiH.l\PR 2 no-
na,AalOT B l(linIiH.l\p 1 Ii OCe.l\aIOT Ha ero
BHYTpeHHeH nOBepXHOCTH. OCTaBJI.H.H CJIe.fi B
BIi.l\e Y3KOH nOJlOCKH, napaJlJIeJIbHOH enli.
Ecnli IiJIHH.l\pbl HenO.l\BH>KHbI, TO nOJlOCKa pacnOJIOmeHa HanpOTIiB ll\eJIIi (TO'I-
Ka B Ha pliC. 6.8. a) Ii IiMeeT O.l\IiHaKOBYIO TOJIll\HHY.
IIpli paBHOMepHOM Bpall\eHIiIi IiJIIiH.l\pa C yrJIOBOH CKOpOCTblO 00 nonOCKa
CMell\aeTCSI B CTOPOHY, npOTIiBOnOnO>KHYIO Bpall\eHIiIO, Ha paCCTO.HHlie S OTHO-
CIiTeJIbHO TO'IKIi B (pIiC. 6.8, 6). Ha TaKoe paCCTOSlHlie CMeCTIiJIaCb TOqKa B
IJ;HJIIiH.l\pa 1 3a BpeMSI t, KOTopoe He06xo.l\HMO, qTo6bJ aTOMbI cepe6pa npo-
IllJIIi paCCTO.HHMe. paBHoe R - r,
r.l\e R Ii r - pa.l\MYCbI HJIIiH.l\-
pOB 1 M 2.
o
1
2
10
1
Puc. 6.7
a
6
4
w = const
B
Puc. 6.8
134
S = VI t = ill Rt,
r.l\e VI - JIIiHeHHa.H CKOpOCTb TO-
'IeK nOBepXHOCTIi IJ;MJIIiH.l\pa 1.
OTCIO.l\a
S
t =-.
ooR

CKOPOCTb aTOMOB cepe6pa
_ R-r _ w(R-r)R
v--- .
t s
3Ha.R R, r, W H H3MepHB 3KCnepHMeHTaJlbHO S, no 3TOR cpopMYJle MO:iKHO
paCCqHTaTb cpeAHlO1O CKOpOCTb ABHmeHH.H aTOMOB cepe6pa B onbITe IIITepHa
M
v == 650 -. 3TO 3Ha'leHHe COBna,AaeT C TeOpeTH'leCKHM 3Ha'leHHeM cpeAHeH KBaA-
C
paTH'lHOn CKOpOCTH MOJIeKyn. 3TO CJIY:iKHT 3KcnepHMeHTaJIbHblM AOKa3aTenb-
CTBOM cnpaBeAJIHBOCTI1 cpOPMYJIbI (6.12), 8 CJIeAOBaTeJIbHO, H cpOPMYJIbI (6.11).
B onblTe IIITepHa 6bIJIO 06H8PY:iKeHO, 'lTO
IIHipHHa nOJIOCKH Ha nOBepXHOCTH BpaalOe-
roc.H I1,HJII1Hpa ropa3110 60JIbllle reOMeTpl1'IeCKO-
ro H306pameHH.H eJIH 11 TOJII1Ha ee B pa3-
HblX MeCT8X HeOAI1HaKOBa (pl1c. 6.9, a). 8'1'0
MO:iKHO 06'bSICHHTb TOJIbKO TeM, 'ITO aTOMbl ce-
pe6pa ABHmYTC.R C pa3JII1'IHbIMH CKOpOCT.HMH.
ATOMbl, JIeTSIHe C HeKoTopon CKOpOCTblO, no-
rraAalOT B TO'lKY B'. ATOMbl, JIeTSIHe 6bICT-
pee, nonaAalOT B TO'lKY, JIe:iKaylO Ha PHCYH-
Ke 6.8 BbIllle TOqKH B', a JIeT.HHe MeAJleH-
Hee, - HH:iKe TO'IKH B'. TaKHM 06pa30M,
Ka:iKAOn TO'IKe H306pa:iKeHH.H COOTBeTcTByeT onpeAeJIeHHa.H CKOpOCTb, KOTOPYIO
llOCTaTO'IHO npOCTO onpeAeJIHTb 113 onbITa. 3THM H 06'b.HCH.ReTC.H TO, qTO TOJI-
I1Ha CJIOR aTOMOB cepe6pa, oceBlllHX Ha nOBepXHOCTH I1,HJI11 Hllpa, He Be3Ae
OllHH8KOBa. HaH60JIblllaSI TOJIHHa B cpellHen tIaCTH CJIOR, a no KpaRM TOJI-
II.\HHa YMeHblllaeTCSI.
M3yqeHHe cpOpMbI Ce'leHH.H nonOCKH oceBlllero cepe6pa C nOMOll.tblO MHKpO-
CKona nOKa3aJIO, qTO OHa HMeeT BH, npl1MepHO COOTBeTCTBYIOHn H306pa-
meHHOMY H8 pHCYHKe 6.9,6. TIo TOJIHHe OTJIomHBlllerOC.R CJIO.H MomHO CY-
11I1Tb 0 pacnpeAeJIeHHH aTOMOB cepe6pa no CKOpOCT.HM.
Pa306beM BeCb I1HTepBaJJ H3MepeHHbIx Ha onbITe CKopocTen aTOMOB cepe6pa Ha
MaJIble t1 v. rIYCTb V - OAHa H3 CKopocTeH
3Toro HHTepBaJIa. TIo nJIOTHOCTH CJIO.R nOA-
C'lHTaeM 'lHCnO t1N aTOMOB, HMelOHX CKO-
pOCTb B I1HTepBaJle OT v llO v + t1 v, H nOCTpo-
a
u
6
Puc. 6.9
f(v)
HM rpaQJHK cpYHKI1,HH f (v) = I1N . rAe N -
AV l v. AV 2 AVa V N I1v
06ee tIHCJIO 8TOMOB cepe6pa, oceBlllHX H8
Puc. 6.10 nOBepxHocTH I1,HJIHHApa. rIOJIY'IHM KPHBYIO,
H306p8:iKeHHYIO Ha pHCYHKe 6.10. OHa Ha3bI-
BaeTC.R c/JYliKl.{UeU pacnpeoe.n.eliUSl Mo.n.eKY.n. no cKOpOCm.RM.
u t1N t1N
rrJIOaAb 3alllTpHXOBaHHOH nJIOaAKH p8BH8 -l1v = -, T.e. paBH8
N t1v N
OTHOCHTeJIbHOMY 'lHCJIY 8TOMOB, HMelOHX CKOpOCTb B npeAeJIaX v + 6.v. MbI
BI111I1M, 'lTO 'lHCJIa 'laCTHI1" HMelOHX CKOpOCTb H3 pa3HbIX HHTepBaJIOB VI + t1Vlt
V2 + t1V2' V3 + 6.V3' pe3KO pa3JIH'lHbl. CyecTByeT KaKaSI-TO CKOpOCTb, OKOJlO 3Ha-
135

1.IeHH.H KOTOpOR HaXO.HTC.H CKOpOCTH, C KOTOpbIMH BHmeTC.H HaH60JIbillee 'lHC-
JIO MOJIeKYJI. OHa Ha3bIBaeTC.H liau60Jlee 6epORmnou CKopocmblO VB' H eR CO-
OTBeTcTByeT MaKCHMYM Ha pHCYHKe 6.10. 8Ta KpHBa.H XOpOIllO COOTBeTcTByeT
KpHBOR. nOJIY1.IeHHOR LIm. MaKCBeJIJIOM. KOTOPbIR. nOJIb3Y.HCb CTaTHCTH'leCKHM
MeTO,l{OM. TeOpeTHqeCKH ,l{OKa3an. qTO B ra3ax. HaXO,l{.HlUHXC.H B COCTO.HHHH Tep-
MOHHaMHqeCKOrO paBHOBeCH.H. YCTaHaBJIHBaeTC.H HeKOTopoe. He MeH.HIOIlleeC.H co
BpeMeHeM, pacnpeneJleHHe MOJleKyn no CKOpOCT.HM, KOTopoe nO,l{'lHH.HeTC.H BnOJl-
He onpe.ueneHHOMY CTaTHCTHqeCKOMY 3aKOHY. rpRcpH1.IeCKH H306pamaeMOMY KPH-
BOR f(v). HaH60JIee BepO.HTHa.H CKOpOCTb, KaK nOKR3RJI MaKCBeJIJI, 3aBHCHT OT
2RT
TeMnepaTypbI H MaCCbI ero MOJIeKYJI no cpopMYJIe (VB) = M'
TaKOMY pRCnpeeJIeHHIO nOqHH.HIOTC.H MOJIe-
KYJIbI BceB03MomHbIX BelUeCTB B pa3JIH1.IHbIX CO-
CTO.HHH.HX npH aHHO:H TeMnepaType. ECJIH YBe-
JIH1.IHTb TeMnepaTYPY (T 2 > Td, TO KpHBa.H (pHC.
. 6.11) CMeCTHTC.H Bnp8BO, HaH60JIee BepO.HTHa.H
V
CKOpOCTb B03paCTeT. nO.HBHTC.H 60Jlbille 6bICT-
pbIX 1.IaCTHU. YMeHblllHTC.H 'lHCJlO MeJIeHHbIX
qaCTHl.\ H ,Aame Tex, KOTopbIe BHmYTC.H co
CKOpOCT.HMH, 6JIH3KHMH K HaH60Jlee BepO.HTHoR. rrJIOlUaH no KpHBbIMH 6y-
YT OHHaKOBbIMH, TaK KaK 06lUee 1.IHCJIO 1.IaCTHl.\ N He H3MeH.HeTC.H.
3Ha.H 3aKOH H3MeHeHH.H cpYHKl.\HH pacnpeeJIeHH.H MOJIeKYJI no CKOpOCT.HM
f(v), MomHO paCC1.IHTaTb cpeHIOIO, HJIH cpeHIOIO apH(})MeTHqeCKYIO. CKOpOCTb
MOJIeKY JlbI:
Puc. 6.11
1 00
(v): - J vdN = J vF(v)dv.
N 0
rrOCTaBJI.H.H clOL(a 3Ha1.IeHHe F(v) H HHTerpHpy.H (B BHY CJIOmHOCTH F(v)
MbI ee He npHBOHM), nOJIY'lalOT
(v) = lkT = lRT .
nmo nM
CpeHee 3Ha1.IeHHe KBaL(paTa CKOpOCTH
1 00
(v 2 ) = - J v 2 dN = J v 2 F(v)dv.
N 0
CpeH.H.H KBapaTH'lHa.H CKOpOCTb
(v K .) = J0l) = lT .
6.16. BepoHTHoCTh H 3HTpon1lH.
TIOpH!J:OK II 6ecnopH!J:oK B MaKpOCIICTeMax
JII060e MaKpOCKOnH1.IeCKOe Teno COCTOHT H3 orpOMHoro qHCJIa qaCTHl.\ (MO-
JIeKYJI, aTOMOB H p.). Hero MomHO paCCMaTpHBaTb KaK aHCRM6JIb. COCTOH-
lUH:H H3 60JIbillOrO qHCJla npOCTbIX CHCTeM.
136

COCTO.HHHe MaKpOCKOnHqeCKOrO TeJla MOmeT 6bITb 3a,n;aHO C nOMOblO 06'be-
Ma, ,n;aBJleHH.H, TeMnepaTypbI H ,n;pyrHx MaKpOCKOnHqeCKHX napaMeTpOB, KOTO-
pbIe XapaKTepH3YIOT BCe TeJlO B n;eJlOM. OXapaKTepH30BaHHOe TaKHM cnoc060M
COCTO.HHHe Ha3bIBaeTC.H MaKpOCOCTOJlHHeM. STO MaKpOCOCTO.HHHe 3aBHCHT OT CO-
CTO.HHH.H BCeX 06pa3YIOHX TeJlO MOJleKYJI (OT MHKpOCOCTO.HHHH). COCTO.HHHe
MaKpOCKOnHqeCKOrO TeJla, OXapaKTepH30BaHHOe HaCTOJlbKO nO,n;p06HO. qTO OKa-
3bIBalOTCH 3a,AaHHhIMH COCTOHHHH BCeX 06pa3YIOHX TeJlO MOJIeKYJI, Ha3bIBaeT-
CSI MUKPOCOCTOJlHHeM.
BC.HKOe MaKpOCOCTO.HHHe MOmeT 6bITb OCYllteCTBJleHO pa3J1HqHbIMH cnoc06a-
MH, Kam,n;OMY H3 KOTOpbIX COOTBeTCTBYIOT HeKOTOpbIe MHKpOCOCTO.HHH.H TeJla.
"lIHCJlO pa3J1HQHbIX cnoc060B (MHKpOCOCTO.HHHH), KOTOpbIMH COCTO.HHHe CHC-
TeMbI MOmeT 6bITb peaJlH30BaHO, Ha3bIBaeTC.H TepMO,AHHaMH'IeCKOH BepOJlTHOC-
TLIO W ,AaHHOrO COCTOJlHUJI CHCTeMLI.
IIo.HCHHM CMbICJ:I TepMO,n;HHaMHqeCKOH BepO.HTHOCTH COCTO.HHH.H Ha npHMepe.
TIpe,n;CTaBHM ce6e 3aMKHYTbIH COCY)J;, B KOTOpOM HaXO,nHTC.H QeTbIpe O,n;HHaKOBbIe
MOJ:leKYJlbI. Pa3,n;eJlHM MbICJleHHO cocy,n; Ha ,n;Be paBHbIe nOJlOBHHbI. MOJleKYJlbI
BCe BpeM.H ,lI;BHmYTC.H. BbI.HCHHM, KaK OHH paCnpelIeJl.HIOTC.H Mem,ny ,nBYM.H nOJlo-
BHHaMH cocy,n;a. IIpoHYMepyeM MOJleKYJlbI. IIo,n; MaKpOCOCTO.HHHeM 6y,n;eM nOHH-
MaTb COCTO.HHHe, KOTOpOe onpe,n;eJl.HeTC.H QHCJlOM MOJleKYJI B
Kam,n;OH nOJlOBHHe cocy,n;a, a no,n; MHKpOCOCTO.HHHeM - CO-
CTO.HHHe, KOTopoe onpe,n;eJl.HeTC.H TeM, KaKHe HMeHHO MOJleKY-
JlbI H KaK pacnOJlOmeHbI B TOH HJlH ,n;PyroH nOJlOBHHe COCY-
,n;a. OqeBH.nHO, qTO B03MomHbI CJlen;YlOlliHe MaKpOCOCTO.HHH.H:
I - B nepBoH nOJlOBHHe cocy,n;a C06paJlHCb Bce qeTbIpe MO-
JleKYJlbI, BTOpa.H nOJlOBHHa nYCTa; II - B nepBoH nOJlOBHHe
TPH MOJleKYJlbI, BO BTOpOH - o,n;Ha; III - B ,n;BYX nOJlOBH-
Hax no ,n;Be MOJleKYJ:lbI; IV - B nepBoH nOJ:lOBHHe cocy,n;a
o,n;Ha MOJleKYJla, BO BTOpOH - TPH; V - Bce tJ:eTbIpe MOJle-
KYJlbI HaXO)J;.HTC.H BO BTOPOH nOJlOBHHe cocy,n;a, nepBa.H nYCTa
(pHC. 6.12). Kal< BH,n;HM, COCTO.HHHH I H V peaJ:lH3YIOTC.H
TOJlbKO O,n;HHM cnoc060M, Kam,n;oe H3 COCTO.HHHH n H IV -
qeTbIpbM.H cnoc06aMH Ii, HaKOHen;, COCTO.HHHe III - meCTblO
cnoc06aMH. TaKHM 06pa30M, II Ii IV MaKpOCOCTO.HHH.H xapaK-
TepH3yIOTC.H qeTbIpbMSJ MHKpOCOCTO.HHH.HMH, III MaKpOCOCTO.HHHe
xapaKTepH3yeTcSI meCTblO MHKpOCOCTO.HHH.HMH. 06ee qHCJlO
B03MomHbIX MHKpOCOCTO.HHHH N = 16 = 2 4 . "lIHCJlO MaKpococTO-
.HHHH 5. A eCJlH MOJleKYJl 10, TO qHCJlO B03MomHbIX MHKpO-
COCTO.HHHH N = 2 1 °= 1024 (Ta6J1. 6.1). qHCJlO MaKpOCOCTO.H-
HHH paBHO 11. Ho, KaK H B npe,n;bI,n;yeM cJlyqae, eCTb
TOJlbKO O,n;HH Ii3 cnoc060B, Kor,n;a Bce qaCTHn;bI pa3MeeHbI B
O,n;HOH nOJlOBHHe cocy,n;a, H 252 MHKpOCOCTO.HHH.H, Kor,n;a OHH
pacnpe,n;eJleHbI paBHOMepHO no o6eHM nOJlOBHHKaM cocy,n;a.
PaccMaTpHBa.H no,n;06HbIM 06pa30M cJlyqaH )J;J:I.H qHCJla qa-
CTHn;, paBHoro N l' nOJlyqHM, qTO 06ee tJ:HCJlO B03MomHbIX
MHKPOCOCTO.HHHH (qHCJlO B03MomHbIX pa3Mell.\eHHH qaCTHn; Mem-
,n;y ,AByM.H nOJlOBHHaMH cocy,n;a) N = 2 N1 . Ii TOJlbKO O,n;HH CJlY-
'laM H3 3Toro 'IHCJla COOTBeTCTByeT pacnpe,n;eJleHHIO QaCTHn;, npH
KOTOpOM Bce OHB 6Y)J;YT B O,n;HOH nOJlOBHHe cocy,n;a. "lIHCJlO
I;I
I
234 1
. . . .
1 3 4 2
. . . .
1 2 4 3
. . . .
1 2 3 4
. . . .
n
1 2 3 4
. . . .
1 3 2 4
. . . .
1 4 2 3
. . . .
2 3 1 4
. . . .
2 4 1 3
. . . .
3 4 1 2
. . . .
III
1 2 3 4
. . . .
2 1 3 4
. . . .
3 1 2 4
. . . .
4 1 2 3
. . . .
IV
11   i I
V
Puc. 6.12
137

Ta6J1Ha 6.1 me cnoc060B, npH KOTOpOM qaCTHIJ;bl pac-
npeeJleHbI no 06'beMY paBHOMepHO, pac-
TeT C YBeJlHqeHHeM qHCJla qaCTHl.( H TeN
6bICTpee. t.IeM HX 60JlbWe (OHO paBHO t.IHC-
ny COqeTaHHM Co = 10.9.8.7.6 = 252).
1.2.3.4.5
B cTaTHCTHqeCKOM «pH3HKe JlH H3Y-
t.IeHHH npOl.(eCCOB, npOHCXO.HHX B CHC-
TeMe, CocToHeM H3 orpoMHoro MHO>Ke-
CTBa t.IaCTHl.(, npHMeH.HIOTC.H MeTObI Te-
OpHH BepO.HTHOcTeM.
MaTeMaTHt.IeCKa.H BepO.HTHOCTb P Ka-
Koro-nH60 cJlyqaMHOrO C06bITHH onpee-
nHeTC.H KaK OTHoweHHe t.IHCJla cJlYt.Ia-
eB N I' COOTBeTCTBYIOHX H8cTynneHHIO
OmH,AaeMOrO C06bITHH, K 06eMY t.IHcny cnyqaeB N, ecnH Bce cnY'laH paBHO-
B03MO>KHbl:
HOMep I II qHcno
M8KpO- nonOBHH8 nonOBHH8 MHKpO-
COCTO$lHH$I COCYAa cocy,na COCTOSlHHH
1 0 10 1
2 1 9 110
3 2 8 45
4 3 7 120
5 4 6 210
6 5 5 252
7 6 4 210
8 7 3 120
9 8 2 45
10 9 1 10
11 10 0 1
P = N l .
N
PaCCqHTaeM MaTeMaTHt.IecKYIO BepO.HTHOCTb paBHoMepHoro pacnpeeJleHH.H MO-
neKYJl B cocye: P = N l = 252 . BepoHTHoCTb >Ke Toro, tITO Bce 10 MOJleKyn
N 1024
1
OKamYTCH B OHOM nonOBHHe, PI --
- 1024
TaKHM 06pa30M, HaH60Jlee BepOHTHbIM COCTO.HHHeM .HBnHeTC.H paBHOMepHoe
pacnpeeneHHe MoneKyn B cocye. JIMeHHo OHO Bcera H Ha6J1IOaeTC.H B 3KC-
nepHMeHTe. XOT.H CTaTHCTHKa He OTpHl.(aeT B03MomHOCTH, Kora Bce 10 MOJle-
Kyn OKamYTC.H B OHOM nonOBHHe cocY,Aa, HO BepOHTHOCTb 3Toro C06bITHSI
u 1
KpaHHe MaJla (oHa paBHa -).
1024
TepMoHHaMH'leCKa.H BepOSlTHOCTb H3MeHSIeTC.H napanJleJlbHO MaTeMaTHqeCKOM.
OHa MaKCHMaJlbHa (W = 252) nSl paBHOMepHoro pacnpeeJleHH.H MOJleKYJl B co-
cye H MHHHManbHa (W = 1) nH cnyqaSl, KOr.D;a Bce 10 MOJleKyn HaXO.HTCSI
B OHOM nOJlOBHHe cocya. CneOBaTeJlbHO, TepMOHHaMHt.IeCKaSi BepO.HTHOCTb
cnymHT xapaKTepHcTHKOM HanpaBJleHHOCTH TennOBbIX npOneCCOB. CpaBHHB Be-
pO.HTHOCTH BYX COCTOHHHH TepMoHHaMHqeCKOM CHCTeMbI, MO>KHO cpa3Y YCTa-
HOBHTb HanpaBJleHHe npOl.(eCca, KOTOpbIM B03MomeH B aHHoM CHCTeMe: 3TO
6yeT nepexo H3 MeHee BepOSiTHoro B 60nee BepO.HTHOe COCTOHHHe.
COcToSlHHe, ocyecTBn.HeMOe OTHOCHTenbHO ManbIM tIHCnOM cnoc060B, Ha-
3bIBaeTC.H ynop.HJJ;O'leHHLIM. COCTO.HHHe, ocyecTBnHeMoe MHorHMH pa3J1HqHbIMH
cnoc06aMH, Ha3bIBaeTC.H 6ecnOp.HJJ;O'lHLIM (HnH cnyqaHHLlM).
AJI.H KOJlHqeCTBeHHoM OneHKH HeynopSl.D;OqeHHOCTH COCTO.HHH.H CHCTeMbI BBO-
HT «pH3Ht.IecKYIO BeJlHqHHY, KOTopaH Ha3bIBaeTC.H 3HTponHeM H 0603Hat.IaeTCSI
6YKBOM S.
3HTpOnHH -- KOJlHqeCTBeHHa.H Mepa HeynopHOqeHHOCTH:
S=klnW,
138

r,Ae k= 1,38.10- 23 .n:; - nOCTO.HHHa.H BOJlbnMaHa. W - TepMoHHaMHqeC-
Ka.H BepO.HTHOCTb aHHoro COCTO.HHH.H CHCTeMbI.
Kal< BHHM, B paccMoTpeHHbIx BbIllie npHMepax HaH60nee BepO.HTHbIMH OK a-
3anHCb cJlyqaH paCnpe,AeneHH.H MoneKyn C MaKCHMaJlbHOH 3HTpOnHeH.
TenJlOBbIe, TepMOHHaMHqeCKHe npOneCCbI npOTeKalOT, KaK nOKa3bIBaeT
OnbIT, TaKHM 06pa30M, 'liTO npHBO.HT CHCTeMY K paBHOBeCHOMY COCTO.HHHIO,
KOra BCe napaMeTpbI, XapaKTepH3YIOHe CHCTeMY, nOCTO.HHHbI £0 BpeMeHH H
no 06'beMY CHCTeMbI, T.e. pacnpe)J;eneHHe MoneKYJI no 06'beMY CHCTeMbI .HBJI.H-
eTC.H paBHOMepHbIM. a 3TO HeynOp.HOqeHHOe COCTO.HHHe. TaK, HanpHMep, BHYT-
peHH.H.H 3HeprH.H nepeXOHT OT TeJla, 06na,Aaroero 60Jlee BbICOKOH TeMnep8TY-
pOM, K TeJlY, MeHee HarpeTOMY, HO He Ha060pOT. 3TO nepexo OT nOp.HKa K
6eCnOp.HKY: HaJlHqHe pa3HOCTH TeMnepaTYP Y BYX TeJl 03HaqaeT HeKOTOpylO
ynop.HOqeHHOCTb; Kora TeMnepaTypbI BbIpaBHHBaIOTC.H, 3Ta ynop.HOqeHHOCTb
HC'le3aeT. .D:pyro'ii npHMep - H<P<PY3H.H. B npHBeeHHOM B  6.3 npHMepe
nepeMellIHBaHH.H BObI H paCTBopa MeHoro Kynopoca npOHCXOHT nepeXO,A OT
ynOp.HllOqeHHOrO COCTO.HHH.H K HeynOp.HllOqeHHOMY. TaKHM 06pa30M. caMonpOU3-
60JlbHO npomeKalOm maKue npOl{.eCCbl. e KomopblX e03pacmaem lieynopSliJolleIi-
Hocmb, m.e. alimponuSl cmpeMumcSl K c60eMY MaKcuMa.n.blioMY 3lialleliUIO.
olimponuSl, CJleOBaTenbHO, Sle.n.SlemcSl Mepou lieynopSliJolleHHocmu cocmo.
SlHUSl cucmeMbl U xapaKmepucmuKoil lianpa6.n.eHliocmu npOl{.eccoe.
BOllPOCM H CAMOKOHTPOH
1. KaKue onblmbl noiJm6epJICOa1Om oe.n.uMocmb 6eU{ecm6a?
2. KaKue naUMelibwue ttaCmUl{.bl Sl6.n.SllOmcSl 1i0cume.n.SlMU ceoucme XUMU'
lleCKUX a.n.eMelim06 6eU{ecm6a? KaKoebi ux pa3Mepbl?
3. KaK Mo.n.SlpliaSl Macca 6eU{ecm6a C6Sl.1alia c omnocume.n.bfWU MOJleKY-
.n.SlpIiOU Maccou? KaK liaumu ee. ucno.n.b3YSl ma6.n.uqy p. H. M elioe.n.eeea?
4. Om Ilezo 3a6ucum xapaKmep xaomUlleCKOZO oeuJlCeliu.Jl .Mo.n.eKY.n.? UIi-
meHCU6liOCmb oeUJlCeliuSl?
5. KaKue onblmbl .Mozym noom6epoumb lia.n.Ullue cu.n. npUmSlJICeliUSl Me}IC-
oy .Mo.n.eKy.n.a.Mu? cu.n. omma.n.KU6aliUSl .MeJICoy nu.Mu?
6. KaKoea 3aeUCUMocmb Mo.n.eKy.n.SlpliblX CUJl om paccmoSlliuSl MeJICoy MO-
.n.eKYJLaMu? KaK zpaljJullecKu u306paJICaemCSl 3ma 3aeUCU.MOCmb?
7. KaKue Sl6.n.eIiUSl oOKa3bl6a10m liaJlUIlUe npoMeJICymK06 MeJICoy llacmul{.a.Mu?
8. IT OlleMY OuljJcjJY3USl 6 JlCUOKOCm.JlX npoucxooum 3liallume.n.bli0 .MeOJleli-
nee. lle.M OUcjJcjJY3USl 6 za3ax?
9. 0 lle.M c6uoeme.n.bcm6yem 6pOYli06cKoe 06UJlCeliue? KaK 3a6ucum ezo
UHmeliCU61i0Cmb om me.MnepamYPbl?
10. KaK06bt. pa3Mepbt. 6pOYH06CKUX llacmuij? KaK u nOlleMY U3MeliSlemcSl
xapaKmep iJeUJlCeliUSl 6pOYli06CKUX Ilacmuq c yee.n.ulleliueM ux pa3Mepoe?
11. q e.M om.n.UllalOmcSl m6epoble. JlCUOKUe U za3006pa3lible me.n.a no ceoeu
cmpYKmype. no xapaKmepy men.n.060ZO 06UJlCeIiU.Jl ux .Mo.n.eKYJl?
12. llOlleMY za3 3aliUMaem eeCb npeoocmae.n.eliliblu e.MY 06'OeM?
13. ITpu KaKUX YCJl06USlX pea.n.bliblu za3 no ceou.M ceoilcmeaM 6JlU30K K
uoea.n.bliO.MY?
139

14. KaK 060Rcliumb Oa6JleliUe za3a c mOIlKU 3peHuR M KT2
15. II oileMY OaBJleliUe za3a lia cmeliKU cocyoa OOUliaK060 60 6cex lianpaB'
Jl.eliURX? Om Ilezo 3a6ucum 6eJl.UIlUlia OaBJl.eliUSl za3a?
16. fIPOu3Booum JlU za3 OaBJl.eliUe lia cmeliKU cocyoa 6 cocmOSlliUU lieBe-
cOMocmu?
17. q mo npuliuMalOm 3a 060eM UOeaJlbliOZO za3a?
18. KaKaSl 6eJl.UIlUlia xapaKmepu3yem cocmOSlliue menJl.060Z0 pa6li06ecuR?
19. q eM onpeOeJl.SlemCSl meMnepamypa za3a c mOIlKU .3pe liUSl M KT?
20. Om llezO 3a6ucum cpeoliJl,R KUliemUlleCKaSl 3liepZU.R MOJleKYJl UOeaJlbHOZO
za3a? cpeoHSlSl KBaopamUIlHa.R CKopocmb MOJleKYJl.?
21. q eM omJlUllaemCSl WKaJl.a meMnepamyp  eJl.bCUSl om WKaJlbl KeJlb6uHa 2
22. qmo npUHRmo 3a a6COJl.lOmliblU liYJl.b? /(ocmUJlCUM Jl.U Oli?
23. T{ mo nOHUMalOm nOG mepMoounaMullecKou 6epoRmnocmblO?
24. KaK C6Sl3aIi bl MeJICoy c060u mepMoouliaMullecKaSl BepOSlmliocmb U 3nm
ponuR? 3limpOnUSl U 6ecnopSlOOK 6 MaKpocucmeMax?
OCHOBHblE q,OPMY Jlbl
m N V
V=-=-=-
M N A V M
KOJlH'tleCTBO Beru;eCTBa, r,lJ.e m - Mac-
ca Beru;eCTBa, M - MOn.HpHaH Macca,
N - 'tlI1CJlO 'tlaCTI1I..( (aTOMOB, MOJleKYJI)
Beru;eCTBa. N A = 6.10 23 MOJlb -1 - nOCTO-
HHHaH ABOra,ApO, V - 06'beM ra3a,
M 3
V M =22.4.10- 3 - - MOJlHpHblll 06'beM
Monb
ra3a npl1 HopManbHblX YCJlOBHSlX.
m M
mo=-=-
N N A
- Macca MOJleKY JlbI.
(v) = 8kT
nmo
(V KB ) = lkT
mo
(VB) = lkT
mo
- 3aBl1Cl1MOCTl1 Cpe,lJ.Hell apl1cpMeTl1qeC-
KOll (v). Cpe,AHell KBa.n;p8TI1'lHOll (v KS ) H
Hal160Jlee BepO.HTHOll (vJ CKopocTell Ten-
JlOBOrO XaOTI1'tleCKOrO ,lJ.Bl1meHHH MOJleKY JI
OT TeMnepaTypbl T. r.n;e k - nOCTOHH-
Ha.H BOJlbI..(MaHa, mo - Macca MoneKYJlbl.
p = mon(v2). \
p = n(WK)'
3
p = nkT
- OCHOBHoe ypmmeHHe MOJleKYJI.HpHO-KI1He-
TH'tleCKOll: TeopHH, r,lJ.e p - ,lJ.8.BJleHl1e ra3a,
n - KOHI..(eHTpaI..(I1H MOJleKyn, (W K ) -
Cpe,lJ.H.HH KI1HeTl1'leCKaH 3HeprH.H MOJleKY-
JlbI, T - a6cOnlOTHaJl TeMnepaTypa.
m ( v 2 )
( w ) = 0 =kT
K 2 2
- Cpe,lJ.H.H.H KI1HeTI1'leCKa.H 3Heprl1H nocTY-
naTeJlbHoro .n;BI1:>KeHI1H MOJleKY Jlbl O,lJ.HOaTOM-
140

T = t + 273
,r(m
Horo ra3a, re k = 1,38 . 10- 23 K - no-
CTO.HHHa.H BOJIbIJ;MaHa, T - a6COJlIOTHa.H
TeMnepaTypa.
- COOTHOIlIeHHe Memy a6COJllOTHOH TeM-
nepaTYPOH T R TeMnepaTYPOH t no IlIKa-
ne L(eJIbCH.H.
IIPIIMEPbI PElliEHIIH 3AJJ:A q
1. 3a 10 CYTOI< nonHOCTbIO HcnapHnOCb H3 CTaK8Ha 100 r B o.n; 1.1 . CKonb-
KO B Cpe,AHeM BbweTano MoneKyn C nOBepXHOCTH BO,Abl 3a 1 c? HaH,AHTe Maccy
MoneKynbI BO,AbI.
,Z:(8HO: t=8,64.10 5 c. m=0,100 Kr, M=18.10- 3
Kr
Monb
HaHTH: mo, N 1 .
N
PeweHHe. HCKoMoe 'lHCnO MoneKyn N 1 = -, r,Ae N - 'lHCnO MoneKY]]
t
m
SOAbI B .n;aHHoH nopqHH, KOTopoe MOJKHO H8HTH no <l>opMyne N = - N A- S,AeCb
M
M - MOn.HpHaSl Macca BO,AbI, N A - nOCTO.HHHa.H ABOra,Apo. Cne,AOSdTenbHO.
N 1 = m M N t A . , N1 = 4, 0 . 10 18 C -1 .
HCKOMoe qHCnO MoneKyn
Maccy MoneKynbI SO,AbI HaH,AeM no <l>opMyne
M
mo = N A ; mo = 3.0 10- 26 Kr.
2. HaH,AHTe Cpe,AHIOIO KBa.n;paTH'lHYIO CKOpOCTb H cpe.n;HIOIO KHHeTHqeCKYIO
3HeprHIO nocTyn8TenbHoro ,ASHJKeHHSI MoneKyn renHSI npH 27°C. Onpe,AenHTe
nOJIHYIO 3HeprHlO scex MoneKyn 100 r renHSI.
KI'
,Z:(8HO: T=300 K, M=4.0.10- 3
m=0,100
RI'.
Monb
HaHTH: (V KB ), ,W K ), W.
Pew e H H e. Cpe,AH.HSI KHHeTH'leCKaSl 3HeprH.H nocTyn8TenbHoro ,ABHmeHHSI
MoneKynbI paCC'lHTbIB8eTCSI no <l>opMyne
(W K ) =  kT; (W K ) = 6,2.10- 21 ,r(m.
2
m (v 2 )
C ,ApyroH CTOPOHbI. (W K ) = ° 2 ' [',Ae mo - M8CC8 aTOMa reJlH.H;
M
mo = N' r.n;e M - MonSlpH8S1 M8cca renHSI.
A
Cne,AOBaTenbHO,
(W K ) = (V2).
2N A
141

OTCIO,Aa
(v 2 ) = 2N AWK) .
MCKOMaSl KB8,ApanPiH8S1 CKOpOCTb
(v..) = J(V 2 } = 2NAW.) ; (v.,) = 1,4 10" .
3HaSl Maccy m renHSI. MOmHO H8HTIi "IHCnO N aTOMOB renHSI:
m
N =-NA'
M
Tor,Aa nOnHaSl 3HeprHSI BCeX aTOMOB
W = N (W K ) = N A (W K ); W = 9,4.10 4 ,[(m. .
M
3. CKonbKo MoneKyn KHCnOpO,Aa HaXO.l{HTCSI B cocY,Ae 06'beMOM 1.0 n. ecnli
TeMnepaTypa KHCnOpO.l{8 150°C. a AaBneHHe 0.132 TIa?
.D.aHO: V=I,O.10- 3 M 3 . T=423 K. p=0,132 TIa, M=0.032
Kr
Monb
HaHTH: N.
P e ill e H H e. OCHosHoe ypasHeHl1e MKT p = nkT. OTCIO,Aa KOHu.eHTpau.l1S1 MO-
neKyn KI1CnOpO,A8 n = ....!!...... Tor,Aa I1CKOMoe 'll1cno MoneKyn
kT
N = nV = pV . N - 2 0 10
kT' -,' 15.
Toporo r83a, H8XO,ASlIl\erOCSI no,A AaBneHl1eM 50 KTIa, p8BeH 2,02. 10 5
4. Cpe,AHHH KB8,Ap8T CKOpOCTI1 nocTynaTenbHoro ,ABH}KeHIiSl MoneKyn HeKO-
M 2
7'
Bbl'lHCnHTe nnOTHOCTb 3Toro ra3a npl1 ,AaHHbIX ycnoBHSlX.
( 2 ) 5 M 2
JJ:aHO: v =2,02.10 2' p=5,0 10 1 TIa.
C
H a H T Ii: p.
m
P e ill e H H e. TIo onpe,AeneHHIO nnOTHOCTb p = V ' r,Ae m - Macea ra3a. V -
06'beM r838. Maccy r838 MomHO H8HTIi, 3HaSl Maccy MoneKynbl mo 11 "IHCnO
MoneKyn N:
m=moN.
'lfHcno MoneKyn MomHO H8HTI1, 3HaSl KOHu.eHTpau.111O MoneKyn n: N = nV.
CJIe,AOBaTenbHO,
monV
p=-=mon.
V
38nl111IeM OCHOSHoe ypaBHeHl1e MKT B BH,Ae p =  nmo (v 2 ).
OTCIO,Aa n = 3 ( P 2} ' CJIeAOBaTeJILHO, HCKOMaH OJIOTHOCTL ra3.
mo v
3p 3p Kr
p = mo ( 2 ) = ( 2 ) ; P = 0,75 3"'
mo v v M
142

6.17. Y paBHeHHe cOCTOHHHH l1.L1.eaJIbHOrO ra3a
PaBHOBeCHoe COCTO.HH1:Ie naHHOH MaCCbI ra3a npH OTCYTCTBHH BHellIHHX no-
JIeH nOJIHOCTblO onpe,neJIeHO. eCJIH H3BeCTHbI .n;aBJIeHHe P. TeMnepaTypa T H
06'beM ra3a V.
'Y paBHeH1:Ie, CB.H3bIBalOll.\ee TeMnepaTYPY, ,AaBJIeHHe H 06'beM H.n;eaJIbHOrO ra3a
B COCTO.HHHH TenJIOBOrO paBHOBeCH.H, Ha3blBalOT ypaBHellHeM COCTOHIIHH ra3a
3TO ypaBHeHHe 6bIno nOJIyqeHO 3KcnepHMeHTaJIbHO, HO ero MomHO BbIBec-
TH H3 OCHOBHoro ypaBHeHH.H MKT:
P= nkT.
ITo onpe.n;eJleHHIO KOHu.eHTpau.H.H ra3a
N
n =-.
V
r.n;e N - t.IHCJIO MoneKYJI. Tor,na
p = N kT  pV = kN.
V T
(6.13)
ITpH Hell3MeHHOH Macce ra3a q1:lCJIO MoneKYJI B HeM nOCTO.HHHO H npOll3Be-
.neHHe kN = const. CJIe.n;oBaTeJIbHo.
pV = const HJlll .n;JI.H .n;BYX COCTO.HHHH PI VI = P2 V 2 (6.14)
T  
CooTHollIeHHe (6.14) H eCTb ypaBHeHHe COCTO.HHH.H H.neaJIbHOrO ra3a. Ero
Ha3bIBalOT ypaBHeHHeM KJIaneiipOHa. 11M nOJIb3YlOTCH B Tex cnyt.IaHX. Kor.na
Macca ra3a Hero XHMHqeCKHM COCTaB He H3MeHHIOTC.H Ii HymHo cpaBHHTb ,ABa
COCTO.HHH.H ra3a.
B ypaBHeHHH (6.13) t.IHCJIO MOJIeKyn N MomHO BbIpa3HTb 'lepe3 nOCTO.HH-
m
HYIO ABora.npo N = - N A' r.ne m - Macca ra3a. M - ero MOJI.HpHa.H Mac-
M
pV m
ca. Tor.n;a nOJIY'laeM - = - kN A 
T M
m
pV =-RT.
M
(6.15)
3,AeCb R = kN A - YHHBepcanbHaH ra30Ba.H nOCTO.HHHa.H. QHa HMeeT onpe,Ae-
JIeHHbIM cpH31it.IeCKHH CMbIcn (CM.  7.5).
R= 1 38.10- 23 ,n:m . 6 02.10 23 MOJIb- 1 =8 , 31
, K'
,n:m
MOJlb.K
'YpaBHeHHe (6.15) - 3TO Tome ypaBHeHHe COCTO.HHll.H H.n;eanbHoro ra3a. Ero
Ha3bIBalOT ypaBlleHHeM KJIaneiipOHa-MeHl1eJIeeBa. QHO cnpaBenJIHBO nJI.H JIIO-
60il MaCCbl ra3a H CBH3bIBaeT Mem.n;y C060H napaMeTpbI o.n;Horo COCTO.HHHH ra3a.
113 ypaBHeHH.H COCTO.HH1i.H BbITeKalOT ,ABa CJle.n;CTB1i.H:
pV
1) 113 cpopMynbI (6.13) nOJIyt.IHM N = -, OTKY,Aa BH.nHO, t.ITO eCJIH pa3-
kT
nH'lHbIe ra3bI 3aH1iMaIOT npH o,nHHaKOBbIX TeMnepaTypax H o,nHHaKOBbIX ,naB-
JIeH1i.HX paBHbIe 06'beMbI, TO t.I1:ICJlO N MOJIeKYJl y H1:IX Tome o.n;HHaKOBO, T.e.
143

BbITeKaeT YCTaHOBJleHHbIH OnbITHbIM nYTeM 3aKOH ABOra.n;pO: npu pa6liblX aa6-
Jl.eliUSlX U me.Mnepamypax 6 OOUliaK06blX 06oe.Max Jl.106blX a306 coaepJICUmCSl
OOUliaK060e IlUCJl.O .MOJl.eKYJl..
2) IIycTb B cocY,Ae HMeeTC.R CMeCb ra30B, Kam)J;bIH H3 KOTOpbIX npH OT-
CYTCTBHH ,lI;pyrHx OKa3bIBaeT COOTBeTCTBeHHO ,lI;aBJIeHHe PI' P2' ... (napu,Hanb-
HbIe lIaBJleHH.H ra30B). 3anHilleM )J;JI.H Kam)J;oro ra3a ypaBHeHHe COCTO.HHH.H:
PIV=NlkT, P2 V =N 2 kT. ...
H CJlOmHM HX:
(N I + N 2 + ...)kT NkT
PI + P2 + ... = V = V-'
NkT
r,Ae N I + N 2 +... =N - qHCJlO MOJleKYJI CMeCH ra30B. Ho - = p.
V
Cne,ll;OBaTeJlbHO, P=PI+P2+"" T.e. aa6JleliUe C.MeCU za306 pa61W cY.M.Me
napu.uaJl.bliblX aaBJteliUU KaJICaOZO U3 za30B - STO 3aKOH LIaJlLTOHa. OTKpbI-
TbIH HM B 1801 r. SKcnepHMeHTaJlbHO.
6.18. qacTllble CJIyqaH ypaBllellHlI COCTOllHHlI H,aeaJIbHOrO ra3a
IIpou,eCCbI, npH KOTOpbIX O)J;HH H3 napaMeTpOB P, V HJlH T OCTaeTC.H no-
CTOHHHbIM, Ha3bIBalOT U30npoeccaMU. 3aKoHbI, onHCbIBalOHe H30npou,eCCbI B
H)J;eaJlbHOM ra3e. 6bIJlH OTKpbITbI SKcnepHMeHTaJIbHO 3a,ll;OnrO ,AO C03,l1;aHH.H MKT
H,lI;eaJlbHOrO ra3a.
a) 1I30TepMH'leCKUH npOl.{ecc - npou,ecc H3MeHeHH.H COCTO.HHH.H ra3a npH no-
CTO.HHHOH TeMnepaType: T = const. LIJI.H ero OCYLUeCTBJleHH.H Ha,ll;O COCY,lI; C nopllI-
HeM, HanOJlHeHHbIH ra30M. npHBecTH B KOHTaKT C TepMOCTaTOM. T .e. TenOM Ha-
CTonbKO 60JlbillOH MaCCbI. QTO OHa 06eCneqHBaeT nOCTO.HHCTBO TeMnepaTypbI ra3a.
,Aa>Ke Korna OT)J;aeT ra3Y HnH nOJlyqaeT OT Hero HeKOTopoe KOJlHqeCTBO TenJlOTbI.
m
B ypaBHeHHH (6.15) npH m = const H M = const - RT = const. Tor)J;a
M
HMeeM
pV=const HJlH )J;JI.H )J;BYX COCTO.HHHH PI VI =P2V2' (6.16)
<I>OPMYJlb] (6.16) onHCbIBalOT 3aKOH EOHJI.H-MapuoTTa: npu nocmOSlliliOU
meMnepamype. lieU3.MeHIiOU .Macce U HeU3.MeIiIiOM XU.MUlleCKO.M cocma6e za3a
nPOU36eaeliUe OaBJl.eIiUSl lia 06oe.M ecmb 6eJl.UIlUlia nocmOSlliliaSl.
rpacpHKH 3aBHCHMOCTH Mem)J;Y napaMeTpaMH .v;aHHoH MaCCbI npH nOCTO.HH-
HOR TeMnepaType Ha3bIBaIOTC.H H30TepMaMH. Ha pHCYHKe 6.13 H306pa>KeHbI
H30TepMbI B OC.HX (p. V), (V, T). (P. T).
P V
T]
p
V
T 1 T 2
Puc. 6.13
T
T 1 T 2
T
144

6) H306apHblH npoecc - npoecc H3MeHeHH.H COCTO.HHH.H ra3a npH no-
V mR
CTO.HHHOM ,n;aBJleHHH: p = const. M3 ypaBHeHH.H (6.15) - - - OTHOllIeHHe
T Mp
mR
- = const npH m = const. M = const. CJle,n;oBaTenbHo,
Mp
v = const, HJlH Vi = V 2
T 7i T 2
YpaBHeHH.H (6.17) onHCbIBalOT 3aKOH llIapJI.H: 06'Oe.M oaliliou .MaCCbl za3a
npu nOCmOSl1iIiO.M Oa6JLeliUU U nOCmOSl1iliO.M XU.MUl£eCKO.M cocma6e np.R.Mo npo,
nopquoliaJl.eH a6COJl.lOmliOU me.Mnepamype.
3aKoH llIapJl.H MomHO 3anHcaTb qepe3 TeMnepaTYPY t, H3Mep.HeMYIO no IlIKa-
JIe QeJlbCH.H: V = V o (1 + at), re V o - 06'beM ra3a npH 0 °e. a == 23 K- I
(6.1 7)
TeMnepaTYPHbIH K03cpcpHHeHT 06'beMHOrO paCllIHpeHH.H.
rpacpHKH 3aBHCHMOCTH Mem,n;y napaMeTpaMH ra3a npH nOCTO.HHHOH Macce
ra3a H ,AaBneHHH Ha3blBalOT H306apaMH (pHC. 6.14).
v p p
1 P2>P.
. P2 P2
PI PI
, ,
,
T T V
Puc. 6.14
B) H30XOpHLIH npoecc - npoecc H3MeHeHH.H COCTO.HHH.H ra3a npH no-
CTO.HHHOM 06'beMe: V=const. M3 ypaBHeHH.H (6.15) g = mR JI.H ,n;aHHO:H
T MV'
mR
MaCCbl l'a3a m = const H M = const OTHOllIeHHe - = const H
V
P = const, HJlH PI = P2 . (6.18)
T 11 T 2
YpaBHeHH.H (6.18) onHCblBalOT 3aKOH reH-JIlOccaKa: Oa6Jl.eliUe Oa1iliOU .Mac.
Cbl za3a npu nocmOSlliliO.M 06'Oe.Me U nocmOSlIiHO.M XU.MUlleCKO.M cocma6e npSl
MO nponop-quoliaJl.bIiO a6COJl.lOmliOU me.Mnepamype.
EcnH TeMnepaTYPY H3Mep.HTb no llIKaJle IJ;eJlbCH.H, TO 3aKOH reH-JIlOccaKa
3anHllIeTC.H B BHlIe: p = Po(1 + at), rlIe Po - ,AaBJleHHe ra3a npH 0 °e. ex -
TeMnepaTYPHbIH K03cpcpHHeHT' ,n;aBJleHH.H, OKa3aBllIHHC.H OHHaKOBbIM lIJI.H Bcex
1 K - 1
ra30B: a = - .
273
rpacpHKH 3aBHCHMOCTH Me*,AY n8paMeTpaMH ra3a npH nOCTO.HHHOH Macce
ra3a H nOCTO.HHHOM 06'beMe Ha3blBalOT H30xopaMH (pHC. 6.15).
145

p
V 2 > VI
.V2
" ,
p
V
V
VI
T
VI V 2
v
T
Puc. 6.15
TaTeJIbHaR 3KcnepIIMeHTaJIbHaR npoBepKa cOBpeMeHHbIMH MeToaMH nOKa-
3aJIa, '-ITO ypaBHeHIIe COCTORHII.H II,AeaJ:TbHOrO ra3a H BbITeKalOIIe II3 Hero 3a-
KOHbI BOHJI.H-MapIIOTTa, reH-JIlOccaKa II IllapnR ,AOCTaTOqHO TO'-lHO onIIcbIBa-
lOT nOBe.lleHIIe peaJlbHbIX ra30B npII He60JIblllIIX ,AaBneHII.HX H He cnIIlllKOM
HII3KIIX TeMnepaTypax.
BOllPOCM AH CAMOKOHTPOH
1. llpu KaKUX YCJl.06USlX cnpa6eoJtu6 3aKOli BOUJl.Sl-Mapuomma? reu-JIlOc-
caKa? ill ap./l.R?
2. 060Slcliume lia OCli06e MKT. nOlleMY npu u30mepMuilecKoM npou,ecce
Oa6Jl.eHUe oaliliou MaCCbl za3a Y6eJtuiluBaemcSl C YMelibweliueM 06oeMa.
3. HaK MOJfCliO 060RCHumb 3aKOlibl reu-JIlOccaKa U lllap./l.R Ha OCli06e MKT?
4. II olieMY liazpemble .MeoUl.{UliCKUe 6aliKu .. npucacbl6alOmcSl" K meJty?
5. IIpu liaKallUBaHUU pe3ulioBoU Ka.Mepbl B030YXO.M oOliOBpe.MeliliO YBeJtu-
IlUBaemC.R. ezo o6oeM U Oa6Jl.eliue. He npomUBopellum Jl.U 3mo 3aKOliY Bou-
JtSl- M apuomma?
6. Co oHa 6000e.Ma nooliU.MaemCSl 6030YWliblU nY3blpeK. KaK U3.M.eli.R.emCSl
C ZJty6UHOU CUJl.a, 6blmaJl.KU6alOU{aSl ezo U3 600bl? q mo .MOJfCIiO CKa3amb 06
YCKopeliuu nY3blpbKa?
7. IIolle.M.Y oemcKuu 6030YWliblU wap, lianOJl.lieliliblu 6030YXOM, no Mepe
nooliSlmuSl 66epx nocmeneHliO Y6eJl.UIlUBaemcSl 6 06oe.Me?
8. H 3.M.eli.R.emC.R. JlU npu U30XOPHOM npou,ecce nJtomliocmb za3a?
9. JJoKaJfCUme. Ilmo Oa6Jl.eliUe oaHliou MaCCbl za3a npu nocmOSlliliOU meM-
nepamype nponopu,UOliaJl.bliO nJl.omliocmu.
10. Y3KaSl CmeKJl..R.HHa.R. mpy6Ka onyU{elia omKpblmbl.M. KOHU,O.M 6 cocyo C
pmymblO. Cpa6liUme n06eoeliue pmymu 6 mpy6Ke. ecJl.U Bmopou ee KOlieu,
a) 3anaSlIi, 6) omKpblm.
11. KaK 3a6ucum noooeMliaSl CUJl.a aapocmama om meMnepamypbl o KpYJfCa-
lOU{ezo B030yxa?
12. Hliozoa U3 6ymblltKU, lianOJl.lieliliOU za3up06aliliou 6000U. Bblltemaem np06-
Ka, ecJl.U 6ymblJl.Ka nOCmaBJl.eHa B menJl.oe Mecmo. llolle.MY amo npoucxooum?
13. KaK pacnOJl.OJfCelibl lia zpa(jJuKax (p. V). (V. T). (p. T) u.JOmepMbl
OOliOU u muu JlCe .MaCCbl za3a npu pa3liblX me.Mnepamypax? Mozym Jtu OliU
nepeceKambcSl?
14. II Olie.MY U30mep.M.a, pacnOJWJfCenna.R. BbllUe. coomBemcmByem npOl,{eccy
npu 60Jlee BblCOKOU me.M.nepamype?
15. KaK pacnOJl.OJICelibl lia zpa(jJuKax (p. V). (V, T), (p. T) u306apbl 00-
HOU U mou JlCe .MaCCbl za3a npu pa3liblX Oa6Jl.eliuSlx?
146

16. KaK paCnOJlOJICelibL lia zpaljJuKax (p. V), (V, T). (p. T) U30XOPbl 00
nou U moil JlCe .MaCCbL za3a npu pa3libLX 06-oe.Max?
17. I[ e.M OmJl.UllalOmCSl opyz om opyza 06l' U30XOPbL, coom6emcm6YIOUJ,ue
pa3libL.M .Macca.M oonozo U mozo JlCe za3a, 3anUMalOUJ,U.M 00unaK06bLe 06-oe.MbL?
18. H306pa3ume zpatPUllecKU 6 KoopoUliamax (p. V) 06e U30mep.MbL, coom-
6emcm6YIOUJ,ue pa3libL.M MaCCa.M OOIiOZO U mozo JlCe za3a. liaXOOSlUJ,ezocSl npu
00unaK060U me.Mnepamype.
19. BbL6eoume ypa6lieliue cocmOSlliUSl UOeaJl.bIiOZO za3a, UCXOOSl U3 ocn06-
nozo ypa6lieliUSl M KT.
20. I[mo .MOZJl.O npoucxooumb c Za30.M. ecJl.U npu Y6eJl.Ul£enUU ezo 06-oe.Ma
Oa6JleliUe OCma6aJl.OCb nocmOSllilibL.M?
21. llpu nepexooe onpeOeJl.eliliOU .MaCCbL za3a U3 oonozo cocmOSlHUSl 6 opy-
zoe ezo Oa6Jl.eliUe Y.MelibWaemcSl. a meMnepamypa Y6eJl.UIlU6aemcSl. Kal(, U3.Me-
liSlemcSl ezo 06-oe.M?
22. BapO.MempUlleCKaJl mpy6Ka 6Cma6Jl.elia 6 cocyo c pmymblO, npu amo.M
pmymb 6 mpy6Ke 6bLwe. iteM 6 cocyoe. KaK liaoo U3.MenUmb me.Mnepamypy
za3U 6 mpy6Ke. Y6eJl.UI£Umb UJlU Y.MenbUlUmb, I£mu6bl YPU6nu pmymu 6 cucyoe
U mpy6Ke cmaJl.U 00unUK06bl.MU?
23. llpu nepexooe onpeOeJl.eliliOU MaCCbl UOeaJl.bliOZO za3a u3 OOliOZD co-
cmO.Rnu.R 6 opyzoe o6oe.M ezo Jl.uneUliO Y.MelibWaemcSl C n06blWenUe.M me.Mne-
pamypbL. KaK U3.MeIiSlemCSl Oa6Jl.eliUe za3a?
24. H azpe6aliue UJl.U oXJl.aJICoeliue npoucxooum C UOeaJl.blibl.M Za30.M, ecJl.U
b
on pacwupSlemcSl no 3aKoliY p = , zoe bun - lieKomopbLe nocmOSlHHble,
V
npUl£e.M 0 < n  I?
25. BblBeoume 3aBUCU.MOCmb nJlomnocmu UOeaJlbnOZO za3a om me.Mnepamy-
Pbl. nOJlb3Y.RCb ypa6lienUe.M KJl.aneupona- M enOeJleeBa.
26. IfCXOOSl U3 ypaBnenU.R KJl.aneupona-MenoeJl.eeBa, nOJlYllume 3aKonbl
UOeaJlbliOZO za3a: BOUJl.Sl- M apuomma. reu-JI IOccaKa. ill apJl.Sl.
OCHOBHhIE cI»OPMY Jlh]
m
pV=-RT
M
- ypaBHeHHe
(ypaBHeHHe
r l1.e V - 06'beM
COCTO.HHHH Hl1.eaJIhHOro ra3a
KJIaneHpOHa- MeHl1.eJIeeBa),
,n:m
ra3a, R = 8,31
Monh.K
ra30BaH nOCTO.HHHaH.
YHHBepCanhHa.H
pV
- = const npH v = const
T
- ypaBHeHHe Knanei1poHa, rl1.e v
JIH'leCTBO ra3a.
KO-
pV = const npH v = const, T = const - 3aKOH Boi1n.H-MapHoTTa.
.!!.. = const npH v = const, V = const - 3aKOH reH-JIlOccaKa.
T
V
- = const npH v = const. p = const - 3aKOH llIapJl.H.
T
147

METOHqECKHE PEKOMEHAQHH
IIpH pemeHHH 3a.n;aq Ha paCqeT napaMeTpOB COCTO.HHH.H ra3a peKOMeH.n;yeT-
C.H cne.n;YIOa.H nOCJle.n;OBaTeJlbHOCTb:
1. BbISlCHHTb, H3MeHSleTC.H HJlH HeT COCTO.HHHe ra3a. ECJlH B 3a.n;aqe 3a.n;a-
HO O.D;HO COCTO.HHHe ra3a, TO nOJlb3YIOTC.H ypaBHeHHeM KJIaneHpOHa-MeH.n;e-
JleeBa.
2. ECJIH B 3a.n;aqe .naHb! .n;Ba HJlH HeCKOJlbKO COCTO.HHHH ra3a. TO peKOMeH-
.n;yeTC.R napaMeTpbI 3TH X COCTO.HHHH (KaK AaHHble, TaK H HCKOMbIe) 3anHCaTb
B BH.n;e Ta6J1HbI:
1) COCTO.HHHe ra3a - VI="', PI="', V I =..., T I =...;
2) COCTO.HHHe ra3a - V2= ..., P2= ..., Y 2 = ..., T2=....
C.n;eJlaTb, eCJlH B03MomHO, cXeMaTHqeCKHH PHCYHOK.
3. BbI.HCHHTb, H3MeH.HeTC.H JlH Macca ra3a Hero COCTaB, T.e. MOJl.HpHa.H
Macca. ECJlH Macca ra3a H3MeH.HeTC.H HJlH Macca .n;aHa B YCJlOBHH, TO ,AJI.H
Kam,AOrO COCTO.HHH.H 3anHcaTb ypaBHeHHe KJlaneHpOHa-MeH.n;eJleeBa. ECJlH Macca
ra3a He H3MeH.HeTC.H, TO 3anHcaTb ypaBHeHHe KJlaneHpOHa HJlH O.n;HH H3 ra30-
BbIX 3aKOHOB: BOHJI.H-MapHOTTa, reH-JIlOccaKa HJlH IIIapJl.H.
4. IIpe.n;cTaBHTb B pa3BepHYTOM BH.n;e napaMeTpbI (p, Y, T) HaqaJlbHOro H
KOHe"lHOrO COCTO.HHHH ra3a.
5. 3anHcaTb AOnOJlHHTeJlbHbIe ypaBHeHH.H. CB.H3bIBalOI.Q:He HCKOMbIe BeJlHqH-
HbI HJIH napaMeTpbI COCTO.HHH.H, HcnOJlb3Y.H YCJlOBHe 3a.n;aqH.
6. IIpH paCCMOTpeHHH npOn;eCCOB, CB.H3aHHbIX C H3MeHeHHeM COCTO.HHH.H
.n;BYX HJlH Tpex ra30B, BXO.n;.HID;HX B COCTaB CMeCIi HJlH OT.n;eJleHHbIX ,Apyr OT
.n;pyra nOpmH.HMH HJlH neperopO)J;KaMH, Bce YKa3aHHbIe .n;eHcTBH.H HymHo npo-
.n;eJlaTb .n;JI.H Kam.n;oro ra3a OT.n;eJlbHO.
7. PemHTb nOJlyqeHHYIO CHCTeMY ypaBHeHHH.
llPHMEPbI PEIIIEHH.H 3AA q
1. 11306pa3HTe rpa<}>H'leCKH 3aBHCHMOCTh nnOTHOCTH ,AaHHOH MaCChI H.neanb-
Horo r83a OT TeMnep8TypbI npH nOCTOSlHHOM ,A8BneHHH.
P e III e H H e. llpH p = const H m = const BbInonHSleT-
CSI 3aKOH 1lI8pnSl
p

v
- = const. OTCIO,Aa V = const. T.
T
m
llo onpe,AeneHHIO nnOTHOCTh p = -. Cne,AOBaTenbHO,
V
T
m
p = , T.e. HCKOM8S1 38BHCHMOCTh H306pamaeTCSI
T . const
rHnep6oJIOH (pHC. 6.16).
Puc. 6.16
2. Ha pHCYHKe 6.17 H306pameHhI npoeCCbI, npOHCXO,ASlIl.\He C onpeneneHHoH
MaccoH H.n;eanbHoro ra3a. 11306pa3HTe 3TH npoeCChI B OCSIX (p, V) H (p. T).
P e III e H He. 113 pHcYHKa 6.17 BHnHO. 'lTO npouecc 1-2 - H30TepMH'tJeCKoe
paCIIIHpeHHe. 2-3 - H30xopHoe HarpeB8HHe. 3-4 - H30TepMH'leCKOe paCIIIHpe-
HHe. B OCSIX (p, V) H (p, T) 3TH npoeCCbI H306pameHbI Ha pHCYHKe 6.18.
148

V 4 P - P v1
2vA
,. 3
"
"
1 " 2 / 2
/
" "
/
T V T
Puc. 6.17 Puc. 6.18
3. B 3anaSlHHOll C O,AHOrO KOHa CTeKnSlHHOll Tpy6Ke, ,AnHHa KOTOpOll
70 CM, HaXO,AHTCSI cTon6JiK B03,Ayxa. 38nepTbIll cTon6JiKOM PTYTH BbICOTOll
20 CM, ,AOXO,ASIJiM AO BepxHero KpaSl Tpy6KJi. Tpy6KY Me,AneHHO nepeBopa-
'lJiBaIOT. npH 3TOM 'laCTb PTYTH BbInHBaeTCSI. KaKoB8 BbICOTa cTon6JiKa PTY-
TJi, KOTOpbIll OCTaHeTCSI B TPy6Ke, ecnJi aTMocQ>epHoe ,AaBneHHe 100 KTIa?
TIpH K8KOM ,AnHHe Tpy6KJi cTon6HK PTYTH TOll me BblCOTbl BblnbeTCSI Ji3
Tpy6KH nonHOCTblO?
HaHTJi: x, ll'
,Z:(8HO: l=0,70 M, h=0,20 M. Pa=1,00'10 5 TIa, p=1,36 10 4
Kr
3 .
M
x = (Pa + pgl) I (Pa + pgl)2 - 4pg(Pa h - pgh(l- h)) .
2pg
TIO,ACT8BnSlSl 'lJiCneHHbIe 3HaQeHHSI, nonY'lHM Xl = 1.4 M H x2= 0,04 M. I10-
cKonbKY nepBbIll KopeHb npoTHBOpeQHT ycnoBHIO 381I8'lH (XI> l). TO X = 0.04 M.
M3 BbIpameHJiSl (1) BHJJ.HO. 'lTO npH X = 0
,"
"
'\
/ , '
)
I I' 1
/ ,
J \ "-
)
J I '\
/
" ....\.
)
// x
Puc. 6.19
Pew e H H e. [JPJi Me,AneHHOM nepeBOpa'iJiBaHJiH Tpy6-
KJi TeMnepaTypa OCTaeTCSI nocToSlHHOll. TI03TOMY MomHO
3anJiCaTb 3aKOH BOMnSi-MapJioTT8 P I V l =P2V2'
M3 pHCYHKa 6.19 BJiAHO, 'lTO
V 1 =(l-h)S; P1=Pa+pgh; V 2 =(l-x)S; P2=Pa-pgx,
r,Ae p - nnOTHOCTb PTYTH. S - nnOa,Ab Ce'leHHSI
Tpy6KH. TIO,ACTaBHM B aaKOH BOllnSl-MapJioTTa
(Pa t pgh )( l - h) S = (Pa - pgx )( l - x) S 
pgx 2 - (Pa + pgl)x + Pa h - (l- h )pgh = O.
OTCIOJJ.8
(1)
Pa + pgh
Pah - pgh (ll - h) = 0  II = ;
pg
II = 0,95 M.
TaKHM o6pa30M, PTYTb BbIJIbeTCSI npH II  0,95 M.
4. B HnJiH,Ape, nnO8,Ab OCHOB8HJiSl KOToporo 100 CM 2 . HaXO,AJiTCSI B03,AYX
npJi TeMnepaType 7°C. Ha BbICOTe 60 CM OT OCHOBaHHSI JinHH,Apa pacnonomeH
nopmeHb MaccoM 10 Kr. H8 CKonbKO onYCTHTCSI nopmeHb, ecnH H8 Hero nOCTa-
BJiTb rJiplO Maccoll 100 Kr, a B03,AYX B JinHH,Ape HarpeTb JJ.O 27°C? TpeHHeM
nopmHSI 0 CTeHKH JinJiH,Ap8 npeHe6pe'lb. ATMocQ>epHoe ,A8BneHJie 0.10 Mila.
149

.D:aHO: 8=1,00,10- 2 M 2 , T I =280 K. h=0,60 M, m=10 KI', ml=100 KI',
T 2 =300 K, Pa=1,0'105 TIa.
H a H T H: /:1h.
Pew e H H e. Macc8 B03AYX8 B l.J,HnHH,Ape He H3MeHSleTCSI. Cne,AOB8TenbHO,
MO)KHO Hcnonb30BaTb ypaBHeHHe KnaneHpOHa:
PI VI = P2 V 2
T 1 T 2
H8'lanbHblH 06'beM B03ltYX8 VI = h8. KOHetIHblH V 2 = (h - t1h}8.
TaK KaK nopmeHb B HatI8ne H B KOHue npoueCC8 HaXO,AHTCSI B COCTOSlHHH
p8BHOBeCHSI. TO
mg
PI = Pa + S'
(m + m,,)g
P2 = Pa + 8 .
I10,ACTaBnSlSl nonytIeHHbISI 3Ha'leHHSI Vb V 2 . Pit P2 B HCXOltHYIO c:popMyny,
nonY'lHM
(Pa+¥)hS Jpa+(m+;,jg)(h-M)S =>
7i T 2
( (PaS + mg)T2 ]
h = h 1 - ( ( ) ) ; h = 0, 26 M.
PaS + m + m i g T 1
5. Onpe,AenHTe nnOTHOCTb CMeCH, COCTOSleH H3 4,0 r BO,AOpO,A8 H 32 I'
KHCnOpO,Aa npH TeMnepaType 7,0 °C H ,A8BneHHH 93.3 KTI8.
.D:8HO: ml=4.0.10- 3 KI'. m2=0,032 Kr. T=280 K, p=9,33.10 4 fIa.
-3 Kr Kr
M I =2.0'10 M 2 =0,032
Monb
Monb
H 8 H T H: p.
m
Pew e H H e. HCKOM8S1 nnOTHOCTb P = V ' r,Ae m = mi + m2 - M8cca CMeCH,
V - 06'beM CMeCH. TIo 38KOHY .D:8nbToHa P = PI + P2. l',Ae PI H P2 - n8pl.J,H-
anbHble ,A8BneHHSI BO,AOpO,A8 H KHCnOpO,Aa COOTBeTCTBeHHO.
113 yp8BHeHHfl Kn8neHpoHa-MeH,AeneeBa ,AnSI BO,AOpO,Aa H KHCnOpO,Aa HaH,AeM
miRT m 2 RT
PI = M V ' P2 = M V .
I 2
Tor ,Aa
P = RT ( + \ V = RT ( + \
V M 1 M 2 J P M 1 M 2 J
I10,ACTaBHM 3Ha'leHHe V H m B HCXOAHYIO c:popMyny H nonY'lHM
(ml + m 2 ) P
P- .
- RT ( + m2 J .
M 1 M 2
KI'
P = 0.48 3"'
M
150

7. OCHOBW TEPMOAHHAMHKH
TepMO,lJ;HHaMHqeCKHU no,lJ;XO,lJ; K H3yqeHHIO H3HqeCKHX npoeCCOB. TepMo-
,lJ;HHaMHqeCKH napaMeTpLI COCTO.HHHH TeJla. BuyrpeHHHH 3HeprHH TeJla. H3Me-
HeHHe BHYTpeHHeu 3HeprHH. KOJlHqeCTBO TenJlOTLI. Pa60Ta B TepMO,lJ;HHaMHKe.
nepBblu 3aKOH TepMO,lJ;HHaMHKH. npHMeHeHHe nepBoro 3clKOHa TepMO,lJ;HHa-
MHKH K pa3J1HqHLIM TenJlOBLIM npoeccaM. A,lJ;Ha6aTHblu npoecc. TenJlOeM-
KOCTH npH nOCTOHHHOM ,lJ;aBJleHHn It nOCTOHHHOM 06'beMe.
06paTHMLIe n He06paTHMLI npoeCCbl. He06paTHMocTb TenJlOBLIX npoec-
COB. BTOPOU 3aKOH TepMO,lJ;HHaMHKH.
TenJlOBbie ,lJ;BHraTeJlH. npHHHn ,lJ;eUCTBHH TenJlOBLIX ,lJ;BHraTeJleU. Kn.a Ten-
JlOBOrO lIBHraTeJlH H nYTH ero nOBblweHHH. TpeTHU 3aKOH TepMOllHHaMHKH.
POJlb TenJlOBLIX ,lJ;BIUaTeJleH B pa3BHTHH TenJl03HepreTHKH H TpaHcnopTa. Ten-
JlOBLle MaWHHLI H oxpaHa npHpO,lJ;bl.
PeaJlbHLle ra3LI. KpHTHqeCKOe COCTOHHHe BeeCTBa.
CBouCTBa mH,lJ;KOCTeu. nOBepXHOCTHaH 3HeprHH. nOBepXHOCTHoe HaTHme-
Hlle. CMaqHBaHHe. KanllJlJlHpHLle HBJleHH.H.
KpHCTaJlJlllqeCKHe II aMopHLle TeJla. CTpoeHlle KpHCTaJlJlOB. AHII30TpO-
nHH KpIlCTaJlJlOB. nOJlIIMOp1I3M. MOHOKpllCTaJlJlbl II nOJlIIKpIlCTaJlJlLI. npo-
CTpaHCTBeHHaH peweTKa.
AeeKTLI B KpIlCTaJlJlax. AeopMaIIH. BII,lJ;LI ,lJ;eOpMaIIU TBep,lJ;blX TeJl.
HanpHmeHHe. 3aKOH rYKa. MeXaHHqeCKlle CBouCTBa TBep,lJ;LIX TeJl: ynpyrocTb,
npOqHOCTL, nJiaCTllqHOCTb, xpynKoCTL. J];lIarpaMMa paCTHmeHIlH.
ArperaTHLle COCTOHHHH BeecTBa. IlcnapeHlle H KOH,LJ;eHCaHH. HaCbleH-
Hble H HeHaCLIeHHLle napLI. 3aBilCHMOCTb ,lJ;aBJleHIIH II nJlOTHOCTIl lIaCbleH-
Horo napa OT TeMnepaTypbl.
KllneHHe mll,lJ;KOCTeu. 3aBIICilMOCTb TeMnepaTypbl KllneHHH mllIlKOCTIl OT
,lJ;aBJleHIlH.
BJlaJKHOCTL B03,lJ;yxa. TOqKa pOCLI. nCIIXpOMeTp. rllrpOMeTp.
AllarpaMMa COCTOHHIIH BeeCTBa.
7.1. TepMO,lJ;HHaMH'IeCKHe napaMeTpbI COCTOHIIHH TeJIa
B TepMOHHaMIiKe IIMelOT eJlo C TepMOIIHaMH'tJeCKHMII CIiCTeMaMH. rro
TepMO,lJ;HHaMHqeCKOU CIlCTeMOH nopa3YMeBaeTC.H JlI060e KOHe'tJHbIX pa3MepOB
MaKpOTeJlO HJlII cOBoKynHocTb MaKpOTeJI. rro MaKpOTeJIOM IIMeeTC.H B BIIY
TeJlO, pa3MepbI KOToporo BeJlHKH no cpaBHeHlI10 C aTOMHbIMII pa3MepaMII II
KOTopoe COCTOIiT 113 orpOMHoro qlICJIa MIIKPOq8CTIIn;.
B OTJIH'tJlIe OT COCTO.HHIIH MexaHII'tJeCKOM CIiCTeMbI. onpe.n:eJI.HeMOrO COBO-
KynHocTblO KoopIIHaT II CKopOCTeM TeJI, BXOD;.HIIX B Hee, COCTO.HHlIe Tep-
MOIIHaMII'tJeCKO:H CIiCTeMbI onpeenHeTC.H Ha6opOM 3Ha'leHlIM TepMOIIHaMII'tJec-
KIiX napaMeTpOB, T.e. cpH3H'tJeCKIiX BeJIII'lIIH, xapaKTeplI3YIOHX CBOMCTBa CH-
CTeMbI B n;eJIOM. HeKOTopble H3 3TlIX BeJIlIqlIH. KaK, HanpliMep, aBneHlIe II
TeMnepaTypa, He npHMeHHMbI K MHKpO'tJaCTHn;aM, H3 KOTOpbIX COCTOHT Tep-
MOHHaMHqeCKa.H CIiCTeMa. TepMOHHaMH'leCKlie napaMeTpbI .HBJIHIOTC.H MaKpo-
BeJIH'lHHaMH, T.e. TaKHMH BeJIH'tJHHaMH, KOTopbIe MorYT 6bITb JlH60 Henoc-
peCTBeHHO H3MepeHbI C nOMOblO nplI60pOB, JIH60 BbIpa:>KeHbI 'tJepe3 PyrHe,
151

HenocpeCTBeHHO H3Mep.HeMbIe H8 OnbITe BeJlHtIHHbI. 'tJHCJlO napaMeTpOB, on-
peeJl.HIOHX COCTO.HHHe CHCTeMbI, 3aBHCHT OT poa CHCTeMbI H BHelIlHHX B03-
eHCTBHii. COCTO.HHHe npOCTeHIIlHX TepMOHHaMHqeCKHX CHCTeM (OHOpOHbIX
ra30B H mHKOCTeii, He nO,ABepmeHHbIX eHCTBHIO K8KHX-JlH60 [loJIeii) onpe-
,AeJlHeTC.H 8BJIeHHeM, 06eMoM H TeMnep8TYPoH.
7.2. BHYTpeHHJUI 3HeprHH TeJI
COrJIaCHO MKT Bce BeeCTB8 COCTO.HT H3 tI8CTHlI" KOTopbIe HaXO.HTC.H B
HenpepbIBHOM TenJIOBOM BH:>KeHHH H B3aHMoeiicTBYIOT ,lJ.pyr C PyrOM. II03To-
MY  ,lJ.ame eCJlH TeJlO HenO,lJ.BH:>KHO H HMeeT HyneBYIO nOTeHlI,HaJlbHYIO 3HeprHIO,
OHO 06J1a,lJ.aeT 3HeprHeH (BHYTpeHHeH 3HeprHeH), npeCTaBJI.HIOeH c060ii CYM-
MapHYIO 3HeprHIO BHmeHH.H H B38HMO,AeHCTBHH MHKpOqaCTHlI" COCT8BJIHIOHX
TeJIO. B COCTaB BHYTpeHHeH 3HeprHH BXO,lJ..HT: 1) KHHeTHqeCKa.H 3HeprH.H no-
cTynaTeJIbHOrO, BpaaTenbHoro .Ii KOJIe6aTeJIbHOrO BHmeHH.H MOJIeKYJI; 2) no-
TeHlI,H8JIbHaH 3HeprH.H B3aHMoeiicTBH.H aTOMOB H MOJleKYJI; 3) BHYTpHaTOMHa.H
H BHYTpH.H,AepHa.H 3HeprHH.
B TepMOHHaMHKe paCCMaTpHBaIOTC.H npOlI,eCCbI npH TeMnepaTyp8X, npH KO-
TOpbIX He B036ym,AaeTC.H KOJle6aTeJIbHOe ,lJ.BHmeHHe aTOMOB B MOJleKYJlaX, T.e.
npH TeMnepaTypax, He npeBbIlIlalOHx 1000 K. B 3THX npolI,eCcax H3MeHHIOT-
C.H TOJlbKO nepBbIe Be COCTaBJI.HIOHe BHYTpeHHeii 3HeprHH. II03ToMY no,n BHyr-
peHHeH 3HeprHeH B TepMO,lJ.HHaMHKe nOHHMaroT CYMMY KHHeTHqeCKOH 3HeprH"
Bcex MoneKyn " aTOMOB Tena H nOTeHlI,HanbHOH 3HeprHH HX B3aHMo,neHcTBHH.
BHYTpeHHH.H 3HeprH.H Tena onpe,AeJI.HeT ero TennOBoe COCTO.HHHe H H3MeHH-
eTCH npH nepexoe H3 O,AHOrO COCTO.HHH.H B pyroe. B ,D;aHHOM COCTO.HHHH TeJIO
o6JIa,D;aeT SnOJIHe onpe,D;eneHHOH BHYTpeHHeH 3HeprHeH He 38BHC.Heii OT
Toro, B pe3YJlbT8Te KaKoro npOlI,eCca OHO nepelIlJlO B aHHoe COCTO.HHHe. IIo-
3TOMY BHYTpeHHIOIO 3HeprHIO OqeHb qaCTO Ha3bIB81OT c}>YHKlJ,HeH COCTOHHHH
TeJIa.
PaCCqHTaTb BHYTpeHHlO1O 3HeprHIO MomHO TonbKO nH HeaJlbHOrO ra3a.
T8K KaK B He8JIbHOM ra3e MOJleKYJIbI He B3aHMoeiicTBYIOT Me:>Ky C060H,
TO nOTeHlI,HaJIbHa.H 3HeprHH. HX p8BHa HYJIIO H BHYTpeHH.H.H 3HeprH.H HeaJIb-
Horo ra38 npeCT8BJI.HeT c060ii KHHeTHqeCKYIO 3HeprHIO Bcex ero MOJIeKYJI.
Cpe,AH.H.H KHHeTH'leCKa.H 3HeprH.H MOJleKYJlbI (W K ) = kT.
2
m
'tJHCJIO MOJIeKYJI B ra3e N = - N A'
M
CJIeOBaTeJlbHO, BHYTpeHH.H.H 3HeprH.H He8J1bHOrO ra38
( m i
U=N W K )= M NA"2kT.
Y tIHTbIBas, tITO kN A = R - YHHBepCaJIbHa.H ra30Ba.H nOCTOHHH8H, HMeeM
im
U = - - RT BHYTpeHH.H.H 3HeprHH He8nbHoro ra3a. (7 1)
2M
3m
B tIaCTHOCTH, )J;JI.H 0H08TOMHoro ra38 U = --RT.
2M
152

113 3TH X cpOPMYJI BR.n;HO, 'lTO BHyrpeHHJUI 3HeprHH H,ZJ:eaJlLHOrO ra3a 3aBH-
CHT TOJILKO OT TeMnepaTypLI H qHCJIa MOJIeKYJI H He 3aBHCHT HH OT 06'beMa,
Hli OT .n;aBJleHHH. n03TOMY H3MeHeHHe BHYTpeHHeH 3HeprHR R.n;eaJIbHOrO ra3a
onpeeJIHeTCH TOJIbKO R3MeHeHHeM ero TeMnepaTypbI R He 3aBHCRT OT xapaK-
Tepa npoecca, B KOTOpOM ra3 nepexo.n;RT R3 o.n;Horo COCTO.HHR.H B .n;Pyroe:
im
fj,V = V 2 - VI = -- Rfj,T. rAe fj,T = T 2 -71.
2M
MOJleKYJIbI peaJIbHbIX ra30B B3aHMO,AeHcTBYIOT Mem.lI.Y C060H H n03TOMY 06-
na.n;alOT nOTeHHaJIbHOH .3HeprHeH W n' KOTopaH 3aBRCHT OT paCCTOHHHH Mem-
.n;y MOJIeKYJIaMH H, CJIe.n;OBaTeJIbHO, OT 3aHHMaeMoro ra30M 06'beMa
TaKRM o6pa30M. BHYTpeHHHH 3HeprH.H peaJILHOrO ra3a 3aBRCHT OT ero TeM-
nepaTypbI. 06'beMa R CTPYKTYPbI MOJIeKYJI.
JIH perneHH.H npaKTH'leCKHX BonpOCOB CYlI..\eCTBeHHYIO pOJIb I1rpaeT He caMa
BHYTpeHHHH 3HeprHH, a ee H3MeHeHHe l::.V= V 2 - VI' 113MeHeHHe me BHYTpeH-
Hell 3HeprHH paCCqHTbIBaIOT, HCXO.n;H H3 3aKOHOB coxpaHeHH.H 3HeprHH.
7.3. H3MellellHe BIIYTpelllleH 311eprHH
BHYTpeHHH.H 3HeprHH TeJIa MomeT H3MeHHTbCH .n;BYMH cnoc06aMH:
1. IIpH COBeprneHHli MeXaHHqeCKOH pa6oTbI.
a) ECJIH BHernH.HH CHJIa BbI3bIBaeT .n;ecpopMaHIO TeJIa, TO npH 3TOM H3Me-
HHIOTC.H paCCTO.HHH.H Memy qaCTHu.aMH. H3 KOTOpbIX OHO COCTOHT. a CJIeJJ.OBa-
TenbHO, H3MeHHeTC.H nOTeHHaJIbHaH 3HeprH.H B3aHMOlleHCTBH.H qaCTHu.. IIpR He-
ynpyrHx .n;ecpopMaHHX, KpOMe Toro, H3MeH.HeTCH TeMnepaTypa TeJIa, T.e. H3-
MeHHeTC.H KHHeTHqeCKa.H 3HeprR.H TenJIOBOrO .n;BlimeHRH qaCTH. Ho npH .n;e-
<l>opMaHH Tena COBeprnaeTCH pa60Ta. KOTopaH H HBnHeTCH MepOH 113MeHeHHH
BHYTpeHHell 3HeprHH TeJIa.
6) BHYTpeHH.HH 3HeprR.H TeJla li3MeHHeTC.H TaKme npR ero HeynpyroM coy.n;ape-
HIm C APyrHM TeJIOM. KaK MbI BR.D.eJIH paHbrne, npH HeynpyroM coy.n;apeHHH TeJI
liX KHHeTHqeCKaH 3HeprH.H YMeHbrnaeTC.H, OHa npeBpalI..\aeTCH BO BHYTpeHHlO1O (Ha-
npHMep, eCJIH Y.D.apHTb HeCKOJIbKO pa3 MOJIOTKOM no npOBOJIOKe, JIemalI..\eH Ha Ha-
KOBaJIbHe, - npOBOJIOKa HarpeeTCH). MepoH H3MeHeHli.H KHHeTliqecKOH 3HeprHH TeJIa
HBJIHeTC.H, COrJIaCHO TeopeMe 0 KHHeTHqeCKOH 3HeprHH, pa60Ta .u.eHCTBYIOlI..\HX CHJI.
3Ta pa60Ta MomeT cJlymHTb H MepoR H3MeHeHR.H BHYTpeHHeH 3HeprHH.
B) H3MeHeHHe BHYTpeHHeH 3HeprHH Tena npOHCXO.D.HT no.n; .n;eHCTBHeM CHJIbI
TpeHHH. nOCKOJlbKY, KaK H3BeCTHO 113 onbITa, TpeHHe Bcer.n;a conpOBom.n;aeTC.H
M3MeHeHHeM TeMnepaTypbl TPYlIlHXCH TeJl. Pa60Ta CHJIbI TpeHH.H Mo:meT CJIY-
mMTb MepOH H3MeHeHli.H BHYTpeHHeH 3HeprHH.
2. CYlI..\eCTByeT elI..\e cnoco6 R3MeHeHHH 3HeprHH TeJIa, He CB.H3aHHbIH C
pa60TOH CHJI. TaK, eCJIli TeJIO nOMeCTHTb B nJIaM.H rOpeJIKli, ero TeMnepaTypa
M3MeHHTCH, CJIe.n;OBaTeJIbHO, H3MeHliTCH 11 ero BHYTpeHH.HH 3HeprHH. O.n;HaKO
HHKaKa.H pa60Ta 3.n;eCb He COBeprnaJIaCb, H60 He npOHcxo.n;Hno BH.D.HMOrO nepe-
MelI..\eHHH HI1 caMoro TeJIa, HH ero qaCTeH.
113MeHeHHe BHYTpeHHeH 3HeprRR CHCTeMbl 6e3 COBeprneHR.H pa60Tbl Ha3bIBa-
eTCH TenJIo06MeHOM (TenJlOnepe.D.a qeH).
CYlI..\eCTByeT TPR BR.n;a Tennoo6MeHa: TenJlOnpOBo.n;HOCTb, KOHBeKH.H R R3-
nyqeHlie.
153

a) TenJIOnpOBo.a;HOCTLIO Ha3bIBaeTC.H npoecc TenJlo06MeHa Me>K,ny TeJIaMH
(HJIH qaCT.HMH TeJIa) npH HX Henocpe,nCTBeHHOM KOHTaKTe, 06YCJIOBJIeHHbIn Ten-
JIOBbIM XaOTHqeCKHM ,nBHmeHHeM qaCTH Tena. AMnJIHTy,na KOJIe6aHHH MOJIeKYJI
TBep.n;oro TeJIa TeM 60JlbIlIe, 'leM BbIllie ero TeMnepaTypa. TenJIOilpOBO,nHOCTb ra-
30B 06YCJIOBJIeHa 06MeHOM 3HeprHen Mem.n;y MOJIeKyn8MH ra3a I1PH HX CTOJIKHO-
BeHH.HX. B CJIyqae mH,AKOCTen pa60TalOT 06a MexaHH3Ma. TenJIOnpOBo.n;HOCTb Be-
eCTBa MaKCHMaJIbHa B TBep.n;OM H MHHHManbHa B ra3006pa3HOM COCTO.RHHH.
6) KOHBeKHH npe,nCTaBJI.ReT co6on TenJIOnepe.n;aqy HarpeTbIMH nOTOKaMH
>KH,AKOCTH H:JIH ra3R OT O.n;HHX yqaCTKOB 3RHHMaeMoro HMH 06'beMa B ,npyrHe.
B) TenJIo06MeH npH H3JIyqeHHH ocymeCTBJI.ReTC.R Ha paCCTO.HHHH nocpelICTBOM
3JIeKTpOMarHHTHbIX BOJIH.
7.4. KOJIHlIeCTBO TenJIOTbI
KaK H3BeCTHO, npH pa3JIHqHbIX MexaHH'tJeCKHX npoeccax npOHcxo,nHT H3-
MeHeHHe MexaHH'tJeCKon 3HeprHH W Mex' Mepon H3MeHeHH.R MexaHH'tJeCKon 3Hep-
rHH .HBn.ReTC.H pa60Ta CHJI, npHJIO>KeHHbIX K CHCTeMe:
11 WMex=A.
IIpH TenJIo06MeHe npOHcxo,nHT H3MeHeHHe BHYTpeHHen 3HeprHH TeJIa. Mepon
H3MeHeHH.H BHYTpeHHen 3HeprHH npH TenJIoo6MeHe RBJI.ReTC.H KOJlH'tJeCTBO TenJIOTbI.
KOJlHQeCTBO TenJIOTLI - 3TO Mepa H3MeHeHHR BHYTpeHHen 3HeprHH, KOTO-
PYIO TeJIO nonyqaeT (HnH OT,AaeT) B npoecce TenJIo06MeHa.
TaKHM 06pa30M, H pa60Ta, H KOnHqeCTBO TenJIOTbI xapaKTepH3YlOT H3Me-
HeHHe 3HeprHH. HO He TomlIeCTBeHHbI 3HeprHH. OHH He xapaKTepH3YlOT caMO
COCTO.HHHe CHCTeMbI. a onpe,AeJI.RIOT npoecc nepexOlIa 3HeprHH H3 OlIHoro BHlIa
B ,npyron (OT OlIHoro TeJla K lIpyrOMY) npH H3MeHeHHH COCTO.HHH.R H cywe-
CTBeHHO 3aBHC.RT OT xapRKTepa npoecca.
OCHOBHoe pa3JIHqHe Me>K.n;y pa60TOH H KOJIHt.J:eCTBOM TenJIOTbJ COCTOHT B TOM,
ITO pa60Ta xapaKTepH3yeT npoecc H3MeHeHHR BHYTpeHHen 3HeprHH CHCTeMbI.
conpOBom,nalOHnC.H npeBpaeHHeM 3HeprHH H3 o.n;Horo BH.n;a B .n;pyron (H3 Me-
xaHH'tJeCKon BO BHYTpeHHIOIO). KOJlH'tJeCTBO TenJIOTbI xapaKTepH3yeT npou,ecc ne-
pe,na'lH BHYTpeHHen 3HeprHH OT O,AHHX TeJI K .n;pyrHM (OT 60JIee HarpeTbIX K
MeHee HarpeTbIM), He COnpOBO>K.n;alOHnC.H npeBpaeHH.RMH 3HeprHH.
OnbIT nOKR3bIBaeT, qTO KOJIH'leCTBO TenJIOTbI, He06xo,nHMoe ,nJI.H HarpeBa-
HHR TeJIa Maccon m OT TeMnepaTypbI TJ ,no TeMnepaTypbI T 2 , paCCt.J:HTbIBaeTCR
no (j>OPMY JIe
Q = cm(T 2 - T 1 ) = cml1 T,
(7.2)
r.n;e c - y,neJIbHa.H TenJIOeMKOCTb BeeCTBR;
c = Q .
m (T2 -1;.)
E,AHHHen y,neJIbHOn TenJIOeMKOCTH B CM .HBnReTCR .n;>K0YJIb Ha KHJlOrpaMM-
KeJIbBHH r A>K ) .
Kr K
Y.a;eJILHaH TenJlOeMKOCTb c qHCJIeHHO paBHa KOJlHqeCTBY TenJIOTW, KOTopoe
Heo6xo,nHMO C006HTb TeJIY Maccon 1 Kr, 'tJTo6w HarpeTb ero Ha 1 K.
154

TenJIOeMKOCTL TeJla C T qHCJleHHO paBHa KOJIHtJ:eCTBY TenJlOTbI, He06xo)J;H-
MOMY JI.H H3MeHeHHH TeMnepaTypM TeJl8 Ha 1 K:
C = Q = em.
T T 2 -T 1
EilRHRl\eii rennoeMKOCTR TeJJa B CM SlBJUIeTCSI i\>KoynL HB KRnorpaMM r  ].
)J;JI.H npeBpaeHH.H )KHKOCTH B nap npH HeH3MeHHoH TeMnepaType He06xo-
,AHMO 3aTpaTHTb KOJIHqeCTBO TennOTbI
Q=Lm,
(7.3)
r,lIe L - YlIeJlbHa.H TenJlOTa napoo6pa30BaHH.H. IIpH KOHlIeHcau,HH napa BbI,lIe-
n.HeTCH TaKoe me KOJlHqeCTBO TenJlOTbI.
)J;JI.H Toro qT06bI paCnJlaBHTb KpHCTaJlJIH"t{eCKoe TeJIO MaCCOH m npH TeM-
nepaType nJlaBJleHH.H, He06xoHMO TeJlY C006HTb KonHtJ:eCTBO TenJlOTbI
Q= Am, (7.4)
r,lIe A - Y,lIeJlbHaSl TennOTa nnaBJleHH.H. IIpH KpHCTaJlJlH3aU;HIi TeJI8 TaKoe
me KOJlHqeCTBO TenJlOTbI BbI,lIeJIHeTC.H.
KOJlHtJ:eCTBO TenJlOTbI, KOTopoe BbI,lIeJlHeTC.H npH nOJlHOM cropaHHH TOnJlH-
Ba MaCCOH m,
Q=qm, (7.5)
r,lIe q - Y,lIeJlbHa.H TenJlOTa crOpaHH.H.
EHHHu;a yeJlbHbIX TenJlOT napo06pa30BaHH.H, nJIaBJIeHHH H crOpaHH.H B
CM - ,lImOYJIb Ha KHnorpaMM r : ).
7.5. Pa60Ta B TepMO,lJ:HHaMHKe
IIpH paCCMOTpeHHH TepMO,lIHHaMH'leCKHX npou;eCCOB MexaHHtJ:eCKoe nepeMe-
eHHe MaKpOTeJl B u;eJlOM He paCCMaTpHBaeTC.H. IIOH.HTHe pa60TbI 3,lIeCb CB.H-
3bIBaeTC.H C H3MeHeHHeM 06'beMa TeJIa, T.e. nepeMem;eHHeM tJ:aCTeH MaKpOTeJla
,lIPyr OTHOCHTeJlbHO ,lIpyra. IIpou;ecc 3TOT npHBO,lIHT K H3MeHeHHIO paCCTOHHH.H
Mem,lIy qaCTHu;aMH, a TaKme "t{aCTO K H3MeHeHHIO CKopOCTeH HX ,ABHmeHH.H,
CJIe,lIOBaTeJIbHO, K H3MeHeHHIO BHYTpeHHeH 3HeprHH Tena.
IIYCTb B U;HJlHH,Ape C nO,lIBHmHbIM nopllIHeM HaXOHTCSJ ra3 npH TeMnepa-
Type Tl (pHC. 7.1). BY,lIeM Me,AJIeHHO HarpeBaTb ra3 ,AO TeMnepaTypbI T 2 . fa3
6YAeT H306apH"t{ecKH paCllIHp.HTbC.H, H nopllIeHb nepeMeCTHTCSl H3 nOJlOmeHHH 1
B nOJlOmeHHe 2 Ha paCCTOHHHe  l. CHJla ,lI8BJIeHHSJ: ['83a npH aTOM cOBepllIHT
III pa60TY Hall BHellIHHMH TeJIaMH. TaK KaK
p = const, TO H CHJla lIaBneHH.H F = pS Tome
nOCTO.HHHa.H. II03TOMY pa60TY 3TOH CHJIbI
MomHO paCCqHTaTb no «popMYJle
.
.
A =Flll=pSlll =pIlV=P(V 2 -V 1 ), A=pIlV, (7.6)
.
.
. .
. .
. . .
. .
.
2
r,Ae  V - H3MeHeHHe 06'beM8 ra3a. EcnH
06'beM ra3a He H3MeH.HeTCSI (H30XOpHbIH npo-
u;ecc) , TO pa60Ta ra3a paBHa HYJlIO.
Pue.
155

CUJla ,lI;aBJleHUlI ra3a BbInOJlHlIeT pa60TY TOJlbKO B npou,ecce U3MeHeHU.H
06'beMa ra3a.
llpH paCIIHlpeHMH (V > 0) ra3a COBepillaeTC.H nOJIOmMTeJIbHaH pa60Ta
(A? 0); npM cmaTMH (V < 0) ra3a COBepillaeTCSl OTpMl1,aTeJlbHaH pa60Ta
(A <... 0), nonOmMTeJIbHYIO pa60TY cOBepillalOT BHeIllHMe CMnbl A' = -A.? O.
3anMilleM ypaBHeHMe KJIaneHpOHa-MeH,n:eJIeeBa ,n:JIH ,n:BYX COCTO.HHMH ra3a.
m m m
pV I = -R'IJ.; pV 2 = -RT 2 => p(V 2 - Vi) = -R(T2 - Td.
M M M
CJIe,n:oBaTeJlbHO, npM H306apHoM npol1,ecce
m
A = -RT.
M
ECJIM m = M (1 Monb M,n:eaJIbHOrO ra3a), TO npM /). T = 1 K nOJlY'lMM R =A
OTclO,n:a BbITeKaeT cpM3M1.IeCKMH CMbICJJ YHMBepCaJIbHOH ra30BoM nocToHHHOH: OHa
'UICJIeHHO paBHa pa60Te, cOBepillaeMoH 1 MOJIb M,n:eaJIbHOrO ra3a npM ero M30-
6apHoM HarpeBaHMM Ha 1 K.
Ha rpacpMKe p = f(V) npM M306apHoM npOl1,eCce pa60Ta paBHa nJIOlI.{a,n:M 3a-
IllTpMxoBaHHoro Ha pMCYHKe 7.2, a np.HMoyrOJIbHMKa.
ECJIH npol1,ecc He H306apHbIH (pMC. 7.2, 6), TO KPMBYIO P = f(V) MomHO
npe,n:CTaBMTb KaK JIOMaHYIO, COCTO.HlI.{YIO M3 60JIbillOrO KOJIH1.IeCTBa H30XOP M
M306ap. Pa60Ta Ha 1I30XOPHbIX Y1.IaCTKaX paBHa HYJIIO, a CYMMapHaH pa60Ta
Ha Bcex M306apHbIX yqaCTKaX 6y,n:eT
n
A = hm L PiVj I1JII1 A = J p(v)dl-,
6V O i=I
T .e. 6y,n:eT paBHa nJIOlI.{a,n:M 3aIllTpHxoBaHHoH qmrYPbI. llplI M30TepMI1qeCKOM
npou.ecce (T = const) pa60Ta paBHa nJIOlI.{a,n:M 3aIllTpMxoBaHHoH cpMrYPbI, 1I306-
pameHHoH Ha pMCYHKe 7.2, 6.
a
6
6
p
p
p

V
o
VI
o
VI V 2
Puc. 7.2
V 0
VI
V 2
v
Onpe,n:eJIIITb pa6oTY, I1CnOJIb3YH nOCJIe,n:HIOIO cpOPMYJIY, MO}KHO TOJIbKO B TOM
CJIY'lae, eCJIM M3BeCTHO, KaK M3MeH.HeTC.H ,n:aBJIeHMe ra3a npM M3MeHeHHII ero
06'beMa, T.e. 113BeCTeH BM,n: cpYHKl1,MM p(V).
TaKMM 06pa30M, ra3 nplI paCIlll1peHHM cOBepillaeT pa60TY. llpM60pbI M ar-
peraTbI. lleHCTBM.H KOTOpbIX OCHOBaHbJ Ha CBOHCTBe ra3a B npou.ecce paCIllMpe-
HMH cOBepIllaTb pa60TY, Ha3bIBalOTCH nHeBMaTH'IeCKUMU Ha 3TOM npMHllMne
,n:eHCTBYIOT nHeBMaTMqeCKMe MOJIOTKII, MexaHM3MbI ,n:JI.H 3aKpbIBaHMH M OTKpbI-
BaHMH ,n:Bepeft Ha TpaHcnopTe M ,n:p.
156

7.6. fiepBbIH 3aKOH TepMO,ZJ;HHaMHKH
IIepBbIH 3aKOH (nepBoe HaqaJlo) TepMOHHaMHKI1 - 3TO 3aKOH coxpaHeHH.H
H npeBpaeHH.H 3HeprHH B npHMeHeHHH K TenJlOBbIM npoeccaM.
ECJlH MeXaHHqeCKa.H 3HeprH.H CHCTeMbI He H3MeH.HeTC.H, a CHCTeMa He 3aM-
KHYTa H Mem,AY HeH 11 oKPymalOeH cpeOH npOHcxOHT TenJlo06MeH, TO H3-
MeHHeTC.H BHYTpeHH.H.H 3HeprH.H:
/).U = Q + A BH .
(7.7)
YpaBHeHHe (7.7) - nepBLIH 3aKOH TepMOAHHaMHKH, KOTOpbIH <POPMYJlHPY-
eTC.H TaK: U3.MelieliUe 6liympeliHeu aliepzuu npu nepexooe mep.MOOUlia.MUlleCKOu
CUCme.Mbl U3 OOIiOZO cocmOSlliUSl 6 opyzoe pa61i0 pa60me 6lieWliUX CUJl. U KO-
Jl.U'leCm8Y menJl.Ombt, nepeoaliliO.MY mep.MOOU1ta.MU1leCKOU CUcme.Me 8 npOl.{ecce
menJl.006.Melia.
ECJlH BMeCTO pa60TbI BHeIIIHHX CHJI A BH BBeCTH pa60TY A CHCTeMbI Ha,A
BHeIIIHHMH TeJlaMH A = -ABu' TO BbIpameHHe (7.7) 3anHIlleTC.H:
Q=.hU +A.
(7.8)
Tora nepBbIH 3aKOH TepMOHHaMHKH MomHO C<POPMYJlHpOBaTb TaK: KOJl.U-
lleCm60 menJl.Ombl. coot5enlioe mep.Moouna.MU1leCKOU cucme.Me. uiJem Ha U3.Me-
Heliue ee 6Hympelilieu aliepzuu U IiQ C06epweliue pa60mbl Cucme.MOU npomu6
6HeWliUX CUJl..
113 nepBoro 3aKOHa TepMOHHaMHKH BbITeKaeT HeB03MomHOCTb C03aHH.H
Be'tlHOrO BHraTeJI.H nepBoro poa, T.e. TaKoro BHraTeJlH, KOTOpbIH COBepIllaJI
6bI pa60TY 6e3 3aTpaTbI 3HeprHH H3BHe.
.D:eHcTBHTeJlbHO, eCJlH K CHCTeMe He nOBOHTCH 3HeprH.H (Q = 0), TO
A = -.hU H pa60Ta MomeT 6bITb COBepIlleHa 3a CqeT y6bIJlH BHYTpeHHeH 3Hep-
rHH CHCTeMbI. IIocJle Toro KaK 3anac 3HeprHH OKameTC.H HCqepnaHHbIM, BH-
raTeJIb nepeCTaHeT pa60TaTb.
EcnH CHCTeMR 3aMKHYTa (ABU=O) H TennOH30JIHpOB8Ha (Q=O) (CM.  7.8),
TO nepBbIH 3aKOH TepMOJJ.HHaMHKH 6YJJ.eT HMeTb BHJJ.:
.hU = O.
ECJlH B TaKOH CHCTeMe HMeIOTC.H TeJla C pa3JIH'tIHOH TeMnepaTYPOH, TO Mem-
JJ.Y HHMH 6yeT [lpOHCXOHTb TenJlo06MeH: TeJIa, Y KOTOpbIX TeMnepaTypa
BbIIlle, 6YYT OT,AaBaTb 3HeprHIO 11 OXnam,AaTbCH, a TeJla C MeHbIlleH TeMnepa-
TYPOH 6YYT nOJlyqaTb 3HeprHIO H HarpeBaTbC.H. 3TO 6yeT npoHcxoHTb ,AO
Tex nop, nOKa TeMnepaTypbI Y Bcex TeJl He CTaHYT OHHaKOBbIMH. IIpH 3TOM
.hU l + U2+ ... + .hU n = 0
HJIH
Ql+Q2+...+Qn=0.
(7.9)
rIepBbIH 3aKOH TepMOHHaMHKH (7. 9) JI.H 3aMKHYToH H aHa6aTH'tIeCKH H30-
nHpoBaHHoH CHCTeMbI Ha3bIBalOT ypaBHeHHeM TenJIOBOrO 6aJIanca. BbIpame-
HHe (7.9) npeCTaBJI.HeT C060H aJIre6paHqecKyJO CYMMY: CJlaraeMbIe, COOTBeTCTBY-
IOHe HarpeBaHHIO, napo06pa30BaHHIO, nJIaBJleHHIO, nOJlOmHTeJlbHbI, a COOTBeT-
CTBYIOHe OXJIameHHIO, KOHeHcaHH, KpHCTaJIJlH3aIJ;HH, - OTpHaTeJlbHbI.
157

7.7. IIpHMeHeHHe nepBoro 3aKOHa TepMOJJ;HHaMHKH
K pa3JIH1.IHbIM TenJIOBbIM npou;eccaM B HJJ;eaJIbHOM ra3e
1) H30XOpHbIH npoecc.
06'beM He H3MeH.HeTC.H: V = const =:) D. V = 0 - pa60Ta He COBepllIaeTCSI.
IIepBbIH 3aKOH TepMo,nHHaMHKH 6y,neT HMeTb BH,A:
Q = D.U.
llpu U30XOPliOM npOl{ecce 3liepzuSl. c00614aeMaJl za3Y nymeM menJl.006.Melia.
pacxooyemcSl l{eJl.UKO.M. ria Y6eJl.Ulleliue ezo 6liympelilieu 3liepzuu. npUlle.M CO-
ZJl.acliO (7.1)
im
D.U = - - RD.T.
2M
2) H30TepMHQeCKHM npou.ecc.
TeMnepaTypa ra3a He H3MeH.HeTC.H: T = const. Cne,nOBaTenbHO, D.U = O.
I1epBblH 3aKOH TepMOAHHaMHKH HMeeT BH,A:
Q=A.
llpu U30mepMUilecKOM npOl{ecce 6ce n006eOeriliOe K za3Y KOJl.UlleCm60 men
Jl.Ombl uoem lia c06epweliue za30M pa60mbl.
3) J1306apHbIH npoecc.
,naBneHHe He H3MeHSleTCSI: p = const.
IIpH paCllIHpeHHH ra3 COBepllIaeT pa60TY A = pD. V H HarpeBaeTC.H, T .e. H3-
MeHSleTC.H ero BHYTpeHH.HSI 3HeprHSI:
im
D.U = -- R6.T.
2M
IIepBbIH 3aKOH TepMo,nHHaMHKH 3anHllIeTC.H TaK:
Q=A+D.U.
llpu u306apliO.M npoijecce noo6eoelilioe K za3Y KOJl.UlleCm60 menJtOmbl Ilac.
mUllliO uoem lia Y6eJl.Ulleliue ezo 6liympelilieu 3liepZUU, a llaCmUllliO IiG pa60-
my. C06epwaeMY1O za30M 6 npOl{ecce ezo pacwupeliUSl.
7.8. A,aHa6aTHbIH npou;ecc
"a6aTHLI" npoecc - 3TO npoecc. npOHCXO,A.HII:(HM 6e3 TenJIo06MeHa CH-
CTeMbI C oKpymalOII:(eM cpe,noM, T.e. Q = O. IIepBbIH 3aKOH TepMo,nHHaMHKH MMe-
eT BH,n:
6.U + A = 0 =:) A = -6.U.
3TO 3HaQHT, QTO npH a,nHa6aTHoM npoecce CHCTeMa MomeT BbInOJIH.HTb
pa60TY HaA BHellIHHMH TenaMH TonbKO 3a CQeT y6bIJIH cBoeM BHYTpeHHeH 3Hep-
im
rHH. ECJIH A> 0, TO D.U = -A <-. O. T.e. U 2 < U 1 , a TaK KaK U = - - RT TO
2M '
T 2 < TI'
KaK H3BeCTHO, ra3 COBepllIaeT nOJIOmHTeJlbHYIO pa60TY, ecnH 6.V> O.
158

TaKHM 06pa30M, npu aoua6amlio.M pacwupeliuu Za3 C06epWaem pa60my u
ca.M OXJl.aJICOaemCSl. Ha060poT. npu aiJua6amli0.M cJlCamuu A '- 0 liaiJ Za30.M
c06epUlaemC.R pa60ma u za3 nazpe6aemcSl.
IIpH alia6aTHoM npOll;.eCce aBJleHlie li 06'beM CBH3aHbi Me*y C060R ypaB-
HeHlieM
pvY = const,
re y> 1 - nOKa3aTeJlb aHa6aTbI (liJlH K03cpcplill;.lieHT IIyaccoHa, CM. 9 7.9).
3TO ypaBHeHlie Ha3bIBaeTC.H ypaBHeHlieM a.n;lia6aTbl liJlli ypaBHeHlieM IIyaccoHa.
A.n;Ha6aTHoe H3MeHeHHe COCTO.H-
HH.H ra3a MomHO BbIpa3HTb rpa.
cpntIeCKH. rpacpliK 3Toro npOll;.eCca
Ha3bIBalOT a.l1.Ua6aToH. llpli O,AHlfX
H Tex me HaqaJlbHbIX YCJlOBli.HJI:
(Po, V o ) npli alia6aTHoM pacmH-
peHliH aBJleHlie ra3a YMeHblllaeT-
C.fl 6bIcTpee, qeM npli H30TepMli-
qeCKOM (pliC. 7.3), TaK KaK na.
eHHe .n;aBneHH.H BbI3BaHO He Tonb-
KO YBenliqeHlieM 06'beMa (KaK npli
H30TepMHqeCKOM paClllHpeHHH), HO
H nOHHmeHlieM TeMnepaTypbI. IIo-
3TOMY aHa6aTa HeT Hlime li30-
TepMbI H ra3 npli alia6aTHoM
V
pacllllipeHlili cOBeplllaeT MeHblllYIO
pa60TY, qeM npH li30TepMH'leCKOM
pacllllipeHlfli .
IIpH 6bICTpOM cmaTHli (paCllllipeHlili) TenJlo06MeH npOli30HTli He ycneBaeT
H npOn;eCCbI MomHO paCCMaTpHBaTb KaK a.n;Ha6aTHbIe (HepaBHOBeCHbIe). II03TO-
MY nlO60H ra3 npH 6bICTpOM cmaTHR HarpeBaeTCSI (HanpHMep, HarpeBaHlie Ha-
coca npH HaKaqHBaHHH BenOCHne.n;HOH lllHHbI). IIpH CHnbHOM li 6bICTpOM cma-
TlfH B03.n;yxa TeMnepaTypa MomeT nOBbICHTbC.H HaCTOJlbKO, qTO npli HanHqliH
B B03yxe napOB 6eH3HHa OHM BOCnJlaMeH.HIOTC.H. 3TO licnonb3yeTC.H B .n;H3enb-
HblX BliraTeJl.HX JI.H 3amHraHli.H rOplOqeH CMeCli. 3THM 06'bHCH.HeTCH He06xo-
AHMOCTb CneUHaJlbHOrO OXJlamll.eHH.H MOIUHbIX KOMnpeccopOB.
OxnameHHe B03yxa npli a,lIlia6aTHOM paCllllipeHHli BbI3blBaeT 06pa30BaHlie
06naKOB.
p
p'z
V z
V o
Puc. 7.3
7.9. TenJIOeMKOCTH np" IIOCTO.HHHOM .a;aBJIeIlHH
" nOCTO.HHHOM 06'beMe
IIpli co06IUeHHH TeJlY HeKOToporo KOJlHqeCTBa TenJlOTbl H3MeH.HeTC.H ero TeM-
nepaTypa (3a liCKnlOqeHlieM arperaTHbIX npeBpaIUeHliR H Bo06IUe H30TepMli'leC-
KHX npOn;eccoB). XapaKTepHcTHKaMli TaKoro H3MeHeHli.H .HBn.HIOTC.H pa3nHqHbIe
TennoeMKOCTli: TennoeMKOCTb Tena C T , yeJlbHa.H TennoeMKOCTb BeeCTBa c,
MOnHpHa.H TenJlOeMKOCTb C.
IIOH.HTHH TenJlOeMKOCTli Tena H yenbHoH TenJlOeMKOCTH paCCMOTpeHbl B 9 7.4
159

MOJlJlpHa.H TenJlOeMKOCTL C - BenH'lHHa. paBHa.H KOJIHqeCTBY TenJIOTbI.
He06xo,AHMOMY ,nJlH HarpeBaHH.H 1 Monb BeeCTBa Ha 1 K:
Q
C=-.
vl:1T
(7.10)
E,nHHHeH MOJl.HpHOH TenJIOeMKOCTH B Cll .HBJI.HeTCH .n;.>K0YJlb Ha Monb-
KenbBHH r Am ] .
MOJlb.K
Y,neJIbHa.H TennoeMKOCTb CBH3aHa C MOJI.HpHOH COOTHOllIeHHeM
C = cM.
B OTJlHqHe OT TaKOH, HanpHMep, xapaKTepHCTHKH BelI.(eCTBa, KaK ero MO-
JIeKYJIHpHaH Macca M n y.n;eJIbHa.H TennoeMKOCTb BelI.(eCTBa He .HBJI.HeTC.H HeH3-
MeHHbIM napaMeTpOM. Y,AeJIbHaSl TenJIOeMKOCTb MO.>KeT pe3KO H3MeH.HTbC.H npH
nepexo,ne Bell.\eCTBa H3 o,nHoro arperaTHoro COCTO.HHH.H B ,APyroe TaK, Bo,na
B ra3006pa3HOM COCTOHHHH HMeeT y,AeJlbHYIO TenJIOeMKOCTb 2,2. 10 3
Am
-,
Kr.K
a B mH,nKOM 4.19.10 3 Am
Kr.K
TenJlOeMKOCTb 3aBHCHT H OT YCJlOBHH, npH KOTOpMX npOHcxo,nHT nepe,na'la
TenJIOTbI TeJIY. IToCJIe.n;Hee oc06eHHO OTHOCHTCH K ra3aM. HanpHMep. npH H30'
TepMHqeCKOM paCllIHpeHHH ra3a eMY nepe,naeTCH HeKOTopoe KOJIHqeCTBO TenJIO-
TbI Q> 0, a l:1 T = O. CJIe,nOBaTeJIbHO, Y,AeJIbHaH TenJIOeMKOCTb ra3a npH H30.
TepMHqeCKOM npoecce
Q
c=-oo.
ml:1T
llpH a.ZI.Ha6aTHOM cmaTHH (pacllIHpeHHH) ra3 He nOJIyqaeT TennOTbI H He nepe-
,naeT ee oKpymalOll.\HM TeJlaM (Q = 0), a TeMnepaTypa ra3a H3MeHHeTC.H (l:1 T  0).
Cne,nOBaTeJlbHO, y,neJIbHa.H TenJIOeMKOCTb ra3a npH a,nHa6aTHOM npoecce
Q
c = - = O.
ml:1T
HaH60JIbllIHH HHTepec npe,nCTaBJI.HeT TenJIOeMKOCTb ,nJlH CJIyqaeB, KOr,Aa Ha-
rpeBaHHe npOHcxo,nHT npH nOCTOHHHOM 06eMe HJIH npH nOCTOHHHOM .n;aBne-
HHH. B nepBOM CJIyqae TenJIOeMKOCTb Ha3bIBaeTCH TenJIOeMKOCTblO npH no-
CTOHHHOM 06'beMe HJlH H30XOpHOH TenJIOeMKOCTb!O (cv, CV), BO BTOpOM -
TenJIOeMKOCTblO npH nOCTOHHHOM .n;aBJIeHHH HJlH H306apHoli TenJlOeMKOCTb!O
(c p ' C p )'
ECJlH 06eM He H3MeHHeTCH (l:1 V = 0), TO pa60Ta. cOBepllIeHHa.H ra30M, TaK-
.>Ke paBHa HYJlIO (A = 0). COrJlaCHO nepBOMY 3aKOHY TepMo,nHHaMHKH
Q=l:1U H
l:1U
C TV - l:1T ' oTKy,na 6.U = C TV .l:1T = c v ml:1T.
(7.11)
CJle,nOBaTeJlbHO. TenJlOeMKOCTL npH nOCTOHHHOM 06'LeMe paBHa H3MeHeHHIO
BHYTpeHHeii 3HeprHH ra3a npH H3MeHeHHH TeMnepaTypbl Ha 1 K.
160

ECJlH ra3 IIAeanbHbIH, TO B cpopMyne (7.11)
i m
U=--RC::.T.
2M
C y _ _ flU M
TorAa MOJI.HpHaSl TennoeMKOCTb npH nOCTOSlHHOM 06'beMe rAe
T '
i
flU M = - RflT - H3MeHeHHe BHYTpeHHeH ::meprHH 1 MOJIb ra3a. 113 3THX pa-
2
HaSl TenJlOeMKOCTb ra3a npH nOCTOSlHHOM 06'beMe -
im
- C Ty ="2 M R;
i
C v = - R.
2
MOJlSlp-
BeHCTB TenJIOeMKOCTb ra3a lIPH nOCTOSlHHOM 06'beMe
ECJIH ra3 HarpeBaeTC.H npH nOCTO.HHHOM AaBJIeHHH, TO corJIaCHO nepBOMY
3aKOHY TepMO,AHHaMHKH
m
Q = t1U + A, r,n;e A = pV = -RflT.
M
TorAa TenJlOeMKOCTb ra3a npH nOCTO.HHHOM AaBneHHH
Q /::,.U m m i -to 2 m
C Tp = /::,.T = /::,.T + M R = C Ty + M R = 2 M R.
MOJl.HpHa.H TenJlOeMKOCTb npH nOCTO.HHHOM AaBJIeHHH:
C p = C y + R - ypaBHeHHe MaHepa;
i i + 2
C =-R+R=-R.
p 2 2
TaKHM o6pa30M, TenJIOeMKOCTb npH lIOCTOSlHHOM AaBneHHH BCer,Aa 60JIbille
TenJlOeMKOCTH npH nOCTOSlHHOM 06'beMe I1x OTHOllleHHe paBHO
_ C p _ i + 2
y - - - --:--'
C y l.
r.ue y - nOKa3aTeJlb aAHa6aTbi (K03cpcpHHeHT IIyaccoHa).
113-3a MaJlOCTH BeJIHt.J:HHbl K03cpcpHHeHTa 06'beMHOrO paClllHpeHHSI TBepAblX
H mHAKHX TeJI pa6oToii, cOBeplllaeMoH HMH npH HarpeBaHHlI npH nOCTO.HHHOM
,AaBJIeHHH, MomHO npeHe6pet.J:b H C'lHTaTb, t.J:TO TenJIOeMKOCTlI npH nOCTOSlHHOM
06'beMe H nOCTO.HHHOM AaBJIeHHH npaKTHt.J:eCKlI COBnaAaIOT. II03ToMY TenJIOeM-
KOCTb TBepAblX H mHAKHX Ten npll 3aAaHHOH TeMnepaType MO>KeT Ct.J:HTaTbCSI
BnOJIHe onpeAeJIeHHOH BeJlHt.J:HHOH.
7.10. 06paTHMbIe H Heo6paTHMbIe npou;eCCbI.
Heo6paTHMocTb TenJIOBbIX npou;eCCOB.
BTOPOH 3aKOH TepMOHHaMHKH
06paTUMLIM Ha3blBaeTCSI npoecc, KOTOpbIH OTBet.J:aeT CJIeAYlOll.\HM YCJIOBH-
SlM: 1) ero MomHO npOBeCTH B ABYX npOTHBOnOJIOmHblX HanpaBJIeHHSlX; 2) B
, Ka>KAOM H3 3THX cnY'laeB CHCTeMa H OKpY>KalOlUHe ee TeJIa lIPOXO,A.HT 'lepe3
OAHH H Te me npOMe>KYTOt.J:HbIe COCTO.HHH.H; 3) nOCJIe npOBe)J;eHH.H np.HMOrO H
161

06paTHoro npOn;eCCOB CHCTeMa H oKpymalOHe ee TeJla B03BpaaIOTC.H I< HC-
XO,nHOMY COCTO.HHHIO.
BC.HKHH npon;ecc, He y,nOBJleTBOp.HIOIQHH XOT.H 6bI O)J;HOMY H3 3THX YCJlO-
BHH, HBJI.HeTC.H He06paTHMLlM.
TaK, MomHO ,nOKa3aTb, qTO a6conlOTHO ynpyrHH IIIapHK. na,naH B BaKYYMe
Ha a6COJllOTHO ynpyrylO nJlHTY, BepHeTC.H nocJle OTpameHHR B HCXO,nHYIO TOt!-
KY, npOH,AH B 06paTHOM HanpaBJleHHH Bce Te npOMemYTO'lHbIe COCTOHHH.H, KO-
Topble OH npoxo,nHJI npH n3.,neHHH.
Ho B npHpo,ne HeT CTporo KOHcepBaTHBHblX CHCTeM, B nlO60H peaJlbHOH
CHCTeMe lIeHCTBYIOT CHnbI TpeHHH. Ilo3TOMY Bce peaJlbHbIe npoueCCbI B npHpo-
one He06paTHMbI.
PeaJlbHbIe TenJlOBhle npoeCChl TaKme Heo6paTHMbl.
11 p H M e p bI: 1) IlpH ,nHcpcpY3HH BbIpaBHHBaHHe KOHn;eHTpan;HH npOHCXO,AHT
caMonpOH3BonbHO. 06paTHbIH me npon;ecc caM no ce6e HHKor,na He nOH,neT: HH-
Kor,na caMonpOH3BOJlbHO CMeCb ra30B, HanpHMep. He pa3,neJlHTCSl Ha COCTaBJI.HIO-
IQHe ee KOMnOHeHTbI. CJle,nOBaTeJlbHO, ,nHcpcpY3HSl - He06paTHMbIH npon;ecc.
2) Tennoo6MeH. KaK nOKa3bIBaeT onbIT, TaKme .HBnSleTC.H O,nHOCTopOHHe Ha-
npaBneHHbIM npOn;eCCOM. B pe3YJlbTaTe Tennoo6MeHa 3HeprH.H nepe,naeTCfl caMa
no ce6e Bcer,na OT TeJla C 60nee BbICOKOH TeMnepaTYPoH K Teny C 60Jlee HH3-
KOH TeMnepaTYPoH. 06paTHbIH npon;ecc nepe,na"tlH TenJlOTbI OT XOJlo,nHoro Tena
K rOp.HqeMY caM no ce6e HHKor,na He npoHcxo)J;HT.
3) He06paTHMbIM .HBJI.HeTCSl TaKme npon;ecc npeBpaIQeHH.H MeXaHH"tleCKOH
3HeprHH BO BHYTpeHHlO1O npH HeynpyroM Y,Aape HJlH npH TpeHHH.
Mem,ny TeM H3 nepBoro 3aKOHa TepMOlIHHaMHKH HanpaBJleHHOCTb H TeM
caMbIM He06paTHMocTb TennOBbIX npOn;eCCOB He BbITeKaeT. IlepBbIH 3aKOH Tep-
Mo,nHHaMHKH Tpe6yeT nHIIIb, 'lT06bI KOJlH"tleCTBO TennOTbI, OT,naHHoe O,AHHM Te-
nOM B TO"tlHOCTH paBH.HJlOCb KOnH"tleCTBY TennOTbI, KOTopoe nOJlY"tlHT ,nPyroe.
A BOT Bonpoc 0 TOM, OT KaKoro TeJla, OT rop.H"tlero K XOJlO,nHOMY HnH Ha-
060POT. nepeH,neT 3HeprH.H, OCTaeTCSl OTKpbITbIM.
HanpaBJleHHOCTb peaJlbHbIX TenJlOBbIX npOn;ecCOB onpe,nen.HeTC.H BTOpbIM 3a-
KOHOM TepMo,nHHaMHKH, KOTOpbIH 6bIJI YCTaHOBJleH Henocpe,nCTBeHHbIM 0606IQe-
HHeM onbITHbIX cpaKTOB. 8TO nocTYJlaT. HeMen;KHH yqeHbIH P. KJlaY3HYC ,nan
TaKYIO cpopMynHpOBKY BToporo 3aKOHa TepMO,lJ.HHaMHKH: He603MOJICliO nepeeecmu
menJl.O om 60Jl.ee XOJl.OOliOU cucmeMbl K 60Jl.ee zopSllleu npu omcymcmeuu opyzux
OOlioepeMeliliblX U3MelieliUU 6 06eux cucmeMax UJl.U oKpy:JtCalO14UX meJl.ax.
113 BToporo 3aKOHa TepMo,nHHaMHKH BbITeKaeT HeB03MomHOCTb C03,naHHSI
BeqHOrO ,nBHraTeJl.H BToporo po,na, T.e. ,nBHraTen.H, KOTOpbIH 6bI COBepIIIaJl pa-
60TY 3a CqeT OXJlam,neHHSl KaKoro-nH60 o,nHoro TeJla.
7.11. TenJIOBLIe ABHraTeJIH.
IIpHHHn AeHcTBH1I TenJIOBLIX ABHraTeJIeH
TenJlOBLIM ABHraTeJleM Ha3bIBaeTC.H YCTPOHCTBO, COBepIIIalOee MeXaHH"tleC-
KYIO pa60TY 3a C"tleT BHYTpeHHeJ1 3HeprHH TonnHBa.
TennOBbIe ,nBHraTeJlH BeCbMa pa3Ho06pa3HbI KaK no KOHCTpYKn;HH, TaK H
no Ha3HaqeHHIO. 8TO H napOBbIe Typ6HHbI Ha TennOBbIX 3neKTpOCTaHn;HSlX, H
,nBHraTenH BHYTpeHHero crOpaHH.H Ha aBTOM06Hn.HX, TpaKTopax, H peaKTHBHbIe
,nBHraTenH pa3J1HqHbIX THnOB.
162

p 1
Ql
2
c v
Puc. 7.4
Puc. 7.5
Bce TenJlOBble BHraTeJlH 06J1alIalOT 06Il.{HM cBoi1cTBOM - nepHO,AHqHOCTblO
eiiCTBH.H (U.HKJIH'lHOCTLIO), B pe3YJlbTaTe qero pa60qee TeJlO nepHOHH'leCKH B03-
BpaIQaeTC.H B HCXOHoe COCTO.HHHe.
IIpHHHn eHCTBH.H TenJIOBOrO BHraTeJl.H paccMoTpHM Ha npHMepe nopm-
HeBoro BHraTeJl.H.
JII060ii TenJIOBOH BHraTeJlb COCTOHT li3 Tpex OCHOBHbIX qacTeii: pa60qero
Tena, HarpeBaTeJl.R 11. XOJlOHJlbHHKa (pHC. 7.4).
Pa60qee TeJlO (nap HJlH ra3) nOJlyqaeT HeKOTopoe KOJlHqeCTBO TenJlOTbJ QI
OT HarpeBaTeJl.H, y KOToporo 3a CqeT crOpaHH.H TOnJIHBa noepmHBaeTC.H no-
CTO.HHHa.H BbICOKa.H TeMnepaTypa T I' 3TO KOnHqeCTBO TenJlOTbI HeT Ha YBeJlH-
qeHHe BHYTpeHHeii 3HeprHH ra3a H COBepmeHHe HM pa60TbI AI' B pe3YJlbTaTe
ra3, paCmHp.H.HCb, nepeXOHT H3 COCTO.HHH.H 1 B COCTO.HHHe 2 (JlHHH.H 1 a2),
COBepma.H pa60TY AI, paBHYIO nnOIQaH qm:rYPbI Bl a2C (pHC. 7.5).
'llT06bI npoecc 6b[JI HKJIHqeCKHM, nopmeHb He06xoHMO BepHYTb B HC-
XO,AHOe nOJlOmeHHe. ECJlH npOl.{eCC CmaTH.H npOBeCTH B o6paTHoM nOp.HlIKe (JIH-
HH.R 2al), TO pa60TbI ra3a H HaJJ: ra30M 6ynYT OHHHaKOBbI H CYMMapHa.H pa-
60Ta 6Y,AeT paBHa HYJlIO. II03TOMY, tITo6bI pa60Ta C:>KaTH.H A 2 6bIJIa no a6co-
JIIOTHOMY 3HaqeHHIO MeHbme pa60TbI pacmHpeHH.H, HY:>KHO, qT06bI Kam,AOMY
3HaqeHHIO 06'beMa npH cmaTHH COOTBeTCTBOBaJIO MeHbmee aBJIeHHe, qeM npH
pacmHpeHHH (JlHHH.H 2bl). A 3TO B03MomHO ocyecTBHTb, TOJlbKO eCJIH ra3
nepe,A cmaTHeM OXJla,AHTb (CM. pHC. 6.13). ,nJlSl 3Toro pa60qee TeJlO npHBO.HT
B KOHTaKT C TeJlOM MeHbmeii TeMnepaTypbI T 2 < TI (xonO,AHnbHHKoM). Pa60-
'lee TeJlO npH 3TOM OTaeT XOJIOHJIbHHKY HeKOTopoe KOJlHqeCTBO TenJIOTbI Q2,
Ii npH cmaTHH COBepmaeTC.H pa60Ta A2' paBHa.H nJlOaH cpHrYPbI 1 b2CB.
ITone3Ha.H pa60Ta 3a HKJI A =AI - A 2 qHCJleHHO paBHa nJlOaH cpHry-
pbI la2bl.
7.12. K03HU;HeHT nOJIe3HOrO ,lJ;eHCTBHH TenJIOBOrO ,lJ;BHraTeJIH
BaJIaHC 3HeprHH 3a UHKJI MomHO nOJlyqHTb Ha OCHOBe nepBoro 3aKOHB Tep-
MOHHaMHKH. Pa60qeMY TeJlY nepe,AaHO nYTeM TenJlo06MeHa KOJIHqeCTBO Ten-
nOTbI QI' H H HHM COBepIIleHa pa60TB A 2 . Pa60qee TeJlO COBepIIlHJlO pa60TY
npH paCIIIHpeHHH A I H nepeaJlo KOJlHqeCTBO TenJlOTbI Q2 XOJlOHJlbHHKY. AJI.H
H,AeaJIbHOrO TenJlOBOrO BHraTeJl.H li3MeHeHHe BHYTpeHHeii 3HeprHH U = 0, H60
pa60qee TeJlO BepHYJlOCb B HCXO,AHOe COCTO.HHHe.
Ql + A 2 = Q2 + Al . OTCIO8 A = Al - A 2 = Ql - Q2'
163

OTHOlIIeHHe nOJle3HOH pa60TbI K KOJIHqeCTBY TenJlOTbI, KOTOpOe pa60qee
TenO nOJIY'lHnO OT HarpeBaTeJI.SI, Ha3bIBaeTC.H K03<!><!>HIJ.HeHTOM nOJle3HOrO AeH-
CTBH.SI TenJlOBOrO JJ;BHraTeJlJl (KTIn):
1') = ; 1') = Q} - Q2 = 1 _ Q2 .
Q} Q} Ql
113 cpopMynbI (7 .12) BHHO, 'lTO ,Aa)Ke y l1eaJIb-
Horo TenJlOBOrO ,ABHraTen.H T) < 1. HclH60JIblllUH Kn.n;,
KaK nOKa3aJI BnepBble C. KapHo. MO)KHO nOJIyqHTb
JI.H H.n;eaJIbHOrO BHraTeJI.SI B CJIYtIae, eCJIH pa60qee
TeJIO COBeplIIaeT ll,HKJI KapHo C TeM )Ke HarpeBaTeneM
3 H XOJlO.n;HJIbHHKOM. IJ;HKJI KapHo COCTOHT H3 BYX H30-
T 2 =const TepMHtIeCKHX (pHC. 7.6 - KpHBbIe 1-2 H 3-4) 11
,ZJ;BYX 8..AHa6aTHblx (KpHBble 2-3 H 4-1) npOll,eccOB:
T} -T 2
1')max = ----r:-'
1
113 cpOPMYJIbI (7.13) BHHO, qTO JI.H nOBbIlIIeHH.H KTI,n: Heo6xoHMO YBe-
nHtIl1BaTb T 1 H YMeHblIIaTb T 2' TaK KaK XOJIO.n;HJIbHHKOM B 60JIblIIHHCTBe cny-
qaeB CJIY)KHT oKpymalOa.H cpe.n;a, TO OCHOBHOH cnoc06 nOBbIlIIeHH.H KTI,n: CO-
CTOHT B nOBbIlIIeHHH TeMnepaTypbI HarpeBaTeJI.SI.
IJ;HKn KapHO - HeaJIbHbIH ll,HKn. B peaJIbHbIX ll,HKJlaX HeJIb3.H oCYlIJ.e-
CTBHTb HeaJIbHYIO aHa6aTHOCTb H H30TepMHtIHOCTb. KpOMe Toro. He YCTpa-
HHMbI nOTepH Ha TpeHHe. TI03TOMY KTI,n: B peaJlbHbIX TenJIOBbIX ,ABHraTeJI.HX
BCer,Aa MeHblIIe, tIeM paCCtIHTaHHbIM no cpopMYJIe (7.13).
113 CPOPMYJI (7.12) H (7.13) HMeeM
1 - Q2 = 1 _ T 2 => Q2 = T 2 => Ql = Q2 ,
Ql 1] Ql T 1 T} T 2
re Q2 - KOJIHqeCTBO TenJIOTbI, oTaHHoH pa60tIHM TeJIOM XOJIO.n;HJIbHHKY, no-
3TOMY OHO OTpHu;aTeJIbHO.
CJIeOBaTeJIbHO, MO)KHO
(7.12)
p
1
Tl =const
v
Puc. 7.6
(7.13)
3anHcaTb aJIre6paHtIecKYIO CYMMY:
Q} + Q2 = O.
T} T 2
BeJIl1tIHHa Q Ha3bIBaeTC.SI npu6eOeliHbl.M KOJlUlleCm60.M menJl.Ombl.
T
CJleOBaTeJIbHo. B ll,HKJle KapHo cYMMa npHBe,AeHHbIX KOJIH'leCTB TenJlOTbI
paBHa HY JIIO.
CTporHH TeOpeTHqeCKHH aHaJlH3 nOKa3bIBaeT, qTO He TOJIbKO ,AJI.H ll,HKJIa
KapHO, a .n;JI.SI JII06oro 06paTHMoro KpyroBoro npou;ecca CYMM8 npHBeeHHbIx
KOJIHqeCTB TenJIOTbI paBHa HYJIIO. 3TO 3HaqHT, tITO B 06paTHMOM npou;ecce co-
XpaH.SIeTC.H HeKOTOpa.H BeJIHqHHa, H3MeHeHHe KOTOpOH paBHo npHBeeHHoMY KO-
JIHtIeCTBY TenJlOTM:
Q
6.8 =-.
T
KaK nOKa3aJI P. KJIaY3HYc, S - 3TO 3HTpOnH.SI.
CJIeOBaTeJIbHO, B 06paTHMbIx npou;eccax 3HTpOnH.SI He H3MeH.HeTC.SI. TIpH
He06paTHMbIX npOll,eCcax 3HTpOnH.H 3aMKHYTOH H aHa6aTHqeCKH H30JIHpOBaH-
HOM CHCTeMbI B03paCTaeT: SHco6p > O.
164

B 06eM cnyqae: npu Jl.106blX npOlteCcax. npOmeKalOll.{UX 6 3a.MKliymou UJlU
aaUa6amUileCKU U30Jl.UpOealiliOU .MaKpOCKOnUlleCKOU CUCme.Me, ee 3limpOnUSl liP
y6bl6aem. m.e. 8SO.
3TO 3aKJIIO'IeHHe MomHO paCCMaTpHBaTb KaK HaH60JIee 06ylO OPMYJIH-
POBKY BTOpOrO HaqaJIa TepMO,nHHaMHKH.
113 OPMYJlbI (7.13) HMeeM T} = 1 - T2 . Cne,noBaTeJIbHo. BH,nHM. 'ITO 11 <: 1.
'Ii
KII,n: 6bIJI 6bI 11 = 1, ecnH 6bI TeMnepaTypa XOJIO,nHJIbHHKa 6bIna paBHa a6COJlIOT-
HOMY HynlO: T 2 = 0 K. Ho ee B 1912 r. HeMeI1.KHH H3HK B. HepHcT TeopeTH-
'leCKH ,AOKa3aJI HeB03MomHOCTb ,AOCTHmeHHH TaKOH TeMnepaTypbI. 3TO YTBepm,ne-
HHe Ha3bIBalOT mpembU.M HallaJl.O.M mep.MOaUHa.MUKU; a6COJl1OmliblU liYJlb me.Mnepa
mypbl lieaOcmuJlCu.M; K He.MY .MOJtCliO Jl.UWb aCU.MnmOmUlleCKU npu6Jl.uJtCambcSl.
7.13. POJIL TenJIOBLIX JJ:BHraTeJIeH B pa3BHTHH TenJI0311epreTHKH
H TpaHcnopTa. TenJIOBbIe JJ:BHraTeJIH H oxpalla npHpOJJ:LI
TenJlOBble ,nBHraTeJIH He06xo.n:HMbI .n:JI.H nOJIY'IeHH.H 3JIeKTp03HeprHH. .n:JI.H
npHBe,neHHH B .n:BHmeHHe 60JIbllIHHCTBa TpaHcnopTHbIX MallIHH.
HaH60JIbllIee 3Ha'IeHHe HMeeT npHMeHeHHe MOHbIX napOBbIX Typ6HH Ha
3neKTpOCTaHI1.H.HX ,nJI.H BpaeHHH pOTOpOB reHepaTopOB. [lapoBbIe Typ6HHbI YC-
TaHaBJlHBalOT TaKme Ha aTOMHbIX 3JIeKTpOCTaHI1.HHX, r,ne ,nJI.H nOJlY'IeHHH napa
BbICOKOH TeMnepaTypbI HCnOJIb3yeTC.H 3HeprHH aTOMHbIX .H,nep.
Ha COBpeMeHHOM TpaHcnopTe HCnOJIb3YIOTCH Bce BH,nbI TenJIOBbIX ,lJ;BHraTe-
neH. B aBTOM06HJI.HX, TpaKTopax, caMoxo,nHbIX KOM6aHHax, TenJlOB03aX npH-
MeH.HIOTC.H nopllIHeBbIe ,nBHraTeJlH BHYTpeHHero crOpaHH.H, B aBHaI1.HH - ra30-
Bble Typ6HHbI. Ha KOCMH'IeCKHX paKeTax - peaKTHBHbIe ,nBHraTenH.
TenJIOBbIe ,nBHraTeJIH OKa3bIBalOT HeKOTopble Bpe,nHble B03,neHcTBH.H Ha OK-
pymalOYIO cpe,ny:
1) KII,n: TennOBbIX ,nBHraTeJIeH 11 < 50 %, CJIe,nOBaTeJIbHO, 60nbllIaH 'IaCTb
3HeprHH TOnJIHBa pacceHBaeTC.H B oKpymalOeM npOCTpaHCTBe, Bpe,nHO BJlH.HH
Ha o6ylO 3KOJlOrH'IeCKYlO 06cTaHoBKY;
2) TenJIOBbIe 3JIeKTpOCTaHI1.HH H aBTOM06HJIH BbI6pacbIBalOT Bpe,nHble ,nJIH
pacTeHHH. mHBOTHbIX H 'IeJIOBeKa npo,nYKTbI crOpaHH.H TOnJIHB8 (cepHHcTble co-
eAHHeHH.H, OKCH,nbI yrJIepo,na, OKCH,nbI a30Ta H ,np.);
3) nOBbIllIeHHe KOHI1.eHTpaI1.HH yr JIeKHCJIOrO ra3a B aTMOC(pepe YBeJIH'IHBaeT
"napHHKoBbIH 3eKT" 3eMJIH.
B CB.H3H C 3THM BeCbMa BamHOH CTana npo6neMa oxpaHbI npHpo,nbI. ,n:JIH
oxpaHbI oKpymalOeH cpe,nbI He06xo,nHMO 06eCne'IHTb:
1) 3eKTHBHYIO OqHCTKY BbI6pacbIBaeMbIx B aTMocepy oTpa6oTaHHbIx ra-
30B;
2) HCnOJIb30BaHHe Ka'IeCTBeHHoro TOnJIHBa, C03,naHHe YCJIOBHH ,nJI.H 60JIee
nOJIHOrO ero cropaHHH;
3) nOBbIllIeHHe KII,n: TenJIOBbIX ,nBHraTeneH 3a C'IeT YMeHbllIeHH.H IIOTepb
Ha TpeHHe H nOJIHOrO crOpaHH.H TOnJIHBa H ,np.
IIepcneKTHBHo HCnOJIb30BaHHe Bo.n:opo.n:a B Ka'IeCTBe roplOqero .n:JI.H TenJIO-
BbIX ,lI,BHraTeJIeH: npH cropaHHH Bo.n:opo.n:a o6pa3yeTcH Bo,na. l1AYT HHTeHCHB-
Hble HCCJIe,nOBaHHH no C03,naHHIO 3JIeKTpOM06HJIeH, cnoco6HbIX 3aMeHHTb aBTO-
Mo6HJIH C ,nBHraTeJIeM, pa60TalOHM Ha 6eH3HHe.
165

BOllPOCM AR CAMOKOHTPORH
1. B lleM omJl.UIlUP mepMoouliaMullecKozo cnoco(ja u3YIleliuSl ljJu3uIlecKux
npoqecc06 om MOJl.eKYJl.JlpIiO-KUliemullecKozo?
2. l{mo nOliUMalOm noo 6liympelilieu 3liepZUeu C mOIlKU 3peliuSl MKT2
3. KaKue cYl4eCm6YlOm cnoc06bl U3MelieliUSl 6liympelilieu 3liepzuu?
4. B lleM cxoocm60 U pa3Jl.UIlUe pa60mbl U "OJlUlleCm6a menJl.ombd
5. l{mo 06l4ezo Y menJl.onp060ihwcmu, U3Jl.YlleIiUSl U KOH6eKquu?
6. MemaJl.Jl.UlleCKUe npeOMembl 6 KOMliame lia 0l4ynb 6cezoa KaJICymcSl
XOJl.OOliee oepe6SlHliblX UJl.U nJl.aCmMaCC06blX. II olleMY?
7. KaK nepeoaemcSl 3liepZUSl 6 6e36030YWliOM npocmpalicm6e?
8. Bcezoa Jl.U npu co06l4eliuu mepMoouliaMullecKou cucmeMe 3liepZUU ny-
meM menJl.006Melia 6liympeliliSlSl 3liepZUSl 603pacmaem?
9. KaK U nOlleMY U3MeHSlemcSl 6liympelili.RJl 3liepZUSl meJl.a npu nJl.a6Jl.e-
HUU, npu KpUCmaJlJl.U3aquu?
10. l{mo npeOCma6Jl.Slem c060u 6liympeliliSlSl 3liepZUSl UOeaJl.bIiOZO za3a? Om
Ilezo .Ja6ucum ee 6eJl.Ullulia?
11. M OJICIiO Jl.U n06blcumb meMnepamypy meJl.a. lie co06l4aSl eMY KOJl.Ulle-
cm60 menJl.Ombl?
12. llpu KaKUX YCJl.06USlX ea3 MO:HCem C06epwamb pa60my?
13. Co olia 6000eMa nooliuMaemcSl nY3blpeK za3a. C06epwaem Jl.U 3mom
za3 pa60my?
14. KaK paccilumbl6aemCJl pa60ma, c06epwaeMaSl za30M npu pacwupeliuu2
KaK ee MOJICIiO onpeOeJl.Umb no zpa(jJuKY p = f(V)?
15. KaK U3MeliSlemcSl meMnepamypa za3a. ecJl.U Oli c06epwaem nOJl.O:HCUmeJl.b-
IiYIO pa60my?
16. llolleMY U30mepMUileCKoe pacwupeliue za3a 603MOJICii0 mOJl.bKO npu noo-
6eoeliuu K lieMY lieKomopozo KOJl.UlleCm6a menJl.Ombl?
17. COnp060JICOaemCR Jl.U u306aplioe nOliU:HCeliUe meMnepamypbl UOeaJl.bIiOZO
za3a C06epweliueM pa60mbl?
18. B KaKOM CJl.Yllae UOeaJl.bliblU za3 npu oouliaK060M Y6eJl.UlleIiUU 06'OeMa
C06epwaem 60JlbWY1O pa60my: npu u306apiioM UJl.U U30mepMUileCKOM pacwupeliuu?
19. OOIiY U my JlCe Maccy UOeaJl.bliOZO za3a liazpe6a1Om lia 1 °C OOUIi pa3
npu nocmOSlliliO.M. 06'OeMe. opyzou pa3 npu nocmOSlHIiOM Oa6Jl.eliuu. B KaKOM
cJl.Yllae mpe6yemcR 60Jl.bWee KOJl.UlleCm60 menJl.Ombl? II olleMY?
20. M oJICem Jl.U U3MeliSlmbCSl 6liympeliliSlSl 3HepZUJl menJl.OU30Jl.Up06aliliou
cucmeMbl? 3aMKliymou cucmeMbl?
21. l{acmliblM CJlYllaeM KaKOZO 3UKonu RBJl.JlemCJl ypu6HeHue menJl.060Z0 6a-
Jtalica? /(Jl.Sl KaKUX cucmeM OliO UCnOJl.b3yemcSl?
22. B KaKUX CJl.YllaSlx U 00 KaKUX nop MeJICoy meJl.aMU 3aMKliymou U
menJlOU30JlUp06aHliou cucmeMbl 6yoem npoucxooumb menJl.006Meli?
23. AJl.IOMUIiUe6blu U CmaJl.bHOU wapUKU oouliaK060u MaCCbl liazpembl 00
00uliaK060u meMnepamypbl U OmOeJl.bliO onYl4elibl 6 OOUHaK06ble KaJl.OpUMem-
pbl C pa6liblMU MaccaMU 600bl npu pa6HblX meMnepamypax. B "aKOM "aJl.O-
pUMempe ycmaH06umcSl 60Jl.bWaSl meMnepamypa?
24. B oouliaK06blP KaJl.OpUMempbl C pa6liblMU MaccaMU 600bl U npu OOU-
liaK06blX meMnepamypax onYl4elibl .Meolioe U 0Jl.06SlIiIiOe meJl.a pa6liblX Macc.
B KaJl.OpUMempax ycmali06UJl.aCb oouliaK06aSl meMnepamypa. KaKoe U3 meJl
6blJl.0 liazpemo 00 60Jl.bWeU meMnepamYPbl?
166

25. II oileMY lie603MOJICeli 6ellliblU 06UZameJl.b nep60zo poiJa?
26. KaK U3MeliSlemcSl BHympeHH.R.R 3liepZU.R UOeaJl.bHOZO za3a npu U30mep
MUlleCKOM pacwupeliuu? U30XOPliOM liazpeBaliuu? u306apHoM c:HCamuu?
27. H a Ilmo pacxooyemcSl KOJl.UtleCmBO menJl.Ombl, co06w,elilioe UOeaJlblioMY
za3Y npu u30mepMullecKoM. U30XOpliOM. u306aplioM npou,eccax?
28. KaK U3Meli.RemCSl BliympeliliSlSl 3liepZUSl UOeaJlbliOZO za3a npu aoua6am
HOM pacwupeH.uu? 3a cllem KaKOZO npou,ecca npoucxooum 3mo U3MeH.eliue?
29. llpu 6blcmpoM cJICamuu za3a meMnepamypa ezo nOBblCUJl.aCb. MOJfCHO
JLU CKa3amb. Ilmo za3Y co0614elio lieKomopoe KOJl.UlleCm60 menJlOmbl? Ilmo
Bliympeli1iJlJl 3liepZUSl za3a YBeJl.UIlUJlacb?
30. llpu pe3KOM onYCKaliUU nopWliSl 060eM UOeaJLbHOZO za3a 6 U,UJl.Uliope
YMelibWUJl.CSl Bmpoe. M OJICliO Jl.U CKa3amb, Ilmo Oa6JLeHUe za3a B03pOCJlO 6mpoe?
31. HOeaJl.bliblU za3 c HallaJl.bliblM OaBJl.eliUeM PI' 3aliUMalO14UU 06oe.M VI'
pacwupSlemcSl 00 06oe.Ma V 2 . B KaKOM CJlYllae za3 COBepwaem 66JlbWY1O pa60-
my: npu U30mep.MUtleCKO.M UJl.U npu aoua6amHO.M pacwupeHuu?
32. .H 6Jl..RemCSl JlU yoeJlbliaSl menJl.Oe.MKOCmb nOCmO.RliHUU xapaKmepucmu
KOU Be14eCm6u?
33. T.J e.MY paBlia menJl.Oe.MKOCmb za3a B aoua6amliOM npou,ecce? B U30mep
MUlleCKO.M npou,ecce?
34. llolleMY menJl.OeMKOCmb za3a npu nOCmO.RliliOM iJaBJl.eHUU 60Jlbwe. lleM
npu nOCmOSlHH.OM o6?JeMe? KaKoBo ux coomHoweHue?
35. MOJIClio JlU lia OCli06e nep60zo liallaJl.a mep.MOOUliaMuKu onpeiJeJlUmb
HanpaBJleliliOCmb menJl06blX npou,ecc06?
36. J(Jl.Sl Ilezo CJlYJlCam B menJl.OBOM oBUZameJle liazpeBameJlb, pa60llee meJlO,
XOJlOOUJlbHUK? T.Jmo .RBJtSlemcSl liazpe6ameJle.M U XOJlOOUJlbliUKOM B peaKmUB
HOM OBUZameJl.e CaMOJl.ema?
37. llolleMY oJt.R lienpepb16liOZO oeUCmBU.R menJl.OBOZO oBUZameJl.R lie06xo
iJuMO oCY14ecmBUmb u,uKJl.UlleCKUU npou,ecc?
38. KaK no zpacjJuKY u,UKJl.UlleCKOZO npou,ecca 6 KoopOUliamax (p, V) onpe-
OeJl.Umb pa60my menJlOBOZO 06UZameJl.Sl 3a U,UKJl? pa60my. C06epwaeMY1O za30M
npu ezo pucUlupeHuu?
39. M oJICem JlU Kll menJl.06blX iJ6uzameJl.eu cmamb paBliblM 100 %,
eCJlU mpeliue 6 llacmSlX OBUZameJl..R C6ecmu K IiYJlIO?
40. IIpu KaKOM u,UKJl.e KIlJ( UOeaJl.bliOZO menJl.060Z0 06UZameJl..R npUliUMQ
em MaKCUMaJl.bliOe 3H.alleH.Ue? Om Ilezo 3aBucum BeJlUIlUHa llmax?
41. KaKue MeponpUSlmU.R npoBOOSlmC.R B liaweu cmpalie no oxpalie OKPY-
JfCQI014eU cpeobl B CB.R3U C npuMeHeliueM menJl.OBblX oBUZameJleu?
OCHOBHblE «I>OPMY JIbl
im
U = --RT
2M
- BHYTpeHH.H.H 3HeprH.H HeaJIbHOrO ra3a. re i - 'lHCJIO
cTeneHeH cBo6obI (i = 3 - ,AJISI O,AHOaTOMHbIX MOJleKyn,
i = 5 - JI.H BYX- H 60Jlee aTOMHbIX JlHHeHHbIx MOJleKYJI,
i = 6 - JI.H Tpex- H 60nee aTOMHbIX HeJIHHeHHblx MoneKYJI),
m - Macca ra3a, M - MOnSlpHa.H Macca, R - YHHBepCMb-
Ha.H ra30Ba.H nOCTO.HHHa.H, T - a6COJlIOTHa.H TeMnepaTypa.
167

Q = Cm(T2 -21)
Q = { Am
Lm
Q=qm
!!J.U '= AuH + Q,
Q=A+!!J.U
QI + Q2 + ... + Qn = 0
A =p!!J.V
Q
c =-
T !!J.T
Q
c=-
m!!J.T
Q
C = - = cM
v!!J.T
!!J.U = cym!!J.T
i
C y = - R
2
- KOJIHqeCTBO TenJlOTbI. Heo6xonHMoe nJl.H HarpeBaHIf.H
TeJla, r.n;e C - Y)J;eJlbHa.H TenJlOeMKOCTb BeecTBa, m -
Macca TeJla, T l' T 2 - COOTBeTCTBeHHO Ha'qaJIbHa.H If KO-
He'qHa.H TeMnepaTypbI TeJla.
- KOJIlfqeCTBO TenJlOTbI, Heo6xo.n;IfMoe .n;JI.H 1f3MeHeHIf.H
arperaTHoro COCTO.HHIf.H BeeCTBa. r.n;e A - Y,AeJlbHaSI
TenJIOTa nJIaBJleHIfSI; L - y.n;eJIbHaSI TenJIOTa napo06pa-
30BaHIf.H .
KOJlHqeCTBO TenJlOTbI, BbI.n;eJl.HIOeeC.H Uplf cropaHHH
TonnHBa, r.n;e q - y.n;enbHa.H TenJlOTa crOpaHIf.H TOnJlHBa.
- nepBbIH 3aKOH TepMo.n;HHaMIfKIf, r.n;e 6U - H3MeHe-
Hlfe BHYTpeHHeH 3Heprlflf CIfCTeMbI, A BH - pa6oTa, npo-
H3Be.n;eHHa.H BHelIIHHMH CIfJlaMH Ha.n; cHcTeMoH, A - pa-
60Ta, npolf3Be.n;eHHaSI clfcTeMoH Ha,A BHelIIHIfMIf TeJlaMIf.
Q - KOJlHqeCTBO TenJlOTbI, co06eHHoe CIfCTeMe.
ypaBHeHlfe TenJlOBOrO 6aJlaHca.
pa60Ta ra38 npH H306apHoM npoecce, r.n;e p - .n;aB-
JleHHe ra3a, 6 V - H3MeHeHHe 06'beMa ra3a.
- TenJlOeMKOCTb Tena, r.n;e Q - co06w;eHHoe TeJlY KO-
JlHqeCTBO TenJlOTbI, 6T - 1f3MeHeHlfe TeMnepaTypbI Tena.
y.n;eJlbHa.H TenJIOeMKOCTb BeeCTBa.
- MOJl.HpHa.H TenJlOeMKOCTb, r,v;e v - KOJIHqeCTBO Bew;e-
CTBa, M - MOn.HpHa.H Macca.
1f3MeHeHlfe BHYTpeHHeM 3Heprlflf ra3a.
MOJI.HpHaSI TenJlOeMKOCTb ra38 npH nOCTO.HHHOM 06'beMe.
i+2
C = C y + R = - R - MOJl.HpHa.H TenJlOeMKOCTb ra3a nplf nOCTO.HHHOM ,v;aB-
p 2 JleHIfIf.
C p i + 2
y=-=-
C y i
11 =  = QI - Q2
QI Ql
_21- T 2
llmax - 
168
- COOTHOllIeHlfe Mem,v;y MOJl.HpHOR (y,v;eJlbHoH) TenJlOeM-
KOCTblO ra3a npH UOCTO.HHHOM ,v;aBJleHIfIf If npH nOCTO-
.HHHOM 06'beMe (noKa3aTeJIb a,v;lfa6aTbI).
- RITA TenJlOBOrO nBHraTeJl.H. rne A - nOJIe3HaSI pa60-
Ta, cOBepllIaeMa.H TenJlOBbIM .n;BlfraTeJleM, Ql - KOJIIf'qe-
CTBO TenJIOTbI, nOJlyqeHHoe ,v;BlfraTeJleM OT HarpeBaTen.H,
Q2 - KOJlHqeCTBO TenJlOTbI, OT.n;aHHoe XOJlO,v;IfJlbHIfKY.
- MaKCIfMaJlbHOe 3HaqeHlfe KIlL{ TenJlOBOrO ,v;BlfraTeJI.H.
re TI H T 2 - TeMnepaTypbI HarpeBaTeJI.H H xono,v;lfnb-
HHKa COOTBeTCTBeHHO.

METO)].lIqECKIIE PEKOMEH)].AQIIII
PerneHHe 3aatI 3TOH TeMbI OCHOBaHO Ha HCnOJIb30BaHHH nepBoro 3aKOHa
TepMOHHaMHKH. IIpH perneHHH 3aaq peKoMeHyeTC.H:
1. Y CTaHOBHTb, KaKHe TeJIa BXO.HT B paccMaTpHBaeMYIO TepMoHHaMH'le-
CKYIO CHCTeMY.
2. BbIHCHHTb, tITO .HBJISIeTC.H npHtIHHOH H3MeHeHHH BHYTpeHHe:H 3HeprHH TeJI
CHCTeMbI.
3. ECJlH CHCTeMa aHa6aTHtIeCKH H30JIHpOBaHa H 3aMKHYTa. TO He06xo.l1.H-
MO YCTaHOBHTb, Y KaKHX TeJl CHCTeMbI BHYTpeHHSI.H 3HeprH.H YBeJIHqHBaeTC.H. a
Y KaKHX YMeHbrnaeTC.H, 06paTHB BHHMaHHe, npOHCXO,ASIT JIH npH 3TOM arpe-
raTHble npeBpaeHHSI. Y06HO HJlJlIOCTpHpOBaTb 3TO rpaqmKoM 3aBHCHMOCTH H3-
MeHeHH.H TeMnepaTypbI TeJI OT KOJlHtIeCTBa TenJIOTbI, nOJIyqeHHOrO (OTaHHoro)
npH TenJIo06MeHe, T = f(Q).
4. CocTaBHTb ypaBHeHHe TenJIOBOrO 6aJIaHCa (tIaCTHbIH CJlYtIaH nepBoro 3a-
KOHa TepMoHHaMHKH):
n n
L!:lU i = 0 HJIH LQi = O.
i=1 i=1
Cne.l{yeT nOMHHTb. tITO B 3TO:H cYMMe CJIaraeMbIe. COOTBeTCTBYIOII.:(He nnaB-
neHHIO TBepllbIx TeJl HJlH napoo6pa30BaHHIO mH.l1.KOCTeH. 6epYT co 3HaKOM "+",
a CJIaraeMble, cooTBeTcTBYIOHe KpHCTaJlJIH3aHH )KHKOCTeH HJIH KOHeHca-
u:.HH napa, - co 3HaKOM "-".
5. ECJlH npH B3aHMoeHcTBHH .l{BYX TeJI BHYTpeHH.H.H 3HeprH.H H3MeH.HeTC.H
BCJIeCTBHe cOBeprneHHoH pa60TbI, TO HymHo npeme Bcero YCTaHOBHTb, Y Ka-
Koro H3 BYX B3aHMOe:HCTBYIOHX TeJI H3MeHSIeTC.H BHYTpeHH.H.H 3HeprHSI H
qTO SIBJI.HeTC.H npHqHHOH 3Toro H3MeHeHH.H: pa60Ta A, COBeprnaeMa.H caMHM Te-
JIOM. HJIH pa60Ta ASH' COBeprneHHa.H Ha TeJIOM.
6. 3anHcaTb nepBbIH 3aKOH TepMO.l{HHaMHKH, KOTOpbI:H npH OTCYTCTBHH no.l{-
Boa TenJlOTbI H3BHe HMeeT BH:
!:lU + A = 0 HJIH !:lU =AsJI'
7. B 3a.l{aqaX, B KOTOpbIX npH B3aHMoeHcTBHH Tpex HJlH 60Jlee Ten npo-
HCXO.l{HT Tenno06MeH C oKpymalOII.:(eH cpeoH H COBeprnaeTCSI MeXaHHqeCKa.H pa-
6oTa, nepBbIH 3aKOH TepMO.l{HHaMHKH 3anHCbIBalOT B caMOM 06II.:(eM BH.l{e:
!:lU =A SH + Q HnH !:lU + A = Q.
nPHMEPbI PEIlIEHII.H 3A)1A '"I
1. HeKoTopaSl M8cca l111eanbHoro ra38 nepeXOlll1T 113 COCTOHHI1S1 B B COCTO-
HHHe C ,llBYMH pa3nl1'lHbIMH cnoc06aMH: BaC 11 BbC (pHC. 7.7). npH'leM B
npouecce Kamlloro nepeXOlla ra3 COBepmaeT pa60TY 11 npOI1CXO,AI1T Tenno06-
MeH. O,AHHaKOBbI nH npl1 Kam,AOM nepeXO,lle: p860Ta. COBepmaeMaH ra30M? 113-
MeHeHl1e ero BHYTpeHHeH 3Heprl1H? C006II.:(eHHOe ra3Y KOnHtIeCTBO TennOTbl?
PeweHH e. Pa60T8 AI' COBepm8eM8H r830M B npouecce BaC. p8BH8 nJIO-
1.l.\8,ll11 cpHrYPbI V I BaCV 2 . P860T8 A 2 , COBepm8eM8fl B npouecce BbC. p8BH8
nJIOII.:(8,AH <l>HrYPbI V I BbCV 2 . 113 pl1CYHK8 7.7 BH,AHO. tITO AI> A 2 . T8K KaK
169

P B
rrapaMeTpbI Ha'l8nbHOrO H KOHe'lHOrO COCTOSlHHH B
060HX npou.eCC8X O,AMHaKOBbI. TO M3MeHeHHe 3Hep-
rMH
I:1U 1 = I:1U 2'
CornaCHO rrepBOMY 3aKOHY TepMO,AMHaMMKM
Q = I:1U + A, H nocKonbKY AI> A 2 . TO Ql> Q2'
V.
V 2 V
2. Orrpe,AenMTe H3MeHeHHe BHYTpeHHeH 3HeprHM
1,0 Kr M,AeanbHOro r838, oxnam.LJ;alOIl\erOCSI npM
nOCTOSlHHOM 06'beMe. ecnH ero Ha'l8nbH8S1 TeMrre.
,AaBneHHe B pe3ynbT8Te oxnamlleHHSI YMeHbWMnOCb
PUC. 7.7
paTypa paBH8 423 K. 8
OT 8,08 TIa ,AO 2,02 TIa; Cy= 700
>K
Kr.K
m
)J.aHo: m=l,O Kr, TI=423 K. pI=8,08 Oa, P2=2,02 TIa, cy=700
Kr.K
H8HTH: t1U.
Pew e H H e. ROnM'leCTBO TerrnOTbI Q = cyml:1T. COrJI8CHO rrepBOMY 3aKOHY
TepMO,lIHH8MMKH Q = I:1U + A. TaK KaK rrpH V = const p860Ta He cOBepwaeTCSI
(A = 0), TO HCKOMoe H3MeHeHHe BHYTpeHHeH 3HeprMM I:1U = Q, T.e. I:1U = cyml:1T.
TIpH H30XOpHOM npouecce BbInOnHSleTCSI 38KOH lli8pnSl:
PI = P2 => T 2 = p211 ; 1:1 T = T 2 - 11 = (P2 - pd 11.
11 T 2 PI PI
Cne,AOBaTenbHo.
I:1U = c y m (P2 - PI) 7i ; I:1U = -2,2.10 5 )J.m.
PI
3. Bbl'iHCnMTe KOHeqHYIO TeMnepaTYPY H KOHe'lHOe ,A8BneHMe OAH08TOMHoro
r83a, HaXO,ASlIl\erOCSI B 68nnOHe 06'beMOM 1.5 M 3 rrpH TeMnepaType 300 K M
,AaBneHHM 1,8. 10 5 TI8. ecnH 3TOMY ra3Y C006Il\eHO KOnH'leCTBO TennoTbI, p8B-
Hoe 54 Km
8HO: V 1 =1,5 M 3 , T 1 =300 R, PI=1,8.10 5 TIa, Q=5,4.10 4 .lI:m.
H8HTH: T2' P2'
Pew e H H e. TaK KaK V = const, TO A = 0 11 M3 rrepBoro 3aKOHa TepMO,AM-
H8MHKH Q = A + I:1U HMeeM Q = I:1U.
113MeHeHHe BHYTpeHHeH 3HeprHH OAHoaTOMHoro MAeanbHoro ra3a
3m
I:1U = - - RI:1T.
2M
Cne,AOBaTenbHO,
3 m 3 m
Q = --RI:1T = --R(T2 -11).
2M 2M
3anHweM yp8BHeHHe Kn8neHpoHa-MeH,AeneeB8 ,AnSI Ha'l8nbHOro COCTOSlHHSI:
m
PI V = -R7i,
M
OTKYA8
!!!:.... R = PI V .
M '11
170

3 PI V ( 2Q J
Tor,lla Q = -- (T2 - T I ) => T 2 = T 1 1 + - ; T 2 = 340 K.
2 T I 3 PI V
E1- = T I => P2 = PIT2 ; P2 = 2,1.10 5 TI8.
P2 T 2 T 1
3anHilleM 3aKOH IlIapnSl:
4. Onpe.ZJ,enHTe YBenH'leHHe BHYTpeHHeH 3HeprHH O,AHOa'fOMHOrO ra3a, HaXO,ASI-
erocSI no,A ,A8BneHHeM 1,6 .10 5 TIa, eCJlH ero 06'beM H306apHo YBenHQHBaeTC.H Ha
0,60 M 3 . BbP.mCnH"I'e KonWieCTBO TennOTbI, nonyqaeMoe ra30M npH aTOM npou.ecce.
,LJ;aHO: p=I,6'10 5 TIa, 6.V=O,60 M 3 .
HaHTH: 6.U, Q.
P e ill e H If e. YBenHqeHHe BHYTpeHHeH 3HeprHH O.ZJ,HoaTOMHoro ra3a MOJKHO
3m
onpellenHTb no <l>opMyne 6.U = --R6.T.
2M
3anHilleM ypaBHeHlle KnaneHpoHa-MeHlleneeBa .ZJ,nSl ,ABYX COCTO.RHHH:
m
pV 1 = M R7i , (1)
m
pV 2 = M RT 2 . (2)
BbIqHTaSl H3 ypaBHeHHSI (2) yp8BHeHHe (1), nonY'lHM:
m
p6.V = - R6.T.
M
3
Tor.ZJ,8 6.U = - p6. V, 6.U = 1,4 . 10 5 ,LJ;m.
2
TIpl1 H306apHoM npou.ecce COBeplllaeTCSI pa60T8 A = p6. V. Cor naCHO nepBOMY
3aKOHY TepMO,AHHaMHKH KOnH'leCTBO TennOTbI. nonY'laeMOe ra30M,
3 5
Q = 6.U + A ="2 p6.V + p6.V = "2 p6.V; Q = 2,4 10 5 )];JK.
5. B U.HnHH,Ape KOMnpeccopa CJKHM810T 4,0 Monb H.ZJ,e8nbHoro O,AHOaTOMHO-
1'0 ra3a. Onpe,AenHTe, H8 CKonbKO nO,lJ,H.RnaCb TeMnepaTypa ra3a 3a O,AHH XO,A
noplllHSI, ecnH npll 3TOM 6bIna cOBepllleHa pa60Ta 500 .D:JK. TIpou.ecc CqHTaTb
8,AHa68THblM.
.D: 8 H 0: v = 4,0 Monb, A DH = 500 )];JK.
H8HTH: 6.T.
P e ill e H If e. TIocKonbKY npH 8,AHa68THoM npou.ecce Q = 0, TO H3 nepBoro
38KOHa TepMO,AHHaMHKH BbITeK8eT, qTO 6. U = A DH .
3m
TaK KaJ< /1U = - - R6.T, TO
2M
3 m 3
L1 = -- R6.T HnH L1 H = -vR6.T. OTCIO,lJ,a
H 2 M .. '"B 2
6.T = 2H . 6.T = 10 K.
3vR'
6. M,Ae8nbHbIH l'a3, Macca H MOnSlpH8.R Macca KOToporo COOTBeTCTBeHHO
paBHbI m H M, HaXO.ZJ,HTCSI npH TeMnepaType T nocne H30xopHoro oxnaJK.ZJ,e-
HHSI. B pe3ynbT8Te KOToporo .u.aBneHHe YMeHblllHnOCb B n pa3, l'a3 HcnblTbIB8eT
H306apHoe p8ClllHpeHHe, npH'leM B KOHe'lHOM COCTOSlHHH TeMnep8Typ8 paBHa
171

nepSOHaQanbHoi1. 11306pa3liTe rp8qm'leCKIi 3TIi npOlI,eCCbI B OCSIX (p V) H BbI-
'lHCnHTe COBeprneHHYIO ra30M pa60TY.
PI
n a H 0: m, M, TI = T2' VI = V 2 , P2 = -. P2 = P3' 73 = 71.
n
H 8 i1 T H: A.
P
PI
1
Pew e H H e rpa<l>liK npe,ACTaSneH H8 PHCYH-
Ke 7.8. TIocKonbKY npli H30XOpHOM npOlI,eCce pa-
60Ta He coseprnaeTCSI. TO pa60Ta, cOBeprneHHaSl
Ha nepeXO,Ae 1-2-3,
A = P2(V 3 - Vd.
\
,
,
\
\
,
,
"
,
2 '.........._ 3
_ PI
IIo ycnoBli1O P2 - -.
n
IIocKonbKY TO'lKIi 1 Ii 3 ne>KaT Ha O,AHOH
V H30TepMe, TO
P2
VI
V 3
Puc. 7.8
PI VI
PI VI = P2 Va  Va = -
P2
Cne,AOSaTenbHO, IiCKOM8S1 pa60T8
A _PI ( PI V 1 n _ TT ) _ n - 1 V
-- - "I --PI l'
n PI n
3anliIlleM ypaBHeHlie Knanei1poHa-MeHAeneeBa:
m
PIVI = -RT.
M
Tor,na
(n -1)mRT
A= .
nM
7. TeMnepaTypa ra30B, 06pa3YIOHxCSI npli crOpaHIiI1 TonnliBa B I1.linHH,A-
pax ,ABliraTenfl aBToM06HnSl, 897°C; TeMnepaTYP8 BblxnonHblX ra30B 97°C.
CKonbKo KHnOMeTpOB npoi1,AeT aSToM06Hnb, HMelOHi1 B 6aKe 40 n TonnliBa,
,D;m
Y,AenbHafl TennOT8 cropaHHSI KOToporo 3.2. 10 10 , 8 Clin8 conpOTHsneHHSI
M
,ASH>KeHHIO ,ASHr8TenSl paSHa 1,7 KH? nBHraTenb CQHTaTb H,AeanbHbIM, p860-
TalOHM C M8KCHM8nbHO B03MomHbIM KTIn.
,n:aHO: T 1 =1170 K, T 2 =370 K, V=O,040 M 3 , q=3,2.10 10
F = I, 7 . 10 3 H.
Hai1TH: S.
,n:m
3 ·
M
A
Pew e H H e. TIo onpeAeneHli1O KTI,n: TennOBoro ,ABliraTeJISI 11 = -.
Ql
TIone3HaSl pa60T8 B ,AaHHOM cnY'l8e A=FTS=FS. r,Ae F T - CHna TSlrH,
pa3SlisaeMaSl ,ASliraTeneM aBTOM06HnSl, paSHaSl npH paBHOMepHOM ,ABIi>KeHIiIi
CHne COnpOTI1SneHH.SI F. KOnH'leCTSO TennOTbI QI. nonY'leHHOe p860'lHM TenOM
,ASHr8TenSl, paSHO KOnl1'leCTSY TenJIOTbI, Sbl,AeneHHOMY npH cropaHHH TonnHBa,
1i- T 2
Ql = qV. MaKCHManbHo S03MomHbli1 KTI,n: 11 = -
T 1
172

Cne,AOBaTenbHO, 71 - T 2 = FS
71 qV .
OTC IO,A 8
(71 - T 2 )qV
S = . 8=5.0,10" KM
7J.F '
8. Ha P- V ,AHarp8MMe (pHC. 7.9) H306p8>KeH u.HKn, npOBO,AHMbIH C O,llHO-
aTOMHbIM ra30M. Onpe,AenHTe K03<1><I>Hu.HeHT nOne3HOl'0 ,AeHCTBHSI 3Toro llHKna.
A
P e ill e H He. ETI,l], u.HKn8 '1 = ---1!... TIone3H8S1
Ql
pa60Ta An 'lHCneHHO paBHa nnOa,AH TpeyronbHH-
Ka. Orp8HH'leHHOrO u.HKnOM:
113 rpa<}>HKa BH,AHM, 'lTO B npou.ecce 1-2 npOHC-
XO,AHT H30xopHoe HarpeBaHHe, a B npou.ecce 2-3 -
1i3068pHoe p8CIllHpeHHe. Cne,AOBaTenbHo, B 3THX
npou.eccax ['a3 nonY'l8eT 3Heprmo QI = Q123' TIo nep-
BOMY 3aKOHY TepMO)J.HHaMHKH QI23=A I23 + UI23'
Pa60T8 3aBHCHT OT BHlI;8 npOlleCC8. Cne,AOBaTenbHo, A I23 =A12 + A 23 . TaK
K8K npH H30XOpHOM npou.ecce p860Ta ra3a paBHa HynlO (A I2 =0). TO AI23=A23=
= 4po. (2V o - Yo) = 4P o V o .
BHYTpeHHSISI 3HeprHSI SlBnSleTCSI <l>YHKu.HeH COCTOSlHHSI TepMO,AHHaMH'leCKOH CH-
CTeMbI H He 38BHCHT OT Toro. KaKHM 06p830M CHCTeMa nepeillna B 3TO COCTOSl-
HHe. 1l03TOMY H3MeHeHHe BHYTpeHHeH 3Hepl'HH r83a He 38BHCHT OT BH,A8 npou.ec-
COB H paBHO pa3HOCTH ee 3Ha'leHHH B KOHe'lHOM H Ha'l8nbHOM COCTOSlHHSlX:
3 21
U123 = U 3 - U I = -(4po. 2V o - PoV o ) = - PoV o .
2 2
21 29
QI = 4p o V o + - PoV o = - PoV o .
2 2
3
- Po V o
'1 = 2
29
- PoV o
2
P
4po
Po  
I I
Yo 2V o
Puc. 7.9
TOr,A8
1 3
Au = -(4po - Po)(2V o - Yo) = - Po Yo.
2 2
v
I1CKOMbIH EIl,l], 6Y,AeT paBeH
3
= - == 10 %.
29
9. ETI,n: TennoBoH MaIllHHbI p860'lHM TenOM KOTOpOH SlBnSleTCfl v MoneH
0)J.H08TOMHoro H,AeanbHOro 1'83a, p8BeH 1'}. r83 COBepillaeT u.HKn (pHC. 7.10),
COCTOSlIl.\HH H3 H30TepMH'leCKOrO 1-2, H30xopHoro 2-3 H 8l1;H868THoro npo-
u.eCCOB. Pa3HocTb M8KcHManbHoH H MHHHM8nbHoH TeMnep8TYP r838 B llHKne
paBHa T. HaHHHTe p86oTY. COBepIlleHHYIO r830M
B H30TepMH'leCKOM npOllecce.
,I:(aHo: v, 1'}, T.
H a H T H: Al2'
P e ill e H He. KTI,n: u.HKn8 H8H,AeM no q>opMyne
1'} = Ql - Q2 = 1- Q2 . l',Ae Ql H Q2 - KOnH'leCTBO
Ql Ql
P
1

3
Puc. 7.10
v
TennOTbI. nonY'leHHOe ra30M OT H8rpeB8TenSl H OT-
8HHoe XOnO,AHnbHHKY COOTBeTCTBeHHO. 113 rp8<1>HK8
173

SIi,AHM, 'lTO B npou.ecce 1-2 ra3 li30TepMIi'leCKIi paCIlIHpSleTCSI Ii. cnellosaTenb-
HO. nonY'laeT 3HeprHIO, 8 B npou.ecce 2-3 ra3 H30XOpHO OXnam,A8eTCSI. cne,AO-
saTenbHO, OH OT,A8eT 3HeprHIO. TIpli a,Ali868THOM npou.ecce 3-1 Tenno06MeHa
He npOHCXO,AHT. TI03TOMY, Ql = Q12 H Q2 = Q23' no nepSOMY 3aKOHY TepMO,AH-
H8MHKH Ql =A12 + tJ..U 12 =A I2 . TaK K8K npli H30TepMH"'leCKOM npou.ecce (T = const)
U12=O, AHanOrH'lHO Q2=Q23=A23+tJ..U23=ItJ..U231, T8K KaK A 23 =O. 113MeHeHlie
SHYTpeHHeH 3HeprHH (CM. npe,AbI,AylltylO 38,A8'lY) tJ..U 23 = U 3 -U 2 = vR(73 -12).
2
llpli li30TepMli1.IeCKOM npou.ecce TeMnep8Typa nOCTOSlHHaSl, n03TOMY T 1 = T 2 .
llpH li30XOpHOM OXnam,AeHHH TeMnepaTypa YMeHbIlIaeTCSI, T.e. T3 < T 2 . Cne,Ao-
BaTenbHO, B ,A8HHOM u.liKne T 2 - MaKCHManbHaSl TeMnepaTypa, T3 - MHHIi-
M8nbH8.H.
3H8'lHT. T 2 - T 3 =tJ..T.
11 - _ 1- 3vR!!:.T
113 nonY'leHHOrO paseHCTBa HaH,AeM 'I
2A 12
OTCIO,Aa liCKOMaSl p860Ta A _ 3v RtJ.. T
12 - 2 (1- 11)'
7.14. PeaJIbHbIe ra3bI
KaK MbI BH.n;eJlH, H.n;eaJlbHbIH ra3 - 3TO ynporoeHHa.H MOlleJlb peanbHbIX
ra30B. B 3TOH MO,lIeJlH He y'tlHTbIBaIOTC.H 06'beM MOJleKyn Ii CHJlbI B3aHMo.n;eH-
CTBIi.H Mem.n;y HHMH. Mem.n;y TeM MoneKYJlbI peanbHbIX ra30B 3aHHMalOT onpe-
,AeneHHbIH 06'beM H B3aHMo.n;eHCTBYIOT Mem.n;y C060H. IIpH 60nblllHX .n;aBJleHH.HX
Ii HH3KHX TeMnepaTypax CTaHOBHTC.H 3aMeTHbIM BJlH.HHHe c06CTBeHHbIX 06'be-
MOB MOJleKyn H CHn B3aHMo.n;eHCTBH.H Mem.n;y HHMH. IIpH 3THX YCJlOBHSlX ypaB-
HeHHe KnaneMpOHa-MeH,AeJleeBa H 3aKOHbI BOMn.H-MapHOTTa, reH-JIlOccaKa,
IIIapn.H HenpHro.n;HbI .n;JlSJ. onHCaHH.H COCTO.HHH.H peanbHbIX ra30B.
B 1873 r. rOJlJlaH.n;cKHH <pH3HK M. BaH-.n;ep-Baanbc BBeJl B ypaBHeHHe Kna-
neMpoHa-MeH.n;eneeBa nonpaBKH Ha pa3Mep MoneKyn H Ha .n;eHcTBHe CHn npH-
TSI>KeHH.H Mem.n;y HHMH. Ii. BaH-.n;ep-Baanbc npe.n;nO>KHn Mo.n;eJlb peaJlbHOrO ra3a,
B KOTOpOH MOJleKYJlbI npHHHMaIOTC.H 3a TBeplIbIe lllapHKH .n;HaMeTpOM d H 3a-
HHMalOT XOTb ManbIM, HO HeKoTopbIM 06'beM. MOJleKynbI He TOJlbKO OTTaJlKH-
BaIOTC.H npH coy.n;apeHH.HX, HO ellte H npHT.HrHBaIOTC.H .n;pyr K .n;pyry cpaBHH-
TeJlbHO cna6bIMH CHJlaMH Ha paCCTO.HHH.HX, cpaBHHMbIX C pa3MepaMH MOJleKYJl.
3anHllleM ypaBHeHHe KJlaneMpOHa-MeHeJleeBa .n;JI.H o.n;Horo MOJl.H H.n;eaJlb-
Horo ra3a (m = M): P V M = RT, r.n;e V M - MonSlpHbIM 06'beM ra3a.
Y'tleT c06CTBeHHoro 06'beMa MOJleKYJl npHBo.n;HT K TOMY, qTO <}>aKTH1.IeCKHM
CB060.n;HbIH 06'beM, B KOTOpOM MorYT ,ABHraTbCSI MoneKYJlbI peaJlbHOrO ra3a,
6Y,AeT He V M, a (V M- b), r.n;e nOCTO.HHHa.H b paBHa npH6J1H3HTeJlbHO yqeTBe-
peHHOMY c06cTBeHHoMY 06'beMY MOJleKYJl.
.D:eMcTBHe CHJI npHT.H>KeHH.H Me>K,Ay MOJleKynaM:H peanbHoro ra3a npHBo.n;HT
K nO.HBJleHHIO .n;onOJlHHTenbHoro .n;aBJleHH.H Ha ra3. IIpH npH6J1HeHHH HeKOTO-
pOH MOJleKYJlbI K CTeHKe cocy.n;a Bce OCTanbHbIe MOJleKYJlbI OKa3bIBaIOTC.H no
O.n;HY CTOPOHY OT Hee H paBHo.n;eMcTBYlOllta.H CHn npHT.HmeHH.H, .n;eMcTBYIOHX
Ha 3TY MOJleKYJlY, OKa3bIBaeTC.H HanpaBJleHHOM OT CTeHKH cocy.n;a BHYTPb ra3a.
STO npHBo.n;HT K TOMY, tITO YMeHblllaeTC.H HMnYJlbC, nepe.n;aBaeMblH MOJleKYJloM
174

CTeHKe cocy,na. B pe3YJlbTaTe ,naBJIeHHe ra3a p Ha CTeHKH cocy,na YMeHbllIaeTC.H
no cpaBHeHHIO C TeM PHP.' KaKHM OHO 6bIJIO 6bI B OTCYTCTBHe CHJI npHTHme-
HH.H: P = PH/J. - pi =:) PH/J. = P + p'. KaK nOKa3bIBalOT paCqeTbI, 3TO ,nononHHTeJlb-
Hoe ,naBJleHHe 06paTHO nponOpHOHBJIblIO KBa,ApaTY 06'beMa ra3a, T .e.
I a
P =,
V M
r,ne a - nOCTO.HHHa.H.
BBO,n.H 3TH nOnp8.BKH, nOJl}"IHM ypaBHeHHe BaH-Aep-BaaJILCa ,nJl.H O,AHOrO MOJl.H
ra3a (ypaBHeHHe COCTO.HHHJI peaJlLHblX ra30B):
(P+ vf. )(VM -b)= RT.
EcJlH ra3 npOH3BOJlbHOH MaCCbI m 38.HHMaeT 06'beM V, TO em MOJI.HpHbIH 06'beM
V m
V M = -, r,ne KOJlH'tleCTBO r838 v = -. fIO,nCT8BHB 3HaqeHHe MOJI.HpHOrO 06'beM8,
v M
nOJl}"IHM yp8.BHeHHe B8.H-,nep-BaBJIbC8 ,nJISI npOH3BonbHoro qHCna v MOJleH ra38:
I10CTO.HHHbIe )J;nH Kam,noro ra3a nonpaBKH a H b HaxO,nSlT 3KcnepHMeHTaJIbHO.
TaK KaK MOJIeKY JlbI H,ne8J1bHOrO ra38 He HMelOT 06'beM8 H He B3aHMO,AeH-
CTBYIOT Mem.qy C060H, TO HlIeaJIbHbIH ra3 npH JII06bIX H3MeHeHHHX ero nap8-
MeTpOB OCTaeTC.H ra30M.
COBceM HHaqe Be,AYT ce6S1 pe8JIbHbIe r83bI. 't.JT06bI y6e,nHTbCSI B 3TOM. HC-
CJIe,nyeMbIH ra3 nOMe8IOT B np03paqHbIH HJlHH,Ap C no,nBHmHbIM nopllIHeM
(pHC. 7.11), KOTOpbIH nOMeI.,QalOT B TepMocT8T.
I1pH Me,nJleHHOM cmaTHH ra3a TeMnepaTypa
ero He H3MeHHeTCSI (npoecc H30TepMHqeCKHH) H
,A8BneHHe ra38 YBeJlHqHBaeTC.H B COOTBeTCTBHH C
38KOHOM BOHJI.H-MapHOTTa. Ha rp8<}JHKe 3T8
CTa,nH.H onbITa H306pameHa KpHBOH 1-2. Ho,
HaqHHa.H C HeKOToporo 06'beMa, ,naJIbHeHllIee
V YMeHbllIeHHe 06'beMa ra3a yme He npHBo,nHT K
YBeJIHqeHHIO ,naBneHH.H, a Ha CTeHK8X HJIHH,Ap8
nOHBJlHIOTC.H KaneJlbKH mH,nKOCTH. fIPH 3TOM Ha-
xo,nHI.,QHeCH o,nHOBpeMeHHO r83 H :>KH,AKOCTb HMe-
lOT O,AHHaKOBYIO TeMnep8TYPY H HaXOlI.HTC.H no,A
O)J;HHaKOBbIM ,D;aBJIeHHeM, T. e. HaXO,lI.HTCSI B Tep-
MO)J;HHaMHqeCKOM paBHOBeCHH.
ra3, Haxo,n.HI.,QHHCSI B TepMo,nHH8MHqeCKOM
paBHOBeCHH co CBoeH mH,nKOCTblO, H83bIBaeTC.H Ha-
CLIIeHHLIM napOM (CM.  7.29).
TaKHM 06pa30M, yqaCTOK H30TepMbI 2-3 co-
OTBeTcTByeT HaCbIeHHOMY n8py H8,A mH,AKOCTblO
(,AByx<}Ja3Ha.H cpe,Aa). Kor,Aa BeCb ra3 npeBpaTHT-
CSI B mH,nKOCTb, ,n8J1bHeHllIee YMeHbllIeHHe 06'be-
p
o Va
V 2
',"",' . "
" ,""

Puc. 7.11
( m2 a )( m ) m
P+-'- V--b =-RT.
M 2 V 2 M M
175

Ma npHBO,nHT K pe3KOMY B03paCTaHHIO ,naBJleHH.H (y'tlaCTOK KpHBOH 3-4), no-
CKOJlbKY MOJleKYJlbI B mH,AKOCTH ynaKoBaHbI ,nOCTaTOqHO UJlOTHO H ee cmHMa-
eMOCTb MaJla. Bc.H KpHBa.H 1-2-3-4 Ha3bIBaeTC.H u30mepMou peaJl.bliOZO
za3a.
TaKHM 06pa30M, H30TepMbI peanbHoro ra3a 3aMeTHO OTJlH'tlaIOTC.H OT H30-
TepM H,neaJlbHOrO ra3a HaJlHqHeM Y HHX ropH30HTaJlbHbIX yqaCTKOB. COOTBeT-
CTBYIOI.I!HX 06J1aCTH CYI.l!eCTBOBaHH.H ZIByxcpa3HoH CHCTeMbI.
Cne,noBaTeJlbHO, OCHOBHoe OTJlH'tlHe peanbHbIX ra30B OT H,neanbHoro COCTO-
HT B TOM, qTO peaJlbHbIH ra3 MOmeT 6bITb npeBpaeH B mH,nKOCTb.
7.15. KpHTH1.J:eCKOe COCTOHHHe BeI.IJ;eCTBa
CXO,nCTBO CBOHCTB HeHaCbIeHHbIX napOB H ra30B HaTOJlKHYJlO M. <1>apa,ne.H
Ha npe,nnOJlOmeHHe: He .HBJI.HIOTC.H nH r83bI HeHaCbIeHHbIMH aapaMH COOTBeT-
CTBYIOHX mH,nKOCTeH? ECJlH npe,nnOJlOmeHHe BepHO, TO MomHO nonbITaTbC.H
c,nenaTb HX HaCbIeHHbIMH H CKOH,neHCHpOBaTb. ,D;eHcTBHTeJlbHO, cmaTHeM y,na-
nOCb C,AeJlaTb HaCbIeHHbIMH MHorHe ra3bI, KpOMe IlleCTH, KOTopbIe M. <1>apa-
,neii Ha3BaJl "nOCTO.HHHbIMH": 3TO a30T, BO,nOpO,n, B03,nyx, reJlHH, KHCJlOpO,n,
OKCH,n yrJlepo,na CO.
QT06bI nOH.HTb. B qeM 3,neCb ,neJlO, H3yqHM no,np06Hee 1i30TepMHqeCKHH npo-
necc CmaTH.H (paCIllHpeHH.H) napa. MbI BHZIeJlH. qTO H30TepMa peaJlbHOrO ra3a
OT JlHqaeTC.H OT H30TepMbI H,neaJlbHOrO ra3a HanH'tlHeM ropH30HTaJlbHOrO yqacT-
Ka, COOTBeTCTBYIOero 06naCTH cyecTBOBaHH.H ,nByxcpa3HOH CHCTeMbI: HaCbI-
eHHoro napa Ii mH,nKOCTH.
ECJlH npOBo,nHTb onbITbI npH 60Jlee BbICOKHX TeMnepaTypax (T 1 < T 2 < T3 <
< T K < T 4)' TO MomHO 06HapymHTb 3aKOHOMepHOCTb, o6ylO ,nJl.H Bcex BerqecTB
(pHC. 7.12).
p
IIap HeHaCbII.qeHHbIH
p",p -------
JKH)J;KOCTb
JKH)J;KOCTL H
HaCLIIT(eHHbIH
nap
o
I
I
I
I
I
V 2  V'; V",p V'  V.
v
Puc. 7.12
176

Bo-nepBbIX, qeM BbIllle TeMnepaTypa, TeM MeHbllle 06'beM, npH KOTOpOM Ha-
'IHHaeTC.H KOHeHCalJ,H.H ra3a: VI> Vi> Vi' . eCJlH TI < T 2 < T3'
BO-BTOpbIX, 'IeM BbIllle TeMnepaTypa. TeM 60JIbllle 06'beM, 3aHHMaeMbIH *HA-
KOCTblO nOCJIe Toro, KaK BeCb nap KOHAeHcHpyeTCH:
V 2 < V 2 < V 2 ' .
CJIeOBaTeJlbHO, JlHHa np.HMOJIHHeHHOro Y'laCTKa H30TepMbI C pOCTOM TeM-
nepaTypbI YMeHblllaeTC.H.
3TO JlerKO 06'b.HCHHTb: C pOCTOM T AaBJleHHe HaCbIeHHoro napa 6bICTpO
HapaCTaeT, H JI.H Toro, qTo6bI ,AaBJIeHHe HeHaCbIeHHoro napa CpaBH.HJlOCb C
aBneHHeM HaCbIeHHoro, He06xOAHMO YMeHbllleHHe 06'beMa. TIpHqHHa YBeJlH-
'leHH.H 06'beMa V 2 - B TenJIOBOM paClllHpeHHH mHKOCTH npH HarpeBaHHH.
TaK KaK 06'beM VI YMeHblllaeTCH, TO nnOTHOCTb napOB npH YBeJIHqeHHH TeM-
nepaTypbI YBeJIHqHBaeTCH; YBenH'IeHHe 06'beMa V 2 CBH,AeTenbcTByeT 06 YMeHb-
llleHHH nJlOTHOCTH mHKOCTH. 3TO 3HaqHT, qTO pa3JIHqHe Memy mHKOCTblO
H ee HaCbIeHHbIM napOM B npOlJ,eCce TaKoro HarpeBaHH.H CrJIamHBaeTCH H
npH ,o:OCTaTOqHO BbICOKOH TeMnepaType 1l0JImHO HC'Ie3HYTb COBceM.
,l].. MeH.ZIeJleeB YCTaHOBHJI, 'lTO .nJl.H Kamll;oH mH,AKOCTH OJlmHa cyecTBoBaTb
TaKaH TeMnepaTypa, KOTOpa.H 3KcnepHMeHTaJlbHO BnepBbIe 6bIJIa YCTaHOBJleHa JI.H
MHorHX BeeCTB T. 3HpIOCOM H HOCHT Ha3BaHHe KpHTH'IeCKOH TeMnepaTypbI.
p KpHTH'leCKa.H TeMllepaTYPa T Kp - 3TO TaKa.H TeM-
nepaTypa, npH KOTOpOH nJlOTHOCTb >KHKOCTIi H nJIOT-
HOCTb ee HaCNeHHoro napa CTaHOBHTCSI O,AHHaKOBN-
MH (pHC. 7.13).
Ha H30TepMax npH T = T Kp ropH30HTanbHbIH yqac-
TOK npeBpaaeTC.H B TOqKY neperH6a K.
)];aBJIeHHe HaCbIeHHoro napa KaKoro-nH60 BeeCTBa
npH ero KpHTHqeCKOH TeMnepaType Ha3bIBaeTCH KPHTH-
'1eCKHM AaBJleHHeM PKP' OHO .HBJI.HeTC.H HaH60JlblllHM
o t KP t.oC
B03MO>KHbIM aBJIeHHeM HaCbIeHHbIX n8pOB BeeCTB8.
Puc. 7.13 06'beM. KOTOpbIH 3aHHMaeT BeeCTBO npH PKP
H t KP ' Ha3bIBaeTC.H KpHTH'IeCKHM 06'LeMOM V KP' 3TO
HaH60JIblllHH 06'beM. KOTOpbIH MO>KeT 3aHHMaTb HMelOaHC.H Macca BeeCTBa
B mHKOM COCTOSlHHH.
TIPH KpHTH'IeCKOH TeMnepaType pa3JIHqHe Memy ra30M H mHKOCTblO HC-
'Ie3aeT, H n03ToMY yenbHa.H TenJlOTa napo06pa30BaHH.H CTaHOBHTC.H paBHoH
HYJIIO.
COBoKynHocTb TO'leK, cooTBeTcTBYIOHX KpaHM ropH30HTaJIbHOrO Y'IaCTKa
H30TepM (CM. pHC. 7 .12), BbIen.HeT B nJIOCKOCTH p-V 06JIaCTH cyecTBoBa-
HH.H Byxc}>a3HoH CHCTeMbJ H OT,AeJI.HeT ee OT 06J1aCTeH 0,AHoc}>a3HbIx COCTO.HHHH
BeeCTBa. TIorpaHH'IHa.H KpHBaH 06JIaCTIi Byxc}>a3HbIx COCTO.HHHH co CTOpOHbI
60nblllHx 3Ha'leHHH 06'beMa onHCbIB8eT COCTOHHHe HaCbIeHHoro napa H O,AHO-
BpeMeHHO npeCTaBJI.HeT C060R KPHBYIO KOH,ZJ;eHCal.\HH (Ha'lHHaeTCH KOHeHca-
lJ,HH napa npH H30TepMHqeCKOM cmaTHH). TIorpaHH'IHaH KpHBa.H co CTOpOHbI
MeHblllHX 06'beMOB npeACTaBJI.HeT C060H KPHBYIO, Ha KOTOpOH 3aKaH'IHBaeTCH
KOHAeHCalJ,HH npH cmaTHH HaCbIeHHoro napa H HaqHHaeTC.H HcnapeHHe mH-
KOCTH npH H30TepMH'IeCKOM paClllHpeHHH. Ee Ha3bIBalOT KpHBOH HcnapeHHH.
177

CyecTBoBaHHe KpHTHqeCKOH TeMnepaTypbI BeeCTBa 06'b.HCH.HeT, nOqeMY
npH 06blqHbIX TeMnepaTypax O,lJ.HH BeeCTBa MOrYT 6bITb KaK mH,lJ.KHMH, TaK
H ra3006pa3HbIMH, a ,lJ.PyrHe OCTaIOTC.H ra3aMH.
BbIllie KpHTHqeCKOH TeMnepaTypbI mH,lJ.KOCTb He 06pa3yeTC.H ,Aame npH O'leHb
BCOKHX ,lJ.aBJleHH.HX.
IIpHqHHa 3aKJlIOqaeTC1I B TOM, 'ITO 3,lJ.eCb HHTeHCHBHOCTb TennOBoro ,ABHme-
HH.H MOJleKYJl OKa3bIBaeTC.H HaCTOJlbKO 60JlbllIOH, qTO ,lJ.ame npH OTHOCHTeJlbHO
nJlOTHOH HX ynaKoBKe, BbI3BaHHOH 60JlbllIHM ,lJ.aBJleHHeM, MOJleKYJlSlpHbIe CHJlbI
He MorYT 06eCneqHTb C03,AB.HHe ,lJ.WKe 6J1HmHerO, a TeM 60Jlee ,lJ.aJlbHerO nOpSl,lJ.Ka.
TaKHM 06pa30M. BHlIHO. qTO npHHUHnHaJlbHOH pa3HHUbI Memny ra30M H
napOM HeT. 06blqHO ra30M Ha3bIBalOT BeeCTBO B ra3006pa3HOM COCTO.HHHH,
KOr,lJ.a ero TeMnepaTypa BbIllie KpHTH'leCKOH. IIapOM Ha3bIBalOT TaKme Bee-
CTBO B ra3006pa3HOM COCTO.HHHH, HO KOr,lJ.a ero TeMnepaTypa HHme KpHTH'IeC-
KOH. IIap MomHO nepeBeCTH B mH,lJ.KOCTb O,lJ.HHM TOJlbKO YBeJlHqeHHeM ,lJ.aBJle-
HH.H, a ra3 HeJlb3.H.
B HaCTO.Hee BpeMSl Bce ra3bI nepeBe,AeH B mH,lJ.KOe COCTO.HHHe npH O'leHb
HH3KHX TeMnepaTypax. IIOCJle,lJ.HHM B 1908 r. nepeBe,lJ.eH reJlHH (t Kp = -269°C).
7.16. CBoHcTBa jKH,ZJ;KOCTeH
IIo CBOHM «pH3HqeCKHM CBOHCTBaM .mH)J;KOCTH 3aHHMalOT npOMemYTOqHOe no-
JlOmeHHe Mem,lJ.Y peaJlbHblMH ra3aMli II TBep,lJ.bIMH TeJlaMH. IIo,lJ.06HO ra3aM H B
OTJlH'IHe OT TBep,lJ.bIX TeJl mH,AKOCTH He 06J1a,lJ.aIOT onpe,AeneHHOH «POpMOH. a
npHHHMalOT «POPMY cocy,na. B KOTOpOM HaXO,lJ..HTC.H. IIon06Ho TBep,lIbIM TeJlaM Ii
B OTJlH'IHe OT ra30B mH,lJ.KOCTH COXpaH1IIOT CBOH 06'beM. mH,AKOCTH HMelOT ,lJ.0-
CTaToqHO nJloTHYIO ynaKoBKY qaCTHn: nJlOTHOCTb BeecTBa B mH,lJ.KOM COCTO.H-
HHH He HaMHoro OTJlHqaeTC.H OT nJlOTHOCTH B TBep,lJ.OM COCTO.HHHH, H ,lJ.JI.H cma-
TH.H Tpe6YIOTC.H orpOMHbIe YCHJlR.H. Mem,Ay MOJleKY JlaMIi mH,lJ.KOCTH ,lJ.eHCTBYIOT
3Ha1.IHTenbHbIe CHJlbI B3aHMO,lJ.eHCTBH.H, XOT.H OHH H HeCKOJlbKO MeHbllIHe, 'IeM
y TBep,AbIX TeJl. MOJleKYJlbI mH,lJ.KOCTH COBepllIalOT HenpepbIBHbIe 6eCnOpSl,lJ.Oq-
HbIe ,lJ.BHmeHH.H caMbIX pa3J1H'IHbIX THnOB: nocTynaTeJlbHoe nepeMeeHHe MOJle-
KYJl, BpaeHHe MOJleKYJl, Kone6aHHSl aTOMOB BHYTPR MOJleKyn, KOJle6aHH.H MO-
JleKYJl B nOJle COCe,lJ.HHX MOJleKYJl.
mH,AKOCTH 6J1Hme K TBep,lJ.bIM TeJlaM, 'IeM K ra3aM. Ha 3TO YKa3bIBaeT
KOnH'IeCTBeHHa1l 6J1H30CTb HX nJlOTHOCTeH, y,lJ.eJlbHbIX TenJlOeMKOCTeH, K03«P«PH'
lJ,HeHTOB 06'beMHOrO paCllIHpeHHSI. Y,AeJlbHaSl TenJlOTa nJlaBJleHH1I cyecTBeHHO
MeHbllIe TenJlOTbI napo06pa30BaHH.H. B onbITax no pacceHBaHHIO peHTreHOBCKHX
JlY'leH B mHnKOCT.HX YCTaHOBJleH "6J1HmHHH nOp.HnOK" B pacnOJlOmeHHH Mone-
KYJl: ynop.H,lJ.OqeHHOe pacnOJlOmeHHe COXpaH.HeTC.H JlHllIb Cpe,lJ.H 6J1HmaHllIHX CO-
Ce,lJ.eH, T.e. Ha paCCTO.HHH.HX nOp.H,lJ.Ka HeCKOJlbKHX MOJIeKYJl.HpHbIX ,lJ.HaMeTpOB.
7.17. IIoBepxHocTHaH 3HeprHH
HaH60Jlee XapaKTepHbIM CBOHCTBOM mH)J;KOCTH, OTJlH'IaIOHM ee OT ra3a, 1IB-
JI.HeTC1I TO, qTO Ha rpaHHlJ,e C ra30M mH,lJ.KOCTb o6pa3yeT CB060,lJ.HYIO nOBepXHOCTb,
HaJIHqHe KOTOpOH npHBO,lJ.HT K B03HHKHOBeHHIO .HBJleHHH oc06oro pO,lJ.a, Ha3bIBae-
178

MbIX nOBepXHOCTHbIMH. CBOHM B03HHKHO-
BeHHeM OHH 06.H3aHbI OC06bIM CPH3W!ecKHM
YCJlOBH.HM. B KOTOpbIX HaXOlI.HTC.H MOJIeKY-
JIbI B6JIH3H CBo601IHOH nOBepXHOCTH.
Ha K8:>KlIYIO MoneKyny mHlIKOCTH
lIeHCTBYIOT CHJIbl npHT.HmeHH.H co CTOpO-
Hbl OKpY:>KalOHX ee MOJleKYJI, paCnOJIO-
meHHbIX OT Hee HB paCCTO.HHHH nOpSl,lJ.-
K8 10- 9 M (p8,lJ.HYC MOJleKYJI.HpHOrO ,lJ.eH-
CTBH.H). Ha MOJIeKYJIY M I , p8cnonomeH-
HYIO BHYTPH mHlIKOCTH (pHC. 7.14).
.neHcTBYIOT CHJIbI co CTOpOHbI TaKHX :>Ke
MoneKYJI, H paBHO,lJ.eHcTBYIOIU8.H 3THX CHn 6J1H3Ka K HYJIIO.
,lIn.H MOJIeKyn M 2 paBHOlIeHCTBYIOIUHe CHJI OTJlH'IHbI OT HYJI.H H H8npaBJIe-
HbI BHYTPb :>KHlIKOCTH. nepneHlIHKYJI.HpHO  ee nOBepXHOCTH. TaKHM 06pa30M.
BCe MOneKYJlbI mHlIKOCTH. H8XOlI.HHeC.H B nOBepXHOCTHOM cJloe. BT.HrHBaIOTC.H
BHYTPb :>KHlIKOCTH. Ho npOCTpaHCTBO BHYTPH :>KHlIKOCTH 38H.HTO PyrHMH MO-
JIeKYJIaMH. n03TOMY nOBepXHOCTHbIii CJIOH C03,naeT ,naBJIeHUe Ha :lKH.IUWCTb (MO-
JIeKYJI.HpHOe ,naBJIeHUe).
QT06bI nepeMeCTHTb MOneKYJlY M 3 , p8CnOJIO:>KeHHYIO HenocpelICTBeHHO nOlI
nOBepXHOCTHbIM CJIOeM. Ha nOBepXHOCTb, Heo6xOlIHMO COBeplllHTb pa60TY npo-
THB CHJI MOJleKYJI.HpHOrO 1I8BJIeHH.H. CJIelIOBaTeJIbHO, MOJleKynbI nOBepXHOCTHO-
ro CJIO.H mH,llKOCTH 06JIalIalOT lIOnOJIHHTeJIbHOH nOTeHIlHaJIbHOH 3HeprHeH no
cpaBHeHHIO C MOJleKYJlaMH BHYTPH mHlIKOCTH. STY 3HeprHIO Ha3bIBalOT nOBepx-
HOCTHOii 3Heprueii.
O'IeBH,llHO, qTO BeJlH'IHHa nOBepXHOCTHOH 3HeprHH TeM 60Jlblile. 'IeM 60nb-
me nJlOJIIalIb CB0601IHOH nOBepXHOCTH.
IIycTb nJIOIU8,lJ.b CBo6o.nHOH nOBepXHOCTH H:3MeHHJIaCb H8 I1S, npH 3TOM
nOBepXHOCTHa.H 3HeprH.H H3MeHHJlaCb Ha 11 W n = al1S, r,Ae a - K03cpcpHIlHeHT
nOBepXHOCTHoro HaT.H:>KeHH.H.
TaK KaK ,llJl.H 3Toro H3MeHeHH.H Heo6xOlIHMO COBeplllHTb pa60TY
/M'
2
Puc. 7.14
A = Wn' TO A = a. S. OTCIO,AB
A
a=-
I1S
E,llHHHu,eH K03cpcpHu,HeHTa nOBepXHOCTHoro HaT.H:>KeHH.H B Clf .HBJI.HeTC.H ,lJ.moynb
88 KB8I\P8T8L1Ji MeTp ( '; ).
K03Hu.HeHT nOBepXHOCTHoro HaTH:lKeHH.H - BenH'lHHa, qHCJIeHO paBHa.H
pa60Te, COBepllleHHOH MOJleKYJI.HpHbIMH CHJIaMH npH H3MeHeHHH nJIO8,lJ.H CBO-
60JJ;HOH nOBepXHOCTH :>KH,lJ.KOCTH H8 e,lJ.HHHu,y npH H30TepMH'IeCKOM npou,ecce.
TaK KaK Jl106a.H CHCTeMa, npeJJ;OCTaBJleHHa.H caM8 ce6e. CTpeMHTCs:I 38H.HTb
T8Koe nonomeHHe, B KOTOpOM ee nOTeHll,H8J1bHa.H 3HeprH.H HaHMeHbllla.H, TO
>KH,lJ.KOCTb 06H8PymHB8eT CTpeMJIeHHe K COKpaeHHIO CB060,lJ.HOH nOBepXHOCTH.
IIOBepXHOCTHblH CJlOH :>KU,lJ.KOCTH Be,AeT ce6.H nO,lJ.06HO P8CTSIHYTOH pe3HHO-
BOH nJleHKe, T.e. Bce BpeMs:I CTpeMHTC.H COKpaTHTb nnOa,llb CBoeH nOBepXHOC-
TH ,lJ.O MHHHMaJlbHbIX pa3MepOB, B03MomHbIX npH ,lJ.8HHOM 06'beMe.
II p H M e p: KanJl.H .>KH,lJ.KOCTH B COCTO.HHHH HeBeCOMOCTH HMeeT ccpepH'leC-
KYIO CPOPMY.
179

7.18. IIoBepxHocTHoe Ha THeHHe
CBOHCTBO nOBepXHOCTIf mH,lJ;KOCTH COKpaaTbCSI MomHO
HCTOJlKOBaTb KaK cyecTBoBaHHe CHJI, CTpeMHHXC.H COKpa-
THTb STY nOBepXHOCTb. MOJleKYJIa M 1 (pHC. 7.15), pacnOJIO-
meHHaH Ha nOBepXHOCTH :>KH,lJ;KOCTH, B3aHMO,AeHcTByeT He
TOJlbKO C MOJleKYJlaMH, HaXO,ASIHMHC.H BHYTPH mH,lJ;KOCTH, HO
H C MOJleKYJlaMH, HaXO,lJ;.HHMHCH Ha nOBepXHOCTH mH,lJ;KO-
CTH, pacnOJlOmeHHbIMH B npe,lJ;eJlax ccpepbI MOJIeKYJI.HpHOrO
,lJ;eHCTBH.H. ,l(JI.H MOJIeKYJIbI M 1 paBHO,AeHCTBYIOa.H R MOJIe-
KYJlHpHbIX CHJI, HanpaBneHHbIX B,lJ;OJIb CB060,lJ;HOH nOBepXHOC-
TH mH,lJ;KOCTH, paBHa HYJIIO, a ,lJ;JlH MOJleKYJlbI M 2' pacnOJIO-
meHHoii Y rpaHHbI nOBepXHOCTH mH,AKOCTH, R:I: () H R
HanpaBJleHa no HopManH K rpaHHQaM CB060AHOH nOBepXHOC-
TH H no KacaTenLHoii K caMoii nOBepXHOCTH H,ZJ;KOCTH.
PaBHO,lIeiicTByIOIUa.H C1:IJI. ,lJ;eHCTByIOIUHX Ha Bce MOJleKYJlbI. HaXOlIHIUHeC.H Ha
rpaHHe CB060,lJ;HOii nOBepXHOCTH. H eCTb CHJla nOBepXHoCTHoro HaT.HeHHH. B
eJIOM OHa ,lIeiicTByeT TaK, tITO CTpeMHTC.H COKpaTHTb nOBepXHOCTb mHKOCTH.
MomHO npe,lJ;nOJlOmHTb, qTO CHJla nOBepXHOCTHoro HaTH>KeHHH F npHMO npo-
nOpHOHaJIbHa ,lJ;JlHHe l rpaHHbI nOBepXHOCTHoro CJlO.H mH,lJ;KOCTH, Be)J;b Ha Bcex
yqacTKax nOBepXHOCTHoro CJlO.H :>KH,lJ;KOCTH MOJleKYJlbI HaXO,lJ;.HTC.H B O,lJ;HHaKOBbIX
YCJlOBH.HX:
Puc. 7.15
a
6
Puc. 7.16
F-l.
,l(eHCTBHTeJIbHO, paccMOTpHM BepTHKaJIbHbrH
np.HMoyrOJIbHbIii KapKac (pHC. 7.16,a,6).
nO,lJ;BH>KHa.H cTopoHa KOToporo ypaBHoBellle-
Ha. llocJIe H3BJleqeHH.H paMKH H3 pacTBopa
MbIJlbHOH nneHKH nO,lJ;BHmHa.H tIaCTb nepeMe-
aeTC.H H3 nOJlOmeHHH 1 B nOJlOmeHHe 2.
YtIHTbIBa.H. qTO nJleHKa npelICTaBJI.HeT co-
60H TOHKHii cJloii mH,lJ;KOCTH H I1MeeT ABe
cB060,lJ;HbIe nOBepXHOCTH, HaH,lJ;eM pa60TY,
COBeplllaeMYIO npH nepeMeeHHH nOnepetIH-
HbI Ha paCCTOSlHHe h = ala2' A = 2Fh, r,Ae
F - CHna, lIeHCTBYIOaH Ha KapKac co
CTOpOHbI Kam,AOrO nOBepXHOCTHoro CJlO.H.
C ,lJ;PyroH CTOpOHbI, A = aD.S = ex. 2lh.
F
a-
-T'
CJle,lJ;OBaTeJlbHO, 2Fh = a. 2lh; F = al,
OTKY)J;a
COrJlaCHO STOR cpopMYJle e,lJ;HHHeH KOscpcpHHeHTa nOBepXHOCTHoro HaTH-
meHRSI B CM SlBJISleTCSI HhlOTOH HB MeTp ( : l
K03cI>HU:HeHT nOBepXHOCTHoro HaTJIeHHH a tIHCJleHHO paBeH CHJle nOBepx-
HOCTHoro HaT.HmeHH.H, ,lJ;eHcTBYIOeii Ha e,lJ;HHHII;Y ,lJ;JlHHbI rpaHHII;bI CB060,lJ;HOH
nOBepXHOCTH >KH,lJ;KOCTH. KOscpcpHHeHT nOBepXHOCTHoro HaTHmeHH.H 3aBHCHT OT
npHpO,lJ;bI mH,AKOCTH, OT TeMnepaTYPbI H OT HaJIWlH.H npHMeceH. llpH YBeJIHqeHHH
TeMnepaTypbI OH YMeHblllaeTC.H. llpH KpHTHqeCKOH TeMnepaType, Kor)J;a HCqe3a-
eT pa3J1HqHe Mem,lJ;Y mH)J;KOCTblO H napOM, a = O.
180

IIp:HMecH B OCHOBHOM YMeHbIllalOT (HeKOTOpbIe YBen:H'tl:HBaIOT) K03cpcp:HII.HeHT
nOBepXHOCTHOrO HaT.HmeH:H.H.
T8.K:HM 06pa30M, 3a C'tleT C:HJI nOBepXHOCTHOrO HaT.HmeH:H.H nOBepXHOCTHbIH CJlOH
)KHAKOCT1i BCerAa HaXOAHTCS B COCTO.HH:H:H HaT.HmeHH.H. OAHaKO 3TO COCTO.HHHe
CYI..QeCTBeHHO OTJlH'tlaeTC.H OT HaT.HmeHH.H ynpyroH pe3HHOBOR nJleHKH. Ynpyra.H
IL1IeHKa paCTSrHBaeTCSI 3a CqeT YBeJlH'leHH.H paCCTOSIHH.H MemAY qaCTHII.aM:H, np:H
3TOM CHJla HaTSImeHH.H B03paCTaeT, np:H paCT.HmeHHH me m:HAKOH nJIeHK:H pac-
CTOSIHHe MemAY qaCT:HII.aMH He MeH.HeTC.H, a YBenH'tleHHe nOBepXHOCT:H ,AOCT:HraeT-
CSI B pe3YJlbTaTe nepeXOAa MoneKYJI :H3 TOJIIQH m:H,AKOCTH B nOBepXHOCTHbIH CJlOH.
IIo3TOMY npH YBeJIlPIeH:HH nOBepXHOCTH mHlIKOCTH CHJIa nOBepXHOCTHoro HaTSIme-
H:HH He H3MeHHeTCSI (OHa He 3aB:HC:HT OT nJIOI..Q8,AH nOBepXHOCTH).
7.19. CMa1.J:HBaHHe
CMaqHBaHHe - .HBJleHHe, B03HHKalOee BCJIeACTBHe B3a:HMOAeHCTB:H.H Mone-
KYJI mHAKOCTH C MOJIeKYJlaM:H TBep,AbIX TeJl. ECJIH C:HJIbI npHTHmeHH.H MemAY
MoneKYJIaM:H m:HAKOCTH H TBepAoro TeJla 60JIbIlle CHJI npHT.HmeHHH Mem,AY MO-
neKynaMH mHAKOCTH, TO >K:HAKOCTb Ha3bIBalOT CMaqHBalOeH; eCJIH CHJlbI npH-
TSlmeH:H.H mHAKOCTH H TBepAoro TeJla MeHbIlle C:Hn npHT.HmeHH.H MemAY MOJle-
KynaMH mHAKOCTH, TO mHAKOCTb Ha3bIBalOT HeCMa'lHBalOeH 3TO TeJlO.
O,AHa H 1'a me mHAKOCTb MomeT 6bITb CMa'tlHBalOI..QeH H HeCMaq:HBalOeii
no OTHOIlleHHIO :K pa3HbIM TeJlaM. TaK, BOAa CMa'tlHBaeT CTeKJlO H He CMaqH-
BaeT mHpHYIO nOBepXHOCTb, PTYTb He CMaqHBaeT CTe:KJIO, a CMaqHBaeT MeAb.
CM8qHBaHHe H.1IH HecMaqHB8.HHe m:HAKOCTblO CTeHOK COCYAa, B KOTOpOM OHa Ha-
XOA:HTC.H, BJlH.HeT Ha <PoPMY CBo6oAHOH nOBepXHOCT:H mHAKOCTH B COCYAe. EcJIH 6oJIb-
moe KOnlPIecTBO mHAKOCTH H8JIHTO B COCYA. TO cpOpM8 ee nOBepXHOCTH OnpeAeJI.HeT-
CH CH.1IOH THmecTH. KOTOpa.H 06ecne'tl:HBaeT IL1IOCKyro H ropH30HT8JIbHyIO nOBepXHOCTb.
OAHaKO Y C8..MbIX CTeHOK nOBepXHOCTb mHAKoCTH BCe-TaKH HCKpHBJISIeTC.H.
PaCCMOTpHM MOJleKYJIY m:HAKOCTH Y CTeHKH COCYAa (p:Hc. 7.17). IIycTb it -
paBHO,AeHcTBYIOI..Qa.H C:HJI, AeHCTBYIOIQHX Ha 3TY MOJIeKYJlY co CTOpOHbI TBepAoro
TeJI8, F 2 - co CTOpOHbI MOJleKYJI m:HAKOCTH. CHJIa it nepneHA:H:KYJI.HpH8 CTeH-
Ke. F 2 Hanp8BJleHa nOA yrJlOM 45° I< CTeHKe. ,nn.H CMa'tlHBalOeH m:HAKOCTH
(p:Hc. 7.1 7, a) MOAY JlH CHJI F 1 > F 2' PaBH01(eHCTBYIOI..Q8.H 3THX C:HJI F HBJI.HeT-
C.H C:HJIOH MOJIeKYJI.HpHOrO AaBJIeHH.H, IIp:HJlOmeHHOR K MOJIeKYJIaM, :H AOJImH8
6bITb HanpaBJleHa B rJIy6b >K:HAKOCT:H nepneHA:HKYJI.HpHO ee nOBepXHOCTH IIo-
3TOMY nOBepXHOCTb B6J1:H3H CTeHOK He ropH30HTaJIbHa, a HCKp:HBJI.HeTCSI AHa-
nOrH'lHO AOKa3MB8eTCSI :HCKpHBJIeHHe Y CTeHOK CBo6oAHOH nOBepXHOCTH H AJISI
HeCMa'tlHBalOI..QeH mHAKOCTH (p:Hc. 7.17. 6).
a 6
M
PI
F 2 =
Fz=--=-
Puc. 7.17
181

MepOH CMa1JHBaHH.H cJlymHT KpaeBOH yroJl e - 3TO yroJl Mem,lJ;y nJlOCKO-
CTblO. KaCaTeJlbHOH K nOBepXHOCTH mH,lJ;KOCTH, H CTeHKOH (nJlOCKOCTblO nOBepX-
HOCTH TBep,lJ;OrO TeJla). BHYTPH KpaeBOrO yrJla BCer,lJ;8 HaXO,lJ;HTCs.I mH,lJ;KOCTb
(pHC. 7.18, a, 6). ,lJ,JI.H CMaqHBalOeH mH,AKOCTH e - OCTPbIH. ,AJI.H HeCMaqH-
B81OeH e - TynoH. IIpH nOJlHOM CMa1JHB8HHH e = 0, npH nonHOM HeCMR''lHBa-
HHH e = 180 0 . ECJlH CMa1JHBalOas.l >KH,lJ;KOCTb H8XO,lJ;HTCs.I Ha OTKpbITOH nOBepXHOC-
TH TBeplIoro TeJl8 (pHC. 7.19, a), TO npOHCXO,AHT ee paCTeKaHHe no 3TOH nOBepx-
HOCTH. EcnH Ha OTKpbITOH nOBepXHOCTH TBep,lJ;OrO TeJl8 HaXO,lJ;HTC.H HeCMaqHBalO-
Il\as.l mH,AKOCTb, TO OH8 npHHHMaeT cpOPMY, 6J1H3KyIO K lllapOBOH (pHC. 7.19, 6).
CMa'lHBaHHe HMeeT BamHoe 3H81JeHHe KaK B 6bITY, TaK H B npOMbIlllJleHHOC-
TH. Xopolllee CMaqHBaHHe He06xo,lJ;HMO npH KpallleHHH, CTHpKe, 06pa60TKe cpOTO-
M8TepHaJIOB, H8HeceHHH Jl8KOKpaCOqHbIX nOKpbITHH, npH CKJleHBaHHH M8TepHaJIOB,
npH n8HKe, BO cpJlOTau.HoHHbIX npou.eCC8X (o6or8eHHe PY,A u.eHHoH nOpO,AOH).
H Ha060poT, npH coopymeHHH rH,ApOH30Jl.HU.HOHHbIX YCTPOHCTB He06xo,lJ;HMbI
M8TepHaJlbI, He CMaqHR8eMble BO,lJ;OH.
a
6
Puc. 7.18
a
6
/3/ L
ene30
PTYTb

/// - -
TeKJlO M
Puc 7.19
7.20. KanHJIJUlpHbIe HBJIeHHH
a
6
Puc. 7.20
HCKpHBJleHHe nOBepxHocTH mH,lJ;KOCTH y
KpaeB COCY,lJ;8 oc06eHHo OTqeTJlHBO BH,lJ;HO B Y3-
KHX Tpy6Kax. r,Ae HCKpHBJlSleTC.H BCs.I CB060,A-
Ha.H nOBepxHocTb mH,lJ;KOCTH. B Tpy6K8x C Y3-
KHM Ce1JeHHeM 3Ta oBepxHocTb npe,lJ;CT8BJI.HeT
C060H qaCTb ccpepbI, ee Ha3bIB81OT MeHHCKOM.
Y CM81JHB81OIl\eH mH,lJ;KOCTH o6p83yeTcSl BorHY-
TbIH MeHHCK (pHC. 7.20, a), a y HeCMa1JHB8-
lOIl\eH - BbInYKJlbIH (pHC. 7.20, 6). T8K K8K
nnOIl\8,lJ;b nOBepXHOCTH MeHHCK8 60Jlbllle, 1JeM
nJlOIl\8,Ab nOnepe1JHOrO Ce1JeHHs.I Tpy6KH, TO no,lJ;
HCKpHBJleHHa.H nOBepXHOCTb mH,lJ;KOCTH CTpeMHT-
,lJ;eHCTBHeM MOJleKYJlSlpHbIX CHJI
Cs.I BbInp.HMHTbCSl.
CHJlbI nOBepXHOCTHoro H8T.HmeHH.H C03,A81OT ,lJ;OnOJlHHTenbHoe (Jlann8COBO)
,lJ;8BJleHHe no,lJ; MCKpHBJleHHOH nOBepXHOCTblO mH,AKOCTH.
,lJ,J1.H paCqeTa H36bITOqHOrO ,lJ;aBJleHH.H npe,lJ;nOJlOmHM, qTO nOBepXHOCTb mH,lJ;-
KOCTH HMeeT cpOPMY ccpepbI p8,lJ;Hyca R (pHC. 7.21, a), OT KOTOpOH MbICJleHHO
OTCeqeH lllapoBoH cerMeHT, onHp81OHHC.H H8 oKpymHocTb pa,lJ;Hyca r= Rsina.
182

a
M
M 1
o
Puc. 7.21
6
8
y-
- -
-
-I
Ha KambIH 6eCKOHeqHO ManbIH 3neMeHT ,qnHHbl t1l 3Toro KOHTypa eH-
cTByeT KaCaTeJlbHa.H K nOBepxHocTIi cqJepbI cl:IJla nOBepxHocTHoro HaT.HmeHH.H,
MOYJIb KOTOpOH t1F = at1l. Pa3J10m1:lM BeKTop t1F Ha Be COCTaBJlHIOI.I:(l:Ie CHJIbI
[).F 1 1:1 [). F 2. H3 pl:lcYHKa 7.21, a BI:I,qI:lM, 't{TO reOMeTpl:l't{ecKa.H CYMM8 CI:IJI
[). F 2 ,AJIH BYX BbI,qeJIeHHbIX ,Al:laMeTpanbHO npOTHBOnOJlOmHbIX 3J1eMeHTOB t1l
palma HYJIIO. IIo3TOMY CI:IJIa nOBepXHOCTHoro HaT.HmeHI:IH HanpaBJleHa nepneH,qI:lKY-
nHpHO nJlOCKOCTI:I ce't{eHI:I.H BHyrpb mHKOCTH (pHC. 7.21, 6) 1:1 MO,Aynb ee paBeH
  .  r ar  ar 2anr 2
F = L.J M 1 = L.J t1F SIn a = L.J at1l- = - L.J t11 = - . 2nr = - .
R R R R
F
I'136bITo't{Hoe ,AaBJIeHl:le, c03aBaeMoe 3TOH CHJlOH: p = -, re S = 1[,..2 - nno-
S
IIW,Ab OCHOBaHI:I.H cqJepl:l't{eCKoro cerMeHTa. IIo3ToMY
2anr 2 2a
p = R. nr 2 = Ii'
ECJlH nOBepxHocTb jKHKOCTH BorHYTaH. TO CHna nOBepXHOCTHoro HaTH:me-
HI:I.H HanpaBJleHa H3 mH,qKOCTH (pHC. 7.21. 6) H ,qaBJleHHe no.lI. BorHYToH no-
v 2a
BepXHOCTblO ml:l,qKOCTI:I MeHbllIe, qeM no,A nJlOCKOI:I, Ha TY me BeJlI:I't{I:IHY P = -.
R
3Ta qJopMYJIa onpe,qeJI.HeT JIanJIaCOBO ,AaBneHHe JlH CJIY't{a.H cqJepl:l't{ecKoH qJop-
MbI cB06oHOH nOBepXHOCTH :mHKOCTH. OHa .HBJI.HeTC.H qaCTHbIM cnyqaeM qJOp-
MYJIbI JIanJIaca, onpe,qen.HlOI.I:(eH 1:136bITO'lHOe ,qaBJIeHl:le ,AJlSI npOl:l3BOJIbHOH no-
BepXHOCTI:I ml:l,AKOCTI:I .lI.BO.HKOH KpI:lBI:I3HbI:
p = a (  + ) ,
Rl R;.
re RI II R 2 - paI:IYCbI KpHBH3HbI BYX JlI06bIX B3aHMO nepneHHKYJI.HpHbIX
HOpMaJIbHbIX CeqeHHH nOBepXHOCTH :mHKOCTH. Pa,qHYC Kpl:lBI:I3HbI nOJIOmHTeJIeH,
eCJIH II.eHTp Kpl:lBI:I3HbI COOTBeTCTBYIOI.I:(erO CeqeHI:I.H Haxo,qI:lTC.H BHYTPH :mH,qKO-
CTI:I, 1:1 OTpl:lu.aTeJleH. ecnl:l n:.eHTp Kpl:lBI:I3HbI HRXO.lI.I:ITC.H BHe ml:l.lI.KOCTI:I. lIJlH
II.I:IJII:IHpl:lqeCKOH nOBepXHOCTIi (R 1 = l; R 2 = 00) 1:136bITOqHOe ,AaBJIeHl:le p = a .
R
ECJlH nOMeCTHTb Y3KYIO TPy6KY (KanHJlJlHp) OHI:IM KOHII.OM B mHKOCTb,
HaJII:ITYIO B 1lII:IpOKI:IH COcy, TO BCJleCTBl:le HaJlHqHH CHJlbI JlanJIRCOBa RBJIe-
183

a 6 HIUI mH)J.KocTb B Kan1:lJIJI.Hpe nO)J.HHMaeTcH (eCJIH
>KH)J.KOCTb CMa'tlHBalOrna.H) 1:IJIH onycKaeTcH (ecJIIi
>KHllKOCTb HeCMa'iIiBalOrna.H) (pHC. 7.22, a, 6), TaK
t KaK no,A nJIOCKOH nOBepXHOCTblO >KH)J.KOCTH B IIIH-
pOKOM cocY)J.e H36bITO'lHOrO )J.aBJleHH.H HeT.
.HBJleH1:I.H H3MeHeHIi.H BbICOTbI ypOBH.H mH)J.KOCTH
B KanliJIJIHpaX no cpaBHeHHIO C ypoBHeM >KH)J.KO-
CTH B llIHpOKHX cocY)J.ax Ha3bIBaIOTC.H KanHJIJIHp-
HblMH HBJIeHH.HMH.
Puc. 7.22 mH)J.KOCTb B Kan1:lJIJI.Hpe nO)J.HHMaeTC.H HJlH
onycKaeTC.H H8 TaKYIO BbICOTY h, np1:l KOTOpOH
C1:IJIa r1:l)J.pocTaT1:IqeCKOrO ,naBJIeH1:I.H cToJI6a mH,nKOCT1:I ypaBHOBe1lI1:IBaeTC.H C1:I-
nOH 1:I36bITO'tlHOrO )J.aBJIeHH.H. T.e.
2a
- = pgh. OTKY,Aa
R
h = 2a . ECJIH CMa'iliBaHlie He nOJIHOe 8"# 0 ( 8"# 180 0 ) ,
pgR
2a
TO, KaK nOKa3bIBalOT paCqeTbI, h = -cosS.
pgR
KanliJlJI.HpHbIe .HBJIeH1:ISI BeCbMa pacnpOCTpaHeHbl. IIo)J.HHTlie BO)J.bI B nO'tlBe,
C1:ICTeMa KpOBeHOCHbIX COCY)J.OB B nerK1:IX, KOpHeBa.H CHCTeMa y paCTeHIiH, CPH-
THJlb H npOMOKaTeJIbHaSl 6YMara - Kan1:lJIn.HpHbIe C1:ICTeMbI.
7.21. KpHCTaJIJIH'IeCKHe H aMopHbIe TeJIa
B OTJIIi'tlHe OT mH)J.KocTeH TBep)J.oe Teno COXpaH.HeT He TonbKO CBOH 06'beM,
HO 1:1 CPOPMY  06J1a)J.aeT 3Ha'tlHTeJIbHOH npO'lHOCTblO.
Pa3Ho06pa3HbIe TBep)J.bIe TeJIa, C KOTOpbIMH np1:lXO)J.HTC.H BCTpe'tlaTbC.H, MO>K-
HO pa3)J.eJlHTb Ha )J.Be rpynnbI, cyecTBeHHO pa3JI1:I'tlalO1:IeC.H no CBOHM CBOH-
CTBaM: KpHCT8.JIJlH'leCKHe H aMopcpHbIe.
OCHOBHLIe CBOHCTBa KpHCTaJlJlH'IeCKHX TeJI
1. KpHCTaJIJIIi'tleCKHe Tena HMelOT onpeJJ;eJIeH-
IIYIO TeMnepaTypy nJIaBJIeHHH t nn , lie H3MeHHIO-
YIOCH B npou.ecce nJIaBneHH.H npli nOCTO.HHHOM
llaBJleHH1:I (pHC. 7.23. Kp1:lBa.H 1).
2. AJI.H KpHCTaJIJl1:IqeCK1:IX TeJl xapaKTepHO Ha-
JIHqHe npocTpaHcTBeHHoH KpHCTaJIJIHqeCKOH pe-
lUeTKH, KOTOpa.H npe)J.CTaBJI.HeT C060H ynop.H)J.O'tleH-
Hoe pacnOJIO>KeHHe MOJIeKYJI, aTOMOB 1:IJI1:I 1:I0HOB,
nOBTOp.HlOeeC.H no BceMY 06'beMY TeJla (JJ;aJILHHH
nopJJJJ;OK). AJI.H JlI060H KpHCTaJlJIHqeCKOH pellIeT-
KH xapaKTepHO cyecTBoBaHlie TaKoro 3J1eMeHTa ee CTPYKTypbl, MHorOKpaT-
HbIM nOBTopeHlieM KOToporo B npOCTpaHCTBe MomHO nonY'lHTb BeCb KpHCTaJIJI.
3TO MOHOKpHCTaJIJI. IIOJIHKpHCTaJlJl COCTOHT 1:13 MHO>KeCTBa O'leHb MeJIKIiX,
CpOC1lI1:IXC.H Mem)J.Y C060H MOHoKpIiCTaJInOB, KOTopbIe oplieHTlipoBaHbI B npo-
cTpaHcTBe XaOTH'tleCKH.
t
1
tUIi
Puc. 7.23
184
Q

3. <1>H3HqeCKHe CBOHCTBa (MexaHHqeCKHe,
3J1eKTpH'tleCKHe, MarHHTHbIe, TenJIOBbIe HAP.)
MOHOKpHCTaJIJlOB pa3nHqHbI no pa3HbIM HanpaB-
JleHH.HM BHYTPH KpHCTaJIJla. 8TO .HBJleHHe Ha3-
BalOT aHH30TponHeH KpHCTanJIOB. OHO 06'b.HCH.H-
eTCH pa3J1H'tlHeM B nJIOTHOCTH pacnOJIOmeHH.H qa-
CTHII. B KpHCTaJlJlHqeCKOH pelIIeTKe no pa3HbIM
HallpaBJleHH.HM. Ha pHCYHKe 7.24 YCJlOBHO H306-
pameHO paCIIOJlOmeHHe aTOMOB B O)J;HOM H3 nnoc-
KocTeH MOHOKpHCTaJIJla. "lJepe3 Y3J1bI 3TOH IlJIOC-
KOH pelIIeTKH IIpOBeAeHbI pa3JIH'tlHO opHeHTHpo-
7.24 BaHHbIe napaJlJlenbHbIe np.HMbIe (1, 2, 3, 4).
BHAHO, qTO Ha eAHHHII.Y AJIHHbI IIp.HMbIX IIpHXO-
AHTC.H He OAHHaKOBoe KOJlH'tleCTBO aTOMOB. A MeXaHH'IeCKHe CBOHCTBa KpHCTaJIJIa
3aBHC.HT OT nJlOTHOCTH pa3MeeHH.H o6pa3YIOI..QHX ero 'laCTHII..
IIOJIHKpHCTaJIJlbI 1i30TpOIlHbI.
4. KpHCTaJIJlHqeCKa.H CTpYKTypa CB.H3aHa C MHHHMYMOM 1I0TeHIJ.HaJIbHOH 3Hep-
rHH, T .e. IIPH 06pa30BaHHH KpHCTaJIJlOB 'tlaCTHu.bI caMonpOH3BonbHO paCIIOJlaraIOTC.H
TaK, qT06bI HX B3aHMHa.H 1I0TeHu.HaJIbHa.H 3HeprH.H 6bIJIa MHHHMaJIbHOH.
AMopcpHbIe TeJla - 3TO CTeKJlO, CMona, KaHHcpOJlb, MHorHe IlJIaCTMaCCbI, cyp-
ryq, nJlaCTHqeCKaH cepa, .HHTapb, pa3J1HqHbIe nOJlHMepbI - OpraHHqeCKHe aMopcp-
HbIe TeJla (u.eJlJlIOJl03a, KayqYK, Koma, IlJIeKCHrJlaC, lIonH3THneH) HAP.
OCHOBHble CBOHCTBa aMoP4>HbIX TeJI
1. Y aMopcpHbIX TeJl HeT
a 6 KpHCTaJlJlHqeCKOH pelIIeTKH,
Y HHX 06HapymeH TOJIbKO
6JIHjKHHH nopHAoK B pacno-
JlOmeHHH MOJIeKYJI. Ha PH-
CYHKe 7.25 H306pameHB
1IJ10CKa.H cxeMB paCIIOJIome-
HH.H MOJIeKYJI KBapu.a (a) H
KBapu.eBoro CTeKJIa - aMopcp-
Horo TeJla (6).
Puc. 7.25 2. AMopcpHoe TeJlO 06J1a-
AaeT cJla60 BbIpameHHoH TeKY-
'1ecTLIO. TaK, eCJIH BOPOHKY HanOJlHHTb KYCO'tlKaMH BOCKa, TO 'tlepe3 HeKOTopoe
BpeM.H (pa3J1HtIHOe ,ZJ;JI.H pa3HblX TeMllepaTyp) KYCOqKH BOCKa 6YAYT "paCnJlbIBaTb'
CH". BOCK npHMeT CPOPMY BOpOHKH H HaqHeT "BbITeKaTb" H3 Hee.
TeKyqeCTb CB.H3aHa C nepecKoKaMH MOJleKYJI H3 oAHoro nOJlOmeHHH paBHo-
BeCH.H B APyroe.
3. Y aMop4>Hblx TeJl HeT onpe.AeJIeHHOH TeMnepdTyphl nJlaBJleHHH. BeI..Qe-
CTBO B aMopcpHoM COCTO.HHHH IIPH HarpeBaHHH nOCTeneHHO pa3M.HrqaeTC.H H ne-
peXOIIHT B mHIIKOCTb (CM. pHC. 7.23. KpHBa.H 2). BMecTo TeMnepaTypbI lIJ1aB-
neHH.H npHXOAHTC.H rOBopHTb 0 TeMllepaTypHOM HHTepBane pa3M.HrQeHH.H.
4. AMopcpHble TeJIa H30TponHhl, T.e. HX c:pH3HqeCKHe CBoHcTBa 110 BceM
HallpaBJleHH.HM O,AHHaKOB.
185

5. BHYTpeHH.HH 3HeprH.H BeeCTBa B aMop<pHoM COCTO.HHHH 60JlbIlIe, qeM B
KPHCT8JIJlHqeCKOM. IIo3ToMY aMop<l)HLle TeJIa Moryr caMOnpoH3BOJILHO nepeXOAHTL
B KpHCTaJlJIll'lecKoe COCTO.HHHe (npHMep: nOMyrHeHHe co BpeMeHeM CTeKOJl).
7.22. CTpoeHHe KpHCTaJIJIOB. TIpocTpaHcTBeHHaJl pemeTKa.
3JIeMeHTapHaH HqeHKa
llOH.HTHe KpHCTaJlJla 06bIqHO accouHHpyeTC.H Y Hac C MHHepaJlaMH, HMelO-
HMH reOMeTpHqeCKH npaBliJlbHYIO <POPMY, KOTOpa.H O,AHHaKOBa KaK ,AJlSI 60Jlb-
IlIHX, TaK H ,AJlH MaJlbIX KYCKOB MHHepaJlOB. PaCCMaTpHBa.H OT,AeJlbHbIe KpHC-
TaJlJlbI, MomHO y6e,AHTbC.H, qTO OHH OrpaHHqeHbI nJIOCKHMH, KaK 6bI IlIJlH<pO-
BaHHblMH rpaH.HMH B BH,Ae npaBHJlbHbIX MHoroyrOJIbHHKOB.
KpHCTaJlJlbI O,AHOrO H Toro me BeeCTBa MorYT HMeTh pa3J1HqHYIO <POPMY,
TaK KaK OHa 3RBHCHT OT YCJlOBH HX 06pa30BaHH.H. O,AHaKO ,AJI.H KpHCTaJIJlOB
Kam,AOrO BeeCTBa xapaKTepHO nOCTO.HHCTBO yrJlOB Mem,AY HX pe6paMH H CO-
OTBeTCTBeHHO Mem,AY HX rpaH.HMH.
TaK, KpHCTaJlJlbI nOBapeHHOH COJIH BCTpeqaIOTC.H B BH,Ae Ky60B, napaJlJle-
JlenHne,AOB, npH3M H ,Aame TeJl HenpaBHJlbHOH <pOPMbI, HO BCer,Aa B BepIlIHHe
Kam,AOrO KpHCTaJlJla CXO,A.HTC.H TPH pe6pR, 06pa3YIOHe Mem,AY C060H np.HMbIe
yrJlbI. H3MepHB yrJlbI Mem)J;Y BceMH rpaH.HMH HeH3BeCTHoro KpHCT8JIJla. MomHO
no CneltHaJlbHOMY KaTaJlory onpe.neJlHTb. KpHCTaJIJI KaKoro BemeCTBa HCCJle,Ay-
eTC.H (KpHCTaJIJlOXHMHqeCKHH aHanH3).
IIpaBHJIbHa.H reOMeTpHqeCKa.H <popMa KpHCTaJlJlOB .HBJI.HeTC.H pe3YJlbTaTOM
ynopSl,AOqeHHOro pacnOJlOmeHH.H qaCTHu, (aTOMOB, MOJleKYJl, HOHOB») COCTaB-
JlSlIOHX KpHCTaJlJl. TOqHee, 1{aCTHltbI KOJle6J1IOTC.H OKOJlO onpe,AeJIeHHbIX no-
JlOmeHHH paBHOBeCH.H. ECJlH HX MblCJleHHO COe,AHHHTb np.HMbIMH JlHHH.HMH,
TO nOJlyqaeTC.H CBoero pO,Aa "CKeJleT" KpHCTaJlJIa. TaKoe H306pameHHe KpHC-
TaJlJla Ha3bIBaeTC.H KpHCTaJIJIHqeCKOB npOCTpaHCTBeHHOB pellleTKOB. B OCHO-
Be KpHCTaJlJlHqeCKOH pemeTKH JIemHT 3J1eMeHTapHa.H .HqeHKa - <pHrypa HaH-
MeHbIlIero pa3Mepa, nOCJle,AOBaTeJlbHbIM nepeHOCOM KOTOpO MomHO nOCTpo-
HTb BeCb KpHCTaJlJl.
,D;JlH onHCaHH.H CTpOeHH.H JlI060ro KpHCTaJIJlR ,AOCTaTOqHO H3yqHTb CTpoeHHe
3J1eMeHTapHOH .HqeHKH npOCTpaHCTBeHHOH peIlIeTKH 3Toro KpHCTaJlJla.
B 3aBHCHMOCTH OT pO,Aa qaCTHlt, pacnOJlOmeHHbIX B Y3nax KpHCTRJlJlHqe-
CKOH peIlIeTKH. H xapaKTepa CHJI B3aHMo.neHCTBH.H Mem,Ay HHMH KpHCTaJlJlhI
pa3.neJl.HIOT Ha 4 THna: HOHHbIe, aTOMHbIe, MeTaJlJlHqeCKHe H MOJIeKYJI.HpHbIe.
B Y3J1RX HOHHOB peweTKH B onpe,AeJIeHHOM nOpH,A-
Ke qepe,AytOTC.H HOHbI npOTHBOnOJlOmHbIX 3HaKOB, y)J;ep-
mHBalOHeC.H 3J1eKTpOCTaTH1{eCKHMH CHJlaMH (DoHHaH
CBH3L). HOHHa.H peIlIeTKa KaMeHHOH COJIH NaCl npH-
Be,AeHa Ha pHCYHKe 7.26.
B Y3J1ax aTOMHOH pellleTKH HaXO,A.HTC.H HeHTpaJlb-
HbIe aTOMbI, Mem,AY KOTOpbIMH cyecTByeT KOBaJIeHT-
Ha.H CBH3L: npH C6J1HmeHHH aTOMOB HX BaJleHTHble
3J1eKTpOHbl KaK 6bl CTaHOB.HTC.H 06HMH ,AJlH 6J1HmaH-
IlIHX COCe,AeH. Ha pHCYHKe 7.27, a, 6 KOBaJleHTHaH
CB.H3b H306pameHa ,AJI.H repMaHH.H.
cr
Na+
NaCl
Puc. 7.26
186

a
6
Puc. 7.27
B Y3JIaX MeTaJlJlHqeCKOH peweTKH HaXOHTC.H nonOmHTeJlbHbIe HOHbI, B3a-
HMoeHcTBHe Memy KOTOpbIMH ocyeCTBJI.HeTCH qepe3 0606eCTBJIeHHble CBO-
6oJJ;HLle 3JIeKTpoHLI.
B Y3J1aX MOJIeKYJIHpHOH peweTKH HaXO.HTC.H HeiiTpaJlbHble MoneKYJlbI, Y,Aep-
mHBalOHecH MOJleKYJIHpHLIMU CHJlaMU npHTH:iKeHHH. 8TH CHJlbI HeBeJlHKH.
II03TOMY MOJleKYJI.HpHble KpHCTMJIbI JIerKO pa3pYIllaIOTC.H. 8TO KpHCTaJlJlbI CO 2 ,
O 2 , N 2 B TBep,AOM COCTOHHHH. 60nblllHHCTBO OpraHHqeCKHX OeHHeHHH (napa-
<pHH, CnHpT, pe3HHa H p.).
7 .23. IIoJIIfMop1f3M
IIpaKTHqeCKH Bce BemeCTBa B TBepLIOM COCTO.HHHH MorYT cymecTBoBaTb
B ,ABYX HJlH 60JIee KpHCTaJlJlHqeCKHX pa3HOBHHOCT.HX (Mo.uH<pHKaIlH.HX), OT-
nHqalOHXC.H <pH3HqeCKHMH CBOHCTBaMH. 8TO HBJleHHe Ha3bIBaeTC.H nOJIHMOp-
<pH3MOM. TaK, Y yrJlepO8 Be pa3HOBHHOCTH - aJlMa3 H rpacpHT: rpacpHT
OTJIHqaeTC.H M.HrKOCTblO, aJlMa3 TBep, rpa<pIiT - npOBOHHK, anMa3 - H-
sneKTpHK. 113BecTHbI 4 MOH<pHKaU;HH meJle3a, 9 MOH<pHKaHH cepbI H p.
K8ma.H MOH<pHKaU;HH YCTOHqHB8 B onpeeJleHHOM HHTepBaJle TeMnepaTYP
H aBJIeHHH.
7 .24. ):(eeKTbI B KplfCTaJIJIaX
Ae<peKTaMH B KpHCTaJIJlaX Ha3bIBalOT HapYIlleHH.H HeaJlbHOH KpHCTaJlnH-
qeCKOH CTPYKTypbI. TaKoe HapYllleHHe MomeT 3aKJIlOqaTbC.H B 3aMeHe aTOMa
,A8HHOrO BeeCTBa qymHM aTOMOM (aToMoM npHMecH) (pHC. 7.28, a), BO BHe-
,ApeHHH JlHlllHerO aTOMa B Mem,AoY3J1He (pHC. 7.28. 6). B OTCYTCTBHH aTOMa B
a
1
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-
 
,. .,
-
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, J
-
6
r
...,
.
-
-'
J
Puc. 7.28
8
187

Y3JIe (pHC. 7.28, 6). IIop;06HbIe p;ecpeKTbI Ha3bIBRIOT TO'le'lHLIMU. OHH BbI3bI-
BalOT HapYllleHH.H npRBHJIbHOCTH pellleTKH, paCnpOCTpaH.HIOHeC.H HR paCCTO.H-
HHH nOp.Hp;KR HeCKOJlbKHX nepHOp;OB.
KpOMe TOqeqHbIX, cyeCTBYIOT p;ecpeKTbI, cOCpeAOTOqeHHbIe B6JIH3H HeKO-
TOpbIX JIHHUH. Mx Ha3bIBalOT J1UHeHHLIMU ,ZJ;e4>eKTaMu HJlH ,ZJ;UCJlOKaU.HMU.
)];ecpeKTbI TaKoro BHP;R HapYlllalOT npaBHJIbHOe qepep;OBaHHe KpHCTaJIJlHqeC-
KHX nJIOCKOCTeH.
IIpOCTeHlllHMH BHAaMH AHCJIOKau,HH HBJI.HIOTC.H KpaeBa.H H BUHTOBaH P;HC-
nOKau,HH.
KpaeBa.H ,AHCJIOKau,H.H 06YCJIOBJIHBaeTC.H
JIHlllHeH KpHCTaJlJIHqeCKoi1 nOJIynJIOCKOCTblO,
BHep;peHHOH Me:iKp;y P;BYMH COCep;HI1MH CJIO.HMI1
RTOMOB (pl1c. 7.29). BI1HTOBYIO p;I1CJIOKau,HIO
MO:iKHO npep;CTRBHTb KaK pe3ynbTaT pa3pe3a
KpHCTaJIna no nOJlynJlocKoCTH 11 nocnep;YIOero
cp;BHra JIemaHX no pa3HbIe CTOpOHbI pa3pe3a
qaCTei1 peilleTKI1 HaBCTpeqy p;pyr p;pyry Ha Be-
JIl1ql1HY op;Horo nepl1op;a (pHC. 7.30).
)];ecpeKT OKa3BaIOT CHJIbHOe BJII1.HHHe Ha
cpH311qeCKHe CBOHCTBa KpHCTaJIJIOB, B TOM qHC-
Jle H Hd HX npO'lHOCTb.
IIepBoHaqaJIbHO HMeBlllaHC.H p;HCJIOKau,HH nop;
B03,AeHCTBHeM C03AaHHbIX B Kpl1CTanJIe Hanp.H-
meHI1H nepeMemaeTCH BAOJIb KpHCTaJIJIa. IIepe-
MeweHHIO p;I1CJlOKaUI1H npenHTcTByeT HaJIl1ql1e
p;pyrl1x ,llecpeKTOB B KpHCTanJIe, HanpHMep, npH-
CYTCTBHe aTOMOB npHMeCH. )];HcnoKau,HH TOpMO-
3.HTCH TaKme npH nepeCeqeHHH p;pyr C p;pyrOM.
YBeJIl1qeHHe nJlOTHOCTH AHCJlOKRU,HH (1 B03pac-
TaHHe KOHu,eHTpau,HH npHMeceH npI1BOp;HT K
CI1JIbHOMY TopMOmeH111O p;HCJIOKau,I1H 11 npeKpa-
eHI1IO HX p;BH:iKeHH.H. B pe3YJIbTaTe npOqHOCTb
MaTepl1aJIR paCTeT. TaK. HanpHMep, nOBbIllleHHe
npO'lHOCTH :iKeJle3a p;OCTHraeTC.H paCTBopeHHeM B
HeM aTOMOB yrJIepop;a (CTRJlb).
IIJlacTHqeCKa.H ,AecpopMau,H.H conpOBO:iKAaeTCSI
pa3pYllleHI1eM KpHCTaJlJIH't.leCKOH pellleTKH H 06pa30BRHHeM 60JIbillOrO KOJIHqe-
CTBR p;ecpeKTOB, npen.HTCTBYIOI.qHX nepeMeI.qeHI1IO p;HCJIOKau,HH. 3TI1M 06'b.HCH.H-
eTCH ynpOqeHHe MaTepl1aJIOB npH HX XOJlOP;HOH 06pa60TKe.
Puc. 7.29

Puc. 7.30
7.25. Ae«lJopMau.uH. BUD:L1 D:e«lJopMauHH TBepD:bIX TeJI
IIop; BHelllHHM B03p;eHcTBHeM TeJI8 MorYT p;ecpopMl1pOBaTbC.H.
,l(e4>opMauH - H3MeHeHHe cpOpMbI H pR3MepOB Tena. IIpl1ql1Ha p;ecpopMa-
U,HH 3aKJllOqaeTC.H B TOM, 't.lTO pa3JII1'lHbIe 't.lRCTH TeJla COBepillalOT Heop;HHaKO-
BbIe nepeMeI.qeHHH npH p;eHCTBHH Ha TeJIO BHelllHI1X CHJI.
)];ecpopMau,I1H, KOTopbIe nOJIHOCTblO I1Cqe3alOT nOCJIe npeKpaeHI1.H p;eHCTBHSl
CHJIbI, - ynpyrue, KOTopbIe He HCqe3a1OT, - nJiaCTUqeCKUe
188

IIpH ynpyrHx eq)QpMaIJ;H.HX npOHCXOHT H3MeHeHHe paCCTO.HHH.H Memy 'IaC-
THIJ;aMH Tena. B Hee4>opMHpoBaHHOM Tene qaCTHIJ;bI HaXO.HTC.H B OnpeeJleHHbIX
nOJlOmeHH.HX paBHOBeCH.H (paCCTO.HHH.H Me>Ky BbIeJleHHbIMH qaCTHIJ;aMH rO - CM.
pHC. 6.2, 6). B KOTOpbIX CHJlbI OTTaJIKHBaHH.H H npHT.H>KeHH.H, eHcTBYIOHe co
CTOpOHbl pyrHx 'IaCTHli" paBHbI. IIpH H3MeHeHHli paCCTO.HHHSl Me>Ky 'IaCTHIJ;aMH
o.n;Ha H3 3THX CHJI Ha'IHHaeT npeBbIllIaTb .n;pyrylO. B pe3ynbTaTe B03HHKaeT paB-
HO,AeHcTBYIOa.H 3THX CHJI, CTpeM.Ha.HCSI BepHYTb 'IaCTHIJ;y B npemHee nOJlO>Ke-
HHe paBHOBeCHSI. PaBHOl1eHcTBYIOaSl CHJI. l1eHcTBYIOWHX Ha Bce qaCTHUbI .n;ecpop-
MHpOBaHHoro TeJla, H eCTb Ha6J1IOaeMa.H Ha npaKTHKe CHJla ynpyroCTH. TaKHM
06pa30M. CJle.n;CTBHeM ynpyroH .n;e4>opMan;HH HBJlSleTC.H B03HHKHOBeHHe ynpyrHx CHn.
IIpH nJlaCTHqeCKOH Ae<}>opMaIJ,HH. KaK nOKa3aJlH Ha6J1IOeHHSI, CMeeHHSI
'laCTHIJ; B KpHCTaJlJle HMelOT COBceM .n;pyroH xapaKTep, 'leM npH ynpyroH. IIpH
nnaCTHqeCKOH e<l>opMaIJ;HH KpHCTaJlJla npOHCXOHT COCKaJlb3bIBaHHe CJlOeB KPH-
CTanJla OTHOCHTeJlbHO pyr .n;pyra (pHC. 7.31, a, 6). 3TO MomHO YBH.n;eTb C
nOMOblO MHKpOCKona: rJlaKa.H nOBepXHOCTb KpHCTaJlJlH'IeCKOrO CTepmHSI noc-
Jle nJlaCTHqeCKOH e4>opMaIJ;HH CTaHOBHTC.H llIepoxoBaToH. COCKaJlb3bIBaHHe npo-
HCXO,AHT BOJlb CJlOeB, B KOTOpbIX 60JlbllIe Bcero aTOMOB (pHC. 7.32).
o 0 oto 0 .2
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o 0 0 0--6 0 0
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o <yo 0 0 0 0
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6 0 0 0 1 0 0 0 0
o 0 OO 0 0 0
Q
6
Puc. 7.31
Puc. 7.32
rIpH TaKHX cMeeHHHx 'IaCTHIJ; TeJlO OKa3bIBaeTCSI .n;e<l>opMHpoBaHHbIM. HO
Ha cMeeHHbIe qaCTHIJ;bI npH 3TOM He eHCTBYIOT "B03BpaalOHe" CHJlbI, TaK
KaK y Kamoro aTOMa B ero HOBOM nOJlO>KeHHH TaKHe me coce.n;H H B TaKOM
me 'IHCJle, KaK H ,AO CMeeHH.H.
IIpH paC'IeTe KOHCTPYKIJ;HH, MaIlIHH, CTaHKOB. Tex HJlH HHbIX coopY.>KeHHH,
npH 06pa60TKe pa3J1H'lHbIX MaTepHaJlOB BamHO 3HaTb, KaK 6Y,AeT e4>opMHpo-
BaTbC.H Ta HJlH HHa.H eTaJlb no eHCTBHeM HarpY3KH, npH KaKHX YCJlOBH.HX
ee .n;e(j)opMaIJ;li.H He 6yeT BJlH.HTb Ha pa60TY MallIHH B IJ;eJlOM, npH KaKliX Ha-
rpY3Kax HacTynaeT pa3pYllIeHHe l1eTaneH H T ..n;.
,l(e<l>opMaIJ;HH MorYT 6bITb OqeHb CJlOmHbIMH. Ho HX MomHO CBeCTH K BYM
BH.n;aM: paCT.H>KeHHIO (C>KaTHIO) H cBHry.
Ae<}>opMalJ,HH pacTHmeHHH (cmaTHH). JIHHeHHaSl .n;e(j)opMau,HH B03HHKaeT npH
npHJlO.>KeHHH CHJlbJ F BOJlb OCH CTep>KH.H, 3aKpenneHHoro C oHoro KOHu,a
(pHC. 7.33, a, 6). TIpH JlliHeHHbIX e4>opMaIJ;li.HX CJlOH TeJla OCTalOTCSI napaJl-.
neJlbHbIMli .n;pyr ,Apyry, HO 1I3MeHHIOTC.H paCCTO.HHHSI Me>Ky HliMli. JIliHeHHYIO
.n;ecpopMaIJ;HIO xapaKTepH3YlOT a6conlOTHbIM H OTHOCHTeJlbHbIM yJlHHeHHeM.
A6coJllOTHoe YWIHHeHHe /)'l = l-lo, r.n;e l - nHHa e<l>opMHpoBaHHoro TeJla,
lo - I1J1HHa TeJla B Hel1e(j)opMHpoBaHHoM COCTO.HHHH.
189

a
J-L
!.I
Puc. 7.33
Puc. 7.34
t"l
OTHOCHTeJlLHOe Y.l1.J1HHeHHe E = - - OTHOllIeHHe a6COJllOTHOrO YlIJIHHeHH.H
10
K ,lJ,nHHe He,lJ,eq>opMHpoBaHHoro TeJla.
Ha npaKTHKe pacTSlmeHHe HcnbITbIBalOT TpOCbI nO,lJ,'beMHbIX KpaHoB, KaHaT-
HbIX ,lJ,opor, 6YKcHpHbIe TpOCbI, CTPYHbI MY3bIKMbHbIX HHcTpYMeHToB. CmaTHIO
nO,lJ,BepraIOTC.H KOJlOHHbI, CTeHbI H c}>YH,lJ,aMeHTbI 3,lJ,aHHH H T.,lJ..
Aecl>opMaU;HH cABHra B03HHKaeT no,lJ, ,lJ,eHCTBHeM cHn, npHJlOmeHHbIx K ,lJ,BYM
npoTHBOnOJIOmHbIM rpaH.HM TeJla TaK, KaK nOKa3aHO Ha pHCYHKe 7.34. 3TH
CHJlbI BbI3bIBalOT CMeeHHe CJlOeB TeJla, napaJIJIeJlbHbIX HanpaBJleHHIO CHJI. Pac-
CTO.HHHe Mem,lJ,Y CJIO.HMH He H3MeH.HeTC.H. JII060H np.HMoyrOJIbHbIH napaJlneJle-
nHne,lJ,. MbICJIeHHO BbIlIeJleHHbIH B Tene. npeBpaIl.taeTC.H B HaKJIOHHbIH.
MepoH lIec}>opMaHH C,lJ,BHra .HBn.HeTC.H yroJi ClJ.BHra y - yroJl HaKJlOHa Bep-
THKMbHbIX rpaHeH (pHC. 7.35).
.nec}>opMaHIO C,lJ,BHra HcnbITbIBaIOT, HanpHMep, 3aKJlenKH H 60JITbI, COe,lJ,HH.HIO-
He MeTMJlHqeCKHe KOHCTPYKHH. C,lJ,BHr npH 60JlbllIHX yrnax npHBO,lJ,HT K pa3-
PYllIeHHIO TeJIa - cpe3Y. Cpe3 npOHCXO,lJ,HT npH pa60Te HO>KHH, nHJIbI H ,lJ,p.
):(ecl>opMalJ)lH H3rH6a nO,ABepraeTC.H 6aJIKa, 3aKpenJIeHHa.H C O,AHOrO KOHa HJIH
3aKpenJleHHa.H C ,lJ,BYX KOHOB, K Cepe,lJ,HHe KOTOpOH nO,lJ,BellIeH rpY3 (pHC. 7.36).
.nec}>opMaH.H H3rH6a xapaKTepH3yeTC.H CTpeJlOH nporH6a h - CMeeHHeM ce-
pe,lJ,HHbI 6aJlKH (HJlH ero KOHa). IIpH H3rH6e BbInYKJlbIe 'laCTH TeJI HcnbITbI-
BalOT paCT.HmeHHe, a BorHYTbIe - cmaTHe, cpe.n;HHe qaCTH TeJIa npaKTHtIeCKH
He ,Aec}>opMHPYIOTC.H - HeiiTpaJlLHLIH CJlOH. HaJlHtIHe cpe.n;Hero CJlO.H npaKTH-
qeCKH He BJlH.HeT Ha conpOTHBJI.HeMOCTb TeJla H3rH6y, n03TOMY TaKHe ,lJ,eTanH
BbIrO,AHO ,lJ,eJlaTb nOJIbIMH (3KOHOMH.H MaTepHaJla H 3Ha't{HTenbHO CHHmeHHe IiX
MaCCbI). B COBpeMeHHOH TeXHHKe llIHpOKO HCnOJlb3YIOTC.H nOJlbIe 6aJlKH, TPy6-
KH. Y qeJlOBeKa KOCTH Tome Tpy6tIaTbIe.
Puc. 7.35
Puc. 7.36
190

J)11
Puc. 7.37
AeopMau.HIO KpyqeHH.H MomHO Ha6J1IOaTb, ecnH
Ha cTepmeHb, OHH KOHeu; KOToporo 3aKpenJleH, eH-
cTByeT napa CHJI (pHC. 7 .37), JlemaHX B nJlOCKO-
CTH, nepneH.a.HKYJl.HpHOH OCH CTepmH.H. TIpH KpyqeHHH
OT.a.enbHbIe CJlOH TeJla OCTaIOTC.H napaJlJleJlbHbIMH, HO
nOBOpaQHBaIOTC.H .a.pyr OTHOCHTeJlbHO .a.pyra Ha HeKO-
TOpbIH yroJl. ,uecpopMaUH.H KpyqeHH.H npelICTaBJI.HeT co-
60H HepaBHOMepHbIH ClIBHr. ,IJ;ecpopMaUHH KpyqeHH.H
B03HHKalOT npH 3aBHHqHBaHHH raeK. npH pa60Te Ba-
JlOB MaIlIHH.
7.26. HanpJlmeHHe. 3aKoH fyKa
EcnH Teno .lle<}>opMHpOBaHO. TO B JlI060M ero CeqeHHH .a.eHCTBYIOT CHJlbI
ynpyroCTH. npen.HTCTBYIOIUHe pa3pbIBY TeJla Ha qaCTH. 06bIqHO COCTO.HHHe .a.e-
cpOpMHpOBaHHoro TeJla xapaKTepH3YlOT OC060H BeJlHqHHOH. Ha3bIBaeMOH Hanp.H-
meHHeM (MexaHHqeCKHM Hanp.HmeHHeM).
Hanp.HeHHe - BeJlHqHHa. paBHa.H OTHOllIeHHIO MOy JI.H CHJlbI ynpyrocTH
K nnOm;aH nOnepeqHOrO CeqeHH.H TeJla:
F
cr =-.
S
B Cll eJ\HHHIeii HanpH"'eHHH HBJIHeTCH naCKaJIL (ITa =  )-
OnbITHbIM nYTeM YCTaHOBneHO, qTO npu MaJl.blX iJeqJOpMaijUSlX (ynpyzux oe-
cpopMaquSlX) lianpSlJlCeliUe npSlMO nponOpijUOliaJl.bliO omliOCUmeJl.bliOMY YoJl.Ulie-
HUIO (3aKOH rYKa):
!!il
cr = EE = E -,
lo
r.n;e E - MO)J;YJlb ynpyrocTH (MOYJlb IOHra n.H CTepmHeH)
!!il l -lo
EcnH - = 1, T.e. - = 1, TO 210 = l, oTKya E = cr.
10 lo
MOJIL IOHI'a - <l>H3HqeCKa.H BenHqHHa, xapaKTepH3YIOa.H ynpyrHe CBOH-
CTBa MaTepHaJla. qHCneHHO p8BH8S1 HanpHmeHHIO, B03HHK81Om;eMY B 06pa3ue
npH OTHOCHTeJlbHOM y,AnHHeHHH. paBHOM 1 (T.e. Kor.a.a ero lIJlHHa y.a.BOHTC.H).
F
TaK KaK cr = - , TO 3aKOH rYKa MomHO 3anHcaTb HHaqe:
S
F!!il ES ES
- = E -  F = - !!il = k . !!il, re k = - - meCTKOCTb 06pa3ua.
S lo lo 10
191

7.27. MeXaHH'IeCKHe CBoHcTBa TBep.IJ;bIX TeJI.
.l(HarpaMMa paCTHjKeHHH
He06xo.zUiMbIe CBeeHH.H 0 MeXaHHqeCKHX CBOHCTBax pa3J1HqHbIX MaTepHa-
JlOB nOJlyqalOT 3KcnepHMeHTaJIbHO.
J],JI.H HCCJle,AOBaHH.H ,Ae<l>opMaIJ;HH pac-
T.HmeHH.H CTep.>KeHh H3 liccJleyeMoro
E
MaTepHaJIa npH nOMOI.I.1H CneIJ;HaJlbHbIX
YCTPOHCTB nOBepralOT paCT.HmeHHIO
(pHC. 7.38), li3Mep.HIOT yJlHHeHHe 06pa3-
IJ;a 11 B03HHKalOee B HeM Hanp.HmeHHe.
Ha OCHOBaHHH )J;aHHbIX onbITa CTpOHT
rpa<l>HI< 3aBHCHMOCTH Hanp.HmeHH.H OT OT-
HOCHTenbHoro yJlHHeHHH, Ha3bIBaeMbIH
JI.HarpaMMOH paCTHJKeHHH. LIHarpaMMa pa-
E CT.HmeHH.H H306pameHa Ha pHCYHKe 7.39.
IIpH He60nblIIHX HanpHmeHHHX (JIHHeH-
HhIH yqaCTOK OA) OTHOCHTeJIbHOe yJIHHe-
HHe np.HMO npOnOpIJ;HOHaJIbHO HanpH.>KeHHIO, T .e. BbInOJlH.HeTC.H 3aKOH rYKa. IIpH
aJlbHeHlIIeM YBeJlHqeHHH HanpHmeHH.H e<l>opMaIJ;H.H ee OCTaeTC.H ynpyroH, HO
ee 3aBHCHMOCTb OT HanpHmeHHH nepeCTaeT 6bITb JlHHeHHOH (yqaCTOK AB).
HaH60JIblIIee HanpHmeHHe an' npH KOTOpOM CnpaBe,AnHB 3aKOH rYKa, Ha-
3MBalOT npenenOM nponopUHOHanLHOCTH.
HaH60JIblIIee Hanp.HmeHHe cry, npH KOTOpOM ,Ae<l>opMaIJ;H.H ee OCTaeTCSI yn-
pyroH, Ha3bIBaeTC.H npeJI.enOM ynpyroCTH.
I1pH Hanp.HmeHH.HX, npeBbIlIIalOHX npe,AeJl ynpyrocTH, 06pa3eu; nocJle CHSI-
TH.H HarpY3KH He BOCCTaHaBnHBaeT CBOIO <l>OPMY (nYHKTHpHa.H KpHBaSl). 3TO nJIa-
CTHqeCKHe e<l>opMaU;HH. MepOH OCTaTOqHOH e<l>opMaU;HH CJIymHT OTpe30K £OCT'
B 06J1aCTH nJlaCTHqeCKOH )J;e<l>opMaIJ;HH (yqaCTOK CD) e<l>opMaIJ;H.H npOHCXO-
,AHT nO'ITH 6e3 YBeJlHqeHH.H Hanp.HmeHHSI. 3TO HBJIeHHe Ha3bIBaeTC.H TeKyqec-
TLIO. Hanp.HmeHHe crT' npH KOTOpOM MaTepHaJl TeqeT. Ha3bIBalOT npeJI.enOM Te-
KyqecTH. MaTepHaJlM, y KOTOpbIX 06naCTb TeKyqecTH CD 3HaQHTeJIbHa, MorYT
6e3 pa3pYlIIeHH.H BbIepmHBaTb 60JlblIIHe e<l>opMaIJ;HH. EcnH me 06J1aCTb TeKY-
QeCTH MaTepHaJla nOqTH OTCYTCTByeT, OH 6e3 pa3pYlIIeHH.H CMomeT BbIepmH-
BaTb JIHlIIb He60JlblIIHe e<l>opMaIJ;HH. MaTepHaJl, KOTOpbIH pa3pYlIIaeTC.H npH He-
60JlblIIHX e<l>opMaIJ;H.HX, Ha3bIBalOT XpynKHM (cTeKJlO, <l>ap<l>op, qyrYH, 6eToH).
J],JISI AaJIbHeHlIIero YBeJIHqeHH.H Ae<l>opMaIJ;HH He06xoHMO CHOBa YBeJIHqHTb
HanpJimeHHe (YQaCTOK DE). I10cJle TOqKH E KpHBa.H HAeT BHH3 H AaJlbHeH-
lIIaSi ,ne<l>opMaUH.H BnJlOTb .no pa3pbIB8 npOHcxo.nHT npH Bce MeHblIIHX HanpSl-
meHH.HX. HaH60JlblIIee Hanp.HmeHHe cr np ' KOTopoe cnoc06eH BbIAepmaTb 06pa3eu;
6e3 pa3pYlIIeHHH, Ha3bIBalOT npep;enOM npO'!HOCTH
Cnoc06HoCTb MaTepHaJIa conpOTHBJI.HTbCSI pa3pYlIIeHHIO H OCTaTOqHOH Ae<l>op-
MaIJ;HH, B03HHKalOHM B pe3YJIbTaTe BHelIIHHX B03AeHcTBHH, Ha3bIBalOT npo'!-
HOCTLIO MaTepHaJla.
3HaHHe BeJlHqHH crY' crT' cr np MaTepHaJlOB BamHO AJISI HCnOJIb30BaHHSI HX B
npOMbIIIIJIeHHOCTH H CTpOHTenbCTBe.
0'
O'op
&
O'Y
D
EOC'I'
Puc. 7.38
Puc. 7.39
192

TIpH KOHCTPYHPOB8HHH M81IIHH H coopymeHHH HeJlb3SI paCC1.J:HTbIB8Tb Ha HX
pa60TY B YCJlOBH.HX, KOr.n;a HCnbITbIBaeMbIe HMH Hanp.HmeHH.H 6JIH3KH K npe.n;eny
np01.J:HOCTH MaTepHaJIa, H3 KOTOpOrO OHH C.n;eJlRHbI, TaK KaK 3TO BCer.n;a 6y.n;eT
C03.n;aBaTb OnaCHOCTb pR3PYlIIeHHH. Ha np8KTHKe .n;OnYCTHMOe Hanp.HmeHHe BbI6H-
p810T TaK, qT06bI OHO COCT8BJI.HJlO JIHlIIb HeKOTOpylO 1.J:aCTb npe.n;eJl8 npOqHOCTH.
1JHCJlO, nOKa3bIBalOee, BO CKOJIbKO pa3 npe.n;eJl npOqHOCTH O'np npeBOCXO-
,AHT ,AOnYCTHMOe (c}>RKTHqeCKH ,AeHCTBYIOee' H8npSl:>KeHHe, H83bIB81OT aanaCOM
npO'lHOCTH (K03<}J<}JHQ,HeHTOM 6e30naCHOCTH) n:
a np
n=-.
a A
AJI.H nOBbIlIIeHH.H npOqHOCTH M8TepHaJIOB HX nO.n;BepralOT TepMH 1 -IeCKOH 06-
p860TKe H 06pa60TKe .n;aBJIeHHeM HJlH BBO.n;.HT B HHX ynp01.J:H.HIOHe npHMeCH
(nerHpoBaHHe) .
BOllPOCM AJIH CAMOKOHTPOJIH
1. q eM OmJLUl£a10mCSl cocmOSlliUSl MOJl.eKYJl. lia n06epXliocmu U 8liympu
:HCuiJKocmu?
2. KaK06bl oco(jennocmu nOBepxliocmliozo C./WSl JlCuiJKocmu2 KaKoBa moJl.-
U{ulia amozo CJl.OSl?
3 llolleMY C60(JoiJliaSl nOBepxliocmb JlCuiJKocmu cOKpaU{aemcSl? KaK U3Me
1UlemcSl n06epXliOCmliaJl aliepzuSl npu COKpaU{eIiUU nJl.oU{aiJu nOBepxliocmu2
4. KaK 06'OSlcliumb 603liUKli06eliue CUJl n06epXliocmliOZO liamSlJICeliUSl? KaK
nanpa6Jl.elia CUJl.a n06epXliocmliOZO liafflSlJlCeliuSl?
5. Om l£ezo 3a6ucum KoarjJrjJuquelim nOBepxliocmliozo namSlJICeliUSl2 £.[ eM
OmJl.UllalOmCSl CUJl.bl nOBepXliocmliOZO liamSlJICeliUJl om CUJl ynpyzocmu?
6. KaK06a ljJopMa KaneJl.eK 60iJbl. U3 Komopblx cocmoum mYMali?
7. II oileMY UCKpU6Jl.SlemCSl C60(JoiJliaSl n06epxliocmb :HCuiJKocmu y cmeliOK
cocyiJa? KaKYIO rjJOPMY olia UMeem 6 KanUJl.Jl.Rpe?
8. II OlleMY CnJl.IOU{U6alOmCJl (JOJl.bWUe KanJl.U :HCuiJKocmu. nonaiJalOU{ue lia
onopy?
9. KOJl.6a U3 CmeKJl.a 3anOJl.lielia lianOJl.06uliY pmymblO U lianOJl.06UIiY 60-
iJOU. KaKYIO rjJOPMY npuMym JlCUOKOCmu 6 lie6ecoMocmu?
10. II OlleMY 603liUKaem u36blmOillioe iJa6Jl.eliUe 6liympu crjJepullecKou KanJl.u2
11. Om "tezo .:Ja6ucum 6btCOma noiJ-oeMa JlCuiJKocmu 6 KanUJl.Jl.Slpax?
12. B KanUJl.Jl.SlpIiOU CmeKJLRnnOU mpy6Ke. Jl.eJICaU{eu ZopU30limaJLbli0, Haxo-
iJumcSl CmOJl.(JUK 60iJbl. KaK (JyiJem eecmu ce(JSl amom CmOJl.(JuK. ecJl.U OOUIi
KOlieq KanUJl.Jl.Slpa nooozpemb?
13. KaKUMU CBoucmBaMU m6epiJblx meJl. onpeOeJl.SlemCSl ux wupOKoe uc-
nOJl.b306aliue 6 lleJl.06elleCKOU npaKmuKe?
14. q eM O(J-OSlcliSlemcSl cyU{ecm606aliue npa6UJl.bliOU 6lieWlieU rjJOpMbl KpUC-
maJl.Jl.06?
15. q eM omJl.UllaemCSl KpUCmaJl.Jl.UlleCKOe meJl.O om aMoprjJlioZO?
16. q eM MOIiOKpUCmaJl.Jl.bl OmJl.UllalOmCSl om nOJl.UKpUCmaJl.Jl.OB?
17. II oileMY MOlioKpUCmaJl.Jl.bl aliu30mponlibl?
18. II OlleMY npu KpUCmaJl.Jl.U3aquu meJl. 6liympeliliSlSl aliepzUSl YMelibwaemcSl?
19. B lleM omJl.Ullue ynpyzou oerjJopMaquu om nJl.aCmUllecKou?
20. Om Ilezo 3a6ucum ynpyzoe lianpSlJlCeliUe 6 oerjJopMup06aliliOM meJl.e?
193

21. lfmo xapaKmepu3yem MOfJYJl.b ynpyzocmu?
22. q mo lia3b18aemCSl npefJeJWJH ynpyzocmu, npefJeJWM npoilliocmu Mamepua.n.a?
23. II OlleMY npeiJeJl. npOllliocmu peaJl.blibtX meJl. 3liallUmeJl.bli0 HUJlCe ezo me-
opemUlleCKOZO 3lialleliUJl?
F
a=-
l
A=atJ.S
2a
p = -cos8
r
h = 2acos8
pgr
F
cr =-
S
fj.l
cr = E-
lo
fj.l
E=-
to
cr np
n=-
cr,lt
OCHOBHblE <1>OPMY JIbl
K03cpcpHnHeHT nOBepXHOCTHoro HaT .HmeHHH
MO,AYJlb CHJlbI nOBepXHOCTHoro HaT.HmeHHH,
rpaHHnbI nOBepXHOCTH mH,AKOCTH.
- pa60Ta CHJI nOBepxHocTHoro HaT.HmeHH.H npli H30Tep-
MHqeCKOM cOKpaeHHH nOBepxHocTH >KH,lJ;KOCTH Ha fj.S.
r,lJ;e F -
- ,lJ;JlHHa
- H36bITO'tIHOe ,lJ;a.BJIeHHe mKOCTH B KallHJIJI.Hpe (JlB.II.1IaCOBO
,lJ;a.BJleHHe), r,Ae r - pa.n:HYc Kall1:lJlJl.Hpa, e - KpaeBOH yroJI.
- BbICOTa nO,A'beMa cToJl6a :>KH,lJ;KOCTIi B KanHJlJI.Hpe, r,Ae
p - nJlOTHOCTb mH,lJ;KOCTH, g - YCKopeHHe CB060,lJ;HOrO
na,AeHH.H, e - KpaeBOH yroJI.
- MeXaHH'leCKOe Hanp.HmeHHe, r,Ae F - MO,lJ;YJlb CHJIbI
ynpyrocTH, S - nJIOa,lJ;b Ce'tIeHH.H 06pa3na.
- 3aKOH rYKa, r,lJ;e E - MO,lJ;YJlb IOHra, fj.l - a6co-
JlIOTHOe Y,lJ;JlHHeHHe, lo - ,lJ;JIHHa 06pa3na B He,lJ;ecpopMH-
pOBaHHoM COCTOHHHH.
- OTHOCHTeJlbHOe y,AJlHHeHHe.
- 3allaC npo'tIHOCTH (K03qxpHUHeHT 6e30nacHOCTH). rne O'np
npe,lJ;eJl npO'tIHOcTH. O'A - ,lJ;OnycTHMoe Hanp.HmeHHe.
IIPHMEPbI PEIlIEHHH 3 A.lJ: A q
1. DpH nOCTOSlHHOll TeMnep8Type 10 Kanenb PTYTH p8,AHYCOM 0,10 MM
K8m,A8.R cnHsalOTCSI B OAHY. KaKoe KOnH'leCTSO TennOTbI npH 3TOM BbIAenSleTcSI?
H
,l(aHO: n=10, r=1,0.10- 4 M. a=0,47
M
H8HTH: Q.
P e ill e H H e. TIpH cnliSlHli1i HeCKonbKHX K8nenb B O,AHY 1i3MeHSleTCSI nno-
ll\8,Ab nosepXHOCTH, 8 Cne,AOBaTenbHO. H nOBepXHOCTH8S1 3HeprHSI. I1CKOMoe KO-
nH'leCTSO TennOTbI 6Y,AeT paBHO H3MeHeHHIO nOBepXHOCTHOH 3HeprHH:
Q=-fj. W=-(W 2 - Wd=W 1 - W 2 .
TIosepxHOCTH8S1 3HeprHSI K8nenb ,AO cnHSlHHSI
WI = naSI = na . 41tr 2 .
TIocne cnHHHHSI Kanenb W2=aS2=a. 41tR 2 , r,Ae R - p8,AHYC 60nbIIJOH K8n-
JIH PTYTH. Y'lHTbIB8S1, 'lTO CYMM8PH8S1 M8CC8 K8nenb PTYTH He H3MeHSleTCSI,
np!nr 3 = p!nR 3  R = r.
3 3
194

Cne,lJ,OB8TenbHO, Q = 41ta( nr 2 - R2) = 41tar 2 (n - );
Q = 3,1 . 10- 7 )J;m.
2. K8KYIO cHny H8,A0 npHnomHTb ,AnSI OTpbIB8 8nlOMHHHeBoro KOnb1l.8 M8C-
COH 5,0 r H ,AH8MeTpOM 8,0 CM OT nOBepXHOCTH rnHlI.epHH8? rnHlI.epHH non-
HOCTblO CM8'lHBaeT KOnb[(O.
.n8HO: m=5,0.10- 3 Kr,
H
d = 0,080 M. a = 0,060
M
H8HTH: F.
Pewe H He. )J;nSl OTpbIB8 KOnb1l.8 HymHo npHnomHTb cHny, yp8BHOBeWH-
B81OYIO CHJIY T.HmeCTH KOnb1l.8 H CHJIY nOBepXHOCTHoro HaTSlmeHHH.
F=FT+F H .
Y'lHTbIB8S1, 'lTO KOnbll.O K8caeTCSI no BHYTpeHHeMY H BHeWHeMY KP81O, ,lJ,nSl
CHnbI nOBepXHOCTHoro H8TSlmeHHSI HMeeM
F Jf = al = a . 2 . 21tr = 2a1td.
Cne,lJ,OB8TenbHO, HCKOM8S1 CHn8 F = mg + 2a1td; F = 80 MH.
3. C nOMOblO Cnell.H8nbHOH Tpy6KH BbI,AYB8IOT ,ABa
MbInbHbIX nY3bIpSl A H B (pHC. 7.40) p83HbIX p83Me.
pOB. K8K 6Y,AYT H3MeHSlTbCH HX p83MepbI, ecnH 38KpbITb
BMXO,lJ,HOe oTBepcTHe Tpy6KH?
Pew e H H e. )J;8BneHHe, 06ycnoBneHHoe KpHBH3HO
2a
nOBepxHocTH (n8nn8coBo ,A8BneHHe), p8BHO p = - H 38-
R
BHCHT OT p8.AHyca KpHBH3H. Cne,AOB8TenbHO. ,lJ,8BneHHe B
nY3bIpe A 6Y,AeT 60nbwe, qeM B nY3bIpe B. B pe3ynbT8Te
B03,AYX 6Y,AeT BbITeCHHTbCSI H3 nY3bIpSl A B nY3bIpb B,
6Y,AeT YMeHbW8TbCSI B 06'beMe, 8 nY3blpb B YBenH'lHB8TbCSI.
c
Puc. 7.40
n03TOMY nY3bIpb A
4. )J;B8 KanHnnSlpa C ,lJ,H8MeTpoM oTBepcTHH 0,25 MM H 0.50 MM 06p83YlOT
U-06P83HbIH COCY,A. CocY,A 38nonHSlIOT 'leTbIpexxnopHCTbIM yrnepO,AOM. K8KOB8 p83-
HOCTb YPoBHeH mH,AKOCTH B C00681OHXCSI COCY,A8X? B 60nee Y3KHH K8nHnnSlp
BBO,AHT cTon6 BO,AbI BbICOTOH h, nocne 'lero ypOBHH nepBoH mH,AKOCTH B 060HX
KanHnnSlp8X YCT8H8BnHBalOTCSI H8 0,AHH8KOBOR BbICOTe. H8H,lJ,HTe BblCOTY h. C'lH-
T8Tb, 'lTO mH,AKOCTH nonHOCTblO CM8'lHB81OT CTeHKH K8nHnnSlpOB. TIpHHSlTb K03<1>-
H
Q>HlI.HeHT nOBepXHOCTHoro H8TSlmeHHSI H8 rp8HHlI.e p83,Aen8 mH,AKOCTeH a = 0,047 -.
M
J1 8 H 0: d 1 = 2,5. 10- 4 M, d 2 = 5,0. 10- 4 M. PI = 1.63 . 10 3
Kr
3 .
M
P2 = 1 ,0 . 10 3
Kr
3' al =0,027
M
H
a2= 0,072
H
a= 0.047
H
M
M
M
H8HTH: hit h.
Pew e H H e. no 38KOHY C00681OHXCSI COCY,AOB ,A8BneHHe B TO'lK8X 1 H 2
4a 4a
(pHC. 7.41, a) 0,AHH8KOBO. Cne,AOB8TenbHo, Po +Plg -d; = Po -d;' r,Ae Po -
8TMocQ>epHoe ,A8BneHHe.
195

a
OTCIO,A8
5. K Me,AHOH
rpY3 MaccoH 100
Puc. 7.42
TOr,A8
A = 1,3 .D:m.
6
OTCIO,A8
 _ 4a (   )  = 1,3 CM.
- gPl d 1 - d 2 ;
PaccMoTpHM ropH30HT8nbHbIH ypo-
seHb p8311ena BOllbI H 'leTbIpexxnopHC-
Toro yrnepO,A8 (pHC. 7.41, 6). TIo 38-
KOHY C00681OHXCfl COCY,AOB
Po - 4U2 + P2gh - 4a = Po _ 4a 1 .
llt llt 
Puc. 7.41
h =  ( a2 + a - a l J ; h = 17 CM.
P2g  d 2
npOBonOKe ,AnHHOH 1,0 M H pa,AHYCOM 1,0 MM
Kr. qeMY paBH8 p860T8 p8CTSlmeHHSI npoBonoKH?
,I:(8HO: lo=1.0 M, r=1.0.10- 3 M, m=100 Kr.
H8HTH: A.
P e ill e H H e. HCKoMaSl pa60T8
nO,ABeCHnH
TeHHanbHOH 3HeprHH npOBonOKH
6Y,AeT p8BH8 H3MeHeHHIO no-
kx 2
A = -A ynp = . Wn = 2 H8
III
rpY3 ,AeHCTSYIOT CHna TSlmeCTH H CHn8 H8TSlmeHHSI co CTOpO-
HbI npOBonOKH (pHC. 7.42). YcnoBHe paBHOBeCHfl F ynp = mg. TIo
mg
38KOHY rYK8 F ynp = kx. Cne,AOBaTenbHo, kx = mg:=) x =-.
k
"\IJ" eCTKOCTb k = ES = Enr 2
 npoBonoKH  
mglo
X=- H
nr 2 E
Enr2(mg)2 _ m 2 g 2 1o.
A = 2lo (nr2Et - 2nr 2 E '
6. IIpe,Aen npOqHOCTH Ha cmaTHe ,AnSI KHpnH'la 15 MTIa, ,AnSI 6eToHa
60 MTIa. IIOACtIHTaHTe, KaKoH npe,AenbHOH BbICOTbI 3AaHHe MomHO nOCTpOHTb
H3 KHpnH'la H H3 6eToHa npH 38n8ce npO'lHOCTH, paBHOM 6.
,I:(8HO: 0'Inp=1,5.10 7 ITa, 0'2np=6,O'10' ITa, PI=1.8.10 3
Kr
3'
M
3 Kr
P2=2,2'10 3' n=6.
M
H8HTH: hI' h 2 .
P e ill e H He. 1l0,A ,AeHCTBHeM c06cTBeHHoro seca 3,AaHHe 6Y,AeT ,Ae<l>opMHpo-
mg
B8TbCSI, H B HeM 6Y,AeT B03HHKaTb ynpyroe H8npSlmeHHe 0' = -.
S
QTo6bl 3,AaHHe 6bIno H8AemHbIM, 3TO H8npSlmeHHe He ,AOnmHO npeBbIIIIaTb
mg
,AOnYCTHMoro HanpSlmeHHSI O'n' T.e. S = O'n'
196

Macc8 3)J.8HHSl m = phS. TOr,A8 all. = pgh. r)J.e h - BbICOTa 3,AaHHSI. TIo onpe,Ae-
a np a np a np a np
neHHIO 38n8C npO'-IHOCTM n = - => a = -. Cne,AOB8TenbHo, - = ghp => h = - .
all. Ann npg
TIoAcT8BnSl.H '-IHCneHHbIe AaHHbIe. nonY'lHM .nnSl KHpnH'l8 hi = 1.4 . 10 2 M. .nnSl
6eToH8 h 2 = 4.6 . 10 2 M.
7.28. HcnapeHlle II KOHlleHcaUIIH
TIepexo)J. BeeCTBa B ra3006pa3Hoe COCTO.HHHe Ha3bIBaeTC.H napoo6pa30Ba-
HHeM.
COBoKynHocTb MOJleKYJI, BbIJleTeBWHX H3 BeeCTBa, Ha3bIBaeTC.H napOM 3TO-
ro BeeCTBa.
TIpH napo06pa30BaHHIi YBeJlH'-IHBaIOTC.H Cpe,AHHe paCCTO.HHH.H Me:>K)J.Y MOJle-
KynaMH. B pe3ynbTaTe nOTeHHanbHa.H 3HeprH.H B3aHMo)J.ellcTBH.H '-IaCTH YBe-
nH'-IHBaeTC.H (qHCJleHHoe 3Ha'IeHHe ee YMeHbWaeTC.H, HO OHa OTpHaTenbHa). Ta-
KHM 06pa30M, npoecc napo06pa30BaHH.H CBH3aH C YBenH'-IeHHeM BHYTpeHHell
3HeprHH BeecTBa.
TIapo06pa30BaHHe MO:>KeT npOHcxo.n.HTb Henocpe.n.CTBeHHO H3 TBep.n.oro COCTO-
.HHH.H - 3TO B03rOHKa (HnH cy6J1HMaUHH).
TIepexo)J. H3 mH)J.Koro COCTO.HHH.H B ra3006pa3Hoe B03MO:>KeH )J.BYMH pa3nH'I-
HbIMH npoeccaMH: HcnapeHMeM H KHneHHeM
HcnapeHHe -- 3TO napoo6pa30BaHHe, npOHCXO)J..Hee TonbKO co CB060)J.HOll
nOBepXHOCTH mH)J.KOCTH, rpaHH'Iaell C ra3006pa3HOll cpe)J.Oll HnH C BaKYYMOM.
8KcnepHMeHTanbHO YCTaHOBneHbI cne)J.YIOHe 3aKOHOMepHOCTH.
TIpH O,AHHaKOBbIx ycnoBHHx pa3nHqHbIe BeeCTBa HCnap.HIOTC.H C pa3nH'I-
HOll CKOpOCTblO (CKOpOCTb HcnapeHHH onpe)J.eJl.HeTCH 'lHCnOM MoneKYJI, nepexo-
)J..HHX B nap C nOBepXHOCTlf BeeCTBa 3a 1 c).
CKOpOCTb HcnapeHH.H TeM 60Jlbwe:
qeM 60nbwe nJlOa)J.b cB06o)J.HOll nOBepXHOCTH mH,AKOCTH;
'IeM MeHbwe nnOTHOCTb napOB Ha)J. nOBepXHOCTblO mH)J.KOCTH. CKOpOCTb YBe-
nH'lHBaeTC.H npH )J.BH:>KeHHH OKpY:>KalOerO B03)J.yxa (BeTep);
'-IeM 60JlbWe TeMnepaTypa mH)J.KOCTH.
TIpH HcnapeHHH TeMnepaTypa Tena nOHHmaeTCH.
MexaHH3M HcnapeHH.H MO:>KHO 06'b.HCHHTb C TO'IKH 3peHH.H MKT: MoneKynbI,
HaXO)J..HHeC.H Ha nOBepXHOCTH, y.n;ep:>KHBaIOTCH CHnaMH npHT.HmeHH.H co CTOpO-
HbI )J.pyrHx MoneKYJl BeeCTBa. MOJleKyna MomeT BbIneTeTb 3a npe,AenbI mH)J.-
KOCTIi JlHWb Tor.n;a, Kor.n;a ee KHHeTH'IeCKa.H 3HeprH.H npeBbIwaeT 3Ha'IeHHe TOll
pa60TbI, KOTOPYIO He06xo)J.HMO COBepWHTb, qT06bI npeO)J.oneTb CHJlbI MOJleKyn.Hp-
Horo npHT.H:>KeHHH (pa60Ta BbIxo.n;a). TI03TOMY nOKHHYTb BeeCTBO MorYT TOJlb-
KO 6bICTpbIe MoneKynbI. B pe3ynbTaTe cpe)J.H.H.H KHHeTH'IeCKa.H 3HeprH.H OCTaB-
IlIHXC.H MoneKyn YMeHbIlIaeTC.H, a TeMnepaTypa mH.n;KOCTH nOHH:>KaeTC.H.
,Z:(JI.H nO)J.)J.epmaHHSl TeMnepaTypbI HCnap.HlOellC.H mH.n;KOCTH HeH3MeHHOll K
Hell He06xo,AHMO nO)J.BO,AHTb HeKOTopoe KOnHqeCTBO TennOTM.
KOnHqeCTBO TennOTbl Q, He06xo)J.HMoe .n;nH npeBpaeHH.H mH)J.KOCTH B nap
npH nOCTOHHHOH TeMnepaType, Ha3bIBaeTCH TenJlOTOH napoo6pa30BaHHH.
8KcnepHMeHTaJlbHO YCTaHOBJleHO. qTO Q = Lm, r,Ae m -- Macca HcnapHB'
IlIeHCH :>KH.n;KOCTH, L -- y)J.enbHa.H TennOTa napo06pa30BaHH.H.
197

L, m
Kr
2,5 . 10 8
Y AeJlLllaH TenJlO ra nap006pa30Ba-
HH.H - BeJlH''lHHa, qHCJleHHO paBHall
KOnHqeCTBY TenJlOTbI, He06xoHMOMY
,AJllI npeBpaeHH.H B nap )KHKOCTH
eHHHqHOH MaCCbI npH HeH3MeHHOH
TeMnepaType.
YeJlbHa.H TennOTB napo06pa30Ba-
HH.H L 3aBHCHT OT poa mHKOCTH
H BHelliHHX YCJlOBHH. IIpH YBeJlHqe-
HHH TeMnepaTypbI OHa YMeHblliaeTClI
t.oC (pHC. 7.43). 8TO 06'b.HCH.HeTC.H TeM,
't{TO Bce mHKOCTH npH HarpeBaHHH
paClliHpllIOTC.H. PaCCTO.HHH.H Memy MO-
JleKYJlaMH npH 3TOM YBenHqHBaIOTC.H
H CHJlbI MOJleKYJl.HpHOrO B3aHMoeHCTBH.H YMeHblliaIOTC.H. KpoMe Toro, qeM 60Jlb-
we TeMnepaTypa. TeM 60Jlbllie Cpe,AH.H.H KHHeTHqeCKall 3HeprHlI ,ABHmeHH.H MO-
neKYJl H TeM MeHbIlIe 3HeprHH HM HymHo 06aBHTb, 't{T06bI OHH MornH BbIJle-
TeTb 3a npeenbI nOBepXHOCTH mHKOCTH.
MOJleKYJlbI napa xaOTH't{eCKH BHmYTC.H. ll03TOMY CKOpOCTH HeKOTopbIX H3
HHX 6YYT HanpaBJleHbJ B CTOPOHY mHKOCTH. .D:ocTHrHYB nOBepXHOCTH, OHH
BT.HrHBaIOTClI B Hee CHnaMH npHT.HmeHHlI co CTOpOHbI MoneKYJI, HaXO.HHXC.H
Ha nOBepXHOCTH mHKOCTH, H CHOBa CTaHOB.HTC.H MOJleKYJlaMH mHKOCTH. IIpo-
necc nepeXO,A8 BeI..QeCTBa H3 ra3006pa3Horo COCTOllHHlI B mHKoe Ha3bIBaeTClI
KOH,ZJ:eIlCaltHeH.
'tJHCJlO B03BpaTHBIlIHXC.H B mHKOCTb 38 OnpelIeJleHHbIH npOMemYTOI< BpeMe-
HH MOJleKYJI TeM 60JlbIlIe, qeM 60nbille KOHneHTpanHlI MoneKYJI napa, 8 cne-
,AOBaTeJlbHO, qeM 60Jlbille aBJleHHe napa Ha mH,AKOCTblO. KOHeHCanH.H napa
conpOBomaeTC.H HarpeBaHHeM mHKOCTH. IIpH KOH,AeHCanHH BbIen.HeTClI TaKoe
me KOJlHqeCTBO TenJlOTbI, KOTopoe 6bIJlO 3aTpaqeHO npH IIcnapeHHH.
0,5 . 10 8
o
194
374
Puc. 7.43
7.29. HacLIeHHLI H HeHaCLIeHHLle napLI
H HX CBOHCTBa
Ha cB060HOH nOBepXHOCTblO mH)J;KOCTH Bcera HMeIOTC.H napbI 3TOH mH-
KOCTH. ECJlH COcy C mHKOCTblO He 3aKpbIT, TO KOHneHTpa:qHlI qaCTH:q napa
npH nOCTO.HHHOH TeMnepaType MOme'l' H3MeHlITbC.H B lliHpOKHX npe)J;eJlax B CTO-
POHY YMeHbllieHH.H H B CTOPOHY YBeJlHqeHHlI.
IIpoltecc HcnapeHH.H B 3aMKuyToe npOL'TpaHCTBO (3aKpbITbIH COcy.l1 C mH.l1KO-
CTblO) MO:iKeT npH ,lIaHHOH TeMnepa-rype npOHCKOnHTL TOJILKO no onpeneJleHHOrO
npeAeJia. 8TO 06'b.HCH.HeTClI TeM, qTO OHOBpeMeHHO C HcnapeHHeM mHKOCTH npo-
HCXOHT KOH)J;eHcanHlI napa. CHaqaJI8 qHCJlO MOJleKyn, BbIJIeTaIOlI.\HX H3 mHKO-
CTH 3a 1 C, 60Jlbllie qHCJla MOJleKyn, B03BpaalOI..QHXC.H 06paTHo, H nJlOTHOCTb,
a 3HaqHT, H aBJleHHe napa paCTeT. 8TO npHBO,AHT K YBeJlHqeHHIO CKOpOCTH KOH-
eHcanHH. 'tJepe3 HeKOTopoe BpeM.H HacTynaeT HHaMHqeCKOe paBHOBeCHe, npH KO-
TOpOM nJlOTHOCTb napa Ha,zJ; mHKOCTblO CTaHOBHTC.H nOCTO.HHHOH. IIap, HaXO.H-
HHC.H B COCTO.HHHH HHaMHqeCKOrO paBHOBeCH.H co cBoeH mHKOCTblO, Ha3bIBa-
eTC.H HaCLIlI(eHHLIM napoM. IIap, KOTOpbIH He HaXOHTC.H B COCTO.HHHH AHHaMH-
qeCKOrO paBHOBeCHH co CBoeH jKHKOCTblO, Ha3bIBaeTC.H HeHaCLIlI(eHHL.
198

OnbIT nOKa3bIBaeT, qTO HeHaCbII.QeHHbIe napbI nOqHH.HIOTCjJ BCeM ra30BbIM
3aKOHaM (CM.  6.18), H TeM TOqHee, qeM aJlbWe OHH OT HaCbII.QeHHSl
,ZJ;JljJ HaCbIeHHbIX napOB xapaKTepHbl CJleYIOI.QHe cBoiicTBa:
1) nJlOTHOCTb H aBJleHHe HaCbII.QeHHOrO napa npH aHHoH TeMnepaType
3TO MaKCHMaJlbHble nJlOTHOCTb H aBJleHHe. KOTopbIe MomeT HMeTb nap npH
aHHoH TeMnepaType;
2) nJlOTHOCTb H aBJleHHe HaCbIeHHoro napa 3aBHC.HT OT pO,Aa BeI.QeCTBa.
qeM MeHbwe yeJlbHa.H TenJlOTa napo06pa30BaHHjJ )KHKOCTH, TeM 6bIcTpee OHa
HCnap.HeTC.H H TeM 60nbllIe aBJleHHe H nJlOTHOCTb ee napOB;
3) aBJleHHe H nJlOTHOCTb HaCbIeHHoro napa 0H03HaqHO onpe,AeJl.HIOTCjJ
ero TeMnepaTYPOH (He 3aBHC.HT OT Toro, KaKHM 06pa30M nap OCTHr 3TOH TeM-
nepaTYPbI: npH HarpeBaHHH HJlH npH oXJlameHHH);
4) ,AaBneHHe H nJlOTHOCTb napa 6bICTpO B03paCT81OT C YBenHqeHHeM TeMne-
paTypbI (pHC. 7.44, a, 6).
a p
6
1
1
1
1
1
1
1
1
1
1
1
1 "
1 "
I,'
P HaCbIIl\eHHbIH IIap
T
T
Puc. 7.44
OnbIT nOKa3bIBaeT, qTO npH HarpeBaHHH mH)J;KOCTH ypoBeHb mHKOCTH B
3aKpbIToM cocye nOHHmaeTC.H. CJleOBaTeJlbHO, Macca H nJlOTHOCTb napa B03-
pacTalOT. BOJlee CHJlbHOe YBeJlHqeHHe aBJleHH.H HaCbIIl1eHHOrO napa no cpaBHe-
HHlO C HeaJlbHbIM ra30M (3aKoH reH-JIlOccaKB He npHMeHHM K HaCbIeHHOMY
napy) 06'b.HCH.HeTC.H TeM, TO 3eCb npOHCXOHT pOCT aBJleHH.H He TOJlbKO 3a
CqeT pOCTa cpeHeH KHHeTHqeCKOH 3HeprHH MoneKYJI (KaK Y HeaJIbHOrO ra3a),
HO H 3a CqeT YBeJlHqeHH.H KOHeHTpan;HH MOJleKYJI;
5) npH nocTojJHHOH TeMnepaType aBJleHHe H nJlOTHOCTb HaCbIeHHoro napa
He 3aBHC.HT OT 06'beMa. Ha pHCYHKe 7.45 JI.H cpaBHeHH.H npHBeeHbI H30Tep-
MbI HeaJlbHOrO ra3a (a) H Ha-
CbII.QeHHOrO napa (6).
OnbIT nOKa3bIBaeT. qTO npH
H30TepMHQeCKOM pacwHpeHHH
YPoBeHb mHKOCTH B cocye no-
1iHmaeTC.H, npH cmaTHH - nOBbI-
WaeTC.H, T.e. H3MeH.HeTC.H qHCJlO
V MOJleKyn napa TaK, qTO nJlOT-
HOCTb napa OCTaeTC.H nOCTO.HHHOH
a P
6
P

V
Puc. 7.45
199

7.30. KHneHHe :iKHJJ;KocTeH. 3aBHcHMoCTL
TeMnepaTypLJ KHneHHH OT ,lJ;aBJIeHHH
KHneHHe - 3TO napo06pa30BaHHe, npOHCXO,ASIee o.n;HOBpeMeHHO a C nOBepx-
HOCTH, H no BceMY 06'beMY mH.n;KOCTH. OHO COCTOHT B TOM, qTO BCnJlbIBalOT H
nOnalOTC.H MHOrOqHCneHHbIe
nY3bIpbKH, BbI3bIBa.H xapaKTepHoe 6YPJleHHe.
KaK nOKa3bIBaeT onbIT, KHneHHe mH.n;KOCTH npH
3a.n;aHHOM BHeIllHeM .n;aBJleHHH HaqHHaeTCSI npH
BnOJlHe onpe.n;eJleHHOH H He H3MeH.HIOll\eHC.H B npo-
ecce KHneHH.H TeMnepaType H MomeT npOHCXO,AHTb
TOJlbKO npn no.n;Bo.n;e 3HeprHH H3BHe B pe3ynbTaTe
Tenno06MeHa (pHC. 7.46):
Q = Lm,
Q r,l1e L - yJtenbHaSi TennOTa napo06pa30BaHH.H npH
Puc. 7.46 TeMnepaType KHneHH.H.
MexaHH3M KHneHH.H: B >KH.n;KOCTH Bcer.n;a HMe-
eTC.H pacTBopeHHbIH ra3, CTeneHb pacTBOpeHH.H KOToporo nOHHmaeTC.H C pOCTOM
TeMnepaTypbI. KpoMe Toro, Ha CTeHKax cocy.n;a HMeeTCSI a.n;cop6HpoBaHHbIH ra3.
IIpH HarpeBaHHH >KH.n;KOCTH CHH3Y (pHC. 7.47) ra3 HaqHHaeT BbI,AeJlSlTbC.H B
BH,Ae nY3bIpbKOB Y CTeHOK cocy.n;a. B 3TH nY3bIpbKH npOHcxo.n;HT HcnapeHHe
mH,AKOCTH. II03TOMY B HHX, KpOMe B03.n;yxa, Haxo.n;HTCSI HaCbIeHHbIH nap,
.n;aBJleHHe KOToporo C pOCTOM TeMnepaTypbI 6bICTpO YBenHqHBaeTC.H, H nY3bIpb-
KH paCTYT B 06'beMe, a cne.n;OBaTeJlbHO, YBeJlHqHBaIOTC.H .n;eHCTBYIOHe Ha HHX
CHJlbI ApXHMe.n;a. Kor,Aa BbITaJlKHBalOll\aSl CHna CTaHeT 60JlbIlle CHJlbI T.HmeCTH
nY3bIpbKa, OH HaqHHaeT BCnJlbIBaTb. Ho nOKa mH.n;KOCTb He 6y.n;eT paBHOMepHO
nporpeTa. no Mepe BCnJlbITH.H 06'beM nY3bIpbKa YMeHbIllaeTC.H (.n;aBJleHHe HaCbI-
ll\eHHOrO napa YMeHbIllaeTC.H C nOHHmeHHeM TeMnepaTYPbI) H, He .n;OCTHrHYB CBO-
60.n;HOH nOBepXHOCTH. nY3bIpbKH HCqe3alOT (3aXJlOnbIBaIOTC.H) (pHC. 7.47. a), BOT
nOqeMY MbI CJlbImHM xapaKTepHbIH mYM nepe.n; 3aKHnaHHeM. Kor.n;a TeMnepaTY-
pa >KH.n;KOCTH BbIpaBH.HeTCSI, 06'beM nY3bIpbKa npH no.n;'beMe 6y.n;eT B03paCTaTb,
TaK KaK .n;aBJleHHe HaCbIeHHoro napa He a3MeH.HeTC.H, a BHeIllHee .n;aBJleHHe Ha
nY3bIpeK, npe,ACTaBJlSlIOee C060H CYMMY rH,ApOCTaTHqeCKOrO ,AaBneHH.H mH,l1KO-
CTH, Haxo.n;.Hll\eHC.H Ha.n; nY3bIpbKOM, H aTMoccpepHoro, YMeHbIllaeTC.H. IIY3bIpeK
.n;OCTHraeT CB060.n;HOH nOBepXHOCTH mH.n;KOCTH, JlOnaeTC.H, H HaCbIll\eHHbIH nap
BbIXO,AHT Hapymy (pHC. 7.47, 6) - mH.n;KOCTb 3aKHnaeT. .D:aBneHHe HaCbIll\eH-
Horo napa npH 3TOM B nY3bIpbKax npaKTHqeCKH paBHO BHeIllHeMY .n;aBneHHIO.
tt<
t
a
6
",
------ 0 -
! -------......,..-----
-------------
-------------- 0
---------- ------- --
T=const ----- -----
----==- n -----.
- --
----- ----
T
o
Puc. 7.47
200

TeMnepaTypa, npH KOTOPOH )J;aBJleHHe HaCbIeHHoro napa mH,AKOCTH paBHO
BHelliHeMY ,n;aBJleHHIO Ha ee CB060,n;HYIO nOBepXHOCTb, Ha3bIBaeTC.H TeMnepaTY-
poil KHneHHH mH,n;KOCTH.
TaK KaK ,n;aBJleHHe HaCbIeHHoro napa YBeJIHqHBaeTC.H C pOCTOM TeMnepa-
TypbI, a npR KHneHHH OHO ,n;OJlmHO 6bITb paBHO BHelliHeMY, TO npH YBenHqe-
HHH BHelliHero ,n;aBJleHH TeMnepaTypa KHneHH.H YBeJIH'tIHBaeTC.
TeMnepaTypa KHneHH 3aBHCHT TaKme OT HaJlH'lH npHMeceH. 06bIqHO YBe-
nH'tIHBa.HCb C pOCTOM KOHeHTpaHH npHMeceH.
ECJlH npe,n;BapHTeJIbHO OCB060,AHTb mH,AKOCTb OT paCTBopeHHoro B HeH ra3a,
TO ee MomHO neperpeTb, T.e. HarpeTb BbIllie TeMnepaTypbI KHneHH. 8TO Heyc-
TOH'tIHBOe COCTOHHe mH,n;KOCTH. ,L(ocTaTOqHO He60JlbllIHX COTp.HCeHHH H mH,AKOCTb
3aKHnaeT, a ee TeMnepaTypa cpa3Y nOHHmaeTC ,AO TeMnepaTypbI KHneHHH.
7.31. BJIa:IKHOCTb B03.nyxa. TO'lKa pOCLI. nCHXpoMeTp. rHrpoMeTp
B pe3YJlbTaTe Hcn8peHH BO,AbI C MHOrO'tlHCJIeHHbIX Bo,n;oeMOB (MopeH. 03ep,
peK H ..n;p.). a TaKme C paCTHTeJlbHbIX nOKpOBOB B aTMoccpepHoM B03,n;yxe Bce-
r..n;a COlIepmHTC.H BO,n;.HHOH nap. OT KOJlH'leCTBa BO,A.HHOrO napa. CO,AepmaerOC.H
B B03,Ayxe, 3aBHCHT norolta, CaMOqYBCTBHe qeJlOBeKa. c}>YHKIlHoHHpoBaHHe MHO-
rHX ero opraHOB, >KH3Hb pacTeHHH, a TaKme coxpaHHOCTb TeXHHqeCKHX 06'beK-
TOB, apXHTeKTypHbIX coopymeHHH, npOH3Be,AeHHH HCKYCCTB. IT03TOMY O'tleHb
BamHO CJle,AHTb 3a BJlamHOCTblO B03,Ayxa, YMeTb H3Mep.HTb ee.
BO,A.HHOH nap B B03,Ayxe 06bIqHO .HBJI.HeTC.H HeHaCbIeHHbIM. ITepeMeeHHe
B03,AYllIHbIX Macc, 06ycJloBJleHHoe B KOHeqHOM CqeTe H3J1Y'leHHeM COJlHa, npH-
BO,AHT K TOMY, qTO B O,AHHX MeCTax HallieH nJlaHeTbI B ,AaHHbIH MOMeHT I1C-
napeHHe BO..n;bI npe06J1a,n;aeT Ha,A KOH,AeHCaI(HeH, a B ,n;pyrHx, Ha060poT, npeo6-
JIa,AaeT KOH,AeHCaI(H.H.
B03)J;yx, cO,AepmaHH BO,A.HHbIe napbI, Ha3bIBalOT BJIajKHLIM. ,L(JI xapaKTe-
pHCTHKH CO,AepmaHH.H BO,A.HHOrO napa B B03,Ayxe BBO,A.HT p.H,A BeJlH'tIHH: a6co-
nlOTHYIO BJlamHOCTb, ynpyrocTb BO,A.HHOrO napa H OTHOCHTeJlbHYIO BJIamHOCTb.
A6COJIIOTHOH BJIajKHOCTLIO p B03,Ayxa Ha3bIBalOT BeJlH'tIHHY, 'tIHCJleHHO paB-
HYIO Macce BO,A.HHOrO napa. co,n;epmawerOC.H B 1 M 3 B03..n;yxa (T.e. nJlOTHOCTb
BO,A.HHOrO napa B B03,Ayxe npH )J;aHHbIX ycnOBH.Hx).
YnpyrocTL BO)J;.HHOrO napa p - 3TO napHaJlbHOe ,AaBJleHHe BO,A.HHOrO napa,
CO,AepmaerOC.H B B03,Ayxe.
B CM e,AHHHI(aMH a6COJllOTHOH BJlamHOCTH H ynpyrocTH BJI.HIOTC COOT-
BeTCTBeHHO KHJlOrpaMM Ha Ky6H'tIecKHH MeTp [ :; ) H naCKaJlb (ITa).
MHor,Aa I1cnOJlb3YIOTC.H BHeCHCTeMHble e,AHHHI(bJ rpaMM Ha Ky6H'tIecKHH MeTp
( :3 ) H MHJlJIHMeTp pTYTHoro cToJl6a (MM pT. CT.).
A6coJlIOTHa.H BJlamHOCTb H ynpyrocTb BO)J;Horo napa CB.H3aHbI Mem,Ay CO-
60H ypaBHeHHeM COCTO.HHH.H
pV =RT P =RT.
M M
ECJlH H3BeCTHa TOJIbKO a6COJlIOTHa.H BJIamHOCTb HJlH ynpyrocTb BO,A.HHOrO
napa, e:I.Qe HeJlb3.H CY,AHTb, HaCKOJlbKO cyx HJlH BJlameH B03,AYX. ,L(JI onpe,Ae-
201

JleHHSI CTeneHH BnamHOCTH B03.l(yxa He06xo.l(HMO 3HaTb, 6nH30K HnH .l(aneK
BO,A.HHOH nap OT HaCbII.qeHH.H.
OTHOCHTeJlbHOH BJlaHOCTLIO B03.l(yxa q> Ha3bIBalOT BblpameHHoe B npoeH-
Tax OTHOIlleHHe a6conlOTHOH BJla>KHOCTH K nJlOTHOCTH Po HaCbIUI;eHHoro napa
npH .l(aHHOH TeMnepaType (HnH OTHOIlleHHe ynpyrocTH BO.l(.HHOrO napa K aB-
neHHIO Po HaCbIeHHoro napa npH ,AaHH011 TeMnepaType):
q> = .E- . 100 % ; q> = L . 100 % .
Po Po
l.{eM MeHbilie OTHOCHTenbHa.H BnamHOCTb, TeM
.l(aJlbilie nap OT HaCbIUI;eHH.H, TeM HHTeHCHBHee
npOHCXO.l(HT HcnapeHHe. ,I:(aBneHHe HaCbIUI;eHHoro
napa Po npH 3a.l(aHHOH TeMnepaType -- BeJlHqHHa
Ta6nH'lHa.H. YnpyrocTb BO.l(.HHOrO napa (a 3HaqHT.
H a6conlOTHYlO Bna>KHOcTb) onpe.l(en.HIOT no TOqKe
t pOCbI.
IIYCTb npH TeMnepaType t 1 ynpyrocTb BO,A.HHO-
ro napa Pl' COCTOSlHHe napa Ha ,AHarpaMMe p, t
H306pa3HTC.R TOqKOH A (pHC. 7.48).
fIPH H306apHoM OXnam,AeHHH .l(0 TeMnepaTypbI t p
nap CTaHOBHTC.H HaCbIUI;eHHbIM Hero COCTO.HHHe H306-
pa3HTC.R TOqKOH B. TeMnepaTypy t p , npH KOTOpo11
BO.l(.HHOH nap CTaHOBHTC.H HacbIUI;eHHbIM, Ha3bIBalOT
TOOH pOCLI. fIpH OXJlam.l(eHHH HH>Ke TOqKH pOCbI
Ha'lHHaeTC.H KOH,AeHCaHSI n8pOB: nO.HBn.HeTC.H TYMaH,
BbIIIll..l(aeT poca. 3anOTeBaIDT OKHa. TOqKa poCbI no-
3BOn.HeT onpeenHTb ynpyrocTb BO.l(.HHOrO napa PI'
HaXO.l(.HUI;erOC.H B B03.l(yxe npH TeMnepaType t 1 .
,I:(eHcTBHTenbHO, H3 pHCYHKa 7.48 BH.l(HM, qTO
.l(aBneHHe PI paBHO .l(aBneHHIO HaCbIUI;eHHoro na-
pa npH TOqKe pOCbI PI = PO,p' Cne.l(OBaTeJlbHO,
q> = Po,p .100 %.
Po
TOqKY pOCbI onpe.l(eJl.HIOT C nOMOI.I..1blO rHrpo-
MeTpOB.
KOH.l(eHCaIJ.HOHHbIH rHrpOMeTp npe.l(CTaBn.HeT C060H MeTannHqeCKYIO Kop06KY A,
nepe,AH.H.H CTeHKa K KOTOpOH XOpOlliO OTnonHpOBaHa (pHC. 7.49) BHYTPb KO-
P06KH HanHBalOT JlerKO HCnap.HIOUI;YIOC.H >KH.l(KOCTb -- 3qmp -- H BCTaBJI.HIOT
TepMoMeTp. fIponycKa.H Qepe3 Kop06KY B03,nyX C nOMOUI;blO pe3HHoBoH rpYiliH r,
BbI3bIBalOT CHnbHoe HcnapeHHe 3cpHpa H 6bIcTpoe OXna>K.l(eHHe Kopo6KH. IIo
TepMoMeTpy 3aMeqalOT TeMnepaTYPY, npH KOTOpOH nO.HBn.HIOTC.H KaneJlbKH pOCbI
Ha nonHpoBaHHoH nOBepXHOCTH CTeHKH K. ,I:(aBneHHe B 06nacTH. npHneralO-
UI;eH K CTeHKe, MO>KHO CqHTaTb nOCTO.HHHbIM, TaK KaK 3Ta 06nacTb co06UI;a-
eTC.H C aTMoccpepoH H nOHH>KeHHe ,AaBneHH.H 3a CqeT OXnam.l(eHH.H KOMneHCH-
pyeTC.H YBenHqeHHeM KOHeHTpaHH napa. IIoSlBneHHe pOCbI YKa3b1BaeT, 'lTO
BO.l(.HHOH nap CTan HaCbIUI;eHHbIM. 3Ha.H TeMnepaTYPY B03.l(yxa H TOqKY pOCbI,
MO>KHO HaHTH napHaJlbHOe .l(aBJleHHe BO,A.HHOrO napa H OTHOCHTenbHYIO Bnam-
HOCTb.
P
Po
PI
o
I
I
I
t p
Puc. 7.48
r
Puc. 7.49
202
t l

OTHOCHTenbHYIO BJlamHOCTb onpe,neJl.HIOT TaKme C
nOMOblO nCHXpOMeTpa.
IIcHxpoMeTp COCTOHT H3 lIBYX TepMOMeTpOB. II1a-
pHK O,nHOrO H3 HHX 06MOTaH TKaHblO, HHmHHe KOH-
bI KOTOpOH onyeHbI B cocy,n C ,nHCTHJlJlHpOBaHHOH
BO,nOH (pHC. 7.50) CYXOH TepMOMeTp perHcTpHpyeT
TeMnepaTYPY B03,Ayxa, 8 BnamHbIH - TeMnepaTYPY
HCnapSlIOeHC.H BO,AbI. Ho npH HCnapeHHH .>KH,nKOCTH
ee TeMnepaTypa nOHH.>KaeTC.H. qeM cYIl1e B03,nyX
(MeHbIl1e ero OTHOCHTeJlbHa.H BJla.>KHOCTb), TeM HHTeH-
CHBHee HCnap.HeTC.H BO,A8 H3 BnamHOH TKaHH H TeM
HHme ee TeMnepaTypa. Cne,nOBaTeJlbHO, pa3HOCTb no-
Ka3aHHH cyxoro H BJlamHOrO TepMOMeTpOB (TaK Ha-
3bIBaeMa.R. nCHXpOMeTpHlIeCKa.H pa3HOCTb) 3aBHCHT OT
OTHOCHTenbHOH BJlamHOCTH B03,nyxa. 3Ha.H 3TY pa3-
HOC'l'b TeMnepaTYP, onpe,neJl.HIOT OTHOCHTenbHYIO BJlamHOCTb B03,nyxa no cneH-
a,,1:bHbIM nCHxpOMeTpHlIeCKHM Ta6nHu.aM.
60 50
40 40
30 30
20 20
10 10
o 0
60 60
40 40
30 30
20 20
10 10
o 0
o
o
o
Puc. 7.50
o
7 .32. HarpaMMa COCTO.HHH.H BeeCTBa
KaK yme YKa3bIBanOCb O,nHO H TO me BeI.qeCTBO MomeT Haxo,nHTbC.H B o,n-
HOM H3 Tpex arperaTHbIX COCTO.HHHH B 3aBHCHMOCTH OT BHeIl1HHX ycnoBHH -
TeMnepaTypbI H ,naBJleHH.R..
3aBHCHMOCTb arperaTHoro COCTO.HHH.H BeI.qeCTBa OT ,naBJleHH.H H TeMnepaTY-
pbI MomeT 6bITb Harn.H,AHO npe,nCTaBneHa ,nHarpaMMOH COCTO.HHH.H. ,n:JI.H ee no-
CTpOeHH.R., OTKJla,nbIBa.H no OCH op,nHHaT ,AaBneHHe p, a no OCH a6CHCC TeMne-
paTYPY T, OTMeqalOT TOqKaMH TaKHe COlIeTaHH.H ,naBJleHH.R. p H TeMnepaTypbI T,
npH KOTOpbIX Haxo,n.HTC.H B ,nHHaMHlIeCKOM paBHOBeCHH mH,nKOCTb - ra3,
raa - TBep,noe Teno, mH,nKOCTb - TBep,noe TeJlO.
KpHBbIe. 06pa3yeMbIe cOBoKynHocTblO 3THX TOlIeK (pHC. 7.51), Ha3bIBaIOTC.H
COOTBeTCTBeHHO: AK - KpHBa.H HcnapeHH.H (KpHBa.H 3aBHCHMOCTH TeMnepaTypbI
p KHneHH.H mHlIKOCTH OT .naBJle-
B HH.H), CA - KpHBa.H cy6nHMa-
· · LJ;HH (KpHBa.H KpIiCTaJlJlH3aHH H3
· · napa), AB - KpHBa.H nnaBJleHH.H
(K pHBa.H KpHCTanJlH3aHH).
KpHBa.H nnaBJleHHSI H,neT no-
qTH BepTHKaJlbHO, OTKJlOH.H.HCb
HeCKOJlbKO BnpaBO. 8TO CB.H3aHO
C TeM, qTO TeMnepaTypa nnaB-
JleHH.H cna60 3aBHCHT OT ,naBne-
HH.R.. He60nbIl1oe OTKJlOHeHHe
BnpaBO rOBopHT 0 nOBbIII1eHHH
TeMnepaTypbI nnaBJleHHSI C YBe-
nHqeHHeM ,naBJleHH.H. Ha6J1IO,nae-
Moro y 60JlbIl1HHCTBa BeeCTB.
IIOBbIII1eHHe ,naBJleHH.H KaK 6bI
npen.HTCTByeT nepexo,ny OT 60Jlee
o
TYp
Puc. 7.51
. ..
. . .
. ..
. ..
. ..
. . .
. ..
. .
. .
r83
. .
. ..
. .
. .
. .
. .
.9 ·
· 10.
. .
. . .
. ..
. .
. ..
Tap
T
203

nJlOTHOH K MeHee nJlOTHOH ynaKoBKe qaCTH. JI:nH BeeCTB, 06J1a,AalOHX B
TBep)IOM COCTO.HHHH MeHee nJlOTHOH ynaKoBKOH qaCTH, tJeM B mH)IKOM COCTO-
.HHHH (JleA, BHCMYT, CepbIH qyrYH), YBeJlHqeHHe ,AaBJleHHH cnOc06cTByeT nJIaB-
JleHHIO. AJI.H TaKHX BeeCTB KpHBa.H nJiaBJIeHH.H OTKnOHeHa BJIeBO OT BepTH-
KaJlH (KpHBa.H AD).
C YMeHbmeHHeM TeMnepaTypbI BeeCTBa, HaXOA.HerOCSI B KpHCTaJIJlH'tleC-
KOM COCTO.HHHH, YMeHblllaeTCH H AaBJleHHe ero HaCbIeHHoro napa. 3TO OTpa-
meHO B XO,Ae KpHBOH cy6JIHMaHH. KpHBaSi AC He cJlyqaHHO He ,AOBeAeHa ,AO
nepeCe'tleHHH C OCbIO a6CHCC HJlH Op,AHHaT. CTporo rOBOp.H, npH CKOJIb yro,A-
HO HH3KOH TeMnepaType, OTJIH'tlHOH OT a6COJIIOTHOrO HYJI.H. JlI060e BeeCTBO
HcnapHeTC.H H C npH6J1HmeHHeM llaBJIeHH.H K HYJIIO 1l0JI:>KHO nepeHTH B ra3006-
pa3Hoe COCTO.HHHe. OlIHaKO Y 60JIblllHHCTBa XHMHqeCKHX 3JIeMeHTOB H HX CO-
e,AHHeHHH ,AaBJleHHe HaCbIeHHbIX napOB npH HH3KHX TeMnepaTypax O'tleHb
MaJIO, a CKOpOCTb HcnapeHH.H HaCTOJlbKO HH3Ka. tITO BeeCTBO HaXO,AHTC.H B
TBepAOM HJIH mH,AKOM COCTOHHHH npaKTH'tleCKH HeorpaHHtJeHHO ,Aonro.
IIpH 3Ha'tleHHHX AaBJleHHH H TeMnepaTypbI, COOTBeTCTBYIOHX 06J1aCTH, pac-
nOnO:>KeHHOH CJleBa OT KpHBOH CAB, BeeCTBO HaXO,AHTCH B TBep,AOM COCTOH-
HHH, B 06JIaCTH Mem,Ay KpHBbIMH BA H AX BeeCTBO HaXO,AHTC.H B mH,AKOM
COCTOSlHHH, a B 06JIaCTH, pacnOJlOmeHHOH cnpaBa OT KpHBOH CAX, BeeCTBO
HaXO,AHTC.H B ra3006pa3HOM COCTO.HHHH.
Oc060ro BHHMaHH.H 3aCJIymHBaIOT TOtIKH A H K Ha JmarpaMMe COCTO.HHHH.
TOtIKa A, KOTOpa.H COOTBeTcTByeT TaKHM 3HaqeHHHM ,AaBJIeHH.H PTP H TeMnepa-
Typbl T TP' npH KOTOpbIX HaXO,A.HTC.H B ,AHHaMHqeCKOM paBHOBeCHH TPH (j)a3bI -
mH,AKaH, TBep,AaSl H ra3006pa3Ha.H, - Ha3bIBaeTCSI TpOHHOH T01JKOH.
3HaqeHH.H PTP HTTP ,An.H HeKOTopbIX BeeCTB
Ta6JIHa 7.1 npHBe,AeHbI B Ta6J1He 7.1. CyecTBeHHbIM OTJlH-
qHeM AJI.H HHX .HBJISleTC.H pacnOJIO:>KeHHe TpOHHOH
TO'tlKH Ha AHarpaMMe OTHOCHTeJlbHO HOpMaJlbHOrO
aTMoc(j)epHoro .D;aBJIeHH.H. ECJlH .D;aBJIeHHe B TpOH-
HOH TO'tlKe MeHbllle aTMoc(j)epHoro (HanpHMep. llJl.H
BO,AbI 6. 10 2 IIa), TO BeeCTBO OTHOCHTC.H K nJla-
BHHMCSI. OHO npH HarpeBaHHH CHa'tlaJla nJIaBHTC.H
H npeBpaaeTCH B mH,AKOCTb, a nOTOM yme nepe-
XO.D;HT eJlHKOM B nap. ECJIH AaBJleHHe B TpOHHOH TO'tlKe 60JIbllle aTMoc(j)epHO-
ro (HanpHMep, CO 2 - 5,11' 10 5 Ua), TO BeeCTBO OTHOCHTCH K JleTyqHM.
IIpH HarpeBaHHH OHO He nJlaBHTCH, a B03rOH.HeTCSI (nepeXO.D;HT B nap).
,l];HarpaMMa COCTOHHHH AJI.H Kam.D;OrO KOHKpeTHoro BeeCTB8 CTpOHTC.H Ha
OCHOBaHHH 3KcnepHMeHTaJlbHbIX .D;aHHbIX. 3Ha.H .D;HarpaMMY COCTO.HHHH, MomHO
npeACKa3aTb, B KaKOM COCTOHHHH 6YlleT Haxo,nHTbCH BeweCTBO npH pa3J1HtJ-
HbIX YCJlOBH.HX, a TaKme, KaKHe npeBpaeHHH 6YAeT npeTepneBaTb BeeCTBO
npH pa3J1HtIHbIX npoeccax.
HanpHMep, npH YCJlOBHHX, COOTBeTCTBYIOHX TOtIKe 1 (CM. pHC. 7.51), Be-
lI\eCTBO HaXO,AHTCSI B TBep,AOM COCTOHHHH. ECJlH ero nO,ABepraTb H306apHoMY Ha-
rpeBaHHIO, H306pameHHoMY Ha ,AHarpaMMe COCTOHHHH ropH30HTaJIbHOH nYHKTHp-
HOH 1-2-3-4, TO npH TeMnepaType, COOTBeTCTBYIOeH TOqKe 2, BeeCTBO
nJlaBHTC.H, npH 60Jlee BbICOKOH TeMnepaType, cooTBeTcTBYIOeH TOqKe 3, - Ha-
tIHHaeT npeBpaaTbCH B ra3. ECJlH me BeeCTBO HaXOAHTC.H B TBepAOM COCTO-
BeIl.teCTBO TTP' K P Tp . na
CO 2 216.5 5.11 . 10 5
NH 3 195,4 6.10 3
H 2 O 273,16 6.10 2
O 2 54.] 1.3' 10
204

.HHHH, COOTBeTCTBYIOeM TOqKe 5, TO npH H306apHOM HarpeBaHHH (nYHKTHp-
Ha.H np.HMa.H 5-6) KpHCTaJlJI npeBpaaeTC.H B ra3, MHHY.H ,}KH,AKYIO cpa3Y. ECJIH
me BeeCTBO HaXOIIHTCSI B COCTOSlHHH. COOTBeTCTBYlOmeM TOqKe 7. TO npH
H30TepMHqeCKOM cmaTHH (nYHKTHpHaSi npSlMaSi 7-8), OHO npOHlIeT CJIel{Yro-
He TPH COCTO.HHH.H: ra3 - mH):{KOCTb - KpHCTaJlJlHqeCKOe COCTOSlHHe.
TO'lKa K Ha ):{HarpaMMe COCTOSlHHH COOTBeTcTByeT KpHTHqeCKOMY COCTO.H-
HHIO BeeCTBa H Ha3bIBaeTCSl KpHTH'IeCKOH TO'lKOH. IIpH TeMnepaType BbIIlle
KpHTHqeCKOH T KP HH npH KaKHX ):{aBJleHH.HX BeeCTBO He nepeXO):{HT H3 ra30-
06pa3Horo COCTO.HHH.H B :>KH):{Koe. IIpH TeMnepaType BbIIlle KpHTHqeCKOH T KP
H3MeHeHHe cpH3HqeCKHX CBOHCTB BeeCTBa C nOBbIIlleHHeM .n;aBJIeHHSI npOHCXO-
,AHT HenpepbIBHO, 6e3 CKaQKOB. pa3J1HQHe Me,}K.n;y :>KH.n;KOCTblO H ra30M HCQe3a-
eT. IIpeBpaeHHe ra3a B mH.n;KOCTb (HJlH Ha060pOT) 6e3 CKaQK006pa3Horo H3-
MeHeHHSl CBOHCTB BemeCTB8 B03MO:>KHO, HanpHMep, nYTeM, KOTOpbIH nOKR3RH
Ha .n;HarpaMMe JlHHHeH 9-10-11-12-13. IIepexolI me KpHCTaJIJlHQeCKOrO
COCTOSlHH.H B :>KH,AKOe HnH ra3006pa3Hoe MomeT 6bITb TOJIbKO CKaqK006pa3HblM
(nocpe.n;cTBoM cpa30Boro nepexo.n;a).
Bonpocw AH CAMOKOHTPOJIH
1. 'Cf eM omJl.UllaemCSl ucnapeliue om KuneliuSl?
2. Om Ilezo 3a6ucum cKopocmb ucnapeliuSl?
3. KaK 6.nUSlem KOliiJelica4uR napa lia meMnepamypy :HCUOKocmu?
4. KaK U3MeliSlemCR KOli4elimpa4uSl MOJl.eKY.n 60iJSlHOZO napa npu liazpe
6aHUU 3aKpblmozo cocyiJa c 60iJOU?
5. II OlleMY iJa6Jl.eliUe liaCbluteliliOZO napa c Y6e.nUlleliue.M meMnepamypbt
pacmem 6btCmpee, lleM y lieliaCbtuteliliozo?
6. KaK 6yiJem U3MeliRmbCSl 6 .JaKpblmOM cocyiJe iJa6.neliue 60iJSlHOZO napa.
6.nU3KOZO K liaCblll.lteliU1O. ecJl.U ezo C:HCUMamb npu nOCmOJlliHOU me.Mnepamype?
7. KaK npe6pamumb lieliaCbl14eliliblU nap 6 liaCbluteliliblu?
8. 'Cf mo maKoe mOIlKa pOCbl?
9. llpu KaKOM YCJl.06UU npoucxoiJum Kuneliue :HCuiJKocmu?
10. M oJICem Jl.U 60iJa Kunemb npu meMnepamype liUJlCl' 100°C? 6btWe
100°C?
11. II oileMY meMnepamypa KUnSluteu npu iJaliliOM. 6lieWlieM iJa6.neliuu JtCuiJ-
Kocmu ocmaemcSl nocmoJlltlwu?
12. B KOJl.6e liaxoiJumcSl He60Jl.bWOe KOJl.UlleCm60 60iJbt npu KOMHamHOU
me.Mnepamype. 'Cfmo 6yiJem npoucxoiJumb c 60iJOU. ec.nu liallamb liacoco.M omKa-
tLU6amb 603iJyX, liaxOOSlutUUCJl 6 KOJl.6e liao 60iJoU? M oJICem Jl.U 60iJa 3aMep3-
liymb npu amo.M?
13. B 6alie .MOJICliO lia6Jl.100amb. Ilmo OiJliU Kpalibl U mpy6bl nOKpbtmbt
6Jl.aZou. a opyzue npaKmullecKu cyxue. II otLe.MY?
14. II olle.MY 3U.MOU 6 6e36empeliliblU oelib, KoziJa liallUliaem 6blnaiJamb cliez,
3aMemHO menJl.eem?
15. KaK C6Sl3ali0 iJa6Jl.eliUe 603iJyxa c nap4ua.nbliblM iJa6Jl.eliUe.M 6xOiJJlutUX
6 liezo za306 u 60iJSliiOZO napa?
16. KaK U3.Meli.R.emCJl nap4UaJl.bliOe iJa6Jl.eliUe u omliOCUmeJl.bliaSl 6Jl.a:HCIiOCmb
603iJyxa npu ezo liazpe6aliUU 6 3aKpblmO.M cocyoe 2
205

17. KaKoe SlBJl.eliUe lia6Jl.lOiJaemcSl npu me.Mnepamype. pa61i0U (u liuJlCe)
mOIlKU pOCbt? 06pa3yemcSl Jl.U poca npu 6empe?
18. 06a mep.MOMempa B nCUxpO.Mempe ABzycma nOKa3bt6alOm oiJuliaKoBYIO
meMnepamypy. KaK06a omliocume.n.blia.R BJl.aJICIiOCmb 603iJyxa?
19. M oJICem .n.u BeU{ecmBO nepexoiJumb U3 za3006pa31i0ZO cocmOSlIiUSl B
m6epiJoe. .MUliY.R ljJa3Y :HCUiJKOZO cocmo.Rliu.R? M oJICem Jl.U mBepiJoe me.n.o npe-
BpamUmbCSl 6 za3 6e3 nJl.a6.n.eliuSl?
20. q mo nOIiU.MalOm noiJ KpumUlleCKU.M cocmOSlIiUe.M?
21. BceziJa Jl.U, nOBbtWaSl iJa6.n.eliue, .MOJICIiO nepe6ecmu za3 (3 JlCuiJKoe co-
cmoSlliue?
IIPHMEPbI PElliEHIf.H 3AJJ:A q
1. KaKyro Maccy JIb,Aa, l1Meroero TeMnepaTYPY -10°C, MO:>KHO paCTonHTb 3a
10 MHHyT C nOMOblO 3JIeKTpHqecKoro HarpeBaTen.H, eCJIH CHJIa TOKa, npOTeKalO-
ero B HeM, 3,0 A npH Hanp.HmeHHH B ceTH 220 B? KII,Z:( HarpeBaTeJl.H 80 %.
,Z:(aHo: t 1 =-10 °C, 1=600 c, 1=3,0 A, U=220 B, 11=0,80,
c = 2.1 . 10 3 Am A = 3,34 . 105 ,[(:>K
Kr.K ' Kr
HaRTH: m.
Q
Pewe H He. IIo onpeeJIeHHIO KII.D: HarpeBaTen.H T) = ----!!.. B, re Qn - KO-
Q
JlHqeCTBO TenJlOTbI, KOTopoe nOJIY'lHT JIe,A; Q - KOJIHqeCTBO TenJlOTbI, KOTO-
poe BbIeJlSIeTCSI B npOBO,AHHKe npH npoxomeHHH TOKa:
Q = JU1.
JIe,A, nOJIyqa.H 3HeprHIO, 6yeT HarpeBaTbC.H o TeMnepaTypbI nJlaBJleHHfl
tnn=O °C, a 3aTeM nJIaBHTbC.H. CneoBaTenbHo, Qn=cm(tnn-tl)+Am.
cm(t nn - t 1 ) + Am TlIU1
Tor,Aa T} = . OTKY,Aa m = , m = 0,95 Kr.
JUT c(t nn - td + A
2. B 3aKpbITOM JI8TYHHOM KanopHMeTpe MaCCOH 200 r HaXOHTCSI 1,0 Kr
nb,Aa npH TeMnepaType -10°C. B KaJlOpHMeTp snYCKalOT 200 r napa. HMelO-
ero TeMnepaTYPY 110°C. KaKa.H TeMnepaTypa YCTaHOBHTC.H B KaJlOpHMeTpe?
'Y,AenbHYIO TenJIOeMKOCTb napa B HHTepBaJIe OT 100°C .no 110 °C ct{HTaTb
paBHoH 1,7 10 3 ,[(m
Kr.K
t
,[(m
,Z:( a H 0: Cl = 380 , ml =0,200 Kr, m2= 1,0 Kr,
Kr.K
e
,[(m
t 1 =-10°C, c2=2,1.10 3 Kr.K ' m3=0,200 Kr,
Q
t 3 =110°C, c3=1,7 .10 3 ,Z:(:>K , L=2,3 .106 ,Z:(:>K .
Kr.K Kr
HaRTH: e.
Puc. 7.52
206

P e III e H H e. B pe3YJIbTaTe TeIL1I006MeH8 K8JI0pHMeTp H Jle (H nOJlyqeHHa.H H3
Hero BO,lJ;a) 6y,lJ;YT nOJIyqaTb 3HeprHIO, a nap (H nOJIyqeHHa.H 11:3 Hero BO,lJ;8) OT,lJ;a-
BaTb ee. IIpH pemeHH:H TaKHX 38,Aaq nOJle3HO nOJIb30BaTbCH rp8qmKoM 38BHC:HMOCTH
TeMnepaTYPbI TeJla OT KOJlHqeCTBa TellJlOTbI, nOJlyqeHHoro (HJlH OT,lJ;8HHOro) TeJlOM
(pHC. 7.52). KOJl}lqecTBO TellJlOTbl, nOJlyqeHHoe K8JI0pHMeTpoM (rpa<l>HK 1):
Ql = clml(t} - tt>.
JIe,lJ; 6yeT HarpeBaTbC.H ,lJ;0 t nJI , nJlaBHTbC.H, H nOJlyqeHHa.H BO,lJ;a HarpeBaeT-
CH ,AO TeMnepaTypbI 8.
KOJlHqeCTBO TenJIOTbI, nonyqeHHOe Jlb,lJ;OM (rpa<l>HK 2):
Q2 = C2 m 2(t rur - t 1 ) + Am2 + C B m 2(8 - t rur ),
r,lJ;e C2 - y,lJ;eJlbHa.H TenJlOeMKOCTb Jlb,lJ;a, t rur - TeMnepaTypa nJIaBJleHH.H Jlb,lJ;a,
A - y,lJ;eJlbHa.H TenJlOTa nJlaBJleHH.H Jlb,lJ;a, C B - y,lJ;eJlbHa.H TenJlOeMKOCTb BO,lJ;bI.
IIap, OT,lJ;B.Ba.H 3Heprmo, 6YlIeT OXJIa>K,lJ;aTbC.H ,lJ;0 TeMnepaTYPbI KHneHH.H, 3aTeM
KOH,D;eHCHpoBaTbC.H H no.nyqeHHa.H BO,lJ;a 6Y,lJ;eT OXJIam,AaTbC.H ,lJ;0 TeMllepaTYPbI S.
KOJlHqeCTBO TenJlOTbI, OT,lJ;aHHOe napOM:
Qg = cgmg( tJ( - t g ) - Lmg + c B mg(8 - tJ()'
Y paBHeHHe TenJlOBOrO 6aJlaHca: Ql + Q2 + Q3 = O.
Cl Tnt (e - t 1 ) + C2 (t nJI - t 1 ) + Am2 + CB (e - t nJI ) +
+ Cgma (tJ( - t 3 ) - Lma + cBma (e - t K ) = 0 =>
Clint t l - C2 (t nJI - t 1 ) - A + c B m 2 t nJI + Lma + cBmgtJ( + c 3 ma (t g - tJ( )
S- .
- Int++ma '
8 = 37 °e.
3. B TenJlOH30JlHpOBaHHbIH lJ,HJlHH,lJ;p 06'beMOM 10 Jl, cO,lJ;epmaHH a30T npH
TeMnepaType 27 °e H ,naBJleHHH 0.10 MIIa. BHeceH Me,lJ;HbIH map MaccoH 100 r,
HarpeTbIH ,AO 727 °e. KaKaH TeMnepaTypa YCTaHOBHTC.H B lJ,HJlHH,Ape B pe-
3YJlbTaTe TenJlo06MeHa? TenJlOeMKOCTblO lJ,HJIHH,lJ;pa npeHe6peqb.
AaHo: V=0,010 M 3 , T 1 =300 K, p=I,0.10 5 IIa, m=0,100 Kr,
J];m
T 2 = 1000 K, c2=380 -
Kr.K
H a H T H: 8.
P e III e H H e. B pe3YJlbT8Te TenJlo06MeHa a30T 6Y,lJ;eT HarpeBaTbCH OT TeM-
nepaTypbl T] ,lJ;0 S, a Me,AHbIH map 6Y,lJ;eT OCTbIBaTb OT T 2 ,lJ;0 e. A30T nOJlY-
qHT KOJlHqeCTBO TenJlOTbI Q] = l:1U] = !:.. Int R(S -11). A30T - ,lJ;ByxaToMHbIH ra3,
2M
n03TOMY i = 5 H Ql =  Int R(S -71).
2M
Maccy a30Ta HaH,lJ;eM H3 yp8BHeHH.H KJIaneHpoHa-MeH,lJ;eJleeBa:
m 1
pV = -R71,
M
OTKY,A8
pVM
m l =-.
R71
Tor,lJ;a Ql = 5pV (8-71).
271
207

KOJlHqeCTBO TenJlOTbI, OT,naHHOe Me,nHbIM mapOM:
Q2 = C2 m 2(e - t 2 ).
Ha OCHOBaHHH ypaBHeHH.H TenJlOBOrO 6aJlaHCa Ql = Q2' CJle,nOBRTeJlbHO,
5pV (8 - Td = c 2 m 2 (T2 - 8). OTCIO,Aa 8 = 5pV + 2c 2 m 2 T 2 71 ; e = 8,7' 10 2 K.
271 5pV + 2c2m271
4. CMemanH 1,0 M 3 B03,nyxa BJla>KHOCTblO 20 % H 2,0 M 3 BJla>KHOCTblO
30 %, HMelO[QHX o,nHHaKOBYIO TeMnepaTYPY. CMecb 3aH.HJla 06'beM 3,0 M 3 .
Onpe,AeJlHTe ee OTHOCHTeJlbHYIO Bna>KHOCTb.
)XaHo: V 1 =1,0 M 3 , <Pl=0,20. V 2 =2,0 M 3 , CP2=0,30, V=3,0 M 3 .
H a H T H: cpo
P e III e H H e. IIo onpe,neJleHHIO OTHOCHTeJlbHa.H BJlamHOCTb <P =.E.-, r,ne P -
Po
a6COJlIOTHa.H BJlamHOCTb (nJlOTHOCTb BO,A8HOrO napa) B03,nyxa, Po - nJlOTHOCTb
HaCbI[QeHHoro napa npH ,naHHoH TeMnepaType.
OTclO,na P = cpPo. Maccy napa, Haxo,n.Heroc.H B B03,nyxe, MomHO paCCqH-
TaTb, 3Ha.H 06'beM, 3aHHMaeMbIH BJlamHbIM B03,nYXOM: m = P V = CPPo V.
CJle,noBaTeJlbHO, ml = CPI V IPO H m2= CP2 V 2PO'
BBH,ny RMHTHBHOCTH MaCCbI, Macca CMeCH
m= ml + m2=(CPI V I +CP2 V 2)PO'
MCKOMa.H OTHOCHTeJlbHaH BJla>KHOCTb
m
_ P _ 11 _ <PI VI + CP2 V 2 .
cP - - - - - , cP = 0,27 = 27 %.
Po Po V
5 TeMnepaTypa B03,nyxa 24°C, OTHOCHTeJlbHa.H BJlamHOCTb 95 0/0. CKOJlb-
KO pOCbI BbIna.n;eT B 1.0 M 3 B03,nyxa, eCJlH TeMnepaTypa CTaJla 16°C?
,I:(aHO: tl=24 °C, CPI=0,95, V=I,O M 3 , t2=16 °C, POl=21,8.10- 3
Kr
M 3 '
P02= 13.6 . 10- 3
Kr
3 .
M
HaHTH: m.
P e III e H H e ITo onpe.neJleHHIO OTHOCHTenbHa.H BJlamHOCTb cP = .E.-, OTKY,A8
Po
P = cpPo.
CJle,nOBaTeJlbHO, PI = CPPOl'
Macca napa, Haxo,n.Heroc.H B B03,nyxe, ml = PI V = <PIPOl V.
Macca napa, He06xo,nHMa.H ,nJl.H HaCbIeHH.H npH TeMnepaType t 2 :
m2 = CP2P02 V = P02 V,
r,ne <1>2= 1 (npH Bblna,AeHHH pOCbl OTHOCHTenbHa8 BnamHOCTb paBHa 100 %).
CJle.noBaTenbHo, Macca BbInaBmeH pOCbI
m = m 1 -m 2 = (CPIPOI -CPP02)V; m = 7.1 r.
208

3nEKTPOAi4HAMi4 KA
8. 3EKTPOCTATHKA
3J1eKTpHqeCKHH 3ap.HJJ;. OnbITbI A.. 1I0cl>cI>e H P. MHJlJlHKeHa. 3aKOH co-
xpaHeHHS 3J1eKTpHqeCKOrO 3ap.HJJ;a. TO'le'lHLIH H pacnpeAeJleHHLIH 3ap.HJJ;bI.
B3aHMoAeHcTBHe 3apSjKeHHLIX TeJl. 3aKOH KYJlOHa. 3J1eKTpHqeCKOe nOJle.
HanpSljKeHHOCTL, JlHHHH HanpSljKeHHOCTH. 3J1eKTpH'IeCKOe nOJle TO'le'lHLIX 3a-
PH.D;OB. npHHIJ;Hn cynepn03HIJ;HH nOJleH. OAHopoAHoe 3JIeKTpH'IeCKOe nOJIe.
Pa60Ta 3J1eKTpH'IeCKOrO nOJis npH nepeMeJJ.\ellHH 3apH.D;OB. llOTeHIJ;HaJi.
Pa3HOCTL nOTeHIJ;HaJlOB. HanpSjKeHHe. CBS3b MejKAY pa3HOCTbIO nOTeHIJ;HaJlOB
H HanpSjKeHHOCTLIO. llpoBonHHKH B 3J1eKTpHqeCKOM nOJle.
3J1eKTpOeMKocTb. EMKOCTL nJlOCKOrO Koo.ueHcaTopa. .lI.H3J1eKTpHqeCKaSl npo-
HHu;aeMOCTb. AH3J1eKTpHKH B 3J1eKTpH'IeCKOM nOJle. MexaHH3M nOJlSpH3aIJ;HH
AH3J1eKTpHKOB. 3J1eKTpeTLI H cerHeT03J1eKTpHKH llLe303J1eKTpHqeCKHH 3c1>cI>eKT
Hero HCnOJlL30BaHHe B TeXHHKe.
3HeprHS H nJlOTHOCTL 3HeprHH 3J1eKTpHtJeCKOrO nOJis.
8.1. 3JIeKTpH'IeCKHH 3apM
H3 Bcex H3BeCTHbIX CPYH,naMeHTRnbHbIx THnOB B3RHMo,neHcTBH.H HRH60nee pRC-
npOCTpaHeHHbIMH .HBn.HIOTC.H 3neKTpOMRrHHTHbIe. ITpocTeHllIHH npHMep 3THX B3R-
HMo,neHcTBHH - npHT.HmeHHe H OTTRJlKHBRHHe Ha3neKTpH30BRHHbIX Ten.
BnepBbIe ero Ha6J1IO,nRn <I>aJlec (VII B. )J;O H.3.), Hcnonb3Y.H .HHTRPHYIO
naJIOqKY, nOTepTYIO 0 llIepCTb. IT03me B. rHnb6epT (XVI B.) o6HRPymHn,
ttTO CBOHCTBOM npHT.HrHBRTb nerKHe npe,nMeTbI 06nR,naIOT, KpOMe .HHTRP.H,
cpRpcpOp H MHOrlie ,npyrHe TeJla, npe,nBRpHTenbHO HRTepTbIe RomeH HnH ,npy-
rHMH M.HrKHMH MRTepHRJlaMH. 3TO .HBJleHHe B. rHJlb6epT HR3BRn 3J1eKTpH3R-
HeH (electron no-rpetteCKH - .HHTRpb).
o TenRX, cnoc06HbIX K TRKHM B3RHMo.neHcTBH.HM, rOBOp.HT. qTO OHH 3J1eK-
TpHqeCKH 3apSjKeHhI, T.e. HM co06JJ.\eH 3J1eKTpH'IeCKHH 3ap.HJJ;.
3J1eKTpHqeCKHH 3ap.HJJ; q -- 3TO KOnHtteCTBeHHR.H Mepa cnoc06HOCTH Ten K
3neKTpOMRrHHTHbIM B3RHMOD.eHCTBH.HM.
E)lHHHUeH 3J1eKTpHtteCKoro 3Rp.HD.R B CM .HBn.HeTC.H KynOH. 1 KYJlOH eCTb
3neKTpHqeCKHH 3ap.H,n, nepeHocHMbIH ttepe3 nonepettHOe CetteHHe npOBo,nHHKR 3R
1 C npH CHJle TOKR 1 A.
OnbITbI no 3J1eKTpH3aHH TeJl TpeHHeM, B3RHMo,neHcTBHIO He3ap.HmeHHOH
CTRHHOJleBOH rHnb3bl C 3ap.HmeHHOH nRnOqKOH nOKR3bIBRIOT, ttTO:
1) CYLQeCTBYIOT ,nBR po,na 3neKTpHqeCKHX 3ap.H,nOB, KOTopbIe ycnoBHo HR3bI-
BRIOT nonomHTenbHbIMH H OTpHRTenbHbIMH; 3Rp.H,n 3J1eMeHTRpHbIX qRCTH -
npOTOHOB, BXO,n.HLQHX B COCTRB JlI06oro .H,npa, HR3bIBalOT nonOmHTeJlbHbIM. a
3RpH,n 3J1eKTpOHOB - OTpHaTeJlbHbIM;
209

2) npH B3aHMoeHcTBHH O.D:HOHMeHHLle 3apMbi OTI'aJIKHBaIOTCJI, pa3HOHMeH-
Hble npHTJlrHBaIOTCJI. Ha 3TOM CBoHcTBe 3ap.HOB OCHOBaHO ,AeHCTBHe npocTeH-
llIHX npH60poB - 3J1eKTpOCKona H 3J1eKTpOMeTpa;
3) COBpeMeHHa.H <}>H3HKa [IpHBOHT K BbIBOY 0 CYI.D;eCTBOBaHHH 3J1eMeHTap.
Horo 3ap.Ha, .HBn.HlOI.D;erOCSI HeOT'beMneMbIM CBOHCTBOM pSla 3J1eMeHTapHbIX 'tIa-
CTH. BnepBbIe HeMeKHH <}>H3HK r. reJlbMrOJlb B 1881 r. BbICKa3an rHnOTe-
3Y, 06'b.HCH.HIOYIO 3J1eKTpHqeCKHe .HBJleHH.H CYI.D;eCTBOBaHHeM 3neKTpH'tIeCKH 3a-
pSl>KeHHbIX 3J1eMeHTapHbIX qaCTHIJ,. BnOCJleCTBHH 3Ta rHnOTe3a nOTBepHJlaCb
OTKpbITHeM 3J1eKTpOHa (B 1897 r. aHrJlHHCKHM <}>H3HKOM Y. TOMCOHOM) H npo-
TOHa (B 1919 r. 3. Pe3ep<}>opoM).
3neKTpoH - 3TO 3J1eMeHTapHa.H qaCTHa. 3ap.H KOTOpOH e = -1,6 . 10 -19 KJI.
MaCCa me= 9.1.10- 31 KI". IIpoToH HMeeT nOJlO>KHTeJlbHbIH 3apSilI e = 1.6.10 -19 KJI.
a Macca npOTOHa B 1836 pa3 60JlbllIe MaCCbI 3neKTpoHa:
4) 3JIeKTpHqeCKHH 3apM .D:HCKpeTeH (3TO ,AOKa3aHO onbITHbIM nYTeM aMe-
pHKaHcKHM <}>H3HKOM P. MHJlJlHKeHOM H PYCCKHM <}>H3HKOM A. <1J. Mo<}><}>e - CM.
n. 8.2). 3TO 3HaqHT. tITO nlO60H 3apSl, 60JlbllIHH 3neMeHTapHoro, COCTOHT
H3 eJloro tIHCJl8 3neMeHTapHbIX 3apSlOB q=-x.N'e, re N=l, 2, 3...
B TeJle, 3J1eKTpHqeCKH HeHTpaJlbHOM, qHCnO npOTOHOB R 3J1eKTpOHOB OH-
HaKOBO H OHH paBHoMepHo pacnpeeJleHbI [10 06'beMY. ECJlH qHCJlO 3J1eKTpOHOB
B TeJle MeHbllIe tIHCJla npOTOHOB, TO OHO 3ap.HmeHO nOnO>KHTeJlbHO, a eCJlH
3ap.H 06YCJlOBJleH H36bITKOM 3J1eKTpOHOB, TO TeJlO 3ap.HmeHO OTpHaTenbHO.
MMeHHO 3TOT H36bITOtIHbIH 3ap.H,A onpeeJl.HeT C060H 3J1eKTpHqeCKHe CBOH-
CTBa TeJla. Hero Ha3bIBalOT 3apSl,AOM Ten8. q = (N + - N _)e. r,Ae N + - tIHCJIO
npOTOHOB. N _ - tIHCJIO 3neKTpOHOB.
IIpoecc, npHBO.HI.D;H:H K nOSlBJleHHIO Ha Tenax HJlH pa3HbIX tIaCT.HX TeJla
H36bITOtIHOrO 3ap.Ha, Ha3bIBalOT 3J1eKTpH3aHeH;
5) 3apSl HHBapHaHTeH, T.e. He 3aBHCHT OT xapaKTepa BH>KeHH.H 3ap.H>KeH-
HOH tIaCTHbI, HH OT ee B3aHMoeHCTBH.H C PyrHMH tIaCTHaMH, HH OT BbI60-
pa CHCTeMbI OTCtIeTa. 06 3TOM CBHAeTeJIbCTBYIOT MHorHe <}>aKTbI. TaK, HeHOHH-
3HpOBaHHbIH aTOM H MOJleKYJIa HeHTpaJIbHbI: 3ap.HbI 3JIeKTpOHHOH 060JIOtIKH aTO-
Ma H SIpa B TOtIHOCTH paBHbI APyr lIpyry. A Me>KlIY TeM xapaKTep lIBH>KeHH.H
3J1eKTpOHOB H .HAep COBepllIeHHO pa3J1HqeH. KpoMe Toro, npH XHMHtIeCKHX npe-
BpaI.D;eHHSlX ,ABH>KeHHe 3neKTpOHOB B 060nOtIKaX aTOMOB H3MeHSleTCSI, 'lTO npH-
BO,AHT K H3MeHeHHIO xapaKTepa cneKTpOB aTOMOB. ECJlH 6bI 3apSiA 3aBHCeJl OT
CKOpOCTH BHmeHH.H 'laCTH. TO B XHMHtIeCKHX peaKHSlX MOrJlH 6bl nO.HBHTbCSI
HeCKOMneHCHpOBaHHbIe 3J1eKTpHtIeCKHe 3ap.H,AbI, HO 3TO He 06HapymeHO.
8.2. OnLITLJ P. MHJlJlHKeHa H A. «1>. Hocl>cI>e
Puc. 8.1
210
Ha pHCYHKe 8.1 H306pa>KeHa cxeMa
YCTaHOBKH, HCnOJIb30BaHHoH B onbITe
A. <1J. M<><l><}>e. B 3aKpbITOM cocY,Ae, B03YX
H3 KOTOpOro OTKa'laH o BbICOKOro BaKYY-
Ma, H8XO,AHJlHCb Be MeTaJIneCKHe nJIac-
THHbI ll, pacnOJIOmeHHbIe ropH30HTaJIbHO.
113 KaMepbI A tIepe3 OTBepCTHe 0 B
npOCTpaHCTBO Me>KY nJlaCTHHaMH nona-
JlH MeJlKHe 3ap.HmeHHbIe nbIJIHHKH HHKa.
3TH nbIJlHHKH Ha6JIIOaJIH B MHKpocKon.

IIpe):{noJlomHM, qTO nbIJlHHKB 3Bp.HmeHB OTpHaTenbHO. IIo):{ ):{eHCTBHeM
CHnbI T.HmeCTH OHB HaqHHaeT nB):{aTb BHH3. Ho ee na):{eHHe MOmHO 3a):{ep-
maTb, eCJlH HHmHlO1O nJlaCTHHY 3ap.H):{HTb OTpHaTeJlbHbIM 3ap.H):{OM, a Bepx-
HIOIO - nOJlOmHTeJlbHbIM. B 3J1eKTpOCTaTHqeCKOM nOJle Mem):{y nJlaCTHHa-
MH Ha nbIJlHHKY CTBHeT ):{eHCTBOBaTb CHJlB F, KOTOpa.H nponopHOHaJlbHa
3ap.H):{Y nbIJlHHKH. EcnH mg = F")n' TO nbInHHKa 6y):{eT HBXO):{HTbCSI B paBHO-
BeCHH CKOJlb yrO,lIHO ):{oJlro. 3aTeM OTpHaTeJlbHbIH 3ap.H):{ nbInHHKH YMeHbwa-
nH, ):{eHCTBY.H HB Hee YJlbTpacpHOJleTOBbIM CBeTOM. IIbInHHKa HBqHHBna nB):{BTb,
TaR KaK CHJla F, )J;eHCTBOBaBlIIa.H Ha Hee, YMeHblIIaJlaCb. Co06a.H nnaCTH-
HaM ):{onOJlHHTeJlbHbIH 3ap.H,A H 3THM YCHJlHBB.H 3J1eKTpHqeCKOe nOJle Mem):{y
nJlaCTHHaMH. nbIJlHHKY CHOBa OCTaHaBJlHBaJlH. TaK nocTynanH HeCKOJlbKO pa3.
OnbITbI nOKa3aJlH, qTO 3ap.H,lI nbInHHKH H:3MeH.HnC.H BCer,lIa CKaqK006pa3HO,
KpaTHO 3ap.H):{y 3J1eKTpOHa. ll3 3Toro onbITB A. <1>. llocpcpe c):{eJlBn cneYIOHH
BbIBO):{: 3Bp.H):{ nbIJlHHKH Bcer):{B BbIpamBeTC.H enbIMH KpaTHbIMH 3HBqeHH.HMH
3neMeHTapHoro 3ap.H):{B e. MeHbwHX "nopHH" 3J1eKTpHqeCKOrO 3ap.H):{a, cnoc06-
HbIX nepeXO):{HTb OT O):{Horo TenB K ,ApyrOMY, B npHpO,Ae HeT. Ho 3ap.H):{ nbI-
nHHKH YXO,AHT BMeCTe C qaCTHeH BeeCTBa. CJle,AOBaTeJlbHO, B npHpo):{e CY-
ecTByeT TaKa.H qac'rHa BeeCTBa, KOTOpa.H HMeeT caMbIH MaJleHbKHH 3ap.H):{.
):{aJlee yme He):{eJlHMbIH. STY qaCTHy Ha3BaJll1 3JIeKTpOHOM.
3HBqeHHe 3ap.H):{a 3J1eKTpOHB BnepBbIe onpe):{eJlHJI aMepHKaHCKHH cpH3HK
P. MHnJlHKeH. B CBOHX onbITax OH nOJlb30BanC.H MeJlKHMH KanenbKaMH Mac-
na, Ha6J1IO):{a.H 3B HX ):{BHmeHHeM B 3J1eKTpOCTaTHqeCKOM nOJle (pHC. 8.2). B
3THX onbITax H:3Mep.HJlaCb CKOpOCTb ):{BHmeHH.H KaneJlb Macna B O):{HOpO):{HOM
3neKTpOCTBTHqeCKOM nOJle Mem,Ay ,ABYM.H MeTaJlnH'leCKHMH nJlBCTHHKBMH. Kan-
n.H MaCJla, He HMelOa.H 3J1eKTpHqeCKOrO 3ap.H):{a H3-3B conpOTHBneHH.H B03):{Y-
xa H BbITaJlKHBalOeH CHnbI, na):{aeT C HeKOTopOH nOCTO.HHHOH CKOPOCTblO.
TaK KaK mg+F A +Fe =0.
ECJlH Ha CBoeM nYTH KanJlSl BCTpeqaeTC.H C HOHOM H
+ npH06peTaeT 3J1eKTpH'leCKHH 3ap.H):{ q, TO Ha Hee, KpOMe
CHnbI T.HmeCTH mg. Fe H FA' ,AeHcTByeT co CTOpOHbI
3J1eKTpOCTaTHqeCKOrO non.H CHJla F;n' Tor):{a npH YCTaHO-
BHBWeMC.H ):{BHmeHHH mg + Fe + FA + F")n = O. ll3Mep.H.H CKO-
pOCTb KanJlH, MHJlJlHKeH CMor onpe):{enHTb ee 3ap.H):{.
Fe
Puc. 8.2
8.3. 3aKOH coxpaHeHHH 3JIeKTpH'IeCKOrO 3apH,1J;a
B Jl.1060U 3Jl.eKmpOU30Jl.Up06aliliOU cucme.Me aJl.Ze6paUlleCKaSl cY.M.Ma 3Jl.eKm.
pUlleCKUX 3apSl006 ocmaemCJl nocmOSlIiIiOU:
n
Lqi = const .
i=l
STO nOJlOmeHHe H3BeCTHO no):{ Ha3BaHHeM 3aKOHa coxpaHeHHH 3JIeKTpHQeC-
Koro 3apH,ZJ.a.
IIo):{ 3J1eKTpOH30nHpOBaHHOH CHCTeMOH nOHHMalOT B ):{aHHOM cJlyqae CHCTe-
MY, B KOTOPYIO He BXO):{.HT H He BbIXO):{.HT 3ap.HmeHHbIe qaCTHbI.
IIpl1 3J1eKTpH3aHH ):{BYX TeJl TpeHHeM O):{HO H3 HHX Tep.HeT CTOJlbKO me
3neKTpOHOB, CKOJlbKO npH06peTaeT )J;Pyroe. ECJlH ):{Ba TeJla C paBHbIMH no MO,
/
211

,nYJlIO. HO pa3HOHMeHHbIMH 30p.H,naMH npHBeCTH B COnpHKOCHOBeHHe. TO HX
3J1eKTpHqeCKHe 3ap.H,nbI HeiiTpOJlH3YIOTC.H.
3aKOH COXpaHeHH.H 3J1eKTpHqeCKOrO 3ap.H,na ,nOKa3bIBaeTC.H 3J1eKTpOCTaTHqeC-
KOH HH,nYKUHeH. 0 TOK:>Ke B3aHMHbIMH npeBpameHH.HMH 3J1eMeHTOpHbIX 1IOCTHU.
TOK, HOnpHMep. npH B3aHMO,AeHCTBHH nOpbI "3neKTpOH-n03HTpOH" H06J1IO,nO-
eTC.H .HBJleHHe aHHHrHJI.HUHH (HCqe3HOBeHH.H qaCTHUbI H OHTHqOCTHUbI. KOTOpbIe
npeBpaaIOTC.H B lIBO cpOTOHa) C BbIlIeneHHeM 3HeprHH.
8.4. TO'le'lHbIH H pacnpe,n;eJIeHHbIH 3a PMbl
H,neaJlbHOH cpH3HqeCKOH Mo,nenblO 3ap.H,na B 3J1eKTpOCTaTHKe HBJI.HeTC.H TO-
qeqHbIH 3ap.H,n.
TOqeqHLIM 30p.H,nOM Ha3bIBaeTC.H 30p.H,n, cocpe)J;oTOqeHHbIH Ha TeJle, pa3Me-
paMH KOToporo MO:>KHO npeHe6peqb no cpaBHeHHIO C paCCTO.HHHeM ,AO ,ApyrHx
TeJl HnH ,no paccMaTpHBaeMoH TOqKH nOJl.H. MHbIMH CJlOBaMH, T01IeqHbIH 3a-
p.H,A - 3TO MaTepHaJlbHa.H TOqKO, KOTOpO.H HMeeT 3neKTpHqeCKHH 3apSl.D,.
ECJlH 30p.HmeHHOe TeJlO HaCTOJlbKO BenHKO, tlTO ero Henb3.H pOCCMOTpHBOTb
K8K TOqeqHbIH 3ap.H,n, TO B 3TOM cJlyqoe He06xo,nHMO 3HaTb pacnpe.u;eJIeHHe
30p.H,nOB BHYTPH TeJla.
BbI,neJlHM BHYTPH 30p.HmeHHOro TeJlO MaJlbIH 06'beM l::. V H 0603HaqHM qepe3
l::.q 3J1eKTpHqeCKHH 3ap.H.D" HaxO,n.HHHC.H B 3TOM 06'beMe. IIpe,neJl OTHOlIIeHH.H l::.q ,
l::.V
Kor)J;a 06'beM l::. V HeOrpaHHqeHHO YMeHbIilOeTC.H, Ha3bIBOlOT 06'LeMHOH nJIOTHOC-
TLIO 3JIeKTpHQeCKOrO 3apn.na B ,nOHHOH TOqKe. 0603HOqOlOT ee 6YKBOH p:
1 . 6.q
P = 1m-,
vo V
E,lIHHHUeH 06'beMHOH nJlOTHOCTH 3ap.H1I8 p B CM .HBJI.HeTC.H KYJlOH Ha Ky6H-
qeCKHH MeTp ( :: ).
B cnyqoe HepaBHOMepHO 30p.HmeHHOro TeJla nJlOTHOCTb pa3nHqHa B pa3HbIX
T01IKOX. Pacnpe,neJleHHe 3ap.H,na B 06'beMe TeJla 3a,naHO, eCJlH H3BeCTHO p KaK
CPYHKH.H Koop,nHHaT.
B MeTOJlnH1IeCKHX TeJlaX 30p.H,AbI pacnpe,nen.HIOTC.H TOJlbKO BHYTPH TOHKoro
cno.H, npHJlerOIOero K nOBepXHOCTH. B 3TOM cnY'I0e y,n06HO nOJlb30BOTbC.H no-
BepXHOCTHOH llJIOTHOCTLIO 3apH,ZJ;a a, KOTOpBJI npe,nCTaBJI.HeT C060H npe,Aen OTHO'
lIIeHH.H 3ap.H,Aa K nJlOa.nH nOBepxHocTH, no KOTOpOH pacnpe,neneH 3TOT 3ap.H,n:
1 . 6.q
a = Im-,
s-+o S
r,ne 6.q - 3ap.H,n, Haxo,n.HHHC.H Ha yqaCTKe nOBepXHOCTH nJlOo,nblO I1S.
CJle)J;OBOTenbHO, nOBepXHOCTHa.H nJlOTHOCTb 3ap.H,na H3Mep.HeTC.H 3ap.H)J;OM
npHxo,n.HHMC.H Ha e,nHHHY nOBepXHOCTH Tena. Pacnpe)J;eJleHHe 3ap.H,nOB no no-
BepXHOCTH onHCbIBaeTC.H 30BHCHMOCTblO nOBepXHOCTHOH nJlOTHOCTH a(x, y. z) OT
Koop,nHHaT TOqeK nOBepXHOCTH.
E,AHHHeH nOBepXHOCTHOH nJlOTHOCTH 3ap.H,na a B CM .HBJlSleTC.H KYJlOH HO
u r KJI ]
KBo,ZJ;paTHbIH MeTp .
B TOM cJlyqae, eCJlH 3ap.HmeHHOe Teno no cpopMe npe,ZJ;CTaBJI.HeT C060H HHTb
(,nHaMeTp nOnepeqHOrO CeqeHH.H TeJla MHoro MeHblile ero ,nJlHHbI: d« l), y,n06HO
212

HCnOJlb30BaTb JlHHeHHYIO nJIOTHOCTb 3ap.H8 l' = Hrn C1q , re !1q - 3ap.H, Ha-
61-+0 /)'l
XO.HHHC.H Ha nHHe /)'l TeJI8.
E,AHHHI.'(eH JIHHeHHOH nJIOTHOCTH 3ap.Ha l' B CM .HBJI.HeTC.H KYJIOH H8
MeTp r  J.
ECJIH H3BeCTHO pacnpeeJIeHHe 3ap.HOB BHYTPH TeJIa, TO MomHO BbI'l{HCJIHTb
Hanp.HmeHHOCTb 3JIeKTpOCTaTM'l{eCKOrO nOJI.H. C03aBaeMoro 3TMM TeJIOM. llJI.H 3TO-
ro 3ap.HmeHHOe TeJlO MbICJIeHHO pa36HBalOT Ha 6eCKOHe'lHO MaJlbIe 'laCTH H. pac-
CM8TpHB8.H MX KaK TOqe'l{HbIe 3ap.H,AbI, BbI'lHCJI.HIOT Hanp.HmeHHOCTb non.H. C03.n8-
BaeMYIO OT.neJIbHbIMH 'l{aT.HMH TeJla. CYMMapHYIO Hanp.HmeHHOCTb nOJI.H Haxo.n.HT
3aTeM CYMMMpOBaHMeM E i nOJIeH, C03.uaBaeMbIX OT.lIenbHbIMH 'l{aCT.HMM TeJIa, T.e.
n
E = LEi'
i=1
8.5. B3aHMo.a;eHcTBHe 3ap.H:lKellHbIX TeJI. 3aKoH KYJIolla
OnbITbI cppaHuY3cKoro cpH3HKa III. }];lOcpe nOKa3aJlH, 'l{TO TeJIa, HMelOl.l.\Me
38p.H.lIbI npOTMBOnOJIOmHOrO (O.llMHaKoBoro) 3HaKa, B3aMMHO npHT.HrMBaIOTC.H (OT-
TaJIKHBaIOTC.H). fIpH 3TOM CMJIa B3aMMO.lleHCTBH.H Mem.llY Ha3JIeKTpH30BaHHbIMM
TeJIaMH CJIOmHbIM 06pa30M 3aBHCHT OT cpOpMbJ H83JIeKTpM30BaHHbIX Ten H xa-
paKTepa pacnpe,ll;eJIeHH.H 3ap.H8 Ha HHX. II03TOMY He CYl.l.\eCTByeT e,ll;MHOH npo-
CTOH CPOPMYJIbI, onHCbIB81OI.l.\eH 3J1eKTpOCTaTM'l{eCKOe B38HMoeHCTBHe ,lI;JI.H npo-
H3BOJlbHOrO cnY'l{a.H. M TOJIbKO ,lI;JI.H TOqeqHLIX 3ap.H,AOB 3aKOH B3aHMO,ll;eHCTBM.H
3anHCbIB8eTC.H B ,lI;OCT8TO'l{HO npOCTOH cpopMe.
3aKOH B38HMO,ll;eHCTBH.H TO'l{e'lHMX 3JIeKTpll'l{eCKMX 3a?.H,lI;OB 6MJI OTKpMT B
1785 r. III- KYJlOHOM C nOMOl.l.\blO KPYTMJlbHbIX BeCOB. KpYTHJIbHble BeCbI
(pHC. 8.3) COCTO.HT 1'13 ,lI;BYX 0HH8KOBbIX lllapHKOB A 1-1 C; lllapHK A YKpen-
JIeH Ha KOpOMbICJle, CB.H3aHHOM C npOTHBOBeCOM B H HHTblO L, BepxHHH KOHeI.'(
KUTOpOH YKpenJleH Ha KPYTHJlbHOH rOJIOBKe T.
IIIapHK C npH60pa YKpenneH Ha M30nMpOBaHHOM
CTepmHe 1'1 BBOMTC.H BHYTPb npM60pa. IIIapMKH
A H C npMBO.HTC.H B conpMKOCHOBeHHe, 1'1 TaK
K8K lll8pMKll 0HH8KOBble. TO 3apSl,ll; rnapMK8 C
nopOBHY pacnpeeJI.HeTC.H Memy HHMH. IIIapHKH
OTTaJlKMB8IOTC.H Pyr OT pyr8. IIo yrJIY 3aKpy-
'l{MBaHM.H HMTH onpee.TI.HIOT CHJIY B3aHMOAeHCTBH.H
3ap.HmeHHbIX lllapHKOB. PaCCTO.HHHe r MemAY lllR-
pHKaMH H3Mep.HIOT no lllKaJIe, HaHeceHHOH H8 60-
KOBOH nOBepXHOCTH I.'(HnHHApa. H3MeH.H.H r 1'1 q,
F = k Iqlllq21
r 2
Puc. 8.3
F I2 ql
-+---r
.
I
Puc. 8.4
III. KYJIOH HallleJI, 'l{TO
MJIH, B
q2 F 21

BeKTopHOM BHAe, F = k qlq2 . f.., rAe !.. - e,AM-
r 2 r r
HH'lIHbIH BeKTop. CHnbI B38HMOAei1:cTBH.H ,ABYX
O,AHOHMeHHO 3ap.HmeHHbIx lll8pHKOB H306pameHbI
H8 pHCYHKe 8.4.
213

CUJla B3aU.MOOeUCmBUSl .MeJICoy 06YMSl lien006UJlClibt.MU mOllelllibt.MU 3Jl.eKm
pUlleCKU.MU 3apSlOa.MU 8 6aKYY.Me npSl.MO nponopquoliaJl.blia npOU36eOeliUIO 6e-
Jl.UIlUIi 3apSlOOB. 06pamlio nponopquonaJl.blia K6aopamy paCCmOSlliUR. .MeJICoy IiU.MU
U lianpa6Jl.elia 600Jl.b npSlMOU, coeouliSllOw,eu 3mu JapSlObt.
38KOH KYJlOH8 Cnp8Be,nJlHB H ,nn.H 38p.HmeHHbIX llIapOB H8 JlI060M paCCTO.H-
HHH Mem,ny HX eHTpaMH, ecnH 06'beMHa.H HJIH nOBepXHOCTHa.H rrJlOTHOCTb 3a-
P.H,n8 Kam,AOrO H3 HHX rrOCTO.HHH8. (OTMeTHM, 'ITO B OTJlHqHe OT rp8BHTaH-
OHHoro 3J1eKTpOCTaTHqeCKOe B3aHMO,AeHCTBHe MomeT npHBO,nHTb K rrpHT.HmeHHIO
H OTTaJlKHBaHHIO TeJl.)
H'M 2
K03<P<PHI.J;HeHT npOIIOpu,HOH8J1bHOCTH k = 9 . 10 9 -. Q8CTO BMeCTO k HC-
Kn 2
nOJlb3YlOT ,npyrylO nOCTO.HHHYIO, H83bIB8eMYlO 3J1eKTpH'IeCKOH nOCTO.HHHOH:
1 K.'I 2
£0 = - = 8,85 .10- 12 2 ' Tor,Aa 38KOH KynoHa MOmHO 38nHC8Tb T8K:
41tk (II. M)
F = Iql11q21
41t£0 r 2 .
F =  = Iql11q21
41t£0 r 2 '
CHJla B38HMO,neHCTBH.H TO'le'lHbIX HerrO,ABHmHbIX 38p.H,nOB B Cpe,Ae C ,nH3J1eK-
TpH'IeCKOH rrpoHHu,8eMOCTblO E (CM. HHme  8.19)
F = Iqlllq2\
41tEo Er 2 .
,I:(JI.H B8KYYM8 H B03,nyxa £ = 1, B JII060H ,npyroH cpe)J;e CHJIa B38HMo,neHcT-
BH.H YMeHbllI8eTC.fI B £ p83.
8.6. 3JIeKTpOCTaTH1.J:eCKOe nOJIe
38KOH KYJIOHa orrpe,Aen.HeT CHJIY B3aHMo,neHcTBH.H Mem,ny 3neKTpHqeCKHMH
38p.H.naMH. HO He 06'b.HCH.HeT, KaK 3TO B3aHMo,neHCTBHe rrepe.naeTC.H H8 paCCTO-
.HHHe OT O,AHOrO TeJl8 K pyrOMY.
OrrbITbI nOKa3bIB8IOT, qTO 3TO B3aHMo,neHCTBHe H86J1IO,naeTC.H H TOr,A8, Kor-
,na Ha3J1eKTpH30BaHHbIe TeJla H8XO.n.flTC.H B BaKYYMe. 3Ha'lHT, ,nJI.H 3JIeKTpHqeC-
Koro B38HMo.neHCTBH.H He HymHa cpe.na. Do TeopHH, p83BHTOH M. cI>ap8,neeM H
,I:(m. M8KCBeJlJlOM, B npOCTp8HCTBe, r,ne H8XO,AHTC.H 3J1eKTpH'IeCKHH 38p.H,n, cy-
ecTByeT 3J1eKTpHqeCKOe nOJIe.
3JIeKTpOCTaTH'IeCKOe nOJIe - OC06bIH BH,n MaTepHH, ee HCTO'lHHKOM
.HBJI.HIOTC.H Heno,nBHmHbIe OTHOCHTeJIbHO p8CCM8TpHB8eMoH HHepn;H8J1bHOii CHCTe-
MbI OTCqeT8 (MCO) 38p.H,nbl, nocpe,nCTBOM KOTOpOH ocyeCTBJI.fIeTC.H HX B3aH-
Mo.neHcTBHe.
TaKHM 06P830M, 3J1eKTpOCTaTHqeCKOe nOJle - MaTepHaJlbHO. OHO He-
npepbIBHO B npOCTpaHCTBe. llcxo,n.H H3 COBpeMeHHbIX npe,ncTaBJleHHH, He-
no,nBH)KH8.H 3ap.HmeHHa.H qaCTUn;8 .HBJI.HeTC.H HCTO'lHHKOM 3J1eKTpOCT8THqeCKO-
ro nOJl.H, a HaJlHqHe nOJl.H - npH3HaKOM cyecTBOB8HH.H caMOH 38pHmeHHoi1
qaCTHn;bI. B3aHMo.neHcTBHe 3J1eKTpHqeCKHX 3ap.H,nOB CBO,nHTC.H K cJle.nYIOe-
214

MY: nOJle 3apHAa ql AeHcTByeT Ha 3apHA q2' a none 3ap.H,Aa q2 AeHcTByeT
Ha 3ap.HA ql' 3TH B3aHMOAeHCTBH.H nepe,AaIOTC.H He MrHOBeHHO, a C KOHe'l-
HOH CKOpOCTblO, paBHOH CKOpOCTH CBeTa c = 300000 KM . 3neKTpH'leCKOe
C
nOJIe, C03,AaBaeMOe HenOABHmHbIMH 3J1eKTpH'ieCKHMH 3apH,AaMH OTHOCHTeJIb-
HO paccMaTpHBaeMoH MCO Ha3bIBaeTC.H 3J1eKTpOCTaTHqeCKHM.
MbI He MomeM HenocpeACTBeHHO BocnpHHHMaTb 3JIeKTpOCTaTH'ieCKOe nOJIe
C nOMOblO HaIllHX opraHOB 'iYBCTB. 0 cyecTBoBaHHH 3neKTpOCTaTH'leCKOrO
nOJIH MbI MomeM CYJJ.HTb no ero AeHCTBHHM. 3JIeKTpOCTaTH'ieCKOe nOJIe 3apH-
.na .neHcTByeT C HeKOTopOH CHJlOll Ha JII060H l(pyroll 3apHl(. OKR3RBIllHllCH B
none AaHHoro 3ap.H,Aa.
CHna, C KOTOpOH 3JIeKTpOCTaTH'ieCKOe nOJIe AeflcTByeT Ha BHeceHHbIll B Hero
3JIeKTpHqeCKHH 3apHlI, Ha3bIBaeTCH 3J1eKTpHQeCKOH CHJlOH.
AeHcTBHe 3JIeKTpOCTaTH'ieCKOrO nonH Ha 3ap.HA 3aBHCHT OT pacnOJlOmeHHH
3ap.H,Aa B 3TOM none.
ECJIH eCTh HeCKOJlbKO 3ap.HmeHHbIX TeJI, pacnOJIOmeHHblX B pa3JIH'iHbIX TOt.J-
Kax npOCTpaHCTBa, TO B JII060H TOt.JKe 3Toro npOCTpaHCTBa 6YJJ.eT npOHBJIHTb-
CH COBMeCTHoe ,AeHCTBHe Bcex 3apHJJ.OB, T.e. 3JIeKTpOCTaTH'leCKOrO nOJI.H, CO-
3,AaBaeMOro BceMH 3THMH 3Rp.HmeHHbIMH TeJIaMR.
8.7. HanpHmellilocTb 3JIeltTpOCTaTH'IeCKOrO nOJIH.
JIHIIHH lIanpHmellllOCTH
AJIH HCCJIe,AOBaHHH 3J1eKTpOCTaTH'leCKOrO nOfl.H B Hero Heo6xoJJ.HMU BHeCTH
npo6HbIH 3ap.HA qo (3ap.HA qo - TOt.Jet.JHbIH nOJIOJKHTenbHbIll 3ap.HA AOCTaTOt.JHO
ManOll BenH'lHHbI. t.JT06bI ero BHeceHHe B UCCJIeJJ.yeMoe none He HCKan<aJIO 3TO
none, HHa'ie. 3TO 3ap.Hl(. c06CTBeHHoe none KOToporo npeHe6pe.mHMO MaJIO no
cpaBHeHHIO C HCCJIe,AyeMbIM) H H3MepHTb CHJIY, KOTopaH AeHcTByeT Ha Hero B
3aARHHOH TOqKe nOJl.H.
CHJla, AeHCTBYIOa.H Ha 3ap.HJJ. qo, nOMeeHHblfI B 'l'Ot.JKY M HccJleAyeMo-
ro nOJIH (pHC. 8.5), nponopHOHaJIbHa 3TOMY 3apHAY H 3aBHCHT OT CBOHCTB
nonH B 3TOH TOt.JKe: F M - qo' OTHoIlleHHe .me 3TOH CHJIbI K 3ap.HJJ.Y qo He
3aBHCHT OT BHOCHMoro 3ap.HAa H AJISI ARHHOH TOt.JKli nOJIH eCTb BenHt.JHHa
F ----+
nOCTO.HHHRH: - = const;
qo
/",-----------
/ q -----....
f  ...."
I '\
I M \
\ ... I
\ qo F N:
'........ I
.... I
'.... /
........ I
....... /
--------/"
Puc. 8.5
- F
E=-.
qo
3TO OTHOIlleHHe Ha3bIBaeTCH Hanp.HmeHHOCTblO
nOJIH E H HBnHeTC.H xapaKTepHcTHKOH CHJIOBOrO
JJ.eHcTBHH 3JIeKTpOcTaTHt.JecKoro nOJI.H Ha BHeceHHbIe B
Hero 3ap.HmeHHbIe Tena.
B CM eAHHHeH Hanp.HmeHHOCTH nOJlSl SlBJISleTC.H
HLIOTOH HR KYJIOH (  ) "JI" ROJILT HR MeTp ( : )
(CM.  8.16).
215

HanpHJKeHHOCTLIO E B ,ARHHOH TO'tlKe 3neKTpOCTRTlPIeCKOrO nOJISI HR3bIBa-
lOT BeKTOpHYIO cpH3H'tleCKYIO BenH'tlHHY, paBHYIO OTHOIIIeHHIO CHJlbI. .l(eHCTBYIOeH
B ,AaHHOH TOqKe Ha TOqeqHbIH np06HbIH 3apH, K 3TOMY 3ap.HlIY.
3Ha.H Hanp.H>KeHHOCTb B HeKOTOpOH TO'tlKe nOJl.H, BCer.l(a MOmHO Onpejle-
JlHTb CHJIY, .l(eHCTBYIOYIO Ha nlO60H 3ap.H,A q, nOMeeHHbIH B ,AaHHYIO TO'tlKY
nOJI.H: F = qE .
rpacpHqeCKH H306pamaTb 3J1eKTpOCTaTH'tleC-
Koe none Y.l(06Hee He B BH.l(e BeKTOpOB Ha-
npHJKeHIlOCTH. 8 C nOMOblO JlHHHH Hanp.H-
meHHOCTH. 3TO npelInO>KHJI M. <1>apa,AeH. TIpH
nOCTpoeHHH JIHHHH Hanp.HmeHHOCTH He06xOlIH-
MO npH.l(epmHBaTbC.H CJle.l(YIOero:
R) KaCaTeJIbHaH K RHHHli Hanp.H>KeHHOC-
TH B Kam,AOH TOqKe COBna,AReT C BeKTopOM
Hanp.HmeHHOCTH if; (pHC. 8.6);
6) nHHHH Hanp.HmeHHOCTH 3neKTpOCTaTH-
qeCKOrO non.H - He3aMKHYTbIe JlHHHH: OHH
Ha'tlHHaIOTC.H Ha nOBepXHOCTH nOJIOmHTeJIbHO
3Rp.H>KeHHbIX TeJl (HJIH B 6eCKOHeqHOcTH)
(pHC. 8.7. a, 6) H OKaH'tlHBaIOTCH Ha nOBepx-
HOCTli oTpHqaTeJIbHO 3apHmeHHbIx TeJI (HJIH
B 6eCKOHe'tlHOCTH);
B) JIHHHH HanpHmeHHOCTH He nepeceKalOT-
C.H, TaK KaK B Kam.l(OH TO'tlKe nOJl.H BeKTOp
E HMeeT nHIIIb O.l(HO HanpRBneHHe;
r) B JlI060H TOqKe non.H nJlOTHOCTb JIHHHH Hanp.HmeHHOCTH p8BH8 MO.l(YJIIO
Hanp.H>KeHHOCTH nOJl.H B 3TOH TOqKe. TIJlOTHOCTblO JlHHHH Hanp.HmeHHOCTH Ha3bI-
BalOT qHCJlO nHHHH. npOXOlI.HIIIHX CKB03b eHHH'tlHYIO nJlOa.nKY, nepneHlIHKYJI.Hp-
HYlO HM. B CJIae TOqe'tlHOrO 3apH.l(a (HJlH IIIapa) , HRXO.l(HerOCH H8 .l(OCTaTOqHO
60JIbIIIHX paCCTO.HHHHX OT .ltPyrHx TeJI, 3JIeKTpOCTaTH'tleCKOe nOJIe CHMMeTpH'tlHO
OTHOCHTeJIbHO qeHTpa 3ap.H.l(a (pHC. 8.7, a. 6).
E
Puc. 8.6
a
6
Puc. 8.7
a


Puc. 8.8
6
Ha pHCYHKe 8.8, a H306pameHo nJlOCKOe CeqeHHe nOJI.H .l(BYX O.l(HOHMeHHO
3ap.HmeHHbIX IIIapOB, a Ha pHCYHKe 8.8, 6 - .l(BYX pa3HOHMeHHO 3apHmeH-
216

HbIX mapOB, Ha pHCYHKe 8.9, a - 3neKTpOCTaTHtJeCKoe nOJle 6eCKOHeqHO:H
paBHOMepHO 3ap.HmeHHOH nJlOCKOCTH, Ha pHCYHKe 8.9, 6 - nOJle CHCTeMbJ
ABYX pa3HOHMeHHO 3ap.HmeHHbIX nJlOCKOcTeH.
a

6
+ + + + + + +
Puc. B.9
8.8. 3JIeKTpOCTaTII'IeCKOe nOJIe TO'le'lHOrO 3ap.Rna
II paBHOMepHO 3ap.RmeHHOH cepM
IIycTb nOJle C03AaHO TOqeqHbIM 3apSIAOM <J. On-
peAeJlHM HanpSImeHHOCTb nOJl.H B TOqKe M. Haxo-
,A.HeftcSI Ha paCCTO.HHHH r OT 3ap.H,A8 (pHC. 8.10).
IIoMecTHM B 3TY TO'lKY np06HbIH nonOmHTeJlbHbIH
3ap.HA qo. Tor,Aa, COrJlaCHO 3aKOHY KYJlOHa, Ha 3a-
pSI.n. qo co CTOpOHbI 3apSI.n.a q ,AeHcTByeT CHna. MO-
.n.YJlb KOTOpOH F = Iqllqol .
41re o r2
COrJlaCHO onpe,neJleHHIO, MO,nYJlb Hanp.HmeHHOCTH non.H B TotJKe M
q r M
._-------------e
qo
Puc. B.I0
E =  = Iqllqol _ Iql
I I 2 2 .
qo \qo 1 41re o r 41tEo r
ECJlH 3ap.H q Haxo.nHTCSI B cpe,ne C ,nH3J1eKTpHtJecKoH npOHHuaeMOCTblO E,
TO MO.D;YJlb Hanp.HmeHHOCTH nOJl.H. C03aBaeMoro np06HbIM 3ap.H.D;OM q H8 pac-
CTO.HHHH r OT Hero, COCTaBJI.HeT
E = Iql .
41tEoEr2
Hanp8.BJleHHe BeKTopa Hanp.HmeHHOCTH B TOqKe M nOJl.H 3aBHCHT OT 3HaKa
3ap.H,na, C03,nalOero 3TO none, H COHanpaBneHO C CHJlOH, ,neHCTBYIOeH Ha 3a-
p.H.D; qo (pHC. 8.11 a, 6). MomHo CB.H3aTb Ha-
npaBneHHe BeKTopa Hanp.HmeHHOCTH C HanpaBne-
HHeM pa,nHyca-BeKTopa TOqKH M. ECJlH 3ap.H,n,
c03,nalOHH nOJle, q> 0, TO E ii f. T.e. Hanp.H-
meHHOCTb HanpaBJleHa OT 3ap.H.na, C03.nalOlliero
nOJle. ECJlH me q < O. TO IE HanpaBJleHa Ha-
BCTpeqy r, T. e. K 3ap.H,ny, c03,nalOeMY none:
- q r
E= .-
41tEo Er 2 r'
a q>O M E
r ---f-- l F . .
6 q<O E F
M
r « c . i
r
Puc. B.ll
217

IIo 3TOH cpOpMYJle MOmHO paCCqHTbIBaTb Hanp.HmeHHOCTb nOJlH, 06pR30BRH-
HOrO 3apHmeHHOH npOBO.HeH CcpepOH, B TOqKaX, HRXO.HHXC.H HR nOBepX-
HOCTH CcpepbI H BHe ee. BHYTPH me npOBO.HeH Ccpepbl E = 0 TaKHM 06pa-
E = Iql (eCJlH
41tC o r2
30M,
r  R). E = 0 (ecnH r < R). re R - PRHYC Ccpepbl.
8.9. npHHHn cynepno3HHH nOJIeil
q2>O
,
I
,
,
I
,
I
, I
", M/ - -
q3:____------; r:c " Es _ Fa
E 2 EI
F 2 FI
ql>O
..
,
"
,
,
,
,
,
,
Puc. 8.12
EcnH 3J1eKTpOCTRTHecKoe none C03,AaHO
oHoBpeMeHHo HeCKOJlbKHMH 3ap.HaMH (pHC.
8.12), TO pe3YJlbTHPYIOR.H CHJIa, eHCTBYIO-
R.H HR np06HbIH 3Rp.H qo> 0 B TOqKe M
3Toro nOJl.H:
Fp = Pi + F 2 + F3 .
PR3eJlHM 06e QRCTH npeblyero P8-
- F
BeHCTBa Ha qo Ii YTeM. TO E = - :
qo
Fp PI P 2 P 3
- = - + - + - HnH Ep = E 1 + E 2 + E3 .
qo qo qo qo
IIOCJIeHee paBeHCTBO BblpRmaeT npHHHn cynepn03HJJ;HH nOJIeH: HanpSlJICeH
Hocmb B Jl.1060U mOIlKe 3Jl.eKmpUlleCKOZO nOJlSl cucmeMbl 3ap.R806 paBHQ zeo-
MempUllecKOU cYMMe HanpSlJICe1tIiOCmeU nOJl.eu. c038a6aeMblX 6 3mou mOIlKe KaJIC-
8bl.M U3 3ap.R006 6 Om8eJl.bHOcmu:
;q2
'"
/
/
+ql k:f - 2 :_ E
---- - - p
-------- //
---- a //
M .-.
EI
Puc. 8.13
n
Ep = LEi'
i=1
IIYCTb, HRnpHMep, nOJIe C03,ARHO BYM.H
pa3HOHMeHHbIMH 3ap.HaMH +ql H -q2 (pHC.
8.13). IIo npHHn:,Hny cynepn03Hn:,HH
£p = £1 + £2' BeKTOp Hanp.HmeHHOCTH nOJl.H B
TOqKe M HRXO.ZI.HTC.H no npaBHJlY napRJlJleJlO-
rpaMMa, a Moynb Ep - no TeopeMe KOCHHY-
COB:
Ep =  E + E + 2E 1 E 2 cosa.
ECJIH 3J1eKTpOCTRTHqeCKOe nOJle C03B.H0 COBOKynHOCTblO npOT.HmeHHbIx 3Rp.H-
meHHbIX TeJl, TO HX MbICJleHHO pa36HBalOT HR TOqetIHbIe 3RpHbl H Hanp.HmeHHO-
CTH non.H B JII060H TOKe npOCTpaHCTBR HRXO.HT KRK reOMeTpHeCKYIO CYMMY
HRnp.HmeHHOCTeH noneH 3THX TOqeHbIX 3RpHOB:
218
£ =  £. =  qi . 
p L.J, L.J 4 2
i=1 i=1 1tC o Eli 'i

8.10. OHopoHoe aneKTpOCTaTHqeCKOe none
O,aHOpO,L{HLIM 3JIeKTpOCTaTHqeCKHM nOJIeM Ha3hIBaeTC.H nOJIe, B Ka>K.lJ.OH TOt.I-
Ke KOToporo BeKTOp Hanp.HmeHHOCTH E HMeeT O.lJ.HHaKOBoe HanpaBJIeHHe H MO-
--+
.u.ynh, T.e. E = const.
d
Puc. 8.14
O.u.HOpO.u.HOe 3J1eKTpOCTaTH'leCKOe nOJle H306pa>KaeTC.H na.
paJIJlenbHbIMH nHHH.HMH HanpH>KeHHOCTH C 0.u.HH8KO-
BOH nnOTHOCThlO. O.u.HOpO,AHhIM HBJI.HeTCSI 3JIeKTpOCTa-
THqeCKOe none Mem,Ay ,ABYM.H pa3HoHMeHHo 3apHmeH-
HbIMH nnaCTHHaMH (noJIe KOH.u.eHcaTopa) (pHC. 8.14),
ecnH paCCTO.HHHe d Mem.u.y nJlaCTHHaMH HaMHoro
MeHbIlle JlHHeHHbIx pa3MepoB nJlaCTHH (d« l). Ha
KpaHX nJlaCTHH JlHHHH HanpHmeHHOCTH HCKpHBJI.HIOT-
C.H H nOJIe He .HBJIHeTC.H O,AHOpO.u.HhIM.
8.11. fiOTOK BeKTopa HanpHjKeHHOCTH
aneKTpOCTaTHqeCKOrO nonH
BBe,AeM nOHHTHe nOTOKa N BeKTopa HanpHmeH-
HOCTH if CKB03b HeKOTOpylO nOBepXHOCTb nJIOa.u.hlO
S. nYCTb nnOCKa.H nOBepXHOCTh nJIOa.u.hlO S Haxo-
.u.HTCH B O.u.HOPO.u.HOM 3JIeKTpOCTaTHqeCKOM nOJle E
(pHC. 8.15). BeKTop ii - HOpMaJIh K 1I0BepXHOC-
TH. YroJI Me>K.u.Y HanpaBJIeHHeM JIHHH:H HanpH>KeH-
HOCTH H HOpMaJIblO p8BeH a. llOTOKOM N BeKTopa
HanpHJKeHHOCTH E '1epe3 nOBepXHOCTb nJIOa.u.hlO
S Ha3bIBalOT H3Ht.IecKYIO CKaJl.HpHYIO BeJIHqHHY, on-
pe.u.eJI.HeMYIO BbIpa>KeHHeM
N = E S cos a = En S ,
r.u.e En - npOeKI.J,H.H BeKTopa E Ha HanpaBJIeHHe HOpMaJIH ii .
TaK KaK rYCTOTa JIHHHH Hanp.HmeHHOCTH xapaKTepH3yeT Mo.u.ynb Hanp.H>KeH-
HOCTH E, TO MOmHO CKa3aTh, t.ITO nOTOK BeKTOpa HanpHmeHHOCTH t.Iepe3 .u.aH-
HYIO nOBepXHOCTb paBeH nOJlHOMY qHCJlY JIHHHH Hanp.HmeHHOCTH, npOXO.u..HHX
'lepe3 3TY nOBepXHOCTb.
ECnH nOJle Heo.u.HopO,AHO, a nOBepXHOCTb He HBJI.HeTC.H nJIOCKOH, TO B
aTOM cJlYt.Iae .u.JIH onpe.u.eJleHH.H nOTOKa BeKTopa Hanp.H>KeHHOCTH nOBepXHOCTb
pa36HBaeTC.H Ha He60JlbIllHe yqaCTKH, KOTopbIe MO>KHO CqHTaTb nnOCKHMH,
a nOJle B npe.u.eJlax KamlIoro H3 HHX o.u.HOpO,AHbIM. 3aTeM Haxo.u..HT 3J1eMeH-
TapHbIe nOTOKH BeKTopa Hanp.HmeHHOCTH N; t.Iepe3 MaJlble nJlOa.u.KH S; no
opMYJle N; = E;S; cos a;. IIOJlHbIH nOTOK qepe3 nOBepXHOCTb paBeH aJlre6pa-
ii E
Puc. 8.15
Ht.IeCKOH cYMMe 3JIeMeHTapHbIx nOTOKOB t.Iepe3 Bce ee yt.IaCTKH:
n
N = 2,N i .
i=l
219

8.12. TeopeMa OCTporpa,ncKoro- raycca
H ee npHMeHeHHe ,nnH paCqeTa aneKTpOCTaTHqeCKHX noneH
ITOJleHHbIH pe3YJIbTaT 6Y,AeT CnpaBe,AJl1iB 1i ,nn.H
nOBepXHOCT1i npOH3BOJIbHOH cpOpMbI, a TaKme npH mo-
60M pacnOJlOmeH1i1i 3ap.H,Aa BHYTPH 3TOH nOBepXHOCTH. ,Z:(eHCTB1iTeJIbHO, eCJIH OK-
Pym1iTb ccpepy np01i3BOJIbHOH 3aMKHYToH nOBepXHOCTblO (p1ic. 8.16. a - no-
BepXHOCTb 1i306pameHa lllTp1iXaM1i), TO Kam,na.H JIHHH.H Hanp.HmeHHOCT1i, npo-
H1i3bIBalOa.H ccpepy. npOH,neT H CKB03b 3TY nOBepXHOCTb.
ECJI1i 3aMKHYTa.H nOBepXHOCTb np01i3BOJIbHOH <pOpMbI OXBaTbIBaeT 3ap.H.n;
(p1ic. 8.16, 6). TO npH nepeCeqeH1i1i JlI060H BbI6paHHOH JI1iHHH Hanp.HmeHHOC-
T1i C nOBepXHOCTblO OHa TO BXO,A1iT B nOBepXHOCTb, TO BbIXO,A1iT 1i3 Hee. He-
qeTHOe q1iCJlO nepeCeqeH1iH np1i BbI'l1iCJIeH1i1i nOTOKa B KOHeqHOM CqeTe CBO-
.llHTC.H K OllHOMY nepeCeqeH1iIO. TaK KaK nOTOK Cq1iTaeTC.H nOJlOm1iTeJIbHbIM.
eCJlH n1iH1i1i Hanp.HmeHHOCT1i BbIXO,n.HT H3 nOBepXHOCT1i. 1i OTp1iaTeJlbHbIM ,nn.H
JlHH1i1i, Bxo,n.HeH B nOBepXHOCTb. ECJI1i me BHYTPH nOBepXHOCTH nJlOa,nblO
8 1 (CM. p1iC. 8.16) 3ap.H,AbI OTCYTCTBYIOT, TO nOTOK Hanp.HmeHHOCT1i qepe3
3TY nOBepXHOCTb paBeH HynlO (N s= 0).
Ecn1i paCCMaTp1iBaeMa.H nOBepXHOCTb OXBaTbIBaeT
He O,nHH, a HeCKOJIbKO 3neKTp1iQeCK1iX 3ap.H,nOB, TO
no.n; q CJIe.n;yeT nOH1iMaTb aJlre6paHQeCKYIO CYMMY
3T1iX 3apSl,AOB (pHC. 8.17) 1i
a ,,-'\.
", .....
"" \
// \
-- \
",,- \
/ \
/ I
/ I
I I
I I _
I n
\ 1
\ 1-
N= 0 1 n
1, I
" I
'\. --"
,"'"'--",.----
6
Puc. 8.16
G)q3
Puc. 8.17
220
ITYCTb nOJle C03,naeTC.H TOQeqHbIM 3JIeKTp1iQeC-
K1iM 3ap.H.n;oM q. ITpoBe,AeM 3aMKHYTYIO c<pep1iqec-
KYIO nOBepxHocTb nnoa.n;blO 8 (p1iC 8.16), OK-
pymalOYIO 3TOT 3ap.H.n;, eHTp KOTOpO:H COBna.n;a-
eT C TOQKOH HaXOm,AeH1i.H 3ap.H,na. BbIQ1iCJI1iM no-
TOK BeKTopa Hanp.HmeHHOCTH Qepe3 3TY nOBepx-
HOCTb. 3a nOJlOm1iTeJIbHOe HanpaBJIeH1ie HOpMan1i
BbI6epeM HanpaBneH1ie BHelllHeH HopManH n B
3TOM cJlYQae BO Bcex TOQKaX c<pep1iQeCKOH nOBepx-
HOCT1i E = const 1i cosa = 1.
MO.n;YJlb Hanp.HmeHHOCT1i nOJI.H Ha paCCTO.HH1i1i R
OT 3ap.H,na E = q . ITJIOI.lta,nb nOBepXHOCTH
41tE O ER 2
ccpepbI 8 = 4nR 2 .
CJle.n;oBaTenbHo, nOTOK BeKTopa Hanp.HmeHHOCT1i
Qepe3 ccpep1iQecKYIO nOBepXHOCTb
q4nR 2 q
N s = =
41tEOER2 Eo€
N s = ql + q2 + q3 .
EOE
8Ta CPOPMY JIa BbIpamaeT TeopeMY OCTporpaAcKo-
ro-raycca: nomo" 6eKmopa lianpSlJICeliliOCmu llepe3
3aMKliymy1O n06epXliocmb pa6eli aJl.Ze6paUllecKou

CYMMe 3apSl006 paCnOJl.OJICelilibLX 6liympu amou n06epXIiOCmU. iJeJl.eliliOU liQ
npOU36eOeliUe aJl.eKmpUlleCKOU nOCmOSlliliOU U OUaJl.eKmpUlleCKOU npOliUijaeMOC-
mu cpeiJbL.
TIpHMeHHM 3TY TeOpeMY .n;n.H paCqeTa 3J1eKTpOCTaTHQeCKHX nOneH HeKOTO-
pbIX npOBO.n;HHKOB.
1) PaBHoMepHo 3apHeHHaH 6eCKOHeqHaH
ii nJlOCKOCTb.
TIYCTb a - nOBepXHOCTHa.H nnOTHOCTb 3a-
p.H.n;a HB nJlOCKOCTH (pHC. 8.18).
B KaqeCTBe nOBepXHOCTH nJlOa.nblO S BbI6e-
peM I..{HnHH.n;pHqeCKyIO nOBepXHOCTb, 06pa3YIOa.H
KOTOpOH nepneH.n;HKyn.HpHa nnOCKOCTH. OCHoBa-
HHH 3Toro I..{HnHH,Apa pacnOJlOeHbl nepneH.n;HKY-
JI.HpHO nHHH.HM Hanp.HJKeHHOCTH no 06e CTOpOHbI
Puc. 8.18 OT nJlOCKOCTH. TaK KaK 06pa3YIOHe I..{HnHH.n;pa
napaJlnenbHbI nHHH.HM HanpHmeHHocTH (a. = 90°,
cosa = 0), TO nOTOK qepe3 60KOBYIO nOBepXHOCTb l{HnHH.n;pa OTCYTCTByeT, H nOJl-
HbIH nOTOK qepe3 nOBepXHOCTb I..{HnHH,Apa paBeH CYMMe nOTOKOB Qepe3 .n;Ba OC-
HOBaHH.H: N = 2ES. BHYTPH l{HnHH.n;pa 3aKnlOQeH 3ap.H.n; q = as, n03TOMY, CO-
as
rnaCHO TeopeMe OCTporpa.n;CKoro-raycca, 2ES = -, r)J;e E = 1 (,An.H BaKYYMa),
EoE
oTKy.n;a CJle.n;yeT, qTO Hanp.HmeHHOCTb non.H paBHOMepHO 3ap.HmeHHOH 6eCKOHeq-
HOH llJlOCKOCTH
a
E=-.
2EoE
2) BeCKOHeqHaH paBHOMepHO 3apHeHHaH HHTb.
TIYCTb 't - nHHeHHaH nnOTHOCTb 3ap.H,Aa
HHTH. BbI.n;enHM YQaCTOK HHTH ,AJlHHOH f1l H
E oKpymHM ero I..{HnHH,ApHqeCKOH nOBepXHOCTblO,
pacnonomeHHoH T8K, QTO OCb I..{HnHH.n;pa COBna-
t ii .n;aeT C HHTblO (pHC. 8.19).
JIHHHH Hanp.HJKeHHOCTH 3neKTpOCTaTHQeC-
Koro nOJl.H, C03,AaBaeMOro HHTblO B CeqeHHH,
6l nepneH.n;HKyn.HpHOM caMoH HHTH, HanpaBJleHbI
nepneH,AHKYJI.HpHO 60KOBOH nOBepXHOCTH I..{H-
Puc. 8.19 nHH,Apa, n03ToMY nOTOK HanpHmeHHocTH
CKB03b 60KOBYlO nOBepxHocTb N = E .21tRt:J ,
r.n;e R - pa.n;HYc I..{HnHH.n;p8.
t.{epe3 06a OCHOBaHH.H uHnHHApa nOTOK HanpHJKeHHOCTH paBeH HynlO (a. = 90°.
cosa = 0). Tor.n;a nOJlHbIH nOTOK Hanp.HmeHHOCTH qepe3 BbI.lI;eneHHbIH l{HnHH.n;p
N = E . 21tR6l .
3ap.H,A. HaXO.lI..HI.I.1HHC.H BHYTPH 3Toro I..{HnHH,Apa. q = "t. 6l .
CornacHo TeopeMe OCTporpa.n;cKoro-raycca, MomHO 3anHcaTb E. 21tRlll =
till
= -  CJle,AOBaTeJlbHO, Mo.n;ynb Hanp.HJKeHHOCTH nOJl.H, C03.n;aBaeMoro paB-
EoE
221

HOMepHO 3apHmeHHOH 6eCKOHeqHO ,lJ.nHHHOH HHTblO Ha paCCTO.HHIUI R
OT Hee,
E=
27tEOER
't
BOIIPOCbl .D:JI.H CAMOKOHTPOJI.H
1. KaKue 63aU.MoiJeucm6uSl lia3bl.6alOm 3Jl.eKmpOMaZIiUmlibl.Mu? t[mo nOliu-
.MalOm noiJ 3Jl.eKmpUllecKuM 3apSliJoM? KaKou 3apRO lia3bl.6alOm 3Jl.e.Melimap-
libl.M?
2. B KaKOM cJl.Yllae .MaKpOCKonullecKoe meJl.O 3apSlJICeli0? MOJICem Jl.U U3.Me-
liSlmbCSl cYM.Maplibl.U .JapSliJ 3a.MKliymou CUCme.Mbl.?
3. KaK npoucxoiJum 3Jl.eKmpU3al{.UJl meJl. mpeliue.M? riJe UCnOJlb3yemcR Sl6
Jl.eliue dJleKmpU3al{.Uu?
4. KaK cpoP.MYJl.upyemc.R 3aKOH KYJl.olia? KaK060 Hanpa6Jl.eliUe CUJl.bl. 63au-
.MoiJeucm6UJl, iJ8yx mOllelllibl.X 3apSliJ06?
5. t[mo npUliSlmo 3a OCli06liYlO eiJuliuUty 3apSliJa 6 Clf? KaK 3ma eiJuliUUta
yc malia6Jl.U6aemcSl?
6. KaK080 liaU.MeH06aliue 6 OCH06Hbl.X eiJuliUl{.ax CH K03rjJcpUl{.ueHma .. k"
6 3aKOlie KYJl.olia? KaK 6bl.paJICaemCSl K03C/JcpUijUelim .. k" llepe3 3Jl.eKmpUlleCKYIO
nocmoSlHIiYIO?
7. KaK ocyz.qecm6Jl.SlemCSl 63aU.MoiJeucm6ue 3Jl.eKmpOCmamUlleCKUX 3apRiJ06?
8. KaK06bl OCli06lible C60UCm6Q 3Jl.eKmpUlleCKUX nOJl.eu? C KaKOU CKOpOC-
mblO pacnpocmpa1iSllOmcSl 3JteKmp0.MaZliUmlibl.e 63aU.MoiJeucm6uSl?
9. t[ mo lia3bl8aemcSl lianpSlJICeliliOCmblO 3Jl.eKmpUlleCKOZO nOJl.R2 KaK060 lia-
npa6Jl.eliUe HanpSlJICeliliOCmu 3Jl.eKmpOCmamUlleCKOZO nOJl.Sl? KaK paCCIlUmbl.6aem.
CSl lianpSlJICeliliOCmb nOJl.Sl mOllellHOZO 3apRiJa?
10. B lleM cocmoum npUHl{.Un cynepn03UijUU nOJl.eu? KaK paCCIlUmbl.6aem-
CSl HanpSlJICeliliOCmb nOJl.Jl CUCme.MbL mOllelllibLX 3apSliJ06?
11. t[mo lia3bl.6aemCSl Jl,Uliueu lianpSlJICeliliOCmu 3Jl.eKmpOCmamullecKozo
noJl.Sl? KaK .MOJICliO COeJl.amb Jl.UliUU lianpSlJICeliliOCmu "6UOU.Mbl..MU"? KaK no
zycmome JtUliUU lianpSlJICeliliOCmu .MOJICIiO cyiJumb 0 lianpSlJICeliliOCmu nOJlJl,?
12. KaK06a KapmUHa JlUHUU HanpJl,:JICeH1iOCmU nOJl.R mOllel£1iOZO 3apRiJa?
nOJl.Jl iJ6YX mOllellliblX pa3HOU.Melilibl.X (oiJliOU.Melilibl.x) 3apRiJ06?
13. KaKoe nOJl.e 1ta3bl.6alOm OiJliOPOiJlibl..M? KaK06a KapmUlia JtUHUU lianpSl-
JlCeliliocmu maKOZO nOJl.Sl?
14. KaK u306paJICaemCR C nO.Moz.qblO Jl.UliUU lianpSlJICeliliOCmu nOJl.e 3apSl-
JlCeliHOU nJl.OCKocmu? nOJl.e 06YX napaJl.Jl.eJl.bHblX nJl.aCmUIi, 3apRiJbl. Komopblx pa6'
libl. no .MoiJyJl.1O U npomU60nOJl.OJIClibl. no 3liaKY?
15. II 0 KaKOU cpopMYJl.e paCCIlUmbl.6aemCSl HanpRJlCeliliOCmb nOJl.Sl 3apSlJICeli-
HOZO wapa?
OCHOBHbIE c1>OPMY JIb]
q=eN
- 3ap.H,A Tena ('I aCTHIJ:bI), r,lJ.e e - SJleMeHTapHbIH 3a-
p.H,lJ.; N - \lHCnO H36bITOqHbIX sneMeHTapHbIx 3apH,AOB
O,lJ.HOrO 3HaKa.
222

F = k I q1 1l;21 =  I q1 1l;21 - 3aKOH KYJlOHa, re F - MOYJlb CHnbI B3aHMO-
Er 41tE o £r eHCTBII.H HenOBllmHbIX TOqe'lHbIX 3apSlOB ql II
k - _ 9 - 1 0 9 H . M 2 n-n-
q2; - K03't''t'lIl1.lIeHT npOnOpl1. lI O-
KJl 2
HaJlbHOCTH; EO = 8,85.10- 12 q;>
M
- P
E=-
qo
E
E=
E
E=kEL=EL
Er 2 41tE o Er 2
Ep = El + E 2 + ... + En
n
HJlH Ep = L E,
1=1
(a) = 
8
HJlH
1 . !:!.q
a = Im-
68-+0 68
E=
2£oE
3J1eKTpHqeCKa.H
nOCTO.HHHa.H; E - H3J1eKTpHqeCKa.H npOHHl1.aeMOCTb
cpebI; r - paCCTOS!HHe Memy TOqeqHbIMIi 3ap.H-
aMIi.
HanpS!.meHHOCTb 3J1eKTpOCTaTHqeCKOrO non.H, re
F - CHJla, ,AeHCTBYIOa.H Ha nOJlOmHTeJlbHbIH np06-
HbIH 3apS! qo.
- H3J1eKTpHqeCKaSl npOHHl1.aeMOCTb cpebI, re Eo,
E - Hanp.HmeHHOCTb 3neKTpOCTaTII'IeCKOrO nOJl.H B
BaKYYMe H cpee COOTBeTCTBeHHO.
- MOYJlb HanpS!meHHOCTH 3neKTpOCTaTII'IeCKOrO
nOJl.H TOqeqHOrO 3ap.H,qa B HeKoTopoH TOqKe. r.ne
q - 3ap.H,A, C03alOlltliH nOJle; r - paCCTO.HHlle OT
3ap.H,Aa ,AO TOqKII.
- npllHl1.lIn cynepn031111.1I11 noneii, r,qe Ep - Ha-
np.HmeHHOCTb pe3YJlbTlipYlOlltero nonS!; EJ - HanpS!-
.meHHOCTb i-ro nonS!.
- CpeAH.HS! nOBepXHOCTHaS! nnOTHOCTb 3ap.HAa, rAe
q - 3ap.HA; 8 - nJlOllta,Ab nOBepXHOCTIi.
- MOAYJlb Hanp.HmeHHOCTIi nOJl.H, C03AaBaeMoro 6ec-
KOHe'lHOH paBHOMepHO 3ap.HmeHHOH nnOCKOCTblO.
223

METOAlIqECKIiE PEKOMEHAAQIIIi
3aaqH no 3JIeKTpOCTaTHKe Y06HO pa3eJIHTb Ha ,ABe rpynnbI. K nepBO:H:
rpynne MmKHO OTHeCTH 38,AaqH H8 p8CqeT CHJlOBbIX H 3HepreTHqeCKHX xapaK-
TepHCTHK 3J1eKTpOCTaTHqeCKOrO nOJl.H. KO BTOpO:H: - Bce 3alIaqH Ha paBHOBe-
CHe JIH60 lIBHmeHHe 3ap.HmeHHbIX TeJl (qaCTH) B 3neKTpOCTaTHqeCKOM none.
3alIaq:u nepBO:H: rpynnbI CB.H3aHbI C npenCTaBJIeHHHMH 0 cyecTBoBaHHH
3J1eKTpOCTaTHqeCKOrO nOJI.H BOKpyr HenOBH>KHbIX 3apH>KeHHbIX TeJI H CBO.HT-
CH 1< HaXO>K,AeHHIO OCHOBHbIX xapaKTepHCTHK nOJIH: Hanp.H>KeHHOCTH H nOTeH-
HaJlOB. TIp:u HX pelIIeHHH He06xoHMO:
1) C,AeJlaTb PHCYHOK, H306pa3HB JlHHHH HanpH>KeHHOCTH 3neKTpOCTaTHqeC-
Koro nOJI.H H 3ap.HmeHHbIe Tena, nOMeeHHbIe B 3TO nOJIe;
2) npH HaXOm,AeHHH HanpH>KeHHOCTH nOJIH B HeKoTopoE TOqKe CJIe,AyeT
nOMHHTb. qTO 3TO BeKTOpHa.H BenHqHHa H HanpaBJIeHa no KaCaTeJIbHO:H: K JIH-
HHH Hanp.H>KeHHOCTH. ECJlH nOJle 06pa30BaHo TOqeqHbIM 3apSlOM, TO OHa H8-
npaBJIeHa BOJIb np.HMOH. COeHH.HIOe:H: 3ap.H H TOqKY, B KOTOpO:H: onpe,Aen.H-
lOT E, OT 3apHa, eCJIH q> 0, H 1< 3ap.H,Ay, eCJIH q < 0;
3) eCJIH nOJle C03aHO CHCTeMO:H: TOqeqHbIX 3ap.HOB. TO Hanp.H>KeHHOCTb
nOJI.H paCCqHTbIBalOT COrJlaCHO npHHHny cynepn03HHH: Ep = E 1 + E 2 + ... + En'
He 3a6bIBa.H, qTO 3TO reOMeTpHqeCKa.H CYMMa. CJIeyeT npaBHJIbHO onpeenHTb
HanpaBJleHHe E j : JIH nOJIO>KHTeJIbHOrO TOqeqHOrO 3apHa BeKTOp E HanpaB-
neH no paHaJIbHO:H: JIHHHH OT 3ap.Ha, ,AJI.H OTpHaTenbHoro - K 3apHy;
4) eCJIH npH B3aHMOe:H:CTBHH 3ap.H>KeHHbIX TeJI npOHCXOHT nepepacnpeene-
HHe 3apHOB, TO 3anHlIIHTe 3aKOH coxpaHeHHH 3ap.HOB.
PelIIeHHe 3aaq BTOpO:H: rpynnM OCHOBaHO Ha npHMeHeHHH 3aKOHOB Mexa-
HHKH C yqeTOM 3aKOHa KYJIOHa H BbITeKalOHX H3 Hero CJIe,ACTBH:H:. TaKHe
3aqH peKoMeHyeTcH pelIIaTb B CJle,AYIOeM nopHKe.
1. Cena:H:Te PHCYHOK. H306pa3HB Bce CHnbI, lIeHCTBYIOWHe Ha 3ap.HmeHHOe
Teno, H 3anHlIIHTe ,AJI.H Hero YCJIOBHe paBHOBeCHH HJlH OCHOBHoe ypaBHeHHe
HHaMHKH.
2. Bblpa3HTe CHJIbI 3JIeKTpOCTaTHqeCKOrO B3aHMoeHCTBH.H 'lepe3 3apHbI
HnH xapaKTepHCTHKH non.H H nOCTaBbTe 3TH BbIpa>KeHHH B HCXOHoe ypaB-
HeHHe. TIpH 3TOM Ha,AO nOMHHTb. qTO Mem,Ay O,AHOHMeHHbIMH 3ap.H,AaMH ,AeH-
CTBYIOT CHJlbI OTTRJlKHBaHH.H, a Me>Ky pa3HOHMeHHbIMH - CHJIbI npHT.Hme-
HH.H. TIpH BbIqHCJIeHHH B «}>OPMYJlY 3aKOHa KYJIOHa nO,ACTaBJI.HIOT a6COJlIOT-
HbIe 3Ha'leHH.H 3apHoB, 3HaKH 3ap.HOB yqHTbIBalOT TOJIbKO npH onpeene-
HHH HanpaBJIeHH.H CHJI. ECJIH H3BeCTHa Hanp.HmeHHOCTb nOJI.H, TO CHJIY MO>K-
HO onpeeJlHTb no «}>opMYJle F = qE, yqHTbIBa.H, qTO F ii E, eCJlH q> 0 H
F i l E, ecnH q < O.
3. TIpH paCCMOTpeHHH lIBH>KeHH.H 3apHmeHHbIX qaCTH B 3JIeKTpOCTaTHqeC-
KOM none CJIeyeT BbI.HCHHTb xapaKTep ,ABHmeHHH H HCnOJIb30BaTb KHHeMaTH-
qeCKHe H HHaMHqeCKHe yp8BHeHH.H ,ABH>KeHH.H HJlH 3aKOH coxp8HeHH.H 3Hep-
rHH (HJIH TeopeMY 0 KHHeTHqeCKOH 3HeprHH).
4. 3anHlIIHTe BcnOMOraTeJIbHbIe «}>opMynbI, a nonyqeHHYIO CHCTeMY ypaBHe-
HHH pelIIHTe OTHOCHTeJIbHO HeH3BecTHoH BeJlHqHHbI.
224

nPHMEPbI PEIIIEHH.H 3A,l(A q
1. ,l(B8 38pSl,A8 +1,66 HKn H +3,33 HKn H8XO,ASlTCSI B B03,Ayxe Ha paccTo-
SlHHH 20 CM ,Apyr OT ,Apyra. r,Ae H8AO nOMeCTHTb TpeTHH 3apSl,A. 'lT06bI OH
OK838ncSI B paBHoBecHH?
,l(aHo: ql=1,66.10- 9 Kn. q2=3,33.10- 9 Kn. r=0,20 M, E=1.
H 8 H T H: rl'
ql Pz q PI
I: .  .
Tl
r
Puc. 8.20
,l(onycTHM, q -
CHnbI l't H i2 (pHC.
FI H F 2 Ha OCb x:
Pew e H He. 38pSl,A q H8XO,AHTCSI B paBHO-
q2 BeCHH, 3Ha'lHT, ij + F 2 = 6. OTCIO,Aa F 1 = -F 2 ,
J + T. e. CHnbI iI H F 2 ,AOnmHbI 6bITb H8np8Bne.
HbI B,AOnb OD:HOH npSlMOH B npOTHsononom.
HbIe CTOpOHbI. II03TOMY TO'lKa. r,Ae Ha,AO no-
.. MeCTHTb TpeTHH 38pSl,A q (nonomHTenbHbIH
x
HnH OTpHI.'(,8TenbHbIH), ,AOnmHa nem8Tb Ha
nHHHH. COe,AHHSlIOll.\eH 3TH 38pSl,AbI.
nonomHTenbHbIH 3apSl,A. Ha HerO ,AeHCTBYIOT KynoHoBcKHe
8.20) co CTOpOHbI 3apSl,AOB ql H q2' Cnpoeu.HpyeM BeKTopbl
11 - F 2 = 0 HnH Fl = F 2 . Y'lHTblB8S1. 'lTO FI = Iqlllql2 ;
41tEoErl
F 2 = Iqllq21 . nonY'lHM
41tEoE (r - rl )2
v'fQJr . _
r 1 = L1 /i:I' r 1 - 0,08 M.
vl Q1 1 + vl Q2 1
 _ Iqllq21 Iqll _ Iq21
2- ( ) 2=> 2- ( ) 2
41tEOEr} 4m:oE r - r l r l r - r}
OTKY,Aa
2. ,l(Ba OD:HHaKOSbIX He60nblIlHx lIlapHK8 M8CCOH 0.10 r KamD:bIH nO,Ase-
ll1eHbI B s03yxe Ha HHTSIX nHHoH 25 CM. IIocne Toro KaK lIlapHKaM 6bInH
C006ll.\eHbI O,AHHaKOSbIe 3apSl,AbI. OHH pa301I1nHCb Ha paccToSlHHe 5.0 CM. On-
peD:enHTe 3apSlD:bI lIlapHKOB.
)laHO: ml=m2=m=O,10.10- 3 Kr, l}=l2=l=25.10- 2 M, r=5.0.10- 2 M.
E = 1.
H8HTH: q.
Puc. 8.21
F
Pew e H He. H8 K8m,AbIH H3 OTKnOHeUHbIX
lIlapHKOB ,Aei1:cTSYIOT CHnbI: mg - CHna TSI-
meCTH, T - CHna HaTSlmeHHSI HHTH. F
KynoHoBcK8S1 CHna S38HMO,AeHCTBHSI lIlapHKOS
(pHC. 8.21). 3anHlIleM ycnoBHe p8BHoseCHSI
O,AHOrO H3 1I18pHKOB B BeKTopHoH <}>opMe:
mg + t + F = 6.
CnpOeI.'(,HpyeM BeKTopHble BenHt.J:HHbI Ha KO-
Op,AHH8THbIe OCH X H Y:
x -Tsina + F = 0: Tcosa - mg = O.
HCKnlO'l8S1 H3 3THX ypasHeHHH T. nonY'l8eM
tg a =  = Iql2
mg 41tE o Er 2 mg
mg
225

T8K K8K l»!... TO yron ex M8n Ii T8HreHC MomHO 38MeHliTb CHHYCOM:
2
. r T r
tga == Slna = -. Or,A8
2l 2l
Iql 2
I I 21tEoEmgr I I
OTCIO,A8 q = r ; q = 5.2 HKn.
41tEoEr2mg . l
3. nOBepxHocTH8S1 nnOTHOCTb 38pSl,A8 Ha W8pliKe P8,AHYCOM 2.0 CM p8B-
HEn
H8 200 . H8H,AHTe H8npSlmeHHocTb nonSl B TO'lKe. Y,A8neHHOH OT nOBepx-
M
HOCTH wapHK8 H8 2.0 CM.
En
.l! 8HO: R=0.020 M. 0=2.0.10- 7 _ 2 r=0.020 M.
M ·
H8HTH: E.
Pew e H He. MO,Aynb H8npSlmeHHOCTH E = q . S8pSl,A W8pHK8
41tEoE (R + r)2
MomHO H8HTIi. 3HaSl nOBepXHOCTHYIO nnOTHOCTb 38p.H,AOB q = oS = o4nR 2 .
o4nR 2 oR 2 KB
Cne,AOS8TenbHO. E = 2 2 ' E = 5,7
41t£oE(R + r) EoE(R + r) M
4. Mem,Ay ropli30HT8nbHbIMIi nn8CTliHaMIi 38pSlmeHHoro B03,AYWHoro KOH-
B
,AeHC8Top8. H8npSlmeHHOCTb nonSl KOToporo 49
H8XO,AHTCB B paBHOBeCli1i
M
nblnHHK8. HMelOI.l.\8S1 3apSl,A 20 HEn. EaKos8 MaCC8 m nbInHHKH?
)J;8HO: E=49 B
M
Y
A f:e
-I
EI
1
I +
Puc. 8.22
q=2.0.10- 8 Kn.
H8HTIi: m.
Pew e H He. H8 nbInHHKY ,AeHCTBYIOT CHn8 TSlmec-
TH mg Ii CHna P co CTOpOHbI aneKTpOCT8TH'leCKOrO
nonfl (pliC. 8.22)-
Y cnOSHe paBHOBeCHSI mg + F = O.
B npOeKII.HH H8 BepTHK8nbHYIO OCb y:
F - mg = O.
T8R K8K F = qE. TO
qE - mg = 0 => m = qE : m = 1.0 .10- 7 Kr.
g
M
C
5. 3neKTpoH, HMelOI.l.\liH CKOpOCTb 1.8.10 4
BneTaeT B O,AHOpO,AHOe 3neK-
TpOCT8TIi'leCKOe none B BaKYYMe HanpBmeHHOCTblO 3,0 MB
M
THB H8np8BneHHSI nHHHH H8npSlmeHHOCTH. K8KOBbI 6Y,AYT
pOCTb 3neKTpOH8. KOr,Aa OH npOH,AeT p8CCTOSlHHe 7.1 CM?
nOTpe6yeTcSI ,AnSI ,AOCTHmeHHSI 3TOH CKOpOCTH?
H ,ABH>KeTCB npo-
YCKopeHlie H CKO-
CKonbKO BpeMeHH
B
)J;8HO:
M
vo= 1,8 . 10 4 -, E = 0,0030
C
M
m=9.1.10-31 Kr.
226
s = 0,071 M. e = 1,6 .10- 19 Kn,

H8HTH: a. v. t.
PeWeHHe. Ha 3neKTpOH (pHC. 8.23) co CTO-
pOHbl 3neKTpOCT8TH'leCKOrO nonSl .l{eHcTByeT CH-
n8 F, Hanp8BneHH8S1 npOTHB nHHHH H8npSlmeH-
HOCTH. CHnoH TSlmeCTH sneKTpOH8 MomH8 npe-
He6pe'lb, T8K K8K OH8 M8n8 no cp8BHeHHIO C CH-
nOH F. Corn8cHo BTOPOMY 38KOHY HbIOTOH8,
F = mii HnH F=ma. T8K KID< F=eE, l',Ae e - 38-
pSl,A sneKTpOH8, TO
eE 8 M
eE = ma; a = -; a = 5,3 .10 -
m c 2 '
TaK K8K none O,AHOpO,AHOe, TO CHna F = const H 3neKTpOH ,ABH}KeTCH p8B-
HOYCKopeHHO. IIo3ToMY HCKOMYIO CKOpOCTb H BpeMSI H8H,AeM H3 H3BeCTHblX <l>op-
Myn ,AnSI P8BHOYCKopeHHoro ,ABHmeHHSI:
_Fe
a

-+
v o
.---
Puc. 8.23
.
E
..
..
v 2 - v = 2aS => v =  v + 2aS ;
M
t - v - vo. t 4 0
--, =, MKC.
a
v = 2,0.10 4
C
B03YIlIHbIH
KOH,AeHCaTOp
M
6. 8neKTpoH. 06na,A81OHH CKOpOCTblO 6,0. 10 1 -. BneT8eT B nnOCKHH
C
ero nn8CTHHaM. HanpSlmeHHocTb
3neKTpOCT8TH'leCKOrO nonSl
B8HHoe noneM KOH,AeHC8TOpa,
nap8nnenbHO
KB ..
60 -. HaH.l{HTe OTKnOHeHHe sneKTpOHa.
M
ecnH ,AnHHa ero nn8CTHHbi 5.0 CM.
Bb13-
7 M 4 B
,n8HO: vo=6,0 .10 -, E=6,0 .10 l=0,050 M.
C M
Pew e H He. H8 3neKTpOH co CTOpOHbl 3neKT-
pOCT8TH'leCKOrO nonH ,AeHcTByeT CHn8 F3JI' Ha-
np8BneHH8S1 BHH3 (pHC. 8.24), P3JI = eE .
,nBHmeHHe sneKTpOHa MomHO p8CCM8TpHB8Tb
KaK cynepn03Hl:HIO .l{BYX He38BHCHMbiX .l{BHmeHHH,
npOHCXO,ASIHX B I'OpH30HTanbHOM H BepTHK8nb-
HOM H8np8BneHHSlX.
B ropH30HTanbHOM Hanp8BneHHH sneKTpOH 6y-
,AeT ,ABHraTbCSI p8BHOMepHO, T8K KaK B 3TOM H8-
np8BneHHH CHnbl He ,AeHCTBYIOT. O,AHOBpeMeHHO C
3THM no,A ,AeHCTBHeM 3neKTpH'leCKOH CHnbl F OH 6Y,AeT P8BHOYCKopeHHO nepeMe-
aTbCSI BHH3. Tp8eKTopHeH ,ABHmeHHH 6Y,AeT n8pa6on8. Cne,AOBaTenbHO, 3a Bpe-
MSI t sneKTpOH nponeTHT no ropH30HT8nH paCCTOSlHHe l = vot, 8 no BepTHK8nH
at 2
nepeMeCTHTCH H8 p8CCTOHHHe h = -. PeIlI8S1 3TH ypaBHeHHSI COBMeCTHO. nony-
2
H 8 H T H: h.
Puc. 8.24
h
+
'lHM
al 2
h = . TIo BTOPOMY 38KOHY HblOTOH8
2vo
F eE
a =- =-.
m m
Cne,AOBaTenbHO,
eEl 2
h=; h=3.7 MM
2mvo
227

7. .D:Ba TO'le'lHblX 3apSln;8 1,0.10- 8 Kn H -1.0.10- 8 Kn H8XOLl;SlTCSI Ha
paccToSlHHH r= 5.0 CM n;pyr OT n;pyra. Onpe,l1enHTe H8npSlmeHHOCTb nonSl B
TO'lKe, pacnonomeHHoH H8 paccToSlHHH rl = 3.0 CM OT nonomHTenbHoro 3apSl1l8
H r2 = 4,0 CM OT OTpHlJ,aTenLHoro.
.D:8HO: ql=1.0.10- 8 Kn. q2=-l,O.10- 8 En, r=0.050 M, rl=O,030 M.
r2=0.040 M, £=1.
HaHTH: EA'
Pew e H He. TIo npHHlJ,Hny cynepn03HlJ,HH noneH HanpSlmeHHOCTb nonSl B
TO'lKe A (pHC. 8.25) E = El + E 2 , rn;e El H8npSl»<eHHOCTb nonSl. C03,A8H-
Horo 38pSl,AOM qlt B TO'lKe A, 8 E 2 - H8npSlmeHHOCTb nonSl, C03,A8HHOrO
38pSl,AOM q2' TIocKonbKY 38pSl,ALI ql H q2 TO'le'lHble. TO
Q2<O r Ql>O
B,---------------------C
, /
, /
", //
r 2 " /
". // r 1
/
/
/
Puc. 8.25
E 1 = Iqll 2 ' E 2 = Iq21 2 .
47tEoErl · 41tEoEr2
Y'lHTblSaSl, 'lTO CTOpOHbl MBC p8BHbl CO-
OTBeTCTBeHH O 3, 4 H 5 CM, HaXO,AHM ex = 90°,
TOr,Aa E =  E: + E: HnH
E=
Iq11 2 + Iq21 2
( 47tE oEr/ t (47tE o Er 2 2)2
41tEOE
1
2 2
!h. + q2 ;
r1 4 r2 4
E=1.1.10 2
KB
M
8.13. Pa60Ta 3JIeKTpOCTaTH1JeCKOrO DOJIH
npH nepeMeeHHH 3apOB
Ha ::weKTpHqeCKHe 3ap.HbI B SJleKTpOCTa-
THqeCKOM none eHcTBYIOT CHnbI. llOSTOMY,
eCJlH 3ap.HAbI nepeMeaIOTC.H, TO 3TH CHJlbI CO-
BeplUalOT pa6oTY. PaCC'lHTaeM p860TY CHJI O,A-
HopollHoro sJleKTpOCTaTHqeCKOrO non.H npH ne-
peMeeHHH nonomHTenbHoro 3ap.HL{a Qo H3
TOqKH 1 B TOqKY 2 BHaqane no TpaeKTopHH
1  2, a 3aTeM no TpaeKTopHH 1 -? 3 -? 2
(pHC. 8.26).
TaK KaK nOJle O)J;HOPOAHO, TO CHna
----+
F = qoE = const H pa60TY MomHO paCCqHTaTL
no <}JopMYJle
A 12 = Fr12 = qOE(Xl - X2)'
r,Ae Llr 12 - MOYJlb nepeMeeHH.H 3ap.HAa MemAY TOqKaMH 1 H 2.
llPH nepeMeeHHH 3ap.H,Aa qo no TpaeKTopHH 1  3  2 pa60Ta MomeT
6bITb npe,ACTaBJleHa KaK cYMMa pa60T Ha yqaCTKaX 1  3 H 3  2:
A 132 =A13 + A 32 .
+
3 F
+
1
2
x
+
 I I
--_-.!.
I I
I I
I I
XI X 2
+
Puc. 8.26
228
o

Ha yqacTKe 1  3 pa60TB A = Fl1r l3 cos a, re r13 - MOYJIb nepeMe-
lQeHH.H 3ap.Ha Memy TOqKaMH 1 H 3.
M3 pHCYHKa BHHO, qTO l1r l3 cos a = Xl - x 2 ' T. e. A l3 = qOE(XI - X2)'
Ha Y't.J:aCTKe 3  2 pa60Ta paBHa
A 32 = F l1r 32 cos  = F l1r 32 cos 90° = 0,
r,Ae l1r 32 - MOYJlb nepeMeIQeHH.H 3ap.H,Aa Memy TO't.J:KaMH 3 H 2.
CJleOBaTeJIbHO, A 132 = qoE (Xl - X2)'
CpaBHHBa.H nOJlyqeHHble BblpameHH.H, MomHO 3aMeTRTb, tJTO pa60ma no ne-
peMeuteliU10 3apRiJa B 3Jl.eKmpOCmamUlleCKO.M. nOJl.e lie 3aBucum om ljJOp.M.bl mpa-
eKmopuu, no KOTOpO:H BRraJlC.H 3apS! qo, a 3a6ucum mOJl.bKO om ttallaJl.bliO-
ZO U KOHellliOZO nOJl.OJICeIiUU 3apSl8a. 3TO YTBepmeHRe cnpaBeJlHBO H JI.H
HeOHopOHoro 3J1eKTpOCTaTHqeCKOrO nOJIS!.
HaH,AeM pa60TY Ha 3aMKHYTOH TpaeKTopHR 1  3  2  1:
A 1321 = A 13 + A 32 + A 21 = qOE(Xl - X2) + 0 + qOE(XI - x2 )cos180° =
= qoE (Xl - X2) - qoE (Xl - X2) = 0 .
nOJle, pa60Ta CHn KOToporo He 3aBRCHT OT c}>OpMbI TpaeKTopRH R Ha 3aM-
KHYTOH TpaeKTopHH paBHa HYJIIO, Ha3bIBaeTC.H nOTeHHaJlLHLIM.
8.14. nOTeHI.{HaJI
M3 MexaHHKR R3BeCTHO. qTO pa60Ta KOHcepBaTRBHbIX CRJI CB.H3aHa C R3-
MeHeHHeM nOTeHHaJlbHOH 3HeprHH. CHCTeMa "3ap.H - 3J1eKTpOCTaTRtJeCKoe
nOJIe" 06JIa.n;aeT nOTeHHa.n:bHOH 3HeprHeH (3HeprHeH 3JIeKTpOCTaTHtJeCKoro B3aH-
Mo.n;eHcTBHS!). I103TOMY, ecnH He yqHTbIBaTb B3aHMO,AeHCTBRe 3ap.H.n;a C rpaBH-
TaHOHHbIM nOJIeM H oKpymalOlQeH cpe.n;OH, TO pa60Ta, cOBeplIIaeMa.H npR ne-
peMelQeHRH 3ap.H.n;a B 3J1eKTpOcTaTHtJecKoM nOJle, paBHB H3MeHeHRIO nOTeHH-
aJlbHOH 3HeprHR 3ap.H.n;a, B3.HTOMY C npoTHBOnOJIOmHbIM 3HaKOM:
A l2 =-(2-d=1-Wn2'
(8.1)
ECJIH W n2 = O. TO B KamttOH TO't.J:Ke 3J1eKTpOCTBTR'leCKOrO nOJI.H nOTeHu,H-
BJlbHa.H 3HeprHS! 3apS!.n;a qo paBHa pa60Te, KOTOpa.H 6bIJla 6bI COBeplIIeHa npH
nepeMelQeHHH 3ap.Ha qo H3 .n;aHHOH TOqKR B TOqKY C HYJleBOH 3HeprHeH.
I1YCTb 3JIeKTpOCTaTRqeCKOe nOJle C03.n;aHO B HeKOTopOH 06naCTR npOCTpaH-
CTBa nOJlOmHTeJlbHbIM 3ap.H.n;OM q (pHC. 8.27).
By.n;eM nOMelQaTb B TOqKY M 3Toro nOJlS! pa3J1HtJHble
np06HbIe nOJlOmH'reJlbHble 3ap.H.n;bI qo. I1oTeHu,HaJIbHa.H
W
3HeprR.H HX pa3J1HqHa, HO OTHOlIIeHHe ---!!... = const .n;JI.H
qo
naHHoH TOqKH non.H H CJIymHT xapaKTepHcTHKOH non.H
Ha3bIBaeMoH nOTeUUHaJlOM nOJIS! <P B .n;aHHoH To't.J:Ke:
/"--------..............
I ........
I q ............
\ ....,
\  M "
"  J
" qo I
"......... //1
.......-------"
PUC. 8.27
W n
cp=-
qo '
229

E,nHHHneH nOTeHnHRJla B Cll .HBJI.HeTC.H BOJlbT (B) HJlH ,nmOYJlb Ha KY-
nOH r : }
nOTeHaJlOM 3J1eK1'pOCTaTH'IeCKOrO nOJIH B aHHoH TO'IKe Ha3bIBalOT CKa-
JI.HpHYIO cl>H3HqecKYIO BeJlHqHHY, xapRKTepH3YIOYIO 3HepreTHqeCKOe COCTO.H-
HHe nOJl.H B ,D;aHHOH TOqKe npOCTpaHCTBR H qHCJleHHO paBHYIO OTHomeHHIO no-
TeHHaJlbHOH 3HeprHH, KOTOpOH 06J1a.u;aeT np06HbI:H nOJlOmHTeJlbHbIH 3ap.H,D;, no-
MelQeHHbIH B 3TY TOqKY, K 3HaqeHHIO 3ap.H,D;R.
nOTeHnHaJl - 3TO 3HepreTHqeCKa.H xapaKTepHCTHKa nOJl.H B OTJlHqHe OT
Hanp.HmeHHOCTH nOJl.H, .HBJI.HIOeHC.H CHJlOBOH xapaKTepHCTHKOH nOJl.H.
He06xo)J;HMO OTMeTHTL, qTO nOTeHnHaJlbHaa 3HeprH.H 3ap.H,D;a B )J;aHHOH TOq-
Ke nOJl.H, a 3HaqHT, H nOTeHHaJl 3aBHC.HT OT BbI60pa HYJleBOH TOqKH. HYJle-
BOH 3TR TOqKa HR3bIBaeTC.H nOTOMY, qTO nOTeHHaJlbHYIO 3HeprHIO (COOTBeT-
CTBeHHO nOTeHD;HaJI) 3ap.H,D;a, nOMeeHHoro B 3TY TOqKY nOJl.H, YCJlRBJlHBaIOT-
C.H CqHTRTb pRBHOH HYJlIO.
HYJleBOH ypOBeHb nOTeHnHaJIbHOH 3HeprHH BbI6HpaeTC.H npOH3BOJlbHO, n03TO-
MY nOTeHnHaJI MomHO onpe,D;eJlHTb TonbKO C TOqHOCTblO ,D;O HeKOTopOH nOCTO-
.HHHOH, 3HaqeHHe KOTOpOH 3aBHCHT OT Toro, B KaKOH TOqKe npOCTpaHCTBa BbI6-
paHO ero HYJleBOe 3HaqeHHe.
B TexHHKe npHH.HTO CqHTaTb HYJleBOH TOqKOH JlI06yIO 3a3eMJleHHYIO TOqKY,
T .e. coe,nHHeHHYIO npOBo,nHHKOM C 3eMJleH. B cl>H3HKe 3a HaqaJlO OTCqeTa no-
TeHltHaJlbHOH 3HeprHH (H nOTeHltHaJla) npHHHMaeTC.H Jl106a.H TOqKa. 6eCKOHeq-
HO Y,D;aJleHHR.H OT 3ap.H.nOB. C03.nalOI.I.1HX nOJle. ECJlH HYJleBR.H TOqKa BbI6pRHR,
TO nOTeHHaJlbHa.H 3HeprH.H (COOTBeTCTBeHHO H nOTeHUHaJl B .naHHOH TOqKe)
3ap.H,ZIa qo CTaHOBHTC.H onpe.neJleHHOH BeJlHqHHOH.
HR paCCTO.HHHH r OT TOqeqHOrO 3ap.H,D;R q, C03RlOlQerO nOJle, nOTeHnHRJI
onpe)J;eJl.HeTC.H cl>OPMY JlOH
q>= q .
41tE o Er
npH YKa3aHHOM BbIme BbI60pe HYJleBOH TOqKH nOTeHnHaJl B JlI060H TOqKe
nOJl.H, C03)J;aBaeMoro nOJlOmHTeJlbHbIM 3ap.H,D;OM q, nOJlOmHTeJleH, R nOJl.H, CO-
3,D;RBaeMOro OTpHnRTeJlbHbIM 3ap.H,D;OM, OTpHnaTeJleH:
eCJlH q > 0, TO <P > 0; eCJlH q < 0, TO <P < o.
no 3TOH cl>opMYJle MomHO paCCqHTbIBaTb nOTeHnHaJl nOJl.H. 06pa30BaHHoro
paBHOMepHO 3ap.HmeHHOH npOBO)J;.HeH ccl>epoH pa,D;HYCOM R B TOqKaX, HaXO,D;.H-
HXC.H Ha nOBepXHOCTH ccl>epbI H BHe ee. BHYTPH c€}JepbI nOTeHnHaJl TaKOH
me, KaK H Ha nOBepXHOCTH, T.e.
<P = q npH r:S R H q> =  npH r > R .
41tEEoR 41tEEor
ECJlH 3J1eKTpOCTaTHqeCKOe nOJle C031IaeTC.H CHCTeMOH 3ap.H,nOB. TO HMeeT Me-
CTO npHHn cynepno3HHH: nomeliquaJl. 6 Jl.lOtJOU mOIlKe maKOZO noJLSl pa6eli
aJl.Ze6paUlleCKOU CYMMe nomeliijuaJl.06, C030a6aeMbLX 6 amou mOIlKe KaJICObLM
3ap.suJoM 6 OmOeJl.bIiOcmu:
n
<P p = L <P, .
1=1
230

3Ha.H nOTeHHaJl <P nOnH B ,naHHOH TOqKe, MmKHO paCCqHTaTb nOTeHl1.H-
anbHYIO 3HeprHIO 3ap.H,na qo, nOMelQeHHOrO B 3TY TOqKY: W n = qO<p. ECJlH no-
JlO:>KHTb, qTO W n2 =0, TO H3 ypaBHeHH.H (8.1) 6y,neM HMeTb
Aloe = W nl = qO<Pl .
llOTeHl1.HanhHa.H 3HeprH.H 3ap.H,na qo B ,naHHoH TOqKe nonH 6y,neT paBHa
pa60Te CHJI 3J1eKTpOCTaTHqeCKOrO nonH no nepeMelQeHHIO 3ap.H,na qo H3 ,naH-
HOH TOqKH B HYJleBYIO. 113 n:OCJle,AHeH CPOPMYJlbI IIMeeM
A loo
<PI = - .
qo
llOTeHHaJl nOJlH B ,naHHOH TOqKe t{HCJleHHO paBeH pa60Te no nepeMelQe-
HHIO e,nHHHqHOrO nOJlOmHTeJlbHOrO 3ap.H)J;a H3 ,naHHOH TOqKH B HYJleBYIO
(B 6eCKOHeqHOCTh).
llOTeHl1.HaJlbHa.H 3HeprHH 3apH,na qo, nOMelQeHHOrO B 3neKTpOCTaTHqeCKOe
nOJle TOqeqHOrO 3apSl,na q Ha paCCTO.HHHH r OT Hero,
W=
n 41t£oEr .
ECJlH q II qo - o)J;HoHMeHHbIe 3ap.H,nhl, TO W n > 0, eCJlH q H qo - pa3-
HhIe no 3HaKY 3apH)J;bI, TO W n < 0.
OTMeTHM elQe pa3. qTO no 3TOH cpopMYJle MomHO paCC'lHTaTb nOTeHl1.Hanb-
HYIO 3HeprHIO B3aHMo,neHCTBH.H BYX TOqeqHbIX 3ap.H1I0B. eCJlH 3a HyneBoe 3Ha-
qeHHe W p BhI6paHO ee 3HaqeHHe npH r = 00 .
EcnH 3neKTpOCTaTHqeCKOe none 06pa30BaHO CHCTeMOH n TOqeqHbIX 3J1eKT-
pHqeCKHX 3ap.H,nOB, TO nOTeHl1.HaJlhHa.H 3HeprH.H CHCTeMhJ onpe,nenHeTC.H no
cpopMyne
1 n
W = - Lql<Pl.
2 l=l
r,ne <Pi - nOTeHHaJl nOJl.H, C03,naHHoro BceMH 3ap.H)J;aMH, KpOMe 3ap.H,na qi' B
TOH TOqKe nOJl.H, r,ne Haxo,nHTCSJ 3ap.H,n qi'
8.15. Pa3HocTb nOTeHI\HaJlOB. HanpH:tKeHHe
Pa60Ta CHn 3J1eKTpOCTaTHqeCKOrO nOJlH no n:epeMeeHHIO 3ap.H,na qo H3 TOq-
KH 1 B TOqKY 2 n:OJl.H
A 12 = W nl - 12 .
BhIpa3HM nOTeHl1.HanhHYlO 3HeprHIO qepe3 nOTeHl1.HaJlbI nOJl.H B COOTBeTCTBY-
IOHX TOqKax:
W nl = qO<i'I' W n2 = qO<P2 .
Tor ,na
A I2 = qo (<PI - <P2 ) .
TaKHM 06pa30M, pa60Ta onpe,neJl.HeTCH npOH3Be,neHHeM 3ap.H,na Ha pa3HOCTb
nOTeHHaJlOB HaqaJlbHOH H KOHeqHOH TOqeK.
231

A 12
M3 3TOU qJOpMynbI pa3HOCTb nOTeHRaJlOB <PI - <P2 =-.
qo
Pa3HOCTb nOTeHRanOB - 3TO CKan.HpHa.H cpR3R'tIeCKa.H BenRqRHa. 'lRCneH-
HO paBHaH OTHOllleHRIO pa60Tbi CRn non.H no nepeMemeHRIO 3ap.H,Aa Mem.ny
aHHbIMR TO'tlKaMR non.H K 3TOMY 3ap.HY.
B CM eHHReH pa3HOCTR nOTeHRanOB HBn.HeTC.H BonbT (B).
1 B - pa3HOCTb nOTeHRanOB Me>Ky BYM.H TaKHMH TO'tlKaMR 3neKTpo-
CTaTH'leCKOrO nonH, npR nepeMeeHRR Memy KOTOpbIMR 3ap.Ha B 1 KJI CR-
JlaMH non.H COBeplllaeTC.H pa60Ta B 1 .D:m.
Pa3HocTb nOTeHRaJlOB B OTnH'tIRe OT nOTeHRaJla He 3aBHCHT OT BM60pa
HyneBoH TO'tlKR. Pa3HOCTb nOTeHRaJlOB <PI - <P2 'laCTO Ha3bIBalOT 3J1eKTpHqeC-
KHM HanpJleHHeM Memy aHHbIMR TO'tlKaMH nOJlH:
U = <PI - <P2 .
HanpJleHHe Mem,AY ,ABYMH TOqKaMR non.H onpeenHeTC.H pa60TOH CHn 3Toro
nonH no nepeMeeHRIO 3ap.H,Aa B 1 Kn R3 OHOH TO'tlKR B ,APyrylO. B 3neKTpo-
CTaTH'leCKOM none Hanp.HmeHRe Bonb 3aMKHYToro KOHTypa Bcera paBHO HynlO.
Pa60TY CHn 3neKTpHQeCKOrO nOJl.H RHora BbIpamalOT He B ,lpKoyn.HX, a B
3J1eKTpOHBOJILTax. 1 3B paBeH pa60Te, cOBeplllaeMoH CRnaMR non.H npR nepe-
MeeHRR 3J1eKTpOHa (e = 1,6 . 10- 19 Kn) Me>Ky ,lIBYMH TO'lKaMH, HanpHmeHRe
Memy KOTOpbIMR paBHO 1 B.
1 3B = 1,6.10- 19 Kn.l B = 1,6.10- 19 .D:m.
1 M3B= 10 6 3B = 1,6 .10- 13 .D:m.
3neKTpR'leCKOe none rpacpR'tIeCKH MO>KHO H306pa3HTb He TonbKO C nOMOblO
nRHRH Hanp.HmeHHOCTR, HO R C nOMOblO 3KBHnOTeHRanbHbIX nOBepXHOCTeH.
3KBHnOTeHHaJlLHOH Ha3bIBaeTCH Bo06pamaeMa.H nOBepXHOCTb, B KamOH
TOQKe KOTOpOH nOTeHHan O,ARHaKOB. Pa3HOCTb nOTeHRaJlOB Memy ,ABYMH
nlO6blMH TO'tlKaMR 3KBHnOTeHRanbHOH nOBepXHOCTH paBHa HynlO.
CneOBaTeJlbHO, pa60Ta no nepeMeeHHIO 3ap.Ha Bonb 3KBRnOTeHII.R-
aJlbHOH nOBepxHocTIi paBHa O. Ho pa60Ta paCCqRTbIBaeTC.H no cpopMYJle
A = Ftlrcosa = qoEtlrcosa .
cos a = 0 ==> a = 90<' .
CneoBaTenbHo, nHHRR HanpHmeHHocTH nepneHRKYJlHPHbl 3KBHnOTeHHanb'
HblM nOBepXHOCT.HM. IIepBa.H 3KBRnoTeHRaJlbHa.H nOBepxHocTb MeTannHqeCKOrO
npoBoHHKa - 3TO nOBepxHocTb caMoro
3apH>KeHHOrO npoBoHHKa, 'lTO nerKO
npoBepHTb 3neKTpoMeTpOM. OCTaJlbHMe
3KBHnoTeHRanbHbIe nOBepxHocTR npoBo-
HTC.H TaK, QT06bI pa3HocTb nOTeHRa-
nOB Memy BYMH coceHRMH nOBepXHO-
CT.HMR 6bIJla nOCTOHHHOH.
KapTHHbl 3KBRnOTeHHanbHbIX nOBepx-
HOCTeii HeKOTopbIX 3ap.H>KeHHbIX Ten
npHBeeHbI Ha pRC. 8.28.
3KBRnoTeHRanbHbIMR nOBepXHOCT.H-
MR OHopOHoro 3neKTpOCTaTH'tIeCKOrO
a 6
E
»
»
»
.
»
.
<p=const
Puc. 8.28
232
3eCb
<p= const
qo :J: 0 ,
E1;O,
/1r :J: 0 .
3Ha'tlRT,
E

non.H .HBJI.HIOTC.H nJlOCKOCTH, nepneH,AHKY JI.HpHbIe JlHHH.HM Hanp.HmeHHOCTI1
(pHC. 8.28. a).
SKBHnoTeHHaJlbHbIe nOBepXHOCTH nOJl.H TO'lle'llHOrO 3ap.H,Aa npe,ACTaBn.HIOT
C060H c<})epbl, B eHTpe KOTOpblX pacnonomeH 3apSl,A q (pHC. 8.28, 6).
8.16. CBH3L Me:iKJJ;Y pa3110CTLIO nOTeHHaJIOB
H HanpH:iKeHHOCTLIO
llOJlb3Y.HCb KapTHHoH SKBHnOTeHHaJlbHbIX nOBepxHocTeH, MomHO HaHTH
Hanp.HmeHHOCTb nOJl.H B nlO60H TO'llKe. PaCCqHTaeM pa60TY, COBeprnaeMYIO CH-
JlaMH sneKTpOCTaTHqeCKOrO nonSi npH nepeMelQeHHH sneKTpH'lIeCKOrO 3ap.H,Aa C
O,AHOH sKBHnoTeHHanbHoH nOBepXHOCTH <1'1 Ha ,ApyrylO <1'2 no HopManH K STOH
nOBepXHOCTH. llepeMelQeHHe nOJlOmHTenbHoro 3ap.H,Aa qo no,A ,AeHCTBHeM CHn
non.H H3 TOqKH 1 B TOqKY 2 paBHO I1l (pHC. 8.29). Pa60Ta CHn non.H
A = qo (<1>1 - <1>2)' ECJlH paCCTO.HHHe l1l Memy nOBepXHOCT.HMH no HopManH
MaJlO, TO Ha STOM yqacTKe nOJle MomHO C'lIHTaTb OHOpOHbIM. llOSTOMY Mom-
HO BOCnOJlb30BaTbCSI <})OPMYJlOH ,An.H paC'leTa pa60TbI no nepeMelQeHHIO 3ap.H,Aa qo
B O,AHOpO,AHOM SJleKTpOCTaTH'lIeCKOM nOJle ( 8.13): A = Fl1l = qoEI11. llpHpaB-
HHBa.H ,ABa BblpameHH.H ,AJI.H pa60TbI, nonY'llHM:
qo (<PI - <P2) = qE . I1l =) E = <PI - <P2 .
I1l
MOYJlb Hanp.HmeHHOCTH B nlO60H TOqKe
nonSi qHCneHHO paBeH pa3HocTH nOTeHHanoB.
npHXOSllQeHCSI Ha eHHHY nHHbI JlHHHH Ha-
np.HmeHHocTH. 1..JeM MeHbrne l1l. TeM TeCHee
<P2 pacnonomeHbl sKBHnoTeHHanbHble nOBepxHoc-
TH, TeM 60nbrne B STOM MeCTe HanpSlmeH-
HOCTb non.H.
B O,AHOpO,AHOM nOJle I1l MomeT 6bITb JlIO-
60e. EcnH l1l = d, TO
E = <PI - <P2 = _ Ll<p
d d
STa «popMyna BblpamaeT CB.H3b Memy Ha-
npSlmeHHOCTblO H pa3HOCTblO nOTeHl{HanOB O,A-
HOpO,AHOrO 3JIeKTpOCTaTH'leCKOrO nOJISI. Ha OCHOBaHHH STOH «P0PMYJIbI MomHO
YCTBHOBHTb eAHHHI\Y HBnp"",eHHOCTH B CM: BOJlbT HB MeTp [  ).
Puc. 8.29
8.17. IIpOBOJJ;IIHKH B 3JIeKTpH'IeCKOM nOJIe
llpOBO,AHHKaMH Ha3bIBaIOTC.H BelQeCTBa, B KOTOpbIX HMeeTC.H 3Ha'llHTenbHoe
'lIHCJIO cB060HbIX 3ap.H,AOB, T .e. TaKHX :iap.HOB, KOTopbIe MorYT 6e3 3aTpaTbI
SHeprHH nepeMelQaTbC.H no BceMY npOBOHHKY. STO MeTaJlJlbl, 3J1eKTpOJlHTbl H
HOHH3HpOBaHHble ra3bI. B MeTanJIaX CB060,AHbIMH 3ap.H,AaMH SlBJI.HIOTC.H SJIeKT-
POHbI. KOTopbIe nepeMelQalOTCSI Memy Y3J1aMH KpHCTaJIJIH'lIeCKOH perneTKH. 06-
233

pa30BaHHOH HOHaMH MeTanna. 3TH cB060HbIe 3J1eKTpOHbI 06pa3YIOT TaK Ha-
3bIBReMbIH 3J1eKTpOHHbIH rR3. OHH yqaCTBYIOT B TenJlOBOM BHmeHHH nO06HO
MoneKYJlRM ra3R H MorYT nepeMelQaTbC.H no BceMY 06'beMY MeT8J1nR. BHYTPH
3Rp.HmeHHbIX npoBolIHHKOB OTCYTCTByeT 3J1eKTpOCTRTHqeCKOe nOJle. TRK KRK B
npoTHBHOM cnyqRe cB060,AHble 3Rp.H,AbI BHrRnHcb 6bI no,A ,AeHCTBHeM CHn 3TO-
ro nOJl.H.
OTCYTCTBHe 3J1eKTpOCTRTHqeCKOrO nOJl.H BHYTPH npOBoHHKa npHBoHT K
TOMY, qTO HecKoMneHCHpoBaHHbIe 3Rp.H,AbI MorYT pR3Mem;RTbC.H TOJlbKO HR ero
nOBepXHOCTH.
+
+
+
+
+
Puc. 8.30
TIoMecTHM He3ap.HmeHHbIH npoBoHHK B OHO-
pOHoe 3J1eKTpOCTRTHqeCKOe none. TIolI eHcTBHeM
CHn non.H cB060)J;HbIe 3J1eKTpOHbI B npoBo)J;HHKe 6y-
,AYT nepeMeRTbC.H B HanpRBneHHH, npoTHBono-
nomHOM BHeWHeMY nOJlIO. H HRKannHBaTbC.H HR
nOBepXHOCT.HX npoBoHHKR, C03aBR.H 3J1eKTpOCTa-
THqeCKOe none Ecoo, HanpaBJleHHOe HaBCTpeqy
BHeWHeMY nOJlIO. TIepeMelQeHHe 3J1eKTpOHoB 6yeT
npoHcxoHTb o Tex nop, nOKR BHeWHee 3neKTpo-
CTaTHqeCKOe nOJle He CKOMneHcHpyeTC.H nOJleM,
B03HHKRIOm;HM BHYTPH npOBOHHKR (pHC. 8.30).
B pe3YJlbTRTe HRnp.HmeHHOCTb non.H E pC3 BHYTPH
npOBOHHKR 6yeT paBHR HYJlIO:
EPf:J = Eco6 + E Bllew , E peJ = Ec06 - E Bllew = 0 .
3J1eKTpH3RHIO npOBOHHKa BO BHeWHeM 3J1eKTpOCTaTHqeCKOM nOJle pR3e-
neHHeM yme HMelOI..QHXC.H B HeM B paBHbIX KOJlHqeCTBRX nOJlOmHTeJlbHbIX H
OTpHRTeJlbHbIX 3Rp.H1I0B HR3bIBRIOT JIBJleHHeM 3J1eKTpOCTaTH'IeCKOH HH.AYKUHH.
a CRMH nepepacnpeeJleHHbIe 3Rp.HbI - HHpOBaHHbIMH. 3TO .HBneHHe Mom-
HO Hcnonb30BaTb JI.H 3neKTpH3RHH He3ap.HmeHHbIx npoBoHHKOB.
TIoMecTHM Ha Ba oHHaKoBbIx 3neKTpoMeTpa BMeCTO mapHKoB Ba OH-
HaKOBbIX MeTaJlJlHqeCKHX HJlHH,Apa A H B (pHC. 8.31) H npHBeeM HX
OCHOBaHH.H B conpHKOCHOBeHHe. 3a3eMJlHM Kopnyca 060HX 3neKTpOMeTpOB.
rrOHeceM K npOBOHHKY B, He KaCa.HCb ero, nonOmHTeJlbHO 3ap.HmeHHbIH
234

A
B
Puc. 8.31

mapHK C. CTpeJlKH 060HX 3J1eKTpOMeTpOB OTKJlOH.HIOTC.H Ha O,D;HH H TOT
me yroJl. Y6epeM mapHK C, CTpeJlKH nOKamYT "0". IIoJle OTCYTCTByeT, 3a-
p.H,AbI On.HTb pacnpe,D;eJl.HIOTC.H paBHOMepHO. IIpOBe,D;eM ,ApyroH onbIT, aHaJlO-
rHqHbIH nepBOMY onbITY, HO Tenepb HJlHH,D;pbI pa3,D;eJlHM B npHCYTCTBHH
mapHKa C, a aaTeM yme y6epeM H caM mapHK C. B 3TOM cJlyqae CTpeJl-
KH 3neKTpOMeTpOB yme He 6Y,D;YT Ha HYJle. C nOMOblO nonOmHTeJlbHO
3ap.HmeHHOH CTeKJI.HHHOH naJlOqKH MomHO .nOKa3aTb, qTO I.(HJlHH,D;P A 3a-
pSlmeH nOJlOmHTeJlbHO, a IIHJlHHp B - OTpHaTeJlbHO. TaKoH cnoc06
3J1eKTpH3aHH TeJl Ha3bIBaeTC.H 3J1eKTpH3alJ,HeH tlepe3 BJlH.HHHe. IIpHBe,D;eM
Tenepb HJlHH,D;pbI B conpHKOCHOBeHHe, 3ap.H,D;bI HeHTpaJlH3YIOTC.H, CTpeJlKH
3J1eKTpOMeTpOB On.HTb Ha HYJle.
He3ap.HmeHHbIH npOBO,D;HHK MomHO Ha3neKTpH30BaTb nYTeM conpHKOCHO-
BeHH.H C ,D;pyrHM 3apHmeHHbIM npOBO,D;HHKOM.
,I:J;JI.H Toro qT06bI nepe,D;aTb npOBO,D;HH-
KY B BeCb 3ap.H,D; npOBO,D;HHKa A, HymHo
3ap.HmeHHbIR npOBO,D;HHK A BHeCTH
BHYTPb npOBO,D;HHKa B (pHC. 8.32, a) H
3aTeM KOCHYTbCH HM BHYTpeHHeH CTeHKH
npOBO,D;HHKa B (pHC. 8.32, 6).
+ KaKHM 6bI cnoc06oM HH 6bIJI 3apH-
meH npOBo.nHHK, BHYTPH Hero nOJle OT-
CYTCTByeT. 9TO n03BOJl.HeT HCnOJlb30-
BaTb 3a3eMneHHbIe nOJlbIe npOBO,D;HHKH
co CnJlOmHbIMH HJlH CeTqaTbIMH CTeH-
KaMH ,D;JI.H 3neKTpOCTaTHqeCKOH 3aHTbI
OT BHemHHX aneKTpOCTaTHqeCKHX nOJleH. TaK, HanpHMep, 3J1eKTpOMeTp no-
MeIQalOT B MeTaJlJlHqeCKHH «PYTJI.Hp ,D;JI.H Toro, 't.IT06bI HH,D;YlJ,HpOBaHHbIe Ha
HeM BHemHHMH aJleKTpHqeCKHMH nOJl.HMH 3ap.H)J;bI He HCKamaJlH ero nOK8-
38HHH. ,I:J;JI.H 3aI.QHTbI BoeHHbIX CKJla,D;OB, CJlymaHx ,D;JI.H xpaHeHH.H B3pbIB-
tlaTbIX BeeCTB, OT Y,D;apa MOJlHHH HX oKpymalOT 3a3eMJleHHOH npOBOJlOq-
HOH ceTblO.
EaK 6bIJlO OTMeqeHO Bblme, 3ap.H,D;bI pacnpe,D;eJl.HIOTC.H TOJlbKO Ha nOBepx-
HOCTH npOBO,D;HHKa. IIOBepXHOCTHa.H nJlOTHOCTb pacnpe,D;eJleHH.H 3aBHCHT OT KPH-
BH3HbI nOBepXHOCTH H 60JlbIIIe TaM, r,D;e KpHBH3Ha nOBepXHOCTH 60JlbIIIe.
C nOMOblO 3J1eKTpOMeTpa H ManeHbKoro IIIapHKa C (pHC. 8.33) Mom-
HO nOKa3aTb, qTO 3ap.H,D;bI pacnpe,D;eJleHbI Ha nOBepXHOCTH wapa M paBHO-
MepHO (cTpeJlKa 3J1eKTpOMeTpa Bce Bpe-
M.H OTKJlOHeHa Ha O)J;HO H TO me qHC-
JlO ,D;eJleHHH).
TaKHM me cnoco60M MomHO y6e,D;HTbC.H,
tJTO BHyrpH npOBO)J;HHKa nOJle OTCyrCTBYeT.
B pa3J1HqHbIX TOqKaX nOBepXHOCTH
lJ,HJlHH,D;pHqeCKOrO npOBO,D;HHKa C KOHH-
qeCKHM BbIcTynoM Ha O,D;HOM OCHOBaHHH
H TaKoH me Bna,D;HHOH Ha ,APyroM Ha-
np.HmeHHOCTb nOJl.H H nOBepXHOCTH8.H
nJlOTHOCTb 3ap.H,D;a a HeOlIHHaKOBbI. 3TO
MomHO nOKa3aTb C nOMOI.QblO O,AHHaKO-
a
6
+
++
+
+
Puc. 8.32
K 3J1eKTpoMeTpy
C
Puc. 8.33
235

BbIX 6YMamHbIX 3J1eKTpOCKOnOB, npHKpenJleHHbIX K
pa3HbIM TOqKaM npOBO,AHHKa. YroJl paCXOmeHH.H JlH-
CTOqKOB npOnOpHOHaneH nOBepXHOCTHOH nJlOTHOCTH
3ap.Ha (a) B TOH qaCTH npOBOHHKa, C KOTOpOH
conpHKacaeTCSI 3neKTpOCKon. MaKCHManbHoe 3HaqeHHe
nJlOTHOCTH 3ap.Ha a Ha OCTpHe, MeHbwee - Ha 60-
Puc. 8.34 KOBOH nOBepxHocTH H MHHHMaJlbHOe (paBHoe HYJlIO) -
BO BnR,AHHe. T8KHM 06pR30M. qeM 60nbwe KPHBH3-
Ha nOBepxHocTH npOBO,AHHKR. TeM 60nbwHe 3HaqeHHSI HMelOT nOBepxHOCTHa.H
nnOTHOCTb 3ap.H.na H Hanp.HmeHHOCTb nOJl.H (pHC. 8.34).
HanpSlmeHHocTb 3J1eKTpOCTRTH'leCKOrO non.H OKOJlO OCTpH.H 3ap.HmeHHOro
npOBOHHKa MomeT 6bITb CTOJlb 60nbwoH, qTO BbI3bIBaeT HOHH3RHIO MoneKYJl
B03,Ayxa. 1l0HbI, 3ap.HmeHHble OHOHMeHHO C OCTpHeM, OTTaJlKHB8IOTC.H OT Hero,
06pa3Y.H TaK Ha3bIBaeMbIH 3neKTpHqeCKHH BeTep, cnoc06HbIH OTKJlOHHTb nnaMSI
CBeQH. llOHbI npOTHBononomHoro 3HaKa npHT.HrHBaIOTC.H K npOBo.n;HHKY, 6bICT-
po pa3p.Hma.H ero.
ECJlH 3neKTpOMeTp 3a3eMJlHTb, TO OH 6y.n;eT H3Mep.HTb nOTeHIJ,HaJJ npOBO-
HHKa OTHOCHTeJJbHO 3eMJlH. C nOMOblO TaKoro 3neKTpOMeTpa MomHO nOKa-
3aTb, QTO nOTeHHan BO Bcex TOqKaX nOBepXHOCTH npOBo.n;HHKa JlI060H <pop-
MbI OHHaKOB. A TaK KaK BHYTPH npOBOHHKa E = 0, TO Ha OCHOBaHHH <pop-
MynbI E = <PI - <P2 MumHO YTBepmaTb, qTO H BO Bcex TOQKaX BHYTPH npOBO-
d
HHKa q> = const.
TaKHM 06pa30M. onbITbI nOKa3bIBaIOT. qTO:
1) HHyHpOBaHHbIe 3ap.HAbI, KaK H H36bITOQHbIe, pacnpeAen.HIOTC.H Ha no-
BepXHOCTH npOBO,AHHKa;
2) nnOTHOCTb pacnpeeneHHSI 3THX 3ap.H.n;OB 3aBHCHT OT KPHBH3HbI nOBepx-
HOCTH Ii 60nbwe TaM, r.n;e KpHBH3Ha nOBepXHOCTH 60JlbWe;
3) Hanp.HmeHHOCTb non.H BHYTPH npOBo.n;HHKa paBHa HynlO;
4) nOTeHHan nOJl.H BO Bcex TOQKaX npOBo.n;HHKa o.n;HHaKOB H paBeH no-
TeHHaJlY Ha ero nOBepXHOCTH;
5) HH.n;yHpOBaHHbIe 3ap.HbI MomHO pa3.n;enHTb.
8.18. 3JIeKTpOeMKocTb
3JIeKTpOeMKOCTL xapaKTepH3yeT cnoc06HOCTb npOBOHHKOB HnH CHCTeMbI H3
HeCKonbKHX npOBOHHKOB HaKannHBaTb 3neKTpH'leCKHe 3ap.H,AbI, a cne.n;OBaTeJlb-
HO, H 3J1eKTp03HeprHIO, KOTOpa.H B aJlbHeHWeM MomeT 6bITb Hcnonb30BaHa.
HanpHMep, npH cpoTOC'beMKe (BcnbIwKa) H T..n;.
Pa3J1H'laIOT 3J1eKTpOeMKOCTb yeHHeHHoro npOBOHHKa, CHCTeMbI npOBOHH-
KOB (B 'laCTHOCTH. KOH.n;eHcaTopoB).
Ye,llHHeHHbIM Ha3bIBaeTCSI npOBo.nHHK. p8cnonomeHHbIH B,AanH OT .npyrHx
3apSlmeHHbIX H He3ap.HmeHHbIX Ten TaK, qTO OHH He OKa3bIBalOT Ha 3TOT npo-
BOHHK HHKaKoro BJlH.HHH.H.
3JIeKTpoeMKOCTL yeAHHeHHoro npOBO,lJ,HHKa - <pH3HQeCKaSJ BeJlHqHHa, paBHaH
OTHoweHHIO 3J1eKTpHQeCKOrO 3ap.Ha yeHHeHHoro npOBOHHKa K ero nOTeHHaJIY:
C = !l. B Cll eHHHeH 3neKTpoeMKOCTH .HBn.HeTC.H <papa (q».
<p
236

1 <1> - 3TO 3J1eKTpOeMKOCTb TaKOrO npOBO,nHHKa, nOTeHII.HaJl KOTOpOrO H3Me-
H.HeTC.H Ha 1 B npH co06eHHH eMY 3apH,na B 1 Kn. IIOCKOJlbKY 1 <1> - OqeHb
60nblliaH e,nHHHII.a eMKOCTH, npHMeH.HIOT ,nonbHbIe e,nHHHII.bI: 1 n<1> (nHKo<}Ja-
pa,n) = 10- 12 <1>, 1 H<1> (HaHocpapa,n) = 10- 9 <1>. 1 MK<1> (MHKpocpapa,n) = 10- 6 <1>
H T .,n.
3J1eKTpOeMKOCTb npOBo,nHHKa He 3aBHCHT
OT po,na BeeCTBa H 3apH,na, HO 3aBHCHT
OT ero cpOpMbI 11 pa3MepOB, a TaKme 0'1' Ha-
JlHqH.H B6J1H3H ,npyrHx npOBo,nHHKOB HnH ,nH-
3J1eKTpHKOB. .D:eHcTBHTeJIbHO, npH6JIH3HM K
3apHmeHHOMY lliapy, COe,AHHeHHOMY C 3J1eKTpO-
MeTpOM. He3ap.H>KeHHYIO naJIOqKY (pHC. 8.35).
OH nOKameT YMeHblIleHHe nOTeHUHaJla lIlapa.
3ap.H,lJ; q lliapa He H3MeHHJlC.H, CJle,nOBaTenb-
HO, YBeJIHqHnaCb eMKOCTb. 3TO 06.HCH.HeTC.H
TeM, qTO Bce npoBo,nHHKH, pacnOnO>KeHHble
B6J1H3H 3apH>KeHHOrO npOBo,nHHKa, 3JIeKTpH3Y-
IOTC.H qepe3 BJlHHHHe B none ero 3ap.H,Aa H
60nee 6J1H3KHe K HeMY HH,nYII.HpOBaHHble 3ap.H,nbI npoTHBOnOJlOmHOrO 3HaKa
OCna6J1.HIOT nOJle 3ap.H,na q.
EcnH ye,nHHeHHbIM npoBo,nHHKOM .HBn.HeTCS! 3ap.HmeHHa.H ccpepa, TO nOTeHII.HaJI

QJ
db
Puc. 8.35
non.H H8 ee nOBepxHocTH q> = q ,r,ne R - pa,nHYC c<}JepbI. e - ,AH3JIeKTpH-
47tEoER
qeCKaSl npOHHII.aeMOCTb Cpe,AbI, B KOTOpOR Haxo,nHTCSI npoBo,nHHK.
Tor,na
c = !l.. = 47tE o ER -
<P
sneKTpoeMKocTb ye,nHHeHHoro ccpepHqeCKOrO npoBo,nHHKa.
06bIqHO Ha npaKTHKe HMelOT eJIO C ,ABYM.H H 60JIee npOBOlIHHKaMH. Pac-
CMOTpHM cHcTeMY H3 ,nBYX pa3HoHMeHHo 3apHmeHHblx npoBo,nHHKOB C pa3Ho-
CTblO nOTeHII.HanOB <PI - <P2 Mem,ny HHMH. 1.{T06bI YBeJlHqHTb pa3HocTb nOTeH-
II.HaJlOB Mem,ny 3THMH npOBO,AHHKaMH, He06xo,nHMO cOBeplIlHTb pa60TY npo-
THB CHJI 3neKTpOCTaTHqeCKOrO nOJIS! H nepeHecTH ,n06aBOqHbIH OTpHII.aTeJlb-
Hblii 3ap.H,n -q C nOJlOmHTeJlbHO 3ap.H>KeHHOrO npoBo,nHHKa Ha OTpHII.aTenbHo
3apH>KeHHbIH (HnH 3ap.H,A +q C OTpHII.aTeJlbHO 3ap.HmeHHOro npoBo,nHHKa Ha
nonomHTenbHO 3ap.HmeHHbIH). IIpH 3TOM YBeJlHqHBaeTC.H a6COJllOTHOe 3Haqe-
HHe 060HX 3ap.H,lJ;OB: Kal( nOJlOmHTeJlbHOrO, TaK 11 oTpHu;aTeJlbHoro. II03TOMY
B3aHMHoH 3JIeKTpOeMKocTLIO ,ABYX npoBo,nHHKOB Ha3bIBalOT cpH3HqecKYIO Be-
nH'lHHY, qHCJIeHHO paBHYIO 3apH,AY, KOTOpbIH HY>KHO nepeHecTH C o,nHoro npo-
BO,AHHKa Ha ,APyroH, ,AnH Toro qT06bI 113MeHHTb pa3HOCTb nOTeHU;HaJIOB Me>K-
,ny HHMH Ha 1 B:
c= q
<PI - <P2
B3aHMHa.H 3J1eKTpOeMKOCTb 3aBHCHT OT <}JOpMbI H pa3MepOB npOBo,nHHKOB,
0'1' HX B3aHMHoro pacnOJIOmeHH.H H OTHOCHTeJIbHOH ,AH3JIeKTpHQeCKOH npOHHu;a-
eMOCTH cpe,nbI, 3anOJlH.HIOeH npOCTpaHCTBO Mem,ny HHMH.
237

8.19. AH3JIeKTpHKH B 3JIeKTpOCTaTHQeCKOM nOJIe.
MexaHH3M nOJIHpH3al\HH )J;H3JIeKTpHKOB.
AH3JIeKTpHQeCKaS npOHH.....aeMOcTb
3J1eKTpH3aH.H npOBO.lJ;HHKOB H ,AH3J1eKTpHKOB BO BHeIllHeM 3J1eKTpHqeCKOM
none cyecTBeHHO OTJlHqaeTC.H .lJ;pyr OT .lJ;pyra: HH,AyHpOBaHHbIe 3ap.H,AbI H8
nOBepXHOCTH npOBO,AHHKOB B03HHK81OT B pe3ynbTaTe nepeMeeHHH CBo60,lJ;HbIX
3apSl,AOB BO BHeIllHeM none, B TO BpeMSI KaK B ,AH3J1eKTpHKe HeT CB060,lJ;HbIX
3ap.H,llOB, cnoc06HbIX nepeMemaTbC.H no,lJ; .lJ;eHCTBHeM nOJl.H.
3J1eKTpH3aHIO AH3J1eKTpHKOB Ha3bIBalOT nOn.HpH3anHeH. MexaHH3M ee 3aK-
nlOqaeTC.H B nepepacnpe.lJ;eneHHH 3ap.H,lJ;OB BHYTPH HeHTpanbHbIX aTOMOB H Mone-
KYJI no.lJ; .lJ;eHCTBHeM nOJl.H, JlH60 B nOBopOTax .lJ;HnOneH B nOJl.HpHbIX .lJ;H3J1eKT-
pHKax, JlH60 CMeeHHeM nO,lJ;peIlleTOK B npOCTpaHCTBe B KpHCT8JlJlHqeCKOM .lJ;H-
3J1eKTpHKe. J],eno B TOM, qTO KpHCTaJlnHqeCKHe peIlleTKH MHorHX HOHHbIX
,lJ;H3neKTpHKOB THna N aCI MomHO CqHTaTb COCTO.HHMH H3 ABYX BCT8.BJleHHbIX O.lJ;Ha
B ,lJ;pyryro nO,ApeIlleTOK, Ka>K,Aa.H H3 KOTOpbIX 06pa30B8..Ha HOH8MH O.lJ;HOro 3HaKa.
Bo BHemHeM 3J1eKTpHtJeCKOM nOJle H npoHCXO.lJ;HT CMeeHHe 3THX nO,lJ;peIlleTOK.
Bce ,lJ;H3J1eKTpHKH MomHO YCJlOBHO pa3.lJ;eJlHTb Ha TPH rpynnbI: nOJlSlpHbIe,
HenOJl.HpHbIe H HOHHbIe.
HenOJl.HpHbIe .lJ;H3J1eKTpHKH COCTO.HT H3 aTO-
MOB H MoneKY n, Y KOTOpbIX eHTpbI pacnpe,Ae-
JleHHH nOJlOmHTenbHbIX H OTpHaTeJlbHbIX 3ap.H-
1l0B (pHC. 8.36, 6) COBnallalOT (HHepTHbIe ra3bI.
KHCJlOpO.lJ;, 6eH30n, BO.lJ;OpO.lJ; H .lJ;p.).
IIOJlHpHbIe .lJ;H3neKTpHKH COCTO.HT H3 MOJle-
Kyn, Y KOTOpbIX eHTpbI pacnpe,AeneHH.H nOJlo-
mHTeJlbHbIX H oTpHaTenbHbIx 3ap.H.lJ;OB He CO-
Bna.lJ;alOT (pHC. 8.36, a) (cnHpT, BO.lJ;a H )J;p.).
B KaqeCTBe MO.lJ;eJlH MOJleKYJlbI nOJl.HpHOrO .lJ;H-
3J1eKTpHKa HcnOJlb3YlOTCH 3J1eKTpHqeCKHe ,AHnO-
nH - CHCTeMa .lJ;BYX paBHbIX no MO.lJ;YJlIO H npOTHBononomHbIX no 3HaKY TOqeq-
HbIX 3ap.H.lJ;OB, HaXO.lJ;.HHXC.H Ha paCCTOHHHH l .lJ;pyr OT .lJ;pyra (pHC. 8.37, a).
TenJlOBOe .lJ;BHmeHHe MoneKyn npHBO.lJ;HT K XaOTHqeCKOH opHeHTaHH .lJ;HnO-
neH. 113-3a 3Toro Ha nOBepXHOCTH .lJ;H3neKTpHKa, a TaKme H B JlI060M ero
06'beMe, CO.lJ;epmaeM 60JlbllIOe qHCnO MoneKyn (pHC. 8.38, a), 3J1eKTpHqeCKHH
3ap.H.lJ; B Cpe.lJ;HeM paBeH HYJlIO.
IIoMecTHM Tenepb .lJ;H3J1eKTpHK B O,AHOpO,AHOe 3J1eKTpOCTaTHqeCKOe nOJle Mem-
.lJ;y .lJ;BYMH napaJlJleJlbHbIMH MeTannHqeCKHMH nJlaCTHHaMH Co CTOpOHbI nOJlH
6
 (jJ =::e
&-0 +
-e
BO.lJ;a reJlHH
a
Puc. 8.36
a 6 6
F 1 - + qE
. . . .
+q -q -q . +q
F 2 = - q E E
Puc. 8.37
238

Ha Kam,lJ;bIH 3J1eKTpHqeCKHH ,n:HnOJlb 6y,n:YT ,n:eHCTBOBaTh ,lJ;Be CHJlbI, O,n:HHaKO-
BhIe no MO,lJ;YJlIO H npOTHBOnOJlOmHbIe no HanpaBJleHHIO (pHC. 8.37, 6). OHH
C03,lJ;a1OT MOMeHT napbI CHJI, CTpeM.HHHC.H nOBepHYTh ,n:HnOJlb TaK, qT06bI OCb
ero 6bIJl8 HanpaBJleHa no JlHHH.HM Hanp.HmeHHOCTH nOJl.H (pHC. 8.37, 6). IIo-
JlOmHTeJlbHbIe 3ap.H,n:bI CMeaIOTC.H npH 3TOM B Hanp8BJleHHH 3J1eKTpOCTaTHqeC-
Koro nOJl.H, a OTpHn;aTeJlbHbIe - B npOTHBOnOJlOmHYIO CTOPOHY.
XaOTHqeCKOe TenJlOBOe ,n:BHmeHHe MO-
a 6 JleKYJI npen.HTCTByeT C03,n:SHHIO ynop.H,n:o-
qeHHOH opHeHT8n;HH Bcex ,n:HnOJleH. TOJlb-
KO npH S6COJllOTHOM HYJle Bce ,n:HnonH
BbICTpOHJlHCb 6bI B,n:OJlb nHHH:M Hanp.HmeH-
HOCTH. IIo,n: BnH.HHHeM nOJl.H npOHCXO,lJ;HT
JlHllib qaCTHqHa.H opHeHT8n;H.H 3J1eKTpHQeC-
KHX ,n:HnOJleH. STO 03HaqaeT. qTO B cpe,n:-
HeM qHCJlO ,AHnOJleH. opHeHTHpOBaHHbIX
B.nOJlb nOJl.H, 60JlbllIe, qeM QHCJlO ,n:HnOJleH,
opHeHTHpOBaHHbIX npOTHBOnOJlOmHO nOJlIO.
Hs pHC. 8.38, 6 BH,n:HO: y nonOmHTeJIb-
Puc. 8.38 HOH nnSCTHHbI H8 nOBepxHocTH ,n:H3JIeKT-
pHKa nO.HBJI.HIOTC.f.I npeHMyecTBeHHo OT-
pHn;aTeJIbHbIe 3ap.H,n:bI ,n:HnOJleH, a Y OTpHn;STeJlbHO 3ap.HmeHHOH - nOJlOmH-
TeJlbHbIe. B pe3YJlbTaTe Ha nOBepxHocTH ,n:H3JIeKTpHKa B03HHK8eT HeCKOMneH-
cHpoBaHHbIH CB.H3aHHbIH 3ap.H,n:. BHYTPH ,n:H3J1eKTpHKa nOJlOmHTeJlbHbIe H OTPH-
n;aTeJlbHbIe 3ap.H,lJ;bI ,lJ;HnOJle:M KOMneHcHpYIOT ,lJ;Pyr ,n:pyra H cpe,n:HH:M CB.H3SHHbIH
3J1eKTpHqeCKHH 3Sp.H,n: no-npemHeMY paBeH HYJlIO.
HenOJl.HpHbIH ,n:H3J1eKTpHK B 3J1eKTpHqeCKOM nOJle TaKme nOJl.HpH3yeTC.H. no,n:
,n:eHcTBHeM nOJl.H nOJlOmHTeJlbHbIe H OTpHn;STeJlbHbIe 3apSl,n:bI MOJleKYJlbI CMea-
IOTC.H B npoTHBOnOJlOmHbIe CTOpOHbI H n;eHTpbI p8cnpe,n:eJleHH.H nOJlOmHTeJlbHO-
ro H OTpHn;aTeJlbHOrO 3ap.H,n:oB nepecTalOT COBna,n:aTb, KaK H y nOn.HpHOH MO-
JleKYJlbI. TaKHe ,n:e<}JopMHpOB8HHbIe MOJleKYJlbI MomHO paccMsTpHBaTb KaK 3J1eK-
TpHqeCKHe ,n:HnOJlH, OCH KOTOpbIX HanpaBJleHbI B,lJ;OJlb nOJl.H. Ha nOBepXHOCT.HX
,n:H3J1eKTpHKS, npHMbIKalOHX K 3ap.HmeHHbIM nJlaCTHH8M, nO.HBJI.HIOTC.H CB.H3aH-
HbIe 3ap.H,lJ;bI, KaK H npH nOJl.HpH3an;HH nOn.HpHOrO ,lJ;H3J1eKTpHKa.
CB.H3aHHbIH 3ap.H,lJ; C03,n:aeT 3J1eKTpOCTaTHqeCKOe nOJle H8np.HmeHHOCTblO El'
HanpaBneHHOH B ,lJ;H3J1eKTpHKe npOTHB Hanp.HmeHHOCTH BHelliHero nOJl.H 3ap.H-
,n:OB Ha nJlaCTHHax (pHC. 8.39). II03ToMY nOJle BHYTpH ,n:H3J1eKTpHKa ocna6J1.HeT-
C.H. H8np.HmeHHOCTb pe3YJlbTHpYIOero
1: 1 1:0 3J1eKTpOCT8TH'tJeCKoro nOJls
+
+
+
+
Puc. 8.39
E = Eo + El JlH E = Eo - El .
CTeneHb oCJlS6J1eHH.H nOJl.H 38BHCHT OT
CBOHCTB ,n:H3J1eKTpHKa. )];JI.H xapaKTepHcTH-
KH 3neKTpHqeCKHX CBOHCTB ,lJ;H3J1eKTpHKoB
BBO,n:HTC.H oc06a.H BeJlHqHHa, Ha3bIBaeMS.H
,lJ;H3J1eKTpHQeCKOH npoHHn;aeMocTblO.
,lI;H3J1eKTpH'lecKaSl npOaeMOCTL E -
3TO <}JH3HqeCK8.H BeJlHqHHa, nOKS3bIB81O-
a.H, BO CKOJlbKO pa3 MO,n:YJlb Hanp.HmeH-
239

HOCTH SJleKTpOCTaTHqeCKOrO nOJl.H E BHYTPH O)J;HOpO)J;HOrO )J;HSJleKTpHKa MeHb-
lIle MOYJl.H Hanp.HmeHHOCTH nOJl.fl Eo B BaKYYMe, C03aHHoro TeMH me 3ap.H-
aMH:
E
E=.
E
8.20. KOH,ll;eHCaTopLl. EMKOCTb nJIOCKOrO KOH,ll;eHCaTopa.
CoeAHHeHHe KOHAeHcaTopOB
KOH.neHcaTopbI COCTO.HT H3 .nBYX HJlH 60Jlee 6J1H3KO pacno-
JIOmeHHbIX pyr K APyry npOBOHHKOB (06KJlaoK), pa3eJleH-
HbIX CJlOeM ,AHSJleKTpHK8 (pHC. 8.40), npHqeM TOJlHHa CJIO.H
HSJleKTpHKa Memy npOBOHHKaMH 3HaqHTeJlbHO MeHblIle pa3-
MepOB CaMHX npOBOHHKOB.
IIpH He60JlblIlHX pa3Mepax KOHlIeHcaTOp OTJIHqaeTC.H 3HaqHTeJlb-
HOR eMKOCTblO, He 3aBHCSIeR OT HanHqH.H B6J1H3H Hero PyrHx
3ap.HOB HJlH npOBOHHKOB.
06KJlaKaM KOHeHcaTopa co06alOT OHHaKOBbIe no MO-
YJlIO. HO npOTHBOnOJlOmHble no 3HaKY 3ap.HbI, qTO cnoc06-
cTByeT HaKOnJIeHHIO 3ap.HOB, TaK KaK pa3HOHMeHHble 3ap.H-
)J;bI npHT.HrHBaIOTCSI H nOSTOMY pacnOJlaralOTC.H Ha BHYTpeHHHX nOBepXHOC-
T.fIX nJlaCTHH.
IIo,A 3ap.H.ltOM KOH,AeHCaTOpa nOHHMalOT 3ap.H O.ltHOR nJlaCTHHbI.
3JIeKTpOeMKOCTblO KOHlIeHcaTopa Ha3bIBalOT «pH3HqeCKYIO BeJlHqHHY. qHCJleH-
HO paBHYIO OTHomeHHIO 3ap.Ha KOH.neHcaTopa K pa3HOCTH nOTeHllHaJlOB Mem,ny
ero 06KJIa,nKaMH:
Puc. 8.40
c = q
<PI - <P2
IiJlH
c = !!... .
u
IIoJle nJlOCKOrO KOHeHcaTopa MomHO paccMaTpHBaTb KaK cOBoKynHocTb no-
JleR BYX 6eCKOHeqHbIX pa3HoHMeHHo 3ap.HmeHHblx nJlOCKOCTeR (pHC. 8.41, a
H 6). Hanp.HmeHHOCTb nOJl.H (pHC. 8.41, 8) MomHO HaRTH no npHHu,Hny
CJ a
cynepn03Hu,HH: E = 2EI = -; E = - Hanp.HmeHHOCTb non.H KOH,AeHCa-
EoE EoE
240
a
6
6
+0'
+0'
-a
-a
.
-
.
.
-
Puc. 8.41

Topa, re a nOBepXHOCTHa.H nJlOTHOCTb 3ap.H,Aa Ha 06KnaKax KOH-
eHcaTopa.
PaCCqHTaeM eMKOCTb nJlOCKOrO KOHeHcaTopa.
0603HaqHM nJlOab OHOB 06KJlaKH S, paCCTO.HHHe Me>KY HHMH d. TaK
KaK C = !L, q = as, U = Ed, TO C = as r"e E = . CJleOBaTenbHO , eM-
U Ed'''''' EoE
KOCTb nJlOCKOrO KOHeHcaTopa
C = EoES
d
TaKHM 06pa30M, eMKOCTb nJlOCKOrO KOHAeHcaTopa 3dBHCHT OT nJlOII.{a-
H 06Kna,ltOK. paCCTO.HHH.H Memy HHMH H ,ltH3J1eKTpHqeCKOB npOHHaeMOC-
TH H3J1eKTpHKa, 3anOJlH.HlOwerO npOCTpaHCTBO Memy 06KJla,ltKaMH KOH-
eHcaTopa, HO He 3aBHCHT OT MaTepHana. H3 KOToporo 3TH nnaCTHHbI H3-
rOTOBJleHbI.
KOHeHcaTopbI MomHO KJlaCCHqJHHpOBaTb no cneYIOII.{HM npH3HaKaM H
CBOHCTBaM:
a) no Ha3HaQeHHIO - KOHeHcaTopbI nOCTO.HHHOB H nepeMeHHOB eMKOCTH;
6) no 4JopMe 06KJIOK pa3J1HqalOT KOHeHcaTopbI nJlOCKHe, cqJepHqeCKHe,
HJlHHpHqeCKHe Ii p.;
B) no THny AH3J1eKTpHKa - B03YlIIHbIe, 6YMa>KHbIe, CJlIO.HHbIe, KepaMH-
qeCKHe, 3J1eKTpOnHTHqeCKHe H T..
,Z:(JI.H nOJlyqeHH.H Heo6xoHMOH eMKOCTH KOHeHcaTopbI COe,AHH.HIOT Me>Ky
C060B B 6aTapeH, npHMeH.H.H npH 3TOM napaJlJleJlbHOe, nocneOBaTeJlbHOe H
CMelIIaHHoe COe,AHHeHH.H.
C 2
+ -
Puc. 8.42
IIpH napaJlJleJlbHOM coeJ];HHeHHH KOHlIeHcaTopoB
O,ltHH 06KJla.n;KH Bcex KOH.n;eHcaTopoB COeHH.HIOTC.H B
O,llHH Y3eJl, .ztpyrHe - B pyroB Y3eJl. 06WHH 3ap.H,n
paBeH anre6paHqeCKOH cYMMe 3ap.HOB Kam,noB H3 06-
KJla)J;OK OTeJlbHbIX KOHeHcaTopOB (pHC. 8.42):
q=qI + q2+ Q3'
TaK KaK coe,nHHeHHbIe 06KJla.nKH npe.n;CTaBJI.HIOT C060B
OHH npOBO,AHHK, TO nOTeHU;HanbI Bcex COe,AHHeHHbIX B
OHH Y3eJl 06KJla,nOK o,nHHaKOBbI H pa3HOCTb nOTeH-
HaJlOB Memy 06KJlaKaMH Bcex KOHeHcaTopOB 01f-
HaKOBa:
U I = U 2 = U 3 .
Tax Kal< q=CU, qI=CIU, q2=C 2 U, q3=C 3 U, TO CU=C I U+C 2 U+C 3 U=>
n
C = C 1 + C 2 + C 3 ' C = L C t .
i=1
EMKocmb 6amapeu napaJl.Jl.eJl.bliO coeiJulieliliblX KOliiJelicamop06 paBlia CYM'
Me eMKocmeil omiJeJl.bliblX KOliiJencamopOB.
ECJlH napaJlJlenbHO COeHH.HIOT n o,nHHaKOBbIX KOH,neHcaTopOB, TO
C = nC I .
241

£1 8 1
-  8 2 -==--
Puc. 8.43
C 1 C 2 C a
M r--1 r-rll-rl t----y N
I I I I
!. Vi .!. U 2 .!. U a .!
I U I
,... . I
Puc. 8.44
ECJlH Mem,D;Y 06KJla,D;KaMH nJlOCKOrO
KOH,D;eHCaTOpa HaXO,D;.HTC.H ,D;BS PS3J1HqHbIX
,D;H3J1eKTpHKa, npHqeM nepBbIH 3aHHMaeT
qaCTb nJlOa,D;H 8 1 , S BTOpOH - qSCTb
nJlOI.qS,D;H 8 2 (HanpHMep, B03,D;YIIIHbIH
KOH,D;eHCaTOp qaCTHqHO norpymeH B Ke-
pOCHH (pHC. 8.43», TO TSKYIO CHCTeMY
MomHO paCCMaTpHBaTb KaK ,D;Ba napsJI-
JleJlbHO COe,lJ;HHeHHbIX KOH,D;eHCaTopa. Mx
06a.H eMKOCTb
C - C C - EoEI8 1 EoE282
- 1+ 2-+'
IIpH nOCJle)J;OBaTeJlLHOM coe)J;HHeHHH
KOH,D;eHCaTOpOB (pHC. 8.44) nOTeHHSJI co-
e,lJ;HHeHHLIX Mem,D;Y C060H 06KJla,ZJ;OK KOH-
,D;eHCaTOpOB O,lJ;HHaKOB. ECJlH C006I.qHTb 0,lJ;-
HOH H3 06KJla,D;OK nepBoro KOH,D;eHCSTOpa
3ap.H,D; +q, TO Y BTOpOH 06KJla,D;KH 6Y,D;eT
3ap.H1I -q. Y coce.nHeH 06KJla.nKH BToporo
KOH,lIeHCSTOpa 3ap.HH +q H T.,D;. CJle,lJ;OBa-
TeJlbHO,
q=ql =Q2=q3'
HSnp.HmeHHe HS 6aTspee p8BHO CYMMe Hanp.HmeHHH Ha Bcex KOH.neHCSTopax:
U=U 1 +U 2 +U g .
TaK KaK U =!I; U I =!L; U 2 =!L; U 3 =!L, TO !I =.!L +.!L +.!L =>
C C I C 2 C 3 C C I C 2 C 3
1 n 1
- = L,-.
C ;=] C j
BeJl.UIlUlia. 06pamlia.R eMKocmu 6amapeu nOCJl.e006ameJl.bIiO coe8ulielilibtX KOIi'
oelicamop06. pa6lia cYMMe 6eJl.U1lUIi. o6pamlibtX eMKOCm.R..M Om8eJl.blibtX KOlioeli-
camop06.
ECJlH nOCJle,lJ;OBaTeJlbHO COe,lJ;HHeHbI n O,lJ;HHaKOBbIX KOH,lJ;eHCaTOpOB, TO C = C; .
n
8.21. 3JIeKTpeTbI H Cel'HeT03JIeKTpHKH
B HeKOTopbIX ,D;H3J1eKTpHKSX nOJl.HpH3SH.H He HCqe3SeT CPS3Y nocJle BbI-
KnlOqeHH.R BHelliHero 3J1eKTpHqeCKOrO nOJl.H.
,ZJ;H3J1eKTpHKH, ,D;JlHTeJlbHOe BpeM.H COXpaH.HIOHe nOJl.HpH3aHIO nocJle BbI-
KJllOqeHHH BHelliHero 3J1eKTpHqeCKOrO nOJl.H H C03,D;SIOHe B oKpymalOeM npo-
CTpaHCTBe c06CTBeHHoe 3J1eKTpHqeCKOe nOJle, Ha3bIBalOT 3J1eKTpeTaMH. K HHM
OTHOC.HTC.H: CTeKJlO, 360HHT, cepa, rrapacpHH, KaHHcpOJlb, nJleKCHrnac H ,D;p.
242

BpeM.H COXpaHeHH.H nOJl.HpH3aHH 6e3 3aMeTHOrO ee YMeHbIlIeHH.H B pa3HbIX
3J1eKTpeTRX pa3nHqHO. Bo MHOrHX cJlyqa.HX OHO HCqHCJI.HeTC.H ,neC.HTKaMH JleT.
ECJlH paCnJlaBHTb CMeCb BOCKa H CMOJlbI, a 3aTeM Me,nJleHHO OXJla,nHTb ee ,no
MB
KOMHaTHOH TeMnepRTypbI B nOJle C Hanp.HmeHHOCTblO nOp.H,nKa 10 -, TO
M
opHeHTan;H.H MOJleKYJl.HpHbIX ,nHnoneH, COOTBeTCTBYIOI..QRSI mH,AKOMY COCTOSlHHIO,
COXpRH.HeTC.H H B TBep,nOM. TaKOH 06pa3en; cnoc06eH C03,ARTb B oKpymalOI..QeM
npOCTpRHCTBe 3J1eKTpOCTaTHqeCKOe nOJle, cyeCTBYIOee OqeHb nnHTeJlbHOe
BpeM.H .
Cel'lleT03J1eKTpHKH npe,nCTRBJI.HIOT C060H KpHCTaJlJlbI, 06J1a)J;alOI..QHe cnOHTaH-
HOH nOJl.HpH3an;HeH B HeKOTopOM TeMnepaTypHOM HHTepBaJIe.
CnOHTaHHa.H nOn.HpH3aH.H CB.H3aHR C HaJlHqHeM B KpHCTaJlJlaX MHKpO-
CKOnHqeCKHX 06J1aCTeH caMonpOH3BOJlbHOH nOJl.HpH3an;HH - ,nOMeHOB. STO
B03MomHO B TOM cJlyqae, eCJlH B HOHHOM KpHCTanJle eHTpbI pRcnpe,neJle-
HH.H nOJlOmHTeJlbHOrO H OTpHaTeJlbHOrO 3ap.H)J;OB .HOHOB, pacnOJlOmeHHbIX B
O,nHOH 3J1eMeHTRpHOH .HqeHKe, He COBna,naIOT H KRm,nR.H TRKa.H .HqeHKR MO-
meT paCCMaTpHBaTbC.H KaK ,nHnOJlb. B KRm,nOM ,nOMeHe ,nHnOJlH 3J1eMeHTap-
HbIX .HqeeK opHeHTHpOBaHbI o,nHHaKOBO, HO caMH )J;OMeHbI opHeHTHpOBaHbI
XaOTHqHO (pHC. 8.45, a). Ha nOBepXHOCTH KpHCTaJlJlR qepe,nYIOTC.H nonomH-
TeJlbHO H OTpHnaTeJlbHO 3ap.HmeHHbIe yqaCTKH, H B n;eJlOM KpHCTRJIJI He CO-
3,lIaeT B oKpymalOeM npOCTpaHCTBe 3J1eKTpHqeCKOrO nOJl.H. BHeceHHe cerHe-
T03J1eKTpHKR B 3J1eKTpHqeCKOe nOJle npHBo,nHT K opHeHTan;HH ,nOMeHOB B,nonb
nHHHH Hanp.HmeHHOCTH (pHC. 8.45, 6), npHqeM H3MeHeHHe opHeHTRHH ,no-
MeHOB npOHcxo,nHT He nJlaBHO, a CKaqKaMH (3c}>cpeKT BapKraY3eHa). B CHJlb-
HOM 3J1eKTpHqeCKOM nOJle Bce ,no-
MeHbI opHeHTHpOBRHbI B,nOJlb
non.H, HO npH YMeHbIlIeHHH HR-
np.HmeHHOCTH BHeiliHero nOJl.H )J;O
o---e o---e o---e o---e + HYJl.H opHeHTan;H.H HapywaeTC.H.
O,nHaKO B )J;H3J1eKTpHKe HH,nYH-
o-e o---e o---e o--e +
pOBaHHa.H nOJl.HpH3an;H.H qaCTHqHO
o---e o---e o---e o---e + OCTaeTC.H H3-3a Toro, qTO MHorHe
,nOMeHbI COXpaH.HIOT C03,naHHYIO
o---e o---e o---e o---e + BHeWHHM nOJleM opHeHTRn;HIO.
lln.H YCTpRHeHH.H OCTaTOQUOH no-
J1J1pH3aUHH cerHeT03neKTpHK no-
MealOT B 3J1eKTpHqeCKOe nOJle
npOTHBOnOJlOmHOrO HRnpRBneHH.H,
H Ta Hanp.HmeHHOCTb, KOTOpa.H He06xo,nHMa ,nJl.H 3Toro, Ha3bIBaeTC.H K03pH-
THBHOH CHJlOH. TaK, ,nJl.H THnHqHOrO cerHeT03J1eKTpHKa - THTaHaTa 6apH.H
KB
BaTi0 3 K03pn;HTHBHa.H CHJla E=100 -.
M
CerHeT03J1eKTpHKH 06J1anRIOT BbICOKHMH 3HRqeHH.HMH ,AHSJleKTpH''leCKOH npo-
HHuaeMOCTH E - 10 3 - 10 4 H MorYT 6bITb HCnOJlb30BaHbI B H3rOTOBJleHHH MaJlO-
rR6apHTHbIX KOH,AeHCRTopOB.
)J.JI.H cerHeT03J1eKTpHKoB cyecTByeT npe,neJlbHR.H TeMnepRTypa, HR3bIBReMa.H
T01JKOH KIOPH, npH npeBbIweHHH KOTOPOH npOHcxo,nHT HRpyweHHe opHeHTR-
a
6'
Eo

Puc. 8.45
243

HH ,AHnOneH B OMeHe BCJleCTBHe TenJlOBOrO BHmeHH.H H CnOHTaHHa.H nOJl.H-
pH3aH.H HCqe3aeT. B STOM cJlyqae cerHeTosneKTpHK CTaHOBHTC.H 06bIqHbIM H-
sneKTpHKOM. 06'b.HCH.HeTC.H STO TeM, qTO npH TeMnepaType KlOpH npOHCXOHT
nepeCTpOHKa KpHCTaJIJlH'leCKOH CTPYKTypbI.
8.22. llbe303JIeKTpH'IeCKHH 3<}><}>eKT
Hero HCnOJIb30BaHHe B TeXHHKe
ECJlH 06pa3e H3 cerHeTosneKTpHKa cmHMaTb BOJlb HanpaBJleHHSl OCTaTOq-
HOH nOJl.HpH3aHH, TO B STOM HanpaBJleHHH 6YYT YMeHblIIaTbC.H pa3MepbI Bce-
ro 06pa3na H KamlIoH SJleMeHTapHoH .HqeHKH. IIpH STOM paCCTO.HHHe Memn;y
neHTpaMH pacnpeneJleHH.H nOJlOJKHTeJlbHOrO H OTpHnaTeJlbHOrO 3ap.HOB B .HtleHKe
H OCTaTOqHa.H nOn.HpH3aI.J.H.H 06pa3a B eJlOM TaKme YMeHb1II8IOTC.H. B HTore
6yeT YMeHblIIaTbC.H H pa3HocTb nOTeHH8J10B Mem,AY rpaH.HMH 06pa3n;a, nep-
neHHKYJl.HpHbIMH HanpaBJleHHIO eHCTBH.H CHJlbI. IIpH paCT.HmeHHH, Ha060pOT,
OCTaTOtlHa.H nOJl.HpH3an;H.H H cooTBeTcTBYIOJ..qa.H eH nOBepxHOCTHa.H nJlOTHOCTb
3ap.H,AOB Ha rpaH.HX YBeJlHqHB8IOTC.H, qTO BeeT K B03HHKHOBeHHIO pa3HocT11
nOTeHHaJlOB, npOTHBOnOJlOmHOH no 3HaKY.
HBJleHHe B03HHKHOBeHH.H nOJl.HpH3aHoHHbIX 3ap.HOB npH ecpopMaHH Ha-
3bIB81OT npsMLIM nLe30SJleKTpH'IeCKHM seKTOM. OHO 6bIJlO OTKpbITO 6paTb-
.HMH KIOPH Ha KBape, fie .HBJI.HIOJ..qeMC.H cerHeTOSJleKTpHKOM.
OK83bIBaeTC.H, qTO Bce cerHeTOSJleKTpHKH .HBJI.HIOTC.H OHOBpeMeHHO nbe30-
SJleKTpHKaMH, HO aJleKO He Bce nbe30SJleKTpHKH 06J1aaIOT cerHeTOSJleKTpH-
qeCKHMH CBOHCTBaMH.
KOnHtleCTBeHHO nbe30scpcpeKT xapaKTepH3YlOT nbe30MOYJleM d: d =  ' re
q -- B03HHKalOHH 3ap.H, F -- Moynb ecpopMHpYIOeH CHJlbI.
CYJ..qeCTByeT TaKme 06paTHbIH nbe30scpcpeKT -- nO.HBJleHHe ecpopMaHH
nbe30SJleKTpHKa npH nOMeJ..qeHHH ero B sneKTpHqeCKOe nOJle. 3TO .HBJleHHe
06'b.HCH.HeTC.H yJlHHeHHeM Hnl1 YKopaqHBaHHeM sneMeHT8pHbIX .HqeeK no
eHCTBHeM SJleKTpOCTaTHtleCKOrO nOJl.H H, KaK CJleCTBHe, -- H3MeHeHHeM
p83MepoB 06pa3a B pe3ynbTaTe HToroBoH MaKpOCKOnHqeCKOH ecpopMaHH.
,ZJ;ecpopMan;H.H B STOM cJlyqae, KaK nOKa3aJlH HCCJleOB8HH.H, np.HMO npo-
nopHOHaJlbHa npHJlOmeHHOH pa3HOCTH nOTeHHaJlOB H 3aBHCHT OT nbe30-
MOy JI.H.
IIp.HMOH H 06paTHbIH nbe30scpcpeKTbI flalIIJlH lIIHpOKoe npHMeHeHHe B Tex-
HHKe JI.H npe06pa30BaHH.H MeXaHHqeCKHX KOJle6aHHH B SJleKTpHtleCKHe H H8-
060pOT. IIbe30sJleKTpHqeCKHH SJleMeHT. HanpHMep. HaXO,AHTC.H B rOJlOBKe CO-
BpeMeHHoro SJleKTpocpOHa lIJI.H BocnpOH3BeneHH.H rpaM3anHCH.
IIIHpoKoe npHMeHeHHe HalIIJlH nbe30SJleKTpHtleCKHe MHKpOcpOH H TeJlecpOH,
H3MepHTenH BH6paHH, MeX8HHtleCKHX H8np.HmeHHH H T.. KBapeBa.H nnac-
THHKa, HanpHMep, .HBJI.HeTC.H MOJ..qHbIM H3J1aTeJleM YJlbTp83BYKOBbIX BOJlH, qTO
HcnoJlb3yeTC.H B TeXHHKe, 6HOJlOrHH, MeHn;HHe.
IIbe30sJleKTpHtleCKHe Kone6aHH.H npHMeH.HIOTC.H TaKme JI.H cTa6HJlH3an;HH
Q8CTOT reHep8TopOB SJleKTpHQeCKHX KOJle68HHH B pa)J;HOTeXHHKe.
244

8.23. 3HeprHs H nJIOTHOCTL 3HeprHH
3JIeKTpOCTaTHQeCKOrO nOJIS
ECJlH It nJlaCTHHaM 3ap.HmeHHOro KOHeHcaTopa npHcoeHHHTh JlaMnOQKY,
TO OHa BcnbIXHeT, a KOHeHcaTOp pa3p.HHTC.H, H 3JIeKTpOCTaTHqeCKOe nOJle
Memy ero nJlaCTHHaMH HCqe3HeT. CJIeOBaTeJIhHO, SJleKTpOCTaTHqeCKOe nOJIe
KOHeHcaTopa 06J1alIaeT SHeprHeH. KOTOpa.H H npeBpaTHnaCb B CBeTOBYIO.
PaCCqHTaeM SHeprHIO 3apSImeHHoro KOHeHcaTopa. 3ap.H1l KOToporo q. Ha-
np.HmeHHe Ha KOHeHcaTope U, eMKOCTb C.
B npoecce pa3p.H,lJ;KH KOH,lJ;eHCaTOpa pa3HOCTb nOTeHHaJlOB Mem,lJ;Y 06KJIa,lJ;-
KaMH paBHOMepHO y6hIBaeT OT U ,lJ;0 HYJI.H, Cpe,lJ;Hee me 3HaqeHHe pa3HOCTH
U
nOTeHHaJlOB paBHO < U> = - .
2
Tora pa60Ta A, cOBepmaeMa.H sJleKTpHqeCKHM nOJleM npH pa3p.He KOH-
,AeHCaTOpa,
qU CU 2 q2
A =q' <u> =2=2= 2C '
a SHeprH.H, KOTOpOH 06J1a,lJ;aeT 3ap.HmeHHbI:U KOH,lJ;eHCaTOp, paBHa STOH pa-
60Te.
3TO:U SHeprHeH 06JIa,lJ;aeT SJleKTpOCTaTHQeCKOe nOJle KOH,lJ;eHCaTopa. BbIpa-
Cu 2
3HM ee Qepe3 xapaKTepHCTHKH nOJl.H. nOCTaBHB B «POPMYJlY W a = 2 BbIpa-
EoES
meHHe c = - a TaKme U = Ed , HMeeM
d '
w = CE 2 d 2 = EO ESd 2 E 2 = £oEE 2 Sd = EOEE 2 V .
3 2 2d 2 2
W = EEoE2 V.
2
3HeprH.H OHOpOllHOrO nOJl.H nponopUHOHaJlbHa 06'heMY. 3aHHMaeMOMY no-
neM. B CBSI3H C 3THM rOBOp.HT 06 3HeprHH ellHHHUbI 06'heMB nOJl.R ( 06'LeMHOH
nJlOTBOCTH 3BeprHH - w a ). w 3 = W 3 . B CM eHHHu.e:u 06'heMHOH nJIOTHOCTH
V
3HeprHH 'IBnHeTCH l\>Koynb Ha Ky611'IeCKHH MeTp ( ': 1-
Tora
E EE 2
W =-L-
3 2'
nOJlyqeHHa.H «POPMYJlB CnpaBe,lJ;JlHBa He TOJlbKO ,lJ;JI.H OHOpOllHOrO 3J1eKTpO-
CTaTHqeCKOrO nOJl.H, HO H JI.H JlI060ro Pyroro 3JIeKTpOCTaTHqeCKOrO nOJl.H, a
TaKme H JI.H nepeMeHHoro 3J1eKTpHqeCKOrO nOJl.H.
245

BonpOCbI ,l(JIH CAMOKOHTPOJIH
1. KaKue 6e14eCm6a omliOCSlmcSl K np060oHuKaM? q mo npoucxooum C 3apSl-
Oa.MU 6 np0600liUKe npu BlieceliUU ezo 6 3Jl.eKmpOCmamUlleCKOe nOJl.e? KaK MOJIC
HO 3apRiJumb U30JLUp06aliliblil np0600HUK, lie npUKaCa.JlCb K lieMY 3apSlJICeliHblM
meJl.OM?
2. llolleMY lianpSlJICeliliOCmb 3Jl.eKmpOCmamullecKozo nOJl.Sl Bliympu np0600
liUKa paBlia IiYJl.IO? KaK pacnpeOeJl.RlOmCSl 3apRobl B npoBooliuKe B cocmOSlliUU
pa61i06ecUSl?
3. KaKue OU3Jl.eKmpUKu lia3bl6alOm nOJl.SlpliblMU. lienOJl.SlpliblMU? q mo ma
Koe nOJl.SlpU3aUSl OU3Jl.eKmpuKa?
4. q eM 060SlCHJlemCSl U3Melieliue 3Jl.eKmpOCmamUlleCKOZO noJl.Jl npu 6lieCeliUU
6 Hezo OU3JLeKmpUKa? qmo lia3blBa1Om OU3Jl.eKmpUlleCKoil npoHuaeMocmblO?
5. KaKue nOJl.Sl lia3bl6a1Om nomeliUaJl.bHblMU? q mo Ha3bl6a1Om nomeliua
Jl.0.M 3Jl.eKmpOCmamullecKozo noJl.Sl. pa31i0Cmb1O nomeliUaJl.06? KaK MOJICIiO U3-
Mepumb pa3liOCmb nomeliUaJl.OB? q mo npuliSlmo 3a eouHuy nomeliuaJl.a?
6. KaK paCCilumbl6aemcSl nomeliuaJl. noJl..Jl mOllellliOZO JapSloa? 3apSlJICelili0-
zo wapa?
7. KaK paccilumbl6aemcSl nomeHijuaJl.blia.R 3liepZUSl cucmeMbl 63auMooeilcm6Y-
I014UX mOllellHblX 3apSlo06?
8. KaKOBa C6Sl3b lianpRJICeliUSl C nanpSlJlCeliHOCmblO 3.n.eKmpocmamUlleCKOZO noJL.R?
9. KaKaSl nOBepxliocmb Ha3blBaemcSl 3KBunomeIiUUaJl.bHoil? II OlleMY 3K6uno-
meliUaJl.bHble n06epxliocmu nepneliouKYJl.Slplibl Jl.UHUJl.M HanpSlJICeliliocmu?
10. KaKoBa Kapmulia 3KBunomeliUaJl.bHblX n06epxliocmeil OOIiOpOOIiOZO
nOJl.Sl? nOJl.Sl mOlleliliOZO 3apSloa? noJl.Sl 3apSlJICeliliOZO wapa?
11. II olleMY nomeHUaJL Bcex mOlleK 3apR:JICeHH020 npoBoiJHuKa oiJuHaKoB?
12. KaKYIO qJu3uIlecKY1O BeJLUlluny na3blBa1Om 3Jl.eKmpOeMKocmblO np060o-
liuKa? 3Jl.eKmpOeMKocmblO KOlioelicamopa? H a306ume eOUIiUijbl 3Jl.eKmpOeMKoc-
mu?
13. Om Ilezo 3a6ucum 3Jl.eKmpOeMKocmb nJl.OCKOZO KOlioelicamopa? q mo no-
HUMalOm noo 3apSlOO.M KOlioelicamopa?
14. KaK paccilumbl6aemcSl 3liepZWl 3apSlJICeliliOZO KOlioelicamopa?
15. q mo lia3bl6a1Om 06oeMiioil nJl.OmliOCmblO 3liepZUU 3Jl.eKmpOCmamUlieCKO-
zo nOJl.Sl? q eMY olia pa6lia? KaK liailmu 3liepZU1O 3Jl.eKmpOCmamullecKozo nOJl.Jl2
OCHOBHbIE cI»OPMY JIbl
w
<p = -2!..
qo
-- nOTeHHaJl sneKTpOCTaTHqeCKOrO nOJlIl B TOqKe,
re VV n -- nOTeHHanbHaR SHeprH.H np06Horo 3apR-
a qo, HaXO.Heroc.H B aHHoH TOqKe.
A = qo (<1>1 - <P2) = qU
pa60Ta sneKTpOCTaTHqeCKOrO nOJlIl no nepeMee-
HHIO np06Horo 3ap.Ha Memy BYM.H TOqKaMH 1 H
2 nOJl.H, r.ne (<PI - <P2) -- pa3HOCTb nOTeHHanOB
Memy STHMH TO'lKaMH; U -- Hanp.H)KeHHe.
246

q 1 q
cp=k-=--
Er 41tEo Er
W = k qlq2 =
n Er 41tE o
q2
w=
n 81tEEoR
E = <PI - <P2
d
U
d
q
C=-
cp
- q
C--
U
C = EoES
d
cr
E=-
EoE
n
C = L C t
i=1
1 n 1
C =  C t
_ qU _ q2 _ CU 2
w------
2 2C 2
W E 2
W =- =EoE-
V 2
- nOTeHHan nOJl.H TO'ieqHOrO 3ap.H,Aa q B HeKOTO-
pon TOqKe, r - paCCTO.HHHe OT 3ap.Ha o TO'iKH.
- nOTeHHaJlbH8.H 3HeprH.H B38HMO,AenCTBH.H ,ABYX
TOqeqHbIX 38p.HOB ql H q2'
- nOTeHHB..1IbH8.H 3HeprH.H nOJlS! 3ap.HmeHHOro m8pa,
re q - 3ap.H mapa, R - ero pa,AHYC.
- CB.H3b Hanp.HmeHHOCTH OHOpO,AHOrO sneKTpOCTa-
TH'ieCKOrO nOJl.H C pa3HOCTblO nOTeHI.{HanOB (Hanp.H-
meHHeM) BlIOJIb JlHHHH H8np.HmeHHOCTH, r lIe d -
paCCTO.HHHe Memy TOqKaMH 1 H 2 nOJl.H.
- 3neKTpoeMKOCTb yeHHeHHoro npOBO,AHHKa, re
q - 3apSl,A npOBO,AHHKa; q> nOTeHI.{U8J1 npo-
BOHHKa.
- eMKOCTb KOHeHcaTopa, re q - 3ap.H KOH-
)J;eHcaTopa; U - Hanp.HmeHHe Memy ero nJlaCTH-
HaMH.
- eMKOCTb nJlOCKOrO KOHeHcaTopa, r,Ae S - nno-
ab nJlaCTHHbI; d - paCCTO.HHHe Memy nnaCTH-
HaMH; E - H3JIeKTpHqeCKa.H npOHHI.{8eMOCTb Cpe,AbI
Memy nJlaCTHHaMH KOH,AeHCaTopa.
- Hanp.HmeHHOCTb non.H Memy nJlaCTHHaMH nJloc-
Koro KOH,AeHC8TOpa, re cr - nOBepXHOCTHa.H nJlOT-
HOCTb 3ap.Ha.
- 3J1eKTpOeMKOCTb 6aTapeH napaJIneJIbHO COe,AHHeH-
HbIX KOHeHcaTopOB, re C j - eMKOCTb OT,AenbHOro
KOHeHcaTopa.
- <popMyna ,AnSI onpe,AeJIeHH.H eMKOCTH 6aTapeH noc-
JleOBaTeJIbHO COe,AHHeHHbIX KOHeHcaTopOB.
- 3HeprH.H 3J1eKTpOCTaTHqeCKOrO nOJl.H 3ap.HjKeHHOrO
KOH)J;eHcaTopa, re q - 3ap.H KOH,AeHCaTOpa; C -
eMKOCTb KOH,AeHCaTOpa; U - Hanp.HmeHHe Memy
nnaCTHHaMH.
- 06'beMHa.H nnOTHOCTb 3HeprHH sneKTpOCTaTHqeCKO-
ro non.H, re W - 3HeprH.H nOJl.H; V - 06'beM
npOCTpaHCTBa, B KOTOpOM C03aHO nOJle; E - Ha-
np.HmeHHOCTb non.H; E - H3J1eKTpHqeCKa.H npOHH-
aeMOCTb Cpe,AbI.
247

METO,lI.IIQECKIIE PEKOMEHllAUlI1I
1. ITpH onpeeneHHH nOTeHHana non.H TOqeqHOro 3ap.Ha He06xoHMO
nOMHHTb, 'ITO 3HaK nOTeHHaJla Onpe,AeJl.HeTCSJ 3HaKOM 3ap.Ha (eCJlH q> 0,
TO <P> 0 H eCJlH q < 0, TO <P < 0), a nOTeHHaJl non.H, 06pa30BaHHoro He-
CKOJIbKHMH TOtIetIHbIMH 3ap.HaMH, onpe,Aen.HeTC.H KaK anre6paHtIeCKaSI CYMMa:
<P = <p] + <P2 + ... + qJn .
2. ITpH pelIleHHH 3aatI, B KOTOpbIX Tpe6yeTcSI onpeenHTb nOTeHHanbI,
3HeprHH H pyrHe xapaKTepHCTHKH CHCTeMbI TeJl - ccpepHqeCKHX npOBOHH-
KOB, KOH,AeHCaTOpOB cneyeT HMeTb B BHy, liTO npHHHn cynepn03HHH, 3a-
KOHbI coxpaHeHH.H 3neKTpHqeCKOrO 3ap.Ha H 3HeprHH cnpaBenHBbI H 3eCb 11
HX Hao Hcnonb30BaTb.
3. EcnH B 3aatIaX onHCMBaeTC.H pa3BHmeHHe nnaCTHH KOHeHcaTopa HnH
y,AaneHHe H3neKTpHKa, TO Heo6xoHMO oc060e BHHMaHHe 06paTHTb Ha TO, 6bIn
nH 3ap.H>KeHHbIH KOH,AeHCaTOp npeBapHTenbHO OTKnlOqeH OT HCTO'lHHKa Ha-
npSl>KeHH.H (B 3TOM cnyqae COXpaH.HeTC.H 3apSl,A Ha KOHeHcaTope) HJlH KOHeB-
caTOp He OTKnlOqanCSI (B 3TOM cJlyqae HanpSI>KeHHe Me>Ky nJlaCTHHaMll KOH-
eHcaTopa He H3MeHSIeTCSI).
TIPIIMEPbI PEllIEHII.H 3A.L(A q
1. TIpOTOH BbIneTaeT 113 TO'lKH. nOTeHI.l,H8n KOTOpOH 475 B, co CKOpOCTblO
M
190 K8KYIO CKOpOCTb no MO,AynlO OH 6Y,AeT HMeTb B TOtIKe C nOTeHIl.Ha-
C
nOM 450 B?
)J;aHO: <1'1=475 B.
q = 1,6 .10- 19 KJl.
H8HTH: v2'
P e ill e H He. Cor n8CHO
<P2 = 450
B.
VI = 190
M
C
m = 1,67 .10- 27
Kr,
TeopeMe 0 KHHeTH'leCKOH 3HeprHH, WK = A. M3Me-
mv 2 mv 2
HeHHe KHHeTHtIeCKOH 3HeprHH npOTOH8 WK = ----.1. - --1.... By,u.eM npeHe6per8Tb
2 2
rp8BHT81l.HOHHbIM B38HMO,AeHCTBHeM. Tor,Aa pa60T8 CHJJ 3neKTpOCTaTH'leCKOrO
nonSl A = q (<PI - <P2)' Cne,AOBaTenbHO.
mv _ mv = q(<PI _ <P2) => V2 = 2q(<Pl - <P2) + vi: V2= 70 KM
22m C
2. TO'le'lHbIH 3apSI,A 5.0 HKn H8XO,AHTCSI Ha p8CCTOSlHHH 0,45 M OT no-
BepXHOCTH ye,AHHeHHOro lll8P8 p8,AHYCOM 10 CM, 38pSlmeHHoro /1,0 nOTeHIl.1l8-
na 2400 B. K8KYlO p860TY H8,A0 COBeplllHTb, 'lT06bI YMeHblllHTb paCCTOSlHlle
Mem,Ay HHMll Ha 40 CM?
)J;aHO: qo=5,O.10- 9 Kn, rl=O,45 M, R=0.10 M. <p=2400 B, r=0,40 M.
H a H T H: A.
P e III e H He. BY,AeM CtIHTaTb, 'lTO 3apSlmeHHbIH lllap C03,u.aeT 3neKTpOCTaTH'leC-
Koe none, H B 3TOM none nepeMe8eTCSI TO'le'lHbIH 38p.H,A qo. CHnbI nonSl COBep-
lllalOT pa60TY Al = qo (<PI - <P2)' l',Ae <PI = ( ) ; <1>2 = (q ) .
41tEoE R + r 1 41tEoE R + r 1 - r
248

3HaSl nOTeHRan mapa. MomHO HaiiTR 3apSl,ll. q Ha HeM:
<p = q  q  41tEoER<p.
41tEoER
I10,ACTaBRM 3HaqeHHe q B <l>opMynbI ,AnSI <PI Ii <P2:
R<p R<p
<PI = - ; <P2 =
R+r 1 R+rl-/1r
Tor,As pa60Ta
Al = qo (  _ R<p ] = qoR<pllr
R + r 1 R + rl - llr (R + rd (R + rl - llr) .
MCKOM8S1 pa60Ta A, COBepmaeMaSl npOTIiB CHn 3neKTpOCTaTH'leCKOrO nonSI.
A = -A] = 4.8 MK)J,>K.
3. Me>K,AY nnaCTIiHaMH nnOCKoro B03,AymHoro ropH30HTanbHO pacnonomeH'
Horo KOH,AeHCaTOpa HaXOHTCSI 3apSImeHHaSl KannSl M8CCOH 10 MKr. 3apSllI Kan.
nH 1.0.10- 12 Kn. TIPIi pa3HOCTH nOTeHUHanOB 600 B 11 Ha'lanbHOH CKOpOC-
Tli. p8SHOH HynlO. KannSl npOXO.ll.liT HeKOTopoe paCCTOSlHue B 3 pa3a Me,AneH-
Hee. 'leM npH OTCYTCTBRR 3neKTpR'leCKOrO nonSl. HaH.lI.HTe paCCTOSlHHe Mem,AY
nnaCTHHaMH.
,l(8HO: m= 1,0 .10- 9 Kr, q= 1,0 .10- 12 Kn, t 2 =3t}J U=600 B, vo=O.
HaHTH: d.
Pew e H He. B OTCYTCTBlie 3neKTpOCTaTIi'leCKOrO nonSl Ha KannlO ,AeHcTByeT
M
TonbKO CliJI8 T.H>KeCTIi, R KannSl ,ABlimeTCSI C YCKopeHReM g = 9,8 c 2 ' HeKOTO-
poe paCCTOSlHHe /1r OH8 npOXO,AIiT 38 BpeMSI t l' TaK KaK
!:.r = gt: . TO t, = J2l:.r . (1)
2 g
B 3neKTpOCTaTIi'leCKOM none Ha KannlO (pIiC. 8.46) ,AeHcTByeT CHna TSlmeCTH
mg Ii clina F co CTOpOHbI 3neKTpOCTaTli'leCKOrO nonSl. YpaSHeHHe ,ASlimeHIiSI Kan-
nli B 3TOM cnyq8e mii = mg + F. CnpoelipyeM BeKTopHbIe BenH'lHHbI Ha OCb y:
ma=mg-F RnH ma=mg-qE.
U
TaK K8K E = -, TO
d
ma = mg _ qU => d = qU .
d m(g-a)
YCKopeHHe HaH,AeM R3 <popMynbI
2 2
llr = at 2 = a . 9 . t 1 => a = 211r .
2 2 9t 1 2
'Yt.JHTbIBaSl (1), nonY'lHM a = !; d = 9qU ; d = 6,8 CM.
9 8mg
4. TO'le'lHbIe 3apSl,AbI 2,0 MKKJI Ii 5.0 MKKn pacnonomeHbI Ha paCCTO-
SlHHH 0,40 M ,Apyr OT ,Apyra. OnpeenHTe nOTeH1i8nbl nonSl B TOt.JKax B
Ii C (pIiC. 8.47), ecnH d = 0,30 M. Onpe,AenHTe pa60TY no nepeMe1l\eHHIO 3a-
pSl.ll.a 1.0. 10 HKn H3 TO'lKIi B B TO'lKY C.
E t F _a
mg
+
y
Puc. 8.46
249

d
q 1....--------
q2
B
AaHO: ql=2,0.10- 6 En, Q2=5.0.10- 6 Kn,
Qo=1,0.10- 8 En, l=0,40 M, d=0,30 M.
H 8 H T H: <PB, <Pc, ABC'
P e ill e H He. TIo npHHu,Hny cynepn03Hu,HH noneH
<P B = <PIB + <P2B =  + Q2 =
41tfoEd 41tf O f .Jd 2 + l2
... C =  ( q2 + ql ) ; <PB= 150 KB.
Puc. 8.47 41tEoE .J d 2 + l2 d
<Pc = <PIC + <P2C =  ( h + q2 ) ; <PC = 1,9.10 2 KB.
41tfOE d 2 + l2 d
P860Ta 3neKTpOCTaTH'leCKOrO nonSl ABC = qo (<PB - <Pc): A Bc = -0.36 MKAm.
3H8K "-" YK83bIsaeT, 'lTO A8HH8S1 p860T8 cosepm8eTCSI BHemHeH CHnOH npo-
THB 3neKTpH'leCKHX CHn.
C 2
cc
Puc. 8.48
5. OnpeLIenHTe eMKOCTb 6aTapeH KOHLIeHC8TopOB (pHC.
8.48). ecnH eMKOCTb nepBoro KOH,AeHC8TOp8 1.0 MKcI>.
SToporo - 2,0 MKcI>. TpeTbero - 4.0 MKcI>.
A8HO: C I =1,0.10- 6 cI>. C 2 =2.0.10- 6 cI>.
C 3 = 4,0 . 10- 6 cI>.
HaHTH: C.
P e ill e H He. EOH,AeHCaTOpbl C 3 H C 2 COe,AHHeHbl
n8p8nnenbHO, n03TOMY C' = C 2 + C 3 . EOH,AeHCaTOp C 1
BKnlO'leH nOCne,AOS8TenbHO C C', n03TOMY
.!. =  + . C =  C' = C 1 (C 2 + C 3 )
C C 1 C' , C 1 + C' C 1 + C 2 + C 3 .
C = 0,90 MKcI>.
6. ABe nnaCTHHbI nnorq8,AblO 200 CM 2 norpymeHbl B Macno. ,AH3neKTpH-
QeCK8S1 npoHHu,8eMOCTb KOToporo 2,2, H nO,AKnlO'leHbI K HCTO'lHHKY TOK8 C
H8npSlmeHHeM 200 B. E8KYlO p860TY Heo6xo,AHMO COBepmHTb, 'lTo6bI nocne
OTKnlO'leHHfl 68T8peH YMeHbmHTb p8CCTOSlHHe Mem,lJ,y nn8CTHH8MH OT 5.0
,AO 1.0 CM?
AaHO: S=0.02 M 2 , E=2.2, U=200 B, d 1 =0,05 M, d 2 =0.01 M.
H 8 H T H: A.
P e ill e H He. P860T8 3neKTpH'leCKOrO nonSl, COBepmaeM8SI npH c6nHmeHHH nn8-
CTHH. p8BH8 H3MeHeHHIO 3HeprHH KOH,AeHCaTop8: A = -W = WI - '"'2. 3HeprHfl
2 2
KOH,AeHCaTOp8 ,AO c6nHmeHHSI nnaCTHH WI = C 1  I , nocne C6nH)KeHHSI W 2 = C2 2 .
C 1 U 1
TaK K8K 38pSl,A npH 3TOM COXp8HSleTCSI. TO ql =q2 Ii C 1 U 1 = C 2 U 2  U 2 =-.
C 2
2 2 222
TOr,A8 w; - C I U I . A = CIU 1 _ C I U I . Y'lHTbIB8S1, QTO C 1 = EoES ,
2 - 2C 2 ' 2 2C 2 d"
HMeeM A = EofSU 1 2 (d" - ) , A = 3,1 MKAm.
2d,,2
250
C 2 = EoES
c4'

7. )J;Ba III8pa eMKOCTSlMIi 2,0 n<1> Ii 1,5 net> 38pSlmeHbI ,AO nOTeHH8nos
200 B Ii 400 B COOTBeTCTBeHHO. Onpe,AenHTe 38pSl,AbI H8 W8pax 11 IiX nOTeH-
l1.lian nocne COe,AHHeHH.H III8pOB npOBO,AHIiKOM, conpOTIiBneHHeM Ii eMKOCTblO
KOToporo MomHO npeHe6pe'lb.
)J;8HO: C 1 =2,0.10- 12 <1>. C 2 =1,5.10- 12 <1>, <1>1=200 B, <1>2=400 B.
HaHTH: q, q;, <1>.
P e ill e H He. HaH,AeM 38pSl,AbI Ha wap8X ,AO HX COe,AHHeHHSI: ql = C 1 <1>1;
q2 = c;<P2'
ITepeXO,lJ; 3apSl,lJ;OB C O,lJ;HOrO IIIapa Ha ,ApyroH B03MomeH nliIIIb B TOM CJIY-
tlae, ecnli <Pl:1; <P2' Ii npeKpalQ8eTC.R, KOr,Aa nOTeHl1.lianbI III8pOB CTaHYT paB-
HbIMH. COrJI8CHO 3aKOHY coxpaHeHIiSl aneKTpH'leCKOrO 38p.H,Aa, ql + q2 = q + q; ,
r,Ae q = C 1 <1> H q; = C 2 <1> - HOBbIe 38pSl,AbI H8 III8pax, <I> - nOTeH1i8n K8m-
,Aoro III8P8 nocne HX COe,lJ;HHeHHSI.
Cm +Cm =m ( C +C ) m= Cl<1>l +C 2 <P2 . m 29 10 2 B
1 "t"l 2"t"2 "t" 1 2 "t" C 1 +  ' "t" = , . .
, _ Cd C 1 <Pl + C 2 <1>2) .
ql - C 1 + C 2 '
, C 2 (C 1 <1>1 + C 2 <P2) .
q2 = ,
C;+C 2
q = 0,57 HEn.
q = 0,43 HEn.
8. HaH,AIiTe nOTeH1i8nbHYIO 3HeprlilO CIiCTeMbI 'leTbIpex TO'le'lHbIX nono-
mHTenbHbIX 38pSl,AOB q, pacnonomeHHbIX B BaKYYMe B,AOnb O,AHOH npSlMoH H8
p8CCTOSlHHH r APyr OT ,Apyra.
)J;8HO: q, r. k.
HaHTIi: W o .
Pew e H He. rrOTeHlianbH8S1 3HeprHSI B3aHMO,AeHCTBHSI CIiCTeMbl 38pSl,AOB
(pHC. 8.49)
W o =W I2 + W 13 + W 14 + W 23 + W 24 + W 34 ,
1 r 2 r 3 r 4
. . . .
+q +q +q +q
Puc. 8.49
r,Ae
kq2 kq2 kq2
2=--;3=--;4= 3r ;
r 2r
kq2 kq2 kq2
W 23 = --; W 24 = --; W 34 = -- .
r 2r r
Tor ,A8
kq2 kq2 kq2 kq2 kq2 kq2 13k q 2
W =--+--+--+--+--+--=-
o r 2r 3r r 2r r 3r
linli ,
nO,ACT8BnSlH
1 13q2
k=-, IiMeeM W =-
47tEo n 127tEor'
251

9. 3AKOHbl nOCTOHHHOrO TOKA
3J1eKTpH'IeCKHH TOK. CHJla TOKa. 3aKOH OMa )J.JlSI yqaCTKa enH.
ConpOTHBJleHHe npOBO)J.HHKOB. 3aBHC}IMOCTh conpOTHBJleHHSI npOBO)J.HHKa OT
TeMnepaTypbl. 3J1eKTpHqeCKHe enH. llOCJle)J.OBaTeJlbHOe H napaJlJleJlhHOe co-
e,aHHeHHe npOBO.lJ.HHKOB.
3J1eKTpOllBHjKyaH CHJla. 3aKOHLI OMa )J.JlH HeOllHopOllHoro yqacTK3 uenH
H )J.JlH aaMKHyroH enH.
npaBHJla KHpxroa. PaC'IeT pa3BeTBJleHHLIX 3J1eKTpHqeCKHX eneH. UIYH-
Thl H )J.06aBOqHLle conpOTHBJleHHSI
Pa60Ta H MOII\IIOCTh TOKa. 3aKOH .n;jKoYJl.H-JIeHa.
9.1. 3JIeKTpH'IeCKHH TOK. CHJIa II nJIOTHOCTL TOKa
3J1eKTpH'IeCKHM TOKOM Ha3bIBaeTCSI HanpaBJleHHOe (ynop.H,AOqeHHoe) ,ABHme-
HHe 3ap.HmeHHbIX qaCTH.
SneKTpHqeCKHH TOK B npOBO,AHHKaX pa3J1HqHOrO pO,Aa npe.ACTaBJI.HeT C060H
JlH60 H8npaBJleHHOe )J;BHmeHHe 3neKTpOHOB B MeTanJlaJe (npOBO.AHHKH nepBoro
p0JI.a). HMelO1I(HJe oTpHllaTenbHbIH 3ap.H,A. JlH60 H8npaBJleHHOe JI.BHmeHHe 60nee
KpynHbIx qaCTH BeeCTBa - HOHOB, HMelOHX KaK nonomHTenbHbIH, T8K H
oTpHaTenbHbIH 3ap.H,A - B 3neKTpOJlHTaX (npOBO.AHHKH BToporo pO.Aa), nH60
HanpaBneHHoe .ABHmeHHe 3J1eKTpOHOB H HOHOB 060HX 3H8KOB B HOHH3HpOBaH-
HblX ra3ax (npOBO.AHHKH TpeTbero pO.Aa).
3a HanpaBJleHHe 3J1eKTpH'IeCKOrO TOKa ycnoBHo npHHSlTO HanpaBJleHHe ,ABH-
meHH.H nonOmHTeJlbHO 3ap.HmeHHbIX qaCTH.
,I:(JlH ceCTBOBaHHH 3J1eKTpH'leCKOro TOKa B BeeCTBe He06xo,AHMO:
1) HanH'lHe 3ap.HmeHHblX qaCTHll, cnoc06HbiX CBo60,AHO nepeMeaTbCH no
npOBOHHKY no .AeHCTBHeM CHJI 3J1eKTpHqeCKOrO non.H:
2) HanHqHe HCTOQHHKa TOKa, C03,AalOerO H nO,A.AepmHB81Oero B npOBO,A-
HHKe B TeqeHHe ,AnHTenbHoro BpeMeHH 3neKTpHqeCKOe none;
KOnHqeCTBeHHbIMH xapaKTepHCTHKaMH 3J1eKTpHqeCKOrO TOKa .HBJI.HIOTCSI CHna
TOKa I  nnOTHOCTb TOKa j.
CHJla TOKa - CKaJlSIpH8.H (}>H3HqeCKa.H BenHQHHa, onpe.AeJl.HeMaSI OTHome-
HHeM 38p.H,Aa !1q, npOXO,ASIerO qepe3 nOnepeQHOe CeQeHHe npOBO,AHHKa 3a He-
KOTOpblH npOMemYTOK BpeMeHH !1t, K 3TOMY npOMemYTKY BpeMeHH.
(I) = llq ; I = lirn llq .
llt 6t--+0 6.t
E.AHHHeH CHJlbI TOKa B CM .HBJI.HeTC.H aMnep (A).
EcnH CHna TOKa Hero Hanp8BJleHHe co BpeMeHeM He H3MeH.HIOTC.H, TO TOR
H83bIBaeTC.H nOCTOHHHLIM.
E,AHHHa CHnbI TOKa - OCHOBH8SI e,AHHHa B CM 1 A - eCTb CHJl8 T8-
Koro HeH3MeH.HIOerOC.H TOKa, KOTOpbIH. npOXOJI..H no JI.BYM 6eCKOHeQHO JI.nHH-
HbIM napanJleJlbHbIM np.HMOnHHeHHbIM npOBO.AHHKaM OqeHb MaJleHbKOro ceQe-
HH.H, pacnOJlOmeHHbIM Ha p8CCTO.HHHH 1 M ,Apyr OT .Apyra B BaKYYMe, BbI3bI-
252

BaeT CHJlY B3aHMO,AeHCTBH.H Mem.n;y HHMH 2. 10- 7 H H8 Kam.n;bIH MeTp .n;JIHHbI
npOBO,AHHKOB.
PaCCMOTpHM, KaK 3aBHCHT CHJla TOKa OT CKOpOCTH ynOp.HOqeHHOrO .n;BH-
meHH.H cB060.n;HbIX 3apSI.n;oB.
1 2
V
-..
6l
Puc. 9.1
BbI,AeJlHM Y'laCTOK npoBolIHHKa nJIOa.n;blO
CeqeHH.H S H .n;JlHHOH t:>.l (pHC. 9.1). 3apSl,A
Kam.n;oH qaCTHbI qo. B 06'beMe npOBo.n;HHKa. or-
paHHqeHHOM CeqeHH.HMH 1 H 2, co.n;epmHTC.H
nSl1l qaCTH, r.n;e n - KOHeHTpaHSI qaCTH.
Mx 06HH 3ap.H.n; I1q = q o nSl1l .
ECJlH cpe.n;H.H.H CKOpOCTb ynop.H.n;OqeHHOrO .n;BH-
meHHSI cB060.n;HbIX 3ap.H.n;OB (v), TO 3a npOMe-
mYTOK BpeMeHH !J.t = t:J / (V) BCe qaCTHbI, 3aK-
JIlOqeHHbIe B paCCMaTpHBaeMOM 06'beMe, npOH,AYT
qepe3 CeqeHHe 2. TI03TOMY CHna TOKa:
I = I1q = qon(v)Sl1l = qon(v}S .
I1t 6l
TaKHM 06pa30M, CHna TOKa B npOBo.n;HHKe 3aBHCHT OT 3ap.H.n;a, nepeHOCH-
Moro O.n;HOH qaCTHeH, HX KOHeHTpaHH, cpe.n;HeH CKOpOCTH HanpaBJleHHOrO
.n;BHmeHH.H qaCTH H nJlOa.n;H nOnepeqHOrO CeqeHH.H npOBo.n;HHKa.
3aMeTHM, qTO B MeTaJlJlaX MO.n;YJIb BeKTopa cpe.n;HeH CKOpOCTH ynop.H.n;O'leH-
Horo ,ABHmeHH.H 3JIeKTpOHOB (v) npH MaKCHManbHO ,AOnYCTHMbIX 3Ha qeHH.HX
CHJlbI TOKa - 10- 4 : ' B TO BpeMSI KaK cpe.n;H.H.H CKOpOCTb HX TenJlOBOrO .n;BH-
(9.1)
meHH.H _10 6 M .
C
nnoTHocTL TOKa j - 3TO BeKTOpHa.H cpH3HqeCKa.H BeJIHqHHa. MO.n;YJlb KO-
TOpOH onpe.n;eJI.HeTC.H OTHOllIeHHeM CHnbI TOKa I B npOBO,AHHKe K nJIOa.nH S
nOnepeqHOrO CeqeHHSI npOBo.n;HHKa, T.e.
. I
J =-
S'
B Clf el\lnIHl\eH ILJIOTHOCTH TOKa HBJlHeTCH 8.Mnep HR KBaJIP'ITHLlH MeT!' ( :. ).
KaK CJle.n;yeT H3 cpOPMYJlbI (9.1), J = qon(v}. HanpaBJIeHHe BeKTopa
nJlOTHOCTH TOKa j COBna,AaeT C HanpaBneHHeM BeKTopa CKOpOCTH ynop.H,AO-
qeHHOrO .n;BHmeHH.H (v) nOJIOmHTeJIbHO 3ap.HmeHHbIX 'laCTH. TIJIOTHOCTb no-
CTO.HHHOrO TOKa nOCTO.HHHa no BceMY nOnepeqHOMY CeqeHHIO npOBO)J;HHKa.
9.2. 3aKoH OMa ,l];JIH O,nHopo,nHoro Y'laCTKa ....enH
YqaCTOK enH. Ha KOTOpOM He .neHCTBYIOT CTopOHHHe CHJIbI, npHBo.n;.HlI.1He
K B03HHKHOBeHHIO S,nC (pHC. 9.2), Ha3bIBaeTCSI O.n;HOpO.n;HbIM.
253

M I
R
o ..
Puc. 9.2
v
Puc. 9.3
I
o
Puc. 9.4
N 3axOH OMa JI.H OHopo)J;Horo yqaCTK8 en:u
o 6bIJI YCTaHOBJleH 3KcnepHMeHT8JIbHO B 1826 r.
r.OMOM.
COrJl8CHO 3TOMY 38KOHY, cu.na mOKa I 6 00'
liOpOOIiOM MemaJl.Jl.UlleCKOM np0600liuKe npSlMo
nponOpijUOliaJl.blia ftanpSlJICeliUlO U Ita KOliqax
amozo np0600liuKa U 06pamli0 nponopquonaJl.b
lia conpOmU6Jl.eliUlO R amozo np06001iuKa:
U
1= U .
R
H8 pHCYHKe 9.3 H306pameH8 cxeMa 3J1eK-
TpHqeCKOH enH, n03BOn.HIO8.H 3KcnepHMeH-
TaJIbHO npOBepHTb 3TOT 3aKOH. B yq8CTOK
MN en:u nOOqepe,AHO BKJIIOq81OT npOBOHHKH,
06na81OHe pa3nH'lHbIMH conpOTHBJleHHSlMH.
HanpSlmeHHe Ha KOHuax npOBOLIHHK8 H3-
MepSleTC.H BOJIbTMeTpOM H MomeT H3MeHSlTbCSI
C nOMOblO nOTeHHOMeTpa. CHJlY TOKa H3-
Mep.HIOT aMnepMeTpOM, conpOTHBJIeHHe KOToporo
HHqTOmHO M8J10 (R A ::::: 0). fpa<l>HK 3aBHCHMOCTH
CHJlbI TOKa B npOBOHHKe OT Hanp.HmeHHSI H8
HeM - BOJIbT-aMnepH8.H xap8KTepHCTHK8 npo-
BO,AHHKa - npHBeeH Ha pHCYHKe 9.4. YroJI
H8KJlOH8 BOJlbT-aMnepHoH xapaKTepHcTHKH 3a-
BHCHT OT 3neKTpHqeCKOrO conpOTHBJleHHSI
npOBOHHK8 R (HJlH ero 3J1eKTpOnpOBOlIHMOC-
1
TH G): tga = - = G.
U
9.3. ConpoTHBneHHe npOBOHHKOB. YenLHoe conpOTHBneHHe
K8K yme OTMeqanOCb, CHna TOK8 B enH 3aBHCHT He TOJlbKO OT H8npSlme-
HHSJ H8 KOH8X yqacTKa, HO T8Kme H OT CBOHCTB npOBOlIHHK8, BKJIlOqeHHOrO B
enb. 3aBHCHMOCTb CHJlbI TOKa OT CBOHCTB npOBOHHKOB 06'b.HCH.HeTC.H TeM, qTO
pa3HbIe npOBOlIHHKH 06J1a,naIOT p83JIWIHbIM 3J1eKTpHqeCKHM conpOTHBJleHHeM.
3neKTpH'IeCKOe conpOTHBJleHHe R - ct>H3H'tJeCKaSl CKaJI.HpH8.H BeJIH'tJHHa.
xap8KTepH3YIO8.H CBOHCTBO npoBOlIHHKa YMeHbIIIaTb CKOpOCTb ynop.HOqeHHOrO
lIBHmeHH.H CB060HbIX HOCHTeJleH 3ap.HOB B npOBOHHKe.
0603HaqaeTC.H conpOTHBJleHHe 6YKBOH R. B Cll eHHHeH conpOTHBJleHHSI
npoBOlIHHKa SlBJlSleTCSI OM (OM).
1 OM - conpOTHBJleHHe TaKoro npoBolIHHKa, CHna TOKa B KOTOpOM paB-
Ha 1 A npH Hanp.HmeHHH Ha HeM 1 B.
IlpHMeHSIIOTC.H Ii lIPyrHe e,nHHHu.bI: KHJlOOM (KOM). MeraOM (MOM), MHJIJIH
OM (MOM): 1 KOM = 10 3 OM; 1 MOM = 10 6 OM; 1 MOM = 10- 3 OM.
254

<%>H3HqeCKYIO BeJlHqHHY 0', 06paTHYIO COnpOTHBneHHIO, Ha3bIBalOT 3neKTpH-
1
qeCKOH npoBo,nHMOCTblO: G = - .
R
E,nHHHeH 3J1eKTpHqeCKOH npOBo,nHMOCTH B CI1 SlBn.HeTC.H CHMeHC: 1 CM -
3TO npOBo,nHMOCTb npOBO)J;HHKa conpOTHBJleHHeM 10M.
llPOBO,nHHK co,nepmHT He TOJlbKO cBo60,nHbIe 3apSlmeHHbIe qaCTHIJ.bI - 3neK-
TpOHbI, HO H HeHTpaJlbHbIe qaCTHbI H CBSl3aHHbIe 3ap.H,nbI. Bce OHH yqaCTBY-
lOT B XaOTHqeCKOM TenJlOBOM llBHmeHHH. paBHOBepO.HTHOM B nlO6bIX HanpaB-
neHH.HX. llpH BKJllOqeHHH 3neKTpHqeCKOrO nonSl no,n ,neHCTBHeM 3J1eKTpHqeCKHX
CHJI 6y,neT npe06J1a,naTb HanpaBJleHHOe ynopSl,nOqeHHOe ,nBHmeHHe CB060,nHbIX
3ap.H,nOB, KOTopbIe ,nOJlmHbI ,nBHraTbCSl C YCKopeHHeM 1'1 HX CKOpOCTb ,nonmHa
6bIJla 6bI CO BpeMeHeM B03paCTaTb. Ho B npOBo,nHHKax cB060,nHbIe 3ap.H,nbI ,nBH-
mYTC.H C HeKOTopOH nOCTO.HHHOH cpe,nHeH CKOpOCTblO. CJle,AOBaTenbHO, npOBO,n-
HHK OKa3bIBaeT conpOTHBJleHHe ynop.H,nOqeHHOMY ,nBHmeHHIO CB06o,nHbIX 3apSl-
,nOB, qaCTb 3HeprHH 3Toro ,nBHmeHH.H nepe,naeTC.H npOBo,nHHKY, B pe3YJlbTaTe
qerO nOBbImaeTC.H ero BHYTpeHH.H.H 3HeprHSl. 113-3a ,nBHmeHH.H CB060,AHbIX 3a-
p.H,nOB HCKamaeTC.H ,name H,neaJlbHa.H KpHCTaJlJlHqeCKa.H pemeTKa npOBo,nHHKa.
Ha HCKameHHSiX KpHCTanJlHqeCKOH CTPYKTYPbI pacceHBaeTC.H 3HeprH.H ynopSl,no-
qeHHOrO ,nBHmeHH.H CB06o,nHbIX 3ap.H,nOB. llpOBo,nHHK OKa3bIBaeT conpOTHBneHHe
npoxom,neHHIO 3J1eKTpHqeCKOrO TOKa.
ConpOTHBJleHHe npOBOlIHHK8 3aBHCHT OT MaTepHaJla. H3 KOToporo OH H3ro-
TOBJleH, ,nJlHHbI npOBo.nHHKa 1'1 nJlOH nOnepeqHOrO CeqeHH.H. .n:JI.H npOBepKH
3TOH 3aBHCHMOCTH MomHO BOCnOJlb30BaTbCSl TOH me 3neKTpHqeCKOH cxeMOH. qTO
1'1 ,AJlSl npOBepKH 3aKOHa OMa (pHC. 9.3), BKJllOqaSl B yqacToK IJ.enH MN pa3-
nHqHbIe no pa3MepaM npOBo,nHHKH U.HJlHH,npHqeCKOE cpOpMbI) H3rOTOBneHHbIe H3
o,nHoro 1'1 Toro me MaTepHaJla, a TaKme 1'13 pa3HbIX MaTepHaJlOB.
Pe3YJlbTaTbI 3KcnepHMeHTa nOKa3aJlH, qTO conpOTHBneHHe npOBo,nHHKa np.H-
MO nponopHOHaJlbHO ,nJlHHe l npOBo,nHHKa, 06paTHO nponopHOHaJlbHO nno-
lQa,nH S ero nOnepeqHOrO CeqeHH.H H 3aBHCHT OT po,na BeIl\eCTBa, H3 KOToporo
H3rOTOBJleH npOBo,nHHK:
R = pl
S ) r,ne p - y,nenbHoe conpOTHBJleHHe npOBo,nHHKa.
YAenLHoe COnpOTHBneHHe npOBOAHHKa - CKanSlpHa.H cpH3HqeCKa.H BeJlHqH-
Ha, qHCJleHHO paBHa.H conpOTHBJleHHIO O,nHopo,nHoro IJ.HJlHH,npHqeCKOrO npOBO,n-
HHKa, H3rOTOBJleHHOrO 1'13 ,naHHoro BeeCTBa H HMelOero ,nJlHHY 1 M H nno-
ll\a,nb nOnepeqHOrO CeqeHH.H 1 M 2 , HJlH conpOTHBneHHIO Ky6a C pe6poM 1 M.
E,nHHHeH y,neJlbHoro conpOTHBJleHHSI B CI1 .HBJlSleTCSl OM-MeTp (OM' M).
Y,neJlbHOe conpOTHBJleHHe MeTaJlJlHqeCKOrO npOBo,nHHKa 3aBHCHT OT 1) ROH-
eHTpau.HH cB060,nHbIX 3J1eKTpOHoB B npOBo,nHHKe; 2) HHTeHCHBHOCTH pacceH-
BaHHSl cB060,nHbIX 3J1eKTpOHOB Ha HOHax KpHCTaJlnHqeCKOH pemeTKH, COBepma-
IOHX TennOBbIe KOJle6aHH.H; 3) HHTeHCHBHOCTH pacceHBaHH.H cB060,nHbIX 3J1eK-
TpOHOB Ha ,ne<}>eKTax H npHMeC.HX KpHCTanJlHqeCKOH CTPYKTypbI. HaHMeHbmHM
y,neJlbHbIM conpOTHBJleHHeM 06J1alIaeT cepe6po H Mellb. OqeHb BeJlHKO YlIeJlbHOe
conpOTHBneHHe y cnnaBa HHKenSl, meJle3a, xpOMa H MapraHa - "HHxpoMa".
Y,AenbHOe conpOTHBJleHHe KpHCTannOB MeTaJlJlOB B 3HaqHTenbHoH CTeneHH 3a-
BHCHT OT HanHqH.H B HHX npHMeceH. HanpHMep, BBe,AeHHe 1 % npHMeCH Map-
raHa YBeJlHqHBaeT y,neJlbHOe conpOTHBJleHHe Me)J;H B TPH pa3a.
255

9.4. 3aBHcHMoCTb conpOTHBJIeHHH npOBOHHKa OT TeMnepaTYpbI
YeJlbHoe conpOTHBJleHHe, a CJleOBaTeJlbHO, H conpOTHBJleHHe MeTaJlJlOB,
3aBHCHT OT TeMnepaTypbI, YBeJlHqHBa.HCb C ee pOCTOM. TeMnepaTYPHa.H 3aBHCH-
MOCTb conpOTHBJleHH.H npOBOHHKa 06'b.HCH.HeTC.H TeM, qTO 1) B03paCTaeT HH-
TeHCHBHOCTb pacceHBaHH.H (qHCJlO CTOJlKHOBeHHH) HOCHTeJleH 3ap.HOB npH no-
BbIlIIeHHH TeMnepaTypbI; 2) li3MeH.HeTC.H HX KOHIJ;eHTpaIJ;H.H npH HarpeBaHHH
npOBOHHKa. OnbIT nOKa3bIBaeT, qTO npH He CJlHIIIKOM BbICOKHX H He CJlHIII-
KOM HH3KHX TeMnepaTypax 3aBHCHMOCTH yeJlbHoro conpOTHBJleHH.H H conpo-
THBJleHH.H npOBOHHKa 0'1' TeMnepaTypbI BbIpamalOTC.H c}>opMYJlaMH:
Pt =po(l+at), Rt=Ro(l+at),
r)J;e Po, Pt - yeJlbHbIe conpOTHBJleHH.H BeeCTBa npOBOHHKa COOTBeTCTBeH-
HO npli O°C H t °C; Ro, Rt - conpOTHBJleHH.H npOBOHHKa npH O°C Ii t °C,
a - TeMnepaTYPHbIH K03c}>c}>HIJ;HeHT conpOTHBJleHH.H: H3Mep.HeMbIH B CM
B KenbBHHax B MHHYC nepBOH CTeneHH (K- 1 ). ,l];JI.H MeTaJlJlHqeCKHX npOBOHH-
KOB 3TH c}>OPMYJlbI npHMeHHMbI HaqHHa.H C TeMnepaTypbI 140 K H BbIllie.
TeMllepaTYPHLIH K03c1>cI>Hn;HeHT conpOTHBJleHH.H Bew;eCTBa xapaKTepH3yeT 3a-
BHCHMOCTb H3MeHeHH.H conpOTHBJleHH.H npH HarpeBaHHH OT poa Bew;eCTBa. OH
qHCneHHO paBeH OTHOCHTeJlbHOMY H3MeHeHHIO conpOTHBJleHH.H (yeJlbHOrO co-
npOTHBJleHH.H) npOBOHHKa npH HarpeBaHHH Ha 1 K.
1.p
< a> = -. r.n;e <ex> - cpe.n;Hee 3HaqeHHe TeMnepaTypHoro K03Q>c}>HUH-
pT
eHTa conpOTHBJleHH.H B HHTepBane T.
,l];JI.H Bcex MeTaJlJlHqeCKHX npOBOHHKOB a > 0 H cJla60 H3MeH.HeTC.H C H3-
MeHeHHeM TeMnepaTypbI. 'Y qHCTbIX MeTaJlJlOB a =  K- 1 . Y MeTaJlJlOB KOH-
273
u;eHTpaIJ;H.H cB060HbIX HOCHTeJleH 3ap.HOB (3J1eKTpoHoB) n = const H YBeJlHqe-
HHe p npOHCXOHT 6J1aroap.H POCTY HHTeHCHBHOCTH pacceHBaHH.H cB060HbIX
3neKTpOHOB Ha HOHax KpHCTaJlJlHqeCKOH pelIIeTKH.
)J;JI.H paCTBopOB 3J1eKTpOnHTOB ex < 0, HanpHMep, JI.H 10 % -Horo paCTBopa no-
BapeHHoH COJIH ex = -0,02 K- 1 . ConpOTHBJleHHe 3neKTpOJlHTOB C pOCTOM TeMne-
a paTypbI YMeHblIIaeTC.H, TaK KaK YBeJlHqeHHe qHCna
R cB060HbIX HOHOB H3-3a HCCOIJ;HaU;HH MOJleKY JI npe-
BbIlliaeT pOCT pacceHBaHH.H HOHOB npH CTOJlKHOBe-
HH.HX C MOJleKYJlaMH paCTBOpHTeJl.H.
<1>0PMYJlbI 3aBHCHMOCTH p H R OT TeMnepaTyphI
JIJI.H 3J1eKTpOJlHTOB aHaJIOrHqHbI npHBeeHHbIM BbIllie
t c}>opMYJlaM )J;JI.H MeTaJlJlHqeCKHX npOBOHHKOB. He-
06XO)J;HMO OTMeTHTb, qTO 3Ta JlHHeHHa.H 3aBHCH-
MOCTb COXpaH.HeTC.H nHIlIb B He60JlblIIOM Hana30He
H3MeHeHH.H TeMnepaTYP, B KOTOpOM ex=const. llPH
60JlblIIHX me HHTepBaJlaX H3MeHeHH.H TeMnepaTYP
3aBHCHMOCTb conpOTHBneHH.H 3J1eKTpOnHTOB OT TeM-
nepaTypbI CTaHOBHTC.H HeJlHHeHHOH.
rpaqJHqeCKIi 3aBHCHMOCTH conpOTHBJleHH.H MeTaJI-
t JlHqeCKHX npOBOHHKOB Ii 3J1eKTpOJlHTOB OT TeMne-
paTypbI H306pameHbI Ha pHCYHKax 9.5, a, 6.
Ro
o
6
R
o
Puc. 9.5
256

I1pll OtIeHb HII3KIIX TeMnepaTypax, 6nH3KHX K a6COJllOTHOMY HYJlIO (-273°C).
conpOTIIBJleHHe MHorHX MeT8J1nOB CKatIKOM n8,AaeT ,no HYJl.H. 3TO .HBJleHHe
nonY'llino H83B8Hlle CBepxnpOBo,n:HMOCTH (CM. Hllme  10.1). MeT8nJI nepexo-
,nIlT B CBepxnpOBo.n;.Hee COCTO.HHHe.
3aBllcliMoCTb conpOTIIBJleHIISI MeTannOB OT TeMnep8TypbI IiCnOJlb3YIOT B Tep-
MOMeTpax conpOTIIBJleHII.H. 06bItIHO B KatIeCTBe TepMOMeTpllqeCKOrO TeJl8 TaKO-
ro TepMOMeTpa 6ePYT nJI8TIIHOBYlO npOBOJIOKY, 3aBHCHMOCTb conpOTIIBJIeHIISI KO-
TOpOH OT TeMnepaTypbI ,AOCTaTOqHO 113Y'leHa.
06 113MeHeHII.HX TeMnepaTypM Cy.n;.HT no H3MeHeHHIO conpOTIIBJIeHII.H npo-
BOJlOKII, KOTopoe MomHO 113MepliTb. T8KHe TepMOMeTpbI n03BOJI.HIOT 113Me-
p.HTb OtIeHb HII3Klle II OqeHb BbICOKlie TeMnepaTypbI, Kor.n;a 06bltIHble mll.n;-
KOCTHbIe TepMOMeTpbI Henpllro.n;HbI.
9.5. nOCJIe.ZJ;oBaTeJILHOe H napaJIJIeJILHOe
coe.ZJ;HHeHHH npOBOHHKOB
M Rl R 2 N

llOCJ1e,n:OBaTeJIbHbIM Ha3bIBaeTC.H T8Koe co-
e.n;HHeHHe pe3HCTopOB, Kor.n;8 KOHe o.n;Horo
npOBO)1HHKa coe.n;HH.HeTC.H C H8'l8nOM .n;pyroro Ii
T..n;. (pIiC. 9.6). I1pli nOCJle.n;OBaTeJIbHOM Coe)1l1-
HeHIIH CHn8 TOK8 H8 JII060M yqaCTKe 3J1eKT-
pll'leCKOH enll 0,nIlH8KOB8. 3TO 06'bSlCH.HeTC.H
TeM, qTO 38p.H,AbI He MorYT H8K8nJIIIB8TbCSI B
Y3J18X enH. fix HaKOnJleHHe npliBeJlO 6bI K
113MeHeHllIO Hanp.HmeHHOCTIi 3J1eKTpll'leCKOrO non.H, a cne.n;OBaTeJlbHO, II K 113-
MeHeHll10 CHJlbI TOK8. I103TOMY
Puc. 9.6
I = 11 = 1 2 ,
AMnepMeTp A 113Mep.HeT cHny TOKa B enll H 06J1a,A8eT M8J1MM BHYTpeH-
HHM conpOTIIBJleHlleM (R A  0).
BKJIlOtIeHHbIe BOJlbTMeTpbI VI H V 2 H3Mep.HIOT Hanp.HmeHHe U 1 H U 2 Ha
conpOTIIBJleHII.HX R 1 H R 2 . BonbTMeTp V 113Mep.HeT no.n;Be.n;eHHoe K KJleMM8M
M II N Hanp.HmeHlle U. BOJlbTMeTpbI nOK83bIBaIOT, tITO npll nOCJle.n;OBaTeJIb-
HOM COe)1I1HeHIIII H8np.HmeHlie U p8BHO CYMMe H8np.HmeHIIH H8 OTlIeJIbHbIX }"I 8-
CTK8X enll:
U=U t +U 2 .
(9.2)
I1pIlMeH.H.H 38KOH OM8 .n;JI.H K8m.n;oro Y'laCTKa u,enli, nOJIytIHM:
U = IR; U 1 =IR 1 ; U 2 =IR 2 ,
r.n;e R - 06ee conpOTHBneHHe nocne.n;OBaTeJIbHO coe.n;HHeHHoH enll.
I10.n;CT8BJI.H.H U, U l' U 2 B (})OPMYJIY (9.2), IIMeeM
IR=IRI + IR 2  R=Rl + R 2 .
257

ConpOTHBJleHHe enH, COCTO.HeH H3 n nOCJleOBaTenbHO COeHHeHHbIX pe-
3HCTOpOB, paBHO CYMMe COnpOTHBJleHHH 3TH X pe3HCTOpOB:
n
R=Rl +R 2 +... +Rn' HnH R = LRI .
1=1
ECJlH COnpOTHBneHH.H OTeJlbHbIX pe3HCTOpOB paBHbI Me*y C060H, T.e.
Rl = R 2 = ... = Rn' TO 06ee COnpOTHBJleHHe 3THX pe3HCTOpOB npH nOCneOBaTeJlb-
HOM COeHHeHHH B n pa3 60JlbWe COnpOTHBJleHH.H OHOrO pe3HCTOpa: R = nR l .
IIpH nOCJleOBaTeJlbHOM COeHHeHHH pe3HCTOpOB CnpaBenHBO COOTHOWeHHe
VI _ 
V 2 -  ' T.e. Hanp.H*eHH.H Ha pe3HCTOpaX np.HMO npOnOpHOHanbHbI con pO-
THBJleHH.HM.
llapaJIJleJILllblM Ha3bIBaeTC.H TaKoe CO-
e,AHHeHHe pe3HCTopOB, Kora OHH KOH-
bI Bcex pe3HCTopOB coeHHeHbI B O,AHH
Y3en, PyrHe KOHbI - B PyroH Y3eJl
(pHC. 9.7). Y3J10M Ha3bIBaeTCH TOqKa
pa3BeTBneHHoH uenH. B KOTOpOH CXOHT-
C.H 60nee .ABYX npOBOHHKOB. IIpH na-
V panJleJlbHOM coe)J;HHeHHH pe3HCTopOB K
TOqKaM M H N nOKJllOqeH BOJlbTMeTp.
Puc. 9.7 OH nOKa3bIBaeT, qTO HanpHmeHHH Ha
OTeJlbHbIX yqaCTKaX enH C conpoTHB-
JleHH.HMH R 1 H R 2 paBHbI. 3TO 06'bHCHHeTC.H TeM, qTO pa60Ta CHJI cTaHoHap-
Horo 3neKTpHqeCKOrO nOJl.H (CM. HH*e  9.6) He 3aBHCHT OT c})OpMbI TpaeKTO-
pHH:
V 1 =U 2 =U.
AMnepMeTp nOKa3bIBaeT. qTO CHna TOKa I B Hepa3BeTBneHHoH qaCTH enH
paBHa CYMMe CHJI TOKOB I I H 1 2 B napaJlnenbHO COe,AHHeHHbIX npoBoHHKax RI
H R 2 :
1=1 1 +1 2 ,
(9.3)
3TO BbITeKaeT H H3 3aKOHa coxpaHeHH.H 3neKTpHqeCKOrO 3apH,Aa.
IIpHMeHHM 3aKOH OMa .AnSI OT,AeJlbHbIX yqacTKoB nenH H BceH enH C 06-
HM conpOTHBJleHHeM R:
_V. _V. _V
1--, 1 1 --, 1 2 --.
R RI 
IIo,ACT8BJlHH I. II H 1 2 B c})OPMYJlY (9.3), nOJlyqHM:
V U VII 1
-=-+--=-+-.
R I4 R;, R  R;,
BenHqHHa. 06paTHa.H conpOTHBneHHIO uenH, CocToHeH H3 n napaJlJleJlb-
HO coeHHeHHbIX pe3HCTopOB, paBHa CYMMe BeJlHQHH, 06paTHbIx conpOTHBJleHH-
.HM 3THX pe3HCTopOB:
1 n 1
-=L-.
R 1=1 R i
258

EcnH COnpOTHBJleHH.H BCeX n napaJlJleJlbHO COe,nHHeHHbIX pe3HCTOpOB O,nH-
HaKOBbI H paBHbI Rh TO ! =. OTKy,na R =  .
R  n
COnpOTHBJleHHe n;enH, COCTO.HeH H3 n O,nHHaKOBbIX napaJIJleJlbHO COe,nH-
HeHHbIX pe3HCTOpOB, B n pa3 MeHblIIe COnpOTHBJleHH.H Kam,nOrO H3 HHX.
IIpH napaJlJleJlbHOM COe,nHHeHHH pe3HCTOpOB CnpaBe,nJlHBO COOTHOlIIeHHe
1 R
-1.. = , T.e. CHnbI TOKOB B BeTB.HX napaJlJleJlbHO coe,nHHeHHoH enH 06paT-
1 2 
HO nponopn;HOHaJlbHbI conpoTHBneHHHM BeTBeH.
9.6. 3JIeKTpH'IeCKHe enH. 3JIeKTpO,lJ;BHjKyaJl CHJIa
3neKTpHQeCKan enL COCTOHT H3 HCTOqHHKa
TOKa, nOTpe6HTeJleH 3J1eKTp03HeprHH, coe,nHHH-
TeJlbHbIX npOBo,nOB H KJllOqa, cJlymaero ,nJl.H
pa3MbIKaHH.H H 3aMbIKaHH.H n;enH H ,nPyrHx 3J1e-
MeHTOB (pHC. 9.8).
PHCYHKH, Ha KOTOpbIX H306pameHbI cnoc06bI co-
e,nHHeHH.H 3J1eKTpHqeCKHX npH60poB B enb, H83bI-
BalOTCH 3J1eKTpHQeCKHMH CXeMaMH. IIpH60pbI Ha
cxeMax 0603Haqa1OTC.H YCJlOBHbIMH 3HaKaMH.
KaK OTMeqaJIOCb B  9.1 , n.H no,nepm8HH.H B
n;enH 3J1eKTpHqeCKOrO TOK8 He06xoHMO, qT06bI H8
KOHn;ax ee (pHC. 9.9) cyecTBoBana nOCTO.HHHa.H
pa3HOCTb nOTeHn;HaJlOB q> A - q>B' IIycTb B HaqaJlb-
HbIH MOMeHT BpeMeHH <P A > <P B' TOr,na nepeHOC no-
JlmKHTeJlbHOrO 3ap.H,na q H3 TOqKH A B TOqKY B
npHBe,neT K YMeHblIIeHHIO pa3HOCTH nOTeHHanOB
Mem,ny HHMH. .D;n.H coxpaHeHH.H nOCTO.HHHOH pa3HO-
CTH nOTeHn;HaJlOB He06xo,nHMO nepeHeCTH TOqHO Ta-
KOH me 3ap.H,lJ; H3 B B A. EcnH B HanpaBJleHHH
A -+ B 3ap.H,nbI ,nBHmYTC.H no,n ,neHcTBHeM CHJI 3J1eK-
TpOCTaTHqeCKOrO non.H, TO B HanpaBJleHHH B -+ A
nepeMeeHHe 3ap.H,nOB npoHcxo,nHT npOTHB CHJI 3J1eKTpOCTaTHqeCKOrO non.H, T.e.
no,n ,neHcTBHeM CHJI He3neKTpOCTaTHQeCKOH npHpO,AbI, TaK Ha3bIBaeMbIX CTO-
pOHHHX CHJI. 3TO YCJlOBHe BblnOJlH.HeTC.H B HCTOqHHKe TOKa. KOTOpbIH no,A-
,nepmHBaeT lIBHmeHHe 3J1eKTpHqeCKHX 3apH,noB. B 60JlblIIHHCTBe HCTOqHHKOB
TOKa ,lIBHmYTC.H TOJlbKO 3J1eKTpOHbI, B raJlbBaHHqeCKHX 3J1eMeHTax - HOHbI
060HX 3HaKOB.
HCTOqHHKH 3neKTpHqeCKOrO TOKa MorYT 6bITb pa3nHqHbI no CBoeH KOHCT-
PYKn;HH HO B JlI060M H3 HHX cOBeplIIaeTcH pa60Ta no pa3,neJleHHIO nOJlOmH-
TeJlbHO H OTpHaTeJlbHO 3ap.HmeHHbIX qaCTH. Pa3,neJleHHe 3ap.H,noB npoHcxo-
,nHT no,n ,neHcTBHeM CTOpOHHHX CHJI. CTopoHHHe CHJlbI ,neHcTBYIOT JlHlIIb BHYT-
pH HCTOqHHKa TOKa H MorYT 6bITb 06YCJlOBneHbJ XHMHqeCKHMH npOn;eccaMH
(aKKYMYJI.HTOpbI, raJlbBaHHqeCKHe 3J1eMeHTbI), ,neHCTBHeM CBeTa (cpOT03J1eMeHTbI),
H3MeH.HIOHMHCH MarHHTHbIMR nOn.HMH (reHepaTopbI) H T.,n.
JII060H HCTOqHHK TOKa xapaKTepH3YIOT 3J1eKTpo,nBHmYIQeH CHJlOH - 3.D;C.
R3
Puc. 9.8
R
Puc. 9.9
259

aJIeKTpOnBHeH CHJIOH g' HCTOtIHHKa TOKa Ha3bIB81OT Q>H3HqeCKYIO CK8-
JI.HpHYIO BeJlHqHHY, P8BHYIO p860Te CTOpOHHHX CHJI no nepeMeeHHIO eHHHq-
Horo nOJlOmHTeJlbHOrO 3ap.Ha BOJIb 3aMKHYTOH nenH
go = A CTop .
q
EHHHneH 3J1eKTpoBHmyeH CHJlbI B Cll SlBJI.HeTC.H BOJlbT (B).
SAC HBJI.HeTC.H 3HepreTHqeCKOH xap8KTepHCTHKOH HCTOqHHKa TOKa.
B HCTOqHHKe TOKa B npoecce pa60TbI no pa3eJleHHIO 3ap.HmeHHbIX qaCTHn
npOHCXOHT npeBpaeHHe MeX8HHqeCKOH, CBeTOBOH, BHYTpeHHeH H T.n. SHeprHH
B 3JIeKTpHqeCKYIO. Pa3eJIeHHbIe qaCTHnbI H8K8nJIHBaIOTC.H Ha nOJllOcax HCTOq-
HHKa TOKa (MeCT8, K KOTOpbIM C nOMOblO KJIeMM HJlH 3amHMOB nO,ACOeHH.H-
lOT nOTpe6HTeJlH). OHH nOJllOc HCTOqHHKa TOKa 3ap.HmaeTCH nOJlOmHTeJlbHO,
,ApyroH - OTpHaTeJlbHO. Memy nOJIIOCaMH HCTOtIHHKa TOKa C03aeTCH SJleKT-
POCT8THqeCKOe nOJle. ECJIH nOJllOca HCTOqHHKa TOKa coeHHHTb npOBOHHKOM,
TO B T8KOH 3J1eKTpHqeCKOH enH B03HHKaeT sneKTpHqeCKHH TOK. ITpH STOM X8-
paKTep nOJISI MeH.HeTCH, OHO nepeCT8eT 6blTb SJleKTpOCT8TH'leCKHM.
Ha pHCYHKe 9.10 CXeM8THqHO B BHlIe ccpepH-
tIeCKOrO npOBOlIHHKa H306p8)KeHa OTpHUaTeJlbH8.H
KJleMM8 HCTOqHHKa TOK8 H CeqeHHe npHcoeHHeH-
Horo K HeH KOHn8 MeTaJlJlHqeCKOrO npOBoa.
ITYHKTHPOM nOK83aHbi HeKOTopbIe JlHHHH Hanp.H-
)KeHHOCTH nOJlH KJleMMbI o BHeCeHH.H B Hero
npOBoa, 8 CTpenKaMH - CHJlbI, eHCTBYIOHe
Ha cB060HbIe SJleKTpOHbI npOBoa, H8XO.HHe-
C.H B TOqK8X. nOMeqeHHbIX nHy>paMH. SJleKTpOHbI
B p83J1Ht.IHbIX TOqK8X nOnepeqHOrO CeqeHH.H npOBoa no eHCTBHeM KY no-
HOBCKHX CHJI nOJlH KJIeMMbI npH06peT81OT BHmeHHe He TOJlbKO BOJlb OCH
npOBO8. H8npHMep, 3J1eKTpOH, H8XO.HHHC.H B TOt.IKe 1, OK83bIB8eTC.H BOB-
neqeHHbIM B "ToKoBoe" BHmeHHe. Ho B6J1H3H TOqeK 2, 3, 4, 5 3neKTpOHbI
HMelOT B03MomHOCTb CK8nJIHBaTbC.H H8 nOBepXHOCTH npOBoa. IIpHqeM no-
BepXHOCTHoe p8Cnpe,AeJleHHe 3J1eKTpOHOB no ,AJlHHe npOBoa He 6yeT paBHO-
MepHbIM. CJIeOBaTeJlbHO, nOKJIIOt.IeHHe npOBoa K KJIeMMe HCTOqHHK8 TOK8
npHBeeT K TOMY, qTO HeKOTopbIe sneKTpOHbl HaqHYT BHraTbC.H BOJlb npo-
BO8, 8 qaCTb 3J1eKTpOHOB 6yeT CKanJlHBaTbC.H H8 nOBepXHOCTH. Hep8BHO-
MepHoe p8CnpeeJleHHe SJleKTpOHOB Ha ero nOBepXHOCTH 06eCneqHBaeT HeSK-
BHnOTeHnH8J1bHOCTb 3TOH nOBepXHOCTH, HaJlHqHe COCTaBJI.HIOHX Hanp.HmeH-
HOCTH 3J1eKTpHqeCKOrO nOJl.H, H8np8BJleHHbIX BOJlb nOBepXHOCTH npOBOHH-
Ka. STO nOJle nepepacnpeeJIeHHbIX 3J1eKTpOHOB C8Moro npOBOHHKa H 06ec-
net.IHB8eT ynop.HOqeHHOe BH)KeHHe PyrHx 3J1eKTpOHOB. ECJlH pacnpeeJIeHHe
3JIeKTpOHOB no nOBepXHOCTH npOBOHHKa C Tet.IeHHeM BpeMeHH He H3MeH.HeT-
CH, TO T8Koe nOJle H83bIB81OT CTalJ,HOHapHblM 3J1eKTpHqeCKHM nOJIeM. TaKHM
06p830M. rJlaBHYIO ponb B C038HHH CT8l{HOH8pHoro 3J1eKTpHt.IeCKOrO nOJl.H Hr-
palOT 38p.H.D:bI, HaXOlI.HII..tHeC.H Ha nOJIIOC8X HCTOqHHKa TOKa. IIpH 38MbIK8HHH
3neKTpHqeCKOH uenH B3aHMOlIeHCTBHe HMeHHO 3THX 3ap.H1I0B CO CB0601IHbIMH
3ap.HlIaMH npOBOlIHHK8 npHBo,nHT K nO.HBJIeHHIO H8 BceH nOBepXHOCTH npOBO-
HHKa HeCKOMneHCHpOBaHHbIX nOBepXHOCTHbIX 38p.HOB. I1MeHHo 3TH 3ap.HbI CO-
3,Aa1OT CTanHOHapHoe 3JIeKTpHt.IeCKOe nOJle BHYTpH npOBOHHKa no BceH ero JIH-
He. STO nOJle BHYTPH npOBOHHKa OHopOHoe. H JlHHHH Hanp.HmeHHOCTH Ha-
8-
3 ........ -___
...........
Puc. 9.10
260

,
;
.
.
.
I
npaBJleHbI B,nOJlb OCH npOBO,nHHKa (pRC. 9.11).
rIpoecc YCTaHOBJIeHH.H 3J1eKTpHqeCKOrO nOJI.H
Bonb npOBOHHKa npOHCXOHT co CKOpOCTblO
C ::::: 3 . 10 8 M
C
KaK I-f 3J1eKTpOCTaTRqeCKOe nOJle, OHO nOTeH-
RaJIbHO. Ho Mem.l{Y 3TRMR nOJlSlMR RMeIOTC.H CY-
I..QeCTBeHHble OT JIRqR.H:
1) 3J1eKTpOCTaTRqeCKOe none - nOJIe HenO.l{BHmHbIX 3apSl,nOB. I1cToqHHKOM
CTaHOHapHoro 3J1eKTpHqeCKOrO nOJl.H .HBJI.HIOTC.H .l{BHmYI..QHeC.H 3apSl,nbI, npHqeM
o6I..Qee 'lHCnO 3ap.H.l{OB R KapTHHa HX pacnpe,AeneHHSI B .l{aHHOM npOCTpaHCTBe
C TeqeHHeM BpeMeHH He H3MeHSIIOTC.H;
2) 3JIeKTpOCTaTHqeCKOe nOJIe CYI..QeCTByeT BHe npOBO.lIHHKa. HanpSlmeHHocTb
3neKTpOCTaTHqeCKOrO nOJl.H Bcera paBHa 0 BHYTPH 06eMa npOBOHRKa, a B
KaJK.l{OH TOqKe BHemHeH nOBepXHOCTH npOBOHHKa HanpaBJIeHa nepneH.l{HKYJI.Hp-
HO K 3TOH nOBepXHOCTH. CTaHoHapHoe 3JIeKTpRQeCKOe nOJIe CYI..QeCTByeT I-f BHe
Ii BHYTPH npOBo,nHHKa. Hanp.HJKeHHOCTb CTaHOHapHoro 3J1eKTpR'leCKOrO nonSi
He paBHa HYJIIO BHYTPH 06'beMa npOBO,AHRKa, a Ha nOBepxHocTH H BHYTPH
o6eMa HMeIOTC.H COCTaBJI.HIOI..QHe Hanp.HJKeHHOCTH, He nepneH.l{HKyn.HpHbIe K no-
BepXHOCTH npOBO.l{HHKa;
3) nOTeHHaJIbI pa3HbIX TOqeK npOBo,nHHKa, no KOTOPOMY npOXOHT nOCTO-
HHHbIH TOK, pa3HbIe (noBepxHocTb H 06eM npOBOHHKa He 3KBHnOTeHHaJlb-
HbI). I1oTeHHaJlb] Bcex TOqeK nOBepXHOCTH npOBOHHKa, HaXO.l{HI..QerOC.H B
3neKTpOCTaTHQeCKOM none, OHHaKOBbI (nOBepXHOCTb 11 06'beM npOBo,nHHKa 3KBH-
nOTeHHaJlbHbI);
4) 3J1eKTpOCTaTH'leCKOe none He COnpOBOJK,naeTC.H nO.HBJleHHeM MarHHTHoro
nOJI.H, a CTaUROHapHoe 3J1eKTpHQeCKOe none COnpOBOJKllaeTC.H ero nO.HBJleHHeM
H Hepa3pbIBHO C HHM CBSl3aHO.
E
Puc. 9.11
9.7. 3aKoH OMa ;:J;JUI HeO,lJ;HOpO,lJ;HOrO yqacTKa l\enH
I1pR npoxomeHRH 3J1eKTpHqeCKOrO TOK8 B 3aMKHYToH enH Ha cB060-
Hble 3ap.HbI ,AeHcTBYIOT CHJIbI co CTOpOHbI CTaROHapHoro 3JIeKTpR'leCKOrO
nOJI.H H CTopOHHHe CHJIbI. rIpH 3TOM Ha OTenbHbIX Y'laCTKaX 3TOH enH
TOK C03aeTC.H TonbKO CTaROHapHbIM 3neKTpHqeCKRM noneM. TaKHe yqaCTKH
enH Ha3bIBaIOTC.H O.ll.HOpO.ll.HLIMH. Ha HeKOTopbIX YQaCTKaX 3TOH enR, KpO-
Me CRJI CTaHOHapHoro 3JIeKTpRQeCKOrO nOJlH. eHCTBYIOT H CTopOHHRe CHJlbI.
Y'laCTOK enH, Ha KOTOpOM ,neHCTBYIOT CTopOHHHe CRJlbI, Ha3bIBalOT HeO.ll.HO-
pO.ll.HLIM yqacTKoM enH.
AnSI Toro qT06bI BbI.HCHHTb. OT Qero 3aBHCHT CHna TOKa Ha 3TRX yqaCT-
Kax. He06xo,nHMO YTOqHHTb nOH.HTHe HanpSlmeHRSI.
PaccMoTpHM BHaqaJle O,nHOPO,AHbIH yqaCTOK u.enH (pRC. 9.12, a). B 3TOM
cnYQae pa60TY no nepeMell1eHHIO 3ap.H,na COBepmalOT TOJIbKO CHJIbI CTaHO-
HapHoro 3JIeKTpHQeCKOrO nonSl, H 3TOT Y'laCTOK xapaKTepH3YIOT pa3HOCTblO
nOTeHHaJIOB . Pa3HocTb nOTeHHaJIOB Ha KOHax YQaCTKa
YJl A 6
<P = <PI - <P2 = -, re KYJI - pa OTa CHJI CTaHOHapHoro 3J1eKTpHQeCKOrO
q
261

a
6
1 R 2
o-------C::J---
I
1 g' +
 p:-;
R
2
t---o
I
Puc. 9.12
nOJUI. HeoHopoHbIH yqacToK enH (pHC. 9.12, 6) coepmHT B OTJlHqHe OT
OHopoHoro yqacTKa HCTOqHHK 3,l{C, H K pa60Te CHJI 3J1eKTpOCTaTHqeCKOrO
nOJl.H Ha 3TOM y-qaCTKe 06aBJlHeTCH pa60Ta CTOpOHHHX CHJI. no onpeeJle-
HRIO, J1 = <PI - <P2, rIle q - nOJlOmHTenbHbIH 3apSl.q, KOTOpbIH nepeMeaeTCH
q
Mem,Ay JII06bIMR BYM.H TOqKaMH enR; <PI - <P2 - pa3HOCTb nOTeHRaJlOB TO-
qeK B HaqaJle H KOHe paccMaTpHBaeMoro yqaCTKa; Acr = E. Tora rOBOp.HT 0
q
Hanp.HmeHHH JI.H HanpHmeHHocTH: ECTan.s.n = ES/cTaT.n + E CTOP ' Hanp.HmeHRe U
Ha yqaCTKe enR npeIlCTaBn.HeT C060H cpR3RqecKYlO CKaJlSlpHYIO BeJlHqRHY, paB-
HYIO cYMMapHoH pa60Te CTOpOHHRX CRJI H CRJI 3JIeKTpOCTaTR'leCKOrO nOJI.H no
nepeMeeHHIO e,AHHHqHOrO nOJIOmHTeJlbHOrO 3apHa Ha 3TOM yqacTKe:
yn p (p
U=-+-=<Pl-<P2+ 0 .
q q
M3 3TOH cpOPMYJlbI BRHO, qTO B 06eM cnyqae Hanp.HmeHRe Ha aHHOM
yqacTKe enH paBHo aJlre6paHQeCKOH CYMMe pa3HocTH nOTeHHaJlOB H 3,l{C H8
3TOM YQaCTKe. ECJlH me Ha yqaCTKe eHcTBYIOT TOJlbKO 3J1eKTpHQeCKHe CHJIbi
(g:' = 0), TO U = <PJ - <P2' TaKHM 06pa30M, TOJIbKO JI.H OHopOHoro YQaCTK8
enH nOH.HTH.H HanpHmeHH.H H pa3HOCTH nOTeHUHaJIOB COBna,lIaIOT.
3aKOH OMa JI.H HeO,AHOpOHoro y-qaCTKa enR RMeeT BH,A:
I = U = <PI - <P2 + g
R R '
r,Ae R - 06mee conpOTHBJIeHHe HeO,AHOpOHoro yqaCTKa.
S,l{C g:' MomeT 6bITb KaK nOJIOmRTeJIbHOH. TaK H OTpHuaTenbHOH. STO
CB.H3aHO C nOJl.HpHOCTblO BKJIlOqeHH.H S,l{C B YQaCTOK: eCJIH HanpaBJIeHHe, CO-
3)J;aBaeMoe HCTOqHHKOM TOKa, COBnaaeT C HanpaBJIeHHeM TOKa, npOXO.Hero
B yqacTKe (HanpaBJleHHe TOKa Ha yqaCTKe COBnaaeT BHYTPH HCTOQHHKa C Ha-
npaBJleHHeM OT OTpHaTeJlbHOrO nOJllOca K nonOmHTeJIbHOMY), T.e. SJ(C cno-
c06cTByeT BRmeHHIO nOJIOmHTeJIbHbIX 3ap.HOB B aHHOM HanpaBneHRH, TO
g:' > 0, B npOTRBHOM cnyqae, eCJlR S,l{C npenHTcTByeT BHmeHHIO nOJlOmRTenb-
HbIX 3ap.HOB B aHHOM HanpaBJleHHH, TO g:' < O.
9.8. 3aKoH OMa JI.H 3aMKHYTOH l.\enH
3aMKHYTa.H uenb \pHC. 9.13) COCTOHT H3 IlBYX QaCTeH - BHYTpeHHeH H BHellI-
HeH. BuyrpeHHJUI qaCTL u.enH npeCTaBJlHeT C060H HCTO'lHHK TOKa, 06JIaalO-
262

 r
1 + -2
I.QHH BHYTpeHHHM conpOTHBneHHeM r; BHeUlHJIJI - P83-
JIHqHbIe nOTpe6HTeJlH. coeJtHHHTeJlbHbIe npOBOJta, npH60pbI
H T.,D;. 06I.Qee conpOTHBJleHHe BHemHeH qaCTH 0603HaqaeT-
C.H R. Tor,D;a nOJIHOe conpOTHBneHHe enH paBHO r + R.
TIo 3aKOHY OMa ,D;JI.H BHemHero yqaCTKa enl1 1  2
HMeeM:
I
Puc. 9.13
<PI - <P2 = I R .
BHYTpeHHHH yqaCTOK enH 2  1 .HBJI.HeTC.H HeO,D;HOpO,D;HbIM. COrJIaCHO 3a-
KOHY OMa. <P2 - <PI + g' = Ir. CJlOmHB 3TH paBeHCTBa. nOJIY'lHM
g'=IR+ Ir.
(9.4)
OTCIO,D;a
g'
1=-, (9.5)
R+r
TIoCJle,D;H.H.H cpopMYJla npe,D;CTaBJI.HeT C060H 3aKOH OMa .D:J1J1 3aMKuyTOH ....enH
nOCTOSlHHoro TOKa. CUJl.a mOKa B ijenu npSl.MO nponOpijUOHaJl.blia 8J(C ucmoll'
HUKa U 06pamli0 nponOpijUOHaJl.bHa nOJl.IiO.MY conpOmUBJleliUIO ijenu.
TaK KaK ,D;JI.H O,D;HOpO,D;HOrO yqaCTKa enH pa3HOCTb nOTeHHaJlOB eCTb Ha-
np.HmeHHe, TO <PI - <P2 = IR = U 11 cpopMyny (9.4) MomHO 3anHcaTb:
g'=U+Ir U=g'-Ir.
M3 3TOH <POPMYJIbI BH,D;HO, liTO Hanp.H>KeHHe Ha BHemHeM YlIaCTKe YMeHb-
maeTC.H C YBeJlHqeHHeM CHJlbI TOKa B u;enH npH g' = const.
TIO,D;CTaBHM B nOCJle,D;HIOIO cpOPMYJlY CHJIY TOKa (9.5), nOJlyqHM
U=g' ( l- \.
R+ r J
TIpoaHaJlH3HPyeM 3TO BbIpa>KeHHe ,D;JI.H HeKOTopbIX npe)J;eJIbHbIX pe>KHMOB pa-
60TbI enH.
a) TIpH pa30MKHYTOH enH (R  oo)U = g', T.e. HanpSlJICeHUe lia nOJl.lOcax
ucmOIlHUKa mOKa npu pa30.MKllymou ijenu paBIlO :3J(C ucmOllllUKa mOKa.
Ha 3TOM OCHOBaHa B03MO>KHOCTb npH6J1H3HTeJlbHOrO H3MepeHH.H SAC HC-
TO'tlHHKa TOKa C nOMOI.QblO BOJlbTMeTpa, conpOTHBJleHHe KOToporo MHoro 60nb-
me BHYTpeHHero conpOTHBJleHH.H HCTOtIHHKa TOKa (Rv» r). AJI.H 3Toro BOJlbT-
MeTp nO,D;KJllOqalOT K KJleMMaM HCTO'tlHHKa TOKa.
6) ECJII1 K KJIeMMaM HCTOtIHHKa TOKa nO,D;KnIOIIHTb npOBO,D;HHK, conpOTHB-
neHHe KOToporo R«r, TO R+r=r, TOri\a U=g(l-;J=O' a CHna TOKa
g'
I = - ,D;OCTHraeT MaKCHMaJIbHOrO 3HatIeHH.H.
r
TIO,D;KJIlOqeHHe K nOJlIOCaM HCTO'lHHKa TOKa npOBO,D;HHKa C HH'lTOmHO Ma-
nbIM conpOTHBJIeHHeM Ha3bIBaeTC.H KOpOTKHM 3aMbIKaHHeM, a MaKCHMaJlbHYIO
,An.H ,D;aHHOrO HCTOqHHKa CHJlY TOKa Ha3bIBalOT TOKOM KopOTKoro 3aMbIKaHH.H:
g'
I =-
K.3
r
263

Y HCTOqHHKOB C MaJlbIM 3HaqeHHeM r (HanpHMep, Y CBHH:U;OBbIX aKKYMY-
JI.HTOpOB r= 0.1- 0,01 OM) CHJla TOKa KOpOTKOrO 3aMbIKaHHSI OqeHb BenHKa.
OC06eHHO OnaCHO KOpOTKOe 3aMbIKaHHe B OCBeTHTenbHbIX CeT.HX, nHTaeMbIX OT
nO.n;CTaHlJ,HH (g> > 100 B), I K . 3 MomeT .n;OCTHrHYTb TbIC.HqH aMnep. QT06bJ H3-
6emaTb nomapOB, B TaKHe lJ,enH BKJllOqalOT npe,AOXpaHHTeJlH.
3anHlIIeM 3aKOH OMa .n;n.H nOJlHOH lJ,enH B cJlyqae nocne.n;OBaTeJlbHOrO H na-
paJlnenbHOro COe,AHHeHH.H HCTOqHHKOB TOKa B 6aTapelO. IIpH nOCJle,AOBaTenbHOM
coe.n;HHeHHH HCTOllHHKOB "-" o.n;Horo HCTOqHHKa coe.n;HH.HeTCSl C "+" BToporo, "-"
BToporo C "+" TpeTbero H T..n;. (pHC. 9.14, a). ECJlH g>1=g>2=g>3, a rl=r2=r3'
TO g> 6 = 3g> I' r6 = 3rl' B 3TOM cnyqae 3aKOH OMa .n;JI.H nOJlHOH :u;enH HMeeT BH.n;:
W  
J =  = , HnH .n;JI.H n o.n;HHaKOBbIX HCTO'lHHKOB I = 1
R + r6 R + 3r l R + nr 1
a
6
WI W 2 W.
o----=J 1+ -11+ -I 
W 2
Puc. 9.14
IIocJle.n;oBaTeJlbHoe coe)J;HHeHHe npHMeH.HIOT B TOM cJlyqae, Kor.n;a BHelliHee
nW I
conpOTHBJleHHe R > > nrl' TOr,Aa I = - H: 6aTape.H MomeT lIaTb CHJlY TOKa,
R
B n pa3 60JlblIIYIO qeM CHJla TOKa OT o.n;Horo HCTOllHHKa.
IIpH napaJlJleJlbHOM coe)J;HHeHHH HCTOqHHKOB TOKa Bce "+" HCTOllHHKOB co-
e.n;HHeHbI BMeCTe H "-" HCTOllHHKOB - TaKme BMeCTe (pHC. 9.14, 6). B 3TOM
cnyqae
Q:> Q:> rl I _ W 6 _ WI
06 =01' r6 =-. OTKy.n;a ----
3 R + r 6 R + r l .
3
.lJ;n.H n o.n;HHaKOBbIX HCTOqHHKOB I =  .
R+ rl
n
IIapaJlJleJlbHoe coe.n;HHeHHe HCTOqHHKOB TOKa npHMeH.HIOT Tor.n;a, Kor.n;a
HymHo nOJlyqHTb H:CTOqHHK TOKa C ManbIM BHYTpeHHHM conpOTHBneHHeM HnH
KorlIa .lIn.H HOpMaJlbHOH pa60TbI nOTpe6HTeJl.H 3J1eKTp03HeprHH B u.enH .lIon-
meH npOTeKaTb TOK. 60JlblIIHH. qeM .lIOnycTHMbIH TOK O.lIHOrO HCTOqHHKa.
IIapaJlJleJlbHoe coe.n;HHeHHe BbIro.n;HO, KOr,A8 R HeBeJlHKO no cpaBHeHHIO C r.
MHor.n;a npHMeH.HIOT CMelIIaHHoe coe.n;HHeHHe HCTOllHHKOB.
264

9.9. npaBHJIa KHpxrollJa.
PaC'IeT pa3BeTBJIeHHLIX aJIeKTpHQeCKHX ene:H
Puc. 9.15
3J1eKTpHqeCKa.H enb npecTaBnSleT C060H cOBoKynHocTb HC-
TOqHHKOB TOKa, npOBO,AHHKOB H nOTpe6HTeneH aneKTp03Hep-
rHH. 3J1eKTpHqeCKa.H enb qae Bcero .HBJI.HeTC.H pa3BeTBneH-
HOH (cnomHoH) H CO,AepmHT Y3nbI (pHC. 9.15). PaCqeT pa3-
BeTBJleHHOH aneKTpHQeCKOH enH 3aKnWQaeTC.H B TOM. QT06bI
no 3a):(aHHbIM conpOTHBJleHH.HM YQaCTKOB enH H S,D;C HaHTH
CHnbl TOKOB H Hanp.HmeHH.H Ha Kam):(OM yqaCTKe enH.
,D;JI.H paCQeTa p83BeTBneHHbIX eneH nOCTO.HHHOrO TOKa npHMeH.HIOT npaBH-
KHpxroc}>a.
COrJIaCHO nepBOMY npaBHJlY KHpxroa:
aJl.Ze6paUlleCKaJl cY.M.M.a CUJl m01W6. CXOaSlUXCSl 6 Y3Jl.e. pa6lia liYJl.IO:
 1J.
II M
13
Jla
n
L Ii = 0 ,
i=1
r,Ae n -- qHCJlO npoBo):(HHKOB, 06p83YIOHX Y3eJl.
TIpH 3TOM TOKH CqHT8IOTC.H nOJlOmHTenbHbIMH, eCJlH OHH BXO):(.HT B Y3eJl.
H oTpHaTenbHbIMH, ecnH BbIXO):(.HT H3 Y3na. ,D;n.H Y3J1a, H306pameHHoro HB
pHCYHKe 9.15, II - 1 2 - 1 3 = O.
COrJIaCHO BTOpOMY npaBHJIY KHpxroa:
6 Jl.1060M npocmOM 3aMKliymoM KOlimype. npOU360Jl.b1i0 6bl6upaeMOM 6 pa3
6emBJleliHOU 3JleKmpUileCKOU ijenu, aJlZe6paUileCKaSl CYMMa npOU3BeaeliUU CUJl
mOK06 lia conpOmUBJl.eliU.R coomBemcm6YIOUX YllacmKOB paBlia aJl.ze6paullec
KOU CYMMe 3J(C. UMelOUXC.R 6 KOlimype:
n m
L l kRk = L g i ,
k=l I=}
r):(e m - QHCJlO HCTOqHHKOB B KOHType, n -- qHCnO conpOTHBJleHHH B HeM.
EcnH HanpaBneHH.H TOKOB COBna,AaIOT C Bbl6p8HHbIM HanpaBneHHeM 06XO):(8
KOHTypa, TO CHJlbI TOKOB I k CQHTaIOTC.H nonOmHTeJlbHbIMH. 3,D;C Si CqHTaIOTC.H
nOJlOmHTenbHbIMH, eCJlH OHH C03):(a1OT TOKH. COHanpaBneHHble C HanpaBJleHHeM
o6xo,Aa KOHTypa.
TIpaBHna KHpxro<pa He BbIpamalOT HHKaKHX HOBbIX CBOHCTB CT8HOHapHO-
ro 3J1eKTpHQeCKOrO nOJl.H B npOBO):(HHKax C TOKOM no cpaBHeHHIO C 38KOHOM
OMa. TIepBoe H3 HHX .HBn.HeTC.H cne):(CTBHeM 3aKOHa coxpaHeHH.H 3J1eKTpH'leC-
KHX 3ap.H):(OB, BTopoe -- cne):(CTBHeM 3aKOHa OMa ):(n.H HeO):(HopO):(Horo yqaCTKa
enH. O):(HaKO HX HCnOJlb30BaHHe 3HaqHTeJlbHO ynpoaeT paCqeT TOKOB B pa3-
BeTBneHHbIX en.Hx.
PaC'leT pa3BeTBJleHHOH aneKTpH'leCKOH enH nOCTO.HHHOrO TOKa BbInOnH.HeT-
C.H B CJleYlOlIleM nOp.H):(Ke:
a) npOH3BOJlbHO BbI6HpalOT HanpaBJleHHe TOKOB BO Bcex yqaCTKaX uenH:
6) 3anHCbIBalOT n - 1 He3aBHCHMbIX ypaBHeHHH, cornaCHO nepBOMY npaBHny
KHpxroc}>a, r):(e n -- KOJlHqeCTBO Y3J10B B enH;
B) BbI6HpalOT npOH3BonbHO aaMKHYTble KOHTypbI TaK, 'lTofibI Kam):(bIH HO-
BbIH KOHTYP CO,AepmaJl XOT.H 6bI O,AHH yqacToK enH, He BXO):(.HHH B paHee
BbI6paHHbIe KOHTypbI. 3anHCbIBaWT )J;JI.H HHX BTopoe npaBHno KHpxrQ(pa.
265

B pa3BeTBJleHHOH enH, coepmaeH n Y3J10B Ii m yqacTKoB enH Memy
COCeHHMH Y3J1aMH, qHCJlO He3aBHCHMhIX ypaBHeHHH, COOTBeTCTBYlOlQHX npaBH-
JlY KOHTypOB, COCTaBJI.He'I m - n + 1.
R8 OCHOBe npaBHJI KHpxro<l>a COCTaBJI.HIOT CHCTeMY ypaBHeHHH, pelIIeHHe
KOTOpOH n03BOJlSleT HaHTH CHJIhI TOKOB B BeTB.HX u;enH.
9.10. lliYUTLI H Ao6aBO'lHLIe conpOTHBJIeHHH
CHny TOKa B enH H3MepSllOT aMnepMeT-
pOM. AMnepMeTp npeCTaBJlSleT C060H 06blq-
HhIH ranhBaHOMeTp, IIIKaJla KOToporo nporpa-
YHpOBaHa B aMnepax. BRJllOqaeTC.H aMnep-
MeTp B enb nocneOBaTeJlbHO (pHC. 9.16).
AMnepMeTp He onmeH H3MeH.HTb cHny TOKa
B enH, H n03TOMY ero conpOTHBneHHe .o;onmHO 6bITb OqeHh MaJlbIM, T.e.
R A «R. B uenH. COCTOSlI.l.!eH H3 nOCJle.n:OBaTenhHO coe.n:HHeHHbIX npOBo.n:HH-
KOB, 3ap.H HHre He HaKannHBaeTC.H H HHre He HCtle3aeT. 8TO 3HaqHT,
qTO CHna TOKa B TaKOH enH Be3)J;e OHHaKOBa Ii aMnepMeTp MomHO BKJlIO-
qaTb B JlI060H yqaCTOK enH, COCTO.HlQHH H3 nocneOBaTeJlhHO coeHHeH-
HbIX npOBo.n:HHKOB.
KambIH aMnepMeTp paCCqHTaH Ha HeKOTOpylO MaKCHM8.JIbHYIO CHJlY TOKa,
npH npeBbIlIIeHHH ROTOpOH npH60p MomeT neperopeTb.
,nJl.H paclIIHpeHH.H npeeJlOB H3MepeHHSI aMnepMeTpa npHMeH.HeTC.H WYHT -
onOnHHTeJlbHOe conpOTHBJleHHe I nO,AKnlOtlaeMOe napanJlenbHO aMnepMeTpy.
RaH,AeM conpoTHBneHHe Rw IIIYHTa, KOTOpbIH He06xo,AHMO nOKJllOtlHTb K
aMnepMeTpy .n:n.H H3MepeHH.H CHnhI TOKa B uenH. B n pa3 npeBbIlIIalOlQeH cHny
TORa, Ha KOTOPYIO paCCqHTaH npH60p: 1= nl A .
ConpOTHBJleHHe aMnepMeTpa 0603HatlHM qepe3
R A . TIPR nOKnlOtleHHH WYHTa qaCTb H3MepSle-
MOH CHJlbI TOKa I w nOH,AeT no HeMY. lJepe3
aMnepMeTp onmeH HTH TOR, He npeBbIlIIalO-
lQHH l A (pHC. 9.17).
CHJla TOKa I A MeHhlIIe H3Mep.HeMOH B n pa3
Puc. 9.16
l A
A
R A
I
Rm
1m
I
Puc. 9.17 ( I A = )- CJIellOBBTeJILHO, eHB lIeJIeHH" npH-
60pa B03pacTeT B n pa3 n.H cJlyqaSl, ecnH WKaJla npH60pa paBHoMepHaSl,
T.e. OTKJlOHeHHIO CTpeJlKH Ha OHO eneHHe 6yeT COOTBeTCTBOBaTb B n pa3
60JlblIIaSl CHJla TOKa. IiHatle rOBopSl, tlYBcTBHTeJlbHocTb aMnepMeTpa YMeHbwHT-
CSI B n pa3: npH nOKJllOqeHHH IIIYHTa CTpeJlKa npH60pa OTKnOHHTCSI Ha yron,
B n pa3 MeHhIIIHH, qeM 6e3 Hero.
TIpH napaJlneJlbHOM coeHHeHHH 1= nI A =l A + l w ' a Hanp.HmeHHe Ha IIIYHTe
H aMnepMeTpe OHHaKOBO H, COrJlaCHO 3aKOHY OMa, paBHo: lARA =lwRw. Iic-
KJllOqa.H CHJlY TOKa l A H3 ,ABYX nOCJle,AHHX ypaBHeHHH, nOJlyqHM
266
R = R A
w .
n-l

AJI.H H3MepeHH.H Hanp.H>KeHH.H Ha yqaCTKe nenH
npHMeH.HIOT BOJlLTMeTpLI. BKJlIOq81OT BOJlbTMeTp n8-
panneJlbHO TeM TOqK8M uenH. H8np.HmeHHe MemlIY
KOTOpbIMH HaO H3MepHTb (pHC. 9.18). BonbTMeTp
He On>KeH H3MeH.HTb H8np.H>KeHHe H8 H3Mep.HeMOM
yq8CTKe nenH t n03TOMY CHn8 TOKa, npOXO.Hl..Qero
qepe3 BOJlbTMeTp, OJlmHa 6bITb MHoro MeHbme, qeM
CHn8 TOKa B H3Mep.HeMOM yq8CTKe.
ECJlH conpOTHBJleHHe BOJlbTMeTpa R y , TO nocJle BKnlOqeHH.H ero B nenb
M
R
N
Puc. 9.18
conpOTHBJleHHe yqacTKa 6yeT Y>Ke He R, 8 R' = RR y
R+R y
R
R < R.
1+-
Rv
BCJleCTBHe 3Toro H3MepseMoe Hanp.HmeHHe Ha yqacTKe :nenH YMeHbIlIHTC.H.
An.H Toro "lT06bI BOJlbTMeTp He BHOCHJI 38MeTHbIX HCK8meHHH B H3Mep.HeMOe
HanpsmeHHe, ero conpOTHBneHHe OJl>KHO 6bITb 60nbllIHM no cpaBHeHHIO C CO-
npOTHBJleHHeM yqaCTK8 nenH, Ha KOTOpOM H3MepseTCSJ H8npsmeHHe, T.e.
Ry» R.
JII060H BOJlbTMeTp paCCqHTaH Ha npeenbHoe H8np.HmeHHe U y. Ho C nOMO-
I..QblO nOKJlIO"leHHS nocneOB8TeJlbHO C BOJlbTMeTpOM .n;068BOqHOrO conpOTHBne-
HH.H Rp, MO>KHO H3MepSTb B n pa3 60JlbllIHe Hanp.H>KeHH.H: U = nU y . H8HlIeM
.n;o6aBOqHOe conpOTHBneHHe, He06xoHMoe )J;n.H H3MepeHHSJ Hanp.HmeHHH, B n pa3
60nbllIHX Tex, H8 KOTopbIe paCCqHT8H npH60p.
IIpH BKJllOqeHHH B nenb BOJlbTMeTpa ,A06aBOq-
Horo conpOTHBJleHH.H BOJlbTMeTp no-npemHeMY 83-
MepseT H8npsmeHHe U y, HO 3TO COCTaBJI.HeT JlHllIb
1 U
qaCTb H3Mep.HeMOrO H8np.HmeHH.H U y =-
n n
Hanp.H>KeHHe H8 068BOqHOM conpOTHBJleHHH
Up, = U - U y (pHC. 9.19). TI03TOMY npeenbI H3Mepe-
HH.H YBeJlHqHBalOTCS B n pa3 t H BO CTOJlbKO me
pa3 YBeJlHqHBaeTCS l.\eHa ,AeJleHH.H BOJlbTMeTpa, a
cne.n;OBaTenbHO, YMeHbllIaeTC.H ero 'iYBCTBHTenbHOCTb.
B BOJlbTMeTpe H ,A06aBOqaOM conpOTHBJleHHH CHn8 TOK8 0,AHH8KOB8, T8K
KaK OHH BKnlOqeHbI nOCJleOBaTeJlbHO. II03ToMY U y= IR y , Up, = I Rp, H
U=nUy=nIR y .
TIpH nOCJle.n;OBaTeJlbHOM coeHHeHHH Hanp.HmeHHe H8 yqaCTKe paBHO CYMMe
HanpS>KeHHH Ha OTenbHbIX pe3HCTop8X yqacTKa, T.e. U = U y + Un' Cne.n;OBa-
Rv R",
Puc. 9.19
TenbHO, InR y = IR y + 1Rll' OTclO.n;a
Rp, = (n - I)R y .
9.11. Pa60Ta H MOIl.\HOCTL TOKa
3J1eKTpHqeCKa.H 3HeprH.H nerKO npe06p83yeTC.H B ,ApyrHe BHbI 3HeprHH -
MeXaHHqeCKYIO, XHMHqeCKYIO, CBeTOBYIO, BHYTpeHHlO1O 3HeprHIO Bel..QeCTBa, qTO
IlIHpOKO npHMeH.HeTC.H B npOMbIllIJleHHOCTH H B 6bITY.
267

MepOH U3MeHeHHJI 3HepI'HH 3J1eKTpHqeCKOrO TOKa cnymHT pa60TB HCTOq-
HHKa TOKa, C03,lJ,alOlQerO H nO,lJ,,Aep>KHBalOlQerO sneKTpHqeCKOe none B enH.
CTaHOHapHoe SJleKTpHqeCKOe nOJle, nepeMelQalOlQee 3ap.H,lJ,bI no npOBO,lJ,-
HHKY. COBepwaeT pa60TY. STY pa60TY Ha3bIBalOT pa60ToH TOKa. Pa60Ta
sneKTpHqeCKOrO TOKa Ha yqaCTKe enH, KaK Cne,lJ,yeT "3 onpe,lJ,eneHH.H Ha-
np.H>KeHH.H ,
A = qU,
r,Ae q - sneKTpHqeCKHH 3ap.H,lJ" npOXO,lJ,.HlQHH no yqaCTKY enH, a U - Ha-
np.H>KeHHe Ha yqaCTKe.
YqHTbIBa.H, qTO q = It, r,lJ,e I - CHJla TOKa B npOBO,AHHKe. a t - BpeM.H
npOXO>K,AeHH.H sneKTpHqeCKOrO TOKa. ,An.H pa60TbI TOKa nonyqHM
A = IUt.
EcnH R - conpOTHBJleHHe O.n;Hopo.n;Horo yqacTK8 enH, TO, Hcnonb3Y.H 3aKOH
OMa .n;n.H yqacTKa enH, MomHO nonyqHTb <popMyny .n;n.H paCqeTa pa60Tbi TOKa:
2 U 2
A = I Rt = - t .
R
EcnH yqacToK enH He .HBn.HeTC.H O,lJ,HOpO,AHbIM, TO pa60TY COBepwaeT He
TonbKO CTaHOHapHoe sJleKTpHqeCKOe nOJle, HO H CTopOHHHe CHJlM, " nOJlHa.H
pa60Ta onpe,lJ,en.HeTC.H no <popMYJle
A = I (<PI - <P2 :t go) t .
IIo BbIwenpHBelleHHbIM cpopMYJlaM MO>KHO paCCqHTaTb nonHYIO pa60TY TOKa
Ha ,AaHHOM yqaCTKe uenH.
EcnH B enH eCTb 3neKTpO,lJ,BHraTenb. TO 3HeprH.H 3J1eKTpHqeCKOrO TOKa,
Bo-nepBbIX. pacxo.n;yeTC.H Ha COBepweHHe MeXaHH'leCKOH pa60TbI - nOJle3Ha.H
pa60Ta A MCX ' BO-BTOpbIX, 3aTpaqHBaeTC.H Ha HarpeBaHHe 06MOTOK 3neKTpO,lJ,BH-
raTen.H H coe.n;HHHTenbHblx npOBO,lJ,OB - Tep.HeMa.H SHeprH.H. B STOM cJlyqae
Kosq)<pHHeHT nOJle3HOrO ,lJ,eHCTBH.H MomHO paCCqHTaTb KaK
11 = A..tex ...d =..d + Q' 11 = A..tex = A,.lex
..d ,.I. ;-01(8 e)l.. , W ..d + Q
UKa 3 ex
rOBOpSl 0 KOS<P<PHHeHTe nOJle3HOrO ,lJ,eHCTBH.H HCTOqHHKa TOKa, nO,A nOne3-
HOH pa60TOH nO.n;pa3YMeBa1OT pa60TY, COBepWaeMYIO BO BHeWHeH enH nOCTO-
.HHHOrO TOKa:
 = IUt = 1 2 Rt .
3aTpaqeHHa.H me pa60Ta "CTOqHHKa TOKa paBHa pa60Te CTOpOHHHX CHn:
Puc. 9.20
A 3 = qg'= Ig't, r.n;e g'=I(R+r).
TOr,lJ,B A3 = 1 2 (R + r)t.
KIIJJ; HCTOqHHKa 11 =  = IUt = U = . r,Ae
 19ot go R + r
U - Hanp.H>KeHHe BO BHewHeH enH (Hanp.H>KeHHe
Ha nonlOcax HCTO'lHHK8 TOKa).
R rpa<pHqeCKa.H 3aBHCHMOCTb 11 = f ( R) npH r = const
npHBe,lJ,eHa Ha pHC. 9.20.
11HCT
1 -----
o
268

E,AHHHu.a pa60TbI 3neKTpHqeCKOrO TOKa B CM - mOYJlb (.lJ.m). 1 ,l{:JK
npenCTaBJI.HeT pa60TY TOKa. 3KBHBaneHTHYlO MeXaHHqeCKOH pa60Te B 1 nm.
1 ,l(m = KJI . B = A . B . c.
M3Mep.HIOT pa60TY 3neKTpHqeCKOrO TOKa CqeTqHKaMH.
CKOpOCTb COBepIIIeHHSI pa60TbI TOKa Ha aHHOM Y'laCTKe u.enH xapaKTepH-
A
3yeT MOI.,QHOCTb TOKa. MOI.,QHOCTb TOKa onpeen.HIOT no <l>opMyne P = - HnH
t
P = [V.
McnoJlb3Y.H 3aKOH OMa lIJI.H yqaCTKa u.enH, MomHO 3anHcaTb HHaqe <l>OPMY-
U 2
ny n.H MOI..qHOCTH TOKa: P = [2 R = -. B 3TOM cnyqae peqb HeT 0 Tenno-
R
BOM MOI.,QHOCTH.
EHHHu.a MOI.,QHOCTH TOKa - BaTT: 1 BT = ,l(m . OTCIO,Aa ,l(m = BT' C.
C
KpoMe TOro, npHMeH.HIOT BHecHcTeMHbIe e,AHHHUbI: KHnOBaTT-qaC HJIH reKTOBaTT-
qac: 1 KBT.q=3,6.10 6 )];m=3,6 M)];m; 1 rBT.q=3,6.10 5 )];m=360 K)];m.
,l(n.H H3MepeHHSI MOHOCTH TOKa CYl.,QeCTBYIOT cneu.HanbHbIe npH60pbI -
BaTTMeTpbI.
9.12. 3aKoH .n:moYJIH-JIeHa
B 3J1eKTpHqeCKOH u.enH npH npoxomeHHH TOKa npOHCXOHT p.H npeBpa-
l.,QeHHM 3HeprHH. Bo BHeiliHeM Y'laCTKe enH pa60TY no nepeMel.,QeHHIO 3apSI,Aa
COBepilialOT CHnbI CTaIJ.HOHapHoro 3neKTpHqeCKOrO nonSl H 3HeprHSI 3Toro nonSI
npeBpal.,QaeTCSI B pyrHe BH,AbI: MeXaHHqeCKYlO, TennOBYlO, XHMHqeCKYIO. B 3Hep-
rHIO 3neKTpOMarHHTHoro H3nYQeHH.H. CJlenOBaTenbHO. nOnHa.H pa60Ta TOKa Ha
BHeiliHeM yqacTKe u.enH
A-rOK8 = W Mex + W XHM + W H3JI + Q.
EcnH me Ha yqacTKe u;enH no eHcTBHeM 3neKTpHQeCKOrO non.H He COBep-
IIIaeTC.H MeXaHHqeCKa.H pa60Ta H He npOHCXO.HT XHMHQeCKHe npeBpal.,QeHHSI.
TO pa60Ta 3J1eKTpHqeCKOrO TOKa npHBOHT TOJlbKO K HarpeBaHHIO npOBO,AHHKa.
B 3TOM cnyqae KOnHqeCTBO BbIenHBIIIeMC.H TenJlOTbI paBHO pa60Te, COBep-
IIIaeMOM TOKOM.
KOnHqeCTBO TennOTbI Q, BbIen.HeMOH TOKOM l 3a BpeM.H t H8 YQaCTKe
u.enH conpOTHBJleHHeM R, paBHO Q = [2 Rt.
3Ta <l>opMyna BbIpamaeT 33KOH ,lI.mOYJIH-JIeHI\3, YCTaHOBJleHHbIM onbIT-
HbIM nYTeM B XIX B. BYM.H yqeHbIMH (aHrnHMcKHM - )];m. )];moyneM H
PYCCKHM 3. X. JIeHu.eM).
llpu npoxoJICoeliuu 3Jl.eKmpUllecKozo mOKa no np0600liUKY KOJl.UlleCm60 men-
Jl.OmbL, 6bLOeJl.SlIOU{eilcSl 8 np06001iUKe. npSlMO nponopquoliaJl.bliO K6aopamy CUJl.bL
mOKa, conpOmU6Jl.eliUIO np06001iUla U 6peMeliU npOXOJICOeliUSl mOKa.
Ha 3aKOHe .lJ.moyn.H-JIeHu.a OCHOBaHO eMCTBHe MHorHX 3neKTpOHarpeBa-
TeJlbHbIX npH60pOB. 3TO YTlOrH, 3neKTpOnJlHTbI, 3J1eKTpOqaHHHKH, KHn.HTHnb-
HHKH. na.HJlbHHKH, 3neKTpOKaMHHbI H T..
OCHOBHOM qaCTblO nlO60ro 3J1eKTpOHarpeBaTeJlbHOrO npH60pa .HBnSIeTCH
HarpeBaTeJIbHblH 3JIeMeHT (npOBO,AHHK C 60nbIIIHM yeJlbHbIM conpOTHBne-
269

HHeM HaMaTbIBaeTC.H Ha nJlaCTHHKY H3 mapOCTOHKOrO MaTepHaJla: CnlObI,
KepaMHKH).
BbIlIIenpHBeeHHYIO «}>OPMYJlY 3aKOHa )J;moYJI.H-JIeHa Y06HO npHMeH.HTb
npH nOCJleOBaTeJlbHOM COeHHeHHH pe3HCTOpOB, TaK KaK CHJla TOKa BO BCeX
yqacTKax nOCJleOBaTeJlbHO COe,AHHeHHOH enH O,AHHaKOBa. ECJlH nOCJleOBaTeJlb-
HO coeHHeHbI ,ABa pe3HCTopa C conpOTHBJleHH.HMH Rl H R2' TO Ql =[2R 1 t,
Q2= [2R 2 t, oTKya Ql =!!L, m.e. KOJl.UlleCm60 menJl.OmbL. 6bLOeJl.SleMOU mOKOM
Q2 
6 YllacmKax nOCJl.e006ameJl.bIiO coeoulieliliou ijenu. nponopquoliaJl.bliO conpomU8-
Jl.eIiUSlM amux YllacmK08.
Cor JlaCHO 3aKOHY OMa, JI.H OHopOHoro yqaCTKa nenH nOCTO.HHHOrO TOKa
U U 2
I = -. Tora Q = - t .
R R
STY «}>OPMYJlY Y06HO HCnOJlb30BaTb npH napanJleJlbHOM coeHHeHHH pe3Hc-
TOpOB, TaK KaK Hanp.HmeHHe Ha KamOH BeTBH TaKOH enH oHHaKoBo. EcnH
napaJlJleJlbHO coe)J;HHeHbI B8 pe3HCTopa C conpoTHBneHH.HMH Rl H R2' TO
u 2 U 2
Ql = -t; Q2 = -t, oTKya
 
Ql _
Q2 - Rl '
m.e. KOJl.UlleCm60 menJl.OmbL. 6btOeJl.ReMOU mOKO.M 8 8em8SlX napaJl.Jl.eJl.bIiO co.
eOUlieJtliOU qenu. 06pamHO nponopquoliaJl.bliO conpOmU6Jl.eltURM pe3ucmop08.
6KJllOI£eHHbLX 8 amu 6em6U.
BonpOCbI J(JIH CAMOKOHTPOJIH
1. IJmo lia3bL6alOm aJl.eKmpUlleCKUM mOKoM? IJmo npuliuMalOm 3a lianpa6-
Jl.eHUe mOKa?
2. KaKue oeucm6uR MOJICem OKa3bL6amb aJl.eKmpUlleCKUU mOK?
3. IJ mo maKoe CUJl.a mOKa? H a306ume eouliuqbl. CUJl.bl. mOKa.
4. KaKo8bL YCJl.08UR cY14ecm608aliUJl aJl.eKmpUllecKozo mOKa 8 npo800liuKe?
5. KaK (jJOPMYJl.UpyemcSl 3aKOli OMa OJl.Sl YllacmKa uenu? IJmo nOIiU.MalOm
noo 60Jl.bm-aMnepliou xapaKmepucmuKou np060oliuKa?
6. IJmo maKoe conpOmU6Jl.eliUe np060oliuKa? Ha306ume eOUIiUijbL conpo-
mU8Jl.e IiUSl.
7. Om llezo 3a6ucum aJl.eKmpUllecKoe conpOmU6Jl.eliUe np060oliuKa?
8. IJ mo maKoe yoeJl.bliOe conpOmU6Jl.eliUe npo60oliuKa? H a308ume eOUliUijbL
yoeJl.bliOZO conpOmU6Jl.eIiUSl.
9. KaK liaumu 0614ee conpOmU6Jl.eliUe 06YX pe3ucmop06. coeoulielilibLx no-
CJl.e006ameJl.blio? napaJl.Jl.eJl.blio?
10. KaKue npu60pbL u3MepSllOm CUJl.y mOKa 6 qenu? lianpSlJICeliUe lia Yllac-
mKe qenu? KaK amu npu60pbL 6KJl.lOllalOmCSl B qenb 2
11. KaKU.M OOJl.JlCIiO 6bLmb 6liympeHliee conpOmU6Jl.eliUe aMnepMempa?
60Jl.bmMempa? KaK pacwupumb npeOeJl.bl. u3MepeliUSl amux npu60p06?
12. IJ mo lia3bL8a1Om pa60mou mOKa? II 0 KaKUM (jJopMYJl.aM paCCllumbL8a-
emcSl pa60ma mOKa? M014IiOCmb mOKa?
270

13. II O'leMY npoucxooum liaZpe6aliUe np060aliUKa npu npOXOJICOeHUU no
lieMY 3Jl.eKmpU'leCKOZO mOKa? KaK (jJOPMYJl.UPyemcSl 3aKOli J(JlCoYJl.Sl-JIeliu,a?
14. l{mo maKoe cmopoliliue CUJl.bt? KaK06a pOJl.b ucmOllliUKa mOKa 6 3aM
Kliymou 3Jl.eKmpUllecKOu u,enu?
15. l{ mo lia3bt6aemcSl 3Jl.eKmpOa6UJlCyeu CUJLOU (oJ{C) ucmOllliUKa mOKa?
16. l{mo maKoe 6lieWliUU U 6liympeliliuu YllacmKU 3Jl.eKmpUllecKou u,enu?
IIoa aeucmBueM KaKUX CUJl. aBU::HCymcSl 3Jl.eKmpUlleCKue 3 ap.R.abt Ita BlieWlieM
( BliympeHlieM) YllacmKe u,enu?
17. KaK ljJoPMYJLUpyemc.R. 3aKOli OMU OJl..R. 3aMICHymou u,enu?
18. KaK C6Sl3alia opC ucmOIlHUKa c lianp.R.JlCeliUeM lia 6lieWlieM YllacmKe
u,enu? H 3MeliSlemcSl Jl.U 3liaileliUe oJ(C ucmOllliUKa mOKa npu U3MeHeliUU lia-
npSlJICeliUSl 60 6lieWlieU ijenu?
19. l{mo lia3bt6alOm meMnepamyplibtM K03(jJ(jJuijUelimOM conpOmUBJl.eliU.R.?
H a306ume eaUliUU,bt meMnepamypliozo K03(jJ(jJUijUelima conpOmU6Jl.eliuSl.
20. KaK U3MeliSlemcSl conpOmU6Jl.eHUe np060aliuKa npu n06btWeliUU meMne-
pamYPbt? KaK zpa(jJullecKu u306paJICaemCSl ama 3aBucuMocmb?
OCHOBHblE cI»OPMY Jlbl
g = Acrop
q
- 3J1eKTpo.u;BHmYll(a.H CHJla HCTOqHHKa, r)J;e
A CTOP - pa60Ta CTOpOHHHX CHJI no nepeMe-
ll(eHHIO nOJlOmHTeJlbHOrO 3ap.H.u;a q B 3aMK-
H}'TOH u.enH.
U = (A.o-n + p) = (<Pi - <P2) + g> - Hanp.HmeHHe Ha yqaCTKe I.\enH, r.u;e A Kyn -
q pa60Ta CHJI 3J1eKTpOCTaTHeCKoro nOJl.H no ne-
peMeeHHIO 3ap.H.u;a q; (<Pi - q>2) - pa3HOCTb
nOTeHI.\H8JI0B.
I==en(v}S
t
- CHJla nOCTO.HHHOrO TOKa, r.u;e q - 3ap.H.u;,
npOXO)J;HHH 'Ie pea nOnepeqHOe CeqeHHe npo-
BO.u;HHKB 3a BpeMH t; e - 3J1eMeHTapHbIH 3a-
p.H.u;; n - KOHI.X.eHTpau,H.H CB0601IHbIX 3ap.H1I0B
B npOBo.u;HHKe; (v) - cpe.u;H.H.H CKOpOCTb Ha-
npaBJleHHOrO .u;BHmeHHH CB060.u;HbIX 3ap.H.u;OB;
S - nJlOa.u;b nOnepeqHOrO CeqeHH.H npOBO.u;-
HHKa.
j=i=en(v}
S
- MO.u;YJlb nJlOTHOCTH nOCTO.HHHOrO TOKa.
l
R =p-
S
- conpOTHBJleHHe npOBo.u;HHKa, r.u;e p -
y.u;eJlbHoe conpOTHBJIeHHe; l - .u;JlHHa npOBO,A-
HHKa; S - nJlOa.u;b nOnepeqHOrO CeqeHH.H.
U
I = - = GU
R
- 3aKOH OMa .u;JlH yqacTKa I.\enH, r.u;e G -
npOBO)J;HMOCTb npOBo.u;HHKa.
271

R = Ro (1 + at)
n
R = LR i
i=1
1 n 1
-=L-
R i=1 R i

1=-
R+r
I=
R+nr

J=-
R+r
n
u 2 t 2
A = qU = - = J Rt = JUt
R
Q = [2 Rt
U 2
P = IU = [2 R = -
R
2
A = qZ'= [Z't =-t
R+r
Z'2
P = JZ' =- = J2(R+r)
R+r
272
- conpOTHBJleHHe MeTannHqeCKOrO npOBOAHH-
Ka npH TeMnepaType t, rAe Ro - conpOTHB-
JleHHe npH 0 DC; a - TeMnepaTYPHbIE K03<p-
<pHu.HeHT conpOTHBJleHH.H.
- conpOTHBJleHHe yqaCTKa u.enH nocne.n;OBa-
TeJlbHO COe,AHHeHHbIX npOBOLIHHKOB. r JIe R j -
conpOTHBJleHHe i-ro npOBOAHHKa; n - 'tJHcno
npOBOAHHKOB.
- <popMyna AJI.H paCqeTa 06lI\ero conpOTHBne-
HH.H Y'tJaCTKa u.enH npH napanneJlbHOM coeAH-
HeHHH n npOBO,AHHKOB.
- 3aKOH OMa ,AJlSI 3aMKHYTOH u.enH, rJIe
rg - S.n;C HCTOqHHKa; R - BHelIlHee conpo-
THBJleHHe (conpOTHBneHHe nOTpe6HTeneH); r -
BHYTpeHHee conpOTHBneHHe (conpOTHBJleHHe HC-
TO'tJHHKa).
- CHJla TOKa B u.enH, 3aMKHYTOH Ha 6aTa-
pelO n OAHHaKOBbIX nocne)J;OBaTeJlbHO coe)J;HHeH-
HbIX HCTO'tJHHKOB, rAe rg, r - S.n;C H BHYT-
peHHee conpOTHBneHHe OAHoro HCTO'tJHHKa CO-
OTBeTCTBeHHO.
- CHna TOKa B u.enH, 3aMKHYTOH Ha 6aTa-
pelO n O,AHHaKOBbIX napaJlJleJlbHO coe.n;HHeHHbIX
HCTOqHHKOB.
- pa60Ta Ha yqaCTKe u.enH, rAe t - BpeMSI
npoxO>KAeHH.H TOKa, R - conpOTHBneHHe y'tJa-
CTKa u.enH (pe3HCTopa).
- 3aKOH AmoYJl.H-J!eHu.a, rLIe Q - KonH'tJe-
CTBO TennOTbI, BbI.neJlSleMOH Ha pe3HCTope CO-
npOTHBneHHeM R npH npoxom.n;eHHH no HeMY
TOKa, [ - CHJla npOXOA.HerO TOKa.
- MOlI\HOCTb TOKa Ha yqaCTKe u.enH.
pa60Ta HCTo'tJHHKa TOKa B 3aMKHYToH
u.enH.
- MOlI\HOCTb HCTOqHHKa TOKa.

t'I=  = U =
A g' R+r
- K HCTOqHHKa TOKa, re An - nOne3Ha.H pa-
60Ta, COBepmaeMaH HCTO'lHHKOM TOKa, A - pa60TB
HCTO'lHHKa TOKa.
METOAHqECKHE PEKOMEHAAQHll
3aaqH aHHoro pa3ena Y,A06HO pa3eJlHTb Ha TPH THna: 3aaqH Ha BM-
'tJ:HcneHHe conpOTHBneHHH, CHn TOKOB HnH Hanp.HmeHHH Ha KaKOM-nH60 yqaCT.
Ke IJ;enH; 3aaqH Ha pa60TY, MOI.,QHOCTb H TennOBoe eHCTBHe TOKa H 3aaqH
Ha 3J1eKTpOJlH3.
TIpH pemeHHH 3aa1J nepBoro THna peKoMeHyeTC.H:
1. Ha'tJ:epTHTL cxeMY H YKa3aTb Ha HeH Bce 3J1eMeHTbI IJ;enH: HCT01JHHKH
TOKa, pe3HCTopbI H p., YCTaHOBHTb npeBapHTenbHO, 0 KaKOH IJ;enH rOBopHT-
CH B ycnoBHH 3aa1JH: 3aMKHYTOH HnH 0 KaKOM-nH60 yqaCTKe IJ;enH.
2. YCTaHOBHTb, KaKHe 3neMeHTbI IJ;enH BKnlO'leHbI nOCJle,AOBaTenbHO, KaKHe
napaJlJlenbHO.
3. Ha KamOM yqaCTKe enH onpeeJlHTb HanpaBJleHHS TOKOB H 3anH-
caTb ,l\JI.H Ka:>K,l\OH TOqKH pa3BeTBJleHHH (eCJlH OHH eCTb), ypaBHeHH.H TOKOB
H ypaBHeHH.H, CB.H3bIBaIOWHe Hanp.H>KeHH.H Ha Y1JaCTKaX uenH H 8)]C (npa-
BHna KHpxrocpa).
4. TIpH paC1JeTe conpOTHBneHHH cnomHbIX coeHHeHHH Ha,l\O OTbICKaTb B
cxeMe pe3HCTopbI, KOTopbIe coeHHeHbI Memy C060H nocneOBaTenbHO HnH
napaJlneJlbHO, 3aMeHHTb HX 3KBHBaJleHTHMM conpOTHBJleHHeM H nonY'tJ:HTb yn-
pOl.,QeHHYIO cxeMY. B03MomHO, 3TOT npHeM Hao 6yeT npHMeHHTb HeCKOJlb-
KO pa3.
5. EcnH B cxeMe HeT HH nOCJle,l\OBaTenbHO, HH napaJlnenbHO COe,AH-
HeHHbIX pe3HCTopOB, nH BbI'lHCJleHH.H 06I.,Qero conpOTHBneHH.H HymHo no-
nMTaTbC.H HaHTH T01JKH C O,l\HHaKOBbIMH nOTeHIJ;HaJlaMH. OHH BCer,l\a eCTb
B cxeMax, 06na,l\alOl.,QHX OCblO HnH nJlOCKOCTblO CHMMeTpHH OTHOCHTeJlbHO
To'tJ:eK nOKJllOqeHH.H HCTOqHHKa nHTaHHH. ECJlH cxeMa CHMMeTpHqHa OTHO-
CHTeJlbHO OCH, npOXO,l\.HeH 1Jepe3 TOqKH BXO,l\a H BbIXOa TOKOB, TO TO'tJ:-
KH O,l\HHaKOBOrO nOTeHIJ;Hana HaXO,A.HTC.H Ha KOHu,aX CHMMeTpH'lHMX pe3HC-
TOpOB. ECJlH CXeMa CHMMeTpH'tJ:Ha OTHOCHTenbHO OCH. nepneHllHKyn.HpHOH
nHHHH, Ha KOTOpOH nemaT T01JKH BXO.n;B H BbIXO.n;a TOKOB, TO O,lIHHaKO-
BbIM nOTeHIJ;HanOM 06J1aaIOT BCe TO'tJ:KH, nemal.,QHe HB nepeCe'tJ:eHHH 3TOH
OCH C npOBOHHKaMH.
HaH,l\.H TOqKH C OHHaKOBbIMH nOTeHIJ;HanaMH, HymHo COeHHHTb HX (eCnH
OHH 6bInH pa3'LeHHeHbI) HnH pa3'Le,l\HHHTb (eCnH TOqKH 6MJlH COeHHeHbI),
nocJle qerO nOJlyqalOT 3KBHBaJleHTHYlO CXeMY, COCTaBJleHHYIO H3 nOCJleOBaTeJlb-
HO H napaJIneJlbHO COeHHeHHbIX pe3HCTopOB.
6. McnOJlb3Y.H 3aKOH OMa, YCTaHOBHTb CB.H3b Memy TOKaMH, HanpHmeHH-
.HMH H 8,AC.
7. PemHTb nOJlY'leHHYIO CHCTeMY ypaBHeHHH OTHOCHTenbHO HCKOMOH BeJIH'IHHbI.
TIpH pemeHHH 3aaq BToporo THna HymHO:
1. 06paTHTb BHHMaHHe Ha BbI60p HCXOHOH c}>OPMY nbI JlH p8C'tJ:eTa MOHO-
CTH. AHaJIH3HpYH YCJlOBHe 3B.l\aqH, He06xo,AHMO npe:>K,l\e Bcero YCT8HOBHTb, H,l\eT
273

( 2 U2 1
nH peqb 0 MOHOCTH, BbIeJl.HeMOH Ha yqacTKe nenH P = IU = I R = Ii' ' HJlH
o MOHOCTH. BbI,AenSleMOH HCTO'lHHKOM ( p = Ig' =  ) , HnH 0 MOHOCTH
R+r
( g'2 R )
BO BHeIlIHeH lIaCTH nenH HCTOIIHHKa P = IU = 2 .
(R + r)
2. IIpH peIlIeHHH 3aaq Ha TennOBoe eHcTBHe TOKa HCnOJlb30BaTh 3aKOH
J],moy JI.H -JIeHu;a.
3. B 3a,Aa'laX Ha cpaBHeHHe KOJlHqeCTB TenJlOTbI, BhIeJl.HeMOH B pa3HhIX
pe3HCTOpSX, He06xoHMO BhI.HCHHTh, qTO HS 3THX yqSCTKSX O,AHHSKOBO: CHJlS
TOKS HnH HSnp.HmeHHe, H B 3SBHCHMOCTH OT 3Toro Hcnonh30BaTb <t>opMyny
u 2
HJlH Q = - t .
R
4. EcnH no YCJlOBHIO 3S'tIH npOHCXOHT npeBpaIQeHHe 3J1eKTpH'tIeCKOH 3Hep-
rHH B MeXaHHqeCKYIO, TennOBYIO H XHMH'tIeCKYIO, 8 T8Kme B 3neKTpOMarHHT-
HYIO 3HeprHIO H3nyqeHH.H, TO He06xoHMO HCnOJlb30BaTb 3aKOH coxpaHeHH.H H
npeBpaeHH.H 3HeprHH:
Q = 1 2 Rt
IU=W Mex + W H3J1 + WXHM+Q.
IIpH peIlIeHHH 38,Aaq Ha 3neKTpOJlH3 H8O npHMeHHTh 38KOHbI <1>8pa,Ae.H.
nPIIMEPbI PEIIIEHIIH 3A)I.A 
1. llpoBonoK8 C ,AH8MeTpoM nOnepe'lHOrO Ce'leHHfl D, Y,AenbHbIM conpoTHs-
neHHeM p Ii nnOTHOCTblO d HMeeT M8CCY m. qeMY p8BHO conpoTHBneHHe npo-
BonoKH?
.II: 8 H 0: D, p, d. m.
H8HTH: R.
Pew e H He. K8K H3seCTHO, conpOTHBneHHe onpe,AenSleTCSI no <t>opMyne
pl nD 2
R = -. TIn01I.\8,Ab nOnepe'lHOrO Ce'leHHSI S = -. .II:nHHY npOBO,AHHK8 HSH-
S 4
,AeM. 3H8S1 ero M8CCY:
m = dV = dlS => l = .!!:.. .
dS
llO,lICT8BHB 3H8'leHH.H l H S B <t>opMyny Iln.H conpoTHBneHH.H. nonYIIHM
R _ m _ 16pm
- p dS 2 - dn 2 D 4 .
2. Y'l8CTOK uenH npeCT8BnSleT C060H TPH nspsnnenhHO COe,AHHeHHbIX pe-
3HCTOp8. ConpoTHBneHHSI ,ABYX pe3HCTopOB p8BHbI COOTBeTCTBeHHO 6 OM Ii
4 OM. qeMY p8BHO conpOTHBneHHe TpeTbero pe3HCTop8, ecnH 061I.\8S1 CHna TOK8
274

Rt
Ra
H8 yq8CTKe 9 A, a CHna TOKa, npOXO,ASlIl.\e.
ro no STOPOMY pe3HCTOpy, p8BHa 3 A?
CKonbKo 3neKTpOHOB npOH,AeT 'lepe3 Y'l8CTOK
3a 20 c?
.II.8HO: R 1 =6 OM, R 2 =4 OM, 1=9 A.
1 2 =3 A, t=20 C.
H8HTH: R3' n.
Pew e H He. H8pHcyeM cxeMY
KOH uenH (pHC. 9.21). MCKoMoe
3neKTpH'leC-
conpOTHBne-
U
HHe R3 HaH,AeM H3 38KOHa OM8: R3 = -
13
Cne.u.o-
npoxo-
I
1 2
R 2
Puc. 9.21
TIpH n8pannenbHOM COe,AHHeHHH pe3HCTopOB H8npSlmeHHe O,AHHaKOBoe.
B8TenbHO, U = 1 2 R 2 . CHn8 TOKa I Ha Y'l8CTKe paSH8 CYMMe CHn TOKOB.
,ASlIl.\HX no OT,AenbHblM pe3HCTop8M, I = II + 1 2 + 13' OTCIO,Aa
U 1 2 R 2
13 = I - II - 1 2 = 1- - - 1 2 = 1- - - 1 2 ,
R 1 R 1
Cne,AOB8TenbHO. R3 = 12R1 ; R = 3 OM.
(I - 1 2 )R 1 - 12
38 t C 'lepe3 Y'l8CTOK npOH,AeT 38pSlA q = It. MCKoMoe 'lHCnO 3neKTpOHOB
n = 9.. = It , r,Ae e= 1,6 .10- 19 Kn - 38pHA 3neKTpOHa, n= 1.10 21 .
e e
3. 3neKTpH'leCKHH KHnSlTHnbHHK paCC'lHT8H H8 H8npSlmeHHe 120 B npH
CHne TOK8 4 A. KaKoH AnHHbl H nnOIl.\8blO nOnepe'lHOrO Ce'leHHSI HymHo
B3S1Tb HHXpOMOBblH npOSO,AHHK )InSl H3rOTOBneHHSI H8rpeB8TenbHoro 3neMeHTa
A
KHn.HTHnbHHK8, eCJlH ,AOnycTHM8S1 nnOTHOCTb TOK8 10,2 , 8 y.u.enbHoe CO-
MM
npOTHBneHHe HHXpOM8 npH p860Te KHnSlTHnbHHK8 1,3 .10- 6 OM' M?
,lJ.8HO: U=120 B. 1=4 A, j=1,02.10 7
A
2'
M
p= 1,3.10- 6 OM' M.
H8HTH: l, S.
Pew e H He. .II.nHHY npOSO,A8 H nnOIl.\8,Ab ero Ce'leHHH MomHO HaHTH. HC-
nonb3YSl Q>opMynbl:
R = pl . I _ U .
S' - R '
I = J 'S => l = U .
. ,
PI
S = !..; S = 0,4 MM 2 .
j
l = 9 M.
4. 3neKTpH'leCK8S1 naMnO'IKa C Bonb<l>P8MoBoH HHTblO BKnlO'leH8 B D.enb
npH TeMnepaType 25°C, npH 3TOM BonbTMeTp nOK83blB8eT 10 MB, aMnep-
MeTp - 4,0 MA. B pa60'leM COCTOSlHHH H8npSlmeHHe Ha naMnO'IKe 120 B a
CHn8 TOK8 4,0 A. Onpe,AenHTe TeMnep8TYPY n8MnO'IKH B p860'leM COCTOSlHHH.
TeMnep8TYPHblH K03Q>Q>HD.HeHT conpOTHBneHH.H BonbQ>paM8 a = 0,0042 K- 1 .
.II.8HO: t 1 =25 °c, U 1 =1.0.10- 3 B, 1 1 =4.0.10- 3 A, U 2 =120 B,
a=4,2.10- 3 K- 1 , 1 2 =4,0 A.
HaHTH: t2'
275

P e ill e H " e. 3aBHCHMOCTb COnpOTHBneHHSI OT TeMnepaTypbI
R 1 = Ro (1 + at l )  Ro =  .
1 + at l
M3 3aKOHa OM8 R 1 = U 1 . CneoBaTenbHo, Ro = U I TeMnepaTYPY
II I I (1 + atd
t2 H8H,AeM H3 <l>opMynbI R 2 = Ro(1 + at 2 ). IIo,AcTaBHB BMeCTO R 2 = U 2 H Y'IH-
1 2
TbIBaSl H8H,AeHHOe nfl Ro BblpameHHe, nonY'lHM
U 2 I I (1 + at l ) - 1 2 U I
t 2 = . t2= 2,8. 10 3o C.
I 2U I a '
5. H8H,AIiTe TOK KopOTKoro 3aMbIK8HHSI B enH C HCTO'IHHKOM, S.D.C KOTO-
poro paBHa 1,3 B, ecnH npH BKnlO'leHHH BO BHelliHlO1O enb pe3HCTopa co-
npOTHBneHHeM 3,0 OM CHn8 TOK8 COCT8BnSleT 0,40 A.
.D.8HO: g:'=1,3 B R=3,0 OM, 1=0,40 A.
H a H T H: I K.3'
g
P e ill e H" e. MCKoMaSl CHn8 TOK8 KopoTKoro 3aMbIK8HHSl I K.3 = - .D.nSl Ha-
r
XOm,AeHHSI BHYTpeHHero conpOTHBneHHSI Bocnonb3yeMcSI 38KOHOM OM8:
g> g> - IR g> I
1=-. OTKY,A8 r = . Cne,AOBaTenbHo, I K . 3 = g> IR ; I K . 3 = 5,2 A.
RTr I -
6. JImpT M8CCOM 2,40 T nO,AHHM8eTCSI H8 BbICOTY 45.0 M B Te'leHHe 40,0 c.
C K8KOii MOHOCTblO p860TaeT sneKTpO,ABHr8Tenb, npHBO,ASIHH B ,ABHmeHHe
mt<l>T, ecnH KTIL( YCTpoiicTB8 60.0 %? Onpe,AenHTe cHny TOK8 B 06MoTKe 3neKTpo-
BHr8TenSl npH HanpSlmeHHH 380 B.
.D. a H 0: m = 2,40. 10 3 Kr. h = 45,0 M, t = 40,0 c. 11 = 60.0 % = 0,600.
U=380 B.
H8MTH: P, I.
P e ill e H" e. Kos<l><PHHeHT none3Horo ,AeHCTBHfl onpe,AenSleTCSl <POPMY nOH
T} = L\.c:x = Pt.tc:x . IIone3HaSl MOHOCTb P = L\.cx = mgh
W p MCX t t
38Tp8'leHH8S1 MOHOCTb P=IU.
Cne,AOBaTenbHo, T} = mgh = mgh  p = mgh . I = mgh . 1= 118 A' P= 45 0 KBT.
Pt I U t T} t' T} U t · "
7. OT HCTO'lHHKa TOKa C HanpSlmeHHeM 750 B He06xo,AHMO nepe,A8Tb no-
Tpe6HTenlO MOHOCTb 5.0 KBT. KaKoe H8H6onbIlIee conpoTHBneHHe MomeT
HMeTb nHHHfl nepe,Aa'lH, 'lT06bI nOTepH 3HeprHIi B HeM He npeBbIllianH 10 %
OT MOHOCTIi nOTpe6HTenSl?
.D.aHO: U=750 B. P I =5,0.10 3 BT, 11 =0,10.
H a H T H: R np '
P e ill e H" e. TIepe,A8B8eM8S1 OT HCTO'lHHK8 TOKa MOHOCTb P = IU. OHa 6y-
,AeT p8BHa CYMMe MOHOCTH, BbI,AenSleMOH H8 npOBO,Aax. Ii MOHOCTH, OIlIeA-
IlIeH ,AO nOTpe6HTenSl. T.e. P = p] + Pnp. Ho P np = 0, 1P I .
CJIe,AOBaTenbHo, P = 1,1P I = IU. OTKY,Aa 1= 1,1 .
U
276

C ,Apyroii CTOpOHbl, MOll.\HOCTb, TepSleMaSl H8 npOBOAaX,
2 0, 111 U 2
P np = 0,111 = I R np ' OTCIOA8 p = J2 = 12,111 ; R np =0,93 OM.
8 .[(Ba HCTO'lHHK8 TOK8, 3.[(C KOTOpbIX 1,9 B
H 1,1 B. BHYTpeHHlie conpOTHBneHliSl 0.80 OM H
B 0,10 OM COOTBeTCTBeHHO. n8p8nnenbHO COe,AHHeHbl
H 38MKHYTbl Ha pe3HCTOp conpOTHBneHHeM 10 OM
(pHC. 9.22). Onpe,AenHTe cnny TOKa BO BHelllHeii l\enH.
C ,l],aHo: Z'1=1,9 B. Z'2=1.1 B, r]=0,80 OM,
r2=0,10 OM, R=10 OM.
HaiiTH: 1.
P e III e H He. 3neKTpl1IeCl{afl lenb COAep)KHT ,ABa
D Y3n8 B TO'lK8X K 11 C H TPH HeH3BeCTHbIX TOK8
(pHC. 9.22).
TIpHMeHSISI nepBoe np8BHno KHpxro<pa ,AnSI Y3na
K, Haii,AeM 1 1 +/ 2 -1=0.
TIpliMeHSISI BTopoe npaBHno KHpxro4>a ,AnSI 3aMKHYToro KOHTypa ABDM KA
(06XO,AHM KOHTYP npoTHB xO,Aa 'lacoBoii CTpenKIi), nonY'lHM I]r 1 + I R = g> l'
AHanOrH'lHO, npHMeHSISI BTopoe npaBHno Kl1pXrOQ>8 ,AnSI 38MKHYToro KOH-
Typa ABCKA (06XOAHM KOHTYP npOTHB XO,A8 'lacoBoii CTpenKH), no.nY'lHM
I]r] - 1 2 r 2 = g> 1 - g> 2' PelllaSl COBMeCTHO CI1CTeMY nonY'leHHbIX ypaBHeHHii, Haii-
A
II WI
+ -
K
1 2 f?'2
+ -
I
R
M
Puc. 9.22
.aeM cHny TOK8:
1= g'1 r 2 +g'2 r l ; 1=0.12 A.
R(rl + r2) + r 1 r 2
9. B ceTb H8npSlmeHHeM 220 B nocnellOB8TenbHO BKnlOQeHbl lIBe n8MnOQ-
KI1 MOll.\HOCTblO 60,0 BT H 250 BT, P8CC'lHT8HHble H8 HanpSlmeHHe 110 B
K8m,A8S1. K8Koe KOnHQeCTBO TennOTbI BbIAenl1TCSI 3a 30,0 MHH ropeHHSI B K8)K-
Aoii H3 n8Mn?
,Z:(aHo: U=220 B. Pl=60,0 BT, P 2 =250 BT, U o =110 B, t=1800 c.
H a ii T H: Ql. Q2'
P e 111 e H He. no 3aKOHY ,Z:(moynSl-JIeHu8 Ql = If R 1 t H Q2 = It. 3HaSl
MOll.\HOCTH n8Mn Ii HanpSlmeHHSI,
u 2
conpOTHBneHlie naMn: R] =-.J!..
'
KOHOM OM8 AnSi Y'laCTKa l\enH:
Ha KOTopbIe OHH P8CC'lHTaHbl. MO)KHO H8iiTH
R;. = U . CHny TOK8 H8ii,AeM, 1I0nb3YSlCb 38-

Cne,AOB8TenbHO.
AHanOrHQHO,
1= U
+
U
U 2 U 2
-.J!.. + -.J!..
11 
UP2
U (11 + )
U 2 R 2 R 2 U 2 t U 2 R R 2 t
Q - 1 2 0 - 1 2 . Q] = 250 K,Z:(m.
1 - U (fl + P 2 )2  - U (11 + )2 ·
Q _ U 2 P22t
2 - U (PI + P 2 )2
Q2= 70,0 KAm.
277

10. SneMeHT 3aMbIKaeTCSI pe3HcTop8MH: O,AHH p83 - conpoTHBneHHeM
4 OM, ,ApyroH pa3 - conpOTHBneHHeM 9 OM. B TOM H ,ApyroM CnY'l8e KOnH-
'leCTBO TennOTbI, Bbl,AenSlIOll\eHCSI B pe3HCTOpaX 3a O,AHO H TO me BpeMSI, OK8-
3blS8eTCSI O,l],HH8KOSbIM. qeMY p8BHO BHYTpeHHee conpOTHBneHHe 3neMeHT8?
,naHO: Rl=4 OM, R 2 =9 OM, Ql=Q2, tl=t 2 .
HaHTH: r.
PeweHHe. CornaCHO 3aKOHY .nmoynSl-JIeH8, Ql = I;R 1 t H Q2 = 1:t
,nnSl H8XOm,AeHHSI CHnbI TOK8 npHMeHHM 38KOH OM8 ,AnSI nonHOH u.enH:
- g .
II - -,
 +r
g
1 2 = -.
+r
Tor ,Aa
 = g2 R1t ;
( + r )2
(1)
Q _ g2 R2t
2 -
(R2 + r )2
TIpHp8BHHB8S1 (1) H (2). nonY'lHM
(2)
g2 Rl t
( + r )2
g2t
2 .
(+ r)
H3BneqeM H3 06eHX 'laCTeH KB8,ApaTHbIH KopeHb H peWHM YP8BHeHHe OT-
HOCHTenbHO r:
JR;R; (JR; - JR;) fDD
r = .jii; - ,JR; = V R I R 2; r= 6 OM.
278

10. 3JIEKTPHQECKH:A: TOK B PA3JIHQHbIX CPEAX
3JIeKTpOHHaJi npOBOAHMOCTL MeTaJIJIOB. CBepxnpoBoAHMOCTL.
3JIeKTpH'leCKHH TOK B paCTBopax H paCnJIaBaX 3JIeKTpOJlHTOB. 3aKOH
3JIeKTpOJIH3a. OnpeAeJIeHHe 3apJlAa 3JIeKTpOHa. fipHMeHeHHe 3JIeKTpOJlH3a B
TeXHHKe.
3JIeKTpH'leCKHH TOK B ra3ax. HecaMOCTOJlTeJILHLIH H caMOCTOJlTeJILHLIH
pa3pLJ B raaax. BLI caMOCTOJlTeJILHOrO pa3pa. TeXHH'leCKOe HCnOJIL30-
BaHHe ra30Boro pa3pa. nOHJlTHe 0 nJIa3Me.
TOK B BaKYYMe. 3JIeKTpOHHaJi 3MHCCHJI. AByx3JIeKTpOAHaJi JIaMna. BOJILT-
aMnepHaa xapaKTepHCTHKa AHOAa. 3JIeKTpOHHLle nyqKH H HX CBOHCTBa. 3JIeK-
TpoHHo-nyqeBaJi Tpy6Ka.
3JIeKTpHqeCKHH TOK B nOJIynpoBoAHHKax. 3J1eKTpH'leCKaJl npOBOAHMOCTL
nonynpoBoAHHKOB H ee 3aBHCHMOCTL OT TeMnepaTypLI H OCBel1.{.eHHOCTH. C06-
CTBeHHaJi H npHMeCHaH npOBOlJ;HMOCTH nOJlynpoBollHHKOB. TepMo- H 4>oTope-
3HCTOpLI. 3JIeKTpOHHO-ALlpO'lHbJH nepeXOA. nonynpoBoAHHKOBLIH AHOA. TpaH-
3HCTOp. npHMeHeHHe nonynpOBOAHHKOBLIX npH6opoB.
10.1. 3JIeKTpOHHaSl npOBO,lJ;HMOCTL MeTaJIJIOB. CBepXnpOBO,lJ;HMOCTL
3neKTpOHH8H npOBOHMOCTb MeTanJlOB 6bIna BnepBbIe 3KcnepHMeHT8J1bHO
OKa3aHa HeMeu.KHM ij>H3HKOM 3. PHKKe B 1901 r. Qepe3 TPH nnOTHO npH:>Ka-
TbIX Pyr K ,Apyry OTllOJIHpOBaHHbIX II.HJIHHpa - MeHbIH, 8JIIOMHHHeBbIH H
CHOBa MelIHbIH - lIJlHTeJlbHOe BpeM.H (B TeqeHHe rOlIa) nponYCKaJIH 3JIeKTpHqeC-
KHH TOK. 06WHH 3ap.HD;, npowelIwHH 3a 3TO BpeM.H, 6bIJI paBeH 3.5.10 6 KJI.
ITOCKOJlbKY MaCCbI aTOMOB Me,AH H aJlIOMHHH.H cyecTBeHHo OTJlHqalOTCH APyr
OT pyra, TO MaCCbI II.HJlHHpOB OJl:>KHbI 6bIJlH 6bl 3aMeTHO H3MeHHTbCSI,
eCJlH 6bI HOCHTeJl.HMH 3ap.Ha 6bIJlH HOHbI. Pe3YJlbT8TbI OnblTOB nOKa3aJlH,
'lTO MaCCa KamOrO H3 II.HJlHHpOB OCTaJIaCb HeH3MeHHOH. B COnpHKaCalOHX-
CH nOBepXHOCT.HX 6blJlH 06HapymeHbI JlHWb He3HaqHTeJlbHbIe CJlebI B3aHMHO-
ro npOHHKHOBeHHH MeTaJIJlOB, KOTopbIe He npeBbIWaJlH pe3YJlbTaTOB 06bIqHOH
Hcpij>Y3HH aTOMOB B TBepbIX TeJlax. CJleOBaTeJlbHO, CB06oHbIMH HOCHTen.H-
MH 3apH)J;a B MeTaJlJlaX HBJlHIOTCH He HOHbI, a TaKHe qaCTHII.bI, KOTopbIe
OHHaKOBbI H B MeH, H B anIOMHHHH. T8KHMH qaCTHn;aMH MOrJlH 6bITb
TOJlbKO 3J1eKTpOHbI.
Puc. 10.1
I1p.HMOe H y6e)J;HTeJlbHoe )J;oKa3aTenbCTBO
CnpaBe,AJlHBOCTH 3Toro npenOJlO:>KeHHH 6bIno
nOJlyqeHo B onbITax. nOCTaBneHHbIX B 1913 r.
JI. H. M8HeJlbWT8MOM H H.,n. IT8naneKcH H
B 1916 r. T. CTlOapToM H P. TOJlMeHoM.
Ha KaTYWKY HaMaTbIBalOT npOBOJlOKY,
KOHn;bI KOTOpOH npHnaHB810T K BYM MeTaJl-
JlHqeCKHM HCKaM, H30nHpOBaHHblM pyr OT
pyra (pHC. 10.1). K KOHII.aM ,AHCKOB C no-
MOblO CKOJlb3.HHX KOHTaKTOB npHCOeHH.H-
lOT raJIbBaHOMeTp.
279

KaTymKY npHBO,ASIT B 6bICTpOe BpaeHHe, a 3aTeM pe3KO OCTaHaBnHBaIOT.
IIocJle pe3Kmi OCTaHOBKH KaTymKH CB06o,AHbIe 3apSI>KeHHbIe qaCTHbI 6Y,AYT He-
KOTopoe BpeM.H ,ABHraTbC.H B,AOJlb npOBO,AHHKa no HHepHH, H, CJle,AOBaTeJlbHO,
B KaTymKe B03HHKHeT SJleKTpHqeCKH:M TOK. TOK 6Y,AeT cyecTBoBaTb KOPOT-
Koe BpeMSI, TaK KaK H3-3a conpOTHBneHH.H npOBO,AHHKa 3apSImeHHbIe 'tIaCTHbI
TOpM03.HTC.H H ynop.H,AOqeHHOe ,ABHmeHHe tJaCTH npeKpaaeTC.H.
HanpaBneHHe TOKa rOBopHT 0 TOM, qTO OH C03,AaeTC.H ,ABHmeHHeM OT-
pHaTenDHO 3ap.HmeHHbIX qaCTH. IIepeHocHMbIH npH STOM 3apSI,A nponop-
HOHaJleH OTHomeHHIO 3ap.H,Aa 'tlaCTH, C03,AaJOHX TOK, K HX Macce, T.e.
6.q - qo . llOSTOMY, H3MepSI.H 3apSl,A, npOXO,A.HHH tJepe3 ranbBaHOMeTp 3a
m
Bce BpeMSI cyecTBOBaHH.H TOKa B enH, y,AaJlOCb onpe,AenHTb OTHomeHHe
q 
-.!l. OHO OKa3aJlOCb paBHbIM 1,8.10 11 -. 3Ta BeJlHtJHHR COBna,AaeT C
m Kr
OTHomeHHeM 3ap.H,Aa sneKTpOHa K ero Macce, HaHIIeHHbIM paHee H3 ,ApyrHx
onbITOB.
TaKHM 06pa30M, SJleKTpHtJecKHH TOK B MeTanJlaX C03,AaeTC.H ,ABHmeHHeM
OTpHaTenbHO 3ap.HmeHHbIX qaCTH SJleKTpOHOB. CornacHo KJlaCCHqeCKOH sneK-
TpOHHOH TeopHH npOBO,AHMOCTH MeTannOB (ll. ,l];pY,Ae, 1990 r., X. JIopeH,
1904 r.), MeTaJlJlHtJecKHH npOBO,AHHK MomHO paCCMaTpHBaTb KaK <pH3HqecKYIO
CHCTeMY COBOKynHOCTH ,ABYX nO,ACHCTeM: 1) CB060,AHbIX SJleKTpOHOB C KOHeH-
TpaHeH _10 28 M- 3 H 2) nOnO>KHTenbHO 3apSImeHHbIX HOHOB, Kone6nlOHxcSI
OKOJlO nonOmeHH.H paBHOBeCH.H.
IIo.HBneHHe CB060,AHbIX sneKTpOHOB B KpHCTaJlJle MomHO 06'b.HCHHTb cne,Ay-
IOHM 06pa30M.
llPH 06'be,AHHeHHH aTOMOB B MeTaJlnHqeCKHH KpHC-
TaJln cJla6ee Bcero CB.H3aHHbIe C .H,ApOM aTOMa BHemHHe
SJleKTpOHbI OTpbIBaIOTC.H OT aTOMOB (pHC. 10.2). llo-
STOMY B Y3nax KpHCTanJlHqeCKOH pemeTKH MeTanna pac-
nOJlaralOTCSI nonomHTenbHbIe HOHbI, a B npOCTpaHCTBe
Mem,Ay HHMH ,ABHmYTC.H SJleKTpOHbI, He CB.H3aHHbIe C .H,A-
paMH CBOHX aTOMOB. 3TH sneKTpOHbI Ha3bIBaIOTC.H CBO-
60JJ;HLIMH HnH SJleKTpOHaMH npOBOJJ;HMOCTH. OHH COBep-
malOT XaOTHqeCKOe ,ABHmeHHe, nO,A06HOe ,ABHmeHHIO MO-
neKYJl ra3a. TIOSTOMY cOBoKynHocTb CB060,AHbIX sneKTpOHOB B MeTaJlJlaX Ha-
3bIBalOT SJleKTpOHHLIM ra30M.
ECJ!H K npOBO,AHHKY npHnomeHO BHemHee SJleKTpH'tleCKOe nOJle, TO Ha
6eCnOpSI,AOtJHoe xaOTHtJeCKoe ,ABHmeHHe CB060,AHbIX sneKTpOHOB HaKna,AbIBa-
eTC.H HanpaBJleHHOe ,ABHmeHHe no,A ,AeHCTBHeM CHJI sneKTpHtJeCKoro nOJl.H,
'lTO H nopO>K,AaeT sneKTpHtJeCKHH TOK. CKOPOCTb ,ABHmeHHSI caMHX sneKT-
pOHOB B npOBO,AHHKe -- HeCKonbKO ,AOneH MHnnHMeTpa B ceKYH,Ay, O,AHaKO
B03HHKalOee B npOBO,AHHKe SJleKTpHqeCKOe nOJle pacnpOCTpaH.HeTC.H no
BceH ,AJlHHe npOBO,AHHKa CO CKOpOCTblO, 6nH3KOH K CKOpOCTH CBeTa B Ba-
M
KYYMe (3.10 8 -).
C
TaK KaK sJleKTpHqeCKHH TOK B MeTannax 06pa3YIOT cB060,AHbIe sneKTpo-
HbI, TO npOBO.lI.HMOCTb MeTaJlnHqeCKHX npOBO,AHHKOB Ha3bIBaeTCSI SJleKTpOHHOH
npOBO,lJ,HMOCTLIO.
0G)0G)0G)0
G)0 m °(D°G)
0(D0G)0G)0
° 0 °
Puc. 10.2
280

3neKTpOHbI nO,A BJIH.HHHeM nOCTO.HHHOH CHJlbl. p;eHCTBYIOIUeH co CTOpOHbI
3JIeKTpHqeCKoro nOJl.H, npH06peT81OT Onpep;eJleHHYIO CKOpOCTb ynOp.Hp;OqeHHOrO
,lJ;BHmeHH.H (ee Ha3bIB81OT ,lJ;peHcpOBOH). 3Ta CKOpOCTb He YBeJIHqHB8eTC.H B p;aJIb-
HeHlIIeM co BpeMeHeM, T8K K8K npH CTOJlKHOBeHHH C HOHaMH KpHCTaJlJlHqeC-
KOH pemeTKH 3J1eKTpOHbl nepe,A8lOT KHHeTHqeCKYIO 3HeprHIO, npH06peTeHHYIO B
3JIeKTpHqeCKOM nOJle, KpHCTaJIJlHqeCKOH pelIIeTKe. B nepBOM npH6J1HmeHHH Mom-
HO CqHT8Tb. qTO H8 ,lJ;JlHHe CB060,lJ;HOrO np06er8 A (3TO paCCTO.HHHe, !COTopoe
3JIeKTpOH npOXO,lJ;HT Mem,lJ;Y ,lJ;BYM.H nOCJle,lJ;OBaTeJlbHbIMH CTOJIKHOBeHH.HMH C HOHa-
eE
MH) 3J1eKTpOH ,lJ;BHmeTC.H C YCKopeHHeM a = - Hero ,lJ;peHcpOBa.H CKOpOCTb
m
eEl
nHHeHHo B03pacTaeT co BpeMeHeM: v = at = -. B MOMeHT CTOJlKHOBeHH.H 3JIeK-
m
TpOH nepe,AaeT KHHeTHqeCKYIO 3HeprHIO KpHCTaJlJlHqeCKOH pelIIeTKe. IIoToM OH
onHTb YCKOp.HeTC.H. H npouecc nOBTOp.HeTC.H. B pe3YJlbTaTe Cpe,lJ;H.H.H CKOpOCTb yno-
p.H,lJ;OqeHHOrO ,lJ;BH>KeHH.H 3J1eKTpOHoB nponOpUHOHaJIbHa Hanp.H>KeHHOCTH 3JIeKTpH-
qeCKOrO nOJl.H B npOBO,AHHKe (v) - E Ii, CJIe,lJ;OBaTenbHO, pa3HocTH nOTeH[(Ha-
U
JIOB H8 KOHuax npOBO,lJ;HHKa, TaK K8K E = -, r)J;e l - p;nHH8 npOBO,lJ;HHK8.
1
ll3BecTHo, qTO CHJla TOKa B npOBO,lJ;HHKe nponopUHOHaJlbHa CKOpOCTH yno-
p.H,lJ;OqeHHOrO p;BHmeHH.H qaCTHu (CM.  9.1): I = en < v > S, a 3HaqHT, corJlac-
HO npe,lJ;bI,lJ;yeMY, CliJI8 TOK8 npOnOp[(HOHaJlbH8 p83HOCTH nOTeH[(HaJIOB H8 KOH-
[(8X npOBO,lJ;HHKa: I -U. B 3TOM COCTOHT KaqeCTBeHHoe 06'bHCHeHHe 38KOH8 OMa
Ha OCHOEe KJI8CCHqeCKOH 3J1eKTpOHHOH TeopHH npOBO,lJ;HMOCTH MeTaJIJlOB.
Op;HaKo B paMKax 3TOH TeopHH B03HHKJlH TPY,lJ;HOCTH. ll3 TeopHH CJle,nOB8-
JlO, qTO y,neJlbHoe conpOTHBJleHHe ,nOJlmHO 6bITb nponOpUHOHaJIbHO KOpHIO KB8JI-
paTHOMY H3 TeMnep8TypbI (p -.JT), Memp;y TeM, COrJl8CHO onbITY, p - T. Kpo-
Me Toro, TenJlOeMKOCTb MeTaJlJlOB, COrJlaCHO 3TOH TeopHn, )J;OJImHa 6bITb 3H8-
qHTeJlbHO 60JlblIle TenJIOeMKOCTH O,lJ;HOaTOMHbIX KpHCTaJlJlOB. B ,AeHCTBHTeJIbHO-
CTH TenJlOeMKOCTb MeTaJlJIOB MaJIO OT JIHqaeTC.H OT TenJlOeMKOCTH HeMeT8JInn-
qeCKHX KpHCTaJlJlOB. 3TH TPY,lJ;HOCTH 6bIJlH npeO,lJ;OJleHbI TOJlbKO B KB8HTOBOH
TeopHH.
B 1911 r. rOJlJl8H,lJ;CKHH CPH3HK r. KaMepJlHHr-OHHec, H3yqa.H H3MeHeHHe
3JIeKTpHQeCKOrO conpOTHBJleHHH PTYTH npn HH3KHX TeMnepaTypax, 06H8PY>KHJI,
'lTO npH TeMnep8Type OKOJlO 4 K (T.e. npH -269°C) yp;enbHoe conpOTHBJleHHe
CK8QKOM YMeHb1II8eTC.H (pHC. 10.3) npaKTHqeCKH ,lJ;0 HYJl8. 3TO HBJleHHe 06P8-
I.QeHHH 3J1eKTpHQeCKOrO conpOTHBneHHSI B HyJlb r. K8MepnHHr-OHHec Ha3B8J1
CBepxnpOBOJJ.HMOCTLIO.
B .n;8JIbHeHlIIeM 6bIJlO Bbl.HCHeHO. qTO 60Jlee 25 XH-
MHQeCKHX 3neMeHTOB - MeTaJlJlOB npH OqeHb HH3KHX
TeMnep8Typax CTaHOB.HTC.H CBepXnpOBO,lJ;HHK8MH. ¥
K8m,lJ;OrO H3 HHX CBO.H KpHTHqeCKa.H TeMnepaTypa ne-
peXO,A8 B COCTOHHHe C HYJIeBbIM conpOTHBneHHeM. C8-
Moe HH3Koe 3H8qeHHe ee y BOJIbcppaMa - 0,012 K,
caMoe BbICOKoe y HH06H.H - 9 K.
T. K CBepXnpOBO,lJ;HMOCTb Ha6J11Op;8eTC.H He TOJlbKO y qH-
CTblX MeTaJlJlOB, HO H Y MHorHX XHMHqeCKHX COe,lJ;H-
R
o 4,2
Puc. 10.3
281

HeHHH H CnJlaBOB. IIpH 3TOM CaMIi 3J1eMeHTbI, BXO,AHHe B COCTaB CBepXnpO-
BO,AHerO COe,AHHeHH.H, MOrYT H He HBJlHTbCH CBepXnpOBO,AHHKaMH. HanpHMep,
NiBi, AU2Bi, PdTe, PtSb H ,ApyrHe.
BeecTBa B CBepxnpoBO,A.HeM COCTO.HHHH 06J1a,AaIOT Heo6blt{HbIMH CBOHCTBaMH:
1) 3J1eKTpHqeCKHH TOK B CBepXnpOBO,AHHKe MomeT cyecTBoBaTb ,AJlHTeJlb-
Hoe BpeM.H 6e3 HCTOqHHKa TOKa;
2) BHYTPH BeeCTBa B CBepXnpOBO,AHeM COCTOHHHH HeJlb3.H C03,AaTb Mar-
HHTHoe nOJle:
3) MarHHTHoe nOJle pa3pYllIaeT COCTO.HHHe CBepxnpOBo,nHMOCTH. CBepxnpoBo-
,AHMOCTb - .HBJleHHe, 06'bHCH.HeMOe C TOqKH 3peHHH KBaHToBoH TeopHH. ,Z:{ocTa-
TOqHO CJlOmHOe ero onHcaHHe BbIXO,AHT 3a paMKH llIKonbHoro Kypca <}>H3HKH.
llIHPOKOMY npHMeHeHHIO CBepXnpOBO,AHMOCTH ,AO He,AaBHerO BpeMeHH npen.HT-
CTBOBaJlH TPY,AHOCTH, CBH38HHble C He06xo,AIIMOCTblO OXnam,AeHHH ,AO CBepXHH3-
KHX TeMnepaTYP, ,An.H qero HCnOJlb30BaJlCH mH,AKHH reJlHH. TeM He MeHee, He-
CMOTpS1 Ha CJlOmHOCTb 060pY,AOBaHHH, ,ne<}>HnHTHOCTb H ,noporOBH3HY reJlH.H, C
60-x rO,AOB XX BeKa C03,AaIOTC.H CBepXnpOBO,A.HHe MarHHTbI 6e3 TenJlOBbIX
nOTepb B HX 06MOTKax, qTO C,AenaJlO npaKTHIeCKH B03MomHbIM nonyqeHHe
CHnbHbIX MarHHTHbIX nOJleH B cpaBHHTeJlbHO 60nbllIHX 06'beMax. llMeHHo Ta-
KHe MarHHTbI Tpe6YIOTC.H ,nn.H C03,naHH.H YCTaHOBOK ynpaBJI.HeMOrO TepMO.H,Aep-
Horo CHHTe3a C MarHHTHbIM y,nep>KaHHeM nna3MbI, ,nnH MOHbIX YCKopHTeneH
3apH>KeHHbIX qaCTHn. CBepxnpOBo,nHHKH HcnOJlb3YIOTC.H B pa3J1HqHbIX H3MepH-
TenbHbIX npH60pax, npem,ne Bcero B npH60pax ,nn.H H3MepeHH.H OqeHb CJla6bIX
MarHHTHbIX nOJleH C BbICOqaHllIeH TOqHOCTblO.
B HaCTOHee BpeM.H B JlHHH.HX 3neKTpOnepe,AaqH Ha npeo,noneHHe conpo-
THBneHH.H npOBo,nOB YXO,nHT 10-15 % 3HeprHH. CBepxnpoBo,n.HHe JlHHHH HnH
XOT.H 6bI BBO,AbI B KpynHbIe ropO,Aa npHHeCYT rpOMa,AHYIO 3KOHOMHIO. ,Z:{Pyra.H
06J1aCTb npHMeHeHH.H CBepXnpOBO,AHMOCTH - TpaHcnopT.
Ha OCHOBe CBepXnpOBO,A.HHX nneHOK C03,AaH p.H,A 6bIcTpo,neHcTBYIOHX no-
rHqeCKHX H 3anOMHHalOHX 3neMeHTOB ,AJI.H CqeTHO- pellIalOHX YCTPOHCTB.
IIpH KOCMHqeCKHX HCCne,AOBaHH.HX nepcneKTHBHO HCnOJlb30BaHHe CBepxnpOBO-
,AHHX coneHOH,nOB ,AJI.H pa,AHan;HOHHOii 3aHTbI KOCMOHaBTOB, CTbIKOBKH KO-
pa6neH, HX TopMomeHHSI H opHeHTanHH, ,AJI.H nJla3MeHHblX paKeTHbIX ,ABHra-
TeneH.
B HaCTOHee BpeM.H C03,naHbI KepaMHqeCKHe MaTepHaJlbI, 06J1a,nalOHe
CBepXnpOBO,AHMOCTblO npH 60Jlee BbICOKOii TeMnepaType - CBbIllIe 100 K, TO
eCTb npH TeMnepaType BbIllIe TeMnepaTypbJ KHneHH.H a30Ta. B03MomHOCTb OX-
nam,naTb CBepxnpOBo,nHHKH mH,AKHM a30TOM, KOTOpbIH HMeeT Ha nopHOK 60-
nee BbICOKYIO TennOTY napo06pa30BaHH.H, cyecTBeHHO ynpoaeT H y,nellIeBJI.H-
eT Bce KpHoreHHoe o60pY,AOBaHHe, 06eaeT orpoMHbIH 3KOHOMHqeCKHH 3<}><}>eKT.
10.2. 3JIeKTpH'IeCKHH TOK B paCTBopax
H paCnJIaBaX 3JIeKTpOJIHTOB
KaK H TBep,AbIe TeJla, mH,AKOCTH MorYT 6bITb ,AH3J1eKTpHKaMH H npOBO,AHH-
KaMH. )1HcTHnnHpOBaHHa.H BO,Aa, H8npHMep, - ,AH3J1eKTpHK, a He60nbllIoe KO-
nHqeCTBO COJlH, 1I06aBJleHHOH B lIHCTHJlJlHpOBaHHYIO BO,ny, ,neJlaeT ee npOBO,n-
HHKOM 3neKTpHqeCKOrO TOKa - 3J1eKTpOJlHTOM.
282

3J1eKTpOJlHTaMH Ha3bIBaIOTC.H paCTBOpbI H paCnJlaBbI COJleH, OCHOBaHHH H
KHCJlOT, .HBn.HIOHeC.H npOBo.n;HHKaMH 3J1eKTpHqeCKOrO TOKa.
3ap.HmeHHbIe qaCTHbI, 06eCneqHBalOIl1;He CYIl1;eCTBOBaHHe 3J1eKTpHqeCKOrO TOKa
B 3J1eKTpOJlHTaX, 06pS3YIOTC.H B pe3YJlbTSTe 3J1eKTpOJlHTH'IeCKOH 1];HCCOHa-
HH - pacnalla MOJleKY JI paCTBOp.HeMOrO BemeCTBS Ha HOHbI nOlI lIeHCTBHeM
MOJleKYJI paCTBOpHTeJl.H. TaK. HanpHMep. npH paCTBopeHHH B BOlle COJIH CYJlb-
«l>aTs Me.n;H MOJleKYJla Cu80 4 )J;HCCoHHpyeT HS )J;BS HOHS: Cu 2 i- H 80l-:
Cu80 4 p Cu 2 + +80/-.
TIpH aTOM nOJlOmHTeJlbHO 3ap.HmeHHbIMH OKa3bIBaIOTC.H HOHbI MeTaJIJlOB, a OTPH-
aTenbHO 3ap.HmeHHbIMH - JlH60 KHCJlOTHbIe ocTaTKH, JlH60 r1:l)J;pOKC1:IJIbHa.H rpynna.
TIpH OTCYTCTBHH BHeillHero 3J1eKTpHQeCKOrO nOJl.H HOHbI BMeCTe C Hepac-
naBIllHMHC.H MOJleKYJlaMH Haxo.n;.HTcSJ B TenJlOBOM XaOTHqeCKOM .n;BHmeHHH.
HOHbI npoTHBOnOJlOmHOrO 3HaKa npH BCTpeqe BHOBb MorYT 06pa30BaTb HeHT-
panbHYIO MOJleKYJlY. 3TOT npoecc Ha3bIBaeTC.H peKOM6HHal(HeH HOHOB (npo-
u.ecc, 06PSTHbIH )J;HcCOu.HanHH). Me>K)J;Y nponeccsMH peKOM6HHau.HH H )J;HCCOnH-
au.HH YCTaHaBJlHBaeTC.H lIHHaMHqeCKOe (nOlIBHmHoe) pSBHoBecHe, npH KOTOpOM
onpe)J;eJleHHa.H qaCTb MOJleKYJI paCTBOp.HeMOrO BeecTBa HaXOlIHTC.H B lIHCCOUH-
HpoBaHHoM COCTO.HHHH. CTeneHb .n;HCCOHSHH ex onpe.n;eJl.HeTC.H OTHOIlleHHeM
QHCJla pacnaBIllHXC.H Ha HOHbI MOJleKYJI K HX 06Il1;eMY qHCJlY.
TIpH nOMeeHHIf B 3J1eKTpOJlHT )J;BYX 3J1eKTpo)J;oB, npHcoe)J;HHeHHbIx K pa3-
nHqHbIM nOJlIOCaM HCTOqHHKa TOKa (pHC. 10.4), Ha XaOTHqeCKOe .n;BHmeHHe
HOHOB HaKJIbIBaeTC.H HX ynop.H)J;OqeHHOe )J;BHmeHHe K co-
OTBeTCTBYIOIl1;HM 3neKTpo.n;aM: nOJlOmHTeJlbHbIX HOHOB K
KaT01];y (OTpHSTeJlbHO 3Sp.HmeHHbIH 3J1eKTpo)J;) H OTpHa-
TeJlbHbIX HOHOB - K aHo1J:Y (nOJlOmHTeJlbHO 3ap.H>KeHHbIH
3J1eKTpO,A), H B 3J1eKTpOJlHTe B03HHKHeT 3J1eKTpHqeCKHH
TOK, npe,ACTSBJlSlIOIl1;HH C060H .n;Ba nOTOKS HOHOB 060HX
3HaKOB B npOTHBOnOJlOmHbIX HanpaBJleHH.HX. TaKHM 06pa-
30M, xapaKTep npOB01];HMOCTH 3J1eKTpOJlHTOB - HOIDILIH.
):(JI.H 3J1eKTpOJlHTOB TaKme cnpaBe.n;JlHB 3aKOH OMa
U
6e3 YQeTa nOJl.HpH3aHH 3J1eKTpOnHTOB: [= - H 3SKOH
R
K.:!1 -
Puc. 10.4
):(moy JI.H -JIeHa: Q = [2 Rt.
10.3. 3aKoH 3JIeKTpOJIH3a.
OnpeeJIeHHe 3apa 3JIeKTpOHa
3aKOH apB.1Ie.H )J;JI.H 3J1eKTpOnH3a CB.H3bIBaeT MSCCY BbIlIeJl.HIOerOC.H Beme-
CTBa C npOIlle.n;IllHM qepe3 3J1eKTpOJlHT 3J1eKTpHqeCKHM 3ap.H,AOM. TIpH npOXom,Ae-
HHH 3J1eKTpHqeCKOrO TOKS qepe3 3J1eKTpOnHT npOHcxo.n;HT BbI)J;eneHHe HS 3neKT-
po.n;ax COCTaBHbIX qaCTeH 3J1eKTpOJlHTa. STO .HBJleHHe Ha3bIBaeTC.H 3J1eKTpOJlH30M
(OT rpeqeCKOrO "JlHO" - pa3.n;eJl.HIO). SJleKTpOJlH3 CB.H3aH C npoeccaMH 06MeHS
3ap.H.n;aMH Mem.n;y HOHaMI1 H 3J1eKTpO)J;aMH. Hs aHO)J;e OTpHaTeJlbHO 3ap.HmeH-
HbIe HOHb] (aHHoHbI) OT.n;aIOT CBOH JlHIllHHe 3J1eKTpOHbI (OKHCJlHTeJlbHa.H peSK-
H.H), S Ha KaTO)J;e nOJlOmHTeJlbHbIe HOHbI (KSTHOHbI) nOJlyqslOT He)J;OCTalOHe
3J1eKTpOHbI (BOCCTsHOBHTeJlbHaSJ peaKH.H). OT.n;SB CBOH H36bITOqHbIe 3Sp.H.n;bI,
HOHbI npeBpaaIOTC.H B HeHTpaJlbHbIe aTOMbI. 3TH aTOMbI (HJlH 06pS30BSHHbIe
283

H3 HHX MOJleKynbI BbI,n;eJl.HIOTC.H H8 3neKTpO,n;aX, nH60 BCTynalOT B peaKIJ;HIO C
3neKTpO,n;aMH HnH paCTBOpHTeJleM. XHMH'tIeCKHe peaKIJ;HH, B KOTOpbIe BCTynalOT
HeHTpaJIH30BaBillHeC.H HOHbI, Ha3bIBaIOTC.H BTOpH'tIHbIMH. IIpo,n;YKTbJ BTOPHqHblX pe-
aKIJ;HH BbI,n;enSIIOTC.H Ha 3J1eKTpO,n;aX HJlH nepeXO,n;.HT B paCTBOp.
PaCCMOTpHM HeCKOJlbKO npHMepOB, BbIpamalOrqHx CYTb 3J1eKTpOJlH3a.
npHMep 1. IIycTb B 3neKTpOJlHTH'tIeCKOH BaHHe HaXO,n;HTC.H BO,n;HbIH paCTBOp
COJISIHOH KHCJlOTbI. MOJleKYJlbI COn.HHOH KHCJlOTbI npH paCTBOpeHHlt B BO,n;e no-
qTH nOnHOCTblO ,n;HCCOIJ;HHPYIOT Ha HOHbI BO,n;OpO,Aa II 1I0HbI xJlopa:
HCI p H-r+Cr.
IIpH npOXOmlIeHHH 3J1eKTpHqeCKOrO TOKa qepe3 paCTBOp Ha 3J1eKTpO,n;aX npo-
HCXOHT HeHTpaJlH3aUH.H HOHOB.
IIo,n;oH,n;.H K aHO,n;y, cr OT,n;aeT H36bITO'tlHbIH 3neKTpOH, npeBparqaeTC.H B HeHT-
paJIbHbIH aTOM CI, aTOMbI CI cpa3Y me 06'be,n;HHSIIOTC.H nonapHO B MOJleKynbI CI 2 :
CI-+ cr- 2e CI 2 ,
H Ha aHo,n;e BbI,n;en.HeTC.H MoneKynSIpHbIH XJlOp.
ATOMbI Bo,n;opOlIa, HeHTpanH30BaBillHecSI y KaTo,n;a, 06'be,n;HH.HIOTC.H nonapHO
B MoneKynbI H 2 :
H++H++2e H 2o
H Ha KaTo,n;e BbI,n;en.HeTCSI MOJleKYJlSIpHbIi1 BO,n;OpO,n;.
TaKHM 06pa30M, B pe3YJlbTaTe npoxom,n;eHH.H 3J1eKTpHqeCKOrO TOKa Qepe3
BO,AHbIH paCT BOp HCI Ha KaTo,n;e BbI,n;en.HeTC.H BO,n;OpO,A, Ha aHo,n;e - xnop.
BTOPHQHbIX peaKIJ;HH He npOHcxo,n;HT.
npHMep 2. MOJleKYJlbI cepHoH KHCJIOTbI npH paCTBopeHHH B Bo,n;e ,n;HCCOIJ;H-
HPYIOT Ha HOHbI BOllopOlla H HOHbl KHcnOTHoro OCTaTKa:
H2804 p 2H+ + 8ol-.
IIplt npoxom,n;eHHH 3J1eKTpH'tIeCKOrO TOKa qepe3 paCTBOp Ha 3J1eKTpo,n;ax npo-
TeKalOT cne,AYIOHe peaKIJ;HH:
2H+ + 2e  H2'
80 4 2 - - 2e  80 4 ,
Bo,AOpo,n; BbI,AeJl.HeTCSI B BH,n;e nY3bIpbKOB Ha KaTo,n;e. HeHTpanbHa.H rpynna
804 XHMH'tIeCKH O'tleHb aKTHBHa H BCTynaeT BO BTOpH'lHYIO peaKIJ;HIO. EcnH
3neKTpo,n;bI H3rOTOBJleHbI, HanpHMep, H3 nnaTHHbI 1inH HHKenSI, TO 80 4 pearH-
pyeT C BO,AOH:
280 4 + 2H 2 0  2H 2 80 4 + 02'
MoneKYJla cepHOH KHcnOTbI nocTynaeT B pacTBop, KHCJlOpO,n; BbI,n;en.HeTC.H B
BH,Ae nY3bIpbKOB Y aHo,n;a, T.e. BTOpH'lHa.H peaKIJ;H.H npoTeKaeT C pacTBopHTeneM.
npHMep 3. IIycTb B 3J1eKTpOJIHTH'tIeCKOH BaHHe Haxo,n;HTC.H BO,n;HbIH paCTBOp
CYJlb<paTa Me,n;H. IIo,n; ,n;eHcTBHeM paCTBopHTenSl MoneKynbI cynb<paTB Me,n;H ,n;HC-
COIJ;HHPYIOT Ha HOHbI Me,n;H H HOHbI KHCJlOTHOrO OCTaTKa:
CU804 p Cu 2 + + 80l-.
IIpH npoxom,n;eHHH 3neKTpH'tIeCKOrO TOKa 'tIepe3 paCTBOp Ha KaTo,n;e HOHbI
Me,n;H npeBparqalOTcSl B HeHTpsnbHbIe aTOMbI Cu++ + 2e  Cu, II Me,n;b OTKJla,n;bI-
BaeTCSI Ha KSTo,n;e. Ha SHo,n;e npOHcxo,n;HT HeHTpaJIH3SIJ;HH KOMnJIeKCa 80 4 --:
80 4 -- - 2e  80 4 ,
284

ECJlH 3J1eKTpO,nbI Me,nHbIe, TO HeiiTpaJlbHa.H rpynna 80 4 npe,nnOqTHTenbHee
BCTynaeT B peaKHIO C Me,AblO, qeM C BO,nOH: 80 4 + Cu ---t Cu80 4 .
MOJIeKYJla Cu80 4 nocTynaeT B pacTBop. BTOpHqHa.H peaKIJ;H.H npoTeKaeT C
3neKTpo,noM H3 Me,nH. TIPOHCXO,nHT OTJlOmeHHe Me,AH Ha KaTO,Ae H pacTBopeHHe
aHO,Aa. XHMHqeCKHH COCTaB 3J1eKTpOJIHTa npH 3TOM OCTaeTC.H HeH3MeHHbIM.
He06xo,l1HMbIM YCJloBHeM 3JIeKTpOJIH3a HBJIHeTC.H npoxomneHHe qepe3 3JIeKT-
ponHT nOCTO.HHHOrO 3J1eKTpHqeCKOrO TOKa.
Maccy BeI.,QeCTBa, BbI,l1eJIHBIllerOC.H npH 3JIeKTpOJlH3e, MomHO paCCqHTaTb,
nonb3YHcb aneKTpoHHOH TeopHeH. 3Ta Macca onpe,l1eJI.HeTC.H MaccoR Bcex N
HOHOB, KOTopbIe ocenH Ha 3JIeKTpo,ne: m = mON, r,ne mo - Macca o,nHoro HOHa,
N - ({HCJIO HOHOB, KOTopoe MomHO paCCqHTaTb, 3Ha.H Maccy BeI.,QeCTB8 m H
m
ero MOJl.HpHYIO Maccy M: N = - N A' r,ne N A - qHCJlO ABOra,npO.
M
C ,npyroH CTOpOHbI, qHCJlO HOHOB MOmHO BbIpa3HTb qepe3 38p.H,n q, npo-
Ille,nIllHH qepe3 3JIeKTpOnHT. H 3apH,n O,nHOrO HOHa qH: N = !L.
qH
q m qM
CJIe,nOBaTeJlbHO, - = - N A :::::) m = -.
q M qHNA
3ap.H,n HOHa JII060ro BeI.,QeCTBa KpaTeH 3apH,ny O,AHOBaJIeHTHOrO HOHa, T.e.
3ap.H,ny 3JIeKTpOHa qH = ne, r,ne n - BaJIeHTHOCTb HOHa.
TaKHM 06pa30M,
M
m=-q.
neN A
BeJIHqHHbI M H n nOCTO.HHHbI ,l1JI.H naHHOrO BemeCTBa. T103TOMY BbIpame-
M
HHe - = k - BeJIHqHHa nOCTOHHHaH ,nJl.H ,AaHHOrO BeI.,QeCTBa. Ee Ha3bIBa-
neN A
lOT 3JIeKTpOXHMHqeCKHM
3KBHBaJIeHTOM. Tor,na m = kq.
TIpHBe,neHHa.H 3aBHCHMOCTb BnepBbIe 6bIJI8
3KCnepHMeHTaJIbHO YCTaHOBJIeHa M. 4>apa.AeeM B
30-x ro,nax XIX BeKa. 3Ta qJopMYJla BbIpa)Ka-
eT 3aKOH apeH: .Maeea 6eeem6a. 6bLOeJl.U6-
4 wezoeSl lia 3Jl.eKmpooe. npSl.MO nponOpijUOliaJl.b'
lia 3apSlOY q, npoweowe.MY llepe3 3Jl.eKmp0Jl.Um:
m = kq.
8neKTpOXHMHQeCKHH 3KBHBaJleHT QHCJIeHHO
paBeH Macce BeI.,QeCTBa, BbI,neJIHBIllerOC.H Ha
3J1eKTpo,ne npH npoxom,neHHH Qepe3 paCTBOp
3JIeKTpOJlHTa 3ap.H,na B 1 KJI. B cn e,nHHHu;eii
3JIeKTPQXHMHQecKoro 3KBHBaneHT8 .HBJIHeTCH KHJlO-
Puc. 10.5 rpaMM Ha KynoH ( : ).
Y6e,l1HTbC.H B cnpaBenJlHBOCTH 3aKOHa 4>apane.H MomHO Ha onbITe. C06epeM
YCTaHOBKY, nOKa3aHHYlO Ha pHCYHKe 10.5. Bce TPH 3JIeKTpOJIHTHqeCKHe BaH-
HbI 3anonHeHbI O,nHHM H TeM me paCTBopOM 3JIeKTpOJIHTa, HO TOKH, npoxo,nH-
He Qepe3 HHX, pa3J1HqHbI. 0603HaQHM CHJlbI TOKOB Qepe3 1h 1 2 , 13' Tor,na
1 1 = 1 2 + 13' H3Mep.H.H MaCCbI ml, m2, m3 BbI,neJIHBIllHXC.H Ha 3J1eKTpo,nax Be-
1 2

II

=
(10.1)
285

eCTB B pa3HbIX BaHHaX. MOmHO y6e.l1HTbC8. 'liTO OHH npOnOpUHOHaJlbHbI CO-
OTBeTCTBYIOHM CHJlaM TOKa Ih 1 2 , 13'
3aKoH <I>apa.n;eSJ CbIrpaJl BamHYIO pOJIb B HCTOpHH pa3BHTH.H cpH3HKH. 'HMeH-
HO 3aKoH, nOJlyqeHHbIH <I>apa,AeeM onbITHbIM nYTeM, nocnymHn TOJlqKOM K
BbI.n;BHmeHHIO rHnOTe3bI 0 cyecTBoBaHHH B npHpo.n;e 3J1eMeHTapHoro 3neKT-
pHqeCKOrO 3ap8.n;a. 3aKoH <I>apa.n;eSJ n03BOJI.HeT onpe.n;eJlHTb 3ap8.n; O,AHOBaJIeHT-
Mq Mq Mlt
Horo HOHa (3ap.H,lJ; 3J1eKTpOHa). H3 CPOPMYJlbI (10.1) e = - = - = -,
mNAn mN A mN A
T .e. Ha.n;o H3MepHTb Ha onbITe TOJIbKO Maccy 6bLOeJl.U6WeZOcSl 6el4ecm6a m II
CHJlY TOKa B uenH I.
10.4. TIpHMeHeHHe 3JIeKTpOJIH3a B TeXHHKe
3neKTpOJIH3 Haxo.n;HT llIHpOKoe npHMeHeHHe B TeXHHKe.
01lHCTKa HJlH pa4>HHHpoBaHHe MeTaJlJlOB. IIpouecc npOHcxo.n;HT B 3neKT-
POJlHTHqeCKOH BaHHe. AHO.n;OM cJlymHT MeTaJIJI, no.n;JIemaHH OqHCTKe, KaTO-
.n;OM - TOHKa.H nJIaCTHHKa H3 'lHCTOrO MeTanJla, a 3J1eKTpOJlHTOM - pa-
CTBOp COJIH .n;aHHoro MeTanna, HanpHMep, npH pa<pHHHpoBaHHIi Me.n;H - pa-
CTBOp Me.n;Horo Kynopoca. B 3arp.H3HeHHbIX MeTaJIJlaX MorYT co.n;epmaTbC.H ueH-
HbIe npHMeCH. TaK, B Me.n;H qaCTO co.n;epmHTC.H HHKenb H cepe6po. ,l(Jl8 Toro
qT06bI Ha KaTo.n;e BbI.n;en.HJIC8 TOJIbKO qHCTbIH MeTaJln, He06xo.n;HMO yqHTbI-
BaTb, qTO BbI.n;eneHHe Kam.n;oro BeeCTB8 HaqHHaeTC.H nHllIb npH HeKoTopoH
onpe.l1eJIeHHOH. pa3HOCTH nOTeHUHaJlOB Mem,Ay 3neKTpO,AaMH, Ha3bIBaeMOH "no-
TeHUHaJIOM pa3JIOmeHH8". IIpH Ha.l1nemaeM ee BbI60pe H3 paCTBopa Me.l1HOrO
Kynopoca Ha KaTo.n;e BbI.n;eJl.HeTC.H qHCTaSl Me.n;b, a npHMeCH BbIna.n;alOT B BH,Ae
OCa,AK8 HJlH nepeXO,A.HT B paCTBOp.
3J1eKTpOMeTaJlJIYPrHH. HeKoTopbIe MeTaJIJIbI, HanpHMep, aJIIOMHHHH, nony-
qalOT MeT0.l10M 3J1eKTpOJlH3a H3 paCnJlaBJleHHOH py.n;bI. 3JIeKTpOJlHTHqeCKOR BaH-
HOR H O,lIHOBpeMeHHO KaT0.l10M CJIymHT meJIe3HbIR .HlIlHK C yrOJIbHbIM nOJIOM,
a aHOJJ;OM - yronbHbIe CTepmHH. TeMnepaTypa py.n;bI (OKOJIO 900°C) no.n;JJ;ep-
mHBaeTC.H npOTeKalOHM B HeR TOKOM. PaCnJlaBJleHHbIH RJlIOMHHHH onycKaeT-
C8 Ha ,AHO .HHKa, oTKy.n;a ero qepe3 oc060e oTBepcTHe BblnycKalOT B cpOpMbI
,An8 OTJlHBKH.
raJILBaHOCTerHH - 3J1eKTpOJlHTHqeCKHH cnoc06 nOKpbITH.H MeTannHqeCKHX
H3.n;enHH CJIOeM 6J1arOpOJJ;HOrO HnH JJ;pyroro MeTaJIn8 (30JIOTa, nJIaTHHbI), He
nO,AJJ;alOerOC.H OKHCJleHHIO. HanpHMep, npH HHKeJlHpOBaHHH npe.n;MeTa OH caM
cJlymHT KaTo.n;OM, KYCOK HHKeJl8 - aHo.n;OM. IIponycKa8 qepe3 3JIeKTpOJlHTH-
qeCKYIO BaHHY B TeqeHHe HeKOToporo BpeMeHH 3JIeKTpHqeCKHH TOK, nOKpbIBa-
lOT npe.n;MeT CJlOeM HHKeJI.H HymHoH TOJIHHbI.
raJILBaHOnJIaCTHKa, HJIH 3J1eKTpOJIHTHqeCKOe OCam,AeHHe MeTaJIJI8 Ha nOBepx-
HOCTM npe.n;MeTa .n;JI.H BocnpOH3Be.n;eHH.H ero <pOPMbI, 6bIna H306peTeHa B 1837 r.
PYCCKHM yqeHbIM B. C. .HK06H, npe,AnOmHBllIHM Hcnonb30BaTb 3J1eKTpOJlH3 .n;n8
nonY'leHHSI MeTaJlJlH'leCKHX OTne'laTKOB peJlbe<pHbIx npe,AMeTOB (MeJJ;aJleH, MO-
HeT H ,Ap.). C npe,AMeTa CHHMalOT CJIenOK H3 BOCK8 HnH BbIpe3810T BblnYKnoe
H306pameHHe Ha JJ;epeB.HHHOH JJ;OCKe H JJ;eJlalOT ero npOBO,A.HHM, nOKpbIBa.H CJIO-
eM rpa<pHTa. 3aTeM onYCKalOT CJlenOK HnH JJ;OCKY B KaqeCTBe KaTOJJ;8 B 3JIeKT-
pOJlHT. AHO.n;OM CJIymHT KYCOK MeTaJIJla, HcnoJlb3yeMoro .n;JI.H ocam.n;eHH.H. 3THM
cnoc060M H3rOTOBJI.HIOT, HanpHMep, THnorpacpCKHe KJIHllIe.
286

3neKTpOnHTHqeCKHM nYTeM nonyqalOT T.H>KeJlYIO BO,ny (D 2 0), B KOTOpOH
aTOMbI BO,lIOpOlIa 3aMeHeHbI aTOMaMH ero H30TOna - lIeHTepHH (D) C aToMHoH
MaCCOH 2.
10.5. 3JIeKTpH'IeCKHH TOK B ra3ax
B 06bIqHbIX YCJlOBH.HX ra3bI .HBn.HIOTCH H30Jl.HTOpaMH (B HHX HeT ,nOCTaTOq-
Horo KOJlHqeCTBa cB060,nHbIX 3ap.H,noB) H CTaHOB.HTC.H npOBo,nHHKaMH JlHIlIb Tor-
,Aa, Kor,n8 OHH KaKHM-TO 06pa30M HOHH3HpOBaHbI. HOHH3aTopaMH MorYT CJlY-
>KHTb peHTreHOBCKHe nyqH, KOCMHqeCKHe nyqH, YJlbTpacpHoJleToBbIe nyqH, pa-
,AHOaKTHBHoe H3nyqeHHe, HHTeHCHBHoe HarpeBaHHe H ,np.
IIpoecc HOHH3aHH ra30B 38KJllOqaeTC.H B TOM, qTO noon ,neHCTBHeM HOHH-
38TOpOB OT aTOMOB OTpbIBaeTC.H O.nHH MJlH HeCKOJlbKO 3J1eKTpOHOB. B pe3YJlb-
TaTe 3Toro BMeCTO HeHTpaJlbHOrO aTOM a B03HHKalOT nOJlOmHTeJlbHbIH HOH H
3neKTpOH. qacTb 06pa30B8BIlIHXC.H 3J1eKTpOHOB MomeT 6bITb npH 3TOM 3aXBa-
qeHa ,npyrHMH HeHTpanbHbIMH aTOMaMH, H Tor,na nO.HBn.HIOTC.H OTpHaTenbHO
3apHmeHHbIe HOHbI. OTpbIB 3neKTpOHa OT aTOMa Tpe6yeT 3aTpaT onpe,AeneHHOH
3HeprHH - 3HeprHH HOHH3aHH W H , KOTopaH H3MepHeTCSI pa60TOH npOTHB
CHnbI npHT.H>KeHH.H 3J1eKTpOHa .H,npOM aTOM8: W H = eU H' r,ne e - 3apSl,n 3J1eKT-
pOHa, U H - nOTeHHan HOHH3aHH ,nJlSl ,naHHoro BeeCTBa.
9HeprH.H MOHH3aHH 3aBHCHT OT XHMHqeCKOH npHpo,nbI ra3a H 3HepreTHqeC-
Koro COCTOHHH.H 3J1eKTpOHa B 8TOMe. TaK, ,nn.H OTpbIBa nepBoro 3neKTpOHa OT
aTOMa a30T8 3aTpaqHBaeTCH 3HeprH.H 14,5 3B, a ,An.H OTpbIBa BToporo 3neKTpo-
Ha - 29,5 3B, ,An.H OTpbIBa TpeTbero - 47,4 3B.
9J1eKTpOHbI H nOJlOmHTeJlbHbIe HOHbI, B03HHKIlIHe BO BpeM.H ,neHcTBH.H HOHH-
3aTopa, He MorYT ,noJlro cyecTBoBaTb pa3,AenbHO H npH CTonKHOBeHH.HX BHOBb
06pa3YlOT HeHTpanbHbIe aTOMbI HJlH MoneKynbI. 9TO .HBJleHHe Ha3bIBaeTC.H pe-
KOM6HHal.tHeH. II03TOMY nocne npeKpaIIteHHH neHCTBHSI HOHH3aTopa 3neKTpH-
'leCKHH TOK B ra3e HCqe3aeT. IIpOBOltHMOCTb ra3a, C03,naHHYIO BHeIlIHHMH HOHH-
3aTopaMH, HO He CB.H3aHHYIO C 3J1eKTpHqeCKHM nOJleM, Ha3bIBalOT HecaMOCTO.R-
TeJlLHoH.
ECJlH npH HOHH3aHH ,nJl.H oTpbIBa 3J1eKTpOHa OT aTOMa He06xo,nHMa 3Hep-
rH.H, TO npH peKOM6HHall;HH 3Ta 3HeprH.H BbI,AeJl.HeTC.H 60nbIlIeH qaCTblO B BH,ne
CBeTOBoro H3nyqeHH.H. IIpH ,AOCTaTOqHOH HHTeHCHBHOCTH peKOM6HHaHH 3neK-
TpHqeCKHH TOI< B ra3ax conpOBom,naeTC.H 3aMeTHbIM CBeqeHHeM. IIpH Henpe-
pbIBHOM ,AeHCTBHH HOHH3aTopa H OTCYTCTBHH 3J1eKTpHqeCKOrO nOJl.H B ra3e
YCTaHaBJlHBaeTC.H no,nBHmHoe paBHOBeCHe Me>K,ny HOHH3aHeH MOJleKY JI H pe-
KOM6HHall;HeH HOHOB, xapaKTepH3YIOeeC.H onpe,neJleHHOH KOHeHTpall;HeH
HOHOB.
IIpH nOMeeHHH HOHH3HpOBaHHoro ra3B B 3neKTpHqeCKOe nOJle Ha CB060,Zl;-
HbIe 3ap.H.nbI .neHcTBYIOT 3J1eKTpHqeCKHe CHJlbI H OHH .npeHCPYIOT napaJlJlenbHO
nHHH.HM HanpHmeHHocTH: 3J1eKTpOHbI H oTpHUaTeJlbHbIe MOHbI - K aHOlty, no-
nOmHTeJlbHbIe HOHbI - K KaTo,ny. Ha 3J1eKTpo,nax HOHbI npeBpaaIOTC.H B HeH-
TpaJlbHbIe aToMbI, oT,naBa.H HnH npHHHMa.H 3J1eKTpOHbI, TeM caMbIM 3aMbIKaH
enb. B ra3e B03HHKaeT 3J1eKTpHqeCKHR TOK. 9J1eKTpHqeCKHH TOK B ra38X Ha-
3bIBaeTCSI ra30BLIM pa3p.R)];OM. TaKHM 06pa30M, npOBO,l);HMOCTL ra30B HMeeT
HOHHO-3JIeKTPOHHLIH xapaKTep.
287

10.6. HecaMOCTO.HTeJILHLIH H caMOCTO.HTeJILHLIH pa3p.H,IJ;LI B ra3ax
C06epeM 3J1eKTpHqeCKYIO n;enb, co.n;epmaylO
HCTOqHHK TOK8, BOJlbTMeTp v, aMnepMeTp A Ii
.n;Be MeTaJlJlHqeCKHe nJl8CTHHbI, pa3.n;eJleHHbIe
B03.n;YIIIHbIM npOMemYTKOM (pHC. 10.6).
ECJlH nOMeCTHTb B6nH3H B03.n;YIIIHoro npo-
MemYTKa HOHH3aTOp D. TO B n;enH B03HHKHeT
3neKTpHqeCKHH TOK. HCqe3810IllHH C npeKpame-
HHeM .n;eHCTBH.H HOHH3aTOpa. PeKOM6HHaUH.H
HOHOB npOHCXO.n;HT H BO BpeM.H ,AeHCTBH.H HOHH.
3aTOpa, HO npH 3TOM HacTynaeT .n;HH8MHqeCKOe
p8BHOBeCHe Mem,Ay B03HHK81OHMH H peKOM6H-
HHPYIOHMH HOH8MH, KOr.n;8 qHCJlO nap HOHOB
B e)J;HHHn;e 06'beM8 He H3MeH.HeTC.H.
3J1eKTpHqeCKHH TOK B ra3e C HecaMOCTO.H-
TenbHOH npOBO,AHMOCTblO Ha3bIBaeTCSI HecaMOC-
TO.HTeJlbHbIM ra30BLIM pa3pH,IJ;OM.
rpa<}>HK 3aBHCHMOCTH CHnbI pa3p.H.n;Horo
TOKa OT pa3HOCTH nOTeHn;HanOB (Hanp.HmeHH.H)
Mem.n;y 3J1eKTpo.n;aMH npH HeH3MeHHOH HHTeHCHB-
U HOCTH HOHH38Topa - BOJlbT-8MnepHa.H xapaK-
TepHCTHK8 ra30Boro pa3p.H.n;a - npHBe.n;eH H8
pHCYHKe 10.7. I1HTeHcHBHoCTb HOHH3al1HH H3-
Mep.HeTC.H KOnHqeCTBOM HOHOB, B03HHKIIIHX nOlI
,AeHCTBHeM HOHH38Topa B e.n;HHHn;e 06'beMa napa 3a e,AHHHn;y BpeMeHH. H8 rp8-
<}>HKe MomHO BbI.n;eJlHTb HeCKOJlbKO xap8KTepHbIX 06J1aCTeH.
H8 yqaCTKe Oa C06J1IO.n;aeTC.H 38KOH OMa, 3aTeM nponopn;HOHanbHOCTb Ha-
pYIll8eTC.H (yqacToK ab), H H8qHH8S1 C U H CHJl8 TOKa He H3MeH.HeTC.H. HaH-
60JlbIllYlO CHJlY TOKa, B03MomHYlO npH .n;eHCTBHH .n;aHHoro HOHH38Topa, Ha3bI-
B810T TOKOM HaCLIIu;eHHH [Hac' llpH TOKe HaCbIeHH.H Bce B03HHK81OHe HOHbI
.n;OJleTaIOT )J;O 3J1eKTpO)J;OB, He ycneB peKOM6HHHpOB8Tb. 3HaqeHHe CHJlbI TOKa
H8CbIeHHSI onpe.n;eJl.HeTCSI CBOHCTBaMH HOHH38TOp8. AJlSi YBenHQeHH.H Iuac Hym-
HO YBeJlHqHTb HHTeHCHBHOCTb HOHH38n;HH.
PaCCMOTpHM YQaCTOK cd. llpH .n;OCTaTOqHO 60nbIIIHX Hanp.HmeHH.HX KHHe-
THQeCKa.H 3HeprH.H 3J1eKTpOH8 B03p8CT8eT H8CTonbKO, qTO npH ero coy.n;apeHHH
C HeHTp8J1bHOH MOJleKYJlOH ra3a OT Hee OTmenn.HeTC.H BHeIIIHHH 3J1eKTpOH. 3TO
.HBJleHHe H83bIB8eTC.H YAapHOH HOHH3al.\HeH MoneKyn ra38. Pa3HOCTb nOTeHltH-
aJlOB, KOTOPYIO .n;OJlmeH npOHTH 3neKTpOH B YCKOp.HlOeM 3J1eKTpHQeCKOM nOJle
.n;n.H Toro, 1.{T06bI npH06pecTH .n;OCTaTOqHYIO .n;JI.H pa60TbI HOHH3an;HH 3HeprHIO,
Ha3bIBalOT nOTeHl.\HaJlOM HOHH3al.\HH. OCB060.n;HBIIIHeC.H 3J1eKTpOHbI YCKOp.HIOTC.H
B 3J1eKTpHqeCKOM nOJle H B CBOIO OQepe,Ab HOHH3HPYlOT CT8J1KHB81OHecSI C
HHMH HeHTp8J1bHbIe MOJleKynbI ra3a. qHCJlO 3J1eKTpOHOB H HOHOB B ra3e n8-
BHHo06pa3Ho p8CTeT, 8 BMeCTe C HHM paCTeT H pa3p.H.n;HbIH TOK. llpH ee
60JlbIIIHX 3HaqeHH.HX Hanp.HmeHHSI y.n;apHYIO HOHH3an;HIO BbI3bIB81OT H HOHbI.
Tenepb K 060HM 3J1eKTpO)J;aM .n;BHmYTC.H JlaBHHbI: K KaTO.n;y - nonOmHTeJlbHa.H
HOHHa.H, K aHO.n;y - 3J1eKTpOHH8.H. 3TH JlaBHHbI 06YCJlOBJlHBaIOT caMOCTOJITeJIL-
HYIO npOBoJJ;HMOCTL ra3a. YqaCTOK cd rpa<}>HKa xap8KTepH3yeT caMOCTOSl-
v
+1\-
Puc. 10.6
I
I.u
b
1
3
4
2
Puc. 10.7
288

TeJIbHLlH ra30BblH pa3pHJJ:, KOTOpbIH MO>KeT cyrqecTBoBaTb 6e3 BHelllHerO HOHH-
3aTOpa. HenpepbIBHa.H y6bInb HOCHTeJIeH TOKa, rrpOHCXO.n;.Hrqa.H BCJIe.n;CTBHe HeH-
TpaJIH3au,HH HOHOB y 3JIeKTpO.n;OB, nOrJIorqeHH.H 3J1eKTpOHOB aHO.n;OM H peKOM-
6HHau,HH HOHOB. BOCnOJlH.HeTC.H B 3TOM cJlyqae 3a CqeT 3J1eKTpHqeCKOrO nOJI.H,
a He 3a C'leT HOHH3aTopa.
TIpH BC.HKOM caMOCTO.HTeJIbHOM pa3p.H.n;e HMeeT MeCTO HOHH3au,H.H ra3a 3JIeK-
TpOHHbIMH y.n;apaMH, HO HaJIHqHe TonbKO HOHH3au,HH 3JIeKTpOHHbIM Y,AapOM erqe
He npHBO,AHT K caMOCTO.HTeJIbHOMY pa3p.H.n;y. .n:JI.H cyrqecTBOBaHH.H caMOCTO.H-
TeJlbHOrO pa3p.H.n;a He06xo.n;HMO, qT06bI B ra3e npOHCXO,AHJIH H .n;pyrHe npou,ec-
CbI, rrpOH3BO,A.HHe HOBbIe 3JIeKTpOHbl B3aMeH Yllle.n;lllHX Ha aHO.n;. TaKHMH npo-
u,eccaMH MorYT 6bITb BTOPHqH8.H 3MHCCH.H 3J1eKTpOHOB C KaTo.n;a (BbI6HBaHHe
3JIeKTpOHOB H3 KaTo.n;a pa30rHaHHbIMH B 3JIeKTpHQeCKOM none nonOmHTeJIbHbI-
MH HOHaMH), HOHH3au,HH HOHaMH H BHYTpeHH.H.H CPOTOHOHH3au,H.H (H3JIYQeHHe.
conpOBom.n;alOII.1ee peKOM6HHau,HIO HOHOB. MO>KeT 6bITb nOrJlOII.1eHO HeHTpaJIb-
HblMH aTOMaMH, B pe3YJIbTaTe Qero HeKOTopbIe H3 HHX HOHH3HPYIOTC.H).
10.7. BH,lI,bl caMOCTOHTeJlbHOrO paap.HJJ:a
H HX TeXHHQeCKOe HCnOJlb30BaHHe
B 3aBHCHMOCTH OT )];aBJIeHH.H ra3a, HanpHmeHH.H, rrpHJIO>KeHHOrO K 3JIeKT-
po.n;aM, cpOpMbI H xapaKTepa pacnOJIOmeHH.H 3JIeKTpo.n;OB pa3JIH'lalOT CJIe.n;ylO-
rqHe THnbI caMOCTO.HTeJIbHOrO pa3pH.n;a: TJIelOrqHH, KOpOHHbIH. .n;yroBoH H HC-
KpOBOH.
TJIelOnH pa3pHJJ: Ha6JIIO.n;aeTC.H npH nOHHmeHHbIX .n;aBJIeHH.HX ra3a (nop.H.n;-
Ka 0,1 MM pT. CT.). ECJIH K 3JIeKTpo.n;aM, BnaSlHHbIM B CTeKJI.HHHYIO Tpy6KY,
npHJlOmHTb nOCTO.HHHOe Harrp.H>KeHHe B HeCKOJIbKO COT BOJIbT H 3aTeM nOCTe-
neHHO OTKaQHBaTb B03.n;yx H3 Tpy6KH, TO Ha6JIIO.n;aeTC.H cne.n;YlOlqee .HBJIeHHe:
npH YMeHbllleHH1i .n;aBJIeHH.H ra3a B HeKOTopbIH MOMeHT B Tpy6Ke B03HHKaeT
pa3p.H.n;, HMelOrqHH B1i.n; CBeT.HrqerOC.H lllHypa, CO-
 e)];HH.HlOrqero aHO.ll. H KaTO)]; Tpy6KH (pHC. 10.8).
\  TIpH J(aJIbHeihneM YMeHbllleHHH J(aBJIeHH" 3TOT
4 lllHYP paClllHp.HeTC.H H 3anOJIH.HeT Bce CeQeH1ie
Tpy6KH, a CBeQeHHe B6JIH3H KaTo.n;a OCJIa6eBaeT.
OKOJIO KaTO)];a 06pa3yeTC.H nepBoe TeMHoe npo-
CTpaHCTBO 1, K KOTOPOMY npHJIeraeT HOHHbIH CBe-
T.HrqHi1c.H cnoH 2 (TJIelOrqee CBeQeHHe). KOTOpbIH HMeeT pe3KYlO rpaHHu,y co CTO-
pOHbI KaTo.n;a H nOCTeneHHO HCQe3aeT co CTOpOHbI aHo.n;a. 3a TJIelOlqHM CBe'tIeH1i-
eM Ha6JIIO,AaeTCH onHTb TeMHbIH npOMe>KYTOK 3. Ha3bIBaeMbIH cpapeeBbIM HJIH
BTOpbIM TeMHbIM npOCTpaHCTBOM. 3a HHM Jle>KHT CBeT.Hrqa.HC.H 06JIaCTb 4, npo-
CTHpalOlqa.HC.H )];0 aHo.n;a, HJlH nOJIOmHTeJIbHbIH CTOJI6.
Oc060e 3HaQeHHe B TnelOrqeM pa3p.H.n;e HMelOT TOJlbKO .n;Be ero QaCTH -
KaTo.n;Hoe TeMHoe npOCTpaHCTBO II TJIelOrqee CBeQeHHe, B KOTOpbIX npOHcxu.n;.HT
OCHOBHbIe npou,eCCbI, no.n;.n;ep>KHBalOrqHe pa3p.H.n;. 3JIeKTpOHbI, HOHlI3HpYlOlqne ra3,
B03HHKalOT B pe3YJIbTaTe cpOT03MHCCHH C KaTo.n;a H CTOJIKHOBeHHH nOnO>KH-
TenbHbIX nOHOB C KaTo.n;OM Tpy6KH.
B HaCTO.Hrqee BpeM.H TJIelOII.1HH pa3p.Hlt lllHpOKO ncnoJlb3yeTC.H B KaQeCTBe
HCTOQHHKa CBeTa B pa3nHQHbIX ra30BbIX Tpy6Kax B HCTOQHHKaX ltHeBHOro CBe-
Ta pa3p.H,A 06bIqHO npOHCXO,AHT B napax PTYTH. ra30BbIe Tpy6KH npHMeH.HIOT-
C.H TaKme .n;JI.H peKJIaMHbIX H .n;eKopaTHBHbIX u,eJIeH.
-71/ 1 , \
+'7 \
1 2 3
Puc. 10.8
289

T JlelOHH pa3p.H,n HCnOJlb3YIOT ,nJl.H K8TO,nHOrO paCnbIJleHH.H
MeTaJIJlOB, TaK KaK BeeCTBO KaTO,na B TnelOeM pa3p.H,ne no-
CTeneHHO nepeXO,nHT B ra3006pa3HOe COCTO.HHHe H OCe,naeT B
BH,ne MeT8J1J1HqeCKOH nbIJlI:I Ha CTeHK8X Tpy6KH. TIoMe8.H B
TnelOIIJ;HH p83p.H,n p83J1H'IHbIe npe,nMeTbI, nOKpbIB81OT HX P8B-
HOMepHbIMH H npOqHbIMH CnO.HMH MeT8nJla. STHM cnoc060M
nOJlb3yIOTC.H ,nn.H H3rOTOBneHHSl MeTanJlH'IeCKHX 3epKaJI BbICO-
KOrO KaqeCTBa.
IICKpOBOH pa3pM, q8CTO Ha6J1IO,naeMbIH B npHpO,ne. --
MOJlHH.H. MOJlHH.H -- aTO pa3p.H,n Mem,ny lIBYM.H 3ap.HmeH-
HbIMH 06J1aKaMH HJlH MemlIY 06naKOM H 3eMJleH. HOCHTe-
JI.HMI:I 38p.H,nOB B 06J1aKaX .HBn.HIOTC.H 38p.HmeHHbIe K8neJlb-
KH BO,nbI HnE: CHemHHKE:.
B na60paTOpHbIX ycnOBH.HX HCKpOBOii pa3p.H,n MOmHO no-
JlY'IHTb, eCJlH nOCTeneHHO YBeJlH'IHB8Tb Hanp.HmeHHe Mem,ny
,nBYM.H 3neKTpO,naMH, H8XO,n.HHMHC.H B aTMOCcpepHOM B03,ny-
xe H HMelOIIJ;HMH TaKYIO cpOPMY, qTO aJleKTpHqeCKOe nOJle
Mem,ny HHMH MaJlO OTJlH'I8eTC.H OT O,nHopo,nHoro. TIpH HeKO-
TOpOM Hanp.HmeHHH B03HHKaeT aJleKTpH'IeCKa.H HCKpa. TIPH
aTOM HCKpOBOH pa3p.H,n C orpOMHOH 6bICTpOTOH npOHH3bIBaeT
pa3p.H,nHbIH npOMemYTOK, raCHeT H BHOBb B03HHKaeT. .HPKO
CBeT.HIIJ;HHC.H H30rHYTbIH K8H8n HCKpbI coe,nHH.HeT 068 aJleKTpo,na H HMeeT cnom-
Hoe p83BeTBJleHHe (pHC. 10.9). CBeqeHHe B HCKpe -- pe3ynbT8T HHTeHCHBHbIX
npOneCCOB HOHH3anHH. 3BYKoBbIe acpcpeKTbI, conpOBOm,A81OIIJ;He HCKPY, nopom-
,AaIOTC.H nOBbIllIeHHeM ,naBJleHH.H (,no COTeH aTMoccpep) BCJlelICTBHe HarpeBaHHSI
r838 (,AO 10 5o C) B MeCTax npoxomlIeHHSI pa3p.H1I8. I1cKpa B03HHKaeT B TOM cny-
qae, eCJlH Hanp.HmeHHOCTb aJleKTpH'IeCKOrO nOJl.H B ra3e ,nOCTHraeT HeKOTopOH on-
pe,neJleHHOH BeJlHqHHbI, KOTOpa.H 38BHCHT OT po,na ra3a Hero COCTO.HHH.H.
ECJlH, OCTaBJI.H.H HanpSImeHHe nocToSIHHbIM, YMeHbllIHTb paCCTO.HHHe Mem,ny
aJleKTpO)J;aMH, TO Hanp.HmeHHOCTb non.H B r830BOM npOMemYTKe 6y,neT YBenH-
'IHBaTbCSl. TIpH HeKOTopOM ee 3Ha'leHHH npOH30H,neT HCKpOBOH pa3p.H,n. qeM
BbIllie 6y,neT npHJlOmeHHOe Hanp.HmeHHe, TeM 60nbllIe 6y,neT paCCTO.HHHe Mem-
,Ay aJleKTpo,n8MH, npl:l KOTOpOM npOH30H,neT HCKpOBOH pa3p.H,A. TIpHHU;Hn ,AeH-
CTBHSI HCKpOBoro BOJlbTMeTpa -- npH60pa ,nJl.H H3MepeHH.H O'leHb BbICOKHX Ha-
np.HmeHHH -- OCHOB8H K8K pa3 Ha aTOM .HBJleHHH.
AyrOBOH pa3PM MomHO Ha6J11O,n8Tb npH cJle,AYIOIIJ;HX ycnOBH.Hx: ecnH noc-
Jle 38mHraHH.H HCKpoBoro pa3p.H,A8 nOCTeneHHO YMeHbllIaTb conpOTHBJleHHe n;emi,
TO CHJl8 TOK8 B HCKpe 6y,neT YBeJlH'IHBaTbC.H. Kor,n8 conpOTHBJleHHe n;enH CT8-
HeT 1I0CTaTOqHO MaJlbIM. B03HHKHeT HOB8.H cpopMa ra30Boro pa3p.HlIa. Ha3bIBae-
Moro ,nyroBbIM. TIpH aTOM CHna TOKa pe3KO YBenHqHBaeTC.H, ,nOCTHra.H )J;eCSIT-
KOB H COTeH aMnep, a Hanp.HmeHHe H8 pa3p.H)J;HOM npOMemYTKe YMeHbllIaeTCSI
,nO HeCKOJlbKHX ,neC.HTKOB BonbT. STO nOKa3bIBaeT, qTO B pa3p.H,ne B03HHKalOT
HOBbIe npOn;eCCbI, C006alOIIJ;He ra3Y OqeHb 60nbllIYlO aneKTponpOBo,nHOCTb.
B HaCTO.HIIJ;ee BpeM.H aneKTpH'leCKYIO ,nyry, rop.HIIJ;YIO npH aTMoccpepHOM ,naB-
JleHHH, q8IIJ;e Bcero nOJlY'lalOT Mem,ny cnen;HanbHbIMH yronbHbIMI:I aneKTpO,Aa-
MH. HaH60Jlee rOp.HqHM MeCTOM ,nyrH .HBJI.HeTC.H yrny6neHHe, 06pa3YIOIIJ;eeC.H H8
nonOmHTeJlbHOM aneKTpo,ne H Ha3bIBaeMoe KpaTepOM ,nyrH. Ero TeMnepaTypa
npH aTMoccpepHOM ,naBJleHHH OKOJlO 4000°C.
Puc. 10.9
290

3JIeKTpHqeCKa.H yra .HBJI.HeTC.H MOHbIM (fCTO'lHHKOM CBeTa H IIIHpOKO npH-
MeH.HeTC.H B npOeKn;HOHHbIX, npOmeKTOpHbIX H PyrHx OCBeTHTeJIbHbIX YCTa-
HOBKaX. BCJleCTBHe BbICOKOH TeMnepaTypbI yra llIHpOKO npHMeH.HeTC.H ,AJI.H
CBapKll II pe3KH MeTaJIJIOB. BbICOKYIO TeMnepaTYPY yrH llCnOJlb3YlOT TaKme
npll YCTpOHCTBe yroBbIx 3J1eKTpHqeCKHX neqeH, HrpalOHX BamHYIO pOJIb B
COBpeMeHHOH 3J1eKTpOMeTanJlyprHH.
KOpOHHLlH pa3PM Ha6J1IOaeTC.H npll CpaBHHTenbHO BbICOKHX aBJleHH.HX
ra3a (HanpHMep, npH aTMOCQ>epHOM aBJIeHHH) B pe3KO HeOHOpOHOM 3J1eKT-
pHqeCKOM nOJle. ,Z:(JI.H nOJlyqeHH.H 3HaQHTeJIbHOH HeOHopOHOCTH nOJI.H 3JIeKT-
pObI ,AOJImHbI HMeTb pe3KO pa3J1HQalOHeC.H nOBepXHOCTH, T.e. OHH 3JIeKT-
pO,A - OqeHb 60JlbllIYlO nOBepXHOCTb, a pyroH - O'leHb MaJlYIO' TaK, Ha-
npHMep, KOpOHHbIH pa3p.H MomHO nerKO nOJIYQHTb, pacnOJIara.H TOHKYIO npo-
BOJIOKY BHYTPH MeTaJIJIHQeCKOrO UHnllHpa, panHYC KOToporo 3HaQHTenbHO 60Jlb-
llIe paHyca npOBonOKH.
Hanp.HmeHHOCTb nOJI.H B6nH3H npOBOJlOKH HMeeT Hall60JIbllIee 3HaqeHHe.
MB
Kora Hanp.HmeHHOCTb nOJI.H ,AOCTHraeT 3HaqeHH.H E:::: 3 -, Memy npOBOJIO-
M
KOH H n;HJIllHpOM 3a>KllraeTC.H pa3p.H,A, H B n;enH nO.HBJI.HeTC.H TOK. TIpH 3TOM
B03ne npOBOJlOKH Ha6JIIOaeTC.H CBeQeHHe, HMelOee Bll 060JIOqKH HJIH KOpO-
HbI, oKPY>KalOeH npOBonOKY, OTKY,Aa H npOH3011Ino Ha3BaHHe pa3p.Ha.
KOpOHHbIH pa3p.H B03HHKaeT KaK npH OTpHn;aTenbHOM nOTeHn;HaJIe Ha npo-
BOJIOKe (oTpHn;aTeJIbHa.H KopoHa). TaK H npH nOnO>KHTeJIbHOM (nOJIO>KHTeJIb-
Ha.H KopoHa), a TaK>Ke npH nepeMeHHOM Hanp.HmeHHH Me>K.n;y npOBOJIOKOH H
n;HJIHH.n;pOM.
KOpOHHbIH pa3p.H Hcnonb3yeTC.H B TeXHHKe n.H YCTpoHcTBa 3JIeKTpo<})HJIb-
TpOB, npe.n;Ha3HaqeHHbIX ,An.H OQHCTKll npOMbIllIneHHbIX ra30B OT TBep.n;bIX H
mH,AKHX npHMeceH.
B npHpO,Ae KOpOHHbIH pa3p.H,A B03HllKaeT HHora no eHcTBHeM aTMoc<})ep-
Horo 3JIeKTpHqeCKOrO nOJI.H Ha BeTKax epeBbeB, BepXYlliKax MaQT (TaK Ha3bI-
BaeMbIe orHH CB.HTOrO 3JIbMa). KOpOHHbIH pa3p.H MO>KeT B03HHKHYTb Ha TOH-
KHX npOBo.n;ax, Haxo.n;.HHXC.H no,A Hanp.H>KeHHeM. B03HllKHoBeHHeM KopOHHoro
pa3p.Ha H8 OCTpH.HX npOBo.n;HHKOB 06'b.HCH.HeTC.H .n;eHcTBHe rpOMOOTBo.n;a, 3aH-
alOero 3.n;aHH.H II JIHHHH nepe.n;aq OT y.n;apOB MonHHH.
10.8. IIoHHTHe 0 nJIa3Me
nJla3Ma - 3TO QaCTHQHO RJIll nonHOCTblO llOHH3HpoBaHHbIH ra3, B KOTOpOM
nJIOTHOCTIi nOJlO>KHTeJIbHbIX H OTpHn;aTenbHbIX 3ap.HOB npaKTHQeCKH o,n:HHaKO-
BbI. Ilo3ToMY B n;eJIOM nJIa3Ma HBJI.HeTC.H 3JIeKTpllQeCKH HeHTpaJIbHOH CHCTeMOH.
CTeneHb HOHHaallHH DJla3Mbi a onpe.n;eJI.HeTC.H OTHOllIeHHeM 4:HCJI8 llOHH3H-
N
pOBaHHbIX aTOMOB K HX 06eMY QHCJIY: a = --!!... B 3aBHCHMOCTH OT CTeneHH
N
(fOHH3an;HH nJIa3Ma nO,Apa3eJI.HeTC.H Ha cJla60 1f0HHaHpOBallHYIO (a - ,n:OJIH
npOn;eHTa). qaCTHQHO HOHH3HpOBaliHYlO (a - HeCKonbKO npOn;eHToB) H OOJl-
1I0CTblO nOllH3HpOBaUHyro (a::::; 100 %). CJla60 (fOHH3HpoBaHHoH nJla3MOH .HBJI.H-
eTC.H HOHoc<})epa - BepxHHH CJIOH 3eMHOH aTMocQ>epbI. B COCTO.HHHH nOJIHOC-
TblO HOHll3HpoBaHHoH nJIa3MbI HaXOHTC.H COJIHn;e, rOp.HQHe 3Be3.n;bI. ConHn;e II
291

3Be3bI npeCTaBJI.HIOT C060H rHraHTCKHe CrYCTKH rOp.HqeH nJla3MbI, re TeMne-
paTypa OqeHb BbICOKa.H, nOp.HKa 10 6 -10 7 K. l1CKYCCTBeHHO C03aHHoH nJIa3-
Moii pa3J1HqHOH CTeneHH HOHH3aHH .HBJI.HeTC.H nJla3Ma B ra30BbIX pa3p.HaX,
ra30pa3p.HHbIX JIaMn8X.
CyecTBoBaHHe nJIa3MbI CB.H3aHO JIH60 C HarpeBaHHeM ra3a, JIH60 C H3JIY-
qeHHeM pa3J1HqHOrO pO,Aa, JIH60 C 60M6apHpOBKOH ra3a 6bICTpbIMH 3ap.HmeH-
HbIMH qaCTHaMH.
P.H CBOHCTB nJla3MbI n03BOJI.HeT paccMaTpHBaTb ee KaK oc060e COCTO.HHHe
BeecTBa. IlJIa3Ma - caMoe pacnpocTpaHeHHoe COCTO.HHHe BeecTBa. fiJIa3-
MB cyecTByeT He TOJIbKO B KaqeCTBe BeeCTBa 3Be3,A H COJIHa, OHa 3a-
nOJIH.HeT H KOCMHqeCKOe npOCTpaHCTBO Memy 3Be3aMH H raJIaKTHKaMH.
BepXHHH CJIOH aTMocc:pepbI 3eMJIH TaKme npeCTaBJI.HeT C060H cJla60 HOHH-
3HpOBaHHYlO nJIa3MY. l..JacTHl.{bI nJIa3MbI HHTeHCHBHO B3aHMO,lIeHCTBYlOT C BHe-
lllHHMH 3JIeKTpHqeCKHMH H MarHHTHbIMH nOn.HMH: H3-3a 60nblllOH no.nBH.>K-
HOCTH 3ap.HmeHHbIe qaCTHbI nna3MbI JIerKO nepeMealOTC.H no eHCTBHeM
3neKTpHqeCKHX H MarHIITHbIX nOJleH. Il03TOMY JI1060e HapYllleHHe 3J1eKTpH-
qeCKOH HeHTpaJIbHOCTH OTeJIbHbIX 06JIaCTeH nJIa3MbI, BbI3BaHHoe CKOnJleHH-
eM qaCTH C 3ap.HOM OHoro 3HaKa, 6bICTpO HCqe3aeT. B03HHKalOHe 3neK-
TpHqeCKHe nOJI.H nepeMealOT 3ap.HmeHHbIe qaCTHb] O Tex nop, nOKa 3JIeK-
TpHqeCKa.H HeHTpaJIbHOCTb He BOCCTaHaBJIHBaeTC.H H 3neKTpHqeCKOe nOJIe He
CTaHOBHTC.H paBHbIM HYJIIO.
Me.>K,AY 3ap.HmeHHbIMH qaCTHaMH nJIa3MbI eHCTBYlOT KYJlOHOBCKHe CHJIbI,
cpaBHHTenbHO MeJIeHHO y6bIBalOHe C paCCTO.HHHeM. Kama.H qaCTHa B3aH-
MoeHcTByeT cpa3Y C 60JIblllHM KOJIH"t!eCTBOM oKpymalOHx "t!aCTHI.{. BJIaroap.H
3TOMY Hap.Hy C xaOTH"t!eCKHM TenJlOBbIM ,ABH.>KeHHeM qaCTHbI nJIa3MbI MorYT
Y'laCTBOBaTb B pa3Hoo6pa3HbIX ynop.HAo"t!eHHbIX ABH.>KeHHSlX. B nna3Me nerKO
B036Y.>KaIOTC.H pa3Horo poa Kone6aHH.H H BOJIHbI.
IlpoBo.nHMOCTb nJIa3MbI YBeJIHqHBaeTC.H no Mepe pOCTa CTeneHH HOHH3al.{IUI.
3JIeKTpOnpOBOAHOCTb H TenJIOnpOBOHOCTb nOJlHOCTblO HOHH3HpOBaHHOH nJIa3-
MbI 3aBHC.HT OT TeMnepaTypbI no 3aKOHaM: _T 3 / 2 H _T 5 / 2 COOTBeTCTBeHHO. IlpH
BbICOKOH TeMnepaType nonHOCTblO HOHH3HpOBaHHa.H nJIa3Ma no CBoeH npOBOH-
MOCTH npH6J1HmaeTC.H K CBepxnpOBOHHKaM.
l1oHH3aH.H aTOMOB Mem3Be3HOH cpebI npOH3BO,AHTC.H H:3nyqeHHeM 3Be3 H
KOCMH'leCKHMH JIyqaMH - nOTOKaMH 6bICTpbIX "t!aCTH, npOHH3bIBalOHMH npo-
CTpaHCTBO BCeJIeHHOH no BceM HanpaBJIeHH.HM. B OTnH"t!He OT rOp.H'leH nJIa3MW
3Be3 TeMnepaTypa Mem3Be3HOH nJIa3MbI OqeHb MaJIa.
YnpaBneHHe BHmeHHeM nJIa3MbI B 3JIeKTpH'leCKHX H MarHHTHbIX no-
JI.HX .HBJI.HeTC.H OCHOBOH ee HCnOJlb30BaHH.H KaK pa60"t!ero TeJIa B pa3JIH1.I-
HbIX BHraTeJI.HX JI.H HenocpeCTBeHHoro npeBpaeHH.H BHYTpeHHen 3Hep-
rHH B 3J1eKTpH"t!eCKYIO - nJIa3MeHHbIe HCTO"t!HHKH 3JIeKTp03HeprHH, MarHH-
TorHpo.nHHaMHqeCKHe reHepaTOpbI. LI,JI.H KOCMHqeCKHX Kopa6JIeii nepcneK-
THBHO HCnOJIb30BaHHe MaJIOMOHbIX nJIa3MeHHbIX ABHraTeJleii. MOHa.H
CTPY.H nJIOTHOH nna3MbI. nOJIyqaeMa.H B nJIa3MOTpOHe, lllHpOKO Hcnonb3Y-
eTC.H ,AJI.H pe3KH H CBapKH MeTaJlJIOB, 6ypeHH.H CKBamHH, YCKopeHH.H MHO-
rHX XHMHqeCKHX peaKHH. IlpOBOA.HTC.H lllHpOKOMaCIIITa6HbIe HCCJIeOBaHHSI
no npHMeHeHHIO BbICOKOTeMnepaTYPHOH ITJla3MbI ,AJI.H C03aHH.H ynpaBn.He-
MbIX TepMO.HepHbIX peaKHH.
292

10.9. 3JIeKTpOHHaH 3MHCCHH
B Y3J1aX KpHCTaJlJlHqeCKOH pellIeTKH MeTanJlOB HaXO.HTC.H nOJlOmHTenbHbIe
HOHbI, 8 Memy HHMH CB060HO BHmYTC.H 3neKTpOHbI. OHH K8K 6bI nn8B81OT
no BceMY 06'beMY npOBOHHKa, Tal< KaK CHJlbI npHT.HmeHH.H K nOJlOmHTeJlb-
HbIM HOHaM pellIeTKH. eHCTBYIOHe Ha cB06oHbIe 3J1eKTpOHbI, HaXO.HHeC.H
BHYTPH MeTaJlJla, B cpeHeM B3aHMHO ypaBHoBeIlIHBaIOTC.H. AeHCTBHe CHJI npH-
T.H>KeHH.H co CTOpOHbI nOJlOmHTenbHbIX HOHOB Ha 3J1eKTpOHbI MelliaeT nOCJle,A-
HHM BbIHTH 3a npeeJlbI nOBepXHOCTH MeTaJlJla. JIHllIb HaH60Jlee 6bICTpbIe 3'ueK-
TpOHbI MorYT npeO)J;oneTb 3TO npHTSImeHHe H BbIneTeTb H3 MeTaJlna. OHaKO
COBceM nOKHHYTb MeTaJlJl 3J1eKTpOH He MomeT, TaK KaK npHT.HrHBaeTC.H nono-
mHTenbHbIM nOBepXHOCTHbIM HOHOM H TeM 3ap.HOM, KOTOpbIH B03HHK B Me-
Tanne B CB.H3H C nOTepeH 3neKTpOHa. PaBHoneHCTBYlOlU8.H 3THX CHn npHT.H>Ke-
HH.H He paBHa HYJlIO, a HanpaBJleHa BHYTPb Me-
TanJla nepneHHKyn.HpHO ero nOBepXHOCTH (pHC.
10.10). 1.Jepe3 HeKOTopoe BpeMSI 3neKTpOH no
,AeHCTBHeM 3THX CHJI MomeT B03BpaTHTbC.H B Me-
TanJl. CpeH 3J1eKTpOHOB, HaXO,A.HIQHXCSI B6nH3H
nOBepXHOCTH MeTanJla, HaHeTC.H 60nbllIoe 'llHC-
JlO TaKHX, KOTopMe BpeMeHHO 6YYT nOKH,AaTb
MeTaJlJl, a 3aTeM B03BpaaTbC.H o6paTHo. 3TOT
npoecc HanOMHHaeT HcnapeHHe mHKOCTH. B
KOHe KOHIJ;OB YCTaHaBJlHBaeTC.H HHaMH'lleCKOe
paBHOBeCHe Memy nOKHalOHMH H B03BpaIQa-
IOIQHMHC.H 3J1eKTpOHaMH. TaKHM 06pa30M, H8
rpaHHe MeTaJlna C BaKYYMOM B03HHKaeT ,ABOH-
HOH CJlOH 3neKTpH'leCKHX 3apSI,AOB, none KOTO-
poro non06Ho nOJlIO nJlOCKOrO KOHneHcaTopa.
3neKTpH'lleCKOe nOJle 3Toro CJlO.H MO>KHO C'llHTaTb
OHOpOHbIM (pHC. 10.11). Pa3HocTb nOTeHHa.
JlOB <PKO.I B 3TOM cJloe Ha3bIBaeTC.H KOHTaKT-
HOH pa3HOCTLIO nOTeHlJ,HaJlOB Memy MeTaJlnOM
H BaKYYMOM.
3TOT BOHHOH 3J1eKTpH'leCKHH cnOH He C03aeT nOJl.H EO BHelliHeM npo-
CTpaHCTBe, HO npen.HTCTByeT BbIXOY 3J1eKTpOHOB H3 MeTanna.
KaK nOK83bIBalOT paCqeTbI H cneHanbHO nOCTaBJleHHbIe onbITbl, TOnIQHHa
3Toro CJlO.H Mana H: paBHa npHMepHO 10- 10 M.
TaKHM 06pa30M, qT06bI nOKHHYTb MeTanJl H YHTH B oKpymalOIQYIO cpe,AY,
3JIeKTpOH OJlmeH COBepllIHTb pa60TY A B npOTHB CHJI npHT.HmeHH.H co CTOpOHbI
nOJIOmHTeJlbHOrO 3ap.Ha MeT8J1JIa H npOTHB CHJI OTTaJIKHBaHH.H OT OTpHll,a-
TenbHO 3ap.HmeHHOro 3J1eKTpOHHoro 06J1aKa. OHa npH6J1H3HTeJlbHO paBHa
As = eL\q>KOH' re e - 3apSI 3neKTpoHa. AJISI 3Toro 3J1eKTpOH onmeH 06JIa-
aTb OCTaTOqHOH KHHeTHqeCKOH 3HeprHeH.
MHHHMaJlbHYIO pa60TY AB' KOTOPYIO onmeH cOBepllIHTb 3neKTpoH 3a CqeT
cBoeH KHHeTHqeCKOH 3HeprHH ,AJI.H Toro, 'llT06bI BbIHTH H3 MeTaJlJl8 H He Bep-
HYTbC.H B Hero, Ha3bIBalOT pa60TOH BblxOJJ;a.
Pa60Ta BbIxoa 3aBHCHT TOJlbKO OT poa MeTaJlna Hero qHCTOTM. Pa60TY
BbIxoa npHH.HTO H3Mep.HTb B 3neKTpOHBonbTax (3B).
o
(f)
Puc. 10.10
-+++++++++++
:tt
Puc. 10.11
293

.D:JI.H qHCTbIX MeTaJIJIOB AD COCTaBJI.HeT HeCKOJIbKO 3J1eKTpOHBOJIbT. TaK, Ha-
npHMep, JlSI u;e3H.H ee 3HaqeHHe paBHO 1,81 sB, n.H nJlaTHHbI 6,27 3B.
BbIXO cB060HbIX SJIeKTpOHOB H3 MeTaJlna Ha3bIBaeTCSI 3MHccHeH 3J1eKTpO-
HOB. IIpH HOpMaJlbHbIX BHelIIHHX ycnOBH.HX 3JIeKTpOHHa.H 3MHCCH.H BbIpameHa
cna60, TaK KaK CpeH.H.H KHHeTHqeCKa.H 3HeprHSI XaOTHeCKOrO TennOBOrO ,ABH-
meHHSI 60JlblIIHHCTBa CB06oHbIX 3neKTpOHOB B MeTaJIJlaX ropa30 MeHblIIe pa-
60TbI BbIxoa. .D:JISI nOBbIlIIeHHSI HHTeHCHBHOCTH 3MHCCHH CJIeyeT YBeJIHHTb
KHHeTHqeCKYIO 3HeprHIO CB060HbIX 3JIeKTpOHOB O 3HaeHHH, paBHbIX HJIH 60nb-
IIIHX 3HaqeHHSI pa60TbI BbIxoa. ::horo MomHO OCTHrHYTb pa3J1HqHbIMH cnoco-
6aMH. Bo-nepBbIX, C03aHHeM 3JIeKTpHqeCKOrO nOJlSI OqeHb 60JIblIIOH Hanp.H-
>KeHHOCTH [E -10 6 C ). cnoco6Horo BLIpBaTL 3JIeKTpOHbl H3 MeTaJIJIa, - XQ-
J10AllaJl 3MHCCHJI. TaKa.H 3MHCCH.H Hcnonb3yeTC.H B 3neKTpOHHbIX MHKponpoeK-
Topax. BO-BTOpbIX, 60M6apHpOBKOH MeTaJIJIa 3JIeKTpOHaMH, npeJJ;BapHTeJIbHO pa-
30rHaHHbIMH 3JIeKTpH'leCKHM nOJIeM O OqeHb 60JIblIIOH CKOpOCTH, - BTOpH1J-
HaJl 3J1eKTpOHHaJl 3MUCCHJI. B-TpeTbHX, HHTeHCHBHbIM OCBeeHHeM nOBepXHOC-
TH MeTaJIJIa - c}>OT03MHCCHJI Ha .HBJIeHHH cpOT03MHCCHH OCHOBaH BHelIIHHH cpO-
T03cpcpeKT H YCTPOHCTBO BaKYYMHoro cpOT03neMeHTa. B-qeTBepTbIX. HarpeBaHHe
MeTaJlJla - TepM03J1eKTpOHHaJl 3MHCCHH. 8JIeKTpOHbI, HcnYCKaeMble HarpeTblM
TeJIOM, Ha3bIBaIOTC.H TepM03J1eKTpOHaMH, a caMO 3TO TeJIO - 3MHTTepOM.
10.10. 3JIeKTpH"IeCKHH TOK B BaKYYMe
nO.n: BaKYYMOM nOHHMalOT TaKoe COCTOSIHHe ra38 B cocye, npH KOTOpOM
JIHHa cBo60Horo np06era 3ap.HmeHHbIX 'tIaCTHU; npeBbIlliaeT pa3MepbI cocy.n;a,
r,Ae HaXOHTCSl ra3.
BaKYYM - HeaJlbHbIH H30Jl.HTOp. TaK KaK B HeM HeT CB060HbIX HOCHTe-
JIeH 3apSIa. .D:JI.H Toro qTo6bI 'tIepe3 npOCTpaHCTBO, B KOTOpOM C03aH BbICO-
KHH BaKYYM, nOlIIeJI TOK, HymHo HCKYCCTBeHHO BBeCTH B 3TO npOCTpaHCTBO
HCTO'lHHK CB060HbIX 3apSIOB. 8TO MomHO C,AeJIaTb C nOMOblO TepM03JIeKT-
pOHHOH 3MHCCHH, nOMea.H B BaKYYM MeTaJIJIHqeCKYIO npOBOJIOKY, KOTOpylO
MomHO BKJIlO'tIaTb B 3neKTpHqeCKYIO u;enb. IIpH nponYCKaHHH qepe3 Hee 3JIeKT-
pH'leCKOrO TOKa npOBonOKa HarpeBaeTCSI H cBo60.n;HbIe 3J1eKTpOHbl MeTanna npH-
06peTalOT 3HeprHIO, OCTaTO'lHYIO .n;JISI COBeplIIeHHSI pa60TbI BbIxoa. H, nOKH-
,AaSl MeTann, 06pa3YIOT B6JIH3H Hero 3neKTpOHHoe 06naKO. IIpoBoJloKa npH 3TOM
3apHmaeTC.H nonOmHTeJIbHO, Ii no BJIH.HHHeM 3JIeKTpH'tIeCKOrO nOJI.H 3JIeKTpO-
HbI H3 06JIaKa 'tIaCTH'tIHO B03BpaI..QaIOTC.H Ha 3JIeKTpo. B paBHOBeCHOM COCTOSl-
HHH 'tIHCJIO 3JIeKTpOHOB. nOKHHYBIIIHX 3JIeKTpOJl: B ceKYHY. paBHO 'tIHCJIY 3JIeK-
TpOHOB, B03BpaTHBIIIHXC.H H8 3J1eKTpO 3a 3TO BpeMSI. qeM BbIllie TeMnepaTypa
MeTanna, TeM BbllIIe nJIOTHOCTb 3JIeKTpOHHoro 06naKa.
.D:JI.H B03HHKHOBeHH.H TOKa He06xoHMO OnOJIHHTeJlbHOe YCJIOBHe - C03aHHe
3JIeKTpH'tIeCKOrO non.H, no eHCTBHeM KOToporo 3neKTpOHbI 6YYT ,ABHraTbCSI
HanpaBJIeHHO.
TOK B BaKYYMe npeCTaBJI.HeT C060H nOTOK 3J1eKTpOHOB. Pa3JIH'tIHe Mem.n;y
rOpH'lHM H XOJIOHbIM 3JIeKTpOaMH, Bna.HHHbIMH B COcy, npHBOHT K O.n;HO-
CTopOHHeH npOBo.n;HMOCTH 3JIeKTpH'tIeCKOrO TOK8 Memy HHMH. IIpH nOKnlOqe-
HHH 3JIeKTpO,AOB K HCTO'lHHKY TOKa Mem.n;y HHMH B03HHKaeT 3JIeKTpH'leCKOe
nOJIe. ECJIH nOJIOmHTeJIbHblH nOJIIOC HCTOqHHKa coeHHeH C XOJIO.n;HbIM 3neKT-
294

pO.n;OM (aHO.n;OM), 8 OTpRaTeJlbHbIH - C HarpeTbIM (KaTO.n;OM), TO Hanp.HmeH-
HOCTb 3neKTpRQeCKOrO nOJUI H8np8BJleHa K HarpeTOMY 3J1eKTpO.n;y. IIo.n; .n;eH-
CTBReM 3TOrO nOJl.H 3J1eKTpOHbI qaCTRqHO nOKH.n;SIOT 3J1eKTpOHHOe 06J1aKO H
,lJ;BRmYTC.H K XOJlO.n;HOMY 3J1eKTpO.n;y. 3J1eKTpRQeCKS.H uenb 38MbIKSeTC.H. R B HeH
YCT8H8BJIRBaeTC.H 3J1eKTpRqeCKRH TOR. ITPH npOTHBOnOnOmHOM BKJIlOqeHRR RC-
TOQHHK8 Hanp.HmeHHOCTb nOJl.H HanpaBJleH8 OT KaTO.n;8 R 8HO.n;y. 3JIeKTpRqeC-
KOe nOJle OTTaJlKRBaeT 3J1eKTpOHbI 06J1aK8 Ha3a.n; K KaTO.n;y. IJ;enb OKa3bIBaeT-
Cfl pS30MKHYTOH, H: TOR B enH OTCYTCTByeT. CJle.n;OBaTeJlbHO, .n;RO.n; 06J1a.n;seT
O.n;HOCTopOHHeH npOBo.n;HMOCTblO.
10.11. 3JIeKTpOHHaH JIaMna - ,lJ;HO,lJ;.
BOJIbT-aMnepHaH xa paKTepHCTHKa JJ.HO.ll.a
O.n;HOCTOpOHH.H.H npOBo.n;RMOCTb RCnOJIb3yeTC.H B 3J1eRTpOHHbIX npH60p8X C
,lJ;BYM.H 3J1eKTpo.n;aMR - BaKYYMHbIX .n;Ho.n;ax.
COBpeMeHHbIH BaKYYMHbIH .n;RO,A npe.n;CTSBJI.HeT C060H 6aJIJIOH R3 CTeKJlS HJlH
MeTSJlJlOKepaMHKR, H3 KOToporo OTK8Q8H B03.n;yx .n;o .n;aBJIeHR.H 10- 6 - 10- 7 MM
pT. CT. BHYTPH HerO pa3MeeHbI .n;BS 3J1eKTpo.n;s (pRC. 10.12). O,AHH H3
HHX - KaTO.n; - HMeeT BH.n; BepTHKaJlbHOrO MeTaJlJlHqeCKOrO HJlHH.n;pa, no-
KpbIBaeMoro 06bIqHO CJlOeM OKCH.n;S eJlOqHO-3eMeJlbHbIX MeTSJlnOB - 68pR.H,
CTpOHR.H, KaJlbR.H. TaKoH KaTO.n; Ha3bIBalOT OKCH.ZI.HLIM. ITPR HsrpeBaHRR no-
BepXHOCTb OKCH.n;Horo KaTo.n;a HcnYCKaeT ropa31I0 60JlblIIe 3neKTpOHOB, qeM no-
BepXHOCTb KaTO,Aa R3 qRCTOrO MeTaJln8. BHYTPR K8To.n;a pacnOJIOmeH R30nRpo-
BaHHbIH npOBO,AHRK, H8rpeBaeMbIH nOCTO.HHHbIM HJlH nepeMeHHbIM TOKOM. Ha-
rpeTbIH K8TO.n; HcnYCKaeT 3JIeKTpOHbI, .n;OCTRralOI.qHe aHo.n;a. AHO.n; JlSMnbI npe.n;-
CT8Bn.HeT C060H KpyrJIbIH HJIH OBaJlbHbIH URJIRH.n;p. HMelOIURH 06IUYIO OCb C
KaTO,lIOM. CXeMaTHQeCKOe H306pameHHe .llHO,n8 nOKa38HO Ha pHCYHKe 10.13
(H306pameH .n;HO.n; C RaTO,AOM np.HMOrO HaK8JIa).
BamHeHlIIeH xapaKTepRcTRKOH ,AHo.n;a .HBJI.HeTC.H ero BOJILT-aMnepHan xapaK-
TepHCTHRa - 3aBRCRMOCTb CHJlbI TOKa OT Hanp.HmeHR.H Mem.n;y 3J1eKTpo.n;aMR
I a = f(U a) apR nOCTO.HHHOM Hanp.HmeHHH HaKaJla U H = const. ,ZJ;n.H nonY'leHR.H
BOJlbT-aMnepHOH xapaKTepRCTHKR aHo.n;a MomHO BOCnOJlb30BaTbC.H 3J1eKTpHqeC-
KOH enblO, npRBe,AeHHOH Ha pRCYHKe 10.14, r.n;e npRMeH.HeTC.H .n;RO.n; C KaTO-
.n;OM KOCBeHHoro HaKana.
KaTO
AHO
8
EM
Puc. 10.12
Puc. 10.13
Puc. 10.14
295

1.
I.a ------ -
2
T 2 >Tl
BonbT-8MnepHa.H xapaKTepliCTIiKa ,Alio,n:a
C MeT8JInliQeCKliM KaTO,AOM (pIiC. 10.15)
.HBJI.HeTC.H HeJIIiHeHHOH B OTJlliQ1:le OT BOJIbT-
I.,
1
Tl
aMnepHOH X8p8KTepliCTliKIi MeT8JIJIliQeCKOrO
npOBo,n:HIiK8. IIpli Hanp.HmeHIiIi Mem,n:y KaTO-
,n:OM Ii aHo,n:OM, paBHOM HYJIIO. BbIJIeTeBllllie
1i3 K8To,n:a 3JIeKTpOHbI 06p83YIOT BOKpyr Hero
3neKTpOHHoe 06JIaKO (npOCTpaHCTBeHHbIH OT-
pliIl.aTeJIbHbIH 3ap.H,n:), OTT8JIKIiBalOlIl.ee BbIJIe-
TalOlIl.lie 1i3 KaTo,n:a ::weKTpOHbI. BOJIblllIiH-
CTBO 3JIeKTpOHOB B03BpalIl.aeTC.H Ha KaTO,n:. Ii
Jllillib He3HaQliTeJIbHOe IiX q1:lCJIO ,n:OCTliraeT
aHo,n:a. C YBeJIIit.IeHlieM U a QIiCJIO 3JIeKTpOHOB, ,AOCTliralOlIl.IiX aHO,n:a. YBeJIIit.IIi-
BaeTC.H Ii 3JIeKTpOHHOe 06JIaKO nOCTeneHHO YMeHblllaeTC.H. Kor,n:a me Bce Tep-
M03JIeKTpOHbI, BbIJIeTalOlIl.lie 1i3 KaTo,n:a, nona,n:alOT Ha aHO,n:, CIiJIa aHo,n:Horo
TOKa ,n:OCTliraeT HaCbIlIl.eHIi.H Iuac (Ha rpa<}:>IiKe pliCYHKa 10.15 - ropli30HTaJIb-
HbIH YQaCToK). B 06JIaCTIi 1 rpa<}:>IiKa, T.e. npli YBeJIIit.IeHIiIi aHo,n:Horo H8np.H-
meHIi.H OT 0 ,AO U llac , TOK t.Iepe3 ,n:1i0,n: B03paCTaeT, HO He npOnOpIl.IiOHaJIbHO
3
Hanp.HmeHIiIO, a no 3aKOHY Ia = k . U a 2. 3TO BbIpameHlie Ha3bIBalOT 4JOPMY-
JlOH BoryCJlaBCKoro-JIeHrMlOpa IiJIIi 3aKOHOM "Tpex BTOpbIX". B 06JIaCTIi 2
,n:aHHoro rpa<}:>IiKa, T .e. np1:l U  U llae , TOK 'lepe3 .L\IiO,n: He 3aBliCliT OT Hanp.H-
meHH.H. TaKOH TOK Ha3bIBalOT TOKOM HaCbIeHIiH. CliJI8 TOKa HaCbIlIl.eHIi.H on-
pe,Aen.HeTC.H qJOPMYJIOll Iliac = en. r,n:e n - 'llicno 3JIeKTpOHOB, BbIJIeTalOlIl.IiX B
e.L\HHIiIl.Y BpeMeHIi C nOBepXHOCTH KaTo,n:a; e - 3ap.H,lI. 3JIeKTpOHa. IIpli nOCTO-
.HHHOH TeMnepaType KaTo,n:a C1:IJIa TOKa B Mem3JIeKTpo,n:HOM npOMemYTKe 3aBH-
CHT OT aHo,n:Horo Hanp.HmeH1:I.H
,lIJI.H YBeJlIit.IeHH.H TOKa HaCbIlIl.eHIi.H HymHo nOBbICIiTb TeMnepaTYPY KaTo,n:a,
YBeJIIit.I1:IB C1:IJIY TOKa HaKaJIa.
,lIHO,n: nponYCKaeT TOK TOJIbKO B O,n:HOM HanpaBJIeHIiIi. 3TO ero CBOHCTBO
HCnOJIb3yeTC.H ,n:JI.H BbInp.HMJIeHIi.H nepeMeHHoro TOKa. )J,HO,n:, ,AellcTBYIOlIl.HH B
Kat.IeCTBe BbInp.HMHTeJl.H, Ha3bIB81OT KeHOTpOHOM. Ha pliCYHKe 10 16, a nOKa-
3aHbI cxeMbI o,n:HonOJlynepIiO,AHOrO, a Ha pliCYHKe 10.16, 6 - ,n:ByxnOJIynepH-
o,n:Horo Bbmp.HMJIeHH.H nepeMeHHoro TOKa.
o
U.
U.
Puc. 10.15
a
6
K
Puc. 10.16
296

Ha pHCYHKe 10.17, a, 6 nOKa3aHbI COOTBeTCTBYIOHe rpacpHKH 3aBHCHMO-
CTH CHJlbI BbInp.HMJleHHOrO TOKa, npOXO.HlQero qepe3 conpOTHBJleHHe R. OT
BpeMeHH.
a
6
I
I
t
t
Puc. 10.17
B OTJlHqHe OT nHoa TpeX3J1eKTpOnHa.H 3J1eKTpOHHa.H naMna -- TpHon --
coepmHT, KpOMe KaTona H aHo.qa, elQe H TpeTHH 3J1eKTpO -- ynpaBn.HlOlQYIO
ceTKY. 06bIQHO ceTKa npeCTaBn.HeT C060H cnHpanbHYIO npOBOJlOqKY, oKpyma-
lOlQYIO np.HMOJlHHeHHbIH KaTO,A, a OCb HnHHpHqeCKOrO aHoa COBna,AaeT C
OCblO KaToa H ceTKH (pHC. 10.18, a); YCJlOBHOe H306pameHHe TpHoa npHBe-
eHO Ha pHC. 10.18, 6.
a
6
K
I
A
-=- B.
C
R
K
Puc 10.19
A
c
K
Puc. lU.18
CxeMa BKnlOQeHHH TpHoa B enb JI.H YCHneHH.H aHO,AHOrO TOKa npHBee-
Ha Ha pHC. 10.19. 3ecb A aHO,Zl, JlaMnbI; K -- ee KaTO,A; C -- ceTKa;
Ba -- aHOHa.H 6aTape.H; Bc -- CeTOqHa.H 6aTape.H, C03alOa.H Hanp.HmeHHe
Memy ceTKoH H KaTOOM; R - nOTpe6HTenb TOKa.
CeTKa pacnOJlOmeHa 6J1Hme K KaTOY, qeM aHO, H Ha nYTH KaTO --
ceTKa Ha 3neKTpOHbJ eHcTByeT CYMMapHoe nOJle, C03aBaeMoe Mem,AY aHOOM
H KaTOOM (<p A - <PK) H Memy ceTKOH H KaTOOM (<PC - <PK ). Bo BpeM.H pa-
60TbI JlaMnbI JlHIllb qaCTb 3J1eKTpOHOB nonaaeT Ha ceTKY H BHmeTC.H K Ka-
TOY no BHeIIIHeH enH, 06pa3Y.H CeTOqHbIH TOK Ic.
ECJlH nOTeHHaJl ceTKH BbIllie nOTeHHana KaToa, TO BHmeHHe 3J1eKTpO-
HOB OT KaToa K aHOY YCKOp.HeTC.H, CHna aHOHoro TOKa paCTeT. EcnH me
nOTeHlI.HaJl ceTKH MeHbIlle nOTeHlI.Hana KaTona, TO BHmeHHe 3neKTpOHOB K
aHO.n;y 3aMenJl.HeTC.H. H CHna aHOnHoro TOKa YMeHbIllaeTC.H. IIpH OCTaTOqHO
60nbIIIOM no a6COJllOTHOMY 3HaqeHHlO OTpHuaTenbHOM nOTeHUHane ceTKH aHO,A-
HbIH TOK nonHOCTblO npeKpalQaeTC.H -- B 3TOM cnYQae rOBOp.HT, qTO "naMna
3anepTa" .
297

,Z:{JI.H ynpaBneHHSI TOKOM BHYTPb SneKTpOHHOH JlaMnbI BBO.HT OnOJlHHTeJlb-
HbIe CeTKH. JIaMny C BYM.H CeTKaMH Ha3bIBalOT TeTpO.u;OM, T.e. qeTbIpeXSJleK-
TpOHOH, C TpeM.H - neHTO.u;OM (n.HTHSJleKTpOHOH).
llO.HBneHHe SJIeKTpOHHbIX JlaMn H pa3H006pa3HbIX YCTPOHCTB. OCHOBaHHbIX
Ha HX npHMeHeHHH, CbIrpaJlO OrpOMHYIO pOJlb B pa3BHTHH pa,AHO.
10.12. 3JIeKTpOHHLIe nyqKII II IIX CBoHcTBa
3J1eKTpOHHLle "yqKH npe)J;CTaBJI.HIOT C060H nOTOK 6bICTpO JIeT.HHX sneKT-
pOHOB. 3J1eKTpOHHbIe nyqKH 06pa3YIOTC.H B SJleKTpOHHOH JlaMne Ii pa3J1HqHbIX
ra30pa3p.HHbIX YCTpoHcTBax.
3neKTpoHHbIe nyqKH 06J1aaIOT cJleYIOHMH CBoHcTBaMH:
1) BbI3bIBalOT CBeqeHHe HeKOTopbIX TBep,AbIX H JKH,AKHX TeJl (cTeKJla, CYJlb-
cpHOB IJ;HHKa H KaMH.H). B HaCTO.Hee BpeM.H npHMeH.HIOTC.H nlOMHHocpOpbI. y
KOTOpbIX 1I0 25 % SHeprHH SJIeKTpOHHoro nyqKa npeBpaII.1aeTC.H B CBeTOBYIO:
2) npH TopMomeHHH 6bICTpbIX SJleKTpOHHbIX nyqKOB B BeeCTBe B03HHKa-
eT peHTreHOBCKoe H3nYQeHHe. 3TO Hcnonb3yeTcSI B peHTreHOBCKHX Tpy6Kax;
3) SJleKTpOHHble nyqKH OTKJlOH.HIOTC.H B SJleKTpHqeCKHX non.HX, HanpHMep,
B none nnOCKoro KOHeHcaTopa npOHCXOHT CMeeHHe SJleKTpOHHoro nyqKa K
nonOmHTeJlbHO 3ap.HmeHHOH nnaCTHHe;
4) SJleKTpOHHble nyqKH OTKnOHSlIOTC B MarHHTHbIX non.HX BCJleCTBHe eH-
CTBH.H Ha SJleKTpOHbI CHJlbI JIopeHIJ;a. llpOneTa.H Ha ceBepHbIM nOJlIOCOM Mar-
HHTa, SJleKTpOHbI OTKnOH.HIOTC.H B OHY CTOPOHY, a npOJleTaSl Ha IOmHbIM no-
JlIOCOM - B npOTHBOnOJlOmHYIO CTOPOHY. OTKJlOHeHHe SJleKTpOHHbIX nOTOKOB,
H)J;yHX OT ConHn;a, B MarHHTHOM none 3eMJlH npHBoHT K TOMY, qTO OHH
orH6alOT nOBepXHOCTb 3eMnH H JlHllIb B nOJlSlpHbIX 06J1aCTSlX He60JlbllIaSl
qaCTb STHX qaCTHIJ; BTopraeTC.H B BepXHHe CJlOH aTMoccpepbl H BbI3bIBaeT CBe-
qeHHe ra30B aTMoccpepbI y nOJlIOCOB (ceBepHoe cHSlHHe);
5) npH nona.n:aHHH Ha BeeCTBO SJleKTpOHHbIe nyqKH HarpeBalOT ero H
OKa3bIBalOT MeXaHHqeCKOe .neHCTBHe. HarpeBaHHe, KOTopoe BbI3bIBaeT sneKTpOH-
HbIH nyqoK, nonaaSl Ha KaKOe-JIH60 TeJlO, HcnOJlb3YlOT JI.H nJlaBKH CBepXqH-
CTbIX MeTaJlJlOB B BaKYYMe;
6) SJleKTpOHHbIH nyqOK npli nonaaHHH Ha cpoTonJleHKY BbI3bIBaeT ee no-
TeMHeHHe.
BnaroapSl B03MomHOCTH ynpaBJI.HTb SJleKTpOHHbIM nyqKOM C nOMOblO
sneKTpHqeCKOrO HJlH MarHHTHoro nOJl.H H CBeqeHHIO nOKpbIToro JlIOMHHOcpOpOM
3KpaHa no eHcTBHeM nyqKa ero npHMeHSlIOT B sJleKTpoHHo-JlY'leBOH Tpy6Ke.
10.13. 3JIeKTpOHHO-JIyqeBaSl Tpy6Ka
Puc. 10.20
B OCHOBY pa60TbI sJleKTpoHHo-nYQe-
BOH Tpy6KH nOJlOmeHO OTKJlOH.HIOee
.n:eHcTBHe sJleKTpHqeCKOrO H MarHHTHoro
F nOJl.H Ha SJleKTpOHHbIH nyqOK. CxeM8
YCTpoHcTBa 3JIeKTpOHHo-JlyqeBoH Tpy6KH
npHBe,AeHa Ha pHC. 10.20.
B ee Y3KHH KOHen; BMOHTHpOBaH8
SJleKTpOHHa.H nYIliKa ll, COCTO.Ha.H H3
298

KaToa K, aHoa A H HeCKOJlbKHX MeTaJ":JlHqeCKHX KOJleu. (YCKOp.HIOHX
aHooB). <1>OPMY, nOJIOmeHHe aHOOB H HalIp.HmeHHe Ha HHX Bb16HpalOT TaK,
qT06bI OHOBpeMeHHO C YCKopeHHeM 3J1eKTpOHOB IIpOHCXOlIHJIa H OKycHpoBKa
3J1eKTpOHHoro nyqKa. T.e. YMeHbIlleHHe IIJlOalIH 1I0nepeqHOrO CeqeHH.H
3J1eKTpOHbI BbIJleTalOT H3 KaTona. HarpeBaeMoro 3J1eKTpHqeCKHM TOKOM, a
3neKTpHQeCKOe nOJle MemAY KaTOOM H MeTaJIJlHqeCKHMH KOJIbnaMH (<}>OKYCH-
PYIOero YCTpOHCTBa) CBOHT HX B Y3KHH nOK - 3JIeKTpOHHblH JIY'!. lliH-
pOKoe HO :3 3J1eKTpoHHo-JlyqeBOH TPy6KH nOKpbITO CJlOeM <}>JlyopeCnHPYIOero
BeeCTBa H CJlymHT 3KpaHOM. IIo eHCTBHeM Y)J;apOB nOlIaalOHX Ha Hero
3neKTpOHOB 3KpaH CBeTHTC.H, H B TOM MeCTe, Kya nonaaeT 3J1eKTpOHHbIH
nYQ, nO.HBJI.HeTC.H 06bIqHO 3eJleHOe CBeTJIOe II.HTHbIillKO F. Memy 3J1eKTpOHHOH
IIYIllKOH H 3KpaHOM nOMeeHbI YlipaBJI.HIOHe 3JIeKTpObI, npeCTaBJI.HIOHe CO-
60H CHCTeMY ropH30HTaJlbHO H BepTHKaJlbHO pacnOJlOmeHHbIX lIap nJlaCTHH, T .e.
KOHeHcaTopOB C 1 H C 2 . 8J1eKTpHqeCKHe nOJI.H 3ap.HmeHHbIX KOHeHcaTopOB
B3aHMHO nepneH,AHKYJI.HpHbI. IIoJIe KOH,AeHCaTOpa C} OTKJlOH.HeT Jlyq B ropR-
30HTaJlbHOM HanpaBJleHHH, nOJle KOH,AeHCaTOpa C 2 -- B BepTHKaJlbHOM. H3Me-
H.HR Hanp.HmeHHe Ha IIJlaCTHHKax K8moro H3 KOHeHcaTopOB, MomHO OTKJlO-
HHTb 3J1eKTpOHHbIH Jlyq B JII060M HalipaBJleHHH TaK. QTO n.HTHbIIllKO B03HHKa-
eT Ha 3KpaHe Ha pa3JIHqHbIX paCCTO.HHH.HX OT ero ueHTpa.
B u.eHTp 3KpaHa 3J1eKTpOHbI lIonaaIOT, KOr,Aa KOH)J;eHcaTopbI He 3ap.HmeHbI.
3JIeKTpOHHLIM OCU;HJIJIOrpa4>oM Ha3bIBalOT 3J1eK-
TpOHHo-JlyqeBYIO Tpy6KY, npHMeH.HeMYIO JI.H HCCJle-
OBaHH.H 6bICTponpOTeKalOHX 3J1eKTpH'leCKHX npo-
u.eCCOB. CJIOBO OCnHJlJlorpa<}> 03HaqaeT "3aIIHCbIBalO-
HH KOJle6aHH.H". Ha nepBbIH KOH,AeHCaTOp C 1 OC-
nHJlJlorpa<}>a HaKJlabIBaeTC.H H3MeH.HIOeeC.H BO Bpe-
MeHH nHJIo06pa3Hoe Hanp.HeHHe (pHC. 10.21). Ha
npOT.HmeHHH Kamoro nepHoa OHO CHaqaJla nJlaB-
HO paCTeT, a 3aTeM MrHOBeHHO naaeT. IT03TOMY
II.HTHbIIllKO Ha 3KpaHe BHmeTC.H CHaqaJIa CJleBa Ha-
npaBO, a IIOTOM MrHOBeHHO B03BpaaeTC.H B HCXO,A-
Hoe nonomeHHe, a TaK KaK qaCTOTa KOJle6aHHH HalIp.HmeHHR BeJlHKa, TO rJIa3
Bce BpeM.H BHlIHT ropH30HTaJlbHYlO CBeTJlYIO np.HMYIO. ECJlH, HanpHMep, Ha IIJIa-
CTHHbI BToporo KOHeHcaTopa C BepTHKaJIbHO HanpaBJIeHHbIM nOJleM nO,AaTb Ha-
IIp.HmeHHe CHHycOH,AaJlbHOrO nepeMeHHoro ropocKoro TOKa (v = 50 ru.). TO npH
O,AHOBpeMeHHOM eHCTBHH KOHeHcaTopOB 3neKTpOHHbIH Jlyq onHIlleT Ha 3Kpa-
He CHHYCOHy, npeCTaBJI.HIOYIO C060H OCnHJlJlOrpaMMY HCCJleyeMoro HanpR-
meHH.H .
B HeKOTopbIX THnax 3J1eKTpoHHo-JlyqeBbIX Tpy60K OTKJIOHeHHe 3J1eKTpOHHO-
ro nyqKa IIpOH3BO)J;HTC.H MarHHTHbIM nOJIeM. IIpH 3TOM BMeCTO OTKJlOH.HIOHX
IIJIaCTHH eHCTBYIOT )J;Be B3aHMHO nepneHHKYJI.HpHble napbI .KaTYIlleK, pacno-
JIOmeHHbIe cHapymH Tpy6KH. Kama.H napa KaTYIlleK C03aeT nepneH,AHKYJI.Hp-
Hoe Jlyqy MarHHTHoe nOJle.
8J1eKTpOHHO-JlY'ieBble Tpy6KH HMeIOT orpOMHoe npaKTHqeCKOe 3HaqeHHe. Hx
IIpHMeH.HIOT B palIHOJlOKanHOHHbIX YCTaHOBKax. TeJleBH30pax. 3J1eKTpOHHbIX
MHKpOCKonax H pyrHx npH6opax. Be3 3JIeKTpOHHoro OCUHJlJlOrpa<}>a He 06xo-
JIHTC.H HH onHa <}>H3HqeCKa.H Jla60paTOpH.H, HM IllHpOKO nonb3YIOTC.H B MeHnH-
He, 6HOJIOrHH H T..
u
o
Puc. 10.21
299

10.14. 3JIeKTpH'IeCKHH TOK B nOJIynpoBO,ZJ;HHKax
IIoJIynpoBO,AHHKH 3aHHMalOT npOMemYTOqHOe nOJIOmeHHe no 3J1eKTpOnpo-
BO,AHOCTH (HJIH no yeJIbHOMY conpoTHBneHHIO) Mem,AY npOBO,AHHKaMH H H-
3neKTpHKaMH. O,AHaKO 3TO ,AeJIeHHe Bcex BeeCTB no HX CBoi:f.cTBY 3J1eKTpO-
npOBOHOCTH HBJI.HeTC.H YCJIOBHblM t TaK KaK no ,AeHCTBHeM p.H,Aa npHqHH (npH-
MeCH, 06JIyqeHHe, HarpeB8HHe) 3JIeKTpOnpOBO)J;HOCTb H Y)J;eJlbHOe conpOTHBJIe-
HHe Y MHorHX BeeCTB BeCbM8 3HaqHTeJlbHO H3MeH.HIOTC.H, oc06eHHO Y nOJlY-
npOBOHHKOB .
B CB.H3H C 3THM nOJIynpoBoHHKH OT MeTaJIJlOB OTJlHq81OT no lJ,enOMY p.HY
npH3HaKOB:
1) yeJlbHOe conpOTHBJleHHe Y nOJIynpoBoHHKOB npH 06blqHbIX YCJlOBHHX
rOpa3,A0 60JIbllIe, qeM Y MeTaJIJlOB;
2) Y,AeJIbHOe conpOTHBneHHe qHCTbIX nOJIynpoBoHHKOB YMeHbllIaeTCH C po-
CTOM TeMnepaTypbl (y MeTanJIOB OHO p8cTeT);
3) npH OCBeI.l{eHHH nOJlynpoBoHHKOB I1X conpOTHBneHHe 3HaqHTeJlbHO
YMeHbIlI8eTC.H (Ha conpOTHBJleHHe MeTaJlJlOB CBeT nOqTH He BJIH.HeT):
4) HHQTOmHOe KOJlHQeCTBO npHMeceH OKa3b1BaeT CHJlbHOe BJIH.HHHe H8 co-
npOTHBJIeHHe nOJlynpoBoHHKOB.
K nOJlynpoBoHHKaM npHHaJIemaT 12 XHMHqeCKHX 3neMeHTOB B cpeHeH
QaCTH Ta6JIHlJ,bI MeH,AeJleeBa - B, C, Ge, Sn. P, Te, Se, 1, Sb. As, S, Si,
coeHHeHH.H 3J1eMeHTOB TpeTbeH rpynnbl C 3J1eMeHTaMH n.HTOH rpynnbI, MHorHe
OKCHbI H CYJlbcpHbl MeT8J1JIOB, pSl,A pyrHx XHMHQeCKHX coeHHeHHH, HeKOTO-
pble OpraHHqeCKHe BeI.l{eCTBa. HaH60nbllIee npHMeHeHHe ,AJI.H HaYKH H TeXHHKH
HMelOT repMaHHH Ge H KpeMHHH Si.
IIoJlynpoBoHHKH MorYT 6bITb QHCTbIMH H C npHMeCHMH. CooTBeTcTBeHHo
pa3JIHQalOT co6CTBeHIfYIO H npHMeclfYIO npOBO):(HMOCTL nOJIynpoBoHHKOB. IIpH-
MeCH B CBOIO OQepeb eJI.HT Ha OHopHLle 11 aKl.\enTOpHLle.
8JIeKTpHQeCKHH TOK B nonynpoBoHHKax C03aeTC.H ynop.HOQeHHbIM ,ABHme-
HHeM 3JIeKTpOHOB H lIbIpOK B lIBYX npOTHBOnOJlOmHbIX HanpaBJIeHH.HX. qTO 6y-
eT paCCMOTpeHO HHme.
10.15. Co6cTBeHHaH 3JIeKTpH'IeCKaH npOBOIJ:HMOCTb
nOJIynpOBOl{HMKOB H ee 3aBMCHMOCTb
OT TeMlIepaTypbI H OCBeeHHOCTH
.I(JI.H nOHHMaHHH MexaHH3Ma 3J1eKTpH'leCKOH npOBO,AHMOCTH B nOJIynpOBO,A-
HHKax paCCMOTpHM CTpoeHHe nOJIynpoBoHHKOBbIX KpHCTaJlJIOB H npHpoy CBH-
3eH, yepmHBalOHX aTOMbI KpHCTaJIJla pyr
B03JIe pyra.
KpHCTaJlnbI repMaHH.H R ,ApyrHx nOJlynpo-
BO,AHHKOB HMelOT aTOMHYIO KpHCTaJIJIHQeCKYIO
pellieTKY. TInoCKa.H cxeMa CTPYKTypbI repM8-
HH.H nOKa3aHa Ha pHCYHKe 10.22.
repMaHHH - qeTblpexBaneHTHbIH 3ne-
MeHT, BO BHellIHeH 060JlOQKe aTOMa eCTb Qe-
Tblpe 3J1eKTpOHa, cJla6ee CBH3aHHbiX C .HpOM,
Puc. 10.22 qeM OCTaJIbHble. lIHCJlO 6J1HmaHIlIHx coceeH
300

Ka)KJJ;oro aTOM a repMaHH.H TaK)Ke paBHO 4. B3aHMoJJ;eHcTBHe rrapbI coceJJ;HHX
aTOMOB ocyeCTBJI.HeTC.H rrocpeJJ;CTBOM rrapHoaneKTpoHHOH - KOBaJIeHTHOH
CBH3H.
B ee 06pa30BaHlui yqacTByeT no OJJ;HOMY aneKTpOHY OT KamJJ;oro 8TOM8. 3TH
KOBaJIeHTHble CBH3H rrpH HH3KHX TeMnepaTypax JJ;ocTaTO'lHO npo'tJHbl, H CBo6o..Q-
HbIX aneKTpOHOB B KpHCTanne nOt.lT:H HeT. II03TOMY nonynpOBO..QHHKH npH HH3-
KOM TeMnepaType He npOBO,A.HT aneKTpHt.leCKHM TOK. Yt.laCTBYlOlliHe B CBH3H Ba-
neHTHble aneKTpOHbI rrpOt.lHO npHB.H3aHbI K KpHCT8J1nH'tJeCKOM pemeTKe, H BHem-
Hee aJleKTpH'leCKOe nOJIe nOqTH He BJIHHeT Ha HX JJ;BHmeHHe.
3neKTpOnpOBOJJ;HMOCTb XHMH'tJeCKH 'tJHCToro nonynpOBOJJ;HHKa B03MO)KH8 B
TOM cnY'lae, KorJJ;a KOBaJIeHTHbIe CB.H3:H B KpHCTaJlJIaX pa3pbIBalOTCH H nOHB-
JIHIOTCH cB060JJ;HbIe aneKTpOHbI. HanpHMep, HarpeBaHHe JJ;ame JJ;O He60nbllIHX
TeMrrepaTYP npHBOJJ;HT K pa3pbIBY KOBaJIeHTHbIX CBH3eM, nO.HBJIeHHIO CB060..Q-
HbIX 3J1eKTpOHOB H B03HHKHOBeHHIO co6CTBeHHoii aJIeKTpOHHOH npOBOJ].HMOCTH
'lHCTOrO rrOJIynpOBOJJ;HHKa (npOBOJJ;HMOCTH n-THn8). IIocJIe yxo..Qa aJIeKTpOHa co
CBoero MeCTa B aTOM 06JI8CTH KpHCTaJIJIa HapYllIHnacb ero HeHTpaJIbHOCTb. B
TOM MeCTe, OTKYJJ;a ymen aneKTpOH, B03HHKaeT H36bITOt.lHbIM nOJIO)KHTeJIbHbIH
3apHJJ; - o6pa3yeTcH rrOJIO)KHTenbHaH "JJ;bIpKa" (pHC. 10.22). OHa 06JIa,AaeT
rrOnO)KHTeJlbHbIM 3apHJJ;OM, paBHbIM no MO..QynlO 3apH.D;y aJIeKTpoHa. Ha oCB060-
..QHBllIeeC.H OT aneKTpoHa BaKaHTHoe MeCTO - ,AbIPKY - MO)KeT nepeCKOqHTb
.n:pyroM aneKTpoH. a aTO aKBHBaneHTHO nepeMeIIIeHHIO JJ;bIPKH B HanpaBneHHH,
rrpoTHBOnOJIO)KHOM HanpaBJIeHHIO JJ;BH)KeHH.H 3JIeKTpoHa. B OTCYTcTBHe BHellIHe-
ro 3JIeKTpH'tJecKoro rrOJl.H 3TH cB06oJJ;HbIe aneKTpOHbI H JJ;bIPKH JJ;BH)KYTCH B KPH-
CTaJIJIe nOJIyrrpOBOJJ;HHKa xaOTH'tJeCKH.
Bo BHelliHeM aneKTpHt.leCKOM none aneKT-
pOHbI nepeMealOTCH B CTOPOHY, npOTHBono-
JIOmHYIO HanpaBneHHIO HanpH)KeHHOCTH 3neKT-
pH'tJeCKoro nonH. IIOnO)KHTenbHbIe JJ;bIPKH ne-
peMe8IOTCH B HarrpaBJIeHHH HanpH)KeHHOCT:H
aneKTpHt.leCKOrO non.H (pHC. 10.23). I1poecc
nepeMeeHH.H aneKTpOHOB H JJ;bIPOK BO BHellI-
HeM nOJIe npOHCXOJJ;HT no BceMY 06'beMY no-
nynpOBOJJ;HHKa.
8JIeKTpOnpOBO..QHOCTb t.lHCTOrO rrOJIynpOBO..Q-
HHKa, 06YCJIOBneHHaH ynopH..QOt.leHHbIM nepeMe-
eHHeM JJ;bIPOK, Ha3bIBaeTCH c06CTBeHHOM JJ;bI-
pO'lHOH npOBOJJ;HMOCTblO (npOBO..QHMOCTb p-THna). 06aH yJJ;eJIbHaH 3JIeKTpOnpo-
BOJJ;HOCTb nonynpOBOJJ;HHKa CKJIa,AbIBaeTCH H3 npOBOJJ;HMOCTeM n-THna H p-THna.
I1pH aTOM y 'tJHCTbIX nonynpOBOJJ;HHKOB 'tJHcno aneKTpOHOB npOBOJJ;HMOCTH Bce-
r.n:a paBHO t.lHCJlY JJ;bIpOK.
,[(ononHHTenbHa.H aHeprH.H, KOTopaH JJ;onmHa 6bITb 3aTpat.leHa, t.lTo6bI pa-
30pBaTb rrapHOaJIeKTpOHHYIO CB.H3b H CJJ;eJIaTb aneKTpOH CBo60JJ;HbIM, Ha3bIBaeTCH
3HeprlfeH aKTHBaHH. I10JIy'tJHTb aTY aHeprHIO aJIeKTpOHbI MorYT npH HarpeBa-
HHH KpHCTaJIJIa, npH 06JIY'leHHH ero BbICOKO'tJaCTOTHbIMH 3neKTpOMarHHTHbIMH
BonHaMIf H T .JJ;.
C nOBbIllIeHHeM TeMnepaTypbJ B03paCTaeT t.lHCJIO pa3pbIBOB KOBaJIeHTHbIX CBH3eM
H YBeJIHt.lHBaeTCH KOJlH'tJeCTBO CB06o..QHbIX aneKTpOHOB H ..QbIpOK B KpHCTaJIJIaX t.lHC-
TbIX nOJlynpoBOJJ;HHKOB, a CJIeJJ;OBaTeJlbHO, B03paCTaeT yJJ;eJIbHa.H aneKTponpOBO..QHOCTb
"
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.
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Puc. 10.23
301

p
o
Puc. 10.24
H yMeHbllIaeTC.H Y,AeJlbHOe conpOTHBJleHHe tIHCTbIX nOJlY-
npoBoHHKOB. rpaqmK 38BHCHMOCTH yeJIbHOrO conpoTHB-
JleHH.H qHCTOrO nOJIynpoBoHHK8 OT TeMnepaTypbJ npHBe-
eH H8 pHC. 10.24.
KpoMe H8rpeBaHH.H, p83PbIB KOB8neHTHbIX CB.H3eH
H, K8K CJIeCTBHe, B03HHKHOBeHHe c06cTBeHHoH npo-
BO,AHMOCTH nOJlynpOBO,AHHKOB H YMeHbllleHHe Y,AeJIbHO-
t ro conpOTHBJIeHHSI MorYT 6bITb BbI3B8HbI OCBeeHHeM
(<l>oTonpoBoHMOCTb nOJlynpOBO,AHHK8), a T8Kme eii-
CTBHeM CHJIbHbIX 3JIeKTpH'tleCKHX nOJIeH.
10.16. npHMecHaH npOBO,ZJ;HMOCTL nOJIynpOBO,ZJ;HHKOB
C06CTBeHH8.H npOBOHMOCTb nonynpoBoHHKOB 06bIqHO HeBeJlHK8, T8K K8K
qHCJlO cB060HbIX 3J1eKTpOHOB, H8npHMep, B repMaHHH npH KOMHaTHOH TeMne-
p8Type nOp.HK8 3. 10 13 CM- 3 . B TO me BpeM.H 'tlHCJIO 8TOMOB repM8HHfl B
1 cM 3 -10 23 . IIpoBoHMocTb nOJIynpoBoHHKOB YBeJlHqHB8eTC.H C BBeeHHeM npH-
Meceii, KOr,A8 Hap.H,Ay C c06cTBeHHoii npOBOHMOCTblO B03HHK8eT OnOJIHHTeJlb-
H8.H npHMeCH8.H npOBOHMOCTb.
IIpHMecHoH npoBoAHMOCTbJO nOJlynpoBoHHKOB H83bIB8eTC.H npOBOHMOCTb,
06YCJIOBJIeHH8.H H8JIHqHeM npHMeceii B nOJIynpOBO,AHHKe.
IIpHMecHbIMH eHTp8MH MorYT 6bITb:
a) 8TOMbI HJIH HOHbI XHMHqeCKHX 3J1eMeHTOB, BHepeHHbIe B pellieTKY no-
JIynpOBO,AHHK8;
6) H36bITO'tlHbIe 8TOMbI HJlH HOHbI, BHe,ApeHHbIe B Mem,AOY3JIH.H pellleTKH;
B) P83JIHqHOrO poa pyrHe e<l>eKTbI 1-1 HCK8meHH.H B KpHCTaJIJIHqeCKOii pe-
llleTKe: nYCTbIe Y3J1bI, TpeHHbI, C,ABHrH, B03HHK81OHe npH ,Ae<l>opM8HSIX KPH-
CTaJlJlOB, H p.
113MeH.HSI KOHeHTpaHIO npHMeceii, MomHO 3Ha'tlHTeJlbHO YBeJlH'tlHBaTb qHC-
JIO HOCHTeneH 38pSIOB Toro HJIH HHoro 3H8K8 H C038BaTb nonynpoBoHHKH
C npeHMyecTBeHHOH KOHeHTpaHeH JlH60
OTpH8TenbHO, JlH60 nOJlOmHTeJlbHO 3ap.HmeH-
HbIX HOCHTeneii.
IIpHMecH MomHO p83eJIHTb H8 .n;OHopHLle
(OT,AalOI..UHe) H aKu.enTopUble (npHHHMalOI..UHe).
PaCCMOTpHM MeX8HH3M 3JIeKTpOnpOBOHOCTH
nOJIynpoBoHHK8 C OHOpHOH n.HTHB8neHTHoii
npHMeCblO MbIlllb.HKa As5+, KOTOPYIO BBO.HT B
KpHCTBJIJI, H8npHMep, KpeMHH.H. IISITHB8neHT-
HbIH aTOM MbllllbSIK8 OTaeT 'tleTbIpe BaJleHT-
HbIX 3J1eKTpOHa H8 06pa30B8HHe KOBaJleHTHbIX
CB.H3eH, a n.HTbIH 3JIeKTpOH OKa3bIBaeTC.H He38-
H.HTbIM B 3THX CB.H3.HX (pHC. 10.25).
3HeprH.H OTpbIB8 (3HeprH HOHH38HH) n.HTOrO BaneHTHoro 3JIeKTpOH8 MbI-
lllbSIK8 B KpeMHHH paBH8 0,05 3B = 0.08. 10- 19 ,l];m, qTO B 20 p83 MeHbllle
3HeprHH OTpbIBa 3JIeKTpOH8 OT aTOMa KpeMHHSI. II03ToMY yme npH KOMH8THOii
TeMnep8Type nOqTH Bce 8TOMbI MbllllbHK8 Tep.HlOT OHH H3 CBOHX 3JIeKTpOHoB
H CT8HOB.HTC.H nOJIOmHTeJlbHbIMH HOH8MH. IIOJlOmHTeJIbHbIe HOHbI MbIlllb.HK8 He
Puc. 10.25
302

MorYT 3aXBaTHTb ::meKTpOHbI COCeHHX aTOMOB, TaK KaK BCe qeTbIpe CB.H3H Y
HHX yme YKOMnJleKTOBaHbl 3J1eKTpOHaMH. B 3TOM cJlyqae nepeMeI..QeHH.H 3J1eKT-
pOHHOH BaKaHCHH - "bIpKH" He npOHCXOHT H .lIbIpOqHa.H npOBO.lIHMOCTb OqeHb
Mana, T.e. npaKTHqeCKH OTCYTCTByeT. He60JlblIIaSl qaCTb c06CTBeHHbIX aTOMOB
nOJlynpOBo.nHHKa HOHH3HpOBaHa, H qaCTb TOKa 06pa3yeTC.H ,AbIpKaMH, T.e. O-
HopHbIe npHMeCH - 3TO npHMeCH, nOCTaBJI.HIOI..QHe 3J1eKTpOHbI npOBOHMOCTH
6e3 B03HHKHOBeHH.H paBHoro KOJlH'leCTBa nO,ABHmHbIX ,AbIpOK. B HTore MbI no-
nyqaeM nOJIYIlpOBOJJ:HHK C npeHeCTBeHHO 3J1eKTpOHHOH npOBO,AHMOCTblO, Ha-
3bIBaeMbIH nOJIYIlpOBOJJ:HHKOM n-TlUla.
B cJlyqae aKn;enTopHOH npHMeCH, HanpH'
Mep, TpeXBaJleHTHOro HHH.H In 3 + aTOM npH-
MeCH MomeT aTb CBOH TPH 3J1eKTpOHa )J;JI.H
OCYI..QeCTBneHH.H KOBaJleHTHOH CB.H3H TOJlbKO C
TpeM.H coceHHMH aTOMaMH KpeMHH.H, a O-
Horo 3J1eKTpOHR «He,AOCTaeTI) (pHC. 10.26).
OHH H3 SJleKTpOHOB coceHHX aTOMOB KpeM-
HH.H MomeT 3anOJlHHTb STY CB.H3b. TOr.lIa
aTOM In CTaHeT HenO.lIBHmHbIM OTpHn;aTeJlb-
HbIM HOHOM, a Ha MeCTe YlIIe,AlIIero OT O-
Puc. 10.26 Horo H3 aTOMOB KpeMHH.H 3J1eKTpOHa o6pa-
3yeTC.H bIpKa. AKIJ;enTOpHbIe npHMecH, 3ax-
BaTbIBa.H 3neKTpOHbJ H c03)J;aBa.H TeM caMbIM nO)J;BHmHbIe )J;bIPKH, He YBeJlH-
'1HBalOT npH 3TOM '1HCJla 3J1eKTpOHOB npOBO,AHMOCTH. OCHOBHbIe HOCHTeJlH 3a-
p.Ha B nonynpoBoHHKe C aKn;enTopHOH npHMeCblO - bIpKH, a HeOCHOB-
HbIe - SJleKTpOHbl.
rrOJlynpoBoHHKH, y KOTOpbIX KOHIJ;eHTpan;H.H ,AbIpOK npeBbIlliaeT KOHn;eHT-
pan;HIO 3J1eKTpOHOB npOBOHMOCTH, Ha3bIBaIOTC.H nOJlynpoBOJJ:HHKaMH p-THna.
He06xoHMO oTMeTHTb, '1TO BBeeHHe npHMeceH B nonynpoBoHHKH, KaK H
B JlI06bIX MeTaJlJlaX, HapYlliaeT cTpoeHHe KpHCTaJlJlHqeCKOH pelIIeTKH H 3aT-
pYH.HeT BHmeHHe 3J1eKTpoHoB. OHaKO conpOTHBJleHHe He YBenHqHBaeTC.H H3-
3a Toro, qTO YBeJlHqeHHe KOHn;eHTpan;HH HOCHTeneH 3ap.H.lIOB 3HaqHTeJlbHO
YMeHblIIaeT conpOTHBJleHHe. TaK, BBel(eHHe npHMeCH 60pa B KOJlH'IeCTBe 1 aTOM
Ha CTO TbIC.Hq aTOMOB KpeMHH.H YMeHblIIaeT yenbHoe 3J1eKTpHqeCKOe conpo-
THBJleHHe KpeMHH.H npH6J1H3HTeJlbHO B TbIC.Hqy pa3, a npHMeCb O,AHOrO aTOMa
HHH.H Ha 10 8 - 10 9 aTOMOB repMaHH.H YMeHblIIaeT yenbHoe 3J1eKTpH'IeCKOe CO-
npOTHBJleHHe repMaHH.H B MHJlnHOHbI pa3.
B03MomHocTb ynpaBJleHH.H yeJlbHbIM conpOTHBJleHHeM 6J1aroap.H BBeeHHIO
npHMeceH HcnoJlb3yeTC.H B nOJlynpoBoHHKOBbIX npH60pax.
AbIpOqHa.H npOBOHMOCTb He .HBJI.HeTC.H HCKJllOqHTeJlbHOH oc06eHHOCTblO no-
nynpoBoHHKOB. Y HeKOTopbIX MeTanJlOB H HX CnJlaBOB cyecTByeT CMelIIaH-
Ha.H 3J1eKTpOHHO-bIpOqHa.H npOBOHMOCTb 3a CqeT nepeMeeHHH HeKOTopOH qa-
CTH HeKOJlJleKTHBHpOBaHHbIX BaJleHTHbIX 3J1eKTpOHOB. HanpHMep, B IJ;HHKe, 6e-
pHJlJlHH, KaMHH, CnJlaBaX MeH C OJlOBOM hlpOqHa.H COCTaBJI.HIOa.H SJleKTpH-
'1eCKOrO TOKa npe06J1aaeT Ha 3J1eKTpOHHOH.
ECJlH B nOJlynpOBO.lIHHK O.lIHOBpeMeHHO BBO.lI.HTC.H H OHopHbIe H aKn;enTOp-
HbIe npHMeCH, TO xapaKTep npOBOHMOCTH (n- HJlH p-THn) onpeeJl.HeTC.H npH-
MeCblO C 60Jlee BbICOKOH KOHn;eHTpaIJ;HeH HOCHTeJleH TOKa - 3J1eKTpOHOB HJlH
bIpOK.
303

10.17. TepMo- H <}>oTope3HcTopLI
3J1eKTpHeCKoe conpOTHBJleHHe nOJlynpoBonHHKOB B 3HaqHTeJlbHOH CTeneHH
3aBHCHT OT TeMnepaTypbI. 3TO CBOHCTBO HCnOJlb3YIOT JI.H H3MepeHII.H TeMnepa-
TYPbI no CIIJle TOKR B lJ,enH C nOJlynpOBO,AHIIKOM. TRKHe npll60pbI HR3bIBRIOT
TepMope3HCTopaMH HJlII TepMHCTopaMH. llOJIynpoBoHHKoBoe BeeCTBO nOMe-
IIJ,ReTC.H B MeTaJlJlHeCKHH 3aIlJ,HTHbIH qeXOJl, B KOTOpOM IIMeIOTC.H 1130JlHpO-
BRHHbIe BbIBObI ,AJI.H BKJllOqeHH.H TepMope3HCTopa B 3JIeKTpHqeCKYlO enb. He-
KOTopbIe TepMope3HCTopbI He HMelOT CnelJ,HaJlbHOH 3aHTHoii 060JlOqKII, nOJlY-
npOBO,AHHKOBbIH MaTepllaJI B HHX JlHIIIb nOKpbIT CJIOeM JIaKR.
M3MeHeHHe conpOTHBJleHH.H TepMope311CTopOB npH HarpeBaHIIH HJlH oXJlam-
eHHH n03BOJl.HeT HCnOJlb30BaTb HX B npH60pRX JI.H H3MepeHH.H TeMnepaTypM,
,AJI.H nO,AepmRHIISI nOCTO.HHHOH TeMnepaTypM B RBTOMaTHqeCKHX YCTpOHCTBax --
B 3aKpbITbIX KaMepRX-TepMOCTRTax, JI.H 06ecneeHH.H npOTHBonomapHOH CHr-
HaJlH3alJ,HH H T ..
TepMIICTOpbI -- O,AHII 113 caMbIX npOCTbIX nOJlynpoBoHHKOBbIX npH60pOB.
BbInYCKaIOTC.H TepMHCTOpbI B BH,Ae CTepmHeH. Tpy60K. ,AHCKOB. IIIRH6 H 6YCH'
HOK pa3MepOM OT HeCKOJlbKHX MHKpOMeTpOB no HeCKOJlbKHX caHTHMeTpOB.
,[(Hana30H H3Mep.HeMbIX TeMnepaTYP 60nbIIIHHCTBa TepMHCTOpOB JlemHT B HH-
TepBaJle OT 17U o 570 K. Ho CYIIJ,eCTBYIOT TepMHCTOpbI JI.H H3MepeHH.H KaK
oeHb BbICOKHX (T == 1300 K). TaK H OqeHb HII3KHX (T == 4 - 80 K) TeMnepa-
TYp.
3JIeKTpHqeCKa.H npOBOHMOCTb nOJlynpoBoHHKOB nOBbIlliaeTC.H He TOJlbKO
npH HRrpeBaHIIH, HO H npH OCBeIlJ,eHHH.
B 3TOM MomHO y6e,AHTbC.H C nOMOIlJ,blO
YCTaHOBKH, cxeMR KOTOpOH H306pRmeHR
HR pHCYHKe 10.27. MomHo 3RMeTHTb,
qTO npll OCBeIlJ,eHHH nOJlynpoBoHHKa
(pHC. 10.27, 6) TOK B u.enH 3RMeTHO B03-
paCTaeT. 3TO YKa3bIBReT HR YBeJIllqeHlle
npOBonHMOCTH (YMeHbIIIeHHe conpOTHBne-
HH.H) nOJlynpoBoHHKOB no eHcTBHeM
CBeTa. ,[(aHHbIH 3cpcpeKT He CB.H3aH C HR-
rpeBaHHeM, TaK KaK MomeT Ha6JIIOaTbC.H
H npH HeH3MeHHOH TeMnepaType.
3J1eKTpH'leCKR.H npOBOHMOCTb B03paCTa-
eT BCJle,ACTBlle pa3pbIBa CB.H3eH H 06pa30-
BaHH.H CB060HbIX 3J1eKTpOHOB H bIpOK 3a ceT 3HeprHH CBeTa, naalOIIJ,ero
HR nOJIynpOBO,AHHK. 3TO .HBJleHHe Ha3bIBRIOT BHYTpeHHHM cpOT03JIeKTpHqeCKIIM
3cpcpeKTOM.
llPH60PbI, B KOTOpbIX HCnOJlb3YIO'f BHYTpeHHHH cpOT03J1eKTpHQeCKHH 3cp-
cpeKT B nOJIynpoBoHHKax, Ha3bIBRIOT <}>OTope3HCTopaMH HJlH <}>OToconpOTHB-
JIeHIUIMII. ct>OTope3HCTopbI H3rOTRBnIlBRIOTC.H B BHe TOHKIIX CJIOeB nOJlynpo-
BOHIIKOBoro BeweCTBa. HaHeceHHMX Ha nO.l1JIOmKY H30Jl.HTOpa. MaTepHRJlaMII
,AJI.H H3rOTOBJIeHII.H cpOTope3HCTopOB cnymaT coenHHeHHSI TllnR CdS. CdSe.
PhS H p.H .l1pyrHX.
MHHHRTlOpHOCTb H BbICOKR.H QYBCTBHTeJlbHOCTb cpoTope3HcTopOB n03BOn.H-
lOT HCnOJlb30BaTb HX B caMbIX pa3J1HQHbIX 06J1aCT.HX HaYKH H TeXHHKH nH
a
6
Puc. 10.27
304

perHCTpalJ;lUI H H3MepeHH.H CJIa6bIX CBeTOBbIX nOTOKOB. C nOMOll\blO cpOTOpe-
3HCTOpOB onpe.n;eJl.HIOT KaqeCTBO nOBepXHOCTeH, KOHTpOJlHPYIOT pa3MepbI H3.n;e-
nHH H T..D;.
10.18. 3J1eKTpOHHO- ,lILIpO'lHLIH nepexon
8neKTpOHHO-.D;bIPOqHbIH nepeXO.D; (coKpall\eHHO n-p-nepexo,A) B03HHKaeT B no-
nynpOBO.D;HHKOBOM KpHCTanJIe, HMelOll\eM o.n;HOBpeMeHHO 06JIaCTH C n-THna (co-
.D;epmHT .D;OHOpHbIe npHMecH) (-I p-THna (c aKlJ,enTOpHbIMH npHMeC.HMH) npOBO-
,AHMOCT.HMH Ha rpaHHlJ,e Mem.D;y 3THMH 06J1aCT.HMH.
JJ:onycTHM. y Hac eCTb KpHCTaJIn, B KOTOpOM
cnpaBa Haxo.n;HTC.H 06JIaCTb nOJIynpOBO.D;HHKa C ,AbI-
POQHOH, a CJleB8 - C 3JIeKTpOHHOH npOBo.n;HMOCTblO
(pHC. 10.28). BJIaro.D;ap.H TenJIOBOMY .D;BHjKeHHIO
npH 06pa30BaHHH KOHTaKTa 3JIeKTpOHbI H3 nony-
npOBO,AHHKa n-THna 6Y,AYT .n;H<P<PYH.n;HpoBaTb B 06-
JlaCTb p-THna. I1pH 3TOM B 06JIaCTH n-THna OCTa-
HeTCH HecKoMneHcHpoBaHHbIH nOJlOjKHTeJIbHblH HOH
.nOHopa.
I1epeH.D;H B 06nacTb C .D;bIpO'lHOH npOBO,AHMOCTblO,
3JIeKTpOH OQeHb 6bICTpO peKoM6HHHpyeT C ,AbIpKOll,
npH 3TOM B 06JIaCTH p-THna 06pa3yeTC.H HeCKOMneH-
cHpoBaHHblH HOH aKlJ,enTopa.
AHaJlOrHqHO 3JIeKTpOHaM .D;bIPKH H3 06JIaCTH
p-THna .D;H<PCPYH.n;HPYIOT B 3JIeKTpOHHYIO 06JIaCTb,
OCTaBJI.H.H B .D;bIPOQHOH 06J1aCTH HecKoMneHcHpoBaHHbIH OTpHlJ,aTenbHO 3apH-
jKeHHbIH (-IOH aKlJ,enTopa. I1epeH.D;.H B 3JIeKTpOHHYlO 06JIaCTb, ,AbIpKa peKOM-
6HHHpyeT C 3J1eKTpOHOM. B pe3YJlbTaTe 3Toro B 3JIeKTpOHHOH 06naCTH 06-
pa3yeTcH HecKoMneHcHpoBaHHbIH nOJIOjKHTeJIbHbIH HOH .D;OHopa.
JJ:H<P<PY3H.H OCHOBHblX HOCHTeJleH qepe3 nepexo.n. C03,l\aeT 3JIeKTpHQeCKHH TO1\.
IocJ{' HanpaBJleHHblH H3 p-06JIaCTH B n-06JIaCTb.
B pe3YJIbTaTe .llH<P(PY3HH Ha rpaHHue MemD;Y 3THMH 06JIaCT.HMH 06pa3yeTC.H
,l\BOHHOH 3JIeKTpHQeCKHH CJIOH pa3HOHMeHHO 3apHjKeHHblX HOHOB, TOJIlllHHa l
KOToporo He npeBbllllaeT ,l\OneH MHKpOMeTpa.
MejK.n;y CJIO.HMH HOHOB B03HHKaeT 3J1eKTpHQeCKOe nOJle C Hanp.HjKeHHOCTblO
Ej' 8TO none npen.HTCTByeT ,AanbHeHllIeH ,l\H<P<PY3HH OCHOBHblX HOCHTeJIeH:
3neKTpOHOB H3 n-06JIaCTH H .nbIpOK H3 p-06naCTH.
He06xo.n;HMO 3aMeTHTb, QTO B n-06JIaCTH Hap.H,l\Y C 3JIeKTpOHaMH HMeIOT-
C.H HeOCHOBHbIe HOCHTeJIH - ,l\blpKH, a B p-06naCTH - 3JIeKTpOHbI. B nOJIY-
npOBO,l\HHKe HenpepbIBHO npOHCXO,l\.HT npOlJ,eCCbI pOjK,l\eHHH H peKOM6HHalJ,HH
nap. llHTeHcHBHoCTb 3Toro npOlJ,eCca 3aBHCHT TOJIbKO OT TeMnepaTypbI H O,l\H-
HaKOBa BO BceM 06'beMe nOJIynpOBo.n;HHKa. I1pe.n.nonOjKHM, 1.J:TO B n-06naCTH
B03HHKJIa napa "3JIeKTpOH-,l\bIpKa". JJ:bIpKa 6Y,l\eT XaOTH1.J:eCKH nepeMell\aTbCR
no n-06JIaCTH ,l\0 Tex nop, nOKa He peKOM6HHHpyeT C KaKHM-JIH60 3neKTpo-
HOM. O,l\HaKO ecnH napa B03HHKaeT .n;OCTaTOqHO 6J1H3KO K nepeXO,l\y, TO npejK-
,Ae, 'leM npOH30H.n;eT peKOM6HHaIJ,H.H, .D;bIPK8 MomeT OKa3aTbCR B 06JIaCTH, r,l\e
CYll\eCTByeT 3JIeKTpHQeCKOe nOJIe, Ii no,l\ ero ,l\eHCTBHeM OHa nepeH.D;eT B
n
I I
I I
. ..i + - 1 0 .°0.000
.. .0 : JOOOOo.o
. 0 ..0 + - 1 0 o. 0 0
. ... 100 0 0
. .. 0 .: + - I o. 0 o. 0
o · I I 00
....1+ _100 0 °
.. ° .: :0 0 ..
. . . 1 + I . 0 0 0 00
.. 0 , -I 0
0. .. I i 1 °0. 0 ·
I:
IE, 1
p
Puc. 10.28
305

p-06J1aCTb, T.e. 3J1eKTpHqeCKOe none nepexo.n;a cnoc06cTByeT nepeXO,AY He-
OCHOBHbIX HOCHTeneH B COCe,AHIOIO 06naCTb. COOTBeTCTBeHHO, C03,AaBaeMbn1 HMH
TOK lueocH Man. TaK KaK HeOCHOBHbIX HOCHTeJleH Mano.
TaKHM 06pa30M, B03HHKHOBeHHe 3J1eKTpH"C.IeCKOrO nOJlSl E i npHBo.n;HT K no-
.HBneHHIO HeOCHOBHoro TOKa I ueOCH' HaKonneHHe 3ap.H.n;OB OKOJlO nepeXO,Aa 3a
C"C.IeT .n;HCPCPY3HH H YBenH'leHHe E i 6y.n;YT npO,AOn)KaTbCSI ,AO Tex nop. nOKa
TOK IHeocl1 He ypaBHoBeCHT TOK locH (/HeocH=/oCH) H pe3ynbTHpYIOIl..\HH TOK qepe3
3J1eKTpOHHO-.n;bIPOqHbIH nepeXO,A CTaHeT paBHbIM HYJlIO.
EcnH K n-p-nepexoy npHnomHTb pa3HOCTb nOTeHQ,HBJIOB, TO BHelliHee 3neK-
TpH'leCKOe none EHc:r CKna.n;bIBaeTC.H C noneM E i . Pe3ynbTHpYIOIl..\ee none, CY-
Il..\eCTBYIOIl..\ee B 06J1aCTH nepexo.n;a, E = E HCT + E i . TOKH lOCH H IneOCH COBepIlIeH-
HO pa3nHqHO Be.n;YT ce6S1 no OTHOIlIeHHIO K H3MeHeHHIO nOJlSl B nepexo.n;e. IHeOCH
C H3MeHeHHeM nonSl O'leHb cna60 H3MeH.HeTCSI. TaK KaK OH 06ycnoBneH KOJlH-
"C.IeCTBOM HeOCHOBHbIX HOCHTeneH, a OHO B CBOIO O"C.Iepenb 3aBHCHT TonbKO OT
TeMnepaTypbI.
IOCR (.n;HCPCPY3H.H OCHOBHbIX HOCHTeJleH) O'leHb qYBCTBHTeJleH K nOJlIO Hanp.H-
)KeHHOCTblO E. locH 6bICTpO YBenH"C.IHBaeTC.H C ee YMeHbIlIeHHeM H 6bICTpO na-
.n;aeT npH YBeJlH"C.IeHHH.
IIycTb KneMMa "+" HCTOqHHKa TOKa coe.n;HHeHa C n-06naCTblO. a "-" - C
p-06J1aCTblO (06paTHoe BKnlO"C.IeHHe (pHC. 10.29, a». CYMMapHoe none B nepe-
xo.n;e YCHJlHBaeTC.H: E > E HCT . H OCHOBHOH TOK YMeHbIlIaeTC.H. EcnH E ,AOCTa-
TO'lHO BenHK8, TO lOCH < < I Heocn H TOK qepe3 nepeXO,A C03.n;aeTC.H HeOCHOBHbIMH
HOCHTeJl.HMH. ConpOTHBJleHHe n-p-nepexo.n;a BeJlHKO, TOK MaJl.
a
6
EI
EI
.
.
n
+
p
n
P
+
EacCT
.
EacCT
.II
Puc. 10.29
EcnH BKnlO"C.IHTb HCTO"C.IHHK TaK, "C.IT06bI 06nacTb n-THna OKa3aJlaCb nO,AKnlO-
qeHa K "-", a 06J1aCTb p-THna K "+" (pHC. 10.29, 6), TO BHelliHee none
6Y,AeT HanpaBneHO HaBCTpeqy Et H E = E i + E HCT  E = E i - E HCT < Ei' T.e.
none B nepeXO,Ae ocna6nSleTCSI. IIoTOK OCHOBHbIX HOCHTeneH qepe3 nepexo.n pe3-
KO YBenHqHBaeTC.H, T.e. locH pe3KO B03paCTaeT.
306

I
-u ------ 0
Puc. 10.30
+
Puc. 10.31
u
TaKoe BKJllOqeHHe Hoa Ha3bIBaeTC.H
np.HMbIM. TaKHM 06pa30M, KpHCTaJlJl C
sneKTpoHHO- bIPOqHbIM nepexooM 06J18,Aa-
eT O,AHOCTOpoHHeH npoBoHMOCTblO H MO-
>KeT CJlymHTb llJl.H BblnpSlMneHHSI nepeMeH-
Horo TOKa. BOJlbT-aMnepHa.H xapaKTepHc-
THK8 TaKoro Hoa MeeT BH,A, npe,ACTaB-
JleHHblH Ha pHCYHKe 10.30. CnJlOIIIHaSl
KpHBa.H COOTBeTcTByeT np.HMOMY BKJllOqe-
HHIO, a nYHKTHpHa.H - 06paTHoMY.
JI.H CHSlTH.H 3TOH xapaKTepHCTHKH
MO>KHO BOCnOJlb30BaTbC.H 3J1eKTpHqeCKOH
enblO, cxeMa KOTOpOH npHBeeHa Ha PH-
CYHKe 10.31.
TaKHM 06pa30M, n-p-nepexo no OT-
HOlIIeHHIO K TOKY OKa3bIBaeTC.H HeCHMMeT-
PHqHbIM: B np.HMOM HanpaBJleHHH conpo-
THBJleHHe nepeXO,Aa 3HaqHTenbHO MeHblIIe,
qeM B o6paTHoM.
10.19. nOJlynpoBoHHKOBbIH HO
Cnoc06HOCTb n-p-nepexoa nponYCKaTb TOK B OHOM HanpaBJleHHH HCnOJlb-
3yeTC.H B nonynpoBoHHKOBbIX npH60pax, Ha3bIBaeMbIX HoaMH.
rrOJlynpoBoHHKoBbIe HObJ H3rOTaBJlHBalOT H3 repMaHH.H, KpeMHH.H, ceJle-
Ha H pyrHx BeeCTB.
n-p-rrepexo He yaeTc nOJlyqHTb nYTeM MeXaHHqeCKOrO coeHHeHH.H BYX
nOJlynpoBoHHKOB C pa3J1HqHbIMH THnaMH npOBOHMOCTH, TaK KaK npH 3TOM
nonyqaeTcSI CJlHIIIKOM 60JlblIIOH 3a30p Me>KY nonynpoBoHHKaMH. TOJlHHa me
n-p-nepexoa He OJl>KHa npeBbIlIIaTb Me>KaTOMHbIe paCCTOHH.H. fI03TOMY B
OHY H3 nOBepxHOCTeH repMaHHeBoro 06pa3ua BnJlaBJI.HIOT HHllHH.
BCJleCTBHe H<l><l>Y3HH aTOMOB HHHSI B
rJly6b MOHOKpHCTaJlJla repMaHH.H Y nOBepx-
HOCTH repMaHHSI 06pa3yeTC.H 06naCTb C npo.
BOHMOCTblO p-THna. OCTaJlbHa.H qaCTb 06-
pa3a repMaHH.H, B KOTOpylO aTOMbI HHHSI
He npOHHKJlH, no-npemHeMY HMeeT npOBO-
HMOCTb n-THna. Memy BYM.H 06J1aCT.HMH
C npOBOHMOCT.HMH pa3HbIx THnOB H B03-
HHKaeT n-p-nepexo (pHC. 10.32, a). B
nOJlynpoBoHHKOBOM ,AHO,Ae repMaHHH CJlY-
>KH'I' KaTOOM, a HHHH - aHOOM.
nSl npe,AOTBpaeHH.H Bpe,AHbIX B03eH-
CTBHH B03,Ayxa H CBeTa KpHCT8J1J1 repM8HHSI
nOMealOT B repMeTH'leCKHH MeTaJlJlHqeCKHH
Kopnyc. CXeMaTHqeCKOe H306pa>KeHHe ,AHO-
a
n
Puc. 10.32
6
307

a npHBelIeHO Ha pHCYHKe 10.32, 6. IIoJlynpoBoHHKoBbIe HO,AbI .HBJI.HIOTCfl
OCHOBHbIMH 3J1eMeHTaMIi BbInp.HMHTeJleH nepeMeHHoro TOKa. IIpH BKJIlOqeHHH
HO,Aa B nenb nepeMeHHoro TOKa (pHC. 10.33, a) TOK Ha HarpY30qHOM conpo-
THBJIeHHH R 6Y,AeT npaKTHqeCKR nOCTO.HHHbIM no HanpaBJIeHHIO. Ha npOT.Hme-
HHH nOJlOBHHbI nepHoa, Kora nOTeHnHaJl nOJIynpOBO,AHHKa p-THna nOJlo.mH.
TeJIeH. TOK cBo60HO npOXOHT qepea n-p-nepeXO,A. B cne,AYIOYID nOJlOBHHY
nepHO,lta TOK npaKTHqeCKH paBeH HYJlIO (pHC. 10.33. 6).
MomHo ocyecTBHTb H lIByxnoJlynepHO,AHOe BbInp.HMJleHHe nepeMeHHoro TOKa
(pHC. 10.33, B, z). IIoJIynpOBOLJ;HHKOBbIe BbInp.HMHTeJlH HaemHb] H ,AOJIrOBe't.l-
RbI, HMelOT BbICOKYIO MexaHH't.IeCKYIO npOqHOCTb H KII,l]" HO 3aTO MorYT pa60-
TaTb JIHWb B OrpaHH't.IeHHOM HHTepBaJle TeMnepaTYP (OT 203 o 398 K).
a
6
R
I
8
z
It
or
t
Puc. 10.33
IIOJlynpOBO,ltHHKOBbIe BbInpSlMHTenH npH Tex .me 3HaqeHH.HX BbInpSlMJIeHHO-
ro TOKa 60Jlee MHHHaTlOpHbI, 't.IeM 3neKTpoHHbIe JIaMnbI. BCJle,ltCTBHe 3Toro pa-
,AHOYCTpoHcTBa, c06paHHbIe Ha nOJIynpOBO,ltHHKaX, KOMnaKTHee.
OTMeqeHHbIe npeHMYI.l.\eCTBa nOJlynpOBO,ltHHKOBbIX 3JIeMeHTOB oc06eHHo cy-
I.l.\eCTBeHHbI npH HCnOJlb30BaHHH HX B l1CKyccTBeHHbIx cnYTHHKax 3eMJIH, KOC-
MH't.IeCKHX Kopa6JI.Hx, 3J1eKTpOHHO-BbI't.Il{CJIHTeJlbHbIX MaWHHax.
10.20. TpaH3HcTop
CBOHCTBO O,AHOCTOpoHHeH npOBO,ltHMOCTH n-p-nepeXo,lta B nOJlynpoBoHHKax
MomeT 6bITb HCnOJIb30BaHO ,AJISI YCHJleHH.H H reHepanHH 3JIeKTpHqeCKHX KOJIe6a-
HHR, ,ltJI.H qero npHMeH.HIOTC.H nOJlynpOBO,ltHHKOBbIe TpHO,ltbI HnH TpaH3HCTopbI.
)J;JI.H H3rOTOBJleHH.H TpaH3HCTopa H3 MOHOKpHCTaJIJIa repMaHH.H C 3JIeKT-
pOHHOH npOBO,ltHMOCTblO B Hero C ,ltBYX npOTHBOnOJlo.mHbIX CTOpOH BBO,ltHTCSI
npHMeCb aTOMOB HH,ltH.H. )J;Be 06JIaCTH MOHOKpHCTaJlJla repMaHH.H C npHMeCb10
HH,ltH.H CTaHOB.HTCSI nOJIynpOBO,AHHKaMH C
bIPOqHOH npOBO,ltHMOCTblO, a HR rpaHHne
conpHKOCHOBeHHSI HX C OCHOBHbIM KpHC-
TaJlJlOM B03HHKalOT lIBa n-p-nepeXOlIa.
Cpe,nH.H.H 06naCTb KpHCTaJlJla Ha3bIBa-
eTC.H 6a30H TpaH3HCTopa, a ,ltBe KpaHHHe
06J1aCTH KpHCTaJIJIa, 06J1a,ltalOI.l.\He npOBO-
,ltHMOCTblO npOTHBOnOJlOmHOrO 6a3e THna.
Ha3bIBaIOTC.H KOJlJIeKTOpOM H 3MHTTepOM
Puc. 10.34 (pHC. 10.34). TpaH3HcTopbI, B KOTOpbIX
308

3MHTTep H KOJIJleKTOp 06na.n,alOT .n,bI-
pO'llHOH npOBo.n,HMOCTblO, a 6a3a -
3JIeKTpOHHOH. Ha3bIBaIOTC.H TpaH3HC-
TopaMH p-n-p-nepeXO,Aa.
TpaH3HCTOpbI n-p-n-nepexo.n,a
n-p-n
HMelOT aHaJIOrH'llHOe YCTPOHCTBO.
TOJIbKO MaTepHaJI 6a3bI B HHX 06-
Jla.n,aeT .n,bIpO'llHOH npOBo.n,HMOCTblO, a
KOJIJIeKTOp H 3MHTTep - 3JIeKTpOH-
HOH. ¥CJIOBHOe 0603Ha'lleHHe TpaH-
3HcTopa Ha cxeMax npe.n,cTaBJIeHO Ha
pHCYHKax 10.35, a, 6.
TaKHM 06pa30M, B TpaH3HCTope nO.HBJlSlIOTCSI .n,Ba n-p-nepexo.n,a, npSlMble
HanpaBJIeHH.H KOTOpbIX npOTHBOnOJIOmHbI. TpH BbIBo.n,a OT 06nacTeH C pa3JIH'l-
HbIMH THnaMH npOBo.n,HMOCTH n03BOJI.HIOT BKJIIO'llHTb ero B enb, cxeMa KOTO-
pOH H306pa:>KeHa Ha pHcYHKe 10.36. B COOTBeTCTBHH C TeXHOJIOrHeH npOH3-
BO,ACTBa KOHeHTpaH.H aTOMOB npHMeCH (a 3Ha'llHT, H KOHeHTpaH.H OCHOB-
HblX HOCHTeJIeH TOKa) Y 3MHTTepa H KOJIJIeKTOpa .n,OJI:>KHa 6bITb pa3JIH'llHoH.
06JIaCTb C caMoH BbICOKOH KOHeHTpaHeH HOCHTeJIeH TOKa (B HaIlleM cny-
'llae .n,bIpOK) Ha3bIBaeTCSI 3MHTTepOM, 06J1aCTb C MeHbIlleH KOHeHTpaHeH HO-
CHTeJleH TOKa (TO:>Ke .n,blpOK B HaIlleM cnY'l8e) - KOJIJIeKTOpOM - - cpe.n,HSI.H
'll8CTb (c npOBo.n,HMOCTblO n-THna) - 6a30H. TOJIIUHHa 6a3bl COCTaBJI.HeT
10-50 MKM, a KOHI.I.eHTpaI.I.HSI HOCHTeneH TOKa B 6a3e BO MHoro pa3 MeHb-
Ille, qeM B KOJIJIeKTope.
a
6
K
K
B
B
p-n-p
:3
:3
Puc. 10.35
WI
+ -
B2
+ t---1
Puc. 10.36
I1pH .n,aHHOM BKJIIO'lleHHH JleBblH p-n-nepexo.n, .HBJISIeTC.H npSlMbIM H OT.n,eJISI-
eT 6a3Y OT 06J1aCTH C npoBo.n,HMOCTblO p-THna, Ha3bIBaeMoH 3MHTTepoM. ECJIH
6bJ He 6blJIO npaBoro n-p-nepexo.n,a, B en1i "3MHTTep-6a3a" CYll\eCTBOBaJI 6bI
TOK, 3aBHC.Hll\HH OT HanpSl:>KeHMSI HCTOtJHHKOB (6aTapeH B 1 H HCTOqHHKa TOKa
nepeMeHHoro HanpSl:>KeHHSI) H conpOTHBJIeHHSI enH, BKJIIO'la.H Manoe conpOTHB-
JIeHMe np.HMOrO nepexo.n,a "3MHTTep - 6a3a".
BaTape.H B 2 BKJllOtJeHa TaK. 'lTO npaBblH n-p-nepeXOl1 B cxeMe (pMC. 10.36)
.HBJI.HeTC.H 06paTHb1M. OH OTl1enSieT 6a3Y OT npaBoH 06J1aCTH C npOBOl1HMOCTblO
p-TMna, Ha3blBaeMOR KOJlJIeKTopOM. EcnH 6bl He 6bIJlO neBoro p-n-nepexo.n,a,
CHJIa TOKa B enH KOJIJIeKTOpa 6bIJIa 6b1 6JIM3Ka K HYJIIO, TaK KaK conpOTHB-
309

JleHHe 06paTHOrO nepexoAa OqeHb BeJlHKO. IIpH CYll\eCTBOBaHHH :me TOKa B
neBOM p-n-nepexoAe nO.HBJI.HeTC.H TOK Ii B u,enH KOJlJleKTOpa, npHqeM CHJla
TOKa B KOJlJleKTOpe JlHIlIJ:. HeMHoro MeHbllIe CHnbI TOKa B 3MHTTepe. 8TO MomHO
06'b.HCHHTb TeM, qTO npH C03,AaHHH Hanp.HmeHH.H MemAY 3MHTTepOM Ii 6a30H
OCHOBHbIe HOCHTeJlH nonynpoBoAHHKa p-THna - AbIpKH - npOHHKalOT B 6a3Y,
rAe OHH .HBJI.HIOTC.H yme HeOCHOBHbIMH HOCHTeJl.HMH. IIoCKOJlbKY TOJlHHa 6a3bI
OqeHb MaJla H qHCJlO OCHOBHbIX HOCHTeneH (3J1eKTpOHOB) B HeH He BeJlHKO,
nonaBIlIHe B Hee AbIpKH nOqTH He peKOM6HHHPYIOT C 3J1eKTpOHaMR 6a3bI H
npOHHKalOT B KOJlJleKTOp 3a CqeT AHc}>c}>Y3HH. IIpaBbIH n-p-nepexo,lJ; 3aKpbIT AJI.H
OCHOBHbIX HOCHTeJleH 3ap.HAa 6a3bI - 3neKTpOHOB. HO He AJI.H AbIpOK. B KOJl-
JleKTOpe lIbIPKH YBJleKaIOTC.H 3neKTpHqeCKHM non eM H 3aMbIKalOT nenb. CliJla
TOKa, OTBeTBn.HlOerOC.H B u,enb 3MHTTepa H3 6a3bI, OqeHb Mana, TaK KaK
nJlOaAb CeqeHHSI 6a3bI B ropH30HTaJlbHOH (no pHC. 10.36) nJlOCKOCTH MHoro
MeHbllIe nJlOll\alIH CeqeHH.H B BepTHKaJlbHOH nJlOCKOCTH.
CHJla TOKa B KOJlneKTOpe, npaKTHqeCKH paBHaSl CHJle TOKa B 3MHTTepe,
H3MeH.HeTC.H BMeCTe C TOKOM B 3MHTTepe. ConpOTHBJleHHe pe3HCTopa R Mano
BJlH.HeT Ha TOK B enH KOJlJleKTOpa, H 3TO conpOTHBneHHe MomHO CAeJlaTb
AOCTaTOqHO 60JlbIlIHM. YnpaBJI.H.H TOKOM 3MHTTepa c nOMOblO HCTOqHHKa TOKa
nepeMeHHoro Hanp.HmeHH.H, BKnlOqeHHOrO B ero u,enb, MbI nOJlyqHM CHHXPOH-
Hoe H3MeHeHHe Hanp.HmeHH.H Ha pe3HCTope R. IIpH 60JlbllIOM conpOTHBJleHHH
pe3HCTopa H3MeHeHHe Hanp.HmeHH.H Ha HeM MomeT B AeC.HTKH TbIC.Hq pa3 npe-
BbIllIaTb H3MeHeHHe Hanp.HmeHH.H CHrHaJla B u,enH 3MHTTepa. 8TO 03HaqaeT
YCHJleHHe Hanp.H:meHHSI. OAHoBpeMeHHo H MOHOCTb, BbIAen.HlOa.HC.H H8 H8-
rpY3Ke R, 6YAeT 3HaqliTenbHO npeBbIllIaTb MOHOCTb, pacxoAyeMYIO B u,enli
3MHTTepa. IIPOHCXOlIHT YCHJleHHe MOIUHOCTH. TpaH3HCTOpbI nonyqHnH IlIHpoKoe
npHMeHeHHe B TeXHHKe (CM.  10.21).
BOTIPOCbI J(JUI CAMOKOHTPOJUI
1. H a KaKue zpynnbl noOpa30eJl.SllOmCSl 6e14ecm6Q no ux np0600U.MOCmu?
2. KaKue Onbtmbt nOKa3aJl.U. Ilmo np0600U.MOCmb .MemaJl.Jl.06 06YCJl.06Jl.elia
06UJlCeliUeM c80600libtX 3Jl.eKmpoIi08?
3. lJ mo maKoe 3Jl.eKmpOJl.Umbt? KaKou npoqecc na3bt6a1Om 3Jl.eKmpOJl.Umu-
lleCKOU ouccoquaqueu? Om Ilezo 3a6ucum cmenenb ouccoquaquu?
4. KaKou npoqecc lia3bt8aemcSl 3Jl.eKmpOJl.U30.M? KaK ljJoP.MYJl.UpyemcSl 3a.
KOIi c1>apaoeSl OJl.Sl 3Jl.eKmpOJl.U3a?
5. KaK06 ljJu3uIlecKuu C.MbtCJl. 3Jl.eKmpOXU.MUlleCKOzo 3K6U6aJl.elima? H a306ume
eouliuqbt 3Jl.eKmpOXU.MUlleCKOzo 3K6U6aJl.elima. Om Ilezo 3a6ucum ezo 3lialleliue?
6. KaK .MOJICIiO onpeOeJl.Umb 3apSlo 3Jl.eKmpolia?
7. KaKue 3apSlJICelilibte llaCmUijbt 06ecnellu6a1Om np060ouMocmb za306?
8. KaKou pa3pSlO lia3bt6aemcSl lieCa.MOCmOSlmeJl.blibtM? llpu KaKUX YCJl.06USlX
Oli na6Jl.100aemcSl?
9. KaKOU mOK 6 za3e lia3bt8aemcSl mOKO.M liaCbt14eIiU.R.?
10. TIpu KaKUX YCJl.06USlX 6 za3e npoucxooum yoapliaA uOIiU3aijUSl?
11. KaKou pa3p.RiJ lia3bt8aemcSl ca.MOCmOSlmeJl.blibt.M? llpu KaKUX YCJl.06USlX
Oli lia6Jl.100aemcSl?
12. lJmo npeOCma6Jl..R.em c060u .MOJl.IiUSl?
310

13. lfmo maKoe OYZ060U pa3pSlo? llpu KaKUX YCJl.06U.Jl,X Oli lia6Jl.100aemCSl2
roe npU.Meli.RemcSl?
14. f{ mo lia3bl6a1Om pa60mou 6blxooa 3Jl.eKmpOlia U3 .MemaJl.Jl.a?
15. f{mo maKoe mep.M03Jl.eKmpOliliaSl 3.MUCCUSl?
16. KaK nOJl.Yllumb 3Jl.eKmpUllecKuu mOK 6 6aKYY.Me?
17. BblnOJl.IiSlemCSl Jl.U 3aKon O.Ma OJl.Sl mOKa 6 6aKYY.Me?
18. Kale ycmpoen 6aKYY.MnblU OUOO. napucyume 3JleKmpUileCleYHJ cxe.MY OJl..R
cnSlmuSl ezo 60Jl.bm-a.MnepIiOU xapaKmepucmUKU.
19. KaK06a pOJl.b cemKU 6 mpuooe?
20. If e.M OmJl.UllalOmCSl nOJlynpoBooliUKU om npoBooliUKOB?
21. If e.M 06YCJl.06Jl.elia c06cm6eliliaSl np0600U.MOCmb nOJl.ynp060onuK06? Kale
U3.MeliSlemCSl conpOmU6Jl.eliUe nOJlynp060oliUK06 c U3.MelienUe.M me.MnepamYPbl?
II olle.MY?
22. KaK npu.Mecu 6Jl.USllOm lia np0600U.MOCmb nOJl.ynp060oliUK06? If mo ma
Koe OOlioplible U aKqenmoplible npu.Mecu?
23. lfmo maKoe np0600U.MOCmb p-muna? n-muna?
24. KaK060 C60ucm60 n-p-nepexooa? roe OliO UCnOJl.b3yemcSl? II olle.MY npSl
.MOU mol\, llepe3 n-p-nepexoo 3liallUmeJl.bliO 60Jl.bWe 06pamliozo?
25. KaKou 6UO U.Meem 60Jl.bm-a.MnepliaSl xapaKmepucmuKa nOJl.ynp060oliUKO
60Z0 ouooa? Bb£nOJl.liSlemCSl Jl.U 3aKOli O.Ma npu npoxoJICoeliuu mOKa llepe3
n-p-nepexoo?
26. IIolle.MY npu 6lieCeliUU 6 .MaZliumnoe nOJl.e nJl.aCmUliKU c np0600U.MOC-
mblO n- UJl.U p-muna. no Komopou npoxooum mOK. lia ee npomUBOnOJl.OJICliblX
zpanSlX nO.Jl,BJlSlHJmc.Jl, 3apsWbL pa3nbLx 3naKoB?
27. KaK06bl npeu.MyU{ecm6a nOJl.ynp060oliUK06blX OU0006 U mPUOa 0 6 no
Cpa6lieliU1O C 3Jl.eKmpOlilibl.MU Jl.a.Mna.Mu?
28. Ha lle.M OCli06ali npuliqun oeUCmBU.R mep.MUCmOp06 U fjJomope3ucmop06?
OCHOBHhIE cl>OPMY JIhI
m=kq
1M
k=--
F n
- 3aKOH <l>apa,ne.H ,nJl.H ::meKTpOJlH3a, r,ne m -
Macca BeI..QeCTBa, BbI,nenHBlIIerOC.H Ha 3J1eKTpo,ne;
k - 3neKTpOXHMH'tIeCKHll 3KBHBaJleHT, q - 3a-
p.H,n, npolIIe,nlIIHll qepe3 3neKTponHT, F - qHCJlO
<l>apa,ne.H, n - BaJleHTHOCTb, M - aTOMHa.H
Macca BeI..QeCTBa.
All = e<l>l(
- pa60Ta HOHH3aHH aTOMa, r,ne <1>11 - nOTeH-
HaJl HOHH3aHH; e - 3apH,lJ; 3J1eKTpoHa.
IIPHMEPbI PEillEHHH 3A,I(A q
1. IIpli 3neKTponH3e p8CTBOp8 cepHoH KlicnOTbI P8CXO,AyeTcSI MOHOCTb
37 BT. Onpe,AenHTe conpoTHsneHHe 3neKTponHT8, ecnH 3a 50 MHH SblAenSleT-
CSI 0.3 r BO,AOpO,Aa.
Kr
aHo: P=37 BT, t=3,O.10 3 c, m=3,0.10- 4 Kr, k=0.01.10- 6 En'
HaHTH: R.
311

PeweHHe. l1cKoMoe COnpOTHBneHHe HaHAeM H3 <l>opMynbI
2 P
P=I R=:)R=-
1 2 .
.I{nSl HaXOm,AeHHSI CHnbI TOKa socnonb3yeMcSI 3aKOHOM <t>apa,AeSl ,AnSI sneK-
m h 2 t 2 P
TponH3a: m = hIt  I = -. Cne,AOB8TenbHo, R = ; R=0,4 OM.
ht m
2. ITpH HHKenHpOBaHHH H3,AenHSI ero nOBepXHOCTb nnOa,AblO 100 CM 2 no-
KpbIsaeTCSI cnoeM HHKenSl TonHHoH 50 MKM. Onpe,AenHTe Cpe,AHIOIO cHny
TOKa, ecnH HHKenHpOBaHHe ,AnHTCSI 2,5 'I.
.I{aHO: h=5,0.10- 5 M, t=9,0.10 3 c. 8=0,010 M 2 , d=8,9.10 3
k=3,04.10- 7  .
H8HTH: I.
Pewe H He. .I{nSl H8XOm,AeHHSI CHnbI TOKa I npHMeHHM 3aKOH <t>apa,AeSl
,Anfl sneKTponH3a m = kIt (1), r,Ae k - sneKTpOXHMH'leCKHH 3KBHB8neHT. Mac-
ca OCa,AHBlllerOC.H HHKenSl m = dV = d8h (2), r,Ae d - nnOTHOCTb HHKenSl;
8 - nnOa,Ab nOBepXHOCTH H3,AenHSI.
TIO,ACT8BnSlSl BbIpameHHe (2) B ypaBHeHHe (1), HaH,AeM cHny TOKa:
Kr
M 3 '
dh8
I = -. 1= 1 6 A.
tk' ,
3. KaKOH MHHHM8nbHOH CKOpOCTblO ,llOnmeH 06n8LIaTb 3neKTpOH LInSl Toro.
'lT06bI HOHH3HpOBaTb aTOM BOLIOpO,A8? IToTeHu.H8n HOHH3au.HH aTOMa BO,AOpOAa
13.5 B.
.I{aHo: e=1,6.10- 19 Kn. m=9,1.10- 31 Kr, U , =13.5 B.
HaHTH: v.
Pc III e H He. .I{nSl HOHH3al.\HH 8TOMa SOAOpO,Aa sneKTpoH ,AOnmeH 06na,AaTb
mv 2
KIiHeTH'leCKOH sHeprHeH, paBHOH 3HeprliH IiOHIi3al.\HH: W K =  linli 2 = eU; .
r,Ae e H m - 38pSl,A Ii Macca sneKTpOH8.
OTCIO,Aa
2eU. M
v = ; v = 2,2 . 10 6 C
4. B ra30pa3pSl,AHOH Tpy6Ke Mem,Ay nnOCKHMH 3neKTpO,AaMIi nnOa,AblO
10 CM 2 , pacnonomeHHbIMIi Ha paCCTOSlHHIi 10 CM ,Apyr OT ,Apyra, CHn8 TOKa
HaCbIeHHSI paBHa 1,0 MKA. Pa3pSl,A Hec8MocToSlTenbHbIH. KaKoe 'lHCnO sne-
MeHTapHbIX 38pSlAOB 060HX 3H8KOB C03,A8eTCSI emeceKYH,AHO B 1,0 CM 3 ?
.I{aHo: 8=1.0.10- 3 M 2 , d=0.10 M, 11\=1,0,10- 6 A, V 1 =1,0.10-6 M 3 .
t=l,O c.
HaHTH: n.
Pc III e H He. 3a 1 C 'lepe3 nOnepeQHOe Ce'leHlie sneKTpO,AOB npOXO,AHT 3a-
pH,A q = lilt = ne8d. r,Ae ne8d - 3apSl,A. KOTOpbIH B03HHKaeT 3a t C BO BceM
06'beMe V=8d, nOSTOMY
n = IHt . n = 6.3. 10 10 CM- 3 .
e8d'
312

11. MArHIITHOE IIOJIE
B3aHMO.D;eHCTBHe npOBO.D;HHKOB C TOKaMH. MarHHTHoe nOJIe TOKa. MarHHT-
HaR HH.LJ;YKHH. JIHHHH MarHHTHUH HH.LJ;YKHH. MarHHTHoe nOJIe npHMoro TOKa,
KpyroBoro TOKa, COJIeHOH,.l];a.
,l{eHcTBHe MarHHTHoro nOJIH Ha npOBO.D;IIHK C TOKOM. CHJIa AMnepa. 3JIeK-
TpO.D;BHraTeJIb nOCTO.HHHOrO TOKa. IIpHHuHD neHCTBHH 3JIeKTpOH3MepHTeJIbHbiX
npH6opoB. rpOMKoroBOpHTeJIL.
CHJIa JIopeHa. .II:BHmeHHe 3JIeKTpH'IeCKHX aap.H.D;OB B 3JIeKTpHqeCKOM H
MarHHTHOM nOJIHX.
Macc-cneKTporpa4>. Y CKOpHTeJIH 3apHmeHHbiX qaCTH. MarHHTIILle CBOH-
CTBa BeIIJ;eCTBa. MarHHTHaH npOHHaeMOCTL. eppoMarHeTH3M. MarHHTHaH 3a-
nHCL HH4>opMaHH.
11.1. B3aHMOJJ;eUCTBHe npOBOJJ;HHI{OB C TOKaMH
113BeCTHbIe C .n;peBHHX BpeMeH .HBneHHSI npHT.HmeHHSI pa3HOHMeHHbIX H OT-
TaJIKHBaHH.H o.n;HOHMeHHblX nOJlIOCOB MarHHTa HanOMHHalOT .HBneHHSI B3aH-
Mo.n;eHcTBHSI pa3HOHMeHHbIX H o.n;HOHMeHHbIX 3JIeKTpHqeCKHX 3apSI.n;OB. O.n;HaKo
MHOrOqHCJleHHble nonbITKH yqeHbIX YCTaHOBHTb CB.H3b Mem.n;y 3JIeKTpHlIeCKHMH
H MarHHTHbIMH .HBJleHH.HMH Ha npOT.HmeHHH MHorHX CTOJleTHH OCTaBanHCb
6e3pe3YJlbTaTHbIMH. 06 3TOH CBSI3H rOBopHT TaKme 3aMeqeHHbIH <l>aKT HaMar-
HHlIHBaHH.H meJIe3HbIX npe.n;MeTOB 11 nepeMarHHlIHBaHH.H KOMnaca 80 BpeM.H
rp03bI.
BnepBbIe 3Ta CBSI3b 6bIJIa 06HapymeHa X.3pCTe.n;OM H A. AMnepOM B 1820 r.
A. AMnep nOKa3aJ1, 'lTO ,ABa napaJlJleJIbHbIX npOBO,AHHKa C TOKaMH npWr.HrH-
BaIOTC.H HJIH OTTaJlKHBaIOTC.H B 3aBHCHMOCTH OT HanpaBJleHHSI TOKa B HHX
(pHC. 11.1. a. 6). 3TO B3aHMo.n;eHCTBHe He MomeT 6bITb BbI3BaHO 3JIeKTpO-
CTaTHqeCKHM noneM no cne.n;YIOHM npH'tlHHaM. Bo-nepBbIX, npH pa3MbIKaHHH
u.enH (Ha pHCYHKe 11.1  B nepeMbIlIKa Mem.n;y BepXHHMIi KJleMMaMH OTcoe.n;H-
HeHa) B3aHMo.n;eHcTBHe npOBo.n;HHKOB
npeKpaaeTC.H, XOT.H 3ap.H.n;bI Ha
npOBo.n;HHKax Ii HX 3J1eKTpOCTaTH-
qeCKHe nOJl.H OCTaIOTC.H. BO-BTOpbIX,
O,AHOHMeHHbIe 3ap.H.n;bI (3neKTpoHbI B
npoBo.n;HHKe) Bcer.n;a TonbKO OTTan-
KHBaIOTC.H .
B onbITe X. 3pcTep;a npOBop;HHK
pacnOJlararoT Ha.n; MarHHTHoH CTpeJI-
KOH (HJlH no.n; HeH) napaJIJleJIbHO ee
OCH (pHC. 11.2). IIpH nponYCKaHHIi
TOKa no npOBo.n;HHKY CTpeJlK8 OT-
KJlOH.HeTC.H OT CBoero nepBOHaqaJIb-
Horo nOJlOmeHH.H. IIpH pa3MbIKaHHIi
u.enH MarHHTHa.H CTpeJlKa B03Bpaa.
eTC.H B CBoe nepBOHalIanbHOe nOJIO-
a
6
B
+
+
Puc. 11.1
313

meHHe. STOT onbIT nOKa3bIBaeT, qTO B npOCTpaHCTBe, oKpymalOeM npOBO-
HHK C TOKOM, eHCTBYIOT CHnbI. BbI3bIBalOHe nOBopOT MarHHTHOH CTpeJlKR,
TO eCTb CHnbI, no06HbIe TeM, KOTopbIe eHCTBYIOT Ha Hee B6nH3H nOCTO.HH-
HbIX MarHHTOB.
Puc. 11.2
J];eHCTBHe MarHHTHbIX CHJI o6HapymeHo B npocTpaHcTBe BOKpyr OT,AenbHO
BHmYHXC.H 3ap.HmeHHbIx qaCTHI.1,. TaK, A. <1>. Ho<}><}>e B 1911 r. Ha6J1IOan
OTKnOHeHHe MarHHTHbIX CTpenOK, pacnonomeHHbIX B6nH3H nyqKa BHmYHXCH
3neKTpOHOB. CxeMa ero onbITa npeCTaBneHa Ha pHCYHKe 11.3. Ha,A 11 no
Tpy6KOH HaXOHnHCb Be OHHaKOBbIe, HO npOTHBOnOJlOmHO HanpaBJleHHbIe Mar-
HHTHbIe CTpenKH. YKpenneHHbIe H8 06meM Konbue, nOlIBerneHHOM Ha ynpyroH
HHTH. IIpH npoxomlIeHHH B Tpy6Ke nOTOKa 3neKTpOHOB MarHHTHbIe CTpenKH
nOBOpa'llHBanHCb.
C
-L-------E]-
EO
+
Puc. 11.3
EcnH qaCTb rH6Koro npoBoHHKa, npHcoeHHeHHoro K OHOMY nOJllOcy I1C-
TO'llHHKa, 8 3Ha'llHT, 3apSlmeHHoro, nOMeCTHTb B6nH3R yro06pa3Horo MarHHTa
(pHC. 11.4, a), TO eHCTBHe nonH MarHHTa Ha npOBOHHK He Ha6nlOaeTC.H.
OHaKO nocne 3aMbIKaHHH uenR (pRC. 11.4, 6, 6) npOBOHRKH npRXO,AHT B
BRmeHRe. TaKHM 06pa30M, MarHRTHbIe CHnbI ,AeHCTBYIOT TOJlbKO H8 BHmy-
ReCH 3apHbI.
314

a
6
6
Puc. 11.4
11.2. MarHHTHoe nOJIe TOKa
Cnoco60M nepeLJ;at.Ili MarHHTHoro B3aliMOLJ;ellCTBli.H HBJI.HeTCH MarHliTHoe nOJIe.
MHOrOt.IHCneHHbIe onbITbI npliBeJIli eHbIX K BbIBon;y. ({TO BOKpyr JII06oro
npOBOlIHliKa C TOKOM. T.e. BOKpyr llBU:iKYll\UXCH 3JIeKTpUqeCKUX 3apJlllOB, CY-
ll\eCTByeT MarHUTHoe nOJle. B OTJIUqlie OT 3J1eKTpUt.IeCKOrO nonH, KOTopoe Aell-
cTByeT H Ha HenOLJ;BUmHbIe 3apHLJ;bI, li Ha ,ABli>KYll\lieCH, MarHliTHoe nOJIe LJ;eH-
cTByeT TOJIbKO Ha ,ABli>KYll\lieC.H 3apSl,AbI (TOKli).
MarHliTHoe nOJle MomHO o6HapymliTb Ii liCCJIe,AOBaTb C nOMOll\blO mene3-
HbIX OnliJIOK, MarHliTHOll CTpeJIKli, a TaKme He60JlbllIOrO KOHTypa C TOKOM
(paMKli C TOKOM) (pHC. 11.5), npli'lleM c06cTBeHHoe MarHliTHoe none KOHTypa
LJ;OJI>KHO 6bITb CJla6bIM no cpaBHeHHIO C liCCJleLJ;yeMbIM.
I1pOBOLJ;HHKli, nO.ZJ,BO,A.HUe TOK K KOHTYPY, LJ;OJImHbI 6bITb pacnOJlO>KeHbI
B6JIli3H LJ;Pyr LJ;pyra liJIU CnJIeTeHbI Me>KLJ;y C060H (TOrLJ;a HX MarHHTHbIe nOJI.H
B3aHMHO KOMneHclipYlOTC.H). OpHeHTalJ,liH TaKoro KOHTypa xapaKTepH3yeTcH H8-
npaBJIeHlieM HOpM8JIli ii K KOHTYPY. B KaqeCTBe nOJIO:iKUTeJILHOrO HanpaBJIe-
HUH HOpMaJIU npliHHMaeTC.H HanpaBJIeHHe, CB.H3aHHOe C TOKOM np8BHJIOM npa-
Boro BHHTa (6ypaBt.IliKa): ronOBKY BliHTa nQBopa'iliBalOT no HanpaBJIeHlilO TOKa
B KOHType. nocTynaTeJIbHOe LJ;BlimeHlie OCTpH.H BliHTa YKa3bIBaeT HanpaBJIeHHe
nOJIO>KHTeJIbHOH HOpMaJIH ii.
OnbIT nOKa3bIBaeT, 'IlTO eCJIli TaKOH KOHTYP nOLJ;BeCHTb Ha rH6KHX npOBO.ZJ,HH-
Kax B MarHHTHOM nOJIe, TO OH nOBepHeTCH H YCTaHOBHTCH onpeLJ;eJIeHHbIM 06pa-
30M, TO eCTb MarHHTHoe nOJIe OK83bIBaeT Ha KOHTYP C TOKOM opHeHTHpYlOll\ee
LJ;eHCTBlie. I1pH 3TOM HanpaBJIeHHe nOJIO>KliTenbHOH HOpMaJIli ii K nJIOCKOCTR KOH-
Typa COBnaLJ;aeT C HanpaBJIeHHeM npO.ZJ,OJlbHOH OCH MarHHTHOH CTpeJIKH (T.e. OT S
K N BHYTPli ee), nOMell\eHHOll B TY me TO'lKY MarHHTHoro nOJIH (pHC. 11.5).
I103TOMY 3a Hanp8BJIeHHe MarHHTHoro nOJI.H npHHHMalOT HanpaBJIeHMe OT
lO.lKHOrO nOJIlOca S K ceBepHOMY N no OCM CB06oLJ;HO YCTaHoBliBweHcH B Mar-
HHTHOM nOJIe CTpeJIKH HJIH Hanp8BJIeHHe nOJIO.IKMTeJIbHOll HOpMaJIH K 3aMKHY-
TOMY KOHTYPY C TOKOM.
315

B
SN
Puc. 11.5
11.3. MarllHTHan HHJJ;YKH1J
,ZJ;JI.H KOJlHqeCTBeHHoro onHCaHH.H MarHHTHoro nOJI.H MomHO BOCnOJlb30BaTb-
CH KOHTypOM C TOKOM. TaK KaK KOHTYP C TOKOM HcnbITbIBaeT opHeHTHpylO-
r..qee ,AeHCTBHe nOJlH, TO Ha HerO B MarHHTHOM nOJle ,l.{eHcTByeT napa CHJI,
KOTOpa.H C03,l.{aeT MOMeHT CHJI OTHOCHTeJIbHO HeKOTopOH HenO,ABHJKHOH OCH. Bpa-
r..qalOHH MOMeHT CHJI 3aBHCHT KaK OT CBOHCTB nOJIH B ,l.{aHH011 TOqKe, TaK H
OT CBOHCTB KOHTypa. ,ZJ;JI.H llJIOCKOrO KOHTypa C TOKOM I BeJIHqHHa, paBHaH
npOH3Be,l.{eHHIO CHJIbl TOKa I Ha nJlOa,Ab S, OrpaHHqeHHYIO KOHTypOM, Ha3bI-
BaeTC.H MarHHTHLIM MOMeHTOM KOHTYPa Pm'
MarHHTHbIH MOMeHT - BeKTOpHa.H BeJlHqHHa. Ero HanpaBJIeHHe COBnanaeT
C HanpaBJleHHeM nOJIOmHTeJlbHOH HopManH K KOHTYPY.
Pm = ISii ,
r,l.{e ii e,l.{HHHqHbIH BeKTOp HOpMaJIH K nJIOCKOCTH KOHTypa.
OnbIT nOKa3bIBaeT, qTO BpaalOHH MOMeHT 3aBHCHT OT pacnOJlOmeHH.H KOH-
Typa B MarHHTHOM nOJIe. BpaalOHH MOMeHT paBeH 0, eCJlH MarHHTHoe none
nepneH,l.{HKYJI.HpHO nJIOCKOCTH KOHTypa (pHC. 11.6, a), Ii MaKCHMaJleH, eCJlH HOp-
MaJIb K KOHTYPY nepneH,l.{HKYJI.HpHa MarHHTHOMY nOJlIO (pHC. 11.6, 6).
MaKCHMaJlbHbIH BpaalOHH MOMeHT, KaK nOKa3bIBaeT onbIT, nponopn;HO-
HaJleH CHJle TOKa I H nJlOa,AH KOHTypa paMKH C TOKOM, T.e.
M max - IS .
a
:Il, _
N
316
6
'«
Puc. 11.6

ECJlIi B .n;aHHYIO TOtlKY MarHliTHOrO nOJI.H nOMeaTb KOHTypbI C pa3HbIMIi
MarHliTHbIMIi MOMeHTaMH, TO H8 HHX 6y.n;YT ,AeHCTBOBaTb pa3J1litlHble BpaalO-
M
He MOMeHTbI, O.n;HaKO OTHOIlleHlie  .n;n.H BCeX KOHTypOB O.n;HO H TO me
Pm
Ii n03TOMY MOmeT cnymliTb XapaKTepliCTHKOH MarHHTHOrO nOJI.H. Ha3bIBaeMOH
MarHliTHOH liH.n;YKlieH.
MarHHTHaH HH.ZJ.YKHH - 3TO BeKTOpHaSl <pli3litleCKa.H BeJIIi'lliHa, .HBJI.HIO-
a.HCSI CliJlOBOH XapaKTepliCTliKOH MarHliTHOrO nOJI.H, tlliCJleHHO paBHaSi MaK-
CliMaJlbHOMY BpaalOeMY MOMeHTY, .n;eHCTBYIOeMY Ha KOHTYP C e.n;liHlitlHbIM
MarHHTHbIM MOMeHTOM, H HanpaBJleHHaSi B.n;Onb nOnOmHTenbHOH HOpManH K
KOHTYPY.
MO.n;YJIb MarHliTHOH HH.n;YKHIi paBeH
B - M max _ M max
-IS-p;:-'
E.n;HHIieH MarHliTHOI1 liH.n;YK1i1i B CM HBJI.HeTC.H Tecna (T JI).
R'M H
1 TJI----
- A. M 2 - A. M .
1 TJI - MarHliTHa.H liH.n;YKH.H TaKoro O.n;Hopo.n;Horo nOJl.H, B KOTOpOM Ha
KOHTYP C MarHHTHbIM MOMeHTOM 1 A. M 2 .n;eHcTByeT BpaalOliH MOMeHT
1 H'M.
MarHHTHa.H HH.n;YKH.H B nonHOCTblO xapaKTepH3yeT MarHHTHoe nOJIe B
Kam.n;oH TOqKe MomeT 6bITb HaH.n;eH ee MO.n;YJlb H HanpaBJIeHlie.
none, B Kam.n;oH TOqKe KOToporo Mo.n;ynb H HanpaBJIeHHe MarHliTHOH liH-
IIYKI<"" OIl"HBKOBLl [ 13 .;;;;;;t l HB3LIBBeTCH Ol\HoP01\HL1M ManUITHLIM nOJleM
ECJIIi MarHHTHoe nOJle 06pa30BaHo clicTeMoH n npOBo.n;HliKOB C TOKa-
MIi, TO, liMeeT MeCTO npHHHn cynepno3HHH MarHHTHLIX nOJle": MarHliT-
HaSi HH,AYKHSI nOJISI CHCTeMb] TOKOB paBHa reOMeTplitleCKOH CYMMe Mar-
HliTHbIX liH,AYKliH nOJleH Kam.n;oro li3 TOKOB B OT.n;eJlbHOCTIi:
Puc. 11.7
n
E = B) + E 2 + ... + En; B = L B; .
;=1
11.4. JIHIIHH MarHHTHOH HH,ZJ;YKl.\HH
MarHliTHoe none MomHO li306pamaTb rpa<pHqecKIi C no-
MOblO JIliHliH MarHHTHOH HH.n;YKHH, no.n;o6Ho TOMY, KaK
3J1eKTplitleCKOe nOJIe H306pamalOT C nOMOblO nliHliH Ha-
npSlmeHHOCTIi .
JlHHHeu MarHHTHOU HHJl.YKHU Ha3bIBaeTCSI JIHHH.H, Ka-
caTenbHaSi K KOTOpOH B Kam.n;oH TO'lKe nOJISI COBn8,AaeT C
BeKTopOM MarHHTHOH liHYKUliH (pliC. 11.7). TaK KaK B
Kam.noH TOtlKe MarHHT_Hoe nOJIe xapaKTepli3yeTC.H onpe.ne-
neHHbIM 3Ha'leHHeM B, TO qepe3 Kam.n;ylO TOqKY nOJI.H
MomHO npOBeCTH nUHHIO MarHHTHOH HH.n;YKHIi 11 TOJIbKO
O.n;HY, JIHHHH MarHHTHOH HH.n;YKliH lie nepeceKalOTCH.
317

llpH H306pameHHH MarHHTHOrO nOJl.H JlHHHH MarHHTHOH HH.n;YKn,HH npOBO-
JUITC.H TaK. qT06bI HX rYCTOTR 6bIJla npOnOpn,HOHaJlbHa MO.n;YJIIO MarHHTHOii HH-
.n;YKn,HH.
Har JI.HlIHOe npe.n;CTaBJIeHHe 0 JlHHH.HX MarHHTHOH HH,AYKIJ;HIi MOmHO no-
JlYt.lHTL, eCJlH Ha JIHCT CTeKJla, CKB03b KOTOpbIR npOXO.n;HT npOBO.n;HHK C TO-
KOM, HaCbInaTb meJle3HbIe OnHJIKH H BCTp.HXHYTb HX. OnHJlKH HaMarHHqH-
BaIOTC.H, CTaHOB.HTC.H MaJleHbKHMH MarHHTHbIMH CTpeJIKaMH Ii paCnOJlaralOT-
C.H B.n;OnL B.
HCCJle.n;OBaHHe pa3JIHqHbIX MarHHTHbIX nOJleii nOKa3aJlO, qTO JIHHHIi Mar-
HHTHOH HH.n;YKIJ;HH B OTnHqHe OT JlHHHH Hanp.HmeHHOCTH 3neKTpOCTaTHqeCKOrO
nOJl.H .HBn.HIOTC.H 3aMKHyrLIMH JIHHH.HMH. 3aMKHYTOCTb nHHHH MarHHTHOH HH-
.n;YKn,HH npe.n;CTaBJI.HeT C060H «PYH.n;aMeHTaJlLHOe CBOHCTBO MarHHTHOrO nOJl.H. OHO
CBH,AeTenLcTByeT 0 TOM, qTO MaruHTHLIX 3ap.HJJ;OB, no,o;:o6HLIX 3JIeKTpHQeCKHM,
B npHpo,o;:e neT. HCTOqHHKOM MarHHTHoro nOJl.H .HBJI.HIOTCS! ,ABHmYHeC.H 3ap.H-
,ALl H nepeMeHHLIe 3J1eKTpHqeCKHe nOJl.H.
MarHHTHoe nOJIe - BHxpeBoe nOJIe.
11.5. MarHHTHoe nOJIe npHMOJIHHeHHOro npOBOAHHKa C TOKOM
JI.H nonyqeHH.H cneKTpa MarHHTHoro nOJIS np.HMOrO npOBo.n;HHKa C TOKOM
npOBO,AHHK nponYCKalOT CKB03L JIHCT KapTOHa. Ha KapTOH HaCbInalOT TOHKHH
CJlOH meJIe3HLIX OnHJlOK, H onHnKH CJlerKa BCTp.HXHBaIOT. llo,A ,AeHCTBHeM
MarHHTHoro nOJlS! meJle3HbIe OnHJlKH pacnonaralOTC.H no KOHIJ;eHTpHqeCKHM OK-
PymHOCT.HM. rIo KaCaTeJlbHbIM K HHM pacnOJIOmaTC.H H MarHHTHLIe CTpenKH
BOKpyr TaKoro npOBo.n;HHKa C TOKOM.
TaKHM 06pa30M, JlHHHH MarHHTHOH HH,AYKIJ;HH MarHHTHoro non.H np.HMO-
JIHHeHHOro TOKa npe.n;CTaBJI.HIOT C060H KOHIJ;eHTpHQeCKHe OKpYjKHOCTH. pac-
nonomeHHbIe B nJlOCKOCTH. nepneHllHKyn.HpHOH K npOBo.n;HHKY, C IJ;eHTpOM
Ha OCH npOBO,AHHKa. HanpaBJleHHe nHHHH HH.n;YKIJ;HH onpe.n;en.HeTCSI npaBH-
JIOM npaBoro BHHTa: ecJl.U n060pallU6amb ZOJl.06KY 6Ulima maK. Ilm06bl no-
cmynameJl.bliOe 06UJlCeliUe ocmpuSl 6Ulima npoUCXOOUJl.O 600Jl.b mOKa 6 npo-
600liUKe. mo lianpa6Jl.eliUe 6paU{e-
liUSl ZOJl.06KU YKa3bl6aem lianpa6Jl.e-
liue Jl.UIiUU MaZliumHOU UHOYKU,UU
nOJl.Sl npJlM.OZO np0600liUKa c mOKOM.
Ha pHCYHKe 11.8, a np.HMOnH-
HeHHLIH npOBo.n;HHK C TOKOM pacno-
JlOmeH B nJlOCKOCTH pHCYHKa, nH-
HHH HH,AYKIJ;HH - B nJlOCKOCTH,
nepneH,AHKYJlS!pHOH PHCYHKY. Ha
pHCYHKe 11.8, 6 H306pameHo ceqe-
HHe npOBo.n;HHKa, pacnOJIOmeHHOro
nepneH,AHKYJI.HpHO nJlOCKOCTH PHCYH-
Ka, TOR B HeM HanpaBneH OT Hac
(3TO 0603HaqaeTC.H KpeCTHKOM "x"),
JlHHHH HH.n;YKn,HH pacnOJlaralOTC.H B
nnOCKOCTH pHCYHKa. KaK nOKa3LI-
a
6
Puc. 11.8
318

BalOT paCqeTbI, MO,AYJlb MarHHTHOH HH,AYKHH nOn.H npJlMOJlHHeHHOrO TOKa
MOmeT 6bITb paCCqHTaH no «popMyne
I
B = o - ,
21tr
r,Ae  - MarHHTHaH npOHHaeMOCTb Cpe,Abl, o = 41t .10- 7  - MarHHTH8H
A 2
nOCTO.HHHa.H, I - CHJlB TOKa B npOBO,AHHKe, r - paCCTO.HHHe OT npOBO,AHHKa
,AO TOqKH, B KOTOpOE BbI'IHCn.HeTCH MarHHTHa.H HH,AYKH.H.
MarHHTHaJi npOHHl(aeMOCTb cpe,ZJ;:bI - 3TO «pH3HqecKa.H BenHQHHa, nOKa3bI-
BalOa.H, BO CKonbKO pa3 MO,Aynb MarHHTHoH HH,AYKHH B non.H B O,AHOPO,D.-
HOH cpe,Ae OTnHqaeTC.H OT MO,AynH MarHHTHoH HH.n;YKIJ;HH Bo B TOH me TOQKe
nOJl.H B BaKYYMe:
B
 = Bo .
MarHHTHoe none np.HMOrO npOBO,AHHKa C TOKOM - nOJle Heo.n;Hopo.n;Hoe.
11.6. MarHHTHoe nOJIe KpyroBoro TOKa

Puc. 11.9
B eHTpe KpyroBoro TOKa MO.n;YJlb
QHCneH no «popMyne
MCCne,AOBaHH.H nOKa3MH, 'lTO nHHHH
MarHHTHoE HH.n;YKIJ;HH non.H KpyroBoro
TOKa He .HBn.HIOTC.H npaBHnbHbIMH OK-
pymHOCT.HMH (pHC. 11.9). HO OHH 3a-
MbIKaIOTC.H. 06XOlIH npoBoJJ.HHK. no KO-
TOPOMY H,AeT TOK. HanpaBneHHe JlHHHH
MarHHTHoH HH,AYKHH MomHO onpe,Ae-
nHTb C nOMOblO npaBHJla npaBoro
BHHTa (npaBHJlO 6ypaBQHKa): eCJlU zo°
Jl.06KY 6Ulima 6paU{amb 6 lianpa6Jl.eliUU
mOKa 8 np060i)liuKe. mo nocmynameJl.b-
lioe 06UJlCeliUe ocmpuSl 8UHma nOKa-
JlCem lianpa6Jl.eliUe MaZHumliOU Ulii)YK-
L{UU 6 lfenmpe KPYZ080Z0 mOKa.
MarHHTHOH HH.n;YKHH MomeT 6bITb Bbl-
I
B = o 2R '
r,Ae R - pa,AHYC KpyroBoro npOBo.n;HHKa.
MarHHTHoe none KpyroBoro TOKa - Heo.n;HopO,AHOe none.
11.7. MarHHTHoe nOJIe COJIeHOHJ];a
COJleHOH,ZJ;OM Ha3bIBalOT KaTYIliKY HnHH,ApHQeCKOH <pOpMbI H3 npOBonOKH,
BHTKH KOTOpOH HaMOTaHbI BnnOTHYIO B O.n;HOM HanpaBneHHH, a .n;nHHa KaTYIlI-
KH 3Ha'lHTenbHO 60nbllIe pa,AHyca BHTKa.
319

MarHIlTHoe none COJIeHOIl.D:a MomHO
npeCTaBIlTb KaK pe3YJlbTaT CJlOmeHIlSI
nOJIell, C03.D:aBaeMbIX HeCKonbKIlMIi KPY-
rOBbIMIi TOKaMll, IlMelOIlMM o6ylO
OCb. Ha pllCYHKe 11.10 BH.D:HO, qTO
BHYTPIi COJIeHOIl,Aa JlIlHIlIi MarHHTHoll
IlH.D:YKll;1l1i Kam.D:OrO OT ,AeJIbHOrO BIlTKa
IlMelOT O,AIlHaKOBoe HanpaBJIeHlle, TOr.D:a
KaK Mem):{y coce):{HIlMH BHTKaMI1 OHII
IlMelOT npOTIiBOnOJlOmHOe HanpaBJIeHlle.
Il03ToMY npH ):{OCTaTOqHO nJIOTHOll Ha-
MOTKe COJIeHOH):{a npOTHBOnOJIOmHO Ha-
npaBJIeHHbIe yqacTKIl JIIiHllll M8rHIlTHOH
HH.D:YKll;HI1 COCe,AHHX BIlTKOB B3allMHO
YHMqTo:maIOTCSI, a O.D:HHaKOBO HanpaB-
N JIeHHbIe yqaCTKIl COJIbIOTC.H B 06ylO
JIIiHIlIO MarHHTHoll IlHnYKUIlIl, npoxo-
,ASIYIO BHYTPIi COJIeHOIl):{a II OXBaTbI-
BalOYIO ero cHapy:mll. H3yqeHIle 3Toro
non.H C nOMOblO OnllJIOK nOKa3aJIO,
qTO BHYTPH COJIeHOIl.D:a nOJle .HBJISIeTC.H
O):{HOpO):{HbIM. MarHIlTHbIe JlIlHIlH npe):{-
CTaBJI.HIOT C060ll np.HMbIe JlIlHMIl, napan-
neJIbHbIe OCII COJIeHOIl.D:a, KOTopbIe pac-
XO.D:SlTC.H Ha ero KOHll;ax II 3aMbIKaIOT-
C.H BHe COJleHOIl.D:a (pIiC. 11.11).
HeTPY):{Ho 3aMeTIlTb CXO):{CTBO Mem-
):{Y MarHIlTHblM nOJIeM COJIeHOIl):{a (BHe
ero) II MarHIlTHbIM nOJIeM nOCTOSlHHO-
1'0 CTepmHeBoro MarHIlTa (pIiC. 11.12).
KOHell; COJleHOlina. 113 KOToporo MarHIlT-
HbIe JIIlHI1Il BbIXO):{SlT, aHaJIOrllqeH ce-
BepHOMY nonlOcy MarHI!Ta N, .D:PyroH
:me KOHell; COJIeHOH.D:a, B KOTOpbIH Mar-
HIiTHbIe JIIiHIlIi BXO,ASlT, aHanOrllQeH
IOmHOMY nOJllOcy MarHHTa S.
IIOJIIOCbI COJIeHOH):{a C TOKOM Ha onbITe nerKO onpe):{eJIIlTb C nOMOblO
MarHIlTHOH CTpeJIKH. 3HaSi me HanpaBJIeHHe TOKa B BHTKe, 3TI1 nOJIIOCbI MO>K-
HO onpe.D:enllTb C nOMOblO npaBlina npaBoro BIlHTa: BpaaeM rOJIOBKY npa-
Boro BIlHTa no TOKY B BHTKe, Tor):{a nocTynaTeJIbHOe ):{BllmeHHe OCTpll.H BHHTa
YKameT HanpaBJIeHHe MarHIlTHoro nOJISI COJIeHOIl.D:a, a CJIe.D:OBaTeJIbHO, II ero
ceBepHOI'O nOJIlOca. MO.D:YJIb MarHMTHOH MH.D:YKll;IlM BHYTPIi OAHOCJIOHHOrO COJIe-
HOHAa BbIqIlCJlSleTC.H no <}JopMYJIe
Puc.ll.l0
+
Puc. 11.11
-----...
. /-..... I
\ I  _---  I
\  r ___  \ J /1
\L ,,, I -  \ I 1# 
" ,\ /,//
"
-./:;:: ./ -:; :\ I )..
I r  ---  1 \ ",
! I ..........----'/ J J
......--..,...,..
Puc. 11.12
I
NI
B = JlJlo - = JlJlonl
l
r):{e N - qllCJIO BHTKOB B COJIeHOIl):{e, l - ):{JIIiHa coneHoll):{a, n - qllcno
BHTKOB. npHxo):{SIeeC.H Ha e):{HHHll;Y ):{JIIlHbI COJIeHOIl):{a.
320

11.8. eiicTBHe MarHHTHoro nOJUI Ha npOBOllHHK C TOKOM.
CHJIa AMnepa
OnbITbI AMnepa, KaK MbI BH,neJUI (CM. 11.1), nOK83aJIll, qTO ,nBa npoBO,A-
HllKa npllT.HrllBaIOTC.H llJIH OTTaJIKllB81OTC.H B 3aBllCllMOCTll OT H8npaBJIeHll.R
TOKa B HllX. 3TO B3allMOJleHCTBlle 06'b.HCHSIeTC.H TeM. qTO CllJla. KOTOPY1O llC-
nbITbIBaeT K8m,AbIH 113 npOBo.nHllKOB, 06YCJlOBneHa MarHllTHblM nOJIeM. C03.na-
B8eMbIM TOKOM ,Apyroro npOBo,nHllKa.
Bo061l\e, MarHOTHOe nOJIe ,11.eHcTByeT C HeKOTOpoO COJIOO Ha JII06011 npo-
BO,11.HHK C TOKOM, HaXO,11.nH"CSI B 3TOM nOJIe.
.n;eHCTBllTeJlbHO. paCnOJIOJKllM npOBo,nHllK C TOKOM (CM. pllC. 11.13) TaK,
'lT06bI TOJIbKO O,nllH npSlMOJlHHeHHbIH yqaCTOK ero ab OK838JICSI B CHJIbHOM
M8rHllTHOM nOJle (MeJK,ny nOJlIOCaMll nO,AKOBOo6pa3Horo MarHHT8), a OCT8nb
HbIe qaCTH enH Haxo,nHJIHCb B 06JIaCT.HX npOCTpaHCTBa, r,ne MarHHTHoe nOJIe
cJIa60e II ero ,neHcTBlleM Ha 3Tll qaCTH enFl MOJKHO npeHe6peqb.
OnbIT nOKa3bIBaeT, qTO B 3aBHCHMOCTll OT HanpaBJleHH.H TOKa 11. OT pac no-
nOmeHll.H non1OCOB MarHllTa npOBo,nHHK ab ,ABllmeTC.H BnpaBO llJIH BJIeBO. BBepx
llJIll BHH3. Ha npOBo,nHllKll, pacnOJIOJKeHHbIe B,nOJIb HanpaBJleHH.H MarHHTHoro
nOJI.H, CHJIbI He ,neHCTBYIOT.
Puc.11.13
(FA -1) B npOBO,AHHKe. qeM
oplleHTaUHll npoBo.nHHKa B
b
CllJIY. JleHcTBYlOll\YIO Ha npOBO,AHllK C TOKOM
B MarHllTHOM nOJIe, H83bIBa1OT CHJlOn AMnepa.
HanpaBJIeHHe CllJIbI AMnepa MOJKHO onpe,neJIFlTb,
nOJIb3Y.HCb npaBHJIOM JleBO" PyKH:
PYKY pacnoJlazaJom maK, Ilm06bt nOpMaJtbliaSl K
np060iJnuKY COCma6Jl.RIOaSl Mazliumnou uniJYK-
ltuu 6xoiJUJl.a 6 Jl.aiJolib, Ilembtpe 6btmSlliymbtX naJl.b
lta 6btJlU nanpa6Jlelibt no mOKY: moziJa omozliY-
mbtU na 90° 60Jl.bUlOU naJl.elt YKaJICem lianpa6Jle
liue iJeucm6YlOeu lia np060iJliuK CUJlbt AMnepa
(pllC. 11.13).
AMnep YCTaHOBllJI 3KcnepllMeHTaJIbHO, qTO MO-
,nYJIb 3TOH CHJIbI TeM 60JIblIIe, qeM CHJIbHee Mar-
HHTHoe nOJle (FA - B), 'leM 60JIbille CHJla TOKa
60nbIlie JlnHHa npOBOJlHHKa (FA - l) II 3aBHCHT OT
MarHHTHOM nOJIe:
FA = BI III sina ,
r,ne a - yroJl MeJK,ny HanpaBJIeHHeM TOKa B npOBo,nHllKe H BeKTopOM Mar-
HllTHOH I1H,nYKllll.
3Ta cpopMYJla HBJI.HeTC.H M8TeMaTllqeCKllM BblpaJKeHlleM 3aKOHa AMnepa. E10
MOJKHO nOJIb30BaTbCSI TOJIbKO Tor,na, Kor,na AJIllHa npOBOJlHllKa TaKOB8, qTO
HH,nYKll.H BO Bcex TOqKaX npOBo,nHHKa MOJKeT CqllTaTbCSI OJlHHaKOBoH, HO eCJIH
MarHllTHoe nOJle O,AHOpo,nHoe, TO ,nJIHHa npOBo,nHllK8 MOJKeT 6bITb JI1060H, HO
npll 3TOM npOBOJlHllK ueJlHKOM JlOnJKeH Haxo,nHTbC.H B nOJIe.
I1HTepec npe,nCTaBn.HeT BpaeHHe npSlMoyrOJIbHOH paMKll C TOKOM B O,nHO-
pOJlHOM MarHllTHOM nOJIe.
CllJIbI JleHcTBYlOll\lle Ha npOBo,nHHK C TOKOM B MarHllTHOM nOJIe, lIIHpOKO
HcnOJlb3YIOTC.H B TeXHllKe. 3JIeKTpo,nBllraTeJlH H reHepaTopbI, YCTpoHcTBa ,nJI.H
321

3anHCH 3BYKa B MarHHTOcpOHaX, TeJlecpOHbI H MHKpOcpOHbI - BO BCeX 3THX H
BO MHOmeCTBe ,ApyrHx npH60pOB H YCTPOHCTB HCnOJlb3yeTC.H B3aHMO,AeHCTBHe
TOKOB, TOKOB H MarHHTOB H T.,A.
11.9. 3JIeKTpOnBHraTeJIb nOCTOHHHoro TOKa
SJleKTpO,ABHraTeJlb - 3TO ,ABHraTeJlb, cJlymaHH ,AJlH npe06pa30BaHH.H 3J1eK-
TpHqeCKOH 3HeprHH B MeXaHHqeCKYIO.
OCHOBHa.H qaCTb 3J1eKTpO,ABHraTeJl.H - 3TO KOHTYP (paMKa, KaTYillKa) C
TOKOM, pacnOJlOmeHHbIH B CHnbHOM MarHHTHOM nOJle (pHC. 11.14). Ha KOH-
TYP B MarHHTHOM nOJle ,AeHcTByeT BpaalOHH
MOMeHT, B pe3YJlbTaTe tJero KOHTYP nOBOpaqHBa-
eTC.H H OCTaHaBJlHBaeTCSI B nOJlOmeHHH paBHOBe-
CH.H, T.e. B nonomeHHH, B KOTOpOM ero MarHHT-
HbIH MOMeHT HanpaBJleH napaJlJlenbHO MarHHT-
HOH HH,AYKHH Pm 11 B (nJlOCKOCTb KOHTypa
nepneH,AHKY n.HpHa nHHH.HM HH,AYKHH MarHHTHO-
ro nOJl.H). ECJlH npH npOXOm,AeHHH KOHTypa qe-
pe3 nonomeHHe paBHOBeCH.H HanpaBneHHe TOKa
H3MeHHTC.H Ha npOTHBOnOJlOmHOe, TO H3MeHHTC.H
H HanpaBJleHHe MarHHTHOrO MOMeHTa. IIpOH,A.H no
HHepUHH nOJlOmeHHe paBHOBeCH.H. KOHTYP c.nena-
eT ee non-060poTa. EcnH nepHOHqeCKH H3Me-
H.HTb HanpaBJleHHe TOKa, TO KOHTYP npH,AeT BO
BpaaTeJlbHOe ,ABHmeHHe. H3MeHeHHe HanpaBJle-
HH.H TOKa ocyecTBn.HeTC.H aBTOMaTHtJeCKH C no-
MOblO YCTpOHCTBa, KOTopoe Ha3bIBaeTC.H KOJlJleK-
TOpOM. KOJlJleKTOp COCTOHT H3 .n;BYX MeTaJlJlHqeC-
KHX nonYI1.HnHH,ApOB, K KOTOpbIM npHcoe.n;HHeHbI
KOHbI KOHTypa. t.J:epe3 HHX H CKOJlb3.HHe KOH-
'l'aKTbI (eTKH) KOHTYP npHCOe,AHH.HIOT K HCTOq-
HHKY TOKa.
HaH60JlbillHH MOMeHT AeHcTByeT Ha KOHTYP,
nJlOCKOCTb KOToporo napaJlJleJlbH8 MarHHTHOH
HH,AYKHH B. Cne.n;OBaTeJlbHO, ecnH pacnono-
mHTb ,ABa KOHTypa nepneH,AHKYJl.HpHO ,Apyr K
.n;pyry H BbIBeCTH HX KOHbI Ha qeTBepTbKonb-
eBOH KOnJleKTOp (pHC. 11.15), TO BpaalO-
HH MOMeHT pe3KO B03paCTeT H YBenHqHTC.H
nnaBHOCTb XO,Aa nO,ABHmHOH tJaCTH ,ABHraTen.H
(pOTopa).
B npOMbIillJleHHbIX .nBHraTeJl.HX MarHHTHoe nOJle C03aeTC.H 06MOTKOH 3neK-
TpOMarHHTa: B pOTope .nenalOT na3bI. B KOTopble YKJla.nbIBaIOT MHoro BHTKOB
O,AHOH ceKHH (BMeCTO paMKH); pa3nHqHbIe ceKHH YJlomeHbI no,A yrJloM APyr
K ,Apyry, H HX KOHl1.bI BbIBe,AeHbI Ha npOTHBOnOJlOmHbIe 60Ka KOJlJleKTOpa, K
KOTOPOMY npHmHMaIOTC.H eTKH, COe,AHHeHHble C HCTOtJHHKOM TOKa. OT HCT01.J-
HHKa TOKa Hanp.HmeHHe no.n;aeTC.H B 3J1eKTpOMarHHTbI CTaTopa (HenO,ABHmHOH
@
+
Puc. 11.14
- +
Puc. 11.15
322

'laCTH ,ABHraTeJl.H). lIo Kam,AOH CeKH11 H)J;eT TOK TOJlbKO TOr,Aa, KOr,Aa ee nJla-
CTHHbI KaCaIOTC.H eTOK, T.e. KOr,Aa nJlOCKOCTb 3TOH CeKHH napaJlJleJlbHa BeK-
TOPY MarHHTHOH HH,AYKHH. lIpH 3TOM CeKHH nOO'lepe,AHO C03,Aa1OT CaMbIH
60nbllIOH BpaalOHH MOMeHT.
MarHHT IinH 3J1eKTpOMarHHT, KOTOpbIH C03,AaeT MarHHTHOe nOJle, Ha3bIBa-
lOT qaCTO HH,AYKTOPOM, a paMKY (06MOTKY), 'lepe3 KOTOpylO nponYCKalOT 3neK-
TpH'tJeCKHH TOK, - .HKOpeM.
OCHOBHOH pa60qeH xapaKTepHCTHKOH 3J1eKTpO,ABHraTeJl.H .HBJI.HeTC.H BpaalO-
llUIH MOMeHT M. C031IaBaeMbIH Ha BaJlY LIBHraTen.H CHnOH AMnepa, LIeHCTBYIO-
eH Ha 06MOTKH HKOp.H:
M = 2FrN = 2IBlrN,
r,Ae I - CHJla TOKa B 06MoTKe. B - HH.AYKUH.H MarHHTHoro non.H, l -
.nnHHa npOBOLIHHKa, r - panHYC pOTopa, N - qHCJlO BHTKOB B 06MOTKe.
TaKHe .ZlBHraTeJlH nOCTO.HHHOrO TOKB Hcnonb3YIOT Ha TpaHcnopTe (B 3J1eKT-
pOB03ax, TpaMBa.HX, TpOJlJleH6ycax), Ha nO,A'beMHbIX KpaHax, BO MHorHX 6bI-
TOBbIX 3J1eKTpHqeCKHX YCTpoHcTBax (3J1eKTp06pHTBbI, MarHHTo<}>oHbI 11 .Zlp.).
C nOMOblO 3J1eKTpO.ZlBHraTeJl.H nOCTO.HHHOrO TOKa - CTapTepa - ocyecT-
BJI.HeTC.H 3anYCK .ZlBHraTeJl.H aBTOM06Hn.H.
11.10. 3JIeKTpOlf3MeplfTeJILHLIe nplf60pLI
B IlIKOJlbHbIX Jla60paTOpH.HX 'tJae Bcero HCnOJlb3Y-
IOTC.H 3J1eKTpOH3MepHTenbHbIe npH60pbl .ZlBYX CHCTeM:
MarHHT03J1eKTpHqeCKOH H 3J1eKTpOMarHHTHoH.
a) MarHHT03J1eKTpHqeCKHe npH60pLl. B MarHHTO-
3J1eKTpH'tJeCKOM npH60pe Hcnonb3yeTcSI .HBneHHe B3aHMO-
,AeHCTBH.H MarHHTHoro non.H nOCTO.HHHOrO M8rHHTa 11
npOBO.ZlH11Ka, no KOTOPOMY npOXO,AHT TOK.
Ha pHCYHKe 11.16 nOKa3aHa cxeMa YCTpoHcTBa on-
Horo H3 TaKHX npH60poB. B Y3KOM npOCBeTe Mem.ZlY
nonlOCHbIM11 HaKOHeqHHKaMH N H S CHJlbHOrO nOCTO-
.HHHOrO MarHHTa .Ii u.HnHH.ZlpOM 2 nOMeeHa JlerKa.H
aJlIOMHHHeBa.H paMKa 1, Ha KOTOpOH HaMOTaHa KaTYIli.
Ka H3 TOHKOH H30nHpoBaHHoH npOBonOKH. PaMKa YK-
penJleHa Ha .ZlBYX nonyoc.HX; C nepe,AHeH nOJlYOCblO CB.H-
3aHa CTpeJlKa, ,ABHralOa.HC.H B,AOnb IlIKanbI npH nOBO-
pOTe paMKH. PaMKa C KaTYIliKOH Y.ZlepmHBaeTC.H B on-
pe.ZleneHHOM nOJlOmeHHH .ZlBYM.H cnHpanbHbIMH npY>KHHa
MH 3. 8J1eKTpH'leCKHR TOK qepe3 npY>KHHbI 3 H nony-
OCH nO.ZlBO,AHTC.H K KaTYIlIKe, KOTOpa.H np" npOXOJK,Ae-
HHH TOKa nOBOpaqHBaeTC.H B MarHHTHOM none, H npH
3TOM npymHHbI 3aKpyqHBaIOTC.H. PaMKa nOBOpaqHBaeTC.H LIO Tex nop. nOKa MO-
MeHT napbI CHJI FA' BpaalOHH ee, He ypaBHoBellIHBaeTCSl npOTHBO,AeHCTBYIO-
HM MOMeHTOM, C03,AaBaeMbIM CHJlaMH ynpyrocTH 3aKpyqeHHblx npY>KHH qeM
60nbllIe CHna TOKa, TeM 60JlbllIe yroJl nOBopOTa paMKH.
raJlbBaHOMeTpbI .ZlaHHOH CHCTeMbI O'tJeHb qYBCTBHTeJlbHbI, seCbMa TOqHbI H
HMelOT paBHOMepHYIO IlIKaJlY. pa60TalOT B u.en.HX nOCTO.HHHOrO TOKa.
Puc. 11.16
323

Puc. 11.17
Puc. 11.18
Y CJlOBHOe 0603Ha'IeHHe npH60poB MarHHTO-
3J1eKTpHqeCKOB CHCTeMbI: C'I - nOKoBo06pa3HbIB
MarHHT C He60JlbllIHM np.HMoyrOJlbHHKOM (KaTYllI-
KOH) Mem,AY nOJIIOCaMH.
6) 3JIeKTpOMa.rHlfTIlLIe npH60pLI. B OCHOBe YCT-
pOHCTBa 3JIeKTpOMarHHTHoro npH60pa JlemHT .HBJle-
HHe B3aHMoeHCTBH.H MarHHTHoro nOJl.H KaTyuIKH H
HaMarHHqeHHOrO elO KycKa meJle3a. CxeMa YCTPOH-
CTBa TaKOrO npH60pa nOKa3aHa Ha pHCYHKe 11.17.
meJIe3Ha.H nJlaCTHHKa B nOMeeHa B6J1H3H KaTym-
KH A. IIpH nponYCKaHHH no KaTyuIKe TOKa nJIac-
THHKa BT.HrHBaeTCSI B KaTYllIKY, nOBOpaqHBa.H CTpeJI-
KY C. IIHJIHHp D. B KOTOpOM BHmeTC.H nopllIeHb
E, CJlY»<HT .nJl.H ycnoKOeHH.H KOJle6aHHH CTpeJIKH.
CBSl3aHHa.H C OCblO nJlaCTHHbI B npymHHa, 3aKpy-
qHBaSlCb, C03aeT npoTHBoeHcTBYIOHH MOMeHT.
IIpH60pbI 3J1eKTpOMarHHTHoH CHCTeMbI MorYT
npHMeH.HTbCSI JI.H H3MepeHHB B en.HX nOCTO.HH-
Horo H nepeMeHHoro TOKa. YCJlOBHOe 0603Haqe-
HHe t - np.HMOH 3JIeKTpOMarHHT.
11.11. rpOMKoroBOpHTeJlb
3
rpOMKorOBOpHTeJIb npHMeH.HeTC.H JI.H npeo6pa-
30BaHHSI 3J1eKTpHQeCKHX KOJle6aHHH B 3BYKOBbIe.
B rpOMKorOBOpHTeJle KaTYlliKa 1 (pHC. 11.18) H3
MeHoro npOBO,Aa coeHHeHa C rH6KoH MeM6paHoH
2 H KOHHqeCKHM H<P<PY30POM 3. KaTYlliKa Ha-
XOHTCSI B MarHHTHOM nOJle nOCTO.HHHOrO MarHH-
Ta 4. IIpH nponYCKaHHH rrepeMeHHoro TOKa Ka-
TYlliKa no eHcTBHeM H3MeH.HIOI.QeHC.H CHJIbI AM-
nepa KOJIe6JIeTCSI C qaCTOTOH, paBHoH qaCTOTe KO-
Jle6aHHH CHJlbI TOKa. KaTYlliKa 3aCTaBJI.HeT KO-
Jle6aTbC.H C TaKoH me qaCTOTOH MeM6paHY H H<p-
<PY30p. 3TH KOJle6aHHSI C03,Aa1OT KOJIe6aHHSI ,AaB-
JleHH.H B03yxa, TO eCTb 3BYKOBble BOJlHbI.
11.12. CHJla JIopeHu,a
Ha npOBOHHK C TOKOM B MarHHTHOM nOJIe eHcTByeT CHJla AMnepa, MO-
YJIb KOTOpOH
FA = IB!::t.lsina .
CYI.QeCTBOBaHHe 3TOB CHJlbI JIopeH 06'b.HCHHJI TeM, qTO MarHHTHoe none
eHcTByeT Ha OTeJIbHbIe BHmYI.QHeC.H 3ap.HmeHHbIe qaCTHbI B npOBOHHKe c
TOKOM. CHJIY AMnepa MomHO paCCMaTpHBaTb KaK paBHoeHcTBYIOYIO CHJI, ,AeH-
CTBYIOI.QHX Ha Bce cB060HbIe 3ap.HmeHHbIe qaCTHbI, )J;BH»<YI.QHeC.H B npOBO,n-
HHKe npH npoxomeHHH B HeM TOKa.
324

CHJla, C KOTOpOH MarHHTHOe nOJle ,neHcTByeT Ha ,lI;BHmyIl..tYIOC.H B HeM 3a-
p.HmeHHYIO qaCTHy, Ha3bIBaeTC.H CHJlOH JIopeHua F JI. MOI.{VJIb CHJIbI JIopeHua
FA
Fn = N '
r,ne N - qHCJlO CB060,nHbIX 3ap.HmeHHbIx qaCTH B npoBo,nHHKe.
EcnH 3ap.H,A O,nHOH qaCTHbI q, TO CYMMapHbIH 3ap.H,n Bcex qaCTH /).q = Nq.
ITo onpeneJleHHIO CHJlbI TOKa I = /).q , rne /)'t - BpeM.H npOXOmneHH.H TOKa.
/).t
Tor,Aa I = N q . ITo 3aKOHY AMnepa FA = I Bl1l sin a = N q I1l B SIn a. Ho I1l = v
I1t I1t I1t
MO,nYJlb CKOpOCTH 3ap.HmeHHOH cB060,nHOH qaCTHbI.
CJIe,noBaTeJlbHO, MO,nYJlb CHJlbI JIopeHIJ;a
r,ne a - yrOJl Mem,ny HanpaBJleHH.HMH CKOpOCTH V H MarHHTHoH HH,nYKHH B.
HanpaBJleHHe CHJIbI JIopeHa onpe,neJl.HIOT no npa-
X BHJlY J1eBOH PyKH: Jl.aOOlib Jl.e60U PYKU pacnOJl.azalOm
maK. Ilm06bt nepneliouKYJtSlpliaSl K CKopocmu 3apSlJICeli-
X lioil llaCmUijbt COCma6Jl.SlIOl4aSl MaZliumlioil UIiOYKijUU
6XOOUJl.a 6 liee, Ilembtpe 6btmSlliymbtX naJl.bija 6btJl.U lia.
X npa6Jl.elibt 600Jl.b CKopocmu 06UJlCeliUJl nOJtOJICUmeJl.bIiO
.lapSlJICelilioil (npomU6 CKopocmu 06UJlCeliUH ompUija-
X meJtbHO 3apH:JfCe1i1iOU 1laCmuqbt). mozoa omozHymbtil Ha
90° 60Jl.bWOU naJl.eq YKaJICem nanpa6Jl.eliUe CUJl.bl JIo-
X peliija (pHC. 11.19).
ITocKonbKY CHJla JIopeHa HanpaBJleHa no yrnoM
X 90° K CKOpOCTH ,ABHmeHH.H 3ap.HmeHHOH qaCTHIJ;bI B
Kam,noH TOqKe TpaeKTopHH, TO pa60Ta CHJlbI JIopeHa
X npH ,nBHmeHHH 3ap.HmeHHOH qaCTHIJ;bJ B MarHHTHOM
nOJle paBHa HYJlIO: A FJI = O.
CornaCHo TeopeMe 0 KHHeTHqeCKOH 3HeprHH, H3Me-
HeHHe KHHeTHqeCKOH 3HeprHH 3TOH 3ap.HmeHHOH qac-
THbl /).WK=AFA=O. CJle,AOBaTeJlbHO, WK=const, T.e.
KHHeTHqeCKa.H 3HeprH.H qaCTHIJ;bI, ,ABHmYIQeHC.H B MarHHTHOM nOJle, He H3Me-
H.HeTC.H, a 3HaqHT, 3ap.HmeHHa.H qaCTHa B MarHHTHOM nOJle ,nBHmeTC.H C
nOCTO.HHHOH no Mo,nYJIIO CKOpOCTblO, a HanpaBneHHe CKOpOCTH H3MeH.HeTCSl
HenpepbIBHO.
,UeHCTBHe CHJlbI JIopeHIJ;a Ha6J1IO,naeTC.H H B npHpO,Ae, H 80 MHorHX TeXHH-
'leCKHX YCTpOHCTBax. HanpHMep. CHna JIopeHua OTKJIOH.HeT 3ap.HmeHHbIe qac-
THbI, BTOpralOlQHeC.H H3 KOCMoca H nOnaI.{alOlQHe B MarHHTHoe nOJle 3eMJlH.
K nOJl.HpHbIM 06J1aCT.HM, r,ne OHH BbI3bIBalOT nOJI.HpHble CH.HHH.H. .D:eHcTBHe Mar-
HHTHoro nOJl.H Ha ,nBHmYHeC.H 3ap.HmeHHbIe qaCTHbJ HcnoJlb3yeTC.H ,nJI.H yn-
paBJleHH.H ,nBHmeHHeM 3J1eKTpOHOB B TeJleBH3HOHHbIX Tpy6Kax, B YCKopHTen.HX
H T .,n.
x
X
x
B
F
X 1X X
Xv
X X v
XX X
X -+ X X
F 2
X
X
x
x
x
X
X
x
Puc. 11.19
F J1 =   Fn = qBvsina,
X
x
325

11.13. )I,BHmeHHe 3apHmeHHbIX qaCTH
B 3JIeKTpH'IeCKHX H MarHHTHbIX IIOJIHX
Ha 3ap.HmeHHYIO qaCTlu1.Y B 3J1eKTpOCTaTHqeCKOM nOJIe ,Aei1cTByeT KynoHOB.
CK8.H CHJI8. KOTOpylO MomHO HallTH, 3H8.H H8np.HmeHHOCTb nOJISl B .n;aHHoll TOqKe:
F = qE. 3Ta CHJI8 co06aeT YCKopeHHe ii = F = qE , r.n;e m - Macca 3ap.H-
m m
meHHOll qaCTHI.I.bI. KaK BH.n;HO, H8npaBJIeHHe YCKopeHH.H 6y.n;eT COBna.n;aTb C H8-
npaBneHHeM E, eCJlH 3ap.H.n; qaCTHn;bI nonOmHTeJleH (q> 0), H 6y.n;eT npOTHBO-
nOJIOmHO E. eCJlH 3ap.H.n; OTpHI.I.aTeJIeH (q < 0).
EcnH 3J1eKTpOCTaTH'leCKOe none O,lJ;HOpO,lJ;HOe (E = const), TO YCKopeHHe
ii = const H qaCTHI.I.a 6Y,AeT COBepIIIaTb P8BHOYCKopeHHoe .n;BHmeHHe (pa3YMeeTcSl,
npH OTCYTCTBHH .n;PyrHx CHJI). BH.n; TpaeKTopHH qaCTHI.I.bI 3aBHCHT OT HaqaJIbHbIX
YCJIOBHll. ECJlH BHa'laJle 3ap.HmeHH8Sl q8CTHI.I.a nOKOHJlaCh (vo= 0) HJIH ee Ha-
q8JIbHaSl CKOpOCTb COHanpaBJIeHa C YCKopeHHeM (v o ii ii), TO qaCTHI.I.a 6y.n;eT co-
BepIIIaTb P8BHoycKopeHHoe npSlMOJIHHeHHOe .n;BHmeHHe B.n;OJIb nOJISI H ee CKO-
pOCTb 6y.n;eT paCTH. ECJlH Vo i J, ii, TO qaCTHI.I.a 6y.n;eT TOpM03HTbC.H B 3TOM nOJle
ECJIH yrOJI Mem.n;y HaQ8JIbHOH CKOpOCTblO H YCKopeHHeM OCTpbIH 0 < ex < 90 0
(HnH TynOH), TO 3apSlmeHHa.H qaCTHn;a B TaKOM 3JIeKTpOCTaTHqeCKOM nOJIe 6y-
.n;eT .n;BHraTbC.H no napa60JIe (CM.  8.10, 38,A8qH 5, 6).
Bo Bcex CJIYQ8SlX npH .n;BHmeHHH 3apSlmeHHoll qaCTHI.I.bl B 3J1eKTpOCTaTH-
qeCKOM nOJIe 6Y,AeT H3MeHSlTbCSl MO.n;YJIb CKOpOCTH, a CJIe.n;OBaTeJIbHO, H KHHe-
TH'leCKa.H 3HeprH.H qaCTHI.I..
CyecTBeHHoe OTJIH'lHe M8rHHTHoro nOJl.H OT 3J1eKTpOCTaTHQeCKOrO COCTO-
HT, Bo-nepBbIX, B TOM, 'lTO MarHHTHoe none He ,lJ;eHcTByeT Ha nOKOSlYIOCSl
3apSleHHyro qaCTHY. MarHHTHoe nOJIe .n;eHcTByeT TOJlbKO Ha .n;BHmYHecSl B
nOJIe 3ap.HmeHHble qaCTHI.I.bI. BO-BTOPbIX, CHna JIopeHI.I.8. .n;eHCTBYIOa.H Ha 3a-
pSlmeHHbIe qaCTHI.I.bI B M8rHHTHOM nOJle. Bcer.n;a nepneH.n;HKYJlSlPHa CKOpOCTM MX
.n;BHmeHMSl. II03TOMY MOl\YJIb CKOpOCTH B MarHMTHOM nOJIe He H3MeH.HeTCSl. He
M3MeH.HeTC.H, CJIe.n;OB8TeJlbHO, H KMHeTH'leCK8.H 3HeprMSl QaCTHI.I.bI. BH,A TpaeKTO-
pMM 3apSlmeHHoll QaCTMI.I.bI B MarHMTHOM nOJIe 38BMCMT OT yrJla Mem.n;y CKOpO-
CTblO BJIeTalOell B nOJIe QaCTMI.I.bl M MarHMTHoll
HH.n;YKI.I.HeH. B03MomHbI TPM pa3JIHqHbIX CJI8Sl.
1. 3apSlmeHHaSl QaCTMI.I.a BJIeTaeT B MarHMT-
Hoe nOJIe CO CKOpOCTblO V, HanpaBJIeHHOll B.n;OJIb
nOJl.H (v ii B) !lJIM npOTMBononomHO H8npaBJIe-
HMIO MarHMTHOH MH.n;YKI.I.MH non.H (v i J, B ).
B 3TMX CJIY'l8.HX CMJIa JIopeHua FJI = 0 M 'laCTM-
I.I.a 6Y,AeT npo.n;onm8Tb .n;BHraTbCSl paBHOMepHO
npHMOJlHHeHHo.
2. 3ap.HmeHHa.H qaCTMI.I.a .n;BMmeTC.H nepneH-
.n;MKYJl.HpHO JlHHH.HM MarHHTHOll HH.n;YKI.I.HH
(pMC. 11.20), Tor.n;a CHJIa JIopeHI.I.8 FJJ= qBv. 8
CJIe.n;OB8TeJIbHO, M coo6aeMoe YCKopeHMe 6y.n;YT
B ·
. .+q
/- .........
. "", . .',\ .
/ ii"
. , . ,.
\
. /. .\ .
· I . FIC · \ .
I
. \ . . I ·
. ,. ./ .
/
. \ . I.
"-
.",. . /.
" ./
. .- ....- ...- -';
v
Puc. 11.20
326

nOCTO.HHHbI no MoynlO H nepneH,AHKYJI.HpHbJ K CKOpOCTH qaCTHbI. B pe3ynb-
TaTe qaCTHa 6yeT BHraTbC.H no OKPYJKHOCTH. pa,AHYC KOTOpOH MomHO HaH-
TH Ha OCHOBaHHH BToporo 3aKOHa HbIOTOHa:
mv 2 mv
F n =ma,,; q Bv=-=>R=-
 R qB .
OTHOllIeHHe !L Ha3bIBalOT YAeJlLHLIM 3apJl.llOM qaCTHULI.
m
IIepHo Bpall\eHH.H qaCTHU;bI
h
I
T = 21tR = 2nm
v qB '
TO eCTb nepHO,A Bpall\eHH.H He 3aBHCHT OT CKOpOCTH qa-
CTHbI H paHyca TpaeKTopHH. Ha 3TOM OCHOBaHO eH-
CTBHe U;HKnOTpOHa (CM. HH>Ke  11.16).
3. CKOpOCTb 3ap.H>KeHHOH qaCTHbI HanpaBJleHa no
yrJloM a. K BeKTOpy B (pHC. 11.21). )];BHmeHHe qaCTH-
bI MomHO npe,ACTaBHTb B BHe cynepn03HHH paBHO-
MepHoro np.HMOnHHeHHoro BHmeHH.H Bonb non.H co
CKOpOCTbJO v n = v cos a. H BH>KeHH.H no OKPY>KHOCTH C
nOCTO.HHHOH no MoynlO CKOpOCTblO v1. = v sin a. B nnoc-
KOCTH, nepneH,AHKyn.HpHOH nonlO. Pa,nHYC OKPY>KHOCTH
onpenen.HeTC.H aHaJlOrHqHO npenbIlIyweMY cnyqalO. Tonb-
KO Hao 3aMeHHTb v Ha v 1. = v sin ex, TO eCTb
R = mvsina. .
qB
B pe3ynbTaTe cnomeHHSI 3THX ,ABHmeHHH B03HHKaeT BHmeHHe no BHHTO-
BOH J1HHHH, OCb KOTOpOH napaJlJleJlbHa MarHHTHOMY nonlO liar BHHTOBOH nH-
Puc. 11.21
HHH
2nm v cos a.
h = vII . T = v cos a. . T =
qB
HanpaBneHHe, B KOTOpOM 3aKpHBaeTC.H CnHpaJlb, 3aBHCHT OT 3HaK8 3ap.H-
a 'laCTHbI.
EcnH CKOpOCTb 3ap.H>KeHHOH 'laCTHU;bI COCTaBn.HeT yron a. C HanpaBJIeHHeM
BeKTopa B HeO,lJ;HOpO,lJ;HOI'O MarHHTHoro nOJl.H, HHYKH.H KOToporo B03pacTaeT
B HanpaBJleHHH ,ABHmeHH.H qaCTHbl, TO R H h YMeHbllIaIOTC.H C pOCTOM B.
Ha 3TOM OCHOBaHa <POKYCHpOBKa 3ap.HmeHHblX 't{aCTH B MarHHTHOM nOJle.
EcnH Ha BHmYll\YIOC.H 3ap.H>KeHHYIO qaCTHy nOMHMO MarHHTHoro non.H C
HH,AYKHeH B eHcTByeT OHoBpeMeHHO H 3J1eKTpOCTaTHqeCKOe nOJle C Hanp.H-
>KeHHOCTblO E, TO paBHoeHCTBYIOll\a.H CHJla, npHJlOmeHHaSl K qaCTHe, paBHa
BeKTopHOH CYMMe 3J1eKTpHqeCKOH CHnbI H CHJIbI JIopeHa: F = F3 + Fn. XapaK-
Tep BH>KeHH.H H BH,A TpaeKTopHH 3aBHC.HT B aHHOM cnY'lae OT COOTHOllleHH.H
3THX CHJI H OT HanpaBJleHHSI 3neKTpOCTaTHQeCKOrO H MarHHTHoro noneH.
327

11.14. IIcnOJIb30BaHue CUJIbI JIopeHa. Macc-cneKTporpa
.n:eHCTBHe, OKa3bIBaeMoe MarHHTHbIM nOJIeM Ha BHmYlI..\HeC.H 3ap.HmeHHbIe
qaCTHl.{bI. OqeHb IIIHpOKO Hcnonb3YIOT B TeXHHKe.
HanpHMep, OTKJIOHeHHe 3JIeKTpOHHoro nyqKa B KHHeCKonax TeJleBH30pOB
OCYlI..\eCTBJI.HIOT C nOMOlI..\blO MarHHTHoro nOJl.H, KOTopoe C03aIOT CneII.HaJIbHbI-
MH KaTYlliKaMH. B p.He ::meKTpOHHbIX npH60poB MarHHTHoe none HCnOJIb3yeT-
CSl ,An.H Q>OKYCHPOBKH nycIKOB 3ap.HmeHHbIX qaCTHII..
B C03aHHbIX B HaCTO.HlI..\ee BpeM.H 3KcnepHMeHTaJIbHbIX YCTaHOBKax n.H
OCYlI..\eCTBJIeHH.H ynpaBJI.HeMOH TepMO.HepHOH peaKII.HH eHcTBHe MarHHTHoro
nOJI.H Ha nna3MY HCnOJIb3YIOT JI.H cKpyqHBaHH.H ee B IIIHYP, He KaCalOlI..\HHC.H
CTeHOK pa60qeH KaMepbI. .n:BHmeHHe 3ap.HmeHHbIx qaCTHII. no oKpymHocTH B
OHOpOHOM MarHHTHOM nOJIe H He3aBHCHMOCTb nepHoa TaKoro ABHmeHH.H OT
CKOpOCTH qaCTHII.bI HCnOJIb3YIOT B II.HKJlH"t.leCKHX YCKopHTeJl.HX 3ap.HmeHHbIX qa-
CTHII. - I..\HKJIOTponax.
.n:eHCTBHe CHJIbI JIopeHII.a HCnOJIb3YIOT Ii B npH6opax, Ha3bIBaeMbIX Macc-
CneKTpOrpa4>aMH. KOTopbIe npe,AHa3Ha"t.leHbI AJI.H pa3.ZJ.eJleHH.H 3ap.HmeHHbIX 'lac-
THII. no HX YAenbHbIM 3ap.HaM.
CxeMa npOCTeHIIIero Macc-cneKTporpaQ>a
nOKa3aHa Ha pHCYHKe 11.22. B KaMepe 1,
H3 KOTOpOH OTKaqaH B03AYX, HaXOAHTC.H HC-
TO'lHHK HOHOB 3. KaMepa nOMelI..\eHa B OA-
HopOAHoe MarHHTHoe nOJIe, B KamoH TO'lKe
KOToporo HHAYKII.H.H B nepneH.ZJ.HKYJI.HpHa
nnOCKOCTH "t.lepTema H HanpaBJIeHa K HaM
(Ha pHCYHKe 11.22 3TO nOJle 0603Ha"t.leHO
KPymOqKaMH). MemAY 3JIeKTpOaMH A H B
npHJIOmeHO YCKOp.HlOlI..\ee Hanp.HmeHHe. no,l(
eHcTBHeM KOToporo HOHbI, BbIneTalOlI..\He H3
HCTO"t.lHHKa, pa3rOH.HIOTC.H Ii C HeKoTopoH CKO-
pOCTblO nona,nalOT B MarHHTHoe nOJIe nepneH-
AHKYJI.HpHO nHHH.HM HHYKII.HH. .n:BHra.HCb B
MarHHTHOM nOJIe no yre oKpymHocTH, HOHbI nonaalOT Ha Q>oTonnacTHHKY 2,
qTO n03BOJI.HeT onpe,AeJlHTb paHYc R 3TOH yrH. 3Ha.H HH.ZJ.YKII.HIO MarHHTHoro
nOJI.H B H CKOpOCTb V HOHOB, no Q>opMYJIe
1,
B __--.-...
0/
/2
A
B
!
K BaKYYMHOMY
Hacocy
Puc. 11.22
!L=
m RB
MomHO onpe,AeJIHTb yeJIbHbIH 3ap.H HOHOB. A eCJIH 3ap.H HOHa H3BeCTeH.
MomHO BbIqHCJIHTb ero Maccy.
11.15. YCKopHTeJIU 3apH:lKeHHblX 'IaLTHQ
YCKopHTeJlH - YCTpollcTBa, CJIymalI..\He JI.H nOJIyqeHH.H nyqKa qaCTHII. (3JIeK-
TpOHOB, npOTOHOB, HOHOB) c 60JIbIIIHMH 3Heprn:.HMH.
328

He06xoHMbIM 3JIeMeHTOM YCKOpHTeJl.H .HBJI.HeTC.H 3J1eKTpOCTaTHqeCKOe nOJIe
C BbICOKOH pa3HOCTblO rrOTeHIJ;HaJlOB U. 8TO nOJle H YCKOp.HeT 'laCTHIJ;bI, co06-
a.H HM KHHeTHqeCKYIO 3HeprHIO
WJ<=qU,
re q - 3ap.H qaCTHIJ;bl.
OHaKO C03aHHe 3J1eKTpHqeCKHX nOJleH C pa3HOCTblO nOTeHIJ;HaJlOB nOp.HA-
Ka MHJlJlHOHa BOJlbT H 60Jlee CB.H3aHO C KOJIOCCaJIbHbIMH TPY,AHOCT.HMH. no-
3TOMY He06xo,AHMYIO 3HeprHIO 'laCTHIJ;bI nOJlyqalOT, eCJlH OHH HeCKOJlbKO pa3
npOH,AYT YCKOp.HlOmee 3neKTpHQeCKOe nOJle. lIJI.H 3Toro QaCTHua 1l0Jl.>KHa llBH-
raTbC.H no 3aMKHYTOH TpaeKTopHH HJIH CnHpaJIH. 8TO B03MomHO, eCJlH BH.>KY-
YIOC.H qaCTHIJ;Y nOMeCTHTb B MarHHTHoe
nOJle. YCKopHTeJlH, pa60TalOHe no TaKO-
MY npHHIJ;Hny, Ha3bIBaIOTC.H HKJlHqeCKHMH
YCKopHTeJlHMH (U;HKJIOTpOHaMH). CxeMa
IJ;HKJIOTpOHa H306pameHa Ha pHCYHKe 11.23.
B BaKYYMHoH KaMepe (,AaBJleHHe OKO-
no 10- 5 -10- 6 MM pT. CT.) pa3MemalOTC.H
Ba nOJIYIJ;HJlHHpa, Ha3bIBaeMbIX JI.yaHTa-
MH. B6J1H3H IJ;eHTpa KaMepbI pacnOJIara-
eTC.H HCTOQHHK T.HmeJlbIX 3ap.HmeHHbIX qa-
CTHIJ; (npOTOHOB, nOJlO.>KHTeJlbHbIX HOHOB).
K yaHTaM nOllBO,AHTC.H BCOKOQaCTOTHoe
(-1-10 fJ.lrIJ;) nepeMeHHoe Hanp.HmeHHe U
Puc. 11.23 3HaqeHHeM OT HeCKOJlbKHX COT O HeCKOJIb-
KHX ,AeC.HTKOB TbIC.Hq BOJlbT.
KaMepa nOMeaeTC.H Mem,AY nOJlIOCaMH MOHoro 3J1eKTpOMarHHTa, KOTOpbIH
C03aeT MarHHTHoe nOJle C HHYKIJ;HeH 1,3 -1,6 TJI. nOJlOmHTeJIbHbIH HOH (HnH
npoToH), nonaB B 3a30p Memy yaHTaMH, npH06peTaeT KHHeTHqeCKYIO 3Hep-
mv 2
rHIO WJ< = -----L = qU. no eHcTBHeM MarHHTHoro nOJl.H 3ap.HmeHHa.H qaCTHIJ;a
1 2
Rl = m qB v l - - B l 2m q U .
BH.>KeTC.H BHYTPH yaHTa no OKPY.>KHOCTH paHYCOM
ECJlH B TOT MOMeHT, KOr,Aa 3ap.HmeHHa.H QaCTHIJ;a On.HTb nonaeT B npoMemy-
TOK Mem,AY ,AyaHTaMH, 3J1eKTpH'leCKOe nOJle H3MeHHT CBOIO nOJl.HpHOCTb, TO 'la-
CTHu;a nOJIyqHT el.Qe O,AHY nopU;HIO 3HeprHH. Ee KHHeTHQeCKaH 3HeprHH CTaHeT
o I t mu
paBHOH W K = 2qU H pa)IHYC nOJlYOKpymHocTH Rl = - -. CeJIaB N 060-
2 B q
pOTOB, qaCTHua npOHlleT YCKOp.HlOmee nOJle 2N pa3. H ee KOHeQHa.H 3HeprH.H
6Y,AeT WJ< = 2N qU .
):(JI.H Toro QT06bI QaCTHIJ;a HenpepbIBHO YCKOp.HJIaCb, He06xoHMO. QT06bI ne-
pHOJI. H3MeHeHHH nOJUIpHOCTH YCKopmoero 3J1eKTpH'leCKoro noJUI COBna.n:.aJl C ne-
pHOJI.OM 06paeHHH qaCI. 8TO ycnoBHe Ha3bIBaeTC.H YCJlOBHeM CHHXpoHH3cUJ,HH:
21tm
TnoJul = T HoHa = q B
329

QHKnOTpOH npHrOeH nH YCKOpeHH.H qaCTH TOJlbKO O CKOpOCTeH. MHOrO
MeHbIIIHX CKOpOCTH CBeTa. TaK KaK npH CKOpOCT.HX, 6nH3KHX K CKOpOCTH CBe-
Ta, MaCCa 'IaCTH He OCTaeTC.H nOCTO.HHHOH H YCJlOBHe CHHxpOHH3aIJ;HH 6yeT
HapYIllaTbCSI. I103TOMY IJ;HKnOTpOH COBepIlleHHO He npHroeH JI.H YCKopeHH.H
3neKTpOHOB H HCnOJlb3yeTC.R JlH YCKopeHHH OTHOCHTeJlbHO T.HmenbIX HOHOB H
npOTOHOB.
11.16. MarHHTHble CBoHcTBa BeIl.\eCTBa.
MarHHTHaH npOHHlJ,aeMOCTb
MHOrOqHCneHHbIe onblTbI CBHeTeJlbCTBYIOT 0 TOM, qTO Bce BeI..QeCTBa, no-
MeI..QeHHbIe B MarHHTHoe none, HaMarHHqHBalOTC.H Ii C03aIOT c06CTBeHHoe Mar-
HHTHoe nOJle, eHCTBHe KOToporo CKJlabIBaeTCH C eMCTBHeM BHeIIIHero Mar-
HHTHoro nOJl.H:
B=Bo+Bl'
r,Ae B - MarHHTHa.H HHYKIJ;H.H non.H B BeI..QeCTBe; Bo - MarHHTHa.H HH-
YKH.H nOJl.H B BaKYYMe, Bl - MarHHTHa.H HHYKIJ;HH nOJlH. B03HHKIllero
6naroapH HaMarHH'IHBaHHIO BeI..QeCTBa. I1pli 3TOM BeI..QeCTBO MO:>KeT nH60 YCH-
nHBaTb, JlH60 OCJla6J1HTb MarHHTHoe nOJle. BnHHHHe BeI..QeCTBa Ha BHeIIIHee
MarHHTHoe nOJle xapaKTepH3yeTC.H BenH'IHHOH J.l, KOTOpa.H Ha3bIBaeTC.H MarHHT-
HOH npOHHaeMOCTblO BeI..QeCTBa
B
J..l=-
Bo
MarHHTHaH npOHHIJ;aeMOCTb - 3TO cpH3H'IeCKa.H CKan.HpHa.H BenH'IHHa, no-
Ka3bIBalOa.H, BO CKonbKO pa3 HHYKH.H MarHHTHoro nOJl.H B aHHOM Be-
I..QeCTBe OTJlH'IaeTC.H OT HHYKHH MarHHTHoro non.H B BaKYYMe.
BeI..QeCTBa. OCna6n.HIOI..QHe BHeIIIHee MarHHTHoe none. Ha3bIBalOT JI.HaMarHe-
THKaMH (BHCMYT. a30T. renHH. yrneKHcnOTa. Bona. cepe6po, 30nOTO, IJ;HHK,
KaMHH H p.).
BeI..QeCTBa, YCHnHBalOI..QHe BHeIIIHee MarHHTHoe nOJle, - napaMarHeTHKIf
(anIOMHHHH, KHcnopo, nnaTHHa, Meb, KanbHH, XPOM, MapraHe, conH KO'
6anbTa H p.).
.LI;JI.H ,AHaMarHeTHKOB J.l < 1, JI.H napaMarHeTHKOB J..l > 1. Ho B TOM H PY-
rOM cnY'Iae OTnH'IHe J.l OT 1 HeBenHKO (HecKonbKo eC.HTHTbIC.HqHbIX Hnli CTO'
TbIC.H'lHbIX ,AOneH eHHHIJ;bI). TaK, HanpHMep. Y BHCMYTa J.l = 0.9998 = 1,000.
HeKoTopbIe BeI..QeCTBa (:>KeJle30, K06anbT, HHKeJlb, raOJlHHHH H pa3nHqHbIe
cnnaBbI) BbI3bIBalOT O'leHb 6onbIlloe YCHneHHe BHeIIIHero nOJl.H. Mx Ha3bIBalOT
<l>eppoMarHeTHKaMH. .nn.H HHX Jl == 10 3 -10 5 .
BnepBbIe 06'b.HCHeHHe npH'IHH, BCJleCTBHe KOTOpbIX TeJla 06J1a,AaIOT Mar.
HHTHbIMH CBOHCTBaMIi, aJl AMnep. COrJlaCHO ero rHnOTe3e, BHYTPH MoneKyn
H aTOMOB HpKynHpYIOT 3neMeHTapHble 3neKTpH'IeCKHe TOKH. KOTopble Ii onpe.
,nen.HIOT MarHHTHbIe CBOHCTBa JlI06oro BeI..QeCTBa.
B HaCTO.HI..Qee BpeM.H YCTaHOBJleHO, qTO Bce aTOMbl H 3J1eMeHTapHbIe qaCTH.
bI eHCTBHTenbHO 06naalOT MarHHTHbIMH CBOHCTBaMH. MarHHTHble CBOHCTBa
aTOMOB B OCHOBHOM onpeenHIOTC.H BXO.HI..QHMH B HX COCTaB 3J1eKTpOHaMH.
330

COrJlaCHO nOJlYKJlaCCHqeCKOH MOeJlH aTOMa, npeJlOmeHHOH S. Pe3epq)QpOM
H H. BOpOM (CM. nOp06Hee HHme 9 20.3-20.5), 3neKTpOHbI B aTOM ax BH-
mYTC.H BOKpyr .Hpa no 3aMKHYTbIM op6HTaM (B nepBOM npH6nHmeHHH MomHO
CqHTaTb, qTO no KpyroBbIM). .D:BHmeHHe 3J1eKTpOHB MomHO npeCTaBHTb KaK
3neMeHTapHbIH KpyroBoH 1'0K i = e . v, r,Ae e - 3apSl 3J1eKTpOHa, v - qaCTO-
Ta BpaI.QeHH.H 3J1eKTpOHa no op6HTe. STOT TOK 06pa3yeT MarHHTHoe none,
KOTopoe xapaKTepH3yeTC.H MarHHTHbIM MOMeHToM, MOYJlb ero onpe,AeJl.HeTCSl
c}>OPMYJlOH Pm = is = evS, re S - nnOI.Qab Op6HTbI.
MarHHTHbIH MOMeHT 3J1eKTpOHa. 06YCJlOBneH-
HbIH ero BHmeHHeM BOKpyr .Hpa, Ha3bIBalOT op-
6HTaJlLHLIM MarHHTHLIM MOMeHTOM. Op6HTaJlbHbIH
MarHHTHbIH MOMeHT - 3TO BeKTOpHa.H BeJlHqHHa,
Ii HanpaBJleHHe Pm Onpe,Aen.HeTC.H no npaBHny
npaBoro BHHTa. EcnH 3neKTpoH BHmeTcSl no
xoy qaCOBOH CTpenKH (pHC. 11.24). TO TOKH Ha-
e npaBJleHbI npOTHB XOlIa qaCOBOH CTpenKH (no Ha-
npaBneHHIO ,qBHmeHH.H nonOmHTeJlbHOrO 3ap.Ha),
Ii BeKTOp Pm nepneHHKYJl.HpeH nJlOCKOCTH Op6H-
TM. TaK KaK B aTOMe nJlOCKOCTH Op6HT pa3J1Hq-
HbIX 3J1eKTpOHOB He COBnaaIOT. TO HX MarHHT-
HbIe MOMeHTbI HanpaBJleHbI nOlI pa3HbIMH yr JlaMH
APyr K pyry. Pe3Y nbTHpYIOHH Op6HTaJlbHbIH
MarHHTHbIH MOMeHT MHOr03J1eKTpOHHoro aTOMa paBeH BeKTopHOH CYMMe op6H-
TaJlbHbIX MarHHTHbIX MOMeHTOB oTenbHbIx 3J1eKTpoHoB.
HeCKOMneHCHpOBaHHbIM Op6HTaJlbHbIM MarHHTHbIM MOMeHTOM 06J1aaIOT aTO-
MM C qaCTHqHO 3anOJlHeHHbIMH 3J1eKTpOHHbIMH 060JlOqKaMH. B aTOMax C 3a-
nonHeHHbIMH 3J1eKTpOHHbIMH 060JlOqKaMH OH paBeH O.
KpOMe Op6HTaJlbHOrO MarHHTHoro MOMeHTa, 3J1eKTpOH 06naaeT eI.Qe co6-
CTBeHHLIM (cnHHOBLIM) Mal'HHTHLIM MOMeHTOM P ms ' qTO BnepBbIe YCTaHoBHnH
O.IlITepH H B. repJlax B 1922 r. CYI.QeCTBOBaHHe MarHHTHoro nOJl.H y 3J1eKT-
pOHa 6MJlO 06'bSlCHeHO ero BpaeHHeM BOKpyr c06CTBeHHoH OCH, XOT.H H He
cneyeT 6YKBaJlbHO ynolI06J1.HTb 3J1eKTpOH BpawalOweMYCSl 3ap.HmeHHOMY III a-
PHKY (BOJlqKY).
.D:ocToBepHO YCTaHOBneHO, qTO MarHHTHoe nOJle 3neKTpOHa .HBJI.HeTC.H Ta-
KHM me HeOT'beMJleMbIM CBOHCTBOM, KaK ero Macca H 3ap.H. SneKTpOH, B BeCb-
Ma rpy60M npH6J1HmeHHH, MomHO npeCTaBHTb
KaK OqeHb ManeHbKHH IIIapHK, oKpymeHHbIH 3J1eK-
TpHqeCKHM H MarHHTHbIM nOn.HMH (pHC. 11.25).
MarHHTHbIe nOJlSl y Bcex 3J1eKTpOHOB OHHaKOBbI,
KaK OHHaKOBbI HX MaCCbI H 3ap.H,AbI. CnHHoBbIH
MarHHTHbIH MOMeHT Pms - BeKTOp, HanpaBneH-
HbIH BOJlb OCH BpaeHH.H. OH MomeT opHeHTH-
pOBa TbC.H TOJlbKO BYM.H cnoco6aMH: JlH60 no.. . ,
JlH60 npOTHB... ECJlH B TOM MeCTe, r)J;e Haxo-
HTC.H 3neKTpOH, eCTb BHelliHee MarHHTHoe nOJle,
TO nH60 no nonlO, JlH60 npoTHB nOJl.H. KaK no-
Ka3aHO B KBaHTOBOH c}>H3HKe, B OHHaKoBOM 3Hep-
Pm
-_ i
--,
v
Puc. 11.24
Puc. 11.25
331

reTHqeCKOM COCTO.HHIiIi MOrYT HaXOlIIiTbC.H TOnbKO lIBa 3J1eKTpOHa, CnliHOBbIe
MarHIiTHbIe MOMeHTbI KOTOpbIX npOTIiBOnOJIO>KHbI (npIiHlin rIaYJIIi; CM. HIi>Ke
 21.5).
'Y MHOr03J1eKTpOHHbIX aTOMOB cnliHOBbIe MarHIiTHbIe MOMeHTbI OT,AeJIbHbIX
3JIeKTpOHOB. KaK Ii Op6liTaJlbHbIe MOMeHTbI, CKna}:(bIBaIOTC.H KaK BeKTopbI. ITpli
3TOM pe3YJlbTIiPYIOIill: cnHHOBbIll: MarHIiTHbIll: MOMeHT aTOM8 y 8TOMOB C 3a-
nOJIHeHHbIMH 3neKTpOHHbIMl1 060JlO'lK8MIi paBeH O.
061ii1 MarHIiTHbIll: MOMeHT aTOMa (MOJIeKYJIbI) Pa paBeH BeKTOpHOll: CYM-
Me MarHIiTHbIX MOMeHTOB (op61iTaJIbHbIX H cnHHOBbIX) BXO}:(.HII.1IiX B aTOM (MO-
JIeKYJIY) 3JIeKTpOHOB:
liaMarHeTIiKIi COCTO.HT 1i3 aTOMOB, KOTopbIe B OTCYTCTBlie BHelilHero Mar-
HIiTHoro nOJIS! He IiMelOT c06CTBeHHbIX MarHIiTHbIX MOMeHTOB, TaK KaK Y HIiX
CKOMneHClipOBaHbI Bce cnliHOBbIe Ii Bce Op6liTaJlbHbIe MarHIiTHbIe MOMeHTbI.
BHelilHee MarHIiTHoe none He }:(ell:CTByeT H8
BeCb aTOM ,AliaMarHeTIiKa, HO }:(ell:CTByeT Ha OT-
.n.eJIbHbIe 3JIeKTpOHbI aTOMa. MarHIiTHbIe MOMeH-
TbI KOTOpbIX OTJIliqHbI OT HYJI.H. ITYCTb B }:(aH-
HbIll: MOMeHT CKOpOCTb V 3JIeKTpOHa COCTaBJI.HeT
HeKoTopbIi1 yron a (pIiC. 11.26) C MarHHTHoi1
HH}:(YKluleH Bo BHelilHero nOJIR. BJIarOapR co-
CTaBJIRIOII.1ell: V 1. H8 3JIeKTpOH 6y}:(eT }:(eHcTBoBaTb
CIiJIa JIopeHua Fll (HanpaBJIeHHa.H K HaM H8
pHC. 11.26), KOTOp8.H BbI30BeT }:(onOJIHIiTeJIbHOe
(KpoMe }:(Pyrlix }:(BIi>KeHIiH, B KOTOpbIX yt.J:aCTByeT
3JIeKTpOH npli OTCYTCTBIiH nOJI.H) }:(BlimeHHe no
oKpymHoCTH. Ho 3TO }:(BlimeHlie npe}:(CTaBJI.HeT
C060H .n;onOJIHIiTenbHbIll: KpyroBOll: TOK, KOTOpbIH
C03}:(aCT MarHIiTHoe nOJIe, xapaKTepli3yeMoe Mar-
HIiTHbIM MOMeHTOM Pm (HaBe}:(eHHbIM), HanpaBJIeHHbIM no npaBIiJIY npaBoro
BHHTa HaBCTpe'tly 130. B pe3YJIbTaTe }:(HaMarHeTIiKH OCJIa6JIRIOT BHelilHee Mar-
HHTHoe nOJIe.
ITapaMarHeTHKH COCTORT H3 aTOMOB, y KOTOpbIX pe3YJlbTIiPYIOII.1Hll: Mar-
HIiTHbIH MOMeHT aTOMa pa:t O. B OTCYTCTBHe BHelilHero nOJIR 3TH MOMeH-
TbI opHeHTHpOBaHbI xaOTHt.J:eCKH H BeeCTBO B enOM He C03}:(aeT BOKpyr
ce6.H MarHIiTHoro nOJI.H. npH nOMeII.1eHHH napaMarHeTIiKOB B MarHHTHoe
nOJIe npOIiCXO}:(IiT npeHMyecTBeHHaJi oplieHTaH.H BeKTopOB Pa no nOJIIO
(3TOMY npen.HTCTByeT TenJIOBOe }:(BH>KeHHe t.J:aCTH). TaKIiM 06pa30M, napa-
MarHeTIiK HaMarHHt.J:HBaeTC.H, C03,AaBa.H c06CTBeHHOC MarHHTHoe nOJIe, CO-
Bna}:(alOee no HanpaBJIeHli10 C BHelllHIiM nOJIeM H YCHJIHBalOII.1ee ero. STOT
3cpcpeKT Ha3bIBaeTC.H napaMarHIiTHbIM. ITpH OCJIa6neHHH BHelilHero MarHHT-
Horo nOJI.H }:(O HYJI.H oplieHTaHR MarHHTHMX MOMeHTOB BCJle}:(CTBHe Tenno-
Boro }:(BH>KeHHR HapYlilaeTCR H napaMarHeTHK pa3MarHHt.J:HBaeTCR. B nap8-
MarHeTHKax Ha6JIIO}:(aeTCR H }:(HaMarHIiTHbIll: 3cpcpeKT, HO OH 3Hat.J:HTeJIbHO
cna6ee napaMarHHTHoro.
'"
\
\
* \V.L___ 1 V
I
_ a. I _
Pm _ Bo
I V.
ill I
Ii
/
/
Puc. 11.26
332
Pa = L Pm + L Pms .

11.17. <l>eppoMarHeTH3M
KaK YKa3aHO BbIIlIe, <l>eppoMarHeTHKH - 3TO BeeCTBa, Y KOTOpbIX Mar-
HHTHa.H npOHHllaeMOCTb Jl OqeHb BeJlHKa (Jl» 1). KpoMe Toro. <l>eppoMarHe-
THKH 06JIap;a1OT elQe H ,ApyrHMli CBoHcTBaMH, CYlQeCTBeHHO OTJlHqalOHMIi HX
OT p;Ha- H napaMarHeTHKoB. Oc06bIe CBoHcTBa <l>eppoMarHeTHKoB 06YCJIOBJlHBa-
lOTC.H )J;BYM.H <l>aKTopaMH: 1) HaJlHqHeM HeCKOMneHCHpOBaHHbIX MarHHTHbIX MO-
MeHTOB B He)J;OCTpoeHHbIX 3J1eKTpOHHbIX 060JIOqKaX; 2) OC060H KpHCTaJIJIHtIeC-
KOH CTPYKTYPOH <l>eppoMarHeTHKoB.
..
B
»
Puc. 11.27
Cnen;HaJlbHbIe onbITbI. npoBe.neHHbIe .ne raa30M. nOKa3aJIH. qTO ctJeppoMarHe-
TH3M 06YCJlOBJIeH cnHHOBbIM MarHHTHbIM MOMeHTOM 3JIeKTpoHoB. a He HX Op6H-
TaJlbHbIM ,ABHmeHHeM. IIpHqeM B 3THX BelQeCTBaX 06pa3YIOTC.H I:(eJIbIe 06naCTH
()J;OMeHbI), B KOTOpbIX HeCKOMneHCHpOBaHHbIe cnHHOBbIe MOMeHTbJ opHeHTHpOBaHbI
B O)J;HOM HanpaBJleHHH. IIpH OTCYTCTBHH MarHHTHoro nOJI.H )J;OMeHbI opHeHTHpOBa-
HbI XaOTHqeCKH, a npli HaJlOmeHHH BHelliHero MarHHTHoro nonSl opHeHTHpYIOTC.H
B,AOnb Hero (pHC. 11.27). 8THM 06"b.HCH.HIOTCg CBoHcTBa <l>eppoMarHeTliKoB:
1) <l>eppoMarHHTHbIe CBoHcTBa BelQeCTBa npO.HBJI.HIOTC.H TOJlbKO TOr,Aa, Kor)J;a
COOTBeTCTBYlOlQee BelQeCTBO HaXO)J;HTC.H B KpHCTaJlJIH'IeCKOM COCTOJlHHH;
2) P;Jlg Kam)J;oro BelQeCTBa HMeeTC.H onpe,lJ,eJIeHHaH TeMnepaTypa (TOtIKa
KlOpH), BbIllie KOTOpOH <l>eppoMarHHTHbIe CBOHCTBa HCqe3alOT H <l>eppoMarHeTHK
npeBpalQaeTCH B 06bIqHbIH napaMarHeTHK;
3) MarHHTHaJl npOHHL\aeMOCTL cpepPoMar-
HeTHKa 3aBHCHT OT BHeWHero MarHHTHoro
J.1 nOJIJI (pHC. 11.28). BHa'laJle Jl paCTeT C
J.1", YBeJIHqeHHeM Bo, 3aTeM. .D,OCTHrag MaKCHMY-
Ma, HaqHHaeT YMeHbIlIaTbC.H, CTpeM.HCb B
CJIyqae CHJIbHbIX nOJIeft K 1 (npH 3HaqeHliH
MarHHTHoH HHP;YKI:(HIi B Bce .D,OMeHbI OPH-
eHTHpoBaHbI B)J;OJIb nOJI.H Ii npH .D,aJIbHeHIlIeM
YBeJlHqeHHH Bo MarHHTHa.H HH)J;YKI:(H.H B 1 B
06pa311e nepeCTaeT H3MeH.HTbCH. a Bo YBe-
1 JlHqHBaeTC.H H MarHHTHa.H npOHHllaeMOCTb
B'
o
Bo
Puc. 11.28
B B+n n
Jl=-= 0  =1+..=1.-
Bo Bo Bo
YMeHbIlIaeTC.H);
333

4) q,eppoMarHeTHKH HaMarHHqHBaIOTC.H J1.0 HaCbleHH.fI B cna6bIx MarHHT-
HbIX nOJl.Hx. Ha pHCYHKe 11.29 nOKa3aHO, KaK H3MeH.HeTC.H MarHHTHa.H HH-
,nYKIJ;H.H B B CTaJIH C H3MeHeHHeM BHeWHero nOJl.H Bo. AJI.H cpaBHeHH.H: napa-
MarHeTHK HaMarHH'tIHBaeTC.H 1I0 HaCbII..QeHH.H B nOJI.HX C Bo= 100 + 1000 Tn;
B.TJI
B
A
1
2
3
Be
10- 3 Tn
Be
1,5
A'
Puc. 11.29
Puc. 11.30
5) y q,eppoMarHeTHKoB Ha6JIIO,naeTC.H OCTaTO'lHaH HaMarHH'IeHHOCT:L. ECJIH,
HanpHMep, q,eppoMarHHTHbIH CTepmeHb nOMeCTHTb B COJleHOH,n, no KOTOPOMY
npoxo,nHT TOK, H HaMarHHTHTb ,no HaCbII:QeHH.H (TO'tIKa A) (pHC. 11.30), a
3aTeM YMeHbwaTb TOK B COJIeHOH,Ae, a BMeCTe C HHM If Bo, TO MomHO 3aMe-
THTb, qTO HH,nYKIJ;H.H nOJl.H B CTepmHe B npOIJ;eCce ero pa3MarHHqHBaHH.H OCTa-
eTC.H Bce BpeM.H 60JlbllIeH, 'tIeM B npou;ecce HaMarHHqHBaHH.H. Kor,na Bo = 0 (TOK
B COJleHOH,ne BbIKJIlOqeH), HH,AYKIJ;H.H 6y,neT paBHa Br (ocTaTO'tIHa.H HH,nYKIJ;H.H).
CTepmeHb MomHO BblHYTb H3 COJleHOH,na H HCnOJlb30BaTb KaK nOCTO.HHHbIH Mar-
HHT. 't.JT06bI OKOHqaTeJlbHO pa3MarHHTHTb CTepmeHb, HY*HO nponYCTHTb no
COJIeHOH,ny TOK npOTHBOnOJlOmHOrO HanpaBJleHH.H. T.e. npHJIOmHTb BHeWHee
MarHHTHoe nOJle C npOTHBOnOJlOmHbIM HanpaBneHHeM Bo. YBeJIH'tIHBa.H Tenepb
no Mo,nynJO HH,nYKIJ;HJO 3Toro nOJI.H, p83M8rHHqHBalOT CTepmeHb (B = 0). Mo-
,nYJIb Boc HH,nYKU;HH MarHHTHoro nOJI.H, pa3MarHHQHBalOI:QerO HaMarHHqeHHbIH
q,eppoMarHeTHK, Ha3bIBalOT K03pHTHBHOH CHJlOH. IIpH ,nanbHeHweM YBeJIHQe-
HHH Bo MomHO HaMarHHTHTb CTepmeHb ,no HaCblI:QeHH.H (TOqKa A').
YMeHbWa.H Tenepb Bo ,no HYJI.H, nOJlY'tIaIOT On.HTb nocToSlHHbIH MarHHT, HO
C HH,AYKIJ;HeH -Br npOTHBononomHoro HanpaBJIeHH.H. 't.JT06bI BHOBb pa3MarHH-
THTb CTepmeHb, HymHo CHOBa BKJIlOtIHTb B COJleHOH,n TOK nepBOHaqaJIbHOrO
HanpaBJleHH.H, H CTepmeHb pa3MarHHTHTC.H, Kor,na HH,nYKIJ;H.H Bo CTaHeT paB-
HOH Boc. IIpo,noJIma.H YBeJIHtIHBaTb Bo, CHOBa HaMarHHtIHBalOT CTepmeHb ,no
HaCbII:QeHH.H (TOqKa A).
TaKHM 06pa30M, npH HaMarHHQHBaHHH H pa3MarHH'tIHBaHHH eppoMarHe-
THKa HH,nYKIJ;H.H B OTCTaeT OT Bo. 8TO OTCTaBaHHe B OT Bo Ha3bIBaeTC.H
.HBJleHHeM rHCTepe3Hca. H306pameHHa.H Ha pHCYHKe 11.30 KpHBa.H Ha3b1-
BaeTC.H neTJIeH rHCTepe3Hca. BenHQHHbI Bn Boc H Jl m onpe,neJI.HIOT 06JIaCTb
npHMeHeHH.H eppoMarHeTHKoB ,nJI.H npaKTHtIeCKHX IJ;eJIeH. eppoMarHeTHKH
C Y3KOH neTJleH rHCTepe3Hca (MaJIOe 3HaQeHHe Br H Boc) (pHC. 11.31, a)
Ha3bIBalOT M.fIrKHMH, HX HCnOJIb3YIOT ,nJI.H H3rOTOBJIeHH.H cep.n;eqHHKOB TpaHC-
334

a
B
Bo
Puc. 11.31
6
B
Bo
opMaTopOB, reHepBTopoB, ABHrBTe-
JleH, rAe eppoMBrHeTHK nOABeprBeT-
C.H tIaCTbIM nepeMarHHqHBaHH.HM. <I>ep-
pOMarHeTHKH C WHpOKOH neTneH rHC-
Tepe3Hca (60nbwoe 3HatIeHHe Br H
Boe) (pHC. 11. 31, 6) HB3bIBBIOT jKeCT-
KHMH H HcnOJlb3YlOT ,AnSI H3rOTOBne-
HH.H nOCTO.HHHbIX MBrHHTOB;
6) npou.ecc HaMarHHtIHBaHH.H cpeppo-
MBrHeTHKOB conpOBOm,ABeTCSI H3MeHeHH-
eM HX JlHHeHHbIX pa3MepOB H 06'beMB.
STO HBJleHHe Ha3bIBaeTCH MaI'HHTOCTpHK-
lJ,HeH.
11.18. MarHHTHaH 3anHCL HH4>opMaHH
npOH3Be,AeHH.H 3BYKa.
C nOMOlUblO MHKpooHa 1 3BYKO-
Bble Kone6aHH.H npe06pB3YIOTCH B KO-
ne6aHH.H sneKTpHtIeCKOrO TOKB, H STOT
H3MeH.HIOHHC.H B COOTBeTCTBHH co
3BYKOBbIMH Kone6aHH.HMH TOK nponyc-
KBeTC.H no KBTYWKe 3neKTpOMarHH-
Ta 2 C ,Ayro06pB3HbIM Cep,AetIHHKOM H
Y3KHM Mem,AynonlOcHbIM npOMemYTKOM
(3anHcbIBalOaH ronoBKa). MHMO ro-
JlOBKH npOXOAHT ABHmYB.HCSi no CH-
CTeMe KaTyweK 3 (c nOMOblO SJleKTp0,ABHraTeJl.H) MarHHTHB.H neHTa 4. STO
TOHKa.H neHTB H3 nOJlHMepHOro MBTepHBna (au.eTHnu.ennlOn03bI), HB KOTO-
PYIO TOHKHM cnoeM HaHeceH MBrHHTHbIH nopOWOK H3 OKCHAa mene3a
(Fe203). pa3Be,AeHHOrO B naKe.
TIpOXO,A.H MHMO nonlOCOB MarHHTa 2. nopOWHHKH HaMarHH'lHBaIOTC.H B co-
OTBeTCTBHH C TeM TOKOM, KOTOpbIH B STOT MOMeHT npOXOAHT no KaTYWKe. H
STY HBMarHHtIeHHOCTb COXpaH.HIOT (OCTBTO'lHB.H HHD;YKU.H.H) CKonb yro.nHO .uonro
(He HaAO TOJlbKO CHJlbHO rpeTb JleHTY H nOABeprBTb ee ,AeHCTBHIO CHJlbHbIX
MarHHTHbIX noneH).
n.H BocnpOH3BeAeHHH 3anHcaHHoro 3BYKa Hcnonb3yeTcSi ronOBKa 5 (Bocnpo-
H3BOA.HBH ronoBKB), YCTpoeHHa.H npHMepHO TaK me, KBK H 3BnHCbIBBIOBH
ronOBKB. 3anHcbIBalOB.H ronOBKa 2 npH STOM AonmHa 6bITb OTKnlO'IeHB.
KorAB JleHTa C 3anHCBHHbIM Ha MBrHHTHOM SI3bIKe 3BYKOM npOXOAHT MHMO
SJleKTpOMarHHTa, B ero KaTYWKe B03HHKBeT SC, 3HBtIeHHe KOTOpOH pB3nHtI-
HO AnSI pa3HbIx Y'lBCTKOB JleHTbI nOA nOJlIOCaMH MBrHHTB rOJlOBKH, nOToMY
'lTO pa3J1HqHble Y'laCTK1:I neHTbI no-pa3HoMY HBMBrHH'leHbI. STB H3MeH.HIO8H-
C.H SC c03AaeT nepeMeHHbIR TOK B KaTywKe rpOMKorOBOpHTen.H 6.
6
5
)
......7
Puc. 11.32
\
4
1
meCTKHe eppoMBrHeTHK1:I Hcnonb-
3YIOTC.H ,AnSI 3BnHC1:I 1:1 xpaHeHHSi 3BYKa
HnH APyroH HHOPMBU.HH. HB pHcYHKe
11.32 nOKB3aHB cxeMB 3BnHCH H BOC-
2

3
335

3anl1CaHHbIR 3BYK MOmHO 1:1 CTepeTb. ,[(JUI 3TOrO cnyml1T CTl1palOll\a.H ro-
nOBKa 7. 8TO Tome 3J1eKTpOMarHI1T, B KaTYlliKY KOToporo nOaeTC.H TOK BbI-
COKOR qaCTOTbI, HO CI1JIa TOKa B HeM nOCTeneHHO y6bIBaeT. TaKoR nOCTeneH-
HO YMeHbllIalOll\I1RC.H nepeMeHHbIR TOK nOJlHOCTblO pa3MarHI1ql1BaeT HaMarHI1-
qeHHble nopOlIIHHKI1, 11 JIeHTa rOTOBa K HOBOR 3anHCI1.
BonpOCbI ,lJ;JUI CAMOKOHTPOJUI
1. KaKue B3au.Jrwoeucm6uSl lia3bl6a1Om MazliumHblMu? KaKue CUJl.bl lia3bl-
6a1Om MazliumliblMu? KaK 63auMooeucmBY1Om MeJICoy c060u nOJlIOCbl .JrtaZHU-
mOB?
2. f{PM 06YCJl.06Jl.eIi0 cyU{ecm606aliue MaZliumliOZO nOJl.Sl? KaKuMu npucno.
C06JleHUJl.MU nOJlb3YlOmCJl. OJl.Jl. uCCJleOOBaHu.n C60ucmB MaZliumHOZO nOJlJl.?
3. KaK zparjJullecKu MOJICem 6blmb u306paJICeliO MaZliumlioe nOJl.e? f{mo Ha.
3blBaKJm Jl.UliUJl..M.U .M.aZliUmIiOU UIiOYKqUU? KaK .M.OJICIiO COeJl.amb KapmuliY JlU
HUU .M.aZliUmHOU UIiOYKquU "BUOU.M.OU"?
4. KaK .M.OJICIiO pa3Mazliumumb .M.azlium?
5. 0 lleM c6UOemeJlbCmByem onblm 3pcmeoa?
6. KaK B3auMooeucmBY1Om npSlMble np0600liUKU C mOKOM OOliOZO (npomu
60nOJlOJICHOZO) Hanpa6Jl.eHUJl.? KaK ocyU{ecmBJl.SlemCJl 3mo 63auMooeucmBue?
7. KaKoe oeucmBue OKa3blBaem MaZHumlioe nOJl.e lia nOMeU{eHHble 6 Hezo
paMKY C mOKOM? MaZliumliYKJ CmpeJl.KY?
8. KaK ycmaHa6Jl.UBaemCJl 6 MaZliumliOM nOJl.e paMKa C mOKOM? II OlleMY
np0600a. nOOBOOSll4Ue mOK K paMKe, CJlYJlCal4eu UHOUKamopOM MaZliumliOZO noJl.R.
pacnOJl.aZalOmCJl. 6Jl.U3KO opyz K oPyzy UJlU cnJl.emaKJmCSl BMecme? KaKue OOJl
JfCHbl 6blmb pa3Mepbl maKOU paMKu?
9. f{ mo na3bl6a1Om nOJlOJICUmeJlbliOU liOp.M.aJl.bKJ " nJl.OCKOCmu paMKU C mo-
KOM?
10. f{ eJl.OBeK KallaemCJl lia KalleJl.SlX. oepJICa 6 PYKe 3apJl.JlCeliliblU wap. Co-
30aem Jl.U 3apSlJICeliliblU wap MaZliumlioe nOJl.e omliOCUmeJl.bliO 3mozo lleJl.06e-
Ka? omHOCUmeJl.bliO lleJl.06eKa, cmOJl.U{ezo lia 3eMJl.e?
11. KaK06a KapmuHa Jl.UliUU MaZliumliOU UliOYKlJ,UU nOJl.Sl npSlMOZO mOKa?
KamywKu C mOKO.M.? KaK onpeOeJl.SllOm ux lianpaBJl.eliue? Om Ilezo 3a6ucum
MOOYJl.b MaZliumliOU UIiOYKqUU?
12. KaKou xapaKmep UMeem MaZHumlioe nOJl.e 6liympu U 6lie KamywKu C
mOKoM? KaK onpeOeJlJl.lOm nOJlIOCbl maKou KamywKu? KaK 6JlUJl.em JfCeJle3liblU
CmepJICeHb, BCmaBJl.eliliblU B KamywKY C mOKO.M., Ha .M.aZliUmHoe nOJl.e. c030a6ae-
Moe KamywKou?
13. B lleM cocmoum npUlilJ,un cynepn03UlJ,UU MaZliumliblX nOJl.eu?
14. TpaMBauliblu np0600 pacnoJl.oJICeli maK, Ilmo no neMY npoxooum mOK
C IOza Ha ce6ep. Kyoa OmKJl.OHSllOmCSl ce6eplible KOlilJ,bl MaZliumliblX CmpeJl.OK.
pacnOJlOJICeHliblX liao npoBoooM U noo npOBOOO.M.? Om6em 060cliyi1me.
15. KaK onpeOeJl.SllOm lianpaBJl.eHUe CUJlbl. oeucm6YIOl4eU Ha npJl.MOJl.UlieU-
HblU npoBooliUK c mOKOM 6 MaZHumHOM nOJl.e? Om Ilezo 3a6ucum MOOYJlb
3mou CUJl.bl?
16. Ecmb JlU pa3JlUilUe MeJICoy 63auMooeucmBuSlMu OBYX napaJl.JleJl.bliblX
3Jl.eKmpOliliblX nYIlKOB U 06YX napaJl.Jl.eJl.bliblX npoBooliUKOB C mOKaMU npoxo-
OSll4UMU 6 OOliOM lianpa6Jl.eHUU?
336

17. B OCli06e oeUCm6U.R KaKUX npu60p06 Jl.eJICum 8paw,eliue pa.MKU 6
.MaZliUmliOM nOJl.e? 0Jl.eKmpOU3.MepUmeJl.blible npu60pbl KaKUX Cucme.M 6bl
3liaeme?
18. KaKoe Sl811eliue Jl.eJICum 6 OCli08e ycmpoUCm6Q 3Jl.eKmpOU3.MepUmeJl.bliO-
ZO npu60pa .MaZHUm03Jl.eKmpUlleCKOU CUCme.Mbl? llolle.MY UX 80 8pe.M.R U3.Mepe-
liUU lie CJl.eiJyem pacnOJl.aZamb 6Jl.U3KO opyz K iJpyzy?
19. KaK ycmpoeli 311eKmpOU3.MepUmeJl.bliblU npu60p 311eKmp0.MaZliUmHou cu-
Cme.Mbl? 3a8ucum Jl.U npum.RJlCeliue nJl.aCmUliKU K 3Jl.eKmp0.MaZHumy 6 zallb8a-
HO.Mempe om lianpa6Jl.eliUJl mOKa?
20. KaKYIO CUJl.y lia3bl6alOm CUJl.OU JIopem-,a? Om Ilezo 3a8ucum ee Ha-
npa8Jl.eliUe U no KaKO.MY npa8UJl.Y ezo onpeiJellSllOm?
21. 'C.Je.MY pa6lia pa60ma. c06epWae.Ma.R. CUJl.OU JIopeliu,a? H3.MeHSlemCSl JlU
KUliemUlleCKaSl 3liepZUSl 3ap.JlJICeHHOU llaCmuu,bl. iJ8uJlCyw,eUC.Jl 6 .MaZHUmHO.M
nOJl.e?
22. IT 0 KaKU.M mpaeKmOpUSl.M .MOJICem iJ6uzambcSl 3ap.RJlCeliliaSl Ilacmuu,a 8
.MaZliumliO.M nOJl.e?
23. KaKou ljJaKm c8uiJemeJl.bCm8yem 0 cyw,ecm806aliuu .MaZHUmliOZO nOJl.Sl
3e.MJl.u? riJe liaxoiJ.JlmCSl .MaZliUmlible nOJl.IOCbl 3e.Mllu?
24. H a lle.M OCli08alia 803.MOJICliOCmb opuelima4uu Ha .Mecmliocmu npu no-
.MOW,U KO.Mnaca? MOJIClio Jl.U Ha JIYlie opuelimup08ambcSl c nO.Mow,blO KO.Mna-
ca? B KaKO.M .Mecme 3e.MIiOZO wapa .MaZIiUmliaSl CmpeJl.Ka 060U.MU KOH4a.MU
nOKa3b18aem na IOZ?
25. B lle.M OCli06liOe pa3Jl.UIlUe 6 oeucm6USlX 3Jl.eKmpUlleCKOZO U .MaZlium.
HOZO nOJl.eu lia 3ap.RJlCeliliYIO llacmu4Y?
26. 3a6ucum Jl.U nepuoiJ 06paw,eliuSl 3apSlJICeliliOU llaCmU4bl no OKPYJICHOC-
mu 8 .MaZliumliO.M nOlle om CKopocmu 1laCmU4bl? om paiJuyca OKPYJICHocmu?
riJe 3mo UCnOJl.b3yemC.R?
27. KaK no iJ8UJlCeHU1O 3apSlJICeHHoiJ. 1laCmU4bl 6 .MaZliUmHO.M nOlle .MO:JICIiO
onpeiJeJl.Umb ee yiJeJl.bliblU 3apSlO? 060SlCliume npUli4un iJeucm8uSl .MaCC-CneKm-
pozpaljJa.
28. H3.Meli.RlOm Jl.U 8ew,ecm6a. nO.Mew,ae.Mbte 8 .MaZliumlioe nOlle. UliiJYK4UIO
3mozo noJl..R?
29. KaKue 8ew,ecm6Q lia3bl6alOm iJua.MaZliemUKa.Mu? napa.MaZHemUKa.Mu?
ljJepp0.MaZliemUKa.Mu? KaKU.MU C60UCm6a.MU OliU 06Jl.aiJalOm?
30. l{ e.MY pa6eli .MaZliUmliblU .MO.Melim amO.M06 iJua.MaZliUmHozo 8ew,ecm6a?
H.MelOm llU .MaZliUmliblU .MO.Melim amO.Mbl napa.MaZliumliozo 6ew,ecm8a?
31. l{e.M 06YCJl.08Jl.elibl .MaZliUmlible C80ucm8a ljJepp0.MaZliemuK08? l{mo ma-
Koe iJO.Melibl?
32. KaKue ljJaKmbl YKa3bl8a1Om lia pewalOw,ylO pOJl.b 83aU.MOOeUCm8U.R amo.
.M06 6 npO.R6Jl.eHUU ljJepP0.MaZIiUmliblX c8oucm8?
33. llolle.MY .MaZliUmlia.R. npOHUijae.MOCmb ljJepP0.MaZliemUK06 Ollelib 6eJl.UKa?
34 KaK 06o.RCHumb. Ilmo ljJeppO.MaZHemUKu COXpaH.RlOm .MaZHUmHoe nOlle
nOCJl.e ycmpaHeHuSl 6lieWlieZO nOJl.Sl?
35. KaKue mpe606aliU.R npeiJo.R811.RlOmCSl K .MaZliUmHbl.M .MamepuaJl.a.M. uiJy-
W,U.M lia U3Z0m06Jl.e1iUe nocmOSlHliblX .MaZlium08?
337

B = M max
IS
n
E = LEi
i=l
I
B = JlJlo -
2nr
B
Jl=-
Bo
I
B = J.1J.1o -
2R
B = JlJlonI
FA = BIlsina
M = BISsina
Fo = qBv sin a
338
OCHOBHblE cI»OPMY Jlbl
- MO)];ynb HH,AYKIJ;HH MarHHTHoro nOJlS!, r)];e M rnax -
MaKCHMaJlbHbIH Bp8alOHH MOMeHT, )];eHcTBYIOHH Ha
KOHTYP C TOKOM; I - CHna TOKa; S - nnOa)];b, orpa-
HH'leHHaS! KOHTypOM C TOKOM.
- npHHIJ;Hn cynepn03HIJ;HH MarHHTHbIX noneH, r)];e 13
HH,AYKIJ;H.H pe3Y JlbTHpYIOero MarHHTHoro non.H; 13; - HH-
)];YKIJ;HS! i-ro MarHHTHoro nonS!.
- MO)];ynb HYKHH MBrHHTHOro nOJl.H np.HMOrO TOKa, r,Ae
4 10 -7 rH
Jl - MBrHHTHa.H npOHHIJ;aeMOCTb Cpe)];bI; J.1o = 7t. --
M
MarHHTHaS! nOCTO.HHHaS!; r - paCCTO.HHHe OT npOBO,AHH-
Ka )];0 TOqKR, B KOTOpOH onpe,Aen.HIOT MarHHTHYIO HH-
)];YKIJ;HIO.
- MarHHTHa.H npOHHIJ;aeMOCTb Cpe,AbI, r)];e B - MO)];ynb
MarHHTHOH HH)];YKIJ;HH nOJl.H B cpe)];e; Bo - MO)];ynb Mar-
HHTHOH HH)];YKIJ;HH nOJl.H B BaKYYMe.
- MO.D;YJlb HH,AYKIJ.HH MarHHTHoro nOJlSl B IJ.eHTpe Kpyro-
Boro npOBO)];HHKa C TOKOM. r)];e R - pa.nHYC KpyroBoro
npOBO,AHHKa.
- MO,AYJlb HH,AYKIJ.HH MarHHTHoro nOJlS! BHYTPH COJleHOH-
N
)];a, r)];e n = - qHCnO BHTKOB Ha e)];HHHIJ;Y )];nHHbI;
l
N - 06ee qHCnO BHTKOB; l - )];nHHa COJleHOH)];a.
- MO)];YJlb CHnbI AMnepa. r)];e l - )];nHHa npOBO)];HHKa;
a - yroJl Mem.ny HanpaBneHHeM TOKa H BeKTopOM Mar-
HHTHOH HH)];YKIJ;HH.
- MO,AYJlb BpaalOero MOMeHTa, )];eHCTBYIOero Ha KOH-
TYP C TOKOM, r)];e S - nnOa,Ab, OrpaHHqeHHa.H KOHTY-
pOM, a - yron Mem)];y HopManblO K KOHTYPY H BeKTopOM
MarHHTHOH HH,AYKHH.
- MO)];YJlb CHJlbI JIopeHIJ;a, r,Ae q - 3apS!,A )];BHmyeH-
C.H 'laCTHIJ;bI; v - CKOpOCTb qaCTHIJ;bI; a - yroJl Mem,AY
HanpaBneHHeM CKOpOCTH H BeKTopOM MarHHTHOH HH)];YK-
IJ;HH.

METOAlIqECKIiE PEKOMEHAAQIIII
B lIaHHOR TeMe MOmHO BbI,nenHTh TPH THrra 3a,naq:
1) Ha paCqeT rrOJleR (BbI'lHCJleHHe MarHHTHOR HH,nYKHH B KaKOR-JIH60 TOq-
Ke MarHHTHoro nOJISI);
2) 0 CHJIOBOM ,neRcTBHH MarHHTHoro rron.H Ha rrpOBo,nHHKH (KOHTYP) C TO-
KOM;
3) 0 CHJIOBOM .neRcTBHH MarHHTHoro H 3J1eKTpHqeCKOrO nOJIeR Ha .nBHmy-
HecSi B HeM 3apSlmeHHhIe qaCTHhI.
ITpH pemeHHH 3a,naq nepBoro THna CJle,nyeT nOMHHTb. qTO CHJlOBOR xapaK-
TepHCTHKOH MarHHTHoro rrOJISI .HBn.HeTC.H MarHHTHa.H HHL1;YKH.H B. KOTOpa.H B
Kam,nOR TOqKe rrOJl.H HarrpaBJleHa no KaCaTeJlbHOH K nHHHH MarHHTHOR HH,nYK-
HH. IT03TOMY He06xo,nHMO:
1) c,neJlaTh CXeMaTHqeCKHH PHCYHOK. YKa3aB Ha HeM rrpOBo.nHHK C TOKOM.
C03,nalOHH MarHHTHoe nOJIe;
2) qepe3 ,naHHYIO TOqKY npOBeCTH nHHHIO MarHHTHoR HH,AYKHH. onpe,nenHB
ee HanpaBJleHHe. nOJlb3Y.HCb rrpaBHJIOM rrpaBoro BHHTa (CM.  11.6- 11.8);
3) H306pa3HTb MarHHTHYIO HHL1;YKUHIO B TOqKe Ha6JIIO,AeHH.H B. HarrpaB-
JIeHHYIO no KaCaTeJlhHOH K JlHHHH MarHHTHOR HH,nYKHH;
4) paCCqHTaTb MO,nYJlb MarHHTHOH HH,nYKHH, HCrrOJlb3Y.H CPOPMYJlbI ,nJI.H Mar-
HHTHoro nOJl.H:
I
np.HMOrO TOKa - B = JlJlo -. KpyroBoro TOKa
2nr
COJleHOH.na - B = JlJlonI ;
5) rrpH HaXOm,AeHHH MarHHTHOH HH.nYKHH rrOJI.H. C03,AaHHOrO HeCKOJlb-
KHMH TOKaMH) CJle,nyeT Hcrronb30BaTh rrpHHHn cynepn03HHH rroneH
I
B = JlJlo - .
2R
- - - -
B =  + B 2 + ... + Bn. npH 3TOM 06.H3aTeJlhHO HymHo nOMHHTb. qTO 3TO BeK-
TOpHa.H CYMMa.
ITpH pemeHHH 3a,naq BToporo THna He06xo.nHMO:
1) c,neJlaTb cXeMaTHqeCKHH qepTem. YKa3aB Ha HeM JIHHHH HH,nYKHH Mar-
HHTHoro nOJl.H. 'tJ:aCTO JlHHHH HH,AYKHH H306pamalOT B rrJIOCKOCTH qepTema. B
HeKOTopbIX cJlyqa.HX y,n06HO HX H306pamaTh nepneH.nHKYJl.HpHO rrnOCKOCTH qep-
Tema ("x" - OT Ha6J1IO.naTeJI.H. "." - K Ha6nlO,naTeJlIO);
2) H306pa3HTh KOHTYP C TOKOM. HaXO,A.HHHC.H B 3TOM none. OTMeTHTb
yrJlbI Mem,ny HarrpaBJIeHHeM non.H H OT,AenbHhIMH 3neMeHTaMH KOHTypa. ecnH
nOCJIe)J;HHH COCTOHT H3 HeCKonhKHX npSlMbIX rrpOBo,nHHKOB;
3) HCrrOJlh3YE npaBHJIO JIeBOH PYKH. OrrpelIeJIHTh HarrpaBJIeHHe CHJI AMrre-
pa, lIeRcTBYIOIUHX Ha KamlIbIR 3neMeHT KOHTypa. H H306pa3HTb HX Ha qepTe-
me;
4) 3anHcaTh cpopMynbI ,AJlE CHJI AMrrepa HJIH BpaalOero MOMeHTa. co-
3,naBaeMoro 3THMH CHJlaMH. FA = BIlsina HJIH M = BISsina. H HaHTH H3
HHX HCKOMYIO BeJIHqHHY;
5) eCJIH B 3a.naqe paCCMaTpHBaeTCSI paBHOBeCHe rrpOBO,AHHKa HJIH ero
.nBHmeHHe B MarHHTHOM nOJle. TO. KpOMe CHnbI AMnepa. HymHo YKRaRTb H
Bce OCTaJlbHbIe CHJIbI, ,neRCTBYlOlQHe Ha npOBo,nHHK. H 3arrHcaTb YCJlOBHSI
339

n
ero paBHOBeCHSI L  = 0 ,
i=l
n
L M i = 0 HJlH OCHOBHoe ypaBHeHHe ,AHHaMHKH
;-1
n
L F; = mu 11, cnpoeHpOBaB BeKTopHbIe BeJlH1.lHHbI Ha OCH X H y, HaHTH HC-
i=l
KOMYIO BeJlH1.lHHY.
IIpH pelIIeHHH 3a,lJ,aq TpeTbero THna He06xo,lJ,HMO:
1) C,lJ,eJlaTb 1.lepTem, YKa3aB Ha HeM nHHHH HH,AYKHH MarHHTHoro nOJl.H H
JIHHHH Hanp.HmeHHOCTH 3JIeKTpHqeCKOrO nOJI.H, HaqaJlbHYIO CKOpOCTb qaCTHbl
H OTMeTHTb 3HaK ee 3ap.H,lJ,a;
2) eCJlH CKOpOCTb qaCTHbI HanpaBJleHa no,lJ, yrJlOM K JlHHHH HH,lJ,YKHH
MarHHTHoro nOJI.H, ee CJIe,lJ,yeT pa3J10mHTb Ha ,ABe COCTaBJI.HIOHe, O,lJ,Ha H3 KO-
TOpbIX 1I0n.}KHa 6blTb nepneH,lJ,HKYJl.HpHa MarHHTHOH HH,lJ,YKHH 13, BTopaSI na-
pannenbHa eH. TaKoe pa3J10:>KeHHe n03BOJl.HeT npe,ACTaBHTb CJlOmHOe ,lJ,BH.}KeHHe
B BH,lIe IIBYX 60Jlee npOCTblX 11 B 3HaqHTeJlbHoH Mepe ynpOIuaeT 3a,Aa1.lY. no-
CKonbKY npH ,lJ,BHmeHHH qaCTHbl BHOJlb JlHHHH MarHHTHoro nOJl.H CHJIa JIo-
peHa HB qaCTHy He ,AeHcTByeT;
3) H306pa3HTb CHJlbl, ,lJ,eHcTBYIOHe Ha 3ap.HmeHHYIO qaCTHY. IIpH HaXO:>K-
,lJ,eHHH HanpaBJIeHHSI CHnbI JIopeHa no npaBHny JIeBOR PYKH CJle,lJ,yeT 06pa-
THTb oc060e BHHMaHHe Ha 3HaK 3ap.H,lJ,a qaCTHbI;
4) YKa3aB CHJlbI, HymHo nonbITaTbC.H onpe,lJ,eJlHTb BH,lJ, TpaeKTopHH qaCTHbI;
5) 3anHcaTb OCHOBHoe ypaBHeHHe ,lJ,HHaMHKH, BbI6paTb OCH KOOp,AHHaT 11
cnpoeHpOBaTb BeKTopHble BeJIHqHHbI Ha OCH x H y;
6) HcnOJlb3Y.H cpOPMYJlbl 3JIeKTpOCTaTHKH H <}>OPMYJIY CHJIbl JIopeHa, no,lJ,-
CTaBHTb B nOJIyqeHHbIe yp8BHeHH.H 3Ha'leHH.H CHn, ,lJ,06aBHTb npH He06xo,AHMO-
CTH K ypaBHeHH.HM .ltHHaMHKH ee <}>OPMYJlbI KHHeMaTHKH H pelIIHTb nOJlyqeH-
HYIO CHCTeMY ypaBHeHHH OTHOCHTenbHO HCKOMOH BeJIH'lHHbI.
nPIIMEPbJ PElliEHIIH 3A,[(A q
1. IIo qeTblpeM AnHHHbIM npSlMblM napannenbHbIM npOBO,AHHKaM. npOXO,ASI-
lI.\HM 'lepe3 BepIIIHHbl KBa,ApaTa (CTOpOH8 KBa,Ap8Ta 30 CM), nepneH,AI1KynSlpHo
ero nnOCKOCTH. npOXO,ASlT O,AI1HaKOBble TOKI1 no 10 A, npH'leM no TpeM npOBO,A-
HHKaM npOXO,ASlT TOKH B O,AHOM H8npaBneHHH. a no qeTBepTOMY - B npOTH-
BononomHOM. Onpe,AenHTe HH,AYKHIO MarHHTHoro nonH B eHTpe KBa,ApaT8.
)J;aHo: a=30 CM, 1=10 A.
HaHTH: B.
P e III e H He. [10 npHHHny cynepn03HHH HH,AYKHfl MarHHTHoro nonSl B
eHTpe KBa,lJ,paTa .8 = .8 1 + 13 2 + 8 3 +.8 4 , r,Ae Bl' Bz. B3' B4 - M8rHHTHbie HH-
,AYKHI1 B u.eHTpe KBa,ApaTa noneH. 06pa30B8HHblX COOTBeTCTBeHHO TOKaMH
11' 1 2 , 1 3 , 14'
)J;nSl Toro 'lT06bI onpe,AenHTb H8npaBneHHe BeKTopOB 8 1 , B2' 8 3 , B4' npo-
Be,AeM oKpymHocTH (nHHHH MarHHTHoH HH,AYKHH) 'lepe3 eHTp KBa,ApaT8, pa-
,l\HYCLI KOTOpLlX paBHLI nOJlOBHHe ,l\HaroHaJlH KBa,l\paTa (T.e. r = a;: ). a ueH-
340

1 2
II
,,' '/
"\ f /
".1/
 , /
'I X / -
- '/....
v
Bj-; _
--- ,--- '\.. -
/ I B. \ BI
/ I \"
/. ,"
. I.
Puc. 11.33
I,
TpbI nemaT Ha OCIi COOTBeTCTBYIOllU1X npOBO,AHH-
KOB. TIo npaBHny npaBOrO BHHTa Onpe,lJ,enHM Ha-
npaBneHHe 3THX lIHHHi1 MarHHTHOii I1H,AYKI.I.HH H
npOBe,AeM no K8caTenbHbIM K HHM COOTBeTCTBYIO-
He BeKTopbI B 1 , B 2 , B 3 , 13 4 (pHC. 11.33). Mo-
,lJ,ynH MarHHTHbIX HH,AYKI.I.Hii BI = B 2 = B3 = B 4 , T8K
K8K CHnbI TOKOB H paCCTO.HHl1e OT npOBO,AHHKOB
,AO U.eHTp8 KBa,ApaT8 O,AHHaROBbI.
113 pHcYHKa 11.33 BH,AHO. 'lTO 13 3 H 13 1 npo-
THBononomHO H8npaBneHbI, n03TOMY 13 1 + B3 = O.
BeKTopbI B 2 H 13 4 COHanpaBneHbI. Cne,AOB8TenbHO,
JloI JloI 2Jl o I :>
B = B 2 + B4 = 2Bl = 2- = - = -,::; B = 1,9.10- TJI = 19 MKTn.
2nr nr na v 2
2 COneHOH,lJ, ,AnHHOii 40 CM H ,lJ,118MeTpOM 4,0 CM CO,lJ,epmHT 2,0. 10 3 BHT-
KOB npoBonoKH conpOTHBneHHeM 150 OM (pl1c. 11.34). Onpe,AeJII1Te MarHI1T-
HYIO I1H,lJ,YKI.I.111O non.H BHYTpl1 KaTYllIKH, ecnl1 K
KaTYlliKe nOABeAeHO H8npSlmeHI1e 6.0 B.
AaHo: l=0,40 M, d=0,040 M, N=2.0.10 3 .
U=6,0 B, R=150 OM.
S N HaiiTH: B.
P e III e Hue. BHYTPH COJIeHOH,Aa none O,lJ,HOpO,A-
Hoe, H MO,Aynb BeKTopa MarHI1THOii HH,AYKI.I.HH Ha-
XO,AflT no <}JopMyne
Puc. 11.34 JloIN
B =oIn =-Z'
r,Ae
N
n=-
l
- 1.1I1CJIO BHTKOB Ha e,AHHHI.I.Y ,lJ,nHHbI. CHny TOKa HaH,AeM no 38-
KOHY OMa
U
I = -. r,Ae R - COnpOTI1BneHHe 06MOTKI1 COneHOH,Aa. Cne,AOBa-
R
TenbHO,
1-..
a
mg
FA
Puc. 11.35
r
UN
B = Jlo Rl ; B = 2,5 MT n .
3. I1pSlMOH npOBO,lJ,HHK ab (pHC. L 1.35) ,zWH-
HOH 0,2 M H MaccoH 5 r nO,lJ,BellieH ropH30HTanb-
HO Ha lIBYX HeBeCOMblX HHT.HX B OllHOpOllHOM Mar-
HHTHOM none. BeKTop MarHI1THOii HHlIYKI.I.1111 nep-
neH,lJ,HKynSlpeH npOBO,AHHKY H paBeH no MO,AynlO
49 MTn. KaKOH TOR Ha,AO nponYCTHTb 'lepe3 npo-
BO,AHHK, t.IT06bI O,AHa H3 HHTeH pa30pBanaCb, ecnH
HHTb p83pbIBaeTCSI npH H8rpY3Ke, paBHOH Hnl1 npe-
Bbl1lI81Oeii 39,2 MH?
AaHo: l=0.2 M. m=0.005 Kr. B=49.10- 3 Tn,
F m =39,2.10- 3 H.
HaiiTH: I.
P e III e Hue. Ha npOBO,AHI1K ,AeiiCTBYIOT ,ABe
paBHble CHnbI H8TSlmeHI1S1 Fl' CHna TSlmeCTH
b
341

mg H CHn8 AMnepa FA' YcnoBHe p8BHoBecHSI B npoeKHH Ha OCb y 3anH-
meTCSI :
mg + FA
2Fl - mg - FA = 0 => Fl =
2
II:nSl Toro 'lT06bI HHTb pa30pB8n8Cb. He06xo,AHMO, 'lT06bI
mg + FA  Fm .
2
no 3aKOHY AMnepa FA = / Bl. Cne,AOBaTenbHO.
2F - mg
mg + /Bl  2Fm => /  m ; /  3 A.
Bl
4. H8H,AHTe cHny B38HMO,AeHCTBHSI, npHXO,ASlIQYlOCSI Ha e,AHHHy ,AnHHbI
npOBO,AOB B03,AymHoH nHHHH ::meKTpOnepe,Aa'lH, ecnH CHna TOKa B nHHHH 3neK-
TpOnepe,Aa'lH 500 A. 8 p8CCTOSlHHe Mem,Ay npOBO,AaMH 50 CM.
A8HO: l=1 M. /=500 A. d=0,50 M.
HaHTH: F 2 .
Puc. 11.36
F 2
P e ill e H He. TIYCTb npOBO,A8 pacnonomeHbI
nepneH,AHKynSlpHo nnOCKOCTH pHCYHKa. TOKH B
HHX npOXO,ASlT B npOTHBOnOnO)KHbIX H8np8Bne-
HHSlX (pHC. 11.36). MomHo C'lHT8Tb. qTO npo-
BO,AHHK 1 06pa3yeT M8rHHTHoe none. B KOTOpOM
HaXO,AHTCSI npOBO,AHHK 2. TIpoBe,AeM nHHHlO M8r-
HHTHOH HH,AYKHH 3Toro nonSl, npOXO,ASlIQYlO 'le-
pe3 TOqKY 2. TIo np8BHny 6YP8B'lHK8 onpe,AenSl-
eM ee Hanp8BneHHe H HanpaBneHHe BeKTopa .
IIo npaBHny neBoH PYKH onpeJlenSleM H8npaBne-
HHe CHnbI F 2 . ,AeHcTBYlOIQeH co CTOpOHbI Mar-
HHTHoro nonSl H8 BTOPOH npOBO,AHHK. ITo 3aKO-
HY AMnepa
F 2 = /B1lsin 90° = /l.
AnSI M8rHHTHoro nonSl npSlMoro TOKa
R. = J..lo/
LI} 2nd'
Cne,AOBaTenbHO,
J 2 l
F 2 = J..lo 2nd ; F 2 = 0,] H.
M
npOTOH BneTaeT co CKOpOCTblO 1.0.10 3
C
(pHC. 11.37) no,A yrnoM 60 0 K nHHHSlM M8rHHTHOH HH,AYKHH.
p8,AHYC H mar BHHTOBOH nHHHH, no KOTOpOH 6Y,AeT ,ABHraTbCSI
M8rHHTH8S1 HH,AYKHSI nonSl p8BHa 10 MTn.
M
5
none
nHTe
ecnH
AaHo: v=1.0.10 3
q=1,6.10- 19 Kn.
HaHTH: R. h.
342
B O,AHOpOJlHOe M8rHHTHoe
Onpe,Ae-
npOTOH.
a = 60 0 ,
B=1.0.10- 2 Tn. m=1.67.10- 27 Kr,
C

{
Puc. 11.36
P e ill e H" e. H8 npoToH B M8rHHTHOM
none ,AeHcTsyeT CHn8 JIOpeHU.8 FJ1 = qBv SIn a .
8T8 CHn8 H8np8sneH8 nepneH,AHKynSlpHo B
H V. P83nomHM seKTOp V H8 ,Ase COCT8S-
nSlIOll\He: B,AOnb nonSl V n H nepneH,AHKynSlp-
HO eMY V.l'
TOI',A8 ,ABH.>KeHHe npoToHB MomHO p8ccMaT-
pHBaTb KaK cynepn03Hu.HIO ,ABYX ,ABHmeHHH:
,ABHmeHHSI no oKpymHocTH C nocToSlHHOH no
MO,AynlO CKOpOCTblO v.l = vsina B nnocKocTH.
nepneH,AHKynSlpHoH B. 11 paBHOMepHOI'O npSl-
MonHHeHHoro ,ASHmeHHSI B,AOnb nl1HHH M8I'HHT-
HOH HH,AYKU.HH co CKOpOCTblO VII = V cos a. B
pe3ynbT8Te npOTOH 6Y,AeT ,ABHl'aTbCSI no SHH-
TOBOH nl1HI1I1.
IIo BTOPOMY 38KOHY HblOTOH8 F n = mau.'
eHTpOCTpeMHTenbHoe YCKopeHHe
v 2 v 2 sin 2 a
a =-='=-=
U R R
Cne,nOB8TenbHO.
B . mv 2 sin 2 a R mvsin a
q VSlna = =) = R=9.0 MM.
R qB
ID81' BHHTOBOH nHHl111
HHSI.
Cne,AOS8TenbHO.
h = vllT. l',Ae
T = 21tR = 21tm
V.l qB
- nepl10,A 06p8ll\e-
h _ _ 21tmv COS a . .
h = 3. 2 CM.
qB
343

12. 3EKTPOMArHHTHAH HHllYKUHH
3JIeKTpOMal'HHTHaH HYKU;HH. OnblTbl <I>apa)J.e.H. llpaBHJIO JIeHu;a. 3,lI;C
HYKU;HH. 3aKOH 3J1eKTpOMaI'HHTHOH HYKU;HH. 3,lI;C HII.JJ.YKU;HH B )J.BHYIU;HX-
C.H npOBO)J.HHKax. BHxpeBoe 3JIeKTpHqeCKOe nOJle.
.HBJIeHHe caMOHYKU;HH. IIYKTHBHOCTb. 3HeprH.H MarHHTHoro nOJIH Ka-
TYlliKH C TOKOM.
3Hepl'HH Mal'HHTHOI'O nOJI.H. IIJlOTHOCTb 3HeprHH.
OTHOCHTeJIbHOCTb 3JIeKTpHqeCKOl'0 H MarHHTHoro nOJIeR. llOHHTHe 06 3JIeK-
TpOMarHHTHOM nOJIe.
Mal'HHTHOe nOJle 3eMJIH. PalIHaU;HoHHble nOHca 3eMJIH.
12.1. nOTOK MarHHTHoH HHJ];yKl.\HH
MarHHTHLIH nOTOK <I> 'tIepe3 nJlOCKYIO nOBepx-
HOCTb - CKaJl.HpHa.R cpH3HqeCKa.R BeJlH'tIHHa, 'tIHC-
neHHO paBHa.H npOH3Be.l(eHHIO MO.l(YJI.H MarHHTHOH
HH.nYKIlHH B Ha nJlOIIla.nh nOBepXHOCTH S H Ha
KOCHHYC yr Jla a. Mem.l(Y HOpMaJlblO K nOBepXHOCTH
a MarHHTHOH HH.l(YKHeH (pHC. 12.1):
<I> = BScosa..
B Cll e,AHHHl.\eH MarHHTHoro nOTOKa .HBn.HeT-
C.H Be6ep (B6).
1 86 - MarHHTHhIH nOTOK O.l(HOpO.l(HOrO Mar-
HHTHoro nOJl.H C HH.l(YKI.\HeH 1 TJI 'tIepe3 nepneH-
.l(HKYJI.HpHYIO eMY nOBepxHocTb nnOIIJ;a.l(blO 1 M 2 .
1 B6 = 1 T JI . M 2 .
Puc. 12.1 EcnH qepe3 nOBepxHocTh nJlOIIJ;a.l(hlO 1 M 2 ,
nepneH.nHKYJI.HpHYro nHHH.HM MarHHTHoH HH.nYK-
nHH, npOBO.l(HTh qHCnO JIHHHR MarHHTHOH HH.l(YKI.\HH, paBHoe MO.l(YJlIO B.
TO MarHHTHbIH nOTOK qepe3 nnOIIJ;a,Ah nOBepXHOCTH S paBeH qHcny nHHHH
MarHHTHOH HH.l(YKHH, nepeceKalOHX 3TY nOBepXHOCTh.
n
12.2. 3JIeKTpOMarHHTHall HH,lJ;yKl.\HlI. OnbITbI <l>apa,lJ;eSl
MhI BH.l(eJlH, qTO BOKpyr npOBO.l(HHKa C TOKOM BCer.l(a CYIIJ;eCTByeT MarHHT-
Hoe nOJle.
A Henb3.H JlH C nOMOhlO MarHHTHoro non.R C03.l(aTh TOK B npOBO.l(HHKe?
::hy 3a.l(aqy peIIIHJI M. <I>apa.l(eH. llocne Hanp.HmeHHhIX HCKaHHH, 3aTpa-
THB MHoro TpY.l(a H H306peTaTenhHocTH. OH npHIIIen K BbIBO,AY: TonhKO
MeH.HIOIIleeC.H co BpeMeHeM MarHHTHoe none MomeT nOpO.l(HTh 3neKTpHqeC-
KHH TOK.
OnhITbI <I>apa.ne.R COCTO.HJlH B cne.nyroIIleM. EcnH nOCTO.HHHhIH MarHHT B.nBH-
raTh BHYTPh KaTYIIIKH, K KOTOpOH npHCOe.l(HHeH raJlhBaHOMeTp (pHC. 12.2. a),
TO B enH B03HHKaeT 3J1eKTpH'tIeCKHH TOK. ECJlH MarHHT BM.l(BHraTb H3 KaTym-
344

a
 N 0
c.
. .
.
6
0
..,.
. .
.
6 C'
0
 p.
l
[.]
. .
.
Puc. 12.2
+ -
:A
B
Puc. 12.3
t
KH, raJIbBaHOMeTp TaKme nOKa3bIBa-
eT TOK, HO npoTHBOnOJlOmHOrO Ha-
npaBJIeHH.H (pHC. 12.2) 6). 3J1eKT-
pHqeCKHH TOK B03HHKaeT H B TOM
CJlYt.J:ae, KOr,Aa MarHHT HenO,ABHmeH,
a BHmeTC.H KaTYlliKa (BBepx HJIH
BHH3). KaK TOJIbKO BHmeHHe npe-
KpaaeTC.H, TOK TOTt.J:ac me HC'le3a-
eT. OHaKO He npH BC.HKOM BHme-
HHIi MarHHTa (HJlH KaTYlllKH) B03-
HHKaeT 3J1eKTpHt.J:eCKHH TOK. EcnH
BpaaTh MarHHT BOKpyr BepTHKaJIh-
HOH OCH (pHC. 12.2, 8). TOR He
B03HHKaeT. raJlhBaHOMeTp nOKameT
HaJIH'lHe TOKa B KaTYlliKe B npH
OTHOCHTenhHOM nepeMeeHHH ee H
KaTYlliKH A C TOKOM (pHC. 12.3, a)
B MOMeHT 3aMhIKaHH.H HJIH pa3MhI-
KaHH.H KJIlOqa K HJlH npH H3MeHe-
HHH CHJIbI TOKa B u.enH KaTYlliKH
A (npH nepeBHmeHHH BHmKa peo-
CTaTa. pHC. 12.3, 6).
HeTPY,AHO 3aMeTHTh, t.J:TO TOK B
KaTYlliKe B03HHKaeT BC.HKHll pa3,
Kora H3MeH.HeTC.H MarHHTHhIll no-
TOK, npoHH3hIBalOHll KaTYlliKY.
.HBJIeHHe B03HHKllOBeHlfH 3,[(C B
npOBO.HeM KOHType (ToKa, eCJIH
KOHTYP 3aMKHYT) "PH H3MelleHHH
MarHHTHoro nOTOKa, npOHH3LIBalO-
ero KOHTYP, Ha3LIBaeTCH HBJIeHH-
eM 3J1eKTpOMarHHTHoii nIlAyK....HH.
IIonY'leHHhIll TaKHM cnoc060M TOK
Ha3hIBaeTCSl HHYKU.HOHHhIM TOKOM,
a C03alOa.H ero 3,[(C - 3,lI,C HH-
AyK....HH.
12.3. npaBHJIO JIella
MHYKU.HOHHhIH TOK B KaTYlliKe C03aeT c06CTBeHHoe MarHHTHoe nOJIe Ha
pHCYHKe 12.4 cxeMaTHt.J:eCKH nOKa3aHO HanpaBJIeHHe HHYKu.HOHHoro TOKa
(cTpeJIKH B u.enH raJIbBaHOMeTpa) H HanpaBJleHHe BHmeHH.H MarHHTa (BepTH-
KaJIhHhIe CTpeJlKH).
345

a
v
6
Vt
6
V
PUC. 12.4
Z
Vt
llOJlb3Y.HCb npaBHJlOM 6ypaBqHKa, MomHO onpeeJlHTb nOJlIOCbI KaTYIlI-
KH C HHYKHOHHbIM TOKOM. OHH 0603HaqeHbI Ha pHCYHKe 12.4 6YKBaMH
N H S. 113 pHCYHKa BHHO, qTO KOra MarHHT npH6J1HmalOT K KaTYIliKe
(pHC. 12.4, a, 6), TO B03HHKaeT HHYKHOHHbIH TOK TaKOrO HanpaBJleHH.H,
qTO Ha 6J1H:>KaHllIeM KOHe KaTYIliKH nO.HBJI.HeTC.H OHOHMeHHbIH MarHHTHbIH
nOJllOc. B cnyqae, H306pa:>KeHHOM Ha pHC. 12.4, a, BeKTOp MarHHTHOH HHYK-
HH B nOJl.H nOCTO.HHHOrO MarHHTa HanpaBJleH BHH3, a BeKTOp MarHHTHOH
HHYKHJ.f B i nOJI.H B03HHKllIero HHYKHOHHoro TOKa - BBepx, TaK KaK JIH-
HHH MarHHTHoH HHYKHH nOJl.H KaTYIliKH BbIXO.HT H3 ceBepHoro nOJllOca. 8TO
3HaqHT, qTO npH YBeJlHqeHHH MarHHTHoro nOTOKa, npOHH3bIBalOero KaTYIlIKY,
B HeH B03HHKaeT RHYKHOHHbIH TOI< TaKoro HanpaBJIeHH.H, qTO ero MarHHTHoe
nOJle HanpaBJleHO HaBCTpeqy MarHHTHOMY nonJO, nopomalOeMY STOT TOK. II
H806opOT, KOr,Aa MbI ya.n.HeM MarHHT (pHC. 12.4, 6, z), TO Ha 6J1HmaHllIeM
KOHe KaTYIliKH B03HHKaeT MarHHTHbIH nOJlIOC, npOTHBOnOJIOmHbIH nOJIlOcy no-
CTO.HHHOrO MarHHTa, T.e. On.HTb MarHHTHoe none HHYKHOHHoro TOKa npen.HT-
cTByeT H3MeHeHHIO MarHHTHoro nOJI.H, nopomalOero STOT HHYKHOHHbIH TOI<.
TaKHM 06pa30M. npOCJle,AHB 3a B3aHMoeHCTBHeM Mem,AY nOJlIOCaMH I<a-
TYIliKH H MarHHTa BO Bcex CJIyqa.HX H cpaBHHB ero C HanpaBJleHHeM )IBJ.f-
meHH.H MarHHTa, MomHO BHeTb, qTO B3aHMoeHCTBHe Mem)J;Y nOJlIOCaMH
Bcera npen.HTCTByeT BHmeHHIO MarHHTa. 8. X. JIeHy yaJlOCb 0606HTb
STY 3aKOHOMepHOCTb H C«}>OPMYJIHpOBaTb npaBHJlO. HaHeHHYIO RM CB.H3b Ha-
3bIBalOT npaBHJIOM JIeHu;a: OJ(C UliaYKquu C03aaem 6 3aMKliymoM KOlimy
pe UliaYKqUOlilibtU mOl< maKOZO lianpa6Jl.eIiUSl, Ilmo C03aalilioe UM MaZliumnoe
nOJl.e npomU60aeucm6yem U3MelieliUIO MaZliumliOZO nomOKa. 6bt3bt6alOU{.ezo
3mom mOK.
e nOMOblO npaBHJla JIeHu;a Bcera MO:>KHO onpeeJlHTb HanpaBJIeHHe HH-
.lI:YKU;HOHHoro TOKa. AJI.H SToro He06xoHMO: 1) BbI.HCHHTb npHqHHY B03HHKHO-
346

BeHH.H HHYKHOHHOrO TOKa (YBeJlHqHBaeTC.H HJlH YMeHbliIaeTC.H MarHHTHbIH no-
TOK qepe3 KOHTYP); 2) onpeeJlHTb HanpaBneHHe BeKTopa 13 HHYHpYIOm;ero
MarHHTHoro non.H; 3) HaRTH HanpaBJleHHe BeKTopa Bj MarHHTHoro non.H HH-
,AYKHOHHoro TOKa (ecnH Ll4> > O. TO B i i! B. eCJlH Ll4> < O. TO B i ii .8);
4) no HanpaBJleHHIO B i onpeeJlHTb. nOJlb3Y.HCb npaBHnOM 6ypaBqHKa, HanpaB-
JleHHe HHYKHOHHoro TOKa.
12.4. a,l];C HHJJ;YKl.{HH. 3aKoH 3JIeKTpOMarHHTHoii HHJ.(YXlJ;HH
BbIllie paCCMOTpeHHbIe onbITbI nOKa3aJlH, 'ITO B 3aMKHYTOM KOHType B03HH-
KaeT HH)J;YKHOHHbIR TOK npH H3MeHeHHH MarHHTHoro nOTOKa, npOHH3bIBalOm;e-
ro nOBepXHOCTb, orpaHH'tJeHHYIO KOHTYPOM. KaK H3BeCTHO, TOK B npOBO,AHHKe
B03HHKaeT B TOM cJly'tJae, eCJlH Ha cB06oHbIe 3ap.H,AbI npOBO,AHHKa eRCTBYIOT
CTopOHHHe CHJlbI. Pa60TY 3THX CHn npH nepeMem;eHHH eHHH'tJHoro 3ap.Ha B,AOnb
3aMKHYToro npOBOHHKa Ha3bIBalOT 3neKTpO,ABHmyrneH CHJlOH. CJle.nOBaTeJlbHO.
npH H3MeHeHHH MarHHTHoro nOTOKa Qepe3 nOBepXHOCTb. OrpaHHqeHHYIO KOHTY-
pOM, B HeM nO.HBJI.HIOTC.H CTopOHHHe CHnbI (npHpoy HX BbI.HCHHM HHme:  12.5,
12.6), eHcTBHe KOTOpbIX xapaKTepH3yeTC.H 3,IJ;C, Ha3bIBaeMoH 3):(C HH.AYKHH.
KaK nOKa3bIBaeT onbIT, 3HaQeHHe HHYKHOHHoro TOKa (a 3HaqHT, H gi)
He 3aBHCHT OT npH'tJHHbI H3MeHeHH.H MarHHTHoro nOTOKa (H3MeH.HeTC.H JlH nno-
ab, OrpaHHqeHHa.H KOHTYPOM, HJlH ero opHeHTaH.H B npOCTpaHCTBe, H:3Me-
H.HeTC.H nH HHYKH.H MarHHTHoro non.H npH nepeMem;eHHH ero HCTOqHHKOB
HJlH 3a CqeT H3MeHeHH.H Cpeb] H T .11..). Cyw,eCTBeHHoe 3HaqeHHe HMeeT nHlIIb
cl>
CKOpOCTb H3MeHeHHJI MarHHTHoro nOTOKa - (TaK, CTpeJlKa raJlbBaHOMeTpa
Llt
B onbITax 4>apae.H OTKJlOH.HeTC.H TeM 60nblIIe, qeM 6bICTpee BBHraeTC.H Mar-
HHT B KaTYlIIKY).
Llcl>
(gi )= -t;t. (12.1)
3Ta <popMYJla BbIpamaeT 3aKOH 4>apa)J;e.H )J;JI.H 3J1eKTpOMarHHTHoH HH)J;YKHH:
cpeiJliee 3liaileliUe oJ(C UliiJYK'L{UU 6 np060iJSl14e.M KOlimype nponop'L{UOnaJl.b
liO CKopocmu U3.Melie1iUSl .MaZIiUmliOZO nomOKa 1lepe3 n06epXliocmb. ozpanUlleli
IiYIO KOlimypO.M. M Zli08elilioe 3liailenUe oJ(C UliiJYK'L{UU pa6liO 83Slmou C npo-
mU80nOJlOJIClibt.M 3IiaKO.M nep60U npOU.J60iJnuu um Maznumnozo nomOKa no 6pe
MeIiU. m.e. gi = cl>'(t).
3HaK "-" yqHTbIBaeT npaBHJlO JIeHa, cor naCHO KOTOPOMY npH YBeJlHQeHHH
MarHHTHoro nOTOKa (  > 0) SAC HH)\YKI\HH OTpHl\aTenbHaSl (jf. < 0) H. Ha-
6 r  <I> t < 0] 3 TT C
o OpOT, npH YMeHblIIeHHH MarHHTHoro nOTOKa L1  HHYKHH no-
JlOmHTeJlbHa.H (gj > 0).
CHJla HHYKHOHHoro TOKa B 3aMKHYTOM KOHType paCC'lHTbIBaeTC.H no 3a-
I - gj
KOHY OMa: . - -, re R - conpOTHBJleHlIe KOHTypa.
I R
347

MH,nYKIJ,HOHHbIll TOK B03HHKaeT He TOJlbKO B nHHe:HHbIX npOBo,nHHKax, HO
H B MaCCHBHbIX CnJIOIllHbIX npOBo,nHHKax, nOMeeHHbIX B nepeMeHHoe Mar-
HHTHoe nOJIe. B COOTBeTCTBHH C 3aKOHOM ::meKTpOMarHHTH0I1 HH,nYKIJ,HH JII06bIe
H3MeHeHH.H MarHHTHoro nOTOKa, npOHH3bIBalOero npOBo,n.HI.IJ;ee TeJIO, conpo.
BO>K,AaIOTC.H B03HHKHOBeHHeM B HeM HH,nYKIJ,HOHHbIX TOKOB. STH TOKH OKa3bI-
BaIOTC.H 3aMKHYTbIMii B TOJIe npOBo,nHHKa H n03TOMY Ha3bIBaIOTC.H BHxpeBLI-
MH (a TaK:Ke TOKaMH YKO). TOKH <1>YKO, KaK H HH,nYKIJ,HOHHbIe TOKH B
nHHe:HHbIX npOBo,nHHKax, no,n'lHH.HIOTCSJ npaBHJIY JIeHIJ,a: HX MarHHTHoe nOJIe
HanpaBJIeHO TaR, qT06bI npOTHBo,ne:HCTBOBaTb H:3MeHeHHIO MarHHTHoro nOTOKa,
HH,nYIJ,HpYIOero BHxpeBbIe TOKH. TOKH <1>YKO MO>KHO 06HapY>KHTb HB onbITe
C Ma.HTHHKOM (npOBo,n.HeH nJIaCTHHO:H), KOJIe6JIIOeMC.H B 3a30pe Me>K,ny no-
JIIOCaMH 3J1eKTpOMarHHTa. }J;o BKJIlOqeHH.H Ma.HTHHK COBepillaeT npaKTHqeCKH He-
3aTyxalOHe KOJIe6aHH.H. IIpH nponYCKaHHH TOKa 'lepe3 KaTYIllKY 3JIeKTpOMar-
HHTa Ma.HTHHK H:CnbITbIBaeT CHJIbHOe TopMomeHHe H OqeHb 6bICTpO OCTaHaBJIH-
BaeTCH. TOpMO>KeHHe Ma.HTHHKa 06'b.HCHHeTC.H ,Ae:HCTBHeM MarHHTHoro non.H Ha
HH,AYKIJ.HOHHble TOKH, B03HHKalOHe B nnaCTHHe npH ee ,ABH>KeHHH B MarHHT-
HOM nOJle. ECJlH B nJlaCTHHe c,neJlaTb pa3pe3bl, TO BHxpeBble TOKH oCJla6n.H-
IOTC.H H: TOpMO>KeHHe nOqTH OTCYTCTByeT. STOT cpaKT TopMOmeHH.H HCnOJIb3yeT-
C.H ,nnH ycnoKOeHH.H no,nBH>KHbIX qaCTe:H pa3J1HqHbIX npH60pOB.
TOKH <1>YKO BbI3blBalOT HarpeBaHHe npOBo,nHHKOB (HKOp.H reHepaTopOB (CM.
 14.7) H cep,neqHHKOB TpaHccpopMaTopoB (CM.  14.16», BbI,neJl.HeMa.H TOKa-
MH <1>YKO TenJlOTa HCnOJIb3yeTC.H B HH,nYKIJ,HOHHbIX MeTaJIJIyprHqeCKHX neqax
H B pyrHx CJIyqa.Hx.
IIo 3aKOHY <1>apa,Ae.H (12.1) onpe,nen.HeTC.H 8JJ:C HH,nYKIJ,HH. B03HHKalOa.H
H B ,nBH>KyeMC.H npOBo,nHHKe, H B Heno,nBHjKHOM (CM. onblTbI, onHcaHHbIe B
 12.1). Ho MexaHH3M npOlicxom,neHIi.H SJJ.C IiH,nYKIJ,HH B 3TliX CJIY'laSlX pa3-
JlH'leH.
12.5. 3,l1;C HH,IJ;YKIJ,HH B ABHJKYW;HXCH npOBOAHHKax
Bbl.HCHHM MexaHH3M B03HHKHOBeHH.H S,D;C HH,nYKIJ,HH B npOBo,nHHKe, ,nBH-
>KYI.IJ;eMC.H B nOCTO.HHHOM MarHHTHOM nOJIe.
IIYCTb MeTaJIJIHqeCKHH npOBo,nHHK ,nBH:iKeTCH co CKOpOCTblO v, nepneH-
,AHKYJI.HPHOH BeKTOpy MarHHTHOH HH,nYKIJ,HH 13 (pHC. 12.5). BMecTe C npo-
BO,nHIiKOM, eCTeCTBeHHO, ,nBlimYTC.H cB060,nHbIe 3neKTpOHbl. Cne,noBaTeJlbHO,
Ha 3JIeKTpOHbI 6y,neT ,neHcTBoBaTb clina JIopeHIJ,a. IIo,n ,ne:HCTBHeM 3TOll
CHnbI cB060,nHble 3J1eKTpOHbI nepeMeaIOTC.H K O,nHOMY H3 TOpIJ,OB npOBo,nHHKa
(Ha pHC. 12.5, a K JleBOMY TOPIJ,Y), B pe3ynbTaTe npOHcxo,nHT pa3,neJIeHlie
3ap.H,nOB: nOJlOmHTeJlbHbIe 3ap.H,nbl HaKanJlHBalOTCSI Ha OHOM KOHu.e npOBO,A-
HHKa, oTpHllaTeJlbHbIe - Ha lIpyrOM. 8TH 3ap.HlIbI C031IaYT BHYTPH npo-
BOAHHKB 3JIeKTpOCTaTHqeCKOe nOJIe. IIepepaCnpeAeJIeHHe 3ap.HAOB 6YAeT npo-
HCXOIiTb ,no Tex nop, nOKa 3JIeKTpHqeCKa.H CHJIa F3 = qE He CKOMneHCHpy-
eT cliny JIopeHu,a.
IIYCTb CpelIH.H.H CKOpOCTb ynop.H,nO'leHHOrO BHmeHIi.H 3JIeKTpOHOB. 06YCJIOB-
JIeHHa.H ,ne:HCTBlieM CHJIbl JIopeHlla, B CHCTeMe OTCqeTa. CB.H3aHHa.H C npOBO,Z:{-
HHKOM, paBHa ii. Tor,na CKOpOCTb V o ABHmeHH.H 3J1eKTpOHOB B HHepu,liaJIbHOll
CHCTeMe OTCqeTa. B KOTOpO:H cyecTByeT MarHHTHoe nOJle, 6yeT paBHa
348

a
6
D
D'
Puc. 12.5
V o = ii + v, npRqeM BeKTop Vo COCTaBJI.HeT yroJI a co CKOpOCTblO V ,ABRmeHR.H
npoBo.n;HRKa. CRJla JIopeHa 6y.n;eT rrepneH,ARKYJI.HpHa Vo (pRC. 12.5, 6), R ee
MO.n;YJIb FJI=qBvo. Pa3J10mRM BeKTop FJI Ha .n;Be COCTaBJI.H101..Q1:Ie: HanpaBJIeH-
HbIe B,AOJIb npoBo.n;HRKa F JIII R nepneH.n;RKYJI.HpHO npoBo.n;HRKY FJI.L'
IIpo.n;oJlbHa.H COCTaBJI.HIOa.H F JIII 1:1 BbInOJIH1:IT pa60TY no pa3.n;eJIeH1:I1O 3a-
p.H.n;OB.
113 pRC. 12.5, 6 BR,ARM, qTO F JIII = FJI cos<p = qBvo cos<p, a Vo cosq> = v.
CJIe.n;oBaTeJlbHO, F JIII = qBv. IIepeMeI..QeH1:Ie 3JIeKTpOHoB npeKpaTRTC.H, Kor.n;a
F JIU = F3' T.e. qBv = qE. Hanp.HmeHHOCTb 3JIeKTpOCTaTR'leCKOrO nOJI.H CB.H3aHa
U
C pa3HocTblO nOTeHRaJIOB E = -. 113 3T1:IX ypaBHeH1:IH HaH.n;eM U = Bvl. Ta-
l
KRM 06pa30M, Me»<.n;y KOHaMR npOBo.n;HRKa B03HRKJIa pa3HOCTb nOTeHRaJIOB.
Cne,AOBaTeJlbHo. CTopoHHen CRJIOn. BbI3bIBalOIUeii pa3.n;eJIeHRe 3ap.HJ{OB. .HBJI.HeT-
C.H npoJJ.OJlbHaH COCTaBJUIIOm;aJl CHJILI JIopeHa. Pa60Ta, KOTOPYIO COBeprnaeT
3Ta CRJla npR nepeMel..QeHRR 3neKTpRqeCKOrO 3ap.H.n;a (3J1eKTpoHa) B.n;OJIb npo-
A
Bo.n;HRKa, Ai = FJIlIl = qBvl. 8,lJ,C RH.n;YKRR B 3TOM cJlyqae g'; = - = Bvl.
q
ECJIR TaKoH npoBo.n;HRK 3aMKHYTb, TO B u.enR noii.n;eT RH,AYKHOHHbIH TOK,
HanpaBJleHRe KOToporo MO»<HO onpe.n;eJIRTb no npaB1:IJlY npaBOH PYKH (OHO .HB-
n.HeTCSI qaCTHbIM CJIyqaeM npaBRJIa JIeHa):
ecJl.U npaBYIO PYKY pacnOJl.OJICUmb BiJOJl.b npOBOOIiUKa maK. Ilm06bt nepneH
iJUKYJl.SlpliaSl COCma6Jl.SlIOU{aSl .MaZliUmliOu UliOYKquu Bl 6XOOUJl.a 6 Jl.aOOHb. a
omozliymbtU 60Jl.bWOU naJleq nOKa3bt6aJl. Hanpa6JleliUe iJ6UJlCeliUSl np060iJHuKa.
mo Ilembtpe BbtmSlliymbtX naJl.bqa YKaJICym lianpa6Jl.eliUe UHiJYKqUOliliOZO mOKa
6 npoBoiJliuKe.
ECJIR V He nepneH.n;RKyn.HpHO B, TO
=Bvlsina,
r,Ae a - yrOJI Mem.n;y V 1:1 B.
349

IIycTb npOBO.n;HHK nepeMeaeTC.H B MarHHTHOM nOJle B HanpaBJleHHH .n;eH-
CTBH.H BHelIIHeH CHJlbI F. 3a BpeM.H t, B TeqeHHe KOToporo 3J1eKTpOH, .n;BHmy-
HHC.H co CKOpOCTblO ii B.n;OJlb npOBo.n;HHKa, npOH.n;eT paCCTO.HHHe 1 = ut, npo-
1
BO.n;HHK nepeMeCTHTC.H Ha I1r. I1r = vt = v-. Pa60Ta, KOTOPYIO COBeplIIHT 3a
u
3TO BpeM.H nepneH,AHKyn.HpHaH COCTaBn.HlOa.H CHJlbI JIopeHa,
A 2 = F JI .ll1r cos 180 = -F JI .ll1r = -FJI sin <P .l1r = -qBvo sin<p .l1r.
TX 12 5 . u T B u 1
n3 pHCYHKa ., 6 Sin <Po = -. or.n;a  = -q V o - v - = -qBvl
V o V o u
TaK KaK A 2 < 0, TO npOHcxo.n;HT TopMomeHHe .n;BHmeHH.H npOBo.n;HHKa B
MarHHTHOM nOJle. AJI.H Toro 'tJ:T06bI npOBo,nHHK lIBHranC.H paBHOMepHO. K HeMY
He06xo,nHMO npHnomHTb BHelIIHlO1O CHJlY. no MO.nYJlIO paBHYIO nepneHLIHKY-
n.HpHOH COCTaBJI.HIOeH CHJlbI JIopeHa. IIoJlHa.H pa60Ta CHJlbI JIopeHIla
A = A] - A 2 = qBvl- qBvl = O. B .n;aHHOM cJlyqae HMeeT MeCTO npeBpaeHHe
MeXaHHqeCKOH 3HeprHIi HCTOqHHKa BHelIIHeH CHJlbI F B 3HeprHIO HH.n;YKIJ;HOH-
Horo TOKa. STO npeBpaeHHe npOHcxo,nHT B COOTBeTCTBHH C 3aKOHOM coxpa-
HeHH.H H npeBpaeHH.H 3HeprHH.
12.6. BHxpeBoe 3JIeKTpH'IeCKOe nOJIe
ECJlH 3aMKHYTbIH npOBo,nHHK. Haxo.n;.HIUHHCH B MarHHTHOM nOJle. HenOlIBH-
meH, TO 06'b.HCHHTb B03HHKHOBeHHe SAC HHLIYKIlHH LIeHCTBHeM CHJlbI JIopeHua
Henb3.H, TaK KaK OHa .n;eHcTByeT TOJlbKO Ha .n;BHmYHeC.H 3ap.H.n;bI.
M3BeCTHO, qTO .n;BHmeHHe 3ap.H,AOB MomeT npOHcxo.n;HTb TaKme no.n; .n;eH-
CTBHeM 3J1eKTpHqeCKOrO nOJl.H CJle.n;OBaTeJlbHO, MomHO npe.n;nonomHTb, 'tJ:TO
3JIeKTpOHLI B HenOABHHOM npOBOAHHKe npHBOAHTCH B ABHeHHe 3JIeKTpH-
'1eCKHM nOJIeM, H 3TO nOJIe HenocpeACTBeHHO nopmKAaeTCH nepeMeHHLIM Mar-
HHTHLIM nOJIeM. K 3TOMY BbIBO,AY BnepBbIe npHlIIeJl Am. MaKCBeJlJl.
SneKTpHqeCKOe nOJle, C03.n;aBaeMoe nepeMeHHbIM MarHHTHbIM nOJleM, Ha3bI-
BaeTC.H HHpOBaHHLIM 3JIeKTpH'IeCKHM nOJIeM. OHO C03.n;aeTC.H B JlI060H TO'tJ:-
Ke npOCTpaHCTBa, r.n;e HMeeTC.H nepeMeHHoe MarHHTHoe none, He3aBHCHMO OT
Toro, HMeeTC.H nH TaM npOBO,AHliH KOHTYP HJlH HeT. KOHTYP n03BOJl.HeT JlHIllb
06HapymHTb B03HHKalOee 3neKTpH'tJ:eCKoe nOJle. TeM caMbIM Am. MaKCBenn
0606IUHJI npelICTaBJleHHH M. <1>apa.n;e.H 0 HBJIeHHH 3J1eKTpOMarHHTHoH HH.n;YKIJ;HH,
nOKa3aB, 'tJ:TO HMeHHO B B03HHKHOBeHHH HH,AYHpOBaHHoro 3J1eKTpH'tJ:eCKoro
nOJl.H, BbI3BaHHoro H3MeHeHHeM MarHHTHoro non.H, COCTOHT <pH3H'tJ:ecKHH CMbIcn
.HBJleHH.H 3J1eKTpOMarHHTHoH HH.n;YKHH.
MH,nYHpOBaHHoe 3neKTpH'tJ:eCKoe none OTnH'tJ:aeTC.H OT H3BeCTHbIX 3J1eKTpO-
CTaTH'tJ:eCKoro H CTaHOHapHoro 3neKTpH'tJ:eCKoro nOJleH.
1. OHO BbI3BaHO He KaKHM-TO pacnpe,AeneHHeM 3ap.H.n;OB, a nepeMeHHbIM Mar-
HHTHbIM nOJleM.
2. B OTJlHqHe OT JlHHHH Hanp.HmeHHOCTH 3J1eKTpOCTaTHqeCKOrO H CTaHOHap-
Horo 3J1eKTpH'tJ:eCKoro nOJleH, KOTopbIe HaqHHaIOTC.H Ha nOJlOmHTeJlbHbIX 3ap.H.n;ax
Ii 3aKaH'tJ:HBaIOTC.H Ha OTpHaTeJlbHbIX 3ap.H,AaX, JIHHHH HanpHeIDlOCTH HH-
pOBamlOro nOJID - 3aMKHYTLIe JIHHHH. II03ToMY 3TO nOJle - BHxpeBoe nOJIe.
350

MCCJleOBaHH.H nOKa3aJlH, qTO nHHHH HH,AYKHH M8rHHTHOrO nOn.H H
nHHHH H8np.HmeHHOCTH BHXpeBOrO SJleKTpHqeCKOrO nOn.H paCnOJlOmeHbI BO
B3aHMHO nepneHHKYJI.HpHMX nnOCKOCT.HX. BHXpeBOe sneKTpH'leCKOe none
CB.H3aHO C H8BO,A.HHM ero nepeMeHHbIM MarHHTHbIM noneM np8BHJlOM J1e-
Boro BHHTa:
ec.n.u ocmpue .n.e60Z0 6UHma nocmyname.n.bliO 06UJlCemCSl no lianpa6.n.eliUIO
tJ3. mo n060pom ZO.n.08KU 6Ulima YKaJICem Hanpa6.n.eliue .n.UliUU HanpSlJICeIiIiOC-
mu uliayqup06aHHOZO 3.n.eKmpUl£eCKOZO no.n..R (puc. 12.6).
tM
B
E
4-
B
E
Puc. 12.6
3. HHYHpoBaHHoe sneKTpHqeCKOe nOJle He HBJlHeTCH nOTeHHaJlLHLIM.
Pa3HOCTb nOTeHHaJlOB Memy nlO6bIMH BYM.H TOqK8MH npOBOHHKa, no KO-
TOPOMY npOXOHT HH,AYKHOHHbIH TOK, paBHa O. Pa60Ta, COBepmaeMa.H STHM
non eM npH nepeMeeHHH 3apSla no 38MKHYTOH TpaeKTopHH, He paBHa HynlO.
8.lI.C HHYKHH Ii eCTb pa60Ta HHYHpOBaHHoro sneKTpHlIeCKOrO nOJl.H no ne-
peMeI..UeHHIO e,AHHHqHOrO 3apHlta no paCCMaTpHBaeMOMY 3aMKHYTOMY KOHTYPY.
T.e. He nOTeHHaJl, a 8.lI.C HHl{YKlIHH .HBJlSleTC.H sHepreTHlIeCKOH xapaKTepHC-
THKOH HHYHpOBaHHoro non.H.
12.7. HTHBHOCTL. HBJIeHHe CaMOHH,IJ;YKl.\HH
8J1eKTpHqeCKHH TOK. npOXO,A.HHH no KOHTYPY. C03.n;aeT BOKpyr Hero Mar-
HHTHoe nOJle. MarHHTHbIH nOTOK <I> lIepe3 KOHTYP SToro npOBOltHHKa (ero Ha-
3bIBalOT c06CTBeHHbIM MarHHTHbIM nOTOKOM) nponopHOHaneH MoynlO HH,AYK-
HH B MarHHTHoro non.H BHYTPH KOHTypa, a HHYKH.H MarHHTHoro non.H B
CBOIO 01lepeb nponopHOHanbHa CHne TOKa B KOHType. CneOBaTeJlbHO, c06-
CTBeHHbIH MarHHTHbIH nOTOK qepe3 KOHTYP npSlMO nponopHOHaneH CHne TOKa
B KOHType:
l/J - I,
l/J = LI.
KOSHHeHT nponopHOHaJlbHOCTH L Mem,AY CHnOH TOKa I B KOHType H
MarHHTHbIM nOTOKOM <I>. C03aBaeMbIM STHM TOKOM, Ha3bIBaeTC.H HHJJ.YKTHBHOC-
TLIO KOHTY}>a. I1HYKTHBHocTb KOHTypa 3aBHCHT OT pa3MepoB H OpMbI KOH-
Typa, OT MarHHTHbIX CBOHCTB cpebI, B KOTOpOH HaXOHTC.H KOHTYp. TaK, HH-
351

YKTHBHOCTb OHOCJlOHHOrO COJleHOHa MOmHO paCCqHTaTb no cpopMYJle
L = ON2S
I
re 
MarHHTHa.H npOHHaeMOCTb Cep,AeqHHKa, o - Mar-
HHTHa.H nOCTO.HHHa.H, N - qHCJlO BHTKOB COJleHOHa, S - nJlOa,Ab BHTKa,
I - JlHHa COJleHOHa.
ElIHHHeH .HHYKTHBHOCTH B CM HBJI.HeTC.H reHpH (fH). STa e,AHHHa
<1>
onpeen.HeTC.H Ha OCHOBaHHH cpopMynbI L = -.
I
llHn;YKTHBHocTb KOHTypa paBHa 1 fH. ecnH rrpH CHne nOCTO.HHHOrO TOKa
1 A MarHHTHblH nOTOK qepe3 KOHTYP paBeH 1 B6:
lfH=B
lA'
ECJlH B KOHType npOXOHT nOCTO.HHHbIH TOK, TO BOKpyr KOHTypa cYIne-
cTByeT nOCTOHHHoe MarHHTHoe none. C06CTBeHHbIH MarHHTHbIH nOTOK, rrpOHH-
3bIBalOInHH KOHTYP, He H3MeH.HeTC.H C TeqeHHeM BpeMeHH, .H S,ZJ;C HHYKHH B
3TOM KOHType He B036ymaeTC.H.
EcnH me TOK, npOXOHHH B KOHType, 6yeT H3MeH.HTbC.H co BpeMeHeM,
TO COOTBeTCTBeHHO H3MeH.HIOHHC.H c06CTBeHHbIR MarHHTHbIH IIOTOK, cor JlaCHO
3aKoHy 3J1eKTpOMarHHTHOH HHYKHH, C03aeT B KOHType 8,ZJ;C. B03HHKHOBeHHe
SAC HHYKHH B KOHType, KOTopoe BbI3BaHO H3MeHeHHeM MarHHTHoro nonH
TOKa, npOXOJI:.HII1ero B 3TOM me KOHType. Ha3bIBalOT HBJIeHHeM CaMOHIIlI.YKl1HH.
a nOHBJI.HIOInYIOC.H 8,ZJ;C - 8,ZJ;C caMOHH.n;YKHH i' 8,ZJ;C caMOHHn;YKHH co-
3,AaeT B KOHType TOK CaMOHH,AYKI.\HH.
HanpaBneHHe TOKa caMOHHYKI.\HH onpeen.HeTC.H no npaBHny JIeHl.\a: TOK
caMOHHYKI.\HH Bcera HanpaBneH TaK, qTO OH npoTHBoeHcTByeT H3MeHeHHID
OCHOBHoro TOKa. EcnH OCHOBHOH TOK B03paCTaeT, TO TOK caMOHHYKI.\HH Ha-
npaBneH npOTHB OCHOBHoro TOKa, ecnll YMeHbllIaeTCH. TO HanpaBneHH.H OCHOB-
Horo TOKa H TOKa caMOHHYKI.\HH COBnaaIOT.
ITo 3aKOHY 3J1eKTpOMarHHTHOH HHYKI.\HH cpeHee 3HaqeHHe S,ZJ;C caMOHH-
YKI.\HH, B03HHKalOeH B KOHType C HHYKTHBHOCTblO L (HH,AYKTHBHOCTb KOH-
Typa He H3MeHHeTC.H),
L\ <1> l::11
<g', > = -- = -L- MrHOBeHHoe 3HaqeHHe SITC g', SI ' = -LI' ( t ) .
SI L\t L\t . /J;
3J(C caMOUlij)YKijUU npSlMO nponopquolia.n.blia UIiOYKmUBliocmu KOlimypa u
cKopocmu U3MelieliUSl CU.n.bl mOKa 6 KOlimype.
113 3TOH cpOPMYJlbI CJleyeT, qTO HIIlI.YKTHBHOCTL - cpH3HqeCKa.H BenHqHHa,
'lHCneHHO paBHa.H S.IJ;C caMOHHYKI.\HH, B03HHKalOIneH B KOHType npH H3MeHe-
HHH CHJlbI TOKa Ha 1 A 3a 1 c.
McnoJlb3Y.H 3TO BbIpameHHe, MomHO aTb BTopoe onpe,AeJleHHe eHHHl.\bl HH-
YKTHBHOCTH: 3J1eMeHT 3J1eKTpH'leCKOH enH 06naaeT .HHYKTHBHOCTblO B 1 fH,
eCJlH npH paBHOMepHOM H3MeHeHHH CHnbI TOKa B l.\enH Ha 1 A 3a 1 C B HeM
B03HHKaeT 8D;C caMOHH.n;YKUHH 1 B.
M3MeHHTb MarHHTHbIH nOTOK MomHO TaKme nYTeM H3MeHeHH.H HHYKTHB-
HOCTH KOHTypa.
352

B 06[QeM cJlyqae MOmeT H3MeH.HTbC.H KSK CHJla TOKa B KOHType, TaK H
ero HH,l(YKTHBHOCTb. Tor)J;a Cpe,l(Hee 3HaqeHHe 8,I:{C CSMOHH,l(YKn;HH
<g',.> = _ /1 (LI) = - ( L /1I + I /)'L )
SI /1t /1t t'
a ee MrHOBeHHoe 3HaqeHHe i = - (LI)' =- (LI'(t) + I L'(t».
IIpHMepaMH CSMOHHYKUHH .HBJI.HIOTC.H 3KCTpaTOKH 3aMbIKSHH.H
KSHH.H .
C06HpalOT 3J1eKTpHqeCKYIO n;enb
TblO, pe3HCTopa t )J;BYX O,l(HHaKOBbIX
H pa3Mbl-
H3 KSTYlliKH C 60JlbllIOH HH,l(YKTHBHOC-
JlaMn HaKaJlHBaHH.H H HCTOqHHKa TOKa
(pHC. 12.7). Pe3HcTop ,l(OnmeH HMeTb
TaKoe me 3J1eKTpHQeCKOe conpOTHBJlCHHe,
KaK H KaTYlliKS. OnblT nOKS3bIBseT. qTO
npH 3aMbIKSHHH n;enH 3J1eKTpHqeCKS.H
JlaMna, BKJllOqeHHS.H nOCJlelIOBaTeJlbHO C
KaTYlliKon, 3arOpSeTC.H HeCKOJlbKO n03-
me. qeM JlSMna, BKJllOqeHHa.H nOCJle,AO-
BaTeJlbHO C pe3HCTopOM. HspacTsHHIO
TOKa B n;enH KaTYlliKH npn 3aMbIKSHHH
npen.HTCTByeT 8IlC CaMOHH,l(YKn;HH, B03-
HHKSIO[QS.H npH B03pSCTSHHH MarHHTHO-
ro nOTOKS B KSTYlliKe. IIpH OTKJllOqe-
HHH HCTOqHHKB TOKa BcnbIXHBSIOT 06e
JISMnbI. B 3TOM cJlyqse TOK B n;enH
nO,A,AepmHBaeTC.H 3,I:{C CaMOHH,l(YKn;HH, B03HHKalO[QeH npH y6bIBaHHH Mar-
HHTHoro nOTOKa B KaTYlliKe. Hs pHCYHKe 12.8 H306psmeHbI rps<pHKIi H3-
MeHeHH.H TOKS qepe3 lISMnOqKY 1 npH 3SMbIKSHHH (a) H pa3MbIKaHHH (6)
n;enH. HBJleHHe CaMOHH,l(YKn;HH C03,l(aeT HCKpy B TOM MeCTe, r,l(e npOHCXO-
,l(HT pa3MbIKaHHe n;enH. ECJlH B n;enH HMeIOTC.H MOHbIe 3J1eKTpOMarHHTbI,
TO HCKpa MomeT nepeHTH B ,l(yroBoH pa3p.H,l( H HcnopTHTb BbIKJllOqaTeJIb.
,I:{JI.H pS3MbIKaHH.H TSKHX n;eneH HS 3JleKTpOCTSHn;H.HX nOJlb3YIOTC.H MaCJI.H-
HbIMH BbIKJlIOQSTeJl.HMIL B JlHHH.HX 3JleKTpOnepe,l(aqR npe,l(YCMaTpHBaIOTC.H
Cnen;HaJlbHbIe SBTOMSTHqeCKHe BbIKJlIOQaTeJlH, pS3Mew;eHHbIe B OT,l(eJlbHbIX
311aHH.HX H cHa6meHHbIe HCKporSCHTeJlbHbIMH YCTpOHCTBaMH.
1
2
Puc. 12.7
a
6
I _p
lS:
1_ ------ -------
..
t.e
..
t.e
Puc. 12.8
353

12.8. 3HeprHH MarHHTHoro nOJUI KarylliKH C TOKOM.
3HeprHH MarHHTHoro nOJIH
BOKpyr KOHTypa, no KOTOPOMY npOXOHT 3neKTpHqeCKHH TOK, BCer,Aa cy-
ecTByeT MarHHTHoe none t npHqeM MarHHTHoe none B03HHKaeT H HCqe3aeT
BMeCTe C B03HHKHOBeHHeM H HCqe3HOBeHHeM TOKa. CJleOBaTenbHo, qaCTb 3Hep-
rHH TOKa HeT Ha C03aHHe MarHHTHoro nons, KOTopoe, no06Ho 3neKTpHQeC-
KOMY, .HBn.HeTCS HOCHTeJleM 3HeprHH. ECTeCTBeHHO npenOJlOmHTb, qTO 3Hep-
rHS MarHHTHoro nOJls paBHa pa60Te, KOTOpa.H 3aTpaqHBaeTC.H TOKOM Ha co-
3.ZJ;aHHe 3Toro nOJl.H. OHa paBHa pa60Te npOTHB 3):{C CaMOHH,AYKHH, B03HHKa-
lOeH npH 3aMbIKaHHH enH. OnpeenHM 3TY pa60TY.
IIOKJlIOQHM K HCTOQHHKY TOKa npOBO.HHH KOHTYP C HHYKTHBHOCTblO L.
lipH 3aMblKaHHH o;enH 3a BpeM.H I1t CHJl8 TOKa YBenHqHTC.H OT HYJI.H o HeKO-
Toporo 3H8'leHH.H I. IIpH 3TOM MarHHTHbIH nOTOK, C03,AaBaeMbIH 3THM TOKOM,
B03paCTeT OT Hyn.H ,AO 3HaqeHHS  = LI. Cpe,AHee 3HaQeHHe 3nC CaMOHH,AYK-
11<D
HH, npensTcTBYIOeH YBeJlHQeHHIO TOKa B KOHType, g'si = - -. 3a BpeMS
I1t
I
I:1t qepe3 KOHTYP nepeHOCHTCS 3ap.H.ZJ; q=<I>l1t. r.ZJ;e <1>= - - cpe.ZJ;Hee 3H8-
2
qeHHe CHnbI TOKa 3a BpeM.H I1t npH paBHOMepHOM ero B03paCT8HHH. ECJlH H3Me-
HeHHe TOKa npOHCXO,AHT He paBHOMepHO, TO He06xo,AHMO paCCMaTpHBaTb Ma-
nbIe npOMemYTKH BpeMeHH, B TeqeHHe KOTOpbIX MO>KHO CqHT8Tb 8):{C CKOpOCTb
M
H3MeHeHHS - - nOCTO.HHHOH).
I1t
IIpH nepeHoce 3ap.Ha q HCTOqHHK TOKa COBeprnaeT pa60TY
A =<-g'Si>q = <-g'Si> <I> I1t,
11<1> LI I LI 2
A = - < I> I1t = - . -l:1t = - .
I1t I:1t 2 2
Pa60Ta t COBeprnaeMas HCTOqHHKOM TOKa npOTHB 3):{C CaMOHH,AYKHH, H
6Y.ZJ;eT paBHa 3HeprHH VV MarHHTHoro nons:
VV M = LI 2 = <1>1 = <1>2
2 2 2L
ECJlH MarHHTHoe nOJle C03aHO TOKOM, npOXO.ZJ;.HIl1HM B COJleHOHe. TO
N 2
L = J.lJ.lo -8,
I
BI
=>1=-.
J.lJ.lo N
MarHHTHa.H HHYKH.H non.H BHYTPH COJleHOHa B = J.lJ.loNI =>
l
IIo,AcTaBHB 3TH 3HaqeHHS L H I B c}>opMyny .ZJ;JlS 3HeprHH, nonyqHM
W 1 N 2 8 B 2 l2 =.!. B 2 81.
M = '2 J.lJ.lo - (J.lJ.lO)2 N 2 2 J.lJlo
B 2
TaR KaK 81 = V - 06'beM COJleHOHa, TO W = - V - meprH.H MarHHT-
M 2J.lJ.lo
Horo non.H coneHOHa C TOKOM.
354

COrJlaCHO TeOpHH 6J1H3KO.neHCTBH.H, 3HeprH.H MarHHTHOrO nOJlH (KaK H 3Hep-
rH.H 3eKTpHqeCKOrO nOJlH) pacnpeeJleHa no BceMY 06eMY V npOCTpaHCTBa, B
KOTOpOM cyecTByeT MarHHTHoe nOJle.
W
BeJlHqHHa W M = , paBHa.H ::meprHH MarHHTHoro nOJl.H, 3aKJlIOqeHHOH B
V
eHHHHOM o6eMe 3Toro nOJIH. Ha3bIBaeTCH 06'heMHOH nJlOTHOCTblO 3HeprHH
MarHHTHoro nOJls. Ee MomHO paCCHTRTb no cpopMYJIe
B 2
W M = - ,
2JlJlo
re B - MO)J;YJlb HH)J;YKn;HH MarHHTHoro nOJI.H,  - MarHHTHa.H npOHHn;ae-
MOCTb cpebI, Jlo - MarHHTHa.H nOCTO.HHHa.H.
12.9. OTHOCHTeJILHOCTL 3JIeKTpH'IeCKOrO " MarHHTHoro nOJIeH.
lloHHTHe 06 3JIeKTpOMarHHTHOM nOJIe
KaK 6bIJIO OTMeqeHO, COrJIaCHO rHnOTe3e ,JJ;m. MaKcBeJlJla, B JlI060H o6J1ac-
TH npOCTpaHCTBa, r,ZJ:e CYIIJ;eCTByeT nepeMeHHoe MarHHTHoe nOJle, nOHBJlSeTCH
BHxpeB02 3JIeKTpHqeCKOe nOJle, JlHHHH Hanp.HmeHHOCTH KOToporo CB.H3aHb] C
JIHHH.HMH MarHHTHOH HH,AYKnHH nOpmK,AalOerO ero MarHHTHoro nOJl.H npaBH-
JIOM JleBOrO BHHTa ( 12.6, pHC. 12.6).
B Cepe,AHHe 60-x rOOB XIX B. ,JJ;m. MaKCBeJlJI npHllIeJI K BbIBOY, tlTO
Hap.HY C npOn;eCCOM nO.HBJIeHH.H BHxpeBoro 3J1eKTpHqeCKOrO nOJl.H npH H3MeHe-
HHH MRrHHTHoro nOJl.H CYI.QeCTByeT H 06paTHbIH npon;ecc, COCTOHHH B TOM,
'1TO nepeMeHHoe 3J1eKTpHqeCKOe nOJle BLI3LIBaeT nOHBJleHHe nepeMeHHoro Mar-
HHTHoro nOJls, JIHHHH HHYKn;HH KOToporo ox-
BaTbIBalOT nHHHH Hanp.HmeHHOCTH nepeMeHHO-
M t ro 3J1eKTpH'IeCKOrO nonH (pRC. 12.9) H CB.H-
3aHbI C HHMH npaBHJIOM npaBoro BHHTa. Co-
BOKynHocTb Hepa3pbIBHO CBH3aHHblX nepeMeH-
HbIX BHxpeBbIX 3J1eKTpHqeCKOrO H MarHHTHoro
nOJIeH HR3bIBalOT 3J1eKTpOMarHHTHbIM nOJleM.
OTeJlbHOe pRCCMOTpeHHe 3J1eKTpHtleCKOrO H
MarHHTHoro noneH HMeeT OTHOCHTe.TIbHbIH
CMbICJI. TaK, eCJlH 3neKTpOCTaTHqeCKOe nOJle
C03aeTCH CHCTeMOH HenO,ABHmHbIX 3Rp.H1I0B, no
3TH 3ap.HbI, .HBJI.H.HCb HenOBH.>KHbIMH OTHOCH-
TeJlbHO OHOH HHepn;HRJIbHOH CHCTeMbI OTCqe-
Ta, ,ABHmYTC.H OTHOCHTeJlbHO pyroH H, CJleOBaTeJIbHO, 6Y,AYT nOpO.>K,AaTb He
TOJIbKO 3J1eKTpHqeCKOe, HO H MarHHTHoe nOJle. AHaJlOrHqHO, HenOBHmHbIH OT-
HOCHTeJlbHO OHOH HHepn;HaJlbHOH CHCTeMbI OTCqeTR KOHTYP C nOCTO.HHHbIM TO-
KOM, B036ym,Aa.H B Kam,AOH TOqKe npOCTpaHCTBa nOCTO.HHHOe M8rHHTHoe nOJIe,
BH.>KeTC.H OTHOCHTeJIbHO pyrHx HHpenHaJlbHbIX CHCTeM OTCqeTR, R C03,naBae-
Moe HM nepeMeHHoe MRrHHTHoe nOJle B036ymlIaeT BHxpeBoe 3JIeKTpH'leCKOe
none. B npHpoe cyecTByeT eHHoe 3J1eKTpOMarHHTHoe nOJle.
E
13
Puc. 12.9
355

12.10. MarHHTHoe DOJIe 3eMJIH.
PaAHaHoHHMe DORca 3eMJIH
M3BeCTHO, qTO MarHHTHa.H CTpeJlKa. nOBeIlleHHa.H Ha HHTH (HJlH YKpen-
JleHHa.H Ha OCTpHe), YCTaHaBnHBaeTC.H B Kam,AOH TOqKe B6J1H3H 3eMHOH nOBepx-
HOCTH onpeeJleHHbIM 06pa30M. STOT $aKT CBHeTeJlbCTByeT 0 TOM, qTO BOK-
pyr 3eMnH CYIQeCTByeT MarHHTHoe nOJle. 0 ero cyecTBoBaHHH JlIOH 3HanH
aBHO, OHH C aBHHX BpeMeH HCnOJlb30BaJlH n.H opHeHTaIJ,HH KOMnaCbI, CTpen-
Ka KOTOpbIX OHHM KOHOM nOKa3bIBaeT Ha ceBep, PyrHM - Ha lOr.
MarHHTHa.H CTpeJlKa YCTaHaBJlHBaeTC.H BOJlb nHHHH MarHHTHOH HHYKIJ,HH
nOJl.H 3eMnH, KOTopoe Ha3bIBalOT reOMarHHTHbIM. Halli 3eMHOH llIap - 3TO
OqeHb 60JlbllIOH MarHHT. MCCJleOBaHH.H nOKa3aJlH, 'ITO MarHHTHoe none 3eMJlH
HanOMHHaeT MarHHTHoe nOJle rpOMa,AHOrO CTepmHeBoro MarHHTa (pHC. 12.10, a).
CXeMaTHqeCKIi 3TO nOJle H306pameHo Ha pliC. 12.10. 6. KaK BHHO (3TO Bnep-
BbIe YCTaHOBHJI Y. rHJlb6epT B 1600 r.), MarHHTHble nOJlIOCbI (8 H N) He
TOJlbKO He COBnaaIOT C COOTBeTCTBYIOIQHMH reorpa$HqeCKHMH nonlOcaMIi (10
Ii C), HO pacnOJlOmeHbI B pa3HbIX nonYlliapH.HX: IOmHbIH MarHHTHbIH nOJllOc
pacnOJlOmeH Ha 75-M rpayce ceBepHOH llIHpOTbI H Ha 99-M rpayce 3anaHOH
onrOTbl, a ceBepHbIH MarHHTHbIH nOJllOc - Ha 66-M rpayce IOmHOH IllHpO-
TbI H 141-M rpayce BOCTO'llHOH onrOTbI. MarHHTHbIe nOJlIOCbI, KpOMe Toro,
HaXO.HTC.H He Ha caMOH nOBepXHOCTH 3eMJlH, a no HeR.
a
c
10
Puc. 12.11
356
6
Puc. 12.10
BepTHKaJlbHa.H nJlOCKOCTb, B KOTOpOH pacnOJlaraeT-
C.H OCb MarHHTHoH CTpeJlKH, Ha3bIBaeTC.H nJIOCKOCTLIO
MarHHTHOrO MepHaHa AaHHoH TOQKH 3eMHoH nOBepx-
HOCTH. Y rOJl q> (pHC. 12.11) Memy reorpa$HqeCKHM
H MarHHTHbIM MepH)J;HaHaMH )J;aHHoH MeCTHOCTH Ha3bI-
BalOT MarHHTHLIM CKJIOHeHHeM. MarHHTHoe CKJlOHeHlie
Ha3blBalOT 3anaHbIM HJlE BOCTOqHbIM B 3aBHCHMOCTE aT
Toro, K 3anay HJlH BOCTOKY OT nJlOCKOCTH reorpa$H-
qeCKOrO MepHHaHa OTKJlOH.HeTC.H ceBepHblH nOJllOc Mar-
HHTHOH CTpeJlKH.
YroJl i, KOTOpbIH MarHHTHa.H CTpenKa 06pa3yeT C
ropH30HTOM, Ha3bIBaeTC.H MarHHTHblM HaKJIOHeOHeM.

CKJlOHeHHe. HaKJlOHeHHe H 'tJHCJleHHOe 3HaqeHHe rOpH30HTaJlbHOH COCTaBn.H-
IOeH MarHHTHOH HHYKl1.HH (3J1eMeHTbI 3eMHOrO MarHHTHOrO nOJl.H) nOJlHOC-
TblO xapaKTepH3YIOT no BeJlHqHHe H HanpaBneHHIO MarHHTHoe nOJle 3eMnH B
aHHOM MeCTe.
COCTaBn.HIOTC.H CneIJ;HaJlbHbIe MarHHTHbIe KapTbI, Ha KOTOpbIX HaHeceHbI H30-
rOHbI (JlHHHH OHHaKOBoro CKJlOHeHH.H) H H30KnHHbI (JlHHHH OHHaKOBoro Ha-
KnOHeHH.H). n03BOJl.HIOHe lllTypMaHaM Kopa6J1.H HJlH caMoneT8 opHeHTHpOBaTb-
C.H C nOMOblO KOMnaca.
Ha 3eMHOM wape BCTpeqaIOTC.H 06J1aCTH, B KOTOpbIX xapaKTepHCTHKH 3eM-
Horo MarHHTHoro nOJl.H CHJlbHO OTJlHqaIOTC.H OT HX 3Ha'tJeHHH B coce,qHHX Mec-
THOCT.HX. TaKHe 06J1aCTH Ha3bIBalOT MaI'HHTHLIMH aHOMaJIHHMH TIpHqHHOH Mar-
HHTHOH aHOMaJlHH HBn.HeTC.H HanHqHe no nOBepXHOCTblO 3eMnH 60JlblllHX 3a-
nemeH M8rHHTHOH meJle3HOH PYbI.
HOBeHlllHe HCCJleOBaHH.H reOMarHHTHoro nOJl.H nOKa3aJlH, qTO OHO HMeeT
60Jlee CJlOmHYIO CTPYKTYPY, qeM Ta, KOTOpa.H H306p8meHa Ha pHC. 12.10. OK a-
38nOCb, qTO Ha reOMarHHTHoe nOJle cyecTBeHHO BJlH.HeT TaK Ha3bIBaeMbIH con-
HeqHbIH BeTep, KOTOpbIH rrpe,qCT8BnHeT C060H nOTOK 6b1CTpblX 3apH:>KeHHbIX q8-
CTHl1. (3J1eKTpOHOB H npOTOHOB), KOTopble .HBJI.HIOTC.H pe3YJlbT8TOM npOl1.eCCOB,
rrpOHcxo,n.HIltHX Ha COJlHl1.e. BJlaro.nap.H 3TOMY BnH.HHHIO MarHHTHOe nOJle HB
.nHeBHOH CTopOHe 3eMHoro wapa CHJlbHO OTJlH'tJaeTC.H OT MarHHTHoro nonSl HB
HOqHOH CTopOHe. eHCTBHTenbHaSl
KapTHHa nHHHH MarHHTHOH HH-
YKIJ;HH reOMarHHTHoro nOJl.H no-
Ka3aHa Ha pHCYHKe 12.12. COJl-
He'lHbIH BeTep KaK 6bI cmHMaeT
nOJle c HeBHoH CTOpOHbI o
8 + 14R (R - paHYC 3eMJlH),
npH STOM Ha HOqHOH CTopOHe nH-
HHH nOJl.H OKa3bIBaIOTC.H oT6po
llleHHbIMH aJleKO OT 3eMnH, 06-
pa3Y.H nHHHbIH "XBOCT" (HaMeT-
pOM -40R H JlHHOH 60Jlee
900R). Ha'tJHH8.H C paCCTO.HHH.H
-8R STOT XBOCT pa3eJleH Ha qa-
CTH nJlOCKHM HeHTpaJlbHbIM CJlO-
eM, B KOTOpOM MarHHTHa.H HH.nYK-
IJ;H.H nOJl.H 6J1H3Ka K HYJlIO.
reOMarHHTHoe none 06pa3yeT MarHHToccpepy - "6pOHeBOH no.Hc", oKpyma-
IOHH 3eMnlO Hap.H,ny C aTMoccpepOH H HOHoccpepOH. rlIe HH.nYKl1.H.H reOM8rHHT-
Horo nOJl.H cyecTBeHHO npeBbIlllaeT HH,nYKIJ;HIO MemnJl8HeTHOro non.H. MarHH-
Toccpepa He no,nnYCKaeT K 3eMJle nOTOKH KOCMHqeCKHX qaCTHIJ;.
QTo6bJ nOH.HTb STO, BcnOMHHM, tlTO 3ap.HmeHHbIe qaCTHl1.bI B OHOpO,AHOM
M8rHHTHOM nOJle ,nBHmYTC.H no BHHTOBOH JlHHHH (CM.  11.13), pa,nHYC Ii lllar
KOTOpOH 38BHC.HT OT 3H8'tJeHH.H M8rHHTHOH HH,AYKIJ;HH B H KOTOpa.H KaK 6b1
HaMOTaHa Ha JlHHHIO MarHHTHoH HH,nYKIJ;HH:

p.

(1)
I:Q -..
1::s1
:a
::r:
:J'
-..
t:::
8
Puc. 12.12
R = mvsina. ; h = 2nmv cosa. .
qB qB
357

ECJlH me 3ap.HmeHHaS qaCTHa ABHmeTC.H B HeOAHOpO,AHOM MarHHTHOM nOJle,
TO OHa Tome 6Y,AeT onHCbIBaTb BHHTOBYIO JlHHHIO BOKpyr JlHHHH HHAYKHH.
OAHaKO no Mepe nepeMeeHHs qaCTHbI BAonb BHHTOBOH nHHHH pa,AHYC H IlIar
ee He OCTaIOTC.H HeH3MeHHbIMH. ECJlH qaCTHa, nepeMea.HCb, nOna,AaeT B 06-
JlaCTb 60Jlee CHJlbHOrO nons, TO ee paAHYC H IlIar BHHTOBOH JlHHHH YMeHb-
IlIalOTCSI (pHC. 12.13, a); npH nepeMeI..qeHHH me qaCTHbI B HanpaBJleHHH, B
KOTOpOM nOJle ocna6eBaeT. ee paAHYC H IlIar BHHTOBOH JlHHHH B03paCTalOT
(pHC. 12.13. 6). TIpe,AnOnomHM. 'lTO HaqaJlbHa.H CKOpOCTb qaCTHbI HanpaBJle-
Ha TaKHM 06pa30M. qTO qaCTHa. onHCbIBa.H BHHTOBYIO nHHHIO. nepeMeIQaeTC.H
B 06nacTb 60nee CHnbHoro non.H. B aTOM cnyqae OHa 6YAeT BCTpeqaTb npOTH-
BOAeHCTBHe CO CTOpOHbI non.H: Ha qaCTHY 6Y,lIeT ,lIeHCTBOBaTb CHJla. CTpeM.H-
a.HC.H BepHYTb ee Ha3aA. B pe3YJlbTaTe IlIar BHHTOBOH JlHHHH 6YAeT YMeHb-
IlIaTbC.H AO Hyn.H, nocJle qero qaCTHa, npOAOnma.H ABHmeHHe no BHHTOBOH
JlHHHH. HaqHeT nepeMeI..qaTbC.H 06paTHO - B 06J1aCTb 60Jlee cJla60ro nOJlSl.
Tenepb YKa3aHHa.H CHJla HaqHeT nOArOH.HTb qaCTHy, BCJle,ACTBHe qero IlIar
BHHTOBOH JlHHHH 6YAeT B03pacTaTb.
a
6
Puc. 12.13
CJleAOBaTeJlbHO, nOTOK aneKTpoHoB H npoToHoB, npHHocHMbIH COnHeqHbIM
BeTpoM, AOCTHra.H 3eMJlH, 3aXBaTbIBaeTC.H reOMarHHTHbIM noneM, H qaCTHb] Aa-
nee ABHmYTC.H no BHHTOBbIM
Puc. 12.14
358
c
nHHHSM BOKpyr nHHHH MarHHTHoH HHAYKHH
(pHC. 12.14), COBepillas KOJle6aHH.H OT OAHO-
ro nonlOca K ,lIpyrOMY 3a 0,1-1 c. 3Ta 06-
naCTb Ha3bIBaeTC.H paJ];HaHOHHLlM nOBCOM
3eMJlH - 06J1aCTb, COCTO.HI..qa.H H3 3aXBaqeH-
HbIX qaCTH COnHeqHOrO BeTpa. TIO.HC OXBa-
TbIBaeT 3eMnlO co Bcex CTOpOH. KpOMe npH-
nOJl.HpHbIX 06nacTeH. Ero ycnoBHo pa3Aen.H-
lOT Ha ABa: BHYTpeHHHH H BHeIlIHHH. BHYT-
peHHHH no.HC HaXOAHTC.H Ha BbICOTe 500-
1000 KM, BHeIlIHHii - Ha BbICOTe 10000-
15000 KM.
BJlaro,Aap.H reOMarHHTHOMY nonlO qaCTHbI,
npHHOCHMbIe "COJlHeqHbIM BeTpoM", orH6alOT
3eMJlIO, H JlHilib B nOn.HpHbIX 06naCT.HX He-
60nbIlIa.H HX qaCTb BTopraeTC.H B BepXHHe
cnOH aTMoccpepbI H3 pa,AHaHOHHoro nO.Hca
3eMnH H, CTanKHBa.HCb C MoneKYJlaMH KHcno-
10

po,na H a30Ta, B036Y>K,naeT HX HJlH HOHH3HpyeT. TIpH 06paTHOM nepexo,ne B
HeB036Y>K,neHHOe COCTOHHHe aTOMbI KHCJlOpO,na H3JIY'laIOT cpOTOHbI C A = 0,56 MKM
H A = 0,63 MKM, a HOHH3HpOBaHHbIe MOJleKYJlbI a30Ta npH peKOM6HHalJ,HH
BbICBeqHBalOT CHHHe II cpHOJleTOBbIe nOJlOCbI cneKTpa. TIpH 3TOM B03HIIKaeT
He06bIqaHHO BenHqeCTBeHHoe 3penHe - nOn.HpHOe CH.HHHe. Oc06eHHo ,AIIHa-
MHqHO H .HpKO ceBepHoe CH.HHHe BO BpeM.H MarHHTHbIX 6YPb.
M3BecTHo. qTO Ha COJIHlJ,e npOHcxo,nHT Bce BpeM.H 6bICTpO npOTeKalOHe
npOlJ,eCCbI: npOHcxo,n.HT BcnbIlliKH. nO.HBJI.HIOTC.H, nepeMealOTCH H IICqe3alOT
nHTHa. B 3TOM npO.HBJI.HSTC.H COJlHeqHa.H aKTHBHOCTb. BpeMeHaMH COJIHeqHaH
aKTHBHOCTb pe3KO YCHJIHBaeTC.H. 8TO npHBOlIHT K pe3KOMY YBeJIHqeHlllO nJIOT-
HOCTH H CKOpOCTH "COJIHeqHOrO BeTpa", B03,AeHcTBYIOero Ha MarHHToccpepy
3eMJlH. B pe3YJlbTaTe MarHIIToccpepa npeTepneBaeT cyecTBeHHbIe B03MyeHH.H,
HaqHHaeTC.H nepHO,n CHJlbHbIX MarHHTHbIX 6YPb Ha 3eMJle.
BonpOCbI ,[(JUI CAMOKOHTPOJUI
1. KaKue l/JaKmbl. c6uoeme.n.bcm6YlOm 0 C6Jl3U 3.n.eKmpullecKux U M.aZlium-
libl.X SlB.n.eIiUU?
2. fJ mo nOIiUM.alOm noo M.aZHUmHbI.M nomoKoM? KaKUM.U cnoc06aM.u M.OJIC-
HO U3M.eliJlmb M.aZliUmlibl.U nomoK?
3. OnUUlume U nOJlCHUme C nOM.OblO PUCYHKOB 3KCnepUM.eHmbl., B Komo-
pbl.X 06liapYJICuBaemCJl Sl6.n.eliue 3.n.eKmp0M.aZliUmliou UliOyKlJ,UU. B lleM. cyw.-
liocmb 3mozo Sl6.n.eliUJl?
4. Onuwume U nOJlCliume C nOMOU{blO PUCYliK06 3KcnepUMeHmbl.. 6 KO-
mOpbl.X UCCJleoyemcSl 3aBUCUMocmb HanpaBJleliUJl UliOYKU,UOHHOZO mOKa om
xapaKmepa U3MelieliUSl MaZliumliOZO no.n.Sl. KaK l/JoPM.y.n.upyemcSl npa6U.n.0
JI eliu,a?
5. fJmo lia3bl6alOm 3J(C UHOYKU,UU? KaKaSl ljJopM.y.n.a 6bl.paJICaem OCli06liOU
3aKOli 3.n.eKmpoMazliumliou UliOYKijUU?
6. KaK06a npupooa oJ(C UHOYKU,UU 6 np06001iUKe. o6uJlCyU{eMCJl B Maz-
liumliOM no.n.e 2 IT 0 KaKOU l/JopM.y.n.e M.OJICIiO ee paccilumamb?
7. B lleM cocmoum npa6u.n.0 npa60u PYKU? J(.n.Sl Ilezo OliO c.n.YJICum?
8. KaKOBa npupooa :3J(C UIiOYKU,UU, 6031iUKalOU{eu 6 lienOOBUJICliOM. KOlimy'
pe npu U3M.eHeliUU MaZHumHOZO nOJlSl. 6 KomopOM liaxooumcSl 3mom KOlimyp?
9. KaKuMu om.n.Ullume.n.blibI.MU C60ucm6aMU 06.n.aoaem ulioyu,upoBalilioe 3.n.eK
mpUlleCKoe no.n.e no Cpa6lieliU10 C 3.n.eKmpocmamUlleCKUM. U CmalJ,UOHaplibl.M.
3.n.eKmpUlleCKUM.U nOJl.SlMu?
10. Ifmo maKoe 6UXpe6bl.e mOKU (moKu CPYKO)? ITpu6eoume npUM.epbl. uc
no.n.b306aliWl ux B meXHUKe. B KaKUX CJlYIla.RX C HUMU npuxoiJumcSl 60pOmbCJl?
11. 3aMKliymbl.U 6umOK np0600a 1iaXOOUmcSl y np0600liuKa C mOKOM.
(puc. 12.15). Byoem .n.u 603liUKamb 6 6umKe UHOYKijUOlilibl.U mOK. ec.n.u: a)
BpaU{amb BumOK 60KPYZ ocu. npOXOOSlU{eu llepe3 np060oliUK; 6) BpaU{amb
BOKPYZ ocu. napa.n..n.e.n.bliou np06001iUKY; 6) BpaU{amb BOKPYZ ocu. nepneliou-
Ky.n.SlpHOU npOBOOHUKY; z) 06uzamb nocmyname.n.bliO napaJl.n.e.n.bHO np06001iU-
KY; 0) OBuzamb nocmyname.n.bliO nepneliouKy.n.JlpliO np060oliuKy2
12. B MaZliumliOM no.n.e, .n.UHUU UliOYKU,UU Komopozo u306paJICelibl. lia
pUCYliKe 12.16. 06UJlCemCJl np0600liUK co cKopocmblO V. Onpeoe.n.ume lianpa6-
.n.eliue UliOYKU,UOIiHOZO mOKa B np060oliuKe.
359

I
B
Ii
D[.D
Puc. 12.15
Puc. 12.16
Puc. 12.17
13. Ha pucYHKe 12.17 CmpeJl.KOU nOKa3aliO lianpaBJleliUe UliOYKlJ,UOHHOZO
mOKa B npoBooHuKe, Iwmopblu aBUJlCemC.R Me:HCoy nOJlIOCaMU nOCmO.RHHOZO Maz-
liuma nepneliouKYJl.RpHO nJl.OCKOCmu pUCYliKa Ha llumameJl.R. KaKou nOJl.IOC Maz-
liuma u306paJICeH Ha pucYliKe CJl.eBa om npoBooHuKa?
14. KOJl.blJ,O. cocmOSlUl,ee U3 HeCKOJlbKUX 6umKOB U30JlUpOBaHHOZO npoBooa.
c60600liO nooBewelio. llJl.OCKOCmb KOJlblJ,a 6epmUKaJl.blia. JIo 6umKaM npoxo-
oum mOK. C mou cmopolibl KOJl.blJ,a. omKyoa lianpaBJl.eliUe npoxOJICOeliUSl mOKa
cOBnaoaem C lianpa6Jl.eliUeM 06UJlCeHU.R llaC060U cmpe.TlKu. K lieMY nooliocSlm
ceBepliblu nOJl.IOC ZopU.JOHmaJlbHO pacnoJlo-
B JfCeliliOZO npSlMozo MaZHuma. l{ mo npou30u-
oem C KOJl.blJ,OM? l{nw npou30uoem, eCJtU Ha-
npaBJleHUe mOKa B KOJlblJ,e U3.MeHUmb HQ
npomUBOnOJlOJfClioe?
15. OnpedeJlume lianpa8Jl.eHUe UliOYKlJ,U
OHHOZO mOKa 6 CnJl.OWliOM KOJlblJ,e A, K KO
mopoMY nOOHOC.Rm MaZHum (puc. 12.18).
KaK 6yoem o6UZambC.R KOJl.blJ,O? l{mo 6yoem.
ecJl.U MaZHum nooliocumb K KOJl.bU,Y B?
16. CUJl.a mOKa 6 np0600liUKe 00
(puc. 12.19) y6bl6aem. Hauoume lianpa6Jle
liue UliOYKlJ,UOliliOZO mOKa B HenOOBUJICHOM
KOlimype ABCD U HanpaBJl.eliUSl CUJl.. oeucm6Y1OUI,UX Ha KaJICoylO U3 cmopOH
KOlimypa. KaKo80 Hanpa6JleliUe UliOYKlJ,UOliliOZO mOKa B KOHmype. eCJlU ezo npu-
6Jl.UJlCamb K npoBooliUKY OO?
17. PSlOOM pacnOJl.OJICelibl o6a OJl.UliliblX npoBooliuKa 1 U 2 (puc. 12.20).
OnpeoeJl.ume HanpaBJl.eliUe UliOYKlJ,UOHliOZO mOKa 6 npoBooliuKe 2 npu YBeJlU-
lleHUU (YMelibweliuu) CUJl.bL mOKa B npoBooliuKe 1.
A
;/;
Puc. 12.18
0
ADB
D C
0
Puc. 12.19
360
II
1
Puc. 12.20

18. H a 6epmuKa.n.blio pacno.n.OJICeliliOU KamYWKe .n.eJICum .Mema.n..n.UlleCKUU
npeo.Mem. II olle.MY 3mom npeo.Mem liazpe6aemcSl. Kozoa no 6UmKa.M KamYWKU
npoxooum nepe.MeltltbtU nWK. U ocmaemcSl XOJtOOltbt.M npu nOCmOSl1tItO.M mOKe?
19. B lle.M cocmoum Sl6.n.eliue Ca.MOUlioYKUtUU? Om Ilezo 3aBucum ol1C ca-
.MOUliOYKqUU?
20. l{ mo lia3bt6a1Om ulioYKmU61i0Cmb1O KOlimypa? Om Ilezo olia 3a6ucum?
21. 06o.R.cltume. nOlle.MY O.n.Sl C030a1tU.R. mOKa B npoBooliuKe oOJtJIClia 6btmb
C06epwelia pa60ma. 3a Cilem KaKOU 3ltepzuu olta c06epwaemcSl?
22. IT 0 KaKOU ljJop.My.n.e onpeoe.n.SllOm 3liepZU1O .MaZliumliozo nOJtSl npoBoiJ-
IiUKa c mOKO.M?
23. l{ mo lia3bt6a1Om 06oe.Mliou n.n.omliocmblO 3liepZUU .MaZHUmIiOZO no.n.Sl?
KaK ee paccilumamb?
24. KaKou ljJaKm C6uoemeJtbcm6yem 0 cYl4eCm606aliuu .MaZliUmIiOZO no.n.Sl
3e.M.n.u? roe liaxooSlmcSl .MaZliumlibte no.n.IOCbt 3e.M.n.u?
25. H a lte.M OCIiOBalia 603.MOJICIiOCmb opuelimaquu lia .Mecmliocmu npu no.
.M0l4U KO.Mnaca? MOJIC1to .n.u lia JIYlie opuelimup06ambcSl npu nO.M0l4u KO.Mna-
ca? B KaKo.M .Mecme 3e.MHOZO wapa .MaZltumliaR. cmpe.n.KQ 060U.MU KOliqa.MU
nOKa3btBaem Ha IOZ?
26. C lle.M CBSl3aliO 603liUKli06eHue .MaZliumlibtX 6YPb lia 3e.M.n.e?
27. II Olle.MY ce6eplibte CU.R.IiUSl lia6.n.100alOmcSl 6 OCIiOBIiO.M 6 no.n.SlplibtX pau-
oliax 3e.M1tOZO wapa?
<1> = BS COsa
< g',. > = _ 6<1>
I f1t
g', = -<1>'
 = vlBslna
<1> = LI
OCHOBHbIE (})OPMY JIbI
MarHH:THbIR nOTOK O.D:HOpO.D:HOrO nOJl.H 'lepe3 nJlOCKYIO
nOBepXHOCTb, r,ll;e S - nJlOa.nb nJlOCKOH nOBepXHOCTH.
npOHH:3bIBaeMOR JlHHH:SlMH: MarHHTHOR H:H,AYKH:H:; ex - yrOJJ
Mem,ll;Y HanpaBJleHH:eM HopManlf K nOBepXHOCTH: H: MarHH:T-
HOR RH,lI;YKHeR.
- 3aKOH 3J1eKTpOMarHHTHOR RH)J;YKHH, r,ll;e < g'i > H 
cpe)J;Hee H MrHOBeHHoe 3HaqeHH.H SAC HH,lI;YKHH COOT-
f1<1>
BeTCTBeHHO; t;t - CKOpOCTb H:3MeHeHH.H MarHHTHoro no-
TOKa; f1<1> - H3MeHeHH:e MarHHTHoro nOTOKa; f1t - npo
MemYTOK BpeMeHH, 3a KOTopoe 3TO H3MeHeHHe npOH:30Ill-
JlO; <1>' - npOH3BO,ll;Ha.H OT MarHHTHoro nOTOKa no BpeMe-
HR.
- SAC HH,lI;YKHH B ,lI;BHmYlQeMC.H npOBO,ll;HH:Ke, r,ll;e V
CKOpOCTb ,lI;BHmeHH.H npOBO,ll;HHKa; l - ,lI;JlHHa npOBO,ll;HH-
Ka; a - yroJl Mem,ll;Y CKOpOCTblO npOBO,ll;HH:Ka H Mar-
HH:THOR HH,lI;YKHeR.
- c06CTBeHHbIii MarHHTHblR nOTOK qepe3 KOHTYP, r,ll;e L
HH,lI;YKTHBHOCTb KOHTypa, I - CHJla TOKa B KOHType.
361

I
<W S1 > = -L-
t
W si = - LI (t)
L = JlJl O N 2 S
1
LI 2
W =-
M 2
B 2
W =-v
M 2JlJlo
B 2
W =-
M 211110
- cpeHee 3HaqeHHe SAC caMOHHYKHH, re L - HH-
M
,AYI<THBHOCTb npOBO,AHHl<a; !::.t - CKOpOCTb H3MeHeHH.H
TOKa B npOBOHHKe.
- MrHOBeHHoe 3HaqeHHe SAC caMOHHYKlHH, re I'(t) -
npOH3BO,AH8H OT CHnM TOK8 no BpeMeHH.
- HHYKTHBHOCTb OHocnOMHoro COJleHOHa, re Jl
MarHHTHa.H npOHHaeMOCTb BeI:QeCTBa BHYTPH coneHOHa;
o - MarHHTHa.H nOCTOHHHa.H; N - qHCJlO BHTKOB; 1 -
nHHa COJleHOH,Aa; S - nJlOI:Qab BHTl<a COJleHOH,Aa.
- 3HeprH.H MarHHTHoro non.H KOHTypa C TOKOM. re L -
HHYKTHBHOCTb KOHTypa, 1 - CHna TOKa B HeM.
- 3HeprH.H MarHHTHoro nOJlH coneHOHAa c TOKOM, re V -
06eM coneHOHa, 3aHHTMM nOJleM.
- 06eMHRH nnOTHOCTb 3HeprHH MarHHTHoro nOJl.H
METOAHqECKHE PEKOMEHAAQIIH
ITpH pemeHHH 3aaq Ha 3aKOH 3J1eKTpOMarHHTHOH HHYKHH peKOMeH-
,lI,yeTC.H:
1. AHaJlH3HPYH ycnoBHe 3a,lI,a'tlH, Heo6xo,AHMO npemAe Bcero YCTaHOBHTb
npHqHHbI H3MeHeHH.H MarHHTHoro nOTOKa, npOHH3MBalOI:QerO KOH'l'YP, H onpe-
enHTb, l<aKaH H3 BenH'tIHH, B, S HnH a, BXO.HI:QHX B BblpameHHe nSl Mar-
HHTHoro nOTOKa <1> = BScoso., H3MeHHeTCH C TeqeHHeM BpeMeHH. ITocne 3Toro
3anHcaTb 3aKOH 3neKTpoMarHHTHoM HH,lI,YI<HH:
 $  '
< 0i > = --, HnH OJ = -<1> (t).
t
EcnH B 3a,Aa'tle pe'lb HeT 0 nocTynaTenbHOM BHmeHHH npHMonHHeMHoro
npOBO,lI,HHl<a, TO S.D;C HHYKHH onpe,lI,eJl.HIOT no cpopMyne  = viB sin a.
2. 3aTeM BblpameHHe nH <1> Hao npe,ACTaBHTb B pa3BepHYTOM BH,lI,e. .D;nH
3Toro BbI6HpalOT Ba MOMeHTa BpeMeHH t 1 H t 2 H n.H Ka>KOrO H3 HHX onpe-
,AeJlHIOT nOTOKH <1>1 H <1>2' CBH3aHHbIe C ,AaHHbIM KOHTypOM. M3MeHeHHe Mar-
HHTHoro nOTOKa 3a BpeM.H t = t 2 - t 1 B 3aBHCHMOCTH OT YCJlOBH.H 3ana'tlH 6y-
eT paBHO <1> = (B2 - B1)S cos a, ecnH H3MeHHeTCSl MarHHTHa.H HHYKHH
nOJl.H, HJlH <1> = BS cos a, r.n.e !::..S - nnOI:QR,Ab, onHCaHHa.H B npocTpaHcTBe
362

,ABHmYHMC.H npOBOHHKOM; HJIH <1> = BS (cos a2 - cos a 1 ), eCJIH H3MeH.HeTC.H
nOJIOmeHHe paMKH B nOJle. TaK HaXO.HT <I> npH paBHOMepHOM H3MeHeHHH
MarHHTHoro nOTOKa. MrHOBeHHoe 3HaqeHHe S,n:c HHYKHH HaXO.HT, B3.HB npo-
H3BOHYIO OT MarHHTHoro nOTOKa no BpeMeHH: g', = -$' .
3. .D.aJIee Hao nOCTaBHTb BbIpameHHe JI.H <I> B HCXOHYIO cpOPMYJlY 3a-
KOH8 3neKTpoMarHHTHoR HHYKHH H, 3anHcaB ,AOnOJIHHTenbHble ycnOBH.H, pe-
IlIHTb nOJlyqeHHbIe ypaBHeHH.H OTHOCHTeJIbHO HCKOMOR BeJIHqHHbI.
4. IIpH paC'leTe 3JIeKTpHQeCKHX eneH. HMelOHX HCTO'lHHKH nOCTO.HHHOrO
TOKa H Ha OHOM H3 yqacTKoB KOTOpOR B03HHKaeT S,n:C HHYKHH, BbI3BaHHa.H
,ABHJKfHHeM npOBOHHKa, He06xoHMO HaqHHaTb c onpeeJleHH.H 3HaqeHHH H 3Ha-
Ka 3TOR S,n:C (HanpaBJleHH.H HHYKHOHHoro TOKa), nOCJIe Qero 3aaQa CBOHT-
C.H K paCQeTY 06bIqHOR enH nOCTO.HHHOrO TOKa C HeCKOJIbKHMH HCTOqHHKaMH
S,n:C, coeHHeHHbIMH Memy C060H nOCJleOBaTeJIbHO HJIH napanJIeJlbHO.
5. llpH pellIeHHH p.H,Aa 3a,Aaq nOJle3HO HCnOJlb30B8Tb 3aKOH coxpaHeHH.H H
npeBpameHH.H 3HeprHH.
nPHMEPbI PEIlIEHHH 3A.L(A q
1. TIpSlMoyronbHblH KOHTYP ABCD nepeMell\aeTCSI nocTyn8TenbHO B Mar-
HHTHOM none npHMonHHeHHoro ,AnHHHOro npOBO,AHHK8 C TOKOM (pHC. 12.21).
Onpe,AenHTe H8np8sneHHe H:H,AYKl.{HOHHOro TOK8 B KOHType, ecnH KOHTYP Y,Aa-
nSleTCSI OT npOSO,Aa. KaKHe CHnbl AeHcTBYIOT H8 KOHTYP?
P e ill e H He. TIpH Y,A8neHHH KOHTypa OT npOBO,AHHKa 'lHCnO nHHHH HHAYK-
I.\HH, npOHH3bIB81Oll\HX KOHTYP, YMeHblllaeTCSI. BCne,ACTSHe HsneHHSI 3neKTpo-
M8rHHTHOH H:H,AYKl.{HH B KOHType B03HHK8eT HH-
,AYKI.\HOHHbIH TOK T8Koro Hanp8sneHHSI. 1.ITO ero
M8rHHTHoe none H8np8sneHO TaK me (cornaCHO
npaBHny JIeHI.\a). K8K H M8rHHTHoe none npH-
Moro TOK8. AnSI onpe,AeneHHSI H8npasneHHSI Mar-
HHTHoro nonH npSlMoro TOKa B MeCTe p8cnono-
meHHSI KOHTypa npOSe,AeM 'lepe3 npOH3BonbHYIO
TO'lKY 0 nOBepXHOCTH. Orp8HH'leHHOH KOHTypOM
(pHC. 12.21). JlHHHIO HH,AYKUHH MarHHTHoro nonSi
npSlMoro TOK8 I. 8TO oKpymHocTb C I.\eHTpOM H8
OCH npOBO,AHHK8. TIo npaBuny 6ypaS'lHK8 AnSI
npSlMoro TOK8 onpeAenHM H8np8sneHHe nHHHH
HHAYKI.\HH. Cne,AOBaTenbHo, nHHHH HH,AYKl.{HH,
npoHH3blB81Oll\He KOHTYP. Hanp8BneHbi OT Hac.
3Ha'lHT. H BeKTOp Bj H:H,AYKl.{HOHHOro TOK8 H8-
np8sneH OT H8C. TIo np8BHny 6YP8B'lHK8 ,AnSI
KpyroBoro TOt<a, 3H8S1 H8npaBneHHe MarHHTHoro nonH B j , MO:>KHO onpe,AenHTb
H8np8BneHHe C8Moro HH,AYKl.{HOHHOro TOKa. TOK B KOHType 6Y,AeT H8np8BneH
no XO,Ay 'l8COBOH CTpenKH.
H8 Y'l8CTKH KOHTYPa. AB. BC, CD, DA co CTOpOHbI M8rHHTHoro nonSl
TOK8 I 6Y,AYT AeHcTBoB8Tb CHnb] Ii. 12, Fa H F4' H8np8BneHHSI KOTOpbIX
onpe,AenHM no np8sHny neBoH PYKH. OHH YK838Hbi Ha pHC. 12.21.
2 113 npOBO,Aa ,AnHHOH 2,0 M C,AenaH KB8,AP8T. KOTOpblH p8cnonomeH ro-
pH30HTanbHO. K8KoH 3apH,A npOH,AeT no npOBO,AY, ecnH ero nOTSlHYTb 38 Ase
F 2
I
A
D
F4
Puc. 12.21
363

,lUiarOHanbHO npOTHBOnOnOmHble BepIlHiHbl T8K. 'lT06bJ OH CnmKl1nCSI? Conpo-
THBneHHe npOBOlla 0,10 OM. BepTHK8nbH8S1 COCTaBnSlIOIl.(aSl HHAYKUHI1 M8rHI1T-
HOrO nonSl 3eMnH 50 MKTn.
J],8HO: l=2.0 M, R=0,10 OM. B=5,0.10- 5 Tn.
H8HTH: q.
P e 111 e H Ii e. TIo 3aKOHY <1>8p8,AeR cpe.n;Hee 3Ha'leHHe SJ],C I1H,AYKUHI1, B03HH-
<I>
K81OI.I..teH B KOHType, < g. > = --, r,Ae <I> = BS 113MeHeHHe M8rHHTHoro
I t
l2 _l2
nOTOKa 38 speMSI t. H3MeHeHI1e nnOI.I..ta.n;H KOHTYP8 f18 = 8 2 - 8, = 0 - - =-
16 16
Bl 2
Tor.n;8 < g > = - ITo 3aKOHY OMa ,AnSI 3aMKHYToH u.enH
I 16t
g. Bl 2
<I> = ---!.. =><1> =-
R 16Rt
I1CKOMblH 3ap.H.n;, npOllle,AlllHH no KOHTYPY,
Bl 2
q =<I>t = 16R ' q=0,13 MKn.
3. TIpOBO,AHHK MaccoH m 11 ,AnI1HOH 1 nemHT Ha ,ABYX ropH30HTanbHblX
npOSO,ASlI.I..tHX penbC8X. 38MKHYTblX Ha pe311CTOp COnpOTI1BneHl1eM R. BcSI CI1C-
TeMa H8XO,AHTCSI B BepTHK8nbHOM M8rHI1THOM none, MO,Aynb HH,AYKUHI1 KOTO-
poro B. K03<1><lmL\HeHT TpeHHSI Mem,Ay penbC8MH H npOBO,AHHKOM )1. Onpe,Ae-
nHTe MO,Aynb Cl1nbl. H8np8BneHHoH no,A yrnoM a K ropH30HTY, KOTOPYIO cne-
,AyeT npHnO:>KHTb K npOSO,AHHKY, 'lT06bl OH llSHr8ncSI ropH30HTanbHO co CKO-
pOCTblO v?
J],8HO: m. t, R, B, J..l. a. v.
H8HTH: F.
e
mg
Pue. 12.22
364
x
P e 111 e H He. ITpl1 .n;SlImeHHH npo-
SO,AHI1K8 (pHC. 12.22) S,AOnb OCH OX
B M8rHI1THOM none B HeM B03HI1KaeT
SJ],C HH,AYKIlHH  = Bvl. KOTopaSl co-
311aeT H B 3aMKHYTOM KOHType abed
HH,AYKUHOHHbIH TOK, Hanp8sneHHbIH
corn8CHO np8sHny np8soH PYKH OT a
K b. TIo 3aKOHY OMa
I. =  = Bvl .
I R R
Ha npOBO,AHI1K C TOKOM 6Y,AeT ,AeH-
CTBOBaTb CHna AMnepa FA, HanpaBne-
HHe KOTOpOH onpe,AenSlIOT no np8BHny
nesoH PYKH. MO,Aynb 3TOH CHnbI
F A = /.Bl = B 2 vl 2
I R
KpoMe CHnbI AMnepa. Ha nposoll-
HRK ,AeHcTsyeT CRna TSlmeCTH mg,

Clinbl pe8Ku,HH N p , Clinbl TpeHHSI F TP Ii CHna F TIpOBO,AHHR 6Y,AeT ,ABHr8TbCSI
C nOCTOSlHHOH CKOPOCTblO. ecnli 3TH CHnbl 6Y,AYT CKOMneHCHpOB8Hbl, T.e.
F Tp, + F Tp + N PI + N pz + FA + F + mg = 6 .
IIpoeUHPYfl BeKTOpHble BenH'lliHbl Ha OCH x H y. nOnY'lHM
{ -2F TP - FA + Fcos a = 0:
2N p + Fsina -mg = O.
PelliliB nonY'leHHYIO CIiCTeMY ypaBHeHIiH C Y'leTOM Toro. 'lTO F Tp = JlN p ,
nonY'lliM
F = FA + Jlmg
cosa + Jlsina .
TIO.n;CT8BHM 3H8'leHHe CHnbl FA H HaH,AeM HCKOMYIO cHny
F = B 2 l2V + J,1mgR
R( cosa + J..lsina) .
4. TIpOBO,AHHK ,AnIiHOH 1 M nemliT Ha rn8,AKIiX T1pOBO,ASIIiX ropli30H-
TanbHblX 1lI1iH8X, p8cnonomeHHbIX B BepTHK8nbHOM M8rHliTHOM none C IiH-
,AYKu,HeH 0, 1 Tn. C KaKoH YCTaHoBHBllIeHcSI CKOpOCTblO Ii B K8KOM H8npaB-
neHHH 6Y,AeT ,ABHr8TbCSI npOBo.n;HHK. ecnH K llIHHaM nO,AKnlOqHTb HCTO'lHHK
TOKa C S}].C 0,5 B (pHC. 12.23). BHYTpeHHHM conpOTHBneHHeM HCTO'lHHK8
npeHe6pe'lb.
}]. a H 0: l = 1 M. B = 0.6 Tn, g:'= 0,5 B.
H8HTH: v.
KOHTypa.
B M8rHHTHOM none H8 npOBO,AHHK C TO-
KOM H8'lHeT .n;eHcTBoB8Tb CHna AMnep8, Ha-
npaBneHH8H, corn8CHO np8BHny neBOH PYKIi,
BneBO. IIo 3aKOHY AMT1ep8 FA = I Bl. TIoA
,AeHCTBHeM CHnbl AMnepa npOBO,AHHK Ha'lHe'f
YCKopeHHO ,ABlir8TbCSI BneBO. TIpli 3TOM nno-
8,Ab, OrpaHIi'leHHaSl KOHTypOM abed, 6Y,AeT
YMeHbllIaTbCH, H BCne,ACTBlie HBneHIiSl 3neKT-
pOM8rHHTHoH HH,AYKU,IiH B npOBO,AHHKe B03HliKHeT S}].C HH,AYKU,HIi :5j, KOTopaSl
C03,A8CT B HeM HH,AYKU,HOHHbIH TOK, H8np8BneHHbIH, corn8CHO np8BHny np8BOH
PYRH, OT b K a.
XCX X X xbx X X
X
X
X
X
X
X
X t
X _X
B
X X
X

x
..)<
F
X
X
I
X
X
X
X
x
X
X
X
X
xd X
X
X
Xa X
X
Pue. 12.23
PeweHHe. IIpH nO,AKnlO'leHIiH HCTO'lHIi-
K8 TOKa B npOBO,AHIiKe nOH,AeT TOK, Ha-
npaBneHHblH OT a K b. Clin8 TOK8 no 3a-
KOHY OMa
g
I = Ii' r,Ae R - conpOTHBneHHe
X
X
X
Zi= Bvl;
I, = Bvl
R
365

Pe3ynbTHPYIOI.I..tHH TOK B npOBO,AHHKe 6Y,AeT YMeHbIII8TbCSI. H. Cne,AOB8Tenb-
HO, 6Y,lJ.eT YMeHbIIIaTbCH CHn8 AMnep8. CKOpOCTb nepeCTaHeT paCTH. KOr,A8
FA = O. T.e. I = Ii' Cne,AOB8TenbHO,
g' = Bvl => v = g' . V = 5 M
R R BI ' C
5. Qepe3 ,AnHHHbIH O,AHOCnOHHbIH COneHOH,A. HH,AYKTHBHOCTb KOToporo
0,40 MfH H nnOl.I..ta,Ab Ce'leHHfl 10 CM 2 . npOXO,AHT nocToSlHHblH TOK. CHn8
TOK8 0,50 A. K8KOB8 HH,AYKI4HSI nonSl BHYTPH COneHOH,lJ.a, ecnH OH CO,AepmHT
100 8HTKOB? KaKoB8 3HeprHSI MarHHTHoro nonSl 3TOH K8TYIIIKH?
,I:{aHo: L=O,40 .10- 3 fH. S= 10- 3 M 2 , 1=0,50 A, N= 100.
H8HTH: B, W.
P e ill e H He. TIo onpe,AeneHHIO M8rHHTHblH nOTOK
<1> = N BScosa. = N BS,
TaK K8K a. = 0 H cosa. = 1. C ,ApyroH CTOpOHbl. <1> = LI. CnelloB8TenbHo.
N BS = LI => B = LI . B = 2 0 MT n.
NS' ,
SHeprHH MarHHTHoro nonH
LI 2
W = -, W = 50 MK,I:{m.
2
6. K8TYIIIK8 HH,AYKTHBHOCTblO 0,3 fH, BHTKH KOTOpOH C,AenaHbl H3 Ton-
CToro Me,AHOrO npOBO,Aa, COe,AHHeH8 n8p8nnenbHO C pe3HCTopOM conpOTHBneHH-
eM R H nO,AKnlO'leHa K HCTO'lHHKY TOKa C S,I:{C 4 B H BHYTpeHHHM conpOTHB-
JIeHHeM 2 OM. K8Koe KOnH'leCTBO TennOTbl Bbl,AenHTCSI B K8TYlliKe H B pe3HC-
Tope nocne OTKJIlO'leHHSI HCTOt{HHK8 TOK8?
,I:{aHO: L=0.3 fH, g'=4 B. r=2 OM.
H8HTH: Q.
P e ill e H He. TaK KaK BHTKH KaTYlliKH C,AenaHbl H3 TonCToro Me,AHOrO npo-
BO,A8, TO ee 3neKTpH'leCKOe conpOTHBneHHe Mano H 6Y,AeM C'lHT8Tb, 'lTO OHO
H8MHoro MeHbIIIe conpOTHBneHHH pe3HCTopa R. TI03TOMY CHn8 TOK8, npoxo-
,ASlI.l.\ero 'lepe3 K8TYlliKY. B YCT8HOBHBIIIeMCSI pemHMe 6Y,AeT paBH8, corn8C-
HO BTOPOMY np8BHny KHpxro<l>8, ,AnSI KOHTYP8 abed
g'
c (pHC. 12.24) 1=-. Cne,lJ.OB8TenbHo, 3HeprHSI M8rHHT-
r
Horo nonSl K8TYlliKH
b
L
f
e
LI 2 Lg'2
W=-=-
2 2r 2 '
TIpH pa3MblK8HHH I4enH B KOHType beef Bcne,ACTBHe
SlBneHHSI CaMOHH,AYKI4HH 6Y,AeT npOXO,AHTb TOK caMOHH-
,AYKI4HH, H 3T8 3HeprHSI H3paCXO,AyeTcSI H8 p860TY npo-
THB CHn BHxpeBoro 3neKTpH'leCKOrO nonSl H npeBp8THT-
d CH BO BHYTpeHHlO1O 3HeprHIO. T8KHM 06p830M, Bbl,lJ.enHB-
IIIeecSI KOnH'leCTBO TennOTbl
R
a
r 
Pue. 12.24
Lg'2
Q = ; Q = 0,6 ,I:{m.
2r
366

KOnE6AHi4S1
i4 BOnHbl
13. MEXAHllqECKIIE KOJIEBAHIIH
KOJIe6aTeJILHOe ,lJ;BHeHHe. CB060,lJ;HLle KOJIe6aHHH B H,ZJ;eaJILHLIX KOJIe6a-
TeJILHLIX CHCTeMax. rapMOHH'IeCKHe KOJle6aHHH. llepHO,lJ;, 'IaCTOTa t aMnJlHTY-
.na, cl>a3a.
rpacl>H'IeCKOe npe,lJ;CTaBJIeHHe rapMOHH'IeCKHX KOJle6aHHH. BeKTopHLIe ,lJ;Ha-
rpaMMLI. npHHHn cynepno3HHH. CJIOeHHe rapMOHH'IeCKHX KOJIe6aHHH_ He-
rapMOHH'IeCKHe KOJIe6aHHH. rapMOHH'IeCKHe H HerapMOHH'IeCKHe KOJIe6aHHH
B npHpO,lJ;e H TeXHHKe.
MaTeMaTH'leCKHH Ma.HTHHK. llepHoA KOJle6aHHH MaTeMaTH'IeCKOrO MaHTHHKa.
KOJIe6aHHH rpY3a Ha npymHHe. llpeBpall\eHHe 3HeprHH npH rapMOHHqeC-
KHX KOJIe6aHH.HX.
3aTYXalOll\He KOJIe6aHHH. BLIHy.neHHLle KOJIe6aHHH. Pe30HaHC.
13.1. KOJle6aTeJlLHOe J].BH:m:eHHe.
CBo6oAHhle KOJIe6aHHH B HAeaJIhHhIX KOJIe6a TeJIhHhIX CHCTeMax
.D:BHmeHHe, np1t KOTOpOM COCTOHHH.H ,ABHmeHHSI TeJla C TeqeHHeM BpeMeHH
nOBTopHIOTCH, npHqeM TeJlO npOXO,AHT qepe3 nOJlO>KeHHe YCTOHtIHBOrO paBHO-
BeCHH nOOqepe,AHO B npOTHBOnOnO>KHbIX HanpaBneHHHX, Ha3bIBalOT MeXaHHqeC-
KHM KOJIe6aTeJIbHLIM .nBHeHHeM.
EcnH COCTOHHH.H ,ABHmeHHH TeJla nOBTOp.HIOTCH 'lepe3 onpe,AeJleHHbIe npo-
MemYTKH BpeMeHH, TO Kone6aHHH .HBn.HIOTCSI nepHo.nH'IeCKHMH.
<I>H3HqeCKYIO CHCTeMY (Teno), B KOTOpOH npH OTKJlOHeHHH OT nOJlO)KeHH.H
paBHOBeCH.H B03HHKalOT H cymeCTBYIOT KOJle6aHH.H. Ha3bIBalOT KOJIe6aTeJIbHOH
CHCTeMOH.
KOJle6aTeJlbHbIH nponecc B CHCTeMe MomeT npOHCXO,AHTb no,A eHCTBHeM
KaK BHeIlIHHX, TaK H BHYTpeHHHX CHn.
KOJle6aHH.H, npOHCXO,AHHe B CHCTeMe no,A ,AeHCTBHeM TonbKO BHYTpeHHHX
CHn, Ha3bIBaIOTC.H CBo60,lJ;HLIMH.
a
6
-+ 6
N
.D:nH Toro tITo6bI B CHCTeMe B03HHK-
JlH CB060,AHbIe KOJle6aHH.H, He06xo,AHMO:
1. HanHtIHe nOJlOeHH.H YCTOH1JH-
Boro paBHOBeCHH CHCTeMLI.
TaK, CB060,AHbIe Kone6aHH.H B03-
HHKHYT B CHCTeMe, H306pameHHOH Ha
pHCYHKe 13.1, a; B cJlyqaHX l5 H 6
OHH He B03HHKHYT.
Puc. 13.1
367

2. HaJlHqHe Y MaTepHaJlbHOH TOqKH H36bITOqHOH MeXaHHqeCKOH 3HeprHH
no cpaBHeHHIO C ee sHeprHeH B nOJlOmeHHH YCTOHqHBOrO paBHOBeCHSI. TaK, B
CHCTeMe (pHC. 13.1. a) HallO. HanpHMep, TeJlO BbIBeCTH 113 nOJlOmeHHSJ paBHO-
BeCH.H: T.e. C006HTb H36bITOJ< nOTeHUHanbHOH SHeprHH.
3. ,[(eHcTBHe Ha MaTepHaJlbHYIO TOqKY B03BpaIU;alOIU;eu CHJILI, T.e. CHJlbI.
HanpaBJleHHOH Bcera K nOJlOmeHHIO paBHOBeCH.H. B CHCTeMe, H306pa:>KeHHOH
Ha pHC. 13.1, a, B03BpaalOeH CHJlOH .HBJI.HIOTCSJ paBHoeHcTBYIOa.H CHna
T.HmeCTH H CHJla HOpMaJlbHOH peaKnHH N onopbI.
4. B HeaJlbHbIX KOJle6aTeJlbHbIX CHCTeMax CHJlbI TpeHH.H OTCYTCTBYIOT. H
B03HHKlIIHe KOJle6aHH.H MorYT npOOJlmaTbC.H onro. B peaJlbHbIX YCJlOBH.HX KO-
Jle6aHHSI npOHCXO.HT npH HaJlHqHH CHJI conpOTHBJleHH.H. 't.J.T06bI KOJle6aHHe
B03HHKJlO H npOOJlmaJlOCb, H36bITOqHa.H SHeprH.H, nOJlyqeHHa.H MaTepHaJlbHOH
TOqKOH npli CMeeHHH H3 nOJlO:>KeHH.H YCTOHqllBOrO paBHOBeCH.H. He OJlmHa
6bITb nOJlHOCTblO paCXOOBaHa Ha npeO)J;OJleHHe conpOTHBJleHH.H npH B03Bp8-
eHHH B STO nOJlOmeHHe.
13.2. rapMOHH'IeCKHe KOJIe6aHHH
TIpocTeHlIIHM BHOM KOJle6aHHH
.HBJI.HIOTC.H rapMOHH'IeCKHe KOJle6a-
HHH - KOJle6aHH.H, npH KOTOpbIX
CMeeHHe KOJle6nlOeHC.H TOqKH
OT nOJlO:>KeHH.H paBHOBeCH.H H3MeHSJ-
eTC.H C TeqeHHeM BpeMeHH no 3aKO-
HY cHHyca HnH KocHHyca.
TaK, npH paBHOMepHOM Bpae-
HHH lIIapHKa no oKpymHocTH ero
npOeKnH.H (TeHb B napaJlnenbHbIX
Jlyqax CBeTa) COBeplIIaeT Ha BepTH-
KanbHOM SKpaHe (pHC. 13.2) rapMO-
HHqeCKOe Kone6aTenbHoe llBHmeHHe.
CMeeHHe OT nOJlOmeHH.H paBHO-
BeCHSJ npll rapMOHHqeCKHX Kone6a-
HH.HX onHCbIBaeTCSI ypaBHeHHeM (ero
Ha3bIBaIOT KHHeMaTH"tIeCKHM 3aKOHOM
rapMOHHqeCKoro BHmeHH.H) BHa:
x = A COs(  t + Ipo ) HJ1H X = A Sin(  t + Ip; )-
re x - CMemeHHe - BenHqHHa. xapaKTepH3YlOma.H nonomeHHe KOJle6J11O-
eHC.H TOqKH B MOMeHT BpeMeHH t OTHOCHTeJlbHO nonOmeHH.H paBHOBeCH.H H
H3Mep.HeMa.H paCCTO.HHHeM OT nonOmeHH.H paBHOBeCH.H ,AO nOJlOmeHHSI TOqKH B
3aaHHbIH MOMeHT BpeMeHH; A - aMnnHTya KOJle6aHHH - MaKCHManbHoe
CMeeHHe TeJla H3 nOJlOmeHH.H paBHOBeCHSI; T - nepHO,Z:{ KOJle6aHHH - Bpe-
M.H COBeplIIeHH.H OHoro nonHoro Kone6aHH.H; T.e. HaHMeHblIIHH npOMemYTOJ<
BpeMeHH, no HCTeqeHHH KOToporo nOBTOp.HIOTC.H 3HaqeHHSI <}>H3HqeCKHX BenH-
2n
qHH, xapaKTepH3YIOHX KOJle6aHHe; <Po - HaqaJlbHa.H <}>a3a; <P = - t + <Po
T
Puc. 13.2
368

cpa3a KOJle6aHHH B MOMeHT BpeMeHH t. <1>a3a KOJle6aHHH - STO aprYMeHT ne-
pHO)J;H'IeCKOH CPYHKltHH, KOTOpbIH npli 3a,naHHOH aMnnHTy,ne KOJle6aHHH onpe-
,nen.HeT COCTO.HHHe KOJle6aTeJlbHOH CHCTeMbl (CMell\eHHe, CKOpOCTb, YCKOpeHHe)
TeJla B JlI060H MOMeHT BpeMeHH.
ECJlH B HaqaJlbHbIH MOMeHT BpeMeHH t o = 0 KOJle6nlOa.HC.H TOqKa MaKCH-
MaJlbHO CMeeHa OT 110JlOmeHHSJ paBHOBeCH.H, TO <Po = 0, a CMell\eHHe TOqKH OT
nOJlOmeHHH paBHOBeCH.H H3MeH.HeTC.H no 3aKOHY
21t
x = Acos-t.
T
EcnH KOJle6J1lOll\a.HC.H TOqKa npH to = 0 Haxo,nHTC.H B nOJlOmeHHH YCTOHqH-
Boro paBHOBeCH.H, TO CMell\eHHe TOqKH OT nOJlOmeHH.H paBHOBeCH.H H3MeH.HeT-
C.H no 3aKOHY
A . 21t
X = Sln-t
T
BeJlHqHHY v, 06paTHYIO nepHO,ny H paBHYIO qHCJlY nOJlHbIX KOJle6aHHH, CO-
BepmaeMblX 3a 1 c, Ha3blBalOT qaCTOTOH KOJle6aHHH:
1
v = - (B Cll e,nHHHn;eH qaCTOTbI .HBJI.HeTC.H repn;, 1rlt = Ic- I ).
T
ECJlH 3a BpeM.H t TeJlO COBepmaeT N nonHblX KOJle6aHHH, TO
t N
T--' V=-.
- N' t
21t
BenHqHHY ro = 2nv = -, nOKa3bIBa101l\YIO, CKOJlbKO KOJle6aHHH COBepmaeT
T
TeJlO 3a 21t C, Ha3bIBalOT HKJlH'IeCKOH (KpyroBOH) '1aCTOTOH.
KHHeMaTHqeCKHH 3aKOH
a
x
A
0
-A
t5 V x
roA
0
6
ax
(02A
0
Puc. 13.3
rapMOHH'IeCKOrO ,nBHmeHH.H MomHO 3anHcaTb B BH,ne:
x = A cos (2nvt + <Po), x = A cos (rot + <Po)'
rpacpH'IeCKH 3aBHCHMOCTb CMeeHH.H
KOJle6J1lOll\eHC.H TOqKH OT BpeMeHH H306pa-
maeTC.H KOCHHYCOH,nOH (HJlH CHHYCOHlIOH).
Ha pHCYHKe 13.3, a npe,nCTaBJleH rpa-
CPHK 3aBHCHMOCTH OT BpeMeHH CMeeHH.H
KOJle6J1IOeHC.H TOqKH OT nOJlOmeHH.H paB-
HOBeCH.H ,nn.H cJlyqa.H <Po = 0, T.e. x =Acosrot.
BbI.HCHHM, KaK H3MeH.HeTC.H CKOpOCTb
KOJle6J1lOll\eHC.H TO'IKH co BpeMeHeM. ,D;JI.H
SToro HaH)J;eM npOH3Bo,nHYIO no BpeMeHH OT
t SToro BbIpameHH.H:
V x = x' = -(JJA sin rot = roA cos( rot + )-
r,ne wA = Ivxl m - aMnJlHTy,na npOeKltHH
CKOpOCTH Ha OCb x.
t 8Ta cpopMYJla nOKa3blBaeT, qTO npH rap-
MOHHqeCKHX KOJle6aHH.HX npOeKltH.H CKOpO-
CTH TeJla Ha OCb x H3MeH.HeTC.H Tome no
T
t
369

rapMOHHqeCKOMY 3aKOHY C TOH me qaCTOTOH, C APyroH aMnJlHTYAOH H onepe-
7t
maeT no <pa3e CMeweHRe Ha - (pRC. 13.3. 6).
2
,nJlSl BbI.HCHeHRSI 3aBHCHMOCTH YCKopeHHSI ax(t) HaH,AeM npOH3BO,AHYIO no
BpeMeHH OT npoeKnHH CKOpOCTH:
ax = v = _ro 2 A cos rot = ro 2 A cos (rot + 1t),
rAe ro 2 A = laxl m - aMnnHTYAa npoeKnHH YCKopeHH.H Ha OCb x.
IIpH rapMOHHqeCKHX KOJle6aHHSlx npOeKnH.H YCKopeHHH onepemaeT CMee-
HHe no <pa3e Ha 7t (pHC. 13.3. 6).
AHaJlOrHqHO MomHO nOCTpOHTb rpa<pHKH 3aBHCHMOCTeH x(t), vx(t) H ax(t),
ecnH x=Asinrot npH <Po=O.
YqHTbIBaSl, qTO Acoswt = X, <POPMYJlY ,AnSI YCKopeHHSI MomHO 3anHcaTb
2
ax = -ro x,
T.e. npH rapMOHHqeCKHX KOJle6aHHSlX npOeKnH.H YCKopeHHSI npSlMO nponopnR-
OHaJlbHa CMeeHHIO H npOTHBOnOJlOmHa eMY no 3HaKY, T.e. YCKopeHHe Ha-
npaBneHO B CTOPOHY, npOTHBononomHYIO CMeIr(eHHIO.
TaK, npOeKnH.H YCKopeHHSI - 3TO BTopaSi npOH3BOAHaSl OT CMeIQeHH.H
a x = x", TO nonyqeHHOe COOTHOllleHHe MomHO 38nHcaTb B BHAe:
a x +w 2 x=0 HnH x"+w 2 x=0.
llOCJleAHee paBeHCTBO Ha3bIBalOT ypaBHeHHeM rapMOHHQeCKHX KOJle6aHHH.
(f>H3H'leCKYIO CHCTeMY, B KOTOpOH MorYT cyecTBoBaTb rapMOHHqeCKHe KO-
ne6aHH.H, Ha3bIBalOT rapMOHHqeCKHM OCnHJlJlHTOpOM, a ypaBHeHRe rapMOHHqeC-
KHX Kone6aHHH - ypaBHeHHeM rapMOHHQeCKOrO OCnHJlJlHTopa.
13.3. rpa4>H'IeCKOe npe,lJ;CTaBJIeHHe rapMOHH'IeCKHX KOJle6aHHH.
BeKTopHble ,lJ;HarpaMMbl
HaH60nee qaCTO ynoTpe6n.HIOTCSI TPH cnoc06a rpa<pHqeCKOrO npeACTaB1'eHHSI
Kone6aHRH.
1. 3a.n:aHH rpa4>HKa KOJle6aHHH x = f(t) B np.HMoyrOJlbHOH CHCTeMe KOOp-
AHHaT. IIo OCH a6cnRCC OTKnaAbIBaeTCSI BpeM.H t, R no OCR 0PAHHRT - 3Ha-
qeHRe H3MeH.HIOeHC.H BenRqRHbI x (CMeeHHSl, CKOpOCTR, YCKopeHR.H HAP.).
,nnH rapMOHHqeCKHX Kone6aHHH 3TOT rpa<pHK - KOCHHYCOHAa (CM. pHC. 13.3)
RnR CRHYCORA8.
2. CneKTpaJlbHblH cnoc06. IIo OCH opAHHaT OTKJlaAbIBaeTCSI aMnJlHTYAa. a
no OCH a6CnHCC - qaCTOTa rapMOHHqeCKHX KOJle6aHHH. TaK, HRnpHMep.KOJle-
6aTeJlbHbIH nponecc x = 5cos4t (M) 6YAeT
npe,ACTaBneH B 3TOM cnyqae BepTHKaJlbHbIM
OTpe3KOM npSlMOH AJlHHOH 5 M, npOBe,AeH-
HbIM OT TOqKH C KooPAHHaToH w = 4 rn Ha
OCH a6CnHCC (pRC. 13.4). 3TOT cnoc06 He
,AaeT HHKaKOH HH<popManHH 0 <pa3e Kone-
6aHH.H .
3. Cnoc06 BeKTopHblX .n;HarpaMM. IIycTb
BenHqHHa x H3MeH.HeTCSI co BpeMeHeM no
3aKOHY x =Acos(w t + <Po).
A
5
4
3
2
1
1 2 345
W,rn
Puc. 13.4
370

Ha nJIOCKOCTH Bbl6HpalOT npOH3BOJlbHO HanpaBJIeHHYIO KOOp.n;HHaTHYIO OCb Ox.
113 HaqaJIa KOOp.n;HHaT no.n; yrJIOM <Po, paBHblM Ha'laJIbHOH cpa3e KOJIe6aHHH, npo-
BO.n.HT BeKTOp A. MO.n.YJIb KOToporo paBeH aMnJIHTy.n.e rapMOHH'IeCKOrO Kone-
6aHHH A (pHC. 13.5). ECJlH BeKTOp A BpalI(aeTC.H BOKpyr TOqKH 0 C nOCTO.HH-
HOH yrJlOBOH CKOpOCTblO 0) npOTHB 'IaCOBOH CTpeJlKH, TO yroJl <P Mem,AY BpalI(a-
IOlI(HMC.H BeKTOpOM H OCblO Ox B JlI060H
MOMeHT BpeMeHH Onpe,AeJIHTC.H BblpameHH-
eM <P = (0) t + <po). IIpoeKltHH KOHlta BeKTopa
A 6Y,AeT nepeMelI(aTbC.H no OCH OX H npH-
HHMaTb 3HaqeHHH OT -A .n;o +A, a KOJle6-
JIIOlI(aHCH BeJlH'IHHa 6Y,AeT H3MeHHTbCH co
BpeMeHeM no 3aKOHY x = A cos (rot + <Po ).
TaKHM 06pa30M, rapMOHHqeCKOe KOJle-
6aHHe MomHO npe,ACTaBHTb npOeKIJ;HeH Ha
x HeKOTOpylO npOH3BOJlbHO BbI6p8HHYIO OCb
BeKTopa aMnJIHTy.n;bI A, OT JlOmeHHOro OT
npOH3BOJlbHOH TO'lKH OCH no.n; yrJIOM <Po,
paBHbIM Ha'laJlbHOH cpa3e, H BpalI(alOlI(e-
rOC.H C yrJIOBOH CKOpOCTblO 0) BOKpyr
3TOH TOqKH.
A
o
x = Acos<po
Puc. 13.5
A
13.4. npHHQHn cynepn03HQHH. CJIOmeHHe KOJIe6aHHH
'tJaCTO TeJlO yqacTByeT B .n;BYX HnH HeCKOJIbKHX KOJIe6aHH.HX. HanpHMep,
rpY3, no.n;BerneHHblH Ha npymHHe K nOTOJIKY peccopHoro BarOH8, KOJIe6J1eTC.H
OTHOCHTeJlbHO TO'lKH no.n;Beca, KOTOpa.H B CBOIO O'lepe.n;b COBeprnaeT KOJIe6aHHH
BMeCTe C BarOHOM Ha ero peccopax. TaKHM 06pa30M, rpY3 6y.n;eT COBeprnaTb
,ABHmeHHe, CKJIa.n;bIBalOeeC.H H3 .n;BYX KOJle6aHHH. B TaKHX cJlyqa.HX He06xo.n;H-
MO HaHTH pe3YJIbTHpYIOlI(ee KOJle6al!He, HHbIMH CJI()BaMH, KOJle6aHHH He06xo-
.n;HMO CJIOmHTb. ITo npHHIJ;Hny cynepn03HIJ;HH 3TH KOJIe6aHH.H paCCMaTpHBaIOTC.H
KaK He3aBHCHMbIe, H pe3YJlbTHpYIOee CMelI(eHHe Haxo.n;HTCH B Kam.n;blH MOMeHT
BpeMeHH KaK BeKTOpHa.H CYMMa CMelIJ.eHHH OT.n;eJlbHbIX KOJIe6aHHH. B CJIyqae
CJlOmeHH.H Kone6aHHH, HanpaBJleHHbIX no O.n;HOH npHMOH, pe3YJlbTHpYIOee CMe-
lI(eHHe paBHO aJIre6paHqeCKOH CYMMe CMelI(eHHH OT .n;eJIbHblX Kone6aHHH.
IIpH CJlOmeHHH KOJle6aHHH MomHO nOJlb30BaTbCH aHanHTH'leCKHM, rpacpH-
'IeCKHM MeTo.n;aMH H MeTo.n;OM BeKTopHbIX .n;HarpaMM.
CJlOmHM rapMOHHqeCKHe KOJle6aHH.H o.n;Horo HanpaBJleHH.H H o.n;HHaKOBOH
'IaCTOTbI:
Xl = Al COS ( O)t + <pd;
x 2 =  cos(rot + <P2)'
1. AHaJIHTH'IeCKHH MeTO,lJ;. Pe3YJIbTHpYIOlI(ee CMelI(eHHe
X = Xl + X2 = Al COS (rot + <l>d + A 2 cos (rot + <1>2 ) .
IIoKameM, t.lTO pe3YJlbTHpYlOrnee KOJle6aHHe TaKme .HBJI.HeTCH rapMOHHqeC-
KHM KOJle6aHHeM TOH me qaCTOTbI, T.e. t.lTO
x = Acos(rot + cp),
371

r,ne A
Tor ,na
aMnJlHTy,na pe3YJlbTHpYIOero KOJle6aHH.H; <P - ero Ha'tlanbHaR c}>a3a.
A cos (wt + <p) = Al COS (wt + <PI) + A 2 cos (wt + <P2 ).
McnoJlb3Y.H c}>OPMYJlY KocHHyca CYMMbI ,nBYX yrnoB, nony'tlHM
A cos wt cos <P - A sin wt sin <P = Al cos wt cos <PI -
- Al sin wt sin <PI + A 2 cos wt COS <P2 - A 2 sin wt sin <P2'
3TO ypaBHeHHe 6y,neT TOmlteCTBOM OTHOCHTenbHO t, ecnH K03<pc}>HII.HeHTbI
npH cos w t If sin w t B ero JleBOH 'tIaCTH paBHbl COOTBeTCTBYIOHM K03c}>c}>HII.H-
eHTaM B npaBoH qaCTH B RI060H MOMeHT BpeMeHH:
A cos <P = Al COS <PI -t A 2 cos <P2 '
A sin <P = Al sin <PI + A 2 sin <P2'
PellIHB 3TY CHCTeMY ypaBHeHHH. nonY'lHM
A =  Af + A + 2AIA2 cos( <P2 - <pd
t Al sin <PI + A 2 sin <P2
gcp= .
Al COS <PI + A 2 cos <P2
2. rpa4>HQeCKHH MeTO. CnomeHHe
CBO,nHTCR K CYMMHpOBaHHIO op,nHHaT B
Kam,nbIH MOMeHT BpeMeHH ('tIeM 60JlbllIe
TO'tleK, TeM TO'lHee) - PHCYHOK 13.6.
3. MeToJJ; BeKTopHLIX JJ;HarpaMM. IIo-
CTpOHM BeKTopHbIe ,nHarpaMMbI 3TH X 1<0-
ne6aHHH (pHC. 13.7). TaK KaK BeKTopbI
Al 11 A 2 BpaaIOTC.H C O,AHHaKOBoH yr-
nOBOH CKOpOCTblO, TO yroJl Mem,ny HHMH,
a 3Ha'tlHT, H pa3HOCTb c}>a3 (<P2 - <PI) CO-
CTaBn.HIOHX KOJle6aHHH, OCTaIOTC.H nOCTO-
RHHbIMH. Pe3YJlbTHpYIOee CMeeHHe
x = Xl + X2' r,Ae Xl T X2 - CYMMa npoeK-
II.HH BeKTopOB Al H A 2 Ha OCb Ox. M3
MaTeMaTHKH H3BeCTHO, qTO CYMMa npoeK-
II.HH BeKTopOB paBHa npOeKII.HH BeK-
Topa CYMMbI, T.e. Alx+A2x=Ax, r,ne
A = Al + A 2 . CJle)J;OBaTeJlbHO, aMnJlHTy,na
o X pe3YJlbTHpYIOero KOJle6aHHH paBH8 MO,AY-
Puc. 13.7 JlIO BeKTopa A. M3 pHCYHKa 13.7 BH,AHO.
'tITO no T eopeMe KOCHHYCOB
A =  Af + A: - 2AIA2 cos(n - (<P2 - <pd) =  Af + A: + 2AIA2 COS(<P2 - <pd.
YroJl HaKnOHa <P BeKTopa KOCH OX paBeH Ha'tlaJlbHOH c}>a3e pe3YJlbTHPYIO-
ero Kone6aHH.H. M3 pHCYHKa 13.7 BH,nHO, qTO
Ay Al sin <PI + A 2 sin <P2
tg <P = - =
Ax Al COS <PI + A 2 cos <P2
X
Puc. 13.6
rA
372

TaKHM 06pa30M, TeJlO, yqaCTBYfl B .n;BYX rapMOHHqeCKHX KOJIe6aHH.HX O.n;-
HOrO HanpaBJleHH.H H O.n;HHaKOBOH qaCTOTbI, TaKme COBepmaeT rapMOHHqeCKOe
KOJle6aHHe B TOM me HanpaBneHHH H C TOH me qaCTOTOH, qTO H CKJIa,AbIBae-
MbIe KOJle6aHH.H. AMnJlHTY,A8 pe3ynbTHpYIOero KOJIe6aHH 3aBHCHT OT pa3-
HOCTH <l>a3 <1'2 - <Pl' IIpH CJlOmeHHH rapMOHHqeCKHX KOJle6aHHH o.u.Horo HanpaB-
neHH.H, HO pa3HbIX qaCTOT. pe3YJlbTHpYIOee KOJle6aHHe He flBJlfleTC.H rapMO-
HH'leCKHM. Ee 60Jlee CJlOmHbIMH OKa3bIBaIOTC.H pe3Y JIbTHpYIOHe KOJIe6aHHSI,
. nOJIyqeHHbIe npH CJIOmeHHH KOJle6aHHH, npOHcxo.n;flJ..QHX B pa3HbIX HanpaBJle-
HH.HX.
13.5. HerapMOHH'IeCKHe KOJIe6aHHH. rapMOHH'IeCKHe
H HerapMOHH'IeCKHe KOJIe6aHHH B npHpO,ZJ;e H TeXHHKe
PeaJIbHbIe KOJIe6aHH.H He npo-
HCXO.HT B TOt.IHOCTH no rapMOHH-
qeCKOMY 3aKOHY. HH O,lJ.HH Kone-
6aTenbHbIH npon;ecc B npHpO,Ae H
TeXHHKe He npo.n;OJlmaeTCfl 6eCKO-
HeqHO .n;OJIro, a HMeeT HaqanO H
KOHen; BO BpeMeHH. A KOJIe6aTenb-
HbIH npon;ecc, OrpaHHqeHHbIH BO
Puc. 13.8 BpeMeHH, He flBJI.HeTC.H rapMOHH-
qeCKHM. IIIapHK, nO,ABemeHHbIH
Ha HHTH, H rpY3, no.n;BemeHHbIH Ha npymHHe, COBepmalOT Kone6aHHfl, KOTO-
pbIe JlHmb 6J1H3KH K rapMOHHqeCKHM. ConpOTHBJIeHHe B03.n;yxa, He06paTHMbIe
nOTepH 3HeprHH Ha HarpeBaHHe HHTH H npymHHbI npH HX ,Ae<l>opMan;HH npH-
BO.n;.HT K TOMY, ({TO aMnnHTy.n;a KOJle6aHHH YMeHbmaeTC.H. H KOJIe6aHHfl CTaHO-
B.HTCfl HerapMOHHqeCKHMH.
B npHpo.n;e BCTpeqaIOTC.H Kone6aTeJlbHbIe npOn;eCCbI, npOTeKalOHe BeCbM8
.n;JIHTeJIbHOe BpeM.H. rIpHMepoM KOJIe6aHHH TaKoro po.n;a MorYT CJlymHTb nepH-
o.n;Ht.IeCKHe H3MeHeHHfl H8np.HmeHH.H Mem.n;y p83J1H'lHbIMH yqaCTKaMH qeJIOBe-
t.IeCKOrO TeJIa, B03HHKalOll.\He B pe3YJlbTaTe pa60TbI cep,lJ.ua. rpacpHK 3aBHCH-
MOCTH OT BpeMeHH "BbIpa6aTbIBaeMoro" cep.neqHOH MbImueH Hanp.HmeHH.H Ha-
3bIBaeTC.H 3J1eKTpOKapAHorpaMMoi. 8JIeKTpOKap.n;HorpaMMa t.IeJIOBeKa OqeHb
MaJlO noxoma Ha CHHYCOH.n;y, T.e. KOJle6aHHfl 6HOTOKOB .HBJI.HIOTCfl HerapMOHH'
'leCKHMH (pHC. 13.8).
)];pyroH npHMep HerapMOHHt.IeCKHX KOJIe6aHHH, npOHcxo.n;Slll.\HX B npHpO,lJ.e, -
KOJle6aHHfl ypOBHfl BO.n;bI B OTKpbITbIX MOp.HX H OKeaHax. Kone6aHH.H ypOBH.H
BO,lJ.bI BO MHorHX MOpCKHX nopTax HaCTOJIbKO 3HaqHTeJIbHbI, qTO TOt.IHOe npe.n;-
CKa3aHHe OTJlHBOB H npHJlHBOB OKa3bIBaeTC.H BamHO" npaKTHqeCKOH 3a.n;a'leH:
rJIy6HHa oca.n;KH COBpeMeHHbIX MOpCKHX Cy.n;OB BeJIHKa, H MHorHe MopCKHe
nopTbI MorYT npHHHMaTb HX JIHillb B qaCbI npHJIHBa.
B Kam.n;OM nopTY npOH3Bo.n;.HT perYJISlpHbIe Ha6JIIO.n;eHHSI npHJIHBOB H Ha
OCHOBe MHOrOJleTHHX H86JIIO,AeHHH onpe.n;eJlSlIOT nepHo.n;HqHOCTb H3MeHeHH.H
ypOBH.H BO.n;bI. 3aBHCHMOCTb BbICOTbI npHJIHBa OT BpeMeHH OKa3bIBaeTCfl Ot.IeHb
CJlOmHOH H Bblpa3HTb ee KaKOH-TO O.n;HOH cpOPMYJIOH .n;OBonbHO Tpy.n;HO.
IIpH CJlOmeHHH rapMOHHqeCKHX KOJIe6aHHH pa3HbIX qaCTOT CYMMapHoe KO-
ne6aHHe HOCHT 60Jlee CJlOmHbIH xapaKTep. MomHO nOCTaBHTb 06paTHYlO 3a.u.a-
373

qy: B03MOmHO nH CJlOmHOe KOJle6aHHe npe.n;CTaBHTb K8K CYMMY rapMOHHqeC-
KHX Kone6aTe.TlbHblX .n;BHmeHHH?
TeOpeTHqeCKHe p8CqeTbI nOK83bIB8IOT, qTO JlI060e nepHO,AHqeCKOe KOJle68-
HHe MOmeT 6bITb M8TeM8THqeCKH npe,ACT8BJleHO K8K CYMM8 r8pMOHHQeCKHX
Kone68HHii K pa THbIX Q8CTOT 00, 200 , 3w, . .., noo, npHqeM 8MnnHTY,lJ;bl r8 pMO-
HHK aToro pSI,lJ;8 (p.H,lJ;8 YPbe) C YBenHQeHHeM HOMep8 n YMeHbWaIOTC.H. T8K,
H8 pHCYHKe 13.9. iJ H306pameHo CJlOmHOe KO-
ne68HHe, COCTO.HIr(ee H3 qeTbIpex rapMOHHqeC-
KHX (pHC. 13.9):
a) Xl = 10sinoo t;
6) x2=1,5sin3oot;
6) x3=0,6sin5oot;
z) X4 = 0,3sin 7oot.
CneKTp aToro CJlOmHOrO Kone6aHHSI npe,lI-
-
t CTaBneH Ha pHCYHKe 13.10.
a
x
6 x tv\Jvrv
6 X 
"""""""
vvvvvvvv
z X 
...............................A..#IIo.
W'W''W''V'"W''V'¥''V'........
iJ X
Puc. 13.9
t
..
t
A
10,0
.
t
t
1,5
0,3
00
300
700 00
500
Puc. 13.10
13.6. MaTeMaTH'leCKHH MaSlTHHK.
TIepHon Kone6aHHH MaTeMaTH'leCKOrO MaSlTHHKa
MaTeMaTH'IeCKHM MaHTHHKOM H83bIBalOT MaTepHaJlbHYIO TOQKY, no.n;BeweH-
HYIO H8 HeBeCOMOH H Hep8cTSImHMoH HHTH, npHKpenJleHHOH K no.n;Becy H H8-
XO,lJ;SIIr(eHCSI B nOJle CHnhl TSImeCTH (HnH HHOH cHnbl).
JlcCJle,lJ;yeM KOJle68HHSI M8TeM8THqeCKOrO M8SITHHK8 B HHepIJ;.H8J1hHOH CHCTe-
Me OTC'leTa, OTHOCHTenbHO KOTOpOH TOqK8 ero nO,ABeC8 H8XO,lJ;HTC.H B nOKoe HnH
,lJ;BHmeTC.H p8BHOMepHO npSIMOJlHHeHHO. CHnOH conpOTHBJleHH.H B03,lJ;YX8 6Y,lJ;eM
npeHe6peraTb (H,lJ;eanbHbIH MaTeMaTHqeCKHH MaSITHHK). llepBOH8'laJlbHO MaSITHHK
nOKOHTC.H B nonomeHHH paBHOBeCH.H C. llpH aTOM .n;eHCTBYIOHe Ha Hero CHn8
'l'.HmeCTH F H CHJla ynpyrocTH F ynp HHTH B38HMHO KOMneHCHpYIOTCSI
BbIBe.n;eM Ma.HTHHK H3 nOJlOmeHH.H paBHOBeCH.H (OTKJlOHHB ero, HanpHMep,
B nOJlOmeHHe A) H OTnYCTHM 6e3 H8'lanbHOH CKOpOCTH (pHC. 13.11). B aTOM
CJlY'l8e CHnbI F H Fynp He ypaBHoBewHB81OT pyr ,Apyra. T8HreHUHanhH8SI CO-
CT8BnSIIOa.H CHJlbI TSImeCTH Fe, .n;efrcTBY.H H8 M8SITHHK, co06aeT eMY TaH-
reHIJ;.H8J1bHOe YCKopeHHe aT (COCTaBn.HlOIr(aSI nOJlHOrO YCKopeHHSI, Hanp8BneH-
HaSI B,lJ;Onb K8C8TeJlbHOH K Tp8eKTopHH ,lJ;BHmeHH.H M8TeM8THqeCKOrO M8SITHH-
Ka), H Ma.HTHHK HaqHHaeT ,lJ;BHraTbCSI K nOJlOmeHHIO p8BHOBeCH.H C B03paCT81O-
374

F F" F=mg F
n
eH no MO.n;YJlIO CKOpOCTblO. TaHreH-
:U;HaJIbHa.H COCT8.BJI.HIOa.H CHnbI T.H-
meCTH F y HBnHeTCH, TaKHM 06pa30M,
B03BpaI:u;aIOeH CHJlOH. HOpMaJIbHa.H
COCTaBnHIOaH Fn CHnbI T.HmeCTH
HanpaBneHa B.n;OJlb HHTH npOTHB
CHJlbI ynpyrOCTH F ynp . PaaHo.n;eiicTBY-
1Oa.R CHJI Fn H Fynp co06aeT Ma-
.HTHHKY HOpMaJIbHOe YCKopeHHe an'
KOTOpoe H3MeH.HeT npH 3TOM HanpaB-
neHHe BeKTOpa CKOpoCTH, H Ma.HTHHK
,l!BHmeTCH no n;yre ABCD.
'tJeM 6nH>Ke nO,AXOn;HT Ma.HTHHK
F K nonomeHHIO paBHOBeCH.H C, TeM
-n
MeHblIIe CTaHOBHTCH 3Ha'leHHe T8H-
reH:U;HaJlbHOH COCTaBJlHIOIIteH
F
Ft = F sin a . B nOnO>KeHHH paBHOBe-
CH.H OHa paBHa HynlO, a CKOpOCTb
.n;OCTHraeT MaKCHMaJlbHOrO 3HaqeHHH, H Ma.HTHHK .n;BHmeTC.H no HHepIHH n;anb-
Puc. 13.11
lIIe, no.n;HHMaHCb no .n;yre BBepx. TIpH 3TOM COCTaBn.HlOa.H  HanpaBneHa
npoTHB CKOpOCTH. C YBeJlHQeHHeM yrJla OTKnOHeHHH a MOn;YJlb CHnbI F y YBe-
JlHtIHBaeTC.H, a MO.n;YJlb CKOpOCTH YMeHblIIaeTcH, H B TO'lKe D CKOpOCTb MaSlTHHKa
CTaHOBHTCH paBHoH HYJlIO. Ma.HTHHK Ha MrHOBeHHe OCTaHaBnHBaeTC.H, a 3aTeM Ha-
qHHaeT .n;BHraTbCH B 06paTHoM HanpaBneHHH K nOJlOmeHHIO paBHOBeCH.H. BHOBb
npon.n;.H ero no HHep:U;HH, Ma.HTHHK, 3aMe,An.H.H .n;BH>KeHHe, .n;onn;eT .n;o TO'lKH A
(TpeHHe OTCYTCTByeT). T.e. cOBeplIIHT non Hoe Kone6aHHe. TIocJle 3Toro .n;BHmeHHe
MaHTHHKa 6y.n;eT nOBTOp.HTbCH B Y>Ke onHcaHHOH nocne.n;OBaTeJlbHOCTH.
TIonyqHM ypaBHeHHe, onHCbIBalOee cB060.n;HbIe Kone6aHH.H MaTeMaTHQeC-
Koro Ma.HTHHKa.
TIYCTb Ma.HTHHK B .n;aHHbIH MOMeHT BpeMeHlf Haxo.n;HTCH B TOQKe B. Ero
CMeeHHe S OT nOJlOmeHH.H paBHOBeCHH B 3TOT MOMeHT paBHO .n;nHHe .n;yrlf
CB (T.e. S = ICBI). 0603HaQHM .n;nHHY HHTH no.n;Beca i, a Maccy Ma.HTHHKa - m.
S
M3 pHCYHKa 13.11 Blf:,AHO. qTO Ft=Fsina. r,l!e a=-. TIpH MaJlbIX yrnax
l
(ex < 10°) OTKJlOHeHH.H Ma.HTHHKa sin ex = ex, n03TOMY
S S
F = -F- = -mg-.
T l l
3HaK MHHYC B 3TOll tPopMYJle CTaB.HT nOTOMY, qTO TaHreH:U;HaJlbHaH COCTaB-
nHIOa.H CHJlbI T.HmeCTH HanpaBneHa K nOJlOmeHHIO paBHOBeCH.H, a CMeeHHe
OTC'lHTbIBalOT OT nOnO>KeHHH paBHOBeCH.H.
Cor JlaCHO BTOPOMY 3aKOHY HblOToHa ma = mg + F ynp . Cnpoe:U;HpyeM BeKTOp-
HbIe BenHqHHbI 3Toro ypaBHeHHH Ha HanpaBJleHHe KaCaTeJlbHOH K TpaeKTopHH
.n;BHmeHHSl MaTeMaTHqeCKOrO Ma.HTHHKa
Ft = ma'[.
375

ll3 3THX y paBHeHHH nOJlyqHM
I'" = -fs 1 - I\HHBMHQeCKOe ypBBHeHHe I\BH>KeHHH MBTeMBTHQeCKOrO MB-
HTHHKa. TaHreHHaJlbHOe YCKOpeHHe MaTeMaTHqeCKOrO Ma.HTHHKa npOnOpHO-
HanbHO ero CMell.\eHHIO H HanpaBJleHO K nOJlOmeHHIO paBHOBeCH.H. 3TO ypaB-
g
HeHHe MomHO 3anHcaTb B BH.n;e: aT + is = O. CpaBHHBa.H ero C ypaBHeHHeM
rapMOHHqeCKHX KOJle6aHHH ax + w 2 x = 0 (CM.  13.3), MomHO c.n;eJlaTb BbIBO.n;,
qTO MaTeMaTHqeCKHH Ma.HTHHI< COBepmaeT rapMOHH'tIeCKHe KOJle6aHH.H. A TaR
KaK paCCMOTpeHHble KOJle6aHHH MaHTHHKa npOHcxo.n;HnH no.n; .n;eHCTBHeM Tonb-
KO BHYTpeHHHX CHn, TO 3TO 6bInH cB060.n;HbIe Kone6aHHH MaHTHHKa. CJle.n;OBa-
TenbHO, cBo6oAHLle KOJIe6aHHSI MaTeMaTHqeCKOI'O MaSlTHHKa "PH MaJILIX OT-
KJIOHeHHSlX SlBJI.HIOTCSI rapMOHHqeCKHMH.
0603Ha'tlHM ! = CJ)2. OrKy.n;a CJ) = f! - HKJIl{'qecKa.H qaCTOTa Kone6aHHH MqTHHKa.
l Vi
2n
IIepHo.n; KOJle6aHHH Ma.HTHHKa T = -. Cne.n;OBaTeJlbHO,
m
I T = 2n.1
STO BbIpameHHe Ha3bIBalOT OPMYJlUd flOHreHca. OHO onpe.n;eJl.HeT nepHO.n;
CB060,AHbIX Kone6aHHH MaTeMaTH'tIeCKOrO Ma.HTHHKa. H3 cpopMynbI cne.n;yeT, 'tITO
npH MaJIbIX yrnax OTKJlOHeHH.H OT nOJlOmeHH.H paBHOBeCHH nepHO.n; Kone6aHHH
MaTeMaTH'tIeCKOrO Ma.HTHHKa: 1) He 3aBHCHT OT ero MaCCbI II aMnnllTy.n;bI Kone-
6aHHH; 2) npOnOpIJ;HOHaJIeH KOpHIO KBa.n;paTHOMY H3 .n;nHHbI Ma.HTHHKa H 06paT-
HO npOnOpIJ;HOHaJIeH KOpHIO KBa.n;paTHOMY H3 YCKopeHH.H cB060)J;Horo na.n;eHHH. STO
cornacyeTcH C 3KcnepHMeHTaJIbHbIMH 3aKOHaMH ManbIX Kone6aHHH MaTeMaTH-
'tIeCKOrO Ma.HTHHKa, KOTopHe 6bIJlH OTKpbITbI f. raJlHneeM.
IIo,A'tIepKHeM, 'tITO 3TY CPOPMYJlY MomHO Hcnonb30BaTb .n;JI.H paCqeTa nepHO-
.n;a npH o.n;HoBpeMeHHoM BbInOJlHeHHH lIBYX YCJlOBHH: 1) KOJle6aHHH Ma.HTHHKa
,lIOJlmHbI 6bITb ManbIMH; 2) TO'lKa no.n;Beca Ma.HTHHKa .n;OJlmHa nOKOHTbC.H HnH
,lIBHraTbC.H paBHOMepHO np.HMOJlHHeHHO OTHOCHTenbHO HHepUHaJlbHOH CHCTeMbI
OTCqeTa, B KOTOpOH OH HaXOllHTC.H.
ECJlH TOqKa no.n;Beca MaTeMaTH'tIeCKOrO Ma.HTHHKa .n;BHmeTC.H C YCKopeHHeM
ii, TO npH 3TOM H3MeH.HeTCSI CHna HaT.HmeHH.H HHTH, qTO npHBo.n;HT K H3Me-
HeHHIO H B03BpaalOeH CHJlbI, a cne.n;OBaTeJlbHO, qaCTOTbI H nepHo.n;a Kone-
6aHHH. KaK nOKa3bIBalOT paC'tleTbI, nepHO,A Kone6aHHH Ma.HTHHKa B 3TOM cny-
'lae MomHO paCC'tIHTaTb no cpopMYJle
T = 2n Jf .
r,Ae g' - "3cpcpeKTHBHoe" YCKopeHHe MaHTHHKa B HeHHepIJ;HaJlbHOH CHCTeMe
OTC'tIeTa. OHO paBHO reOMeTpH'tIeCKOH CYMMe YCKopeHH.H cB060.n;Horo na.n;eHHH
g H BeKTopa, npOTHBOnOJlOmHOrO BeKTOpy ii, T.e. ero MomHO paCC'tIlITaTb no
cpopMYJle
g' = g + (-a).
376

a
13.7. npymHHHbIH MaHTHHK
IIpY>KHHHbIH Ma.HTHHI< - 3TO Kone-
6aTenbHaH CHCTeMR, COCTO.Ha.H 1'13 Ma-
TepH8.JIbHOH TO'lJ:KH MaCCOH m 1'1 npY>KH'
HbI. PaCCMOTpHM rOpH30HT8.JIbHbIH npy-
mHHHbIH MB..HTHHK (pHC. 13.12, a). OH
npe.nCTRBJI.HeT C060H MaCCHBHoe Teno.
npOCBepneHHoe nocepe.nHHe 1'1 Ha.neToe
HR ropH30HTaJlbHbIll CTepmeHb, B.nOJlb
KOToporo OHO MO>KeT CKonb3HTb 6e3
TpeHHH (H)J;eRnbHa.H KOJle6aTeJlbHRH
cHcTeMR). CTep>KeHb 3RKpenneH Mem-
.nY .nBYM.H BepTHKaJlbHbIMH onopaMH.
K Teny O.nHHM KOHOM npHKpenneHR
HeBeCOMaH npymHHR. ,Z:(pyroH ee KO-
He 3aKpenneH Ha onope, KOTopaH B
npocTeHllIeM CJlY'lRe Haxo.nHTC.H B no-
Koe OTHOCHTeJlbHO HHepHanbHoH CH-
CTeMbI OTC'lJ:eTa. B KOTOpOH npOHCXo.n.HT
Kone6aHHH MaHTHHKR. B Ha'lJ:Rne npy-
mHHa He .necpOpMHpOBRHa. H Teno Ha-
XO.nHTC.H B nOJlO>KeHHH paBHOBeCHH C.
EcnH, paCT.HHYB HnH cmaB npymHHY,
BbIBeCTH Teno H3 nOnO>KeHIf.H paBHO-
BeCH.H. TO CO CTOpOHbI .necpOpMHpOBaH-
HOH npymHHbI Ha Hero nR'lJ:HeT .neH-
CTBOBaTb CHna ynpyroCTM. Bcer.na HR-
npaBneHHR.H K nOJlO>KeHHIO pRBHOBe-
CHSI. IIycTb MbI C>KanM npymMHY, ne-
peMeCTHB Teno B nonomeHMe A, M
OTnYCTHnM (vo = 0). IIo.n .neHcTBHeM
CHnbl ynpyrocTH OHO CTaHeT .nBMraTbCH YCKopeHHO. IIpH 3TOM B nOnO>KeHMM
A HR Teno .neHcTByeT MaKCHManbHRH CHn8 ynpyrocTH, TRR KRK 3,AeCb a6co-
nlOTHoe y.nnHHeHHe x m npY>KMHbI HRH60JlbllIee. Cne.noBaTenbHo, B 3TOM no-
nO>KeHHH YCKopeHMe MaKCMManbHoe. IIpM .nBMmeHMM TenR K nonomeHMIO paB-
HOBeCH.H a6COJllOTHOe y.nJlHHeHHe npymMHbI YMeHbllIReTCH, R cne.nOBaTenbHO.
YMeHbllIaeTC.H YCKopeHHe, co06aeMoe CMJlOH ynpyrocTH. Ho TRK KaK YCKope-
HMe np" .nRHHOM .nBHmeHMM cOHanpaBJleHO co CKOpOCTblO, TO CKOpOCTb MaHT-
HMKa YBenH'lJ:HBaeTC.H 1'1 B nonomeHHH pRBHOBeCH.H OHR 6y.neT MaKCHManbHa.
,Z:(OCTHrHYB nOJlOmeHH.H pRBHOBeCH.H C, TeJlO He OCTRHOBHTC.H (XOT.H B 3TOM no-
nomeHHH npymHHa He .necpopMHpOBRHR, H CHnR ynpyrocTH paBHR HynlO), a
06nR.nRH CKOpOCTblO, 6y.neT no HHepH" .nBHrRTbC.H )J;anbllIe, paCT.HrHBR.H npy-
mHHY. B03HHKRIORH npH 3TOM CHna ynpyrocTH HRnpRBneHR Tenepb npOTHB
,ABHmeHH.H TeJla 1'1 TOpM03HT ero. B TO'lJ:Ke D CKOpOCTb TeJlR OKRmeTC.H paB-
HOH HynlO, a YCKopeHHe MRKCHManbHO, Teno H8 MrHOBeHHe OCTRHOBHTCH, noc-
ne qero no.n .neHCTBHeM CHJlbI ynpyrocTH Ha'iHeT .nBHraTbC.H B 06pRTHYlO CTO-
POHY, K nonomeHHIO pRBHOBeCH.H. BHOBb npOH.nH ero no HHepHH, Teno, C>KH-
6
o [ ______ X-9___
x
x
F YDPm
x
X m
mg
Puc. 13.12
377

Ma.H npymHHY H 3aMe.ZJ;nSUI ,ABHmeHHe, .ZJ;OH,AeT ,AO TOqKH A (Tal< KaK TpeHHe
OTCYTCTByeT), T.e. COBeplllHT nOJlHOe KOJle6aHHe. nOCJle 3Toro .ZJ;BHmeHHe Tena
6Y,AeT nOBTOp.HTbC.H B onHcaHHOH nOCJle.ZJ;OBaTeJlbHOCTH. lfTaK, npH'lHHaMH CBO-
60.ZJ;HbIX Kone6aHHH npymHHHoro Ma.HTHHKa .HBJI.HIOTC.H .ZJ;eHCTBHe CHJlbI ynpy-
rOCTH, B03HHKalOeH npH ,Ae(}>opMaII.HH npymHHbI, H HHepTHOCTb TeJl8.
no 3aKOHY rYKa Fx= -kx. no BTOPOMY 3aKOHY HblOToHa Fx = ma x ' Cne-
.ZJ;OBaTenbHO, ma x = -kx. OTclO.n;a
lax = -  xl HJIH   - ilHHBMH'leCKOe yp8.lmeHHe ilBH>KeHHH npy-
mHHHoro Ma.HTHHKa.
BH.ZJ;HM. qTO YCKopeHHe np.HMOnpOnopUHOHaJlbHO CMemeHHIO H npoTHBono-
nomHO eMY HanpaBJleHO. CpaBHHBa.H nOJlyqeHHOe ypaBHeHHe C ypaBHeHHeM
rapMOHH'tIeCKHX Kone6aHHH a x + ro 2 x = 0, BH.ZJ;HM, liTO npymHHHbIH Ma.HTHHK CO-
BeplIlBeT rBpMOHH'leCKHe KOJIe6BHHH C HKJIH'leCKOH 'lBCTOTOH (I) = Jf . TBK
21t
KaK T = -, TO
(t)
I T = 21! 1 - nepHOil KOJIe6BHHH npY>KHHHOrO MBHTHHKB.
no 3TOH me (}>opMyne MomHO paCC'tIHTbIBaTb H nepHO.ZJ; Kone6a.HHH BepTHKMb-
HOro npymHHHoro Ma.HTHHKa (pHC. 13.12. 6). .lIeHcTBHTenbHo. B nOJlOmeHHH paB-
HOBeCH.H 6naro,Aap.H .ZJ;eHcTBHIO CHnbI T.HmecTH npymHHa yme paCT.HHyra Ha HeKO-
TOpyro BenH'llHHY Xo, onpe.n;en.HeMyro COOTHOllleHHeM mg = kxo. npH CMeeHHH Ma-
HTHHKa H3 nonOmeHH.H paBHOBeCH.H 0 Ha x npOeKII.H.H CHnbI ynpyrocTH F' ynpx=
= -k(xo + x) H no BTOPOMY 3aKOHY HblOToHa ma x = -k(xo + x) + mg. nO.ZJ;CTaBnHH
k
clO.ZJ;a 3Ha'tIeHHe kxo = mg, nonyqHM ypaBHeHHe .n;BHmeHHSi Ma.HTHHKa ax + - x = O.
m
cOBnMaroee C ypaBHeHHeM .ZJ;BHmeHH.H ropH30HTMbHoro Ma.HTHHKa.
13.8. npeBpaeHHe 3HeprHH npH rapMOHH'IeCKHX KOJIe6aHH.HX.
3aTyxalOHe KOJIe6aHH.H
PaCCMOTpHM npOII.eCC npeBpaeHHH 3HeprHH nplf rapMOHHqeCKOM KOJle6a-
TeJlbHOM ,ABHmeHHH Ha npHMepe H,AeaJlbHOrO (F TP = 0) ropH30HTaJlbHOrO npy-
mHHHoro Ma.HTHHKa. BbIBO.n;.H TeJlO H3 nOJlOmeHH.H paBHOBeCH.H, HanpHMep cmlf-
Ma.H npymHHY Ha x =A. MbI co06aeM eMY HeKoTopbIH 3anac nOTeHII.HaJlbHOH
kA 2
3HeprHH W n = - (ropH30HTaJlbHbIH ypOBeHb, Ha KOTOpOM HRXO.ZJ;HTC.H MRHT-
o 2
HHK, BbI6HpaeM 3a HYJleBoH YPOBeHb OTCQeT8 nOTeHIJ;HaJlbHOH 3HeprHH MRHT-
HHKa B nOJle CHJlbI T .HmeCTH, Tor.ZJ;a W n = 0). n pH .ZJ;BHmeHHH TeJlR K nOJlome-
kx 2
HHIO P aBHOBeCH.H ero nOTeHII.HaJlbHa.H 3HeprH.H W = - y6bIBaeT, a KHHeTH-
n 2
mv 2
qeCKa.H W K = 2 B03paCTaeT, TaK KaK .n;e(}>opMaI1,H.H npymHHbI YMeHblllaeTC.H, a
378

t Tena ero KHHeTHqeCKaH 3HeprH.H
paBH8 HynlO, a nOTeHHanbHaH -
MaKCHManbHa. TaKHM 06pa30M, npH
KOJle6aHH.HX nepHO,AHqeCKH npOHC-
XOHT nepeXO,A nOTeHHanbHOH
3HeprHH B KHHeTHqeCKYIO H 06paT-
HO. IIonHaH MeXaHHqeCK8.H 3HeprHH
t npymHHHoro Ma.HTHHKa pRBHR CYM-
Me ero KHHeTHqeCKOH Ii nOTeHH-
anbHOH 3HeprHH W = W K + \-\1'
EcnH CMelI(eHlie MaTepHanbHOH
TOqKH, COBepmalOlI(eH rRpMoHH-
qeCKHe Kone6RBH.H, H3MeH.HeTCH C
TeqeHHeM BpeMeHH no 38KOHY
t x=Acoswt, TO npoeKHSl CKOpO-
CTH Ha OCb x V x = -w A sin wt ( CM.
 13.2). CneoBRTenbHo, KHHeTH-
qeCKa.H 3HeprH.H B nlO60H MOMeHT BpeMeHH MomeT 6bITb 3RlIaHa CPYHKuHeH
mv 2 molA 2 sin 2 wt mro 2 A 2
W K = ----:!. = = (1 - cos 2rot ), a nOTeHHanbHaSl 3HeprH.fI -
224
W _ kx 2 .'lA 2 cos 2 rot _ mw 2 A 2 (1 2 ) 2 k
CPYHKHeH - - = - + cos wt, TRK KRK W = - ,
n 2 24m
x
+A
-A
W K
mal A 2
2
o
mw 2 A 2
2
o
mol A 2
W
2
o
T
4"
ll'
4
I.
2
T
Puc. 13.13
TO k = mw 2 .
t
CKOpOCTb .z:r.BHmeHH.H TeJlR YBenWlRBR-
eTCSl. B MOMeHT npOXOmeHR.H Te-
JlOM nonOmeHR.H pRBHOBeCHSl ero no-
TeHHRnbHa.fl 3HeprHSl pRBHa HYJlIO.
2
a KRHeTHqeCKR.H W = mv rnHx
Ko 2
MRKCHMRJlbHa. IIocne npoxom.z:r.eHH.H
nonomeHHSl pRBHOBeCRSl CKOpOCTb
TeJIa YMeHbmReTCSl, R npymHHR pa-
CTSlrHBaeTC.H. CJle.z:r.oBRTeJlbHO, KHHe-
THqeCKaSl 3HeprHSl 'I en8 y6bIBRe r, 8
nOTeHHaJIbHR.H - B03paCTaeT. B
TOqKe MaKCRMRJlbHOrO OTKnOHeIlR.H
mw 2 A 2 sin 2 wt mw 2 A 2 cos 2 wt mw 2 A 2 kA 2
IIoJlHaSl 3HeprHSl W = 2 + 2 2 - 2'
113 3TH X CPOPMYJl BHlIHO. qTO W K H W n R3MeH.HIOTCSl Tome no rapMOHH-
mw 2 A 2
qeCKOMY 3aKOHY, C O.z:r.HHRKOBOH aMnJlRTYlIOH
R B npOTRBo<})a3e lIPyr
4
C ,APyroM H C qRCTOTOH 2w (pHC. 13.13), 8 nonHRSl MeXRHR'leCK8Sl 3HerHH
He H3MeHSleTC.H CO BpeMeHeM. OHa pRBHR JIH60 nOTeHHRnbHOH 3HeprHH Tena
379

B MOMeHT MRKCHMRJlbHOrO OTKJlOHeHH.H, JlH60 ero KHHeTHqeCKOH SHeprHH B
MOMeHT npOXOmlIeHH.H nOJlO>KeHH.H paBHOBeCH.H:
kA 2 mv 2 mro 2 A 2
'" = - = ---!!!... =
222
B peaJlbHbIX YCJlOBH.HX H8 Ma.HT-
HHK BcerlIa lIeHcTBYIOT CHJlbI conpo-
THBJleHH.H, nOSTOMY nOnHa.H SHeprH.H
YMeHbllIaeTC.H, H cB060,AHbIe KOJle6a.
HHSI Ma.HTHHKa C TeqeHHeM BpeMeHH
t 3aTyxalOT. T.e. HX aMnJlHTYlIa YMeHb-
llIaeTC.H )J;O HYJI.H (pHC. 13.14).
KOJle6aHH.H, aMnnHTYlIa KOTOpbIX
co BpeMeHeM YMeHbllIaeTC.H, Ha3bIBa-
Puc. 13.14 IOTC.H 3aTyxalOHMH KOJIe6aHH.HMH.
x
o
13.9. BLIHym.ZJ;eHHLIe KOJIe6aHHH. Pe30Hallc
CB0601IHbIe Kone6aHH.H BcerlIa 3aTyxalOT 3a TO HJlH HHoe BpeM.H H nOSTO-
MY He HaXOlI.HT npaKTHqeCKOrO npHMeHeHH.H. HaH60nee npOCTOH cnoc06 B03-
6ymlIeHH.H He3aTyxalOHx KOJle6aHHH COCTOHT B TOM, qTO Ha CHCTeMY lIeH-
CTBYIOT BHellIHeH nepHOlIHqeCKOH CHJlOH, B036ymlIalOeH KOJle6aHH.H, KOTopbIe
CHCTeMa caMa He COBepllIaJla 6bI.
KOJle6aHH.H, npOHCXOlI.HHe nOlI lIeHCTBHeM BHellIHeH nepHolIHqecKOH cHnbI,
Ha3bIBaIOTC.H BLIHY)KJJ;eHHLIMH KOJIe6aHH.HMH.
ECJlH BHeIlIH.H.H nepHolIHqeCKa.H CHJla ,AeHcTByeT Ha cHcTeMY, KOTOpa.H cno-
c06Ha cOBepllIaTb H cB0601IHbIe Kone6aHH.H, TO KpOMe F Buew 6YlIYT lIeHCTBO-
BaTb ee H CHJla TpeHH.H, H B03BpaalOa.H CHJla. Pa60Ta BHellIHeH CHJlbI
Hall CHCTeMOH 06eCneqHBaeT npHTOK SHeprHH K HeH H3BHe, KOTOpbIH He ,AaeT
Kone6aHH.HM 3aTyxaTb, HeCMOTp.H Ha ,AeHCTBHe CHJI TpeHH.H.
OCHOBHoe OTJIHqHe BLIHY)KJJ;eHHLIX KOJIe6aHHH 01' CB060JJ;HLIX COCTOHT B
TOM, qTO npH CB0601IHbIX KOJle6aHH.HX CHCTeMa nOJlyqaeT SHeprHIO TonbKO
OlIHH pa3, KorlIa OH8 BbIBO,AHTC.H H3
nOJlOmeHH.H paBHOBeCH.H. a npH BbI-
HymlIeHHbIX KOJle6aHH.HX SHepn:l.H no-
nOnH.HeTC.H nOCTO.HHHO 3a CqeT pa60TbI
BbIHymlIalOeH CHJlbI.
BbIHym)J;eHHbIe KOJle6aHH.H HMeJOT
P.HlI oc06eHHOCTeH. BHaqane, B npoec-
ce YCTaHOBneHH.H BbIHym)J;eHHbIX KOJle-
6aHHH, OHH HOC.HT CJlOmHbIH xapaKTep:
npOHCXOlIHT HaJlO>KeHHe CB0601IHbIX 3a-
TyxalOHx, a TaK>Ke BbIHymAeHHbIX
Kone6aHHH. TIocJle Toro KaK CB060,AHbIe
KOJle6aHH.H npeKpaT.HTC.H, OCTaHYTC.H
TonbKO BbIHY>KAeHHbIe KOJle6aHH.H. rpa-
<pHK BbIHymlIeHHbIX KOJle6aHHH H306pa-
meH Ha pHCYHKe 13.15.
x
,/
,/
-...
t
"
.....
Y CT8HOBneHHe
Kone6aHHH:
Puc. 13.15
380

1. BHeiliHee B031IeHCTBHe HaB.H3bIBaeT CHCTeMe CBOH 3aKOH KOJle6aHHU: TaK.
eCnH F BHeW H3MeH.HeTC.H no 3aKOHY cHHyca (HJIH KocHHyca) F = Fm cos 2nvt, TO
BbIHym,neHHbIe KOJle6aHH.H rapMOHHqeCKHe:
x = A cos (2nvt -4- <p).
2. 'tJacToTa BbIHymlIeHHbIX KOJle6aHHH paBHa 'laCTOTe H3MeHeHHH BbIHym-
.ZIalOeH CHJlbI.
3. Mem,ny BbIHymlIeHHbIMH KOJle6aHHHMH H KOJle6aHHHMH BHeIIIHeH CHnbI
F BHew eCTb pa3HOCTb cpa3 <po
4. AMnJIHTy,na BbIHym)J;eHHbIX KOJle6aHHH TeM 60JIbIIIe, qeM 60JIbilie aMn-
JIHTy,na BbIHym,nalOeH CHJlbI.
5. AMnJIHTy,na BbIHymlIeHHbIX Kone6aHHH 3aBHCHT OT qaCTOTbI BbIHym,na-
lOero B03,neHcTBH.H, OHa ,AOCTHraeT MaKCHMaJlbHOrO 3HaqeHH.H npH COBnae-
HHH qaCTOTbI BbIHym,AeHHbIX KOJle6aHHH v C c06CTBeHHOH qaCTOTOH (qaCTOTOH
CB060,nHbIX KOJle6aHHH) CHCTeMbI Yo. TIpH v  V o aMnJlHTy,na A  ax' TIpH
npH6J1HmeHHH qaCTOTbI BbIHym,neHHbIX KOJIe6aHHH K qaCTOTe c06CTBeHHbIX aM-
nJlHTY)J;a KOJle6aHHH B03paCTaeT.
.HBJleHHe pe3Koro B03paCTaHHH aMnJIHTy,nbI BbIHym,neHHbIX KOJle6aHHH npH
npH6J1HmeHHH v K Vo Ha3bIBaeTC.H pe30HaHCOM.
'tJaCTOTa. npH KOTOpOH aMnJlHTY,Aa BbIHym,neHHbIX KOJIe6aHHH MaKCHMRJIb-
Ha, Ha3bIBaeTC.H pe30HaHCHOH. rpacpHK 3aBHCHMOCTH A(v) Ha3bIBaeTC.H pe30HaH-
CHOU: KpHBOU: (pHC. 13.16).
6. AMnJlHTy,na BbIHym,neHHbIX KOJle6aHHH 3aBHCHT OT CHJlbI TpeHH.H. OHa
YMeHbiliaeTC.H C YBeJlHqeHHeM CHJIbI TpeHH.H, a pe30HaHCHbIe KpHBbIe CTaHO-
BSlTC.H 60Jlee nOJIOrHMH (OCTpbIH H TynoH
pe30HaHc). Ha pHCYHKe 13.16 1i306pa-
meHbI pe30HaHCHbIe KpHBbIe ,nJl.H
A
o
V o
Puc. 13.16
F TP1 < F TP2 < F TPa . ECJlH CHJla TpeHH.H
OqeHb MaJIa (F TP  0), TO aMnJIHTy,na
BbIHym,neHHbIX KOJle6aHHH, KaK nOKa3bI-
BalOT paCqeTbI, onpe,nen.HeTC.H no cpOPMY-
v
F. 2
Jle A = 2 0 2 ' r,ne Wo H <.0 - HK-
000 - 00
JlHqeCKHe qaCTOTbI CB060,nHbIX Ii BbI-
Hym,neHHbIX KOJle6aHHH CHCTeMbI, F 0 -
aMnJlHTy,na BHeIIIHeH nepHo,nHqeCKOH
CHJlbI. TIpH CTpeMJIeHHH <.0 K qaCTOTe
CBo60,nHbIX KOJIe6aHHH <.00 aMnJlHTy,na A
BbIHym,AeHHbIX KOJle6aHHH CTpeMHTC.H K
6eCKOHeqHOcTH. 3TO nOKa3aHO Ha PHCYH-
Ke 13.16 nYHKTHpOM.
381

BOIIPOCbI .D;JI.H CAMOKOHTPOJUI
1. q mo Sl6.n.Sle mCJl liau60.n.ee xapaKmeplibtM npU3liaKOM Ko.n.e6ame.n.bH.OZO
iJ6UJlCeIiUJl?
2. IIpu KaKUX YCJW6USlX 6 cucmeMe 603liUKa1Om c6060iJlibte Ko.n.e6anuJl?
3. II pu KaKUX YC.n.06u.RX pea.n.blibte Ko.n.e6aliUSl MOJICliO paCCMampU6amb KaK
zapMoliullecKue?
4. KaK06a C6Sl3b MeJICoy Ilacmomoil U nepuoiJo.'rt Ko.n.e6aliuil? KaK C6SlJalia
Z;UKJLUlleCKaJl Ilacmoma C nepuoiJoM zapMoliullecKux Ko.n.e6aliuu?
5. q eM om.n.UllalOmcSl zpaljJuKu zapMOIiUlleCKUX Ko.n.e6aliUU, y KomOpbtX pa3-
libte aMn.n.umYObt? llaCmOmbt? Halla.n.blibte ljJa3bt?
6. KaK aMn.n.umyiJa CKopocmu zapMoliullecKux Ko.n.e6aliUU 3a6ucum om aM
n.n.umyiJbt CMeU.{eliUSl? nepuoiJa? B lleM om.n.ullue zpaljJuK06 CMeU.{eliUSl U CKO-
pOC mu?
7. KaK C6.R3alibt YCKopeliue U Koop8uliama npu zapMoliullecKux Ko.n.e6aHU.R.X?
8. q eMY pa6eli CiJ6UZ ljJa3 MeJICiJy YCKopeliueM KOJLe6J/.IOU.{eucSl mOIlKU U CKO-
pocmblO?
9. Om Ilezo 3a6ucum yCKopenue meJLa. Ko.n.e6JLIOw,ezOCJl noo iJeucm6ueM
CU.n.bt ynpyzocmu? YCKopeliue MameMamUlleCKOZO MaSlmliuKa? q mo 06w.ezo 6
ypa6neliUSlX iJ6UJlCeliUSl 3mux cucmeM?
10. B KaKUX mOIlKax cKopocmb (ycKopeliue) KOJLe6.n.IOU.{ezOCJl MaJlmHUKa
taKCuMa.n.blia?
11. H 6.n.SlemcSl .n.u iJ6UJlCeliUe MameMamUlleCKOZO MaSlmliUKa K no.n.OJICeliUIO
pa6li06eCUSl pa61i0ycKopeliHbtM?
12. Om 'Ilezo 3a6ucum nepuoo c6060iJlibtX Ko.n.e6aliuu MameMamUlleCKOZO Ma-
SlmnuKa? npYJICuliliozo MaJlmliUKa?
13. II OlleMY U3MeliumcSl nepuoo Ko.n.e6aliUU MameMamUlleCKOZO MaJlmliU Ka,
eCJLU noiJliSlmb ezo lia 6blCOKY1O zopy?
14. Byoym .n.u om.n.UllambCSl nepuoiJbl Ko.n.e6aliUU meJL pa3liOU MaCCbl lia
liumSlX OOUliaK060U iJ.n.Ulibt? lia npYJICUHax oiJuliaK060U :HCeCmKocmu?
15. qmo npou30uiJem C Ilacmomou Ko.n.e6a1iUU me.n.a, ec.n.u npYJICUIiY, lia
Komopou alia nuiJ6eweno, YKopomumb?
16. ByiJem JLU npoiJo.n.JlCamb Ko.n.e6aHUSl MameMamUlleCKUU MaSlmliUK npu
c6060iJliOM naiJeliuu?
17. M OJICliO .n.u Cllumamb, Ilmo npu zapMoliullecKux Ko.n.e6aHUSlX 3a6UCU
Mocmb KUliemUlleCKOU U nomeH4UaJLb1WU 3Hepzuu om BpeMeliU JlBJLJlemCJl zap-
MOliullecKoli? q eMY paBeli nepuoiJ U3MelieHUSl 3mux 3liepzuil?
18. B KaKUX no.n.OJICeIiUSlX Ko.n.e6.n.IOU.{eucSl mOIlKU MaKCUMa.n.blia KUliemu
lleCKaSl ,JliepzuSl? nomeli4ua.n.bliaSl 3HepzuSl?
19. KaK Ja6UCSlm MaKCUMa.n.blible 3liaileliUSl KUliemUlleCKOU U nomeli4ua.n.b'
HOU 3liepZUU npu zapMoliullecKux Ko.n.e6aliUSlX om aMn.n.umyiJbt? om nepuoiJa?
20. CoxpaliSlemCJl flU npu zapMoliullecKux Ko.n.e6aliUJlX no.n.liaSl MexaliUllec-
KaR 3liepzuSl?
21. B lleM cocmoum om.n.ullue 6btliYJICiJelilibtX Ko.n.e6aliUU om c6060iJHblX 2
q eM om.n.UllalOmcSl 6blliYJICiJelilible Ko.n.e6akUSl om 3amyxalOU.{ux?
22. llpu KaKUX YCJL06UJlX 603liUKaem Jl6.n.enue pe30nanca? Om Ilezo .Ja6U
cum "ocmpoma" pe30lialicliou KpU60U? qeM onpeoe.n.SlemcSl aMn.n.umyiJa Ko.n.e-
6aHuu npu pe30lialice 2
382

23. KaKue cnoc06bl zpacpU'leCKOZO npeOCma6Jl.eliUSl Zap.MOliUlleCKUX KOJl.e6a
liUU 6bl 3liaeme? KaK npeOCma6Jl.SllOmCSl KOJl.e6aliUSl C nO.MOlqblO 6eKmopliblX
aUaZpa.MM?
x" + w 2 x = 0
x = A cos ( wt + <Po )
x = Asin(wt + <Po)
t
T=-
N
1
V=-
T
w = 21tV
v =
x
= u m cos( rot + <Po +  )
V m = <.uA
OCHOBHblE «1>OPMY JIbl
ypaBHeHHe rapMOHHqeCKHX KOJle6aHHH.
3aBHCHMOCTH CMeeHH.H x OT BpeMeHH t npH
rapMOHHqeCKHX Kone6aHH.HX, r.z:\e A -- aMnnHTY.z:\a.
U) - HKnHqeCKa.fl qaCTOTa, <P = w t ;- <Po -- cpa3a KO-
Jle6aHHH, <Po - HaqaJlbHa.fl cpa3a.
- nepHO.n; KOJle6aHHH, r.z:\e N -- qHCJlO nonHbIX KO-
Jle6aHHH 3a BpeMSI t.
-- qaCTOTa KOJle6aHHH.
-- HKJlHqeCKa.H qaCTOTa.
- 3aBHCHMOCTb npoeKHH CKOpOCTH V x OT BpeMeHH
npH rapMOHHqeCKHX Kone6aHH.fIx.
- MO.z:\YJlL aMnJlHTY.z:\Horo 3HaqeHH.H npoeKHH CKO-
pOCTH Ha OCb X.
ax = am cos (wt + <Po + 1t) - 3aBHCHMOCTb npoeKHH YCKopeHH.fI OT BpeMeHH
npH rapMOHHqeCKHX Kone6aHH.fIx.
a = w 2 A
m
a = _w 2 x
x
Fx = -kx
k
a = --x
x m
-- MO,AYJlb aMnJlHTY,AHOrO 3HaqeHH.H npoeKllHH YC-
KOpeHH.H Ha OCb x.
-- CB.H3b YCKopeHH:.H co CMeeHH:eM np" rapMOHH:-
qeCKHX KOJle6aHH.Hx.
-- npOeKH.fI KBa3HynpyroH (B03BpaalOeH) CHnbI
Ha OCb x, r)J;e k = mw 2, m -- Macca KOJle6J1IOerOC.H
TeJla.
- .z:\HHaMHqeCKOe ypaBHeHHe .z:\BHmeHH.H npymHHHo-
ro Ma.HTHHKa, r.z:\e ax - npOeKII.H.H YCKopeHH.H Ha
OCb x, x - CMeeHHe, k - meCTKOCTb npymHHbI,
m - Macca Ma.HTHHKa.
383

T = 21t *
g
a = --8
t l
T = 21t
T = 21t 
g' = g + (-ii)
mo} A 2
w= =
2
kA 2 mu 2
= _ = -1!!..
2 2
- nepHO.D, KOJle6aHHH npymHHHoro Ma.HTHHKa.
- .D,HHaMHqeCKOe ypaBHeHHe .D,BHmeHH.H MaTeMaTH-
qeCKOrO MaHTHHKa, r.D,e at - KaCaTeJlbHOe YCKope-
HHe, T.e. npOeKII.H.H YCKopeHH.H Ha OCb 't (Ha HanpaB-
JleHHe KaCaTeJlbHOH K TpaeKTopHH), l - ,nJlHHa Ma-
.HTHKa, g - MOllY JIb YCKopeHHSI cB060,nHoro na.ne-
HH.H, 8 - CMeeHHe.
- nepHO.D, KOJle6aHHH MaTeMaTHqeCKoro Ma.HTHHKa.
- nepHO,n MaJIhIX KOJle6aHHH MaTeMaTHqeCKOrO Ma-
.HTHHKa, Haxo,n.HerOCH B HeHHepu.HaJIbHOll CHCTeMe
OTCqeTa, rlIe g' - "3cpcpeKTHBHoe" YCKopeHHe Ma.HT-
HHKa B HeHHepUHaJIbHOH CHCTeMe OTCqeTa.
- "3cpcpeKTHBHoe" YCKopeHHe Ma.HTHHKa B HeHHep-
II.HaJlbHOH CHCTeMe OTCqeTa, r,ne g - YCKopeHHe
cB060,nHoro na,neHH.H B HHepII.HaJIbHOH CHCTeMe OT-
CqeTa, ii - YCKopeHHe HeHHepII.HaJlbHOll CHCTeMbI
OTCqeTa OTHOCHTeJlbHO HHepII.HaJIbHOll CHCTeMbl.
- nOJlHa.H 3HeprH.H rapMOHHqeCKHX KOJle6aHHll
TeJla. rlIe £.t) - UHKJlHqeCKa.H qaCTOTa. A - aMll-
kA 2
JlHTY.D,a KOJle6aHHH, W =-
n 2
MaKCHMaJIbHa.H llO-
2
TeHu.HaJlbHa.H 3HeprH.H TeJla, W = mUm - MaKCH-
K 2
MaJIbHa.H KHHeTHqeCKa.H 3HeprH.H TeJla.
METO.n;HqECKHE PEKOMEH.n;A:QHH
3a,naqH 0 KOJle6aTeJlbHOM ,nBHmeHHH MaTepHaJIbHOH TOqKH MomHO pa3,ne-
JlHTh Ha .D,Be rpynnbl: 3a)J;aqH, Tpe6YIOHe npHMeHeHH.H 06HX ypaBHeHHll
rapMOHHqeCKHX KOJle6aHHH, H 3a.n.aqH 0 Ma.HTHHKaX H Ma.HTHHKOBbIX qacax.
IIpH pemeHHH 3a.n.aq nepBoH rpynnhI peKOMeH.D,yeTC.H CJle.D,YIOa.H nOCJle.D,O-
BaTeJlbHOCTh .D,eHCTBHH:
1. 3anHcaTb 3a.n.aHHoe ypaBHeHHe H ypaBHeHHe rapMOHHqeCKHX KOJle6aHHll
B 06eM BH.D,e.
2. COnOCTaBHB 3TH ypaBHeHH.H, onpe.D,eJlHTb BeJlHqHHbI, XapaKTepH3YIOHe
KOJle6aHH.H (aMnJlHTY,ny, qaCTOTY, nepHO.D" cpa3Y) H .D,pyrHe BeJlHqHHbI B COOT-
BeTCTBHH C YCJlOBHeM 3a,naqH, nOCTpOHTb rpacpHK KOJle6aHHH.
384

3. B HeKOTopbIX 3aqax, Ha060poT, no H3BeCTHbIM BeJlHqHHaM, xapaKTe-
PH3YIOHM KOJle6aTeJlbHYlO CHCTeMY, CJle,AyeT 3anHcaTb ypaBHeHHe rapMOHH-
qeCKHX KOJle6aHHH.
3a.n;aqH BTOpOH rpynnbI Tpe6YIOT ,AeTaJIbHOrO aHaJlH3a cpH3HqeCKOrO .HBJle-
HH.H .Ii HCnOJlb30BaHH.H OCHOBHbIX CPOPMYJl:
T = 2nff T = 2n R .
MHorHe 3a.n;aqH 3TOH rpynnbI peillalOTC.H UYTeM MCnOJlb30BaHH.H 3aKOHa co-
XpaHeHH.H H npeBpaeHH.H 3HeprHH. .D;JI.H peIlleHH.H He06xo,AHMO:
1. C,AeJlaTb qepTem.
2. BbI6paTb HYJleBoH ypOBeHb OTCqeTa UOTeHu.HaJIbHOH 3HeprHH.
3. YCTaHOBHTb HaqaJIbHOe H KOHeqHOe nOJlOmeHH.H CHCTeMbI.
4. YCTaHoBHTb, KaKHe npeBpaeHH.H 3HeprHH npOHCXO,A.HT B YKa3aHHbIX B
3a.n;aqe npou.eccax.
5. CocTaBHTb ypaBHeHHe 3aKOHa COXpaHeHH.H MeXaHHqeCKOH 3HeprHH.
llPHMEPbI PEIIIEHH.H 3A,l(A q
1. -qepe3 KaKoH MHHHM8JIbHbIH npOMeJKyroK BpeMeHH nocne Ha'l8JIa Kone6a-
HHH CMeeHHe TOQKH H3 nOJIOJKeHHSI paBHOBeCHSI 6Y,AeT paBHO nonOBHHe aMnnH-
TY,AbI, ecnH nepHO,A KOJle6aHH.H 24 c, Ha'l8JIbHaSl Q>a3a paBHa HynlO?
1
.n; a H 0: xl = - A, T = 24 c, <Po = O.
2
HaHTH: t l .
Pew e H H e. BbI6epeM 3a HaQ8JIbHbIH MOMeHT BpeMeHH to = 0 MOMeHT npoxom-
)J,eHHSI M8TepH8nbHoH TO'lKOH nonomeHHSI p8BHoseCHSI. 3anHIIleM yp8SHeHHe r8p-
MOHHqeCKOrO Kone6aHHSI:
A . 2n
X = Sln-t.
T
ITo ycnosHlO
A . 2n 1 A
Xl = slnTt l = '2 ' HnH. COKp8a.H Ha A, nonY'lHM
21t 1 1 1t 21t 1t
. A H t
s1n T t 1 = '2 . 0 CHHYC paBeH 2' KOr,A8 ero 8prYMeHT p8seH 6' T.e. T 1=6'
T
OTKY)J,8 t l - 12 ' t l = 2 c.
2. 38nHIIlHTe ypaBHeHHe r8pMOHH'leCKHX Kone6aHHH Tena, ecnH ero M8KCHManb-
M
Hoe YCKopeHHe 1,58 2' nepHO,A Kone6aHHH 1 c, 8 CMeeHHe "3 nonomeHHSI paB-
C
HOBeCHSI B Ha'l8nbHbIH MOMeHT BpeMeHH 2 CM.
M
)l 8 H 0: am = 1.58 c 2 ' T = 1 c. xo= 0.02 M.
H8HTH: x=f(t).
Pew e H He. TIo ycnosHlO Teno cosepIIl8eT r8pMOHH'leCKOe Kone6aHHe, T.e.
X =Acos(oot + <Po).
385

OnpeAenliM H8'lanbHYlO <l>a3Y Kone6aHIiH. ITo ycnoBli1O npli t = 0 x = XO, T .e.
Xo
Xo = A COS<PO. OTKY,A8 COS <Po = A '
M8KcHM8nbHoe 3H8'leHHe YCKOpeHHSI
2 47t 2 A = a m T2 . A 4
= w A = _ 2 A. OTCIO.na 2 ' = CM.
T 47t
a8'l8nbHafl <l>83a
x4 7t
cos<Po = ---L-; cos <PO = 0,512 => <Po = - .
a T 2 3
m
am
Cne,AOBaTenbHO.
27t 27t
IT HKnliQeCK8SI 'l8CTOT8 w - -' w = - = 27t.
......... - T ' 1
TIO,ACT8BnSlSl 3Ha'leHHSI A, W H <Po, 38nHllleM yp8BHeHHe r8pMOHH'leCKHX Kone
68HHH:
x = 4 . 10- 2 COS ( 2ltt + i) M.
3. Teno MaccoH 10,0 r COBeplllaeT r8pMOHH'leCKHe Kone6aHHSI C 8MnnHTY,AOH
30,0 MM H QaCTOTOH 10.0 rq. Onpe,AenHTe nonHYIO 3HeprHIO Kone6aHHH, 8 T8Kme
MrHOBeHHbIe 3Ha'leHliSl npoeKqHH CKOpOCTIi Ii YCKopeHHSI H KOOp,AHHaTbl Tena, ecnli
B Ha'l8nbHbIH MOMeHT BpeMeHH OHO H8XO,AHTCSI B nonomeHHIi paBHOBeCIiSl.
,[(aHO: m=0.0100 Kr. A=0,0300 M. Y=10,0 rq. xo=O.
H 8 H T H: W. v x' ax. x.
P e ill e H He. TIo ycnoBHIO Kone68HliSl Ten8 SlBnSlIOTCSI rapMOHH'leCKHMH. H T8K
KaK npH to = 0 Teno HaXO,AHTCSI B nonomeHHH paBHOBeCIiSl. TO Kone68Hlifl npOHC-
XO,ASlT no 38KOHY CIiHYC8:
x =Asin(21tYt + <Po).
T8K K8K Xo=O. TO 0 = Asin<po => <Po=O.
CnellOBaTenbHO. x =Asin 21tYt.
T8K K8K CKOpOCTb - 3TO npOH3BO,AH8S1 OT CMemeHIiSl no BpeMeHH. TO
V x = x' = 27tVAcos27tVt.
YCKopeHHe ax = V = _47t 2 y2 A sin 27tVt.
mv 2 mw 2 x2
TIonH8S1 3HeprHSl Tena W = --L + -
2 2
TIO,ACT8BHM 3Ha'leHHe V x Ii x, nonY'lHM
w = m 47t 2 y2 A 2 cos 2 27tVt + m 47t 2 y2A2 sin 2 27tVt =
2 2
2 2 2
47t yAm 2 2 2 A 2
= 7t Y m.
2
TIoAcTaBHB 'lliCneHHbIe 3HaqeHHSI. nOJIY'lHM
x = 0,03 sin 20ltt (M); v x = 1,88 cos 20 Itt ( : ); a x = -115 sin 207tt ( :. ); w = 18,0 M.l(>K.
4. Ha rnMKOM ropH30HTaJIbHOM CTone ne)KIiT 1118p M8CCOH 0,35 Kr. 1lI8p npli-
KH
KpenneH K npymHHe meCTKOCTblO 5 -. TIynSl MaccoH 10 r, neTSlll't8S1 ropli30H-
M
386

M
T8nbHO co CKOpOCTblO 600 nOn8.LJ.aeT B map H 3aCTpeB8eT B HeM. Onpe,AenHTe
C
8MnnHTY,Ay Kone6aHHH map8 C nyneH. M8cCOH npymliHbI Ii conpOTliBneHHeM B03-
.u.YX8 MomHO npeHe6pe'lb.
.n8HO: ml=O,Ol Kr, m=0,35 Kr, k=5.10 3 H , v=600 M
M C
H8HTH: A.
N p
r/:/
mg
 v' _
////////
P e m e H H e. P8CCMOTpHM CHCTeMY "m8p -
nynSl" (pHC. 13.17). ST8 CHCTeM8 He3aMKHYTaSl.
T8K KaK ,AeHcTBYIOT BHemHHe CHnbl (cHn8 TSlme-
CTH Ii CHn8 ynpyrocTH npymliHbI). Ho T8K KaK
BpeMSI B38HMo.u.eHcTBH.R M8no (t::::: 0). TO
CYMM8PHbIH HMnynbc CHCTeMbI Coxp8HSleTCSI.
T.e.
TnJV = (m + TnJ) V'. OTCIO,l(a u' = m1v
m+ml
B pe3ynbTaTe m8p C nyneH nonY'lHnH KH-
HeTli'leCKYIO 3HeprHIO:
= (m + mdv,2 _ m:v 2
WI( 2 - 2 (m + m 1 ) .
Puc. 13.17
Bcne,ACTBHe 3Toro li8p 6Y,AeT cmHM8Tb npymHHY. H8'tiHeT .u.eHcTBoBaTb CHna ynpy.
rOCTH. H8npaBneHH8S1, cornaCHO 3aKoHY rYK8. npoTHBononomHo CMell.\eHHIO. H T8K K8l<
conpOTHBneHHeM B03,Ayxa npeHe6peraeM, TO map C nyneH H8'lHYT COBepm8Tb r8p-
MOHH'leCKlie Kone6aHHSI. KHHeTH'leCK8S1 3HeprHSI 6Y,AeT npeBp8ll.\aTbCSI B nOTeHllli-
8nbHYlO H H8060pOT. nOJIH8S1 MeX8HH'leCKaSl 3HeprH.R 6Y.lleT COXP8H.RTbCSI. Cne.llOB8'
TenbHO.
m:v 2
W"m = W nm =>
2(m + md
OTKY,Aa liCKOMaSl 8MnnHTY,Aa A = TnJv . A = 3 CM.
J k(m + TnJ)'
5. K ,AHH8MOMeTpy nO,ABeCHnH rpY3, BblBe-
nIl ero li3 COCTOSlHHSI paBHOBeCHSI H OTnYCTH-
nH. TIpH 3TOM B03HliKnH Kone68HHSI. 'laCTOTa
KOTOpbIX 2 r. H8 K8KOM p8CCTOflHHH OT HY-
J1eBOrO nonomeHHSI OCT8HOBHTCSI YK838Tenb .u.H-
HaMoMeTpa nocne npeKpall.\eHHSI Kone68HliH?
M8ccy npymHHbI He y'lHTbIB8Tb.
.n a H 0: v = 2 r.
H a H T Ii: l.
P e ill e H He. IIocne npeKp8ll.\eHHSI Kone68-
HHH rpY3 H8XO,AHTCSI B p8BHOBeCHH. H8 Hero
,AeHCTBYIOT ,ABe CHnbI: CHn8 TSlmeCTH mg Ii
CHn8 ynpyrocTH P (pHC. 13.18), KOTopble
ypaBHoBemHBalOT ,Apyr ,Apyr8, T.e. F= mg. TIo
38KOHY rYK8 F = kl1l. Cne.u.oBaTenbHo. kl1l = mg.
mi
mi
Puc. 13.18
kA 2
2
387

)KecTKocTb npymHHbI k H8H,AeM H3 <}>OpMynbI nepHO,A8 Kone6aHHH npymHHHorO Ma-
SlTHHK8:
t 1 2 2
T = 2n - = - => k = 4n v m.
k v
T8KHM 06p830M, HCKOMoe p8CCTOSlHHe
g
6.l = ; 6.l = 6 CM.
4n v
6. MaHTHHKOBbIe 'laCbI, TO'lHO H,AyHe B ropax (Ha SbICOTe 6200 M H8JJ. ypOB-
HeM MOpH), nepeHecnH H8 ypOBeHb MOpS!.. Ha CKonbKO npH 3TOM 6Y,AYT cneWHTb
3TH 'l8CbI 3a CYTKH?
A8HO: R=6.4.10 6 M. h=6.2.10 3 M, t=8,64'10 4 C.
H a H T H: 6.t.
P e ill e H H C. TIpH nepeMeeHHH M8S1THHKOBbIX Q8COS C ropbI Ha ypoBeHb MOpSl
H3MeHSleTCSI nepHO,A Kone6aHHH MaSlTHHK8, T8K K8I< YCKopeHHe cB060,AHOrO n8,Ae-
HHSI 3aSHCHT OT BbICOTbI. TIo ycnosHlo ,AaHHOH 3a,A8'lH 'laCbJ nepeHocSlT H8 60nee
HH3KHH yposeHb. n03ToMY YCKopeHHe g ysenH'lHSaeTcSI. nepHolI Kone68HHH MaSlT-
HHKa 'laCOB YMeHbIIJaeTCSI, H 'laCbI CT8HYT cneWHTb.
TIYCTb to - nOKa3aHHSI 'l8COB Ha BbICOTe h. t - nOKa3aHHH 'laCOB H8 ypoBHe
MOpSl. TorlIa /),t = t - to. TIoK83aHH.H 'laCOB nponopUHoHanbHbJ 'lHcny Kone6aHHH Ma-
t N
SlTHHKa - = - N ' r,Ae No H N - 't.lHcno Kone68HHii MaSlTHHKa 38 CYTKH COOTBeT-
to 0
CTBeHHO H8 BMCOTe h H Ha ypOBHe MOpSl.
N
OTCIO)J.a t = to -. H8 SblCOTe h 'l8CbI 38 CYTKH C)J.enaIOT 'lHCnO Kone6aHHH
No
No   , rile To  21t Jf nepHoll KOJle6aHHij ..aCOB, go - YCKopeHHe cB0601l-
Horo na.neHH.H Ha SbICOTe h.
AH8JlorH..Ho Ha ypoBHe MOP" N  ; ; T  21t.
v N To If
M3 3THX ypaBHeHHH - = - = -.
No T go
M M
Ho YCKopeHHe CS060,AHOrO n8)J.eHHSI g = G R 2 ; go = G (R + h)2 .
g (R + h)2
Cne,AOBaTenbHO, -
go = R 2
N R+h
Tor.n.a ---
No R
N R+h
B pe3Y nbT8Te nOK838HHSI 'l8COS Ha ypOBHe MOpSl t = to - = - to.
No R
MCKOM8H pa3HOCTb nOKa3aHHH 'laCOB
R+h h
6.t = t - to = - to - to = - to; 6.t = 70 c.
R R
388

14. 3JIEKTPOMArHMTHbIE KOJIEBAHM.H
CB060Hble 3J1eKTpOMarIlHTHble KOJIe6aHHH B KOHType. npeBpaeHHe 3Hep-
rHH B KOJIe6aTeJIbHOM KOHType. Co6CTBeHHaH '1aCTOTa KOJIe6allHH B KOHType.
3aTyxalOHe 3JIeKTpH'IeCKHe KOJIe6aHH.H.
ABTOKOJIe6aHHH. reHepaTop He3aTyxalOHx KOJle6aHHH (Ha npHMepe TpaH-
3HCTopa ).
BblHymeHHble 3JIeKTpH'IeCKHe KOJIe6aHH.H. nepeMeHHblH TOK. reHepaTop
nepeMeHHoro TOKa. reHepaTop nOCTOJlHHOrO TOKa.
.n.eHcTBYIOHe 3Ha'leHHJI HanpJlmeHH.H H CHJIbl TOKa. AKTHBHoe, eMKOCT-
Hoe H HII)];YKTHBHoe conpOTHBJIeHHJI. 3aKOH OMa JIJI 3JIeKTpH'IeCKOH enH
nepeMeHHoro TOKa. Pa3HOCTb cl>a3 Memy KOJle6aHHJlMH CHJlbI TOKa H Hanp.H-
meHH.H. Pe30HaHC B 3J1eKTpH'IeCKOH enH. MOHOCTb B enH nepeMeHHoro
TOKa.
TpaHccl>opMaTOp. Tpexcl>a3HblH 3JIeKTpH'IeCKHH TOK. npOH3BOCTBO, nepe-
.o:aqa H HCnOJIb30BaHHe 3JIeKTpH'IeCKOH 3HeprHH B npOMblWJIeHHOCTH, CeJIbC-
KOM X03J1HCTBe, Ha TpaHcnopTe. np06JIeMbi COBpeMeHHOH 3HepreTHKH H OX-
palla npHpO,lIbl.
14.1. CBo6o,l:(HbIe 3JIeKTpOMarHHTHbIe KOJIe6aHHH B KOHType.
IlpeBpaweHHe 3HeprHH B KOJIe6aTeJIbHOM KOHType
3J1eKTpOManIHTHble KOJIe6aHHJI - 3TO nepHo.n;H'leCKHe H3MeHeHHSI co Bpe-
MeHeM 3JIeKTpH'leCKHX H MarHHTHbIX BeJIHtIHH (3apSI.n;a, CHnbI TOKa, HanpSI-
meHH.H, HanpSImeHHOCTH, MarHHTHOH HH.n;YKHH Ii ,Ap.) B 3JIeKTpHtIeCKOH enH.
,Z:(JI.H B036ym,AeHHSI H no.n;.n;epmaHHSI 3JIeKTpO-
MarHHTHbIX Kone6aHHH Tpe6YIOTC.H onpe.n;eJIeHHbIe
CHCTeMbI, npocTeHllIeH H3 KOTOpbIX SIBJISIeTCSI KO-
JIe6aTeJIbHblH KOHTYP - enb, COCTOSILQa.H H3
BKnlOqeHHbIX nOCJle.n;OBaTenbHO KaTYlliKH HH,AYK-
THBHOCTblO L, KOH.n;eHcaTopa eMKOCTblO C H pe3H-
CTopa conpOTHBneHHeM R (3TO MomeT 6bITb conpo-
THBneHHe npOBo.n;a KaTYlliKH H npOBO,AOB, coe.n;H-
HSIIOIl.tHX KaTYlliKY C KOH.n;eHcaTopoM) (pHC. 14.1).
lIJI.eaJIbHblH KOHTYP TOMCOHa - Kone6aTenbHbIH
KOHTYP 6e3 aKTHBHoro conpOTHBJIeHHH (R = 0).
PaccMoTpHM CBo6o,lJ.HbIe 3J1eKTpOMarHHTHLle KOJle6aHH.H - Kone6aHH.H, npo-
HCxo.n;SIIl.'(He B H.n;eaJlbHOM KOJle6aTeJlbHOM KOHType 3a CqeT pacxo,nOBaHH.H co-
06Il.'(eHHOH 3TOMY KOHTYPY 3HeprHH, KOTOpa.H B ,nanbHeHllIeM He nOnOnH.HeTC.H.
PHCYHOK 14.2 HJlJlIOCTpHpyeT xapaKTepHbIe CTa.n;HH KOJle6aHHH B KOHType 38
O,AHH nepHO,n.
OTCtIeT BpeMeHH t MbI HatIHHaeM C MOMeHTa no.n;KnlOtIeHH.H K KOHTYPY 3a-
pSImeHHoro KOH.n;eHcaTopa. B 3TOT MOMeHT (pHC. 14.2, a) HanpSImeHHOCTb 3neK-
TpHtIeCKOrO nOJl.H E B KOH,neHcaTope (HanpaBJleHHa.H cBepxy '8HH3), a TaKme
Hanp.HmeHHe U Ha 06Kna.n;Kax KOH,AeHCaTOpa MaKCHMaJlbHbI, a TOKa B KOHType
eIl.'(e HeT, cne,nOBaTeJlbHO, OTCYTCTByeT H MarHHTHoe none. ITpH 3TOM BC.H 3Hep-
c
L
R
Puc. 14.1
389

rHfl W KOJle6aTeJlbHOrO KOHTypa 3aKJIIO'tIeHa B 3J1eKTpH'tIeCKOM nOJIe KOHJJ.eHca-
Topa, T.e.
cu2 q2
W - w - 0_0
- 30 -2- 2C '
T
B npOMemYTKe BpeMeHH OT 0 JJ.O - (pHC. 14.2, 6) KOH,AeHCaTOp, pa3pSI-
4
maSlCb, C03.naeT qepe3 KOHTYP TOK I. HLtymHH no qaCOBOH CTpeJlKe. TIpli 3TOM
cornaCHO npaBHJIY JIeHu,a B KaTYllIKe B03HHKaeT S,n:C caMOHHJJ.YKU,HH, npen.HT-
CTBYIOIl\a.H HapaCTaHHIO 3Toro TOKa. TIPH pa3p.HJJ.e KOHD;eHcaTopa YMeHbllIalOT-
C.H Hanp.HmeHHOCTb 3J1eKTpH'tIeCKOrO nOJISI E (coxpaHSI.H npe*Hee HanpaBneHHe)
H Hanp.HmeHHe U Me:>KD;Y ero 06KJlaJJ.KaMH. CJIeJJ.OBaTeJlbHO. YMeHbllIaeTC.H 3Hep-
rHSI 3JIeKTpHqeCKOrO nOJl.H B KOHJJ.eHcaTope. CHJIa TOKa I H HHJJ.YKU,H.H B Mar-
HHTHoro nOJl.H, C03JJ.aBaeMoro 3THM TOKOM, YBeJIH'tIHBaIOTC.H, T.e. B03paCTaeT
3HeprHSI MarHHTHoro nOJl.H B KaTYllIKe HH,AYKTHBHOCTH. CJleJJ.oBaTeJlbHO, 3Hep-
rH.H 3JIeKTpOCTaTHqeCKOrO nOJI.H KOHJJ.eHcaTopa npeBpaIl\aeTC.H B 3HeprHIO Mar-
HHTHoro nOJI.H KaTYllIKH.
E E
1 1
T T T T 3
-<t<- t =- - < t <-T
4 2 2 2 4
:He
E
10 1
3 3 t = T
t =-T -T < t < T
4 4
Puc. 14.2
390
E
t=O
E
1
T
O<t<-
4
6
10
T
t =-
4

T
K MOMeHTY BpeMeHH t = - (pHC. 14.2, 6) KOH.neHCaTOp nOJIHOCTblO pa3pg-
4
maeTC.H, Hanp.H>KeHHe U Mem.ny ero 06KJla,AKaMH CTaHOBHTC.H paBHbIM HynlO,
H 3neKTpHqeCKOe nOJle B HeM OTCYTCTByeT (E = 0). K 3TOMY BpeMeHH TOK 1
B KOHType H HH,AYKII.H.H B MarHHTHoro nOJlg 3Toro TOKa .nOCTHraIOT MaKCH-
MaJlbHbIX 3HaqeHHH. CJle.noBaTeJIbHO, BC.H 3HeprHg KOHTypa 3aKJlIOQeHa B 3TOT
MOMeHT B ero MarHHTHOM nOJle, T.e.
L1 2
W = W Mo =--.JL
2
1 1
B npOMemYTKe BpeMeHH OT - T .no - T npH YMeHbllleHHH TOKa B KaTYlll-
4 2
Ke B03HHKaeT 3,D;C caMOHH.nYKII.HH H HH,AYKII.HOHHbIH TOK, HanpaBneHHe KOTO-
poro, COrJI8CHO npaBHJlY JIeHII.a, COBna,AaeT C HanpaBJleHHeM y6bIBalOero pa3-
p.H):(Horo TOKa. B pe3YJIbTaTe KOH.neHcaTOp nepe3apgmaeTCg: HH>KHgg 06KJla.nKa
KOH.neHcaTopa nOJlyqaeT H36bITOqHbIH nOJlOmHTeJIbHbIH 3ap.H,n, a BepXH.H.H - OT-
pHII.aTeJIbHbIH. CJIe,noBaTeJlbHO, B KOH,neHcaTope nO.HBJI.HeTC.H 3J1eKTpHQeCKOe nOJIe,
Hanp.HmeHHOCTb KOToporo HanpaBJIeHa CHH3Y BBepx. B YKa3aHHOM HHTepBaJle
BpeMeHH CHna TOKa 1 B KOHType H HH,nYKII.Hg B MarHHTHoro nOJl.H 3Toro TOKa
y6bIBa1OT, a HanpSlmeHHOCTb E 3JIeKTpHqeCKOrO nOJl.H H HanpgmeHHe U MemJJ:Y
o6Kna,nKaMH KOH,neHcaTopa B03paCTaIOT. 3HaqHT, 3HeprH.H MarHHTHoro nOJI.H Ka-
TYIIIKH npeBpaaeTC.H B 3HeprHIO 3JIeKTpHQeCKOrO nOJlSl KOH,AeHCaTop8.
1
K MOMeHTY BpeMeHH t = - T (pHC. 14.2, a) TOK B KOHType npeKpaaeTCSI,
2
cne,nOBaTeJIbHO, HCQe3aeT MarHHTHoe nOJle (.8 = 0). Hanp.H>KeHHOCTb 3JIeKTpH-
QeCKOrO nOJlg E H Hanp.HmeHHe U KOH,neHcaTopa MaKCHMaJIbHbI. TaKHM 06-
pa30M, BC.H 3HeprH.H KOJle6aTeJIbHOrO KOHTypa 3aKJIIOqeHa Tenepb B ero 3neK-
TpHQeCKOM nOJIe, T.e.
CU 2
W=W a =.
o 2
1 3
B npOMemYTKe BpeMeHH OT - T ,AO - T (pHC. 14.2, e) KOH.neHcaTOp BHOBb
2 4
pa3p.H>KaeTC.H H C03,naeT B KOHType TOK. O,nHB.KO Tenepb nOJlOmHTeJlbHO 3apg>KeHR
HH>KHgg 06KJIKa KOH.neHcaTopa, n03ToMY HanpaBJleHHe TOKa 1 B KOHType MeHg-
eTC.H Ha npOTHBOnOJlOmHoe. MeH.HeTcg H HanpaBJIeHHe HH,nYKII.HH jj C03,naBaeMO-
ro HM MarHHTHoro nOJl.H. STOT TOK He MomeT cpa3Y ,nOCTHrHYTb MaKCHMaJlb-
Horo 3HaqeHH.H, TaK KaK B KaTYIIIKe B03HHKaeT SLIC caMOHH,nYKII.HH, npengT-
CTBYIOa.H 6bICTPOMY HapaCTaHHIO TOKa. B YKa3aHHOM npOMemYTKe BpeMeHH
CHJla TOKa I H HH,nYKU.Hg B MarHHTHoro nOJl.H 3Toro TOKa YBeJlHqHBaIOTCg, a
Hanp.HmeHHOCTb 3J1eKTpHqeCKOrO nOJlg E If Hanp.HmeHHe U Mem,ny 06KJIa,nKa-
MH KOH,AeHCaTOpa YMeHblllaIOTC.H. CJle,noBaTeJlbHO, 3J1eKTpHqeCKa.H 3HeprHg ne-
pexo,nHT B MarHHTHYIO.
391

3
K MOMeHTY BpeMeHH t = - T (pHC. 14.2, JIC) KOH,AeHCaTOp nOJlHOCTblO pa3-
4
pH.>KaeTCSI, HanpHmeHlle U Mem,AY ero 06KJIa,AKaMIi na,AaeT ,AO HynH, 3J1eKT-
pli''leCKOe nOJle IIC'-Ie3aeT (E = 6), a TOK I B KOHType H IIH,AYKU,HH f3 Mar-
HIiTHoro nOJIH B 3TOT MOMeHT MaKCIiMaJIbHM. Bc.H 3neKTpll'-leCKan 3Heprlln
KOHTypa npeBpaTllnaCb B 3HeprlllO MarHIITHoro nonn, T.e.
LI 2
W - TIT - 0
- t'YM --.
o 2
3
B npOMemYTKe BpeMeHH OT - T ,AO T (pHC. 14.2, 3) CIiJIa TOKa YMeHbllla-
4
eTcn, a B03HIIKlllaSl B KaTYlllKe 3,Z:{C CaMOIIH,AYKU,HH npennTcTByeT 3TOMY. Ha
BepxHeH nJlaCTIiHe nO.HBJIHIOTCH 1136MTO'-lHble nOJlO.>KIITeJIbHbIe 3apH,AbI, a Ha
HII.>KHeH - oTpHu,aTenbHbIe. B KOH,AeHCaTOpe nO.HBJI.HeTCH 3JIeKTpll'-leCKOe none,
Hanp.HmeHHOCTb E KOToporo HanpaBJleHa Tenepb cBepxy BHII3. B YKa3aHHOM
npOMe.>KYTKe BpeMeHII CIiJIa TOKa I B KOHType II IIH,AYKU,H.H 13 MarHHTHoro
nOJI.H y6MBaIOT, a HanpH.>KeHHOCTb E 3neKTpll'-leCKOrO non.H B KOH,AeHCaTOpe
H Hanp.H.>KeHHe U Mem,Ay ero 06KJIa,AKaMIi B03paCTaIOT. CJIe,AOBaTenbHO, Mar.
HIiTHaH 3HeprHH npeBpaI.I.:(aeTC.H B 3neKTpH'leCKYIO.
K MOMeHTY BpeMeHII t = T (pIiC. 14.2, u) TOK B KOHType npeKpamaeTC.H.
HCtIe3aeT MarHIITHoe nOJIe, a Hanp.H.>KeHHOCTb E 3JIeKTplitIeCKOrO nonH KOH-
,AeHCaTOpa II HanpH.>KeHlle U Me.>K,AY ero 06KJIa,AKaMIi MaKCIiMaJIbHbI.
3HatI1IT, BCH 3HeprllSl KOJIe6aTenbHoro KOHTypa 3aKnlO'-IeHa Tenepb B ero
3JIeKTplitIeCKOM nOJIe, T.e.
CU 2
W-W - 0
- 3 0 - 2
BTopaH nepe3ap.H,AKa B03Bp8I.I.:(aeT KOHTYP B IICXO,AHOe COCTOHHlle.
TaKIiM 06pa30M. 3aBeplllllJIOCb nOJlHOe KOJIe6aHlie. B ,AaJlbHeHllleM npOllecc
nOBTOpHeTC.H B yme onllcaHHOM nOp.HDKe.
14.2. Co6cTBeHHaH qaCTOTa KOJIe6aHHU B KOHType
B lIJJ;eaJIbHOM KOJle6aTeJlbHOM KOHType R = O. II03ToMY nonHaH 3HeprH.H '"'
OCTaeTC.H nOCTOHHHOH B Te'-leHlle Bcero BpeMeHH KOJle6aHIIH:
q2 LI2
W=WS+W M =-+-,
2C 2
r,Ae q 1-1 I - MrHOBeHHbIe 3Ha'-leHII.H 3ap.H.Aa KOH.AeHCaTOpa II ClinbI TOKa
B ROHType. TIpoIl3BO.AHaH no BpeMeHH W' (t) = 0 (TaR Kal\ W = const) CJIe.AOBa-
TenbHO,
( L + LI 2 ) ' = 2qq' + L 211' = O. OTCIO,Aa II' = -qq'.
2C 2 2C 2 LC
392

Ho I = q/. 3Ha qHT. [' = q". II03ToMY
/ " 1/" 1
qq =--qq q =--q.
LC LC
1 2
0603Haq:HM - = lOO' TOra
LC
q" = -(1)q :HJI:H q" + (1)q = 0 - ypaBHeH:He CBo6oHbIX 3JIeKTpOMarH:HTHbIX KO-
JIe6aH:H:H B HeanbHOM KOJIe6aTeJlbHOM KOHType.
CpaBH:HBa.H 3TO ypaBHeH:He C ypaBHeHHeM X" + (1)2X = 0, OnHCbIBalO:HM rap-
MOH:HqeCKHe Kone6aH:H.H, MomHO C.n;enaTb BbIBO,A: CB060,AHble 3J1eKTpOMarH:HT-
HhIe KOJle6aH:H.H B KOHType (npH R = 0) .HBJI.HIOTCH rapMOHH'IeCKHMH.
PellleH:He 3Toro ypaBHeH:H.H :HMeeT B:H,A: q = qu cosw ot, re qu - HaqaJlbHOe
(aMnJl:HTY,AHOe) 3HaqeH:He 3ap.H,Aa, C006eHHOrO KOHeHCaTOpy; Wo - co6-
CTBeHHaH l'(:HKJI:H'leCKaH qaCTOTa CB060,AHbIX 3JIeKTpOMarH:HTHbIX Kone6aH:H:H B
KOHType
"'0 =  1 = . TaK KaK T = 27t , TO
LC -fLC (1)0
T = 2n-fLC - cpOPMYJl8 TOMCOHB (nepHO.n; CB060HbIX 3JIeKTpOMarH:HTHbIX KO-
ne6aHHH B KOHType)
o
t
IIpoHcpcpepeHl'(:HpOBaB no BpeMeHH BbIpa-
>KeHHe .n;JI.H 3apHa, Ha:H,AeM, 'lTO
1 = q' = -qo"'" sin "'ot = 10 Cosr "'ot + i)-
re 1 0 = qowo - aMnJI:HTYHoe 3HaqeH:He C:HJlbJ
TOKa. CJIeOBaTeJIbHO, CHJIa TOKa I B KOJIe-
6aTeJIbHOM KOHType COBepillaeT TaKme rapMO-
H:HqeCKHe Kone6aH:H.H C TO:H me qaCTOTO:H Wo,
q.1
...,,1
Puc. 14.3
n
HO no cpa3e OH:H CMeeHbI Ha - OTHOCHTeJlb-
2
HO KOJle6aH:H:H 3ap.H,Aa (p:Hc. 14.3).
14.3. 3aTyxalO"e 3JIeKTpH'IeCKHe KOJIe6aHHH
B peaJlbHbIX KOJle6aTeJlbHbIX KOHTypax Bcera :HMeeTC.H aKT:HBHOe conpo-
T:HBJleHHe (R  0). n03TOMY qaCTb 3Hepr:HH KOHTypa npeBpaaeTC.H BO BHYTpeH-
HIOIO (npoBOH:HKH HarpeBaIOTC.H). qacTb 3Hepr:H:H TepHeTC.H BCJIeCTB:He :H3J1Y-
qeHHH. KpoMe Toro. qaCTb 3HeprHH Tep.HeTCSJ Ha nepeMarHHqHBaH:He cepeq-
HHKa H :H3MeHeH:He nOJI.Hp:H3al'(:HH H3JIeKTp:HKa. II03ToMY nOJIHaH 3Hepr:HH
KOHTypa C TeqeH:HeM BpeMeHH YMeHblllaeTC.H, B pe3YJlbTaTe YMeHblllaeTC.H H aM-
nJlHTy.n;a KOJle6aHH:H.
KaK 6bIJlO OTMeqeHO paHee ( 13.8), KOJle6aHH.H, aMnJl:HTya KOTOpbIX C
TeqeHHeM BpeMeH:H YMeHblllaeTC.H, Ha3bIBaIOTC.H 3aTyxalOHMH KOJle6aHH.HMH.
393

3aTyxaHHe HapYlliaeT nepHOHqHOCTb KOne-
6aHHH, n03TOMY 3aTyxalOHe Kone6aHH.H He
HBJlHIOTC.H nepHOHqeCKHMH H. CTporo rOBOp.H,
K HHM HenpHMeHHMO nOH.HTHe nepHO,Aa. O,AHa-
KO, eCJlH 3aTyxaHHe MaJlO, MomHO YCJlOBHO
t nonb30BaTbC.H nOH.HTHeM nepHo,na 1<aK npOMe-
mYTKa BpeMeHH Me>K,Ay ,ABYM.H 6J1HmaHllIHMH
MaKCHMYMaMH KOJle6J1IOeHC.H BenHlIHHbI.
TaKHM 06pa30M, cB060HbIe KOJle6aHH.H B
peaJlbHOM KOJle6aTenbHOM KOHType Bcera HB-
JlHIOTC.R 3aTyxalOHMH (pHC. 14.4).
ECJlH cpaBHHTb npou,eCCbI, npOHCXO.HHe B KOJle6aTeJlbHOM KOHType, Ha-
npHMep, C KOJle6aHHHMH npY>KHHHOrO MaHTHHKa, TO 06HapymHBaeTC.H aHaJlO-
rHH Mem,ny HHMH. IIpOu,eCCbI onHCbIBalOTCH O,AHHaKOBbIMH ypaBHeHH.HMH. 3TO
n03BOJl.HeT nepeHOCHTb 3aKOHOMepHOCTH, nonyqeHHble npH H3yqeHHH OHoro
BH,na KOJle6aHHM, Ha KOJle6aHH.H pyroM npHpO,AbI.
q
qo
Puc. 14.4
COOTBeTCTBHe Mey 3JIeKTpOMarHHTHblMH
H MeXaHHqeCKHMH BeJIHqHHaMH
MeXaHH'IeCKHe BenH'lHHbI 3neKTpoMarHHTHLle BenH'IHHLI
KOOp,AHH8T8 (cMeIl\eHHe) x 88pSlA q
CKOPOCTb v x = x' CHn8 TOKa l=q'
mecTKocTb npymHHb] k BenH'lRHa 06P8TH8S1 eMKOCTH 1/C
Macc8 m MHAYKTHSHOCTb L
K03<l>«pHllHeHT TpeHRSI  ConpOTHBJIeHRe R
YCKopeHRe a x = v CKOPOCTL H3MeHeHRSI CHn TOK8 /),1 I /).t
rroTeHH8nbHaSl 3HeprHSI W o SneKTpH'leCKa 3HeprHSI KOHAeHC8Top8 W 3
KHHeTH'leCK8S1 3HeprR W K SHeprRSI M8rHHTHoro non K8TYlliKH W M
CHn8 F S.ZI;C C8MORHAYKllRH 'i
14.4. ABTOKOJIe6aHHH. feHepaTop He3aTyxalOI.I.(HX KOJIe6aHHH
(Ha TpaH3HCTope)
CB060,AHbIe 3neKTpOMarHHTHbIe KOJle6aHH.H B peaJlbHOM KOJle6aTeJlbHOM KOH-
Type Bcera 3aTyxalOHe. .D:JI.H Toro 't{T06bI OHH 6blnH He3aTyxalOHMH, Hym-
HO C03)J;aTb YCTPOHCTBO, C nOMOblO KOToporo KOMneHCHpOBanHCb 6bI nOTepH
3HeprHH npH KamOM nOJlHOM Kone6aHHH B KOHType. lIIHpoKO npHMeHHMbI
TaK Ha3bIBaeMbIe aBTOKOJIe6aHHH - He3aTyxalOHe KOJle6aHH.H, nOlIep>KHBa-
eMbIe B CHCTeMe 3a CqeT nOCTOHHHoro BHelliHero HCTOlIHHKa 3HeprHH, npHlIeM
caMa CHCTeMa ynpaBJI.HeT HM, 06eCneqHBa.R cornaCOBaHHOCTb nocTynJleHH.H 3Hep-
rHH onpeeJleHHbIMH nopu,H.HMH B HY>KHbIH MOMeHT BpeMeHH.
JI106a.H aBTOKOJle6aTeJlbHa.H CHCTeMa COCTOHT H3 cneyJOHX qeTbIpex qaCTeM
(pHC. 14.5): 1) Kone6aTenbHaH CHCTeMa; 2) HCTOlIHHK 3HeprHH, 3a ClIeT KOToporo
KOMneHCHpyJOTC.H nOTepH; 3) KJlanaH - HeKoTopbIM 3J1eMeHT, perynHpYIOHH no-
cTynJleHHe 3HeprHH B KOJle6aTenbHYlO CHCTeMY onpe,AeneHHblMH nopu.H.HMH B HY>K-
HbIH MOMeHT; 4) 06paTHa.H CB.H3b - ynpaBJleHHe pa60ToH KnanaHa 3a CqeT npo-
u,eCCOB B caMoM KOJle6aTeJlbHOH CHCTeMe.
394

Puc. 14.5
reHepaTop Ha TpaH3HcTope - npHMep aB-
TOKOJle6aTeJlbHOH CHCTeMbI. H8 pHCYHKe 14.6
npHBe,AeHa ynpoeHHa.H cxeMa TaKoro re-
HepaTopa, B KOTOpOM pOJIb "KJIanaHa" HrpaeT
TpaH3HCTOp. KOJle6aTenbHbIH KOHTYP nOKJlIO-
'IeH K HCTO'lHHKY TOKa nOCJIe,AOBaTeJIbHO C
TpaH3HCTopOM. 8MHTTepHbIH nepexo TpaH3H-
CTopa qepe3 KaTYIIlKY LCD HHYKTHBHO CB.H3aH
C Kone6aTeJlbHbIM KOHTypOM. STY KaTYIIlKY
Ha3bIBalOT KaTYIIlKOH 06pa'IHOH CB.H3H.
I1pH 38MhlK8HHH enH "lIepe3 Tp8H3HCTOp
npOXOHT HMnYJlhC TOKa, KOTOpblH 3ap.HmaeT
KOHeHCaTOp C Kone6aTenbHOrO KOHTypa. B pe-
3YJIbTaTe 'IerO B KOHType B03HHKaJOT CB060,lJ;HbIe
3JIeKTpOMarHHTHbIe KOJIe6aHH.H MaJIOH aMnJIHTY-
,lIb!. TOK, npOTeKalOlIlHH no KOHTYPHOH KaTYIIlKe L, HHltYIJ,HPYeT Ha KOHuax Ka-
TYIIlKH 06paTHOH CB.H3H nepeMeHHoe Hanp.HmeHHe. I10 ,lJ;eHCTBHeM 3Toro Hanp.H-
>KeHH.H 3JIeKTpWIeCKoe nOJIe 3MHTTepHoro nepexona nepHonWIeCKH TO YCHJlHBaeT-
C.H. TO ocna6J1.HeTC.H, a TpaH3HCTOp TO OTKpbIBaeTC.H, TO 3anHpaeTC.H. B Te npOMe-
>KYTKH BpeMeHH, Korna TpaH3HCTOp OTKpbIT. qepe3 Hero npoXOltSlT HMnYJlbChI TOKa.
EcnH KaTynIKa LCD nO,AKnlO'IeHa npaBHJlbHO (nOJIO>KHTeJIbHa.H 06paTHaSl CB.H3b), TO
'IaCTOTa HMnYJlbCOB TOK8 COBna.n.aeT C qaCTOTOH KOJIe6aHHH. B03HHKlIlHX B KOHTY-
pel H HMnYJlbChI TOKa npHXO,lJ;.HT B KOHTYP B Te MOMeHTbI. Kor.n.a KOH.n.eHC8TOp
3ap.Hm8eTC.H (Kor.n.a BepXH.H.H nJlaCTHHa KOH.n.eHcaTopa 3ap.H>KeHa nOJlO>KHTeJIbHO).
Il03TOMY HMnYJIhCbI TOK8, npOXO,A.HI..QHe 'Iepe3 Tp8H3HCTOp, nO,A38p.H>K8IOT KOH,AeH-
caTOp H nOnOJIHSIIOT 3HeprHIO KOHTypa, 11 KOJIe6aHH.R B KOHType He 3aTyxalOT.
ECJlH npH nOJIO>KHTeJlbHOH 06paTHOH CBH3H Me.n.neHHO YBeJIH'IHBaTb paCCTO-
.HHHe Me)KY KaTYIIlKaMH LCD H L, TO C nOMOl..QblO OCHJInOrpacpa MomHO 06-
H8pymHTh, qTO aMnJIHTY8 aBTOKOJle6aHHH YMeHblIlaeTC.H, H aBTOKOJIe6aHH.R
MorYT npeKpaTHTbC.H. 3TO 3HaqHT, 'lITO npH CJla60H 06paTHOH CB.H3H 3HeprHSI.
nocTynalOl..QaSl B KOHTYP, MeHblIle 3HeprHH, He06paTHMO npe06pa3yeMoH BO
BHYTpeHHIOIO. TaKHM 06pa30M, 06paTHa.H CB.H3b On>KHa 6bITb TaKOH, 'IT06bI:
1) Hanp.H>KeHHe Ha 3MHTTepHOM nepexoe H3MeH.HJlOCb cHHcpa3Ho C Hanp.H>Ke-
HHeM Ha KOH,AeHCaTOpe KOHTypa - 3TO cpa30Boe ycnoBHe caMoB036ym,AeHHfl
reHepaTopa; 2) 06paTHa.H CB.H3b 06eCneqHBaJla 6bI nocTynJIeHHe B KOHTYP
CTonbKO 3HeprHH, CKOJlbKO ee He06xo,lJ;HMO JI.H KOMneHcaHH nOTepb 3HeprHH
B KOHType - 3TO 8MnJlHTY,AHOe YCJIOBHe caMoB036Y>KeHH.H.
"tlacToTa aBTOKOJle6aHHH paBHa qaCTOTe CB060HbIX KOJIe6aHHM B KOHType
H 3aBHCHT OT ero napaMeTpOB.
YMeHblIla.H L H C, MO>KHO nonyqHTb BbICOKO'IaCTOTHbIe He3aTyxalOI..QHe KO-
ne6aHH.H, HcnoJlh3yeMbIe B paHOTeXHHKe.
AMnJIHTya YCTaHOBHBIIlHXC.H aBTOKone6aHHH, KaK nOKa3bIBaeT onbIT, He
3aBHCHT OT HaqaJIbHbIX YCJlOBHH H onpeeJl.HeTC.H napaMeTpaMH aBTOKOJle6a-
TeJIbHOH CHCTeMbI - Hanp.H>KeHHeM HCTO'IHHKa, paCCTO.HHHeM Me>Ky LCD H
L, conpOTHBJleHHeM KOHTypa.
HCTO'IHHK 3HeprHH
Puc. 14.6
C
395

14.5. BLIHY:IKeHHbIe 3JIeKTpH'IeCKHe KOJIe6aHHH. nepeMeHHLIH TOK
ECJlH B u;enb KOHTypa BKJIlOqHTb BHeWHlO1O
nepeMeHHYIO SAC (pHC. 14.7), TO HanpSI>KeH-
HOCTb nOJI.H B npOBO.lI.HHKe KaTYWKH H npo-
BO.n;aX, COe,AHHSlIOHX 3neMeHTbI KOHTypa Mem-
.lI.Y C060H, 6Y.lI.eT nepHO.lI.H'leCKH H3MeHSlTbC.H, a
3HaqHT, 6Y.lI.eT nepHO.lI.HqeCKH H3MeH.HTbC.H H
CKOp::>CTb ynopSl.lI.OqeHHOro .lI.BHmeHH.H CB060.lI.-
HbIX 3ap.H.lI.OB B HRX, B pe3ynbTaTe 6Y.lI.eT ne-
pHo.n;HqeCKH H3MeH.HTbCSI CRJla TOKa B KOHType,
qTO B30BeT nepRo.n;RqeCKRe R3MeHeHRSI pa3HO-
CTR nOTeHU;RaJ10B Mem,Ay 06Kna.n;KaMH KOH.n;eH-
caTopa H 3ap.HlIa Ha KOHlIeHcaTope. T.e. B uenH B03HRKHYT BbIHymlIeHHbIe
3neKTpR'leCKHe KOJle6aHHSI.
BLIHy.u;eHHLle 3J1eKTpHqeCKHe KOJleOaHH.H - 3TO nepHO.lI.HqeCKHe H3MeHe-
HRSI CHJ1 TOKa B KOHType R .lI.pyrHx 3J1eKTpH'leCKHX BeJIH'lHH no.n; .lI.eHCTBR-
eM nepeMeHHoH SAC OT BHeWHero HCTO'lHHKa.
HaH60JIee WRpOKoe npHMeHeHHe B cOBpeMeHHoH TeXHHKe H B 6bITY Ha-
weJI CHHycOH.lI.aJIbHbIH nepeMeHHbIH TOK qaCTOTOH 50 fu;.
nepeMeHHblH TOK - 3TO TOK, nepHO,AHqeCKH H3MeH.HIOHHC.H co BpeMe-
HeM. OH npe.n;CTaBJlSleT C060H BblHym.n;eHHbIe 3J1eKTpR'leCKRe Kone6aHR.H, npo-
RCXO,ASIHe B 3J1eKTpHqeCKOH u.enH no.n; ,AeHCTBHeM nepHo.lI.R'leCKH R3MeHSlIO-
eHcSI BHewHeH SIIC. nepHO,lJ;OM nepeMeHHoro TOKa Ha3blBaeTCSI npoMemYToK
BpeMeHR, B Te'leHRe KOToporo CRJIa TOKa cOBepwaeT O.n;HO nOJIHOe K0J1e6aHRe.
qacToToH nepeMeHHoro TOKa Ha3blBaeTCSJ qHCnO K0J1e6aHRH nepeMeHHoro TOKa
3a ceKYH,AY.
QT06bI B u;enH cyecTBOBaJI CRHYCOR.lI.anbHbIH TOK, RCTOqHRK B 3TOH u;enH
.lI.OJlmeH C03.lI.aBaTb nepeMeHHoe 3JIeKTpRqeCKOe nOJIe, R3MeHSJIOlI.\eecSJ CRHYCOR-
,AanbHo. Ha npaKTRKe CRHycOR.lI.aJ1bHa.H S,n:C C03,AaeTC.H reHepaTopaMH nepeMeH-
Horo TOKa. pa60TalOlI.\HMH Ha 3JIeKTpOCTaHU;H.Hx.
c
R
L
Puc. 14.7
14.6. reHepaTop nepeMeHHoro TOKa
SJIeKTpRqeCKHe MaWHHbI, npe06pa3YIORe MeXaHHqeCKYIO 3HeprHIO B 3JIeK-
TpRqeCKYIO. Ha3bIBalOT reHepaTopaMH. B cOBpeMeHHoH 3HepreTRKe npHMeHSJ-
lOT RH.lI.YKU;HOHHbIe reHepaTopbI nepeMeHHoro TOKa, B KOTOpbIX HcnoJlb3yeTcSI
.HBJ1eHRe 3JIeKTpOMarHHTHOH HH,AYKURR. OHH n03B0J1.HIOT nonyqaTb 60nbWRe
TOKR npH .lI.OCTaTOqHO BbICOKOM Hanp.HmeHHR.
IIpocTeHweH MO.lI.eJIblO TaKoro reHepaTopa MomeT CJ1ymHTb paMKa abed
(pHC. 14.8), BpaalOa.HC.H B O.lI.HOpO.lI.HOM MarHHTHOM none BOKpyr cBoeH OCM
00', nepneH.lI.RKYJ1.HpHOH RH,AYKU;HR MarHHTHoro nOJl.H.
IIycTb B HaqaJlbHbIH MOMeHT BpeMeHH t = 0 nJ10CKOCTb paMKH nepneH.lI.R-
KYJlSlpHa BeKTOpy MarHHTHoH HH.lI.YKU;RH (pRC. 14.8, a), ee npOHH3blBaeT MaK-
CHMaJ1bHbIH MarHHTHbIH nOTOK <1>m(a = 0). IIpH paBHOMepHOM BpaeHHH paM-
KH npoHH3bIBalOI.I..!HH ee MarHHTHbIH nOTOK YMeHbWaeTC.H. BCJIe.lI.CTBRe 3Toro,
cornaCHO 3aKOHY 3JIeKTpOMarHRTHoH RH.lI.YKU;HH, B paMKe B03HHKaeT S,n:C RH-
396

a
6
6
c
z
)II
I I
t I t
at I )II
-:-
- I .....
B I )II B
$
o a
Ii
a
Puc. 14.8
,nYKD;HH . Kor,na nnOCKOCTb paMKH CTaHeT napaJlneJlbHa BeKTopy HH,nYKD;HH
MarHHTHoro nOJl.R (pHC. 14.8, 6), npoHH3bIBalO:I.QHH ee MarHHTHbIH nOT OK CTa-
ReT paBRLIM RYJIlO (ct> = 0, a = ). CKOpOCTL me H3MeReRH" MarHHTROrO no-
TOKa npH npoxom,neHHH paMKH qepe3 3TO nOJlOmeHHe HaH60JlbWaH, TaK KaK
npOBO,AHHKH paMKH ab H cd ,nBHmYTCH B 3TOT MOMeHT nepneH,nHKY JlHpHO JlH-
HH.HM HH,nYKD;HH. B03HHKalO:I.Qa.H 3)J;C HH,nYKD;HR, nponopD;HOHaJlbHaH CKOpOC-
TH H3MeHeHHH MarHHTHoro nOTOKa, 6y,neT MaKCHManbHa.H, H c03)J;aBaeMbIH elO
B 3TOM cnyqae HH,nYKD;HOHHbIH TOK HanpaBJleH (cor JlaCHO npaBHJlY npaBoH
PYKH) OT b E a H OT d K C. rIpH )J;aJlbHeHWeM nOBopOTe paMKH MarHHTHbIH
nOTOK YBeJlHqHBaeTC.H, 3)J;C, He H3MeH.H.R CBoero 3HaKa, 6y,neT YMeHbWaTbC.H
no BeJlHqHHe H B nOJlOmeHHH (pHC. 14.8, 6) <1> = <1>o,  = 0, TaK KaK npH
npoxom,neHHH qepe3 3TO nonomeHHe npOBo,nHHKH BHTKa ab H cd CKOnb3.HT
B,nonb JlHHHH nOJlH, He nepeCeKa.H HX. CJle,nOBaTenbHo, 3):(C HH,nYKD;HH, 803-
HHKalO:I.Qa.H B KOHType B 3TOM cJlyqae, g';= O. IIpH ,lJ;aJlbHeHWeM BpaIlteHHH paM-
EH MarHHTHbIH nOTOK YMeHbwaeTCH. CKOpOCTb H3MeHeHH.H MarHHTHoro nOTOKa yae-
JlHqHBaeTC.H H 3)J;C HH,AYKD;HH B03paCTaeT. COrJlaCHO PHCYHKY 14.8, Z <1> = 0,
1 - 1 - ( - ) H I g; I = I giO I, HO HanpaBJleHHe HH,nYKD;HOHHoro TOKa B BHTKax
-  0
(COrJlaCHO npaBHJlY npaBOH PYKH) COBna,AaeT C HanpaBneHHeM OT a K b H OT
C K d (npOTHBOnOJlOmHO TaKOBOMY H8 pHC. 14.8, 6). 3TO HanpaBneHHe 6y,neT
COXpaH.HTbC.H H npH ,naJlbHeHWeM ,nBHmeHHIt paMKH H g I HaqHeT y6bIBaTb,
TaK KaK MarHHTHbIH nOTOK XOT.H H YBeJlHqHBaeTC.H, HO CKOpOCTb H3MeHeHHH
ero YMeHbwaeTcH.
IIpH nOCJle,lJ;YIOIltHx o60poTax paMKH Bce 3TH .HBneHH.H 6y,nYT nOBTOp.HTbC.H
BHOBb. TaKHM 06pa30M, 3)J;C HH,nYKD;HH BO BpaalO:I.QeHC.H paMKe 3a O,nHH 060-
pOT H3MeH.HeTCH OT -O ,no +g'IO'
BbI.HCHHM, no KaKOMY 3aKOHY 6y,neT H3MeH.HTbC.H 9,[(C HH,nYKD;HH B paMKe
nJlOa,ZJ;b1O S, eCJlH paMKa BpaeTC.H C nOCTOJlHHOH yr JlOBOH CKOpOCTblO 00
(00 = const) B O,nHOpOlIHOM MarHHTHOM nOJle C HH,lJ;YKUHeH B BOKpyr OCH, pac-
nOJlOmeHHOH B nJlOCKOCTH paMKH H nepneH,nHKYJI.HpHOH BeETOpy B.
397

3a BpeM.H t paMKa nOBepHeTCSI Ha yrOJI cp == w t. H yroJl Memy HOpMaJIblO
K paMKe H BeKTOpOM MarHHTHOM HHYKHH 6yeT a = (J.) t. MarHHTHbIM nOTOK.
npOHH3bIBalOWHH paMKY. B MOMeHT BpeMeHH t paBeH <1> = BScos(J.)t.
COrJlaCHO 3aKOHY <1>apa.neSl, MrHOBeHHoe 3HaqeHHe 8,z::{C HH,AYKHH, B03HH-
KalOe:H B paMKe, paBHO nepBOM npOH3BOHOM MarHHTHoro nOTOKa no BpeMe-
HH == -<1>'. II03ToMY
g t == (J.) BS sin (J.) t = g osin (J.) t,
r.n;e g 0= (J.)BS - aMnJlHTY,AHOe 3HaqeHHe 8,z::{C, KOTopoe, KaK BH,AHO, TeM 60Jlb-
me, qeM 6bIcTpee BpaaeTC.H paMKa, qeM 60Jlbme nJlOa.n;b paMKH, TeM 60Jlb-
we MarHHTHa.H HH.n;YKH.H non.H.
<1>, g'" It
g'o
10
- - - -""1'
- I'
I \ g'
IV
\
\
TaKHM 06pa30M, npH paBHOMepHOM Bpae-
HHH paMKH B O.n;HOpO.n;HOM MarHHTHOM nOJIe B
HeM B03HHKaeT CHHycOH.n;aJIbHaH 8JJ:C C qaCTO-
TOM KOJle6aHHM. paBHOM qaCTOTe BpaweHHSI
paMKH, KOTOpa.H 6y.n;eT c03.n;aBaTb CHHycoH.n;anb-
HbIM TOK B nenH TOM me qaCTOTbI, HO cpa3a KO-
JIe6aHHSI TOKa He 06H3aTeJlbHO .n;OJl:>KHa COBn8-
.n;aTb C cpa30M H3MeHeHHSI 8JJ:C. IIo3TOMY B 06-
eM cJlyqae MrHOBeHHoe 3HaqeHHe CHJlbI TOKa
I==Iosin«(J.)t4-cp), r,Ae cp - pa3HOCTb cpa3 Mey
KOJle6aHHSlMH C1:IJIbI TaKa H 8,z::{C (pHC. 14.9). 1.JTO-
6bI HCnOJlb30BaTb nepeMeHHbIM TOK, paMKH Coe,AH-
HSiIOT C BYM.H H30nHpOBaHHbIMH KOJIbaMH, K
KOJlbaM npH>KHM8.IOT eTKH, K KOTOpbIM npHCO-
e.n;HH.HIOT KOHTaKTbI BHemHeM enH (pHC. 14.10).
MbI paCCMOTpeJlH npHHHn pa60TbI reHepaTopa nepeMeHHoro TOKa. YCTpOH-
CTBO reHepaTopa nepeMeHHoro TOKa ropa3,n0 CJlOmHee. B HaCTO.Hee BpeMSI
HMeeTCSI MHoro pa3nHqHbIX THnOB HH,AYKHOHHbIX reHepaTopOB. O,AHaKO B
Ka>K,nOM H3 HHX eCTb O,nHH H Te me OCHOBHbIe qaCTH:
1) 3J1eKTpOMarHHT (HJIH nOCTO.HHHbIH MarHHT), c03,nalOHH MarHHTHoe
nOJle. OH Ha3bIBaeTCSI HYKTOPOM;
2) 06MoTKa, B KOTOpOH HH)J;YHpyeTC.H 8.L(C. 8Ta qaCTb HOCHT Ha3BaHHe
.HKOpL;
«X>o
o
o
Puc. 14.10
398
I
I
I
I
1 /
,1-'
t
Puc. 14.9

3) COCTO.HHH H3 MeTaJlJlHtIeCKHX KOJlelJ, KOJlJIeKTOp;
4) meTKH. COe.D:HH.HIOIUHe HenO.D:BHmHbIe npOBO.D:HHKH C BpaIUaIOIUHMHC.H
npoBo,nHHKaMH.
HenO.D:BHmHa.H qaCTb reHep8Topa Ha3bIBaeTC.H CT8TOpOM. B onHC8HHOM CJlY-
'lae CTaTopOM .HBJI.HeTC.H HH.D:YKTOp. I10.D:BHmHa.H qaCTb reHepaTopa - pOTOp.
B paccMoTpeHHoM cJlyqae pOTOpOM .HBn.HeTC.H .HKOpb. ,I:(n.H nOJlyqeHH.H SL{C HH-
,AYKII.HI1 BamHO OTHOCHTeJlbHOe nepeMeIUeHHe npOBO,AHHKa H M8rHHTHoro non.H.
I103TOMY Ha npaKTHKe HH,nYKTOP ,nenalOT BpaalOHMC.H. a .HKOpb - HenO,A-
BHmHbIM. STO BbI3BaHO TeM, qTO C nOMOblO no,nBHmHbIX KOHTaKTOB npaKTH-
qeCKH HeB03MomHO OTBO,nHTb OT reHepaTopa TOKH BbICOKoro Hanp.HmeHH.H
(6onbwoH aMnJlHTY.D:bI) H3-3a CHnbHoro HCKpeHH.H B nO.D:BHmHbIX KOHTaKTax.
IfH,AYKTOPOM me .HBJlSleTC.H 3neKTpoMarHHT, AJI.H nHTaHH.H KOToporo HymeH
cpaBHHTeJlbHO cna6bIH nOCTO.HHHbIH TOK, H npH TaKOM CJla60M TOKe CKOnb3.H-
He KOHTaKTbI XopOWO pa60TalOT.
,I:(JI.H Toro qT06bI YBeJlHtIHTb aMnJlHTY.D:Y S,I:(C, He06xo.D:HMO YBenHtIHTb Mar-
HHTHbIH nOTOK qepe3 BHTKH .HKOp.H. A 3TO MomHO c,neJlaTb. CKOHII.eHTpHpOBaB
MarHHTHbIH nOTOK B TOM MeCTe, r,Ae HaXO.D:HTC.H .HKOpb. I103TOMY MarHHTHYIO
CHCTeMY reHepaTopa H3rOTaBnHBalOT B BH.D:e 3aMKHYTOH II.enH, COCTO.HeH 83
,ABYX cep,netIHHKOB. C.D:enaHHbIX H3 mene3a. 06MOTKH HH.D:YKTopa pa3MeeHbI
B na3ax o,nHoro H3 Cep.ZIetIHHKOB, a 06MOTKH .HKOp.H - B na3ax ,nPyroro. O.ZIHH
H3 cep,netIHHKOB, 06bItIHO BHYTpeHHHH, BpaaeTCSJ BMeCTe C HH,nYKTOpOM. ,I:(PY-
rOH cep,netIHHK C .HKOpeM B na3ax Heno,nBHmeH (CTaTOp). 3a30p Mem,ny cep-
,Ae'lHHK8MH CTaTopa H pOTopa ,AenalOT KaK MomHO yme. STHM .l(OCTHraeTC.H
HaH60nbwee 3Ha'leHHe nOTOKa MarHHTHOH HH,AYKII.HH.
Cep.D:etIHHKH H3rOTaBJlHBalOT H3 TOHKHX H30nHpOBaHHbIX .D:Pyr OT ,npyra
mene3HbIX nJlaCTHH, qT06bl YMeHbWHTb TOKH <1>YKO (CM.  12.4), npH npOTe-
KaHHH KOTOpbIX npOHCXO.D:HT HarpeBaHHe Cep.D:e'lHHKOB.
,I:(n.H YBeJlHtIeHH.H aMnJlHTY.ZIHOrO 3HatIeHH.H S,I:(C HHYKllHH .HKOpb H3rOTaB-
JlHBalOT H3 60JlbWOrO qHCJl8 BHTKOB. AMnnHTY,Aa B03HHKalOeH S,I:(C HH,AYK-
II.HH 6y,neT paBHa B 3TOM cnyqae cWo = N<t>ow, r,Ae N - qHCJlO BHTKOB.
Kor,na Y pOTopa HMeeTCH OAHa napa MarHHTHbIX nonlOCOB, TO qaCTOTa Bpa-
eHHSI COBna,naeT C qaCTOTOH nepeMeHHoro TOKa. B HaweH CTpaHe H CTpaHax
CHr HCnOJlb3yeTC.H npOMbIWJleHHbIH nepeMeHHbIH TOK qaCTOTOH 50 ru.. B CJlY-
tIae N nap nOJlIOCOB qaCTOTa nepeMeHHoro TOKa B N pa3 60nbwe qaCTOTbI
BpaIUeHH.H pOTOp8. O.D:HY n8py nOJlIOCOB .D:eJlalOT Y Typ60reHepaTopoB. pOTOpbI
KOTOpbIX npHBOD.HTCSJ BO BpaeHHe napoBoH Typ6HHOH, a THXOXODHbIe MHoro-
nonlOCHbIe reHepaTopbI YCTaH8BnHBaIOTC.H Ha rH.D:pOCTaHu.H.HX. TaK, Ha BpaT-
CKOH rsc - 24, Ha BonmcKoH rsc - 44, Ha YrnHtICKOH Ha Bonre -
48 nap nOJlIOCOB.
14.7. feHepaTop nOCTOHHHoro TOKa
KaK 6bIJlO nOKa3aHO B npe.D:bI,AYIUeM naparpa<t>e, SJJ:C B p8MKe, BPaIU81O-
eHC.H B MarHHTHOM nOJle. .D:Bam.D:bI MeH.HeT CBoe HanpaBneHHe 3a O.D:HH 060-
pOT paMKH. ,I:(JI.H nonyqeHH.H BO BHeWHeH II.enH nOCTOSlHHoro no HanpaBJle-
HHIO TOKa npHMeH.HIOT oc06oe MeXaHHqeCKOe nepeKnlOtIalOee YCTPOHCTBO -
KOJIJleKTOp.
399

6 I
0
, I t
'-/
I
8
0
t
Puc. 14.11
B npocTeHIIIeM cJlyqae KOnJleKTOp
npe,ACTaBJI.HeT C060H .ABa H30JlHpOBaH-
HX .Apyr OT .Apyr8 nOJlYKOJlbl(a, K
KOTOpbIM npHKpenJl.HIOTC.H KOHl(bI P8M-
o KH. IIoJlYKoJlbl(a YKpenJl.HIOTC.H H8 OCH
H BpaaIOTC.H BMeCTe C p8MKOH, K8C8-
.HCb npH 3TOM HenO.ABHmHbIX w;eTOK
(pHC. 14.11, a).
B Te MOMeHTbI, Kor)J;a TOK B paMKe
MeH.HeT HanpaBneHHe, nOJlYKOJlbl(a Me-
H.HIOT eTKH. II03ToMY BO BHeIIIHeH
1(enH TOK Bce BpeM.H 6YlIeT HMeTh O.AHO
H TO me H8npaBJleHHe, HO OH 6y.n;eT Me-
H.HTbC.H no BeJlHqUHe (nYJlbCHPYIOWHH
TOK). rpacpHK H3MeHeHH.H TOKa BO
BHeIIIHeH l(enH. COe,AHHeHHOH C reHepa-
TOpOM, H306pameH Ha pHCYHKe 14.11, 6.
IIYHKTHpHoH CHHYCOH.AOH nOKa3aH TOK
B paMKe; CnJlOIIIHOH JIHHHeH - TOK BO
BHewHeH nenH.
IIpHMeH.H.H BMeCTO O.AHOH paMKH
60nbIIIoe qHCJlO paMOK (BHTKOB), pacno-
JlOmeHHblX no.A yr nOM .Apyr K .Apyry,
MomHO 3HaqHTenbHO CrJla.n;HTb nYJlbca-
1(HIO TOKa, T.e. nOJlHTb TOK, 6J1H3KHH
K nOCTO.HHHOMY, rp8cpHK KOToporo npe.A-
CTaBJI.HeT C060H nOqTH np.HMYIO, napaJl-
JleJlbHYIO OCH BpeMeHH (pHC. 14.11, 6).
KOJlJleKTOp B 3TOM cJlyqae 6Y.AeT COCTO-
HTb H3 MHorHX H30JlHpOBaHHbIX .APyr OT
.n;pyra nnaCTHH.
14.8. AKTHBHoe conpOTHBJIeHHe
R
-U
Puc. 14.12
400
P8CCMOTpHM yqaCTOK nenH, COCTO.H-
W;HH H3 COe.AHHHTeJlbHbIX npoBo.n;HHKOB
Ii pe3HcTopa C MaJlOH HH.AYKTHBHOCTbIO
Ii 3HaqHTeJlbHbIM conpOTHBJleHHeM R
(pHC. 14.12), Hanp.HmeHHe H8 KOTOpOM
H3MeH.HeTC.H no rapMOHHqeCKOMY 3aKOHY
U = Uosinwt. MrHoBeHHoe 3HaqeHHe
CHJlbl TOKa COrJlaCHO 3aKOHY OMB
U U o sinwt .
I = - = = 10 SIn wt .
R R
OTKY.Aa I = 10 sin wt,

U o
r,l\e [0 = - - aMnJIHTYHOe 3HaqeHHe CHJIbI TOKa, U o - aMnJIHTy.n;HOe 3Ha-
R
qeHHe Hanp.HmeHH.H.
TaKHM 06pa30M, B .n;aHHOM enH MrHOBeHHbIe 3HaqeHH.H CHJlbI TOKa H Ha-
np.HmeHH.H COBna.n;aIOT no qJa3e. Ha pHcYHKe 14.13 H306pameHbI COOTBeTCTBYIO-
He rpaqJHKH MrHOBeHHbIX 3HaqeHHM CHJlbI TOKa H Hanp.HmeHH.H (a) H BeKTOp-
HbIe ,l\HarpaMMbJ (6). ConpoTHBneHHe npo-
BO,AHHKa. B KOTOpOM H3MeHeHHSI HanpSlme-
HH.H H CHJIbI TOKa COBnaIOT no qJa3e, Ha-
3bIBaeTC.H aKTHBHL conpoTHBneHHeM.
MrHoBeHHYIO MOHOCTb MomHO pac-
CqHTaTb no qJopMYJIe
P _ [ 2 R _ [ 2 R . 2 _ [ 2 R 1 - cos 2wt
- - 0 SIn wt - 0
2
KaK BH,AHO H3 3TOM qJOPMYJlbI, MO-
HOCTb, BbI,l\eJl.HeMa.R Ha aKTHBHOM con po-
THBJIeHHH, H3MeH.HeTC.H Tome no rapMOHH-
qeCKOMY 3aKOHY, HO C ,l\BOHHOH qaCTOTOH:
Ha pHCYHKe 14.14 BH,AHO, 'lTO MOHOCTb
COBepillaeT O,l\HO nOJlHOe KOJle6aHHe 3a
T
BpeM.H t = -. Cpe,D;H.H.H MOHOCTb. BbI,l\e-
2
JI.HeMa.H 3a nepHOA
< P >=< [2 > R =< ([0 sinwt)2 > R = [5R < sin 2 wt > =
2 1 - cos2wt [5 R
= [ R < >= -, TaK KaK < cos2wt >= O.
2 2
a
1. U
U o
10
T
t5
10
..
U OR
.. ----------
Puc. 14.13
P
15 R
[2R

2
o
I.
<I
1:. II T
2 "
Puc. 14.14
t
TaKHM 06pa30M. aKTHBHoe conpoTHB-
JleHHe HrpaeT ,l\BO.HKYIO pOJlb B nenH ne-
peMeHHoro TOKa:
1) OHO OrpaHH'lHBaeT cHny TOKa
[ 10 :  ):
2) Ha aKTHBHOM conpOTHBJleHHH npo-
HCXO,l\HT 6e3B03BpaTHoe npeBpaeHHe
3J1eKTpHqeCKOH 3HeprHH B ,D;PyrHe BH,l\bI
(BO BHYTpeHHIOIO).
14.9. }J;eHcTByroHe 3Ha'leHHH CHJIbJ TOKa H HarrpHmeHHH
llYCTb nepeMeHHbIH TOK npOXO,D;HT no npOBO)J;HHKY ,AaHHOrO conpOTHBJle-
HH.H H B Kam,D;YIO ceKYH,D;Y B npOBO,D;HHKe BbI,D;eJl.HeTC.H HeKOTopoe KOJlHqeCTBO
TenJlOTbI. OqeBH,D;HO, no 3TOMY npOBO,D;HHKY MomHO nponYCTHTb TaKOH nOCTO-
401

SlHHbIH TOK, qT06bI B CeKYH)J;Y BbI,l'(eJlHJlOCb TaKoe me KOJlHqeCTBO TenJlOTbI,
K8K H B cJlyqae nepeMeHHoro TOKa.
llo 3aKOHY ,I(:>KoYJI.H-JIeHa QnocT=[2Rt. B cJlyqae nepeMeHHoro TOKa STO
KOJIHqeCTBO TenJlOTbI MomHO paCCqHTaTb no «l>opMYJle Qnep= < P >t, r,Ae < P > -
Cpe,AHSI.H MOI..QHOCTb 38 nepHO,n. KaK MbI BH,AeJlH B npe,nbI,AYI..QeM n8p8rp8«l>e,
[2
< P >=< [2 > R = ..J!.R.
2
IIpHpaBH.HB QnocT = Qnep, nOJlyqHM
[2 Rt =< [2 > Rt =  Rt. OTCIOAa [2 =< [2 >=} [A = ,J< [2 > . [A =  .
BeJlHqHHY, paBHYIO KOpHIO KBa,l'(paTHOMY H3 Cpe,l'(HerO 38 nepHO,l'( 3HaqeHH.H
KB8,l'(paTa CHJlbI nepeMeHHoro TOK8, H83bIB81OT AeHCTByroHM (s4>4>eKTHBHLIM)
3Ha'leHHeM nepeMeHHoro TOKa. ,I(eHCTBYIOee 3H8qeHHe CHJlbI nepeMeHHoro
TOKa [11. paBHO CHJle TaKoro nOCTO.HHHOrO TOKa, KOTOpbIH BbI,AeJl.HeT B O,l'(HOM
H TOM me npOBO,AHHKe 38 o,nHHaKOBoe BpeM.H TO :>Ke KOJlHqeCTBO TenJlOTbI, qTO
H nepeMeHHbIH TOK.
,I(eHCTBYIOee 3H8qeHHe Hanp.H:>KeHH.H 11 3,I(C Haxo,n.HT aH8J10rHqHbIM 06P8-
30M:
U A = ,J < U 2 >. U A =  ; g A = ,J< g2 >; g A =  .
r,ne U o H g'o - aMnnHTy,nHbIe 3HaqeHH.H Hanp.HmeHH.H H 3,I(C.
Cne,nOBaTeJlhHO, KOJlHqeCTBO TenJlOTbI, KOTopoe BbI,AeJl.HeTCSI H8 pe3HCTope
conpOTHBJleHHeM R npH npOXO:>K,l'(eHHH no HeMY nepeMeHHoro TOKa, MomHO
P8CCQHTbIB8Tb no 38KOHY ,I(moYJI.H-JIeHn;a
Q = [;Rt,
r,ne [11. - ,l'(eHCTBYIOee 3H8qeHHe CHnbI nepeMeHHoro TOKa.
AMnepMeTpbJ H BOJlbTMeTpbI nepeMeHHoro TOKa rp8,l'(YHpYIOT no ,neHCTBYIO-
I..QHM 3H8qeHH.HM CHJlbI TOKa H Hanp.HmeHH.H. B TeXHHqeCKHX nacnopTax Bcex
sneKTpHqeCKHX MamHH, annapaTOB H npH6opOB nepeMeHHoro TOKa TaKme YKa-
38HbI ,l'(eHCTBYIOHe 3H8QeHHSI CHJlbI TOK8 H Hanp.H:>KeHHSl.
14.10. EMKocTHoe conpOTHBJleHHe
OnbIT nOK83bIB8eT, QTO eCJlH nOCne,l'(OBaTeJlbHO C JlaMnOqKOH coe)J;HHHTh
KOH,l'(eHCaTOp If nOKJllOqHTb HX K reHepaTOpy nOCTOSlHHoro Hanp.HmeHH.H. TO
n8MnOqKa He ropHT. 3TO nOH.HTHO, T8K K8K 06KJl8,nKH KOH,l'(eHCaTOpa pa31Ie-
neHbI ,l'(HSJleKTpHKoM, H n;enb OK83bIBaeTCSI pa30MKHYTOH. IIpH nO,l'(KJllOqeHHH
KOH,l'(eHCaTOpa K HCTOQHHKY nOCTOSlHHoro TOKa B03HHKaeT KpaTKOBpeMeHHbIH
HMnYJlbC TOK8, KOTOpbIH 3ap.H,l'(HT KOH,l'(eHCaTOp ,l'(0 Hanp.HmeHH.H RCTOqHHKa, 8
38TeM TOK npeKpaTHTC.H. Ho ecnH STY n;enb nO,l'(KJllOqHTb K HCTOqHHKY nepe-
MeHHoro Hanp.HmeHH.H, TO naMnOqKa ropHT. llepeMeHHbIH TOK npe)J;CTaBJI.HeT
C060H BbIHym,l'(eHHble sneKTpOMarHHTHble KOJle6aHH.H, npoHcxO,l'(SIHe no,l'( ,l'(eH-
CTBHeM nepeMeHHoro SJleKTpOMarHHTHoro nOJl.H reHepaTopa. IIpH BKJllOqeHHIf
KOH,neHcaTopa B enb nepeMeHHoro TOK8 npon;ecc ero 3ap.H,l'(KH ,nJlHTC.H QeT-
402

BepTb nepHO)J;a. IIocJle )J;OCTHmeHH.H aMnJlHTYHOrO 3HaqeHH.H Hanp.HmeHHe
Me*y 06KJlaKaMH KOHeHCaTOpa YMeHbmaeTC.H, H KOH,AeHCaTOp B TeqeHHe
qeTBepTH nepHOlIa pR3p.HmaeTC.H. B CJlelIYIOI.I1YIO qeTBepTb nepHO,na KOH,neHCR-
TOp CHOBR 3ap.HmReTC.H, HO 3HaK 3ap.H,na Ha ero 06KJlR)1KaX H3MeH.HeTC.H HR
npOTHBOnOJlOmHbIH H T.. '1lepe3 H3J1eKTpHK, pR3eJl.HIOHH 06KJlR,nKH KOH-
eHCRTopa, KaK H B n;enH nOCTO.HHHOrO TOKa, 3J1eKTpHqeCKHe 3ap.HbI He npo-
XO.HT. Ho no npOBORM, COe,AHH.HlOlQHM 06KJla,AKH KOHeHcaTopa C HCTOqHH-
KOM HRnp.HmeHH.H, TeqeT nepeMeHHbIH TOK pa3p.HKH H 3ap.HKH KOHeHcaTO-
pa. ll03TOMY JlaMnOqKa, BKJllOqeHHa.H nOCJleOBaTeJlbHO C KOH)J;eHcaTopOM, 6y-
eT ropeTb HenpepbIBHO. ECJIH Tenepb KOHeHcaTOp OTcoeHHHTb, TO JlaMnO'1-
KR ropHT .Hpqe. CJlelIOBaTeJlbHO, KOHlIeHCRTOp OKa3bIBReT nepeMeHHOMY TOKY
conpOTHBJleHHe, KOTopoe Ha3bIBaeTC.H eMKOCTHLIM CO-
npOTHBJleHHeM.
PaCCMOTpHM Jenb (pHC. 14.15), COCTO.HYIO H3
KOH,AeHCaTOpa Ii nOBO.HInHX npOBO,AOB. conpOTHB-
JleHHe KOTOpbIX npeHe6pemHTeJlbHO MaJlO, H reHepa-
Topa nepeMeHHoro Hanp.H*eHH.H.
IIycTb Hanp.HmeHHe Ha KOHeHcaTope H3MeH.HeT-
C.H no 3aKOHY U = U osinw t. KaK H3BeCTHO, 3ap.H Ha
06KJlaKaX KOHeHcaTopa MomHO onpeeJIHTb no
4>OPMY Jle q = CU = CU osin w t. CHJlR TOKR 1 = q'. CJle-
OBaTenbHO,
1
T C
l u
r
Puc. 14.15
I = -roCU o cosrot = roCu o Sin( rot + )- OTClO1\R 1= 10 Sin( rot + )-
re 10=wCU o - aMnJlHTYHoe 3HaqeHHe CHJlbI TOKa:
U U o 1
10 = _ 1 0 ; 10 = X ' r)J;e Xc =-.
c wC
roC
BbIpa3HB aMnJlHTYHbIe 3HaqeHH.H qepe3 eHcTBYlOlQHe 10 =.J21 H U 0 = .J2u,
U
nOJlYQHM 1 = X ' T.e. lIeHcTBYIOI.I1ee 3HaqeHHe CHJlbI TOKa CBSl3RHO C ,neHcTBY-
c
IOInHM 3HaqeHHeM Hanp.HmeHH.H HR KOHlIeHcaTope TOqHO TaK me, KaK CB.H3R-
HhI COrJlaCHO 3RKOHY OMR CHJlR TOKR H Hanp.HmeHHe Ha yqRcTKe uenH nOCTO-
.HHHOrO TOKa. 9TO n03BOJI.HeT paCCMaTpHBaTb BeJlHqHHY Xc KaK conpOTHBJle-
HHe KOHeHcaTopa nepeMeHHOMY TOKY:
1
Xc = - - eMKOCTHoe conpOTHBJleHHe.
wC
B CM eHHHn;eH eMKOCTHoro conpOTHBJIeHH.H .HBJI.HeTC.H OM (OM).
KaK BHHO H3 nOJlyqeHHOH BbIme 4>OPMYJlbI, eCJlH B enH BKJllOqeHO TOJlb-
KO eMKOCTHoe conpOTHBJleHHe, KOJle6aHH.H CHJIbI TOKa B 3TOH n;enH onepema-
1t
lOT no 4>a3e KOJle6aHH.H Hanp.HmeHH.H Ha KOHlIeHcaTope Ha qTO H306pRme-
2
HO Ha rpa4>HKe H Ha BeKTopHoH )J;HarpaMMe (pHC. 14.16).
403

I. U. P
o
t

U oc
10
.
Puc. 14.16
MrHOBeHHaSl MOHOCTb
P = IU = 10 Sin[ rot +  )U O sin rot = 10U o sin rot cos rot = loo sin 2M,
T.e. MOHOCTb nepHo.n;HqeCKH H:3MeHHeTCH C .n;BORHOR qaCTOTOR, a cpe.n;Hee 3Ha-
'leHHe MOHOCTH - 3a nepHO.n; < P > = 0, TaK KaK < sin2wt > = O. IIepBYIO H:
TpeTblO qeTBepTH nepHo.n;a, Kor.n;a KOH.n;eHcaTOp 3ap.HmaeTC.H, OH nOJlyqaeT 3Hep-
rHIO OT reHepaTopa, a BTOpylO H qeTBepTYIO qeTBepTH nepHO,Aa, Kor.n;a KOH-
,AeHCaTOp pa3pH»<:aeTCSI. OH OT,AaeT 3HeprHIO reHep8TOpy.
TaKHM 06pa30M, TaK >Ke, KaK aKTHBHoe conpOTHBJIeHHe, eMKOCTHoe con po-
THBJleHHe OrpaHHqHBaeT CHJlY TOKa B enH, HO B OTJlH'lHe OT aKTHBHoro co-
npOTHBJleHHH Ha eMKOCTHOM conpOTHBJleHHH 3neKTpHqeCKa.H 3HeprH.H He npe-
Bp8aeTCH Heo6paTHMO B ,ApyrHe BH.n;bl 3HeprHH.
14.11. IIH.D;yKTHBHOe conpOTHBJIeHHe
ECJIH nOCJIe.n;OBaTeJlbHO C JlaMnOqKOR BKnlO'lHTb KaTYlliKY HH.n;YKTHBHOCTH
H nOOqepe.n;HO no.n;KJllOqaTb K reHepaTOpy nOCTOHHHoro Hanp.HmeHHH H nepe-
MeHHoro Hanp.H>KeHH.H (npH 3TOM nOCTOHHHoe HanpH>KeHHe paBHO .n;eRcTBYIO-
eMY 3HaqeHHIO nepeMeHHoro Hanp.HmeHHH), TO B nepBOM CJIyqae naMnOqKa
ropHT .Hpqe, qeM BO BTOpOM. CJle.n;OBaTeJlbHO, KaTYlliKa HH.n;YKTHBHOCTH OKa-
3BaeT .n;ononHHTenbHoe conpOTHBJleHHe nepeMeHHOMY TOKY, KOTopoe Ha3Ba-
eTCH HH)J;YKTHBHbIM conpOTHBJleHHeM.
PaCCMOTpHM yqacToK enH, B KOTOpbIR BKJllOqe-
Ha H.n;eaJIbHaH KaTYlliKa C 3JIeKTpHqeCKHM conpOTHB-
JleHHeM npOBo.n;a, paBHM HYJlIO, H reHepaTOp nepe-
MeHHoro HanpH>KeHHSl (pHC. 14.17). IIycTb nepeMeH-
_ U Hoe HanpHmeHHe H3MeHHeTCH no 3aKOHY U = Uosinwt.
Tor.n;a no yqaCTKY 6y.n;eT npoxo.n;HTb nepeMeHHbIR
TOK, B pe3YJIbTaTe 'lero B03HHKHeT S,n:C caMOHH-
.n;YKHH, MrHOBeHHoe 3HaqeHHe KOTOpOH
g. = _ LI' = - L dI .
Sl dt
L
Puc. 14.17
404

ITo 3aKOHY OMa .D;JI.H HeO,AHOpO.D;HOrO yqacTKa enH HMeeM
V o sin rot - L d1 = O. OTCIO,Aa d1 = V o sinwtdt.
dt L
ITpoHHTerpHpOBaB J d1 =  J sin rotdt, nonyqHM
- V o V o . ( n ) ( n )
1 = - cos rot = - SIn rot - - = locos rot - - ,
roL roL 2 2
v U
r.D;e 10 =  =. BenH'tIHHa XL=wL - HH.D;YKTHBHoe conpOTHBJIeHHe.
roL XL
KaK BH.D;HO H3 q>OPMY nbI, KOJle6aHHJI CHJlLI TOKa Ha Y'laCTKe enH, COCTO-
JIeH TOJILKO H3 HYKTHBHoro conpOTHBJleHHJI, OTCTaeT no 4>a3e OT KOJle6a-
1t
HHH HanpneHHJI Ha -, qTO H nOKa3aHO Ha rpaq>HKe H Ha BeKTopHOH .D;Ha-
2
rpaMMe (pHC. 14.18).
MrHOBeHHa.H MOlqHOCTb
P = IV = -IoUocoswtsinwt = - IoU o sin2wt.
2
Cpe.D;Hee 3Ha'tleHHe MOlqHOCTH, nOTpe6n.HeMOH H.D;eanbHOR KaTYIIlKoH OT :ac-
TO'tlHHKa nepeMeHHoro TOKa 3a nepHO,A. < P> = o.
KaK H B npe,AbI.D;yeM cJlY'lae. npH HanH'tIHH B enH TonbKO HH.D;YKTHBHO-
ro conpOTHBneHH.H qaCTOTa H3MeHeHHJI MOlqHOCTH B.D;BOHHe 60nbllIe 'tIaCTOTbI
H3MeHeHH.H CHJIbI TOKa, H Ha HH,AYKTHBHOM conpOTHBJIellHH 3neKTpH'leCK8JI
3HeprHJI He npeBpalqaeTC.H He06paTHMO B .D;PyrHe BH.D;bI. IT!')li B03paCTaHHH
TOKa 3HeprHJI OT reHepaTopa nonallaeT Ha KaTYllIKY, KaTYIIlKa pa60Tf.eT B pe-
mHMe nOTpe6HTen.H, 3anaCaJI 3HeprHIO MarHHTHoro nOJlJI, a npl1 ero y6bIBa-
HHH - B pemHMe reHepaTopa, i303BpalqaJI reHepaTOpy 3anaceHHYIO 3HeprHIO.
IT03TOMY eMKOCTHoe H HH.D;YKTHBHoe conpOTHBneHHJI B OTngqHe OT aKTHBHoro
Ha3bIBalOT peaKTHBHLIMH.
1. V. P
Po
V o
10 f "
10
..
0 t
Puc. 14.18
405

14.12. 3aKoH OMa JJ;JIJI Y'laCTKa enH nepeMeHHoro TOKa.
Pa3HUcTb f}>a3 MeJJ;Y KOJIe6aHHJlMH CHJIbI TOKa H HanpJleHHJI
PaCCMOTpHM yqaCTOK enH, coep-
maIQHH pe3HCTOp conpOTHBJleHHeM R,
KaTYillKY HH,AYKTHBHOCTblO L H KOH,AeH-
caTOp eMKOCTblO C, BKJllOqeHHbIe nocJle-
OBaTeJlbHO (pHC. 14.19). Ha yqaCTOK
nOaeTC.H nepeMeHHoe Hanp.HmeHHe
V = Vosinoot, B pe3YJlbTaTe B03HHKaeT
nepeMeHHbIH TOK. TaK KaK SJleKTpOMar-
HHTHbIe B3aHMoeHCTBH.H pacnpOCTpaH.H-
IOTC.H co CKOpOCTblO CBeTa. TO BO BCeX
nOCJle,lIOBa TeJlbHO BKJllOqeHHbIX SJleMeH-
TaX enH H3MeHeHH.H CHJlM TOKa npOHCXO.HT npaKTHQeCKH OHOBpeMeHHO. Ko-
JIe6aHH.H me MrHOBeHHbIX 3HaqeHHH Hanp.HmeHHH Ha KamOM H3 SJleMeHTOB
V R, V L' V c He COBnaaIOT no ct>a3e C KOJle6aHH.HMH CHJlbI TOKa. B JlI060H
MOMeHT BpeMeHH CYMMa MrHOBeHHbIX 3HaqeHHH HanpSlmeHHH Ha nOCJIeOBa-
TeJIbHO BKJllOqeHHbIX SJIeMeHTax enH paBHa MrHOBeHHOMY 3HaqeHHIO npHJIO-
meHHoro HanpH>KeHH.H:
U=UR+UL+U C '
YqHTbIBaH, qTO Ha aKTHBHOM conpOTHBJIeHHH KOJIe6aHH.H CHJIbI TOKa co-
BnaaIOT. Ha eMKOCTHOM onepemalOT. Ha HHYKTHBHOM OTCTalOT OT Kone6a-
HHR Hanp.HmeHHH ( 14.8-14.11). nOCJlelIHee paBeHCTBO MomHO 3anHcaTb
V o cosrot = V OR cosrot + Voc cos[ rot -i)+ VOL cos[ rot + i)-
AMnJlHTYY KOJle6aHHH Hanp.H>KeHH£
B enH MomHO BbIpa3HTb Qepe3 aMnnH-
TYHbIe 3HaqeHH.H Hanp.HmeHH.H Ha OT-
eJIbHbIX ee SJleMeHTax, BOCnOJlb30BaB-
lllHCb MeTOOM BeKTopHbIX )J;HarpaMM.
Ha pHCYHKe 14. 20 npeCTaBJleHa BeK-
TOpHa.H ,nHarpaMMa aMnJlHTY Hanp.Hme-
U OR HHR Ha pe3HCTope (U OR ) , KOH,neHCRTope
(Vod H KaTYlliKe (VOL)' AMnJlHTY,AHOe
3HaqeHHe CHJlbI TOKa 10 B enH COBna-
aeT no ct>a3e C aMnJlHTy,nHbIM 3HaQe-
HHeM Hanp.HmeHH.H V OR Ha pe3HCTope R.
AMnJIHTya (U 0) npHJlOmeHHOro Hanp.HmeHH.H OJlmHa 6bITb paBHa reOMeTpH-
qeCKOH CYMMe STHX aMnJlHTY. YrOJI q> onpeeJl.HeT pa3HOCTb ct>a3 Memy Ha-
np.H>.,eHHeM Ii CHJlOR TOKa 1 = 10 cos(oot -<1». 113 pHcYHKa 14.20 BHHO, QTO
R
C
L
V
Puc. 14.19
VOL
V oc
10
Puc. 14.20
t VOL -V oc
gcp = =
V OR
1
ooL--
ooC
R
406
V o

IIo TeOpeMe nmparopa
2 2 2 2 2 2 ( 1 ]
U o = (U OL - UOc) + U OR => U o = (loR) + 10 roL - roC 2. OTKY.IIB
1 - U o 0
- 3aKOH Ma .n;JlH yqacTKa u,enH nepeM£ HHoro TOKa.
o -  R2 + ( wL - :C r
BeJlH'lHHB Z =  R 2 + ( roL -  r HB3LIBBeTCH nOJlHLlM conpOTHBJle"'HeM
nenH, (WL - :c ) - peaKTHBHLlM conpOTHBJleH1feM.
14.13. Pe30llallc B 3J1eKTpHlIeCKOii u.enH
1 a u
W pea =  . aTY qaCTOTY Ha3bIBalOT pe30HaHCHOH.
...;LC
IIPH 3TOM YCJIOBHH Z = R, T.e. nOJlHOe conpOTHB-
(J) neHHe u,enH c-raHOBHTC.H HaHMeHbIllHM. aMnJIHTy.n;a
CHJIbI TOKa npH .n;aHHOM Hanp.H>KeHHH npHHHMaeT
HaH60nbIllee 3HaeHHe (pHC. 14.21). B 3TOM CJIY-
qae aMnJIHTy.n;a Hanp.H>KeHHH Ha aKTHBHOM conpOTHBJleHHH paBHa aMnJlHTy.n;e
BHeIllHero Hanp.H>KeHHH, npHJIOmeHHOro K yqaCTKY u,£:nH (U OR = U o ), a HanpSl-
meHHSI Ha KaTYIllKe HH,AYKTHBHOCTH H KOH.n;eHcaTope o.n;HHaKOBbI no Mo.n;ynlO
H npOTHBOnOJlOmHbI no <}>a3e;
( U OL ) = 1 0w pe aL = 10L  = 10 FL ;
pea -V LC VC
(U oc ) = 10 = 10 JLC = 10 fL ;
pe3 W pe3 C C Vc
( u ) = ( u ) - 1 FL _ U o FL
OL pea OC pea - °VC - R VC'
10
o
(J)pe3
Puc. 14.21
1. Pe30HaHC HanpHjKeHHH. ECJlH B u,enli l CM.
pliC. 14.19) no.n;06paTb TaK eMKOCTb C KOH.n;eHcaTO-
1
pa H HH.n;YKTIiBHOCTb KaTYIllKIi, qT06bI wL = --, TO
wC
pa3HOCTb <}>a3 Me>K.n;y KOJle6aHHHMH CIiJIbI TOKa Ii
Hanp.HmeHHH cp = O. T.e. H3MeHeHIUJ TOKa H HanpH-
>KeHH.H npOHCXO,A.HT cHHcpa3HO. 113 3TOrO paBeHCTBa
TIPH 3TOM ( u OL ) = (U OC ) ,I{ OHI{ MOrYT 3HaqHTeJlbHO npeBbIIllaTb U O.
pea pea
STO HBneHHe Ha3bIBaeTC.H pe30HaHCOM HanpHjKeHHH. Pe30HaHC HcnoJlb3yeTcH
B pa.n;HO- H 3J1eKTpOTeXHHKe .n;JlH YCHJleHH.H KOJle6aHHH Hanp.HmeHH.H KaKOH-JIH60
407

onpe,neJleHHOH qaCTOTbI. Ero Ha,nO HTbIBaTb npH paCqeTe H30Jl.HI.J;HH 3J1eKT-
pHqeCKHX nHHHH, co,nepmaHX KaTYlliKH HH,nYKTHBHOCTH H KOHl1eHCaTOpa.
HHaqe MOmeT Ha6J1lOl1aTbC.H HX np060H.
2. Pe30HaHc TOlWB. PaccMoTpHM yqacToK I.J;enH nepeMeHHoro TOKa, co,nep-
maHH napaJIJleJlbHO BKJIlO'leHHbIe KOH,AeHCaTOp eMKOCTbIO C H KaTYlliKY HH-
,nYKTHBHOCTblO L (pHC. 14.22). TIYCTb aKTHBHoe conpOTHBJleHHe MaJlO, HM
MomHO npeHe6peqb. ECJlH npHJlOmeHHOe Hanp.H-
meHHe R3MeH.HeTC.H no 3aKOHY V = V osi:ru.o t, TO
B BeTBH lC2 npoxo,nHT TOK
II
C
I
1 1 2
2
II = 1 01 Sin( rot + ). rile 1 01 = o .
roC
L
B BeTBH lL2 npoxo,nHT TOK
1 2 = 1 02 sin( rot -  1 , r,ne 1 02 = V o .
l 2 roL
TaKHM 06pa30M. pa3HOCTb «}>a3 TOKOB B BeTB.HX lC2 H lL2 paBHa 1t,
T.e. KOJIe6aHH.H TOKOB B BeTB.HX npOTHBOnOJlOmHbI no «}>a3e. AMnJlHTy,na TOKa
V
Puc. 14.22
BO BHemHeii (Hepa3BeTBneHHoii) enH loe = 1 1 01 - 1 02 1 = U o [ roC - :L 1- EcnH
1
ro = ro pe3 = .J'LC ' TO IOl = 1 02 H 1 0 = O. AMnJlHTy,na CHJlbI TOKa 10 OKa3aJlaCb paB-
HOH HYJlIO nOTOMY, qTO aKTHBHbIM conpOTKBJleHHeM yqaCTKa npeHe6peraJIH.
ECJlH yqeCTb conpOTHBneHHe R, TO pa3HOCTb «}>a3 He 6y,neT paBH8 7t H Io 0,
HO 10 npHMeT HaHMeHbllIee B03MomHoe 3HaqeHHe, a aMnJlHTy,nbI CHn TOKOB
1 01 H 1 02 MorYT 3HaqHTeJlbHO npeBbImaTb aMnJIHTYAY CHJlbI TOKa 10'
.HBJleHHe pe3Koro YMeHbllIeHH.H aMnJlHTy,nbl CHJIbI TOKa BO BHellIHeH I.J;enH,
nHTalOeH napaJlJleJIbHO BKJllOqeHHbIe KOH,neHcaTop 11 KaTYlliKY HH,nYKTHBHOC-
TH, npH npH6J1HmeHHH qaCTOTbI ro npHJlOmeHHOro Hanp.HmeHH.H K pe30HaHC-
HOH qaCTOTe 00 pea Ha3bIBaeTC.H pe30HaHCOM TOKOB (napaJIJIeJILHbIM pe30HaHCOM).
3TO .HBJleHHe HCnOJIb3yeTC.H B pe30HaHCHbIX YCHJlHTeJl.HX, n03BOJl.HIOHX BbI-
,neJl.HTb O,nHO onpe,neJleHHOe KOJle6aHHe H3 CHrHaJla CJlOmHOH «}>OPMbI, a TaKme
B HH,nYKI.J;HOHHbIX neqax, r,ne napaJIJleJlbHO HarpeBaTeJlbHOH KaTYlliKe BKJlIO-
qaeTC.H KOH,neHcaTOp, eMKOCTb KOToporo no,n6HpaeTC.H TaK, qT06bI npH qaCTO-
Te reHepaTopa nOJIyqHJlC.H pe30HaHC TOKOB. B pe3ynbTaTe CHJla TOKa qepe3 Ka-
TYlliKY 6y,neT ropa3,n0 60JIbllIe. qeM CHJla TOKa B no,nBOlI.HIUHX npOBOl1ax.
14.14. MOHOCTh B enH nepeMeHHoro TOKa
ECJlH Hanp.HmeHHe B I.J;en11 nepeMeHHoro TOKa V = Vosinoot, TO MrHOBeHHoe
3HaqeHHe MOHOCTH nepeMeHHoro TOKa paBHO npOH3Be,AeHHIO MrHOBeHHbIX 3Ha-
qeHHH CHJlbI TOKa H Hanp.HmeHH.H P = IV= 1 osin(oo t - q» V osinoo t. BbInOJlHHB
npeo6pa30BaHHe TpHroHOMeTpHqeCKOH «}>YHKI.J;HH sin (rot - q», nOJlyqHM
P = 1 0 U 0 (sin 2 rot cos <p - sin rot cos rot sin <p) .
408

TIpaKTHqeCKHH HHTepeC npeCTaBn.HeT cpeHee no BpeMeHH 3HaqeHHe MOlQ-
. 2 1
HOCTH. ECJlH yqecTb, qTO < SIn rot >= - H < sinrotcosrot >= 0, TO cpe.n;Hee 3Ha-
2
'leHHe MOlQHOCTH 3a nepHO,A 6Y,AeT paBHO
IV
< P >= cos<p => < P > = IIJ.UIJ. COS<p,
2
r,Ae IJJ. H UJJ. - eHcTBYlOlQHe 3HaqeHH.H CHnbI TOKa H HanpSlmeHH.H. MHomH-
Tenb cos <p Ha3bIBalOT K03<1>4>HU;HeHToM MOIl.\HOCTH.
TIoCJle.n;H.H.H Q>opMYJla nOKa3bIBaeT. qTO M0ll1HOCTb. BbI,nen.HeMaSi Ha yqaCT-
Ke enH. COCTO.Hll1eM H3 nocne.n;OBaTenbHO coe,nHHeHHbIX pe3HCTopa. KaTYlliKH
H KOHeHcaTopa nepeMeHHoro TOKa. 3aBHCHT He TOJlbKO OT CHnbI TOKa H Ha-
npSlmeHHSI, HO H OT C,lJ;BHra Q>a3 Memn;y HHMH. 113 pHcYHKa 14.20 BH.llHO. 'lTO
U lR R
cos<p =  =  =-.
U o loZ Z
TIO,ACTaBHM B Q>opMyny
lUR
HOCTH < P >= . Ho no
Z
Cpe,AHeH MOlQHOCTH 3HaqeHHe K03<l>Q>HHeHTa MOlQ-
U
3aKOHY OMa 1 JJ. = ..2.. Cne.n;oBaTenbHo,
Z
2 15
< P > = 1 R = - R.
JJ. 2
TaKHM 06pa30M, cpe.n;HSlSI MOlQHOCTb B enH nepeMeHHoro TOKa paBHa cpe.n;-
HeH MOlQHOCTH, BbIenSleMoH Ha aKTHBHOM conpOTHBneHHH R (CM.  14.9).
8TO aKTHBHaSl MOlQHOCTb. T.e. cpe,nHSISI M0ll1HOCTb He06paTHMbix npeBpalQeHHH
B enH nepeMeHHoro TOKa.
YMeHbIlIeHHe cos<p npHBO,AHT He TonbKO K HenonHOMY Hcnonb30BaHHIO
MOHOCTH reHepaTopa 3J1eKTpOCTaHHH (qaCTb 3HeprHH B03BpalQaeTCSI B ceTb -
3TO peaKTHBH8.H MOlQHOCTb). HO R K YBenHqeHHIO TennOBbIX nOTepb B enH
< P > 2 < p2 > R
(TaK KaK 1 = U COS<P' TO t:J> = 1 R = U 2 cos 2 <p), TIoBbIIlIeHHe K03Q>Q>HHeHTa
MOHOCTH 3J1eKTpHqeCKHX YCTaHOBOK SlBJI.HeTCSI BamHoH HapolIHOX03.HHCTBeH-
HOH 3a.n;aqeii.
»
:.:
IS:
=
IJ'
o
Eo<
()
Is:

:=JIC
Puc. 14.23
14.15. TpaHccJ.>opMaTop

a::

i
1:1

TpaHc4>opMaTop - YCTPOHCTBO, cnymalQee
.n;n.H npe06pa30BaHHSI CHnbI H HanpSlmeHH.H ne-
peMeHHoro TOKa npH HeH3MeHHOH qaCTOTe.
OH 6bIJI H306peTeH TI.H..H6J10qKOBbIM B
1878 r., 8 TeXHHqeCKHH TpaHCQ>opMaTOp Bnep-
Bble C03,AaJl 11.q>.YcarHH B 1882 r.
Pa60Ta TpaHcQ>opM8Topa OCHOBaHa Ha .HBne-
HHH 3J1eKTpOMarHHTHoH HHYKHH. rIpocTeH-
IlIHH TpaHcQ>opMaTop (pHC. 14.23) npe,ACTaBJlSleT
C060H Be H30JlHpOBaHHbIe .n;pyr OT pyra KaTYIlI-
KH (06MOTKH), HaMOTaHHbIe H8 06lQHH 3aMKHY-
TbIH CepeqHHK. TIo O.n;HOH H3 06MOTOK (nepBHq-
409

HOH) npOnYCKaeTC.H npe06pa3yeMbIH nepeMeHHbIH TOK, a BTopH'tJHa.H 06MOTKa
COe,ll;HH.HeTC.H C nOTr,e6HTeJleM. TOK B nepBH'tJHOH 06MOTKe C03,l1;8eT B cep,ll;e'tJ-
HHKe nepeMeHHbIH MarHHTHbIH nOTOK, KOTOpbIH B036ym,AaeT SllC caMOHH.n;YK-
HH g = - <I> B Kam,AOM BHTKe nepBH'lHOH KaTYWKH (<1> - H3MeHeHHe Mar-
t
HHTHoro nOTOKa 'tJepe3 OlIHH BHTOK 3a BpeM.H t). STOT me MarHHTHbIH no-
TOK npOHH3bIBaeT BHTKH BTOPHqHOH KaTYWKH H C03aeT B Kam.n;OM ee BHTKe
S,lJ;C HHYKHH (g. ECJIH nepBH'tJHa.H 06MOTKa HMeeT N 1 BHTKOB, a BTopH'tJHa.H
N 2 BHTKOB, TO B 06MOTKax HHYHpYIOTC.H (6e3 yqeTa nOTepb Ha pacceHBa-
HHe MarHHTHoro nOTOKR) COOTBeTCTBeHHO 3JIeKTpoBHmylltHe CHJIbI
(ff N
a HX OTHoweHHe --.l = -1, T.e. B03HHKalOHe
(ff2 N 2
(caMOHHlIYKHH) nponopHOHaJlbHbI qHCJlY BHT-
<I> <1>
lif 1 = -N 1 -, ?i'2 = -N 2 -,
t t
B KaTYW:KaX S,lJ;C HH,AYKHH
KOB B HHX:
k = N 1 ; (g 1 = k.
N 2 g' 2
OTHoweHHe 'tJHCJIa BHTKOB B nepBH'tJHOH 06MOTKe K 'tJHCJIY BHTKOB BO BTO-
pH'tJHOH Ha3bIBalOT K034>4>HHeHTOM TpaHc4>opMaJ..{HH k.
ECJlH N 2 > N 1 (k < 1), TO TpRHc<popMa TOp Ha3bIBaeTC.H nOBLlwaJ{)HM, a
eCJIH N 2 < N 1 (k > 1) - nOHHaJ{)JJ..(HM.
K03cp<PHuHeHT TpaHc<popMaUHH onpeeJl.HeTC.H 06bI'lHO npH XOJlOCTOM XOlle
TpaHc<popMRTopa, T .e. npH pa30MKHYTOH uenH BTOPHqHOH 06MOTKH. B 3TOM
cnY'tJae B nepBHqHOH 06MoTKe npOXO,ll;HT TaK Ha3bIBaeMbIH TOK XOJlOCTOrO xoa,
eHcTBYlOlltee 3HaqeHHe KOToporo Ix. Ha OCHOBaHHH 3aKOHa OMa JI.H 3aMK-
HYTOH enH eHcTBYlOlltHe 3HaqeHH.H Hanp.HmeHH.H U 1, npHJlOmeHHOro K nep-
BHqHOH 06MOTKe, 3,lJ;C caMOHHYKHH (g 1 H CHJla TOKa Ix B nepBHqHOH 06-
MOTKe CB.H3aHbI Memy C060H COOTHoweHHeM U 1 - (g 1 = I xRl' r e R 1 - aKTHB-
Hoe conpOTHBJleHHe nepBHqHOH 06MOTKH. 3HRK MHHYC 06YCJlOBJleH TeM, 'tJTO
S,lJ; C (g 1 cor JlaCHO npaBHJlY JIeHa npOTHBo<pa3Ha U l' TpaHc<popMaTOp npoeK-
THpyeTC.H TaK, 'tJT06bI B OTCYTCTBHe H8rpY3KH nOTpe6JI.HeMbIH 113 ceTH TOK 6bIn
He3Ha'lHTeJlbHbIM. STO .n;OCTHraeTC.H BbI60pOM Manoro aKTHBHoro conpOTHBJIe-
HH.H R 1 11 OCTaTOqHO 60JlbWOrO HH.n;YKTHBHoro conpOTHBJleHH.H roL. ,lJ;JI.H YBe-
nH'IeHH.H HHYKTHBHOCTH KaTYWKH B Hee BBO.n;.HT CTaJlbHOH cep.n;eqHHK H Ha-
MaTbIBalOT .n;OCTaTO'tJHO 60JlbWOe 'lHCJIO BHTKOB N l' Tor.n;a CHna TOKa
I - 1 6yeT MaJla H BeJlHqHHOH I xRl MomHO npeHe6peqb. Cneo-
x  R 2 + ( roL )2
BaTeJlbHO. U 1 =(ffl'
IJ,enb BTOPHqHOH 06MOTKH npH XOJlOCTOM xO)J;e pa30MKHYTa, BCJlelICTBHe
'tJero B HeH TOKa HeT, H Hanp.HmeHHe HR 3amHMax BTOPHqHOH 06MOTKH paB-
HO HH.n;YHpOBaHHOH B Hei! S,lJ;C HH,AYKHH U 2 =fC 2 . Il03TOMY K03<P<PHHeHT
TpaHc<popMaHH MomHO HaHTH, H3MepHB Hanp.HmeHH.H Ha KOHax KaTywel\
npH XOJlOCTOM pemHMe
k = g'1 = U 1 .
g' 2 U 2
410

IIpH BKJllOqeHHH BO BTOPHqHYIO I(enb KRKOH-JIH60 HarpY3KH {pa60'lHH XO,A
TpaHcpOpMaTOpa) B HeH HaqHHaeT npOXO.n;HTb TOI< HarpY3KH 1 2 (nepeMeHHbIH,
TaKOH me qaCTOTbI). TOK 1 2 C03.n;aeT B cep.n;eqHHKe MarHHTHbIH nOTOK, Ha-
npaBneHHbIH no npaBHJIY JIeHI(a HaBCTpeqy nOTOKY nepBHqHOH 06MOTKH. B
pe3YJlbTaTe CYMMapHbIH nOTOK MarHHTHOH HH,AYKI:(HH B nepBHqHOH KaTYIilKe
YMeHbmaeTC.H, YMeHblilaeTC.H H 3.lI.C Z'1' a Cne,AOBaTeJlbHO, CHJIa TOKa
I - U 1 - Z'1
1 -

YBenHqHBaeTC.H. YBeJIH'leHHe TOKa B nepBHqHOH I(enH npHBonHT
I< YBeJIHqeHHIO MarHHTHoro nOTOKa, 3.lI.C HH.n;YKI(HH R CHnbI TOKa BO BTOPHq-
HOH I(enH. Ho YBeJlHqeHHe TOKa BO BTOPHqHOH I(enH conpOBo.m,AaeTCH YBeJIH'
qeHHeM TOKa caMOHH)J;YKI(HH H. CJle.n;OBaTeJlbHO, YMeHbIlleHReM MarHHTHoro no-
TOKa (KOTOpbIH TOJIbKO ({TO B03paCTaJI). B KOHne KOHI(OB npH nOCTOHHHOH H8-
rpY3Ke YCTaHaBJlHBaIOTC.H onpe.n;eneHHble MarHHTHbIH nOTOK, 3.n;C HH.n;YKI(HH BO
BTOPHqHOH I(enH, TOK lIB nepBHqHOH I(enH (11 > 1 x), TaKHM 06pa30M. YBeJlH-
qeHHe TOKa 1 2 BO BTOPHqHOH I:(enH 8BTOMaTl'.qeCKH npHBo.n;HT K YBeJlH'leHMIO
TOKa II B nepBHqHOH I(enH, T.e. TpaHccpopMaTOp aBTOM8THqeCKH perYJIHpyeT
nOTpe6J1eHHe 3HeprHH B 3aBMCHMOCTH OT HarpY3KH BO BTOpH'lHOR nenH.
IIpH pa60qeM xo.n;e TpaHccpopMaTopa npOHCXO,AHT HenpepblBHaH nepe,Aaqa
3HeprHH H3 nepBHqHOH I(enH BO BTOPHqHYIO. MOI.l.(HOCTb. nOTpe6n.HeMaH B nep-
BHt.IHOH nenH PI =I 1 U 1 COS<Pl1 8 BbI.n;enHeMaH Ha HarpY3Ke P 2 = I 2 U 2 cos<P2' Ko-
3cpcpHUHeHT nOJIe3HOrO JIeHCTBH.H TpaHccpopM8Topa
P2 I 2 U 2 cos<P2
T}=-=
11 II U 1 COS <PI
He BCH 3HeprH.H. nOTpe6J1.HeMajJ OT reHep8Topa, nepe.n;aeTC.H nOTpe6RTenlO. IIpH
pa60Te TpaHccpopMaTopa HMelOTCH nOTepH Ha HarpeBaHHe 06MOTOK TpaHccpopMa-
Topa, Ha pacceHBaHHe MarHHTHoro nOTOK8 B npOCTpaHCTBO, Ha BHxpeBble TOKH
<I>YKO (CM.  12.4) B cep.n;eqHHKe Hero nepeMarHHqHBaHHe. .n;JlH YMeHbIlleHHH
3THX nOTepb npHHHMalOTCH CJIe,AYIOI.I.(He MepbI: 1) o6MoTKa HH3Koro HanpH.meHHH
.n;eJlaeTC.H 60JlbIllerO CeqeHH.H, TaK KaK no HeH npOXO,AHT TOK 60JlblileH CHJIbl;
2) CepAeqHHK AeJIalOT 3aMKHYTbIM, qTO YMeHbIll8eT p8cceHBaHHe MarHHTHoro no-
TOKa; 3) cep.n;eqHHK .n;enalOT H3 H30JIHpOBaHHbiX nnaCTHH )lJI.H YMeHblLeHH.H TO-
KOB <l>YKO HAP. BnarOA8pH 3THM Mep8M KII.lI. COBpeMeHHblX Tp8HCcpOpM8TOPOB
,AOCTHraeT == 95-99 %, C,ABHrH cpa3 Mem,AY KOJIe6aHHSlMH CHJIbI TOKa H HanpH-
meHH.H 6nH3KH K HYJIIO (cos<p == 1).
ECJlH HHorAa MomHO npeHe6peqb nOTepHMH B TpaHccpopMaTope, T.e. CqH-
o II U 2
TaTb T}=100 Yo, TO 1l =P 2 IIUl =I 2 U 2 -=-, 3TO 3HaqHT, YBeJlH'lHBa.H
1 2 U 1
C nOMoblO TpaHccpopMaTopa HanpHmeHHe, MbI BO CTOJIbKO me pa3 YMeHbIlla-
eM CHJIY TOKa H Ha060pOT.
IIpH pa60qeM pe.mHMe TpaHccpopMaTopa Hanp.H.meHHSI H8 ero 06MOTKax yme
He 6Y,AYT paBHbI 3.lI.C. Y'lHTbIBajJ nOTepH TOJlbKO Ha aKTHBHbIX conpOTHBJle-
HHHX, HanpHmeHH.H U 1 H U 2 MO.mHO paCCqHTaTb, RCXO.n;.H H3 3aKOHa OMa AJISI
3aMKHYTOH I(enH.
IIo 3aKOHY OMa An.H 3aMKHYTOH nepBHqHOH I(enH TpaHccpopMaTopa anre6-
paHqeCKa.H CYMMa nO,ABOAHMOro K TpaHccpopMaTOpy HanpHmeHH.H U 1 Ii B03HH-
411

KalOI.,QeH 8)J:C CaMOHH,nYKlJ,HH  I paBHa na,neHHIO Hanp.HmeHH.H B lJ,enH (Ha aK-
THBHOM COnpOTHBneHHH nepBHt.J:HOH 06MOTKH R I ):
U I - I = IIRI' OTCIO,Aa U I = g:'1 + IIRi'
)J:JI.H nO,nKJlIOt.J:eHHOH HarpY3KH R pOJlb HCTOqHHKa TOKa BbInOnH.Herr BTO-
pHqHa.H 06MOTKa, 3)J:C Is KOTOpOH 2' OHa OJl:>KHa 6bITb paBHa na,neHHIO Ha-
npH:>KeHH.H BO BTOpHt.J:HOH u,enH (Ha HarpY3Ke R H Ha aKTHBHOM conpOTHBne-
HHH R 2 BTOpHt.J:HOH 06MOTKH):
c%'2 = 1 2 ( + R), HO 1 2 R = U 2 .
CJle,noBaTeJlbHO, 2 = U 2 + I 2 R 2 . OTKY,Aa U 2 = 2 - I 2 R 2 .
14.16. Tpex<}>a3l1LIH 3JIeKTpH'IeCKHH TOK
PaCCMOTpeHHbIH HaMH BbIIIIe nepeMeHHbIH TOK C03,naBaJlC.H O,nHOH 8)J:C. B03-
HHKllIeH B reHepaTope. TaKOH TOK Ha3bIBalOT 0.nHocpa3HbIM. O,nHaKo HaH60nee
BbIro,nHbIe YCJlOBH.H nepe,nat.J:H 3neKTpHqeCKOH 3HeprHH no npOBo,naM o6ecne-
qHBaeT TaK Ha3bIBaeMaSi Tpexcpa3HaH CHCTeMa nepeMeHHoro TOKa, C03,naHHa.H
B KOHu,e XIX B. PYCCKHM 3J1eKTpOTeXHHKOM M.O. )J:onHBo-,lJ:06poBonbcKHM.
Tpexa3HLIM TOKOM Ha3bIBaeTC.H CHCTeMa Tpex 0,nHocpa3HbIx TOKOB, C03,na-
BaeMbIX TpeMH 8)J:C, HMelOI.,QHMH o,nHHaKOBbIe aMnnHTy,nbI H 'laCTOTY, HO C,nBH-
HYTbIMH o,nHa OTHOCHTenbHO ,npyroH no cpa3e Ha 120 0 .
Kam.nylO OT.nenbHYIO u,enb Tpexcpa3HoH CHCTeMbI COKpaI.,QeHHO Ha3bIBalOT
a30H.
B reHepaTopax Tpexcpa3Horo TOKa TPH CTaTopHbIe 06MOTKH (HKOpH) COB Me-
l.,QalOT B O.nHOM Kopnyce CTaTopa, CMeI.,QaH HX ,npyr OTHOCHTenbHO ,npyra Ha
1
- oKpymHocTH. BHYTPIi reHepaTopa BpaI.,QaeTC.H 06I.,QHH ,nnH Bcex .HKOpeH HH-
3
,AYKTOP (pHC. 14.24). HaqaJla 06MOTOK Ha pHCYHKe 14.24 0603HaqeHbI 6YKBa-
MH A. B, C, KOHIJ;bI - COOTBeTCTBeHHO X, Y, Z. IIpH BpaI.,QeHHH pOTopa BO
Bcex 06MoTKax .HKOpeH HH,nylJ,HpyeTC.H nepeMeHHa.H 8)J:C O.nHHX H Te:x me t.J:a-
B
"
/il
I
120 0 120 0
A

y

z
x
Puc. 14.24
412

1
3HaqeHHfl CBHHYTbI Ha - nepHoa
3
pyr OTHOCHTeJlbHO pyra (pHC. 14.25).
Kam,Aa.H 06MOTKa Tpex«!>a3Horo re-
HepaTopa .HBn.HeTC.H caMOCTOflTeJlbHbIM
reHepaTopOM TOKa H K KamoH H3
HHX MomHO nocoeHHHTb nOTpe6H-
TeJlb. ,lJ,JIfl HX nOCOe)IHHeHH.H nOTpe-
60BaJlOCb 6bI IlleCTb npOBOOB C e-
JIblO YMeHbIlleHHSJ qHCJIa npOBO,AOB
06MOTKH reHepaTopa coeHHHIOT Me:>K-
Y C060H. TaKoe COe,AHHeHHe MO:>KeT
6bITb BbInOJlHeHO lIBYM.H cnoco6aMH:
3Be3oH H TpeyrOJIbHHKOM.
Coe,ZJ;HHeHHe 3Be3,ZJ;oii. TIpH coeH-
HeHHH 3Be3oH (pHC. 14.26) TPH KOH-
a CTaTopHbIX 06MOTOK X, Y, Z co-
eHHflIOT B OHY TOqKY 0, Ha3bIBae-
MYIO HyneBoH TO''lKOH reHepaTopa. K
HaqaJIaM A, B, C Tex me 06MOTOK
nOnKnlOqalOTC.H npOBona nHHHH. Ha3bI-
BaeMbIe JIHHeHHbIMH npOBO)IaMH. TIpo-
BOJJ;, HYlQHH OT HYJleBOH TOqKH 0
Ha3bIBalOT HY JleBbIM npOBOOM.
TaKa.H qeTblpexnpOBOHaH CHCTeMa
HMeeT .ABa pa3HbIX Hanp.HmeHH.H: 4>a3-
Hoe HanpHeHHe U 41 - HanpHmeHHe
Memy HaqaJIOM H KOHOM KamoH 06-
MOTKH cTaTopa, H JlHHeiiHoe HanpHe-
HUe Un - HanpHmeHHe Mem,AY HaqaJIaMH ,ABYX «!>a3HbIX 06MOTOK. AMnJlHTY,AbI
(COOTBeTCTBeHHO ,AeHCTBYlOlQHe 3Ha'leHH.H) «!>a3HbIX H nHHeHHbIX HanpHmeHHH CB.H-
3aHbI Mem.AY c060ii COOTHOIlleHHeM (3TO MomHO .AOKa3aTb. nOJlb3Y.HCb MeTOOM
BeKTopHblX ,AHarpaMM ,AJI.H CJIOmeHH.H rapMOHH'leCKHX KOJle6aHHH): Un = J3u <1>'
PaCqeT nOKa3bIBaeT, qTO npH O,AHHaKOBOH HarpY3Ke TOK B HyneBoM npo-
BO,Ae paBeH HYJlIO (10) H MO:>KHO nepeHTH K 60Jlee 3KOHOMHOH TpeXnpOBO,AHOH
CHCTeMe. Ho eCJIH HarpY3Ka HeOHHaKOBa.H, TOK 10 OTJIHqeH OT HYJI.H, H HY-
neBOH npOBO 06.H3aTenbHO OJI:>KeH 6bITb, HO ero MO:>KHO CeJIaTb 60JIee TOH-
KHM, qeM JIHHeHHbIe npOBoa, TaK KaK 1 0 ropa30 cJIa6ee 11' 1 2 , 13'
B HaCTO.HlQee BpeM.H BO Bce mHnbIe oMa BBOHTC.H Tpex«!>a3HbIH TOK C
nHHeHHblM Hanp.HmeHHeM 220 B HJlH 380 B. 3J1eKTpHqeCKHe naMnOqKH no,A-
KJIlOqalOT Memy JlHHeHHbIM npOBOOM H HYJleBbIM Ha HanpHmeHHe COOTBeT-
I
T
3"
T
3"
T
Puc. 14.25
A
I.
B
1 2
b&
&
b
Ia
Puc. 14.26
CTBeHHO
CTOTbI H aMnJIHTY,AbI, HO npH 3TOM MO-
MeHTbI npoxOmeHH.H 3THX 3,lJ,C qepe3
MaKCHMaJIbHble (HJIH qepe3 HYJleBbIe)
t
u    127 B II U    220 B.
413

A
Z
Puc. 14.27
a
B
y
CoeAHHeHHe TpeyroJlLHHKOM (pHC. 14.27). IIpH
TaKOM coeHHeHHH KOHe nepBo 06MOTKH X COe,AH-
H.HIOT C HaqaJlOM B BTOpOH. KOHe Y BTOpOH co-
eHH.HIOT C HaqaJlOM C TpeTbeH 06MOTKH. KOHen Z
TpeTbeH - C HaqaJlOM A nepBoH.
IIpH COe,AHHeHHH TpeyronbHHKOM nHHeHHoe H
«l>a3Hoe HanpHmeHH.H O,AHHaKOBbI:
U lI = Uet>.
He3aBHcHMo OT Toro, KaK coeHHeHbI 06MOTKH
reHepaTopa, nOTpe6HTeJIH MorYT 6bITb coeHHeHbI
KaK 3Be3OH, TaR Ii TpeyrOJIbHHKOM (pHC. 14.28).
1 11
6
B
2 3
1 11
C C 2
A A
III
Puc. 14.28
IIpH COe,AHHeHHH nOTpe6HTeJIe TpeyronbHHKoM (pHC. 14.28. a) Kam,AbIH
nOTpe6HTeJIb HaXOHTC.H no,A JlHHeHbIM Hanp.HmeHHeM, a npH coeHHeHHH 3Be3-
OH (pHC. 14.28, 6) - no,lI HanpHmeHHeM, B J3 pa3 MeHbllIHM. CJlelIOBaTeJIb-
HO, npH nepeKJllOqeHHH nOTpe6HTeJIeH co 3Be3bI Ha TpeyronbHHK Hanp.Hme-
HHe Ha KamoH HarpY3Ke, a cneOBaTeJlbHO. H TOK. nOBbIllialOTCH B .J3 pa3.
lIpH HeO,lIHHaKOBbIX nOTpe6HJI.HX 3TO COOTHOllIeHHe He BbIllOJIH.HeTC.H. Ho 3KOHOMH-
e He HCqepnbIBaeTCH npeHMeCTBO Tpexcpa3Horo TOKa. C ero nOMOblO yaeTcSI
nonyqHTb BpalOeecSI MarHHTHoe nOJIe, KOTopoe HCnOJIb3yeTcSI JIH pa60TbI HaH-
60nee pacnpOCTpaHeHHbIX npOMbIllIJleHHbIX aCHHxpOHHbIX 3JIeKTpo.nBHraTeJleH.
14.17. TIepe.na'la 3JIeKTp03HeprHH
SJleKTpHqeCKYIO 3HeprHIO npOH3Bo.nHT Ha 3JIeKTpOCTaHHHX. Ee Ha,AO ne-
peaTb nOTpe6HTeJl.HM, qaCTO HaXOlI.HII.1HMCH OqeHb .naJIeKO OT CTaHHH. ,IJ:JIH
3Toro Memy CTaHII.HeH H nOTpe6HTeJleM CTpO.HT JIHHHH 3JIeKTponepe.naq (JISlI).
lIpH nepeaqe 3J1eKTp03HeprHH HeH36emHbI nOTepH, CBH3aHHbIe C HarpeBa-
HHeM npOBOOB. B03HHKaeT npo6neMa YMeHbllIeHH.H 3THX nOTepb.
lIo 3aKOHY ,IJ:moYJIH-JIeHa KOJlH'leCTBO TenJlOTbI, BbIenHeMoe B npOBO,A-
HHKe npH npoxomeHHH TOKa, paBHO
Q = I;Rt.
QTo6bI YMeHbllIHTb nOTepH B JISlI, Heo6xo.l1HMO, KaK BHlIHO H3 3aKOHa,
YMeHbllIHTb conpOTHBJleHHe R HJlH CHJIY TOKa 11J. B He. ConpOTHBJIeHHe npo-
414

l
BO,nOB R = P - 6y,neT MeHbllIe, eCnH YMeHbllIHTb l, HO ,nJlHHa onpe,neJl.HeTC.H
S
paCCTO.HHHeM, Ha KOTOpOe Ha,nO nepe,naBaTb 3neKTp03HeprHIO MomHo YBenH-
'tlHTb nJlOID;a,nb nOnepeqHOrO CeqeHHH S. Ho 3TO Be,neT K nepepacxo,ny ,nOpO-
rOCTO.HerO BeTHOrO MeTanna Ii B03HHKHOBeHHIO Tpy,nHOCTeH npH 3aKpenJle-
HHH npOBO,AOB Ha cToJl6ax. TI03TOMY TaKOH cnoco6 CHHmeHH.H nOTepb npaK-
TH'tleCKH HeB03MOmeH.
J];pyroH nYTb 3aKJllOqaeTC.H B YMeHbllIeHHH CHJlbI TOKa B JlHHHH nepe,naqH.
Ho npH ,naHHOH MOID;HOCTH P = I JJ.U JJ.cos CP YMeHbllIeHHe CHJlbI TOKa B03MomHO
nHllIb npH YBeJlHqeHHIf Hanp.HmeHH.H.
TaKHM 06pa30M, npH nepe,naqe 3neKTp03HeprHH Ha 60JlbllIHe paCCTO.HHH.H He-
06XO,nHMO nonb30BaTbC.H BbICOKHM Hanp.HmeHHeM. TaK, aneKTp03HeprH.H Bonm-
CKOH rsc nepe,naeTC.H B MOCK BY npH Hanp.HmeHHH 500 KB, OT Ca.HHo-Illy-
llIeHcKoH rsc - npH Hanp.HmeHHH 750 KB.
Ha 3J1eKTpOCTaHH.HX reHepaTopbI BbIpa6aTbIBalOT 3J1eKTpHqeCKYIO 3HeprHIO
npH Hanp.HmeHH.HX, He npeBbIllialOID;Hx 20 KB. TI03TOMY Ha 3J1eKTpOCTaHH.HX
YCTaH8BnHBalOT nOBbIllIalOHe Tp8Hc<popMaTopbI, 8 Ha MeCTe nOTpe6neHHH -
nOHHmalOID;He TpaHc<popMaTopbI. Ha pHCYHKe 14.29 npe,ACTaBJleHa 6J10K-CXeMa
JlHHHH nepe.naqH nepeMeHHoro TOKa. TaK KaK TpaHc<popMaTopbI 06J1a,naIOT
60JlbllIHM HH,nYKTHBHbIM conpOTHBneHHeM, KOTopoe npHBo,nHT K c,nBHry <}>a3
Mem,ny TOKOM H Hanp.HmeHHeM, TO ,nJl.H YBenH'tleHH.H K03<l><PHHeHTa MOID;HOC-
TH B enb BKJllOqalOT KOH,neHcaTopbI.
reHep8TOp
nepeMeHHoro
TOKa
TIOSbI1l18-
IOID;HH TpaHC-
<l>OpM8TOp
KOH,AeH-
C8TOpbI
TIoHHm8-
IOHH Tp8HC-
<l>OpM8TOp
IIoTpe6HTenb
Puc. 14.29
IIoTepH npH nepe,naqe nOCTOJlHHOrO TOKa 6bInH 6bI MeHbllIe npHMepHo B
nonTopa pa3a (HeT nOTepH Ha nepeMarHHqHBaHHe, nOTepn Ha peaKTHBHoM co-
npOTHBJleHHH). Ho nOKa HeT cnoc060B TpaHc<popMaHH nOCTO.HHHOrO TOKa. Ae-
JlaeTC.H nonbITKa npOMbIllIJleHHOH nepe,naqH nOCTO.HHHOrO TOKa BbICOKoro Ha-
np.HmeHH.H Ha 60JlbllIHe paCCTO.HHH.H, HO TpaHc<popMHpyeTC.H nepeMeHHbIH TOK,
KOTOpbIH 3aTeM npH BbICOKOM Hanp.HmeHHH BbInp.HMn.HeTCJI C nOMOblO nOJlY-
npoBo,nHHKOBbIX npH60poB. IIocJle nepe.naqH nOCTOJlHHbIH TOK npe06pa3yeTcH
B nepeMeHHbIH (B HHBepTopax), KOTOpbIH 3aTeM CHOBa TpaHc<popMHpyeTcSI.
Tpy,nHOCTH npe06pa30BaHHH TOKa B TaKoH JlHHHH nepe,na'lH He n03BOJl.HIOT nOKa
llIHpOKO HCnOJlb30BaTb 3TOT 3KOHOMH'tlHbIH MeTO,n nepe,naqH 3J1eKTp03HeprHH. Ha
pHCYHKe 14.30 nOKa3aHa 6J10K-CXeMa JlHHHH nepe,naqH nOCTO.HHHOrO TOKa.
feHepaTOp
nepeMeHHoro
TOKa
IIOBbIll1a-
IOHH TpaHC-
cpOpM8TOp
BbInpSl-
MHTenb
- MHBep-
TOp
IIoHHma-
IOHH Tp8HC-
<l>OpM8TOp
IIoTpe6H-
Tenb
Puc. 14.30
415

BOIIPOCbI jJ;JUI CAMOKOHTPOJI.H
1. H 3 KaKUX 3Jl.e.Melim06 cocmoum KOJl.e6ameJl.bHblU lwlimyp? M ozym Jl.U
803liuKliymb KOJl.ef5aliuSl 6 KOlimype, coiJepJICaU{e.M KOliiJelicamop U pe3UCmOp?
KamYWKY uHiJYKmU81i0CmU U pe3ucmop?
2. KaKue ljJu3uIlecKue 6eJl.UIlUHbl U3MeliSl1OmcSl npu KOJl.ef5aliuSlx 6 KOJl.ef5a
meJl.bliO.M KOlimype?
3. q eM OmJl.UllalOmCSl KOJl.ef5aliuSl 3Jl.eKmpUlleCKOZO 3apSlaa U CUJl.bl mOKa B
KOJl.ef5ameJl.bliOM KOlimype? KaK no zpaljJuKY 3a6UCU.MOCmu 3apSloa om 6pe.Me'
liU nocmpoumb zpaljJuK 3a8UCU.MOCmu CUJl.bl mOKa om 6pe.Meliu?
4 Om Ilezo 3a8ucum nepuoiJ 3Jl.eKmpOMazliumliblx KOJLef5aliuu 6 KOlimype?
M oJICem Jl.U nepuoiJ KOJl.ef5aliuu 6 KOlimype ocma6ambCSl nOCmO.Rlilibl.M npu U3-
.MelieliUU uliiJYKmU81i0cmu U eMKocmu?
5. KaK U3.MeHUmC.R nepuoa 3JLeKmp0.MaZHUmHblx KOJLe6aHUU B KOJLef5ameJl.b-
HOM KOHmype. eCJLU Y6eJLU1lUmb IlUCJLO 8umK06 KamywKu? 8Hecmu B Kamyw.
KY }fCeJl.e3HblU cepiJellHuK?
6. KaKue npe6paU{eliUSl 3liepZUU npoucxoiJ.Rm 6 KOJLef5ameJl.bliO.M KOlimype?
1
roe cocpeoomollelia 3liepZUSl npu C60f50iJliblX KOJLef5aHuSlx 6 KOHmype llepe3 8'
1 1
- nepuoiJa nOCJl.e liallaJl.a pa3pSliJKu KOliiJelicamopa?
4' 2
7. M ozym Jl.U f5blmb c60f5oolible KOJl.ef5aliuSl 6 peaJl.bliO.M KOlimype lie3amy-
xalOU{uMu?
8. q e.M OmJl.UllalOmCSl iJpyz om iJpyza c60f5oiJHble KOJl.ef5aliuSl 6 iJ6YX KOIi-
mypax C OOUliaK08bl.MU napa.Mempa.MU, ecJl.U UX KOHiJelicamopbl f5blJl.U 3apSl'
JlCelibl om f5amapeu C pa3libl.MU oJJC?
9. IIpu KaKUX YCJl.06USlX 6 .JJLeKmpUlleCKOU qenu 603liUKa1Om 6blliYJICiJeH-
lible 3Jl.eKmpUlleCKue KOJl.ef5aliuSl?
10. KaK iJeucm6YIOU{ee 3liaileliUe CUJl.bl mOKa (lianpSlJICeliuSl) C6Sl3ali0 C a.M-
nJl.UmyiJlibl.M 3lialleIiUe.M? KaKue 3liaileliUSl CUJl.bl mOKa U lianpSlJICeliUSl U3.Me-
pSllOm a.Mnep.Mempo.M U 60Jl.bm.MempO.M 6 qenu nepeMeliliOzo mOKa?
11. H a KamywKY nooaemcSl oouliaK060e lianpSlJICeliUe - CliallaJl.a om cemu
nOCmOSlltIiOZO, a 3ame.M om cemu nepe.Meliliozo mOKa. B KaIW.M cJl.Yllae OHa
liazpe6aemcSl CUJLbHee?
12. II Olle.MY 6 qenu, Cooep:HCaw,eu KOlioelicamop, MO:JfCem npoxooumb mOJLb-
KO nepe.MeliHblU mOK?
13. B lleM cxoiJcm6o U 6 lleM omJLUllue aKmU6HOZO, eMKocmHOZO U UHiJYK-
mU61i0ZO conpOmU6Jl.eliUU 6 L(,enSlX nepe.MeliliOZO U nocmOSlliliOZO mOK06?
14. B oC6emUmeJl.bliYIO cemb nepeMeliliozo mOKa nOCJl.e006ameJl.bliO 8KJl.lOlle-
libl Jl.a.Mna U KOliiJelicamop. KaK U3MeHumcSl liaKaJl. Jl.a.Mnbl npu nOOKJl.lOlle-
liUU eU{e OOliOZO KOlioelicamopa napaJl.Jl.eJl.bHO nep60.MY? nocJl.eo06ameJLbliO C
HUM?
15. M OJICIiO Jl.U 6 ijenb nepe.MeliliOZO mOKa lianpSlJICeHUeM 220 B 6KJl.IOIlUmb
KOlioelicamop. npof5oulioe lianpSlJICeliUe Komopozo 250 B?
16. llpu KaKUX YCJl.06USlX 603liUKaem pe30lialic lianpSlJICeliUU 6 3Jl.eKmpu'
lleCKOU ijenu? Om KaKUX 6eJl.UIlUIi npu pe30liaHce 3a6ucum a.MnJl.umyoa CUJLbl
mOKa? aMnJl.UmyiJa lianpSlJlCeliUU lia KOliiJelicamope U KamYWKe uliiJYKmUBHO-
cmu?
416

17. n olleJ'tty 6 paMKe, pa6liOMepii0 6pa14alOU{eUCSl 6 MaZIiUmliOM nO./l.e. 603-
IiUKaem ol1C? T{ eM omJl.UllaemCSl npOMblWJl.eliliblU zeliepamop om C60eu npo-
cmeuweu MOOeJl.U (paMKU. 6paU{alOU{eucSl 6 MaZliumliOM nOJl.e)?
18. KaKuM 06pa30M nOJl.YllalOm nepeMeliliblu mOle c Ilacmomou v = 50 r
npu MeOJl.eIiIiOM 6paU{eliuu pomopa?
19. Jl.Sl Ilezo CJl.YJlCum mpalicljJopMamop? H a306ume ezo OCli06lible llacmu2
KaK06 npuliun oeucm6uSl mpalicljJopMamopa?
20. KaKuM 06pa30.M npou360oSlm npe06pa306aliuSl lianpSl:HCeHUSl U CUJl.bl
mOKa 8 ijenSlX nepeMeliliOZO mOlea? M OJICIiO Jl.U C06epwamb maKue npeo6pa30-
6aliUSl 6 enSlX nocmOSlliliOZO mOKa?
21. T{ eM omJl.UllaemCSl XOJl.ocmou peJICUM pa60mbl mpaHccpop./ttamopa om pa-
60"l.ezo? KaK C6Sl3aIi bl Me;){(oy c060u 3HalleliU.Jl CUJl. mOKOB U HanpSlJICeliUU 6
06MomKax mpalicl/JopMamopa?
22. 3a cllem Ilezo 6 mpalicl/JopMamope npOUCXOOSlm nomepu 31lepZUU npe.
06pa3yeMozo mOKa? KaK06bl nymu nOBblWeliUJl KIll( mpaHcfjJopMamopa?
23. fIolleMY OJl.Jl nepeoa'l.{u 3Jl.eKmpUllecKou 3Hepzuu lia 60Jl.bWUe paccmOJl-
IiUSl UCnOJl.b3yemcR nepeMeliliblu mOK? II OlleMY nepeoallY .:JJl.eKmp03IiepZUU 6bl'
ZOOliO ocyU{ecm6./l.Slmb npu BblCOKOM lianpSlJICeliuu?
24. KaK06bl npeu.'KyU{ecm6a Jl.UHUU 3Jl.en:mponepeoallu mpexljJa3liOZO m01ca?
25. KaK06bl coomlioweliUSl MeJICoy rjJa3liblMU U Jl.UlieUliblMU lianpSl:HCeHU,R-
MU (CU.ll.aMU mOK06) npu coeoulieliuu ljJa3 zeliepamopa 36e300u (mpeyzOJl.bIiU-
ICOM)? npu pa3Jl.UllliblX cxeMax BKJl.lOlleIiUJl HazpY3Ku?
" 2 0
q T rooq =
T = 2n.JLC
OCHOBHblE c1>OPMY JIbl
- ypaBHeHHe CB060JJ;HbIX 3JIeKTpOMarHHTHbIX KOJle6a-
HHR B H,AeaJIbHOM Kone6aTeJIbHOM KOHType, r,Ae q-
3apH,A Ha KOHJJ;eHcaTope, 000 - co6cTBeHHa.H lI.HKJIIec-
Ka.H qaCTOTa 3J1eKTpOMarHHTHbIX KOJle6muIH KOHrrypa.
- «popMYJla TOMcoHa, l'JJ;e T - nepHOJJ; C06CTBCH-
HbIX KOJle6aHHR B KOHType, L - HHJJ;YKTHBHOCTb
KOHTypa, C - eMKOCTb KOHTypa.
2 LI 2 2 L12
W =  + - = !!9- = ---.!L - nOJIHa.H 3J1eKTpOMarHHTHaH 3HeprH.H KOJle6aTeJlb-
2C 2 2C 2 Horo KOHTypa, rJJ;e q - MrHOBeHHoe 3HaqeHHe 3a-
p.HJJ;a Ha KOHJJ;eHcaTope, 1 - MrHOBeHHoe 3HaqeHHe
CHJlbI TOKa B KOHTYpe, qo - aMnJlHTYJ:(Hoe 3Haqe-
HHe 3ap.HJ:(a Ha KOHJ:(eHcaTope, 10 - aMnJlHTYJ:(Hoe
3HaqeHHe CHJlbI TOKa.
g' = N BSrosin rot =
= g' 0 sin rot
- MrHOBeHHoe 3HaqeHHe S,D:C HHJ:(YKI1.HH, B03HHKa-
lOeH B paMKe, paBHOMepHO BpaalOeHC.H B Mar-
HHTHOM nOJle C lI.HKJlHqecKOH qaCTOTOH w, rJ:(e N -
'IHCJIO BHTKOB B paMKe, B - MO,n;YJlb MarHHTHOH
HH,n;yJ<I1.HH, S - nJlOmMb, OXBaTbIBaeMa.H paMKOH,
g' 0 - B.MnJlHTYJ:(Hoe 3HaqeHHe S,D:C.
417

1 = 10 sin 2nvt
 10
1 = v< 1- > = -
J2
.J 2 U o
U= <U >=-
.J2
XL = roL
1
Xc=-
roC
Up,
1 =
·  R2+(WL- :c J
z =  R 2 + ( XL - Xc )2
t XL - Xc
g<p =
R
1
v =
pe3 21tJLC
- MrHOBeHHoe 3HaqeHHe CHJlbI nepeMeHHoro TOKa.
re 10 - aMnJlHTYHoe 3HaqeHHe CHJlbI TOKa, v -
qaCTOTa nepeMeHHoro TOKa.
- eHCTBYIOee 3HaqeHHe CRJIbI TOKa, r,Ae </ 2 > -
cpeHee 3HaqeHHe KBapaTa MrHOBeHHOH CRJIbI TOKa,
10 - aMnJIRTYHoe 3HaqeHRe CRJIbI TOKa.
- eHCTBYIOee 3HaqeHRe Hanp.HmeHRH, rlIe U 0
aMnJlHTY,AHOe 3Ha'IeHHe Hanp.H:>KeHHH, <U 2 > - cpel1-
Hee 3Ha'IeHRe KBa,ApaTa MrHOBeHHoro Hanp.HmeHR.H.
- RH,AYKTRBHoe conpOTRBJleHHe. re L - RH,AYK-
TRBHOCTb KaTYIlIKH, (0 - RKJIRqeCKaH qaCTOTa ne-
peMeHHoro TOKa.
- eMKOCTHoe conpOTHBneHHe, re C - 3J1eKTpOeM-
KOCTb KOH,AeHCaTOpa, (0 - RKnRqeCKaSl qaCTOTa ne-
peMeHHoro TOKa.
- 3aKOH OMa .n:n.H enR nepeMeHHoro TOKa. CO,Aep-
maeH nOCJle,AOBaTeJlbHO BKJllOqeHHbIe pe3RCTOp, Ka-
TYIIIKY HH.n:YKTHBHOCTH R KOHeHcaTOp, r.n:e R -
aKTRBHoe conpOTRBneHlie uenR, Ip, Ii Up, - ,AeHCTBY-
IOHe 3HaqeHH.H CHJlbI TOKa H HanpHmeHH.H B enR.
- nonHoe conpOTRBneHRe enR, COCTOHI..QeH 1i3 noc-
JleOBaTenbHO coe.n;HHeHHbIX pe3HCTopa, KaTYIIIKIi
HH,AYKTHBHOCTH Ii KOHeHcaTopa, re R - conpo-
TliBneHHe pe3HCTopa, XL - HHYKTHBHoe conpOTliB-
JleHRe KaTYIlIKR, Xc - eMKOCTHoe conpOTRBneHlie
KOH.neHcaTopa.
- CBHr «}>a3 Me:>K.n;y Kone6aHHHMH HanpHmeHHH Ii
CRJlbl TOKa B enR, COCTOHeH R3 nOCne,AOBaTeJlbHO
coeHHeHHbIX aKTHBHoro, HHYKTHBHoro H eMKOCT-
Horo conpOTHBJleHRH.
- pe30HaHCHaH 'IaCTOTa nepeMeHHoro TOKa, r,Ae L
R C - RH,AYKTRBHOCTb KaTYIlIKR H eMKOCTb KOH-
,AeHCaTOpa, BKJllOqeHHbIX Ha }T'IaCTKe enH nocne.n;o-
BaTeJlbHO C pe3RCTopOM.
(U OL )pe3 = (U oc )pe3 = 10 {L - Hanp.H:>KeHHe Ha KOHeHcaTope H Ha KaTYIlIe HH-
'Ie .n;YKTHBHOCTH B cnY'Iae pe30HaHca HanpHmeHHR.
418

Q = I:Rt oo
IU
< P >= cos<p =
2
I 2 R
=  = IUcos<p
2
k = N I = g'1
N 2 g' 2
U I = g' I + II RI ;
U 2 = g' 2 - 1 2 R 2
R
T1=
PI
- 3aKoH J(>KoYJl.H-JIeHa
peMeHHoro TOKa. r.ne I JJ.
CHnbI TOKa Ha yqacTKe,
neHHe yqaCTK8.
.n;JI.H yqacTKa enH ne-
.neHcTBYIOIltee 3HaqeHHe
R - aKTHBHoe conpOTHB-
- aKTHBHa.H (cpe.nH.H.H) MOIltHOCTb, BbI.neJl.HeMa.H B
enH nepeMeHHoro TOKa, l'.n;e cosq> - K03«l>«l>HHeHT
MOIltHOCTH, q> - pa3HOCTb «l>a3 Mem.ny KOJle6aHH.H-
MH Hanp.HmeHH.H H CHJlbI TOKa.
- K03«l>«l>HHeHT TpaHc«l>opMaHH TpaHc«l>opMaTopa,
r.n;e N 1 H N 2 - qHCJlO BHTKOB B nepBHqHOH H BTO-
pHttHOH 06MOTKaX TpaHc«l>opMaTopa, g> I H g> 2 -
S,n:C HH.nYKnHH B HHX.
- Hanp.HmeHH.H Ha nepBWlHOH: H BTOpWlHOH 06MOTKax
TpaHc<J.x>PM8TOpa, r.n;e I I H 1 2 - CHJIbI TOKa B 06MOT-
K8.X, R] II R 2 - aKTHBHbIe COnpoTHBJIeHH.H 06MOToK.
- KII,n: TpaHc«}>opMaTopa, r)J;e PI MOID;HOCTb, no-
Tpe6J1.HeMa.H B nepBHqHOH nenH, P 2 - MOHOCTb,
nepe.n;aB8eMa.H nOTpe6HTeJlIO.
METOAllqECKIIE PEKOMEHAAIIIIII
B 3TOH TeMe MomHO BbI.nenHTb TPH rpynnbI 3a,naq; 06 3J1eKTpOMBrHHTHbIX KO-
ne6aHH.HX B KOJle68TeJlbHOM KOHType; 0 nepeMeHHOM TOKe H 0 TpaHc«}>opMaTopax.
PerneHHe MHorHX 3a.n;aq nepBOH rpynnbI CB.H3aHO C HaXOm,AeHHeM nepHo.na
(HnH c06CTBeHHOH qaCTOTbI) KOJle6aHHH KOHTypa C Hcnonb30BaHHeM «}>OPMYJlbI
TOMCOHa. IIpH perneHHH HeKOTopbIX 3a)J;att HcnOJlb3YIOTC.H 3aKOHbI coxpaHeHHH
H npeBpaeHHH 3HeprHH B KOJle6aTenbHOM KOHType.
PeKOMeH.n;anHE K perneHHIO TaKHX 3a.n;aq TaKHe me, KaK H K COOTBeTCTBY-
IOIltHM 3a.n;aqaM no TeMe "MexaHHtteCKHe KOJle6aHH.H".
IIpH perneHHH 3a,naq no TeMe "IIepeMeHHbIH TOK" CJle.nyeT nOMHHTb. qTO 3TO
BbIHym.neHHbIe 3J1eKTpHtIecKHe Kone6aHH.H, n03TOMY peKOMeH,AaHH K HX perneHHIO
Te me, qTO Ii .n;JI.H MeXaHHtteCKHX KOJle6aHHH. He06xo,AHMO yqHTbIBaTb TonbKO OT-
nHtIHe .n;eHCTByIOIlterO 3HaqeHH.H CHnbI TOKa (Hanp.HmeHH.H) OT aMnJlHTY.n;Horo.
IIpH paCqeTe neneH nepeMeHHoro TOKa Heo6xo.n;HMO nOMHHTb, qTO eMKOCTb
Ii HH)J;YKTHBHOCTb npe.n;CTaBJlHIOT C060H .n;ononHHTeJlbHOe peaKTHBHoe conpOTHB-
JIeHHe B OTnHqHe OT aKTHBHoro R. 3aKOH OMa BbInOnH.HeTC.H .n;n.H aMnnHTY.n-
HbIX H .n;eHCTBYIOIltHX 3HaqeHHH CHnbI TOKa H HanpHmeHH.H:
U
1 - 0 .
o -  R" +( ooL -  r
U
I = JJ. .
  R2 +( ooL-  r
TenJlOTa BbI,neJlHeTC.H TOJlbKO Ha 8KTHBHOM conpOTHBneHHH. 3aKoH ,ZJ;moy-
n.H-JIeHa 3anHCbIBaeTC.H TaK: Q = I:Rt, r.ne I1J. - .n;eHcTBYIOIltee 3HatteHHe
CHJlbI nepeMeHHoro TOKa.
419

nPIIMEPbI PEllIEHII.H 3A,Z:(A q
1. H.n;e8nbHbIH Kone6aTenbHbIH KOHTYP COCTOHT 1i3 KaTYllIKH HH,AYKTHBHOCTIi H
.n;BYX 0,AHH8KOBbIX KOH,AeHC8TOpOB, BKnlO'leHHblX n8p8nnenbHO. TIepHO,A Kone68HHH
KOHTYP8 20 MKC. t.{eMY 6Y,AeT paBeH nepHO.n;. ecnH KOH,AeHC8TOpbI BKnlO1.£IiTb no-
Cne,AOB8TenbHo?
,Z:( 8 H 0: C 1 = C 2 , TI = 20 MKC.
H8HTH: T 2 .
P e ill e H He. CornaCHO cpopMyne TOMCOH8. nepHO)J. Kone68HliH B KOHType
T = 21t.JLC. TIpli n8p8nnenbHOM coe.n;HHeHIiIi KOH,AeHCaTOpOB eMKOCTb 6aTapeH paB-
H8 C = C 1 + C 2 = 2C I' Cne,AOBaTenbHO. 7i = 21t  L . 2C; .
IIpH nOCne,AOBaTeJIbHOM COe,AHHeHHH ,ABYX o.n;HHaKOBbIx KOH,AeHCaTOpOB eMKOCTb
6aTapeH paSHa C = C , . IIo3TOMY T 2 = 27t L C "
2 2
T, 2 T,
Tor,Aa .-!. = 1 = 2 => 72 = --1; 72 = 10 MKC.
T 2 2 L i 2
2. KOH,AeHC8TOp eMKOCTblO 2,0 MK<1> 38pSl.n;Hnli ,AO H8npSl)KeHIiSl 100 B, 8 38-
TeM 3aMKHynH Ha K8TYllIKY C IiH,AYKTHBHOCTblO 5,0 MfH. Onpe,AenHTe 38pSl.n; H8
KOH,AeHC8TOpe 'lepea O,0257t MC nocne a8MbIKaHHSI.
JJ. 8 H 0: C = 2,0' 10- 6 <1>, U o = 100 B, L = 5.0' 10- 3 fH, t l = 0,0251t. 10- 3 C.
H a Ii T H: ql'
P e ill e H H e. B KOHType B03HliK81OT rapMOHH'leCKIiC Kone68HIiSl. 38pSl,A H8 06-
Kn8,AK8X KOH.n;eHC8Top8 H3MeHSleTCSI no 38KOHY
21t
q = qocos-t,
T
r.n;e qo=CU o - aMnnHTY,AHOe 3Ha'leHlie 3apSl,Aa.
ECJIH B Ka'leCTBe MO,Aenli KOHTyp8 Bb16paTb H,AeanbHbIH Kone6aTenbHbIH KOH-
TYP, TO nepHO,A 3JIeKTpOMarHHTHbIX Kone6aHHH MO)KHO onpe,AeJIHTb no <l>opMYJIe
TOMCOHa:
T = 21tJLC.
1 1
Cne,AOBaTenbHo, q = CUo cos r;-;; t. B MOMeHT t = t l 38PH,A ql = CUo cos r;-;:; t l
vLC vLC
Ii 1.£IiCneHHO paBeH ql = 0,14 MKn.
3. M8KCliMaJIbHbIH aapSl.n; H8 06Kn8,AKax KOH.n;eHcaTopa li.n;eaJIbHOrO Kone68TenbHoro
KOHTyp8 1,0 MKKn. AMnnHTY,AHOe 3H8'1eHHe ClinbI TOKa B KOHType 1,0 MA. Onpe,Ae-
nHTe nepHo.n; Kone6aHIiH.
,Z:( a H 0: qo= 1,0. 10- 6 Kn, 1 0 = 1,0. 10- 3 A.
H8HTH: T.
P e ill e H H e. IIonHaSl 3HeprHSI B H,Ae8nbHOM Kone6aTenbHoM KOHType COXP8HSI-
eTCH. TI03TOMY M8KCHM8nbH8H 3HeprHSI 3neKTpH'leCKOrO nonSl KOH.n;eHC8Top8 p8BHa
M8KCliManbHOH 3HeprliH MarHIiTHoro nOJISI K8TYllIKH, T.e.
q 2 LI2 2
o - 0 0 LC = q0 2 .
2C - 2""' TCIO,Aa 10
420

no cpopMyne TOMCOHa neplioA 3neKTpOMarHliTHbIX KOJle68HHH B KOHType
T = 21tJLC; T = 21t qo ; T = 6.3 MC.
10
4. KBa,ApaTH8S1 paMK8 nnO8,AblO 625 CM 2 C 3aMKHYToH 06MOTKOH H3 MeAHoro
npOBO,A8 Bp8aeTCSI B O.D;HOpO,AHOM M8rHHTHOM none C HH,AYKlI.HeH 1,0. 10- 2 Tn
BOKpyr OCH. ne)KaeH B nnOCKOCTH paMKH H nepneH.D;HKynSlpHoH nonlO COBepwaSi
1200 060pOTOB B MHHYTY. Onpe,AenHTe H3MeHeHHe TeMnep8TypbI 06MOTKIi 38
1,0 MliH B npe,AnOnOmeHHIi. 'lTO Tennoo6MeH C oKpym81OweH cpe.noH OTCYTCTByeT
,[(8HO: S=6.25.10- 2 M 2 , p=1,75.10- 8 OM'M, d=8800 K , c=378 ,[(m
M Kr . K '
B=I,0.10- 2 Tn, y=20 c- 1 , t=60 C.
H a H T Ii: l:1t.
P e ill e H H e. TIpJi paBHoMepHoM Bp8eHIiIi paMKIi MarHliTHbIH nOTOK, npOHIi-
3bIB81OHH ee nnOa,Ab. li3MeHSleTCSI no rapMOHIi'IeCKoMY 38KOHY
ct>=BScoswt, r,Ae w=27ty.
TIo 38KOHY 3neKTpoM8rHliTHoH HH,AYKU.JiH B paMKe B03HHK8eT S,[(C HH,AYKU.liH,
MrHOBeHHoe 3H8'leHHe KOTOpOH
iff; = -ct>' = w BS sin w t.
Cne,AOB8TeJlbHo. 8MnnHTY,AHOe 3H8'leHHe S,[(C
Fe 0 = wBS = 27tyBS.
Ecnli C'IHTaTb, 'ITO conpoTliBneHlie p8MKH SlBnSleTCSI aKTliBHbIM, TO 8MnnliTYA-
Hoe 3H8'leHHe CHnbl TOKa fo = o = 21tV:S , r,Ae R - conpOTHBneHHe npOBO,AHH-
1
K8, li3 KOToporo C,Aen8H8 p8MK8. ,[(eHcTBYIOee 3Ha'leHHe CHnbI TOK8 1 =  =
JJ. J2
J21tV BS
R
KOnH'leCTBO TennOTbI. BbIAenSlIOeecSi B p8MKe npH npOXO)KAeHHH TOK8,
no 38KOHY )lmoynSl-JIeHlI.8
2 21t 2 y2 B 2 S2 21t 2 y2 B 2 S 2 t
Q = 1 Ri: = Ri: = .
JI. R 2 R
TaK KaK nOTepli 3HeprHH B oKpymalOYIO cpe,Ay OTCYTCTBYIOT, TO 3TO KOnH'le-
CTBO TennOTbI nOHAeT H8 H3MeHeHlie BHYTpeHHeH 3Heprlili p8MKIi. Cne,AOBaTenbHO,
21t 2 y2 B 2 S2t I 4.JS
Q = cml:1t, TOr,A8 = cml:1t. 3,AeCb R = p- = p-, I - ,AnliH8 npOBO,AHHK8
R SI 
p8MKH, SI - nnO8,Ab ero nOnepe'lHOrO Ce'leHHSI. M8cc8 06MOTKH m = dlS I . Tor,Aa
1t 2 y2 B 2 St
l:1t = . {j,t = 32°C.
8cdp' ,
5. TIpH BKnlO'IeHHIi K8TYWKH B u.enb nOCTOHHHoro TOK8 C H8npHmeHHeM 12 B
8MnepMeTp nOK838n cHny TOK8 4,0 A. TIpH BKnlO'leHHH TOH me KaTYWKIi B u.enb
nepeMeHHoro TOK8 C Q8CTOTOH 50 rU. Ii HanpSlmeHHeM 12 B 8MnepMeTp nOKa3an
cl1ny TOK8 2.4 A. Onpe,AenHTe HH,AYKTHBHOCTb K8TYWKH. 1.JeMY 6Y,AeT p8BHa 8K-
THBHaSi MOHOCTb B u.enli. ecnJi nOCne,AOBaTenbHO C KaTywKoH BKnlO'lJiTb KOH,AeH-
C8TOp eMKOCTblO 394 MKct>.
,[(aHo: U 1 =12 B. 1 1 =4,0 A. y=50 rU., U 2 =12 B, 1 2 =2,4 A.C=3,94.10- 4 ct>.
HaHTH: L, P.
421

U I
Pew e H H e ITo 38KOHY OM8 ,AnSI Y'l8CTK8 enli nOCTOSlHHoro TOK8 II = -
R
Cne,AOB8TenbHO. 8'KTHBHOe conpOTHBneHlie K8TYWKIi R = U 1 . TIpli BKmO'leHHH 3TOH
II
U 2
K8TYWKH B enb nepeMeHHOI'O TOKa CHn8 TOKa 1 2 corn8CHO 3aKoHY OM8 1 2 = Z'
z =  R 2 + (WL)2 =  R 2 + (2nv L)2 .
U 2
TOr,A8 1 2 = .
 R 2 + (2nv L )2
OTKy.a8
L = 2i: r -R" =  ( : r -( : J.
ITO,ACT8BnHH 'lHCneHHbIe 3H8'leHIiH, H8H,AeM L = 13 MfH.
ITocne nO,AKnlO'leHHSI KOH,AeHCaTOp8 8KTHBHYIO MOHOCTb B enH MomHO p8C-
C'lHT8Tb no <l>opMyne P = I:R, r,Ae 13 - Clin8 TOK8, TeKyeI'O no 3TOH enH. ITo
38KOHY OM8 .anSl nepeMeHHoro TOK8
I - U 2 U 2
3 - J R" + ( wL - w r = J R" + ( 2nv L - 2C r
Cne,lIOB8TenbHO. (iCKOM8S1 MOWHOCTb
p = U:R . p = 17 BT.
2 '
R2+ ( 21tVL- )
21tVC
6. Tp8HC<I>OpM8TOP. CO,Aepm8HH B nepBH'lHOH 06MoTKe 300 BHTKOB, BKnlO'leH
B ceTb C H8npSlmeHHeM 220 B Bo BTOpli'lHYIO enb TpaHc<I>opM8TOP8. HMelOYIO
165 BIiTKOB, BKnlO'leH nOTpe6HTeJIb conpoTIiBneHHeM 50,0 OM. K8KOB8 CHn8 TOKB
BO BTOpH'lHOH qenH. ecnH H8npSlmeHHe H8 nOTpe6HTene 50,0 B? qeMY p8BHO co-
npoTHBneHlie BTOpH'lHOH K8TYWKIi?
.n;8HO: N I =300, U 1 =220 B, N 2 =165, R=50,O OM, U 2 =50,O B.
H 8 H T Ii: 1 2 . R 2 .
Pew e H H e. K03cp<pHHeHT Tp8HC<I>OpM8UliH Tp8HC<l>OpM8TOp8 k = N 1 . C ,APY'
N 2

rOH CTOpOHbI, k = : ' I'e g'1 = U] (noTepSlMIi B nepBli'lHOH 06MoTKe npeHe6per8'
eM) - S.n;c C8MOIiH,AYKHIi B nepBH'lHOH 06MOTKe, g' 2 - S.n;C HH,AYKHH BO BTO-
pH'lHOH 06MOTKe.
422

ITo 3aKOHY OMa Ann BTOpli'lHOH 06MOTKli :5'2=1 2 R 2 + V 2 . Cne,AOB8TenbHo,
N 1 = VI Cliny TOKa 80 BTOpli'lHOH n:enH HaHAeM li3 3aKOHa OMa AnSI Y'l8C-
N 2 U 2 + 1 2 R;,
V
TK8 n:enH 1 2 = ; ; 1 2 = 1 A.
N I VI _ R(N 2 U I - N 1 U 2 ).
N 2 V ' OTCIOAa R;. - N I V 2 ' R;. = 710 OM.
V 2 + R;.
R
7. JIliHliSl 3neKTpOnepe,A8"lli HMeeT conpOTliBneHHe 250 OM. KaKoe H8npSl>Ke-
HHe .lJ.OnmeH liMeTb reHep8TOp, 'lT06b1 npH nepe,A8Qe no 3TOH nHHHH K nOTpe6HTe-
nlO MOIl.tHOCTli 25 KBT nOTepli nHHlili He npeBbI1ll8nH 4 ,0% nepe,A8BaeMOH nOTpe-
6HTenlO MOIl.tHOCTH?
.n8HO: R np =250 OM, P=2,5.10 4 BT, 11=0,040.
HaHTH: U.
Pew e H H e. Cor n8CHO ycnoBHIO, nOTepH MOIl.tHOCTH H8 npOBO,AaX P nOT = 11P. Cne-
,AOBaTeJlbHO, MOIl.tHOCTb, nOTpe6nSleMaSl OT reHepaTopa. 6Y,AeT p8BH8 CYMMe
Pnar+ P=11J.U1J.' r,Ae 111. - ,AeHcTBYIOIl.tee 3Ha'leHlie CHnbl TOK8 B n:enH, U1J. - ,AeH-
CTBYIOIl.tee 3H8"leHlie HanpSlmeHHH H8 reHep8Tope. IToAcTaBliM 3H8'leHHe P nOT' no-
nY'lHM 11J.V1J.=(11 + l)P. MOIl.tHOCTb, BbI,AenSleM8.H H8 npOBO,AaX,
Po", = I;R"p. OrcIOl\a 1. = : =  .
CJIeI\OBBTeJIbHo. J I]P U = (I] + 1) P.
R np
HCKOMoe HanpH>KeHHe U = (I] + 1 ) J P p ; U = 13 KB.
423

15. -BOJIHbI
MeXaHH'IeCKHe BOJIUbI. PacnpocTpaueHHe KOJIe6allHH B ynpyrHx cpe.aax.
IIOnepe'lllLle H npo.aOJILHble BO"lIbI. JI.TJ:HHa BOJIIILI. CBll3L rUIHHLI BOJIIILI co
CKOpOCTLJO ee pacnpOCTpaHeHHn. YpaBHeHHe BO"HLI.
3BYKOBLIe BOJIHLI. CKOpOCTL 3BYKa. rpOMKOCTL 3BYKa H BblCOTa TOHa.
3JIeKTpOMarllHTHble BOJIHLI H OCHOBLI pa.ll;HOTeXHHKH. 3JIeKTpOMarHHTHoe
nOJIe. 3J1eKTpoMaruHTHbIe BOJIllbl H CKOpOCTL HX pacnpOCTpanCHHll. YpaBHe-
HHe BOJIHbI. CBOHCTBa 3J1eKTpOMarIlHTHLIX BOJIH. 3HeprHn 3JIeKTpOMarllllTHoH
BOJIULI. IIJIOTHOCTL nOTOKa H3JIyqeHHn (noBepxlIocTHan).
IIpHeM H H3JIY'1ellHe 3"TJ:eKTpOMaruHTIILIX BOJlll. H306peTeHHe p311HO A.C. no-
nOBLIM. npHIIlIn pa.ll;HOTeJIc<}>oIlHOH CBll3H. MOAYJInHn II ;J.eTeKTllpOBaHHe.
II pocTeihulIH pa.ll;HOnpIlCMIIHK.
Pa.ll;Ho"oKal.\Hn. TeJIeBH.i1.eHHe. Pa3BHTHe cpeJJ:CTB CBll3H.
15.1. PacnpocTpaHeHue KOJIe6allHH B ynpyrllx cpeAax
Cpe,lJ.a Ha3bIBaeTC.H ynpyrOH, eCJIM Mem,lJ.Y ee qaCTMlJ,aMU CYllJ,eCTBYIOT CHJIbI
B3aliMO,AeMCTBlul. npen.HTCTBYIOllJ,Me KaKoM-JIM60 ,AecpopMalJ,MH 3TOM Cpe,AbI. Kor-
,Aa KaKOe-JlM60 TeJIO COBeplllaeT KOJIe6aHM.H B ynpyroH cpelle. TO OHO B03)IeH-
cTByeT Ha t.JaCTlilJ,bI Cpe,AbI, npMneralOllJ,Me K TeJIY, li 3aCTaBJlSleT MX COBep-
lllaTb BblHym,AeHHble KOJle6aHH.H. Cpe,Aa B6nH3M KOJIe6JIIOllJ,erOC.H TeJI8 ,Aecpop-
MlipyeTCSl. M B HeM B03HMKalOT ynpyrHe CHnbI. 8TH CMJIbI B03,AeMCTBYIOT Ha
Bce 60JIee Y,AaJIeHHbIe 0'1' TeJla 'LIaCTHlJ,bI Cpe,AbI, BbIBO,ASl liX li3 nOJIOmeHliSl
paBHOBeCliSl. IIocTeneHHo Bce t.JaCTMlJ,bI Cpe,AbI BOBJIeKaIOTC.H B KOJIe6aTenbHoe
,lJ.BHmeHMe.
TeJln, KOTopbIe BbI3bIBalOT pacnpOCTpaH.HIOllJ,HeCSl B cpe,lJ.e ynpyrlie BonHbI,
SlBJI.HIOTC.H HCTO'lllHKaMH BOJIH (KoJIe6JIIOII.1MeC.H KaMepTOHbI, CTPYHbI MY3bIKaJIb-
HbIX MHCTpYMeHTOB).
YnpyrHMH BOJIllaMH Ha3bIBalOTCSl MexaHlit.JeCKHe B03MYllJ,eHIISl (,AecpopMa-
lJ,HII) , npOII3BO,AIIMbIe HCTOt.JHMKaMH, KOTopbIe pacnpOCTpaHSlIOTCSl B ynpyroM
cpe,Ae. Ynpynle BonHbI B BaKYYMe pacnpOCTpaHSlTbCSl He MorYT.
IIpli onlicaHMM BOJIHOBOrO npOlJ,eCca cpe,lJ.y CqMTalOT cnnOlllHOM H Henpe-
PbIBHOM, a ee qaCTHlJ,aMH SlBJI.HIOTC.H 6eCKOHeqHO MaJIbIe aneMeHTbI 06'beMa (,lJ.0-
CTaTOt.JHO MaJIbIe no cpaBHeHHIO C ,AJIMHOH BOJIHbl), B KOTOpbIX HaXO,lJ.MTC.R
60JIblllOe KOJIHqeCTBO MOJIeKYJI. IIpH pacnpOCTpaHeHHII BOJIHbI B CnJIOlllHOH cpe-
,lJ.e t.JaCTIIlJ,bI Cpe,lJ.bI. yqaCTBYlOllJ,lIe B KOJIe6aHIISlX, B Kam,lJ.bIH MOMeHT BpeMe-
HH HMelOT onpe,lJ.eneHHbIe <l>a3bI KOJIe6aHH.H.
reOMeTpMqeCKOe MeCTO TOqeK Cpe,lJ.bI, Kone6JIIOllJ,MXC.H B O,AMHaKOBbIX <}>a-
3ax, o6pa3yeT BOJIHOByJO nOBepXHOCTL.
BOJIHOBYIO nOBepXHOCTb, OT,AeJISlIOllJ,YIO KOJIe6nlOllJ,lIeCfl t.JaCTI1lJ,bI Cpe,AbI OT
qaCTMlJ" ellJ,e He HaqaBllll1X KOJIe6aTbCSl. Ha3bIBBlOT <}>pOHTOM BOJlHbl B 3a-
BHCHMOCTH 0'1' cpOpMbI <}>pOHTa BOJIHbI pa3JIM'lalOT BOJIHbI nJIOCKl1e, c<}>epH-
qeCKHe 11 ,Ap.
424

-
-
-
-
-
-
-
-
Puc. 15.1
Puc. 15.2
JIHHH.H, npOBe,AeHHa.H nepneH,n;HKY JI.HpHO
BOJIHOBOMY POHTY B HanpaBJleHHH pacnpo-
cTpaHeHH.H BOJlHbI. Ha3bIBaeTC.H JlyqOM. JIyq
YKa3BaeT HanpaBJleHHe pacnpocTp8HeHH.H
BOJlHbI.
B nJlOCKOH BOJlHe BOJlHOBbIe nOBepxHo-
CTB npe,ACTaBn.HIOT C060H nJlOCKOCTB. ncp-
neH,AHKYJl.HpHbIe K HanpaBJleHHIO pacnpocT-
paHeHH.H BOJlHbI (pHC. 15.1). TIJlOCKHe BOJl-
HbI MomHO nOJlyqHTb Ha nOBepXHOCTH BO,n;bI
B nnocKoH B8HHo'tJKe C nOMOI1.1blO KOJle6a-
HHH nJlOCKOrO CTepmH.H.
B CepBqeCKOH BonHe BOJlHOBbIe nOBepx-
HOCTB npe,n;CTaBJlSlIOT C060H KOHeHTpHqeC-
KHe cepbI. C(pepHqecKYIO BOJIHY MomeT
C03,n;aTb nynbcBPYIOI1.1HH B O,AHOpO.n;HOH yn..
pyroH cpe.n;e llIap. TaKa.H BOJlHa p8cnpOCT-
paH.HeTC.H C 0.n;HH8KOBOH CKOpOCTblO no BceM
HanpaBJleHIUIM. JIyqaMB .HBJI.HlOTC.H pa.n;HYCbI
c(pep (pHC. 15.2).
15.2. IIOnepe'lHbIe H npOOJIbHLIe BOJIHbI
Pa3J1Bqa1OT npO,AOJlbHbIe H nOnepeqHbIe
BOJlHbI. BOJlHa Ha3bIBaeTC.H nOnepeqHoH.
eCJlH qaCTBII.bI cpe,n;bI COBepllIalOT KOJle6a-
r HH.H B HanpaBJleHHH, nepneH.n;HKYJl.HpHOM K
HanpaBJleHHID pacnpOCTpaHeHHSl BOJlH
(pHC. 15.3). TIonepeqHa.H BOJlHa pacnpOCT-
paH.HeTC.H. HanpHMep, B,AOJlb HaT.HHYToro ro-
pH30HTaJlbHOrO pe3HHOBoro llIHypa, O,n;HH
H3 KOHOB KOToporo 3aKpenneH, a .n;pyroH
npBBe,AeH B BepTHKaJlbHOe KOJle6aTeJlbHOe
,ABHmeHHe.
PaccMoTpBM nO,Ap06Hee npoecc 06pa30-
B8HH.H nonepe'tJHbIX BOJIH. B03bMeM B Ka'tJeCTBe Mo.n;eJlH peaJlbHOrO llIHypa e-
nOqKY llIapHKOB (MaTepHaJlbHbIX TOqeK), CB.H3aHHbIX .n;pyr C ,n;pyroM ynpyrB-
MH CHJl8MH (pHC. 15.4, a). Ha pHCYHKe 15.4 H306pameH npOlJ,eCC pacnpOCT-
paHeHH.H nOnepeqHOH BOJlHbI If nOKa3aHbI nOJlOmeHH.H 1lI8pHKOB qepe3 nocne-
,AOBaTeJlbHbIe npOMemYTKH BpeMeHH, paBHb!e qeTBepTH nepHo,n;a.
B HaqaJlbHbIH MOMeHT BpeMeHH (t 1 = 0) Bce TO'tJKH Haxo.n;.HTC.H B COCTO.HHHH
paBHOBeCH.H (pHC. 15.4, a). 3aTeM BbI3bIBaeM B03MYI1.1eHHe, OTKJlOHHB TO'tJKY 1
OT nOJlOmeHHH paBHOBeCHH Ha BeJlHqHHY A H 1-SI TOqK8 HaqHH8eT KOJle-
6aTbC.H. 2-.H TOqKa, ynpyro CB.H3aHHa.H C i-H. npHXO,AHT B KOJle6aTeJlbHOe ,ABH-
meHHe HeCKOJlbKO n03me, 3-.H - eI1.1e n03me H T.,n;. 'tJepe3 qeTBepTb nepH-
Hanp8BJIeHHe
Kone68HHH:
H8npaBneHHe
pacnpOCTpaHeHH$I
BonHbI
Puc. 15.3
425

Olla (t 2 = : 1 KOJle6aHHH pacnpOCTpaHHTCH 110 4- i\ TO'lKH. 1-H TO'lKa ycneeT
OTKnOHHTbC8 OT CBoero nOJIOmeHH.H paBHOBeCH.H Ha MaKCHMaJlbHOe paCCTO.HHHe,
p8BHoe aMnJlHTy,ne KOJle6aHHH A (pHC. 15.4, 6). Qepe3 nonnepHo,na 1-8 TOq-
Ka. ,nBHra8cb BHH3, B03BpaTHTC.H B nOJlOmeHHe paBHOBeCH.H, 4-8 OTKJIOHHJIaCb
OT nOJlOmeHH.H paBHOBeCH.H Ha paCCTO.HHHe, paBHoe aMnnHTy,ne Kone6aHHH A
(pHC. 15.4, 6). BOJIHa pacnpOCTpaHHJlaCb ,nO 7-H TOqKH 11 T.,n.
K MOMeHTY BpeMeHI1 t s = T 1-8 TOqKa, cOBepllIHB nOJlHOe KOJIe6aHHe, npo-
XO,nHT qepe3 nOJIOmeHHe paBHOBeCH.H, a KOJIe6aTeJIbHOe ,nBHmeHHe pacnpocT-
paHHTc8 ,no 13-H TOqKH (pHC. 15.4, 0). Bce TOqKH OT 1-11 ,no 13-H pacnoJlo-
meHbl TaK, qTO 06pa3YIOT nOJIHYIO BOJIHY, COCT08YIO H3 Bna,lJ.HHbl H rop6a.
a
x
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 t = 0
1
6
1 2
A
5 6 7 8 9 10 11 12 13 14 15 16 17 18 T
t 2 =T
8
8 9 10 11 12 13 14 15 16 17 18 t =.1:..
8 2
8
z
10
11 12 13 14 15 16 17 18 3T
t.="'4
a
13
L4 15 16 17 18
t 5 =T
e
17 18 5T
t =-
6 4
A.
o
r
Puc. 15.4
426

HanpaBJIeHHe
.
Kone6aHHH
BOJIHa Ha3bIBaeTCSI npO.u.OJlLHOH, eCJlH
qaCTHnbI Cpe,AbI COBepIllalOT KOJle6aHH.H B
HanpaBJleHHH paCnpOCTpaHeHH.H BOJlHbI
(pHC. 15.5).
npOOJlbHyro BonHY MomHO Ha6nlO,naTb
Ha JlHHHOH M.HrKOH npymHHe 60JlbIIIOrO
HaMeTpa. Y,AapHB no OHOMY H3 KOHnOB
npymHHbI. MomHO 3aMeTHTb, KB.K no npy-
)KHHe 6y,nyr pacnpoCTpaH.HTbC.H nocneOBaTenbHbIe crYIneHHSI H pa3pemeHHSI ee BHT-
KOB, 6erYInHe pyr 3a pyrOM. Ha pHCYHKe 15.6 TO'IKaMH nOKa3aHO nOJlOmeHHe
BHTKOB npY)KHHbI B COCTO.HHHH nOKOSI. a 3aTeM nOJlO)KeHH.H BHTKOB npymHHbI qe-
pe3 nOCJIeOBaTeJIbHbIe npOMe)KyrKH BpeMeHH. paBHbIe 'IeTBepTH nepHOl{a.
.
HanpaBneHHe
pacnpOCTpaHeHH$I
BonHbI
.
Puc. 15.5
v

o "'. · · · · · · ·
+ A "' Cz
0--.,......"'. ·
Pa3 "'
q,..../ · · · J
/-A Pa3/
. . . ."". .
'- Cz /'
0..............
· · · · · · · · · · · · · · · · · t 1 = 0
T
· · · · · t 2 = 4"'
Cz
· · · · · · '" · · · · · · · · · · · · · t s = 
Pa3 "' CZ 3T
· · · · J........ "'. · · · · · t. = 4'
P a 3 ./. . :' a 3. . \.. 9! . . . .
tr, = T
Puc. 15.6
TaKHM 06pa30M, npoonbHaSi BOJlHa B paccMaTpHBaeMoM cJIae npecTaBnSI-
eT C060H qepe,AYIOHeC.H crYIneHHSI (Cz) H pa3pe)KeHH.H (Pa3) BHTKOB npymHHbI.
BH BOJIHbI 3aBHCHT OT BHa ec}>opMaI.UIH cpebI. ITpo,AOJIbHbIe BonHbI 06YCJIOB-
JleHbI ,Ae<}>opMB.nHeH C)KaTHSI - pacTSImeHH.H. nOnepe'IHbIe BonHbI - ec}>opMB.nHeH
CBHra. IT03TOMY B ra3a.x H iKKOCT.HX, B KOTOpbIX ynpyrHe CHJIbI B03HHKaJOT TOnb-
KO npH cmaTHR, pacnpocTpaHeHHe nonepe'IHbIX BOJIH HeB03MO)KHO. B TBepbIx Te-
nax ynpyrHe CHnbI B03HHKaIOT H npH CTa)KIDI (paCT.HmeHHH) H npH CHre, n03TO-
MY B HHX B03MomHO pacnpocTpaHeHHe KB.K npoOJIbHbIX, TaR H nonepeqHbIX BOJIH.
KaK nOKa3bIBalOT pHCYHKH 15.4 H 15.6, H B nOnepe'lHOR H B npOOJIbHOH
BOJIHaX Ka)Ka.H TOqKa cpebl KOJle6J1eTC.H OKOJlO CBoero nOJlOmeHHSI paBHOBeCH.H H
CMeInaeTC.H OT Hero He 60JIee qeM Ha aMnnHTY,ny, a COCTO.HHHe ecpoManHH cpe,nbI
nepe,AaeTC.H OT O,AHOH TO'IKH Cpe,AbI I< ,ApyrOH. BamHoe OTnlNHe ynpyrHx BonH B
cpee OT JlI060ro ,npyroro ynopSI,nOqeHHOrO ,nBHmeHH.H ee qaCTHn 3aKJllOqaeTC.H B
TOM, qTO pacnpOCTpaHeHHe BOJIH He CB.H3aHO C nepeHOCOM BeeCTB8 Cpe,AbI.
CJleOB8TeJlbHO, npH pacnpOCTpaHeHHH BonH npOHcxo,nHT nepeHOC 3HeprHH
ynpyroH ,necpopManHH H HMnYJlbCa 6e3 nepeHoca BeeCTBa. 3HeprH.H BOJlHbl B
ynpyroH cpee COCTOHT H3 KHHeTH'IeCKOH 3HeprHH COBepIll81OInHX KOJIe68HHSI
'IaCTHn H H3 nOTeHnH8JIbHOH 3HeprHH ynpyroH ecpopManHH cpe,nbI.
PaCCMOTpHM. HanpHMep. npOOJlbHYIO BOJIHY B ynpyroH npymHHe. B CPHK-
CHpOBaHHbIH MOMeHT BpeMeHH KHHeTHqeCKa.H 3HeprH.H pacnpe.neJIeHa no npy-
)KHHe HepaBHOMepHO. TaK KaK O.nIlH BHTKH npY)KHHbI B 3TOT MOMeHT nOKO.HT-
C.H. a l{pyrHe. HanpOTHB. .nBH)KYTCSI C MaKCHM8J1bHOH CKOpOCTblO. To me C8-
Moe cnpaBe,nJIHBO H ,nJISI nOTeHnHaJlbHOH 3HeprHH, TaK KaK B 3TOT MOMeHT
KaKHe-TO 3neMeHTbI npY)KHHbI He ,necpopMHpOBaHbI, ,npyrHe )Ke ,AecpopMHpOB8-
HbI MaKCHMaJlbHO. n03TOMY npH paCCMOTpeHHH 3HeprHH BOJIHbI BBO,ASIT T8KYlO
xapaKTepHCTHKY, KaK nJlOTHOCTb W KHHeTH'IeCKOH H nOTeHnHaJlbHOH 3HeprHH
427

w
(w = 11 - SHepnUI. npRXOLI.HW8.HC.H Ha eLIRHRIJ:.Y 06'beMa). fIJlOTHOCTb SHeprRR
BOJIHbI B Ka.m,nOH TO'lKe cpeLIbI He OCTaeTCSI nOCTOSJ:HHOH. a nepHOlIR'leCKH R3Me-
HSJ:eTC.H npR npoxomn;eHRR BonHbI: SHeprRSJ: pacnpOCTpaH.HeTCSJ: BMeCTe C BOJIHOH.
JII060H RCTO'lHRK BOJIH 06JIa,lIaeT SHeprReH W, KOTOpVIO BOJlHa npR CBoeM
pacnpOCTpaHeHHR nepe,naeT Q8CTRII.aM Cpe,AbI.
llHTeHcHBHoCTb BOJIHbl I nOKa3bIBaeT, KaKYIO SHeprRIO B cpe,nHeM nepeHO-
CHT BonHa 3a e,nHHRII.Y BpeMeHR Qepe3 e,nRHRII.Y nJIOlI..\a,AH nOBepXHOCTH, nep-
W
neHLIRKYJI.HpHOH K HanpaBJIeHRIO pacnpocTpaHeHHSJ: BonHbI: I = tS '
B Cll e,nRHRII.eH RHTeHCHBHOCTH BonHbI .HBJI.HeTCSJ: BaTT H3 KBa,npaTHbIH MeTp
( MC =  )-
3HeprRSI R HHTeHCRBHOCTb BOJIHbI np.HMO npOnOpIJ:.ROHanbHbI KBa,npaTY ee
aMnnRTYLIbI I - A 2 .
15.3. )];JlHHa BOJlHbI. CBH3b JlHHbI BOJlHbI co CKOpOCTblO
ee pacnpOCTpaHeHHH. YpaBHeHHe BOJlHbI
x 
\ ...v
°1 I. '/).. .1 \
KaK y.me OTMe'laJIOCb, B03MYWeHRe, C03,Aa-
BaeMoe KOJIe6JIIOlI..\HMCSI B ynpyroH cpe,ne Te-
nOM, nepe,naeTC.fl OT O,nHOH TOqKR cpe,nbI K
,ApyrOH. 3TO npORCXO,ART He MrHOBeHHO, a C
onpe,neJIeHHOll CKOpOCTblO. CKOpOCTblO pacnpo-
CTpaHeHRSJ: BOJIHbI Ha3bIBaeTC.H cpR3RtleCKa.H Be-
JIHtlHHa. onpe,neJl.HeMaSJ: paCCTO.HHReM, KOTopoe
npoxo,nRT JII06aSl TOtiKa (ppOHTa BOJIHbI 3a eLIli-
HHIJ:.Y BpeMeHR. BeKTOp CKOpOCTli V HanpaBJIeH no HOpMaJIli K BOJIHOBOH nOBepx-
HOCTH B CTOPOHY pacnpOCTpaHeHli.H BonHbI R B O,nHOpO.ijHOH li30TpOnHOH. cpe,ne co-
BnaAaeT C HanpaBJleHReM nyqa. CJle,nyeT OTJIlitlaTb CKOpOCTb pacnpOCTpaHeHHSJ: BOJI-
Hbi OT CKOpOCTli KOJIe6aHRSI tlaCTliIJ:. CpeLIbI OKOJIO CBORX nOJIo.meHliH paBHOBeCliSl.
fIYCTb BOJlHa pacnpOCTpaH.HeTC.fl B,AOnb ropH30HTaJIbHOH OCR (HanpliMep.
B,AOnb ynpyroro ropR30HTaJIbHOrO wHypa). B ,AaHHbIH MOMeHT BpeMeHli cpop-
Ma BOJIHbJ nOBTOp.HeTCSJ: B npOCTpaHCTBe B,nonb wHypa tlepe3 onpe,AeneHHbIe
OTpe3KH. Ha pHCYHKe 15.7 nOKa3aH rrpocpliJIb BOJlHbJ B onpe,neJIeHHbIH MOMeHT
BpeMeHR. C TeqeHlieM BpeMeHR BCSJ: 3Ta KapTHHa rrepeMewaeTCSJ: co CKOpOC-
TblO v CJleBa HanpaBO (WTpHXOBa.H JlHHHSJ:).
PaCCTO.HHHe Mem,ny 6J1HmaHWRMH TOqKaMR, KOJIe6JIIOWHMRC.H B o,nRHaKO-
BbIX c}:>a3ax (pHC. 15.7), Ha3bIBaeTC.H nJlHHOH BOJIHLI. J].JIHHa BOJIHbI A paBHa
paCCTOSlHHIO, Ha KOTopoe pacnpOCTpaH.HeTC.H cppOHT BonHbI 3a BpeMSJ:, paBHoe
nepRO,ny T KOJIe6aHliH RCTOtlHRKa BOJIH: A = vT.
Puc. 15.7
1 UJ
TaK KaK T = -; v = -, TO CKOpOCTb BOJIHbI CBSJ:3aHa C 'laCTOTOH KOJle6a-
v 2n
v 2nv
HHH ypaBHeHHeM v = A v. OTclO,na A = - = -, r,ne ill - II.HKJIHtleCKa.H qaCTOT3
v UJ
Kone6aHHH, v R ill 3aBHC.HT TOJlbKO OT CBOHCTB HCTOqHRKa BOnHbI.
428

IIpH B03HHKHOBeHHH BOJIH B cpeAe HX qaCTOTa OnpeAeJl.HeTC.H 'IaCTOTOH KO-
JIe6aHHH HCTOqHHKa, a CKOpOCTb paClIpOCTpaHeHHSI BOJIHbI 3aBHCHT OT CBOHCTB
cpeAbI. II03ToMY BOJJHbI OAHOH H TOH me qaCTOTbI HMelOT pa3JIH'lHYIO AJlHHY
B pa3HbIX cpeARX.
BbIBeAeM ypaBHeHHe nJIOCKOH BOJlHbI.
IIycTb HCTO'lHHK COBepmaeT rapMOHH'IeCKHe KOJIe6aHH.H IIO 3aKOHY
x =Asinrot,
r,Ae x - CMeeHHe, A - aMnJlHTYAR, ro - IJ;HKJlH'IeCKa.H qaCTOTa, t - BpeMSl.
B TOqKaX, OTCTO.HI.QHX Ha paCCTO.HHHH r OT HCTOqHHKa, KOJle6aHH.H qaCTHIJ;
Cpe,AbI BOJlHOBOrO cppoHTa 6Y,AYT TaKme rapMOHH'IeCKHMH, C TOH me tIaCTO-
TOH, HO 6YAYT OTCTaBaTb BO BpeMeHH OT Kone6aHHH HCTO'lHHKa Ha t1t = !...
v
Tor,Aa CMeI.QeHHe TO'lKH Cpe,Abl C KooPAHHaToH r B MOMeHT BpeMeHH t
x = A sin ro[ t -; 1- HJlH X = ASin[ rot _ ro; )-
3TO H eCTb ypaBHeHHe llJIOCKOH 6erymeH MOHoxpOMaTHqeCKOH BOJIHbl (lIpH
3TOM npeltnOnaralOT, tITO 3aTyxaHHeM BOJIHbI B IIpOIJ;eCce ee paClIpOCTpaHeHH.H
MomHO IIpeHe6pe'lb). CMeeHHe JII060H TOqKH cpeAbI H3 paBHoBecHoro nOJIO-
meHHSI IIPH npoxomAeHHH BOJIHbI .HBJI.HeTC.H cpYHKIJ;HeH ABYX lIepeMeHHbIX: Bpe-
MeHH t H paCCTO.HHH.H r AO paBHoBecHoro nOJlOmeHH.H TO'lKH cpeAbI.
113 ypaBHeHH.H BOJlHbI cJleAyeT:
1. AMnJIHTY,Aa nJIOCKOH He3aTyxalOI.QeH BOJlHbl B AaHHoH TO'lKe cpeAbI IIO-
CTO.HHHa H paBH8 aMnJIHTYAe KOJIe6aHHH HCTOtIHHKa.
2. JI106a.H TO'lKa cpelIbI COBepmaeT rapMOHH'leCKHe Kone6aHH.H. Ha'laJlbHaSl
cpa3a KOTOpblX 3aBHCHT OT YAaJIeHH.H r AaHHoH TO'lKH OT HCTOqHHKa KOJle6aHHH:
wr
x = Asin(rot + a). r,Ae a = --.
v
3. IIoJIomeHH.H KOJIe6nlOI.QHXC.H TO'leK cpeAbI B HeKoTopbIH cpHKcHpoBaHHbIH
MOMeHT BpeMeHH OIIHcbIBaroTC.H ypaBHeHHeM
x = ASin[ 0;; + ). rAe  = -rot
HMeeT pa3Hoe 3Ha'leHHe B pa3HbIe MOMeHTbI BpeMeHH.
Ha pHCYHKe 15.8, 6 IIpHBe,AeH rpacpHK 3TOH cpYHKIJ;HH IIPH t = 0, npeA-
CTaBJI.HIOHH C060H "MOMeHTRJIbHbIM CHHMOK" BonHbI, T.e. 1I0JlOmeHHe Bcex 'la-
CTHIJ; Cpe,Abl, B KOTOpOM paClIpOCTp8H.HeTC.H BOJIH8, B MOMeHT BpeMeHH t = 0 -
rpacpHK BOJIHbI. B OTJlH'IHe OT rpacpHKa KOJIe6aHHH (pHC. 15.8. a), KOTOpbIM
IIOKa3blBaeT 3aBHCHMOCTb CMeI.QeHH.H OlIHOH 'IaCTHUbl OT BpeMeHH B AaHHoH
x
T
6
x
A
a
o
o
Puc. 15.8
429

TOqKe CpebI, rpacpHI< BOJlHbI (pHC. 15.8, 6) nOKa3bIBaeT 3aBHCHMOCTb CMee-
HHJI Bcex qaCTH cpebI H3 nOJIOmeHHJI paBHOBeCH.H OT paCCTOJIHH.H r ,AO HC-
TOqHHKa KOJle6aHHH B HeKoTopH cpHKcHpoBaHHbIH MOMeHT BpeMeHH.
HaH,AeM pa3HOCTb cpa3 KOJle6aHHH ,ABYX TO'leK cpebI, HaXO.HHXC.H Ha
paCCTOJIHHH r= r2- rl Pyr OT pyra (pHC. 15.8, 6). KOJle6aHHJI 3TH X TOqeK
OnHCbIBaIOTC.H ypaBHeHHJIMH:
xl = A sin co(t -  ): x 2 = A sin (ill t -  l
Pa3HOCTb «}>a3 <Pl = <PI - <P2 6yeT paBHa
l rl ] l r2 ) w ( ) 21t ( ) 21t
<P = w t - - - w t - - = - r 2 - r 1 = - r 2 - r 1 = - /).r.
v v v Tv A
EcnH Be TOqKH HaXOJITCJI Ha paCCTOJIHHH r = A, TO pa3HOCTb «}>a3 KOJle-
6aHHH 3THX TO'leK <P = 21t, qTO COOTBeTcTByeT aHHOMY BbIwe onpeeJleHHIO
,AJIHHbl BOJlH.
15.4. 3BYKoBbIe BOJlHbI. CKOpOCTb 3BYKa
3BYKOBbie BOJlHbI (3BYK) - 3TO ynpyrHe npOOJlbHble BOJlHbl, KOTopbIe,
B03eHcTBY.H H8 CJIYXOBOH annapaT qeJIOBeKa, BbI3bIBaJOT onpeeJIeHHbIe (CJIY-
xOBbIe) ow;yeHH.H.
qeJIOBeqeCKOe yxo BocnpHHHMaeT B BH,Ae 3BYKa ynpyrHe KOJIe6aHH.H, qac-
TOTa KOTOpbIX HaXOHTC.H B npe,Aenax OT 16 o 20 000 r. TaKHe KOJle6a-
HHJI Ha3blBalOTCJI aKYCTHqeCKHMH. AKycTHKa - pa3eJI CPH3HKH, B KOTOpOM
paCCMaTpHBalOTCJI CBOHCTBa 3BYKOBbIX BOJIH, 3aKOHOMepHOCTH HX B036Y>KeHH.H,
pacnpOCTpaHeHH.H H )J;eHcTBH.H Ha BCTpeqHe npen.HTCTBH.H.
3BYKoBbIe KOJle6aHH.H C qaCTOTaMH, MeHbWHMH 16 r. Ha3bIBaIOTC.H HH-
pa3BYKaMH, a C qaCTOTaMH, 60JlbWHMH 20 000 r, - YJILTpa3BYKaMH. JII060e
Teno (TBepoe, mHKoe HJIH ra3006pa3Hoe), KonoonJOw;eec.H co 3BYKOBOH qaCTOTOH,
C03aeT B oKpymalOeH cpee 3BYKOBYlO BOJIHY.
B BaKYYMe 3BYKOBbIe BOJlHbI pacnpOCT-
paH.HTbC.H He MorYT. .D:JlSI OKa3aTeJlbCTB8
3Toro 3J1eKTpHqeCKHH 3BOHOK HymHo nOMe-
CTHTb no,A KOnOKOJl B03,AYWHoro Hacoca
(pHC. 15.9). IIo Mepe Toro KaK aBJleHHe
B03)J;yxa no KOJIOKOJIOM YMeHbwaeTC.H, 3BYK
OCJIa6eBaeT, nOKa He npeKpaaeTCJI COBceM,
XOT.H KOJIOOaHH.H 3BOHKa npOHCXO.HT.
ITJloxo npOBOJIT 3BYK TaKHe MaTepHa-
JI, KaK BOHnOK, nopHCTbIe naHeJlH, npec-
COBaHH8.H np06Ka H T.. 8TH M8TepHanbi
/ HcnOJlb3YlOT n.H 3BYKOH30Jl.HHH, T.e. ,AJIJI
 3aHTbI nOMeeHHH OT npoHHKHOBeHHJI B
HHX nOCTopoHHHX 3BYKOB.
3BYKOBbie BOJIHbJ pacnpOCTpaHJIIOTC.H C
KOHeQHOH CKOpOCTbID, KOTOp8JI 3aBHCHT OT
oc06eHHoCTeH cpebl: nJIOTHOCTH, ynpyro-
rMI
III
;; .
A
!.
Puc. 15.9
430

CTH, TeMnepaTypbI. HanpHMep, CKOpOCTb 3BYKa
M M
BHSX paBHa 331 -, B Bo,nopo,ne - 1270
C C
CKOpOCTb pacnpOCTpaHeHHH 3BYKOBMX BOJIH
cpOPMY ne
B B03,nyxe npH HOpMaJIbHbIX ycno-
M
a B yrneKHcnOM ra3e - 258 -.
C
B ra3ax MO>KHO paCCqHTaTb no
v = y r;: ,
r,ne y - nOKa3aTeJlb a,nHa6aTbI (CM.  79) (,AJI.H B03,Ayxa y = 1.4), R - YHH-
BepCaJlbHa.H ra30Bas nOCTO.HHHa.H, T - a6COJllOTHaS TeMnepaTypa, M - MO-
nspHaSl Macca ra3a.
CKOpOCTb 3BYKa B Bo,ne 60JlbllIe, qeM B B03,nyxe. IlpH 8 °C CKOpOCTb 3BY-
M
Ka B BO,Ae paBHa 1435 -. B TBep,nbIX TeJlaX OHa elQe 60JlbllIe, qeM B >KH,A-
C
M
KOCTSX. HanpHMep, B CTanH CKOpOCTb 3BYKa npH6nH3HTeJlbHO paBHa 5200
C
M M
B Me,nH - 3910 -. B anlOMHHHH - 4880 -.
C C
CKOpOCTb pacnpOCTpaHeHH.H npo,nOJlbHbIX 3BYKOBMX BOJIH B ynpyroM CTep-
"'He V npo . =  , rile E - MOilynb IOHra, p - nnOTHOCTb MaTepHana CTeP"'HH.
15.5. rpOMKOCTb 3BYKa H Bbl<'OTa TOHa
3BYKOBble BOJlHbI, KaK H ,npyrHe BOJlHbI, xapaKTepH3YIOTC.H TaKHMH 06'beK-
THBHbIMH BenHqHHaMH, KaK qaCTOTa, aMnnHTY,Aa, <l>a3a KOJle6aHHH. CKOpOCTb
pacnpOCTpaHeHHSI, HHTeHCHBHOCTb 3BYKa H ,ApyrHMH. Ho. KpOMe 3Toro, OHH
onHCMBaIOTC.H TpeM.H cy6'beKTHBHbIMH xapaKTepHCTHKaMH. 3TO - rpOMKOCTb
3BYKa, BbICOTa TOHa H TeM6p.
'1JYBcTBHTeJlbHocTb qeJlOBeqeCKOrO yxa pa3J1H'lHa ,Ans pa3HbIX qaCTOT. ,L:(nSl
Toro qT06bI BbI3BaTb 3BYKOBoe OYlQeHHe, BOJlHa )J;on>KHa 06J1a,naTb HeKOTO-
pOH MHHHManbHOH HHTeHCHBHOCTblO, HO eCJlH 3Ta HHTeHCHBHOCTb npeBbIlliaeT
onpe,neneHHMH npe,nen, TO 3BYK He CJlMllieH H BM3MBaeT TonbKO 60neBoe oy-
eHHe. TaKHM 06pa30M, ,nn.H Ka>K,nOH qaCTOTM KOJle6aHHH cyecTByeT HaH-
MeHbllIa.H (nopor CJlLIWHMOCTH) H HaH60JlbllIaH (nopor 60JleBOl'0 Oll.\Yll.\eHH.H)
HHTeHCHBHOCTb 3BYKa, KOTopas cnoc06Ha BM3BaTb 3BYKoBoe oyeHHe. Ha
pHCYHKe 15.10 npe,nCTaBneHa
3aBHCHMOCTb noporOB CJlbIlllH-
MOCTH H 60JleBOrO OlQyeHHH
OT qaCTOTM 3BYKa. 06nacTb,
pacnOnO>KeHHaS Mem,ny 3THMH
,nBYM.H KPHBMMH, SBJlSeTC.H 06-
J1aCTLIO CJlLIWHMOCTH. HaH60nb-
llIee paCCTO.HHHe Me>K,ny KpHBbI-
MH npHxo,nHTC.H Ha qaCTOTbI, K
10 5 v, r KOTOpbIM yxo HaH60nee qyBCTBH-
TenbHO (1000-5000 r).
I, BT
M 2
10- 1
10- 5
10- 11
10
10 2
10"
10 3
Puc. 15.10
431

ECJlH HHTeHCHBHOCTb 3BYKa - BeJIH'tIHHa. 06'beKTHBHO XapaKTepH3YIOa.fl
BOnHOBOH npoecc, TO cy6'beKTHBHOH XapaKTepHCTHKOH 3BYKa .HBJI.fIeTC.fI rpOM-
KOCTb rpOMKOCTb 3aBHCHT OT HHTeHCHBHOCTH 3BYKa, T.e. onpe,nen.HeTC.H KBa,n-
paTOM aMnnHTy,nbI KOJle6aHHH B 3BYKOBOH BOnHe Ii 'tIYBCTBHTeJlbHOCTblO yxa
(<pH3HOnOrH'tIeCKHMH OC06eHHOCTHMH). TaK KaK HHTeHCHBHOCTb 3BYKa I - A 2 .
TO qeM 60JIbllIe 8MnJlHTy,n8 KOJIe68HHH, TeM rpOM'tIe 3BYK.
BhlCOTa TOHa - Ka'tleCTBO 3BYKa, Onpe,AeJl.HeMOe qeJIOBeKOM cy6'beKTHBHO Ha
CJIYX H 3aBHCHee OT 'tIaCTOTbI 3ByI<a. "tJeM 60JIblIIe 'laCTOTa TeM BbIllie TOH 3ByI<a.
3BYKOBe Kone6aHH.H, npOHcxo,n.HHe no rapMOHHqeCKOMY 3aKOHY, C on-
pe,neJleHHOH 'tIaCTOTOH, BocnpHHHMaIOTC.H 'tIenOBeKOM KaK onpe.n;eJleHHbIH MY3hl-
KaJlhHhlH TOH. KOJle6aHH.H BbICOKOH 'tIaCTOTbI BocnpHHHMaIOTC.H KaK 3BYKH Bhl-
COKoro TOHa, 3BYKH HH3KOH qaCTOTbI - KaK 3BYKH HH3Koro TOHa. .n:Hana30H
3BYKOBbIX KOJIe6aHHH, COOTBeTCTBYIOHH H3MeHeHHIO qaCTOTbI KOJle6aHHH B
.n;Ba pa3a, Ha3bIBaeTCH OKTaBoii. TaK. HanpHMep, TOH "JIH" nepBOH OKTaBbI CO-
OTBeTcTByeT qaCTOTe 440 r, TOH "JlH" BTOpOH OKTaBbI - 'tIaCTOTe 880 r.
MY3bIK8JIbHbIM 3BYKaM COOTBeTCTBYIOT 3BYKH, H3,AaB8eMbIe rapMOHHqeCKH
KOJle6J1IOIl.{HMCSI TeJlOM.
OCHOBHhlM TOHOM CJIO>KHOrO MY3bIKaJIbHOrO 3BYKa Ha3bIBaeTCH TOH, COOT-
BeTcTBYIOIUHH HaHMeHblIIeH 'tIaCTOTe. KOTopaSJ liMeeTCH B Ha60pe qaCTOT .n;aH-
Horo 3BYKa. TOHbI, COOTBeTCTBYIOHe OCTaJIbHbIM 'tIaCTOTaM B COCTaBe 3BYKa.
Ha3bIBaIOTC.H o6epToHaMH. EcnH 'tIaCTOTbI 06epToHoB KpaTHbI 'tIaCTOTe Vo OCHOB-
Horo TOHa, TO 06epTOHbI Ha3bIBaIOTC.H rapMOHHqeCKHMH, npH'tIeM OCHOBHOH TOH
C qaCTOTOH "0 H83bIBaeTC.H nepBoii rapMOHHKOii, 06epTOH CO cJle.n;YIOIl.{eH qac-
TOTOH 2v o - BTOpoii rapMOHHKOii H T.,A.
MY3bIKaJIbHbIe 3BYKH C O,nHHM 11 TeM me OCHOBHbIM TOHOM pa3J1H'tIaIOTCH
TeM6pOM. KOTOpbIH onpe,neJl.HeTC.H H8J1H'tIlieM 06epTOHOB - liX 'tIaCTOT8MH 11
aMnJlHTy,naMIi, xapaKTepOM HapaCTaHH.H 8MnJIHTY.n; B HaqaJIe 3B-qaHI1.H Ii HX
cna.n;OM B KOHe 3By'tlaHH.H.
lIpH O,AHOH BbICOTe TOHa 3BYKIi, 113,AaBaeMbIe, HanpHMep, CKpHnKOH Ii nli-
aHliHO, OTJlHqaIOTC.H TeM6pOM.
BocnpHHTHe 3BYKa opraHaMH CJIyxa 3aBHCHT OT Toro, KRKlie qaCTOT BXO-
,A.HT B COCTRB 3BYKOBOH BonH.
llIYMbl - 3TO 3BYKH, 06pR3YIOHe CnJIOlliHOH cneKTp, COCTO.HHH H3 HR-
60pR 'tIRCTOT, T.e. B lIIYMe npHCYTCTBYIOT KOJle6aHI1.H BceB03MO>KHbIX qaCTOT.
15.6. 3JIeKTpOMarHIITHoe nOJIe.
3JIeKTpOMarHHTHLIe BOJIHLJ
KRK 6bIJIO YKR3aHO pRHee ( 12.6), B npOCTpaHCTBe, r.n;e H3MeHHeTCH Mar-
HHTHoe nOJle, Bcer.n;a nO.HBJI.HeTCH BHxpeBoe 3JIeKTpH'tIeCKOe nOJIe (HH.n;yHPO-
BaHHoe 3JIeKTpH'tIeCKOe nOJle). lIpH 3TOM 3aMKHYTbIH KOHTYP n03BOJI.HeT ero
TonbKO 06HRpymHTb, nOCKOJlbKY nOJIe CYIl.{eCTByeT He38BHCHMO OT HaJIHqH.H 3TO-
ro 3aMKHYToro KOHTYPR. JIHHHH HanpHmeHHOCTIi BHxpeBoro 3JIeKTpHqeCKOrO
nonH Bcer.n;R 3RMKHYTbI, H HX HanpaBJleHHe CB.H3RHO C li3MeHeHHeM HRBO.n;.H-
ero (liH,nyHpYIOero) MarHHTHoro non.H npaBHnOM JIeHa (npaBliJIOM JleBO-
1"0 BHHTa) (pHC. 15.11, a), T.e. 3TO nOJle HOCHT BHxpeBoH, HenoTeHHaJlbHbIH
XRpRKTep, no.n;06Ho MarHI1THOMY. .n:>K. MRKcBeJlJl B CBH3H C 3THM 06CTO.HTeJIb-
432

a
6
B
tB
E
LE
B
Puc. 15.11
CTBOM BbICKa3aJl MbICJlb 0 B03MomHOM paBHonpaBHH nOJleH: npH H3MeHeHHIi
MarHHTHoro non.H B03HHKaeT 3J1eKTpHqeCKOe nOJle (pHC. 15.11, a) Ii Ha060-
pOT, npH H3MeHeHHH 3neKTpHqeCKOrO ,AOJlmHO B03HliKaTb MarHIiTHoe nOJle
(pHC. 15.11, 6). MaKcBeJlJl TeOpeTliqeCKH ,AOKa3aJl CBoe npe.nnOJlo.meHlie. co-
3JJ;aB TeoplilO 3J1eKTpOMarHIiTHoro nOJl.H Ha OCHOBe .nBYX nocTynaTOB:
1) nepeMeHHoe MarHHTHoe none C03aeT B oKpy.malOeM ero npocTpaHcTBe
BlixpeBoe 3J1eKTpliqeCKOe none, nliHli1i Hanp.H.meHHOCTIi KOToporo npeJJ;CTaBn.H-
lOT C060H 3aMKHYTbIe JlHHHH, oXBaTbIBalOHe JlHHHH liHJJ;YK:U;HH MarHHTHoro
non.H (pliC. 15.11. a). HanpaBJleHHe BeKTopa Hanp.HmeHHOCTH E BHxpeBoro
3neKTpHqeCKOrO nOJl.H onpe.nen.HeTC.H no npaBHny JleBOrO BHHTa: ecnH oCTplie
neBoro BHHTa nocTynaTenbHO JJ;BHmeTC.H no HanpaBJleHli10 BeKTopa H3MeHeHIi.H
MarHHTHoH HHJJ;YK:U;HH !lB, TO nOBopOT rOJlOBKH BHHTa YKa3bIBaeT HanpaBne-
Hlie RIiHIiH Hanp.HmeHHOCTJif BHxpeBoro 3J1eKTpli'tleCKOrO nonSl;
2) nepeMeHHoe 3neKTpli'tleCKOe none C03JJ;aeT B oKpymalOl.QeM ero npOCTpaH-
CTBe BlixpeBoe MarHHTHoe none. JlHHIiH HHJJ;YKII.HIi KOToporo OXBaTbIBalOT nH-
HIiH Hanp.H.meHHOCTH nepeMeHHoro 3J1eKTpHqeCKOrO nOJl.H (pliC. 15.11, 6). Ha-
npaBJleHHe IiHJJ;YK:U;HH jj MarHHTHoro nOJl.H, "nopo.mJJ;eHHoro" nepeMeHHbIM
3J1eKTpHqeCKHM nOJleM, onpeJJ;eJl.HeTC.H no npaBHJlY npaBoro BHHTa: eCJlH OCT-
pHe npaBoro BHHTa nocTynaTeJlbHO JJ;BH.meTC.H no HanpaBJleHHIO BeKTopa H3-
MeHeHH.H Hanp.HmeHHOCTH 3J1eKTpHqeCKOrO non.H /:1£, TO nOBopOT rOJlOBKH BHH-
Ta YKa3bIBaeT HanpaBJleHHe JlHHHH HHJJ;YK:U;HH MarHHTHoro non.H. IIpJif 3TOM
'leM 60Jlbilie CKOpOCTb H3MeHeHIi.H Hanp.HmeHHOCTIi 3J1eKTpliqeCKOrO nOJl.H, TeM
60Jlee CHJlbHOe B03HHKaeT MarHIiTHoe nOJle, CB.H3aHHOe C 3J1eKTpliqeCKHM:
/1E
B - - TOqHO TaK me, qeM 60Jlbilie CKOpOCTb H3MeHeHIi.H HHD;YKUIiH Mar-
6.t .
6.B
HHTHoro nOJl.H, TeM 60Jlee CHJlbHOe B03HHKaeT 3J1eKTpH'tIeCKOe nOJle: E - M
BHxpeBoe 3neKTpH'tIeCKOe H MarHHTHoe nOJl.H "c:u;enJleHbI" JJ;pyr C JJ;Py-
rOM, cyeCTBYIOT OJJ;HOBpeMeHHO, B3aliMHO nopomJJ;alOT ,Apyr JJ;pyra. Henb3.H
C03JJ;aTb nepeMeHHoe MarHHTHoe none 6e3 Toro, 'tIT06bI B npOCTpaHCTBe He
B03HHKJlO nepeMeHHoe BHxpeBoe 3J1eKTpHqeCKOe nOJle. He MeHee BamHO TO 06-
CTO.HTenbCTBO, qTO 3J1eKTpHqeCKOe nOJle 6e3 MarHHTHoro, H Ha060poT, MorYT
cyecTBoBaTb JlHllib no OTHOIIIeHHIO K onpe)J;eJleHHbIM CHCTeMaM OTCqeTa. TaK,
433

nOKO.HHHC.H 3ap.H C03aeT TOJlbKO 3J1eKTpOCTaTHQeCKOe nOJle. Ho Beb 3a-
p.H nOKOHTC.H JlHlllb OTHOCHTeJlbHO onpeeJleHHOH CHCTeMbI OTCqeTa, a OTHO-
CHTenbHO pyroH OH 6yeT BHraTbC.H H, CJle)J;OBaTenbHO, C03aBaTb MarHHT-
Hoe nOJle.
COBoKynHocTb Hepa3pbIBHO CB.H3aHHbIX pyr C ,ApyroM H3MeH.HIO:I.qHXC$:I
3J1eKTpHqeCKOrO H MarHHTHoro nOJleH npeCTaBnSleT C060H 3neKTpOMarHHTHoe
none. lIo cBoeH npHpoe 3J1eKTpOMarHHTHoe nOJle He OCTaeTC.H JlOKaJlH30BaH-
HbIM B MeCTe 3apOmeHH.H, a pacnpOCTpaH.HeTC.H B npOCTpaHCTBe.
PacnpOCTpaH.HIOeeC.H B npOCTpaHCTBe nepHOHqeCKH H3MeH.HIO:I.qeeC.H aJleK-
TpOMarHHTHoe nOJle H npeCTaBJI.HeT C060H 3neKTpoMarHHTHYIO BonHY.
PacnpOCTpaHeHHe 3J1eKTpOMarHHTHOH BOJlHbI CB.H3aHO C HaBeeHHeM 3J1eKT-
pOMarHHTHoro non.H B nocJleYIOIl\HX TOQKaX H YHHQTOmeHHeM B npebIyHX
(npOH,AeHHbIx) TOqKaX npocTpaHcTBa (pHC. 15.12). ECJlH B KaKOH-JlH60 06J1ac-
TH npOCTpaHCTBa cyecTByeT nepeMeHHoe MarHHTHoe nOJle, TO OHO BbI3bIBaeT
nO.HBJleHHe B coceHHX 06J1aCT.HX npocTpaHcTBa BHxpeBoro aJleKTpH'leCKOrO
nOJl.H, JlHHHH Hanp.HmeHHOCTH KOToporo OXBaTbIBalOT nHHHH HH,AYKHH 3Toro
MarHHTHoro nOJl.H. B CBOIO OQepeb, nepeMeHHoe BHxpeBoe 3J1eKTpHQeCKOe nOJle
BbI3bIBaeT nO.HBJleHHe B coceHHx 06J1aCT.HX npocTpaHcTBa nepeMeHHoro Mar-
HHTHoro nOJl.H, nHHHH H:HYKHH KOToporo OXBaTbIBalOT lIHHHH Hanp.HmeHHO-
CTH aHHoro BHxpeBoro 3J1eKTpHqeCKOrO nOJl.H, H T.. STOT npoecc pacnpOCT-
paH.HeTC.H B npOCTpaHCTBe no BceM HanpaBneHH.HM.
l!B
->0
l!t
B
!!J.E
\ !!J.t l!B
'--
l!t

\
J
\ j
'-
Puc. 15.12
B OTJlHqHe OT MeXaHHqeCKHX BOJlH, KOTopbIe pacnpOCTpaH.HIOTC.H TOJlbKO B
ynpyroH cpee, 3J1eKTpOMarHHTHbIe BOJlHbI MorYT pacnpOCTpaH.HeTbC.H H B Ba-
KYYMe.
15.7. CKOpOCTL pacnpOCTpaHeHHH 3JIeKTpOMarHHTHLIX BOJIH
CBoHcTBa 3JIeKTpOMarHHTHLIX BOJIH
1. 113 TeopHH MaKCBenJla BbITeKaeT, qTO eCJlH B KaKOH-JlH60 MaJIOH o6na-
CTH npOCTpaHCTBa nepHO)J;HqeCKH H3MeH.HTb 3J1eKTpHqeCKOe H MarHHTHoe non.H,
TO 3TH H3MeHeHH.H )J;OJlmHbI nepHO)J;HqeCKH nOBTOp.HTbC.H H BO Bcex pyrHx
TOqKaX npOCTpaHCTBa, npHqeM B Kam)J;OH nocJleYIOIl\eH HeCKonbKO n03me, qeM
B npebIYIl\eH, T.e. OT HCTOqHHKa 3J1eKTpOMarHHTHbIX KOJle6aHHH )J;OJlmHbI BO
434

Bce CTOpOHbI paCnpOCTpaH.HThC.H 3J1eKTpOMarHHTHbIe BOJlHhI C onpe,neneHHOH
CKOpOCTblO. BbIBO 0 KOHeqHOCTH CKOpOCTH paCnpOCTpaHeHH.H 3J1eKTpOMRrHHT-
HhIX BOnH - OqeHh BRmHOe CJle,nCTBHe H3 TeOpHH MaKCBeJlJla.
.n;m. MRKCBeJlJl qHCTO MaTeMaTHqeCKH nOKa3aJl, qTO CKOpOCTb pRCnpOCTpaHe-
HH.H 3J1eKTpOMarHHTHOrO nOJl.H B BaKYYMe pRBHa CKOpOCTH CBeTR C == 3 . 10 8 M
c
a B cpe,ne 3TR CKOpOCTh v MeHbwe H 3RBHCHT OT CBOHCTB Cpe,AbI:
C
V=-
'
r,ne E - ,nH3J1eKTpHqeCKa.H npoHHD;aeMOCTb cpe,nbI, J.l - MarHHTHa.H npoHHD;a-
eMOCTb cpe,nbI.
2. IIpH pacnpOCTpaHeHHH 3J1eKTpOMarHHTHhIX BOJIH B Kam,noH TOqKe npo-
CTpRHCTBR npOHcxo,n.HT nepHo,nHqeCKH nOBTOp.HIOHeC.H H3MeHeHH.H 3J1eKTpHqeC-
Koro H MRrHHTHoro nOJleH. 3TH H3MeHeHH.H Y,A06HO H306pamaTb B BHlIe KOJle-
6aHHH BeKTopOB Hanp.HmeHHOCTH 3neKTpHqeCKOrO nOJl.H E H HH,nYKD;HH Mar-
HHTHoro non.H 13 B Kam,nOH TOqKe npOCTpaHCTBa. 3neKTpOMarHHTHa.H BOJlHR -
nOnepeqHa.H BOJlHa, TaK KRK
E 1. V H B 1. V.
3. KOJle6aHH.H BeKTopOB E H 13 B Kam,noH TOqKe 3J1eKTpOMRrHHTHoH BOJl-
HbI npOHCXOlI.HT B O,AHHRKOBbIX cpa3ax H no lIBYM B3aHMHO nepneHlIHKY JI.Hp-
HbIM HanpRBJleHH.HM: E -L 13 B Kam,noH TOqKe
z
v
x
npOCTpaHCTBR.
4. BeKTOphJ E Ii 13 06-
pa3YIOT C BeKTopOM CKOpOCTH
pRcnpOCTpRHeHH.H V npRBO-
BHHTOBYIO CHCTeMY (pHC.
15.13): eCJlH rOJlOBKY npaBO-
ro BHHTa paCnOJlOmHTb B
nnOCKOCTH BeKTopOB E II B
H nOBOpaqHBaTb ee B HanpaB-
JleHHH OT E K B no KPRT-
OCTpH.H BHHTR YKRmeT HRnpaB-
E
Puc. 15.13
qRHweMY nYTH, TO nocTynaTeJlbHOe ,ABHmeHHe
neHHe BeKTopa V B MOMeHT BpeMeHH t.
5. IIepHolI 3J1eKTpOMRrHHTHoH BOJlHbI (qRCTOTR) pRBeH nepHOY (qaCToTe)
KOJle6aHHH HCTOqHHKB 3J1eKTpOMRrHHTHbIX BOJIH. .n;JI.H 3neKTpOMarHHTHbIX BOJIH
cnpaBe)J;JlHBO COOTHoweHHe
v
A = vT; A = - .
v
c
B BaKYYMe Ao = - = cT - ,nJlHHa BonHhI HRH60JlhWa.H no cpaBHeHHIO C J...
v
B ,nPyroH cpe,ne, TaK KRK v = const H H3MeH.HIOTCSI TOJlbKO v H A npH nepexo-
one OT O,nHO" cpe)J;bI I< ,nPyroH.
6. 3J1eKTpOMarHHTHa.H BOJlHR, KRK H ynpyrR.H, .HBJI.HeTC.H HOCHTeJleM 3HeprHH,
npHqeM nepeHOC 3HeprHH COBepWaeTC.H B HanpaBJleHHH pRcnpOCTpRHeHH.H BOJlHhI.
3HeprHIO W 3M 3J1eKTpOMarHHTHOH BOJlHbI MomHO PRCCqHTRTb no cpopMYJle
435

'--,
EE E 2 B 2 2 B 2
W SM =W S +W M =V+-V=EOEE V =-V,
2 211110 11110
r,Ae V - 06'beM cpeJJ;bI, B KOTOpOM cocpeJJ;OTO'leHa 3neKTpOMarHHTHa.H BOJIHa
(CM. HHme  15.9).
lIepeHocHMa.H SHeprHH nponopHOHaJlbH8 qeTBepToH CTeneHH qaCTOTM. 1I0-
3TOMY HCTOqHHKOM HHTeHCHBHbIX 3JIeKTpOMarHHTHbIX BOJlH, cnoc06HbIX nepe-
HOCHTb 3JIeKTpOMarHHTHYlO 3HeprHIO Ha 3HaqHTeJlbHbIe paCCTO.HHHH, JJ;OJlmHbI
6bITb 3J1eKTpOMarHHTHbIe Kone6aHHH OqeHb BbICOKOH qaCTOTbI (nopHJJ;Ka MHJI-
JlHOHa rep). 1I0HHTHo, qTO HHKaKHe MeXaHHqeCKHe reHepaTopbI He MorYT CO-
3,AaTb nepeMeHHMH TOK qaCTOTOH -10 6 r (JJ;nH 3Toro .HKOpb JJ;OJlmeH 6bIn 6bI
cOBepmaTb 10 6 060pOTOB B 1 c). HCTOqHHKOM 3JIeKTpOMarHHTHbIX BOJIH Ta-
KOH qaCTOTbI MomeT 6MTb TonbKO KOJle6aTenbHbI:H KOHTYp.
7. 3J1eKTpOMarHHTHbIe BOJlHbI pacnpOCTpaHHIOTC.H np.HMOJlHHeHHO B OJJ;HO-
pO,AHOH cpeJJ;e, HcnbITbIBalOT npeJIOMJIeHHe npH nepexoJJ;e H3 O,AHOH cpeJJ;bI B
APyrylO, OTpamalOTC.H OT nperpaJJ;. ,lJ;n.H HHX xapaKTepHbI HBJleHH.H JJ;Hc»paK-
HH H HHTepc»epeHHH.
15.8. YpaBHeHHe 3JIeKTpOMarHHTHoH BOJIHLI
KaK OTMeqaJIOCb B npeJJ;bIJJ;yeM naparpacpe, 3neKTpOMarHHTHa.H BOJIHa xa.
paKTepH3yeTcH HanpHmeHHOCTblO E 3JIeKTpHqeCKOrO nOJIH H MarHHTHOH HH-
JJ;YKHeH 13 MarHHTHoro non.H.
IIycTb B,AOJlb OC:H OX (pHC. 15.13) pacnpOCTpaH.HeTCH nnOCKaH 3neKTpOMar-
HHTHaH BonHa, npHqeM BeKTOp Hanp.HmeHHOCTH E COBepmaeT KOJle6aHHH
BJJ;OJlb OCH Oz, a BeKTOp HHJJ;YKHH B - BJJ;OJlb OCH Oy.
IIpeJJ;nonomHM, qTO B HaqaJle KOOp,AHHaT KOJIe6aHHH BeKTopa E npOHCXO-
,AHT no 3aKOHY Ez= Eocos(oot + CPo), rJJ;e Eo - aMnJlHTYJJ;a KOJle6aHHH BeKTopa
HanpHmeHHOCTH, 00 = 21tv - HKJIHqeCKa.H qaCTOTa, CPo - HaqaJlbHaSl cpa3a
BOnHbI. TorJJ;a B HeKoTopoH TOqKe, OTCTO.HeH OT HaqaJl8 KoopJJ;HHaT Ha pac-
CTOHHHe OA = x, KOJle6aHHe npOH30HJJ;eT C 3ana3JJ;bIBaHHeM l:1t = , r,Ae c -
c
CKOpOCTb 3neKTpoM8rHHTHoH BonHbI (CKOpOCTb CBeT8 B B8KYYMe). CJle,AOB8Tenb-
HO, KOJIe6aHHe Hanp.HmeHHOCTH B TOqKe A npOHCXOJJ;HT no 3aKOHY
Ex = 0; Ey = 0; Ez = Eo cos( rot - ro; + <Po )-
TaK KaK Kone6aHH.H BeKTopOB HHJJ;YKHH B H HanpHmeHHOCTH E B 3JIeKTpO-
MarHHTHOH BonHe npOHCXOJJ;.HT B OJJ;HOH H TOH me cpa3e (CHHcpa3HO), TO, aHa-
JIOrH1:lHO paccymJJ;a.H, nOJIyqHM, qTO B TOqKe A KOJle6aHHe MarHHTHOH HHJJ;YK-
H:H npOHCXO,AHT no 3aKOHY
Bx = 0; B, = 0; By = Eo cos[ rot - rocx + 'Po ),
rJJ;e Bo - aMnJlHTYJJ;a Kone6aHHH HHJJ;YKHH MarHHTHoro nOJlH B 3JIeKTpOMar-
HHTHOH BOJlHe, CPo - HaqaJIbH8.H cpa3a KOJIe6aHHH. 1I0JlyqeHHbIe ypaBHeHHH H
eCTb ypaBHeHH.H 3JIeKTpOMarHHTHoH BOJlHbI.
436

MO,l\YJlH BeKTOpOB E H B B nJlOCKOH 3J1eKTpOMarHMTHOH BOJIHe CBSl3aHbI
COOTHOWeHHeM
2 B 2
EOEE =-.
J..1oJ..1
B BaKYYMe E = J..1 = 1 M B = E ,JEoJ..1o = E . r.ne Eo M J..1 0 - 3JIeKTpMtIeCKaSl M
C
MarHMTHa.H nOCTOSlHHble, C - CKOpOCTb CBeTa B BaKYYMe.
15.9. 3HeprHH 3JIeKTpOMarHHTHOH: BOJIHLI.
TIJIOTHOCTL rrOTOKa H3JIyqeHHH (rroBepxHocTHaH)
KaK OTMetIaJlOCb B  15.2. MeXaHHtIeCKHe ynpyrHe BOJIHbI nepeHOC.HT Me-
XaHHQeCKYIO 3HeprHIO, 3J1eKTpOMarHHTHbIe BonHbI Tome nepeHOCSlT 3HeprMIO,
HO 3HeprHIO 3neKTpOMarHMTHYIO. 8HeprHSl nepeHOCHTC.H B HanpaBJIeHHH pac.
npOCTpaHeHM.H BOJlHbI, T.e. B HanpaBneHHM BeKTopa v.
8HeprH.H 3neKTpOMarHHTHoro nOJISl tIMCneHHO paBHa CYMMe 3HeprMA ero
3neKTpMtIeCKOH H MarHMTHoA COCTaBn.HIOI..QMX:
WSM=WS+W M ;
E EE 2 B 2
W SM =V+-V.
2 2J..1oJ..1
B TeopMM MaKCBenJIa ,l\OKa3aHO, qTO Cpe,l\HMe 3HatIeHH.H 3JIeKTpMtIeCKOA H Mar-
HHTHOA COCTaBJISllOl..QMX 3HeprMM 3JIeKTpOMarHMTHoro nOJISl paBHbI MeiK,l\Y co6oA:
(W 3 ) = (W M ).
CJIe,l\OBaTeJIbHO, 3HeprH.H 3JIeKTpOMarHMTHoA BOJIHbl paBHa
2 B 2
W SM = 2(W 3 ) = 2(W M ). H03TOMY "M = EoEE V = - V.
J..1oJ..1
06'beMHaSl llJlOTHOCTb 3HeprHM 3neKTpOMarHMTHOH BOJIHbI COOTBeTCTBeHHO
paBHa
vv. 2 B 2
W - 3M. W E ,.. L'
3M - V ' 3M = oc..L:J = J..1oJ..1
8HeprH.H 3J1eKTpOMarHMTHOH BonHbI B JII060H 06JIaCTH: npOCTpaHCTBa 113Me-
H.HeTC.H nepMO,l\HqeCKH C TeqeHHeM BpeMeHH (CM. BbIwe  15.6).
MHTeHcHBHoCTb 3JIeKTpOMarHHTHoro H3J1yqeHH.H (noBepxHOCTHa.H nJIOTHOCTb
nOTOKa H3J1YtIeHH.H) 1 = W SM . TaK KaK WSM=WSMV, a V=Svilt, TO I=wsMV'
Silt
15.10. 1I3JIYQellHe H rrpHeM 3JIeKTpOMarHHTIILIX BOJIH
MCTOqHHKOM rapMOHHqeCKHX 3JIeKTpOMarHHTHbIX BonH .HBJISleTCSl OTKpbITbIA
KOJle6aTeJlbHbIA KOHTYP. M3J1yqeHHe paCCMOTpeHHoro Bblwe KOJle6aTeJlbHOrO
KOHTypa TOMCOHa ( 14.1) - 3aKpbIToro KOJIe6aTeJlbHOrO KOHTypa - MaJlO,
TaK KaK OH H3JIYQaeT B oKpymalOee npocTpaHcTBo HHtITOmHYIO tIaCTb 3Hep-
437

rHH (pHC. 15.14). STO 06'b.HCH.HeTC.H
TeM, qTO 3TOT KOHTYP npe.lJ;CTaBJI.HeT
C060H nOqTH 3aMKHYTYIO 3J1eKTpHqeC-
KYIO D;enb. B 3TOH D;enH CHJla TOKa B
.lJ;aHHbIH MOMeHT BpeMeHH O,AHHaKOBa
BO Bcex ee yqacTKax. TaKoH KOHTYP
OqeHb cJla60 H3J1yqaeT 3neKTpOMarHHT-
HbIe BOJlHbI. Kam.lJ;OMY yqaCTKY BHTKa
KaTYWKH KOHTypa COOTBeTcTByeT 6J1H3-
KO pacnOJlOmeHHbIH yqacTol< Ha npOTH-
BOnOJlOmHOM KOHD;e .lJ;HaMeTpa BHTKa.
no KOTOPOMY TOK npOXO.lJ;HT B npOTH-
BOnOJlOmHOM Hanp8BJleHHH (pHC. 15.15).
Ha 60nbwoM paCCTO.HHHH OT BHTKa
3TH yqaCTKH C03.lJ;a1OT MarHHTHble
nOJl.H I HH.lJ;YKD;HH KOTOpbIX nOqTH paB-
HbI no MO.lJ;YJlIO H HanpaBJleHbI B npo-
THBOnOJlO>KHble CTOpOHbI. B pe3YJlbTa-
Te B)J;aJlH OT KOHTypa nOJl.H OCJla6n.HIDT
.n;Pyr .n;pyra, TaK t{TO MarHHTHoe none
OKa3bIBaeTC.H cocpe.n;OTOqeHHbIM nHWb
BHYTPH KaTYWKH.
TaK me 06CTOHT .n;eno C 3J1eKTpHQeCKHMH nOJl.HMH 3ap.H.nOB Ha 06Knal{KaX
KOH.n;eHcaTopa. 3ap.H.n;bI paBHbI no MO,AYJlIO H npOTHBOnOJlOmHbI no 3HaKY. IIo-
qTH Bce 3J1eKTpHqeCKOe none cocpelIOTOqeHO Me>K)J;Y nJlaCTHHaMH, a B.n;aJlH OT
HHX nonSl 3ap.H.n;OB npOTHBononomHMX 3HaKOB nOqTH D;enHKOM KOMneHCHpYIDT
.n;Pyr ,Apyra.
TaKHM 06pa30M, 60nbWa.H QaCTOTa KOJle6aHHH B KOJle6aTeJlbHOM KOHType
elI(e He rapaHTHpyeT HHTeHCHBHoe H3J1yqeHHe 3neKTpOMarHHTHbIX BonH, XOTSI
3HeprHH H3J1yqeHH.H W - CO 4 .
He06xo.n;HMO nepeHTH OT 3aKpbITO-
ro KOJle6aTeJlbHOrO KOHTyp8 K OTKpM-
TO MY . ECJlH KOHTYP COCTOHT H3 Ka-
TYWKH H .nBYX nJlaCTHH, He napan-
JleJlbHbIX .n;pyr .n;pyry (pHC. 15.16), TO
qeM no,n: 60JlbWHM yr JlOM pa3BepHVTbi
3TH nnaCTHHbI, TeM 60Jlee CB060,AHO
3neKTpOMarHHTHbIe BonHbI H3J1YQaIOTCSI
B oKpymalOee npOCTpaHCTBO. rIpe-
.n;eJlbHbIM cJlyqaeM HBJI.HeTC.H pa3Hece-
HHe nnaCTHH H8 npOTHBOnOJlOmHbIe
CTOpOHbI KaTYWKH. IIpH TaKOM npeo6pa30BaHHH KOHTypa 6y.n;eT H3MeH.HTbCSI
1 E ES
qaCTOTa 3J1eKTpOMarHHTHbIX BOJIH. KaK H3BeCTHO, V = JLC ' r,Ae C =.
21t LC d
ECJlH pa3.lJ;BHraTb nJlaCTHHbI KOH.n;eHcaTopa, TO ero eMKOCTb npH 3TOM 6y.n;eT
YMeHbWaTbC.H, TaK KaK YBeJlHqHBaeTC.H d. MO>KHO YMeHbwHTb H HH.n;YKTHB-
HOCTb KaTywKH, eCJlH 3aMeHHTb ee npHMbIM npOBo.n;oM (pHC. 15.16, a, 6, 8).
+
E
I
.... - - _,B
,.., J
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Puc. 15.14
C/)
Puc. 15.15
Puc. 15.16
438
B

'tfacToTa Kone6aHHH B KOHType B03paCTaeT. TaKHM 06pa30M, MbI nonyqHJlH -
aHTeHHY. B aHTeHHe 3apH,lJ;bJ yme He COCpe,lJ;OTOqeHbI Ha KOHax, a pacnpe,lJ;e-
neHbI no BceMY npOBO,lJ;HHKY. TOK B )J;aHHbIH MOMeHT BpeMeHH BO Bcex ceqe-
HH.HX npOBO,lJ;HHKa HanpaBJleH B O,lJ;HY H TY >Ke CTOPOHY, HO CHJla TOKa He
o,n;HHaKOBa B pa3J1HqHbIX CeqeHH.HX npOBO,AHHKa. Ha KOHuax OHa paBHa HYJlIO.
a nOCepe,l1HHe ,l10CTHraeT MaKCHMYMa. (HanOMHHM, qTO B 06bI'tJHbIX en.HX ne-
peMeHHoro TOKa CHJla TOKa BO Bcex CeqeHH.HX B ,lJ;aHHbIH MOMeHT BpeMeHH O,n;H-
HaKOBa. )
BnepBbIe aHTeHHY H306peJl A.C. IIonoB B
1895 r., 3aTeM OH ee YCOBepllIeHCTBOBan, HC-
nOJlb3YH .HBJleHHe pe30H8Hca. A.C. IIonoB OCTaBHJI
KOHTYP HeH3MeHHbIM, HO O,n;HH KOHe KaTYIliKH
3a3eMJlHn. a K ,n;PyrOMY KOHY npHcoelIHHHn
BepTHKaJlbHbIH npOBO,n; A (pHC. 15.17) co CB060,lJ;-
HbIM BepXHHM KOHOM. STOT BepTHKaJlbHbIH npo-
BO,lJ; npHHHTO Tenepb Ha3bIBaTb CHH>KeHHeM. Ko-
ne6aTeJlbHbIH KOHTYP c aHTeHHOH Ha3bIBaeTC.R OT-
KpLITLIM KOJle6aTeJlLHLIM KOHTypOM. ,Z:(JI.H Xopo-
llIero H3J1Y'tJeHH.H H npHeMa ,lJ;nHHa aHTeHHbI ,lJ;OJl-
mHa 6bITb paBHa HJlH KpaTHa nOJlOBHHe ,n;nHHbI
BOJlHbI. B 3TOM cnyqae aHTeHHa 6Y,lJ;eT HaCTpoe-
Ha B pe30HaHC C reHepaTopOM Kone6aHHH.
ECJlH Ha nYTH BonH, H3nyqeHHbIX aHTeHHOH reHepaTopa Kone6aHHH nepe-
,AaT'lHKa, paCnOJlOmHTb KOJle6aTeJlbHbIH KOHTYP, nepeMeHHoe 3neKTpOMarHHT-
Hoe nOJle B036Y,lJ;HT B KOH"rype nepeMeHHbIe 3J1eKTpHqeCKHe TOKH, T.e. B HeM
6Y,AYT npOHCXO,AHTb BbIHym,AeHHbIe 3J1eKTpOMarHHTHble KOJle6aHH.H. Kor,n;a qa-
CTOTa 3J1eKTpOMarHHTHbIX BOJIH ,lJ;aJleKa OT 'laCTOTbI c06cTBeHHLIX Kone6aHHH
KOHTypa, aMnnHTY,lJ;a BbIHym,lJ;eHHbIX KoneoaHHH OqeHb Maw'la. JIHllIb npH CO-
Bna,lJ;eHHH qaCTOTbI BbIHym,lJ;eHHbIX KOJle6aJlllH c co6CTBeHHOH qaCTOTOH KOHTY-
pa B03HHKalOT 3aMeTHbIe 3J1eKTpOMarHHTHbIe Kone6aHHH. T.e. Ha6nlO,lJ;aeTC.H
3neKTpHqeCKHH pe30HaHC.
BnepBbIe 3J1eKTpOMarHHTHbIe BOJlHbl 3KcnepHMeHTaJlbHO nonyqHJI, nepe-
,n;aJl H8 paCCTO.HHHe (npaB,lJ;a, B npe,n;eJlax CTona) H npHH.Hn reHpHX rep.
B KaqeCTBe KOJle6aTeJlbHbIX KOHTypOB OH Hcnonb30BaJl TaK Ha3bIBaeMbIe ,lJ;H-
nOJlIi repa (BH6paTopbI repa): ,n;Ba CTep>KH.H C llIapHKaMH, Me:tK,AY KOTO-
pbIMH OCTaBJleH onpe,n;eJleHHbIH 3a30p. K llIapHKaM OT HH,AYKHOHHOH Ka-
TYIliKH no,n;Bo,n;HJlH ,lJ;OCTaTOqHO BbICOKoe HanpH>KeHHe, H Me>K,n;y HHMH npo-
CKaKHBaJla HCKpa - BbICOKOqaCTOTHbIH pa3p.H.lI: (nepeMeHHLIH TOK). CJlelIO-
BaTeJlbHO, B npOCTpaHCTBe B03HHKaJlO 6bICTponepeMeHHoe 3J1eKTpOMarHHTHoe
nOJle, pacnpOCTpaH.HnaCb 3J1eKTpOMarHHTHa.H BonHa, HHTeHCHBHOCTb KOTOpOH
6bIJla HaH60JlbllIeii B HanpaBJleHHH, nepneH,lJ;HKYJlHpHOM BeKTOpy Hanp.HmeH-
A
Puc. 15.17
HOCTH E.
IIpHeMHHK 3J1eKTpOMarHHTHbIX BOJIH r. rep C,lJ;eJlaJl aHaJlOrH'lHbIM 06pa-
30M, TOJlbKO paCCTO.HHHe Me>K,lJ;Y llIapHKaMH 6bIJlO YMeHbllIeHO. rep Ha6nlO-
,lJ;aJl 3J1eKTpOMarHHTHbIe KOJle6aHHH B npHeMHOM lSH6paTope no HCKopKaM, npo-
CKaKHBalOHM Mem)J;y npOBO,lJ;HHKaMH npHeMHoro BH6paTopa.
439

r. repIJ; He TOJlbKO nonYtIHn 3neKTpOM8rHHTHbie BOJIHbI, HO H 06HapymHJI,
qTO OHH BeYT ce6S:1 nO,A06HO pyrHM BH,A8M BOJIH. OB H86JIIO8JI OTp8meHHe
3J1eKTpOM8rHHTHbIX BonH OT MeTaJIJlHQeCKOrO JIHCT8, YCT8HOBHJI, qTO OHE p8C-
npOCTp8H.HIOTC.H np.HMOnHHeHHO B OHoponHOH cpene. HcnbITbIB810T npeJIOMJIe-
HHe Ha rpaHHIJ;e np03patIHbIX cpe, Ha6JIIO,AaJI HHTepcpepeHIJ;HIO H H8 OCHOBa-
HHE 3KcnepHMeHTaJlbHbIX .naHHbIX Onpe,AnHJI JlHHY 3J1eKTpOMarHHTHOH BOJlHbI
B onbIT8X fepIJ;8 A = 66 CM. A npH H3BecTHoH q8CTOTe MomHO BbItIHCJIHTb CKO-
pOCTb pacnpOCTpaHeHHS:I 3JIeKTpOMarHHTHbiX BOJIH v = AV. OH paCCtIHTan, qTO
8 M
V = 3 . 10 -, T.e. OHa paBH8 CKOpOCTH CBeT8 B B8KYYMe.
c
OnblTbI repIJ;a lIOKa3bIBaJIH B03MomHOCTb nepe,A8'lH H npHeMa 3J1eKTpOMar-
HHTHoro H3J1ytIeHH.H. OlIH8KO Ha npaKTHKe T8KOH cnoc06 nepen8tIH H npHeMa
BOJIH He npHMeH.HeTC.H, TaK KaK npH 3TOM B03HHK81OT 3aTYX81OHe KOJle6a-
HH.H, 8 BOJlHbI nepeHOC.HT HHITomHYIO 3HeprHIO, KOTOpylO MomHO YJlOBHTb
JIHlllb H8 M8JIOM paCCTO.HIU1H.
15.11. 1I306peTeHHe panHO A. C. TIOIlOBbIM
OnbITbI fepIJ;8 nOKa3aJIH, 'lTO C nOMOblO
3J1eKTpOM8rHHTHbIX BOJIH MomHO OTnp8BJI.HTh H
A npHHHM8Tb CHrHaJIbI, HO Bce 3TO ,AeJIaJIOCb H8
O'leHb M8J10M p8CCTO.HHHH, B npeeJl8X CTOJl8
Jla60p8TopHH. llpoBeSI B8mHblH ,AJI.H H8YKE 3K-
cnepHMeHT, repu, He YBHeJl np8KTHqeCKOH
IJ;eHHOCTH HCnOJIb30B8HH.H 3J1eKTpOMarHHTHbIX
BOJIH H ame caM OTpHIJ;aJl B03MomHOCTb HX
npHMeHeHH.H .
OHaKO 3TH onbITbI 38HHTepeCOB8J1H cpH3HKOB
Bcero MHp8. B POCCHH O,AHHM H3 nepBbIX 3aH.HJI-
CSI H3Y'leHHeM 3J1eKTpOM8rHHTHbIX BOJIH npenona-
BaTeJlb BbIClllero YQe6HOrO 3aBeeHH.H B KpOlllll-
T8Te A.C. llonoB, C038BlllHH B anpeJIe 1895 r.
nepBbIH B MHpe paHonpHeMHHK, B KOTOpOM
npHeM CHrH8J10B perHCTpHpOB8JIC.H C nOMOblO
3J1eKTpHqeCKOrO 3BOHK8.
CxeMa nepeaTtIHKa IIonoB8 H306pameHa H8
pHCYHKe 15.18. KOJIe6aTeJIbHbIH KOHTYP COCTOHT
H3 HHYKTHBHOCTH BTOPHqHOH 06MOTKH HHYKIJ;HOHHOi1 KaTYlllKH L, nHTa-
eMoH 6aTapeeH B, H eMKOCTH - HCKpOBoro npOMemYTKa ab. EcnH HamaTb H8
KJllOq K, TO B HCKpOBOM npOMemYTKe KaTYlllKH npOCK8KHBaeT HCKpa, npeCTaB-
JI.HIOa.H C060H BbICOKOtIaCTOTHbIH pa3p.H H BbI3bIBalO8.H 3JIeKTpOMarHHTHbIe KO-
ne6aHHS:I B aHTeHHe A.
AH1'eHHa .HBJI.HeTC.H OTKpbITbIM BH6p8TOpOM H H3J1yqaeT 3neKTpOMarHHTHbIe
BOJlHbI, KOTopbIe, OCTHrHYB aHTeHHbI npHeMHOH CTaHI.J;HH, B036ym,Aa1OT B Hell
3J1eKTpHQeCKHe KOJIe6aHH.H.
,lJ.JI.H perHCTpaI.J;HH npHH.HTbIX BonH A.C. llonoB npHMeHHJI CneI.J;HaJIbHbIll
npH60p - Korepep K, COCTO.HHH H3 CTeKJI.HHHOH Tpy6KH, B KOTOpOH Haxo-


K
Puc. 15.18
440
a
b

A
,A.HTC.H MeTaJlnHTleCKHe OnHnKH
(pHC. 15.19). B J'IeBbIH KOHe
-rPy6KH BBe,n;eHa MeTannHqeCKa.H
nnaCTHHKa B, B npRBbIH - npo-
BO,A C, conpHKaCalOntHHCSI C OnHJI-
KaMH. B 06bI'lHbIX YCJlOBHSIX co-
npOTHBneHHe onHnOK BenHKO. HO
no,A ,AeHCTBHeM 3neKTpH'leCKHX KO-
N JIe6aHHH MemJJ;Y HHMM npOCKaKH-
BalOT Ma.TIeHbKHe HCKOpKH, onHnKH
cnHnalOTCSI, H conpOTHBneHHe KO-
repepa pe3KO YMeHbillaeTC.H. Ecnli
BCTp.HXHYTb TPy60qKY HnH Cnel'Ka
y,n;apHTb no HeH, TO onHnKH pac-
na,AaIOTC.H, H HX conpOTHBneHHe
CHOBa B03paCTeT. A.C. [JonOB
BKJllOqHn Korepep B enb. COlIep-
.>KaII(YIO MCTOqHHK S,n:C B H 3BOHOK, MOJlOTOqeK KOToporo npH ,AeHCTBHH 3BOH-
Ka Mor y,n;ap.HTb no pe3HHOBOH Tpy6Ke T. KorA8 conpOTHBJIeHHe Korepepa Be-
nHKO, CHJla TOKa, nOCTO.HHHO HAynterO B llenH BBCNB. He,AOCTaTOqHa AnSI
npHT.HmeHH.H .HKOp.H B peJle. C nO.HBJleHHeM 3neKTpOMal'HHTHOH BOJIHbI conpo-
THBneHHe Korepepa naaeT, CHJla TOKa B u,enH BBCNB YBenHTlHBaeTcSI. SIKOpb S
pene 3aMbIKaeT B TOqKe Q enb 3J1eKTpOMarHHTa M l' BKnlOqeHHOrO napan-
neJlbHO enH Korepepa, H MOJlOTO'leK 3BOHKa CHI'HanH3HpyeT 0 npHXOAe Bon-
HbI. ITpH 3TOM enb 3J1eKTpOMaI'HHTa }v[ pa3MbIKaeTCSI, H MOJlOTOqeK y,n;ap.HeT
no KOl'epepy. ConpOTHBJleHHe KOl'epepa YBeJlHqHBaeTC.H, II peJle pa3MbIKaeT
llenb 3BOHKa. Ha O,n;HHOqHYIO BOJIHY npH60p OTBeqaeT KOpOTKJiM 3BOHKO, a
Ha HenpepbIBHO npHHHMaeMbIe BOJlHbI - QaCTbI:\1H 3BOHKaMH 'lepe3 paBHbIe
npOMemYTKH BpeMeHH.
Pene n03BOnHJlO A.C.ITonoBY He pel'HCTpHpOBaTb HenocpeACTBeHHO npHHH-
MaeMbIe aHTeHHO:H BOJlHbI, a Hcnonb30BaTb HX ManylO 3Hepl'HIO An.H ynpaBJIe-
HH.H HCTOqHHKOM 3HeprHH. KOTOpbIH nHTaeT annapaT, peI'HCTpHpYIOII(I1H no-
SIBJleHHe 3TH X BOJIH.
B HIOHe 1895 1'. A.C. ITonoB YCOBepIlleHCTBOBaJl CBOH npHeMHHK. A06aBliB
K HeMY lInSI nOBbIIlleHHSI qYBCTBHTeJlbHOCTH BepTHKaJlbHbIH npOBO,n; - npHeM-
HYIO aHTeHHY, a B MapTe - TeJlel'paQ>HbIH annapaT AnSI npHeMa CJ'IOBeCHOro
TeKcTa, H nonyqHJI B03MomHOCTb 3anHCbIBaTb npHHHMaeMbIe CHrHanbI Ha Te-
nerpat}>HYIO JleHTY. 24 MapTa 1896 1'. 6bIJlH nepe,n;aHbI nepBbIe CJlOBa "feHpHx
fep" C nOMOII(blO a36YKH Mop3e.
)]:06HBillHCb ycnexa, A.C. IIonoB npO,AOJlmaJl onbITbI no YBeJlHqeHHIO ,lXanb-
HOCTH pa,n;HOCB.H3H. OH Hcnonb30Ban .HBJleHHe pe30HaHca, ,lXn.H qero npHMeHIIJI
B CBOHX npH60pax 3J1eMeHTbI HacTpoHKH Ha onpeAeneHHYIO ,AJlHHY BOJlHbI, H
yme B 1898 r. A.C. IIonoB ocyecTBHJI pa)J;HOCBSI3b Mem)J;y ,n;BYM.H Kopa6J1.H-
MH Ha paCCTO.HHHe 5 KM.
B 1899 r. ero yqeHHK IT.H. PbI6KHH o6HapymHJI B03MomHOCTb npHeMa pa-
,lXHOTeJlerpat}>HbIx CHI'HaJlOB "Ha cJlYx". BCKope nocJle 3Toro A.C. IIonoB CKOH-
CTpYHpOBaJI nepBbIH CnellHRJIbHbIH paJJ;HonpHeMHHK H TeM caMbIM nOJlOmHn Ha-
qaJlO pa3BHTHIO pa,lXHOTeJlecpoHHoH CB.H3H.
M
IIIIII
B
K 1
Puc. 15.19
441

XOT.H COBpeMeHHbIe pa,nHOnpHeMHHKH OqeHb MaJIO HanOMHHalOT npHeMHHK
IIonoBR. OCHOBHbIe npHHII.HnbI HX ,neRCTBH.H Te me. B JlI060M npHeMHHKe HMe-
eTC.H aHTeHHa, B KOTOpOR npHXOlIHIII.a.H BOJlHa BbI3bIBaeT OqeHb cna6bIe 3J1eKT-
pHqeCKHe KOJle6aHHH. 3TH CJla6bIe CHrHaJlbI ynpaBJlHIOT HCTOqHHKaMH 3Hep-
rHH, nHTalOIII.HMH nocJleAYIOIII.He II.enH.
15.12. npHHlJ;Hn paHOTeJIe4>oHHoH CBH3H
3J1eKTpOMarHHTHbIe BOJlHbI B HaCTO.HI..qee BpeM.H llIHpOKO npHMeHHIOTCH B
pa3nHQHbIX 06naCT.HX pa,nH03J1eKTpOHHKH (oTpacnb HaYKH H TeXHHKH, Hcnonb-
3YIOI..qaH 3J1eKTpOMarHHTHbIe BOJlHbJ ,nJl.R nepe,naqH, npHeMa H npe06pa30BaHH.H
HHQJopMaII.HH): pa,nHOCB.H3b, pa,AHOBeI..qaHHe, TeJleBH,AeHHe, pa,AHOJlOKaII.HH. pa-
,AHoynpaBneHHe, pa,AHOTeJleMeTpH.H, pa,AHOMeTeOpOJlOrH.H, pa,nHopa3Be,AKa H ,np.
OCTaHOBHMCH Ha npHHII.HnaX cOBpeMeHHoR pa,nHoTeneQJoHHoR CB.H3H.
Pa,AHOnepeAaqH CTanH B03MomHbI nocne C03,AaHH.H reHepaTopa He3aTyxalO-
I..qHX Kone6aHHR. IIpH pa,nHOTeneQJOHHOR CB.H3H Kone6aHH.R ,AaBne-
HHH B03,nyxa B 3BYKOBOR BonHe C nOMOI..qblO MHKPOQJOHB npeBpaI..qaIOTC.H B
3neKTpHqeCKHe Kone6aHH.H TOR me QJOPMbI. Tpy,nHOCTb nepe,naqH 3BYKoBoro
CHrHaJla COCTOHT B TOM, qTO ,nnH pa,nHOCB.H3H He06xo,nHMbI KOJle6aHHH BbICO-
KOR qaCTOTbI, a KOJle6aHH.H 3BYKOBoro ,nHana30Ha - HH3KOqaCTOTHbIe KOJle6a-
HHH, ,nJl.H H3J1YQeHHH KOTOpbIX HeB03MomHO nOCTpOHTb 3QJQJeKTHBHbIe aHTeH-
HbI. IIo3TOMY KOJle6aHH.H 3BYKOBOR 'l8CTOTbI npHXOAHTCH TeM HJlH HHbIM cno-
C060M HaKna,nbIBaTb Ha Kone6aHH.H BbICOKOR qaCTOTbI. KOTopbIe yme nepeHO-
C.HT HX Ha 60JlbllIHe paCCTOHHHH.
PalIHonepe,nalOI..qee YCTPORCTBO CO,AepmHT
cne,nYIOI..qHe OCHOBHbIe 3neMeHTbJ (pHC. 15.20):
r - 3a,AalOIII.HH reHepaTOp Kone6aHHR BbI-
COKOR qaCTOTbI npe06pa3YIOHR 3HeprHIO HC-
TOqHHKa nOCTO.HHHOrO HanpHmeHH.H B rapMoHH-
'leCKHe KOJle6aHHH BbICOKOR qaCTOTbI. 1..JacToTY
3THX Kone6aHHR Ha3bIBalOT HecYI..qeR. OHa ,non-
mHa 6bITb CTporo nOCTOHHHOR;
MK - npeo6pa30BaTenb C006I..qeHHR B 3J1eK-
TpH'leCKHR CHrHaJI, Hcnonb3yeMbIR ,£J.JI.H MO,AynSl-
II.HR KOJle6aHHR HeCYI..qeR qaCTOTbI. BH,n npeo6-
pa30BaTeJl.H 3aBHCHT OT QJH3HQeCKOH npHpo,nbI nepe,naBaeMoro CHrHaJla: npli
3BYKOBOM CHrHqe npe06pa30BaTeJleM .HBn.HeTC.H MHKPOQJOH, npH nepe,naqe H306-
pameHHH - nepelIalOllla.H TeJleBH3HOHHaH Tpy6Ka:
M - Mo,nyn.HTOp, B KOTOpOM npOHcxo,nHT MOAYJl.HII.H.H BbICOKOqaCTOTHoro
CHrHaJla B COOTBeTCTBHH C qaCTOTOH 3BYKOBoro CHrHana, HeCYI..qero HHopMa-
II.HIO, nO,AJlemaIII.YIO nepe,Aaqe;
YBIf - 06bIQHO HMeeTC.R O,nHH HnH ,ABa KaCKa,na YCHJlIiTen.H MOI..qHOCTIi
MO,AYJlHpOBaHHoro CHrHana;
Al - H3J1yqalOIII.a.H aHTeHHa, npe,nHa3HaqeHHa.H ,nJl.H H3J1yqeHH.H 3J1eKTpO-
MarHHTHbIX BOJIH B oKpymalOee npOCTpaHCTBO.
Pa,nHonpHeMHoe YCTPOHCTBO npe,nHa3HaqeHO ,AJI.H npHeMa HHQJopMaII.HH. ne-
pe,AaBaeMOH C nOMOI..qblO 3neKTpOMarHHTHbIX BOJlH, H3nyqaeMbIX nepe,AalOI..qeR
Puc. 15.20
442

aHTeHHOH pa.ZJ;HOnepe.ZJ;aTqHKa. Pa-
.ZJ;HOnpHeMHOe YCTPOHCTBO CO,Aep-
mHT CJle.ZJ;YIOHe OCHOBHbIe 3ne-
MeHTbI (pHC. 15.21):
A 2 - npHeMHaSl aHTeHHa CJlY-
mHT .ZJ;JI.H ynaBJlHBaHH.H 3neKTpo-
MarHHTHbIX KOJle68HHH. BbIBalOT
aHTeHHbI, paCCqHTaHHbIe Ha npR-
Puc. 15.21 eM Kone6aHHH CTporo onpe.ZJ;eJleH-
HOH qaCTOTbI (HaCTpoeHHbIe 8H-
TeHHbI). H aHTeHHbI. He HaCTpoeHHbIe Ha onpe,AeneHHYIO qaCTOTY (BceBonHO-
BbIe aHTeHHbl). B nOCne,lIHeM cnyqae B aHTeHHe B03HHKalOT BbIHym.ZJ;eHHbIe MO-
,AY JlHpOBaHHbIe KOJle6aHH.H, B036ym.ZJ;8eMbIe pa3J1HqHbIMH pa,nHOCTaHH.HMH;
P K - pe30HaHCHbIH KOHTYP, HaCTpaHBaeMbIH Ha onpe.n;eJleHHYIO QaCTOTY.
KOTOpbIH H3 MHomeCTBa npHHSlTbIX aHTeHHOH CHrHanOB Bbl,AenSieT none3HbIH
CHrHan;
YB£f - B PK B pe3YJlbTaTe pe30HaHca npOHCXO.ZJ;HT YBenHqeHHe aMnJlHTY-
,AbI Hanp.HmeHH.H npHH.HTbIX Kone6aHHH. O.ZJ;HaKO npH 3TOM .ZJ;onOJlHHTenbHa.H
BbICOKOqaCTOTHaSi 3HeprH.H He C03.n;aeTC.H Ii MOHOCTb npHHSlToro CHrHana He
B03paCTaeT. Bonee Toro, OHa .n;ame HeCKonbKO YMeHbllIaeTC.H H3-38 HeH36em-
HbIX nOTepb 3HeprHH H8 aKTHBHOM conpOTHBneHHH BXO,AHOH enH. MOHOCTb
npHHSlToro CHrH8.JIa HCKnlOQHTeJlbHO Mana. II03TOMY B YCHJlHTene BbICOKOH Qa-
CTOTbI nOBbIlliaeTCSI HanpSlmeHHe npHH.HTOrO CHrHana H YBenHqHB8eTCSI ero
MOHOCTb;
J( - .n;eTeKTopHbIH KaCKa.n;. 3.n;eCb YCHJleHHbIH MO,AYJlHpOBaHHbIH BbICOKO-
qaCTOTHbIH CHrHaJl npeo6pa3yeTC.H H H3 Hero BbI,AeJlSleTC.H Mo,nynHPYIOHH CHr-
H8.JI, HeCYI..qHH nepe,AaBaeMYIO HH<popMaHIO. Cne,nOBaTenbHO, ,AeTeKTHpOBaHHe -
npoecc, 06paTHbIH MO.ZJ;YJI.HHH. B KaqeCTBe ,neTeKTopa liCnOJlb3YIOT npH60pbI
C HenHHeHHOH xap8KTepHCTHKOH - 3neKTpOHHble JlaMnbI H nonynpOBo,nHHKO-
BbIe npH60pbI;
YH£f - YCHnHTeJlb HH3KOH qaCTOTbI. BbI.n;eJleHHOe B .n;eTeKTopHOM KaCKa-
,Ae MO.n;YJlHpYIOee HanpSlmeHHe HH3KOH qaCTOTbI MaJlO Hero YCHnHBalOT B
YCHnHTeJle HH3KOH qaCTOTbI;
rB - nocne YCHneHH.H HH3KOqaCTOTHbIH CHrHan nocTynaeT Ha rpOMKoro-
BOpHTeJlb (Tene<pOH).
PK
YBQ
15.13. MOAYJUIl\HH
rapMOHHqeCKOe KOJle6aHHe E = Eosin(wt + CPo) xapaKTepH3yeTcSI TpeMSI no-
CTO.HHHbIMH BO BpeMeHH napaMeTpaMH: aMnnHTy.n;oH Eo, qaCTOTOH w H HaqaJlb-
HOH <p a30H CPo.
O.n;HaKo cyeCTBYIOT Kone6aHHSI, Y KOTOpbIX O.n;HH H3 3THX napaMeTpOB He
nOCTO.HHeH, a H3MeH.HeTC.H no onpe,AeJleHHOMY 3aKOHY. TaKHe KOJle6aHHSI Ha-
3bIBaIOTC.H MO]Q'J1HpOBaHHbIMH.
MO,nYJlHpOBaHHble KOJle6aHH.H 06blqHO nonyqalOT H3MeHeHHeM rapMOHH'leC-
Koro KOJle6aHHSI B COOTBeTCTBHH C KOJle6aHHeM, KOTopoe .n;n.H 3TOH enH BbI-
6Hpa1OT. STOT npoecc nOJlyqHJI Ha3BaHHe MOAYJlSlHH.
443

TaKHM 06pa30M, MOYJI.Hn;H.H - npOn;ecc H3MeHeHH.H OHOrO H3 napaMeT-
pOB rapMOHHqeCKOrO Kone6aHH.H B COOTBeTCTBHH C PyrHM Kone6aHHeM. Pa3-
JlH'tIalOT aMnnHTYHYIO, qaCTOTHYIO H «Pa30BYIO MO,AYJI.Hn;HH.
PaCCMOTpHM npHHn;Hn aMnJlHTYHOH Moyn.Hn;HH Kone6aHHH. IIpH aMnJlH-
TYHOH MOYJI.Hn;HH co 3BYKOBOR qaCTOTOH H3MeH.HIOT aMnnHTY,AY BbICOKOqaC-
TOTHbIX KOJle6aHHH. AMnJlHTY,AHYIO MO,AYJlS:ll1.HIO MOmHO OCYIlteCTBHTb cJlen;ylO-
IltHM cnoc060M.
B CaMOM npOCTOM YCTpOHCTBe ,AJlSI peaJlH3al1.HH aMnJlHTYHOii Moyn.Hn;HH
nOCJle,nOBaTeJlbHO C HCTO'tlHHKOM nOCTO.HHHOrO Hanp.HmeHH.H BKnlO'tIalOT 1I0nOn-
HHTeJlbHbIH HCTOqHHK nepeMeHHOrO Hanp.HmeHH.H HH3KOR 'tIaCTOTbI. 3THM HC-
TOqHHKOM MOmeT 6bITb, HanpHMep, BTOpHqHa.H 06MOTKa TpaHC«pOpMaTOpa, ecnH
no ero nepBHqHOH 06MOTKe npoxo,nHT TOK 3BYKOBOH qaCTOTbI. AJI.H SToro
(pHC. 15.22) B nepBH'tIHYIO n;enb TpaHc«popMaTopa BKnlOqalOT HCTOqHHK nOCTO-
Puc. 15.22
a
I
1!'" !, ii"""'!
0""""""""" t
6
: .t
6
1
Puc. 15.23
444
e
@
I Cu,.
t
.HHHOrO Hanp.HmeHHS:I H MHKPO«pOH.
MHKpO«POH npeCTaBJI.HeT C060H nepe-
MeHHoe conpOTHBJleHHe, BeJlH'tIHHa
KOToporo 3aBHCHT OT aBJleHH.H Ha
ero MeM6paHY. Kor,Aa nepe MHKpO-
«pOHOM HeT HCTOqHHKOB 3BYKa, ero
conpOTHBJleHHe nOCTO.HHHO, H B n;enH
MHKpo«poHa (nepBH'tIHa.H n;enb TpaHc-
«popMaTopa) HeT CJla6bIH nOCTO.HH-
HbIH TOK. reHepaTop c03aeT He3aTY-
xalOHe Kone6aHH.H BbICOKOR 'tIaCTO-
TbI (pHC. 15.23, a). Kor,na nepe,A
MHKpO«pOHOM BocnpOH3BO.HTC.H Ka-
KHe-HH6Yb 3BYKH, ero conpOTHBne-
HHe H3MeH.HeTC.H co 3BYKOBOH 'tIaCTO-
TOR. AHaJlOrHqHO H3MeH.HeTC.H TOK B
n;enH MHKpo«poHa, H 6J1aroap.H .HBJle-
HHIO SJleKTpOMarHHTHOH HHYKn;HH BO

BTOPHqHOH enH TpaHC<pOpMaTOpa B03HHKaeT nepeMeHHOe Hanp.HmeHHe 3BYKO-
BOH qaCTOTbI, KOTOpOe COOTBeTCTBeHHO YBenHqHBaeT HnH YMeHblllaeT cHny TOKa
B enH KOJlneKTOpa.
AMnJlHTy,/:{a KOJle6aHHH HeCYI.I:(eH 'tlaCTOTbI Ha'tlHHaeT H3MeH.HTbC.H co 3BYKO-
BOH qaCTOTOH - nOJlyqalOTC.H MO,/:{YJlHpOBaHHbIe KOJle6aHH.H, KOTOpble H3ny'tla-
IOTC.H B npOCTpaHCTBO.
BpeMeHHYIO pa3BepTKY MO,AYJlHpOBaHHbIX Kone6aHHH MOmHO HenOCpe,/:{CTBeH-
HO Ha6J1IO,/:{aTb Ha SKpaHe OCHnnOrpa<pa, eCnH nO,/:{aTb Ha HerO Hanp.HmeHHe
C KOJle6aTenbHoro KOHTypa.
TaK, eCJlH 3BYKOBa.H BonHa HMeeT BHlI. H306pameHHbIH Ha PHCYH-
Ke 15.23, 6, TO aHTeHHa 6Y,AeT H3J1yqaTb Mo,/:{ynHpoBaHHbIe BOJlHbI, H30-
6pameHHbIe Ha pHCYHKe 15.23, 6.
15.14. .lJ:eTeKTHpoBaHHe. IIpocTeiimHii paJJ;HorrpHeMHHK
L
Puc. 15.24
I
o
Puc. 15.25
I
o
Puc. 15.26
C 1
ITO,/:{ B03,AeHCTBHeM 3neKTpoM8rHHT-
HbIX BonH, H3J1yqaeMbIX aHTeHHaMH ne-
pe,/:{aT'lHKOB, B aHTeHHe pa,/:{HonpHeMHH-
Ka (pHC. 15.24) B03HHKalOT BbIHymlIeH-
HbIe MO,/:{y nHpOBaHHbIe Kone6aHH.H.
O,/:{HH H3 3neMeHTOB KOHTypa, HanpH-
Mep KOHeHcaTOp C, ,/:{eJlaeTC.H nepeMeH-
HOH eMKOCTH ,/:{JI.H HacTpoHKH B pe30-
HaHC C KaKOH-TO cTaHHeH. B KOHType
B03HHKHYT MO,/:{y JlHpOBaHHbIe BbICOKO'tla-
CTOTHble KOJle6aHH.H (pHC. 15.25). STOT
BbICOKO'tlaCTOTHbIH MO,/:{y nHpOBaHHbIH
CHrHaJl He cnoc06eH Henocpe,/:{CTBeHHO
BbI3BaTb Kone6aHH.H MeM6paHbI Tene<po-
Ha CO 3BYKOBOH qaCTOTOH. OH TOJlbKO
B036Y,/:{HT BbICOKOqaCTOTHbIe KOJle6aHH.H,
He BocnpHHHMaeMbIe HalllHM YXOM.
ITpH 60nblllOH qaCTOTe MeM6paH8 BCne,A-
CTBHe HHepTHOCTH He 6Y,AeT ycneBaTb
CMeI.I:(8TbC.H CKonbKO-HH6Y,Ab 3Ha'tlHTenb-
HO 3a Manoe BpeM.H. paBHoe nepHO,/:{y
BbICOKOqaCTOTHbIX KOJle6aHHH. ITOSTOMY
STOT MO,/:{y nHpOBaHHbIH CHrHaJl nOl1aeT-
C.H Ha ,/:{eTeKTOp (BaKYYMHbIH HnH KpH-
CTannH'tleCKHH nOJlynpOBO,/:{HHKOBbIH
,/:{HO,/:{) , 06na,/:{alOI.I:(HH O,/:{HOCTopOHHe:H
npOBO,/:{HMOCTblO. ITocne npoxom,/:{eHHH
.o;eTeKTopa nonyqalOTC.H ,AeTeKTHpOBaH-
HbIe KOJle6aHH.H, npe,ACTaBJI.HIOI.I:(He CO-
KOJle6aHHH BbICOKOH H HH3KOH qaCTOT
t
t
60H cynepn03HHIO BbInp.HMneHHbIX
(pHC. 15.26).
nSl OKOH'tlaTenbHoro OT,/:{eneHH.H 3BYKOBOH q8CTOTbI OT BbICOKOH napannenb-
HO C TeJle<poHoM BKJllOqalOT 6noKHpOBO'tlHbIH KOH,AeHCaTOp C l' Ero eMKOCTb
445

no6HpaeTCSl TaK, qT06bI eMKOCTHOe CO-
npOTHBJleHHe (  I )J;JI.H HH3KHX qaC-
· wC I J
t
TOT 6bIJlO rOpa30 60JlblIIe, qeM HH,AYK-
THBHOe COnpOTHBJleHHe TeJlecpoHa. A Te-
JIecpoH npeCTaBJI.HeT 60JlblIIOe HHYK-
THBHoe conpOTHBJleHHe (wL) JI.H BbICOKOH 'tIaCTOTbI. Tora TOKH BbICOKOH 'la-
CTOTbI nOHYT qepe3 KOHeHcaTOp, a HH3KOH - qepe3 TeJlecpoH. MeM6paHa
TeJlecpoHa 6yeT KOJle6aTbC.H aHaJlOrHqHO MeM6paHe MHKpocpoHa, H MbI YCJlbI-
illHM TOqHO TaKoH me 3BYK, KaK nepe MHKpOcpOHOM (pHC. 15.27).
J{eTeKTopHbIH npHeMHHK OqeHb npOCT, HaemeH, He Tpe6yeT HCTOqHHKOB
nHTaHH.H. 3TO OTCYTCTBHe HCTO'tlHHKa .HBn.HeTC.H H He.nOCTaTKOM ero. OH npH-
HHMaeT CHrHaJlbI TOJlbKO OT 6JIH3KHX H
OqeHb MOHbIX pa,AHOCTaHHiL ECJIH
He06xoHMO npHH.HTb CJIa6bIH CHrHaJI,
TO ero HymHo npe,ABapHTenbHO YCH-
nHTb. J{JI.H 3Toro HCnOJlb3YIOT, HanpH-
Mep, TpHO, KOTOpbIH HrpaeT B HRlIIeH
cxeMe pOJlb eTeKTopa H YCHJIHTeJl.H O-
HOBpeMeHHO. CxeMa rrpOCTeHlIIero npH-
eMHHKa C JlaMnOBbIM YCHJlHTeJleM nOKa-
3aHa Ha pHCYHKe 15.28.
MO,AYJlHpOBaHHbIe BbIHymeHHbIe KO-
JIe6aHH.H KOHTypa no.naIOTC.H Tenepb Ha
ceTKY TpHoa H BbI3bIBaIOT 3Ha'tlHTeJlbHbIe H3MeHeHH.H CHJlbI TOKa B ero aHoHoH
u.enH, KOTopbIe npOHCXO,AHT B TaKT C KOJle6aHH.HMH, H3J1yqReMbIMH aHTeHHOH ne-
pe.naT'tIHKa. A ,AaJlee YCHJIeHHbIH .neTeKTHpoBaHHbIH CHrHaJl nona.naeT Ha Tene-
cpOH T H cpHJlbTp C h r.qe HH3Ka.H qaCTOTa OKOH'tlaTeJlbHO OT,lIeJIHeTCH OT BbICOKOH.
,AJI.H 3HR'tIHTeJlbHOrO YCHJIeHHH HCnOJlb3YIOTC.H MHorOKaCKaHble npHeMHHKH.
I: 
Puc. 15.27
t--I
T
Puc. 15.28
C 1
15.15. PanUOJIOKa:U:UJI
B cOBpeMeHHoH TeXHHKe .HBJleHHe OTpameHHSI paHOBOJlH pa3JIHqHbIMH npe-
n.HTCTBH.HMH HaXO)J;HT lIIHpOKoe npHMeHeHHe. BbICOKOqYBcTBHTeJlbHbIe npHeMHH-
KH YJlaBJlHBalOT I-t YCHJlHBalOT oTpameHHbIH CHrHan C u.eJlblO nonyqHTb HHcpOp-
Mau.HIO 0 TOM, re HRXOHTC.H TOT npe,AMeT, OT KOToporo OTpa3HJlaCb BOJlHa.
06HapymeHHe H TOqHOe onpe)J;eJleHHe MeCTOHaXO)KeHH.H 06'beKTOB C nOMO-
blO pa,AHOBOJlH Ha3bIBRIOT pa.lJ;HOJIOKaIJ,HeH. Pa,AHOJlOKau.HOHH8H YCTaHOBK8 -
paJl.HOJIOKaTOp (pa.nap) - COCTOHT H3 nepe.nalOeH H npHeMHOH qaCTeH. B pa-
)J;HOJlOKaHH HCnOJlb3YIOT 3J1eKTpH'tIeCKHe KOJle6aHH.H CBepXBbICOKOH qaCTOTbI
(CB"tJ) (10 8 -10 11 r). MOHbIH reHepRTop CB"tJ CB.H3RH C RHTeHHoH, KOTopaSl
H3J1yqaeT oCTpoHanpRBJIeHHYIO BOJIHY. B paHOJlOKaTOpax, pa60TalOHx Ha
JlHHaX BOJIH nOp.HKa 10 CM H MeHblIIe, TaKa.H BOJlHa C03aeTCH aHTeHH8MH
B BHe napa60JlHqeCKHX 3epKaJl. ,AJI.H BOJIH MeTpOBoro HanR30Ha aHTeHHbI
HMelOT BH CJlOmHbIX CHCTeM BH6paTopoB. TIpH 3TOM OCTpa.H HanpaBJleHHOCTb
H3JIyqeHH.H nOJlyqaeTC.H BCJleCTBHe HHTepcpepeHu.HR BOJIH. AHTeHHa YCTpoeHa
TaK, qTO BOJlHbI, nOCJlaHHbIe Ka)KbIM H3 BH6paTopoB, npH CJlOmeHHH B3aHM-
446

HO YCHJlHBalOT )J;Pyr ,lJ;pyra nHIllb B 3a,lJ;aHHOM HanpaBJleHHH. B OCTaJlbHbIX Ha-
npaBJleHH.HX npH CJlOmeHHH BOJIH npOHCXO)J;HT nOJlHOe HJlH qaCTHqHOe HX B3a-
HMHOe raIlleHHe.
OTpameHHa.H BOJlHa YJlaBJlHBaeTC.H TOE: me H3J1yqalOWeH aHTeHHOH JlH60
,lJ;PyroH, Tome OCTpOHanpaBJleHHOH npHeMHOH aHTeHHOH.
AJI.H onpe,lJ;eJleHH.H paCCTO.HHH.H ,lJ;0 eJlH npHMeH.HIOT HMnYJlbCHbIH pemHM
H3J1yqeHH.H. reHepaTOp H3J1yqaeT BOJlHbI KpaTKOBpeMeHHbIMH HMnYJlbCaMH.
AJlHTeJlbHOCTb Kam,lJ;OrO HMnYJlbCa COCTaBJI.HeT MHJlJlHOHHbIe ,lJ;OJlH ceKYH,lJ;bI, a
npOMemYTOK Mem,lJ;Y HMnYJlbCaMH npHMepHO B 1000 pa3 60JlbIlle. Bo BpeMSJ
naY3 npHHHMaIOTC.H oTpameHHble BOJlHbI.
Onpe)J;eJleHHe paCCTO.HHH.H R npOH3BO,lJ;HTC.H nYTeM H3MepeHH.H 06ero Bpe-
MeHH t npOXOm,lJ;eHH.H pa,lJ;HOBOnH ,lJ;0 eJlH H 06paTHo. TaK KaK CKOpOCTb pa-
JIHOBOJIH v = 3 . 10 8 M B aTMoc«pepe npaKTHqeCKH nOCTO.HHHa. TO
C
BCJle,lJ;CTBHe BceB03MomHbIX nOTepb pa,lJ;HOBOJlH ,lJ;0 npHeMHHKa ,lJ;OXO,lJ;HT
JlHlllb HH'ITOmHa.H qaCTb TOH 3HeprHH, KOTOpylO H3J1Y'IaeT nepe,lJ;aTqHK. llo-
3TOMY npHeMHHKH pa,lJ;HOJlOKaTOpOB YCHJlHBalOT npHH.HTbIH CHrHaJl B 10 12 pa3.
TaKOH qYBCTBHTeJlbHbIH npHeMHHK, pa3YMeeTC.H, )J;OJlmeH 6bITb OTKJllOqeH Ha
BpeM.H nOCbIJlKH HMnYJlbCa nepe,lJ;aTqHKOM.
AJI.H «pHKcaHH nocnaHHoro H OTpameHHO-
ro CHrHaJlOB HCnOJlb3YIOT 3J1eKTpoHHo-JlyqeBYro
TPy6KY. B MOMeHT nOCbIJlKH HMnYJlbCa CBeT-
Jla.H TOqKa, paBHOMepHO ,lJ;BHmya.HC.H no 3K-
paHY 3J1eKTpoHHo-JlyqeBOH Tpy6KH, OTKJlOH.HeT-
C.H. Ha 3KpaHe nO.HBJI.HeTC.H "nCnJleCK" OKO-
JlO HYJleBOH OTMeTKH IllKaJlbI )J;aJlbHOCTH
(pHC. 15.29). CBeT.HeeC.H n.HTHbIIllKO Ha 3K-
paHe npO,lJ;OJlmaeT paBHOMepHO ,lJ;BHraTbC.H
B)J;onb III KaJlbl H B MOMeHT npHeMa cJla60ro
OTpameHHoro CHrHaJla CHOBa OTKJlOH.HeTC.H.
PaCCTO.HHHe Mem,lJ;Y "BcnnecKaMH" Ha 3KpaHe
nponopHOHanbHO BpeMeHH npOXOm,lJ;eHH.H CHr-
Hana H. Cne,lJ;OBaTeJlbHO. nponopHOHaJlbHO
paCCTO.HHHIO R 1I0 ueJlH. 8TO n03BOJl.HeT npo-
rpa,lJ;YHpOBaTb IllKaJlY HenocpellCTBeHHO B KH-
JlOMeTpax. Pa,lJ;HOJlOKaHOHHbIe YCTaHOBKH 06HapymHBalOT Kopa6J1H H CaMOJle-
TbI Ha paCCTO.HHH.HX ,lJ;0 HeCKOJlbKHX COT KHJlOMeTpOB. Ha HX pa60TY MaJlO
BJlH.HIOT YCJlOBH.H no ro,lJ;bI H BpeM.H CYTOK. B 60JlbIllHX a3ponopTax nOKaTopbI
CJle)J;.HT 3a B3neTalOHMH H H,lJ;YHMH Ha nOCa,lJ;KY CaMOJleTaMH. Ha3eMHa.H
cnym6a nepe)J;aeT no pa,lJ;HO nHJlOTaM He06xo)J;HMbIe YKa3aHH.H H TaKHM 06pa-
30M 06eCneqHBaeT 6e30naCHOCTb nOJleTOB. Kopa6J1H H CaMOJleTbI TaKme cHa6-
meHbI pa,lJ;HOJlOKaTOpaMH, CJlymaHMH )J;JI.H HaBHraHOHHbIX eJleH. C nOMO-
blO JlOKaTOpOB Ha6J1IO,AaIOT MeTeopbI B BepXHHX CJlO.HX aTMoc«pepbI. JIoKaTo-
pbI HCnOJlb3YIOTCSJ CJlym60H nOrO,lJ;bI ,lJ;n.H Ha6J1IO,lJ;eHH.H 38 06J1aKaMH. H8KOHe,
JlOKaTOpbl HCnOJlb3YIOTC.H B KOCMHqeCKHX HCCJle)J;OBaHH.HX.
1
Puc. 15.29
R = vt .
2
r
447

15.16. TeJleBH,ZJ;eHHe
IIpHHIJ,Hn nepeJJ;aqH H306pRmeHHH Ha pRCCTO.HHHe COCTOHT B CJle,AYIOeM.
HR nepeJJ;RIOeH CTRHIJ,HH npOH3BOJJ;HTC.H npeo6pR30BRHHe H306pameHH.H B no-
CJleJJ;OBRTeJlbHOCTb sneKTpHtIeCKHX CHrHRJlOB. 8THMH CHrHRJlRMH 3aTeM MOJJ;Y-
JlHPYIOT KOJle6RHH.H. BblpR6RTbIBReMbIe reHepRTopOM BbICOKOH 'tIaCTOTbI. Mo,AY-
nHpOBaHHR.H sneKTpOMRrHHTHR.H BOJlHR nepeHOCHT HHcpOpMRIJ,HIO Ha 60JlbllIHe
paCCTO.HHH.H. B npHeMHHKe npOH3Bo.n;HTC.H o6paTHoe npe06pa30BaHHe. BbICOKO-
tIRCTOTHbIe MO,AYJlHpOBaHHbIe KOJle6aHH.H .n;eTeKTHpYIOTC.H, R nonyqeHHbIH CHr-
Han npe06pa3yeTC.H B BH,AHMOe H306pRmeHHe. .D:n.H nepe.n;atIH .n;BH:>KeHH.H HC-
nonb3YIOT npHHIJ,Hn KHHO: HeMHoro OTnHqRIOHeC.H .n;Pyr OT .n;PyrR H306pame-
HH.H .n;BHmyerOC.H 06'beKTa (Ka.n;pbr) nepe.n;alOT .n;eC.HTKH pa3 B ceKYH.n;y.
IIpe06pR30BaHHe H306pameHH.H Ka.n;pa B cepHIO sneKTpHtIeCKHX CHrHanOB
npOH3BOlIHTC.H C nOMOblO nepelIRlOweH BRKYYMHOH SJleKTpOHHOH TPy6KH -
HKOHOCKonR (pHC. 15.30). BHYTPH HKOHOCKona pacnOnO:>KeH M03RHqHbIR SK-
06'beKTHB
M03aHqHbIA
3KP8H
. =-=
.. .
- -
 -
8neKTpOHHa$l
nYWK8
R

e::

:s:
e::
:s:

»
::c=
+
--L-
Puc. 15.30
paH, Ha KOTOpbIH C nOMOblO OnTH'leCKOH CHCTeMbI npOelJ,HpyeTcSI H306pR:>Ke-
HHe 06'beKTR. KRmJJ;a.H .HtIeHKa M03aHKH 3ap.H:>KReTC.H, npHtIeM ee 3ap.H.n; 3RBH-
CHT OT nR.n;RIOeH HR .HtIeHKY CBeTOBOH SHeprHH. 8TOT 3Rp.H.n; H3MeH.HeTC.H npH
nOn8,ARHHH HR .HtIeHKY SJleKTpOHHoro nY'lKR. C03,ARBReMOro 3J1eKTpOHHOH nyw-
KOH. 8neKTpOHHbIH nyqoK nOCJlelIOBRTeJlbHO nonalIaeT Ha Bce sneMeHTbI CHR-
tIaJla O.n;HOH CTPOqKH M03aHKH, 3aTeM ,ApyroH CTPOqKH H T..n;. OT Toro, HR-
CKonbKO CHJlbHO MeH.HeTC.H 3ap.H.n; .HqeHKH, 3aBHCHT CHna TOKa B HMnYJlbCe,
npOTeKalOeM no pe3HCTOpy R. IIoSTOMY Hanp.HmeHHe HR pe3HCTope R H3Me-
H.HeTC.H npOnOplJ,HOHaJlbHO H3MeHeHHIO OCBeeHHOCTH B,AOnb CTpOK KR.n;pa.
B HeKOTopbIX COBpeMeHHbIX TeneBH3HOHHbIX nepe.n;RTqHKaX (HRnpHMep, B
BH.n;HKOHe) HcnOJlb3YIOTC.H BH.n;OH3MeHeHHbIe sneKTpoHHo-JlY'leBble Tpy6KH C MRr-
HHTHbIM ynpRBJleHHeM, r.n;e BMeCTO CBeT.HerOC.H SKpRHa YCTaHOBJleH CBeTOtIYB-
CTBHTenbHbIH nonynpOBo.n;HHKOBbIH SKpaH. BJlRro.n;ap.H 3RBHCHMOCTH conpOTHB-
neHH.H nOJlynpOBOJJ;HHKOB OT OCBeeHHOCTH, KOJle6aHH.H OCBeeHHOCTH npe06-
pa3YIOTC.H B SJleKTpHqeCKHe Kone6aHH.H, KOTopbIe MorYT 6bITb HCnOJlb30BRHbI
.n;JI.H MO.n;y JI.HIJ,HH BbICOKOtIaCTOTHbIX KOJle6aHHH.
448

T8KOH me CHrH8J1 nOJIY'tlaeTC.H B TeJIeBH3HOHHOM npHeMHHKe nOCJIe .n;eTeK-
THpOB8HH.H. 3TO BH.n;eOCHrHan. OH npeo6pa3yeTcH B BH.n;HMoe H306pameHHe H8
3Kp8He npHeMHOH BaKYYMHOH 3J1eKTpOHHOH TPy6KH -- KHHeCKona. 3J1eKTpOH-
HaH nYlIlKa TaKOH Tpy6KH CH86meH8 3J1eKTpo.n;OM, ynpaBJI.HIOHM KOJlH'leCTBOM
3J1eKTpOHOB B ny'tlKe H, CJIe,AOB8TenbHO, CBe'tleHHeM 3KP8H8 B MeCTe nOn8,A8-
HHH JlY'tla. CHCTeMbI K8TYlIleK ropH30HTaJlbHOrO H BepTHK8J1bHOrO OTKJlOHeHH.H
38CT8BJI.HIOT 3J1eKTpOHHbIH nyq CK8HHpOB8Tb BeCb 3KP8H TOqHO T8KHM me 06-
p830M, K8K 3J1eKTpOHHbIH Jlyq cK8HHpyeT M038H'lHblH 3Kp8H B nepe,A8101I1eH
Tpy6Ke. CHHXpOHHOCTb .n;BHmeHHH JIyqeH B nepe.n;81OeH H npHeMHoH Tpy6-
Kax .n;OCTHraeTC.H nOCbIJlKOH CneL\H8J1bHbIX CHHXpOHH3HPYIOHX CHrH8nOB.
TeJIeBH3HOHHbIe pa.n;HocHrH8JIbI MorYT 6bITb nepe.n;8HbI TOJIbKO B .n;H8n830-
He YJIbTp8KOpOTKHX (MeTpOBbIX) BOJIH. T8KHe BOJIHbI p8cnpOCTpaH.HIOTCH 06bI'tl-
HO nHlIlb B npe.n;eJIax np.HMOH BH.n;HMOCTH aHTeHHbI. 1l03TOMY .n;JI.H OXB8Ta Te-
JIeBH3HOHHbIM BeaHHeM 60JlblIlOH TeppHTopHH He06xo,AHMO pa3MeaTb TeJle-
nepe,A8T'lHKH 'l8e H no.n;HHM8Tb HX aHTeHHb] BbIlIle. OCT8HKHHCK8.H TeneBH-
3HOHHa.H 6alIlH.H B MOCKBe BbICOTOH 540 M o6eCneqHB8eT YBepeHHbIH npHeM
TeJlenepe.n;8'tl B p8,AHyce 120 KM. 30H8 YBepeHHoro npHeM8 TeJleBH,AeHHSl He-
npepbIBHo YBeJlH'tlHB8eTCSI B CBSl3H C nO.HBJIeHHeM H HCnOJIb30B8HHeM peTp8HC-
nHL\HOHHbIX cnYTHHKOB.
lloJIY'tleHHe L\BeTHOro H306p8meHHH ocyeCTBJIHeTC.H 38 C'tleT nepe,A8qH BH-
,AeOCHrH8J10B. HeCYWHX KOMnOHeHTbI H306p8meHHH. COOTBeTCTBYIOWHe OCHOB-
HbIM UBeTaM cneKTpa -- KpaCHOMY, 3eJleHOMY, CHHeMY.
15.17. Pa3BHTHe cpe.u;CTB CBH3H
B H8CTO.Hee BpeM.H P83JIHqHbIe cpe.n;CTB8 CB.H3H p83BHB8IOTC.H H COBeplIleH-
CTBYIOTCH B yme OCBoeHHbIX 06JI8CT.HX, a TaKme HaXO,A.HT H HOBbIe 06JIaCTH
npHMeHeHH.H. TaK, ee He.n;8BHO Mem.n;yropo.n;HHH Tene<}>OHH8H CB.H3b ocye-
CTBJIHJI8Cb TOJIbKO no B03.n;YlIlHbIM JIHHHSlM CBSl3H.
H8 ee H8.n;emHOCTb BJIHHJIH rp03bI H B03MomHOCTb
06JIe.n;eHeHHH npOBO,AOB. B H8CTOHee me BpeMH
lIlHpOKO npHMeH.HIOTC.H Ka6eJIbHbIe H p8.n;HopeneH-
HbIe JIHHHH, COTOB8.H M06HJIbH8S1 CBH3b, nOBbI1Il8eT-
CH ypOBeHb aBTOMaTH38L\HH CB.H3H.
P8.n;HopeJleHHbIe JlHHHH CB.H3H HCnOJIb3YlOT ynb-
TpaKopOTKHe (.n;eL\HMeTpOBble H C8HTHMeTpOBble)
BOnHbI. 3TH BOJlHbl p8cnpoCTp8HSllOTCSI B npe,Aen8X
npHMOH BHJ1HMOCTH. 1l03TOMY JlHHHH COCTOHT H3
L\enO'tlKH M8J10MOHbIX p8.n;HOCT8HL\HH, K8m.n;8.H H3
KOTOpbIX nepe.n;aeT CHrH8JIbI K coce.n;HeH K8K 6bJ no
3CT8<}>eTe. T8KHe CT8HL\HH HMelOT M8'tlTbI BbICOTOH
60--80 M, Haxo.n;HHeC.H H8 paCCTO.HHHR 40--60 KM
.n;pyr OT .n;pyra. Y cneXH B 06J1aCTH KOCMH'tleCKOH
p8,AHOCBH3H n03BOJIHJIH C03,AaTb HOBYIO CHCTeMY
CB.H3H, H83B8HHYlO "Op6HT8". B 3TOH CHCTeMe HC-
nOJlb3YIOTC.H peTp8HCJI.HI.{HOHHbIe cnYTHHKH CBSl3H
Puc. 15.31 (pHC. 15.31). CnYTHHKH CB.H3H cepllH "MOnHH.H"
449

3anYCKalOTCH Ha CHJlbHO BbITHHYTbIe Op6HTbl. nepHO,A HX 06pal..QeHH.H COCTaB-
JI.HeT OKOJlO 12 q.
C03.n;aHbI MOHbIe H Ha.n;emHbIe CHCTeMbl, 06eCneqHBalOI..QHe TeJleBH3HOHHbIM
BeaHHeM paHoHbI CH6HpH H ,D;aJlbHerO BOCTOKa H n03BOJl.HIOHe ocyecTBHTb
Teneq:>oHHO-TeJlerpaq:>HYIO CB.H3b C OT.n;aJleHHbIMH paHoHaMH CTpaHbI.
HOBbIe cnYTHHKH CB.H3H cepHH "Pa,Ayra" 3anYCKaIOTC.H Ha OP6HTY pa.n;HY-
COM OKOJlO 36000 KM. Ha STOH op6HTe nepHO,A 06p8eHHH cnYTHHKa paBeH
24 'I, nOSTOMY cnYTHHK Bce BpeM.H Haxo.n;HTC.H Hall OJtHO:H H TOH me TOqKOH
nOBepXHOCTH 3eMJlH.
COBepmeHCTBYIOTC.H H Haxo.n;.HT HOBbIe npHMeHeHHH H TaKHe cpaBHHTeJlbHO
CTapbIe Cpe,ACTBa CB.H3H, KaK TeJlerpaq:> H q:>OTOTeJlerpaq:>.
o pa3Maxe, KOTOpbIH nOJlyqHJl8 nepe.n;aqa Heno.n;BHmHbIX H306pameHHH no
q:>OTOTenerpaq:>y, MomHO CY,AHTb no TaKHM Hq:>paM: B ro.n; no q:>OTOTeJlerpaq:>y
nepe.n;aeTC.H .n;o 70 TbIC.Hq ra3eTHbIX nOJlOC, C KOTOpbIX neqaTaeTC.H CBbIme
3 MHJlJIH8PJtOB SK3eMnJl.HpOB ra3eT.
BOllPOCbI ,l(JI.H CAMOKOHTPOJI.H
1. lfmo maKoe 60Jl.Ha? t[mo lia3bt6aemcSl cKopocmblO pacnpocmpalieliUSl
60Jl.libt?
2. KaKU.MU C80UCmBa.MU oOJl.JtClia 06Jl.aOamb cpeoa. Ilm06bt B lieu .MOZJl.U
603liUKliymb .MeXaliUlleCKue 60Jl.libt?
3. KaKue 60Jl.libt lia3bt6alOmcSl nonepelllibt.Mu? npoiJOJl.blibt.MU?
4. llpoucxooum Jl.U nepelioc 6el4eCm6a 60Jl.HOU? nepeHoc 3liepzuu?
5. KaK08a C8.R3b .MeJICoy OJl.UIiOU 80Jl. Ii bl U C1wpocmblO ee pacnpocmpaHeliUJl?
6. Mozym Jl.U 36YK06bte 60Jl.libt pacnpOCmpa1iSlmbCSl 6 8aKYY.M.e?
7. K KaKO.M.Y 6UOY 80Jl.1i (nonepelllibtX UJl.U npOOOJl.blibtX) omliOCSlmcSl 3Jl.eK-
mp0.M.aZliumlibte 60Jl.libt? KaK C6Sl3alibt MeJICoy c060u 6 80Jl.lie lianpa6Jl.e'
liue CKopocmu pacnpocmpalieliUSl U lianpa6Jl.eliUSl KOJl.e6aIiUU 8eKmop06
E u B?
8. II OlleMY npu 603liUKli06eliUU 3Jl.eKmpOMaZliumlibtX KOJl.e6aliUU 6 3aKpbtmO.M.
KOJl.e6ameJl.bliOM KOlimype U3Jl.Ylleliue 3Jl.eKmp0.M.aZliumlibtX 60Jl.H .MaJl.o? KaK U3-
MeHJlemCSl UlimeliCU81i0Cmb U3Jl.YlleliUR. 3Jl.eKmp0.M.aZliumlibtX 60Jl.1i npu nepexooe
K omKpbtmOMY KOJl.e6ameJl.bIi0.M.Y KOlimypy? t[ mo maKoe 6u6pamop repija?
9. KaK 3liepzuSl. U3Jl.YllaeMaSl 6u6pamopOM 6 eOUIiUl.{Y 6peMeliU. 3a8ucum
om lI.aCmOmbt KOJl.e6aliuu?
10. KaKue CBoucmBa 3Jl.eKmp0.MaZ1iUm1ibtX 80Jl.1i 6btJl.U 06IiapY:HCe1ibt B Onbt.
max repija? KaKuM 06pa30.M. 6 3mux OnbtmaX 6btJl.U nOOm6epJICOelibt npeOCKa-
3aliUSl J(JIC. M aKC8eJl.Jl.a omliOCUmeJl.bliO CKopocmu pacnpocmpalieliUSl 60Jl.1i?
11. KaK08bt OCli06libte 3Jl.e.M.elimbt paouOnpUe.M.liUKa A.C. II on06a, U KaK060
ux lia3lialleliue? B lle.M cxoocm80 npuliijuna oeucm8uSl C06pe.M.eli1iOZO paouo-
npueMliuKa C npulil.{uno.M. oeUCm6UJl paouOnpUe.M.HUKQ A.C. II on08a?
12. B lle.M. cocmoum cnoc06 a.M.nJl.umyoliou .M.ooYJl.Sll.{uu? J(Jl.Sl 'Ilezo olia npU.M.e-
liSlemcSl? If mo maKoe oemeKmup06aHue? M OJICliO Jl.U Cilumamb, Ilmo oemeKmupo-
Baliue U 6btnpJl.MJl.eliUe nepe.M.eliliOZO mOKa .Jl8Jl.SllOmCSl oouliaK08bt.M.U npOijeCCaMu?
13. KaKU.M. 06pa30M MOJICliO oCYl4eCm6umb a.M.nJl.umyoIiYIO MOOYJl.SlijUIO? Ka-
K06a npu 3mOM pOJl.b zeliepamopa lie3amyxalOl4UX KOJl.e6aliuu? l.{enu. 8 Komo.
poit C030aemcSl IiU3KOilacmomlibtU CUZliaJl.?
450

14. KaK pa60maem oemeKmop? KaKUM 06pa30.M 6 npUeMliUKe 6blOe.n.Slem-
CR liU3KOilaCmOmliblU cuzlia.n.?
15. KaKuMu C60ucm6aMU 06.n.aoalOm 3.n.eKmpOMaZliumlible 60.n.Hbl 2 KaK 3mu
C60ucm6a Mozym 6blmb 06liapYJICelibl 6 COBpeMeliliOM OeMolicmpau,UOliliOM 3KC-
nepUMe lime?
16. C{mo maKoe paouo.n.oKaijUSl? KaKUM 06pa30M oocmuzaemcSl 1tanpa6.n.eli-
Hocmb U3.n.YlleIiUSl paOUOJLOKamopa?
17. KaK MOJICIiO Bbt1luc.n.umb paccmoSlHue om paouo.n.oKamopa 00 u,e.n.u2 KaK
amo pac'cmoSlliue tPuKcupyemcSl lia 3Kpalie paouo.n.oKau,uolilioU ycmaHoBKu?
A = vT
V=AV=A
2n
x = ASinro ( t -  I
v}
2n (r2 - rd
L\q> = q>1 - q>2 =
A
C
V=-

OCHOBHbIE OPMYJIbI
- JIHHa BOJlHbI, re v - CKOpOCTb pacnpOCTpaHe-
HH.H BOJlHbI; T - nepHO KOJle6aHHii HCTOqHHKa.
- CBSl3b Mem,AY JlHHOH BOJIHbI, CKOpOCTblO ee pac-
npOCTpaHeHHSI H qaCToToH. r,Ae v - q8CTOT8 KOJIe-
6aHHH HCTo't{HHKa; 00 - u,HKnHqeCKaSi qaCTOTa.
ypaBHeHHe nnocKoH BOJIHbI, rAe x - CMeeHHe
OT nOJlOmeHHSI paBHOBeCH.H TO't{KH cpebI, OTCTO.HeH
OT HCTO't{HHKa KOJle6aHHH Ha paCCTOSlHHH r B Ha-
npaBJIeHHH pacnpOCTpaHeHH.H BOJIHbI; A - aMnnH-
Tya KOJle6aHHH TO't{eK cpebI.
- pa3HOCTb <»a3 KOJle6aHHH BYX TOqeK cpebI. Ha-
XO,ASIHXCSI Ha paCCTOSlHHSlX rl u. r2 COOTBeTCTBeH-
HO OT HCTO't{HHK8 KOJIe6aHHH.
CKOpOCTb pacnpocTpaHeHHSI 3JIeKTpOMarHHTHoH
BOJlHbI B cpe,Ae, r,Ae c - CKOpOCTb CBeT8 B B8KYY-
Me; E. Jl - OTHOCHTeJlbHbIe .lIH3J1eKTpHqeCKa.H H
MarHHTHa.H npoHHu,aeMOCTH cpelIbI COOTBeTCTBeHHO.
METO,l(HqECKHE PEKOMEH,l(Al\HH
3aaqH B 3TOH TeMe MomHO pa3,AeJlHTb Ha Be rpynnbI: 1) 3a.aaqH, yqH-
TbIBalOHe CKOpOCTb pacnpOCTpaHeHH.H BonH B pa3JIH'lHbIX cpeax; 2) 3aaqH
Ha npHMeHeHHe ypaBHeHH.H nJIOCKOH BOJlHbI.
I1pH pemeHHH 3aatJ BToporo THna Heo6xoHMO:
1. OnpeenHTb xapaKTep BOJIHbI u. 3anHcaTb cooTBeTcTBYIOee ypaBHeHHe.
2. BbI.HCHHTb CBOHCTBa cpebI, B KOTOpOH pacnpOCTpaH.HeTCSI BOJIHa, H ce-
naTb COOTBeTCTBYIOee 3aKnlOqeHHe 0 CKOpOCTH ee pacnpOCTpaHeHH.H. BamHo
T8K:>Ke pa3JIH'l8 Tb MrHOBeHHYIO CKOpOCTb Kone6nlO:l.QeHC.H TOqKH Cpe,AbI
(v x = vocos(oot + <Po» H nOCTO.HHHYIO CKOpOCTb p8cnpOCTp8HeHH.H BOJIHbI B ,A8H-
HOH cpe.ne (v = AV).
3. YqeCTb, qTO pa3HOCTb <}>a3 KOJIe6aHHH ,ABYX TOqeK, OTCTO.HHX ,Apyr OT
Pyra Ha paCCTO.HHHH, paBHOM JlHHe BOJlHbI, COCTaBJlSleT 27t.
451

4. YqeCTb, qTO 3aKOHbI pacnpOCTpaHeHH.H 3J1eKTpOMarHHTHbIX BOJIH aHaJlO-
rHqHbI 3RKOHRM pRcnpOCTpRHeHH.H ynpyrHx BOJIH. OHaKo Ha,AO 06paTHTb BHH-
MaHHe Ha TO, 't{TO 3J1eKTpOMarHHTHbIe BOJlHbI MOrYT paCnpOCTpaH.HTbC.H H B
M
BaKYYMe CO CKOpOCTblO CBeTa, paBHoH 3. 10 8 -. CKOPOCTb pacnpocTpRHeHH.H
C
3J1eKTpOMarHHTHbIx BOJIH B cpelle pRBHa CKOpOCTH CBeTa B lIRHHOH cpee
c
v=-
.
5. HCnOJlb3Y.H YCJlOBHe 3aaqH, COCTaBHTb CHCTeMY ypaBHeHHH H pemHTb
ee OTHOCHTeJlbHO HCKOMOH BeJlHqHHbI.
nPIiMEPbI PElliEHIIH 3A,ZJ;A q
1. Onpe,AenHTe Q8CTOTY Kone68HHH B B03,Ayxe (t = 20 °C), ecnH paCCTOSlHHe Mem-
,AY 6nHmaHIllHMH TOtlKRMH 3BYKOBOH BOJIHbI, OTnH'lRIOHMHCSI no «l>R3e HR 1t,
M
COCTRBnHeT 25,0 CM. CKOpOCTb 3BYKR B B03,Ayxe npH A8HHOH TeMnepRType 340
C
M
,I{RHO: r=0,250 M, v s =340 cp=1t.
C
H8HTH: v.
v
Pew e H H e. Q8CTOTR 3BYKOBbiX Kone68HHH sonHbl v = i' r,Ae v - CKOpOCTb
pRcnpOCTp8HeHHSI BonHbI npH t= 20 OCt A - AnHH8 sonHbI.
<1>83bI Kone6RHHH ,ABYX TO'leK BonHbI, p8CcToSlHHe Mem,Ay KOTOpbIMH r, OTnH-
tlRlOTCSI HR
2r 21tr
cp = -. OTCIO,AR A = -
A cp
v
TIO,ACT8SnSl.H 3TO BbIp8meHHe B «l>opMyny v = -. nonytlHM
A
vq> V1t V
v=-; v=-=-; v=680 ru..
21tr 2r 2r
2. Bo CKonbKO pR3 H3MeHSleTCSI ,AnHHR 3SYKOBOH BonHbI npH nepeXO,Ae 1'13 B03,AYX8
M M
B BO,AY? CKOpOCTb pRcnpOCTp8HeHHSI 3BYKR B B03,Ayxe 340 B BO,Ae - 1450
C
C
M M
,I{ 8 H 0: VI = 340 V2= 1450
C C
H u A 2
RHTH: -.
Al
Pew e H He. lIpH nepexoAe BonHbI H3 B03,AYX8 B BO,Ay CKOpOCTb v 1'1 ,AJIHHa
BonHbI A H3MeHSlIOTC.H, 8 'laCTOTa v HeH3MeHHa. Y'lHTbIB8S1, 'lTO V =  (B B03,Ay-
Al
xe), v =  - (B BOAe), H npHpaBHHBaSl 3TH p8BeHCTBa, HaXO,AHM
A 2
v =  = . OTKY,AR 1.. 2 = V 2 ; A 2 = 4.3.
Al A2 Al VI Al
452

3. CMel.QeHHe OT nonomeHHSI p8BHOBeCHSI TO'lKH. H8XO,AHl.QeHCSI HS pSCCTOSlHHH
4.0 CM OT HCTO'lHHK8. 'lepe3 t =! p8BHO nonOBHHe SMnnHTY,AbI. HaH,AHTe ,AnHHY
6
BonHbI.
T A
,L(8HO: r=0,040 M, t=-, X=-.
6 2
H 8 H T H: A.
Pew e H H e. Yp8BHeHHe BOJIHbI X = A sin OJ( t - ; 1- Y'lHTblBIISI, '1TO OJ =  H
A = vT, nonY'lHM
X = ASin(  t - ,,; ) = ASin(  t _ 2r )-
nOcTaBHM ,ASHHbIe ycnoBHH 38,A8"1H:
 = ASin(  .  - 2 ). OTKYA8 Sin(  - 2 ) = i.
1t 21tr 1t
CJleOBaTeJlbHO, aprYMeHT CHHYC8 '3 - T - '6' OTCIO8 HCKOMa.H JlHH8
BOJlHbI A = 12r; A = 0, 48 M.
4. TIonepe'lHbIe BonHbI pscnpOCTpaHSlIOTCSI B,AOnb H8TSlHYToro IllHYP8. P8CcToSlHHe
Mem,AY 6nHmaHIllHMH TO'lKaMH, HMelOl.QHMH O,AHH8KOBble OTnH'lHbIe OT HynSl CMel.Qe-
HHSI, p8BHO 25 CM. H8H,AHTe nHHY BonHbI, ecnH <l>83bI Kone68HHH TO'lel< B 3TOT
MOMeHT OTHOCSlTCSI K8K 1: 2.
,L(aHo: X1=X2, 6.r=0,25 M, 'P1:<P2=1:2.
H a H T H: A.
P e 1II e H He. Corn8CHO ycnoBHIO 388"1H
<PI =!
<P2 2
(1)
38nHmeM ypaBHeHHe nnocKoH BonHbI ,AnSI 3THX TO'leK:
XI = A sin <PI;
x2 = A sin <P2'
T8K K8K no ycnoBHIO
XI = x 2 . OTCIO,A8 sin <PI = sin<P2' 3H8'lHT. <P2 = 1t - <Pl'
TIO,ACT8BHM (2) B (1):
(2)
<PI _ 1 _ 1t . 21t
- - -  <PI - -, <P2 = -.
1t - <PI 2 3 3
PS3HOCTb a3 Kone68HHH 3THX TO'leK
1t
6.<p = 'P2 - <PI = - .
3
HCKOM8fl ,AnHHS BonHbI pSBHa
A = 21t6.r . A = 15M.
6.<p , ,
5. P8,AHOnOKaTop MomeT o6HapymHTb n;enb, H8XOSlI.QYIOCSI HS p8CCTOSlHHH OT
100 M ,AO 100 KM. Onpe,AenHTe nHTenbHOCTb nOCbIn8eMbIX HMnynbcoB H 'l8CTOTY
HX Cne,AOBaHHSI.
453

):(8HO: lmm=100 M; lmax=I,00'105 M; v=3,0.10 8 M
C
H 8 H T H: t, v.
P e ill e H He. CHrH8n, OTpameHHblH OT enH, HaXO,ASIeHCSI Ha 6nHmHeM p8C.
CTOSlHHH, MOJKeT 6bITb 38<pHKCHpOB8H nHlIIb B TOM cnYQ8e, ecnH BpeMSI ero ,ABH-
meHHSI TY,A8 H 06paTHO 6Y,AeT 60nblIIe HnH p8BHO ,AnHTenbHOCTH HMnynbCOB. T.e.
2 1 .
t=; t=0.70 MKC.
V
BpeMSI ,nBHmeHHSI CHrHana TY,Aa H 06p8THO npH OTp8JKeHHH OT uenH. H8XOIUI-
l.QeHCSI H8 ,A8nbHeM paCCTOSlHHH. He ,AOnmHO 6blTb 60nblIIe nepHO,Aa cne,nOBaHHSI
HMnynbCOS. n03TOMY
t 1 = 2L.nax = T = !,
V v
V
OTKY,Aa v - -; v = 1,5 Kr.
- 2l.nax
6. P83HOCTb nOTeHH8nOB H8 06Kna,nK8x KOH,AeHCaTOp8 H,AeanbHOro Kone6aTenb-
Horo KOHTypa H3MeHSleTCSI no 3aKOHY U = 50cos(10 4 nt)B. EMKOCTb KOH,neHcaTop8
0,90 MK<1>. H8H,AHTe HH,AYKTHBHOCTb KOHTyp8, a T8Kme ,AnHHY sonHbI. H8 KOTO-
PYIO pe30HHpyeT 3TOT KOHTYp.
M
):(aHo: U=50cos(10 4 nt)B, C=9,O.10- 7 <1>, v=3,0.10 8
C
H 8 H T H: L, A.
P e ill e H He. 113 <l>opMynbI TOMCOH8 T = 2n H8H,AeM HH,AYKTHSHOCTb KOHTyp8
T 2
L=-
4n 2 C
HnH. Y'lHTbIBaH, 'lTO T = 2n . nonY'lHM
w
1
L=-
w 2 C'
113 ypaBHeHHSI U = 50cos(104nt)B Cne,AyeT (0 = 10 4 n P8.lI.
C
Cne,AOS8TenbHo, HH,AYKTHSHOCTb KOHTyp8 6Y,AeT paSHa L = 1,1 . 10- 3 rHo
):(nHH8 BonHbI
2nv
A = vT = -; A = 6,0 . 10 4 M.
W
454

OnTKA
16. fEOMETPHqECKAH OllTHKA
reOMeTPll'leCKaH onTHKa. OCHOBHLIe nOHHTHH I'eOMeTpHqeCKOH onTHKH. 3a-
KOHLI reOMeTpHqeCKOH OnTHKH (npHMOJlHHeHHOI'O paCnpOCTpaHeHHH, oTpaace-
HH.H, npeJlOMJleHHH).
llJlOCKOe 3epKaJlO. C4>epHqeCKHe 3epKaJla. llOJlHOe oTpaaceHHe. XOA J1yqeH
B npH3Me.
JIHH3a. llOCTpOeHHe H306paaceHHH B J1HH3aX. «I»opMYJla TOHKOH J1HH3LI.
YBeJlll'leHHe J1HH3LI.
r J1a3 KaK OnTHqeCKaH CHCTeMa. )Ie4>eKTLI apeHHH. OqKH.
OnTHqeCKHe npH6opLI. «I»oToannapaT. npOeKUHOHHbIH annapaT. nyna. MHK-
pOCKOn.
3J1eMeHTLI 4>oToMeTpHH. 3aKOHLI OCBeeHHOCTH.
OnTHKa - pa3eJl <pH3HKH, B KOTOpOM H3yqa1OTC.H CBOHCTBa CBeTa, 3aKO-
HOMepHOCTH ero H3J1yqeHH.H, pacnpOCTpaHeHH.H H B3aHMoeHCTBH.H c Bee-
CTBOM, BbI.HCHHeTC.H npHpoa CBeTa.
COBpeMeHHa.H onTHKa 6a3HpyeTC.H Ha 3J1eKTpOMarHHTHOH TeopHH CBeTa, CO-
rJlaCHO KOTOpOH CBeT - 3TO 3J1eKTpOMarHHTHbIe BOJlHbI onpe)J;eJleHHOrO Ha-
na30Ha C qaCTOTaMH OT 1.5 .10 11 o 3 .10 16 r.
16.1. feOMeTpHQeCKaH onTHKa.
OCHOBHMe nOHHTHH reOMeTpHQeCKOH onTHKH
BnepBbIe .n;m. MaKCBeJlJl B 1865 r. npHWeJl K BbIBOY, 'liTO CBeT - 3TO
3J1eKTpOMarHHTHbIe BOJlHbI. 3J1eKTpOMarHHTHa.H TeOpH.H CBeTa OKa3aJlaCb HCKJlIO-
qHTeJlbHO nJlOOTBOpHOH. OHa 06'b.HCH.HJla MHorHe CBeTOBbIe .HBJleHH.H H 3aKO-
HbI (CM. HHme rJl. 17): HanpHMep, 3KcnepHMeHTaJlbHO OTKpbITbIe 3aKOHbI OT-
pameHH.H H npeJlOMJleHH.H CBeTa H T.p,.
OHaKo B onTHKe HMeeTC.H Kpyr 3a)J;aq, Ha peWeHH€ KOTOpbIX BOJlHOBa.H
npHpoa CBeTa nOqTH He CKa3bIBaeTC.H. 3TO BonpocbI, CB.H3aHHbIe C H3yqeHH-
eM 3aKOHOB pacnpOCTpaHeHH.H CBeTa B cpeax, a TaKme C nOCTpoeHHeM H306-
pameHHH B OnTHqeCKHX np:u60pax Ii C pa3pa60TKOH Ha:u60Jlee COBepweHHbIX
KOHCTPYKU:UH 3THX np:u60pOB. OHH paCCMaTpHBaIOTC.H B pa3p,eJle <pH3HK:U "reo-
MeTpHqeCKaSi onTHKa".
OCHOBHbIMH nOH.HTH.HMH reOMeTpHqeCKOH onTHKH .HBJI.HIOTC.H CBeTOBOH ny-
qOK H CBeTOBOH Jlyq.
455

CBeTOBOH nyqOK -- 06J1aCTb npOCTpaHCTBa. B npeenax KOTOpOH paCnpOCTpa-
H.HeTC.H CBeT. Ha npaKTIiKe CBeTOBbIe nKIi nOnalOT C nOMObIO lia<pparM-
OTBepCTHH. Ho nonyqliTb CKOnbKO yroHo TOHKIiH nyqO CBeTa Henb3.H. 3TO
06'b.HCH.HeTC.H ,A1i<ppaKlieH CBeTa. Pa3nliqalOT napanJlenbHbIH. paCXO.HIiHC.H H
CXO.HIiHC.H CBeTOBble nKIi (pHC. 16.1).
KaK nOKa3bIBaeT onbIT. CBeTOBbIe
nyqKIi .HBn.HIOTC.H He3aBUCUMLIMU: Kam-
bIH cBeToBoH nyqOK npli B3aliMHOM ne-
peCeqeHHH Be,AeT ce6.H caMOCTO.HTenbHO,
He3aBHCHMO OT .o;pyrlix nyqKOB Ii He
OKa3blBaeT HHKaKoro Bnli.HHIi.H Ha .o;py-
i'He nyqKH CBeTa (3TO cnpaBenHBO B
TOM cnyqae. ecnH peqb li.o;eT 0 He OqeHb 60nbllIliX IiHTeHCHBHOCT.HX CBeTa).
HanpaBJleHlie pacnpOCTpaHeHIi.H nlO6bI:X BonH, B TOM qlicne H CBeTOBbIX,
onpeen.HeTC.H C nOMOblO nyqeH -- RHHIiH. nepneHIiKyn.HpHbIX BonHOBbIM
nOBepXHOCT.HM Ii YKa3bIBalOIiX HanpaBneHlie pacnpOCTpaHeHIi.H 3HeprHIi
BOJlHbI.
CBeTOBOH JIY'I - 3TO J1UHUH, YKa3LIBalOUJ,aH HanpaBJIeHUe pacnpOCTpa-
HeHUH CBeTa, a He TOHKIiH CBeTOBOH nyqOK. QTo6bI onpeenHTb 3TO Ha-
npaBneHue, MbI BbI.o;en.HeM Y3Klie CBeTOBble nyqKIi, liaMeTp KOTOpbIX Bce
me ,AOnmeH npeBOCXO.D.IiTb .o;JlHHY BOnHbI. OCb 3Toro nyqKa H eCTb CBeTO-
BOH nyq.
feOMeTpliqeCKa.H onTIiKa 6a3HpyeTC.H Ha cne.o;YIOlix 3aKOHax: 3aKOH npSl-
MOJlIiHeHHOro pacnpOCTpaHeHIi.R CBeTa. 3aKOHbI OTpameHIi.H Ii npenOMneHIi.H CBe-
Ta. npliHHn 06paTliMOCTIi CBeTOBbIX nyqeH.


..

Puc. 16.1


16.2. 3aKoH rrpHMOJlHHeHHOro pacrrpOCTpaHeHHH CBeTa
OnbITHbIM nYTeM YCTaHOBJleH aaKUH npHMOJIUHeHIIOro pacnpOCTpaHeHUH
CBeTa: B OHOpOHOH np03paqHOH cpee CBeT pacnpOCTpaH.HeTC.H np.HMOnH-
HeHHo.
I1MeHHo np.HMOnliHeHHocTblO pacnpOCTpaHeHIi.H CBeTa 06'b.HCH.HeTC.H 06pa30-
BaHlie TeHIi. reOMeTpRqeCKIi nO,A06HOH OCBeaeMOMY Henp03paqHOMY npe.o;Me-
TY. TeHblO Ha3bIBalOT TY qaCTb npOCTpaHCTBa 38 HenpOHHaeMbIM npe,AMeTOM,
Ky.o;a He npOHIiKaeT CBeT.
Puc. 16.2
456
Bce HCTOqHHRH CBeTa RMelOT onpe.o;eneHHbIe
pa3Mepbl. Ho ecnH pa3MepbI HCTOqHHKa ManbI
no cpaBHeHHIO C paCCTO.HHHeM. Ha KOTOpOM ou.e-
HIiBaeTC.H ero .o;eHcTBHe, ero MomHO paccMaTpH-
BaTb KaK CBeT.HYIOC.H TOqKY (TOqeqHbIH IiCTOq-
HIiK cBeTa).
B npocTpaHcTBe 3a Henp03paqHbIM llIapoM, Ha
KOTOpblH na.o;aeT CBeT OT TOqeqHOrO IiCTOqHIiKa
(pliC. 16.2), 06pa3yeTC.H KOHyco06pa3Ha.H TeHb, a
Ha 3KpaHe nO.HBIiTC.H TeHb B BIi,Ae Kpyra.
<I>opMa pe3KoH TeHIi Ha 3KpaHe nO,A06Ha
<}>opMe HeKOToporo CeqeHHSI npe.o;MeTa, napaJl-

JleJlbHOrO nJlOCKOCTH 3KpaHa; pa3MepbI me TeHM onpe,neJl.HIOTC.H B3aHMHbIM pac-
nOJlOmeHHeM HCTOqHHKa, npe,nMeTa H 3KpaHa.
B cJlyqae HCTOqHHKOB KOHeqHbIX pa3MepOB Ha 3KpaHe 06pa3yeTC.H pe3Ka.H
TeHb, OrpaHHqeHHa.R nOJlyreHLIO (pHC. 16.3) - qaCTHqHO oCBeeHHoH 06J1ac-
TblO npOCTpaHCTBa.
TIp.HMOJlHHeHHOcTblO pacnpOCTpaHeHH.H CBeTa 06'b.HCH.HIOTC.H pa3J1HqHbIe <pa3bI
JIYHbI, COJlHeqHbIe If JlYHHbIC 3aTMeHIf.H, a TaKme B03MomHOCTb nOJlyqeHH.H
H306pameHlfH c nOMOblO MaJlOrO OTBepCTH.H (pHC. 16.4).
Puc. 16.3
Puc. 16.4
11306pameHHe XOpOllIO BocnpoH3Bo,nHT <POPMY npe,nMeTa. eCJIH oTBepcTHe
,nocTaTOqHO MaJlO. C YMeHbllIeHHeM pa3MepoB oTBepCTH.H pe3KocTb YJlyqllIaeT-
C.H, HO ,no onpe,neJleHHOrO npe,neJla. npH OqeHb MaJIbIX pa3Mepax, Kor,na
D  .J?i., l",ne D - ,nHaMeTp oTBepCTH.H, r - paCCTO.HHHe OT npe,nMeTa ,no OT-
BepCTH.H. "A. - ,AnHHa BOJlHbl na,AalOero CBeTa, Ha6nlO,naeTC.H .HBJleHHe ,AH-
«ppaKUIfH. H 3aKOH np.HMOJIHHeHHOro pacnpocTpaHeHH.H CBeTa He BbInOJlH.HeTC.H.
3aKOH np.HMOJlIfHeHHOro CBeTa MomHO nOJIy'tJIfTb, HCnOJlb3Y.H npHHUHn
<1>epMa. STOT npHHIJ,Hn, C<POPMYJIHpOBaHHbIE B 1660 r. TI. <1>epMa, .HBn.HeTC.H
o606eHHeM 3aKOHOB reOMeTpHqeCKOH onTHKH. OH rJIaCHT: 6 npocmpalicmee
.MeJICay a6YMSl mOIlKa.MU C6em pacnpocmpaliSlemcSl no mo.MY nymu. 6aOJlb KO-
mopozo 6pe.MSl ezo npOXOJfCaeliUSl .MUHU.MaJlbliO.
B O,nHOpO,nHOH cpe,ne CKOpOCTb CBeTa nOCTO.HHHa. TI03TOMY MHHHMaJIbHOMY
BpeMeHH pacnpOCTpaHeHH.H CBeTa Mem,ny ,nBYM.H TOqKaMH COOTBeTcTByeT pac-
npOCTpaHeHHe no KpaT'IaHllIeMY paCCTO.HHHIO. T.e. no np.HMOH JlHHHH.
16.3. OTpameHHe CBeTa. 3aKoHLI OTpameHHH
Puc. 16.5
CBeT pacnpOCTpaH.HeTC.H np.HMOJlHHeHHO TOJlb-
KO B O,nHOpO)J;HOH cpe,ne. ECJIH CBeT no,nxo,nHT K
rpaHHue pa3,neJla ,ABYX cpe,n, OH H3MeH.HeT Ha-
npaBJleHHe pacnpocTpaHeHH.H.
KpoMe Toro, qaCTb CBeTa B03BpaeTC.H B nep-
BYIO cpe,ny. 3To .HBJIeHHe Ha3bIBaeTC.R OTpeHHeM
CBeTa. JIyq CBeTa. H,AyI.l\HH K rpaHHue pa3,neJIa
cpen B nepBOH cpene (pHC. 16.5). Ha3bIBaeTC.H na-
.n;alOIl\HM (a). JIyq. oCTalOHHC.H B nepBoH cpelle
nOCJIe B3aHMO,AeHCTBH.H Ha rpaHHlJ,e pa3,neJla cpe,n,
Ha3LIBaerell OTpeIDlLIM (b).
457

YroJl ex Mem,AY nanalOIUHM JlyqOM H nepneH.lI.HKYJI.HpOM. BOCCTaBJleHHbIM R
OTpamalOeH nOBepXHOCTH B TOqKe na,neHH.H JIyqa, Ha3bIBaeTC.H yrJlOM neHHH.
YrOJI Y Mem,ny OTpameHHbIM JIyqOM H TeM me nepneH,nHKYJI.HpOM Ha3bIBa-
eTC.H yrJlOM OTpaeHHH.

,,\.'
I I I
.t..
,  '\

". '.'"
'1 r (i>-j;.
..,.. '"It': ';
\. ,:...'" \.
,,\I'\'f,,;"t
I I  \ \
Puc. 16.6
A
I
 '1/
I I I
. O!.'¥'
f :"1:.> ) 
.;f
11,:t.-q'ij
'1;i.
. '1(
'. 
x
L
Puc. 16.7
maHCb OT TOqeK 0 H 0'. BpeM.H.
Ee B III B. ,AO H.3. ,ApeBHerpe-
qeCKHM yqeHbIM EBKnH.lI.OM onbITHbIM
nYTeM 6bIJlH OTKpbITbI 38KOHbI OTpa-
meHH.H. B COBpeMeHHbIX YCJlOBH.HX
npOBepKY 3Toro 3aKOHa MomHO npo-
BeCTH C nOMOblO OnTHqeC'KOH waH-
6bI (pHC. 16.6), COCTO.HeH H3 .AHC-
Ka, no oKpymHocTH KOToporo HaHe-
ceHbI ,neJIeHHH, H H3 HCTOqHHKa CBe-
Ta, KOTOpbIH MomHO nepeMeaTb no
KpalO .lI.HCKa. B IJ.eHTpe .lI.HCKa 3a-
KpenJl.HIOT OTpamalOYIO nOBepXHOCTb
(nJlOCKOe 3epKano). HanpaBJI.H.H CBeT
Ha OTpamalOYIO nOBepXHOCTb, H3Me-
pHIOT yrJIbI na,AeHHH H yrnbI OTpa-
meHHH.
3aKoHLI OTpaeHHH:
1. JIyqH na,AalOHH, oTpameHHbIH
Ii nepneH,nHKyn.Hp, BOCCTaBJIeHHbIH K
rpaHHIJ.e ,ABYX cpe,A B TOqKe na.n.eHH.H
Jlyqa, nemaT B O,nHOi1 nJIOCKOCTH.
2. Y ron OTpameHH.H paBeH yr ny
na,neHH.H:
y = a.
3aKoHbI OTpameHH.H MomHO BbIBe-
CTH TeOpeTHqeCKH, nonb3Y.HCb npHH-
IJ.HnOM <1>epMa.
rIYCTb Ha 3epKaJlbHYlO nOBepx-
HOCTb na,AaeT CBeT H3 TOqKH A. B
TOqKe Al co6HpalOTCH JlyqH, oTpa-
meHHbIe OT 3epKaJla (pHC. 16.7).
llpe,nnOJlOmHM. qTO CBeT MomeT pac-
npOCTpaH.HTbC.H .n.BYM.H nyrHMH, OTpa-
KOTopoe nOTpe6yeTC.H CBeTY. qT06bI npoHTH
AO OA I
nYTb AOA I , MomHO HaHTH no qJopMYJle t = - + -, r,ne v - CKOpOCTb pac-
v v
npOCTpaHeHH.H CBeTa.
KpaTt.IaHWee paCCTO.HHHe OT TOqKH A )J;O 3epKanbHoH nOBepxHocTH 0603Ha-
qHM qepe3 l, a OT TOqKH Al - qepe3 ll'
M3 pHCYHKa 16.7 H8H,neM
AO = .J l 2 + x 2 ; OA I =  (L - xl + if .
458

Tor,Aa
 l2 +X 2 +  (L_X)2 +1i
t= .
v
HaH,neM npOH3BO,AHYIO
1 [ x
=   l2 + X 2
, 1 ( 2x 2 (L - x )( -1) J
t x = -;; 2 ,1 /2 + x 2 + 2  (L _ X)2 + If =
L-x 1 1 ( XL-X )
 ( L _ X)2 +  =  AO - OA 1 .
x .
M3 P HC Y HKa BH,nHM, qTO - = SIn a'
AO '
C ' 1 ( . . )
Jle,nOBaTeJlbHO, t x = - SIn a - SIn y .
v
.n;JI.H Toro qT06bI BpeM.H t 6bIJlO MHHHMaJlbHO, npOH3Bo,nHaSl ,nOJlmHa 6bITb
paBHa HYJIIO. TaKHM 06pa30M, !.(SIna-Siny) = O. OTCIOAa sin a = siny, a TaK
v
KaK yrJIbI a H y - OCTpbIe, TO OTCIO,na CJle,AyeT paBeHCTBO yrJlOB: y = a.
Mb] nOJIyqHJlH COO'IHOIIIeHHe, BblpamalOee BTOpOH 3aKOH OTpameHHSi. 113
npHH:qHna <t>epMa BbITeKaeT H nepBbIH 3aKOH OTpameHH.H: OTpameHHbIH Jlyq
nemHT B nnOCKOCTH, npOXO,ASIeH Qepe3 na,AalOHH nyq H HOpMaJII:. K OTpa-
malOeH nOBpXHOCTH, TaK KaK ecnH 6bI 3TH JlyqH nemanH B pa3HbIX nnOCKO-
CT.HX, TO nYTh AOA 1 He 6bIJI 6bI MHHHMaJIbHbIM.
na,.u;alOHH H OTpameHHbIH Jlyq:a 06paTHMbI, T.e. eCJIH na,.u;alOHH Jlyq Ha-
npaBHTb no nYTH OTpameHHoro Jlyqa, TO OTpameHHbIH Jlyq nOH,neT no nYTH
na,.u;alOero - 3aKOH 06paTHMOCTH CBeTOBbIX JlyqeH.
B 3aBHCHMOCTH OT CBOHCTB rpaHH:qbI pa3,nena cpe,n O'l'pameHHe CBeTa MO-
meT 6bITb 3epKaJIbHbIM H ,nHc}Jc}JY3HbIM (paCCe.HHHbIM).
3epKaJlLHLIM Ha3blB8eTC.H OTpameHHe, npH KOTOpOM na,AalOHH Ha nJloc-
KYIO nOBepXHOCTb (pHC. 16.8) napaJlJIeJlbHbIH nyqoK nyqeH nocJle OTpameHH.H
OCTaeTC.H napaJlJleJIbHbIM.
lliepoXOBaTa.H nOBepXHOCTb OTpamaeT napannenbHbIH na,nalOH" Ha Hee
nyqOK CBeTa no BceB03MomHbIM HanpaBJleHH.HM (pHC. 16.9). TaKoe OTpame-
HHe CBeTa Ha3bIBalOT AH4>4>Y3HbIM.
L-x .
- = SIn y.
OA 1
Puc. 16.8
Puc. 169
459

COOTBeTCTBeHHO pa3J1HqalOT 3epKanbHble H MaTOBbIe nOBepXHOCTH.
CJle.n;yeT OTMeTHTb, qTO 3TO OTHOCHTeJIbHbIe nOHSlTHSI. llOBepXHOCTeH, OT-
pamalOHX TOJIbKO 3epKanbHO, He cyecTByeT. B 60nblliHHCTBe cJlyqaeB HMe-
eTC.H JlHllib MaKCHMYM OTpameHH.H B HanpaBJleHHH yr Jla 3epKanbHoro OTpame-
HHSI. 8THM 06'b.HCHSIeTC.H TO, qTO MbI BH,AHM 3epKaJIO H ,ApyrHe 3epKaJIbHO OT-
pamalOWHe nOBepXHOCTH CO Bcex CTOpOH. a He TOJlbKO B O.lI.HOM HanpaBJIe-
HHH. B KOTOpOM OHH OTpamalOT CBeT.
O.n;Ha H Ta me nOBepXHOCTb MomeT 6bITb 3epKanbHOH H MaTOBOR B 3aBH-
CHMOCTH OT .n;JIHHbI BOJIHbJ na,AalOero CBeTa.
ECJIH rpaHHa HMeeT BH,A nOBepXHOCTH, pa3Mepbl d HepOBHOCTeH KOTOpOH
MeHbllie ,AnHHbI BOJlHbI CBeTa A, TO OTpameHHe 6Y,AeT 3epKaJIbHbIM (nOBepx-
HOCTb KannH pTyrH, OTnOJIHpOBaHHaSi MeTaJlJIHqeCKa.H nOBepXHOCTb H T ..n;.), eCJIH
d »A, OTpameHHe 6y.n;eT .lI.HcpCPY3HbIM. qeM JIyqllie 06pa60TaHa nOBepXHOCTb,
TeM 60nbllia.H .n;OJI.H na,AalOero CBeTa OTpamaeTCSI B HanpaBJIeHHI1 yr na aep-
KaJlbHOrO OTpameHH.H, a MeHbllia.H - pacceHBaeTCH.
PaCCe.HHHbIH CBeT B03HHKaeT BCJIe,ACTBHe MeJIKI1X ,Aecl>eKToB nonHpOBKI1. a-
panHH. MeJIbqaHlliHX nbIJIHHOK, HMelOHX BeJIH'tIHHY nOp.H.n;Ka HeCKOJIbKHX MHK-
pOHOB.
lloBepxHocTb, KOTOpa.H paBHOMepHO pacceHBaeT na,AalOl.I..tHH CBeT BO Bce CTO-
pOHbI. Ha3bIBalOT a6COJll()THO MaTOBOH. A6COJIIOTHO MaTOBbIX nOBepxHocTeH TaK-
me He cyecTByeT. K a6COJIIOTHO MaTOBbIM nOBepXHOCTSIM 6nH3KH nOBepXHO-
CTH HerJla3ypOBaHHoro cpapcpopa. 'tIepTe.mHoii 6YMarH, CHera.
,I:(ame .n;nSl O,AHOrO H Toro me H3nyqeHHH MaTOBaH nOBepXHOCTb MomeT
CTaTb 3epKanbHoH, eCJIH YBeJIH'tIHTb yroJl na.n;eHHSI. ,I:(HcpCPY3HO OTpamalOHe
nOBepXHOCTH MorYT OTJIHtJaTbC.H H no BenHqHHe K03cpcpHHeHTa OTpameHH.H
W OTP
p = W'" nOKa3bIBalOero, J<:aKYIO qaCTb 3HeprHH W na.n;alOero Ha nOBepx-
HOCTb CBeTOBoro nY'tIKa COCTaBJI.HeT 3HeprHSI W OTp OTpameHHoro CBeTOBoro
nyqKa.
BeJlaH 6YMara .n;JIH pHCOBaHHSI HMeeT K03cpcpHHeHT OTpameHHSI. paBHbIH
0,7-0,8. OqeHb BbICOKHH K03cpcpHHeHT OTpameHH.H .n;n.H nOBepxHocTeii, no-
KpbITbIX OKHCblO MarHH.H, - 0,95  OqeHb ManblH ,AJIH tJepHoro 6apxaTa -
0.01-0,002.
3aMeTHM, qTO 3aBHCHMOCTb OTpameHHH H nornoeHHH OT qaCTOTbI KOJle-
6aHHB qae Bcero HMeeT H36HpaTeJlbHbIH xapaKTep.
16.4. IIJIocKoe 3epKaJIO. IIocTpoelllle 11306pamellHH B nJIOCKOM 3epKaJIe
llJlOCKOe 3epKaJlO - 3TO nJlOCKa.H nOBepXHOCTb, 3epKaJIbHO OTpamalOa.H
CBeT.
llocTpoeHHe H306pameHH.H B 3epKaJlaX OCHOBbIBaeTC.H Ha 3aKOHax np.HMO-
JIHHeHHOro pacnpOCTpaHeHHSI H OTpameHHH CBeTa.
llOCTpOHM H306pameHHe TO'tIeqHOrO HCTotJHHKa S (pHC. 16.10). OT HCTOq-
HHKa CBeT H.n;eT BO Bce CTOpOHbI. Ha 3epKaJlO na.n;aeT nyqOK CBeTa SAB, H
H306pameHHe C03)J;aeTC.H BceM nyqKOM. Ho .n;n.H nocTpOeHH.H H306pameHHH .n;o-
CTaTOqHO B3.HTb KaKHe-JIH60 .n;Ba JIyqa H3 3Toro nytlKa, HanpHMep SO H SC.
460

Puc. 16.10
JIyq SO na)J;aeT nepneH.nHKYJI.HpHO nOBep-
XHOCTH 3epKaJlB AB (yroJl na,n;eHHSI pa-
BeH 0), n03ToMY oTpameHHbIH noii,n;eT B 06-
paTHOM HanpaBJleHHH OS. JIyq SC OTpa-
3HTC.H no,n; yr JlOM Y = a. OTpameHHbIe
JlyqH OS H CK pacxo,n;.HTC.H H He nepece-
KaIOTC.H, HO ecnH OHH nona,n;alOT B rJla3
B qeJlOBeKa, TO qenOBel< YBH,n;HT H306pame-
HHe Sl' KOTopoe npe)J;CTaBnSleT C060H TOq-
KY nepeCeqeHHSI npO)1.0JImeHHJI OTpameH-
HbIX JIyqeH.
M306pameHHe, nOJlyqaeMOe Ha nepece-
qeHHH OTpameHHbIX (HJlH npeJlOMJleHHbIX)
nyqeH. Ha3bIBaeTCSI lIeHCTBHTeJILHLIM H306-
pameHHeM.
H306pameHHe, nonaeMoe npH nepece-
qeHHH He caMHX OTpameHHbIX (HnH npe-
npO,AOJlmeHHH, Ha3bIBaeTC.H MHHMLIM H306pame-
A
nOMJleHHbIX) Jlyqeii, a HX
HHeM.
TaKHM 06pa30M, B nJlOCKOM 3epKaJle H306pameHHe Bcer,n;a MHHMoe.
MomHo ,n;OKa3aTb (paCCMOTpHTe TpeyronbHHKH SOC Ii SIOC), qTO paCCTO.H-
HHe SO = SIO, T.e. H306pameHHe TOqKH Sl Haxo,n;HTC.H OT 3epKaJla Ha TaKOM
me paCCTO.HHHH, KaK Ii caMa TOqKa S. OTCIO,n;a BbITeKaeT, qTO ,n;JI.H nOCTpOeHH.H
H306pameHHSI TOqKH B nJlOCKOM 3epKaJle ,n;OCTaTOqHO onYCTHTb Ha nJlOCKOe
3epKaJlO H3 3TOH TOqKH nepneH,n;HKYJl.Hp H npo,n;OJlmHTb ero Ha 'l'8KOe me pac-
CTO.HHHe 3a 3epKaJlO (pHC. 16.11).
lIpH nOCTpoeHHH H306pameHH.H KaKOrO-JlH60 npe,n;MeTa nocne,n;HHH npe,A-
CTaBJI.HIOT KaK cOBoKynHocTb TOqeqHbIX HCTOqHHKOB CBeTa. IIo3ToMY ,n;OCTaTOq-
HO HaHTH H306pameHHe KpaHHHx TOqeK npe.nMeTa.
M306pameHHe AIBI (pHC. 16.12) npe,n;MeTa AB B nJlOCKOM 3epKaJle Bcer)J;a
MHHMoe, np.HMOe, Tex me pa3MepOB, qTO H npe,n;MeT, H CHMMeTpHqHOe OTHOCH-
TeJlbHO 3epKaJIa.
S
S.
-------
A I _. B
I I
I I
I I
"" 'i.." ',:", ""'" '''' ,
I I
I I
I I
A. L______ B.
Puc. 16.11
Puc. 16.12
16.5. C4>epH'IeCKHe 3epKaJIa
CcpepHqeCKOe 3epKano npe,n;CTaBnSleT C060H ccpepHqeCKHH cerMeHT, 3epKaJlb-
HO oTpamalOll.\HH CBeT.
461

C<}>epHqeCKHe 3epKaJl8 6bIBalOT BOrHYTbIe (pHC. 16.13, a) - y HHX OTpa-
JKalOee nOKpbITHe HaHeceHO Ha BHYTpeHHlO1O nOBepXHOCTb, H BhInYKJlbIe
(pHC. 16.13, 6) - Y HHX OTpamalOee nOKpbITHe HaHeceHO Ha BHeIIIHlO1O no-
BepXHOCTb.
a
P
Puc. 16.13
feOMeTpHqeCKHH neHTp 0 c<}>epHqeCKOH nOBepxHocTH 3epKaJla pa.u;HYCOM R
Ha3bIBaeTC.H }\eHTpOM 3epKaJIa, a TOqKa P, .HBJI.HIOa.HC.H BepIIIHHoH c<}>epHqeC-
Koro cerMeHTa - nOJIIOCOM 3epKaJIa. JII06aSl npSlMaSl (HanpHMep, OM HOP),
npoxo)J;.Ha.H Qepe3 eHTp 0 3epKaJla, Ha3bIBaeTC.H OIITHqeCKOH OChlO. OnTH-
qeCKa.H OCb OP, npOXO,A.Ha.H qepe3 nOJllOc 3epK8JI8, Ha3bIB8eTC.H rJIaBHOH onTH-
lIecKoH OChlO, Bce OCTaJlbHbIe OCH - n060'IHbIMH OnTHlIeCKHMH OCHMH. .HCHO,
qTO Jl106a.H OnTHqeCKa.H OCb B TOqKe nepeCeqeHH.H C nOBepXHOCTblO 3epKaJl8
.HBJI.HeTC.H HOpMaJlblO K nOCne,AHeH (Jl1060H pa,AHYC nepneH)J;HKYJI.HpeH K K8C8-
TeJlbHOH K nOBepXHOCTH c<}>epbI). TOqKa F Ha rJlaBHOH OnTHqeCKOH OCH, qepe3
KOTOPYIO npOXO,A.HT nocJle OTp8meHH.H OT 3epKaJl8 JlyqH (HJlH HX npO,AOJlme-
HH.H). na,AalOlltHe Ha 3epK8J10 n8p8J1J1eJlbHO rJl8BHOH OnTHqeCKOH OCH, Ha3bIBa-
eTC.H 4JOKYCOM aepKaJIa. Y BorHYToro 3epKaJla cl>OKYC ,AeHCTBHTeJlbHbIH. Y BhI-
nYKnoro 3epK8J18 cl>OKYC MHHMhIH. PaCCTO.HHHe OT <}>OKyca c<}>epHqeCKOrO 3ep-
Kana ,AO ero nOJllOca PF Ha3hIBaeTC.H OKYCHLIM paCCTOHHHeM. Ero npHH.HTO
0603HaqaTb TaKme 6YKBOH F. TIJlOCKOCTb KL, npOXO,A.Ha.H qepe3 q-JOKYC nep-
neH,AHKYJI.HpHO K rJlaBHOH OnTHqeCKOH OCH, Ha3bIBaeTC.H 4JoKaJIbHOH DJIOCKO-
CTblO. B c1>OK8J1bHOH nJlOCKOCTH nepeceKalOTC.H nocJle OTp8meHH.H OT 3epKan8
JlyqH (HJlH HX npO,AOJlmeHH.H), na,AalOHe Ha 3epKaJlO napaJlJleJlbHO KaKOH-JlH60
n060qHOH OnTHqeCKOH OCH.
Onpe)J;eJlHM nOJlOmeHHe <}>OI<yca ccI>epHqeCKOrO 3epKa-
Jla. TIYCTb Ha 3epKaJlO (pHC. 16.14) na)J;aeT Jlyq NM, na-
C N paJlJleJlbHbIH rJlaBHOH OnTHqeCKOH OCH. OTpameHHbIH OT
3epKaJl8 Jlyq MF npOH,AeT qepe3 <}>OKYC F. JIyq N M co-
CTaBJI.HeT C p8,AHYCOM OM yroJl a. Yron OTp8.)KeHHSI
LOMF = a H LMOF = a KaK H8KpeCT JlemaHe npH na-
p8JIJlenbHbIX np.HMbIX MN H PO H ceKyeH MO. CJle,AOB8-
TeJlbHO,  OMF - p8BH06e,ApeHHbIH (FO = MF). Yron
MFE = 2a (yroJl BHeIIIHH:H no OTHoweHHIO K  MOF).
BY,AeM paCCMaTpHBaTb TonbKO T8K Ha3bIBaeMble n8p8-
Puc. 16.14 KCHaJlbHbIe nyqKH, T.e. Y3KHe nyqKH, COCTaBJI.HIOHe C
OnTHqeCKOH OCblO 3epKaJla OqeHb MaJlble yrJlbI (B WHpO-
KOM nyqKe TPH Jlyqa, 06pa3YIOHe 3HaqHTenbHbIe yrnbI
462

pyr C .ApyroM, He nepeceKalOTC.H B O)J;HOH TOqKe). Tor)J;8 PE« R, EF =:: PF
Ii tg a =:: sin a = a.
. h h h It
M3 !:J. MOC SIn a = -  a=:: -. 0:3 !:J. MEF tg 2a = -  2a ==-.
R R EF PF
h h
OTCIO,lX8 2 - = -
R PF
R
 PF = -. T8KHM 06pa30M. TOqKa F JlemHT Ha rn8B-
2
HOH OnTHqeCKOH OCH H eJlHT paHYC 3epKaJla OP H8 Be O)J;HHaKOBbIe qac-
R
TH. 3HaqHT, <2>oKycHoe paCCTO.HHHe F = 2'
AHaJlOrHqHO MomHO .AOKa3aTb, qTO <2>OKYC BbInYKJIOrO c<2>epHqeCKOrO 3epKa-
JI8 JlemHT Ha rJIaBHOH OnTHqeCKOH OCH 38 3epK8J10M Ii Y8JIeH OT nonlOC8
3epK8J1a Ha paCCTO.HHHe, paBHoe nOJIOBHHe paHyca 3epKaJla. <1>oKYCHoe p8C-
CTO.HHHe BbInYKJloro 3epK8J1a npHHHTO CqHTaTb OTpHLT.aTeJlbHbIM (TaK KaK y
R
BbInYKJloro 3epKaJla <2>OKYC MHHMbIH), T.e. Y BbInYKnoro 3epKaJla F = --.
2
<1>opMYJIa c4>epHlleCKOrO 3epKaJIa. llYCTb TOqeqHhIH
HCTOqHHK CBeTa S (pHC. 16.15) pacnOJlOmeH Ha rJIaB-
HOH OnTHqeCKOH OCH 3epKaJla Ha paCCTOHHHH SP = d.
YroJl n8JJ;eHHH Jlyq8 SM H8 nOBepXHOCTb 3epK8J18
LSMO = a. OTp8meHHbIH Jlyq nepeceKaeT rJlaBHYIO on-
S THqeCKYIO OCb B TOqKe S'. YroJl OTpameHH.H LOMS' = a
(no 38KOHY OTpameHH.H). 0603H8qHM yron H8KJlOH8 n8-
alOero Jlyqa K rJl8BHOH OnTHqeCKOH OCH LMSO = cp,
yrOJl H8KJlOH8 OTpameHHoro Jlyqa LMS'P = y, yroJl Ha-
KJlOHa pal1Hyca LMOP = (3, paCCTO.HHHe OT TOQKH M 110
rJlaBHOH OnTHqeCKOH OCH qepe3 ME = h.
YroJl p - BHellIHHH no OTHOllIeHHIO K !:J. OMS. IIo-
3TOMY (3 = ex + <p.
YroJl Y - BHellIHHH no OTHOllIeHHIO K tJ. 8'OM. l103TOMY Y = a + (3.
M3 3THX paBeHCTB nOJlyqaeM
P
d
Puc. 16.15
y + <p = 2(3.
h h
M3  8'EM HaXOJJ;HM tgy = E8' == f'
h h
M3 !:J. SEM HMeeM tg<p = SE =:: d'
TaK KaK MbI paCCMaTpHBaeM TOJlbKO napaKCHaJlbHhIe JlyqH. TO TaH-
reHCbI yrJloB MomHO 3aMeHHTb 3HaqeHH.HMH caMHX yrJloB B paHaH8X.
(16.1)
h h
M3 !:J. OME HMeeM tgp = OE =:: R '
CJleOB8TeJlbHO,
h h h
y=,; (3= R ; <P=d'
llOCT8BHM B (16_1), nOJlyqHM
+=2!!:  !+!.=. R
f d R d f R A TaK KaK F = 2' TO MomHO 3anHcaTb
463

] 1 1
-+-=-. (16.2)
d I F
STO BbIpameHHe Ha3bIBalOT «POPMYJlOH c«pepHqeCKOrO 3epKan8. cI>OPMYJlY
(16 2) MomHO npHMeH.HTb H )J;JI.H BbInYKnbIX c«pepHqeCKHX 3epKaJl, eCJlH HC-
nOJlb30BaTb npaBHJlO 3HaKOB: CqHTaTb 3H8KH BeJlHqHH d, I, R H F nOJlOmH-
TeJlbHbIMH, eCJlH 3TH paCCTO.HHH.H H3MepeHbI OT nOJllOca 3epKaJl8 B TY CTOpO'
HY, OTKY,Aa Ha 3epKaJlO na,AaeT CBeT OT npe,AMeTa, H OTpH8TeJlbHbIMH, eCnH
OHH OTCqHT8HbI OT nonlOca 3a 3epK8J10. ,Z:(nSi BbInYKnbIX 3epK8J1 d> 0, a R < 0,
F < O. ECJlH H306p8meHHe MHHMoe, TO 1<0.
T8K KaK B «POPMYJlY (16.1) He BXO,ASIT 3H8qeHHSI h H yrJl8 q>, TO 3TO 03-
HaqaeT, qTO JlI060H Jlyq, BbIXO,n.HHH H3 S, npOH,AeT qepe3 TOqKY S'. CJle,nOBa-
TeJlbHO, TOqKa S' .HBJI.HeTC.H i306pameHHeM TOqKH S.
16.6. IIocTpoeHHe H306paeHHJI npeMeTa B c4>epH'IeCKHX 3epKaJIaX
OT JlI060H TOqKH npe,nMeT8, pacnOJlOmeHHOro nepe,n 3epK8nOM. n8,AaeT 6ec-
KOHeqHOe MHomeCTBO nyqeH - p8CXO,ASIHHCSI CBeTOBOH nyqOK. ,Z:(nSi nOCTpo-
eHH.H H306p8meHH.H 3TOH TOqKH ,AOCT8TOqHO BbI6paTb TOJlbKO ,ABa K8KHX-JlH60
nyqa, HO Jlyqme T8KHe, XO,n KOTOpbIX nocJle 3epKaJla 3apaHee H3BeCTeH. K T8-
KHM nyqaM OTHOC.HTCSI (pHC. 16.16): 1 - nyq, n8paJlJleJlbHbIH rJl8BHOH on-
THqeCKOH OCH. OTpameHHbIH Jlyq (HnH npo,nOJlmeHHe OTp8meHHoro Jlyq8) npo-
XO,AHT Qepe3 rJlaBHbIH «POKYc; 2 - nyq, npoxo,n.H[QHH qepe3 rJlaBHbIH «POKYc
(HJlH ero npO,AOJlmeHHe npoxo,nHT qepe3 rn8BHbIH «POKYc), oTpameHHbIH JlY'l
H,neT napanJleJlbHO rJlaBHOH OnTHqeCKOH OCH; 3 - nyq (HJlH ero npo,noJlme-
HHe), npoxo,n.HHH qepe3 lJ,eHTp 0 3epKan8. OTpameHHbIH nyq B03Bpa[QaeTCSI
B npOTHBOnOJlOmHOM Hanp8BneHHH B,AOnb TOH me npSIMoH; 4 - nyq, npoxo-
,A.H[QHH Qepe3 nonlOC, nocJle OTp8meHHSI nOH,neT CHMMeTpHqHO OTHOCHTenbHO
r JlaBHOH OnTHqeCKOH OCH 3epKaJla.
a
--
o
Puc. 16.16
EcnH npe,nMeT nnOCKHH, TO ,nOCTaTOqHO nOCTpOHTb H306p8meHHe ero KpaH-
HHX TOqeK. H8 PHCYHK8X 16.17-16.19 npHBe,neHbI nOCTpOeHH.H H306pameHHH
,nJlSl pa3HbIX nOJlOmeHHH npe,nMeTa AB nepe,n 3epK8nOM. K8K BH,nHO, ecnH
npe,nMeT pacnOJlOmeH Ha p8CCTO.HHHH d > R (pHC. 16.17) (,Aanbme lJ,eHTpa BO-
rHYToro 3epKaJla), TO 06pa3yeTC.H nepeBepHYToe, ,AeHCTBHTeJlbHOe, YMeHbmeHHoe
H306p8meHHe npe,AMeTa. ECJlH F < d < R (pHC. 16.18), TO H306pameHHe B BO-
rHYTOM 3epKaJle nepeBepHYToe, ,AeHCTBHTeJlbHOe, YBenHqeHHoe. ECJlH d < F
(pHC. 16.19), TO H306pameHHe npSIMoe, MHHMoe, YBeJlHqeHHoe.
464

Puc. 16.17
Puc. 16.18
p
Puc. 16.19
B BbInYKJlOM 3epKaJle (pHC. 16.20) H306pameHHe Bcerna MHHMoe. np.HMoe.
YMeHbmeHHoe.
C yqeToM npaBHJla 3HaKOB MomHO 3anHcaTb cpopMyny ccpepHqeCKOrO 3ep-
1 1 1
KaJla ,lJ;n.H CJlyqaeB, H306pameHHbIx Ha pHCYHKax 16.17 Ii 16.18: d + f = F j
1 1 1 1 1 1
Ha pHCYHKe 16.19: d - f - F ' Ha pHCYHKe 16.20: d - f - - F '
--
o
Puc. 16.20
Puc. 16.21
6
,l];JI.H nocTpOeHH.H H306pameHH.H TOqeqHOrO npe,lJ;MeTB A (pHC. 16.21), pBC-
nOJlOmeHHOro Ha rnaBHOH OnTHqeCKOH OCH, H3 3Toro npeMeTa HanpaBn.HIOT
HB 3epKaJlO JlI060H Jlyq AM, npOBOn.HT napanJlenbHO 3TOMY Jlyqy n060qHYlO
OnTHqeCKYIO OCb ON H cpoKaJlbHYIO nnOCKOCTb. HaXOlI.HT n060qHbIH CPOKYC F'
(TOqKa nepeCeqeHH.H n060QHOH OllTHtIeCKOH OCH C cpoKanbHoH nJlOCKOcTblO). OT-
pameHHbIH Jlyq npoHeT qepe3 n060qHbIH cpOKYC F H nepeCeqeT rJlaBHYIO on-
THqeCKYIO OCb B TOqKe Alt KOTOpa.H H .HBJI.HeTC.H H306pameHHeM TOqKH A.
16.7. JIHHeHHoe YBeJIH'IeHHe CepH'IeCKOrO 3epKaJIa
KaK BHHO H3 PHCYHKOB 16.17-16.20, pa3MepbI H306pameHH.H npe,lJ;MeTB,
,lJ;aBaeMOro ccpepHqeCKHM 3epKaJlOM, B OTJlHqHe OT nJlOCKOrO 3epKaJla 3aBHC.HT
OT nOJlOmeHH.H npenMeTa OTHOCHTeJlbHO 3epKana.
IIo,lJ; JIHHeHHLIM YBeJIH'leHHeM ccpepHqeCKOrO 3epKaJla nOHHMalOT OTHome-
HHe JlHHeHHbIX pa3MepOB H306pameHH.H K JlHHeHHbIM pa3MepaM npe,AMeTa:
r- Al = H
- AB h .
465

M3 nO,A06H.H f1BPA H f1B 1 PA I (pHC. 16.17) Haxo,nHM, qTO H = L
h d
f
CJle,noBaTeJlbHO, r = d'
16.8. 3aKoHLI npeJIOMJIeHHJI CBeTa
CBeT pacnpOCTpaH.HeTC.H np.HMOJlHHeHHO TOJlbKO B np03paqHOH O)J;HOpO,nHO
cpe,Ae. lIpH nepexo,ne H3 O,AHOH O,nHOPO,nHOH Cpe,AbI B ,npyrylO (HanpHMep, H3
B03,nyxa B Bony) CBeT Ha rpaHHue pa3neJla H3MeH.HeT CBoe HanpaBJleHHe.
M3MeHeHHe HanpaBJleHH.H pacnpOCTpaHeHH
CBeTB Ha rpaHHLT.e pa3,neJIa ,nBYX cpe,n npH ero
nepexo)J;e H3 O,nHOH Cpe,AbI B )J;pyrylO Ha3bIBaeT-
C.H npeJlOMJIeHHeM CBeTa. JIyq OB (pHC. 16.22),
npoxo,n.HHH BO BTOPYIO cpe,ny, Ha3bIBaeTCH
npeJIOMJIeHHLIM JlyqOM.
N YroJl  Mem,ny npeJlOMJleHHbIM JlyqOM J1
nepneH,nHKYJl.HpOM OD K rpaHHLT.e pa3,neJla cpe,n
B TOqKe na,neHH.H Jlyqa Ha3bIBaeTC.H yrJIOM npe-
JlOMJIeHHH.
OnbIT nOKa3bIBaeT, qTO eCJlH na,nalOHH JI}"!
nepneH,AHKYJl.HpeH K rpaHHLT.e pa3,neJla (ex. = 0), TO
yroJl npeJlOMJleHH.H paBeH 0 (= 0), T.e. Jlyq
H)J;eT He npeJlOMJI.H.HCb. ECJIH YBeJIHqHBaTb yroJI
naneHH.H. TO yroJl npeJlOMJleHH.H 6yneT TaKme paCTH. HO OH 6yneT MeHbllle HJlH
60Jlbllle yrJla na,neHH.H.
3KcnepHMeHTaJlbHO YCTaHOBJleHbI cJle,nYIOHe 3aKOHLI npeJlOMJleHHH CBeTa.
1. JIyq na,nalOHH, Jlyq npeJIOMJleHHbIH H nepneH,nHKYJlHp, BOCCTaBJleHHbI
K rpaHHue )J;BYX cpe,n B ToqKe na,neHH.H Jlyqa, JIemaT B O,nHOH nJlOCKOCTH
2. OTHollleHHe cHHyca yrJlB na.neHH.H K CHHYCY yrJla npeJlOMJleHH ,nJI.H
)J;aHHbIX )J;BYX cpe,n eCTb BeJIHqHHa nOCTO.HHHa.H, Ha3bIBaeMa.H OTHOCHTeJlbHbIM
nOKa3aTeJleM npeJlOMJleHHH BTOpOH no Xo,ny Jlyqa cpe,nbJ OTHOCHTeJlbHO nep-
BOH:
IC
I
I
I
I
I
M
Puc. 16.22
sin ex.
--:-- = n21'
SIn 13
3aKOHbI npeJlOMJleHH.H TeOpeTHqeCKH MorYT 6bITb BbIBe,neHbI Ha OCHOBaHHH
npHHLT.Hna <t>epMa.
lIYCTb B cpe,ne I CKOpOCTb pacnpOCTpaHeHH.H CBeTa VI' a B cpe,ne II - V2'
lIpe,nnOJlOmHM, qTO CBeT H3 TOqKH Al na,AaeT Ha nOBepXHOCTb pa3,neJla ,nBYX
cpe,n I H II H nOna,AaeT B TOqKY A 2 . BbI6epeM H3 Bcex B03MomHbIX TpaeKTO-
pH pacnpOCTpaHeHH.H CBeTa TY. KOTOpOH COOTBeTcTByeT MHHHM8..1IbHOe BpeMSI
pacnpOCTpaHeHH.H CBeTa.
,lJ;JI.H npoxom,neHH.H CBeTa H3 TOqKH Al B TOqKY A 2 6Y)J;eT 3aTpaQeHO
BpeM.H
A10 O
t=-+-.
VI V2
466

A 1 X
l1
Puc. 16.23
TIYCTb KpaTqaHllIee p8CCTO.HHHe OT TOqKH Al ,no
nOBepxHocTH pa3,neJla cpe,n lI' a OT TOqKH A 2 ,no
Hee - l2'
11:3 pHcYHKa 16.23 B H.nHM, qTO
AIO =  Ii + x 2 ; O =  Ii + ( L - X)2 .
Tor,na
 t; + x 2  Ii + (L - X )2
t= + .
VI V2
ECJlH 3TO MHHHMaJlbHOe BpeM.H. TO npOH3BOlIHa.H
t = O. HaH.neM npOH3Bo.nHYIO t:
, 2x 2 ( L - x )( -1 )
t x = 2Vl .J  + x 2 + 2V2   + (L _ X)2
x L-x
VI . AIO V2' OA 2
x . L - X A l:t
H3 pHCYHKa 16.23 Haxo.nHM, qTO - = SIn a; - = sin....., r.ne a H ..... -
AIO O
yrnbI na.neHH.H H npeJlOMneHHSI COOTBeTCTBeHHO.
T sin a sin (3 0
or.na --- = .
VI V2
sin a VI
OTKy.na - =-.
sin J3 V2
T8K K8K CKOpOCTb CBeT8 B Kam.noH cpe.ne eCTb BeJlHqHHa nOCTO.HHH8.H, TO
VI = const. 0603H8qHB ee n2I' nOJlyqaeM 3aKOH npenOMJleHH.H. CJle.nOB8TeJlbHO,
V2
VI
n2I = -.
V2
OrHocHTeJlLHLIH nOKa3aTeJIL npeJlOMJleHHJI nOKa3bIBaeT, BO CKOJlbKO pa3 CKO-
pOCTb CBeTa B nepBOH no XO)J;Y nyqa cpe.ne OTJlHqaeTC.H OT CKOpOCTH p8cnpo-
CTpaHeHH.H CBeTa BO BTOPOH cpe.ne.
c
n =-,
V
CBeTa B BaKYYMe. V - CKOpOCTb CBeT8 B naHHOM BeeCTBe.
TIoK83aTeJlb npeJlOMneHH.H .naHHoro Bem;eCTBa OTHOCHTeJlbHO B8KYYM8 Ha-
3bIBaeTC.H a6COJIlOTHbiM nOKa3aTeJleM npeJlOMJleHHJI. OH nOK83bIBaeT, BO CKonb-
KO p83 CKOpOCTb CBeTa B BaKYYMe 60JlbllIe, qeM B )J;aHHOM BeeCTBe.
Tor.na .nJl.H .n8HHbIX )J;BYX cpe.n
ECJlH CBeT na.n8eT H3 B8KYYM8 B BeeCTBO, TO
r,Ae C - CKOpOCTb
_ c . _ c 0 n _ VI _ 
n i - -, n2 - -. TKy.n a 21 - - - -.
VI V2 V2 
Cpe,na, Y KOTOpOH a6COJllOTHbIH nOK838TeJlb 60nbllIe, H83bIB8eTC.H OnTHqeC-
KH 60Jlee nJlOTHOH.
467

IIpeJlOMJleHHeM CBeT8 06'b.HCH.HeTC.H TOT <}>aKT, qTO rJly6HHa BO.n;OeMa
(pHC. 16.24, a) KameTC.H HaM MeHbIIIe, qeM H8 caMOM ,AeJle, a npe.n;MeT,
p8CCMaTpHBaeMbIH qepe3 nJlOCKOnapaJlJleJlbHYIO nJl8CTHHKY HJlH npH3MY
(pHC. 16.24. 6. 6), 6y,neT K838TbC.H CMeeHHbIM OTHOCHTeJlbHO CBoero HCTHH-
Horo nOJlOmeHH.H. MbI BHlIHM He C8M npelIMeT. a ero MHHMoe H306p8meHHe.
TaK KaK B TOqKe 8 1 nepeceKalOTC.H He caMH npeJlOMJIeHHbIe JlytlH, S HX npo-
,nOJlmeHH.H .
a
4J
-=- s:
s
6
Puc. 16.24
8

16.9. ITOJIHOe OTpameHHe CBeTa
I
1 I
I
-+----
a n PI -_ -_
Puc. 16.25
KaK sin (3 He MomeT 6bITb
M3 3aKOHa npeJlOMJIeHH.H CJle,nyeT, ({TO
'A n 1 , E A
SIn..., = -SIn a. CJlH nl> n2' TO ..., > a, T.e. eCJlH
n2
CBeT nepeXO,AHT H3 Cpe,AbI. OnTHqeCKH 60Jlee
nJlOTHOH, B cpe,ny, OnTHqeCKH MeHee nJIOTHYIO,
TO yroJl npeJlOMJleHH.H 60JlbIIIe yrJls nS,AeHH.H
(pHC. 16.25). IIPH YBeJlHqeHHH a YBenHqHBa-
eTC.H H . IIpH HeKOTopOM yrJle a = a np npe-
JlOMJleHHbIH Jlyq 2 6y.n;eT CKOJlb3HTb B,nOJlb rpa-
HHbI pa3.D;eJla (= 90°). IIpH a > a np CBeT He
npOH,neT BO BTOPYIO np03paqHYlO cpe,ny (TaK
> 1) OH nOJlHOCTblO OTpa3HTC.H. STO .HBJleHHe Ha-
3
3bIBaeTC.H nOJIHLIM oTpaaceHHeM CBeTa.
Y rOJl n8,neHH.H ex. op' npH KOTOpOM npeJlOMJleHHbIH JI CKOJlb3HT B,nOJlb no-
BepXHOCTH pa3,neJla cpell. Ha3bIBaeTC.H npe.neJILHLIM yrJIOM nOJIHOrO oTpaace-
HRH.
M3 3aKOHa npeJlOMJleHH.H CJle)J;yeT, qTO B aTOM cJlyqae
sin anp = n 2 . . n 2
OTKy.n;a SIn anp = - = n 21 .
sin 90° nl nl
468

IIoJlHoe OTpameHHe MOmHO Ha6J1IO,ZJ;aTb B paBH06e,ZJ;peHHOH np.HMoyrOJlbHOH
CTeKJI.HHHOH npH3Me (pHC. 16.26), KOTOpa.H IlIHpOKO HcnoJlb3yeTcH B nepHCKO-
nax, 6HHOKJI.HX H IIp.
a t5 6
1
1 2 2
1
2 2 1
2
1
1 2
Puc. 16.26
Puc. 16.27
CBeT na,ZJ;aeT nepneH,ZJ;HKYJI.HpHO nepBoH
rpaHH (pHC. 16.26. a) H n03TOMY 3.z:r.eCb He
npOHcxo.z:r.HT npeJlOMneHH.H (a = 0 H 13 = 0).
'YroJl na)leHHH H8 BTOPYIO rpaHb a = 45°, T.e.
60JlbllIe a np (,ZJ;JI.H CTeKJIa a op = 42°). IIo3To-
MY Ha 3TOH rpaHH CBeT HcnbITbIBaeT nOJlHOe
OTpameHHe. 3TO nOBopOTHaH npH3Ma. OHa
nOBOpaqHBaeT Jlyq Ha 90°.
B cJlyqae, H306pameHHoM Ha pHCYHKe
16.26. 6, BHYTPH npH3MbI CBeT HcnbITbIB8eT
yme ,ABYKpaTHoe nonHoe OTpameHHe. 3TO
Tome nOBopOTHaH npH3Ma. OHa nOBop8t.JHBa-
eT Jlyq Ha 180°.
Ha pHCYHKe 16.26, 6 H306pameHa 060POT-
H8.H npH3M8. DpH BbIxo.z:r.e H3 npH3MbI JlYt.JH
BepXHHH lIa,A81OH" nyq BbIXO,AHT 1f3 npH3MbI
napaJlJlenbHbI n8.z:r.81OII.J;HM, HO
HHme. a HHmHHH - BbIllie.
B H30rHYTOM CTeKJI.HHHOM CTepmHe (PHC. 16.27) B03HHKaeT MHorOKpaT-
Hoe non Hoe OTpameHHe CBeTa. 3TO HcnoJlb3yeTcSI B cBeToBoax. BOJlOKHHCTbIe
cBeToBo,ZJ;bI CJIymaT ,ZJ;JI.H nepe,AaqH He TOJlbKO CBeTa, HO H H306pameHH.H npe,ZJ;Me-
Ta. KaqeCTBO H306pameHH.H, nepe,ZJ;8BaeMoro no CBeTOBO,ZJ;y, 3aBHCHT OT ,ZJ;HaMeT-
p8 BOJlOKOH H nJlOTHOCTH HX YKJIa,ZJ;KH. OHH IlIHpOKO H:CnOJlb3YIOTCH B Me,ZJ;H-
HHe )J;JI.H OCBeeHH.H H Ha6JIIO)J;eHH.H Tex HJlH HHbIX yqacTKoB BHYTpeHHHX op-
raHOB, a TaKme B COBpeMeHHOH TeXHHKe H B npOMbIIlIJIeHHOCTH.
16.10. Xo.n; JlY'leH B npH3Me
MOHoxpOMaTHqeCKHH CBeT na,ZJ;aeT Ha rpaHb AB CTeKJI.HHHOH npH3MbI
(pHC. 16.28), H8xo.z:r..HeHCH B B03,Ayxe, S101 - na..n;alOHH Jlyq, a1 - yrOJI
na.neHH.H. 0 1 0 2 - npeJlOMJleHHbIH Jlyq. 131 - yroJl npeJlOMJleHH.H. TaK K8K
CBeT nepeXO,ZJ;HT H3 cpe,ZJ;bI OnTHqeCKH MeHee nJlOTHOH B OnTHqeCKH 60Jlee nJlOT-
Hyro, TO 131 < al' IIpOH,ZJ;.H qepe3 npH3MY, CBeT na..n;aeT Ha ee rpaHb AC. 3,AeCb
OH CHOBa npeJlOMJlHeTC.H: a2 - yroJl n8,ZJ;eHH.H, 132 - yrOJ1 npeJlOMJleHH.H. Ha
469

)J;aHHOH rpaHH CBeT nepeXOHT H3 Cpe,A on-
A THqeCKH 60Jlee nJlOTHOH B OnTHqeCKH MeHee
nJlOTHYIO. ll03'fOMY P2> a2'
rp8HH BA H CA, H8 KOTOpbIX npOHCXO-
,AHT npeJlOMJleHHe CBeT8, Ha3bIBaIOTC.H npe-
J10MJlJIJOHMH rp8HJlMH. Y ron <p Memy npe-
JlOMJlHIOHMH rpaH.HMH H83bIBaeTCH npeJlOM-
J1HIOHM yrJlOM npH3M. YroJl 0, 06pa30B8H-
HbIH H8npaBJIeHHeM Jlyqa, BXO.Hero B npH-
3MY, H HanpaBneHHeM Jlyq8, BbIXO.Hero H3
Hee, Ha3B8IOT yrJlOM OTKJlOHeHHJI. rp8Hb,
Puc. 16.28 JIem88.H npoTHB npeJlOMJI.HIOll{erO yrJla, H8-
3bIB8eTC.H OCHOB8HHeM npH3MLI.
An.H npH3MbI Cn?8Be.nJlHBbI cJle)1YJOHe COOTHoweHHH:
1) AJI.H nepBoH npenOMJI.HIOWeH rp8HH 3aKOH npenOMneHH.H CBeTa 3anHllleT-
C.H T8K:
sin al
-=n
sin r3 1 '
re n - OTHOCHTeJIbHbIH nOKa3aTeJlb npeJlOMJleHH.H BeeCTBa, H3 KOToporo
C,AeJl8H8 npH3M8.
2) AJI.H BTOpOH rp8H:
sin a 2 =.!..
sin r32 n .
3) llpeJJOMJI.HIDHH yroJl npH3MbI:
<p = a 2 + r3 1 .
YroJl OTKJIOHeHH.H JlYQ8 npH3M OT nepBOH8q8J1bHOro H8np8BJleHH.H:
o = al + f32 - <po
CJle)J;oBaTeJlbHo. eCJlH OnTHqeCK8H nJlOTHOCTb BeeCTBa npH3ML] 60Jlbllle,
qeM oKpymalOeH Cpe,AbI, TO JIy't.l CBeT8, npOXOHHH qepe3 npH3MY, OTKJIO-
H.HeTC.H K ee OCHOB8HHIO. HeCJlOmHO nOKa3aTb, qTO eCJIH OnTHqeCKa.H nJlOT-
HOCTb BeeCTBa npH3Mbl MeHbllle, 'leM OKpY:>K81OeH cpebI. TO JIyq CBeT8
nocJle npoxomeHHH Qepe3 npH3MY OTKJlOHHTC.H K ee BeplllHHe.
BonpOCM AAH CAMOKOHTPOAH
1. Bcezoa.n.u C6em pacnpocmpaHSlemcSl npSlMo.n.ulieulio? llpu KaKUX yc.n.o
6USlX HapywaemcSl 3aKOli npSlMOJlUHeUHOzo pacnpocmpalieliuSl c6ema?
2. lj mo Ha3btBalOm mOlle'llibtM ucmOIlIiUKO.M CBema? KaK no.n.Yllumb ma-
KOU UCmO'lHuK?
3. llOileMY 06pa3YIOmcSl me lib u no.n.ymelib? KaK no.n.Yllumb om npeOMe-
ma mo.n.bKO no.n.ymelib? pe3KYlO melib 6e3 no.n.ymeliu? Om Ilezo 3a6uCSlm pa3'
Mepbl U ljJopMa meHU om oaliliozo npeOMema lia 3Kpalie? KaK 6.n.uSllOm pa3-
Mepbl ucmOllliUKa C6ema Ha WUPUIiY 06.n.acmu nOJlymeliu?
4. M eJICoy mOllellhbtM ucmOIlHUKOM C6ema U 3Kpa1iO.M nOMe14eli OUCK. II pu
KaKOM no.n.OJICeIiUU oucKa ezo melib 6yoem UMemb ljJOPMY Kpyza? KaK06a ljJop.
Ma ezo meliu npu opyzux no.n.OJICeIiUSlX oucKa?
470

5. H a lleM OCli06alia 603MOJICliOCmb no.n.Yllenu.R u306paJICenuu C nO.MOblO
.Ma.n.ozo om6epcmuSl?
6. KaKoe Sl6.n.enue lia3bl6aemcSl ompaJICenueM. c6ema 2 q eM. om.n.UllaemcSl
iJuljJljJY31i0e ompaJICeliue C6ema om 3epKa.n.bliozo?
7. CljJopMy.nupyume 3aKOlibl ompaJICenuSl C6ema.
8. qmo nOliUMalOm noo u306paJICeliUeM mOllellliOO npeiJMema? ICaKoe
u306paJICeliUe lia3bl6alOm iJeucm6ume.n.bnblM? MliUMblM?
9. riJe U KaKoe no.n.YllumcSl u306paJICeliUe C6em.RlqeUcJl mOIlKU 6 n.n.OCKOM
3epKa.n.e? npOmSlJICeliliOZO npeiJM.ema 6 lieM?
10. II olle.MY u306paJICeliUe .n.1060zo npeiJ.Mema B n.n.OCKOM 3epKa.ne MOJIC-
no c(jJomozpaljJup06amb, a no.n.Yllumb ezo u306paJICeliUe na 3Kpalie lieB03-
MOJICliO?
11. II epeiJ n.n.OCKUM 6epmuKa.n.bliblM 3epKa.n.oM cmounz. lle.n06eK U 3aMella.
em. Ilmo Oli lie .MOJICem Y6uoemb C60e ompaJICeliue no.n.liocmblO. Y6uiJum JlU
lle.n.06eK lw.n.liocmblO C60e u306paJICeliue. ec.nu omouiJem iJa.n.bwe UJlU noiJou-
iJem 6.n.uJlCe K 3epKa.n.y?
12. C nO.MOlqblO c(jJepullecKozo 3epKaJlU na 3Kpalie nOJlYlluJLU Y.MenbUlennoe
u306paJICeliUe npeiJMema. KaKoe 3mo 3epKa.n.0? rile pacno.n.oJICen npeiJMem om-
1i0cume.n.bliO 3epKa.na?
13. npeiJ.Mem nepeMelqalOm U3 6eCKoneilliocmu " ljJOKYCY 60zliymozo 3epKa-
Jia 6iJO.n.b z.n.a6liOU onmUlleCKOU ocu. KaK 6yiJem nepeMelqambCSl u306paJICenue
npeiJMema?
14. B lle.M 3aK.nlOllaemcSl .R6.n.enue npe.n.O.M.n.enuSl c6ema? KaKU.MU .JaKOna.MU
ono onucbl6aemCJl?
15. q mo nOIiU.MalOm noiJ omliocume.n.bnbl.M nOKa3ame.neM npe.n.oMJlenuSl? a6-
COJLlOmnblM nOKa3ame.n.eM npe.noM.n.eliuSl? KaK06a C6Sl3b MeJICay liU.MU?
16. KaKYIO cpeiJy CllumalOm onmUlleCKU 60.n.ee n.n.omliou? onmUlleCKU .Meliee
n.n.omliou? KaK a6co.n.lOmliblU nOKa3ame.n.b npe.n.O.M.n.eliU.R cpeobl 6blpaJICaemCSl
llepe3 cKopocmb C6ema 6 3mou cpeoe?
17. C6em nepexoiJum U3 onmU'leCKU 60.n.ee n.n.omnou cpeObl 6 onmUlleCKU
Meliee n.n.omliYlO. KaKOu yzo.n. 60.n.bwe: yzo.n. naiJenuSl UJlU yzo.n. npe.n.oMJlenuSl?
18. B IlCM cocmoum Sl6JleliUe no.n.liozo ompaJICeliUSl c6ema? llpu KaKUX YC-
Jl06USlX OliO 603MO:HCliO? KaKou yzo.n. na3bl6a1Om npeiJe.n.blibl.M yz.n.O.M no.n.liozo
ompaJICeliuSl? Om Ilezo 3a6ucum 6e.n.UIlUlia 3mozo YZw'la?
19. KyiJa omK.n.onUmCJl npoweiJwuu llepe3 npU3MY .n.YIl CBerna, eCJlU a6co-
JllOmliblU nOKa3ame.n.b npe.n.oM.n.eliuSl npu3Mbl 60Jlbwe. lle.M a6co.n.lOmliblU nOKa-
3ame.n.b npe.n.oM.n.eliuSl oKpYJICalOlqeU cpeiJbl?
20. B KaKOM c.n.Yllae .n.YIl C6ema 6 mpexzpaliliOU npU3Me 6yoem omK.n.OliSlmb-
CoR K ee 6epwune?
21. J( 6a lia6.n.lOoame.n.Sl oiJli06peMelili0 onpeoe.n.SllOm Una z.n.a3" 6blcomy COJl,-
nija liaiJ ZopU30IimOM. 1i0 OiJUIi liaxooumcSl noo 60iJOU. a opyzou - lia no-
6epXliocmu 3e.M.n.u. Ko.MY U3 liUX Co.n.lie 6yoem Ka3ambCSl 6blwe?
22. AK6a.n.alizucm. n.n.a6alOlqUU noiJ 60iJoe, 6ceziJa .H.OJICem 6uiJemb pbl6a-
Ka. liaXOOSllqeZOCSl lia 6epezy. Pbl6aK JlCe. CUOSllqUU lia 6epezy .n.UWb 6 pea.
KUX c.n.Y'laSlX MO:HCem Y6uiJemb aKBa.n.alizucma. n.n.a6alOlqezo noiJ 60iJOU. II 0-
lleMY?
471

y=a
1 1 1 2
-+-=-=-
d f F R
sin a
--=-- = n21
sIn
n21
 
n 1 V2
.
a llp = arCSIn -
nl
OCHOBHblE cl>OPMY JIbl
- 3aKOH OTpameHH.H CBeTa, r,Ae y - yroJl OTpa:>Ke-
HRSI a - yrOJI na,AeHH.H.
- Q>opMYJla ccpepR't:IeCKOrO 3epKaJla, r,ne d H f -
paCCTO.HHR.H OT 3epKana lIO npelIMeT8 H H306pa:>Ke-
R
HH.H COOTBeTCTBeHHO, F = - - cpOKYCHoe paCCTO.H-
2
HHe 3epKaJla, R - pa,nHYC 3epKaJla.
- 3aKOH npeJlOMJleHR.H, r,Ae P yroJl npeJlOMJleHH.H,
n21 - OTHocHTenbHbIR nOKa3aTeJlb npenOMneHH.H
BTOpOH Cpe,AbI OTHOCHTenbHO nepBoR.
- OTHOCHTenbHblH nOKaaaTenb npeJlOMJleHH.H. r,Ae nl
II (12 - a6conlOTHbIe nOKa3aTeJlH npeJlOMJIeHH.H nep-
BOll H BTOpOH no XO,Ay nYlIa cpelI, vI H V2 - CKO-
pOCTH CBeTa B 3THX cpe,Aax.
- npe,AenbHbIH yroJl nOJlHOrO oTpameHH.H CBeTa.
METO.l(lIqECKlIE PEKOMEH.l(AQlIlI
3a,AaQH Ha 3aKOHbl oTpameHHH - 3TO 3a,A8't:1H Ha onpe,AeneHHe pa3MepoB H
B3aHMlIoro pacnOJlOmeHHH H306pameHHH, npelIMeToB H 3epKBJI. HeKOTopbIe 3Ma-
qH Tpe6YlOT npoBecTH TonbKO rpacpHqeCKOe nOCTpoeHHe, ,ApyrHe - paCqeTHbIe.
PemeHHe 3a,Aa1.1 HaqHHalOT C BblnOnHeHH.H nOCTpOeHH.H. ,I:(JlH 3Toro HymHO:
1. M306pa3HTb 3epKano.
2. YKa3aTb caM npe,AMeT.
3. J]:JI.H nOCTpoeHHH 1I306pameHHH npe,AMeTa .nOCTaTOqHO HaRTH H306pa:>Ke-
HHe ,ABYX ero rpaHHqHblX TOqeK.
4. ,I:(n.H nOCTpoeHHH H306pa:>KeHH.H TOqKH H3 Hee Ha 3epKano Ha,AO Hanpa-
BHTb ,ABa Jlyqa, npOBeCTH HOpMaJIb K nOBepXHOCTH 3epKana B TOQKe na,AeHHSI
nyqa, nOCTpOHTb no yrJlY na,AeHH.H yron OTpameHHH, npOBeCTH caM JIyq. B
TOM MeCTe, r,Ae nepeceKYTC.H OTpameHHble nyqH HnH HX npO,AOnmeHH.H, Haxo-
,AHTC.H HCKOMoe H306pameHHe TOqKH.
IIpH pemeHHH 3a,Aaq 0 npenOMneHHH CBeTa Ha nnOCKOH rpaHHe ,ABYX
cpe,A, BKJllOq8.H nnOCKOnapaJlJleJlbHble nnaCTHHKH II npH3MbI, peKOMeH,AyeTC.H:
1. C,AenaTb cXeMaTHqeCKHH PHCYHOK, YKa3aB Ha HeM XO.A JlyqeH, H)J;yHX
H3 O,AHoii Cpe,AbI B ,ApyrylO.
2. BoccTaBHTb nepneH,nHKYJl.HpbI K rpaHHaM pa3,Aena ,ABYX cpe,A B TO't:l-
Kax na,AeHH.H JIYQeH.
3. OTMeTHTb yrJlbI na,neHH.H H npeJlOMneHH.H. IIepe,A TeM, KaK qepTHTb npe-
nOMneHHbIH nYQ, He06xo,AHMO YCTaHOBHTb, nepeXO,AHT JIH OR H3 OnTHqeCKH Me-
Hee nnOTHOR Cpe,AbI B 60Jlee nnOTHYIO HnH Ha060poT. B 3aBHCHMOCTH OT 3TO-
ro nyq HJIH npH6JIH:>K8eTC.H K HOpMaJlH B TOqKe na,AeHH.H, HnH y,Aan.HeTCSI
OT Hee.
472

4. 3anHcaTb 3aKOH npeJlOMneHH.H CBeTa n.H cpe.
5. 3anHcaTb HeOCTalOHe ypaBHeHH.H, HcnOJlb3Y.H HCXOHbIe aHHbIe 38-
a'tni.
6. PelliHTb nOJlyqeHHbIe ypaBHeHH.H.
TIPlIMEPbI PEIIIEHlI.H 3AJ.(A q
1. JI:HaMeTp <poToc<pepbI COJlHu.a 1,39. 10 6 KM. paCCTO.HHHe OT ConHu.a ,AO
3eMnH H3MeHHeTC.H He3HaqHTenbHO H B Cpe,AHeM COCTaBn.HeT 1,50. 10 8 KM.
PaccToHHHe OT eHTpa JIYHbI o nOBepXHOCTH 3eMJIH H3MeH.HeTC.H OT 3,57. 10 5 KM
o 3,99. 10 5 KM. OnpeenHTe, npH KaKHX ycnOBH.HX COJlHeqHOe 3aTMeHlle 6yeT
nOJlHbIM H npH KaKHX Konbu.e06pa3HbIM, eCJlH HaMeTp JIYHbI 3,48. 10 3 KM.
JI:aHo: D= 1,39.10 9 M, L = 1,50.10 11 M,
II = 3,57 . 10 8 M, l2= 3,99 . 10 8 M,
d = 3,48 . 10 6 M.
HaRTH: l.
P e ill e H H e. Ha pHCYHKe 16.29 H306-
pameH CJlyqaH, Kora COnHet.IHOe 3aTMeHHe
6yeT nonHbIM (I - COJlHu.e. II - JIYHa,
III - TOqK8 Ha nOBepXHOCTH 3eMnH).
A
I
E
L
Puc. 16.29
113 nO06H.H
D L
!lACE 'Ii !l BCF cneyeT, qTO d -T'
Tora
TaKHM 06pa30M, eCJlH paCCTO.HHHe Memy JIYHOR H 3eMJleH MeHbllie
3,75. 10 5 KM, TO COJlHeqHOe 3aTMeHIIe 6yeT nonHbIM, B npOTHBHOM cny-
qae - Konbu.e06pa3HbIM.
2. 'tJeJlOBeK CMOTpHTC.H B 3epKaJlO. BHC.Hee Ha CTeHe C He60JlblliHM HaKJlO-
HOM. IIocTpoHTe H306pameHHe qeJlOBeKa B 3epKaJle. KaKYIO qaCTb CBoero TeJla
6yeT BHeTb qenOBeK? IIpli nOCTpoeHHH
MomHO H306pa3HTb qeJlOBeKa B BHe OT-
pe3Ka AB, pacnonomHB r Jla3 B TOqKe C.
P e ill e H H e. IIocTpoHM H306pameHHe
KpaHHHx TOqeK qeJlOBeKa A H B. .D;n.H
3Toro Ha nJlOCKOCTb 3epKaJla onYCKaeM
nepneH)J;HKyn.HpbI AD Ii BE (pliC. 16.30)
H npOOJlmaeM HX 3a 3epKaJlO Ha paCCTO-
.HHH.H, paBHbIe COOTBeTCTBeHHO AID =AD H
BIE=BE.
AIBI - MHHMoe H306pameHHe qeJlOBe-
Ka AB. 'tJeJlOBeK YBH)J;HT TOJlbKO TY qaCTb
TeJla, JlyqH OT KOTOpOH nocJle oTpameHH.H
nonaYT B rJla3 C, T.e. KILl=KL.
3. Ha CKonbKO rpaYCOB OTKJlOHHTC.H oTpameHHbIR OT 3epKaJla nyq, eCJlli
3epKaJlO nOBepHYTb Ha 15°?
l = L!!..; l = 3. 75 . 10 5
D
KM.
A D, At
-r - "'1
K - -,- ;""K 1
\
C \
\
\
\
" \
" \
" \
L-__\Ll
---, \
, \
, \
, _l
--E- Bl
L
B
Puc. 16.30
473

.D;8HO: <p= 15°.
HaHTH: .
P e ill e H H e. IIycTb AO (pHC. 16.31) -
A na,nalOHH Jlyq, MN - nnOCKoe 3epK8-
JlO. YroJl AOC = a - yroJl n8,neHH.H Jlyqa,
yrOJJ COB ="{ - yroJl oTpameHH.H. IIo 38KO-
HY oTpameHH.H y = a. CJle,noBaTeJlbHO,
LAOB = 2a.
IIoBepHeM 3epK8J10 BOKpyr TOqKH 0 H8
yrOJI <po M'N - HOBoe nOJlOmeHHe 3epKaJl8.
OC' - nepneH,nHKYJl.Hp K 3epKaJlY. YrJlbI
C'OC H MOM' - yrJlbI C COOTBeTCTBeHHO nepneH,nHKYJl.HpHbIMH CTopOHaMH.
Cne,noBRTeJlbHO, LC'OC = <po IIocJle nOBopOTa 3epKaJla LAOC' = a l - yroJl na-
,neHH.H, LC'OB' = "{1 - yroJl OTpameHHSI, a l = Yl' ll3 pHCYHKa BH)J;HO, qTO
Puc. 16.31
a l = a - <po CJle,noBaTeJlbHO, LAOB' = 2a 1 = 2 (a - <p).
llCKOMbIH yroJl 13 = LB'OB = 2a - 2a l = 2a - 2 (a - <p) = 2<p; 13 = 30°.
4. KaKoB palIHYC BorHyroro 3epKaJla, Haxo,n.Heroc.H Ha paCCTO.HHHH 2,0 M
OT npe,nMeTa, ecnH ero np.HMOe H306pameHHe B nonTopa pa3a 60JlblIIe, qeM B
nJlOCKOM 3epKaJle, Haxo,n.HeMC.H Ha TOM me paCCTO.HHHH OT npe,nMeTa?
r l
.D;aHO: d=2,0 M, -=1,5.
r 2
HaHTH: R.
B}
....;'"1
........;... I
........ ... I
............ ......... I
........... IH
............ .......... I
---...-.... B 2 :
h I
A 2 Al
d+
Puc. 16.32
P e ill e H H e. TaK KaK H306pa-
meHHe, ,naBaeMoe BorHYTbIM 3ep-
KaJlOM, np.HMOe H YBeJlHqeHHOe,
TO npe,nMeT AB Haxo,nHTC.H Mem-
,ny cl>OKYCOM H nOJlIOCOM 3epKaJla
(pHC. 16.32). 3anHlIIeM cl>0PMYJlY
ccl>epHqeCKOrO 3epKaJla:
1 1 1 2
-----=-
d f F R
Ha pHCYHKe 16.32 A1Bl -
H306pameHHe npe)IMeTB B ccl>epH-
qeCKOM 3epKaJle, A 2 B 2 B
nJlOCKOM 3epKaJle.
(1)
JIHHeHHoe YBeJlHqeHHe ccl>epHqeCKOrO 3epKaJla
r 1 = H = L.
h d
(2)
Pa3MepbI H306pameHH.H B nJlOCKOM 3epKaJIe Bcer,na paBHbI pa3MepaM npe,n-
r
MeTa, T.e. r 2 = 1. CJle,nOBaTeJlbHO, --1. = r 1 = 1,5.
r 2
Tor,na f = r ld.
474

IIOACTaBHB 3HaqeHHe f B OPMYJlY CepHqeCKOrO 3epKaJla, nOJlyqHM
1 1 2 2r d
----- OTK y ,na R=' R=12 M.
d rId R r 1 -1'
5. Ha HHmHlO1O rpaHb nJlOCKOnapaJlJlenbHOH CTeKn.SIHHO nJlaCTnHKIi HaHe-
ceHa u;apanHHa. Ha6J1IOAaTeJlb, rn.HASI cBepxy, Bn,AnT u;apanHHY Ha paCCTOSl-
HnH 4 CM OT BepxHeH rpaHH nJIaCTHHKn. KaKoBa TonnHa nnacTnHKn?
,D;aHo: h 1 =O,04 M; n=1,5.
HaHTH: h.
Pew e H H e. IIycTb u;apanHH8 HaXQ,nHT-
CH B TOqKe A HH:>KHeH nOBepXHOCTn CTeK-
JISlHHOH nJIaCTHHKH. IIocTpoHM H306pa:>Ke-
HHe TOqKH A. KOTopoe BH,nHT Ha6nlOnaTeJlb
(pHC. 16.33). ,I:{JISI 3Toro paCCMOTpHM ,nBa
nyqa: AC - nY'l. na,AalOHH nepneH,AnKY-
JISipHO Ha BepXHlO1O nOBepXHOCTb nJlaCTnH-
KH; AD - Jlyq, na,AalOnH Ha BepXHlO1O
nOBepXHOCTb no,A MaJIblM yr JlOM a. 113 PH-
CYHKa BH,nHO, qTO TOt.:lKa B 6Y,AeT MHHMbIM
H306pameHHeM TOqKH A.
CD
PaccMoTpHM  ACD : AC = -, HJln, nOCKonbKY AC = h Ii CD = d, TO
tg a
d
f;
f;
f;
f;
Puc. 16.33
d
h = - . OTpe30K d HaHAeM H3  BCD: CD = CB tg 13, HJIH. yqliTblBa.H, qTO
tga
tg 13
CD = d H CD = , liMeeM d = hI tg 13. Tor,na h = hI -. IIoCKOJIbKY yrnbI a
tga
H 13 MaJIbl, OTHOllleHHe TaHreHCOB 3TH X yrJloB MomHO 38MeHHTb OTHOllleHneM
tg 13 sin J3 sin J3
HX cnHYCOB, T.e. -::::: -. CJIeAOBaTeJIbHO, h = hI -. Ho no 38KOHY
tg a sin a sin a
sin a n B 1
npeJlOMJleHH.H - = - = -, TaK KaK a6COJIIOTHblH nOKa38Tenb npenOMJIe-
sin J3 nc nc
Hn.H B03Ayxa n B = 1. Tor,na h = nc; h = 0,06 M.
6. JIyq CBeTa na,naeT Ha nepe,nHIOIO rpaHb Tpexrp8HHoH npn3MbI C npe-
nOMnSlIOHM yrJloM <po IIocne npeJIOMJIeHn.H OH nOna,A8dT H8 3a,AHIOIO rpaHb
npn3MbI Ii Bblxo,nnT B B03,AYX. Onpe,neJIHTe
yroJl OTKJlOHeHH.H CBeTOBoro Jlyqa 0, ecnH
yron na,neHH.H JIyqa Ha nepeAHlO1O rpaHb pa-
BeH a, 8 a6conlOTHbIH nOK83aTeJIb npeJIOM-
JIeHH.H MaTepH8na npH3Mbl n.
.n8HO: <p, a, n.
HaHTH: O.
Pewe H He. XOA JIyqa B npn3Me nOKa-
3aH Ha pnCYHKe 16.34. KaK BHeIIIHHH yron
K iLEC yroJl OTKJlOHeHHSI 0 paBeH
B
Puc. 16.34
o = (a - (3) + (131 - ad.
(1)
475

KpOMe TOrO, LABC = LADK = cp KaK yrJlbI, 06pa30BaHHble B3aHMHO nep-
neHHKYJl.HpHbIMH CTOpOHaMH. YroJl ADK .HBJI.HeTCH BHelIIHHM yrnoM K 1J.ADC,
n03TOMY
cp = 13 T a l .
113 ypaBHeHHH (1) H (2) nOJlyqHM
o = a - cp + 13 1 .
Cor naCHO 3aKOHY npeJlOMJleHH.H MomHO 3anHcaTb
sin a _ sinal _ 1
--n ---
sin 13 'sin 131 n .
113 ypaBHeHH.H (2) a l = cp - 13, n03TOMY
sin 131 = n sin ( cp - (3) = n (sin cp cos 13 - sin 13 cos cp).
IIocTaBHM B 3TY <}>OPMYJIY 3HaqeHHe sin p H3 (4):
. " sin a " J l sin 2 a
slnp = --;;- H cosp = -.
sinJ3 1 =sin cp-J n 2 -sin 2 a -coscpsina.
C yqeTOM ypaBHeHHH (3) H (5) nOJl yqaeM BbI pameHHe JI.H yrJla 0:
o = a - cp + arcsin (sin cp .J n 2 - sin 2 a - cos cpsin a).
7. Ha ,AHe cocya, 3anOJIHeHHOro BO,AOH o BbICOTbI h, HaXOHTC.H TOqe'l-
HbIH HCTOqHHJ\ CBeTa. Ha nOBepXHOCTH BObI nJlaBaeT J\pyrJlbIH .D.HCJ\ TaJ\, qTO
ero eHTp Haxo.nHTC.H Ha.n HCTOqHHKOM. IIp11 J\aKOM MHH11MaJlbHOM pa,AHyce
HCKa JlyqH OT HCTOqHHJ\a He 6YYT BbIXOHTb H3 BObI?
.naHO: h, n.
HaHTH: R.
P e III e H H e. JIyqH OT HCTOqHHJ\a He 6y-
YT BbIXO)J;HTb H3 BObI, eCJl11 HCJ\ 3aKpo-
eT TY qaCTb CBeTOBoro nyqKa, KOTOpbIH
Mor 6bI BbIHTH H3 BObI, T.e. qaCTb JlyqeH.
yroJl naeHH.H JI.H KOTOpbIX a < (lop' JIyqH,
.nJl.H KOTOpbIX (l  (lop' HcnblTblBalOT non-
Hoe OTpameHHe OT rpaHH:qbI Boa-B03YX.
IIpeeJlbHbIH yroJl nOJlHOrO OTpameHHH
HaHeM H3 3aKOHa OTpameHH.H, yqHTbIBaH,
Tora
s
Puc. 16.35
sin (lnp
sin 90°
(2)
(3)
(4)
(5)
qTO yroJl npeJlOMJleHH.H J3 = 90°, T.e.
1
O . 1
TKya SIn (lnp =-,
n
--.
n
re n - a6COJllOTHbIH nOJ\a3aTeJlb npeJlOMJleHH.H BObl.
113 pHcYHKa 16.35 BHHO, qTO
476

sin a. up sin a. up 1 h
R = h tg a. = h = h = h = .
up cosa.  2 R 
n 1-- n _ 1
up 1 - sin a. up
n 2
TaKHM 06pa30M, HCKOMbIH pa.nHYC )J;HCKa
h
R= .
../n 2 -1
16.11. JIHH3LI. «I>oKycHoe paCCTOJlHHe JIHH3LI
JIHH3LI - 3TO np03paqHbIe
TeJI8, OrpaHHqeHHbIe KpHBOJIHHeH-
HbIMH nOBepXHOCTHMH. q8e HC-
nOJlb3YIOT c«}>epHqeCKHe JIHH3bI. Y
KOTOpbIX OrpaHHqHBalOHMH no-
BepXHOCT.HMH .HBJI.HIOTCSI ccl>epbI
HJIH c«}>epa H nJIOCKOCTb. Pa3JIH-
qalOT JIHH3bJ BblnYKJlbIe H BorHY-
Tble (pHC. 16.36). Y BbInYKJIbIX
JlHH3 cepe)J;HHa TOJle, qeM Kpa.H,
y BorHYTbIX, Ha060poT, cepe.n;HH8
TOHbllIe KpaeB.
BbInYKJlbIe JlHH3bI: a
IIJlOCKOBbInYKJla.H, 6 - .n;BO.HKOBbI-
nYKJIa.H, 6 - BorHYTOBbInYKJla.H.
BorHYTbIe JlHH3bI: Z - nJloc-
KOBorHYTa.H, 0 - BbInYKJlOBorHY-
Ta.H, e - .n;BO.HKOBorHYTa.H.
BbInYKJlYIO JlHH3Y MomHO
npe.n;CTaBHTb B BH.n;e cOBoKynHoc-
TH nJlOCKOnapaJlJlenbHOH nJlac-
THHKH (B cepe.n;HHe) H yceqeHHbIX
npH3M (pHC. 16.37, a), paCllIHp.H-
IOHXC.H K cepe.n;HHe npH3MbI, a
BorHYTYIO - KaK cOBoKynHocTI:.
nJlOCKOnapaJlJleJIbHOH nJlaCTHHKH
H yceqeHHbIX npH3M, paCllIHp.HlO-
HXC.H K Kpa.HM (pHC. 16.37, 6). ECJlH npH3M8 c.n;eJIaHa H3 MaTepHaJla, onTH-
qeCKH 60Jlee nJlOTHOrO, qeM oKpymalOa.H cpe.n;a, TO OHa OTKJlOH.HeT Jlyq K
OCHOBaHHIO. II03TOMY napaJIJleJlbHbIH nyqOK nocJle npeJlOMJleHH.H B BbInYKJloH
(BorHYToH) JlHH3e CTaHeT CXO.n;.HHMCH (pacxo.n;.HHMC.H). TaKHe JIHH3bI Ha3bI-
BaIOTC.H c06HpalOI.Q.HMH (pacceHBalOI.Q.HMH).
ABO.HKOBbInYKJlYIO JIHH3Y MomHO nOJlyqHTb nYTeM nepeCeqeHH.H nOBepXHO-
CTeH .n;BYX np03paqHbIX llIapOB (pHC. 16.38). C 1 H C 2 - eHTpbI llIapOBbIX
nOBepxHocTeH. Y TOHKHX JlHH3 TOqKH 0] H O 2 CJlHBaIOTC.H B TOqKY 0, KOTO-
pas Ha3bIBaeTC.H OnTHqeCKHM eHTpOM JlHH3bI.
a
 6 0 8
a e
[ ] K
Z
Puc. 16.36
a
6
B

'V
,
,
,
,
,
....
............ C

Puc. 16.37
Z
477

Puc. 16.38
Puc. 16.39
F
Puc. 16.40
OnTH1JeCKHH eHTp J1HH3LI - TOqKa,
qepe3 KOTOpylO JlyqH H,nYT, He npeJlOM-
JI.HHCb. JI106a.H np.HMaH, npoxo,n.HaH qe-
pe3 OnTHqeCKHH eHTp, Ha3bIBaeTCH on-
TH'leCKOH OCLIO. OnTHqeCKa.H OCb, npoxo-
.n.Ha.H qepe3 eHTpbI cc})ep C 1 H C 2 , Ha-
3bIBaeTC.H rJlaBHOH ODTH1JeCKOH OCLIO.
Bce OCTaJlbHbIe - n060qHbIMH OnTHqeC-
KHMH OC.HMH.
TOHKYIO c06HpalOYIO JlHH3Y YCJlOBHO
0603HaqalOT TaK, KaK YKa3aHO Ha PHCYH-
Ke 16.37, 6.
JIyqH. napannenbHbIe rnaBHoH onTH-
qeCKOH OCH. nocne npeJlOMJleHH.H nepece-
KYTC.H B O,AHOH TOqKe F, Jlemaeii Ha
rJlaBHOH OnTHqeCKOH OCH (pHC. 16.39).
3Ta TOqKa Ha3bIBaeTC.H rJlaBHLIM 4>OKY-
COM nHH3bI. Y JlHH3bI ,nBa rnaBHbIX c})o-
Kyca: nepe,nHHH H 3a,nHHH, TaK KaK OHa
npenOMnHeT CBeT, na,nalOHH Ha Hee C
,nBYX CTOpOH. 06a 3TH c})OKYCbl pacnono-
meHbI CHMMeTpHqHO OTHOCHTenbHO TOq-
KH O.
BorHYTa.H nHH3a, C,AeJlaHHa.H H3 Ma-
TepHaJla, OnTHqeCKH 60nee nnOTHOI'O,
qeM oKpymalOllIaH cpella. pacceHBa-
eT CBeT. YCJlOBHOe 0603HaqeHHe pacce-
HBalOeH nHH3bl YKa3aHO HB PHCYH-
Ke 16.37, Z. «bOKYC pacceHBalOeH
J1HH3L1 - TOqKa (pHC. 16.40), B KOTO-
pOH nepeceKalOTC.H npo,nOJlmeHH.H npe-
JlOMneHHbIX JlyqeH. eCJlH nalIalOHe
nyqH 6bInH napaJlnenbHbI I'JlaBHOH on-
THqeCKOH OCH. Y pacceHBalOllIeH nHH-
HeM nepeceKalOTC.H He caMH npenOMneHHble
3bI CPOKYC MHHMLIH, TaK KaK B
JlyqH, a HX npo,nOJlmeHH.H.
PaCCTO.HHHe OF OT OnTHqeCKOrO eHTpa ,no cpoKyca Ha3bIBaeTC.H OKYCHblM
paCCTOJlHHeM F.
16.12. llocTpoeHHe H306pameHHH B JIHH3aX
CBeTOBbIe nyqH, HyHe B6J1H3H rJlaBHOH OnTHqeCKOH OCH. Ha3bIBalOT na-
paKCHanbHbIMH (npHoceBbIMH). MbI 6Y,AeM paCCMaTpHB8Tb TonbKO TaKHe
JlyqH.
OnbIT nOKa3bIBaeT, tITO napaKCHaJlbHbIe nyqH CBeTa, BbIxo,nHHe H3 O,AHO:H
cBeTHeHC.H TOqKH, nocJle npoxom,neHH.H qepe3 nHH3Y CXO,n.HTC.H TaKme B o,n-
HOH TOqKe, KOTOpa.H .HBJI.HeTC.H H306pameHHeM CBeT.HeHC.H TOqKH. II03ToMY
478

.n;JI.H nOCTpOeHH.H H306pameHH.H TOqKH )J;OCTaTOqHO B3S1Tb .n;Ba JlI06bIX JIyqa, HO
nyqllIe Te, XO.n; KOTOpbIX nocJle npeJlOMJleHH.H 3apaHee H3BeCTeH: 1 - JIyq,
H)J;yHH qepe3 OnTHqeCKHH ueHTp; 2 - JIyq. napaJIJleJlbHbIH rJlaBHOH onTH-
qeCKOH OCH; 3 - Jlyq, npoxo.n;.HHH qepe3 nepe.n;HHH CPOKYC c06HpalOeH JlHH-
3bI (HJlH npo.n;OJlmeHHe Jlyqa 3 npOXOlIHT qepe3 3a.n;HHH CPOKYC pacceHBalOeH
nHH3bI) (pHC. 16.41).
a
Puc. 16.41
6
1
IIOJlOmeHHe H306pameHH.H .n;eHCTBHTeJlbHOrO npe,AMeTa Hero pa3MepbI 3a-
BHC.HT OT nOJlOmeHH.H npe,AMeTa OTHOCHTeJIbHO JlHH3bI. IIYCTb d - paCCTOSl-
HHe OT npe.n;MeTa .n;o JlHH3bI, f - paCCTO.HHHe OT JIHH3bI .n;o H306pameHH.H.
IIoCTpoHM H306pameHHe nJlOCKOrO npe.n;MeTa AB, pacnOJlOmeHHOro Ha pa3J1Hq-
HbIX paCCTO.HHH.HX d OT nHH3bI.
ECJlH JIHH3a c06HpalOa.H, TO npH d> 2F (pHC. 16.42) H306pameHHe lIeH-
CTBHTeJIbHOe, nepeBepHYToe, YMeHbllIeHHoe, F < f < 2F.
lIpH F < d < 2F (pHC. 16.43) H306pameHHe lIeHCTBHTeJlbHOe, nepeBepHYToe,
YBeJlHqeHHOe, f> 2F.
Puc. 16.42
F<d < 2F
Puc. 16.43
IIpH d < F (pHC. 16.44) H306pameHHe MHHMoe, np.HMOe, YBeJIHqeHHOe, Haxo-
.n;HTC.H C TOH me CTOpOHbI OT JIHH3bI, qTO H caM npe,AMeT, HO .n;aJlbllIe npe,A-
MeTa (f> d).
B pacceHBalOeH JlHH3e (pHC. 16.45) H306pameHHe .n;eHcTBHTeJlbHOrO npe.n;-
MeTa Bcer.n;a MHHMoe, np.HMOe, YMeHbllIeHHoe, HaXO,AHTCSI Mem,AY JlHH30H H ee
CPOKYCOM co CTOpOHbI H306pamaeMoro npe)J;MeTa.
479

d<F
Bit,
, ........
: ',............
I ,........
I B
I
I
Al F
A
Puc. 16.44
F
Puc. 16.45
IIocTpoeHHe H306pameHH.H 8 1 TOqeqHOrO HCTOqHHK8 CBeTa 8, pacnOJlOmeH-
Horo Ha rn8BHOH OnTHqeCKOH OCH, nOKa38HO Ha pHCYHKe 16.46.
a
B
F'
Puc. 16.46
6
F
"
"
"
"
S
"
I "
"
I "
"
F'
'tJT06bI HaHTH, re 06pa3yeTcH ero H306p8meHHe Sit npoBeeM H3 TO"t!KH
S Ba Jlyq8: Jlyq SO BOJlb rJIaBHOH OnTHqeCKOH OCH (OH npOXOHT qepe3
OnTHqeCKHH u,eHTp nHH3bI, He npeJIOMJI.HHCb) H Jlyq SB, naalOHH Ha JIHH-
3Y B npOH3BOJlbHOH TOqKe B. IIpoBeeM n060qHYlO OnTHqeCKYIO OCb, napan-
JlenbHYIO Jlyqy SB (nOK838Ha lIITpHXOBOH JIHHHeH). Ha"t!epTHM 38HIOIO CPO-
KanbHYIO nJlOCKOCTb B cJly"t!ae C06HpalOI.I.teH JlHH3bI (nepe,AHIOIO cpOKaJIbHYIO
nnOCKOCTb B cnyqae pacceHBalOeH nHH3bI). II060qHa.H OCb nepeCeqeT cpO-
KaJIbHYIO nJlOCKOCTb B n060qHOM «poKyce F'. 'tJepe3 3TOT n060qHbIH CPOKYC
npOH)J;YT Bce nap8nJIeJIbHbIe aHHOH OnTHqeCKOH OCH nyqH nOCJIe npeJlOMne-
HH.H B c06HpalOeH JlHH3e (HJlH npoonmeHHH npeJlOMJleHHbIX JlyqeH B pac-
ceHBalOeH JIHH3e), CJIeOB8TeJIbHO, H JIyq SB. IIpeJIOMJleHHbIH JIyq (HJIH ero
npOOJlmeHHe) nepeCeqeT rJl8BHYlO OnTHqeCKYIO OCb B TOqKe Sit KOTopaH H
HBnSIeTC.H H306pameHHeM TOqeqHOrO npe,ZJ;MeTa S.
16.13. OnTH'IeCKaH CHJIa JIHH3bl
OnbIT nOKa3bIBaeT, qTO JIHH3bI C 60Jlee BbInYKnbIMH nOBepXHOCT.HMH npe-
nOMJI.HIOT CHJlbHee H alOT 60nblIIee YBeJlH"t!eHHe, qeM nHH3bI C MeHblIIeH KPH-
BH3HOH. Y JlHH3bI C 60JlblIIeH KpHBH3HOH CPoKycHoe paCCTOHHHe MeHblIIe.
IIpenOMJIHIOYIO cnoco6HOCTb JIHH3bI xapaKTepH3YlOT OnTHqeCKOH CHnOH
JIHH3bI.
480

OllTH'leCKaH CHJla JlHH3LI D - BeJlHqHH8, 06paTH8.H <POKYCHOMY p8CCTO.HHHIO:
D _ 1.
- F '
B CM e,nHHHn;eH OnTHqeCKOH CHnbJ .HBn.HeTC.H ,nHonTpH.H (,nnTp). 1 ,nnTp -
OnTHqeCKa.H CHJl8 nHH3, oKycHoe p8CCTO.HHHe KOTOpOH paBHO 1 M.
OnTHqecKYIO CHJlY c06HpalOeH JlHH3bI YCJlOBHJlHCb CqHT8Tb nOJlOmHTeJlb-
HOH (<p0KYc ,neHCTBHTeJlbHbIH), a OnTHqeCKYIO cHny p8cceHBalOeH JlHH3bI (<po-
KYC MHHMbIH) - OTpHn;8TenbHoH.
OnTHqecKYIO CHJlY JlHH3bI MomHO paCCqHTaTb no <popMYJle
1 ( 1 1 )
D = - = (n21 -1) - + - ,
F RI R;.
n2
r,ne n21 = - OTHOCHTenbHbIH nOK838TeJlb npeJlOMJleHH.H, n2 - a6COJllOTHbIH
nl
nOKa3aTeJlb npeJlOMJleHH.H BeeCTBa JlHH3bI. nl - a6COJllOTHbIH nOKa38TeJlb
npeJlOMJIeHH.H cpe,nbI, RI H R 2 - PMHYCbI KpHBH3HbI nOBepxHocTeH JlHH3bI.
Pa,nHycbI BbInYKJlbIX nOBepxHocTeH CqHTaIOTC.H nOJlOmHTeJlbHbIMH. BorHYTbIX -
OTpHn;aTeJlbHbIMH.
OnbIT nOKa3bIBaeT, tlTO 06a.H OnTHqeCK8.H CHJla ,nBYX HJlH 60Jlee JlHH3.
CJlOmeHHbIX BMeCTe, paBHa aJIre6paHqeCKOH C}'MMe OnTHqeCKHX CHJI 3THX JlHH3:
D = DI + D 2 + ... .
16.14. (f>0pMYJIa JIHH3LI. JIHHeHHoe YBeJIH'IeHHe JIHH3LI
M3 no,n06H.H TpeyrOJlbHHKOB AIBIF H OCF (pHC. 16.47) CJleyeT
AIBI = AIF AHaJIOrHqHO H3 no,n06H.H ABO Ii AIBIO All\. = AI YqHTbI-
00 . AB 
BaH, qTO AB = CO, H3 3THX paBeHCTB nOJlyqHM
1 1 1
+- =+-+-
-F -d-f
B HeH F 6epYT co 3HaKOM "+" AJI.H co-
6HpalOeH JlHH3bJ H co 3HaKOM "-" - ,nn.H
pacceHBalOeH JlHH3bI. f - co 3H8KOM "+",
eCJlH H306p8meHHe ,neHcTBHTeJlbHOe, H co 3HaKOM "-", eCJlH OHO MHHMoe,
d - co 3HaKOM "+" JI.H eHCTBHTeJlbHOrO npe,AMeTa H "-" - ,nJl.H MHHMoro
npe,nMeTa.
h
At
A
F
d
Puc. 16.47
AIF = AIO  f-F =L  i-I =L.
OF OA F d F d
P83,neJlHM Bce CJlaraeMbIe H8 f, no-
nyqHM
1 1 1
---+-
F-d f'
B 06eM cnyq8e
H
<popMYJla TOHKOH nHH3bI.
481

JIHHeHHoe YBeJIHlleHHe JIHH3LI r - OTHOllIeHHe RHHeHHbIX pa3MepOB H306-
pa»<eHHSI H K RHHeHHbIM pa3MepaM npe,nMeTa h:
H
r=-.
h
11.3 TpeyronbHHKOB ABO H AIB10
AIBI = A10
AB AO
H f
Tor,na - --
h d
f
CJle,AOBaTeRbHO, YBeJlHqeHHe nHH3bI r = -.
d
16.15. OnTHqeCKHe npH60pbl
OnTHqeCKHe npH60pbI. npe,nCTaBRSlIOI..U:He C060H cOBoKynHocTb HeCKORbKHX
npH3M HnH nHH3, HeCKORbKHX 3epKan HRH O,AHOBpeMeHHO JIHH3, npH3M H 3ep-
KaJl, npe,nHa3HaqeHbI ,nJl.H npe06pa30BaHH.H CBeTOBbIX nyqKOB. C HX nOMOblO
MorYT H3MeH.HTbC.H HanpaBJleHH.H XO,ll;a CBeTOBbIX JlyqeH, llRH TeJleCHbIe yrJIbI,
B npe,neJIax KOTOpbIX pacnpOCTpaH.HIOTC.H CBeTOBbIe nyqKH. IlocJIe,nHee 06CTO.H-
TenbCTBO CB.H3aHO C nOJIyqeHHeM H306pa»<eHHH, pa3MepbI KOTOpbIX OTnHqaIOT-
CSI OT pa3MepOB npe,ll;MeTOB.
IlepBoe, Ha qTO HymHo 06paaTb BHHMaHHe IIPH aHanH3e ,neHcTBH.H onTH-
qeCKOH CHCTeMbI, - STO Ha3HaqeHHe H peaJlbHbIe YCJIOBH.H ee pa60TbI. r,ne
MomeT pacnOJIaraTbC.H npe,nMeT nepe,n CHCTeMOH? KaKoe H306pameHHe (YBeJIH-
qeHHOe, YMeHbllIeHHoe. 06paTHoe HJIH npSlMoe) ,AOnmHa ,naBaTb CHCTeMa? C no-
MOblO qero perHcTpHpyeTcSI nonyqeHHoe H306pameHHe (Ha SKpaHe, <poTonneH-
Ke. paCCMaTpHBaeTC.H HeBoopy»<eHHbIM rJla30M HJIH rJIa30M qepe3 KaKYIO-HH-
6y,nb JIHH30BYlO cHcTeMY)?
<I>oToannapaTOM Ha3bIBaeTC.H onTHKO-MeXaHHqeCKHH npH6op, npe,ll;Ha3HaqeH-
HbIH ,lI;JlSI nOJIyqeHH.H Ha <poTOnJIeHKe HnH <poTOnJIaCTHHKe H306pameHHSI <po-
Torpa<pHpyeMoro npe,nMeTa.
BamHeHIlIHMH SReMeHTaMH <poToannapaTa .HBJlSlIOTCSI Henp03paqHa.H KaMe-
pa H 06eKTHB. 06eKTHB <poToannapaTa npe,nHa3HaqeH HJI.H nOJIyqeHHSI qae
Bcero YMeHbllIeHHbIX H306pa»<eHHH npe,nMeTOB Ha <poTonneHKe. Ilpe,AMeT Ha-
XO,nHTC.H OT 06eKTHBa Ha paCCTOSlHHH d > 2F
(pHC. 16.48). IlosToMY H306pa»<eHHe npe,nMeTa
6y,neT ,lI;eHCTBHTeJIbHbIM, YMeHbllIeHHbIM. Haxo-
,nSlHMCSI Mem,ny F H 2F. CJIe,noBaTenbHo,
06eKTHB <poToannapaTa ,lI;OJImeH BbInOJlHSlTb
<PYHKltHH c06HpalOI..U:eH nHH3bI. TaK KaK pac-
CTOSlHHe Me»<,Ay 06eKTHBOM H nJIeHKOH B <po-
ToannapaTe OrpaHHqeHO, TO B KaqeCTBe 06eK-
THBa HCnOJlb3YlOT KOpOTKO<POKYCHYIO JIHH3Y
(qae CHCTeMY JIHH3).
AJI.H nOJIyqeHH.H KaqeCTBeHHOrO H306pa»<e-
HH.H 60JIbiliOe 3HaqeHHe HMeeT "HaBOAKa Ha
2F
Puc. 16.48
482
16.16. <l>oToannapaT

pe3KOCTL" - COBMeeHHe H306pameHHSI H nJIeHKH. AJI.H 3TOrO B CTapbIX cpO-
TOannapaTaX enaJlaCb nOBHmHOil 3aHSI.H CTeHKa (cpOTOKaMepa B BHe rap-
MOIllKH), B HOBbIX - C nOMOblO BHHTOBOH Hape3KH B Ty6yce nepeBHraeTCSI
06'beKTHB, CB.H3aHHbIiI C )J;aJlbHOMepOM. AJI.H YBenHqeHH.H "rny6HHLI pe3KOCTH"
HCnOnb3YIOT ,AHacpparMY. C ee nOMOI.l.\blO pery nHpYIOT TaKme OCBeeHHOCTb cpO-
TOIIJIeHKH.
Bo BpeM.H cpOTOrpacpHpOBaHH.H 06'beKTHB OTKpbIBalOT C nOMOI.l.\blO CneHaJIb-
HOro 3aTBOpa.
OCBeeHHOCTb H306pameHH.H, JI.H KOTOpOrO HCTOqHHKOM CBeTa SlBJI.HeTCSI
06'beKTHB, np.HMO npOnOpHOHaJIbHa nnOI.l.\a,zJ;H ero OTBepCTH.H, KOTopaSl, B CBOIO
O'lepe,Ab, nponopHOHaJIbHa KBapaTY HaMeTpa d 2 06'beKTHBa. OCBeeHHOCTb
TaKme 06paTHO nponopHOHaJlbHa KBa,n;paTY paCCTOSlHH.H OT HCTOqHHKa O H306-
pameHH.H, B HallieM cJlyqae nOqTH KBa,n;paTY CPOKYCHoro paCCTO.HHH.H F 2 . MTaK,
d 2
OCBeeHHOCTb npOnOpIJ,HOHaJIbHa p06H 2' KOTOPYIO Ha3bIBalOT CBeTOCHJlOH
F
06'LeKTHBa. KopeHb KBapaTHblii H3 CBeTOCHJIbI Ha3bIBalOT OTHOCHTeJILHLIM OT-
BepCTHeM.
16.17. I1poeKD.HOHHLIH annapaT
npoeKHOHHbIiI annapaT (npoeKTop) IIpeHa3-
HaqeH JI.H nOJIyqeHHSI Ha 3KpaHe eilCTBHTeJIb-
Horo YBeJIHqeHHOrO H306pameHHSI. CJIeOBaTenb-
HO, H 3eCb 06'beKTHB npeCTaBJI.HeT c060il co-
6HpalOYIO nHH3Y. TOJIbKO npeMeT nOMealOT
Memy F H 2F (F < d < 2F), a H306pameHHe no-
nyqaeTC.H Ha paCCTO.HHHH, 60JlbIlleM 2F (f> 2F)
(pHC. 16.49).
IIpoeKHoHHbIe annapaTbI - 3TO XOpOIllO H3-
BeCTHble cpHnbMocKonbI, 3nHnpoeKTopbI, Hanpo-
eKTopbI, 3nHHaCKonbI, KHHoannapaTbI, KoaCKO-
nbI H p.
CxeMa OnTHqeCKOrO YCTpoHcTBa HanpoeKTopa H306pameHa Ha PHCYH-
Ke 16.50. a. rnaBHa.H qaCTb annapaTa - 06'LeKTHB O. IIpe,AMeToM CJIymHT
np03paqHbIiI PHCYHOK HJIH cpOTOCHHMOK Ha CTeKn.HHHOil nJlaCTHHKe (HJIH npo-
3paqHOil nJleHKe) J(. TaKYIO nJlaCTHHKY Ha3bIBalOT AHan03HTHBOM.
Pa3MepbI Han03HTHBa 06bIqHO 60JlbIlle pa3MepOB 06'beKTHBa. n03TOMY 'lTO-
6b! HanpaBHTb B 06'beKTHB seCb CBeT, HYHH OT Han03HTHBa. npHMeHHIOT
KOH.Zl:eHCOp K, KOTOpblii npeCTaBn.HeT c060il KOpOTKOCPOKYCHYIO CHCTeMY JlHH3
3HaqHTeJlbHoro pa3Mepa. PacnOJlaralOT KOHeHCOp TaK, qT06bI CBeT OT Hero
CXOHnCH B OnTHqeCKOM u,eHTpe 06'beKTHBa. B KaqeCTBe HCTOqHHKOB CBeTa JI
HCnOJlb3YIOT MOIlIHble (300. 500 H 1000 BT) JlaMnbI HaKaJIHBaHHSI (HnH yroBbIe
naMnbI) C pecpJIeKTOpOM P, HCTOqHHK CBeTa nOMeeH B cpoKyce pecpJIeKTopa.
An.H npOel(HpOBaHH.H Ha 3KpaH Henp03paqHbIX npeMeTOB (qepTemeii, PHCYH-
KOB H3 KHHr H p.) HCnOJlb3YIOT 3mmpoeKTop (pHC. 16.50, 6). npeMeT OCBe-
2F
Puc. 16.49
483

6
8
aeTCSI C60KY CBe'l'OM, OTpameHHbIM OT
BOrHYTOrO 3epKana, B qJOKyce KOTOpOrO
paCnOJIOmeH HCTOqHHK CBeTa JI. OTpa-
meHHbIH OT npe,nMeTa CBeT C nOMOblO
nJIOCKOrO 3epKaJla 3 HanpaBJI.HeTC.H Ha
06'beKTHB O.
AnnapaTbI, B KOTOpbIX YCTPOHCTBO
06bIKHOBeHHoro npoeKnliOHHoro annapaTa
(,nliacKona) Ii 3nliCKona COBMeeHO, H8-
3bIBalOT 3nH,ll.HaCKOnaMH (pliC. 16.50, 8).
3nH,nHaCKOn HMeeT ,ABa 06'beKTHB8 0
Ii Oh OTKH,nHYIO IllHPMY ill, OTp8m8-
Tenb H CTOJIHK C ,nJI.H Henp03paqHbIX npe,nMeTOB. Kor,na IllHpMa III onyeH8
(pHC. 16.54, 6), 3nli,nHaCKon ,neHcTByeT KaK npoeKnHOHHbIX annapaT.
TIpH no,nHSlTHIi IllHpMbI 3aKpbIB8IOTC.H KOH,neHcaTOp K Ii 06'beKTliB 0, Ii
OTKpbIBaeTC.H CTOJIliK C, Ha KOTOpOM nOMealOT Henp03paqHbIH npe,nMeT, oc-
BeaeMbIH TeM :me nOBepHYTbIM liCTOqHHKOM CBeT8 JI C peqJneKTopOM P. CBeT,
OTp8meHHbIH OT npe,nMeTa. na,AaeT H8 3epKaJlO 3 Ii OT Hero OTpamaeTCSI Ha
BTOpOH 06'beKTliB 0 1 ,
Puc. 16.50
16.18. rJIa3 KaK OnTHQeCKaH CHCTeMa
Puc. 16.51
484
rJIa3 qeJlOBeKa - 3TO nOqTIi Illapo06-
pa3Hoe TeJIO, KOTopoe Ha3bIBalOT rJIa3HLIM
S6JIOKOM. BHeIllH.H.H TBep,na.H 060JlOqKa 1
(pliC. 16.51), Ha3bIBaeMaSi CKJIepOH, 3ali-
aeT BHYTpeHHOCTb r Jla3a OT MeXaHliqeC-
KliX nOBpem,AeHHH. Ha nepe,AHeH nOBepx-
HOCTIi CKnepa nepeXO,AliT B TOHKYIO npo-
3paqHYIO poroBHny 2; OCTaJIbH8.H ee qaCTb
Henp03paqHa, liMeeT 6eJlbIH nBeT Ii Ha3bI-
BaeTC.H 6eJIKoM. C BHYTpeHHeH CTOpOHbI K

CKJlepe npHJleraeT COCY,lJ;HCTa.H 060JlO'lKa, COCTO.Ha.H H3 KpOBeHOCHbIX COCY-
AOB, nHTalOHX rJla3. STa 060JlOqKa B nepe)J;HeH qaCTH nepeXOAHT B PMY'K-
HYJO o6oJlo'IKY 3, OKpameHHYIO Y pa3HbIX JlIOAeH B pa3JIH'tIHbIH BeT. Pa,nymHa.H
060JlOqKa HMeeT B cepeAHHe OTBepCTHe, Ha3bIBaeMoe 3pa'IIKOM 5, AHaMeTp KOTO-
poro MomeT H3MeH.HTbC.H peq>JIeKTOpHo (6e3 yqaCTH.H c03HaHH.H) B 3aBHCHMOC-
TH OT KOJlHqeCTBa na,AalOero CBeTa: npH .HpKOM OCBeeHHH 3paqOK YMeHb-
lllaeTC.H, npH CJla60M - YBeJlHqHBReTC.H. Ha BRYTpeHHeH nOBepXHOCTH COCY-
AHCTOH 060JlOqKH pacnOJlOmeHR CeT'IaTaJl 060JlO'lKa 9 (ceTqRTKa). npeACTaBJI.H-
lOa.H C060H pa3BeTBJleHH.H 3pHTeJlhHOrO HepBa 10 C HepBHbIMH OKOHqaHH.H-
MH B BHAe naJlOqeK H KOJl60qeK, KOTopbIe .HBJI.HIOTCfl CBeTOqYBCTBHTeJlbHbIMH
3J1eMeHTaMH.
Mem)J;Y poroBHeH H paAymHoH 060JlOqKOH HaXOAHTC.H BO,lJ;JlHHCTaJl H,.lJ;-
KOCTh 4. 3a 3paqKOM pacnOJIOmeH xpYCTaJlHK 6 - np03paqHOe ynpyroe TeJlO,
HMelOee cl>OPMY ABO.HKOBbInYKJloH JlHH3bI. [(HJlJlHapHaJl MhIwa 7 MomeT H3-
MeH.HTb cl>OPMY xpYCTanHKR, a 3Ha'tlHT, Hero OnTHqeCKYIO CHJlY. OCTaJlbHYIO
qaCTb rJla3a 11.0 3alIHeH CTeHKH (rJIa3HOrO lIRa) 3RHHMaeT np03paqHOe nOJlY-
mHAKoe CTeKJIOBH,.lJ;HOe TeJlO 8.
CBeT, [1aAalOHH B rJla3, npeJIOMJI.HeTC.H Ha nepeAHeii nOBepXHOCTH rJla-
3a, B pOrOBHl..{e, xpYCTaJlHKe H CTeKJlOBHAHOM TeJle (T.e. B OnTHqecKOJ1 CH-
CTeMe rJla3a), H Ha CeTqaTKe 06pa3yeTC.H AeHCTBHTeJlbHOe, YMeHbllIeHHoe, ne-
peBepHYToe H306pRmeHHe paCCMaTpHBaeMbIX npeAMeTOB (pHC. 16.52). Pa3-
ApameHHe OKOHqaHHH 3pHTeJlbHOrO HepBa Ha CeTqaTKe nepeAaeTCSI B M03r,
KoppeKTHpyeTC.H HM, H B pe3YJIbTaTe npeAMeT BocnpHHHMaeTC.H 'leJlOBeKOM
np.HMbIM.
Pa3Mep H306pameHH.H HR CeTqaT-
Ke 3aBHCHT OT pa3MepOB npeAMeTa H
paCCTO.HHH.H OT Hero AO OnTHqeCKO-
ro eHTpa 0 rJla3a, T.e. 3aBHCHT OT
yrJla q>, nOA KOTOpbIM paccMaTpHBa-
eTC.H npe,AMeT. STOT yroJl Ha3bIBalOT
yrJlOM 3peHHJI. C npH6J1HmeHHeM K
HaM npelIMeTa yroJl 3peHH.H B03pac-
TaeT, H H306pameHHe Ha CeTqaTKe
Tome YBeJlHqHBaeTC.H. YCTaHOBJleHO, qTO eCJlH yroJl 3peHH.H Mem,Ay ABYM.H pa3-
,AeJlbHbIMH TOqKaMH paCCMaTpHBaeMoro npeAMeTR q> > 1', H306pameHHe 3THX TO-
qeK Ha CeTqaTKe rJla3a nOna,AaeT Ha pa3HbIe HepBHbIe OKOHqaHH.H, H MbI 3TH
TOqKH BHAHM pa3)J;eJIbHbIMH. ECJlH me q> < 11, TO H306pameHH.H ABYX pa3HbIX
TOqeK npeAMeTa nonaAalOT Ha OAHO HepBHoe OKOHqaHHe, H MbJ HX BHAHM
CJIH BlllHMHC.H.
C OnTHqeCKOH TOqKH 3peHH.H rJla3 HanOMHHaeT cl>oToannapaT (CM.  16.16).
POJlb 06'beKTHBa BblnOJIH.HeT xpYCTaJlHK, AHacl>parMoH .HBJI.HeTC.H 3paqOK, KpbIlll-
KOH 06'beKTHBa cJlymaT BeKH. IIOA06Ho COBepllleHHbIM COBpeMeHHbIM cl>oToan-
napaTaM, rJla3 aBTOMaTHqeCKH HaBOAHTC.H HR pe3KOCTb - H3MeHeHHeM KpH-
BH3HbI xpYCTaJlHKa (aKKoMOAaH.H), aBTOMaTHqeCKH AHacl>parMHpyeTC.H - CY-
meHHeM 3paqKa. rJla3 AaeT BeTHoe H306pameHHe, "3anoMHHaeT" 3pHTeJlbHbIe
06pa3bI .
A Az
I
B Bz 0 
1
3
Puc. 16.52
485

MbIIIIIJ;a, cpHKCHpYIOa.H XpYCTaJlHK, MOmeT H3MeHHTb ero KPHBH3HY. M3-
MeHeHHe KpHBH3HbI xpYCTaJIHKa onpe.n;eJl.HeT cnoc06HOCTb I'Jla3a K aKKOMo.n;a-
IJ;HH. AKKOMO.z:J:aU;Hen Ha3bIBaeTC.H H3MeHeHHe OnTHqeCKOM CHJIbI I'JIa3a.
Cnoc06HOCTb I' JIa38 K aKKOMo,n8IJ;HH o6ecnet.JHBaeT B03MomHOCTb nonyqe-
HH.H Ha CeTqaTKe pe3KHX H306pameHHM npe.nMeTOB. Haxo.n.HHXC.H Ha pa3-
nHqHbIX paCCTOSlHH.HX. HOpMaJlbHbIH rJla3 npH paCCJIa6neHHOH IJ;HJIJlHapHOM
MbIlIIIJ;e .n;aeT Ha CeTqaTKe OTqeTJlHBOe H306pameHHe y.n;aJIeHHbIX npe.n;MeTOB.
C nOMOblO MbIIlleqHOI'O YCHJIHSI, YBenHqHBalOel'O KPHBH3HY xpYCTaJIHKa,
I'JIa3 ocyeCTBJISleT HaBO,AKY Ha HymHoe paCCTO.HHHe. Ho aKKOMo.n;aIJ;H.H I'JIa3a
HMeeT npe,AeJ1. HaHMeHblIIee paCCTO.HHHe (paCCTO.HHHe HaHnyqlIIero 3peHHSI) do,
Ha KOTOpOM MomHO paCCMaTpHBaTb ,AeTaJlH 6e3 60JlblIIOrO Hanp.HmeHH.H, y
HOpMaJlbHOrO rJIa3a paBHO 25 CM.
16.19. )J;e4>eKTLI 3peHHn. O'lI\:H
TOt.JKa, KOTOPYIO BH.n;HT I' JIa3 npH pacCJIa6JIeHHOM u,HJlJIHapHOM MbIIIIIJ;e,
Ha3blBaeTC.H .z:J:aJlbHen TOllKOn, a npH MaKCHMaJIbHOM HanpSlmeHHH - 6J1H-
He" TOqKon. )],JI.H HOpMaJlbHOI'O I'JIa3a .n;anbH.HSI TOt.JKa JIemHT 6eCKOHeqHO ,Aa-
JIeKO, 6J1HmH.H.H - B cpe,nHeM Ha paCCTO.HHHH 15-20 CM. IIoJlo:meHHe 6nH:m-
HeM TOt.JKH H3MeHSleTC.H C B03pacToM: B ,AeTCTBe (,no 10 JIeT) - Ha paCCTO.H-
HHH 7-8 CM, B 30 JIeT - 15 CM. K 50 l'o.n;aM - 50 CM.
Y HeKOTopbIX JIIO.n;eM B HeHanpSlmeHHOM COCTO.HHHH I'Jla3a H306pameHHe y,na-
neHHbIX npe.n;MeTOB o6pa3yeTC.H He Ha CeTqaTKe, a nepe,n HeM (pHC. 16.53, a).
TaKHe JIIO.n;H He MorYT BH,AeTb Y,AaJleHHbIe npe.n;MeTbI. STOT .n;ecpeKT Ha3bIBaeT-
C.H 6JIH30PYKOCTblO. rIpH 6J1H30PYKOCTH paCCTO.HHHe HaHnyqmero 3peHH.H MeHb-
me 25 CM. BJIH30PYKOCTb BbI3bIBaeTCSI JIH60 BbITSlHYTOCTblO rJIa3HOl'O Sl6JIOKa,
JIH60 cna3MOM IJ;HnJlHapHOM MbImIJ;bl. BnH30PYKOCTb HCnpaBJI.HIOT OqKaMH C pac-
CeHBalOI.IJ;HMH nHH3aMH (pHC. 16.53, 6).
a 6
s8
Puc. 16.53
a
6
Puc. 16.54
)],ecpeKT 3peHHSI, npH KOTOpOM H306pameHHe y.n;aJIeHHbIX npe.n;MeToB nony-
qaeTC.H 3a CeTqaTKoM. H83bIBaeTC.H .z:J:aJILH030pKOCTLIO (pHC. 16.54, a). .l1.aJlbHO-
30pKOCTb BbI3BaHa JlH60 YKopOqeHHOCTblO rJla3HOrO .H6J10Ka, nH60 CJla60M aK-
KOMo.n;aIJ;HeM. Y.n;aJleHHbIe npe.n;MeTbI TaKoM I'Jla3 BH.n;HT xopomo (HO He Jlyq-
me, qeM HOpMaJIbHbIM I'JIa3). KOppeKIJ;H.H .n;aJlbH030pKOCTH npOH3Bo,nHTC.H C no-
MOblO OqKOB C c06HpalOHMH JlHH3aMH (pHC. 16.54, 6).
486

16.20. JIyna
QT06bI YBH,D;eTb MeJlKHe ,D;eTaJlH npe,D;MeTa, HX HymHo paCCMaTpHBaTb nOl{
60JIbIIlHM yrnoM 3peHH.H, HO YBeJIHqeHHe 3Toro yrJIa OrpaHHqeHO npe,D;eJIOM aK-
KOMO,D;8l.\HOHHbIX B03MomHOCTeH rJla3a. YBeJlHqHTb yroJl 3peHH.H (coxpaH.H.H pac-
CTOHHHe HaHnyqIIlerO 3peHH.H do) MomHO, HcnOJlb3Y.H OnTHqeCKHe npH60pbI
(JIynhl, MHKpocKonbl).
JIynOH Ha3bIB81OT KOpOTKOcpOKYCHYIO C061ip8-
IOII.1;YIO JIHH3Y IiJIIi CHCTeMY JlHH3, ,D;eHcTBYIOII.1;IiX
KaK O,D;Ha c06HpalO8.H JIHH3a (06bIqHO cpOKYC-
Hoe paCCTO.HHHe Jlynb] He npeBbIIIl8eT 10 CM).
XO,D; JIyqeH B Jlyne nOK838H H8 pHCYHKe
16.55. JIyny nOMeII.1;8IOT 6JIH3KO K rJIa3Y, 8 p8C-
Bl B 2 CM8TpHBaeMbIH npe,D;MeT AB =AIBI pacnOJIaralOT
Mem,Ay JIynoH H ee nepe,D;HIiM cpOKYCOM, qYTb
6nHme nOCJle,D;Hero. IIo,D;6Hpa1OT nOJlOmeHHe JIY-
TIbI Mem,D;y rJla30M R npe,D;MeTOM TaK, qT06bI BH-
,AeTb pe3Koe H306pameHHe npe,D;MeTa. 3TO H306-
pa)KeHl1e A 2 B 2 nOJIyqaeTC.H MHI1MbIM, npHMbIM,
YBeJIl1qeHHbIM H HaXO,D;HTC.H Ha paCCTO.HHHH HaHJIyqIIlero 3peHH.H O = do OT
Puc. 16.55
rJla3a, a caM rJla3 HaXOHTC.H HenOCpe,D;CTBeHHo nepe,D; JlynoH.
I1cnOJlb30BaHHe Jlynbl npHBO,D;HT I( YBeJIHqeHHIO yrJla 3peHI1.H, no,D; KOTO-
pbIM rJla3 paccMaTpHBaeT npe,D;MeT. .lI.eHcTBHTeJlbHO, KOr,D;a npe,D;MeT HaXO,AHJI-
CH B nOJlOmeHHH AIBI H paccMaTpHBaJlC.H HeBoopymeHHblM rJIa30M, yroJl 3pe-
HH.H 6bIJI <Pl' IIpe,D;MeT nOMeCTHJlH Mem,D;y cpOKYCOM H OnTl1qeCKHM l.\eHTpOM
JIynbI B nOJlOmeHl1e AB, H yron 3peHI1.H CT8J1 <P2' IIOCKOJlbKY <P2> <Pit TO C
nOMOblO JlynbI Ha npe,D;MeTe MomHO paCCMOTpeTb 60nee MeJlKl1e ,AeTaJlH, qeM
HeBoopymeHHbIM r Jla30M.
113 pHcYHKe 16.55 BH,D;HO TaKme, qTO JlHHeHHOe YBeJlHqeHl1e nynbI
r - A 2 B 2 _ OB 2
- AB - OB .
TaK KaK OIl;, = do, a OBI:::: F (F - cpoKycHoe paCCTO.HHHe JlynbI), TO
r = do r,D;e do = 25 CM. CJle,D;OBaTenbHO, yueJIH'leHHe, ,lJ;aBaeMoe JIynOH, paB-
F'
HO OTHOlIleHHIO paCCTOHHHH HaHJIyqlIlero 3peHHH K OKYCHOMY paCCTOHIIHIO
JIynLI.
16.21. MHKpocKon
MHKpOCKonOM Ha3bIBalOT OnTliqeCKl1n npH60p, cnymaHH ,D;n.H paCCMaTpli-
B8HI1H OqeHb MenKIiX npe,D;MeTOB (B TOM qHCJIe HeBIi,AI1MbIX HeBoopymeHHbIM
rJI830M).
MHKpOCKon COCTOHT H3 ,D;BYX C06I1palOII.1;HX KOpOTKOcpOKYCHbIX JlI1H3 - 06'beK-
THBa 11 OKYJIHpa, paCCTO.HHHe Me)K,D;y KOTOpbIMI1 MO)KeT H3MeH.HTbCH.
XO,D; JlyqeH B MHKpOCKone nOKa38H Ha pHCYHKe 16.56. 06'beKTHB C03,D;8eT
,D;eHCTBHTenbHoe, nepeBepHYToe. YBenl1qeHHOe npOMemYTOqHOe H306pameHlie
AIBI npe,D;MeTa AB.
487

OK

06
B
A 2
' B '
,,' 1 ;,'
, ,
, ,
, ,
,,' ,.,;'
, ,
, ,
, '
" "
" ,;'
, ,
, ,
" ;'
""",
, ,
, ,
" "
tr!/
B 2
f2
..
Puc. 16.56
C nOMoblO MHKpOMeTpHqeCKOrO BHHTa OKYJI.Hp nOMealOT OTHOCHTeJIbHO
06'beKTHB8 TaKHM 06pa30M, qT06bI 3TO npoMemYTOqHOe H306p8meHue AIBI OKa-
3aJIOCb Mem.n;y nepe.n;HHM <POKYCOM F 2 H OllTHtJeCKHM u.eHTpOM O 2 OKYJI.Hpa. Tor-
,Aa OKYJI.Hp CT8.HOBHTC.H JlynOH H C03,AaeT MHHMOe, np.HMOe (OTHOCHTeJlbHO npOMe-
mYTOqHOrO) H yBenHtJeHHOe H306pameHHe A 2 B 2 npe,AMeTa AB. llcnonb30BaHHe
MUKpOCKOna npHBO.n;HT K 3HaqUTeJlbHOMY YBenHqeHHIO yrJla 3peHH.H, no.n; KOTO-
pbIM rJla3 paCCMaTpUBaeT npe,AMeT (<1>2» <1>1), qTO n03BOJl.HeT BH.n;eTb lIeTanu,
He BH,AHMbIe HeBoopymeHHbIM rJla30M. JIHHeHHoe YBeJl1:l1ieHHe MHKpOCKona
r - A 2 B 2 _ A 2 B 2 Al B 1
- AB - Al . AB .
A 2 B 2
TaK KaK Al = r OK
Al
JlHHeHHOe YBeJlHqeHHe oKYJI.Hpa H - = roo -
AB
JlUHeHHOe YBeJlHqeHHe 06'beKTHBa. TO JlUHeHHOe YBeJIUqeHHe MHKpOCKona
r = roo' r OK'
113 P Hc Y HKa 16.56 BH.n;HO, qTO r - Al - 01 A l
06- AB - OlA '
r.n;e 0 1 Al = 0 1 F 1 + F 1 F 2 + F2 A l'
0603HaqHM 8 paCCTO.HHHe Mem.n;y 3a.n;HHM cpOKYCOM 06'beKTHBa H nepe.n;HHM
cpOKYCOM OKYJl.Hp8, T.e. 8 = F 1 F 2 . TaK K8K 8» 01 F I H 8» FiAl' TO
0 1 Al ::: 8. IIoCKOJIbKY 0 1 A ::: F 06' r.n;e F 06 - cpoKycHoe paCCTO.HHHe 06'beKTH-
Ba, nOJlyqaeM
8
roo = -.
F06
488

JIHHeHHoe YBeJlHqeHHe OKYJI.Hpa onpeeJl.HIOT no TOH me cpopMYJle, qTO H
d
YBeJlHqeHHe JlynbI, T.e. r OK =--2-, r)J;e do - paCCTO.HHHe HaHJlyqIIIerO 3peHH.H,
F OK
F OK - cpOKYCHoe paCCTO.HHHe OKYJI.Hpa.
f> - do
Tora nOJlHM r = F. . F. - JlHHeHHOe YBeJlHqeHRe MRKpOCKona.
06 OK
16.22. 3JIeMeHTbI <}>oToMeTpHH. Cy6'beKTHBHbIe H 06'beKTHBHbIe
xapaKTepHCTHKH H3JIyqeHH
OTOMeTpHH - pa3en onTHKH, B KOTOpOM H3yqa1OTC.H MeTObI H npH-
eMbI H3MepeHH.H CBeTOBOH 3HeprHH.
OeHHBaTb SHeprHIO CBeTOBoro H3J1yqeHH.H MomHO BH3yaJILHo (rJla30M) HJlH
C nOMOblO KaKorO-TO cneijuaJl.bliOZO npu60pa (cpoT03J1eMeHTa, 60JlOMeTpa H
T.n.). TIpH 3TOM cJleAyeT nOMHHTb. qTO qYBCTBRTeJlbHOCTb qenOBeqeCKOrO rna-
3a HeOHHaKOBa K H3J1yqeHH.HM pa3HoH )J;JlHHbI BOJlHbI. TaK, JI.H 3eJleHOrO CBe-
Ta qYBCTBHTeJlbHOCTb rJla3a B 100 pa3 BbIIIIe, qeM n.H KpaCHoro. CJleOBa-
TenbHO, cy6'beKTHBHa.H On;eHKa ::meprHH H3J1yqeHH.H (no 3pHTeJlbHOMY oye-
HHIO) H 06'beKTHBHa.H (npH60pOM) He COBnaaIOT. TI03TOMY B CBeTOBbIX H3Me-
peHH.HX npHMeH.HIOT Be CHCTeMbI 0603HaqeHHB H Be CHCTeMbI eHHHn;: O,AHa
H3 HHX OCHOBaHa Ha 3HepreTHqeCKOH On;eHKe CBeTa 6e30THOCHTeJlbHO K ero
AeBCTBHIO Ha npHeMHHKI1 H3J1yqeHH.H, pyra.H - Ha On;eHKe CBeTa no 3pH-
TenbHOMY OYIUeHHIO. OHa xapaKTepH3yeT CPH3HOJlOrHqeCKHe lIeHCTBH.H CBeTa.
1. 3HepreTll'leCKHe BeJIHQHHLI.
a) n omOK U3Jl.YlleIiUSl. BbIAenHM MbICJleHHO Ha nYTH
CBeTa, pacnpOCTpaH.HIOerOC.H OT KaKOrO-JlH60 HCTOqHH-
Ka, He60JlbIIIYlO nJloaAKY S (pHC. 16.57). t.{epe3 3TY
nJlOa,AKY 3a BpeM.H t npoHeT HeKOTopoe KOJlHqeCTBO
cBeToBoH 3HeprHH W.
W
OTHollieHRe p = -, nOKa3bIBalOee, KaKoe KonH-
t
Puc. 16.57 qeCTBO 3HeprHH npoxoHT Qepe3 nJlOa,AKY 3a eHHH-
y BpeMeHH, Ha3bIBaeTC.H nOTOKOM H3JIyqeHHH P Qepe3
nJlOaAKY S.
B CIf eHHHeH nOTOKa H3J1yqeHH.H .HBJI.HeTCSJ BaTT
6) H 3Jl.YllameJl.bIiOCmb Re - BeJlHqHHa, paBHa.H OTHOIIIeHHIO nOTOKa H3ny-
qeHH.H, HcnYCKaeMoro nOBepXHOCTblO, K nJlOaH S CeqeHH.H, qepe3 KOTopoe
3TOT nOTOK npOXO)J;HT:
P
R =-
e S
B CM eJ\HHeH H3JIY'lBrenLHOCTH smnHeTCSI BaTT HB KBIIJIPIITIILIH MeTp [ :; ).
1f3J1yqaTeJlbHocTL npe)J;CTaBJI.HeT C060H nOBepXHOCTHYIO nnOTHOCTb nOTOKa
H3J1yqeHHH. STOT TepMHH aHanOrHqeH HHTeHCHBHOCTH BonHbI.
489

B) CUJl.a U3Jl.YlleIiUSl. PaCCMOTpHM TOqe'IHbIH H30TpOnHbIH HCTO'IHHK CBe-
Ta - T.e. HCTOqHHK CBeTa, pa3MepaMH KOToporo no cpaBHeHHIO C paCCTO.HHH-
.HMH ,no MeCTa Ha6J1IO,AeHH.H MomHO npeHe6pe'lb H KOTOpbIH paBHOMepHO H3J1Y-
qaeT CBeT no BceM
Puc. 16.58
nJIOa,AH KBa,ApaTa
TeJleCHbIH yroJl
HanpaBJleHH.HM.
HanpaBJleHHe CBeTOBoro H3J1yqeHH.H 3a,AalOT c nOMO-
blO TeJIeCHOrO yrJla. TeJleCHLIM yrJlOM Ha3bIBalOT 06-
JlaCTb npOCTpaHCTBa, OrpaHHqeHHYIO KOHHqeCKOH nOBepx.
HOCTblO (pHC. 16.58). 3HaqeHHe TeJleCHOrO yrJlB onpe,AeJl.H-
S
eTC.H no <})opMYJle 00 = "'2' r,Ae S - nnoa)J;b UIapOBoro
r
cerMeHTa, Ha KOTOpbIH onHpaeTC.H TeJleCHbIH yroJl, r -
pal1HYC ccl>epbI. ElIHHHu.eH TeJleCHOrO yrJla .HBJI.HeTC.H CTe-
paHaH ( cp ).
1 cp - TeJleCHbIH yron C BepUIHHoH B II,eHTpe c<}JepbI,
BbIpe3alOHH Ha nOBepXHOCTH ccpepbI nJlOa,Ab, paBHYIO
co CTOpOHOH, paBHoH pa,AHYCY ccl>epbI.
000' cyecTBYIOHH BOKpyr TOqKH 0 (T.e. onHpalOHHC.H
Ha BCIO nOBepXHOCTb ccl>epbI nJIOa,AblO S = 4nr 2 ), H83bIBalOT nOJlHLIM TeJleC-
HLIM yrJlOM:
4nr 2
000 = - = 4n Cpo
r 2
CHJla H3J1yqeHHH [e - <})H3HqeCK8.H BeJlHqHH8, p8BHa.H OTHOUIeHHIO nOTO-
Ka H3J1yqeHH.H K TeJleCHOMY yr JlY , B npe,AeJIax KOToporo STO H3JIyqeHHe pac-
npOCTp8H.HeTC.H:
P
1 =-.
e 00
B CM ellHHlUeii CHJILI H3JI}"leHH" "BJI"eTC" BaTT Ha CTepallHaH r ; ).
2. CBeToBLle BeJlH'IHHLI.
a} C6em060u nomOK <lJ - cl>H3HqeCKa.H BeJIHqHHa, KOTOpa.H xapaKTepH3Y-
eTC.H (no 3pHTeJlbHOMY oyeHHIO) OTHOUIeHHeM cBeToBoH SHeprHH, nepeHO-
CHMOH qepe3 KaKYIO-JlH60 nJlOa,AKY B e,AHHH:U;y BpeMeHH:
W
<lJ =-,
t
T.e. cBeToBoH nOTOK - STO MOHOCTb CBeTOBoro H3J1yqeHH.H, o:u;eHHBaeMaSl
BH3yaJlbHO. E,AHHH:u;a CBeTOBoro nOTOKa - JlIOMeH (JIM).
6} CUJl.a C6ema - CKaJl.HpHa.H cl>H3HqeCKa.H BeJIHqHHa, paBHa.H OTHOUIeHHIO
CBeTOBoro nOTOKa K TeJleCHOMY yrJIY, B KOTOpOM STOT nOTOK pacnpe)J;eJl.HeTC.H:
<I>
[- -
ro
E,AHHH:u;eH CHJlbI CBeTa B CM .HBn.HeTC.H KaHneJla (K,A).
KaH,AeJla - OCHOBHaSl e,AHHHI:(a H3MepeHH.H. KaH,AeJla BocnpOH3BO,AHTC.H C
nOMOblO CneI:(HaJlbHOrO STaJIOHHOrO HCTOqHHKa CBeTa. 1 K,A - CHJla CBeT8,
490

licnYCKaeMoro C nOBepXHOCTH nJlOWalIblO
1
600 OOOn
M 2 nOJlHOrO li3J1yqaTeJl.H
B nepneH,AliKYJI.HpHOM HanpaBJleHHH npH TeMnepaType li3J1yqaTeJl.H. paBHOI1:
TeMnepaType 3aTBeplIeBaHliH nJlaTliHbI npH lIaBJleHHli 101 325 lla.
113 onpe,neJleHli.H CliJlbI CBeTa cne,nyeT, qTO CBeTOBOM nOTOK  paBeH
 = I w. OCHOBHOH e,nHHlieM CBeTOBoro nOTOKa B CIf .HBJlHeTCH J1IOMeH (JIM).
JIM = K,A . Cpo
JhOMeH - CBeTOBOM nOTOK, KOTOpbIM li3J1yqaeTC.H TOqeqHbIM liCTOqHliKOM
CliJlOM CBeTa 1 K,n BHYTPli TeJleCHOrO yrJla, paBHoro ] Cpo
CBeToBoM nOTOK o, pacnpOCTpaH.HIOHHC.H BHYTPli nonHoro TeJleCHOrO yrJla,
o= 41tI
Ha3bIBaeTC.H nOJlHLIM CBeTOBLIM nOTOKOM. OH xapaKTepli3yeT nOJlHYIO CBeTO-
BYIO 3HeprHIO (oeHliBaeMYIO no 3pliTeJlbHOMY oyeHHIO), li3J1yqaeMYIO liC-
TOqHliKOM CBeTa B e,nHHliy BpeMeHH no BceM HanpaBJleHHHM.
B) OCBel4eliliOCmb n06epXliocmu E - qJH3liqeCKa.H CKaJl.HpHaH BeJlliqliHa.
paBHa.H OTHOIlIeHHIO CBeTOBoro nOTOKa, na,naIOero Ha KaKYIO-Jlli60 nOBepx-
HOCTb, K nnOa,Ali S 3TOM nOBepXHOCTli:
<I>
E=-.
S
flM
E,nHHliueM OCBeeHHOCTli B CIf .HBJI.HeTC.H J1IOKC (nK). flK = 2'
M
BHeCHCTeMHa.H e,nliHlia H3MepeHH.H OCBeeHHOCTH - qJOT (qJT)
nM
qJT =  = 10 4 J1K.
CM
ECJlH HCTOqHHKH CBeTa He TOqeqHbIe, a npOT.H>KeHHbIe, TO ,nJl.H liX xapaK-
TepliCTliKli BBO,n.HTC.H BenHqliHbI: CBeTliMOCTb H .HpKOCTb.
r) CBemU.MOCmb R xapaKTepH3yeT CBeTOBOM nOTOK, H3J1yqaeMbIM nOBepx-
HOCTblO CBeT.HerOCSI TeJla e,nHHHqHOM nJlOa,nH no BceM HanpaBJleHH.HM, T.e.
E,nHHHeM
MeTp r : ) .
on) HpKocmb ucmOIlIiUKa. fIycTb CBeT.Ha.HCSI nOBepXHOCTb HaXO,AliTC.H B
npe,AeJlax TeJleCHOrO yrJla W (pHC. 16.59), a - yrOJl Mem,ny HopManblO K 3TOM
nOBepXHOCTH H HanpaBJIeHHeM Ha6JIIO,AeHHH CBeTOBoro nOTOK8, Sn - npoeK-
H.H nJlOIlta,AKH S Ha nJlOCKOCTb, nepneH-
,nHKYJl.HpHYIO HanpaBneHHIO Ha6J1lOlIeHliH
(Sn = S cos a).
.f.[PKOCTb B xapaKTepli3yeT CBeTOBOH
nOTOK, H3J1yqaeMbIM nJlOa,nKoM e,nHHliq-
HOM nJlOa,Ali B nepneH,AliKyn.HpHOM Ha-
npaBJleHliH B npe,nenax e,nHHliqHOrO TeJleC-
Horo yrna:
<I>
R=-.
S
H3MepeHH.H CBeTHMOCTH B CIf HBJI.HeTC.H J1IOKC Ha KBa,npaTHbIH
Ii
-
o
Puc. 16.59

B=-=
wS n wS cos a
4>
491

I
cI>
T8K KaK I = -, TO B =
(0 8cosa
KA
B Cll eAHHHeH .HpKOCTH .HBJI.HeTC.H HHT (HT). HT = 2' BHeCHCTeMH8.H eAH-
M
KA
HHl.\a .HpKOCTH - CTHnb6 (C6). C6 =  = 10 4 HT.
CM
I1CToqHHKH CBeTa C .HpKOCTblO B > 16 C6 BbI3bIBalOT 60Jle3HeHHoe oye-
HHe B r Jla3Y .
16.23. 3aKoHLI OCBeeHHOCTH
H86J1IOAeHH.H nOKa3bIB8IOT, qTO OCBeeHHOCTb,
C03,A8B8eM8.H TOqeqHbIM HCTOqHHKOM CBeT8, 38BH-
CHT OT CHJlbl CBeT8 I 3Toro HCTOqHHK8, p8CCTO.H-
HH.H r OT HCTOqHHKa ,AO oCBe8eMoH nOBepXHOC-
TH H yrn8 n8AeHHSI CBeTOBblX nyqeH (yrn8 Mem-
y n81IalOHM nyqOM H nepneHlIHKynSlpoM K 3TOM
nOBepxHocTH). IIpHqeM 3TO YTBepmAeHHe BepHO
An.H JlI060ro HCTOqHHKa.
IIycTb Eo - OCBeeHHOCTb nOBepXHOCTH npH
nepneH,AHKYJlSlPHOM na,u;eHHH Ha Hee CBeTOBblX ny-
qeH, E - npH nlO60M APyroM yrne na,u;eHH.H. OnH-
meM BOKpyr TOqeqHOrO HCTOqHHKa C cHnoH CBeT8 I cc}>epHqecKYIO nOBepXHOCTb
pa,u;HYCOM r (pHC. 16.60). OCBeeHHocTb BHYTpeHHeH CTOpOHbI 3TOH nOBepx-
HOCTH Be3Ae OAHHaKOB8. H JlyqH na,nalOT nepneHAHKYJlSlPHO nOBepXHOCTH cc}>epbI.
OCBeeHHocTb Eo = <1>0 , rAe <1>0 - nonHbIH CBeTOBOH nOTOK, S - nnO8)J;b
8
Puc. 16.60
4nI I
nOBepxHocTH cc}>epbI, <1>0 = 41tl, S = 41[,2 . II03ToMY Eo = -  Eo = -
4nr 2 r 2
nepBLIH 3aKOH oCBew,eHHOCTH: oceer.qeliliocmb no-
6epXliocmu Jl.Ylla.MU, naaalOUI,u.Mu Ha liee nepnen-
aUKYJlSlpliO, npJUtO nponop"+UOliaJlblia CUJl.e C6ema
mOllellliOZO ucmOllliUKa U 06pamli0 nponopu.UOliaJl.b-
lia Keaapamy paccmOSlHUSl om ucmOllliUKa do oc-
6ew,aeMou noeepxliocmu.
IIycTb nap8nnenbHblH paBHOMepHO p8cnpe-
AeneHHblH nyqoK CBeTa naAaeT H8KnOHHO H8 no-
BepXHOCTb 8 (pHC. 16.61). Ee OCBeeHHOCTb
Puc. 16.61
492
cI>
E = -. CnpOel.\HpyeM nOBepXHOCTb 8 Ha nnoc-
8
KOCTb, nepneHAHKYJl.HpHYIO n8A81OInHM Jlyq8M.
nonyqHM HOpM8J1bHYIO K Jlyq8M npoeKIlHIO 8 0 , Ee
OCBeIl.\eHHOCTb Eo = <1>0 . 113 pHcYHKa 16.61
S

lo loh So E So
cosa=-=-=-. Tora -=-=cosa.
l lh S Eo S
OTKya E = Eo cos a - BTOpOH 3aKOH OCBeeHHOCTH: OC6el4eHliocmb no-
6epXliocmu napaJl.Jl.eJl.blibl.M C6em06blM nYIlKoM npSl.MO nponopquoliaJl.bHa KOCU-
liYCY YZJl.a naiJeIi USl.
I
TaK KaK Eo =-
r 2 '
I
TO E = - cos a - o606eHHLIH 3aKOH OCBeeHHOCTH.
r 2
BOIIPOCbI J],JUI CAMOKOHTPOJI.H
1. l{ mo lia3bl6aemcSl Jl.UIi30U? KaKue Jl.Uli3bl lia3bl6alOmcSl 6blnYKJl.blMU?
60zHymbUf.u? c06UpalOl4UMu? pacCeU6alOl4UMu?
2. Bcezoa Jl.U 60zliymaJl Jl.Uli3a R6.n.RemCSl pacCeU6alO14eu?
3. TpexzpaliliaSl npU3.Ma OmKJl.01iSlem C6em06ble Jl.YIlU K 6epWUlie C06upa-
lOl4eu UJl.U pacCeU6alOl4eU 6yiJem 6blnYKJl.aSl Jl.UIi3a. U3Z0m06Jl.eliliaSl U3 amozo
3ICe MamepUaJl.a?
4. l{ mo lia3bl6aemcSl onmUlleCKU.M qelimpoM Jl.UIi3bl? ZJl.a61i0U onmUlleCKOU
OCblO?
5. l{ mo nOIiU.MalOm noo CPOKYCO.M Jl.Uli3bl? KaKou CPOKYC lia3bl6aemCJl iJeu-
Cm6UmeJl.blibl.M? .MIiUMblM? KaKaSl Jl.U1i3a UMeem .MIiU.MblU CPOKYC?
6. l{mo maKoe onmUlleCKaSl CUJl.a Jl.UIi3bl? B KaKO.M CJl.Yllae onmUlleCKaSl
CUJl.a nOJl.OJICUmeJl.blia? ompUijameJl.blia? B KaKUX eiJuliuijax OHa U3MepSlemcSl?
Om Ilezo 3a6ucum onmUlleCKll.J1 CUJl.a Jl.UIi3bl?
7. KaKuMu Jl.Ylla.Mu yo061i0 nOJl.b306ambCSl npu nocmpoeliuu uJ06paJICeHUR
mOllellliOZO npeiJ.Mema 6 Jl.UH3e? KaKoe .MUIiU.MaJl.bIiOe IlUCJl.O UX liaiJo 63Slmb?
8. Om llezo 3a6UCSlm nOJl.OJICeHUe U pa3.Mepbl u306paJICeliUSl npeiJMema 6
Jl.UIi3e?
9. KaK 3anUCbl6aemcSl tpopMYJl.a mOIiKOU Jl.Uli3bl? KaKoe npa6UJl.O 3liaK06
cJl.eiJyem Yllumbl6amb npu 3anucu amou CPOPMYJl.bl?
10. l{ mo lia3bl6aemcSl Jl.UlieUlibl.M Y6eJl.UlleliUe.M? KaK paCCilumbl6aemcSl Jl.U-
lieUlioe Y6eJl.UlleliUe Jl.UIi3bl?
11. riJe U KaKoe nOJl.YllumcSl u306paJICeliUe oeuCm6UmeJl.bliOZO npeoMema. lia-
XOiJSll4eZOCSl Ja iJ60Ulibl.M CPOKYClibl.M paccmOSlIiUe.M c06upalOl4eu Jl.UIi3bl? llpu
KaKO.M YCJl.06UU c06upalOl4aSl Jl.U1i3a iJaem .MliU.Moe u306paJICeliue? H a KaKO.M
paCcmOJlIiUU liaiJo nO.MeCmUmb npeiJ.Mem nepeiJ c06upalOl4eu Jl.Uli30U. Ilm06bl no
Jl.Yllumb Y6eJl.UlleHHOe u306pa:HCeliUe npeiJ.Mema?
12. KaKUM 06pa30.M 603liUKaem u306paJICeliUe 6 ZJl.a3Y lleJl.06eKa? l{ mo ma-
Koe aKKO.MoiJaijUJl ZJl.a3a?
13. KaKue iJecpeKmbl 3peliuSl 6bl 3liaeme? KaKue Jl.UH3bl UCnOJl.b3YlOmcSl 6
OIlKax OJl.Sl UCnpa6Jl.eliUSl amux oecpeKm06 3pe1iUSl?
. 14. l{mo lia3bl6alOm YZJl.OM 3peliUSl? KaKoe 3liaileliUe UMeem YZOJl. 3peliUSl 6
3pUmeJl.bliOM 60cnpUSlmuu? llpu KaKOM YZJl.e 3peliUSl MeJICiJy paCCMampU6ae.Mbl-
MU mOIlKaMU npeiJ.Mema ZJl.a3 6uoum amu mOIlKU pa30eJl.bHo?
15. KaK ycmpoeli u iJ.n.R Ilezo CJl.Y3ICum cpomoannapam? npoeKquoHHblU an.
napam? H a KaKOM paccmOSlliUU nOMel4alOm npeiJMem no omlioweliulO K ux
06'OeKmu6aM?
493

16 C KaKOU Jl.Uli30U .MOJICH.O Cpa6H.Umb oeucm6ue 06'OeKmU6a tpomoanna-
pama? KaK060 lia3liaileliUe ouatppaZ.MbL tpomoannapama? 3am60pa? II OlleMY
Y.MelibWeliue ouatppaZ.MbL tpomoannapama YJl.Yllwaem KallecmBO CIiU.MK06?
17. B lte.M cocmOSlm cxoocmBo ZJl.a3a c ljJomoannapamo.M U pa3Jl.UIlUe .MeJIC-
oy IiUMU?
18. J( Jl.Sl ttezo CJl.YJlCum KOlioeHCOp 6 npOeKijUOIiIiO.M annapame? l[ eM omJl.U-
llaemcSl 3nunpoeKmop om ouanpoeKmopa?
19. l[ mo lia3bL6a1Om omIiOCUmeJl.blibL.M omBepCmUe.M 06'OeKmuBa? CBemocu-
Jl.OU 06'OeKmU6a ljJomoannapama?
20. l[mo lia3bLBa1Om Jl.ynou? J(Jl.Sl Ilezo olia npeOlia3Iialle'lta? no KaKOU ljJop-
.MYJl.e onpeOeJl.SlemCJl Jl.UlieUlioe Y6eJl.UlleliUe Jl.ynbL?
21. KaKue cyecm6YlOm cnoc06bL oqeliKu 3liepZUU C6em060ZO U3Jl.YlleliuSl2
B lle.M cocmoum oc06eliliocmb 6u3yaJl.bliOU OijeliKu?
22. l[ mo lia3bL6a1Om CBem06bL.M nOmOKO.M? CUJl.OU c6ema? B KaKUX eouH.u'
qax OIiU 6bLpaJICalOmcSl?
23. l[mo maKoe oC6eeliH.OCmb? B iteM cocmOSlm 3aKOlibL oC6ew,eH.H.Ocmu?
1 [ 1 1 1
D=-=(n-l) -:t-
F Rl R;,
D = Dl + D 2
1 1 1
+-+-=+-
-d-' -F
r= H =1..
h d
r= do
F
r=r 6 .r = Bdo
o OK F. F.
06' OK
w
p=-
t
494
OCHOBHblE «I>OPMY JIbl
- OnTHqeCKa.H CHJla nHH3bI, r,z:te F - <POKYC-
Hoe paCCTO.HHHe JlHH3bI, n - OTHOCHTeJlbHbIH
nOKa3aTeJlb npeJlOMneHH.H, Rl Ii R 2 - pa,AliYCbI
KpHBH3HbI nOBepXHOCTeH nHH3bI.
- OnTHqeCKa.H CHJla CHCTeMbI ,z:tBYX JlHH3, CJlO-
meHHbIX BnJlOTHYIO.
- <popMYJla TOHKOH nHH3bI, r)J;e d Ii , - pac-
CTO.HHH.H OT nHH3bI )J;O npe,z:tMeTa H )J;O H306pa-
meHH.H COOTBeTCTBeHHO.
- JlHHeHHOe YBeJlHqeHHe JlHH3bI, r,z:te H H h -
JlHHeHHbIe pa3MepbI H306pameHH.H H npe,z:tMeTa.
- YBeJlHqeHHe nynbI, r,z:te do - paCCTO.HHHe
HaHJlyqmero 3peHH.H, F - <PoKycHoe paCCTO.H-
HHe JlynbI.
- JlHHeHHOe YBenHqeHHe MHKpocKona, r,z:te r 06
H r OK - YBenHqeHHe 06'beKTHBa H OKYJI.Hpa.
F 06 H F OK - <POKYCHbIe paCCTO.HHH.H 06'beKTHBa
H OKYJI.Hpa, B - paCCTO.HHHe Memn;y 3a,n;HHM
<POKYCOM 06'beKTHBa H nepe,z:tHHM c1>OKYCOM OKY-
n.Hpa.
- nOTOK H3nyqeHH.H Qepe3 nJlOa,z:tKY S, r,n;e
W - CBeTOBa.H 3HeprH.H, npome,z:tma.H qepe3 nno-
a,z:tKY 3a BpeM.H t.

P
R =-
e S
P
I =-
e
ro
W
<1>=
t
<1>
1=-
ro
<1>
E=-
S
«t>
R=-
S
<1>
B ---
- -
roS n ro S cosa
E = I
r 2 cos a
- H3nyqaTeJlbHOCTb, r,n;e P - nOTOK H3J1yqeHH.H, Hcnyc-
KaeMoro nOBepXHOCTblO, S - nJlOa,n;b CeqeHH.H, qepe3
KOTopoe 3TOT nOTOK npoxo,n;HT.
- CHna H3J1yqeHH.H. r,Ae ro - TeJleCHbIH yron. B npe,n;e-
Jlax KOToporo pacnpOCTpaH.HeTC.H nOTOK H3J1yqeHH.H.
- CBeTOBOH nOTOK, r,n;e W - CBeTOBa.H 3HeprH.H, nepe-
HOCHMa.H qepe3 KaKYIO-JlH60 nJloa,n;KY 3a BpeM.H t, one-
HHBaeMa.H BH3yaJlbHO.
- CHJla CBeTa, rJIe <1>
HbIH yron.
CBeTOBOH nOTOK. ro - TeJleC-
- OCBeeHHOCTb nOBepXHOCTH, r,n;e <1> - CBeTOBOH no-
TOK, na,n;alOHH Ha nOBepXHOCTb nnoa,n;blO S.
- CBeTHMOCTh, r,n;e <1> - cBeToBoH nOTOK, H3J1yqaeMbIH
nOBepXHOCTblO CBeT.Hw;erOC.H TeJla, S - nJlOa,n;b 3TOH
nOBepXHOCTH.
<1>
- .HpKOCTb, r,n;e Sn = S cos ex - npOeKnH.H nJlOa,n;KH S
Ha nJlOCKOCTb, nepneH,n;HKY n.HpHYIO HanpaBneHHIO Ha6J11O-
,n;eHHSI, 0 - yroJl Mem,n;y HOpMaJlblO K nOBepXHOCTH H
HanpaBneHHeM Ha6J1IO)J;eHHSI, CP - H3J1yqaeMbIH CBeTOBOH
nOTOK.
- 0606w;eHHbIH 3aKOH OCBeeHHOCTH, r,n;e r - paCCTO.H-
HHe OT HCTOqHHKa )J;O oCBeaeMoH nOBepXHOCTH, E -
OCBeeHHOCTb nOBepXHOCTH, ex - yroJl na,n;eHH.H CBeTO-
Boro nyqKa, I - CHJla CBeTa HCTOqHHKa.
METO,lUlqECKIiE PEKOMEHAAQIIII
ITpH pemeHHH 3a,n;aq Ha nOCTpoeHHe Ii paCqeT H306pameHHH B J1HH3aX pe-
KOMeH)J;yeTC.H:
1. AJI.H Kam)J;oro nOJlOmeHH.H npe,n;MeTa HymHo nOCTpOHTb H306pameHHe.
.n:JI.H 3Toro qae Bcero 6epYT JlyqH, napaJlnenbHbIe rnaBHoH OnTHqeCKOH OCH
(npeJlOMJleHHbIe H,n;YT qepe3 3aHHH CPOKYC c06HpalOeH JlHH3bI HnH npo,n;oJl-
meHH.H npeJlOMJleHHbIX JlyqeH npoxo,n;.HT qepe3 nepeHHH CPOKYC pacceHBalOllIeH
JlHH3bI), Ii JlyqH. H,n;yHe qepe3 OnTHqeCKHH neHTp (HX HanpaBJleHHe He Me-
H.HeTC.H).
2. 3anHcaTb CPOPMYJlY nHH3bl, 06paTHB oc060e BHHMaHHe Ha BbI60p 3HaKOB
nepe,ll. d, f Ii F; CJle,n;yeT nOMHHTb, qTO Bce paCCTO.HHH.H ,n;o MHHMbIX TOqeK
Ha,n;o 6paTb co 3HaKOM "MHHYC".
495

3. 3anHcaTb cpOPMYJlY nHHeHHoro YBeJlHqeHH.H.
4. 3anHcaTb COrJlaCHO ycnoBHIO 3aaqH onOnHHTeJlbHbIe ypaBHeHH.H, KO-
TopbIe, KaK npaBHJlO, CB.H3bIBaIOT paCCTO.HHH.H, BXO.HHe B OCHOBHbIe ypaB-
HeHH.H, H MorYT 6bITb JlerKO nOJlyqeHbI H3 aHanH3a qepTema.
5. PellIHTb COBMeCTHO COCTaBneHHbIe ypaBHeHHH OTHOCHTenbHO HCKOMOH
BenHqHHbl.
TIPIIMEPbI PEllIEHII.H 3A.n;A q
1. Ha pHCYHKe 16.62 nOK838H XO nY'l8 o H nocne ero npenOMneHHSI B p8cce-
HBalOll.\eH nHH3e. HaH,AHTe nOCTpoeHHeM nonomeHHe rnaBHblX <l>OKYCOS nHH3bl.
PeweHHe. IIpoBe,AeM n060'lHYIO OCb M'M ' (pHC. 16.63), n8pannenbHYIO na-
,AalOll.\eMY nyqy AB. B pacCeHS81Oll.\eH nHH3e Bce nY'lH. napannenbHble 8HHOH OCH
M'M', nocne npenOMneHHSI nOH,AYT TaK. 'lTO HX npO,AOn:>KeHHSI nepeceKYTCSI B O,A-
HOH TO'lKe - n060'lHOM <l>oKyce F'. Cne,AOBaTenbHo, ecnH npO,AOnmHM npenOMneH-
HblH nyq BC ,AO nepeCe'leHHSI C n060'lHOH OnTH'leCKOH OCblO M'M', TO H8H,AeM no-
60'lHbIH <l>OKYC F'. rrpoBeeM 'lepe3 Hero (l>oKanbHYIO nnOCKOCTb, nepneH,AHKynSlp-
HYIO rnaBHOH OnTH'leCKOH OCH M M, H TO'lKa nepeCe'leHHSI 3TOH nnOCKOCTH C rnaB-
HOH OCblO H eCTb rn8BHbIH nepe,AHHH <l>OKYC nHH3bI F. TaK KaK Bce <l>OKYCbI ne-
maT B <l>oKanbHOH nnOCKOCTH, a rnaBHbIH (l>OKYC - Ha rn8BHOH OnTH'leCKOH OCH.
A
B
M' F'- B---- F"
M M --- M"
-- --- ---
-- ---
M F --- --. F M
M" M'
Puc. 16.62
Puc. 16.63
,ZJ;nSl HaxomlleHHSI 3allHero <l>oKyca npOSOHM n060'lHYlO OCb M" M". n8pannenb-
HYIO npenoMneHHoMY nyqy BC. TIpO,AOn:>KHM na,A81Oll.\HH nyq AB ,AO nepeCe'leHHSI
C OCblO M"M" H HaXO,AHM cooTBeTcTBYIOll.\HH n060'lHb1H (l>OKYC F U . TIPOBO,AHM (1)0-
KanbHYIO nnOCKOCTb H HaXO,AHM 38,AHHH n060'lHbIH (l>OKYC F.
2. IIpeAMeT BbICOTOH 4 M HaXO,AHTCH Ha paCCTO-
SlHHH 6 M O'I' OnTH'leCKOrO eHTpa c06HpalOll.\eH nHH-
3bl C (l>OKYCHbIM p8CCTOSlHHeM 2 M. Onpe,AenHTe Sbl-
COTY H306pameHHSI npe,AMeTa. IIocTpoHTe XOll nY'leH.
h ,ZJ;aHo: h=4 M, d=6 M, F=2 M.
H8HTH: H.
Pew e H He. IIocTpoeHHe H306pameHHSI H npe,A-
MeT8 h npHBe,AeHO Ha pHCYHKe 16.64. <1>opMyna
1 1 1 Fd
TOHKOH nHH3bI F = d + f' OTKY,Aa f = d _ F ' YBe-
Puc. 16.64
496

nH'leHHe nHH3bI r - H C ,ApyroH CTOpOHhI, r = - d f. Cne,AOBaTenhHO, H f I1o-
- h . h d
CT8BHM CIO,A8 HaH,AeHHOe 3Ha'leHHe f, nonyqHM
H F
J;- d-F '
F
Puc. 16.65
F
hF
OTK y "a H=-' H=2 M.
,... d - F'
3. I1pe,AMeT HaXO,AHTCSI Ha p8CCTOSlHHH
10 CM OT nHH3bJ C OnTH'leCKOH cHnoH
-5 ,AnTp. Ha K8KOM paccToSlHHH 'lenOseK.
rnSl,ASI Qepe3 3TY nHH3Y, YBH,AHT H306p8meHHe
npe,AMeT8 H K8KHM OHO 6Y,AeT?
,IJ;aHo: D=-5 ,AnTp, d=0,10 M.
HaHTH: f. r.
PeweHHe. TaK KaK D<O, TO nHH3a -
p8cceHBalO8S1 (pHC. 16.65).
,IJ;nfl pacceHBalOeH nHH3bI q,OPMY n8 TOH-
KOH JIHH3bl HMeeT BH,A
1 1 1
--- = -- =-D
d f F
OTCIO,A8 paccToSlHHe OT nHH3bl ,AO H306pameHHSI
d
f = 1 + dD .
IIo,AcTaBHB 'lHCnOBble 3H8'leHHSI. nonY'lHM f = 6 CM. JIHHeHHoe ysenH'leHHe
f
r = - ; r = 0.6.
d
4. Tpe6yeTcSI cq,oTorpa<l>HpoBaTb KOHhK06ema, np06eralOero nepe,A annap8-
M
TOM CO CKOpOCTblO 10 -. Onpe,AenHTe MaKCHM8nbHO ,AOnYCTHMYIO 3Kcn03HHIO npH
C
ycnoBHH, 'lTO p83MblTOCTb H306p8meHHSI He ,AOnmH8 npeSblW8Th 0,2 MM. <1>OKYC-
Hoe paCCTOSlHHe o6eKTHBa 10 CM, paCCTOSlHHe OT KOHbKo6ema ,AO annapaTa 5 M.
B MOMeHT <l>oTorpaq,HpoBaHHSI rnaBHaSl OnTHqeCK8S1 OCh o6eKTHsa 8nn8paTa nep-
neH,AHKynSlpHa K Tp8eKTopHH ,ABHmeHHSI KOHbK06em&..
M
,lI;aHo: v=10 -, H=0,2.10- 3 M, F=0,10 M, d=5 M.
C
HaHTH: t.
Pew e H He. 38 BpeMSI 3Kcno3HHH TO''lK8 A CMeCTHTCH OTHOCHTenhHO rnaBHOH
OnTHt.IeCKOH OCH Ha p8CCTOSlHHe h = v t. Tor,Aa. H306p8meHHe Al TO'lKI1 A CMe-
CTHTCSI Ha p8CCTOSlHHe H. KOTopoe no ycnoBHIO H::5 0,2 MM. TaKHM 06pa30M,
H - 3TO H306pameHHe h. YBenH'leHHe 06eKTHBa  = 
H f
IIo3TOMY - = -
vt d
113
1 1 1 Fd
q,opMynbI nHH3bI - = - + - HaHeM f = -. TOr,A8
F d f d-F
H Fd F H (d - F)
- - - - OTKY,A&' t = ; t =: 0.001 c.
vt - (d - F)d - d - F' Fv
497

D 1 = (n -1 ) ( .!.. - .!.. J = - n -1 .
00 R R
nJIOCKoBbInYKJIOH D 2 = (n -1 >( : +  J = n; 1 ,
Puc. 16.66
)J;J! S!
5. JIHH3bl 1 H 2 CJJ;enaHbl H3 OJJ;Horo copTa CTeKna
(pHC. 16.66). HaH,AHTe OIITH'leCKYlO cHny nHH3bl 1. 3HaSl,
'lTO J1HH3a 2 HMeeT OnTH'leCKYlO cHny 2,25 ,AnTp.
)J; 8 H 0: D 2 = 2.25 JJ;nTp.
HaHTH: DI'
P e ill e H He. M3 pHCYHKa 16.66 BH,AHO, 'lTO BlInOTHYlO
cnomeHbl ,ABe nHH3bl: 1 - IInOCKOBorHYTafl 11 2 - nnoc-
KOBbmYKnaSl C O,AHHaKOBbIMH pa,AHycaMH KpHSH3Hbl c€}>epH-
'leCKHX 1I0BepxHocTeH.
)J;nSl IInocKoBorHYToH nHH3 OIITH'leCKaSl CHna paCC'lHT-
B8eTCSI no <l>opMyne
OTClO,Aa
D
-!. = -1'
n '
2
D 1 = -D 2 = -2.25 ,AnTp. 3a,A8'lY MomHO pelllHTb HHa'ie. ecnH Y'leCTb, 'lTO OIITH'leC-
KaSl CHna CHCTeMbl nHH3 D CHCT = D 1 + D 2 ; T8K KaK o6e nHH3bl o6pa3YlOT nnOCKO-
lIapannenbHYlO nnaCTHHKY c D CHCT = 0, TO D 1 + D 2 = 0 H D 1 = -D 2 = 2,25 ,AnTp.
6. TIoMecTHS H8 paCCTOSlHHH 20 CM OT nHH3bl CBe'lY. 1I0nY'lHnH Ha 3Kpa.
He H306pameHHe CBe'lH, ysenH'leHHOe B 10 pa3. KaKoe 6Y,AeT YBenH'leHHe, ecnH
BnnOTHYlO K ,A8HHOH lIHH3e npHnomHTb nHH3Y C OnTH'leCKOH cHnoH 2.5 lInTp?
-2,5 ,AnTp?
)J;aHo: d=0.20 M. r=10. D 1 =2.5 lInTp. D 2 =-2.5 1I1ITp.
HaHTH: r 1 . r 2 .
1 1
P e ill e H H e. IIo €}>opMyne TOHKOH nHH3bl Do = d + f' l',Ae Do - OIITl1'leCKaSi
nepsoH nHH3bI. JIHHeHHoe YBenH'leHHe f = 1.... OTClO,Aa f = fd
d
cl1na
Tor1I8
1 1
Do = - + - . OIITHQeCKaSl CHn8 ,ABYX c06HpalOHx nHH3 D' = Do + Dl' II03TOMY
d rd
1 1
Do + Dl = - + -. ITO,ACTaBHM BMeCTO Do HaHJJ;eHHoe 3Ha'leHl1e;
d rId
1 1 IlL
-+- + Dl = -+-. OTKY,Aa f) =
d rd d rId d Ud +D. J
r
r 1 = I, 7.
1 + rdD l '
OIITH'leCKaSl CHna BO BTOpOM cnyqae D" = Do - D 2 , l'JJ;e D 2 - MOJJ;ynb OIITJ.i-
'leCKOH CJ.ilIbl. TOI'JJ;a
1 1 1 1 r
D" = - + -' Do - D 2 = - + -. Cne,AOB8TenbHo, r 2 = r 2 = 2.5.
d r 2d ' d r 2d 1 - rdD 2
498

7. K8KOH He,AOCT8TOK apeHHH Y 'lenOBeK8. ,AnSI KOToporo p8CCTOSlHHe H8Hny'l'
Illero apeHH.H 12,5 CM? K8K 3TOT He,AOCT8TOK MomHO Hcnp8BHTb?
.D:8HO: d}=0,125 M, d o =0,25 M.
H8HTH: D.
Pew e H H e. T8K K8K d} < do. r,Ae do - p8CCTOSlHHe H811nY'lillerO 3peHHSI y HOp'
M8nbHoro rn838, TO y 3Toro 'lenOBeK8 6nI130pYKOCTb.
1 1
TIpHMeHI1M K rn83Y <l>opMyny TOHKOH nHH3bl  + f = D 1 , rlle , - p8CCTOSl-
HHe OT OnTHqeCKOrO eHTp8 rn838 ,AO CeT'l8TKH. DI - OnTI1'leCK8S1 CHn8 rn838.
QT06bl BOCnOnHI1Tb He,AOCT8TOK 3peHI1S1, 'lenOBeKY HY>KHbl O'lKI1 C p8CCeI1B81Oll.\ei'i
nl1H30H T8KOH OnTH'leCKOH CHnbl, 'lT06bl nY'lI1, n8,A81Oll.\l1e OT TO'leK, y,v;8neHHblx
H8 p8CCTOHHI1e do, cpOKycl1poBanHcb CI1CTeMOH "O'lKH-rn8a" H8 CeTqaTKe. Cne,v;o-
1 1
B8TenbHO, - + - = D 1 + D, r,v;e D - OnTH'leCK8S1 CHn8 nl1H3bl O'iKOB. Bbl'lHT8S1
do ,
113 BToporo YP8BHeHHSI nepsoe, nonY'lHM
1 1
- - - = D.
do 
ITO,ACT8BHB 'll1CnOBbie 3H8'leHHSI. nonyqHM D = -4 ,AnTp. T8KHM 06P830M. He06-
XO,AHM8 p8cCeHB81Oll.\8S1 nl1H38 C OnTHqeCKOH cHnoH D = -4 ,v;nTp_
8. C nOMOli..\blO <poT08nn8p8T8 C 06'beKTI1BOM, OnTH'leCK8S1 CI1n8 KOToporo
10 ,AnTp, <l>oTorp8<1>HPYIOT npe,AMeT, H8XO,ASlli..\HHCSI H8 ,v;He Bo,v;oeM8, rny6HH8 KO-
Toporo 1,2 M. K8KOBO p8CCTOSlHHe Mem,v;y 06'beKTHBOM H nneHKoH? 06'beKTHB p8C-
nOnO>KeH H8 P8CCTOSlHHH 0,50 M OT nOBepXHOCTH SO,Abl.
.D: 8 H 0: D = 10 ,AnTp, h = 1,2 M, h 2 = 0,50 M.
H8HTI1: ,.
Pew e H He. <1>oT08nn8p8T 6Y,AeT <l>oTorp8<1>Hpo-
B8Tb He C8M npellMeT 8. 8 ero H306p8meHI1e 8 1
(pHC. 16.67). KOTopoe H8XOllHTCSI H8 p8CCTOSlHHH
Puc. 16.67
h
 =-,
n
(CM. pellleHHe
p8CCTOSlHHe OT
r,v;e n - nOK838Tenb npenOMneHI1H SO,Abl
38,A8'lH 5 113  16.8). Cne,AOB8TenbHo,
8 1 ,v;o 06'beKTI1B8 <l>oT08nn8p8T8
h h+n
d=+h2=-4-h2= .
n n
lid
TIo <popMyne TOHKOH nHH3bI d + f = D ::::) f = dD -1 ' ITO,ACT8SI1B aH8qeHHe d,
nonY'lHM
h + nh 2
f = ; f = 10 CM.
D(h + n)- n
9. C K8Koro H8H60nbillero p8CCTOSlHHSI p83Be,A'lHK MomeT 38MeTI1Tb HOqblO oro-
HeK n8nHpOCbl HenpHHTenbCKoro COn,A8T8, ecnH CHn8 CBeT8 n8nl1pOCbl npH CHnb-
HOM 3aTSlrHS8HI111 2.5. 10- 3 K,A, H8HMeHblllHH cBeTosoH nOToK, BocnpHHHM8eMbIH
rn830M, p8seH 1. 10- 13 nM H nnOli..\8,Ab nOBepXHOCTH 3p8'lK8 rn838 B TeMHOTe
0,4 CM 2 ?
499

,I:(8HO: 1=2,5 .10- 3 K, <1>=1.10- 13 JIM. 8=0.4.10- 4 M 2 .
HaHTH: r.
t'I>
Pew e H H e IIo onpe,AeJIeHHIO OCBeeHHOCTb E = S' OTCIO,Aa cBeToBoH nOTOK,
1
nOna,A8IOHH B rna3 <I> = E8. IIo nepBOMY 3aKOHY OCBeeHHOCTH E = 2' Cne,Ao-
r
BBTeJIhHO, ct> =( :. )8. OrKY,IIB r = I : ; r= 1 .10' M.
10. Ha KaKoH BbICOTe Ha.A 'lepTemHoH ,AOCKOH Cne,AyeT nOBeCHTb naMny MOHO-
CTbIO 200 BT, 'lT06bI nonyqHTb OCBeeHHOCTb ,AOCKH no,A n8MnoH, paBHYIO 50 nK?
nM
CSeTOOT,A8'la naMnbI paBH8 12 BT ' Yron HaKnOHa ,AOCKH K ropH30HTY a = 30°.
nM
,I:(aHo: P=200 BT, E=50 nK, L=12 BT ' a=300.
H8HTH: h.
Pew e H H e. CBeTOOT,AaqeH naMnbI H83bIB8IOT OTHOllIeHHe nonHoro CBeTOBoro
<I>
nOTOK8 <1>. H3nYQ8eMOrO JlaMnOH, K ee sneKTpH'leCKOH MOHOCTH L = -. JIaMny
P
MomHO p8CCM8TpHB8Tb KaK TOQe'lHbIH IiCTO'lHHK CBeT8. TI03TOMY nonHbIH cseTO-
1
BOH nOTOK <I> = 41tl. 11:3 o606eHHoro 3aKOHa oCBeeHHocTH E = 2 cos a. r,Ae
h
a - yron Mem,Ay HanpaBneHHeM nY'leH, na.AaIOHX Ha ,AOCKY, H HOpMMbIO K HeH,
p8BHbIH yrny HaKnOHa ,AOCKH (KaK yrnbI C COOTBeTCTBeHHO nepneH,AHKynSlpHbIMH
cTopoHaMH). IIO,ACTaBHM <I> B nepBYIO «}>opMyny: L = 47tl  1 = LP . IIO,ACT8BHM
P 47t
3Ha'leHHe CHnbI CBeTa B 3aKOH OCBeeHHOCTH:
LP
E =-cosa
47th 2
1
A
Puc. 16.68
500
2
LPcosa
Cne,AOBaTenbHo, h = ; h = 1,8 M.
47tE
11. H8,A ropH30HT8nbHbIM CTonOM Ha BbICOTe
2,0 M H Ha paCCTO.HHHH 1.0 M ,Apyr OT ,Apyra no-
MeeHbI ABa HCTO'lHHKa CBeTa, ,AaIOHe CBeTOBbIe
nOTOKH no 300 nM KamAbIH. Onpe,AenHTe OCBeeH-
HOCTb H8 CTone B TO'lKaX no,A HCTO'lHHKaMH CBeT8
H H8 Cepe,AHHe p8CCTOSlHHSI Mem,Ay HHMH.
,I:(8HO: D=2.0 M, l=I,O M, <1>1=<1>2=<1>=
= 300 nM.
H8HTH: EA' E s .
Pew e H H e. OCBeeHHocTb E A B TO'lKe A
(pHC. 16.68) p8SH8 CYMMe oCBeeHHocTeH, C03,A8-
BaeMbIX K8m,AbIM H3 HCTO'lHHKOB B OT,AenbHOCTH
E A = E Al + E A2' OT HCTO'lHHKa 1 B TO'lKY A CBeT

I
naeT HopM8nbHO, n03TOMY E A. = 2' OCBeeHHocTL B TO'lKe A, C03,A8B8eM8S1 HC-
D
I
TO':lHHKOM 2, corn8CHO 06'Le,AHHeHHoMY 3aKOHY OCBeeHHOCTH p8BH8 EA, = 2 cos a,
r
r,Ae p8CCTOSlHHe r OT HCTO'lHHKa 2 /1,0 TO':lKH A MomHO H8HTH no TeopeMe IIHcpa-
rop8:
r 2 = Ii- + l2 H cos a = D = D .
r  D 2 + l2
I I D
Tor,Aa E A = 2 + 2 2' .J . TaK K8K HCTO':lHHKH CBeT8 TO'le'lHLIe, TO
D D + l D 2 + l2
nonHbIe CBeTOBbIe nOTOKH, ,AaB8eMLIe 3THMH HCTOqHHK8MH, c1> = 41tl. OTKY,A8
c1>
1=-.
4n
Cne,AOB8TenbHO,
E A = <t> [  + D 3 ] ; EA = 10 nK. OCBeeHHocTb B
4n D (D 2 + l2)"2
TO'lKe B p8BHa
I
E B = E + EBI = 2E = 2 ,.,2 cos,
1
r,Ae
T.=FGJ=
4D2 + l2
2
D 2D
cos = r 1 = 4D2 + l2
E - _ 2 c1> 2D
TOr,A8
B 4D 2 +l 2 / 4D 2 +l 2
4n "
4
4c1>D
E B = 11 nK.
3 '
n(4D2+l2)"2
501

17. CBETOBbIE BOJIHbI
CBeT KaK 3JIeKTpOMarHHTHaH BOJIHa. CKOPOCTL CBeTa. HHTepepeHuH
CBeTa, KorepeHTHOCTL. CneKTpaJILHOe pa3JIOjKeHHe npH HHTeP<l>epeHHH. Me-
TO,abl Ha6JIIO,aeHHH HHTep4>epeHHH CBeTa. UHTep4>epeHJJ;HH CBeTa B TOHKHX
nJIeHKax. npHMeHeHHe HHTepepeHUHH B TeXHHKe.
,lI;H4>paKHH CBeTa. npHHuHn rIOHreHca-<lJpeHeJIH. MeTon 30H <lJpeHeJlH.
.n:H4>paKHoHHaH peWeTKa. .n:H4>paKHoHHbIH cneKTp. nOHHTHe 0 roJIOrpa4>HH.
nOJlHpH3aHH CBeTa. 3aKOH MaJllOca. ,lI;HcnepCH.H H nOrJIOeHH CBeTa.
3aKOH Byrepa. )];HcnepcHoHHLIH cneKTp. CneKTpocKon. CneKTpLI HcnYCKaHHH
H nOrJIOeHHH. CneKTpaJILHLIH aHaJIH3.
illKaJIa 3JIeKTpOMarHHTHLIX BOJIH. 3JIeKTpOMaI'HHTHble H3JIyqeHHH pa3HLIX
,aJIHH BOJIH - pa,aHOBOJIHbl, HHpaKpacHoe H3JIyqeHHe, BH,aHMoe H3JIY'leHHe,
YJILTpaHOJIeTOBOe H peHTreHOBCKoe H3JIY'leHHe, raMMa-H3JIyqeHHe. CBOHCTBa
H npHMeHeHHe 3THX H3JIY'leHHH.
17.1. CBeT KaK 3JIeKTpOMarHHTHaH BOJIH3
K KOHY XIX CT. cpH3HKa HaKOnHJIa 1I0CTaTOqHO MHOrO 3KCnepHMeHTanb-
HbIX aHHbIX, CBHeTenbCTBYIOHX 0 B3aHMOCB.H3H CBeTOBI:.IX, 3J1eKTpHqeCKHX H
MarHHTHblX .HBJleHHiL 3TO n03BOJlHnO B 70-x rr. XIX B. .D:m. MaKcBeJlJlY co-
3aTb 3neKTpOMarHHTHYIO TeopHIO nOJl.H. OH ,AOKa3aJI, '1TO B npHpoe onmHbI
cyecTBoBaTb 3J1eKTpOMarHHTHbIe BOJlHbI. MaKcBeJlJl paCCqHTaJl CKOpOCTb pac-
npOCTpaHeHH.H 3J1eKTpOMarHHTHI:.IX BOJIH B BaKYYMe H B cpee:
c
v=  '
re c:::: 3.10 8
M
- CROpOCTb HX pacnpocTpaHeHH.H B BaKYYMe, £ H Jl - H-
C
3J1eKTpHqeCKa.H H MarHHTHa.H npOHHlJ,aeMOCTH cpebI. COBna,AeHHe 3HaqeHH.H
CKOpOCTH pacnpocTpaHeHH.H 3JIeKTpOMarHHTHblX BOJIH co CKOpOCTblO CBeTa npH-
BeJlO MaKcBeJlJla K MbICJlH, qTO CBeT - 3TO 3neKTpoMarHHTHbIe BOJlHbI. n03-
>Ke r. replJ, Ha onbITe nOJlyqHJI H HCCJle,AOBaJl 3J1eKTpOMarHHTHble BOJIHbI, Ii
pe3YJIbTaTbI 3THX onbITOB nO,ATBep,AHnH TeopHIO MaKcBeJlna. OKoH'.IaTenbHO
3J1eKTpOMarHHTHa.H TeOpH.H YTBeplIHJIaCb nocJle 06HapymeHH.H n.H. Jle6elIeBbIM
,AaBJleHH.H CBeTa Ha TeJIa, pacnOJlO>KeHHbIe Ha nYTH HX pacnpocTpaHeHH.H.
TaKHM 06pa30M, BOJlHOBa.H TeOpH.H rIOHreHca-<1>peHeJl.H 0 npHpoe CBeTa
3BOJlIO[(HOHHpOBaJla B 3JIeKTpOMarHHTHYIO TeopHIO CBeTa. Cor JIaCHO 3TOH Teo-
pHH, CBeT - 3TO 3J1eKTpOMaI'HHTHble BOJIHLI onpe,aeJIeHHOrO OnTH'IeCKOrO ,lI;H-
ana30Ha 3 .10 9 - 3.10 19 flJ,. OnTllqeCKOe H3J1yqeHHe B npeenax JlHH BonH
OT 760 HM o 380 HM cnoc06HO HenOCpe,ACTBeHHO BI:.I3bIBaTb 3pHTeJlbHOe OIlJ,Y-
lIJ,eHHe B qeJlOBeqeCKOM rJIa3Y. CJIeOBaTenbHo, OHO .HBJI.HeTC.H BH,AHMbIM. OnTH-
qeCKOe H3JIyqeHHe C A > 760 HM Ha3bIBaeTC.H IIHcppaKpacHbIM, a C A < 380 HM -
Y JIbTpacpHOJIeTOBbIM.
KaK JII06bIe 3J1eKTpOMarHHTHbIe BOJIHbI, CBeTOBbIe BOJlHbI xapaKTepH3YIOTC.H
- -
Hanp.H>KeHHOCTblO E 3J1eKTpHqeCKOrO nOJI.H H MarHHTHOH HH.n;YKIJ,HeH B Mar-
502

HHTHoro nOJI.H BOJlHbI. ijo, KaK nOKa3aJIH liCCJIe.n:OBaHH.H. BCe OnTliqeCKlie .HB-
JIeHli.H BbI3bIBaIOTC.H .n:ei%cTBlieM 3JIeKTpHqeCKOH COCTaBn.HlOeH 3neKTpOMarHHT-
HOrO nOJI.H Ha 3J1eKTpOHbI aTOMOB BeeCTBa, n03TOMY CBeTOBbIe BOnHbI Onli-
CbIBaIOTC.H 06bIqHO ypaBHeHHeM
E = Eo COS( wt _ 2r )-
r.n:e Eo - aMnJIliTy.n:a Hanp.HmeHHOCTli, (0 - U:liKnliqeCKaSl qaCTOTa, A - .n:nli-
Ha BOJlHbI CBeTa, r - paCCTOSlHHe .n:o liCTOqHliKa CBeTa.
17.2. CKOpOCTh CBeTa
CKOpOCTb CBeTa B BaKYYMe - o.n:Ha H3 HaH60JIee BamHblX cpli3HQeCKliX
KOHCTaHT. Ee onpe.n:eJIeHHe HMeJlO orpOMHoe 3HaqeHlie .n;JI.H no.n:TBepm.n;eHli.H
3neKTpoMarHliTHoH Teoplili CBeTa. A. 3HHIllTeHH npH C03.n:aHHH TeopHH OTHO-
CliTeJlbHOCTH OCHOBbIBaJlC.H Ha He3aBHCHMOCTli CKOpOCTli CBeTa B BaKYYMe OT
CKOpOCTli .n:BlimeHli.H liCTOqHHKa (CM. Hli:>Ke  19.1).
TIocKonbKY CKOpOCTb pacnpOCTpaHeHli.H CBeTa OqeHb BeJlliKa. CBeT 3aTpaQli-
BaeT 3aMeTHoe BpeM.H JlliIllb Ha npoxom.n:eHlie OqeHb 60JIbIllliX paCCTO.HHliH.
CJIe,AOBaTeJlbHO, n.H onpe.n:eJleHliSl CKOpOCTli CBeTa CJle.n;yeT H3Mep.HTb JIli60
OQeHb MaJIbIe npOMemYTKli BpeMeHli, JIli60 aCTpOHOMliQeCKHe paCCTOSlHli.H.
BnepBbIe CKOpOCTb CBeTa li3MepHJI .n:aT-
CKHH aCTpOHOM O. PeMep B 1676 r., Ha-
6JIIO.n;a.H 3aTMeHHe cnYTHHKa IOnHTepa -
110 (pliC. 1 7.1). TIepBoe Ha6J1IO,AeHHe 6bIno
npOBe.n;eHO B TO BpeMSI, KOr.n:a 3eMJI.H, .n:Bli-
ra.HCb BOKpyr COJlHu,a, HaXO.n:linaCb 6JIlime
BCerO K IOnliTepy. TIOBTopHOe Ha6JIIO.n:e-
HHe, npOBe.n:eHHOe qepe3 6 MeC.Hu,eB, Kor-
.n:a 3eMJI.H y.n:aJIHJIaCb OT IOnliTepa npli-
MepHO Ha .n:liaMeTp CBoeH Op6HTbI, nOKa-
3aJIO. qTO 110 on03.n:aJl nO.HBliTbC.H H3 TeHH
IOnHTepa Ha 22 MliH. 3TO 3ana31IbIBaHlie
BbI3BaHO TeM, qTO no PeMepy CBeT TpaTliT
22 MHH Ha npoxom.n;eHHe paCCTO.HHli.H,
npHMepHO paBHoe .n;HaMeTpy 3eMHoH Op6HTbI. Pa3.n;eJlliB 3TO paCCTO.HHHe Ha Bpe-
KM
M.H 3ana3.n:bIBaHliSl, PeMep HaIlleJl CKOpOCTb CBeTa (215000 -). OIllli6Ka 6bIna
C
HaxomlIeHHeM ,AHaMeTpa Op6HTbI. Bnocne.n;cTBHH 6bInH
60JIee TOqHble MeTO,AbI Jla60paTopHbIX li3MepeHHH CKO-

_.-. r
. ...., .-""""...1....
,{ . \ Op6HT8 cnYTHHK8 .
Y " IOnHTep8 
. I
(t ...-  ,
 I =
'3eMJIfl . ::
\ <tI' '" '"
" .... -' -" 0 6 OP6RT8;
p HT8 IOnHTep9:'t
3eMnH
I
\
It-
-.-
"/I
Puc. 17.1
CB.H3aHa C HeTOqHbIM
pa3pa60TaHbI .n:PyrHe,
pOCTH CBeTa.
B 1881 r. A. MaHKeJlbCOH onpe.n:enHJI CKOpOCTb CBeTa C nOMOlIlblO BpalIlalO-
eHC.H BocbMHrpaHHoH 3epKanbHoH npH3MbI (pHC. 17.2). JJ:n.H CBOHX li3Mepe-
HHH MaHKeJlbCOH Bocnonb30BaJlC.H .n:BYM.H ropHbIMli BepIllliHaMli: AHTOHRO H
BHJlbCOH (B KaJIlicpopHRli). paCCTO.HHHe Mem.n:y KOTOpbIMli (35,426 KM) 6bIJIO
TaTenbHO H3MepeHO. Ha BepIllliHe ropbI BHnbcoH 6bIJI YCTaHOBJIeH CHJIbHbIH
HCTOqHHK S, CBeT OT KOToporo, npoxo.n;.H qepe3 eJlb, na.n;aJI Ha BOCbMHrpaH-
HYIO 3epKaJIbHYlO npH3MY A. OTpameHHbIH OT 3epKaJIbHOH rpaHH npH3MbI CBeT
503

nona,AaJI Ha BorHYToe 3ep-
Kano B, YCTaHOBJIeHHOe Ha
BeplIIHHe ropbI AHTOHHO
,D;aJlee CBeT naaJl Ha 3ep-
KaJlO m H. OTpama.HCb OT
HerO, na,naJI Ha ,ApyrylO Toq-
KY 3epKaJla B, nocne tlero
nonaaJl Ha BTOpylO rpaHb
3epKanbHOH npH3MbI A H
OTpamaJlC.H. OTpameHHbIH
CBeT YJlaBJlHBaJIC.H c nOMO-
m;blO 3pHTeJIbHOM Tpy6bI C.
BbIlIIe,AIIIHH H3 eJlH CBeT
Mor nonaCTb B 3pHTeJlbHYlO
TPy6y TOJlbKO npH YCJlOBHH,
eCJlH 3a BpeM.H npoxome-
HH.R CBeTOM paCCTO.HHH.H OT
pacnOJlOmeHHH 3epKaJl HHqerO He H3Me-
B
m
Puc. 17.2
OHOH ropbI o pyroH Ii 06paTHo B
HHJlOCb.
3epKaJlbHaSi npH3Ma A C nOMOI.ltblO MOTopa npHBOJI.HJIaCb BO BpaI.lteHHe. npH-
qeM CKOpOCTb MOTopa perYJlHpOBanaCb TaK, tlT06bI tlepe3 3pHTeJlbHYIO TPy6y
eJlb S 6bIJla BHHa HenpepbIBHO. 3TO MOrJlO 6bITb TOJIbKO npH YCJlOBHH, eCJIH
]
3a BpeM.H nOBopOTa npH3MbJ Ha '8 060poTa CBeT npOXOHJI nYTb, paBHbIH ,ABOM-
HOMY paCCTOSlHHIO Memy BeplIIHHaMH rop. 3Ha.H qHCnO 060pOTOB 3epKaJla B
ceKYH,Ay H npoHeHHbIM CBeTOM nYTb, A. MaMKeJIbCOH HalIIen, '{TO CKOpOCTb
CBeTa B B03yxe v paBHa CKOpOCTH CBeTa B BaKYYMe (c):
c = (299 796 :t 4 ) KM == 3 . 108 M .
C C
KaK nOKa3bIBalOT onbITbl, CKOpOCTb CBeTa B pa3nHqHbIX
Bem;eCTBax HeoH-
KM
HaKOBa. B BOJI.e. HanpHMep. CKOpOCTb CBeTa OKOJlO 225 000 -. B CTeKJIe -
C
KM
== 200 000 -. CKOpOCTb CBeTa B JlI060M BeeCTBe MeHblIIe CKOpOCTIi CBeTa
C
B BaKYYMe. OnTHqeCKHe CBoMcTBa Bem;eCTBa xap8KTepH3yeT a6COJllOTHLIH no-
Ka3aTeJlb npeJlOMJleHUH BeeCTBa. OH nOKa3bIBaeT, BO CKOJlbKO pa3 CKOpOCTb
CBeTa B aHHoM cpee MeHblIIe CKOpOCTH CBeTa B BaKYYMe:
c
n =-.
v
17 .3. "HTepepeHHH CBeTa. KorepeHTHocTb.
CneKTpanbHoe pa3noeHHe npH HHTepepeHHH
MHTep<})epeHHSI .HBJI.HeTC.H xapaKTepHbIM npH3HaKOM BOJlHOBbIX npoeCCOB
JlI060H npHpObI.
MHTep<})epeHH.R: B03HHKaeT npH HaJlOmeHHH KorepeHTHblX BOJIH. Kore-
peHTHble BOJlHbl - 3TO BOJlHbI OJI.HHaKOBoH qaCTOTbI (Vi = V2), C nOCTOSlHHOM
504

pa3HOCTLIO cl>a3 (<P2- <PI = const), KOJIe6aHHSI BeKTopOB Hanp.HmeHHOCTH B KO-
TOpX npOHCXO.HT B OHOH nJlOCKOCTH.
IIycTb B TOqeqHbIX KorepeHTHbIX HCTOqHHKaX 8 1 H 8 2 KOJle6aHH.H BeKTO-
pOB E npOHcxOHT B O,AHHaKOBbIX cpa3ax. BOJIHbl OT 3THX HCTOqHHKOB O TOq-
KH M (pHC. 17.3) pacnpOCTpaH.HIOTC.H B OHOH cpee. OT HCTOqHHK8 8 1 npH-
XO,AHT B TOqKY A1 BOJIHa
M
, r 2
D.r "
8 2
Puc. 17.3
x
E = Eo, cos( wt - 2;,r' )-
OT HCTOqHHKa 8 2 npHxoHT BOJlHa
E = E 02 cos ( wt _ 2';.'"2 )-
re rl Ii r2 - paCCTO.HHH.H OT HCTOqHHKOB 8 1
H 8 2 o TOqKH A1. .l(n.H HaxomeHHH aMnJlH-
TYbI Eo pe3YJlbTHPYIOeH cBeToBoH BOJlHbI B
TOqKe M, B03HHKalOeH B pe3Y JIbTaTe Hanome-
HH.H 3THX BonH, BOCnOJlb3yeMC.H MeTO,AOM BeK-
TOpHbIX ,AHarpaMM (pHC. 17.4):
E = E1 + E2 + 2E o1 E o2 cos (<PI - <P2)' (17.1)
AMnJIHTY,A8 pe3YJlbTHpYIOIltero KOJIe6aHH.H 38-
BHCHT OT pa3HOCTH CP83 (Q)1 - <P2)' P83HOCTb cp83
<Pi - <P2 = l wt - 2';.'", J -l wt - 2';.'"2 J =
2n (r2 - rd
= = D.<p
).
3aBHcHT, KaK BHHO, OT reOMeTpHqeCKOH pa3HocTH xoa (r2 - rl) (eCJIH BOJIHb]
pacnpOCTpaH.HIOTCH B pa3HbIx cpeax, H8,A0 yqHTbIB8Tb OnTHqecKYIO pa3HocTb
xoa. OnTHqeCKOH JlHHOH nYTH Ha3bIBaeTC.H npOH3BeeHHe reOMeTpHqeCKoi1
JIHHb] r nYTH cBeToBoH BOJIHbI B ,AaHHOH cpe,Ae H8 86conlOTHbIH nOK838Tenb
npeJlOMJleHH.H 3TOH cpebI: L = nr. OnTHqeCKa.H pa3HocTb XO8 BOJIH
D. = r2 - nl rl' re n2 Ii n] - a6COJllOTHbIe nOK83aTeJIH npenOMJleHH.H 3THX
cpe). TaK KaK BOJlHbI KorepeHTHbI, TO D.q> HMeeT nOCTO.HHHOe BO BpeMeHH (HO
CBoe ,AJI.H KamOH TOqKH npOCTpaHCTBa) 3HaqeHHe. HHTeHCHBHOCTb CBeTa I npo.
nOpI(HOHaJlbHa I - E 2 , n03TOMY yp8BHeHHe (1 7.1) MomHO nepenHcaTb B BH,Ae
I = II + 1 2 + 2 ..)11 1 2 cos D.<p.
.l(n.H HeKorepeHTHLIx BOJIH pa3HocTb cpa3
Puc. 17.4
D.<p = ( - wd t - 2n [ !L -  l
Al A 2
HenpepBHo H3MeH.HeTCH co BpeMeHeM, n03ToMY cpeHee 3H8qeHHe 38 nepHO
< cos D.q> > = 0, H HHTeHCliBHOCTb pe3Y JIbTHpYIOeH BonHbI BCIOY 0HH8KOB8 Ii
paBHa I = II + 1 2 , H eCJIH II = 1 2 , TO 1=21 1 ,
505

,l];n.H KOrepeHTHLIX BOJIH B TOqKaX npOCTpaHCTBa, r,ne COSD.q> > 0, I> II + 1 2 ,
r,ne COSD.q> < 0, HHTeHCHBHOCTL I < II + 1 2 , CJle,noBaTeJlbHO, npH HaJlOmeHHH
,nBYX KorepeHTHbIx CBeTOBbIX BonE npOHcxo,nHT npOCTpaHCTBeHHoe nepepacn-
pe,neneHHe 3HeprHH [10 BOJIHOBOMY cppOHTY. B pe3YJlbTaTe qero B npOCTpaH-
CTBe 06pa3yeTC.H YCTOHqHBa.H KapTHHa qepe,AOBaHH.H 06naCTeH MaKCHMYMOB H
MHHHMYMOB HHTeHCHBHOCTH. Ho cpe,nHee 3HaqeHHe 3HeprHH BO Bcex TOqKaX
Ha6J1IO,naeMOH KapTHHbI paBHO, KOHeqHO, CYMMe 3HeprHH, npHHOCHMbIX 06eH-
MH BOJlHaMH.
21t (r2 - rd
ECJlI1 pa3HocTb cpa3 D.<p = = m . 21t, TO
A
r 2 -r i =mA, r,Ae m=O, 1,2, (17.2)
TaKHM 06pa30M, eCJlH pa3HOCTb XO,Aa B BaKYYMe paBHa Q,eJlOMY qHCJlY ,nJlI1H
BonH, TO cosD.q> = 1 11 I = II + 1 2 -r 2 ..JIII2 ' EcnH II = 1 2 , TO I = 41 1 , HHTeH-
CI1BHOCTb CBeTa 6y,neT MaKCI1MaJIbHa (KoJle6aHH.H, B036ym,naeMble B TOqKe M o6en-
MH BOJlHaMH, Haxo,n.HTC.H B o,nHHaKoBblx cpa3ax Ii YCHJlHBalOT ,npyr .qpyra).
PaBeHcTBo (1 7.2) eCTb YCJlOBHe HHTep4>epeHHoHHoro MaKCHMYMa.
21t(r 2 - rd
D.<p = = (2m + 1 )1t,
A
ECJlH
TO COSD.q> =-1
Ii
I = II + 1 2 - 2 ..JIII2 '
ECJlH II = 1 2 , TO 1=0, T.e. IiHTeHCHBHOCTb CBeTa MHHHMaJlbHa (KoJle6aHHSl,
B036ym,AaeMbIe B TOqKe M 06e11M11 BonHaMH, Haxo,n.HTC.H B npoTHBocpa3e H oc-
Jla6n.HIOT ,APyr ,nPyra). :3TO 6Y,AeT B TOM cJlyqae, ecnH pa3HOCTb XO,Aa B BaKY-
YMe 6y,neT paBHa HeqeTHOMY qHCJlY nonYBonH:
r,ne m = 0,
HHM)'Ma_
TaKHM 06pa30M. HHTep4>epeHHSI CBeTa - HaJIOeHHe CBeTOBLIX KorepeHT-
HLIX BOJIH, B pe3YJILTaTe KOToporo Ha6JIIO,ZJ;aeTCSI YCTOHqHBaSi BO BpeMeHH
KapTHHa qepe,ZJ;OBaHHSI MaKCHM)'MOB H MHHHM)'MOB HHTeHCHBHOCTH CBeTa.
ECJlH KorepeHTHbIMH HCTOqHI1KaMH
CBeTa .HBJI.HIOTC.H ,nBe Y3KHe napanneJlbHbIe
eJln (pHC. L 7.5), TO Ha 3KpaHe HHTep-
cpepeHUl10HHa.H KapTHHa npe,nCTaBJI.HeT CO-
X m 60H qepe,nOBaHHe CBeTJlbIX H TeMHbIX no-
noc, napaJlJleJlbHbIX ,npyr ,npyry KaK no-
Ka3bIBalOT paCqeTbI (CM. HHme "TIpl1MepbI
pemeHHSl 3a,naq" , 3a,naqa 2), nonomeHHe
CBeTJlbIX nOJloc OTHOCHTeJlbHO TOqKI1 0
onpe,Aen.HeTC.H cpopMynoH
A
r2 - r l = (2m + 1) - ,
2
I, 2, ... PaBeHcTBo (17.3) eCTb YCJlOBl1e HHTep4>epeHHoHHoro MH-
(17.3)
sl
II
SLD
c
:3
mAD
x m =-.
l
Puc. 17.5
506

PaCCTOHHlie Mem,Ay COCe,AHHMIi MaKCHMYMaMIi
AD
!lx = x -x 1 =-.
m m- l
KaK BIi,AHO, /).x He 3aBliCliT OT nOpH,AKa liHTepcpepeHqlioHHoro MaKCliMY-
Ma H .HBJI.HeTC.H nOCTO.HHHOH ,AJI.H ,AaHHbIX A, D H t.
rJIaBHbIH MaKCliMYM, cooTBeTcTBYIOliH m = 0, HaXO,AliTC.H B TOqKe 0 BBepx
Ii BHli3 OT Hero, Ha paBHbIX paCCTO.HHIi.HX ,APyr OT ,Apyra pacnOJIO>KeHbI MaK-
CHMYMbI (MliHliMYMbI) nepBoro (m = 1), BToporo (m = 2) 11: T.,A. nOp.H,AKOB.
3Ta KapTliHa CnpaBe,AJlliBa JlHIIIb npli OCBeeHHH eneH MOHoxpOMaTliqeC-
KliM CBeTOM. ECJlIi liCnOJlb30BaTb 6eJlbIH CBeT, TO HHTepcpepeHl1.HOHHbIe MaK-
CHMYMbI ,AJI.H pa3J1liqHbIX ,AnliH BOJIH cornaCHO cpopMYJIe (17.4) 6Y,AYT CMee-
HbI ,Apyr OTHOCliTenbHO ,Apyra Ii 6Y,AYT liMeTb BIi,A pa,AymHbIx 110JIOC. TonbKO
,AJI.H m = 0 MaKCHMYMbI Bcex ,AJIHH BOJIH COBna,AaIOT H B Cepe,AHHe 3KpaHa 6y-
,AeT Ha6J1IO,AaTbC.H 6eJIa.H nOJlOca, no 06e CTOpOHbI OT KOTOpOH CHMMeTpHqHO
pacnOJlaralOTCH cneKTpaJlbHO OKpaIIIeHHbIe nOJlOCbI MaKCliMYMOB I-ro, 2-ro Ii
T .,A. nOp.H.lI.KOB.
(17.4)
17.4. MeTO.ll;bI Ha6J1IO,lJ;eHHH "HTep4>epeHHH CBeTa
I1CToqHliKIi KorepeHTHbIX BOJIH TaKme Ha3bIBalOTCSJ KorepeHTHbIMIi. ,[(Ba He-
3aBliCliMbIX liCTOqHHKa (KpoMe Jla3epoB) HeKorepeHTHbI. 3TO 06'b.HCH.HeTCH TeM,
qTO li3nyqeHlie TaKliX liCTOqHliKOB npe.lI.CTaB-
JIHeT c060ii CYMMapHoe li3nyqeHlie OT.lI.eJlb-
HbIX aTOMOB, a aTOMbI H3JIyqalOT 6ecnop.H-
.lI.OqHO, Ii n03TOMY HeT KaKliX-JIH60 nOCTO.HH-
HbIX COOTHOIIIeHliH MemJJ;Y cpa3aMH BOJIH, 11:3-
JlyqaeMbIx liCTOtIHliKaMH. TIpOl1.eCC H3JIyqe-
HIi.H aTOMa JJ;JIliTC.H KopOTKoe BpeM.H
('t == 10- 8 C). 3a TaKoe BpeMH aTOM licnYCKa-
eT OrpaHHqeHHbIn l1.yr BOJIH (06pbIBOK CliHY-
COHJJ;bI (pliC. 17.6» HeKoTopoH JJ;n Ii HbI l.
JIliIIIh B TetIeHlie 3Toro BpeMeHIi BonHbI, liC-
nYCKaeMbIe aTOMaMIi, HMelOT npli6JIli3liTenb-
HO nOCTOHHHble aMnJIliTY,AY Ii cpa3Y KOJle6a-
Hlill. 3a 3TO BpeMH (BpeMH KorepeHTHocTIi) BonHa npOXO,AHT paCCTO.HHHe
M
L = c't = 3 10 8 - .10- 8 C =3 M, Ha3bIBaeMoe AJIUHO" KorepeHTHOCTU. Cne,AOBa-
C
TeJIbHO, Ha6nlOJJ;eHlie liHTepcpepeHl1.HH CBeTa B03MO>KHO JIHIIIb npli OnTHtIeCKOll
pa3HOCTH xOJJ;a 6. $ L. TIpH pa3HOCTH XO,Aa 6. > L BCTpeqaTbC.H 6YJJ;YT BonHbI
pa3HbIX l1.yroB, H KapTHHa HaJlOmeHIi.H BOJIH 6YJJ;eT HenpepbIBHO li3MeHHTbCH.
,[(0 nO.HBJleHH.H Jla3epOB BO Bcex npli60pax ,AJI.H Ha6nlO,AeHIi.H liHTepcpepeH-
l1.HIi CBeTa KorepeHTHbIe BOJIHbI nOJlyqaJlIi pa3JJ;eJIeHlieM Ii nocneJJ;YIOliM CBe-
JJ;eHlieM CBeTOBbIX BOJIH, npeJJ;CTaBn.HlOliX qaCTIi O.lI.HOn Ii TOll me BOJlHbI,
H,AYlie OT eJJ;HHCTBeHHoro liCTOtIHliKa Ii JJ;OCTliralOlie TOqKIi Ha6nlOJJ;eHH.H
pa3HbIMH nYT.HMH. TIpaKTliqeCKH 3TO MomHO ocyecTBliTb C nOMOblO 3Kpa-
HOB Ii eJIeH, 3epKaJI Ii npeJlOMJI.HIOliX TeJl.
E
o
x
Puc. 17.6
507

Puc. 17.7
8 2
C 2
A I
Puc. I 7.8
:3
I A
a) Memoo IOliza.
MCTOqHHKOM CBeTa cJlymHT .HpKO OCBe-
m;eHHaSl eJlb 8 (pHC. 17.7), OT KOTOpOH
q>pOHT BOJlHbI na,AaeT Ha .n;Be Y3Kl'le paB-
Hoy.n;aJleHHbIe w;eJlI'I 8 1 1'1 8 2 , napaJIJleJlb-
HbIe II(eJlI'I 8. TaKI'IM 06pa30M. w;eJlI'I 8 1
H 8 2 I'IrpalOT pOJlb KorepeHTHbIx HCTOqHH-
KOB. MHTepq>epeHD;HOHHa.H KapTI'IHa Ha6J11O-
.n;aeTC.H Ha 3KpaHe, pacnOJlOmeHHOM napaJl-
JleJlbHO w;eJI.HM, B 06J1aCTH BC. B D;eHTpe
pacnOJlOmeHa CBeTna.H nOJlOca, 8 no Kpa-
.HM pa.n;ymHble nOJlOCKH.
6) 3epKaJl.a 4>pelieJl.Sl.
CBeT OT HCTOqHI'IKa 8 (pHC. 17.8) pac-
XO.n;.HHMCSl nyqKOM nona.n;aeT Ha .n;B8
nJlOCKI'IX 3epKaJIa, pacnOJlOmeHHbIX .n;pyr
OT .n;pyra no.n; yrJlOM, 6J1H3KHM K 180°.
CBeToBbIe nyqKH, OTpa3HBIIIHeC.H OT 060HX
3epKaJl, MomHO CqHTaTb BbIXO.n;.HI'IMH OT
MHHMbIX I'ICTOqHI'IKOB 8 1 1'1 8 2 , .HBJI.HIOI'IX-
C.H MHHMbIMH H306pameHH.HMH HCTOqHHK8
8 B 3epKaJlax. MHI'IMbIe HCTOqHHKH 8 1 H
8 2 KorepeHTHbI, H I'Icxo.n;.Hl'Ie 1'13 HI'IX CBe-
TOBbIe JIH, BCTpeqa.HCb .n;pyr C .n;pyroM,
HHTepq>epHpYIOT B 06J1aCTH B3aHMHoro ne-
peKpbITI'I.H. MHTepq>epeHD;HOHHa.H KapTHHa
Ha6J1IO.n;aeTC.H H8 3KpaHe :3, 3aHeHHOM
OT np.HMOrO nona.n;aHH.H CBeT8 I'ICTOqHI'IK8
8 Henp03paqHbIM 3KpaHOM K K.
17 .5. IIHTepepeHHH CBeTa B TOHKHX nJIeHKaX
Cpella 1
Cpe.n;a 2
C
Puc I 7.9
B03.n;yX napaJlJleJlbHO BOJlHe 1'.
508
H86J1IO.n;aeMoe B npl'lpo.n;e pa.n;ymHoe
OKpalIIHBaHHe TOHKHX nJleHOK (MaCJI.HHbIe
nJleHKH Ha Bo.n;e, MbIJlbHbIe nY3bIpH, OK-
CH.n;HbIe nJleHKH H8 MeTaJlJlax) 06'b.HCHH-
eTC.H I'IHTepq>epeHHeH CBeTa, B03HHKaIO-
eH B pe3YJlbTaTe OTpameHH.H CBeTa OT
nepe.n;HeH H 38.n;HeH nOBepxHocTeH nneH-
KH. Ha pHCYHKe 17.9 H306pameHa TOH-
Ka.H nJlOCKOnapaJlJleJlbHa.H np03paqH8H
nJleHKa, Ha KOTOPYIO na,AalOT Jlyql'l CBeT8.
B TOqKe 0 CBeT 'laCTl'lqHO OTpa3I'1TC.H OT
BepxHeH nOBepXHOCTI'I nJleHKH (BOJlHa 1'),
a qaCTl'lqHO npeJIOMI'ITC.H 1'1 OTpa31'1TCH OT
3a.n;HeH ee nOBepXHOCTH B TOqKe C H,
npeJlOMHBIIIHCb B TOqKe B, BbIH,AeT B
BOJlHbI l' H 1" KorepeHTHbI.

ECJlH H8 nYTH 3THX Jlyqe:H nOCTaBHTb C06HpalOYIO JlHH3Y, TO OHH 6y,nYT
HaKJla,lJ;bIBaTbC.H B ee <pOKaJlbHO:H nJlOCKOCTH H ,naBaTb HHTep<pepeHHOH-
HYIO KapTHHY, KOTOpa.H Onpe,lJ;eJl.HeTC.H HX OnTHqeCKO:H pa3HOCTblO XO,lJ;a
A A
6 = n2 (OC + CB) - nl .0A:t -, r,ne qJleH - 06YCJlOBJleH nOTepe:H nOJlYBOJlHbI
2 2
npH OTpameHHH CBeT8 OT rpaHHbI p83,neJla OnTHqeCKH 60Jlee nJlOTHO:H cpe-
,lJ;bI. ECJlH n2> nh TO nOTep.H nOJlYBOJlHbI npOH30:H,neT B TOqKe 0, H 3TOT qJleH
6YileT CO 3H8KOM "ITJIIOC": 11 = n 2 (DC + CB) - lit ( OA - i)- ECJIH II:! < nt. TO
nOTep.H nOJlYBOJlHbI npOH30:H,neT B TOqKe C, H OH 6y,neT HMeTb 3Hal<: "MHHYC":
11 = n2 ( OC + CB - i ) - ntOA.
IIycTb TOHKa.H nJleHKa Haxo,nHTC.H B B03,nyxe (nl = 1). 113 pHcYHKa 17.9 BH-
,nHM, qTO OA = OBsina; OB = 2htgf3; OC = CB =. r,ne a H J3 - yrJlbI
cos I-'
na,neHH.H H npeJlOMJleHH.H COOTBeTCTBeHHO. Tor,lJ;a <pH3HqeCKa.H pa3HOCTb
xo,na
!1 = 2hn2 - 2htgJ3 sin a + ?:.
cos J3 2
. A sin a Y A I 2 A t A sinf3 II
slnp = -. 'ITeM, qTO cosp = ",1- sin ..." a g...,= . oCJle HeCJlOm-
"2 .J 1-sin 2 f3
113 38KOH8 npeJlOMJleHH.H H8:H,neM, liTO
HbIX M8TeM8THqeCKHX npe06pa30BaHH:H nOJlyqHM:
A  2 . 2 A
!1 = 2hn2 cos f3 + - HJlH 6 = 2h n 2 - sIn a + - .
2 2
1. IIoJIOCLI paBHoro HaKJIOHa.
KaK BH,nHO H3 nOCJle,nHe:H <POPMYJlbI, HHTep<pepeHHOHHa.H K8pTHHa onpe-
,nenSieTC.H 3HaqeHH.HMH ,lJ;JlHHbI BOJlHbI A, TOJlHHbI nJlaCTHHKH h, nOKa3aTeJlSl
npeJlOMJleHH.H n H yrJla n8,neHH.H a. JI.H ,naHHbIX A, n H h K8m,nOMY H8KJlO-
HY COOTBeTcTByeT CBO.H HHTep<pepeHHOHH8.H nOJlOC8. MHTep<pepeHHoHHbIe no-
JlOCbI, B03HHKalOHe B pe3ynbTaTe HaJlOmeHHSI Jlyqe:H, na,nalOHX H8 nnOCKO-
napaJlJleJlhHYIO nJleHKY no,n o,nHHaKOBbIMH yrJlaMH, Ha3bIBaIOTC.H nOJIOCaMH
paBHoro HaKJIOHa.
JIyqH 3, HaKJlOHeHHbIe no,lJ; ,lJ;pyrHM yrJlOM, c06epYTC.H B ,npyroH TOqKe P'.
ECJlH OnTHqeCKa.H OCb JlHH3bI nepneH,lJ;HKYJl.HpHa nOBepXHOCTH UJlaCTHHKH, no-
JlOCbI paBHoro HaKJlOHa 6y,nYT HMeTh BH,lJ; KOHeHTpHqeCKHX KOJle C eHTpOM
B <poKyce JlHH3bI.
2. IIoJIOCLI paBHoH TOJIHHLI.
Ha pHCYHKe 1 7 .1 0 H306pameHa I<:JlHHOBH,nHa.H nJlaCTHHKa (yroJl a Mem,ny
60KOBhIMH rpaHHMH MaJl). Ha Hee na,naeT CBeT, H8npaBneHHe pacnpOCTpaHe-
HH.H KOToporo COBnalIaeT C napaJlJleJlbHbIMH JlyqaMH 1 H 2.
KorepeHTHbIe JlyqH l' H 1" ,nalOT HHTep<pepeHHoHHYIO KapTHHY B TOqKe A'.
KOTOpa.H onpe,neJl.HeTC.H pa3HOCTblO xo,na 3TH X JlyqeH H 6y,neT 3aBHceTb OT TOJl-
509

HHbI h KJlHHa B MeCTe naeHH.H Ha
HerO Jlyqa. JIyqH 2' H 2" c06HpalOT-
C.H JlHH30H B TOqKe A". OnTHqeCKa.Sl
pa3HOCTb XO,Aa OnpeDeJl.SleTC.H TonlUH-
HOH h'. TaKHM 06pa30M. Ha 3KpaHe
Ha6nlOaIOTC.SI HHTepqJepeHlI.HOHHbIe
nOJlOCbl. Kama.H H3 nonoc B03HHKa-
eT 3a CqeT oTpameHH.H OT MeCT nna-
CTHHKH, HMelOHX OHHaKOBYIO Ton-
HHY .
llHTepqJepeHlI.HOHHbIe nOJlOCbl Ha
nOBepXHOCTH nneHKH HMelOT O.lI.HHaKO-
BYIO OCBeeHHOCTb Ha Bcex TOqKaX
nOBepXHOCTH, COOTBeTCTBYIOHX O,AH-
HaKOBbIM TOJlHHaM nneHKH, H Ha3bI-
BaIOTC.SI nOJlOCaMH paBHoH TOJlHLI.
IIpHMepoM nonoc paBHoH TOJlHHbI .HBn.HIOTC.SI KOJlblI.a HblOToHa, KOTopbIe
Ha6nlOaIOTC.H npH oTpameHHH CBeTa OT B03,AYIliHoro 3a30pa, 06pa30BaHHoro
nnocKOnapaJlnenbHOH nnacTHHKoH H conpHKacalOeHC.H C HeH nJlOCKOBbInYK-
JlOH nHH30H C 60nblIIHM paHycoM KpHBH3HbI (pHC. 17.11, a). B03YIIIHa.H
npocnoHKa nOCTeneHHO YTOJlaeTC.H OT TOqKH conpHKOCHOBeHH.SI nHH3bI K Kpa-
.HM. OTpameHHbIe OT BepxHeH H HHmHeH rpaHHlI. B03YIIIHOH npOCJlOHKH CBe-
TOBbIe BOJlHbI HHTepqJepHPYIOT Mem,AY C060H. llpH 3TOM nOJlyqaeTC.H cne,AYIO-
a.Sl KapTHHa: B TOqKe conpHKOCHOBeHH.H Ha6nlO.n.aeTC.SI qepHOe n.SlTHO. OKPY-
meHHoe p.HD;OM KOHlI.eHTpHqeCKHX CBeTOBbIX H qepHbIX Konell y6bIBalOlUeH IIIH-
pHHbI (pHC. 1 7.11, 6).
Puc. 1 7.10
a
d
6
Puc. 17.11
KaK JI.H nonoc paBHoro HaKJlOHa, TaK H n.H nonoc paBHoH TOJlHHbI,
nonOmeHH.H MaKcHMYMOB 3aBHC.SIT OT A. II03ToMY npH oCBeeHHH MOHOXPOMB-
THqeCKHM CBeTOM nOJlyqaeTC.SI CHCTeMa TeMHbIX H .HpKHX nonoc, a npH HB-
6nlOeHHH B 6enoM CBeTe HHTepqJepeHlI.HOHHa.Sl KapTHHa npHo6peTaeT pa.n.ym-
HYIO oKpacKY. llHTepqJepeHlI.HIO MomHO Ha6nlO,AaTb H B npoXO.HeM CBeTe,
npHqeM B aHHOM cnyqae He npOHCXOHT nOTepH nOJlYBonHbI. B pe3YJlbTaTe
MaKCHMYMbI HHTepqJepeHlI.HH B OTpameHHOM CBeTe COOTBeTCTBYIOT MHHHMYMaM
B npOXO.HeM Ii Ha060poT.
510

17.6. TIpHMeHeHHe HHTepepeHHH B TeXHHKe
.HBJleHHe HHTepc}:>epeHHH llIHpOKO HCnOJIb3YIOT ,An.H C03,AaHH.H pa3JIHqHbIX
H3MepHTeJlbHbIX H KOHTpOJlHPYIOHX YCTpoiicTB.
1. CyecTBYIOT Cnel.\HaJIbHbIe npH60pbI - HHTep<}JepoMeTpbl, ,AeiiCTBHe KO-
TOpbIX OCHOBaHO Ha .HBJleHHH HHTepc}:>epeHl.\HH. llx Ha3HaqeHHe - TOqHOe H3-
MepeHHe ,nJlHH BOJIH, nOKa3aTeJleii npeJIOMJleHH.H. K03c}:>c}:>Hu,HeHToB JlHHeiiHOro
paCllIHpeHH.H H ,Ap.
AeiicTBHe Bcex HHTepc}:>epoMeTpoB OCHOBaHO Ha O,AHOM H TOM me npHHl.\H-
ne, H HHTepc}:>epOMeTpbI pa3nHQa1OTC.H JlHllib KOHCTPYKTHBHO. Ha PHCYH-
Ke 17.12 npe,ACTaBJleHa ynpoeHHa.H cxeMa HHTepc}:>epoMeTpa MaiiKeJIbCoHa.
MOHoxpOMaTHqeCKHii nyqOK CBeTa OT HCTOq-
HHKa S na,AaeT no,A yrJloM 45° Ha nnOCKonapan-
JIeJIbHYIO nJlaCTHHKY Pl' CTopOHa nJlaCTHHKH.
Y,AaJleHHa.H OT S, nOKpbITa TOHKHM cnoeM cepe6-
pa C TaKHM paCqeTOM. qTO OH nonOBHHY CBeTO-
Boro nyqKa nponYCTHT, a nOJlOBHHY OTpa3HT
(noJlynp03paqHa.H nn8cTHHKa), T.e. 3,AeCb ny1.J
pa3,AeJl.HeTC.H Ha ,ABe qaCTH: JIyq 1 OTpa»<aeTC.H OT
nocepe6peHHoro CJlOS!, nyq 2 npOXO,AHT Qepe3
Hero. JIYQ 1 OTpa»<aeTC.H OT 3epKaJla M I H, B03-
Bpaa.HCb 06paTHO, BHOBb npOXO,AHT qepe3
nJIaCTHHKY PI (Jlyq 1'). JIyq 2 H,AeT K 3epKaJIY
M 2, OTpamaeTC.H OT Hero, B03BpaaeTCH 06paT-
HO H OTpamaeTCH OT nJlaCTHHKH PI (JlYQ 2').
TaK KaK nepBbIii Jlyq npOXO,AHT nJlaCTHHKY PI
,ABam,AbI, TO ,AJI.H KOMneHCal.\HH B03HHKIlleM pa3HOCTH xo,na Ha nYTH BTO-
poro Jlyqa CTaBHTC.H nJlaCTHHKa P 1. (TOqHO TaKa.H me, KaK H PI' TOJlbKO
He nOKpbITa.H CJlOeM cepe6pa).
JIyqH l' H 2' KorepeHTHbI, CJle,AOBaTeJlbHO, 6Y,AeT Ha6J1IO,AaTbCH liHTepc}:>e-
peHl.\H.H, pe3YJlbTaT KOTOpoii 3aBHCHT OT OnTHqeCKOii pa3HOCTH XO,Aa nyqa 1
OT TOqKH 0 .n;o 3epKaJla M I H Jlyqa 2 OT TOqKH 0 ,no 3epKaJIa M 2' ITpH
/////// M 2
Puc. 17.12
2'
A
nepeMeeHHH O)];HOrO H3 3epKaJI Ha paCCTO.HHHe - pa3HOCTb xo,na 060HX JlY-
4
A
Qeii H3MeHHTCH Ha 2"' H B HHTepc}:>epeHu,HoHHOii KapTHHe MaKCHMYM C)];BH-
HeTC.H Ha MeCTO MHHHMYMa. H Ha060poT, T.e. HHTepc}:>epeHl.\HOHHbIii MaKCHMYM
C)];BHHeTC.H Ha nOJlOBHHY paCCTO.HHH.H Mem.n;y nOJlOCaMH. TaKoii C,ABHr nOJloc
Ha6J1IO)];aTeJlb OTqeTJlHBO YBH,AHT. CJle)];OBaTeJlbHO, no He3HaqHTeJlbHOMY CMee-
HHID HHTepcpepeHu,HoHHOii KapTHHbI MO»<HO Cy,nHTb 0 MaJlOM nepeMeeHHH o,n-
Horo H3 3epKaJI H HCnOJlb30BaTb HHTepc}:>epoMeTp ,AJI.H .n;OCTaTOqHO TOqHbIX
(- 10- 9 M) H3MepeHHH )];JlHH ()];JlHHbI TeJl, )];JlHHbI cBeToBoii BOJlHbI, on pe)];eJIe-
HHM TeMnepaTypHoro K03c}:>c}:>Hu,HeHTa JlHHeiiHOro pacllIHpeHH.H Ii .n;p.).
2. llcnoJIb3Y.H HBneHHe HHTepc}:>epeHl.\HH, MO»<HO Ol.\eHHTb Ka'leCTBO 06pa-
60TKH nOBepXHOCTH H3)];eJlH.H C TOqHOCTblO )];0 10- 6 CM. AJIH 3Toro HY»<HO CO-
3)];aTb TOHKYIO KJlHHOBH)J;HYIO npOCJloiiKY B03)];yxa Me»<.n;y nOBepXHOCTblO 06pa3-
511

a lJ OqeHb r JIa,D;KOH 3TaJIOHHOH nnac-
THHKOH. HepOBHOCTH nOBepXHOCTH BbI-
30BYT 3aMeTHbIe HCKpHBJIeHH.H HHTep-
cpepeHUHOHHbIX nOJlOC, 06pa3YIOHXC.H
npH OTpameHHH CBeTa OT npOBep.HeMOH
nOBepXHOCTH H HHmHeH rpaHHbI 3Ta-
JlOHHOH nnaCTlJHKH. Ha pHCYHKe 17.13
npHBe,D;eHbI Ha6nlO,AaeMbIe HHTepcpepeH-
HOHHbIe KapTHHbI npH oTcTynJIeHHlJ
OT Tpe6yeMoH TOqHOCTH 06pa6oTKH Ii
npH ,AOCTHmeHHH He06xol1HMOH TOqHO-
CTH 06pa60TKH nnocKoH nOBepXHOCTlJ
,D;eTanH J(.
3. npOCBeTJleHHe onTHKH. OTnOJlH-
pOBaHHa.H nOBepXHOCTb CTeKJla OTpama-
eT OKono 4 % nepneH,AHKYJI.HpHO na.n;a-
lOero Ha Hee CBeTa. COBpeMeHHbIe on-
THqeCKHe npH60pbI COCTO.HT H3 6onbIIIo-
ro qHCJla OnTHqecKHX CTeKOJl - JIHH3,
npH3M H T .,D;. ll03TOMY 06He nOTepH
CBeTa B 06'beKTHBe cl>oToannapaTa co-
CTaBJI.HIOT OKOJIO 25 %, B MHKpOCKO-
ne - 50 % H T.,A. B pe3YJIbTaTe oc-
BeeHHOCTb H306pameHH.H nOJIyqaeTCSI
MaJlOH, YXY,AIIIaeTC.H TaKme KaqeCTBO
lJ306pameHlJ.H.
qacTb CBeTOBoro nyqKa nOCJIe
MHorOKpaTHoro OTpameHH.H OT BHYT-
peHHHX nOBepxHocTeH Bce me npoxo-
,D;HT qepe3 OnTHqeCKHH npH60p, HO
pacceHBaeTC.H H yme He yqacTByeT B
C03,AaHHlJ qeTKOrO H306pameHlJ.H, a HB
cpOTorpacplJH 06pa3yeTC.H "ByaJlb".
,lJ:n.H YMeHbIIIeHH.H CBeTOBbIX nOTepb Ha nOBepXHOCTb OnTHqeCKOrO CTeKJIa Ha-
HOC.HT TOHKYIO nJleHKY C a6conlOTHbIM nOKa3aTeJleM npeJlOMneHlJ.H nn. MeHbIlIHM.
qeM a6COJllOTHbIH nOKa3aTeJlb npeJlOMJIeHH.H CTeKJIa nc (pHC. 17.14). TIpH OTpa-
meHHlJ CBeTa OT rpaHH pa3,D;eJla B03,D;yx-nJIeHKa H nneHKa-CTeKJlO B03HHKaeT
HHTepcl>epeH.H KorepeHTHbIX BOJIH 1 H 2. TonHHY nJleHKH h H nOKa3aTeJIb
npenOMJleHH.H nn no.n.6HpalOT TaK, qT06bI HHTepcl>epHPYIOHe BOJlHbI raCHJlH .n.pyr
.n.pyra. CqHTa.H, qTO CBeT na,n:aeT HOpMaJIbHO (a = 0) H yqHTbIBa.H, qTO nOTep.H
nonYBOJIHbI npOHCXO,D;lJT Ha 06eHx nOBepXHOCT.HX. TaK KaK nc> nn> n 003 1J,' 6y-
a
6
fee
- -
8
-
Puc. 17.13
i?i 2
-  -- -
- - -
- - -
- nn
  
   
  
 
 nc
  
Puc. 17.14
(2m + 1)A-
,D;eM HMeTb 2 nnh =
2
A- A-
H npH m=O 2n n h = -. OTKY,Aa h = -. B
2 ( 4n n )
pe3YJIbTaTe raIIIeHH.H OTpameHHbIX BOJIH npOHCXO,D;HT YCHJleHHe BOJIHbI, KOTO-
pa.H npOXO,D;HT B CTeKJlO.
512

TaK KaK 06bIqHO Ha nOBepXHOCTb CTeKna nallaeT 6eJIbIH CBeT. TO oCYll1e-
CTBHTb rallIeHHe OTpameHHbIX BOnH BCeX qaCTOT HeB03MOmHO. TOnHHY nJIeH-
KH nO6Hpa1OT TaK, qT06bI nOJIHOe rallIeHHe HMeJlO MeCTO JI.H BOJIH CpeHeH
qaCTH CneKTpa (3eJleHbIH BeT).
rallIeHHe KpaCHbIX H cpHOJleTOBbIX qaCTeH CneKTpa npOHCXO,AHT He3HaqH-
TeJIbHO. I103TOMY 06'beKTHB C npOCBeTJIeHHOH OnTHKOH HMeeT CHpeHeBaTbIH OT-
TeHOK.
4. .HBJleHHe HHTepcpepeHHH IiCnOJlb3yeTcH nH nOJlY'leHHH BbICOKOOT-
pamalOHX nOKpbITHH. B 3TOM CJIyqae Hcnonb3YlOT TOHKYIO nneHKY Ton-
A
HHOH h = _ 2 H3 MaTepHaJIa, a6conlOTHbIH nOKa3aTeJIb npeJIOMJleHH.H KO-
nn
Toporo nn 60JlbllIe a6COJllOTHOrO UOKa3aTeJl.H npeJlOMJleHH.H CTeKJla nc'
B 3TOM cJlyqae OTpameHHe OT nepeHeH rpaHH npOHCXOHT C nOTepeH no-
JlYBOJlHbI, TaK KaK no> n B03A ' a OTpameHHbIe OT 3aHeH rpaHHbI - 6e3
nOTepH nOJlYBOJlHbI. B pe3YJlbTaTe D. =A H OTpameHHbIe BOJIHbI YCHJI.HT Pyr
,APyra.
17.7. ,l(H4>paKHH CBeTa
a I
6 I
6 I IIIII
Puc. 17.15
HcppaKI.\HeH CBeTa Ha3bIBaeTCH COBO-
KynHocTb OnTHqeCKHX .HBJIeHHH, 06YCJIOB-
JleHHbIX BOJIHOBOH npHpooH CBeTa H Ha-
6J1IOalOHXC.H npH ero pacnpocTpaHeHHH
B cpe,Ae C pe3Ko BbIpameHHbIMH HeOHO-
pOHOCT.HMH. B pe3YJlbTaTe npoHcxollHT
orH6aHHe BOJlHaMH npenHTcTBHH. BCTpeqa-
IOHXCH Ha HX nYTH. HnH B 60Jlee llIH-
pOKOM CMbICJIe - .HBJIeHHe OTKnOHeHH.H
OT np.HMOnHHeHHoro Hanp8BneHH.H pacnpo-
nrn'9""'-.n....n......'I'.1"r n..."n.rnn .... "yy......"......." ft.....".... n,.. Tl'1'\..
"'J.pan.1[;;n..n.n "'.D1[;;J.a D VAn.VPVAn.V6J. "'P1[;;A1[;;.
BJlarO,AapH HcppaKHH BOJIHbI MorYT nOna,AaTb B 06JIaCTb reOMeTpHqeCKOH
TeHH, orH6aTb npenHTCTBH.H, pa3MepbI KOTOpbIX cpaBHHMbl C ,AJIHHOH BonHbI
CBeTa, npoHHKaTb qepe3 He60JIbllIHe oTBepCTH.R B 3KpaHax H T ..
Ha pHCYHKe 17.15 H306pameH TOqeqHbIH HCTOqHHK CBeTa. OT KOToporo
BOJIHa npOXOHT qepe3 eJIb, pa3MepbI KOTOpOH MomHO H3MeH.HTb I1pH UIH-
pH He eJIH 1-2 MM Ha 3KpaHe BHHa .HpKa.H CBeTJla.H nOJIOCKa C qeTKO
OqepqeHHbIMH KpaHMH (pHC. 17.15, a). I1pH YMeHbllIeHHH llIHpHHbI eJIH
qeTKOCTb KpaeB nOJlOCKH Ha 3KpaHe HapYllIaeTC.R H npH 3HaqHTeJIbHOM
YMeHbllIeHHH llIHpHHbI eJIH cnpaBa Ii CJIeBa OT oCBeeHHoH nOJIOCKH nOHB-
nHIOTC.H cna60 3aMeTHbIe pa,AymHbIe nonOCbI. EcnH 3a IiCTOqHHKOM CBeTa no-
CTaBHTb cBeTocpHnbTp. paymHbIe nOJIOCbI CTaHOB.HTCH OHOBeTHbIMH. OnbIT
CBH)leTeJIbCTByeT 0 HcppaKuHH CBeTa: CBeT orH6aeT Henp03paqHbIe KpaH
eJlH H 3aXO,AHT B 06JIaCTb TeHH.
Ha pHCYHKe 17.16 H306pameH He60JIbllIOH nJIOCKHH ,AHCK, OCBeaeMbIH TO,
qeqHbIM HCTOqHHKOM CBeTa, pacnOJlOmeHHbIM Ha OCH, npOBeeHHOH nepneH-
513

a
S
:0---------
,AHKynSIpHo K II.eHTpy ,nHCKa rIpH ,n0-
CTaTOQHO 60JIblIIOM paCCTO.HHHH OT
,AHCKa ,no 3KpaHa B II.eHTpe TeHH Ha-
6J1IO,naeTC.H CBeTJlOe nSITHO, oKpymeH-
Hoe TeMHbIMH 1'1 CBeT JlbIMH KOJIbII.aMH
(pHC. 17.16. 6).
6
Puc. 17.16
17 .8. npHHHn rlOHreHca-<I>peHeJUI
.HBJIeHHe ,nHcppaKII.HH 06'b.HCH.HeTC.H C
nOMOblO npHHII.Hna rlOHreHca, COrJIaCHO
KOTOPOMY Kam,na.H TOqKa, .n;o KOTOpOH ,no-
XO,AHT BOJIHa, .HBJI.HeTC.H II.eHTpOM BTOPHq-
HbIX ccpepHqeCKHX BOJlH, a orH6alOaSI
3THX BonH ,AaeT nOJlOmeHHe BonHOBoro
cppoHTa B CJle,AYIOlI.\HH MOMeHT BpeMeHH.
TaK. Ha pHCYHKe 17.17 nOKa3aHa nJlOCKaSI
BOJIHa, na,nalOlI.\a.H Ha nperpa,ny C OTBepCTH-
eM. HOBbIH cppOHT BOJlHbI .HBn.HeTC.H nJIOC-
KHM TonbKO B cpe.n;HeH qaCTH, a Y rpaHHII.
OTBepCTHSI npOHcxo,nHT 3arH6aHHe BOJIHOBO-
ro cppOHTa, T.e. BonHa npOHHKaeT B 06-
JlaCTb reOMeTpHqeCKOH TeHH, orH6a.H Kpa.H npen.HTCTBH.H.
C nOMOblO npHHII.Hna rlOHreHca MomHO pelIIHTb 3a.n;aqy 0 HanpaBJleHHH
pacnpOCTpaHeHH.H BOJlHOBOrO cppoHTa, HO HeJIb3.H BbI.HCHHTb, OT qero 3aBHCHT
aMnJIHTy,na, a CJle.n;OBaTeJIbHO, HHTeHCHBHOCTb BOJIH, pacnpOCTpaHSIIOlI.\HXC.H no
pa3HbIM HanpaBJIeHHSIM. <I>peHeJlb pa3BHn 3TOT npHHII.Hn ,nanblIIe, .n;onOJlHHB
ero H.n;eeH 06 HHTepcpepeHII.HH BTOPHqHbIX BOJIH. COrJlaCHO npHHHny rlOHreH-
ca-<I>peHeJlH, Kam,AaSi TOqKa <l>poHTa BOJ!HbI .HBJlSIeTC.H HCTOqHHKOM BTOpH'I-
HLIX ccI>eplNeCKHX KorepeHTHblX BOJIH. HOBbIH cppOHT BOJlHbI 06pa3yeTC.H B pe-
3YJIbTaTe HHTepcpepeHuHH BTOPHqHbIX BOJIH.
YQeT aMnJIHTY.n; 1'1 cpa3 BTOPHqHbIX BOJIH n03BOn.HeT B Kam.n;OM KOHKpeT-
HOM cJlyqae HaRTH aMnJlHTY.n;y (HHTeHCHBHOCTb) pe3YJIbTHpYIOeH BOJlHbI B
nlO60H TOqKe npOCTpaHCTBa.
\..)()()()()()()(./
,,-JlJtJlJLJl.Y
Puc. 17.17
17.9. MeTO)]; 30H <l>peHeJlH
,l1.JI.H Haxom.n;eHH.H pe3YJlbTaTa HHTepcpepeHII.HH BTOPHqHbIX BOJIH <I>peHenb
npe.n;JIOmHJI MeTO.n; pa36HeHH.H BOJlHOBOrO cppoHTa Ha 30HbI, Ha3bIBaeMbIe 30Ha-
MH <I>peHeJl.H.
514

IIpenoJlomHM, qTO HC-
TOqHHK CBeTa S (pHC. 17.18)
TOqeqHbIH H MOHoxpOMaTH-
qeCKHH, a cpea, B KOTOpOH
pacnpOCTpaH.HeTC.H CBeT, H30-
TpOnHa.H. BOJIHOBOH <}>pOHT B
npOH3BOJlbHbIH MOMeHT Bpe-
MeH:H 6y eT :HMeTb <}>OPMY
c<}>epbI pa)J;HYCOM r=ct. K8m-
a.H TOqKa Ha 3TOH c<}>epHqeC-
KOH nOBepXHOCTH HBJI.HeTC.H
BTOPHqHbIM CTOqHHKOM BOJIH.
KOJle6aHH.H BO Bcex TOqKaX
BOJIHOBOH nOBepXHOCTH npo-
HCXO.HT C O:HHaKOBOH qaCTO-
TOll H B OHHaKOBOH <}>a3e. CneOBaTeJlbHO, Bce 3TH BTOp:HqHbIe HCTOqHHKH
KorepeHTHbI. AJI.H HaXOmeHH.H aMnnHTYbI KOJle6aHHH B TOqKe M He06xoH-
MO npOH3BeCT:H CJlOmeHHe KorepeHTHbIX Kone6aH:HH OT Bcex BTOP:HqHbIX HC-
TOqHHKOB Ha BOJIHOBOH nOBepXHOCTH.
<1>peHeJlb pa36HJI BOJIHOBYIO nOBepXHOCTb <1> Ha KOJlbu.eBbIe 30HbI TaKoro pa3-
S
Puc. 17.18
A
Mepa, qT06bI paCCTO.HHHSI OT KpaeB 30HbI o TOqKH M OTJlHqaJ1HCb Ha "2' T .e.
A
P I M-P o M=P 2 M -PIM -= 2'
A
TaK KaK pa3HOCTb xoa OT BYX coceHHX 30H paBHa "2' TO Kone6aHH.H
OT HHX npHXO.HT B TOqKY M B npOTHBOnOJlOmHbIX <}>a3ax H IIp:H HanomeHHH
3T:H KOJle6aHH.H 6YYT B3aHMHO OCJla6J1.HTb Pyr Pyra. II03TOMY aMnnHTYlIa
pe3YJlbTHpYIOero CBeTOBoro KOJle6aHHSI B TOqKe M 6yeT paBHa
A = Al -  + A3 - A4 +...:t  ' (17.5)
re Al A 2 , ..., Am - aMnnHTYbI Kone6aHHH, B036ym,AaeMbIX I-fl, 2-H,
m-H 30HaMH.
<1>peHeJlb npenonomHn TaKme,
qTO eHCTBHe OT eJlbHbIX 30H B
TOqKe M 38BHCHT OT HanpaBne-
HH.H pacnpOCTpOHeHH.H (OT yrJla m
(pHC. 17.19) Memy HopManblO ii
K nOBepXHOCTH 30Hb] H HanpaBne-
M HHeM Ha TOqKY M). C YBeJlHqeHH-
eM  m eHCTBHe 30H y6bIBaeT H
npH yrnax q>m  90° aMnJlHTya
B036ym,AaeMbIX BTOPHqHbIX BOJIH
paBHa O. KpoMe Toro, HHTeHCHB-
HOCTb H3J1yqeHH.H B HanpaBneHHH
TOqKH M YMeHbWaeTC.H C pOCTOM m
S
Puc. 17.19
515

H BCJleCTBHe YBeJlHqeHH.H paCCTO.HHH.H OT 30HbI O TOqKH M YqHTbIBa.H 06a
cpaKTOpa, MOmHO 3anHCaTb, qTO
Al > A 2 > A3 > ...
1. 06'LJlCHeHHe npJlMOJlHHeHHOcTH pacnpOCTpaHeHHJI CBeTa.
06ee qHCJlO 30H 4>peHeJl.H, BMealOHXC.H Ha nOJlyccpepe paHycoM SPo,
paBHbIM paCCTO.HHHIO OT HCTOqHHKa CBeTa S o cppoHTa BOJlHbI, OqeHb BeJlHKO.
llOSTOMY B nepBOM npH6J1HmeHHH MomHO CqHTaTb, qTO aMnJlHTya KOJle6a-
HHH Am OT HeKOTopOH m-H 30HbI paBHa cpeHeMY apHcpMeTHqeCKOMY OT aMn-
JIHTY,n. npHMbIKalOIIUiX K HeH 30H. T .e.
 = Am-I + Am+l .
2
Tora BbIpameHHe (17.5) MomHO 3anHcaTb B BHe
A= I +[ ' -Ao+ 3 j+[ 3 -+ i j+...I m .
Am
2
TaK Kal< BbIpameHH.H, CTO.HHe B cK06Kax, paBHbJ 0, a
HHqTOmHO
MaJla, TO
A - Al +  _ Al
-2-2-2'
(17.6)
TaKHM 06pa30M, aMnJIHTY,Aa KOJle6aHHH, C03aBaeMa.H B npOH3BOJlbHOH TOq-
Ke M CcpepHqeCKOii BOJIHOBOH nOBepXHOCTblO, paBHa nOJlOBHHe aMnJlHTYbI, CO-
3aBaeMOH OHOH u,eHTpaJlbHOH 30HOH. 113 pHCYHKa 17.19 pa,AHYC r mono" 30HbI
MaJIa, TO
r m = (b + J -(b + h m )".
r m =  b+  J _b 2
TaK KaK h m « b H )J;JlHHa BOJlHbI CBeTa
4>peHeJl.H
m2}...2
mb}'" + - = mb}"' . 3HaqHT, paHYC nepBOH
4
30HbI 4>peHeJI.H r l =.Jb}:.. YqHTbIBa.H, qTO }... JlHHa BOJlHbI MomeT HMeTb 3Ha-
bl
300 ,no 860 HM. nOJlyqHM r l « b. CJlelIOBaTeJlbHO. pacnpOCTpaHe-
OT S K A1 npOHCXOHT TaK, 6YTO CBeTOBOH nOTOK pacnpOCTpaH.HeT-
C.H BHYTPH OqeHb Y3Koro KaHaJIa BOJlb SM, ,AHa-
MeTp KOToporo MeHbllIe paHYCB nepBOH 30HbI 4>pe-
HeJI.H, T .e. npJIMOJlHHeHHO.
2. )lH<}>paKI..{HH Ha KpyrJIOM OTBepCTHH.
C<})epHqeCKa.H BOJlHa, pacnpOCTpaH.HIOa.HC.H H3
TOqeqHOrO HCTOqHHKa S. BCTpeqaeT HB CBoeM nYTH
SKpaH C KpyrJlbIM OTBepCTHeM (pHC. 17.20). BH
HcppaKu,HOHHOH KapTHHbI 3aBHCHT OT QHCJla 30H
4>peHen.H, YKJIa,AbIBalOHXC.H B OTBepCTHH. CorJIac-
HO (17.5) H (17.6) B TOqKe B aMnJlHTya pe3ynb-
THpYIOero KOJle6aHH.H
'S
I
I
I
qeHH.H OT
HHe CBeTa
B
:3
Puc. 17.20
516

A = Al + Am
2 - 2 '
b+.b r,Ae 3HaK "nJIIOC" COOTBeTcTByeT HeqeTHbIM m, a
2 3HaK "MHHYC" - qeTHbIM m.
Kor,u,a OTBepCTHe OTKpbIBaeT HeqeTHOe "LIHCJIO 30H
<I>peHeJISI. TO aMnnHTY,Aa KOJIe6aHHH B TO"LIKe B
6y,u,eT 60nbIlle, qeM npH OTCYTCTBHH 3KpaHa.
EcnH B OTBepCTHH YKJla,u,bIBaeTC.H o,u,Ha 30Ha <I>pe-
HeJI.H, TO B TOqKe B aMnJlHTy,u,a A=A}, T.e. B,u,Boe
60JlbllIe, qeM B OTCYTCTBHe Henp03pa'tIHoro 3Kpa-
Ha. ECJlH B OTBepCTHH YKJIa,u,bIBaIOTCSI ,u,Be 30HbI
<I>peHeJl.H. TO HX lIe:HCTBHe B TOqKe B npaKTHqeC-
KH YHHqTOmaeT Pyr pyra H3-3a HHTepcpepeHIlHH.
TaKHM 06pa30M, ,u,HcppaKlJ,HOHHan KapTHHa OT KpyrJIOro OTBepCTHSI B6JIH3H TOq-
KH B 6y,u,eT HMeTb BH,u, "LIepe,u,YIOHXCSI TeMHbIX H CBeTJIbIX KOJIelJ, C Q,eHTpa-
MH B TOqKe B (eCJIH m - "LIeTHoe. TO B u,eHTpe TeMHoe KOJIbU,O, eCJIH m -
HeqeTHOe - CBeTJIOe KOJIbU,O), npHqeM HHTeHCMBHOCTb MaKCHMYMOB y6bIBaeT
C paCCTOSlHHeM OT u,eHTpa KapTHHbI.
3. AH4>paKHH Ha ,lJ,HCKe.
nYCTb ,u,HCK (pHC. 17.21) 3aKpbIBaeT m nepBbIX 30H <I>peHeJlSl. Tor,u,a aMn-
JIHTy,u,a pe3YJIbTHpYIOero KOJle6aHH.H B TOqKe B paBHa
,8
I
I
I
I
B
Puc. 17.21
A A A A A m + l ( Am+l A Am+3 ,
= "'+l - "'+2 + "'+3 +... = + -"'+2 + J+'"
HJlH A = +l ,
2
TaK KaK BbIpameHH.H, CTOSlMe B cK06Kax, paBHbI O.
CJIe,u,oBaTeJIbHO, B TO"LIKe B BCer,lJ;a Ha6JIIO,u,aeTcSI CBeTJlOe nHTHO. COOTBeT-
CTBYIOee nOJIOBHHe ,u,e:HCTBH.H nepBO:H OTKpbITO:H 30Hbl <1>peHeJISI. eHTpaJlb-
HbI:H MaKCHMYM oKpymeH KOHu,eHTpHqeCKHMH C HHM TeMHbIMH H CBeT JIbIMH
KOJIblJ,aMH, a HHTeHCHBHOCTb y6bIBaeT C paCCTO.HHHeM OT lJ,eHTpa KapTHHbI.
17 .10. ,llHpaKl{HOHHa.H pellIeTKa. ,lIH«ppaKl{HOllllbIH CIIeKTp
Ba:mHYIO pOJIb B npHKJla,u,HoH onTHKe HrpalOT .HBJIeHH.H ,AHcppaKu,HH Ha OT-
BepCTH.HX B cpopMe eJIH C napaJlJleJIbHbIMH Kpa.HMH. TIpH 3TOM HCnOJIb30Ba-
HHe ,u,McppaKu,HH CBeTa Ha O,u,HO:H eJIH B npaKTHqeCKHX lJ,eJI.HX 3aTpy,u,HeHO
H3-3a cJla60:H BHlIHMOCTH lIMcppaKIlHOHHO:H KapTHHbI. llIHpOKO HCnOJIb3YIOTCSI
,u,HcppaKlJ,HOHHbIe peIlleTKM.
AII4>paKHoHHaH pemeTKa - cneKTpaJIbHbI:H npH60p, CJIymaH:H ,u,JI.H pa3-
Jlo:meHHSI CBeTa B cneKTp H H3MepeHH.H ,u,JIHHbI BonHbI. Pa3JIHqalOT np03paq-
HbIe H OTpamalOHe peIlleTKH. ,Z:(HcppaKlJ,HOHHa.H peIlleTKa npe,u,CTaBJISleT C060:H
cOBoKynHocTb 60JIbllIOrO "LIHCJIa napanneJlbHbIX IIITpHXOB O,AHHaKOBO:H cpOpMbI,
HaHeceHHbIX Ha nJIOCKYIO HJIH BorHYTYIO nonHpOBaHHYIO nOBepXHOCTb Ha O,u,H-
HaKOBOM paCCTO.HHHH ,u,Pyr OT ,u,pyra.
B np03paQHO:H nJlOCKO:H ,u,HcppaKlJ,HOHHO:H pelIIeTKe (pHC. 17.22) IIIHpHHa
np03paqHOrO IIITpHXa paBHa a, IIIHpHHa Henp03paqHOrO npOMemYTKa - b. Be-
517

1
JlHqliHa d = a + b = - Ha3bIBaeTC.H no-
N
CTOHHHOH (nepHooM) HpaKHoHHOH
pemeTKH. re N - qliCJIO IllTpliXOB Ha
eIiHIiY JIIiHbI pellleTKH.
I1YCTb nJlOCKa.H MOHoxpOMaTliqeCKa.H
BOJIHa na,AaeT HOpMaJIbHO K nJIOCKOCT1:I
pellleTKH (pIiC. 17.22). 110 npliHl.\1:Iny
rlOHreHca -<1>peHeJI.H Kama.H eJlb SlB-
JI.HeTC.H HCTO'-IHIiKOM BTOplit.JHbIX BOJIH.
cnoco6HbIX IiHTepcpeplipoBaTb pyr C
pyrOM. 110JIyqIiBIllYIOC.H ,AlicppaKl.\IiOH-
HYIO KapT1:IHY MomHO Ha6J1IO,AaTb B cpo-
KaJlbHOH nnOCKOCTIi JIHH3W, Ha KOTOpylO
na,AaeT ,AlicpparlipoBaHHbIH nY'-IOK.
)J;onycTIiM. qTO CBeT Hcppar1:lPyeT Ha eJI.HX no yrJloM <po TaK KaK enI1
HaXO.HTC.H pyr OT pyra Ha OIiHaKOBbIX paCCTO.HHIi.HX. TO pa3HOCTIi xoa
nyt.JeH, 1:IyI1X OT BYX COCe,AHI1X eJIeH, JI.H aHHoro HanpaBneHH.H <p 6YYT
OIiHaKOBbIMIi B npe.neJIax BceH I1cppaKIlI10HHOH pellleTK1:I:
J_ t _ Y _ t _{ _,_ J_ t _ Y _ t_{
M
N
D
C
MN=a
NC=b
B
Puc. 1 7.22
S
/). = CF = (a + b) sin <p = d sin <p.
B Tex HanpaBJIeHIi.HX, JI.H KOTOpWX pa3HOCTb XO,Aa paBHa qeTHOMY q1:lcny
nonYBonH, Ha6JIIOaeTC.H IiHTepcpepeHl.\HOHHWH MaKCIiMYM. Hao60pOT, JI.H Tex
HanpaBJleHIiH, re pa3HOCTb xoa paBHa HeqeTHOMY q1:lCJlY nOJIYBOJIH, Ha6nlO-
,AaeTC.H IiHTepcpepeHl.\HOHHbIH MIiHI1MYM. TaKIiM 06paaOM, B HanpaBJIeHI1.HX, nSl
KOTOpbIX yrJIbI <p y,AOBneTBOp.HIOT ycnOBI11O
d sin <p = mA (m = 0, 1, 2, ...).
Ha6JIIOaIOTC.H rJlaBHbIe MaKC1:IMYMbI 1:IcppaKl.\1:I0HHOH KapTIiHbI. STY cpopMyny
t.JaCTO Ha3bIBalOT OPMYJlOH HpaKHOHHOH pemeTKH. B HeH m Ha3bIBaeTCSI
nOp.HKOM rnaBHoro MaKC1:IMYMa. Memy rJIaBHbIMI1 MaKCI1MYMaMll pacnOJIa-
raeTC.H (N - 2) cna6bIX no60t.JHbIX MaKCI1MYMOB, HO Ha cpoHe .HpKIiX rJIaBHbIX
MaKCI1MYMOB OHIi npaKTIit.JeCKH He BIiHbI. I1pli YBeJlI1qeHI11i '-Ilicna IllTpI1XOB N
(meJIeH) rJIaBHbIe MaKCHMYMbI. OCTaBa.HCb Ha npemHI1X MeCTax. CTaHOB.HTC.H Bce
60JIee pe3KIiMH.
I1pli Ha6JIIOeH1:I1:I I1cppaKl.\1:I1i B HeMOHOXpOMaTI1qeCKOM (6eJIOM) CBeTe Bce
rJlaBHbIe MaKCHMYMbI, KpOMe HyneBoro l.\eHTpaJIbHOrO MaKCI1MYMa, OKpallleHbI.
STO 06'b.HCH.HeTC.H TeM, qTO, KaK B1:IHO H3 cpOPMYJlbI sin <p = mA , pa3n1:lqHbIM
d
,AJIIiHaM BOJIH COOTBeTCTBYIOT pa3J1liqHbIe yrJIbI, Ha KOTOpbIX Ha6J1IO,AaIOTC.H I1H-
TepcpepeHl.\IiOHHbIe MaKCI1MYMbI. PaY>KHa.H nOJlOCKa, coep>Kaa.H B 06eM
cnyqae ceMb l.\BeTOB - OT cp1:l0neTOBoro ,AO KpaCHoro (CqIiTaeTC.H OT l.\eHTpanb-
Horo MaKcIiMYMa), Ha3bIBaeTC.H ,AlicppaKl.\IiOHHbIM cneKTpOM.
UUI1p1:lHa cneKTpa 3aBliCHT OT nOCTO.HHHOH pellleTK1:I H YBeJl1:lq1:lBaeTC.H npI1
YMeHbllleHIiH d. MaKCIiMaJlbHbIH nOp.HOK cneKTpa onpeeJI.HeTC.H 1:13 ycnOBI1S1
d 1
A NA
sin <p S 1, T.e. 111max
518

17.11. TIoH.HTHe 0 rOJIOrpa4>HH
IIpH 06bIqHOM <poTorpaqmpoBaHHH 06'beMHbIX npe,AMeTOB HX H306pameHH.H,
C03aBaeMbIe 06'beKTHBOM, <PHKCHPYIOTC.H Ha nnOCKOH <poTonneHKe. Bnaroap.H
nepcneKTHBe H TeH.HM OT 60KOBoro OCBeeHHSI npe,AMeTOB, a TaKme "npHBbIq-
Ke" BHeTb 06'beMHble npeMeTbI nJlOCKHMH (onTH'IeCKa.H CHCTeM8 r na3a H <po-
ToannapaT8 OHHaKOBbl: JlHH3a--<poTonnacTHHKa, xPycTanHK--CeTqaTKa) npH
paCCMaTpHBaHHH <poTorpa<pHH MbJ nOJlyqaeM npeCTaBneHHe 0 npOCTpaHCTBeH-
HOM pacnonomeHHH npeMeTOB.
OHaKo 06bIqHa.H <poTorpa<pH.H eCTb BecbMa YCJlOBHOe H306pameHHe 06'beMHOrO
npe,AMeTa. B 3TOM H306pameHHH He,AOCTaeT 3HaqHTeJlbHoii 'IaCTH Toii HH<popMaHH
o npeMeTe, KOTOpyro MomHO nonyqHTb npH ero HenocpeCTBeHHOM paCCMaTpHBa-
HHH (Henb3.H. HanpHMep, YBHeTb, KaK npeMeT BbIrnSlHT C60KY, C3a,Z:(H).
8TO CBSl3aHO C TeM, 'ITO <poTonnacTHHK8 perHcTpHpyeT TonbKO OCBell.{eH-
HOCTb, T.e. KBal(paT aMnJlHTYbI CBeTOBblX BonH. OTpameHHbIX OT pa3nHqHbIX
TO'IeK npe.nMeTa. 1.JeM Slpqe TOqKa npe,AMeTa, TeM 60nbIlIe aMnnHTY,Aa BeKTO-
pa 3J1eKTpH'IeCKOH Hanp.HmeHHOCTH, TeM CBeTnee H306pameHHe 3TOH TO'IKH H8
n03HTHBe. B TO me BpeM.H CBeTOBa.H BonHa B 3TOH TOqKe npOCTpaHCTB8 xa-
paKTepH3yeTC.H He TOJlbKO aMnnHTYOH, HO H <pa30ii Kone6aHHH. A Ha noqep-
HeHHH <poTonnacTHHKH cpa3a Kone6aHHH He CKa3bIBaeTCSI. ct>a38 me KOJIe6a-
HHH 3aBHCHT OT paCCTO.HHH.H O COOTBeTCTBYIOll.{eH TOqKH npeMeTa. CJIeOBa-
TenbHO, <pOTorpa<pH.H He HeceT HHKaKOH 06'beKTHBHOH HH<popMaHH 0 paCCTO.H-
HHH O pa3nHqHbIX TOqeK 06'beMHOrO 06'beKTa C'beMKR.
BnepBbIe 60Jlee nonHYIO, qeM B npOCTOH <pOTorpa<pHH, perHCTpaHIO pac-
Ce.HHHOrO npeMeTOM CBeTa yanocb nonyqHTb B 1948 r. aHrnHHcKoMY <pH3H-
KY .n. fa60PY. OH Ha3BaJl npe,AJlOmeHHbIH HM cnoc06 I'onorpa<pHeH (OT rpe-
'leCKOrO "roJloc" -- eJlblH, BeCb).
rOJlOrpaH.H -- OC06bIii cnoc06 3anHCH Ii nocneYIOero BOCCTaHOBJleHH.H
BonHOBoro nOJl.H, OCHOBaHHbIH Ha perHCTpaHH HHTep<pepeHHOHHOH KapTHHbI.
IIepBble nonblTKH nonyqeHH.H ronorpaMM npenpHHHMaJIHCb .n. fa60pOM H
ero COTpYHHKaMH C Hcnonb30BaHHeM H3J1yqeHH.H PTYTHOH naMnbl. HCTHHHbIH
nepeBopOT B pa3BHTHH H Hcnonb30BaHHH rOJlOrpa<pHH npOH30IlIeJl B 1960 r.
C nO.HBneHHeM Jla3epOB (CM.  20.11).
IIpH 3anHCH na3epHoH rOJIOrpaMMbI Ha nJIeHKe JIa3epHbJH nyqOK c nOMO-
Il.\blO nOJlynocepe6peHHoro 3epKaJla 3 enHTC.H Ha Be 'IaCTH (pHC. L 7.23).
OHa qaCTb nyqKa HanpaBJI.HeTC.H HenOCpe,ACTBeHHO Ha nJleHKY (onopHa.H BOJl-
Ha), a pyra.H Ha <poTorpa<pHpyeMbIH 06'beKT H, OTpama.HCb OT Hero, nona,Aa-
eT Ha nneHKY (CHrHanbHa.H BonHa). OnopHaSl H CHrHaJIbHa.H BonHbI, .HBJI.H.HCb
KorepeHTHblMH H HaKJIabIBa.HCb Pyr Ha Pyra, 06pa3YlOT Ha <pOTonnaCTHHKe
HHTep<pepeHHOHHYIO KapTHHY. IIocJle npO.HBneHH.H <pOTonnaCTHHKH H nOJlyqa-
eTC.H TaK Ha3bIBaeMa.H rOJlOrpaMMa -- 3aperHCTpHpOBaHHa.H H8 <pOTOnJIaCTHH-
Ke HHTep<pepeHHOHHa.H KapTHHa, 06pa30BaHHa.H npH cnomeHHH onopHOH H
CHrHanbHOH BOJIH. fOJlOrpaMMfi coepmHT HH<popMaHIO He TOJlbKO 06 aMnnH-
Tye BOJIH, HO H 06 HX <pa3e, TaK KaK aMnJIHTya pe3ynbTHpYIOero KOJle6a-
HH.H (CJleOBaTeJlbHO, H OCBell.{eHHOCTb <poTOnJlaCTHHKH H ee np03paqHOCTb noc-
ne n03HTHBHOH 06pa60TKH) 3aBHCHT OT aMnnHTyn CKnanbIBaeMbIX BOJIH H pa3-
HOCTH HX <pa3:
A 2 = A; +  + 2Al cos(a 2 - ad.
519

JI83epHoe
H3nY'leHHe
OnopHbIH
nY'lOK
q>OTonneHK8
3
Puc. 17.23
TaKliM 06pa30M, 6JIarO,D.apH llHTepcpepeHllH npOllCXO,D.HT npeo6pa30BaHlie
cpa30BbIx CooTHoIIIeHllH Mem,AY BonHaMli B COOTHOIIIeHll.H aMnnHTYlI. B pe3YJlb-
TaTe qero cpa30BbIe COOTHOIIIeHH.H Mem,u.y BOJlHaMH 6Y,D.YT 3acpHKCHpOBaHbJ B
BH,Ae COOTBeTCTBYIOHX H3MeHeHHll np03paqHOCTli cpOTOnJlaCTllHKH.
TIpH rOJIOrpacplitIeCKOM cnoco6e 3anliCH H306pameHH.H He npllMeH.HIOTcsr HH
JIllH3bI, HH ,u.pyrlle YCTpOllCTBa, cpOpMHpYIOHe H306pameHlle. Kam,Aa.H TOqKa
npe,u.MeTa, paCCeHBa.H CBeT, nOCbIJIaeT ero Ha BCIO rOJIOrpaMMY, H Kam,u.a.H TOq-
Ka rOJlOrpaMMbI nOJlyqaeT CBeT OT Bcero npe,u.MeTa. rOJlorpacpHqeCKOe H306pa-
meHHe npe,D.MeTa HH 'B Koell Mepe He COOTBeTcTByeT ero BHelliHeMY BH,AY.
TIpoecc nOJlyqeHHH H306pameHH.H C nOMOblO rOJIOrpaMMbI Ha3bIBaeTCH
BOCCTanOBJleHHeM.
J{nSl BOCCT8HOBJIeHHH H306paJKeHHH (pHC. 17.23, 6) rOJlOrpaMMa nOMea-
eTC.H B TO me caMoe nOJlOmeHHe, r,D.e OHa HaXO,D.HJIaCb ,D.O perHCTpaHH. Ee
OCBeaIOT onopHbIM nyqKOM Toro me JIa3epa (BTOpa.H tIaCTb Jla3epHOro nytIKa
nepeKpbIBaeTCH ,D.llacpparMOll). B pe3ynbTaTe ,u.HcppaKHll CBeT8 Ha HHTepcpe-
peHl'lHOHHOH CTpYKType rOJIOrpaMMbI BOCCTaHaBJIHBaeTC.H KOnH.H npe,D.MeTHOii
BOJIHbI, 06pa3YIOa.H 06'beMHOe MHHMoe H306pameHHe. pacnOJIOmeHHOe B TOM
MeCTe, r,u.e npe,u.MeT HaXO,D.HJIC.H npH rOJIOrpacpHpOBaHHH.
PaCCMaTpHBasr H3 pa3HbIX nOJIOmeHHH 06'beMHOe H306pameHHe npe,u.MeTa,
,D.aBaeMOe ronorpaMMOH. MomHO YBH,D.eTb 60nee y,D.aJIeHHbJe npe,D.MeTbI, 3a-
KpbITbIe 60Jlee 6JIH3KHMH 113 HHX. rOJlorpaMMY MomHO pa3,D.eJlllTb Ha HeCKonb-
KO qacTeii. Ho ,D.ame MaJIa.H ee qaCTb BOCCTaHaBnHBaeT 11306pameHlle nOJIHO-
CTblO .
Ha O,D.HOH cpOTOnJlaCTHHKe MomHO nOOqepe,u.HO 3anHcaTb HeCKOJIbKO H306-
paJKeHHH pa3nlltIHbIX npe,D.MeTOB. J{JI.H 3Toro nocne 3KcnOHHpOBaHll.H o,u.Horo
520

npe,AMeTa ,ApyroH paCnOJIaralOT B 11:HOM MeCTe 11:JI11: npOCTO MeHHIOT HanpaBJIe-
H11:e OnOpHOrO nyqKa CBeTa. Kamoe H306pameH11:e BOCCTaHaBJl11:BaeTC.H 6e3 no-
MeX co CTOpOHbI pyrHx H306pameH11:H.
IIpHMeHeH11:.H rOJlOrpacpHH pa3H006pa3HbI, HO Ha11:6oJlee BamHbIM11: .HBn.HIOT-
C.H 3an11:Cb 11: XpaHeH11:e 11:HcpOpMal.\11:11:. MeTObI rOJIOrpacp11:11: n03BOJI.HIOT 3an11:-
CbIBaTb B COTH11: pa3 60JIbllIe CTpaH11:l.\ neqaTHOrO TeKCTa, qeM MeTO,AbI 06bl'1-
HOH M11:KpOcpOTOrpa<p11:11:.
17.12. TIOJI.HpH3aUH.H CBeTa. 3aKoH MaJIlOca
KaK OTMeqaJlOCb BbIllIe, CBeT - 3TO nOnepeqHa.H 3neKTpOMarH11:THaH BOJI-
Ha: BeKTopbI Hanp.HmeHHOCTH E 3JIeKTpHqeCKOrO nOJl.H H HHYKl.\HH B Mar-
HHTHoro nOJl.H BOJIHbI B3aHMHO nepneH,AHKyn.HpHbI H KOJle6J1IOTC.H nepneHH-
KYJI.HpHO BeKTOpy CKOpOCTH V pacnpOCTpaHeHH.H BOJlHbI. IIonepeqHocTb CBeTO-
BbIX BOJIH HapYlliaeT HX oceBYIO CHMMeTpHIO OTHOCHTeJlbHO HanpaBJIeHH.H pac-
npOCTpaHeHH.H, T.e. CBeTOBa.H BOJIHa HMeeT nOnepeqHYIO aH11:30TponHIO. <I:>11:-
3HqeCKOH xapaKTepHcTHKOH nOnepeqHOH aHH30TponHH .HBn.HeTC.H 11:X nonHpH-
3al.\11:.H. IIJloCKOCTb, B KOTOpOH npOHCXO,A.HT KOJIe6aHH.H BeKTopa E, Ha3bIBaeT-
C.H nJIOCKOCTblO KOJIe6aHHH, a nJlOCKOCTb, B KOTOpOH npOHCXO,AHT KOJle6aHH.H
nJlOCKOCTLIO nOJI.HpH3aHH.
CBeT. H3J1yqaeMbIH KaKHM-TO HCTOqHHKOM,
npeCTaBJIHeT C060H CYMMapHoe 3neKTpOMarHHT-
Hoe 11:3J1eH11:e MHomeCTBa aTOMOB. ATOMbI me
11:3JIaIOT CBeTOBble BOJlHbI He3aB11:C11:MO D;pyr OT
APyra, n03TOMY CBeTOBa.H BonHa, H3JIaeMaH Te-
JIOM B l.\eJlOM, xapaKTepH3yeTC.H BceB03MOmHbIMI1
paBHOBepO.HTHbIM11: HanpaBJleHH.HMH KOJle6aHHH
CBeTOBoro BeKTopa E (pHC. 17.24. a). CBeT co
BceB03MomHbIMH paBHOBepOHTHbIMH opHeHTal.\H.H-
M11: BeKTopa E OTHOC11:TeJlbHO OC11: pacnpOCTpaHe-
HHH Ha3bIBaeTC.H eCTeCTBeHHLIM (HenOJI.HpH30BaH-
HbIM). CBeT, B KOTOpOM Ha6JIIOaeTC.H npeHMyrn;ecTBeHHoe HanpaBJleHHe KOJIe6a-
HHH BeKTopa E (HO He HCKJIlOqHTeJIbHoe!) (pHC. 17.24, 6), - qaCTHqllO no-
JI.HpH30BaHHLIH. CBeT, B KOTOpOM BeKTOp E KOJIe6J1eTCSI B onpe,AeneHHOH
nJIOCKOCTH (pHC. 17.24, 6), Ha3bIBaeTC.H nOJl.HpH30BaHHLIM (JlHHeHIIO nOJI.HpH30-
BaHHbIM). QaCT11:qHO nOJlHp11:30BaHHbIH CBeT npe,ACTaBn.HeT C060H COqeTaHHe ec-
TeCTBeHHoro H JlHHeHHO nOJl.HpH30BaHHoro CBeTa. lIon nOJI.HpH3al.\HeH CBeTa no-
HHMalOT BbIeJleHHe H3 eCTeCTBeHHoro CBeTa CBeTOBbIX KOJle6aH11:H C onpeeJIeH-
HbIM HanpaBJleHHeM 3J1eKTpHqeCKOrO BeKTopa.
Ha6J1IOeHH.H nOKa3bIBaIOT, qTO npHqHHbI nonHpH3al.\HH CBeTa MorYT 6bITb
pa3JIHqHbIMH, HO HaH60JIee qaCTO nOJl.HpH3al.\H.H npOHCXOHT B Tpex CJIyqaHX:
a) npH npOXOmeH11:H CBeTa qepe3 HeKOTopbIe KpHCTaJlJlbI (HanpHMep, TypMa-
nHH); 6) npH OTpameHHH H npeJlOMJleHHH CBeTa Ha rpaHHl.\e pa3ena BYX
H3J1eKTpHKOB; B) npH BOHHOM JIenpeJlOMJleHHH CBeTa.
BeKTopa B,
a
6
*
Puc. 17.24
6
t
521

1. ECJIH HanpaBHTb eCTeCTBeHHbIH CBeT nepneH,D;HKYJI.HpHO nJlaCTHHKe TYP-
MaJlHHa T 1 (pHC. 17.25), BbIpe3aHHoH napaJIJleJlbHO OnTHlIeCKOH OCH 00' (Ha-
npaBJleHHe B KpHCTaJlJle, OTHOCHTeJlbHO KOTOpOrO aTOMbI KpHCTaJlJlHqeCKOH pe-
meTKH paCnOJlOmeHbI CHMMeTpHqHO), H BpaIUaTb KpHCTaJlJl T 1 BOKpyr HanpaB-
JleHH.H JIa. TO HHKaKHX H3MeHeHHH HHTeHCHBHOCTH npomen;mero qepe3 TYP-
MaJIHH CBeTa He Ha6J1IOn;aeTC.H. ECJlH Ha nYTH Jlyqa nOCTaBHTb BTOPYIO nJlac-
THHKY TypManHHa T 2 H BpaaTb ee BOKpyr HanpaBJleHH.H JIyqa, TO HHTeH-
CHBHOCTb CBeTa, npome,D;mero qepe3 nJlaCTHHKH, MeH.HeTC.H B 3aBHCHMOCTH OT
yrJla a Mem,D;y OnTHqeCKHMH OC.HMH KpHCTaJlJlOB no 3aKOHY MaJllOca:
I = 10 cos 2 a,
r,D;e 10 H I - COOTBeTCTBeHHO HHTeHCHBHOCTH CBeTa, na,D;alOero H8 BTOpOH
KpHCTaJlJl H Bblme,D;mero H3 Hero.
ECTeCTseHHbIH
cseT
0'
I
TIonSlpH30s8HHbIH
cseT
0'
Puc. 17.25
Pe3YJlbTaTbI onbITOB MomHO 06'bHCHHTb TaK: nepBa.H nJlaCTHHKa TypMaJIHHa
nponycKaeT KOJle6aHH.H TOJlbKO onpe.n;eJIeHHOrO HanpaBJleHH.H (Ha pHC. 17.25 3TO
HanpaBJleHHe nOKa3aHO CTpeJlKOH AB). T.e. npe06paayeT eCTeCTBeHHbIH CBeT B
nJIOCKOnOJl.HpH30BaHHbIH (n03ToMY 3TY nJIaCTHHKY Ha3bIBalOT nOJl.HpH3aTopoM).
Kone6aHH.H, BeKTOp E KOTOpbIX nepneH,D;HKYJlHpeH OnTHqeCKOH OCH. nJlaCTHH-
Ka npaKTHqeCKH He nponYCKaeT - BOJlHa CHJlbHO norJlOaeTC.H. 3aBHCHMOCTb
nOKa3aTeJl.H nOrJlOeHH.H BeeCTBa OT HanpaBJIeHH.H KOJIe6aHHH BeKTopa E
Ha3bIBaeTC.H AHXPOH3MOM. BTOpa.H me nnaCTHHKa TypManHHa B aaBHCHMOCTH
OT ee opHeHTaHH H3 nOJI.HpH30BaHHoro CBeTa nponycKaeT 60JIbmyIO HJlH MeHb-
mylO ero qaCTb, KOTOpa.H cooTBeTcTByeT KOMnOHeHT€ E,
napaJIJI€JlbHOH OCH BTOpOH nJlaCTHHKH TypMaJlHHa. Ma
pHCYHKa 17.26 BH,D;HO, qTO
E = Eo cosa.
OTclOa CJIe,D;yeT, qTO E 2 = E5 cos 2 a. TaK KaK HH-
'I'eHCHBHOCTb CBeTa nponopHOHaJIbHa KBa,D;paTY Ha-
np.HmeHHOCTH 3JIeKTpHqeCKOrO nOJl.H: I - E 2 , 1'0
I = 10 cos 2 a. I1oCJIe,D;Hee BbIpameHHe H npe,D;CTaBJI.HeT
C060H 3aKOH MaJIlOca.
I1J1aCTHHKa T2' cJlyma.ru;a.H ,lJ,JI.H aHaJIH3a creneHH nOJl.H-
pHHH CBeTa, HaabIBaeTC.H aHaJIH3aTopoM.
Puc. 1 7.26
522

TIOJIHpH3aTOp no CBoeH KOHCTPYKHH HHqeM He OTJIHqaeTC.H OT aHaJIH3aTO-
pa. Pa3HHa B «PYHKH.HX: nOJl.HpH3aTOp BbI.n;eJl.HeT H3 eCTeCTBeHHoro CBeTa ny-
qOK C OlIHHM HanpaBJleHHeM KOJle6aHHH BeKTopa if;, a aHaJlH3aTOp onpe.n;eJlH-
eT, KaKOBO HanpaBJleHHe 3THX KOJle6aHHH. MMeHHO n03TOMY nOJlHpH3aTopbI H
aHaJlH3aTOpbI HOC.HT 06ee Ha3BaHHe nOJUlpOLI.
2. OnbIT nOKa3bIBaeT, qTO npH OTpameHHH 11
npeJlOMJleHHH CBeTa Ha rpaHHe pa3.n;en8 .n;BYX ,AH-
3J1eKTpHKOB nOJIHpH3YIOTC.H H oTp8meHHbIH. H npe.
JIOMJleHHbIH JlyqH. Xap8KTep me HX nOJIHpH38HH
pa3J1HqeH: oTp8meHHbIH CBeT nOJI.HpH3yeTC.H npe-
HMym;ecTBeHHO B nJlOCKOCTH, nepneH.n;HKYJI.HpHOH
nnOCKOCTH na,AeHHH (H8 pHC. 17.27 OHH 0603H8-
qeHbI TOqK8MH). 8 npenOMJIeHHbIH CBeT nOn.HpH3Y-
eTC.H npeHMyecTBeHHO B nJIOCKOCTH naneHH.H (Ha
pHC. 17.27 3TH Kone6aHH.H H306pameHbI CTpen-
KaMH). CTeneHb nOJlHpH3aHH 3aBHCHT OT yrJIa
na.n;eHH.H JlyqeH H nOKa3aTeJl.H npeJIOMJleHHH.
.n. BplOCTep YCTaHOBHJI, qTO cyecTByeT TaKOH
yroJl na,AeHHH as, npH KOTOpOM OTpameHHbIH CBeT
CTaHOBHTC.H nOJlHOCTblO nOJl.HpH30BaHHbIM B nJloc-
KOCTH, nepneH.n;HKY JIHpHOH nJlOCKOCTH na.n;eHHH.
TI03TOMY yrOJl as Ha3bIBaIOT yrJlOM nOJDIOH nOJlHpH3aH (HJlH yrJloM BpIOCTe-
pa). YCTaHOBJleHO, qTO yrOJI as BMecTe C COOTBeTCTBYIOHM eMY yrJloM npeJlOM-
JleHH.H J3 COCTaBJI.HeT B CYMMe 90°, T .e.
Puc. 17.27
1t
as + J3 = - .
2
MCnOJIb3Y.H 38KOH npenOMJleHH.H, nonyqHM
sina B sin as t
"21 =  = Sin(  _ as ) = g as,
I 1 = tg as 1 - 38KOH BplOcTepa.
TIpeJlOMJleHHbIH Jlyq npH yr Jle na.n;eHH.H as nOJlHpH3yeTC.H MaKCHMaJlbHO,
HO He nOJlHOCTblO.
3. HBneHHe ,nBoHHoro JlyqenpeJloMJleHHH. T.e. p83,ABaHBaHH.H Kam.n;oro na-
.n;81Oero H8 .n;BoHKOnpeJlOMnHIOHH KpHCT8nn CBeTOBoro nyqK8, Ha6JIIO,AaeTCH
npH nepeXOlIe CBeTa 1-13 H30TponHOH cpe.n;bI B aHH30TponHYIO.
3TO HBJleHHe BnepBbIe 06H8pymHJI y HCJl8H.n;CKoro IllnaTa (K8JIbIlHTa C8C0 3 )
3. BapTOJIHH. ECJlI1 Ha TOJICTbIH KpHCTaJIn HCJIaH.n;CKoro IllnaTa H8npaBHTb
B.n;OJIb ero OnTHqeCKOH OCH Y3KHll nyqoK CBeTa, TO H3 KpHCTanna BbIH,AYT .n;Ba
npOCTpaHCTBeHHO pa3.n;eJIeHHbIX JIyqa, napaJlJIeJlbHbIX .n;pyr .n;pyry H na.n;alOe-
MY Jlyqy (pHC. 17.28). .name B TOM CJIyqae, Kor.n;a nepBHqHbIH nyqOK na.n;a-
eT Ha KpHCTaJlJI HOpM8J1bHO, npeJlOMJIeHHbIH nyqOK pa3.n;eJI.HeTC.H Ha .n;B8
(pHC. 17.29), npHqeM O.n;HH H3 HHX HBJI.HeTCH npo.n;OJImeHHeM nepBHqHOrO
(06bIKHoBeHHbIH nyq - 0), a BTOpOH OTKJIOH.HeTCH (He06bIKHoBeHHbIH nyq - e).
BbIIlle,AIllHe H3 KpHCTaJIJI8 JIyqH nnOCKOnOJl.HpH30BaHbI BO B3aHMHO nepneH.n;H-
KYJl.HpHbIX nJlOCKOCT.HX: 06bIKHoBeHHbIH JIyq B nJlOCKOCTH, nepneH.n;HKYJl.HpHOH
523

Puc. 17.28
e
o
Puc. 17.29
r naBHoMY CeqeHHIO KpHCTaJlJla, He06bIKHoBeHHbIH Jlyq - B r JlaBHOH nJlOCKO-
CTH. 06bIKHoBeHHbIH Jlyq no.c;qHH.HeTC.H 3aKoHY npeJlOMJleHH.H, a )J;JI.H He06bIK-
HOBeHHoro nyqa 3TOT 3aKOH He BbInOJlH.HeTC.H: nOKa3aTeJlb npeJlOMJleHH.H He-
06bIKHOBeHHoro Jlyqa .HBJI.HeTC.H nepeMeHHOH BeJlHqHHOH, 3aBHC.HeH OT yrJla na-
.c;eHH.H. IIocJle BbIxo.c;a H3 KpHCTaJlna, eCJlH He npHHHMaTb BO BHHMaHHe non.H-
pH3aHIO BO B3aHMHO nepneH,AHKYJl.HpHbIX nJlOCKOCT.HX, 3TH ,ABa nyqa HHqeM
.c;pyr OT )J;pyra He OTJlHqaIOTC.H.
HBneHHe nOJl.HpH3aHH CBeT8 HMeeT 60JlbllIOe np8KTHqeCKOe npHMeHeHHe
B MallIHHOCTpoeHHH Ii CTpOHTeJlbHOH HH.c;YCTPHH .HBneHHe nOn.HpH3anHM
CBeTa HcnOJlb3YlOT .c;JI.H HCCJle.c;OBaHH.H Hanp.HmeHHH, B03HHK81OHX B Y3J1ax
MallIHH H CTpOHTenbHbIX KOHCTPYKHH. B cpOTorpacpHIi nOJl.Hpli3anHOHHbIe
CBeTocpHJlbTpbI npHMeH.HIOTC.H .c;JI.H rallIeHH.H 6J1HKOB, B03HHK81OHX npH 3ep-
KanbHOM OTpameHHH CBeTa OT cpOTorpacpHpyeMbIx 06'beKTOB. McnOJlb3Y.H .HB-
JleHHe nOJl.HpH3anHH, MomHO nJlaBHO perynHpoBaTb HHTeHCHBHOCTb CBeTOBO-
ro H3J1yqeHH.H. .LI:JI.H 3Toro nepe.c; HCTOqHHKOM CBeTa CTaB.HT nOJl.HpH3aTOp H
aHaJlH3aTOp, OTHOCHTeJlbHbIM nOBopOTOM KOTOpbIX .c;OCTHraIOT He06xo.c;HMoro
3cpcpeKTa.
HBJleHHe nOJl.HpH3aHH CBeTa npHMeH.HIOT T8Kme B )J;eKopaTHBHbIX nen.Hx:
npH ocpopMneHHH BHTPHH, YCTpOHCTBe TeaTpaJlbHbIX .c;eKopanHH H T..c;.
3HaqeHHe a6COJllOTHOrO nOKa3aTeJl.H npeJlOM-
neHH.H cpe.c;bI onpe.c;eJl.HeTC.H B OCHOBHOM CBOH-
CTBaMH 3TOH cpe.c;bI; O)J;HaKO OHO 3aBHCHT ee
OT .c;JlHHbI BonHbI (qaCTOTbI) CBeTa.
IIooqepe.c;Ho nponYCKa.H qepe3 TpexrpaHHYIO
npH3MY nyqKH MOHoxpOMaTHqeCKOrO CBeTa pa3-
HOH BeTHOCTH, HanpaBneHHbIe Ha rpaHb npH-
3MbI no.c; O.c;HHM H TeM me yr JlOM na.c;eHH.H
(pHC. 17.30, r.c;e m - eJlb, CP - cpHJlbTp,
S :3 - 3KpaH), MomHO 06HapymHTb, qTO cpHOJleTO-
BbIe JlyqH OTKJlOH.HIOTC.H OT nepBOHaqanbHOrO Ha-
npaBJleHH.H CHnbHee, qeM KpaCHbIe. CJle,AOBaTenb-
HO, yron npenOMneHH.H KpaCHbIX JlyqeH PK 60Jlb-
Pit> (PK>PI)J)' M3 3aKOHa npeJlOMJleHH.H
I
I
m;<t>
I
I
I
I
I
I
I
'">---,
I
I
a
Puc. 1 7.30
llIe, qeM <pHOJleTOBbIx
17 .13. HcnepcHH CBeTa
sin a
-=n
sin J3
CJle)J;yeT, qTO nil> > n K . A TaK KaK a6COJllOTHbIH nOKa3aTeJlb npeJlOMJleHH.H
524

c
n = -, re C H V - CKOpOCTH CBeTa COOTBeTCTBeHHO B BaKYYMe H Cpe,Ae TO
V
OTCIOa BbITeKaeT, qTO KpaCHbIH CBeT paCnpOCTpaH.HeTC.H B cpee 6bICTpee, qeM
qmoneTOBbIH:
V K > VIt>.
TIOCKOJlbKY BeT, BOCnpHHHMaeMbIH rJla30M. OnpeeJl.HeTC.H TOJlbKO qaCTO-
TOH CBeTOBOH BOJlHbI. TO BeT npH nepexoe H3 BaKYYMa B BeeCTBO HnH H3
OHOrO BeeCTBa B Pyroe He H3MeHHeTCH.
3aBHCHMOCTb CKOpOCTH pacnpOCTpaHeHH.H CBeTOBbIX BOJIH B cpe)J;e (nOKa3a-
TeJl.H npeJlOMJleHH.H cpebI) OT qaCTOTbI ()J;nHHbI BOJlHbI) CBeTa Ha3bIBaeTC.H ABC-
nepcHeit: CBeTa.
n
200
600
Puc. 17.31
npenOMneHH.H YBenHqHBaeTCH
A..HM
.D:HcnepcHH CBeTa npeCTaBn.HeTCH B BH,Ae 3a-
BHCHMOCTH n = f ( v) . OnbIT nOKa3bIBaeT, qTO n.H
60nbIlIHHcTBa BeII.1;eCTB nOKa3aTeJlb npeJlOMneHHH
YMeHbIlIaeTC.H C YMeHbIlIeHHeM qaCTOTbI (c YBe-
nHqeHHeM A). .D:HcnepcHIO TaKoro pO,Aa Ha3bIBa-
lOT HOpMaJlLHOH. KpHBa.H 3aBHCHMOCTH n = f (A)
n.H CTeKna (pHC. 17.31) - KpHBa.H Hcnep-
CHH - nOKa3bIBaeT, qTO 3Ta 3aBHCHMOCTb HenH-
HeHHaH. TIoKa3aTeJlb npenOMJleHH.H CTeKna B 06-
JlaCTH KOpOTKHX ,AJlHH BonH H3MeHHeTC.H 6bICT-
pee, qeM B 06naCTH nHHHbIX.
B napax HO,Aa H B HeKOTopbIX mHlUWCTHX Ha-
6nlOanH aHOMaJlbHYIO HBcnepCHIO: nOKa3aTenb
C YBenHqeHHeM A, T.e. B HHX CKOpOCTb pacnpOCT-
paHeHHH v K < vlt>.
.D:HcnepcHIO MomHO 06'bHCHHTb C TOqKH 3peHH.H 3neKTpoMarHHTHoH 'l'eOpHH.
TaK KaK CKOpOCTb CBeTa B BaKYYMe He 3aBHCHT OT qaCTOTbI, a HCnepCH.H
Ha6nlO,AaeTC.H TonbKO B BeII.1;eCTBe, TO OHa CBH3aHa CO CTpoeHHeM BeII.1;eCTBa
(.AJlH 06'b.HCHeHHH Hcnonb3yeM 3neKTpOHHYIO TeopHIO CTpOeHH.H BeII.1;eCTBa, pa3-
pa60TaHHYIO X. JIopeHuoM B 1880 r.).
TIocKonbKY aTOMbI H MoneKYJlbI caMH Moryr .HBnHTbCH HCTOqHHKaMH 3neKT-
pOMarHHTHbIX Kone6aHHH, OHH He OCTaIOTC.H 6e3yqacTHbIMH, Kora Ha HHX B03-
,AeHcTByeT BHeIlIH.HH 3J1eKTpOMarHHTHaH BOJlHa (CBeT). B BeII.1;eCTBe B03HHKalOT BbI-
HymeHHbIe 3neKTpOMarHHTHbIe Kone6aHH.H 3neKTpOHOB B aTOMax. ATOMbI Ha-
qHHalOT H3J1yqaTb 3neKTpOMarHHTHbIe BOJlHbI, KOTopbIe HaKJlabIBaIOTC.H Ha BHeIll-
HIOIO BOJIHY. qacToTbI BbIHymeHHbIX Kone6aHHH COBna,AaIOT C qaCTOTOH BHeIll-
HeH BOJlHbI, HO HX cpa3bI MorYT OTnHqaTbC.H OT cpa3bI BHeIllHeH BonHbI (B 3aBH-
CHMOCTH OT CTPYKTypbI qaCTHU BeII.1;eCTBa. HX opHeHTaUHH H T .J1..). 3TO npHBO-
.AHT K TOMY, qTO CKOpOCTH npOXOmeHH.H pe3Y nbTHpYIOeH 3J1eKTpOMarHHTHOH
BonHbI qepe3 aHHoe BeeCTBO npH pa3HbIX qaCTOTax 6YYT HeOHHaKOBbIMH.
Pa3J1HqHe Memy CKOpOCT.HMH TeM 60JlbIlIe. qeM CHnbHee BbIHym,AeHHbIe KO-
ne6aHH.H 3J1eKTpOHOB, T.e. qeM 6J1Hme qaCTOTa CBeTOBOH BOJlHbI K c06CTBeH-
HOH qaCTOTe Kone6aHHH 3J1eKTpOHOB. TI03TOMY CKOpOCTb CBeTa B BeII.1;eCTBe 3a-
BHCHT OT qaCTOTbI CBeTOBOH BOJlHbI. .D:m. MaKCBeJlJl nOKa3aJI, qTO
525

c
U=-
Jii.'
r,ne E H  - ,nH3neKTpHqeCKa.H H MarHHTHa.H npOHHD;aeMOCTH BeeCTBa. B
pe3ynbTaTe BbIHYJK,neHHbIX Kone6aHHH MOJleKYJl Cpe,AbI H3MeH.HeTC.H nOn.HpH3Y-
eMOCTb MOJleKYJl H. COOTBeTCTBeHHO. E.
17.14. ,n:HcnepcHoHHLIH cneKTp
Cne,nCTBHeM ,nHcnepCHH .HBn.HeTC.H pa3nomeHHe CBeTa npH npOXOJK,neHHH ero
qepe3 npH3MY Ha COCTaBJI.HIOHe C pa3nHqHbIMH ,AnHHaMH BonH (MOHoxpoMa-
THqeCKHe JlyqH).
llpH pa3J10meHHH 6eJloro CBeTa, T.e CBeTa B BH-
J],HMOM ,nHana30He, CO,Aepmaero ,nnHHbI BonH B ,nH-
ana30He 380-760 HM, B03HHKaeT pa,nymHa.H nOJloc-
Ka (pHC. 17.32), KOTOpylO Ha3bIBalOT cneKTpOM.
<1> BnepBbIe no,np06HO HCCJle,nOBaJl ,nHcnepCHIO CBe-
Ta 11. HbIOTOH. OH ,nOKa3aJl, qTO He npH3Ma OKpa-
IIIHBaeT CBeT, a 6eJlbIH CBeT - CJlOHLIH, OH CO-
CTOHT H3 npOCTbIX (MOHoxpOMaTHqeCKHX) JlyqeH, KO-
TopbIe npH npoxom,neHHH qepe3 npH3MY OTKnOH.HIOT-
C.H, HO He pa3J1araIOTC.H, H TonbKO B cOBoKynHocTH MOHoxpOMaTHqeCKHe JlyqH
,nalOT oyeHHe 6eJloro CBeTa.
.u;pyroH BamHbIH BbIBOJ]" KOTOpbIH ClIeJlan 11. HbIOTOH: JlyqH pa3J1HqHOrO
D;BeTa npeJIOMJI.HIOTC.H no- pa3HOMY .
TaKHM 06pa30M, C nOMOblO npH3MbI KaK H C nOMOblO ,nH<ppaKD;HOH-
HOH peIlleTKH, MomHO nOJlyqHTb cneKTp HeKOToporo H3J1yqeHH.H.. B ,nHcnepCH-
OHHOM H ,AH<ppaKD;HOHHOM cneKTpax HMeIOTC.H pa3J1HqH.H: 1) ,nJl.H ,nH<ppaKD;H-
OHHoro cneKTpa MomHO C03,naTb paBHOMepHYIO IllKany no 'A.. A TaK KaK <PYH-
KD;H.H n = f (A) He JlHHeHHa.H., TO ,nJl.H ,nHcnepCHOHHoro cneKTpa 3Toro c,nenaTb
HeJlb3.H, ,nHcnepCHOHHbIH cneKTp He paBHOMepHbIH, OH cmaT B KpaCHOH 06na-
CTH H paCT.HHYT B <pHoneToBoH; 2) B ,nHcnepCHOHHOM cneKTpe 60JlbIllee OT-
KJlOHeHHe OT nepBOHaqaJlbHOro HanpaBJleHH.H HcnbITbIBalOT <pHOJleToBbIe JlyqH,
B ,AH<ppaKD;HoHHOM - KpaCHbIe; 3) B ,AH<ppaKD;HoHHOM cneKTpe Ha6nlO,naeTC.H
HeCKOJlbKO nOp.HJ],KoB cneKTpa, B )J;HcnepcHoHHOM - O,nHH.
Puc. 17.32
17.15. CneKTpocKon
.flBJleHHe ,nHcnepCHH lIemHT B OCHOBe YCTpOHCTBa npH3MeHHbIX cneKTpaJlb-
HbIX npH6opOB: cneKTpOCKonOB H cneKTporpa<poB, KOTopbIe cJlymaT ,nJl.H no-
nyqeHH.H H Ha6J1IO,neHH.H cneKTpOB. Xo,n nyqeH B npOCTeHIlleM cneKTporpa<pe
nOKa3aH Ha pHCYHKe 17.33. Oocne oCBeaeMoH HCTO'llHHKOM CBeTa enH m,
nOMeeHHoH B <poKyce nHH3bI L] KOJlJlHMaTOpa, H,neT nyqoK pacxo,n.HHXC.H
nyqeH, KOTOpbIH, npoH,n.H JlHH3Y L 1 (KOJlnHMaTOpHbIH 06'beKTHB) CTaHOBHTC.H
napaJlJleJlbHbIM nyqKoM Jly-qeH. STOT napanJleJlbHbIH nyqoK, npeJlOMJI.H.HCb B
npH3Me, pa3J1araeTC.H Ha nyqKH CBeTa pa3Horo II.BeTa (pa3HOH 'A.), KOTopbIe CO-
526

6HpaIOTC.H Jl1:IH30H £2 (KaMep-
HbIM 06'beKT1:IBOM) B ee <p0-
KaJIbHOH nJlOCKOCTH, 1:1 BMe-
CTO O.n;HOrO H306pameHH.H
eJlH nOJlyqaeTC.H eJlbIH
p.HlI H306pameHHH. KamlIOH
qaCTOTe (Y3KOMY cneKTpaJlb-
HOMY HHTepBany) COOTBeT-
cTByeT CBoe H306pameHHe.
COBoKynHocTb :3T1:IX H306pa-
Puc. 1 7.33 meHHH H npe,ACTaBJlHeT co-
60H cneKTp.
CneKTp MomHO Ha6J1IO.n;aTb qepe3 OKYJl.Hp, HcnoJlb3yeMbIH B KaqeCTBe JlynbI.
TaKoH np1:l60p Ha3bIBalOT cneKTpocKonoM. ECJlH HymHo nOJIyq1:lTb <poTorpa<pHIO
cneKTpa, TO cpOTOnJIaCTHHKY nOMealOT B cpOKanbHOH nJIOCKOCTH KaMepHoro
06'beKTHBa. np1:l60p lIJI.H cpOTorpacpHpOBaH1:IH cneKTpa Ha3bIBalOT cneKTpo-
rpacJ>OM.
17.16. llOrJIOll\eHHe CReTa. 3aKOH Byrepa
norJIOJ.eHHeM (a6cop6HeH) CBeTa Ha3bIBaeTC.H .HBJIeHHe nOTepH :3HeprHH
cBeToBoH BonHoH, npoxo.n;HrqeH qepe3 BerqecTBo. B pe3ynbTaTe nornoeHHH
HHTeHC1:IBHOCTb CBeTa npH npOXOm,AeHHH qepe3 Ber.qeCTBO YMeHblIIaeTcH.
nOrJlOeHHe :3HeprHH CBeTa B BeecTBe onHCbIBaeTC.H 3aKOHOM Byrepa:
r,Ae 10 H I - HHTeHCHBHOCTH nJlOCKOH MOHO-
XpOMaT1:I1IeCKOH cBeToBoH BonHbI Ha BXOlIe H
BbIXOn;e cno.H nornorqalOrqero BeeCTBa TOJIIIlH-
Hoil x, a - K03cJ>cJ>HHeHT nOrJIOeIlHH, 3aBH-
C.Hrq1:l:H OT .n;JI1:IHbI BOJlHbI CBeTa, XHM1:IqeCKOH
npHpo.n;bI H COCTO.HH1:IH BerqecTBa H He 3aBHCH-
rqHH OT HHTeHCHBHOCTH CBeTa np1:l CJIa6bIX CBe-
TOBbIX nOTOKax.
Ha pHCYHKe 17.34 npe.n;CTaBJIeHa THnHqHa.H
3aB1:ICHMOCTb K03cpcpHIJ,HeHTa nor JIor.qeHHH OT
,llJIHHbI BOJlHbI CBeTa. 3aBHC1:IMOCTblO K03cpcpH-
IJ,HeHTa nOrJIOeHH.H OT A 06'b.HCH.HeTC.H OKpa-
lIIeHHOCTb nOrJIOalOHX TeJI. Hanp1:lMep, CTeKno, cJla60 norJIoalOrqee Kpac-
HbIe 1:1 opaHmeBbIe Jlyq1:l H C1:IJlbHO nornoalOee 3eJleHbIe 1:1 CHH1:Ie, np1:l
OCBeeHH1:I 6eJlbIM CBeTOM 6Y,AeT Ka3aTbC.H KpaCHbIM. EcnH H8 TaKoe CTeKno
HanpaB1:ITb 3eneHbIH H C1:IH1:IH CBeT, TO 1:13-38 CHJIbHOrO nornOeHH.H CBeT8
:3THX ,AnHH BOJIH CTeKJlO 6Y,AeT K83aTbCH qepHbIM. 3aBHCHMOCTb KO:3cpcpHIJ,H-
eHTa nOrJlor.qeHH.H OT lIJlHHbI BOJlHbI CBeTa Hcnonb3yeTcH .lI.JlH H3rOTOBneHHH
CBeTocJ>HJIbTPOB. KOTopbIe B COOTBeTCTBHH C XHMHqeCKHM COCTaBOM nponyc-
K810T CBeT TOJIbKO onpe.n;eJleHHbIX )J;n1:lH BOJlH, norJloa.H OCT8JIbHbIe.
a.
Puc. 17.34
1= Ioe- ax ,
A
527

17.17. illKaJIa 3JIeKTpOMarHHTHLIX BOJIH.
3JIeKTpOMarHHTHLIe H3JIyqeHHH pa3HLIX ,l);JIHH BOJIH.
CBoHcTBa H npHMeHeHHe 3THX H3JIyqeHHH
I1cCJle,nOBaHIUI, npOBo,nHBIIIHeC.H B caMbIX pa3Ho06pa3Hblx 06J1aCT.HX cpH3H-
KH, n03BOJIHJIH YCTaHOBHTb, qTO ,AHana30H qaCTOT (HJIH ,nJIHH BOJIH) 3JIeKTpo,
MarHHTHbIX BOJIH qpe3BblqaHHO IIIHpOK. 113 TeOpHH MaKCBenJIa CJIe,nyeT, t.ITO
pa3JIHqHbIe 3J1eKTpOMarHHTHbIe BOJIHbI, B TOM qHCJIe H CBeTOBble, HMelOT 06-
YIO npHpo,ny. II03ToMY HX y,n06HO npe,nCTaBHTb B BH,ne e,AHHOii IIIKaJlbI, HMe-
lOm;eH ,nHana30H qaCTOT OT HeCKonbKHX repu. ,no 10 22 rU., liTO COOTBeTcTByeT
,nJIHHaM BOJIH OT TbIC.Hq KHJlOMeTpOB ,nO 10- 14 M.
MCKJllOqHTeJlbHbIM ycnexoM 3J1eKTpOMarHHTHOH TeOpHH MaKCBeJIJla .HBHJIOCb
C03,AaHHe IIIKaJlbI 3J1eKTpOMarHHTHbIX BOJIH. B,AOJlb IIIKaJIbI CJleBa HanpaBO He-
npepbIBHO B03paCTaeT O,nHa BeJIHqHHa - qaCTOTa (YMeHbIIIaeTCH ,nJlHHa Bon-
HbI), a ee YBeJlHqeHHe npHBo,nHT K nO.HBJIeHHJO KaqeCTBeHHO pa3J1HqHbIX 11:3-
nyqeHHH.
B BHY orpOMHoro pa3J1Ht.IH.H lIJIHH BOJIH 3Ta IIIKaJIa nOCTpoeHa B JIora-
pH(I>MHqeCKOM MaCIllTa6e: MeTKH Ha IIIKaJle COOTBeTCTBYIOT ,l\JIHHaM. Kam,na.H
H3 KOTOpbIX OTJlHqaeTC.H B 10 pa3 OT coce,nHeH. Ha IIIKaJIe YKa38HbI yqaCTKH
,nJlHH BOJIH (HJlH A), 38HHMaeMbIe pa3JIHqHbIMH THnaMH 3J1eKTpOMarHHTHbIX
BOJIH. Pacnpe,neJleHHe 3JIeKTpOMarHHTHbIX BOJIH no THnaM c,neJIaHO B COOTBeT-
CTBHH co cnoc06aMH I1:X reHepau.HH. C H3MeHeHHeM ,nJlHHbI 3neKTpOMarHHT-
HblX BOJIH H3MeH.HeTC.H Ii HX B3aHMo,neHcTBHe C Bem;eCTBOM, n03TOMY MeTO,AbI
HX perHCTpall,HH H H3yqeHH.H pa3J1HqHbI. Pa3J1Hqa1OT cne,nYIOHe yqaCTKH Ha
IIIKaJle:
1) 3J1eKTpOMarHHTHbIe Kone6aHH.H HH3KOH qaCTOTbI ( 3 . 10 4 M < A < 00);
2) pa,nHoBOJIHbI (1.10- 4 M < A $ 3.10 4 M);
3) HHcppaKpacHoe H3J1yqeHHe (7,6,10- 7 M < A $ 1 .10- 4 M);
4) BH,AHMbIii CBeT (4.10- 7 M < A $ 7,6.10- 7 M);
5) YJIbTpacpHoJleToBoe H3nyqeHHe (6.10- 9 M < A $ 4.10- 7 M);
6) peHTreHOBCKoe H3J1yqeHHe (10- 12 M < A $10- 8 M);
7) y - H3J1yqeHHe (A < 10- 11 M).
I1epBbIH yqacToK WKaJIbI CO,AepmHT BOJIHbI, B036y»<:,naeMbIe HH3KOqaCTOTHbI-
MH 3J1eKTpOMarHHTHbIMH KOJIe6aHH.HMH, npOHcxo,n.HHMH B YCTpoHcTBax. 06na-
,nalOHX 60JlbIIIOH HH,AYKTHBHOCTblO H eMKOCTblO (B reHepaTopax nepeMeHHoro
TOKa). TaKHe BOJlHbI npaKTHqeCKH He H3J1yqaIOTC.H B oKpy»<:alOee npOCTpaH-
CTBO H 6bICTpO 3aTyxalOT.
BTOpOH yqacToK IIIKanbI - pa,nHOBOJIHbI. OH, B CBOIO Oqepe,nb, ,neJIHTC.H Ha
,nBe lIaCTH. K nepBOH 11:3 HHX OTHOC.HTC.H: JIHHHble (3. 10 3 M < A .$ 3 . 10 4 M),
cpeHHe (2 .}02 M < A $ 3 .10 3 M) H KopOTKHe BOJlHbl (10 M < A $ 2 .10 2 M).
528

3TH BOJlHbI H3J1yqaIOTC.H OTKpbITbIMH KOJle6aTenbHbIMH KOHTypaMH H paCnpOCT-
paH.HIOTC.H B npOCTpaHCTBe. ,Z:(JlHHHbIe BOJlHbI CnOC06HbI orH6aTb 3eMHYlO no-
BepxHOCTb, a KopOTKHe BOJlHbI pacnpOCTpaH.HIOTC.H, noOqepe,AHO OTpama.HCb OT
HOHo-ccpepbI H nOBepxHOCTH 3eMJlH.
Ko BTOpOH qaCTH ,AaHHOrO yqaCTKa llIKanbI OTHOC.HTC.H YJlLTpaKOpoTKHe
(MeTpOBbIe) paAHOBOJlHLI, caHTHMeTpOBLIe H MHJlJlHMeTpOBLIe BOJlHLI. 8TH
BOJlHbI 1i3J1yqa1OTC.H cneu.HaJlbHbIMH sneKTpOMarHHTHbIMH BH6paTopaMH H pe-
rHCTpHpYIOTC.H pa,AHOTeXHHqeCKHMR YCTpOHCTBaMR. TaKHe BOJlHbI pacnpOCT-
paH.HIOTC.H np.HMOJlHHeHHO, qepe3 HOHoccpepy OHI1 cnoc06HbI YXO,AHTb B KOC-
MOC. Hx IiCnOJlb3YIOT ,AJI.H KOCMHqeCKOH CB.H3H, nepe,AaqH TeJleMeTpHqeCKOH
HHcpopMaUHH. a Ha 3eMJle (B YCJlOBH.HX np.HMOH BHJJ.HMOCTH) - B TeJleBH-
,AeHHH H pa)J;HOJlOKau.HH. (Cne,AyeT OTMeTHTb, qTO BonHbI, OTHOC.HIu.HeC.H K
nepBOMY H BTOPOMY yqaCTKaM llIKaJlbI SJleKTpOMarHHTHbIX BonH, H3J1YQaIOT-
C.H CB060,AHbIMH 3ap.H,AaMH, ,ABHmYHMHC.R YCKopeHHO, H npe,ACTaBJI.HIOT CO-
60H SJleKTpOMarHHTHoe H3J1yqeHHe, nOJlyqaeMOe C nOMOblO KOJle6aTeJlbHbIX
KOHTYPOB H MaKpOCKOnHqeCKHX BH6paTopOB.)
C TpeTbero yqacTKa llIKaJlbI SJleKTpOMarHHTHbIX BOJIH HaqHHaIOTC.H BonHbI,
KOTopbIe H3J1yqaIOTC.H aTOMaMH H MOJleKYJlaMH BeeCTBa. YqaCTKH TpeTHH,
qeTBepTbIH H n.HTbIH ,AaHHOH IIIKaJlbI (T.e. HHcppaKpaCHoe, BH,AHMOe H YJlbTpa-
cpHOJleTOBOe H3J1yqeHH.H) OTHOC.HTC.H K OnTHqeCKOMY H3J1yqeHHIO. 3TH BOJlHbI H3-
nyqalOTC.H BHYTpHaTOMHbIMH SJleKTpOHaMH. OHH pacnpOCTpaH.HIOTC.H np.HMOJlH-
HeHHo (npH OTCYTCTBHH ,AHcppaKIJ;HH).
BH,AHMbIH CBeT BocnpHHHMaeTC.H rJla30M. HHcppaKpaCHoe H3J1yqeHHe HBn.H-
eTC.H npeHMymecTBeHHO TenJlOBbIM H3J1yqeHHeM. Ero perHcTpHpYIOT TenJlOBbI-
MH MeTO,AaMH, a TaKme qaCTHqHO cpOTOSJleKTpHqeCKHMH H cpOTorpacpHqeCKH-
MH MeTO,AaMH. Y JlbTpacpHOJleTOBOe H3nyqeHHe XHMHqeCKH H 6HOJlOrHqeCKH aK-
THBHO. OHO BbI3blBaeT .HBJleHHe cpOToscpcpeKTa, cpnyopecu.eHIJ.HIO H cpoccpopec-
u.eHIJ;HIO (CBeqeHHe) p.H)J;B BeeCTB. Ero perHCTpHpYIOT cpOTorpacpHqeCKHMH H
cpOTOSJleKTpHqeCKHMH MeTO,AaMH. ,Z:(JI.H nOJlyqeHH.H cneKTpa YJlbTpacpHOJleTOBOro
H3J1yqeHH.H HcnOJlb3YlOT npH3MbI H ,AHcppaKu.HOHHbIe pellIeTKH 1i3 KBapu.eBoro
CTeKJla. ,Z:(JI.H H3yqeHH.H YJlbTpacpHOJleTOBoro H3J1yqeHH.H, ,AJlHHa BOJlHbI KOTOpO-
ro KOpOqe 2. 10- 7 M, Hcnonb3YlOT BaKYYMHbIe cneKTporpacpbI, nOToMY qTO STO
H3J1yqeHHe CHJlbHO norJlOaeTC.H B03,AYXOM.
IIIecToH yqaCTOK llIKaJlbI SJleKTpOMarHHTHbIx BOJIH 06pa3yeT peHTreHoBcKoe
H3J1yqeHHe. PeHTreHOBcKoe H3J1yqeHHe B03HHKaeT npH B3aHMO,AeHCTBHH 6bICT-
pbIX SJleKTpOHoB C aTOMaMH TBep,AbIX TeJl H 06ycnOBJleHO nepeXO,AaMH SJleKT-
pOHOB Ha BHYTpeHHHx 060JlOqKaX aTOMOB. Ero nOJlyqalOT C nOMoblO cneu.H-
aJlbHbIX peHTreHoBcKHX Tpy60K. PeHTreHoBcKoe H3J1yqeHHe 06na,AaeT 60nbllIOH
npoHHKalOeH cnoc06HOCTblO. Ero perHcTpHpYIOT cpoTorpacpHqeCKHMH. cpnlOo-
porpacpHqeCKHMH H HOHH3au.HoHHbIMH MeTO,lIaMH.
Ce)J;bMoH yqacToK llIKanbI sneKTpoMarHHTHbIX BOJIH - raMMa-H3nYQeHHe.
OHO B03HHKaeT B pe3YJlbTaTe npou.eCCOB, npOHCXO,A.HHX B aTOMHbIX H,ApaX,
Ii conpOBOm,AaeT .H,AepHbIe peaKIJ;HH. raMMa-H3nyqeHHe 06J1a)J;aeT rpOMa,AHOH
npOHHKalOeH cnoc06HoCTblO. Ero perHcTpHpYIOT HOHH3au.HoHHbIMH MeTO,AaMH.
raMMa-H3J1yqeHHe HcnoJlb3YlOT B ,AecpeKTOCKOllHH.
C YMeHbllIeHHeM ,AJlHHbI ::meKTpoMarHHTHbIx BOJIH Bce CHJlbHee npO.HBJI.H-
IOTC.H KBaHTOBbIe CBOHcTBa H3J1yqeHH.H H Bce c 60JlbllIHM OCHOBaHHeM BMeCTO
CJlOBa "BOJlHbI" MomHO HCnOJlb30BaTb BbIpameHHe "noToK cpOTOHOB".
529

17.18. CneKTphI HcnYCKaHHJI
CneKTpOM HcnYCKaHHH Ha3bIBalOT cneKTp. nOJlyqaeMbIH npH pa3J10meHHH
CBeTa, H3J1yqeHHoro caMOCBeT.HHMHC.H TeJlaMH.
CneKTpaJIbHbIH COCTaB H3J1yqeHHH pa3J1HqHbIX BeeCTB BeCbMa pa3J1HqeH. Ho
MHOrOqHCJleHHbIe Ha6J1IOeHHSI H «l>oTorpa«l>HH cneKTpOB nOKa3bIBaIOT, qTO Bce
cneKTpbI HcnYCKaHH.H MomHO pa3eJlHTb HB TPH CHJIbHO OTJlHqalOHXC.H pyr
OT )J;pyra THna, KOTopbIe onpeeJl.HIOTCSI COCTOHHHeM CBeT.Hw.erOC.H 06'beKTa.
1. CnJIOIUHLIe (HenpepbIBHbIe) cneKTpLI.
B CnJIOlIlHOM cneKTpe npeCTaBJleHbI Bce u,BeTa (JlHHbI BOJlH), npHqeM ne-
pexo OT o.n;Horo u,BeTa K pyrOMY COBeplIlaeTC.H nOCTeneHHO (HenpepbIBHo)
CnJlOlIlHbIe cneKTpbI H3nyqalOTC.H paCKaJleHHbIMH TBepbIMH H mHKHMH
BeeCTBaMH, a TaKme ra3aMH, HaXO.HHMHCSI no 60JlblIlHM aBJleHHeM.
STO YKa3bIBaeT Ha TO, qTO BH HenpepbIBHoro cneKTpa H caM «l>aKT ero
cyecTBOBaHH.H onpeeJl.HIOTC.H He TOJlbKO CBOHCTBaMH OTeJlbHbIX H3J1yqa-
IOHX aTOMOB, HO H B CHJlbHOH CTeneHH 3aBHC.HT OT B3aHMoeHCTBH.H aTO-
MOB pyr C pyrOM. OCHOBHYIO pOJlb B H3J1yqeHHH HrpaeT B036ymeHHe aTO-
MOB H MOJIeKYJl npH XaOTHqeCKOM )J;BHmeHHH 3TH X qaCTHu" 06YCJlOBJleHHoe
BbICOKOH TeMnepaTYPOH. COJIHu,e aeT CnJlOlIlHOH cneKTp. TaK KaK ero 060-
JlOqKa COCTOHT H3 nJla3MbI BbICOKOH nJlOTHOCTH.
2. JIHHeHqaTLIe cneKTpLI.
JIHHeHqaTbIM Ha3bIBalOT cneKTp, COCTO.HHH H3 OTeJIbHbIX pe3KO OqepqeH-
HbIX BeTHbIX JlHHHH, OTeJIeHHbIX pyr OT pyra lIlHpOKHMH TeMHbIMl1 npo-
MemYTKaMH. Kam)J;a.H JlHHH.H npe.n;CTaBJI.HeT C060H H306pameHHe w.eJlH cneKT-
porpa«l>a B onpeeJleHHOM Y3KOM HHTepBane .n;JlHH CBeTOBbIX BonH.
TaKHe cneKTpbI nOJlyqalOTC.H OT CBeT.HHXC.H aTOMapHbIX ra30B HJIH napOB.
JIHHeHqaTbIe cneKTpbI 06YCJIOBJleHbI nepexoaMH BaJIeHTHbIX 3J1eKTpOHOB B03-
6ymeHHbIx aTOMOB Ha 60Jlee HH3KHe 3HepreTHqeCKHe ypOBHH.
JIHHeHqaTbIe cneKTpbI pa3J1HqHbIX XHMHqeCKHX 3JIeMeHTOB OTJIHqaIOTC.H
BeTOM (A), nOJlOmeHHeM H tIHCJlOM OTeJIbHbIX CBeT.HHXC.H JIHHHH.
3. UOJIOCaTLIe cneKTpLI.
COCTO.HT H3 OTeJlbHbIX nOJIOC, pa3.n;eJIeHHbIX TeMHbIMH npOMemYTKaMH. C
nOMOblO OqeHb xopOlIlero cneKTpaJlbHOrO annapaTa MomHO 06HapymHTb, qTO
Kam,Aa.H nOJloca npeCTaBJI.HeT C060H cOBoKynHocTb 60nblIloro qHCJla OqeHb Tec-
HO pacnOJlOmeHHbIX lIHHHH.
IIoJlocaTbIe cneKTpbI H3J1yqaIOTC.H OTeJlbHbIMH B036ymeHHbIMH MOJleKY-
JlaMH (MOJleKYJI.HpHbIH ra3). H3J1yqeHHe BbI3BaHO KaK 3J1eKTpOHHbIMH nepe-
XO,AaMH B aTOMax, TaK H KOJle6aTeJlbHbIMH BHmeHH.HMIi caMHX aTOMOB B
MoneKYJle.
17.19. CneKTphl nOrJlOeHHJI
CneKTpbI nOrJlOeHH.H nOJlyqalOT, nponYCKa.H CBeT OT HCTOqHHKa, .AalOero
CnJIOlIlHOH cneKTp, qepe3 BeeCTBO, aTOMbI H MOJleKYJlbI KOToporo HaXO.HTC.H
B HeB036ymeHHoM COCTO.HHHH.
CneKTpbI nOrJlOeHH.H TBep.AbIX Ii mHKHX TeJl 06bIqHO HMelOT BH.A UlH-
pOKHX TeMHbIX nOJlOC, 3aKpb1BalOHX qaCTb CnJlOlIlHOrO cneKTpa HCTOqHHKa,
530

a B cJlyqae aTOMapHbIX ra30B H napOB OHH COCTO.HT H3 OT,AeJlhHbIX qepHbIX
JlHHHH, BHHMbIX Ha (}>OHe CnJlOlliHOrO cneKTpa. IIponYCTHB, HanpHMep, CBeT
OT 3neKTpHqeCKOH JlaMnhI qepe3 cocy C napaMH HaTpH.H, MbI nOJlyqHM Ha
CnJlOlliHOM cneKTpe JlaMnhI ,ABe Y3KHe qepHbIe nHHHH B menTO:H 06naCTI1
cneKTpa.
,I:{JI.H cneKTpOB nornOeHH.H OTKpbIT 3aKOH 06paTHMOCTH cneKTpaJlbHbIX nH-
HHH KHpxro(}>a: J1HHHH nOrJlo:u.-.eHHJI COOTBeTCTByroT J1HHHJlM HcnYCKaHHJI, T.e.
aTOMbI MeHee HarpeToro BemeCTBa nornoalOT H3 cnnOlliHoro cneKTpa KaK pa3
Te qaCTOTbI, KOTophIe OHH B PyrHx YCJlOBH.HX HcnYCKaIOT.
IIpHMepOM cneKTpa nOrJlOeHH.H MoeT cnYHTh cneKTp nornOeHH.H COJl-
He'lHOH aTMoc(}>epbI. COJlHe - paCKaJleHHbIH ra30BbI:H lIIap (TeMnepaTypa no-
BepXHOCTH 6000 °C), re ra3 HaXOHTC.H no 60JlbllIHM aBJleHHeM. CJle,AOBa-
TeJlbHO, OHO HcnYCKaeT cnnOlIIHO:H cneKTp. IIpH pacnpOCTpaHeHHH B HanpaB-
neHHH K 3eMJle COJlHeqHbIe nyqH npOXO.HT qepe3 aTMoc(}>epy COJlHa (KOpO-
HY), r,Ae TeMnepaTypa 2000-3000 °C. KopoHa norJlOaeT H3 cneKTpa onpee-
JleHHbIe qaCTOThI (20000 JlHHHH), H MbI Ha 3eMJle (}>HKcHpyeM COJlHeqHbIH
cneKTp nOrJlOeHH.H: Ha CnJlOlliHOM cneKTpe TOHKHe TeMHbIe JlHHHH (<I>paYH-
ro(}>epoBbI nHHHH). IIo 3THM JlHHH.HM 6bIJI onpeeJleH COCTaB aTMoc(}>epbI COJl-
Ha (H, Ca, Na, Fe Ii p.) H BnepBbIe OTKpbIT reJlHH.
17.20. CneKTpaJILHLIH aHaJIH3
CneKTpaJlLHLIM aHaJlH30M Ha3hIBalOT MeTO HCCJleOBaHH.H XHMHqeCKOrO
COCTaBa pa3J1HqHbIX BeeCTB no HX cneKTpaM.
AHaJlH3, npoBOHMbI:H no cneKTpaM HcnYCKaHH.H, Ha3bIBalOT 3MHCCHOHHbIM,
a no cneKTpaM nOrJlOeHH.H - a6cop6HOHHbIM cneKTpaJlbHbIM aHaJlH30M.
B OCHOBe 3MHCCHOHHoro cneKTpanbHoro aHaJIH3B JlemaT cJleYIOHe (}>aKThI:
1. KambIH 3J1eMeHT HMeeT CBOH cneKTp (oTJlHqaeTC.H qHCJlOM nHHHH, HX
pacnonomeHHeM H nHHaMH BOJlH), KOTOphIH He 3aBHCHT OT cnoc060B B036y-
eHH.H.
2. HHTeHcHBHoCTh cneKTpanbHbIX JlHHHH 3aBHCHT OT KOHeHTpaHH 3ne-
MeHTa B aHHOM BeeCTBe.
,I:{JI.H BbInOnHeHH.H cneKTpanbHoro aHaJlH3a BeeCTBa C HeH3BeCTHbIM XH-
MHqeCKHM COCTaBOM Heo6xoHMO ocyecTBHTb ,ABe onepaHH: 3aCTaBHTb Ka-
KHM-TO 06pa30M aTOMbI 3Toro BemeCTBa H3J1yqaTb CBeT C JlHHeHqaTbIM cneK-
TpOM, 3aTeM pa3nomHTb 3TOT CBeT B cneKTp H onpe.nenHTh .nnHHbI BonH Ha-
6J1IOaeMbIX B HeM JlHHHH. CpaBHHBa.H nOJlyqeHHbIH JlHHeiiqaTbIH cneKTp C H3-
BeCTHbIMH cneKTpaMH XHMHqeCKHX 3J1eMeHTOB Ta6nHI.J:.bI MeHeneeBa, MOHO
onpeeJlHTb, KaKHe XHMHqeCKHe 3neMeHTbI HMeIOTC.H B COCTaBe HccJleyeMoro
BeeCTBa. IIYTeM cpaBHeHH.H HHTeHCHBHOCTH pa3nH'lHbIX JIHHHH cneKTpa MO-
HO onpeeJlHTb H OTHOCHTenbHoe coepmaHHe pa3J1HqHbIX 3neMeHTOB B 3TOM
BeeCTBe.
CneKTpanhHhIH aHanH3 MomeT 6bITb KaqeCTBeHHbIH H KOnHqeCTBeHHhIH.
ECJlH Hccne.,nyeMoe BeeCTBO Haxo.,nHTC.H B ra3006pa3HOM COCTO.HHHH. TO JI.H
B036ymeHH.H aTOMOB BeeCTBa 06hlqHO npHMeH.HeTCSJ HCKpOBO:H pa3p.H,A. Hc-
CJle,AyeMbIM ra30M 3anOJlH.HeTC.H Tpy6KB C BYM.H 3J1eKTpOaMH Ha KOHax. Ha
3TH 3J1eKTpObI nOaeTC.H BbICOKoe Hanp.HmeHHe H B Tpy6Ke B03HHKaeT
531

3neKTpHqeCKHH pa3pH.n;. Y.n;apbI 3J1eKTpOHOB, pa3rOHHeMbIX 3J1eKTpHqeCKHM no-
JIeM, npHBO.n;HT K HOHH3aD;HH H B036ym.n;eHHIO aTOMOB HCCJIe.n;yeMoro ra3a. IIpH
nepeXO,Aax B036ym.n;eHHbIX aTOMOB B HOpMaJlbHOe COCTOHHHe 1i3nyqalOTC.H KBaH-
TbI CBeTa, XapaKTepHbIe .n;JI.H .n;aHHOrO 3J1eMeHTa.
AJI.H onpe.n;eJleHH.H XHMHqeCKOrO COCTaBa BeeCTBa, HaXO.n;.HerOCSI B
TBep,AOM IiJlH mli.n;KOM COCTO.HHHH, no ero cneKTpy H3nyqeHIi.H He06xo.n;HMO
CHaqaJl8 nepeBeCTH IiCCJle,AyeMoe BeeCTBO B ra3006pa3Hoe COCTO.HHHe Ii 3a-
CTaBHTb KaKHM-TO 06pa30M 3TOT ra3 IicnYCKaTb CBeT. 06blqHO .nn.H npOBe-
.n;eHH.H cneKTpaJlbHOrO aHaJlH3a 06pa3D;OB BeeCTBa B TBep.n;OM COCTO.HHHH
Hcnonb3YlOT .n;yroBoH pa3p.H.n;. B nJla3Me .n;yrH npOHcxo.n;HT npeBpaeHHe Be-
eCTBa B nap, B036ym.n;eHHe Ii HOHH3aD;li.H aTOMOB. 3J1eKTpo.n;bI, Mem.n;y KO-
TOpbIMIi 3amHraeTC.H .n;yroBoH pa3pH.n;, 06bIqHO H3rOTaBJlHBaIOTC.H H3 HCCJle-
.n;yeMoro BeeCTBa (eCJIH OH MeTaJlJI) HJlH H3 rpacpHTa IiJlH Me,AH. YrJlepo.n.
H Me.n;b BbI6lipalOTCH no TOH npHqHHe, qTO cneKTpbI H3J1yqeHH.H HX aTOMOB
B BH.n;HMOH 06JIaCTH HMelOT He6oJlblIIOe llliCJlO JlHHHH H, CJle.n;OBaTenbHO, He
C03,Aa1OT cepbe3HbIX nOMex ,AJIH Ha6J1IO,neHH.H cneKTpa HCCJle.n;yeMoro Beme-
CTBa. B yrJly6JIeHlie HlimHero 3J1eKTpo,na nOMeaeTC.H nopOIliOK HCCJIe.n;ye-
Moro BeeCTBa.
BOllPOCbI AJIH CAMOKOHTPOJIH
1. B lleM cocmoum cyrqliocmb 3Jl.eKmpOMazHumliou meopuu c6ema? KaKue
3KcnepUMeHmaJlbHble cpaKmbl ee 060CHoBblBalOm? KaKue 3JleKmpOMazliumHble
60Jl.Hbl 6bl3blBalOm C6em060e orqyrqeliue?
2. KaKue .R.6JleliUSl nOOm6epJICOalOm 60Jlli06ble C60ucmBa cBema?
3. B lleM cocmoum Sl6Jl.eliUe ulimepcpepeliijuu c6ema? l{mo npeOCma6Jl.Slem
c060u ulimepcpepeliquoliliaJl Kapmulia? llpu KaKUX YCJl.06USlX olia 603liUKaem?
4. KaKue ucmOllliUKU C8ema .R6Jl.SllOmCSl KozepelimliblMu? KaKuMu cnoc06a
.MU MOJICIiO nOJl.Yllumb Kozepelimlible C8em08ble 80Jl.libl? llolle.MY 06blillible uc-
mOIlHUKU c6emalie.M01i0Xp0.MamUlllibl U lieKozepeHmlibl? l{mo maKoe "OJlUlia
KozepeHmHocmu"?
5. llpu KaKUX YCJl.06USlX 6 ulimepcpepeliquoliliou KapmUlie nOSl6Jl.SllOmC.R
MaKCUMY.Mbl U .MUIiU.MY.Mbl UlimeliCU6liOCmU c6ema?
6. l{ eM 06-OSlCIiSlemcSl OKpaCKa mOIiKUX nJl.eIiOK? B KaKO.M CJl.Yllae 8031iU-
KalOm KOJl.bqa H blOmolia?
7. B onblme IOliza oOlia U3 rqeJl.eu 3aKpblma CUIiUM C6emocpUJl.bmpO.M, a
6mOpa.R KpaCHbl.M. B03liUKaem Jl.U npu 3mO.M ulimepljJepeliquolilia.R KapmUHa?
II Olle.MY ij6em OKpaCKU OOliOZO U mozo JlCe .Mecma .MblJl.bIiOZO nY3btpSl lienpe-
pbt6liO U3.MeliSlemcSl?
8. KaK C nOMorqblO ulimepljJepeliijuu MOJICIiO Oijeliumb Kallecm60 06pa60m-
KU n08epxHocmeu?
9. l{ mo maKoe npOC6emJl.eliUe onmuKu? KaK 6bl6upaemcSl mOJl.rquliQ nJl.eli
KU liQ 06-oeKmU8e npu60pa C npOC6emJl.eHliOU onmuKou? II OlleMY 06-oeKmU6 C
npOC6emJl.eliliOU onmUKOU 6 OmpaJICeliliO.M C6eme U.Meem cupelie6blU ommelioK?
10. B lle.M cocmoum .R.6Jl.eHUe ouljJpaKquu c6ema? llpu KaKUX YCJl.06USlX OHa
603liUKaem?
532

11. l[mo npeiJcmaBJl.R.em c060il ouljJpaKijUOHliaSl pewemKa? fJmo nOIiU.Ma
10m noo nepuoiJo.M pewemKu? KaK no nepuoiJy pewemKu onpeiJeJl.Umb IlUCJl.O
wmpux06, npuxoiJSl14UXCSl lia 1 .M.M ee iJJl.Ulibl?
12. KaKoBo YCJl.OBUe 603liUKH06eliUSl ZJla6liblX .MaKCU.MY.M06 6 iJuljJpaKlJ,UOH-
1i0U KapmUlie. nOJl.YlleIiIiOU npu npOXOJICiJeliuu C6ema llepe3 iJuljJpaKlJ,UOliliYIO
pewemKY?
13. II Olle.MY B03HUKaem cneKmp npu npoxoJICiJeliuu 6eJl.ozo CBema llepe3
iJuljJpaKuuoHIiYIO pewemKY? KaKou .MaKCU.MaJlbliblU nOp.R.iJoK cneKmpa B iJu
ljJpaKlJ,UOIiIiOU Kapmulie? l[ e.M iJuljJpaKlJ,UOHHblU cneKmp omJl.Ullaemc.R. om ouc-
nepcuoliliOZO cneKmpa?
14. KaK06bl HeiJocmamKu 06blilliOZO ljJomozpaljJullecKozo .MemoiJa nOJlYlleliUSl
u306paJICeliuSl?
15. KaK nOJl.YllalOm zOJlozpa.M.MY? l[mo olia c060u npeiJcma6Jl.Slem? KaK
.MOJICliO ee Boccmali06umb?
16. KaKoBo omJl.Ullue zOJl.OZpaljJullecKozo u306paJICeliUSl om 06blilHOZO ljJomo-
zpaljJullecKozo?
17. KaKou CBem Ha3blBaemC.R. nJlOCKOnOJl.SlpU30BaHHbLM? l[ eM omJlUllaemc.R.
ecmecm6eliliblU C6em om nOJl.SlpU306aHliozo?
18. II Olle.MY U3.MeIiSlemCSl UlimeliCU6liOCmb CBema, npoweiJwezo llepe3 iJBe
nJl.aCmUliKU U3 OiJliOOCliOZO KpUCmaJl.Jl.a npu n060pome OiJliOU U3 liUX?
19. KOJle6aliUSl KaKOZO 6eKmopa - E UJl.U B - U.MelOm B 6uiJy, KoziJa
z060pSlm 0 lianpa6Jl.eliUU KOJl.e6aliUU 6 CBem060U 60Jl.He?
20. KalCu.Mu cnoco6a.Mu ecmecm6eliHblU C6em .MOJICliO npe6pamumb 6 nJl.Oc,
KOnOJlSlpU306ali Ii blU?
21. IIpu KaKOM YCJl.06UU Ompa:HCeHHblU om ZpaHUlJ,bL pa3iJeJla iJ6YX OU3Jl.eK-
mpUK06 C6em .MaKCU.MaJl.bliO nOJl.SlpU30Bali? q mo nOliU.MalOm noiJ YZJl.O.M BpIOC-
mepa?
22. 06blKli06eliliblu U lie06blKli06eliliblu Jl.YIlU nOJl.YllalOmcSl nyme.M pa3Jl.0-
JlCeliUSl OOHOZO U mozo JfCe nYIlKa ecmecm6eliliOZO C6ema, CJl.e006ameJlbliO, Y CO-
omBemcm6YI014UX 60Jl.H ljJa3bl a6COJl.lOmliO oiJuliaK06ble. B03liUKliem Jl.U Kap.
mUlia ulimepljJepeliquoliliblX .MaICCU.MY.M06 U .MUHU.MY.M06, ecJl.U C6ecmu 06a Jl.Ylla
6.MeC me?
23. 3a6ucum Jl.U nOKa3ameJlb npeJl.O.MJleHU.R. om CBoucmB cpeiJbL? If mo ma-
Koe oucnepcuSl c6ema?
24. KaKou C6em lia3bl6aemcSl .MOliOXp0.MamUlleCKU.M?
25. J(Jl.Sl KaKUX Jl.Ylleu - "paCliblX UJl.U ljJuOJl.em06blx - nOKa3ameJlb npe-
JlO.MJl.eliUSl CmeKJl.a 60Jl.bwe? Y KaKUX Jl.Ylleu clCopocmb pacnpocmpalieHuSl C6e-
ma 6 CmeKJl.e 60Jl.bwe?
26. II olle.MY nYIlOIC 6eJlozo C6ema npu npOXOJICiJeHuu llepe3 npU3.MY pa3Jla-
zaemcSl lia MOliOXp0.MamUlleCKue nYIlKu? M OJICliO Jl.U nOJl.Yllumb 6eJl.bLU C6em
npu liaJlO:HCeHUU M.OHOXp0.MamUlleCKux nYIlKoB?
27. H a6Jl.100aemCSl Jl.U iJucnepcuSl C6ema npu npoxoJICiJeliuu llepe3 6a-
KYY.M?
28. H a KaKue YllacmKU noOpa30eJl.SlemCSl WKaJl.a 3Jl.eKmp0.MaZHUmliblX
60JlH? l[ e.M OliU omJl.UllalOmCSl iJpyz om iJpyza?
29. M OJIC Ii 0 Jl.U 3azopemb Y 3aKpblmozo OKlia. llepe3 Komopoe npoxoiJum
SlpKUU COJl.liellliblU c6em?
533

c c
v=-=-
.JEV.n
( 21tr )
E = Eo cos wt - T
L=nr
l:!1 = rl - r2
l:!1
l:!1cp = 21t -
A
OCHOBHblE OPMY JIbl
- CKOpOCTb CBeTa B cpe.n;e, r.n;e c - CKOpOCTb
CBeTa B BaKYYMe, E H J.1 - .n;H3J1eKTpHqeCKaSl H
MarHHTHa.H npOHHIJ;aeMOCTH cpe.n;bI, n - a6COJlIOT-
HbIH nOKa3aTeJlb npeJlOMJleHH.H cpe.n;bI.
- ypaBHeHHe cBeToBoH BOJlHbI. rlIe E - Hanp.H-
meHHOCTb 3J1eKTpHqeCKOrO nOJl.H BOJlHbI B MO-
MeHT BpeMeHH t B TOqKe, Haxo.n;.HeHC.H Ha pac-
CTO.HHHH r OT HCTOqHHKa CBeTa, Eo - aMnJlH-
Ty.n;a Hanp.HmeHHOCTH 3J1eKTpHqeCKOrO nOJl.H Bon-
HbI, A - ,AJlHHa BOJlHbI.
- OnTHqeCKa.H .n;JlHHa nYTH Jlyqa, r.n;e r - reo-
MeTpHqeCKa.H .n;JlHHa nYTH Jlyqa B cpe.n;e C a6co-
JlIOTHbIM nOKa3aTeJleM npeJlOMJleHH.H n.
OnTHqeCKa.H pa3HOCTb xo.n;a .n;BYX JlyqeH.
pa3HOCTb cpa3 KOJle6aHHH, CKJla.n;bIBaeMbIX B
TOqKe Ha6J1IO.n;eHH.H.
l:!1 = !mA, m = 0, 1, 2... - YCJlOBH.H HHTepcpepeHIJ;HOHHOro MaKCHMYMa,
r.n;e m - nOp.H.n;KoBbIH HOMep MaKCHMYMa.
A
l:!1 = I(2m + 1)-, m = 0, 1,2... - YCJlOBHe HHTepcpepeHIJ;HOHHOro MHHHMYMR.
2
1
d =a+b =-
N
d sin cp = :tmA, m = 0, 1.. .
tg Cl>B = n
I = 10 cos 2 a
534
- nOCTO.HHHa.H ,n:HcppaKIJ;HOHHOH pemeTKH, r,Ae
a - IIIHPHH8 enH, b - IIIHpHHa Henp03paq-
Horo IIITpHxa, N - qHCJlO eneH peIIIeTKH,
npHxo,n:.Heec.H Ha e,n:HHHIJ;Y )J;JlHHbI peIIIeTKH.
cpopMYJla .n;HcppaKIJ;HOHHOH peIIIeTKH, r.n;e
<p - yroJl Mem.n;y HOpMaJlblO K nOBepXHOCTH pe-
IIIeTKH H HanpaBJleHHeM .n;HcpparHpoBaHHbIx JlY-
qeH, m - nOp.H.n;KoBbIH HOMep ,n:HcppaKIJ;HOHHOro
MaKCHMYMa.
- 3aKOH BplOcTepa, r)J;e <PB - yroJl na.n;eHHH
Jlyqa (yroJl BplOcTepa) Ha nOBepXHOCTb pa3.n;en8
.n;BYX .n;H3J1eKTpHqeCKHX cpe.n;, n - OTHOCHTeJlb-
HbIH nOK838TeJlb npeJlOMJleHH.H BTOpOH cpe.n;bI OT-
HOCHTeJlbHO nepBoH.
- 3aKOH ManlOca, r.n;e 10 H I - RHTeHCHBHOC-
TH CBeTa, na,n:alOI.qero Ha aHaJlH3aTOp H npoxo-
,n:.Hero Qepe3 Hero COOTBeTCTBeHHO, a - yron
Mem)J;y HanpaBneHHeM KOJle6aHHH BeKTopa E
CBeToBoH BOJIHbI, na.n;alOeH Ha aHaJlH3aTOp, H
OnTHqeCKOH OCblO aHaJlH3aTopa.

1 = 1 0 e-ax
- 3aKOH Byrepa, re 10 H 1 - HHTeHCHBHOCTH naa-
lOero CBeTa H npOllIellIero B nor JloalOeii cpee
nYTb X, ex - K03<P<PHlJ;HeHT nor JlOeHHS.
METQ,[(II"tJECKHE PEKOMEH,[(AQIIII
llPH pellIeHHH 3aaq H8 HHTep4>epeHHIO CBeTa peKoMeHyeTC.H:
1. CeJlaTb cooTBeTcTBYIOHii qepTem, YKa3aB Ha HeM XO Jlyqeii.
2. BbISCHHTb npHqHHbI nO.HBJleHH.H OnTHqeCKOii pa3HOCTH xoa Memy HH-
Tep<pepHpYIOHMH JlyqaMH.
3. OnpeeJlHTb OnTHqeCKYIO pa3HOCTb xoa nyqeii.
4. 3anHcaTb HJlH. eCJlH He06xo,AHMO, BblBeCTH YCJlOBHe MaKCHMYMa HnH MH-
HHMYMa HHTep<pepeHuHH.
5. PellIHTb nOJlyqeHHOe ypaBHeHHe, npOH3BeCTH BbIqHCJIeHH.H H OlJ;eHHTb pe-
aJlbHOCTb nOJlyqeHHoro pe3YJlbTaTa.
B 3aaqaX Ha ,l];H4>paKHIO CBeTa peKoMeHyeTC.H:
1. CeJlaTb qepTem, YKa3aB Ha HeM XO,A Jlyqeii.
2. 3anHcaTb YCJlOBHe MaKCHMYMa HJlH MHHHMYMa ,AH<ppaKlJ;HH Ha COOTBeT-
CTBYIOHX 06'beKTax.
3. B CJIyqae Heo6xoHMOCTH Bocnonb30BaThC.H He,AOCTalOHMH ypaBHeHH.H-
MH, npHBJleKaS .nonOJlHHTeJlbHbIe aHHbIe 3a.naqH.
4. PellIHTb nOJlyqeHHbIe ypaBHeHHS (cHcTeMY ypaBHeHHH), OlJ;eHHTb peaJlb-
HOCTb nOJIyqeHHOrO pe3YJlbTaTa.
llpH pellIeHHH 3aaq Ha ,l];HCnepCHIO CBeTa peKoMeH)J;yeTC.H:
1. CeJlaTb qepTem, YKa3aB Ha HeM XO COOTBeTCTBYIOHX JIyqeii.
2. 3anHcaTh Ha OCHOBaHHH 3aKOHOB npenOMJIeHHSJ OCHOBHbIe ypaBHeHH.H.
3. 3anHcaTb, HcnOJlb3Y.H ,AOnOJlHHTenbHbIe YCJlOBH.H, HeOCTalOHe ypaBHeHH.H.
4. PellIHTb nOJlyqeHHbIe ypaBHeHH.H.
1. Ha nYTH nY'la. ",AYI.l{.ero B B03,Ayxe. nOCTaSHnH CTeKnSlHHYIO nn8CTHHKY Ton-
I.qHHOH 1.0 MM. KaK H3MeHHTCSI OnTH'leCKaSl nHH8 nYTH nY'la, ecnH ny'l 6Y,AeT
na,AaTb Ha nnaCTHHKY: 8) HopM8nbHo, 6) no yrnoM 30°?
,l];aHo: h=1.0 MM, n=1,5, al=O, a2=300.
HaHTH: f1 h f1 2 .
Pew e H H e. a) ny'l 1 HeT He npenOMnSlSlCb
(pHC. 17.35). OnTH'leCKafl ,AnHHa nYTH H3MeHSleTCSI H8
BenH'lHHY l = nh - h = (n -l)h; l = 0,50 MM;
6) OnTH'leCK8fl ,AnHHa nYTH nY'l8 2 (pHC. 17.35)
H3MeHHTCH H8 BenH'lHHY
2
h
"
"
Puc. 17.35
TIPIIMEPbI PEIlIEHHH 3AJ{A"tJ
1
f1 2 = n . OA + AC - OB.
h
113  010A HaXO,AHM AO = .
COS....
535

M3 /:10l0B HaXO,AHM OB =
cosa
h
M3 f1ABC HMeeM AC = ABsina, r,Ae AB = 0IB-OlA = htg a-htg (3.
Tor,Aa /:1 2 =  + h sin a(tg a - tg (3) -.
cos (3 cos a
TIo 3aKOHY npenOMneHHSI
sin a . sin all 0
- = n, oTKY,Aa sln = - = - = - = 19 .
sin  n 2 . 1,5 3
TIO,ACTaBnSlSl 'lHCneHHbIe ,n8HHbIe, nonY'lHM 2= 0,45 MM.
2. B onbITe IOHra HCTOqHHKaMH CBeTa cnymaT ,nBe Y3KHe enH 8 1 H 8 2 . pac-
nOnO>KeHHbIe Ha paCCTOSlHHH 1,0 MM ,Apyr OT ,npyra (pHC. 17.36). Ha p8CCTOSlHHH
3.0 M OT HCTO'lHHKOB p8cnonomeH 3KpaH.
):(nHH8 sonHbI CBeTa, H3nY'l8eMOrO HCTO'l.
HHKOM S, 0.60 MKM. Onpe,nenHTe nono-
x m )KeHHe H8 3KpaHe Tpex nepBbIX CBeTnblX
nonoc.
,I:(8HO: l=I,0.10- 3 M, D=3,0 M,
A=6,0.10- 7 M.
HaHTH: Xl' X2, X3'
Pew e H H e. TaK KaK Ha enH na,Aa-
eT CBeT OT O,AHOrO H Toro me HCTO'lHH-
K8 S (pHC. 17.36), TO enH HsnSl10TCSI
KorepeHTHbIMH HCTO'lHHKaMH CBeT8, H Ha
3KpaHe B03HHK8eT HHTepepeHHOHHaSl
KapTHHa. Pe3ynbTaT HHTep<l>epeHHH CBeT8 B npOH3s0nbHOH TO'lKe C onpe,AenSleT.
CSI pa3HOCTb1O XO,Aa nyqeH 88 l C H 8S 2 C. ):(nSl eHTpanbHOH TO'lKH 3KpaHa (TO'lK8 A)
pa3HocTb XO,n8 nyqeH p8BHa Hyn1O, H TaM 6Y,AeT Ha6n1O,AaTbCSI CBeTnaSl nonOC8, KO-
TOp8fl C'lHTaeTCSI HHTepepeHHOHHOH nonOCOH HyneBoro nOpSl,AKa.
TIpH y,A8neHHH OT TOqKH A Bsepx HnH BHH3 H3MeHSleTCSI pa3HocTb XO,A8 ny-
qeH, B pe3ynbTaTe 'lero CBeTnaSl nonoca CMeHSleTCSI TeMHoH (HHTep<l>epeHHoHHbIH
MHHHMYM), 3aTeM CHOBa B03HHKaeT CBeTnaH nonoca (HHTep<l>epeHHoHHbIH MaKCH-
MYM) H T.,A.
):(onycTHM, 'lTO B TO'lKe C 3KpaHa Ha6n1O,AaeTCSI m-:H HHTep<l>epeHHOHHbI:H MaK-
CHMYM. Pa3HocTb XO,Aa nY'leH ,AnSI 3TOH TOqKH f1 = r2 - r 1 . Tor,Aa, Hcnonb3YSl YC-
nOBHe M8KcHMYMa HHTep<l>epeHHH. MomHO 3anHC8Tb
r 2 - r 1 = mA.
Puc. 17.36
(1)
M3 PHCYHK8 BH,AHO, qTO
J 2
2 2 l
r 2 = D + ( X m + "2
( J 2
2 2 l
r l = D + X m - '2
Cne,AOSaTenbHO,
rl-r{ = (r2 -r 1 )(r 2 + rd = 2lx m . (2)
nOCKonbKY nepSbIe TPH CBeTnbIe nonOCbI 6Y,AYT Ha6JI1O,naTbcH B6nH3H eHTpa
3KpaHa. TO MomHO CqHTaTb, qTO r 2 + r 1 == 2D. Tor,na H3 (2) Cne,AyeT, 'lTO
lX m
r2- r l=-'
D
(3)
536

fIo,AcTaBHB 3Ha'leHHe (r2 - rd B (1), H Bblpa3HB X m , nonY'lHM
mAD
x m --Z'
AD
TIonomeHHe nepBoro (m = 1) MaKCHMYM8 x] = 1=1,8 MM, BToporo (m = 2)
x2=3,6 MM H TpeTbero (m=3) x3=5.4 MM.
3. HaH,AHTe Bce ,AnHHbI BonH BH,AHMOrO CBeTa (OT 0,76 1l.0 0.38 MKM), KOTopble
B pe3ynbTaTe HHTepcpepeHl1.HH npH pa3HOCTH xo.n:a HHTep<I>epHpYIOIl\HX nY'leH
1,8 MKM 6Y,AYT: 1) MaKCHManbHO YCHneHbI; 2) MaKCHManbHO ocna6neHbI.
,Z:(a H 0: A] =0.76 MKM. A2= 0,38 MKM,  = 1.8 MKM.
HaHTH: A.
P e ill e H He. 113 ycnoBHSI MaKCHMYMa HHTeHCHBHOCTH HHTepcpepHPYIOIl\HX BonH
A
1:::.=2m-
2
nonyqHM nnHHbI BonH. nnfl KOTOpbIX 6Y)1eT H86nlO,AaTbCSI MaKCHMYM YCHneHHH:
A = = 1,8 MKM.
m m
B 3a.n:aHHOM ,AHana30He ,AnHH BonH MaKCHMYMbI HHTeP<l>epeHl1.HH COOTBeTCTBY-
lOT .n:nHHaM BonH A]=0,6 MKM H A2=0,45 MKM npH m=3; 4.
AHanOrH'iHO H3 ycnoBHSI MHHHMYMa YCHneHHSI
A
 = (2m + 1)"2
2 3. 6
A=-=- MKM.
2m+l 2m+l
B 3TOM cnyqae ycnoBHIO 38,Aa'lH YIl.OBneTBopSlIOT BonHbI C ,AnHHOH
A3=0,72 MKM; A4=0,51 MKM; A5=0.40 MKM (cooTBeTcTBeHHo m=2; 3; 4).
4. Ha TonCTYIO CTeKnSlHHYIO nnaCTHHKY (n CT = 1,5). nOKpbITYIO O'leHb TOHKOH
nneHKoH. a6conlOTHbIH nOKa3aTenb npenOMneHHSI BeIl\eCTBa KOTOpOH paBeH
1,4. na,AaeT HopManbHO napannenbHblH nY'lOK nY'leH MOHoxpOMaTHqeCKOrO CBeTa
(A = 0.6 MKM). Onpe,AenHTe TOnIl\HHY nneHKH, nplf KOTOpOH oTpameHHbIH CBeT MaK-
CHManbHO ocna6neH BCne,ACTBHe I1HTep<l>e-
peHUHH.
,Z:(aHo: n2= 1,4. n3 = 1.5, A = 6,0 .10- 7 M.
HaHTH: h.
P e ill e H H e. 113 nY'lKa napannenbHbIX
nY'leH. na.n:alOIl\HX Ha nneHKY. BblnenHM
O,AHH ny'l SA. Xon 3Toro nY'la B cnY'lae,
KOr,A8 yron na,AeHHSI a 1 =F- 0, nOK83aH Ha
pHCYHKe 17.37. B TO'lKaX A H B na,l{alO-
Il\HH nyq 'laCTH'lHO oTp8maeTcSI H 'laCTH'l-
HO npenOMnSleTCSI. OTp8meHHble nY'lH AS]
H BCS 2 nanalOT Ha C06HP81OI.l.lYIO nHH3Y
L, nepeceK810TcSI B ee cpoKyce H HHTep-
cpepHpYIOT Mem,AY C060H.
nonY'lHM
S
h
Puc. 17.37
537

TaK KaK a6COnIOTHbl:H nOKa38Tenb npenOMneHHSI B03,Ayxa (n} = 1) MeHbwe a6co-
nlOTHoro nOKa3aTenH npenoMneHHSI Bel.I..l;eCTBa nneHKH (n2 = 1,4), KOTOpblH B CBOIO
O'lepe,Ab MeHbwe 86conlOTHoro nOKa38TenSl npenOMneHHSI CTeKna (n3 = 1,5), TO B 060-
HX cnyqaSlx OTpameHHe npOHcxonHT OT cpeAM. OnTH'leCKH 60nee nnOTHO:H, 'leM Ta
cpe,Aa, B KOTOpO:H HneT n8,AalOll.tHH ny'l. TI03TOMY <l>a3a Kone6aHHSI nyqa AS} npH
OTp8meHHH B TO'lKe A H3MeHSleTCSI Ha 7t p8,A H TO'lHO TaK me Ha 7t p8,A H3MeHHeT-
CSI <l>83a Kone68HHH nyq8 BCS 2 npH OTp8meHHH B TO'lKe B. Cne.n;oB8TenbHo, pe-
3ynbTaT HHTep<l>epeHHH 3THX nyqeH npH nepeCe'leHHH B <l>oKyce F nHH3bl 6Y,AeT
TaKoH }Ke, KaK ecnH 6bl HHKaKoro H3MeHeHHH <l>a3bl Kone6aHH:H HH Y Toro, HH y
,Apyroro nyqa He 6Mno.
113 PHCYHK8 17.37 BH,AHO, 'lTO OnTH'leCKaR p83HOCTb XO,Aa nyqeH SADS} H
SABCS 2
 = (AB + BC)n2 - ADn 1 .
Cne.n;oBaTenbHo, ycnoBHe MaKCHM8nbHoro ocn86neHHfl CBeT8 npHMeT BH,A
(2m + l ) A
(AB+BC) -AD = .
2
(2m + 1) A
TIpH a = 0 reOMeTpH'leCKaSl pa3HOCTb xo.n:a AB + BC = 2h Ii  = 2hn2 = 2 '
OTKY,Aa HCKOMaSl TOnl.I..l;HHa nneHKH
h = (2m + 1 )A .
4n 2
TIon8raSl m = O. 1, 2, 3, ..., nonY'lHM pSl,A B03MomHblX 3Ha'ieHIi:H Ton I.I..I;H Hbl
nneHKH:
A 3A
ho = - = 0,11 MKM; hI = 4 n- = 3ho = 0,33 MKM H T.,A.
4n 2 '-1
5. H8 ,AH<I>P8KHOHHYIO peweTKY HopManbHO K ee nOBepXHOCTH na,AaeT napan-
nenbHblH nY'lOK lIY'leH C ,AnHHOH BonHbl 0,50 MKM. TIepHO,A .n:H<I>paKHOHHO:H pe-
weTKH 4.95 MKM. Onpe,AenHTe, CKonbKO MaKCHMYMOB ,AalOT ,AH<I>paKHoHHaSl pe-
weTK8 H MaKcHManbHblH yron OTKnOHeHHfl nY'le:H, COOTBeTCTBYlOl.I..I;HX nOCne,AHeMY
,AHpaKHoHHOMY MaKCHMYMY.
)J;aHo: A=5,O.10- 7 M, d=4,95.10- 6 M
Ha:HTH: m, <Pm'
Pewe H H e. OqeBH,AHO, 'lTO ,AH<I>paKHoHHaSl pellleTKa He MomeT OTKnOHHTb
nyqH 60nbwe, qeM H8 90°. TI03TOMY, 38MeHHB sin <P B <l>opMyne ,AH<I>paKHoHHOH
peweTKH d sin <P = mA H8 1. nonyqHM 3H8'leHHe
d
m = -; m = 9,9.
A
QHcno m 06S138TenbHO nonmHO 6MTb enbIM, HO OHO He MomeT 6blTb paB-
HbIM 10, TaK KaK npH 3TOM 3HaqeHHe sin<p CTaHeT 60nbwe 1. Cne,AOBaTenbHo, B
06e CTOpOHbI OT eHTpanbHoro MaKCHMYMa, COOTBeTCTBYlOl.I..I;erO m = 0, 6Y,AeT H86mo-
,AaTbCSI no N = 9 MaKCHMYMOB.
Hcxo,ASI H3 MaKCHManbHoro 3H8'leHHSI m = 9, nerKO onpe,AenHTb <p m' ITo <l>OPMY-
ne ,AH<I>paKHoHHOH peweTKH
sin <Pm = 9A :::: 0,9091. Tor,Aa <Pm :::: 65°.
d
538

6. ,I:(H<I>paKHoHHaSl peIlIeTKa, HMeIO:[,QaSl 100 IlITpHXOB H8 1,0 MM ,AnHHbI, no-
Me:[,QeHa H8 paCCTOSlHHH 2,0 M OT 3KpaHa H OCBe:[,QaeTCSI nY'lKOM lIY'leii 6enoro CBe-
T8. Onpe,AenHTe IlIHPHHY ,AH<I>paKHoHHoro cneKTpa nepBoro nOpSl,AK8, nonY'leHHO-
ro Ha 3KpaHe. rpaHHa BH,AHMOrO cneKTp8 no ,AnHHaM BonH: ,AnSI <l>HOJIeTOBOro CBe-
Ta 0.40 MKM, ,AnSI Kp8CHoro 0,68 MKM.
LL
Puc. 17.38
,I:(aHO: N=I,O .10- 5 M- I , D=2,0 M,
m= 1, A4>=4,0 .10- 1 M, AJ(p=6.8 .10- 1 M.
HaiiTH: I::1x.
Pew e H H e. I1YCTb nepBbIii MaKcHMYM
,AnSI KpacHoro CBeTa OTKnOHeH OT eHTpa
3KpaHa (TO'lKa 0) Ha paccToSlHHe X2,
.n;nSl <pHoneTOBOI'O - H8 paccToSlHHe xI
(pHC. 17.38). B 38.n;8'1e Tpe6yeTcSI onpe,Ae-
nHTb IlIHPHHY ,AH<PP8KHoHHoro cneKTp8
nepBoro nopSl.n;K8, T.e H8iiTH pR3HOCTb
I::1x = x 2 - xl' 113 <popMynbl .n;H<I>P8KHOHHOR
pemeTKH d sin <p = mA, nonomHB m = 1, Haii-
. A
.n;eM Slncp = d' TaK KaK yron OTKnOHeHHSI cp Man (x« D). TO MomHO CqHTaTb, 'lTO
. t x
Slnq>= gq>= n '
AD
C Y'leTOM 3Toro nonY'IHM x = - = ADN. OTKY,AR
d
XI = ADN, 8 X2= AJ(pDN.
Tor,Aa I::1x = DN(AJ(p - A4»; I::1x = 76 MM.
T8KHM 06pa30M. B 06naCTH nepBoro MRKCHMYMR cneKTp BH,AHMOrO CBeTa 6Y,AeT
paCTSlHYT Ha 76 MM.
7 CBeToBaSl BonHa C ,AnHHOii BOJIHbI 700 HM pacnpOCTpaHSleTCSI B B03,Ayxe. Ka-
KOBa ,AnHHa BonHbl B BO,Ae?
)J;aHo: AI=7,00.10- 1 M, n=I.33.
HaRTH: A2'
Pew e H H c. )J;nHHbI CBeTOBbIX BonH Al H A2 B B03.n;yxe H B Bo.n;e CBSl38Hbl co
CKOpOCTSlMH VI H V2 p8cnpOCTpaHeHHSI 3THX BonH B B03,Ayxe H B BO,Ae cne.n;YIO:[,QH-
MH COOTHomeHHSlMH:
Al = VI H A2 = V2 .
V v
r.n;e v - 'laCTOTa CBeTOBbIX Kone68HHii. KOTOp8S1 He H3MeHSleTCSI npH nepexo.ne CBe-
Ta H3 O,AHOR Cpe,Abl B ,ApyryIO. Pa3.nenHB nO'lneHHO ypaBHeHHSI (1), nonY'lHM
Al = VI
A 2 V2
CKOpOCTH pacnpOCTpaHeHHSI CBeTa B B03,Ayxe H B Bo.ne CBSl3aHbl C a6conIOTHbl-
MH nOKa3aTenSlMIi npenOMneHHSI nl Ii n2 3THX cpe,A COOTHomeHHeM
= 
v 2 n l
(1)
(2)
(3)
Al = n 2
CpaBHHB8S1 MemAY co6oii Bblp8meHHSI (3) H (2). H8XO.nHM A 2 n 1 ' OTKY,A8
A 2 = Aln 1 ; A 2 =5,26.10- 1 M =526 HM

539

18. 3EMEHTbI TEOPHH OTHOCHTEbHOCTH
IIocTYJlaTLI TeopHH OTHOCHTeJlLHOCTH. OCHOBHLle CJleACTBH.H TeopHH OT-
HOCHTeJlLHOCTH H ux 3KcnepHMeHTaJlLHa.H npOBepKa. PeJl.HTHBHCTCKHH 3aKOH
CJlOeHH.H CKopocTeH. CKOpOCTL CBeTa B BaKyy.Me KaK npeAeJlLHa.H CKOpOCTL
nepeAa'lH cHruaJla. 3aBHCHMOCTL MaCCLI TeJla OT CKOpOCTH. 3aKOH B3aHMO-
CB.H3H MaCCbI H 3HeprHU.
18.1. IIocTYJIaTLI TeopHH OTHOCHTeJILHOCTH. CKOpOCTb CBeTa
B BaKYYMe KaK npeeJILHaH CKOpOCTL nepeaqH CHrHaJIa
Bce 3aKOHbI H3HKH onHCbIBaIOT, KaK npOTeKalOT Te HJlH HHbIe H3HqeC-
KHe .HBJleHH.H. B MexaHHKe H3yqaeTC.H MeXaHHqeCKOe ,lI;BHmeHHe. BC.HKoe ,lI;BH-
meHHe M01KHO onHcaTb C nOMOblO npocTpaHcTBeHHo-BpeMeHHbIx xapaKTepHC-
THK. TaK KaK JlI060H 06'beKT HMeeT onpe,AeJleHHOe nonomeHHe. KOTopoe npH
,lI;BHmeHHH H3MeH.HeTCSI co BpeMeHeM.
B OCHOBe KJlaCCHqeCKOH H3HKH nemaT CJle,ll;YIOHe npe,ll;CTaBJleHH.H 0 CBOH-
CTBax npOCTpaHCTBa H BpeMeHH:
a) TpexMepHOCTb npOCTpaHCTBB H O,ll;HOMepHOCTb BpeMeHH;
6) 06paTHMocTb npOCTpaHCTBa H He06paTHMocTb BpeMeHH;
B) O,ll;HOpO,ll;HOCTb npOCTpaHCTBa H BpeMeHH, qTO 03HaqaeT paBHonpaBHe Bcex
TOqeK npOCTpaHCTBa H Bcex MOMeHTOB BpeMeHH;
r) H30TponHOCTb npOCTpaHCTBa 03Ha'IaeT, qTO B npOCTpaHCTBe paBHonpaB-
HbI He TonbKO Bce TOqKH, HO H Bce HanpaBneHH.H;
,lI;) a6COJllOTHOCTb npOCTpaHCTBa H BpeMeHH nOKa3bIBaeT, qTO Bce BbIllie ne-
peqHCneHHbIe CBOHCTBa npOCTpaHCTBa Ii BpeMeHH nOCTOSlHHbI, He 3aBHC.HT OT
,lI;BHmeHH.H MaTepHaJlbHbIX TeJl, He CB.H3aHbI ,lI;pyr C ,lI;pyroM H C ,lI;BHmeHHeM
MaTepHH.
OCHOBHbIM npHHl1HnOM KJlaCCHqeCKOH MexaHHKH .HBJI.HeTC.H npHHUHn OTHO-
CHTenbHOCTH raJlHJleSl, COrJlaCHO KOTOPOMY Bce HHepIJ;HanbHbIe CHCTeMbI OTCqe-
Ta paBHonpaBHbI no OTHOlIIeHHIO K MeXaHHt.:IeCKHM .HBneHH.HM. BJIaro,ll;apSi 3TO-
MY paBHonpaBHIO OPMYJlHpOBKH 3aKOHOB MexaHHKH BO Bcex HHepl1HaJlbHbIX
CHCTeMax OTCqeTa O,ll;HHaKOBbI.
TeopHJI OTHOCHTeJlLHOCTH npe,ll;CTaBJlSleT C060H HOBoe yqeHHe 0 npOCTpaH-
CTBe H BpeMeHH, npHllIe,ll;llIee Ha CMeHY CTapOMY KJlaCCHqeCKOMY yqeHHIO.
nO.HBJleHHe TeopHH OTHOCHTeJlbHOCTH 6Mno CB.H3aHO C pa3BHTHeM 3J1eKTpO-
,lI;HHaMHKH. nOcne Toro KaK Am. MaKCBeJlnOM BO BTOpOH nOJlOBHHe XIX B.
6bIJlH COPMYJlHpoBaHbI OCHOBHbIe 3aKOHbI 3J1eKTpO,ll;HHaMHKH, BCTaJl Bonpoc 0
npHMeHHMOCTH MeXaHHqeCKOrO npHHIJ;Hna OTHOCHTenbHOCTH K 3J1eKTpOMarHHT-
HbIM H OnTHqeCKHM .HBJleHH.HM. A ,lI;JI.H 3Toro HymHo 6bIJlO BbI.HCHHTb, H3Me-
HSiIOTC.H JlH OCHOBHbIe 3aKOHbI 3J1eKTpO,ll;HHaMHKH npH nepeXO,ll;e OT O,AHOH
HHepUHanbHoH CHCTeMbI OTC'leTa K ,lI;PyroH HnH. KaK H 3aKOHbI HblOToHa. oc-
TaIOTC.H HeH3MeHHMMH (HHBapHaHTHbIMH).
Cor naCHO KnaCCHqeCKOMY 3aKOHY CJIOmeHH.H CKopOCTeH, KOTOpbIH SlBJlSleT-
CH CJle,ll;CTBHeM H3 npe06pa30BaHHH raJIHne.H, CKOpOCTH CBeTa B BaKYYMe B
nO,ll;BHmHOH H HenO,ll;BHmHOH HHepIJ;HaJlbHbIX CHCTeMax OTCqeTa )J;OJlmHbI 6bITb
pa3HbIMH:
540

c = C' + V, OTCIO.v;a C = c'1: v, r.v;e c - CKOpOCTb CBeTa B BaKYYMe B HenO-
,ABlDKHOH H c' - B nO,ABHmHOH CHCTeMe OTCqeTa; v - CKOpOCTb .v;BHmeHH.H no-
.v;BHmHOH CHCTeMbI OTCqeTa OTHOCHTeJlbHO HenO,ABHmHOH. 113 ypaBHeHHH MaKC-
BeJIJIa CJle.v;OBaJIO, qTO CKOpOCTb CBeTa (CKOpOCTb pacnpOCTpaHeHHH 3JIeKTpO-
MarHHTHbIX BonH) B BaKYYMe o.v;HHaKOBa no BceM HanpaBJleHHHM H paBHa
c = 3.108 M .
C
B03HHKIIIee npoTHBOpeqHe Me*.v;y 3J1eKTpo.v;HHaMHKoH H KJlaCCHqeCKOH Me-
xaHHKoH MomHO YCTpaHHTb, eCJlH npHHHTb O,AHY H3 Tpex rHnOTe3:
a) CqHTaTb, qTO npHHHn OTHOCHTeJIbHOCTE HenpHMeHHM K 3J1eKTpOMarHHT-
HbIM HBJIeHH.HM;
6) HeBepHbI ypaBHeHHH MaKCBeJlJIa H HX HymHo COOTBeTCTBYlOlI..{HM o6pa-
30M H3MeHHTb, qT06bI OHH y.v;OBJleTBOp.HJlH npe06pa30BaHHHM ranHJIe.H;
B) HeBepHbI npe06pa30BaHH.H raJIHJIeH H HX Cne,AyeT 3aMeHHTb ,ApyrHMH.
P.H,A yqeHbIX, B TOM qHCJle H X. JIopeH, CqHTaJlH, qTO [lpHHHn OTHOCH-
TeJlbHOCTH HenpHMeHHM K 3JIeKTpOMarHHTHbIM H OnTHqeCKHM .HBJIeHHHM, 11 qTO
3JIeKTpOMarHHTHbIe npoeCCbI npOHCXO,A.HT B OC060H Heno.v;BHmHOH cpe-
.v;e - 3qmpe, 3anOJIH.HIOeH Bce npOCTpaHCTBO. IIHepHaJlbHaH CHCTeMa OT-
CqeTa, nOKO.Ha.HCH OTHOCHTeJIbHO 3cpHpa, HBnHeTC.H npHopHTeTHoli no cpaB-
HeHHIO C .v;pyrHMH HHepHaJlbHbIMH CHCTeMaMH, H TonbKO B 3Toli CHCTeMe CKO-
pOCTb CBeTa OHHaKOBa no BceM HanpaBJIeHHHM.
O.D;HaKO 3KcnepHMeHTbI, nOCTaBneHHbIe B 1881 r. A. MalIKeJlbcoHoM H
S. MOPJlH, He nO.D;TBep,AHJlH 3Toro.
MalIKeJIbcoH H MOPJlH C nOMOblO HHTepcpepoMeTpa CneHaJIbHOH KOHCT-
PYKHH cpaBHHBaJlH Mem,Ay C060H 3HaqeHHH CKOpOCTH CBeT8 B CHCTeMe OT-
CqeTa, CB.H3aHHOH C 3eMJlelI, B HanpaBJIeHHH ,ABHmeHH.H 3eMJlH H B nep-
neH.D;HKYJlHPHOM eMY HanpaBneHHH. 113MepeHHH MHoroKpaTHo nOBTOpHJIHCb B
pa3J1HqHbIe BpeMeHa ro.v;a H B pa3Hoe BpeMH CYTOK. OKa3aJIOCb, '({TO CKOpOCTb
CBeTa B CHCTeMe OTCqeTa. cB.H3aHHoli C 3eMJleli. no BceM HanpaBneHHHM OlIH-
HaKOBa. TaKHM o6pa30M, 3KcnepHMeHTbI MalIKeJlbcoHa H MOpJIH (H .v;pyrHe
3KcnepHMeHTbI, nOCTaBJleHHbIe C eJlblO 06HapymeHH.H ,ABHmeHH.H 3eMJlH OT-
HOCHTenbHO 3cpHpa) nOKa3anH, qTO lIBHmeHHe 3eMJIE HHKal< He BJIH.HeT Ha CKO-
pOCTb pacnpOCTpaHeHH.H CBeTa. TeM caMbIM 6bIJla onpOBeprHYT8 He TOJlbKO
H.D;eH 0 cyecTBoBaHHH npeHMyecTBeHHOH CHCTeMbI OTCqeT8, CB.H3aHHOIi C
3cpl1pOM, HO E rl1nOTe3a 0 cyecTBoBaHHH caMoro 3cpHpa.
B 1905 r. A. 3lIHIIITeliH npHIIIeJl K BbIBO.v;y, qTO npOTHBOpet.IH.H Mem.v;y
TeopHeli H 3KcnepHMeHTOM. B03HHKalOII..{He npH npHMeHeHHH MeXaHHqeCKOrO
npHHHna OTHOCHTeJIbHOCTH K 3JIeKTpOMarHHTHbIM .HBJIeHH.HM, MomHO npeo.D;O-
JIeTb, eCJlH CqHTaTb H npHHHn OTHOCHTeJlbHOCTH Ii ypaBHeHH.H MaKCBenJIa
CnpaBe,AJIHBbIMH, a npeo6pa30BaHHH raJIHJleH HeTOqHbIMH H 3aMeHHTb HX
,ApyrHMH npe06pa30BaHH.HMH, OTHOCHTeJlbHO KOTOpbIX HHBapHaHTHbIMH HBJI.H-
IOTC.H He ypaBHeHHH KJIaCCHqeCKOIi MexaHHKH, a nlO6bIe ypaBHeHH.H, MOlIenH-
pYIOHe cpH3HqeCKHe 3aKOHbI.
SIIHIIITeliH nOKa3aJI, '({TO npe06pa30B8HHH ranHne.H BepHbI nl1IIIb B TOM
cnyqae. KorlIa CKOpOCTH ,ABHmeHHH Ten H CHCTeM OTCt.IeTa MHoro MeHbIIIe CKO-
pOCTH CBeTa B BaKYYMe. ECJIH me 3TH CKOpOCTH 6JIH3KH K CKOpOCTH CBeTa.
541

TO 3aKOHbI KnaCCHqeCKOH MeXaHHKIi ,AOnmHbI 6bITb 3aMeHeHb] 60Jlee 06liMIi
3aKOHaMH, yqliTbIBalOliMI1 OC06eHHOCTIi TaKOrO ,ABlimeHH.H.
3HHWTeHH BnepBble npHWeJl K BbIBOY 0 3aBliCliMOCTH CBOHCTB BpeMeHIi
Ii npOCTpaHCTBa OT lIBlimeHIi.H MaTepHaJIbHbIX 06'beKTOB. C KOTOpbIMH CB.H3bI-
BaIOTC.H liHepUliaJlbHbIe CliCTeMbI OTCqeTa. IIepecMoTp KJlaCCliqeCKliX npeCTaB-
neHHE n03BOJlHn 8HHWTeHHY C03aTb HOBOe eHlie 0 npOCTpaHCTBe, BpeMeHIi
Ii BHmeHIiIi - Cnel.\lianbHYIO TeOpHIO OTHOCHTenbHOCTH (CTO).
CTO 6a3lipyeTC.H Ha BYX nOCTYJlaTaX. IIepBbIH nOCTYJlaT Ha3bIBaeTC.H npliH-
linOM OTHOCliTeJlbHOCTH 8HHWTeHHa. OH npeCTaBJI.HeT C060H 0606eHHe Me-
XaHliqeCKOrO npHHHna OTHOCHTenbHOCTH Ha BCe 6e3 liCKJlIOt.zeHHSI cpH3Ht.zeC-
Klie .HBneHIi.H Ii MOmeT 6bITb CcpOpMynlipoBaH TaK: npH OHliX Ii Tex me YC-
nOBH.HX Bce cpli3HqeCKlie SIBneHH.H B m060H HHepl1.lianbHOH CliCTeMe OTCqeTa
npOliCXO,ASIT COBepweHHO O)1liHaKOBO, 8 cpli3HqeCKlie 3aKOHbI He 3aBliC.HT OT
BbI60pa HHepl1.liaJlbHOH CHCTeMbI OTCqeTa.
8TO 3 Hat.z liT , t.zTO HliKaKliMIi 3KcnepHMeHTaMIi (MexaHHqeCKHMH, 3J1eKTpO-
MarHliTHbIMIi, onTlit.zeCKliMIi Ii ,Ap.), nOCTaBJleHHbIMIi BHYTPIi liHepl1.liaJlbHOH CIi-
CTeMbI, HeB03MomHO YCTaHOBliTb, nOKOliTCSI 3Ta CliCTeMa linli BlimeTC.H paB-
HOMepHO Ii npSIMOJlliHeHHO.
M3 aHHoro nOCTYJlaTa cneyeT, t.zTO npH nepexoe OT O,AHoii liHepl.1.lianb-
HOH CliCTeMbI OTCqeTa K ,APyroH MaTeMaTliqeCKlie BblpameHliSI 3aKOHOB CPli31i-
Kli He .nonmHbI H3MeH.HTbC.H.
BTOpOH nOCTYJlaT Ha3bIBalOT npliHl1.linOM nOCTOSIHCTBa (liJlIi npliHl1.linOM
liHBapHaHTHOCTH) CKOpOCTH CBeTa. OH MomeT 6bITb CCPOPMYJlHpOBaH TaK:
CKOpOCTb CBeTa B BaKYYMe He 3aBliCliT OT CKOpOCTIi BHmeHIi.H HCTOqHliKOB
Ii npHeMHHKOB CBeTa H BO Bcex HHepHaJlbHbIX CHCTeMax OTCqeTa OIiHa-
KOBa.
B3aHMOe"CTBHH Memy TenaMH B npHpoe He MOryT pacnpOCTpaHHTLCSI
C 6eCKOHe'lHO 60JILIIIOH CKOpOCTblO. CKOPOCTb CBeT8 B BaKYYMe SIBnSIeTCSI
npe.nenbHOH CKOpOCTblO nepe.naqH CHrHana H B3aliMo.neiicTBliii Bo06IUe B npli-
poe .
IIpeenbHbIii xapaKTep CKOpOCTH CBeTa He .HBnSIeTC.H nOCTYJlaTOM CTO, O-
HaKO OH lirpaeT TaKYIO me ponb, KaK Ii caMH nOCTYJlaTbI, Ii nO,ATBepm,AaeTCSI
MHorOt.zlicneHHbIMIi 3KcnepHMeHTaMIi.
18.2. OCHoBHLIe CJIe]J;CTBHJJ TeopHH OTHOCHTeJILHOCTH
H HX 3KcnepHMeHTaJILHan npOBepKa
CornaCHO nepBOMY nocTynaTY CTO Bce cpH3liqeCKlie 3aKOHbI He 3aBliCSIT
OT KOHKpeTHoro BbI60pa liHepl1.HanbHOH CHCTeMbI OTCqeTa. n03TOMY ypaBHeHliSI.
npeCTaBnSIIOlie MaTeMaTHqeCKYIO 3anHCb cpH3liqeCKHX 3aKOHOB, )J;OJlmHbI
6bITb HHBapliaHTHbI OTHOCliTenbHO npe06pa30BaHliH JIopeHl1.a (npe06pa30BaHliSI
JIopeHa YCTaHaBJlHBalOT COOTHOllieHIi.H Memy KoopliHaTaMH Ii BpeMeHeM O,A-
Horo H Toro me C06bITliSI B ,ABYX liHepl1.liaJlbHbIX CIiCTeMax OTCqeTa - HenO,A-
BHmHOH K Ii nOBlimHOH K', BHmyeHcSl OTHOCHTenbHO CHCTeMbI K CO CKO-
pOCTblO v).
B npe06pa30BaHH.HX JIopeHa H3MeH.HIOTC.H He TOJlbKO KoopliHaTbI, HO II
BpeM.H, KOTopoe He CqliTaeTC.H TeKYliM OliHaKOBO BO Bcex liHepl1.HaJlbHbIX
542

IIpH )J;BHmeHHH TeJl II CHCTeM OTCqeTa CO CKO-
pOCT.HMH, MHOrO MeHbWHMH, qeM CKOpOCTb CBeTa B
BaKYYMe, npe06pa30BaHHH JIopeHa nepeXO)J;.HT B npe06pa30BaHH.H raJlHJle.H.
ECJlH V« C, TO « 1. TOr1I8 X' = X - vt; y' = y; z' = z; t = t'.
c
CJle,AOBaTenbHo, KnaCCHqeCKa.H (HbIOToHOBCKa.H) MexaHHKa HBJI.HeTC.H qaCT-
HbIM cJlyqaeM CTO. IIpH V  C BeJlHqHHbl X' II t' Tep.HIOT cpH3HqeCKHH CMblcn
(cTaHOB.HTC.H MHHMbIMH). 9TO HaXO,AHTC.H B COOTBeTCTBHH C TeM. qTO JJ;BHme-
HHe CO CKOpOCTblO V > C HeB03MomHO.
M3 nocTYJlaTOB CTO BbITeKaeT HeCKonbKO OCHOBHbIX CJle,D;CTBHH.
1. OTHOCHTeJlLHOCTL O,ll;HOBpeMeHHocTH.
COrJlaCHO KJlaCCHqeCKOH MexaHHKe )J;Ba C06bITH.H, o)J;HoBpeMeHHbIe B O)J;HOH
HHepHaJlbHOH CHCTeMe OTCqeTa, 6Y,AYT O,D;HOBpeMeHHbI H B JlI060H ,ApyroH
HHepHaJlbHOH CHCTeMe OTCqeTa, TaK KaK XO,A BpeMeHH O,D;HHaKOB BO Bcex TOq-
Kax npOCTpaHCTBa H BO Bcex HHepHaJlbHbIX CllCTeMax OTCqeTa.
BTOPOH nOCTYJlaT CTO OTBepraeT rHnOTe3Y a6-
COJlIOTHOH o.nHOBpeMeHHOCTH pa3HOMeCTHbIX C06bI-
THH. O,D;HOBpeMeHHocTb npOCTpaHCTBeHHO pa3,AeJleH-
HbIX C06bITHH OTHOCHTeJlbHa: C06bITH.H, .HBJI.HIOH-
ec.H O,AHOBpeMeHHbIMH B O,AHOH HHepHaJlbHOH CH-
CTeMe OTCqeTa, He O,AHOBpeMeHHbI B ,APyrHx HHep-
HaJlbHbIX CHCTeMax, ,D;BHmYHXC.R OTHOCHTeJlbHO
nepBOH. 't.JT06bI y6e,D;HTbC.R B 3TOM, paCCMOTpHM
CJleJJ;YIOIl.tIlH MbICJleHHbIH 3KcnepHMeHT. 1I0nycTHM,
qTO KOCMOHaBT XOqeT cpaBHHTb XO,A qaCOB A liB,
YCTaHOBneHHbIX H8 npOTHBOnOJlOmHbIX KOHax KOC-
MHqeCKOrO Kopa6J1.H - CHHXpOHH3HpOBaTb qaCbI (pHC. L8.2). OqeBH,AHO, 3TO
npoe Bcero C,D;eJlaTb C nOMOblO CBeTOBoro CHrHaJla. MbICJleHHO BbInonHHM
Ha6J1IO,D;eHHe 3a 3THM npoeCCOM C TOqKH 3paHH.H ,D;BYX MCO - K', CBSl3aH-
HOH C Kopa6J1eM, H MCO K, OTHOCHTeJlbHO KOTOpOH Kopa6J1b ,ABHmeTC.H. AJI.H
3Toro C nOMOblO HCTOqHHKa, HenO,ABHmHOro OTHOCHTeJlbHO Kopa6J1S1 H pac-
nOJlOmeHHOro B ero Cepe,AHHe. KOCMOHaBT npOH3BO,AHT BcnbIWKY CBeTa. CBeT
O,AHOBpeMeHHO JJ;OCTHraeT 060HX qaCOB. ECJlH nOKa3aHH.H qaCOB B 3TOT MOMeHT
O,llHHaKOBbI. TO qaCbI HYT CHHXpOHHO.
Ho TaK 6YlIeT JlHWb OTHOCHTeJlbHO CHCTeMbI K'. CB.H3aHHOH C Kopa6J1eM.
B CHCTeMe me OTCqeTa K. OTHOCHTenbHO KOTOpOH Kopa6J1b ,ABHmeTC.H. nOJlo-
meHHe HHoe. 't.JacbI Ha HOCY Kopa6J1.H Y)J;aJl.HIOTC.H OT Toro MeCTa, re npo-
-
X X'
PUC. 18.1
Y ,!
I K'
  
B A
L_____------_
,'0' C X'
I
;Z'
PUC. 18.2
CHCTeMax OTCqeTa. B npoCTeHweM cJlyqae, Kor)J;a
CHCTeMa K' ,D;BHmeTC.H paBHoMepHo H np.HMOJlHHeH-
HO B,AOnb OCH X (pHC. 18.1), npe06pa30BaHH.H JIo-
peHua )J;n.H KOOp,D;HHaT H BpeMeHH CJle,AYIOlIe:
vx
( ) t----
, X - vt , , , 2
X = g 2 ; y = y; z = z; t = g C v2 .
1-- 1--
c 2 c 2
(18.1)
543

H30Illna BCnbIIIIKa CBeTa HCTOqHHKa, H, qT06bI .n;OCTHrHYTb qaCOB A, CBeT .n;on:>KeH
npeo.n;OJleTb paCCTO.HHHe, 60JIbIllee nonOBHHbI ,lVIHHbI Kopa6J1.H (pHC. 18.3, a, 6).
HanpoTHB, qaCbI B Ha KopMe npH6J1HJKaIOTC.H K MeCTY BcnbIIIIKH, H nYTb CBe-
TOBoro CHrHaJIa MeHbIlle nonOBHHbI ,AnHHbI Kopa6JI.H. II03ToMY Ha6nlO.n;aTeJIb
B CHCTeMe K npH,AeT K BbIBO.n;y, "lTO CHrHaJlbI ,AOCTHraIOT 060HX qaCOB He-
o.n;HOBpeMeHHO. 3TO MomHO .n;OKa3aTb, HCnOJlb-
3Y.H npe06pa30BaHH.H JIopeHa. ECJIH C06bITHSI
B CHCTeMe K npOCTpaHCTBeHHO pa3.n;eJleHbI
(Xl :I: X 2 ), HO O.n;HOBpeMeHHbI (t l = t 2 = t), TO B
a YY' K
,I,
-0-
','
At'9
c
XX'
CHCTeMe K ' COrJlaCHO npe06pa30BaHH.HM JIo-
peHa (18.1)
x;= g ;
1--
c 2
6
Y !Y'
......
K  ' ,',
I
B I
o /0' C
I
Z ;Z'
...
X X'
A 
VX l
t--
I 2
t - C .
1- g 2'
1--
c 2
, x2 - vt
x2 = g 2 ;
1--
c 2
t _ VX2
I c 2
t 2 = g 2 .
1--
c 2
OTKy.n;a X:I: x, t l ':# t, T .e. C06bITH.H. oc-
Pue. 18.3 TaBa.HCb npocTpaHcTBeHHo pa3,AeJIeHHbIMH, OKa-
3bIBaIOTC.H H Heo.n;HoBpeMeHHbIMH.
,l],Ba JII06bIX C06bITH.H B TOqKaX A H B. O,AHOBpeMeHHbIe B CHCTeMe K, He-
O,AHOBpeMeHHbI B CHCTeMe K. Ho B CHJlY npHHHna OTHOCHTeJlbHOCTH CHCTe-
MbI K' H K COBepIlleHHO paBHonpaBHbI. HH O.n;HOH H3 STHX CHCTeM Henb3.H
OT.n;aTb npe.n;nOqTeHHe. IIosToMY MbI H BbIHym,AeHbI npHHTH K 3aKJIlOqeHHIO.
qTO 0JI.1I0BpeMeHHOCTb npOCTpaHCTBeHHO pa3JI.eJIeHHbIX C06LITHH OTHOCHTeJIbHa.
IIpH"lHHOH OTHOCHTeJlbHOCTH o.n;HOBpeMeHHOCTH HBJI.HeTCH KOHet.IHOCTb CKOpOC-
TH pacnpOCTpaHeHH.H CHrHaJIOB.
2. OTHOCHTeJILHOCTL npOMe:)KyrKoB BpeMeHH.
OTHocHTenbHocTb o.n;HOBpeMeHHOCTH C06bITHH npHBo.n;HT K TOMY, t.ITO .n;nH-
TeJIbHOCTb (npOMeJKYToK BpeMeHH) B pa3HbIX HHepHanbHbIX CHCTeMax OTCqe-
Ta TaKme 6y.n;eT pa3nHqHa.
AJIHTeJlbHOCTLIO (npOMe:)KyrKoM BpeMeHH) Mem.n;y .n;BYM.H C06bITH.HMH Ha3bI-
BaeTC.H BpeMH, npOIlle.n;Illee MeJK,AY STHMH C06bITH.HMH. H3MepeHHoe qaCaMH.
pacnOJIOmeHHbIMH B )J;aHHoH CHCTeMe OTCqeTa. IIycTb B TOqKe A, Heno.n;BHJK-
HOH OTHOCHTeJlbHO CHCTeMbI K', B MOMeHTLI BpeMeHH t 1 H t 2 npOH30IIIJlH .n;Ba
C06bITH.H. HanpHMep, MaTeMaTHqeCKHH Ma.HTHHK ,ABam,AbI npOIlleJl nOJlOmeHHe
paBHOBeCH.H. TIpoMemYToK BpeMeHH to Mem.n;y 3THMH C06bITH.HMH 'to = t; - t.
r.n;e t H t - BpeM.H, OTCqHTbIBaeMoe no qaCaM CHCTeMbI K'. BpeMH, H3Me-
pHeMoe Ha t.Iacax B TOH HCO. r,Ae OHH nOKO.HTC.H, Ha3bIBaeTC.H c06cTBeHHLIM
BpeMeHeM. C06cTBeHHoe BpeM.H OTCqHTbIBaeTC.H no qaCaM, .n;BHmYHMCSJ BMec-
Te C cHcTeMoH OTCqeTa. B CHCTeMe K, OTHOCHTenbHO KOTOpOi1 ,ABHmeTC.H CHC-
TeMa K ' . npOMemYTOK BpeMeHH t Mem.n;y ,ABYM.H C06bITH.HMH, npOHCIlle.n;IIIHMH
544

B TOqKe A, paBeH 't = t 2 - t I , r)J;e t l H t 2 - BpeM.H, OTCqHTbIBaeMoe no qaCaM
CHCTeMbl K.
COrJlaCHO
npe06pa30BaHH.HM JIopeHu.a
VX2
t2-
, C '
t 2 = g ; tl
v 2
1--
c 2
t l _ vXI
= g '
v 2
1--
c 2
Tora
( ) v (X2 - xd
t 2 - t l - 2
, , C
'to ==t 2 -t l == g
v 2
1--
c 2
==
V(X2 -xd
't-
c 2
g 2
1--
c 2
TaK KK x 2 - Xl eCTb CMeeHHe BOJlb OCH OX 3a BpeM.H 't B CHCTeMe
OTqeTa K, B KOTOpOH npOHCXO,AHJIH 3TH C06bITH.H, TO X2 - Xl = v't. ll03TOMY
[7'
'to = 'tV 1 -  ' OTCIOa
 = 1: v 2 .
c 2
TaKHM 06pa30M, B BHmYHXC.H OTHOCHTeJIbHO Pyr pyra CHCTeMax OT-
CqeTa co CKOpOCTblO, 6J1H3KOH K CKOpOCTH CBeTa B BaKYYMe, Ha6J1IO,AaeTC.H 3cp-
cpeKT 3aMe.r.tJIeHH.H BpeMeHH. T.e. 't> 'to.
,l(JlHTeJlbHOCTb .HBJleHH.H, npOHCXO.HIT(ero B HeKOTopOH TOqKe A npOCTpaH-
CTBa, 6Y)J;eT HaHMeHbIIIeH B TOH HHepu.HaJlbHOH CHCTeMe OTCqeTa, OTHOCHTeJlbHO
KOTOPOH 3Ta TOqKa HenO)J;BHmHa. STO 3HaqHT, qTO qaCbI, BHmYHeC.H OTHOCH-
TeJlbHO HHepu.HaJlbHOH CHCTeMbI OTCqeTa, HYT MeJIeHHee HenO,ABHmHbIX qa-
COB 11 nOKa3bIB81OT MeHbIIIHH npOMemYTOK BpeMeHH Memy C06bITH.HMH. ECJlH
v 2
me v« c, TO - ---7 0 H 't == 'to' T.e. npH BHmeHHH C MaJlbIMH CKOpOCT.HMH
c 2
3cpcpeKTOM 3aMe,AJIeHH.H BpeMeHH MO>KHO npeHe6peqb.
CKOpOCTH, C KOTOpbIMH HaM npHXOHTC.H CTaJlKHBaTbC.H B nOBCe,AHeBHOH
mH3HH, ropa30 MeHbIIIe CKOpOCTH CBeTa c, n03TOMY HeYHBHTeJlbHO. qTO MbI
He Ha6J1IOaeM 3aMeJleHHe BpeMeHH Ka:K peaJlbHbIH 3cpcpeKT. ,l(JI.H npOBepKH
3aMeJleHH.H BpeMeHH 6bIJlH nOCTaBJIeHbI cneu.HaJlbHbIe 3KcnepHMeHTbI. HanpH-
Mep, B 1971 r. CBepXTOqHbIe aTOMHbIe qaCbI COBepIIIHJIH KpyrocBeTHbIH nOJIeT
Ha 60PTY peaKTHBHoro CaMOJleTa. TaK Ka:K CKOpOCTb peaKTHBHoro CaMOJleTa
KM
(1 . 10 3 -) ropa30 MeHbIIIe CKOpOCTH CBeTa, norpeIIIHocTb xoa q8COB OJI-
'I
mHa 6bIJla COCT8BJI.HTb HeCKOJlbKO HaHoceKYH, n.H Toro qT06bI MomHO 6bIJIO
o6HapymHTb 3cpcpeKT 3aMeJleHH.H BpeMeHH. 't.JacbI 06JIaaJlH Tpe6yeMoH TOqHOC-
TblO, H npecKa3aHH.H 3HIIITeHHa 0 3aMe,AJleHHH BpeMeHH 6bInH nO,ATBepm,AeHbI.
(18.2)
545

ScpcpeKT 3aMe,nJleHH.H BpeMeHH nOTBepmaeTC.H T8Kme Ha6J1IOeHHHMH H8.A
SJleMeHTapHbIMH qaCTHaMH. TaK t H8npHMep, YCT8HOBJleHO, qTO CpeHee Bpe-
M.H mH3HH MIOOHa B COCTOHHHH nOKO.H paBHO 2,2 MKC. T8TeJlbHhIe SKcnepH-
MeHTbI nOKa3anH, qTO Kor,na MIOOH neTHT co CKOpOCTblO, 6J1H3KOH K CKOpOC-
TH CBeTa, ero BpeM.H mH3HH YBeJlHqHB8eTC.H T8K, K8K npe,nCK83bIBaeT cpOPMY-
Jla 38Me)J;JleHH.H BpeMeHH (18.2).
3. OTHOCHTeJILHOCTL AJIHH (paccTosHHii).
B KJlaCCHqeCKOH MeX8HHKe C'IHT8eTC.H HeonpOBepmHMbIM TOT CP8KT, qTO
p8CCTO.HHHe Memy )J;BYM.H TOqKaMH TeJla (HanpHMep, ,lJ,JIHH8 CTepmH.H) 0HH8-
KOBO BO Bcex HHepltHanbHbIX CHCTeMax OTCqeT8. B TeopHH me OTHOCHTeJlbHO-
CTH p8CCTO.HHHe He .HBJI.HeTC.H a6COJIIOTHOH BeJlHqHHOH: OHO 3aBHCHT OT CKOpO-
CTH BHmeHH.H TeJla OTHOCHTeJIbHO paCCMaTpHBaeMOH CHCTeMbI OTCqeT8.
nYCTb lo - JlHHa HenOBHmHoro CTepmH.H B HHepHaJlbHOH CHCTeMe
OTCqeTa K' - co6CTBeHHas AJIHHa CTepBS (,AJlHHa CTepmH.H B CllCTeMe OT-
CqeT8, OTHOCHTeJlbHO KOTOpOll TeJIO nOKOHTCH). ,AnHH8 l SToro me CTepmHH
B CllCTeMe OTCqeTa K, OTHOCllTeJlbHO KOTOpOH CTepmeHb ,ABHmeTC.H co CKO-
pOCTblO v, napanJleJlbHOH OCH CTepmH.H, p8BH8
(7
l = loV 1 -  = lov 1 - 13- .
(18.3)
11:3 nOCJleHeH cpOPMYJlbI CJleyeT, qTO l < lo, T.e. B HHepHaJIbHbIX CllCTe-
Max OTCqeTa, BHmYHXC.H Pyr OTHOCHTeJIbHO pyra CO CKOpOCTblO, 6J1H3KOM
K CKOpOCTH CBeTa B BaKYYMe, Ha6J1IO,naeTC.H scpcpeKT COKpaeHH.H JIllHbI Ten.
IIpHqeM COKpaeHHe JlHHbI TeJla Ha6J1IOaeTC.H TOJlbKO B HanpaBneHHH ero
BHmeHH.H, a nOnepeqHbIe pa3MepbI TeJla OCTaIOTC.H HeH3MeHHbIMH.
COKpaeHHe JlHHbI TeJla He CB.H3aHO C eHCTBHeM K8KHX-JlH60 CHJI, "C8B-
JlHBalOHX" CTepmeHb BlIOJlb ero JlHHbI. STO - KHHeMaTHqeCKHH scpcpeKT.
BbITeKalOHH H3 npe06pa30BaHHH JIopeHa. AeHCTBHTeJlbHO, c06CTBeHHa.H )J;JlH-
Ha CTepmH.H B CHCTeMe K - STO pa3HOCTb KoopHHaT KOHOB CTepmHH
to = X - X , a JlHH8 I BHmyerOC.H CTepmH.H B CHCTeMe K p8BH8 pa3HOCTH
KoopHHaT ero KOHOB l = X2 - Xl'
nOCTaBHM 3HaqeHH.H X H X; no cpopMYJle (18.1):
to = x2 - vt _ Xl - vt = X2 - Xl = l .
r;7r;7Gr;7
V- V- V- -
11:3 npe06p830B8HHH JIopeHa CJleyeT T8Kme, qTO y - y = Y2 - YI H
Z - Z = Z2 - Zl' T.e. nOnepeqHbIe pa3MepbI TeJIa He 3aBHC.HT OT CKOpOCTH )J;BH-
meHH.H.
ECJlH me v« C, TO 13  0 H l::::: 10, T.e. npH BHmeHHH C MaJlbIMH CKO-
pOCT.HMH scpcpeKTa COKpaeHH.H nHHbI He H86nlOaeTC.H.
P8CCMOTpeHHbIe BbIllie scpcpeKTbI H83bIBalOTCH peJISTHBHCTCKHMH. T8K K8K
OHH Ha6J1IOaIOTC.H npH CKOpOCT.HX BHmeHH.H, 6nH3KHX K CKOpOCTH CBeT8, 11
onHCbIBaIOTC.H CTO, HO He MorYT 6bITb 06'b.HCHeHbI C n03HltHH KJlaCCHqeCKOM
cpH3HKH.
(7
OTCIOa l = lovl-.
546

18.3. PeJUITHBHCTCKHH aaKOH CJIOmeHHH CKopocTen

X X'
HOBbIM peJl.HTHBHCTCKHM npe,l\CTRBJleHH.HM 0
IIpOCTpaHCTBe H BpeMeHH COOTBeTcTByeT HOBLIH 3a-
KOH CJlOmeHH.H CKopocTeH.
3aIIHIIIeM 3aKOH CJlOmeHHSI CKopocTeH ,l\JlH qaCT-
Horo cJlyqa.H, Kor,na TeJIO M ,l\BHmeTCH B,l\OJlb OCH
X' CHCTeMbI OTCqeTa K', KOTopaH, B CBOIO Oqepe,Ab,
,nBHmeTC.H co CKOpOCTblO V OTHOCHTeJlbHO CHCTeMbI
OTCqeTa K. I1pHqeM B npoQ,ecce lIBHmeHHSI KOOp,nH-
HaTHbIe OCH X H X' Bce BpeM.H COBnalIalOT. a KOOp-
,l\HHaTHLIe OCH Y H Y', Z H Z' OCTaIOTC.H napannenb-
HbIMH (pHC. 18.4).
0603HaqHM MO,nYJlb CKOpOCTH TeJIa OTHOCHTeJlL-
HO K' qepe3 VI' a MO,l\YJlb CKOpOCTH 3Toro me TeJla
V2' Tor,l\a peJlHTHBHCTCKHH 3aKOH CJlOmeHH.H CKOpOC-
VI
M
Puc. 18.4
OTHOCHTeJlbHO K qepe3
TeH 6Y,l\eT HMeTL BH,l\
VI + V
v 2 = .
1 + Vi V
c 2
3aMeTHM, qTO <popMYJla (18.4) npHMeHHMa TOJlbKO B TOM CJIyqae. eCJIH Bce
TPH BeKTopa V, Vi H V2 HanpaBJleHbI B.nonb O.D.HOH np.HMOH. B 06eM CJlY-
qae 3TOT 3aKOH HMeeT 60Jlee CJlOmHbIH BH,n. O,l\HaKO npH JlI060H <}JopMe 3anH-
CH 3aKOHa ero CYHOCTb 3aKnlOqaeTC.H B TOM, qTO CKOpOCTb C CBeTa B BaKYY-
Me SlBJI.HeTCH npe.u.eJlbHOH CKOpOCTblO nepe,naqH CHrHRnOB.
AeHCTBHTeJlbHO, nYCTL VI = c. HaH,neM CKOpOCTb V2:
(US.4)
C+V
V 2 = - = c.
1 + cv
c 2
I1pe,l\nOnOmHM, qTO TeJlO ,l\BHmeTC.H co CKOpOCTLIO VI = C OTHOCHTeJlbHO CHC-
TeMLI K', KOTopaH B CBOIO Oqepe,l\L ,nBHmeTC.H CO CKOpOCTLIO V = C OTHOCHTenL-
c+c
HO CHCTeMbI K. Tor,na V2 = = c.
1 + c.c
c 2
CJle,l\OBaTeJlbHO, npH JlI06bIX CKOpOCT.HX v] H V pe3YJlbTHpYIOIIJ;a.H CKOpOCTb
V2 He npeBbIIIIaeT C.
VI V
ECJlH v« C H VI« C, TO qneHOM c 2 B 3HaMeHaTeJIe MomHO npeHe6peqL
H BMeCTO (18.4) nOJlyqHM KJlaCCHqeCKHH 3aKOH CJlOmeHH.H CKopocTeH:
v 2 = vI + v. 3TO cornacyeTC.H C npHHQ,HIIOM COOTBeTCTBHH, cornaCHO KOTOPOMY
HOBa.H cpH3HqeCKa.H TeOpH.H He OTBepraeT Q,eJlHKOM npe,AIIIeCTBYIOYIO TeopHIO.
OHa YKa3bIBaeT npelIeJl IIpHMeHHMOCTH CTapOH TeopHH.
547

18.4. Macca, 3Hepl'HH H HMnyJILC
B KJIaCCH'IeCKOH MexaHHKe H B CTO
B KJlaCCHqeCKOH MexaHHKe BamHOH lIHHaMHqeCKOH BeJlHqHHOH .HBJI.HeTC.H
M8cca TeJla (CM.  2.4). H3 BToporo 3aKOHa HblOToHa CJleJJ;yeT. qTO Macca TeJla
.HBJI.HeTC.R MepoH ero HHepTHbIX CBOHCTB. B cpopMYJlax. CB.H3bIB81Oll.{HX HMnYJlbC
TeJla Hero CKOpOCTb. a TaKme KHHeTHqeCKYIO SHeprHIO H CKOpOCTb TeJl8, M8C-
ca TeJl8 TaKme BbIcTynaeT KaK Mepa HHepTHOCTH.
M8CC8, BXOA.Ha.H B 38KOH BceMHpHoro T.HrOTeHHSI, .HBJI.HeTC.H MepOH rp8-
BHTanHOHHoro B3aHMOAeHCTBH.H. HHepTHOCTb Ii cnoc06HOCTb K rpaBHTanHOH-
HbIM B3aHMOAeHCTBH.HM npeACTaBJI.HIOT C060H COBepIl.IeHHO pa3J1HqHbIe npO.HB-
JleHH.H CBOHCTB MaTepHH. OAHaKO TO, qTO MepbI STHX P83J1HqHbIX npO.HBJleHHH
0603H8q81OTC.H OAHHM H TeM me CJlOBOM, He CJlyq8HHO, a 06YCJlOBJleHO TeM.
qTO 06a CBoikTB8 BcerA8 cyeCTBYIOT COBMeCTHO H BcerAa APyr APyry npo-
nopnHOHaJlbHbI. MHOrOqHCneHHbIe SKcnepHMeHTaJlbHbIe cpaKTbJ )J;OKa3bIB8IOT,
qTO HHepTHa.H H rpaBHTanHOHHa.H Macc paBHbI.
B KJlaCCHqeCKOH MexaHHKe Macca paCCM8TpHBaeTC.H H KaK Mepa KOJlHqe-
CTB8 BeeCTBa: qeM 60JlbIl.Ie OTAeJlbHbIX q8CTHn (aTOMOB) COAepmHT CPH3H-
qeCKa.H CHCTeMa, TeM 60JlbIl.Ie ee Macca.
B CnenHaJIbHOH TeopHH OTHOCHTeJlbHOCTH B OTJlHqHe OT KJlaCCHqeCKOH MeX8-
HHKH MaCC8 TeJl8 He .HBJI.HeTC.H MepOH KOJlHqeCTB8 BeeCTBa. 9TO 06'b.RCH.HeTC.H
TeM, qTO B peJl.HTHBHCTCKOH TeopHH B nOH.HTHe MaTepHH BKJllOqaeTC.H He TOJlbKO
BeeCTBO (npOTOHbI, SJleKTpOHbI, HeHTpOHbI), HO Ii H3J1yqeHHe (cpOTOHbI). 3AeCb
yqHTbIBaeTC.H, qTO SHeprH.H OTAeJlbHbIX qaCTHn OqeHb BeJlHKa, Macca 6YAeT on-
peAeJl.HTbC.H He CTOJlbKO '1HCJlOM qaCTHn, CKOJlbKO HX SHeprH.HMH H B3aHMHOH
opHeHTanHeH HMnYJlbCOB.
B CTO Macca ABHmerOC.H TeJla He .HBJI.HeTC.H MepOH ero B3aHMOAeHCTBH.H C
rpaBHTanHOHHbIM nOJleM, TaR KaR STO nOJle 38BHCHT OT SHeprHH H HMnynbca TeJl8.
Macca TeJla, )J;BHmyerOC.H co CKOpOCTblO, 6nH3KoH K CKOpOCTH CBeTa, He
.HBJI.HeTC.H MepOH ero HHepTHOCTH. H3 nOCTYJl8TOB TeopHH OTHOCHTeJlhHOCTH BbI-
TeKaeT, qTO CKOpOCTb qaCTHnbI MomeT 6bITb JlH60 paBH8 CKOpOCTH CBeT8 BO
Bcex HHepnH8JIbHbIX cHcTeM8x, JlH60 MeHbIl.Ie CKOpOCTH CBeT8. CooTBeTcTBeHHo
p8CCMaTpHBaIOTC.H ABa KJl8CCa qaCTHn. 't.JaCTHnbI, ABHmYHeC.H C 86COJllOTHOH
CKOpOCTblO, OTJlHqaIOTC.H npeAenbHOH HHepUHOHHOCTblO: OHH Bcerlla llBHmYTC.R
TOJlbKO no HHepUHH H He MorYT 6bITb HH 3aMellJleHHbI. HH YCKopeHHbI. H8
OCHOBaHHH SToro npeAnOJlOmHJlH. qTO Macca TaKHX q8CTHn paBH8 HYJlIO. 9TH
qaCTHnbI Ha3bIBalOT 6e3MaCCOBLIMH. B SKcnepHMeHTe TaKHe q8CTIUbI 06H8PY-
meHbI - STO cpOTOHbI (CM.  19.6) H HeHTpHHo (CM.  23.2-23.3).
TIpH B3aHMoAeHcTBHH 6e3M8CCOBbIX q8CTHn C qaCTHnaMIi BeeCTBa BbInOJl-
H.HIOTC.H 38KOHbI coxp8HeHH.H SHeprHH H HMnYJlbCa, 0 qeM CBH)J;eTeJlbCTBYIOT
onbITbI no B3aHMoAeHcTBHIO «l>OTOHOB H HeHTpHHo C aTOMaMH, .HApaMH H sne-
MeHT8pHbIMH qaCTHnaMH. :3TO n03BOJl.HeT YTBepmAaTb, '1TO 6e3MaCCOBbIe qac-
THnbl 06J1aAa1OT sHeprHeH w* H HMnYJlbCa p*. TIpH STOM BO Bcex HHepnH-
aJlbHbIX CHCTeMax OTCqeTa BeKTOp HMny JlbCa TaKOH qaCTHnbI H ee SHeprH.H
OTnHqHbI OT HYJl.H, TaK KaK qaCTHny, ABHmyyIOC.H co CKOpOCTblO CBeTa c,
OCTaHOBHTb HeJlb3.H. B03MomHO JIHIl.Ib O)J;HOBpeMeHHoe 06paeHHe B HYJlb p*
H W* 6e3MaCCOBOH q8CTHnbI, '1TO 6YAeT 03HaqaTb npeKp8eHHe ee cyecTBo-
BaHHH (HanpHMep, B pe3YJlbTaTe nOrJlOeHH.H qaCTHneH BeeCTBa). J];JI.H 6e3-
MaCCOBbIX qaCTHn HMeeT MeCTO COOTHOIl.IeHHe
W*2 _p*2C 2 = O.
548

3TO BbIpameHHe CnpaBe)J;nHBO AJI.H 6e3MaCCOBbIX qaCTHII. BO BCeX HHepII.H-
aJlbHbIX CHCTeMaX OTCqeTa.
B OTJlHqHe OT 6e3MaCCOBbIX qaCTHu., ABHmYHXC.H C a6COJllOTHOH CKOpOC-
TblO, cyeCTBYIOT "laCTHII.bI (HanpHMep, qaCTHII.bI BeeCTBa), KOTOpbIe BCerAa
ABHmYTC.H co CKOpOCTblO V < C. CKOpOCTH TaKHX qaCTHu. B 3aBHCHMOCTH OT
HX HHAHBHAyanbHbIx KaqeCTB H B3aHMOAeHCTBHH C APyrHMH qaCTHII.aMH MO-
rYT H3MeH.HTbC.H B llIHpOKHX npeAeJlax OT HYJI.H AO JlI060ro 3HaqeHH.H v < C.
3TO CBOHCTBO qaCTHII. OnpeAeJl.HeTC.H HaJlHqHeM y HHX MaCCbI. llx Ha3bIBalOT
MaCCOBbIMH qaCTHu.aMH. Macca eCTb HX HHAHBHAyanbHa.H XapaKTepHCTHK&,
MaCCa qaCTHII.bI a6COJlIOTHa: OHa He 3aBHCHT OT BbI60pa HHepl{HaJlbHOH CHCTe-
MbI OTCqeTa. a 3HaqHT, H OT CKOpOCTH )J;BHmeHH.H qaCTHu.bI.
B CTO peJl.HTHBHCTCKHH HMnYJlbC CB060AHO ABHmyeHC.H qaCTHl{bI p (HJlH
) ... _ vW _
CHCTeMbI qaCTHl{, HJlH TeJla onpeAeJl.HeTC.H <}>OPMYJlOH p =, rAe v - CKO-
C
pOCTb qaCTHl.{bI. W - ee 3HeprH.H, onpelIen.HeMa.H H3 COOTHOllIeHH.H Mem)J;Y
3HeprHeH H HMnYJlbCOM: W 2 - p 2 C 2 = m 2 c 4 , rAe m - Macca qaCTHII.bI. TIpH
3TOM 3HeprH.H H HMnYJlbC H3MeH.HIOTC.H npH nepeXO,Ae OT OAUOH MCO K ,APY-
rOH (COrJlaCHO npe06pa30BaHH.HM JIopeHu.a), a Macca He H3MeH.HeTC.H.
MHBapHaHTHOH OTHOCHTeJlbHO npeo6pa30BaHHH JIopeHl{a (18.1) .HBJI.HeTC.H
pa3HOCTb w 2 - p 2 C 2 . 3HeprH.H H HMnYJlbC OAHOBpeMeHHO 06paTHTbC.H B HYJlb
MorYT TOJlbKO )J;JI.H 6e3MaccoBbIX qaCTHu..
ECJlH TeJlO MaccoH m nOKOHTC.H, p = 0, TO 3HeprH.H TeJla wl- 0 = m 2 c 4 He
06paaeTC.H B HYJlb. W o = mc 2 . 3TO 3HeprH.H nOKO.H, ee Ha3bIBalOT c06CTBeH-
HOH 3HeprHeH qaCTHII.bI.
ECJlH 'B <}>OPMY JlY COOTHOllIeHH.H MemAY
w 2 _ p 2 C 2 = m 2 c 4
3HeprHeH H
vW
no.ncTaBHTb 3HaqeHHe MOAYJI.H HMnYJlbCa p = .
C
W 2 2 2 [ 2 1
w 2 - c c = m 2 c 4 ; W 2 1- 2 = m 2 c 4 .
HMny JlbCOM
TO nOJlyqHM
OTKY)J;a nOJlHa.H 3HeprH.H TeJla
w=
mc 2
W= g 2'
1--
c 2
(18.5)
IIo)J;cTaBHM 3HaqeHHe W B <}>OPMYJlY )J;JI.H HMnYJlbCa
v mc 2
p=-' .
c 2 g 2'
1--
c 2
mv
P= g 2'
1--
c 2
IIOCJleAH.H.H <}>opMYJla onpeAeJl.HeT peJl.HTHBHCTCKHH HMnYJlbC TeJla, ,ABHmy-
eroc.H co CKOpOCTblO v, 6J1H3KOH K CKOpOCTH CBeTa c.
(18.6)
549

HymHo OTMeTHTb. qTO KaK H B KnaCCHqeCKOH MeXaHHKe. 3HeprH.H 06J1a-
,AaeT a)J;,AHTHBHOCTblO ,AJI.H CB060,AHbIX Ten (B OTCYTCTBHe B3aHMO,AeHCTBHH), a
HMnYJlbC BCer.Aa a.A.AHTHBeH, T. e. nOJlHbIe HMnynbc H 3HeprH.H CHCTeMbI n CBO-
n n
60,lIHbIX TeJl jJ = L P" W = L . B cJlyqae 3aMKHYTOH cpH3HqeCKOH CHCTeMbI
1=1 1..1
HBJI.HIOTC.H CnpaBe.AnHBbIMH 3aKOHbI coxpaHeHH.H HMnYJlbca H 3HeprHH.
KHHeTHqecKR.H 3HeprH.H qaCTHnbI W K B HeKOTopOH CHCTeMe OTCqeTR onpe.n;eJl.H-
eTC.H KaK pa3HOCTb Mem,qy ee nOJlHOH 3HeprHeH If 3HeprneH nOKO.H W K = W - W o .
mc 2 2 2 [ 1 ]
W K = g 2 -me =me g 2 -1 .
1-- 1--
e 2 e 2
TIpH CKOpOCTH v < < C KHHeTHqeCKa.H 3HeprH.H TeJla
(18.7)
2 [ 1 ] 2 [ 1 J mv2
W =me g -l =me -1 ::::-
I< v 2 1 v 2 2 '
1-- 1--.-
e 2 2 e 2
T .e. onpe.AeJl.HeTC.H no cpopMYJle KHHeTHqeCKOii 3HeprnH B HbIOTOHOBCKOii cpH3HKe
B pen.HTHBHCTCKOH cpH3HKe Macca cOBoKynHocTH qaCTHn He paBHa CYMMap-
HOH Macce OT.AenbHbIX B3aHMO,AeHCTBYIOII1;HX qaCTHn.
BonpOCbI AnH CAMOKOHTPOnH
1. KaK (jJOPiKYJl.UpyemcSl npuliijun omiiOCUmeJl.bliOCmU oUliwmeulia? B lleM
omJl.U1lue 3mozo npUliijuna om MexaliUlleCKOZO nplll-lijuna omliOeUmeJl.bHOCmU
KJzaecullecKOU (jJU3UKU?
2. KaK (jJOPMYJl.UpyemcJl npuliijun noemOSllicm6a CKopocmu C6ema 6 6aKY-
YMe? II OlleMY CKopocmb C6ema 6 6aKYYMe Sl6Jl.SlemC.R npeiJeJl.bliOU CKopocmblO
paenpocmpalieHU.R B3auMoiJeucm6uu?
3. llolle.MY meopu.R omliOCUmeJl.bliOemU He COZJl.acyemcSl C KJl.aCCUlleCKU.MU
npeOCma6Jl.eIiUJl.MU 0 npocmpaHcmBe U 6peMeliu?
4. II 0 KaKOU (jJopMYJl.e paCCllumbwaemcSl nOJl.lia.R 3liepZUSl meJl.a B CTO?
5. Mozym Jl.U oiJli06peMelili0 6blmb pa6liblMU IiYJl.1O nOJl.liaSl 3liepZUSl U UM
nYJl.bC llaCmUijbl .MaCCOU m?
6. B lleM omJl.Ullue peJl.SlmU6UCmCKOZO UMnYJl.bCa om U.MnYJl.bCa meJl.a 6
KJl.aCCUlleCKOU .MeXaHU Ke? B KaKOM CJl.Yllae pa3Jl.UIlUe MeJICOY YKa3aliHblMU UM-
nYJl.bCaMU liecyU{ecm6eliIi0?
7. 3anuUlume eoomlioweHue MeJICoy uMnYJl.bCOM U 3liepZUeu 6 meopuu om-
HOCUmeJl.bHOcmu.
8. If mo Ha3blBaemeSl 3liepzuei1 nOKoSl? B KaKO.M CJl.Yllae 3liepZUR nOKOSl paB-
Ha liYJl.IO? KaKou (jJaKm Sl6JlSlemCSl noiJm6epJICOeliUeM cyU{ecm606aHuSl 3liepZUU
no KOSl?
9. KaK paCCilumbl6aemcSl KUliemUlleCKa.R 3liepZUR peJl.SlmUBUCmcKou llaCmUijbl 2
550

g 2
l =  1--
c 2
't = g 't O V2
1--
c 2
VI +V
V2 =
1 VIV
+-
c 2
w 2 _ p 2 C 2 = m 2 c 4
mc 2
W= g 2
1--
c 2
W o = mc 2
mv
P= g 2
1--
c 2
WI( =W-W o
OCHOBHblE <l>OPMY Jlbl
- .n;JlHHa CTepmH.H, .n;BHJKyerOC.H OTHOCHTeJlbHO HHepH-
anbHoH CHCTeMbI OTCqeTa co CKOpOCTblO V, HanpaBJleHHOH
B,AOJIb npo.n;OJIbHOH OCH CTepJKH.H; lo - C06CTBeHHa.H .n;JIH-
Ha CTepmHH; C - CKOpOCTb CBeTa B BaKYYMe.
npOMemYTOK BpeMeHH MeJK.n;y C06bITHHMH, H3MepeHHbIH
B HenO.n;BHJKHOH CHCTeMe OTCqeTa, r.n;e 'to - C06CTBeHHOe
BpeM.H, H3MepeHHOe B CHCTeMe OTC'leTa, .n;BHmyeHC.H co
CKOpOCTblO V OTHOCHTeJlbHO HenO.n;BHmHOH CHCTeMbl OTCqeTa.
- peJl.HTHBHCTCKHH 3aKOH CJlOJKeHHH CKOpOCTeH, r.n;e V2
- CKOpOCTb TeJla OTHOCHTeJlbHO HenO.n;BHmHOH CHCTeMbl OT-
CqeTa, VI - CKOpOCTb TeJIa OTHOCHTeJlbHO nO.n;BHJKHOH CH-
CTeMbl OTCqeTa (CKOpOCTH VI H V2 COHanpaBJleHbl), V -
CKOpOCTb nO.nBHmHOH CHCTeMbl OTCqeTa OTHOCHTeJlbHO He-
nO)J;BHmHOH.
- COOTHOmeHHP Mem.n;y nOJIHOH 3HeprHefi W TeJla H HM-
nYJlbCOM p, r.n;e m - MaCCa TeJla, C - CKOpOCTb CBeTa B
BaKYYMe.
- nOJlHa.H 3HeprH.H TeJla, r.n;e m - MaCCa TeJla, .n;BHJKY-
erOC.H CO CKOpOCTblO V, 6J1H3KOH K CKOpOCTH TeJla.
- 3HeprHH nOKO.H Tena (C06CTBeHHa.H 3HeprHH).
peJl.HTHBHCTCKHH HMnYJlbC TeJla, r,Ae m - MaCCa TeJla,
.n;BHmyerocH CO CKOpOCTblO V, 6JIH3KOH K CKOpOCTH CBe-
Ta B BaKYYMe.
KHHeTHqeCKaH 3HeprH.H TeJla, .n;BHJKyerOC.H CO CKOpOC-
TblO V, 6nH3KOH K CKOpOCTH CBeTa C, W - nOnHa.H 3Hep-
rH.H TeJla, W o - ero 3HeprH.H nOKO.H.
METO,[(WIECKHE PEKOMEH,[(A[(Hli
fIpH pemeHHH 3a.n;aq no 3TOH TeMe B pH.n;e cJlyqaeB MomHO BOCnOJlb30BaTb-
C.H CJle,AYIOIl1HM aJlrOpHTMoM:
1. BbI6paTb J{Be CHCTeMbI OTCqeTa, O,AHY CB.H3aB C ,ABHmYHMCH npe.n;Me-
TOM, .n;pyrylO - C nOKOHIl1HMC.H Ha6JIIO,AaTeJIeM.
2. 3anHcaTb ypaBHeHH.H J{BHmeHH.H H OCHOBHbIe COOTHOmeHH.H CTO ,AJI.H
B3aHMOCB.H3H HMnYJlbca H 3HeprHH, JlHHeHHbIx pa3MepoB H HHTepBaJlOB BpeMe-
HH co CKOpOCTblO .n;BHmeHH.H.
551

3. J];onHcaTb HeOCTalOHe ypaBHeHH.H.
4. PeIIIHTb nOJlyqeHHbIe ypaBHeHIUI (cHcTeMY ypaBHeHHH), npOH3BeCTH BbI-
qHCJleHH.H H oeHHTb peaJlbHOCTb nonyqeHHoro pe3YJIhTaTa. IIPH He06xoHMO-
CTH cpaBHHTb pe3YJlbTaT C aHHbIMH, nOJlyqalOHMHC.H Ha OCHOBe RJlaCCHqeC-
RHX npeCTaBJleHHH.
nPIIMEPbI PElliEHII.H 3A,n;A q
1. jKecTKMH cTepmeHh ,AnMHOH 2,5 M nOKOHTC.H B CHCTeMe K' (pHC. 18.5).
CTepmeHb opHeHTHpOB8H no,A yrnoM <Po = 30° K OCM x'. Onpe,AenHTe ,AnMHY CTep-
mHSI H yron ero opHeHTall,HH B CHCTeMe OTC'leT8 K , ABHmyeHcSI OTHOCHTenbHO
CHCTeMbI K' co CKOpOCTblO V = 0,95 c.
y
y'
v
,Z:(aHO: 10 = 2,5 M. <Po = 30°, v = 0,95 c.
HaHTH: I, cpo
P e m e H H e. MOAynb npoeKUHH CTepmHSI Ha OCM
x' H y' B CHCTeMe K' paBeH lox' = 10 cos <Po;
K'
0'
x'
loy' = 10 sin CPo. MO,Aynb npOeKlI,HH BepTHK8nbHoH
cocTaBnSlIOeH CTepmHfl He H3MeHSleTCSI, T.e.
1,/ = loy' = 10 sin <Po' PenSlTHBHCTCKoe COKpaeHMe
,AnHHbI ropH30HTanbHoH cocTaBnSlIOeH CTep>KHSI B
CHCTeMe OTC'IeTa K Ix = t"x. l- : = t,, 1- : . cosCPo.
,ZJ;nHHa CTepmHSI, H3MepeHHaSl B CHCTeMe K.
K
o
x
Puc. 18.5
1= I + I =  sin 2 CPo + (1 <: )cos 2CPo =
V2 cos 2 <p v 2
(sin2<po+cos2<po)- 2 0 =10 1-"2cos2<p; l=1,4 M.
c C
B CHCTeMe K CTepmeHb 6YAeT opHeHTHpOBaH no,A yrnoM
t - Ly _ 10 sin <Po tg<po
gcp - Ix - t,, 1- v 2 cosCPo - 1- v 2 .
c 2 c 2
tgcp = 1,8237. CneAoBaTenhHo, <P = 61°.
2. Me30H, ,ABHmYHHcSI co CKOpOCTblO v = 0,99 c, nponeTen B CMCTeMe OT-
C'leTa. CBSl3aHHOH C 3eMneH, OT MeCTa csoero pomAeHHSI AO TOqKH pacnaA8 paCCTO-
SlHHe 4.7 KM. Onpe,AenHTe c06cTBeHHoe BpeMSI mH3HH Me30Ha. KaKoe paccToSlHHe
nponeTen 6bI Me30H B CO, cBSl3aHHoH C 3eMneH, ecnH 6bI penSlTHsHcTCKHH 3<1><I>eKT
OTHOCHTenbHO npoMemYTKa BpeMeHH He HMen MeCTa?
552

aHO: v=0,99 C, l=4,7.10 3 M.
H a H T H: to' t.
l
Pew e H He. BpeMSI ,ABHmeHHSI Me30Ha BCD. cBSl3aHHoH C 3eMneH. t = -.
v
t
C06cTBeHHoe BpeMSI mH3HH Me30Ha H8H,AeM H3 <l>opMynbI t = g o .
v 2
1--
c 2
OTKY -
I\a 10 = t 1- v 2 ; 10 = i 1- v 2 ; 10 = 2.2.10-6 c.
c 2 V c 2
EcnH He yqHTbIB8Tb penSlTHBHCTCKHe 3<1>4>eKTbI. T.e. C't{HT8Tb. 'lTO BpeMSI mH3-
HH nOKOSlIl'(erOCSI Me30H8 p8BHO BpeMeHH mH3HH ,ABHmYIl'(erocSI. TO Me30H npone-
Ten 6bI paCCTOSlHHe B CO 3eMnH
l' = vt o ; l' = 0,65 KM.
3. Ba HCTOqHHKa CBeTa ,ABHmYTC.H C penHTHBHCTCKHMH CKOpOCTSlMH vI H V2
H8BCTpe'lY ,Apyr APyry B HHepI1.HanbHOH CHCTeMe OTCqeT8 (MCO) K. OnpeAenHTe
CKOpOCTb c6nHmeHHSI HCTO'lHHKOB H CKOpOCTb ,ABHmeHHSI OAHoro HCTO'lHHKa OTHO-
CHTenbHO ,Apyroro.
8HO: VI' V2'
HaHTH: v, V12'
PeweHHe. MO,Aynb CKOpOCTH c6nHmeHHSI HCTO'lHHKOB B MCO K paBeH
v = vI + v 2 ' TaK K8K OHH ,ABHmYTC.H H8BCTpe'lY ,Apyr ,Apyry.
B MCO K 10 CBSl38HHOH CO BTOpblM HCTO't{HHKOM. BTOpOH HCTO't{HHK nOKOHTCH.
a nepBbIH ,ABHmeTCH no HanpaBneHHlO K HeMY co CKOpOCTblO V12' 3anHlIIeM penSl-
THBHCTCKHH 38KOH cnomeHHH CKopOCTeH
V I 2 - V 2
vI =
1- vI2 . v2
c 2
OTKYAa MOAynb CKOpOCTH nepBoH 'laCTHI1.bI OTHOCHTenbHO BTOpOH
_ VI + v2
VI2 - .
1 + VI v2
c 2
4. IIpH KaKoH CKOpOCTH KHHeTH'leCKBJI SHeprHH n1060H 'laCTHI1.bI p8BHa ee 3Hep-
rHH nOKoH?
M
,II 8 H 0: W K = W Q. c = 3 .0 . 10 8
C
HaHTH: v.
PemeHHe.
IIonHaSl
SHeprHSI
qaCTHI1.bI
w = W o + W K = 2W o ,
r,Ae
2
W = g mc v 2 ' SHeprHSI nOKOSl
1--
c 2
'l8CTHI1.bI
W o = mc 2 .
553

mc 2
Cne,AOBaTenbHO, g = 2mc 2 .
v 2
1--
c 2
OTKYI\B 1 - :  . CJIel\OBBTeJIbHO, v  c-;s ; v  2,6 10 8 .
5. 3neKTpoH pa3rOHHeTCSI B sneKTpH'leCKOM none C HanpSlmeHHOCTblO
B
3,0. 10 6 -. H8M,AHTe CKOpOCTb sneKTpoH8 'lepe3 1,0 HC. K8KOB8 6bIJIa 61.1 ero
M
CKOpOCTb, ecnH He Y'lHTbIB8Tb penSlTHBHCTCKHX 3<p<peKTOB?
B
)J;8H 0: E = 3,0.10 6 t= 1,0. 10- 9 C.
M
H 8 M T H: v, v'.
P e III e H H e. Ha sneKTpOH B 3neKTpH'IeCKOM none ,AeMcTByeT CHna
F=qE. (1)
IIo BTOPOMY 38KOHY HblOTOH8
6.p = Ft, (2)
r,Ae H3MeHeHHe HMnynbc8 3neKTpOH8 6.p = P - Po = p, T8K K8K BH8'l8ne sneKTpOH
nOKOHncSI.
Cne,AOB8TenbHo.
p = Ft.
(3)
(4)
mv
I1Mny nbC penSlTHBHCTCKoro sneKTpOH8 p = g .
u 2
1--
c 2
Pelll8fl COBMeCTHO YP8BHeHHSI (1) - (4), nonY'IHM
g = qEt. OTCIOI\B m 2 v 2 = (qEt)2 ( 1 <: )-
c 2
O,Aynb HCKOMOM CKOpOCTH 3neKTpOH8
qEt qEt
v  m2+( qt J = m  ;
u = 2,6 . 10 8 M
C
EcnH He }"-IHTbIB8Tb penSlTHBHCTCKHX 3<p<peKTOB, TO CKOpOCTb sneKTpOH8
qEt 8 M
u = -; u = 5,3 . 10
m C
'lTO 3H8'IHTenbHO npeBOCXOAHT CKOpOTb CBeT8 H HeB03MOJKHO C TO'lKH 3peHHSI Teo-
pHH OTHOCHTenbHOCTH 3MHllITeMH8.
554

KBAHTOBASI
Cl>M3MKA
19. CBETOBbIE KBAHTbI. JI.EiICTBIiE CBET A
B03HHKHOBeHHe }"IeHHSI 0 KBaHTax. «I>oToa(}>(}>eKT Hero 3aKOHLI. YpaBHe-
HHe 3HHIlITeHHa JJ.JllI (}>OToa(}>(}>eKTa. «I>OTOH, ero aHeprHSI H HMnYJlLC. npllMe-
HeHHe OT03(}>(}>eKTa B TeXIIHKe.
,l{aBJleHHe CBeTa. OnLITLJ n. H. JIe6e.neBa. XHMllqeCKOe .neHCTBHe cneTa II
ero npHMeHellHe. KopnYCKYJlSlPIIO-BOJlIIOBOH yaJlH3M.
19.1. B03HHKHoBeHHe yqeHHH 0 KBaUTax
TeOpeTH'leCKHe HCCJle,nOBaHHSI ,l];m. MaKCBeJIJIa nOKa3anH, qTO CBeT eCTb
3JIeKTpOMarHHTHbIe BOJlHbI onpe,neJIeHHOrO ,nHana30Ha. TeOpH.H MaKCBeJIJIa no-
nHJla 3KcnepHMeHTaJlbHOe no,nTBepm,neHHe B onbITax r. repl(a. 113 TeopHH
MaKCBeJIJIa CJlenOBaJlO. qTO CBeT, na,ZIaSI Ha JII060e TeJlO, OKa3bIBaeT Ha Hero
AaBneHHe. 3TO ,ZIaBJIeHHe Y.lIaJIOCb 06HapymHTb n. H. JIe6e.lIeBY. OnbITbI JIe6e-
AeBa nO.lITBep,nHJlH 3..'IeKTpoMarHliTHYIO TeopHIO CBeTa. COrJIaCHO p86o'l'aM MaK-
cBenna, nOKa3aTeJlb npeJlOMJIeHH.H BellteCTBa onpe,neJl.HeTC.H cpopMynoH n =,
T.e. CB.H3aH C aJleKTpH'leCKHMH H MarHHTHbIMH CBoHcTBaMH 3Toro BellteCTBa (E
H J.l - COOTBeTCTBeHHO OTHOCHTeJlbHbIe .lIH3JIeKTpHqeCKa.H H MarHHTHaSI npo-
HHl(aeMOcTH BeeCTBa). Ho TaKoe SIBJIeHHe, KaK .lIHCnepCH.H (3aBHCHMoCTb no-
Ka3aTeJl.H npeJlOMJleHH.H OT ,nJIHHbI cBeToBoH BOJlHbI), TeOpH.H MaKcBeJlJla 06'bSIC-
HHTb He CMOrJla. 3TO 6bIJlO C.lIeJlaHO X. JIopeHl(eM, C03.lIaBIlIHM 3J1eKTpoHHYIO
TeopHIO B3aHMO.lIeHCTBH.H CBeTa C BellteCTBoM. JIopeHl( npe,nnOJIOmHJI, qTO 3JIeK-
TpOHbI no,n .lIeHCTBHeM 3JIeKTpH'leCKOrO nOJI.H 3J1eKTpOMarHHTHo BOJIHbI cOBep-
IIl810T BbIHym,neHHbIe Kone6aHH.H C qaCTOTOH v, KOTopaSI paBHa 'laCTOTe 3neK-
TpOMarHHTHOH BOJlHbl, a ,AH3neKTpHQeCKa.H npOHHl(aeMOCTb BellteCTBa 3aBHCHT
OT QaCTOTbI H3MeHeHHH 3J1eKTpOMarHHTHoro nOJlSI, CJle.lIOBaTeJlbHO, H n = f{v).
O.lIHaKO npH H3eHHH cneKTpa HcnYCKaHHSI a6COJllOTHO qepHOrO TeJIa. T.e.
TeJIa, KOTopoe norJlOlltaeT Bce na.zxalOlltHe Ha Hero H3JIyqeHHSI JlI060H QaCTOTbl,
H3HKa He CMOrJla B paMKax 3J1eKTpOMarHHTHOH TeopHH 06'bSICHHTb pacnpe,ne-
neHHe aHeprHH no ,nJlHHaM BOJIH. Pacxom,neHHe Mem.lIY TeOpeTHQeCKO (nYHK-
THpHaSI) H 3KcnepHMeHTaJIbHOH (cnJlOlliHaSI) KpHBbIMH pacnpe.lIeJleHH.H nJlOTHOC-
TH MOlltHOCTH H3J1Y'leHHSI B cneKTpe a6COJllOTHO qepHOrO TeJla (pHC. 19.1),
T.e. pa3J1HQHe Mem.lIY TeopHeH H onbITOM, 6bIJlO TaK 3HaQHTeJlbHO, qTO ero
Ha3BaJlH "y JlbTpaqJHOJleTOBOH KaTacTpocpoH" 3J1eKTpOMarHHTHaSI TeOpH.H He
MOrJla TaKme 06'b.HCHHTb B03HHKHOBeHHe JIHHeHQaTbIX cneKTpOB ra30B H 3aKO-
HbI qJOToaqJcpeKTa.
555

OJ. BT/M 8
4.10 11
I
I
I
I
I
I
I
I
\
,
\
\
,
"
.....
HOBa.H TeOpl:UI CBeTa 6bIJla BLI-
BHHYTa M. IIJlaHKoM B 1900 r.
COrJlaCHO rHnOTe3e M. IInaHKa,
3J1eKTpOHLI aTOMOB H3J1}T'1aIOT CBeT
He HenpepLIBHO, a OT enbHbIMIi
nOplI.IiSIMIi - KBaHTaMH. 3HeprHSI
KBaHTa VV npOnOplI.HOHaJlbHa qaCTO-
Te KOJle6aHHH v:
10.10 11
8.10 11
6.10 11
2.10 11
VV = hv,
re h - K03cpcpHlI.HeHT nponoplI.HO-
HaJILHOCTH, Ha3bIBaeMbIH nOCTOSIHHOH
IIJlaHKa:
h = 6,62 . 10- 34 Am. c.
o 1
2 3 4 5 6 7 8 9 A, MKM
TaK KaK H3J1yqeHlie HcnYCKaeT-
CSI nOplI.HSIMH, TO 3HeprHSI aTOM8
HJlH MOJleKynLI (OClI.HnJl.HTOpa) MO,
meT npHHIiMaTb JlHllIb onpeeJleHHbIH HCKpeTHbIH p.H 3HaqeHHH, KpaTHbIX lI.e-
JlOMY QHcny 3J1eKTpOHHbIX nOPlI.Hii w, T.e. 6bITb paBHOH hv, 2hv, 3hv Ii T.p'.
He cyecTByeT KOJle6aHHH, 3HeprHSI KOTOpbIX HMeeT npOMemYTOqHOe 3HaqeHHe
Me>KY BYM.H nOCJleOBaTenLHLIMH lI.enLIMH qHCJl8MH, KpaTHLIMH hv. 3TO 03-
H8qaeT, qTO H8 8TOMHO-MOJleKYJlSlpHOM ypOBHe KOJle6aHH.H npOHCXOSIT He C
nlO6bIMH 3HaQeHH.HMH aMnnHTY,A. nonYCTHMble 3Ha'leHH.H aMnnHTY.n onpe.nenSl-
IOTC.H qaCTOTOH KOJle6aHHH.
lfCnOJlb3YSI 3TO npenOJlOmeHHe H CTaTHCTHQeCKHe MeTObI, M. IIJlaHK CY-
MeJl nonyqHTb cpopMyny pacnpeeneHHSI 3HeprHH B cneKTpe H3nY'leHIiSI, COOTBeT-
CTBYIOYIO 3KcnepHMeHTanLHLIM 8HHLIM (CM. pHC. 19.1).
KBaHToBbIe lIpeCTaBneHH.H 0 CBeTe, BBeeHHbIe B HaYKY IIJlaHKoM, pa3BHn
aJlee A.3HHIIITeHH. OH npHIIIeJl K BLIBOY, qTO CBeT He TOJILKO H3J1yqaeTC.H,
HO H pacnpOCTpaH.HeTCSI B npOCTpaHCTBe, H nornoaeTCSI BeeCTBOM B BHe
KBaHTOB.
KBaHTOB8SI TeOpH.H CBeT8 nOMorna 06'L.HCHHTL pSl .HBneHHH, Ha6nlOaeMLIX
npH B3aHMoeHCTBHH CBeTa C BeeCTBOM.
Puc. 19.1
19.2. cl»oToa«l>«I>eKT Hero 3aKOHId
<I>oT03cpcpeKT B03HHKaeT npH B3aHMO)J;eHCTBHH BeeCTBa C nornoaeMbIM
3neKTpOMarHHTHLIM H3nyqeHHeM.
Pa3nHqalOT BHeIIIHHH H BHYTpeHHHii cpOT03cpcpeKT.
BHeWHHM 4>oTo34>4>eKToM Ha3bIBaeTC.H HBJleHHe BbIpbIBaHH.H 3J1eKTpOHOB H3
BeeCTBa no eHCTBHeM naalOero Ha Hero CBeTa.
BHyrpeHHHM 4>oTo3«1>«I>eKToM Ha3bIBaeTC.H .HBJleHHe YBeJlHQeHH.H KOHlI.eHTp8-
lJ;HH HOCHTeneH 3ap.Ha B BeeCTBe, a cneOBaTenLHO, H YBeJlHqeHHSI 3neKTpo-
npOBO,AHOCTH BeeCTBa no.n; eHCTBHeM CBeTa. 1..JacTHbIM cJlyqaeM BHYTpeHHero
cpOT03cpcpeKTa .HBJlSIeTCSI BeHTHJlLHLIH 4>oTo3«1>«I>eKT - .HBneHHe B03HHKHOBeHHSI
no eHCTBHeM CBeTa 3J1eKTpoBHmyeH CHJlbI B KOHTaKTe BYX pa3J1HqHbIX
nOJlynpoBoHHKOB HJlH nonynpoBoHHKa H MeTaJlJla.
556

BHelllHHH cpOT03cpcpeKT 6bIJI OTKpbIT B
1887 r. r. repeMt a RCCJleOBaH e-
TaJlbHO B 1888-1890 rr. A. r. CTone-
TOBbIM. B onbITax C 3J1eKTpOMarHHTHbI-
MH' BOJIHaMH r. rep 3aMeTHJI t qTO npo-
CKaKHBaHHe HCKpbI Memy HHKOBbIMH
mapHKaMH pa3p.HHHKa npOHCXOHT npH
MeHbllleH pa3HOCTH nOTeHHanOB, eCJIH
O.lJ.HH R3 HHX OCBeTHTb ynbTpaqmOJIeTOBbI-
MH JlyqaMH. IIpH HCCJleOBaHHH 3TOro .HB-
JleHHSl CTOJleTOBbIM HCnOJlb30B8JIC.H nJIOC-
KHH KoeHcaTOp, O)J;Ha H3 nnaCTHH KOTO-
poro (HHKoBaSl) 6bIna CnJIOlllHOH, a BTO-
paSl - BbIllOJIHeHa B BH.Ae MeT8JIJl1{qeCKOH
ceTKH (pHC. 19.2). CnJIOlllHa.H nnaCTHHa
COe.lJ.HH.HJlaCb C OTpHaTeJlbHbIM nOJlIOCOM
HCTOqHHKa TOKa, a CeTqaTaSl - C nOJlo-
mHTeJlbHbIM. BHyrpeHHSl.H nOBepXHOCTb OT-
pHaTeJlbHO 3ap.HmeHHOH nnaCTHHbI KOH.lJ.eHCaTOpa OCBeI.q8JIaCb CBeTOM OT 3JIeKTpH-
qeCKOH yrHt B CneKTp8JIbHblii COCTaB KOTOpOH BXO.lJ.SlT YJlbTpaqmoJleToBble nyqH.
IIoKa KOH,AeHCaTOp He OCBeI.q8JIC.H t TOKa B enH He 6bInO. IIpH OCBeI.qeHHH HH-
KOBOH nJlaCTHHbI K YJlbTpacpHOJleTOBbIMH JlyqaMH r8JIbBaHOMeTp G cpHKCHpoB8JI Ha-
JlHqHe TOKa B enH. B TOM cnyqae t eCJIH KaTO.lJ.OM CTaHOBHJlaCb CeTKa At TOKa B
enH He 6bInO. CJleOBaTenbHOt I.{HHKOBaSl nnaCTHHa no,D. .lJ.eHCTBHeM CBeTa HCnyC-
K8JIa OTpHaTeJIbHO 3ap.HmeHHbIe qaCTHbI. K MOMeHTY 06HapymeHHSI cpOT03cpcpeK-
Ta erqe He 6bIJlO HHqero H3BecTHO 06 3J1eKTpOHaX t OTKpbITbIX m. TOMCOHOM TOJlb-
KO 10 JleT cnYCT.H, B 1897 r. IIoCJIe OTKpbITHSI 3JIeKTpOHa <1>. JIeHap,AOM 6bIno
OKa3aHO. qTO BbIneTaIOIUHMH no )J;eHcTBHeM CBeTa OTpHuaTenbHO 3apSImeHHbIMH
qaCTHaMH SlBJI.HIOTCSl 3JIeKTpOHbl t Ha3BaHHble 4>0T03J1eKTpOHaMH.
CTOJleTOB npOBOHJI onbITbI C KaToa-
MH H3 pa3HbIX MeT8JIJlOB Ha YCTaHOBKe,
cxeMa KOTOpoH nOKa3aHa Ha pHCYHKe 19.3.
B CTeKJISlHHbIH 6aJlJlOH t H3 KOToporo
BbIKa'laH B03,AYX t BnaHB8JIHCb .lJ.Ba 3J1eK-
Tpoa. BHYTPb 68JIJIOHa 'lepe3 KBapl.{eBOe
"OKOlllKO u t np03paqHOe JI.H YJlbTpacpHoJle-
TOBoro H3JIeHHSlt nOn8,AaeT CBeT Ha Ka-
TO K. IIoaBaeMoe Ha 3JIeKTpObI Ha-
np.HmeHHe MomHO H3MeH.HTb C nOMorqblO
nOTeHHOMeTpa H H3MepSlTb BOJlbTMeTpOM
V. IIOlI lIeHcTBHeM CBeTa KaTO,A Hcnyc-
KaJl 3J1eKTpOHbI, KOTopbIe aaMbIKaJIH
enb Memy 3J1eKTpoaMHt H aMnepMeTp
cpHKcHpOBaJl HaJlHqHe TOKa B enH. 1I3-
MepHB TOI< H Hanp.HmeHHe, MomHO nOCT-
pOHTh rpacpHK 3aBHCHMOCTH CHnbJ cpOTO-
TOKa OT Hanp.HmeHH.H Memy 3neKTpoa-
K
s
G
+ q II
v
Puc. 19.2
-
+
Puc. 19.3
557

I
MH 1= I(U) (pHC. 19.4). 113 rpacpHKa cne-
,D;yeT, qTO:
1. I1pH OTCYTCTBHH Hanp.HmeHH.H Mem-
lty 3J1eKTpO,D;aMH cpOTOTOK OT JlHqeH OT
HYJlSI, qTO MomHO 06'b.HCHHTb HanH'lHeM y
cpoT03neKTpoHoB npH BMneTe KHHeTeCKOH
3HeprHH.
2. I1PH HeKOTopOM 3HaqeHHH HanpSlme-
HHSI Mem,D;Y 3J1eKTpO)J;aMH U II CHJla cpOTOTO-
U Ka nepeCTaeT 3aBHceTb OT Hanp.HmeHHSI,
T.e. ,D;OCTHraeT HaCbIeHH.H 1M'
CHna CPOTOTOKB HaCbIeHH.H III = qmax ,
t
r,Ae qmax - MaKCHMaJlbHbIH 3ap.H,D;, nepeHocHMbIH cpoT03neKTpoHaMH. OH paBeH
qmax = net, r.ne n - qHCJlO cpoT03J1eKTpOHoB, BbIJleTalOHX C nOBepXHOCTH oc-
BeaeMoro MeTaJlJla 3a 1 c, e - 3ap.H,A 3J1eKTpOHa. Cne,D;OB8TeJlbHO, npH <}>o-
TOTOKe HaCbIeHH.R Bce 3neKTpOHbI, nOKHHYBlliHe 3a 1 C nOBepXHOCTb MeTaJl-
na, 38 3TO me BpeM.H nOna,D;alOT Ha aHO,D;. I103TOMY no CHJle cpOTOTOKa HaCM-
eHHSI MomHO CY,D;HTb 0 qHCne cpOT03J1eKTpOHOB, BbIJleTalOHX C KaTO,D;a B
e,D;HHHY BpeMeHH.
3. ECJlH KaTO,D; COe.AHHHTb C nOJlOmHTenbHbIM nonlOCOM HCTOqHHKa TOKa,
a aHO,D; - C OTpHaTeJlbHbIM, TO B 3J1eKTpOCTaTHQeCKOM none Mem,D;Y 3J1eKT-
pO,AaMH cpOT03J1eKTpOHM 6Y,D;YT TOpM03HTbCSI. a CHn8 cpOTOTOK8 YMeHblliaTbCH
npH YBeJlH'leHHH 3HaQeHHSI 3Toro OTpHuaTeJlbHOrO Hanp.HmeHHSI. llpH HeKOTO-
pOM 3HaqeHHH OTpHaTeJlbHOrO Hanp.HmeHH.H U 3 (ero Ha3bIBalOT 3a,D;epmHBalO-
HM HanpSlmeHHeM) cpOTOTOK npeKpaaeTCSI.
COrJlaCHO TeopeMe 0 KHHeTHqeCKOH 3HeprHH, pa60Ta 3epmHBalOero 3J1eK-
TpHqeCKOrO nOJl.H paBHa H3MeHeHHIO KHHeTHqeCKOR 3HeprHH cpOT03JIeKTpoHOB:
U 3
o
Us
Puc. 19.4
mv 2 mv 2
.d = - eU ' AW = - CJle"OBaTeJlbHO eU = 
 3' Ll K 2''''' · 3 2'
3TO BbIpameHHe nOJlyqeHO npH YCJlOBIUi, qTO CKOpOCTb v« c, r,D;e C -
CKOpOCTb CBeTa.
CJIe,D;OBaTeJlbHO, 3Ha.H U 3 , MomHO HaRTH MaKCHMaJIbHYIO KHHeTHqeCKYIO 3Hep-
rHIO cpOT03J1eKTpOHOB.
Ha pHCYHKe 19.5. a npHBe.ll.eHbI rpacpHKH 3aBHCHMOCTH 1tt>(U) lIJlSI pa3J1Hq-
HMX CBeTOBbIX nOTOKOB, na,D;alOnx Ha cpoToKaTO,D; npH nOCTO.HHHOR qaCTOTe
CBeTa. Ha pHCYHKe 19.5, 6 npHBe)J;eHbI rpacpHKH 3aBHCHMOCTH 14>( U) )J;JI.H no-
CTOSlHHoro CBeTOBoro nOTOKa H pa3J1HqHbIX QaCTOT na,D;alOero Ha KaTO,D; CBeTa
AHaJIH3 rpacpHKoB Ha pHCYHKe 19.5, a nOKa3MBaeT, qTO CHJIa cpoToToKa
HaCbIeHH.H YBeJlHqHBaeTC.H C YBeJlHqeHHeM HHTeHCHBHOCTH na,D;alOero CBeTa.
ECJlH no 3THM ,D;aHHMM nOCTpOHTb rpacpHK 3aBHCHMOCTH CHJIbI TOKa HaCMe-
HH.H OT HHTeHCHBHOCTH CBeT8, TO nOJlyqHM np.HMYIO, KOTOpa.H npOXO,D;HT Qepe3
Ha'IaJlO KOOp,D;HHaT (pHC. 19.5, 6). CJIe,D;OBaTeJIbHO. CHna cpOTOHa HaCbIeHHSI
nponopUHOHaJlbHa HHTeHCHBHOCTH CBeTa. na,D;alOIU,ero Ha KaTO.n: Itt>- I.
KaK CJIe.AyeT H3 rpacpHKoB Ha pHCYHKe 19.5, 6, BeJIH'IHHa 3a.nepmHBalOe-
ro Hanp.HmeHH.H YBeJlHqHBaeTC.H C YBeJIHQeHHeM qaCTOTbI nMalOero CBeTa. I1pH
558

a
v = const
<1>. > <1>2 > <1>1
<1>.
<1>2
<1>1
Us
o
U
8
I.H
o
6
I.
I =const
v.>V 2 >V 1 >V O
U
2
U a
I
o
v
Puc. 19.5
YMeHblIIeHIUi qaCTOThI na,A81Oll1erO CBeTa U a YMeHbIII8eTC.H. H npH HeKoTopoH
Q8CTOTe Vo 38.n;epmHB81Oee H8np.HmeHHe U aO = O. IIpH V < Vo <pOT03<pcpeKT He
H86nlO,A8eTC.H. MHHHM8J1hH8SJ Q8CTOTa Vo (M8KCHManhH8S1 .n;nHHa BonHhI Ao) n8-
,D;alOero CBeTa, npH KOTOpOH ee B03MomeH <pOT03<p<peKT, Ha3bIBaeTC.H Kpac-
Hon rpaHHu;eH 4JoTo34J4JeKTa. Ha OCHOBaHHH ,D;8HHbIX rpa<pHKa 19.5, 6 MO)K-
HO nOCTpOHTh rpa<pHK 38BHCHMOCTH Ua(v) (pHC. 19.5, z).
H8 OCHOB8HHH 3TH X 3KcnepHMeHTaJIhHbiX ,A8HHbIX 6bIJlH c<popMynHpoB8HhI
38KOHbI <pOT03<p<peKTa.
3aKOHLI 4JoT034J4JeKTa
1. 4HCJlO <poT03J1eKTpOHoB, BbIphIBaeMbIX 38 1 C C nOBepXHOCTH K8TO,D;8,
nponopn;HOHaJlhHO HHTeHCHBHOCTH CBeTa, na,D;alOero H8 3TO BeeCTBO.
2. KHHeTHqeCKa.H 3HeprHSI <pOT03J1eKTpOHOB He 38BHCHT OT HHTeHCHBHOCTH
na,D;alOero CBeTa, a 3aBHCHT JlHHeHHO OT ero q8CTOTbI
3. Kp8cHaSl rp8HHn;a <poT03<p<peKTa 3aBHCHT TonhKO OT pO,D;8 BeeCTBa Ka-
To.n;a.
4. <1>OT03<p<peKT np8KTHqeCKH 6e3HHepn;HoHeH, TaK K8K C MOMeHT8 06nyqe-
HH.H MeT8J1J18 CBeTOM /1.0 BbIJleTa 3J1eKTpOHOB npOXO,D;HT BpeMSI ::;::; 10- 9 C.
559

19.3. YpaBHeHHe 3HHlliTeHHa AJUI «I>oToa«l>«I>eKT3
lIonbITaeMC.H 06'b.HCHHTb 3KcnepHMeHTMbHble 3aKOHbl t}>oT03t}>t}>eKTa, HCnOJlb-
3Y.H 3J1eKTpOMarHHTHYIO TeopHIO MaKCBeJlJla. 3J1eKTpOMarHHTHa.H BOJlHa 3aCTaB-
JlSleT 3J1eKTpOHbI COBepmaTb 3J1eKTpOMarHHTHbIe Kone6aHH.H. lIpH nOCTO.HHHOH
aMnJlHTy,ne BeKTopa Hanp.HmeHHOCTH 3J1eKTpHqeCKOrO nOJl.H KOJlHqeCTBO 3Hep-
rHH, nOJlyqeHHoH B 3TOM npou,ecce 3J1eKTpOHOM, nponopu,HOHaJlbHO qaCTOTe Bon-
HbI H BpeMeHH "paCKa'lHBaHH.H". B 3TOM cJlyqae 3HeprHIO, paBHYIO pa60Te BbI-
xO,Aa. 3J1eKTpOH ,nOJlmeH nOJlyqHTb npH JlI060H qaCTOTe BOJlHbl. HO 3TO npOTH-
BOpeqHT TpeTbeMY 3KcnepHMeHTanbHOMY 3aKOHY t}>OT03t}>t}>eKTa. lIpH YBeJlHqe-
HHH qaCTOTbl 3J1eKTpOMarHHTHOH BOJlHbI 60Jlbme 3HeprHH 3a e,nHHHu,y BpeMeHH
nepe.naeTC.H 3J1eKTpOHaM, H t}>oT03neKTpOHbl .nOJlmHbl BbIJleTaTb B 60JlbWeM KO-
JlHqeCTBe, a 3TO npoTHBOpe'lHT nepBoMY 3KcnepHMeHTaJlbHoMY 3aKoHY. TaKHM
06pa30M, 3TH t}>aKTbI 06'b.HCHHTb B paMKax 3J1eKTpOMarHHTHOH TeopHH MaKC-
BeJlJl8 6bIJlO HeB03MomHO.
B 1905 r. IlJl.H 06'b.HCHeHH.H HBJleHH.H t}>oT03c}>t}>eKTa A. 3HHwTeHH HCnOJlb-
30BaJl KBaHTOBble npe,nCTaBJleHH.H 0 CBeTe, BBe.neHHble B 1900 r. lInaHKoM, H
npHMeHHJI HX K nOrJlOeHHIO CBeT8 BeeCTBOM. MOHoxpOMaTHqeCKOe CBeTOBoe
H3nyqeHHe, na.nalOee Ha MeTaJln, COCTOHT H3 CPOTOHOB. <1>OTOH - 3TO 3J1e-
MeHTapHa.H qaCTHa, 06J1a,AalOa.H 3HeprHeH W 0 = hv. 3J1eKTpOHbI nOBepXHOCT-
Horo CJlOS! MeTaJlJla nOrJlorqalOT 3HeprHIO 3THX cpOTOHOB, npH 3TOM O,nHH 3J1eK-
TpOH norJlOaeT u,eJlHKOM 3HeprHIO OllHoro IIJ1H HeCKOJlbKHX t}>OTOHOB.
EcnH 3HeprH.H t}>oToHa W 0 paBHa HJlH npeBbIwaeT pa60TY BbIXO,Aa, TO 3J1eK-
TpOH BbIJleTaeT H3 MeTaJlJla. lIpH 3TOM qaCTb 3HeprHH cpOTOHa TpaTHTC.H H8
COBepweHHe pa60TbI BbIxo,na As, a OCTaJlbHa.H qaCTb nepeXO,AHT B KHHeTH'leC-
KYIO 3HeprHIO t}>oT03J1eKTpoHa:
mu 2
-w; =A +
o B 2'
mu 2
hv = Au + ;ax - ypaBHeHHe 3HHWTeHHa AJI.H «}>oTo3«}>«}>eKTa.
OHO npe,nCTaBJlS!eT C060H 3aKOH coxpaHeHH.H 3HeprHH B npHMeHeHHH K t}>o-
T03t}>t}>eKTY. 3TO ypaBHeHHe 3anHcaHO ,AJI.H O,AHOt}>oTOHHoro <l>oT03t1><l>eKTa, KOr,Aa
peqb H,AeT 0 BbIpbIBaHHH 3J1eKTpOHa, He CB.H3aHHOro C aTOMOM (MoneKYJlOH).
Ha OCHOBe KBRHTOBbIX npe,ACTaBJleHHH 0 CBeTe MomHO 06'b.HCHHTb 3aKOHbI
t}>OT03<l>t}>eKTa.
W
ll3BeCTHO, qTO HHTeHCHBHOCTb CBeTa 1 = St ' r,Ae W - 3HeprH.H na,AalOe-
ro CBeTa, S - nJlOa.nb nOBepXHOCTH, H8 KOTOPYIO na)J;aeT CBeT, t - Bpe-
M.H. COrJlaCHO KBaHToBoH TeopHH, 3Ta 3HeprH.H nepeHOCHTC.H <l>oToHaMH. Cne-
.nOBaTeJlbHO, W = N ItJhv, r.ne N  - qHCJlO cpOTOHOB, na,AalOHX Ha BeeCTBO.
OqeBH,AHO. liTO qHCJlO 3J1eKTpOHOB N e' BbIpBaHHbIX H3 BerqecTBa. npOnOplT,HO-
HanbHO qHCJlY <l>OTOHOB, na,AalOrqHx Ha BerqecTBo, T.e. N e - N ItJ , a CJle,AOBa-
TeJlbHO, N e - I. TaKHM 06pa30M, MbI 06'b.HCHHJlH nepBbIH 3aKOH cpoT03t}>t}>eKTa.
560

H3 ypaBHeHH.H 9HHillTeHHa CJle,nyeT, qTO
mv 2
;ax = hv - H AB=hvQ.
OTCIO,Aa BH,nHO, qTO MaKCHMaJIbHa.H KHHeTHqeCKaJl 3HeprHJI cpOT03J1eKTpOHOB
RHHeHHO 3aBHCHT OT qaCTOTbI na,nalOero CBeTa, a KpaCHa.H rpaHHu.a cpOT03cp-
cpeKT8 - OT pO,Aa BeeCTB8 KaTo,n8 (BTOpOH H TpeTHH 38KOHbI cpOT03cpcpeKT8).
19.4. «!»OTOH, ero 3HeprHH H HMnYJILC
ITOpHJI CBeTOBoro H3J1yqeHH.H -- KBaHT CBeTa -- 06J18,n8eT KopnYCKYJI.Hp-
HbIMH CBOHCTBaMH H MomeT paCCMaTpHBaTbC.H KaK 3J1eMeHTapH8.H qaCTHa, Ha-
3bIB8eM8.H c}»OTOHOM. cI>OTOHbI .HBJlJlIOTC.H HOCHTeJl.HMH CBOHCTB 3ReKTpOM8rHHT
Horo nOJl.H. "lleM BbIille qaCTOTa H3J1yqeHH.H, TeM CHJlbHee npO.HBJI.HIOTC.H KOp-
nYCKYJI.HpHbIe (KBaHToBbIe) CBOHCTBa CBeT8.
CBeTOBbIe qaCTHbI -- cpOTOHbI -- 06J1a,naIOT 3HeprHeH
he
W o =hv=-:;:,
r,ne h -- nOCTO.HHHa.H ITJlaHKa, v -- qaCTOTa cBeToBoH BORHbI, A -- ee ,ARH-
Ha, C -- CKOpOCTb CBeTa.
ct>OTOH Bcer,na )J;BHmeTC.H co CKOpOCTblO CBeTa, H HeT HHKaKOH CHCTeMbI
OTCqeTa, B KOTOPOH 6bI OH nOKOHJlC.H. 3HaqHT, ero Macca m = 0 H COOTBeT-
CTBeHHO (CM.  18.4) 3HeprH.H cpoToHa Hero HMnYJlbC CB.H3aHbI COOTHOIIIeHH-
eM W_p2C2= O. OTKy,na HMnYJlbC cpOTOHa
_Wo_hv_h
P---I'
To, qTO cpOTOH o6J1a,lIaeT HMnYJlbCOM. 3KcnepHMeHTaJlbHO nO,ATBepm,naeTC.H
OTKpbITHeM CBeTOBoro ,naBJleHH.H (CM. HHme  19.6).
ECRH CBeT .HBJlJleTCJI MOHoxpOMaTHqeCKHM, TO Bce cpOTOHbI HMelOT 0,lIHH8-
KOBbIe 3HeprHIO H HMnYJlbC.
ct>OTOHbI B03HHKalOT (H3J1yqaIOTC.H) npH nepexo,nax aTOMOB, MOJleKYJI, HOHOB
11 aTOMHbIX .H,nep 113 B036ym,neHHbIX 3HepreTHqeCKHX COCTO.HHHH B COCTO.HHHJI C
MeHbilleH 3HeprHeH. cI>OTOHbI H3J1yqaIOTC.H TaKme npH YCKopeHHH H TopMome-
HHH 3apJlmeHHbIX qaCTHu., npH pacna,nax HeKOTopbIX q8CTH H YHHqTOmeHHH
(npH aHHHrHJI.HHH) napbI 3J1eKTpOH -- n03HTpOH.
ITpou.ecc nOrJlOeHH.H CBeTa BeeCTBOM CBO,nHTC.H K TO MY , qTO cpOTOHbI u.e-
RHKOM nepe,nalOT CBOIO 3HeprHIO qaCTHaM BeeCTBa. IIpoecc nOrJlOeHH.H CBe-
Ta paCCMaTpHBaeTCJI B KBaHTOBOH cpH3HKe KaK ,lIHCKpeTHbIH 11 BO BpeMeHH, H
B npOCTpaHCTBe.
19.5. I1pHMeHeHHe €}>oTo34>4>eKTa B TeXHHKe
ITpH60PbI, npHHu.Hn ,neHcTBH.H KOTOpbIX OCHOBaH Ha .HBJleHHH cpoT03cpcpeK-
Ta, Ha3bIBalOT c}»oT03J1eMeHT8MH. cI>oT03J1eMeHTbI, ,neHcTBHe KOTOpbIX OCHOBaHO
Ha BHeillHeM cpoT03cpcpeKTe, HMelOT cJle)J;YIOee YCTPOHCTBO (pHC. 19.6). BHYT-
peHH.H.H nOBepXHOCTb CTeKJI.HHHOrO 6aJlJlOHa, H3 KOToporo BbIKaqaH B03,nyx, no-
561

KpbITa CBeTOqYBCTBHTeJlbHbIM CJlOeM K C He60nbllIHM
np03paqHbIM JIJI CBeTa yqaCTKOM - "OKHOM" 0 n.H ,AO-
cTyna CBeTa BHYTPb 6aJlJlOHa. B eHTpe 6anJlOHa HaXO-
HTC.H MeTaJlJlHqeCKOe KOJlbO A. OT 3J1eKTpOOB K H A
ClIeJlaHbI BbIBObI lIJI.H nOKJllOqeHH.H cpoT03J1eMeHTa K
3J1eKTpHqeCKOH enR. B KaqeCTBe CBeTOqYBCTBHTeJlbHoro
CJlO.H 06bIqHO HcnOJlb3YlOT HanbIJleHHbIe nOKpbITH.H H3
eJlOqHbIX MeTanJlOB, HMelOHX MaJlYIO pa60TY BbIxoa.
T.e. tIYBCTBHTeJlbHbIX K BH,AHMOMY CBeTY (H3rOTOBJI.HIOT
H cpoT03J1eMeHTbI, qYBCTBHTeJlbHbIe TOJlbKO K YJlbTpacpHo-
JleTOBbIM JlyqaM).
<t>oT03J1eMeHTbI. lIeHcTBHe KOTOpbIX OCHOBaHO Ha
BHelliHeM cpoT03cpcpeKTe, npeo6pa3YIOT B 3J1eKTpHqeCKYIO 3HeprHIO JlHllIb He3Ha-
qHTeJlbHYIO qaCTb 3HeprHH H3J1yqeHH.H. II03ToMY B KaqeCTBe HCTOqHHKOB 3J1eK-
Tp03HeprHH H:X He Hcnonb3YIOT, 3aTO IlIHpOKO npHMeH.HIOT B pa3J1HqHbIX cxe-
Max aBTOMaTHKH JI.H ynpaBJleHHSI 3J1eKTpHqeCKHMH en.HMH c nOMOblO CBe-
TOBbIX nyqKOB.
B KaqeCTBe npHMepa paCCMOTpHM npHHIJ.Hn eHCTBH.H cpOT03J1eKTpHtIeCKOrO
peJle, cpa6aTbIBalOero npH npepbIBaHHH CBeTOBoro nOTOKa, n8lIalOero Ha cpO-
T03J1eMeHT (pHC. 19.7, a). <t>oTopene COCTOHT H3 cpoT03J1eMeHTa tP, YCHJlHTenH
cpoToToKa, B KaqeCTBe KOToporo HCnOJlb3YIOT nOJlynpoBoHHKOBbIH TpHO,lJ; (TpaR-
3HCTOp) T, H 3J1eKTpOMarHHTHoro peJle, BKJllOqeHHOrO B enb KOJlJleKTOpa TpaR-
3HCTopa. Hanp.HmeHHe Ha cpoT03J1eMeHT noalOT OT HCTOqHHKa TOKa E l' a Ha
TpaH3HCTOp - OT HCTOqHHKa TOKa E 2 . Memy 6a30H H 3MHTTepOM TpaH3HC-
Topa BKJllOqeH HarpY30qHbIH pe3HCTOp R.
Puc. 19.6
a
6
3 1 3a
3I1eKTpOMarHTHoe
pene
4J
K
3 2
A
Puc. 19.7
Kora cpoT03J1eMeHT OCBeI.QeH, B ero enH, cO,lJ;epmaI.QeH pe3HcTop R, H,lJ;eT
cna6bIH TOK, nOTeHHaJl 6a3bI TpaH3HcTopa BbIllie nOTeHHaJla 3MHTTepa, H
TOK B KOJlJleKTOpHoH enH TpaH3HcTopa OTCYTcTByeT.
ECJlH me nOTOK CBeTa, naalOHH Ha «poT03J1eMeHT, npepbIBaeTC.H, TOK B
ero enH cpa3Y npeKpaaeTC.H, nepexo 3MHTTep - 6a3a oTKpbIBaeTC.H nH
OCHOBHbIX HOCHTeJleH, :Ii qepe3 06MOTKY peJle, BKJllOqeHHOrO B enb KOJlJleKTO-
pat nOHeT TOK. PeJle cpa6aTbIBaeT. Hero KOHTaKTbJ 3aMbIKalOT HCnOJlHH-
TeJlbHYIO IJ.enb. Ee «PYHKIJ.H.HMH MorYT 6bITb OCTaHOBKa npecca, B 30HY eH-
562

CTBH.H KOTOpOrO nonaJla PYKa qenOBeKa, BbI.n;BHraHHe nperpa,AbI B TypHHKeTe
MeTpO, aBTOMaTHqeCKOe BKJllOqeHHe OCBeeHHH Ha YJlHlJ,aX. <1>oTosneMeHTbI npH-
MeHHIOTC.H B BOeHHOM ,nene B CaMOHaBO,nHIl1HXC.H CHap.H,naX. ,nnSl CHrHanH3aUHH
H JlOKaUHH HeBHHMbIMH JlyqaMH (HHcppaKpaCHbIMH).
C nOMOlI.\blO <pOTOSJleMeHTOB OCYlQeCTBnSleTC.H BOCnpOH3Be,AeHHe 3BYKa, 3a-
nHCaHHOrO Ha KHHOnJIeHKe, a TaKme nepe.n;a'la ,nBH>KYlI.\HXC.H H306pameHHH (Te-
JIeBH,neHHe) .
KOM6HHaHH .HBJleHHH <pOToscpcpeKTa co BTOPHqHOH SJIeKTpOHHOH SMHCCHeH
npHMeH.HeTC.H B cpOTosneKTpOHHbIX YMHO>KHTeJl.HX (<I>8Y) (pHC. 19.7, 6), npe,n-
CTaBJI.HIOHX C060H BaKYYMHYIO Tpy6KY C cpoToKaTo.n;oM K H aHo.n;OM A, Mem-
,ny KOTOpbIMH pacnOJlOmeHO HeCKonbKO SJleKTpo.n;OB-SMHTTepOB. 8JIeKTpOHbl, BbIp-
BaHHbIe C cpoToKaTo.n;a no.n; .n;eHcTBHeM CBeTa, nona.n;alOT Ha SMHTTep 01' npoH-
,ASI YCKOpSIlOlQYIO pa3HOCTb nOTeHIJ,HaJlOB Mem.n;y K H 01' 113 SMHTTepa 3 1 BbI-
6HBaIOTC.H sneKTpoHbI. Y CHJleHHbIH sneKTpOHHbIH nOTOK HanpaBJISIeTCSI Ha SMHT-
Tep 3 2 H npolJ,ecc YMHO>KeHHSI nOBTopSIeTcSI Ha Bcex nocne,nYIOlI.\HX sMIITTe-
pax. Y CHJIeHHe 9- KaCKa.n;Horo <I>8Y .n;OCTHraeT 10 6 , T. e. Ha BbIxo.a;e H3 cpOTO-
YMHomHTeJISI CHJl8 TOKa B MHJlnHOH pa3 npeBocxo,nHT nepBHqHbIH cpOTOTOK.
H8 SlBJIeHHH BHYTpeHHero cpoToscpcpeKTa OCHOBaHa pa60Ta cpoToconpOTHBJIe-
HHU. npOcTeHlIIee cpoToconpoTHBneHHe (pHC. 19.8) - STO nnaCTHHKa H3 ,nHS-
neKTpHKa, nOKpbITa.H TOHKHM cnoeM nOJlynpOBo.n;HHKa, Ha nOBepXHOCTH KOTO-
poro YKpenneHbI TOKOnpOBO,lJ;.HlI.\He sneKTpo.n;bI. TIpH OCBelI.\eHHH nnaCTHHKH B03-
HHKaeT cpoTonpoBo.n;HMOCTb, H B enH. r,Ae BKnlOqeHbI cpoToconpoTHBneHHSI. li,neT
TOK. <I>oToconpOTHBJleHH.H npHMeH.HIOTC.H B 3BYKOBOM KHHO, .n;n.H CHrHaJlH3aIJ,HH,
B TeneBH,neHHH, aBTOMaTHKe H TeJleMeXaHHKe.
<I>oTosneMeHTbI npHMeH.HIOTC.H ,nJl.H COpTHpOBKH MaCCOBbIX H3.n;enHH no pa3-
MepaM H OKpaCKe. TIyqOK CBeTa n8.n;aeT Ha cpoTosneMeHT, OTp83HBlIIHCb OT cop-
THpyeMbIx H3,neJlHH, KOTopbIe HenpepblBHO no.n;alOTCSI Ha KOHBeMep. OKpacKa H3-
,AeJIHSI HJlH ero pa3Mep onpe,Aen.HIOT cBeToBoH nOTOK, nona.n;alOll1HH Ha cpOTO-
SJleMeHT, H CHJlY cpoToToKa. B 3aBHCHMOCTH OT CHJlbI cpoToToKa aBTOMaTHqeC-
KH npOH3Bo.n;HTC.H COpTHpOBKa H3.n;enHH.
v.
F.,-Q,,0
:r:
Q)


Ha pHCYHKe 19.8 H306pa-
>KeHa cxeMa YCTPOHCTB8 4>OTO-
SJIeMeHTa C 3anHpalOIlJ,HM CJI0-
OKCH,lJ; MeTaJIJl8 eM (BeHTHJIbHblH cpOT03JIeMeHT).
(nOJlynpOBo.n;BHK) ,l(Be conpHK8CalOlI.\HeCSI .n;pyr C
.n;pyroM nnaCTHHKH, H3rOTOB-
neHHbIe H3 MeT8J1J1a Hero
OKCH.n;a (nonynpOBo.n;HHK8), no-
KpbITbI cBepxy TOHKHM npo-
3paqHbIM cnoeM MeTanJIa. TIo-
rpaHHqHbIH CJlOH Me>K.n;y Me-
TannOM Hero OKCH.n;OM IiMeeT
O,lJ;HOCTOpOHHIOIO SJleKTpOnpOBo.n;HOCTb - sneKTpOHbI MorYT npoxo.n;HTb nHlUb B
HanpaBJIeHHH OT OKCH,A8 MeTaJlJla K MeTaJlny. TIOTOK sneKTpOHOB, H,AylI.\HH B
STOM HanpaBJleHHH. C03.n;aeTCSI no,A ,neHCTBHeM CBeTa 6e3 BCSIKoro BHelUHero
HCTOqHHK8 Hanp.HmeHH.H. BeHTHnbHbIH cpOTosneMeHT HenOCpe,ACTBeHHO npeBpa-
lQaeT SHeprHIO CBeTOBOH BonHbI B SHeprHIO sneKTpHqeCKOrO TOKa. T.e. SIBn.H-
eTC.H HCTOqHHKOM TOKa. Ha STOM npHHIJ,Hne OCHOB8HO .n;eHCTBHe COnHe'lHbIX
CBeT
II p03paqHbIH
CJlOH
.-
. '" MeTanJl
Puc 19.8
563

6aTapeH, KOTOpbIe YCT8HaBJlHBaIOTC.H Ha KOCM1:IqeCKHX Kopa6J1.Hx. TaKHe c}>OTO-
3neMeHTbI .HBJI.HIOTC.H OCHOBHOH qaCTblO JlIOKCMeTpOB - npH60pOB JI.H H3Mepe-
H1:I.H OCBeeHHOCTH, a TaKme c}>OT03KcnOHOMeTpOB.
19.6. ,l(aBJIeHHe CBeTa. OnbITLl n. H. Jle6e,IJ;eBa
.l(aBJIeHHeM CBeTa Ha3bIBaeTC.H )J;aBJleHHe, KOTopoe npOH3BOA.HT 3J1eKTpOMar-
H1:ITHbIe CBeTOBbIe BOJlHbI, naAalO1:Ie H8 nOBepXHOCTb KaKOrO-Jl1:I60 TeJla. Cy-
eCTBOB8H1:Ie lIaBJleHH.H 6bIJlO npelICKa3aHO JIm. MaKCBeJlJlOM B ero 3J1eKTpO-
M8rH1:ITHOH TeOp1:lH CBeT8.
ECJlH, Hanp1:lMep, 3J1eKTpOMarH1:ITH8.H
BOJlHa na)J;aeT Ha MeTaJlJl (pHC. 19.9),
TO nOA AeHCTB1:IeM 3J1eKTpHqeCKOrO nOJl.H
BOJlHbI C H8np.HmeHHOCTblO E 3J1eKTpO-
HbI nOBepXHOCTHoro CJlOH MeTaJlJl8 6y-
lIYT lIBHraTbCH B H8np8BJleHHH. npOTH-
BOnOJlOmHOM BeKTOpy E, co CKOpOCTblO
V = const. MarHHTHoe nOJle BonHbI C
HH)J;YKr(HeH B )J;eHcTByeT Ha )J;B1:ImYHe-
C.H 3J1eKTpOHbI C C1:IJlOH JIopeHr(a F JI B
HanpaBJleHH1:I. nepneH1I1:IKYJl.HpHOM nOBep-
XHOCTH MeTanJla (cor JlaCHO npaB1:IJlY Jle-
BOR PYKH). aBJleH1:Ie p, OK83bIB8eMoe
BonHOH H8 nOBepXHOCTb MeTMJl8, MO:>KHO paCCqHTaTb K8K OTHOIIIeHHe p8BHO-
AeRCTBYIOeR CHJI JIopeH8, eRCTBYIOHX Ha cB060HbIe 3J1eKTpOHbI B nOBep-
XHOCTHOM cJloe MeTaJlJla, K nJlOaAH nOBepXHOCTH MeTaJlJla:
/

Fn
A
''1
I" I.
/
v
Puc. 19.9
B
n
L J;JI
p _ i=1
--So
Ha OCHOBaHHH 3J1eKTpOMarHHTHoH TeopHH M8KcBeJlJl nOJlyqHJI c}>OPMYJlY lIJI.H
CBeTOBoro A8BJleHH.H. C ee nOMoblO OH paCCq1:lTM aBJleH1:Ie COJlHeqHOrO CBe-
Ta B .HPKHH nOJl)J;eHb Ha a6COJllOTHO qepHOe TeJlO, p8CnOJlOmeHHOe nepneH,A1:I-
KYJI.HpHO COJlHeqHbIM Jlyq8M. 9TO AaBJleH1:Ie OKa3MOCb paBHbIM 4,6 MKIIa:
p = (1 + p) J .
c
rAe J - 1:IHTeHCHBHOCTb CBeTa, p - K03c}>c}>Hr(1:IeHT oTpameHH.H CBeTa
(CM.  16.3), C - CKOpOCTb CBeTa B BaKYYMe. JI.H 3epKaJlbHbIX nOBepXHOC-
TeH p = 1, np1:l nOJlHOM nOrJlOeH1:IH ()J;JI.H 86cOJllOTHO qepHOrO TeJla) p = 0
C TOqK1:I 3peHH.H KBaHToBoR TeopHH, AaBJleHHe .HBJI.HeTC.H CJleACTBHeM Toro,
hv
qTO Y c}>OTOH8 HMeeTC.H HMnYJlbC Pi}) = -. IIycTb CBeT naAaeT nepneHAHKYJI.Hp-
C
HO nOBepXHOCTH TeJla 1:1 3a 1 C Ha 1 M 2 nOBepXHOCT1:I naaeT N c}>OTOHOB.
qacTb 1:13 H1:IX nOrJlOT1:ITC.H nOBepXHOCTblO TeJla (Heynpyroe COY)J;8peHHe), 1:1
Kam)J;bIR H3 nOrJlOeHHbIX c}>OTOHOB nepeAaeT 3TOR nOBepXHOCTH CBOH HMnYJlbC
564

hv
P4.> = -. 'tJaCTb me cpOTOHOB OTpa3MTC.H (ynpyroe cOY,AapeHMe). OTpameHHbIll
C
cpOTOH nOJleTHT OT nOBepXHOCTH B npOTMBOnOJlOmHOM HanpaBJleHMM. TIonHhIll
HMnYJlhC, nepeaHHbIii nOBepXHOCTM OTpameHHbIM cpOTOHOM, 6Y,AeT paBeH
6P4.> = P4.> - (-P4.» = 2pq, = 2 hv .
c
,Z:(aBJleHHe CBeTa Ha nOBepXHOCTb 6Y,AeT paBHO HMnynhCY, KOTOphIH nepealOT
3a 1 C Bce N cpOTOHOB, nap;aroMx Ha 1 M 2 nOBepXHOCTH TeJla. EcnH p -
KOscpcpHHeHT oTpameHH.H CBeTa OT npOH3BOJlhHOH nOBepXHOCTH, TO p' N -
STO qMCJlO OTpameHHhIX cpOTOHOB, a (1- p)N qMCJlO nOrJlOeHHhIX cpOTO-
HOB. CJle)J;OBaTeJlhHO, ,AaBJleHHe CBeTa
hv hv hv
p = 2pN - + (1- p)N - = (1 + p)N-.
c c c
TIpoM3Be,AeHHe npe,ACTaBJI.HeT C060H SHeprMIO Bcex <pOTOHOB, na,AalOMX Ha
1 M 2 nOBepXHOCTH 3a 1 c. 9TO eCTb MHTeHCMBHOCTb CBeTa (noBepxHOCTHa.H
nnOTHOCTh nOTOKa M3J1yqeHM.H na,AalOero cBeTa):
W
Nhv = - = I.
S .t
I
TaKHM 06pa30M, ,AaBJleHHe CBeTB P = (1 + p) -.
c
TIpe,ACKa3aHHoe MaKCBeJlnOM CBeTOBoe ,AaBneHMe 6bInO SKcnepMMeHTaJIhHO 06-
HapymeHo H H3MepeHO PYCCKHM cpH3MKOM II. H. JIe6e,AeBbIM. B 1900 r. OH H3-
MepHJI aBJleHHe CBeTa Ha 'l'BepbIe Tena, a B 1907-1910 rr. - ,AaBJleHMe
CBeTa Ha ra3hI.
Puc. 19.10
TIpH60p, C03,AaHHbIH JIe6e,AeBhIM .AJI.H M3MepeHH.H
aBJleHH.H CBeTa. npeCTaBJI.Hn C060H OqeHh qYBCTBM-
TeJlbHbIH KPYTHnhHhIH HHaMoMeTp (KpYTHnbHhIe
BechI). Ero nOBMmHoH qaCTblO .HBn.HJlaCh nO,ABellIeH-
Ha.H Ha TOHKOH KBapueBoH HHTM nerKBH paMKa C YK-
penJleHHhIMH Ha HeH KpbIJlbIlliKaMH - CBeT nbIMM M
qepHbIMH HCKaMM TonMHoll o 0,01 MM. KphIJlbIllI-
KH enaJlM M3 MeTannMqeCKOH f:}>onhrM (pMC. 19.10).
PaMKa 6hIna nOBellIeHa BHYTPH cocya, H3 KOToporo
OTKaqaJlH B03YX.
CBeT, naa.H Ha KphIJlbIllIKH, OKB3hIBan H8 CBeT-
nbIe H 'tlepHhIe ,AHCKH pa3Hoe aBJleHHe. B pe3YJlbTaTe
Ha paMKY )J;eHcTBOBaJl BpaalOHii MOMeHT, KOTOphIH
nOBeca. TIo yrny 3aKpyqHBaHH.H HHTH onpeen.HJlOCb aBne-
e
3aKpyqMBaJl HHTh
HHe CBeTa.
TpYHOCTM H3MepeHH.H CBeTOBoro aBJleHHJl BhI3hIBanHCb ero HCKJllOqMTeJlb-
HO MRJlbIM 3HR"'leHMeM H cyw;ecTBoBaHHeM HBJleHMH, CHnhHO BnHHIOW;HX HB TO'q-
HOCTh M3MepeHMH. K HX 'lMCJlY OTHOCHnaCb HeB03MomHOCTb nOJlHOCTblO OTKR-
qaTb B03YX H3 cocya, qTO npHBOlIHJlO K B03HHKHOBeHHIO TaK Ha3hIBaeMoro
pHOMeTpHqeCKOrO s4><I>eKTa.
CYW;HOCTh SToro .HBJleHHH B cJleyrow;eM. CTopoHa KpbInbIllieK, 06pReHHa.H
K MCTOqHHKY CBeTa, HarpeBaeTC.H CHnhHee npoTHBOnOJlOmHOH CTOpOHbI. TIOSTOMY
565

MOJleKYJlbI B03yxa, OTpamalOLQHec.H OT 60Jlee HarpeTOH CTOpOHbI, nepealOT KpbJ-
JlbIWKY 60JlblIIHH HMnYJlbC, qeM MOJleKYJlbI, OTpamalOLQHeC.H OT MeHee HarpeTOH
CTOpOHbI. TaK nO.HBJI.HeTC.H )J;OnOJlHHTeJlbHbIH BpaLQalOLQHH MOMeHT.
CXeMa YCTaHOBKH JIe6e,AeBa JI.H H3MepeHH.H
)J;aBJleHH.H CBeTa Ha ra3bI H306pameH8 Ha pHCYHKe
19.11. CBeT, npOXO.HLQH:H CKB03b CTeKJI.HHHYIO
CTeHKY A, eHcTByeT Ha ra3, 3aKJlIOqeHHbIH B H-
nHHpHqeCKOM KaHaJIe B. IIo.n; aBJleHHeM CBeTa
ra3 H3 KaHMa B nepeTeKaeT B C006LQaIOLQHHC.H C
HHM KaHaJl C. B KaHaJle C HaXOllHTC.H nerKHH
nOJ];BHmHbIH noplIIeHb D, no.zlBelIIeHHbIH Ha TOH-
KOH ynpyroH HHTH E, nepneHHKYJlSlpHOH nJloc-
KOCTH qepTema. CBeToBoe aBJleHHe paCCqHTbIBa-
nOCb no yrJlY 3aKpyqHBaHH.H HHTH.
XOTSI CBeTOBoe aBJleHHe OqeHb MaJIO B 06blq-
HbIX YCJlOBH.HX, ero eHcTBHe, TeM He MeHee, MomeT OKa3aTbC.H CYLQeCTBeHHbIM B
PyrHx YCJlOBH.HX. BHYTPH 3Be3 npH TeMnepaType B lIeCKOJlbKO eCSITKOB MHn-
nHOHOB KenbBHH aBJleHHe SJleKTpOMarHHTHoro H3J1yqeHHSI onmHO OCTHraTb rpo-
MHoro 3HaqeHH.H. CHJlbI CBeTOBoro aBJleHH.H Hap.HY C rpaBHTalJ,HOHHbIMH CH-
JlaMH HrpalOT cyecTBeHHyro pOJlb B npOlJ,eCcax, npOHCXOSlLQHX BHYTPH 3Be3.
11I.I'I#II.lI.I.lII.I.I.l1
-----..C
, E
(Q}--+:[]
11111.l1'.I.1I.1.1.1.1I.1.1III.1
A
Puc. 19.11
19.7. XHMH'IeCKOe ,lJ;eHCTBHe CBeTa Hero npHMeHeHHe
XHMHqeCKOe eHCTBHe CBeTa npOHBn.HeTC.H B TOM, qTO CYLQeCTByeT lJ,eJlbIH
pHll XHMHt.IeCKHX npeBpaLQeHHH, npOHCXOLISlWHX TOJlbKO no.zr lleHCTBHeM CBeTa.
XHMHqeCKHe peaKlJ,HH, npOTeKalOHe no eilCTBHeM CBeTa, Ha3bIBalOT c}>OTO-
XHMH'IeCKHMH.
<1>OToXHMH'IeCKHe peaKHH MorYT npOTeKaTb nH60 no nYTH CHHTe3a (06pa-
30BaHHe no eHCTBHeM CBeTa H3 MOJleKYJl HCXOHbIX BeLQeCTB 60nee CJlO.>K-
HbIX MOJleKYJI), JlH60 no nYTH pa3J10eHHJI (06pa30B8HHe no eHCTBHeM CBe-
Ta npOCTbIX MOJleKYJI H3 60Jlee CJlOmHbIX).
<1>OToXHMHqeCKHe peaKlJ,HH qaCTO conpOBomaIOTC.H BTOPHqHbIMH XHMHqeC-
KHMH npeBpaLQeHH.HMH.
YCTaHOBJleHO, qTO Macca BeeCTBa, yq8CTBYIOLQel'0 B xoe nepBHqHOH <pO-
TOXHMHqeCKOH peaKlJ,HH, npOnOplJ,HOHanbHa SHeprHH CBeTOBoro H3J1yqeHH.H, no-
rnOLQeHHOrO BeLQeCTBOM.
.D:JlSI KamOH «pOTOXHMH'leCKOH peaKlJ,HH CYLQeCTByeT nopOrOBaJl qaCTOTa v,
KOTOPYIO Ha3bIBalOT KpaCHOH rpaHHl.{eH aHHOH peaKI.{HH. CBeT C qaCTOTO:H
v < Vo He MomeT BbI3BaTb .zraHHYIO <pOTOXHMHqeCKYIO peaKlJ,HIO.
3aKOHOMepHOCTH <pOTOXHMHqeCKHX peaKlJ,HH 06'b.HCH.HIOT Ha OCHOBe KB8HTO-
BOH TeopHH: aTOMbI B MOJleKYJlaX yepmHBaIOTC.H 6J18t'0,Aap.H XHMHqeCKHM CB.H-
3S1M. ECJlH SHeprHH, nOrJlOeHHO:H MOJleKYJlOH <pOTOHa, XBaTaeT JI.H pa3pbIB8
XHMHqeCKOH CB.H3H (v  Vo), TO <pOTOXHMHqeCKa.H peaKlJ,H.H npOHCXOHT, eCJlH me
v < Vo, TO peaKlJ,H.H He npOHCXO)J;HT.
MHorHe <pOTOXHMHqeCKHe peaKHH HrpalOT 60JlblIIYIO pOJlb B npHpoe Ii
TeXHHKe.
BamHeHIIIHe <pOTOXHMHqeCKHe peaKHH npOHCXO.HT B 3eJleHbIX JlHCTbSlX )J;epe-
BbeB H TpaBbI, B HrJlaX XBOH Ii BO MHorHX MHKpoopraHH3Max. JIHCTb.H norJlo-
566

IqalOT H3 B03]1;yxa yrJleKHCJlbIH ra3 H paCI.I..tenJIHIOT ero MOJleKYJIbI Ha COCTaBHble
qaCTH: yrJlepo,lJ; H KHCJlOpO.ll. IIPOHCXOlIHT 3TO, KaK YCTaHOBHJI PYCCKHH 6HOJlor
K. A. THMHp.H3eB, B MOJleKYJlaX XJlOpmpHJlJla no,n ,neHCTBHeM KpaCHhIX JlyqeH COJl-
HeqHOrO cneKTpa. IIpHcTpaHBa.H K yrJlepo,nHOH I.1;enOqKe RTOMbI ,npyrHx 3neMeH-
TOB, H3BneKReMbIX KOpH.HMH H3 3eMnH, paCTeHHH CTpOHT MOJleKYJlbI 6eJlKOB, >KH-
pOB H yrJleBo,nOB - nHI.I..tY ,nn.H qeJlOBeKa H mHBOTHbIX. Bce 3TO npOHcxo,nHT 3R
CqeT 3HeprHH COJlHeqHbIX JlyqeH. IIpHqeM 3,neCb oc06eHHO BamHa He TonhKO caMR
3HeprH.H, HO H Ta <t>opMa, B KOTOpOH OH8 nOCTyIIaeT. MTaK. «}JOTOCHHTe3 - 3TO
npOI.1;eCC 06pa30BaHH.H yrJleBo,nOB no,n .n;eHCTBHeM CBeTa C BbI,neJleHHeM KHcnopo,n8
B paCTeHH.HX H HeKOTophIX MHKpOOpraHH3Max.
XHMHqeCKOe ,neHCTBHe CBeTa JIemHT H B OCHOBe «}JOTorpa«}JHH. 'tJYBCTBH-
TeJlbHbIH CJlOH <t>OTOnJlaCTHHKH COCTOHT H3 ManeHbKHX KpHCTanJlHKOB 6pOMHJJ;8
cepe6pa (AgBr), BKpanJleHHbIX B menaTHH. IIonaJJ;aHHe <t>OTOHOB B KpHCT,aJl-
JlHK npHBo,nHT K OTPblBY 3J1eKTpOHOB OT OT,nenbHblX HOHOB 6pOMa. 3TH 3neK-
TpOHbI 3aXBaTbIBaIOTC.H HOHaMH cepe6pa, H B KpHCTanJlHKe o6pa3yeTC.H He60nb-
llloe KOJlHqeCTBO HeHTpBJIbHbIX aTOMOB cepe6pa.
Pacna.n; MOJleKYJlbI npoHcxOJJ;HT no cxeMe
AgBr + hv  Ag* + Br+ + e-.
r.n;e Ag* - 3HepreTH'lleCKH B036Y>K,AeHHbIH aTOM cepe6pa. Br+ - nOJlO>KHTenh-
HblH HOH 6poMa, e- - 3J1eKTpOH. IIo,n ,lJ;eHCTBHeM npOHBHTeJlH 6pOM paCTBO-
pHeTC.H, a OCTaBWHeC.H aTOMbI cepe6pa ,lJ;alOT HeraTHBHoe H306pa>KeHHe J],aJlee
HeraTHB 06pa6aTbIBaeTC.H B 3aKpenHTeJle, KOTOpbIH paCTBOp.HeT HeKopearHpOBaH-
HbIH 6pOMHJJ; cepe6pa.
<1>oTorpa<t>H.H nOJlY'llHJla lllHpOKoe pacnpOCTpaHeHHe B HaYKe H TeXHHKe. Co-
BpeMeHHhIe JJ;OCTHmeHH.H n03BOJl.HIOT npOH3Bo,nHTh <}>OTorpa<}>HpOBaHHe He TOJlb-
KO npH BHJJ;HMOM CBeTe, HO H B TeMHOTe (B HH<t>paKpaCHbIX Jlyqax). <1>oTorpa-
<}>HIO npHMeH.HIOT TaKme ,nJlH 3anHCH 3BYK8 B KHHO.
BbII.1;BeTaHHe TKaHeH Ha COJlHI.1;e H 06pa30BaHHe 3arapa - Tome npHMepbI
XHMH'lleCKOrO ,neHCTBHH CBeTa.
19.8. KOpnyCKYJUlpUO-BOJIHOBOH AyaJIH3M
KaK cneJJ;yeT H3 H3J10meHHOro Bblllle, CBeTY npHCYI.I..tH KopnYCKYJlHpHhIe
CBOHCTBa. Ha HX OCHOBaHHH 06'h.HCHHeTC.H I.1;eJIblH PHJJ; Ha6nlOJJ;aeMhlx <}>H3H'leC-
KHX .HBneHHH - npOHcxom,neHHe nHHeHqaTblX cneKTpOB, <}>oT03<}><}>eKT.
Ho TaKHe .HBJleHH.H, KaK HHTep<}>epeHI.1;H.H, JJ;H<}>paKI.1;H.H, nOJlHpH3aI.1;H.H, C TOq-
KH 3peHH.H KBaHTOBOH TeopHH 06'h.HCHHTb 3aTpy,nHHTeJlbHO. OHH HBJI.HIOTCH no,lJ;-
TBep>K)1.eHHeM BOJlHOBbIX CBOHCTB CBeTa.
Ha nepBblH B3rJl.H.n KameTC.H, qTO ,nBe TOqKH 3peHHH - BOnHOBa.H H KBaH-
TOBa.H - B3aHMHO HCKnlO'llaIOT ,npyr ,npyra. P.HJJ; npH3HaKOB BonH H 'llaCTHI.1;
JJ;eHcTBHTeJlbHO npOTHBOnOJlO>KHbl. HanpHMep, ,ABHmYHeC.H qaCTHI.1;bI (<}>OTOHbI)
Haxo,n.HTC.H B onpe.n;eJleHHbIX TOqKaX npOCTpaHCTBa, a pacnpOCTpaHHIOI.I..tYIOCH
BOJIHY HymHo paCCMaTpHBaTh KaK "pa3Ma3aHHYIO" B npOCTpaHCTBe, H Henb3H
rOBopHTb 0 npe6bIBaHHIi BOJlHbI B HeKoTopoH onpeJJ;eneHHoH TO'llKe.
Pa3BHTHe onTHKH, BC.H cOBoKynHocTb OnTHqeCKHX HBneHHH nOKa3aJlH, qTO
CBOHCTBa HenpepbIBHOCTH, xapaKTepHbIe ,lJ;JlH 3J1eKTpOMarHHTHoro nOJl.H CBeTO-
567

BOll BOJlHbI, He CJleyeT npOTHBonOCTaBJI.HTb CBOllCTBaM HCKpeTHOCTH, xapaK-
TepHbIM n.H <pOTOHOB.
CBeT 06J1aaeT BOHCTBeHHOH KopnYCKYJI.HpHO-BOJlHOBOH npHpOOH. no CO-
BpeMeHHbIM B033peHH.HM, CBeT 06J1aeT RaR BOJIHOBLIMH, TaR H RopnYCKY-
J1.HpHLIMH CBOHCTBaMH. ,l(BoHcTBeHHOCTb CBOHCTB CBeTa HaxOHT CBoe BbIpame-
HHe H B <popMYJlax:
hv hv
Wo=hv; m =2:; p =-.
C C
KopnYCKYJI.HpHbIe xapaKTepHCTHKH <pOTOHa - 3HeprH.H W o , Macca m, HM-
nYJlbC P - CB.H3aHbI C BOJIHOBOH xapaKTepHCTHKOH - qaCTOTOH V.
CBeT, 06J1a)J;a.R OHOBpeMeHHO KopnYCKYJI.HpHO- BOJlHOBbIMH CBOHCTBaMH, 06Ha-
PymHBaeT onpeeJIeHHbIe 3aKOHOMepHOCTH B HX npO.HBJIeHHH. TaR, BOJlHOBbIe
CBoHcTBa CBeTa npO.HBJI.RIOTC.H B 3aKoHoMepHOCT.RX ero pacnpocTpaHeHH.H, HHTep-
cpepeHU;HH, .ltH<ppaKU;HH, nOJI.HpH3aU;HH. a KopnycKYJI.HpHbIe - B nponeccax B3a-
HMO.lIeHCTBH.R CBeTa C BemeCTBOM. qeM 60Jlbme lInHHa BOJlHbI, TeM MeHbme 3Hep-
rH.H H HMnYJlbC <poTOHa H TeM TpYHee 06HapymHBa1OTC.H KBaHTOBbIe CBoHcTBa
CBeTa. Ha060pOT, qeM MeHbme JIHHa BOJlHbI, TeM 60JIblIIe 3HeprH.R H HMnYJlbC
<poTOHa H 'reM 'rpYHee 06HapymHBa1OTC.R BOJIHOBbIe CBOHCTBa CBeTa. npH pac-
npOCTpaHeHHH CBeTa npO.HBJI.HIOTC.H ero BOJlHOBbIe CBOHCTBa, a npH B3aHMoeH-
CTBHH C BeeCTBOM (H3J1yqeHHH H nOrJlOeHHU) - KopnYCKYJI.HpHbIe.
HHJIbC Bop C<POPMYJlHpOBaJl npuliijun OOnOJl.IiUmeJl.bIiOcmu: OJl.Sl nOJl.IiOZO no.
IiUMaliUSl npupoiJbl C6ema lie06xoiJu.M.0 Yllumbl6amb KaK 60Jl.Ii06ble. maK U KOp
nYCKYJl.Slplible C60ucm6a C6ema: OIiU 63aU.M.1i0 iJonOJlliSllOm iJpyz iJpyza. O,AHaKO
JI.H 06.RCHeHH.R KaKOrO-JlH60 3KcrrepHMeHTa CJleyeT HCrrOJIb30BaTb JIH60 BOJI
HOBbIe, JlH60 KopnycKYJI.HpHble npecTaBneHH.R 0 npHpoe cBeTa, HO He Te H
,ApyrHe O,AHOBpeMeHHO.
BonpOCM RH CAMOKOHTPORH
1. C KaKUMU mpyoliocmJUtu CmOJl.KliYJl.aCb 3Jl.eKmpoiJuliaMuKa M aKC6eJlJl.a
npu 060SlClieliUU 3aKOliOMepliocmu pacnpeOeJl.eliUSl 3liepZUU 6 cneKmpax menJl.O-
60Z0 U3Jl.YlleIiUSl? B lleM cY14IiOcmb zunome3bl II Jl.aIiKa?
2. KaK U3Jl.YllaemcSl U nOZJl.014aemcSl C6em amOMaMU 6e14eCm6a?
3. 'C{mo lia3bl6aemC.R tPom03q)(jJeKmoM? B iteM cocmOSlm 3aCJl.YZU CmOJl.emo-
6a 8 U3YlleliUU tPom03tPtPeKma U UCnOJl.b306aIiUU ezo lia npaKmuKe?
4. C nO.M.014blO KaKOU ycmaH06KU M03fCIiO np06ecmu U3Y'lleliue 3a6ucUMOC-
mu tPomomOKQ om lianpSl3fCeIiUSl?
5. KaK 3a6ucum CUJl.a tPomomoKa om lianpSlJlCeliUSl? 'C{ mo nOIiUMalOm noiJ
3aiJepJICU6alO14UM lianpSlJlCeliueM? 'C{ eM onpeOeJl.SlemCSl ezo 6eJl.U'IlUlia?
6. Om Ilezo 3a6ucum IlUCJl.O tPom03Jl.eKmpoli06. 6blpbl6aeMblx 6 eouliuu,y
6pe.M.eHU C n06epXliocmu MemaJl.Jl.a?
7. Om Ilezo 3a6ucum CKopocmb tPom03Jl.eKmpoIi06?
8. 'C{ mo maKoe KpaCliaSl zpaliuija ljJom03ljJtPeKma? Om Ilezo 3a6ucum ee 6e-
Jl.Ullulia?
9. H a Ilmo pacxooyemcSl 3liepZUSl K6alima. nOZJl.014eliliozo tPom03JleKmpO-
liOM? KaK 3anUCbl6aemcSl ypa6lieliue OUliwmeUlia iJJl.Sl tpom03tPtPeKma? KaK
060SlCliumb 3aKOlibl tPom03tPtPeKma C K6alim060il mOIlKU 3peliUSl?
568

10. IJmo maKoe ljJomoli? MOJICem Jl.U Oli cY14eCm606amb 8 cocmOJlliUU no-
KOSl? q: eMY pa6lia ezo CKopocmb 6 .ruo60u cpeiJe?
11. KaK onpeiJeJl.Umb 3liepZUIO U uMnYJl.bC ljJomolia?
12. IJmo npeiJcma8JUllOm c060il ljJom03Jl.eMeH.mbt? riJe OIiU npU.Me1iSllOmcSl?
13. IJmo maKoe 8liympeliliuu ljJom03ljJljJeKm? £[mo npeiJcma6JtSllOm c060u
ljJomope3ucmOpbt?
14. I.J e3ue6btu ljJom03Jl.eMelim OC6e14alOm CUliUM C6emO.M. KaK U3.MeliUmCSl
cu.n.a ljJomomoKa. ecJl.U ezo OC6e14amb YJl.bmpaljJuOJl.em06bt.M U3Jl.Y1leliUe.M mou JlCe
ulimeliCU8liOCmu?
15. H 306pa3ume zpaljJullecKu 3aeUCU.M.OCmb .MaKCU.MaJtbH.OU KUH.emUlleCKOU
3liepZUU ljJom03Jl.eKmpOlia om llaCmOmbt C6ema. KaK U3 zpaljJuKa onpeiJeJl.Umb
nocmoSlliliYIO II Jl.aIiKa?
16. llolleMY iJa6Jteliue C6ema Ita 1lepliYIO n06epXH.OCmb .MelibUle. lle.M H.a
6eJl.YIO? OiJuliaK060 Jl.U 6yiJem iJa6Jl.eliUe C6ema lia n06epxHocmb. ecJl.U OJl.UHa
60Jl.libt naiJalO14ezo C6ema 6yiJem u3MeliSlmbcSl?
17. II olle.MY X80cm KO.Membt 6ceziJa H.anpa6Jl.eH om COJl.Iil,a. xomJl .MeJICiJy
6e14ecm60.M KO.Membt u COJl.liqeM cY14eCm6yem zpa6umaquolilioe npUmSlJICeliue?
he
W o = hv = -
A
_ W o _ hv
p----
e c
mv 2
hv = A + 
s 2
2
mv max = eU
2 3
OCHOBHbIE cbOPMY Jlbl
3HeprH.H q>OTOHa, rlle h = 6.62' 10- 34 ,lIm. c - nOCTO-
M
.HHHa.H TIJlaHKa, v - 'laCTOTa, e = 3. 10 8 - CKOpOCTb
c
CBeTa B BaKYYMe, A - .n;JlHHa BOJlHbI.
- HMnYJlbC q>OTOH8.
- ypaBHeHHe 3MHIllTeHHa .n;n.H BHeIllHero <pOT03q><peKTa, r.n;e
As - pa60Ta BbIxo.n;a 3JIeKTpOHa, m - Macca 3JIeKTpOHa,
V max - MaKCHMaJlbHa.H CKOpOCTb <pOT03JIeKTpOH8.
- MaKCHMaJIbHa.H KHHeTH'lecKaH 3HeprH.H cpOT03J1eKTpOHa, r.n;e
e - 3ap.H.n; 3J1eKTpOHa, U a - 3a.n;epmHB8IOIl\ee H8np.HmeHHe.
V _As. '1 _ he Th ThTh
o - h' /\'0 - As - KpaCH8.H rpaHHn.a 'i'OT03'i''i'eKTa.
1(1 + p)
p=
C
- CBeTOBoe .n;aBJIeHHe, r.n;e I - HHTeHCHBHOCTh CBeTa, p -
K03<P<PHII.HeHT oTpameHH.H CBeTa.
METOHqECKHE PEKOMEHAUHH
TIpH peIlleHHH 38.n;aq no 3TOM TeMe He06xollHMO:
1) nOMHHTb 0 B3aHMOCB.H3H MemllY BOJIHOBbIMH H KBaHTOBbIMH xapaKTepH-
CTHKaMH qaCTHn.;
2) 3HaTb, qTO B3aHMO,AeHCTBHe cpOTOHOB C BeeCTBOM no.n;qHH.HeTC.H 3aKO-
HaM coxpaHeHH.H 3HeprHH H HMnYJlbca. YpaBHeHHe 3HHllITeHHR .HBJI.HeTC.H cne.n;-
CTBHeM 3aKOHa coxpaHeHH.H 3HeprHH (.n;JI.H BHeIllHero cpOT03q>q>eKTa), a <POPMY-
569

na AJI.H paCqeTa CBeTOBOrO AaBJleHH.H BbITeKaeT H3 3aKOHa COXpaHeHH.H HM-
nYJlbCa.
llPHMEPbJ PEllIEHHH 3A,l(A'l
1. 3neKTpoH, npOH,ASI p83HOCTb nOTeHH8nOB 4,9 B, CT8nKHB8eTCSI C 8TO-
MOM PTYTH H nepeBO,AHT ero B nepBoe B036ym,AeHHOe COCTOSlHHe. K8KYlO ,AnH-
HY BonHbI HMeeT Q>OTOH, cooTBeTCTBYIOHH nepexo,Ay STOMa PTYTH B HopManb-
Hoe COCTOSlHHe?
JJ:8HO: U=4,9 B, h=6,62'10- 34 JJ:m.c, e=1,6.10- 19 Kn. e=3,0'10 8
M
C
H8HTH: A..
P e 111 e H He. Onpe,AenHM CH8'l8n8 3HeprHIO W, KOTOPYIO npH06peTaeT aneK-
TpOH, npOH,ASI B aneKTpOCT8TH'leCKOM none p83HOCTb nOTeHH8nOB U: W=eU.
3TY aHepI'HIO aneKTpOH np!1: CTonKHOBeHHH nepe,A8eT 8TOMY PTYTH, HaxO,ASIe-
MYCfl B HopManbHOM COCTOSlHHH. H nepeBO,AHT ero B B036ym,lIeHHOe COCTOSlHHe.
IIpH 06paTHOM nepeXOlle 6Y,lIeT H3nY'l8TbCSI <pOTOH TaKOH me 3HeprHH:
W = hv = he , OTCIO"a A = he = hc . A = 2 , 4,10-7 M.
A ,.... W eU'
2. H8H60nbIII8S1 ,AnHH8 BonHbI CBeT8, npH KOTOpoii npOHCXO,AHT <POT03<}J-
<}JeKT ,AnSI Bonb<ppaM8, paBH8 275 HM. Haii,AHTe pa60TY BXO,Aa aneKTpOHOB
H3 Bonb<PP8M8 H H8H60nbIIIYIO CKOpOCTb aneKTpOHOB, BblpblBaeMblX H3 Bonb<p-
p8M8 CBeTOM C ,AnHHOH BonHbl. paBHoii 180 HM.
M
JJ: 8 H 0: h = 6.62. 10- 34 JJ:>K' C, A.o= 2,75. 10- 7 M. e = 3,0.10 8
A.=1,8.10- 7 M.
H 8 H T H: AD' V rnaX '
P e 111 e H He. 113 onpe,AeneHHSI KpaCHoii rpaHHbI <}JOToaQ>Q>eKT8 Cne,AyeT, 'lTO
A.=hVo= e ; A B =7.2.10- 19 n>K.
o
TIo <popMyne 3iiHIIITeHH8 ,AnSI CPOT03Q>Q>eKTa. Y'lHTB8.H, 'lTO
C
c
V=-
A'
OTCI01I8
U rnax =   (  - A. ).
HMeeM
hc =...1 + mVax
A "' 2'
5 M
v=9,1.10 -.
C
3. HaH,AHTe aHeprHIO H HMnynbc Q>OTOH8, ecnH COOTBeTCTBYIOa.H eMY nl1-
H8 BonHbl p8BHa 1,6' 10- 12 M.
M
JJ:aHo: )..=1,6'10- 12 M, e=3.10 8 -, h=6,62.10- 34 ,[(>K'c.
C
H8HTH: W, m, p.
P e 111 e H H e. 3HeprHIO <POTOH8 MomHO H8HTH, Hcnonb3YSI <popMyny
he
W = hv = -. W= 1,2,10- 13 JJ:m.
A
W Kr'M
I1Mnynbc Q>oToHa p = -, p = 0,42 . 10- 21
e C
570

4. Onpe,AeJIHTe 3a,AepmHBaI0Iee HanpSlmeHHe Mem,AY 3neKTpO,A8MH cpOT03ne-
MeHTa npH OCBeeHHH ero CBeTOM C ,AJlHHOH BonHbI 200 HM, eCJIH p860T8
SbIXO,Aa 4,0 3B?
,ZJ;aHo: e=1,6.10- 19 En, m=9,1.10- 31 Kr, A=2,00.10- 7 M,
As= 6,4 . 10- 19 ,ZJ;m.
HaHTH: U 3 .
Pew e H H e. 38nHIIIeM yp8BHeHHe 8HHIIITeHHa lInSl <l>oT03<t><I>eKTa
mv 2 e
hv =  +, 1',Ae v=-.
2 A
QT06bI <}>OT03neKTpOHbI He ,AOCTHrnH aHO,Aa, Mem,AY KaTO,AOM H 8HO,AOM H8,A0
npHnomHTb p83HOCTb nOTeHH8nOB, npH KOTOpoii pa60T8, COBeplIIaeMaSl 3neK-
TpH'leCKHM noneM no TopMomeHHIO 3neKTpOHa, 6Y,AeT p8BH8 H8'lanbHOH KHHe-
2
U mVrnax
TH'leCKOH 3HeprHH <l>oT03neKTpOHa, T.e. e 3= -. IIo,AcTaBHB 3TO BbIpame-
2
he
HHe B <}>opMyny 8HHIIITeHHa, HMeeM T =AB + eV a .
OTCIO,Aa 38,AepmHBalOaSl pa3HOCTb nOTeHHaJIOB
U = he -  . V 2 2 B
Ae ' 3=' .
5. rna3 'lenOBeKa BocnpHHHMaeT CBeT ,AnHHOH BonHbI 0.50 MKM, ecnH MHHH-
M8nbH8fl MOHOCTb H3nY'leHHSI COCTaBnHeT 20,8.10- 18 ,ZJ;m B ceKYH,AY. EaKoe
KOnH'leCTBO cpOTOHOS npH 3TOM emeceKYH,AHO nOn8,A8eT Ha CeT'laTKY rn838?
,ZJ; a H 0: A = 0,50. 10- 6 M, WI = 20,8. 10- 18 ,l(m.
HaHTH: n.
he
Pew e H He. 8HeprHSI O,AHOrO cpOTOHR W = T' EOnH'leCTBO <l>OTOHOB, eme-
N = Wi = WiA . N 53
ceKYH,nHO nOn8,A81OIl.tHX H8 CeT'l8TKY rn838, W he' = .
6. Onpe,AenHTe H3MeHeHHe HMnynbc8 MeT8nnH'leCKOH nnaCTHHKH npH BbIneTe
H3 Hee O,AHOrO 3neKTpOH8, ecnH Ha ee nOBepXHOCTb n8,n8eT KB8HT CBeT8 C ,AnH-
HOH BonHbI 250 HM. P860Ta BbIXO,A8 3neKTpOH8 p8BHa 4.5 3B. C'{UTaTb. '{TO
CBeT na,A8eT H8 nnaCTHHKY HOpMaJIbHO, a 3neKTpOH BbIneT8eT nepneH,AHKynSlpHo K
nnaCTHHKe.
,l(8HO: 1..=2,5,10- 7 M, A s =7,2.10- 19 ,ZJ;m.
HaHTH: 6.p.
Pew e H H e. EcnH C'lHT8Tb cHcTeMY (MeT8nnH'leCK8S1 nJI8CTHHKa, <}>OTOH H
3neKTpoH) 38MKHYTOH, TO 3aKOH coxpaHeHHSI HMnynbca MomHO 38nHcaTb TaK:
PJUt 
it
P2
Pue. 19.12
Pnal{ = PICOH' Pun = PI + 6,
Cne,nOBaTenbHO.
tJ.Pl = Pon - 6 = Pnn,
[',Ae PI H P2 - HMnynbCbJ <l>OTOH8 H 3neKTpOH8
(pHC. 19.12). CnpoeHpyeM seKTopHbIe SenH'lHHbJ
Ha HopManb K nnaCTHHKe, nOnY'l8eM:
-PI =P2- Pun' oTKy,na Pnn= PI + P2'
PICOH = P2 + Pon .
571

h
I1Mnynbc <l>OTOHR PI = i' MO,Aynb HMnynbcR aneKTpOHR P2=ml v l, r,Ae mt H
VI - MRCCa H MO.Aynb CKOpOCTH <l>oT03neKTpOHa.
MO,Aynb CKOpOCTH 3neKTpOH8 H8H,AeM H3 yp8SHeHHSI 8HHIlITeHH8 ,AnSI <l>OTO-
a<l><I>eKTR
V= J2( A. 1.
Tor.aa
P2 =  2m,[  -.4. ).
TaK KaK BHa'l8ne nnRCTHHKR nOKOHnaCb, TO HCKOMoe H3MeHeHHe liMnynb-
C8 nnaCTHHKH
60Pt = Pn:J - 6 = Pnn.
TIO,ACTRBJISISI BbIp8meHHSI ,AnSI PI H P2 B yp8SHeHHe ,AnSI 6op. nonY'lHM
h  ( he ') -12 M
60p = i + 2l11J. T -  f 60p = 3.7.10 Kr. .
7. 110HoxpoMaTHQecKHii nYQOK CBeTa C AnHHoH BonHbI 490 HM, nR,ARSI HOp-
ManbHO Ha 3epK8nbHYlO nnOIltRAKY, npOH3BO,AHT Ha Hee ,AaBneHHe 9,8. 10- 7 TIR.
K8Koe KOnH'leCTBO <l>OTOHOB n8,AReT Ha nOBepxHocTb nnOIlta,AblO 2,0 MM 2 38
SpeM.H, paBHoe 10 c?
.L(aHo: A=4,9.10- 7 M, p=9,8.10- 7 TIa, 6ot=10 c. 8=2,0,10- 6 M 2 .
HaHTH: N.
Pew e H H e. TIOCKOJlbKY nnOIltR,AK8 3epKRJIbH8S1. TO Bce H3nY'leHHe OTpR-
maeTCSI OT ee nOBepXHOCTH. B aTOM cnY'l8e Q>OTOH npH B3RHMO,AeHCTBHH C
BeIlteCTBOM nOBepXHOCTH MeHSIeT H8npaBneHHe CBoeH CKOpOCTH HR npOTHBono-
nomHoe. TIoaTOMY no BTOPOMY 3aKOHY HblOTOHR Ft = jJ06 B npoeKD;HH HR
HopM8nb K nnOIlta,AKe
hv
F6.t = 2N-,
e
r,Ae F - CHn8 CBeTOBoro ARBneHHSI, N - 'lHCnO <}>OTOHOS. Y'lHTbIB8S1. 'lTO
CHna ,ARBneHHSI CBeTa F = pSt r,Ae P - ,ARBneHHe CBeTa, nOJlY'lReM
P S60t = 2Nh oTK y "a N = pS6otA . N = 7 , 0 ,1015.
A'  2h ·
MOAY JIb H HMny JlbC H3MeH.HIOTC.H Ha 2 hv . a H3MeHeHHe MO.l(y JI.H HMny JlbCa
e
Bcex N «}>OTOHOB
60p = 2N hv = 2Nh .
e A.
572

20. H3HKA ATOMA
HBJIeHHSI, nO,nTBep2KJl;alOe CJI0HOe CTpoeHHe aTOMa. 0nL1T 3. Pe3epcl>0p]J:a
no pacceHBaHHIO a-'1aCT. epHasI MO]J:eJlh aTOMa.
llocTYJIaTLI H. Bopa. MO]J:eJIL aTOMa BO]J:opO]J:a no Bopy. OnLIT 1(. c1>paHKa
Hr. repua. TpY]J:HOCTH TeopHH Bopa.
IIcnycKaHHe H nOrJIOll(eHHe CBeTa aTOMOM. CneKTp 3HepreTH'leCKHX COCTOSl-
HHH aTOMa. llpOHcxoeHHe JIHHew.aTLlx cneKTpOB. HenpepblBHLle cneKTpLI.
CnoHTaHHoe H BLIHY)J;eHHoe H3JIY'1eHHe.
JIa3epLJ H HX npHMeHeHHe B TeXHHKe.
20.1. HBJIeHHH, no.n;TBep.n;alOHe CJIOHOe CTpoeHHe aTOMa
ATOMbI npeCTaBJlSlIOT C060H OqeHb npO'lHbIe CHCTeMbI, HeCOH3MepHMO 60-
Jlee YCTOHqHBbIe, qeM COCTaBJleHHbIe H3 aTOMOB MOJleKYJlbI. TaKHe BHelIIHHe
B03eHcTBHSI, KaK HarpeB, H3MeHeHHe )J;aBJleHHH, MOHbIe 3J1eKTpHqeCK1:Ie PR3-
p.HbI H T.,A.. npHBOlISlT JlHlIIb K He3HaqHTenbHbIM 1:I3MeHeHH.HM RTOMOB: OHH
MorYT HOHH30BaTbC.H. II03ToMY ,lI0 KOHa XIX B. aTOMbI Cq1:lTaJl1:l npOCTeHIII1:I-
MH He)J;eJlHMbIMH qaCTHIJ;aM1:I BeeCTBa. OHaKo nocne)J;YlOlQee pa3B1:IT1:Ie HaYK1:I
onpOBeprJlO 3TY TOqKY 3peH1:I.H. BbIJlo YCTaHOBJleHO, qTO aTOMbI npeCTaBJI.HIOT
C060H CJlOmHbIe 06pa30BaHH.H. 06 3TOM CB1:IeTeJlbCTBYIOT cJleYIO1:Ie <}>aKTbI.
1. 3JIeKTpOMarHHTHaSl TeopHSI CBeTa CBHeTeJlbCTByeT, 'tITO RTOMbI coep-
maT B ce6e 3J1eKTpHqeCKHe 3ap.HbI, cnoc06HbIe nepeMelQaTbC.H. .D:eHCTB1:ITeJlb-
HO, COrJlaCHO TeopHH MaKcBeJlna, 3J1eKTpOMarH1:ITHbIe BOJlHbI H3nyqalOT YCKO-
peHHO .l1BHmymHecSI 3ap.H.l1bI. Ho aTOMbI. KaK nOKa3bIBRJI onbIT. cnoc06Hbi 1:IC-
nYCKaTb CBeT, npH'leM He TOJlbKO OnT1:IQeCKOrO ,lI1:Iana30Ha - 3J1eKTpOMRrHHT-
HbIe BOJlHbI. CJleOBaTeJlbHO, B COCTaB aTOMOB BXO.HT YCKopeHHO BHmYlQHe-
C.H 3ap.HbI.
2. lIayqeHHe 3JIeKTponpOBO)J;HOCTH MeTaJlJlOB nOKa3ano, QTO B COCTaB aTO-
MOB BXO.HT OTp1:laTeJlbHO 3apHmeHHbIe qaCT1:IbI (3J1eKTpoHbI).
3. 3aKOH 3JIeKTpOJlH3a, OTKpbITbIH M. <I>apa.AeeM, CB1:IeTeJlbCTBOBaJl B nOJlb3Y
cyecTBOBaH1:I.H MeJlbqaHlII1:IX He)J;eJl1:lMbIX HOC1:ITeJleH 3J1eKTp1:lqeCKOrO 3ap.Ha.
IIp1:l 3neKTpOJl1:l3e 1 Monb JlI060ro n-BaJleHTHOro BemeCTBa nepeHOCHT 3a-
p.H,A Iql = nF = 96500 . n KJI, re F - nOCTO.HHHaSl ct>apa.l1eH. Ha O.l1HH HOH npH-
XOHTCH 3ap.H Iql = Fn = 1,6. 10- 19 n En. re N A - nOCTO.HHHaSl ABorapo.
N A
STY 3aKOHOMepHOCTb JlerKO nOH.HTb, eCJl1:I npHH.HTb, qTO 3ap.H
lei = 1,6. 10- 19 KJI .HBJI.HeTC.H MeJlbQaHlIIeH nOp1:IeH 3ap.H)J;a - 3J1eMeHTapHbIM 3a-
p.HOM. 3Haq1:lT. B COCTaB aTOMOB 3J1eKTpOJl1:lTOB BXO)J;.HT qaCT1:IIJ;bl C 3ap.HOM e.
4. B 1879 r. npH nponYCKaHHH 3J1eKTp1:l'leCKOrO TOKa qepe3 pa3p.HmeHHbIH
ra3 B TPy6Ke 6bIJlH 06HapymeHbI TaK Ha3bIBaeMbIe KaTO)J;Hble nyqH, npeCTaB-
JI.HIO1:Ie C060H qaCTHIJ;bI, JleT.HHe OT KaToa K aHOY. OHH OTKJlOH.HIOTC.H B
3J1eKTp1:lqeCKOM 1:1 MarHHTHOM nOJl.HX, alOT TeHb <}>HrypHoro aHO,AR Ha CBeT.H-
eMC.H CTeKJle 1:1 T.. B 1897 r. .D:m. TOMCOH OKa3an, qTO KRTOHbIe JlyqH
npeCTaBJI.HIOT C060H nOTOK 3J1eKTpOHOB. IIp1:lq1:lHR 1:IX nO.HBJleH1:I.H - 60M6ap-
573

JJ;llpOBKa KaTOJJ;a llOHaMll pa3pe>KeHHOrO ra3a, YCKOpeHHbIMll B 3neKTpH"IeCKOM
none Me>KJJ;Y KaTOJJ;OM II aHOJJ;OM. CJleJJ;OBaTeJlbHO, B COCTaB aTOMOB KaTOJJ;a
BXOJJ;.HT 3J1eKTpOHbI.
5. B. PeHTreH OTKpbIn TaK Ha3bIBaeMbIe X-nyqH (peHTreHoBcKHe J1Y'lH).
KOTopbIe B03HHKaJlH npH TopMomeHHH 6bICTpbIX 3J1eKTpOHOB Y aHOJJ;a. Cye-
CTBOBaHHe peHTreHOBCKllX nY"IeH CBH,AeTenbCTBOBano 0 KaKHX-TO npoeccax,
npOHCXOJJ;.HHX BHYTPH aTOMa, 0 CJlOmHOM CTpoeHHH caMoro aTOMa.
6. PeIlIalOee JJ;OKa3aTenbCTBO Toro, qTO B COCTaB JlI06oro aTOMa BXOJJ;HT
H nOJIO>KHTeJlbHO, H OTpHaTenbHO 3ap.HmeHHbIe qaCTHbI, nOJlY'IHn 4>paHIJ;Y3c-
KllH y"IeHbIH A. BeKKepeJlb. B 1896 r. OE 06HapymHJI HBneHHe eCTecTBeHHoH
pa,IJ;HOaKTHBIIOCTH, npH KOTOpOM H3 aTOMOB BbIJleTalOT 3J1eKTpOHbI H nOJlO>KH-
TenbHO 3ap.H>KeHHbIe "IaCTHbI. nonyqHBIlIHe H83BaHHe a-"I8CTH.
TaKHM 06pa30M. Bce npHBeIl.eHHbIe 4>aKTbI. a TaKme MHorHe JJ;pyrHe .HBJle-
HllH CBll,l1eTenbCTBYIOT 0 CnO>KHOH CTpYKType aTOMa. B pe3YJlbTaTe B03HHKJla
He06xo,AHMOCTb C03JJ;aHHH MOJJ;eJlH 8TOMa.
nepBYIO MOJJ;enb aTOMa npeJJ;nO>KHn Am. TOMCOH. COrJlaCHO 3TOH Mo,neJlH.
aTOM npeJJ;CTaBJI.HeT C060H nonomHTenbHO 3apH>KeHHbIH IlIap, BHYTPH KOToporo
HaXO,AHTCH TaKoe KOJlH'leCTBO 3J1eKTpOHoB, qTo6bI aTOM B eJlOM 6bIn 3J1eKT-
pH"IeCKH HeHTpaJleH. 8TH 3neKTpOHbI MorYT cOBepIlIaTb Kone6aHHH, qTO n03BO-
n.HeT 06'bHCHHTb H3J1Y"IeHHe aTOMOM 3neKTpOMarHHTHbIx BOJIH.
8Ta Mo,nenb HMena PHJJ; HeJJ;OCT8TKOB.
OHa He MOrJla 06'b.HCHHTb HaJlH'tlHe 60JlbIlIOrO '1HCJla JlHHHH B cneKTpax
aTOMOB.
Mo,neJlb He n03BOn.Hna 06'b.HCHHTb cneKTpaJlbHbIe 3aKOHOMepHOCTH B cneKT-
pe aTOMa Bo,nopo,na (CM.  20.9), HaHJJ;eHHbIe 3MnllpH"IeCKH.
Mo,nenb TOMCOHa He ,naBana B03MO>KHOCTH nOH.HTb, 'leM onpe,neJl.HIOTCH pa3-
MepbI aTOMOB.
rJlaBHbIH me 4>aKTOp. 3aCTaBHBIlIllH OTKa3aTbC.H OT Mo,neJlH TOMCOHa.
COCTO.Hn B TOM. "ITO 3Ta MOLIenb OKa3anaCb B nonHOM npOTHBOpe"IHll C onbI-
TOM 8. Pe3epcpop)J;a.
20.2. OnbIT Pe3ep4>op,ZJ;a no pacceHBaIlHIO a-'1aCTHl.\
5
3
2
Puc. 20.1
574
BHYTPll repMeTH'lHOH KaMepbI (pllC. 20.1). B
KOTOpOH 6bln C03LIaH BbICOKHH BaKYYM. HaXOLIHn-
C.H CBHHOBbIH KOHTeHHep 2 C pa,nlloaKTHBHbIM
3neMeHTOM 3, HcnYCKaBIDHM a-"IaCTHbI. Qepe3
MaJIeHbKOe OTBepCTHe B CTeHKe KOHTeHHepa Y3-
KHH nY'IOK 4 a-qaCTll n8AaJI nepneH,nHKynHpHo
Ha nOBepXHOCTb MeTanJlH"IeCKOH (30noToH) 4>onb-
rH 5, HMeBllieH TOJlHHY OKOJlO 1 MKM.
PerHCTpall.H a-"IaCTH npOH3Bo,nHnaCb no
BcnbIlliKaM CBeTa (CHHTIfJlJlJlH.HM), BbI3bIBae-
MbIM HMH Ha 3KpaHe 6, nOKpblTOM nIOMHHocpO-
pOM. 8KpaH 6bIn YKpenneH nepe,n 06'beKTHBOM H8
Kopnyce MHKpOCKona 7, C nOMOblO KOToporu
Bll3Y8J1bHO Ha6nlO,AaJIH ClI.llHTHnnHlI.llll H no,nC'lH-
TbIBanH HX qllCno. TaK onpeJJ;enHnH KOnH'IeCTBO

a-'t{aCTH, )J;BHmYHXC.H B aHHOM HanpaBJleHHH nocJle HX B3aHMOeHCTBH.H C
aTOMaMH BeeCTBa (}>OJlbrH.
MHKpOCKOn BMeCTe C 3KpaHOM MOr BpalQaTbC.H BOKpyr BepTHKaJlbHOH OCH,
npOXO.HlQeH qepe3 eHTp KaMephI, re nOMeaJIaCb (}>OJlbra. 9TO aBaJlO B03-
MOmHOCTb 06HapymHBaTb H nOCqHThIBaTh a-qaCTHbl, paCCe.HHbIe nO,A pa3HbI-
MH yrJlaMH npH HX npOXOmeHHe qepe3 (}>OJlhry.
OnbITbI nOKa3aJlH, qTO: 1) 6onLwHHcTBO a-'IaCTH npOXOHT He OTKnOHH-
HCL; 2) C YBenH'IeHHeM yrna pacceHBaHHH 'IHCnO OTKJlOH.HIOHXCH a-qaCTH
pe3KO YMeHLwaeTCH; 3) HMelOTCH OTeJlhHbIe a-qaCTHbI, pacceHBaeMbIe Ha
O'leHL 60nLwHe yrnLI (== 180°), T.e. oT6pacbIBaeMbIe aTOMaMH Ha3a.
1.JeM me 06'b.HCHHTb pacceHBaHHe a-qaCTHu? BnOJlHe eCTeCTBeHHO npe.nnono-
mHTb, qTO OHO npOHCXOHT BCJleCTBHe B03,neHCTBHH Ha HHX qaCTH. H3 KO-
TOpbIX COCTOHT Bel1.\eCTBO. Macca a-qaCTHIJ, nO'lTH B 8000 pa3 60JlbIlle MaCCbI
3J1eKTpOHa. 3HaqHT, H3MeHeHHe HanpaBJleHHH BHmeHH.H a-qaCTHIJ, BhI3BaHO OT-
HIOb He 3J1eKTpOHaMH, BXOSlIQHMH B COCTaB aTOMOB 30JlOTOH (}>OJlbrH, a no-
JlOmHTeJlbHO 3ap.HmeHHbIMH qaCTHaMH aTOMa.
BOJlbIIIHe OTKJlOHeHHH, OqeBH)J;HO, Ha6J1IOaIOTC.R. Tora, Kora a-qaCTHa
6J1H3KO nO)J;XO)J;HT K nOJlOmHTeJlbHO 3ap.HmeHHOH qaCTH aTOMa: Ha MaJlbIX pac-
CTO.HHH.HX CHJlbI OTTaJlKHBaHH.H Memy OHOHMeHHO 3ap.HmeHHOH a-qaCTHeH H
qaCTHaMH aTOMR BeJlHKH, H 3TO BbI3bIBaeT pe3Koe OTKJlOHeHHe a-qaCTH OT
HX nepBOHaqaJlbHOro HanpaBJleHH.H.
Pe3KHe OTKJlOHeHH.H Ha6J1IOanHCb BeCbMa peKO. II03TOMY Pe3ep(}>op,A C,Ae-
JlaJl 3aKJlIOqeHHe, qTO JlHIllb OqeHb MaJla.H qaCTb aTOMa npeCTaBJI.HeT npenHT-
CTBHe 1!JI.H npoxom.neHHH a-qaCTHII.: BepO.HTHOCTb nOna,AaHH.H a-qaCTHII.bI B Ma-
JlYIO no 06'beMY 3ap.HmeHHYIO qaCTb aTOMa HeBeJlHKa.
Ha OCHOBe aHaJIH3a 3THX onbITOB Pe3ep(}>op npeJlo-
mHJI CBOIO MOeJlb aTOMa - nnaHeTapHYIO MOAenL (no-
06Hyro CTpoeHHIO nJlaHeTHOH CHCTeMbI): aTOM COCTOHT H3
nOJlOmHTeJlbHO 3ap.HmeHHOro .Hpa, B KOTOpOM cocpeOTOqe-
Ha OCHOBHaH Macca aTOMa. 3ap.H .Hpa q=Zlel, re Z -
nopHKoBbIH HOMep 3J1eMeHTa B Ta6nHe MeHeJleeBa, e -
3apH 3J1eKTpOHa. Pa3MepbI .Hpa 10- 15 _ 10- 14 M. BOKpyr
.Hpa no 3aMKHYTbIM op6HTaM BHmeTC.H Z 3J1eKTpOHOB
(pHC. 20.2). AToM B enOM HeHTpaJIeH. 9J1eKTpOHbI 06H3a-
TeJlbHO )J;BHmYTC.H, HHaqe H3-3a KYJlOHOBCKHX CHJI npHTHme-
HH.H Mem1!y .HpOM H 3J1eKTpOHaMH OHH ynaJIH 6bI Ha .HpO
H aTOM npeKpaTHJI 6bI CBoe CYlQeCTBOBaHHe.
IIpH BceH CBoeH y6e)J;HTeJlbHOCTH nJlaHeTapHa.H MOeJlb aTOMa OKa3aJlaCb
BHYTpeHHe npOTHBOpe'lHBoil. IIo 3aKOHaM 3J1eKTpo.nHHaMHKH BHmYI.l!HHC.H YC-
KopeHHO 3ap.H H3J1yqaeT 3J1eKTpOMarHHTHbIe BonHbI. CJleOBaTenbHO, aTOM OJl-
meH HenpepLlBHO H3J1yqaTb 3J1eKTpOMarHHTHhIe BOJlHbI. M3J1yqeHHe CBH3aHO C
YMeHbIlleHHeM 3HeprHH 3J1eKTpOHa B nOJle .Hpa. CJleCTBHeM 3Toro OJlmHO
.HBHTbC.H YMeHbIlleHHe CKOpOCTH ,ABHmeHH.H H qaCTOThI BpaeHH.H 3J1eKTpOHOB
BOKpyr .Hpa. ECJlH 'laCTOTa HenpepbIBHo H3MeH.HeTC.H, TO cneKTp H3nyqeHHH
aTOMa AoneH 6LITL cnnOWHLIM. YMeHbIlleHHe CKOpOCTH 3neKTpOHa OJlmHO
v
Puc. 20.2
20.3. H.ZJ;epHaH MO.ZJ;eJIb aTOMa
575

npHBeCTH K TOMY, qTO 3J1eKTpOH 6yeT )J;BHraTbCSI no CnHpaJlH K .HPY H B
TeqeHHe KopOTKoro BpeMeHH yna,AeT Ha Hero, T.e. B 3TOH MOeJlH aToM OJI-
meH 6bITb HeYCTOH'IHBbIM.
3TO HaXOHTC.H B nOJIHOM npOTHBOpeqHH C 3KcnepHMeHTaJlbHbIMH cpaKTaMH,
KOTopbIe CBHeTeJlbCTBYIOT 0 TOM, qTO:
a) aTOM .HBJI.HeTC.H OCTaTOqHO YCTOHtIHBOH cHcTeMoH;
6) aTOM H3J1YtIaeT 3JIeKTpOMarHHTHbIe BOJlHbI nHIIIb npH onpeeJleHHbIX YC-
JlOBH.HX, a He HenpepbIBHO;
B) cneKTpbI H3J1yqeHHSI aTOMOB .HBJI.HIOTCSI JlHHeHqaTbIMH.
3TH npOTHBOpeqH.H B03HHKJIH nOTOMY, tITO K 3J1eKTpOHaM B aTOMax npH-
MeH.HJlH 3aKOHbI KJlaCCHqeCKOH CPH3HKH, a B MHKpoMHpe eHCTBYIOT CBOH 38-
KOHbI t OTJlHtIHble OT 3aKOHOB MaKpOMHpa.
20.4. IIocTYJIaTLI Bopa
IIepBbIM npH3HaJl HeB03MomHOCTb npHMeHeHH.H 3aKOHOB KJlaCCHqeCROH cpH-
3HKH K aTOM aM aTCKHH yqeHbIH HHJlbC Bop. Bop BBeJI 3J1eMeHTbI KBaHTOBOH
TeopHH B MOeJlb aTOMa Pe3epcpopa Ii B 1913 r. C03,AaJl HeKJlaCCHtIeCRYIO
TeopHIO aTOMa. B OCHOBe 3TOH TeopHH JlemMa He.H CBSl3aTb B e)J;HHoe eJIoe
TPH pe3YJIbTaTa, nOJIyqeHHbIe B cpH3HKe K TOMY BpeMeHH:
1) 3MnHpHtIeCKHe 3aKOHOMepHOCTH JlHHeHqaTOrO cneKTpa aTOMa Boopoa;
2) .HepHYIO MOeJlb aTOMa Pe3epcpopa, He onYCKalOl.QYIO RJlaCCHtIeCKOrO
HCTOJlKOBaHHSI ;
3) KBaHTOBbIH xapaKTep H3J1yqeHH.H H nOrJlOmeHH.H.
,D;n.H pemeHH.H 3TOH 3aaqH Bop, COXpaH.HSI KJlaCCHqeCKHH no.AXO K onH-
caHHIO nOBeeHH.H 3J1eKTpOHa B aTOMe, BbIBHHYJI Ba nOCTYJlaTa, KOTopbIe Ha-
3bIBaIOTC.H nOCTYJlaTaMH Bopa.
Cpa3Y me 3aMeTHM, qTO cpH3HtIeCKHH CMbICJI 3THX nOCTYJIaTOB He TOJlbKO He
Mor 6bITb 06'b.HCHeH B KJlaCCHtIeCKOH cpH3HKe, HO, 60Jlee Toro, HaXOHJlC.H B rJIY-
60KOM npOTHBOpeqHH C KJIaCCHtIeCKHM OnHC8.HHeM BHmeHH.H 3J1eKTpOHa B aTOMe.
IIOJIHHHbIH CMbIcn H 3HaqeHHe nOCTYJlaTOB Bopa BCKpbIJlHCb n03Hee, nocJle
C03aHH.H KBaHTOBOH MexaHHKH.
nepBLIH nOCTYJlaT (nOCTYJlaT CTaHoHapHbIX COCTO.HHHH): cyU{ecm6YlOm cma-
UOliaplibte cocmOSlIiUSl amO.Ma. liaxoiJSlCb 6 KomOpbtX Oli lie U3Jl.Yllaem 3liep.
ZUU. 3THM CTaHOHapHbIM COCTO.HHH.HM COOTBeTCTBYIOT onpeeJleHHbIe CTaHO-
HapHbIe Op6HTbI, no ROTOpbIM BHmYTCSI 3J1eKTpOHbI. IIpH BHmeHHH no CTa-
HOHapHbIM op6HT8M 3J1eKTpOHbI He H3J1yqa1OT 3J1eRTpOMarHHTHbIe BOJlHbI. He-
CMOTp.H Ha YCKopeHHoe lIBHmeHHe. B KamlIOM CTailHOHapHOM COCTO.HHHH aTOM
06J1aaeT onpeeJleHHbIM RBaHTOBaHHbIM 3HaqeHHeM 3HeprHH.
BTOpol1 nocryJIaT (npaB1:IJI0 qaCTOT): npu nepexoiJe amo.Ma U3 oOliOZO cmauo-
napliOzo COCmOJlIiUSl 6 iJpyzoe U3Jl.YllaemCSl UJl.U noZJWU{llemcSl K6alim 3liepzuu.
ATOMbl H3J1yqa1OT Ii nOrJlOlQalOT 3HeprHIO HCKpeTHbIMH nopHSlMH - RBaH-
TaMH t 3HaqeHHe KOTOpbIX W o = hv paBHO pa3HOCTH 3HeprHH Tex CTaHOHapHbIX
COCTO.HHHH, Memy KOTOpbIMH npOHCXOHT aHHbIH nepexo, T.e.
(WOk - WOn)
hVkn = WOk - WOn' HJlH Vkn = ,
h
re V - qaCTOTa H3JIyqeHH.H; WOn H WOk - 3HeprHH CTHOHapHbIX COCTO.HHHH.
576

AJI.H 06.HCHeHH.H 3MnHpHqeCKHX 3aKOHOMepHOCTeH B CneKTpe aTOMa BOO-
poa 3TH )J;Ba nOCTYJlaTa 6bIJlH )J;OnOJlHeHbJ npaBHJlOM KBaHTOBRHH.H Op6HT, KO-
TOpOe HHOr)J;a Ha3bIBaIOT TpeThHM nOCTYJlaTOM H. Bopa.
TpeTHH nOC-ryJIaT (npaBHJlO KBaHTOBaHH.H Op6HT): B CTan;HOHRpHOM COCTOSl-
HHH RTOMa 3J1eKTpOH. BHmYI.QHHC.H no KPyroBoH op6HTe. OJlmeH HMeTh KBRH-
TOBRHHbIe (,nHCKpeTHbIe) 3Ha'tleHH.H MOMeHTa HMnYJlbCR, KOTopbIe y,nOBJleTBOp.H-
JOT YCJlOBHIO
h
mvr n = n- = nli
21t '
r)J;e m - Macca 3J1eKTpOHa, v - CKOpOCTb 3J1eKTpOHa Ha n-H op6HTe, r n -
pa)J;HYC CTan;HOHapHOH Op6HThI, n = 1, 2, 3,... - HOMep Op6HTbI 3J1eKTpOHa.
20.5. Mo.u;eJlL aTOMa Bo.u;opo.u;a no Bopy
Rcxo,A.H H3 3THX nOCTYJlaTOB H HCnOJlb3YH nnaHeTapHYIO MOeJlb CTpOeHH.H
aTOMR, H. Bop pa3pa60TaJl KOJlHqeCTBeHHYIO TeopHIO aTOMa Bonopona. OH pac-
CqHTaJl PHYCbI CTRn;HOHRpHbIX Op6HT 3J1eKTpOHa B RTOMe Bo,n;opo,n;a H BbIqHC-
nHn COOTBeTCTBYIOIl.\He HM 3HaqeHH.H 3HeprHH.
PaClIeT PMHYCOB Op6HT. 3J1eKTpOH ,n;BHmeTC.H BOKpyr .H)J;pa B aTOMe BO,n;OpO-
,n;a no KPyroBoH op6HTe no,n; ,n;eHcTBHeM KYJlOHOBCKOH CHJlbI, KOTopaH C006lQReT
eMY n;eHTpOCTpeMHTenhHoe YCKopeHHe. no BTOPOMY 3RKOHY HhlOTOHR F = mii.
v 2
IJ;eHTpocTpeMHTeJlbHOe YCKopeHHe a 14c = - 3neKTpOHY co06ReT KYJlOHOB-
r
e 2
KR.H CHJla npHT.HmeHH.H co CTOpOHhI .Hpa F = . CJleOBRTeJlhHo.
41t€or2
mv 2 e 2 2 e 2
- OTKY)J;a mv =-
r - 41t€or2 ' 41tE o r
(20.1 )
COrJlaCHO III nOCTYJlaTY Bopa:
h
mvr = n -, OTCIO,n;a
21t
nh
V=-.
21tmr
(20.2)
mn 2 h 2 e 2
R3 ypaBHeHHH (20.1) H (20.2) nOJlyqHM -.
41t 2 m 2 r 2 41tE o r
n 2 h 2 E
r= 0
1tme 2
BbIpa.meHHe ,n;nH PRHYCOB pR3pelIleHHbJX CTB.:u;HOHapHbIX Op6HT 3J1eKTpOHa B aTO-
Me BO,AOpoa. 3eCb n - HOMep Op6HTbI, pa,n;HYC KOTOpOH r, Eo - 3J1eKTpHqeC-
KR.H nOCTOHHHRH, h - nOCTO.HHHR.H nJlaHKR, m - MRCCR 3J1eKTpOHR, e - 3R-
OTKY)J;a
(20.3)
pH 3J1eKTpOHa.
577

Puc. 20.3
3
MbI BH,lJ;HM. qTO pa,[l;HycLI cTaHoHapHLlx
YCTOHqHBLIX Op6HT B03pacTalOT nponopHo-
HaJiLHO KBa,z:r;paTaM HOMepoB Op6HT (pHC. 20.3):
rl: r2: r3'" = 1 2 : 2 2 : 3 2 If T.,lJ;. ECJlH aJleKTpOH B
aTOMe BO,lJ;OpO,lJ;a HaXO,lJ;HTC.H Ha O,lJ;HOH H3 CTa-
HoHapHbIx Op6HT, TO aTOM 06J1a,lJ;aeT onpe,lJ;e-
JleHHbIM 3HaqeHHeM 3HeprHH, onpe,lJ;en.HeMOH
3HeprHeH 3J1eKTpOHa:
mv 2 e 2
W=w +W =---
K n 2 4
1tE o r
(3HaK "-" nepe,lJ; nOTeHHaJlbHOH 3HeprHeH 03HaqaeT, qTO 3a W n = 0 npH-
H.HTO TO 3H8QeHHe, KOTopoe COOTBeTcTByeT r= 00). IIo,lJ;cTaBHB B 3TY cpopMyny
3HaqeHH.H v H r H3 cpOPMYJI (20.2) H (20.3), nOJlyqHM:
TaKHM 06pa30M, 3HeprH.H 3J1eKTpOHa, HaXO,lJ;.HIr(erocn Ha n-H op6HTe,
me 4
W=
8E 2 n 2 h 2 .
o
R3 3TOH cpOPMYJlbI BH,lJ;HO, qTO 3HaqeHHSI
3HeprHH aTOMa BO,lJ;OpO,lJ;a KBaHTOBaHbI H, qeM
60nbme n, TeM 60Jlbme 3Heprlf.H W n . JJ:Jln Ha-
rn.H,lJ;HOrO npe,ACTaBJleHH.H B03MomHblX 3Hepre-
THqeCKHX COCTonHHH aTOMOB HCnOJlb3YIOTC.H
3HepreTHqeCKHe ,lJ;HarpaMMbI, Ha KOTOpbIX Kam-
,lJ;oe CTaHOHapHoe COCTO.HHHe aTOMa OTMeqaeT-
C.H ropH30HTaJIbHOH JlHHHeH, Ha3bIBaeMoH 3Hep-
reTHqeCKHM ypOBHeM (pHC. 20.4). HHme Bcex
Ha ,lJ;HarpaMMe pacnOJlaraeTC.H 3HepreTHqeCKHH
ypOBeHb, cooTBeTcTBYIOHH OCHOBHOMY COCTon-
HHIO (COCTO.HHH.H C MHHHManbHOH aHeprHeH).
SHepreTHqeCKHe YPOBHH B036ym,lJ;eHHbIx COCTon-
HHH pacnonaralOTC.H Ha,lJ; OCHOBHbIM ypOBHeM Ha
paCCTOSlHHSlX, nponopHOHaJlbHbIX pa3HOCTH 3HeprHH B036ym,lJ;eHHOrO H OCHOB-
Horo COCTO.HHHH. IIepexo,lJ;bI aTOMa H3 O,lJ;HOrO COCTO.HHH.H B ,lJ;pyroe H306pama-
IOTC.H BepTHKaJlbHbIMH nHHH.HMH Mem,lJ;y COOTBeTCTBYIOHMH ypOBH.HMH Ha ,lJ;H-
arpaMMe. HanpaBJleHHe nepeXO,lJ;a 0603HaqaeTCSI cTpenKoH. IIpH nepeXO,lJ;e 3J1eK-
TpOHa C k-H Ha n-IO Op6HTY H3J1YQaeTCSI cpOTOH C qaCTOTOH
m n 2 h 2
w=-
2 4n 2 m 2 r 2
W
n=oo
n=4
n=3
n=2
n=1
Puc. 20.4
e 2
n 2 h 2 rrm 2 e 4
8n 2 mn 4 h 4E
=
me 4
&n2h2 .
e 2 nme 2
41tEon2h2Eo
41tE o r
(20.4)
W k - W n me 4 ( 1 1 J
v kn = h = 8Eh3 n 2 - k 2 .
CpaBHHBa.H 3TO BbIpameHlfe C 3MnHpHqeCKOH cpopMynoH
vkn = R( :2 - :2 ].
578
(20.5)

me 4
BH,AHM, qTO nOCTO.HHHaH PH,A6epra R - -' Cne,AOBaTeJIbHO, B opMYJIe (20.5)
- 8Eh3 '
k - HOMep Op6HTbI, C KOTOpOH npOHCXOHT nepexo 3J1eKTpOHa B RTOMe, n -
HOMep Op6HTbI, Ha KOTOPYIO nepeXO,AHT 3J1eKTpOH.
20.6. OnLIT <I>paHKa H repa
H3 MO,AeJlH aTOMa H. Bopa CJle,AOBaJlO, qTO MHHHMaJlbHa.H 3HeprHSI, KOTO-
PYIO MomeT nOrJlOTHTb aTOM npH B3RHMO,AeHCTBHH C .l{pyrHM aTOM OM HJlH 3J1eK-
TpOHOM, paBHa pa3HOCTH 3HeprHH B ero nepBOM B036ym,AeHHOM If OCHOBHOM
COCTO.HHH.HX. CJle,AOBaTeJIbHO, ecnH nponYCKaTb Qepe3 ra3 nyqoK 3JIeKTpOHOB.
B KOTOpOM 3HeprH.H KRm,AOrO 3JIeKTpOHR MeHbIIIe pR3HOCTH 3HeprHH RTOMR B
nepBOM B036ym,AeHHOM H OCHOBHOM COCTO.HHH.HX, TO npH CTOJlKHOBeHHH C aTO-
MRMH 3TH 3J1eKTpOHbI He ,AOJlmHbI Tep.HTb 3HeprHIO. ECJIH me 3HeprH.H 3J1eKT-
pOHOB paBHa YKa3aHHOH pa3HOCTH, TO OHH npH cOY,AapeHHHX C aTOMRMH CMO-
rYT OT,AaBRTb CBOIO KHHeTHqeCKYIO 3HeprHIO HR B036ym,AeHHe aTOMOB (Heynpy-
roe CTOJlKHOBeHHe).
BnepBbIe Heynpyme CTOJlKHOBe-
HHSI 3J1eKTpOHOB C aTOMRMH PTY-
TH 6bInH 06HRpymeHbI B onbITRX
.D:. <t>paHKa Hr. repu,R B 1913 r.
B 3THX onbITax npHMeH.HJlRCb CTeK-
JI.HHHa.H Tpy6KR, 3RnOJlHeHHaH nR-
pRMH pTYTH (pHC. 20.5). KRTO,A K
HarpeBReTCH 3J1eKTpWleCKHM TOKOM.
8neKTpoHbl, BbIJIeTeBIIIHe H3 KRTO,AR,
YCKOp.HIOTC.H 3J1eKTpWlecKHM nOJleM
Me>K.AY KaTO,AOM H ceTKOH C. Mem-
I1.Y ceTKOH C H B.HO,AOM A 3JIeKTpo-
RbI TOpM03SITC.H 3JIeKTp1{qecKHM nOJIeM, C03,AR-
BaeMbIM 6RTapeeH B 2 . I1oTeHu,HaJI ceTKH C
BbIIIIe nOTeHanR B.HO,AB HR 0,5 B.
B onbITRX .D:. <t>pRHKa Hr. repu.a HC-
CJle,AOBaJlaCb 3aBHCHMOCTb CHnbI TOKa B
u,enH RHO,AB OT HRnpHmeHH.H Mem,Ay KaTO-
,AOM H ceTKOH. TaK KaK nOTeHu.Han ceT-
KH 6bIJI BbIllie nOTeHu,HaJla aHO,Aa Ha 0,5
B, TO .l{OCTHrHYTb aHO,Aa MOrJIH TOJIbKO
3neKTpOHbI, KHHeTHQeCKRH 3HeprH.H KOTO-
pbIX npeBbIIIIaJlR 0,5 3B.
Pe3YJlbTRTbI, nOJl}"IeHHbIe B O,AHOM H3
onbITOB, npe,ACTaBJleHbI Ha pHC. 20.6. I10Ka
Hanp.HmeHHe Ha ceTKe He npeBOCXO,AHT
4,9 B, CHna TOKa B u.enH nOCTOHHHa.
U. B Pe3Koe YMeHbIIIeHHe CHnbI TOKa B
RHO.nHOH uenH aHO,Aa npH .nOCTHmeHHH Ha-
np.HmeHH.H Mem,Ay KRTO,AOM H ceTKoH, paB-

Puc. 20.5
I.MA
300
250
200
150
100
50
0 5
Puc. 20.6
579

HOM 4,9 B, 3aCTaBJI.HeT c,neJlaTb BbIBO,n, qTO 3J1eKTpOHbI C KHHeTH'ieCKOM 3Hep-
rHeH 4,9 3B nOJlHOCTblO Tep.HIOT ee npH cOY)J;apeHHH C BTOMaMH PTYTH B6J1H-
3H ceTKH. TaK KaK KHHeTHqeCKa.H 3HeprH 3neKTpOHOB nocJle TaKoro coy,nape-
HH.H OKa3bIBaeTC.H 6J1H3KOH K 0, ,name cJla60e BCTpeqHOe nOJle Mem,ny ceTKOH
H aHo,nOM He nponYCKaeT HX K aHO,ny H CHJlB TOKa B enH YMeHbwaeTCSI.
B COOTBeTCTBHH C TeopHeH H. Bopa MomHO c,neJlaTb BbIBO)J;, qTO pa3HOCTb
3HeprHH Mem,ny nepBbIM B036ym,neHHbIM CTaHOHapHbIM COCTO.HHHeM aTOMB PTY-
TH 11 ero OCHOBHbIM COCTOSlHHeM paBHa 4,9 3B: W 2 - WI =4,9 3B.
3To nO.ATBepaeTC.H TaK:me Ha6J1IO,naeMbIM npH aTOM HcnycKaHHeM napaMH PTY-
(W 2 - Wd
TH CBeTa C 'iaCTOTOH v 21 = , KOTOpoe npoHcxo,nHT npH caMOIIpoH3BOJlb-
h
HOM nepexo,ne aTOMOB PTYTH H3 B036ym,neHHoro COCTO.HHH.H 2 B HeB036ym,neHHoe 1.
20.7. TpYAHOCTH TeopHH Bopa
Mo,neJlb aTOMB H. Bopa 6bIJla KpynHbIM warOM B pa3BHTHH aTOMHOH «}>H3H-
KH H .HBHJlaCb BamHbIM 3TanOM B C03,naHHH KBaHTOBOH MexaHHKH. O,nHaKo 3Ta
Mo,neJlb 8TOM8 BHYTpeHHe npOTHBOpeqHBa: C O,nHOH CTOpOHbI, npHMeH.HeT 3aKO-
HbI KJlaCCHqeCKOH «}>H3HKH, c ,npyroH - OCHOBbIBaeTC.H Ha KBaHTOBbIX nocTyna-
Tax. C ee 110MOblO y,naJIOCb 06'b.HCHHTb OCHOBHbIe 3aKOHOMepHOCTH B cneKTpax
aTOMOB Bo,nopo,na H BO)J;opo,nono,n06HbIX CHCTeM H BbIqHCJlHTb qaCTOTbI cneKT-
paJIbHbIX JlHHHH. OCTaBaJlOCb. o,nHaKO, He.HCHbIM, OT 'iero 3aBHCHT HHTeHCHB-
HOCTb H3J1Y'ieHH.H Tex HJlH HHbIX 'iaCTOT. Be3 OTBeTa OCTaJlC.H Bonpoc, nOqeMY
COBepWaIOTC.H Te HJlH HHbIe nepeXO)J;bI. Cepbe3HbIM He,nOCTaTKOM Mo,nenH aTOMa
Bopa 6bIJla HeB03MomHOCTb OIIHcaTb C ee nOMOblO aTOM renH.H - O,nHH H3
npOCTeMwHX aTOMOB, Henocpe)J;CTBeHHO cJle,nYIOHH 3a aTOMOM Bo,nopO)J;a.
Mo,neJlb aTOMa Bopa .R:BHJlaCb nepexo,nHbIM 3TanOM Ha nYTH K C03,naHHIO
COBpeMeHHOH TeopHH aTOMHbIX 11 .H,nepHbIX .HBJleHHH. B 1925-1927 rr. Ha
CMeHY Mo,neJlH aTOMB Bopa npHWJla KBaHTOBaa MexaHHKa. KOTOpa.H BHJlaCb
CTporOH HenpOTHBOpeqHBOH TeopHeH H HMeJla CBOH c06CTBeHHbIH MaTeMaTHqeC-
KHH annapaT.
20.8. llcnycKaHHe H nOrJIOeHHe CBeTa aTOMOM
PaCCMOTpHM aTOM Bo,nopo,na.
COrJlaCHO TeopHH Bopa, npH )J;BH:meHHH 3J1eKTpOHa 110 6J1HmaHmeH K .H,npy
cTaHoHapHoH op6HTe aTOM HaXO)J;HTC.H B OCHOBHOM COCTO.HHHH, .HBJI.HIOeMC.H
HaH60Jlee YCTOHqHBbIM. B OCHOBHOM COCTO.HHHH aTOM MomeT HaXO)J;HTbCSI Heo-
rpaHHqeHHO ,noJlroe BpeM.H, IIOCKOJlbKY 3TO COCTO.HHHe COOTBeTcTByeT HaHMeHb-
meMY B03MomHOMY 3HaqeHHIO 3HeprHH aTOMa.
Kor,na 3J1eKTpOH ,nBHmeTC no KaKOH-JlH60 )J;pyroH H3 pa3pemeHHbIX op-
6HT, COCTO.HHHe aTOMa Ha3bIBaeTC.H B036y)];eHHLIM H .HBJI.HeTC.H MeHee YCTOH-
'iHBbIM. qeM OCHOBHoe COCTO.HHHe. 'lJepe3 He60JlbWOH npOMemYTOK BpeMeHH (no-
p.H,nKB 10- 8 c) aTOM caMOIIpOH3BOJlbHO nepexo.nHT H3 B036ym,neHHoro COCTO.H-
HH.H B OCHOBHoe, H3J1yqa.H npH 3TOM KBaHT 3HeprHH (pHC. 20.4):
hVkn=W k - W n .
580

TaK KaK B B036ym.l{eHHOM COCTO.HHHH 3HeprH.H aTOMa 60JlbWe, qeM B OC-
HOBHOM, TO npOH3BOJlbHO aTOM B B036ym.l{eHHOe COCTO.HHHe nepeHTH He MO-
meT. Cnoc06bI B036ym.l{eHH.H aTOMOB BeeCTBa MorYT 6bITb CRMbIMH pa3J1Hq-
HbIMH: y.l{ap 06 aTOM KaKOH-TO qaCTHbl, XHMHQeCKHe peaKHH, B03.l{eHCTBHe
CBeTa H T ..l{. Ho OHH OKa3bIBaIOTC.H 3cpcpeKTHBHbIMH .l{JI.H B036ym)J;eHH.R TOJlbKO
TOr.l{a, Kor)J;R nOCTaBJI.HIOT 3HeprHIO KBaHTaMH, KOTopbIe B COCTO.HHHH B036y-
.l{HTb ,AaHHble aTOMbI. ECJlH 3Ta 3HeprH.H He.l{OCTRTOqHa .l{JlSI nepe6pocKH aTOMR
C HH3wero 3HepreTHqeCKOrO ypOBHSI Ha 60Jlee BbICOKHH, TO 8TOM TRKYIO 3Hep-
rHIO npHMeT H npH 3TOM B03pRCTeT, HRnpHMep, 3HeprH.H ero TenJlOBOrO XRO-
THqeCKOrO .l{BHmeHH.H. HO B B036ym.neHHoe COCTO.HHHe aTOM He nepeH.neT.
3HeprH.H cpOTOHR, nOrJlOReMOro RTOMOM npH
nepeXO.l{e H3 O.l{HOrO COCTO.HHH.H B .l{pyroe, B TOq-
n=4 HOCTH paBHa pa3HOCTH 3HeprHH RTOMa B 3THX
.l{BYX COCTOSIHH.HX (pHC. 20.7):
n=3
t
t
n=2
hV21 = W 2 - Wit hV31 = W 3 - Wit
HHbIMH cnOBaMH, OH nor JlOaeT CBeT TOJlb-
KO TRKOH qRCTOTbI, KOTOPYIO CRM MomeT Hcnyc-
n=1 KaTb (3RKOH nOrJlOeHH.H H HcnYCKaHH.H CBeTa,
nOJlY'leHHbIH 3KcnepHMeHTaJlbHO r. KHpxrocpoM).
HCKJllOqeHHe COCTaBJI.HeT CJlyqaH, KOr.l{a BHeWHee
B03)J;eHcTBHe MomeT co06aTb aTOMY 3HeprHIO 60JlbWe TOH, KOTopaSI He06xo-
.l{HMa AJI.H ero HOHH3aHH. IIpH 3TOM QaCTb 3HeprHH BHeWHero B03.neHcTBH.H
TpaTHTC.H Ha HOHH3a:U;HIO RTOMa, R H36bITOK 3HeprHH nepe.naeTC.H BbIpBaHHOMY
3J1eKTpOHY B BH.ne ero KHHeTHqeCKOH 3HeprHH. IIocne.nH.H.H MomeT HMeTb npo-
H3BOJlbHYIO BeJlHqHHY.
HTaK. MomHO c.neJlaTb CJleYIOWHe BbIBO.l{bI.
1. CB060.l{HbIH aTOM norJlOaeT H H3J1y'.raeT 3HeprHIO TonbKO :u;eJlbIMH KBaH-
Puc. 20.7
TRMH.
2. IIpH nepexo)J;e B B036ym.l{eHHOe COCTOHHHe aTOM norJlOaeT TOJlbKO Ta-
KHe KBaHTbI, KOTopbIe MomeT caM HcnYCKaTb.
20.9. CneKTp 3HepreTH'IeCKHX COCTOHHHH aTOMa.
IIpoHcxom.n;eHHe JIHHeHQaTbIX cneKTpOB.
HenpepbIBHble cneKTpbJ
HCCJle.l{OBaHH.H cneKTpOB H3J1yqeHH.H pa3pemeHHbIX ra30B (T.e. cneKTpOB H3-
JlyqeHH.H OT.l{eJlbHbIX aToMoB) nOKa3aJlH, qTO KRm.l{OMY ra3Y npHCY BnonHe
onpe.l{eJleHHbIH JlHHeHqaTbIH cneKTp, COCTOSIHH H3 OT.l{eJlbHbIX cneKTpaJIbHbIX
nHHHH HJlH rpynn 6J1H3KO pacnOJlOmeHHbIX JlHHHH. HRH60Jlee H3YQeHHbIM .HB-
JI.HeTC.R cneKTp caMoro npOCToro aTOMa - aTOMa BO.l{OpO.l{R. BnepBble H. Banb-
Mep B 1885 r., H3yqa.H cneKTpbI aTOMR BOAOpO.l{a, 06HapymHn 3aKOHOMepHOCTb
B pacnOJlOmeHHH JIHHHH B BH)J;HMOH qaCTH cneKTpa, qaCTOTbI KOTOpbIX MorYT
6bITb onpe.l{eJleHbI H3 COOTHOmeHH.H
v = R ( :2 - 2 ). r.<le m = 3. 4. . ... R nOCTOJIHHaJI PH.<I6e pra
(R = 3,2 . 10 15 c- 1 ).
581

Bce aTH JlHHHH CneKTpa BO)J;OpO)J;a COCTaBJI.HIOT CepHIO BaJILMepa. II03me
6bIJlH YCTaHOBJleHbI:
1) B ynLTpa<l>HoneToBoH o6nacTH CepHJI JIaHMaHa: v = R( 1 - 2 l r,!te
m=2, 3, 4, ...;
2) B HH<ppaKpacHoH 06J1aCTH 3 cepHH:
CepH.H llameHa: v = R (  -  1 , rlIe m = 4. 5. .
3 2 m 2 . . .
CepHJI BpeKKeTa V =R ( --.!... l ' r,ne m=5, 6,
4 2 m 2 ...,
CepHJI n<!>YllJl a : V = R( :2 - 2 ). r,!te m = 6. 7. ....
TaKHM 06pa30M, Bce cneKTpaJIbHbIe nHHHH B cneKTpe Bo,nopo,na MO)l<'HO pac-
CqHTaTb no O,nHOM qJopMYJIe, Ha3bIBaeMOH o606eHHoii <popMynoH BaJlbMepa:
R( 1 1 ) ......
v kn = n 2 - k 2 ' r,ne n HMeeT B Kam,nOH ,naHHOH cepHH nOCTO.HHHOe 3Haqe-
HHe, n = 1, 2, 3, 4, ..., a k npHHHMaeT n;eJlOqHCJIeHHbIe 3HaqeHH.H HaqHHaSl
C n+1.
HaJIHqHe JIHHeHqaTbIX cneKTpOB aTOMa BO,AOpo,na MomHO 06'b.HCHHTb Ha oc-
HOBaHHH MO)J;eJlH aTOM8 Bopa. Kam,na.H cneKTpaJIbHa.H JIHHH.H 1I0JlyqaeTC.H B pe-
3YJlbTaTe Toro, qTO aTOM HcnYCKaeT OTOH npH nepeXO,Ae H3 OlIHoro 3HepreTH-
qeCKOrO COCTO.HHH.H B ,npyroe. IIpH aTOM pa3HOCTb Mem..o;y aHeprH.HMH aTOMa B
HaqaJIbHOM H KOHeqHOM COCTO.HHH.
.HX onpe..o;eJI.HeT qaCTOTY aJleKTpOMar.
n = 00 HHTHoro H3J1YQeHH.H, a CJle,nOBa.
TeJlbHO, H nOJlOmeHHe ..o;aHHOH nH.
HHH B cneKTpe. IIpH,naBa.H n B
<popMYJle (20.4) pa3J1HqHbIe 3Haqe.
HH.H, MomHO nOJlyqHTb p.HlI 3Haqe-
HHH aHeprHH CTaI.J.HOHapHbIX COCTO-
.HHHH aTOMa Bo,nopo,na (pHC. 20.8).
IIpH B03paCTaHHH qHCJIa aneK.
TpOHOB B aTOMe YBeJIHqHB8eTC
qHCJlO B03MomHbIX aHepreTH'leCKHX
COCTO.HHHH aTOMa, 60raqe CTaHO-
BHTC.H ero cneKTp.
OnTHQeCKHe QaCTOTLI (aJleKTpO-
MarHHTHoe H3nyqeHHe OnTHqeCKOrO ,nHana30Ha) CB.H3aHbI C nepexo,naMH 3JIeKT-
pOHOB, Haxo,n.HHXC.H H8 BHeIIIHHX 060JIOqKaX (BaneHTHbIe aJIeKTpoHbI).
IIpe,ncT8BHM ce6e, qTO B MHOrOaJIeKTpOHHbIH T.HmeJIbIH aTOM KaKHM-TO 06pa30M
nonaJI aJleKTpOH 60JIbIIIOM aHeprHH, npOJleTalOHH B6J1H3H .H,npa (pHC. 20.9) (Ha
cxeMe nOKa3aHbI He B03MomHbIe Op6HTbI ,nBHmeHH.H aJIeKTpOHOB, a aneKTpOHHble
CJIOH). OH MomeT BbI6HTb O,nHH H3 aJIeKTpOHOB rJly6HHHoro CJIO.H. 110H C BaKaHT.
HbIM MeCTOM B rJIy6HHHOM 3JIeKTpOHHOM CJIoe aHepreTHqeCKH HeBbIro,neH Ii ,nonro
E n.aB
I
t 
! CepH.H IIameHa
CepH.H BaJlbMepa
-0,54
-0,85
-1,51
-3,39
-13,6
CepH.H JIaHMaHa
Puc. 20.8
582
5
4
3
2
1

CyJ.QeCTBOBaTb He 6y,neT. Ha MecTO BbI6HToro 3JIeKT-
pOHa npH,neT KaKOH-TO ,npyrOH 3J1eKTpOH ,naHHoro aTO-
Ma. TIo,no6Ha.H 3aMeH8 MomeT npOH30HTH TaK: Ha Me-
CTO YlIIe,AlIIerO 3JIeKTpOHa npHXO,AHT 3J1eKTpOH H3 CO-
ce,nHero CJlOH, Ha ero MeCTO - 3J1eKTpOH H3 coce,n-
Hero C HHM CJlO.H H T.,n. (nepexo,nbI 1, pHC. 20.9).
Kam,nbIH H3 T8.KHX nepexo,nOB conpOBom,naeTCH COOT-
BeTCTByIO BbI,ZJ;eJleHHeM 3HeprnH B BH,ne c}>oToHa qa-
CTOTbI, 6J1H3KOH K BH,nHMbIM. T8.KHe n01{TH o,nHOBpe-
MeHHO BbIlleJlHIOIUHecH <pOTOHbI lIeHCTBHTeJlbHO Ha6J1lOlIaJOTC.H.
Ho B03MomeH H ,npyroH BapHaHT 3aMeeHH.H BaKaHTHoro MeCTa, Kor,na
Ha Hero nepexo,nHT O,nHH H3 ,naJIeKHX nepH<pepHHHbIX 3J1eKTpOHOB (nepexo,nbI 2,
pHC. 20.9). TIpH 3TOM 3HeprH.H aTOMa cpa3Y BeCbMa 3Ha1{HTeJIbHO H3MeH.HeTCH,
HcnYCKaeTC.H O,nHH <pOTOH, HO ropa3,n0 60JlblIIeH 3HeprHH. 8TO xapaKTepHCTH-
1{eCKOe peHTreHOBCKoe H3nyqeHHe.
BOJlee CJlOmHbIH cneKTp. qeM aTOMbI, ,nalOT MOJleKYw'lbI. 8HeprH.H MOJleKYJlbI
CKJla,AbIBaeTC.H H3 Tpex COCTaBJI.HIOHX: 3HeprHH 3JIeKTpOHOB, 3HeprHH Kone6a-
TeJlbHOrO ,ABHmeHHSI .H,nep aTOMOB H 3HeprHH BpaeHH.R H,nep OTHOCHTenbHO
06ero eHTpa T.HmeCTH. Bce 3TH COCTaBJI.HIOHe ,nHCKpeTHbI, H HX H3MeHe-
HHe HMeeT KBaHTOBbIH xapaKTep. TIpH pa3J1H1{HbIX C01{eTaHH.HX 3THX Tpex
KBaHTOBaHHbIX COCTaBJI.HIOHX 3HeprHH o6pa3yeTC.H BeCbMa 60JlblIIOe 'lHCJlO B03-
MomHbIX 3HepreTH1{eCKHX YPOBHeH MoneKYJlbI. OTCIO,AR 60JlblIIOe 1{HCnO B03Mom-
HbIX nepeXO,AOB MOJleKYJlbI H3 OlIHoro COCTOHHH.H B ,npyroe. TaK nOJlY1{aIOTCH
nOJlOCaTbIe cneKTpbI MOJleKYJI, B KOTOpbIX Kam,AaH H3 nOJloc COCTOHT H3 60Jlb-
lIIoro 1{HCJl8 6J1H3KO pacnOJIOmeHHbIX nHHHH.
B mH,nKHX H TBep,nbIX TeJlaX, r,ne qaCTHnbI CHJIbHO B3aHMOlIeHCTBYIOT ,Apyr
C ,nPyroM, 3HeprH.H Kam,nOH qaCTHnbI BKJllOqaeT B ce6H H 3HeprHIO ee B3aHMO-
,neHCTBH.H C ,npyrHMH qaCTHaMH. TaK KaK 3HeprH.H B3aHMO)J;eHCTBH.H qaCTHn
MomeT HMeTb caMbIe pa3Ho06pa3HbIe 3HaqeHHH, TO BMeCTO OT,neJlbHbIX 3Hepre-
THqeCKHX YPOBHeH 06pa3YIOTC.H CnJlOlIIHbIe nOJIOCbI B03MomHbIX 3HepreTHqeCKHX
COCTOHHHH. TI03TOMY H BenHqHHa KBaHTOB H3J1yqeHH.H MomeT 6bITb caMOH pa3-
JlH1.IHOH, H cneKTp nOJlyqaeTC.H CnJlOlIIHbIM (HenpepbIBHbIM).
Puc. 20.9
20.10. CnoHTaHHoe H BLIHYjKJJ;eHHOe H3JIyqeHHe
ATOM, HaXO,A.HHHC.H B B036ym,neHHOM COCTOHHHH, caMonpOH3BOJlbHO nepe-
XO,nHT B OCHOBHoe COCTO.HHHe HJlH Ha 60Jlee HH3KHH 3HepreTHqeCKHii: ypOBeHb.
H3J1yqeHHe, HcnYCKaeMoe npH caMonpOH3BOJlbHOM nepexo,ne H3 O,AHOrO CO-
CTO.HHH.R B ,npyroe, Ha3bIBaeTC.H cnOHTaHHLIM H3JIyqeHHeM. CnOHTaHHoe H3ny-
qeHHe pa3J1HqHbIX aTOMOB npOHcxo,nHT HH,nHBH,nyaJlbHO, TaK KaK Kam)J;bIH aTOM
Ha1{HHaeT H 3aKaHqHBaeT H3J1yqeHHe He3aBHCHMO OT ,npyrHX.
B 1916 r. 8HHlIITeHH npe,nCKa3M, tITO nepexo,n 3JIeKTpOHa B aTOMe C Bepx-
Hero 3HepreTHqeCKOrO YPOBH.H Ha HHmHHH H conpOBom,nalOee 3TOT aKT H3JIY-
qeHHe MorYT npOHcxo,nHTb TaKme noon BJlH.HHHeM BHelIIHero 3J1eKTpOMarHHTHoro
nOJlH. TaKoe H3nyqeHHe Ha3bIBalOT BLmyeIDlLIM, HJlH HHpOBaHHLIM.
BepO.HTHOCTb HH,nynHpoBaHHoro H3J1yqeHHH pe3KO B03paCTaeT npH COBna,ne-
HHH qaCTOTbI 3J1eKTpOMarHHTHoro nOJIH C c06CTBeHHOH qaCTOTOH H3nyqeHHSI
B036ym,neHHoro aTOMa.
583

W2W2
hv
JV\./'\,,-
B pe3Y nbT8Te B3aHMO,lJ;eHCTBHSI B03-
6ym,lJ;eHHOrO aTOMa, rOTOBoro HcnYCTHTb
Q>OTOH C 3HeprHeH hv = W 2 - W 1. C Q>O-
TOHOM hv, BbIHym,lJ;81OHM 3TOT nepe-
XO,lJ;, nOJlyq8IOTC.H ,lJ;Ba COBepmeHHO O,AH-
H8KOBbIX no 3HeprHH Ii H8np8BJleHHIO
,lJ;BHmeHH.H Q>OTOHa-6J1H3Hea (pHC. 20.10).
TIpOJleT81OHH Q>OTOH KaK 6bI "CTP.HXH-
BaeT" C B036ym,lJ;eHHOrO 8TOMa nO)J;06-
HbIH ce6e cpOTOH, He 38TpaqHBafl Ha 3TO 3HeprHIO. C TOqKH 3peHH.H BonHOBOH
TeopHH aTOM H3J1yqaeT 3J1eKTpOM8rHHTHYIO BOJlHY, COBepllIeHHO O)J;HHaKOBYIO no
HanpaBJleHHIO pacnpOCTp8HeHH.H, qaCTOTe, cp83e H nOn.HpH3aHH C TOH, KOTO-
p8S1 BbIHY,lJ;HJl8 8TOM H3J1Y'l8Tb. B HTore nonyq8eTcSI pe3YJlbTHPYlOllt8.H BOJlHa
C 8MnJlHTYlIOH 60JlbmeH. qeM y n8,A8101IteH. Oc06eHHOCTblO HH,AYUHPOB8HHoro
H3J1yqeHH.H .HBJlSleTCSI TO. qTO OHO MOHOXPOM8THqHO H KorepeHTHO.
TIpH XaOTHqeCKOM TenJlOBOM ,lJ;BHmeHHH p8cnpe,[(eJleHHe 3HeprHH cpe.uH 8TO-
MOB Hep8BHOMepHO. HeKOTOp8.H q8CTb aTOMOB B036ym,lJ;eHa, qTO COOTBeTcTByeT
HX Haxom)J;eHHIO H8 60Jlee BbICOKHX, qeM OCHOBHOH, YPOBHSlX 3HeprHH. B YCJlO-
BHSlX TenJlOBOrO paBHOBeCH.H H npH OTCYTCTBHH BHemHero 3J1eKTpOMarHHTHoro
nOJl.H 60JIbmaH qaCTb aTOMOB H8XO,lJ;HTC.H B HeB036ym,lJ;eHHOM COCTO.HHHH. 06pa3-
HO rOBOp.H, H8CeJleHHOCTb BepXHHX ypOBHeH MeHbme HaCeJleHHOCTH HHmHHX.
TIO,lJ; BJlH.HHHeM BHellIHHX B03)J;eHCTBHH - nOBbImeHH.H TeMnepaTypbI, OCBe-
eHH.H, 60M6ap,lJ;HpOBKH 6bICTpbIMH 't{8CTHaMH - ,AOJlfl B036ym,lJ;eHHbIX 8TOMOB
B03p8CTaeT. T .e. H8CeJleHHOCTb BepXHHX YPOBHeit YBenHqHBaeTC.H.
Ka3aJlOCb 6bI, no Mepe nOBbIllIeHH.H TeMnep8TypbI MomHO nOJlyqHTb TaKoe
p8Cnpe,lJ;eJleHHe q8CTHU no ypOBH.HM, npH KOTOpOM HaCeJleHHOCTb BepXHHX YPOB-
HeH 60nbme. qeM HHmHHX. Ho 3TO He TaK. Be,Ab B036ym,lJ;eHHOe COCTO.HHHe
HeYCTOHqHBO. TIo Mepe YBeJlHqeHH.H 38CeJleHHOCTH BepXHHX YPOBHeH YBeJlHqH-
B8eTC.H BepO.HTHOCTb cnOHTaHHbIX nepeXO,[(OB. conpOBom.n.aeMbIX H3J1yqeHHeM.
B 1939 r. B. A. 86pHKaHT BbICKa38n npe,lJ;nOJlOmeHHe 0 B03MomHOCTH co-
3,lJ;8HH.H TaKoro pacnpe,lJ;eJleHH.H q8CTH no 3HeprH.HM, npH KOTOpOM qHCJlO B03-
6ym,lJ;eHHbIX aTOMOB 60Jlbme 'l{HCJla aTOMOB, HaXO,lJ;.HHXCfl B OCHOBHOM COCTO-
.HHHH. TaKoe COCTO.HHHe Ha3bIBalOT COCTOSlHHeM C HHBepcHoii 3aCeJIeHHOCTbIO
YPoBHeii. EMY npHCYH oc06bIe CBOHCTB8.
TIPH pacnpOCTp8HeHHH CBeT8 B BelQeCTBe 06bIqHO Ha6J1IO,lJ;aeTC.H norJloe-
HHe CBeTa. 8TO npOHCXO,lJ;HT nOTOMY, qTO B COCTO.HHHH TepMO,lJ;HHaMHqeCKOrO
p8BHOBeCH.H qHCJlO HeB036ym,lJ;eHHbIX aTOMOB B BeeCTBe MHoro 60Jlbme, qeM
qHCJlO B036ym,lJ;eHHbIX, H, c.ne,lJ;OBaTeJlbHO, cpOTOHbI qalQe B3aHMO,lJ;eHcTBYIOT C
HeB036ym,lJ;eHHbIMH 8TOM8MH, T.e. nOrJlOaIOTC.H BelQeCTBOM.
B BeeCTBe me C HHBepcHoH HaceneHHOCTblO YPoBHeH qHCJlO B036ym)J;eH-
HbIX 8TOMOB 60nbllIe qHCJla HeB036ym,lJ;eHHbIx. IIpH 3TOM YMeHbllIaeTC.H Bepo-
.HTHOCTb BCTpeqH Q>OTOHOB C HeB036ym.n.eHHbIM 8TOMOM. T.e. YMeHbm8eTC.H Be-
pO.HTHOCTb nOrJlOeHH.H Q>OTOHOB. BeecTBo CT8HOBHTC.H 60Jlee np03p8qHbIM
HJlH .n.8me cnoc06HbIM YCHJlHB8Tb CBeT. AeHCTBHTeJlbHO, eCJlH B HeM J{BHmeTCfl
Q>OTOH, 3HeprH.H KOToporo B TOqHOCTH p8BHa p83HOCTH 3HeprHH 8TOMOB B CO-
CTO.HHH.HX W 2 H WIt TO, B3aHMO,lJ;eHCTBY.H C B036ym,lJ;eHHbIM aTOMOM, TaKOR
cpOTOH BbI30BeT HH,lJ;yHpOBaHHoe H3J1yqeHHe. B pe3YJlbTaTe nO.HBHTC.H BTOpOll
T8KOH me Q>OTOH. B3aHMO,lJ;eHCTBY.H C ,lJ;pyrHMH ,ABYM.H B036ym,lJ;eHHbIMH aTOM a-
hv
JV\./'\,,-
hv
JV\./'\,,-
WI
WI
Puc. 20.10
584

MH, 3TH )J;Ba t}>OTOHa BbI30BYT "BbICBeqHBaHHe" ee )J;BYX aTOMOB. B KOHeqHOM
CqeTe BMeCTO OHOrO t}>OTOHa H3 Bel..QeCTBa BbIHeT MHOrO t}>OTOHOB, T .e. 6yeT
HMeTb MeCTO YCHJleHHe CBeTa.
Cpea Ha3bIBaeTC.H aKTHBHOii, eCJlH B HeH 'tJHCJlO HHYHpOBaHHblX t}>OTO-
HOB npeBbIlliaeT qHCJlO nOrJlOllteHHbIX.
C03aTb HHBepCHYIO HaCeJleHHOCTb B CHCTeMe, COCTOSleH H3 BYX 3Hepre-
THqeCKHX ypOBHeH nYTeM JlHTenhHOrO OCBel..QeHH.H BeeCTBa MOHhIM CBeTO-
BbIM HMnYJlbCOM, HeB03MOmHO. BO-nepBbIX. C YBeJlHqeHHeM KOHeHTp8HH q8C-
TH Ha BepXHeM ypOBHe B03paCTaeT BepO.HTHOCTb CnOHTaHHhIX nepeXOOB; BO-
BTOpbIX, B036Y)KalOee H3J1yqeHHe BbI3bIBaeT HHYHpOB8HHOe H3J1yqeHHe, nepe-
BO.Hl..Qee q8CTHbI H8 HHmHHH YPOBeHb.
HHbIe B03MOmHOCTH OTKpbIBaIOTC.H npH HCnOJlh30B8HHH Bel..QeCTB. CpeH 3Hep-
reTHqeCKHX COCTO.HHHH KOTOpbIX HMeIOTC.H TaKHe B036ym)J;eHHbIe COCTO.HHH.H, B KO-
TOpbIX aTOMbI MOrYT HaxOHTbC.H 3HaqHTenbHO )J;OJlbllIe, qeM 10- 8 C. TaKHe co-
CTO.HHH.H H COOTBeTCTByIOl..QHe HM ypOBHH 3HeprHH H83bIBaIOT MeTaCTa6HJIL11bIMH.
llYCTb ypOBeHb 2 C 3HeprHeH W 2 (pHC. 20.11, a) MeTaCTa6HJlhHblH, X8-
paKTepH3yeMbIH "BpeMeHeM )KH3HH" B 1000 pa3 60JlbllIe, qeM "BpeM.H mH3HH"
ypOBH.H 3 W 3 ('t - 10- 8 c).
W a
6
3 W a . . . 3
W 2 2
2 
  
 
1 WI 1
Puc. 20.11
a
W 2
WI
llpH HaJIH'tJHH ypOBH.H 2 B03MomHbI cnOHTaHHbIe nepeXObI H3 COCTO.HHH.H 3
He TOJlbKO B OCHOBHoe COCTOSlHHe 1, HO H H8 MeTaCTa6HJlbHbIH ypOBeHb 2.
06a nepexoa conpOBomaIOTC.H H3J1yqeHHeM 3J1eKTpOHOB COOTBeTCTByIOeH 3Hep-
rHH:
hV3I =W 3 - WI'
hV32 = W 3 - W 2 .
BOJlbllIOe pa3J1HqHe BO BpeMeHax mH3HH B COCTOSlHH.HX 3 H 2 npHBOHT 1<
TO MY , qTO no )J;eHCTBHeM B036ymalOl..QerO t}>OTOH8 C 3HeprHeH hv 31  W 3 - WI
aTOMbI nepeXO.HT CHaqaJla B COCTOSlHHe 3, a 38TeM, H3J1Y'la.H t}>OTOH C 3Hep-
rHeH hV32 = W 3 - W 2 , - B COCTO.HHHe 2. B pe3YJlbTaTe npOHCXOHT HaKOnJle-
HHe aTOMOB B MeTaCTa6HJlbHOM COCTO.HHHH 2. Co BpeMeHeM HX qHCJlO OK83bI-
BaeTC.H 60JlhllIHM qHCna aTOMOB B OCHOBHOM COCTO.HHHH 1. 3HaqHT, B Bel..Qe-
CTBe C TaKoH cHcTeMoH YPoBHeH eHcTBHeM B036ymalOl..QerO H3J1yqeHHSI
hV31 = W 3 - WI MomeT 6bITb C03)J;aHa HHBepCHa.H HaCeJleHHOCTb YPOBHeH.
585

EcnJ1 HanpaBHTb B npeBapHTeJlbHO nOrOTOBJleHHOe TaKHM 06pa30M Bee-
CTBO KBaHT 3neKTpOMarHHTHOrO H3J1eHH.H hV21W2- WI' TO BCJleCTBHe 1IBne-
HH.H HHynHpoBaHHOrO H3J1yqeHH.H npOH30HeT YCHJleHHe CBeTa, qTO YCJlOBHO H
nOKa3aHO Ha pHC. 20.11, tJ. "tJHCJlO cpOTOHOB YBeJlHqHBaeTC1I B reOMeTpHqeCKOH
nporpeCCHH. Ha BbIXO,Ae YCHJlHTen1l CBeTa cpOTOHOB 60nbme, 'leM Ha Bxoe.
CHcTeMa aTOMOB C HHBepcHoH HaCeJleHHOCTblO YPoBHeH cnoc06Ha He Tonb-
KO YCHnHBaTb, HO H reHepHpOBaTb 3neKTpOMarHHTHoe H3J1yqeHHe. .lIJl1I pa60TbI
B pemHMe reHepaTopa He06xo.D:HMa nOJlOmHTenbHa.H 06paTHa.H CB.H3b. npJ1 KO-
TOpOH qaCTb CHrHaJla C Bxoa YCTpOHCTBa nOaeTC.H Ha ero BXO. JlSI 3TO-
ro aKTHBHaH cpe,Aa, B KOTOpOH C03aeTCH HHBepCHa.H HaCeJleHHOCTb YPOBHeH,
pacnOJlaraeTC1I B pe30HaTope, COCTO.HeM H3 BYX napaJlneJlbHbIX 3epKaJl.
B pe3YJlbTaTe O)J;Horo H3 cnOHTaHHX nepeXOOB aTOMOB C MeTaCTa6HJlbHO-
ro YPOBHSI Ha OCHOBHOH B03HHKaeT cpOTOH C 3HeprHeH hV21W2- WI' TIpH BH-
meHHH B CTOPOHY OHoro H3 3epKM 3TOT cpOTOH BbI3bIBaeT HH,AYIJ;HpOBaHHoe
H3J1yqeHHe H K 3epKany nOXOHT nenaSl naBHHa cpOTOHOB C 3HeprHeH hV21
Ka>KbIii. nocne OTpameHH.H OT 3epKaJla JlaBHHa cpOTOHOB BHmeTC.H B npOTH-
BOnOJlOmHOM HanpaBJleHHH, nonYTHO 3aCTaBn.H1I BCBeqHBaTbCSI C MeTaCTa6Hnb-
Horo YPOBH.H B036ymeHHbIe aTOMbl, OCTaBIIIHeC.H B 3TOM COCT01lHHH.
PaCCMOTpeHHbIH npHHIJ;Hn YCHJleHH.H J1 reHepaIJ;HH 3neKTpOMarHHTHoro H3-
JlyqeHH1I 6bIJI npeJlOmeH COBeTCKHMH cpH3HKaMH H. r. BaCOBbIM H A. M. TIpo-
XOpOBbIM H O,AHOBpeMeHHO aMepHKaHCKHM cpH3HKOM "lJ. TaYHCOM.
20.11. JIa3ephI H HX npHMeHeHHe B TeXHHKe
JIa3ep - 3TO KBaHTOBbIH reHepaTOp 3J1eKTpOMarHHTHbIX BonH B OnTHqeC-
KOM Hana30He (BH,AHMOM). KBaHTOBbIH reHepaTOp - HCTOqHHK CBeTa, pa60-
TalOHH Ha npHHIJ;Hne BbIHymeHHoro H3nyqeHH.H.
OnTHqeCKHH KBaHTOBbIH reHepaTOp COCTOHT 113 .D;BYX OCHOBHblX qaCTeH: aK-
THBHOH Cpe,AbJ J1 pe30HaTopa.
B nepBbIX na3epax aKTHBHOH cpeOH 6bIn KpHCTaJlJl Py6HHa (AI 2 0 3 ) C npH-
MeCblO OKono 0,05 % xpOMa (pHC. 20.12). STOT OCHOBHOH 3J1eMeHT Jla3epa
06bIqHO HMeeT cpOPMY nHJlHHpa 1. TOpIJ;bI IJ;HJlHHpa 3 H 4 cTporo napan-
nenbHbI, Ha HHX HaHeceH cnoH cepe6pa. OHa H3 3epKanbHbIX nOBepxHocTeH
nOJlynp03paqHa.H: 92 % CBeTOBoro nOTOKa OTpamaeTCSl OT Hee J1 OKOJlO 8 %
CBeTOBoro nOTOKa nponYCKaeTCSI eID.
Py6HHOBbIH CTepmeHb nOMeIlteH BHYTpH HMnYJlbC-
1 HOR KceHoHoBoH CnHpBJIbHOR naMnbl 2, nHTaeMoH HM-
nYJlbCaMH BbICOKoro Hanp.HmeHHH OT 6aTapeH KOHeH-
caTopOB eMKOCTblO ,AO 10 4 MK. 3ap.H>KaeMOH .no Ha-
np.H>KeHH1I B HeCKOJlbKO TbIC1Iq BOJlbT. TIpH pa3p1le
Qepe3 JlaMny KOH)J;eHcaTopbI 6aTapeH oT.nalOT 3HeprHID
B COTHH TblC.Hq mOYJleH. JlHTeJlbHOCTb BCnbIIIIKJ1
COCTaBn.HeT 10- 3 c, a MOHOCTb JlaMn npeBbIlliaeT
10 7 BT. JIaMna .HBJlSIeTC.H HCTOQHHKOM B036ym.naIOero
H3neHH.H. ATOMbI xpoMa, norJloa.H H3J1eHHe C
.nJlHHOH BOJlHbI 560 HM, coepmaeec.H B cneKTpe
KceHOHOBOH JlaMnbI, nepeXO1IT C OCHOBHoro ypOBH.H
Puc. 20.12 WI Ha B036Y>KeHHbIH ypOBeHb W 3 (CM. pHC. 20.11).
586

BpeM.H mH3HH aTOMOB XpOMa Ha B036ym)J;eHHOM ypOBHe W 3 MaJlO. JI.H
nepeXOOB W 3 ---+ W 1 OHO COCTaBJI.HeT 10- 5 c, a JI.H nepexoa W 3 ---+ W 2 OHO
MeHbme 10- 7 C. II03ToMY 60Jlbma.H qaCTb B036ymeHHbIx aTOMOB C YPOBH.H
W 3 COBepmaeT nepeXObI Ha BTOpOH B036ymeHHbIH YPOBeHb W 2'
BpeM.H mH3HH aTOM a xpOMa Ha ypOBHe W 2 cpaBHHTeJlbHO BeJlHKO - no-
p.HKa 10- 3 C. 3TOT ypoBeHb .HBJI.HeTC.H MeTaCTa6HJlbHbIM. ECJlH MOHOCTb JlaM-
nbI-BcnbImKH ,AOCTaTOqHO BeJlHKa, TO HaCeJleHHOCTb MeTaCTa6HJlbHOrO ypOBH.H OK a-
meTC.H 60JlblIIe, qeM HaCeJleHHOCTb OCHOBHoro YPOBH.H. IIpH OCTHmeHHH HHBepc-
HOH HaCeJleHHOCTH YPOBHeH KpHCTaJIn py6HHa CTaHOBHTC.H aKTHBHOH cpeOH.
IIpoecc nepeBoa aTOMOB H3 OCHOBHoro B B036ymneHHoe COCTO.HHHe Ha3bI-
BalOT HaKaqKOH. COOTBeTCTBeHHO HcnoJlb3yeMYlO JI.H 3Toro HMnYJlbCHYIO Kce-
HOHOBYIO JlaMny Ha3bIBalOT JlaMnOH HaKaqKH.
OCTaTOqHO OHOMY aTOMY xpOMa COBepmHTb cnOHTaHHbIH nepexo C Me-
TaCTa6HnbHoro ypOBH.H W 2 Ha OCHOBHOH W 1 C HcnYCKaHHeM cpOTOHa C qaCTO-
(W 2 - WI)
TOH v = , KOTOpOH COOTBeTcTByeT JlHHa BOJlHbI KpaCHoro CBeTa
h
A = 694,3 HM, KRK B03HHKReT JIaBHHa CPOTOHOB. OHa BbI3BaHa HH,AYHPOBRH-
HbIM H3J1yqeHHeM aTOMOB XpOMR. HaXOn.HI..UHXC.H B MeTaCTa6HJlbHOM COCTO.HHHH.
ECJlH HanpaBJleHHe BbIJleTa nepBHqHOrO <}>OTOHa 6bIJlO nepneHl{HKY JI.HpHO nJloc-
KOCTH 3epKaJl pe30HaTopa, TO 1'13 nOJlynpo3paqHOrO 3epKaJla pe30HaTopa BbIpbI-
BaeTC.H MOHoxpOMaTHqeCKOe, KorepeHTHoe H OCTpOHanpaBJleHHOe H3J1yqeHHe C
JlHHOH BOJlHbI A=694,3 HM.
JIa3epHbIe HCTOqHHKH 06J1aaIOT p.HOM npeHMyeCTB:
1) OHH C03a1OT OqeHb Y3KHe nyqKH CBeTa;
2) H3J1yqaeMbIH HMH CBeT 06J1a,AaeT HCKJllOqHTeJlbHoH MOHoxpOMaTHqHOCTblO
C nOCTO.HHHOH <}>a30H BOJlHbI;
3) Jla3epbI - OqeHb MOI..UHbIe HCTOqHHKH CBeTR (- 10 13 BT Ha 1 CM 2 ).
JIa3epHbIe HCTOqHHKH HamJlH mHpOKoe npHMeHeHHe B TeXHHKe: nJl.H CB.H3H
(B KOCMHqeCKOM npocTpaHcTBe), JI.H (iCnapeHH.H MaTepHaJlOB B BaKYYMe, nS:l
CBapKH, B xHpyprHqeCKHX onepaH.HX, JI.H nonyqeHH.H 06'beMHOrO H306pame-
HH.H (B rOJlOrpa<}>HH), B BoeHHOM eJle (cBeTOJlOKaTOpbI H p.). OHH BbI3bIBalOT
pa3J1HqHbIe XHMHqeCKHe peaKHH; Jla3epbI HcnOJlb3YlOT JlS:I ocyecTBJleHH.H yn-
paBn.HeMOH TepMO.HepHOH peaKHH H p.
BonpOCM RH CAMOKOHTPORH
1. KaKue Sl6Jl.eIiUSl nOOm6epJICiJaKJm CJl.OJIClioe cmpoeliue amo.Ma?
2. Onuwume .MOOeJl.b amO.Ma TO.Mcolia II olle.MY ama .MOOeJl.b amO.Ma OKa3a-
Jl.aCb liecoCmOSlmeJl.bliou?
3. B lte.M 3aKJl.KJltaJl.CSl onblm Pe3epljJopoa 2 0 lle.M c6UOemeJl.bCmeyem om-
KJl.OlieliUe a.-llacmUij lia YZJl.bl. 60Jl.bWUe 90°?
4. lJmo npeOCma6Jl.Slem c060il nJl.aliemapliaSl .MOOeJl.b amoMa 2 KaKoe cmpoe-
Hue UMeem amo.M 60iJopoiJa COZJl.aCIiO amoil MOOeJl.u?
5. lJ mo nOIiU.MaKJm noiJ aliepzueu amo.Ma? KaKoe cocmOSlliue amO.Ma lia3bl-
6aKJm OCli06libl.M? 6036yJICoelilibt.M?
6. lJmo ym6epJICiJaKJm nocmYJl.ambl Bopa?
587

7. KaKuMu cnoc06aMu MOJICIiO nepe6ecmu amOM 6 6036YJlCiJelilioe cocmoSl-
liue? M oJICem Jl.U amOM npu nepexooe 6 603t5YJlCOeIiIiOe cocmOSlHue nOZJl.OmUmb
npOU360Jl.b1iYIO nopijUIO 3liepzuu? q mo Ha3bl6aemcSl 3Hepzueil uOliu3aquu?
8. KaKoB .M.eXaliU3M U3Jl.YlleliUJl, U nOZJl.014eIiUSl 3Jl.eKmpOMaZliumHblX 60Jl.1i
amOMaMU no MOOeJl.U amOMa Bopa?
9. 3anuwume 6blpaJICeliUe OJl.Jl, 3liepZUU amOMa 6 cmaquoliapliblX cocmoSl-
IiUSlX. 6blmeKalO14ee U3 MOOeJl.U amOMa Bopa. H apucyume cxeMY 3liepZemU1lec.
KUX YP06lieU amOMa 6000pooa U 06'OSlcliume. 6CJl.eOCm6Ue KaKUX nepexoo06
nOJl.YllalOmcSl cepuu cneKmpaJl.bliblX Jl.Uliuil amOMa 60oopoiJa.
10. KaK rjJOPMYJl.UpyemcSl npa6UJl.0 K6alim06aliUSl op6um? Tlo KaKOU rjJop.
MYJl.e paccilumbl6alOmcSl paouycbl 60p06CKUX op6um?
11. B lleM cocmoum iJucKpemliblil xapaKmep U3Jl.YlleIiUSl? Tlo KaKOU rjJop-
MYJl.e .MOJICliO paccllumamb B03.M.OJIClible Clacmombl U3Jl.YlleIiU.R amOMa 60oopooa?
12. KaKue npuHqunua.nbH.ble lieoocmamKu Mooe.nu amOMa H. Bopa?
13. KaK06 nop.RiJoK BpeMeliU JlCU3liU amOMa 6 6036YJICoeHliOM cocmoSlliuu?
14. B lleM 3aKJl.lOllaJl.UCb onblmbl CPpaliKa U repija? 0 lleM OIiU c6uiJe.
meJl.bC m606aJl.u?
15. KaKoe U3Jl.YlleHue Ha3blBaemCJl UHOYijUp06aliliblM? OnuUlume npUHijUn
zeHepup06aliuSl U YCUJl.eIiUSl 3Jl.eKmpOMazliumliblx 60Jl.1i.
e 2
W=--
81t£or
hv = W 2 - WI
v =R ( - )
n 2 k 2
nh
mv r =-
n n 2n
h 2 n 2 E
r. =
n nme 2
OCHOBHblE cI»OPMY JIbl
3HeprH.H 3neKTpOHa, HaXOAHerOC.H Ha cTaHoHapHoH
op6HTe B aTOMe BOAOpO,Aa, rAe e - 3apHA 3J1eKTpOHa,
Eo - 3J1eKTpHqeCKR.H nOCTO.HHHa.H, r - pa)J;HYC 60pOBCKOH
Op6HTbI 3J1eKTpOHa.
- 3HeprH.H, H3nyqeHHa.R HJlH nOrJlOeHHR.H RTOMOM BOAO-
pO)J;a, r,Ae h - nOCTO.HHHaSl IIJlaHKa, v - qRCTOTR H3J1Y-
qeHH.H, Wi H W 2 - 3HeprHH RTOMB B COOTBeTCTBYlOlltHX
CTaHOHapHbIX COCTO.HHHHX.
- c}>OPMYJlR BRJlbMepa, r)J;e v - qaCTOTR BOJlHbI, H3ny-
qeHHOH aTOMOM, R - nOCTOHHHa.H PHA6epra, n - HOMep
Op6HTbI, Ha KOTOpylO nepeXOAHT 3J1eKTpOH, k - HOMep op-
6HT, C KOTOPOH nepeXO)J;HT aneKTpOH.
- YCJlOBHe KBaHTOBaHH.H CTaHOHapHbIX Op6HT, rAe m -
Macca 3J1eKTpOHa, V n - CKOpOCTb 3J1eKTpOHa Ha n-H Op6H-
Te, r n - paAHYC n-H Op6HTbI.
- PRAHYC n-H CTaHOHapHOH Op6HTbI 3J1eKTpOHa B aTOMe
BOAOPO.v;R.
METOAHQECKHE PEKOMEHAAHH
IIpH pemeHHH 3aAaq no 3TOH TeMe CJle)J;yeT nOMHHTb, qTO )J;BHmeHHe 3J1eK-
TpOHa B aTOMe paCCMaTpHBaeTC.H C TOqKH 3peHH.H KJlaCCHqeCKOH c}>H3HKH, no-
588

3TOMY MOmHO HCnOJlh30BaTb cpOPMYJlbI H3 MeXaHHKH H 3J1eKTpOCTaTHKH Ho
npH 3TOM 06.H3aTeJlhHO yqHTbIBaTh, tITO, COrJlaCHO nOCTYJlaTaM Bopa, paAHYChI
CTaUHOHapHMX Op6HT 3J1eKTpOHa H 3HeprHH aTOMa MOrYT npHHHMaTh TOJlbKO
)J;HCKpeTHbIe 3HaqeHH.H. B p.Hne cJlyqaeB CJlelIyeT yqHTbIBaTb. qTO 3J1eKTpOH,
ABHmym;HHCSI no op6HTe, KpOMe KHHeTHqeCKOH, 06J1anaeT H nOTeHUHaJlbHOH
3HeprHeH, 06YCJlOBJleHHOH KYJlOHOBCKOH CHROH npHT.H)KeHH.H ero .HAPOM aTOMa.
IIpH peweHHH 38Aaq peKOMeH,AyeM npHAepmHBaThCH CJleAYIOm;ero aJlrOpHTMa:
1) 3anHcaTh OCHOBHbIe MaTeMaTHqeCKHe COOTHOWeHH.H, BbITeKalOHe H3 no-
CTYJlaTOB Bopa;
2) onpeAeJlHTh r JlaBHbIe KBaHTOBbIe qHCJla )J;JI.H HccJleAyeMbIx COCTO.HHHH
aTOMa;
3) 3anHcaTb BbIpameHH.H AJI.H 3HeprHH H HMnYJlhca 3J1eKTpOHa B paCCMaT-
pHBaeMhIX CTan;HOHapHbIX COCTO.HHHHX aTOMa;
4) AonHcaTb HeAOCTalOHe ypaBHeHH.H. Hcnonh3Y.H ypaBHeHH.H H3 MexaHH-
KH H 3J1eKTpOCTaTHKH;
5) peWHTb nOJl}"leHHYIO CHCTeMY ypaBHeHHH.
nPHMEPLI PEIlIEHHH 3A,l(A'l
1. Ha CKonbKO H3MeHHn8Cb 3HeprHSI 3J1eKTpOH8 B 8TOMe BO,AOpO,Aa npH
H3neHHH aTOMOM <l>oToHa C ,AnHHOH BOnHbJ 486 HM?
)J; a H 0: h = 6,62 . 10- 34 )J;m. c, e = 3,0. 10 8 , A = 4,86. 10- 7 M.
HaHTH: L\W.
Pew e H H e. H:3MeHeHHe 3HeprHH 8TOMa npH H3nY'leHHH <l>oToHa p8BHO
e he
6W=hv, HnH TaK K8K V=):, TO L\W="I'; 6W=4,1.10- 19 )J;m.
2. PMHYC nepBoH 60POBCKOH Op6HTbI B aTOMe SO,AOpO,l(8 p8BeH 0,53 .10- 10 M.
Onpe,AenHTe nHHeHHYIO H yrnOByK> CKOpocTIi ABHmeHHSI 3neKTpoHa Ha 3TOH op6HTe.
)J;8HO: rl=0,53.10- 10 M, m=9,1.10- 31 Kr, e=1.6'10- 19 Kn.
H8HTH: 0), v.
Pew e H H e. .n;BHmeHHe 3neKTpOH8 no KpyroBoH op6HTe npOHCXO,AHT nOA
,AeHCTBHeM KynoHoBcKOH CHnbI npHT.HmeHHSI co CTOpOHbI SlApa. no BTO-
e 2 mv 2 e
POMY 3aKOHY HblOTOH8 F = mii.  = -. Cne,AOBaTenbHo. v = ;
4m:or r  4m:omr
v=2,2.10 6
M
C
YrnOB8S1 CKOpOCTb
v
ro = -;
r
0) = 4,1 . 10 16 PM
C
3. npOTOH, neTSlm;HH ropH30HTanbHO co CKOpOCTblO, MO,Aynb KOTOpOH
M
4,6. 10 4 -, CTaJlKHBaeTC.H C HenO,ABlimHbIM CBo6o,AHbIM SI,ApOM reRHSI. nOcne
C
. v
y,A8P8 npOTOH OTCK8KHBaeT co CKOPOCTblO. MO,Aynb KOTOpOH -, a aTOM nepe-
2
XO,AHT B B036ym,AeHHOe COCTO.HHHe. OnpeAeJIHTe ,AnHHY BOnHbJ CBeTa, KOTOpbIH
H3naeT aTOM renHSI, B03Bp8m;aSCh B nepBOH8'l8nbHOe COCTO.HHHe.
589

,n8HO: m p =I,67'10- 27 Kr, v=4,6.10 4
H8:HTH: A.
Pew e H H e.
M
C
V
VI = -.
2
3anHIlIeM 38KOH
renH CY" . . V
po .n. mpv = -m p 2" + mau.
pOCTb 8TOMa renHSI. OTCIOA8
coxp8HeHHSI HMnynbc8 ,AnSI CHCTeMbl "npoToH+SI,A-
r,Ae M8cca 8TOMa renHSI ma = 4mp, U - CKO-
V 3u
v = -- + 4u => U =-.
2 8
IIo 3aKOHY coxpaHeHHSI 3HeprHH KHHeTHQeCKaSl SHeprHSI npOTOHa (SIAPO re-
nHSI nOKoHnocb) ,AO CTonKHOBeHHSI paBHa CYMMe KHHeTH'leCKHX 3HeprH:H npo-
TOHa H SI,Apa renHSI nocne CTonKHOBeHHSI
m p v 2 = mp ( v 2 ) + 4m p u 2 + W.
2 2 4 2
3v u 6
u = - H H8HAeM 3HeprHIO B03 ym,AeHHSI
8
IIo,AcTaBHM 3HaQeHHe
m [ 2 9 2 ) 0 1875 v 2 m
W = ---.1!. v 2 --4 = ' p . W = 33.10- 19 TT m .
2 4 64 2' p' oM.
TIocne nepexo,Aa aTOMa renHSI B HeB036ym,AeHHOe COCTOSlHHe H3nY'l8eTCSI <po-
he
TOH, 3HeprHSI W KOToporo 6Y,AeT p8SH8 3HeprHH s036Y»<,AeHHSI W = WcJ> =-.
A
Cne,AOB8TenbHO
HCKOMaSl ,AnHHa BonHbl
A= hc .
W'
=6,0' 10- 7 M
4. Onpe,AenHTe HOMep 3neKTpOHHO:H Op6HTbI, Ha KOTOPYIO nepeXO,AHT sneK-
TpOH B 8TOMe BO,AOpOAa, H8XO,AHBIlIHHCSI H8 BTOpOM 3HepreTHQeCKOM YPOBHe,
npH nOrnOIl\eHHH <poToHa C 3HeprHe:H 1,88 sB? 3HeprHSI aTOMa B OCHOBHOM
COCTOSlHHH p8BH8 -13,53 3B.
,naHo: W I =-13,53 3B, WcJ>=1,88 sB, nl=2.
H 8 :H T H: n.
Pew e H He. 3HeprHIO 3neKTpOHa H8 n-M YPOBHe MomHO paCC'lHTaTb no
<POPMY ne
rAe WI - 3HeprHSI 8TOM8 B
WI
eM W 2 = --
4
W
W n = W 2 + WcJ> = _---1 + WcJ>'
4
WI
W n = ----"2'
n
OCHOBHOM COCTOSlHHH.
(1)
Tor,Aa npH nl = 2
HMe-
TIocne
nor nOIl\eHHSI
<l>OTOH8 3HeprHSI
8TOM8 6YAeT p8BH8
  
TIO,ACT8SHM B (1). nonY'lHM --t = _---1 + WcJ> => n =  I ; n = 3.
n 4 --1.-W
4 cJ>
590

21. 3EMEHTM KBAHTOBOR MEXAHHKH
rHnOTe38 .n;e BpOHJlS. BOJIHOBLle CBOHCTB8 3JIeKTpOHa. AH4>P8KH.H 3JIeKT-
pOHOB.
COOTHOllleHHe HeOnpe.n;eJIeHHOCTeH reH3eH6epra.
OCHOBLI KB8HTOBO-MeXaHH'leCKHX npe.n;CT8BJIeHHH. AHCKpeTHOCTL 3HeprHH H
.n;pyrHX 4>H3H'leCKHX BeJIH'lHH.
KBaHTOBO-MeX8HH'leCK8.H MO.n;eJIL 8TOMa BO.n;OpO.n;a.
IIpHHHn T03t.n;eCTBeHHOCTH 'laCTH. IIpHHHn 3anpeT8 IIaYJIH.
KBaHTOBO-MeXaHH'leCKOe 060CHOB8HHe nepHO.D;H'leCKOH CHCTeMLI MeH.n;e-
JIeeBa.
MOJIeKYJILI. IIpHpO.D;8 XHMH'leCKOH CBH3H.
rp8HH npHMeHHMOCTH 38KOHOB KJIaCCH'leCKOH H3HKH.
21.1. rHnOTe3a Ae BpOHJUI. BOJIHOBble CBOHCTBa 3JIeKTpOHa
)J;H4>paKHH 3JIeKTpOHOB
B 1923 r. JIYH l1.e BPOHJlb BbIl1.BHHYJI rHnOTe3Y 06 YHHBepCaJlbHOCTH KOp-
nYCKYJlHpHO-BOJlHOBOrO l1.YaJIH3Ma. OH npel1.nOJlOmHJI, qTO COqeTaHHe BOJlHOBbIX
H KopnYCKYJI.HpHbIX CBOHCTB .HBJI.HeTC.H 06HM CBOHCTBOM JlI06bIX MaTepHRJlb-
HbIX 06'beKToB.
STO 3HaqHT, qTO JlI060e TeJlO MaccoH m, l1.BHmyeeC.H co CKOpOCTblO V, MO-
meT 6bITb oxapaKTepH30BaHO He TOJlbKO KopnYCKYJI.HpHbIMH napaMeTpaMH (KO-
Opl1.HHaTOH B npOCTpaHCTBe, IiMnYJlbCOM, 3HeprHeH), HO H BOJlHOBbIMH (qaCTOTOH
H l1.JIHHOH BOJlHbI). no npel1.nOJlOmeHHIO l1.e BPOHJI.H. l1.JIHHa BOJIHbI l1.JI.H JlI06oro
MaTepHaJIbHOrO 06'beKTa CB.H3aHa C ero HMnYJlbCOM. a qaCTOTa - C 3HeprHeH
TaKHMH me COOTHOmeHH.HMH, KaK H )J;JI.H CPOTOHOB.
h h W
AB =-=-; V=-.
P mv h
h
A B = - - 4>OPMYJI8 .n;e BPOHJIH.
mv
B HaCTO.Hxnee BpeM.H BOJlHbI. KOTopbIe conOCTaBJI.HIOT JlI060MY MRTepHRJlbHOMY
06'beKTY (qaCTHue), Ha3bIBaIOTC.H BOJIHaMH lIe BPOHJI.H. a lIJIHHa BOJlHbI AB -
)J;e6pOHJleBCKOH l1.JlHHOH BOJlHbI.
ECJIH CKOpOCTb v qaCTHbI 6J1H3Ka K CKOpOCTH CBeTa, TO nol1. p nOHHMRIOT
peJl.HTHBHCTCKHH HMny JlbC:
mv
Ppen = g 2
1--
c 2
BCKope rHnOTe3a l1.e
1927 r. K. ,ZJ;eBHccoH H
h l- v 2
c 2
H )J;JlHHa BOJlHbI )J;e BPOHJI.H AB =
mv
BpOHJI.H 6bIJla nO)J;TBepm)J;eHa 3KcnepHMeHTaJlbHO. B
JI. ,ZJ;mepMep (CillA) , a B 1928 r. n. C. TapTaKoBc-
591

KHH (JIeHHHrpaCKHH YHHBepCHTeT) H r. TOMCOH (AHrJlH.H) Ha6J1IOaJlH ,lJ;H-
cppaKHIO 3J1eKTpoHOB.
B OnbITaX aMepHKaHCKHX yqeHLlx Y3KHH nyqoK aJleKTpOHOB, HMelOHX on-
peAeJleHHYIO CKOpOCTb, HanpaBJI.HJlC.H Ha MOHOKpHCTaJlJJ HHKeJl.H H OTpamaJIC.H OT
Hero. CneHaJILHbIM npHeMHHK perHCTpHpOBan OTpameHHbIe 3J1eKTpOHbI. OmHAa-
eMoro 3epKaJlLHOrO OTpameHH.H 3J1eKTpOHOB He Ha6J1IO,lJ;aJIOCL: OHH OTpamaJIHCL
OT nOBepXHOCTH KpHCTaJIJla no,lJ; pa3JIHqHLIMH yrJlaMH. IIpH aTOM 6LIJla 06Hapy-
meHa CJlOmHaa 3aBHCHMOCTb HHTeHCHBHOCTH oTpameHHoro nyqKa OT yrJl8 pac-
Ce.HHH.H 3J1eKTpOHOB. STY 3aBHCHMOCTL YaJIOCb 06'bHCHHTb KaE CJleACTBHe .HBJle-
HHH AHcppaKHH aJleKTpOHOB Ha aToMax KpHCT8..1IJlHtJeCKOH pemeTKH HHKeJl.H (KaK
Ha AHcppaKHOHHOH pemeTKe). )IHCPPaKHH 3J1eKTpOHOB CTaHOBHJlaCb 3aMeTHOH
JlHmb npH TaKHX CKOpOCT.HX lIBHmeHH.H 3J1eKTpOHOB, npH KOTOpbIX ,lJ;e6pOHJleB-
CKaH lIJlHHa BOJlHbI cpaBHHMa C nepHO,lIOM KpHCTaJIJlHqeCKOH pemeTKH.
TOMCOH H TapTaKOBCKHH Ha6J1IOl{aJIH AHcppaKHIO 6LICTpbIX aJleKTpOHOB npH
npoxomAeHHH CKB03b TOHKYIO (TOJlHHOH 1 MKM) MeTaJIJlHqeCKYIO cpOJlbry, HMe-
IOYIO nOJlHKpHCTanJlHqeCKYIO CTPYKTYPY. Ha «poTOnJlaCTHHKe BOKpyr eHTpaJlb-
Horo n.HTHa 6LIJIH nOJlyqeHbI qepe,lJ;YIOHeC.H CBeTJlbIe H TeMHLIe KOJlbn;a. 110
paAHycaM KOJle onpeAeJlHJlaCL ,lJ;JlHHa BOJlHbI AB H npOBep.HJlaCL CnpaBe,lJ;JlH-
BOCTb «POPMYJlbI ,lJ;e BpoHJI.H.
B 1949 r. COBeTCKHe cpH3HKH JI. M. BH6epMaH, H. r. CymKHH H
B. A. <I>a6pHKaHT npOBeJlH onbITbI no AH«ppaKHH O,lJ;HHOqHbIX nOOqepeHO JleT.H-
HX 3J1eKTpOHOB: npOMemYTOK BpeMeHH Mem,lJ;y nOn8AaHH.HMH Ha 3KpaH ,lJ;BYX
3J1eKTpOHOB, JleT.HHX APyr 3a ApyrOM, 60Jlee qeM B 10 4 pa3 npeBbImaJl BpeM.H
npoxomAeHHS:I npH60pa KambIM 3J1eKTpoHoM. IIpH MHoroKpaTHo nOBTOp.HlOeM-
C.H onbITe Ha6J1IOAaJIH T8KYro me AHcppaKHoHHYIO KapTHHY, KaE H npH AH-
cppaKHH nOTOKa 3J1eKTpoHoB. BepO.HTHOCTb nOna,lJ;aHH.H aJleKTpOHoB, npomeAmHx
CKB03h BeecTBo, B TY HJlH HHYIO TOqKY 3KpaHa onpe,lJ;eJl.HeTC.H BOJlHOBbIMH CBOH-
CTBaMH ,lJ;BHmYHXC.H 3J1eKTpoHoB: 3aBHCHT OT ,lJ;JlHHbI ,lJ;e6pOHJleBCKOH BOJlHbI.
BnOCJleACTBHH BOJlHOBbIe CBoHcTBa 6bIJlH 06HapymeHbI H Y HeHTpOHOB, npo-
TOHOB, Y aTOMOBIi MOJleKYJI.
SKcnepHMeHTaJlLHOe oKa3aTeJlbCTBO HaJlHqH.H BOJIHOBLIX CBOHCTB MHKpOqa-
CTH npHBeJlO K BbIBO,lJ;Y 0 TOM, qTO COqeTaHHe BOJlHOBbIX H KopnYCKYJI.HpHbIX
CBOHCTB - 3TO 06ee CBOHCTBO MaTepHH.
SKcnepHMeHTaJlLHO 06HapymHTb BOJlHOBbIe CBoHcTBa MaKpOCKOnHqeCKHX
06'beKTOB (HanpHMep, neCqHHOK, TeHHHCHbIX mapOB H: AP.) He YAaJIOCL. 3TO
06'b.HCH.HeTC.H TeM, qTO AJI.H TaKHX 06'beKTOB AB OqeHb MaJIa.
PaCCqHTaeM, HanpHMep, AB lIJlH TeJla MaccoH 1 r. lIBHmyerOC.H co CKOpO-
M
CTblO 0,5
C
AB =  = 6,62.10- 34 Am. C == 10- 30 M.
mv 10- 3 Kr. 0 5 M
, C
3HaqeHHe A.B nOJlyqHJlOCb Ha 20 nOp.H,lJ;KOB MeHLme pa3MepOB aToMal ITepH-
OAHqeCKHX CTPYKTYP C nepHOAOM d -10- 30 M He cyecTByeT. II03TOMY CqHTa-
eTC.H, qTO MaKpOCKOnHqeCKHe TeJla npO.HBJI.HIOT TOJlbKO OHY CTOPOHY CBOHX
CBOHCTB - KopnYCKYJI.HpHYIO - H He npO.HBJI.HIOT BOJlHOBYro.
HaJlHqHe BOJlHOBbIX CBOHCTB Y MHKpOqaCTHn; n03BOJlHJlO yqeHbIM onHCLI-
BaTb HX .Q;BHmeHHe B BHAe BOJlHOBOrO npoecca. M. BopH ,AaJI BepO.HTHOCTHYIO
592

TpaKTOBKY BOJIH l1.e BpOHJI.H: K6aOpam aMn.n.UmYObt oe6pOU.n.e6CKOU 60.n.libl 6
iJaliliou mOIlKe npOCmpaliCm6a Sl6Jl.SlemCSl MepOU 6epOSlmliOCmU mozo, Ilmo Ilac-
mUija liaxooumcSl 6 amou mOIlKe.
21.2. CooTHomeHHe Heonpe.n;eJIeHHOCTeii B. reii3eH6epra
06HapymeHHe BOJlHOBbIX CBOHCTB qaCTH[( npHBeno K OTKpbITHIO c}>YHl1.aMeHTaJIb-
Horo 3aKOHa, ynpaBJI.HIOero BceMH 8BJleHH.HMH MHpa MHKPOq8CTH[(, - COOTHO-
meHH.H HeOnpel1.eJleHHOcTeH. OHO 6bIno OTKpbITO B 1927 r. B. reH3eH6eproM.
PaccMoTpHM lIHcppaKuHIO 3J1eKTpOHOB Ha lUeJlH.
llYCTb nOTOK 3J1eKTpoHOB npoXOI1.HT qepe3 Y3KyID enb
illHpHHOH f).. x, pacnOJlOmeHHyID nepneHI1.HKY JI.HpHO H8-
npWJJleHHIO HX I1.BHmeHH.H (pHC. 21.1). TaK KaK 3neK-
TpOHbI 06J1MaIOT BonHOBbIMH CBOHCTBaMH, TO npH p83-
Mepax enH, cpaBHHMbIX C I1.nHHOH BOJlHbI l1.e BpOH-
JI.H 3neKTpoHa, BO BpeMS:I HX npoxom.n;eHH.H qepe3 enb
H86J1lOl1.aeTC.H I1.Hc}>PaK[(HSI. )1;Hc}>p8K[(HOHHa.H K8pTHHa,
H86J1lOl1.aeM8S1 Ha 3KpaHe 3, xapW<TepH3YeTCS:I rn8BHbIM
M8KCHMYMOM, pacnonomeHHbIM CHMMeTpHqHO OCH Y
(n060qHbIe MaKCHMYMbI Ha pHC. 21.1 He YKa3aHbI B
B1:IJ1.y HX OTHOCHTeJlbHO MaJIOH HHTeHCHBHOCTH).
)1;0 npoxom)J;eHH.H qepe3 eJlb 3J1eKTpOHbI I1.BH-
raJlHCb BlIOJlb OCH y, n03TOMY COCT8BJI.HlOlUa.H HM-
nYJlbCa Px= 0, TaK 'ITO llpx= 0, a KOOpl1.HH8Ta x qa-
CTH[(bJ .HBnS:leTC.H COBepmeHHO HeOnpel1.eneHHoH. B
MOMeHT npOXOml1.eHH.H 3J1eKTpOHOB qepe3 eJlb HX nOJlOmeHHe B HanpaBJleHHH
OCH OX onpel1.en.HeTC8 C TOqHOCTblO 11.0 mHpHHbJ eJlH AX. B 3TOT MOMeHT
BCJlel1.CTBHe l1.HcppaK[(HH 3J1eKTpOHbI OTKJlOH.HIOTC.H OT nepBOHaqaJlbHOro Hanp8B-
neHH.H It I1.BHmYTC.H B npel1.eJlax yrna 2cp (cp - yroJl. cooTBeTcTBYIOHH nep-
BOMY l1.HcppaK[(HOHHOMY MHHHMYMY). CJle)J;oBa TenbHO, nO.HBJI.HeTC8 HeOnpel1.eneH-
HOCTb B 3HaqeHHH COCTaBJI.HIOeH HMnYJlbCa Bl1.0Jlb OCR OX, KOTOpa.H, K8K BHl1.-
HO H3 pHCYHKa, paBHa
Y
x
--
-I llx '--
iIt
Puc. 21.1
:3
/).Px = P sin <po
h h .
A T8K KaK p =:;-, TO /lpx = -Sln<p.
.l\.B A.B
113 TeopHH I1.Hc}>paKH H3BeCTHO, qTO nepBblli MHHHMYM COOTBeTCTBYeT yrny <po
Y 11.0BJleTBOp.HlOII\eMY YCJlOBHIO
Axsin<p == A.B' OTKYI1.8 Ax =.
sin cp
H3 3THX cpOPMY JI HMeeM llx. IIp = h .
)1;JI.H HeKoTopoH, XOT.H H He3HaqHTeJlbHO:U qaCTH 3J1eKTpOHOB, nOnal1.alOHX
3a npe)J;eJlbI rJlaBHOrO MaKCHMYMa, BeJlHqHH8 /lpx > Px sin cpo CJleI1.0BaTeJlbHO, B
o6eM cJlyqae I1x .l1p  h. XOT.H 3TO COOTHomeHHe nonyqeHO ,An.H ,AHc}>P8K[(HH
3J1eKTpOHOB Ha Ol1.HOH eJlH, 3KcnepHMeHTbI nOK83bIBaIOT, qTO OHO BepHO ,An.H
JlI06bIX qaCTH[(. llpOBel1.S:I 60Jlee TaTeJlbHbIH aHaJIH3, reH3eH6epr 06HapymHJI,
593

qTO B nyqmeM cJlyqae HeOnpe)J;eJleHHOCTH l!x H 6.px y.n;OBJleTBOp.HIOT YCJlOBHIO
h h
I1x . I1px  -; - = Ii; TOr,Aa I1X' I1px  11.
21t 21t
CMbICJI nOJlyqeHHoro COOTHOmeHHSI, KOTOpOe Ha3bIBaeTC.H COOTHOWeHHeM HeO-
npeAeJIeHHOCTeil, 3aKJllOqaeTC.H B cJle.n;YIOeM. llpH JlI060H nOnbITKe BCe 60Jlee
TOqHOrO onpe.n;eJleHH.H nOnOmeHH.H qaCTHbI B npOCTpaHCTBe (ee KOOp,AHHaTbI)
06.H3aTeJlbHO 6y.n;YT BCe MeHee onpe.n;eJleHHbIMH CBe.n;eHH.H 06 HMnynbce qaCTH-
II.bI. ECJlH me KaKHM-JlH60 cnoc060M 60Jlee TO'lHO onpe.n;eJl.HTb 3HaqeHHe liM-
nYJlbca qaCTHbI, TO Bce MeHee TOqHbIMH 6Y)J;YT CBe)J;eHH.H 0 Koop.n;HHaTax.
llpHHHnHaJlbHO BamHO, qTO COOTHomeHHe Heonpe.n;eJleHHOCTeH He CB.H3aHO
C HeCOBepweHCTBOM npHMeH.HeMbIX H3MepHTeJlbHbIX npH60pOB, a OTpamaeT OCO-
6eHHOCTH <pH3HqeCKOH npHpO,AbI 06'beKTOB MHKpOMHpa.
llpHHII.Hn HeOnpe,AeneHHOCTeH CB.H3bIBaeT TaKme Heonpe.n;eJleHHOCTb 3HeprHH
06'beKTa (qaCTHUbI) l! W H BpeMeHH B3aHMOlIeHCTBH.H 06'beKTa (qaCTHUbI) C H3-
MepHTeJlbHbIM npH60pOM l! t. AJI.H qJOTOHa, H3J1yqaeMoro aTOMOM, HMeeM
l!W
I1px = -, t1x = cl1t.
c
llO,ACTaBHM B COOTHoweHHe Heonpe.n;eneHHOCTeH
l!W
- cl!t  Ii => I1W .l!t  Ii,
c
r.n;e l! W - Heonpe.n;eJleHHOCTb 3HaqeHH.H 3HeprHH qaCTHII.bI B COCTO.HHHH C 3Hep-
rHeH W, l! t - Heonpe.n;eJleHHOCTb BpeMeHIi t npe6bIBaHH.H qaCTHII.bI B ,AaHHOM
COCTO.HHHH.
COOTHoweHHe Heonpe.n;eJleHHOCTeH n03BOJl.HeT oeHHTb, HanpHMep, B KaKOH
Mepe MomHO npHMeH.HTb nOH.HTH.H KJlaCCHqeCKOH MexaHHKH K MHKpOqaCTHaM.
B qaCTHOCTH, C KaKOH CTeneHblO TOqHOCTH MomHO rOBopHTb 0 TpaeKTOpH.HX
MHKpOqaCTHII.. H3BeCTHO, qTO )J;BHmeHHe no TpaeKTopHH xapaKTepH3yeTC.H BnOJl-
He onpe,AeJleHHbIMH 3HaqeHH.HMH KOOp,AHHaT H CKOpOCTH.
OII.eHHM XapaKTep lIBHmeHH.H 3J1eKTpOHa B 3J1eKTpoHHo-nyqeBoH Tpy6Ke. 3eCb
V x = 1,0. 10 8 M IIycTb CKOpOCTb onpe,AeJleHa C TOqHOCTblO .n;o O,Olv x = 10 6 M
C C
Ii 6,62. 10- 34 -8
Tor.n;a /).x = - = 31 (j3 == 7,3.10 M, T.e. KpaHHe MaJIa no
ml!v x 2.3,14.9,1.l(f .1
cpaBHeHHIO C pa3MepaMH n.HTHa H8 3KpaHe. Cne,AOBaTeJlbHO, B .n;aHHOM cnyqae
MomHO rOBopHTb 0 .n;BHmeHHH 3J1eKTpOHa no onpe.n;eneHHoH TpaeKTopHH.
PaccMoTpHM 3J1eKTpOH, .n;BHmYHHC.H B aTOMe Bo.n;opo.n;a. llYCTb l!x ==
10 Ii 6 M
== 1,0 . 10- M (nOp.H.n;oK pa3Mepa CaMoro aTOMa). Tor)J;a l!v x = - == 7,3 .10 - .
m/).x C
PaCqeTbI nOKa3bIBaIOT, qTO CKOpOCTb 3J1eKTpOHa B aTOMe no op6HTe pHyca
r=0,5.10- 1O M paBHa 2,3 .10 6 . TaKHM 06pa30M, l!v x > vxl OqeBH,AHO, qTO
C
B .n;aHHoM cJlyqae HeJlb3.H nOJlb30BaTbC.H ypaBHeHH.HMH KJlaCCHqeCKOH qJH3HKH.
594

21.3. OCHOBLI KBaHTOBO-MeXaHH'IeCKHX npeACTaBJIeHHH.
.n;HcKpeTHocTL 3HeprHH H APyrHx H3H'IeCKHX BeJIH'IHH
SKcnepHMeHTaJlbHOe nO):{TBepm):{eHHe rHnOTe3bI Ae BpOHJI.H 06 YHHBepCaJlb-
HOCTH KopnYCKYJI.HpHO-BOJlHOBOrO ):{yaJlH3Ma, OrpaHHqeHHOCTb npHMeHeHH.H KJlaC-
CHqeCKOH MexaHHKH K MHKpo06eKTaM, ):{HKTyeMa.H COOTHomeHHeM Heonpe):{e-
JleHHOCTeH, a TaKme npOTHBOpeqH.H Mem):{y pe3YJlbTaTaMH neJloro p.H):{a 3Kcne-
pHMeHTOB H npHMeH.HeMbIMH ):{JI.H HX 06.HCHeHH.H TeOpH.HMH npHBeJlH B HaqaJIe
XX B. K C03):{aHHIO KBaHToBoH cpH3HKH.
KBaHTOBa.H cpH3HKa - pa31Ien cOBpeMeHHoH TeOpeTHqeCKOH cpH3HKH. H3yqalO-
HH 3aKOHbI ,nBHmeHH.H qaCTHU B 06J1aCTH MHKpOMHpa. 06eKTaMH H3yqeHH.H
KBaHToBoH MexaHHKH .HBJI.HIOTC.H KpHCTaJIJlbI, MOJleKYJlbI, aTOMbI, aTOMHbIe .H):{pa
H 3J1eMeHTapHble qaCTHnbI (3J1eKTpOHbI, npOTOHbI, HeHTpoHbI, Me30HbI H ):{p.).
q>H3HqeCKHMH OCHOBaMH KBaHToBoH cpH3HKH .HBJI.HIOTC.H:
1) npe):{CTaBJleHH.H M. TIJlaHKa 0 KBaHTax 3HeprHH;
2) npe)J;CTaBJleHH.H A. SHHmTeHHa 0 cpoToHax;
3) H):{eH ):{e BpOHJI.H 0 BOJlHOBbIX CBOHCTBax qaCTHn.
OnHcaHHe ):{BHmeHH.H -qaCTHn B KBaHTOBOH cpH3HKe HOCHT cBoe06pa3HbIH,
HeKJlaCCHqeCKHH, xapaKTep. B npHMeHeHHH K qaCTHnaM nOH.HTH.H onpe)J;eJleH-
Horo MeCTOnOJlOmeHH.H H TpaeKTopHH Tep.HIOT CMbICJI. KBaHTOBa.H cpH3HKa ):{aeT
CTaTHCTHqeCKOe onHcaHHe COCTO.HHH.H qaCTHn. OHa n03BOJl.HeT JlHllIb npe):{CKa-
3aTb, C KaKOH BepO.HTHOCTblO qaCTHna MomeT 6bITb 06HapymeHa B pa3J1HqHbIX
TOqKaX npOCTpaHCTBa.
B KBaHTOBOH cpH3HKe COCTO.HHHe MHKpoqaCTHnbI onHCbIBaeTC.H C nOMOblO TaK
Ha3bIBaeMOH BOJIHOBOH CPYHKI.\HH 'V (nCH-cpYHKI.\H.H). OHa .HBJI.HeTC.H OCHOBHbIM HO-
CHTeJleM HHcpOpMH 0 KopnYCKYJI.HpHbIX H BOJlHOBbIX CBOHCTBax MHKpOqaCTHn.
OCHOBHbIM ypaBHeHHeM KBaHTOBOH MexaHHKH, onHCbIBB.1OHM )J;BHmeHHe MHK-
pOqaCTHn B pa3J1HqHbIX CHJlOBbIX nOJl.HX, .HBJI.HeTC.H ypaBHeHHe llipe)J;HHrepa:
v 2 \jJ + 2';' (w - u) \V = 0,
h
r):{e CHMBOJlOM V2'V 0603HaqeHa CYMMa BTOpbIX qaCTHbIX npOH3BO)J;HbIX BOJlHO-
BOH cpYHKnHH no Koop):{HHaTaM, W - nOJlHa.H 3HeprH.H qaCTHnbI B CTanHO-
HapHOM COCTO.HHHH, m - Macca qaCTHnbI, U - nOTeHnHaJlbHa.H 3HeprH.H qac-
h
THnbI, Ii = -.
21t
Yp8.BHeHHe UIpe)J;HHrepa, KaK H Bce OCHOBHbIe ypaBHeHH.H CPH3HKH (HanpH-
Mep, ypaBHeHH.H HblOTOHa B KJIaCCHqeCKOH MexaHHKe H vpaBHeHH.H MaKCBeJlJla
)J;JI.H 3J1eKTpOMarHHTHoro nOJl.H), HeJlb3.H BbIBeCTH H3 KaKHX-JlH60 H3BeCTHbIX pa-
Hee COOTHomeHHH. OHO nOCTYJlHpyeTC.H. IIpaBHnbHocTb 3Toro ypaBHeHH.H no):{-
TBepm)J;aeTC.H COrJlaCHeM C onbITOM nOJlyqaeMbIx C ero nOMOblO pe3YJlbTaTOB.
Oc060e 3HaqeHHe HMeeT ypaBHeHHe llipe):{HHrepa ):{JI.H onHCaHH.H ):{BHmeHH.H
3J1eKTpOHOB B aTOMax, MoneKYJlax H KpHCTaJlJlaX TBep):{bIX TeJl.
PemeHHe 3Toro ypaBHeHH.H BbIXO):{HT 3a paMKH mKOJlbHOH nporpaMMbI. Oc-
TaHOBHMC.H TonbKO Ha HeKOTopbIX OCHOBHbIX CJle):{CTBH.HX, BbITeKalOHX H3 pe-
meHH.H ypaBHeHH.H, KOTopbIe OTpamalOT 06eKTHBHo cyeCTBYIOHe 3aKOHOMep-
HOCTH. YpaBHeHHe llipe)J;HHrepa n03BOJl.HeT HaHTH BepO.HTHOCTb Haxom)J;eHH.H
qaCTHnbI B pa3J1HqHbIX TOqKaX npocTpaHcTBa.
595

KBaHTOBaHHe 3HeprHH. C nOMOblO 3TOrO ypaBHeHH.H MOmHO .n;OK838Tb
O)J;HH H3 cpYH)J;8MeHTaJlbHbIX BbIBO.n;OB COBpeMeHHOH CPH3HKH 0 TOM. liTO 3liep-
ZUU 3.n.eKmp01i06 B aTOMaX, MOJleKYJlaX H KpHCTaJlJlaX He MorYT HMeTb npo-
H3BOJlbHble 3HaqeHH.H. OHH MorYT npHHHMaTb JlHWb onpeiJe.n.elilibte. iJucKpem.
libte 31ialleliUSl. m.e. K6alimYlOmcSl. KBaHTOBaHHbIe 3HaqeHH.H 3HeprHH W n Ha-
3bIBaIOTC.H YPOBH.HMH 3HeprHH. 8 qHCJlO n. onpe)J;eJl.HIOee 3HepreTHlieCKHe YPOB-
HH qaCTHIJ;bI. - z.n.a6libtM K6alim06bt.M 'lUC.n.OM.
TaKHM 06pa30M. H.n;e.H KB8HTOBaHHOCTH 3HepreTHqeCKHX COCTO.HHHH, KOTO-
pa.H Y Bopa cpOPMYJlHpOBaJIaCb B KaqeCTBe nOCTYJlaTa. OKa3bIBaeTC.H CJle)J;CTBH-
eM ypaBHeHH.H UUpe.n;HHrepa.
KBaHTOBaHHe MOMeHTa HMnYJILca. B KJlaCCHqeCKOH cPH3HKe qaCTHIJ;a. KOTO-
pa.H Bpa:I.Q:aeTC.H BOKpyr HeKOTopOH OCH. xapaKTepH3yeTC.H MOMeHTOM HMnYJlbCa
i = lro. Hanp8BJleHHbIM B.n;OJlb OCH BpaeHH.H. r.n;e 1 - MOMeHT HHepIJ;HH.
<.0 - yrJlOBa.H CKOpOCTb Bp8eHH.H. IIpH 3TOM H3BeCTeH MO.n;YJlb MOMeHT8 HM-
nYJlbca L = mur H TPH ero npOeKIJ;HH Lx. Ly. Lz Ha OCM KOOp.n;HH8T.
B KB8HTOBOH MeX8HHKe Jl106a.H Bpa81Oa.HC.H CHCTeMa MOmeT 6bITb oxapaK-
TepH30BB.Ha o.n;HOBpeMeHHO TOJlbKO .n;BYM.H napaMeTpaMH MOMeHTa HMnYJlbC8 - ero
Mo.n;YJleM H O.n;HOH H3 Tpex npoeKHH BeKTopa MOMeHTa HMnYJlbca. HanpHMep Lz.
.D:Be .n;pyrHe npoeKHH Lx H Ly He MorYT 6bITb onpe.n;eJleHbI B CHJlY COOTHOIIIe-
HH.H Heonpe.n;eJleHHOCTeH. TaK KaK. 3acpHKCHpOBaB Lz. MbI BHOCHM Heonpe.n;eJleH-
HOCTb B Lx H Ly. II03TOMY B KB8HTOBOH MeX8HHKe 381I8HHe L 2 H OJJ;HOH H3
npoeKUHH MOMeHTa HMnYJlbca nOJlHOCTblO OnpelIeJl.HeT BpamaTeJlbHOe COCTO.HHHe
TeJl8 (liaCTHUbI).
H3 peIIIeHH.H ypaBHeHH.H UUpe.n;HHrep8 BbITeK8eT. qTO MO.n;YJlb MOMeHT8 HM-
nYJlbca JlI060H CHCTeMbI He MomeT 6bITb npOH3BOJlbHbIM. a npHHHMaeT TOJlbKO
.n;HCKpeTHbIe 3HaqeHH.H. onpe.n;eJl.HeMbIe cpOPMYJlOH
L 2 =h21(1+1),
h
r)J;e h = - - nOCTO.HHHa.H IIJlaHKa, )J;eJleHHa.H Ha 21[, a 1 Ha3bIBaeTC.H op6u.
21t
ma.n.bH.btM K6alim06bt.M 'lUC.n.O.M. 3TO paBeHCTBO onpe.n;eJl.HeT 3aKOH KBaHTOBa-
HH.H MOMeHT8 HMnYJlbca. T.e. .n;JlHHY BeKTopa i. IIpH 3a.n;aHHOM n qHCJlO 1
MomeT npHHHMaTb 3HaqeHH.H 1 = 0. 1.... . (n -1). T.e. Bcero n 3H8qeHHH.
113 peIIIeHH.H ypaBHeHH.H UUpe.n;HHrepa CJle.n;yeT TaKme. qTO BeKTOp i MomeT
HMeTb JIHIIIb TaKHe opHeHT8HH B npOCTpaHCTBe. npH KOTOpbIX ero npOeKIJ;H.H
Lz Ha H36paHHoe HanpaBJleHHe (HanpHMep, Ha HanpaBJleHHe BHeIIIHero MarHHT-
L Horo nOJl.H) npHHHMaeT KBaHTOBaH-
z
Lz HbIe 3HaqeHH.H. KpaTHbIe 11.
Lz = mh, r)J;e m - MarHHTHoe
KB8.HTOBOe qHCJlO. KOTOpoe npH 3a-
.n;8.HHOM 1 MOmeT npHHHMaTb 3H8qe-
HH.H m = 0. :tl. :t2. ..., :tl, T.e.
Bcero (2l + 1) 3HaqeHHH.
Ha pHCYHKe 21.2 nOKa3aHbI
B03MomHbIe OpHeHTan;HH BeKTopa i
npH 1 = 1 H 1 = 2.
596
Puc. 21.2

21.4. KBaHToBo-MeXaHH'IeCKaH MO,lJ;eJlb aTOMa BO,lJ;OpO,lJ;a
COrJlaCHO KBaHTOBOH TeopHH, B IJ,eHTpe aTOM a BO)];OpO)];a HaXO)];HTC.H .H)];PO
MaJlbIX pa3MepOB - npOTOH - C nOJlOmHTeJlbHbIM 3J1eMeHTapHbIM 3apH,AOM e.
MaccR npOTOHa 3HaqHTeJlbHO 60Jlbme MaCCbI 3J1eKTpOHa, nO:3TOMY SI,ApO OTHO-
CHTeJlbHO IJ;eHTpa aTOMa MomHO CqHTaTb HenO)];BHmHbIM. 9J1eKTpOH )];BHmeTC.H
B :3J1eKTpHqeCKOM nOJle .H)];pa.
IToTeHIJ;HaJlbHa.H :3HeprHH B3aHMoeHcTBHH 3J1eKTpOHa C .H)];POM onpe.D;eJl.HeT-
2
C.H q>OPMYJlOH W n = - , r)];e r - paCCTO.HHHe Memy :3J1eKTpOHOM H H,A-
41t£or
pOM. rpaq>H'tleCKH 3aBHCHMOCTb Wn(r) H306pameHa
Ha pHCYHKe 21.3 >KHpHOH KpHBOR - rHnep60-
JlOR. PemeHHe ypaBHeHH.H UUpe)];HHrepa n03BOJlH-
JlO paCCqHTaTb nOJlHYIO :3HeprHIO aTOMa BO,AOpO)];a:
1 me 4
W = -2' r)];e (n = 1, 2, 3, ...).
n 8R Eo
9TO BbIpameHHe COBn8.)];aeT C q>OPMYJlOH, HaR-
,AeHHOR BOpOM ,AJI.H :3HeprHH aTOMa BO)];OpO)];a. 0)];-
HaKO )J;JI.H ero nOJlyqeHH.H He HCnOJlb30BaJIHCb
KnaCCHqeCKHe npe)];CTaBJIeHH.H 0 )];BH>KeHHH :3J1eK-
TpOHa no onpe)];eJleHHOH op6HTe, npHBOHHe K
Hepa3peIIIHMbIM npOTHBOpeqH.HM.
B03MomHbIe 3HaqeHH.H :3HeprHH WI' W 2 . W 3 ...
nOKa3aHbI Ha pHCYHKe B BH)];e ropH30HTaJIbHbIX
np.HMbIX (:3HepreTHqeCKHe YPOBHH). CaMbIH HHmHHH ypOBeHb Wit COOTBeTCTBYIO-
HH MHHHMaJIbHO B03MomHOH :3HeprHH, - OCHOBHOH, Bce OCTaJIbHbIe - B03-
6ym)];eHHbIe. IIpll W < 0 )];BH>KeHHe :3J1eKTpOHa HBJIHeTCH CB.H3aHHbIM: HaxO)];HTC.H B
npe)];eJIax aTOMa. B COOTBeTCTBHH C q>OpMOH KpHBOR Wn(r) rOBOp.HT, liTO :3neKTpOH
HaxO,AHTC.H BHYTPH rHnep60JlWIeCKOH "nOTeHIJ;HanbHOR .HMbI", T .e. ero )];BH>KeHHe
OrpaHHqeHO B npOCTpaHCTBe B6J1H3H .H)];pa. H3 pHCYHKa cJle)J;YeT, 't{TO no Mepe pOC-
Ta rJl8.BHOrO KBaHTOBOro qHCJIa n 3HepreTWIeCKHe ypoBHH paCnOJlRr81OTCH TecHee H
npM n  00 W _ = O. II pH W> 0 )];BHmeHHe :3J1eKTpoHa .HBJI.HeTC.H CB060)];HbIM - OH
MomeT yHTH B 6eCKOHeqHOCTb. 06J1aCTb HenpepbIBHoro cneKTpa W> 0 (3aIllTpHXO-
BaHR Ha pHCYHKe) COOTBeTCTBYeT HOHH3HpoBaHHOMY aTOMY.
H3 ypaBHeHH.H UUpe)];HHrepa CJle)J;yeT, qTO aTOM BO)J;opO)];a MomeT HMeTb
O)];HO Ii TO me 3HaqeHHe :3HeprHIi W n , HaXO)J;.HCb B HeCKOJlbKHX COCTO.HHM.HX,
KOTOpbIe OTJlHqaIOTC.H 3HaqeHH.HMH KBaHTOBbIX qHCeJl l H m. COOTBeTCTBYlOlltMX
pa3J1HqHbIM 3HaqeHH.HM MO)J;YJI.H MOMeHTa HMnYJIbCa :3J1eKTpOHa H npOeKIJ;HH MO-
MeHTa HMnYJlbCa Lz.
)J;anbHeHIIIee pa3BHTHe q>H3HKH npHBeJlO K OTKpbITHIO Y :3J1eKTpOHa c06cm6eli-
1i0ZO MexaliUlleCKOZO MOMelima uMny.n.bca, He CB.H3aHHOro C )];BHmeHHeM :3JIeKT-
pOHa B npOCTpaHCTBe, - cnulia. BHaqaJIe CqHTaJIOCb, qTO cnHH 06YCJlOBJleH
BpaeHHeM :3J1eKTpOHa BOKpyr CBoeH OCH, qTO npHBeJlO K IJ,eJlOMY p.H)];Y npOTH-
BOpeqHH. B HaCTO.Hee BpeM.H YCTaHOBJleHO, qTO cnHH .HBJI.HeTC.H HeOT'beMJleMbIM
CBOHCTBOM :3J1eKTpOHR, nO)];06HO ero 3ap.H)J;Y H Macce. CnHH :3J1eKTpOHa - KBaH-
TOBa.H BeJlHqHHa, Y Hee HeT KJlaCCHqeCKOrO aHaJlOra.
W u
o
fW a
W, W 2
lw
I
Puc. 21.3
597

MO,lJ; YJlb CnH HB 8J1eKTpOHa MOmeT npHHHMaTb TOJlbKO ,nHCKpeTHbIe 3HaqeHH.H:
Ls = I1  S(S + 1), r,ne s - cnHHOBoe KBaHTOBoe 1IHCJlO 8J1eKTpOHa - HMeeT
1
TOJlbKO O,nHO 3HaqeHHe s = -.
2
IIpoeKn;H.H cnHHa 8J1eKTpOHa Ha BbI6paHHYIO Koop,nHHaTHYIO OCb MomeT npH-
HHMaTb TOJlbKO ABa 3HaqeHH.H:
11 11
L Sz = 2' H L Sz = - 2"
MTaK, COCTO.HHHe 8J1eKTpOHa B aTOMe xapaKTepH3yeTC.H qeTbIpbM.H KBaHTO-
BbIMH qHCJlaMH: rJlaBHbIM n, Op6HTaJlbHbIM l, MarHHTHbIM m H cnHHOBbIM S.
,I:(JI.H 0603HaqeHH.H pa3J1HqHbIX COCTO.HHHH 8J1eKTpOHa B aTOMe HcnOJlb3YlOT
3aHMCTBOBaHHbIe 3 cneKTpOCKonHH YCJlOBHbIe 0603HaqeHH.H.
COCTO.HHH.H 8J1eKTpOHa co 3HaqeHH.HMH l = 0 Ha3bIBalOT S-COCTO.HHH.HMH (8J1eK-
TpOH B 8TOM COCTO.HHHH - S-8J1eKTpOHOM), l = 1 - p-COCTO.HHH.HMH, l = 2 -
d-COCTO.HHH.HMH, l = 3 - f-COCTO.HHH.HMH Ii T..lI.. 3HaqeHHe rJlaBHOrO KBaHTOBO-
ro qHCJla YKa3bIBaeTC.H nepe,lJ; YCJlOBHbIM 0603HaqeHHeM Op6HTaJlbHOrO KBaHTO-
Boro t.J:HCJla. HanpHMep, 8J1eKTpOHbI B COCTO.HHH.HX n = 2 H l = 0  1 0603Haqa-
IOTC.H COOTBeTCTBeHHO 2s H 2p.
B03MomHbI CJle,nYIOHe COCTO.HHH.H 8J1eKTpOHa:
Is
2s, 2p
38, 3p, 3d
4s, 4p, 4d, 4f H T.,lJ;.
06ee qHCnO B03MO]KHbIX COCTO.HHHH 8J1eKTpOHa C O,nHHM H TeM me 3Ha-
1IeHHeM rnaBHoro KBaHTOBoro qHCna n, HO pa3J1HqHbIMH 3HaqeHH.HMH l, m, S
I=n-l
paBHO 2 L (2l + 1) = 2n 2 .
1=0
IIocKonbKY npH ,lJ;BHmeHHH 8J1eKTpOHa B aTOMe cyecTBeHHbI ero BOJlHOBbIe
CBOHCTBa 8J1eKTpOHa, KBaHTOBa.H MeXB.HHKa OTKa3bIBaeTC.H OT KJlaCCHqeCKOrO npe,A-
CTaBJleHH.H 06 8J1eKTpOHHbIX op6HTax. COrJlaCHO KBB.HTOBOH MexaHHKe, Kam,AOMY
8HepreTHqeCKOMY COCTO.HHHIO COOTBeTcTByeT BOJlHOBa.H cpYHKH.H. KBa.u.paT ee MO-
,nyn.H onpe,neJl.HeT BepO.HTHOCTb 06HapymeHH.H 8J1eKTpOH8. B e,nHHHe 06'beMa.
BepO.HTHOCTb o6HapymeHH.H 8J1eKTpOHa B pa3J1HqHbIX qaCT.HX aTOM a pa3JIH1I-
Ha. 8J1eKTpOH npH CBoeM ,nBH:>KeHHH KaK 6bI "pa3Ma3aH" no BceMY 06'beMY,
06pa3Y.H TaK Ha3bIBaeMoe 8J1eKTpOHHoe 06JIaKO.
O.ll.ex:mpOlilibtM 06.n.aKOM Ha3bIBalOT onpe,neJleHHOe pacnpe,neJleHHe B aTOMe
,ne6pOHJleBCKOH BOJlHbI 8J1eKTpOHa. IIJlOTHOCTb ee pacnpe,lJ;eJleHH.H xapaKTepH3Y-
eT BepO.HTHOCTb HaXOm,lJ;eHH.H 8J1eKTpOHa B pa3J1Ht.J:HbIX TOqKaX 06'beMa aTOMa.
B KBaHTOBOH MexaHHKe npHH.HTO CqHTaTb, qTO KBaHTOBbIe t.J:HCJla n H 1
XapaKTepH3yIOT pa3Mep H cpOPMY 8J1eKTpOHHoro 06J1aKa, a KBB.HTOBOe '{HCJlO m
opHeHTan;HIO 8J1eKTpOHHoro 06J1aKa B npOCTpaHCTBe.
AToM Bo,nopo,na B lS-COCTO.HHHH MomHO npe,nCTaBHTb B BH,ne ccpepbI. B ee
n;eHTpe Haxo,nHTC.H .H,lJ;pO, oKpymeHHoe ccpepHt.J:eCKH CHMMeTpHqHbIM 8J1eKTpOHHbIM
06JIaKOM C MaKCHMaJlbHOH llJlOTHOCT:b1O Ha paCCTO.HHHH r OT .H,lJ;pa, paBHOM
pa,nHYCY nepBOH 60pOBCKOH Op6HTbI (pHC. 21.4, a).
598

B nepBOM B036ym,neHHOM 2S-COCTO.HHHH 3neKTpOHHOe 06J1aKO CcpepHqeCKH
CHMMeTpHqHO, KaK H B OCHOBHOM Is-cOCTO.HHHH. O,nHRKO HRH60Jlee BepO.HT-
HOe paCCTO.HHHe 3J1eKTpOHa OT .H,npa OKa3bIBaeTC.H B 5 pa3 60Jlbllie pa,AHyca
nepBoH 60POBCKOH Op6HTbI (pHC. 21.4, 6).
a Z <1>OpMR 3neKTpOHHoro 06n8.KR B 2p-COCTOHHHH
HanOMHHaeT raHTenb (pHC. 21.5). Ha pHC. 21.6
nOKa3aHO pacnpeeJleHHe nJlOTHOCTH 3J1eKTpOH-
Horo 06J1RKR B 3S-COCTO.HHHH
it.  ......
..
y
z
z
x
6
z
,
:it- '
x
y
y
.
y
: .)-
....\
t "
x
x
Puc. 21.4
Puc. 21.5
Puc. 21.6
21.5. llpHHHn TOjK)];eCTBeHHOCTH '1aCTH.
llpHHHn 3anpeTa llaYJIH
B aTOMe BO,nOpO,nR TOJlbKO O,nHH 3neKTpOH. OH MomeT HRXO,nHTbC.fI B pa3-
JlHqHbIX COCTOHHH.HX. B OCHOBHOM COCTO.HHHH ero 3HeprHSl MHHHMRJlbHR, H B
3TOM COCTOHHHH OH MomeT npe6bIBaTb HeOrpaHHqeHHO ,noJlro. B B036ym,neH-
HbIX COCTOHHH.HX RTOM cyecTByeT OrpRHHtleHHOe BpeM.H (-10- 9 -10- 7 c) H ca-
MonpOH3BOJlbHO nepexo,nHT B OCHOBHoe COCTOHHHe C H3nyqeHHeM cpOTOHR.
AToM C HeCKonbKHMH 3J1eKTpOHaMH npe,nCTaBnHeT C060H CJlomHYIO CHCTeMY
B3aHMo,neHCTBYIOHX ,npyr C ,npyroM 3J1eKTpOHOB, ,nBHmYHXCH B KYJlOHOBCKOM
nOJle .H,nPR. IIoJle, ,neHCTBYIOee H8 HeKOTopbIH 3J1eKTpOH, 3aBHCHT OT COCTO.HHH.H
Bcex OCTaJIbHbIX 3J1eKTpOHOB. II03TOMY COCTO.HHH.H Bcex 3J1eKTpOHOB B RTOMe ,nOJI-
mHbI onpe,AeJlHTbCH o,nHOBpeMeHHo. CornRcHo KBaHTOBOH TeopHH, npH OTCYTCTBHH
BHelliHHX B03,neHCTBHH TaKa.fl CHCTeMR MomeT HRXO,AHTbCH B onpe,neJleHHbIX CTR-
lI.HOHRpHbIX COCTO.HHH.HX, xRpRKTepH3yeMbIx ,AHCKpeTHbIMH 3HRtleHH.HMH 3HeprHH.
COCTO.HHHe 3J1eKTpOHOB B MHor03neKTpOHHbIX RTOMax onHCbIBReTC.H TaKHM
me Ha60pOM KBaHTOBbIX qHCeJl. tlTO H Y aTOMR BOJJ:opOJJ:a. OJJ:HaKO 3JJ:ecb 3Hep-
rH.H 3J1eKTpOHa 3aBHCHT He TonbKO OT rJlRBHOrO KBaHTOBoro qHCJlR. YKR3bIBR-
lOm;ero Ha pa3Mep 3J1eKTpOHHoro 06JIaKa B JJ:aHHOM COCTO.HHHH, HO H OT .npy-
rHX KBaHTOBbIX qHCeJl, xapaKTepH3YIOHX KOHcpHrypauHIO 3Toro 06J1RKR.
QT06bI nOJlyqHTb npe,nCTRBneHHe 060 BceH 3J1eKTpOHHOH 060nOqKe RTOMR,
HymHo paCCMOTpeTb, KaKHM 06pa30M npOHcxo,nHT 3anOJIHeHHe 3J1eKTpOHaMH
599

pa3pellIeHHbIX CTaHOHapHbIX COCTO.HHHH OT,l{eJlbHbIX 3J1eKTpOHOB. B HeB036ym-
,l{eHHOM aTOMe 3J1eKTpOHHbIe COCTO.HHH$I ,l{OJlmHbI 6bITb 3anOJlHeHbI T8KHM 06-
pa30M, qT06bI 3HeprH.H BceH 3J1eKTpOHHOH 060JlOqKFI HMeJla HaHMeHbllIee 803-
MomHoe 3H8qeHHe. KOHeqHO, H8HMeHbllIee 3HaqeHHe 3HeprHH nOJlyqHJloCb 6bI,
eCl1H 6bI Bce 311eKTpOHbI HaXO,l{HJlHCb B H8HHH3llIeM B03MOmHOM COCTO$lHHH,
T.e. B COCTO.HHHH C n = 1. Ho OnbITHbIe ,l{aHHble CBH,l{eTeJlbCTBYIOT, qTO 3anOJl-
HeHHe 3J1eKTpOHHOH 060JlOqKH aTOMa npOHCXO,l{HT HHaqe. 3,l{eCb npO.HBJI.HIOTC$I
oc06bIe CBoHcTBa MHOr03J1eKTpOHHbIX CHCTeM, He HMelOHe aH8J10ra B Kn8CCH-
'tieCKOH cpH3HKe. C HHMH MbI H n03HaKOMHMC.H.
B KJl8CCHqeCKOH MexaHHKe ,Aame O,l{HHaKOBbIe qaCTHnbI B npHHHne MomHO
p83J1H'tiHTb no nOJlOmeHHIO B npOCTpaHCTBe H HMny JlbCaM. ECJlH 'l8CTHnbI B
K8KOH-TO MOMeHT BpeMeHH npOHYMepOBaTb, TO B CJle,AYIOHe MOMeHTbI BpeMe-
HH MomHO npOCJle!1HTb 3a TpaeKTopHeH JlI060H H3 HHX. KJl8CCHqeCKHe 'laCTH-
nbI, T8KHM 06pa30M, 06J1a,l{8IOT HHHBH,l{yal1bHOCTblO.
B KBaHTOBOH MexaHHKe nOJlOmeHHe HHoe. COCTO.HHHe qaCTHbI onHCbIB8eTC.H
BonHOBOH cpYHKHeH. OHa n03B0l1.HeT BblqHCJI.HTb l1HllIb BepO.HTHOCTb H8XOm,l{e-
HH.H MHKPOq8CTHbI B OKpeCTHOCT.HX TOH Hl1H HHOH TOqKH npOCTpaHCTBa. EcnH
me BOJlHOBbIe cpYHKnHH )J;BYX O,l{HHaKOBbIX (HMelOHX O,l{HHaKOBYIO Maccy, 311eK-
TpH'tieCKHH 3ap.H,l{, cnHH H ,l{PyrHe xapaKTepHCTHKH FI Ha3bIBaeMbIX TOm,l{eCTBeH-
HbIMH) qaCTHn B npOCTpaHCTBe nepeKpbIBalOTC.H, TO pa3rOBOp 0 TOM, KaKa.H qa-
CTHa HaXO,l{HTC.H B ,AaHHOH 0611aCTH, Bo06e JlHllIeH CMbICJla: MomHO JlHllIb
rOBopHTb 0 SepO.HTHOCTH Haxom,l{eHH.H B ,l{aHHOH 06J1aCTH O,l{HOH H3 TOm,l{eCTBeH-
HbIX q8CTH. TaKHM 06pa30M, B KBaHTOBOH MexaHHKe TOm,l{eCTBeHHble q8CTHbI
nonHOCTblO Tep$llOT CBOIO HHJJ;HBH,l{Y8J1bHOCTb H CT8HOBSITCH Hepa3J1HqHMbIMH.
He06bI'tiHbIe CBOHCTBa CHCTeMbI O,l{HHaKOBbIX TOm,l{eCTBeHHbIX qaCTH npO.HB-
n.HIOTCS! B tjJynoaMelima.n.bnOM npHHHne KB8HTOBOH MeX8HHKH - npuli",une
liepa3.n.UIlUMocmu mOJICiJecm6elilibtX Itacmuij. COZJLaCIiO eMY. lie603.M.OJIC1i0 3Kcne-
pUMelima.n.bliO pa3.n.Ullumb mOJICiJecmBelilibte ltaCmUijbt.
B COOTBeTCTBHH C npHHHnOM MHHHMYM8 3HeprHH MomHO 6bI110 OmH,A8Tb,
'tiTO Bce 3J1eKTpOHbI MHOr03J1eKTpOHHoro aTOMa 6Y,l{YT HaXOJJ;HTbC.H B O,l{HOM H
TOM me COCTO.HHHH C KBaHTOBbIMH qHCJlaMFI n = 1, l = o. Ho B ,l{eHCTBHTeJlbHO-
CTH H3}"1eHHe cneKTpOB pa311HqHbIX 8TOMOB nOK83aJlO, qTO B HeHTpaJlbHOM He-
B036ym,l{eHHOM aTOMe, CO,l{epmaeM Z 3J1eKTpOHOB, 1IHllIb ,l{Ba H3 HHX Haxo-
,A.HTC.H 8 COCTO$lHHH C MHHHM811bHOH 3HeprHeH. OCTaJlbHbIe 3J1eKTpOHbI Haxo-
,l{$lTC$I H8 60Jlee BbICOKHX 3HepreTHqeCKHX YPOBH.HX H caMonpOH3BOJlbHO He ne-
peXO,l{.HT Ha 60Jlee HH3KHe ypOBHH HeOrpaHHqeHHO ,l{OJlroe BpeM.H. TaKoe nOBe-
,l{eHHe 3J1eKTpOHOB B aTOM ax 06'b.HCHHJI B 1925 r. B. IIaYJlH.
OH CcpOPMYJlHpOBaJl ee O,l{HH cpYH,l{aMeHTaJlbHbIH 3aKOH KB8HTOBOH MeX8-
HHKH, KOTOpbIH H83bIB8eTC.H npUliijUnOM II ay.n.u: 6 .n.lOtJOU K6alim060U cucme-
Me iJ6a U tJo.n.ee 3.n.eKmpOlia lie Mozym liaxoiJumbcSl oiJlioBpeMeliliO B OiJIiO.M. U
mOM JlCe K6alim060M cocmOSlIiUU.
COCTO.HHHe 3J1eKTpOH8 B KB8HTOBOH CHCTeMe onpe,l{eJl$leTC.H qeTbIpbM$I KBaH-
TOBbIMH qHCJl8MH: n, l, m, s. CJle,l{OBaTeJlbHO, 6 KBanmOBOU cucmeMe ne .M.O-
zym cyU{ecm606amb 3.n.eKmpOlibt. y KomOpbtX 6bt.n.U 6bl oiJuliaKOBbtMU 6ce Ite-
mbtpe K6alim06btX Ituc.n.a.
IIpHHHny naYJlH nO)J;qHH.HIOTC.H He TOJlbKO 3J1eKTpOHbI, HO H JlI06bIe )J;PY-
rHe 'I8CTHnbI C nOJlyneJlbIM cnHHOM.
600

21.6. KBaHToBo-MeXaHHQeCKOe 060CHOBaHHe
nepHO,lJ;HQeCKOH CHCTeMLJ 3JIeMeHTOB J1,. H. MeH,L'(eJIeeBa
B 1869 r. )];. M. MeH,ZJ;eJleeB OTKpbIJI nepHoLIWIecKHH 3aKOH H3MeHeHH.H XHMWIec-
KHX H «pH3WIecKHX CBOHCTB 3J1eMeHTOB B 3EiBHCHMOCTH OT nOp.H)J;KOBOro HOMep8 Z.
TaK KaK XHMHqeCKHe II' HeKOTopbIe «pH3HlIeCKHe CBoHcTBa 3J1eMeHTOB 06'h.HC-
H.HIOTC.H nOBeeHHeM BHeWHHX (BaJleHTHbIx) 3J1eKTpOHOB B 8TOM8X, TO nepHO-
)J;HqHOCTb CBOHCTB XHMHqeCKHX 3J1eMeHTOB OJlmHa 6bITb CB83aHa C nepHOHq-
HOCTblO B p8CnOJlOmeHHH 3J1eKTpOHOB B aTOMaX.
TeOpeTHqeCKOe HCTOJlKOBaHHe nepHOHqeCKOrO 3aKOHa MeHeJleeB8 aeT
KBaHTOB88 MeXaHHKa. COBpeMeHHa.H TeOpH.H OCHOBbIBaeTC.H H8 'tIeTbIpeX OCHOB-
HbIX nOJlOmeHH.HX:
a) nOp.H)J;KOBbIB HOMep XHMHqeCKOrO 3J1eMeHTa p8BeH 06eMY qHCJlY 3J1eKT-
pOHOB B aTOMe aHHoro 3J1eMeHTa;
6) COCTO.HHHe 3neKTpOHOB B aTOMe onpeeJl.HeTC.H Ha60poM HX KB8HTOBbIX
qHCeJl n. l. m. s;
B) pacnpeeJleHHe 3J1eMeHTOB B aTOMe no 3HepreTHqeCKHM COCT08HH8M ,Aon-
mHO YOBJleTBOp.HTb npHHUHny MHHHMYMa nOTeHI:(H8J1bHOH 3HeprHH - C B03-
paCTaHHeM qHCJla 3J1eKTpOHOB KamLIbIH CJleYIOHH 3J1eKTpOH LIOJlmeH 3aH.HTb
B03MomHoe 3HepreTHqeCKOe COCTO.HHHe C HaHMeHbIIIeH 3HeprHeH;
r) 3anOJlHeHHe 3J1eKTpOHaMH 3HepreTHqeCKHX COCTO.HHHH B aTOMe OJlmHO
npOHCXO)J;HTb B COOTBeTCTBHH C npHHI.J;HnOM IIaYJlH.
KaK MbI BeJlH. MaKCHMaJIbHOe qHenO 3J1eKTpOHOB. HaxO.HHXC.H B pa3nHlIHbIX
COCTO.HH1:l.HX, onpe)J;eJl.HeMbIX 8.HHbIM rJlEiBHbIM KB8.HTOBbIM 'tfHenOM n, pEiBHO 2n 2 .
COBoKynHocTb 3J1eKTpOHOB B MHOr03J1eKTpOHHOM aTOMe, HMelOHX OHO H
TO me rJlaBHOe qHCJlO n, Ha3bIBaeTC.H 3.n.eKmpOIiIiOU 060.n.OIlKOU. B K8mOB H3
060JlOqeK 3J1eKTpOHbI nopa3eJl.HIOTC.H no nO060JlOqK8M, COOTBeTCTBYIOHM aH-
HOMY 3H8qeHHIO l. 'tJHCJlO no060JlOqeK paBHO nOp.HKOBOMY HOMepy N 060JlOlI-
KH. KOJlHqeCTBO 3J1eKTpOHOB B no060JlOqKe onpeeJl.HeTC8 MarHHTHbIM H cnH-
HOBbIM KBaHTOBbIMH qHCJlaMH: MaKCHManbHoe qHCJlO 3J1eKTpOHOB B no060JlOqKe
C ,lIaHHbIM l pEiBHO 2(2l + 1). 0603HaqeHH.H 060JlOqeK. 8 T8Kme p8CnpeeJleHHe
3J1eKTpOHOB no 060JlOqK8M H no060nOqKaM npeLICTaBJleHbI B T86J1Hue 21.1.
Ta6J1Hl.J;a 21.1
Pacnpe,ZJ;eneHHe 3JIeKTpOHOB no 060JIO'lKaM H no nO,ZJ;060JIO'lKaM
rnaBHoe KBaHTO- 2 3 5
Boe 'IRCnO n 1 4
CHMBon 060nO'IKH K L M N 0
M8KcHM8nbHoe
'lHCnO 3neKTpOHOB 2 8 18 32 50
B 060nO'lKe
(N 1 = 2n2)
Op6HTMbHoe 0 0 1 0 1 2 0 1 2 3 0 1 2 3 4
KB8HTOBoe 'lHCnO l
CHMBOJJ 1s 2s 2p 3s 3p 3d 4s 4p 4d 4f 5s 5p 5d 5f 5g
nO,l(060nO'lKH
MSKCHM8nbHoe
'lHCnO 3neKTpOHOB 2 2 6 2 6 10 2 6 10 14 2 6 10 14 18
B nO,l(060nO'IKe
(N 2 =2(2l+ 1»
601

rrOp.HOK 3anOJlHeHH.H 3J1eKTpOHHbIX 060JlOqeK aTOMOB XHMH'tleCKHX 3JIeMeHTOB
C nOp.HKOBbIMH HOMepaMH OT Z = 1 o Z = 36 npeCTaBJleH B Ta6J1Hne 21.2.
Ta6JIHna 21.2
nOpJJ.nOK 3anOJIHeHHH nOAo6oJIolleK aTOMOB 3JIeKTpOHaMH
IIepHOA Z 3neMeH K L M N
18 28 2p 38 3p 3d 48 4p 4d 4{
I 1 H 1
2 He 2
3 Li 2 1
4 Be 2 2
5 B 2 2 1
II 6 C 2 2 2
7 N 2 2 3
8 0 2 2 4
9 F 2 2 5
10 Ne 2 2 6
11 N8 2 2 6 1
12 Mg 2 2 6 2
13 Al 2 2 6 2 1
III 14 Si 2 2 6 2 2
15 P 2 2 6 2 3
16 S 2 2 6 2 4
17 C] 2 2 6 2 5
18 Ar 2 2 6 2 6
-
19 K 2 2 6 2 6 - 1
20 C8 2 2 6 2 6 - 2
21 Sc 2 2 6 2 6 1 2
22 Ti 2 2 6 2 6 2 2
23 V 2 2 6 2 6 3 2
24 Cr 2 2 6 2 6 5 1
25 Mn 2 2 6 2 6 5 2
IV 26 Fe 2 2 6 2 6 6 2
27 Co 2 2 6 2 6 7 2
28 Ni 2 2 6 2 6 8 2
29 Cu 2 2 6 2 6 10 1
30 Zn 2 2 6 2 6 10 2
31 Ga 2 2 6 2 6 10 2 1
32 Ge 2 2 6 2 6 10 2 2
33 As 2 2 6 2 6 10 2 3
34 Se 2 2 6 2 6 10 2 -1
35 Br 2 2 6 2 6 10 2 5
36 Kr 2 2 6 2 6 10 2 6
MbI BH.lUIM. qTO CHaqaJla 3anOJlHSlIOTC.H COCTOSIHHSI C HaHMeHbIlIeH B03MO.>K-
HOH 3HeprHeH, a 3aTeM COCTO.HHHSI co Bce 60JIbIlIeH 3HeprHeH. AJI.H MHorHX
aTOMOB 3TOT nOp.HOK cooTBeTcTByeT TOMY, 'tlTO CHa'IaJla 3anOJIH.HIOTCSI 060-
JIOqKH C MeHbIlIHM 3Ha'IeHHeM rJIaBHOrO KBaHTOBoro 'IHCJla n, a 3aTeM 060-
JlO'IKH C 60JIblI1HM n. B npeeJIax OHOH 060JIOt{KH CHaqaJIa 3anOJIH.HIOTC.H co-
602

CTO.HHH.H C l = O. a 3aTeM COCTO.HHH.H C 60nbllIHM l, BnnOTb .n;o l = n - 1. Ho
HHor.n;a 3Ta nOCJle.n;OBaTenbHOCTb HeCKonbKO HapYlliaeTC.H. TaK, H8npHMep, y Ka-
JlH.H K, KOTOpbIH no CBOHM XHMHqeCKHM H OnTH'tleCKHM CBOHCTBaM cxom C aTO-
MaMH JlHTH.H Li H HaTpH.H Na, liMelOHMIt BHellIHHH B8JIeHTHbIH 3neKTpOH B
S-COCTO.HHHH, 3anOJlHeHHe N -060JlOqKH Ha'tlHHaeTC.H npH He3anOJlHeHHOH M -060-
nOqKe. 3TO 06'b.HCH.HeTC.H TeM, 'tlTO B pe3YJlbTaTe B3aHMo.n;eHcTBH.H 3neKTpOHOB
COCTO.HHHe n = 4, l = 0 HMeeT MeHbllIYIO 3HeprHIO, tleM COCTO.HHHe n = 3, l = 2.
BnH30cTb XHMHqeCKHX CBOHCTB Fe, Co, Ni 06'b.HCH.HeTC.H o.n;HHaKOBbIM CTpO-
eHHeM BHellIHeH N-060JlOqKH, cO)J;epmaeH no .n;Ba 3J1eKTpOHa B 4S-COCTO.HHHH.
OTnHqaIOTC.H .n;pyr OT .n;pyra aTOMbI 3THX Tpex XHMH'tleCKHX 3neMeHTOB TOJlbKO
KOnHqeCTBOM 3J1eKTpOHOB Ha BHYTpeHHeH A1-060nO'tlKe, npHqeM Bce 3TH 3neKT-
pOHbI 3anOJlH.HIOT O.n;HY 3d-060nOtiKY.
TaKHM 06pa30M, OTKpbITa.H MeH.n;eJleeBbIM nepHo.n;HqHOCTb B XHMH'tleCKHX
CBOHCTBax 3J1eMeHTOB 06'b.HCH.HeTC.H nOBTOp.HeMOCTblO B CTpYKType BHellIHHX 060-
nOqeK Y aTOMOB pO,ACTBeHHbIX 3J1eMeHTOB. TaK, HHepTHbIe ra3bI (He, Ne, Ar,
Kr H .n;p.) HMelOT o.n;HHaKOBble BHellIHHe 060nOqKH C 3anonHeHHbIMH s- H
p-COCTOSlHH.HMH; BO BHellIHeH 060nO'tlKe enOqHbIX MeTanJlOB (Li, Na, K H .a;p.)
HMeeTC.H nHllIb O,AHH BHellIHHH S-3J1eKTpOH; BO BHellIHeH 060JlO'tlKe eJlOqHO-
3eMeJlbHbIX MeT8JIJlOB (Be, Mg, Ca It ,Ap.) HMeeTC.H 2 S-3J1eKTpOH8 H T..n;.
21.7. MOJIeKYJIbI. llpHpoJJ;a XHMHlIeCKOH CBH3H
MoneKY na - HaHMeHbma.H qaCTHI(8 BeeCTBa, 06na.n;alOaSl ero OCHOBHbIMH
XHMH'tleCKHMH CBOHCTBaMH. MoneKynbI COCTO.HT H3 O,AHHaKOBbIX HnH pa3nHtlHbIX
aTOMOB, coe.n;HHeHHbIX Mem.n;y C060H XHMH'tleCKHMH CB.H3.HMH. XHMHqeCKHe CB.H3H
06ycnoBJleHbI B3aHMo.n;eHcTBHeM BHeIlIHHX, BaneHTHbIX, 3neKTpOHOB aTOMOB. BHyT-
peHHHe 3neKTpOHbI, pacnOJlOmeHHbIe HaH60Jlee 6J1H3KO K .H.n;PY aTOMOB, B 06pa-
30BaHHH MOJleKY n He yqacTBYIOT. 06 3TOM rOBopHT pe3Koe H3MeHeHHe OnTH'leC-
Koro cneKTpa aTOMOB npH BCTynneHHH HX B XHMH'tleCKHe coe.n;HHeHH.H.
HaH60nee 'tlaCTO B MOJleKYJlax BCTpe'tlaIOTC.H .n;Ba THna CB.H3H - HOHHa.H Ii
KOBaJleHTH8.H.
THnH'tlHbIMH npHMepaMH MoneKyn C HOHHbIMH CB.H3.HMH aTOMOB .HBn.HIOTC.H
MOJleKYJlbI eJlOqHO-3eMeJlbHbIX coneH - NaCI, RbBr, CsI H .n;p., - 06pa30-
B8BIlIHeC.H npH coe.n;HHeHHH 8TOMOB 3neMeHTOB 1-H H 7 -H rpynn nepHo.n;H'tleC-
KOH CHCTeMbI. B3aHMo.n;eHcTBYIOHe aTOMbI npH 3TOM npeBpaaIOTC.H B HOHbi.
O.n;HH H3 aTOMOB (HanpHMep. Na). KOTOpbIH HMeeT cpaBHHTenbHO He60nbllIYlO
BeJlHqHHY nOTeHUHana HOHH3aUHH (5.1 3B), OT.n;aeT O.n;HH HnH HeCKonbKO 3neK-
TpOHOB H npeBpamaeTC.H B nOJlOmHTeJlbHbIH HOH. pyroH aTOM (HanpHMep. CI)
npHcoe.n;HH.HeT 3TH 3neKTpOHbl, npH06peTaeT OTpHuaTeJlbHbIH 3ap.H1I H CTaHOBHT-
C.f.I OTpHUaTeJlbHbIM HOHOM. 3J1eKTpOCTaTH'leCKOe npHT.HmeHHe npOTHBononomHO
3ap.HmeHHblX tlaCTHI( npHBo.n;HT K HX c6nHmeHHIO. Ha BeCbMa ManbIX paCCTO.H-
HH.HX Mem.n;y HOHaMH Ha'lHH81OT .n;eHCTBOBaTb CHnbl OTT8.JIKHBaHH.H, KOTopbIe BbI3-
BaHbI r JlaBHbIM 06pa30M OTTaJlKHBaHHeM Mem.n;y .H.n;paMH H Mem.n;y BHYTpeHHHMH
3neKTpOHaMH HOHOB Ha MaJlbIX paCCTO.HHH.HX Mem.n;y HHMH. B KOHI(e KOHI(OB
HOHbI pacnOJlaralOTC.H Ha paBHOBeCHOM p8CCTO.HHHH ro .n;pyr OT .n;pyra. COOTBeT-
CTBYIOeM ypaBHoBellIHBaHHIO CHJI npHT.HmeHH.H H CHJI OTTaJlKHBaHH.H.
I1TaK, CB.H3b, 06ycnoBJleHHa.H aneKTpOCTaTHtieCKHM npHT.HmeHHeM HOHOB,
06pa3YIOHXC.H npH nepexo.n;e 3J1eKTpOHOB OT o.n;Horo aTOMa K ,ApyrOMY, Ha3bI-
BaeTC.H UOliliOU C8Sl3b.
603

IIpH 06pa30BaHHH MOJIeKYJI H3 HeHTpaJIbHbIX, qaCTO O,AHHaKOBbIX aTOMOB (H 2 ,
O 2 , N 2 , a TaKme HF, NO, NH 3 , CH 4 H )J;p.) ocyweCTBJI.HeTCSI Koea.n.elimliaJl
C6Sl3b. Ee npHpO)J;Y YAaJIOCb YCTaHOBHTb TOJlbKO Ha OCHOBe KBaHTOBOH MexaHHKH.
B OCHOBY KBaHTOBO-MeXaHHqeCKOrO 06.HCHeHHSI KOBaJIeHTHOH CB.H3H nOJIOmeHbI:
1) BOJIHOBble CBoHcTBa ::meKTpOHa H pacnpe,AeJleHHe BepOSiTHOCTH 06Hapy-
meHH.H 3J1eKTpOHa B aTOMe;
2) npHHnHaJIbH8.SI Hepa3JIHtlHMOCTb 3J1eKTpOHOB - T.e. HX TomAeCTBeHHOCTb.
)::(Ba 3J1eKTpOHa B MOJIeKYJIe BOAopOAa, ABHmYlI.\HeCSI
Ka>KAbIH BOKpyr CBoero HApa, HHtleM He OTJlHtJaIOTCSI APyr
OT APyra. Y HHX OAHHaKOBbIe 3ap.HAbI, MaCCbI nOKO.H, cnH-
Hbl. OTCIOAa CJIeAyeT, 'liTO ecJIH 06a 3J1eKTpOHa MeH.HIOTCSI
MeCTaMH (pHC. 21.7), TO B COCTO.HHHH MOJleKYJlbl HHqero
He H3MeHHTCSI. TaKoH o6MeH MomeT npoH30HTH, eCJlH 06a
aTOMa BOAopOAa AOCTaTOqHO C6J1H3.HTC.H, npl1 3TOM "3J1eKT-
pOHHbIe 06J1aKa" 060HX aTOMOB 6YAyr nepeKpbIBaTbC.H. CJIH-
.HHHe 06J1aCTeH, rAe Moryr HaXOAHTbC.H 06a 3J1eKTpOHa B
MOJleKYJle H 2 , 03Ha'llaeT B03HHKHOBeHHe MemAY HHMH oco-
6oro OO.M.eIiHOZO 63aUMooeucm6USl: Ka>KAbIH H3 3J1eKTpOHOB
MomeT npHHa,nJlemaTb nonepeMeHHO TO O,AHOMY, TO APyro-
MY .H.APY, 3JIeKTpOHbl HenpepbIBHO MeH.HIOTC.H MecTaMH.
MOJleKYJla .HBJI.HeTC.H KBaHTOBOH CHCTeMoiL OHa onH-
CbIBaeTC.H ypaBHeHHeM IlIpeAHHrepa, KOTopoe Y'IHTbIBaeT Bce BHAbI ABHmeHHH,
npOHCXOA.HHe B MOJleKYJle: ABHmeHHe 3JIeKTpOHOB B MOJleKYJle, KOJIe6aHH.H aTO-
MOB MOJleKYJlbI, BpalI.\eHHe MOJleKYJlbI. IIoJIHa.H 3HeprH.H MOJIeKYJlbI W paBHa
W = W nOCT + W BPSut + W J(OJI + W aJI + W $fA' r,ne W nOCT - 3HeprH.H nocTynaTeJlbHOrO ,ABH-
meHHH eHTpa Macc MOJleKYJlbI; W BPSut - 3HeprHH BpalI.\aTeJlbHOrO )J;BHmeHHH
MOJleKYJlbI KaK eJloro BOKpyr HeKOTopbIX oceR; W J(OJI - 3HeprH.H KOJle6aTeJlb-
Horo ,nBHmeHH.H .H,nep aTOMOB, BXO,n.HID;HX B MOJleKYJlY; W aJI - 3HeprH.H ,nBH-
meHHH 3J1eKTpOHOB B aTOMax MOJleKYJIbI; W RA - 3HeprH.H H,nep aTOMOB B MO-
JIeKYJIe. SHeprH.H W' MOJleKYJlbl, onpeAeJIHIOlI.\a.H ee cneKTp, COCTOHT H3:
W' = W aJI + W J(OJI + W Bpaut'
onbITaMH H TeOpeTHqeCKHM paCqeTaMH, 'liTO
1
2
Puc. 21.7
2
1
IIpH 3TOM )J;oKa3aHo
W 3JI» W J(OJI» W Dpfil.q'
KamAa.H H3 3TH X 3HeprHH H3MeH.HeTC.H AHCKpeTHO. 113 TeopHH H 3KcnepH-
MeHTa CJle,nyeT, 'tITO paCCTO.HHH.H MemAY BpaaTeJlbHbIMH ypOBHHMH 3HeprHH
6. W BPSut ropa3,n0 MeHbllIe paCCTOSlHHH Mem,AY KOJle6aTeJlbHbIMH ypOBHSlMH 6. W J(OJI
(pHC. 21.8, a), KOTopoe, B CBOIO Oqepe,nb, MeHbllIe paCCTO.HHH.H Mem)J;Y 3J1eKT-
pOHHbIMH YPOBHHMR (pHC. 21.8, 6).
6
a
I E
I I
SneKTpOHHhle Kone68TenbHhle
ypOSHH ypOSHH
Kone68TenbHhle Bp8.II:(8TenLHhle
ypOBHH ypOSHH
604
Puc. 21.8

CTpoeHHe MOJleKYJI H CBOHCTBa HX 3HepreTHqeCKHX YPOBHeH npO.HBn.HeTC8
B MOJleKYJI.HpHbIX CneKTpBX.
1JHCJlO B03MOmHbIX nepeXO)J;OB pe3KO YBeJlH1J:HBaeTC.H, B03HHKaeT OrpOMHOe
qHCJlO JlHHHH CneKTpa, CJlHBalOIr(HXC.H B WHpOKHe nOJlOCbI.
21.8. rpaHHLI npHMeHHMOCTH 3aKOHOB KJIaCCH'IeCKOH 4>H3HKH
«1>H3HKa He .HBJI.HeTC.H 3aBepweHHOH HBYKOH. B KOTOpOH Bce .HCHO.
3aKOHbI CPH3HKH onHCbIBBIOT. KBK npOTeKalOT Te HJlH HHble .HBJleHH.H, CTa-
palOTC.H 06'b.HCHHTb. nOqeMY OHH TBK npOTeKalOT. IIo Mepe HaKOnJleHH.H 3Kcne-
pHMeHTaJlbHOrO MBTepHBJIa B03HHKalOT np06J1eMbI 06'bSlCHeHH.H HeKOTopbIX 4>BK-
TOB, KOTopbIe He COOTBeTCTBYIOT cyeCTBYlOmHM B033peHH.HM. YqeHble BcerlIa
CJleYIOT npHHuHny: eCJlH 3KcnepHMeHTaJlbHble cpaKTbI HaXO,lISlTC.H B npOTHBope-
qHH C npebIYHMH B033peHH.HMH, TO HBO MeH.HTb He CP8KTbI, a B033peHH.H.
TaK B03HHKJlB CneHaJlbHB.H TeOpH.H OTHOCHTeJlbHOCTH -- COBpeMeHHoe yqeHHe
o npOCTpaHCTBe H BpeMeHH, KOTopoe KopeHHbIM 06pB30M H3MeHHJlO CTapble B03-
3peHHSI Ha npOCTpBHCTBO H BpeM.H, npHH.HTbIe B KJlBCCHqeCKOH MeXBHHKe.
rrOTBepmeHHB.H 3KcnepHMeHTBJIbHO rHnOTe3B e BpOHJI.H 0 KopnYCKyn.HpHO-
BOJIHOBOM YBJIH3Me CBOHCTB BeIr(eCTBa nOJlHOCTblO H3MeHHJla npeCTaBneHH.H 0
CBOHCTBax MHKp006'beKTOB. BceM MHKp006'beKTaM npHCYIQH H KopnYCKyn.HpHbIe,
H BOJlHOBble CBOHCTBa. B TO me BpeMSI JlI06ylO H3 MHKpOqaCTH HeJlb3.H CqH-
TaTb HH qaCTHeH, HH BOJIHOM B KJlaCCHqeCKOM nOHHMaHHH. IIona,AaSl B pa3-
nHqHbIe ycnoBHSI) MaTepHBJIbHbIe 06'beKTbI npO.HBJI.HIOT CBOH BOJlHOBbIe u.n.u KOp-
nYCKYJlSlpHbIe CBOHCTBB. AJI.H 06'b.HCHeHH.H BHYTpeHHHX npoeCCOB B cnomHbIX
aTOMHbIX CHCTeMax nOHa06HJlOCb C03,AaTb nocneOBaTenbHYIO KBaHTOBYIO Teo-
pHIO, B03HHKJlH KBaHTOBa.H MexaHHKa H KBaHTOBaSl 3J1eKTpOHHaMHKa.
KBaHTOBa.H MexaHHKa, B OTJlHqHe OT MeXBHHKH KJlaCCHqeCKOH, yqHTbIBaeT
He TOJlbKO KopnYCKYJlSlpHbIe, HO H BOJlHOBbIe CBOHCTBa '1J:aCTH. Y1J:eT BOJlHO-
BbIX CBOHCTB nOTpe60BaJl OTKa3a OT p.Ha nOH.HTHH KJlaCCHqeCKOH MexaHHKH.
B 1J:aCTHOCTH, npHWJlOCb OTKa3aTbC.H OT BHmeHH.H qaCTH no TpaeKTOpH.HM.
nOTep.HnO CMbICJI nOH.HTHe 3J1eKTpOHHbIX Op6HT. B OT nH1J:He OT KnaCCH1J:eCKHX
npe,ACTaBJleHHH, BMeCTO 0H03Ha'1J:HOH npeonpeeJleHHOCTH B KBaHTOBOH Mexa-
HHKe Bce C06bITH.H HOC.HT BepO.HTHOCTHbIH xapaKTep.
OHaKO KBaHTOBa.H H KJlaCCHqeCKa.H MexaHHKa OTHIOb He OTropomeHbI ,APyr
OT Pyra Henp06HBaeMOM CTeHOM. Pa3BHTHe KBBHTOBOH CPH3HKH He npHBeno K
onpOBepmeHHIO OCHOBHbIX cpYHaMeHTBJIbHbIX 3aKOHOB CPH3HKH KJlaCCHqeCKOH -
3aKOHOB coxpaHeHHSI 3HeprHH, HMnynbca, MOMeHTOB HMnynbca, 3J1eKTpHqeCKO-
ro 3ap.Ha.
KBaHTOBa.H cpH3HKa, .HBJI.H.HCb 60Jlee 06eH no cpaBHeHHIO C KJlaCCHqeCKOH
TeopHeH, BKJllOqaeT B ce6.H 3aKOHbI KJlaCCHqeCKOH cpH3HKH. IIpH onpeeJleHHbIX
YCJlOBH.HX MomHO 06HBPymHTb B03MomHOCTb nJlaBHOrO nepexoa, 6J1H3KOrO
CXOmeHH.H KBaHTOBbIX H KJlaCCH1J:eCKHX npeCTaBJleHHH 0 CBOHCTBax MHKpOCH-
CTeM, B03MomHOCTb nJlaBHOrO nepexoa OT KBaHTOBbIX ypaBHeHHH lIBHmeHH.H
K HblOTOHOBCKHM ypaBHeHH.HM. Mx CXOCTBO MomHO YBHeTb, HanpHMep. B
cJlYQBe nepexoa aTOMa Boopoa H3 OHoro B036ymeHHoro COCTO.HHH.H B PY-
roe npH 60JlbWHX 3HaqeHH.HX rJlaBHOrO KBaHTOBoro QHCJla. Tora ypOBHH 3Hep-
rHH KBaHTOBbIX CTaHOHapHbIX COCTO.HHHH C6J1HmaIOTC.H HBCTOJlbKO, qTO nepe-
XO aTOMa H3 OHoro KBaHTOBoro COCTO.HHH.H B coceHee CTaHOBHTC.H npaKTH-
qeCKH HeOTJlHqHMbIM OT npOl.\eCca C HenpepbIBHbIM H3MeHeHHeM qaCTOTbI.
605

CHTyanHH 3eCb aHaJlOrHqHa TOM, KOTOpa.H HMeeT MeCTO B TeopHH OTHOCH-
TeJlbHOCTH. TaM npH CKOPOCTHX, MHoro MeHbllIHX CKOpOCTH CBeT8 B BaKYYMe,
peJIHTHBHCTcKHe COOTHOmeHH.H nepeXO.HT B H3BeCTHbIe <l>OPMYJlbI HbIOTOHOBC-
KOH MeX8HHKH.
CJleOBaTeJlbHO, HH Cnen;H8J1bH8H TeOpH.H OTHOCHTeJlbHOCTH HH KBaHTOB8S1
<l>H3HKa He OTBepralOT 3aKOHOMepHOCTH KJIaCCHqeCKOM <l>H3HKH, OHH JlHmb YK8-
3bIBalOT Ha rpaHHIJ;bI ee npHMeHHMOCTH. 06J18CTblO npHMeHeHHSI KJIaCCHQeCKOH
MexaHHKH .HBJlHIOTC.H MaKpOCKOnHqeCKHe .HBJIeHHH.
3TO HaXOHTC.H B nOJlHOM COrJlaCHH C c<l>opMynHpoB8HHbIM BOpOM npUliijU-
nOM coom6emcm6USl: 6CSlKaJl 1i06aSl. 60.n.ee 0614aJl meopUSl. Sl6Jl.SlI014aSlCSl pa36U-
mueM K.n.aCCU1leCKOU. lie om6epzaem ee no.n.liocmblO. a 6K.n.lOllaem 6 ce6Sl K.n.ac-
CU1leCKYIO meopulO. YKa3bl6aSl ZpaliUijbl ee npUMelieliUSl. npUlleM 6 onpeiJe.n.en-
liblX npeiJe.nb1tblX c.n.Y1laSlX 1tOBaSl meopuSl nepexoiJum 6 cmapylO.
BonpOCM AH CAMOKOHTPOH
1. B lleM cymb zunome3bl iJe BpOU.n.Sl?
2. KaKue 3KcnepUMelima.n.blible fjJaKmbl c6uiJeme.n.bcm6YlOm 0 CY14eCm60Ba-
IiUU 60.n.li06blX C60ucmB Ilacmuij 6e14eCm6a?
3. KaK06 fjJu3uIlecKuu CMblC.n. coomlioweliUSl lieonpeiJe.n.eliliocmeu iJ.n.Sl KO-
opiJuliambl U uMny.n.bca?
4. KaK06 fjJu3uIlecKuu CMblC.n. coomlioweliUSl lieonpeiJe.n.eliliocmeu iJJl.Sl 3liep-
zuu U 6peMeliu?
5. B lleM cocmoum npUliijUnua.n.blioe om.n.ullue K6alim060-MexaIiUlleCKOZO
onucaliu.R cocmOSlIiUSl cucmeMbl om K.n.aCCUlleCKOZO onucaliUSl?
6. KaKue fjJu3uIlecKue xapaKmepucmUKU amOM06 onpeiJe.n.SllOmcSl z.n.a6liblM
K6alim06blM IlUC.n.OM? Op6uma.n.bliblM K6alim06blM IlUC.n.OM? M aZliumliblM K6ali-
m06blM IlUC.n.OM?
7. q mo nOIiUMalOm noiJ 3.n.eKmpOliliblM 06.n.aKoM? KaKou BUiJ UMelOm 3.n.eK-
mpolilible 06.n.aKa B pa3.n.ullliblX cocmOSlIiUSlX amOMa 60iJopoiJa?
8. KaKue K6alim06ble Iluc.n.a onpeiJeJl.SllOm pa3Mepbl 3.n.eKmpOliliOZO 06.n.a-
Ka? 4JOPMY 3.n.eKmpOlia 06.n.aKa? OpuelimaijUIO ezo B npocmpalicmBe?
9. q mo maKoe cnu1t 3.n.eKmpolia? KaKue Jlialleliu.R Oli MOJICem npUIiU.Mamb?
10. B lle.M cocmoum npuliijun II ay.n.u? KaKue llaCmUijbl noiJlluliSllOmCJl
npuliijuny II ay.n.u?
11. KaKU.MU npa6u.n.aMu onpeiJe.n.SlemC.R nopSliJoK 3ano.n.lieliUSl 3.n.eKmpOliliblX
060.n.01leK amO.M06?
12. q e.M o6'oSlCliSlemcSl nepuoiJulleCKaJl n06mOp.ReMOCmb XUMUlleCKUX C60ucmB
3.n.eMelim06?
OCHOBHbIE cI»OPMY bI
h
A B =-
mv
- <l>OPMYJl8 e BPOHJISI, r,Ae AB - )J;JIHH8 BOJlHbI e BpOH-
JI.H, h - nOCTO.HHHa.H IIn8HK8, m - Macca qaCTHn;bI, v -
CKOpOCTb qaCTHn;bI.
6.x . 6.Px  Ii
COOTHomeHHe HeOnpeeJleHHOCTeH reH3eH6epra re
tJ.x - HeOnpeeJleHHOCTb KOOp,AHHaTbI, tJ.px - HeOnpe,AeJleH-
606

Ll W . Llt  Ii
L = Ii l (l + 1)
Lz = mli
Ls = Ii  s{s + 1)
e 2
w=--
II 41tEor
1 me 4
W = ---
n 2 8h2E
HOCTb npOeKIJ;HH HMnYJlhCa,
h
Ii =-
2n'
h - nOCTO.HHHa.H
IIJlaHKa.
- COOTHOlIIeHHe HeOnpeeJleHHOCTeH, r,Ae 6. W - Heonpee-
JleHHOCTh 3HaqeHH.H 3HeprHH CHCTeMbI, 6.t - HeOnpeeJleH-
HOCTh npOMemYTK8 BpeMeHH npe6hIB8HH.H CHCTeMbI B 8H-
HOM COCTOJIHHH.
- npOeKIJ;H.H MOMeHTa HMnYJlbC8 CHCTeMhI, l = O. 1, ....
(n - 1) - Op6HTaJlhHOe KBaHTOBoe qHCJlO, n - rJI8BHOe
KBaHTOBoe qHCJIO.
- npOeKH.H MOMeHTa HMnYJIbCa Ha HanpaBJleHHe z, re
m = 0, Z 1, ..., z l - MarHHTHoe KBaHTOBoe qHCJlO.
1
- MOYJlb cnHHa 3J1eKTpOHa, I'e s = - - cnHHOBoe KBaH-
2
TOBoe qHCJlO.
- nOTeHIJ;HaJlhHaJI 3HeprH.R B3aHMoeHCTBH.R 3J1eKTpOH8 C
.RpOM B aTOMe Boopoa, re e - 3J1eMeHTapHbIH 38p.H,
Eo - 3JIeKTpHqeCKa.H nOCTO.HHHa.R, r - paCCTO.HHHe Memy
3J1eKTpOHOM M .RpOM.
- nOJlHa.H 3HeprH.H 3J1eKTpOHa B aTOMe BO)J;opO)J;a, re m -
MaCC8 3J1eKTpOHa, n == 1, 2, 3, r JlaBHOe KB8HTOBoe
qHCJlO.
llPHMEPbI PEllIEHH.H 3A,n;A q
1. 8neKTpoH BMmeTCSI B M8rHMTHOM none C IiHYKl1;lieH 8 MTn no OK-
pymHocTH P8,AMYC8 0,5 CM. Onpe,AenHTe ,AnHHY sonHbI ,Ae BpOHnSl.
)J;8HO: q=I.6.10- 19 Kn: B=8.10- 3 Tn: R=5.10- 3 M.
H 8 H T H: AB'
Pew e H H e. H8 3neKTpoH B M8rHHTHOM none ,AeHcTByeT CHna JIopeHIJ;8
F 11 = qBv. TIo STOPOMY 38KOHY HblOTOH8 mau.= F 11'
mv 2
OTKY,A8 - = qBv. Cne,ll;OB8TenbHo, mv = qBR.
R
BonHa ,Ae BpoHnSl onpe,ll;enSleTCSI no cpopMyne
h h
As = - = -; As = 0.1 HM.
mv qBR
2. ,lJ;nHHa BonHbI ,Ae BpoHnSl ,AnSI 3neKTpOH8, npOlIIe,AlIIerO HeKOTOpylO YC-
KOPSlIOlI.\YIO p83HOCTb nOTeHl1;H8nOB, p8BHa 0,04 HM. t.JeMY p8BH8 YCKOpJIIOlI.\8S1
p83HOCTb nOTeHu,H8noB?
,lJ;8HO: As=0,4.10- 10 M, m=9,1.10- 31 Kr, e=1,6.10- 19 Kn.
HaHTH: U.
607

h
Pewe H He. BOJlHR ,Ae BpOHnSi PRCCqHTbIBReTCSI no <l>opMyne A.B = -. OT-
mv
h
CIO,A8 CKOpOCTb aneKTpOH8 v = -. no TeopeMe 0 KHHeTH'leCKOH 3HeprHH
mA B
mv 2
l:1 W K = A, r,Ae WK = W K - 0 = -.
2
mv 2 h 2
P860TR CHn aneKTpOCTRTH'lecKOro nonS! A = eU :::) - = eU. OTCIO,A8 U = -'
2 2meA '
U = 1 KB.
3. 3neKTpoH C KHHeTHqeCKOH aHeprHeH 15 3B HRXO,AHTC.H B MeTRnnHqeC-
KOH nblnHHKe ,AH8MeTpOM 1 MKM. OeHHTe (B npoeHT8x) HeOnpe,AeneHHOCTb
3H8'leHHSI CKOpOCTH aneKTpOHa.
,D; 8 H 0: W K = 2,4. 10- 18 ,D;m; d = 1 . 10- 6 M.
H - v x
RHTH: -.
v
Pew e H H e. 3H8S1 KHHeTH'lecKYIO 3HeprHIO 3neKTpoH8, H8H,AeM CKOpOCTb
aneKTpoHR:
W K = mv 2  v = 2WK .
2 m
h
3RnHIIIeM COOTHomeHHe HeOnpe,AeneHHOcTeH B. reH3eH6eprR I:1x. Px  - , r,Ae
21t
HeOnpe,AeneHHOCTb HMnynbc8 I:1px= ml:1v x ' H3 3THX t}>opMyn H8XO,AHM aeonpe,Ae-
h
neHHOCTb 3HR'leHHSI CKOpOCTH v = . Heonpe,AeneHHOcTb KOOp,AHH8Tbl
x 21tmx
C'lHTaTb paBHoH ,AHRMeTpy nbInHHKH 1:1 x ::::: d. TOr,A8
3neKTpOH8
h
I:1v x =-.
21tmd
HCKOMR.H
MomHO
OTHOCHTenbHRSI HeonpeAeneHHOCTb 3H8'leHHSI CKOpOCTH
I:1v x _ h _ h . I:1v x = 0,005 %.
 - 21tmvd - 21td .J 2mWK' v
608

22. cI>H3HKA ATOMHOrO HAPA
COCTaB JI,lJ;pa aTOMa. H30TonLI. HJ1:epHLle CHJILI. 3HeprHJI CB.H3H aTOMHLIX
JI,lJ; ep.
Pa,nHoaKTHBHoCTL. 3aKOH pa,nHoaKTHBHoro pacna,na. Pa,nHOaKTHBHOCTL H ee
BH.IJ:LI. AJILcj)a- H 6eTa-pacna,n. raMMa-H3JIyqeHHe npH aJILcj)a- H 6eTa-pacna-
one. HeHTpHHo. UCKyccTBeHHa.H pa,nHoaKTHBHOCTL. n03HTpOH. 3KcnepHMeHTaJIL-
HLle MeTo,nLI perHCTpal.\HH 3apJl:iKeHHLIX '1aCTHIJ;. H.ri;epHLle peaKIJ;HH. 3Hepre-
TH1JeCKHH BLIXO,n JI,lJ;epHLIX peaKIJ;HH.
)J;eJIeHHe JIlIep ypaHa. .HlIepHLIH peaKTOp. TepMoJln;epHLle peaKI.\HH. Co-
3,naHHe H y,nep:iKaHHe BLicoKoTeMnepaTypHoH nJIa3MLI. TOKaMaK. YcneXH H
nepcneKTHBLJ pa3BHTH.H JI,lJ;epHOH 3HepreTHKH.
nOrJIOeHHaJl ,n03a H3JIyqeHHJI H ee 6HOJIOrH1JeCKOe ,neHCTBHe. OCHOBLI
,n03HMeTpHH. 3aHTa OT H3JIyqeHHH.
22.1. CocTaB HJJ;pa aTOMa
IIocne C03aHH.H .HepHOH MO)J;eJlH aTOMa Bonpoc 0 COCTaBe aTOMHoro .Hpa
CTaJl OHHM H3 OCHOBHbIX B .HepHOH cpH3HKe. M3 qero COCTOHT aTOM Hoe HpO?
KaKHe CHJlbI yepmHBalOT COCTaBHbIe qaCTH .H)J;pa )J;pyr B03J1e pyra? KaKHe
npeBpaeHHJI H,lJ;ep B03MO>KHbI?
OTBeTbI Ha 3TH BonpOCbI cpH3HKa CMOrJla l1aTb TOJlbKO no Mepe HRKOnJle-
HH.H CBeeHHH 0 pa3J1HqHbIX CBOHCTBax HlIep, B oco6eHHOCTH CBelIeHHH 0 3a-
pSl,Ae If MRcce .H,APR.
3apH .Hpa qSl = Ze (B CM H3MepHeTCH B KYJlOHax). B eHHHax 3J1eMeH-
TapHoro 3apH8 3ap.H .Hpa paBeH Z. Maccy Hpa BbIpRmRIOT 06bIqHO B aTOM-
HbIX eHHHax MRCCbI. 3a aTOMHYIO e,AHHHy MaCCbI (a.e.M) npHHSlTR 1/12
MRCCbI aTOMR yrJlepoa lC.
= 1,6605402 . 10- 27 Kr.
12 1110
I: C
TRK KRK MRcca 3J1eKTpOHR OqeHb MRJlR, TO MRCCR .H,APR aTOMR npH6J1H3H-
TeJlbHO paBHa ero aTOMHOH Macce.
IIpH TOqHOM H3MepeHHH aTOMHbIX Macc Bcex H30TonOB 6bIJlO 06HRpymeHo,
qTO MaCCLI H30TonOB, BbIpR>KeHHbIe B R.e.M., 6JIH3KH K u.eJILIM '1HCJIaM
eJloe 'IlHCJlO. 6J1HmaHllIee K 3Ha'lleHHIO aTOMHOH MRCCbI. BbIpa>KeHHOH B
a.e.M., Ha3bIBaeTCH MaCCOBbIM 1JHCJlOM A.
Y CJlOBHJlHCb .H)J;pO 0603HR'IlaTb XHMHqeCKHM CHMBonOM RTOMR, KOTOPOMY OHO
npHHanemHT, C BYM.f.I HHeKcaMH; BBepxy - MaCCOBoe qHCJlO, BHH3Y - 3a-
pH B eHHHu;ax 3J1eMeHTapHoro 3apH,lJ;a. Ha3bIBaeMbIH HHora 3ap.HOBbiM 'IlHC-
nOM: 1x.
HanpHMep, HpO aTOMa KHcnopoa 19O, yrJlepO,Aa lC.
MCCJleOBaHH.H nOKa3aJIH, qTO aTOMHbIe MaCCbI H30TonOB TeM 6nHme K e-
JlbIM tlHCJlaM, qeM Jlertle H30Ton, T.e. 'IleM MeHbllIe aTOMHR.H Macca. 3TO HaBe-
JlO Ha MbICJlb, qTO .HpO COCTOHT H3 qaCTHIJ., aTOMHbIe MRCCb] KOTOpbIX 6J1H3-
KH K 1. IIOCKOJlbKY 3TOMY ycnoBHIO XOpOlliO yOBneTBOp.HJlO .HpO aTOMa BO-
1 a.e.M. =
1
609

,nopoa (aTOMH8.H MaCCa == 1,008 a.e.M.), TO npe,nnOJlOmHJlH, qTO B COCT8B BCeX
.H)J;ep BXOHT .H)J;pO BOOpO)J;8 -- npOTOH.
lH = Ip -- npOTOH - 3J1eMeHT8pHa.H qaCTHa. 3ap.H KOTOpOH nOJlom1:l-
TeJleH H paBeH 3J1eMeHTapHOMY 38p.HY + e = + 1,6. 10- 19 EJI. Macca
m p = 1,6726. 10- 27 Kr= 1.00728 8.e.M. == 1 a.e.M.
B 1919 r. 3. Pe3ep4>op H II. M. C. BJleKKeT, OCYl.QeCTBHB nepBYIO .Hn;ep-
HYIO peaK1:I1O, Ha onbITe 06Hapym1:lJlH npOTOH. IIpH 38XBaTe a-qaCTHbI .HpO
18 p
a30Ta npeBpaaJlOCb B COCTaBHoe .H)J;pO 4>Topa 9 ' KOTOpoe HaXOHJlOCb B
B036ymeHHoM COCTO.HHHH H B TeqeHHe 10- 16 __10- 12 C npeBpaaJlOCb B KO-
HeqHOe .HpO H30Tona K1:ICJlOpOa IJ 0, np1:l 3TOM BbI)J;eJl.HJlC.H npOTOH:
*
lN + He  C:F) -+ IO + fp.
ECJlH 6bI B COCT8B .Hep CXOHJlH TOJlbKO npOTOHbI. TO 38p.H,A .Hpa. BbIp8-
meHHbIH B 3J1eMeHT8pHbIX 38p.HaX. 6bIJI 6bI qHCJleHHO p8BeH M8CCe .Hn;pa. BbI-
p8meHHbIH B aTOMHbIX eHHHuaX M8CCbI.
BbIJlO 06HapymeHo, qTO M8CCbI BCeX 8TOMOB, KpOMe lH, BbIp8meHHbIe B
a.e.M., npeBbIIII810T q1:lCJleHHO 3ap.HbI 1:IX .Hep, BbIpameHHbIe B e1:IH1:I8X 3J1e-
MeHT8pHOrO 3ap.H8: A> Z, np1:lqeM no Mepe YBenHlIeH1:I.H Z 3TO p83nHqHe B03-
paCTaeT. 113 3Toro CJleOBaJIO, qTO npOTOHbI He MorYT 6bITb eHHCTBeHHbIM1:I
qaCTHaMH, COCTaBJI.HIOI.QHMH .Hpo.
B 1920 r. Pe3ep4>op BbICKa3an npe)J;nOJlOmeH1:Ie, qTO B Hpax aTOMOB
HMeIOTC.H K8KHe-TO 3J1eKTpHqeCKH HeHTpaJIbHbIe qaCTHbI C MaccoH, n p1:l6J1 1:1 31:1-
TeJlbHO p8BHOH M8cce npOTOH8. B 1932 r. COTPYHHK Pe3ep4>op8 .nm. qe-
BHK 06HapymHJI HX Ha onbITe, 60M6ap1:IPY.H aTOMbI 6ep1:lJlJl1:l.H a-qaCT1:IaMH.
STY qaCTHY H83BaJI1:I HeHTpOHOM An. HeHTpOH -- 3J1eMeHTapHa.H q8CTHa. 38pSl,A
KOTOpOH paBeH 0, a Macca 6J1H3Ka K M8cce npOTOHa m p = 1,6749 .10- 27 Kr=
= 1,00866 a.e.M. == 1 8.e.M.
*
:Be + He  (l:C)  l:C + n.
BCKope nocJle OTKpbITH.H HeHTpoHa B 1934 r. cOBeTcKHH yqeHbIH
.n. H. HBaHeHKO H HeMeK1:IH 4>H31:1K B. reH3eH6epr He38BHC1:IMO pyr OT n;pyr8
npeJlOmHJlH npOToHHO-HeilTpoJlHYlO MOeJIL .H.npa, COrJl8CHO KOTOpOH aTOMHoe
.HpO COCTOHT H3 npOTOHOB 1:1 HeHTpoHoB (HX 06ee Ha3B8H1:Ie -- HYKJlOHbI).
q1:lcJlo npOTOHOB B Hpe paBHO Z -- 3ap.HOBOMY qHCJlY, KOTopoe paBHO nop.H-
KOBOMY HOMepy XHMHqeCKOrO 3J1eMeHTa B Ta6J1He MeHeJleeBa. TaK KaK aTOM
B eJlOM HeHTpaJIeH, TO qHCJlO Z onpeeJl.HeT O)J;HOBpeMeHHO H q1:lCJlO 3J1eKTpO-
HOB B aTOMe, H HX pacnpeeJleHHe no 060nOqKaM, TaK KSK 3TO pscnpen;eJleHHe
38BHCHT OT HX 06Iltero qHCJla. MaCCOBoe lIHCJlO A onpen;eJl.HeT 06Iltee qHCJlO
HYKnOHOB. CJlen;OBaTeJlbHO, q1:lcno HeHTpoHoB B .H)J;pe paBHO N =A - Z.
ECJlH npOCJle1:ITb 3a pacnpeeJleH1:IeM q1:lCJla npOTOHOB Z 1:1 HeHTpoHoB N B
.Hpax pa3J1WlHbIX ::meMeHTOB nep1:l01:IqeCKOH Ta6J11:1bI MeHeJleeBa, TO MomHO 3a-
MeT1:ITb, qTO .n.n.H .Hep 3J1eMeHTOB BllJIOTb O cepe1:IHbI Ta6J11:1bI q1:lcno HeHTpOHOB
A-Z
npHMepHO pRBHO qHcny npoTOHOB, TaK qTO - = 1. IIo Mepe YT.HmeJleHH.H ep
Z
A-Z
ROJlH1leCTBO HeHTpoHoB B03paCTaeT 1:1 B KOHIJ,e Ta6J11:1bI ---z- = 1,6.
610

22.2. HaOTonhI
llpH H3yqeHHH pa,nHoaKTHBHbIX npeBpameHHH 6bIJlO YCTaHOBJleHO, qTO CY-
eCTBYIOT XHMHqeCKHe 3J1eMeHTbI, aTOMbI KOTOpbIX 06J1a,naIOT O,nHHRKOBbIMH
XHMHqeCKHMH CBOHCTBRMH, HO pRcna,naIOTC.H no-pR3HOMY. OT,neJlHTb 3TH aTOMbI
,npyr OT ,npyra HeB03MOmHO HHKaKHMH XHMHqeCKHMH MeTO,naMH. B 1911 r.
q>. CO,n,nH npe,nJlOmHJI HR3bIBRTb TaKHe pa3HOBH,nHOCTH RTOMOB o,nHoro H Toro
me XHMHqeCKOrO 3J1eMeHTa H30TonaMH (qTO no-rpeqeCKH 03HaqaeT "paBHoMecT-
HbIe"), TaK KaK no CBOHM XHMHqeCKHM CBoHcTBaM OHH ,nOJlmHbI 6bITb nOMe-
eHbl B O)J;HY H TY me KneTKY Ta6J1HbI MeH,neneeBa.
llPH HCCJle,nOBaHH.HX, npOBe,neHHbiX C nOMOblO Macc-cneKTporpa4>oB, 6bIJlO
YCTaHOBJleHO, qTO H30TonbI o,nHoro H Toro me 3J1eMeHTa 06J1a)J;aIOT pa3J1Hq-
HOH MaccoH, npHqeM MaCCbI H30TonOB JlHlIIb He3HaqHTeJlbHO OTJlHqRIOTCSI OT
eJlbIX qHCeJl. CJle,noBaTeJlbHO, H30TonaMH HBJUIIOTCH aTOMhl C O,lJ;HHaKOBLIM
3apJlJl;OBhlM '1HCJlOM, 80 pa3J1H'IHblMH MaCCOBhlMH '1HCJlaMH. llocJle C03,naHHSI
npOTOHHO-HeHTpOHHOM Mo,neJlH SI,npa CTaJlO .HCHO, qTO H30Tonbl - 3TO aTO-
MbI, Y KOTOpblX H,npR HMelOT O,lJ;HHaKOBoe '1HCJlO npOTOHOB H pa3J1H'IHOe '1HC-
J10 HeHTpOHOB.
B HaCTO.Hee BpeM.H H3BeCTHO, qTO H30TOnbJ HMeIOTC.H Y Bcex XHMHqeCKHX
3J1eMeHTOB.
Y Bo,nopo,na TPIi H30Tona: npOTHH tH, ,neHTepHH D H TpHTHH fT.
YpaH (Z = 92) HMeeT 12 H30TonOB C MRCCOBbIMH qHCJlaMH OT A = 228 ,no
A= 239, Y TeXHeH.H - 16, y npOMeTH.H - 14.
Pa3J1HqaIOTC.H H30TonbI cTa6HJlLHhle, COXpaH.HIOHeC.H HeH3MeHHbIMH CKOJlb
yro,nHO ,nJlHTeJlbHOe BpeMSI, H HeCTa6HJlLHhle, KOTopble co BpeMeHeM npeBpR-
RIOTCSI B SI,npa ,npyrHx 3J1eMeHTOB.
XHMHqeCKHe 3J1eMeHTbI, cyeCTBYIOue B npHpo,ne, npe)J;CTaBJlSlIOT c060ii
CMeCH H30TonOB. ll03TOMY RTOMHR.H MaCCR XHMHqeCKOrO 3J1eMeHTR npe,nCTRBJlSI-
eT C060H cpe,nHee 3HaqeHHe aTOMHbIX Macc Bcex ero H30TonOB. HanpHMep, y
xJlopa ,nBR cTa6HJlbHbIX H30Tona C aTOMHbIMH MaccaMH 35 H 37, OTHOCHTeJlb-
Hoe co,nepmaHHe KOTOpbIX 75 H 25 %. [103TOMY cpe,nHee 3HaqeHHe aTOMHOH
MaCCbI 3J1eMeHTa xJlopa paBHo (35.0,75) + (37.0.25) = 35.5.
22.3. epHhIe CHJIhI
.H,npa RTOMOB - OqeHb YCTOHqHBbIe 06pR30BRHH.H, 60Jlee YCTOHqHBbIe, qeM
caMH aTOMbI. TaK, HRnpHMep, qT06bI pa3,neJlHTb .H,npo reJlH.H HR OT,neJlbHbIe
HYKnOHbl. He06xOlIHMO 3aTpaTHTb B COTHIi TbIC.Hq pa3 60JlblIIe 3HeprHH. qeM
,nJl.H OTpbIBR 060HX ero 3J1eKTpOHOB OT .H,npR. STO 06'b.HCH.HeTC.H TeM, qTO Me3t-
II.Y HyKJIOHaMH HMeeT MeCTO HOBoe, BHYTpHSI,nepHoe, B3aHMo,neHcTBHe, T.e. ,neH-
CTBYIOT oc06bIe, JlJl;epHhle, CHJlhl. .H,nepHbIe CHnbI 06HapymHBRIOT cJle)J;YIOHe
cneH4>HqeCKHe CBOHCTBa.
1. 3TO KOpOTKO,lJ;eHCTBYIOe CHnbI. OHH )J;eHcTBYIOT Ha paCCTO.HHH.HX Mem,ny
HYKJlOHaMH, nopH,nKa 10- 15 M, H pe3KO y6bIBalOT npH YBeJlHtJeHHH paCCTO.HHH.H;
npH paCCTO.HHH.HX 1.4. 10- 15 M OHH yme npaKTHqeCKH paBHbI O.
2. STO caMbIe MOIIlHhle CHJlLI H3 Bcex. KOTOpbIMH pacnOJlaraeT npHpo,nR.
ll03TOMY B3RHMo,neHcTBHe qaCTH B SI,npe qaCTO HR3bIBalOT CHJlhHhlMH B3aHMO-
,lJ;eHCTBHHMH.
611

3. .H,l\epHhIM CHJlaM CBOHCTBeHHO H8CLIm;eHHe. STO 3HaqHT, qTO HYKJlOH B3a-
HMO,l\eHcTByeT He co BceMH OCTaJIbHbIMH HYKnOHaMH, a JlHlIIb C HeKOTopbIMH
6JIHmaHlIIHMH COCe,l\.HMH.
4. .H,AepHbIM CHJlaM CBoHcTBeHHa 3apHJJ:OB8J1 HeaaBHCHMOCTL. STO 3HaqHT,
qTO C O,l\HHaKOBoH no MO,l\YJlIO CHJlOH npHTSlrHBaIOTC.H ,APyr K ,Apyry H 3ap.H-
meHHbIe, H He3ap.HmeHHbIe qaCTHIJ.bI, T.e. CHJla npHTSlmeHHSI Fpp Mem,Ay BY-
M.H npOTOHaMH paBHa CHJIe npHT.HmeHHSI F nn Mem,AY ,ABYMSI HeHTpoHaMH H paB-
Ha CHJle npHTSlmeHH.H F pn Mem,l\y npOTOHOM H HeHTpoHoM.
5. .H,nepHbIe CHJlbI He JlBJIJlIOTCB QeHTp8JILHLIMH, T.e. OHH He HanpaBJleHbI
B.nOJIh np.HMOH, COelIHH.HlOmeH neHTpbI 3THX 3apH,l\OB.
6. .H,l\epHbIe CHJlbI .HBn.HIOTC.H TaK Ha3bIBaeMbIMH 06MeHHbIMH CHJI8MH. 06-
MeHHbIe CHJIbI HMelOT KBaHToBbIH xapaKTep, y HHX HeT aHaJlOra B 06bIqHOH
ij)H3HKe. B3aHMo,neHcTBHe Mem,ny HYKJIOHaMH B03HHKaeT BCJle,ACTBHe 06MeHa
Mem,ny HHMH HeKoTopoH TpeTbeH 'IaCTHIJ;eH. STY 'IaCTHIJ.Y Ha3bIBalOT 1[-Me30-
HOM, HJlH nHOHOM. MaccR 1[-Me30HOB npHMepHO B 280 pa3 60JlblIIe MaCChI
3J1eKTpOHa. IIHOHhI 6bIBalOT Tpex BH,nOB: nOJlOmHTeJIbHbIe 1[+, OTpHIJ;aTeJlbHbIe
1[- H HeHTpaJIbHbIe 1[0. B3aHMo,neHCTBHe Mem,l\Y O,l\HOpO,nHbIMH HYKJlOHaMH OCY-
J..QeCTBJI.HeTCH HeHTpaJlhHbIMH 1[°-Me30HaMH, a B3aHMO,AeHCTBHe Mem,AY pa3J1Hq-
HbIMH HYKJlOHaMH - 3ap.HmeHHbIMH 1t-Me30HaMH.
22.4. 3HeprHH CBH3H aTOMHbIX HlIep
nOon 3HeprHeii CBJl3H aTOMHoro .H,npa nOHHMalOT 3HeprHIO, KOTOPYIO HymHo
3aTpaTHTb, 1.{T06bI paCJ..QenHTb .H,npo Ha OT,neJIbHbIe HYKJlOHbI. TaKa.H me 3HeprHSI
BbI)XeJlHeTC.H npH 06pa30BaHHH .H,l\pa H3 CB060,AHbIX HYKJIOHOB. Ee MomHO paCC'IH-
TaTb, nOJlh3Y.HCb ij)OPMYJlOH A. SHHlIITeHHa, CB.H3hIBalOeH Maccy qaCTHIJ;hI H
3HeprHIO:
W = mc 2 .
IIocJle C03,naHH.H Macc-cneKTporpac}>a MomHO 6bIJIO C 60JlblIIOH TOqHOCThlO
(,no 0,01 %) H3MepHTh MaCCbI 'Bcex H30TonOB 3JIeMeHTOB Ta6J1HIJ.bI MeH,neJlee-
Ba, qTO H c,neJlaHO yqeHbIMH.
AHaJlH3 3THX ,naHHbIX nOKa3bIBaeT, qTO ,nJl.H Bcex 3JIeMeHTOB Macca nOKOSl
H,npa MeHhlIIe, qeM CYMMa Macc nOKOSi COCTaBJI.HIOJ..QHX ero HYKJlOHOB, eCJlH
nOCJle,nHHe Haxo,n.HTCH B cB060,nHOM COCTO.HHHH. STO pa3JIHqHe MomeT 6bITb
oxapaKTepH30BaHO BeJlHqHHOH
m = Lm H - mil = Zmp -r (A - Z)m n - mil'
KOTOpa.H HOCHT Ha3BaHHe .nec}>eKT8 Macc. YMeHblIIeHHe MaCCbI npH 06pa30Ba-
HHH H,l\pa H3 cB060,nHbIX qaCTHIJ; 03HaqaeT, qTO npH 3TOM YMeHhmaeTCH 3Hep-
rH.H 3TOH CHCTeMbI qaCTHIJ. Ha BeJIHqHHY 3HeprHH CB.H3H
WCB = 11 mc 2 = (Zm p + (A - Z)m n - m ll )c 2
SHeprH.H CB.H3H onpe,neJl.HeTC.H BeJlHqHHOH pa60TbI, KOTOPYIO HymHo COBep-
lIIHTh ,nJl.H paCenJleHH.H .H,npa Ha COCTaBJlHIOHe ero HYKJlOHbI. Ho Ky,na pac-
xo,nyeTC.H 3Ta 3HeprH.H?
IIpH 06pa30BaHHH SI,npa H3 HYKJlOHOB nOCJle,nHHe 3a CqeT ,neHCTBHH .H,nep-
HbIX CHJI Ha MaJIbIX paCCTO.HHH.HX YCTpeMJI.HIOTC.H ,npyr K ,npyry C orpOMHhIMH
612

YCKopeHH.HMH. M3J1yqaeMbIe npH 3TOM 'Y-KBaHTbI 06J1a,n;aIOT 3HeprHeH CB.H3H W CB ,
T.e. npH 06pa30BaHHH .H,n;ep H3 H}'KJlOHOB 3Ta 3HeprH.H CB.H3H BbI,n;eJl.HeTC.H. 3Hep-
rH.H CB.H3H OqeHb BeJlHKa (ee 06bIqHO BbIpamalOT B M3B: 1 M3B = 10 6 3B =
= 1,6 . 10- 13 .D:m). 06 3TOH BenHqHHe MomHO Cy,n;HTb no TaKOMY npHMepy: 06-
pa30BaHHe 4 r reJlH.H conpOBOm,AaeTC.H BbI,n;eJleHHeM T8KOH me 3HeprHH, KaK
npH crop8HHH 5--6 BarOHOB KaMeHHoro yrJl.H.
B8mHOR xapaKTepHcTHKOH R,n;pa cJlymHT Cpe,AH.H.H 3HeprH.H CB.H3H .H,n;pa, npH-
XO,n;.H8.HC.H Ha O,n;HH HYKJlOH (TaK Ha3bIB8eM8.H y)];eJILHaH aHeprH.H CBH3H Mpa),
'tleM OHa 60JlbIlIe, TeM CHJlbHee CB.H3aHbI Mem,n;y C060H HYKJlOHbI, TeM IIPOq-
Hee .H,n;po. STY y,neJlbHYIO 3HeprHIO CB.H3H WCB BCer,n;a MOmHO nO,n;CqHTaTb. Pe-
3YJlbTaTbI IIOKa3bIBaIOT, tlTO ,nn.H 60nblllHHCTBa H,Aep W CB == 8 M3B H YMeHb1ll8-
eTC.H ,nJl.H OqeHb JlerKHX H OqeHb T.HmeJlbIX .H,nep.
ITo Mepe YBenU'leHH.H qHCJla HYK-
weB' MaB JlOHOB B .HApe B03p8CTalOT KYJlOHOB-
CKHe CHnbI OTTaJlKHBaHH.H Mem.n;y
8,7 MaB npOTOHaMH, OCJla6J1.HIOHe CB.H3H B
.H,npe, H BeJlHqHH8 WCB y T.HmenbIX
.H,nep YMeHblllaeTC.H. BeJlHqHHa WCB
M8KCHMaJIbHa Y .H,nep Cpe,n;HeH MaCCbI
(A = 50.. .60), CJle,n;OBaTeJlbHO, OHH
OTJlHtl8IOTC.H H8H60nbllleH npOqHOC-
TblO (pHC. 22.1).
ITpoeccbI ,AeneHH.H T.fImeJlbIX
.H.nep H CHHTe3a JlerKHX .HBn.HIOTC.H
3HepreTHqeCKH BbIro,nHbIMH, nOTOMY
80 120 160 200 240 A qTO conpOBOm,AaIOTC.H B03p8CTaHHeM
3HeprHH CB.H3H, T .e. BbI,neJleHHeM
Puc. 22.1 3HeprHH. Ha 3TOM OCHOBaHO, K8K
MbI YBH,n;HM HHme, nOJlyqeHHe aTOM-
HOH 3HeprHH npH ,n;eJleHHH T.HmeJlbIX .H,nep H TepMO.H,nepHOH 3HeprHH -- IIPH
CHHTe3e nerKHX.
9
8
7
6
5
4
3
2 H
1
0 40
WCB
WCB =-.
A
22.5. Pa,nHoaKTHBHOCTb. 3aKOH pa,nHoaKTHBHoro pacna,na
ECTeCTBeHHOH pHoaKTHBHOCTLIO Ha3bIBaeTC.H caMonpOH3BOJlbHOe IIpeBpae-
HHe .H,nep HeYCTOHqHBbIX H30TOIIOB o,nHoro XHMHqeCKOrO 3J1eMeHT8 B H,n;pa H30-
TonOB ,npyrHx XHMHqeCKHX 3J1eMeHTOB.
Pa,nHoaKTHBHOCTb BnepBbIe 06HapymeHa A. BeKKepeJleM y COJleR ypaHa. Ec-
TeCTBeHHa.H pa)J;HoaKTHBHOCTb, KaK npaBHno, Ha6J1IO)J;aeTC.H y .H)J;ep T.HmeJlbIX
3J1eMeHTOB, paCIIOJlaralOHXC.H B Ta6J1He MeH,neJleeBa 3a CBHHOM. CyeCTBY-
lOT H nerKHe pa,AHOaKTHBHbIe .H,npa K, lC, Rb H ,np.
CKOpOCTb paCIIa,na pa3J1HqHbIX pa,n;HOaKTHBHbIX 3J1eMeHTOB HeO,n;HHaKOBa 1:1
XapaKTepH3yeTC.H nepHO,nOM nOJlypacna)J;a.
613

N
No
No
2"
I
I
I
==========----
I
o
T 2T 3T
Puc. 22.2
N - Nt _ No _ No
2------.
2 4 2 2
T
llepHOA nonypacna - 3TO npoMemYToK
BpeMeHH T, B TeqeH1:Ie KOToporo pacnaaeTC.H
nOnOB1:IHa nepBOHaqaJlbHOro KOJl1:lqeCTBa .Hep
(pHC. 22.2).
EcnH B HaqaJlbHbIH MOMeHT BpeMeHH t o = 0
6bIJlO No paHoaKTHBHbIX SIep, TO '1epe3 npo-
MemYTOK BpeMeHH, paBHbIH nep1:l0y nOJlypac-
naa tt = T, q1:lCnO HepaCnaB1ll1:lXC.H .H,Aep
NO
Nt = -. CnycTSI eIQe TaKoH me npOMemYTOK
2
BpeMeHH t 2 = 2T q1:lCJlO HepaCnaB1ll1:lXC.H .Hep
rIo HCTeqeHHH t = nT pa1:I0aKTHBHbIX SIep OCTaHeTCSI
N = N o . IIoCKOJlbK Y n = .!. TO
2 n T'
t
N=N o 2 T.
3Ta cpOPMY Jla SlBJI.HeTCSI aHan1:lT1:IQeCKHM BbIpameH1:IeM 3aKOHa paHoaKTHB-
Horo pacnaa, YCTaHOBJleHHOrO q>.CO1:I.
q1:lcJlo pacnaBlllHXC.H SIep 3a BpeMSI 1:1 t = t - to = t 6Y,AeT paBHO 1:1 N = No - N =
N+-2}
AKTHBHOCTLIO A HYKJl1:Ia B PMHOaKT1:IBHOM HCTOqH1:IKe Ha3bIBaeTCSI BeJl1:l-
Q1:lHa, paBHR.H OTHOllleH1:I1O q1:lCJla paCnaB1lI1:IXCSI .Hep KO BpeMeH1:I, 3a KOTopoe
pacna.n: npOH30111eJl. T.e. 3TO q1:lCJlO pacna.n:OB B e.n:1:IH1:I[(Y BpeMeH1:I
MJ
A=-.
I:1t
B CM e,AHHHu.eH aKT1:IBHOCT1:I SlBJI.HeTCSI 6eKKepeJlb. 1 BI< - 3TO aKTHB-
HOCTb PM1:IOaKTHBHoro npenapaTa, B KOTOpOM np01:lCXO1:IT pacna O)J;Horo SIpa
3a 1 c.
BHecHCTeMHa.H e)J;HH1:Iu.a a1<T1:IBHOCT1:I - 1 KlOpH = 3,7. 10 10 BK. t.feM MeHb-
llIe nepHo nOJlypacnaa T, TeM 6bIcTpee np01:lCXO)J;1:IT pacna, TeM a1<T1:IBHee
3neMeHT.
qacTo pa.n:HoaKTHBHbIH pacna.n: xapaKTep1:l3YlOT He nepHo.n:OM nOJlypacna.n:a
T, a nOCTO.HHHOH pa1:I0aI<T1:IBHOrO pacnaa A, KOTopaSi CBSl3aHa C nepHOOM
.. 1 In 2 II
cpOPMYJl01:l I\, = -. OCTOSlHHaSi pa,AHOaI<T1:IBHOrO pacnaa onpeenSieT Bepo-
T
.HTHOCTb pacnaa OHoro SI)J;pa 3a 1 c.
Tora 3aKOH PM1:I0aKTHBHoro pacnaa 6Y)J;eT HMeTb BH:
N = Noe-J...t.
614

1
BeJlHqHHa, 06paTHa.H nOCTO.HHHOH pacna,na 't = I' Ha3bIBaeTC.H cpe.n;HHM Bpe-
MeHeM mH3HH pa.n;HoaKTHBHoro 3J1eMeHTa.
B03HHKalOHe B pe3YJlhTaTe pa.n;HoaKTHBHoro pacna.n;a .H.n;pa MorYT 6bITb, B
CBOIO O'lepe.n;h, pa.n;HoaKTHBHbIMH. 8TO npHBo.n;HT K B03HHKHOBeHHIO o;enOllKH,
HJlH pH,ZJ;a, pa.lJ;HOaKTHBHLIX npeBpam;eHHH, 3aKaH'IHBalOI.QHXC.H cTa6HnbHbIM
H30TonOM. COBoKynHocTb .H,nep. 06pa3YIOI.QHX TaKYIO enOqKY, Ha3bIBaeTCSI pa-
.lJ;HOaKTHBHLIM ceMeHCTBOM. 1f3BeCTHbI TPH pa,AHOaKTHBHbIX ceMeHCTBa: ceMeH-
CTBO ypaHa (OT 2 U ), ceMeHCTBO TOpHSI (OT 2 Th) H ceMeHCTBO aKTHHHSI (OT
2Ac). Bce ceMeHCTBa 3aKaH'IHBaIOTCSI cTa6HJlbHbIMH H30TonaMH CBHHa
2gPb; 2gPb; 2gPb COOTBeTCTBeHHO.
22.6. PaJJ;HOaKTHBHoCTL H ee BH.n;L1
Pa,nHOaKTHBHOCTh conpOBOmlIaeTCH H3JIY1.IeHHeM. TIPWIeM. KaK nOKa3aJIH onbI-
TbI, HHTeHCHBHOCTb H3J1yqeHHSI He 3aBHCHT OT BHeIllHHX ycnoBHH. aHa nOCTO.HHHa
npH H3MeHeHHH TeMnepaTYPhI, ,n8.BJIeHHSI, arperaTHoro COCTOSIHHSI, He 3aBHCHT OT
B1:I.lJ;a XHMH'lecKoro coe,nHHeHHSI, B COCTaB KOToporo BXO,nHT pa,nHOaKTHBHbIH H30-
Ton, He H3MeHSIeTCSI no,lJ; ,lJ;eHCTBHeM 3J1eKTpHqeCKHX H MarHHTHbIX nOJleH H ,np.
Pa.n;HoaKTHBHoe H3J1yqeHHe HMeeT CJlOmHbIH COCTaB. B MarHHTHOM nOJle Y3-
KHH nyqOK pa,nHoaKTHBHoro H3J1yqeHH.H paCI.QenJl.HeTC.H Ha TPH KOMnOHeHTa
(pHC. 22.3): a-, p- H Y-JlyqH. TIo.n;po6Hoe HCCJle,nOBaHHe 3THX KOMnOHeHTOB no-
3BOJlHJlO BbI.HCHHTb HX npHpo,ny H OCHOBHbIe CBOHCTBa. Ha pHCYHKe 22.3:
1 - TOJlCTOCTeHHbIH COCY)J. H3 CBHHa, 2 - pa)J.HoaKTHBHbIH HCTOqHHK.
a-JIY'lH npe)J.CTaBJlSIIOT C060H nOTOK 6hICTpO )J.BHmYI.QHXC.H qaCTH, 3apSI)J.
KOTOpbIX paBeH 21el, a Macca paBHa Macce reJlHSI, T.e. a-lIaCTHo;a - ABY-
KpaTHO HOHH3HpOBaHHLIH aTOM reJIHH a = He. OHH BbIJleTalOT co CKOpOCTblO
::; (1,4 - 2). 10 7 M , T.e. 06J1a,naIOT 60nhilioH 3HeprHeH (4-9 M3B), a TaKme
C
BbICOKOH HOHH3HpYIOI.QeH H MaJlOH npOHHKaIOI.QeH cnoc06HoCThlO (nornOI.QaIOT-
C.H, HanpHMep, cnoeM aJlIOMHHH.H. TOJlI.QHHOH 0,05 MM).
COrJlaCHO COBpeMeHHhIM npe,nCTaBJleHHSIM, a-qac-
THIJ,hl 06pa3YIOTC.H B MOMeHT pa)J.HoaKTHBHoro pacna-
,na npH B3aHMo,neHcTBHH )J.BHmYI.QHXC.H BHYTPH SI,npa
nBYX npOTOHOB H ,lJ;BYX HeHTpoHoB
2lH + 2An  He.
Y
a
Puc. 22.3
a-pacna,n Ha6J1IO,naeTC.H TOJlbKO Y .H)J.ep C 60nbIllHM
MaCCOBbIM qHCnOM A (A> 200). a-pacna,n npOHcxo,nHT
B COOTBeTCTBHH C 3aKOHaMH COXpaHeHH.H 3HeprnH 3J1eK-
TpH'leCKOrO 3ap.HJJ.a H '1HCJla HYKJIOHOB
A x a-pacn8,A 4 A-4 y
Z ) 2 a + Z-2 '
r,ne :X - .H)J.PO, HcnbITbIBalOI.Qee a-pacna,n (MaTe-
pHHCKoe .H.n;pO) , 1-=-y - B03HHKmee .H)J.pO (.n;OqepHee
.H,lJ;PO ).
1
2
,
"" J3
I'
I'
l/
615

B pe3YJlbTaTe a-paCnalIa XHM1:IqeCK1:IH 3J1eMeHT nepeMelIIaeTC.H B Ta6J11:1e
MeHeJleeBa Ha ltBe KJleTOqKH 6J11:1me K HaqaJlY (npaB1:IJlO CMeeH1:I.H).
-JIY'lH npeCTaBJI.HIOT C060H nOTOK 6bICTpbIX (u::: 0,99 c) 3JIeKTpOHOB. OHH
CHJlbHee OTKJlOH.HIOTC.H B MarHHTHOM nOJle, qeM a-qaCTHbI. Mx 1:I0H1:I31:1pylO-
a.H cnoc06HoCTb 3Haq1:lTeJlbHO MeHbme (npHMepHo Ha Ba nOp.HKa), a npo-
H1:IKalOa.H cnoc06HoCTb ropa30 60Jlbme (nOrJlOaIOTC.H CJlOeM Al TOJlHHOH
OKOJlO 2 MM), qeM Y a-qaCT1:I.
llpH - -pacnae 01:IH 1:13 HeHTpOHOB .H,Apa B MOMeHT pacnaa npeBpaaeT-
C.H B npOTOH C OHOBpeMeHHbIM 06pa30BaHHeM 3J1eKTpOHa 1:1 BbIJleTOM 3J1eKT-
pOHHoro aHT1:IHeHTp1:lHO 8v e
1 1 0 0-
on  IP+ -l e + OV e .
llpH - -paCna,Ae XHMHqeCKHH 3J1eMeHT nepeMeaeTC.H B Ta6J11:1e MeHeJlee-
Ba Ha OHY KJleTOqKY BnpaBO (npaBHJlO CMeeH1:I.H)
A x  .pacn8A 0 A y
Z ) -l e +Z+l .
llp1:l 1:ICKYCCTBeHHbIX npeBpaI.QeH1:I.HX .Hep (CM. CJle,AYI01II1:IH naparpaCP) MO-
rYT B03HHKHYTb 1:I30TOnM. 1:ICnYCKalO1II1:Ie pan1:l0aKT1:IBHOe 1:I3J1yqeHHe n03HTpOH-
Hoe - +-1:I3J1yqeH1:Ie. OHO npeCTaBn.HeT C060H nOTOK n03HTpoHOB. T.e. qac-
T1:I, OTJl1:IqalO1:Ixc.H OT 3J1eKTpOHOB TOJlbKO 3HaKOM 3ap.Ha 1:1 B03H1:IKalO1:IX
B pe3YJlbTaTe npeBpaeH1:I.H npoToHa B HeHTpOH.
1 1 0 0 0
IP  on + +1 e + 0 V e , re OV e - 3J1eKTpOHHoe HeHTpHHO.
3J1eKTpOHHoe HeHTp1:lHO 1:1 aHTHHeHTpHHO .HBn.HIOTC.H qaCT1:IaMH, He 1:IMelO-
HM1:I 3ap.HlIa. C BeCbMa MaJlOH MaccoH nOKO.H, nOqTH paBHOH HYJlIO. OHH
06J1a.n;aIOT orpOMHOH npOH1:IKalOI.QeH cnoc06HOCTblO.
- -H3J1yqeH1:Ie Ha6J1lOlIaeTC.H Y .HlIep C OTHOCHTeJlbHbIM 1:I36bITKOM HeHTpo-
HOB:
90Sr
38
-pacn8A
90 y 0 0-
) 39 + _Ie + OV.
+ -1:I3nyqeH1:Ie Ha6J1IOaeTC.H Y .Hep C OTHOC1:ITeJlbHbIM H36bITKOM npOTOHOB:
30 p 30 S ' 0 0
15  14 1 + +1 e + 0 V.
raMMa-JIyqH He OTKJlOH.HIOTCsr HH 3J1eKTpHqeCKHM, HH MarHHTHbIM nOJleM.
raMMa-JlyqH npeCTaBJI.HIOT C060H KOpOTKOBOJlHOBOe 3J1eKTpOMarH1:ITHOe 1:I3J1yqe-
H1:Ie C qpe3BbIqaHHO MaJlOH lIJlHHOH BOJlHbI A < 10- 10 M 1:1 BCJleCTB1:Ie 3Toro -
C srpKO BbIpameHHbIM1:I KopnYCKYJlsrpHbIM1:I CBoHcTBaMH, T.e. .HBJI.HIOTC.H nOTO-
KOM 't{aCTH - 'Y-KBaHToB. OH1:I 06J1aaIOT CpaBH1:ITeJlbHO HeBbICOKOH HOH1:I31:1-
pYIOeH cnoc06HOCTblO 1:1 BbICOKOH npOH1:IKalOeH cnoc06HoCTblO (npoXO.HT qe-
pe3 CJlOH CBHHa TOJl1:IHOH 5 CM). llp1:l 'Y-1:I3nyqeH1:IH MaCCOBoe q1:lCJlO A 1:1
3ap.H Z .Hpa He H3MeH.HIOTC.H. 'Y-H3nYQeH1:Ie He .HBJlJleTC.H caMOCTO.HTeJlbHbIM
BHOM pacnaa, a TOJlbKO conpOBomaeT a- H -pacnabI, a TaKme B03H1:IKaeT
npH JI)J;epHbIX peaK1:I.HX, np1:l TopMOmeH1:I1:I 3ap.HmeHHbIX qaCTH, 1:IX pacnae
Ii T ..
B HaCTOJlee BpeM.H YCTaHOBJleHO, qTO 'Y-1:I3nYQeHHe HcnYCKaeTC.H ,AOqepH1:IM
JIpOM, KOTopoe B MOMeHT 06pa30BaH1:IJI H3 MaTep1:lHCKOrO .Hpa .HBJI.HeTC.H, KaK
npaB1:IJlO, B036ymeHHbIM H 3a BpeM.H 't::: 10- 13 _10- 14 C (3HaQ1:lTeJlbHO MeHbme
616

BpeMeHH mH3HH B036ymeHHorO aTOMa 't::= 10- 8 C) nepexoHT B OCHOBHoe co-
CTOSlHHe C HcnycKaHHeM Y-H3J1yqeHH.H. B03BpaaSlCb B OCHOBHoe COcToSlHHe, B03-
6ymeHHoe .HpO MomeT npOHTH qepe3 p.H npoMemYTOqHbIX COCTO.HHHH, n03TO-
MY H3J1yqeHHe OHOrO H Toro me pa,nHoaKTHBHoro H30Tona MomeT coepmaTb
HeCKOJlbKO rpynn Y-KBaHTOB, OTJlHqalOHXC.H OHa OT pyrOH CBoeH 3HeprHeH.
22.7. HCKyccTBeHHas pa)];HOaKTHBHOCTL
IIcKyccTBeHHaJl pHoaKTHBHOCTh - pacna H30TonOB. nOJlyqeHHbIX HCKYC-
CTBeHHbIM nYTeM (B pe3YJlbTaTe .H,AepHbIX peaKHH).
H,Apa, 06J1a.AalOHe HCKyccTBeHHoH paHoaKTHBHOCTblO, nOJlyqalOT nYTeM 60M-
6apHpoBKH cTa6HJlbHbIX .Hep T.HmeJlbIMH qaCTHaMH: a-qaCTHaMH, HeHTpoHa-
MH, peme npOTOHaMH H p. IIpli 3TOM .HepHbIe npeBpaeHH.H npOHCXO.HT B
Ba 3Tana. CHaqaJla 60M6apHpYIOa.H qaCTHa, nonaa.H B .HpO, BbI3bIBaeT
ero npeBpaeHHe B npOMemYTOqHOe HeCTa6HJlbHOe (paHoaKTHBHoe) .Hpo. 3a-
TeM 06pa30BaBllleeC.H pa.AHOaKTHBHoe .HpO caMonpOH3BOJlbHO HcnYCKaeT qaCTHY
H npeBpaaeTC.H JlH60 B cTa6HJlbHoe. JlH60 B HOBoe pa,AHOaKTHBHoe HpO.
IIpHHltHnHaJlbHOrO pa3J1HqH.H Mem.l1Y eCTeCTBeHHOH H HCKyccTBeHHoH pa.l1HO-
aKTHBHOCTblO HeT, H saKOHbI pa):(HoaKTHBHoro npeBpaeHH.H B 060HX cJlyqa.HX
OHHaKOBbI.
22.8. epHLle peaKHH.
3HepreTH'IeCKHH BLIXO)]; JI,D;epHLIX peaKHH
epHhlMH peaKHJlMH HaSbIBalOT npeBpaeHH.H SI,Aep npH B3aHMoeHcTBHH
HX C 3J1eMeHTapHbIMH qaCTHaMH HJlH pyr C pyrOM.
H.l1epHbIe peaKHIi 06bIqHO ocyecTBJlSlIOTC.H nYTeM 60M6apHpoBKH TSlme-
J1bIX SIep 3J1eMeHTapHbIMH qaCTHaMH HJlH JlerKHMH .HpaMH, YCKopeHHbIMH 1I0
60JlblllHX 3HeprHH. Hpo 3aXBaTbIBaeT 60M6apHpylOYIO qaCTHy, norJlOaeT
ee 3HeprHIO H nepeXO,AHT B B036ymeHHoe COCTO.HHHe. Hpo CTaHOBHTC.H Heyc-
TOHqHBbIM H npeBpaaeTC.H B HOBoe .HpO C HcnYCKaHHeM pyroH qaCTHbI.
B JlI060H .H)J;epHoH peaKHH BbInOJlH.HIOTC.H 3aKOHbl coxpaHeHHH 3J1eKTpH-
'1eCKOrO 3apH,ZJ;a H qHCna HYKJlOHOB: CYMMbI 3apSlOB H MaCCOBbIX qHCeJl .Hep
H qaCTH, BCTynalOHx B .HepHYIO peaKHIO, paBHbI CYMMaM 3ap.HOB H Mac-
COBbIX qHCeJI KOHeqHbIX npOYKTOB (.Hep H qaCTH) peaKHH. BbInOJlHJJIOTC.H
TaKme 3aKOHbI coxpaHeHHH 3HeprHH. HMnynhca H MOMeHTa HMnYJlbca.
HepHa.H peaKltH.H xapaKTepH3yeTC.H 3HeprHeH .H.1IepHOH peaKl(HH t!! W (3Hep-
reTH'IeCKHM BhIXO.n;OM peaKHH)t paBHoH pa3HOCTH 3HeprHH nOKO.H .Hpa Ii qa-
CTH )J;O peaKHH H nocJle Hee, T.e.
W=c2rm'-tmjl
re L m , - CYMM8 Macc qaCTH o peaKHH, L m/
I j
- CYMMa Macc qaCTH
nocJle peaKHH.
ECJlH t!! W > 0, TO peaKHSI HeT C BbIeJleHHeM 3HeprHIi H Ha3bIBaeTC.H
3K30TepMHqeCKOH, eCJlH /). W < 0, TO peaKHSI HeT C nOrJlOeHHeM 3HeprHII H
Ha3bIBaeTC.H 3H.ZJ:OTepMHqeCKOH.
617

H,AepHbIe peaKHH KJIaCcH4>HHpyroTCH no CJIeYIOI..QHM npH3HaKaM:
1) no po,a;Y Y'laCTByroHX B HHX 'IaCTH: peaKHH no eHcTBHeM HeHT-
pOHOB; peaKHH no ,neHcTBHeM 3ap.HmeHHbIX qaCTH (npoToHoB, eHTpoHoB,
a-qaCTH); peRKnHH no eHcTBHeM Y-KBRHTOB;
2) no 3Hepl'HH BLI3LIBalOHX HX 'IaCTH: peaKHH npH MaJIbIX 3HeprH.HX
(nOp.HKa 3J1eKTpOH-BOJIbT), npOHCXO.H1T(He B OCHOBHOM C yqaCTHeM HeHTpo-
HOB; peRKnHH npH cpeHHX 3HeprH.HX (O HeCKOJlbKHX Mera3J1eKTpOH-BOJIbT,
npOHCXO,A.HIl!He C yqaCTHeM Y-KBaHTOB H 3ap.HmeHHbIX qaCTHn (npOTOHOB.
a-qaCTHn); peaKHH npH BbICOKHX 3HeprH.HX (COTHH H TblCHqH Mera3J1eKTpOH-
BOJlbT), npHBO.H1T(He K pomeHHIO OTCYTCTBYIOI..QHX B CB060HOM COCTOHHHH
3J1eMeHTSpHbIX qaCTHn H HMelO1T(He 60JIbliiOe 3HaqeHHe ,AJIH HX H3yqeHH.H;
3) no po,a;y Y'laCTBYIOHX B HHX .H,a;ep: peaKHH Ha JlerKHX .Hpax
(A < 50); peaKnHH HR cpeHHX Hpax (50 <A < 100); peaKHH HR T.HmeJlbIX
SI,ApSX (A> 100);
4) no xapaKTepy npOHCXO,a;HX HJJ;epHLlX npeBpaeHHH: peaKnHH C Hcnyc-
KaHHeM HeHTpoHoB; peaKHH C HcnYCKRHHeM 3apHmeHHbIX qaCTHn; peaKnHH 3RX-
BaTa (B cJlyqae 3THX peaKHH COCTaBHoe .HpO He HcnYCKseT HHKaKHX qaCTH, S
nepeXOHT B OCHOBHoe COCTO.HHHe, H3J1yqaH OHH HJlH HeCKOJlbKO Y-KBaHTOB).
IIepBa.H B HCTOpHH H)J;epHaH peaKHH OCYI..QeCTBJleHa 3. Pe3ep<}>opoM (1919)
npH 60M6RpHpoBKe Hep a30TS a-qaCTHRMH, HcnYCKseMbIMH pS,AHOSKTHBHbIM
HCTOqHHKOM:
*
lN + He  (lF)  lO + H.
HepHbIe peaKnHH no eHcTBHeM a-qaCTHn npHBenH K OTKpbITHIO HeHTpo-
Ha (.D:m. eBHKoM):
*
Be + He  CC)  lC + In.
IIpHMepoM HepHoH peSKnHH, oCY1T(eCTBJI.HeMOH npH 60M6spHpoBKe YCKO-
peHHbIMH npOTOHSMH, HBn.HeTCH peaKH.H:
7 L ' I H 4 H 4 H
3 1 + 1 --+ 2 e + 2 e.
IIo eHCTBHeM MeJleHHbIX HeHTpOHOB:
*
2U + In --+ (2U) --+ _e + 2Np.
22.9. 3KcnepHMeHTaJILHLle MeTOLI
perHCTpal(HH 3ap.HeHHLIX 'IaCTHl(
IIpH60pbl JlH perHCTpaHH 3ap.H>KeHHbIX qSCTH HS3bIBaIOTC.H AeTeKTOps-
MH. Cy1T(eCTByeT Ba OCHOBHbIX BHS eTeKTopOB:
1) iJucKpemlible (CqeTHbIe H onpeeJl.HI01T(He 3HeprHIO qaCTH): CqeTqHK reH-
repa, HOHH3aHOHHa.H KaMepa H p.;
2) mpeK06ble (alO1T(He B03MomHOCTb Hs6J1IOSTb H (}>oTorps(}>HpoBSTb cne-
)J;bI (TpeKH) qSCTH B pa60qeM 06'beMe eTeKTops): KSMepa BHJlbCOHS, nY3bIpb-
KOBaH KRMepa, TOJlCTOCJlOHHbIe (}>OT03MYJlbCHH H ,np.
1. ra30pa3pHLIH C'IeT'IHK reHl'epa. .D:JlH perHCTpaHH 3J1eKTpOHOB H
Y-KBaHToB (<}>OTOHOB) 60JlbliiOH 3HeprHH HcnoJlb3yeTcH CqeTqHK reHrepa-MIOJI-
618

CTeKJI.HHHa.H KOJl6a
nepa. OH COCTOHT H3 CTeKJI.HHHOH TPy6KH
(p1:lc. 22.4), K BHYTpeHHHM CTeHKaM KO-
TOpOH npHJleraeT KaTO K - TOHKHH Me-
TaJlJlHqeCK1:IH I1.1:IJl1:IH.u.p; aHo.u.OM A cJlymHT
TOHKa.H MeTaJlJlH"leCKa.H npOBOJlOKa. HaT.H-
HYTa.H no OCH CqeTq1:lKa. Tpy6Ka 3anOJl-
H.HeTC.H ra30M, 06bI'lHO aprOHOM. CqeTqHK
BKJlIO'IaeTC.H B per1:lCTpHpylOYIO cxeMY. Ha
Kopnyc nOaeTC.H OTpHI1.aTeJlbHbIH nOTeHI1.1:I-
aJl, Ha HHTb - nOJlOmHTenbHbIH. IIocJle-
OBaTeJlbHO C'IeTtJHKY BKJllOqaeTC.H pe3HCTOp
R, C KOToporo CHrHaJl noaeTCH K per1:lc-
Tp1:lpYIOeMY YCTPOHCTBY.
eHCTB1:Ie CqeT'IHKa OCHOBaHO Ha yap-
HOH HOHH3aI1.HH. IIycTb B CqeT"lHK nonaJla qaCT1:II1.a, C03aBma.H Ha CBoeM nYT1:I
XOT.H 6bI O.u.HY napy: "HOH + 3J1eKTpoH". 3J1eKTpOHbI, )J;B1:IraHCb K aHOy
(HHTH), nOna.z4alOT B nOJle C HapaCTalOe:H HanpHmeHHOCTblO (Hanp.HmeHHe Mem-
y A 1:1 K -1600 B), HX CKOpOCTb CTpeM1:ITeJlbHO B03paCTaeT, H Ha CBoeM
nYTH OH1:I C03.u.a1OT 1:I0HHYlO JlaB1:IHY (B03H1:IKaeT y.u.apHa.H HOH1:I3aI1.1:I.H). 1l0naB
Ha H1:ITb, 3J1eKTpOHbI CHHmalOT ee nOTeHIlHan, BCJlenCTB1:Ie qero no pe3HCTOpy
R nOHeT TOK. Ha ero KOHI1.aX B03HHKaeT HMnYJlbC Hanp.HmeHH.H, KOTOpbIH H
nocTynaeT B perHCTpaI1.1:I0HHOe YCTPOHCTBO.
Ha pe31:1CTOpe np01:lCXO1:IT na.u.eHHe Hanp.HmeH1:I.H, nOTeHI1.1:IaJl aHoa YMeHb-
llIaeTC.H, H Hanp.HmeHHOCTb nOJlH BHYTp1:l CqeTq1:lKa y6bIBaeT, BCJleCTB1:Ie qel'O
YMeHbllIaeTC.H KHHeT1:IqeCKag 3HeprH.H 3J1eKTpOHOB. Pa3p.H.u. npeKpaaeTC.H. Ta-
KHM 06pa30M, pe3HCTOp HrpaeT pOJlb COnpOT1:IBJleHH.H, aBTOMaTH'IeCKH raC.HlIJ.e-
1'0 JlaBHHHbIH pa3p.H. IIOJlOm1:lTeJlbHbIe 1:I0HbI CTeKalOT K KaTOY B TeqeHHe
t:::: 10- 4 C nocJle Ha'laJla pa3p.Ha.
CqeTqHK reHrepa n03BOJl.HeT pel'HCTp1:lpOBaTb 10 4 qaCTHI1. B ceKYH)J;Y. OH
npHMeH.HeTC.H B OCHOBHOM )J;JlB perHCTpaI1.1:IH 3J1eKTpOHoB H Y-KBaHToB. O)J;HaKo
HenocpecTBeHHo Y-KBaHTbI BCJle)J;cTBHe cBoeH MaJlOH HOHH3HpYIOeH cnoc06Ho-
CT1:I He perHCTp1:lpYIOTC.H. llJl.H 1:IX 06HapymeHHH BHYTpeHHlO1O CTeHKY Tpy6K1:I
nOKpbIBalOT MaTep1:laJlOM, H3 KOTOpOI'O Y-KBaHTbI BbI61:1BalOT 3J1eKTpOHbI. IIp1:l
perHCTpaI1.HH 3J1eKTpOHOB 3«}>«}>eKTHBHOC'I'b CqeTqHKa 100 %,
a npH perHCTpaI1.HH Y-KBaHToB - JlHllIb OKono 1 %.
PerHCTpaI1.H.H T.HmenbIX CX-qaCT1:II1. 3aTpy.u.HeHa, TaK KaI<
CJlOmHO CeJlaTb B CqeTqHKe OCTaTOqHO TOHKoe "OKOllIKO",
np03paqHOe )J;JI.H 3T1:IX qaCTHIJ..
2. KaMepa BHJILCOHa.
B KaMepe 1:IcnoJlb3yeTC.H cnoc06HoCTb qaCTHIJ. 60JlbllI1:IX
3HeprHH HOHH3HpOBaTb aTOMbI l'a3a. KaMepa B1:InbCOHa
(pHC. 22.5) npe.u.CTaBJI.HeT C060H I1.HJl1:IHp1:lqeCKHH cocy.u. C
nopllIHeM 1. BepXH.H.H qaCTb I1.1:IJlHH)J;pa ,CeJlaHa H3 npo-
3paqHOrO MaTepHana, B KaMepy BB01I1:ITC.H He60JlbllIOe KO-
JlH'IeCTBO BO.u.bI HJlH cnHpTa, JI.H qero CHH3Y cocy no-
KpbIT CJlOeM BnaHoro 6apxaTa HnH CYKHa 2. BHYTP1:l :Ka-
MepbI 06pa3yeTcH CMeCb HaCLlJ.Q.eHHLIX napOB H B03,Ayxa.
IIpH 6LICTpOM onycKaHHH nOplllH.H 1 CMeCb aHa6aTHqeCK1:I
K perHCTpH-
pYIOeMY
YCTPOHCTBY
Puc. 22.5
A
Puc. 22.4
619

paCillHp.HeTC.H, qTO COnpOBOm)J;aeTC.H nOHHmeHHeM ee TeMnepaTypbI. 3a CtIeT OX-
Jlam.n;eHH.H nap CTaHOBHTC.H nepeCLIIU.eHHLIM.
ECJIH B03.n;yx OtIHID;eH OT nbIJlHHOK. TO KOH.n;eHcaD;H.H napa B mH.n;KOCTb
3aTpy.n;HeHa H3-3a OTCYTCTBH.H eHTpOB KOH.n;eHcaD;HH. O.n;HaKO n;eHTpaMH KOH-
.u;eHcan;HH MorYT CJlymHTb H ROHbI. n03TOMY eCJlH qepe3 KaMepy (BnYCKaIOT
tIepe3 OKOillKO 3) npOJleTaeT 3ap.HmeHHa.H qaCTHD;a. HOHH3HpYlOlQa.H Ha CBoeM
nYTH MOJleKYJlbI, TO Ha D;enOqKe HOHOB npOHcxo.n;HT KOH.n;eHcaD;H.H napOB H
TpaeKTOpH.H lIBHmeHH.H qaCTHD;bl BHYTPH KaMepbI 6JIaro.n;ap.H oceBillHM MaJIeHb-
KHM KaneJlbKaM mH.n;KOCTH CTaHOBHTC.H BH.n;HMOH. QenOqKa 06pa30BaBillHXC.H
KaneJlb mH.n;KOCTH 06pa3yeT TpeK qaCTHD;bI. TenJlOBOe .n;BHmeHHe MOJleKYJI 6bIC-
TpO pa3MbIBaeT TpeK 'laCTHD;, H TpaeKTopHH tIaCTHD; BH,AHbI OTtIeTJlHBO JIHillb
OKOJlO 0,1 C, qTO, OltHaKO, .AOCTaTOqHO )J;JI.H OTorpaHpOBaHH.H.
BH.n; TpeKa Ha OTOCHHMKe qaCTO n03BOJI.HeT CY-
.n;HTb 0 npHpo.u;e qaCTHD;bI H BeJIH'IHHe ee 3HeprHH.
TaK, <x-qaCTHD;bI OCTaBJI.HIOT cpaBHHTeJlbHO TOJICTbIH
cnnOillHOH CJle,A, npOTOHbI - 60Jlee TOHKHH, a 3JIeKT-
pOHbI - nYHKTHpHbI:M (pHC. 22.6). nO.HBJI.HlOlQeeCSI
paClQenJleHHe TpeKa - "BHJIKH" CBH.n;eTeJlbCTByeT 0
npOHCXO,A.HlQeH peaKD;HH.
q T06bI nO,ArOTOBHTb KaMepy K .n;eHcTBHIO H O"tiHC-
THTb ee OT OCTaBillHXC.H HOHOB, BHYTPH Hee C031IalOT
3JIeKTpHqeCKOe nOJle, npHT.HrHBalOID;ee ROHbI K 3J1eKT-
po.n;aM, r.n;e OHH HeHTpaJlH3YIOTC.H.
COBeTcKHe H3HKH n. JI. KanHD;a H )];. B. CK06eJIbD;bIH npe.n;JlOmHJIH pa3Me-
ID;aTb KaMepy B MarHHTHOM nOJIe, no,A .n;eHcTBHeM KOToporo TpaeKTopHH tIac-
THD; HCKpHBJI.HIOTC.H B TY HJlH HHYIO CTOPOHY B 3aBHCHMOCTH OT 3HaKa 3ap.H-
.n;a. no pa.n;HYCY KpHBH3HbI TpaeKTopHH R HHTeHCHBHOCTH TpeKOB onpe.n;eJl.HIOT
3HeprHIO H Maccy tIaCTHI.I,bI (y.n;eJlbHbIH 3ap.H.n;).
3. nY3LlpLKOBa.H KaMepa. B HaCTO.HID;ee BpeM.H B Hay"t{HbIX HCCJle.n;OBaHH.HX
HCnOJlb3yeTC.H nY3bIpbKOBa.H KaMepa. Pa60tIHH 06'beM B nY3bIpbKOBOH KaMepe
3anOJlHeH mH.n;KOCTblO no.n; BbICOKHM .n;aBJIeHHeM, npe.n;oxpaH.HIOID;HM ee OT 3a-
KHnaHH.H, HeCMOTp.H Ha TO, qTO TeMnepaTypa mH.n;KOCTH BbIille TeMnepaTypbI
KHneHH.H npH aTMocepHOM .n;aBJleHHH. npH pe3KOM nOHHmeHHH ,AaBJleHH.H )KH,A-
KOCTb OKa3bIBaeTC.H neperpeTOH R B TeqeHHe He60JIbillOrO BpeMeHH Haxo.n;HTC.H
B HeYCTOHqHBOM COCTO.HHHH. ECJlH qepe3 TaKYIO mH.n;KOCTb npOJleTHT 3ap.HmeH-
Ha.H qaCTHa, TO B.l(OJlb ee TpaeKTopHH mH.l(KOCTb 3aKHnHT, nOCKOJlbKY 06pa-
30BaBillHeC.H B mH.l(KOCTH HOHbI cJlymaT D;eHTpaMH napo06pa30BaHH.H. IIpH 3TOM
Tp8eKTOpH.H qaCTHD;bI OTMe'laeTC.H D;enOtIKOH nY3bIpbKOB napa, T.e. .l(eJlaeTC.H BH-
,lIHMOH. B KaqeCTBe mH.l(KOCTeH HCnOJlb3YIOTC.H rJlaBHbIM 06pa30M mHlIKHH BO-
.l(Opo.n; H nponaH C3HS' )];JlHTeJlbHOCTb pa60qero D;HKJla nOp.H.l(Ka 0,1 c.
npeHMyecTBo nY3bIpbKOBOH KaMepbI nepe.l( KaMepOH BHnbcoHa 06YCJlOBne-
HO 60JlbilleH nJIOTHOCTblO pa60qero BeID;eCTBa, Bcne.n;CTBHe qero qaCTHD;a Tep.H-
eT 60Jlbille 3HeprHH, qeM B ra3e. np06erH qaCTHD; OKa3bIBalOTCSI 60Jlee KOpOT-
KHMH, H qaCTHD;bI .n;ame 60JlbillHX 3HeprHE 3aCTpeBalOT B KaMepe. 3TO n03BO-
JI.HeT rOpa3.l(0 TOqHee onpe.n;eJlHTb HanpaBJleHHe ,ABHmeHH.H tIaCTHbI H ee 3Hep-
rHIO, Ha6J1IO.l(aTb cepHIO nOCJle,AOBaTeJlbHbIX npeBpaID;eHHH tIaCTHD;bI H BbI3bIBae-
Mble elO peaKHH.
!  /
 \ t/
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)/  ,.11 \
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/: ---,! "-
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 . 1
Puc. 22.6
620

4. MeToA TOJICTOCJIOHHLIX 4JOT03MYJILCHH pa3pa60TaH JI. B. MbICOBCKHM Ii
A. II. )K,AaHOBbIM.
OH OCHOBaH Ha HCnOJlb30BaHHH nOqepHeHH.H cPoTorpacPHqeCKOrO CJlOSl no,lJ;
,neHcTBHeM npoxo,n.HLQHX qepe3 cPOT03MYJlbCHIO 6:bICTpbIX 3apJimeHHbIX qaCTH.
TaKaJi qaCTHua BbI3blBaeT pacnall MoneKYJI 6poMHcToro cepe6pa Ha HOHbI Ag
H Br- H nOqepHeHHe cPOT03MYJlbCHH B,nOJlb TpaeKTopHH ,nBHmeHH.H, 06pa3YJi
CKpblToe H306pameHHe. IIpH npOJiBJleHHH B 3TH X KpHCTaJlJlHKaX BOCCTaHaBJlH-
BaeTCSI MeTaJlJlHqeCKOe cepe6po H 06pa3yeTCJi TpeK qaCTHbI. IIo ,nJlHHe H TOJl-
HHe TpeKa CY,lJ;.HT 06 3HeprHH H Macce qaCTHbI.
AJI.H H3yqeHH.H CJle,lJ;OB qaCTH, 06J1a,ZJ;alOHx OqeHb BbICOKOH 3HeprHeH H ,na-
IOHX ,lJ;JlHHHbIe CJle,nbI, 60JlbWOe KOJlHqeCTBO nnaCTHHOI< CKJla,ZJ;bIBaeTCSI B cTony.
CyecTBeHHbIM npeHMyecTBoM MeTo,na cPoT03MynbcHH, nOMHMO npOCTOT:bI
npHMeHeHHJi, .HBJlJieTC.H TO, qTO OH ,AaeT HeHCqe3alOHH CJIeA qaCTHIJ.bI, KOTO-
pbIH 3aTeM MO>KeT 6bITb TLQaTenbHO H3yqeH. 3TO npHBeJlO K WHPOKOMY npH-
MeHeHHIO ,naHHOrO MeTO,na npH HCCJle,nOBaHHH HOBbIX 3neMeHTapHbIX qaCTHIJ..
3THM MeTO,nOM C ,n06aBJleHHeM K 3MYJlbCHH COe,nHHeHHH 60pa HJlH nHTH.H MO-
rYT 6bITb H3yqeHbI CJle,lJ;bI HeHTpOHOB, KOTOpbIe B pe3YJlbTaTe peaKHH C SI,npa-
MH 60pa H JIHTH.H C03,lJ;a1OT a-qaCTHbI, BbI3bIBalOHe nOqepHeHHe B cJloe .H,nep-
HOH 3MynbCHH. IIo CJle,naM a-qaCTH ,neJlaIOTCJi BbIBO,nbI 0 CKOpOCTH H 3HeprH-
JiX HeHTpOHOB, BbI3BaBlliHX nOJiBJleHHe a-qaCTH.
22.10. AeJIeHHe SI,lJ;ep ypaHa
,lleJleHHe SI,Aep ypaHa 6bIno OTKpbITO B 1938 r. HeMeIJ.KHMH Y'leHbIMH O. fa-
HOM H «1>. IllTpaCCMRHOM. 11M y,naJlOCb YCTaHOBHTb. qTO npH 60M6ap,nHpOBKe
SI,nep ypaHa HeHTpOHaMH 06pa3YIOTCJi 3J1eMeHTbI CpelIHeH qaCTH nepHOlIHqeCKOH
CHCTeMbI: 6apHH, KpHnTOH H p. IIpaBHJlbHOe TOnKOBaHHe 3TOMY cPaKTY ,lIanH
aBCTpHHCKHH cPH3HK JI. MeHTHep H aHrJlHHCKHH <pH3HK O. «1>pHW. OHIi 06'b.HC-
HHnH nO.HBneHHe 3THX 3neMeHTOB paCna,nOM .H,nep ypaHa Ha ,nBe npHMepHO paB-
HbIe qaCTH. STO .HBJleHHe nOJlyqHJlo Ha3BaHHe ,neJleHH.H .H,nep, a 06pa3YIOHeC.H
.H,npa -- OCKOJIKOB ,neneHHJi.
.n
I
a .
(J 
8

.n
n e ..  w"
2 ..............
"-
en
Puc. 22.7
06'b.HCHHTb 3TY peaKHIO ,neJleHH.H MomHO OCHO-
BbIBaJiCb Ha KaneJlbHOH Mo,neJlH .H,Apa. B 3TOH MO-
,neJlH Ji,npo paccMaTpHBReTC.H KaK Kann.H 3neKTpH-
qeCKH 3apSlmeHHoH HecmHMaeMO:H mH,AKOCTH. KpoMe
Ji,nepHbIx CHJI, ,neHcTBYIOLQHX Me>K,Ay BceMH HYKJlO-
HaMH .H,npa, npoToHbI HcnbITbIBalOT ,nonOJlHHTenbHoe
3J1eKTpOCTaTHqeCKOe OTTaJIKHBaHHe, Bcne,nCTBHe KO-
TOporo OHM pacnOJlaralOTCJi Ha nepHcPepHIi .H,npa. B
HeB036ym,lJ;eHHOM COCTOJiHHH CHJlbI 3J1eKTpOCTaTHqeC-
KOro OTTanKHBaHHJi CKOMneHCHpOBaHbI, n03TOMY
Ji,npo HMeeT CcPepHqeCKYIO <}>OPMY (pHC. 22.7, a).
II 235 U u 6
oCJle 3aXBaTa Ji,ApOM 92 HeHTpOHR 0 pa3Y-
eTCJi npOMemYTOqHOe Ji,npo (2:: u ).,. , KOTopoe HaxO-
,nHTC.H B B036ym,neHHOM COCTOJiHHH. IIpH 3TOM 3Hep-
rHJi HeHTpOHa paBHOMepHO pacnpe,nen.HeTC.H Me>K,ny
621

BceMH HYKJlOHRM1:I, R CRMO npOMemYTOqHOe .H,ZJ;pO ,netl>opM1:IpyeTCJI H Haq1:lHaeT
KOJle6aTbC.H. ECJlH B036ym,neH1:Ie HeBeJl1:lKO, TO .H,npO (pHC. 22.7, 6). oCB060m-
,AR.HCb OT 1:I3J1HWKa 3Hepr1:lH nYTeM HcnYCKRH1:ISI 'Y-KBaHTR 1:IJIH He:HTpOHa, B03-
BpalQReTC.H B YCTOHq1:lBOe COCTO.HH1:Ie. ECJIH me 3HeprHJI B036ym,neHH.H ,nOCTR-
TOqHO BeJlHKa, TO ,netl>opMRHJI .H,npR npH KOJle6aHH.HX MomeT 6bITb HRCTOJlbKO
60JIbIlIO:H, qTO B HeM 06pa3yeTcJI nepeT.HmKR (P1:lc. 22.7, 6), RHaJlOrHqHR.H ne-
peT.HmKe Mem,ny ,nBYM.H QaCT.HM1:I pa3,ABa1:lBRIOe:HC.H KanJl1:l mH,AKOCT1:I. Hep.
Hble CHJlbI, e:HCTBYIOHe B Y3KOH nepeT.HmKe, yme He MorYT npOTHBOCTOJlTb
3Haq1:lTeJlbHOH KYJlOHOBCKO:H CHJle OTTaJlKHBaHH.H QaCTe:H .H,ZJ;pa. TIepeT.HmKR pa3-
pbIBReTC.H, 1:1 .H,npo pacna,ZJ;aeTC.H Ha ,nBa "OCKOJlKa" (P1:lc. 22.7 z), KOTopbIe
pR3JIeTaIOTC.H B npOT1:IBOnOJlOmHbIe CTOpOHbI.
OCKOJlK1:I ,neJleHHJI pe,nKO 6bIBalOT O,nHHRKOBbIMH: tJ:RlQe Bcero O,nHH 1:13 H1:IX
npHMepHO B 1,5 pR3R T.HmeJlee ,APyroro (MRCCbI OCKOJIKOB OTHOCJlTC.H KRK 3: 2).
TIPO,nYKTbI eJleHH.H .H,npa ypRHa MorYT 6bITb pR3Ho06pR3HbIMH (Kp1:lnTOH H
6apHH, IJ;e3H:H H py6H,nH:H H T..u..). T1:InH'lHRSl peaKH.H ,AeJleHHJI
*
2U + In -4 (2::U) -4 l::Ba + gKr + 3Jn.
TIpH ,neJleH1:I1:I .H,nep T.HmeJlbIX aTOMOB (2g U) BbI,D.eJl.HeTC.R OqeHb 60JIbIlIRJI
3Hepr1:l.H - OKOJlO 200 M3B npH eJleH1:I1:I Kam,noro .H,ZJ;PR. 3TO 06'b.HCH.HeTC.H TeM,
qTO y,neJlbHR.H 3HeprH.H CB.H3H B "OCKOJlOqHbIX" JI,npax 3Haq1:lTeJlbHO 60JlblIle, qeM
B H,npe ypaHR. OKOJlO 80 % 3TO:H 3Hepr1:lH BbI,nenSleTCJI B BH,Ae KHHeTHtJ:eCKO:H
3Hepr1:lH OCKOJIKOB; OCTaJIbHbIe 20 % np1:lXo,n.HTC.H Ha 3HeprHIO pa,n1:l0RKT1:IBHOrO
H3J1yqeHH.H OCKOJIKOB H KHHeT1:IQecKyIO 3HeprHIO MrHOBeHHblX He:HTpOHoB.
Oc06eHHo BRmHO, qTO B pe3YJlbT8-
Te pacna,na JI,npa ypRHa BbI,neJl.HeTC.H
2-3 HOBbIX He:HTpoHR. 3TO 06'bJlCHJI-
eTCJI TeM, qTO B T.HmeJIbIX RTOMHbIX
.H,npRX npoeHTHoe co,nepmRH1:Ie He:HT-
pOHOB 3HatJ:HTeJIbHO 60JlbIlIe, qeM B
.H,npax 3J1eMeHTOB cpe,nHe:H qaCTH Ta6-
n1:lIJ;bI MeH,lleJleeBa. 3T1:I BTOp1:ltJ:HbIe
He:HTpOHbI MorYT BbI3BaTb ,neJleHHe
,npyr1:lx .H,nep H np1:lBeCTH K enHO:H
pe8K1:IH ,neJIeHHJI: qHCJlO ,AeJI.HlQHXCJI
.H,nep OqeHb 6bICTpO YBeJIHqHBReTCJI
(P1:lc. 22.8). [(enHRJI peaKH.H 6Y,AeT
npH onpe,neJleHHbIX YCJlOBH.HX no,n,nep-
m1:lBRTbC.H 6e3 BHelIlHero 06J1YQeH1:I.H
YPRHR He:HTpOHaM1:I.
QenHaJl peaKl\HJI - .H,nepHaJi pe-
aK1:I.H, B KOTOpO:H tJ:RCTHbI, BbI3bIBR-
IOlQHe peRK1:I1O, 06pR3YIOTC.H KRK npo-
,nYKTbI 3TO:H peRK1:I1:I.
AeJIeH1:Ie .H,Aep npHpo,nHoro ypRHR
npOHcxo,nHT 6J1Rro,nRpJl 1:I30Tony 2U,
KOTOpbIH B eCTeCTBeHHOM ypaHe co-
Me.n;JIHTeJlbHbIH HeHTpoH
BbICTpbIe HeiiTpoHbI
.. _.......1...... J......1.. ..........
j 3aMe.n;JlHTeJlb j
"..... - -/.... -.. .,- - _...."\........
Me.n;JleHHbIe HeiiTpoHbI
I
622
BbICTpbIe HeHTpOHbI
. .l. .... ......... .l........ ...... .\..
: 3aMe.n;JlHTeJlb :
I............. ....-r............. ....\
Puc. 22.8
CTaBn.HeT Bcero
1
140
,nOJlIO OT 60nee

pacnpOCTpaHeHHoro 2U. llpH 3TOM ,ZJ;eJleHHe 2U BbI3bIBalOT Me,nJleHHbIe HeH-
TpOHbI.
OKa3aJlOCb, 'lTO H,ZJ;pa ypaHa 2U TOme MorYT ,neJlHTbCH, HO ,nJlH 3Toro
HymHbI 6bICTpbIe HeHTpOHbI, 06J1a,nalOHe 3HeprHeH 60Jlee 1,1 M3B. HHaqe
3HeprHSI B036ym,neHH.H o6pa30BaBWHXC.H .H,nep 2U OKa3bIBaeTC.H He,nOCTaTOq-
HOH ,nJl.H ,neJleHHH, H Tor,na BMeCTO ,AeJleHH.H npOHcxo,n.HT .H,nepHbIe peaKHH:
2U + In -) 2U -pacnAA ) 2Np + _reo
H30Ton ypaHa 2U -pa,nHoaKTHBeH, nepHO,n nOJlypacna,ZJ;a 23 MHH H30-
Ton HenTYHHH 2Np Tome pa,nHoaKTHBeH, nepHOlI nOJlypacna,na OKono 2 ,nHeH.
239N p p-pacnAA ) 0 e + 239pU
93 -1 94 .
H30Ton nJlYTOHHSI 2:PU OTHOCHTenbHO cTa6HJleH, nepHO,n nonypacna,na
24000 JleT. BamHeHwee CBOHCTBO nJlYTOHH.H COCTOHT B TOM, qTO OH ,lI.eJlHTCSI
no,lI. BJlH.HHHeM HeHTpOHOB TaK me, KaK 2U. l103TOMY C nOMOblO 2:PU
MomeT 6bITb ocyeCTBJleHa enHa.H peaKD;H.H.
PaCCMOTpeHHaH BbIwe cxeMa D;enHOH peaKHH npe,nCTaBJISleT C060H H,neaJIbHbIH
CJlytIaH. B peaJIbHbIX YCJIOBH.HX He Bce 06pa3yIOHecSl npH ,neJIeliliH HeHTpoHbI yqa-
CTByIOT B ,ZJ;eJleHHH ,npyrnx SI,nep. qacTb HX 3axBaTbIBaeTC.H He,neJI.HHMHC.H SI,npaMH
nOCTopOHHHX aTOMOB. ,npyrne BbIJIeT8lOT H3 ypaHa Hapymy (YTe'1Ka HeHTpoHOB).
ll03TOMY D;enHaSl peaKUH.H lIeJleHH.H TSlmeJlbIX HJ1ep B03HHKaeT He BCerJ1a H
He npH JlI060H Macce ypaHa.
Pa3BHTHe D;enHOH peaKD;HH XapaKTepH3yeTC.H TaK Ha3bIBaeMbIM K03c1>cI>HuHeH-
TOM pa3MHomeHHSI HeHTpOHOB K, KOTOpbIH H3Mep.HeTCSI OTHoweHHeM qHCJla N i
HeHTpOHOB, BbI3bIBalOHX ,neneHHe .HlIep BeeCTBa Ha OJ1HOM H3 3TallOB peaKHH.
K qHCJlY N l HeHTpOHOB, BbI3BaBWHX ,neJleHHe Ha npe,nbI,nym;eM s-rane peaKHH:
K =!!.L.
N l - 1
K03c1>cI>HD;HeHT pa3MHOmeHH.H 3aBHCHT OT pSl,na cl>aKTopoB, B qaCTHOCTH
OT npHpo,nbI H KOJlHqeCTBa ,neJl.HerOCSI. BeeCTBa, OT reOMeTpH'1eCKOH cl>op-
MbI 3aHHMaeMoro HM 06'beMa. O,AHO H TO me KOJlH'1eCTBO ,naHHoro Bee-
CTBa HMeeT pa3Hoe 3HaqeHHe K. K MaKCHMaJlbHO, eCJlH BeeCTBO HMeeT wa-
poo6pa3HYlO cl>OPMY, nOCKOJlbKY B 3TOM cJlyqae nOTep.H MrHOBeHHbIX HeHTpo-
HOB qepe3 nOBepXHOCTb 6y,neT HaHMeHbweH.
Macca ,neJlHerOC.H BeeCTBa, B KOTOpOM enHaH peaKHH H,neT C K03c1>-
cl>HD;HeHToM pa3MHomeHHH K = 1, Ha3bIBaeTC.H KpHTH'IeCKOH MaccoH.
3HaqeHHe KpHTHqeCKOH MaCCbI onpe,neJl.HeTCH reoMeTpHeH cl>H3HqeCKOH CHC-
TeMbI, ee CTPYKTYPOH H BHeWHHM oKpymeHHeM. TaK, ,ZJ;JI.H wapa H3 '1HCTOrO
ypaHa 2: U KpHTHqeCKa.H Macca paBHa 47 Kr (wap ,nHaMeTpOM 17 CM). Ho
eCJlH Mem,ny CJlO.HMH ypaHa Haxo,nHTC.H MHOrOqHCJleHHbIe TOHKHe nOJlH3THJleHO-
BbIe llJleHKH H wap oKpymeH 6epHJlJlHeBOH 060JlOqKOH, TO KpHTHqeCKaSl Macca
CHHmaeTC.H ,no 242 r (wap ,nHaMeTpOM 3 CM).
llpH K03c1>cI>HHeHTe pa3MHomeHHSI K = 1 '1HCnO ,neJl.HHXCH SI,nep no,n,nep-
mHBaeTC.H Ha nOCTO.HHHOM ypOBHe. TaKoH pemHM 06eCneqHBaeTCSI B .H,AepHbIX
peaKTopax.
623

ECJlH MaCCa .HepHOrO TOnJlHB8 MeHLme KpHTHqeCKOH MaCCbI, TO KOSc}>c}>H-
nHeHT pa3MHOmeHHSJ K < 1; K8moe HOBoe nOKOJleHHe BbI3bIBaeT Bce MeHLmee
H MeHLmee qHCJlO eJleHHH, H peaKIJ;H.H 6e3 BHemHero HCTOqHHK8 HeHTpoHoB
6bICTpO 3aTYX8eT.
ECJlH me M8cca .HlIepHOro TOnJlHBa 60JlLme KpHTHqeCKOH. TO KOSc}>c}>HIJ;HeHT
pa3MHOmeHH.H K > 1 H Kamlloe HOBoe nOKOJleHHe HeHTpoHoB BbI3LIB8eT Bce
60JlLmee 'IHCJlO eJleHHH. QenH8.H peaKnH.H JlaBHH006pa3Ho H8paCTaeT H HMeeT
xapaKTep B3pLIBa, conpOBomalOerOC.H orpOMHLIM BbIeJleHHeM SHeprHH H no-
BLImeHHeM TeMnepaTypbI oKPymalOeH cpebI o HeCKOJlbKHX MHJlJlHOHOB rpa,lJ;Y-
COB. QenHa.H peaKIJ;H.H TaKoro poa npOHCXOHT IIPH B3pLIBe aTOMHOH 60M6bI.
22.11. .HepHLlii peaKTOp
.H,AepHbIM peaKTopOM Ha3LIBalOT peaKTOp, B KOTOpOM ocyecTBJI.HeTC.H yn-
paBJlSJeM8.H IJ;enHa.H .HepH8.H peaKIJ;H.H ,lJ;eJleHH.H. B HaCTO.Hm;ee BpeMSI cym;ecTByeT
O'leHL MHoro pa3J1HqHbIX THnOB .HepHbIX peaKTopOB pa3HoH MOlltHOCTH. KOTO-
pbIe pa3J1HqaIOTC.H IIO BeJlH'IHHe SHeprHH HcnoJlb3yeMbIx HeHTpoHoB. no THny
HcnoJlL3yeMoro .HlIepHOro TOnJlHBa, no CTpYKType aKTHBHOH 30HbI peaKTopa,
no THny 3aMeJlHTeJl.H, TenJlOHOCHTeJl.H H T.,lJ;. KpHTH'IeCKHe pa3MepbI aKTHB-
HOH 30HbI peaKTopa H, COOTBeTCTBeHHO, KpHTHqeCKa.H Macca eJl.HerOC.H Be-
m;eCTBa onpeeJl.HIOTC.H BHOM HcnoJlb3yeMoro B HeM .H,lJ;epHOrO TOnJlHBa, THnOM
3aMeJlHTeJl.H H KOHCTPYKTHBHLIMH oc06eHHOCT.HMH peaKTopa.
M
PeaKTopLI, pa60TalOHe Ha TenJlOBLIX HeHTpOHax (HX CKOpOCTH - 2 . 10 3 c),
COCTO.HT H3 cJleYIOHx OCHOBHbIX qaCTeH:
8) eJl.Hm;erOC.H BeeCTBa, B KaqeCTBe KOToporo HcnOJlb3YlOT H30TonbI ypa-
Ha (2U. 2U), TOpH.H eTh) HJlH nnYTOHHSJ (2:PU, 2PU, 2lpu);
6) 3aMeJlHTeJl.H HeHTpoHoB, KOTOpbIM cJlymHT rpac}>HT, T.HmeJla.H HJlH 06bIq-
Ha.H Boa;
B) OTpamaTeJl.H HeHTpOHOB, B KaqeCTBe KOToporo 06bIqHO HcnOJlL3YlOT Te
me Bem;eCTBa, qTO H JlSI 3aMe,lJ;JleHH.H HeHTpOHOB;
r) TenJlOHOCHTeJl.H, npeHa3HaqeHHOrO nSJ OTBO8 TenJlOTbI H3 aKTHBHOH
30HbI peaKTopa. B KaqeCTBe TenJlOHOCHTeJlSJ HcnOJlb3YlOT BOY. mHlIKHe MeTan-
JlbI, HeKOTopbIe OpraHHqeCKHe mHKOCTH;
) perYJlHpYIOm;HX CTepmHeH:
e) CHCTeMbI ,lJ;03HMeTpHqeCKOrO KOHTpOJl.H H 6HOnOrHqeCKOH 3am;HTbI OKPy-
malOm;eH cpebI OT nOTOKOB HeHTpOHOB H 'Y-H3J1yqeHH.H, B03HHKalOHX B aK-
THBHOH 30He peaKTopa.
Ha pHC. 22.9 H306pameHa ynpoeHHa.H cxeMa YCTpoHcTBa ypaH-rpac}>HTo-
Boro peaKTopa, HcnoJlL3yeMoro Ha aTOMHLIX SJleKTpOCTaHIJ;H.HX.
B HeM .HepHLIM rOplOqHM .HBJI.HeTC.H 060raeHHbIH ypaH, 3aMeJlHTeJleM HeH-
TpOHOB - rpac}>HT, a TenJlOHOCHTeJleM - 06bIqHa.H Boa. Ha pHCYHKe npHH.H-
TbI cJleYIOHe 0603HaqeHH.H: A - aKTHBHa.H 30Ha peaK TOp a , B - OTp8ma-
TeJlL HeHTpOHOB, C - 6HOJlOrHqeCKa.H 3am;HTa, 1 - TenJlOBLIeJl.HIOm;He sJle-
MeHTbI - TB3JILI (MeTaJlJlHqeCKHe neHaJIbI, coepmaHe ypaH-235), 2 - 6J1o-
KH 3aMe,lJ;JlHTeJlSJ H3 rpac}>HTa, 3 - ynpaBJI.HIOHH CTepmeHb, 4 - aBapHH-
HbIH CTepmeHb. (3TH CTepmHH H3rOTOBJleHbI H3 60pa HJlH Ka,lJ;MH.H, T.e. Be-
624

3 n n 4 sr
BO,lJ;a
p
, ' , ' ntn T
, ' , '
" I '
2 " , '
, . "
, .
1 B
BO,lJ;a
Puc. 22.9
lQeCTB, CHJlbHO nOrJlOlQalOlQHX HeiiTpoHbI. Hx BBe,neHHe YMeHblIIaeT KOJlHqe-
CTBO HeiiTpoHoB B aKTHBHoH 30He.)
TenJlOTa, BbIlIeJl.HeMa.H B .H,nepHoM peaKTope P npH uenHoH peaKUHH lIeJle-
HH.H .H,nep, YHOCHTCS TenJlOHOCHTeJleM ----- BOOH, Haxo,n.HeHC.H no,lJ; lIaBJleHHeM
10 MIIa. BCJle,nCTBHe qero Bo,na HarpeBaeTC.H ,lJ;0 270 °C He 3aKHna.H. JJ;aJlee
BO,lJ;a nocTynaeT B TenJlo06MeHHHK T, r,ne OT,naeT 3HaqHTeJlbHYIO qaCTb CBoeH
BHYTpeHHeH SHeprHH Bo,ne BToporo KOHTypa H C nOMOlQblO HaCOCOB H BHOBb
nona,naeT B aKTHBHYIO 30HY peaKTopa. Bo,na BToporo KOHTypa B TenJlo06MeH-
HHKe npeBpaxqaeTC.H B nap, KOTOpblH nocTynaeT B napOBYIO Typ6HHY llT, npH-
BO,n.HxqylO B ,neHcTBHe SJleKTpOreHepaTOp 3r. BTOpOH KOHTYP, KaE H nepBbIH,
.HBJI.HeTCS 3aMKHYTbIM. IIocJle Typ6HHbI nap nona,naeT B KOH,neHcaTOp K, r,lJ;e
3MeeBHK OXJlam,naeTC.H XOJlO,nHOH npOTOqHOH BO,nOH. 3,necb nap npeBpaaeTCS
B BO,ny H C nOMOblO HaCOCOB H BHOBb nona,naeT B TenJlo06MeHHHK T. Ha-
npaBJleHHe ,lJ;BHmeHH.H BO,lJ;bI B KOHTypax TaKOBO, qTO B TenJlo06MeHHHKe nOTO-
KH BO,nbI B 060HX KOHTypax ,lJ;BHmYTCS HaBCTpeqy ,npyr ,lJ;pyry. Pa3,neJlbHbIe
KOHTypbI He06xo,nHMbI H nOTOMY. qTO B nepBOM KOHType Bo,na, npoxo,n.H qepe3
aKTHBHYIO 30HY peaKTopa, CTaHOBHTCS pa,nHoaKTHBHoH. Bo BTOpOM me KOHTY-
pe nap H BO,lJ;a npaKTHqeCKH Hepa,nHoaKTHBHbI.
IIo CBoeMY Ha3HaqeHHIO .H,nepHbIe peaKTopbI ,neJl.HTC.H Ha cJle,nYIOHe THnbI:
a) HCCJle,nOBaTeJlbCKHe ----- C HX nOMOblO nOJlyqalOT MOHbIe nyqKH HeHT-
pOHOB ,nJl.H HayqHbIx eJleH;
6) SHepreTHqeCKHe ----- npe,nHa3HaqeHbl ,nJlSl nOJlyqeHH.H sJleKTpHqeCKOH SHep-
rHH B npOMbIlIIJleHHbIX MaCIIITa6ax;
B) TenJlQ(l>HKaHoHHble ----- B HHX nOJlyqalOT TenJlOTY ,lJ;JI.H Hym,lJ; npOMbIIII-
JleHHOCTH H TenJlOHKaHH;
r) BocnpOH3BO,lJ;.HHe ----- cJlymaT ,lJ;JlSI nOJlyqeHH.H H3 ypaHa 2U H TOpH.H
2Th 11eJl.HHXCS MaTepHaJlOB nJlYTOHH.H 2:: PU H ypaHa 2:U;
11) TpaHcnopTHbIe ----- HX HcnoJlb3YlOT B ,nBHraTeJlbHbIX YCTaHOBKax Kopa6-
JleH H n011Bo,nHbIX JlO,nOK;
e) peaKTopbI ,lJ;JI.H npOMbIIIIJleHHOrO nOJlyqeHH.H H30TonOB pa3J1HqHbIX XHMH-
qeCKHX SJleMeHTOB, 06J1a11alOHX HCKYCCTBeHHOH pa,nHoaKTHBHoCTblO.
625

22.12. TepMOH,ZJ;epHLIe peaKI(HH.
C03,n;aHHe H y,n;ep3taHHe BLIcoKoTeMnepaTypHoii nJIa3MLI. TOKaMaK.
Y cneXH H nepcneKTHBLI pa3BHTHJI H,ZJ;epHoii aHepreTHKH
TepMOepHa.H peaKHJI - aTO peaKH.H CHHTe3a JlerKHX .H,D;ep B 60Jlee
T.HmeJlble.
)J;JI.H ee ocyeCTBJleHH.H He06xo,AHMO, qT06bI Hcxo,ZJ;HbIe HYKnoHbI HJlH Jler-
KHe .H,D;p8 C6J1H3HJlHCb ,D;O paCCTO.HHH:H, paBHbIx HJlH MeHbillHX pa,D;Hyca c<l>epbI
,D;eHCTBH.H H,D;epHbIX CHn npHT.HmeHH.H (T.e. ,ZJ;O paCCTO.HHHH _10- 15 M). TaKOMY
B38HMHOMY C6J1HmeHHIO .H,D;ep npen.HTCTBYIOT KYJlOHOBCKHe CHJlbI OTTanKHBaHH.H,
JleHCTBYIOWHe Mem,D;Y nOJlOmHTeJlbHO 3ap.HmeHHbIMH .H,D;paMH. )J;JI.H B03HHKHOBe-
HH.H peaKHH CHHTe3a He06xo,D;HMO HarpeTb BeeCTBO 60JlbillOH nnOTHOCTH ,D;O
cBepxBbICOKHX TeMnep8TYP (nOp.H,D;Ka COTeH MHJlJlHOHOB KeJlbBHH), qT06bI KHHe-
THqeCKa.H aHeprH.H TenJlOBOrO ,D;BH»<eHH.H .H,D;ep OKa3aJlaCb ,D;OCTaTOqHOH ,D;JI.H npe-
O,D;OneHH.H KYJlOHOBCKHX CHJ] OTTaJlKHBaHH.H. IIpH TaKHX TeMnepaTypax Bee-
CTBO cecTByeT B BH,D;e nJla3MbI. IIOCKonbKY CHHTe3 MomeT npoHCXO,D;HTb TOJlbKO
npH OqeHb BbICOKHX TeMnepaTypax, .H,D;epHbIe pe8KHH CHHTe3a H nOJlyqHJlH H8-
3B8HHe TepMO.H,D;epHbIX peaKHH (OT rpeq. therme "TenJlO, map").
B TepMO.H,D;epHbIX pe8KH.HX BbI,D;eJl.HeTC.H orpOMHa.H aHeprH.H. H8npHMep, B
peaKHH CHHTe3a ,D;eHTepH.H C 06pa30BaHHeM reJlH.H D + D -+ He + In BbI,D;e-
JI.HeTC.H 3,2 MaB aHeprHH. B peaKHH CHHTe38 ,D;eHTepH.H C 06pa30BaHHeM TPHTH.H
D+ D  T+ p BbI,D;eJl.HeTC.H 4,0 MaB aHeprHH, a B peaKHH D + T 
-+ He + n BbI,D;eJl.HeTC.H 17,6 MaB aHeprHH. Oc06eHHo 60JlbillOe npaKTHqeC-
Koe 3H8qeHHe HMeeT TOT c»aKT, qTO npH TepMO.H,D;epHbIX pe8KH.HX H8 K8mJlbIH
HYKJlOH BbI,D;eJl.HeTC.H 3HaqHTeJlbHO 60Jlbilla.H aHeprH.H, qeM npH enHbIx .H,D;ep-
HbIX peaKH.Hx. HanpHMep, npH CHHTe3e .H,D;ep reJlH.H H3 .H,D;ep BO,D;OpO,D;a Ha
O,D;HH HYKJlOH BbI,D;eJl.HeTC.H aHeprH.H - 6 MaB, B TO BpeM.H KaK npH ,D;eJleHHH
.H,D;pa ypaHa 2U Ha O,D;HH HYKJlOH BbI,D;eJl.HeTC.H 3HeprH.H - 0,9 MaB.
CaMOnO,D;,D;epmHBalOHeC.H TepMO.H,D;epHbIe pe8KHH npOHCXO,D;.HT B He,ZJ;pax
3Be3,D; (B TOM qHCJle COJlHa) H Hrp810T BamHeHillYIO pOJlb B cyecTBoB8HHH
H pa3BHTHH BCeJleHHOH.
Ha 3eMJle nepBa.H TepMO.H,D;epHa.H peaKH.H 6bIJla ocyeCTBJleHa npH B3pbIBe
BO,D;OpO,D;HOH 60M6bI. BbICOKYIO TeMnepaTYPY. He06xOlIHMYlO lIJI.H Haq8J1a TepMO-
.HJlepHOH peaKHH, B BO,D;OpO,D;HOH 60M6e nOJlyqaJlH B pe3YJlbT8Te B3pbIB8 BXO-
,D;.HeH B ee COCT8B aTOMHOH 60M6bI, HrpalOeH pOJlb ,D;eTOHaTopa. TepMO.H,D;ep-
HbIe peaKHH, npOHCXO,D;.HHe npH B3pbIBax BO,D;OpO,D;HbIX 60M6, .HBJI.HIOTC.H
HeynpaBJlJleMbIMH.
ECJlH 6bI B 3eMHbIX YCJlOBH.HX 6bIJla B03MomHOCTb ocyeCTBJI.HTb JlerKO yn-
p8Bn.HeMbIe TepMO.H,ZJ;epHbIe peaKHH, qeJlOBeqeCTBO nOJIyqHJlo 6bI npaKTHqeCKH
HeHCqepnaeMbIH HCTOqHHK aHeprHH, TaK KaK 3anaCbI BO,D;OpO,D;a H8 3eMJle or-
pOMHbI. O,D;H8KO Ha nYTH ocyeCTBJleHH.H aHepreTHqeCKH BbIrO,D;HbIX ynpaBJI.He-
MbIX TepMOH,D;epHbIX pe8KHH CTO.HT 60JlbillHe TeXHHqeCKHe TpYlIHOCTH. IIpemlIe
Bcero He06xoJlHMO C03aBaTb TeMnepaTypbI nOp.H,D;Ka 10 8 K. TaKHe cBepxBbIcO-
KHe TeMnepaTypbI MorYT 6bITb nOJlyqeHbI nYTeM C03,D;aHH.H B nJla3Me 3J1eKTpH-
qeCKHX pa3p.H,ZJ;oB 60JlbillOH MOHOCTH.
626

STOT MeTO,A HCnOJlb3yroT B YCT8HOBKax THna "ToKaMaK", BnepBbIe C03,AaHHbIX
B HHCTHTYTe 8TOMHOH SHeprHH HM. H. B. KypqaTOBa. B T8KHX YCTaHOBKax nJla3-
MY C03,Aa1OT B TOpOH,AanbHOH K8Mepe, .HBJI.HIOeHC.H BTOPHqHOH 06MOTKOH MO-
Horo HMnynbcHoro TpaHccpopMaTopa. Ero nepBHqH8.H 06MOTKa nO,AKJllOqeHa I< 6a-
Tapee KOH,AeHCaTOpOB OqeHb 60JlblIIOH eMKOCTH. KaMepy 38nOJlH.HIOT ,AeHTepHeM.
llpH pa3p.H,Ae 6aTapeH KOH,AeHCaTOpOB qepe3 nepBHqHYIO 06MOTKY B TOpOH,A8J1b-
HOH KaMepe B036ym,AaeTC.H BHxpeBoe SJleKTpHqeCKOe nOJle, BbI3bIB81Oee HOHH38-
II.HIO ,AeHTepH.H H nO.HBJleHHe B HeM MOHoro HMnYJlbC8 sneKTpHQeCKOrO TOK8,
qTO npHBO,AHT K CHJlbHOMY HarpeBaHHIO ra3a H 06p830B8HHIO BbICOKOTeMnep8-
TYPHOH nJla3MbI, B KOTOPOH MomeT B03HHKHYTb TepMO.H,AepH8.H peaKII.HS.
rJlaBHa.H TPY,AHOCTb 3aKJllOqaeTC.H B TOM, qT06bI Y,Aepm8Tb nJl83MY BHYTPH
KaMepbI B TeqeHHe 0,1-1 C 6e3 ee KOHT8KT8 co CTeHKaMH K8MepbI, nOCKOJlb-
I<Y He cyrnecTByeT MaTepHaJlOB. cnoc06HbIX BbIl1epmHB8Tb CTOJlb BbICOKHe TeM-
nepaTypbI. STY TPY,AHOCTb y,AaeTC.H qaCTHqHO npeO,AOneTb C nOMOblO TOPOH-
,AaJlbHOrO M8rHHTHoro nOJl.H, B KOTOpOM H8XO,ZJ;HTC.H KaMep8. llo,A ,AeHCTBHeM
MarHHTHbIX CHJI nJla3M8 cKpyqHBaeTC.H B lIIHYP H KaK 6bI "BHCHT" Ha JlHHH.HX
HH,AYKII.HH MarHHTHoro nOJl.H, He KaCa.HCb CTeHOK KaMepbI.
O,ZJ;HaKO nJla3Ma B M8rHHTHOM nOJle OqeHb HeYCTOHqHBa H nJl83MeHHbIH
lIIHYP paCna,AaeTC.H npem,Ae, qeM Y,AaeTC.H HarpeTb nJla3MY ,AO HymHoH TeMne-
paTypbI. llOKa Y,AaJlOCb nonyqaTb nJla3MY C TeMnepaTYPoH 1,3. 10 7 K H Y,Aep-
mHBaTb ee B TeqeHHe 60-80 MC H8 YCTaHOBKe "TOKaMaK-10". )J;JI.H YBeJlHqe-
HH.H npO,AOJlmHTeJlbHOCTH cyecTBOBaHH.H ynp8BJI.HeMOH TepMO.H,AepHOH peaKII.HH
He06xo,ZJ;HMO YBeJlHqHBaTb p83MepbI YCTaHOBKH, nOSTOMY B HaCTO.Hee BpeM.H
CTpOHTC.H HOB8.H 60Jlbma.H YCT8HOBKa "ToKaMaK-20".
Hcnonb30BaHHe YCTaHOBOK THna "TOKaMaK" (B KOTOpbIX ,AJI.H nOJlyqeHH.H H
HarpeBaHHSI nJl83MbI HCnOJlb3YeTC.H MOrnHbIH SJleKTpHqeCKHH p83pSl.n;, 8 ,AJI.H Y,Aep-
maHH.H nJla3MbI - MarHHTHoe nOJle) .HBJI.HeTC.H O.l1HHM H3 B03MomHbIX nYTeH
ocymecTBJleHH.H ynpaBJI.HeMbIX TepMO.H,ZJ;epHbIX peaKII.HH. )J;pyrHM nYTeM ,ZJ;OCTHme-
HH.H STOH u.eJlH .HBJI.HeTC.H J1aaepHLIH TepMOJIJJ:epHbIH CHHTe3. CYHOCTb T8Koro
MeTO,Aa COCTOHT B CJle,AYIOeM. 3aMopomeHHYIO CMeCb ,AeHTepH.H H TPHTH.H, npH-
rOTOBJleHHYIO B BH,Ae mapHKOB ,ZJ;HaMeTpOM MeHee 1 MM, p8BHOMepHO 06J1yq81OT
co Bcex CTOpOH MOHbIM na3epHbIM H3J1yqeHHeM. 3TO npHBO,AHT K HarpeBaHHIO
H HcnapeHHIO BeeCTBa C nOBepXHOCTH lIIapHKOB. llpH STOM ,AaBJleHHe BHYTPH
1II8pHKOB B03p8CTaeT ,AO BeJlHqHH nOp.H,AK8 10 15 ll8. fio,A ,AeHCTBHeM TaKoro
,A8BJleHH.H npOHCXO,A.HT YBeJlHqeHHe ITJlOTHOCTH H CHJlbHOe HarpeBaHHe BeeCTBa
B II.eHTp8J1bHOH qaCTH lllapHKOB H HaqHHaeTC.H TepMO.H,AepHa.H peaKII.H.H.
B HaCTO.Hee BpeM.H BO MHorHX CTp8Hax MHpa Be,AYTC.H HHTeHCHBHbIe pa-
60TbI no ocyeCTBJleHHIO ynpaBJI.HeMOH TepMO.H,AepHOH peaKI(HH. HMeIOTC.H 060-
CHOBaHHbIe npe,ZJ;nOJlOmeHH.H, qTO STa np06J1eM8 6Y,AeT pellieHa B TeqeHHe 6J1H-
maHlIIHx 20 JleT.
22.13. nOrJIOeHHaH A03a H3J1}"1eHHH H ee 6HOJIOrH1.IeCKOe 1J;eHCTBHe.
OCHOBLI A03HMeTpHH
Pa,nH08KTHBHbIe H3J1yqeHH.H BbI3bIBalOT HOHH38II.HIO aTOMOB H MOJleKYJI Bee-
CTB8, nOSTOMY HX q8CTO H83bIBalOT HOHH3HPYJOH H3JIY'leHHJlMH Bce BH,AbI
p8.AHOaKTHBHbIX H3J1yqeHHH OKa3bIBalOT CHJlbHOe ,ZJ;eHcTBHe H8 mHBbIe opraHH3MbI.
Xap8KTep SToro ,AeHCTBH.H 3aBHCHT OT BH,Aa H3J1yqeHH.H Hero HHTeHCHBHOCTH.
627

MexaHH3M 6HOJlOrHqeCKOrO ,ZJ;eHCTBH.H pa.n;HOaKTHBHbIX H3J1yqeHHH CJlOmeH.
Ero OCHOBY COCTaBJI.HIOT npOeCCbI HOHH3aHH H B036ym.n;eHH.H aTOMOB H MO-
neKyn B mHBbIX TKaH.HX. npOHCXO,n.HII1He npH nOrJlOII1eHHH HMH HOHH3HPYIO-
HX H3J1yqeHHH. 3TH npOecCbI .HBJI.HIOTC.H JlHWb HSqaJlbHhIM 3TanOM B cJlom-
HOH enH 6HOXHMHqeCKHX npeBpaeHHH, BbI3BaHHbIX .n;eHCTBHeM pa.n;HSIIHH.
MOHH3aH.H TKaHeH npHBO.n;HT K pa3pbIBY MOJleKYJI.HpHbIX CB.H3eH H H3Me-
HeHHIO CTPYKTypbI XHMHqeCKHX COe.n;HHeHHH, BXO.n;.HHX B COCTaB TKaHeH.
qSCTh aTOMOB H MOJleKYJl BeeCTBa TKaHeH npH nOrJlOeHHH pa.n;HOsKTHBHbIX
H3J1yqeHHH He HOHH3HpyeTC.H, a TOJlbKO nepeXO,AHT B B036ym.n;eHHoe COCTOSJ-
HHe. qepe3 onpe.n;eJleHHOe BpeM.H 3TH aTOMbI B03BpaaIOTC.H B HeB036ym.n;eHHoe
COCTO.HHHe, OT.n;aBa.H H36bITOK 3HeprHH B BH.n;e KBaHTOB YJlbTpac:l>HoneToBoro H3-
nyqeHH.H. fIOA .n;eHCTBHeM 3Toro H3J1yqeHH.H, norJlOaeMOro TKaH.HMH opraHH3-
Ma, npOHcxo.n;HT 6HOXHMHqeCKS.H peaKIJ;H.H, npHBo.n;.Ha.H K pa3pyweHHIO MOJle-
Kyn HYKJleHHOBbIX KHCJlOT H 6eJlKa.
IIepBHqHbIM pe3YJlbTaTOM .n;eHCTBH.H pa.n;HaHH Ha mHBOH opraHH3M .HBJI.HeT-
C.H nopameHHe KJleTOK. KaK H3BeCTHO, KJleTK8 - 3TO CJlOmHa.H CTPYKTypa,
Hecnoc06HS.H K npo,nOJlmeHHIO HOpMSJlhHOH ,ne.HTeJlhHOCTH npH nOBpem,AeHHH ee
qaCTeH. IIo.n; ,neHCTBHeM 06nyqeHH.H npem,ne Bcero nopamalOTC.H .H,Apa KneTOK.
KOTophIe rOpa3,A0 qYBCTBHTenhHee ee HTonna3MhI. HspywalOTcSI cnoco6HOCTh
KJleTOK K HopMaJlhHOMY ,AeneHHIO H 06MeH BeeCTB B KneTKax.
PS3J1HqHbIe opraHbI qeJlOBeqeCKOrO TeJla pearHpYIOT Ha 06J1yqeHHe no- pa3-
HOMY, HaH60Jlee qYBCTBHTeJlhHbI K pa.n;HaHH KpOBeTBopHbIe OpraHbJ (KOCTHbIH
M03r, CeJle3eHKa, nHMc:l>aTHqeCKHe meJle3bI) H 3nHTenHH CJlH3HCThIX 060JlOqeK
(B qaCTHOCTH, KHWeqHHKa). B pe3YJlbTaTe .n;eHCTBH.H pa.n;HoaKTHBHbIX H3J1yqe-
HHH HS opraHH3M B03HHKalOT T.HmeJleHWHe 3a60JleBaHH.H: JlyqeBa.H 60Jle3Hb,
3J10KaqeCTBeHHbIe onYXOJlH (qaCTO co CMepTeJlbHbIM HCXO.n;OM).
06J1yqeHHe OKa3bIBaeT CHJlbHOe BJlH.HHHe Ii Ha Hacne.n;CTBeHHOCTb, qTO MO-
meT npHBeCTH K nO.HBJleHHIO nOTOMCTBa C ypo.n;nHBbIMH OTKJlOHeHH.HMH HJlH
Bpom,neHHbIMH 3a60JleBaHH.HMH.
CneHc:l>HqeCKa.H oc06eHHOCTb pa.n;HoaKTHBHbIX li3nyqeHHH 3aKJllOqaeTC.H B
TOM, qTO OHH He BocnpHHHMaIOTC.H opraHaMH qYBCTB qeJlOBeK8 H .n;ame npH
CMepTeJlbHbIX ,A03aX He BbI3hIBalOT Y Hero B MOMeHT 06J1yqeHH.H 60JleBbIX oy-
eHHH. STO ,neJlSeT pa,nHsqHIO oc06eHHO onSCHOH.
Ha opraHH3M BJlH.HeT TOJlhKO Ta qaCTh pa,nHOSKTHBHoro H3J1yqeHH.H. KOTO-
pa.H norJlOaeTC.H ero TKaH.HMH. II03TOMY 6HOJlOrHqeCKOe lIeHCTBHe H3J1yqeHHH
npHH.HTO xapaKTepH30BaTb nornoeHHOH A030H HanyqeHH.H.
I1ornoeHHOH A030H H3JIY'1eHHH D HS3bIBalOT BeJlHqHHY, paBHYIO OTHowe-
HHIO 3HeprHH HOHH3HpYIOero H3J1yqeHH.H W, nor JlOeHHOH 06J1yqaeMhIM Be-
eCTBOM, K Macce m 3Toro BeeCTBa.
W
D=-.
m
B CM e.n;HHHeH nOrJlOeHHOH .n;03bI H3nyqeHH.H .HBJI.HeTC.H rp3H (rp). 1 rp
paBeH nOrJlOeHHOH .n;03e H3J1yqeHH.H, npH KOTOpOH 06J1yqaeMoMY BeeCTBY Mac-
COH 1 Kr nepe,ZJ;aeTC.H 3HeprH.H HOHH3HpYIOero H3J1yqeHH.H 1 .nm:
.nm
rp= -.
Kr
BHecHCTeMHa.H e.n;HHHa nornoeHHoH .n;03hI H3J1yqeHH.H D - 1 pa.n; = 10- 2 rp.
628

MOOCTL nOrJlOeHHOii ,IJ;03LI H3JIY'1eHHH N - 3TO BeJIHqHHa, paBHag
nOrJlOm;eHHOH 03e H3J1yqeHH.H, OTHeCeHHOH K eHHHe BpeMeHH.
D
N =-.
t
B Cll eHHHeH MOHOCTH nOrJlOm;eHHOH 03bI H3nyqeHHg SlBJI.HeTCg rp3H
( r p rp BT ] B - -
Ha ceKYHY -; - = -. HeCHCTeMHOH eHHHeH MOm;HOCTH nor JlOm;eH-
C C Kr
HOB A03M H3nyqeHHH HBnHeTCH PRAHaH Ha ceKYHAY ( p ).
)J;JI.H H3MepeHH.H nOrJlOm;eHHOH ,ZJ;03bI Hcnonb3yIOTC.H CneHaJIbHbIe npH60pbI -
,IJ;03l1MeTpbI. H8.1:I60Jlblllee pacnpOCTpaHeHHe HMelOT 03HMeTpbI, B KOTOpbIX aT'tlH-
KaMH .HBJI.HIOTC.H HOHH3aHoHHbIe KaMepbI. B HeKoTopbIX 03HMeTpax B KaqeCTBe
a TtJHKOB HCnOJlb3yIOT CqeT'tlHKH qaCTH, «}>oTOnJleHKY HJlH CHHTHJlJI.HTOpbI.
)J;eHcTBHe H3J1yqeHH.H H8 BeecTBo MomHO oeHHBaTb no 3«}>«}>eKTY HOHH3a-
HH cyxoro aTMoc«}>epHoro B03yxa. 3Kcno3HHoHHaB ,IJ;038 H3JIyqeHHH - 3TO
BenHqHHa, paBHaB OTHOllleHHIO cYMMapHoro 3ap.Ha HOHOB oHoro 3HaKa, 06-
pa30BaBIIIHXCg no eHcTBHeM H3JIY'tleHHg B CYXOM aTMoc«}>epHoM B03yxe, K
Macce HOHH3HpOBaHHoro B03yxa:
Do = !L .
m
B Cll eHHHeH 3Kcn03HHoHHOH 03bI .HBJI.HeTCg KYJlOH Ha KHJlOrpaMM
( : ). BHecHcTeMHIiJI eAHHHa - peHTreH (P).
1 P - 3Kcn03HHOHHa.H 03a H3J1yqeHH.H, npH KOTOpOH B 1 CM 3 cyxoro
B03yxa npH HOpMaJIbHOM aTMoc«}>epHoM aBneHHH 06pa3yeTC.H OKOJIO 2.10 9 nap
HOHOB: 1 P = 2.58 . 10-4 KJI
Kr
Do
MOOCTL 3Kcn03HolUloii ,IJ;03bl H3JIY'IeHH.H B Cll No = -. B Cll eHHH-
t
eH MOII1;HOCTH 3Kcn03HHoHHOH 03bI H3J1yqeHH.H gBJI.HeTCg aMnep Ha KHJlOrpaMM
( A ) PAP A
- . BHecHcTeMHbIe eHHHbI: 1 - = 2,58. 10- 4 - H 1 - = 7,1 77. 10- 8 -.
Kr C Kr 'I Kr
B03eHcTBHe Ha opraHH3M H3J1yqeHHH pa3J1HqHOH npHpobI npH OHO" H
TOH me 03e Do HeOHHaKOBO.
3KBHBaJIeHTHa.H ,IJ;03a H3JIY'1eHH.H (6HOJlOrH'tleCKa.H 03a H3JIyqeHH.H) D3 -
3TO BeJIHqHHa, onpeeJl.HIOag B03eHcTBHe H3JIyqeHH.H Ha opraHH3M:
D3 = kD,
re k - K03«}>«}>HHeHT KaqeCTBa, 3aBHC.Hm;HH OT BHa H3JIyqeHH.H. ,[(JI.H peHT-
reHOBCKHX Jlyqeii, Y-JlyqeH H 3J1eKTpOHOB k = 1, JIg MeJIeHHbIX HeHTpoHoB
k = 3; JI.H 6bICTpbIX HeHTpoHoB H 6bICTpbIX npOTOHOB k = 10, ,ZJ;JI.H npOYKTOB
pacnaa H a-qaCTH k = 20.
E,AHHHe" 3KBHBaJleHTHOH 03bI H3J1yqeHHH D3 B Cll gBJI.HeTC.H 3HBepT (3B).
1 3B paBeH 3KBHBaJIeHTHOH 03e, npH KOTOpOH nOrJlOII1;eHHa.H 03a paBHa 1 rp
H K03«}>«}>HHeHT Ka'tleCTBa paBeH e,ZJ;HHHe.
629

BHeCIiCTeMHa.SI e,nHHII8 - 63P (61I0JlOrlIqeCKIiH 3KBIIBaJleHT peHTreHa),
1 63p = 10- 2 3B.
rrpeenbHO OnYCT1:IMOH ,n030H 06J1Y'IeH1:I.H Cq1:lT8eTC.SI TaK8.SI nOrJlOeHH8.SI
,no3a, I<OTOp8.SI no nOp.SI,nKY BeJl1:lqHHbI COBn8,AaeT C eCTeCTBeHHbIM p8,nH08KTHB-
HbIM CPOHOM. cyecTBYIOIIM H8 3eMJle 1:1 06YCJlOBJleHHbIM B OCHOBHOM KOCMH-
'IeCKHM 1I3nyqeH1:IeM II PalI1:I0aKT1:IBHOCTblO 3eMnH.
C 3TOH TOqKH 3peHH.H, npe,neJlbHO ,nonYCTHM8.H ,n03a ,nn.H qeJlOBeKa B ,n1:l8-
na30He peHTreHOBCKoro, p- II 'Y-1:I3nyqeH1:IH COCTaBn.SIeT OKono 10- 3 rp B ro,lJ;.
,[(JI.H TenJlOBbIX HeHTpoHoB 3Ta 03a B 5 pa3 Hlime. a ,nn.H 6bICTpbIX HeHTpo-
HOB, npOTOHOB H (X-qaCTHn - B 10 p83 H1:I>Ke.
Mem,AYHapo,nHoR KOM1:ICC1:Ieii no p8,n1:la1:I0HHOH 3aHTe ,nJl.SI nlO,neH, nOCTO-
.SIHHO pa60T81OHX C HCTOqHHKaMIi pa,lJ;1I08KTHBHbIX H3J1Y'IeHHH, YCTaHOBneH8
npe.nenbHO 1I0nYCTHM8.SI ,n038 He 60Jlee 10- 3 rp B He.neJlIO. T.e. ORono 0.05 rp
B ro,ll. ,l].038 CBbIllie 3-6 rp. nOJlyqeHHa.H 3a KopOTKoe BpeM.H, JlSI 'IeJlOBeKa
CMepTeJlbH8.
22.14. 3aHTa OT H3JIyqeHHH
IIpH pa60Te C JlI06bIM1:I HCTOqH1:IK8M1:I p8,nHoaKT1:IBHbIX H3J1Y'leH1:IH (.H,nepHbIe
pe8KTopbI, 1:I30TOnbI H T.n.) ,nOJl>KHbI 6bITb np1:lH.HTbI MepbI pa,n1:lan;HOHHOH 38-
HTbI nlO,neH.
H8H60Jlee npOCTOH cnoco6 3aIQ1:ITbI - y,AaJIeH1:Ie pa60T81O1:IX OT HCTOqH1:I-
K8 1I3J1yqeH1:I.H Ha ,nOCTaTO'lHO 60nbllIoe paCCTO.HH1:Ie (TaK K8K HHTeHCHBHOCTb
nlO60ro 1:I0HH3lIpYIOero H3J1yqeH1:I.H y6bIB8eT C YBeJl1:l'leHHeM p8CCTO.HHH.SI OT ero
HCTO'IH1:IKa). IIp1:l pa60Te1:lcnonb3YlOT p83n1:lqHbIe YCTpOHCTBa C ,n1:lCT8HIIOHHbIM
ynp8BJleHHeM, M8HlInyn.HTOpbI T1:In8 "MeXaHlIqeCKHe PYKII" II T.n.
ECJl1:I >Ke y,naJIeHHe JlIO,neH OT 1:ICTOqH1:IKa p8.AHanH1:I Ha Tpe6yeMoe p8CCTO.HH1:Ie
HeB03MomHO, TO .nJl.SI 3aIQHTbI OT 1:I3nyqeHHH Hcnonb3YlOT p83J1H'lHbIe nperp8lIbI
1I3 M8TeplIaJIOB, nOrJlO81O1:IX H3J1Y'leH1:Ie. TOJIIIJ;HHa 3THX nperp8,n 38BIICIIT OT
npOH1:IKaIOIIJ;eH cnoc06HOCT1:I ,n8HHoro BH,na H3J1yqeH1:I.H Hero HHTeHCIIBHOCTH.
HalI60Jlee npOCTOH .HBJI.HeTC.H 3aIIJ;HTa OT BHelliHero 06J1y't{eHH.SI (x-'I8CTHn8-
M1:I, T8K K8K OH1:I HMelOT M8J1YIO ,nJlHHY np06era (Jl1:ICT 6YMarH cnoc06eH nOJI-
HOCTblO nOrJlOTHTb 1:IX). O,nH8KO npOH1:IKHOBeH1:Ie (X-qacr1:l BHYTPb opr8HH3Ma
(H8np1:lMep, C n1:leH) .SIBJI.HeTC.H OqeHb onaCHbIM.
IIp06er p-qaCTHn 3aBHCHT OT 1:IX 3Hepr1:lH. Hanp1:lMep, npli 3Hepr1:l1:l nop.SI,n-
Ka 3 M3B p-qaCTHnbI npOJIeT81OT B B03,lJ;yxe ,no 3 M. O,nHaKo OH1:I He npoxo-
,n.HT qepe3 CJlOH nJlaCTMaCCbI TOJlHHOH 60Jlee 3 MM.
Ha1:l60Jlee CJlOmH8 3aIIJ;1:ITa OT 'Y-1:I3J1yqeH1:I.SI H HeHTpoHoB, 06J1a,lJ;alO1:Ix
60JlbllIOH npOH1:IKalOeH cnoc06HoCTblO. JlyqllIliM nOrnOT1:ITeJleM 'Y-1:I3J1Y'IeHH.SI
.HBn.SIeTC.H CB1:IHen;. II03ToMY .nJl.H 381:ITbI OT 'Y-1:I3J1yqeHH.H P8,A1:I08KT1:IBHbIe npe-
n8paTbl nOMealOT B CB1:IHn;OBbIe KOHTeHHepbI. CaMbIMIi Jlyq1lI1:IM1:I nornOTHTe-
JI.SIMH Me,nJleHHbIX (TennoBbIx) HeHTpoHoB .HBn.HIOTC.H 60p H Ka,AMIiH. M3roToB-
neHHbIe H3 3T1:IX 3neMeHTOB 3aIITHbIe nperpa,nbI 06bI'IHO oKpym81OT TaKme
CJlOeM CBHHn8 ,nJl.H 3aHTbI OT BTOp1:l'IHOrO 1I3nyqeHH.H. BbIcTpbIe HeHTpoHbI
npe,nBapHTenbHO 3aMe,nJl.HIOT C nOMOIIJ;blO rpacplIT8.
,[(JI.H 3aIIJ;1:ITbI OT MOHbIX 1:1 KpynHbIx 1:ICTOqH1:IKOB pa,n1:lanHH (H8npHMep,
.H,AepHbIX peaKTopoB) KpOMe nepeq1:lCneHHbIX cnoc060B 38IQ1:ITbI 1:IcnOJlb3YlOT 6e-
TOHHbIe CTeHbI MHoroMeTpoBoH TonHHbI.
630

He06xo,ZJ;HMo OTMeTHTb, qTO qeJlOBeK He TOJlbKO 3aHaeTC.H OT pa,AHOaK-
THBHbIX H3J1yqeHHH, HO H HCnOJlb3yeT HX. He KaCa.HCb Bonpoca 0 pa3Ho06pa3-
HbIX npHMeHeHH.HX, OTMeTHM TOJlbKO, qTO pa,AHOaKTHBHbIe H3J1yqeHH.H (rJlaB-
HhIM 06p830M Y-H3J1yqeHHe) HcnOJlb3YIOTC.H ,AJI.H nO,A8BJleHH.H 3J10KaqeCTBeHHbIX
onyxoJleH.
BonpOCM H CAMOKOHTPORH
1. KaKue l£aCmUijbt 6XOOSlm 6 cocma6 sliJpa amoMa?
2. 'Cfmo maKoe MaCC060e IlUCJl.o? KaK MOJICIiO onpeOeJl.Umb cocma8 SliJpa
amOMa Jl.1060Z0 aJl.eMelima. UCnOJl.b3YSl ma6Jl.uijY M eIiOeJl.ee6a?
3. KaK06bt OCli06libte C60ucm6a SloeplibtX CUJl.?
4. f{ mo lia3bt8alOm oetjJeKmoM Macc?
5. f{ mo maKoe aliepzuSl C6Sl3U Slopa? II 0 KaKOU tjJopMYJl.e paCCilUmbt6aemcSl
aliepzUSl C6Sl3U sliJpa? f{ mo lia3bt6alOm yoeJl.bliOU aliepzueu C6Sl3U?
6. f{mo maKoe paouOaKmU61i0Cmb? KaK 6btJl. o6liapYJICeli CJl.OJIClibtU cocma6
paoUOaKmU6liOZO U3Jl.YlleliuSl? f{mo npeOCma6Jl.SllOm c060u a-, p-, Y-Jl.Yllu?
7. KaK U.JMeliSllOmcSl Macca U 3apSlo Slopa amOMa paouOaKmU61i0ZO aJl.eMen-
ma npu a-pacnaoe? p-pacnaoe?
B. f{ mo lia3bt6alOm nepuoooM nOJl.ypacnaoa paouOaKmU61i0ZO aJl.e.Melima?
3anuwume tjJOPMYJl.Y. KomopaSl Sl6Jl..RemCSl MameMamUlleCKUM 8btpaJICeliUeM 3a-
KOlia paouOaKmU6liOZO pacnaoa.
9. f{ mo maKoe sliJepliaSl peaKijUSl? KaKue 3aKOlibL coxpaneliUSl UCnOJl.b3YlOm-
CSl npu COCma6Jl.eIiUU ypa8lieliuu SloeplibLx peaKijUu?
10. f{mo lia3bt6alOm aliepZemUlI.eCKUM 6btXOOO.M .Roepliou peaKuu? Kozoa
peaKUSl npomeKaem C 8btOeJl.eIiUeMaliepzuu.KozoaCnozJl.o14eIiUe.Maliepzuu?
B lleM omJl.Ullue SloeplibtX peaKijUU lia lieumpoliax om SloepHbtX peaKl.{UU, 6bt3
8alilibtX 3apSlJICelilibtMU "lac mUl.{a.Mu?
11. KaKU.M za30M 3anOJl.IiSlemCSl CmeKJl.SlliliaSl mpy6Ka 6 CllemllUKe reuzepa?
KaKoe oeucm6ue OKa3bt6aem 3apSlJlCelilia.R llacmUl.{a. npOJl.ema.R 6 cllemlluKe?
12. llolle.MY mOK 6 CllemllUKe reuzepa. 6bt36anlibtU UOIiU3al.{UeU za3a, Sl6-
Jl.SlemCSl KpamKo8peMelilibtM?
13. H a lle.M oc 1i06aH npUliijun oeucm8uSl KaMepbt BuJtbcolia? f{ mo npeo-
Cma6Jl.Slem co6ou cJl.eo llaCmUbt. lia6Jl.100ae.MbtU 6 KaMepe?
14. H a lleM OCIiOBali npuliun oeucm6uSl nY3btpbKOBOU KaMepbt? B lleM
npeU.MY14eCm60 nY3btpbK060U KaMepbt nepeo KaMepou BUJl.bCona?
15. B lleM cY14IiOCmb Memooa mOJl.cmOCJl.OUHbtX tjJomoaMYJl.bCuU?
16. KaK C nOM014blO KaneJl.bliOU MOOeJl.U Slopa 06'OSlCliSlemC.R. oeJl.eliue amo.M-
1i0ZO Slopa ypalia, nOZJl.OmU6Wezo lieumpoli? H a KaKue OCKOJl.KU pacnaoaemcSl
SlOpO ypalia? KaKue llaCmUijbt npu amOM ucnycKalOmcSl?
17. KaK 603liUKaem ijenliaJl peaKijUSl OeJl.eliUSl Sloep ypalia? q eM 06'OSlCIiSl-
emcSl 6btOeJl.eliUe aliepzuu npu amo.M?
lB. II 00 8Jl.USlIiUeM KaKUX lieumpolio8 OeJl..RmCSl Slopa 2U, U?
19. II 01leMY lie803MOJICiia ijenliaJl peaKijUSl (' UCnOJl.b:W6aliueM "lucmozo U30-
mona 2U?
20. q mo maKoe KoatjJtfJUijUelim pa3MIiOJICeliuSl? KaKue tjJaKmOpbt 6Jl.USllOm
lia 6eJl.UIlUIiY KoatjJtjJuijUelima pa3MIiOJICeIiUSl?
631

21. lJmo lie06xoiJu.M0 iJJl.JI. cmaquoliapliozo melI.enuJI. qenlioil SliJeplioil peaK-
quu?
22. H a306ume OCli06libte 3Jl.eMelimbt SliJepliozo peaKmopa. KaK080 ux lia-
3lialleliue?
23. lJmo maKoe KpumUlleCKast Macca staepliozo mOnJlU8a? KaK MO:HCno
YMelibwumb KpumullecKYIO .Maccy?
24. KaK ocyw,ecm6Jl.SlemCSl ynpa6Jl.eliUe xoiJO.M peaKijUU aeJl.eliUSl SliJep 8 pe-
aKmope?
25. II Olle.MY peaKijUU CJl.USlIiUSl Jl.eZKUX SliJep Mozym npoucxoiJumb mOJl.bKO
npu 6btCOKUX me.Mnepamypax? KaK lia3bt6alOm maKue peaKijUu?
26. B KaKUX peaKijUSlX (aeJl.eIiUSl UJl.U CUlime3a SliJep) 6btaeJl.SlemCSl 60Jl.bWe
3liepZUU lia oauli IiYKJl.OIi?
27. B lle.M npeU.Myw,ecm6a amOMlibtX 3Jl.eKmpocmaliquu no Cpa8lieliUIO C
menJl.06bt.MU 3Jl.eKmpOCmaliijUJl..Mu?
28. riJe UCnOJl.b3YlOmcSl paaUOaKmU8libLe u30monbL?
632

23. 3JIEMEHT APHbIE qACTHQbl
3J1eMeHTapHLle qaCTHLI. AHTHqaCTHLI. npeBpaIl(eHHe napLI "3J1eKTpOH-
n03HTpoH" B raMMa-H3J1yqeHHe H o6paTHo. B3aHMHLIe npeBpaIl(eHHJI 3J1eMeH-
TapHLIX qaCTH.
23.1. 3JIeMeHTapHbIe qaCTHbI
3J1eMeHTapHLIMH Ha3bIBaIOTC.H qaCTHII,bI, KOTOpbIM (Ha ,AaHHOM 3Tane pa3-
BHTHS:I cpH3HKH) HeJlb3.H npHnHcaTb HHKaKOH BHYTpeHHeH CTpYKTypbI.
OCHOBHbIe qaCTHII,bI, Bxo,n.HHe B COCTaB aTOMa, - 3J1eKTpOHbI, npOTOHbI
H HeHTpoHbI - BHaqaJle CqHTaJlHCb Hecnoc06HbIMH K npeBpaeHH.HM H Ka-
KHM-JlH60 H3MeHeHH.HM. n03TOMY HX H Ha3BaJlH 3J1eMeHTapHbIMH. O,ZJ;HaKo B
,naJlbHeHlIleM 6bIJlO nOKa3aHO, qTO TepMHH "3neMeHTapHa.H qaCTHII,8" BeCbMa YC-
JlOBeH. TaK, HanpHMep, y cB060,nHoro HeHTpoHR BpeMS:I mH3HH OKOJlO 15 MH-
HYT, a 3aTeM OH pacna,naeTC.H Ha npOTOH, 3J1eKTpOH H aHTliHeHTpHHO:
1 1 0 0-
on  IP+ -le+ OV e .
113 BCeX OTKpbITbIX B HaCTO.Hee BpeM.H 3J1eMeHTapHbIX qaCTHII, JlHlIlb cpO-
TOH, 3J1eKTpOH, npOTOH Ii HeHTpHHO COXpaH.HJlH 6bI CBOIO HeH3MeHHOCTb, eCJlH
6bI Kam,na.H 3 HHX 6bIJla O,nHa B oKpymalOeM npOCTpaHCTBe.
9J1eMeHTapHbIe qaCTHII,bI nO,nQHH.HIOTC.H 3aKOHaM KBaHTOBOH cpH3HKH.
B OCHOBY COBpeMeHHOH KJlaCCHcpHKaII,HH 3J1eMeHTapHbIX qaCTHl1. nOJlOmeHbI
HX OCHOBHble CBOHCTBa: MaCCa, 3J1eKTpHqeCKHH 3ap.H,n, CnHH H BpeM.H mH3HH,
a TaKme JlenTOHHbIH H 6apHOHHbIH 3apSl,nbI.
B Ta6J1HII,e 23.1 npHBe,AeHbl HeKOTopble CBe,neHHS:I 0 CBoHcTBax 3J1eMeHTap-
HbIX qaCTHII, co BpeMeHeM mH3HH 60Jlee 10- 20 C. qacTHUbI B Ta6nHue pacno-
JlOmeHbI no B03pacTaHHIO HX MaCCbI.
B Ta6J1HII,y 3J1eMeHTapHblx qaCTHII, He BKJllOqeHbI Bce KopOTKomHBYHe qa-
CTHII,bI-pe30HaHCbI, B qaCTHOCTH, "OqapOBaHHbIe" qaCTHII,bI. He BKJllOqeHbI TaK-
me nepeHOCqHKH CJla6blX B3aHMo,neHcTBHH - BeKTopHbIe 6030HbI. B pe3YJlbTa-
Te nOJlyqaeTC.H 39 qaCTHII,.
Ta6J1HII,a OTKpbIBaeTC.H CPOTOHOM. q>OTOH, OCTaBa.HCb B o,nHHOqeCTBe, 06pa-
3yeT nepBYIO rpynny. q>OTOHbI npe,nCTaBJlS:lIOT C060H KBaHTbI 3J1eKTpOMarHHTHO-
ro nOJl.H (cBeTa, 'Y-H3J1yqeHHS:I H T.,n.), He HMelOT COOTBeTCTBYIOHX aHTHqaC-
THII" T.e. HBJI.HIOTC.H CBOHMH c06CTBeHHbIMH aHTHqaCTHII,aMH.
CJle,nylOYIO rpynny 06pa3YlOT JlerKHe qaCTHII,bI - JIenTOHLI. B Hee BXO-
,nHT ,ZJ;BeHa,nII,aTb qaCTHII, (BKJllOqa.H aHTHqaCTHII,bI). 9TO 3J1eKTpOH e-, MIOOH J..l-
(OTKpbIT B KOCMHqeCKHX Jlyqax B 1937 r. - 3TO TS:lmeJlbIH aHaJlOr 3J1eKTpO-
Ha, HMelOHH Maccy npHMepHO B 200 pa3 60JlblIlYlO MaCCbI 3J1eKTpoHa) H
't-JlenTOH (TaoH 't - HMeeT Maccy. npHMepHO B 3500 pa3 npeBbIlIlalOYIO Maccy
3J1eKTpoHa). Kam.ua.H H3 3THX Tpex qaCTHII, HMeeT CBoe HeHTpHHo, KOTopoe
conpOBom,naeT CBOIO c06cTBeHHYlO 3ap.HmeHHYIO qaCTHII,y B pa3Ho06pa3HbIx B3a-
HMonpeBpaeHH.HX: 3J1eKTpOHHoe HeHTpHHo pom,naeTC.H BMeCTe C 3J1eKTpOHaMH,
MIOOHHoe HeHTpHHo - BMeCTe C MIOOHaMH, 't-JlenTOHHoe - BMeCTe C 't-JlenTO
HaMH. XOT.H 't-JlenTOH HMeeT OqeHb 60JlblIlYlO Maccy, OH BKJlIOQeH B rpynny
633

Ta6J1Ha 23.1
Ta6J1HQa 3J1eMeHTapHLIX qaCTHQ
CHMBon Maccs B 3neK-
aHTH- 3neK- TpHqe-
HaHMeHOBaHHe qaCTHl.\ qac- qac- TpOHBLlX CKHH BpeMJI :/KH3HH, c
THl.\a THa Maccax 3a pJIJ(
q>OTOH 'Y 'Y 0 0 CT86HneH
HeHTpHHo 3neKTpOH- V e v e 0 0 CT86HnbHO
Hoe
:Q HeHTpHHO MIOOHHoe VJ.L 0 0 CTa6HnbHO
== VJ.L
0
t- T8Y-HeihpHHO V-r V't 0 0 CT86HnbHO
t:
C1.> 3neKTpOH e+ 1 -1 CTa6HneH
I=:: e
MIOOH J.L - J.L" 207 -1 2.2. 10- 6
T8y-nenToH '[- '[+ 3492 -1 1.46. 10- 12
TIH-Me30HbI nO nO 264.1 0 1,83. 10- 16
(nHOHbI) n+ n - 273,1 1 2,6' 10- 8
:Q K+ 1,2. 10- 8
== K8-Me30Hbl K- 966.4 1
0
r? (K80HbI) K O KO 974,1 0  - 8,9. 10- 11
C1.>
 K!l - 5.2. 10- 8
3T8-Hynb-Me30H 1')0 1')0 1074 0 2,4 . 10- 19
Hy- TIPOTOH P jJ 1836.1 1 CT86HneH (1)
:a KJIO- HeihpoH n n 1838.6 0 10 3
==
0 HbI
Po
I::i rHnepoHnSlM6n8 1\0 ;"0 2183.1 0 2.63 . 10- 10

:a rHnepoHcHrM8 1;'" t+ 2327,6 1 8 . 10- 11
== 1;0 to
0 :a 2333.6 0 5,8 . 10- 30
:s:
Po == 1.48 . 10- 10
= 0 I- t- 2343,1 -1
 Po
C1.>
t: rHnepoHKcH :::0 EO 2572.8 0 2,9' 10- 10
:s:
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- - 2586,6 -1
OMer8MHHYc- Q Q 3273 -1 8.2. 10- 11
rHnepOH
JlenTOHOB, nocKonhKY no BceM ,ZJ;pyrHM CBoHcTBaM OH 6J11:130K K HHM. rJl8BHOe
CBOHCTBO, KOTopoe ero pOHHT C OCTaJlbHbIMH JlenTOHaMH, COCTOHT B TOM, qTO
3Ta qaCT1:Ia, KaK H pyr1:le JlenTOHbI, He yqaCTByeT B C1:IJlbHbIX B3a1:lMOeH-
CTBHSlX. JIenToHhI xapaKTepH3YIOTC.H JlenTOHHhIM 38p.HOM.
JIenToHHLIH aapHJJ: - 3TO JlenTOHHoe KB8HTOBoe q1:lCJlO, 3H8qeHHe KOTOpO-
ro ,AJlSI JlenTOHOB +1, ,AJI.H 8HT1:IJlenTOHOB -1, ,AnSI HeJlenTOHOB O. JI.JlSI nen-
TOHHoro 38pSl.n8 cnp8Be.nJlHB 38KOH COXp8HeH1:I.H: B peaKltH.HX C qaCTHaM1:I CYM-
M8pHbIH nenToHHbIH 38p.H He H3MeHSleTC.H. MMeHHO n03TOMY npH p--p8cn8.ne
nO.HBn.HeTC.H HeiiTpHHO. O,AH8KO np1:l p+ -p8cn8.ne B03MO>KH8 1:1 pe8Klt1:l.H, onH-
C8HH8.H BbIllie (no npeBpaeH1:I1O HeHTpOHa B npOTOH C 1:IcnYCKaHHeM aHT1:IHeH-
TpHHO).
634

,l{anee CJle,AYIOT Me30HhI. 3Ta rpynn8 COCTOHT 1:13 BOCbMH qaCTH. HaH60Jlee
JlerKHe 1:13 H1:IX 1t-Me30HbI: nOnOm1:lTenbHbIe, OTp1:laTeJlbHbIe H HeHTpaJlbHbIe. Hx
MaCCbI COCTaBnHIOT 264,1 (1t 0 ) 1:1 273,1 (1t T , 1t-) 3J1eKTpOHHbIX Macc. II 1:I0H bI
.HBnHIOTCH KB8HT8MH H,AepHOrO nOJlH, nO,A06HO TOMY K8K «pOTOHbI - KB8HTbi
3neKTpOMarH1:ITHOrO nOJl.H. Ew;e 1:IMelOTCH 'leTbIpe K-Me30Ha H O,AHH ll o -Me30H.
IIocJle,AH.H.H rpynna - 6apHOHLJ - CaMa.H 06lli1:lpHaH. B Hee BXO,AHT 18
qaCTH H3 39. CaMbIMH JlerKHMH 1-13 6apHOHOB HBn.HIOTCSI HYKJIOHbI - npOTO-
HbI II HeHTpOHbI. 3a HHMH CJle,AYIOT TaK Ha3bIBaeMbIe r1:lnepOHbI. Bc.H Ta6J11:1l{a
3aMbIK8eTC.H Q - (OMera-M1:IHyc)-qaCT1:Il{eH, OTKpbITOH B 1964 r. Ee Macca B
3273 pa38 60JlbWe MaCCbI 3J1eKTpOHa. BapHoHbI xapaKTep1:l3YIOTC.H 6apHoHHbIM
38pH,AOM.
BapHoHHLIH 3apSlJ( - 3TO 68p1:l0HHOe KBaHTOBoe qHCnO, paBHoe ,AJI.H 68pHO-
HOB +1, ,AJI.H He6ap1:l0HOB 0, ,An.H aHT1:I6apHOHoB -1. ,[(n.H 6ap1:l0HHOrO 38p.H,Aa
CnpaBe,An1:lB 3aKOH COXpaHeH1:I.H: B peaKl{HHX C qaCT1:II.I.aMH cYMMapHbIH 6apHoH-
HbIH 3apSI,A He H3MeH.HeTCH. II03TOMY npH B03HHKHOBeHHH, HanpHMep, aHTHnpo-
TOHa B03HHKaeT H npOTOH. TaK. np1:l CTOJlKHOBeH1:I1:I npOTOHa C 3HeprHeH 60nee
6 r3B C HenO,D;B1:I>KHbIM npOTOHOM B03HHKaeT npOTOH-aHTHnpOTOHHaH napa:
I I I 1 1- I
IP+ IP  IP+ IP+ -IP+ IP'
Me30HbI 1:1 6apHoHbI npe,ACTaBn.HIOT C060H Knacc ClJ];pOlIOB qaCTHl{, yqa-
CTBYIOHX B C1:IJlbHbIX B3a1:lMO,AeHCTB1:I.HX. pOHbI pa3,AeJlHIOTCH Ha "cTa61:1nbHbIe n
qaCTHl{bI CO BpeMeHeM >KH3H1:I »10- 23 C H Ha pe30H8HcbI. BpeMSI >KH3HH KOTO-
pbIX == 10- 23 c. T.e. cooTBeTCTByIOT BpeMeHH C1:IJlbHOrO B3a1:lMo.n:eHCTBH.H. L(n1:lHa HX
npo6era C MOMeHT8 pom.n:eH1:I.H ,AO MOMeHTa pacnana COCT8.BJI.HeT OKono 10- 15 M. H
B ,AeTeKTOpax 3T1:I qaCTHl{bI He OCTaBJlSIIOT HHKaKHX TpeKOB. OHH npO.HBJlHIOTCSI
B BH,Ae nHKOB Ha rp8«pHKax 3aBHCHMOCT1:I TaK Ha3bIBaeMbIX CeqeH1:IH paCCeHHH.H
OT 3HeprH1:I. Pe30HaHCbI paCna.,AalOTCH 3a C'leT CHJlbHOrO B3a1:lMO,AeHCTBHSI, CTa-
61:1nbHbIe qaCTHl{bI - 3a CqeT 3J1eKTpOMarHHTHoro 1:1 cJla60ro B38HMO,AeHCTB1:IH.
Pa3,AeJleH1:Ie 3J1eMeHTapHbIX qaCT1:Il{ H8 rpynnbI onpe,AeJl.HeTC.H He TOJlbKO pa3-
Jl1:IqHeM B Maccax, HO H ,Apyr1:lMH BamHbIM1:I CBOHCTBaM1:I, Hanp1:lMep, cnHHOM.
1
JIenToHbI H 6apHoHbI HMelOT Cn1:lH. paBHbIH Cn1:lHbI Me30HOB p8BHbI O. 8
2
cnHH <poToHa paBeH 1.
Mem,Ay 3J1eMeHTapHbIMH qaCT1:Il{aMH cyw;ecTByeT QeTbIpe THna B3a1:lMO,AeH-
CTB1:IH - rpaB1:ITal{1:I0HHOe, 3J1eKTpOMarHHTHoe, CHnbHoe 1:1 cJla60e.
CHJILHOe B3aHMoe:iiCTBHe CBoHcTBeHHo T.HmeJlbIM qaCTHl{aM, HaqHHa.H C nH-
OHa. HaH60Jlee H3BeCTHoe ero npO.HBJleHHe - H,AepHbIe CHJlbI, 06eCneQ1:lBalO-
llJ;1:Ie cyecTBoBaH1:Ie aTOMHbIX H,Aep.
B 3JIeKTpOMarHHTHoM B3aHMoeHcTBHH HenOCpe,ACTBeHHo yqacTBYIOT TonbKO
3J1eKTpHQeCK1:I 3ap.HmeHHbIe qaCT1:Il{bI 1:1 <pOTOHbI. Ha1:l6onee H3BeCTHoe ero npo-
.HBJleHHe - KYJlOHoBcK1:Ie C1:IJlbI. 06ycJloBnHBalOW;1:Ie cyw;ecTBoBaH1:Ie aTOMOB.
HMeHHo 3J1eKTpOMarHHTHoe B3a1:lMO,AeHCTBHe OTBeTCTBeHHO 3a nO,AaBJI.HIOee
60JlblliHHCTBO MaKpOCKOnHQeCK1:IX CBOHCTB Bew;eCTBa. OHO me BbI3bIBaeT aHHH-
rHJlSIl{HIO 3J1eKTpOHHo-n03HTpOHHOH napbI H MHOr1:le ,ApyrHe MHKpOCKOnHQeCKHe
npOl{eCCbI.
CJIa60e B3aHMoeHcTBHe xapaKTepHO JlH Bcex qaCTHl{, KpOMe <pOTOHOB.
HaH60Jlee H3BeCTHoe ero npOSIBneH1:Ie - 13- pacna,A HeHTpoHa 1:1 l{eJloro pH,Aa
aTOMHbIX H,Aep.
635

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$IBn.H.HCb B BHe C1:IJI BCeMHpHOrO T.HrOTeH1:I.H. 8T1:I CHnbI 06eCneqHBalOT cywe-
CTBOBaH1:Ie 3Be3, nJlaHeTHbIX C1:ICTeM H T.n. rpaBHTaUHOHHOe B3a1:lMO.neHCT-B1:Ie
.HBJI.HeTC.H npelteJlbHO CJla6bIM H He 1:IrpaeT cyecTBeHHOH pOJlH B MHpe 3J1e-
MeHTapHbIX 'IaCTH npH 06bIqHbIX 3HeprH.HX. B MHpe 3neMeHTapHbIX qaCTH
rpaBHTa1:I.H CTaHOBHTC.H cyecTBeHHOH np1:l KOJlOCCanbHbIX 3HeprH.HX nop.H-
Ka 10 22 M3B, KOTopbIe COOTBeTCTBYIOT CBepXMaJlbIM paCCTO.HHH.HM nop$lltKa
10- 35 M.
8neMeHTapHbIX qaCTH B HaCTO$lee BpeM.H HaCq1:lTbIBaeTC.H OqeHb MHoro
(60Jlee 350). II03ToMY BCTaeT Bonpoc: eCTb JlH qTO-TO 06ee B CTpYKType
ST1:IX qaCT1:I? MomHo Jl1:I 1:IX Cq1:lTaTb SJleMeHTapHbIMH?
B 1963 r. M. reJlJl-MaHH 1:1 .IJ:m. l(BeHr BbIB1:IHYJl1:l rHnOTe3Y 0 cyecTBo-
BaHHH B npHpoe HeCKOnbK1:IX qaCTH, Ha3BaHHbIX KBapKaMH. CornacHo STOH
r1:lnOTe3e, Bce Me30HbI, 6apHOHbI H pe30HaHCbI - T.e. altPOHbI, COCTO.HT 1:13
KBapKOB H aHT1:IKBapKOB, KOM6HHau,H1:I KOTOpbIX pa3J1HqHbI.
IIepBoHaqaJlbHO 6bIJla BBe,AeHa rHnOTe3a 0 cYWecTBoBaHHH Tpex KBapKOB
(H COOTBeTCTBeHHO Tpex aHTHKBapKOB). KBapKH 0603Haqa1OTC.H 6YKBaMH U. d,
S. OHH JIOJlmHbI 1:IMeTb p06HbIe sneKTp1:lqeCK1:Ie 3ap$lJIbI. IIepBbIH 1:13 HHX -
2
U-KBapK HMeeT 3ap.H + - e, a d- H S-KBapKH HMelOT 01:IHaKOBbIe 3ap$lbI,
3
1
paBHble - '3 e, re e - MOYJlb 3ap.Ha SJleKTpoHa. IIpoToH COCTOHT 1:13 BYX
U-KB8pKOB H OHoro d-KBapKa; nHOHbI COCTO$lT H3 KOM6HHau,HH KBapK - aH-
THKBapK H T. . "CTpaHHbIe" qaCTHu,bI (KaOHbI 1:1 rHnepoHbI) CO,AepmaT 60Jlee
T.HmenbIH S-KBapK. Ha3bIBaeMbIH "CTpaHHbIM".
BbIJlO npelICKa3aHO cyecTBoBaHHe qeTBepToro KBapKa, C-KBapKa, Ha3BaH-
Horo "OqapOBaHHbIM". 3aTeM 3KcnepHMeHTanbHO 6bInH 06HapymeHbI qaCT1:IbI,
coepma1:Ie STOT KBapK. Macca C-KBapKa npeBbIlliaeT Maccy S-KBapKa. Bnoc-
Jle,ACTB1:I1:I 6bIJlH npeCKa3aHbI, a 3aTeM 1:1 OTKpbITbJ ee 60Jlee T.HmenbIe b- 1:1
t-KBapKH.
KBapKH Hap.Hy C JlenTOHaM1:I Cq1:lTaIOTC.H 1:ICT1:IHHO SJleMeHTapHbIM1:I qaCT1:I-
aMH. B cB060HOM COCTOSlHH1:I KBapKH nOKa He HaH,AeHbI, 1:1 CeH'IaC BbICKa3a-
HbI npenOJlOmeHH.H 0 HeB03MomHOCTH pa3eJleH1:I$I 'IaCT1:I Ha KBapKH. B oc-
HOBe ST1:IX npenOJlOmeHHH Jlem1:lT YTBepmeHHe 0 TOM, qTO C1:IJlbI B3aHMoeH-
CTB1:I.H Memy KBapKaM1:I He y6bIBalOT C paCCTO$lH1:IeM, nOSTOMY H3BJleqb KBap-
KH 1'13 'laCTH Henb3.H. TaK, Hanp1:lMep, Ha HOH1:I3aHIO aTOMa 3aTpaqHBaeTCSJ
SHepr1:l$l nOp.HKa 10 sB. Ha paCenJleH1:Ie $Ipa Tpe6yeTcH SHepr1:l.H HeCKOJlb-
KO MsB. Y aJleH1:Ie me OHoro KBapKa Ha paCCTO.HHHe 3 CM OT npOTOHa Tpe-
6yeT 10 13 MsB. OHaKO 3alIOJlrO 1I0 SToro Ha'IHeT lIeHCTBOBaTb OC06bIH Mexa-
H1:I3M pom,ZJ;eH1:I.H qaCT1:I: Kora np1:l yaJIeHHH KBapKa 1:13 HYKJlOHa nOTeH1:I-
aJlbHa.H SHeprH$I CTaHeT OCTaTOqHO BbICOKOH, Ha'IHeT 06pa30BbIBaTbC.H 3a CqeT
STOH SHepr1:lH napa KBapK - aHT1:IKBapK. KBapK OCTaeTCH B HYKJlOHe H BOC-
CTaHaBJlHBaeT STY qaCT1:Iy, a aHTHKBapK 06'be1:IH.HeTC.H C yaJl$leMbIM KBap-
KOM H 06pa3yeT Me30H. BMeCTO yaJleHH$I KBapKa H3 HYKJlOHa npOHCXOlt1:lT
pomeHHe Me30Ha.
IIp1:l CTOJlKHOBeH1:I.HX qaCTH BbICOKOH SHepr1:lH, Hanp1:lMep SJleKTpOHa C no-
3HTpOHOM, 06pa3yeTc$I napa KBapK - aHT1:IKBapK. KBapK H aHTHKBapK pa3J1e-
636

TaIOTC.H B npOTHBOnOnOmHbIe CTOpOHbI, Ii Kam,nbIH H3 HHX pOm,naeT MHOmeCTBO
pOHOB (npeHMeCTBeHHO nHOHOB). TIyqKH a,npOHOB Ha3bIBaIOTC.H CTpYHMH.
Ha6J1IO,neHHe CTPyi{ cnymHT ee O,nHHM ,ZJ;OKa3aTeJlbCTBOM peaJIbHOCTH KBapKOB
B3aHMO,AeHCTBHe KBapKOB OCYl..QeCTBJlHeTCSI nOCpe,ACTBOM 06MeHa OC06bIMH
qaCTHnaMH - rJlIOOHaMH (OT aHrn. glue - KJleH). rJlIOOHbI CKJleHBalOT KBap-
KH BOe,ZIHHO. TIolI06HO cpoToHaM. rJlIOOHbI He HMelOT MaCCbI H 3J1eKTpHqeCKOrO
3ap.H,na.
OTKpbITHe CTPYKTypbI a,npOHOB no CBOHM MacmTa6aM He YCTynaeT OTKpbI-
THIO H,npa H CJlOmHOrO CTpOeHH.H aTOMa. TIocTpoeHa e,nHHa.H TeOpH.H 3neKTpo-
MarHHTHbIX H CJla6bIX B3aHMo,neHCTBHH (TeOpH.H 3J1eKTpOCJla6bIX B3aHMO,ZJ;eH-
CTBHH). BbI,nBHHYTa H Haxo,nHTC.H B CTa,nHH nOCTpOeHH.H HOBa.H TeOpeTHqeCKa.H
Mo,neJlb, nOJlyqHBma.H Ha3BaHHe "BeJlHKOrO 06'be,nHHeHH.H". 3Ta TeOpH.H 06'be-
,nHH.HeT 3J1eKTpOMarHHTHbIe, CJla6bIe H CHnbHbIe B3aHMO,ZJ;eHCTBH.H B OCHOBe ee
JlemaT KBapK-JlenTOHHa.H CHMMeTpH.H (OH8 BbIpamaeTC.H B TOM, qTO B npHpO,Ae
BCTpeqaeTC.H meCTb JlenTOHOB, a Bce CHJlbHO B38HMo,neHCTBYIOI..QHe qaCTHnbI CO-
CTO.HT H3 meCTH KBapKOB) H TeOpH.H 3J1eKTpOCJla60ro B3aHMo,neHCTBH.H. HOB8.H
TeopHSI npe,nCKa3bIBaeT cyecTBoBaHHe MHomeCTBa HOBbIX tlaCTH H HeCTa6Hnb-
HOCTb npOTOHa. Cpe,nHee BpeM.H mH3HH npOTOH8 OKa3bIBaeTC.H p8BHbIM 10 30 _
10 32 JleT. TIoKa 3TH npeJJ;CK83aHH.H He nOJlYtlHJlH 3KcnepHMeHT8J1bHOrO no,n-
TBepmlIeHH.H .
B ee 60Jlee rpaH,ZJ;H03HOM 06061..QeHHH, Ha3BaHHOM "cynepcHMMeTpHeH", ,ne-
JlaeTC.H nonbITKa 06'be,nHHHTb Bce qeTbIpe CPYH,naMeHTaJlbHbIx B3aHMo,neHcTBH.H,
BKJllOqHTb rpaBHTaHIO B paMKH e,nHHoH TeopHH CHJI H nOCTpOHTb 0606eHHe
TeopHH T.HrOTeHH.H - cyneprpaBHT8nHIO. TIoKa npe,nnomeHHbIe B8pHaHTbI Teo-
pHH JJ;aneKH OT peaJlbHOH JJ;eHCTBHTeJlbHOcTH.
23.2. AHTH'IaCTHl\LI. IIpeBpall\eHHe napLl
"3J1eKTpOH - n03HTpoH" B raMMa-H3J1yq:eHHe H o6paTHo
B HaCTO.Hee BpeM.H H3BeCTHO 0 CYI..QeCTBOB8HHH aHTHqaCTHn, KOTopbIe HMe-
IOTC.H nOqTH Y Bcex H3BeCTHbIX qaCTHn. HeKoTopbIe X8paKTepHCTHKH qaCTHn H
aHTHqaCTH OJJ;HHaKOBbI, ,npyrHe - npOTHBOnOJlOmHbI.
BnepBbIe B 1938 r. TI. TIHpaK TeOpeTHqeCKH 060cHoBan. qTO B npHpO,Ae
JJ;OJlmHa cyecTBoBaTb qaCTHna C MaccoH, p8BHOH Macce 3J1eKTpOHa, H C 3a-
p.HJJ;OM, npOTHBOnOJlOmHbIM no 3HaKY, HO paBHbIM no MO,nYJlIO. TaKa.H qaCTH-
8 - n03HTpOH - 6bIJla 06HapymeH8 B 1932 r. B 1933 r. 4>. H H. mo-
nHo-KlOpH 06H8pymHnH pomJJ;eHHe napbI "3J1eKTpOH - n03HTpoH" H3 Y-KBaHTa
C 3HeprHeH w::: 1,02 M3B.
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cooTBeTcTByroeH qaCTHnbI npHCTaBKH "aRTH". HCKJllOqeHHe COCTaBJlHeT aRTHqaC-
THna 3J1eKTpOHa - n03HTpOH. B peaKnH.HX aHTHqaCTHnbI 0603Haqa1OTC.H BonHHC-
TOH tlepTOH H CHMBOJlOM. HanpHMep, In - HeHTpoH, In - aHTHHeHTpoH.
AJI.H qaCTHn H aHTHq8CTHn MaCCbI. cnHHbI, BpeMeHa mH3HH B B8KYYMe P8B-
HbI; 3HaKH 3J1eKTpHqeCKoro. JlenTOHHoro. 6apHoHHoro 3apHlIOB Ii HanpaBJleHH.H
MarHHTHbIX MOMeHTOB npOTHBOnOJlOmHbI.
B qaCTHOCTH, npOTOH H aHTHnpOTOH p83J1HQaIOTC.H He TOJlbKO 3neKTpHqeC-
KHM 38p.H,nOM, HO H 6apHOHHbIM 3ap.H,ZJ;OM H HanpaBJleHHeM MarHHTHoro MO-
637

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JlHqaIOTC.H 6apHOHHbIM 3ap.H,AOM H HanpaBJleHHeM MarHHTHoro MOMeHTa.
Bce H3BeCTHbIe qaCTHlJ,bI 3KcnepHMeHTaJlbHO 06HapymeHbI II HCCJle.ZJ;OBaHbI.
Y 3J1eKTpOHHoro HeHTpHHO V e HMeeTCH aHTHHeHTpHHO V e , pom.ZJ;aeMoe B
npOIJ;eCce 06bIqHOrO 13- -pacn.ZJ;a. CaMH HeHTpHHO yqacTBYIOT B npOIJ;eCce 13+ -pac-
na.na, B TepMO.H.ZJ;epHbIX peaKlJ,H.HX H T. .ZJ;.
Y HeHTpHHO cnHH HanpaBneH npOTHB HMnYJlbCa, Y 8HTHHeHTpHHO - B.ZJ;onb
HMnYJlbca. AHTHqaCTHlJ,en no OTHOllIeHHIO K nonomHTenbHOMY nHOHY 1t- cny-
mHT OTpHlJ,aTeJlbHbIH nHOH It. He3apHmeHHbIH nHOH nO - HCTHHHO HeHT-
paJlbHa.R tIaCTHlJ,a (KaK H q,OTOH).
MomHo nOCTpOHTb aTOM aHTHBO.ZJ;opO.ZJ;a, 3aCTaBHB .ZJ;BHraTbC.H n03HTpOH BOK-
Pyr aHTHnpOTOHa. JIerKo npe.ZJ;CTaBHTb ce6e (XOT.H II OqeHb TpY,AHO HX coxpa-
HHTb) aHTHaTOMbI JlI06bIX .ZJ;pyrHx aHTH3neMeHTOB, B KOTOpbIX n03HTpOHbI .ZJ;BH-
mYTC.H BOKpyr .H,lJ;ep, B COCTaB KOTOpbIX BXO,A.HT aHTHnpOTOHbI H aHTHHeHTpo-
HbI. 113 TaKoro aHTHBellJ,eCTBa MomeT COCTOHTb lJ,enaH raJlaKTHKa HJlH rpynna
raJlaKTHK - 3Be3,lJ;HbIH OCTpOB. B CCCP B 1969 r. BnepBbIe 6bIn nOJlyqeH
aHTHreJlHH. He06xo,AHMO OTMeTHTb, qTO pa3,lJ;eJleHHe Bcex MHKp006'beKTOB Ha
qaCTHIJ;bI H aHTHqaCTHUbI 1I0CTaTOqHO YCJlOBHO. BnOJlHe B03MomHO CqHTRTb aH-
THnpOTOHOM TO, t:lTO MbI npHHHMaeM 3a npOTOH. H Ha060pOT. .D:eno 3aKJllOqa-
eTCH B TOM, qTO npOTOHOB B HallieH BceneHHoH HeH3MepHMO 60JlbIlle, tIeM
aHTHnpOTOHOB H T ..ZJ;.
OCHOBHOM oc06eHHOCTblO napbI "'laCTHlJ,a - aHTHtIaCTHlJ,a" HBJlSleTCSJ cnoc06-
HOCTb aHHHrHJlHpOBaTL npH BCTpe'le, nOJlHOCTblO npeBpallJ,a.HCb B .ZJ;pyrHe q,OpMbI
MaTepHH ('lallJ,e Bcero B H3J1yqeHHe) C C06J1IO.ZJ;eHHeM 3B.KOHOB coxpaHeHH.H. 3Kcne-
pHMeHTaJlbHO YCTaHOBJleHO: eCJlH BCTpetIaIOTC.H qaCTHlJ,a CO CBoeH aHTHtIaCTHu;eH,
06e npeBpallJ,alOTCH B .ZJ;Ba q,OTOHa COOTBeTCTByIOllJ,eH qaCTOTbI, pa3J1eTaIOllJ,HeCH OT
MeCTa npOHCllIeCTBH.H B pa3HbIe CTOpOHbI. TaKHe npeBpallJ,eHHH 6bIJlH 3aq,HKCHpO-
BaHbI npH BCTpeqe 3J1eKTpOHa H n03HTpOHa, a n03.ZJ;Hee - npOTOHa H aHTHnpo-
TOHa. IIpH TaKOM npeBpallJ,eHHH .ZJ;eq,eKT MaCCbI .ZJ;OCTHraeT MaKCHMYMa, BbI,lJ;eJlHeT-
CSI MaKCHMaJlbHO B03MomHoe no 3aKOHY B3aHMOCBSJ3H MaCCbI H 3HeprHH KOJlHqe-
CTBO nOJlHOH 3HeprHR 2mc 2 , r.ZJ;e m - Macca O.ZJ;HOH qaCTHlJ,bI. 11, Ha060pOT, npH
3aTpaTe COOTBeTCTBYIOllJ,eH 3HeprHH TaKa.H napa MomeT B03HHKHYTb.
II03HTpoH, KaK H 3J1eKTpOH, caM no ce6e a6COJllOTHO cTa6HneH. Ho, BCTpe-
qaHCb .ZJ;pyr C .ZJ;pyroM, 3J1eKTpOH H n03HTpOH aHHHrHJlHpYIOT ("YHHqTOmaIOT-
C.H") , nOpOm,lJ;aH .ZJ;Ba q,OTOHa:
o 1 2
-le+1-Ie---7 y.
Pa3YMeeTcSI, HHKaKoro YHHqTOmeHH.H MaTepHH 3,lJ;eCb He npOHCXO,lJ;HT: O,lJ;HH
BH.ZJ; MaTepHH - 3apSlmeHHbIe MaCCHBHbIe qaCTHlJ,bl - nepeXO,lJ;HT B .ZJ;pyroH
BH,lJ; MaTepHH - B HeHTpaJlbHbIe 6e3MaCCOBbIe qaCTHlJ,bI. COOTBeTCTBeHHO, 3Hep-
rHH nOKO.H 3J1eKTpOHa H n03HTpOHa npeBpallJ,aeTC.H B 3HeprHIO ,lJ;BHmYllJ,HXCH CO
CKOpOCTblO CBeTa q,OTOHOB. ECJlH aHHHrHJI.HUHH npOHCXOlIHT H3 COCTO.HHH.H no-
KO.H, TO 4>OTOHbI pa3J1eTaIOTC.H B npOTHBOnOJlOmHbIe CTOpOHbI C O,lIHHaKOBbIMH
HMnYJlbCaMH H O,AHHaKOBbIMH 3HeprH.HMH: hv = m('c 2 = 0,511 M3B.
qacTo (HO He Bcer.ZJ;a) n03HTpOH 06pa3YeTC.H COBMeCTHO C 3J1eKTpOHOM. Pom-
.ZJ;eHHe napbI (_?e, +e) npOHCXO,lJ;HT npH CTOJlKHOBeHHH €pOTOHa C 3ap.HmeHHOM
qaCTHIJ;eH, B KaqeCTBe KOTOpOH 06bIqHO BbICTYnaeT aTOM Hoe .H,lJ;pO X:
X 0 0
y+ X ---7 + -le+ +le.
638

&ra peaKII.H.H B03MOmHa TOJlbKO npH ,nOCTaTOqHO 60JlblIIOH 3HeprHH (l>OToHa:
hv = mf'c 2 = 1,022 M3B.
He cJle,ZJ;YeT AYMaTb, qTO 3J1eKTpoH H n03HTpOH, BCTpeqa.HCb .n;pyr C .n;pyrOM,
Bcer,ZJ;a aHHHrHJlHpyK>T, npeBpaaJlCb B ,ZJ;Ba (l>OToHa. TaK npOHcxo,nHT JlHlIIb B
TOM cJlyqae, Kor.n;a B CHCTeMe OTCqeTa, CB.H3aHHOH C HX II.eHTpOM Macc, KHHe-
THtJecKa.H 3HeprH.H He CJIHlIIKOM BeJIHKa. ECJlH me 3J1eKTpOH H n03HTpOH .n;BH-
myrC.H HaBCTpeqy ,npyr .n;pyry, 06J1a.H OqeHb 60JlblIIHMH 3HeprH.HMH, TO npH
HX CTOJlKHOBeHHH MOryT nopOmlIaTbC.H C8..MbIe pa3Ho06pa3Hble qaCTHII.bI (BnJlOTb
.n;o H8U60Jlee T.HmeJlOH - HnCHJlOH-Me30Ha). HCnOJlb30BaHHe BCTpe1JHLIX nyqKoB
3J1eKTpoHOB H n03HTpOHOB - O.n;HH H3 caMbIX 3(1)<l>eKTHBHbIX MeTO.n;OB reHepa-
II.HH HOBblX qaCTHII.. OH lIIHpOKO npHMeH.HeTC.H B COBpeMeHHOH (l>H3HKe BblCOKHX
3HeprHH. HBJleHHe aHHHrHJI.HII.HH .HBHJlOCb ee O.n;HHM .n;OK83aTeJlbCTBOM B3aHMO-
CB.H3H npeBpa.I.D;eHHH 1JaCTHo;a + aHTH1JaCTHo;a H 3J1eKTpOMarHHTHOe nOJle. Pom-
.n;eHHe 3J1eKTpOHHO- n03HTpOHHOH napbI TaKme Har JI.H.n;HO nOKa3bIBaeT, "'ITO .n;Be
(l>OpMbI MaTepHH - BemeCTBO 1:1 nOJle - He .HBJI.HIOTC.H pe3KO p83rpaH1:IqeHHbI-
M1:I, TaK KaK B03MomHbI B3aHMHbIe npeBpaeHH.H MaTep1:l1:l 1:13 O.nHOH cpOpMbI B
,npyryro.
23.3. B3aHMHLIe npeBpaeHHR 3JIeMeHTapHbIX '1aCTH
C 3J1eMeHTapHbIMH qaCTHnaMH B03MomHbI pe8KnH1:I npeBpaeHH.H, C1:IHTe3a,
pacna,ZJ;a, no.n;q1:lH.HIO1:IeC.H 3aKOHaM coxpaHeHHH.
O.n;HO 1:13 H81:160Jlee 061:IX cpYH,naMeHTanbHbIX CBOHCTB 3J1eMeHTapHbIX qac-
T1:In - HX npeBpalQeHHH ,ZJ;pyr B .n;pyra. TIp1:l 3TOM 06pa3Y101Q1:IeC.H q8CT1:InbI He
BXO.n;.HT B COCTaB liCXO,ZJ;HbIX qaCT1:IIJ;, a pom.n;aIOTC.H Henocpe.n;CTBeHHO B nponec-
cax liX coy.n;apeHHH liJlH pacna,nOB. CJle.n;yeT 3aMeTHTb, qTO cpOTOH T8Kme He
BXO.n;1:IT B COCT8B aTOMa, a pomlIaeTCH Henocpe.n;CTBeHHO B nponecce nepexo.n;a
3J1eKTpOHa B aTOMe C o.n;Horo 3HepreTHqeCKOrO ypOBH.H Ha .n;pyrOH. TonbKO Ta-
K1:IM1:I npeBpameHH.HMH MomHO 06'b.HCH1:ITb J3-pacn. np1:l KOTOpOM 3J1eKTpOHbI
1:In1:l n03HTpOHbI BblJleTalOT H3 .H,npa. B KOTOpOM 1:IX He 6blno. OlIHa H3 TaKHX
1 1 0 0 0 0
peaKU1:lH 1:I1IeT no cxeMe IP  on+ +l e + oV' r.ne OV - HeHTp1:lHO. ..Ie
n031:1TpOH, Ip - npOTOH, In - HeHTpOH. B3a1:lMo.n;eHCTB1:I.H Mem.n;y 3J1eMeH-
TapHbIMH qaCTHIJ;aMH np1:lBo.n;.HT K II.enOMY p.H.D;y B3a1:lMOnpeBpalQeH1:IH 1:1 .D;pyr1:lx
npOIJ;eCCOB, KOTopble MomHO pa3.D;eJl1:lTb Ha Tp1:l rpynnbI: ynpyroe paCCe.HH1:Ie,
HeynpyrHe npOIJ;eCCbl 1:1 paCna.D;bI.
TIpH ynpyrOM paCCeJlHHH qaCTHII.bI He npeTepneBalOT npeBpaeHHe, a npo-
CTO 1:I3MeH.HIOT COCTO.HH1:Ie CBoero ,ZJ;B1:ImeH1:I.H. IIp1:lMepoM MomeT cnym1:lTb pac-
Ce.HH1:Ie CX-qaCTHIJ; aTOMHbIM1:I .H,ZJ;paMH B onbITax 3. Pe3epcpop.n;a.
B HeynpyrHx npo:Q,eccax (peaKnH.Hx) npOHCXO,AHT CTOJlKHOBeHHe .n;BYX qac-
THIJ;, conpOBom.nalOeeC.H HX npeBpaeH1:IeM B qaCT1:IUbI ,npyroro copTa. COOT-
BeTcTBYlOlQHH npHMep .D;aeT aHH1:IrHJI.Hn1:l.H 3neKTpoHHO-n031:1TpOHHOH napbI B .D;Ba
cpoToHa.
QaCT1:IIJ;bl, pom,ZJ;alOHeC.H B npOIJ;eCcax paCCe.HH1:I.H, 3a pe,ZJ;K1:IMH liCKnlOqeH1:I-
.HMH .HBJI.HIOTC.H HeCTa6HJlbHbIM1:I Ii npeTepneBalOT pacnaALI. OHH m1:lBYT nocJle
pOm.D;eHH.H OqeHb MaJlbIe npoMemyrKH BpeMeH1:I, npeBpalQa.HCb 3aTeM B .D;pyr1:le
qaCT1:InbI. CaMa.H YCTOHqHBa.H 1:13 HeCTa6HJlbHbIX "'IaCT1:III. - HeHTpOH, 06J1a.D;alO-
lQ1:IH cpe,nH1:IM BpeMeHeM m1:l3HH 't = 898::t 16 c.
639

BOllPOCM ARB CAMOKOHTPORB
1. £[mo maKoe 3Jl.eMelimaplibte llaCmUijbt? KaK06bt ux c60ilcm6a?
2. PaCCKaJICUme 06 omKpbtmUU n03umpOlia.
3. £[ eM OmJl.UllalOmCSl 3Jl.eMelimaplibte llaCmUijbt iJpyz om iJpyza?
4. q mo maKoe alimullaCmuijbt? KaKue c60ilcm6a y Ilacmuij U alimUllac-
mUl{ pa3Jl.Ulllibt. a KaKue oiJuliaK06bt?
5. g mo maKoe aIiIiUZUJl.Sl4USl 'lacmuij?
6. PaccMompume npUMep aliIiUZUJl.SlijUU 3Jl.eKmpOlilio-n03umpolilioil napbt.
7. II OlleMY llaCmUijbt. pOJICiJalOl4UeCSl npu 63auMoiJeilcm6uSlx. lieJl.b3Sl pac.
cMampu6amb KaK COCma6libte Ilacmu 63auMoiJeilcm6YIOl4UX Ilacmuij?
8. H 3Yllume ma6Jl.uijY 3Jl.eMelimaplibtX Ilacmuij (CM. c. 634). CKOJl.bKO 3Jl.e-
MelimaplibtX llacmu4 Sl8Jl.SllOmCSl Cma6UJl.bHbtMU? H a306ume HeKomOpbte U3 IiUX.
!J.m = Zmp + Nm n - mil
W =!J.m. c 2
CD
W
w =
CD A
t
N = N o 2 T = Noe-).,t
1
1: =-
A
A = !J.N
!J.t
OCHOBHbIE cI»OPMY Jlbl
,ne<}>eKT MaCCbI aTOMHoro .H,npa, r,ne Z - qHCJlO
npOTOHOB, mp - Macca npOTOHa, N - q1:lcno HeHTpo-
HOB, m n - Macca HeHTpOHa, mg - Macca .H,npa.
- 3Hepr1:l.H CB.H31:1 aTOMHoro .H,npa. r,ne c - CKOpOCTb
CBeTa.
- y,neJlbHa.H 3Hepr1:l.H CB.H3H, r,ne A - MaCCOBoe qHC-
JlO, paBHoe q1:lcny HYKJlOHOB B .H,npe.
- 3aKOH pa,n1:l0aKT1:IBHOrO pacna,na, r,ne N - q1:lC-
JlO aTOMOB B HeKoTopbIH MOMeHT BpeMeHH t, No -
q1:lCJlO aTOMOB B HaqaJlbHbIH MOMeHT BpeMeH1:I,
In2 0,693
T = T = --:;:- - nep1:l0,n nOJlypacna,na, A - nOCTO-
.HHHa.H pa,n1:l0aKT1:IBHOrO pacna,Aa.
- CpellHee BpeM.H m1:l3H1:I pa1l1:l0aKT1:IBHOrO .Hllpa.
- aKT1:IBHOCTb pa,AHOaKT1:IBHOrO npenapaTa, r,Ae !J.N -
qHCJlO pacnaBIlIHXC.H .H,nep 3a BpeM.H !J. t.
METOAHqECKHE PEKOMEHAAQHH
IIpH pellIeHHH 3a,naq no TeMe "<I>H31:1Ka aTOMHoro .H,npa" Heo6xo,n1:lMO no-
MHHTb. qTO:
1) np1:l JlI06bIX .H,ZJ;epHbIX peaKH.HX BbInOJlH.HIOTC.H 3aKOHbI COXpaHeH1:I.H 3Hep-
r1:lH, HMnynbca H MOMeHTa HMnYJlbCa, CYMMapHoro 3J1eKTp1:l'tJeCKoro 3ap.H,na 1:1
q1:lCJla HYKJlOHOB, a TaKme npaB1:Ina CMeeHH.H, .HBJI.HIO1:IeC.H CJle,nCTB1:IeM 3a-
KOHOB COXpaHeH1:I.H 3ap.H,na 1:1 q1:lcna HYKJlOHOB;
640

2) p,n.H HaXOmp,eHH.H 'IHCJla HepaCnaBIll1:lXC.H K MOMeHTY BpeMeHH t aTOMOB
HY)KHO 1:ICnOJlb30BaTb 3aKOH pap,1:I0RKTHBHoro pacnRp,a.
IIPHMEPbI PEllIEHH.H 3A.l(A q
1. HRH,AHTe 3HeprHIO CBSl3H SI,Apa H30Tona nHTHfl Li.
,IJ; 8 H 0: m p = 1,6726,10- 27 Kr, m n = 1,6749,10- 27 Kr,
m Jl =11.6475.10- 27 Kr.
HaHTH: WCB'
Pew e H H e. 8HeprHSI CBSl3H SI,Apa BbIpRm8eTCSI CPOPMY nOH
WCB =(Zmp+(A-Z)mn -mJl)c 2 .
113 CHMBOnH'leCKOH 3anHCH H30Tona nHTHSI Li Cne,AyeT, 'lTO A = 7, Z = 3.
Cne,AOBaTenbHo,
WCB =(3,1,6726,10- 27 +4.1,6749.10- 27 -11,6475.l(f27). 9.l(j6 = 6,201. 1d 2 m.
2. K8Koe KOnH'leCTBO ypaHa 2 U P8CXO,AyeTcSI B CYTKH Ha 8TOMHOH 3neK-
TpOCTaHIJ.HH MOIl{,HOCTbIO 5,0 MBT? KII.l( npHHSlTb paBHbIM 17 %. C'lHTaHTe,
'lTO npH K8m,AOM aKTe p8Cn8,Aa BbI,AenSleTCSI 3,2' 10- 11 .l(m 3HeprHH.
,l(8HO: P=5,O.10 6 BT, 11=0,17, W 1 =3,2.10- 11 ,l(m, t=2.64.10 4 c,
M=0,235
Kr
MOJlb
HaHTH: m.
w
Pewe H H e. KTI 3neKTpOCTaHIJ.HH T\ = ;nea , r,Ae Wnonea - none3H8S1 3Hep-
rHSI, W - 3HeprHSI, BbI,AenSleMaSl npH ,AeneHHH SI,Aep. Wnonea=Pt. r,Ae P -
MOIl{,HOCTb CT8HIJ.HH. BbI.v;enSleMRSI 3HeprHSI W = W 1 N, r,Ae N - 'lHCnO SI,Aep,
paCnaBIIIHXCSI aR CYTKH, KOTOpOe MomHO HaHTH, 3H8S1 MOnSlpHYIO M8CCY M.
N = N m , r,Ae N A = 6,02,10 23 Monb -1 - nOCTOSlHHaSl ABOra,ApO. Cne,AO-
BaTenbHO,
PtM
T}=
WlmN A
PtM
 m = . m = 31 r.
T}W1N A '
3. Onpe,AenHTe nepHO,A nonyp8Cn8,AR paAOH8, eCnH 38 O,AHH CYTKH H3
1,0' 10 6 8TOMOS pacna.v;aeTCfl 175000 8TOMOB
,l(RHO: t=8,64.10 4 c, N o =I,0.10 6 . 6N=l,75.10 5 .
H8HTH: T.
Pew e H H e. TIo 38KOHY p8,AH08KTHBHoro paCn8,A8
t
N = N o .2 T.
'IHCJlO pacnaBwHxc" "!lOp l>N = No - N = No f 1- 2 - 1- OTClO!la
2 = No :=) T = t 19 2 T = 3,3.10 5 C.
No - MV 19 No
No-!J.N
641

4. KaKoe KOnH'leCTBO SHeprH1:I BLI,AeJISleTCSI B pe3ynLTaTe TepMOSl,AepHOH pe-
8Kl{HH CHHTe38 1,0 r reJIHH H3 AeHTepHSI H TpHTHSI?
.n; a H 0: m = 1,0 . 10- 3 Kr, maH = 2,01474 a.e.M., ma H = 3,01700 8.e.M.,
I I
m 4He = 4,00387 a.e.M., m n = 1.00866 8.e.M.
a
HaHTH: W.
P e III e H H e. TepMOSl,AepHaSl peaKIlHSI CHHTe3a Sln;ep renHSI H3 AeiiTepHSI H
TPHTHSI npOTeKaeT no cne,AYIOeMY ypaBHeHHIO:
2 H 3 H 4 H 1
1 + 1  2 e + 0 n.
Onpe,AeJIHM ,Ae<l>eKT MaCCLI peaKl{HH
11m = (mH + mH) - (m: He + m n )
m = (2,01474 + 3,01700) - (4,00387 + 1,00866) = 0.01887 8.e.M.
1 a.e.M. COOTBeTcTByeT SHeprHSI 931 MsB, Cne,AOBaTenbHO, SHeprHSI, BbI,Ae-
JIHBlliaSlCH npH CHHTe3e 1 SI,Apa reJIHSI.
WI = 931.0,01887 = 17,6 MsB.
mN A
B 1 r renHSI CO,AepmHTCSI N = - 8TOMOB, r,Ae N A - nOCTOSlHH8S1 ABO-
M
r8APO; M - MonSlpH8S1 M8cca renHSI. TIonH8S1 3HeprHfl
W - NW; - WIN Am . W = 26,5. 10 23 MsB = 4,2. lOll .n;»<.
- 1- M '
5. KapM8HHLIH .o;03HMeTp p8,AH08KTHBHoro 06nY'leHHSI, npe,ACTaBnSlIOHH co-
60H MHHHaTlOpHYIO HOHH381{HOHHYIO K8Mepy eMKOCTLIO 3.0 n<1>. 3apSlmeH ,AO no-
TeHl{Hana 180 B. IIo,A BnHSlHHeM 06nYQeHHH nOTeHnHan CHH3HncSI ,AO 160 B.
CKonbKo peHTreH nOK8meT .o;03HMeTp, ecnH .0;0 3Toro OH 6bIJ1 nOCTaBneH Ha
HynL, 8 06'LeM B03,AYX8 B K8Mepe 1,8 CM 3 ?
.n;aHo: C=3,O.10- 12 <1>, q>1=180 B, q>2=160 B, V=1.8 CM 3 , e=1.6.10- I9 Kn.
HaHTH: D.
P e III e H H e. IIoTeHl{H8n ,A03HMeTpa YMeHLlliHncSI B pe3ynLTaTe HeHTp8nH-
38l.\HH 'laCTH ero 38pH,Aa HOHaMH, B03HHKlliHMH B K8Mepe npH 06nY'leHHH. On-
pe,AenHM BenH'lHHY 06pa30BaBllierocSI aapSl,Aa: q = CU. 3TOMY 3apH,Ay COOTBeT-
cTByeT KOnH'leCTBO nap HOHOB N = .!L, r,Ae qi = e - 38pH.D; 0,AHOB8JIeHTHOro
ql
HOH8. OTCIO.D;a N = C (CPt - CP2 ) :
e
N = 0,38 .10 9 (nap HOHOB).
O,nHOMY peHTreHY COOTBeTcTByeT .11038 06nY'leHHSI. npH KOTOpOH B 1 CM 3
B03,AYX8 B03HHKaeT 2.082.10 9 n8p HOHOB. Cne,AOBaTenbHo, 38perHcTpHpOB8H-
H8S1 .D;03HMeTpoM .I103a 06nY'leHH.H paBH8
D= 0,38.10 9 =0,1 P.
1, 8 . 2,082 10 9
642

CI1P ABOqHbIE MA TEPHAJIbI
1. OCHOBHLle 4JH3HQeCKHe nOCTOHHHLIe (KOHCTaHTLI)
rpaBHTaHOHH8H nOCTOSlHHaSl
TIocToSlHHaSl ABOr8,ApO
IIocToHHHa.H BOnbMaHa
YHHBepC8nbHaSl ra30BaSl nOCTOHHHaSl
MonSlpHbIH 06'beM H,Ae8nbHOrO r838
npH HOpManbHbIX ycnoBHSlX
SneMeHT8pHbIH 3apSl,A (38PSl,A SneKTpOHa)
M8cc8 3neKTpOH8
Macca HeHTpOH8
Macca npOTOH8
SneKTpH'leCK8S1 nOCTOSlHH8S1
MarHHTHaH nOCTOHHHa.H
CKOpOCTb CBeTa B B8KYYMe
IIoCTOSlHH8S1 TIn8HKa
IIocToSlHHaSl PH,A6epr8
M 3
G = 6.672 . 10- 11
Kr.c 2
N A =6,022.10 23 MOnb- 1
k = 1,381 . 10-23 A>K
K
R=8,314 _ Am
Monb . K
3
V M =2,241' 10- 2 
Monb
e = 1.602. 10- 19 Kn
m e =9,10953.10- 31 Kr
m n = 1,67495 . 10- 27 Kr
m p = 1.67265' 10- 27 Kr
£0= 8,85 . 10- 12 <1>
M
rH
Jlo= 41t. 10- 7
M
c = 2.998. 10 8
M
C
h = 6,626 . 10- 34 Am. C
R = 10967758 M- 1
jKHAKOCTH
2 H3HQeCKHe CBoHcTBa HaH60Jlee pacnpOCTpaHeHHLIX BeeCTB
BenHqHH8
IInoTHocTb Y,nenbH8$1 TeMnep8Typa Y,nenbH8$1 TennOT8 K03ct>ct>HH-
npH HOp- TennoeM- KHneHH$I npH n8poo6p830B8HH$I eHT nosepx-
BeeCTBO M8nbHbIX KOCTb. HopM8nbHOM npH TeMnepaType HOCTHoro
ycnOBHJIX K,[(m ,naBneHHH. °C KHneHH$I H HOp- HaT$lmeHH$I,
x 10 3  M&nbHOM ,nasneHHH. MH
Kr.K -
M 3 K)J;m M
Kr
BOtt8 1,00 4,19 100 2260 73
Bott8 MOpCK8S1 1.03
BO,A8 MbInbH8S1 43
BeH3HH 0.70 2.05 70+205 300 29
BeH30n 0,90 3,0
KepocHH 0,80 2.14 150+300 209+230 24
M8cJIO 090 1.67 33
He(}>Tb 0.80 1.67+2.09 30
PTYTb 13.60 0,13 357 293 472
QeTblpex-
xnopHcTblH 1,63 27
yrnepo,A
643

TBEP ,Z:(bIE TEJIA
BeJIHqHH8
IInoT- Y,ZJ.eJIb- TeMne- Y,nenb- MO,nynb IIpe,nen Y,neJIb- TeMnep8TYP-
HOCTb. Ha$l paTypa H8JI IOHr8. npoq- HOe HbIH K03ct>-
BeIlteCTBO x10 3 TennoeM nn8B- TennOT8 x 10 10 HOCTH conpo- ct>HHeHT
Kr KOCTh. JIeHHJI . nn8BJIe- IIa npH THBne- conpOTHB-
M 3 K,[(m °c HH$I, paCT$I- HHe. neHH$I.
Kr K K,[(m meHJ.lH, x 10- 8 X 10- 3 K- 1
Kr rII8 OM'M
AnlOMHHHH 2,7 0.92 660 380 7 0,25 2,8 4,2
BeToH 2,2 0,06
Bon b <PP8M 19.3 0,15 3420 184 5,5 5,0
epeBo (cocHa) 0,5 2,70 1.0
jKene30 7.8 0,46 1539 270 20 0,29 9.8 6,2
30nOTO 19,3 0,13 1064 66.6
KHpnH'l 1,8 0.75 2.8 0,0015
+0,06
KOHCT8HT8H 8,9 1260 50.0 0,05
JI8TYHb 8.5 0,38 1000 11,0 7.0 0,1+ 0,4
JIe 0,9 2,10 0 340
M8Hr8HHH 8.5 43,0 0.01
Meb 8.9 0,38 1083 175 12 0,24 1.7 4.3
HHKenb 8,9 0.50 1455 7,3 6,5
HHXpOM 8.3 1400 10.0 0,1
OnoBo 7.3 0,25 232 58 0,027 12,0 4.9
CSHHe 11.4 0.13 327 25 1.7 0,016 21,0 3,7
Cepe6po 10.5 0.25 960 87 0.14 1,6 4,1
CT8nb 7.8 0.46 1400 84 21 0,8+1,0 11,0 1+4
CTeKno 2,5 0,84 0,06+
0.12
Yp8H 19.1
HHK 7,1 0.40 420 112 8 0,11 6,1 4,2
qyrYH 7.4 0,54 1150 97 11,5 0,025+ 50+80 1,0
0,55
r A3hI
BenHqHH8
IInOTHOCTb npH Y,nenbHa$l TennoeMKOCTb TeMnep8Typa KOH,neHC8
BeIlteCTBO HOpMaJIbHbIX npH nOCTO$lHHOM J.lH npH HOpM8JIbHOM
Kr K.D;m ,ZJ.8BneHHH. °C
YCJIOBHJIX. M 3 ,ZJ.8BJIeHHH.
Kr.K
A30T 1.25 1.0 -196
ADrOH 1.78
BOlIODOlI 0.09 14 -253
B031IVX 1.29 1 0
renHH o 18 5 2
KHcnopo,n; 1.43 0,92 -183
HeoH 0.90
Y r neKHcnbIH ra3 1 98
644

MAm
3. Y.n;eJILHaSi TenJIOTa cropaHHSI TOnJIHBa q,
Kr
BeH3liH 46 KepocliH 44
BYPbIH yronb 9,3 M83YT 40
J(li3enbHoe TonnliSO 42 ITopox 3,8
J(peBecHbIH yronb 31.5 CnHpT 27
J(POBR CYXHe 8,4 Top«p 15
4. J(aBJIeHHe H nJIOTHOCTL HaCLIIIeHHLIX napOB BO.n;LI
npH pa3JIHllHblX TeMnepaTypax
r r r
t. °C p. KIIa p. "3 t. °C P. KIIa P. 3 t. °C p. KIIa p. -3
M M M
0 0,611 0,84 11 1,31 10,0 22 2,64 19,4
1 0.656 5,22 12 1,40 10,7 23 2,81 20,6
2 0.705 5,60 13 1,50 11,4 24 2,99 21.8
3 0,757 5.98 14 1,59 12,1 25 3,17 23,0
4 0.813 6,40 15 1,70 12.8 30 4,24 30,0
5 0,872 6.84 16 1.81 13,6 40 7.37 51.2
6 0,934 7.3 17 1.94 14.5 50 12,3 83.0
7 1,01 7,3 18 2,06 15.4 60 19.9 130
8 1,07 8.3 19 2.19 16,3 70 31,0 198
9 1.15 8,8 20 2.34 17,3 80 47.3 293
10 1.23 9,41 21 2,48 18.3 90 70.1 424
5. KpHTHqeCKHe napaMeTpLI HeKOTopLIX Bem;eCTB
KpHTHqeCK8$1 KpHTHtJecK8$1 KpHTHqeCKOe ,lt8BneHHe
Kr
BeeCTBO TeMnepaTypa. °C nJIOTHOCTb. "3 Mil8 8TM
M
A30T -147.1 311 3.39 34.6
AMMHaK 132,4 235 11,5 117
Al..(eTliJleH 35.7 231 6.24 63,7
BOA8 374.2 307 22,13 225.65
BO,AOPOA -239.9 31.0 1,30 13,5
B03AYX -140,7 350 3.77 38.5
reJIliH -267,9 69.3 0,23 2,3
KliCJlOPOA -118,8 430 5,04 51.4
OKCliA yr nepO,A8 CO -139 301 3.5 36
J(SYOKliCb yrnepoA8 31.0 460 7,35 75,0
CnlipT 243.5 276 6.38 65.2
XJlOp 144.0 573 7,70 78,5
:3 <1m p 193.8 260 3,60 37,0
645

6. ]J;H3J1eKTpH'IeCKaJl npOHHuaeMOCTb BemeCTB8 £
BeH3HH
B8KYYM
BO8
BOOPOA
B03YX
2,3
1
81
1 ,0003
1.0006
rnHIJ.epHH
KepocHH
M8cno Tp8HC(}>OpM8TopHoe
n8p8(}>HH
n8p8(}>HHHPOB8HH8S1 6YM8r8
39
2,0
2,2
2.2
2.0
CJIIO8
CTeKJIO
ct>8P(}>OP
360HHT
.HHT8Pb
6+9
5+ 10
4+7
2,7
2.8
7. 3J1eKTpOXHMHqeCKHe 3KBHBaJleHTLI BeeCTB
3neKTpOXHMHqeCKHH 3neKTpOXHMHqeCKHH
AHHOHbI 3KBHB8neHT. K8THOHbI 3KBHB8neHT.
Kr Kr
x 10- 11 - X 10- 11 -
Kn Kn
Cl- 0,367 Af!.... 1 118
CO- A1 3 + 0.0932
0.311 Au 3 + 0681
N0 3 0.643 Cu 2 + 0.329
0 2 - 0.0829 Fe 3 + o 123
H+ 0.1045
OH 0,177 Hf!.+ 2,079
8 2 - 0.167 N8+ 0238
Ni 2 + 0304
80 0.499 Zn 2 + 0,339
8. UOK8aaTeJIL npeJlOMJleHHJI HeKOTopLIX BeeCTB )J;JlJI )J;JlHHLI BOJlHLI
589 HM (eJlTaJl JlHHHJI)
AnM83 2.42 r nHuepHH 1,47 CTeKno 1,5 + 1,9
BeH3HH 1,41 JIe 1,31 .HHT8pb 1.55
BO8 1.3 CnHpT 1.33
9. llpe)J;eJlLHLle yrJlLI nOJlHOrO OTpaeHHJI
AnM83 24° r nHIJ.epHH 43° 3 <1m p 47°
BeH3HH 45° CnHpT 47°
BO8 49° CTeKJIO 30°+ 42°
10. Pa6OT8 BLIXO)J;8 3J1eKTpOHa A, x 10- 19 Am
BOJIb(}>paM 7.2 OKCH 68pHJt 1.6 Cepe6po 6.9
repM8HHH 7.7 OKCH MeH 8.3 Ue3HH 2,9
MOJIH6eH 6,9 IIn8THH8 8,5
HHKenb 7.2 Py6HHH 3,5
11. U;BeT8 H cooTBeTcTBYJOHe HM HHTepB8J1LI AJlHH BOJIH H '1aCTOT
I..{BeT cneKTp8
KP8CHbIH
8HmeBbIH
jKenTbIH
3eneHbIH
rony6oH
HHHH
ct>HoneTOSbIH
,[(nHHS BOJIHbI, HM
1.JacToT8, Tr
llpUMe'laH.Ue. 06n8cTb BH,nHMOH q8CTH cneKTp8 38KnlOqeH8 B rp8HH8X BonH npH6nH3H-
TenbHO OT 760 ,no 380 HM. rp8HHbI BeTOB cneKTp8 T8Kme onpe,neJIJIIOTCJI JIHWb ycnoBHO.
646

12. H30TonhI
ATOMHbIH HOMep XHMH'ieCKHH aneMeHT CHMBon H30Ton8 M8CC8 H30Ton8.
aneMeHT8 8.e.M.
1 Bo.u.opo,A IH 1.007825
2H 2.014102
3H 3.016040
2 renHH 3He 3,016049
4He 4,002603
3 JIHTHH IiLi 6.015125
7Li 7.016004
4 BepHJInHH 9Be 9,012186
5 Bop lOB 10,012939
6 YrJIepOA 12C 12.000000
13C 13.003354
l4C 14,003240
7 A30T 101N 13.005370
14N 14,003074
I:iN 15.000107
8 KHCnOpOA 16 0 15,994915
nO 16,999139
11 H8TpHH 22N8 21.994430
23N8 22.989771
12 M8rHHH 4:4Mg 23.985042
4:::IMg 24.985839
2liMg 25,982539
13 .AnIOMHHHH 24AI 23.999940
27AI 26,981539
15 q>oc<}>op 30p 29.978310
011p 30,973765
32p 31.973900
16 Cep8 32 8 31.972074
34 8 33,967864
01::18 34.969030
36 8 35,967090
17 Xnop 3:iCI 34.968851
"ICI 36,965898
25 M8pr8He 54Mn 53.940350
26 mene30 54Fe 53,939617
56Fe 55.934936
29 MeAl. 1i3CU 62.929592
65CU 64.927786
30 UHHK 65Zn 64.929230
42 MOnH6,AeH 95Mo 94.905854
49 HHAHH 113In 112,904089
H5In 114.903871
50 Onoso 1128n 111.904835
1138n 112.905180
1148n 113.902773
92 Yp8H 235U 235.11750
238U 238,12376
64'/

13. IIepHO,lJ;LI nonypacna,lJ;a pa,lJ;HOaKTHBHLIX H30TonOB
AKTHHHH 2: Ac 10 CYTOK P8,AHH 2 R8 1620 neT
Mo,n 1I 8 CYTOK Pa,nOH 2Rn 3,8 ,AHSI
MpH,AHH lIr 75 CYTOK CTPOHUHH gSr 28 neT
K068JIbT Co 5.3 rO,A8 TopHH 2Th 7 000 neT
M8rHHH Mg 10 MHHYT Yp8H 2U 7,1 . 10 8 neT
TInYTOHHH 2pu 24,4 rO,A8 Yp8H 2U 4.5' 10 9 neT
Pa,AHH 2: R8 10- 3 C <t>OC<}>OP p 14.3 CYTOK
14. nOrJlOeHHaH ,lJ;03a H3J1yqeHHH
,[(038 ,[(038
MCTOqHHK H3nyqeHH$I H3nyqeHHJI. HCTO'lHHK H3nyqeHHJI H3nyqeHH$I,
rp 10- 6 rp
10- 6 - -
ro,n ro,n
Me,nHUHHCK8S1 ,AH8rHOCTHK8 1500 Pa,nH08KTHBHbIe npO,AYKTbI
CTpoHTenbHbIe M8TepH8nbl SllleDHbIX B3DbIBOB 25
(npe6bIB8HHe B nOMelI{eHHSlx) 1050 TeneBH,AeHHe (4 'l8C8 B ,AeHb) 10
ECTecTBeHHbIH p8,AH8UHOHHbIH YronbHbIe 3neKTpOCT8HUHH 2
d>OH 1000 AToMHble 3neKTpOCT8HUHH 0.2
llpUMe'laH.Ue. B Ta6JIHe npHBe,neHbl opHeHTHpOBOqHbIe 3H8qeHH$I MOll\HOCTH nornoeH-
HOH ,ZJ.03bI peHTreHOBCKoro, HJIH 'Y-H3JIyqeHHJI, KOTOPYIO nOJIyqaeT opr8HH3M qenOBeK8 B Te'le-
HHe ro,n8 OT p83nHqHbIX HCTOqHHKOB.
15. BHOJlOrHqeCKOe 1I;eiiCTBHe pa3J1HqHLIX nOrJlOIU,eHHLIX 11;03 H3.nyqeHH.H
,[(038 A038
H3JIyqeHH$I. IIocne,ncTBHJI 06nyqeHHJI H3JIyqeHH$I. IIocne,ncTBHJI 06JIyqeHH$I
rp rp
,AO 0,25 K8KHx-nH60 nOCne,ACTBHH B 2.5 + 4 JIY'leB8S1 60ne3Hb Cpe,AHeH TSlmec-
COCTOSlHHH 3,AOpOBbSl 'leno- TH
BeK8 He 06H8DV>KHB8eTCSI
0,5 + 1 He3H8'lHTenbHbIe 06p8THMbIe TSlmen8S1 <}>OpM8 nY'leBOH 6one3HH;
npou.eCCbI 4+6 6e3 neqeHHSI B03MomeH CMepTenb-
HbIH HCXO,A (B 50 % cnYQ8eB)
1 + 2,5 JIerK8SI <}>OpM8 nyqeBoH 60- 6+ 10 Kp8HHe TSlmen8.R cpOpM8 nY'leBOH
ne3HH 60ne3HH
llpUMe'laH.Ue. IIOrnOll\eHH8$1 11038 H3nyqeHH$I. npHBo,nJIll\8$1 B 50 % cnyq8eB K rH6eJIH 06.ny-
qeHHbIX opr8HH3MOB. p8BH8 (B rp3ftX):
,nn$l KpynHoro poraTOro CKOT8
nTH
rpbI3YHOB
1.5+2.7
6+ 14
8+ 15
HaceKOMbIX
BHPYCOB
68KTepHH
580 + 2000
62 + 4600
17 + 3500
648

IIPHJIOJKEHHE
TECT I
At. rp8<1>HKH 38BHCHMOCTH KOOp,AHH8Tbl ,ABYX Ten OT BpeMeHH npe,ACT8B-
neHbI H8 pHCYHKe 1. KHHeM8TH'leCKHe yp8SHeHHSI ,ABHmeHHfl 3THX Ten HMe-
lOT BH,A:
X.M
180
120
60
20 40 60 t. C
Puc. 1
1) Xl =60 + t; X2= 180 - 2t;
2) Xl = 60 - t; X2 = 180 + 2t;
3) Xl = t; X2= 180 - 2t;
4) xl=t; x2=120-2t;
5) xl = 2t; X2= 120 + 2t.
A2. TInoTHocTb YP8H8 p= 19,10
r
CM 3 '
MonSlp-
H8S1 M8CC8 M = 0,238
Kr
EcnH B KYCKe YP8H8
Monb
CO,AepmHTCfl N = 1,44' 10 24 MoneKyn, TO ero 06'beM
V p8seH:
1) 36 CM 3 ; 2) 30 CM 3 ; 3) 24 CM 3 ;
4) 18 CM 3 ; 5) 12 CM 3 .
A3. ,nB8 TO'leQHblX 38pSl,ll;8 ql =2,0 HKn H q2 = 4,0 HKn H8XO,ll;SlTCSI H8 p8CCTO-
SlHHH r = 20 CM ,Apyr OT ,Apyr8. ,nnSl Toro 'lT06bl TpeTHH TO'le'lHblH 38pSl,A H8-
XO,ll;HnCSI B p8BHOBeCHH, ero H8,l1;O nOMeCTHTb H8 p8CCTOSlHHH riOT nepBoro
38pSl,A8, p8BHOM:
1) 12 CM; 2) 10 CM; 3) 8.0 CM: 4) 6.0 CM: 5) 4.0 CM.
A4. EcnH 8HO,AHbIH TOK 1=1,0 MA, TO KOnHQeCTSO 3neKTpOHOS n, emece
KYH,AHO BbIneT81OIl\HX H3 K8TO,ll;8 aneKTpOHHOH J18Mnbl, p8BHO:
1) 7,1' 1015; 2) 6,3' 1015; 3) 6,0'1014; 4) 3,8' 1015; 5) 2,6' 10 15 .
A5. B HenO,ABHmHOH K8TYlliKe 38 BpeMSI I!!t = 0,20 C 3HeprHSI M8rHHTHoro
nonSl YMeHblllHn8Cb B k = 4 p838. MH,AYKTHBHOCTb K8TYlliKH L = 0,16 rHo H8-
Q8nbH8S1 CHn8 TOK8 1 1 = 8,0 A. a,nc C8MOHH,lI;YKII.HH t B K8TYlliKe paSH8:
1) 3.2 B: 2) 3.0 B: 3) 2.8 B: 4) 2.6 B: 5) 2.4 B.
A6. EcnH npH OnTH'leCKOH p83HOCTH XO,A8 ,ABYX KorepeHTHblX sonH
I!!r= 1,40 MKM OHH 06p83YIOT ,AH<PP8KII.HOHHbIH M8KCHMYM BToporo nOpSl,AK8, TO
,AnHH8 BonHbI A p8SH8:
1) 700 HM; 2) 600 HM; 3) 550 HM; 4) 500 HM; 5) 400 HM.
A 7. EcnH penSlTHSHCTCKoe COKp8Il\eHHe ,AnHHbl ,ABHmYIl\erOCSI Ten8 COCT8B-
nSleT 11 = 25.0 %, TO CKOpOCTb v Ten8 p8BH8:
1) 1,81 10 8 ; 2) 1,65.10 8 ; 3) 1.98.10 8 ; 4) 2,35'10 8 ;
C C C C
5) 2,58.10 8 M .
C
649

A8. H8H60nblIIaSl p;nHHa BOnHbI Amax CneKTp8nbHOH nHHHH B SH,AHMOH
06naCTH CneKTp8 8TOMapHOrO BO,AOpO,Aa p8BHa (nOCTOSlHH8S1 PH,A6epr8
R = 3,29' 10 15 C- 1 ):
1) 4,17. 10- 7 M; 2) 5,83. 10- 7 M; 3) 6,21' 10- 7 M; 4) 6,57' 10- 7 M;
5) 6,84' 10- 8 M.
A9. M8cca Hp;pa H30Tona 1Li mil = 7,0160 8.e.M. ECJIH MaCCbI npOTOH8
H HeHTpoHa COOTBeTCTBeHHO mp = 1,0076 8.e.M., m n = 1,0090 a.e.M., a
1 a.e.M. = 1,66054 x 10- 27 Kr, TO 3HeprHSI CBH3H WCB SI,Apa H30TOn8 Li p8BHa:
1) 6,8' 10- 12 ,l(m; 2) 6,2' 10- 12 ,l(m; 3) 5,4' 10- 12 ,l(m; 4) 4,7' 10- 12 ):{m;
5) 3,8. 10- 12 ):{m.
AIO. Teno na,A8eT 6e3 H8'l8nbHOH CKOpOCTH C SbICOTbI hI = 5,0 M. OAHO-
BpeMeHHO C HHM C BbICOTbl h 2 = 20 M 6pOC81OT BepTHK8nbHO BHH3 ,Apyroe Teno.
EcnH 068 Tena ynaJIH H8 3eMnlO O,AHOBpeMeHHO, TO H8'laJIbH8fl CKOpOCTb V2 BTO-
poro Tena p8BHa:
M M M M M
1) 5,0 c; 2) 7.5 c; 3) 10 c; 4) 15 c' 5) 20 C'
All. rOpH30HTaJIbHaH nJIaT<popMa ,AnHHOH l = 2 M C nOKOHHMCH Ha ee KpalO
rpY30M p;B1:IraJIaCb co CKOpOCTblO VI 1:1 BHe3anHO OCTaHOBHnaCb. B MOMeHT COCKaJIb-
3bIBaHHSI rpY38 C npOTHBOnOnOmHOrO KpaSl nnaT<popMbI ero CKOpOCTb V2 B Tpl1
p838 MeHblIIe nepBOH8'laJIbHOH. ECnH K03<P<PHIJ;HeHT TpeHHH Memp;y rpY30M H nn8T-
<l>OPMOH Jl = 0,2, TO CKOpOCTb VI nn8TcpopMbI nepe,A OCTaHOBKOH p8BH8:
M M M M M
1) 1 C; 2) 2 C; 3) 3 c; 4) 4 C; 5) 5 C'
ma
Puc. 2
A12. CHCTeM8 H3 Tpex rpY30B (pHC. 2) Ha-
XO,AHTCfl B p8BHOBeCHH. TpeHHe OTCYTCTByeT.
M8CCbI ml H m2 rpY30B H yroJI a HaKnOH8
nnOCKOCTH K ropH30HTY H3BeCTHbI. EcnH yron
Memp;y BepTHKanblO Ii HHTblO, CBSl3bIB81OeH
rpY3bI M8ccaMH ml H m2, p8BeH p, TO M8cca
m3 TpeTbero rpY3a p8BHa:
1) mlsin(a - p) - m2sina;
2) mlsina - m2sin(a - p);
3) mlsin(a - p) - m2cos(a - p);
4) mlsin(a - P) - m2cosp;
5) mlcosa- m2cosp.
A13. ECJlH Cpep;H.HSI KBa,ApaTH1iHaSl CKOpOCTb MoneKyn Hp;e8nbHoro ra38
(V KB ) = 300  , a ero DnOTHOCTb p = 1,3 K , TO p;aBJIeH1:Ie p r83a paBHO:
M
1) 29 £Ia; 2) 29 KIIa; 3) 9,0 KIIa; 4) 39 KIIa; 5) 45 KIIa.
A14. B BepTHKanbHOM IJ;HnHH,Ape no,A noplIIHeM H8XO,AHTCSI B03,AYX no,A
,A8BneHHeM PI = 200 KII8 npH TeMnep8Type t 1 = 27 °C. IInO8,Ab noplIIHSI
S = 30 CM 2 . QT06bI nOCJIe HarpeB8HH.H B03,Ayxa Ha 11 T= 23 K ero 06'beM He
H3MeHHncSI, Ha noplIIeHb H8,A0 nOJIOmHTb rpY3 MaCCOH m, p8BHOH:
650
1) 2,0 Kr; 2) 2,5 Kr; 3) 3,0 Kr; 4) 3,5 Kr; 5) 4,6 Kr.

A15. ECnH nOTeHH8JI B eHTpe 3apSlmeHHOH H30nHpOBaHHOH CepepbI q> = 100 B,
8 Ha paCCTOSlHHH r = 30 CM OT eHTpa nOTeHHan q>1 = 50 B, TO P8AHYC cepe-
pbI R p8BeH:
1) 0,02 M; 2) 0,08 M; 3) 0.15 M; 4) 0,25 M; 5) 0,28 M.
A16. ,IJ:B8 KOH,AeHC8TOpa eMKOCTHMH C I = 100 n<1> H C 2 = 200 n<1> COe,AHHe-
HbI nOCne,AOBaTenbHO H 3apSlmeHbI ,AO p83HOCTR nOTeHHanOB t.q> = 600 B. KOH-
,AeHC8TOpbl, He p83pSlm8S1, p83'beAHHSlIOT H COe,AHHSlIOT O,AHOBpeMeHHO 38pHmeH-
HbIMH nnaCTHH8MH. P83HOCTb nOTeHHanOB t.q>h YCTaHOBHBW8.HCSI npH COe,AH-
HeHHH, p8BH8:
1) 267 B; 2) 221 B; 3) 211 B; 4) 187 B; 5) 133 B.
A17. KOnH'leCTBO N 3neKTpon8Mn, BKnlO'leHHbIX napannenbHO Ii paCC'lH-
TaHHbIX H8 H8npSlmeHHe U = 100 B, MOHOCTblO P = 50 BT K8m1l8.H, npH
nHTaHHH HX OT 68T8peH C 9JI:C lC= 120 B H BHYTpeHHHM conpOTHBneHHeM
r= 10 OM paBHO:
1) 3; 2) 4; 3) 5; 4) 6; 5) 7.
A18. KBa,ApaTHaSl p8MKa co CTOpOHOH a = 5,0 CM nO,ABeWeHa B MarHHTHOM
none TaK, 'lTO HopManb K HeH COCTaBnHeT yron q> = 90 0 C nHHHSlMH MarHHT-
HOH HH,AYKHH. CHn8 TOK8 B p8MKe 1 = 10 A, HH,AYKliSl MarHHTHoro nonSl
B =0,10 Tn. MOMeHT M CHn, ,AeHCTBYIOHX H8 P8MKY. paBeH:
1) 1,5,10- 3 H'M: 2) 2.5.10- 3 H'M: 3) 3.5.10- 3 H'M: 4) 4.5.10- 3 H'M;
5) 5.5 '10- 3 H'M.
A19. BHTOK. Orp8HHqHB81OHH nOBepXHOCTb nnOa,AblO S = 2,0 CM 2 , p8C-
nonomeH nepneH,AHKynSlpHo K nHHHSlM HH,AYKHH O,AHOpO,AHOrO MarHHTHoro
nonSl. EcnH 38 BpeMSI t.t = 0,050 C M8rHHTH8S1 HH,AYKHSI p8BHoMepHo y6bIB8-
eT OT B 1 =0,50 Tn ,AO B 2 = 0,10 Tn, TO 9,IJ:C HH,AYKHH g; B BHTKe p8BHa:
1) 1,9 MB; 2) 1,6 MB; 3) 1,4 MB; 4) 1,2 MB; 5) 1,1 MB.
A20. KaTywKa HH,AYKTHBHOCT1:I BKnlOQeHa B enb nepeMeHHoro TOK8 C '18-
CTOTOH V = 50 r. AMnnHTY,AHoe 3H8QeHHe CHnbI TOK8 1 0 = 2,5 A. llpoMemy-
TOK BpeMeHH t.t, 'lepe3 KOTOpbIH nocne npOXOm,AeHHSI Qepe3 HyneBoe 3H8'leHHe
CHn8 TOK8 6Y,AeT paBH8 1,25 A, COCTaBnSleT:
1) 0,80 MC; 2) 1,0 MC; 3) 1,2 MC; 4) 1,7 MC; 5) 2,2 MC.
A2l. P8CcToSlHHe Bllonb OCH Mem,Ay npellMeTOM Hero npSlMbIM H306p8me-
HHeM l = 5.0 CM. EcnH nHHeHHoe YBenH'leHHe r = 0,50, TO epOKYCHoe paCCTOH-
HHe F nHH3bI p8BHO:
1) 30 CM; 2) 25 CM; 3) 20 CM; 4) 15 CM; 5) 10 CM.
A22. HMnynbc p epoToHa C QaCTOTOH v = 3.0. 10 15 r p8BeH:
6.6. 10- 27 Kr'M 6,8. 10- 27 Kr'M 7,2 . 10- 27 Kr'M
1) , 2) , 3) ,
C C C
7,5 . 10- 27 Kr'M 7,8' 10- 27 Kr.M
4) , 5)
C C
651

A23. 1118pHK M8CCOH m = 50 r, npHKpenneHHbIH K npymHHe, Bp8ll\8eTCSI.
CKonb3S1 no rJl8,AKOH ropH30HT8nbHoH nOBepXHOCTH. JKecTKocTb npymHHbI
H
k = 350 M ' AnHH8 B HeAe<l>opMHpoB8HHOM COCTOSlHHH lo= 30 CM. EcnH npY)KH-
H8 y,AnHHHn8Cb H8 1:1l = 5,0 CM, TO 'l8CTOT8 v Bp8ll\eHHSI mapHK8 p8BH8:
1) 6.0 c- 1 ; 2) 5,5 c- 1 ; 3) 5,0 c- 1 ; 4) 4,5 c- 1 ; 5) 4.0 c- 1 .
A24. TInoTHocTb BO,AbI p = 1
r
CM 3 ' nnOTHOCTb np06KH Pnp = 0,2
r
CM 3 ' TIp06-
KOBbIH m8pHK, nOABemeHHbIM B B03,Ayxe K HeBecoMoH npymHHe, p8CTSlrHBaeT
ee H8 1:1l = 1 CM. EcnH 3T8 npymHH8, 38KpenneHH8S1 H8 AHe, 6Y,AeT y,AepmH-
B8Tb TOT me m8pHK B BO,Ae T8K, 'lT06bI OH nonHOCTblO 6bIn norpymeH B
BO,Ay, TO OHa p8CTSlHeTCSI H8 1:1l" p8BHoe:
1) 1 CM; 2) 2 CM; 3) 3 CM; 4) 4 CM; 5) 5 CM.
A25. LI:H8rp8MM8 II.HKnH'leCKOrO npOII.eCC8 ,AnSI v = 0,2 Monb H,AeanbHOro r83a
B KOOp,AHHaT8X (P. T) 06p830B8H8 npHMbIMH, COe,AHHSlIOll\HMH TOtJKH (24,9 KTI8,
180 K), (24,9 KTI8, 270 K) H (49,8 KTI8, 270 K). P83HOCTb 1:1 V Mem,Ay
M8KCHM8nbHbIM H MHHHM8nbHbIM 3Ha'leHHSlMH 06'beM8 r838 paBHa:
1) 4,5 ,AM 3 ; 2) 6,0 ,AM 3 ; 3) 7.5 ,AM 3 ; 4) 9,0 ,AM 3 ; 5) 13,5 ,AM 3 .
A26. renHH B KOnH'leCTBe v = 1,0 Monb H3 COCTOSlHHSI C TeMnep8TYPoH
T] = 98 K p8cmHpSleTCH T8K. tJTO ero TeMnep8Typ8 H3MeHSleTCSI no 38KOHY
T = av2, rIle a - nOCTOSlHH8.H BenH'lHH8. EcnH a6conlOTHaSl TeMnepaTypa YBeJIH-
'tiHB8eTCSI B k = 4 P838. TO renHH COBepm8eT p860TY A, P8BHYIO:
1) 0,40 KLI:m; 2) 0,60 KL{m; 3) 1,2 KL{m; 4) 2,5 KL{)K; 5) 3,7 KLI:m.
A27. B cocY,Ae, H3 KOToporo BbIK8'lHB8eTCSI B03,AYX, H8XO,AHTCH He6onbmoe
KonHtJeCTBO BO,AbI npH T = 273 K. 38 ctJeT HHTeHCHBHoro Hcn8peHHSI npOHCXQ-
KLI:m
AHT 38Mop8mHB8HHe BO,AbI. Y,AenbH8S1 TennOT8 nn8BneHHSI nbA8 A = 340
Kr
YIlenbH8fl TennOT8 napoo6p830BaHHSI BO,AbI L = 2.26
M,I(m
Kr
TIpH 3TOM B nell
m
npeBp8ll\8eTCSI 'l8CTb BO,AbI  paBH8S1:
m
1) 0,91; 2) 0.89; 3) 0,87; 4) 0,85; 5) 0,83.
A28. .LJ:B8 38pSlmeHHbIX mapHKa C O,AHH8KOBbIMH P8l\lIYC8MH H M8CC8MH
nO,ABemeHbI H8 HHTSIX 0IlHH8KOBOH IlnHHbI H onymeHbI B >KHIlKHH jIH3neKT-
pHK, nnOTHOCTb KOToporo P H D;H3neKTpH'leCK8S1 npOHHU8eMOCTb E. EcnH yron
p8CXOm,AeHHSI HHTeH B B03Ayxe H AH3neKTpHKe O,AHHaKOB, TO nnOTHOCTb PI
M8TepH8n8 m8pHKOB p8BH8:
1) EP' 2) . 3) ( E - 1 ) p; 4) (E - 1) p . 5) EP
, E-1' E E+1
652

A29. MaSlTHHK C nepHO,AOM Kone6aHHH T = 1,0 C npe,ACTaBnSleT C060H W8pHK
M8CCOH m = 5,0 r, nO,ABeWeHHbIH Ha HenpOBO,ASll.I..\eH HHTH. 1lI8PHKY C0061.I..\81OT OTpH-
8TenbHbIH 38pSl,A q = -10 HEn H nOMel.I..\alOT B sneKTpOCTaTH'leCKOe none, nHHHH
HanpSlmeHHocTH KOToporo Hanp8BneHbI BepTHK8nbHo BBepx. TIepHon Kone58HHH
M8S1THHK8 CT8JI Tl = 0.95 c. H8npSlmeHHocTb E sneKTpH'leCKOrO nonSl p8BH8:
MB MB MB
1) 0,54 -; 2) 0,51 -; 3) 0,49 -; 4) 0.43
M M M
MB
MB
M
5) 0,41
M
A30. B cxeMe, npHBe,AeHHOH H8 pHCYHKe 3, 8,lI,C H BHYT-
peHHee conpOTHBneHHe nepBoro HCTO'lHHK8 paBHbJ COOTBeT-
CTBeHHO g'l = 2,0 B H rl = 1,0 OM. BToporo HCTO'lHHKa -
g'2=1,0 B H r2=0,50 OM. BHeWHee conpOTHBneHHe
R=3,5 OM. CHn8 TOK8 I B L\enH p8BH8:
1) 0,2 A: 2) 0.4 A; 3) 0.5 A; 4) 0.7 A: 5) 0.9 A.
Puc.3
A3I. 8neKTpoH, KHHeTH'leCK8S1 3HeprHSI KOToporo
W K = 1,2.10 3 sB. ,ABHmeTCSI no oKpymHocTH B O,AHOpO)J.-
HOM MarHHTHOM none C HHAYKHeH B = 50 MTn. Pa,AHYc R Tp8eKTopHH sneK-
TpOHa p8BeH:
1) 1,9 MM; 2) 2,3 MM; 3) 2,8 MM; 4) 3,1 MM; 5) 4,1 MM.
A32. AMnnHTY,A8 r8pMOHH'leCKHX Kone68HHH M8TepH8JIbHOH TO'lKH A = 2,0 CM,
nOJIH8S1 SHeprHSI ee Kone6aHHH W = 0,30 MK,lI,m. H8 MaTepH8nbHYIO TO'lKY ,AeH-
cTByeT CHn8 F = 2,25.10- 5 H npH CMel.I..\eHHH x OT nonomeHHSI p8BHoBecHSI,
paBHoM:
1) 0,015 M; 2) 0,025 M; 3) 0.035 M; 4) 0,045 M; 5) 0,055 M.
A33. Ha ,AHe PY'lbSl JIemHT KaMeweK. Manb'lHK XOqeT TonKHYTb ero nan-
KOH. TIpHL\enHBaSlcb, OH ,AepmHT n8nKY no,A yrnoM <p = 45°. IIOK838Tenb npe-
nOMneHHSI BO,AbI n = 1,33. EcnH rny6HH8 PY'lbSl h = 50 CM, TO nanKa BOTKHeT-
CSl B ,AHO PY'lbSl H8 p8CCTOSlHHH S OT K8MeWK8, p8BHOM...
1) 19 CM; 2) 17 CM; 3) 15 CM; 4) 13 CM; 5) 10 CM.
K8Tep npOXO,AHT p8CCTOSlHHe MemLIY LIBYMSI npHCT8HSlMH H8 peKe
CTOpOHbI 3a 6.t = 1,4 'I. EcnH CKOpOCTb K8Tep8 B CTOSl'leH BO,Ae
KM KM
1.1 = 35, 0 8 CKOpOCTb Te'leHHSI peKH V2 = 5,00 -, TO p8C-

BI.
B 06e
ml
m2
Puc. 4

CTOSlHHe S MemAY npHcTaHSlMH paBHo
(KM).
t ii
B2. ABToM06Hnb MaccoH m = 2,0 T, cpeAHSISI MOI.I..\HOCTb
MOTopa KOToporo N = 40 KBT, TporaeTCSI C MeCTa H ,ABHmeTCfl
KM
paBHOYCKopeHHO. CKOPOCTH v = 72 aBToMo6Hnb ,AOCTHraeT

'lepe3 BpeMSI t, p8BHoe ... (c).
B3. Qepe3 HeBeCOMbIH 6nOK (pHC. 4). ,ABHmYI.I..\HHCfl C YCKO-
CM
peHHeM a = 200 ' H8np8BneHHbIM BBepx, nepeKHHYTa HHTb
653

C rpY3aMH, MaCCbI KOTOpbIX ml =m H m2=3m, 3aKpenneHHblMH H8 ee KOH8X.
ECJlH CHna ,AaBneHHH Ha OCb 6J10Ka F = 28.8 H. TO MaCC8 ml MeHbWerO rpy-
3a paBHa ... (r).
B4. K HCTO'1HHKY TOKa C SHYTpeHHHM conpOTHsneHHeM r = 2 OM BHa-
'laJle nO,AKnlO'laeTCSI pe3HCTOp C conpOTHBneHHeM R = 2 OM. 3aTeM n8p8JI-
nenbHO nO,AKnlO'1aeTCSI TaKOH me pe3HCTOp. KOnHqeCTBO TennOTbl Q, BbI-
,lI;eJIHeMOH B nepBOM pe3HCTope nocne nO,AKnlO'leHHSI BToporo pe3HCTopa, 3R
TaKoe me BpeM.H YMeHbwaeTCH B . . . p83.
B5. KOH,AeHC8TOp eMKOCTblO C = 2,00 MK<1> 3apSl,AHnH OT IfCTO'1HHKa C Ha-
npSlmeHHeM U = 100 B. 8 38TeM aaMKHynlf H8 KaTYWKY C HH,AYKTHBHOCTblO
L = 5,00 MrH. 'tJepe3 BpeMSI t = 0,625 1t MC nocne 3RMblKaHHSI 3apSl,A q Ha KOH-
,AeHCaTOpe paBeH ... (MKKn).
B6. He60JlbWOe Teno COCKanb3bIBaeT C BepxHeH TO'lKH nOJIyct}>epH'1eCKOH
rna,AKOH nOBepXHOCTH. EcnH OHO OTpbIsaeTCSI OT nOBepXHOCTH H8 SbICOTe
h = 20 CM OT BepWHHbl nonyc<I>epbl, TO pa,AHYC R
nonyc<I>epHqeCKOH nOBepXHOCTH paBeH ... (CM).
V
V 3
----------- 3
1 2
1
1
B7. Ha V-T-,AH8rpaMMe (pHC. 5) H306pameH
HKn, cOBepwaeMblH H,AeaJIbHblM O,AHOaTOMHblM ra-
30M, B3.HTbIM B KOJlH'1eCTBe v = 1,0 MOJlb. B npo-
ecce H30TepMH'leCKOrO paCWHpeHH.R ra3 COBepWHn
p860TY A = 9,0 KAm. B Ha'l8nbHOM COCTOSlHHH
TeMnepaTypa TI = 670 K. EcnH COOTHoweHHe o6'he-
v:
MOB B COCTOSlHHSlX 3 H 2 paBHO ....!. = 2, TO KTIA
V 2
VI
1
/1
/ 1
// 1
o
Tl
Puc. 5
T 2 T
TJ HKJla paBeH ... (%).
654

TECT II
2
AI. I'p8<pHKH a8BHCHMOCTH KOOp,AHH8TbI x
,ABYX Ten OT BpeMeHH npe,ACTSBneHbl H8 PH-
CYHKe 6. H8 CKonbKO OT nH'l8eTCSI npOeKl'lHSI
CKOpOCTH v x 1 nepBoro Ten8 OT npoeKUHH
CKOpOCTH v x 2 SToporo Ten8?
X,M
6
4
o
2
4
6
t, C
1) Bonbme 2.5 M
H8 -'
,
C
2) 2,5 M
MeHbme HS
C
3) OlIHHSKOBbIe;
M
4) 60nbme HS 1,5 -;
C
M
5) MeHbme H8 1,5
C
Puc. 6
A2. ECJIH MonHpH8fl M8CC8 KHCnOpO,As p8BHS M = 32
r
, TO MSCCS mo
Monb
MoneKynbI KHCnOpO,AS pSBH8:
1) 5,3. 10- 26 Kr; 2) 5,3' 10- 25 KI'; 3) 5,3' 10- 24 Kr; 4) 5,3' 10- 23 KI';
5) 5,3. 10- 22 KI'.
A3. .nSS M8neHbKHX IIlSpHKS M8CCOH no m = 1.0 r KSm,AbIH HSXO,ASlTCSI B
S03,Ayxe. 3spSl q, KOTOpbIH HymHo co06mHTb m8pHKY nSl Toro, tJT06bI CHn8
S38HMHoro OTT8nKHS8HHSI 38pSl,AOS YP8sHOBecHn8 rp8BHT8l'lHOHHYIO cHny B38-
HMHoro npHTSI)KeHHSI W8pHKOB, p8seH:
1) 7,8' 10- 14 En; 2) 8,0. 10- 14 En; 3) 8,6. 10- 14 En; 4) 9,2. 10- 14 En;
5) 9,8. 10- 14 Kn.
A4. TIoTeH1J;HSJI HOHH381J;HH 8TOMS renHSI <PI = 24,5 B. P860T8 HOHH3S1J;HH
A, pSBHS:
1) 35,8. 10- 19 .nm; 2) 36,2. 10- 19 .n)K; 3) 37,2. 10- 19 .nm;
4) 38,1. 10- 19 .nm; 5) 39.2.10- 19 .nm.
AS. 8neKTpoMsrHHT C HH.n:YKTHSHOCTblO L = 50 rH IIO,AKnlO'leH K HCTO'lHH-
KY TOK8. EcnH CHn8 TOK8 B l'leDH y6bIB8eT co CKOpOCTblO  = 8,0  TO
8.nC caMOHH.n:YK1J;HH g'sl pSBHS:
1) 60 B; 2) 50 B; 3) 40 B; 4) 30 B; 5) 20 B.
A6. TIepBbIH ,AH<PPSK1J;HOHHbIH M8KCHMYM, nOJIyqeHHbIH C nOMOblO ,AH<p-
p8Kl'lHOHHOH pemeTKH C nepHO,AOM d = 0,0020 MM, OTCTOHT H8 3Kp8He H8 p8C-
CTOSlHHH X = 30,0 CM OT 1J;eHTp8nbHoro M8KCHMYM8. EcnH p8CcToSlHHe OT pe-
meTKH ,AO 3Kp8H8 l = 1.50 M, TO ,AnHH8 BonHbI A n8,A81OerO H8 pemeTKY
CBeTS (npHMHTe ,Anfl M8nbIX yrnoB sina =tga) p8BHS:
1) 400 HM; 2) 450 HM; 3) 500 HM; 4) 550 HM; 6) 600 HM.
655

A7. KHHeTH'leCKaSl SHeprHSI penHTHBHCTCKOrO SneKTpOHa 6Y,AeT paBHa
mc 2
2
(c CKOpOCTb CBeTa B B8KYYMe) npH CKOpOCTH v. p8BHOH:
1 .J2 .J2 ..J3 .J5
1) gC; 2) TC; 3) gC; 4) TC; 5) gC.
A8. 8HeprHSI 8TOM8 SO,AOpO,A8 B OCHOBHOM COCTOSlHHH WI = -13,55 3B. MH-
HHM8nbH8S1 SHeprHfl W. KOTOpylO onmHbI HMeTb sneKTpOHhI, nH Toro tITo6bI
npH B036ym,AeHHH 8TOMOB BOAOPO,A8 YAap8MH STHX sneKTpOHOB cneKTp BOAOPO-
,Aa HMen TonhKO O,AHY cneKTpanhHYIO nHHHIO, paBHa:
1) 8,4 sB; 2) 8,7 sB: 3) 9,5 sB: 4) 9.8 sB; 5) 10.2 3B.
A9. Macc8 SI,Apa ,AeHTepHSI H mJl= 2.0141 8.e.M. EcnH M8CCbI npOTOH8
H HeHTpoH8 p8BHbI COOTBeTCTBeHHO m p =1.0076 8.e.M.. m n =1,0090 a.e.M.,
TO Ae<}>eKT M8CCbI 11m SI,Apa ,AeHTepHSI H paBeH:
1) 0.0018 a.e.M.; 2) 0,0205 a.e.M.; 3) 0,0315 a.e.M.; 4) 0.0418 a.e.M.;
5) 0.0503 a.e.M.
A10. Teno. ABHraHCL BAOJIh OCH OX C YCKopeHHeM, npoeKqHSI KOToporo
M M
a x =-0,50 c 2 ' OCT8H8.BnHB8eTCSI. EcnH npoeKqHSI H8'l8nhHOH CKOpOCTH vo x =5,0
C
M
H8'l8nbH8S1 KOOp,AHHaT8 Xo= 2.0
C
TO BpeMSI t ,ABHmeHHSI, npOH,AeHHhIH nYTb S
H KOHe'lHaSl KOOp,AHH8T8 x paBHhI COOTBeTCTBeHHO:
1) 10 c; 25 M; 27 M; 2) 10 c; 25 M; -23 M; 3) 10 c; 25 M; 25 M;
4) 1 c; 2,5 M; -0.5 M; 5) 1 c; 2,5 M; 2,5 M.
All. C rophI, ,AJIHHa CKnOHa KOTOpOH l = 50 M 11 BbICOTa h = 10 M, Ha
sepeBKe H8'lHH81OT cnYCK8Tb C8HKH MaCCOH m = 60 Kr, COXP8HSISI cHny H8TSI-
meHHSI BepeSKH nocToSlHHOH. Y nO,AHO>KHSI ropbI CKOpOCTh caHOK v = 5,0 M .
C
EcnH MO,Aynb CHnbl TpeHHSI Mem,Ay caHKaMH H nosepXHOCTblO ropbl COCT8BnSl-
eT T1 = 10 % CHnhl TSlmeCTH caHOK, TO CHn8 HaTH>KeHHSI F BepeBKH paBHa:
1) 75 H; 2) 60 H; 3) 45 H; 4) 30 H; 5) 20 H.
A12. ,D;Ba mapHKa pa,AHycaMH R 1 =2,0 CM H R 2 =3,0 CM 1:1 MaccaMH
ml = 100 r H m2=400 r COOTBeTCTBeHHO coeAHHeHbI HeBeCOMbIM CTepmHeM ,Ann-
HOH l = 20 CM. [(eHTp TSlmeCTH CHCTeMbl H8XO,AHTCH OT qeHTpa MeHblllero W8-
pHK8 H8 p8CCTOSlHHH r, p8BHOM:
1) 22 CM; 2) 20 CM; 3) 17 CM; 4) 15 CM; 5) 12 CM.
HSI.H KB8,ApaTH'lHaSl CKOpOCTb ero MoneKyn B03pocna C < V KB1 > = 400
A13. TIpH nOBblllleHHH TeMnep8Typbl HAe8nbHoro r838 Ha I1T 1 = 180 K cpe,A-
M
,AO
C
656

< V KB2 > = 500
M
C
q T06bI YBenH'lHTb Cpe,[J;HIOIO KB8,ApaTH'lHYIO CKOpOCTb MOne-
M M
Kyn C < V KB2 > = 500 - ,AO < U KB > = 600 -. r83 He06xo,AIiMO HarpeTb Ha /1T 2 :
C 3 C
1) 180 K: 2) 200 K; 3) 220 K; 4) 240 K; 5) 260 K.
A14. TIp" H30TepMH'leCKOM cmaTHH ra3a ero 06'beM YMeHblllHncSI H8 /1V 1 = 2.0 n,
8 ,AaBneHHe B03pocno Ha 111 = 50%. EcnH nepBOHa'l8JIbHbIH 06'beM ra3a YMeHblllHTb
Ha /1V 2 = 1.0 n npli TOH )lte TeMnep8Type, TO ,AaBneHlie r838 YBenH'lHTCfl H8 112:
1) 10%; 2) 20%; 3) 30%; 4) 40%; 5) 50%.
A15. Pa3MepHOCTb nOTeHqHBJIa B OCHOBHbIX e,AHHliq8X CIf MomeT 6bITb npe,[J;'
CT8BneHa:
2 A 3 2 Kr . M 2
1) Kr. M ; 2) ; 3) Kr. M . 4) Kr 5)
A . c 3 Kr . M 2 A 2 . c 3 ' A . c 2 ; 
A16. ABa mapa eMKOCTSlMH C 1 =2.00 ncI> H C 2 =1,50 ncI> 38pSl>KeHbI ,AO
nOTeHqlianoB <PI = 200 B H <P2= 400 B COOTBeTCTBeHHO. TIocne COe,AHHeHHH 1118-
pOB npOBO,AHIiKOM, conpOTHBneHHeM Ii eMKOCTblO KOToporo MomHO npeHe6pe'ib,
nOTeHqHanbI <P Ha wapax paBHbI:
1) 212 B; 2) 224 B; 3) 245 B; 4) 286 B; 5) 292 B.
A17. HarpeBaTeJlb 3neKTpO'l8HHHK8 HMeeT ,ABe ceKqliH. IIpH BKnlO'ieHHH
O,AHOH H3 HHX BO,Aa B q8HHHKe 38KHn8eT Qepe3 BpeMSI t l = 15 MHHYT, npH
BKnlO'ieHHH ,APyroH - 'lepe3 BpeMSI t2=30 MHHYT. EcnH BKnlO'lHTb 06e ceK-
qHH nOCne,AOB8TenbHO, TO BO,A8 B 'laHHHKe 38KHnHT 'lepe3 BpeMSI t, p8BHoe:
1) 10 MHH; 2) 25 MHH; 3) 38 MHH; 4) 42 MHH; 5) 45 MHH.
AIS. IIo ropH30HT8nbHOMY npOBO,AHHKY ,AnHHOH l = 20 CM H M8CCOH m = 2 r
npoxo,[J;IiT TOK 1=5 A. IfH,[J;YKqHSI B MarHIiTHoro nonH. B KOTopoe HY>KHO
nOMeCTHTb npOBO,AHIiK, 'iT06bl OH BHcen, He n8,A8S1, paBH8:
1) 0,001 Tn; 2) 0.01 Tn; 3) 0,02 Tn; 4) 0,04 Tn; 5) 0,05 Tn.
A19. B KaTYlllKe, COCTOSleH H3 N = 1000 BHTKOB, npli paBHOMepHOM HC'le3-
HOBeHli1i MarHIiTHoro nonSl B Te'leHHe /1t = 0,1 C B03HHKaeT SAC HH,AYKlJ,HH
g,= 10 B. B H8'l8nbHbIH MOMeHT BpeMeHH MarHIiTHbIH nOTOK <I> Qepe3 nOBepx-
HOCTb, OrpaHli'leHHYIO O,[J;HHM BIiTKOM. paBeH:
1) 5 MB6; 2) 4 MB6; 3) 3 MB6; 4) 1 MB6; 5) 0,5 MB6.
A20. K8TYlliKa c HH,AYKTHBHOCTblO L = 45,0 MfH H aKTHBHbIM conpOTHB-
neHHeM R= 10 OM BKnlO'leHa B qenb nepeMeHHoro TOK8 c 'laCTOTOH v = 50,0 flJ,
H HanpSlmeHlieM U = 220 B. CHna TOKa I B KaTYlliKe p8BHa:
1) 12.7 A; 2) 10,5 A; 3) 9,80 A; 4) 8,70 A; 5) 7,60 A.
A21. PHCYHOI< B KHHre IiMeeT BbICOTY h = 5,0 CM, 8 Ha 3Kp8He H = 0,95 M.
EcnH p8CCTOSlHHe OT 06'beKTHBa 3nli,AH8CKOn8 ,AO 3Kp8Ha f = 4,0 M, TO onTH-
'leCKaSl clina D 06'beKTIiB8 p8BH8:
1) 5.0 ,AnTp; 2) 4,0 ,AnTp; 3) 3,5 ,AnTp; 4) 3.0 ,AnTp; 5) 2.5 ,AUTp.
657

A22. HMnynbc P cpOTOHa peHTreHOBCKOrO H3nY'leHHSI C ,AnHHOH BOnHbl
A= 5.5.10- 8 CM paBeH:
1.4 . 10- 26 Kr'M 1,2 . 10- 26 Kr'M 1,0. 10- 26 Kr'M
1) -; 2) -; 3) -;
C C C
1.4 . 10- 24 Kr'M 1.2 .10- 24 Kr'M
4) -' 5)
.
C C
A23. P8,AHYC 3eMnH R3 = 6400 KM, YCKopeHHe cBo6oltHOrO na,AeHHSI Ha no-
M
moce g = 10,0 "2" . JJ;nSl Toro 'lT06bl Teno Ha 3KsaTope He HMeno Beca, CYTKH T
C
Ha 3eMne ,AOnmHbl 6blTb npoAonmHTenbHocTbJO:
1) 84,0 MHH; 2) 126 MHH; 3) 168 MHH; 4) 214 MHH; 5) 252 MHH.
A24. B C0068JOHecSI COCY,Abl. nnOaltH Ce'leHHSI KOTOpblX OTHOCSlTCSI KaK
r
1 : 2, H8nHT8 PTYTb (nnoTHocTb PTYTH PI = 13, 6 ) TaK, tJTO ,AO BepX8 CO-
CM
CY,AOB OCT8eTCSI h = 10,0 CM. EcnH WHpOKHH COCY,A HanonHHTb ,AOBepxy BO,AOH
r
(nnOTHOCTb BO,Abl P2 = 1,0 --s)' TO ypOBeHb PTYTH B Y3KOM cocY,Ae nO,AHH-
CM
MeTCSI Ha I1h:
1) 1,0 MM; 2) 1,3 MM; 3) 5,3 MM; 4) 1,9 CM; 5) 2,2 CM.
A25. TIpoqecc, npOHCXO,ASIHH C ,AaHHOH M8CCOH H,AeanbHOro ra3a, onHCbl-
BaeTCSI 3aKOHOM pV 2 = const. EcnH 06'heM ra3a YBenH'lHnC.H B k = 3 pa38, TO
TeMnep8Typa ra38 T:
1) He H3MeHHnacb; 2) YMeHbWHnaCb B 3 pa38; 3) YBenHqHnaCb B 3 pa3a;
4) YMeHbWHnaCb B 9 pa3; 5) YBenH'lHnaCb B 9 p83.
A26. H,Ae8nbHblH ra3 H8XO,AHTCSI no,A ,A8BneHHeM P = 100 KTI8 H H3068pHO
pacwHpHeTcSI OT t l = 17,0 °C ,AO t 2 = 104°C. EcnH nepBOHaq8nbHblH 06'heM ra38
VI = 10.0 n, TO OH COBepW8eT npH paClliHpeHHH pa60TY A, paBHYJO:
1) 100 11>K; 2) 200 11m; 3) 300 11m; 4) 200 Kllm; 5) 300 KJJ;m.
A27. 38 C'leT KOnH'leCTBa TennoTbI, BblAenHBweHcSI npH KOH,AeHc8qHH n8pa
npH TeMnepaType t l = 100°C H nocne,AYJOIQeM OXnam,AeHHH BO,Abl ,AO TeMnepa-
Typbl T 2 =273 K, paCT8S1no mJ1=8,00 Kr nb,Aa. YAenbH8S1 TennoeMKOCTb BOAbl
Am KJJ;m
c = 4200 - Y,AenbHaSl TennOTa nnaBneHHSI nb,Aa A = 340 -, Y,AenbH8SJ
Kr . K ' Kr
MJJ;m
TennOTa napo06pa30B8HHSI BO,Abl L = 2,30 -. Macca napa m n p8BHa:
Kr
1) 200 r; 2) 400 r; 3) 600 r; 4) 800 r; 5) 1,00 Kr.
A28. JJ;B8 3apSlmeHHblx wapHKa C O,AHHaKOBbIMH p8AHycaMH H MaccaMH
nO,ABeWeHbI Ha HHTHX O,AHHaKOBoH ,AnHHbI H onyeHbl B mH,AKHH AH3neKT-
pHK, nnOTHOCTb KOToporo p. IInoTHocTb MaTepHana wapHKOB Pl' EcnH yron
658

p8CXOm,AeHHSI HHTeH B B03,Ayxe H B ,AH3neKTpHKe 0,AHH8KOB, TO ,AH3neKTpH'leC-
K8S1 npOHHI.t8eMOCTb E ,AH3neKTpHK8 p8BH8:
1) P 2) 2.. 3)  4)  5)-EL
P - PI PI PI - 1 PI - P P - 1
g't
rt
R
g'2
r 2
A29. M8S1THHK C nepHO,AOM Kone68HHH T I = 1 C
npe,ACT8Bn.HeT C060H m8pHK M8CCOH m = 5 r, no,A-
BemeHHbIH H8 HenpOBO,ASllI.teH HHTH. 1lI8PHKY co06-
lltHnH OTpHI.t8TenbHbIH 38pSl,A H nOMeCTHnH B 3neK-
KB
TpOCT8TH'lecKoe none C H8npSl:>KeHHOCTblO E = 540 - .
M
H8np8BneHHoe BepTHK8nbHO BBepx. IIepHo,A Kone68-
HHH MaSlTHHK8 CT8n T 2 =0,95 c. MO,Aynb 38pH,A8
q m8pHK8 p8BeH:
1) 1. 10- 8 Kn; 2) 1. 10- 6 Kn; 3) 1 - 10- 5 Kn;
4) 1. 10- 4 Kn; 5) 1. 10- 3 Kn.
Puc. 7 A30. B cxeMe, npHBe,AeHHOH H8 pHCYHKe 7,
SAC H BHYTpeHHee conpOTHBneHHe nepBoro HCTOtJ-
HHK8 p8BHbI COOTBeTCTBeHHO g'1 = 2 B H rl = 10M, BToporo HCTO'lHHK8
g'2= 1.0 B H r2=0,50 OM. BHemHee conpOTHBneHHe R=3,5 M. H8npSlmeHHe U
H8 38mHM8X nepBoro HCTO'lHHK8 p8BHO:
1) 1,6 B; 2) 1,8 B; 3) 2,0 B; 4) 2,2 B; 5) 2,4 B.
A31. llpOTOH BneT8eT B O,AHOpO,AHOe M8rHHTHoe none C HH,AYKI.tHeH
B = 20.0 MKTn nepneH,AHKynSlpHo nHHHSlM HH,AYKI.tHH M8rHHTHoro nonSl. 38
6.t = 1000 MC npOTOH B M8rHHTHOM none COBepmHT 'lHCnO 060pOTOB N, p8BHoe:
1) 305; 2) 325; 3) 345; 4) 350; 5) 400.
A32. TO'lK8 COBepm8eT r8pMOHH'leCKHe Kone68HHSI no 38KOHY x = 50sin2t (MM).
B HeKoTopbIH MOMeHT BpeMeHH CHn8. ,AeHcTBYIOIlt8S1 H8 M8TepH8nbHYIO TO'lKY.
F = 5,0. 10- 3 H H ee nOTeHI.tH8nbH8S1 3HeprHfl W n = 100 MKAm. KHHeTH'leCK8S1
3HeprHSI TO'lKH W K B 3TOT MOMeHT BpeMeHH p8BH8:
1) 0,56. 10- 3 JJ:m; 2) 0.56. 10- 4 JJ:m; 3) 4.5. 10- 4 Am;
4) 2,5. 10- 4 JJ:m; 5) 1.3. 10- 4 JJ:m.
A33. P8CcToSlHHe B B03,Ayxe OT n8MnbI ,AO nOBepxHocTH BO,AbI d = 2.2 M.
IIoK838Tenb npenOMneH1:ISI BO,AbI n = 1,33. EcnH H86nlO,A8Tenb HaXO,AHTCSI H8
rny6HHe h = 60 CM B BOAe no,A n8MnOH, TO OH YBH,AHT n8Mny H8 p8CCTOSlHHH I
OT ce6S1, p8BHOM (C'lHT8HTe, 'lTO npH M8nbIX yr-
JI8X sinn:::: tga):
1) 3,5 M; 2) 3 M; 3) 2,8 M; 4) 2.2 M;
5) 1,6 M.
Puc. 8
Bl. Koneco K8THTCSI 6e3 npOCK8nb3bIB8HHSI
no ropH30HT8nbHoH opore co CKOpOCTblO
M
V = 10 OTHOCHTenbHO 3eMnH (pHC. 8). CKO-
C
659

pOCTb TO'lKH 1 060,ll8 KoneC8 OTHOCH-
TeJIbHO IIoporH paBHa ... ().
B2. CHn8 TSlrH TennOB038 F= 235 KH.
nOTpe6nSleM8S1 MOHOCTb N = 3.0 MBT.
Kn.D; T} = 80 %. EcnH p8CCTOSlHHe Me»<,AY
,ABYMSI CT8HHSlMI1 S = 11 KM, TO nOe3,A
npOH,AeT ero 38 BpeMSI t, p8BHOe
(MHH).
Puc.9
B3. 'tJepe3 HenO,ABl1mHbIH 6noK nepeKHHYT8 HHTb, K KOTOpOH nO,ABeIlleHbI
Tpl1 0,AHH8KOBbIX rpya8 M8CCOH m = 0,60 Kr K8m,AbIH (pHC. 9). CHn8 ,A8Bne-
HHfl F H8 OCh 6nOK8 p8BH8 ... (H).
B4. K HCTO'lHHKY TOK8 C BHyTpeHHHM conpOTHBneHl1eM r= 2 OM BHa-tJane nO,A-
KnlO'l8eTClI pe3HCTOp C COnpOTHBneHHeM R = 2 OM, 38TeM n8paJInenbHO nO,AKnlO'l8-
eTCSI T8KOH me pe3I1CTOp. KOnH'tIecTBo TennOTbI Q, BbIAenSleMOH B nepBOM pe3HCTope
nocne nO,AKnlO'tIeHHSI BTOporo pe3HCTOp8, 38 T8Koe me BpeMSI YMeHbIll8eTCSI B ... p83
B5. TIpH BKnlO'leHI1H K8TYIllKI1 B enb nOCTOSlHHoro TOK8 C H8npSlmeHI1eM
U = 12 B aMnepMeTp nOK838n cHny TOK8 h = 4,0 A. TIpH BKnlO'leHHH TOH me
K8TYlliKH B enb nepeMeHHoro TOK8 C 'tI8CTOTOH v = 50 r H H8npSlmeHHeM
U = 12 B 8MnepMeTp nOK838n cl1ny TOK8
1 2 = 2.4 A. MH,AYKTHBHOCTb L K8TYIllKH
p8BH8 ... (MfH).
Puc. 10
p
1
/1
/ I
// I
1 3
I
I
o
Tl
Puc 11
T 2
660
H
B6. no H8KnoHHoH nnOCKOCTH C yr-
nOM H8KnOH8 K ropH30HTY al = 45° C BbI-
COTbI H = 1,0 M 6e3 H8'l8nbHOH CKOpOCTH
COCK8nb3bIB8eT Teno (pHC. 10). .D;oCTl1r-
HYB TO'lKH A. Teno H8'lHH8eT nO,AHHM8Tb-
CSI BBep:x no H8KnoHHoH nnOCKOCTH C yr-
nOM H8KnOH8 a2 = 30°. TIepexo,A C nnoc-
KOCTH H8 nnOCKOCTb nn8BHbIH H rn8,AKI1H.
Ecnl1 K03<pq)HHeHT TpeHHSI Ten8 0 nep-
BYlO nnOCKOCTb Jll = 0, 10, 8 0 BTOpylO
Jl2 = 0,20, TO BbICOT8 h nO,A'beM8 Ten8 H8
BTOPYlO nnOCKOCTb p8BH8 ... (CM).
T
87. H8 p-T-,A118rp8MMe (pHC. 11)
11306p8meH I1Kn. npOBO,AHMbIH C HAe8nb-
HbIM 0,AH08TOMHbIM r830M, B3S1TbIM B KO-
nl1'leCTBe v = 1,0 Monb. B npoecce H30-
TepMH'tIeCKOrO p8CIllHpeHHSI r83 COBepIllHJI
p860TY A = 9,0 K.D;m. EcnH TeMnep8Typb]
T 2 =2T] 11 T 1 =670 K. TO Kn.D; T} HKn8
p8BeH ... (%).

M
V""C
6
4
2
-2
-4
-6
Puc. 12
TECT III
AI. Ha pHCYHKe 12 npellCT8BneH
rpa<pHK 3aBHCHMOCTH npoeKu,HH CKOpOC-
TH Tena H8 OCb OX OT BpeMeHH. OT-
HoweHHe MO,AynSl npoeKu,HH nepeMel.l.{e-
t!..r.
HHSI K npOH,AeHHoMY nYTH ---2... 38 nep-
S
Sble t = 6 C ,ABHmeHHSI p8BHO:
t. C
1) .!:.
3
4) ; 5) 3
1 1
2) !
2
3) !
1
Kr
A2. EcnH nnOTHOCTb Me,AH p = 8900 3'
M
r
MonSlpHaSl M8CC8 Me,AH M = 63 -,
Monb
TO 06'beM, KOTOpblH 38HHM8eT Me,Ab B
KOJIHtJeCTse V = 100 Monb. p8seH:
1) 35 MM 3 ; 2) 35 CM 3 ; 3) 71 MM 3 ;
4) 71 CM 3 ; 5) 0,71 ,AM 3 .
A3. ,lIBa pa3HOHMeHHO 3apSlmeHHblX wapHKa HaXO,ASlTCSI B Macne Ha pac-
CTOSlHHH rl = 5.0 CM. 8TH wapHKH B3aHMO,AeHCTSYIOT C T8KOH me CHnOH. KaK
B B03,Ayxe Ha paCCTOSlHHH r2 = 11,2 CM, ecnH ,AH:3neKTpH'leCKaSl npoHHu,aeMOCTb
E M8cna p8BH8:
1) 4,7; 2) 5.0; 3) 5,5; 4) 5,8; 5) 6,1.
A4. TIoTeHu,H8n HOHH38U,HH aTOMa BO,AOpO8 <Pi = 13,5 B. YCKopSlIOl.l.{aSl pa3-
HOCTb nOTeHu,H8nOB t!..<p, KOTOpylO llonmeH npoHTH :3neKTpOH. 'lT06bl HOHH3HpO-
B8Tb 8TOM BO,AOpOlla. paSH8:
1) 11,2 B; 2) 12,3 B; 3) 10,5 B; 4) 13,5 B; 5) 14,5 B.
A5. IIo KaTYWKe C aKTHSHblM conpOTHBneHHeM R = 5,0 OM H HH,AYKTHS-
HOCTblO L = 50 MrH npOXOIIHT TOK I = 17 A. CHn8 TOK8 U8'lHHaeT B03paCT8Tb
/::J A
co CKOpOCTblO - = 100 -. Qepe3 BpeMSI l:1t = 0,10 C nocne H8tJ8na B03paCTa-
l:1t C
HHSI CHnbl TOK8 H8npSlmeHHe U Ha KOHU,8X KaTYWKH 6YlleT 'P8BHO:
1) 80 B; 2) 85 B; 3) 90 B; 4) 130 B; 5) 135 B.
A6. EcnH Ha ,AH<PP8KUHOHHOH peweTKe nnHHoH 1 = 2.5 CM HaHeceHO 12500
WTPHXOB, TO nOCTOSlHH8S1 IIH<PP8KUHOHHOH peweTKH p8BH8:
1) 0.50 MKM; 2) 1,0 MKM; 3) 1,5 MKM; 4) 2,0 MKM; 5) 2,5 MKM.
A 7. ,lI;nSl Toro 'lT06bI CKOpOCTb :3neKTpOH8 COCT8SHn8 95 % CKOpOCTH cse-
Ta B BaKYYMe, OH ,AOnmeH npoHTH YCKOpSlIOl.l.{YIO p83HOCTb nOTeHu,HanOB l:1<p.
paBHYIO:
1) 1,1 MB; 2) 0,8 MB; 3) 0.7 MB; 4) 0.6 MB; 5) 0,5 MB.
661

A8. j:{nHH8 BonHbI A, COOTBeTCTBYIOIn8S1 TpeTbeH cneKTpsnbHoH nHHHH
B BH,AHMOH 06n8CTH cneKTps STOM8 BO,AOpO,A8 (aHeprHSI OCHOBHoro COCTOSlHHSI
STOMS BO,AOpO,A8 WI = -13,55 aB), p8BH8:
1) 410 HM; 2) 436 HM; 3) 485 HM; 4) 544 HM; 5) 561 HM.
A9. TIocne --p8Cn8,A8 H,Ap8 U o6ps3yeTcSI H30Ton:
1) 4Th; 2) 2ps; 3) 2Np; 4) 2 Pu; 5) 2gU.
A10. TIpoeKlI.HSI CKOpOCTH MSTepHaJIbHOH TO'lKH, ,ABHmyll.teHCSI B,AOnb OCH x,
H3MeHHeTCH co BpeMeHeM no 3aI<OHY v,= 14 - 2t [ : ). MOl\ynb nepeMelI1eHHH
TO'lKH 6.r x 3S BpeMSI OT t 1 = 1,0 C ,AO t2= 3,0 C p8BeH:
1) 13 M; 2) 20 M; 3) 26 M; 4) 32 M; 5) 33 M.
All. BpYCOK M8CCOH m=3,00 Kr TSlHYT pSBHOMepHO no ropH30HT8nbHoH
)10CKe C 1I0MOll.tblO npymHHbI. Koacp<l>HI1.HeHT TpeHHSI Mem,Ay 6PYCKOM H 1I0CKOH
J.l = 0.250. EcnH npymHH8 y,AnHHHn8Cb npH 3TOM H8 x = 50.0 MM. TO ee meCT-
KOCTL k p8BH8:
H
1) 50 -; 2) 100
M M
H
H
H
5) 250
H
3) 150
4) 200
M
M
M
A12. j:{soe PS60QHX HeCYT 6peBHo, M8CC8 KOToporo m = 50,0 Kr, S ,AnHH8
l = 600 CM. O,AHH p860'lHH nO,A,AepmHB8eT 6peBHO H8 p8CCTOSlHHH 8 = 100 CM
OT ero KOHlI.8, BTOpOH - npoTHBononOmHbl:H KO-
HelI. 6peBH8. CHn8 F, C KOTOpOH 6pesHo ,A8BHT
HS BToporo p86o'lero, p8BH8:
1) 500 H; 2) 400 H; 3) 300 H;
4) 200 H: 5) 100 H.
P
2po h3
Po
11
I
I
0 Vi Va V
Puc. 13
A13. ECJlH :0 COCTOSlHHH 1 (pHC. 13) cpe,A-
HSISI KB8,Ap8THtJHSSI CKOpOCTb ,ABHmeHHSI MoneKyn
H,Ae8nLHOro r838 < V KB1 > = Va. TO nocne ocyll.teCTB-
neHHSI npOlI.eCC8 1  2  3 cpellHSISI KBaLtP8THQH8S1
CKOpOCTL < V KB3 > CT8HeT P8BHOH:
1) va; 2) J2vo; 3) J3vo; 4) J6vo; 5) 3vo.
A14. EcnH npH H30XOpHOM OXn8m,AeHHH H,Ae8nbHOro rS38 H8 6.T = 290 K
ero A8BneHHe YMeHLIll8eTCSI B,ABoe. TO HS'lSnLH8S1 Tl H KOHe'lHSSI T 2 TeMnep8-
TypbI r838 p8BHbI COOTBeTCTseHHO:
1) 580 K; 290 K; 2) 290 K; 580 K; 3) 870 K; 580 K;
4) 580 K; 870 K; 5) 870 K; 290 K.
A15. EcnH p8,AHYC ye,AHHeHHOro 38pSlmeHHoro I118p8 R = 1.0 CM. 8 nosepx-
MKKn
HOCTH8S1 nnOTHOCTb aneKTpH'IeCKOrO 38PH1I8 H8 HeM CJ = 0.1 0 . TO H8
M
p8CCTOSlHHH r= 10 CM OT I1.eHTpa I118pS nOTeHI1.H8n <P pSBeH:
662
1) 8,7 B; 2) 9,4 B; 3) 10,8 B; 4) 11,3 B; 5) 13,1 B.

A16. .l{Be nJI8CTHHbI nJIOa,AblO S = 200 CM 2 K8m,A8S1 norpymeHbI B M8cno,
,AHSneKTpH'leCKaSl npOHHaeMOCTb KOToporo E = 2,2, H nO,AKnlO'leHbI K HCTO'lHH-
KY C H8npllmeHHeM U = 200 B. OTKnlO'lHB nn8CTHHbI OT HCTO'lHHK8, YMeHbIll8-
lOT p8CCTOSlHHe Mem,Ay HHMH C d 1 = 5,0 CM ,AO d 2 = 1,0 CM. TIpH STOM COBepIll8-
eTCSI p860Ta A, paBHaSl:
1) 0,12 MK.l{m; 2) 1.8 MK.l{m; 3) 2,1 MK.l{m; 4) 2,5 MK.l{m; 5) 3,1 MK.l{m.
A17. HarpeB8Tenb 3neKTpO'l8HHHK8 HMeeT ,ABe cnHp8nH. TIpH BKnlO'leHHH
OAHOH H3 HHX BO,Aa B 'laHHHKe 38KHnaeT 'lepe3 BpeMSI t 1 = 15 MHH. npli SKnlO-
'leHHH ,ApyroH - 'lepe3 BpeMSI t 2 = 30 MHH. EcnH BKnlO'lHTb 06e cnHpanH na-
p8nnenbHO, TO BO,A8 B qaHHHKe 38KHnHT 'lepe3 BpeMSI t, paBHoe:
1) 20 MHH; 2) 10 MHH; 3) 45 MHH; 4) 25 MHH; 5) 30 MHH.
A18. IIo ,ABYM )InHHHbIM npSlMOJlHHeHHbIM npOBO)IHHKaM, HaxonSlHMClI B
B8KYYMe Ha paCCTOSlHHH 5,0 CM ,Apyr OT APyra. npOXO,ASlT B npOTHBononom-
HbIX HanpaBneHHSlX TOKH II =1 2 = 10 A. B TO'lKe. y,AaneHHOH Ha paCCTOllHHe
rl = 4,0 CM OT O,AHOrO npOBO,AHHKa H Ha p8CCTOSlHHe r2 = 3,0 CM OT APyroro,
MOAynb M8rHHTHoH HH,AYKHH p8BeH:
1) 35 MKTJI; 2) 25 MKTJI; 3) 17 MKTJI; 4) 8,3 MKTJI; 5) O.
A19. KB8,AP8TH8S1 paMKa co CTOpOHOH a = 10 CM nOMeeH8 B OAHopOAHoe
MarHHTHoe none. HOpMaJIb K nnOCKOCTH paMKIi COCTaBn.HeT C nHHHlIMH HH.
,AYKHH MarHHTHoro nonSl yron ex. = 60° . EcnH B p8MKe npH BbIKnlOqeHHH nonH
B TeTleHHe I1t = 0.01 C HH,AyHpyeTcfl Cpe,AHSISI 3.l{C g' = 50 MB, TO HH,AYKHSI
MarHHTHoro nonH B p8BH8:
1) 0,1 Tn; 2) 0,2 Tn; 3) 0,3 Tn; 4) 0,4 Tn; 5) 0,5 Tn.
A20. EcnH HeOHOBYIO naMny nO,AKnlO'lHTb H8 BpeMSI tl = 60 C B l.{enb ne.
peMeHHoro TOKa C ,AeHCTBYIOHM H8npSlmeHHeM U = 120 B H 'l8CTOTOH v = 50 r
(n8Mna 3amHr8eTCSI H r8CHeT npH HanpSlmeHHH U 3 = 84 B), TO BpeM.H t 2 rope-
HHSI naMnbI paBHO:
1) 20 c; 2) 30 c; 3) 40 c; 4) 50 c; 5) 60 c.
A21. TIpe)IMeT nOMeeH nepen p8cceHB81OeH nHH30H H8 p8CCTOSlHHH
d = 40 CM. EcnH nonY'leHHOe H306pameHHe B k = 4 pa38 MeHbIlle C8Moro npen-
MeTa, TO OnTH'leCKaSl CHna D nHH3bI p8BH8:
1) -7.5 ,AnTp; 2) +7,5 ,AnTp; 3) -5,0 ,AnTp; 4) +5,0 ,AnTp; 5) -4.0 ,AnTp.
A22. IIpH OCBeJ..QeHHH KaTO,Aa CBeTOM C 'laCTOTOH Vb a 3aTeM - C 'laCTO-
TOU V2 OKa3aJlOCb. qTO 3a,AepmHBalOee HanpSlmeHHe H3MeHHnOCb B k pa3.
IIpH 3THX YCJIOBHlIX pa60Ta BbIxona A shlx 3JIeKTpOHa H3 KaTO,A8 paBHa:
kv - v h ( VI - V 2 ) h ( kv 1 - V 2 )
1) 1 2 ; 2) (VI-V2.k); 3) ; 4) h(kv1-V2); 5)
k+1 k k-l
A23. M8neHbKHu I118PHK, nO,ABeIlleHHbIH K nOTonKY H8 HesecoMoH HHTH.
onHCbIBaeT B ropH30HTanbHoH nnOCKOCTH oKpymHocTb. EcnH p8CCTOllHHe OT
663

nOTonKa ,AO rOpH30HT8nbHOH nnOCKOCTH l = 1.0 M, TO nepHO,n T 06p8IUeHliSl
m8pliK8 paBeH:
1) 1 c; 2) 2 c; 3) 3 c; 4) 4 c; 5) 5 c.
Kr
A24. ,lJ.epeBSlHHbIH (nnoTHocTb ,AepeBa p = 800 M 3 ) map M8CCOH m = 100 r
n8,A8eT 6e3 H8tJanbHoH CKOpOCTIi C BbICOTbl hI = 10.0 M B BO,AY H norpym8eT-
C.H B Hee H8 rny6HHY h 2 =2,OO M. EcnH conpo-
p TIiBneHHeM B03,nyxa npeHe6pe'lb, TO Cpe,AH.HSI clina
conpOTHBneHHSI Fe BO,nbI ,ABHmeHHlO mapHKa paBHa:
Po
-------71 2
___/ i
11
I
I
1) 1,25 H; 2) 2,50 H; 3) 3,75 H;
4) 4,75 H; 5) 5.00 H.
3po
A25. B COCTO.HHHIi 1 (pHC. 14) TeMnepaTypa
ra38 TI = 200 K. EcnH npH ocyecTBneHIiH npo-
II.eCca 1  2 M8CC8 r838 YMeHbmaeTCSI B,ABOe, TO
TeMnep8Typa r83a T 2 B KOHII.e npOII.eCCa paBHa:
o
V o
3V o V
1) 3600 K; 2) 1800 K; 3) 900 K;
4) 450 K; 5) 225 K.
Puc. 14 A26. B XO,Ae 1i3068pHoro npOII.eCca 06'beM
v = 1.0 Monb o,nHoaTOMHoro li,AeanbHOro ra38 YBe-
nlitIlinCSI Ha 6V= 10 n. Ecnli nepBOHatI8nbHbIH 06'beM ra38 VI = 5,0 n. a KO-
HetIH8S1 TeMnepaTypa T 2 = 501 K. TO BHYTpeHHSI.H 3HeprH.H U I r83a B HatJ8nb-
HOM COCTO.HHHIi p8BHa:
1) 8,4 K,lJ.m; 2) 4.2 K,lJ.m; 3) 2,1 K,lJ.m; 4) 8.4 ,lJ.m; 5) 4,2 ,lJ.m.
A27. B BO,ny M8CCOH ml = 12,0 Kr BnYCKalOT nap MaccoH m2 = 0,70 Kr,
Am
B3S1TbIH npH TeMnepaType t 2 = 100°C. Y,AenbHaSl TennoeMKOCTb BO,AbI 4200 -----y,
Kr'fi
y,nenbH8S1 TennOT8 n8po06pa30B8HH.H BO,AbI L = 2,3
MAm
EcnH TeMnepaTypa
Kr
BO,AbI nO,AHSlnaCb ,AO E> = 70°C. TO nepBOH8'lanbHaSl TeMnepaTyp8 BO,AbI t l 6bIna
paBHa:
1) 40°C; 2) 36 °C; 3) 30°C; 4) 25 °C; 5) 20°C.
A28. AB8 38p.HmeHHbIX m8pHK8 C O,AHHaKOBbIMIi p8,AHycaMH H MaCC8MI1
no,nBemeHbI Ha HHTSIX O,AIiHaKOBoH ,nnliHbI H onyeHbI B mH,nKIiH ,A1i3neKT-
pHK, ,AHSneKTpHtIeCKaSl npOHHII.8eMOCTb KOToporo E. TInoTHocTb M8Tepl1an8 ma-
pHKOB Pl' EcnH yron p8CXOm,AeHlifl mapliKOB B B03,nyxe H B ,A1i3neKTpHKe
0,AIiH8KOB, TO nnOTHOCTb P2 ,AHSneKTpHKa paBHa:
1) pdE-1); 2) . 3) P.(E-l) ; 4) ; 5) PIE.
E-l' E [-I
A29. MaSlTHHK C nepliO,AOM Kone68HliH TI = 1.0 C npe,ACTaBnSleT C060H ma-
pliK MaccoH m = 5,0 r, nO,ABemeHHbIH H8 HenpOBO,ASIeH HHTIi. llIapliKY co06-
alOT OTpHII.aTenbHbIH 3apSl,n q = -1. 10- 8 Kn Ii nOMealOT B 3neKTpli'leCKOe
664

g'l
r 1
R
g'2 r 2
Puc. 15
KB
none HanpSlmeHHocTbIO E = 540 HanpaBneHHoe
M
BepTHK8nbHO BBepx. IIepHo,A Kone68HHii MaSlTHH-
K8 T CTan paBeH:
1) 0,70 c; 2) 1.3 c; 3) 0,95 c; 4) 0,58 c;
5) 0,75 c.
A30. B cxeMe. npHBe.neHHoH H8 pHCYHKe 15.
S.n:C H BHYTpeHHee conpOTHBneHHe nepBoro HCTOtJHH-
Ka paBHbI COOTBeTCTBeHHO l = 2.0 B H rl = 1,0 OM,
BToporo HCTO'lHHK8 - = 1.0 B H r2=0,50 OM.
BHelliHee conpOTHBneHHe R = 3,5 OM. H8npHmeHHe
U 2 H8 3amHM8X BToporo HCTO'IHHK8 paBHO:
1) 2,1 B; 2) 1,7 B; 3) 1,4 B; 4) 1,1 B;
5) 0,8 B.
A31. QacTH1I.8. 38pSl,A KOTOpOH p8BeH 38pSl,AY 3neKTpOH8. BneT8eT B O,AHO-
pO,AHOe MarHHTHoe none no,A yrnoM a = 45° K nHHHSlM MarHHTHOH HH,AYK1I.HH
11 ,ABH:>KeTCH no BI1HTOBOH nHHI1H C IllarOM h = 2,0 CM. HHAYK1I.HH MarHHTHoro
nonH B = 10 MTn. HMnynbc 'l8CTH1I.bl p paBeH:
5.4.10- 24 Kr'M 6.8.10- 24 Kr'M 7.2.10- 24 Kr'M
1) -' 2) . 3) -'
. .
C C C
6.1.10- 23 Kr'M 5,3,10- 23 Kr'M
4) -; 5)
C C
A32. MaSlTHHK ,AnHHOH l = 80 CM. nO,ABeIlleHHbIH B caMoneTe, JleTHeM ro-
pI130HTaJIbHO, COBepIllaeT N = 10 Kone68HHH 3a t = 15 c. C8MoneT neTHT C YCKO-
peHHeM a. paBHbIM:
1) 23
M
2;
C
2) 15


o

--...

'4':.
"
Puc. 16
M
"2; 3) 10
C
M
2;
c
5) 5.0
M
2'
C
M
2;
C
4) 7,5
t,)," I
,
A33. B CTeKJ1e C nOKa38TeneM npenOMneHHSI
n = 1,6 HMeeTCSI c<I>epH'IeCK8S1 nonOCTb paAHYCOM
R = 2,0 CM, 3anonHeHHa.H BO,AOH. nOK83aTenb npe-
nOMneHHSI KOTOpOH nl = 1,33 (pHC. 16). Ha nonOCTb
n8,A8IOT napannenbHble nY'lH CBeT8. P8,AHYC CBeTOBO-
ro nY'lKa r. KOTOpbIH npOHHK8eT B nonOCTb. paBeH:
1) 2.0 CM; 2) 1.9 CM; 3) 1,8 CM; 4) 1,7 CM;
5) 1 ,5 CM.
81. SneKTpOnOe3,A ,ABHmeTCSI H8 ceBep co CKOpO-
KM
CTbIO VI = 161 TIHnoT BepToneT8, nponeTaIOe-
'I
ro H8,A 3J1eKTpOnoe3AOM, 06H8pymHB8eT. tJTO sneKTpOnOe3,A ,ABHmeTCfl nepneH-
M
,AHKynSlpHo K HanpaBneHHIO ero ,ABHmeHHSI co CKOpOCTblO V2= 20 MO,Aynb
C
CKOpOCTH v BepTOJIeTa OTHOCHTeJIbHO nOBepXHOCTH 3eMJIH paaeH ... ()-
665

M
B2. C8MOneT npH B3neTe ,AOJlmeH HMeTL CKOpOCTL v = 25 -. ,nnHH8 npo-
C
6era nepe,A B3neTOM L = 100 M. ECJIH M8CC8 caMoneTa m = 1.0 T H K03<l><t>HHeHT
conpOTHBneHHSI Jl = 0,020. TO MOlQHOCTb N MOTOpOB npH B3neTe pasH8 ... (KBT).
Puc. 17
B3. Qepe3 HenO,ABHmHLIH 6noK nepeKHHYT8
HHTb. K KOTOpOH nO,ABeWeHLI TPH 0,AHH8KOBblX
rpY3a MaccoH m = 0.6 Kr K8>K,AbIH (pHC. 17). CHna
HaTSlmeHHSI F HHTH. CBSl3blBalOlQeH rpY3bI 2 HI,
p8BH8 ... (H).
B4. K HCTOqHHKY TOKa C BHYTpeHHHM conpo-
THBneHHeM r= 2.0 OM BH8'l8ne nO,AKnlO'l8eTCSI pe-
3HCTOp C COnpOTHSneHHeM R = 2.0 OM. 3aTeM n8-
pannenbHO nO,AKnlO'laeTCSI TaKOH me pe3HCTOp. KonH-
'leCTBO TennOTbI Q, BbI.AenSleMOH B nepBOM pe3HCTope
nocne nO,AKnlO'leHHSI BTOporo pe3HCTopa, 3a TaKoe me
BpeMSI YMeHLwaeTCSI B ... (p83).
B5. B enb nepeMeHHoro TOK8 C 'l8CTOTOH v = 50 r H HanpSlmeHHeM
U = 12 B BKJIIO'IeH8 KaTYWKa C HH,AYKTHBHOCTblO L = 12,7 MrH H 8KTHBHbiM
conpOTHBneHHeM R = 3,0 OM nOCne,AOBaTenLHO C KOH,AeHCaTOpOM eMKOCTblO
C = 394 MK<%>. MOlQHOCTL P, Bbl,ZJ;enSleM8S1 B IJ;enH, 6Y,AeT p8SHa ... (BT).
B6. TIYJI$I M8CCOH m = 100 r, JleTeBW8S1 ropH30HTanbHO, nOna,AaeT B no,A-
BeweHHblH Ha HHTH wap MaccoH ml = 2,0 Kr H. np06HB ero, BbIneT8eT co
M
CKOpOCTblO V = 400 -. ECJIH m8p npH 3TOM no,A-
C
HHM8eTCSI Ha BLICOTY h =0,20 M, TO npH Y,A8pe BO
BHYTpeHHlO1O 3HeprHIO nepemno ... ., % KHHeTH'leC-
KOH 3HeprHH nynH.
p
1
3
Puc. 18
666
2
87. O,AHH MOJIL 0,AH08TOMHoro H,ZJ;e8nLHoro ra38
cosepW8eT IJ;HKn, COCTOSllQHH H3 a,AH868THoro pac-
WHpeHHSI 1-2, H30TepMH'leCKOrO cmaTHSI 2-3 H
H30xopHoro npoecca 3-1 (pHC. 18). P86oT8, co-
BepmeHH8S1 ra30M B H30TepMH'leCKOM npOIJ;eCce,
A = 600 ,Um. EcnH p83HOCTL Me>K,AY MaKCHMaJILHOH
H MHHHManLHoH TeMnepaTypaMH ra38 B IJ;HKne paB-
H8  T = 60 E, TO En,n T) HKna paBeH ... (%).
v

X,M
180
120
60
o
20 40
60
Puc. 19
TECT IV
AI. fp8<}>HKH 38BHCHMOCTH KOOp.nHH8TbI
,ABYX Ten OT BpeMeHH npe,ACT8BneHb] H8 pH-
CYHKe 19. PaccToSlHHe S Mem,AY 3THMH TeJIa-
MH H3MeHSleTCSI co BpeMeHeM no 3aKOHY:
1) He H3MeHSleTCSI co BpeMeHeM;
2) S = 1180 - 3tl; 3) S = 1180 - tl;
4) s = 1180 + tl; 5) s = 1180 - 2tl.
2
A2. B cocY,Ae H8XO,AHTCSI KHCnOpO,A M8CCOH
m = 20 r. EcnH MonSlpHaSl Macca KHCnOpO,Aa
M = 0,032 Kr 8 KOHeHTp8HSI MoneKyn
Monb
KHcnopo.na n = 7.5' 10 25 M- 3 , TO 06'beM V CO-
cY,Aa paseH:
1) 2,5 CM 3 ; 2) 5 CM 3 ; 3) 2,5 n;
4) 5 n; 5) 7,5 11.
t, C
A3. P8CCTOSlHHe r, H8 KOTOpOM B B03,Ayxe H8XO,ASlTCSI AB8 p8BHbIX p83HO-
HMeHHbIX TOqe'lHbIX 3apSl,A8 ql =q2= 1,0 MKJI, B38HMO,AeHcTBYIOIl.tHX C CHnOH
F = 9.0 MH, paBHO:
1) 1000 M; 2) 100 M; 3) 10 M; 4) 1,0 M; 5) 0,1 M.
A4. EcnH KOHeHTpaHSI O,AHOBaneHTHbIX HOHOB B HOHH38HOHHOH K8Mepe
A
n = 1 . 10 3 CM -3. 8 nnOTHOCTb TOKa H8CbIIl.teHHSI j = 1 . 10- 12 TO cpeAHSISI
MM 2 '
1) 9
KM
CKOpOCTb < v> H8npaBneHHoro ,ABH)KeHHH HOHOB p8BH8:
KM
KM
5) 5 -.
C
C
KM KM
2) 8 -; 3) 7 -; 4) 6
C C
C
A5. CHna TOKa B KaTYWKe YMeHbWHJIaCb OT 1 1 = 12 A .no 1 2 = 8 A. npH
3TOM 3HeprHSI M8rHHTHoro nOJIH YMeHbWHn8Cb Ha 6. W = 211m. HH,AYKTHBHOCTb
L K8TYWKH p8BHa:
1) 0,03 fH; 2) 0,05 rH; 3) 0,3 rH; 4) 0,4 fH; 5) 0,5 fH.
A6. Ha ,AH<}>paKuHoHHYIO peweTKY nepneH)IHKynSlpHo K ee nosepXHOCTH
naAaeT CBeT. YroJl ,AH<ppaKHH B cneKTpe nepBoro nopSlKa nSl nHHH1:I C
A] = 600 HM COCTaBnSleT <PI = 30°. )1nHHa BOJlHbI A2 nHHHH B cneKTpe BToporo
nOpSl,AKa. yron .nH<}>paKUHH .nnSl KOTOpOH <P2 = 45°. p8BH8:
1) 424 HM; 2) 652 HM; 3) 708 HM; 4) 756 HM; 5) 804 HM.
A 7. IIonHaSl 3HeprHSI Me30Ha B
CKOpOCTH Me30Ha v, paBHoH:
1) 2,985. 10 8 M, 2) 2,285. 10 7
C
M
5) 1,875' 10 8
C
k=10 pa3 60JIbWe 3HeprHH nOKOSl npH
M 2,160 . 10 7 M 2,000 . 10 8 M
, 3) , 4) ,
C C C
667

A8. TIpH H3nYQeHHIi aTOMOM <l>oToHa nOnH8S1 3HeprHSI 8TOMa H3MeHHnaCb
H8 6 W = 2.56 3B. ,Z:(n1iH8 BonHbl A 1i3nY'l8eMOrO CBeTa p8BH8:
1) 315 HM; 2) 380 HM; 3) 410 HM; 4) 440 HM; 5) 485 HM.
A9. TIpH cnHSlHHIi ,ABYX SI,Aep renHH He 8bl,AenSleTCfl npOTOH. H,Apo, 06-
p830B8HHoe B pe3ynbTaTe pe8KHH, npe,ACT8BnSleT C060H:
1) H; 2) rH; 3) Be; 4) gLi; 5) Li.
AIO. EcnH, TporaSlCb C MeCT8, nOe3,A 38 BpeMSI t = 10 C p83BliB8eT CKO-
KM
pOCTb v=36 -, TO nYTb 8=450 M nOe3,lJ, npOH,AeT 3a BpeMH t 1 nocne HatJa-
'I
n8 ,ABlimeHHSI, p8BHoe:
1) 15 c; 2) 20 c; 3) 28 c; 4) 30 c; 5) 40 c.
All. Ha Teno M8CCOH m = 100 Kr, nemaee Ha HaKnoHHoH nnOCKOCTIi.
cocTaBnSlIOeH C ropH30HTOM yron ex = 30°, B ropli30HTanbHOM HanpaBneHHIi
,lJ,eHcTByeT clina F = 1330 H. EcnH K03<l><l>IilieHT TpeHIiSl J.l = 0, 10, TO TenO
,ABHmeTCSI BBepX C YCKopeHHeM a, p8BHbIM:
1) 0,50
M
2";
C
2) 2.0
M
2";
C
3) 2,5
M
2";
C
4) 4,0
M
'2;
C
5) 5,0
M
c 2 .
A12. ll3 O,AHOPO,AHOH KpyrnoH nnaCTIiHbI pa,AIiYCOM R Bblpe3anli KB8,AP8T,
,AHarOHanb KOToporo COBn8,lJ,aeT C p8,AHYCOM Ii paBH8 eMY. l(eHTp TSlmeCTH no-
nY'leHHOH nJlaCTIiHbI H8XO,AIiTCH OT eHTpa KpyrnoH nnaCTIiHbl H8 paCCTO.HHHH
r. paBHOM:
1 ) R ; 2 ) R 3) R . 4) R 5) 2R
2 2 (2n - 1) , 2n - 1 ' 2 ( n - 1)' 2n - 1
A13. B 6annoHe pa,AHOJIaMnbI 06'beMOM V = 100 CM 3 H8XO,AIiTCSI N = 4,00. 10 14
MoneKyn B03,Ayxa. EcnH ,A8BneHHe B naMne p = 13,3 MTIa, a MonSlpHaSl Macca
B03,Ayxa M = 2,90. 10- 2
Kr
Monb
MoneKyn B 6annOHe paBHa:
TO cpe.nH.H.H KB8.np8TH'lHaSl CKOpOCTb < V KB >
M
1) 259
2) 359
M
C
C
M
3) 455 -; 4) 469
C
MM
5) 555
MM
-;
-;
C
C
A14. Ecnli npli H30TepMH'leCKOM npoecce ,A8BneHHe ra3a YMeHblllHnOCb H8
6.p = 50,0 RITa, 8 06'beM YBenH'lHnCSI B k = 6 pa3, TO KOHe'lHOe ,lJ,aBneHHe p
r83a p8BHO:
1) 10 Kna; 2) 12 Kna; 3) 13 KTIa; 4) 14 KTIa; 5) 15 KTIa.
A15. llIapHR MaCCOH m = 1,00 r H 38pSl,AOM q = 100 MKKn no,A .neHCTSl1eM
clin 3neKTpOCTaTIi'leCKOrO nonSl nepeMeaeTCSI H3 TO'lKIi A, nOTeH1i8n KOTO-
668

pOH <PI= 600 B. B TO'lKY B. nOTeHqHaJJ KOTOpOH (j)2=0. EcnH CKOpOCTb llIa-
M
pHKa B TO'iKe B V2 = 0.20 TO ero CKOpOCTb VI B TO'lKe A paBHa:
C
1) 0.31
M
2) 0.25
CM
M
3) 0.25
4) 0.17
CM
M
5) 0.17
-'
.
C
C
C
C
C
A16. EcnH npocTp8HCTBO Mem,AY 06Kn8,lJ;KaMH nnOCKoro B03,AYllIHoro KOH,AeH-
caTopa, nO,AKnlO'leHHOrO K HCTO'lHHKY, 3anonHHTb ,AH3neKTpHKOM C ,AH3neKTpH-
'leCKOH npOHHu.aeMOCTblO E, TO HanpSlmeHHOCTb nonSl E npH 3TOM:
1) YBenH'lHTCSI B £ p83; 2) YMeHbllIHTCSI B £ pa3; 3) YBenH'lHTCfl B 2£
p83; 4) YMeHbllIHTCSI B 2£ p83; 5) He H3MeHHTCSI.
A17. 3IlC HCTO'lHHK8 TOKa rc = 30 B, BHYTpeHHee conpOTHBneHHe r= 5.0 OM.
BHellIHSlSI qenb nOTpe6nSleT MOI..I1.HOCTb P = 8.0 BT. HaHMeHbllIee 3Ha'leHHe CHnbI
TOKa B qenH paBHO:
1) 0,28 A; 2) 0.38 A; 3) 0.48 A; 4) 0,58 A; 5) 0,68 A.
A18. B ropH30HTanbHOM O,AHOpO,lJ;HOM M8rHHTHOM none C HH,AYKqHeH B = 10 MT n
nO,ABellIeH nepneH,AHKynSlpHo M8rHHTHOMY nonlO H8 ,ABYX nerKHX HHTSIX ropH30H-
T8nbHbIH npOBO,AHHK ,lJ;nHHOH l = 10 CM. EcnH no npOBO,AHHKY nponycTHTb 3neKT-
pH'leCKHH TOK 1=10 A, TO CHn8 H8TSlmeHHSI F KIDK,lJ;OH H3 HHTeH H3MeHHTCSI Ha:
1) 5,0' 10- 3 H; 2) 4.0' 10- 3 H: 3) 3.0' 10- 4 H: 4) 2.0' 10- 4 H:
5) 1.0. 10- 4 H.
A19. O,lJ;HOCnOHHa.H K8TYllIK8 nnOa,AblO Ce'leHHSI S = 10 CM 2 , CO,lJ;epm8a.H
N = 100 BHTKOB npOBO,A8. nOMeeH8 B O,lJ;HOpO,AHOe M8rHHTHoe none C HH,AYK-
qHeH B = 8,0 MTn napannenbHO nHHHSlM M8rHHTHOH HH,AYKqHH. KaTYllIK8 38M-
KHYT8 H8KOpOTKO, ee conpOTHBneHHe R = 10 OM. EcnH OTKnlO'lHTb M8rHHTHoe
none, TO no K8TYllIKe npOH,AeT 38pSl,A q, paBHbIH:
1) 90 MKKn; 2) 80 MKKn; 3) 70 MKKn; 4) 60 MKKn; 5) 50 MKKn.
A20. KaTYllIKa C aKTHBHbIM conpoTHBneHHeM R = 15 OM H HH,AYKTHBHOC-
TblO L = 52 MfH BKnlOtJeHa B u.enb nepeMeHHoro TOK8 C 'l8CTOTOH v = 50 fu.
nOCne,AOBaTenbHO C KOH,AeHCaTOpOM eMKOCTblO C = 120 MK<I>. EcnH HanpSl>KeHHe
B ceTH U = 220 B, TO CHn8 TOK8 I B u.enH paBHa:
1) 8 A; 2) 10 A; 3) 12 A; 4) 14 A; 5) 15 A.
A21. AB8 TO'le'lHbIX HCTO'lHHKa HaXO,ASlTCSI H8 p8CCTOSlHHH l = 24 CM ,lJ;Pyr
OT ,APyra. Mem,AY HHMH Ha paccToSlHHH d l = 6,0 CM OT O,AHOrO H3 HHX nOMe-
eH8 C06Hp81OaSl nHH38. EcnH H306p8meHHe 060HX HCTO'lHHKOB nonYtJHnOCb
B O,AHOH H TOH me TO'lKe, TO <POKYCHoe p8CCTO.HHHe F nHH3bI p8BHO:
1) 24 CM; 2) 18 CM; 3) 12 CM; 4) 9,0 CM; 5) 6.0 CM.
A22. P8,AHYC nepBoH 60pOBCKOH Op6HTbI 3neKTpOH8 B 8TOMe BO,AOpO,A8
r= 0,528 .10- 10 M. ,[.(nHHa BonHbI ,Ae BpoHnSi AB ,An.H 3neKTpOHa Ha 3TOH
op6HTe p8BH8:
1) 3,3,10- 10 M; 2) 2,9,10- 10 M; 3) 2,5,10- 10 M; 4) 2,1,10- 10 M;
5) 1,8. 10- 10 M.
669

A23. llIapHK M8CCOH m=200 r, nO,ABeWeHHbI:H Ha HeBeCOMOH HHTH, Ka'l8-
eTCSI B BepTHK8nbHOH nnOCKOCTH T8K, 'lTO ero YCKopeHHSI B Kp8HHeM H HHm-
HeM nonomeHHSlX p8BHbI no MO,AynlO ,Apyr ,Apyry. EcnH yron OTKnOHeHHSI
HHTH OT BepTHKanH B Kp8HHeM DonomeHHH paseH q> = 30 0 , TO CHna H8TSlme-
HHSI F HHTH B HHmHeM nonomeHHH p8BH8:
1) 1 H; 2) 2 H: 3) 3 H; 4) 4 H; 5) 5 H.
A24. IIpH nOl{'beMe rpY3a M8CCO:H m = 30 T C nOMOI.l.tblO rH,ApaBnH'leCKOrO
npecca ManbIH nopweHb C,Aenan N = 100 XO,AOB B Te'leHHe t = 1,25 MHH, onyc-
K8S1Cb 38 O,AHH XO,A H8 h = 30 CM. OnOI.l.t8,AH Ce'leHHSI nopwHeH OTHOCSlTCSI
K8K 8 1 : 8 2 = 1 : 100. EcnH K03<P<PHHeHT none3Horo ,Ae:HCTSHSI npecca 11 = 80 %,
TO MOI.l.tHOCTb P ,ABHraTenSl npeCC8 paBHa:
1) 1,0 KBT; 2) 1,5 KBT; 3) 2,0 KBT; 4) 1,0 MBT; 5) 1,5 MBT.
A25. IIpoecc, npOHCXO,ASIHH C ,AaHHOH MaCCO:H H,AealIbHOrO ra3a, onHCbl-
B8eTCSI 38KOHOM TV 3 = const. EcnH 06'beM ra3a YMeHbW8eTCSI B k = 2 p838, TO
ero ,AaBneHHe p:
1) He H3MeHSleTcSI; 2) YMeHbmaeTCSI B 8 pa3; 3) YBenHHB8eTCSI B 8 p83;
4) YMeHbW8eTCSI B 16 p83; 5) YBenH'lHB8eTCSI B 16 p83.
A26. }:(nSl Toro 'lT06bI BO,AOpO,A M8CCO:H m = 6.5 r C HaanbHoH TeMnepaTYPO:H
T 1 = 300 K H306apHo paCWHpHTb ,AO BBoe 601IbWerO 06'beMa, eMY He06xo,AHMO
C006I.l.tHTb KonWIeCTBO TennOTbI Q, paBHoe (MonSlpH8JI M8cca BO,AOpo8 2, 0  ):
Monb
1) 8,0 K}:(m; 2) 20 K}:(m; 3) 28 K}:(m; 4) 36 K,D;m; 5) 48 K,D;m.
A27. B Me,AHbIH cocy M8CCOH ml = 300 r. H8rpeTbI:H ,AO t 1 = 350 °C, onYCTH-
nH ne, HMelOI.l.tHH TeMnep8TYPY T 2 = 263 K. TIocne YCTaHOBneHHSI TennOBoro P8B-
HOBeCHSI B cocY,Ae OK838JI0Cb m = 520 r nb,Aa H SO,A8. Y,AenbH8S1 TennoeMKOCTb
Am Km
Me,AH Cl = 380 -, y,AenbH8S1 TenlIOeMKOCTb nb8 c2 = 2,10 -, y,Aenb-
Kr.K Kr.K
" _ _ 330 K,D;m
H8JI TennOTa nnaBneHHSI nba I\,
Kr
Ha'lWIbH8S1 M8cca m2 nba paBH8:
1) 980 r; 2) 922 r; 3) 902 r; 4) 800 r; 5) 600 r.
A28. 1lI8pHK M8CCOH m = 1 r H 38pSl,AOM ql = 1. 10- 7 Kn nO,ABeWeH B B03-
,Ayxe Ha TOHKOH wenKOBO:H HHTH, KOTopaSi COCTaBlISleT yron q> = 45 0 C BepTH-
K8nblO B TOM cnY8e, eClIH Ha p8CCTOSlHHH r = 3 CM OT nepBoro wapHKa no-
MeCTHTb BTOpOH mapHK C 3apSl,AOM npoTHBOnOlIOmHOrO 3H8Ka T8K, T06bI
06a wapHKa HaXO,AHnHCb Ha O,AHO:H ropH30HTanH. 3apSl q2 BToporo wapHKa
p8BeH:
1) 1. 10- 8 KlI; 2) 1. 10- 7 Kn; 3) 1. 10- 6 Kn; 4) 1. 10- 5 Kn;
5) 1. 10- 4 Kn.
Kr
A29. AnIOMHHHeBbI:H (nnOTHOCTb anlOMHHHSI PI = 2.7. 10 3 3) wapHK Mac-
M
COH m = 9,0 r, HecyI.l.tHH 38pSl,A q = 1,0 . 10- 7 KlI, nOMeI.l.teH B Macno nnOTHOCTblO
670

Kr
p = 900 3 H C ,AH3neKTpH'IeCKOH npOHHaeMOCTblO E = 2,2. llIapHK H8XO,AHTCSI
M
BO B3BeweHHOM COCTOSlHHH, eCJIH COCY,A C M8cnOM nOMeIlteH BO SHeWHee BepTH-
KMbHoe 3JIeKTpOCT8TH'leCKOe none, HanpSlmeHHOCTb E KOToporo paBH8:
4 B B 5 B 6 B
1) 5,1. 10 - ; 2) 4,7. 10 4 -; 3) 4, 7 . 10 - ; 4) 1.3. 10 - ;
M M M M
5) 1,6' 106 B .
M
A
A30. 1JT06bl npH nJIOTHOCTH TOKa j = 2,5.10 5 - B Me,AHbIX (Y,AenbHOe
M 2
conpOTHBneHHe Me,AH p = 1,7.10- 8 OM' M) npOBO8X ,AByxnpOBO,AHOH nHHHH
3JIeKTpOnepe,A8'1H nOTepH B nHHHH COCT8BnSlnH 1') = 1,0 % OT nepe,A8B8eMOH MOIlt-
HOCTH, nepe,A8B8Tb 3neKTp03HeprHIO H8 paCCTOSlHHe s = 5.0 KM H81I0 nOlI Ha-
npSlmeHHeM U. p8BHbIM:
1) 3,8 KB; 2) 4,1 KB; 3) 4,3 KB; 4) 3,5 KB; 5) 3,1 KB.
A31. 3J1eKTpOH npOJIeT8eT 'lepe3 cnoH M8rHHTHoro nonSl TOnIltHHOH l. CKO-
pOCTb 3neKTpOHa V nepneH,AHKYJISlpHa K8K HH,AYKHH nonSl iJ, T8K H rp8HH'
aM cnoSl 3neKTpOH BbIneTaeT H3 MarH1:ITHOrO nonH nOA yr nOM <P K nepBO-
Ha'lanbHOMY HanpaBJleHHIO, paBHbIM:
1) eBI 2) mv 3) . mv ) . eBl 5) mv
arccos -; arccos -; arCSIn -; 4 arCSIn - ; arctg -.
mv eBI eBl mv eBl
A32. MaSlTHHK ,AnHHOH l YKpenneH H8 TSlmenOH TenemKe. C'be3>K81OIl\eH
6e3 TpeHHSI C HaKnoHHoH nnOCKOCTH, yron H8KnOH8 KOTOpOH K ropH30HTY p8'
BeH a. IIepHo,A T Kone68HHH M8S1THHK8 paBeH:
1) 27t II g l; 2) 211 JlCO g SCt ; 3) 211 lSi g nCt ; 4) 21I J I . 5) 211 J I .
Vg gsina' gcosa
A33. Ha npH3MY c npenOMnSlIOIltHM yrnoM <P = 30° (pHC. 20) nepneH,AHKY-
nSlpHO 60KOBOH rp8HH n8,AaeT ny'l CBeT8. JIy'l He BbIH,AeT H3 npH3Mbl 'lepe3
BTOPYIO 60KOBYlO rp8Hb, ecnH MHHHManbHbIH no-
K83aTeJlb npeJIOMneHHH n npH3Mbl paBeH:
Puc. 20
1) 1,5; 2) 1,7; 3) 1,8; 4) 2,0; 5) 2,2.
81. BepToneT nponeTen C ceBepa H8 lOr pac-
CTOHHHe s= 137 KM. Bo BpeMH peHca ,Ayn BeTep C
3ana,Aa H8 BOCTOK. EcnH CKOpOCTb BepToneT8 OT-
KM
HOCHTenbHO BeTpa Vi = 180 - H CKOpOCTb BeTp8
'I
M
v2=20 TO BpeMH noneTa t paBHO ... (MHH).
C
671

F,H
12
10
8
6
4
2
o
2 4 6 8
Puc. 21
X, CM
m 1
m 2
t a
B2. H8 pHCYHKe 21 H306p8meH8 38BHCHMOCTb
MO,AynSl CHnbI ynpyrocTH OT a6conlOTHoro y,AnHHe-
HHSI npymHHbI. .lI.nSl Toro 'lTo6bI P8CTSlHYTb He,Ae-
<l>0PMHPOS8HHYIO npymHHY H8 x = 80,0 MM, Heo6-
XOAHMO cosepwHTb pa60TY A, P8BHYlO ... (M.lI.m).
B3. Qepe3 HesecoMbIH 6noK nepeKHHYT8 HHTb
C rpY3aMH ml = 1 Kr H m2= 3 Kr. 3aKpenneHHbl-
MH H8 ee KOH8X (pHC. 22). CHn8 AaBneHHSI H8
OCb 6JIOK8 F = 36 H. TpeHHeM B 6noKe npeHe6-
petib. BnoK ,ASHmeTCSI BBepx C YCKopeHHeM a,
paBHLlM ... (  )-
B4. K HCTO'lHHKY TOKa C BHYTpeHHHM con po-
THsneHHeM r = 2 OM BH8'lane nO,AKnlO'l8eTCSI pe-
3HCTOp C conpOTHBneHHeM R = 2 OM, 38TeM n8-
p8nnenbHO nO,AKnlOtJ8eTCSI T8KOH me pe3HCTOp.
KOJIH'leCTBO TennOTbJ Q, BbJAenSleMOH B nepBOM
pe3HCTope nocne nO,AKnlO'leHHH BToporo pe3HCTO-
pa, 38 TaKoe me BpeMH YMeHbW8eTCH B ... p83.
B5. Ha nepBH'lHYIO 06MOTKY TP8HC<I>OpM8TOp8
Puc. 22 nO,AaeTCSI HanpSlmeHHe U 1 = 3500 B. Ero BTOpH'l-
H8S1 06MOTK8 COe,AHHeH8 nO,ABOASIHMH npOSO,Aa-
MH C nOTpe6HTeneM, H8 KOTOpOM H8npSlmeHHe U 2 = 220 B. MOHOCTb TOKa,
SbIn;enSleM8H H8 nOTpe6HTene. P = 25 KBT H K03<1>4>HuHeHT MOlUHOCTH cos<p = 1.
K03<P4>HuHeHT Tp8Hc<I>opMaUHH k = 8. TIoTepSlMH 3HeprHH B 06MOTKax TpaHc<I>op-
M8TOp8 npeHe6pe'lb. ConpoTHBneHHe R nO,ABO,ASl1I.1:HX npOSO,AOB paBHO ... (OM).
B6. .lI.Ba ManeHbKHX W8P8, ,ASHr8BWHXCSI nocTyn8TenbHO no B38HMHO nep-
neHAHKynHpHblM Hanp8sneHHSlM, cTonKHynHCb. JIerKHH W8p MaCOH m2 npH
3TOM OCT8HOBHncSI. EcnH npH Y,Aape BO sHYTpeHHlO1O 3HeprHIO nepellJJIO
11 = 50 % KHHeTH'leCKOH 3HeprHH nerKoro mapa, TO OTHOllJeHHe MaCCbJ TSlme-

noro llJ8p8 K Macce nerKoro llJapa p8BHO ...
m2
B7. TIO,A nOpllJHeM H,AeanbHOH TennOBOH MallJHHbI H8XOAHTCSI r83. A6co-
nIOTH8S1 TeMnepaTyp8 HarpeBaTenSl B AB8 p83a BbIllJe TeMnepaTypbI xonOAHnb.
HHKa. 38 O,AHH HKn M8llJHH8 nO,AHHMaeT nopmeHb M8CCOH m = 5 Kr Ha Bbl.
COTY h = 20 M H cmHMaeT npH 3TOM npymHHY meCTKOCTblO k = 625 KH H8
M
x = 8 CM. KOnHtieCTSO TennOTbl Ql> KOTopoe nonY'laeT r83 OT HarpeBaTenSl 3a
O,AHH HKn, p8BHO ... (K.lI.m).
672

TECT V
X,M
2
AI. rpacpHKH 3aBHCHMOCTH KOOPAHH8TbI
ABYX Ten OT speMeHH npe,ACT8BJIeHbI H8 PH-
CYHKe 23. OTHolIIeHHe MO,AyneH CKOpOCTH Ten
VI p8BHO:
V2
60
o
1
40
20
111
1) "3' 2) 2' 3) 1
2
4) 1
5) 3
1
t, C
A2. EcnH MonSlpH8S1 M8cca renHSI _ =
r
= 4.0 -. TO M8cca mo aTOM8 renHSI p8SH8:
Monb
-20
Puc. 23
1) 6.7. 10- 28 Kr; 2) 6,6. 10- 27 Kr;
3) 6,7. 10- 26 Kr; 4) 6,7. 10- 25 Kr;
5) 6.7. 10- 24 Kr.
A3. Pa3MepHOCTb 3neKTpH'leCKOrO 38pSl,A8 B OCHOBHbIX e,AHHHu.ax Cll:
Kr-M 2 . Kr'M 2 . Kr'M 3 . Kr
1) c 2 . A · 2) A. C; 3) A 2 . c 3 ' 4) A 2 . c 3 ' 5) A. c 2 .
A4. P8CcToSlHHe Memy KaTO,AOM H 8HO.nOM .nHo.na r= 10 MM. EcnH 8HO.n-
Hoe H8npSlmeHHe U = 440 B. TO speMSI 6.t paBHOYCKopeHHoro .nBHmeHHSI 3neKT-
pOH8 OT K8TO,Aa K 8HO,AY p8SHO:
1) 1,1 HC; 2) 1,2 HC; 3) 1.3 HC; 4) 1.6 HC; 5) 1,9 HC.
CKOpOCTblO
KaTYlliKe C 8KTHBHbIM conpOTHBneHHeM R = 5.0 OM H HH,AYKTHBHOC-
MrH npOXOAHT TOK 1= 17 A. CHna TOK8 Ha'lHH8eT B03p8CT8Tb co
M A
- = 100 -. HanpSlmeHHe U H8 KOHu.ax K8TYlliKH npH aTOM p8BHO:
6.t C
A5. Ilo
TblO L = 50
1) 60 B; 2) 70 B; 3) 80 B; 4) 90 B; 5) 95 B.
A6. Ha aKp8He. Y,A8neHHOM OT ,AH<I>paKlJ,HOHHOH pelIIeTKH H8 p8CCTO.RHHe
l = 1,0 M. paCCTOSlHHe Mem,AY MaKCHMYM8MH nepBoro nOp.RAK8 x = 14 CM. EcnH
lIITp
AHcppaKlJ,HOHHaSl pelIIeTKa CO,AepmHT N = 120 -, TO ,AnHH8 BonHbI A CBeT8,
MM
n8AalOIl\erO Ha ,AHcppaKlJ,HOHHYIO pellieTKY, p8BHa:
1) 0,45 MKM; 2) 0.50 MKM; 3) 0,55 MKM; 4) 0,58 MKM; 5) 0,68 MKM.
A 7. .n;nfl Toro 'lT06bI npO,AOnbHbIe p83MepbJ nepBOH8'lanbHO nOKOHSlIIero-
CSI aneKTpoHa YMeHblIIHnHCb B k = 2 p83a, OH ,AOnmeH npoHTH YCKOPSlIOIl\YIO
pa3HOCTb nOTeHlJ,HanOB 6.<p, P8BHYIO:
1) 512 KB; 2) 500 KB; 3) 400 KB; 4) 300 KB; 5) 250 KB.
673

AS. 3neKTpoH B 8TOMe BO,AOpo,na nor nomaeT cpOTOH H nepeXO,AHT C nepSOH
CTaHOH8pHOH Op6HTbI H8 TpeTblO. EcnH 3HeprHSI OCHOBHoro COCTOSlHHSI aTOMa
BO,AOpO,Aa WI = -13,55 3B, TO 3HeprHSI W nor JIOIUeHHOro cpOTOHa p8BHa:
1) 10,16 3B; 2) 12,04 3B; 3) 12.70 3B; 4) 13,10 3B; 5) 13,55 3B.
A9. .H.D;po tx, 06p83YIOIUeecSI B pe3ynbT8Te SI,AepHOH pe8KHH fJAJ +2 C -+
-+ :X + In + He. npe,ACTaBnSleT C060H:
1) Na; 2) Cl; 3} Cl; 4) :Na; 5) Cl.
AlO. JIOKOMOTHB, nO,A'be3maSl K cBeTocpopy co CKOpOCTblO v = 72,0
KM
'I
H8'lan TOpM03HTb H8 p8CCTOSlHHH S = 400 M OT cBeTo<l>op8. EcnH npoeKUHSI
CM
YCKopeHHSI npH TopMomeHHH a x = 10.0 "2"' H 38 Ha'l8JIO OTC'leT8 Koop,nHHaTbl x
C
Bbl6p8Tb nonomeHHe CBeTocpop8, TO 'lepe3 I1t = 10,0 C Koop)J;HHaTa x nOKOMO-
THB8 H npoeKHSI nepeMeIUeHHSI I1rx paBHbI COOTBeTCTseHHO:
1) 205 M; -195 M; 2) -205 M; 195 M; 3) 205 M; 205 M;
4) 205 M; 195 M: 5) 195 M; 205 M.
All. C ropbI. .nnHHa CKnOH8 KOTOpOH l = 50 M H BbICOT8 h = 10 M, H8
BepeSKe Ha'lHH81OT cnYCK8Tb C8HKH M8CCOH m = 80 Kr, COXP8HSlfl cHny H8TSI-
meHHSI sepeBKH F= 60 H nOCTOSlHHOH. EcnH MO,Aynb CHnbI TpeHHSI Mem,Ay C8H-
K8MH H nosepXHOCTblO ropbI COCTaBnSleT 11 = 10 % MO,AynSl CHnbI TSI:meCTH C8-
HOK, TO CKOpOCTb v caHOK y nO,AHOmHSI ropbI p8BHa:
M M
1) 10,0 -; 2) 7,5 -; 3) 5,0
C C
M
M
4) 2,5 -; 5) 2,0
C
M
C
C
A12. TO'lK8 onopbI O,AHOpO,AHOrO CTepmHH ,AnHHOH l = 40 CM H M8CCOH
m=2,0 Kr HaXO,AHTCSI Ha paCCTOSlHHH r=10 CM OT ero JIeBOrO KOHa. K 3TO-
MY KOHY nO,ABeWeH rpY3 M8CCOH ml =4,0 Kr. 't.{T06bI CTepmeHb H8XO,AHnCSI B
p8BHoseCHH, K HeMY He06xo,AHMO nO,ABeCHTb rpY3 M8CCOH m2= 1,0 Kr Ha pac-
CTOSlHHH r2 OT TO'lKH onopbI. p8BHOM:
1) 5 CM; 2) 10 CM; 3) 15 CM; 4) 20 CM; 5) 25 CM.
P
2po 1

Po
2 1 I
1 I
I 1
0 V o 3V o V
Puc. 24
674
A13. EcnH B COCTOSlHHH 1 Cpe,AHSISI KB8,Apa-
TH'IHRH CKOpOCTb )J;B1:ImeHHH MOJleKYJI H)J;e8nbHO-
1'0 ra3a Vo, TO nocne OCYIUeCTBneHHSI npoeCC8
1-+2-+3 (pHC. 24) OHa CT8n8 paBHoH:
1) Vo; 2) 1,2vo; 3) 1,4vo; 4) 2.1vo; 5) 2,4vo.
A14. EcnH ,A8SneHHe ra38 B 38KpbITOM cocY,Ae
YMeHbWHnOCb U8 11 = 20 %. TO TeMnep8Typa r838:
1) He H3MeHHnacb; 2) YBenH'lHn8Cb B 1,25
pa38; 3) YMeHbwHnacb B 1,25 p838; 4) YBenH'lH-
JIaCb B 2,5 pa3a; 5) YMeHbWHJlaCb B 2,5 p83a.

A15. EcnH nOTeHHan 3apSlmeHHOrO H30JlHpOBaHHOrO wapa q>1 = 300 B. 8
ero pa,AHYC R= 10,0 MM, TO H8 paCCTOSlHHH r= 10.0 CM OT [(eHTpa 38pSlmeH-
Horo W8pa nOTeHHan <P2 paseH:
1) 250 B; 2) 200 B; 3) 100 B; 4) 30 B; 5) 20 B.
A16. MemAY 06KJlaAK&MH nJlOCKOrO KOH,AeHC&TOpa HaXO,AHTCSI napacl>H-
HOBaSl nnaCTHHKa (E = 2,2). EMKOCTL KOH,AeHC8TOp8 C = 2,0 MK<1>, 38pSl,A
q = 0.40 MKKn. 'tJTo6bI BbIT8Il.{HTb nn8cTHHKY H3 3830pa, HymHo cOBepwHTL
p860TY A, paBHYIO:
1) 10 H.D:m; 2) 20 H.D:m; 3) 30 H.D:m; 4) 38 H.D:m; 5) 48 H.D:m.
A17. a.D:e HCTO'lHHKa TOK8  = 30,0 B, BHYTpeHHee conpOTHBneHHe
r = 5,00 OM. EcnH BHeWHSISI enb nOTpe6nSleT MOIl.{HOCTb P = 8,00 BT, TO co-
npOTHBneHHe R BHewHeH L\enH paBHO:
1) 102 OM; 2) 90,0 OM; 3) 80,0 OM; 4) 70,0 OM; 5) 62.0 OM.
A18. Pa3MepHOCTL HHl\YKUHH B MarHHTHoro nonSl MomeT 6bITL BbIp8meH8
'lepe3 OCHOBHbIe ellHHHUbI eM:
1) A. c ; 2) ; 3) Kr
Kr A . c 2 A. c 2 . M '
Am A .c 2
4) - B ; 5)
Kr
A19. Pe8KTHBHbIH C&MOJleT C pa3M8XOM KpbInLeB l = 50 M neTHT ropH30H-
M
T8nbHO co CKOpOCTblO v = 200 -. BepTHK8nLH8S1 COCTaBnSlIOIl.{aSl M8rHHTHoro
C
nonSl 3eMnH B = 5. 10- 5 Tn. P83HOCTL nOTeHH8nOB llq> Mem,AY KOHaMH KPLI-
nbeB paBHa:
1) 0,1 B; 2) 0.2 B; 3) 0.3 B; 4) 0,4 B; 5) 0,5 B.
A20. KOH,AeHCaTOp eMKOCTblO C, 3apSlmeHHLIH ,AO HanpSlmeHHSI V, p83pSl-
m8eTCSI Ha KaTYWKY C HH,AYKTHBHOCTblO L. K MOMeHTY BpeMeHH, KOI',Aa TOK
B K8TYWKe ,AOCTHraeT MaKC1:IManLHOl'O 3HaqeHHB I, B KaTYWKe BbI,AenHTCSI KO-
nH'leCTBO TennOTbI Q, paBHoe:
LI 2 CV 2 V 2 CU 2 - LI 2
1 ) 12Rt. 2 ) -' 3 ) -' 4 ) _to 5 )
, 2' 2' R' 2
A21. C06HP81OIl.{8S1 nHH38 ,A8eT YBenH'leHHOe B r 1 = 3 pa38 H306p8meHHe
npe,AMeT8. EcnH nHH3Y 3aMeHHTL ,APyroH nHH30H. OnTH'leCK8B CHn8 KOTOpOH
B k = 2 pa3a MeHbwe, a nonomeHHe npe,AMeTa OTHOCHTenbHO nHH3bI COXP8-
HHTL, TO YBenH'leHHe r 2 H306p&meHHSI npe,AMeT8:
1) He H3MeHHTcSI; 2) YBenH'lHTCSI B 2 pa38; 3) YMeHbWHTCSI B 2 p838;
4) YBenH'lHTCSI B 1,5 pa38; 5) YMeHbWHTCSI B 1,5 p838.
A22. .D:n1:lH& BonHbI ,Ae BpoHnSl AB AJISI 3J1eKTpOHa, npoweAwero YCKOp.HlO-
Il.{YIO pa3HOCTb nOTeHHanOB llq> = 100 B, p8BHa:
1) 1,2. 10- 10 M; 2) 1,8' 10- 10 M; 3) 2,3. 10- 10 M; 4) 3,5' 10- 10 M;
5) 3,9' 10- 10 M.
675

A23. MaTeM8TH'leCKHH M8S1THHK HMeeT M8CCY m = 610 r H )];JIHHY l = 50 CM.
EcnH B MOMeHT. Kor.n8 OH 06pa3yeT yron a = 30° C BepTHK8nblO, ero CKO-
CM
pOCTb V = 79 -. TO CHn8 F HaTSI>KeHHSI HHTH paBH8:
C
1) 10 H; 2) 8,0 H; 3) 6,0 H; 4) 5,0 H; 5) 4,0 H.
Kr
A24. He60JIbIlIOH ,AepeBHHHbIH (nnoTHocTb ,AepeB8 P,tt = 600 3) Ky6HK Y,Aep-
M
r
mHBalOT B BO)];e (nnoTHocTb BO,AbI PB= 1,0 ----a) Ha rny6HHe h = 2,0 M. Ky-
CM
6HK OTnYCK8IOT. EcnH conpOTHBneHHe B 0)]; bJ He Y1.lHTbIBaTb, TO Ky6HK ,AOCTHr-
HeT nOBepXHOCTH BO)];bJ 'lepe3 BpeMSI t, paBHoe:
1) 0,40 c; 2) 0.50 c; 3) 0.68 c; 4) 0,77 c; 5) 0,85 c.
A25. B ropH30HT8JlbHOH 38KpbITOH Tpy6Ke C nnOMblO Ce'leHHSI S = 0,4 CM 2
r83 pa3,AeneH Ha ,AB8 06'beMa VI = V 2 = 50 CM 3 cTon6HKOM PTYTH M8CCOH m = 10
r. Tpy6KY nepeBOpa'lHB81OT B BepTHKanbHoe nonomeHHe. EcnH TeMnep8TYP8 r83a
He H3MeHSleTCH, 8 HHmHHH 06'beM ra38 npH BepTHKanbHOM nonomeHHH TPy6KH
V 2 = 40 CM 3 , TO ,A8BneHHe PI ra3a )];0 nOBopOT8 Tpy6KH 6bIno p8BHO:
1) 4 ITa; 2) 6 ITS; 3) 4 KITa; 4) 6 KITa; 5) 8 RITa.
A26. M,Ae8nbHbIH O)];HOaTOMHbIH ra3 paCIlIHpSleTCSI no 38KOHY p 2 V = const
OT ,AaBneHHSI PI ,AO ,AaBneHHSI P2' EcnH nepBOHa'l8nbHblH 06'beM V h TO H3Me-
HeHHe BHYTpeHHeH 3HeprHH ra3a f1U p8BHO:
1) PIVI ( PI -1 ) ;
2 P2
5) 3 PI VI (  -1).
2) PIVi ( P2 -1 ) ; 3) .!.P I V 1 ( PI _ 1 ) ; 4) .!.P 1 V 1 ( P2 -1 ) ;
2 PI 2 P2 2 PI
A27. JIe,A M8CCOH mi = 15 Kr npH TeMnep8Type t 1 = -10°C onYCTHnH B
BO,AY, M8CC8 KOTOpOH m2= 20 Kr, 8 TeMnep8Typ8 t 2 = 50°C. Y,AenbHaSl Tenno-
Km KAm
eMKOCTb BO,AbI C2 = 4,2 - K ' y,AenbHaSl TennoeMKOCTb nb,Aa CI = 2,]
Kr. Kr.K '
Y,AenbHaSl TennOT8 nn8BneHHSI nb,A8 A. = 330
TeMnep8Typa 0, p8BH8H:
1) 0,0 °C; 2) 5,0 °C; 3) 10°C; 4) 20 °C; 5) 30°C.
K,l];m
Kr
B cocY,Ae YCT8HOBHnaCb
A28. IlI8pHK M8CCOH m = 1 r Ii 3apSl,AOM qi = 1 . 10- 7 Kn nO,ABellieH B B03-
,Ayxe H8 IlIenKoBoH HHTH. HHTb COCTaBnSleT yron a = 45° C BepTHKanblO B TOM
cnY'lae, ecnH Ha HeROTopOM p8CCTOSlHHH OT nepBoro IlIapHK8 nOMeCTHTb BTO-
pOH IlIapHK C 38pSl,AOM q2 = -10- 8 Kn TaR, 1.lT06bI 068 IlI8PHR8 H8XO,AHnHCb
H8 O,AHOH ropH30HT8nH. PaCCTO.HHHe r, Ha ROTOpOM HymHo nOMeCT1:ITb BTOpOH
IlIapHK, p8BHO:
1) 2 CM; 2) 3 CM; 3) 4 CM; 4) 5 CM; 5) 8 CM.
676

Kr
A29. AnlOMHHHeBbIH (nnOTHOCTb anlOMHHHSI PI = 2700 "3") 1I18PHK M8CCOH
M
m = 9 I' H8XO,AHTCSI so B3BelIleHHOM COCTOHHHH B M8Cne, nnOTHOCTb KOTOpOI'O
Kr
p = 900 3 H ,AH3neKTpH'leCK8S1 npOHHlI.8eMOCTb E = 2,2. EcnH COCY,A C MacnOM
M
H8XO,AHTCSI BO SHelilHeM BepTHKanbHOM O,AHOpO,AHOM 3neKTpOCT8TH'leCKOM none
B
C H8npSlmeHHocTblO E = 13,2 . 10 5 -, TO 38pSl,A 1I18PHK8 q p8seH:
M
1) 1. 10- 6 Kn; 2) 1. 10- 7 Kn; 3) 5. 10- 7 Kn; 4) 7. 10- 8 Kn;
5) 1. 10- 8 KlI.
A30. JI8MnbI MOHOCTblO P = 300 BT K8m,AaSl, paCC'lHT8HHbIe Ha HanpSl-
meHHe V a = 110 B. nO,AKnlO'l81OT K I'eHep8TOpy. H8npSlJKeHHe H8 KneMM8X KO-
Toporo V = 122 B. EcnH npOSO,AKa OT I'eHepaTopa C,Aen8H8 Me,AHbIM (Y,Aenb-
Hoe conpOTHBneHHe p = 1,7. 10- 8 OM' M) npOBO,AOM ,AnHHOH l = 100 M H nno-
8,AblO nOnepe'lHOrO Ce'leHHSI S = 9,0 MM 2 , TO MomHO nO,ACOe,AHHHTb napan-
nenbHO 'lHCnO N naMn, p8BHoe:
1) 11; 2) 13; 3) 15; 4) 23; 5) 28.
A31. ABa 3neKTpoHa C KHHeTH'leCKHMH 3HeprHSlMH W Kl H W K2 ,ABHmYTCSI
B M8rHHTHOM nOlle. nepneH,AHKynSlpHoM HX CKOpOCTSlM. C'lHTaHTe. 'lTO CKOpOC-
TH 3neKTpoHoB 3Ha'lHTenbHO MeHblile CKOpOCTH CBeT8 B B8KYYMe. OTHolIleHHe
7i
nepHO,AOB HX 06p8eHHSI p8BHO:
T 2
1 1
1) 4; 2) . 3) 1; 4) 2; 5) -
4' 2
A32. B Ka6HHe a3pocT8T8 YCT8HosneHbI M8S1THHKOBbIe 'l8Cbl. A3pOCT8T H8-
CM
qHH8eT nO,AHHMaTbCH BBepx C YCKopeHHeM a = 20,0 2' KOI',A8 no M8HTHHKO-
C
SbIM tJ8C8M npOH,AeT t = 60.0 C. 83pOCTaT 3a 3TO BpeMfl nOllHHMeTCSI Ha BbICO-
TY h, P8BHYIO:
1) 353 M; 2) 331 M; 3) 248 M; 4) 215 M; 5) 208 M.
A33. fIoK838Tenb npenOMneHHH BO,AbI n = 1,33. C'lHT8HTe, 'lTO npH M8-
nbIX YI'nax tga === sino.. EcnH npH onpe,AeneHHH H8 I'n83 no BepTHK8nbHOMY
H8np8BneHHIO I'ny6HH8 peKH K8meTCSI p8BHOH h = 2,0 M, TO ee HCTHHH8S1 rny-
6HHa h] p8BH8:
1) 2.0 M; 2) 2.4 M; 3) 2,7 M; 4) 3,0 M; 5) 3.2 M.
Puc. 25
Bl. Koneco K8THTCH 6e3 npocK81Ib3bIB8HHH no
I'OpH30HT8nbHoH ,AopOI'e co CKOpOCTblO v = 10 ::: OT-
C
HOCHTenbHO 3eMnH. EcnH yrOJJ a = 60° (pHC. 25). TO
CKOpOCTb TO'lKH 1 060,A8 KoneC8 VI OTHOCHTenbHO ,AO-
porH paSHa ... ().
677

!a
m 1 m 2
Puc. 26
B2. MeX8HHqeCK8S1 nonaT8, npHBO,AHM8S1 B
ABHmeHHe pOTOpOM ,ABHraTenSl MOHOCTblO
P = 366 BT, nO,AHHMaeT neCOK MaccoH m = 180 T
B Te'leHHe q8C8. EcnH KTI,n YCT8HOBKH
11 = 82 %, TO neCOK nO,AHHMaIOT H8 BbICOTY h.
paBHYIO ... (CM).
B3. 1.Jepe3 HeBeCOMbIH 6nOK nepeKHHYT8
HHTL C rpY38MH ml = 1 Kr H m2= 3 Kr, 38K-
penneHHblMH H8 ee KOHax (pHC. 26). CHn8
,A8BneHHSI Ha OCL 6nOKa F = 24 H. TpeHHeM
B 6noKe npeHe6pe'lb. BnoK ,ASHmeTCSI BHH3 C
YCKopeHHeM a. pSBHLlM ... r ;. ).
B4. K HCTO'lHHKY TOKa C BHYTpeHHHM conpOTHBneHHeM r = 2 OM BH8'lane nO,A-
KnlO'l8eTCSI pe3HCTOp C COnpOTHBneHHeM R = 2 OM, 38TeM napannenLHO nOAKnlO'l8eT-
CSI T8KOH me pe3HCTOp. KOnH'lecTBo rennOTbl Q. Bbl,AenSleMOH B nepBOM pe3HCTope
nocne nO,AKnlO'leHHSI BTOpOro pe3HCTopa, 3a T8Koe me BpeMSI YMeHLlll8eTCSI B ... pa3.
B5. KOHTYP p8AHonpHeMHHKa HaCTpoeH Ha pa,AHOCTaHHIO, qaCTOT8 KOTO-
pOH V = 48 Mr. 1.JTo6bl H8CTpOHTL KOHTYP Ha ,AnHHY BonHbI A = 50 M,
HaAO YBenHt{HTb eMKOCTb C nepeMeHHoro KOH,AeHC8TOpa Kone68TenbHoro KOH-
Typ8 B ... p83.
p
----------- 2
1 1 3
/1
/ I I
/ I I
/
0 T 1 T 2
Puc. 27
678
B6. ,lIBa ManeHbKHX 1118p8. OTHOllleHHe
m 1 - - 2.0.
Macc KOTOpLIX ,ABHraBIIIHXCSI nocTY-
m 2
n8TenbHO no B38HMHO nepneHAHKynSlpHLIM
HanpaBneHHSlM, cTonKHynHCb. EcnH npH 3TOM
nerKHH 1118p OCT8HOBHncSI, TO BO BHYTpeH-
HIOIO 3HeprHIO nepellino ... % ero KHHeTH-
'leCKOH 3HeprHH.
B7. KTI,n L1.HKn8 (pHC. 27), npOBO,AHMOrO C
O,AHOaTOMHblM H,Ae8nLHLIM ra30M, S3S1TblM B
T KOnH'leCTBe v = 1 Monb, 11 = 20 %. EcnH TeM-
nepaTypLI T2=2Tl Ii T 1 =670 K, TO p860Ta A.
COBeplll8eMaSl npH H30TepMH'leCKOM npoecce,
p8SH8 ... (K,nm).

v ,1!!.
x C
1
10
8
6
4
2
o
1
2
3
TECT VI
Al. Ha pHCYHKe 28 npe,ACT8Bne-
HbI rp8<1>HKH 38BHCHMOCTH OT BpeMe-
HH npoeKn:HH CKOpOCTH H8 OCb x
,AnSI nSlTH Ten.
,L(BHmeHHIO Ten8, nepeMeIl\eHHe KO-
Toporo 8x=t 2 , COOTBeTcTByeT rp8<1>HK:
1) 1; 2) 2; 3) 3; 4) 4; 5) 5.
A2. EcnH B cocY,Ae H8XOAHTCSI BO,AO-
POA B KOnH'leCTBe v = 20 Monb, TO liHC-
no N MOJleKYJl BOAOPO,A8 paBHO:
1) 1,2. 1025; 2) 2.0. 1024;
3) 1,2. 1024; 4) 2.0. 1023;
5) 1,2.10 23 .
4 t, C
A3. Pa3MepHocTb 3neKTpHQeCKOH no-
CTOSlHHOH Eo B OCHOBHbIX e,AHHHl\a:x CM:
Kr M 3 Kr A 2 . C 4 U' r . M 3
1 ) ' , 2 ) 3 ) 4 ) '"' 5 ) A
2 4 A.c 2 ' -'3; ----s-; . c.
A ,C Kr'M C
Puc. 28
A4. ITo npOBO,AHHKY c nnOIl\a,AblO nOnepeQHOrO Ce'leHHH S = 5,0 MM 2 npo-
XOAHT TOK I = 9,0 A. EcnH Cpe,AHSISI CKOpOCTb H8np8BneHHoro ,ABHmeHHSI cso-
MM
60,AHbIX 3neKTpOHOB < v> = 0,28 -, TO KOHn:eHTp8n:HSI n CBo60,AHbIX 3neKT-
C
pOHOB B BeIl\eCTBe npOBO,AHHKa paBH8:
1) 3,6. 1028; 2) 3,8. 1028; 3) 4,0. 1028; 4) 6,8. 1028; 5) 7.1. 10 28 .
A5. Ha K8TYlliKe C 8KTHBHbIM conpoTHBneHHeM R = 5,0 OM nO,AAepmHS8eT-
CSI nOCTOSlHHoe H8npSlmeHHe U = 50 B. MH,AYKTHBHocTb K8TYlliKH L = 25 MrH.
3HeprHH W, KOTopaSl BbI,AenHTCSI npH pa3MbIKaHHH n:enH K8TYllIKH, p8BH8:
1) 12 ,L(m; 2) 1,3 ,L(m; 3) 43 ,L(m; 4) 0,13 ,L(m; 5) 4,3 ,L(m.
A6. EcnH nepHO,A ,AH<ppaKn:HoHHOH pellIeTKH d = 10 MKM, TO QHCnO N ,AH<I>-
p8Kn:HOHHbIX MaKCHMYMOB ,AnSI CBeTa C ,AnHHOH BOJlHbI A. = 671 HM p8SHO:
1) 3; 2) 5; 3) 7; 4) 9; 5) 11.
A7 E "I - _ 330 K,L(m ,
. CJlH YAenbHaSl TennOT8 nnaBneHHSI JlbAa "" TO M8C-
Kr
C8 m = 1,0 Kr JIb,A8 npH nn8BneHHH H3MeHSleTCSI H8:
1) 3,1' 10- 12 Kr; 2) 3.7. 10- 12 Kr; 3) 3.2' 10- 10 Kr; 4) 2.8' 10- 10 Kr;
5) 2,1.10- 11 Kr.
A8. QHcno B03MomHbIX KBaHTOB pa3nHQHOH 3HeprHH, KOTopoe MomeT HC-
nYCK8Tb aTOM BO,AOpO,Aa, ecnH 3neKTpOH H8XO,AHTCSI Ha nSlTOH CT8n:HOH8pHOH
op6HTe, paBHO:
1) 10; 2) 9; 3) 8; 4) 7; 5) 6.
679

A9. TIpH 60M68P,AHpOBKe HeihpOH8MH SI,Apa J Al 06p8ayeTcSI SI,ApO Na H
licnYCK8eTC.H 'l8CTHua:
1) p; 2) lf3; 3) a; 4) n; 5) 'IP.
AI0. TIoe3,A OTOIllen OT CT8HliH H, ,ABHraSlCb paBHOYCKopeHHO, 3a t = 30 C
npOlllen nYTb 8 = 450 M. TIYTb 810 npOH,AeHHbIH nOe3,AOM a8 ,AeCSlTYIO CeKYH,AY,
p8SeH:
1) 4,5 M; 2) 7.5 M; 3) 9,5 M; 4) 15 M; 5) 19 M.
All. BpYCOK M8CCOH m = 5 Kr TSlHyr paBHoMepHo no ropli30HTaJIbHOH ,AOCKe C
H
nOMOblO npY>KHHbI meCTKOCTblO k = 150 EcnH npli 3TOM npymliHa Y,AnliHH-
M
naCb Ha x = 5,0 CM, TO K03<1><I>HHeHT TpeHHSI Mem,AY ,AOCKOH H 6PYCKOM Jl paseH:
1) 0,10; 2) 0,15; 3) 0.20; 4) 0,25; 5) 0,30.
m 3
Puc. 29
A12. CHcTeMa H3 Tpex rpY30B (pHC. 29)
H8XO,AHTCSI B paSHOBeClili, TpeHlie OTCYTCTByeT.
EcnH MaCCbI rpY30B ml Ii m2 Ii yron a H8-
KnOHa nnOCKOCTH K ropH30HTY H3BeCTHbI, TO
M8cca m3 TpeTbero rpY3a p8SHa:
1) ml; 2) m2; 3) ml - m2;
4) (ml - m2)sina; 5) (ml + m2)sina.
A13. EcnH MaCC8 H,AeanbHOro ra3a
m = 6,0 Kr, 8 38HliM8eMbIH 11M 06'beM
V=5,O M 3 npH ,AaSneHHH p=200 KTI8. TO cpe,A-
HSISI KSa,Ap8TH1.J:H8S1 CKOpOCTb < V KB > ,AB11>KeHHSI
MoneKyn ra3a paBHa:
1) 4.1' 10 2 M 2) 5,1' 10 2 M 3) 6,1,10 2 M 4) 7,1' 10 2 M
-'
,
C C C C
5) 8,1 . 10 2 M
C
A14. EcnH npli H8rpes8HHH H,AeanbHOro ra3a B 3aKpbIToM cocy.u;e Ha
l:1T = 1,0 K ,AaBneHHe B03pocno Ha 11 = 0,40 % OT nepBOHaqanbHOro, TO Ha-
'l8nbH8S1 TeMnep8Typa T 1 ra3a 6bIn8 paBH8:
1) 1,3.10 2 K; 2) 2,5.10 2 K; 3) 3,8. 10 2 K; 4) 5,0. 10 2 K;
5) 6,3' 10 2 K.
A15. B nnOCKOM ropH30HTanbHo pacnonomeHHoM KOH.u;eHcaTope, p8CCTO.H-
Hlie Mem,AY nnaCTliH8MH KOToporo d = 1,0 CM, HaXO,AliTCSI 3apSlmeHHan K8nenb-
Ka M8cn8 M8CCOH m = 2,6 r. B OTCYTCTBHe 3neKTpHtJeCKoro nonSl KanenbKa
M
n8,AaeT C nocToSlHHOH CKOpOCTblO VI = 1,1 . 10- 4 npli H8npSlmeHHH Mem,AY
C
680

nnaCTHHaMH U = 150 B CKOpOCTb KanenbKH V2= 4,3 . 10- 4 M EcnH C'tlHTaTb.
C
'lTO CHn8 conpoTHBneHHSI nponopHoH8nbHa CKOpOCTH, TO 3apSl,A q KanenbKH
p8BeH
1) 4,0 MKKn; 2) 4,8 MKKn; 3) 4.5 MKKn; 4) 3,5 MKKn; 5) 3,0 MKKn.
A16. ,ZJ;Ba nOCne,AOBaTenbHO COe,AHHeHHbIX KOH,AeHC8TOp8, eMKOCTH KOTOpbIX
C l = 2.0 MK<I> H C 2 = 4.0 MKq>, 3apSlmeHbI ,AO HanpSlmeHHSI U = 180 B. KOH,AeH-
caTopbI, He pa3pSlmaSl, p83'be,AHH.HIOT H COe,AHHSlIOT O,AHOHMeHHO 3apSlmeHHbIMH
nnaCTHH8MH. HanpSlmeHHe U 1 Ha KOH,AeHC8TOpax CT8no paBHbIM:
1) 80 B; 2) 60 B; 3) 50 B; 4) 40 B; 5) 20 B.
A17. a,ZJ;c HCTO'lHHK8 TOKa g' = 1,3 B. TIpH BKnlO'leHHH BO SHelliHlO1O
enb conpOTHBneHHSI R = 3,0 OM CHna TOK8 1=0,40 A. TOK KopOTKoro 38-
MbIKaHHSI 1a paBeH:
1) 3.5 A; 2) 4,2 A; 3) 4,5 A; 4) 5,2 A; 5) 5,8 A.
AIS. IIpOBO,A8 B03,AYlliHOH nHHHH aneKTponepe,A8'lH H8XO,ASlTCSI Ha paccTo-
HHHH r = 50 CM ,Apyr OT ,Apyra. IIo HHM npOXO,AHT TOK 1= 500 A. CHna F
B3aHMO,AeHCTsHH. npHXO,ASlw.aSlCSI Ha eHHHy ,AnHHbI npOBO,AOB, paBHa:
1) 2,1 H; 2) 1,1 H; 3) 0,50 H; 4) 0,30 H; 5) 0.10 H.
A19. rOpH30HTanbHbIH MeT8nnH'leCKHH CTepmeHb lInHHOH l = 0.50 M Bp81l1a-
eTCSI BOKpyr BepTHK8nbHOH OCH, npOXO,ASIeH 'lepe3 O,AHH H3 ero KOHOB, C
'tlaCTOTOH v = 2,0 C -1. BepTHK8nbH8S1 COCT8BnSlIOw.aSl MarHHTHoro nonSl 3eMnH
B = 5,0. 10- 5 Tn. Pa3HocTb nOTeHHanOB .6.q> Mem,Ay KOHaMH CTepmHSI p8BHa:
1) 7,5' 10- 4 B; 2) 6.4' 10- 4 B; 3) 7,9' 10- 5 B; 4) 7,2' 10- 5 B;
5) 6,8. 10- 5 B.
A20. TpaHc<I>opMaTop, nOBbIllIalOHH HanpSlmeHHe OT U 1 = 100 B ,AO
U 2 = 3300 B, HMeeT 38MKHYTbIH Cep,Ae'lHHK B BH,Ae Konba. Qepe3 KonbO npo-
nyw.eH npOSO,A. TIpHCOe,AHHeHHbIH K ero KOH8M BonbTMeTp nOK83bIBaeT H8-
npSlmeHHe U 3 = 500 MB. QHcno SHTKOB N B nepSH'tlHOH 06MoTKe paBHO:
1) 120; 2) 140; 3) 150; 4) 180; 5) 200.
A21. <l>oKycHoe p8CCTOSlHHe 06'beKTHS8 npoeKHOHHoro annapaTa F = 15 CM,
pa3MepbI AHan03HTHB8 a x b = (9,0 x 12) CM 2 . QTo6bI nonY'tlHTb Ha 3KpaHe H306-
p8meHHe p83MepOM al x b l = (45 x 60) CM 2 , ,AHan03HTHB Ha,AO nOMeCTHTb Ha p8C-
CTOSlHHH d OT 06'beKTHBa, paSHOM:
1) 16 CM; 2) 17 CM; 3) 18 CM; 4) 20 CM; 5) 25 CM.
A22. EcnH KHHeTH'leCK8S1 3HeprHSI <poToaneKTpoHoS W K' 8 pa60T8 BbIXO,A8
HX H3 KaTO,Aa A BhlX ' TO ,AnHH8 BonHbI A CSeTa. BbI3BaBllIero <l>oT03(jJ<I>eKT, paBHa;
hv hv
1) 2"; 2) hv - A BLlX ; 3) Au...x + W K ; 4)
e e
5)
A BLlX + WI(
he
681

A23. H8 3KBaTOpe HeKOTOpOH nnaHeTbI Ten8 BeCSlT B,ABOe MeHbllle, tJeM Ha
Kr
nonlOce. EcnH nnOTHOCTb BeeCTB8 nn8HeTbI P = 3000 TO nepHO,A T 06-
M 3 '
paeHHSI nnaHeTbI BOKpyr c06cTBeHHoH OCH p8BeH:
1) 324 MHH; 2) 243 MHH; 3) 162 MHH; 4) 120 MHH; 5) 81 MHH.
A24. mene3HhIH (nnoTHocTb mene3a Pm = 7,8 ) lllap nnaBaeT BO PTY-
CM
r
TH (nnOTHOCTb PTYTH P p = 13,6 ). EcnH cBepxy H8nHTb cnOH BO,AbI (nnOT-
CM
r
HOCTb BO,AbI Ps = 1,0 -"3)' nonHOCTblO nOKpbIB81OHH 1118p, TO 06'beM V 1118pa,
CM
norpymeHHorO B PTYTb:
1) He H3MeHHTcSI; 2) YMeHblllHTCSI H8 4,3 %; 3) YBenH'lHTCH H&. 4,3 % ;
4) YMeHblllHTcSI H8 5.9 %; 5) YBenHtJHTCSI Ha 5.9 %.
A25. TIpoecc, npoHcxoASIHH C ,A8HHOH M8CCOH H,Ae8nbHOro r838, onHCbI-
B8eTCSI yp8BHeHHeM VT 3 = const. EcnH 06'beM ra3a YMeHblllHnH B k = 2 pa38,
TO ,A8BneHHe P2 r838:
1) YBenH'lHnOCb B 2 p83a; 2) YMeHblllHnOCb B 2 pa38: 3) YBenH'lHnOCb B
2.5 P838; 4) YMeHblllHnOCb B 2,5 pa3a; 5) YBenH'tIHnOCb B 4,0 pa3a.
A26. EcnH npli H3068pHOM p8ClllHpeHHH renHSI 6hIn8 COBepllleH8 pa60T8
A = 156.8 ,D;m, TO eMY 6bIno co06eHO KOnH'leCTSO TennOTbl Q, p8BHoe:
1) 234 ,D;m; 2) 314 ,D;m; 3) 392 ,D;m; 4) 468 ,D;>K; 5) 549 ,D;m.
A27. nOD. KonOKonOM B03,AYlllHoro H8coca H8XOD.HTCSI BOD.a. M8cca KOTOpOH
m = 40 I', a TeMnepaTypa t = 0 °C. 3a C'leT HHTeHCHBHoro Hcn8peHHSI npOHCXO-
AHT 3aMop8mHBaHHe BO,AbI. Y,AenbHaSl TennOTa nnaBneHHSI nbAa A = 340 
Kr
M)];m
y.aenbH8S1 TennOT8 napo06pa30B8HHSI BOD.bI L = 2.3 -. EcnH TeMnep8Typ8
Kr
06pa30BaBlllerocSI nb,Aa t = 0 °C, TO ero M8cca ml paBHa:
1) 35.6 r; 2) 35,2 r; 3) 34,8 r; 4) 34,4 r; 5) 34,0 r.
A28. IIIapHK, HMelOHH 3apSl,A ql = 1 . 10- 7 Kn, nOABellleH H8 lllenKoBoH
HHTH B B03,Ayxe. HHTb COCT8Bn.HeT yron ex = 45° C BepTHKanblO B TOM cnY'l8e,
ecnH H8 paCCTOHHHH r= 3 CM OT nepBoro lllapHKa nOMeCTHTb BTOpOH W8pHK
C 38pSl,AOM q2= 1 . 10- 8 Kn TaK, 'lT06bI 06a 1118pHKa H8XO,AHnHCb Ha O,AHOH
I'OpH30HT8nH. Macc8 m nepBOI'O W8pHK8 paBHa:
1) 1. 10- 1 Kr; 2) 1. 10- 2 Kr; 3) 1. 10- 3 Kr; 4) 1. 10- 5 Kr; 5) 1. 10- 6 KI'.
Kr
A29. B M8cne, nnOTHOCTb KOToporo P = 900 3 Ii ,AHaneKTpH'leCK8S1 npo-
M
HHaeMOCTb E = 2,2, HaXO,AHTCSI BO B3semeHHOM COCTOSlHHH anlOMliHHesbIH (nnoT-
3 Kr
HOCTb anlOMHHHSI Pa = 2,7 10 3) mapHK C 38pSl,AOM q = 1,0' 10- 7 Kn, ecnH
M
682

COCYA C MaCnOM H8XO,AHTCSI BO BHeWHeM BepTHKaJIhHOM 3neKTpH'tJeCKOM none C
B
HanpSl>KeHHOCThlO E = 13,2' 10 5 -. Macca m m8pHKa paBHa:
M
1) 9,0 r; 2) 10 r; 3) 50 r; 4) 80 r; 5) 90 r.
A30. KOJlHqeCTBO naMn N, BKnlO'leHHblX napanneJIbHO, paCC'tJHTaHHblX H8
HanpSl>KeHHe U = 100 B H nOTpe6nSlI01l\HX M01l\HOCTb P = 50 BT K8m,A8S1, cno-
C06HbIX ropeTb nOJIHblM HaKaJIOM npH nHT8HHH HX OT 6aT8peH C 3C g' = 120 B
H BHYTpeHHHM conpOTHBJleHHeM r= 10 OM, p8BHO:
1) 2; 2) 4; 3) 5; 4) 6: 5) 8.
A3l. IIpoTOH BJIeTaeT B O,AHOpO,AHOe MarHHTHoe nOJle C HH,AYKHeH
B = 10 MT JI co CKOpOCThlO V = 1,0 . 1 0 4  nOA yrnoM 60° K nHHHSlM MarHHTHOH
C
HH,AYKHH. IIIar BHHTOBOH nHHHH, no KOTOPOH ,ABHmeTCSI npOTOH, paBeH:
1) 3,3 CM; 2) 4,6 CM; 3) 5,8 CM; 4) 6,5 CM; 5) 8,5 CM.
A32. rpY3 COBepwaeT BepTHKanbHhle Kone6aHHSI Ha pe3HHOBOM wHype C
nepHO,AOM T. ECJIH WHYP CJIOmHTb B,ABOe H nO,ABeCHTb Ha HeM TOT me rpY3.
TO nepHOA Kone6aHHH T I CT8HeT paBHbIM:
1 ) 3T 2 ) 3T . 3 ) T 4 ) 2T , ' 5 ) T
7; 2 ' 4"; 2
A33. sa nnOCKHX 3epK8na p8cnonomeHbI no,A HeKOTopbIM yrnoM ,Apyr K
APyry. EcnH HCTO'tJHHK CBeTa Hero nepBbIe H306p8meHHR B 3epK8n8X H8XO-
,ASlTCSI B BepWHH8X paBHOCTopOHHero TpeyronbHHK8, TO yron q> Mem,Ay 3epK8-
naMH p8seH:
1) 135°; 2) 120°; 3) 90°; 4) 60°; 5) 45°.
BBepx v2= 17
Bl. EcnH CKOpOCTb ,ABHmeHHSI TennOXO,A8 BHH3 no peKe VI = 21
KM
KM
q
TO p8CCTOSlHHe S = 6 KM B STOH peKe nnOT npOH,AeT 38
8
tJ
BpeMSI t, paBHoe ... ('I).
m
Puc. 30
B2. Teno M8CCOH m= 10.0 Kr nOAHHMalOT H8 BepWHHY
H8KnoHHoH nnOCKOCTH ,AnHHOH 1=400 CM H yrnoM H8Kno-
H8 a. = 30°. EcnH K034><l>HIlHeHT TpeHHSI Tena 0 nnOCKOCTh
J.1 = 0,500, TO CHna TpeHH.H F npH 3TOM COBepmaeT p860TY
A, MO,Aynb KOTOpOH p8BeH ... (m).
a
B3. Qepe3 HeBecoMblH 6noK nepeKHHYTa HHTh C rpY3a-
MH ml = 1,0 Kr If m2= 3,0 Kr, 3aKpenneHHbIMH Ha ee
KOH8X (pHC. 30). BnoK ,ABHmeTCSI BHH3 C YCKopeHHeM
M
a = 2,0 2' EcnH npeHe6pe'lb TpeHHeM, TO CHna ,A8BJIeHHSI
C
F Ha OCb 6JIOKa paBHa ... (H).
3m
683

B4. K HCTO'lHHKY TOKa C BHYTpeHHHM COnpoTHBneHHeM r = 2 OM BHa'l8JIe nO,A-
KnlO'l8eTCSI pe3HCTOp C COnpOTHBneHHeM R = 2 OM, 3aTeM n8pannenbHO nO,AKnlO'la-
eTCSI TaKOH me pe3HCTOp. KOnH'leCTBO TennOTbI Q, BbI,AenSleMOH B nepBOM pe3HCTope
nocne nOKnlO'leHHSI SToporo pe3HCTopa, 3a TaKoe me BpeMSI YMeHblll8eTCSI B ... .
B5. ):{syxnpOBO,AHaSl nHHHSI ,AnHHOH l = 1000 KM BbInOnHeH8 H3 Me,AHOrO
(y,AenbHOe conpOTHBneHHe p = 1,70 10- 8 OM. M) npOSO,Aa C nnOIl.\8,AblO nonepeq-
Horo CeqeHHSI S = 17,0 MM 2 . IIpHeMHHKH 3HeprHH rIoTpe6nSllOT MOIl.\HOCTb P = 2,20
KBT npH HanpSlmeHHH U = 220 B. IIoTepH MOIl.\HOCTH PI B npOBO,A8X nHHHH
COCT8BnSllOT ... (BT).
p
2po
: ;?1 :
I I
I I
Po
o
V o
Puc. 31
3V o
684
B6. TIynSl, JleTSlIl.\8S1 ropH30HT8nbHO. no-
na,A8eT B 1118p. nO,ASellleHHbIH H8 nerKOM me-
CTKOM CTepmHe. 11 3aCTpeBaeT B HeM. M8cca
nynH ml = 5,0 r, Macca lllapa m2 = 0,50 Kr,
M
CKOpOCTb nynH VI = 500 EcnH lllap OT
C
y,Aapa nynH nOHHM8eTCfl o sepxHeH TOqKH
oKpymHocTH, TO paCCTOSlHHe l OT TO'lKH no-
Beca O eHTp8 m8p8 paSHO ... (CM).
v
B7. Ha p-V-,AHarp8MMe (pHC. 31) H306pa-
meH HKn, npOBO,AHMbIH C OHoaTOMHbIM H,Ae-
anbHbIM ra30M. Kn):{ HKna 11 paBeH ... (%).

X.N
180
120
60
o
40
60 t. C
20
Puc. 32
TECT VII
Al. rpa<l>HKH 38BHCHMOCTH KOOp,AHHaTbI
,D;BYX Ten OT BpeMeHH npe,D;CT8BneHbI H8 pH-
cYHKe 32. IIYTH 81 H 82' npOH,AeHHble nep-
BbIM H BTOpbIM Ten8MH ,D;O BCTpe'lH, p8BHbI
COOTBeTCTBeHHO:
2
1) 60 M; 120 M;
2) 60 M; 180 M;
3) 120 M; 180 M;
4) 120 M; 60 M;
5) 180 M; 120 M.
r r
A2. EcnH nnOTHOCTb cepe6pa p = 10,5 CM 3 ' MonSlpH8S1 M8CC8 M = 108 Monb '
TO B 06'beMe V = 180 CM 3 cepe6pa CO,AepmHTCSI 'lHCnO N MoneKYJI, p8BHoe:
1) 210. 1024; 2) 105. 1024; 3) 21.0' 1024; 4) 10,5' 1024; 5) 1.05 . 10 24 .
A3. ,Z:(Ba llIapHKa, HMelOHX O,AHHaKOBbIe OTpH8TenbHble 3apSl,D;bI, B38HMO-
,AeHcTBYIOT C cHnoH F = 0,23 MH H8 p8CCTOSlHHH r = 10 CM. "lJHcno N H36bI-
TO'lHbIX sneKTpOHOB H8 K8m,AOM H3 1lI8pHKOB paBHO:
1) 1. 1012; 2) 1. 1011; 3) 1. 1010; 4) 1. 109; 5) 1. 10 8 .
A4. "lJ To6bI n030nOTHTb C ,ABYX CTOpOH TOHKYIO npSlMoyronbHYIO nnacTHHY
C pa3MepaMH a x b = 3 x 4 CM 2 cnoeM TonHHoH h = 10 MKM, 'lepe3 sneKTpo-
r
nHT nponycK81OT TOK 1=1,5 A. EcnH nnOTHOCTb 30nOTa p = 19,3 - MonSlp-
CM 3 '
H8S1 Macca
M = 197
['
-.
Monb
p8BHO:
BaneHTHOCTb Z = 1, TO BpeMSI /:1t nponYCK8HHSI TOKB
1) 3,8 MHH; 2) 4.6 MHH; 3) 5,7 MHH; 4) 6,5 MHH; 5) 7,6 MHH.
A5. EcnH npH CHne TOKR 1=5.00 A M8rHHTHbIH nOTOK BHYTPH coneHOH-
,D;8 cI> = 500 MB6, TO 3HeprHSI W M8rHHTHoro nonSl COneHOH,Aa paBHa:
1) 1.25 ,Z:(m; 2) 2,25 ,Z:(»<; 3) 3.15 ,Z:(m; 4) 3,25 ,Z:(m; 5) 4,15 ,Z:(>K.
A6. 3Kp8H p8cnonomeH Ha p8CCTOSlHHH l = 1,0 M OT ,AH<I>paKHoHHOH pe-
llIeTKH. EcnH MaKCHMYM BToporo nOpSl,AK8 H8 3Kp8He pacnOJIOmeH H8 paccTo-
HHHH Xl = 20 CM OT eHTpanbHoro M8KCHMYM8, TO M8KCHMYM TpeTbero nopSl,A-
K8 pacnonomeH H8 p8CCTOSlHHH X2 OT eHTpanbHoro, p8BHOM:
1) 28 CM; 2) 29 CM; 3) 30 CM; 4) 31 CM; 5) 32 CM.
685

A 7. KIfHeTlf'leCKaH 3Hepr1:lSl W K 3neKTpOHa, ,ABlfmYIl(erOCSI co CKOpOCTblO
0,60 e (e - CKOpOCTb CBeT8 B BaKYYMe), p8BH8:
1) 0.08 M3B; 2) 0,11 M3B; 3) 0.13 M3B; 4) 0,18 M3B; 5) 0,20 M3B.
A8. ,l{nHH8 BonHbl A. <l>oToHa. HcnYCK8eMoro npH nepeXO,Ae 8TOMa H3 B03-
6ym,AeHHOrO COCTOSlHHSI C 3HeprHeH W n B HeB036ym,AeHHOe C 3HeprHeH W 0'
p8BH8:
1)
he
W n - W o
2) W n - W o . 3) hv ; 4) he ; 5) W o - W n
he' e A h
A9. llplf 60M6ap,AHpOBKe SI,Aep 1f30Tona Mg a-'laCTIfeH BbIneTaeT HeHT-
pOH H 06p83yeTcSI H,APO H30Ton8:
1) Si; 2) Si; 3) Si; 4) S; 5) :S.
AIO. B MOMeHT, Kora TpoHyncH nOe3,A, npOBOmalOIl(HH H8'lan paBHoMep-
M
HO 6em8Tb no XO,AY nOe3,A8 co CKOpOCTblO v = 3,50 -. EcnH C'lHT8Tb ,ABHme-
C
HHe nOe3,A8 paBHOYCKopeHHbIM, TO CKOpOCTb nOe3,A8 VI B TOT MOMeHT, KorA8
npOBomaeMbIH nopaBHSlncSI C npOBOmalOIl(HM, paBHa:
M
1) 1,75
C
2) 3.50 M 3) 5,25 M 4) 7,00 M
C C C
M
5) 10, 5
C
All. K Teny MaccoH m=100 Kr, H8XO,ASlIl.{eMYCSI H8 HaKnoHHoH nnOCKOCTH,
cocTaBnSlIOIl.{eH C ropH30HTOM yron 30°, npHnomHnH BHemHlO1O cHny F, H8npaB-
neHHYIO ropH30HT8nbHO. B pe3ynbTaTe Teno H8Q8nO nO,AHHMaTbCSI no H8KnOHHOH
M
nnOCKOCTH C YCKopeHHeM a = 5.0 2' EcnH K03<1><PHHeHT TpeHHSI Mem,Ay TenOM
C
H HaKnOHHOH nnOCKOCTblO J.1 = 0,10, TO MO,Aynb BHelllHeH CHnbl F p8BeH:
1) 1,0 KH; 2) 1,1 KH; 3) 1,2 KH; 4) 1,3 KH; 5) 1,5 KH.
A12. ,ZJ;Boe p860tJHX HeCYT 6peBHo AnHHoH l = 6.0 M. TIepBbIH p86o'lHH
nOA,AepmlfB8eT 6peBHo H8 paCCTOSlHHH S = 1,0 M OT ero KOHa. BTOpOH - npo-
THBononomHblH KOHe 6peBHa. CHnbl ,AaBneHHH F 1
Ha nne'lH nepBoro pa60'lero H CHna A8BneHHSI
F 2 H8 nne'lH BToporo pa60'lero TaKue, tJTO:
P
2po
h3
1) Fl = 2,OF 2 ; 2) F 2 = 2,OFl; 3) 0,AHH8KOBbIe;
4) Fl = 1.5F 2 ; 5) F 2 = 1,5Fl'
Po
o
11
I
I
VI
Pue. 33
A13. EcnH nocne oCYIl(eCTBneHHSI npoeCC8 1-
2-3 (pHC. 33) KB8,AP8T Cpe,AHeH KB8,ApaTH'lHOH
CKOpOCTH )lBHmeHHSI MoneKyn H)leanbHOro r83a
YBenH'lHnCSI B 6 p83. TO 06'beM V 3 60nbille 06'be-
M8 VI B ... p83.
Va V
1) 1; 2) 2; 3) 3; 4) 4; 5) 5.
686

A14. TIpH H306apHOM OXnam,AeHHH HAeaJIbHOrO r838 ero nnOTHOCTb YBenH-
'lHJIaCb B,ABOe ECJlH Macca r83a He H3MeHSleTCSI, 8 nepBOH8q8nbH8Sl TeMnep8-
Typa T 1 = 380 K, TO TeMnep8Typ8 ra3a B npoecce:
1) YMeHbWHJI8Cb H8 100 K; 2) YMeHbWHnaCb Ha 190 K;
3) He H3MeHHnacb; 4) YBeJIH'lHJI8Cb Ha 100 K; 5) YBenH'lHnaCb Ha 190 K.
A15. P8CcToSlHHe Mem,AY ,ABYMSI ropH30HTaJIbHO p8cnonomeHHbIMH nnaCTHH8-
MH d = 2,0 CM, pa3HOCTb nOTeHI.\H8nOS ttq> = 11,6 B. TIPOTOH. npOH,ASl H3 coc-
TOSlHHSI nOKOSl no,A ,AeHCTBHeM CHnbl SJIeKTpOCT8THQeCKOrO nonSl p8CCTOSlHHe
s = 3,0 MM, npHo6peT8eT CKOpOCTb u, paBHYIO:
1) 2,3.10 3 M 2) 2,1.10 3 M 3) 1,8.10 4 M 4) 1,4.10 3 M 5) 1,2.104 M
C C C C C
A16. KOH,AeHCaTOpbI eMKOCTSlMH C 1 = 1,0 MKcI>, C 2 = 2,0 MKcI> Ii C 3 = 3,0 MKcI>
COe,AHHeHbI nOCne,AOBaTenbHO H npHCOe,AHHeHbl K HCTO'lHHKY C H8npSlmeHHeM
V = 110 B. 3apSl,A q Ha KOH,AeHC8TOpe C 2 p8BeH:
1) 60 MKKn; 2) 50 MKKn; 3) 6,0 MKKn; 4) 50 MKn; 5) 5,0 MKn
A17. TIpH YBenH'leHHH BHeWHero conpOTHBneHHSI sneKTpH'leCKOH l.\enH B k
pa3 HanpHJKeHHe Ha HarpY3Ke YBenHQHBaeTCH OT VI AO V 2 . S.AC g' 68T8peH
paBHa:
1) U 2 (k-l). 2) U I U 2 (k-1) . 3) U I . 4) U 1 U 2 '5) Ud k - 1 )
U 1 ' U 1 k-U 2 ' U 2 (k-1)' U I k-U 2 ' U 2 -U 1
A18. TIpOBO,AHHK AnHHoH l H MaCCOH m nO,ABeWeH H8 TOHKHX npOBOnO'l-
K8X. TIpH nponYCKaHHH no HeMY TOKa OH OTKnOHHncSI B O,AHOP0,AHOM BepTH-
KanbHOM MarHHTHOM none TaK. 'lTO npOBOnO'IKH 06pa30B8nH yron a C BepTH-
KaJIblO. ECJIH CHna TOKa I, TO HH,AYKI.\HSI B MarHHTHoro nonSl p8BH8:
1 ) mg . 2 ) mg sin a . 3 ) mg tg a . 4 ) mg cos a . 5 ) mg ctg a
Il ' Il' Il' II' II
A19. )J';Ba MeT8nnH'leCKHX CTepmHSI p8cnonomeHLI BepTHK8nbHO H 38MKHY-
TbI cBepxy npOBO,AHHKOM. TIo CTepmHSlM 6e3 TpeHHSI H HapyweHHH KOHT8KT8
CKonb3HT nepeMbI'lKa ,AnHHOH l = 20 CM H MaccoH m = 1 r. BCH CHCTeMa H8XO-
,AHTCH B O,AHOpO,AHOM MarHHTHOM none C HH,AYKI.\HeH B = 0.01 Tn, nepneH,AH-
KynSlpHoM K ee nnOCKOCTH. 'YCT8HoBHBwaSlcSI CKOpOCTb ,ABHmeHHSI nepeMbl'lKH
u = 1 M . EcnH npeHe6peQb conpoTHBneHHeM CTepJKHeH H nO,ABO,A.HI.ltHX npoBo-
C
,AOB, TO conpOTHBJIeHHe nepeMbI'lKH R paBHO:
1) 0.2 MOM: 2) 0.3 MOM: 3) 0.4 MOM; 4) 0.5 MOM; 5) 0.6 MOM.
A20. AKTHBHoe conpOTHsneHHe R = 150 OM H KOHAeHC8TOp eMKOCTLIO
C = 5 MKcI> SKnlO'leHbI napannenbHO B enb nepeMeHHoro TOKa HanpSlmeHHeM
U = 120 B H QaCTOTOH v = 50 rl.\. )J';eHcTBYIOI.ltee 3H8QeHHe CHnbI TOKa I paBHO:
1) 0,4 A; 2) 0,5 A; 3) 0,6 A; 4) 0,7 A; 5) 0,8 A.
687

A21. P8CCTOSlHHe H8HnytJIllerO 3peHHSI 3,AOpOBOrO rn838 d o = 25 CM. Ecnl1
paccToSlHHe H8HnY'lIllerO 3peHHSI .a;nSl 'lenOBeKa d = 1.0 M. TO eMY Heo6xo,AHMbJ
OiKH C OnTH'leCKOH CHnOH D, paBHOH:
1) -3 ,AnTp; 2) +3 ,AnTp; 3) -2,5 nTp; 4) +2,5 ,AnTp; 5) +2 ,AnTp.
A22. SHepl'HR WI O,AHOrO <l>OTOH8 B k = 9 p83 60nbIlle 3Hepl'HH W 2 BToporo
<l>oTOHa. OTHOIlleHHe MO,AyneH HMnynbcoB PI 3THX <l>OTOHOB p8BHO:
P2
1) 3; 2) 9; 3) 18; 4) ; 5)
1
9
A23. MOTOl1.HKnHCT ,ABH>KeTCSI no BbmYKnOMY MOCTY, p8,AHYC KpHBH3HbI KO-
TOpOI'O R = 120 M. 'llT06bI B BepxHeii TO'lKe Tp8eKTopHH CHna ,A8BneHHfl H8
,Aopory 6bIn8 B k = 3 p838 MeHbIlle, 'leM npH ,ABH>KeHHH H8 ropH30HTanbHOM
Y'l8CTKe, OH ,AOnmeH ,ABHI'8TbCSI co CKOpOCTblO v, paBHoH:
1) 102 KM ; 2) 72 KM ; 3) 60 KM ; 4) 54
q q 'I
KM
'I
5) 36
KM
"l
A24. B nnaB81Oll\eH nb,AHHe npO,AOn6HnH npopy6b.
r
naCb h = 10 M. IInoTHocTb nb8 PlI = 0.90 ---a' nnOTHOCTb BO,AbI
CM
'll To6bI onYll\eHHOe H8 BepeBKe Bepo ,AOCTHI' no
MHHHManbHOH ,AnHHOH l. p8BHOH:
TOnll\HH8 JIb,Aa OKa3a-
r
P B = 1, 0 .
CM
BO,AW, He06xoHMa BepeBKa C
1) 10 M; 2) 9,0 M; 3) 5,0 M; 4) 4,0 M; 5) 1.0 M.
A25. B l1.HnHH,Ape C nopIllHeM HaXO,AHTCSI r83 M8CCOH m = 12,02 Kr, 3a-
HHM81Oll\HH 06'beM VI = 4000 CM 3 npH TeMnepaType T 1 = 450 K. EcnH 8Bne-
r
HHe r838 He H3MeHSleTCSI, TO ero nnOTHOCTb P = 6000 6Y,AeT npH TeMne-
M 3
paType T2' paBHoH:
1) 450 K; 2) 300 K; 3) 225 K; 4) 200 K; 150 K.
A26. l1e8nbHbIH ra3 B KOnHqeCTBe v Monb HarpeB8eTCSI TaK, tJTO ero
TeMnep8Typa H3MeHSleTCSI OT T 1 ,AO T 2 npSlMO nponopl1.HOHanbHO KB8,ApaTY ,A8B-
neHHSI 1'83a. P860T8 A, COBepIlleHH8S1 npH aTOM, p8BH8:
1) vR (T2 - Td; 2) vR (Tl- TI2); 3) 2vR (Tl- TI2); 4) v: (T2 - Td;
5) v: (Tl- TI2).
A27. B CT8JIbHOH K8JIOpHMeTp MaccoH ml = 300 r, cO,Aep>Kall\HH V 2 = 2000 CM 3
BO,AbJ npH TeMnepaType t l =26,O°C, 6pOC81OT MOKpbIH CHel' M8CCOH m3=400 1'.
Am
Y.llenbHaSl TennoeMKOCTb CTanH C 1 = 460 Kr' K ' Y,AenbHaSl TennoeMKOCTb
688

Km KAm
BOAbI C 2 = 4,20 -, Y,AenbHaSl TennOT8 nnaBneHHSI nbAS A = 340 - ,
Kr.K Kr
Kr
nnOTHOCTb BO,AbI p = 1000 3' EcnH B KanopHMeTpe YCTSHOBHnSCb TeMnepa-
M
Typa 0 = 14,0 °C, TO MSCCS m BO,AbI, KOTOpSSI 6bInS B CHere, paBHS:
1) 150 r; 2) 168 r; 3) 178 r; 4) 188 r; 5) 200 r.
A28. TpH O,AHHaKOBbIX TO'le'tJHbIX nonomHTenbHbIX 3SpSl,Aa q nOMeeHbI B
BepWHHax paBHOCTopOHHero TpeyronbHHKa. QTo6bI CHCTeMa HSXO,AHnaCb B paB-
HOBeCHH, B u.eHTpe TpeyronbHHKB HymHo nOMeCTHTb 3apH,A qb paBHbIH:
1) -q,Ja; 2) -3q; 3) -%; 4) - q : 5) +O.3q.
A29. Ba To'tJe'lHbIX 3SpSl,Aa ql = +5q H q2 = -2q HaXO,ASlTCSI HS paCCTOSlHHH
l = 10 CM APyr OT ,Apyra. HanpSlmeHHocTb 3neKTpH'tJeCKoro nOJlSl paBHa HynIO
B TO'lKe, pacnonomeHHoH OT nepBoro 3apSl,Aa H8 paCCTOSlHHH r, paBHOM:
1) 0,27 M; 2) 0,35 M; 3) 0,44 M; 4) 0,51 M; 5) 0,61 M.
A30. ConpoTHBneHHe O,AHOrO H3 nOCne,AOBSTenbHO COe,AHHeHHbIX npOBO,AHH-
KOB B k = 3,0 pa3a 60nbwe conpoTHBneHHSI BToporo EcnH 3TH npOBO,AHHKH
BKnIO'tJHTb nspSJIneJIbHO, a HanpHmeHHe H8 HHX OCTSBHTb npemHHM, TO CHna
TOK8 B u.enH H3MeHHTCSI B ... pa3.
1) 2,7; 2) 3,5; 3) 5,3; 4) 5,8; 5) 6,1.
A31. Ba npOTOHa C KHHeTH'tJeCKHMH 3HeprHSlMH W KI H W K2 COOTBeTCTBeH-
HO B O,AHOpO,AHOM MarHHTHOM none ,ABHmYTCH no oKpymHocTSlM B nnOCKOCTH,
nepneH,AHKynSlpHoH nHHHHM HHAYKII.HH MarHHTHOI'O nonSl. C'lHTaHTe, 'tJTO HX
CKOpOCTH 3Ha'tJHTenbHO MeHbwe CKOpOCTH CBeTa B BSKYYMe. OTHoweHHe nepHo-
AOB HX o6paeHHSI T 2 paBHo:
Tl
111
1) 2; 2) 1; 3) 2' 4) 3' 5) 4'
A32. QTo6bI nepHO Kone6aHHH MSTeMSTH'leCKOrO MSSlTHHKa Ha BbICOTe
h = 10 KM Ha,A nOBepXHOCTbIO SeMnH (pSlIHYC 3eMnH R = 6400 KM) 6bIn paBeH
nepHOAY Kone6aHHH Ha 3eMJIe. HymHo YMeHbWHTb lInHHY MSSlTHHKS HS /)'l.
pSBHoe:
1) 0,003 l; 2) 0,001 l; 3) 0,03 l; 4) 0,06 l; 5) 0,01 l.
A33. Ha6nIO,AaTenb CMOTpHT no BepTHKanH BHH3 Ha npeAMeT, HSKpbITbI:H
nnocKonapannenbHoH cTeKnSlHHoH nnacTHHKoH C nOKS3aTeneM npenOMneHH.fI
n=1,5 Ii TonHHoH h=3,0 CM. C'lHTaHTe, 'lTO AnSI ManbIX yrnoB tga::=sina.
Hs6nIO,AaTenb YBH,AHT H306psmeHHe npe,AMeTS HS paccToSlHHH d OT BepxHeH no-
BepxHocTH nnSCTHHbI. paBHOM:
1) 1,0 CM; 2) 1,5 CM; 3) 2,0 CM; 4) 2,5 CM; 5) 3,0 CM.
689

KM
Bl. ABTOKOnOHHS ,AnHHOH l = 700 M ,ABHmeTCSI co CKOpOCTblO U = 54,0 -.
'I
MOToHKnHcT Bblexan H3 .ronOBbU KOJIOHHbl B ee tXBOCT. H BepHyncH 06-
KM
pSTHO. EcnH CKOpOCTb MOTOltHKnHCTa OTHOCHTenbHO 3eMnH VI = 72,0 -, TO
'tJ
OH BepHyncSI 'lepe3 BpeMfl t, paBHOe ... (C).
m 1
fa
B2. K 6pYCKY MSCCOH m= 1.0 Kr npHKpenneHa ropH-
H
30HTanbHSSI HeBeCOMSSI npymHHa meCTKOCTblO k = 40 -.
M
K03cl>cl>HHeHT TpeHHSI Mem,Ay 6PYCKOM H ropH30HTsnbHoH
nOBepXHOCTblO Jl = 0,80. ECJlH BHs'tJane npymHHs He ,Ae-
cl>OpMHpOBSHS, TO ,AnSI paBHOMepHoro nepeMell\eHHSI 6pyc-
Ka HS S = 2,0 M Heo6xo,AHMO COBeplllHTb MHHHMSJlbHYIO
pa60TY A, PSBHYIO... (,[(m).
m 2
B3. Qepe3 HeBeCOMblH 6noK, nO,ABellleHHblH K npy-
Puc. 34 mHHe ,AHHSMOMeTpa (pHC. 34), nepeKHHYTS HHTb C rpy-
3SMH ml = 1,0 Kr H m2= 3,0 Kr, 3aKpenneHHblMH Ha ee
M
KOHax. BnoK AB1:I>KeTCH BBepx C YCKopeHHeM a = 2,0 . EcnH npeHe6pe'tJb
c 2
MaCCOH AHHaMOMeTpa, TO ero nOKa3aHHe paBHO ... (H).
B4. K HCTO'lHHKY TOKS C BHYTpeHHHM conpOTHBneHHeM r= 2 OM BHS-
'lane nO,AKnlOqaeTCSI pe3HCTOp C conpOTHBneHHeM R = 2 OM, 3aTeM napsn-
nenbHO nO,AKnlO'tJseTCSI TaKOH me pe3HCTOp. KOnH'leCTBO TennOTbl Q, Bbl-
,AenSleMOH B nepBOM pe3HCTope nocne nO,AKnlO'tJeHHH BToporo pe3HCTopa, 3S
TaKoe me BpeMSI YMeHbwaeTCSl B (pS3).
B5. KaTYWKa C HH,AYKTHBHOCTblO L = 2 MrH npHCOe,AHHeHa I( nnOCKOMY B03-
,AYlllHOMY KOH,AeHCSTOpy C nnOll\a,AblO nnSCTHH S = 100 CM 2 . KOHTYP HSCTpoeH
Ha AnHHY BOJIHbl A = 100 M. PaccToSlHHe d MemAY nnaCTHHaMH KOHAeHCSTopa
pSBHO ... (CM).
B6. ,[(Ba rpY3S MaccaMH ml = 3,0 Kr H m2= 1,0 Kr coeAHHeHbI HHTblO' ne-
pe6poweHHoH qepe3 HenO,ABHmHbIH 6noK. H pacnonomeHbl Ha,A CTonOM Ha Bbl-
COTe h = 60 CM. B Ha'lSnbHblH MOMeHT rpY3bl nOKOSlTCSI, 3STeM HX OTnYCKS-
lOT. EcnH npH YAspe 0 CTon rpY3 MaccoH ml npHnHnaeT K cTony, TO Bbl,Ae-
nHTCSI KonH'tJeCTBO TennOTbI Q, paBHoe ... (,[(m).
B7. IIo,lI noplllHeM HAeanbHOH TennOBOH MalllHHbl HSXOAHTCH ra3. 3a OAHH
HKn MaW1:IHa nOJ:tHHMaeT nopllleHb MaCCOH m = 5 Kr Ha BblCOTY h = 20 M H
k - - 625 KH 8 E
cmHMaeT npH 3TOM npymHHY meCTKOCTblO Ha x = CM. cnH ra3
M
nonY'laeT OT HarpeBaTeJISI 3a O,AHH HKJI QI = 6 K.lIm TennOTbl, TO a6conlOT-
HaSl TeMnepsTypa HsrpeBaTenSl Bblllle TeMnepaTypbl XOnO,AHnbHHKa B ... (pS3).
690

TECT VIII
o
1
AI. rpa<}>HKH 3SBHCHMOCTH KOOp,AHHaTbl
,ABYX Ten OT BpeMeHH npe,ACTaBneHbl HS pH-
CYHKe 35. PaCCTOSlHHe S Mem,AY 3THMH Te-
nSMH H3MeHSleTCSI co BpeMeHeM no 3aKOHY:
1) He H3MeHSleTCSI co BpeMeHeM;
2) S = 140 - 2tl;
3) S = 120 - tl;
4) S = 1-20 + 2tl;
5) S = 140 + 20tl.
X,M
2
60
40
20
t, C
A2. ECJIH MOJISlpHSSI
r
KHCnOpO,AS M = 16 -,
MOJlb
pSBHa:
MaCC8 aTOMapHoro
TO Macca mo Mone-
-20
KynbI
KHcnopOAa
1) 2,7' 10- 26
3) 5,3' 10- 26
5) 5,3' 10- 24
Kr;
Kr;
Kr.
2) 2,7' 10- 25
4) 5,3. 10- 25
Kr;
Kr;
Puc. 35
A3. )J;Ba O,AHHaKOBbIX MeTaJInH'leCKHX lliSpHKS HMelOT 3apSl,Abl ql = q H
q2= 5q. illSpHKH npHBenH B conpHKOCHOBeHHe H BHOBb pS3,ABHHynH HS Ta-
Koe me pSCCTOSlHHe. CHJla B3SHMO,AeHCTBHfl F npH 3TOM:
1) YBenH'lHnaCb B 1,8 pa3a; 2) YMeHblliHnaCb B 1,8 pS3a;
3) He H3MeHHnscb; 4) YBenH'lHnSCb B 3 pS3a,
5) YMeHblliHnaCb B 3 pa3a.
A4. EcnH aneKTpOXHMH'leCKHH 3KBHBsneHT HHKenSl k = 0.30. 10- 6 , TO
Kn
npOMemYTOK BpeMeHH 6t, B Te'leHHe KOToporo npH HHKenHpOBaHHH H3,AenHSI
BblenHncSI CJIOH HHKenSl MSCCOH m = 4,0 r npH CHne TOKa 1=3,0 A, pSBeH:
1) 2,2,10 3 c; 2) 3,2. 10 3 c; 3) 3.8. 10 3 c; 4) 4,1. 10 3 c;
5) 4,4. 10 3 c.
A5. B COneHOH,Ae cO,Aep>KSll\eM N = 400 BHTKOB, npH H3MeHeHHH CHnbl
TOKa nOTOK HHYKI.\HH MsrHHTHoro nonSl, 06pS30BsHHoro O,AHHM BHTKOM, YBe-
JIH'lHnCSI HS £1<1>=6,0 MB6 38 BpeMSI 6t=0.10 c. Cpe,AHSISI 3)J;C caMOHH-
,AYKI.\HH <g.,> B COJIeHOH,Ae pSBHa:
1) 12 B; 2) 18 B; 3) 24 B; 4) 26 B; 5) 28 B.
A6 IT N - _ 200 lliTP . co
. ,....H<I>paKHoHHSSI pellieTKa CO,AepmHT I:.IcnH Ha pellieT-
MM
KY na,ASeT HOpMaJIbHO CBeT C nHHoH BonHbI A.= 600 HM, TO HaH60nblliHH
nopSlOK mmax MaKCHMYMa paBeH:
1) 2; 2) 3; 3) 4; 4) 6; 5) 8.
A 7. EcnH KHHeTH'leCKSSI 3HeprHSI W K penSlTHBHcTCKO:U llSCTHl.\bI paBHS ee
3HeprHH nOKO.H W O , TO 'laCTHn;a BH>KeTCSI CO CKOpOCTblO v, pSBHOH:
1)  c; 2)
1 C' 3 ) .J2 C' 4 ) .J5 c; 5) J3 c.
2 · 3' 3 3
691

A8. ,l(nSl HOHH3aIUI aTOMa BO,AOpO,Aa HymHa 3HeprHSI W = 14 3B. ,l(nH-
Ha BonHbI A H3nY'leHHSI, cnoco6Horo BbI3BaTb HOHH3aHIO, paBHS:
1) 8,9' 10- 6 M; 2) 7,5.10- 7 M; 3) 8,9 .10- 8 M; 4) 6,8' 10- 9 M;
5) 5,3. 10- 9 M.
A9. AKTHBHOCTb pa,AHOaKTHBHoro 3neMeHTa YMeHbwaeTcSI B 4 pS3S 3a
8 CYTOK. TIepHO,A nonypacnS,Aa T paBeH:
1) 2 CYTOK; 2) 4 CYTOK; 3) 6 CYTOK; 4) 8 CYTOK; 5) 9 CYTOK.
AIO. TIoe3,A OTOWeJI OT CTaHIIHH H. ,lIBHraSlCb B Te'leHHe t = 20,0 C
paBHOYCKopeHHO. npowen 3a ,AeCSlTYIO ceKYHl\Y nYTb 8 = 9.50 M. ITYTb 81.
npOHlIeHHbIH nOe31IOM 3a 20.0 c, paBeH:
1) 95,0 M; 2) 100 M; 3) 190 M; 4) 200 M; 5) 295 M.
All. CSHKH MSCCOH m = 75 Kr CKaTbIBSIOTCSI C BepWHHbI ropbI BbICOTOH
h = 11 M H ,AnHHOH CKnOHa l = 80 M. EcnH CKOpOCTb CSHOK Y OCHOBSHHSI ropbI
M
v = 7.0 -, TO MO,Aynb CHJIbI conpOTHBneHHSI Fe ,ABHmeHHIO caHOK pSBeH:
C
1) 90 H; 2) 80 H; 3) 70 H; 4) 60 H; 5) 40 H.
A12. Ha KOHax O,AHOpO,AHOrO CTepmHSI MaCCOH m = 4.0 Kr H ,AnHHOH
l=I,O M nO,ABeWeHbI rpY3bI MsccaMH ml=3,0 Kr H m2=6.0 Kr. U;eHTp
TSlmeCTH CHCTeMbI HSXO,AHTCSI OT MeHbwero rpY3a HS pSCCTOSlHHH 8, paBHOM:
1) 31 CM; 2) 34 CM; 3) 50 CM; 4) 62 CM; 5) 69 CM.
A13. B 3aKpbITOM 68JInOHe 06'beMOM V = 20 n npH TeMnepaType t l = -80°C
r
3aKnlO'leHO m = 0,010 r BO,AOpO,D;a. MonSlpHsSI Maccs BO,D;OpO,Aa M = 2,0 -.
MOnl>
EcnH nOBblCHTb TeMnepSTYPY BO,AOpO,Aa TaK, 'ITo6bI ,AaBJIeHHe CTano P2 = 40 KITS,
TO Cpe,AHSISI KBaAPSTH'lHaSl CKOpOCTI> < V KS > MoneKYJI B03paCTeT B ... pS3.
1) 10; 2) 8.0; 3) 6,0; 4) 4.0; 5) 2,0.
A14. IIpH H306apHOM OXnam,AeHHH HeKOTopOH MaCCbI H,Aeanl>HOrO ra3S ero
06'beM YMeHbWHncSI Ha 1') = 5,0 %. EcnH HS'l8JII>HSSI TeMnepaTypa ra3a TI = 400 K,
TO B KOHe npoeccs OHS:
1) YMeHbwHnscb Ha 40 K; 2) YBenH'lHnSCb HS 40 K; 3) He H3MeHHnscL;
4) YMeHbWHnaCb Ha 20 K; 5) YBenH'IHJISCb Ha 20 K.
A15. SneKTpOCTSTH'leCKOe none C03,AaHO ,ABYMSI MeTSnnH'leCKHMH KOHeH-
TpH'leCKHMIi cq>epsMIi, pS,AHYCLI KOTOpLIX Rl = 10.0 CM H R 2 = 20,0 CM, S 3S-
pSl,AbI ql = 10,0 HKn Ii q2= -30,0 HKn. TIoTeHHsn <p nonSl B TO'lKe, HaXO,ASI-
eHcSI HS paCCTOSlHHH: r= 50,0 MM OT eHTps c<I>ep, pSBeH:
1) -450 B; 2) -410 B; 3) +450 B; 4) +410 B; 5) +380 B.
A16. llnSCTHHS H3 cnlOlIbI C lIH3neKTpH'leCKOH npOHHl1aeMOCTblO E = 7 0 Ha-
XO,lIHTCSI Mem,AY 06KJla,D;KSMH nnOCKoro KOHlIeHcaTopa, npHCOe,AHHeHHOro K HC-
692

TO'lHHKY nOCTO.HHHOrO Hanp.H>KeHHSI. 3apSiA Ha 06Kna,AKaX ql = 14 MKKn. lIpH
Y,AaneHHH nnaCTHHbI Qepe3 HCTO'lHHK npOH,AeT 3ap.H,A q2' paBHbIH:
1) 10 MKKn; 2) 12 MKKJI; 3) 14 MKKn; 4) 16 MKKn; 5) 18 MKKn.
A17. lliecTb 3neMeHTOB C SAC W= 1,6 B H BHYTpeHHHM conpOTHBneHHeM
r= 0,4 OM Kam,AbIii COe,AHHeHbI napannenbHO B 6aTapelO H 3aMKHYTbI Ha BHeIlI-
Hee COnpOTHSneHHe R = 0,2 OM. CHna TOKa I BO BHeIlIHeH enH paSHa:
1) 5 A; 2) 6 A; 3) 7 A; 4) 8 A; 5) 9 A
A1S. Ha ropH30HTaJIbHbIX penbcax, paCCTOSlHHe Mem,AY KOTOpbIMH l = 60 CM,
nemHT nepneHA1:IKynSlpHo HM CTepmeHb. PenbCbJ H CTepmeHb HaXO,ASlTCSI B BepTH-
K8JIbHOM O,AHOpO,AHOM MarHHTHOM none C HH,AYKHeH B = 60 MT n. Macca CTepm-
HSI m = 0,50 Kr, K03<P<PHHeHT TpeHHSI CTepmHSI 0 penbCbI Jl = 0,10. CHna TO-
Ka I, KOTOpbIH HymHo nponYCTHTb no CTepmHIO, QT06bI OH Ha'lan ,ABHraTbCSI.
paBHa:
1) 14 A; 2) 12 A; 3) 11 A; 4) 10 A; 5) 8.0 A.
A19. PaMKa H3 N = 1000 BHTKOB, Kam,AbIH H3 KOTOpbIX OrpaHH'lHBaeT
nOBepxHocTb nnOa,AblO S = 5 CM 2 , 3aMKHYTa Ha ranbBaHoMeTp C conpoTHsne-
HHeM R= 10 KOM H nOMeeHa B O,AHOpO,AHOe MarHHTHoe none C HH,AYKHeH
B = 0,01 Tn, npH'leM nHHHH HHAYKHH MarHHTHoro non.H nepneH,AHKynSlpHbI K
ee nnOCKOCTH. EcnH HanpaBneHHe HH,AYKHH MarHHTHoro nonSl nnaBHO H3MeHHTb
Ha npOTHBononomHoe, TO no enH r8JIbBaHOMeTpa npOTeQeT 3apSl,A q, paBHbIH:
1) 1 . 10- 3 Kn; 2) 1 . 10- 4 Kn; 3) 1 . 10- 5 Kn; 4) 1 . 10- 6 Kn; 5) 1 . 10- 8 Kn.
A20. IIoHHmalOHH TpaHc<popMaTop C K03<P<PHHeHToM TpaHc<popMaHH
k = 10 BKnlO'leH B ceTb HanpSlmeHHeM U I = 127 B. ConpoTHBneHHe BTOpH'lHOH
06MOTKR R 2 = 2,0 OM, CHna TOKa B HeH 1 2 = 3,0 A. EcnH npeHe6pe'lb nOTepSl-
MH aHeprHH B nepBH'lHOH 06MOTKe, TO HanpSlmeHHe U 2 Ha BTOpH'lHOH 06MOT-
Ke paBHO:
1) 5,3 B; 2) 5,8 B; 3) 6,7 B; 4) 7,2 B; 5) 7,5 B.
A21. fipe,AenbI aKKOMO,AaHH rna3a 6J1H30PYKoro tJenOBeKa COCTaBn.HIOT
d 1 = 10,0 CM H d 2 = 50,0 CM. EcnH OB Ha,AeHeT O'lKH, C nOMoblO KOTOpbIX
6Y,AeT XOpOIlIO BH,AeTb Y,AaneHHbIe npe,AMeTbI, TO CMomeT 'lHTaTb KHHry Ha
HaHMeHbIlIeM paCCTOSlHHH d 3 OT rna3, paBHoM:
1) 25,0 CM; 2) 20,0 CM; 3) 17,5 CM; 4) 15,0 CM; 5) 12,5 CM.
A22. EcnH QaCTOTa BToporo <l>OTOH8 B k = 3 pa3a MeHbIlIe 1.JaCTOTbI nepso-
ro, TO OTHOIlIeBHe MO,AyneH HMnynbcoB P2 3THX <pOTOHOB paBHO:
PI
1) 1; 2) 2; 3) 3; 4) 4; 5) 5.
M
A23. ABTOM06Hnb MaCCOH m = 1000 Kr ,ABHmeTC.H co CKOpOCTblO v = 8,0 -
C
no BbmYKnOMY MOCTY, npe,ACTaBnSlIOeMY C060H ,Ayry pa,AHYCOM R= 100 M. EcnH
pa,AHYC, npOSe,AeHHbIH B TO'lKY A MOCTa, COCTasnSleT C sepTHKanblO yron 30°.
TO CHna F HopManbHoro ,AaBneHHSI aBTOM06HnSl Ha MOCT B TO'lKe A paBHa:
1) 7,7 KH; 2) 8,0 KH; 3) 9.0 KH; 4) 9.3 KH; 5) 10,0 KH
693

A24. K np06KOBOMY nOnnSBKY M8CCOH ml = 1 Kr npHBSl38H HS HHTH CBHH-
OBLIH rpY3 M8CCOH m2 = 4,7 Kr TSK, 'tlTO nonn8BOK nonHOCTbIO norpYJKeH B
r
SO,Ay, npH 3TOM rpY3 He K8C8eTCSI ,AHS. EcnH nnOTHOCTb BO,AbI PB = 1 -
CM 3 '
r
CBHHa PCB = 11.4 ----g, TO nnOTHOCTL npo6KH Pop p8BH8:
CM
1) 0,1
r
CM 3 '
2) 0, 2
r
CM 3 '
3) 0,3
r
CM 3 '
4) 0,5
r
CM 3 ;
5) 0, 7
r
CM 3 .
A25. ,D;B8 COe,AHHeHHhlX TOHKOH Tpy6KOH 68.J1nOH8 C 06'heM8MH V l = 7,00 n
H V 2 = 12.0 n CO,Aepm8T HeKOTopoe KOnH'leCTBO 830T8. IIepBLIH 6SJlnOH HMeeT
HeH3MeHHYIO TeMnep8TYPY Tl = 273 K. ,I(nSl Toro 'lTo6bI BO BTOpOM 6snnoHe
1
OCTsnOCb TonbKO k = - o611tero KOnH'leCTBS S30T8, ero He06xo,AHMO H8rpeTL
3
,AO TeMnep8TypbI T2' P8BHOH:
1) 312 K: 2) 468 K: 3) 624 K: 4) 936 K: 5) 790 K.
A26. l1,AeanLHbIH O.l{H08TOMHLIH r83 p8CIIIHpHeTCH no 38KOHY pV 2 = const
OT 06'heM8 V l ,AO 06'heM8 V 2 . EcnH nepBOHa'l8nbHOe ,ASBneHHe r838 Pit TO
H3MeHeHHe BHYTpeHHeH 3HeprHH rS3S U p8BHO:
3 ( V l )
1) - Pl Vi - - 1 ;
2 V 2
5) 3 Pl Vi [ i. - 1 ).
2)  Pl Vi ( V2 - 1 J ; 3)!. Pl Vi ( Vi - 1 J ; 4)!. PI Vi ( V2 - 1 J ;
2 V] 2 V 2 2 VI
A27. B COcy Ae TennoeMKOCTLIO C = 200 ,D;m H8XO,AHTCSI BOA8 MSCCOH
K
ml = 1000 r H JleA M8CCOH m2= 1000 r npH TeMnep8Type TI = 273 K. YAenbHSSI
Am
TennoeMKOCTb BO,Abl CI = 4200 -, y,AenbHSSI TennOT8 nJlSBneHHH Jlb,A8
Kr.K
A = 330 KAm ,
KI'
Toro 'lTo6LI B
THTb nsp npH
M,l];)1(
Y,AeJlLH8S1 TennOT8 n8poo6pS30SSHHSI SO,ALI L = 2,30 -. ,I(nSl
Kr
cocY,Ae YCT8HOBHn8Cb TeMnepsTypa e = 296 K, B Hero H8.l{O Bnyc-
TeMnepaType t 2 = 100°C MSCCOH m. PSBHOH:
1) 402 1'; 2) 365 r; 3) 238 r; 4) 201 1'; 5) 164 r.
A28. B BeplIIHHax KBS,ApSTS p8cnonomeHLI OAHH8KOBLle 3SpSl,Abl q. EcnH
CTOpOH8 KBa,Ap8T8 a, TO CHn8 F, ,AeHCTBYIO8H HS K8JK,AbIH aSpSl,A, pSBH8:
1) O,2 q 2 ; 2) O,48 q 2
nE o a 2 n£ o a 2
3) O,lq2 ; 4) L; 5) 0.4 q 2 .
n£oa 2 nE o a 2 Eoa
A29. ,llBS TO'le'lHLIX 3apHmeHHLIX Tens M8CC8MH m H M H 38pSl,ASMH CO-
OTBeTCTBeHHO -q 1:1 +Q HaXO,ASlTCH BO BHelliHeM O,AHOPOAHOM sneKTpH'leCKOM
694

none C HanpSlmeHHOCTblO E HanpaBneHHOH OT OTpHuaTenbHoro 3apSla K no-
nomHTenbHOMY. PaCCTOSlHHe r, Ha KOTOpOM HaxOASlTCSI 3apSl,AbI APyr OT ,Apyra
npH OAHHaKOBOM YCKopeHHH 060HX 3apSlAOB, paBHO:
1) kq Q : 2) k Q
E .J Qm + qM E .J Qm + qM
kqQ(m + M)
5) E(Qm + qM)"
3) lqQ(m + M) ; 4) kq :
E E(Qm + qM)
A30. ConpoTHBneHHe O,AHOrO H3 nocne,AOSaTenbHO COe,AHHeHHbIX npoBoHH-
KOB B n pa3 60nblIIe conpoTHBneHHSI BToporo. EcnH 3TH npOSO,AHHKH BKnlO-
'lHTb nap8JInenbHO. a HanpSlmeHHe Ha HHX OCTaBHTb npemHHM, TO CHna TOKa I
B qenH H3MeHHTCSI B:
(n + 1)2
1)
n
2) n - 1 ; 3) n - 1 ; 4) n T 1 ; 5) (n + 1 )2
n n 2 n n 2
A31. IIpoTOH BneTaeT co CKOpOCTblO V = 1,0.10 3 M B 06nacTb npOCTpaH-
C
CTBa, r,Ae O,AHOBpeMeHHo cyecTBYIOT O,AHOpO,AHbIe B3aHMHO nepneH,AHKynSlpHbIe
3neKTpH'leCKOe 1:1 MarHHTHoe nonSl. HH,AYKqHSI MarHHTHoro nonSl B = 20 MTn.
EcnH npOTOH ,ABHmeTCSI paBHOMepHO H npSlMonHHeHHO. TO HanpSlmeHHOCTb E
3neKTpH'leCKOrO nonfl paBHa:
1) 12 B ; 2) 14 B ; 3) 18 B
M M M
4) 20 B . 5) 28 B .
M' M
A32. B OTKpbITbIe coo6alOHecSI COCY,AbJ C O,AHHaKOBoH nnOa,AblO none-
pe'lHOrO CeqeHHSI S HanHTa PTYTb MaCCOH m. CTon6HK PTYTH B O,AHOM H3
COCY,AOB BbIBenH H3 nonomeHHSI paBHoseCHSI, qTO npHBeno K B03HHKHOBeHHIO
Kone6aHHH PTYTH. IIepHo,A T Kone6aHHH PTYTH paBeH:
1) J m ; 2)  J m ; 3) lPSK ; 4) 21t J m ; 5) 2 n.J3PSK .
3pSg 21t 3pSg m 2pSg m
A33. He60nblIIoH npe,AMeT pacnonomeH Mem,AY ,ABYMSI nnOCKHMH 3epKana-
MH, nOCTaBneHHbIMH no,A yrnoM <p = 30°, Ha paCCTOSlHHH r= 8,0 CM OT nHHHH
nepeCeqeH1:I.H aepKan. IIepBble MHHMble H306pameHHSI npe,AMeTa B 3epKane Ha-
XO,ASlTCSI Ha paCCTOSlHHH rl ,APyr OT ,Apyra, paBHOM:
1) 4,0 CM; 2) 6,0 CM; 3) 8,0 CM; 4) 10 CM; 5) 12 CM.
BI. BepToneT neTHT H3 nYHKTa A B nYHKT B, pacnonomeHHbIH IOmHee
nYHKTa A Ha s = 137 KM. 80 BpeMSI peHca ,AyeT BeTep C 3ana,Aa Ha BOCTOK.
KM
EcnH CKOpOCTb BepToneTa OTHOCHTenbHO BeTpa vI = 180 - H CKOpOCTb BeTpa
'l
M
v2 = 20.0 -, TO BepToneT BepHeTCSI 06paTHO qepe3 npOMemYTOK speMeHH Ilt
C
nocne BbIneTa, paBHbJH ... (MHH).
695

m}
Puc. 36
t a
B2. Teno MaccoH m = 10,0 Kr nO,AHRMaIOT C nOMOI.qblO
HaKnoHHoH nnOCKOCTH ,AnHHOH l = 400 CM H yrnoM HaKno-
Ha <p = 30,0°. EcnR K03cpcpRReHT TpeHRSI Tena 0 nnOCKOCTb
Jl = 0,500, TO ,AnSI nO,AHSlTHSI Tena Ha BeplllHHY HaKnOHHOH
nnOCKOCTH He06xo,AHMO COBeplllRTb MRHHManbHYIO pa60TY A,
paBHYIO ... (,n:>K).
m 2
B3. Qepe3 HeBecoMblH 6nol< nepeKRHYTa HRTb C rpY3aMR
ml = 1,0 Kr R m2=3,0 Kr, 3aKpenneHHblMR Ha ee KOHax
M
(pHC. 36). BnOK ,ABRmeTCSI BBepx C YCKopeHHeM a = 2,0
c 2 .
EcnR npeHe6pe'lb TpeHHeM, TO CRna ,AaBneHH.H F Ha OCb
6noKa paBHa ... (H).
B4. K CTO'lHHKY TOKa C BHYTpeHHRM conpOTHBneHHeM r = 2 OM BHa'lane
nO,AKnlO'laeTC.H pe3RCTOp c conpOTHBneHHeM R = 2 OM, 3aTeM napannenbHO no,A-
KnlO'laeTCSI TaKOH me pe3HCTOp. KOnH'leCTBO TennOTbl Q, BbI,AenHeMOn B nepBOM
pe3HCTope nocne nO,AKnlO'leHRH BToporo pe3RCTopa, 3a TaKoe me BpeM.H YMeHb-
lllaeTC.H B . .. pa3.
B5. Pe30HaHC B Kone6aTenbHOM KOHType C KOH,AeHCaTOpOM eMKOCTblO
C 1 = 1 MK<1> HacTynaeT npH qaCTOTe vI = 400 r. EcnR napannenbHO nepBOMY
KOH,AeHCaTOpy nO,AKnlO'lRTb BTOpOH KOH,AeHCaTOp, TO pe30HaHCHaSl 'laCTOTa CTa-
HeT paBHOH v2= 200 r. EMKOCTb C 2 SToporo KOH,AeHCaTOpa paSHa ... (MK<1».
B6. KorlIa aKpo6aT CTOHT HenOlIBH>KHO Ha paCTHHYTOH ceTKe. OHa nporH-
6aeTcSI Ha x = 50.0 MM. EcnH OH npbIrHeT C BbICOTbl h = 10,0 M Ha paCTSlHY-
TYIO ceTKY, TO MaKCHManbHaSl rny6RHa nporH6a ceTKH paBHa ... (CM). C'lR-
TaHTe. 'lTO ,Ae<popMaRSI ceTKR nO,A'UIH.HeTCSI 3aKoHY fYKa.
B7. O,ARH Monb O,AHOaTOMHoro H,AeanbHOro ra38 COBeplllaeT 3aMKHYTbIH
HKn, COCTOHll\HH R3 ,ABYX H30XOP H BYX H306ap. EcnH MaKCHManbHoe ,AaB-
neHHe B HKne B 2 pa3a 60nbllle MHHRManbHoro, a MaKcRManbHblH 06'beM B
3 pa3a 60nbllle MHHHManbHoro, TO KTI,n: 11 HKna paBeH ... (%).
696

X,M
180
120
60
o
20 40
60
Puc. 37
TEe T IX
AI. rpatPHKH 3aBHCHMOCTH KOOp,AHHaTbI
,ABYX Ten OT BpeMeHH IIpeCT8BneHbI Ha PH-
CYHKe 37. OTHoIlIeHHe MoyneH CKOpOCTH Ten

p8BHO:
V2
1 ) !.; 2 ) 1 3 ) 1 4 ) ; 5 ) 3
3 2' I' 1 )
2
A2. EcnH B cocY,Ae 06'beMOM 5.0 n Haxo-
,AHTCSI 20 r KHCnOpO,A8 C MonSlpHoH M8CCOH
M = 0,032 , TO KOHeHTp8u.HSI MoneKyn B
MOJl!>
cocY,Ae paBHa:
1) 7,5.10 22 M- 3 ; 2) 1,5,10 23 M- 3 ;
3) 7,5.10 25 M- 3 ; 4) 1.5.10 27 M- 3 ;
5) 1,5,10 26 M- 3 .
t, C
A3. ,D;B8 O,AHH8KOBbIX MeT8nnH'leCKHX IlIapHKa C 38pSl,A8MH ql = q H
q2 = -5q npHBenH B conpHKOCHOBeHHe, 8 3aTeM pa3BHHynH H8 p8CCTOSlHHe,
B,ABOe 60nbIlIe nepBOHa'l8nbHoro. CHna F B3aHMO,AeHCTBHSI IlIapHKOB npH 3TOM:
1) YMeHbIlIHn8Cb B 5 pa3: 2) YMeHbIlIHnacb B 4 pa3a: 3) He H3MeHHn8Cb:
4) YBenH'lHnaCb B 3 pa38: 5) YBenH'lHnaCb B 2 pa3a.
A4. Ecnlf 'lepe3 pacTBopbI CUS04 H Cu(N0 3 h npOTeKan oHHaKoBbIH 3a-
PSl. TO OTHOIlIeHHe Macc MeH mI . BbIenHBIlIeHCSI Ha K8TO,AaX B 060HX
m2
cnY'laSlX, paBHO:
1) 1; 2) 2; 3) 3; 4) 4; 5) 5.
A5. COneHOH,A ,AnHHOH l = 12,5 CM, p8CnOnO:>KeHHbIH B B03,Ayxe, HMeeT pa-
,AHYC BHTKOB R= 2.50 CM. TIpH p8BHOMepHOM B03paCT8HHH CHnbI TOKa B COJIe-
HOH,Ae OT 1 1 = 5,00 A ,AO 1 2 = 10,0 A 38 BpeMSI t = 100 MC B03HHKna S,D;C
C8MOHH,AYKHH g'Si= 10,0 B. qHcno N BHTKOB coneHOHa paBHo:
1) 3,19' 103; 2) 3,88' 103; 3) 3,97' 103; 4) 4,13' 103; 5) 4,28' 10 3 .
A6. EcnH npH OCBeeHHH ,AHtPpaKu.HoHHOH peIlIeTKH CBeTOM C ,AnHHOH Bon-
Hbl A = 656 HM M8KCHMYM BToporo nOpSl,AKa BH,AeH no,A yr nOM cp = 15° K HOp-
ManH ,AH<I>P8KHOHHOH peIlIeTKH, TO nepHO,A d AHtPpaKu.HOHHOH peIlIeTKH
p8BeH:
1) 5 MKM; 2) 4 MKM; 3) 3 MKM; 4) 2 MKM; 5) 1 MKM.
A7 llHTepBany co6cTBeHHoro BpeMeHH 6.t o = 1 C 'laCTHbI, ,ABHmyeHcSI
co CKOpOCTblO v = 0,95 c (c - CKOpOCTb CBeTa B B8KYYMe), COOTBeTcTByeT HH-
TepB8n BpeMeHH t no 'laCaM HenO,ABHmHOro H86nlO,AaTen.H, paBHbIH:
1) 2,4 c; 2) 2,8 c; 3) 3,0 c; 4) 3,2 c; 5) 3,8 c.
697

AS. EcnH 3HeprHSI OCHOBHoro COCTOSlHHSI STOMa BO.ltOpO,AS WI = -13.55 3B.
TO 3HeprHSI W 3neKTpOHS, HSXO,ASllQerOCSI Ha TpeTbeH CTSHOHSpHOH op6HTe aTO-
MS, paSHa:
1) 2,40'10- 19 ,l(m; 2) 1,38,10- 19 ,l(m; 3) -2,40' 10- 19 ,l(m;
4) -1,85,10- 19 ,l(m; 5) -1,38.10- 19 ,l(m.
A9. Cpe,AHee BpeMSI >KH3HH pSHOSKTHBHoro K06anbT8 CO t = 7,35 rO,Aa.
llepHO,A nonypscna,Aa T pSBeH:
1) 5,1 ro,AS; 2) 4,8 roa; 3) 4,3 rO,Aa; 4) 3,2 ro,AS; 5) 2.8 ros.
AIO. KHHeMSTHqeCKOe yp8BHeHHe ,ABHmeHHSI MSTepHsnbHoH TO'lKH S,AOJIb
OCH x HMeeT BH: x=2+2t-0,5t 2 . CKOpOCTb TO'lKH pSBHS HYJlIO B MOMeHT
BpeMeHH t l , pSBHbIH:
1) 0.50 c; 2) 1,0 c; 3) 1,5 c; 4) 2,0 c; 5) 3,0 c.
All. rOpH30HTanbHa.H nnaT<popMS AnHHoH l = 2,0 M C nOKOSlI.qHMCSI Ha ee
M
KpalO rpY30M .ltBHmeTCSI co CKOpOCTblO v = 3.0 - H BHe3snHO OCTaHaBnHBaeT-
C
CSI EcnH B MOMeHT COCKsnb3bIBSHHH rpY3a C npOTHBononomHoro KpaSl nnaT-
<pOpMbI ero CKOpOCTb B TPH pS3S MeHbwe nepSOHS'lanbHOH. TO K03<P<I>HHeHT
TpeHHSI J..l Me>K,AY rpY30M H nnST<popMOH paBeH:
1) 0,10; 2) 0.15; 3) 0,20; 4) 0,25; 5) 0.30.
A12. O,AHOpO,AHbIH HnHH,Ap nOCTSBneH Ha HSKnOHHYIO nnOCKOCTb. EcnH
BbICOTS HnHH,Ap8 B,ABOe 60nbwe pS,AHyca ero OCHOBaHHSI, TO MaKCHMsnbHbIH
yron ex HSKnOHS nnOCKOCTH K ropH30HTY, npH KOTOpOM HnHH,Ap He onpOKH-
HeTCSI, pSBeH:
1) arcctg.!.; 2) srcsin.!.; 3) arctg.!.; 4) srctg2; 5) srctgl.
2 2 2
A13. B KOMHSTe ,AnHHOH l = 600 CM, WHpHHOH b = 400 CM H BbICOTOH
h = 250 CM HaXO,AHTCSI m = 72,5 Kr B03,Ayxa. EcnH ,ASBneHHe B03,AYXS B KOM-
HSTe p = 100 KTIa. TO Cpe,AHSISI KBS,ApSTH'lHSSI CKOpOCTb <V K .> ,ABHmeHHSI MO-
neKYJI B03,Ayxa paBHS:
1) 298 ; 2) 398 M ; 3) 498 M ; 4) 560 ; 5) 598 M .
C C C C C
A14. B ropH30HTSJIbHOH cTeKnSlHHoH Tpy6Ke, 3anaSlHHoH C OD;Horo KOHa,
pacnonomeH CTOJI6 B03yxa. 3anepTbIH cTon6HKoM PTYTH. ,l(nHHa cToJl6a B03-
,AYXS B Tpy6Ke npH TeMnepaType T 1 =294 K pSBHa ll=0.120 M. ,l(nSl Toro
'lT06bI ,AnHHS cTon6a B03yxa YBenH'lHnaCb Ha III = 60.0 MM. TeMnepsTypa B03-
YXS ,AOnmHS YBenH'lHTbCSI ,AO:
1) 300 K; 2) 340 K; 3) 380 K; 4) 420 K; 5) 440 K.
A15. 3neKTpOCTaTH'leCKOe none C03,ASHO ,ABYMSI MeTSnnH'leCKHMH KOHeH-
TpH'leCKHMH c<I>epaMH, pS,AHYCbI KOTOpbIX Rl = 10.0 CM H R 2 = 20.0 CM. a 3S-
PSlAbI ql = 10.0 HKn H q2= -30,0 HKn. IIoTeHHsn q> nonH B TO'lKe. HSXO,A$I-
I.qeHCSI HS paCCTOSlHHH r= 15,0 CM OT eHTpa c<pep, pSBeH:
1) -750 B; 2) -520 B; 3) -315 B; 4) +652 B; 5) +518 B.
698

A16. lInOCKHH B03AYWHbIH KOHAeHCaTOp eMKOCThlO C = 20 n<1> 3apSlmeH ,AO
pa3HocTH nOTeHHanOB l\<p = 100 B H OTKnlO'leH OT HCTO'lHHKa HanpSlmeHHSI.
't.JTo6hI BABoe YBenH'lHTh paCCTOSlHHe MemAY 06Kna,AKaMH, HymHo COBepWHTb pa-
60TY A. paBHYIO:
1) 1. 10- 4 Am; 2) 1 . 10- 5 Am; 3) 1 . 10- 6 )J;m; 4) 1. 10- 7 Am; 5) 1. 10- 8 Am.
A17. TIpH 3aMbIKaHHH HCTO'iHHKa TOKa Ha conpOTHBneHHe Rl = 2.3 OM CHna
TOKa B u.enH 11 = 0,56 A, a npH 3aMhIKaHHH 3Toro me HCTO'lHHKa Ha conpo-
THBneHHe R 2 = 1,8 OM CHna TOKa B u.enH 1 2 = 0,7 A. CHna TOKa KopOTKoro
3aMhIKaHHSI I K.3 paBHa:
1) 7 A; 2) 6 A; 3) 5 A; 4) 3 A; 5) 2 A.
AlS. Ha Kam,AbIH MeTp ,AnHHbI npSlMoro .nnHHHoro npOBO,Aa. CHna TOKa B
KOTOpOM 1=1,0 A, co CTOpOHhI BToporo npOBO,Aa C TaKoH me CHnOH TOKa
,AeHcTByeT CHna F= 2,5' 10- 7 H. PaccToSlHHe Mem,AY npOBO,AaMH r= 1.0 M.
MarHHTHaH npOHHaeMOCTb J..l Cpe,AbI, B KOTOpOH HaXO,ASlTCSI npOBO,AHHKH, paBHa:
1) 3,9; 2) 3,2; 3) 2,2; 4) 1,6; 5) 1.3.
A19. PaMKa nnoaAblO S = 300 CM 2 HMeeT N = 200 BHTKOB H Bpa:[,QaeTCSI
C nocToSlHHOH yrJIOBOH CKOpOCThlO B MarHHTHOM none C HH,AYKHeH B = 15 MTn.
OCb Bpa:[,QeHHSI JlemHT B nnOCKOCTH paMKH H nepneH,AHKyn.HpHa nHHHSlM Mar-
HHTHOH HH,AYKHH. MaKCHMaJIhHaSl SAC HH,AYKHH B paMKe go= 14 B. lIe-
pHO,A T Bpa:[,QeHHSI paMKH paBeH:
1) 0,5 c; 2) 0,3 c; 3) 0,2 c; 4) 0,05 c; 5) 0,04 c.
A20. CHna TOKa B nepBH'lHOH 06MoTKe TpaHc<}>opMaTopa 11 = 0,60 A, Ha-
npSlmeHHe Ha ee KOHax U 1 = 120 B. CHna TOKa BO BTOPHqHOH 06MoTKe
1 2 =4,8 A, HanpSlmeHHe U 2 = 12 B. KlIA 11 TpaHc<}>opMaTopa paBeH:
1) 80 %; 2) 70 %; 3) 60 %; 4) 65 %; 5) 50 % .
A21. <1>oKycHoe paCCTOSlHHe pacceHBalO:[,QeH nHH3bJ FI = -4,0 CM. cpOKYC-
Hoe paCCTOSlHHe c06HpalO:[,QeH nHH3bI F 2 = 9,0 CM. QT06bI nY'lOK nY'leH, na-
panJlenbHbIH rnaBHOH OnTH'leCKOH OCH, npOH,ASI 'lepe3 06e nHH3bI, OCTancSI
napaJlJlenhHbIM, JIHH3hI Ha,AO pacnonomHTb Ha paCCTOHHHH l APyr OT ,Apyra,
paBHOM:
1) 13 CM; 2) 9.0 CM; 3) 5.0 CM; 4) 4.0 CM; 5) Ha n1060M.
A22. ECJIH npH 06nY'leHHH MeTanna CBeTOM C 'laCTOTOH VI <poTo3neKTpoHbI
nOJlHOCTblO 3aepmHBa1OTCH Hanp.HmeHHeM U 31' a npH lleHCTBHH CBeTa C qac-
TOTOH V2 - HanpSlmeHHeM U 32 ' TO nOCTOSlHHaH TInaHK8 h paBHa:
V2 - VI ) ( )
1) A.+eU 3 .; 2) ( ) ; 3) U 3 V2- U 3 VI; 4 e U 3 -U 3 ;
. e U -U I 1 I a
3. 31
e(U 3a - U 31 )
5)
V2 - VI
699

M
A23. KOHbK06emell, ,ABHmeTCSI co CKOpOCTblO V = 10 - no OKPY*HOCTH pa-
c
,AHYCOM R = 30 M. QTo6bI coxpaHHTb paBHOBeCHe, OH onmeH HaKnOHHTbCSI K
ropH30HTY no,A yrnoM, TaHreHC KOToporo tga paBeH:
1) 1: 2) 2: 3) 3: 4) 3.5: 5) 4.
A24. TInoa,Ab Manoro nopIlIHSI rH,ApaBnH'leCKOrO npecca B k = 20 pa3 MeHb-
IlIe nnOIl..la,AH 6onbIlIoro nopIlIHSI. TIpH nO,A'beMe rpY3a C nOMOIl..lblO npecca Ma-
nbIH nopIlIeHb nepeMecTHncSI Ha paCCTOSlHHe hI = 400 MM. EcnH Macca rpY3a
m = 1,50 T. TO npH aTOM 6bIna COBepIlIeHa none3HaSl pa60Ta A. paBHaSl:
1) 300 .lXm; 2) 600 .lXm; 3) 900 .D:m; 4) 1,20 K.D:m; 5) 1,50 K.D:m.
A25. AHarpaMMB l.I,HKnH'leCKOrO npOl.l,eCca nSl v = 0,8 Monb H,AeanbHOro
ra3a B KOOp,AHHaTaX (p, V) 06pa3yeT TpeyronbHHK C BepIlIHHaMH B TO'lKaX
(166 KTIa, 12,0 n). (166 KTIa, 24,0 n) H (124 KTIa, 12,0 n). Pa3HocTb 6.T
Mem,Ay MaKcHManbHoH H MHHHManbHoii TeMnepaTYPoH B lI,HKne paBHa:
1) 175 K; 2) 256 K; 3) 375 K; 4) 556 K; 5) 600 K.
A26. fenHH B KOnH'leCTBe v = 1,00 Monb H3 COCTOSlHHSI C TeMnepaTypoii
Tl = 200 K HarpeBaeTCH TaK, t.zTO ero TeMnepaTypa H3MeHHeTCSI no 3aKOHY
T= ap 2. re a - nOCTOSlHHa.R BeJIH'lHHa. EcnH TeMnepaTypa T2=2Tl' TO re-
nHH COBepIlIaeT pa60TY A, paBHYIO:
1) 3.32 K,lIm: 2) 1.66 K,lIm: 3) 831 .D:m: 4) 416 nm: 5) 208 .D:m.
A27. B COCYAe, H3 KOToporo BbIKa'lHBaeTCSI B03,AYX, HaXOHTCSI He6onbIlIoe
KonHt.zeCTBO BOAbI npH TeMnepaType T = 273 K. 3a CqeT HHTeHCHBHoro Hcnape.
HHSI npOHCXOHT 3aMopamHBaHHe BO,AbI. Y,AenbHaSl TennOTa nnaBneHHSI nba
A = 340 K.D: m , y,AenbHBSI TennOTa napoo6pa30BaHHSI BO,AbI L = 2.3 M.D: m . I1pl1
Kr Kr
aTOM HcnapSleTCH 'laCTb BO,AbI. paBHaH:
1) 0.11; 2) 0,12; 3) 0,13; 4) 0,14; 5) 0,15.
A28. B BepIlIHHax KBa,ApaTa pacnonomeHbI O,AHHaKOBble 3apH,AbI q. Q T06bI
BCH CMCTeMB HaXO,AHnaCb B paBHOBeCHH, B lI,eHTpe KBa,ApaTa Heo6xo,AHMO no-
MeCTHTb 3apSl,A ql' paBHbIH:
1) -0.96q; 2) -0.76q; 3) +0.96q; 4) +0.76q; 5) -0.52q.
A29. JIBa TO'let.zHbIX 3apSlmeHHbIX Tena MaccaMI1 m H M M 3apSl,AaMH -q
H +Q COOTBeTCTBeHHO HaXO,ASlTCH Ha paCCTOSlHHH r Apyr OT ,Apyra BO BHeIlI-
HeM O,AHOpOAHOM 3neKTpH'leCKOM none C HanpHmeHHOCTblO. HanpaBneHHOH OT
OTpHl.I,aTenbHoro 3apSl,Aa K nonomHTenbHOMY. EcnH YCKopeHHSI 060HX 3apSl,AOB
O,AHHaKOBbI, TO HanpHmeHHOCTb E BHelliHero O,AHOpOAHoro aneKTpMt.zeCKoro nonSl
paBHa:
1) k q 2 Q ; 2) k(Q -q) ; 3) k;Q(m + M) ; 4) 2 kqQ ; 5) r 2 (m + M) .
r m + M r (Qm + qM) r (Qm + qM) kqQ
700

A30. B CeTb nOCTOSlHHOrO HanpSl>KeHHSI nOCne,AOBaTeJlbHO BKnlO'leHbI ,ABe
naMnO'lKH MOHOCTblO PI = 60,0 BT H P 2 = 250 BT, paCC'iHTaHHbIe Ha O,AHHa-
KOBoe HanpSlmeHHe. OTHoIlleHHe KOnH'leCTB TennOTbI Ql , BbIAeneHHoH B naM-
Q2
nO'lKaX 3a O,AHO H TO me BpeMSI ropeHHSI, paBHO:
1) 4,17; 2) 3,18; 3) 2,75; 4) 2,58; 5) 2,35.
A31. SneKTpoH H npOTOH, YCKopeHHbIe B aneKTpH'leCKOM nOJIe C p83HO-
CTblO nOTeHHanOB U, BJreTalOT B OAHOpO,AHOe MarHHTHoe none C HH,AYKHeH B,
npHtleM HX CKOpOCTH B MOMeHT BneT8 napanJIenbHbI. EcnH MaCCbI aneKTpOHa
R
H npoToH8 paBHbI COOTBeTCTBeHHO me H m p ' TO OTHOIlleHHe pa,AHYCOB ---!. OK-
Rp
pymHocTeH, onHCbIB8eMbIX aTHMH 'laCTH8MH B MarHHTHOM none, paBHO:
1) me ; 2) Jmp ; 3) 1; 4)!.: 5) 2.
mp me 2
A32. Ha rna,AKOM ropH30HT8nbHOM CTone nemHT Illap MaccoH m = 0,35 Kr.
KH
llIap npHKpenneH K npY>KHHe meCTKOCTblO k = 5.0 -. TIynSl MaCCO:H ml = 10 r
M
M
neTHT ropH30HTanbHO co CKOpOCTblO V = 600 -, nOna,A8eT B Illap H 3aCTpeBa-
C
eT B HeM. MaCCOH npymHHbI Ii conpOTHBneHHeM B03,Ayxa npeHe6pe'lb. AMnnH-
TY,Aa A Kone6aHHH Illapa C nyneH paBHa:
1) 14 CM; 2) 7,0 CM; 3) 5,0 CM; 4) 4.0 CM; 5) 3,0 CM.
A33. Y npH3MbI C npenOMnSlIOIl.tHM yrnoM q> = 30° O,AHa
Ha. JIy'i, na,AalOIl.tH:H Ha ,ApyrylO rpaHb no,A yrnoM ex = 60°,
HHSI H OTp8meHHSI OT nocepe6peHHoH rpaHH BepHY ncSI Ha3a,A
npaBneHHIO. EcnH CKOpOCTb CBeTa B BaKYYMe c = 3,0.10 8 M
C
rpaHb nocepe6pe-
nocne npenoMne-
no npemHeMY Ha-
B BeecTBe npH3MbI V p8BHa:
TO CKOpOCTb CBeTa
1) 1,0.10 8 M
C
2) 1,4.10 8 M
C
3) 1, 7 . 10 8 M
C
4) 2 .10 8 M
C
5) 2, 3 . 10 8 M
C
Puc. 38
M
Bl. llpH CKOpOCTH BeTp8 V = 10 KannSl llOm,ASI nallaeT nOlI
C
yrJIOM q> = 30° K BepTHKanH. llo,A yrJloM q>l = 60° K BepTHKanli
KanJI" 6yneT DMaTb npH CKOpOCTH BeTpa v,. paBKO" ... (]-
B2. CaMoneT, MOIl.tHOCTb ABHraTeneH KOToporo COCTaBnSleT
N=3,0 MBT npH CHne TSlrH F=4,5 KH, nponeTaeT paCCTOSlHHe
s = 360 KM 38 t = 30 MHH, KTI,D, 11 ,ABHr8TeneH caMoneTa paBeH ... (%).
B3. Qepe3 HenO,ABHmHbIH 6noK nepeKHHYTa HHTb, K KOTOpOH
nO,ABeIlleHbI TPH O,AHHaKOBbIX rpY3a (pHC. 38). EcnH CHna HaTSI-
701

meHHSI HHTH, CBSl3blB81OIlteH rpY3bI 2 H 1  F H = 5,0 H, TO CHna. ,AaSneHHH F
H8 OCb 6nOK8 p8BH8 ... (H).
B4. K HCTO'lHHKY TOKa C BHYTpeHHHM conpOTHBneHHeM r = 2 OM BHa'l8ne
nO,AKnlO'laeTCSI pe3HCTOp C conpOTHsneHHeM R = 2 OM. 3aTeM n8pannenbHO no,A-
KnlO'l8eTCSI TaKOH me pe3HCTOp. KOnH'leCTBO TennOTbI Q. BbI,AenSleMOH B nep-
SOM pe3HCTope nocne nO,AKnlO'leHHSI sToporo pe3HCTop8, 38 T8Koe me speMSI
YMeHbm8eTC.H B ... p83.
B5. KaTYWKa C HH,AYKTHBHOCTblO L = 30 MKrH npHCOe,AHHeH8 K nnOCKOMY
KOHAeHC8TOPY, nnOIq8,Ab K8m,AOH nn8CTHHbI KOToporo S = 100 CM 2 , p8CcToSlHHe
Mem,AY HHMH d = 0,1 MM. KOHTYP pe30HHpyeT Ha sneKTpOM8rHHTHYIO BonHY
,AnHHOH A = 750 M. ):(HsneKTpH'leCKaSl npoHHu.aeMOCTb E cpeAhI, 3anOnHSlIOIqeH
npOCTpaHCTBO Mem,AY nJIaCTHHaMH, paBHa ...
B6. IIynSl MaccoH ml = 100 r, neTeBwaSl ropH30HTanbHO, nOna,AaeT B no,A-
BeweHHbIH Ha HHTH W8p M8CCOH m2 = 2000 r H, np06Hs ero, SbIneT8eT co CKO-
M
pOCTblO V2 = 400 -. EcnH m8p npH STOM nO,AHHM8-
C
p eTCSI H8 BblCOTY h = 200 MM, TO nyns neTen8 CO
1
3
2
Puc. 39
702
[ M c I .
CKOpOCTblO Vi' p8BHOH... J
v
B7. KTI):( TennOBOH MaWHHbI, pa60'lHM TenOM
KOTOpOi1 SlBnSleTCSI v = 1 Monb 0,AH08TOMHoro H,Ae-
anbHoro r83a, p8BeH 11 = 25,00 %. r83 COBepwaeT
u.HKn, COCTOSIqHH H3 H30TepMH'leCKOrO 1-2, H30-
xopHoro 2-3 H 8,AH868THoro 3--1 npou.eCCOB
(pHC. 39). EcnH pa3HOCTb MemAY M8KCHM8nbHOH H
MHHHManbHOH TeMnepaTYPoH r838 B u.HKne COCT8B-
nSleT I:1T = 100,0 K. TO pa60T8 A, COBepweHHas r8-
30M B H30TepMH'leCKOM npou.ecce, p8BHa ... ():(m).

TECT X
o
AI. rp8<pHKH 38BHCHMOCTH KOOp.n;HH8TLI ,ABYX
Ten OT BpeMeHH npe.a.CT8BneHbI Ha pHcYHKe 40.
Yp8BHeHHSI. onHCbIB81OIltHe ,ABHmeHHSI 3THX Ten.
HMelOT BH.n;:
X.M
60
1
40
1) xI=20+t;
2) Xl = 20 + 2t;
3) Xl =20 - 2t;
4) xl = 20 + t;
5) Xl = 20 - t;
x2=-20 + t;
x2=-20 + 2t;
X2= 20 + 3t;
x2=-20 + 3t;
X2= 20 - 3t.
20
Puc. 40
t, C A2. B cocY,Ae H8XO,AHTCSI renHH M8CCOH
ml = 20 r. EcnH aTOMH8S1 M8cca renHH paBH8
M r = 4, TO KonHtJeCTBO v renHSI B cocY,Ae p8BHO:
1) 0,005 Monb; 2) 0.05 MonL; 3) 0,5 Monb;
4) 5 Monb; 5) 50 Monb.
-20
A3. qeTbIpe O,AHH8KOBbIX nonomHTenLHbIX TOtJetJHblX 3apSl,A8 no q = 1.0 MKKn
Haxo.n;SlTCSI B BepIlIHH8X KB8.n;p8T8 co CTOpOHOH a = 10 CM. CHna F. ,AeHCTBYIO-
1I.\8S1 co CTOpOHbI Tpex 38pSl,AOB H8 'leTBepTLIH, p8BH8:
1) 1,7 H; 2) 1,5 H; 3) 1,4 H; 4) 1,3 H; 5) 1,2 H.
A4. TIpH 3JIeKTpOnH3e p8CTBopa ZnS04 3neKTpH'leCKHH TOK COBepIlIHn pa-
60TY A = 10 KBT' 'l npH H8npSlmeHHH H8 38mHM8X B8HHbI U = 40 B EcnH
k 0 34 10 -6 Kr
3neKTpOXHMH'leCKHH 3KBHB8neHT HHK8 =, K.7I' TO M8CC8 m BbI,Ae-
nHBIlIerocSI HHK8 p8BH8:
1) 0.1 Kr; 2) 0.2 Kr; 3) 0.3 Kr; 4) 0.4 Kr; 5) 0.5 Kr.
A5. EcnH HH.n;YKTHBHOCTL K8TYlliKH YBenHtlHTL H8 111 = 40 %, 8 cHny
TOK8 - H8 112 = 400 %, TO 3HeprHSI W M8rHHTHoro nonSl B K8TYIlIKe:
1) YMeHbIlIHTCSI B 5,0 p83; 2) YMeHbIlIHTCSI B 7,0 pa3;
3) YBenHtJHTCSI B 10 p83; 4) YBeJlH'lHTCSI B 25 p83;
5) YBenH'lHTCSI B 35 p83.
A6. EcnH CBeT npOXO,AHT B BO,Ae (n=1,33) nYTb 81=100 CM, TO 38 3TO
me BpeMSI B B8KYYMe <ppOHT CBeTOBOH BonHLI npOH,AeT nYTb 82, p8BHbIH:
1) 0,670 M; 2) 1,00 M; 3) 1,15 M; 4) 1,20 M; 5) 1,33 M.
A7. IIp0,Aon*HTenbHocTb *H3HH tJaCTHbI, ,ABHmYlI.\eHCSI co CKOpOCTLIO
v = 0,99 c (c - CKOpOCTb CBeT8 B B8KYYMe), 60nLIlIe c06CTBeHHoro BpeMeHH
mH3HH 'l8CTHLI B k p83, paBHoe:
1) 7,5; 2) 7.1; 3) 6,8; 4) 6,5; 5) 6,3.
A8. ATOM BO,lIOpO8 H3nYtJ8eT <POTOH C ,AnHHOH BonHbI A. = 487 HM. ecnH
3neKTpOH B 8TOMe nepeXOlIHT H8 BTOpylO Op6HTY C Op6HTbI. HOMep KOTOpOH
p8BeH (nocToSlHH8S1 PH,A6epra R = 3.29. 10 15 c- l ):
1) 2; 2) 3; 3) 4; 4) 5; 5) 6.
703

A9. rrOKOSlHeCfl HeihpOH H aHTHHeihpOH aHHHrHnHpYIOT, 06pa3YSI ,ABa <1>0-
TOHa. Ecnli Maccs HeHTpOHa m = 1,674' 10- 27 Kr, TO :':meprHSI W Kam,AOrO H3
<POTOHOB paBHa:
1) 870 M3B; 2) 910 M3B; 3) 920 M3B; 4) 940 M3B; 5) 950 M3B.
AI0. OT ,ABlimyerocH nOe3,Aa OTlennSlIOT BarOH. TIoe3,A npO,AOnmaeT ,ABH-
raTbCSI C TOH me Ha'lanbHOH CKOpOCTblO Vo. EcnH BarOH ,ABHmeTCH paBH03a-
Me,AneHHO, TO OTHOllleHHe nYTeH .!L npOH,AeHHbIX nOe3,AOM H BarOHOM K
82
MOMeHTY OCTaHOSKH BarOHa, paBHO:
1) !
2
2 ) . 3 ) !
l' 1
4 ) !.. 5 ) 
3' 1
All. CaHKH MaccoH m = 75 Kr CKaTblsalOTCSI C BeplllliHbJ ropbI BbICOTOH
h = 11 M H ,AnHHOH CKnOHa l = 80 M EcnH MO,Aynb CHnbI conpOTHsneHHSI ,ABH-
meHHIO caHOK Fe = 80 H, TO CKOpOCTb V CaHOK Y OCHOSaHHSI rOpbI paBHa:
M
1) 5
C
M
2) 6
P
3po
1
Po 2 3
I I
I I
I I
0 VI V 3 V
Puc. 41
C
3) 7  ; 4) 8 ; 5) 10 M .
C C C
A12. Boe pa60tJHX HeCYT 6peBHO, Macca KOToporo
m = 50,0 Kr, a )J,JIHHa l = 600 CM. IIepBbIH pa60'lHH
nO,A,Aep)KHBaeT 6peSHO Ha paCCTOSlHHH 8 = 100 CM OT
ero KOHa, BTOpOH - npoTHBononomHbIH KOHe 6peB-
Ha. CHna F. C KOTOPOH 6pesHo ,AaBHT Ha nepBoro pa-
60tJero, paBHa:
1) 500 H; 2) 400 H; 3) 300 H; 4) 200 H;
5) 100 H.
A13. EcnH nocne ocyecTBneHHSI npoecca 1-2-3
(pHC. 41) KBa)J,paT Cpe,AHeH KBa,ApaTH'lHOH CKOpOCTH
,ABHmeHHSI MoneKyn H,AeanbHOro ra3a YMeHblllHncSl B
k = 1,5 pa3a, TO 06'beM ra3a V 3 60nbllle VI B .n paa.
1) 1; 2) 2; 3) 3; 4) 4; 5) 5.
A14. EcnH npH H306apHoM npoecce 06'beM H,AeanbHOro ra3a YBenHtJHncSI
Ha V = 10 n, a TeMnepaTypa YBenHtJHnaCb B k = 3 pa3a, TO Ha'lanbHblH VI Ii
KOHetJHblH V 2 06'beMbI ra3a paBHbI COOTBeTCTseHHO:
1) 5.0 n; 15 n; 2) 15 n; 5,0 n; 3) 10 n; 20 n; 4) 20 n; 10 n;
5) 15 n; 25 n.
A15. Ba npOBO,ASIHX lllapa pa,AHycaMH RI H R2' 3ap.H)KeHHbIX ,AO IIO-
TeHHanOB <PI H <PI' COe,AHHSlIOT npOBO,AHHKOM, eMKOCTblO KOToporo MO*HO npe-
He6pe'lb. 3apSl,A ql nepBoro lllapa nocne COe,AHHeHHSI paBeH:
1) RI + R 2 .
41t€oRI (RI<PI + <P2 ) ,
5) R 1 <PI + R 2 <P2
 +R 2
704
2) ql + q2 . 3) 41t€oRl (RI<PI + <P2) . 4) C1<PI + C 2 <P2 .
2' R 1 +R 2 ' 2'

Al6. TIpH YBenHqeHHH HanpH:meHHH Ha KOHAeHCaTOpe eMKOCTblO C = 20 MK<I>
B k = 2 pa3a 3HeprHSI nonSl B03pocna Ha W= 0,3 ,I:(m. Ha'lanbHOe 3Ha'leHHe
3HeprHH W 1 nonH paBHO:
1) 0,8 ,I:(m; 2) 0,7 ,I:(:m; 3) 0,5 ,I:(m; 4) 0,4 ,l];m; 5) 0,1 ,l];m.
A17. HCTO'lHHK TOKa 3aMbIKaeTCSI npoBonoKoH, B nepBOM cnY'lae conpo-
THBneHHeM Rl = 4 OM, BO BTOpOM - conpOTHBneHHeM R 2 = 9 OM. B 060HX
cnY'laSlX KOnHqeCTBO TennOTbl Q, BbI,AenSleMOH B npOBonOKax, O,AHHaKOBO. BHYT-
peHHee conpOTHBneHHe r HCTO'lHHKa paBHO:
1) 2 OM; 2) 3 OM; 3) 4 OM; 4) 5 OM; 5) 6 OM.
AlS. TIo ABYM 6eCKOHeqHO ,nnHHHbIM np.HMOnHHeHHbIM napannenbHblM npo-
Bo,nHHKaM. pacnonomeHHblM Ha paCCTOSlHHH r = 20 CM APyr OT APyra. npoxo-
,llSlT TOKH 1 1 = 5.0 A H 1 2 = 10 A. HHAYKUHSI MarHHTHoro nonSl B B TO'lKe.
Y,AaneHHOH Ha rl = 12 CM OT nepBoro npOBO,AHHKa H Ha r2= 16 CM OT BTOpO-
ro, paBHa:
1) 8,0 MKTn; 2) 6,0 MKTn; 3) 11 MKTn; 4) 15 MKTn; 5) 16 MKTn.
A19. COneHOH,A, COCTOSlll.\HH H3 N = 1000 BHTKOB npOBO,Aa, HaXO,AHTCSI B
O,AHOpO,AHOM MarHHTHOM none, HH,AYKl1.HH KOToporo H3MeHSleTCSI co CKOpOCTblO
!:J3 MTJI
- = 20,0 -. OCb coneHOHAa COCTaBnSleT C BeKTopOM HH,AYKl1.HH MarHHTHoro
!:J.t C
nonSl yron a=6,00o. PaAHYC coneHOHA8 R=20,0 MM. S,I:(C HH,AYKl1.HH , B03-
HHKalOlI.\a.H B COneHOH,Ae, paBHa:
1) 12.6 MB; 2) 11,8 MB; 3) 10,5 MB; 4) 8,31 MB; 5) 7,61 MB.
A20. KaTywKa HH,AYKTHBHOCTH, KOH,AeHCaTOp H pe3HCTOp COe,AHHeHbl Mem-
,AY C060H nOCne,AOBaTenbHO. ,I:(eHcTBYlOlI.\He HanpSlmeHHSI Ha HHX paBHbl COOT-
BeTCTBeHHO U L' U C H U R' ,I:(eHcTBYlOlI.\ee HanpHmeHHe U Ha BceM yq acTKe paBHO:
1) UR+U L ; 2) UL-U c ; 3) U R 2 +(U L -Uc>2; 4)  U+(UL-Uc>2;
5) UL-U c
U R
A21. Ha nYTH CXO,AHlI.\erOCSI nyqKa nyqeH nOCTaBHnH c06HpalOIl.tYIO nHH3Y
C <POKYCHbIM paCCTOSlHHeM F = 70,0 MM. B pe3ynbTaTe nY'lH cownHCb B TO'lKe
Alt nemall.\eH Ha rnaBHOH OnTH'leCKOH OCH Ha paCCTOHHHH f = 50,0 MM OT
nHH3bl. EcnH nllH3Y y6paTb, TO OHH COH,AYTCSI OT TO'lKH Ai Ha paCCTOSlHHH 1,
paBHOM:
1) 17,5 CM; 2) 16,5 CM; 3) 14,5 CM; 4) 12,5 CM; 5) 10,5 CM.
A22. Pa60Ta BbIXO,Aa 3neKTpOHa 113 KanHSI As=2,20 3B. TIpH OCBell.\eHHH
KanH.H CBeTOM C AnHHoH BonHbl A = 330 HM 3aAepmHBalOlI.\aSl pa3HOCTb nOTeH-
l1.HanOB !:J.q> ,AnSI 3neKTpOHOB paBHa:
1) 1.21 B; 2) 1.35 B; 3) 1.56 B; 4) 1,65 B; 5) 1.86 B.
A23. MOTOl1.HKnHCT COBepmaeT nOBopOT B ropH30HTanbHoH nnOCKOCTH, onH-
CbIBaH Ayry oKpymHoCTH. EcnH K03<P<PHl1.HeHT TpeHH.H pe3HHbI 0 ,Aopory paBeH
705

Jl = 0,5. TO MOTOHKnHCT, ABHraSlCb C MaKCHMaJIbHO B03MOmHOH CKOpOCTblO,
OTKnOHSleTCSI npH aTOM OT BepTHKanH Ha yron <p, KOTaHreHC KOToporo paBeH:
1) 1: 2) 2: 3) 3: 4) 4; 5) 5.
r
A24. B HnHH,ApH'leCKOe Be,ApO HanHT8 BO,Aa nnOTHOCTblO p = 1, 0 cr.! H Mac-
COH m = 12 Kr. EcnH Ha paCCTOSlHHH h = 0,10 M OT ,AHa rH,ApOCTaTH'leCKOe ,AaB-
neHHe H8 60KOBYlO CTeHKY Be,Apa p = 200 TIa, TO nnOa,Ab S ,AHa Be,Apa paBHa:
1) 0,050 M 2 ; 2) 0.10 M 2 ; 3) 0,15 M 2 ; 4) 0,20 M 2 ; 5) 0,25 M 2 .
A25. TIpouecc. npOHCXO,llSlHH C lIaHHoH MaCCOH H,AeanbHOro ra3a, onHCbI-
BaeTCSI 3aKOHOM p 2 V = const. EcnH ,AaBneHHe ra3a YMeHbWHJIOCb B k = 4 pa3a.
TO ero TeMnepaTypa:
1) YMeHbwHnacl:. B 2 pa3a; 2) YBenH'lHnaCb B 2 pa3a;
3) He H3MeHHnaCb; 4) YMeHbwHnacb B 4 pa3a; 5) YBenH'lHnaCb B 4 pa3a.
A26. ABToMo6Hnb MaCCOH m = 1,0 T, Kn,n ,ABHraTen.H KOToporo 11 = 30 %,
,llBHmeTCSI paBHOMepHO CHa'lana BBepx no lIopore C YKnOHOM (sina. = 0,030), a 3a-
TeM no ropH30HTanbHoH lIopore. Ecnlf C'lHTaTb. 'lTO CHJIa TpeHHfl B 060HX cny-
QRJlX OA"HRKOBR " CKOpOCTL nOCTOJlHHRJI. TO PRCXOA 6eH3"HR (q = 46 M'" 1
Ha nYTH S = 1,0 KM npH ,ABHmeHHH no Aopore C YKnOHOM YBenH'lHTCSI no cpaB-
HeHHIO C paCXO,AOM 6eH3HHa npH ,ABHmeHHH no ropH30HTanbHoH Aopore Ha:
1) 0.30 Kr; 2) 0,20 Kr; 3) 0,10 Kr; 4) 20 r; 5) 10 r.
A27. B neqH, Kn,n KOTOpOH 11 = 20 % . pacnnaBnSlIOT 'lyrYH MaccoH
m = 10,0 T, HMelOHH TeMnepaTYPY t l = 25,0 °c. Y,AenbHaSl TennoeMKOCTb 'lY-
,nm KAm
rYHa c = 540 -, Y,AenbHaSl TennOTa nnaBJIeHHSI qyrYHa A = 97,0 - ,
Kr.K Kr
TeMnepaTypa nnaBneHHSI 'lyrYHa t 2 = 1150 °c. EcnH YlIenbHaSl TennOTa cropa-
M.L(m
HHSI KaMeHHoro yrnSl q = 30,0 -, TO Heo6xo,AHMO Cme'lb KaMeHHbIH yronb
Kr
MaCCOH mI' paBHoH:
1) 162 Kr: 2) 1.01 T: 3) 1,17 T: 4) 1.34 T; 5) 2.51 T.
A28. lliapHK MaCCOH m = 150 Mr, nO,ABeWeHHbIH Ha HenpoBoASIeH HHTH,
If MeeT 3apSl,A ql = 1,0. 10- 8 Kn. Ha paccToSlHHH r= 32 CM CHH3Y OT Hero no-
MeaeTCSI BTOpOH ManeHbKHH wapHK. QTo6bI HaTSlmeHHe HHTH YBenH'lHnOCb
B ,ABa pa3a, 3apSl,A q2 BToporo wapHKa ,AOnmeH 6blTb paBeH:
1) 1,7 MKKn; 2) -1.5 MKKn; 3) -1,7 MKKn; 4) -1,3 MKKn;
5) 1,3 MKKn.
A29. MeTannH'leCKOe KonbO pa,AHYCOM R HMeeT 3apSlA q. HanpSlmeHHocTb
nonSl E B L\eHTpe Konba H Ha paCCTOSlHHH a OT eHTpa B,AOnb OCH, nepneH-
,AHKynSlpHoH nnOCKOCTH Konba. paBHa:
kq kaq kq kq . kq kqa
1 ) 0' , 2 ) O' . 3 ) . 4 ) . 0' , 5 ) 0' ,
I 2 2 ' \% · 2' R 2 + a 2 ' R 2 ' R 2 + a 2 .
'" R + a (R 2 + a 2 J 2 R
706

A30. YqaCTOK u;enH npeACTaBnSleT C060H TPH napannenbHO COe,AHHeHHbIX
MeT8nnH'leCKHX npOBO,AHHK8. COnpOTHSneHHSI nepBOrO H BTOpOrO npOBO,AHHKOS
paBHbl COOTBeTCTBeHHO Rl = 6 OM H R 2 =4 OM. ECJIH CHn8 TOK8 H8 Y'l8CTKe
1=9 A, 8 CHna TOK8, npOXOASlIl'(erO 'lepe3 BTOpOH npOBO,AHHK, 1 2 = 3 A, TO
COnpOTHBneHHe R3 TpeTberO npOBO,AHHK8 p8BHO:
1) 10M; 2) 2 OM; 3) 4 OM; 4) 5 OM; 5) 3 OM.
A31. OTHoIIIeHHe P8AHYCOB Ru oKpymHocTeH, no KOTOpblM ,ABHmYTC.H C
Rp
O,AHH8KOBbIMH CKOpOCTSlMH a.-'l8CTHI.\a He Ii npoTOH  H B O,AHOpO,AHOM M8r-
HHTHOM none, p8BHO:
1 1
1) '4; 2) "2; 3) 4; 4) 1; 5) 2.
A32. M8TeM8TH'leCKHH M8S1THHK ,AnHHOH l = 100 CM cosepIII8eT Kone68HHSI
B6nH3H BepTHK8nbHoH CTeHKH. Ha p8CCTOSlHHH S = 30,0 CM OT TOtJKH nO,ASeC8
B CTeHKY B6HT rB031Ib. TIepHoA T Kone68HHH M8S1THHK8 p8seH:
1) 1,52 c; 2) 1,61 C; 3) 1,82 C; 4) 1,92 C; 5) 1,98 c.
A33. Ha6nlO,AaTenb CMOTpHT no BepTHKanH BHH3 Ha npeAMeT, nOKpblTblH
nnocKon8p8nnenbHoH CTeKnSlHHOH nn8cTHHKOH C nOK838TeneM npenOMneHIiSl
n = 1,6. C'lHT8HTe, 'lTO AnSI M8nblX yrnoB tga. === sina.. EcnH H86nlO,A8Tenb BH-
,AHT 1i306pameHHe npe,AMeT8 Ha p8CCTOSlHHH S = 5,0 CM OT sepxHeH nOBepXHOC-
TH nn8CTHHbI, TO TOnIl'(HH8 h nn8CTHHbI p8BHa:
1) 5,0 CM; 2) 6,0 CM; 3) 7,0 CM; 4) 7,5 CM; 5) 8,0 CM.
Bl. B 6e3BeTpeHHYlO noro,Ay KannH ,Aom,ASI OCTaBnSlIOT Ha OKHe p8BHOMep-
HO ,ABHmYIl'(erOCSI aSTo6yca CJIe,A, H8npaBneHHblH no,A yrnoM a. = 30° K BepTH-
KM
K8nH. EcnH CKOpOCTb 8BT06YC8 VI = 52 -, TO CKOpOCTb KannH V2 OTHOCH-
'l
TeJlhHO 3eMnH paBHa ... r K: ).
Puc. 42
B2. ,l];nSl Toro 'lTo6bI nO,AHSlTb p8sHoMepHo rpY3 M8CCOH
m = 500.0 Kr H3 III8XTbI rny6HHoH h = 200,0 M H8 K8H8Te, Mac-
ca K8m,Aoro MeTp8 KOToporo ml = 1500 r, Heo6xo,AHMO COBepIIIHTb
pa60TY A, paBHYIO n. (K,D;m).
B3. 'tJepe3 HenO,ASHmHbIH 6noK nepeKHHYTa HHTb, K KOTOpOH
no,nBeIIIeHbI TPH O,AHHaKOBbIX rpY38 (pHC. 42). EcnH clina ,A8BneHHSI
H8 OCb 6noK8 F = 80 H. TO Macca m KIDK,AOrO rpY38 p8BH8 n. (Kr).
B4. K HCTO'lHHKY TOK8 C BHYTpeHHHM conpOTHBneHHeM r = 2 OM
SH8'l8JIe nO,AKnlO'l8eTCSI pe3HCTOp C conpOTHBneHHeM R = 2 OM, 38-
TeM n8p8nnenbHO nO,AKnlOtJ8eTCSI T8KOH me pe3HCTOp. KOnH'leCTSO
TennOTbI Q, BbI,AenSleMOH B nepBOM pe3HCTope nocne nO,AKnlO'leHHSI
BToporo pe3HCTopa, 3a T8Koe me BpeMSI YMeHbIII8eTCSI B ... p83.
707

L
Puc. 43
P
2po 2 3
]/i
Po
11 I
I I
I I
0 V o 3V o V
Puc. 44
708
B5. P8,lJ;liOnOK8Top nOCbln8eT N = 5000
HMnynbcoB B ceKYHp;y. ,l{nHTenbHocTb
K8m,AOrO HMnynbc8 t = 3,0 MKC. M8KCH-
M8nbH8S1 p;8nbHOCTb Smax pS3Be,AKH pa)1;HO-
nOK8Top8 p8BH8 ... (KM).
B6. ,l{B8 nO,ABHmHbIX KnHH8 C O,AH-
HSKOBOH M8CCOH M HMelOT nnSBHble ne-
peXO,Abl H8 ropH30HTsnbHYlO nnOCKOCTb
(pHC. 43). C nesoro KnHHS COCK8nb3bIS8-
eT w8H6a M8CCOH m C BblCOTbl h= 1,0 M.
TpeHHe OTCYTCTByeT. EcnH M8CC8 KnHH8
M = 4m. TO M8KCHM8nbH8S1 BblCOT8 hi' Ha
KOTOPYIO nO,AHHM8eTC.H W8H6a Ha npSBOM
KnHHe, paBH8 .n (CM).
B7. H8 p-V-,AHSrp8MMe (pHC. 44) H306-
p8meH HKn. npOSO,AHMbIH C O,AH08TOMHbIM
H,AeSnbHblM rS30M. KII.l( 11 HKns pa-
BeH ... (%).

OTBETbJ K TECT AM
HOMep HOMep TeCT8
Bonpoca I II III IV V VI VII VIII IX X
Al 3 2 1 2 1 3 1 2 2 4
A2 4 1 5 4 2 1 4 3 3 4
A3 3 3 2 1 2 3 2 1 1 1
A4 2 5 4 1 4 3 5 5 1 3
A5 1 3 1 2 3 2 1 3 1 5
A6 1 1 4 1 4 2 3 5 1 5
A7 3 5 1 1 1 2 3 1 4 2
A8 4 5 2 5 2 1 1 3 3 3
A9 2 1 3 5 5 3 3 2 1 4
AI0 4 1 2 4 1 3 4 4 4 2
All 3 3 3 5 3 2 4 2 3 3
A12 2 2 4 2 4 4 4 4 5 3
Al3 4 3 4 3 2 4 3 1 3 2
Al4 5 2 1 1 3 2 2 4 5 1
A15 3 1 4 5 4 2 3 1 1 3
Al6 1 4 1 5 5 1 1 2 4 5
Al7 2 5 2 1 1 4 2 2 1 5
Al8 2 3 4 1 2 5 3 1 5 4
Al9 2 4 1 2 5 3 3 4 5 1
A20 4 1 3 3 5 5 5 3 1 4
A21 5 1 1 4 1 3 2 5 3 4
A22 1 2 5 1 1 5 2 3 5 3
A23 3 1 2 3 3 3 1 2 3 2
A24 4 3 4 2 4 4 5 2 1 2
A25 4 2 1 5 4 3 3 4 3 5
A26 3 3 3 2 1 3 4 1 3 4
A27 3 5 2 5 1 3 2 4 3 3
A28 2 4 3 1 2 3 4 2 1 3
A29 1 1 3 4 2 1 1 5 3 2
A30 1 2 4 3 4 2 3 1 1 5
A31 2 1 3 4 3 1 2 4 1 5
A32 1 2 3 5 1 5 1 4 1 3
A33 1 1 4 4 3 2 3 3 3 5
BI 24 14 40 50 17 3 160 100 30 90
B2 10 18 83 480 60 173 17 373 30 1300
B3 800 16 4 2 2 24 36 36 20 3
B4 2 2 2 2 2 2 2 2 2 2
B5 141 13 17 2 64 200 6 3 6 30
B6 60 67 17 2 50 61 12 105 440 64
B7 20 20 20 6 9 10 2 17 1662 9
709

COAep>KaHMe
BBE,L'(EIllIE
MEXAHHKA
3
5
OCHOBLI KHlleMaTHKH . . . . . . . . . . . . . . . . . . . . . 5
1.1. OCHOBHbIe nOH$lTH$I KHHeMaTHKH........... 5
1.2. OTHOCHTenbHOCTb ,nBHmeHH$I CJIOmeHHe CKopocTeH. 9
1.3. P8BHoMepHoe np$lMOJIHHeHHOe ,nBHmeHHe ........... 10
1.4. P8BHoycKopeHHoe np$lMOnHHeHHoe ,nBHmeHHe . . . . . . . . . 11
1.5. CB060,nHoe na,neHHe TeJI. YCKopeHHe cB06o,nHoro n8,neHH1I 13
1.6. .l{BHmeHHe Ten8. 6pomeHHoro BepTHKanbHO ............ 14
1.7. .l{BHmeHHe TeJI8, 6poweHHoro ropH30UTaJIbHO............ 15
1.8. ,D;BH>KeHHe Tena, 6poweHHoro noon yrnoM K ropH30HTY . . . . . . . . . . . . . . . . .. 16
1.9. ,D;BHmeHHe Ten8 no oKpymHocTH C nOCTOJIHHOH no MO,nYJIlO CKOpOCTblO . . . . .. 18
1.10. YCKopeHHe npH ,nBH>KeHHH Ten8 no oKpymHocTH C nOCTO$lHHOH no MO,nYJIIO
CKOpOCTbIO (LteHTpOCTpeMHTenbHoe ycKopeHHe).... . . . . . . . . . . . . . . . . . . . .. 19
1.11. Bp8Il\8TenbHoe ,nBHmeHHe 86cOJIlOTHO TBep,noro Tens BOKpyr
HenO,l(BHmHOH OCH. . . . . . . . . . . . . . . . . . . . . . . . . . . .
BonpOCbL U 3aaaH.UR aJ/.Sl. CaMOKOH.mpo.nR . . . . . . . . . . . . . .. .............
OCH08H.bte tjJOPMYJ/.bt . . . . . . . . . . . . . . . . . .
MemOOU'leCKUe peKoMeHoaquu ............................
llpuMepbl peWeH.UR 3aOa'l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.
2.
OCHOBbl .lI;HUaMHKH. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1. IIepBbIH 38KOH HbIOTOH8... . . . . . . . . . . . . . . . . . . . . .
2.2. HHepLtH8nbH8$1 CHCTeM8 OTC'leTa ...............
2.3. IIpHHLtHn OTHOCHTeJIbHOCTH r8nHne$l . . . . . . . . . . . . . . . . .
2.4. M8cca........................................
2.5. CHJIa....................
2.6. BTOpOH 38KOH HblOTOH8 . . . . . . . . . . . . . . . . . . . . .
2.7. TpeTHH 38KOH HblOTOH8 ... . . . . . . . . . . . . . . . . .
2.8. CHJIbl ynpyroCTH. . . . . . . . . . . . . .
2.9. 38KOH rYK8 . . . . . . . . . . . . . . . . . . .
2.10. CHnbI TpeHH$I. K03ct>ct>HLtHeHT TpeHH$I. . . . . . . . . . . . . . . .
2.11. ,[(BHmeHHe Ten8 C yqeToM CHJIbI TpeHH$I........... .
2.12. rp8BHT8HoHHbIe CHnbl. 38KOR BceMHpHoro TJIrOTeHHJI .
2.13. CHn8 TJImeCTH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.14. ,[(BHmeHHe Ten8 no,n ,neHcTBHeM CHnbI TJImeCTH .....
2.15. Bec Ten8 . . . . . . . . . . . . . . . . .
2.16. HeBecoMocTb...............
2.17. IIepB8$1 KOCMHqeCKa$l CKOpOCTb .......................
2.18. ,[(BHmeHHe HCKYCCTBeHHbIX cnYTHHKOB . . . . . . . . . . . . . . . . . . .
2.19. CHcTeM8 M8TepH8nbHbIX TOqeK. Yp8BHeHHe nocTynaTenbHoro
,nBHmeHHJI CHCTeMbI M8TepH8JlbHbIX TOqeK. eHTp HHepLtHH ..
2.20. HeHHepHanbHbIe CHCTeMbI OTCqeT8 . .
2.21. SlBneHH$I. H86nlO,n8eMbIe B HeHHepH8nbHbIx CHCTeM8X OTCqeTa.
BonpOCbt U JaOaH.UR OJ/.R CaMOKOH.mpo.nR . . . . . . . . . . . . . . . . .
OCH.08H.We tjJOPMYJ/.bt . . . . . . . . . . . . . . . . . . . . . . . .
hfemOOU'leCKUe peKOMeH.Oaquu
fIpuMepw peWeH.UR 3aOa'l " . .
. . . . . . . . . . . . .. 20
21
22
25
25
. . . . . . . . .. 30
. . . . . . . . . .. 30
31
31
31
32
33
34
34
35
36
38
38
39
40
40
41
. 42
42
43
46
47
47
48
49
.. . .. 49
3. 3aKoHbl cOXpaUeHH$I B MexaHHKe . . . . . . . . 54
3.1. MMnYJIbC Ten8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.2. 38KOH coxpaHeHH$I HMnYJIbCa . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.3. Pe8KTHBHoe ,nBHmeHHe ......................................... 55
710

Bonpocw U 3aOaH.UR OAR CaMOKOH.mpOJ/.R.. . . . . . . . . 56
OCH.08H.We tjJOPMYJ/.b£ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
M emOOU'leCKUe peKOMeH.aaquu.................................................. 58
llpuMepbl peWeH.UR 3aOa'l .............. . . . . . . . . . . . . 58
3.4. MeX8HHqeCK8JI pa60T8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 60
3.5. P860T8 CHJIbI T$lmeCTH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.6. Pa60Ta CHnbI ynpyrocTH . . . . . . . . . . . . . . . . 62
3.7. Pa60T8 CHnbI TpeHHJI . . . . . . . . . . . . . . . . . . . . . . 63
3.8. MOIltHOCTb.................................... 63
Bonpocw OJ/.R CaMOKOH.mpoJ/.R................................................... 64
OCH.08H.We tjJOPMYJ/.W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. ............ 65
M emOOU'leCKUe peKOMeH.Oaquu................. . . . . . . . 66
IIpuMeprn peweHuR 3aOa'l ...................................................... 66
3.9. MeX8HHqeCK8JI 3HeprH$I......................................... 69
3.10. KHHeTHqeCK8.f1 3HeprHJI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 69
3.11. IIOTeHl.\UanbH8$1 3HeprH$I Tena npH rp8BHT8HOHHOM B38HMo,neHcTBHH . . . 70
3.12. IIoTeHQ,H8nbH8JI 3HeprH$I ynpyro ,nect>opMHpoBaHHoro Tena. . . . . . . . . . . . . . .. 71
3.13. 38KOH coxp8HeHH$I 3HeprHH B MeX8HHKe .................. 71
3.14. M3MeHeHHe MeX8HHqeCKOH 3HeprHH B He38MKHYTOH CHCTeMe ..... 72
3.15. 38KOH coxp8HeHH$I H npeBp8IlteHH.fI 3HeprHH ................... 72
3.16. YnpyrHe H HeynpyrHe coy,n8peHHJI Ten. . . . . . . . . . . . . . . . . 72
3.17. IIpocTeHwHe Mex8HH3MbI................. 73
3.18. P8BeHcTBo p860T npH Hcnonb30B8HHH npOCTbIX Mex8HH3MOB.... 75
Bonpocw U .JaoaHuR OJ/.R caMoKoHmpoJ/.R......................................... 76
M emOaU'leCKUe peKOMeH.Oaquu... ... _ .............. . . . . . . . . . . . . . . . . . . . . . 77
llpuMepw peWeH.UR 3aOa'l .................. 78
4. 3neMeHTbI cTaTHKH....................................................... 82
4.1. CHn8. CnomeHHe CHn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.2. YcnOBue p8BHOBeCH$I Ten8. He HMelOero OCH Bp8IlteHH$I ............. 83
4.3. YcnoBHe paBHOBeCHJI Tena C 38KpenneHHoH OCbIO BpaIlteHH$I . . . . . 83
4.4. YCJIOBHJI p8BHOBeCHH TBep,noro Ten8. . . . . . . . . . . . . . . . . . . . 84
4.5. CnomeHHe n8p8nneJIbHbIX CHn. eHTp TJImeCTH . . . . 84
4.6. BH,nbI paBHOBeCH$I........................... 85
BonpOcbt OJ/.R CaMOKOH.mpOASl.................................................... 87
OCH.08H.bte tjJOPMYJl.bJ . . . . . . . . . . . .. ................... ................ .. . . . . 88
M emOOU'leCKUe peKOMeH.Oaquu.............. . . . . . . . . . . . . . . . . . . . . . . . . . . 88
llpuMepw peWeH.UR 3aoa14 .. .. 89
5. 3neMeUTbI CTaTHKH H p;HHaMHKH 3tH.nKOCTeH H raaOB. . . 93
5.1. ,[(8BneHHe. CHn8 ,n8BneHH$I .................................... 93
5.2. 38KOH llaCKftn$l ,nn.fl mH,nKocTeH H ra30B... . . . . . . . . .. .............. 94
5.3. rH,np8BnHqeCKHH npecc . . . . . . . . . . . . . . . 94
5.4. rH,npOCT8TUqeCKOe ,n8BneHHe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 95
5.5. C006Ilt81OHeCJI COCy1lbI.................................... 97
5.6. IIInI03bI................................. 98
5.7. ATMocct>epHoe J18BneHHe. OnbIT TOPPHqennH.
H3MeHeHHe 8TMocct>epHoro ,n8BneHHJI C BbICOTOH.. . .
5.8. BapoMeTpbI. MaHoMeTpbI...................
5.9. ApxHMe,noB8 CHn8 ,nn$l mH,ZJ.KocTeA H r830B .....
5.10. YcnOBH$I nn8BaHH$I Ten ....................
5.11. BO,nHbIH TpaHcnopT..................................
5.12. B03,nyxonn8B8HHe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.13. 38BUCHMOCTb ,n8BneHH$I mH,nKOCTH OT CKOpOCTH ee TeqeHH$I .....
5.14. ,[(BHmeHHe TeJl B mHJ1KOCT$lX H r83ax. CHna conpOTHBneHHJI ....
5.15. IIo,n'beMH8JI CHn8 KpbIn8 c8MoneT8.... . . . . . . . . . . . . . . . . . . . . . . .
Bonpocw OJ/.R CaMOKOHmpOJ/.R.................... . . . . . . . . . . . . . .
OCH.08H.We tjJOPMYAW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. ... -......
MemOaU'leCKUe peKOMeH.Oauuu..........................................
llpUMepbt peweH.USI. 3aOa'l......................................................
99
100
102
102
104
105
106
108
110
111
112
113
114
711

3JIEMEHTbI CTATHCTII'lECKO:R cI»H3HKH II TEPMO.l(IIHAMHKH. . . . . . . .. .. 118
6. OCHOBbI MoneKynRpHo-KHHeTHqeCKOH TeopHH.................................. 118
6.1 CTaTHCTHqeCKHH H TepMo,nHR8MHqeCKHH MeTOl\bl onHC8HHJI
MaKpOCKOnHqeCKHX CHCTeM ...................... 118
6.2. OCHoBHbIe nOJIOmeHH$I MoneKYJIJIpRO-KHHeTHqeCKOH
TeOpH" " HX onblTHoe 060CHOBaHHe.. . . . . . . . . . . . 119
6.3. JJ;Hcpcpy3"$I................ 120
6.4. BpOYHoBcKoe ABHmeHHe. . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.5. B38HMo,neHcTBHe Mem,ny 8TOM8MH H MOJIeKYJIaMH BeeCTB8.
CTpoeHHe TBep,nbIX. mH,nKHX H r83006p83HbIX TeJI 122
6.6. Macc8 MoneKyn ........................ 123
6.7. KOJIHqeCTBO BeIlteCTBa. IIOCTO$lHH8JI ABora,npo .. 124
6.8. Pa3MepbI MOJIeKYJI ....................... 125
6.9. M,ne8nLHbIH rS3 ........... 126
6.10. CKOpOCTH MoneKyn ra30B................... 127
6.11. OCHoBHoe ypaBueHHe MKT H.I:r.e8JIbHOrO r838 . . . . . . . . . 128
6.12. TeMnep8Typa H TenJIOBOe paBHOBeCHe CHCTeMbI . . . . . 129
6.13. M3MepeHHe TeMnep8TypbI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.14. A6coJIlOTHaJI TeMnep8TypHa$l WK8na. A6cOJIlOTHbIH Hynb . . . . . . . . . . . . . 132
6.15. M3MepeHHe CKopocTeH r830BblX MOJIeKYJI.
OnbIT llITepH8. P8cnpe,neneHHe M8KcBenn8. . . . . . . . . . . . . . . . . . . . . . . . .. 134
6.16. BepOJITHOCTb H 3HTponH$I. I10P$l,nOK H 6ecnOp$l,noK B M8KpOCUCTeMSX . . . . . . . .. 136
BonpOCbt OJ/.R CaMOKOHmpOJ/.R.................................................. 139
OCH08H.bte tjJOp.MYJlbt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. ........ 140
llpuMepbt peWeH.UR 3aOa'l..... .......................................... 141
6.17. YpaBHeHHe COCTO$lHH$I H,neanbHoro ra38 . . . . . . . . . . . . . . . . 143
6.18. l{8cTHble cnyq8H ypaBHeHH$I COCTOJIHH$I H,ncanbHoro r83R . . . . . . . . . . . . . .. 144
BonpOCbt OJ/.R Ca.MOKOH.mpOJ/.Sl................................................... 146
OCH06H.bte tjJOPMYJlbL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 147
MemOOU'leCKUe peKO.MeH.Oaquu. . . . . . . .. 148
llpuMepbt peWeH.UR 3aOa'l ... 148
7.
OCHOBbI TepMo,nHHaMHKH ................................................
7.1. TepMo,nHH8MH'IecKHe nap8MeTpbI COCTO$lHHJI Ten8 . . . . . .
7.2. BHYTpeHH$I$I 3HeprH$I Ten . . . . . . . . . .. ............
7.3. M3MeHeHHe BHYTpeHHeH 3HeprHH .....................................
7.4. KOnH'leCTBO TennOTbI .............
7.5. P860T8 B TepMo,nHHSMHKe . . . . . . . . . . . .
7.6. nepRbIH 38KOH TepMo,nHH8MHKH. . . . . . . . .
7.7. IIpHMeHeHHe ncpBoro 38KOHa TepMo,nHH8MHKH
K p83nHqHbIM TenJ'lOBbIM npO[teCC8M B H,ZJ.e8nbHOM ra3e...
7.8. A,nna68THbIH npo[tecc . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.9. TenJIoeMKocTH npH nOCTOJIHHOM ,n8BneHHH
H nOCTOJIHHOM 06'beMe. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.10. 06paTHMbIe H He06p8THMbIe npo[teCCbl. He06paTHMocTb TenJIOBblX
npoeCCOB. BTOpOH 38KOH TepMo,nHHaMHKH. . . . . . . . . . . . . . . . . .
7.11. TenJIoBble ,nBHr8TenH. IIpHHlHn ,neHcTBHJI TenJIOBblX ,nBHr8TeneH
7.12. K03CPct>H[tHeHT none3Horo ,neHcTBH$I TenJIOBOrO ,nBHr8TeJIJI . . . .
7.13. Ponb TennOBbIX ,nBHr8TeneH B p83BHTHH Tenn03HepreTHKH
H TpSHcnopT8. TenJIOBbIe ,nBHr8TenH H oxpaH8 npHpO,ZJ.bI.....
BonpOCbt OJ/.R Ca.MOKOH.mpOJ/.R................................ .........
OCH.08Hbte tjJOp.MYJ/.bt .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .................
A1emOOU'leCKUe peKO.MeHoaquU........
llpU.Mepbt pewenUR 3aOa'l .....................................................
7.14. Pe8JIbHbIe r83bI .............................................
7.15. KpHTHqeCKOe COCTO$lHHe BeeCTB8
7.16. CBOHCTBS mH,nKocTeH .........................................
151
151
152
153
154
155
157
158
158
159
161
162
163
165
166
167
169
169
174
176
178
712

7.17. IIoBepxHocTH8$1 3HeprH$I ... . . . . . 178
7.18. nOBepxHocTHoe H8TJImeHHe . . . . . 180
7.19. CM8qHB8HHe . . . . . . . . . . . . . . . . . 181
7.20. KSnHJIn$lpHbie $IBneHH$I............ 182
7.21. KpHCT8JInHqeCKHe H SMopcpHble Ten8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
7.22. CTpoeHHe KpHCTsnnOB. IIPOCTp8HcTBeHH8JI peweTK8. 8neMeHT8pH8JI SlqeHK8. 186
7.23. fIoJIHMopct>H3M ...................................... 187
7.24. Aect>eKTbI B KpHCT8JIJI8X .................. 187
7 .25. ,[(ect>opM8H$I. BH,nbI ,nect>opM8HH TBep,nbIX Ten. . . . . 188
7.26. H8np$lmeHHe. 3aKoH rYK8 ........................... 191
7.27. MeX8HHqeCKHe CBOHCTB8 TBep,nbIX Ten. AH8rp8MMS p8CTJImeHHSI 192
BonpOCbt OJ/.R CaMOKOH.mpoJ/.R. _ . . . . . . . . . . .. ......................... 193
OCH.08H.bte tjJOPMYJ/.bt .. . . . . . . . . . . . . . 194
llpuMepbt peWeH.UR .woa'l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
7.28. licn8peHHe H KOH,neHCSHJI . . . . . . . . . . . . . . . . . . . 197
7.29. H8cbIeHHbIe H HeH8CbInteHHbIe n8pbI H HX CBOHCTB8 ............. 198
7.30. KHneHHe mH,nKocTeH. 38BHCHMOCTb TeMnep8Typbl KlmeHHJI OT ,ZJ.SBJIeHHSI 200
7.31. BJIamHOCTb B03,nYX8 ToqKS pOCbI. llCHxpOMeTp. rHI'pOMeTp ......... 201
7.32. ,[(Hsrp8MM8 COCTO$lHHJI BenteCTB8 ......... 203
Bonpocw OJ/.R CaMOKOH.mpOJZR . . . . . . . . . . . . . 205
IlpuMepbt peWeH.UR 3aOa'l . . . . . . . . 206
3JIEKTPOIUIHAMHKA . 209
8. 3neKTpocTaTIIKa... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
8.1. 3JIeKTpH t iecKHH a8p$l,n. ...................... 209
8.2. OnbITbI P. MHnnHKeH8 H A. ct>.l1ocpct>e . . . . . . . . . 210
8.3. 38KOH coxp8HeHH$I 3neKTpHqeCKOI'0 38pSl,ZJ.8. . . . . . 211
8.4. TO'leqHbIH H p8cnpe,neneHHbIH 38pSl,ZJ.bI . . . . . . . . . . . . 212
8.5. B3sHMo,neHcTBHe 3SpJImeHHblX Ten. 38KOH KynoH8... 213
8.6. 3neKTpOCT8THqeCKOe nOJIe . . . . . .. ............... 214
8.7. H8npJI>KeHHOCTb 3JIeKTpOCT8THqeCKOrO nOJI$I. JIHlIHH HanpSlmeHHOCTH . . 215
8.8. 3JIeKTpHqeCKOe none TOqeqHOrO 38pJI,nS Ii p8BHOMepHO 38pSlmeHHoH ccpepbl 217
8.9. llpHHHn cynepn03HHH nOJIeH . . . . . . .. ............. 218
8.10. OAHopo,nHoe :meKTpHqeCKOe none. . . . . . . . . . . . . . . . . . . . . . . . . 219
8.11. llOTOK BeKTOps HSnp$lmeHHOCTH 3neKTpOCT8TH'IeCKOI'0 non$l . . . . 219
8.12. TeopeMs OCTporp8,ncKoro-r8YCC8 H ee npHMeHeHHe ,nJISI p8CqeTa
3neKTpOCT8THqeCKHX noneH . . . . . . . . . . . . . . . . . . 220
BonpOCbt OJ/.R CaMOKOH.mpOJ/.R...................... 222
OCH08H.ble tjJOPMYJ/.bl .. ... ..... _ . .. .......... 222
M emoiJU'leCKUe peKOMeH.Oaquu.. 224
llpuMepbt peWeH.UR 3aOa'l .............................................. 225
8.13. Pa60Ta 3neKTpOCTSTHqeCKOI'0 nOJIJI npH nepeMeeHHH 38pH.nOB . . . 228
8.14. rrOTeHH8JI......................... ......... 229
8.15. P83HOCTb nOTeHHsnOB. Hanp$lmeHHe ................ 231
8.16 CBJI3b Mem,ny pa3HOCTblO nOTeHHSJIOB H HanpSlmeHHOCTblO 233
8.17. llpOBOAHHKH B aneKTpH'IeCKOM none 233
8.18. 3JIeKTpOeMKocTb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
8.19. ,[(H3neKTpHKH B 3neKTpOCTaTHqeCKOM none. Mex8HH3M nOJI$lpH38HH
,nH3neKTpHKOB. ,[(H3neKTpHqeCK8JI npOHH8eMOCTb . . . . . . . . . . . 238
8.20. KOH,neHc8Topbl. EMKOCTb nnOCKOI'O KOH,neHcaTop8.
Coe,nHHeHHe KOH,neHcaTopOB . . . . . . . . . . . . . . . . . . . 240
8.21. 3JIeKTpeTbI H CerHeToaneKTpHKH....... ........... 242
8.22. IIbe303JIeKTpHQecKHH act>ct>eKT Hero Hcnonb30BaHHe B TeXHHKe . 244
8.23. 3HepI'HSI H nnOTHOCTb aHeprHH aneKTpOCT8THQeCKOrO nonJI 245
BonpocbL OJ/.R CaMOKOH.mpoJ/.R.............. 246
OCH.06H.bte rjJOPMYJ/.bt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
M emOOU'leCKUe peKOMeH.Oaquu..................... . . . . . . . . . . . . . . . . . . . .. 248
llpUMepbt peWeH.UR 3aOa'l .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
713

9. 3aKoHLI nOCTOJlHHOrO TOKa . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
9.1. 3neKTpHqeCKHH TOK. CHJI8 H nnOTHOCTb TOK8 . . . . . . 252
9.2. 38KOH OM8 ,ZJ.JIH O,nHopo,nHoro yq8CTK8 enH. . . . . . . . . . . 253
9.3. ConpoTHBneHHe npOBO,ZJ.HHKOB. Y,nenbHoe conpOTHBneHHe . . . . 254
9.4. 3SBHCHMOCTb conpOTHBneHHH npOBO,ZJ.HHKS OT TeMnepaTypbI . . . . . . .. 256
9.5. IIocne,noBaTenbHoe H n8p8nJIeJIbHOe coe,nHHeHHH npOBo,nHHKOB .. . . . . . . . .. 257
9.6. 3neKTpHqeCKHe enH. 3neKTpo,nBHmY8H CHn8 . . . . . . . . . . . . . . . 259
9.7. 38KOH OMa AJIH Heo,nHopo,nHoro }"I8CTKa enH ...................... 261
9.8. 38KOH OMa ,nnH 38MKHYTOH enH..... . . . . . . . . . . . . . . . . . . . . . . . . . .. 262
9.9. IIpsBHn8 KHpxroct>s. PSCqeT p83BeTBJIeHHbIX aneKTpHqeCKHX eneH........ 265
9.10. IllYHTbI H ,n068BOqHbIe conpOTHBneHHH . . . . . . . . . . . . . . . . . . . . 266
9.11 Pa60T8 H MOHOCTb TOKa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 267
9.12. 3aKoH ,[(moynJI-JIeHa........ 269
Bonpocbl OJ/.R CaMOKOH.mpOJ/.Sl................................................... 270
OCH.08Hbte tjJOPMYJ/.W .. ....................................................... 271
AlemOOU'leCKUe peKoMeHoaquu............................. 273
llpuMepw peWeH.UR 3aOa'l 274
3neKTpHqecKHH TOK B pa3JIWlHltlX cpep,ax. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.1. 3neKTpoHH8H npoBo,nHMOCTb MeTannOB. CBepxnpoBo,nHMOCTb.............
10.2. 3neKTpHqeCKHH TOK B p8CTBOp8X H pacnn8B8X aneKTponHTOB .....
10.3. 38KOH 3JIeKTponH38. Onpe,neJIeHHe 38pH,n8 :3JIeKTpOH8. . . . . . . . . .
10.4. IIpHMeHeHHe aneKTpOJIH38 B TeXHHKe .... . . . . . . . . . . .
10.5. 3JIeKTpHqeCKHH TOK B r838x..........................
10.6. Hec8MocToHTenbHbIH H C8MocToHTenbHbIH p83pH,nbI B r83sx . . . .
10.7. BHAb! caMOCTOHTeJIbHOrO pa3pH,na H HX TeXHH'ieCKOe Hcnonb30B8HHe
10.8. IIOHHTHe 0 nna3Me ...................................
10.9. 3neKTpOHH8JI 3MHCCHH . . . . . . . . . . . .
10.10. 3JIeKTpHqeCKHH TOI< B BaKyyMe.....
10.11. 3neKTpOHH8JI JIaMn8 - ,nHO,ZJ..
BonbT-8MnepHaH x8p8KTepHcTHK8 ,nHo,na. . . . . . .
3neKTpoHHbIe nyqKH H HX CBOHCTBS. .
3neKTpoHHo-nyqeB8H Tpy6K8. . . . . . . . . . . . . . . . .
3neKTpHqeCKHH TOK B nOJIynpoBo,ZJ.HHK8X ................
Co()cTBeHH8$1 3JIeKTpH'ieCK8H npoBo,nHMOCTb nonynpOBo,nHHKOB
H ee 38BHCHMOCTL OT TeMnep8TypbI H OCBeeHHOCTH . . . . . . . 300
10.16. IIpHMecH8H npoBo,nHMOCTb nOJIynpOBo,nHHKOB.... 302
10.17. TepMo, 51 ct>oTope3HcTopbI . . . . . . . . . . . . . . . . 304
10.18. 3JIeKTpoHHO-,nbIPOQHbIH nepexo,n 305
10.19. IIonynpoBo,nHHKOBbIH ,ZJ.HO,n. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 307
10.20. TpaH3HcTop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 308
BonpOCbt OJ/.R CaMOKOH.mpOJ/.Sl.................................. . . . . . . . . . 310
OCH.08H.bte tjJOPMYJ/.bl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
llpuMepw peweH.USI. 3aaall ........ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 311
10.
10.12.
10.13.
10.14.
10.15.
279
279
282
283
286
287
288
289
291
293
294
295
298
298
300
11. MarHHTHoe nOJIe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.1.
11.2.
11.3.
11.4.
11.5.
11.6.
11. 7.
11.8.
11.9.
11.10.
11.11.
11.12.
11.13.
11.14.
B3aHMo,neHCTBHe npOBo,nHHKOB C TOKaMH. . . . . . . . . . . . . .
M8rHHTHoe none TOK8 . . . . . . . . . . . . .
M8rHHTH8H HH,nYKHH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
JIHHHH M8rHHTHoH HH,nYKHH...................................
M8rHHTHoe none npHMonHHeHHoro npOBo,nHHK8 C TOKON . . . . .
MarHHTHoe none KpyroBoro TOKa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
M8rHHTHoe nOJIe COJIeHOH,ZJ.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
,[(eHcTBHe M8rHHTHoro nOJIH H8 npOBo,nHHK C TOKOM. CHJI8 AMnepa . . . . . . .
3JIeKTpo,ZJ.BHr8Tenb nOCTOJIHHOrO TOKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3JIeKTpOH3MepHTenbHbIe npH60pbI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
rpOMKoroBopHTenb . . . . . . . . . . . . . .. ....................
CHna JIopeHa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
,[(BHmeHHe 38pJImeHHbIX Q8CTH B 3neKTpHQeCI<HX H M8rHHTHbIX nonHx....
Mcnonb30B8HHe CHJIbJ JIopeH8. M8cc-cneKTporp8ct> ...................
313
313
315
316
317
318
319
319
321
322
323
324
324
326
328
714

11.15. YCKopHTeJIH 38pHmeHHbIX q8CTH..
11.16. MarRHTHble CBoHcTBa BeIlteCTB8. M8rHHTH8JI npoHH1l8eMocTb
11.17. ct>eppoM8rHeTH3M . . . . . . . . . . . . . . . . . . . . . . .
11.18. M8rHHTH8JI 38nHCb HHQ>OPM811HH .
Bonpocw OJ/.R CaMOKOH.mpoJ/.R . . . . . . . . . . . . .
OCHo8Hbte tjJOPMYJ/.bJ ..............................................
MemOaU'ieCKUe peKoMeHaaquu...................................
llpuMepw peweHuR 3aOa'i .. ....
328
330
333
335
336
338
339
340
12.
3neKTpOMarHHTHaJi HHltYKl.\HJI .......... . . . . . . . . . .
12.1. IIOTOK MarHHTHOH HH,nYKllHH...... . . . . . . .
12.2. 3JIeKTpOM8rHHTH8JI HH,nYKllHJI. OnbITbI ap8,neH . . .
12.3. IIpaBHJIO JIeHll8. . . . . . . . . . . . . . . . . . . . . . . . . . .
12.4. S,[(C HH,nYKl.\HH. 38KOH 3neKTpoM8rHHTHoH HH,ZJ.YKHH .
12.5. 3,[(C HH,nYKHH B ,nBHmYIltHXCJI npOBo,nHHK8X . .
12.6. BHxpeBoe aneKTpHqeCKOe none ............
12.7. MH,nYKTHBHOCTb. SlBneHHe C8MOHH,nYKHH . . . . .
12.8. 3HeprHJI MarHHTHoro nOJUJ K8TYWKH C TOKOM. 3HeprHJI M8rHHTHoro nOJIJI..
12.9. OTHocHTenbHocTb 3neKTpHqeCKOrO H M8rHHTHoro nOJIeH.
IIOHJITHe o() aJIeKTpOM8rHHTHOM nOJIe... . . . . . . . . . . .
12.10 MarHHTHoe nOJIe 3eMJIH. Pa,ZJ.HallHoHHLIe nOJIca 3eMJIH .....
Bonpocw OJ/.R caMoKoHmpoJ/.R . . . . . . . . . . . . . . . . . . .
OCH08H.We tjJOPMYJ/.bt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .
MemOOUileCKUe peKOMeH.Oauuu................................................
llpuMepw peWeH.UR 3aOa'l
344
344
344
345
347
348
350
351
354
355
356
359
361
362
363
KOJIEBAHH.H JI BOJlHbI . . . . .
367
13. MeXaHH'iCCKHe Kone()auHH .. 367
13.1. Kone68TeJIbHOe ,nBHmeHHe. CB060,nHbIe KOJIe68HHH B H,neanbHbIX
KOJIe6aTeJIbHbIX CHCTeM8X . . . . . . . . . . . . . . . . . . . . . . . 367
13.2. r8pMOHHqeCKHe Kone6aHHH . 368
13.3. rpa<pHqecKoe npe,nCTaBJIeHHe r8pMOHHqeCKHX Kone6aHHH.
BeKTopHble ,nH8rpaMMbl.......................... 370
13.4. npHHHn cynepn03HHH. CJIOmeHHe KOJIe6aHHH . 37]
13.5. HerapMOHHqeCKHe KOJIe68HHH. r8pMOHHqeCKHe H Her8pMOHHqeCKHe
Kone68HHH B npHpo,ne H TeXHHKe . . . . . . . . . . . . . . . . . . . . 373
13.6. MaTeM8THqeCKHH M8JITHHK.
IIepHo,n KOJIe68HHH M8TeMaTHQeCKOrO M8HTHHK8 . 374
13.7. IIpY:>KHHHLIH M8JITHHK . . . . . . . . . . . . . . . . . . . . . . 377
13.8. IIpeBp8eHHe 3HeprHH npH r8pMOHHQeCKHX KOJIe68HHJIX.
38TyxaIOHe KOJIe68HHJI. . . . . . . . . . . . . . . . . 378
13.9. BLIHym,neHHbIe Kone68HHJI. Pe30H8HC. 380
BonpOCbt OJ/.R CaMOKOH.mpO,flR. . . . . . . .. .. . 382
OCH.08H.bte tjJ0PMYJ/.W . . . . . . . . . . . . . . . 383
M emOOU'leCKUe peKOMeH.Oaquu. . . . . . . 384
llpuMepbt peweHUR 3aOa'l. .... 385
14. 3neKTpoMarHHTHLle Kone()aHHJI . . . . . . . . . . . . . . . . . . . . . . 389
14.1. CB060,nHbIe 3neKTpoM8rHHTHbIe KOJIe68HHH B KOHType.
IIpeBp8llteHHe aHeprHH B Kone6aTenbHoM KOHType . 389
14.2. C06cTBeHH8 Q8CTOT8 KOJIe68HHH B KOHType . . . . 392
14.3. 3aTyx81OHe aJIeKTpHQeCKHe KOJIe6aHHH. . . . . . . 393
14.4. ABTOKOJIe68HHJI. reHep8Top He3aTYX81OHX Kone6aHHH (Ha Tp8H3HcTope) 394
14.5. BbIHym,neHHbIe aneKTpH'IeCKHe KOJIe68HH. nepeMeHHbIH TOK . . 396
14.6. reHepaTop nepeMeHHoro '1'OK8. . . . . . . . . . . . . 396
14.7. reHep8Top nOCTOJIHHOrO TOK8 .................... 399
14.8. AKTHBHoe conpOTHBneHHe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 400
14.9. ,[(eHcTBYlOlltHe 3HaQeHHH CHnbI TOKa H HanpJImeHHH. . . . . . . . . . . . . . . . . . .. 401
715

14.10. EMKocTHoe conpoTHBneHHe ......
14.11. MH,nYKTHBHoe conpOTHBJIeHHe ...................
14.12. 38KOH OM8 ,nnH yq8CTKS L\enH nepeMeHHoro TOK8.
PS3HOCTb ct>83 Mem,ny KOJIe6sHHHMH CHnbI TOKS H HanpJImeHHJI. . . .
14.13. Pe30H8Hc B aneKTpHqeCKOH ltenH ....................
14.14. MOIltHOCTb B enH nepeMeHHoro TOKa . .
14.15. Tp8Hcct>opM8Top . . . . . . . . . . . . . . . . . . . .
14.16. Tpexct>83HblH aneKTpHqeCKHH TOK . . . .
14.17. IIepe,naqa aneKTp03HeprHH . . . . . . . . . . . . .
BonpOCbt OJ/.R CaMOICOH.mpoJ/.Jl. . . . . . . . . . . .
OCH08H.bte tjJOPMYJ/.bt . . . . . . . .
M emOOU'leCKUe peICOMeH.Oaquu.
llpuMepbt peWeH.UR 3aOa'l
402
404
406
407
408
409
412
414
416
417
419
420
15.4.
15.5.
15.6.
15.7.
PscnpOCTp8HeHHC KOJIe6aHHH B ynpyrHx cpe,nax. . . . .
[IonepeqHble H npo,nonbHble BOJIHbI. . . . . . . . . . . . . . . . . . . . .
,n:JIHHS BOJIHbl. CBJI3L ,ZJ.JIHHbl BonHbl co CKOpOCTblO ee p8cnpOCTp8HeHHSI.
YpaBHeHHe BOJIHbI ........................................
3BYKoBble BOJIHbI. CKOpOCTb 3BYKa . . . . . . . . . . . . . . .
rpOMKOCTb 3BYK8 H BblCOT8 TOH8 . . . . . . . . . . . . . . . .
8neKTpoM8rHHTHoe none. 8neKTpoM8rHHTHbIe BOJIHbl. .
CKOPOCTb p8cnpOCTpSHeHHH aneKTpOM8rHHTHbIX BonH.
CBoHcTBa aneKTpOM8I'HHTHbIX BonH . . . . . . . . . . . .
15.8. YpaBHeHHe aneKTpoM8rHHTHoH BonHbI .... . . . . . . . . . . . . . . . . .
15.9. 8HeprHJI 3neKTpoM8rHHTHoH BonHbI. IInoTHocTb nOTOK8 H3JIyqeHHH
(noBepxHOCTH8JI) ....................... . . . . . . . . . .
15.10 M3JIyqeHHe H npHeM :meKTpOM8I'HHTHbIX BOJIH. . . . .
15.11. M306peTeHHe p8HO A. C. llonoBblM .
15.12. llpHHL\Un p8,nHoTeJlect>oHHoH CBJI3H.
15.13. MO,nYJIJIlI.HH . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.14. ,[(eTeKTHpoBsHHe. IIpocTeHwHH p8,nHonpHeMHHK
15.15. PSHOJIOK8L\HJI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.16. TeJIeBHp;eHHe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.17. Pa3BHTHe cpe,nCTB CBJI3H . . . .
BonpOCbt OJ/.R CaMOICOH.mp0J/.R .
OCHo6Hbte ljJOPMYJ/.bt .......
MemOOU'leCKUe peKoMeHoaquu. . . .
llpuMepb peWeH.UR 3aOa'l ......
424
424
425
15.
BOJIllbt
15.1.
15.2.
15.3.
428
430
431
432
434
436
437
437
440
442
443
445
446
448
449
450
451
451
452
OIITJlKA
455
16. reOMeTpH'ieCKaH OI1THKa ................................................. 455
16.1. reOMeTpHqeCK8JI onTHK8. OCHoBHble nOHJITHH reOMeTpHqeCKOH onTHKH 455
16.2. 38KOH npJIMOJIHHeHHOro p8cnpocTp8HeHHJI CBeT8 . . . . . . 456
16.3. OTp8meHHe CBeT8. 38KOHbI OTp8meHHH. . . . . . . . . . . . . . . . . . . . . . . 457
16.4. IIJIOCKOe 3epK8JIO. IIocTpoeHHe H306p8meHHH B nnOCKOM 3epK8JIe .. 160
16.5. Cct>epHQeCKHe 3epKaJI8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461
16.6. llocTpoeHHe H306p8meHHH npe,nMeT8 B cllJepHQeCKHX 3epK8JI8X. . . 464
16.7. JIHHeHHoe YBenHQeHHe cct>epHQeCKoro 3epK8Jl8 . . . . . . . . . . . . . . . . . . 465
16.8. 3SKOHbI npeJIOMJIeHHJI CBeT8 .............................. 466
16.9. IIonHoe OTp8meHHe CBeT8 .... 468
16.10. Xo,n JIyqeH B npH3Me . . . . . . . 469
BonpOCbt OJ/.R CaMOKOH.mp0J/.R . . . . . . . . . . . . 470
OCH.08H.bte tjJOPMYJ/.bt . . . . . . . . . . . . . . . . . . . 472
Memoau'lecICue peKOMeH.Oaquu. . . . 472
llpuMepbt peweHuR 3aOa'l ........................................... 473
16.11. JIHH3bI. <I>oKycHoe pSCcToHHHe JIHH3LI ............................. 477
716

16.12. IIocTpoeHHe H306p8meHHH B nHH38X. . . . . . . . . . . . . 478
16.13. OnTHqeCKSH CHn8 nHH3bI. . . . . . . . . . . . . . . . . . . . . . 480
16.14. Cl»opMyn8 nHH3bI. JIHHeHHoe YBeJIH'ieHHe nHH3bI .......... 481
16.15. OnTHQeCKHe npH60pbI . . . . . . . . . . . . . . . . . 482
16.16. Cl»oT08nnap8T. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482
16.17. IIpoeKHoHHbIH snn8p8T. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483
16.18. rn83 KSK OnTHqeCK8H CHCTeM8. . . . . 484
16.19. Aect>eKTbI 3peHHH. OqKH ............. 486
16.20. JIyn8 . . . . . . . . . . . . . . . . . 487
16.21. MHKpocKon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487
16.22. 3JIeMeHTbI ct>oToMeTpHH. Cy6'beKTHBHbIe H 06'beKTHBHbIe xsp8KTepHCTHKH
H3nyqeHHH. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489
16.23. 3aKoHbI OCBeeHHOCTH . . . . 492
Bonpocw OJ/.R CaMOKOH.mpOJ/.Sl.. 493
OCIt08H.WE' tjJ0PMYJ/.W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494
M emOOU'leCKUe peKOMeH.Oaquu. . . . . . . . . . . . . . . . . . . . . . . . . . 495
llpuMepbt peWeH.UR 3aOa'l .... 496
3neMeHTbI TeopHH OTHOCHTenbHOCTH. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.1. nocTyn8TbI TeopHH OTHOCHTenbHOCTH. CKOPOCTb CBeTS B B8KYYMe
KSK npe,neJIbH8H CKOpOCTb nepe,n8qH CHrH8JI8. . . . . . . . . . . . . . .
18.2. OCHoBHbIe cneACTBHH TeopHH OTHOCHTeJIbHOCTH H HX
3KcnepHMeHT8nbH8H npOBepK8. . . . . . . . . . . . . . . . . .
18.3. PenHTHBHcTCKHH 38KOR CJIOmeHHH CKopocTeH . . . . . . . . . . . . . .
18.4. M8CCS. 3HeprHH H HMnynbc B Kn8CCHQeCKOH MeX8HHKe H B CTO. . . . .
BonpOCbt OJ/.R CaMOKOH.mp0J/.R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
OCH08H.bte tjJOPMYJ/.bL . . . . . . . . . .
MemOOU'leCKUe peKOMeH.Oaquu. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
llpuMepbt peWeH.UR 3aOa'l ...........................................
17.
CBeTOBLle BOJIHLI. . . . . . . . . . . . . . . . .
17.1. CBeT K8K :meKTpOMsrHHTH8JI BonH8 . . . . . . . .
17.2. CKOpOCTb CBeT8 ........................
17.3. MHTePct>epeHnHH CBeTS. KorepeHTHocTb.
CneKTpsnbHoe pS3nomeHHe npH HHTepct>epeHnHH . . . . . .
MeTo,nbI H86nlO,ZJ.eHHJI HHTepct>epeHHH CBeT8.. . . . . . . . .
MHTepct>epeHHH CBeTa B TOHKHX nneHK8X. . . . . . . .
IIpHMeHeHHe HHTepct>epeHUHH B TeXHHKe . . . . . . . .
AHct>P8KHH CBeT8 .......................
IIpHHnHn rIOHreHC8-peHenH
MeTO,n 30H Cl»peHenH . . . . . . . . . . . . . . . . . . . . . . . . . .
AHct>P8KnHOHH8H peweTK8. nHct>P8KnHOHHbIH cneKTp . . . . .
IIOHHTHe 0 ronorp8$HH. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
IIOnJIpH38nHH CBeTS. 38KOH M8nlOC8 . . . . . . . . . . . . . . . . . .
AHcnepCHH CBeT8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
AHcnepCHOHHbIH cneKTp .....
CneKTpOCKon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
IIornOeHHE' CBeTa. 38KOH Byrep8 . . . . . . . . . . . . . . . . . . . . . . . . .
llIKsn8 aneKTpOM8rHHTHbIX BonH. 3neKTpOM8rHHTHbIe H3nYQeHH$I
p83HbIX ,nnHH BonH. CBOHCTB8 H npHMeHeHHe 3THX H3nYQeHHH ...
17.18. CneKTpbI HcnYCK8HHH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17.19. CneKTpbI nornOeHHJI ...........................
17.20. CneKTpanbHbIH 8H8nH3 . . . . . .
Bonpocw OJ/.R CaMOKOH.mpoJ/.R . . . . . .
OCH08H.bte tjJOPMYJ/.bt
MemOOU'leCKUe peKoMeHoaquu. . . . . .
llpuMepbL peWeH.UR Jaoa'l ......
17.4.
17.5.
17.6.
17.7.
17.8.
17.9.
17.10.
17.11.
17.12.
17.13.
17.14.
17.15.
17.16.
17.17.
18.
502
502
503
504
507
508
511
513
514
514
517
519
521
524
526
526
527
528
530
530
531
532
534
535
535
540
540
542
547
548
550
551
551
552
717

KBAHTOBAH cI»H3HKA .................................................. 555
19.
CBeTOBbIe KBaHTbI. }1.eHcTBHe CBeTa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19.1. B03HHKHoBeHHe yqeHHH 0 KB8HT8X. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19.2 «%>oToact>ct>eKT Hero 38KOHbI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19.3. Yp8BHeHHe 3HHwTeHH8 ,nnH ct>oToact>ct>eKT8
19.4. «%>OTOR. ero aHeprHH H HMnynbc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19.5. IIpHMeHeHHe ct>oToact>ct>eKT8 B TeXHHKe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19.6. ,[(8BneHHe CBeT8. OnbITbI II. H. JIe6e,neBs . . . . . . . . . .
19.7. XHMHqeCKOe ,neHcTBHe CBeT8 Hero npHMeHeHHe . . . . . . . . . . . . . . . . . . . . . . .
19.8. KopnycKynHpHo-BonHoBoH ,nY8nH3M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Bonpocw OAR caMoKoHmpoAR . . . . . . . . . . . . . . . .
OCH.08Hbte tjJ0PMYAbt . . . . . . . . . . . . . . . . . . . . . . .
MemOOU'leCKUe peKOMeH.Oauuu. . . . . . . . . . . . . . . .
llpUMepw peWeH.UR 3aOa'l. ......
555
555
556
560
561
561
564
566
567
568
569
569
570
20. Cl»H3HKa aTOMa ................................... 573
20.1. SlBneHHH. no,nTBepm,n8IOHe cnomHoe CTpoeHHe 8TOM8 . . . . 573
20.2. On bIT 3. Pe3epct>op,ns no pscceHB8HHlO a -q8CTH ............ _ _ 574
20.3. SI,ZJ.epHaH MO,ZJ.enb aTOMa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575
20.4. IIocTyn8TbI Bops . . . . . . . . . . . . . . . . . . . . . . . . . 576
20.5. MO,ZJ.enb aTOM8 Bo,nopo,na no Bopy. _ . - . 577
20.6. OnbIT ct>p8HKa H rep8 . . . . . . . . . . . . . . 579
20.7. TpY,ZJ.HocTH TeOpHH Bops . . . . . . . . . . . . . . . . . . . . . . .. .............. 580
20.8. I1cnycK8HHe H nOrnOIlteHHe CBeT8 aTOMOM . . . . . . . . . . .. .............. 580
20.9. CneKTp aHepreTHqeCKHX COCTOHHHH aTOM8. IIpoHcxom,neHHe
nHHeHq8TbIX cneKTpOB. HenpepbIBHbIe cneKTpbI . . . . . . . . . . . . . . . . . . . . . .. 581
20.10. CnoHT8HHoe H BbIHym,ZJ.eHHoe H3nYQeHHe . . . . . . . . . . . . . . . . . . . .. 583
20.11. JIa3epbI H HX npHMeHeHHe B TeXHHKe . . . . . . . .. ....... 586
BOnpOCbJ OAR CaMOKOH.mpOAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 587
OCH.08H.We 4JOPMYAbt ............. . . . . . . . . . . . .. ............................. 588
MemOOU'leCKUe peKOMeH.Oaquu. . . . . . . . . . . . . . . . . 588
llpuMepw peWeH.UR 3aOa'l ............ 589
21. 3neMeHTbI KBaHToBoH MexaHHKH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 591
21.1. rHnoTeaa,ne BpoHJIH. BOJIHOBbIe CBOHCTBS aneKTpOHa. )J;Hct>paKHH
aneKTpOHOB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591
21.2. CooTHoweHHe Heonpe,ZJ.eneHHoCTeH B. reH3eH6epra . . . . . . . . . . . . . . . 593
21.3. OCHOBbI KBaHTOBO-MeXSHHQeCKHX npe,ncT8BneHHH. )J;HCKPeTHOCTb aHeprHH
H ,npyrHx ct>H3HQecKHX BeJIHQHH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 595
21.4. KB8HTOBO-MeX8HHQeCK8JI MO,ZJ.eJIb aTOMa Bo,nopo,na . . . . . . . . . . . 597
21.5. IIpHHHn Tom,ZJ.eCTBeHHOCTH Q8CTH. IIpHHHn 38npeTa IIaYJIH . . . . . . . . . . .. 599
21.6. KB8HTOBO-MeXSHHQeCKOe 060CHoB8HHe nepHO..ltHQeCKOH CHCTeMbI
aJIeMeHTOB )J;. Ii. MeH,neJIeeB8 . . . . . . . . . . . . . . . . . . . . . . . . . . . 601
21. 7. MOJIeKYJIbI. IIpHpO,n8 XHMH'ieCKOH CBH3H. . . . . . . . . . . . . . . . . . . . . . . . . . .. 603
21.8. rp8HHbI npHMeHHMOCTH 38KOHOB KJI8CCHQeCKOH ct>H3HKH . . . . . . . . . . . . . . .. 605
Bonpocbt OAR CaMOKOH.mpOAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 606
OCH.08H.btE' tjJ0PMYAbt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 606
llpuMepbt peWeH.UR 3aOa'l ........................................... 607
22. Cl»H3HKa aTOMHoro JI.lI:pa .................................................. 609
22.1. COCTSB H,npS aTOMS........................................... 609
22.2. 1130TonbI....... 611
22.3. SLn.epHbIe CHJIbI . . . . . . . . . . . . . . . . . . . . . . . . . . 611
22.4. 3HeprHH CBJI3H 8TOMHbIX H,nep . . . . . . . . . . . . . . . . . . . . . . . . 612
22.5. Pa,nHOSKTHBHOCTb. 38KOH p8,nH08KTHBHoro paCn8,A8. . . . . . . . . . . . . . . . .. 613
22.6. P8,AH08KTHBHOCTb H ee BH,nbI . . . . . . . . . . . . . . . . . . . . . . . . .. 615
22.7. MCKyccTBeHH8H p8,nH08KTHBHOCTb ................... ............ 617
22.8. SI,nepHbIe pe8KHH. 3HepreTHQeCKHH BbIXO,n H,nepHbIX peaK[(HH. . . . . . . . . . . .. 617
718

22.9. 3KcnepHMeHTftnbHble MeTo,nbl perHcTpaHH 3ap$lmeHHbIx qaCTH . 618
22.10. ,ZJ;eneHHe Jl,nep ypaHa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 621
22.11. SLn.epHblH peaKTop. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 624
22.12. TepMOJl,nepHble peaK[(HH. Co3,n8HHe H y,nepm8HHe BblcoKoTeMnep8TYPHOH
nn83MbI. TOK8M8K. YcnexH H nepcneKTHBbl P83BHTH$I $I,nepHoH 3HepreTHKH. 626
22.13. rrornOeHH8$1 ,n038 H3.nyqeHH$I H ee ()HonOriRecKoe ,neHcTBHe. OCHOBbI
,n03HMeTpHH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 627
22.14. 38HTa OT H3JIy'1eHHH ...... 630
Bonpocw OAR CaMOKOH.mpOAR . . . . . 631
23. 3neMeHTapHble qaCTHl.\bI . . . . . . . . . . . 633
23.1. 3neMeHTapHbIe Q8CTHl.\bI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633
23.2. AHTHqaCTHbI. rrpeBp8eHHe napbI "aneKTpoH - n03HTpoH"
B raMM8-H3n}"leHHe H o()paTHo . . . . . . . . . . . . . . . . . . . . 637
23.3. B38HMHbIe npeBp8eHH$I aneMeHTapHblX q8CTH . . . . . . . . . . . . . . . . . . . . .. 639
BonpOCbt OAR CaMOKOH.mpOAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 640
OCH.08H.We tjJ0PMYAW . . . . . . . . . . . .. ............. 640
M emOOU'leCKUe peKOMeH.Oaquu. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 640
npuMepbt peWeH.UR 3aOa'l ...... .................................... 641
ClIP ABO'lHbIE MATEPHAJIbI . 643
IIPHJlOJKEHHE 649
TecT I ............................................ 649
TeCT n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655
TecT III . . . . . . . . . . . . . . . 661
TecT IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 667
TeCT V ............................................ . . . . . . .. 673
TeCT VI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679
TeCT VII. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685
TecT VIII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 691
TeCT IX . . . . 697
TeeT X 703
OTBETbI K TECT AM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 709

Yqe6HOe H3,ZJ;aHHe
AKceHOBH'I JIHJIHH AHTOHOBHa
PaKHHa HaTaJIbH HHKOJIaeBH8
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TIO,AnHC8HO B neq8Tb 15.11.2004. q>OPM8T 70x 100 Ih6.
BYM8r8 o<pceTH8fl :M 1. r8pHHTYP8 IlIKonbH8S1. TIeq8Tb O<PceTH8S1.
Ycn. ne'l. n. 58,05. Y'l.-H3,A. n. 45.5. THp8m 10000 3K3.
S8K833004.
Yqpem,neHHe "M3,naTenbcTBo .. A.AYKabul i BblxaBaHHe".
JIM M 02330/0133264 OT 30.04.2004.
220004, r. MHHCK. yn. Kopon$l. 16.
OTKpblToe aKHOHepHoe o6ecTBo
"rronHrpaqmqeCKHH KOM6HHaT HMeHH H. Konaca".
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