a. gigineiSvili, g. kukulaZe

zogadi fizikis kursi

(magnetizmi, optika, atomis fizika)

II tomi

@`teqnikuri universiteti”@

saqarTvelos teqnikuri universiteti

a. gigineiSvili, g. kukulaZe

zogadi fizikis kursi

(magnetizmi, optika, atomis fizika)

II tomi

damtkicebulia stu-s

saredaqcio-sagamomcemlo sabWos

mier. 27.05.2009, oqmi #5

Tbilisi

2009

© sagamomcemlo saxli ,,teqnikuri universiteti’’, 2009

ISBN 99940-48-92-9 (orive tomi) ISBN 978-9941-14-611-4 (meore tomi) http://www.gtu.ge/publishinghouse/

yvela ufleba daculia. am wignis arc erTi nawili (iqneba es teqsti, foto, ilustracia Tu sxva) aranairi

formiT da saSualebiT (iqneba es eleqtronuli Tu meqanikuri), ar SeiZleba gamoyenebul iqnas gamomcemlis

werilobiTi nebarTvis gareSe. saavtoro uflebebis darRveva isjeba kanoniT.

3

winasityvaoba

winamdebare saxelmZRvanelo gankuTvnilia teqnikuri universitetis

sainJinro specialobis studentebisaTvis. igi warmoadgens zogadi fizikis

kursis ortomeulis meore toms da Seicavs magnetizmis, optikisa da atomis

fizikis saswavlo programiT gaTvaliswinebul sakiTxebs.

saxelmZRvanelos done da stuqtura iseTia, rom students gauad-

vildes masze damoukideblad muSaoba. kursSi mocemulia klasikuri da

Tanamedrove fizikis mxolod arsebiTi sakiTxebi im moculobiT, rom

students gamoumuSavdes damoukideblad azrovnebis saWiro Cveva. rig

SemTxvevebSi studentebs vTavazobT zogierTi sakiTxis damoukideblad

gadawyvetas, rasac, Cveni varaudiT, bevri maTgani interesiT Sexvdeba.

wignSi farTodaa gamoyenebuli petitis teqsti da sqolio _

studentebisaTvis, romlebic moisurveben ama Tu im sakiTxSi ufro Rrmad

Cawvdomas da damatebiTi informaciis miRebas.

avtorebi gulwrfel madlobas uZRvnian wignis recenzentebs, f.m.m.

doqtorebs, Tbilisis saxelmwifo universitetis zust da sabunebis-

metyvelo mecnierebaTa fakultetis profesors a. bibilaSvils da

saqarTvelos teqnikuri universitetis fizikis departamentis srul

profesors k. cxakaias sasargeblo rCevebisaTvis, agreTve akademiur

doqtors, qalbaton l. CaxvaSvils xelnaweris gaformebaSi gaweuli daxma-

rebisaTvis.

saqmiani SeniSvnebi, romlebic xels Seuwyobs wignis Semdgom

gaumjobesebas, avtorebis mier madlierebiT iqneba miRebuli.

4

pirveli nawili

eleqtromagnetizmi

$1. magnetizmis bunebis Sesaxeb warmodgenis ganviTareba

cnobilia, rom bunebaSi arsebobs gansakuTrebuli rkinis madani (mag-

nituri rkinaqva anu magnetiti, 43OFe ), romelsac aqvs rkinisa da zogierTi

sxva liTonis mizidvis unari. amTvisebis matarebel madnis naWers bunebrivi

magniti ewodeba. bunebrivi magnitis magnituri Tvisebebi, mis mier rkinis

naWrebis mizidvis unari unari, cnobili iyo jer kidev Zveli berZnebisaT-

vis∗. maSinve iyo SemCneuli dedamiwis magnituri Tvisebebi, ris gamoc Rero-

sebri magniti, romelsac vertikaluri RerZis irgvliv Tavisufali brunvis

saSualeba hqonda, TavisTavad garkveulad orientirdeboda _ dgeboda TiT-

qmis geografiuli meridiani gaswvriv. am Tvisebaze agebul kompass Cinelebi

jer kidev daaxloebiT 3000 wlis win iyenebdnen.

aRmoCnda, rom bunebrivi magnitis sxvadasxva nawilis mier rkinis naW-

ris mizidvis unari sxvadasxvaa: igi maqsimaluria magnitis boloebSi, cen-

traluri nawilisaken TandaTan mcirdeba da Sua adgilas nulis toli xde-

ba. magnitis im adgilebs, sadac mizidvis Zala udidesia, magnitis polusebi

ewodeba, xolo Sua adgils _ neitraluri zona. magnitis polusebs aRniS-

naven N da S asoebiT da maT Sesabamisad uwodeben CrdiloeT da samxreT

polusebs. aseTi saxelwodeba dakavSirebulia zemoxsenebul faqtTan, rom

Tavisuflad moZravi magnituri isari dedamiwaze orientirdeba ise, rom

misi erTi bolo ( N ) mimarTulia dedamiwis CrdiloeT polusisaken, xolo

meore ( S ) _ samxreTisaken. cdebiT dadgenilia, rom erTsaxeliani magnituri

polusebi erTmaneTs ganizidavs, sxvadasxva saxeliani ki _ miizidavs.

sivrceSi, romelic gars akravs magnits, aRiZvreba magnituri veli. es

veli, eleqtruli velis msgavsad, materialuria; mas gaaCnia energia. am ve-

lis maorientirebeli moqmedeba magnitur isarze saSualebas gvaZlevs mag-

nitur vels mivceT mimarTuleba: magnituri velis mimarTulebad pirobiT

miCneulia magnituri isris CrdiloeT polusze moqmedi Zalis mimarTuleba.

Tu magnitur isars gadavxriT magnituri velis mimarTulebidan, masze

imoqmedebs meqanikuri mabrunebeli momenti, romlis sidide proporciulia

∗ TviT sityva `magnetizmi~ warmodgeba `berZnuli qalaqis (magnesia) saxelwodebidan, romlis maxlobladac Zvelma berZnebma aRmoaCines bunebrivi magnitis naWrebi.

5

gadaxris kuTxis sinusisa da romelic cdilobs moabrunos isari velis

gaswvriv. amrigad, mudmivi magnitebis urTierTqmedeba gansxvavdeba eleqtru-

li muxtebis urTierTqmedebisagan, magram msgavsia eleqtruli dipolebis

urTierTqmedebisa, romlebzec erTgavrovan eleqtrul velSi moqmedebs ma-

rezultirebeli Zalis momenti, da ara Zala. eleqtruli dipolis msgavsad,

mudmivi magniti erTgavrovan magnitur velSi cdilobs Semobrundes velis

gaswvriv, magram ar gadaadgildeba masSi.

arsebiTi gansxvaveba mudmiv magnitsa da eleqtrul dipols Soris

aris Semdegi: eleqtruli dipoli Sedgeba sididiT toli da sawinaaRmdego

niSnis muxtebisagan. am muxtebis gancalkeveba da maTi sxvadasxva sxeulebze

moTavseba SesaZlebelia. magaliTad, Tu dipols gavWriT SuaSi, misi RerZis

marTobulad, mis erT nawilze aRmoCndeba dadebiTi muxti, meoreze _ uar-

yofiTi. es miuTiTebs imaze, rom bunebaSi arsebobs eleqtruli monopoli.

mudmivi magnitis Suaze gaWris Sedegad ki kvlav miiReba mcire zomis ori

magniti, romelTagan TiToeuls aqvs rogorc CrdiloeTi, ise samxreTi po-

lusi. mudmivi magnitis aranairi gayofa ar iZleva saSualebas ganvacalke-

oT misi polusebi, rac imas niSnavs, rom bunebaSi `magnituri muxtebi~

(magnituri monopoli) ar arsebobs.

mudmivi magnitis msgavsebam eleqtrul dipolTan ganapiroba magnitu-

ri movlenebis Seswavlis istoriuli gza, romelic eyrdnoba hipoTezas

elementaruli magnitebis (magnituri dipolebis) Sesaxeb. am hipoTezis Ta-

naxmad yoveli sxeuli Sedgeba elementaruli magnitebisagan, romlebic

siTburi moZraobis gamo qaosuradaa orientirebuli. amis Sedegad maTi mag-

nituri mikrovelebi erTmaneTs abaTilebs da Cveulebriv sxeuli magnitur

Tvisebebs ar amJRavnebs. gare magnituri velis gavleniT elementaruli mag-

nitebi garkveulad orientirdeba velis gaswvriv, maT mier Seqmnili mikro-

velebi ikribeba da jamSi gvaZlevs makroskopul magnitur vels. am Sem-

TxvevaSi amboben, rom sxeuli damagnitda. rogorc vxedavT, am Teoriis

mixedviT sxeulTa damagniteba savsebiT analogiuria xistdipoliani mole-

kulebisagan Sedgenili dieleqtrikis polarizaciisa.

Tavdapirvelad fiqrobdnen, rom eleqtrul da magnitur movlenebs

Soris araviTari kavSiri ar arsebobs. aseT Sexedulebas iziarebda ingli-

seli mecnieri viliam jilbertic, romelmac magnetizmis Sesaxeb arsebuli

monacemebis safuZvelze 1600 wels gamoTqva azri, rom miuxedavd garegnuli

msgavsebisa, eleqtruli da magnituri movlenebis buneba sxvadasxvaa.

6

miuxedavd amisa, ukve XVIII saukunis Sua wlebisaTvis ganmtkicda azri imis

Sesaxeb, rom eleqtruli da magnituri movlenebi erTmaneTTan mWidrodaa

dakavSirebuli, Tumca am kavSiris buneba, denis mZlavri wyaroebis uqonlo-

bis gamo, garkveuli ar iyo.

magnetizmis ganviTarebaSi fundamenturi mniSvneloba hqonda danieli

fizikosis erstedis aRmoCenas, romelmac 1820 wels daadgina, rom gamtarSi

gamavali deni, mudmivi magnitis msgavsad, iwvevs magnituri isris gadaxras,

riTac gadaidga pirveli arsebiTi nabiji eleqtruli da magnituri mov-

lenebis urTierTkavSiris xasiaTis garkvevaSi. Semdeg gei-lusaki da arago

daakvirdnen, rom gamtarSi gamavali deni iwvevs rkinis naWris damagnitebas.

frangma fizikosma amperma eqsperimentulad aRmoaCina da Seiswavla deniani

gamtarebis magnituri urTierTqmedeba.

me-19 saukuneSi sxvadasxva mecnieris mier Catarebulma Semdgomma

cdebma aCvena, rom magnitur Tvisebebs amJRavnebs ara marto liTonis gam-

tarSi gamavali deni, aramed deni airsa da siTxeSic da saerTod, yvela

moZravi eleqtruli muxti. uZravi eleqtruli muxti eleqtruli velis sa-

SualebiT zemoqmedebas axdens sxva eleqtrul muxtze, magram ara magnitur

isarze. magnituri zemoqmedebis uanari gaaCnia mxolod moZrav eleqtrul

muxts (da cvalebad eleqtrul vels).

amrigad, cdebis Sedegad dadginda, rom moZravi eleqtruli muxtebis

(denebis) irgvliv sivrceSi aRiZvreba kidev erTi saxis veli (eleqtrulis

garda) _ magnituri veli, romlis saSualebiTac es muxtebi urTierTqme-

deben mudmiv magnitebTan da sxva moZrav muxtebTan.

magnituri veli aRiZvreba yoveli moZravi muxtis irgvliv da Tu

sxeuli, romelic Seicavs kolosaluri raodenobis elementarul eleqt-

rul muxtebs, romlebic ganuwyvetliv moZraoben da maSasadame, Tavis irgv-

liv qmnian elementarul magnitur velebs, Cveulebriv magnitur Tvisebebs

ar amJRavnebs, aixsneba am muxtebis moZraobis qaosurobiT. moZarobis qaosu-

robis gamo elementaruli magnituri velebis mimarTulebebi qaosuria da

maTi moqmedeba saSualod erTmaneTs abaTilebs. am sxeulis eleqtrul vel-

Si Setanisas muxtebi iwyeben mowesrigebul moZraobas (eleqtruli deni).

axla ukve elementarul magnitur velebs aqvs erTnairi orientacia da

erTmaneTs ki ar abaTileben, aramed piriqiT, ikribebian da sivrceSi gvaZle-

ven garkveuli sididis makroskopul magnitur vels _ eleqtruli denis

magnitur vels.

7

denis magnituri velis aRmoCenis Semdeg garkveul gaorebas hqonda

adgili am sakiTxSi: magnitur vels qmnis, rogorc mudmivi magniti (magni-

turi dipoli), ise eleqtruli deni. mecnierebi cdilobdnen mospobiliyo

es gaoreba da magnituri movlenebi aexsnaT erTi saerTo poziciidan. amis

Sedegi iyo amperis hipoTeza molekuluri wriuli denebis (mikrodenebis)

Sesaxeb (1821w.). am hipoTezis Tanaxmad yovel molekulaSi cirkulirebs Cake-

tili wriuli denebi, romlebic qmnian wriuli sibrtyis marTobul elemen-

tarul magnitur velebs. swores es elementaruli magnituri velebi gar-

kveul piroebebSi sxeulSi qmnis makroskopul magnitur vels da sxeuli

iZens mudmivi magnitis Tvisebebs. amperes hipoTeza, marTalia, magnetizms

xsnida erTi saerTo poziciidan _ mas ukavSirebda eleqtruli muxtebis

moZraobas (dens), magram TviT wriuli molekuluri denebis warmoqmna gau-

gebari iyo, vidre Cveni saukunis dasawyisSi, rezerfordisa da boris Sro-

mebis Sedegad ar dadginda atomis agebuleba. amJamad cnobilia, rom atomi

Sedgeba masiuri birTvisagan, romlis irgvliv (rezerford-boris naxev-

radklasikuri Teoriis Tanaxmad) stacionarul orbitebze moZraoben el-

eqtronebi. eleqtronebis es wriuli moZraoba warmoadgens swored im mol-

ekulur dens, romelic Tavis mxriv ekvivalenturia magnituri dipolisa.

molekuluri denebis Teorias didi warmatebebi hqonda. misi saSua-

lebiT aixsna ara marto is movlenebi, romlebsac xsnida magnituri dipo-

lebis Teoria, aramed iseTebic (magaliTad, diamagnetizmi, zeemanis efeqti),

romlebic dipolebis TeoriiT sruliad gaugebari iyo.

amrigad, sabolood dadgenilia, rom bunebaSi ar arsebobs gansakuT-

rebuli magnituri substancia _ `magnituri masa~ an `magnituri muxti~. yo-

veli magnituri movlena, maT Soris sxeulTa magnituri Tvisebebi, ganpiro-

bebulia eleqtruli muxtebis moZraobiT.

$2. magnituri velis damaxasiaTebeli sidideebi

da maTi sazomi erTeulebi

wina paragrafSi aRniSnuli iyo, rom magnituri velis warmoSobis pir-

velwyaroa moZarvi eleqtruli muxtebi _ eleqtruli deni. dens, romelic

ganpirobebulia gamtarSi Tavisufali muxtebis mimarTuli moZraobiT

(gamtarobis deni), vuwodoT makroskopuli deni (makrodeni), xolo atomSi

an molekulaSi eleqtronebis wriuli moZraobiT ganpirobebul dens _

mikroskopuli deni (mikrodeni).

8

magnituri velis ZiriTadi maxasiaTebelia magnituri induqciis B veq-

tori. igi axasiaTebs marezultirebel magnitur vels, romelic Seqmnilia

yvela makro da mikrodenebis mier; cxadia, mocemuli makrodenis SemTxveva-

Si B –s mniSvneloba damokidebuli iqneba garemos Tvisebebze.

makroskopuli denis mier Seqmnili magnituri velis∗ dasaxasiaTeblad

SemoRebulia damxmare veqtori _ magnituri velis daZabulobis H veqtori,

romelic ukve aRar aris damokidebuli garemos Tvisebebze∗∗. amrigad, eleq-

truli velis msgavsad, magnituri velis dasaxasiaTebladac iyeneben or

veqtors _ B –sa da H –s. amasTan, istoriuli mizezebis gamo, eleqtruli

velis ZiriTadi maxasiaTeblis _ daZabulobis veqtoris ( )E analogiur

sidides magnituri velisaTvis warmoadgens magnituri induqciis ( )B da ara

magnituri velis daZabulobis ( )H veqtori.

polarizaciis veqtoris msgavsad ( )P , romelic axasiaTebs dieleqtri-

kis eleqtrul Tvisebebs, sxeulebis magnituri Tvisebebis dasaxasiaTeblad

SemoRebulia damagnitebis veqtori ( )P . is ganimarteba rogorc elementaru-

li magnituri momentebis veqtoruli jami sxeulis moculobis erTeulSi.

ganmartebidan gamomdinareobs rom P axasiaTebs sxeulSi arsebuli mikro-

denebis mier Seqmnil magnitur vels.

HB, da P veqtoris damakavSirebeli formulebis saxe, iseve, rogorc

eleqtruli da magnituri movlenebis amsaxveli formulebisa, damokidebu-

lia imaze, Tu erTeulTa romeli sistemaa gamoyenebuli.

SI sistemaSi B –s erTeulia tesla ($3)

2m

wmvtl1

⋅= ,

xolo H –is _ amperi metrze ⎟⎠⎞

⎜⎝⎛m

a. am erTeuls specialuri dasaxeleba

ara aqvs ($6). maSasadame vakuumSic ki, sadac B da H fizikurad ar gansxvavdebian,

SI sistemaSi, erTeulTa gansxvavebis gamo, ar SeiZleba Caiweros, rom

∗ makrodenis magnitur vels zogjer gareSe velsac uwodeben. ∗∗ es imas niSnavs, rom misi sidide vakuumSi ( )0H da nivTierebaSi ( )H erTmaneTis tolia,

rasac sazogadod maSin aqvs adgili, roca erTgvarovani izotropuli garemo mTlianad avsebs im sivrces, sadac makrodenis mier Seqmnili magnituri veli nulisagan gansxvavdeba ($31).

9

B =H . saWiroa proporciulobis koeficienti, romelic gaawonasworebs B –

sa da H –is ganzomilebebs. am proporciulobis koeficients aRniSnaven 0μ –

iT da uwodeben vakuumis magnitur SeRwevadobas an magnitur mudmivas. amri-

gad, SI sisitemaSi, kavSirs B –sa da H –s Soris vakuumSi aqvs saxe:

HB 0μ=vak . (2.1)

0μ -is mniSvnelobis dadgena xdeba deniani gamtarebis urTierTqmedebis

kanonis safuZvelze∗ ($11):

m

hn

ma

wmv=

⋅⋅

⋅⋅= −70 104πμ ,

sadac 1 henri=1v.wm/a aris induqciurobis erTeuli SI sistemaSi.

P damagnitebis veqtors SI sistemaSi aqvs H –is ganzomileba, amitom

veqtori B , romelic axasiaTebs yvela makro ( )H da mikro ( )P denebis mier

Seqmnil marezultirebel vels, ganisazRvreba tolobiT:

( )PHB += 0μ . (2.2)

(2.2) tolobis arsi cxadia: makro da mikrodenebis mier Seqmnili veli

maT mier calk-calke Seqmnili velebis veqtoruli jamis tolia.

garda saerTaSoriso sistemis ( )SI erTeulebisa, eleqtrul da magnitur sidideebs

zomaven agreTve absoluturi eleqtrostatikuri ( )CGSE da absoluturi eleqtromagnitu-

ri ( )CGSM sistemebis erTeulebiTac (It, $2). orive sistemis ageba xdeba CGS meqanikuri

sistemis bazaze (e.i. am sistemebSic yvela fizikuri sididis erTeuli gamoisaxeba

gramis, santimetrisa da wamis saSualebiT), magram CGSE sistemaSi gamosaval formulas

warmoadgens kulonis kanoni 221

rqqkF = , romlidanac dgindeba muxtis erTeuli

moTxovniT, rom k proporciulobis koeficienti iyos uganzomilebo erTis toli sidide

( )1=k . CGSM sistemis gamsazRvrel formulas ki warmoadgens denebis urTierTqmedebis

kanoni d

lIIkF 212= , romlidanac dgindeba denis Zalis erTeuli imave moTxovniT, rom pro-

porciulobis koeficienti 1=k .

CGSE sistemis warmoebuli erTeulebi mouxerxebelia magnituri sidideebis gasa-

zomad, amitom, rogorc wesi, mas iyneben mxolod eleqtruli sidideebis gasazomad, amas

gamosaxavs mis dasaxelebaSi aso ` E ~. CGSM sistema, piriqiT _ gamoiyeneba mxolod mag-

∗ 0μ -is ganzomilebas advilad davadgenT (2.1) formulidanac.

[ ] [ ][ ] ma

wmv

ma

mwmv 2

⋅⋅

=⋅

==HB

0μ .

10

nituri sidideebis gasazomad, radgan misi warmoebuli erTeulebi mouxerxebelia eleqtru-

li sidideebis gasazomad. aqedan _ mis dasaxelebaSi aso `M ~.

fizikaSi ufro xSirad iyeneben erTeulTa e.w. gausis sistemas, romelic warmo-

adgens CGSE da CGSM sistemebis kombinacias. am sistemaSi yvela eleqtruli sidide

(muxti, denis Zala, eleqtrostatikuri velis daZabuloba da potenciali, gamtaris wina-

Roba da sxv.) izomeba CGSE sistemis, xolo yvela magnituri sidide (magnituri velis in-

duqcia da daZabuloba, magnituri momenti, TviTinduqciis koeficienti da sxv.) CGSM sis-

temis erTeulebiT. cxadia, rom gausis sistemac `absoluturia~, masSi yvela fizikuri

sididis erTeuli gamoisaxeba gramis, santimetrisa da wamis saSualebiT.∗

SI sistemisagan gansxvavebiT, gausis sistemaSi, vakuumSi, E da D , agreTve B da

H veqtorebi Tanxvdebian erTmaneTs. amasTan, advili misaxvedria, rom es Tanxvedra aris

ara raime SeTanxmebis Sedegi, aramed gamosaxavs am sidideebis WeSmarit igiveobas.

gausis erTeulTaA sistemaSi B izomeba gausobiT ( tl10gs1 4−= ), H _ erstedobiT

( maerst. 310411 ⋅=π

). orive erTeuls erTnairi ganzomileba aqvs (12121 wmsmg −−) da

mxolod dasaxelebiT gansxvavdebian. gaussa da ersteds Soris araviTari gansxvaveba ar

aris. es erTi da igive sididis ori sxvadasxva dasaxelebaa. miuxedavad amisa, arasworia

gamoTqma ` B tolia amdeni erstedisa~, an ` H udris amden gauss~. cxadia, rom P –s aqvs

igive ganzomileba, rac B –sa da H –s. garda amisa, mikrodenebis mier Seqmnili veli am

sistemaSi tolia Pπ4 –si da ara P –si, amitom (2.2) –is nacvlad gveqneba

PHB π4+= .

Tavi I

magnituri movlenebi vakuumSi

$3. magnituri induqciis veqtori

eleqtruli da magnituri velebis aRmoCena da Seswavla xdeba sxeu-

lebsa Tu xelsawyoebze maTi moqmedebis mixedviT.

eleqtruli velis ZiriTadi Zaluri maxasiaTebeli _ eleqtruli ve-

lis daZabulobis E veqtori ganisazRvreba sasinj muxtze moqmedi Zalis

saSualebiT.

∗ sityva `absoluturi~ sistemis dasaxelebaSi daamkvidra gausma, romelmac fizikuri sididis `absolututi zoma~ uwoda am sididis gamosaxvas sigrZis, droisa da masis saSua-lebiT, e.i. im sidideebis saSualebiT, romlebic asaxaven materialuri samyaros ZiriTad cnebebs.

11

magnituri velis ZiriTadi Zaluri maxasiaTeblis _ magnituri velis

induqciis B veqtoris gansazRvra ki SesaZlebelia sami xerxiT:

a) denian gamtarze moqmedi Zalis saSualebiT (amperis Zala $10),

b) moZrav muxtze moqmedi Zalis saSualebiT (lorencis Zala $13),

g) denian brtyel konturze (`denian CarCoze~) moqmedi Zalis momentis

saSualebiT (maorientirebeli moqmedebis saSualebiT $12).

magnituri velis induqciis veqtoris gansasazRvravad gamoviyenoT

deniani CarCo. unda aRiniSnos, rom CarCos rogorc zomebi, ise masSi gamava-

li deni unda iyos Zalian mcire, raTa CarCos yoveli nawili erTnair moq-

medebas ganicdides da CarCos magniturma velma ar gamoiwvios ZiriTadi

(Sesaswavli) magnituri velis Secvla.

cda gviCvenebs, rom Tu msubuq drekad

Zafze Camokidebul denian CarCos SevitanT erT-

gvarovan magnitur velSi, CarCo orientirdeba _

igi garkveuli kuTxiT Semobrundeba∗. magnituri

velis maorientirebeli moqmedeba gamoyenebulia

magnituri induqciis veqtoris mimarTulebis da-

sadgenad: magnituri induqciis veqtoris mimarTu-

lebad mocemul wertilSi miCneulia orienti-

rebuli CarCos dadebiTi normalis mimarTuleba.

CarCos dadebiTi normalis mimarTuleba ganisazRvreba marjvena burRis

wesiT: Tu burRis taris brunvis mimarTuleba emTxveva CarCoSi gamavali

denis mimarTulebas, maSin burRis gadataniTi moZraobis mimarTuleba

gansazRvravs CarCos dadebiTi normalis mimarTulebas. Tu CarCosTan

axlos movaTavsebT magnitur isars, romelsac SeuZlia brunva vertikaluri

RerZis garSemo, igi miiRebs suraTze naCveneb mdebareobas. maSasadame,

magnituri velis mimarTuleba Tanxvdeba magnituri isris CrdiloeT

polusze moqmedi Zalis mimarTulebas (sur. 3.1).

magnituri velSi deniani CarCos Semobruneba miuTiTebs, rom CarCoze

moqmedebs ZalTa wyvili. am ZalTa wyvilis mabrunebeli momenti aRvniSnoT

M –iT. cda gviCvenebs, rom mabrunebeli momentis sidide damokidebulia

CarCos orientaciaze (orientacia ganisazRvreba CarCos normalsa da magni-

∗ amrigad, erTgvarovani magnituri velis moqmedeba denian CarCoze analogiuria erTgvaro-vani eleqtruli velis moqmedebisa eleqtrul dipolze da ara calkeul muxtze, romelic velis moqmedebiT gadaadgildeba.

sur. 3.1.

I

I

n B

S N

12

turi induqciis veqtors Soris moTavsebuli α kuTxis sinusiT), masSi gama-

val denze da mis zomebze (farTobze) da ar aris damokidebuli CarCos

formaze. velis mocemul wertilSi moTavsebul, sxvadasxvanairad orienti-

rebul, sxvadasxva sididis CarCoebze moqmedebs sxvadasxva sididis mabrune-

beli momenti, magram fardoba αsinIS

M damokidebuli ar aris arc deniani

CarCos Tvisebebze da arc mis orientaciaze. maSasadame, am fardobiT SeiZ-

leba daxasiaTdes velis mocemuli wertili. swored es fardobaa miCneuli

velis Zalur maxasiaTeblad _ magnituri velis induqciad:

αsinIS

MB = . (3.1)

amrigad, magnituri induqciis veqtoris mimarTuleba ganisazRvreba CarCos

dadebiTi normalis mimarTulebiT, xolo sidide (3.1) formuliT.

(3.1)-dan

αsinIBSM = . (3.2)

cdebiT dadgenilia, rom M yovelTvis marTobulia n da B veqto-

rebze gamavali sibrtyisa da mimarTulia im burRis gadadgilebis gaswvriv,

romlis tarsac vabrunebT n –dan B –sken (sur.3.2). amitom (3.2) veqtorulad

ase Caiwereba:

[ ]BnISM 0= (3.3) an [ ]BnISM 0= , (3.4)

sadac 0n normalis erTeulovani veqtoria.

veqtors 0nISPm = (3.5)

romlis moduli tolia CarCoSi gamavali

denisa da CarCos farTobis namravlisa da

romelic mimarTulia CarCos dadebiTi

normalis gaswvriv, ewodeba CarCos magnituri

momenti.∗ amitom (3.4) miiRebs saxes:

[ ]BPM m= . (3.6)

Tu CarCo moTavsebulia magnituri induqciis

veqtoris paralelurad, maSin 2πα = da masze

moqmedi mabrunebeli momenti maqsimaluria,

∗ sazogadod, nebismieri formis brtyeli deniani konturis magnituri momenti gamoiTvleba imave (3.5) formuliT. xSirad, sityva konturs amoagdeben xolme da magaliTad, amboben: `wriuli denis magnituri momenti~.

α

sur. 3.2

I

I n

B

μ

13

IBSM =maqs ,

saidanac

IS

MB maqs= (3.7)

(3.7) formulidan davadginoT maqnituri induqciis erTeuli. SI

sistemaSi magnituri velis induqciis erTeulad miRebulia iseTi velis

magnituri induqcia, romelic 1m2 farTobis mqone CarCoze, romelSic gadis

1 a deni, moqmedebs Mmaqs=1n.m mabrunebeli momentiT. am erTeuls ewodeba

tesla (tl):

222 m

wmv1

ma

j1

ma

mn11tl

⋅=

⋅=

⋅⋅

= ;

1 tesla sakmaod didi induqciaa _ mZlavri eleqtromagnitis mier Seq-

mnili velis induqcia daaxloebiT 10 teslas rigisaa. dedamiwis magnituri

velis induqcia ki magnitur polusze aris 41065,0 −⋅ tesla. mcire moculo-

baSi (ramodenime kuburi santimetri) SeiZleba aRiZras xanmokle (wamis mea-

sedi nawilebis xangrZlivobis) veli, romlis induqcia aRwevs 103 teslamde.

cda gviCvenebs, rom magnituri velisaTvis, iseve rogorc eleqtruli-

saTvis, marTebulia superpoziciis (zeddebis) principi: ramodenime denis

mier Seqmnili magnituri veli ( )B calkeuli denebis mier Seqmnili magnitu-

ri velebis ( )iB veqtoruli jamis tolia:

∑= iBB .

$4. magnituri induqciis wirebi. magnituri induqciis nakadi.

magnituri velis grigaluri xasiaTi.

eleqtruli velis msgavsad, magnituri veli grafikulad gamoisaxeba

magnituri Zalwirebis saSualebiT. magnituri Zalwiri, anu magnituri induq-

ciis wiri ewodeba iseT wirs, romlis yovel wertilSi gavlebul mxebs

aqvs am wertilSi arsebuli magnituri induqciis veqtoris mimarTuleba.

magnituri induqciis wiris mimarTulebad pirobiTad miCneulia misi iseTi

mimarTuleba, romelic yovel wertilSi Tanxvdeba magnituri induqciis veq-

toris mimarTulebas.

magnituri induqciis wirebi SeiZleba ise gavavloT, rom maTi

saSualebiT ganisazRvros magnituri velis ara marto mimarTuleba, aramed

sididec. kerZod, magnituri induqciis wirebs avleben iseTi sixSiriT, rom

14

wirebisadmi marTobul farTobis erTeulSi gamavali magnituri induqciis

wirebis raodenoba toli iyos magnituri induqciis veqtoris mniSvnelobisa

am wertilSi. cxadia, rom sadac veli Zlieria, magnituri induqciis wirebi

gaivleba meti sixSiriT da piriqiT.

magnituri induqciis wirebis asagebad unda visargebloT sasinji

deniani CarCoTi an magnituri isriT. magnituri isris CrdiloeTi polusis

mimarTuleba gansazRvravs induqciis B veqtoris mimarTulebas mocemul

wertilSi. magnituri isris gadaadgilebiT am mimarTulebiT usasrulod

mcired daSorebul mezobel wertilSi gavigebT magnituri velis mimarTu-

lebas am wertilSi da a.S. wiri, romelsac aRwers magnituri isari aseTi

gadaadgilebis dros, iqneba swored magnituri induqciis wiri.

am meTodiT agebuli wrfivi da wriuli denebis, solenoidisa da to-

roidis magnituri velebis Zalwirebis (induqciis wirebis) suraTi moce-

mulia 4.1-4.2 suraTebze.

4.1 suraTidan Cans, rom wrfivi denis magnituri induqciis wirebi war-

moadgens koncentrul wrewirebs, romelTa centrebi denian gamtarze mdeba-

reobs, xolo sibrtyeebi am gamtaris marTobulia. kavSiri denis mimarTule-

basa da magnituri induqciis wirebis mimarTulebas Soris mocemulia bur-

Ris wesiT, romelic dadgenilia maqsvelis mier: Tu burRis gadaadgileba

emTxveva denis mimarTulebas, maSin burRis taris brunvis mimarTuleba

gviCvenebs magnituri induqciis wirebis mimarTulebas. suraTidan Cans

agreTve, rom koncentruli wrewirebis sixSire centridan daSorebisas

klebulobs, rac imas miuTiTebs, rom wrfivi denis magnituri veli

araerTgvarovania.

a b sur. 4.1.

I I

15

wriuli denis magnituri induqciis wirebi (sur. 4.2) wesier wrewirebs

aRar warmoadgenen, magram isinic Caketili wirebia an gagrZelebisas aqvT

Caketvis tendencia. burRis wesi, cxadia, gamoiyeneba wriuli denisTvisac,

magram am SemTxvevaSi ufro mosaxerxebelia Tu SevucvliT adgilebs denis

mimarTulebasa da magnituri velis mimarTulebas. marTlac, Tu burRis

tars vabrunebT wriuli denis mimarTulebiT, misi gadaadgileba gviCvenebs

magnituri velis mimarTulebas wriuli denis SigniT.

praqtikaSi farTod gamoiyeneba sole-

noidis magnituri veli (sur. 4.3). solenoidi

ewodeba saerTo wrfivi RerZis mqone wriul

xviaTa erTobliobas. Tu solenoidis sigrZe

bevrad metia mis diametrze, solenoidis

SigniT magnituri induqciis wirebi urTi-

erTparalelur da Tanabrad daSorebul

wrfeebs warmoadgens. es imas niSnavs rom

solenoidis SigniT magnituri induqciis

veqtors yovel wertilSi erTnairi sidide

da mimarTuleba aqvs. iseT magnitur vels,

romlis yovel wertilSi magnituri induqciis veqtorebi tolia, erT-

gvarovani ewodeba.

solenoidis gareT magnituri veli msgavsia mudmivi magnitis magnitu-

ri velisa (sur. 4.3 a,b). solenoidis is bolo, saidanac gamodian magnituri

Zalwirebi, analogiuria mudmivi magnitis N CrdiloeT polusisa, xolo

meore bolo, romelSic Sedian Zalwirebi, analogiuria mudmivi magnitis S

samxreT polusisa.

toroidi ewodeba wriul xviaTa erTobliobas, romelTac saerTo wri-

uli RerZi aqvT (sur. 4.4). toroidis magnituri velis Taviseburebaa is, rom

I

B

sur. 4.2

16

igi mTlianad Tavmoyrilia toroidis SigniT (induqciis wirebi ikvreba

TviT toroidis SigniT) _ toroidis gareT magnituri veli nulis tolia.

magnituri induqciis wirebis suraTi SeiZleba gavxadoT `xiluli~,

Tu visargeblebT rkinis wvrili naqlibiT. rkinis patara namcecebi magni-

tur velSi magnitdebian, gadaiqcevian patara magnitur

isrebad da ganlagdebian ise, rom maTi RerZebis

mimarTuleba daemTxveva magnituri velis mimarTulebas

mocemul wertilSi. miRebul suraTs xSirad magnitur

speqtrs uwodeben, xolo is wirebi, romelTa gas-

wvrivac ganlagdeba patara magnituri isrebis RerZebi,

ganmartebis Tanaxmad warmoadgenen magnituri induq-

ciis wirebs. 4.1-b suraTze mocemulia wrfivi denis

magnituri velis suraTi, romelic miRebulia muyaoze

dayrili rkinis naqlibis saSualebiT.

ganxiluli magaliTebidan naTlad Cans, rom magnituri induqciis wi-

rebi (eleqtrostatikuri velis Zalwirebisagan gansxvavebiT) yovelTvis

Caketilia (Sekrulia), (isini yovelTvis iketebian eleqtruli denis ir-

gvliv), maT ara aqvT arc dasawyisi da arc dasasruli. magnituri induqciis

wirTa Caketiloba magnituri velis fundamenturi Tvisebaa. igi imiTaa gan-

pirobebuli, rom eleqtruli muxtebisagan gansxvavebiT bunebaSi magnituri

muxtebi ar arsebobs da, maSasadame, magnitur vels ara aqvs wyaro. ase-

Ti tipis velebs grigaluri velebi ewodeba∗. amrigad, magnituri veli

grigaluri velia.

raime farTobis gamWoli magnituri induqciis wirebis raodenobas

ewodeba magnituri induqciis nakadi. vinaidan wirebisadmi marTobul erTeu-

lovan farTobs ganWolavs wirebis B raodenoba, 0dS farTobis gamWoli

wirebis raodenoba, anu magnituri induqciis nakadi Φd iqneba:

0BdSd =Φ . (4.1)

zogad SemTxvevaSi, rodesac dS farTobi wirebisadmi marTobuli ar aris,

( )SdBdSBBdSd n ===Φ αcos . (4.2)

sadac αcosBBn = aris B –s gegmili zedapiris normalis mimarTulebaze.

veqtorul sidides ndSSd = , sadac n aris dS farTobis normalis erTeu-

lovani veqtori, ewodeba ` dS farTobis veqtori~. (4.2)-dan Cans, rom imis

∗ grigalur vels solenoidur velsac uwodeben.

sur. 4.4.

17

mixedviT, Tu rogoria α kuTxe (maxvili Tu blagvi), nakadi SeiZleba iyos

dadebiTi an uaryofiTi.

(4.2) gamosaxavs elementarul magnitur nakads _ nakads elementarul

dS farTobSi. sasruli S farTobis gamWoli sruli nakadis misaRebad, un-

da movaxdinoT (4.2)-is integreba:

( )∫ ∫∫ ==Φ=ΦS S

nS

SdBdSBd , (4.3)

erTgvarovani velis SemTxvevaSi constBn = da

SBn=Φ . (4.3)

davadginoT magnituri nakadis erTeuli SI sistemaSi:

( )vb1veberiwm1v1mm

wmv11m1tl 2

2

2 =⋅=⋅⋅

=⋅==Φ SBn .

veberi iseTi magnituri nakadia, romelic ganWolavs marTobulad

dayenebul 1 m2 farTobs 1 tesla magnituri velis induqciis dros.

gausis sistemaSi magnituri nakadis erTeulia maqsveli (mqs)

1 mqs=1 gs. 1 sm2; cxadia, rom 1 vb=108 mqs.

Tu S zedapiri Sekrulia, misi gamWoli magnituri induqciis nakadi

nulis tolia. marTlac, zedapiridan gamosuli nakadi yovelTvis dadebiTia

(kuTxe α zedapiris gare normalsa da Zalwirebs Soris maxvilia da

0cos >α ), zedapirSi Sesuli nakadi ki _ uaryofiTi (kuTxe α blagvia). mag-

nituri induqciis wirebis Caketilobis gamo es nakadebi sididiT erTmane-

Tis tolia, amitom maTi jami anu sruli nakadi nulis toli iqneba:

( ) 0== ∫∫SS

n SdBdSB . (4.4)

(4.4) toloba warmodgens gaus-ostrogradskis Teoremas magnituri

induqciis nakadisaTvis da gamosaxavs im faqts, rom bunebaSi ar arsebobs

magnituri muxtebi, romlebzec daiwyebodnen an damTavrdebodnen magnituri

induqciis wirebi. es Teorema SeiZleba Caiweros diferencialuri formi-

Tac. veqtoris divergenciis ganmartebis Tanaxmad (Itomi, $115), gamosaxuleba

ττ

∫→

SndSB

0lim ,

(sadac τ aris S zedapiriT SemosazRvruli moculoba) warmoadgens B veq-

toris divergencias ( )Bdiυ , amitom (4.4)-is safuZvelze gveqneba:

18

0=Bdiυ . (4.5)

(4.4) da (4.5) formulebi warmoadgens magnituri induqciis wirebis Cake-

tilobisa da, maSasadame magnituri velis grigalurobis maTematikur Cawe-

ras (gamosaxvas).

SedarebisaTvis gavixsenoT, rom eleqtrostatikuri velis SemTxvevaSi

ρε

υ0

1=Edi .

sabolood SevniSnoT, rom, rogorc cdebi gviCvenebs, (4.5) tolobiT

gamosaxuli fundamenturi Sedegi marTebulia ara marto mudmivi, aramed

nebismieri magnituri velisaTvis. igi aris maqsvelis gantolebaTa sistemis

Semadgeneli nawili.

$5. bio-savar-laplasis kanoni

cdebi gviCvenebs, rom mudmivi makroskopuli denis mier Seqmnili mag-

nituri velis induqcia (vakuumSi) damokidebulia denis sidideze ( )I , moce-

muli wertilidan velis Semqmnel denamde manZilze ( )r da deniani gamtaris

formaze.

magnituri velis induqciis damokidebuleba denis Zalaze saerToa

yvela formis denisaTvis _ induqcia proporciulia denis Zalisa:

IB ~ .

rac Seexeba manZilze damokidebulebas, frangma mecnierebma biom da

savarma mravalricxovani cdebis safuZvelze 1820 wels daadgines, rom

grZeli wrfivi denis SemTxvevaSi magnituri velis induqcia ukuproporciu-

lia denian gamtaramde r manZilisa:

rB 1~ .

magnituri velis induqciis deniani gamtaris formaze damokidebule-

bis dadgena praqtikulad SeuZlebelia formaTa mravalferovnebis gamo.

SeiZleboda movqceuliyaviT Semdegnairad: nebismieri formis gamtari dagve-

yo wrfiv mcire dl elementebad, gagvego mocemul wertilSi TiToeuli

elementis mier Seqmnili velis induqcia da miRebuli Sedegebi Segvekriba.

magram aq vawydebiT principul winaaRmdegobas: SeuZlebelia gaizomos

mocemul wertilSi deniani elementis mier Seqmnili velis induqcia ise,

rom danarCeni elementebis moqmedeba gamoiricxos _ magnituri veli, ro-

19

melsac Cven vakvirdebiT mocemul wertilSi, aris yvela elementis erTob-

livi moqmedebis Sedegi. miuxedavad amisa, frangma mecnierma laplasma

cdiseuli monacemebis ganzogadebis safuZvelze daadgina elementaruli

kanoni, romlis gamoyeneba nebismieri formis gamtarisaTvis gvaZlevs eqspe-

rimentTan Tanxvdenil Sedegebs. imis gamo, rom laplasma gamoiyena biosa

da savaris cdebis Sedegebi, am kanons ewoda bio-savar-laplasis kanoni da

mdgomareobs SemdegSi:

elementaruli dB induqcia magnituri velisa, romelsac qmnis I

denis dl elementi∗ misgan r manZilze, (sur. 5.1)

gamoiTvleba formuliT:

2

sinr

IdlkdB α= , (5.1)

sadac α aris kuTxe dl elementsa da r radius-

veqtors Soris, xolo k proporciulobis

koeficientia. (5.1) gansazRvravs dB–s sidides. rac

Seexeba mimarTulebas, dB yovelTvis marTobia lId

da r veqtorebze gavlebuli sibrtyisa da

mimarTulia im burRis gadaadgilebis gaswvriv, romlis tarsac vabrunebT

lId –dan r –sken (sur. 5.1). amitom 5.1 veqtorulad ase Caiwereba:

[ ]

3rrdlIkdB ⋅

= . (5.2)

superpoziciis principis Tanaxmad, mTeli deniani gamtaris mier Seq-

mnili magnituri velis induqcia mocemul wertilSi toli iqneba calkeu-

li veqtorebis geometriuli jamisa:

∑=

=n

iidBB

1.

kerZo SemTxvevaSi, roca yvela dB erTnairadaa mimarTuli, jami Seicvleba

integraliT:

∫∫ ==ll r

dlkIdBB 2

sinα. (5.3)

proporciulobis k koeficientis ricxviTi mniSvneloba damokidebu-

lia formulaSi Semaval sidideTa erTeulebis SerCevaze. gausis sistemaSi

∗ I denis Zalis namravls dl elementze ( )Idl denis elementi ewodeba. igi veqtoria, aqvs

denis mimarTuleba.

sur. 5.1

I

dl α

r

dB

20

ck 1= , sadac c eleqtrodinamikuri mudmivaa da tolia sinaTlis gavrcele-

bis siCqarisa vakuumSi.

SI sistemaSi πμ4

0=k , sadac 70 104 −⋅= πμ hn/m aris magnituri mudmiva.

amrigad, SI sistemaSi bio-savar-laplasis kanoni iRebs Semdeg saxes:

20 sin

4 rIdldB α

πμ

= (5.4)

da

∫=l r

dlIB 2

0 sin4

απ

μ. (5.5)

$6. bio-savar-laplasis kanonis gamoyeneba

ganvixiloT bio-savar-laplasis kanonis gamoyenebis ramdenime magali-

Ti da gamoviTvaloT sxvadasxva formis deniani gamtaris mier Seqmnili mag-

nituri velis induqcia. moqmedebis gza yvela SemTxvevaSi erTi da igivea:

gamtars davyofT mcire dl elementebad, gamoviTvliT TiToeuli elementis

mier Seqmnili dB magnituri velis induqcias da miRebul Sedegebs SevkrebT.

1. sasruli sigrZis wrfivi denis magnituri velis induqcia.

vTqvaT, sasruli sigrZis wrfiv gamtarSi gadis I deni. gamoviTvaloT

am denis mier Seqmnili magnituri velis induqcia misgan R manZilze

(sur.6.1). amisaTvis davyoT gamtari mcire dl elementebad, gamoviTvaloT Ti-

Toeuli elementis mier mier Seqmnili velis induqcia A wertilSi da mi-

Rebuli Sedegebi SevkriboT. am SemTxvevaSi

yvela dB veqtori mimarTulia erTnairad _

naxazis sibrtyis marTobulad Cvengan. amitom

isini algebrulad Seikribeba.

(5.5)-is Tanaxmad

∫=l r

dlIB 2

0 sin4

απ

μ (6.1)

daviyvanoT integralqveSa gamosaxuleba

erT cvladze. dl monakveTis bolo, C werti-

lidan davuSvaT Cd marTobi r radius-veq-

torze. suraTidan Cans,

αd

1α K

C

d α

sur. 6.1.

dl

I

R A

2α

21

αα sindlrdCd == ∗,

saidanac

αα

sin2 rd

rdl

= .

magram Rr =αsin , amitom R

drdl α

=2 . Tu 2rdl

–is miRebul mniSvnelobas

SevitanT (6.1)-Si da sigrZiT integrebas SevcvliT kuTxiT integrebiT,

gveqneba:

( ) ( )210

12000 coscos

4coscos

4sin

4sin

4

2

1

2

1

ααπμ

ααπμ

ααπμαα

πμ α

α

α

α

−=−−=== ∫∫ RI

RI

dRI

RdI

B . (6.2)

(6.2) formula saSualebas gvaZlevs ganvsazRvroT magnituri

velis induqcia kvadratis formis deniani konturis centrSi.

kvadratSi gamavali deni aRvniSnoT I –Ti, xolo kvadratis gverdis

sigrZe a –Ti (sur.6.2), maSin erT-erTi gverdis mier Seqmnili

magnituri velis induqcia O wertilSi (6.2)-is Tanaxmad iqneba

⎟⎠⎞

⎜⎝⎛ =

2aR :

( )210 coscos

42 ααπμ

−=aIB ,

magram ( )121 90,45 ααα +== , amitom 12 sincos αα −= da gveqneba:

( )a

IaI

aI

Bπμ

πμ

ααπμ

22

22

22

2sincos

200

110

1 =⎟⎟⎠

⎞⎜⎜⎝

⎛+=+= .

(6.3) vinaidan kvadratis calkeuli gverdebis mier mis centrSi Seqmnili velis induqciebi erT-

mxrivaa mimarTuli, amitom sruli induqcia

aI

BBπμ0

122

4 == .

analogiuri msjelobiT miviRebT, rom Tu denian konturs wesieri eqsvkuTxedis

forma aqvs, induqcia mis centrSi

aI

Bπμ2

3 0= , (6.4)

sadac a eqvskuTxedis erT-erTi gverdia.

2. usasrulo sigrZis wrfivi denis magnituri velis induqcia.

usasrulo wrfivi denis magnituri velis induqcias martivad gamoviT-

valoT, Tu gamoviyenebT (6.2) formulas. usasrulo wrfivi denisaTvis

180,0 21 == αα , amitom ( )[ ] 211coscos 21 =−−=− αα da

∗ dl -is simciris gamo SeiZleba CavTvaloT, rom rCA ≈ .

2α

1α

sur. 6.2.

I

I

OR

a

22

RI

Bπμ2

0= . (6.5)

praqtikulad, am formuliT SeiZleba sargebloba sasruli, magram

sakmaod grZeli wrfivi denis magnituri velis induqciis gamosaTvlela-

dac, Tu gamtaris l sigrZe gacilebiT metia R manZilze.

3. wriuli denis magnituri velis induqcia.

vTqvaT, R -radiusian wriul gamtarSi gadis I deni. vipovoT magnitu-

ri velis induqcia:

a) wriuli denis centrSi.

am SemTxvevaSi yvela dB mimarTulia erT mxares _ wriuli denis sib-

rtyis marTobulad qvevidan zeviT (sur. 6.3) amitom

dlr

IB

l∫= 2

0 sin4

απ

μ;

magram yvela dl elementi-

saTvis Rr = , xolo 1sin =α ,

amitom

RI

RRI

dlRI

Bl 2

244

02

02

0 μπ

πμ

πμ

=⋅== ∫ (6.5)

e.i. am SemTxvevaSic induqcia pir-

dapirproporciulia gamtarSi ga-

mavali denisa da ukuproporciu-

lia am gamtaridan manZilisa.

b) wriuli denis RerZze.

centridan h manZiliT daSorebul A wertilSi (sur. 6.4), sxvadasxva

dl elementebis mier Seqmnili dB veqtorebi sxvadasxvanairadaa mimarTuli

(isini qmnian maraos centriT A wertilSi), amitom sruli induqcia A

wertilSi

∑= dBB .

dB veqtori paralelogramis wesiT davSaloT or mdgenelad, romelTagan

erTi ( )1dB iqneba OA RerZis paraleluri, xolo meore ( )2dB _ misi marTo-

buli (sur. 6.4). maSin

21 dBdBdB +=

Bdϕ

ϕ

r

sur. 6.3 sur. 6.4

I I

B

R dl dlR

h

A

1Bd

2Bd

o o

23

da

∑∑ += 21 dBdBB

magram advili misaxvedria, rom ∑ = 02dB , vinaidan diametralurad

sawinaaRdego elementebisaTvis 2dB veqtorebi sididiT toli da mimarTu-

lebiT sawinaaRmdegoa. rac Seexeba 1dB mdgenelebs, isini erTnairadaa

mimarTuli, amitom jami SeiZleba Seicvalos integraliT da gveqneba:

∫=l

dBB 1 (6.7)

suraTidan Cans, rom ( ) 21221 sin

hRRdB

rRdBdBdB

+==⋅= ϕ .

bio-savar-laplasis kanonis Tanaxmad

20

20

4sin

4 rIdl

rIdldB

πμα

πμ

== ,

amitom

( ) 2322

03

01 44 hR

IdlRr

IdlRdB+

==πμ

πμ

.

1dB -is mniSvneloba SevitanoT (6.7)-Si:

( ) ( ) ∫∫+

=+

=ll

dlhR

IRdl

hR

IRB 2322

02322

0

44 π

μ

π

μ

an

( ) 2322

20

2 hRIRB

+=

μ. (6.8)

B mimarTulia wriuli denis RerZis gaswvriv. wriuli denis centrisaTvis

0=h da, rogorc unda yofiliyo, (6.8) gadadis (6.5)-Si.

4. solenoidis magnituri velis induqcia

rogorc aRvniSneT, solenoidi aris wrfivi RerZis mqone wriuli de-

nebis erToblioba, amitom induqcia mis RerZze toil iqneba calkeuli wri-

uli denebis induqciaTa jamisa. Sesabamisi gamoTvlebi (ix. petitiT awyobi-

li) gvaZlevs, rom grZeli solenoidis RerZze induqcia

nIB 0μ= , (6.9)

sadac n aris xviaTa ricxvi solenoidis sigrZis erTeulze, xolo I _

solenoidSi gamavali denis Zala.

24

gamovyoT solenoidis mcire dl elementi (sur. 6.5). igi Seicavs ndl xvias. vinaidan

solenoidis TiToeul xviaSi gadis I deni, es elementi SeiZleba ganvixiloT rogorc

wriuli gamtari, romelSic gadis Indl deni. (6.8)-is Tanaxmad, am `wriuli denis~ mier

Seqmnili magnituri velis induqcia l manZiliT daSorebul A wertilSi.

( ) 2322

20

2 lR

IndlRdB

+=

μ. (6.10)

vinaidan yvela elementis mier Seqmnili magnituri velis induqcia A wertilSi

erTnairadaa mimarTuli (RerZis gaswvriv), jamuri induqcia miiReba (6.10)-is integrebiT:

( )∫∫

+==

ll lRdlInR

dBB 2322

20

2μ

. (6.11)

gamovsaxoT dl da 22 lR + cvladebi erTi damoukidebeli cvladiT. A wertilidan

mocemul elementamde gavavloT r radius-veqtori. kuTxe r –sa da solenoidis RerZs

Soris aRvniSnoT β –Ti. suraTidan Cans, rom

βctgRl = ,

saidanac

ββ

2sinRddl −= .

garda amisa,

( )β

β 2

22222

sin1 RctgRlR =+=+ .

Tu dl –isa da 22 lR + –is miRebul mniSvnelobebs

SevitanT (6.10) formulaSi (am formulaSi dl dadebiTi sididea) da sigrZiT integrebas SevcvliT kuTxiT integrebiT, miviRebT:

( )2100

23

2

2

220 coscos

21sin

2

sin

sin2

2

1

2

1

ββμββμ

β

β

βμ β

β

β

β

−==

⎟⎟⎠

⎞⎜⎜⎝

⎛= ∫∫ nld

nld

R

RnlR

B , (6.12)

sadac 1β da 2β solenoidis kidura elementebis Sesabamisi kuTxeebia, aTvlili RerZis

erTi da imave mimarTulebiT.

es Sedegi damokidebulia solenoidis sigrZeze da A wertilis mdebareobaze:

kerZod, Tu solenoidi usasrulod grZelia, maSin πββ == 21 .0 . da

nIB 0μ= ,

rac Tanxvdeba (6.9)-s. praqtikulad, solenoidi SeiZleba CaiTvalos `usasrulod grZlad~

da visargebloT (6.9) formuliT, Tu solenoidis sigrZe bevrad metia xviis R radiusze.

sasruli L sigrZis solenoidis SemTxvevaSi induqcia iqneba naklebi, vidre

usasrulo grZelisa. amasTan, SeiZleba Cveneba, rom am SemTxvevaSi induqcia maqsimaluria

solenoidis Sua wertilisaTvis da tolia

R r β A

dl

sur. 6.5

l

25

220

4 LRnIB

+= μ

.

Tu A wertili imyofeba grZeli solenoidis erT-erT boloSi, maSin, rogorc sura-

Tidan Cans, an 21πβ = da πβ =2 , an 01 =β da

22πβ = . amitom, metad grZeli solenoidis

SemTxvevaSic ki induqcia solenoidis bolosTan ar aRemateba I021 μ –s.

vinaidan HB 00 μ= , bio-savar-laplasis kanoni da misgan gamomdinare

Sedegebi (iseve, rogorc 0B –is Semcveli sxva formulebi) SeiZleba gamoisa-

xos magnituri velis daZabulobis veqtoris saSualebiT.

gveqneba:

bio-savar-laplasis kanoni:

2

sin41

rIdldH α

π= (5.1')

usasrulo wrfivi denis magnituri veli:

R

IHπ2

= (6.5')

wriuli denis magnituri veli centrSi:

RIH

2= (6.6')

da a.S.

es formulebi gviCvenebs, rom magnituri velis daZabulobis ganzomi-

leba udris denis Zalis ganzomilebas gayofils sigrZis ganzomilebaze.

maSasadame SI sistemaSi H –is erTeuli iqneba amperi metrze (a/m). es erT-

euli SeiZleba ganimartos (6.5’) formulis safuZvelze. Tu am formulaSi

davuSvebT, rom 1=I a da π21

=R m, miviRebT =H a/m. e.i. amperi metrze aris

iseTi magnituri velis daZabuloba, romelsac qmnis usasrulo wrfivi

gamtari, romelSic gadis 1 a deni misgan π21

m manZilze.

$7. moZravi muxtis magnituri veli

viciT, rom yoveli deniani gamtaris irgvliv aRiZvreba magnituri

veli, romlis sidide gamoiTvleba bio-savar-laplasis formuliT

20 sin

4 rIdldB α

πμ

= . (7.1)

26

magram eleqtruli deni aris eleqtruli muxtebis mimarTuli moZrao-

ba, amitom, bunebrivia vifiqroT, rom cakleuli moZravi muxtic unda qmni-

des magnitur vels. es mosazreba dasturdeba cdiT.∗

moZravi muxtis magnituri velis induqcias advilad gamoviTvliT, Tu

(7.1) formulaSi I denis Zalas gamovsaxavT q muxtis sididis, υ moZraobis

siCqarisa da 0n denis gadamtani muxtebis koncentraciis saSualebiT.

gamtaris ganikveTis farTobi aRvniSnoT S –iT da gaviTvaliswinoT,

rom denis simkvrive υqni 0= (I tomi, $137), maSin denis Zala

SqniSI υ0== . (7.2)

denis Zalis es mniSvneloba SevitanoT (7.1)-Si:

200 sin

4 rSdlqn

dBαυ

πμ

⋅= . (7.3)

aseTi induqciis vels qmnis dl elementi, anu Sdl moculobaSi moTavsebuli

yvela muxti, romelTa raodenoba

SdlndN 0= .

yvela dN muxti mowesrigebulad moZraobs erTi mimarTulebiT da

erTnairi υ siCqariT, amitom erTi moZravi muxtis magnituri velis induq-

cia iqneba dn –jer naklebi:

20 sin

4 rq

dNdBB αυ

πμ

== . (7.4)

aseTi induqcia magnituri velisa, romelsac qmnis υ siCqariT moZravi

q muxti misgan r manZiliT daSorebul wertilSi. advili misaxvedria, rom

α aris kuTxe υ –sa da r –s Soris.

Tu gaviTvalisiwnebT υ,B da r veqtorebis mimarTulebas (kerZod

imas, rom B yovelTvis marTobia υ da r veqtorebze gavlebuli sibrtyisa

da mimarTulia burRis wesis Sesabamisad), maSin (7.4) toloba veqtorulad

Semdegnairad Caiwereba:

[ ]rrqB υ

πμ

30

4= .∗∗ (7.5)

∗ moZravi muxtis irgvliv magnituri velis arseboba, agreTve misi sidide, eqsperimentu-lad daadgina 1910 wels a. iofem.

∗∗ B -s mimarTulebis sworad gasarkvevad unda gvaxsovdes, rom denis mimarTulebad

miCneulia dadebiTi muxtebis moZraobis mimarTuleba, amitom, Tu 0>q , B -s mimarTuleba

Tanxvdeba [ ]rυ –is mimarTulebas, xolo Tu 0<q , B -s aqvs [ ]rυ –is sawinaaRmdego, e.i.

[ ]υr –s mimarTuleba.

27

$8. magnituri velis induqciis veqtoris cirkulacia

ganmartebis Tanaxmad, Tu mocemuli gvaqvs romelime A veqtoris veli,

am veqtoris cirkulacia nebismier Caketil wirze, ewodeba gamosaxulebas

dlAl

l∫ , sadac αcosAAl = aris A veqtoris gegmili dl –is mimarTulebaze.

I tomis $112-Si naCvenebi iyo, rom eleqtrostatikuri veli warmoad-

gens potenciur vels, rac maTematikurad imiT gamoisaxeba, rom velis daZa-

bulobis cirkulacia nebismieri Caketili wiris gaswvriv aris nulis

toli:

( ) ∫∫ ==l

ll

dlEdlE 0 .

magnituri induqciis wirebis Caketilobis safuZvelze Cven aRvniSneT

($4), rom magnituri veli warmoadgens grigalur (arapotenciur) vels, ami-

tom, magnituri induqciis veqtoris cirkulacia nebismier Caketil wirze,

sazogadod, nulis toli ar unda iyos.

marTlac, gamoviTvaloT magnituri induqciis veqtoris cirkulacia

∫l

l dlB . simartivisaTvis ganvixiloT usasrulo wrfivi deni, romelic mimar-

Tulia naxazis sibrtyis marTobulad Cvengan da gamoviTvaloT B –s cirku-

lacia erT-erTi (R -radiusiani) induqciis wiris gaswvriv (sur. 8.1). wiris

Semovlis mimarTuleba suraTze isriTaa naCvenebi.

vinaidan yovel wertilSi B induqciis wiris mxebia, 1cos,0 == αα , da

RI

BBl πμ2

0== .

amitom

( ) IRRI

dlRI

dlBdlBll

ll

000 2

22μπ

πμ

πμ

=⋅=== ∫∫∫ .

amrigad, magnituri induqciis veqtoris cirkula-

cia Caketili wiris gaswvriv

( ) ∫∫ ==l

ll

IdlBdlB 0μ . (8.1)

formula gviCvenebs, rom cirkulaciis mniSvnelo-

ba ar aris damokidebuli wiris radiusze da ganisazRvreba magnituri

mudmivas namravliT denis Zalaze.

+

sur. 8.1

I R

B

28

(8.1) gamosaxulebis niSani damokidebulia konturis Semovlis mimarTulebaze. Tu es

mimarTuleba denis mimarTulebasTan dakavSirebulia burRis wesiT, maSin 0=α da (8.1)

gamosaxuleba dadebiTia. winaaRmdeg SemTxvevaSi 180=α da uaryofiTia. niSani SeiZleba

gaviTvaliswinoT, Tu davuSvebT, rom I deni algebruli sididea. amasTan, Tu denis

mimarTuleba konturis Semovlis mimarTulebasTan dakavSirebulia burRis wesiT, igi

CaiTvleba dadebiTad, sawinaaRmdego mimarTulebis deni ki uaryofiTad.

SeiZleba Cveneba (ix. petitis teqsti), rom (8.1) marTebulia nebismieri

formisa da nebismierad orientirebuli Caketili konturisaTvis, romelic

moicavs dens. Tu konturi dens ar moicavs, maSin cirkulacia aseTi Cake-

tili wiris gaswvriv nulis tolia. formula marTebulia agreTve ara

marto wrfivi, aramed nebismieri formis denisaTvis.

davuSvaT, nebismieri formis konturi Zevs wrfivi denisadmi marTobul sibrtyeSi

(sur. 8.2). konturis yovel wertilSi magnituri induqciis veqtori mimarTulia am wertil-

ze gavlebuli wrewiris mxebad. cirkulaciis gamosaxulebaSi dlBl SevcvaloT BBdl –Ti,

sadac αcosdldlB = ∗ aris konturis dl elementis gegmili B –s mimarTulebaze. suraTidan

Cans, rom αRddlB = , sadac R aris manZili deniani gamtaridan konturis dl elementamde,

xolo αd is kuTxea, romliTac Semobrundeba radialuri wrfe R konturis gaswvriv dl manZiliT gadaadgilebisas. amrigad,

απμ

RdRI

BdldlB Bl ⋅==2

0 .

am tolobis gaTvaliswinebiT magnituri induqciis cirkulaciisaTvis gveqneba:

IdI

dlBl

l 0

2

0

0

2μα

πμ π

== ∫∫ (8.2)

rac Tanxvdeba (8.1)-s.

∗ simciris gamo Bdl monakveTi SeiZleba Seicvalos wrewiris rkaliT.

B

αd

dl

+ +

R

αd−

αd+a

b sur. 8.2 sur. 8.3

29

Tu konturi ar Zevs denisadmi marTobul sibrtyeze, maSin yoveli dl SeiZleba dai-

Salos or, 1dl da 2dl mdgenelebad, romelTagan 1dl moTavsebuli iqneba denis marTob sib-

rtyeSi, xolo 2dl – mis perperndikularulad: 21 dldldl += . maSin

( ) ( ) ( )∫∫∫ +=lll

dlBdlBdlB 21 .

magram meore Sesakrebi nulis tolia, radgan yoveli 2πα = da 0cos =α , xolo pirveli

Sesakrebi, damtkicebis Tanaxmad, mogvcems l0μ -s. amitom, am SemTxvevaSic

( ) IdlBl

0μ∫ = .

axla ganvixiloT SemTxveva, roca konturi ar moicavs dens (sur.8.3). am Sem-

TxvevaSi konturis Semovlisas radialuri wrfe Semobrundeba jer erTi mimarTulebiT

(ab ubani), Semdeg sawinaaRmdegoTi (ba ubani). amis gamo

∫ =π

α2

0

0d da (8.2)–is Tanaxmad ∫ =l

l dlB 0 .

Tu konturi moicavs ramodenime, nIIII ...,,, 321 dens, maSin, superpozici-

is Tanaxmad ∑=

=n

kkBB

1

, sadac kB aris kI denis mier Seqmnili velis induqcia.

maSin

( ) ( )∑∫∫ ∑∫==

=⎟⎠

⎞⎜⎝

⎛=

n

k lk

l

n

kk

l

dlBdlBdlB11

.

am integralTagan TiToeuli gvaZlevs kI0μ –s, amitom

( ) ∑∫=

=n

kk

l

IdlB1

0μ . (8.3)

(8.3) gamosaxavs sruli denis kanons magnituri velisaTvis vakuumSi:

magnituri velis induqciis cirkulacia Caketili konturis gas-

wvriv tolia magnituri mudmivas namravlisa im denebis algebrul

jamze, romelsac es konturi moicavs.

erTnairi denebis SemTxvevaSi

nIIn

kk =∑

=1

da ( ) nIdlBl

0μ=∫ . (8.4)

Tu gaviTvaliswinebT, rom HB=

0μ, (8.3) toloba miiRebs saxes

( ) ∑∫=

=n

kk

l

IdlH1

. (8.5)

30

$9. cirkulaciis formulis gamoyeneba solenoidisa

da toroidis magnituri velis gamosaTvlelad

$ 6-Si vnaxeT, rom bio-savar-laplasis kanoni saSualebas gvaZlevs

gamoviTvaloT sxvadasxva formis deniani gamtaris mier Seqmnili magnituri

velis induqcia. magram rig SemTxvevebSi (roca adgili aqvs garkveul si-

metrias) igive amocanis gadawyveta martivdeba, Tu visargeblebT ara bio-

savar-laplasis, aramed sruli denis kanoniT (8.3).

sruli denis kanoni iseTive mniSvnelovan rols asrulebs mudmivi

denis magnituri velis gamoTvlaSi, rogorsac gaus-ostrogradskis Teorema

eleqtrostatikuri velis gamoTvlaSi.

ganvixiloT am kanonis gamoyenebis ramodenime magaliTi.

1. solenoidis magnituri velis induqcia

9.1 suraTze mocemulia solenoidis kveTi da magnituri induqciis wi-

rebis ganlageba. solenoidis sigrZe aRvniSnoT l –iT, masSi gamavali deni

I –iT, xolo xviaTa ricxvi N –iT. Caketil konturad avirCioT induqciis

erT-erTi (NABSN) wiri da gamoviTvaloT cirkulacia mis gaswvriv. vinaidan

arCeuli konturi moicavs I denian N gamtars, amitom (8.4)–is Tanaxmad

∫ =NABSN

l NIdlB 0μ (9.1)

magram

∫∫ ∫ +=NABS

lNABSN SN

ll dlBdlBdlB .

induqcia solenoidis SigniT gacilebiT

metia, vidre mis gareT,∗ amitom wina tolobaSi

meore wevri SeiZleba ugulebelvyoT.

garda amisa, vinaidan Caketil konturad

aRebulia induqciis wiri, BBl = da gveqneba:

BldlBdlBNABSN SN

l ==∫ ∫ . (9.2)

(9.1) da (9.2) tolobaTa Sedareba gvaZlevs:

NIBl 0μ= ,

an

nIB 0μ= . (9.3)

sadac lN

n = aris xviaTa ricxvi solenoidis sigrZis erTeulze.

∗ es daSveba miT ufro zustia, rac ufro grZelia solenoidi. SeiZleba Cveneba, rom usasrulod grZeli solenoidis SemTxvevaSi induqcia mis gareT nulis tolia.

31

2. toroidis magnituri velis induqcia

9.2 suraTze gamosaxulia toroidis kveTi. 1R da 2R kveTis Siga da

gare radiusebia. toroidis magnituri velis induqciis wirebi warmoadgens

wrewirebs, romelTa centri O wertilSia. cxadia, rom erTi da igive

induqciis wiris gaswvriv B –s ricxviTi mniSvneloba erTi da igive iqneba.

Caketil konturad avirCioT r radiusiani induqciis wiri da gamoviTvaloT

magnituri velis induqciis cirkulacia am konturis gaswvriv. vinaidan am

SemTxvevaSi BBl = , amitom

rBdlBdlBl l

l π2⋅==∫ ∫ . (9.4)

meore mxriv, Tu toroidi Seicavs N

xvias, xolo masSi gadis I deni, arCeuli

konturi moicavs I –denian N gamtars da

(8.4)-is Tanaxmad

NIdlBl

l 0μ∫ = . (9.5)

(9.4) da (9.5) formulebis Sedareba gvaZlevs

r

NIBπ

μ20= , (9.6)

saidanac Cans, rom toroidSi

magnituri velis induqcia O centridan

daSorebisas klebulobs:

( )dRNI

RNIB

RNIB

+===

10

20

10 22

;2 π

μπ

μπ

μ minmaqs ,

sadac d xviis diametria.

toroidis RerZul wirze ⎟⎠⎞

⎜⎝⎛ +

==2

21 RRRr saS

nIRNIB 00 2

μπ

μ ==saS

saS (9.7)

sadac n aris toroidis Suawiris (RerZuli wiris) sigrZis erTeulze

mosuli xviaTa raodenoba.

davuSvaT 1Rr < , maSin, vinaidan aseTi radiusiani konturi ( 1L

konturi) ar moicavs dens, 0=∫ dlBl

l da 0=B .

Tu 2Rr > , maSin aseTi radiusiani konturi ( 2L konturi) moicavs I –

denian N2 gamtars. magram vinaidan N gamtarSi dens aqvs erTi

�

�

�

�

�

�

�

�

� �

�

++

+

+

+

+

++

+

+

++

�

0 R1

r

R2

L2

sur. 9.2

o L1

32

mimarTuleba, xolo danarCen N –Si sawinaaRmdego mimarTuleba, yvela

gamtarSi gamavali denebis algebruli jami iqneba nulis toli, amitom

0=∫ dlBl

l da 0=B .

amrigad, magnituri veli toroidis gareT nulis tolia, igi mTlianad

lokalizebulia toroidSi.

cxadia, rom rac ufro gavzrdiT toroidis saSualo r radiuss ( d -sa

da n –is mocemuli mniSvnelobebis dros), miT ufro mcire iqneba

gansxvaveba maqsB –sa da minB –s Soris, e.i. miT ufro erTgvarovani iqneba veli

toroidis SigniT.∗ zRvarSi, roca ∞→r , toroidis nacvlad miviRebT

usasrulod grZel solenoids. aseTi solenoidis SigniT veli

erTgvarovania, vinaidan B –s yvela wertilSi erTnairi sidide da

mimarTuleba aqvs.

$10. magnituri velis moqmedeba denian gamtarze.

amperis formula

magnituri velis moqmedeba denian gamtarze aRmoaCina danielma

fizikosma erstedma, male mas Semdeg (1920 w), rac mis mier dadgenil iqna

deniani gamtaris moqmedeba magnitur isarze. es moqmedeba eqsperimentulad

Seiswavla frangma mecnierma amperma, romelmac daadgina, rom F Zala,

romliTac erTgvarovani magnituri veli moqmedebs wrfiv denian gamtarze,

damokidebulia magnituri velis induqciaze, gamtaris sigrZeze, masSi

gamaval denze da gamtaris orientaciaze magnitur velSi. kerZod, moqmedi

Zala maqsimaluria, roca gamtari moTavsebulia magnituri velis

marTobulad da nulis tolia, roca moTavsebulia mis paralelurad. es

damokidebuleba mocemulia amperis formuliT (kanoniT):

αsinlBIkF = , (10.1)

sadac α aris kuTxe B –sa da denis mimarTulebas Soris. formulidan Cans,

rom roca 2πα = (deniani gamtari moTavsebulia B –s marTobulad),

kIBlFF == maqs , xolo roca 0,0 == Fα .

∗ velis erTgvarovnebaze laparaki pirobiTia, vinaidan am dros yvela wertilSi erTnairi

iqneba B –s ricxviTi mniSvneloba, mimarTuleba ki, cxadia, wertilidan wertilamde Seicvleba.

33

proporciulobis koeficientis ricxviTi mniSvneloba damokidebulia

formulaSi Semavali sidideebis erTeulebis SerCevaze. SI sistemaSi 1=k

da

αsinIBlF = . (10.2)

Tu veli araerTgvarovania, xolo gamtari nebismieri formisaa,

gamtari unda daiyos mcire dl elementebad. simciris gamo dl elementi

SeiZleba CaiTvalos wrfivad, xolo veli mis maxloblobaSi (areSi, sadac

moTavsebulia dl ) _ erTgvarovnad. amrigad, amperis kanoni zogad

SemTxvevaSi ase Caiwereba:

αsinIdlBdF = , (10.3)

sadac α aris kuTxe Idl denis elementsa da B –s Soris.

$3-Si aRvniSneT, rom magnituri velis induqcia aris magnituri velis Zaluri

maxasiaTebeli. marTlac (10.3) formulidan, Tu 2πα =

dldF

IB 1= .

e.i. magnituri velis induqcia ricxobrivad im Zalis tolia, romliTac magnituri

veli moqmedebs mis marTobulad moTavsebul erTeulovani sigrZis gamtarze, romelSic

gadis erTeulovani deni.

(10.3) formula gansazRvravs amperis Zalis sidides. rac Seexeba

mimarTulebas, misi gansazRvris ramdenime wesi arsebobs.

marcxena xelis wesis Tanaxmad, Tu marcxena xels gavSliT ise,

rom magnituri induqciis wirebi Sediodes xelis gulSi, xolo oTxi TiTi

emTxveodes denis mimarTulebas, marTi kuTxiT gaSlili ceri gviCvenebs

denze moqmedi Zalis mimarTulebas. am wesiT sargebloba gansakuTrebiT

mosaxerxebelia, roca gamtari moTavsebulia magnituri velis marTobulad.

sxva SemTxvevaSi wesi moiTxovs damatebiT ganmartebas (unda aviRoT

magnituri velis is mdgeneli, romelic gamtarisadmi

marTobulia).

ufro universaluria burRis wesi. am wesis Ta-

naxmad, Tu burRis saxelurs vabrunebT dlI veqtoridan

B veqtorisken, burRis gadataniTi moZraobis mimar-

Tuleba gviCvenebs dF –is mimarTulebas (sur. 10.1).

am wesis gaTvalisiwnebiT (10.3) veqtorulad

Semdegnairad Caiwereba:

FddlI

sur. 10.1

B

34

[ ]BdlIdF ⋅= . (10.4)

magnituri velis moqmedebas denian gamtarze aqvs mravalmxrivi

teqnikuri gamoyeneba. igi safuZvlad udevs mravali xelsawyos moqmedebas,

romelTaganac yvelaze mniSvnelovani da farTod gavrcelebulia

eleqtroZrava. es moqmedeba safuZvlad udevs agreTve eleqtrosazomi

xelsawyoebis mowyobilobas.

$11. denebis urTierTqmedeba

cda gviCvenebs, rom deniani gamtarebi garkveulad urTierTqmedeben,

rac ganpirobebulia TiToeuli denis magnituri velis moqmedebiT meoreze.

amperis formula saSualebas gvaZlevs martivad gamovTvaloT am

urTierTqmedebis Zalis sidide.

simartivisaTvis ganvixiloT ori sakmaod grZeli urTierTparale-

luri gamtari, romlebSic gadis 1I da 2I deni. Tu am denebs erTnairi

mimarTuleba aqvT, maT ewodeba paraleluri, (sur. 11.1.a) winaaRmdeg SemTxve-

vaSi ki antiparaleluri (sur. 11.1.b). TiToeuli gamtaris sigrZe aRvniSnoT

l –iT, xolo maT Soris manZili d –Ti.

ganvixiloT jer paraleluri denebi.

burRis wesis gamoyenebiT davadgenT, rom 1I denis magnituri velis

induqciis 1B veqtori 2I denis areSi mimarTulia suraTis sibrtyis

marTobulad Cvengan da (6.5)-is Tanaxmad sididiT tolia:

dI

Bπ

μ2

101 = .

d d

I1 I2 I1 1B 1B

2B

2B

1F 1F2F 2F

a b sur. 11. 1.

I2

35

amperis formulis Tanaxmad, 1B induqciis magnituri veli l sigrZis

2I denian gamtarze imoqmedebs ZaliT ⎟⎠⎞

⎜⎝⎛ =

2πα

ldII

lBIFπ

μ2

210122 == . (11.1)

marxcena xelis wesis Tanaxmad, am Zalas aqvs suraTze naCvenebi

mimarTuleba, e.i. mimarTulia 2I denidan 1I –sken.

analogiuri msjelobiT miviRebT, rom 2I denis magnituri veli

dI

Bπ

μ2

202 =

imoqmedebs 1I denian gamtarze 1F ZaliT, romelic sididiT tolia

ldII

lBIFπ

μ2

210211 == . (11.2)

e.i. igivea, rac 2F da mimarTulia denidan 2I –sken (sur. 11.1.a).

amrigad, miviReT, rom paraleluri denebi erTmaneTs miizidaven

ZaliT:

ldII

Fπ

μ2

210= . (11.3)

aseve martivad davadgenT, rom antiparaleluri denebi erTmaneTs

ganizidaven imave sididis ZaliT (sur. 11.1.b).

SeiZleba Cveneba, rom Tu deniani gamtarebi erTmaneTs gadakveTen

raRac kuTxiT, maSin maT Soris aRiZvreba magnituri urTierTqmedebis

Zalebi, romlebic cdiloben Semoabrunon gamtarebi da daayenon erTmaneTis

paralelurad ise, rom dens orive gamtarSi hqondes

erTnairi mimarTuleba (sur. 11.2).

(11.3) formula da denis Zalis SI sistemis

erTeulis _ amperis ganmarteba saSualebas gvaZlevs

davadginoT magnituri mudmivas ricxviTi mniSvneloba,

marTlac, Tu 121 == II a, 1=l m da 1=d m, maSin

7102 −⋅=F n, amitom

m

hn

ma

wmva

ma

mn22

777

210 1041041042 −−− ⋅=

⋅⋅⋅

⋅=⋅⋅

⋅=⋅

= ππππμlIIdF

.

I1 I2 α

sur. 11. 2.

36

$12. magnituri velis moqmedeba Caketil denian konturze

$3-Si aRvniSneT, rom magnitur velSi Setanil Caketil denian kontur-

ze moqmedebs mabrunebeli momenti (ris safuZvelzec xdeba magnituri velis

induqciis veqtoris gansazRvra). ganvixiloT es sakiTxi dawvrilebiT.

davuSvaT, nebismieri formis deniani brtyeli da myari Caketili

konturi moTavsebulia B induqciis erTgvarovan magnitur velSi. ganvixi-

loT jer SemTxveva, roca veli mimarTulia konturis sibrtyis mar-

Tobulad Cvengan (sur. 12.1). amperis formulis Tanaxmad konturis TiToeul

dl elementze imoqmedebs IBdldF = Zala ( )2πα = , romelic marcxena xelis

wesis Tanaxmad mimarTuli iqneba konturis SigniT (sur. 12.1a). velis

erTgvarovnebis gamo yvela dl elementze moqmedebs erTnairi sididis Zala,

amitom maTi tolqmedi nulis tolia da konturi mxolod Tanabrad

ikumSeba.

Tu SevucvliT mimarTulebas dens, an magnitur vels, konturze

imoqmedebs Tanabrad gamWimavi Zalebi (sur. 12.1.b).

denisa da magnituri velis mimarTulebis erTdrouli Secvlisas,

cxadia, elementebze moqmedi Zalebi mimarTulebas ar Seicvlis.

amrigad, erTgvarovani magnituri velis marTobulad moTavsebul

konturze moqmedebs gamWimavi an SemkumSveli Zalebi, romlebic kompensir-

deba konturis deformaciis Sedegad aRZruli drekadi ZalebiT.

ganvixiloT axla SemTxveva, roca konturis sibrtye moTavsebulia

magnituri velis paralelurad. gavavloT magnituri velis paraleluri dh

manZiliT daSorebuli ori wrfe, romlebic konturze CamoWrian mopirda-

pire 1dl da 2dl elementebs (sur. 12.1.g). amperis kanonis Tanaxmad am elemen-

tebze imoqmedebs Zalebi

111 sinαIBdldF = da 222 sinαIBdldF = .

magram

dhdIdI == 2211 sinsin αα ,

amitom

IBdhdFdF == 21 .

marcxena xelis wesis gamoyenebiT davadgenT, rom Cvens SemTxvevaSi

1dF mimarTulia naxazis sibrtyis marTobulad Cvensken, xolo 2dF _

37

Cvengan. e.i. konturis 1dl da 2dl elementebze moqmedebs ZalTa wyvili,

romlis mabrunebeli momenti

IBdSxIBdhdM =⋅= , (12.1)

sadac x aris saSualo manZili 1dl da 2dl elementebs Soris, xolo dS _

daStrixuli nakvTis farTobi.

mTel konturze moqmed mabrunebel moments miviRebT (12.1)-is

integrebiT:

BPIBSIBdSdMM mSS

==== ∫∫ , (12.2)

sadac ISPm = konturis magnituri momentia.

suraTidan Cans, rom M mabrunebeli momentis moqmedebiT konturi

Semobrundeba da dadgeba sibrtyiT magnituri velis marTobulad, Tanac

ise, rom konturis mP magnituri momentis mimarTuleba emTxveodes gare

magnituri velis induqciis B veqtoris mimarTulebas.

am mdgomareobaSi konturi imyofeba wonasworobaSi. amasTan, wonasworobis es

mdgomareoba mdgradia, vinaidan am mdgomareobidan gamoyvanili konturi TavisTavad

ubrundeba mas. konturi wonasworobaSia maSinac, roca mP mimarTulia B –s

sawinaaRmdegod, magram advili misaxvedria, rom wonasworobis es mdgomareoba aris

aramdgradi.

zogad SemTxvevaSi, roca konturis sibrtyis n normali adgens nebis-

mier α kuTxes induqciis B veqtorTan, es ukanaskneli paralelogramis we-

siT daiSleba or mdgenelad, romelTagan erTi ( )1B mimarTulia normalis

gaswvriv, xolo meore ( )2B _ mis marTobulad (sur. 12.2). 1B mdgeneli gamo-

1dF

1dl2dl

I I I

dF

B

2dF

dh

a b g

sur. 12. 1. a, b, g.

+ +

38

iwvevs konturis gaWimvas an SekumSvas, 2B

mdgeneli ki mas Semoabrunebs da daayenebs

velis marTobulad. mabrunebeli momenti

αα sinsin2 BPIBSIBM m=== (12.3)

an veqtorulad,

[ ]BPM m= . (12.4)

ufro rTulia SemTxveva, roca deniani

konturi moTavsebulia araerTgvarovan mag-

nitur velSi. am dros elementaruli dF Zalebis tolqmedi nulisagan

gansxvavebulia, amitom konturi ara marto deformirdeba, aramed

gadaadgildeba am tolqmedis gaswvriv.

gamoviTvaloT es tolqmedi marTkuTxa CarCos SemTxvevaSi, roca igi

moTavsebulia naxazis sibrtyis marTobulad araerTgvarovan velSi, rom-

lis induqcia izrdeba ox RerZis gaswvriv wrfivi kanoniT, ise rom (sur. 12.3)

bxBBB∂∂

=− 12 ,

sadac xB∂∂

aris B –s cvlilebis siCqare

ox RerZis gaswvriv. maSin 12 FF > da

ISxBIab

xBIaBIaBFFF

∂∂

=∂∂

=−=−= 1212 ;

magram

mPIS = ,

amitom

xBPF m ∂∂

= . (12.4)

suraTidan Cans, rom roca BPm ↑↑→

(e.i. konturi mdgrad wonasworobaSia),

Zala mimarTulia velis zrdis mxares, e.i. konturi Seizideba velSi,

xolo roca BPm ↑↓→

(e.i. konturi aramdgrad wonasworobaSia), F Zala

mimarTulia velis Semcirebis mxares, e.i. konturi gamoizideba velidan.

amrigad, erTgvarovan magnitur velSi denian konturze moqmedebs

mxolod mabrunebeli momenti, romlis moqmedebiTac igi dgeba magnituri

velis marTobulad. araerTgvarovan magnitur velSi ki konturze, garda

mabrunebeli momentisa, moqmedebs kidev F Zala, romelic gadaadgilebs mas

B

2B

1B

n

I

sur. 12. 2.

α

2F

1B 2B

1F

mP

a

b I

o

1F 2F

mP

o

I

sur. 12. 3.

39

velis cvlilebis gaswvriv. suraTi savsebiT analogiuria imisa, rac

gvqonda eleqtruli dipolis Setanisas eleqtrul velSi (I tomi, $ 117).

$13. magnituri velis moqmedeba moZrav muxtze.

lorencis Zala

magnituri velis arsebobis erT-erTi gamovlinebaa misi moqmedeba

moZrav muxtze. es moqmedeba advilad aixsneba. marTlac, amperis kanonis

Tanaxmad, magnituri veli garkveuli ZaliT moqmedebs denian gamtarze, deni

ki mowesrigebulad moZravi muxtebis erTobliobaa. amitom cxadia, rom

denian gamtarze magnituri velis moqmedeba ganpirobebuli unda iyos

moZrav muxtebze magnituri velis moqmedebiT (iseve rogorc deniani

gamtaris magnituri veli ganpirobebulia mowesrigebulad moZravi muxtebis

magnituri velebiT ($ 7).

Zalas, romliTac magnituri veli moqmedebs moZrav muxtze,

lorencis Zala ewodeba. am Zalis gamosaTvlelad visargebloT amperis

formuliT, romlis Tanaxmad ($ 10) B induqciis magnituri veli I deniani

gamtaris dl elementze moqmedebs ZaliT

αsinIBdldF = , (13.1)

sadac α aris kuTxe Idl –sa da B –s Soris (sur. 13.1) magram (7.2)-is Tanaxmad

SqnI υ0= ,

amitom

αυ sin0 SBdlqndF = . (13.2)

aseTi Zala moqmedebs I denis dl elementze, anu Sdl moculobaSi

moTavsebul yvela muxtze, romelTa raodenoba

SdlndN 0= .

yvela es muxti mowesrigebulad moZraobs erTi

mimarTulebiT da erTnairi υ siCqariT, amitom erT

moZrav muxtze moqmedi Zala, anu lorencis Zala

iqneba dN –jer naklebi:

αυαυ sinsin0

0 BqSdln

SBlqndNdFF === , (13.3)

sadac α aris kuTxe υ –sa da B -s Soris.

lorencis Zalis mimarTuleba, iseve rogorc

amperis Zalisa, ganisazRvreba marcxena xelis an

α

sur. 13. 1.I

S

dl

B

40

burRis wesiT, amitom veqtorulad

[ ]BqF υ= (13.4)

lorencis Zalis mimarTulebis sworad gansazRvrisaTvis unda

gvaxsovdes, rom denis mimarTulebad miCneulia dadebiTi muxtebis moZra-

obis mimarTuleba. amitom F –is mimarTuleba Tanxvdeba [ ]Bυ –s mimar-

Tulebas (e.i. marcxena xelis wesiT gansazRvrul mimarTulebas) maSin,

roca 0>q . uaryofiTi muxtis (mag.eleqtronis) SemTxvevaSi F –s aqvs [ ]υB –s

mimarTuleba.

(13.3) formulidan Cans, rom magnitur velSi moZrav muxtze moqmedi

Zalis sidide∗ damokidebulia α –kuTxeze. es Zala minimaluria (nulis to-

lia) roca 0=α , e.i. roca muxti moZraobs magnituri velis paralelurad

da maqsimaluria, roca 2πα = , e.i. roca muxti moZraobs velis marTobulad.

maqsimaluria Zalis sidide

BqF υ=maqs . (13.5)

garda amisa, radgan lorencis Zala marTobia nawilakis moZraobis

siCqarisa, am Zalis muSaoba nulis tolia, e.i. igi ar cvlis siCqaris

sidides,∗∗ cvlis mxolod mis mimarTulebas, anu warmoadgens centrskenul

Zalas

RmF

2υ= ,

sadac R traeqtoriis simrudis radiusia.

davuSvaT, υ siCqaris q muxti SeiWra erTgvarovan magnitur velSi

( )constB = mis marTobulad ⎟⎠⎞

⎜⎝⎛ =

2πα . muxtze imoqmedebs lorencis

maqsimaluri Zala, romelic ricxobrivad (13.5) formuliT ganisaRvreba. es

Zala mimarTulia υ da B veqtorebis marTobulad, amitom muxti imoZravebs

∗ Tu muxti uZravia, e.i. 0=υ , maSin 0=F . e.i. uZrav muxtze magnituri veli saerTod ar moqmedebs. ∗∗ xazgasmiT unda aRiniSnos, rom magnituri veli ar cvlis nawilakis siCqaris sidides da maSasadame mis energias mxolod im SemTxvevaSi, roca veli mudmivia (drois mixedviT ar icvleba). cvladi magnituri veli, eleqtromagnituri induqciis kanonis Tanaxmad (ix. $21), aRZravs grigalur eleqtrul vels, romelic cvlis muxtis siCqaris ara marto mimarTulebas, aramed sididesac. cvladi magnituri velis moqmedebiT grigaluri eleqt-ruli velis aRZvra safuZvlad udevs eleqtronebis induqciuri amaCqareblis _ betatronis moqmedebas.

41

B –s marTobul sibrtyeSi. moZraobis traeqtoriis simrudis R radiuss

gavigebT pirobidan:

RmBq

2υυ = ,

saidanac

qBmR υ

= , (13.6)

radgan constB = da, garda amisa, lorencis Zala ar cvlis υ –s

sidides (e.i. const=υ ), amitom traeqtoriis simrudis radiusic mudmivi

iqneba, e.i. nawilaki imoZravebs wrewirze. maSasadame, erTgvarovan

magnitur velSi magnituri Zalwirebis marTobulad moZravi nawilakis

traeqtoria warmoadgens wrewirs, romlis radiusi (13.6) tolobiT

ganisaRvreba. formulidan Cans, rom rac metia nawilakis siCqare, miT met

radiusian wrewirs Semowers igi mocemul magnitur velSi.

brunvis periodi, anu dro, romlis ganmavlobaSic damuxtuli

nawilaki Semowers R –radiusian wrewirs

qB

mRT πυπ 22

== . (13.7)

rogorc (13.7) gviCvenebs, brunvis periodi ar aris damokidebuli

nawilakis moZraobis siCqareze (maSasadame, wrewiris radiusze) da

ganisazRvreba mxolod magnituri velis B induqciiT.∗ brunvis periodis

damoukidebloba moZraobis siCqareze safuZvlad udevs damuxtuli

nawilakis cikluri amCqareblis _ ciklotronis muSaobas.

ganvixiloT zogadi SemTxveva, roca damuxtuli nawilaki moZraobs erTgvarovan

magnitur velSi ise, rom misi siCqaris veqtori adgens magnituri induqciis veqtorTan

nebismier (maxvil) α kuTxes. davSaloT siCqaris veqtori or mdgenelad, romelTagan erTi

( )1υ aris magnituri velis paraleluri, xolo meore ( )2υ _ misi marTobuli: (sur. 13.2.a)

αυυ cos1 = ,

αυυ sin2 = .

1υ mdgenelze magnituri veli araviTar moqmedebas ar axdens, 2υ mdgenels ki ucvlis

mxolod mimarTulebas. nawilaki monawileobas iRebs erTdroulad or moZraobaSi: igi

Tanabrad brunavs 2υ siCqariT wrewirze, romlis radiusi

qBm

qBmR αυυ sin2 == (13.8)

∗ miRebuli Sedegi marTebulia mxolod im SemTxvevaSi, roca nawilakis siCqare gacilebiT naklebia sinaTlis siCqareze, da, maSasadame, misi masa siCqareze damokidebuli ar aris.

42

da gadaadgildeba magnituri velis gaswvriv (e.i. brunvis sibrtyis marTobulad) mudmivi

1υ siCqariT. Sedegad nawilaki imoZravebs xraxnul wirze, romlis RerZi Tanxvdeba

magnituri velis induqciis wirs (sur. 13.2.b). xviis radiusi ganisaRvreba (13.8) formuliT,

xolo xraxnis biji

απυυ cos21 qB

mTh == .

Tu nawilaki moZraobs eleqtromagnitur velSi, masze, garda

magnituri Zalisa, imoqmedebs eleqtruli Zalac qEF = . am SemTxvevaSi

nawilakze moqmedi marezultirebeli Zala miiReba eleqtruli da

magnituri Zalebis geometriuli SekrebiT:

[ ] [ ]( )BEqBqEqF υυ +=+= .

am Zalas ewodeba lorencis ganzogadebuli Zala.

dabolos, ganvixiloT moZravi muxtebis magnituri urTierTqmedebis sakiTxi. ori (an

ramdenime) muxtis erTdrouli moZraobisas, maT Soris, garda eleqtruli urTierTqmedebis

Zalisa ⎟⎟⎠

⎞⎜⎜⎝

⎛= 2

2

041

rqFe πε

, aRiZvreba magnituri urTierTqmedebis Zalac ( )αυ sinBqFm = , ro-

melic imiTaa ganpirobebuli, rom TiToeuli muxti sivrceSi qmnis magnitur vels,

romelSic moZraobs meore muxti. gamoviTvaloT magnituri urTierTqmedebis Zala da

SevadaroT igi eleqtruls.

vTqvaT 1q da 2q erTniSna (dadebiTi) mux-

tebi moZraoben r manZiliT daSorebul parale-

lur 'aa da 'bb wrfeebze erTi da imave υ siC-

qariT (sur. 13.3). 1q muxtis mier Seqmnili velis

induqcia misgan r manZilze ( 2q muxtis moTav-

sebis adgilas) 7.4 formulis Tanaxmad ( )1sin =α

tolia

210

1 4 rq

Bπ

υμ= -is

2B

+ +

υ υ 1B

mF mF

1q 2q

'a 'b

a b

r

sur. 13. 3.

B

B

υ 2υ

1υ q

α

h

r

a b

sur. 13. 2.

43

da mimarTulia naxazis sibrtyis marTobulad Cvengan. es veli imoqmedebs 2q muxtze

lorencis ZaliT ( )1sin =α

2

2210

12 4 rqq

BqFm πυμ

υ == , (13.9)

romelsac aqvs naxazze naCvenebi mimarTuleba.

cxadia, rom sawinaaRmdegod mimarTuli igive sididis Zala imoqmedebs 1q muxtze.

amrigad, paralelur traeqtoriebze moZrav erTsaxela muxtebs Soris moqmedebs

magnituri urTierTmizidvis Zala.

magram garda am Zalisa, am muxtebs Soris moqmedebs eleqtruli ganzidvis Zala.

221

041

rqq

Fe πε= . (13.10)

amitom muxtebi erTmaneTs miizidaven an ganizidaven imis da mixedviT, Tu romelia

meti, mF Tu eF .

am ZalaTa SedarebisaTvis (13.9) gavyoT (13.10)-ze

200 υεμ=

e

m

FF

(13.11)

magram

ma

wmv

⋅⋅

⋅= −70 104πμ da

mv

wma

⋅⋅

⋅= 90 1094

1π

ε ,

amitom

228

16001

103

11091

c=

⎟⎠⎞

⎜⎝⎛ ⋅

=⋅

=

wmmm

wm2

2

εμ ,

sadac 8103 ⋅=c m/wm sinaTlis siCqarea vakuumSi da sabolood gveqneba:

2

2

cFF

e

m υ= (13.12)

(13.12) formulidan Cans, rom Tu c<υ , maSin em FF < . Cveulebriv c<<υ , amitom

em FF << , e.i. paralelurad moZrav erTsaxela muxtebs Soris Warbobs ganzidvis Zala.

marTlac, cnobilia, rom eleqtronebis paraleluri kona TandaTan farTovdeba, rac

miRebuli Sedegis eqsperimentuli dadasturebaa.

$14. holis efeqti

damuxtuli nawilakebis gadaxra magnitur velSi, romelic Sedegia

lorencis Zalis moqmedebisa, farTod gamoiyeneba teqnikaSi (damuxtuli

nawilakebis amCqarebeli, eleqtronuli mikroskopi, televizori, rxeviTi da

sxva swrafcvladi procesebis Sesaswavli gamzomi xelsawyoebi da sxv.)

44

magnituri veli moqmedebs im eleqtronebzec, romelTa mimarTuli

moZraoba ganapirobebs dens myar sxeulSi. es udevs safuZvlad amerikeli

fizikosis holis mier 1880 wels aRmoCenil efeqts, romelic agreTve

farTod gamoiyeneba teqnikaSi.

holis efeqti mdgomareobs SemdegSi. Tu magnitur velSi movaTavsebT

firfitas (Cveulebriv mas amzadeben marTkuTxa paralelepipedis saxiT),

romelSic magnituri velis marTobulad gadis deni, maSin firfitis or

mopirdapire A da C waxnags Soris warmoiSveba potencialTa sxvaoba

CA VV − , romlis sidide proporciulia I denis Zalisa, B magnituri velis

induqciisa da ukuproporciulia firfitis d sisqisa (sur. 14.1):

dIBRVV CA =− (14.1)

proporciulobis R koeficients ewodeba

holis koeficienti an holis mudmiva. misi ricxvi-

Ti mniSvneloba damokidebulia firfitis masalaze,

amasTan zogi nivTierebisaTvis igi dadebiTia,

zogisTvis uaryofiTi.

lorencis Zala saSualebas gvaZlevs ara

marto avxsnaT holis efeqti, aramed miviRoT

gamosaxuleba holis mudmivasaTvis.

marTlac, vinaidan liTonSi eleqtrul dens ganapirobebs eleqtro-

nebis mimarTuli moZraoba, amitom denis naxazze naCvenebi mimarTu-

lebisaTvis, TiToeul eleqtronze imoqmedebs qvevidan zeviT mimarTuli

lorencis Zala. am Zalis moqmedebiT eleqtronebi gadaixrebian qvevidan

zeviT, ris Sedegadac zeda waxnagi daimuxteba uaryofiTad, qveda ki _

dadebiTad.∗ warmoqmnili eleqtruli veli ewinaaRmdegeba eleqtronebis

Semdgom gadaxras, da es gadaxra Sewydeba maSin, rodesac veli miaRwevs

iseT E mniSvnelobas, romlis drosac eleqtronze moqmedi eleqtruli

Zala ( )eEFe = gaawonasworebs lorencis Zalas ( )BeF υ= :

FFe =

an

BeeE υ= da BE υ= (14.2)

magram

∗ cxadia, rom Tu deni firfitaSi ganpirobebulia dadebiTi muxtebis mimarTuli moZraobiT, waxnagebis niSnebi Seicvleba sawinaaRmdegoTi.

B

I

d

A

l

C

CV

AV

sur. 14. 1.

45

lVV

E CA −= ,

sadac l _ waxnagebs Soris manZilia,

xolo (7..2) tolobidan

eldnI

eSnI

00

==υ ,

sadac ldS = firfitis ganikveTis farTia,

amitom (14.2)-dan miviRebT

ldenIB

lVV CA

⋅=

−

0

,

saidanac

dIB

enVV CA

0

1=− . (14.3)

Teoriulad miRebuli (14.3) tolobis Sedareba holis eqsperimentul

(14.1) formulasTan gvaZlevs, rom holis mudmiva

en

R0

1= . (14.4)

(14.4) formula gviCvenebs, rom Tu eqsperimentuli gazomviT davadgenT

R –s, martivad ganvsazRvravT myar sxeulSi denis gadamtani muxtebis

(Tavisufali muxtebis) koncentracias.

garda amisa, holis efeqtis niSani saSualebas gvaZlevs davadginoT,

Tu rogori damuxtuli nawilakebis _ dadebiTis Tu uaryofiTis mimarTuli

moZraoba ganapirobebs dens firfitaSi.∗

dabolos, holis mudmiva SeiZleba davukavSiroT myari xseulis iseT

mniSvnelovan maxasiaTebels, rogoricaa Tavisufali muxtebis

(eleqtronebis an xvrelebis) Zvradoba anu maTi saSualo siCqare

erTeulovani daZabulobis velSi. marTlac, omis diferencialuri

kanonidan kuTri eleqtrogamtaroba

Ei

=γ ;

magram

υeni 0= ,

amitom

eunEen

00 ==υγ ,

∗ denis gadamtan dadebiT nawilakebs myar sxeulSi warmoadgens e.w. `xvrelebi~.

46

sadac E

u υ= eleqtronebis Zvradobaa.

meore mxriv, holis mudmiva

enR

0

1= ,

amitom

Ru

=γ da γRu = (14.5)

miRebuli damokidebuleba saSualebas gvaZlevs R –isa da γ –s

gazomviT ganvsazRvroT u .

$15. eleqtronis kuTri muxtis gansazRvra

$ 13-Si vnaxeT, rom eleqtromagnitur velSi moZraobisas damuxtul

nawilakze moqmedebs lorencis ganzogadebuli Zala

[ ]( )BEqF υ+= .

Tu nawilakis masaa m , am Zalis moqmedebiT igi iZens aCqarebas

[ ]( )BEmq

mFa υ+== . (15.1)

15.1 gviCvenebs, rom nawilakis moZraobis xasiaTi, misi traeqtoria

damokidebulia ara nawilakis q muxtze an m masaze cal-calke, aramed mq

fardobaze. rac naklebia es fardoba, miT naklebia a da, maSasadame miT

nakleb cvlilebas ganicdis nawilakis siCqare velSi moZraobisas.

nawilakis muxtis Sefardebas mis masasTan kuTri muxti

ewodeba. eleqtronis kuTri muxti ⎟⎠⎞

⎜⎝⎛

me

pirvelad gazoma ingliselma mec-

nierma tomsonma 1897 wels. tomsonis meTodis arsi mdgomareobs eleqtro-

nebis nakadze magnituri da eleqtruli velebis erTdroul moqmedebaSi,

romelTa sidide da mimarTuleba isea SerCeuli, rom eleqtronebi gadaxras

ar ganicdian.

cdis sqema aseTia (sur. 15.1): eleqtronul-sxivur milakSi gaxurebuli

K kaTodidan gamosuli eleqtronebi gadian A anodis xvrelSi, viwro

konis saxiT ecemian mafluorescirebel E ekrans da gvaZleven masze mnaT

C wertils. SevqmnaT milakSi erTgvarovani magnituri veli, romlis

induqcia mimarTuli iqneba naxazis sibrtyis marTobulad Cvengan (suraTze

47

magnituri velis moqmedebis are gamoyofilia wyvetili wrewiriT). maSin

eleqtronuli kona gadaixreba zevidan qveviT da mnaTi laqa ekranze

gadaadgildeba C wertilidan 1C –Si.

eleqtronebis gadaxras iwvevs maTze lorencis [ ]BeF υ= Zalis

moqmedeba. es Zala mimarTulia zevidan qveviT, xolo misi sidide, radgan

B⊥υ , gamoiTvleba formuliT BeF υ= .

me-13-e paragrafSi Catarebuli msjelobis safuZvelze davaskvniT, rom

vinaidan Cvens SemTxvevaSi lorencis Zala warmoadgens centriskenul

Zalas, amitom

RmBe

2υυ = ,

saidanac

RBm

e υ= . (15.2)

aq R im wrewiris radiusia, romlis rkalsac warmoadgens eleqtro-

nis moZraobis traeqtoria milakSi. misi gansazRvra xdeba mnaTi laqis

gadaxris saSualebiT. magnituri velis B induqcia SeiZleba uSualod

gavzomoT da maSasadame, kuTri muxtis gasagebad saWiroa eleqtronebis

siCqaris povna.∗ υ -s gasagebad tomsoni eleqtronul konaze erTdroulad

moqmedebda eleqtruli veliTac, romelic iqmneboda milakSi moTavsebuli

brtyeli kondensatoris firfitebiT da ise iyo mimarTuli, rom

eleqtronebs xrida qvevidan zeviT, e. i. magnituri velis sapirispirod.

amasTan eleqtruli velis daZabuloba ise SeirCeoda, rom mnaTi laqa isev

C wertilSi dabrunebuliyo. am SemTxvevaSi eleqtronebze moqmedi magni-

turi da eleqtruli Zalebi sididiT erTmaneTis tolia, e. i.

eEBe =υ ,

∗ xelsawyoSi uzrunvelyofilia `monoqromatuli konis~ miReba, e.i. iseTis romelSic yvela eleqtronis siCqare erTnairia.

1C

C

E

A K +

_

sur. 15. 1.

48

saidanac

BE

=υ .

υ -s mniSvnelobis Casma (15.2)-Si gvaZlevs:

2RBE

me= .

zusti gazomvebiT eleqtronis kuTri muxtisaTvis miRebulia Semdegi

sidide:

gkg

k qCGSEme 1711 1027,51076,1 ⋅=⋅= .

radgan eleqtronis muxti SeiZleba didi sizustiT gaizomos

(magaliTad milikenis meTodiT), amitom eleqtronis kuTri muxtis gazomva

saSualebas gvaZlevs ganvsazRvroT misi masa. eleqtronis muxtis sidide

1910 106,1108,4 −− ⋅=⋅= qCGSEe k, amitom eleqtronis masa 3128 101,9101,9 −− ⋅=⋅= gm kg.

eleqtronis siCqaris gansazRvra sxva meTodidac SeiZleba. kerZod, Tu davuSvebT,

rom eleqtronis siCqare kaTodidan amosvlis momentSi nulis tolia, maSin υ siCqariT

moZravi eleqtronis kinetikuri energia amaCqarebeli velis muSaobis tolia, e. i.

( )21

2

2ϕϕυ

−= em,

saidanac 212 ϕϕυ −⋅=me

, romlis CasmiT (15.2)-Si:

( )2

212RBm

e ϕϕ −= .

es meTodi nakleb zustia, vidre tomsonis meTodi.

eleqtronebisa da ionebis kuTri muxtis gazomvisas aRmoCenili iqna

mikronawilakis axali Tviseba. kerZod, 1900 w. germanelma mecnierma

kaufmanma daadgina, rom eleqtronis kuTri muxtis sidide damokidebulia

misi moZraobis siCqareze _ siCqaris zrdiT me mcirdeba. es eqsperimentuli

faqti, romelic miuTiTebda eleqtronis masis damokidebulebas siCqareze,

aixsna 1905 wels ainStainis fardobiTobis TeoriiT, romlis Tanaxmad

nawilakis masa

2

2

0

1c

mm

υ−

=

sadac c _ sinaTlis siCqarea, 0m _ eleqtronis uZraobis masaa, xolo

m _ υ siCqariT moZravi eleqtronis masa. formulidan Cans, rom υ –s

zrdiT m izrdeba, rac iwvevs kuTri muxtis Semcirebas.

49

$16. ionebis kuTri muxtis gansazRvra.

masspeqtrografi

wina paragrafSi ganxiluli eleqtronis kuTri muxtis gansazRvris

meTodi ionebisaTvis ar gamodgeba. es imiTaa gamowveuli, rom eleqtro-

nebisagan gansxvavebiT, romlebic Cndebian vakuumSi sivrcis erTsa da imave

adgilas (kaTodis zedapirTan) da amitom eleqtruli velis moqmedebiT

erTnair siCqares iZenen, ionebi warmoiqmneba ganmuxtvis milis sxvadasxva

adgilas, amitom amaCqarebeli Zabvis mniSvneloba maTTvis sxvadasxvaa, ris

gamoc isini sxvadasxva siCqariT moZraoben. sxvadasxva siCqaris ionebi

eleqtrul da magnitur velebSi sxvadasxvanairad gadaixrebian, rac

aZnelebs maT fokusirebas _ miiReba didi zomis, `warecxili~ laqa da

gazomvebi xdeba SeuZlebeli.

ionebis kuTri muxtis gasazomad tomsonma 1907 wels daamuSava para-

bolebis meTodi, romelic gamoricxavda siCqareTa sxvadasxvaobiT gamow-

veul siZneles. meTodis arsi mdgomareobs ionTa konaze urTierTparalelu-

ri (mudmivi) eleqtruli da magnituri velebis erTdroul moqmedebaSi, ris

Sedegadac (ix. petitis teqsti) erTnairi kuTri muxtis, magram sxvadasxva

siCqaris mqone ionebi ekranze parabolur wirze lagdebian. Tu ekranis nac-

vlad movaTavsebT fotografiul firfitas, gamJRavnebis Semdeg miviRebT

parabolas, romlis parametris gazomviT ganisazRvreba ionis kuTri muxti.

radgan ionis muxtis gazomvis ramodenime zusti meTodi arsebobos pa-

rabolebis meTodi saSualebas iZleva didi sizustiT ganisazRvros ionis

masa.

davuSvaT, dadebiTi ionebis kona, romelic miiReba ganmuxtvis milSi warmoqmnili

dadebiTi ionebis kaTodis xvrelSi gasvlis Sedegad, moZraobs urTierTparalelur

eleqtrul da magnitur velebSi, am velebis marTobulad. vTqvaT, eleqtruli da magnituri

velebi mimarTulia z RerZis paralelurad, xolo ionebis moZraoba swarmoebs y RerZis

gaswvriv (sur. 16.1). ganvixiloT mocemuli siCqaris

ionze am velebis moqmedeba cal-calke.

1. E daZabulobis elqtruli veli q muxtis

mqone ionze imoqmedebs

EqFe ⋅=

ZaliT, romelic mimarTulia qvevidan zeviT ( )0>q da

aniWebs mas

mqE

mF

a ez ==

υ x

X

Y

Z

E B

OBF

eF

sur. 16.1.

z

P

50

aCqarebas, sadac m ionis masaa. vinaidan veli mudmivia, es aCqarebac aris mudmivi, amitom

ioni imoZravebs z RerZis gaswvriv TanabaraCqarebulad da raime t droSi gaivlis

mqEttaz z

22

22

==

manZils. magram amave droSi nawilaki inerciiT ganagrZobs moZraobas y RerZis gaswvriv

da gaivlis ty ⋅= υ manZils, sadac υyt = da

2

2

21

υyE

mqz ⋅

= .

mocemuli cdis pirobebSi 2

21 Ey mudmivi sididea; aRvniSnoT igi 1C –iT.

maSin

211υm

qCz = . (16.1)

2. B induqciis magnituri veli υ siCqariT moZrav muxtze imoqmedebs lorencis

ZaliT, romelic mimarTulia marcxnidan marjvniv da romlis sidide (vinaidan 2πα = )

BqFB υ= .

am Zalis moqmedebiT ioni imoZravebs x RerZis gaswvriv

mBq

mFa B

xυ

==

mudmivi aCqarebiT da imave t droSi am mimarTulebiT gaivlis

υυ 2

22

21

22By

mqt

mBqta

x x ==⋅

=

manZils. aqac Tu SemoviRebT aRniSvnas

22

21 CBy = ,

miviRebT, rom

υ1

2 mqCx = (16.2)

(16.1) da (16.2) formulebidan Cans, rom eleqtrul da magnitur velebSi ionis gadaxrebi

damokidebulia mis mq

kuTr muxtze da υ siCqareze.

ionebi, romelTac mq

erTnairi aqvT, magram moZraoben sxvadasxva υ siCqariT, o

wertilidan y manZiliT daSorebul xoy sibrtyis paralelur P sibrtyeze moxvdebian

sxvadasxva wertilebSi. am wertilTa geometriul adgils miviRebT, Tu (16.1) da (16.2)

formulebidan gamovricxavT υ siCqares. amisaTvis (16.2) aviyvanoT kvadratSi da gavyoT

(16.1)-ze miviRebT:

51

zmq

CCx

1

222 = .

es parabolis gantolebaa, romlis mq

CCp

1

22= parametris gazomviT ganisazRvreba ionis

kuTri muxti.

Tu ionebis kona Seicavs sxvadasxva kuTri muxtis mqone ionebs, foto-

grafiul firfitaze miviRebT ramodenime parabolas, romelTagan TiToe-

uls mq kuTri muxtis garkveuli mniSvneloba Seesabameba. Tu kona Sedgeba

erTnairi muxtis ionebisagan, maSin cxadia, sxvadasxva parabolaze sxvada-

sxva masis ionebi dalagdebian. qimiurad sufTa neonze cdis Catarebisas

tomsonma aRmoaCina, rom garda intensiuri parabolisa, romelsac Seesabame-

boda atomuri masa 20, fotofirze miiReboda mkrTali parabola, romlis

Sesabamisi atomuri masa iyo 22. am faqtis axsnam mecnierebi miiyvana dask-

vnamde, rom arsebobs neonis atomebis ori saxesxvaoba, romelTac qimiuri

Tvisebebi erTnairi aqvT, magram gansxvavdebian masiT. dRevandeli termino-

logiiT mocemuli nivTierebis atomebis aseT saxesxvaobebs izotopebi

ewodeba. amrigad, tomsonis meTodiT, atombirTvis agebulebis garkvevamde

gacilebiT adre, eqsperimentulad iqna dadgenili izotopebis arseboba.

izotopebis atomuri masebis

ufro zusti gansazRvra xdeba as-

toni xelsawyos saSualebiT, ro-

melsac masspeqtrografi ewo-

deba. astonis masspeqtrografSi

ionebis kona gadis Tanmimdevro-

biT urTierTmarTob eleqtrul da magnitur velebSi, romelTa mimarTule-

bebi isea SerCeuli, rom ionebs xrian urTierTsawinaaRmdego mimarTulebiT.

rogorc eleqtrul, ise magnitur velSi erTnairi kuTri muxtis mqone

ionebi miT ufro Zlier ixrebian, rac naklebia maTi siCqare. amis gamo 1DD

diafragmebis saSualebiT miRebuli ionebis viwro kona eleqtrul velSi

gavlisas iSleba, xolo magnitur velSi gavlisas kvlav ikribeba,

`fokusirdeba~, viwro zolad, romelic diafragmis xvrelis gamosaxulebas

warmoadgens. sxva kuTri muxtis mqone ionebi sxva zols gvaZleven

(sur. 16.2). Tu ekranis nacvlad movaTavsebT fotofirs, gamJRavnebis Semdeg

52

masze miiReba sxvadasxva zolebi, amasTan TiToeuls Seesabameba mq–is gar-

kveuli mniSvneloba. fotofirze miRebuli suraTi analogiuria optikuri

wiriTi speqtrisa, amitom astonma mas masaTa speqtri uwoda, xolo

xelsawyos _ masspeqtrografi. masspeqtografis saSualebiT naCvenebi iqna,

rom qimiuri elementebis umravlesoba izotopebis narevs warmoadgens.

garda masspeqtografisa, damuxtuli nawilakebis moZraoba eleqtrul

da magnitur velebSi safuZvlad udevs mTeli rigi xelsawyoebisa da

danadgarebis moqmedebas, rogorebicaa eleqtronuli oscilografi,

eleqtronuli mikroskopi, eleqtronuli mikroproeqtori, ciklotroni,

magnitohidrodianmikuri generatori da sxv. gavecnoT mokled eleqtronuli

mikroskopisa da magnitohidrodinamikuri generatoris moqmedebis princips.

eleqtronuli oscilografi∗ eleqtronuli oscilofragi aris xelsawyo, romelic gamoiyeneba swrafad cvladi

eleqtruli rxevebis Sesaswavlad, am rxevebis vizualuri dakvirvebisa da registraciisaT-

vis.∗∗ misi mTavari nawilia eleqtronul-sxivuri milaki, romlis sqemac mocemulia 16.3

suraTze. minis vakuumuri balonis viwro nawilSi moTavsebulia cilindruli kaTodi (2),

romlis fskeri dafarulia eleqtronis gamosvlis mcire muSaobis mqone liTonis Txeli

feniT. (1) spiralSi denis gatarebis Sedegad igi varvardeba da axurebs kaTods, ris gamoc

misgan amoityorcneba eleqtronebi. (3) diafragma amotyorcnili eleqtronebidan gamoyofs

viwro konas – eleqtronul sxivs.∗∗∗ (2) kaTodsa da (4) anods Soris modebulia maRali Zab-

va, amitom eleqtronebi iZenen did siCqares ⎟⎠⎞

⎜⎝⎛

wmm710~ da (7) ekranze dacemisas,

∗ warmodgeba sityvebisgan oscillo (laT.) _ rxeva, virxevi da grapho (berZn.) _ vwer. ∗∗ mniSvnelovania aRiniSnos, rom oscilografis saSualebiT SesaZlebelia ara-eleqtruli sidideebis (mag. wnevis, temperaturis, simkvrivis da a.S.) rxevis Seswavlac, Tu am sidideebs specialuri gadamwodebis saSualebiT winaswar gardavqmniT eleqtrulad. ∗∗∗ sinamdvileSi, (3) diafragma warmoadgens uaryofiTad damuxtul cilindrs, romlis kedlebidan ganzidvis Sedegad eleqtronebi ikvreba viwro konad. mas mafokusirebeli cilindri ewodeba.

7

O

6

2

1

5

5

sur. 16. 3.

64

3

53

romelic dafarulia fluorescirebadi nivTierebiT, iwveven mis wertilovan naTebas.

anodidan gamosuli eleqtronuli sxivis gzaze moTavsebulia (5-5) da (6-6) brtyeli

kondensatorebi, romelTa firfitebi urTierTmarTobulia. pirveli kondensatori gadax-

ris sxivs horizontaluri mimarTulebiT (marcxniv da marjvniv), meore ki vertikaluri

mimarTulebiT (zeviT da qveviT). amitom maT xSirad uwodeben horizontalurad da verti-

kalurad gadamxrel firfitebs.

vTqvaT, vertikalurad gadamxrel firfitebze movdeT mudmivi Zabva, maSin

eleqtronuli sxivis mimarTuleba Seicvleba da mnaTi O wertili gadaadgildeba vertika-

lurad wiris gaswvriv. Tu firfitebze modebuli Zabva periodulad icvleba, maSin mnaTi

wertili daiwyebs rxevas vertikaluri wiris gaswvriv da Tu rxevis sixSire sakamaod di-

dia, ekranze davixavT naTel vertikalur zols, romlis sigrZe damokidebulia modebul

Zabvaze. maSasadame, am zolis sigrZis mixedviT SeiZleba ganisazRvros modebuli Zabvis

sidide, anu oscilografi SeiZleba ganvixiloT rogorc swrafmoqmedi voltmetri.

imisaTvis, rom rxevis xasiaTi gaxdes TvalsaCino, unda movaxdinoT misi gaSla

drois mixedviT. amisaTvis horizontalurad gadamxrel firfitebze unda movdoT e.w.

xerxiseburi Zabva, romlis dros garkveul Sua-

ledSi (sanam sxivi moZraobs horizontalurad ek-

ranis erTi kididan meoremde) izrdeba Tanabrad

(e.i. drois proporciulad), Semdeg ki myisierad

ubrundeba sawyis mdebareobas (sur. 16.4). aseT Zab-

vas gamSleli Zabva ewodeba. cxadia, mxolod gam-

Sleli Zabvis moqmedebiT mnaTi wertili imoZ-

ravebs Tanabrad horizontaluri wiris gaswvriv

ekranis erTi kididan meoremde, Semdeg ki myisi-

erad dabrundeba sawyis mdgomareobaSi.

davuSvaT, eleqtronul sxivze erTdroulad moqmedebs orive kondensatoris veli.

maSin, vertikalur rxevebs, romelsac iZleva (6-6) kondensatori, (5-5) gaSlis horizonta-

luri mimarTulebiT anu droSi (vinaidan gamSleli Zabva drois proporciulad izrdeba),

ris gamoc, Tu (6-6) kondensatorze modebuli Zabva sinusis kanonis mixedviT icvleba, mnaTi

wertili ekranze aRwers sinusoidas. didi sixSiris SemTxvevaSi mnaTi wertilis moZraobas

ver vamCnevT da ekranze davinaxavT uwyvet sinusoidalur wirs. rogorc wesi, vertika-

lurad gadamxrel firfitebze mosdeben xolme gamosakvlev xu Zabvas da oscilografis

ekranze miiReba am Zabvis droSi cvlilebis grafiki, romlis mixedviTac msjeloben

eleqtruli rxevebis xasiaTze.

Tu horizontalurad gadamxrel firfitebzec movdebT imave sixSiris cvlad sinu-

soidalur Zabvas, maSin gveqneba urTierTmarTob rxevaTa Sekreba (I tomi, $39) da imisda

mixedviT, Tu rogoria fazaTa sxvaoba, ekranze miiReba sxvadasxvanairad orientirebuli

wrfe an elifsi, xolo toli amplitudebis SemTxvevaSi _ wrewiri.

eleqtronis umniSvnelo masis gamo, eleqtronul sxivs praqtikulad ar gaaCnia

inercia Zalian swrafad cvladi Zabvis SemTxvevaSic ki (sxivis gadaxra zustad miyveba ve-

lis cvlilebas), rac saSualebas gvaZlevs oscilografis saSualebiT SeviswavloT swra-

fad cvladi eleqtruli procesebi. esaa eleqtronuli oscilografis mTavari Rirseba.

u

t

sur. 16.4

54

magnitohidrodinamikuri generatori

magnitur velSi damuxtuli nawilakebis moZraoba safuZvlad udevs magnito-

hidrodinamikuri generatoris (mhd generatoris) moqmedebas, romelic eleqtruli energiis

SedarebiT axali da sruliad gansxvavebuli wyaroa. misi gamartivebuli sqema mocemulia

16.5 suraTze. maRal temperaturamde (~3000K) gaxurebul da maSasadame Zlier ionizirebul

airis nakads atareben specialur arxSi, magnituri velis marTobulad (Cvens naxazze

magnituri velis induqciis wirebi mimarTulia naxazis sibrtyis marTobulad Cvensken).∗

lorencis Zalis moqmedebiT dadebiTi da uaryofiTi muxtebi (ZiriTadad dadebiTi ionebi

da Tavisufali eleqtronebi) moZrao-

ben airis nakadis marTobulad urTi-

erTsawinaaRmdego mimarTulebiT _ el-

eqtronebi 1 eleqtrodisaken, dadebiTi

ionebi _ 2-sken. Tu am eleqtrodebs

mivuerTebT raime momxmarebels, masSi

gaivlis eleqtruli deni.

mhd generatori metad perspeq-

tiuli denis wyaroa, vinaidan igi

uSualod gardaqmnis siTbur energias

eleqtrul energiad (meqanikur energi-

ad gardaqmnis gareSe).

Tavi II

eleqtromagnituri induqcia

$17. eleqtromagnituri induqciis movlena.

faradeis cdebi. lencis wesi.

eleqtromagnituri induqciis movlena aRmoaCina 1831 wels ingliselma

mecnierma faradeim da igi miekuTvneba yvela drois yvelaze ufro RirSesa-

niSnav aRmoCenaTa ricxvs. sakmarisia aRiniSnos, rom es aRmoCena safuZvlad

udevs msoflios yvela eleqtrosadguris generatorTa mowyobilobas, ro-

melTa meSveobiTac meqanikuri energia gardaiqmneba eleqtruli denis

energiad.

am movlenis aRmoCena SemTxveviTi ar yofila. mecnierebi, pirvel

rigSi faradei, msjelobdnen ase: Tu eleqtruli deni qmnis (aRZravs)

magnitur vels, xom ar SeiZleba arsebobdes am movlenis Sebrunebuli mxare

∗ ionizirebuli airis moZraoba emorCileba hidrodinamikis kanonebs, rac ganapirobebs gene-ratoris dasaxelebas.

airis

nakadi

1

2 sur. 16.5

55

_ magniturma velma Seqmnas eleqtruli deni. faradeim aCvena, rom garkveul

pirobebSi magnituri velic qmnis eleqtrul dens. kerZod, mravalricxovani

cdebis safuZvelze man daadgina:

yovelTvis, rodesac icvleba Caketili gamtari konturiT Se-

mosazRvruli farTobis gamWoli magnituri nakadi, konturSi

aRiZvreba eleqtruli deni, romelic iarsebebs magnituri nakadis

cvlilebis mTeli procesis ganmavlobaSi. am dens ewodeba

induqciuri deni∗ , xolo movlenas _ eleqtromagnituri induqciis

movlena.

ganvixiloT faradeis klasikuri cdebi, ro-

melTa saSualebiTac dadgenil iqna es movlena:

1. koWaSi, romelic galvanometrTanaa mierTe-

buli, vamoZraoT mudmivi magniti∗∗ (sur. 17.1). koWaSi

aRiZvreba induqciuri deni, romlis mimarTuleba

damokidebulia magnitis moZraobis mimarTulebaze,

xolo sidide _ magnitis moZraobis siswrafeze: rac

ufro swrafad vamoZravebT magnits koWaSi, miT

ufro didi induqciuri deni aRiZvreba masSi. rodesac magniti gaCerebulia

koWaSi, masSi deni ar aRiZvreba. es cda gviCvenebs, rom koWaSi deni aRiZvre-

ba mxolod misi xviebis gamWoli magnituri nakadis cvlilebis SemTxvevaSi.

kerZod, roca magniti Segvaqvs koWaSi, gamWoli magnituri nakadi izrdeba,

xolo roca gamogvaqvs _ mcirdeba. Tu magniti gaCerebulia koWaSi, mas

mudmivi magnituri nakadi ganWolavs da deni ar aRiZvreba.

2. aviRoT ori koWa, A da B . A koWa mierTebulia galvanometrTan, xo-

lo B koWa _ denis wyarosTan (sur. 17.2).∗∗∗ k CamrTvelis CarTvis an amor-

Tvis momentSi, agreTve masSi gamavali

denis gaZlierebis an Sesustebis Sem-

TxvevaSi, A koWaSi aRiZvreba induqciuri

deni. am SemTxvevaSic denis aRZvra da-

kavSirebulia gamWoli magnituri naka-

∗ induqcia (laT) _ aRZvra, aRgzneba. ∗∗ sainteresoa aRiniSnos, rom analogiur cdebs atarebda faradeis Tanamedrove, Sveica-rieli fizikosi koladoni, magram igi ver gaxda am didi aRmoCenis Tanaavtori imitom, rom galvanometri, romelTanac mierTebuli iyo koWas boloebi (galvanometris isris magnitis moqmedebisagan dacvis mizniT), gatanili iyo mezobel oTaxSi. ∗∗∗ faradeis cdebSi koWebi erTmaneTSi iyo Cadgmuli.

S

N

G

sur. 17.1

A B

G

sur. 17. 2

k

56

dis cvlilebasTan: k CamrTvelis CarTvisas deni B koWaSi izrdeba nu-

lidan raRac sididemde. Sesabamisad izrdeba mis mier Seqmnili magnituri

nakadi. es mzardi magnituri nakadi ganWolavs A koWis xviebs da aRZravs

masSi induqciur dens. k CamrTvelis amorTvisas xdeba piriqiT.

rogorc zemoT aRvniSneT, induqciuri denis sidide damokidebulia

magnituri nakadis cvlilebis siCqareze, magram ar aris damokidebuli

am cvlilebis mizezze. cvlileba SeiZleba gamowveuli iyos mag-

nitis moZraobiT, denis cvlilebiT, konturebis urTierTgadaadgi-

lebiT, konturis deformaciiT da sxv.

konturSi erTgvarovani magnetikis Setanisas induqciuri deni μ –jer

izrdeba, riTac dasturdeba, rom induqciur dens ganapirobebs ara H –is

cvlileba, romelic araa damokidebuli im garemos Tvisebebze, romelSic

magnituri veli iqmneba, aramed B –s cvlileba, romelic garemos μ

magnituri SeRwevadobis proporciulia ($25).

induqciuri denis mimarTuleba ganisazRvreba lencis wesiT. am wesis

Camosayalibeblad ganvixiloT Semdegi magaliTi. vTqvaT, mocemulia

erTmaneTSi Cadgmuli ori koWa, A da B (sur. 17.3; suraTze zedxedia gamo-

saxuli).

A koWaSi gavataroT 0I deni. rogorc viciT, am denis yovelgvari

cvlilebisas B koWaSi aRiZvreba induqciuri deni ( )I . cda gviCvenebs, rom

roca A koWaSi gamavali (induqciis gamomwvevi) deni izrdeba, B koWaSi

aRZrul induqciur dens aqvs misi sawinaaRmdego mimarTuleba (sur. 17.3 a),

xolo roca A koWaSi gamavali deni mcirdeba, B koWaSi aRZrul

induqciur dens aqvs misi Tanxvedrili mimarTuleba (sur. 17.3 b).

gavaanalizoT cdis Sedegi: A koWaSi gamavali denis zrdisas izrdeba

misi magnituri nakadic. am dros B koWaSi aRiZvreba sawinaaRmdego mimar-

Tulebis deni, romelic Ta-

visi magnituri nakadiT ewina-

aRmdegeba (akompensirebs) A

koWaSi gamavali denis magni-

turi nakadis zrdas. da piri-

qiT, A koWaSi gamavali denis

Semcirebisas mcirdeba misi

magnituri veli (magnituri na-

II

A A

B B

0I izrdeba 0I mcirdeba

a b sur. 17. 3

57

kadi), ris Sedegadac B koWaSi aRZrul induqciur dens aqvs iseTive

mimarTuleba, da maSasadame, Tavisi magnituri veliT ewinaaRmdegeba

(akompensirebs) A koWaSi gamavali denis magnituri velis Semcirebas. e.i.

induqciur dens aqvs

yovelTvis iseTi mimarTu-

leba, rom Tavisi magnituri

veliT ewinaaRmdegeba (akom-

pensirebs) im magnituri na-

kadis yovelgvar cvlile-

bas, romelic am denis aR-

Zvris mizezia.

17.4 suraTze gamosaxuli cdis Sedegi, romlis gaanalizebasac

mkiTxvels vTavazobT, saSualebas gvaZlevs lencis wesi Semdegnairad Ca-

movayaliboT (gavixsenoT, rom erTsaxeliani magnituri polusebi erTmaneTs

ganizidavs da piriqiT): induqciur dens aqvs yovelTvis iseTi

mimarTuleba, rom Tavisi magnituri veliT ewinaaRmdegeba mis

gamowvev moZraobas.

$18. induqciis eleqtromamoZravebeli Zala. faradeis kanoni

eleqtromagnituri induqciis movlenaSi induqciuri deni meoradi

efeqtia. pirveladia induqciis eleqtromamoZravebeli Zala, romelic aRiZv-

reba magnituri nakadis cvlilebis Sedegad. eleqtromagnituri induqciis

movlenis aRmoCenisTanave faradeim daadgina, rom induqciis em Zalis

sidide magnituri nakadis cvlilebis siCqaris proporciulia:

dtd

iφε ~ .

germanelma mecnierma helmholcma Teoriulad, energiis mudmivobis

kanonis safuZvelze, miiRo induqciis em Zalis gamosaTvleli formula.

gamoviyvanoT es formula.

vTqvaT mocemulia acbd Caketili konturi, romlis erTi gverdi _ ac

aris moZravi (sur. 18.1). k CamrTvelis CarTvisas konturSi gaivlis I deni

da denis wyaro raRac dt droSi Seasrulebs dtIdA ε= muSaobas. es muSaoba

ixarjeba gamtarebis gaTbobaze. gamoyofili siTbos raodenoba gamoiTvleba

joul-lencis kanoniT: RdtIdQ 2= . energiis mudmivobis kanonis Tanaxmad

S

N G

sur. 17. 4

S

N

N

SG

S

N

58

dQdA = , anu RdtIdtI 2=ε ,

saidanac R

I ε= , sadac R wredis

sruli winaRobaa.

ganvixiloT axla SemTxveva, ro-

ca konturi moTavsebulia erTgvaro-

van magnitur velSi, romlis induq-

ciis B veqtori mimarTulia konturis sibrtyis marTobulad Cvengan. maSin

konturis TiToeul gverdze imoqmedebs amperis Zala, magram vinaidan

moZravi ac gverdia, am Zalis moqmedebiT (romlis mimarTulebac ganisa-

zRvreba marcxena xelis wesiT) mxolod igi gadaadgildeba dt droSi ra-

Rac dx manZiliT da daikavebs ''ca mdebareobas. amperis Zalis sidide

IBacF = ( )90=α , amitom am gadaadgilebaze Sesrulebuli muSaoba

φIdIBdSdxIBacdxFdA ==⋅=⋅=' ,

sadac dxacdS ⋅= aris gamtaris gadaadgilebisas mis mier Semowerili

(daStrixuli) farTobi, xolo BdSd =φ _ am farTobis gamWoli magnituri

nakadi.

amrigad, am SemTxvevaSi denis wyaros muSaobis mxolod nawili

ixarjeba joulis siTbos gamoyofaze, nawili ki xmardeba gamtaris

gadaadgilebaze Sesrulebul 'dA muSaobas. amitom energiis mudmivobis

kanoni Semdegnairad daiwereba:

'dAdQdA += , an φε IdRdtIdtI += 2 ,

saidanac

R

dtd

I

φε −= . (18.1)

aseT saxes iRebs omis kanoni Caketili wredisaTvis. rogorc vxedavT,

mricxvelSi gaCnda axali wevri dtdφ

− . mas aucileblad unda qondes eleq-

tromamoZravebeli Zalis ganzomileba anu unda izomebodes voltebiT∗ da

radgan igi dakavSirebulia magnituri nakadis cvlilebasTan, bunebrivia

davuSvaT, rom swored is gamosaxavs induqciis eleqtromamoZravebel Zalas.

amrigad,

∗ amis uSualo dadgena SeiZleba magnituri nakadis erTeulis _ veberis ganmartebis safuZvelze ($ 4).

−

+ε

k

F

daa '

bcc '

I B⊕ dx

sur. 18.1

59

dtd

iφε −= . (18.2)

(18.2) formula gamosaxavs faradeis kanons. igi eleqtromagnituri

induqciis ZiriTadi kanonia. (18.2) formula Cven miviReT im kerZo

SemTxvevisaTvis, roca gamWoli magnituri nakadi Seicvala konturis

deformaciis Sedegad (acbda konturi dt drois ganmavlobaSi gadaiqca

''' bdaca konturad). magram SeiZleba damtkicdes, rom igi universaluria da

ar aris damokidebuli nakadis Secvlis mizezze.

(18.2) formulaSi minusi niSani lencis wesis gamomxatvelia. marTlac,

Tu nakadi izrdeba ⎟⎠⎞

⎜⎝⎛ > 0

dtdφ , aRZruli induqciis em Zala uaryofiTia ( )0<iε

da konturSi iseTi deni gaCndeba, romlis magnituri veli Seasustebs

(akompensirebs) am nakadis zrdas. da piriqiT _ Tu nakadi mcirdeba ⎟⎠⎞

⎜⎝⎛ < 0

dtdφ

,

aRZruli induqciis em Zala dadebiTia ( )0>iε da konturSi iseTi deni

gaCndeba, romlis magnituri veli am nakads aZlierebs (akompensirebs

nakadis Sesustebas).∗ amrigad, SeiZleba iTqvas, rom (18.2) formula

gamosaxavs faradeis da lencis gaerTianebul kanons.

Tu Caketili konturi Sedgeba N xviisagan da TiToeuls ganWolavs erTidaigive φ

magnituri nakadi, maSin sidides φψ N= nakadSebma (nakadTgadabmuloba) ewodeba. am

SemTxvevaSi eleqtromagnituri induqciis ZiriTadi kanoni iRebs saxes:

dtdN

dtd

iφψε −=−= .

Tu sxvadasxva xviebis gamWoli magnituri nakadebi gansxvavdebian erTamneTisagan, maSin

nφφφφψ ++++= ...321 . maSasadame, nakadTgadabmuloba aris sxvadasxva xviebis

gamWoli magnituri nakadebis jami.

faradeis kanoni saSualebas gvaZlevs mivceT meorenairi ganmarteba

magnituri induqciis nakadis erTeuls SI sistemaSi _ vebers:

veberi konturiT SemosazRvruli farTobis gamWoli iseTi

magnituri induqciis nakadia, romlis cvlilebiT 1 wamSi

konturSi aRiZvreba 1 voltis toli induqciis em Zala.

∗ faqtiurad, (18.2) formulaSi minusi niSani miuTiTebs imas, rom induqciis emZ mimarTulia im emZ-is sapirispirod, romelsac gvaZlevs deni wyaro.

60

$19. induqciis eleqtromamoZravebeli Zalis aRZvris meqanizmi

gavarkvioT induqciis eleqtromamoZravebeli Zalis buneba, misi

aRZvris meqanizmi.

induqciis eleqtromamoZravebeli Zala aRiZvreba or SemTxvevaSi:

1. roca gamtari moZraobs mudmiv magnitur velSi (Zalwirebis marTobulad)

da

2. roca gamtari uZravia, magram igi moTavsebulia cvlad magnitur velSi.

1. ganvixiloT jer pirveli SemTxveva, vinaidan igi araviTari axali ideis

Semotanas ar saWiroebs da misi garkveva martivad xdeba Cvens mier kursis

aTvisebis procesSi ukve SeZenili codnis safuZvelze. vTqvaT, l sigrZis

liTonis ab Rero moZraobs ox RerZis gaswvriv υ siCqariT mudmiv

erTgvarovan magnitur velSi, romelic mimarTulia oz RerZis gaswvriv

(sur. 19.1). liTonTa eleqtrogamtarobis klasikuri eleqtronuli Teoriis

Tanaxmad, igi Sedgeba dadebiTad damuxtuli ionebisagan, romlebic

moTavsebulia kristaluri mesris kvanZebSi (`bmuli muxtebi~) da

Tavisufali eleqtronebisagan, romlebic qaosurad moZraoben kvanZebs

Sorisis sivrceSi (Itomi; $137). gamtarTan erTad imave υ siCqariT

imoZraveben misi dadebiTi da uaryofiTi muxtebi, amitom TiToeul maTganze

imoqmedebs lorencis Zala ($13).

[ ]BqF υ= .

es Zala dadebiTi ionebis moZarobas

gamtarSi, cxadia, ver gamoiwvevs. eleqtronebi ki

daiwyeben moZraobas a -dan b -sken.∗ amis Sedegad

b boloze aRmoCndeba maTi siWarbe, xolo a

boloze _ maTi nakleboba, anu dadebiTi muxtebis

siWarbe. es gamoiwvevs gamtaris boloebs Soris

potencialTa sxvaobis warmoqmnas, rac induqciis em Zalis zoma iqneba.

gamoviTvaloT induqciis em Zala. lorencis Zalis moqmedebiT

gamowveuli dadebiTi da uaryofiTi muxtebis gancalkeveba gamtarSi warmo-

Sobs a bolodan b -sken mimarTul eleqtrul vels, romlis daZabuloba

lE ba ϕϕ −= . amitom TiToeul eleqtronze imoqmedebs eleqtruli Zala

∗ eleqtronebis qaosuri moZraoba am TvalsazrisiT araviTar efeqts ar iZleva, vinaidan maTze moqmedi lorencis Zalis saSualo mniSvneloba nulis tolia.

-

+

__

++

+

-

B b

aυ

sur. 19. 1.

61

EqF = , romelic mimarTulia magnituri Zalis (lorencis Zalis)

sawinaaRmdegod. wonasworoba maSin damyardeba, roca eleqtruli Zala

gautoldeba magniturs:

[ ]BqEq υ= , anu [ ]BE υ= ,

xolo sididiT ( )90== αυBE .

lorencis Zala, axdens ra dadebiTi da uaryofiTi muxtebis

gancalkevebas, amoZravebs eleqtronebs eleqtruli Zalebis moqmedebis

sawinaaRmdegod, anu igi warmoadgens gare Zalas. ganmartebis Tanaxmad, ab

ubanze moqmedi em Zala (romelic gamtaris moZraobiTaa gamowveuli da

maSasadame, warmoadgens induqciis em Zalas) tolia im muSaobisa, romelsac

gare Zala asrulebs am ubanze erTeulovani dadebiTi muxtis gadasatanad

eleqtrostatikuri Zalebis moqmedebis sawinaaRmdegod e.i.

BlElabi υϕϕε −=−=−= .

magram gamtaris moZraobis siCqare dtdx

=υ , amitom

dtdSBl

dtdxBi −=⋅−=ε ,

sadac ldxdS = aris gamtaris mier dt droSi Semowerili farTobi, xolo

φdBdS = _ induqciis nakadi am farTobSi. amitom, sabolood

dtd

iφε −= ,

rac gamoxatavs eleqtromagnituri induqciis ZiriTad kanons.

Tu ab gamtari warmoadgens Sekruli konturis nawils, maSin

magnitur velSi misi moZraobis dros konturSi gaivlis induqciuri deni.

2. ganvixiloT axla meore SemTxveva, roca gamtari (konturi) uZravia,

xolo magnituri veli _ cvladi. am SemTxvevaSi induqciuri denis

warmoSobis asaxsnelad zemoT ganxiluli msjeloba ar gamodgeba, radgan

uZrav muxtebze lorencis Zala ar moqmedebs. magram uZrav muxtebze

moqmedebs eleqtruli veli, rac gvafiqrebinebs, rom cvladi magnituri

veli warmoSobs eleqtrul vels, romelic moqmedebs eleqtrul muxtebze,

mohyavs isini mimarTul moZraobaSi da ganapirobebs induqciur dens. am

daskvnamde pirvelad ingliseli mecnieri maqsveli mivida, riTac gamoav-

lina velis axali fundamenturi Tviseba _ droSi cvlilebisas

magnituri veli warmoqmnis eleqtrul vels. es axali idea

62

safuZvlad daedo maqsvelis saxelganTqmul eleqtromagnitur Teorias, _

warmoadgens mis pirvel ZiriTad debulebas.

xazgasmiT unda aRiniSnos, rom magnituri ve-

lis cvlilebis Sedegad aRZrul eleqtrul vels

aqvs sul sxva struqtura, vidre eleqtrostatikurs,

anu uZravi eleqtruli muxtebis mier Seqmnil eleq-

trul vels. igi ar aris uSualod dakavSirebuli

eleqtrul muxtebTan da maSasadame misi Zalwirebi

ar SeiZleba maTze iwyebodes da bolovdebodes. isi-

ni saerTod arsad ar iwyebian da ar bolovdebian,

aramed, magnituri induqciis wirebis msgavsad, Sek-

rul wirebs warmoadgenen. aseT velebs, romelTa

Zalwirebi Sekruli wirebia (da romelTac, maSasa-

dame, ar gaaCniaT wyaro) Cven grigaluri velebi

vuwodeT ($4). amrigad, magnituri velis cvlilebis Sedegad miRebuli

eleqtruli veli grigaluri velia. grigaluri velis Zalwirebi

moicavs magnituri induqciis wirebs (sur. 19.2). maTi mimarTuleba

ganisazRvreba lencis wesiT. kerZod, magnituri induqciis zrdisas ⎟⎠⎞

⎜⎝⎛ > 0

dtdB

eleqtruli Zalwirebis mimarTuleba B -s mimarTulebasTan qmnis marcxena

xraxns (dakavSirebulia marcxena burRis wesiT)

eleqtrostatikurisagan gansxvavebiT, grigalur velSi Caketil wirze

Sesrulebuli muSaoba nulis toli ar aris, vinaidan gzis yvela ubanze

Zalisa da gadaadgilebis mimarTuleba erTmaneTs emTxveva da muSaobis

erTi da igive niSani aqvs.

rogorc viciT (Itomi, $129), Caketil konturSi (wredSi) eleqtromamoZ-

ravebeli Zala aRiZvreba mxolod maSin, roca konturSi muxtebze moqmedebs

araeleqtrostatikuri bunebis anu gare Zala. gare ZalTa velis daZabu-

lobis cirkulacia Caketili konturis gaswvriv konturSi moqmedi em Zalis

tolia.

uZrav gamtar konturSi magnituri nakadis cvlilebiT aRZruli griga-

luri eleqtruli veli, cxadia, araeletrostatikuri, gare ZalTa daZabu-

lobis velia. Tu am velis daZabulobas aRvniSnavT E -Ti, misi cirkulacia

aRniSnuli konturis gaswvriv konturSi moqmedi induqciis em Zalis toli

iqneba:

0>dtBd

sur. 19. 2.

B

E

63

( )t

dlEldEl

ll

i ∂∂

−=== ∫∫φε ∗ (19.1)

fukos deni. zedapiruli efeqti. induqciuri deni aRiZvreba ara marto xazovan gam-

tarebSi, aramed masiur gamtarSic, Tu mas vamoZarvebT (mudmiv) magnitur velSi an

movaTavsebT cvlad magnitur velSi. masiur gamtarebSi aRZruli induqciuri denebi maTi

Semswavleli frangi mecnieris fukos saxels atarebs. fukos denebis wirebi gamtaris

SigniT Sekrulia, amitom maT grigalur denebsac uwodeben.

fukos deni, iseve, rogorc induqciuri deni xazovan gamtarebSi, emorCileba lencis

wess: misi magnituri veli isea mimarTuli, rom ewinaaRmdegena missave gamomwvev moZraobas,

anu amuxruWebs mas. marTlac, Tu CaurTveli eleqtromagnitis polusebs Soris spilenZis

masiuri qanqara asrulebs praqtikulad miulevad rxevebs, eleqtromagnitis CarTvis Semdeg

masze iwyebs moqmedebas damamuxruWebeli Zala da igi swrafad Cerdeba. es faqti

gamoiyeneba sxvadasxva xelsawyoebis moZravi nawilebis (vTqvaT, galvanometris isris)

dasawynareblad (dempfirebisaTvis). Tu spilenZis firfitas gavukeTebT radialur

Wrilebs (savarcxliseburad), fukos denebi Sesustdeba da damamuxruWebeli efeqtic mcire

iqneba.

vinaidan masiuri gamtaris winaRoba mcirea, masSi aRZruli fukos denebi did

mniSvnelobas aRwevs. Sesabamisad, didia gamoyofili joulis siTboc. garkveuli pirobebis

dacvisas mas liTonis gadnobac SeuZlia. es faqti gamoyenebulia sainduqcio

metalurgiul RumelSi. am Rumelis upiratesobaa is, rom dnoba SeiZleba warimarTos

vakuumSi, rac iZleva zesufTa masalebis miRebis SesaZleblobas.

xSir SemTxvevaSi fukos denebis moqmedeba mavnea, vinaidan masiuri gamtaris

gaTbobaze daxarjuli energia fuWia. am SemTxvevaSi fukos denebis Sesamcireblad vTqvaT,

transformatoris gularebs an generatoris Ruzebs amzadeben ara masiuri liTonisagan,

aramed erTmaneTisagan ganmxoloebuli Txeli firfitebisagan an Reroebisagan. am teqno-

logiuri uxerxulobis Tavidan asacileblad, anu masiuri Ruzebisa da gularebis da-

samzadeblad, bolo aTwleulebSi farTod iyeneben feritebs,∗∗ rac imiTaa ganpirobebuli,

rom maTSi SesaniSnavi magnituri Tvisebebi (didi magnituri SeRwevadoba da sxv.)

Serwymulia maRal eleqtrul winaRobasTan (107-1011-jer ufro meti, vidre fero-

magnetikebSi). es ki minimumamde amcirebs fukos denebs. aqve SevniSnoT, rom feritebi war-

moadgenen myar xsnarebs, romelic Sedgeba rkinis oqsidisa ( )32OFe da erTi an ramodenime

liTonis oqsidisagan. esaa upirveles yovlisa magnetiti 32OFeOFe . garda amisa, feritis

SemadgenlobaSi SeiZleba Sediodes liTiumis, TuTiis, nikelis, kobaltis, tyviis,

∗ magnituri nakadi, romelic figurirebs eleqtromagnituri induqciis ZiriTad kanonSi SeiZleba Seicvalos sxvadasxva gziT: konturis deformaciiT, misi mdebareobis SecvliT,

magnituri velis cvlilebiT da sxva. sruli warmoebuli dtdφ

yvelafer amas iTvaliswi-

nebs. kerZo warmoebuli t∂

∂φ miuTiTebs, rom nakadis cvlileba gamowveulia mxolod mag-

nituri velis cvlilebiT. ∗∗ feritebis teqnikuri gamoyenebis are gacilebiT farToa, romelzedac Cven ar SevCerde-biT.

64

spilenZis, manganumisa da sxva liTonebis Jangeulebi (oqsidebi). martivi feritebis zogad

formulas aqvs saxe 32OMeOFe , sadac Me _ erTerTi zemoT CamoTvlili liTonTagania.

grigaluri denebi aRiZvreba sadenebSic, Tu maTSi gadis cvladi deni. amasTan, es

denebi, iseve rogorc yovelgvari induqciuri deni, cxadia, emorCileba lencis wess. am

wesidan gamomdinareobs, rom denis zrdisas da Semcirebisas, gamtarSi aRZrul grigalur

denebs aqvs sur. 19.3-ze naCvenebi mimarTuleba. suraTidan Cans, rom orive SemTxvevaSi

grigaluri denebi isea mimarTuli, rom ewinaaRmdegebian ZiriTadi denis cvlilebas

gamtaris SigniT da xels uwyoben mis cvlilebas gamtaris zedapiris axlos. maSasadame,

cvladi denisaTvis gamtaris Siga nawilebis winaRoba ufro metia vidre gare nawilebis.

amitom, cvladi denis simkvrive gamtaris ganikveTis sxvadasxva nawilebSi sxvadasxvaa _

igi maqsimaluria gamtaris zedapirze da minimaluria mis RerZze. Tu denis Zala icvleba

sakmaod didi sixSiriT, maSin deni gamtaris SigniT praqtikulad moispoba da mTlianad

gaivlis mis zedapirze (ufro zustad, Txel zedapirul fenaSi). am movlenas ewodeba

zedapiruli efeqti an skinefeqti (inglisurad skin _ tyavi). winaRobis gamosaTvleli

formula SlR ρ= am SemTxvevaSi gamousadegaria, vinaidan masSi Sedis mTeli gamtaris

ganikveTi. radgan skinefeqtis gamo gamtaris Siga nawilebi maRali sixSiris denebis

gatarebaSi monawileobas ar iReben, aseTi denebis gadasacemad mTlian gamtars cvlian

milisebri gamtariT, an didi raodenobis izolirebuli wvrili mavTulebisagan dawnuli

sadeniT, romlis sasargeblo zedapiris fardoba usargeblo moculobasTan didia.

masiur gamtarSi maRali sixSiris denis xanmokle gatarebisas, skinefeqtis gamo,

gamtaris zedapiruli fena Zalian xurdeba, maSin rodesac gamtaris danarCeni nawili

rCeba civi. amazea damyarebuli zedapiruli wrTobis meTodi: Tu maRali sixSiris deniT

gaxurebul foladis detals, vTqvaT, kbilanas an

muxla lilvs swrafad gavacivebT (CavuSvebT

zeTSi), detalis Txeli zedapiruli fena

aRmoCndeba nawrTobi, xolo Siga nawili _

uwrTobi, plastikuri. Sedegad mTlianad detali

iqneba medegi rogorc zedapiruli cveTisadmi,

aseve dartymiTi datvirTvebisadmi. zedapiruli

wrTobis es meTodi imiTaa SesaniSnavi, rom

cvladi denis sixSiris SecvliT SeiZleba

movaxdinoT wrToba nebismier saWiro siRrmemde

(zedapiruli fenis sisqe, romelSic gadis deni,

ukuproporciulia kvadratuli fesvisa denis

cvlilebis sixSiridan).

0>dtdI

I I

0<dtdI

a b

sur. 19.3

65

$20. urTierTinduqcia

urTierTinduqcia (iseve, rogorc TviTinduqcia) eleqtromagnituri in-

duqciis kerZo saxea.

vTqvaT, mocemul konturSi gadis deni. am denis yovelgvari cvlile-

bisas Seicvleba misi magnituri nakadic, romelic ganWolavs sxva (mezobel)

konturebs da eleqtromagnituri induqciis kanonis Tanaxmad aRZravs maTSi

induqciur dens. am dens ewodeba urTierTinduqciis deni, xolo movlenas _

urTierTinduqcia.

amrigad, urTierTinduqcia ewodeba mocemul konturSi denis

cvlilebis Sedegad sxva (mezobel) konturebSi induqciuri denis

aRZvras.∗

ganvixiloT erTmaneTis maxloblad mdebare ori Caketili konturi

(sur. 20.1). vTqvaT pirvel konturSi gadis 1I deni. igi qmnis magnitur vels

( )1B , romlis ZalZirebis nawili ganWolavs meore konturs (meore konturiT

SemosazRvrul farTobs). meore konturis gamWoli es magnituri induqciis

nakadi aRvniSnoT 21φ -iT. igi, garda misi Semqmneli 1I denisa, damokidebulia

konturebis formaze, zomaze, urTierTganlagebaze da im garemos magnitur

Tvisebebze ( )μ , romelSic isinia

moTavsebuli. 1I denis yovelgvari

cvlilebisas Seicvleba 21φ -ic da

eleqtromagnituri induqciis kanonis

Tanaxmad, meore konturSi ariZvreba

urTierTinduqciis em Zala

dt

d 2121

φε −= . (20.1)

radganc 21ε -is aRZvra gamowveu-

lia 1I denis cvlilebiT, upriania igi gamovsaxoT am cvlilebis saSuale-

biT: magnituri induqciis nakadis ganmartebis Tanaxmad 121 ~ Bφ , xolo bio-

savar-laplasis kanonis Tanaxmad 11 ~ IB an

12121 IL ⋅=φ . (20.2)

proporciulobis 21L koeficients ewodeba pirveli da meore konturebis

urTierTinduqciis koeficienti an urTierTinduqciuroba. igi damokide-

∗ urTierTinduqciis movlenasTan Cven ukve gvqonda saqme faradeis mier eleqtromagnituri induqciis movlenis aRmoCenasTan dakavSirebuli cdis ganxilvis dros (sur. 19.2).

1I

1 2

sur. 20.1.

66

bulia konturebis formaze, zomaze, urTierTganlagebaze da garemos

magnitur SeRwevadobaze ( )μ . Tu zemoT CamoTvlili parametrebi mudmivia,

maSin constL =21 da

( )dtdILIL

dtd 1

2112121 −=−=ε . (20.3)

davuSvaT axla, rom pirvel gamtarSi deni ar gadis, xolo meore gam-

tarSi gamavali denis Zala aris 2I . zemoT Catarebuli msjelobis zustad

gameorebiT miviRebT, rom 2I denis yovelgvari cvlilebisas pirvel

konturSi aRiZvreba urTierTinduqciis em Zala

dtdIL 2

1212 −=ε (20.4)

aqac, 12L pirveli da meore konturebis urTierTinduqciis koeficientia.

mtkicdeba, rom Tu konturebi moTavsebulia araferomagnitur

garemoSi,

2112 LL = .

urTierTinduqciis koeficientis fizikuri arsis gasagebad SeiZleba

visargebloT (20.3) an (20.4) formuliT. Tu (20.3)-Si davuSvebT, rom wm

a11 =dtdI

,

miviRebT: 2121 L−=ε . maSasadame, ori konturis urTierTinduqciis koe-

ficienti ricxobrivad im eleqtromamoZravebeli Zalis tolia, romelic

aRiZvreba erTerT konturSi, rodesac meoreSi denis Zala icvleba 1

amperiT wamSi.

urTierTinduqciis koeficientis erTeuli SI sistemaSi aris henri.

misi dadgena xdeba imave (20.3) formulidan. Tu am formulaSi davuSvebT

rom wm

a11 =dtdI

da v121 =ε , maSin ( )hnhenri1a

wmv1 =

⋅=21L . maSasadame, henri

aris iseTi ori konturis urTierTinduqciis koeficienti,

romelTagan erT-erTSi denis Zalis Secvla wm

a1 -iT meoreSi

aRZravs 1 volt urTierTinduqciis eleqtromamoZravebel Zalas.

urTierTinduqciis movlenas didi gamoyeneba aqvs teqnikaSi. kerZod es

movlena safuZvlad udevs eleqtro da radioteqnikaSi farTod

gavrcelebul eleqtrul mowyobilobas transformators, romelic

gamoiyeneba cvladi denis Zabvis Sesacvlelad.

67

$21. TviTinduqcia

mocemul konturSi denis cvlilebisas warmoqmnili cvladi

magnituri nakadi ganWolavs ara marto mezobel konturebs (rac

ganapirobebs urTierTinduqcias), aramed TviT am kontursac (Tavis Tavsac)

da masSic aRZravs induqciur dens. am dens, romelic aRiZvreba konturSi

masSive gamavali denis cvlilebis Sedegad, ewodeba TviTinduqciis deni,

xolo movlenas _ TviTinduqcia, amrigad,

TviTinduqcia ewodeba mocemul konturSi denis cvlilebis Sedegad

amave konturSi induqciuri denis aRZvras.∗

TviTinduqciis eleqtromamoZravebeli Zala, cxadia gamoiTvleba imave

formuliT, riTac urTierTinduqciis em Zala, magram vinaiadan axla ori

konturis nacvlad gvaqvs erTi konturi, indeqsebi aRar aris saWiro da

gveqneba:

dtd

Sφε −= (21.1)

aqac, Tu gaviTvaliswinebT, rom B~φ , xolo IB ~ , gveqneba I~φ an

LI=φ (21.2)

L -s uwodeben TviTinduqciis koeficients an konturis induqciurobas.

igi damokidebulia konturis zomaze, formaze da im garemos magnitur SeR-

wevadobaze, romelSic konturi imyofeba,∗∗ Tu es parametrebi mudmivia,

maSin constL = da miviRebT:

( )dtdILLI

dtd

S −=−=ε (21.3)

(21.3) formula gviCvenebs, rom TviTinduqciis emZ proporciulia gam-

tarSi gamavali denis cvlilebis siCqarisa. amave formulidan davadgenT

induqciurobis fizikur arss:

Tu wm

a1=dtdI

, maSin SL ε= . maSasadame, konturis induqciuroba ricxob-

rivad im em Zalis tolia, romelic aRiZvreba konturSi masSi gamavali

denis wm

a1 -iT cvlilebisas.

induqciurobis erTeuli SI sistemaSi aris henri (hn). (21.3)-dan gamom-

dinareobs rom

∗ TviTinduqciis movlenasTan Cven ukve gvqonda saqme skinefeqtis ganxilvis dros: gamtarSi denis cvlilebisas aRZruli wriuli denebi TviTinduqciis denebia. ∗∗ am TvalsazrisiT induqciuroba analogiuria gancalkevebuli gamtaris tevadobisa.

68

1a

wm1vhn1

⋅= .

maSasadame, henri aris iseTi konturis induqciuroba, romel-

Sic aRiZvreba 1 voltis toli TviTinduqciis em Zala, Tu masSi

deni 1 wm-Si icvleba 1 amperiT.

TviTinduqciis movlena did rols TamaSobs eleqtroteqnikasa da

radioteqnikaSi. wredis induqciuroba arsebiT gavlenas axdens wredSi

cvladi denis gavlaze ($37).

gamoviTvaloT grZeli solenoidis induqciuroba. davuSvaT, mocemulia solenoidi,

romlis sigrZe aris l , xolo xviaTa ricxvi N . CavTvaloT, rom solenoidis sigrZe

bevrad metia xviis radiusze. (6.9) formulis Tanaxmad, aseTi solenoidis RerZze magnituri

velis induqcia nIB 0μ= , sadac 0μ aris magnituri mudmiva, I _ solenoidSi gamavali

denis Zala, xolo lNn = _ xviaTa ricxvi sigrZis erTeulze. $25-Si Cven vnaxavT, rom

nivTierebis Setana magnitur velSi, am ukanasknels μ -jer zrdis, sadac μ _ mocemuli

nivTierebis magnituri SeRwevadobaa. amitom, Tu solenoidi `avsebulia~ nivTierebiT, maSin

magnituri velis induqcia mis RerZze nIB μμ0= . am SemTxvevaSi solenoidis TiToeuli

xviis gamWoli magnituri induqciis nakadi nISBS μμφ 0' == , xolo mTeli solenoidis

gamWoli sruli nakadi (nakadTSebmuloba) nISNN μμφφ 0'== .

Tu solenoidSi deni icvleba, Seicvleba misi gamWoli magnituri nakadic, ris

gamoc solenoidSi aRiZvreba TviTinduqciis em Zala

dtdInSN

dtd

S μμφε 0−=−= . (21.4)

am formulis Sedareba (21.3)-Tan gvaZlevs:

VnnSNL 200 μμμμ == , (21.5)

sadac SlV = _ solenoidis moculobaa. rogorc miRebuli formula gviCvenebs,

solenoidis induqciuroba damokidebulia mis formasa da zomaze ( )Vn, da im nivTierebis

magnitur SeRwevadobaze, romelic avsebs solenoids. (21.5) formulidan gamomdinareobs

agreTve, rom denis Zalis cvlilebis erTnairi siCqaris dros didi raodenobis xveulebis

Semcveli koWaSi ufro didi TviTinduqciis em Zala aRiZvreba, vidre imave koWadan

damzadebul erT xveulSi. amitom, didi raodenobis xveulebis Semcvel koWas, rogorc

wesi, sainduqcio koWas uwodeben. Tu eleqtruli wredi Seicavs sainduqcio koWas (koWebs)

amboben, rom mas gaaCnia induqciuroba.

TviTinduqciis dens (em Zalas), lencis wesis Tanaxmad, yovelTvis

iseTi mimarTuleba aqvs, rom igi ewinaaRmdegeba konturSi gamavali

ZiriTadi denis rogorc gazrdas, aseve Semcirebas. amitomaa, rom wredSi,

69

romelsac gaaCnia induqciuroba, CarTvisas denis Zala uceb ki ar aRwevs

Tavis maqsimalur mniSvnelobas, aramed TandaTan izrdeba da garkveuli tΔ

drois Semdeg xdeba maqsimaluri (sur. 21.1. a). aseve, gamorTvisas, denis

Zala myisierad ki ar xdeba nulis toli, aramed garkveuli tΔ drois

Semdeg (sur. 21.1. b).

analogiur movlenas hqonda adgili meqanikaSi sxeulis inerciis sa-

xiT. SeuZlebeli iyo sxeulis siCqaris myisierad gazrda an Semcireba. maSa-

sadame, eleqtrul wredSi TviTinduqciis movlena analogiuria inerciis

movlenisa meqanikaSi. amboben, rom magnituri veli inerciiT aris

aRWurvili.

wredis CarTvisa da amor-

Tvisas aRZrul TviTinduqciis

denebs CarTvisa da amorTvis eq-

stradenebs uwodeben.∗ amorTvis

eqstradeni iwvevs wredis gan-

rTvad ubanze (amomrTvelze,

Camrazze) naperwklis warmoq-

mnas.∗∗

TviTinduqciis movlenas davakvirdeT Semdegi cdiT: SevadginoT wredi mudmivi denis

wyarosa da ori paraleluri Stosagan. erT StoSi mimdevrobiT CavrToT R winaRoba da

naTura (1), meore StoSi ki _ L koWa da iseTive naTura (2) (sur. 21.2). k CamrTveliT

wredis Sekvrisas pirveli naTura erTbaSad miaRwevs maqsimalur sikaSkaSes, meore naTura

ki _ TandaTan. es imitom xdeba, rom L koWaSi aRiZvreba TviTinduqciis em Zala, ris gamoc

∗ Tu wredis induqciuroba didia, amorTvis eqstradeni SeiZleba bevrad meti aRmoCndes gamorTvamde wredSi gamaval (ZiriTad) denze. ∗∗ wredis didi induqciurobis SemTxvevaSi, misi swrafi ganrTvisas, aRiZvreba didi TviT-induqciis em Zala, romelic iwvevs amomrTvelis kontaqtebs Soris arsebuli dieleq-trikis (sahaero RreCos) garRvevas (Itomi, $111) da naperwklis warmoqmnas. naperwkali azianebs kontaqtebs, amitom mZlavri denebis SemTxvevaSi, amorTvis win dens TandaTanobiT asusteben, TviT amomrTvels ki aTavseben karg dieleqtrikSi, magaliTad zeTSi.

t

I I

ttΔ tΔ

a b

sur. 21.1

L

1

2

R

k

L

'I R

k I A

b c

da

sur. 21.2. sur. 21.3.

I

70

denis Zala masSi maqsimalur sidides aRwevs garkveul tΔ drois Semdeg.

wredis ganrTvisas TviTinduqciis movlenis dasakvirveblad isev ganvixiloT

mudmivi denis wyaros da ori paraleluri Stosagan Sedgenili wredi. erT StoSi CavrToT

L koWa, meoreSi _ R winaRoba da ampermetri. k CamrTveliT wredis ganrTvisas denis

wyaros kavSiri StoebTan wydeba. am dros ikvreba zeda ( abcd ) wredi, romelsac `kvebavs~

L koWaSi aRZruli TviTinduqciis em Zala. igi xels uwyobs pirvandeli denis

SenarCunebas, amitom ganrTvis momentSi ampermetrSi gaivlis 'I deni (wyvetili isari),

romelsac eqneba ganrTvamde arsebuli I denis sapirispiro mimarTuleba (uwyveti isari).

Tu ad da bc StoebSi CavrTavT naTurebs, isini abcd wredSi mimdevrobiT aRmoCndeba

CarTuli, amitom orive naTura nelnela, magram erTdroulad Caqreba.

$22. denis Zalis cvlileba wredis CarTvisa da amorTvis dros

wina parafrafSi aRvniSneT, rom wredis CarTvisa da amorTvis dros aRiZvreba

TviTinduqciis denebi (CarTvisa da amorTvis eqstradenebi), ris gamoc denis Zalis

cvlileba wredSi gansxvavdeba imisagan, rac iqneboda TviTinduqcias rom ar qondes

adgili.

davadginoT denis Zalis cvlilebis kanoni wredis CarTvisa da amorTvis dros.

ganvixiloT wredi, romelSiac CarTulia L induqciuroba, R winaRoba da denis

wyaro, romlis em Zalaa ε (sur. 22.1). k CamrTvelis saSualebiT denis wyaro SeiZleba

CavrToT an amovrToT ise, rom wredi darCes Sekruli.

davadginoT jer denis Zalis cvlilebis xasiaTi wredis ganrTvis

dros.

vTqvaT, k CamrTveli mierTebulia a kontaqtze, maSin wredSi gadis deni

R

I ε=0 . (22.1)

(denis wyaros Siga winaRoba ugulebelvyofilia). sawyis 0=t momentSi

gamovrToTYdenis wyaro, risTvisac k CamrTveli mivuerToT b kontaqts. denis wyaros

gamorTva gamoiwvevs 0I denis Semcirebas da wredSi aRiZv-

reba TviTinduqciis em Zala Sε , romelic ganapirobebs

amorTvis eqstradens (romelsac ZiriTadi denis mimar-

Tuleba aqvs).

RdtdIL

RI S −==

ε.

cvladTa gancalkevebis Semdeg gveqneba:

dtLR

IdI

−=

movaxdinoT integreba 0-dan t -mde (roca t =0, maSin 0II = )

k

b

a

L

R

ε

sur. 22.1.

71

tLR

IIdt

LR

IdI tI

I

−==→−= ∫∫00

ln0

,

saidanac, potencirebis Semdeg miviRebT:

tLR

eII⋅−

= 0 (22.2)

(22.2) formulis grafiki (sur. 22.2a) savsebiT eTanxmeba Cvens Tvisobriv daskvnas

wredis amorTvisas denis TandaTan da ara mkveTrad Semcirebis Sesaxeb. am formulis

Tanaxmad, amorTvisas deni eqsponencialurad mcirdeba, anu xdeba nulis toli usasrulod

didi drois Semdeg. sinamdvileSi ki, Cveulebrivi wredebis SemTxvevaSi, deni mcirdeba

sakmaod swrafad da miT ufro, rac metia R da naklebia L . kerZod, Tu wredis

induqciuroba 0=L , maSin 0=I , e.i. denis Zala am SemTxvevaSi nulamde ecema myisve.

dro τ , romlis ganmavlobaSic denis Zala Semcirdeba orjer ⎟⎠⎞

⎜⎝⎛ =

20I

I , (24.2)-is

Tanaxmad tolia:

RL

RL 7,02ln ≈=τ .

Tu magaliTad, 1,0=L henris, xolo 7,0=R oms, maSin 1,0=τ wams. winaRobis

gazrdisas, an L -is Semcirebisas, es dro sagrZnoblad mcirdeba.

axla davadginoT denis Zalis cvlilebis xasiaTi wredis CarTvis

dros. k CamrTveli gadavrToT a kontaqtze anu wredSi CavrToT denis wyaro, romlis em

Zalaa ε . vidre deni wredSi ar miaRwevs (22.1) formuliT gamosaxul damyarebul

mniSvnelobas, masSi, garda CarTuli eleqtromamoZravebeli Zalisa, imoqmedebs denis

zrdis Sedegad aRZruli TviTinduqciis em Zala Sε , amitom, omis kanonis Tanaxmad, denis

Zala wredSi iqneba:

dtdI

RLI

RI S −=

+= 0

εε,

saidanac

dtLR

IIdI

=−0

,

vinaidan constI =0 , amitom ( )IIddI −−= 0 da sabolood gveqneba:

( )dt

LR

IIIId

−=−−

0

0 ,

romlis integreba gvaZlevs:

( ) CtLRII +−=−0ln . (22.3)

integrebis C mudmivas mniS-

vnelobas davadgenT sawyisi pirobe-

bidan, kerZod, roca 0=t , maSin

0=I ; amitom 0ln IC = . C - am mniS-

vnelobis CasmiT (22.3)-Si gveqneba:

I

t

0I

a b

sur. 22.2.

t

I

0I

72

tLR

III

−=−

0

0ln ,

saidanac, potencirebis Semdeg miiReba:

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

− tLR

eII 10 . (22.3)

aseTia denis Zalis cvlilebis xasiaTi wredis CarTvis dros. misi grafiki

mocemulia 22.2 b suraTze. (22.3) formulidan Cans, rom denis Zala aRwevs Tavis maqsimalur

0I mniSvnelobas ara myisve, aramed TandaTan, sanam TviTinduqciis deni t

LR

eI−

0 ar

moispoba. es ki xdeba miT male, rac metia R da naklebia L .

wredis amorTvisas aRZruli TviTinduqciis eqstradeni gamoyenebul iqna zegamtaro-

bis Sesaswavlad (Itomi, $130). zegamtar mdgomareobaSi 0=R , amitom, (22.2) formulis

Tanaxmad 0II = , anu deni iarsebebs wredSi ragind didxans.

holandieli mecnieri kamerling-onesi, romelic iTvleba dabali temperaturebis

fizikis mamamTavrad, cdas atarebda Semdegnairad: koWa, romlis boloebi erTmaneTTan

mokledaa SeerTebuli, moTavsebulia mZlavri eleqtromagnitis polusebs Soris da

civdeba Txevadi heliumiT gadasvlis kritikul kT temperaturaze ufro dabla.

eleqtromagnitis amorTvisas solenoidSi aRiZvreba induqciuri deni, romelic miedineba

koWaSi ramdenime dRis ganmavlobaSi sagrZnobi Semcirebis gareSe.

$23. denis magnituri velis energia

rogorc viciT ($1), magnituri velis pirvelwyaros warmoadgens eleq-

truli deni. igi warmoiqmneba, icvleba da ispoba eleqtruli denis warmo-

qmnasTan, cvlilebasTan da mospobasTan erTad. magnitur vels, iseve

rogorc eleqtruls, gaaCnia energia. ra warmoadgens denis magnituri velis

energiis zomas?

am kiTxvaze pasuxis gasacemad davuSvaT, rom L induqciurobisa da

R omuri wianaRobis mqone wredSi gadis mudmivi I deni. am SemTxvevaSi

denis wyaros mier dt droSi Sesrulebuli muSaoba dtIdA ε= (sadac ε

wredSi CarTuli mudmivi denis wyaros em Zalaa) mTlianad ixarjeba

gamtarebis gaTbobaze, anu gamoiyofa siTbos saxiT. joul-lencis kanonis

Tanaxmad, dt droSi gamoyofili siTbos raodenoba RdtIdQ 2= . energiis

mudmivobis kanonis Tanaxmad

RdtIdtI 2=ε . (23.1)

sul sxvanairadaa saqme, rodesac wredSi gadis cvladi deni, vTqvaT,

zrdadi deni, rasac adgili aqvs, kerZod, wredis CarTvis dros. am dros

73

aRiZvreba TviTinduqciis em Zala Sε , romelic, lencis wesis Tanaxmad,

ewinaaRmdegeba denis zrdas. amitom deni Tavis mudmiv (maqsimalur) I

mniSvnelobas aRwevs ara myisve, aramed raRac t drois Semdeg. am drois

ganmavlobaSi denis wyaros garkveuli damatebiTi muSaobis Sesruleba

uxdeba TviTinduqciis em Zalis dasaZlevad, amitom, cxadia, (23.1) formuliT

warmodgenili energiis mudmivobis kanoni samarTliani aRar aris.

imis gasarkvevad, Tu ra saxes iRebs am SemTxvevaSi energiis mudmi-

vobis kanoni, davweroT omis kanoni Caketili wredisaTvis. wredis CarTvis

momentidan, vidre denis Zala ar miaRwevs mudmiv I mniSvnelobas, masSi

garda CarTuli denis wyaros ε eleqtromamoZravebeli Zalisa, moqmedebs

TviTinduqciis em Zala dtdILS −=ε , amitom denis Zala wredSi iqneba

RdtdIL

RI S

−=

+=

εεε.

tolobis orive mxare gavamravloT IRdt sidideze:

LIdIdtIRdtI −= ε2 ,

saidanac

LIdIRdtIdtI += 2ε . (23.2)

aseT saxes iRebs energiis mudmivobis kanoni. (23.2) formulidan Cans,

rom drois ganxilul ( )dt−0 periodSi wyaros mier Sesrulebuli dtIε

muSaobis mxolod nawili xmardeba joulis siTbos ( )RdtI 2 gamoyofas, meore

nawili ki dakavSirebulia induqciur movlenebTan da xmardeba denis Zalis

gazrdas dI sididiT. es muSaoba

LIdIdA = . (23.3)

sruli muSaoba, romelic sruldeba denis gasazrdelad nulidan mis

maqsimalur I mniSvnelobamde, miiReba (23.2) tolobis integrebiT:

2

2

0

LILIdIdAAI

=== ∫ ∫ .

vinaidan denis gazrdisas izrdeba misi magnituri veli, bunebrivia

CavTvaloT, rom A warmoadgens denis magnituri velis Seqmnaze

Sesrulebul damatebiT muSaobas, anu denis magnituri velis energiis

zomas:

2

2LIAWm == (23.4)

74

miRebuli formula msgavsia kinetikuri energies formulisa

⎟⎟⎠

⎞⎜⎜⎝

⎛=

2

2υmW . masSi masis magivrobas asrulebs induqciuroba (romelic

warmoadgens konturis inerciulobis zomas), xolo siCqaris magivrobas _

denis Zala. aseTi msgavsebas adgili aqvs magnetizmisa da meqanikis sxva

formulebs Sorisac.

(23.4) formulaSi denis magnituri velis energia gamosaxulia gamtaris

damaxasiaTebeli sidideebiT ( ), IL . gamovsaxoT es energia TviT velis

damaxasiaTebeli sidideebiT. amisaTvis gavixsenoT, rom grZeli solenoidis

SigniT veli erTgvarovania da mis RerZze magnituri velis induqcia

(formula 6.9) InB μμ0= , xolo aseTi solenoidis induqciuroba (formula

23.5) VnL 20μμ= ( n aris xviaTa ricxvi solenoidis sigrZis erTeulze, V _

solenoidis moculoba). Tu L -is mniSvnelobas SevitanT (23.4)-Si miviRebT,

rom solenoidis SigniT erTgvarovani velis energia

VInWm22

021 μμ= .

magram

μμ0

BnI = ,

amitom

VBWm μμ0

2

21

= . (23.5)

magnituri velis induqcia HB μμ0= ($27), amitom denis magnituri

velis energia SeiZleba daiweros Semdegi sami ekvivalenturi formiT,

romlebic am energiis gamosaxaven magnituri velis damaxasiaTebeli

sidideebiT ( )VHB ,, :

BHVVHVBWm 21

21

21 2

00

2

=== μμμμ

. (23.6)

vinaidan veli erTgvarovania, misi energia mW ganawilebulia velis

mTel moculobaSi Tanabrad. energiis moculobiTi simkvrive anu

moculobis erTeulis energia

BHHBV

Wmm 2

121

21 2

00

2

==== μμμμ

ω . (23.7)

aRsaniSnavia, rom iseve, rogorc eleqtruli velis SemTxvevaSi (23.7)

formulebi samarTliania araerTgvarovani velisaTvisac. gansxvavebaa isaa,

75

rom erTgvarovani velis SemTxvevaSi mω sivrcis yvela wertilSi

erTnairia, araerTgvarovani velis SemTxvevaSi ki wertilidan wertilamde

icvleba.

Cven ganvixileT wredis CarTvis SemTxveva da miviReT gamosaxuleba

denis magnituri velis energiisaTvis. wredis ganrTvisas deni ispoba da

denis mier momaragebuli energia gamoiyofa ama Tu im saxiT. es mJRavndeba

sakmaod mZlavr naperwkalSi, romelic didi induqciurobis wredSi

warmoiqmneba. Tu wredi isea Sedgenili, rom denis wyaros amorTvisas xdeba

koWas boloebis mierTeba R winaRobaze (sur. 22.1, k CamrTveli gadairTo a

konturidan b -ze), maSin am koWaSi aRZruli amorTvis eqstradenis energia

gamoiyofa R winaRobaze joulis siTbos saxiT.

Tavi III

magnituri veli nivTierebaSi

$24. magnetikebi

aqamde Cven vixilavdiT magnitur movlenebs vakuumSi. davuSvaT axla,

rom sivrce, romelSic deniani gamtaria moTavsebuli, Sevsebulia garemoTi.

cda da Teoria gviCvenebs, rom bunebaSi arsebuli yvela sxeuli magni-

turad aqtiuria im gagebiT, rom magnitdeba gareSe magnitur velSi Seta-

nisas da iwvevs mis cvlilebas. magniturad aqtiur sxeulebs zogjer

magnetikebsac uwodeben.

magnituri aqtivobis TvalsazrisiT ansxvaveben sustmagnitur da Zli-

ermagnitur sxeulebs. sustmagnitur sxeulebs miekuTvneba paramagnituri

da diamagnituri sxeulebi (paramagnetikebi da diamagnetikebi), Zlier-

magniturs ki _ feromagnituri sxeulebi (agreTve antiferomagnituri

da ferimagnituri sxeulebi. ferimagnitur sxeulebs, romlebsac, naxevar-

gamtaruli Tvisebebic aqvT, feritebi ewodeba).

gavixsenoT jer, ra movlenas hqonda adgili dieleqtrikis Setanisas

eleqtrostatikur velSi. velSi Setanisas dieleqtriki polarizdeba, ris

Sedegadac mis zedapirze warmoiqmneba bmuli eleqtruli muxtebi garkveu-

li 'σ zedapiruli simkvriviT (simartivisaTvis vuSvebT, rom dieleqtriki

erTgvarovania). es muxtebi qmnian Tavis eleqtrostatikur vels, romelic

ikribeba Tavdapirvel velTan _ Tavisufali muxtebis mier Seqmnil velTan.

76

Tu Tavisufali muxtebis mier Seqmnili velis daZabulobas aRvniSnavT 0E -iT,

bmuli muxtebisas '

E -iT, maSin jamuri velis daZabuloba sxeulis SigniT

'0 EEE += .

aRsaniSnavia, rom '

E yovelTvis 0E -is sawinaaRmdegodaa mimarTuli, rac

imas niSnavs, rom dieleqtrikis Setanisas eleqtrostatikur velSi veli

sustdeba.

analogiurad, magnetikis Setanisas gareSe (makrodenebis mier Seqmnil)

magnitur velSi, igi modis gansakuTrebul mdgomareobaSi _ magnitdeba.

damagnitebuli magnetiki qmnis (aRZravs) sakuTar magnitur vels, romelic

ikribeba gareSe magnitur velTan da cvlis mas. Tu gareSe magnituri velis

induqcias aRvniSnavT 0B -iT, damagnitebuli magnetikis mier Seqmnili velis

induqcias '

B -iT,∗ maSin, superpoziciis principis Tanaxmad, jamuri velis

induqcia sxeulis SigniT

'

0 BBB += . (24.1)

an, vinaidan '

000 , BHBHB +== μμ .

dieleqtrikisagan gansxvavebiT, '

B -s SeiZleba hqondes 0B –is rogorc

sawinaaRmdego, ise Tanxvdenili mimarTuleba. maSasadame, magnetikis

Setanisas magnitur velSi es ukanaskneli SeiZleba rogorc Sesustdes, ise

gaZlierdes. iseT magnetikebs, romlebic iwveven magnituri velis Sesus-

tebas, diamagnetikebi ewodeba, xolo iseT magnetikebs, romlebic aZliereben

gareSe magnitur vels _ paramagnetikebi. rogorc dia, aseve paramagnituri

sxeulebi gareSe magnituri velis ararsebobis SemTxvevaSi magnitur

Tvisebebs ar amJRavnebs.

aRsaniSnavia, rom yvela diamagnituri sxeulisTvis, agreTve paramagni-

turi sxeulebis didi umravlesobisTvis, 'B mcirea 0B _Tan SedarebiT. maSa-

sadame, am sxeulebis Setana gareSe magnitur velSi umniSvnelod cvlis mas.

magram paramagnituri sxeulebisagan gamoiyofa mcirericxovani, magram me-

tad didi praqtikuli mniSvnelobis mqone jgufi sxeulebisa, romelTaT-

visac 0' BB >> , e.i. am sxeulebis Setana magnitur velSi iwvevs am

∗ damagniteba dakavSirebulia sxeulSi mikrodenebis arsebobasTan, amitom, sxeulSi aRZrul damatebiT vels zogjer mikrodenebis vels uwodeben.

77

ukanasknelis mkveTr zrdas. maT, jgufis mTavari warmomadgenlis _ rkinis

mixedviT, feromagnetikebi ewodeba.

dabolos SevniSnoT, rom damagnitebuli sxeulis sakuTari magnituri

veli ( )'B , rogorc sxeulis SigniT, ise mis gareT, damokidebulia ara mar-

to nivTierebis gvarobaze, aramed sxeulis formazec (Rero, birTvi, rgoli

da sxv.). magram aRmoCnda, rom aseTi damokidebulebis ugulebelyofa SeiZ-

leba, Tu damamagnitebeli veli erTgvarovania da gamosakvlevi nivTiereba

mTlianad avsebs mas. aseTi pirobebis ganxorcieleba SesaZlebelia, Tu,

magaliTad, gamosakvlev nivTierebas mivcemT cilindris formas da

movaTavsebT mas grZeli solenoidis SigniT.

$25 damagnitebis veqtori. nivTierebis magnituri SeRwevadoba

sxeulis damagnitebis xarisxis dasaxasiaTeblad Semotanilia

damagnitebis veqtoris cneba. damagnitebis veqtori ewodeba magnetikis

erTeul moculobaSi moTavsebul molekulaTa magnituri

momentebis veqtorul jams. Tu VΔ moculobis magnitur moments

avRniSnavT ∑=

N

imiP

1

iT, maSin, ganmartebis Tanaxmad,

V

PP

N

imi

Δ=∑=1 , (25.1)

sadac iP mi -uri molekulis magnituri momentia, xolo N _ molekulebis

saerTo ricxvi VΔ moculobaSi. VΔ unda iyos imdenad mcire, rom mis

farglebSi veli SeiZleba CaiTvalos mudmivad. Tu magnetiki erTgvarovani

da izotropiulia, mis yvela molekulas aqvs erTnairi magnituri momenti

mP , maSin (najerobamde damagnitebis SemTxvevaSi)

m

N

imi PNP =∑

=1 da

mm Pn

VPNP == , (25.2)

sadac VNn = molekulebis koncentraciaa.

wriuli denis magnituri momentis ganmartebidan gamomdinareobs ($3),

rom SI sistemaSi damagnitebis veqtoris sazomi erTeulia a/m.

78

vinaidan sxeulis damagniteba Sedegia mikrodenebze gareSe magnituri

velis ( )H moqmedebisa, cxadia, rom damagnitebis xarisxi ( )P damokidebuli

iqneba am velis sidideze. cda gviCvenebs, rom sustmagnituri sxeulebisTvis

P –sa da H –s Soris wrfivi damokidebuleba arsebobs da izotropuli

garemos SemTxvevaSi aqvs Semdegi saxe:

HkP m= ∗ (25.3)

an (2.1) –is gaTvaliswinebiT

0

0

Bk

P m

μ= , (25.4)

mk proporciulobis koeficients nivTierebis magnituri amTvisebloba

an damagnitebis koeficienti ewodeba. igi uganzomilebo sididea, vinaidan

P da H erTi da imave erTeulebiT izomeba. mk damokidebulia nivTierebis

gvarobaze, amasTan cxadia, rom paramagnetikebisaTvis igi dadebiTia (P -s

aqvs H –is mimarTuleba), diamagnetikebisaTvis ki _ uaryofiTi.

(25.3) formulis fizikuri arsi naTelia. marTlac, rac ufro Zlieria gareSe

magnituri veli, paramagnituri sxeulebis SemTxvevaSi miT ufro ukeTesad axdens

mikrodenebis orientacias (romlis xelSemSleli faqtori siTburi moZraobaa), xolo

diamagnituri sxeulebis SemTxvevaSi miT ufro did inducirebul moments warmoqmnis ($27).

amitom gareSe velis zrdiT proporciulad izrdeba mikrodenebis saSualo velic.

mikrodenebis veli kolinearulia (mimarTulia imave wrfis gaswvriv) makrodenebis velisa

da misi proporciulia.

cxadia, rom unda arsebobdes gar-

kveuli kavSiri damagnitebis P veqtorsa

da damagnitebuli sxeulis (magnetikis)

sakuTar (mikrodenebis) '

B magnitur vels

Soris. davadginoT es kavSiri.

davuSvaT, rom erTgvarovani izo-

tropiuli magnetiki mTlianad avsebs im

sivrces, sadac gareSe magnituri velis

∗ aseT proporciulobas adgili aqvs diamagnituri sxeulebisaTvis da im SemTxvevaSi, Tu

H ar aris Zalian didi, xolo temperatura Zalian dabali _ paramagnituri sxeulebisaTvisac. feromagnituri sxeulebisaTvis aseT proporciulobas adgili ara aqvs.

garda amisa, Tu magnetiki anizotropulia, P da H veqtorebis mimarTulebebi SeiZleba

arc daemTxves erTmaneTs. aseTi magnetikisaTvis kavSiri P -sa da H -s Soris xorcieldeba magnituri amTviseblobis tenzoris saSualebiT.

sur. 25. 1

79

induqcia nulisagan gansxvavebulia. simartivisaTvis davuSvaT, rom

magnetiki warmoadgens cilindruli formis sxeuls, romelic moTavsebulia

misi msaxvelis paralelur magnitur velSi, magaliTad, grZeli solenoidis

SigniT. vinaidan solenoidis SigniT veli erTgvarovania ( )constB =0 ,

magnetikic erTgvarovnad damagnitdeba ( )constP = . solenoidis velis

moqmedebiT wriuli molekuluri denebis magnituri momentebi orientir-

debian cilindris RerZis gaswvriv, TviT wriuli denebi ki am RerZis mar-

Tobulad, rogorc es naCvenebia 25.1 suraTze. ganvixiloT magnetikis

erTerTi ganivikveTi. radgan constP = , SeiZleba CaiTvalos, rom am kveTaSi

yvela molekuluri deni erTnairia, rac gamoiwvevs maT

urTierTgabaTilebas kveTis SigniT. gaubaTilebeli rCeba denebi mxolod

kveTis gare konturze, amitom SeiZleba iTqvas, rom sxeulis sakuTari mag-

nituri veli '

B Seqmnilia cilindris RerZis marTobuli wriuli denebiT,

romlebic cilindris gare zedapirze miedinebian. es denebi msgavsia

solenoidSi gamavali denisa, amitom isini sxeulis SigniT qmnian magnitur

vels, romlis 'B induqciis gamosaTvlelad SeiZleba visargebloT (6.9)

formuliT solenoidis magnituri velis induqciisaTvis. Tu erT-erTi

wriuli denis sidide aris 'I , xolo cilindris erTeul sigrZeze maTi

ricxvi n , maSin (6.9) formulis Tanaxmad

'0

' nIB μ= . (25.5)

meore mxriv, ganvixiloT magnetikis mcire lΔ elementi, romlis

moculoba lSV Δ=Δ , sadac S nimuSis ganikveTis farTia. wriul denTa rao-

denoba am elementze iqneba lnΔ , xolo misi magnituri momenti

''

1SlInP

N

imi Δ=∑

=

.

damagnitebis P veqtoris sidide

''

1 nIlSSlIn

V

PP

N

imi

=Δ

Δ=

Δ=∑= . (25.6)

(25.5) da (25.6) formulebis Sedareba gvaZlevs

PB 0' μ= . (25.7)

damagnitebis ganxiluli meqanizmidan cxadia, rom '

B da P veqtorebi

paralelurni arian, amitom (25.7) tolobas veqtoruli xasiaTic eqneba:

PB 0

'μ= . (25.8)

80

SevitanoT (25.8) formulaSi P –is mniSvneloba (25.4)-dan

00

00

'BkB

kB m

m ==μ

μ

da '

B –is miRebuli mniSvneloba CavsvaT (24.1)-Si

( )mkBBBB +=+= 10'

0 ,

an

0BB μ= . (25.9)

uganzomilebo sidides mk+= 1μ ewodeba nivTierebis fardobiTi

magnituri SeRwevadoba, an ubralod magnituri SeRwevadoba. magnituri

SeRwevadoba nivTierebis mniSvnelovani magnituri maxasiaTebelia. igi gviC-

venebs, Tu ramdenjer metia (an naklebia) makrodenis mier Seqmnili magnitu-

ri velis induqcia mocemul nivTierebaSi ( )B vidre sicarieleSi ( )0B .

diamagnituri sxeulebisaTvis 0<mk , amitom 1<μ ; paramagnituri

sxeulebisaTvis 0>mk da 1>μ ;

unda aRiniSnos, rom rogorc dia-, ise paramagnituri nivTierebebisaT-

vis mk metad mcirea (Sesabamisad 10-9 da 10-7 rigisa), ris gamoc maTTvis μ

umniSvnelod gansxvavdeba erTisagan. feromagnituri sxeulebisaTvis 1>>μ .*

Tu (25.9) formulaSi gaviTvaliswinebT, rom HB 00 μ= , gveqneba:

HB μμ0= . (25.10)

miviReT metad mniSvnelovani Sedegi: aRmoCnda, rom rig SemTxvevebSi

magnituri velis ZiriTadi maxasiaTeblis B induqciis veqtoris gansazRvra

SesaZlebelia mikrodenebis velis codnis gareSe _ sakmarisia

ganvsazRvroT makrodenebis magnituri veli da vicodeT garemos magnituri

SeRwevadoba. magram unda gvaxsovdes, rom (25.9) da, maSasadame (25.10)

formuliT gamosaxuli martivi kavSiri B –sa da 0B –s, B -sa da H –s Soris

samarTliania mxolod erTgvarovani, izotropuli magnetikebisaTvis,

romlebic mTlianad avseben im sivrces, sadac gareSe veli 0≠H . yvela

sxva SemTxvevaSi B –s gasagebad saWiroa mikrodenebis magnituri velebis

saTanado gaTvla, rac gacilebiT rTulia.**

* 25.9 Tanafardoba marTebulia feromagnetikebisaTvisac, magram am sxeulebSi μ mudmivi ki

ar aris, aramed damokidebulia damamagnitebeli velis sidideze (§32).

** nebismieri garemosaTvis vargisi universaluri kavSiri B -sa da H -s Soris mocemulia §31-Si.

81

§26. eleqtronebisa da atomebis magnituri momenti

rogorc § 24-Si aRvniSneT, magnitur aqtivobas amJRavnebs uklebliv

yvela sxeuli, amitom bunebrivia davuSvaT, rom sxeulis magnituri Tvisebe-

bi ganpirobebulia elementaruli nawilakebiT, romlebisganac Sedgeba yove-

li atomi. yvela atomisaTvis saerTo aseTi elementaruli nawilakebia

eleqtronebi, protonebi da neitronebi. gamokvlevebiT dadginda, rom

protonebisa da neitronebis magnituri momentebi TiTqmis 1000-jer naklebia

eleqtronis umcires magnitur momentze, amitom pirvel miaxloebaSi

birTvis magnituri momenti SeiZleba ugulebelvyoT da CavTvaloT, rom

atomis magnituri Tvisebebi mTlianad ganpirobebulia eleqtronebiT.

SevafasoT eleqtronis magnituri momenti. ganvixiloT umartivesi Sem-

Txveva, roca erTi eleqtroni, romlis masaa m , xolo muxti e , moZraobs

uZravi dadebiTi birTvis garSemo wriul orbitaze (wyalbadis atomi).

eleqtronis es moZraoba ganapirobebs wriul dens, romlis magnitur

moments eleqtronis orbituli magnituri momenti ( )0p ewodeba. ganmartebis

Tanaxmad, ISp =0 , sadac I aris denis Zala, xolo S _ orbitis farTi.

evI = , sadac T

v 1= brunTa ricxvia erT wamSi, xolo 2rS π= , sadac r

wriuli orbitis radiusia. Tu eleqtronis wirul siCqares aRvniSnavT υ –

Ti, r

vπυ

2= , amitom magnituri momentisTvis gveqneba:

rerr

eISp υππυ

21

22

0 === . (26.1)

eleqtrons, orbituli magnituri momentis garda, gaaCnia impulsis

orbituli meqanikuri momenti 0L .

[ ] rmLrmL υυ == 00 ; .

magnituri momentis Sefardebas Sesabamis meqanikur momentTan ewodeba

giromagnituri fardoba ( )G da orbituli momentisaTvis igi toli:

me

rmre

LpG

21

21

0

00 ===

υυ

, (26.2)

saidanac

00 21 L

mep = . (26.3)

82

Tu gaviTvaliwinebT 0p –is da 0L –is mimarTulebebs (sur.26.1), toloba

(26.3) veqtorulad Semdegnairad Caiwereba:

000 21 L

meLGp −=−= . (26.4)

(26.4) formula erTmaneTTan akavSirebs eleqtronis orbitul magni-

tur da meqanikur momentebs. am formulaSi ar Sedis eleqtronis orbitis

nomeri. maSasadame, orbituli giromagnituri

fardoba nebismieri orbitisaTvis erTnairia

da udris me

21

-s.

SeiZleba Cveneba, rom (26.4) formula,

romelic miviReT wriuli orbitisaTvis,

zogadia da marTebulia eleqtronis nebismieri

elifsuri orbitisTvisac.

viciT, rom boris postulatis Tanaxmad,

romelic gansazRvravs stacionaluri

orbitis radiuss, eleqtronis impulsis orbituli momenti π2h

ricxvis

jeradia:

π2hnLon ⋅= ,

sadac ...,3,2,1=n gansazRvravs orbitis nomers, xolo 341062,6 −⋅=h j.wm

plankis mudmivaa.

Tu onL –is mniSvnelobas SevitanT (26.3)–Si, wyalbadis atomis

orbituli magnituri momentisaTvis miviRebT:

mehnpon π4

= .

orbituli magnituri momentis umciresi mniSvneloba miiReba, roca

eleqtroni moZraobs pirvel orbitaze ( )1=n . am elementarul magnitur

moments boris magnetoni ewodeba da aRiniSneba Bμ –Ti:

tl

j2331

3419

10927,0101,914,34

1062,6106,14

−−

−−

⋅=⋅⋅⋅⋅⋅⋅

==m

ehB π

μ ;

birTvis garSemo brunvis gamo eleqtroni waagavs bzrialas, es

garemoeba safuZvlad udevs giromagnitur anu magnitomeqanikur movlenebs _

magnetikis damagniteba iwvevs mis brunvas da piriqiT, magnetikis brunva

0p

I

0L

υ

sur. 26.1

83

iwvevs mis damagnitebas. pirveli movlenis arseboba eqsperimentulad

daamtkices ainStainma da de-haasma, meoresi_barnetma.

1. barnetis cda (1914 w).

Tu rkinis patara cilindrs swrafad vabrunebT sakuTari RerZis gar-

Semo, cilindri damagnitdeba. am efeqtis mizezia is garemoeba, rom nivTie-

rebis atomebs gaaCnia rogorc magnituri, ise meqanikuri momentebi, romle-

bic, rogorc es vaCveneT eleqtronis orbituli brunvis magaliTiT, erT-

maneTTan xistadaa dakavSirebuli (formula 26.4) da, maSasadame, erTis

cvlilebam unda gamoiwvios meores cvlileba. cilindris brunvis dros,

`eleqtronuli bzrialebi”, giroskopis mzgavsad, cdiloben miiRon iseTi

orientacia, rom maTi meqanikuri momentebi cilindris brunvis RerZis

paraleluri iyos. Cveulebriv, meqanikuri momentebi qaosuradaa orienti-

rebuli da maTi aseTi mowesrigeba iwvevs magnituri momentebis saTanado

mowesrigebas, ris Semdegac cilindri magnitdeba.

unda aRiniSnos, rom barnetis efeqtiT gamowveuli damagniteba Zalian

mcirea: magnituri velis daZabuloba, romelsac qmnis cilindri, roca igi

wamSi 100 bruns akeTebs, 1000-jer naklebia dedamiwis magnituri velis daZa-

bulobaze. amitom am efeqtis dakvirveba SeiZleba mxolod Zliermagnitur

sxeulebze _ feromagnetikebze.

2. ainStainis da de-haasis cda (1915 w).

am cdaSi rkinis cilindri (Rero) Camokidebulia solenoidis SigniT

kvarcis wvril Zafze (sur. 26.2). solenoidSi denis gatarebisas Rero Semob-

rundeba garkveuli kuTxiT, romlis sididesac gviCvenebs sarkeze aTinaTis

gadaadgileba.

am movlenasac safuZvlad udevs atomis magnitur da meqanikur

momentebs Soris uerTierTkavSiri. solenoidSi denis gatarebisas rkinis

Rero damagnitdeba, e.i. atomebis magnituri momentebi orientirdeba gareSe

(solenoidis) magnituri velis mimarTulebiT. (26.4) formulis Tanaxmad

aseTive orientacias, oRond velis sawinaaRmdegod miiRebs `eleqtronuli

bzrialebis” impulsis oiL momentebi, ris Semdegadac jamuri meqanikuri mo-

enti ∑ oiL nulisagan gansxvavebuli iqneba. magram impulsis momentis mudmi-

vobis kanonis Tanaxmad, izolirebuli sistemis _ `Rero-eleqtronebi” _

84

moZraobis raodenobis momenti unda iyos mudmivi, amitom Rero iZens _

∑ oiL -is tol moZraobis raodenobis moments da iwyebs brunvas.

Reros mobrunebis kuTxe Zalian mcirea, amitom, efeqtis gasaZliereb-

lad iyeneben rezonansis movlenas: solenoidSi atareben cvlad dens, rom-

lis sixSire sistemis rxevis sakuTari sixSiris tolia.

Reros Semobrunebisas kvarcis Zafi igrixeba, ris saSualebiTac ga-

nisazRvreba Reros impulsis momenti, romelic elqtronebis impulsis

momentebis geometriuli jamis tolia. Semdeg Reros damagnitebis

saSualebiT gaigeba eleqtronebis jamuri magnituri momenti da amrigad, ga-

nisazRvreba giromagnituri Tanafardoba 0G . mosalodneli iyo, rom es Ta-

nafardoba me

2-is toli iqneboda. cdis Sedegad ki miRebuli iqna

me, e.i.

orjer meti. es gansxvaveba fizikosebisTvis sruliad gaugebari iyo TiT-

qmis 10 wlis manZilze, vidre haudsmitma da ulenbekma ar gamoTqves mosaz-

reba, rom eleqtrons, garda orbituli magnituri da meqanikuri momentebi-

sa, gaaCnia e.w. spinuri magnituri da meqanikuri momentebi, romlebic

dakavSirebulia eleqtronis brunvasTan sakuTari RerZis garSemo.

spinuri momentebisaTvis giromagnituri fardoba tolia:

me

Lp

Gs

sS == .

e.i. zustad Tanxvdeba cdis Sedegebs. amrigad, ainStainisa da de-haasis

cdaSi miRebuli giromagnituri fardoba warmoadgens spinur da ara

orbitul giromagnitur fardobas.

giromagnituri fardobisaTvis aseTive sidide iqna miRebuli a.iofesa

da p.kapicas cdebSi (1917 w.). am cdebs safuZvlad udevs cnobili faqti, rom

yoveli feromagnituri sxeulisaTvis arsebobs gansakuTrebuli tem-

peratura, e.w. `kiuris wertili”, romelzedac damagniteba ispoba (sxeuli

kargavs feromagnitur Tvisebebs da xdeba paramagnituri). rkinisaTvis

kiuris temperatura C770 –is tolia. damagnitebul rkinis Reros hkideben

kvarcis wvril Zafze da swrafad axureben kiuris temperaturamde. damagni-

tebul ReroSi `eleqtronuli bzrialebi” orientirebulia RerZis gaswvriv.

maSasadame, maT gaaCniaT jamuri impulsis momenti ∑ iL0 (sur. 26.2,a). ganmag-

nitebisas xdeba `eleqtronuli bzrialebis” RerZebis dezorientacia

(sur. 26.2,b), ris Sedegadac maTi jamuri impulsis momenti nulis toli

85

xdeba. impulsis momentis mudmivobis kanonis Tanaxmad Rero iZens _

∑ iL0 -is tol impulsis moments da iwyebs brunvas.

zemoaRwerili giromagnituri

cdebi, romelic SemdgomSi mraval-

jer iqna gameorebuli sxva mecnier-

Ta mier, giromagnituri fardobisaT-

vis yovelTvis iZleoda erTi da igi-

ve sidides _ orjer mets imasTan

SedarebiT, rac eleqtronis orbi-

tuli brunvidan gamomdinareobs. uf-

ro zusti radiospeqtroskopuli

meTodis gamoyeneba, romlis saSu-

alebiTac xdeba amJamad giromagni-

turi fardobis gansazRvra, gvaZlevs

igive Sedegs.

amrigad, cdebiT sabolood

dadgenilia, rom ZiriTad faqtors, romelic gansazRvravs rkinis (da

saerTod, feromagnituri sxeulis) magnetizms, warmoadgens eleqtronis ara

orbituli moZraoba, aramed spini.

rogorc aRvniSneT, Tavdapirvelad spinis Sesaxeb gamartivebuli war-

modgena hqondaT da mas ukavSirebdnen eleqtronis brunvas sakuTari RerZis

garSemo. SemdgomSi dadginda, rom aseTi warmodgena swori ar aris, spini

eleqtronis iseTive ganuyofel, fundamentur Tvisebas warmoadgens, ro-

gorc masa da muxti da misi dayvana sxva raime ufro martivamde ar

SeiZleba.

amrigad, eleqtrons, garda orbituli magnituri ( )0p da meqanikuri

( )0L momentebisa, gaaCnia spinuri magnituri ( )sp da meqanikuri ( )sL

momentebi, amasTan spinuri giromagnituri fardobis sidide ( )sG orjer

metia orbituli fardobis sidideze ( )0G :

meG

Lp

Gs

ss === 02 .

cdiT dadgenilia, rom spinuri magnituri momenti boris magnetonis

tolia:

* S * S

a b sur. 26.2.

86

mehp Bs π

μ4

== .

atomis (molekulis) sruli magnituri momenti miiReba misi yvela

eleqtronis orbituli da spinuri magnituri momentebis geometriuli

SekrebiT (birTvul momentebs ugulebelvyofT):

∑∑==

+=z

isi

z

iia ppP

110 .

§27. diamagnetizmisa da paramagnetizmis buneba*

wina paragrafSi aRvniSneT, rom elementarul nawilaks, romelic Zi-

riTadad ganapirobebs nivTierebis magnitur Tvisebebs, warmoadgens

eleqtroni da gamovTvaleT eleqtronis magnituri momenti. vTqviT, rom

atomis (molekulis) magnituri momenti miiReba misi yvela eleqtronis

magnituri momentebis geometriuli SekrebiT. cxadia, rom imis mixedviT, Tu

rogoria atomSi Semavali eleqtronebis magnituri momentebis (rogorc

orbitulis, aseve spinuris) orientacia, atomis magnituri momenti iqneba

nulisagan gansxvavebuli an nulis toli. amis Sesabamisad bunebaSi arsebuli

nivTierebebi iyofa or jgufad:

1. nivTierebebi, romelTa Semadgeneli atomebis (molekulebis) magni-

turi momentebi nulisagan gansxvavdeba.

Cveulebriv pirobebSi (gareSe magnituri velis ararsebobis SemTxveva-

Si) es elementaruli atomuri magnituri momentebi qaosurad arian orienti-

rebulni da erTmaneTs abaTileben, amitom sxeuli magnitur Tvisebebs ar

amJRavnebs. gareSe (makrodenis) magnitur velSi Setanisas xdeba

elementaruli magnituri momentebis orientacia, ris Sedegadac sxeuli

iZens makroskopuli sididis magnitur moments _ igi magnitdeba.

damagnitebuli magnetiki qmnis sakuTar magnitur vels. amasTan, cxadia, rom

am SemTxvevaSi magnetikis sakuTari velis (mikrodenebis mier Seqmnili

velis) 'B induqcias eqneba gareSe velis 0B induqciis mimarTuleba da

gaaZlierebs mas, e.i. sxeuli iqneba paramagnituri.

maSasadame, paramagnetizmi damaxasiaTebelia iseTi nivTierebebisaTvis,

romelTa Semadgenel atomebs Cveulebriv (gareSe magnituri velis ararse-

bobis) pirobebSi gaaCniaT nulisagan gansxvavebuli magnituri momenti. para-

* am paragrafSi paramagnetizmi da diamagnetizmi axsnilia Tvisobrivad. sakiTxis raodenobrivi mxare ganxilulia §28-30-Si.

87

magnetikis damagnetebis meqanizmi daiyvaneba gareSe magnituri velis

moqmedebiT calkeuli atomebis elementaruli magnituri momentebis orient-

taciamde.

paramagnetikebs miekuTvneba tute liTonebi, iSviaT-miwaTa elementebi,

perioduli sistemis gardamaval elementTa jgufi ( )PtMnCr ,, , zogierTi

airi (mag. Jangbadi), zogierTi marili da maTi wyalxsnarebi da sxv.

paramagnetikis damagniteba gareSe velis mimarTulebiT iwvevs imas,

rom araerTgvarovan magnitur velSi masze moqmedebs velis zrdis gaswvriv

mimarTuli Zala, romlis moqmedebiTac is Seizideba Zlieri velis areSi.

2. nivTierebebi, romelTa Semadgeneli atomebis (molekulebis) magni-

turi momentebi nulis tolia (amas maSin aqvs adgili, roca atomis cal-

keuli eleqtronebis magnituri momentebi iseTnairad arian orientirebul-

ni, rom erTmaneTs abaTileben). atomis magnitur velSi Setanisas mis

TiToeul eleqtronze moqmedebs lorencis Zala, rac iwvevs brunvis kuT-

xuri siCqaris Secvlas. es tolfasia damatebiTi wriuli denis warmoqmnisa

(inducirebisa), romelsac (lencis wesis Tanaxmad) iseTi mimarTuleba aqvs,

rom misi Sesabamisi magnituri momenti yovelTvis gareSe magnituri velis

sawinaaRmdegodaa mimarTuli. amrigad, magnituri velis moqmedebiT atomis

TiToeuli eleqtroni iZens damatebiT mipΔ magnitur moments da vinaidan

atomis yvela eleqtronisaTvis mipΔ -s erTnairi (gareSe velis sawinaaRmde-

go) mimarTuleba aqvs, cxadia, rom maTi veqtoruli jami iqneba nulisagan

gansxvavebuli da imave mxares mimarTuli. calkeuli atomebis magnituri

momentebis Sekrebis Sedegad sxeuli iZens makroskopuli sididis magnitur

moments _ igi magnitdeba.

damagnitebis ganxiluli meqanizmidan naTelia, rom am SemTxvevaSi

sxeulis sakuTari magnituri velis 'B induqcia mimarTuli iqneba gareSe

magnituri velis 0B induqciis sawinaaRmdegod da Seamcirebs mas, e.i. sxeu-

li iqneba diamagnituri.

maSasadame, diamagnitizmi damaxasiaTebelia iseTi nivTierebebisaTvis,

romelTa Semadgenel atomebs gareSe magnituri velis ararsebobis

pirobebSi magnituri momenti ar gaaCniaT, diamagnetizmis damagnitebis

meqanizmi daiyvaneba gareSe magnituri velis moqmedebiT calkeul atomebSi

velis sawinaaRmdegod mimarTuli magnituri momentebis induqcirebamde.

88

diamagnetikebs miekuTvneba nivTierebaTa didi umravlesoba: yvela in-

ertuli airi, wyali, mina, faifuri, naxSirorJangi ( )2CO , organuli naerTe-

bis umravlesoba, agreTve tyvia, vercxliswyali, naxSirbadi, germaniumi,

siliciumi, spilenZi, vercxli, oqro, TuTia.

paramagnituri da diamagnituri sxeulebis damagnitebis ganxiluli

meqanizmidan gamomdinareobs, rom paramagnetikis damagniteba damokidebuli

unda iyos temperaturaze, vinaidan magnituri velis maorientebeli moqme-

debis xelSemSleli faqtors warmoadgens siTburi moZraoba, diamagnetikis

damagniteba ki temperaturaze damokidebuli ar aris.

diamagnetikis damagniteba

gareSe velis sawinaaRmdegod

iwvevs imas, rom araerTgvarovan

magnitur velSi masze moqmedebs

velis Semcirebis gaswvriv mi-

marTuli Zala, romlis moqmede-

biTac is gadaadgildeba velis

Semcirebis mimarTulebiT. 27.1

suraTze naCvenebia araerT-

gvarovani magnituri velis moq-

medeba para-(a) da diamagnitur

(b) sxeulebze.

dabolos, SevniSnoT Sem-

degi: radgan diamagnitizmi ganpirobebulia birTvis irgvliv mbrunav

eleqtronebze lorencis Zalis moqmedebiT, igi warmoadgens universalur

efeqts, e.i. damaxasiaTebelia yvela atomisa da maSasadame, yvela

nivTierebisaTvis. magram paramagnetikebSi es efeqti gadaifareba ufro

Zlieri, orientaciis efeqtiT. amitom sufTa saxiT diamagnitizmi gvxvdeba

mxolod iseT nivTierebebSi, romelTa Semadgenel atomebs Cveulebriv piro-

bebSi magnituri momenti ar gaaCniaT.*

* sityvebi `sufTa saxiT~ SemTxveviT ar aris naTqvami. saqme imaSia, rom atoms Cveulebriv pirobebSi SeiZleba gaaCndes nulisagan gansaxvavebuli magnituri momenti, magram imdenad mcire, rom igi gadafaros gareSe veliT inducirebulma magniturma momentma. aseTebia, magaliTad, oqros, vercxlis, spilenZis atomebi da amiT aixsneba am nivTierebebis diamagnetizmi. sazogadod, nivTierebis dia- an paramagnetizms ganapirobebs is, Tu romeli efeqti sWarbobs masSi _ orientaciuli Tu induqciuri. Tu orientaciuli efeqti sWarbobs, nivTiereba paramagnituria da piriqiT.

S N

N S

a

N S

N S

b

sur. 27.1.

89

§28. atomi magnitur velSi. larmoris Teorema

ganvixiloT ra gavlenas moaxdens gare magnituri veli eleqtronis orbitul moZ-

raobaze. simartivisaTvis davuSvaT, rom mocemulia wyalbadis atomi, romelSic eleqtroni

brunavs wriul orbitaze. gare magnituri veli mivmarToT orbitis sibrtyis marTobulad.

gare magnituri velis ararsebobis SemTxvevaSi birTvsa da eleqtrons Soris

moqmedi kulonis urTierTqmedebis Zala ( )kF warmoadgens centriskenul Zalas ( )cF . am

Zalis moqmedebiT eleqtroni brunavs birTvis garSemo 0ω kuTxuri siCqariT. maSasadame,

rmFF kc20ω== ,

sadac r orbitis radiusia.

gare magnituri velis CarTvisas eleqtronze, garda kulonis Zalisa, imoqmedebs

lorencis Zala

0BeFB υ= .

imis mixedviT, Tu rogoria gare magnituri velis mimarTuleba, lorencis Zalis mi-

marTuleba sawinaaRmdegoa (sur. 28.1, a) an Tanxvdeba (sur. 28.1,b) centriskenuli Zalis

mimarTulebas. pirvel SemTxvevaSi eleqtronis siCqare Semcirdeba. es tolfasia eleqtro-

nis moZraobisa υΔ siCqariT sawinaaRmdego mimarTulebiT (an brunvisa ωΔ kuTxuri siC-

qariT sawinaaRmdegod mimarTulebiT). amis Sedegad aRiZvreba damatebiTi ( pΔ ) magnituri

momenti, romelic mimarTulia velisa da orbituli magnituri momentis sawinaaRmdegod. am

moments inducirebuli momenti ewodeba. meore SemTxvevaSi eleqtronis siCqare izrdeba.

am drosac aRiZvreba damatebiTi (induqcirebuli) pΔ magnituri momenti momenti, romlis

mimarTuleba Tanxvdeba orbituli momentis mimarTulebas,

magram kvlav sawinaaRmdegoa gareSe magnituri velis

mimarTulebisa.

gamoviTvaloT velis moqmedebiT gamowveuli siCqaris

es cvlileba. magnituri velis CarTvis Semdeg centris-

kenuli Zala warmoadgens kulonisa da lorencis Zalebis

tolqmeds:

Bkc FFF ±=

CavsvaT mniSvnelobebi:

020

2 Bermrm υωω ±=

magram ωωω Δ±= 0 , amitom

( ) 020

20 Bermrm υωωω ±=Δ±

020

20

20 2 Bermrmrmrm υωωωωω ±=Δ+Δ±

Tu gaviTvaliswinebT, rom ωωω Δ<<Δ 02

, miviRebT:

002 Berm υωω ±=Δ± ,

saidanac, radgan r0ωυ = , miviRebT:

pΔ

υ

0B

0p

pΔ

BF

kF

a

υ

0B

0p

BF

kF

b

sur. 28. 1 a, b

90

00

0

22B

me

rmBe

==Δωυ

ω . (28.1)

gamoyvanis dros igulisxmeboda, rom orbitis radiusi r ucvlelia.

SevniSnoT, rom ωΔ warmoadgens wriuli sixSiris mcire danamats: 0ωω <<Δ .

marTlac, es piroba SeiZleba Caiweros Semdegnairad:

.2 0

0 em

Bω

<< (28.2)

SevitanoT 0,, ωme -is ricxviTi mniSvnelobebi:

tlk

wmkg 419

11531

0 10106,1

10101,92≈

⋅⋅⋅⋅

<< −

−−

B .

dRemde miRebuli maqsimaluri sididis magnituri veli

ar aRemateba 103 teslas, ase rom (28.2) piroba kargad

sruldeba.

ganxiluli SemTxveva (eleqtronis orbita mar-

Tobulia magnituri velisa) larmoris Teoremis kerZo

SemTxvevaa.

ganvixiloT zogadi SemTxveva, roca orbita velis

maTobuli ar aris. mivmarToT meqanikur models _

gavixsenoT bzrialas (giroskopis) brunva. gaSvebis

Semdeg bzrialaze, Tu misi RerZi mkacrad vertikaluri ar aris, imoqmedebs simZimis Zalis

mabrunebeli momenti, ris gamoc is, garda sakuTari RerZis garSemo swrafi brunvisa, Seas-

rulebs kidev damatebiT brunviT moZraobas, romelsac precesia ewodeba. precesia

bzrialas RerZis vertikalis garSemo brunvaa. romelic sakuTari RerZis garSemo

brunvasTan SedarebiT mcirea. precesiis Sedegad bzrialas OA RerZi, romlis gaswvrivac

mimarTulia misi impulsis momenti L , aRwers konuss verti-

kalis mimarT (sur. 28.2). SevniSnoT, rom vertikalis

mimarTuleba Tanxvdeba simZimis Zalis mimarTulebas.

rogorc aRvniSneT, eleqtroni, birTvis garSemo swra-

fi brunvis gamo, SeiZleba warmovidginoT rogorc Tavi-

seburi bzriala. amitom ganxiluli SemTxvevis analogiurad,

atomis moTavsebisas gareSe magnitur velSi, eleqtronul

orbitaze imoqmedebs [ ]00 Bp mabrunebeli momenti, ris Se-

degadac adgili eqneba orbitis precesias. am momentis moq-

medebiT 0p veqtori da maSasadame L –ic precesirebs 0B veq-

toris mimarT (sur 28.3). mtkicdeba, rom eleqtronis monawi-

leoba precesiul moZraobaSi tolfasia misi damatebiTi

brunvisa 02B

me

=Δω kuTxuri siCqariT, rac, larmoris Teo-

remis Sesabamisad, iwvevs velis sawinaaRmdegod mimarTuli

pΔ

υΔ

υ

e−

e−

0p

0B

r

'r

sur. 28.3

O P

A

sur. 28. 2

91

damatebiTi (inducirebuli) magnituri momentis warmoSobas.

1895 wels ingliselma mecnierma larmorma daamtkica Torema, romlis arsi

eleqtronis orbituli brunvis mimarT daiyvaneba Semdegze: eleqtronis orbitul brunvaze

magnituri velis moqmedebis erTaderTi Sedegi mdgomareobs imaSi, rom imis mixedviT, Tu

rogoria magnituri velis mimarTuleba, eleqtronis brunvis kuTxuri siCqare izrdeba an

mcirdeba erTi da imave 02B

me

=Δω sididiT. amasTan eleqtronuli orbita da 0p

veqtori ganicdian precesias 0B veqtoris mimarT.

mniSvnelovania aRiniSnos, rom larmoris precesiis sixSire mocemuli atomis yvela

eleqtronisaTvis erTnairia, vinaidan is damokidebuli ar aris orbitis radiusze, eleq-

tronis moZraobis siCqaresa da gareSe magnituri velis mimarT orbitis daxris kuTxeze

(formula 28.1).

gamovTvaloT damatebiTi magnituri momenti, romelic, rogorc aRvniSneT, yovel-

Tvis gareSe velis sawinaaRmdegodaa mimarTuli: eleqtronis damatebiTi brunva ωΔ ku-

Txuri siCqariT warmoSobs damatebiT dens πω

2' Δ

=Δ= eveI ; 2'' rS π= , sadac

'S aris eleq-

tronis orbitis S farTobis gegmili 0B veqtoris marTobul sibrtyeze. amitom

0

2'22'02'''

4222B

mre

mreB

ereSI =⋅

=⋅Δ

= ππω

;

veqtorulad

0

2'2

4B

mrep −=Δ . (28.3)

sadac niSani `_” gamosaxavs mravaljer aRniSnul faqts, rom pΔ yovelTvis mimarTulia

0B –is sawinaaRmdegod.

(28.3) formula gamosaxavs erTeleqtroniani atomis inducirebul magnitur moments;

Tu atomi Seicavs Z eleqtrons, romelTa orbitis gegmilis '

ir radiusebi sxvadasxvaa,

maSin jamuri inducirebuli magnituri momenti apΔ toli iqneba:

∑=

−=ΔZ

iia r

mBep

1

2'02

4. (28.4)

(28.4) formula gviCvenebs, rom mravaleleqtroniani atomis an molekulis inducirebuli

magnituri momenti, romelic proporciulia eleqtronebis radiusebis kvadratebis jamisa,

ZiriTadad ganisazRvreba centridan meti manZiliT daSorebuli eleqtronebis orbituli

moZraobiT.

sabolood SevniSnoT, rom gareSe velis sawinaaRmdegod mimarTuli inducirebuli

magnituri momentis warmoqmna warmoadgens universalur efeqts, romelic damaxasiaTebelia

nebismieri nivTierebis atomebisaTvis.

92

§29. diamagnetikebi

diamagnetizmi warmoiSoba gareSe magnituri velis moqmedebiT eleqtronis orbitu-

li moZraobis Secvlis Sedegad. maSasadame, is damaxasiaTebelia yvela nivTierebisaTvis,

magram xSirad misi gamovlineba ar xdeba, vinaidan gadaifareba xolme ufro Zlieri para-

da feromagnetizmiT. sufTa saxiT diamagnetizmi mxolod iseT nivTierebebSi gvxvdeba,

romelTa atomebisa da molekulebis jamuri magnituri momenti gareSe magnituri velis

ararsebobis SemTxvevaSi nulis tolia. aseTi atomebisa da molekulebisagan Sedgenil

nivTierebebs diamagnetikebi ewodeba. maTi damagniteba, romelsac adgili aqvs mxolod

gareSe magnitur velSi, yovelTvis mimarTulia velis sawinaaRmdegod, amitom araerTgva-

rovan magnitur velSi xdeba diamagnetikebis gadaadgileba velis Semcirebis

mimarTulebiT.

imisaTvis, rom atomis an molekulis magnituri momenti nulis toli iyos, saWiroa,

adgili hqondes masSi Semavali eleqtronebis orbituli da spinuri magnituri momentebis

srul kompensacias. aseT kompesacias yovelTvis aqvs adgili iseT atomebSi, romlis eleq-

tronuli Sreebi ( )NMLK ,,, mTlianad aris Sevsebuli eleqtronebiT. amitom yvela iner-

tuli airi ( )RnXeKrArNeHe ,,,,, diamagnetikia. diamagneturia agreTve yvela ioni, rom-

lis eleqtronuli konfiguracia iseTivea, rogorc inertuli airis atomisa. aseTi ionebia,

magaliTad, −+ ClNa , da sxv.

spinuri magnituri momentebis sruli kompensaciisaTvis saWiroa atomi an molekula

Seicavdes eleqtronebis wyvil raodenobas. magram bevri elementi 514729 ,, SbAgCu , rom-

lebic Seicaven eleqtronebis kent raodenobas, amJRavnebs diamagnitur Tvisebebs. es imiT

aixsneba, rom am nivTierebebSi paramagnituri efeqti imdenad mcirea, rom ver faravs

diamagniturs.

garda CamoTvlili nivTierebebisa, diamagnituri Tvisebebis matarebelia ,,, CPbZn

,Hg ,Si 2,, COSGe wyali, mina, faifuri da organuli naerTebis umravlesoba.

diamagnituri atomis induqcirebuli magnituri momentisaTvis miviReT gamosaxuleba

(28.4)

∑=

−=ΔZ

iia r

mBep

1

2'02

4.

erTgvarovani izotropuli diamagnetikis SemTxvevaSi yvela atomis induqcirebuli

magnituri momenti erTnairia da velis sawinaaRmdegodaa mimarTuli, amitom moculobis

erTeulis magnituri momenti toli iqneba:

∑∑==

−=−=Δ=Z

ii

Z

iia r

mHenr

mBenpnP

1

2'02

1

2'02

44μ

,

sadac n atomebis ricxvia diamagnetikis moculobis erTeulSi. meore mxriv, (25.3)-is

Tanaxmad HkP m= amitom

∑=

−=Z

iim r

me

nk1

2'02

4μ

. (29.1)

93

rogorc (29.1) gamosaxuleba gviCvenebs, diamagnetikis magnituri amTvisebloba

damokidebuli ar aris arc temperaturaze, arc magnitur velze da elementis Z rigiTi

nomris proporciulia.

magnituri amTviseblobis temperaturaze damoukidebloba advili gasagebia. diamag-

netikis damagnitebis meqanizmi ganpirobebulia eleqtronis Siga atomuri moZraobis cvli-

lebiT (larmoris precesia), romelzec siTburi moZraoba TiTqmis ar axdens gavlenas.

(29.1) formulis marjvena nawilSi Semavali yvela sidide dadebiTia, amitom mk yov-

elTvis arsebiTad uaryofiTia, 0<mk . aqedan gamomdinare diamagnituri nivTierebis mag-

nituri SeRwevadoba naklebia erTze:

11 <+= mkμ da 00 BBB <= μ .

SevafasoT wyalbadis magnituri amTviseblobis sidide: orbitis radiusi 10105,0 −⋅≈r m,

atomebis koncentracia (normalur pirobebSi) 325107,2 −⋅= mn , amitom

931

202738225

102,1101,94

105,0104106,1107,2 −−

−−−

⋅−≅⋅⋅

⋅⋅⋅⋅⋅⋅⋅=

πmk .

cda gvaZlevs, rom wyalbadis magnituri amTvisebloba 91095,1 −⋅−=mk . rogorc vxe-

davT, Tanxvedra sakmaod kargia (miTumetes, rom formulaSi CavsviT r da ara 'r ).

vinaidan sxvadasxva nivTierebebis atomebis orbitebis radiusebi Zlier ar gansxvav-

deba erTmaneTisagan, amitom erTi da imave agregatul mdgomareobaSi maTi magnituri

amTviseblobebic Zlier ar gansxvavdebian erTmaneTisagan. marTlac, magaliTad,

argonisaTvis 91009,1 −⋅−=mk .

siTxeebsa da myar sxeulebSi atomebis koncentracia daaxloebiT 1000-jer metia, ami-

tom metia maTi magnituri amTviseblobac. magaliTad, wylisaTvis 6109 −⋅−=mk , xolo

vercxlisaTvis 61025 −⋅−=mk .

§30. paramagnetikebi

Cven aRvniSneT, rom nivTierebebi, romelTa atomebs an molekulebs ar gaaCnia mud-

mivi magnituri momenti, diamagnetikebia. nivTierebebi, romelTa atomebs da molekulebs ga-

reSe magnituri velis ararsebobis SemTxvevaSi gaaCnia mudmivi magnituri momenti, imis

mixedviT, Tu rogoria calkeul atomebs Soris urTierTqmedebis xasiaTi, SeiZleba iyos

paramagnetikebi an feromagnetikebi (agreTve antiferomagnetikebi da ferimagnetikebi).

ganvixiloT paramagnetikebi. gareSe magnituri velis ararsebobis SemTxvevaSi

atomebis magnituri momentebi qaosuradaa orientirebuli da erTmaneTs abaTilebs, ris

gamoc paramagnetikis magnituri momenti nulis tolia. gareSe magnituri velis CarTvisas

xdeba atomuri magnituri momentebis upiratesi orientacia velis gaswvriv da sxeuli iZens

nulisagan gansxvavebul jamur magnitur moments, romlis mimarTuleba emTxveva gareSe

94

velis mimarTulebas∗. sxeulis damagniteba am SemTxvevaSi ganisazRvreba statistikuri

wonasworobiT velis maorientirebel moqmedebasa da siTburi moZraobis madezorientire-

bel moqmedebas Soris. es ukanaskneli efeqti miT Zlieria, rac maRalia sxeulis

temperatura. amitom cxadia, rom diamagnetikisagan gansxvavebiT, paramagnetikis magnituri

amTvisebloba damokidebuli unda iyos temperaturaze.

paramagnetizmis klasikuri Teoria Camoayaliba frangma mecnierma lanJevenma jer

kidev 1905 wels. am TeoriaSi daSvebuli iyo, rom atomebi ar urTierTqmedeben da ar iyo

gaTvaliswinebuli magnituri momentebis sivrculi dakvantva, e. i. paramagnetiki ganxiluli

iyo rogorc idealuri airi magnituri momentebisa (isrebis), romlebic erTmaneTTan ar

urTierTqmedeben. paramagnetikis damagnitebis gamoTvla xdeboda statistikuri fizikis me-

TodebiT. amasTan igulisxmeboda, rom siTburi moZraobis gamo atomuri magnituri

momentebis orientacia qaosuri iyo.

gareSe magnituri velis moqmedebiT momentebi orientirdeba velis gaswvriv, amasTan,

atomis magnituri momentis proeqcia H –is mimarTulebaze

Θ= cosaaH pp , (30.1)

sadac Θ aris kuTxe aP -sa da H -s Soris.

Tu magnituri velis ararsebobis SemTxvevaSi Θ kuTxis kosinusis saSualo mniSvne-

loba Θcos nulis toli iyo axla ukve atomuri magnituri momentebis upiratesi orient-

taciis gamo 0cos ≠Θ . lanJevenma atomuri magnituri momentebis wonasworuli gana-

wilebisaTvis mimarTulebebis mixedviT gamoiyena bolcmanis kanoni da miiRo, rom Θcos

tolia magnitur velSi atomuri magnetikis potenciuri energiis ( )HpE a μμ0=p naxevaris

fardobisa siTburi moZraobis kinetikur energiasTan ⎟⎠⎞

⎜⎝⎛ = kTE

23

k :

kT

Hp

kT

Hpa

a

323

21

cos 00 μμ

==Θ , vinaidan 1=μ . (30.2)

aRsaniSnavia, rom (30.2) formula miRebulia daSvebiT: kTHpa <<0μ . Zlieri magnitu-

ri velisa da Zalian dabali temperaturis dros myardeba magnituri gajereba, p -sa da

H –s Soris wrfiv kavSirs adgili aRara aqvs da (30.2) formuliT gamosaxuli kanonzomie-

reba irRveva.

velis mimarTulebaze magnituri momentis proeqciis saSualo mniSvnelobisaTvis

gveqneba:

kTHppp a

aaH0

2

31cos μ

=Θ= ,

xolo damagnitebis veqtorisaTvis

∗ rogorc wina paragrafSi iyo aRniSnuli, orientaciuli (paramagnituri) efeqtis garda nebismieri nivTierebis atomebSi adgili aqvs induqcirebul (diamagnitur) efeqts. paramagnetikis damagniteba am ori urTierTsawinaaRmdego efeqtis moqmedebiT xdeba. amasTan, masSi orientaciuli efeqti ufro Zlieria, vidre induqciuri, amitom sxeuli magnitdeba gareSe velis mimarTulebiT.

95

aHpnP = ,

sadac n atomebis koncentraciaa.

magnituri amTviseblobisaTvis miiReba sidide:

TC

kTpn

HPk a

m ===3

20μ , (30.3)

sadac kpnC a

3

20μ= damokidebulia nivTierebis gvarobaze da mas kiuris mudmiva ewodeba.

TCkm = Tnafardobas kiuris kanoni ewodeba, vinaidan es kanonzomiereba eqsperimen-

tulad dadgenili iyo kiuris mier 1895 wels (lanJevenis Teoriis Seqmnamde), Tumca mas

Teoriuli dasabuTeba ar hqonda. am kanonzomierbam Teoriuli dasabuTeba hpova lanJevenis

SromaSi. (30.3) Tanafardobas mTlianad lanJevenis gantoleba ewodeba.

paramagnetizmis kvanturi Teoria, romelic iTvaliswineb atomis magnituri momentis

sivrcul dakvantvas, magnituri amTviseblobis temperaturaze damokidebulebisaTvis iZle-

va (30.3) Tanafardobis analogiur gamosaxulebas.

SevafasoT paramagnituri nivTierebebis magnituri amTvisebloba oTaxis temperatu-

raze ( )CT 293= . airisaTvis 325107,2 −⋅≈ mn , ap -s nacvlad CavsvaT misi umciresi mniS-

vneloba _ boris magnetoni. gveqneba:

723

746225

103,22931038,13

10410927,0107,2 −−

−−

⋅≈⋅⋅⋅

⋅⋅⋅⋅⋅=

πmk .

cda gvaZlevs, rom oTaxis temperaturaze Jangbadis amTvisebloba 71086,1 −⋅=mk ,

xolo azotis Jangisa ( ) 7108,0 −⋅=mkNO .

myari paramagnetikebis magnituri amTvisebloba gacilebiT meti unda iyos, rac das-

turdeba cdiT.

amrigad, paramagnetikebis magnituri amTvisebloba daaxloebiT 100-jer metia diamag-

netikebisaze. amitomaa, rom paramagnitur nivTierebebSi ar xdeba diamagnituri Tvisebebis

gamovlena. aq diamagnituri efeqti gadaifareba ufro mZlavri paramagnituri efeqtiT.

dabolos, SevniSnoT, rom liTonis magnitur Tvisebebze gavlenas axdens is

garemoeba, rom birTvidan yvelaze ufro daSorebuli savalento eleqtronebi, romelTa

erToblioba qmnis kvantur kanonebs daqvemdebarebul eleqtronul airs, faqtiurad Ta-

visufalia. gareSe magnituri veli orgvar moqmedebas axdens Tavisufal eleqtronebze.

erTi mxriv, magnituri veli amrudebs eleqtronebis moZraobis traeqtorias (romelic

dajaxebebs Soris wrfivi iyo) da aiZulebT maT imoZraon xraxnul wirebze, romelTa

RerZebi Tanxvdeba gareSe magnituri velis mimarTulebas. l. landaum aCvena, rom,

eleqtronebis aseTi moZraoba iwvevs liTonis damagnitebas velis sawinaaRmdegod e.i.

warmoSobs diamagnetizms. mas landaus diamagnetizmi ewodeba. meore mxriv, el-

eqtronebi, romlebsac gaaCnia spinuri magnituri momentebi, magnituri velis moqmedebis

Sedegad ganicdian maorientirebel moqmedebas. es iwvevs liTonis damagnitebas gareSe

velis mimarTulebiT, e.i. warmoSobs paramagnitur efeqts.

landaum Teoriulad gamoiTvala, rom Tavisufali eleqtronebis paramagnituri

efeqti samjer metia diamagniturze, amitom eleqtronuli airi mTlianad paramagnituria.

96

§31. magnituri velis cirkulacia nivTierebaSi

§8-Si gamovTvaleT magnituri velis 0B induqciis veqtoris cirkula-

cia vakuumSi. ganvazogadoT miRebuli Sedegi nivTierebisaTvis. viciT, rom

nivTierebaSi '

0 BBB += , sadac 'B sxeulSi mikrodenebis mier Seqmnili

velis induqciaa. amitom cirkulacia

( ) ( ) ∫∫∫ ⎟⎠⎞⎜

⎝⎛+=

lll

ldBldBldB'

0

(8.2)-is Tanaxmad

( ) ∑∫ == .000 makroIIldB kl

μμ

analogiurad '

B veqtoris cirkulaciisaTvis SeiZleba daiweros

.0

'

mikroIldBl

μ=⎟⎠⎞⎜

⎝⎛∫

makroI da mikroI Sesabamisad im makroskopuli, anu gamtarobis da mikro-

skopuli, anu molekuluri denebis algebruli jamia, romlebsac moicavs

Caketili konturi (anu is marezultirebeli makro- da mikro- denebia,

romlebic gadis l konturiT SemosazRvrul S zedapirSi).

maSasadame, nivTierebaSi jamuri magnituri velis induqciis veqtoris

cirkulaciisaTvis gveqneba:

( ) ( )mikromakro IIldBl

+=∫ 0μ . (31.1)

gamovTvaloT mikroI . SevniSnoT, rom molekuluri denebis algebrul

jamSi wvlils gvaZlevs mxolod is molekuluri denebi, romlebic `asxmu-

li~ arian l konturze (sur. 31.1a). marTlac, molekuluri denebi, romlebic

ar akmayofilebs am pirobas, an saerTod ar gadakveTen S zedapirs (deni

mol'I ), an gadakveTen orjer urTierTsawinaaRmdego mimarTulebiT (deni mol

''I ).

gamovyoT l konturis mcire dl elementi da davuSvaT, rom kuTxe P

damagnitebis veqtorsa da dl -s Soris aris α (sur. 31.1 b). suraTidan Cans,

rom dl elementze `asxmulia~ mxolod is molekuluri denebi, romelTa

centrebi moTavsebulia irib cilindrSi, romlis simaRlea dl da fuZe molS ,

sadac molS _ molekuluri deniT SemosazRvruli farTobia. am cilindris

moculoba αcos⋅= dlSdV mol . Tu molekulebis koncentracias aRvniSnavT n –

iT, maSin molekulebis raodenoba cilindrSi

97

αcos. ⋅⋅= dlSndn mol

xolo jamuri dI mikrodeni, romelsac `moicavs~ dl elementi

.cosα⋅⋅= dlSnIdI mol.mol.mikro

magram apSI =mol.mol. aris calkeuli molekulis magnituri momenti,

xolo anpSIn =⋅ mol.mol. _moculobis erTeulis magnituri momenti, anu damag-

nitebis veqtori P . amitom

( )dlPPdldI =⋅= αcosmikro . (31.2)

jamuri mikrodeni ( )mikroI , romelsac moicavs l Caketili konturi mi-

iReba (31.2) gamosaxulebis integrebiT:

( )∫=l

dlPImikro .

mikroI -s miRebuli mniSvneloba SevitanoT (31.1) formulaSi:

( ) ( )∫∫ +=ll

dlPIldB 00 μμ makro ,

saidanac yvela wevris 0μ –ze gayofisa da mcire gardaqmnebis Semdeg mivi-

RebT:

makroIdlPB

l

=⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⋅⎟⎟⎠

⎞⎜⎜⎝

⎛−∫

0μ. (31.3)

⎟⎟⎠

⎞⎜⎜⎝

⎛− PB

0μveqtors uwodeben magnituri velis daZabulobis veqtors da

aRniSnaven H –iT:

PBH −=0μ

(31.4)

da sabolood gveqneba:

a b sur. 31.1

P

l

'mol.I

''mol.I

dl

α

98

( ) makroIdlHl

=∫ . (31.5)

(31.5) formula warmoadgens (8.3) formulis ganzogadebas da gamosaxavs

sruli denis kanons nivTierebaSi: magnituri velis daZabulobis

veqtoris cirkulacia Caketili konturis mimarT tolia im makro-

skopuli denebis algebruli jamisa, romelsac es konturi moi-

cavs. rogorc vxedavT, damxmare veqtoris _ magnituri velis daZabulobis

veqtoris Semotanam saSualeba mogvca gamogvericxa cirkulaciis Teore-

midan mikrodenebi (rac niSnavs imas, rom H . ar aris damokidebuli niv-

Tierebis gvarobaze).

erTgvarovani, izotropuli garemos SemTxvevaSi HkP m= , amitom

HkBH m−=0μ

,

( )0

1μBkH m =+ ,

da

μμ0

BH = (31.6)

sadac mk+= 1μ _ nivTierebis magnituri SeRwevadobaa.

davuSvaT, makrodeni vakuumSi qmnis erTgvarovan vels, romlis induq-

ciaa 0B . vTqvaT, es aris solenoidis magnituri veli. (31.6)–is Tanaxmad am

velis daZabuloba

0

00

μBH = . (31.7)

SevitanoT solenoidSi cilindruli formis erTgvarovani, izotropu-

li magnetiki. igi damagnitdeba, amasTan P damagnitebis veqtori kolenia-

luri iqneba 0B veqtorisa. viciT ($25), rom magnetikSi aRZruli sakuTari

velis induqcia PB 0

'μ= , amitom jamuri veli magnetikSi

PBBBB 00'

0 μ+=+= . (31.8)

B -s miRebuli mniSvneloba SevitanoT (31.4)–Si

0

0

0

0

00

0

HBPPB

PBH ==−+

=−=μμ

μμ

.

miviReT, rom magnituri velis daZabuloba magnetikSi ( )H Tanxvdeba

velis daZabulobas vakuumSi ( )0H . miRebuli Sedegi samarTliania mxolod

99

maSin, rodesac erTgvarovani, izotropuli magnetiki mTlianad avsebs im

sivrces, sadac 00 ≠H . Tu es pirobebi ar sruldeba, maSin PBH −=0μ

ar

udris 0

00

μBH = -s.

§32. feromagnetikebi

rogorc 24-e paragrafSi iyo aRniSnuli, paramagnituri sxeulebidan

gamoiyofa mcirericxovani, magram didi ptaqtikuli mniSvnelobis jgufi

sxeulebisa, romlebSic mikrodenebis mier Seqmnili magnituri veli

(sakuTari veli) asjer da aTasjer SeiZleba sWarbobdes damamagnitebel

gareSe vels (makrodenebis vels). maT feromagnetikebi ewodebaT. feromag-

netikebia: rkina, nikeli, kobalti, mTeli rigi Senadnobebi da sxv.∗ fero-

magnetizmi maTSi Tavs iCens mxolod kristalur mdgomareobaSi.

para- da diamagnetikebisagan gan-

sxvavebiT (romlebisTvisac μ erTisa-

gan umniSvnelod gansxvavdeba), fero-

magnituri sxeulebisaTvis magnituri

SeRwevadoba Zalian did mniSvnelo-

bebs aRwevs. I cxrilSi moyvanilia

zogierTi feromagnetikis magnituri

SeRwevadobis maqsimaluri mniSvnelo-

bebi.

feromagnituri masalebis far-

To gamoyeneba mxolod me-19-e saukunis bolos daiwyes, roca warmoiSva

eleqtruli manqanebisa da magnituri wredis aparatebis Seqmnis aucileblo-

ba, romlebsac eqneboda rac SeiZleba mcire magnituri winaRoba.

feromagnetikebis eqsperimentuli Seswavla dawyebul iqna stoletovis

mier 1871 wels. aRmoCnda, rom feromagnituri sxeulebisaTvis damaxasiaTebe-

lia magnituri SeRwevadobis ara marto didi mniSvneloba, aramed misi

damokidebuleba magnitur velze: H –is gazrdiT μ Tavdapirvelad swrafad

izrdeba, aRwevs maqsimums, Semdeg mcirdeba da (Zlier didi velebis

∗ feromagnituri Tvisebebis matarebelia agreTve gadoliniumi, erbiumi, dispoziumi, tuliumi, golmiumi da Terbiumi. amasTan, ukanaskneli xuTi iSviaTmiwaTa elementebis feromagnetizmi Tavs iCens mxolod metad dabal temperaturebze.

cxrili I

feromagnituri nivTiereba μ

kobalti ~150

nikeli ~250

Tuji ( )CFe %3+ ~2000

rkina ~5000

kaJiani rkina ( )SiFe %3,3+ ~10000

permaloi ( )FeNi %28%78 + ~100000

100

SemTxvevaSi) miiswrafvis erTisaken∗ (sur. 32.1 a). amitom, Tumca formula

HB μμ0= marTebulia feromagnetikebisTvisac, am sxeulebSi magnituri

induqcia ukve aRar aris damamagnitebeli velis proporciuli: velis

daZabulobis SedarebiT mcire mH mniSvnelobis dros induqcia aRwevs did

(gajerebis) mB mniSvnelobas, ris Sedegadac is icvleba ukve nela, H -is

proporciulad (sur. 32.1, b), e. i. daaxloebiT ise, rogorc paramagnetikSi.

feromagnetikis damagnitebis mrudi gamosaxulia 32.2 suraTze. aRsaniSnavia,

rom damamagnitebeli velis ukve ~(10 _100) a/m mniSvnelobebis dros adgili

aqvs gajerebas.∗∗ SevniSnoT, rom ( )HfP = mrudis gansakuTrebiT swrafi

cvlileba Seesabameba ( )Hf=μ mrudis maqsi-

mums.

feromagnituri sxeulebisaTvis damaxa-

siaTebelia magnituri histerezisis movlena _

feromagnetikis gadamagnitebis sruli ciklis

Sesrulebis Sedegad miiReba Caketili mrudi

(sur. 32.3), romelsac histerezisis maryuJi

ewodeba. aseTi mrudis miRebas ganapirobebs

is garemoeba, rom cvlad magnitur velSi

feromagnetikis damagnitebis cvlileba Camor-

∗ marTlac,

( ).11

0

0

0

→+=+

==HP

HPH

HB

μμ

μμ roca ∞→H .

∗∗ ( )HfB = mrudis saxe eTanadeba ( )HfP = mruds: gajerebis momentSi constP = da

magnituri induqcia ( )PHB += 0μ izrdeba H -is proporciulad.

mH H

mP

P

O

sur. 32.2.

H

1

H mH

mB

Bμ

O O

a b sur. 32.1

101

Ceba magnituri velis cvlilebas.∗ histerezisis mrudis ZiriTadi maxasia-

Teblebia rP narCeni magnetizmi da kH koercitiuli Zala.∗∗ visargebloT

32.3 suraTiT da mivyveT ( )HfP = mrudis cvlilebas: Tavdapirvelad, H -is

zrdiT izrdeba P –c da mH –is raRac mniSvnelobis dros miiReba gajereba

mP . es procesi mimdinareobs OA mrudis

Sesabamisad, romelsac Tavdapirveli

damagnitebis mrudi ewodeba. Tu amis

Semdeg daviwyebT H –is Semcirebas, Sem-

cirdeba damagnitebac, magram P –s Sem-

cireba CamorCeba H –is Semcirebas, ris

gamoc, roca rr PPPH ;0,0 ≠== gansa-

zRvravs narCen magnetizms. mis mosax-

snelad saWiroa sxeulze movdoT sawi-

naaRmdego mimarTulebis veli, romlis raRac kH mniSvnelobis dros rP

xdeba nulis toli. kH -s ewodeba koercitiuli Zala. am velis Semdgomi

zrda gamoiwvevs feromagnetizmis damagnirebas sawinaaRmdego mimarTu-

lebiT, romlis - mH mniSvnelobis dros aRwevs najerobas ( )mP− , xolo

misi nulamde Semcirebisas miiReba narCeni magnetizmi ( )rP− . magnituri ve-

lis dadebiTi mniSvnelobis ganmeorebiTi gazrda gamoiwvevs sxeulis damag-

nitebas _ APr mrudis Sesabamisad da mrudi Seikvreba. amrigad, rogorc

suraTidan Cans, feromagnetikis damagnitebis

veqtori ar warmoadgens damamagitebeli velis

calsaxa funqcias. P –s mniSvneloba damokidebu-

lia ara marto H –is mniSvnelobaze, aramed

damagnitebis winaistoriazec: velis erTsa da

imave 0H mniSvnelobas Seesabameba damagnitebis

sami sxvadasxva mniSvneloba ( )321 ,, PPP . histerezi-

sis maryuJiT SemosazRvruli farTobi gansa-

zRvravs feromagnetikis gadamagnitebis muSao-

bas, romelic sxeulSi gamoiyofa siTbos saxiT.

∗ histerezisi berZnuli sityvaa da niSnavs CamorCenas. ∗∗ sityva `koerticiuli~ laTinuri warmoSobisaa da niSnavs Semakavebels.

cxrili II

nivTiereba

kiuris wertili

C kobalti 1127

rkina 770

nikeli 357

gadoliniumi 16

Terbiumi -43

disproziumi -168

tuliumi -222

erbiumi -253

102

cxadia, am muSaobis sidide damokidebulia narCen magnetizmze da

koercitiul Zalaze. es udevs safuZvlad feromagnetikebis dayofas rbil-

magnitur da xistmagnitur sxeulebad. rbilmagnituri sxeulebisaTvis rP da

kH mcirea (mcirea, maSasadame, gadamagnitebis dros gamoyofili siTbos

raodenobac) da isini gamoiyeneba. magaliTad, transformatoris gularebis

dasamzadeblad. xistmagnituri sxeulebisaTvis, piriqiT, rP da kH didia da

isini gamoiyeneba mudmivi magnitebis dasamzadeblad.

yoveli feromagnetikisaTvis arsebobs gansakuTrebuli temperatura,

e.w. kiuris wertili, romlis zeviTac sxeuli kargavs feromagnitur

Tvisebebs da iqceva ise, rogorc Cveulebrivi paramagnetiki. II cxrilSi

moyvanilia zogierTi feromagnetikis kiuris wertili.

§33 feromagnetizmis axsna

avxsnaT feromagnetizmi, gavarkvioT misi buneba. feromagnetizmis Ta-

viseburebani ar SeiZleba aixsnas ubralo klasikuri warmodgenebis safuZ-

velze. am Taviseburebebis asaxsnelad frangma fizikosma veisma 1907 w. wamo-

ayena ori hipoTeza, romelic safuZvlad daedo feromagnetizmis naxevrad

fenomenologiur Teorias.

pirveli hipoTezis Tanaxmad yovel feromagnetikSi kiuris wertilze

dabla adgili aqvs gareSe magnituri velisagan damoukidebel spontanur

(TavisTavad) damagnitebas.

am hipoTezidan gamomdinareobs, rom yoveli feromagnituri sxeuli

yovelTvis unda iyos damagnitebuli, rac cdiT ar dasturdeba.

Seqmnili winaaRmdegobis gadasalaxavad veisma wamoayena meore hipo-

Teza, romlis Tanaxmadac yoveli feromagnetiki kiuris temperaturaze dab-

la iyofa mcire sididis ( )sm210−≈ areebad (domenebad), romelTa fargleb-

Si adgili aqvs spontanur damagnitebas najerobamde.∗ amasTan, gareSe magni-

turi velis ararsebobis SemTxvevaSi domenebis magnituri veqtorebis

orientacia qaosuria (sur. 33.1,a), ris gamoc sxeulis jamuri damagniteba

udris nuls.

gareSe magnituri velis moqmedebiT xdeba domenebs Soris sazRvrebis

gadaadgileba da maTi moculobis Secvla. is domenebi, romlebic orienti-

∗ am TvalsazrisiT feromagnetiki hgavs segnetoeleqtriks, romelSic agreTve aris TavisTavadi polarizaciis areebi.

103

rebulia velis gaswvriv, izrdeba sxvagvarad orientirebuli domenebis xar-

jze da sxeuli magnitdeba (sur. 33.1, b). gareSe magnituri velis garkveuli

mH mniSvnelobis dros domenebs Soris sazRvrebis gadaadgilebis procesi

damTavrebulia, yvela domeni Semotrialebulia velis gaswvriv da sxeuli

warmoadgens erT mTlian domens (sur. 33.1, g). am mdgomareobas Seesabameba

feromagnetikis damagniteba najerobamde.

veisis pirveli hipoTeza safuZvlad udevs TavisTavadi damagnitebis

Teorias, romelic feromagnetizmis bunebis axsnis gasaRebs warmoadgens.

meore hipoTeza safuZvlad udevs damagnitebis mrudis Teorias, ro-

melic xsnis feromagnetikis qcevas gareSe magnitur velSi moTavsebisas.

feromagnetizmis bunebis gasarkvevad saWiroa pasuxi gaeces Semdeg

kiTxvebs:

1. eleqtronebis romeli magnituri momentebi _ orbituli Tu spinuri

_ ganapirobebs feromagnetizms?

2. ra aris spontanuri damagnitebis mizezi?

3. ra aris feromagnetikis domenebad dayofis mizezi?

dasmul kiTxvebze sruli pasuxis gacema eyrdnoba rTul kvantur-meqa-

nikur gamoTvlebs da scildeba Cveni kursis farglebs. ganvixiloT am

gamoTvlebidan gamomdinare mokle pasuxebi.

1. pirvel kiTxvaze pasuxis gacema xdeba giromagnituri cdebis safuZ-

velze, romelic adre ganvixileT. cdiT miRebuli giromagnituri fardobis

saSualebiT dadginda, rom feromagnetizms ganapirobebs eleqtronis

spinuri da ara orbituli momenti.

magram spini gaaCnia ara marto rkinis, nikelis, kobaltis atomebis

eleqtronebs, aramed, spilenZis, vercxlis, oqros atomebis eleqtronebsac

a b g sur. 33. 1

0=H H mH

104

da saerTod, yvela eleqtrons. ratomaa, rom spilenZi, vercxli da oqro

araferomagnituri arian?

feromagnituri nivTierebebis atomebis eleqtronuli struqturis gan-

xilva gviCvenebs, rom maT aqvT Seuvsebeli Siga eleqtronuli qveSreebi,

romelTa spinebi arakompensirebulia. mTlianad Sevsebuli qveSre yovel-

Tvis magniturad neitraluria, vinaidan adgili aqvs eleqtronuli spinuri

momentebis srul kompensacias.

amrigad, SeiZleba davaskvnaT, rom nivTierebebis feromagnetizmis

arsebobisaTvis aucilebeli pirobaa mis atomebSi Seuvsebeli qveSrisa da

maSasadame arakompensirebuli spinuri momentebis arseboba.

2. nivTierebis feromagnetizmisaTvis mis atomebSi arakompensirebuli

spinebis arseboba warmoadgens aucilebel, magram arasakmaris pirobas.

marTlac, Seuvsebeli Siga eleqtronuli Sre aqvT ara marto fero-

magnituri nivTierebebis atomebs, aramed sxva elementebis atomebsac. aseTe-

bia rkinis jgufis gardamavali elementebi ( )NiCoFeMnVCrTiSc ,,,,,,, . maSasa-

dame, saWiroa kidev atomebs Soris adgili hqondes iseT urTierTqmedebas,

romelic ganapirobebda spinebis paralelur orientacias da maSasadame,

spontanur damagnitebas.

sabWoTa fizikosebis i.dorfmanisa da i.frenkelis, agreTve germaneli

fizikosis haizenbergis Sromebis Sedegad dadginda, rom feromagnetikis

kristalur meserSi sxvadasxva atomebis Seuvsebeli qveSris eleqtronebs

Soris adgili aqvs e.w. gacvliT urTierTqmedebas, romelic energetikulad

ufro xelsayrels xdis spinebis paralelur orientacias, rac ganapirobebs

spontanur damagnitebas.

3. feromagnetizmis domenebad dayofis mizezi unda veZioT aseTi

struqturis warmoqmnis energetikul xelsayrelobaSi.

marTlac, ukve viciT, rom gacvliTi urTierTqmedeba qmnis im pi-

robebs, rom mTel feromagnetikSi Seiqmnas erTgvarovani spontanuri damage-

niteba. amis Sedegad, marTalia, gacvliTi urTierTqmedebis energia iqne-

boda minimumi, magram sasruli zomis feromagnetikis boloebze

warmoiqmneboda magnituri polusebi, e.i. miiReboda mudmivi magniti

(sur. 33.2,a). feromagnetiis aseT mdgomareobas Seesabameba mis mier aRZruli

magnituri velis mniSvnelovani sididis energia. feromagnetikis domenebad

dayofisas magnituri veli da masTan dakavSirebuli energia mcirdeba

(sur. 33.2, b). landaum da lifSicma aCvenes, rom energetikulad yvelaze uf-

105

ro xelsayrelia iseTi domenuri struqtura, rodesac spontanur damage-

nitebasTan dakavSirebuli magnituri nakadi ikvreba feromagnetikis SigniT

(sur. 33.2, g).

rac Seexeba gacvliTi urTierTqmedebis energias, is yvela eleqtroni-

saTvis, garda domenebis sazRvrebze myofisa, ucvleli rCeba. aRmoCnda, rom

feromagnetikis domenebad dayofisas, erTi mimarTulebis domenidan meore-

ze mkveTri gadasvla energetikulad xelsayreli ar aris, amitom warmoiq-

mneba sasazRvro fena, romlis saSualebiTac xdeba erTi domenidan meoreze

TandaTanobiTi gadasvla. es, cxadia, zrdis gacvliTi urTierTqmedebis

energias, magram gamoTvlebi gviCvenebs, rom es efeqti mcirea magnituri

velis energiis SemcirebasTan SedarebiT, amitom saboloo jamSi feromag-

netikis domenebad dayofa energetikulad xelsayrel process warmoadgens.

da bolos SevniSnoT, rom sasazRvro fenaSi myofi atomebis gacvliTi

energia SeiZleba ganvixiloT, rogorc zedapiruli energia, vinaidan is

proporciulia domenebis gamyofi zedapiris farTobisa. feromagnetikis

domenebad dayofisas gamyofi zedapirebis farTobi izrdeba. masTan erTad

izrdeba gacvliTi urTierTqmedebis sruli energiac, xolo magnituri

velis energia mcirdeba. domenebad dayofa Sewydeba maSin, roca magnituri

da gacvliTi energiebis jami minimumi gaxdeba. minimumis es piroba gansa-

zRvravs domenebis zomas.

arsebobs domenuri struqturi eqsperimentu-

li dakvirvebis ramdenime meTodi, romelTagan yve-

laze martivi da TvalsaCinoa akulov-biteris me-

TodiT. 1931 wels sabWoTa mecnierebma n.akulovma

da m.dextiarma da maTgan damoukideblad amerikel-

ma mecnierma s.biterma mogvces domenebis uSualo

N

N S

SS

N

SP

a b g sur. 33. 2

sur. 33.3

106

dakvirvebis meTodi: feromagnituri kristalis zedapiri gulmodgined po-

lirdeba da masze daaqvT feromagnituri nivTierebis (magaliTad, magnetiti

32OFe ) metad wvril marcvlovani fxvnilis wylis suspenzia. nawilakebis

ganlageba zedapirze iRebs seleqciur xasiaTs: magnituri Zalebis

moqmedebiT isini moZraoben domenebis sazRvrebisken (sadac xdeba magnituri

velis Zalwirebis gamosvla), iq ileqebian da amrigad, gvaZleven domenuri

struqturis moxazulobas, romelic kargad Cans mikroskopSi (sur. 33.3).

§34. feromagnetizmis damagnitebis meqanizmi

Cven gavarkvieT, rom feromagnetizmis movlenas safuZvlad udevs spontanuri

damagniteba. gareSe magnituri velis ararsebobis SemTxvevaSi feromagnituri sxeuli iyofa

domenebad, romelTa magnituri momentebi orientirebulia ioli damagnitebis mimarTule-

bebiT, Tanac ise, rom kuTxe mezobeli domenebis magnitur momentebs Soris an 180-ia

(antiparaleluri orientacia) an 90°. domenebis aseTi orientacia energetikulad xelsay-

relia, vinaidan spontanur damagnitebasTan dakavSirebuli magnituri nakadi ikvreba

feromagnetikis SigniT.

amrigad, gareSe magnituri velis ararsebobis SemTxvevaSi domenebis orientacia ise-

Tia, rom feromagnetikis jamuri magnituri momenti udris nuls (sur. 34.1,a). gareSe magni-

tur velSi Setanisas feromagnetikis iZens nulisagan gansxvavebul magnitur moments _ is

magnitdeba. feromagnetikSi momdinare fizikuri movlenebis xasiaTis mixedviT damagnitebis

procesi sam stadiad iyofa: domenebs Soris sazRvrebis gadanacvlebis procesi, damag-

nitebis veqtoris brunvis procesi da paraprocesi. damagnitebis mrudze (sur. 34.2) am pro-

cesis Sesabamisi monakveTebia: ABOA, da BC . ganvixiloT TiToeuli procesi calcalke.

1. gadanacvlebis procesi. SevitanoT feromagnetiki garkveuli mimarTulebis mcire

sididis magnitur velSi. maSin feromagnetikis saerTo energias daemateba misi magnitur

velTan urTierTqmedebis energia. am SemTxvevaSi energiis minimumis piroba, cxadia, ganxor-

cieldeba domenebis magnituri momentebis sxva orientaciisaTvis, rac gamoiwvevs feromag-

netikis damagnitebas. marTlac, im domenebis energia, romelTa magnituri momentebi velis

mimarTulebasTan adgens did kuTxes, meti gaxdeba im domenebis enrgiaze, romelTaTvisac

es kuTxe SedarebiT mcirea. amis Sedegad moxdeba energetikulad xelsayreli orientaciis

mHH < mH

a b g d sur. 34.1

H

107

domenebis zrda mezobeli domenebis xarjze. es zrda xorcieldeba domenebs Soris

sazRvrebis gadanacvlebiT, amitom process ewodeba gadanacvlebis procesi.

Cvens naxazze energetikulad yvelaze ufro xelsayreli orientacia aqvs zeda

domens. amitom igi gaizrdeba mezobeli domenebis xarjze (sur. 34.1b). gareSe velis zrda

agrZelebs gadanacvlebis process, romelic velis garkveuli mHH < mniSvnelobis dros

mTavrdeba. am dros feromagnituri sxeuli warmoadgens erT did domens (sur. 34.1 g). da-

magnitebis mrudze (sur. 34.2) gadanacvlebis process Seesabameba OA monakveTi. am monakve-

Tis dasawyisSi ( )'OO , H -is mcire mniSvnelobebis dros, damagniteba mimdinareobs Tanda-

Tan da process aqvs Seqcevadi xasiaTi _ magnituri velis moxsnisas damagniteba mTlianad

ispoba. magnituri velis Semdegomi zrdisas damagnitebis procesi Seuqcevadia ( AO '

monakveTi).

2. brunvis procesi. gadanacvlebis procesis

damTavrebis Semdeg feromagnetikis mP magnituri mo-

menti (spontanuri damagnitebis veqtori) maqsimalu-

ria, magram is adgens garkveul kuTxes H –is mimar-

TulebasTan. H –is Semdgomi zrda iwvevs damag-

nitebis veqtoris TandaTanobiT Semobrunebas velis

mimarTulebiT. Semobrunebis es procesi mTavrdeba

H –is mH mniSvnelobis dros. feromagnetikis damag-

niteba aRwevs teqnikur najerobas (sur. 34.1 d).

3. paraprocesi. teqnikuri najerobis miRwevis

Semdeg, H -is zrdiT feromagnetikis damagniteba Zalian nela, magram mainc izrdeba (BC

monakveTi). es imiT aixsneba, rom absoluturi nulisagan gansxvavebul temperaturaze spon-

tanurad damagnitebuli areebis yvela spini urTierTparaleluri ar aris. siTburi moZra-

obis gamo atomebis nawilis spinebs antiparaleluri orientacia aqvs (sur. 34.3). Zlieri

magnituri velis moqmedebiT xdeba aseTi spinebis Semobruneba 180-iT, rac iwvevs sxeuli

damagnitebis zrdas. teqnikur najerobamde damagnitebul feromagnetikis damagnitebis

zrdis process paraprocesi ewodeba.

davuSvaT, feromagnetiki imyofeba teqnikuri naje-

robis mdgomareobaSi. daviwyoT magnituri velis Semcireba.

es gamoiwvevs magnituri urTierTqmedebis Semcirebas da

spontanuri damagnitebis veqtoris gadaxras H –is mimar-

Tulebidan. H –is SemcirebiT damagnitebis veqtori sul

ufro da ufro Sordeba H –is mimarTulebas da

uaxlovdeba uaxloesi ioli damagnitebis mimarTulebas. rodesac veli Semcirdeba

nulamde, sxeulis damagniteba nulis toli ar iqneba, vinaidan atomebis spinuri magnituri

momentebi umTavresad orientirebuli iqneba ioli damgnitebis mimarTulebiT. amiT aixsneba

narCeni damagnitebis warmoSoba. mis mosaspobad saWiroa kristalze movdoT sawinaaRmdego

C B

A

'O

mH H

mP

P

O

sur. 34.2.

sur. 34. 3.

108

mimarTulebis veli, romelic zemoT aRwerili sqemis mixedviT gamoiwvevs domenebis

moculobis cvlilebas _ zogs gazrdis, zogs ki Seamcirebs.

feromagnetikis damagnitebis ganxiluli sqema, Tumca sworad asaxavs damagnitebis

calkeul etaps, magram idealizebulia. realur pirobebSi adgili aqvs gadanacvlebis,

brunvis da paraprocesebis zeddebas. dabejiTebiT SeiZleba mxolod imis mtkiceba, rom

susti velebis SemTxvevaSi sWarbobs gadancvlebis procesi, xolo Zlier velebSi _

brunvisa da paraprocesebi.

Tavi IV

cvladi deni

$35. cvladi denis miReba

cvladi deni warmoadgens iZulebiT eleqtromagnitur rxevebs. vnaxoT,

rogor xdeba misi miReba.

mudmiv da erTgvarovan magnitur velSi movaTavsoT mavTulis abcd

xvia (CarCo). am xviis Tanabari brunvisas (magnituri velis marTobuli) 'OO

RerZis irgvliv (sur. 35.1), misi farTobis gamWoli magnituri induqciis

nakadi ganuwyvetliv icvleba rogorc sididiT, ise mimarTulebiT. eleq-

tromagnituri induqciis kanonis Tanaxmad, xviaSi aRiZvreba sididiT da mi-

marTulebiT cvladi induqciis em Zala. Tu xvias gavwyvetT da mis boloebs

mivuerTebT raime momxmarebels, momxmarebelSi idens sididiT da

mimarTulebiT cvladi deni.

amrigad, magnitur velSi mbrunavi xvia warmoadgens cvladi denis

wyaros, xolo zemoT aRwerili mowyobiloba _ cvladi denis umartivesi

(erTfaziani) generatoris models.

gamoviTvaloT xviaSi aR-

Zruli induqciis em Zala.

vTqvaT, sawyis momentSi ( )0=t ,

xviis sibrtye marTobulia

magnituri induqciis wirebis:

B -sa da CarCos normals So-

ris kuTxe 0=α . maSin misi

gamWoli magnituri induqciis

nakadi 0Φ maqsimaluria da ga-

moiTvleba formuliT:

const=ω

d

b

a

O

'O

c constB =

sur. 35.1

109

BS=Φ 0 , sadac S _ xviiT SemosazRvruli farTobia. Tu xvia brunavs ω

kuTxuri siCqariT, maSin t droSi is Semobrundeba tωα = kuTxiT,

induqciis nakadi Semcirdeba da gaxdeba

tBS ωα coscos0 =Φ=Φ . (35.1)

(35.1) gviCvenebs, rom xviis brunvisas misi gamWoli magnituri nakadi

periodulad icvleba. (18.2)-is Tanaxmad, xviaSi aRZruli induqciis em Zala

tSitSinBSdtd ωεωωε 0==Φ

−= , (35.2)

sadac ωε BS=0 _ em Zalis maqsimaluri mniSvnelobaa, (romelsac igi iRebs

roca 1=tSinω ).

amrigad, mudmiv magnitur velSi (gamtari) xviis Tanabari brunvisas,

masSi aRiZvreba cvladi em Zala, romelic sinusis kanoniT icvleba. es em

Zala momxmarebelSi warmoqmnis sinusoidur cvlad dens

tItRR

I ωωεε sinsin 0

0 === . (35.3)

sadac R

I 00

ε= denis maqsimaluri mniSvnelobaa, xolo R _ momxmareblis wina-

Roba. (35.3) gviCvenebs, rom cvladi deni warmoadgens rxeviT process

(harmoniul rxevas), amitom, rxeviTi procesis damaxasiaTebeli sidideebis

dasaxeleba (tomi I, $36), ZalaSi rCeba cvladi denis SemTxvevaSic. saxel-

dobr, 0ε -s ewodeba em Zalis amplituda, 0I -s _ denis Zalis amplituda, ω -s

_ denis wriuli sixSire, tω -s _ cvladi denis faza. cvladi deni

xasiaTdeba agreTve periodiT ( )T da sixSiriT ( )v , amasTan vTt

ππαω 22=== .

yofili sabWoTa kavSirisa da evropis mTel rig qveynebSi

sayofacxovrebo da teqnikuri miznebisaTvis iwarmoeba cvladi deni,

romlis sixSire 50=v hercs. aSS-Si samrewvelo denis sixSire 60 hercia.

Tu em Zala ω sixSiriT icvleba, denis Zalac imave sixSiriT

Seicvleba. magram aucilebeli ar aris rom denis cvlilebis faza emTxve-

odes em Zalis cvlilebis fazas. amitom, zogad SemTxvevaSi

( )ϕω += tII sin0 , (35.4)

sadac ϕ aris fazaTa sxvaoba (fazaTa Zvra) denis Zalasa da em Zalas

Soris.

110

omis kanoni da misgan gamomdinare kirhofis kanonebi dadgenili iyo mudmivi

(stacionaruli) denisaTvis. magram es kanonebi marTebulia cvladi denisa da

ZabvisaTvisac, Tu maTi cvlileba ar xdeba Zalian swrafad. kerZod, Tu τ dro, romelic

saWiroa eleqtromagnituri SeSfoTebis∗ misaRwevad wredis yvelaze daSorebul ubnebamde,

gacilebiT naklebia cvladi denis T periodze, maSin denis Zalisa da Zabvis myisi

mniSvnelobebi wredis yvela kveTSi iqneba praqtikulad erTnairi da SesaZlebeli iqneba

maTTvis mudmivi denis kanonebis gamoyeneba. dens, romelic am pirobas akmayofilebs,

kvazistacionaruli deni ewodeba.

amrigad, denis kvazistacionarobis piroba Semdegnairad daiwereba:

T<<τ (35.5)

eleqtromagnituri SeSfoTebis gavrcelebis siCqare Zalian didia (vakuumSi

sinaTlis c siCqaris tolia). Tu manZils wredis yvelaZe daSorebul ubnamde (anu wredis

xazovan zomas) aRvniSnavT l -iT, maSin cl

≈τ da (35.5) piroba ase daiwereba:

Tcl<< , anu cTl << . magram =cT , amitom <<l

teqnikuri denisaTvis 50=v hc, 02,0=T wm; sinaTlis siCqare 5103 ⋅=c km/wm da

kvazistacionarobis piroba am SemTxvevaSi iqneba: 600010302,0 5 =⋅⋅<<l km. e.i. teqnikuri

deni CaiTvleba kvazistacionalurad, Tu wredis sigrZe ramdenime aseuli kilometriT

ganisazRvreba.

kvazistacionarobis piroba SeiZleba Sesruldes maRali sixSiris denis

SemTvevaSic, Tu wredis zoma mcirea. magaliTad, rxeviT konturSi SeiZleba aRiZvras deni,

romlis sixSire 610=v hc, e.i. periodi

610−=T wm. Tu konturis (koWas gragnilis da

SemaerTebeli sadenebis sigrZe 3=l m, maSin wm

wmm

m 88

10103

3 −=⋅

==clτ da (35.5) piroba

sruldeba.

$36. cvladi da mudmivi denis generatorebi

manqanas, romelic meqanikur energias gardaqmnis eleqtrul energiad,

eleqtromanqanuri generatori ewodeba.∗∗ mis moqmedebas safuZvlad udevs

eleqtromagnituri induqciis movlena.∗∗∗ rogorc viciT ($35), mudmiv magnitur velSi

∗ ase uwodeben eleqtromagnituri velis cvlilebas, romelic ganpirobebulia denis Zali-sa da muxtebis ganawilebis SecvliT. ∗∗ warmodgeba laTinuri sityvisagan generator – mwarmoebeli. sazogadod, generatori ewodeba yovelgvar mowyobilobas, aparats an manqanas, romelic awarmoebs raime pro-duqcias (mag. karbidis generatori, orTqlis generatori), gamoimuSavebs eleqtroenergias (mag. eleqtromanqanuri generatori, magnitohidrodinamikuri generatori, galvanuri elementi, Termoelementi) an warmoqmnis eleqtrul, eleqtromagnitur, optikur, bgeriT signalebs _ rxevebs, impulsebs (mag. radiosignalebis generatori, kvanturi generatori, ultrabgeris generatori). ∗∗∗ amitomaa, rom mas xSirad induqciur generators uwodeben.

111

liTonis CarCos Tanabari brunvisas, CarCoSi aRiZvreba sididiT da mimarTulebiT cvladi

induqciis em Zala. Tu CarCos boloebze liTonis rgolebs mivarCilavT, am

ukanasknelidan, mosriale a da b musebis saSualebiT, CarCoSi aRZruli cvladi deni

SeiZleba gadaviyvanoT gare wredSi. cxadia, rom CarCos erTi brunvis dros musebis

polaroba orjer icvleba.

am principze agebuli umartivesi generatoris sqema mocemulia 36.1. suraTze.

generatorebis momWerebze maRali Zabvis misaRebad iReben ara erT xvias, aramed

mimdevrobiT SeerTebul xviaTa did ricxvs _ gragnils.

cvladi denis generatoris zemoaRwerili tipi,

romelSic magnituri sistema (induqtori) uZravia, xo-

lo gragnili, romelSic aRiZvreba cvladi deni (Ruza)

moZravi, didi simZlavris misaRebad ar gamoiyeneba. saqme

imaSia, rom didi Zabvis SemTxvevaSi mosriale musebsa da

rgolebs Soris adgili aqvs Zlier naperwklur gan-

muxtvas, romelsac SeuZlia mwyobridan gamoiyvanos gene-

ratori. amitom, cvladi denis generatorebSi, gragnili

(Ruza) magrdeba uZravad, xolo magnituri sistema

(induqtori) mohyavT brunviT moZraobaSi. manqanis uZrav nawils statori ∗ ewodeba, xolo

moZravs _ rotori. ∗ ∗ unda aRiniSnos, rom induqtori, rogorc wesi, eleqtromagnits

warmoadgens, romelic mudmivi deniT ikvebeba. amitom mosriale musebi da rgolebi

aqac saWiroa. magram eleqtromagnitis damamagnitebeli deni mniSvnelovnad sustia

generatoris mier gamomuSavebul denze, amitom kontaqtebis mwyobridan gamosvlis

albaTobac bevrad naklebia.

Tu Cvens mier ganxilul manqanaSi rgolebs SevcvliT naxevarrgolebiT, gare

wredSi miviRebT mudmivi mimarTulebis dens. aseT manqanas mudmivi denis generatori ewo-

deba. misi sqema mocemulia 36.2 suraTze.

suraTidan Cans, rom im momentSi, rodesac CarCoSi deni icvlis mimarTulebas, naxe-

varrgolebi musebs Seicvlian. amitom gare wredSi dens mudam erTi da igive mimarTuleba

eqneba. denis sidide ki mudam icvleba ise, rogorc ic-

vleba sinusis funqcia 0 -dan 180 -mde. aseT dens

gamarTuli pulsirebuli deni ewodeba. 36.3 suraTze

naCvenebia gare wredSi gamavali aseTi pulsirebuli

denis grafiki. punqtir-sinusoidiT gamosaxulia denis

grafiki CarCoSi.

zemoaRwerili wesiT CarCos boloebze mirCi-

luli naxevarrgolebi umartives koleqtors warmoad-

gens, xolo misi saSualebiT cvladi denis gamarTvas

koleqtoruli gamarTva ewodeba. amrigad, mudmivi

denis generatoris umartivesi modeli cvladi denis

∗ warmodgeba laTinuri sityvisagan stator _ dgoma. ∗∗ warmodgeba laTinuri sityvisagan rotore _ brunva.

112

generatorisagan gansxvavdeba mxolod gansakuTrebuli avtomaturi (meqanikuri)

gadamrTveliT _ koleqtoriT.

denis pulsacia Semcirdeba anu ufro `gasworebul~ dens miviRebT, Tu erTi xviis

nacvlad aviRebT 2,3,4 da a.S. xvias. Sesabamisad mogvixdeba koleqtoris ara erTi, aramed

2,3,4 da a.S. wyvilis gamoyeneba. 36.4 suraTze mocemulia denis cvlilebis grafiki (msvili

mrudi) im SemTxvevisaTvis, rodesac aRebulia mimdevrobiT SeerTebuli ori marTkuTxa

CarCo, romelTa gverdebi Ruzaze Tanabradaa ganawilebuli. denis cvlilebis grafiki Ti-

Toeul xviaSi gamoisaxeba erTi da imave mrudiT (suraTze wvrili wirebi), magram TviT

mrudebi wanacvlebulia erTmaneTis mi-

marT periodis meoTxediT. denis saer-

To sidide yovel momentisaTvis miiRe-

ba orive mrudis ordinatebis Sekre-

biT.

mudmivi denis Tanamedrove gene-

ratoris Ruza metad rTuli, Caketili

sistemaa. igi Sedgeba didi raodenobis

(ramdenime aTeuli) calke naxvevebisa-

gan _ seqciebisagan, romlebic Cawyobi-

lia Ruzis doluri gularis kiloeb-

Si da firfitovani koleqtorisagan.

koleqtoris firfitebis ricxvi udris

seqciebis ricxvs. koleqtoris yovel

firfitasTan mierTebulia erTi seqci-

is bolo da meoris dasawyisi. seqcie-

bis didi raodenoba uzrunvelyofs

praqtikulad mudmiv dens momxmarebelSi.

$37. cvladi denis sruli wredi

ganvixiloT cvladi denis sruli wredi. es eleqtruli wredis

yvelaze zogadi SemTxvevaa, vinaidan igi Seicavs wredis yvela elements _

R omur winaRobas,∗ L induqciurobis koWas da C tevadobis kondensators

(sur. 37.1). vTqvaT, wredSi CarTuli cvladi denis wyaro icvleba

harmoniuli kanoniT:

tωεε sin0=

miviRoT denis Zalis gamosaTvleli formula. rodesac K CamrTvels

CavrTavT, wredSi gaivlis I deni, romelic gamoiwvevs Zabvis vardnas R

∗ omur winaRobas aqamde ubralod winaRobas vuwodebdiT. misi axali saxelwodeba dakavSirebulia imasTan, rom cvladi denis wredSi gvxvdeba sxva saxis winaRobebic.

t

I

sur. 36. 3

113

winaRobaze ( )IR da C kondensatorze ( )21 ϕϕ −=u . kirxhofis meore kanonis

Tanaxmad, Caketil wredSi ZabvaTa vardnis jami wredSi moqmed em ZalTa

jamis tolia. Cvens wredSi moqmedebs ori em Zala: 1. wredSi CarTuli denis

wyaros em Zala tωεε sin0= da 2. denis cvlilebis Sedegad L koWaSi

aRZruli TviTinduqciis em Zala dtdILS −=ε .

amrigad, kirxhofis meore kanonis Tanaxmad

gveqneba:

dtdILtUIR −=+ ωε sin0 .

magram kondesatoris tevadobis ganmartebis

Tanaxmad CqU = , amitom

dtdILtq

CIR −=+ ωε sin1

0 .

gavawarmovoT miRebuli toloba droiT da gaviTvaliswinoT rom

Idtdq

= , gveqneba:

tICdt

dIRdt

IdL ωωε cos102

2

=++ (37.1)

miviReT meore rigis diferencialuri gantoleba, zustad iseTi,

rogoric gvqonda iZulebiTi meqanikuri rxevebis ganxilvis dros (tomi I,

$41). amitom, mis amoxsnas (romlis Sesrulebasac mkiTxvels vTavazobT) Cven

ar gavimeorebT. ubralod mivuTiTebT amoxsnis gzas:

davuSvaT, (37.1) gantolebis amoxsnas aqvs saxe:

( )ϕω −= tII sin0 , (37.2)

sadac 0I aris cvladi denis amplituda, xolo ϕ _ fazaTa sxvaoba denis

Zalasa da em Zalas Soris. maT mniSvnelobebs vipoviT moTxovnidan, rom

roca (37.2) funqcias CavsvamT (37.1) gantolebaSi, igi unda gadaiqces

igiveobad. amisaTvis (37.2) gavawarmooT orjer, I -s da misi warmoebulis

mniSvnelobebi SevitanoT (37.1) gantolebaSi da movaxdinoT ( )ϕω −tsin -s da

( )ϕω −tcos -s gaSla.∗ miviRebT raRac gantolebas. es gantoleba rom

igiveobas warmoadgendes, saWiroa tolobis orive mxares tωsin -sTan da

tωcos -sTan mdgomi koeficientebi iyos erTmaneTis toli. es moTxovna

∗ ( ) ( ) ϕωϕωϕωϕωϕωϕω sinsincoscoscos;sincoscossinsin ⋅+⋅=−⋅−⋅=− tttttt .

k 21 ϕϕ

c

L~ε

sur. 37.1.

R

114

mogvcems or gantolebas, romelTa amoxsniT miiReba Semdegi mniSvnelobebi

fazaTa sxvaobisa da amplitudisaTvis:

RC

Ltg ω

ωϕ

1−

= (37.3) 2

2

00

1⎟⎠⎞

⎜⎝⎛ −+

=

ωω

ε

CLR

I (37.4)

Tu fazaTa sxvaobisa da amplitudis miRebul mniSvnelobebs SevitanT

(37.2)-Si, miviRebT cvladi denis srul wredSi denis Zalis gamosaTvlel

formulas:

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛ −−

⎟⎠⎞

⎜⎝⎛ −+

=R

CL

arctgt

CLR

I ωω

ω

ωω

ε1

sin1 2

2

0 ∗

ganvixiloT kerZo SemTxvevebi.

1. cvladi denis wredi Seicavs mxolod omur winaRobas, e.i.

0,0 =≠ LR da ∞=C . es ukanaskneli gamomdinareobs im faqtidan, rom, Tu

wredi ar Seicavs kondensators, rac niSnavs, rom igi Secvlilia uwyveti

gamtariT, maSin damatebiTi potencialTa sxvaoba ar warmoiqmneba, anu

021 =−= ϕϕu . magram Cqu = da radgan 0≠q , maSasadame, ∞=C .∗∗

(37.3) tolobidan gamomdinareobs, rom am SemTxvevaSi 0=ϕtg da 0=ϕ .

maSasadame, rodesac wredi Seicavs mxolod omur winaRobas, fazaTa sxvao-

ba em Zalasa da denis Zalas Soris nu-

lis tolia, anu isini irxevian erTna-

ir fazebSi, rac niSnavs imas, rom erT-

droulad aRweven amplitudur da nu-

lovan mniSvnelobebs (sur. 37.2). garda

amisa, (37.4) tolobis Tanaxmad, am dros

RI 0

0ε

= da (37.2)-dan miviRebT ( )0=ϕ :

∗ gavixsenoT, rom xarctg aris kuTxe, romlis tangensi x -is tolia. Cveulebriv, am

Canaweris dros, kuTxe gamosaxeba ara gradusul, aramed radialur zomebSi. ∗∗ igive Sedegi SeiZleba Semdegnairadac aixsnas: kondensatoris firfitebis erTmaneTTan miaxloebiT kondensatoris tevadoba izrdeba da rodesac isini erTmaneTs Seexeba (anu

rodesac kondensatori, rogorc aseTi, ispoba) maT Soris manZili 0=d da ∞==d

SC

εε 0 .

t

I

sur. 37. 2

I

ε ε

115

RRt

tII εωεω ===

sinsin 0

0 ,

e.i. sruldeba omis kanoni im saxiT, rogorc es mudmivi denisaTvis

gvqonda.

aqve aRvniSnoT, rom rodesac cvladi denis wredi Seicavs

mimdevrobiT SeerTebul wredis yvela elements, omis kanoni sruldeba

denis Zalis mxolod amplituduri (da rogorc Semdeg vnaxavT, efeqturi)

mniSvnelobisaTvis (formula 37.4), sadac sidides 2

2 1⎟⎠⎞

⎜⎝⎛ −+=

ωω

CLRZ

wredis sruli winaRoba an impedansi∗ ewodeba.

2. cvladi denis wredi Seicavs mxolod L induqciurobis koWas,∗∗

e.i. 0,0 =≠ RL da ∞=C .

(37.3)-dan ∞==R

Ltg ωϕ da 2πϕ = . amitom ⎟

⎠⎞

⎜⎝⎛ −=

2sin0

πω tII .

maSasadame, rodesac wredi Seicavs mxolod induqciurobis koWas,

denis Zala CamorCeba eleqtromamoZravebel Zalas 2π-is toli faziT.∗∗∗ Ca-

morCena 2π faziT grafikulad mocemulia 37.3 suraTze.∗∗∗∗

(37.4)-dan gamomdinareobs, rom

omis kanons am SemTxvevaSi aqvs

Semdegi saxe: ωεL

I 00 = , anu winaRobis

rols asrulebs sidide ωL . mas

ewodeba induqciuri winaRoba da

aRiniSneba LR -iT:

∗ impedansi, romelic cvladi denis wredis sruli winaRobis moZvelebuli saxelwodebaa, warmodgeba inglisuri sityvidan impedance _ laT. imedio _ vewinaaRmdegebi. ∗∗ SemaerTebeli gamtarebis R omuri winaRoba imdenad mcirea, rom SeiZleba misi ugulebelyofa.

∗∗∗ Tu wredi Seicavs omur winaRobasac, maSin R

Ltg ωϕ = , e.i. am dros CamorCena

RLarctg ωϕ = -is tolia, rac

2π

-ze naklebia. maSasadame, omuri winaRobis arseboba

amcirebs fazaTa sxvaobas.

∗∗∗∗ gavixsenoT, rom tt ωπω cos2

sin ∓∓ =⎟⎠⎞

⎜⎝⎛

.

116

ω⋅= LRL (37.6)

rogorc misi gamosaxuleba gviCvenebs, induqciuri winaRoba sixSiris

proporciulia. mudmivi denis SemTxvevaSi 0=ω da induqciuri winaRobac

nulis tolia. cxadia, LR -s aqvs winaRobis ganzomileba ⎟⎟⎠

⎞⎜⎜⎝

⎛==amperi

volti

0

0

IRL

ε

da izomeba omebiT. magaliTad, induqciuri winaRoba 1 henri induqciurobis

mqone koWasi, romelic CarTulia 50=v herci sixSiris cvladi denis

wredSi, tolia: omi.wm

1

a

vwm3143142 ==== vLLRL πω 5000=v herci sixSiris

dros imave koWas induqciuri winaRoba izrdeba 31400 omamde.

3. cvladi denis wredi Seicavs mxolod C tevadobis

kondensators, e.i. 0, =∞≠ RC da 0=L .

(37.3)-dan −∞=−=R

Ctg ωϕ

1

da 2πϕ −= . amitom ⎟

⎠⎞

⎜⎝⎛ +=

2sin0

πω tII .

maSasadame, rodesac wredi Seicavs mxolod tevadobas, denis Zala win

uswrebs em Zalas 2π faziT, an droSi

4T-iT.

winswreba 2π faziT grafikulad mocemulia (37.4) suraTze.

winswreba faziT Semdegnairad aixsneba:

kondensatoris denis wyarosTan mier-

TebisTanave ( 0=t , Zabva kondensatoris

Semonafenebze 0=u ), imis gamo, rom

SemaerTebeli gamtarebis winaRoba

praqtikulad nulis tolia, deni maSinve iRebs

maqsmimalur mniSvnelobas ( )0II = . iwyeba kon-

densatoris damuxtva. kondensatoris damux-

tvasTan erTad Zabva mis Semonafenebze

TandaTan izrdeba, ewinaaRmdegeba ra muxtebis Semdgom modinebas. amitom, deni wredSi

TandaTan mcirdeba, miuxedavad imisa, rom wyaros em Zala TandaTan izrdeba. rodesac 4T

drois Semdeg em Zala miaRwevs maqsimalur mniSvnelobas ( )0εε = , Zabva kondensatoris

Semonafenebze maqsimaluria da tolia 0ε -is ( )0ε=u . cxadia, am dros muxtebis modineba

Semonafenebze Sewydeba da deni gaxdeba nulis toli. amrigad, sawyis momentSi deni iyo

117

maqsimaluri, em Zala ki minimaluri ( )0=ε . 4T

drois Semdeg denis Zala minimaluria

( )0=I , em Zala ki _ maqsimaluri. amis Semdeg iwyeba kondensatoris ganmuxtva, deni

TandaTanobiT izrdeba (sapirispiro mimarTulebiT, em Zala ki _ mcirdeba. 2T

drois

Semdeg deni maqsimaluria, em Zala ki minimaluri da a.S.

(37.4)-dan gamomdinareobs, rom omis kanons am SemTxvevaSi aqvs saxe:

ω

ε

CI

10

0 = , anu winaRobis rols asrulebs sidide ωC1

. mas ewodeba

tevaduri winaRoba da aRiniSneba CR -Ti:

ωCRC

1= (37.7)

rogorc misi gamosaxuleba gviCvenebs, tevaduri winaRoba sixSiris ukupro-

porciulia. mudmivi denisaTvis 0=ω da ∞=CR . e.i. kondensatori mudmiv

dens ar atarebs. cxadia, SI sistemaSi tevaduri winaRobac izomeba omebiT.

magaliTad tevaduri winaRoba fmkf 51010 −==C tevadobis kondensatorisa,

romelic CarTulia 50=v hc sixSiris cvladi denis wredSi, tolia:

31810314

1211

5 ≈⋅

=== −vCCRC πω

omi. 5000=v hc sixSiris dros imave kondensa-

toris tevaduri winaRoba mcirdeba ~ 3 omamde.

unda aRiniSnos, rom induqciuri da tevaduri winaRobebi analogiuria

omuri winaRobisa mxolod denis Zalis amplitudaze moqmedebis

TvalsazrisiT. gansaxvaveba ki mdgomareobs imaSi, rom:

1. induqciuri da tevaduri winaRobebi warmoqmnian fazaTa sxvaobas

denis Zalasa da em Zalas Soris, agreTve damokidebuli arian sixSireze.

omuri winaRoba ki ar warmoqmnis fazaTa sxvaobas da arc sixSirezea

damokidebuli.

2. omur. winaRobaze xdeba energiis gamoyofa joulis siTbos saxiT,

induqciur da tevadur winaRobebze ki energia ar gamoiyofa.∗ amis gamo

∗ saqme imaSia, rom energia, romelic periodulad ixarjeba kondensatoris Semonafenebs Soris eleqtruli velis Seqmnaze (misi damuxtvis dros), imave raodenobiT da imave periodulobiT brundeba wredSi am velis mospobisas (misi ganmuxtvis dros). zustad aseve, energia, romelic periodulad ixarjeba magnituri velis Seqmnaze induqciurobis koWaSi (denis zrdis dros), imave raodenobiT da imave periodulobiT brundeba wredSi am magnituri velis mospobisas (denis Semcirebis dros). sinamdvileSi energiis mcire danakargebs mainc aqvs adgili kondensatoris eleqtruli velisa da koWas magnituri velis energiis gabnevis gamo, koWas omur winaRobaze (romelic mcire, magram nulisagan gansxvavebulia) joulis siTbos gamoyofis gamo da sxv.

118

omur winaRobas uwodeben aqtiur winaRobas, induqciur da tevadur

winaRobebs ki _ reaqtiuls.

(37.4)-dan gamomdinareobs, rom Tu wredi Sedgeba mimdevrobiT Seer-

Tebuli L induqciurobis koWasa da C tevadobis kondensatorisagan, maSin

wredis reaqtiuli winaRoba induqciuri da tevaduri winaRobebis sxvaobis

tolia: ω

ωχC

L 1−= .

$ 38. ZabvaTa rezonansi

(37.3) da (37.4) formulebi gadavweroT Cvens mier SemoRebuli

aRniSvnebis Sesabamisad:

R

RRtg CL −=ϕ . (38.1)

( )22

00

CL RRRI

−+=

ε. (38.2)

(38.1) gviCvenebs, rom Tu CL RR > , maSin 0>ϕtg da 0>ϕ ⎟⎠⎞

⎜⎝⎛ <<

20 πϕ ,

xolo Tu CL RR < , maSin 0<ϕtg da 0<ϕ ⎟⎠⎞

⎜⎝⎛ <<− 0

2ϕπ

. e.i. denis Zala

garkveuli faziT CamorCeba an win uswrebs em Zalas imis mixedviT, Tu

romeli winaRobaa meti _ induqciuri Tu tevaduri. gansakuTrebiT

sainteresoa SemTxveva, roca

CL RR = . (38.3)

maSin, (38.1)-dan 0=ϕtg da 0=ϕ . e.i. fazaTa sxvaoba denis Zalasa da em

Zalas Soris ara marto maSin aris nulis toli, roca wredSi CarTulia

mxolod omuri winaRoba, aramed maSinac, roca wredi Seicavs (mimdevrobiT

SeerTebul) yvela elements da sruldeba (38.3) piroba.

garda amisa, am dros wredis reaqtiuli winaRoba ( )CL RR −=χ nulis

tolia, sruli winaRoba Z minimaluria ( )RZZ == min , xolo denis Zalis am-

plituda R

I 00

ε= _ maqsimaluri. am movlenas eleqtruli rezonansi ewo-

deba. rezonansis pirobas gamosaxavs (38.3) toloba, saidanac gavigebT

rezonansul sixSires ( )rezω , Tu masSi SevitanT LR -sa da CR -s

mniSvnelobebs:

119

rez

rez ωω

CL 1

= , saidanac LC1

=rezω . (38.4)

rac naklebia wredis aqtiuri winaRoba ( )R , miT metia rezonansuli ampli-

tudis sidide, an, rogorc amboben xolme, miT ufro mkveTria rezonansi

(sur. 38.1).

denis Zalis gazrdasTan er-

Tad rezonansis dros mkveTrad iz-

rdeba Zabva (Zabvis vardna) induqci-

urobis koWaze da kondensatorze.

maqsimaluri Zabva induqciurobis

koWaze da kondesatorze (roca

rezωω = ) Sesabamisad tolia:

;0000 C

LR

LIRIu LLε

ω === rez

;1 0000 C

LRC

IRIu CCε

ω===

rez

rogorc vxedavT, Zabvis amplitudebi induqciurobis koWaze da

kondensatorze erTmaneTis tolia, magram fazebi aqvT sawinaaRmdego (Zabva

induqciurobaze denis Zalas uswrebs 2π faziT, xolo Zabva kondensatorze

CamorCeba denis Zalas 2π faziT), ris gamoc isini erTmaneTs akompensireben

da mTel Caketil wredSi Zabvis vardna ganisazRvreba Zabvis vardniT

aqtiur winaRobaze ( )RIuR 0= , romelic wredSi CarTuli em Zalis tolia.∗

Tu wredis aqtiuri winaRoba mcirea ⎟⎟⎠

⎞⎜⎜⎝

⎛<<

CLR , maSin Zabvis amplitu-

dam induqciurobis koWaze da kondensatorze SeiZleba bevrad gadaaWarbos

wyaros em Zalis amplitudas ( 00 ε>>Lu da 00 ε>>Cu ).

wina paragrafSi moyvanili magaliTebidan Cans, rom sixSire 50=v hc ( )hc314=ω ,

warmoadgens rezonansul sixSires wredisaTvis, romelic Sedgeba mimdevrobiT SeerTebuli

=L 1hn-is toli induqciurobis koWasagan da =C 10mkf-is toli tevadobis

∗ kirxhofis meore kanonis Tanaxmad CLR uuu ++=ε . rezonansis dros Lu da Cu

erTmaneTs akompensirebs, amitom Ru=ε .

O ω

0I

sur. 38. 1

123 RRR <<

1R

rezω

2R 3R

120

kondesatorisagan. am SemTxvevaSi 315≈= CL RR om. Tu wredis aqtiuri winaRoba =R 10oms,∗

xolo wredSi CarTuli denis wyaros em Zalis amplituda =0ε 180volts (es niSnavs, rom

em Zalis efeqturi mniSvneloba =efε 127v. ix. Semdegi paragrafi), maSin denis Zalis

amplituda 1800 ==

RI

εa, xolo Zabvebis amplitudebi induqciurobis koWaze da

kondensatorze toli iqneba:

5670315180000 =⋅==== CLCL RIRIuu v.

aqtiuri winaRobis SemcirebiT denis Zalis amplituda da Sesabamisad ZabvaTa

amplitudebi reaqtiul winaRobebze izrdeba. mag. Tu =R 1oms, maSin =0I 180v da

5670000 == CL uu v.

vinaidan ganxilul SemTxvevaSi Zabvis vardna induqciurobis koWaze

da kondensatorze mkveTrad izrdeba da amasTan isini wredSi mimdevrobiT

arian CarTuli, eleqtruli rezonansis am

saxes ewodeba ZabvaTa rezonansi (mimdevrobi-

Ti rezonansi).

arsebobs eleqtruli rezonansis meore saxe,

romelsac denebis rezonansi (paraleluri rezonansi)

ewodeba. amas maSin aqvs adgili, roca induqciurobis L

koWa da C tevadobis kondensatori paraleluradaa Car-

Tuli (sur. 38.2). rezonansuli sixSire am SemTxvevaSic

(38.4) tolobidan ganisazRvreba. paraleluri rezonansi

rezonansis gansakuTrebuli saxea, romelsac antirezonansic ki SeiZleba ewodos. saqme

imaSia, rom am dros paralelur StoebSi gamavali denebis amplitudebi aRweven udides

mniSvnelobas, magram imyofebian sawinaaRmdego fazebSi, amitom gare wredSi amplituda im

I denisa, romelic calkeul StoebSi denebis algebruli jamis tolia, aris minimaluri.

idealur SemTxvevaSi, roca Stoebis aqtiuri winaRoba nulis tolia, iqneba nulis toli.

swored am movlenas, anu gare wredSi mkveTr Semcirebas denis Zalisa, romelic kvebavs

paralelurad CarTul kondensatorsa da induqciurobis koWas, rodesac modebuli Zabvis

sixSire uaxlovdeba rezonansul sixSires, ewodeba denebis rezonansi (paraleluri

rezonansi). sazogadod, paraleluri rezonansi (iseve rogorc mimdevrobiTi) miT ufro

mkveTrad gamoisaxeba, rac naklebia wredis aqtiuri winaRoba.

∗ am SemTxvevaSi sruldeba piroba CLR << .

L

sur. 38. 2

c tωεε sin~ 0=

121

$39. simZlavre cvladi denis wredSi

eleqtruli simZlavre aris fizikuri isdide, romelic axasiaTebs

eleqtruli energiis gadacemis an gardaqmnis siCqares.

rogorc viciT (tomi I, $133), mudmivi denis Caketil wredSi gamoyofi-

li simZlavre denis Zalisa da em Zalis namravlis tolia:

εIP = . (39.1)

drois Zalian mcire intervalSi cvladi denic SeiZleba CaiTvalos

mudmivad, amitom cvladi denis myisi simZlavre imave (38.1) formuliT

ganisazRvreba, oRond ( )ϕω −= tII sin0 da tωεε sin0= . maSasadame, cvladi

denis myisi simZlavrisaTvis gveqneba:

( )ϕωωε −⋅= ttIP sinsin00 . (39.2)

praqtikuli miznebisaTvis sainteresoa vicodeT simZlavris ara myisi,

aramed saSualo mniSvneloba mravali periodis Semcveli didi drois gan-

mavlobaSi. amisaTvis sakmarisia ganvsazRvroT saSualo simZlavre erTi

periodis ganmavlobaSi, radgan momdevno periodebSic simZlavre cxadia,

igive iqneba.

am amocanas advilad gadavwyvetT, Tu gamoviyenebT ori sinusis

namravlis formulas:

( ) ( )[ ]βαβαβα +−−=⋅ coscos21sinsin .

Cven SemTxvevaSi ( )ϕωβωα −== tt, , amitom (39.2) formula

Semdegnairad gadaiwereba:

( )[ ]ϕωϕε −−= tIP 2coscos21

00 . (39.3)

(39.3) formulaSi droze mxolod frCxilebSi moTavsebuli meore

wevria damokidebuli. periodis ganmavlobaSi misi saSualo mniSvneloba

udris nuls: ( ) 02cos =−ϕω t . amitom periodis ganmavlobaSi saSualo

simZlavrisaTvis miviRebT:

ϕε cos21

00IP = . (39.4)

visargebloT dayvanis cnobili formuliT: ϕ

ϕ21

1costg+

= da gavixse-

noT, rom R

RRtg CL −=ϕ . maSin

( )22cos

CL RRR

R

−+=ϕ . ϕcos -s miRebuli mniSvne-

122

loba SevitanoT (39.4)-Si da gaviTvaliswinoT, rom (38.2)-is Tanaxmad

( )0

022

IRRR CL

ε=−+ . miviRebT: RIP 2

021

= . SemoviRoT aRniSvna 20I

I =ef , maSin

RIP 2ef= . (39.5)

Tu deni wredSi ar asrulebs meqanikur muSaobas, maSin saSualo

simZlavre gamoiyofa aqtiur winaRobaze siTbos saxiT.

sasrul t droSi denis mier gamoyofili siTbos raodenobisaTvis

gveqneba:

RtItPQ 2ef== . (39.6)

miRebuli formula SevadaroT mudmivi I denis mier imave R aqtiur

winaRobaze imave t droSi gamoyofili siTbos raodenobis formulas

(joul-lencis kanoni):

RtIQ 2' = . (39.7)

am Sedarebidan Cans, rom Tu 'QQ = , maSin II =ef . efI -s ewodeba

cvladi denis Zalis moqmedi anu efeqturi mniSvneloba. am dasaxelebas

safuZvlad udevs is faqti, rom igi iZleva iseTive energetikul efeqts,

rogorc misi toli mudmivi deni. amrigad, cvladi denis Zalis moqmedi

anu efeqturi mniSvneloba ewodeba iseTi mudmivi denis Zalas,

romelic imave winaRobaze da imave droSi gamoyofs iseTive siT-

bos raodenobas, rogorsac mocemuli cvladi deni.

analogiurad ganisazRvreba em Zalisa da Zabvis efeqturi

mniSvnelobebi:

2;

200 u

u == efef

εε .

amitom cvladi denis saSualo simZlavrisaTvis (39.4) formulis

safuZvelze sabolood gveqneba:

ϕε cos⋅⋅= efefIP . (39.8) miRebuli formula gviCvenebs, rom saSualo simZlavre, garda denis

Zalisa da em Zalisa, damokidebulia kidev fazaTa sxvaobaze I -sa da ε -s

Soris. simZlavre maSinaa maqsimaluri, roca 1cos =ϕ , anu roca 0=ϕ . amas

ki, rogorc viciT, adgili aqvs or SemTxvevaSi:

1. roca wredSi CarTulia mxolod omuri winaRoba. es SemTxveva

praqtikul interess ar warmoadgens.

2. roca wredSi CarTulia yvela elementi, magram sruldeba rezonansis

piroba CL RR = .

123

Tu wredi Seicavs mxolod reaqtiul winaRobas ( )0=R , maSin 0cos =ϕ da saSualo

simZlavre nulis tolia, ra sididisac ar unda iyos denis Zala da em Zala. es imiT

aixsneba, rom energia, romelsac wyaro awvdis wreds periodis meoTxedis ganmavlobaSi

(vTqvaT, kondensatoris damuxtvisas), ukanve ubrundeba wyaros periodis meore meoTxedSi

(kondensatoris ganmuxtvisas).

vinaidan cvladi denis saSualo simZlavre ϕcos -s proporciulia,

ϕcos -s uwodeben simZlavris koeficients. gamoTqma `gaizarda wredis ϕcos ~

niSnavs, rom gaizarda cvladi denis wredSi gamoyofili simZlavre. cvladi

denis wredebis proeqtirebisas induqciur da tevadur datvirTvebs

iseTnairad arCeven, rom ϕcos axlos iyos erTTan. ϕcos -s umciresi

dasaSvebi mniSvneloba daaxloebiT 0,85-ia.

Tu (38.2) formulis orive mxares gavyofT 2 -ze, amplituduri

mniSvnelobebi Seicvleba efeqturiT:

( )22

CL RRRI

−+= ef

ef

ε (39.10)

maSasadame, cvladi denis SemTxvevaSi omis kanoni sruldeba denis

Zalis da em Zalis ara marto amplituduri, aramed efeqturi

mniSvnelobebisaTvisac.

gavarkvioT, Tu ratom avirCieT cvladi denis dasaxasiaTeblad misi siTburi

moqmedeba. saqme imaSia, rom cvladi denis daxasiaTeba misi saSualo mniSvnelobiT ar

SeiZleba, vinaidan saSualo mniSvneloba periodis ganmavlobaSi nulis tolia. sazogadod,

cvladi denis daxasiaTeba ar SeiZleba arc erTi iseTi movleniT, romlis damaxasiaTebeli

fizikuri sidide icvlis niSans denis Zalis mimarTulebis Secvlis dros (mag. magnitur

velSi deniani konturis gadaxra, eleqtrolizis dros gamoyofili nivTierebis raodenoba

da sxva). cvladi denis daxasiaTeba misi amplituduri mniSvnelobiT principSi

SesaZlebelia, magram praqtikulad Znelia iseTi xelsawyos ageba, romelic uSualod am

sidides gazomavda. maSasadame, cvladi denis dasaxasiaTeblad unda gamoviyeneoT misi

iseTi moqmedeba, romelic denis mimarTulebaze ar aris damokidebuli da mudmiv densac

axasiaTebs. aseTia denis siTburi moqmedeba _ gamoyofili siTbos raodenoba denis Zalis

kvadratis proporciulia da e.i. denis mimarTulebaze damokidebuli ar aris. amitomaa,

rom denis siTburi moqmedebis safuZvelze SemoviReT misi damaxasiaTebeli sidide _ denis

Zalis efeqturi mniSvneloba. roca laparakia cvladi denis Zalis, Zabvis an em Zalis

konkretul mniSvnelobebze, am sidideebis efeqturi mniSvnelobebi igulisxmeba.

cvladi denis eleqtrosazomi xelsawyo gviCvenebs swored cvladi denis Zalis,

Zabvis an em Zalis efeqtur mniSvnelobas, romelic 2 -jer naklebia mis amplitudur

mniSvnelobaze. amitom xelsawyos gaangariSeba saWiroa im varaudiT, rom

zemoTCamoTvlili sidideebi calkeul momentebSi aRwevs 2 -jer met mniSvnelobebs.

124

$40. samfaziani deni

Cven mier $35-Si ganxilul cvlad dens (romelic generatoris erTi eleqtro-

mamoZravebeli Zalis mieraa Seqmnili) erTfaziani deni ewodeba. erTfazianma denma, rigi

mizezebis gamo, romlebzec qvemoT gveqneba saubari, ver hpova iseTi farTo gamoyeneba,

rogorc e.w. samfazianma denma, romelic dReisaTvis eleqtroenergiis warmoebis, gadace-

misa da gamoyenebis denis ZiriTad sistemas warmoadgens.

samfaziani deni ewodeba iseTi sami erTfaziani denis erTobliobas,

romlebsac toli amplitudebi da sixSireebi aqvT, fazebiT ki wanacvle-

bulni arian erTimeoris mimarT 120° -iT ⎟⎠⎞

⎜⎝⎛ −iTπ

32

, an droSi _ 31 periodiT.

aseTi xasiaTis denebi miiReba denis samfaziani generatoris saSualebiT, romlis

sqema mocemulia 40.1 suraTze: generatoris statorze (4) moTavsebulia sami erTnairi

gragnili (`polusuri gragnilebi~) 1, 2, 3, romlebsac generatoris fazebi ewodeba da

romlebic wanacvlebulia erTmaneTis mimarT 120°-iT (anu wrewiris mesmediT). rotors,

isevev rogorc erTfaziani generatoris SemTxvevaSi, warmoadgens mudmivi magniti an

eleqtromagniti (5). rotoris brunvisas TiToeul gragnilSi aRiZvreba cvladi em Zala,

romelTac erTnairi sixSire da apmlituda aqvT, magram wanacvlebulni arian erTmaneTis

mimarT π32

faziT, an droSi _ periodis mesamediT:

⎟⎠⎞

⎜⎝⎛ −=⎟

⎠⎞

⎜⎝⎛ −== πωεεπωεεωεε

34sin,

32sin,sin 030201 ttt . (40.1)

vTqvaT, TiToeuli gragnilis wredSi CarTulia toli sididis 321 ,, RRR omuri

winaRobebi (`simetriuli datvirTva~), maSin am wredSi miviRebT toli sididis denebs,

romlebic fazebiT saTanado em Zalebs emTxveva:

⎟⎠⎞

⎜⎝⎛ −=⎟

⎠⎞

⎜⎝⎛ −== πωπωω

34sin,

32sin,sin 030201 tIItIItII . (40.2)

amrigad, samfaziani denis generatori warmoadgens sami Cveulebrivi (erTfaziani)

cvladi denis generatoris erTobliobas. misi TiToeuli gragnili warmoadgens

eleqtroenergiis damoukidebel wyaros da SeiZleba SeerTebul iqnes calkeul momxmare-

belTan, risTvisac saWiro iqneba eqvsi gamtari _ sam gamtarSi

deni midis denis wyarodan momxmarebelTan, danarCen samSi ki

ukan brundeba. aseTi damoukidebeli sistemebi (rodesac

generatoris calkeuli fazebi erTmaneTTan eleqtrulad

dakavSirebulni ar arian) praqtikaSi TiTqmis ar gamoiyeneba,

vinaidan aRmoCnda, rom samfaziani generatoris gragnilebis

garkveuli wesiT SeerTebisas, miiReba sami an oTxgamtariani

sistemebi, rac ekonomikurad gacilebiT xelsayrelia. arse-

bobs gragnilebis SeerTebis ori wesi: `varskvlaviseburad~ da

~samkuTxedad~.

125

varskvlaviseburad SeerTebas miviRebT, Tu generatoris samive gragnilis

boloebs SevaerTebT erT saerTo A wertilSi, romelsac nulovani an neitraluri werti-

li ewodeba, xolo dasawyisebs miv-

uerTebT momxmarebelTan (Cven Sem-

TxvevaSi 321 ,, RRR winaRobebTan)

mimyvan I, II, III sadenebs. Tu winaRo-

bebsac SevaerTebT varskvlavisebu-

rad, miviRebT sur. 40.2-ze gamosaxul

sqemas.

momxmarebelTan (datvirTvebTan)

SemaerTebel I, II, III sadenebs xazis

sadenebi ewodeba, xolo nulovani

wertilebis SemaerTebel AB sadens

_ nulovani (neitraluri) sadeni.

Zabvebs xazis sadenebs Soris da am sadenebSi gamaval denebs xazurs uwodeben, xolo

dens, romelic gadis generatoris fazis gragnilebsa da datvirTvis fazebSi, agreTve maT

Zabvebs _ fazurs. amrigad fazuri Zabva generatoris fazis Zabvaa (romelic sur. 40.2-ze

moyvanili sqemis Tanaxmad nulovan sadensa da nebismier xazis sadens Soris arsebuli

Zabvis tolia), xolo xazuri Zabva generatoris fazebs Soris arsebuli Zabvaa. suraTidan

Cans agreTve, rom varskvlaviseburi SeerTebis dros xazuri da fazuri denebi erTmaneTis

tolia ( )fx II = , xolo xazuri da fazuri Zabvebi erTmaneTisagan gansxvavdeba.

gamoviTvaloT nulovan sadenSi gamavali 321 ,, III denebis jami im SemTxvevi-

saTvis, roca datvirTvis winaRobebi erTmaneTis tolia. maSin 321 ,, III denebic sididiT

toli iqneba da gamoiTvleba (40.2) formulebiT. SevkriboT jer 1I da 2I . amisaTvis

gamoviyenoT trigonometriidan cnobili ori kuTxis sinusis jamis formula:

2cos

2sin2sinsin βαβαβα −

⋅+

=+ ; Cven SemTxvevaSi πωβωα32, −== tt , amitom

⎟⎠⎞

⎜⎝⎛ −=⎟

⎠⎞

⎜⎝⎛ −=⎟

⎠⎞

⎜⎝⎛ −+=+

3sin

3cos

3sin2

32sinsin 000021

πωππωπωω tItItItIII .

davumatoT am jams 3I :

02

cos65sin2

34sin

3sin 000321 =⋅⎟

⎠⎞

⎜⎝⎛ −=⎟

⎠⎞

⎜⎝⎛ −+⎟

⎠⎞

⎜⎝⎛ −=++

ππωπωπω tItItIIII .

e.i. am SemTxvevaSi AB sadenSi deni nulis tolia (amitom uwodeben mas nulovan

sadens) da igi saWiro aRar aris. amrigad, roca datvirTvis winaRobebi erTmaneTis tolia

generatoris momxmarebelTan samfaziani denis misayvanad eqvsis nacvlad sami sadenia

sakmarisi. igive Sedegi ( 0321 =++ III ) martivad miiReba veqtoruli diagramis gamoyenebiT

(sur. 40.3).

2 II 2I

1I

3R

321 III ++ B fu 2R 1R xu I

III

3

1

3I

sur. 40 .2

126

samfaziani denis ZravaSi samive gragnilis winaRoba erTnairia, amitom aseTi Zravas

wredSi CarTvisas sami sadenia saWiro.

Tu datvirTvis winaRobebi erTmaneTis toli ar aris,

maSin 321 ,, III denebis jami nulis toli ar iqneba da saWiroa

meoTxe sadenis gamoyenebac. sacxovrebeli saxlebis (da saerTod

Senobebis) eleqtroficirebisas, saWiroa oTxi sadenis Seyvana,

radgan nakleb savaraudoa, rom samive fazaSi denis Zala

erTnairi iyos. garda amisa, nulovani sadeni imitomac aris saWi-

ro, rom igi uzrunvelyofs samfazian wredSi erTfaziani datvir-

Tvis (televizoris, macivris da a.S.) CarTvis SesaZleblobas.

rogorc aRvniSneT, varskvlaviseburad SeerTebis dros

fazuri da xazuri Zabvebi ( )xf da uu erTmaneTisagan

gansxvavdeba. davadginoT maT Soris damokidebuleba. meti

sicxadisaTvis davuSvaT, rom wredi ganrTulia ( ∞=== 321 RRR ), maSin, ganmartebis

Tanaxmad, ,sin01 tu ωεε ==f xolo 21 εε −=xu . visargebloT (40.1) formulebiT da ori

kuTxis sinusis sxvaobis formuliT:

2sin

2cos2sinsin βαβαβα −+

=− ,

sadac α da β igive sidideebia, rac wina SemTxvevaSi. gveqneba:

⎟⎠⎞

⎜⎝⎛ −=⎟

⎠⎞

⎜⎝⎛ −−=−=

3cos

3sin2

32sinsin 00021

πωπεπωεωεεε tttux .

magram 23

3sin =

π, xolo ⎟

⎠⎞

⎜⎝⎛ −=⎟

⎠⎞

⎜⎝⎛ +−=⎟

⎠⎞

⎜⎝⎛ −

6sin

23sin

3cos πωππωπω ttt ,

amitom

⎟⎠⎞

⎜⎝⎛ −=−=

6sin3 021

πωεεε tux .

miviReT, rom xazuri Zabvis, anu fazebs Soris Zabvis amplituda 3 -jer metia

fazuri Zabvis amplitudaze ( 0ε -ze). cxadia, igive Sedegs miviRebT Tu gamoviTvliT

(romlis Sesrulebasac mkiTxvels vTavazobT) ( )32 εε − -s, an ( )31 εε − -s.

saqarTvelos eleqtroqselis TiToeul fazaSi Zabva 220 voltia, fazebs Soris Zabva

ki 38022073,12203 =⋅=⋅ volti.

Tu SevajamebT zemoT naTqvams SeiZleba davaskvnaT:

samfaziani denis erTi upiratesoba erTfazianTan Sedare-

biT isaa, rom eqvsi mimyvani sadenis nacvlad saWiroa sami

(maqsimum oTxi) sadeni, rac mniSvnelovand amcirebs masalis

danaxarjs.

samfaziani denis meore da SeiZleba iTqvas, yvelaze

mniSvnelovani upiratesoba isaa, rom igi warmoqmnis mbru-sur. 40. 4

sur. 40.3

127

nav magnitur vels. rogorc viciT, samfaziani deni miiReba rotoris mudmivi magnituri

velis brunviT statoris SigniT. cda da Teoria gviCvenebs, rom SesaZlebelia Sebru-

nebuli procesic: Tu samfaziani generatoris fazebs CavrTavT samfazian eleqtrul qsel-

Si, statoris SigniT Cndeba mudmivi mbrunavi magnituri veli. amaSi advilad avrwmundebiT,

Tu statoris SeigniT movaTavsebT magnitur isars _ igi daiwyebs brunvas. samfaziani denis

es Tviseba safuZvlad udevs samfaziani asinqronuli Zravas moqmedebas.

samfaziani asinqronuli Zravas umartivesi sqema mocemulia 40.4 suraTze. misi

statori iseTivea rogorc generatoris, rotorze ki moTavsebulia ori mokled CarTuli

1K da 2K polusuri koWebi.∗ statoris 1, 2, 3 gragnilebis samfazian wredSi CarTvisas,

masSi warmoiqmneba mbrunavi magnituri veli, ris gamoc 1K da 2K koWebs ganWolavs cvla-

di magnituri nakadi da aRZravs maTSi induqciur dens. am denebis urTierTqmedebis Sede-

gad mbrunav magnitur velTan, rotori daiwyebs brunvas (yovelgvari mosriale kontaqtebis

gareSe). lencis wesis Tanaxmad, induqciuri deni ewinaaRmdegeba mis gamomwvev mizezs. aseTi

mizezia magnituri velis brunva rotoris mimarT, rac iwvevs rotoris koWebis gamWoli

magnituri nakadis cvlilebas. maSasadame, induqciuri deni winaaRmdegobas gauwevs am

brunvas, magram vinaidan magnituri velis brunvis Seferxeba mas ar SeuZlia (igi gareSe

faqtoriTaa ganpirobebuli), winaaRmdegobis gaweva am SemTxvevaSi imiT gamoixateba, rom

rotori daiwyebs brunvas magnituri velis brunvis mimarTulebiT, rac gamoiwvevs

gamWoli magnituri nakadis cvlilebis Semcirebas. rotoris brunvis zrdasTan erTad es

cvlileba sul ufro mcirdeba. amis gamo mcirdeba masSi aRZruli induqciuri deni da

mabrunebeli momentic. Tu davuSvebT, rom rotoris brunva gautolda magnituri velis

brunvas, nakadis cvlileba da mabrunebeli momentic gaxdeba nulis toli. amitom rotori,

romelzec yovelTvis moqmedebs winaaRmdegobis Zalis (mag. xaxunis) momenti, daiwyebs

gaCerebas, damuxruWebas. brunvis siCqaris Semcireba Tavis mxriv gamoiwvevs gamWoli

magnituri nakadis cvlilebas da kvlav gaCndeba mabrunebeli momenti, romelic rotoris

Tanabari brunvis dros yovelTvis winaaRmdegobis Zalis (Zalebis) momentis tolia.

davuSvaT axla, rom winaaRmdegobis Zalis momenti (mamuxruWebeli momenti) iqmneba

rotoris meqanikuri datvirTviT. maSin, amiT gamowveuli rotoris brunvis Senelebis Sede-

gad, mis koWebSi aRiZvreba induqciuri deni da rotorze imoqmedebs mabrunebeli momenti,

romelic SesaZleblobas miscems Zravas daZlios datvirTvis mamuxruWebeli momenti e.i.

Seasrulos meqanikuri muSaoba. zemonaTqvamidan cxadia, rom aRniSnuli Zravas brunTa

ricxvi ar iqneba mudmivi da ramdenadme Seicvleba datvirTvis cvlilebasTan erTad.

amrigad, datvirTuli Zravas rotoris brunvis siCqare yovelTvis CamorCeba

magnituri velis brunvis siCqares. am SemTxvevaSi amboben, rom rotoris brunva magnituri

velis mimarT aris asinqronuli (araerTdrouli), ris gamoc aRniSnuli tipis Zravas

asinqronuli Zrava ewodeba.

CamorCena brunvaSi xasiaTdeba e.w. srialis koeficientiT:

%100⋅−

=n

nNK ,

∗ arsebobs sxva konstruqciis rotorebic.

128

sadac N da n Sesabamisad magnituri velisa da rotoris brunTa ricxvebia wuTSi.

normaluri datvirTvebisas K Cveulebriv Seadgens (3-4)%-s.

asinqronuli Zrava xasiaTdeba konstruqciis gansakuTrebuli simartiviT (ar

saWiroebs mosriale kontaqtebs da sxv.) da maRali saimedoobiT, amitom man farTo

gamoyeneba hpova teqnikaSi.

samkuTxedad SeerTebis dros generatoris pirveli gragnilis bolo uerTdeba

meoreis dasawyiss, meoris bolo mesamis dasawyiss da mesamis bolo _ pirvelis dasawyiss.

Tu datvirTvis 321 , RRR da winaRobebsac SevaerTebT samkuTxedad, miviRebT 40.5 suraTze

gamosaxul sqemas. suraTidan Cans, rom samkuTxedad SeerTebis dros iqmneba mxolod sam-

gamtariani sistema, romelSic moqmedebs mxolod erTi Zabva, vinaidan Zabva fazebs Soris

igivea, rac erT fazaSi: xf uu = . rac Seexeba denis Zalas, SeiZleba Cveneba, rom denis Za-

lis amplituda xazis sadenSi 3 -jer metia, vidre fazaSi.

dasasrul, aRvniSnoT, rom 40.2 da 40.5 suraTebze warmodgenilia is SemTxvevebi,

roca generatoris fazebi da dat-

virTvebi SeerTebulia erTnairad _

an varskvlaviseburad, an samkuTxe-

dad. ra Tqma unda, SeiZleba visar-

gebloT kombinirebuli sqemiT,

roca generatoris fazebi SeerTebu-

lia varskvlaviseburad, datvir-

Tvebi ki samkuTxedad, an piriqiT.

§41. eleqtruli energiis gadacema. transformatori.

eleqtruli energiis umTavresi upiratesoba sxva saxis energiebTan

SedarebiT mdgomareobs imaSi, rom SesaZlebelia misi gadacema Sor

manZilze sadenebis saSualebiT. magram sadenSi denis gavlisas igi Tbeba

(joul-lencis kanoni). sadenis gaTbobaze daxarjuli energia fuWad

daxarjuli energiaa. eleqtroenergiis gadacema ekonomikurad xelsayreli

rom iyos, saWiroa sadenebis gaTbobaze daxarjuli energia rac SeiZleba

SevamciroT. rogor movaxerxoT es?

joul-lencis kanoni RtIQ 2= gviCvenebs, rom sadenebis gaTbobaze

daxarjuli energia SeiZleba Semcirdes ori gziT: an sadenebis winaRobis

SemcirebiT, an denis Zalis SemcirebiT. pirveli gza moiTxovs sadenis

ganikveTis gazrdas ⎟⎠⎞

⎜⎝⎛ =

SlR ρ , rac ekonomikurad araxelsayreli (bevri

fx uu = fIfI

sur. 40. 5

xI

129

spilenZi daixarjeba) da teqnikurad mouxerxebelia (mZime sadenebs anZebze

ver davamagrebT).

maSasadame, siTburi danakargebis Sesamcireblad unda gamoviyenoT

meore gza _ denis Zalis Semcireba. es ki moiTxovs Zabvis Sesabamisad

gazrdas, vinaidan simZlavre IuP = .

amrigad, Sor manZilze eleqtroenergiis gadasacemad aucileblad

unda visargebloT maRali ZabviT.

momxmarebelTan miyvanisas maRali Zabvis deni unda gardavqmnaT

dabali Zabvis denad.

naTqvamidan gamomdinareobs, rom cvladi denis gadacemisa da

moxmarebisaTvis unda SegveZlos misi Zabvis regulireba _ gazrda da

Semcireba. amis ganxorcileba xdeba transformatoris saSualebiT.

transformatori∗ ewodeba iseT statikur (uZravnawilebian) eleqtro-

magnitur mowyobilobas (aparats), romlis saSualebiTac SeiZleba cvladi

Zabvis da denis sidideebis gardaqmna _

gazrda an Semcireba (sixSiris Seucvle-

lad). mis moqmedebas safuZvlad udevs

eleqtromagnituri induqciis movlena.

transformatoris gulari Sedgeba

foladis calkeuli firfitebisagan (fu-

kos denebTan brZolis mizniT), romlebic

Sekrulia Caketili CarCos saxiT. gular-

ze moTavsebulia ori gragnili (koWa) 1S

da 2S (sur. 41.1). gragnilebs aqvT mcire

omuri winaRoba da didi induqciuroba. 1S gragnils (romelic mierTebulia

cvladi Zabvis wyarosTan) vuwodoT pirveladi, xolo 2S -s (romelic

mierTebulia momxmarebelTan) _ meoradi. xveulebis ricxvi gragnilebSi

Sesabamisad aRvniSnoT 1N -iT da 2N -Ti.

transformatoris sruli Teoria, gansakuTrebiT Tu igi iTvalis-

winebs gularis histerezisis rolsac, rTulia. Cven SemovifargloT trans-

formatoris muSaobis gamartivebuli aRweriT.

∗ warmodgeba laTinuri sityvisagan transformo _ gardavqmni.

1u 2u

1N 2N

2S1S

sur. 41.1.

130

1S gragnilis boloebs movdoT cvladi 1u Zabva. gragnilSi gaivlis

cvladi 0I deni, romelic daamagnitebs gulars da warmoSobs masSi cvlad

magnitur nakads.

denis TandaTan zrdasTan erTad izrdeba gularSi magnituri nakadi.

magnituri nakadis zrda ki iwvevs 1S gragnilSi TviTinduqciis em Zalis

aRZvras ( )Sε . es ukanaskneli, lencis wesis Tanaxmad, ewinaaRmdegeba denis

zrdas gragnilSi da rogorc ki TviTinduqciis em Zala gautoldeba

modebuli Zabvis sidides, pirveladi gragnilis wredSi denis zrda

Sewydeba. amrigad, pirveladi gragnilis wredSi moqmedebs modebuli Zabva

1u da TviTinduqciis em Zala Sε . 1u metia Sε -ze gragnilSi Zabvis vardnis

sididiT ( )RI 0 ,∗ magram igi Zalian mcirea, vinaidan gragnilis omuri

winaRoba ( )R aris mcire. amitom, miaxloebiT SeiZleba daiweros (sididiT):

1u Sε≈ . (41.1)

transformatoris gularSi warmoqmnili cvladi magnituri nakadi,

ganWolavs ra 2S gragnils, aRZravs masSi induqciis em Zalas 2ε -s. rodesac

meoradi gragnilis wredi ganrTulia, mis boloebs Soris 2u Zabva tolia

induqciis em Zalisa, e.i.

22 ε=u . (41.2)

satransformatoro foladisagan damzadebuli gulari axdens

magnituri velis koncentracias, ase rom magnituri nakadi arsebobs

praqtikulad mxolod gularSi da mis yvela kveTaSi erTnairia.∗∗ vinaidan

1S da 2S gragnilebs ganWolavs erTidaigive magnituri nakadi, cxadia, maT

TiToeul xviaSi aRiZvreba erTidaigive sididis induqciis em Zala. Tu mas

aRvniSnavT e -Ti, gveqneba:

eNeNS

22

1

==

εε

(41.3)

(41.1), (41.2) da (41.3) tolobebis Sedareba gvaZlevs:

KNN

uu

==2

1

2

1 , e.i. (41.4)

∗ pirveladi gragnilis wredi warmoadgens wredis araerTgvarovan ubans, amitom misTvis

omis kanons aqvs saxe: SuRI ε+=0 . Tu 00 ≈RI , maSin 1u Sε−≈

∗∗ sinamdvileSi, nakadis gabnevis gamo, 2S gragnilis gamWoli magnituri nakadi ramdenadme

naklebia, vidre 1S -is.

131

transformatoris pirveladi gragnilebis boloebze modebuli Zabva ise

Seefardeba meoradi gragnilis boloebze miRebul Zabvas, rogorc

pirveladi gragnilis xviaTa ricxvi meoradi gragnilis xviaTa rixvs.

mudmiv K sidides ewodeba transformatoris transformaciis koefi-

cienti. (41.4)-dan Cans, rom Tu ( )112 <> KNN , maSin 12 uu > . aseT

transformators ewodeba samaRlebeli da piriqiT.

vidre meoradi gragnilis wredi ganrTulia (e.i. masSi deni ar gadis),

transformatoris muSaoba fuWia.∗ am dros energiis gadacema pirveladi

gragnilis wredidan meoreSi ar xdeba. fuWi svlis dros transformatoris

mier moxmarebuli energia mcirea, vinaidan 1S koWas didi induqciurobis

gamo gularis damamagnitebeli 0I deni metad mcirea.

davtvirToT transformatori _ meoradi gragnilis wredSi CavrToT

R winaRobis raime momxmarebeli. masSi gaivlis induqciuri deni ( )2I . 2I

deni, lencis wesis Tanaxmad, gamoiwvevs gularSi magnituri nakadis

Semcirebas. magnituri nakadis Semcirebas mohyveba pirvelad gragnilSi

TviTinduqciis em Zalis Semcireba da wonasworoba 1u -sa da Sε -s Soris

dairRveva. amis Sedegad pirvelad gragnilSi deni gaizrdeba, raRac 1I

sididiT da gaxdeba ( )10 II + -is toli. denis gazrda gamoiwvevs magnituri

nakadis gazrdas da darRveuli wonasworoba 1u -sa da Sε -s Soris kvlav

aRdgeba. amrigad, transformatoris datvirTva iwvevs pirveladi

gragnilis wredSi denis gazrdas 1I sididiT. 1I deni gansazRvravs

1S gragnilis datvirTvis dens.

pirvelad gragnilSi denis Zalis gazrda xdeba energiis mudmivobis

kanonis Sesabamisad. kerZod, meoradi gragnilis wredSi momxmareblis mier

daxarjuli energia, tolia pirveladi gragnilis mier qselidan miRebuli

energiisa. Tu transformatoris datvirTva axloa nominalurTan,∗∗ simZlav-

reebi pirveladi da meoradi gragnilis wredSi daaxloebiT tolia:

,2211 uIuI ≈ saidanac 1

2

2

1

II

uu

= , (41.5)

∗ unda aRiniSnos, rom (41.4) toloba zustad mxolod am dros sruldeba. muSa svlis dros tolobaSi Semaval sidideebs Soris damokidebuleba ufro rTulia. ∗∗ warmodgeba laTinuri sityvisagan nominalis _ saxelobiTi, rac aRniSnulia rameze (xelsawyoze, danadgarze da a.S).

132

e.i. Zabvis ramdenjerme gazrda iwvevs denis Zalis imdenjerve

Semcirebas da piriqiT. (41.5)-is Sedareba (41.4)-Tan gvaZlevs: 2

1

1

2

NN

II

= , e.i.

transformatoris pirveladi da meoradi gragnilebis datvirTvis denebi

maTSi xviebis ricxvis ukuproporciulia.

rogorc aRvniSneT (41.5) toloba miaxloebiTia, sinamdvileSi

1122 uIuI < , vinaidan transformatoris muSaobis dros energiis nawili

ixarjeba gragnilebis gaTbobaze, gularSi fukos denebis aRZvraze, adgili

aqvs magnituri nakadis gabnevas da sxv. magram es danakargebi mcirea.

Tanamedrove transformatoris mqk 97-99%-ia.

arsebobs transformatorebis sxvadasxva saxe. laboratoriuli mizne-

bisaTvis farTod gamoiyeneba avtotransformatori. aseTi transforma-

toris meorad gragnils warmoadgens pirveladi

gragnilis nawili (sur. 41.2). aseTi transformatore-

bi sadablebelia da piriqiT _ Tu 1u Zabva modebu-

lia gragnilis nawilze, 2u Zabvis moxsna ki xdeba

mTeli gragnilidan _ avtotransformatori samaRle-

belia. avtotransformatoris erTerT kontaqts

xSirad akeTeben mosriales, rac saSualebas iZleva

mdored vcvaloT meoradi gragnilidan mosaxsneli

Zabvis sidide.∗

amamaRlebel transformatorebs Soris gamoiyofa sruliad gansxvaveuli tipis

trasformatori, romelsac sainduqcio koWa (induqtori, bobina) ewodeba. induqtoris

gansxvavebulba imiT gamoixateba, rom misi saSualebiT xdeba mudmivi denis gardaqmna

maRali sixSiris cvlad denad. magram mudmivi deni aRZravs induqciur dens mxolod misi

CarTvis da amorTvis momentSi. amitom, mudmivi denis transformaciisaTvis aucilebelia

igi gardaiqmnas wyvetil (pulsirebul) denad. es xorcieldeba sxvadasxva

konstruqciis avtomaturi mwyvetebis saSualebiT, romlebsac Cven ar ganvixilavT.

sainduqcio koWas gulari Cauketavia, igi Sedgeba foladis calkeuli Reroebisagan.

masze moTavsebulia ori gragnili (koWa) _ pirveladi da meoradi. pirveladi gragnili

damzadebulia mcire winaRobis mqone msxvili mavTulisagan da Seicavs xviaTa SedarebiT

mcire raodenobas. meoradi gragnili piriqiT _ damzadebulia metad wvrili mavTulisagan

da Seicavs xviaTa Zalian did raodenobas.∗∗

pirveladi gragnili, romlis wredSi CarTulia ama Tu im konstruqciis avtomaturi

mwyveti, ikvebeba mudmivi denis wyarodan, magaliTad, akumulatorebis batareidan (sur. 41.3).

∗ aseTia, magaliTad CvenTan farTod gavrcelebuli rusuli warmoebis laboratoriuli avtotransformatori `ЛАТР~. ∗∗ meoradi gragnilis dasamzadeblad kilometrobiT sigrZis mavTuli ixarjeba.

2u

1u

sur. 41.2

133

meoradi gragnilis boloebi gamotanilia gareT (momxmarebelTan misaerTeblad). pirvelad

gragnilSi denis (`pirveladi denis~) yoveli CarTvis dros meorad gragnilSi aRiZvreba

pirveladi denis sawinaaRmdego mimarTulebis deni, amorTvisas ki _ pirveladi denis mi-

marTulebis deni (lencis wesi). amasTan, vinaidan meoradi

gragnilis xviaTa ricxvi gacilebiT aRemateba pirveladi

gragnilis xviaTa ricxvs, masSi aRZruli em Zala aRwevs

ramdenime aseul aTas volts (sixSiriT 1000-2000 hc).

sur. 41.4-ze mocemuli grafikebi gamosaxavs induq-

toris pirvelad gragnilSi gamavali denis (a) da meorad

gragnilSi aRZruli induqciis em Zalis (b) droze damo-

kidebulebas. 1t Seesabameba drois im moments, rodesac

mwyveti rTavs pirvelad dens, xolo 2t _ roca amorTavs.

41.4 a grafikidan Cans, rom pirvelad gragnilSi denis Zalis zrda CarTvisas mimdinareobs

ufro nela (oa ubani), vidre misi Semcireba amorTvisas ( ab ubani). Sesabamisad, meorad

gragnilSi aRZruli induqciis em Zala CarTvisas ufro naklebia vidre amorTvisas (sur.

41.4 b). amrigad, meoradi gragnilis momxmarebelTan mierTebisas, masSi gadis asimetriuli

cvladi deni, magram, cxadia, rom orive mimarTulebiT gadatanili eleqtrobis raodenoba

iqneba erTi da igive. Tu meoradi gragnilis boloebs Soris davtovebT raime Sualeds

(rogorc es Cvens suraTzea gakeTebuli), maSin CarTvis momentSi aRZruli induqciis em

Zala ar iqneba sakmarisi boloebs Soris arsebuli airis garRvevisa da naperwklis

warmoqmnisaTvis. naperwkali warmoiSoba mxolod pirvelad gragnilSi denis amorTvis

momentSi. am SemTxvevaSi meo-

radi gragnilis wredSi gaiv-

lis pulsirebuli, magram yo-

velTvis erTi da igive mimar-

Tulebis deni.

sainduqcio koWa far-

Tod gamoiyeneba rogorc

laboratoriul praqtikaSi,

aseve teqnikaSi. magaliTad,

induqtoris gamoyenebiT her-

cma, pirvelad msoflioSi, mi-

iRo eleqtromagnituri tal-

Rebi (§49), induqtoris (bo-

binis) meSveobiT xdeba Siga-

wvis ZravaSi sawvavi narevis

aaleba da sxv.

I

t

o

t

ε

a

b

a

b

sur. 41.4

1t 2t

134

Tavi V

eleqtromagnituri rxevebi da talRebi.

maqsvelis gantolebebi

§42. rxeviTi konturi. tomsonis formula.

eleqtromagnitur rxevebs gaaCnia sixSireTa erTob farTo diapazoni.

sxvadasxva sixSiris rxevebi arsebiTad gansxvavdebian erTmaneTisagan, rogorc

TvisebebiT, ise miRebis saSualebebiT. amitom miRebulia maTi dayofa ramode-

nime saxed. aseTi dayofa mocemulia cxrilSi, 49-e paragrafis bolos. cxadia,

rom saxeobebs Soris sazRvari ar aris da arc SeiZleba iyos mkveTri. adgili

aqvs mezobel saxeTa sixSireTa intervalis nawilobriv gadafarvas. mocemul

paragrafSi ganxilulia im maRali sixSiris eleqtromagnituri rxevebis miRe-

ba, romelic gamoyenebulia radiokavSirgabmulobaSi (radiotalRebi). aseTi

rxevebi miiReba rxeviTi konturis saSualebiT, romelic warmoadgens

nebismieri radioteqnikuri mowyobilobis ganuyofel nawils.

rxeviTi kon-

turi ewodeba eleqt-

rul wreds, romelic

Sedgeba C kondensa-

torisa da L induq-

ciurobis koWasagan

(sur. 42.1. a). misi da-

niSnulebaa maRali sixSiris (radiosixSiris) eleqtromagnituri rxevebis

aRZvra. vnaxoT, rogor xdeba es.

K gadamrTveli gadavrToT a kontaqtze da davmuxtoT kondensatori

(sur. 42.1.b). amis Sedegad firfitebs Soris Seiqmneba eleqtruli veli, romlis

energia 22

22 CuC

qW == , sadac q kondensatoris muxtia, C _ misi eleqtroteva-

doba, xolo u _ firfitebs Soris potencialTa sxvaoba. amis Semdeg K gadam-

rTveli gadavrToT b kontaqtze da daviwyoT aqedan drois aTvla ( )0=t . am

momentisaTvis kondensatoris muxti da firfitebs Soris Zabva maqsimaluria,

Sesabamisad maqsimaluria eleqtruli velis energiac. koWaSi ki jer deni ar

bKa

a

LC

sur. 42.1

b

L+−

135

gadis, amitom nulis tolia koWas magnituri velis energia ⎟⎟⎠

⎞⎜⎜⎝

⎛= 0

2

2LI

(sur. 42.3. b). K gadamrTvelis b kontaqtze gadarTvis Semdeg kondensatori

iwyebs ganmuxtvas _ eleqtronebi daiwyeben moZraobas uaryofiTad damuxtuli

firfitidan dadebiTisaken da wredSi gaivlis sawinaaRmdego mimarTulebis

deni. es deni aRZravs magnitur vels, romelic faqtiurad koWas SigniTaa

koncentrirebuli. kondensatoris ganmuxtvisas misi eleqtruli velis energia

gardaiqmneba koWas magnituri velis energiad, msgavsad imisa, rogorc

wonasworobis mdgomareobidan gadaxrili zambariani qanqaras potencialuri

energia ⎟⎟⎠

⎞⎜⎜⎝

⎛2

2Kx, gadadis kinetikurSi ⎟⎟

⎠

⎞⎜⎜⎝

⎛2

2υm burTulas wonasworobis

mdebareobisaken moZarobisas. rogorc viciT (It. §36), burTula, miaRwevs ra

wonasworobis mdebareobas, ar Cerdeba _ igi inerciiT ganagrZobs moZraobas,

romlis drosac xdeba SeZenili kinetikuri energiis gadasvla potenciurSi.

ramdenadme analogiur movlenas aqvs adgili rxeviT konturSi. ufro

kargad rom gaverkveT rxeviT konturSi mimdinare procesebSi, gavwyvitoT igi

da mivuerToT eleqtronul oscilografs (§16). ekranze miviRebT konturSi ga-

mavali denis cvlilebas drois mixedviT (sur. 42.2). grafikidan Cans, rom gan-

muxtvis deni uceb ki ar aRwevs Tavis maqsimalur mniSvnelobas (oa ubani),

aramed izrdeba TandaTan, iseve rogorc TandaTan ganimuxteba kondensatori.

amis mizezia koWaSi aRZruli TviTinduqciis em Zala, romelic lencis wesis

Tanaxmad, ewinaaRmdegeba denis yovelgvar cvlilebas (am SemTxvevaSi _ gazrdas).

rodesac deni miaRwevs maqsimalur mniSvnelobas ( a wertili. es moxda

konturis CarTvidan 4Tt = drois

Semdeg), kondensatoris firfitebs

Soris Zabva nulamde daecema. amri-

gad, 4T drois ganmavlobaSi konden-

satoris eleqtruli velis energia

gardaiqmna koWas magnituri velis

energiad (es zambariani qanqaras

SemTxvevaSi eqvivalenturia imisa,

T43

2T

I

a

b

c

td o

4T T

sur. 42.2

136

rom burTula mivida wonasworobis mdebareobaSi da misi potenciuri energia

mTlianad gadavida kinetikurSi). omis kanonis Tanaxmad ⎟⎠⎞

⎜⎝⎛ =

RuI , rogorc ki

Zabva firfitebs Soris gaxda nulis toli, denis Zalac myisve unda Semcire-

buliyo nulamde. grafiki ki gviCvenebs, rom deni, miedineba ra imave mimarTu-

lebiT, mcirdeba TandaTan ( ab ubani) da mxolod garkveuli drois Semdeg

(kerZod, 4T drois Semdeg) xdeba nulis toli (b wertili). amis mizezia L

koWaSi aRZruli TviTinduqciis em Zala, romelic ewinaaRmdegeba denis swraf

Semcirebas. denis TandaTanobiTi Semcirebisas xdeba kondensatoris firfi-

tebis gadamuxtva. (koWas magnituri velis energia gardaiqmneba kondensatoris

eleqtruli velis energiad). es gadamuxtva mTavrdeba maSin, roca deni koWaSi

gaxdeba nulis toli. am dros Zabva kondensatoris firfitebs Soris

maqsimaluria, magram eleqtrul vels aqvs Tavdapirveli velis sawinaaRmdego

mimarTuleba: axla ukve kondensatoris qveda firfitaze gvaqvs uaryofiTi

muxti, zedaze _ dadebiTi (konturis CarTvidan gavida 2T dro).

amis Semdeg yvelaferi meordeba sawinaaRmdego mimarTulebiT (bcd ubani)

da kidev 2T drois Semdeg, e.i. konturis CarTvidan T drois Semdeg, igi aRmoC-

ndeba Tavdapirvel mdgomareobaSi.

amrigad, rxeviT konturSi adgili aqvs kondensatoris eleqtruli veli-

sa da koWas magnituri velis periodul cvlilebas. eleqtruli da magni-

turi velebis periodul cvlilebas eleqtromagnituri rxevebi

ewodeba.

sur. 42.3-ze mocemulia Sedareba rxeviT konturSi mimdinare processa da zambariani

qanqaras rxevebs Soris.

sur. 42.3 a gamosaxavs im SemTxvevas, roca kondensatori jer ar damuxtula, anu rxeviT

konturs energia ar gadacemia. zambariani qanqaras semTxvevaSi amas Seesabameba misi is mdgo-

mareoba, roca burTula wonasworobis mdgomareobidan jer ar gamouyvaniaT. kondesatoris

damuxtvas Seesabameba burTulas gadaxra wonasworobis mdebareobidan x sididiT.

drois aTvla ( )0=t iwyeba konturis CarTvidan anu K gadamrTvelis b kontaqtze ga-

dayvanidan. qanqaras SemTxvevaSi es Seesabameba wonasworobis mdgomareobidan x manZiliT

gadaxrili burTulas ganTavisuflebas (xelis gaSvebas). am momentisaTvis kondesatoris

137

muxti da Zabva firfitebs Soris maqsimaluria, e.i. maqsimaluria kondesatoris eleqtruli

velis energia, koWaSi denis Zala ki nulis tolia, e.i. nulis tolia magnituri velis energia.

qanqaras SemTxvevaSi amas Seesabameba burTulas maqsimaluri gadaxra, e.i. misi potenciuri

energia ⎟⎟⎠

⎞⎜⎜⎝

⎛2

2Kxmaqsimaluria, xolo kinetikuri energia nulis tolia, vinaidan burTula jer

kidev gaCerebulia (sur. 42.3 b).

4Tt = momentisaTvis kondesatori mTlianad ganmuxtulia, xolo deni koWaSi maqsima-

luria ( )0I . e.i. am droisaTvis kondensatoris eleqtruli velis energia mTlianad gardaq-

mnilia koWas magnituri velis energiad. qanqaras SemTxvevaSi amas Seesabameba burTulas

wonasworobis mdgomareobaSi mosvla, roca misi siCqare maqsimaluria, e.i. maqsimaluria misi

kinetikuri energia; potenciuri energia ki nulis tolia, vinaidan zambaras deformaciaa nuli

(sur. 42.3 g).

am momentidan iwyeba koWaSi denis Semcireba da kondensatoris gadamuxtva. am process

sWirdeba kidev 4T

dro da konturis CarTvidan 2T

drois gasvlis Semdeg gvaqvs: koWaSi deni

nulis tolia, kondensatoris muxti ki maqsimaluria (gadamuxtva damTavrebulia), oRond el-

eqtrul vels kondensatorSi

aqvs Tavdapirvelis sawinaaRmde-

go mimarTuleba. qanqaras SemTx-

vevaSi amas Seesabameba burTulas

gadaxra marcxena ukidures mdgo-

mareobaSi, roca misi potenciuri

energia maqsimaluria, kinetikuri

ki nulis tolia, vinaidan bur-

Tula wamierad gaCerebulia

(sur. 42.3d).

amis Semdeg yvelaferi me-

ordeba sawinaaRmdego mimarTu-

lebiT da kidev 2T

drois Sem-

deg, anu konturis CarTvidan T

drois Semdeg, sistema aRmoCnde-

ba Tavdapirvel mdgomareobaSi

(sur. 42.3b). Sesruldeba erTi

sruli rxeva. amis Semdeg aRweri-

li procesi ganmeordeba.

4Tt =

2Tt =

a L C

sur. 42.3

0=I 0

0

qq

+−

0I

0=t

0=I

b

g

d 0

0

qq

−+

0=υ

0=υ

maqsυ

x

x

138

eleqtromagnituri rxevis periodi ( )T ewodeba drois im

umcires Sualeds, romlis ganmavlobaSic eleqtruli veli konden-

storis Semonafenebs Soris ( )E , an magnituri veli koWaSi ( )B , ic-

vleba ra Tavis romeliRac mniSvnelobidan, kvlav Rebulobs amave

mniSvnelobas rogorc sididiT, ise mimarTulebiT.

cxadia, rom eleqtromagnituri rxevis periodi damokidebuli unda iyos

rxeviTi konturis parametrebze _ kondensatoris tevadobaze da koWas induq-

ciurobaze. kerZod, C -s da L -is gazrdiT T unda gaizardos. marTlac, rac

metia kondensatoris tevadoba, miT met muxts daitevs igi da miT meti dro

iqneba saWiro misi ganmuxtvisaTvis. meores mxriv, rac metia L , miT meti

TviTinduqciis em Zala aRiZvreba koWaSi da miT ufro nela moxdeba masSi

denis gazrda an Semcireba (e.i. miT ufro nela moxdeba kondensatoris gan-

muxtva). Teoria iZleva Semdeg formulas:

LCT π2= , (42.1)

sadac L aris koWas induqciuroba gamosaxuli henrebSi, C _ tevadoba fara-

debSi, T _ periodi wamebSi. (42.1) formulas tomsonis formula ewodeba.

gamoviyvanoT tomsonis formula. amisaTvis unda davuSvaT, rom rxeviTi

konturi idealuria, anu misi omuri winaRoba 0=R . am SemTxvevaSi konturSi

gveqneba erTi Zabvis vardna (kondensatorze) da erTi induqciis em Zala

(koWaSi aRZruli TviTinduqciis em Zala Sε ). amitom kirxhofis II kanoni Sem-

degnairad daiwereba:

Su ε= . (42.2)

magramM ,;dtdIL

Cqu S −== ε xolo

dtdqI = . am sidideebis Casma (42.2)-Si mogvcems:

012

2

=+ qCdt

qdL .

Tu tolobis yvela wevrs gavyofT L -ze da SemoviRebT aRniSvnas

CL12

0 =ω , (42.3)

sabolood miviRebT:

0202

2

=+ qdt

qd ω . (42.4)

139

es gantolrba zustad Tanxvdeba harmoniuli rxevis diferencialur

gantolebas (Itomi, §36). viciT, rom mis amonaxsens aqvs saxe

( )ϕω += tqq 00 cos . (42.5)

amitom

( ) ( )ϕωϕω +=+== tutCq

Cqu 000

0 coscos

da

( )ϕωω +−== tqdtdqI 000 sin , an ⎟

⎠⎞

⎜⎝⎛ ++=

2cos 00

πϕω tII (42.6)

sadac 00 , uq da 0I muxtis, Zabvisa da denis Zalis maqsimaluri anu amplitu-

duri mniSvnelobebia.

miRebuli formulebi gviCvenebs, rom rxeviT konturSi kondensatoris

Semonafenebze muxti da Zabva, xolo koWaSi deni periodulad icvleba, rac

savsebiT eTanxmeba Cvens mier paragrafis dasawyisSi gakeTebul Tvisobriv

daskvnas.

LC1

0 =ω aris konturis rxevis wriuli sixSire. magram viciT (I tomi,

§36) rom 0

2ωπ

=T , amitom LCT π2= , rac Tanxvdeba (42.1)-s.

rxevis sixSire (rxevaTa ricxvi wamSi) ν periodis Sebrunebuli sididea.

amitom

LCT πν

211

== (42.7)

(42.7) formulis Tanaxmad, maRali sixSiris eleqtromagnituri rxevebis misaRe-

bad L da C unda iyos rac SeiZleba mcire. mag. rxeviTi konturi, romelic

Seicavs 10100001,0 −== mkfC faradis tol tevadobas da 0001,0=L hn-is tol

induqciurobas, aRZravs rxevebs, romlis sixSire

hchc 6

104106,1

101014,321

21

⋅≈⋅⋅

==−−LCπ

ν .

cxadia, veraviTari induqciuri generatori aseTi sixSiris rxevebs (cvlad

dens) ver aRZravs, vinaidan amisaTvis saWiro iqneboda rotors Seesrulebina

milioni bruni wamSi.

Tu konturis omuri winaRoba 0≠R , rxevis periodi izrdeba (§43):

140

2

21

2

⎟⎠⎞

⎜⎝⎛−

=

LR

LC

T π.

Cveulebriv, rxeviTi konturis omuri winaRoba mcirea, amitom periodis

gamosaTvlelad, rogorc wesi, sargebloben (42.1) formuliT.

§43. milevadi eleqtromagnituri rxevebi

Cvens mier wina paragrafSi ganxiluli eleqtromagnituri rxevebi rxe-

viT konturSi aRiZvra erTxel miniWebuli energiis xarjze _ kondensatori

erTxel davmuxteT. aseT rxevas, romelic warmoebs erTxel miniWebuli ener-

giis xarjZe, sakuTari anu Tavisufali rxeva ewodeba,∗ xolo Sesabamis six-

Sires _ rxevis sakuTari sixSire. sakuTari imitom, rom igi damokidebulia

merxevi sistemis sakuTar Tvisebebze (C -ze da L -ze). sakuTari eleqtro-

magnituri rxevebi, rogorc amas sur. 42.2-ze moyvanili grafiki gviCvenebs,

yovelTvis milevadia (meqanikuris msgavsad), vinaidan erTxel miniWebuli

energia ixarjeba konturis winaaRmdegobis daZlevaze (realurad konturis

omuri winaRoba yovelTvis gansxvavdeba nulisagan), misi Sevseba ki ar xdeba.

gamoviyvanoT muxtis Zabvisa da denis cvlilebis formulebi realuri

konturisaTvis ( )0≠R . am SemTxvevaSi kirxhofis meore kanoni Semdegnairad

daiwereba:

SIRu ε=+ . (43.1)

radgan dtdqI

Cqu == , , xolo

dtdILS −=ε , miviRebT:

012

2

=++ qCdt

dqRdt

qdL . (43.2)

es diferencialuri gantoleba Tavisi formiT savsebiT analogiuria

zambariani qanqaris milevadi rxevis diferencialuri gantolebisa (Itomi, §40,

40.3 gantoleba):

02

2

=++ kqdtdxr

dtxdm . (43.3)

∗ an, rac igivea, Tavisufali rxeva ewodeba iseT rxevas, romelic aRiZvreba sistemaSi Sinagani Zalebis moqmedebiT, misi wonasworobis mdgomareobidan gamoyvanis Semdeg.

141

(43.2)-Si m masis analogs warmoadgens L induqciuroba, winaRobis r

koeficientisas _ wredis R omuri winaRoba, winaRobis dtdxr Zalisas _

dtdqR ,

zambaris drekadobis K koeficientisas _ C1, xolo x gadaxrisas _ q muxti.

(43.3) gantolebis amoxsnisaTvis, axali cvladis Semoyvanis meTodis

gamoyenebiT, Cven miviReT:

( ) ( )ϕωϕω +=+=−

tAteAxt

mr

coscos20 , (43.4)

sadac t

mr

eAA 20

−= da ϕ _ amplituda da sawyisi fazaa, xolo

2

220 2

⎟⎠⎞

⎜⎝⎛−=−=

mr

mKβωω (43.5)

milevadi rxevis wriuli sixSirea.

(43.4)-is analogiurad (43.3)-is amoxsnas aqvs saxe:

( ) ( )ϕωϕω +=+=−

tAteqqt

LR

coscos20 , (43.6)

sadac

t

LR

eqA 20

−= (43.7)

kondensatoris muxtis rxevis amplitudaa, ϕ - sawyisi faza, xolo ω _

wriuli sixSire, romlisTvisac (43.5)-is analogiurad gveqneba:

222

0 21

⎟⎠⎞

⎜⎝⎛−=−=

LR

LCβωω . (43.8)

(43.7) gviCvenebs, rom realur konturSi muxtis rxevis amplituda drois

mixedviT eqsponencialurad mcirdeba, anu rxeva milevadia. (43.6) warmoadgens

milevadi eleqtromagnituri rxevis gantolebas. rxevis mileva miT ufro

swrafad xdeba, rac metia L

R2

=β -is sidide, anu rac metia R da naklebia L .

β -s ewodeba milevis koeficienti.

milevadi rxevis periodi∗

∗ sazogadod milevadi rxeva perioduli ar aris, vinaidan merxevi sidide TandaTan mcirdeba da aRar ubrundeba sawyis mniSvnelobas. amitom, mkacrad Tu vimsjelebT, misTvis gamousadegaria sixSirisa da periodis cnebebi. magram Tu mileva didi ar aris, SeiZleba pirobiTad periodi vuwodoT drois Sualeds merxevi fizikuri sididis or momdevno maqsimums (an minimums) Soris.

142

2

21

22

⎟⎠⎞

⎜⎝⎛−

==

LR

LC

T πωπ

. (43.9)

(43.8) da (43.9) formulebi gviCvenebs, rom realuri rxeviTi konturis

sixSire naklebia, xolo periodi metia, vidre idealuris, rac gasagebia,

vinaidan winaaRmdegobis Zalis arseboba, cxadia, gamoiwvevs yovelgvari rxevis

(rogorc meqanikuris, aseve eleqtromagnituris) sixSiris Semcirebas.

muxtis cvlilebis (43.6)

kanonzomierebidan advilad da-

vadgenT Zabvisa da denis Zalis

cvlilebis kanonzomierebebs,

radganac Cqu = , xolo

dtdqI = .

(43.1) suraTze mocemulia

muxtis (uwyveti wiri) da

amplitudis (wyvetili wiri)

cvlilebis grafikebi∗ (I tomis

40.2 suraTze milevadi rxevis

grafiki Seesabameba im SemTxve-

vas, roca 0=ϕ ).

(43.9) formula saSualebas gvaZlevs davadginoT piroba, romelsac unda akmayofi-

lebdes konturis damaxasiaTebeli sidideebi ( )RLC ,, imisaTvis, rom konturSi aRiZvras

rxeva. es piroba mdgomareobs imaSi, rom konturs unda gaaCndes namdvili periodi, anu fes-

vqveSa gamosaxuleba unda iyos dadebiTi:

,04

12

2

>−L

RLC

anu 4

2CRL > .

winaaRmdeg SemTxvevaSi kondensatoris rxeva iqneba aperioduli (araperioduli) xasiaTis _

q muxti monotonurad Semcirdeba drois mixedviT.

∗ roca ,,0 00 qAAt === xolo 00 cos qqq <= ϕ .

Aq,

2A 1A ϕcos0q

c

t

teqA β−= 00A

( )ϕωβ += − teqq t cos0

sur. 43.1

143

§44. iZulebiTi eleqtromagnituri rxevebi. TviTrxevebi

ganmartebis Tanaxmad, iZulebiTi rxeva ewodeba iseT rxevas,

romelic warmoebs periodulad cvladi gareSe Zalis moqmedebiT.

iZulebiT eleqtromagnitur rxevebTan Cven saqme gvqonda cvladi denis sruli

wredis ganxilvisas (§37) _ cvladi deni swored dabali sixSiris eleqtro-

magnituri rxevebia. aq periodulad cvlad gareSe Zalas warmoadgenda wredSi

CarTuli cvladi denis wyaros em Zala:

tωεε sin0= .

deni aseT konturSi icvleba kanoniT

( )ϕω −= tII cos0 ,

sadac

22

00

1⎟⎠⎞

⎜⎝⎛ −+

=

ωω

ε

CLR

I da R

CL

tg ωω

ϕ

1−

= .

am SemTxvevaSi konturSi rxeva iarsebebs nebismierad didi drois

ganmavlobaSi, anu sistema Seasrulebs aramilevad rxevebs.

magram es rxevebi, romelic aRiZvreba eleqtrosadgurSi cvladi denis

generatoris mier gamomuSavebuli em Zalis moqmedebiT, rogorc zemoT iyo

aRniSnuli, dabali sixSiris eleqtromagnituri rxevebia. aseT generators ar

SeuZlia radiokavSirisaTvis aucilebeli maRali sixSiris eleqtromagnituri

rxevebis aRZvra, vinaidan amisaTvis saWiro iqneboda rotors Seesrulebina

asiaTasobiT bruni wamSi.

maRali sixSiris eleqtromagnitur rxevebs warmoqmnis rxeviTi konturi,

magram rogorc §43-Si vnaxeT, realuri konturis SemTxvevaSi ( )0≠R , es rxeve-

bi yovelTvis milevadia. rxevebi rom aramilevadi iyos, saWiroa ganuwyvetliv

Seivsos konturis omuri winaRobis daZlevaze daxarjuli energia. gare

perioduli Zalis moqmedebiT amis miRweva, Tu gaviTvaliswinebT rxevis sixSi-

res, praqtikulad SeuZlebelia. magram aRmoCnda, rom arsebobs aramilevadi

rxevebis aRZvris sxva gzac. kerZod, Tu sistema romelSic SesaZlebelia

Tavisufali rxevebis aRZvra, Seicavs energiis wyaros da Tu sistemas

TviTonve SeuZlia areguliros rxeviTi konturisaTvis energiis miwodeba

144

omuri winaRobis daZlevaze daxarjuli energiis sakompensaciod, maSin rxevis

milevas adgili ar eqneba.

sistemas, romelSic warmoiqmneba aramilevadi rxeva sistema-

Sive arsebuli wyarodan energiis miRebis xarjze, TviTmerxevi

sistema ewodeba, xolo aramilevad rxevebs, romelbic arseboben sistemaSi

gare perioduli Zalis moqmedebis gareSe _ TviTrxevebi∗ (avtorxevebi). aRsa-

niSnavia, rom TviTrxevebis amplituda da periodi ganisazRvreba sistemis

TvisebebiT (iZulebiTi rxevisagan gansxvavebiT).

TviTmerxev sistemebs, romelTa saSualebiTac miiReba maRali sixSiris

eleqtromagnituri rxevebi miekuTvneba milakiani da tranzistoruli genera-

torebi. maT ase ewodeba imitom, rom rxeviTi konturisa da energiis wyaros

garda, pirveli Seicavs sameleqtrodian eleqtronul milaks (troids), xolo

meore _ agreTve sameleqtrodian naxevargamtarul xelsawyos, romelsac

tranzistori∗∗ ewodeba (`naxevargamtaruli triodi~ tranzistoris moZvele-

buli saxelwodebaa).

§45. wanacvlebis deni

rogorc viciT (§19), cvlad magnitur velSi velSi moTavsebul uZrav

gamtar konturSi induqciuri denis aRZvris asaxsnelad maqsvelma Camoayaliba

eleqtromagnituri velis Teoriis pirveli ZiriTadi debuleba: magnituri

velis yovelgvari cvlilebisas droSi warmoiqmneba grigaluri eleqtruli

veli.

sxvadasxva eleqtromagnituri procesebis analizis safuZvelze maqsveli

mivida daskvnamde, rom unda arsebobdes Sebrunebuli movlenac: eleqtruli

velis yovelgvari cvlileba droSi iwvevs grigaluri magnituri

velis warmoqmnas. es mtkiceba gamosaxavs eleqtromagnituri velis

umniSvnelovanes Tvisebas da warmoadgens maqsvelis Teoriis meore ZiriTad

debulebas. radgan magnituri veli yovelTvis dakavSirebulia eleqtrul

denTan (eleqtruli deni warmoadgens magnituri velis pirvelwyaros (§2)),

∗ TviTrxevebi aRiZvreba ara marto eleqtrul, aramed meqanikur sistemebSic. umartives meqanikur TviTmerxev sistemas warmoadgens qanqariani saaTi, romelSic energiis wyaros warmoadgens aweuli tvirTis potenciuri energia. ∗∗ warmodgeba inglisuri sityvebidan transfer _ gadatana da resistor _ winaRoba.

145

amitom maqsvelma cvlad eleqtrul vels uwoda wanacvlebis deni,

ganasxvava ra igi gamtarobis denisagan, romelic ganpirobebulia muxtebis

mowesrigebuli moZraobiT.

eleqtruli velis cvlilebis Sedegad warmoq-

mnili grigaluri magnituri velis induqciis wirebi

moicavs eleqtruli velis Zalwirebs, msgavsad imisa,

rogorc grigaluri eleqtruli velis Zalwirebi

moicavs magnituri induqciis wirebs, magram axla,

eleqtruli velis daZabulobis zrdisas ⎟⎠⎞

⎜⎝⎛ > 0

dtdE

,

kavSiri magnituri induqciis wirebis mimarTulebasa

da E veqtoris mimarTulebas Soris mocemulia mar-

jvena burRis wesiT (amboben, rom induqciis wirebis

mimarTuleba E veqtoris mimarTulebasTan qmnis mar-

jvena xraxns) (sur. 45.1).

wanacvlebis denis SemoRebis Semdeg Seicvala Cveni warmodgena cvladi

denis wredis Cauketavobis Sesaxeb. rogorc viciT (I tomi, §127), mudmivi denis

wredi yovelTvis Caketilia. rac Seexeba cvladi

denis wreds, iTvleboda, rom igi SeiZleba iyos

Cauketavic. amas maSin aqvs adgili, roca cvladi

denis wredi Seicavs kondensators. kondensato-

ris damuxtvisa da ganmuxtvis dros denis gadis

Semonafenebis SemaerTebel gamtarSi da ar gadis

Semonafenebs Soris dieleqtrikSi.

wanacvlebis denis cnebis Semosatanad gan-

vixiloT brtyeli kondensatori, romlis A da

B firfitebis (Semonafenebi) damuxtulia σ± ze-

dapiruli simkvriviT (sur. 45.2). vidre wredi gan-

rTulia, firfitebs Soris veli erTgavrovania, magram K CamrTvelis

CarTvisas kondensatori iwyebs ganmuxtvas da velic icvleba (mcirdeba) dro-

is mixedviT. ganmuxtvis I deni, romelic mimarTulia firfitebis SemaerTebel

gamtarSi (`gare wredSi~) A -dan B -sken, gamtarobis denia _ igi

+

+

+

+

-

-

-

- II

σ−σ+

A B

K

dtDd

D

sur. 45.2

0>dtdE

sur. 45. 1.

E

B

146

ganpirobebulia Tavisufali muxtebis moZraobiT. gamtarobis denis Zala

dtdqI = , xolo simkvrive

dtd

Sq

dtd

dtdq

SSIi σ

=⎟⎠⎞

⎜⎝⎛===

1. (45.1)

cxadia, rom gamtarobis denis simkvrivis veqtoric mimarTuli iqneba A -dan B -

sken.

radgan A da B firfitebs Soris dieleqtrikia (an vakuumi), aq muxtebis

moZraobas adgili ara aqvs da gamtarobis deni nulis tolia _ denis wirebi

Cauketavia. denis wirebis uwyvetobis SesanarCuneblad maqsvelma Semoitana wa-

nacvlebis denis cneba:

t momentSi firfitebs Soris velis daZabuloba (I tomi, §120) εεσ

0

=E ,

xolo induqcia (§121) ED εε 0= , saidanac

σ=D . (45.2)

amasTan, vinaidan A firfita damuxtulia dadebiTad, D mimarTulia A -dan B -

sken. ganmuxtvisas, raRac dt droSi firfitis muxti Seicvleba σd -Ti, amitom

A da B firfitebs Soris sivrceSi induqciis veqtoris cvlilebis siCqare

iqneba

dtd

dtdD σ

= . (45.3)

D induqciis, iseve rogorc σ -is cvlileba xdeba drois mixedviT,

amitom (45.3) tolobaSi sruli warmoebuli SeiZleba Seicvalos kerZo warmo-

ebuliT:

ttD

∂∂

=∂∂ σ

. (45.4)

sidides tD∂∂ maqsvelma uwoda wanacvlebis denis simkvrive.

tDi∂∂

=wan (45.5)

rogorc (45.3) toloba gviCvenebs, igi gamtarobis denis simkvrivis tolia

ii =wan , e.i. nebismier momentSi firfitebs Soris wanacvlebis denis simkvrive

iseTia, rogorc gare wredSi gamtarobis denis simkvrive.

147

vinaidan ganmuxtvisas σ mcirdeba, Semcirdeba D -c, amitom tD∂∂ uaryofi-

Tia da tD∂∂

veqtors eqneba D veqtoris sawinaaRmdego mimarTuleba (sur. 45.2).

magram

wanitD=

∂∂ (45.6)

aris wanacvlebis denis simkvrivis veqtori. e.i. wanacvlebis denis simkvrivis

veqtori firfitebs Soris mimarTulia B -dan A -sken, rac imas niSnavs, rom

gare wredSi gamtarobis denis wirebi uwyvetad gadadian firfitebs

Soris wanacvlebis denis wirebSi (gamtarobis deni ikvreba wanacvlebis

deniT)∗ sruli simkvrive

tDiiii∂∂

+=+= wansr . (45.7)

sruli simkvrive uwyvetia rogorc sididiT, ise mimarTulebiT. amitom, Tu

eleqtruli denis qveS vigulisxmebT srul dens, aRmoCndeba, rom bunebaSi

yvela eleqtruli denebi Sekrulia. es mniSvnelovani daskvna gaakeTa

maqsvelma.

dieleqtrikSi induqciis veqtori ori nawilisagan Sedgeba (Itomi, §121):

PED += 0ε . (45.8)

meore wevri, P polarizaciis veqtori, marTlac axasiaTebs bmuli muxtebis

wanacvlebas dieleqtrikis polarizaciis procesSi. dieleqtrikSi wanac-

vlebis denis simkvrive (45.8)-is Sesabamisad ori nawilisagan Sedgeba:

tP

tEi

∂∂

+∂∂

= 0εwan ;

pirvel Sesakrebs ewodeba wanacvlebis denis simkvrive vakuumSi, meores

_ polarizaciis denis simkvrive.

∗ kondensatoris damuxtvis SemTxvevaSi, roca A firfitis dadebiTi muxti izrdeba, deni

firfitebs Soris SemaerTebel gamtarSi miamarTulia B -dan A -sken. magram am dros tD∂∂

dadebiTia da tD∂∂

veqtors aqvs D -s mimarTuleba. e.i. am SemTxvevaSic gamtarobis deni

ikvreba dieleqtrikSi wanacvlebis deniT.

148

rodesac gamtarSi gadis cvladi deni, gamtaris SigniT warmoiqmneba

cvladi eleqtruli veli, maSasadame, wanacvlebis denic da denis sruli sim-

kvrive ganisazRvreba (45.7) formuliT. magram Cveulebriv, Tu cvlilebis

sixSire (denis sixSire) Zalian didi ar aris, wanacvlebis deni mcirea da

gaTvlebisas mas mxedvelobaSi ar iReben.

Tu gamtarobis deni Sekruli ar aris, rasac adgili hqonda Cvens mier

ganxilul magaliTSi, rogorc es gamomdinareobs (45.3) tolobidan, gamtaro-

bisa da wanacvlebis denebi, ubralod, erTmaneTis tolia. miuxedavad wanac-

vlebis denis mniSvnelovani sididisa, mTeli rigi gaTvlebi tardeba misi gaT-

valiswinebis gareSe. marTlac, wanacvlebis deni, kravs ra kondensatoris

firfitebs Soris gamtarobis dens, am sivrceSi qmnis iseTive magnitur vels,

rogoric Seiqmneboda gamtarobis dens rom gaevlo gauwyvetel wredSi. amitom

wanacvlebis denis arseboba gavlenas ar axdens magnituri velis, TviT-

induqciis koeficientis da a.S. gaTvlaze.

dasasrul, unda aRiniSnos, rom termini `wanacvlebis deni~ ar aris

mTlad mosaxerxebeli. dieleqtrikSi am terminis arsebobas kidev aqvs gar-

kveuli safuZveli, vinaidan eleqtruli velis moqmedebiT iq marTlac xdeba

bmuli muxtebis wanacvleba. magram termini gamoiyeneba vakuumisaTvisac, sadac

araviTari muxtebi ar aris. termini istoriulia da SesaZloa, dakavSirebulia

im drosTan, roca eleqtruli velis cvlileba vakuumSi ganixileboda,

rogorc hipotezuri garemos _ eTeris nawilakebis wanacvleba. erTaderT

gamarTlebad, tD∂∂

sidides rom wanacvlebis denis simkvrive uwodes, SeiZleba

CaiTvalos is, rom mas aqvs denis simkvrivis ganzomileba.

wanacvlebis dens ar axasiaTebs gamtarobis denis arc erT

Tviseba (siTburi, qimiuri da sxv.) garda erTisa _ igi qmnis mag-

nitur vels.

§46. maqsvelis gantolebebi

wanacvlebis denis cnebis SemoRebam maqsvels saSualeba misca Seeqmna

(me-19 saukunis 60-ian wlebSi) eleqtruli da magnituri movlenebis erTiani

Teoria, romelSic gadawyvetilia eleqtrodinamikis ZirTadi amocana: eleq-

149

truli da magnituri velebis damaxasiaTebeli sidideebis ( )HDBE ,,, povna

maTi Semqmneli muxtebisa da denebis saSualebiT.

maqsvelis Teoria eleqtromagnituri velis fenomenologiuri Teo-

riaa, rac niSnavs imas, rom TeoriaSi ar ganixileba garemos (romelSic

arsebobs eleqtromagnituri veli) molekuluri aRnagoba da masSi mimdinare

Siga procesebi. garemos eleqtruli da magnituri Tvisebebi xasiaTdeba sami

parametriT ( )γμε ,, , romelTa mniSvnelobebi cdebiTaa dadgenili.

maqsvelis Teoria makroskopuli Teoriaa, vinaidan igi ganixilavs mak-

roskopul eleqtrul da magnitur velebs, romlebic Seqmnilia makroskopuli

muxtebisa da denebis mier∗.

maqsvelis Teoriis maTematikur gamosaxulebas warmoadgens maqsvelis

oTxi gantoleba, romelTa Cawera miRebulia ori formiT _ integraluriT da

diferencialuriT.

maqsvelis gantolebebi bunebis fundamenturi gantolebebia, romlebic

iseTive rols TamaSobs eleqtromagnetizmis moZRvrebaSi, rogorsac niutonis

kanonebi meqanikaSi. maqsvelis gantolebebma ara marto axsna im droisaTvis

cnobili yvela cdiseuli faqti, (rac TavisTavad metad mniSvnelovania), ara-

med iwinaswarmetyvela mTeli rigi axali movlenebi, romelTa arseboba Sem-

degSi dadasturda cdiT. maqsvelis gantolebebidan gamomdinare Sedegebs So-

ris yvelaze mniSvnelovania daskvna eleqtromagnituri talRebis

arsebobis Sesaxeb, romelTa gavrcelebis siCqare sasrulia da

sinaTlis siCqaris tolia.∗∗ am talRebis Teoriuli gamokvlevis Sedegad

maqsvelma Seqmna sinaTlis eleqtromagnituri Teoria.

xazgasmiT unda aRiniSnos, rom maqsvelis gantolebebis gamoyvana ar

SeiZleba. isini miRebulia eleqtrostatikisa da eleqtromagnetizmis iseTi

mniSvnelovani kanonebis ganzogadebis safuZvelze, rogorebicaa gaus-ostro-

gradskis Teorema, sruli denis kanoni da eleqtromagnituri induqciis Ziri-

Tadi kanoni. gavecnoT am kanonebis ganzogadebas.

∗ e.i. iseTi uZravi da moZravi muxtebis mier, romelTa zomebi gacilebiT metia atomis zomaze. ∗∗ im droisaTvis fizosa da maikelsonis cdebis Sedegad cnobili iyo sinaTlis gavrcelebis siCqare.

150

1. maqsvelis pirveli gantoleba. eleqtromagnituri induqciis ZiriTadi

kanonis Tanaxmad cvlad magnitur velSi moTavsebul uZrav gamtar

konturSi aRZruli induqciis em Zala gamoiTvleba formuliT:

( )t

ldE∂Φ∂

−=∫ (46.1)

erTi SexedviT SeiZleba mogveCvenos, rom eleqtromagnituri induqciis

movlenaSi mTavar rols gamtari TamaSobs. magram maqsvelis mosazrebis Tanax-

mad, es ase ar aris _ gamtari mxolod xels gviwyobs, gvexmareba am movlenis

aRmoCenaSi. eleqtromagnituri induqciis movlenis WeSmariti arsi ki mdgo-

mareobs imaSi, rom droSi cvalebadi magnituri veli sivrceSi, damo-

ukideblad imisa aris iq gamtari Tu ara, warmoSobs grigalur eleqtrul

vels (maqsvelis eleqtromagnituri Teoriis pirveli ZiriTadi debuleba). ase

rom (46.1) formula samarTliania ara marto gamtari konturisaTvis, aramed

cvlad magnitur velSi azrobrivad gamoyofili nebismieri Caketili kon-

turisaTvis. aseTnairad ganzogadebuli (46.1) formula warmoadgens maq-

svelis pirvel gantolebas integraluri formiT. ikiTxeba Semdegnairad:

eleqtruli velis daZabulobis cirkulacia nebismieri (uZravi)

Caketili l konturis gaswvriv tolia am konturis momWimavi

zedapiris (konturze dayrdnobili zedapiris) gamWoli magnituri

nakadis cvlilebis siCqarisa Sebrunebuli niSniT.

magram

( )∫=ΦS

SdB da ( ) ∫∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛⋅

∂∂

=∂∂

=∂Φ∂

SS

SdtBSdB

tt.

amitom maqsvelis pirveli integraluri gantoleba SeiZleba aseTi saxiTac

daiweros:

( ) ∫∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛⋅

∂∂

=Sl

SdtBldE . (46.2)

2. maqsvelis meore gantoleba. rogorc viciT (§31), sruli denis kanons

nivTierebaSi aqvs saxe:

( ) makroIldHl

=∫ (46.3)

151

wanacvlebis denis cnebis SemoRebam (rasac safuZvlad udevs maqsvelis

eleqtromagnituri velis Teoriis meore ZiriTadi debuleba) saSualeba misca

maqsvels ganezogadebina sruli denis kanoni. maqsvelis Tanaxmad, vinaidan

magnitur vels qmnis rogorc makroskopuli (gamtarobis), ise wanacvlebis

deni, amitom (46.3) gantolebis marjvena mxares unda daematos wnacvlebis deni:

( ) wan.makro IIldHl

+=∫ (46.3)

(46.3) formula, romelic gamosaxavs sruli denis kanonis ganzogadebul saxes,

warmoadgens maqsvelis meore gantolebas integraluri formiT. ikiTxe-

ba Semdegnairad: magnituri velis daZabulobis cirkulacia nebismieri

(uZravi) Caketili l konturis gaswvriv, tolia im makrodenebisa da

wancvlebis denis algebruli jamisa, romelsac es konturi moicavs

(an, rac igivea, romelic gadis konturis momWimav zedapirSi).

magram denis simkvrivis ganmartebis Tanaxmad

( )∫∫ ==lS

n SdiidSImakro da ( ) ∫∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛⋅

∂∂

==S

SdtDSdiI wan.wan. ,

sadac i da wani Sesabamisad gamtarobisa da wanacvlebis denebis sim-

kvrivis veqtorebia, amitom maqsvelis meore integraluri gantoleba SeiZleba

aseTi saxiTac daiweros:

( ) SdtDildH

Sl

⋅⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

+= ∫∫ , (46.4)

e.i. daZabulobis veqtoris cirkulacia nebismieri Caketili l konturis gas-

wvriv tolia sruli denis simkvrivis veqtoris nakadisa am konturis momWimavi

S zedapiris mimarT.

3. maqsvelis mesame gantoleba. rogorc viciT (I tomi, §121), gaus-

ostogradskis Teoremas eleqtrostatikuri (stacionaruli) velisaTvis∗

dieleqtrikSi aqvs saxe:

( ) qSdDS

=∫ , (46.5)

sadac ( ) NSdDS

=∫ aris induqciis nakadi S Caketili zedapiris mimarT, xolo

∑= iqq aris am zedapiris SigniT moTavsebuli Tavisufal muxtTa algebruli

∗ eleqtrostatikuri veli stacionaluri velia, vinaidan igi drois mixedviT ar icvleba.

152

jami. maqsvelma ganazogada es Teorema, gamoTqva ra varaudi, rom igi samara-

Tliania nebismieri _ rogorc stacionaruli, ise cvladi _ eleqtruli

velisaTvis. aseTnairad ganzogadebuli (46.5) formula warmoadgens maqsve-

lis mesame gantolebas integraluri formiT. ikiTxeba Semdegnairad:

eleqtruli induqciis nakadi eleqtromagnitur velSi azrobrivad

gavlebuli nebismieri Caketili zedapiris mimarT, tolia am

zedapiris SigniT moTavsebuli Tavisufal muxtTa algebruli

jamisa. magram dVqV∫= ρ , sadac ρ aris S zedapiriT SemosazRvruli V mocu-

lobis SigniT arsebuli Tavisufal muxtTa simkvrive. amitom maqsvelis mesame

integraluri gantoleba SeiZleba Semdegi saxiTac daiweros:

( ) dVSdDVS∫∫ = ρ (46.6)

4. maqsvelis meoTxe gantoleba: gaus-ostrogradskis Teoremas

magnituri velisaTvis vakuumSi aqvs saxe (§4):

( ) 0=∫S

SdB (46.7)

maqsvelma ganazogada es Teoremac, gamoTqva ra varaudi, rom igi

samarTliania nebismieri magnituri velisaTvis (rogorc vakuumSi, aseve ga-

remoSi, rogorc stacionaruli, aseve cvladisaTvis), aseTnairad ganzoga-

debuli (46.7) toloba warmoadgens maqsvelis meoTxe gantolebas integra-

luri formiT. ikiTxeba Semdegnairad:

magnituri induqciis nakadi eleqtromagnitur velSi azrobri-

vad gavlebuli nebismieri (uZravi) Caketili zedapiris mimarT

nulis tolia.

maqsvelis gantolebebi ar aris simetriuli eleqtruli da magnituri

velebis mimarT, rac imis Sedegia, rom bunebaSi eleqtruli muxti arsebobs,

magnituri muxti (magnituri monopoli) ki _ ara.

rogorc aRvniSneT maqsvelis Teoria fenomenologiuri Teoriaa, masSi

ar ganixileba garemoSi mimdinare Siga anu molekuluri procesebi, romlebic

ganapirobeben eleqtruli da magnituri velebis warmoSobas. amitom saWiroa

maqsvelis gantolebebi SevavsoT e.w. materialuri gantolebebiT, romle-

153

bic Seicaven garemos individualuri Tvisebebis damaxasiaTebel sidideebs (ε -

s, μ -s, γ -s). es gantolebebi CvenTvis cnobilia kursis sxvdasxva nawilidan:

ED εε 0= (I tomi, §121); HB μμ0= (§25); Ei γ= (I tomi, §137).

§47. maqsvelis diferencialuri gantolebebi

maqsvelis integraluri gantolebebi amyarebs damokidebulebas eleq-

truli da magnituri velis damaxasiaTebeli sidideebs Soris sivrcis sxvada-

sxva wertilebisaTvis (Caketili konturis wertilebi da am konturis

momWimavi zedapiris wertilebi, an Caketili zedapiris wertilebi da am zeda-

piris SigniT arsebuli wertilebi). maqsvelis integraluri gantolebebidan

SeiZleba gadavideT diferencialur gantolebebze, romlebic am kavSirs

amyareben sivrcis erTidaimave wertilebisaTvis. es gadasvla xdeba

veqtoruli analizis ori Teoremis _ gausis Teoremisa da stoqsis Teoremis

saSualebiT. gavecnoT am Teoremebs.

gausis Teorema. romelime velis damaxasiaTebeli A veqtoris nakadi

am velSi azrobrivad gavlebuli nebismieri S Caketili zedapiris mimarT to-

lia A veqtoris divergenciis integralisa, aRebuli am zedapiriT SemosazR-

vruli V moculobiT:

( ) VdAdiSdAVS∫∫ = υ , sadac

zA

yA

xA

Adi zyx

∂∂

+∂

∂+

∂∂

=υ (47.1)

stoqsis Teorema. romelime velis damaxasiaTebeli A veqtoris cirku-

lacia am velSi azrobrivad gavlebuli nebismieri Caketili l konturis gas-

wvriv tolia Arot veqtoris nakadisa am konturis momWimavi S zedapiris mimarT:∗

( ) ( )∫∫ ⋅=Sl

SdArotldA . (47.2)

∗ rotori maTematikaSi veqtoruli A velis `brunviTi mdgenelis~ veqtoruli maxasiaTebelia,

romlis gegmilebia (koordinatebia) x

Az

Az

Ay

A zxyz

∂∂

−∂∂

∂

∂−

∂∂ , da

yA

xA xy

∂∂

−∂

∂, e.i.

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂

∂+⎟

⎠⎞

⎜⎝⎛

∂∂

−∂∂

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂

∂−

∂∂

=y

Ax

Ak

xA

zA

jz

Ay

AiArot xyzxyz , sadac ji, da k erTeulovani

veqtorebia. zog SemTxvevaSi rotors veqtoruli velis grigalsac uwodeben.

154

1. maqsvelis pirveli diferencialuri gantoleba. stoqsis (47.2)

Teoremis Tanaxmad

( ) ( )∫∫ ⋅=Sl

SdErotldE . (47.3)

am formulis Sedareba maqsvelis pirvel integralur gantolebasTan

( ) ∫∫ ⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

=Sl

SdtBldE , gvaZlevs:

tBErot∂∂

−= . (47.4)

es aris maqsvelis pirveli diferencialuri gantoleba.

2. maqsvelis meore diferencialuri gantoleba. stoqsis Teoremis

Tanaxmad

( ) ( )∫∫ ⋅=Sl

SdHrotldH .

am formuli Sedareba maqsvelis meore integralur gantolebasTan

( ) SdtDildH

Sl

⋅⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

+= ∫∫ , gvaZlevs:

tDiHrot∂∂

+= . (47.5)

es aris maqsvelis meore diferencialuri gantoleba.

3. maqsvelis mesame diferencialuri gantoleba. gausis (47.1) Teoremis

Tanaxmad

( ) VdDdiSdDVS∫∫ = υ .

am formulis Sedareba maqsvelis mesame integralur gantolebasTan

( ) VdSdDVS∫∫ = ρ , gvaZlevs maqsvelis mesame diferencialur gantolebas:

ρυ =Ddi . (47.6) 4. maqsvelis meoTxe diferencialuri gantoleba. gausis Teoremis

Tanaxmad

( ) VdBdiSdBVS∫∫ = υ .

am formulis Sedareba maqsvelis meoTxe integralur gantolebasTan

( ) 0=∫S

SdB , gvaZleva maqsvelis meoTxe diferencialur gantolebas.

155

0=Bdiυ . (47.7)

davweroT maqsvelis diferencialur gantolebaTa sistema:

tBErot∂∂

−= , tDiHrot∂∂

+= , ρυ =Ddi , 0=Bdiυ .

am oTxi gantolebidan ori veqtorulia, oric _ skalaruli (asea inte-

graluri gantolebebis SemTxvevaSic). Tu maT CavwerT koordinatebSi,

miviRebT rva gantolebas. es rva gantoleba akavSirebs 16 sidides: TxuTmeti _

iHBDE ,,,, veqtorebis mdgenelebi da skalari ρ . cxadia, rom 16 sididisaTvis

(ucnobisaTvis) rva gantoleba sakmarisi ar aris. swored amitom umateben maq-

svelis gantolebebs materialur gantolebebs: ED εε 0= ; HB μμ0= da Ei γ= ,

romlebic amasTan, Seicavs garemos individualuri Tvisebebis damaxasiaTebel

γμε ,, mudmivebs.

§48. eleqtromagnituri veli

ganvixiloT sivrcis iseTi nawili, sadac ar aris Tavisufali muxtebi

da maSasadame, eleqtruli deni ( )0=i . maSin, maqsvelis pirvel da meore dife-

rencialur gantolebebs, marjvena nawilebSi niSnis sizustiT, aqvT simet-

riuli saxe:

tBErot∂∂

−= , tDHrot∂∂

= . (48.1)

gansxvaveba niSanSi adasturebs imas, rom tD∂∂

da H veqtorebis mimarTu-

lebebi qmnian marjvena xraxnul sistemas (sur. 48.1 a), xolo tB∂∂

da E veqto-

rebis mimarTulebebi _ marcxena xrax-

nul sistemas (sur. 48.1 b).

maqsvelis (48.1) gantolebebidan ga-

momdinareobs metad mniSvnelovani das-

kvna imis Sesaxeb, rom eleqtrul da

magnitur velebs Soris arsebobs

mWidro kavSiri: magnituri velis

EH

a b sur. 48.1

tD∂∂

tB∂∂

156

cvlileba iwvevs garemomcvel sivrceSi cvladi eleqtruli velis

gaCenas; analogiurad, eleqtruli velis cvlileba iwvevs garemom-

cvel sivrceSi cvaladi magnituri velis gaCenas.

rac ufro didi tD∂∂

siCqariT icvleba eleqtruli veli, miT ufro Zli-

eri magnituri veli aRiZvreba, romelic eleqtrul velTan aris dakavSirebu-

li. zustad aseve, rac ufro didi tB∂∂

siCqariT icvleba magnituri veli, miT

ufro Zlieri eleqtruli veli aRiZvreba, romelic magnitur velTan aris

dakavSirebuli.

cvladi eleqtruli velis da masTan uwyvetad dakavSirebuli

cvladi magnituri velis erTobliobas eleqtromagnituri veli

ewodeba.

orive veli, eleqtrulic da magnituric grigaluria: maTi Zalwirebi Ca-

ketilia da amasTan urTierTgadaxlarTuli _ eleqtruli Zalwirebi `Semo-

xveulia~ magnitur Zalwirebze, xolo magnituri Zalwirebi _ eleqtrulze.

garkveuli warmodgena eleqtromagnituri velis xasiaTze SeiZleba

mogvces 48.2 suraTma, romelzec veli gamosaxulia monacvleobiTi eleqtruli

da magnituri Caketili Zalwirebis Zewkvis saxiT.

eleqtruli da magni-

turi velebis gancalkevebu-

lad warmodgenas aqvs far-

dobiTi xasiaTi. marTlac,

uZravi muxti qmnis mxolod

eleqtrul vels. magram

muxti uZravia garkveuli aTvlis sistemis mimarT. aTvlis sxva sistemebis

mimarT igi moZraobs da maSasadame, qmnis magnitur velsac. zustad aseve,

mudmivdeniani uZravi gamtari qmnis mxolod mudmiv magnitur vels, magram

gamtari uZravia mxolod garkveuli aTvlis sistemis mimarT. aTvlis sxva

sistemis mimarT igi SeiZleba moZraobdes, ris gamoc misi magnituri veli

iqneba cvladi, romelic Tavis mxriv aRZravs grigalur eleqtrul vels.

eleqtruli da magnituri velebi warmoadgens erTi mTlianis _

eleqtromagnituri velis gamovlenas.

EEEE

BBB

sur. 48.2

157

§49. eleqtromagnituri talRa

eleqtromagnituri velis SesaniSnavi Tvisebaa misi sivrceSi yvela

mimarTulebiT gavrcelebis unari. velis swored es Tviseba uzrunvel-

yofs radiokavSiris SesaZleblobas.

vTqvaT, sivrcis romelime wertilSi gaCnda cvladi eleqtruli veli. es

veli sivrcis mezobel wertilebSi induqciiT aRZravs cvlad magnitur vels.

cvladi magnituri veli Tavis mxriv mezobel wertilebSi induqciiT aRZravs

cvlad eleqtrul vels da a.S. cvladi eleqtruli da magnituri velebi moe-

deba sivrcis axal-axal areebs da vrceldeba (rogorc gamoTvlebi gviCvenebs)

sicarieleSi 300000 km/wm siCqariT, e.i. sinaTlis gavrcelebis siCqariT vakuumSi.

eleqtromagnituri velis gavrcelebis procesi warmoadgens

talRur process _ eleqtromagnitur talRas.

Cvens mier Tvisobrivi msjelobis safuZvelze gakeTebuli daskvna eleq-

tromagnituri talRebis Sesaxeb∗ dasturdeba maTematikurad, maqsvelis dife-

rencialuri gantolebebis amoxsniT. rogorc wina paragrafSi aRvniSneT,

maqsvelis gantolebebidan gamomdinare Sedegebidan yvelaze mniSvnelovani iyo

daskvna eleqtromagnituri talRebis arsebobis Taobaze.

CamovTvaloT maqsvelis diferencialuri gantolebebis amoxsniT miRebu-

li ZiriTadi Sedegebi:

1. eleqtromagnituri veli vrceldeba sivrceSi eleqtromagnituri tal-

Ris saxiT sasruli siCqariT.

2. eleqtromagnituri talRis gavrcelebis siCqare garemoSi gamoiTvleba

formuliT εμμε

υ00

1= , sadac 0ε da 0μ Sesabamisad eleqtruli da magnituri

mudmivebia, xolo ε da μ _ garemos fardobiTi dieleqtrikuli da magnituri

SeRwevadobebi.

magram m

f90 1094

1⋅⋅

=π

ε ,

xolo m

hn70 104 −⋅= πμ .

∗ rasac safuZvlad udevs maqsvelis Teoriis orive ZiriTadi debuleba.

158

e.i. wm

m816

9

700

103109

1094104

11⋅=⋅=

⋅⋅⋅

=−

ππμε

. es aris sinaTlis gavrcelebis

siCqare vakuumSi, romelic, Cveulebriv C -Ti aRiniSneba. amitom εμ

υ C= .

nebismieri garemosaTvis 1>εμ , amitom C<υ . sicarieleSi 1== με da C=υ ,

e.i. eleqtromagnituri talRa sicarieleSi vrceldeba sinaTlis siCqariT.

3. eleqtromagnituri talRa aris

ganivi talRa, rac imas niSnavs, rom

talRaSi eleqtruli velis daZa-

bulobisa da magnituri velis induq-

ciis veqtorebi irxevian erTmaneTis

da talRis gavrcelebis υ siCqaris

marTobulad ( BE, da υ veqtorebi qmnian marjvena xraxnul sistemas). amasTan,

rxevebi warmoebs erTnair fazebSi (sur. 49.1)

4. im faqtma, rom vakuumSi eleqtromagnituri talRis siCqare emTxveva

sinaTlis siCqares vakuumSi, afiqrebina maqsvels, rom sinaTle aris eleqtro-

magnituri talRa, romelic sxva eleqtromagnituri talRebisagan gansxvavdeba

mxolod talRis sigrZiT. amiT man Seqmna sinaTlis eleqtromagnituri Teoria.

radgan sinaTlis gavrcelebis siCqare εμ

υ C= , amitom garemos gardatexis ab-

soluturi maCvenebeli εμυ==

Cn . gamWvirvale sxeulebi, rogorc wesi, ara-

feromagnituria, maTTvis μ umniSvnelod gansxvavdeba erTisagan da amitom

miaxloebiT SeiZleba daiweros: ε≈n .

maqsvelma daadgina, rom sxvadasxva dieleqtrikebisaTvis am formuliT gamoT-

vlili gardatexis absoluturi maCvenebeli kargad Tanxvdeba cdiseul Sedegebs.∗

∗ am formuliT sargeblobisas unda gvaxsovdes rom aq ε aris nivTierebis dieleqtrikuli

SeRwevadoba maRali sixSiris eleqtromagnitur velSi (1410~ hc) da amitom misi sidide didad

gansxvavdeba (naklebia) mudmiv eleqtrul velSi misi sidididagan. magaliTad, wylisaTvis

81=ε da misi gardatexis absoluturi maCveneblisaTvis miiReba 9, cdiseuli 1,33-is nacvlad. swrafad cvlad velSi ε -is Semcireba imis Sedegia, rom periodis naxevris ganmavlobaSi wylis dipoluri molekulebi ver aswreben orientacias velis gaswvriv.

159

5. eleqtromagnituri talRa raime zedapirze dacem-

isas awarmoebs garkveul wnevas. marTlac davuSvaT, rom

eleqtromagnituri talRa marTobulad ecema A gamtar

zedapirs∗ (sur. 49.2). talRis eleqtruli mdgeneli iwvevs

gamtarobis denis gaCenas, magnituri mdgeneli ki am denze

moqmedebs F ZaliT, romelic, marcxena xelis wesis

Tanaxmad, mimarTulia talRis gavrcelebis gaswvriv. rad-

gan sinaTle warmoadgens eleqtromagnitur talRas, maq-

svelis Teoriidan gamomdinareobs sinaTlis wnevis arse-

boba, rac eqsperimentulad daadastura rusma mecnierma p. lebedevma 1901 wels.

6. kavSiri eleqtruli da magnituri velis daZabulobebs Soris mocemulia

formuliT:

20

20 HE μμεε =

magram viciT, rom eleqtruli velis energiis simkvrive 2

20 E

Eεε

ω = , xolo

magnituris _ 2

20 H

Hμμω = . maSasadame, eleqtromagnitur talRaSi eleqtruli

energia nebismier momentSi tolia magnituris.

formula εμ=n amyarebs damokidebulebas nivTierebis optikur,

eleqtrul da magnitur Tvisebebs Soris. magram am fromulidan ar

gamomdinareobs, rom n damokidebuli unda iyos talRis sigrZeze. meores

mxriv, cdiseuli faqtebi (magaliTad, dispersia) adasturebs aseTi

damokidebulebis arsebobas. maqsvelis Teoriis es nakli Sedegia imisa, rom

igi ar iTvaliswinebs nivTierebis Sinagan aRnagobas da ixilavs mxolod

makroskopul eleqtrul da magnitur velebs.

maqsvelis eleqtromagnituri velis Teoriam Semdgomi ganviTareba

hpova lorencis klasikur eleqtronul TeoriaSi (1896w). am TeoriaSi

daSvebulia, rom nivTiereba Sedgeba atomebisagan, romlebic Tavis mxriv

Seicavs dadebiT da uaryofiT muxtebs da eleqtruli da magnituri

movlenebis mTeli mravalsaxeoba ganpirobebulia am muxtebisa da

∗ sinaTlis wneva, cxadia, arsebobs aragamtar sxeulebze (maT Soris airebzec), magram misi axsna ise martivi da TvalsaCino ar aris, rogorc gamtari sxeulebis SemTxvevaSi. magaliTad, dieleqtrikSi eleqtromagnituri veli iwvevs muxtebis periodul rxevas, rac agreTve ganapirobebs talRis gavrcelebis gaswvriv mimarTuli Zalis warmoSobas.

gamtI

sur. 49.2

A B

F

160

mikrodenebis urTierTganlagebiT, moZraobiTa da urTierTqmedebiT. am

muxtebisa da denebis mier Seqmnili mikroveli aRiwereba gantolebebiT,

romlebic analogiuria maqsvelis gantolebebisa. es gantolebebi, droiTa da

sivrciT gasaSualoebis Semdeg, gadadis maqsvelis gantolebebSi. es

gasaSualoeba xdeba drois intervalis mixedviT, romelic bevrad aRemateba

Sigaatomuri procesebis periods da velis moculobis mixedviT, romelic

bevrad aRemateba atomebisa da molekulebis moculobas.

§50. eleqtromagnituri talRebis gamosxiveba da miReba

rogorc viciT (§42) rxeviTi konturis daniSnulebaa maRali sixSiris

eleqtromagnituri rxevebis aRZvra, rac Tavis mxriv, uzrunvelyofs radio-

kavSirgabmulobis SesaZleblobas. magram radikavSiris ganxorcilebisaTvis

saWiroa eleqtromagnituri talRebis uwyveti generacia da maSasadame, miule-

vadi eleqtromagnituri rxevebis miReba. rxeviT konturSi aRZruli sakuTari

rxevebi ki milevadia, rac ZiriTadad∗ ganpirobebulia imiT, rom konturisadmi

miniWebuli energia ixarjeba omuri winaRobis daZlevaze da gamoiyofa siTbos

saxiT. rxeviTi konturis am siZneleze Cven ukve gvqonda laparaki §44-Si da

viciT, rom dReisaTvis sakiTxi gadawyvetilia sameleqtrodiani eleqtronuli

milakis an tranzistoris gamoyenebiT (TviTrxevebi).

istoriulad, rxeviTi konturisaTvis energiis periodulad

miwodebisaTvis, germanelma mecnierma hercma gamoiyena sainduqcio koWa (§41)

da 1887 wels, pirvelad kacobriobis istoriaSi, miiRo maqsvelis TeoriiT

nawinaswarmetyvelebi eleqtromagnituri talRebi.

hercma rxeviT konturSi Seitana axali elementi _

sanaperwklo Sualedi (1), romelsac mosdo cvladi

Zabva sainduqcio koWas (2) meoradi gragnilidan

(sur. 50.1). es Zabva TandaTan muxtavda kondensators

da rodesac mis Semonafenebs Soris Zabva miaRwevda

garRvevis mniSvnelobas (I tomi, §153), sanaperwklo

SualedSi Cndeboda (`gaxteboda~) naperwkali, ro-

∗ energiis nawili ixarjeba gamosxivebaze.

1

2

sur. 50.1

161

melic kravda wreds. Sedegad konturSi aRiZvreboda eleqtromagnituri rxe-

vebi da igi asxivebda eleqtromagnitur talRebs. erTdroulad xdeboda sain-

duqcio koWas gamorTva (`damokleba~). sanaperwklo Sualeds didi winaRoba

aqvs, amitom rxevebi male miileoda da wredi wydeboda. magram am dros kvlav

irTveboda sainduqcio koWa, kondensatori imuxteboda, rasac Tan sdevda

axali naperwklis gaxtoma da eleqtromagnituri talRebis generireba da a.S.

rxeviTi konturis meore siZnele mdgomareobs imaSi, rom eleqtruli ve-

lis energia faqtiurad lokalizebulia kondensatoris firfitebs Soris, mag-

nituri velis energia ki _ koWas SigniT.∗ aseTi rxeviTi konturi, romelsac

Caketili ewodeba, eleqtromagnitur talRebs TiTqmis ar asxivebs. gavix-

senoT (§42, formula 42.7), rom rac ufro mcirea kondensatoris tevadoba da

koWas induqciuroba, miT metia rxeviTi konturis mier generirebuli eleqtro-

magnituri rxevis siSire da maSasadame, gamosxivebuli eleqtromagnituri

talRis intensivoba.∗∗ amitom, am meore naklis gamoricxvis mizniT, anu gamos-

xivebis intensivobis gazrdis mizniT, hercma kondensatoris firfitebi daaSo-

ra erTmaneTs, xolo koWa saerTod amoiRo konturidan (ufro sworad, Sec-

vala erTi xviiT). aseTi gardaqmnebis Sedegad (sur. 50.2 a, b, g), man miiRo e.w.

Ria rxeviTi konturi, romelsac SemdgomSi hercis vibratori ewoda. Tu

Tavdapirveli formis konturSi cvladi eleqtruli veli lokalizebulia

kondensatoris firfitebs Soris da mcire moculoba ukavia (sur. 49.2 a), her-

cis vibratorSi eleqtruli veli gars akravs vibrators da bevrad gazrd-

ilia mis mier dakavebuli moculoba (sur. 50.2 g), rac sagrZnoblad zrdis

∗ es imiT aixsneba, rom sxvadasxva niSniT damuxtuli kondensatoris Semonafenebis mier Seqmnili sawinaaRmdego mimarTulebis eleqtruli velebi konturidan moSorebiT sididiT TiTqmis erTmaneTis tolia, amitom isini TiTqmis mTlianad akompensireben erTmaneTs. aseve, koWas xviis mopirdapire elementebSi dens aqvs sapirispiro mimarTulebebi, amitom konturidan moSorebiT maTi magnituri velebi TiTqmis akompensireben erTmaneTs. ∗∗ sixSiris gazrdiT gamosxivebuli energia mkveTrad izrdeba. kerZod, SeiZleba damtkicdes, rom drois erTeulSi gamosxivebuli energia sixSiris meoTxe xarisxis proporciulia.

a b g

sur. 50.2

162

gamosxivebis intensivobas.

eleqtromagnituri talRebis registraciisaTvis hercma gamoiyena

rezonansis movlena: gamomsxivebeli (1) vibratoris maxloblad man moaTavsa

meore (2) vibratori, romelsac hqonda iseTive rxevis sakuTari sixSire,

rogorc pirvels, anu imyofeboda masTan rezonansSi, ris gamoc mas rezo-

natori ewoda. rodesac (1) vibratoris mier gamosxivebuli eleqtromagnituri

talRa aRwevda rezonators, masSi aRiZvreboda eleqtruli rxevebi, rasac Tan

axlda burTulebs Soris naperwklis gaCena (sur. 50.3).

vibratorisa da rezonatoris saSualebiT hercma 1887-91 wlebSi eqsperi-

mentulad Seiswavla eleqtromagnituri talRebis struqtura da gavrcelebis

kanonzomierebebi. kerZod, man daadgina, rom eleqtromagnituri talRa ganivi

talRaa, misi gavrcelebis siCqare haerSi sinaTlis siCqaris tolia. am talRe-

bis arekvla (liTonis zedapiridan), gardatexa ori garemos (dieleqtrikis)

sazRvarze da interferencia iseve warmoebs, rogorc sinaTlis talRebisa da sxv.

eleqtromagnituri talRebis aRZvris

ganxilul meqanizms is nakli hqonda, rom

sainduqcio koWadan konturisTvis (vibrato-

risTvis) energiis miwodebis sixSire bevrad

naklebi iyo konturis rxevis sakuTar

sixSireze. amitom hercis vibratorSi aRZru-

li eleqtruli rxevebi warmoadgenen mile-

vad rxevebs, romlebic meordebodnen erTime-

oris miyolebiT drois mcire intervalis Semdeg (sur. 50.4). rogorc paragra-

fis dasawyisSi iyo aRniSnuli, es nakli SemdgomSi aRmofxvril iqna sameleq-

trodiani eleqtronuli milakis, mogvianebiT ki tranzistoris gamoyenebiT.

herci Tavis vibratoriT iRebda da

ikvlevda eleqtromagnitur talRebs, ro-

melTa sigrZe 102sm-is rigis iyo (rac See-

sabameba 8103 ⋅ hc sixSires). gamosxivebuli

talRis sigrZis Semcireba SesaZlebelia

vibratoris zomis SemcirebiT, vinaidan

am dros mcirdeba vibratoris (liTonis Reroebis) tevadobac da

induqciurobac. am gziT p.lebedevma 1895 wels, gamoiyena ra platinis wvrili

2

sur. 50.3

1

I

t

sur. 50.4

163

Reroebisagan damzadebuli miniaturuli vibratori, miiRo milimetruli sig-

rZis eleqtromagnituri talRebi (6-4mm). kidev ufro mokle talRebi (~0,1 mm

sigrZis) miiRo sabWoTa fizikosma glagoleva-arkadievam 1923 wels e.w.

masobrivi gamomsxiveblis∗ meSveobiT. am talRebma gadafares are infrawiTeli

(siTburi) talRebisa, romlebic miiReba atomebsa da molekulebSi muxtebis

rxevis Sedegad.

paragrafis bolos moyvanilia eleqtromagnitur talRaTa cxrili, rome-

lSic miTiTebulia TiToeuli diapazonis talRebis gamosxivebis wyaro.

cxadia, sazRvari talRaTa calkeul diapazonebs Soris sakmaod pirobiTia.

dabolos, aRvniSnoT, rom maqsveli Tavis genialuri Teoriis triumfs _ eleqtromagni-

turi talRebis arsebobis eqsperimentul dadasturebas ver moeswro. es moxda hercis Sro-

mebis Sedegad mxolod misi gardacvalebidan (1879w) rva wlis Semdeg. gaivlis kidev rva weli

da 1895 wels, radios erTerTi gamomgonebeli, rusi mecnieri popovi gadascems msoflioSi

pirvel radiogramas, romelic Sedgeba mxolod ori sityvisagan _ `henrix herci~. amiT popovma

pativi miago hercis udides damsaxurebas eleqtromagnituri talRebis miRebisa da Seswavlis

saqmeSi.

cxrili

talRis saxe talRis sigrZe,

m talRis sixSire,

hc gamosxivebis wyaro

dabali sixSiris talRebi

410> 4103 ⋅< cvladi denis meqanikuri

generatori.

radiotalRebi 44 1010 −− 124 103103 ⋅−⋅ rxeviTi konturi, hercis vibratori, masobrivi

gamomsxivebeli. infrawiTeli sxivebi

74 107,710 −− ⋅− 1412 104103 ⋅−⋅

sinaTlis sxivebi 77 104107,7 −− ⋅−⋅ 1414 105,7104 ⋅−⋅

ultraiisferi sxivebi

87 10104 −− −⋅ 1614 103105,7 ⋅−⋅

sxvadasxva tipis naTurebi, lazeri

rentgenis sxivebi 118 1010 −− − 1916 103103 ⋅−⋅ rentgenis mili

γ _ sxvivebi 1110−< 19103 ⋅>

radioaqtiuri daSla, birTvuli procesebi

∗ ase ewoda mowyobilobas, romelic eleqtromagnitur talRebs asxivebs (satransformatoro) zeTSi Setivtivebul liTonis umcires nawilakebs (naqlibs) Soris naperwklebis gaxtomis Sedegad.

164

meore nawili

optikis safuZvlebi

$51. optikis ganviTarebis mokle istoria. sinaTlis dualizmi

optika aris mecniereba sinaTlis Sesaxeb. igi Seiswavlis sinaTlis bune-

bas, misi gavrcelebisa da nivTierebasTan urTierTqmedebis kanonebs. tradici-

ulad miRebulia optikis dayofa or nawilad: geometriuli optika da fizi-

kuri optika.

geometriuli optika optikis uZvelesi nawilia. jer kidev 300 wliT

adre Cvens welTaRricxvamde evklides geometriaSi mocemuli iyo sinaTlis

arekvlisa da wrfivi gavrcelebis kanonebi. geometriul optikas didi praqti-

kuli gamoyeneba aqvs, magram igi ar ixilavs sinaTlis bunebas (ar akeTebs ara-

viTar daSvebas imis Sesaxeb Tu ra aris sinaTle), amitom moicavs optikuri

sakiTxebis SedarebiT viwro wres. geometriuli optikis ZiriTadi cnebaa

sinaTlis sxivis cneba, aseTad ki miiCneva sinaTlis gavrcelebis mimarTuleba.

ufro zustad, sinaTlis sxivi aris wiri, romelic uCvenebs sinaTlis energiis

gavrcelebis mimarTulebas.∗ optikuri xelsawyoebis moqmedeba daiyvaneba

sinaTlis sxivebis arekvla-gardatexisa da wrfivi gavrcelebis kanonebze.

fizikuri optika, rogorc mecnieruli disciplina, Camoyalibda gaci-

lebiT gvian, me-17 saukuneSi. fizikuri optika garkveul daSvebas akeTebs si-

naTlis bunebis Sesaxeb da amitom moicavs optikuri movlenebis gacilebiT

farTo wres. unda aRiniSnos, rom fizikuri optika saSualebas iZleva ara

marto gamoviyvanoT geometriuli optikis kanonebi, aramed davadginoT maTi

gamoyenebis sazRvrebic.

optikuri movlenebis sirTulem ganapiroba is, rom istoriulad (TiT-

qmis erTdroulad) Camoyalibda ori urTierTsawinaaRmdego fizikuri Teoria:

sinaTlis korpuskuluri Teoria da sinaTlis talRuri Teoria.

∗ es maTematikuri abstraqciaa, vinaidan, difraqciis gamo SeuZlebelia sinaTlis nakadidan sinaTlis usasrulod wvrili konis (`wiris~) gamoyofa. fizikis TvalsazrisiT sxivi aris sasruli zomis, magram sakmaod viwro sinaTlis kona, romlis miRebis principul SesaZleblobasac iZleva difraqciis movlena (ufro sworad, am viwro konebis RerZebi miiCneva sxivebad). sinaTlis sxivis `danaxva~ SesaZlebelia kvamlian (an mtvrian) oTaxSi, rodesac masSi viwro xvrelidan (WuWrutanidan) Sedis sinaTle.

165

korpuskuluri Teoriis fuZemdebelia didi ingliseli mecnieri niu-

toni. am Teoriis Tanaxmad, sinaTle aris nakadi umciresi nawilakebis – kor-

puskulebisa,∗ romlebic gamoedinebian mnaTi sxeulidan (`gamodinebis Teoria~).

sinaTlis korpuskulebis moZraobis analizisaTvis niutoni iyenebda mis mier-

ve Camoyalibebul meqanikis kanonebs. magaliTad, am nawilakebis inerciiT moZ-

raobiT aixsneboda sinaTlis wrfivi gavrceleba. am Teoriis mixedviT sinaT-

lis arekvla da gardatexa Sedegia mizidvisa da ganzidvis Zalebisa, romle-

bic aRiZvreba ori garemos gamyofi zedapiris Txel fenaSi. amasTan igulis-

xmeba, rom nawilakebi zedapirs ecemian `sxvadasxva faziT~. fazis zogierTi

mniSvnelobisaTvis isini ganizidebian zedapiris mier (arekvla), zogierTi

mniSvnelobisaTvis ki _ miizidebian (gardatexa). gardamtexi garemos mier si-

naTlis korpuskulebis mizidva ara marto ganapirobebs sinaTlis gardatexas,

aramed cvlis kidec mis siCqares. am warmodgenebze dayrdnobiT martivad

miiReba arekvlisa da gardatexis kanonebi, amasTan gardatexis kanonidan

gamomdinareobda, rom sinaTlis siCqare optikurad metad mkvriv garemoSi

ufro metia, vidre naklebad mkvrivSi, rac, rogorc qvemoT vnaxavT, ar Seesaba-

meboda sinamdviles.∗∗ sinaTlis dispersias niutoni xsnida sinaTlis korpus-

kulebis sxvadasxva zomiT. igi Tvlida, rom wiTeli feris sinaTlis korpus-

kula yvelaze didia, iisferis ki – yvelaze patara. sxvadasxva zomis korpus-

kulebi sxvadasxvanairad gadaixrebian samwaxnaga prizmaSi gavlisas, rac gana-

pirobebs speqtris warmoqmnas. magram sinaTlis korpuskuluri Teoria ver

xsnida sinaTlis interferencias, difraqcias, sinaTlis sxivTa damoukideblo-

bis princips (Zneli asaxsneli iyo, Tu ratom ar moqmedeben erTmaneTze siv-

rceSi gadakveTili sinaTlis konebi _ sinaTlis korpuskulebi xom unda eja-

xebodnen erTmaneTs da ganicdidnen gabnevas da sxv.

talRuri Teoriis fuZemdebelia holandieli mecnieri hiugensi. am

Teoriis Tanaxmad, sinaTle warmoadgens drekadi deformaciis talRas, rome-

lic vrceldeba msoflios eTerSi. eTeri hipotezuri garemoa, romelic avsebs

∗ sityva `korpuskula~ laTinuri warmoSobisaa da niSnavs nawilaks. ∗∗ 1850 wlamde cnobili iyo sinaTlis siCqare mxolod vakuumSi, romelic gazoma danielma mecnierma riomerma 1675 wels astronomiuli meTodiT, iupiteris erT-erT Tanamgzavr ios dabnelebaze dakvirvebiT da miiRo daaxloebiT 300000 km/wm. aqve SevniSnoT, rom Tanamedrove monacemebiT planetaSoris sivrceSi 1 sm3-ze modis saSualod erTi atomi, maSin rodesac laboratoriul pirobebSi miRebuli yvelaze maRali vakuumis SemTxvevaSic ki 1 sm3-Si milionze meti atomi an molekulaa.

166

mTel sivrces da aRwevs yvela (gamWvirvale) sxeulSi (ikavebs sxeulis nawi-

lakebs Soris arsebul Sualedebs). mas gaaCnia garkveuli meqanikuri Tvisebebi

_ drekadoba da simkvrive. amasTan eTeris simkvrive metismetad mcirea, riTac

aixsneba planetebis orbitebis praqtikuli ucvleloba.

sinaTlis talRuri TeoriiT advilad miiReboda arekvlisa da gardate-

xis kanonebi, amasTan gardatexis kanonidan gamomdinareobda, rom sinaTlis

siCqare optikurad metad mkvriv garemoSi naklebia, vidre naklebad mkvrivSi.

aseve advilad gasagebi iyo sinaTlis sxivTa damoukideblobis principi _

wylis zedapirze warmoqmnili talRebi xom ar urTierTqmedeben da Tavisuf-

lad gadian erTmaneTSi. Teoria xsnida interferenciisa da difraqciis movle-

nebs da sxva. magram sinaTlis talRuri Teoria ver xsnida sinaTlis wrfiv

gavrcelebas (wylis zedapirze warmoqmnili talRebi xom gars uvlian dab-

rkolebas), ferebis warmoqmnas, sinaTlis polarizacias (romelic, sxvaTa So-

ris, hiugensma aRmoaCina), vinaidan, hiugensis Tanaxmad, drekadi deformaciis

talRa iyo gaswvrivi∗ da sxv.

korpuskuluri Teoriis simartivem (ar saWiroebda hipotezur garemos)

da niutonis didma mecnierulma avtoritetma ganapiroba is, rom Tavdapirve-

lad, xangrZlivi drois ganmavlobaSi, am Teorias ufro meti mimdevari hyavda.

brZola am ori Teoriis momxreebs Soris saukuneze met xans gagrZelda

da dasrulda talRuri Teoriis sruli gamarjvebiT. am gamarjvebaSi didi ro-

li iTamaSa iungis da gansakuTrebiT frenelis Sromebma sinaTlis interfe-

renciasa da difraqciaSi. da mainc, saboloo da gamanadgurebeli dartyma si-

naTlis korpuskulurma Teoriam miiRo 1850-ian wlebSi, roca frangma fiziko-

sebma fizom da fukom daamuSaves sinaTlis siCqaris gazomvis laboratoriu-

li meTodebi da daadgines, rom sinaTlis siCqare wyalSi naklebia vidre vakuumSi.

am droidan moyolebuli, talRuri Teoriis sisworeSi da maSasadame,

msoflio eTeris arsebobaSi eWvi aravis ar epareboda, Tumca eTeris buneba

kvlav rCeboda gaurkveveli. yovelgvari cda, mieweraT eTerisaTvis iseTi

Tvisebebi, romlebic uzrunvelyofda yvela cnobili optikuri movlenis

axsnas, marcxiT mTavrdeboda.

∗ polarizacia damaxasiaTebelia mxolod ganivi talRebisaTvis. drekadi ganivi talRebi ki aRiZvreba da vrceldeba mxolod myar sxeulebSi.

167

msoflio eTeris arsebobis sayovelTao aRiarebasTan erTad mecnierebis

winaSe daisva sakiTxi imis Sesaxeb, waritaceba Tu ara igi moZravi sxeulebis

mier. Zneli iyo iseTi kavSiris warmodgena sxeulisa da eTeris nawilakebs

Soris, romelic uzrunvelyofda am watacebas, wayolas. magram Tu es ase ar

aris, e.i. Tu eTeri (absoluturad) uZravia, maSin sxeulebis moZraobisas masSi

unda warmoiSvas eTeris qari. amerikeli mecnieris maikelsonis mier

Catarebulma metad zustma eqsperimentebma (1881 w. da Semdgom) cxadyo, rom

sxeulebis, kerZod, dedamiwis moZraobisas, araviTari eTeris qari ar warmoiq-

mneba. miRebul iqna Sinagani winaaRmdegoba, ramac aiZula mecnierebi gadaesin-

jaT arsebuli warmodgenebi sivrcisa da drois kategoriebis Sesaxeb. am gada-

sinjvis Sedegi iyo ainStainis mier fardobiTobis Teoriis Seqmna, rac

Tanamedrove fizikis qvakuTxedia.∗

me-19 saukunis meore naxevarSi maqsvelisa da hercis Sromebis Sedegad

dadginda sinaTlis eleqtromagnituri buneba, riTac moxda optikisa da eleq-

trodinamikis Serwyma, sinTezi.

eleqtromagnituri Teoriis Tanaxmad sinaTle aris eleqtromag-

nituri talRa. es talRa aris ganivi, rac niSnavs imas, rom talRaSi eleq-

truli velis daZabulobis da magnituri velis induqciis veqtorebi irxevian

erTmaneTis da talRis gavrcelebis υ siCqaris marTobulad (sur. 49.1).

sinaTlis eleqtromagnitur Teorias didi miRwevebi hqonda. man

brwyinvaled axsna im droisaTvis cnobili optikuri movlenebis umravlesoba

(sinaTlis interferencia, difraqcia, dispersia, polarizacia da sxv.). amitom,

sinaTlis eleqtromagnituri buneba iTvleboda uryevad TiTqmis me-19 saukunis

bolomde, Tumca am droisaTvis dagrovili iyo mTeli rigi sakiTxebi, romel-

sac es Teoria ver xsnida. am sakiTxTa ricxvs miekuTvneba fotoefeqti,

absoluturad Savi sxeulis gamosxivebis speqtrSi energiis ganawileba da sxv.

sakiTxebis es jgufi aixsna sinaTlis kvanturi TeoriiT.

kvanturi Teoriis fuZemdebelia germaneli mecnieri maqs planki, ro-

melmac dauSva (1900 w.) rom nivTierebis atomebi da molekulebi eleqtromagni-

tur energias asxiveben ara uwyvetad, e.i. ara talRis saxiT, aramed wyvetilad,

energiis calkeuli porciebis saxiT. energiis am calkeul porciebs kvantebi

∗ sakmarisia aRiniSnos, rom Tanamedrove teqnikis iseTi dargebi, rogorebicaa birTvuli energetika da elementaruli nawilakebis amaCqareblebis teqnika, am Teoriazea dafuZnebuli.

168

ewoda. mogvianebiT (1905 w), fotoefeqtis asaxsnelad, ainStainma dauSva, rom

nivTierebis mier eleqtromagnituri energiis STanTqmac xdeba calkeuli

porciebis anu kvantebis saxiT. optikuri diapazonis Sesabamis gamosxivebis

kvantebs fotonebi ewoda. amrigad, kvanturi Teoriis Tanaxmad si-

naTle aris fotonebis nakadi. TiToeuli fotonis energia gamoiTvleba

formuliT: hv=ε , sadac 341062,6 −⋅=h j.wm aris plankis mudmiva, xolo v - rxe-

vis sixSire. fotoni sinaTlis gansakuTrebuli nawilakia, romelsac aqvs

energia, impulsi, magram ar gaaCnia uZraobis masa. xazgasmiT unda aRiniSnos,

rom kvanturi Teoriis Seqmna da sinaTlis nawilakebze _ fotonebze warmod-

genis ganviTareba ar aris dabruneba wminda korpuskuluri Teoriisken, vi-

naidan kvantur TeoriaSi SenarCunda talRuri warmodgenebic: energiis kvanti

raodenobrivad gamoisaxeba rxevis sixSiris (talRis sigrZis) saSualebiT.

sinaTlis dualizmi: amrigad, ra aris sinaTle?

arsebobs sinaTlis ori Tamanedrove Teoria _ eleqtromagnituri da

kvanturi. eleqtromagnituri Teoriis Tanaxmad sinaTle aris eleqtromagni-

turi talRa. am TeoriiT aixsneba iseTi optikuri movlenebi, rogorebicaa in-

terferencia, difraqcia, polarizacia da sxv. kvanturi Teoriis Tanaxmad

sinaTle aris fotonebis nakadi. am TeoriiT aixsneba fotoefeqti, energiis ga-

nawileba absoluturad Savi sxeulis gamosxivebis speqtrSi, luminescencia da

sxva (SeiZleba iTqvas, rom gavrcelebis dros sinaTle iqceva talRis msgav-

sad, xolo gamosxivebisa da STanTqmis dros _ nawilakebis nakadis msgavsad).

arcerT am Teorias ara aqvs upiratesoba meoris mimarT, vinaidan calke aRe-

buli vercerTi ver xsnis yvela optikur movlenas. es miuTiTebs imas, rom

sinaTles aqvs rTuli, ormagi buneba. sinaTlis bunebis ormagobas

sinaTlis dualizmi ewodeba.∗

sinaTlis dualizmi, rac ganapirobebs sinaTlis bunebis Sesaxeb ori

Teoriis arsebobas, erTgvari gamowvevaa mecnierebisadmi _ Seiqmnas sinaTlis

erTiani Teoria, romelSic asaxuli iqneba sinaTlis rogorc talRuri, aseve

kvanturi (korpuskuluri) buneba. am mimarTulebiT mimdinareobs intensiuri

muSaoba, Tumca erTiani Teoriis damuSaveba jer damTavrebuli ar aris.

∗ SemdgomSi aRmoCnda, fizikuri bunebis es ormagoba damaxasiaTebelia ara marto sinaTlisa-Tvis, aramed nebismieri mikronawilakebisTvisac ($ ).

169

Tavi VI

geometriuli optikis elementebi

$52. geometriuli optikis ZiriTadi kanonebi

geometriuli (anu sxivuri) optika aris optikis nawili,

romelic Seiswavlis sinaTlis sxivTa gavrcelebis kanonebs. ∗ jer

kidev sinaTlis bunebis garkvevamde cdebiT iyo dadgenili geometriuli opti-

kis oTxi ZiriTadi kanoni. esenia: sinaTlis wrfivi gavrcelebis kanoni, sinaT-

lis sxivTa konebis damoukideblobis kanoni (principi), sinaTlis arekvlis

kanoni da sinaTlis gardatexis kanoni. ganvixiloT mokled es kanonebi.

1. sinaTlis wrfivi gavrcelebis kanoni. Tu Tvalsa da sinaT-

lis wyaros Soris raime gaumWvir sxeuls movaTavsebT, Cven sinaTlis wyaros

ver davinaxavT, rac imaze metyvelebs, rom erTgvarovan garemoSi sinaT-

le wrfivad vrceldeba. sinaTlis wrfivi gavrcelebiT aixsneba Crdilisa

da naxevarCrdilis warmoqmna, agreTve mcire xvretis saSualebiT sagnis gamo-

saxulebis miReba.

Crdili, anu are, sadac ar xvdeba sinaTlis energia, miiReba mcire zomis

(`wertilovani~) S sinaTlis wyaros SemTxvevaSi (sur. 52.1 a), xolo naxevar-

Crdili anu aramkveTri Crdili (romelic gars akravs Crdils) miiReba roca

sinaTlis wyaro didi zomisaa (sur. 52.1 b).

∗ ufro zustad, geometriuli optikis amocanaa sinaTlis sxivis svlis dadgena iseT garemoSi, romlisTvisac cnobilia n gardatexis maCveneblis damokidebuleba koordinatebze da piriqiT _ gamWvirvale da amrekl garemoTa Tvisebebisa da formis dadgena cnobili sxivTa svlis mixedviT.

S S

a b sur. 52.1

170

mcire xvretis saSualebiT sagnis gamosaxulebis miReba naCvenebia 52.2

suraTze. sinaTlis wrfivi gavrcelebis gamo AB sagnis TiToeuli wertili-

dan gamosuli sxivi E ekranze iZleva TiTo mnaT wertils, romelTa erTob-

lioba qmnis sagnis ''BA gamosaxulebas. gamosaxuleba Sebrunebulia.

magram aRmoCnda, rom Zalian mcire zo-

mis xvrelis SemTxvevaSi sinaTlis wrfivi

gavrcelebis kanoni irRveva. marTlac, cda

gviCvenebs, rom yvelaze kargi gamosaxuleba

miiReba, roca xvrelis zoma daaxloebiT

0,5 mm-ia. xvrelis zomis TanadaTanobiTi Sem-

cirebiT gamosaxuleba xdeba sul ufro

metad bundovani, Semdeg kargavs formas da

sagans aRar gavs, xolo roca zoma gaxdeba ≈0,0005 mm-is toli, gamosaxuleba

saerTod qreba da miiReba sustad, magram TiTqmis Tanabrad ganaTebuli

ekrani.∗ cxadia, aRwerili cdis Sedegi SeiZleba aixsnas mxolod imiT, rom Za-

lian viwro xvrelSi gavlisas irRveva sinaTlis wrfivi gavrcelebis kanoni.

erTgvarovan garemoSi sinaTlis wrfivi gavrcelebis kanonis dar-

Rvevas difraqcia ewodeba. difraqciis movlenas Cven ganvixilavT VIII

TavSi.

vinaidan xiluli sinaTlis talRis sigrZe 0,0005 mm-is rigisaa, ganxiluli cdis Sedegi

saSualebas gvaZlevs davadginoT geometriuli optikis gamoyenebis sazRvrebi: difraqciis

movlenis ugulvebelyofa da sinaTlis wrfivi gavrcelebis kanonis gamoyeneba SesaZlebelia

manam, vidre xvrelis (an sagnis, romelsac ecema sinaTle) zoma gacilebiT metia sinaTlis

talRis sigrZeze.

2. sinaTlis sxivTa konebis damoukideblobis kanoni. am kanonis

arsi imaSi mdgomareobs, rom sxivTa konebis gadakveTisas maTi urTierTqmedeba

ar xdeba, e.i. TiToeuli vrceldeba meoresgan damoukideblad ise, TiTqos gan-

saxilveli konis garda sxva konebi ar arsebobdes. marTlac, Tvalis badu-

raze miRebuli sagnis gamosaxuleba ar Seicvleba, Tu sinaTle, romelic qmnis

am gamosaxulebas, Tavis gzaze gadakveTs sinaTlis gverdiT konebs, romlebic

TvalSi ar xvdeba. aRmoCnda, rom es kanoni samarTliania mxolod wrfiv

∗ ekranis srul ganaTebamde miiReba bneli da naTeli koncentruli rgolebis erToblioba ($67).

A

B A′

B′

E sur. 52.2

171

optikaSi anu arc Tu Zalian didi intensivobis sinaTlis konebis SemTxvevaSi.∗

mZlavri lazeruli sxivebis gadakveTisas adgili aqvs maT urTierTqmedebas.

3. sinaTlis arekvlis kanoni. uZvelesi droidan aris cnobili, rom

rodesac sinaTle ecema ori (gamWvirvale) garemos gamyof zedapirs, misi

nawili ukuigdeba zedapiris mier anu airekleba, nawili ki gadadis meore

garemoSi da vrceldeba Secvlili mimarTulebiT anu gardatydeba.

kuTxes dacemul sxivsa da (ori garemos gamyofi zedapirisadmi) dacemis

O wertilSi aRmarTul perpendikulars Soris dacemis kuTxe ewodeba ( )1i , xo-

lo kuTxeebs arekvlil da gardatexil sxivebsa da imave perpendikulars So-

ris _ Sesabamisad arekvlisa da garda-

texis kuTxeebi ( )'1i da ( )2i (sur. 52.3).

arekvlis kanoni optikis erTerTi

uZvelesi kanonia (romelic cnobili iyo

jer kidev 300 wliT adre Cvens wel-

TaRricxvame) da mdgomareobs SemdegSi:

dacemuli sxivi, arekvlili sxivi

da dacemis wertilSi aRmarTuli

perpendikulari erT sibrtyeSi mde-

bareobs;∗∗ arekvlis kuTxe dacemis

kuTxis tolia '11 ii = .

Tu sxivs davcemT OS ' mimarTulebiT, arekvlis kanonis Sesabamisad igi

gavrceldeba 1OS mimarTulebiT. e.i. dacemuli da arekvlili sxivebi

urTierTSeqcevadia.

∗ susti sinaTlis eleqtromagnitur velebTan ⎟

⎠⎞

⎜⎝⎛ ≤

smv10E Catarebulma cdebma aCvena, rom

optikuri movlenebis (mag. STanTqmis) xasiaTi ar aris damokidebuli gamosxivebis intensivobaze. aseT movlenebs wrfivi optikuri movlenebi ewodeba, xolo optikis nawils, romelic Seiswavlis aseT optikur movlenebs _ wrfivi optika. wrfiv optikas safuZvlad udevs is faqti, rom arsebobs wrfivi damokidebuleba polarizaciis veqtorsa da eleqtruli

velis daZabulobas Soris: kEP = . aqve SevniSnoT, rom arawrfivi optika dasabams iRebs 1960 wlidan, da dakavSirebulia sinaTlis mZlavri wyaros _ lazeris SeqmnasTan. lazeris saSualebiT miiReba iseTi sinaTlis talRa, romelSic eleqtruli velis daZabuloba aRwevs

smv610 -s. SedarebisaTvis aRvniSnoT, rom dedamiwis zedapiris maxloblad mzis gamosxivebaSi

eleqtruli velis daZabuloba sm

v7≈E ∗∗ am sibrtyes dacemis sibrtye ewodeba.

N S1 S′ i1 i1

′ I garemo M O N i2 II garemo S2 sur. 52.3

172

ori garemos gamyofi zedapiris Tvisebebis mixedviT arekvlas SeiZleba hqones sarkuli

an gabneuli xasiaTi.

sarkuli arekvla ewodeba iseT arekvlas, rodesac zedapirze dacemuli

paraleluri sxivTa kona airekvleba erTi garkveuli mimarTulebiT (sur. 52.4 a). amas maSin

aqvs adgili, rodesac arekvla xdeba sarkuli zedapiridan, anu iseTidan, romlis uswormaswo-

robaTa zomebi sinaTlis talRis sigrZeze naklebia. Tavisi TvisebebiT sarkuls uaxlovdeba

kargad gaprialebuli liTonis zedapiri, gluvi minis zedapiri da sxv.

gabneuli anu difuzuri arekvla ewodeba iseT arekvlas, roca areklili sxivebi

vrceldeba yvela mimarTulebiT (sur. 52.4 b). amas maSin aqvs adgili, roca arekvla xdeba

mqise, xorkliani zedapiridan, anu iseTidan, romlis uswormasworobaTa zomebi sinaTlis

talRis sigrZeze metia.

difuziuri gabnevis gamoa, rom Cven vxedavT sagnebs, romlebic TviTon ar asxiveben

sinaTles. aseTi ki Cven irgvliv arsebul saganTa udidesi umravlesobaa. mcire gabnevas

adgili aqvs gluvi zedapiridanac, mag. sarkidan da amitomaa, rom vxedavT sarkes.

absoluturad sarkuli zedapiri uxilavia. vxedavT mxolod misgan arekvlil sxivebs,

romlebic sinaTlis wyarodan ecema, anu vxedavT TviT sinaTlis wyaros.

ganvixiloT wertilis gamosaxulebis brtyel sarkeSi, anu iseT sibrtyeSi, romelic

sinaTles sarkulad areklavs. vTqvaT, MN aris brtyeli sarke, xolo S _ mnaTi wertili.

ganvixiloT am wertilidan gamosuli ori sxivi _ SO da 'SO (sur. 52.5). arekvlis kanonis

gamoyenebiT miviRebT, rom arekvlili sxivebi aris ganSladi. am ganSladi sxivebis TvalSi

moxvedrisas adamians eCveneba, rom sinaTle gamodis am sxivebis gagrZelebis gadakveTis 'S

wertilidan. 'S wertili aris S wertilis gamosaxuleba brtyel sarkeSi. am gamosaxulebas

ewodeba warmosaxviTi, vinaidan 'S wertilSi ikveTeba ara TviT arekvlili sxivebi, aramed

maTi gagrZelebebi. sinaTlis energia 'S wertils ar gadaecema.

'SMO da ''MOS tolobidan

gamomdinareobs, rom ,'MSSM = rac niSnavs, rom 'S wertili sarkis mimarT S wertilis

simetriulia. cxadia amitom, rom brtyel sarkeSi sagnis gamosaxulebis misaRebad saWiroa

moinaxos am sagnis TiToeuli wertilix simetriuli wertili. sagnis gamosaxuleba brtyel

sarkeSi sididiT TviT sagnis zomisaa (sur. 52.6).

a b sur. 52.4

173

4. sinaTlis gardatexis kanoni. gardatexis kanoni dadginda

gacilebiT gvian, 1621 wels, holandieli mecnieris sneliusis mier da

mdgomareobs SemdegSi (sur. 52.3): dacemuli sxivi, gardatexili sxivi da

dacemis wertilSi aRmarTuli perpendikulari erT sibrtyeSi

mdebareobs; dacemis kuTxis sinusis fardoba gardatexis kuTxis

sinusTan mocemuli ori garemosaTvis mudmivi sididea da mas

ewodeba meore garemos gardatexis maCvenebeli pirvelis mimarT:

212

1

sinsin

nii= . (52.6)

cda gviCvenebs, rom Tu sxivs davcemT OS2 mimarTulebiT, gardatexis

Semdeg is gavrceldeba 1OS mimarTulebiT. e.i. dacemuli da gardatexili

sxivebi urTierTSeqcevadia. urTierTSeqcevadobis safuZvelze SeiZleba

davweroT:

121

2

sinsin

nii= . (52.7)

(53.1) da (53.2) tolobebis Sedareba gvaZlevs:

1221

1n

n = . (52.8)

mocemuli garemos gardatexis maCvenebels vakuumis mimarT ewodeba am

garemos absoluturi gardatexis maCvnebeli. gardatexis absolutur

maCvenebels Rrma fizikuri Sinaarsi aqvs. igi gviCvenebs, Tu ramdenjer metia

sinaTlis gavrcelebis siCqare vakuumSi ( )C vidre mocemul garemoSi ( )υ :

υCn = .

S B A M O 'O N M N A′ B′ S′ sur. 52.5 sur. 52.6

174

vinaidan haeris gardatexis maCvenebeli erTisagan umniSvnelod gansxvav-

deba ( )0003,1=n , amitom mocemuli garemos gardatexis maCvenebeli haeris mimar-

Tac miCneulia am garemo gardatexis absolutur maCveneblad.

davamyaroT damokidebuleba fardobiT

da absolutur gardatexis maCvenebels So-

ris. SevadginoT naxazi da davweroT gar-

datexis kanonebi:

23

221

2

11

1

1sinsin

;sinsin

;sinsin

nii

nii

nii

=== .

dadgenilia, rom yvela SemTxvevaSi

ii =3 , anu gamosuli sxivi dacemulis para-

leluria. amis gaTvaliswinebiT samive to-

lobis erTmaneTze gadamravlebiT miviRebT:

2

2111n

nn ⋅= , saidanac

1

221 n

nn = ;

miRebuli Sedegis gaTvaliswinebiT gardatexis (52.1) kanoni miiRebs saxes

1

2

2

1

sinsin

nn

ii= . (52.9)

paragrafis bolos kidev erTxel aRvniSnoT, rom geometriuli optikis

kanonebi samarTliania wrfiv optikaSi da mxolod im SemTxvevaSi, roca xvre-

lis an dabrkolebis zoma gacilebiT metia sinaTlis talRis sigrZeze.

$53. sinaTlis sruli Sinagani arekvla da misi gamoyeneba

wina paragrafSi miRebuli gardatexis kanoni (52.9) gadavweroT Semdeg-

nairad:

2211 sinsin inin = (53.1) ori garemos erTmaneTTan Sedarebisas erT-erT maTgans, romlis absolu-

turi gardatexis maCvenebeli ufro metia, ewodeba optikurad metad

mkvrivi.

(53.1) formula gviCvenebs, rom Tu 21 nn < , maSin 12 ii < da piriqiT, e.i. Tu

sxivi gadadis optikurad naklebad mkvrivi garemodan metad mkvrivSi,

gardatexili sxivi uaxlovdeba normals (sur. 53.1a) da piriqiT (sur. 53.1b).

i haeri

1i 1i I garemo

2i 2i

II garemo

3i haeri sur. 52.7

175

ganvixiloT es meore SemTxveva ufro dawvrilebiT, roca sxivi gaadadis

opitikurad metad mkvrivi garemodan naklebad mkvrivSi, vTqvaT, minidan haerSi

(sur. 53.2). suraTidan Cans, rom dacemis kuTxis TandaTanobiTi zrda iwvevs

gardatexis kuTxis Sesabamis zrdas (sur. 53.2 a,b). amitom cxadia, rom dacemis

kuTxis raRac 0i mniSvnelobis dros gardatexis kuTxe gaxdeba 90 –is toli,

anu gardatexili sxivi isrialebs ori garemos gamyof zedapirze (sur. 53.2 g).

Tu dacemis kuTxes kidev gavzrdiT,

gardatexili sxivi saerTod aRar iarse-

bebs, anu sxivi meore garemoSi ar gadava da

mTlianad airekvleba imave garemoSi

(sur.53.2d). am movlenas ewodeba sinaTlis

sruli Sinagani arekvla. dacemis 0i ku-

Txes, romlis Sesabamisi gardatexis kuTxe

90 –ia, ewodeba sruli Sinagani arekvlis zRvruli kuTxe.

amrigad, srul Sinagan arekvlas rom hqondes adgili saWiroa Sesrul-

des ori piroba: 1. sxivi gadadiodes optikurad metad mkvrivi garemodan nak-

lebad mkvrivSi da 2. dacemis kuTxe iyos zRvrul kuTxeze meti.

sruli Sinagani

arekvlis zRvrul kuTxes

advilad gamoviTvliT.

radgan am dros

gardatexis kuTxe 90= ,

amitom gveqneba: n

i 1sin 0 = .

sruli Sinagani

arekvlis movlenas mravalmxrivi gamoyeneba aqvs. magaliTad, es movlena udevs

safuZvlad e.w. sruli Sinagani arekvlis prizmebis moqmedebas, romelTa

saSualebiTac xdeba sxivis mimarTulebis Secvla 90 da 180 -iT, gamosaxulebis

amotrialeba (sur. 53.3 a,b,g).

N N N N

0i 1n 21 nn > 2n 90

a b g d sur. 53.2

1i 1i 1n 21 nn < 1n 21 nn > 2n 2n 2i 2i a b sur. 53.1

176

sinaTlis sruli Sinagani arekvla gamoiyeneba

boWkovan optikaSi sinaTlisa da gamosaxulebis

gadasacemad specialuri Suqsatarebis saSualebiT.

Suqsatari warmoadgens cilindruli (an sxva) for-

mis minis wvril, moqnil Reros _ boWkos, romelsac

SeiZleba mivceT nebismieri forma. boWko dafarulia

gamWvirvale garsis Txeli feniT, romlis gardatexis

maCvenebeli naklebia minis gardatexis maCvenebelze.

mravaljeradi sruli Sinagani arekvlis Sedegad

sinaTle SesaZlebelia mivmarToT nebismierad mrud

gzaze (sur. 53.4), rac gansakuTrebiT mniSvnelovania

Znelad misadgomi ubnebis gasanaTeblad, magaliTad,

adamianis Sinagani organoebis dasamzerad.

boWkos mier gadacemuli sinaTlis energiis gasazrdelad unda gaizardos boWkos

ganivkveTi, magram es gamoiwvevs misi moqnilobis da maSasadame, gamoyenebis aris SezRudvas.

imisaTvis, rom boWkos moqnilobis Seumcireblad gazardon gadacemuli sinaTlis energia,

boWoebs kreben konebad e.w. CaliCebad.

sainteresoa aRiniSnos rom brilianti∗ Tavis gansakuTrebul brwyinva-

lebas unda umadlodes sinaTlis srul Sinagan arekvlas. brilianti warmoad-

gens almasis kristals, romelsac didi gardatexis maCvenebeli ( )42,2=n da

Sesabamisad, mcire zRvruli kuTxe ( )240 =α aqvs. amitom briliantis Sesabami-

si (e.w. saiuveliro) dawaxnagebis SemTxvevaSi∗∗ sinaTlis didi nawili, rome-

lic ecema briliants yvela mxridan, mTlianad airekvleba da gamodis

briliantis zeda waxnagze (waxnagebze), ris

gamoc igi moCans moelvare, mbrwyinavi.

igive principi moqmedebs brolis Sem-

TxvevaSi, romelic warmoadgens Cveulebriv

minas tyviis danamatiT (gardatexis maCve-

neblis gasazrdelad).

∗ sityva franguli warmoSobisaa da niSnavs mbrwyinavs, elvares. ∗∗ aseTi dawaxnagebis SemTxvevaSi brilians aqvs 56 gverdiTi waxnagi.

sur. 53.4

sur.53.3

177

$54. sxivTa svla brtyel paralelur firfitaSi da samwaxnaga prizmaSi

vTqvaT, brtyelparalelur firfitas, romlis sisqea d , A wertil-

Si ecema sxivi 1i kuTxiT. igi gardatydeba ufro naklebi, 2i kuTxiT (sur. 54.1).

vinaidan firfitis waxnagebi paraleluria, gardatexili sxivi qveda waxnags

B wertilSi daecema imave 2i kuTxiT da sxivTa Seqcevadobis gamo gardat-

ydeba 1i kuTxiT. amrigad, brtyel paralelur firfitaSi gavlisas sxivi mimar-

Tulebas ar icvlis, igi mxolod gadaad-

gildeba TavisTavis paralelurad.

gadaadgilebis l sidides advilad

gamoviTvliT. marTlac ABKΔ -dan

( )21sin iiABlBK −== ; xolo ABCΔ -dan

2cos idAB = ; amitom

( )2

21

cossin

iiidl −

= . miRebuli

formula gviCvenebs, rom sxivis gadaadgi-

leba miT naklebia, rac naklebia firfi-

tis sisqe da sxivis dacemis kuTxe ( l ram-

denadme damokidebulia firfitis garda-

texis maCvenebelzec, romelic gansa-

zRvravs gardatexis kuTxes).

axla ganvixiloT sxivTa svla samwaxnaga prizmaSi. mocemulia

ABC samwaxnaga prizma, romlis gardatexis maCvenebelia n , xolo gardamtexi

kuTxe _ α . vTqvaT, prizmis AB wax-

nags ecema sxivi 1i kuTxiT. igi gar-

datydeba 2i kuTxiT, 12 ii < vinaidan

sxivi gadadis naklebad mkvriv ga-

remodan metad mkvrivSi (haeridan mi-

naSi). gardatexili sxivi ecema BC

waxnags '1i kuTxiT da gardatydeba '

2i

kuTxiT (sur. 54.2). '1

'2 ii > , vinaidan

sxivi gadadis metad mkvrivi garemo-

dan naklebad mkvrivSi. amrigad,

N 1i A

2i d 2i K

C B 1i l sur. 54.1

sur. 54.2

178

samwaxnaga prizmaSi gavlisas, sxivi, orgzis gardatexis Sedegad, ixreba fu-

Zisken. advilia Cveneba (ix. petitis teqsti), rom daxris kuTxe ( )1−= nαδ (54.1).

e.i. daxra miT metia, rac metia prizmis gardamtexi kuTxe da gardatexis

maCvenebeli.

marTlac, suraTidan Cans, rom sxivis daxris kuTxe δ ori nawilisagan Sedgeba:

21 δδδ += , sadac 1δ aris sxivis daxris kuTxe AB waxnagze gardatexisas, xolo BC−2δ

waxnagze gardatexisas. magram 211 ii −=δ , xolo '1

'22 ii −=δ , amitom

'1

'221 iiii −+−=δ ;

davweroT gardatexis kanonebi: nii=

2

1

sinsin

; ni

i 1sinsin

'2

'1 = ; davuSvaT, kuTxeebi imdenad mcirea, rom

maTi sinusebi SeiZleba Seicvalos uSualod kuTxeebiT (amas adgili aqvs Txeli prizmis

SemTxvevaSi): nii=

2

1 da nii='

1

'2 . maSin 21 nii = , xolo

'2

'2 nii = da

( ) ( ) ( )( )'12'12 111 iinnini +−=−+−=δ . davakavSiroT

'12 ii + prizmis gardamtex α kuTxesTan.

suraTidan Cans, rom πα =∠+∠+ OBOBOO ''. magram 2

'

2iBOO −=∠

π,

'1

'

2iOBO −=∠

π. amitom

πππα =−+−+ '12 22ii , saidanac

'12 ii +=α da ( )1−= nαδ rac Tanxvdeba (54.1)-s.

$55. linza. linzis fokusi. optikuri Zala.

wertilis gamosaxulebis ageba linzaSi

optikuri linza ∗ ewodeba sinaTlisaTvis gamWvirvale sxeuls

(umTavresad mina, agreTve plastmasa da sxvadasxva kristalebi) ,

romelic ori mxridan SemosazRvrulia sferuli an sxva mrude

zedapiriT.∗∗ erT-erTi zedapiri SeiZleba brtyeli iyos.

linzas, romlis Sua nawili ufro ganieria, vidre napirebi, amozneqili

ewodeba da piriqiT _ Tu linzis napirebi ufro ganieria vidre Sua nawili,

igi Cazneqilia. 55.1 suraTze mocemulia sxvadasxva formis amozneqili (1,2,3)

∗ arsebobs sxva saxis linzebic, magaliTad eleqtronuli linza _ mowyobiloba, romelic gamoiyeneba eleqtronuli konis fokusirebisaTvis; geologiuri linza _ ase ewodeba qanebisa da madneuli sxeulebis formas, romelic kideebisaken swrafad Txeldeba da isoleba. ∗∗ arsebobs cilindruli da sxva mrude zedapirebiT SemosazRvruli optikuri linzebic. Cven maT ar ganvixilavT.

179

da Cazneqili (4,5,6) linzebi. linzis Sua wertils linzis optikuri centri

ewodeba. yvelanairi formis linzis Sua nawili warmoadgens brtyelparale-

lur firfitas, amitom, Txeli linzis SemTxvevaSi, optikur centrze gama-

vali sxivi praqtikulad ar gardatydeba. aqve aRvniSnoT, rom linzas ewodeba

Txeli, Tu misi sisqe gacilebiT naklebia linzis sferuli zedapirebis

simrudis 1R da 2R radiuseb-

ze. SemdgomSi Cven yovelTvis

Txel linzas vigulisxmebT.

nebismier wrfes, rome-

lic gadis linzis optikur

centrze, optikuri RerZi

(optikuri TanaRerZi) ewo-

deba ( 'AOA da 'BOB wrfeebi), xolo optikur RerZs, romelic gadis linzis

sferuli zedapirebis simrudis 1O da 2O centrebze _ mTavari optikuri

RerZi (sur. 55.2). cxadia, rom mTavari optikuri RerZi linzas erTi aqvs, op-

tikuri TanaRerZi ki uamravi).

rogorc aRvniSneT, optikuri RerZis Tan-

xvedrili sxivi linzaSi gardatexis gareSe gaiv-

lis optikuri RerZis paraleluri sxivebi amoz-

neqil linzaSi gavlis Semdeg ikribeba Cazneqil

linzaSi gavlis Semdeg ki _ ganibneva. es imiT

aixsneba rom amozneqili linza warmoadgens fu-

ZeebiT midgmul samwaxnaga prizmebis erTobli-

obas, Cazneqili ki _ wveroebiT midgmuls (sur. 55.3) Cven ki vnaxeT ($54), rom

samwaxnaga prizmaSi gavlisas sxivi fuZisken ixreba.

amrigad, yvela amozneqili linza Semkrebia, yvela Cazneqili ki _

gambnevi. miRebulia Semkreb linzas ewodos dadebiTi, gambnevs ki _ uaryofiTi.

wertils, romelSic ikribeba mTavari optikuri RerZis paraleluri

sxivebi Semkreb linzaSi gavlis Semdeg mTavari fokusi ewodeba, aRiniSneba

F-iT (sur. 55.3 a). manZils linzis optikuri centridan fokusamde fokusuri

manZili ewodeba. fokusuri manZilic F-iT aRiniSneba. linzas meore mTavari

A 'B O

2O 1O B 'A sur. 55.2

180

fokusic aqvs romelic moTavsebulia mis meore mxares da daSorebulia imave

manZiliT (Tu garemo orive mxares erTnairia).

EmTavari optikuri RerZis marTobulad fokusSi gamaval sibrtyes foka-

luri sibrtye ewodeba (MN). optikuri TanaRerZis paraleluri sxivebi linza-

Si gavlis Semdeg ikribebian fokaluri sibrtyis 'F wertilSi, romelsac fo-

kusi (,,Tanafokusi~) ewodeba (sur. 55.4).

gambnevi linzidan gamosul ganSlad sxivebs Tu SevxedavT, mogveCveneba,

rom sinaTle gamodis am sxivebis gagrZelebis gadakveTis wertilidan

(sur. 55.3 b). mas gambnevi linzis mTavari warmosaxviTi fokusi ewodeba

(amiT xazgasmulia misi gansxvaveba `namdvili~ fokusisagan, romelic miiReba

Semkrebi linzidan gamosuli sxivebis gadakveTis Sedegad). Semkrebi linzis

msgavsad, gambnev linzasac aqvs meore warmosaxviTi mTavari fokusi, romelic

mis meore mxaresaa moTavsebuli. Semkrebi linzis fokusuri manZili dadebi-

Tia, gambnevis _ uaryofiTi. cxadia, rom linzis fokusuri manZili damokide-

buli unda iyos linzis sakuTar Tvisebebze _ misi masalis gardatexis maCve-

nebelze da SemomsazRvreli sferuli

zedapirebis 1R da 2R radiusebze. SeiZ-

leba Cveneba, rom es damokidebuleba

mocemulia TanafardobiT:

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+−=

21

1111RR

nF

, an

( )( )21

21

1 RRnRR

F+−

= . (55.1)

(55.1) gviCvenebs, rom rac naklebia

1R da 2R (e.i. rac ufro amozneqilia sur. 55.4

F

F ’

M

N

F

181

linza) da metia n , miT naklebia F .

sxvadasxva linzebis optikuri Tvisebebis dasaxasiaTeblad, maT Soris

saTvalis minebis klasifikaciisaTvis, sargebloben optikuri Zalis cnebiT.

linzis optikuri Zala ewodeba misi fokusuri manZilis Seb-

runebul sidides: F

D 1= .

rac naklebia F , miT ufro Zlierad gardatexs linza sxivebs da miT me-

tia misi optikuri Zala. maSasadame, optikuri Zala aris linzis gardatexis

unaris maCvenebeli.

fokusuri manZilis niSnis Sesabamisad, Semkrebi linzis optikuri Zala

dadebiTia, gambnevis _ uaryofiTi.

optikuri Zalis erTeulia dioptri. dioptri aris iseTi linzis

optikuri Zala, romlis fokusuri manZili erTi metria. (ase magali-

Tad, Semkrebi linzis optikuri Zala, romlis fokusuri manZilia 20 sm, aris 5

dioptri).

ganvixiloT (mnaTi) wertilis gamosaxulebis ageba Semkreb linzaSi. Teo-

riulad mtkicdeba da cdac adasturebs, rom erTi mnaTi wertilidan gamosu-

li paraqsialuri sxivebi∗ linzaSi gavlis Semdeg ikribeba isev erT wertil-

Si, romelsac mnaTi wertilis gamosaxuleba ewodeba. amitom, wertilis gamo-

saxulebis asagebad sakmarisia vipovoT misgan gamosuli Tundac ori sxivis

gadakveTis wertili maTi linzaSi gavlis Semdeg. aseT or sxivad, rogorc

wesi, iReben e.w. `mosaxerxebel~ sxivebs, anu iseT sxivebs, romelTa svla lin-

zaSi cnobilia. esenia: 1. sxivi, romelic vrceldeba optikuri RerZis gaswvriv

da maSasadame, gardatexas ar ganicdis da 2. (mTavari) optikuri RerZis

∗ sxivebi, romlebic iptikur RerZTan mcire kuTxeebs adgenen.

N A S 'F

S F F 'S F F

'S M a b sur. 55.5

182

paraleluri sxivi, romelic linzaSi gardatexis Semdeg gadis (mTavar)

fokusSi. (55.5) suraTze naCvenebia mTavar optikur RerZze (a) da mis gareT (b)

moTavsebuli S wertilis gamosaxulebis ( )'S ageba.

rodesac wertili moTavsebulia mTavar optikur RerZze, saqmes aZnelebs

is, rom orive mosaxerxebeli sxivi erTiandeba da mTavar optikur RerZs

emTxveva. amitom, wertilidan gamosuli sxivis svlis dasadgenad saWiro

xdeba optikuri TanaRerZis gavleba.

$56. linzis formula. gamosaxulebis ageba linzaSi.

rogorc wina paragrafSi aRvniSneT, sagnis romelime erTi wertilidan

gamosuli paraqsialuri sxivebi linzaSi gavlis Semdeg agreTve erT wer-

tilSi ikveTeba. linzis swored es Tviseba ganapirobebs misi saSualebiT sag-

nis nebismieri wertilisa da, maSasadame, mTeli sagnis gamosaxulebis miRebas.

sagnis gamosaxulebis misaRebad sakmarisia avagoT misi ori kidura A

da B wertilis gamosaxuleba (sur. 56.1), risTvisac viyenebT CvenTvis cnobil

mosaxerxebel sxivebs. xSirad sagans aTavseben optikur RerZs zemoT, rogorc

es sur. 56.2-zea warmodgenili.

Txeli linzis Tvisebebi ZiriTadad misi fokusebis ganlagebiT gani-

sazRvreba. es niSnavs, rom Tu viciT linzis fokusuri manZili ( )F da manZili

sagnidan linzamde ( )d , gavigebT manZils gamosaxulebidan linzamde ( )f ise,

rom ar dagvWirdeba sxivebis svlis ganxilva linzaSi. formulas, romelic

amyarebs damokidebulebas Fd , da f sidideebs Soris, linzis formula

ewodeba. miviRoT igi. amisaTvis mivmarToT 56.2 suraTs, romelzedac ''BA aris

AB sagnis gamosaxuleba.

183

suraTidan Cans, rom FOC da '' BFA marTkuTxa samkuTxedebi msgavsia, ris

gamoc SeiZleba daiweros Tanafardoba

''' FAOF

BAOC

= . (56.1)

aseve, ABO da OBA '' marTkuTxa samkuTxedebis msgavsebidan

gamomdinareobs Tanafardoba

OA

AOBA

AB''' = . (56.2)

magram OCAB = , ris gamoc (56.1) da (56.2) tolobebis marcxena nawilebi

tolia, amitom toli iqneba maTi marjvena nawilebic:

OAAO

FAOF

'' = .

magram FOF = , FfFA −=' ; dAO = da fOA =' ; amitom gveqneba

fd

FfF

=−

,

saidanac

fdFdfF =+ .

miRebuli tolobis gayofiT fFd namravlze miviRebT linzis

formulas:

Ffd111

=+ . (56.3)

d , f da F manZilebis aTvla xdeba linzis optikuri centridan.

amozneqili linzis SemTxvevaSi d da F sidideebi yovelTvis dadebiTia, f -is

sidide ki warmosaxviTi gamosaxulebis SemTxvevaSi uaryofiTia. igi aiTvleba

linzis centridan imave mxares, saiTac d . gambnevi linzis SemTxvevaSi

uaryofiTia rogorc f , aseve F .

184

Tu 56.3-Si SevitanT F1-is mniSvnelobas (55.1)-dan, miviRebT linzis

zogad formulas:

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+−=+

21

11111RR

nfd

.

gamosaxulebis wrfivi zomis Sefardebas sagnis wrfiv

zomasTan linzis wrfivi gamadidebloba ewodeba.

ABBAK

''

= , an (56.2)-is Tanaxmad, dfK = .

ganvixiloT amozneqil linzaSi gamosaxulebis agebis sxvadasxva Sem-

Txvevebi. unda aRiniSnos, rom arsebobs sagnis (da, maSasadame, gamosaxulebi-

sac) mdebareobis mxolod eqvsi gansxvavebuli varianti. gamosaxulebis mdeba-

reoba ganvsazRvroT rogorc geometriuli agebiT, ise analizurad _ linzis

formulis gamoyenebiT. amisaTvis linzis formulas vucvaloT saxe:

Ffd111

=+ , an Fd

FddFf

−=−=

111,

saidanac

dF

FFd

Fdf−

=−

=1

. (56.4)

1. sagani moTavsebulia usasruloba-

Si, e.i. ∞=d . maSin 0=dF

da (56.4)-is Ta-

naxmad Ff = . e.i. Tu sagani mdebareobs

usasrulobaSi, gamosaxuleba miiReba fokusSi. gamadidebloba 0==dfK .

marTlac, usasrulod daSorebuli sagnidan wamosuli sxivebi paralelurni

arian, romlebic linzaSi gavlis Semdeg ikveTebian fokusSi.

2. sagani mdebareobs ormag fo-

kuss gareT (Cveulebriv, aseTi mdebare-

oba aqvs sagans fotografirebis dros):

Fd 2> . maSin 21

<dF

, 211 >−

dF

da Ff 2< .

F O

B

F 'A 2F A 2F F O B’

185

1<=dfK . e.i. Tu sagani mdebareobs ormag fokuss gareT, gamosaxuleba miiReba

ormag fokuss SigniT. gamosaxuleba aris namdvili, Sebrunebuli da Semcire-

buli.

3. sagani mdebareobs ormag fokus-

Si: Fd 2= . maSin 21

=dF

, 211 =−

dF

da

Ff 2= . 1==dfK . e.i. Tu sagani mdebare-

obs ormag fokusSi, gamosaxulebac miiReba ormag fokusSi. gamosaxuleba aris

namdvili, Sebrunebuli da imave zomis.

4. sagani mdebareobs fokussa da

ormag fokuss Soris. Fd 2< . maSin

21

>dF

, 211 <−

dF

da Ff 2> . 1>=dfK . e.i.

Tu sagani mdebareobs ormag fokuss

SigniT, gamosaxuleba miiReba ormag fokuss gareT. gamosaxuleba aris

namdvili, Sebrunebuli da gadidebuli (me-4-e SemTxveva me-2-is Seqcevadia).

5. sagani mdebareobs fokusSi: Fd = .

maSin 1=dF

, 01 =−dF

da ∞=f . ∞==dfK .

e.i. Tu sagani mdebareobs fokusSi, gamo-

saxuleba miiReba usasrulobaSi. mar-

Tlac, rogorc agebidan Cans, linzidan

gamodis paraleluri sxivebi, romlebic ar gadaikveTeba (an, rogorc maTema-

tikaSia miRebuli, gadaikveTeba usasrulobaSi).

6. sagani mdebareobs fokuss SigniT:

Fd < . maSin 1>dF

, 01 <−dF

da ∞<f .. es

niSnavs imas, rom gamosaxuleba mdebareobs

linzis imave mxares, sadac sagania moTav-

sebuli, anu igi warmosaxviTia. es kargad

Cans agebidanac. e.i. roca sagani moTavsebu-

B

A F 2F 2F F O 'A 'B

B

F 2F 'A 2F A F O 'B

F F O

'B

B

'A F

F A O

186

lia fokuss SigniT, gamosaxuleba miiReba warmosaxviTi, pirdapiri da

gadidebuli.

gamosaxulebis ageba Cazneqil

linzaSi.

rogorc agebidan Cans, Cazneqil

linzaSi gamosaxuleba yovelTvis (sadac

ar unda mdebareobdes sagani) miiReba

warmosaxviTi, pirdapiri da Semcirebuli.

$57. linzis (optikuri sistemebis) defeqtebi.

wina paragrafSi vnaxeT, rom paraqsialuri sxivebis SemTxvevaSi gamosa-

xuleba miiReba yovelgvari damaxinjebis gareSe, rac imis Sedegia, rom sagnis

yovel wertils (`wertilovan sagans~) Seesabameba wertilovani gamosaxuleba.

aseT gamosaxulebas stigmaturi ewodeba .∗ magram paraqsialur sxivebTan

saqme gvaqvs mcire zomis linzebis SemTxvevaSi. maTi gamoyeneba ki qmnis gar-

kveul uxerxulobas, vinaidan mcire zomis linza atarebs mcire sinaTlis na-

kads da ekranis an fotofiris ganaTeba miiReba susti. amitom praqtikaSi

Cveulebriv iyeneben didi zomis linzebs. es ki iwvevs sxivebis paraqsialuro-

bis darRvevas, ris gamoc gamosaxuleba miiReba sxvadasxva damaxinjebiT. yvela

es damaxinjeba gaerTianebulia erTi saxelwodebiT _ `linzis defeqtebi~.

ganvixiloT zogierTi maTgani.

1. sferuli aberacia.∗∗ am damaxinjebis arsi advilad gaigeba $54-Si

miRebuli formulis saSualebiT:

( )1−= nαδ (57.1)

sadac δ aris samwaxnaga prizmaSi gavlisas sxivis daxris kuTxe, α _

prizmis gardamtexi kuTxe, n _ prizmis masalis gardatexis maCvnebeli. viciT,

rom amozneqili linza warmoadgens fuZeebiT midgmuli samwaxnaga prizmebis

∗ stigma berZnulad niSnavs wertils. stigmaturi, e.i. wertilovani. Sesabamisad astigmaturis niSnavs arawertilovans. gamosaxuleba astigmaturia, e.i. damaxinjebulia. ∗∗ sityva aberacia laTinuri warmoSobisaa da niSnavs gadaxras, gadaxvevas. optikuri xelsawyoebis SemTxvevaSi aberacia niSnavs gamosaxulebis gabundovanebas an damaxinjebas . aqve aRvniSnoT, rom astronomiaSi aberacia niSnavs cis TaRze varskvlavebis moCvenebiT gadaxras, rac gamowveulia dedamiwis moZraobiT.

187

erTobliobas, romelTa gardamtexi kuTxeebi centridan wverosaken moZraobi-

sas ganuwyvetliv izrdeba. es ki iwvevs imas, rom linzis sxvadasxva nawilebSi

dacemuli sxivebi sxvadasxvanairad gar-

datydeba da gamosaxuleba optikur Rer-

Zze miiReba ara wertilovani, aramed

''''SS wrfis saxiT (sur. 57.1). xolo Rer-

Zis marTobulad moTavsebul ekranze

gamosaxulebas eqneba araTanabrad gana-

Tebuli diskos saxe. damaxinjebis am sa-

xes ewodeba sferuli aberacia, Tumca igi damaxasiaTebelia ara marto

sferuli zedapirebisaTvis.

2. qromatuli aberacia. am damaxinjebis aucileblobac gamomdinare-

obs (57.1) formulidan, romlis Tanaxmadac δ damokidebulia gardatexis maCve-

nebelze. cnobilia, rom TeTri sinaTle rTulia da is Sedgeba Svidi

sxvadasxva ferisagan. sxvadasxva feris sxivebi prizmaSi gavlisas sxvadasxva-

nairad gardatydebian _ yvelaze metad gardatydeba iisferi, yvelaze nakle-

bad _ wiTeli. sinaTlis eleqtromagnituri Teoriis Tanaxmad, sinaTlis fers

gansazRvravs talRis sigrZe. e.i. gardatexis maCvenebeli damokidebuli

yofila talRis sigrZeze:

( )λfn = . (57.2) (57.2) formulas ewodeba dispersiis formula, xolo TeTri sinaTlis

ferebad daSlas prizmaSi gavlisas _ dispersiis movlena. cxadia, rom Tu

linzas daeca TeTri sinaTle, moxdeba misi ferebad gaSla _ yvelaze

Zlierad gardatydeba iisferi, yvelaze sustad _ wiTeli (sur. 57.2), ris gamoc

wertilis gamosaxuleba optikur RerZze miiReba ara wertilis, aramed

garkveuli sigrZis Seferadebuli wrfis saxiT, xolo RerZis marTobulad

dayenebul ekranze _ feradi rgolebis saxiT. gamosaxulebis damaxinjebis

ganxilul saxes ewodeba qromatuli aberacia. rogorc sferul, aseve

qromatul aberacias ebrZvian imiT, rom iyeneben ara erT linzas, aramed

sxvadasxva saxis Semkrebi da gambnevi linzebis kombinacias.

S

'''''' SSS sur. 57.1

188

3. astigmatizmi. sferuli abera-

cia aris kerZo SemTxveva linzis (opti-

kuri sistemis) ufro zogadi defeqtisa

(damaxinjebuli gamosaxulebis miRebisa),

romelsac astigmatizmi ewodeba. astig-

matizmi imis Sedegia, rom si-

naTlis talRis sferuli zedapiri

optikur sistemaSi gavlis Semdeg

deformirdeba da kargavs sferulobas.

ansxvaveben astigmatizmis or saxes. pirveli miiReba maSin, roca sxivebi,

romlebic ecemian linzas, mTavar optikur RerZTan adgenen did (30-40°)

kuTxeebs (iribi sxivebis astigmatizmi).

ganvixiloT gardatexa sxivebisa, romlebic gamodian linzis mTavari op-

tikuri RerZidan moSorebiT moTavsebuli S wyarodan (sur. 57.3). gardatexamde

sxivebi vrceldebian S wertilidan radiusebis gaswvriv (homocentruli sxi-

vebi) da maTdami marTobuli talRuri zedapirebi mkacrad sferulia. gardate-

xis Semdeg ki zedapiri kargavs sferul formas da iZens ornair simrudes.

talRuri zedapiris ornairi simrudis gamo linzidan gamosuli sinaTlis ko-

na kargavs homocentrulobas da ikribeba ara erT wertilSi, aramed erTmane-

Tisagan daSorebul urTierT-

marTob 'S da ''S monakveTebze.

Tu 'S da ''S monakveTebs So-

ris, an maT gareT, sxivis mar-

Tobulad movaTavsebT ekrans,

masze miviRebT arawesieri

formis bundovan laqas, romelic gadadis horizontalur 'S monakveTSi, roca

ekrans gadavanacvlebT marcxniv, am wiris moTavsebis adgilas da ''S

vertikalur monakveTSi, roca ekrans gadavaadgilebT marjvniv.

astigmatizmis meore saxes adgili aqvs maSin, roca darRveulia TviT

optikuri sistemis (RerZuli) simetria sinaTlis konis mimarT. amas adgili

aqvs, magaliTad, cilindruli linzis SemTxvevaSi. Tu cilindrul L linzas

davcemT mTavari optikuri RerZis paralelur sxivTa konas, maSin, vinaidan

S wiTeli

iisferi

sur. 57.2

sur. 57.3

189

linza sxivebs gardatexs mxolod cilindris

msaxvelis marTobulad, linzis fokalur si-

brtyeSi moTavsebul Э ekranze miiReba wrfe

da ara wertili, romelic miiReboda linza

rom sferuli yofiliyo (sur. 57.4). Tu aseTi

linzis saSualebiT gamovsaxavT ekranze

wvrili mavTulisagan damzadebul kvadra-

tul ujredebian bades (sur. 57.5 a), ekranze

ufro mkveTrad gamoCndeba cilindris msaxvelis paraleluri wirebi

(sur.57.5b).

asigmatizmis Tavidan asacileblad qmnian rTul optikur sistemebs,∗

romlebic Seicavs iseT linzebs, rom xdeba maT mier warmoqmnili astigmatiz-

mis urTierTkompensireba. astigmatizmisagan Tavisufal aseT sistemebs anas-

tigmatebi ewodeba.

astigmatizmi damaxasiaTebelia adamianis TvalisTvisac, rac imiT das-

turdeba, rom adamiani ver xedavs Tanabrad mkveTrad urTierTmarTobi wirebis

sistemas gamosacdel tabloze. Tvalis astigmatizmi

ganpirobebulia Tvalis Suqmtexi aparatis naklo-

vanebiT da gamowveulia rqovanas, ufro iSviaTad

brolis, uTanabro simrudiT. mxedvelobis am naklis

nawilobrivi an sruli gamosworeba SesaZlebelia

cilindrul minebiani saTvaliT an kontaqturi linzebiT.

$58. Tvali. mxedveloba

58.1. suraTze mocemulia adamianis Tvalis horizontaluri Wrili. Tvals

TiTqmis sferos forma qvs. misi diametri daaxloebiT 2,5 sm-ia.∗∗ Tvalis kaka-

li garedan garSemortymulia sami garsiT. gareTa, myar da mtkice, TeTri fe-

ris, sinaTlisadmi gaumWvir (1) garss ewodeba sklera da igi icavs Tvalis Si-

ga nawils meqanikuri dazianebisagan. wina nawilSi sklera amoburculi da

∗ Txeli linza warmoadgens umartives optikur sistemas. ∗∗ zrdasruli adamianis `Tvalis sigrZe~ (wina-ukana zoma) saSualod 24,3 mm-ia, horizontaluri zoma _ 23,6 mm, xolo vertikaluri _ 23,4 mm.

sur. 57.4

a b sur. 57.5

190

gamWvirvalea da mas rqovana ewodeba (2).

igi TamaSobs erTgvari sarkmlis rols,

romlis gavliTac sinaTle xvdeba xvde-

ba TvalSi. sklerasa da rqovanas saz-

Rvarze mdebareobs e.w. Slemis arxi,∗

romliTac TvalSiga siTxe gareT gamoe-

dineba. am arxis daxSoba iwvevs Tvalis

Siga wnevis momatebas, rasac moyveba

Tvalis mZime daavadeba _ glaukoma.

skleras Sida mxridan akravs sisxlZar-

Rvovani garsi (8), romelic mofenilia

Tvalis masazrdoebeli sisxlZarRvebiT.

es meore garsi Tvalis wina nawilSi ga-

dadis ferad garsSi anu irisSi (3), romelic sxvadasxva adamianebSi sxvadasxva

feradaa Seferili da maSasadame, ganapirobebs adamianis Tvalis fers. feradi

garsis centrSi aris xvreti _ guga (4), romlis diametri ferad garsSi arse-

buli kunTebis saSualebiT refleqsurad (e.i. gonebis Caurevlad, uneburad)

icvleba 2mm-dan (Zlieri ganaTebis dros), 8 mm-mde (susti ganaTebis dros).∗∗

Tvalis mesame, yvelaze SigniTa garsia badura anu badisebri garsi (9),

romelic warmoadgens mxedvelobis nervis (10) ganStoebebs Cxirebis an kolbe-

bis saxis nervuli daboloebebiT. isini warmoadgens sinaTlis aRmqmel ele-

mentebs (receptorebs) _ sinaTlis mier am daboloebaTa gaRizianeba CvenSi iw-

vevs sinaTlis SegrZnebas.∗∗∗ baduras naxevarsferos forma aqvs _ igi faravs

Tvalis mTel fskers, garda misi wina nawilisa. mxedvelobis nervis gamos-

vlis adgilas (11) sinaTlisadmi mgrZnobiare ujredebi ar aris. am adgil brma

laqa (brma xali) ewodeba. brma laqis maxloblad safeTqlis mxrisken, aris

yviTeli laqa (yviTeli xali), romelic centrSi CaRrmavebulia (12). yviTeli

laqa da gansakuTrebiT misi Rrmuli, sinaTlisadmi yvelaze mgrZnobiare adgi-

lebia baduraze (yviTeli laqis horizontaluri zomaa 1-3 mm, vertikaluri _ 0,8 mm).

∗ pirvelad aRwera germanelma anatomma Slemma. ∗∗ es procesi fotoaparatis diafragmis cvlilebis msgavsia. ∗∗∗ receptorebis gaRizianebisas warmoiqmneba eleqtruli impulsebi, romlebic mxedvelobis nervis boWkoebis saSualebiT gadaecema Tavis tvins da Cven SevigrZnobT sinaTles.

sur. 58.1

191

gugis ukan moTavsebulia broli (5) _ linzis msgavsi fenovani gamWvir-

vale sxeuli. misi diametria 8-10 mm, xolo sisqe ~3,6 mm. specialuri kunTi (6),

romelic gars artyia, refleqsurad cvlis brolis formas, xdis ra ufro

amozneqils (e.i. amcirebs fokusur manZils) axlos myofi sagnebis danaxvi-

saTvis da piriqiT.

sivrce rqovanasa da brols Soris, romelsac Tvalis wina kamera

ewodeba, avsebulia wyalwyala siTxiT, xolo Tvalis danarCeni nawili (7) anu

brolis ukana sivrce, romelsac Tvalis ukana kamera ewodeba, ukavia gamWvir-

vale, naxevrad Txevad minisebr sxeuls. wyalwyala siTxisa da minisebri sxeu-

lis gardatexis maCveneblebi erTnairia da udris 1,336-s. rqovanas gardatexis

maCvenebeli 1,376-ia, xolo brolis gardatexis maCvenebeli saSualod udris

1,437-s (brolis sxvadasxva fenebis simkvrive sxvadasxvaa).

pirdapiri xedvis RerZi AB gansazRvravs yvelaze ufro SuqmgrZno-

biare mimarTulebas da gadis brolisa da yviTeli laqis Rrmulis centreb-

ze. igi ramdenadme acdenilia (daxrilia 5 -iT) mTavar optikur RerZs (simet-

riis RerZs), romelic gadis rqovanas, gugis da brolis geometriul centreb-

ze (suraTze naCvenebi ar aris). es imiT aris gamowveuli, rom yviTeli laqa

moTavsebulia ara baduras centrSi, aramed wanacvlebulia safeTqlisaken.

baduraze SuqmgrZnobiare elementebis _ Cxirebis da kolbebis raodeno-

ba sxvadasxvaa (Sesabamisad 130000000 da 7000000). sxvadasxva maTi funqciebic.

Cxirebi ufro SuqmgrZnobiare elementebia, isini aRiqvamen sust sinaTles, xo-

lo kolbebis meSveobiT xdeba Zlieri sinaTlis aRqma da ferebis garCeva.∗

susti ganaTebiT kolbebi ver Riziandeba. amis gamo, bindSi an RamiT mxedve-

loba xorcieldeba CxirebiT. amasTan, sagnebi gansxvavdebian erTmaneTisagan

sikaSkaSiT, xolo maTi ferebis gansxvaveba ar aRiqmeba (Canan erTferovnad).

kolbebis mgrZnobiaroba sxvadasxva ferebis mimarT sxvadasxvaa. iungisa

da helmholcis Teoriis Tanaxmad, arsebobs kolbebis sami saxe, romelTagan

erTi gansakuTrebiT mgrZnobiarea wiTeli feris mimarT, meore _ mwvane feris

mimarT, xolo mesame _ lurji feris mimarT. Tu TvalSi zemoT CamoTvlili

ferebidan erTi romelime feri moxvda, igi gamoiwvevs Sesabamisi kolbebis ga-

∗ Ramis arsebebis _ bu, Ramura, agreTve kata _ badura umetesad Cxirebs Seicavs, dRis arsebe-bis, mag. qaTmis badura ki ZiriTadad kolbebisagan Sedgeba. amitomaa, rom qaTami sibneleSi ver xedavs. aqedan termini _ `qaTmis sibrmave~

192

Rizianebas da Cven aRviqvamT am feris sinaTles. saSualedo feris sxivis mox-

vedrisas ki Riziandeba, amasTan, sxvadasxva intensivobiT, orive ZiriTadi me-

zobeli feris Sesabamisi kolbebi. magaliTad, TvalSi im ferebis moxvedrisas,

romlebic mdebareobs wiTelsa da mwvanes Soris, Riziandeba (yovel calkeul

SemTxvevaSi sxvadasxva intensivobiT) am ZiriTadi ferebis Sesabamisi kolbebi.∗

Tu kolbebi arasworad funqcionireben, adamiani zogierT fers ver ar-

Cevs erTmaneTisagan. yvelaze xSirad aseTi adamiani ver arCevs wiTel fers

mwvanisagan. es movlena Seiswavla cnobilma ingliselma qimikosma daltonma,

romelsac qonda mxedvelobis es nakli. aqedan warmoiSva misi saxelwodeba _

daltonizmi.

baduraze Cxirebi da kolbebi araTanabradaa ganawilebuli: periferiaze,

sadac umeteswilad Cxirebia, maTi ganawilebis simkvrive mcirea da mxedve-

lobis nervis TiToeuli boWko mierTebulia am receptorebis did ricxvTan.

yviTeli laqis mimarTulebiT receptorebis simkvrive TandaTan matulobs

(ZiriTadad kolbebis xarjze) da yviTel laqaze, gansakuTrebiT ki mis Rrmul-

Si, xdeba maqsimaluri. aq, sadac umeteswilad kolbebia, TiToeul receptorze

modis TiTo nervuli boWko.∗∗

sistematuri gamokvlevebis Sedegad dadginda, normaluri Tvalis para-

metrebis saSualo mniSvnelobebi. kerZod, dauZabavi TvalisaTvis, wina da uka-

na fokusuri manZilebisaTvis es mniSvnelobebi Sesabamisad aris 17,1 mm da

22,8 mm. manZilebis gansxvaveba imiT aris gamowveuli, rom garemo, sadac

moTavsebulia sagani (haeri, 1=n ) gansxvavdeba garemosagan, sadac miiReba

gamosaxuleba (minisebri sxeuli, ).336,1=n es manZilebi akomodaciis Sedegad

(ix. qvemoT) icvleba. amitom, mTlianobaSi Tvalis optikuri sistema SeiZleba

ganvixiloT rogorc Semkrebi linza, romelsac aqvs cvladi fokusuri

manZili da ucvleli `siRrme~ (manZili optikuri centridan ekranamde). ekranis

rols asrulebs badura, romelic Tanxvdeba fokalur sibrtyes.

aseTi mcire wina fokusuri manZilis gamo sagani yovelTvis imyofeba

ormag fokusze gacilebiT ufro Sors, praqtikulad usasrulobaSi, amitom

sagnis mkafio gamosaxuleba miiReba fokalur sibrtyeSi anu baduraze. sagnis

∗ Teorias, SesaZloa safuZvlad daedo eqsperimentuli faqti: wiTeli, mwvane da lurji ferebis SereviT miiReba speqtris nebismieri feri. ∗∗ saSualod, mTeli baduris masStabiT, erT nervul boWkoze modis 140 receptori.

193

TvalTan miaxloebisas d mcirdeba, amitom f unda gazrdiliyo da gamosaxu-

leba unda daSoreboda baduras, rac gamoiwvevda mis gabundovnebas, magram es

ase ar xdeba, vinaidan, rogorc zemoT iyo aRniSnuli, sagnis miaxloebisas

specialuri kunTi refleqsurad cvlis brolis formas, xdis mas ufro amoz-

neqils, ris gamoc igi ufro Zlierad gardatexs sxivebs da gamosaxuleba

rCeba baduraze.

Tvalis unars, ise Secvalos brolis simrude, rom sagnis

gamosaxuleba, mudam baduraze darCes da iyos mkafio, akomodacia

ewodeba . ese igi sxvadasxva manZiliT daSorebuli sagnebis mkafio gamosaxu-

lebis miReba akomodaciis anu manZilTan Tvalis Seguebis gamo aris SesaZle-

beli.

axlomxedveloba da Sorsmxedveloba. manZilTan Tvalis Seguebas

aqvs sazRvari. yovel Tvals aqvs mkafio xedvis Soreuli da uaxloesi werti-

lebi. mkafio xedvis Soreuli wertili iseTi manZiliT daSorebuli werti-

lia, romelzedac Tvali sagans xedavs Tvalis kunTis damSvidebul mdgo-

mareobaSi. xolo mkafio xedvis uaxloesi wertili iseTi manZiliT daSo-

rebuli wertilia, romelzedac Tvali sagans xedavs Tvalis kunTis udidesi

daWimvis dros. arsebobs kidev saukeTeso xedvis manZili. es iseTi manZilia

sagnidan Tvalamde, romlis drosac Tvali yvelaze ufro bevr wvrilmans

xedavs da yvelaze naklebad iRleba. am manZilTa sididis mixedviT Tvalebi

sam jgufad iyofa: normaluri, axlomxedveli da Sorsmxedveli Tvalebi.

normaluri ewodeba iseT Tvals, romlisTvisac mosvenebul mdgomareo-

baSi anu Tvalis kunTis dauZabavad, Soreuli wertili mdebareobs usasrulo-

baSi, anu romelic masSi Sesul paralelur sxivTa konas afokusirebs badu-

raze. manZili uaxloes wertilamde asakTan erTad izrdeba: 10 wlamde igi ~10

sm-is tolia, 30 wlisTvis izrdeba 15 sm-mde, xolo 50 wlisaTvis aRwevs 45-

50 sm-s.∗ am movlenas moxucebulobis Sorsmxedveloba ewodeba. misi gamoswore-

ba xdeba saTvalis saSualebiT (ix. qvemoT). normaluri TavlisaTvis saukeTeso

xedvis manZili 25 sm-ia. wvrili detalebis gasarCevad, xanmokle droiT, es

manZili SeiZleba Semcirdes (cxadia akomodaciis sazRvrebSi).

∗ manZilis gazrda imiaTaa gamowveuli, rom asakTan erTad izrdeba brolis fenebis simkvrive, rac amcirebs maT elastiurobas da SekumSvis unars.

194

axlomxedveli Tvali ewodeba iseT Tvals, romelSic Tvalis kunTis

damSvidebuli, dauZabavi mdgomareobis dros fokusi mdebareobs Tvalis kak-

lis SigniT, baduras win (sur. 58.2 a).

aseTi Tvali daSorebul sagnebs xedavs

bundovnad. Tvalis daZabva mdgomare-

obas ki ar aumjobesebs, aramed auare-

sebs, vinaidan es iwvevs fokusis miax-

loebas brolTan da ara badurasTan.

axlomxedveli Tvalis Soreuli werti-

li mdebareobs ara usasrulobaSi, ara-

med sasrul manZilze. saukeTeso xedvis manZili da uaxloesi wertilis man-

Zili misTvis ufro naklebia, vidre normaluri TvalisaTvis (da damokidebu-

lia imaze, Tu ramdenad didia sibece). axlomxedvelobis gamosworeba xdeba

Cazneqillinziani saTvalis tarebiT (sur. 58.2 b).

Sorsmxedveli Tvali ewodeba iseT Tvals, romlis fokusi Tvalis

kunTis dauZabavi mdgomareobis dros mdebareobs Tvalis kaklis gareT

(sur. 58.2 g). aseTi TvalisaTvis Soreuli wertili ar arsebobs, xolo

uaxloesi wertilis manZili daaxloebiT 30 sm-ia. saukeTeso xedvis manZilic

ufro metia, vidre normaluri TvalisaTvis. Sorsmxedvelobis gamosworeba

xdeba amozneqillinziani saTvalis gamoyenebiT (sur. 58.2d).

axlomxedveloba SeiZleba gamoiwvios Tvalis sigrZis metma, xolo

Sorsmxedveloba _ naklebma mniSvnelobam normalurTan SedarebiT. normaluri

mniSvnelobidan erTi milimetris farglebSi gadaxramac ki SeiZleba gamoiw-

vios SesamCnevi axlomxedveloba an Sorsmxedveloba. mxedvelobis es da sxva

naklovanebebi SeiZleba gamoiwvios agreTve rqovanasa da brolis gardamtexi

zedapirebis arasworma simrudem, am zedapirebis arasimetriulobam, brolis

arasworma mdebareobam da sxv.

xedvis kuTxe. mkafio xedvis aucilebeli pirobaa gamosaxulebis

miReba yviTel laqaze da baduras ganaTebis Sesabamisi xarisxi. garda amisa,

saWiroa, rom sagnis xedvis kuTxe zRvrul kuTxeze naklebi ar iyos. xedvis

kuTxe ewodeba Tvalis optikuri centridan sagnis kidura A da B

wertilebamde gavlebul wrfeebs Soris moTavsebul α kuTxes (sur. 58.3).

sxvanairad, es is kuTxea, romliTac Tvalis optikuri centridan Cans sagani.

sur. 58. 2

195

suraTidan Cans, rom xedvis kuTxis gadideba SesaZlebelia an sagnis

`gadidebiT~ an misi TvalTan miaxloebiT. TvalTan mialoeba SezRudulia xed-

vis uaxloesi wertilis manZiliT. es manZili axlomxedvelebisaTvis yvelaze

naklebia, amitom maT yvelaze

ukeT SeuZliaT wvrilad nabeWdi

teqstis wakiTxva da sagnis

wvrilmani detalebis garCeva.

sagnis ori wertilis cal-

calke aRsaqmelad saWiroa, rom

yviTel laqaze maT gamosaxule-

bebs Soris manZili moicavdes

aranakleb or receptors. es zRvruli manZili 0,004 mm-ia. Tu wertilebis

gamosaxulebebs Soris manZili 0,004 mm-ze naklebia, igi Tavsdeba erT nervul

daboloebaze da Tvali maT aRiqvams rogorc erT wertils.∗ am zRvrul

manZils Seesabameba '1 (erTi minuti) kuTxe,∗∗ romelsac xedvis zRvruli

kuTxe ewodeba. es is kuTxea, romliTac 1 sm sigrZis monakveTi Tvalidan 34

metr manZilze moCans. cudi ganaTebisas (mag. bindSi) zRvruli kuTxe SeiZleba

gaizardos 1°-mde.

yviTeli laqis Sesabamisi mxedvelobis are didi ar aris. laqaze

SeiZleba ganTavsdes im sagnis gamosaxuleba, romlisTvisac xedvis kuTxe

horizontaluri mimarTulebiT 8°-ia, vertikaluri mimarTulebiT _ 6 °.

centralur Rrmuls xedvis are kidev ufro mcirea da Seadgens 5,11− °-s. es

imas niSnavs, rom 1 metris moSorebiT mdgari adamianis mTeli figuridan

yviTel laqaze SegviZlia davafiqsiroT, magaliTad, mxolod misi saxe, xolo

centralur Rrmulze _ zedapiri, romelic odnav aRemateba Tvalisas.

sxeulis danarCeni nawili `proeqtirdeba~ baduras periferiaze bundovnad.

xedvis aris aseTi SezRuduloba kompensirdeba imiT, rom cocxal Tvals

aqvs Tvalis budeSi swrafi Semobrunebis unari, ris gamoc mas SeuZlia Zalian

mokle droSi `Semoirbinos~ gansaxilveli didi sagnis xiluli zedapiris

∗ yviTeli laqis RrmulSi manZili kolbebs Soris 0,0025 mm-ia.

∗∗ 10800

11' = radians.

B α 'A A 'B sur.58.3

O

196

yvela wertili. amasTan, Tvalis agebuleba SesaZleblobas iZleva ara marto

miviRoT saerTo warmodgena yvelaferze, rac Tvalis xedvis areSia, aramed ga-

vamaxviloT yuradReba gansakuTrebiT arsebiT detalebze. Tvalis am Tvisebis

gamo xedvis are farTovdeba horizontaluri mimarTulebiT 150, xolo verti-

kaluri mimarTulebiT 120°-mde, rac bevrad aRemateba kargi optikuri

xelsawyoebis monacemebs.

$59. Tvalis SemaiaraRebeli optikuri xelsawyoebi

58.3 suraTidan Cans, rom xedvis kuTxis gadidebisas baduraze miRebuli

sagnis gamosaxuleba izrdeba, rac qmnis sagnis mcire detalebis ukeT garCevis

SesaZleblobas. xedvis kuTxis mniSvnelovani gazrda xdeba Tvalis Semaiara-

Rebeli optikuri xelsawyoebis saSualebiT. rogorc viciT, xedvis kuTxis

gazrda SeiZleba an sagnis `gadidebiT~, an misi TvalTan miaxloebiT. amasTan

dakavSirebiT es xelsawyoebi iyofa or jgufad.

pirvel jgufSi Sedis xelsawyoebi, romlebic gamoiyeneba metad mcire

sagnebis dasakvirveblad (lupa, mikroskopi). es xelsawyoebi TiTqosda

`adideben~ dasakvirvebel sagans.

meore jgufSi Sedis xalsawyoebi, romlebic gamoiyeneba daSorebuli

sagnebis dasakvirveblad (sxvadasxva saxis teleskopebi∗). es xelsawyoebi

TiTqosda `aaxloeben~ dasakvirvebel sagans.

ganvixiloT sxivTa svla am xelawyoebSi.

lupa. Tvalis SemaiaraRebel xelsawyoebs Soris umartivesia lupa. lu-

pa ewodeba moklefokusian

(1-10sm) ormxriv amozneqil

linzas an linzaTa siste-

mas, romelic moqmedebs ro-

gorc erTi Semkrebi linza.

lupis moqmedeba moce-

mulia 59.1 suraTze: mcire

zomis sxeuli AB saukeTe-

∗ berZnulad skopo niSnavs vumzer, mikros _ mcire, tele _ Sors.

sur. 59.1

197

so xedvis manZilidan ( 25=D sm) moCans α kuTxiT.∗

sagnis simciris gamo xedvis α kuTxec aris mcire, ris gamoc Wirs

saganze detalebis garCeva. sagnis TvalTan miaxloebiT ( 11BA mdebareoba) xed-

vis kuTxe gaizarda, ( )αα >1 , magram detalebis garCeva mainc gaZnelebulia,

vinaidan sagani aRmoCnda akomolaciis zRvars miRma (uaxloes wertilze ufro

axlos). magram Tu Tvalsa da 11BA sagans Soris movaTavsebT lupas ise, rom

sagani aRmoCndes fokuss SigniT (fokusTan axlos), maSin lupis mdebareobis

SerCeviT SeiZleba mivaRwioT imas, rom (pirdapiri, warmoasaxviTi da gadide-

buli) gamosaxuleba aRmoCndes ''BA mdebareobaSi, anu Tvali sagans dainaxavs

gadidebuli 1α kuTxiT saukeTeso xedvis manZilze.

ganmartebis Tanaxmad, gamadidebloba

FD

df

BABAK ===

11

''

,

saidanac Cans, rom lupis saSualebiT miRebuli gamadidebloba 2,5-dan

25-mdea.

mikroskopi. mikroskopis optikuri sistema ori linzisagan (linzaTa

sistemisagan) Sedgeba. erTi dasamzeri sagnisken (obieqtisken) aris mimarTuli

da obieqtivi ewodeba ( )1O , xolo meore Tvalis mxridan aris moTavsebuli

da okulari ∗∗ ewodeba ( )2O . orive linza, romelTa optikuri RerZebi Tan-

xvdeba, moTavsebulia liTonis milSi _ tubusSi. obieqtivi moklefokusiani

linzaa, okulari _ grZelfokusiani. manZili obieqtivsa da okulars Soris

(romlebic moTavsebulia tubusis boloebSi), SeiZleba Seicvalos specialuri

xraxnis saSualebiT.

sxivTa svla mikros-

kopSi mocemulia 59.2 suraTze:

mcire zomis dasamzeri sagani

AB moTavsebulia obieqtivis

win, misi 2F fokusidan odnav

moSorebiT, amitom igi gvaZ-

levs namdvil, Sebrunebul da

∗ Tvali linzasTan imdenad axlosaa, rom maTi optikuri centrebi miCneulia Tanxvedrilad. ∗∗ warmodgeba laTinuri sityvisagan ocularis _ Tvalisa.

sur. 59.2

198

gadidebul 11BA gamosaxulebas. es pirveli gamosaxuleba okularisaTvis

sagans warmoadgens. gadidebuli gamosaxulebis xelaxlad misaRebad okulars

ise gadaadgileben, rom 11BA gamosaxuleba okularsa da mis 2F fokuss Soris

moxvdes. maSin, okularis mier mocemuli gamosaxuleba (lupis msgavsad) iqneba

warmosaxviTi, gadidebuli da sagnis mimarT Sebrunebuli. okularis gadaadgi-

lebiT aRweven imas, rom es gamosaxuleba moTavsdes xedvis saukeTeso D

manZilze.

Tu obieqtivisa da lupis fokusebs Soris manZils Δ -Ti aRvniSnavT, ma-

Sin obieqtivis gamadidebloba toli iqneba 1FΔ

-is. okularis gamadidebloba

iseTivea rogorc lupis da tolia 2F

D-is. amitom mikroskopis sruli gamadide-

bloba K toli iqneba:

21FFDK Δ⋅

= (59.1)

mikroskopi iZleva sagnis mniSvnelovan `gadidebas~ da maSasadame, xedvis

kuTxis mniSvnelovan gazrdas. mag. Tu 2=F mm, 151 =F mm da 160=Δ mm, maSin

1330=k . aqve unda aRiniSnos, rom mikroskopis margi gadideba SezRudulia

difraqciuli movlenebiT (igi 1000-s ar aRemateba).

teleskopi. mikroskopis msgavsad, teleskopis mTavari nawilebia obi-

eqtivi da okulari. obieqtivis saxis mixedviT arsebobs sami tipis teleskopi:

linziani (refraqtorebi), sarkiani (refleqtorebi) da sarkian-linziani.

refraqtorebSi obieqtivis rols asrulebs linza da pirveli gamosa-

xuleba miiReba sagnidan wamosuli sxivebis linzaSi gardatexis Sedegad.

refleqtorebSi obieqtivis rols asrulebs Cazneqili sarke da pirveli

gamosaxuleba miiReba sagnidan wamosuli sxivebis sarkidan arekvlis Sedegad∗.

sarkian-linzian teleskopebSi obieqtivis rols asrulebs sarkisa da

linzis erToblioba.

erT-erTi pirveli refraqtori iyo kepleris mier 1611 wels Seqmnili

astronomiuli mili. milis obieqtivi grZelfokusiania, okulari _

moklefokusiani. sxivTa svla kepleris milSi mocemulia 59.3 suraTze.

∗ aqedan warmodgeba teleskopis dasaxeleba: laTinurad refractio niSnavs sinaTlis sxivis gardatexas, xolo reflecto –arekvlas, ukan mimarTvas.

199

siSoris gamo ciuri sxeulis TiToeuli wertilidan wamosuli sxivebi

ptaqtikulad paraleluria, amitom obieqtivis fokalur sibrtyeSi (ufro

zustad, mis marjvniv, masTan Zalian axlos) miiReba sxeulis namdvili, Sebru-

nebuli da Semcirebuli gamosaxuleba 11BA . suraTze marcxniv pirobiTadaa ga-

mosaxuli ciuri sxeuli. am sxeulis diametralurad sawinaaRmdego B da A

wertilebidan gamosuli sxivebi aRniSnulia Sesabamisad erTi da ori isriT.

es sxivebi obieqtivSi xvdebian imave ϕ kuTxiT, romliTac Cven vxedavT sxe-

uls SeuiaraRebeli TvaliT (xedvis kuTxe). es sxivebi, obieqtivSi gardatexis

Semdeg, mis fokalur sibrtyeSi qmnian sxeulis namdvil, Sebrunebul da

(Zalian) Semcirebul gamosaxulebas 11BA -s. gamosaxuleba imdenad mcirea, rom

gaZnelebulia masze detalebis garCeva, magram vinaidan igi namdvilia,

SesaZlebelia misi gadideba, rac xdeba okularis saSualebiT, romelic isea

moTavsebuli, rom misi wina fokusi Tanxvdeba obieqtivis F ukana fokuss.

maSin, zemoTqmulis Tanaxmad, 11BA gamosaxuleba moxvdeba okularis fokuss

SigniT, masTan Zalian axlos da igi imoqmedebs rogorc lupa anu mogvcems ga-

didebul da warmosaxviT gamosaxulebas. igi sxeulis mimarT Sebrunebulia,

magram es ar qmnis uxerxulobas, vinaidan kepleris miliT xdeba ciuri

sxeulebis damzera.

okularidan gamosul TiTqmis paralelur (odnav ganSlad) sxivebs

Soris kuTxe aris γ da damkvirvebeli am kuTxiT xedavs ciuri sxeulis '2

'2 BA

gamosaxulebas. γ kuTxe gacilebiT metia ϕ -ze, rac gansazRvravs teleskopis

gamadideblobas.

200

dedamiwis sxeulebis dasamzerad gamoiyeneba galileis optikuri mili,

romelic man Seqmna keplerze odnav adre, 1609 wels. galileis refraqtometri

keplerisagan imiT gansxvavdeba, rom okulari warmoadgens gambnev da ara

Semkreb linzas. sxivTa svla galileis milSi mocemulia 59.4 suraTze.

AB sxeulidan gamosuli sxivebi, 1O obieqtivSi gavlis Semdeg xdeba

krebadi. am sxivebs unda moeca sxeulis namdvili Sebrunebuli da Semcirebu-

li gamosaxuleba ab , magram gamosaxulebis mocemamde sxivebi xvdeba 2O oku-

larSi, ris gamoc isini xdeba ganSladi. am ganSladi sxivebis damkvirveblis

TvalSi moxvedrisas igi dainaxavs AB sagnis pirdapir da warmosaxviT gamo-

saxulebas 11BA -s. kuTxe γ , romliTac damkvirvebeli xedavs gamosaxulebas,

gacilebiT metia α kuTxeze, romliTac is sxeuls dainaxavda SeuiaraRebeli

TvaliT, rac uzrunvelyofs sxeulze detalebis kargad garCevas.

galileis or SeerTebul mils, romelic saSualebas gvaZlevs sxeuli

erTdroulad orive TvaliT davaTvalieroT, binokli ewodeba (saTeatro

binokli).

pirveli lefleqtori (sarkiani teleskopi) aago niutonma 1672 wels.

misi gamartivebuli sqema mocemulia 59.5 suraTze.

B sferul sarkes ecema Soreuli

ciuri sxeulidan wamosuli sxivebi. sxe-

ulis siSoris gamo sxivebi parale-

luria, amirom B sarkidan arekvlis Sem-

deg isini gvaZleven sxeulis gamosaxu-

lebas sarkis fokalur sibrtyeSi. es iq-

neba ciuri sxeulis namdvili, Seb-

sur. 59. 4

sur. 59.5

201

runebuli da Semcirebuli gamosaxuleba. imisaTvis rom dakvirveba iyos

mosaxerxebeli, fokaluri sibrtyis maxloblad, mis win, aTavseben S brtyel

sarkes, romelic ucvlis mimarTulebas B sarkidan wamosul sxivebs. sferuli

sarkis mier mocemuli gamosaxulebis damzera xdeba L okulariT, romelic

moqmedebs rogorc lupa.

marTalia, sferuli sarke qromatuli aberaciisgan Tavisufalia, magram

igi xasiaTdeba Zlieri sferuli aberaciiT, rac auaresebs gamosaxulebis

xarisxs. sarkuli teleskopis es nakli gamosworebulia sabWoTa mecnieris

maqsutovis mier 1941 wels agebul sarkian-linzian teleskopSi, romlis obieq-

tivi warmoadgens sarkisa da linzis SeTavsebas. am teleskopis moqmedebis

ZiriTadi principi mdgomareobs sferuli sarkis aberaciis gamosworebaSi

linzis sferuli aberaciis daxmarebiT. amisaTvis saWiro iyo iseTi linza,

romelic qromatuli aberaciisgan iqneboda Tavisufali, sferuli aberacia ki

eqneboda imis toli da sawinaarmdego, rac aqvs Cazneqil sarkes da maSasadame

gaabaTilebda mas. aRmoCnda, rom aseTi Tvisebebis matarebelia garkveuli

geometriuli zomebis mqone Cazneqil-amozneqili linza (meniski).

maqsutovis teleskopis sqema mocemulia 59.6 suraTze. ciuri sxeulidan

wamosuli paraleluri sxivebi M me-

niskSi gardatexis Semdeg ecema B sfe-

rul sarkes. sarkidan arekvlili sxi-

vebi ecema meniskis zedapiris S mover-

cxlil nawils, saidanac arekvlis Sem-

deg gadis sferuli sarkis xvrelSi da

ecema L okulars, romelic moqmedebs

rogorc lupa.

obieqtivis damzadeba teleskop-refraqtorisaTvis warmoadgens did siZneles, romelic

swrafad izrdeba obieqtivis zomis zrdisas. mTavari siZnelea didi zomis erTgvarovani

(mudmivi gardatexis maCvenebliT) linzis damzadeba Sinagani daZabulobis gareSe. ukanas-

knelis moxsna xdeba neli gamowviT, romelic TveobiT grZeldeba.

obieqtivis damzadeba teleskop-refleqtorisaTvis gacilebiT iolia, vinaidan

gacilebiT naklebia moTxovna sarkis dasamzadebeli minis erTgvarovnebisadmi, radgan

masSi sinaTle ar gadis da igi mxolod warmoadgens fuZe Sres, razec daaqvT amrekli fena.

am mizeziT arsebuli teleskopebidan yvelaze didia sarkiani _ misi diametri 1 metria

(milis sigrZe 21 metria). teleskopis garCevisunarianoba ki mniSvnelovanad aris damokidebu-

sur. 59.6

202

li mis diametrze. magaliTad, teleskopi, romlis diametri 10 sm-ia, saSualebas gvaZlevs

aRmovaCinoT mTvareze da marsze arxebi, romelTa sigane Sesabamisad 40 m da 5 km-ia. 5

metriani diametris teleskopis SemTxvevaSi es cifrebi mcirdeba erT da as metramde.

$60 fermas principi

geometriuli optikisadmi miZRvnili Tavis bolos ganvixiloT erTi

metad mniSvnelovani principi, romelic Camoayaliba frangma maTematikosma da

fizikosma fermam (1660).

viciT, ($52) rom erTgvarovan garemoSi sxivi wrfivad vrceldeba. ro-

desac sxivi gadadis erTi garemodan meoreSi, romelTac gansxvavebuli gar-

datexis maCveneblebi aqvT, igi gardatydeba, anu icvlis mimarTulebas. vTqvaT,

sxivi vrceldeba araerTgvarovan garemoSi, romlis gardatexis maCvenebeli

wertilidan wertilamde icvleba. maSin sxivi, ganicdis ra ganuwyvetel

gardatexas wertilidan wertilamde, vrceldeba mrud wirze. magram or

wertils Soris uamravi mrudi wiris gavleba SeiZleba. ismis kiTxva _ mainc

ra gziT vrceldeba sxivi araerTgvarovan garemoSi? am kiTxvas pasuxs scems

fermas principi, romelic Semdegnairad gamoiTqmis:

erTi wertilidan meoreSi sxivi vrceldeba iseTi gziT,

romlis gasasvleladac umciresi droa saWiro.∗

fermas principis maTematikuri gamosaxvisaTvis visargebleT gzis opti-

kuri sigrZis cnebiT. gzis optikuri sigrZe ( )l ewodeba gzis geometriuli sig-

rZis ( )S namravlis garemos gardatexis maCvenebelze ( ) .: nSln =

Tu garemo araerTgvarovania AB gza unda davyoT imdenad mcire SΔ

moankveTebad, rom TiToeulis farglebSi n iyos mudmivi. gamoviTvaloT Ti-

Toeuli monakveTisaTvis optikuri sigrZe da Semdeg SevkriboT. iSΔ monakveTis

optikuri sigrZe iii Snl Δ=Δ , xolo mTeli gzis optikuri sigrZe

∑∑==

Δ=Δ=k

iii

k

ii Snll

11

, sadac k _ monakveTebis ricxvia. zRvarSi (roca monakveTebis

sigrZe usasrulod mcirdeba, xolo maTi ricxvi usasrulod izrdeba) jami

gadava integralSi:

∗ am princips umciresi drois principsac uwodeben.

203

∫∫ ==B

A

B

A

ndSdll .

dS monakveTis gasavlelad saWiro dro aRvniSnoT dt -Ti, xolo sinaT-

lis gavrcelebis siCqare mocemul garemoSi (romlis gardatexis maCvenebe-

lia n ) υ -Ti. maSin υdSdt = da A -dan B wertilamde sinaTlis gavrcelebisa-

Tvis saWiro dro toli iqneba:

∫∫∫ ===B

A

B

A

B

A Cdl

CndSdSt

υ

fermas principis Tanaxmad es dro unda iyos umciresi anu minimaluri,

rac maTematikurad niSnavs, rom am drois variacia∗ (warmoebuli raime SerCeu-

li parametriT) unda iyos nulis toli, e.i.

0==== ∫∫∫ Cdl

CndSdSt

B

A

B

A

δδυ

δδ (60.1)

(60.1) toloba warmoadgens fermas principis maTematikur gamosaxulebas.

sinamdvileSi (60.1) toloba ufro zogadia, vidre Tavdapirveli saxiT

Camoyalibebuli fermas principi, vinaidan igi gamosaxavs ara marto minimumis,

aramed maqsimumis (e.i. zogadad eqstremumis) da stacionarulobis pirobasac.

e.i. or wertils Soris gavrcelebisas sxivi `irCevs~ ara marto im gzas, rome-

lic minimalur dros moiTxovs, aramed imasac, romlis gavlac maqsimalur

dros saWiroebs, an gzebs, romelTa gavlasac erTnairi dro sWirdeba.

fermas principi zogadi principia. misi saSualebiT SeiZleba geometriu-

li optikis ZiriTadi kanonebis (wrfivi gavrcelebis, arekvla-gardatexis)

miReba:∗∗

sinaTlis wrfivi garvelebis kanoni. imis gamo, rom umciresi

(minimaluri) manZili or wertils Soris aris am wertilebis SemaerTebeli

wrfe, amitom fermas principidan gamomdinareobs, rom erTgvarovan garemoSi

sinaTle wrfivad unda gavrceldes.

sinaTlis arekvlis kanoni. vTqvaT, A wertilidan gamosuli sxivi,

MN sarkuli zedapiridan arekvlis Sedegad, xvdeba B wertilSi. SeiZleba

∗ warmodgeba laTinuri sityvisgan da niSnavs cvlilebas. ∗∗ ufro metic, fermas principi saSualebas iZleva gamoviyvanoT Txeli linzis zogadi formula gardatexis kanonis gareSe, avxsnaT miraJebis warmoSoba, dRis xangrZlivobis `gazrda~ da sxv.

204

warmovidginoT uamravi gzebi ( BAOAOB 1, da sxv),, romelTa saSualebiTac A

wertilidan gamosuli sxivi arekvlis Semdeg xvdeba B wertilSi. aqedan

mxolod erTia namdvili. kerZod is, romelic akmayofilebs fermas princips

anu romlis gasavleladac minimaluri droa saWiro.

ganvixiloT erT-erTi gza, mag. OBAO + , Tu sinaTlis siCqares mocemul

garemoSi aRvniSnavT υ -Ti, am gzis gasavlelad saWiro dro υυ

OBAOt += , an

suraTze naCvenebi aRniSvnebis Sesabamisad

( ),,,,, 21111111 hBBhAAxaOBconstaBAxOA ==−====

( )υυ

222

221 xahxh

t−+

++

= (60.2)

(60.2) gviCvenebs, rom A da B wertilebis mocemuli mdebareobis dros,

t -s mniSvneloba damokidebulia x -ze (anu O wertilis mdebareobaze). Tu gan-

xiluli gza namdvilia, fermas principis Sesabamisad, t unda iyos minimaluri

anu unda Sesruldes piroba 0=dxdt

. gawarmoebis martivi operaciis Catarebis

Semdeg gveqneba:

( )011

222

221

=⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

−+

−−

+ xah

xa

xh

xυυ

,

saidanac

1sinsin ii = da 1ii = .

amrigad fermas princi-

pidan gamomdinareobs CvenTvis

cnobili arekvlis kanoni.

sinaTlis gardatexis

kanoni. gardatexis kanoni miiReba zustad iseTive msjelobiT, rac CavatareT

arekvlis kanonis miRebisas.

vTqvaT, A wertilidan gamosuli sxivi, ori garemos gamyof MN zeda-

pirze gardatexis Semdeg, xvdeba B wertilSi. SeiZleba warmovidginoT uamra-

vi gza ( BAOAOBBAO 21 ,, da sxv.), romelTa saSualebiTac A wertilidan gamo-

suli sxivi gardatexis Semdeg xvdeba B wertilSi (sur. 60.2). aqedan mxolod

isaa namdvili, romlis gasavleladac, fermas principis Tanaxmad, minimaluri

droa saWiro. ganvixiloT erTerTi gza, mag. OBAO + . Tu pirvel garemoSi

A B x 1h '

1i 2h 1i a M N 1A 1O O 1B sur. 60.1

205

sinaTlis siCqares aRvniSnavT 1υ -iT, xolo

meoreSi 2υ -Ti, am gzis gasavlelad saWiro dro

21 υυOBAOt += , an, Tu gamoviyenebT suraTze naCveneb

aRniSvnebs,

( )2

222

1

221

υυxahxh

t−+

++

= . (60.3)

aqac sinaTlis gavrcelebis dro damo-

kidebulia O wertilis mdebareobaze anu x -ze. Tu ganxilulia gza namdvi-

lia, fermas principis Sesabamisad, t unda iyos minimaluri anu unda

Sesruldes piroba 0=dxdt

. am pirobis Sesruleba gvaZlevs:

( )0sinsin11

2

2

1

122

2222

11

=−=−+

−=

+ υυυυii

xah

xaxh

x,

saidanac viRebT gardatexis kanons:

1

2

2

1

2

1

sinsin

nn

ii

==υυ

. (60.4)

(60.4)-is Tanaxmad, Tu 12 nn > , maSin 12 υυ < . e.i. fermas principidan gamom-

dinareobs daskvna, rom sinaTlis gavrcelebis siCqare meti gardatexis

maCveneblis mqone garemoSi naklebia, rac emTxveva hiugensis Teorias da

ewinaaRmdegeba niutonisas.

vinaidan nebismieri gza A -dan B -mde, romelic ar mdebareobs dacemis

sibrtyeSi, metia dacemis sibrtyeSi mdebare OBAO + gzaze (es exeba rogorc

arekvlas, aseve gardatexas), fermas principidan gamomdinareobs, rom gza, rom-

lis gavlac saWiroebs minimalur dros,

mdebareobs dacemis sibrtyeSi. anu da-

cemuli da gardatexili sxivebi (iseve ro-

gorc dacemuli da areklili) mdebareobs

erT sibrtyeSi da es dacemis sibtryea.

drois stacionarul mniSvne-

lobas Seesabameba sinaTlis arekvla

brunvis elifsoidis Siga Sedapiridan,

romlis erT-erT fokusSi imyofeba mnaTi sur. 60.3

206

wertili P . am wertilis gamosaxuleba Q miiReba meore fokusSi (sur. 60.3) da

vinaidan elifsoidis Tvisebis Tanaxmad OQPO + manZili O wertilis

nebismieri mdebareobisaTvis erTnairia, maSasadame erTnairi iqneba maTi

gavlisas saWiro droc. arekvlas naklebi simrudis mqone zedapiridan (mag.

elifsoidis mxebi MM sibrtyidan) Seesabameba gavrcelebis drois minimumi,

xolo meti simrudis mqone zedapiridan ( NN _ zedapiri) _ maqsimumi.

Tavi VII

sinaTlis interferencia

$61. interferenciis movlena. iungis cda.

sinaTlis talRis sigrZis Sefaseba.

meqnikuri talRebis ganxilvisas Cven vnaxeT (It, $44), rom koherentuli

talRebis zeddebis dros xdeba maTi urTierTgaZliereba da

Sesusteba. am movlenas interferencia ewodeba.

koherentuli ewodeba talRebs, romelTa sixSire erTnairia,

xolo fazaTa sxvaoba mudmivi; Sesabamis wyaroebs koherentuli wyaroebi

ewodeba.

talRebi erTmaneTs aZliereben anu miiReba interferenciuli suraTis

maqsimumi sivrcis im wertilebSi, romelTaTvisac talRebs Soris svlaTa

sxvaoba talRis sigrZis naxevris luwi ricxvis tolia:

λλ⋅==Δ kk

22 , (61.1)

xolo erTmaneTs asusteben, anu miiReba interferenciu-

li suraTis minimumi sivrcis im wertilebSi, romelTaT-

visac talRebs Soris svlaTa sxvaoba talRis sigrZis

naxevris kenti ricxvis tolia:

( )2

12 λ+=Δ k , (61.2)

sadac kk ;...,3,2,1,0= -s ewodeba interferenciis rigi.

svlaTa sxvaoba ( )Δ ewodeba talRebis mier Sexvedramde gavlili manZi-

lebis sxvaobas: 21 xx −=Δ (sur. 61.1).

2x 2O 1x 1O Δ sur. 61.1

207

interferencia talRuri procesisaTvis damaxasiaTebeli movlenaa, ami-

tom, Tu sinaTle talRuri bunebisaa, koherentuli wyaroebis mier gamosxive-

buli sinaTlis konebis gadafarvisas, unda miiRebodes interferenciuli su-

raTi naTeli da bneli zolebis saxiT. cdiT dadasturda, rom sinaTlis

interferencias marTlac aqvs adgili, ramac, Tavis droze, didi roli iTa-

maSa sinaTlis talRuri Teoriis gamarjvebaSi.

sinaTlis interferencias miaqcia yuradReba, daiwyo misi kvleva da

daadgina garkveuli kanonzomierebebi∗ niutonma, magram man, rogorc sinaTlis

korpuskuluri Teoriis mimdevarma, ver axsna igi. koherentuli talRebi miiRo

da sworad axsna talRuri Teoriis mimdevarma, ingliselma mecnierma Tomas

iungma.∗∗ ganvixiloT iungis cdis sqema.

sinaTlisadmi gaumWvir I firfi-

tas, romelsac datanebuli aqvs mcire

S xvreli, ecema paralelur sxivTa ko-

na anu brtyeli talRa (sur.61.2). hiugen-

sis principis Tanaxmad, xvreli TviTon

gadaiqceva sinaTlis wyarod da gamoas-

xivebs naxevarsferul talRebs. Semdeg

sinaTlis es talRa ecema II firfitas,

romelsac datanebuli aqvs ori _ 1S da

2S xvreli. hiugensis principis Tanaxmad, es xvrelebic gadaiqceva meoradi

naxevarsferuli sinaTlis talRebis centrebad. sinaTlis es talRebi iqnebian

koherentulni, vinaidan warmodgebian erTidaigive wyarosagan. amitom, maT

gadafarvis areSi miiReba interferenciuli suraTi naTeli da bneli zolebis

saxiT.∗∗∗ am suraTis dakvirvebisaTvis, 1S da 2S wyaroebidan garkveul L man-

Zilze aTavseben MN ekrans.

ganvixiloT iungis cda ufro dawvrilebiT (sur. 61.3): 1S da 2S wyaroebs

Soris manZili aRvniSnoT d -iT, xolo manZili wyaroebidan MN ekranamde

L -iT. dL >> . dakvirvebas vawarmoebT C wertilSi, romelic ekranis centri-

∗ kerZod is, rom erTi da igive nomris mqone `niutonis rgolebis~ radiusebi izrdeba speqtris iisferi bolodan wiTlisaken _ wiTel rgolebs aqvT maqsimaluri radiusi. ∗∗ sagulisxmoa, rom termini `interferencia~ iungis mieraa SemoRebuli. ∗∗∗ niutonis Teoriis Tanaxmad ekranze unda miRebuliyo ori naTeli zoli.

sur. 61.2

208

dan daSorebulia l manZiliT. Ll << .∗ sigrZeebis arsebuli Tanafardobisas

SeiZleba miaxloebiT daiweros:

Ll

d=

Δ,

saidanac

dLl⋅=Δ . (61.3)

(61.1)-is Tanaxmad C -Si miiReba naTeli zoli,

Tu

λkdLl

=⋅=Δ (61.3)

da (61.2)-is Tanaxmad miiReba bneli zoli Tu

( )2

12 λ+=⋅=Δ kd

Ll

. (61.4)

(61.3)-dan gavigebT im kl manZilebs, romliTac daSorebulia centridan naTeli

zolebi:

Ld

klkλ

= . (61.5)

manZili or mezobel naTel zols Soris (igi iseTivea, rogorc manZili or

mezobel bnel zols Soris):

Ld

l λ=Δ . (61.6)

am manZils ewodeba interferenciuli zolis sigane .

naTeli zolebis mdebareoba SeiZleba ganisazRvros α kuTxiTac. sura-

Tidan Cans, rom αα LtgLl ≈= (vinaidan α mcirea), saidanac Ll

=α . Tu am mniS-

vnelobas SevitanT (61.3)-Si, gavigebT im kα kuTxeebis mniSvnelobebs, romli-

Tac Cans A wertilidan naTeli zolebi:

dkkλα = .

kuTxuri manZili or mezobel naTel zols Soris

dλα =Δ . (61.7)

∗ l ar SeiZleba didi iyos, vinaidan centridan daSorebisas interferenciuli suraTis intensivoba swrafad klebulobs.

N C 2S l d α A L 1S Δ M sur. 61.3

209

(61.7) gviCvenebs: imisaTvis rom αΔ iyos sakmaod didi da Cven SevZloT ori

mezobeli naTeli zolis erTmaneTisagan garCeva, d unda iyos mcire. kerZod,

cdebi gviCvenebs, rom 310−=Δα -s (es Seesabameba zolebs Soris 1mm manZils

roca 1=L m-s) maSin, roca 5,0=d mm-s. am monacemebis safuZvelze SeiZleba

SevafasoT sinaTlis talRis sigrZis rigi:∗

5,010 3 ⋅=⋅Δ= −dαλ mm 43 10510 −− ⋅⋅= m 7105 −⋅= m. (61.8)

Cvens mier ganxiluli interferenciuli suraTi, romelic Sedgeba naTe-

li da bneli zolebis erTobliobisagan, miiReba mxolod monoqromatuli∗∗

anu erTi feris mqone sxivebis interferenciisas. im SemTxvevaSi, Tu es sxivebi

wiTeli ferisaa, naTeli zolebic iqneba wiTeli, Tu mwvane ferisaa, naTeli

zolebi iqneba mwvane da a.S. sul sxvanairadaa saqme rodesac interferireben

TeTri sinaTlis sxivebi, mag. mzis sxivebi. am SemTxvevaSi bneli zolebi saer-

Tod ar miiReba da interferenciul suraTs aqvs Semdegi saxe: ekranis cen-

trSi (O wertilSi) aris TeTri zoli, xolo mis irgvliv sxvadasxva feris

zolebi, romlebic TandaTan, uwyvetad gadadian erTimeoreSi. aseTi suraTis

axsna advilia: jer kidev niutonis mier Catarebuli cdebis Sedegad sinaTlis

dispersiaSi dadginda, rom TeTri sinaTle rTuli sinaTlea _ igi Sedgeba

Svidi sxvadasxva ferisagan (wiTeli, narinjisferi, yviTeli, mwvane, cisferi,

lurji, iisferi). sinaTlis eleqtromagnituri Teoriis Tanaxmad, sinaTlis

fers gansazRvravs talRis sigrZe anu rxevis sixSire.∗∗∗ vTqvaT, ekranis moce-

muli wertilisaTvis ganxorcielda maqsimumis piroba yviTeli ferisaTvis ( 1λ

sigrZis talRisaTvis). maSin mezobel wertilebSi am ferisaTvis Sesruldeba

minimumis piroba (vinaidan Seicvala manZili da unda Seicvalos talRis sig-

rZec). magram am mezobel wertilebSi maqsimumis piroba Sesruldeba narinjis-

feri sxivebisaTvis, romelTac sxva, 2λ talRis sigrZe aqvs da a.S. ekranis

centrSi miiReba TeTri zoli imitom, rom aq svlaTa sxvaobs nulis tolia da

∗ cxadia, talRis sigrZis gasagebad SeiZleba gamoviyenoT (61.6) formulac, amisaTvis saWiro

iqneba mxolod lΔ -is gazomva, vinaidan d da L winaswar cnobilia. ∗∗ warmodgeba berZnuli sityvebisagan monos _ erTi da choma _ feri. ∗∗∗ sinaTlis obieqturi maxasiaTebelia talRis sigrZe, feri ki warmoadgens adamianis subieqturi SegrZnebis Sedegs, misi Tvalis unars sxvadasxva talRis sigrZis sxivebi aRiqvas sxvadasxvanairad. arian cxovelebi, romlebic ferebs ver ansxvaveben (e.i. mxedvelobis nervis daboloebebi ar Seicaven kolbebs) da yvelafers aRiqvamen SavTeTr ferebSi.

210

maqsimumis piroba sruldeba yvela ferisaTvis (nuli luwi ricxvia). Svidive

feris naTeli zolebis zeddebis Sedegad ki miiReba TeTri feri.

cdam aCvena, rom αΔ sxvadasxva feris sxivebisaTvis aris sxvadasxva. igi

yvelaze metia wiTeli ferisaTvis da yvelaze naklebi _ iisferisaTvis. aqedan,

(61.8) formulis safuZvelze vaskvniT, rom yvelaze grZeli talRa aqvs wiTel

fers, yvelaze mokle _ iisfers. dadagenilia, rom sxvadasxva feris sinaTlis

talRis sigrZeebi icvleba Semdeg farglebSi

(1 mikrometri (mkm)= 610− m):

wiTeli ( ) 7103,67,7 −⋅− m ( )63,077,0 −= mkm

narinjisferi ( ) 7100,63,6 −⋅− m ( )60,063,0 −= mkm

yviTeli ( ) 7107,50,6 −⋅− m ( )57,060,0 −= mkm

mwvane ( ) 7100,57,5 −⋅− m ( )50,057,0 −= mkm

cisferi ( ) 7105,40,5 −⋅− m ( )45,050,0 −= mkm

lurji ( ) 7103,45,4 −⋅− m ( )43,045,0 −= mkm

iisferi ( ) 7100,43,4 −⋅− m ( )40,043,0 −= mkm. eleqtromagnitur talRebs, romelTa sigrZe 0,77 mkm-ze metia da 0,40 mkm-

ze naklebia, Cveni Tvali ver aRiqvams. maT Sesabamisad infrawiTeli da

ultraiisferi sxivebi ewodeba.

$62. koherentul talRaTa miRebis meTodebi

yoveli sinaTlis wyaro miliardobiT atomebisagan Sedgeba. sinaTle,

romelsac Cveni Tvali aRiqvams, am atomebis erToblivi moqmedebis Sedegia. si-

naTlis ori damoukidebeli wyaro rom koherentuli iyos, saWiroa, am wyaro-

ebis Semadgeneli atomebis SeTanxmebuli moqmedeba, anu isini ise unda asxiveb-

des sinaTles (talRebs), rom jamur talRebs Soris fazaTa sxvaoba iyos mud-

mivi. es ki SeuZlebelia, vinaidan calkeuli wyaros atomuri gamosxivebis

fazebi swrafad da qaosurad icvleba, ris ganoc jamuri gamosxivebis

(talRis) fazis cvlilebac swrafi da qaosuria. amrigad, sinaTlis ori sxva-

dasxva wyaros mier gamosxivebuli talRebi yovelTvis arakoherentulia.∗

sinaTlis koherentuli talRebis misaRebad iqcevian Semdegnairad: iRe-

ben sinaTlis erT wyaros. misgan gamosxivebul sinaTles (sinaTlis konas) yo-

fen or nawilad, da sxvadasxva manZilebis gavlis Semdeg axvedreben erTma- ∗ es ar exeba lazerul gamosxivebas.

211

neTs. sinaTlis gayofili nawilebi, cxadia, koherentulni iqnebian, vinaidan

isini warmosdgebian erTi da imave wyarodan da maSasadame, dakavSirebulni

arian erTmaneTTan,∗ maTi fazebi erTdroulad (SeTanxmebulad) icvleba, ris

gamoc fazaTa sxvaoba mudmivi iqneba.

sinaTlis konis gayofa or nawilad SeiZleba ganxorcieldes sxvadasxva

saSualebebiT _ xvrelebis, sarkeebis an prizmebis gamoyenebiT.

xvrelebis saSualebiT koherentuli talRebi martivad da maxvilgoniv-

rulad miiRo iungma (iungis meTodi); igi Cven ganvixileT wina paragrafSi

(sur. 61.2). gavecnoT sxva meTodebs.

frenelis sarkeebi . ori brtyeli sarke ganlagebulia erTmaneTisadmi

TiTqmis 180° kuTxiT (sur. 62.1). S wyarodan gamosuli sxivebi airekvleba sar-

keebidan da ecema MN ekrans, romelic daculia S wyaros sxivebis pirdapiri

moxvedrisagan K garsacmiT.

brtyeli sarkidan sinaTlis arek-

vlis kanonis Tanaxmad, pirveli sarkidan

areklili sxivebi TiTqos gamodian 1S

warmoasaxviTi wyarodan, romelic simetri-

uladaa ganlagebuli S wyaros mimarT da

maSasadame, warmoadgens mis gamosaxulebas

am sarkeSi.

analogiurad, meore sarkidan arek-

vlili sxivebi TiTqos gamodian 2S war-

mosaxviTi wyarodan, romelic warmoadgens S wyaros gamosaxulebas am

sarkeSi. 1S da 2S warmosaxviTi wyaroebi cxadia, koherentulni arian, vinaidan

warmosdgebian erTi da igive S wyarodan, amitom maTi gadafarvis areSi

miiReba interferenciuli suraTi, romlis dakvirveba xdeba MN ekranze.

frenelis biprizma. mcire gardamtexi kuTxis mqone ori samkuTxa

prizma erTmaneTze miwebebulia fuZeebiT (sur. 62.2). S wyarodan gamosuli sxi-

vebi gardatydebian am prizmebSi, daixrebian fuZeebisken da gvaZleven ganSlad

konebs. miiReba iseTi suraTi, TiTqos es konebi gamodian 1S da 2S warmosax-

∗ gavixsenoT, rom laTinurad koherentuloba niSnavs kavSirSi yofnas, erTmaneTTan dakavSi-rebuls.

sur. 62.1 M

N

212

viTi wyaroebidan, romlebic araerTxel Tqmuli mizezis gamo, koherentulebi

arian, amitom maTi gadafarvis areSi miiReba interferenciuli suraTi,

romlis dakvirveba xdeba MN ekranze.

loidis sarke. am meTodSi erT

konas qmnis uSualod sinaTlis S wyaro-

dan wamosuli sxivebi, xolo meores _

brtyeli sarkidan (loidis sarkidan) di-

di kuTxiT arekvlili sxivebi, romlebic

TiTqos S wyaros warmosaxviTi 'S gamo-

saxulebidan gamodian (sur. 62.3). amrigad, wina SemTxvevebisgan gansxvavebiT, am

meTodSi koherentul wyvils qmnis sinaTlis pirveladi wyaro da misi

warmosaxviTi gamosaxuleba. garda amisa, Tu wina SemTxvevebSi ekranis

centrisaTvis koherentul talRebs Soris svlaTa sxvaoba iyo nulis toli,

e.i. sruldeboda maqsimumis piroba da miiReboda naTeli zoli, am meTodSi

interferenciuli suraTis centraluri zoli bnelia. aixsneba es imiT, rom

roca sinaTlis talRa irekleba optikurad

metad mkvrivi garemodan (sarkidan), adgili

aqvs naxevartalRis dakargvas, ris gamoc

svlaTa sxvaoba xdeba 2λ-is toli, anu

sruldeba minimumis piroba.

$63. gamWvirvale firfitidan arekvlili sxivebis interferencia.

Tanabari daxrisa da Tanabari sisqis interferenciuli zolebi

interferenciis metad gavrcelebul saxes, romelsac amave dros didi

praqtikuli gamoyeneba aqvs, warmoadgens interferencia sxivebisa, romlebic

miiReba gamWvirvale firfitis zedapirebidan arekvlisas. aq ganixileba ori

SemTxveva:

I. interferencia paraleluri sxivebisa, romlebic miiReba brtyelpara-

leluri firfitis zedapirebidan arekvlisas da

II. interferencia araparaleluri sxivebisa, romlebic miiReba cvladi

sisqis firfitis zedapirebidan arekvlisas.

213

pirvel SemTxvevaSi miiReba e.w. Tanabari daxris interferenciuli

zolebi, xolo meore SemTxvevaSi _ Tanabari sisqis interferenciuli zolebi.

ganvixiloT es sakiTxi dawvrilebiT.

Tanabari daxris inter-

ferenciuli zolebi . P si-

brtyis S manaTobeli wertilidan

gamosuli sxivebi L linzaSi

gavlis Semdeg, 1aa paraleluri

konis saxiT ecema brtyel-

paraleluri firfitis 1AA

zedapirs. dacemuli sinaTlis na-

wili airekleba da gvaZlevs ba '

sxivebs; nawili ki gardatydeba,

ecema firfitis qveda, 1BB zeda-

pirs, airekleba, Semdeg xelmeo-

red gardatydeba 1AA zedapirze da gvaZlevs 1'1ba sxivebs (sur. 63.1). arekvlili

sxivebi urTierTparaleluria, vinaidan firfita aris brtyelparaleluri.

maTi Sekreba xdeba 2L linziT 1P sibrtyis 1S wertilSi da imis da mixedviT,

Tu rogoria svlaTa sxvaoba am sxivebs Soris, 1S wertilSi miiReba bneli an

naTeli laqa. gamoviTvaloT es svlaTa sxvaoba. SevadginoT naxazi (sur. 63.2).

CavTvaloT, rom haeris gardatexis maCvenebeli 1-is tolia, xolo minis _ n -is.

firfitis sisqe aRvniSnoT d -Ti. maSin 'a da '1a sxivebs Soris optikuri svla-

Ta sxvaoba

( )2

'' λ−−+=Δ ODnCOOO 63.1)

aq 2λ gaCnda imitom, rom sinaTlis talRis

arekvlisas optikurad metad mkvrivi garemodan,

xdeba naxevartalRis dakargva. naxazidan Cans, rom

12

'' sin.cos

iOCODi

dCOOO === ;

1L 'a 2L a 1a '

1a b 1b

A 1A B 1B sur. 63.1

11 PSS P

a 'a '1a

11 ii D

1i O C d 2i 2i

'O

sur. 63.2

214

magram 22itgdOC

= ,

amitom 21sin2 itgidOD = . gardatexis kanonis Tanaxmad nii=

2

1

sinsin

, saidanac

21 sinsin ni = .

maSin 2

22

22 cossin2sin2

ii

dnitgidnOD == ;

miRebuli mniSvnelobebi SevitanoT (63.1)-Si:

( )2

cos22

sin1cos2

2sin

cos2

cos2

222

22

2

22

λλλ−=−−=−−=Δ idni

idni

idn

idn

.

magram

122

21

2

22

2 sin1sin1sin1cos innn

iii −=−=−=

da sabolood gveqneba:

2sin2 1

22 λ−−=Δ ind . (63.2)

Tu sruldeba maqsimumis piroba anu

λλ kind =−−=Δ2

sin2 122 , (63.3)

maSin 1S wertilSi miviRebT naTel laqas.

davuSvaT axla, rom S wertili ki ar aris sinaTlis wyaro, aramed

mTeli P sibrtye aris manaTobeli (`ganfenili~ sinaTlis wyaro). interferen-

cias adgili rom ar hqondes, e.i. gamWvirvale firfitis nacvlad rom iyos

sarkulad amrekli zedapiri, maSin 2L linzis fokalur sibrtyeSi miviRebT P

sibrtyis gamosaxulebas 1P -s. magram, imis gamo, rom adgili aqvs sxivebis

interferencias, 1P sibrtyeSi naTel laqas

mogvcems P sibrtyis ara yvela wertili, aramed

mxolod isini, romlebic akmayofileben (63.3)

pirobas. kerZod, is wertilebi, romelTa svlaTa

sxvaoba erTnairia da gvaZleven erTidaigive

0k rigis interferenciul maqsimums, moiZebneba

pirobidan:

constkind ==−− λλ01

22

2sin2 .

N 'a 'b

P a b

11 ii

A 'A

sur. 63.3

P 1P

215

an, vinaidan nd , da λ mudmivia, tolobis marcxena mxare mudmivi iqneba roca

consti =1 . (63.4)

erToblioba im wertilebisa, romlebidanac gamosuli sxivebi akmayofilebs

(63.4) pirobas, miiReba P sibrtyis gadakveTiT im konusTan, romlis RerZs war-

moadgens 1AA zedapiris normali, xolo gaSlis kuTxe 12i -is tolia (sur. 63.3).

wertilTa es erToblioba P sibrtyeze gvaZlevs 1aa rkals. mas 'P sibrtyeze

Seesabameba 'bb naTeli rkali.∗ intereferenciis sxva rigis SemTxvevaSi

miiReba sxva naTeli zoli. naTel zolebs Soris moTavsebulia bneli zolebi.

amrigad, miRebuli interferenciuli suraTi warmoadgens naTeli da

bneli zolebis (rkalebis) erTobliobas. TiToeuli naTeli zoli Seesabameba

dacemis 1i kuTxis mudmiv mniSvnelobas, e.i. miiReba im sxivebis interferenciis

Sedegad, romlebic zedapiris normalis mimarT Tanabrad arian daxrilni. aqe-

dan warmodgeba miRebuli interferenciuli suraTis dasaxeleba _ Tanabari

daxris interferenciuli zolebi.

TiToeuli mnaTi wertili 'P sibrtyeSi miiReba paraleluri sxivebis in-

terferenciis Sedgad. e.i. mkafio interferenciuli suraTis dasamzerad

'P sibrtye unda movaTavsoT 1L linzis fokalur sibrtyeSi anu, moviqceT ise,

rogorc vakeTebT xolme usasrulod daSorebuli sxeulebis gamosaxulebis

misaRebad. amitom amboben, rom Tanabari daxris interferenciuli zolebi

lokalizebulia usasrulobaSi. linza SeiZleba SevcvaloT Tvalis bro-

liT, ekrani _ baduraTi. maSasadame, Tanabari daxris zolebis damzera SeiZ-

leba uSualodac, Tvalis saSualebiT. oRond am SemTxvevaSi Tvali akomodi-

rebuli unda iyos usasrulobaze.

Cven ganvixileT SemTxveva, rodesac firfitas ecemoda monoqromatuli

sinaTle. davuSvaT axla, rom mas ecema TeTri sinaTle, romelic Seicavs

...,, 321 λλλ talRis sigrZeebs. maSin, dacemis mocemuli kuTxis ( )consti =1 Sem-

TxvevaSi, maqsimumis pirobas daakmayofilebs yvela is talRebi, romelTaT-

visac sruldeba piroba:

( ) ( ) ...21 321 =+=+==Δ λλλ kkk (63.5)

∗ Tu firfitas sinaTle ecema yvela mxridan (e.w. gabneuli sxivebi), cxadia, rom is wertilebi,

romlebic akmayofileben consti =1 pirobas, mdebareoben wrewirze. Sesabamisi

interferenciuli suraTic iqneba naTeli rgoli.

216

magram vinaidan sxvasxva talRis sigrZes sxvadasxva feri Seesabameba, in-

terferenciuli suraTi iqneba Seferadebuli: bneli zolebi saerTod aRar iq-

neba, iqneba mxolod Tanmimdevrulad ganlagebuli sxvadasxva feris naTeli

zolebi.

(63.2) formula gviCvenebs, rom rac metia d , miT metia Δ , (63.3)-is Tanax-

mad, rac metia Δ , miT metia k , xolo (63.5)-is Tanaxmad, rac metia k , miT nak-

lebia gansxvaveba ...,, 321 λλλ talRis sigrZeebs Soris. es imas niSnavs, rom

sqeli firfitis SemTxvevaSi interferenciuli zolebi ganlagebuli iqneba

mWidrod, xolo Tu firfita sakmaod sqelia, moxdeba zolebis gadafarva.

amrigad, interferenciuli suraTis misaRebad saWiroa, rom firfita iyos rac

SeiZleba Txeli.

interferenciuli zolebi warmoadgenen wesieri wrewiris nawilebs

mxolod im SemTxvevaSi, Tu firfita mkacrad brtyelparaleluria. brtyel-

paralelurobidan odnavi gadaxrac ki iwvevs zolebis damaxinjebas. amitom,

interferenciuli meTodiT amowmeben firfitis brtyelparalelurobas da

gadaxra SeimCneva maSinac ki, roca igi ar aRemateba 0,00001 mm-s.

Tanabari sisqis interferenciuli zolebi: ganvixiloT axla Sem-

Txveva, roca firfita, romelsac ecema sinaTle, brtyelparaleluri ar aris,

amasTan, misi d sisqe wertilidan wertilamde umniSvnelod icvleba anu Θ

kuTxe 1AA 1BB zedapirebs Soris aris mcire (sur. 63.4).

vTqvaT, sinaTlis wyaro imdenadaa

daSorebuli, rom misgan wamosuli sxivebi

SeiZleba CaiTvalos paralelurad. gan-

vixiloT pirveli sxivi (1), romelic daeca

1AA zedapirs O wertilSi, gardatyda,

airekla 1BB zedapiris D wertilidan,

gardatyda C wertilSi da mogvca '1 sxivi.

cxadia, dacemul sxivTa konaSi moiZebneba

iseTi sxivi (2), romelic firfitas daecema

imave C wertilSi, airekleba da mogvcems

'2 sxivs. es sxivebi, L linzaSi gavlis

2 L 1 '2

1i '1

A O C 'A Θ

B D 'B

sur. 63.4

1P'C

217

Semdeg, gadaikveTebian 'P sibrtyis 'C wertilSi, romelic C wertilis

gamosaxulebas warmoadgens. '1 da '2 sxivebi, cxadia, koherentulni arian da

interferireben, amitom, imis mixedviT, Tu rogoria maT Soris svlaTa sxvaoba,

'C wertilSi miiReba naTeli an bneli laqa. radgan sxivebi CavTvaleT para-

lelurad da firfitis sisqe umniSvnelod icvleba, svlaTa sxvaobisaTvis

SeiZleba visargebloT (63.3) formuliT, romelic Cven miviReT brtyelparale-

luri firfitisaTvis. aqac (iseve, rogorc wina SemTxvevaSi), is wertilebi,

romelTagan gamosuli sxivebis svlaTa sxvaoba erTnairia da gvaZleven

erTidaigive 0k rigis interferenciul maqsimums, moiZebneba pirobidan:

constkind ==−− λλ01

22

2sin2 .

an, vinaidan λ,n da 1i mudmivia, tolobis marcxena mxare mudmivi iqneba roca

constd = .

es imas niSnavs, rom Tu 1AA zedapiris C wertili 'P sibrtyis 'C wer-

tilSi gvaZlevs naTel laqas, maSin naTel laqas mogvcems yvela is wertili,

romelTaTvisac d iseTivea, rogorc C wertilisaTvis. sxvanairad, maqsimumis

pirobas daakmayofilebs 1AA zedapiris is wertilebi, romlebic mdebareoben

Tanabar sisqeze. aqedan warmodgeba interferenciuli suraTis dasaxeleba _

Tanabari sisqis interferenciuli zolebi.

mkafio interferenciuli suraTis misaRebad L linza unda davafokusi-

roT 1AA zedapirze. amitom amboben, rom Tanabari sisqis interferenciuli

zolebi lokalizebulia firfitis zedapirze. linza SeiZleba

SevcvaloT Tvalis broliT, xolo ekrani ( 'P sibrtye) _ Tvalis baduraTi. e.i.

Tanabari sisqis interferenciuli suraTis dakvirveba SeiZleba uSualod

TvaliTac, oRond Tvali akomodirebuli unda iyos firfitis zedapirze.

Tanabari sisqis interferenciuli zolebis forma damokidebulia firfitis

formaze. magaliTad, Tu cvladi sisqis firfitas

aqvs solisebri forma, interferenciuli zolebi

cxadia, iqneba wrfivi zolebi, romlebic solis

wibos paraleluria (sur. 63.5). aqac, Tu firfitas

ecema TeTri sinaTle, interferenciuli zolebi

iqneba sxvadasxva feris, e.i. damzerisas firfitis sur. 63.5

218

zedapiri mogveCveneba Seferadebuli. cxadia, damzeris kuTxis SecvliT inter-

ferenciis piroba Seicvleba da interferenciuli zolebi ferebs Seicvlian.

aseT `ferTa TamaSs~ firfitis zedapirze Cven vakvirdebiT yoveldRiur cxov-

rebaSi, mag. wylis zedapirze, an asfaltze ganTxmuli navTobis Txeli fenis

damzerisas, sapnis buStis damzerisas da sxv.

Tanabari sisqis interferenciuli zolebis saintereso saxes warmoad-

gens wriuli rgolebi, romelsac pirvelad daakvirda niutoni da niutonis

rgolebi ewodeba. niutonis rgolebze dakvirveba xdeba (63.6) suraTze

moyvanili sqemis mixedviT: didi R simrudis radiusis mqone brtyel-

amozneqili linza moTavsebulia brtyelparaleluri firfitis AB zedapirze

da exeba mas O wertilSi (sur. 63.6 a). linzasa da firfitis Soris war-

moiqmneba cvladi sisqis haeris wriuli fena (cvladi sisqis wriuli

`firfita~).

davuSvaT, aseT optikur sistemas daeca

brtyeli talRa, anu paralelur sxivTa kona.

cxadia, rom TiToeuli sxivi mogvecems or arek-

lil sxivs, romelTagan erTi miiReba linzis

amozneqili zedapiridan mina-haeris sazRvarze

arekvlis Sedegad, xolo meore haeri-minis sa-

zRvarze arekvlis Sedegad. es sxivebi koheren-

tulni arian, amitom interferireben da vinaidan

linzasa da firfitas Soris warmoqmnil haeris

fenas aqvs wriuli forma, interferenciuli su-

raTic, iqneba bneli da naTeli wriuli rgole-

bis erToblioba (sur. 63.5 b).

interferenciuli rgolebis radiusebs advilad gamoviTvliT. marTlac

63.7 suraTidan Cans, rom ( )222 dRrrk −−= , sadac kr _ k rigis interferenciuli

rgolis radiusia, xolo d _ haeris fenis sisqe. Tu d -s simciris gamo 2d -s

ugulebelvyofT, gveqneba

dRrk 22 = (63.6)

b b L a

b

219

sxivebs Soris svlaTa sxvaoba 2

2 λ+=Δ d ∗. naTeli

k rigis rgolisaTvis 2

22

2 λλ kd =+ , saidanac

( )2

122 λ−= kd . SevitanoT (63.6)-Si:

( ) λλ RkRkrk ⎟⎠⎞

⎜⎝⎛ −=⋅−=

21

212 ;

aseve martivad miiReba, rom k rigis bneli rgolis

radiusi

λkRrk = .

miRebuli formulebi gviCvenebs, rom Tu gavzomavT interferenciuli rgolis

radiuss da gvecodineba linzis simrudis radiusi ( )R , gamoviTvliT sinaTlis

talRis sigrZes ( )λ da piriqiT, cnobili talRis sigrZis mixedviT SeiZleba

ganvsazRvroT linzis simrudis radiusi.

dabolos, SevniSnoT, rom ganxilul SemTxvevaSi (roca interferireben

areklili da ara optikur sistemaSi gasuli sxivebi) interferenciuli sura-

Tis centrSi miiReba bneli laqa, vinaidan aq 02 =d da 2λ

=Δ , rac minimumis

pirobaa.

$64. interferenciis praqtikuli gamoyeneba

sinaTlis interferencia sinaTlis talRuri bunebis Sedegia, amitom

cxadia, rom interferenciuli suraTis raodenobrivi mxare _ vTqvaT interfe-

renciul zolebs Soris manZili, an interferenciuli rgolebis radiusi,

damokidebuli unda iyos sinaTlis talRis sigrZeze. marTlac, rogorc wina

paragrafSi vnaxeT, zemoT CamoTvlili parametrebis gazomva saSualebas

gvaZlevs ganvsazRvroT sinaTlis talRis sigrZe. interferenciuli suraTi

saSualebas gvaZlevs ganvsazRvroT agreTve gamWvirvale firfitis sisqe,

linzis simrudis radiusi, didi sizustiT SevamowmoT firfitis brtyel- ∗ am SemTxvevaSi geometriul svlaTa sxvaobas _ d2 -s,

2λ

ki ar akldeba, aramed emateba ,

vinaidan optikurad metad mkvrivi garemo, romlidanac xdeba sxivis arekvla, esazRvreba haeris fenis qveda zedapirs.

sur. 63.7

kr

220

paraleluroba, zedapiris damuSavebis xarisxi da sxv. CamoTvlili sakiTxebi

Seadgens interferenciis praqtikuli gamoyenebis pirvel jgufs.

interferenciis praqtikuli gamoyenebis meore, metad mniSvnelovani

sakiTxia e.w. optikis gasxivosneba . saqme isaa, rom rodesac sinaTle ecema

linzas, nawili gadis masSi, nawili ki airekleba. dadgenilia, rom sxivebis

marTobulad dacemis SemTxvevaSic ki airekleba dacemuli energiis 5-8%.

vinaidan Tanamedrove optikuri xelsawyoebi Seicaven linzaTa did raoden-

obas, arekvlili sinaTlis wili sakmaod didia _ xSirad xelsawyos ekranamde

(fotofiramde) aRwevs masSi Sesuli sinaTlis mxolod 10-20%. es, garda imisa,

rom amcirebs gamosaxulebis sikaSkaSes, dakavSirebulia kidev erT usiamov-

nebasTan, kerZod, samxedro saqmeSi: arekvlili sxivis (aTinaTis) saSualebiT

mowinaaRmdeges SeuZlia daadginos SeniRbuli obieqtis adgilmdebareoba.

aRniSnuli usiamovnebebis Tavidan asacileblad saWiroa sinaTlis arek-

vlili nawilis maqsimalurad Semcireba. amis Sedegad gamosaxuleba xdeba uf-

ro naTeli, `gasxivosnebuli~ da aqedan warmodgeba termini `gasxivosnebuli

optika~.

optikis gasxivosneba dafuZnebulia interferenciis movlenaze: linzis

(firfitis, prizmis) wina zedapirs faraven sinaTlisaTvis gamWvirvale Txeli

feniT, ris Sedegadac Cndeba ori arekvlili sxivi. erTi miiReba haerisa da am

fenis gamyofi zedapiridan arekvlisas, meore _ fenis da linzis gamyofi

zedapiridan. (sur. 64.1). arekvlili sxivebi, cxadia, koherentulni arian da ami-

tom interferireben. fenis sisqes da masalas ise arCeven, rom Sesruldes

minimumis piroba, anu arekvlilma sxivebma erTmaneTi Caaqron. am dros qreba

aTinaTic da xelsawyoSi gasuli sinaTlis intensivobac izrdeba, vinaidan in-

terferenciis dros sruldeba energiis mudmivobis kanoni _ is rac akldeba

interferenciul minimumebs, emateba interferenciul maqsimumebs.

fenis d sisqis gamoTvla advilia: amisaTvis

davweroT minimumis piroba: optikuri svlaTa

sxvaoba 2

2 λ=dn , saidanac

nd

4λ

= .

gamoTvlebi gviCvenebs, rom areklili

sxivebi yvelaze srulad maSin aqroben erTmaneTs,

rodesac sruldeba piroba:

2 1 S sur. 64.1

221

lf nn = ,

sadac fn da ln Sesabamisad fenisa da linzis gardatexis maCveneblebia.

interferenciis prqatikuli gamoyenebis mesame jgufi dakavSirebulia

preciziul anu metad zust gazomvebTan. xelsawyoebs, romlebic am mizniT

gamoiyeneba, interferometrebi ewodeba.

ganvixiloT maikelsonis interferomet-

ri. misi sqema mocemulia 64.2 suraTze: sinaTlis S

monoqromatuli wyarodan sxivi (sxivTa kona) 45°-

ian kuTxiT ecema brtyelparalelur gamWvirvale

P firfitas, romlis erTi gverdi dafarulia

naxevradgamWvirvale vercxlis Txeli feniT. am

fenaze xdeba sxivis or nawilad gayofa: nawili

airekleba da gvaZlevs pirvel sxivs (1), xolo

nawili gadis masSi da gvaZlevs meore sxivs (2). 1

sxivi ecema marTobulad dayenebul 1M sarkes,

brundeba ukan, gaivlis firfitaSi∗ da xvdeba T WogrSi (an ekranze). 2 sxivi

airekleba 2M sarkidan, brundeba ukan, Semdeg airekleba vercxlis Txeli

fenidan da isic xvdeba WogrSi (samzer milSi). pirveli sxivi orjer gadis P

firfitis sisqes (win da ukan), meore ki arcerTxel. amitom, amis Sedegad

warmoqmnili svlaTa sxvaobis kompensaciisaTvis meore sxivis gzaze aTavseben

zustad imave zomis da iseTive masalisagan damzadebul 'P firfitas (cxadia,

movercxlili gverdis gareSe). maSin, sxivebs Soris svlaTa sxvaoba

damokidebuli iqneba, mxolod 1M da 2M sarkeebis mdebareobaze. sxivebs

Soris optikuri sxvlaTa sxvaoba ( )212 lln −=Δ , sadac n _ haeris gardatexis

absoluturi maCvenebelia, xolo 1l da 2l _ manZili 1M da 2M sarkeebamde. Tu

1l = 2l , sruldeba maqsimumis piroba da WogrSi daimzireba naTeli laqa. erTerT

sarkis 4λ-iT gadaadgilebisas svlaTa sxvaoba Seicvleba

2λ-iT da WogrSi

daimzireba bneli laqa. cxadia, rom Tu sarkes vamoZravebT gasazomi obieqtis

∗ nawili cxadia, airekleba, romelic Cven ar gvainteresebs.

222

gaswvriv, interferenciuli suraTis cvlilebis mixedviT, didi sizustiT,

kerZod 4λ-is sizustiT, gavigebT am obieqtis zomas.

xelsawyos sizustis gasazrdelad 1M sarkes xrian mcire α kuTxiT. am SemTxvevaSi

interferenciul suraTs eqneba ara erTgvarovani foni (bneli an naTeli), aramed miiReba Ta-

nabari sisqis wrfivi interferenciuli zolebi, romelic Seesabameba 1M sarkisa da 2M sar-

kis warmosaxviT gamosaxulebas Soris Seqmnil sahaero zols. 2S sarkis gadaadgileba gamo-

iwvevs interferenciuli zolebis wanacvlebas, romlis didi sizustiT dakvirveba Zneli ar

aris. am gziT pirvelad iqna gazomili da Sedarebuli sinaTlis talRis standartul sig-

rZesTan sigrZis saerTaSoriso etaloni metri. maikelsonis interferometri gamoiyeneba ag-

reTve gardatexis maCveneblis zusti gazomvisaTvis (interferenciuli refraqtometri),

sinaTlis speqtruli analizisaTvis (interferenciuli speqtrometri) da sxv.

Tavi VIII

sinaTlis difraqcia

$65. difraqciis movlena. hiugens-frenelis principi

difraqcia ∗ ewodeba talRis mier dabrkolebis Semovlas, rac

iwvevs misi gavrcelebis sworxazovnebis darRvevas.

difraqciis movlenas Cven vakvirdebiT yoveldRiur cxovrebaSi. esaa mag.

wylis zedapirze warmoqmnili talRebis mier dabrkolebis Semovla, bgeriTi

an radiotalRebis mier dabrkolebaTa Semovla _ bgera Cven xom maSinac

gvesmis, roca bgeris wyarosa da Cvens Soris raime dabrkoleba, vTqvaT

kedelia moTavsebuli; aseve `gvesmis~ Soreuli sadgurebidan gamosxivebuli

radiotalRebi, romlebsac, vidre Cvenamde (antenamde) moaRwevdes, mravali

dabrkoleba eRobeba win.

difraqcia talRuri procesisaTvis damaxasiaTebeli movlenaa, amitom,

Tu sinaTle talRuri bunebisaa, mis difraqciasac unda qondes adgili, anu,

roca sinaTle ecema gaumWvir sxeuls, unda Semouaros mas da SeiWras geomet-

riuli Crdilis areSi. cda ki gviCvenebs sapirispiros _ sxeulebi, romlebic

∗ iwarmoeba laTinuri sityvisagan diffractus _ gatexili.

223

naTdebian wertilovani wyaros mier gamosxivebuli sinaTliT, qmnian mkveTr

Crdilebs, e.i. sxivebi ar ixrebian Tavdapirveli gavrcelebis mimarTulebidan.

sinaTlis difraqciis damzera gaZnelebulia imiT, rom talRebis dif-

raqcia arsebiTad aris damokidebuli talRis sigrZesa da difraqciis gamom-

wvevi obieqtis zomebs Soris Tanafardobaze. yvelaze mkveTrad difraqcia

maSin vlindeba, roca dabrkolebaTa zoma talRis sigrZis Tanazomadia. zemoT

CamoTvlili talRebisaTvis, romelTa sigrZe santimetrebiT, metrebiT da

kilometrebiT ganisazRvreba, es piroba sruldeba. sinaTlisaTvis ki, romlis

talRis sigrZe santimetris measiaTasedi nawilis rigisaa, `cxovrebiseuli~

obieqtebis zoma metismetad didia. amitomaa, rom sinaTlis difraqcia

yoveldRiur cxovrebaSi ar vlindeba, da mis dasamzerad specialuri

mowyobilobebia saWiro.

difraqciis movlena aixsneba principiT, romelic Camoayaliba sinaTlis

talRuri Teoriis fuZemdebelma holandielma mecnierma hiugensma (1690 w).

hiugensis principis Tanaxmad, sivrcis yoveli wertili, romelamdisac

aRwevs talRuri zedapiri, TviTon xdeba meoreuli sferuli

talRebis wyaro. am meoreuli naxevarsferuli ∗ zedapirebis momv-

lebi warmoadgens axal talRur zedapirs drois momdevno

momentSi.

vTqvaT, sinaTlisadmi gaumWvir AB firfitas, romelsac datanebuli aqvs

ab xvreli, ecema paralelur sxivTa kona anu brtyeli talRa (sur. 65.1). hiu-

gensis principis Tanaxmad, xvrelis TiToeuli wertili gadaiqceva sferuli

talRebis wyarod, romelTa momvlebi warmoadgens xvrelidan gamosuli

talRis zedapirs. suraTidan Cans, rom zedapiri brtyelia mxolod Sua

nawilSi, napirebSi ki mruddeba da vinaidan sxivi marTobulia talRuri

zedapiris, gamodis rom adgili aqvs

sinaTlis SeWras geometriuli

Crdilis areSi, anu difraqcias.

hiugensis principi zogadi

principia. misi saSualebiT miiReba

arekvlisa da gardatexis kanonebi

∗ vinaidan talRebi win vrceldeba, meore naxevris Semoxazva ar aris saWiro.

sur. 65.1

224

talRuri Teoriis safuZvelze, aixsneba ormagi sxivTtexa kristalebSi da sxv.

hiugensis principiT arekvlis kanoni

gamoviyvaneT kursis pirvel tomSi ($43) talRuri

procesebis sakiTxebis ganxilvasTan dakavSirebiT.

miviRoT axla gardatexis kanoni.

davuSvaT, MN gardamtex zedapirs ecema

brtyeli talRa OA . mas Seesabameba paralelur

sxivTa kona, romelic zedapiris normalTan adgens 1i

kuTxes. suraTze oTxi aseTi sxivia gamoyofili

(sur. 65.2). vTqvaT OA talRis A wertilma MN ze-

dapiris B wertils miaRwia t droSi. maSin

tAB 1υ= , sadac 1υ talRis gavrcelebis siCqarea I

garemoSi. hiugensis principis Tanaxmad, MN zedapiris yoveli wertili, masze talRuri

zedapiris miRwevis Semdeg, gadaiqceva meoreuli talRis centrad. cxadia, rom vidre OA

talRis A wertili miaRwevs MN zedapiris B wertils, O wertilSi aRiZvreba da meore

garemoSi gavrceldeba naxevarsferuli talRa, romlis radiusi tOC 2υ= , sadac 2υ aris

talRis gavrcelebis siCqare II garemoSi. sxva wertilebidan ( )'',' OO imave momentisaTvis gav-

rceldebian ufro naklebi radiusis mqone talRebi. am naxevarsferuli zedapirebis momvlebi

BC warmoadgens gardatexili talRis fronts. mas Seesabameba gardatexil sxivTa kona, ro-

melic zedapiris normalTan adgens gardatexis 2i kuTxes. suraTidan Cans, rom

11 sin iOBtAB ==υ da 22 sin iOBtOC ==υ , saidanac 2

1

2

1

sinsin

υυ

=ii

; 2

1

υυ

mudmivi sididea da am-

rigad, miRebuli formula warmoadgens gardatexis kanons. Tu mas SevadarebT gardatexis

cdiseul (52.1) kanonTan, miviRebT 1

2

2

1

nn

=υυ

. Tu pirveli garemo vakuumia ( )1, 11 == nCυ , maSin

22

nC=

υ. magram 12 >n , amitom υ>C . e.i. siCqare vakuumSi metia, vidre raime garemoSi.

amrigad, hiugensis principi warmatebiT wyvets talRis frontis gav-

rcelebis mimarTulebis dadgenis sakiTxs, magram ar ganixilavs sxvadasxva

mimarTulebiT gavrcelebuli talRebis amplitudas da maSasadame, inten-

sivobis ( )2~ AI sakiTxs (e.i. warmoadgens wminda geometriul princips). hiu-

gensis principis es nakli Seavso frangma fizikosma frenelma (1615 w.), da-

umata ra mas idea meoreuli talRebis interferenciis Sesaxeb. amis gamo, hiu-

gensis mier formalurad Semotanilma meoreuli talRebis momvlebma zeda-

225

pirma, romelic ubralo geometriul zedapirs warmoadgenda, SeiZina fi-

zikuri Sinaarsi, rogorc zedapirma, sadac meoreuli talRebis interfe-

renciis Sedegad jamuri talRis intensivoba maqsimaluria. sivrcis nebismier

wertilSi aRZruli rxevac meoreuli talRebis interferenciis Sedegia.

aseTnairad modificirebul hiugensis princips ewoda hiugens-frenelis

principi. igi talRuri optikis ZiriTadi principia da saSualebas iZleva

ganisazRvros sxvadasxva mimarTulebiT gavrcelebuli (jamuri) talRebis

intensivobebi, anu gadaiWras sinaTlis difraqciasTan dakavSirebuli

sakiTxebi. amis Sesabamisad gadawyda sakiTxi sinaTlis wrfivi gavrcelebis

kanonis gamoyenebis sazRvrebis Sesaxeb.

gamovsaxoT hiugensis principi maTematikurad. vTqvaT, wertilovani S

wyaros mier gamosxivebuli talRis zedapiri t momentisaTvis aris σ . davyoT

es zedapiri imdenad mcire σd elementebad,

rom TiToeulis farglebSi r da ψ iyos

mudmivi (sur. 65.3). gamoviTvaloT TiToeuli

elementis mier aRZruli rxevis amplituda

dakvirvebis P wertilSi da Semdeg Sevkri-

boT (imis gaTvaliswinebiT, rom sxvadasxva

σd elementebis mier aRZruli rxevis am-

plitudebis fazebi P wertilSi sxvadasxvaa). cxadia, calkeuli elementebis

mier aRZruli rxevebi koherentulni arian, vinaidan warmodgebian erTidaimave

S wyarodan. TiToeuli elementis mier aRZruli rxevis amplituda miT metia,

rac metia σd farTobi da naklebia r manZili da ψ kuTxe. P wertilSi

aRZruli rxevis gantolebis dasawerad frenelma Semoitana ( )ψk koeficienti,

romelic maqsimaluria, roca 0=ψ da nulis tolia, roca 2πψ ≥ . aRvniSnoT,

rom aseTi koeficientis SemotaniT gamoiricxa meoreuli wyaroebidan ukumi-

marTulebiT (σ zedapiridan S wyarosken) gavrcelebuli talRebis arseboba.

frenelma dauSva agreTve, rom Tu S wyarosa da dakvirvebis P wertils

Soris moTavsebulia sinaTlisadmi gaumWviri firfita xvreliT, firfitis

zedapirze meoreuli rxevebis amplitudebi nulis tolia, xolo xvrelSi

iseTivea, rogoric firfitis ararsebobis SemTxvevaSi.

226

σd elementis mier P wertilSi aRZruli rxevis gantolebis dasawerad

unda gaviTvaliswinoT, rom am rxevis amplituda proporciulia σd far-

Tobisa da ukuproporciulia (rogorc sferuli rxevis amplituda) r

manZilisa. sabolood gveqneba:

( ) ( )00 cos ϕωσ

ψ +−= krtrda

kdx , (65.1)

sadac 0ϕω +t aris rxevis faza σ talRur zedapirze, 0a _ Tanamamravlia,

romelic ganisazRvreba σd ubanze rxevis amplitudiT, xolo ( )rda

kdAσ

ψ 0=

aris elementaruli rxevis amplituda P wertilSi. jamuri rxeva P wer-

tilSi miiReba (65.1)-is integrebiT:

( ) ( ) σϕωψσ

dkrtr

akx ⋅−−= ∫ 0

0 cos . (65.2)

(65.2) warmoadgens hiugens-frenelis principis maTematikur gamosaxvas, romlis

amoxsniT gaigeba x , xolo misi saSualebiT _ rxevis amplituda da maSasadame,

intensivobac.

$66. frenelis zonebis meTodi. sinaTlis wrfivi gavrceleba

zogad SemTxvevaSi (65.2) integralis amoxsna da jamuri amplitudis

povna metismetad rTulia. magram rogorc frenelma aCvena, simetriiT gamor-

Ceul SemTxvevebSi, specialuri, e.w. zonebis meTodis gamoyenebiT, amocana

martivdeba da jamuri rxevis amplitudis gamoTvla daiyvaneba ubralo

algebrul an geometriul Sekrebaze.

frenelis zonebis meTodi ganvixiloT principulad mniSvnelovani

sakiTxis gadawyvetasTan dakavSirebiT: rogor xsnis sinaTlis talRuri

Teoria sinaTlis gavrcelebis praqtikul sworxazovnebas da rogoria am

sworxazovnebaze dafuZnebuli geometriuli optikis gamoyenebis sazRvrebi.

vTqvaT, erTgvarovan izotropul garemoSi moTavsebuli sinaTlis S

wertilovani wyaros mier gamosxivebuli talRis zedapiri drois t momenti-

saTvis aris σ . hiugens-frenelis principis Tanaxmad S wyaros moqmedeba P

wertilSi daiyvaneba am zedapiris TiToeuli wertilis mier gamosxivebuli

meoreuli talRebis jamur moqmedebaze (interferenciaze). talRuri zedapiri

simetriulia SP wrfis mimarT, rac saSualebas iZleva daiyos igi wriul

227

zonebad. freneli amas akeTebda iseTnairad, rom manZilebi momdevno zonebis

kideebidan P wertilamde gansxvavdebodnen 2λ-iT (sur. 66.1). suraTze 4 aseTi

zonaa gamosaxuli, amasTan, kenti zonebi (pirveli da mesame), daStrixulia,

xolo luwi _ dauStrixavi.

manZilebi talRuri zedapiris

O wverodan S wyaromde da

dakvirvebis P wertilamde

aRvniSnoT Sesabamisad a -Ti

da b -Ti∗, maSin mb manZili ne-

bismieri m -uri zonis kididan

P wertilamde toli iqneba

2λmbbm += . (66.1)

aseTnairad agebul frenelis zonebs is SesaniSnavi Tviseba aqvT, rom

mezobeli zonebidan aRZruli rxevebi imyofebian sawinaaRmdego fazebSi da

nawilobriv abaTileben erTmaneTs.

SeiZleba Cveneba (ix. petitis teqsti), rom m -is arc Tu ise didi mniS-

vnelobebisaTvis, frenelis zonebis farTobebi erTnairia da gamoiTvleba

formuliT:

baab+

=Δλπσ . (66.2)

frenelis pirveli, meore da a.S. zonebis mier aRZruli rxevebis ampli-

tudebi aRvniSnoT Sesabamisad mAAAA ...,, 321 -iT. maSin, vinaidan zonebis

farTobebi erTnairia, xolo zonis nomrisTan zrdasTan erTad izrdeba ro-

gorc mb manZili zonis kididan P wertilamde, ise kuTxe ψ zonis normalsa

da mb -s Soris (rac iwvevs ( )ψk koeficientis Semcirebas), SeiZleba davweroT:

mAAAA >>>> ...321 .

rogorc aRvniSneT, mezobeli zonebis mier aRZruli rxevebi imyofebian

sawinaaRmdego fazebSi da erTmaneTs asusteben, ris gamoc kenti zonebis

Sesabamisi amplitudebi SeiZleba CaiTvalos dadebiTad, xolo luwis _

uaryofiTad. maSin jamuri amplitudisaTvis P wertilSi gveqneba:

∗ sferuli talRis SemtxvevaSi a , cxadia, sferuli zedapiris simrudis R radiusia.

sur. 66.1

228

mAAAAAAA ∓...54321 +−+−= , (66.3)

romelic SeiZleba ase gadavweroT:

2...

222225

433

211 mAA

AAA

AAA

A ∓+⎟⎠⎞

⎜⎝⎛ +−+⎟

⎠⎞

⎜⎝⎛ +−+= . (66.4)

vinaidan zonis nomris zrdasTan erTad amplitudebis mniSvnelobebi

monotonurad mcirdeba, SeiZleba miaxloebiT davweroT

211 +− +

= mmm

AAA .

maSin frCxilebSi moTavsebuli gamosaxulebani iqneba nulis toli da jamuri

amplitudisaTvis sabolood miviRebT:

221 mAAA ∓= , (66.5)

sadac dadebiTi niSani Seesabameba m -is kent, xolo uaryofiTi _ luw mniSvne-

lobebs.

Tu davuSvebT, rom are S wyarosa da P wertils Soris mTlianad Tavi-

sufalia (`gaxsnilia~), maSin, rogorc agebidan gamomdinareobs, zonebis ri-

cxvi ( )m Zalian didia,∗ Sesabamisad, mA Zalian mcirea, da SeiZleba daiweros:

21A

A ≈ . (66.6)

(66.6) gviCvenebs, rom jamuri amplituda ( A ), romelic miiReba P

wertilSi σ sferuli zedapiris sxvadasxva ubnebidan wamosuli talRebis in-

terferenciis Sedegad, naklebia vidre centraluri ( )1=m zonis mier gamos-

xivebuli talRis amplituda ( )1A . maSasadame, P wertilze mTeli talRuri

zedapiris moqmedeba daiyvaneba misi mcire ubnis moqmedebaze, romlis farTobi

centraluri zonis farTobze naklebia. centraluri zonis farTobi σΔ ki

sakmaod mcirea. marTlac sinaTlis talRis sigrZe 7105 −⋅≈λ m, amitom, Tu

davuSvebT, rom 1== ba m, maSin (66.2)-is Tanaxmad 226 8,0108,0 mmm =⋅≈Δ −σ .

maSasadame, sinaTlis nakadi S wyarodan P wertilamde vrceldeba Zalian

viwro arxis SigniT, e.i. wrfivad.

ar unda vifiqroT, rom Tu SP wrfis marTobulad movaTavsebT mcire zomis ekrans

(romelic vTqvaT, gadafaravs centarlur zonas), es gamoiwvevs P wertilis Cabnelebas, Tum-

ca misi ganaTebulobis intensivoba odnav moiklebs. marTlac, amis Sedegad amovardeba

∗ ix. petitis teqsti.

229

pirveli wevri niSancvladi (66.2) mwkrividan da miviRebT, rom 2AA < . ekranis zomis Tanda-

TanobiT zrdiT gaizrdeba gadafaruli zonebis ricxvi, rac gamoiwvevs P wertilis ganaTe-

bulobis TandaTan Semcirebas, da mxolod maSin, roca ekrani iqneba sakmaod didi da

gadafaravs frenelis zonebis sakmaod didi ricxvs, igi Cabneldeba, anu amoqmeddeba geo-

metriuli optikis ZiriTadi debuleba imis Sesaxeb, rom sinaTlis gavrcelebis gzaze ekranis

(dabrkolebis) moTavseba iwvevs ekranis ukan geometriuli Crdilis warmoaqmnas (dabnelebas).

amrigad, Catarebuli msjelobis safuZvelze SeiZleba davadginoT geometriuli

optikis gamoyenebis sazRvrebi: difraqciuli movlenebis ugulebelyofa da sinaTlis wrfivi

gavrcelebis marTebulobis daSveba misaRebia manam, sanam ekranis zoma sakmaod didia da fa-

ravs frenelis zonebis did ricxvs. magram frenelis zonis sidide ganisazRvreba talRis

sigrZiT da mcirdeba ukanasknelis SemcirebiT. da vinaidan xiluli sinaTlis talRis sigrZe

Zalian mcirea, Zalian mcire zomis ekranic ki gadafaravs frenelis zonebis did ricxvs.

amitomaa, rom yoveldRiur cxovrebaSi sinaTlis difraqcias ver vamCnevT da mis dasakvirveb-

lad specialuri optikuri xelsawyoebia saWiro.

frenelis zonebis meTodis marTebulobaze miuTiTebs is faqti, rom Tu

davamzadebT da sinaTlis gavrcelebis gzaze movaTavsebT firfitas, romelic

gadafaravs mxolod luw (an mxolod kent) zonebs, maSin (66.2)-is Tanaxmad, P

wertilSi jamuri amplituda ...531 AAAA ++= me-

ti iqneba, vidre gaxsnili sivrcis SemTxvevaSi

e.i. P wertilamde miaRwevs meti sinaTle, vidre

es gvqonda firfitis ararsebobis SemTxvevaSi.

cda mTlianad adasturebs am mosazrebebs: aseT-

nairad damzadebuli firfita, romelsac `zonu-

ri firfita~ ewodeba zrdis P wertilis gana-

Tebulobas da moqmedebs rogorc Semkrebi linza

(sur. 66.2).

gamoviTvaloT frenelis m -uri zonis farTobi mσΔ . misi gare sazRvari sferul

zedapirze gamoyofs mh simaRlis sferul segments (sur. 66.3). am segmentis farTobi aRvniS-

noT mσ -iT. maSin 1−−=Δ mmm σσσ , sadac 1−mσ aris ( )1−m zonis mier gamoyofili sferuli

segmentis farTi.

m -uri zonis gare sazRvris radiusi aRvniSnoT mr -iT, maSin rogorc suraTidan Cans,

( ) ( )22

222

2 mmm hbmbhaar +−⎟⎠⎞

⎜⎝⎛ +=−−=

λ, an

sur. 66.2

230

2

2222 2

42 mmmmm hbhmmbhahr −−+=−=

λλ , (66.7)

saidanac

( )ba

mmbhm +

+=

24

2 λλ.

Tu SemovifarglebiT m -is arc

Tu ise didi mniSvnelobebiT, λ -s

simciris gamo, 4

22 λm SeiZleba ugu-

lebelvyoT:

( )bambhm +

=2

λ (66.8)

analogiurad, ( )1−m -e zonis mier ga-

moyofili sferuli segmentis simaR-

lisaTvis gveqneba:

( )( )ba

bmhm +−

=− 21

1λ

.

cnobilia, rom sferuli segmentis farTobi Rhπσ 2= , sadac R sferos radiusia, h _

segmentis simaRle. Cven SemTxvevaSi aR = , amitom

λπσ mba

abm += ,

xolo m -uri zonis farTobi

λπσσσba

abmmm +

=−=Δ −1 . (66.9)

rogorc vxedavT, farTobi ar aris damokidebuli zonis nomerze, rac niSnavs imas, rom igi

yvela zonisaTvis erTnairia.

m -uri zonis radiusi ganisazRvreba (66.7) tolobidan:

22 2 mmm hahr −= .

aqac, Tu SemovifarglebiT m -is arc Tu ise didi mniSvnelobebiT, segmentis simaRle

ahm << , amitom 2mh SeiZleba ugulebelvyoT da gveqneba:

λmba

abahr mm +== 2 . (66.10)

Tu davuSvebT, rom 1== ba m, 7105 −⋅≈λ m, frenelis pirve-

li zonis radiusi 5,01 =r mm.

dabolos, ganvsazRvroT frenelis zonebis m ri-

cxvi talRuri zedapiris im nawilze, romelic `mimar-

Tulia~ P wertilisaken ( cd nawili, sur. 66.4). c

2λmb +

d

R S

Ra = b P

sur. 66.4

sur. 66.3

231

suraTidan Cans, rom

( ) 222 22

bababamb +=−+=+λ

saidanac

( )λ

bbabm −+=

222.

7105 −⋅≈λ m. Tu davuSvebT 1== ba m, miviRebT

6103 ⋅≈m .

$67. frenelis difraqcia mcire wriul xvrelze da diskoze

frenelis zonebis meTodi saSualebas gvaZlevs avsnaT sinaTlis gavrce-

lebis Taviseburebani, roca mas win xvdeba ama Tu im saxis dabrkoleba. Cven

ganvixilavT or umartives SemTxvevas, roca talRis gavrcelebis gzaze

moTavsebulia:

I. gaumWviri firfita mcire wriuli xvreliT da

II. gaumWviri mcire wriuli disko.

difraqcia mcire wriul xvrelze. monoqromatuli sinaTlis S

wertilovani wyarodan a manZilze movaTavsoT gaumWviri AB firfita, romel-

sac datanebuli aqvs 0r radiusis wriuli xvreli. SO sxivi marTobulia fir-

fitis sibrtyis da gadis xvrelis centrSi. dakvirvebas vawarmoebT AB

frifitis paralelurad moTavsebuli MN ekranis P wertilSi, romelic

xvrelis centridan b manZilzea moTavsebuli (sur. 67.1).

wina paragrafis (66.4) formulis Tanaxmad amplituda P wertilSi

221 mAA

A ∓= (67.1)

Tu xvrelSi moTavsebuli zonaTa ricxvi mcirea (amas ganapirobebs xvre-

lis zomis simcire), mA mcired gansxvavdeba 1A -sagan, amitom, kenti zonebis

SemTxvevaSi 1AA ≈ , xolo luwi zonebis SemTxvevaSi 0≈A . dabrkoleba rom ar

yofiliyo, anu talRuri zedapiri rom yofiliyo `gaxsnili~, (66.5)-is Tanax-

mad, amplituda P wertilSi iqneboda 2

1A. maSasadame dabrkoleba mcire xvre-

liT, romelSic Tavsdeba mcire raodenobis kenti zonebi, ara Tu ar amcirebs

ganaTebas P wertilSi, aramed zrdis amplitudas TiTqmis orjer da

232

maSasadame, intensivobas TiTqmis oTxjer (es imiT aixsneba, rom firfita

`ketavs~ mezobel zonebs, romlebic interferenciis gamo Seasustebdnen

centraluri zonis moqmedebas).

MN ekranze miRebuli difraqciuli suraTis

saxis dasadgenad unda gaviTvaliswinoT xvrelis wer-

tilebis simetriuli ganlageba SP wrfis mimarT, ris

gamoc MN ekranze P wertilidan Tanabrad daSo-

rebuli wertilebis ganaTebuloba erTnairi iqneba.

difraqciul suraTs eqneba naTeli da bneli koncent-

ruli rgolebis erTobliobis saxe centrSi naTeli

laqiT _ Tu P wertilisaTvis zonebis ricxvi kentia.

am SemTxvevas Seesabameba sur. 67.1 a-ze mocemuli inten-

sivobis ganawileba MN ekranze.

Tu ekrans gadavwevT Tavis Tavis paralelurad

(davaSorebT an mivuaxlovebT xvrels), b manZilis

Secvlis gamo Seicvleba xvrelSi moTavsebuli zo-

nebis ricxvi P wertilisaTvis _ igi gaxdeba luwi.

amis gamo P wertili Cabneldeba da difraqciul su-

raTs eqneba, cxadia, isev naTeli da bneli koncen-

truli rgolebis erTobliobis saxe, oRond centrSi

bneli laqiT. am SemTxvevas Seesabameba sur. 67.1 b-ze

mocemuli intensivobis ganawileba MN ekranze.

ufro gasagebi rom gaxdes zemoTqmuli, davuSvaT, rom frenelis zonebis

radiusebis ganmsazRvreli (66.10) formula, romelic Cven miviReT gaxsnili

talRuri zedapirisaTvis, samarTliania xvrelSi moTavsebuli zonebisaT-

visac.∗ maSin

λmba

abr+

=0 ( )mrr =0 ,

saidanac xvrelSi moTavsebuli zonebis m ricxvisaTvis miviRebT:

⎟⎠⎞

⎜⎝⎛ +=

bar

m 1120

λ.

∗ ba, da 0r sidideebis garkveuli Tanafardobis dros aseTi daSveba SeiZleba CaiTvalos

samarTlianad.

sur. 67.1

233

simartivisaTvis davuSvaT, rom xvrels ecema brtyeli talRa.∗ maSin ∞=a da

formula martivdeba:

b

rm

λ

20= . (67.2)

(67.2) gviCvenebs, rom mocemuli xvrelis mocemuli sinaTlis talRiT ganaTe-

bisas zonebis ricxvi damokidebulia b -ze. vTqvaT, P wertilisaTvis b ise-

Tia, rom m aris kenti. maSin am wertilSi miiReba naTeli laqa. magram P -s

mezobeli 'P wertilisaTvis manZili Seicvala, gaxda 'b , Sesabamisad Seic-

vleba zonebis ricxvic _ gaxdeba wyvili da am wertilSi miiReba bneli laqa.

cxadia, rom Cabnelebuli iqneba yvela is wertilebi, romelTaTvisac 'b

iseTivea, rogorc 'P wertilisaTvis. am bneli wertilebis erToblioba mog-

vcems bnel rgols. ''P wertilisaTvis manZili isev Seicvala, ris gamoc

zonebis ricxvi gaxda kenti, amitom es wertili iqneba ganaTebuli, `msgavsi~

wertilebis erToblioba ki mogvcems naTel rgols da a.S. Sedegad,

difraqciuli suraTi warmogvidgeba, rogorc naTeli da bneli rgolebis er-

Toblioba centrSi naTeli laqiT.

Tu MN ekrans odnav movuaxlovebT an davaSorebT AB firfitas, b

manZili Seicvleba, Seicvleba zonebis ricxvic da P wertili Cabneldeba. Se-

sabamisad Seicvleba difraqciuli suraTic. cxadia, igi isev iqneba naTeli da

bneli zolebis erToblioba, magram centrSi bneli laqiT.

Tu xvrelis zoma imdenad mcirea, rom masSi Tavsdeba frenelis mxolod

erTi zona (an zonis nawili), ekranze miiReba erTi didi zomis naTeli laqa,

romlis intensivoba centridan daSorebisas TandaTan moiklebs.

dabolos, SevniSnoT, rom xvrelis TeTri sinaTliT ganaTebisas miiReba

`Seferadebuli~ difraqciuli suraTi.

difraqcia mcire zomis gaumWvir wriul diskoze. monoqromatu-

li sinaTlis S wertilovani wyarodan a manZilze movaTavsoT mcire zomis

0r radiusis gaumWviri disko ise, rom diskos centrze dacemuli SO sxivi

aRmoCndes diskos sibrtyis marTobuli. dakvirvebas vawarmoebT diskos sib-

rtyis paralelurad moTavsebul MN ekranis P wertilSi, romelic xvrelis

centridan b manZiliTaa daSorebuli (sur. 67.2). difraqcias rom adgili ar

∗ aseTi daSveba ar cvlis saqmis arss.

234

hqonoda, MN ekranze unda migveRo wriuli Crdili, romlis zoma damokide-

buli iqneboda diskos zomaze da a da b manZilebze.

difraqciis Sedegad miiReba ufro rTuli suraTi, romelic warmoadgens

naTeli da bneli wriuli rgolebis erTobliobas, amasTan, rac yvelaze ufro

moulodnelia, ekranis centrSi ( P wertili) yovelTvis miiReba naTeli laqa.

am cdiseuli faqtis axsna SesaZleblia frenelis zonebis meTodis gamo-

yenebiT, oRond am SemTxvevaSi unda gamoiricxos diskoTi gadafaruli talRu-

ri zedapiris nawili da frenelis zonebis ageba daiwyos diskos kideebidan.

vTqvaT, disko faravs frenelis m zonas. maSin, P wertilSi aRZruli

jamuri rxevis A amplituda toli iqneba:

2...

222.. 12

211

321++

−++

+−+ =+⎟⎠⎞

⎜⎝⎛ +−+=−+−= mm

mmm

mmmAA

AAA

AAAA , (67.3)

vinaidan frCxilebSi moTavsebuli gamosaxuleba nulis tolia.

Tu m mcirea, maSin 11 AAm ≈+ da 2

1AA ≈ , anu ekranis centrSi gveqneba

TiTqmis iseTive ganaTeba, rogoric gvqonda dabrkolebis ararsebobis Sem-

TxvevaSi. es faqti sinaTlis difraqciis metad TvalsaCino dadasturebaa.

mTeli suraTis simetriuloba SP RerZis mimarT ganapirobebs imas, rom

naTeli laqa monacvleobiTaa garSemortymuli bneli

da naTeli rgolebiT. P wertilidan daSorebisas

rgolebis intensivoba klebulobs (sur.67.2b).

Tu disko didi zomisaa da faravs zonaTa did

raodenobas, maSin 11 AAm <<+ , rac imas niSnavs rom ek-

ranis centri iqneba Cabnelebuli, e.i. amoqmeddeba

geometriuli optikis ZiriTadi kanoni, romlis Tanax-

madac sinaTlis gavrcelebis gzaze dabrkolebis

moTavseba iwvevs mis ukan Crdilis warmoqmnas.

unda aRiniSnos, rom jamuri amplitudis ga-

moTvla MN ekranis nebismieri araRerZuli wertili-

saTvis ( )''' , PP did siZneles warmoadgens, vinaidan am

dros simetriis darRvevis gamo garkveuli nomridan

dawyebuli, maTi Sesabamisi zonebis farTis nawili

235

aRmoCndeba ekranis gaumWvir areSi, rac gamoiwvevs meTodisaTvis saWiro

zonaTa farTobebis tolobis darRvevas.

$68. fraunhoferis difraqcia marTkuTxa xvrelze

wina paragrafSi ganvixileT difraqcia, romlis dasakvirveblad aravi-

Tari optikuri xelsawyo ar gamogviyenebia _ gvqonda sinaTlis wyaro, dab-

rkoleba da ekrani. aseTi saxis difraqcias ikvlevda freneli da mas Cve-

ulebriv frenelis difraqcias uwodeben.

difraqciis gansxvavebuli tipi ganixila germanelma mecnierma

fraunhoferma (1821 w): igi sinaTlis brtyel talRas scemda xvrels, saidanac

difraqciis Sedegad sxvadasxva mimarTulebiT gavrcelebuli paraleluri

sxivebis Sekreba xdeboda linzis fokaluri sibrtyis sxvadasxva wertilSi.

difraqciis am saxes, romlis dasakvirveblad optikuri xelsawyos (linza,

linzaTa sistema) gamoyenebaa saWiro, fraunhoferis difraqcia ewodeba.

linzis gamoyeneba mniSvnelovnad zrdis difraqciuli suraTis sikaSkaSes,

rac aadvilebs mis dakvirvebas.

ganvixiloT frenelis difraqcia usasrulod grZel marTkuTxa xvrelze

(aqve SevniSnoT, rom xvreli iTvleba usasrulod grZlad, Tu misi sigrZe

bevrad metia siganeze. magaliTad, 0,01 mm siganis dros, xvreli, romlis sigrZe

ramodenime milimetria, iTvleba usasrulod

grZlad). davuSvaT, a siganis aseT xvrels mar-

Tobulad ecema brtyeli talRa (sur. 68.1). xvre-

lidan gamosul sxivebs davuxvedroT L Semkrebi

linza. difraqcias rom adgili ar hqondes, xvre-

lidan gamovidoda mxolod Tavdapirveli mimar-

Tulebis sxivebi da linzis fokalur sibrtyeSi

miviRebdiT naTel zols, romelic iqneboda mo-

cemuli xvrelis gamosaxuleba. magram, imis gamo,

rom adgili aqvs difraqcias, xvrelidan gamo-

suli sxivebi vrceldeba yvela mimarTulebiT,

ris gamoc ekranze miiReba ara erTi naTeli

zoli, aramed naTeli da bneli zolebis sur. 68.1

236

erToblioba. SeviswavloT es sakiTxi dawvrilebiT.

rodesac talRis fronti miaRwevs xvrels da daikavebs AB mdebareobas,

xvrelis TiToeuli wertili, hiugensis principis Tanaxmad gadaiqceva wyarod

meoreuli talRebisa, romlebic gavrceldebian win yvela mimarTulebiT. Cven

avirCioT erTi mimarTuleba, kerZod is, romelic Tavdapirvel mimarTu-

lebasTan adgens ϕ kuTxes (sur. 68.1). am sxivebis Sekreba xdeba L linzis MN

fokaluri sibrtyis P wertilSi da imisda mixedviT, Tu rogoria maT Soris

svlaTa sxvaoba, miiReba naTeli an bneli zoli.

A wertilidan sxivis mimarTulebaze davuSvaT perpendikulari AC .

maSin, vinaidan linza ar warmoqmnis damatebiT svlaTa sxvaobas,∗ AD sibrtyi-

dan dawyebuli sxivebs Soris svlaTa sxvaoba aris nulis toli. interferen-

ciis Sedegis ganmsazRvreli svlaTa sxvaoba warmoiqmneba AB sibrtyidan AC

sibrtyemde. davyoT BC wrfe 2λ-is tol monakveTebad. maSin ϕsinaBC =

manZilze moTavsebuli monakveTebis raodenoba toli iqneba:

2

sinλ

ϕam = . (68.1)

TiToeuli monakveTis bolodan gavavloT BC -s paraleluri wrfeebi xvrelis

sibrtyis gadakveTamde. es wrfeebi xvrels dayofen toli siganis zolebad.

agebidan gamomdinareobs, rom es zolebi aris frenelis zonebi. vinaidan

mezobeli zonebidan wamosuli rxevebi imyofebian sawinaaRmdego fazebSi,

amitom, P wertilSi miviRebT bnel zols, Tu m luwia da naTel zols _ Tu

m kentia.

amrigad, difraqciuli minimumebis Sesabamisi kuTxeebi gamoiTvleba

tolobidan:

22sin min

λϕ ka = . (68.2)

difraqciul minimumebs Soris moTavsebuli iqneba difraqciuli maqsi-

mumebi. maT Sesabamis kuTxeebs gavigebT tolobidan:

( )2

12sin maxλϕ += ka . (68.3)

∓∓∓ ,2,1,0=k . . . gansazRvravs difraqciis rigs.

∗ linzis am Tvisebas `tautoqronizmi~ ewodeba.

237

MN ekranze miiReba Semdegi suraTi: ekranis centrSi iqneba naTeli zo-

li (vinaidan aq svlaTa sxvaoba nulis tolia da maSasadame, sruldeba maq-

simumis piroba), mis marjvniv da marcxniv bneli zolebi, Semdeg isev naTeli

zolebi da a.S. centralur naTel zols ewodeba centraluri maqsimumi,

momdevno naTel zolebs ki mTavari maqsimumebi. maTi intensivobebi centridan

daSorebisas TandaTan mcirdeba, vinaidan izrdeba rogorc manZili (MN

ekranamde), ise kuTxe ϕ . intensivobis ganawileba ekranze mocemulia (68.1 b)

suraTze.

zolebis sigane da ricxvi damokidebulia xvrelis siganisa da talRis

sigrZis Tanafardobaze. Cven ganvixiloT ori zRvruli SemTxveva:

1. xvrelis sigane gacilebiT naklebia talRis sigrZeze: λ<<a , maSin,

vinaidan sinusis maqsimaluri mniSvneloba erTis tolia, frenelis zonebis

maqsimaluri ricxvi, (68.1)-is Tanaxmad, toli

iqneba:

2λam =maqx , e.i. 1<<maqsm , rac imas niS-

navs, rom xvreli warmoadgens frenelis er-

Ti zonis metad mcire nawils da misi yvela

wertilidan gavrcelebuli rxevebis fazebi

erTnairi iqneba. minimumis (68.2) piroba ar

SeiZleba Sesruldes maSinac ki roca k min-

imaluria da udris erTs. praqtikulad,

aseTi mcire zomis xvreli warmoadgens si-

naTlis wertilovan (ufro swored xazovan)

wyaros. intensivobis ganawilebas sivrceSi

eqneba (68.2) suraTze mocemuli saxe.

2. xvrelis sigane gacilebiT metia talRis sigrZeze: λ>>a . SevafasoT

pirveli difraqciuli minimumis Sesabamisi kuTxe:

( ) 1arcsin1 <<≈=aaλλϕ min

. maSasadame, pirveli difraqciuli minimumis mdebareoba

Seesabameba sinaTlis wrfivi gavrcelebis mimarTulebidan umniSvnelo

gadaxras. asevea meore difraqciuli minimumic, vinaidan ( ) 12arcsin2 <<=aλϕ min

da

a.S. difraqciul minimumebs Soris iqneba difraqciuli maqsimumebi. amrigad,

sur. 68.2

238

farTo xvrelSi gavlisas talRis frontis centraluri nawili brtyelia, na-

pirebSi ki odnav mruddeba (sur. 68.3 a). Sesabamisad ekranze miiReba xvrelis

geometriuli gamosaxuleba, romelic garSemortymulia metad susti maqsimu-

mebiTa da minimumebiT. intensivobis ganawilebas ekranze aqvs sur. (68.3 b)-ze

mocemuli saxe.

Cven ganvixileT ori ukiduresi SemTxveva. mkafio difraqciuli suraTis

misaRebad saWiroa daculi iqnas iseTi Tanafardoba a -sa da λ -s Soris, rom

m toli iyos (3-5)-is.

(68.2) da (68.3) formulebi gviCvenebs, rom difraqciuli minimumebis da

maqsimumebis Sesabamisi kuTxeebi damokidebulia sinaTlis talRis sigrZeze.

amitom, xvrelis aramonoqromatuli, TeTri sinaTliT ganaTebisas, sxvadasxva

feris Sesabamisi maqsimumebi gaiSleba sivrceSi da ekranze miiReba dif-

raqciuli speqtri. amasTan, rogorc (68.3) formulidan gamomdinareobs, rac

naklebia λ , miT naklebi kuTxiTaa ganTavsebuli Sesabamisi difraqciuli

maqsimumi.

speqtris miReba SeiZleba samkuTxa prizmis saSualebiTac ($71), magram

difraqciul speqtrs is upiratesoba aqvs prizmatulTan SedarebiT, rom aq

ferebis urTierTganlageba ar aris damokidebuli im masalis gvarobaze, rom-

lidanac mzaddeba ekrani da xvreli da calsaxad ganisazRvreba talRis

sigrZiT da xvrelis geometriiT.

speqtris Seswavlas didi

mniSvneloba aqvs. sxvadasxva niv-

Tierebebis gamosxivebisa da

STanTqmis speqtrebis saSuale-

biT dadginda atomis agebulebis

uwvrilmanesi detalebi, energe-

tikuli doneebi atomebSi da sxv.

atomebisa da molekulebis speq-

trebis codna saSualebas gvaZ-

levs CavataroT speqtruli ana-

lizi anu davadginoT gamosakv-

levi sxeulis qimiuri Sedgeni- a b sur. 68.3

239

loba, maT Soris Zlier daSorebuli ciuri sxeulisac.

difraqcia marTkuTxa xvrelze Cven ganvixileT Tvisobrivad. sakiTxis

raodenobrivi ganxilva daiyvaneba sxvadasxva mimarTulebiT gavrcelebuli

meoreuli talRebis jamuri amlitudisa da intensivobis dadgenamde. amisaTvis

saWiroa talRis AB fronti daiyos xvrelis napirebis paraleluri, erTnairi

siganis mcire elementebad (zonebad). gamoviTvaloT calkeuli elementebis

mier aRZruli amplitudebi da SevkriboT isini maTi sidideebisa da fazebis

gaTvaliswinebiT. aseTi gamoTvla gvaZlevs, rom amplituda P wertilSi

ϕλπ

ϕλπ

ϕ

sin

sinsin

0 a

a

AA⎟⎠⎞

⎜⎝⎛

= . (68.4)

(68.4) gviCvenebs, rom roca 0→ϕ (O wertili), maSin 0AA = , vinaidan

1sinlim0

=→ α

αα

.

intensivoba amplitudis kvadratis proporciulia, amitom

2

2

02

sin

sinsin

⎟⎠⎞

⎜⎝⎛ ⋅

⎟⎠⎞

⎜⎝⎛ ⋅

==

ϕλπ

ϕλπ

ϕϕa

a

IAI . (68.5)

miRebuli formulis Tanaxmad, roca 0=ϕ , maSin 200 AII == . e.i. ekranis

centrSi miiReba naTeli zoli. es centraluri maqsimumia.

ϕ kuTxis im mniSvnelobebisaTvis, roca πϕλπ ka

=⋅ sin , anu roca

( )...321,sin ∓∓∓== kka λϕ (68.6)

intensivoba 0=I . amgvarad, (68.6) gansazRvravs ϕ kuTxis im mniSvnelobas,

romelTac Seesabameba bneli zolebi anu difraqciuli minimumebi.

vinaidan mezobel minimumebs Soris aucileblad unda iyos maqsimumi,

gamodis, rom bnel zolebs Soris moTavsebulia naTeli zolebi. am maqsimume-

bis mdebareobis dasadgenad (maT gverdiTi an meoradi maqsimumebi ewodeba)

iyeneben funqciis eqstremumebis povnis meTods. (68.5) funqciis mimarT am

meTodis gamoyenebiT dadgenilia, rom intensivoba aRwevs maqsimalur mniS-

vnelobebs ϕλπ sin⋅

a argumentis Semdegi mniSvnelobebisaTvis: ;46,2;43,1;0 ππ

ππ 47,4;47,3 da a.S.

240

(68.5) formuliT mocemuli damokidebuleba I intensivobasa da ϕ kuTxes

Soris grafikulad mocemulia (68.4) suraTze.

rogorc grafikidan

Cans, gverdiTi maqsimumebis

intensivobebi swrafad kle-

bulobs. gamoTvlebi gviCve-

nebs, rom centraluri da

momdevno (gverdiTi) maqsimu-

mebis mniSvnelobebi ise See-

fardebian erTmaneTs, ro-

gorc 005,0:008,0:047.0:1 da a.S.

e.i. sinaTlis talRis ener-

giis ZiriTadi nawili modis cenralur maqsimumze. energiis daaxloebiT 5%

modis pirvel gverdiT maqsimumebze, ~2% _ meore gverdiT maqsimumebze da a.S.

suraTidan Cans, rom centraluri maqsimumis kuTxuri sigane anu δϕ man-

Zili or minimums Soris ganisazRvreba tolobidan aλδϕ

=2

sin ; Tu λ>>a , maSin

sinusi SeiZleba kuTxiT Seicvalos, e.i. aλδϕ 2

= .

$69. difraqcia or xvrelze. difraqciuli meseri.

praqtikul interess warmoadgens difraqcia difraqciul meserze.

difraqciuli meseri ewodeba paralelurad ganlagebul didi rao-

denobis∗ marTkuTxa xvrelebis erTobliobas, romlebic erTma-

neTisagan gamoyofilia toli siganis gaumWviri SualedebiT

(sur.69.1)

manZils mezobeli xvrelebis Sesabamis wertilebs (mag. xvrelis

napirebs) Soris mesris mudmiva an periodi ewodeba. suraTidan Cans, rom mes-

ris mudmiva bad += , sadac a _ xvrelis siganea, xolo b _ gaumWviri

Sualedis sigane. difraqciuli meseri aris ori tipis _ gamWvirvale da amrek-

li. gamWvirvale meseri mzaddeba minis an kvarcis firfitisagan, romelzedac

∗ ramodenime aTasi 1 mm-ze.

a a a a a a sur. 68.4

241

specialuri damyofi manqaniT (romelic aRWurvilia almasis saWrisiT),

avleben paralelur ganakawrebs. es ganakawrebi warmoadgens mesris gaumWvir

Sualedebs, xolo maT Soris dauzianebeli areebi _ gamWvirvale Sualedebs

anu xvrelebs. amrekl meserSi gamWvirvale Sualedebi Secvlilia sarkulad

amrekli SualedebiT. am SemTxvevaSi ganakawrebis gavleba xdeba kargad

gaprialebul liTonis zedapirze. mesers, romelsac aqvs mudmivi periodi da

erTnairi siganis xvrelebi, regularuli ewodeba.

mesris TiToeuli xvreli

mogvcems misTvis damaxasiaTebel

difraqciul suraTs, romelic ϕ

kuTxiT ganisazRvreba (sur. 69.1),

amasTan calkeuli xvrelebidan

miRebuli suraTebi erTnairia, vi-

naidan kuTxe ϕ yvela Wrili-

saTvis erTnairia. miuxedavad ami-

sa, N raodenobis xvrelis Sem-

TxvevaSi, jamuri intensivoba ϕI

ar udris erTi xvrelis mier gamosxivebul 1ϕ

I intensivobas gamravlebuls N -

ze ( ).1ϕϕ NII ≠ saqme isaa, rom meoreuli talRebis koherentulobis gamo, adg-

ili aqvs sxvadasxva xvrelebis Sesabamisi wertilebidan wamosuli (gamosxi-

vebuli) meoreuli talRebis interferencias,

rac gamoiwvevs zogierT werilebSi maT

urTierTgaZlierebas, zogSi ki _ Sesustebas.

simartivisaTvis ganvixiloT difraq-

ciuli meseri, romelic Sedgeba ori ( AB da

)CD xvrelisagan. davuSvaT, brtyeli mono-

qromatuli talRa marTobulad ecema mesris

sibrtyes. radgan xvrelebi daSorebulni

arian erTmaneTisagan toli manZiliT, amitom

mezobeli zonebis Sesabamisi wertilebidan

wamosuli sxivebis svlaTa sxvaoba erTnairia

da udris (sur. 69.2)

sur. 69.1

242

( ) ϕϕ sinsin dbaCF =+==Δ (69.1)

cxadia, rom im mimarTulebiT ( )ϕ , romlis gaswvrivac arc erTi xvre-

lidan ar vrceldeba sinaTle, ar gavrceldeba ori xvrelis SemTxvevaSic, e.i.

uwindeli minimumebis (romelic gvqonda erTi xvrelis SemTxvevaSi) mdebareoba

ar Seicvleba. vuwodoT maT mTavari minimumebi. maTTvis daculia (68.2) piroba:

( )...,2,1sin ∓∓∓== kka λϕ . (69.2)

magram ori xvrelis Sesabamisi wertilebidan gamosxivebuli sinaTlis

talRebi, interferenciis gamo, garkveuli mimarTulebebiT gaabaTileben

erTmaneTs, rac gamoiwvevs damatebiTi minimumebis warmoqmnas. cxadia, es mimar-

Tulebebi ganisazRvreba pirobidan, rom svlaTa sxvaoba unda iyos

naxevartalRis kenti ricxvis toli:

( ) ( )...3,2,1,0,2

12sin =+= kkd λϕ . (69.3)

da piriqiT, erTi xvrelis moqmedeba gaaZlierebs meoris moqmedebas, Tu

svlaTa sxvaoba talRis sigrZis naxevris luwi ricxvis tolia:

( )...3,2,1,0,2

2sin === kkkd λλϕ . (69.4)

(69.4)-iT gansazRvrul maqsimumebs mTavari maqsimumebi ewodeba.

amrigad, ori xvrelidan miRebuli difraqciuli suraTi ganisazRvreba

Semdegi pirobebidan:

mTavari minimumebi ...,3,2,sin λλλϕ =a

damatebiTi minimumebi ...,25,

23,

21sin λλλϕ =d

mTavari maqsimumebi ...,3,2,sin λλλϕ =d rogorc vxedavT, ori xvrelis SemTxvevaSi, or mTavar maqsimums Soris

moTavsebulia erTi damatebiTi minimumi. es gamoiwvevs maqsimumebis Seviw-

rovebas da maTi intensivobis gazrdas. analogiurad SeiZleba Cveneba, rom

sami xvrelis SemTxvevaSi or mTavar maqsimums Soris moTavsdeba 2 damatebiTi

minimumi, xolo zogadad, N xvrelis SemTxvevaSi _ 1−N damatebiTi minimumi.

damatebiTi minimumebi gamoyofilia erTmaneTisagan susti damatebiTi

maqsimumebiT, romelTa raodenoba or mezobel maqsimums Soris 2−N -ia. dama-

tebiTi maqsimumebis intensivoba Seadgens uaxloesi mTavari maqsimumis

intensivobis %30,4~ -s.

243

Catarebuli Tvisobrivi analizis safuZvelze SeiZleba gavakeToT Semde-

gi daskvna: rac metia xvrelebis raodenoba, miT met sinaTlis energias gaata-

rebs difraqciuli meseri, miT meti damatebiTi minimumebi (Cajdeba) warmoiq-

mneba mTavar maqsimumebs Soris, da maSasadame miT ufro intensiuri da mkveT-

ri iqneba mTavari maqsimumebi∗ xvrelTa didi ricxvis SemTxvevaSi MN ekran-

ze miRebuli difraqciuli suraTi warmoadgens mkveTr, viwro maqsimumebs,

romelTa Soris praqtikulad bneli Sualedebia. TvalsaCinoebisaTvis (69.3)

suraTze SeTavsebulia erTi, ori, oTxi da rva xvrelisagan miRebuli

difraqciuli suraTebi. vinaidan 1sin ≤ϕ , (69.4)-dan gamomdinareobs, rom mTavar

maqsimumTa raodenoba λdk ≤ , anu ganisazRvreba mesris periodisa da talRis

sigrZis TanafardobiT.

sakiTxis raodenobrivi ganxilva gvaZlevs, rom meoreuli talRebis

interferenciis Sedegad intensivoba izrdeba ara xvrelebis N ricxvis pro-

porciulad, aramed am ricxvis kvadratis propor-

ciulad: 1

2ϕϕ INI = , rac Sedegia (interferenciis

gamo) mesris mier gatarebuli sinaTlis mTeli en-

ergiis gadanawilebisa.

(69.4)-dan gamomdinareobs, rom mTavar maqsi-

mumTa mdebareoba (romelic ϕsin -Ti ganisazRvreba),

garda centralurisa ( )0=k , damokidebulia λ

talRis sigrZeze. rac metia λ , miT metia ϕsin ,

anu miT ufro didi difraqciis ϕ kuTxiT miiReba

am talRis Sesabamisi maqsimumi. amitom, meserSi

TeTri sinaTlis gatarebisas, yvela maqsimumi,

garda centralurisa, daiSleba ferebad. miiReba

e.w. difraqciuli speqtri. amasTan, vinaidan tal-

Ris sigrZis zrdiT izrdeba ϕsin da maSasadame ϕ ,

∗ SeiZleba Cveneba, rom centraluri maqsimumis kuTxuri sigane

Ndλϕ 2

=Δ . daaxloebiT

aseTivea yvela mTavari maqsimumis siganec. e.i. rac metia N , miT viwro da maSasadame, mkveTria mTavari maqsimumebi.

sur. 69.3

244

didi difraqciis kuTxiT miiReba didi talRis sigrZis Sesabamisi maqsimumi.

amitom speqtris iisferi nawili yvelaze axlos iqneba difraqciuli suraTis

centrTan, wiTeli ki _ yvelaze Sors (sur. 69.4).

difraqciuli meseri umniSvnelovanesi speqtruli xelsawyoa, romlis

daniSnulebaa sinaTlis speqtrad daSla da talRis sigrZis gansazRvra, rac

martivad xorcieldeba (69.4) formulis saSualebiT. vinaidan mocemuli

mesrisaTvis d cnobilia, amisaTvis saWiroa mxolod gaizomos mocemuli ri-

gis maqsimumis mdebareobis ganmsazRvreli ϕ kuTxe.

$70. rentgenis sxivebis difraqcia.

1895 wels germanelma fizikosma vilhelm rentgenma, kaToduri sxivebis

Seswavlisas aRmoaCina, rom eleqtronebis swrafi damuxruWebis dros Cndeba

uxilavi gamosxiveba, romelsac, manamde ucnobi Tvisebebis gamo (haeris

ionizaciis unari, nivTierebis ganWolvis didi unari, mafluorescirebeli

ekranis naTebis unari da sxv.), X _ sxivebi uwoda. Zalian male am sxivebs, misi

aRmomCenis pativsacemad, rentgenis sxivebi ewoda.∗

rentgenis gamosxivebis warmoSobis meqanizmi sworad iqna axsnili sxva-

dasxva mecnierebisa da TviT rentgenis mier: moZravi eleqtroni warmoadgens

eleqtrul dens, romelic garSemortymulia magnituri veliT. eleqtronis

swrafi damuxruWebisas icvleba (mcirdeba) magnituri veli. maqsvelis Teoriis

Tanaxmad ki, eleqtruli da magnituri velis cvlileba iwvevs eleq-

tromagnituri talRis gamosxivebas. magram Tu rentgenis sxivebi talRuri

∗ sagulisxmoa, rom TviT rentgeni am sxivebs sikvdilamde moixsenebda X _ sxivebad.

iisfer

i mw

vane

wiT

eli

meore pirveli nulovani pirveli meore rigis rigis rigis rigis rigis speqtri speqtri speqtri speqtri speqtri

( )2−=k ( )1−=k ( )0=k ( )1+=k ( )2+=k

sur. 69.4

245

bunebisaa, adgili unda hqondes yovelgvari talRuri procesisaTvis

aucilebel movlenebs _ maT interferenciasa da difraqcias. praqtikulad ki

rentgenis sxivebis difraqciis (iseve, rogorc interferenciis) dakvirveba

(ganxorcieleba) ver xerxdeboda _ isini difraqciul meserSi gadiodnen

yovelgvari gadaxris gareSe. es winaaRmdegoba axsna germanelma mecnierma

lauem 1912 wels: difraqcias rom hqondes adgili saWiroa, rom zoma

xvrelebisa da dabrkolebebisa, romlebic qmnian mesers sxivebis gavrcelebis

gzaze, ar iyos Zalian didi talRis sigrZesTan SedarebiT. marTlac,

formulidan λϕ nd =sin Cans, rom Tu λ>>d , maSin gadaxris ϕ kuTxe iqneba

metismetad mcire, e.i. difraqcias ar eqneba adgili. amitom, rentgenis sxivebis

difraqciis ararseboba, SesaZloa, imiT iyos gamowveuli, rom maTi talRis

sigrZe metismetad mcirea xelovnurad damzadebuli mesris periodTan

SedarebiT. magram Tu SeuZlebelia xelovnurad damzaddes meseri Zalian

mcire periodiT, igi arsebobs bunebaSi kristalis saxiT, romelSic

nawilakebi (atomebi, ionebi, an molekulebi) kanonzomieradaa ganlagebuli da

qmnian kristalur mesers (It, $83). kristalur meserSi manZili atomebs Soris

(xvrelebi) iseTivea, rogorc atomis (gaumWviri Sualedis) zoma da Seadgens

1010~ − m (rac niSnavs rom kristaluri mesris `periodi~ ramodenime rigiT

naklebia difraqciuli mesris periodze). lauem ganaxorciela rentgenis

sxivebis difraqcia kristalur meserze, SeimuSava miRebuli difraqciuli su-

raTis gaTvlis meTodi da gazoma talRis sigrZe. dReisaTvis dadgenilia, rom

rentgenis gamosxivebis talRis sigrZe icvleba ( )118 1010 −− − m-is farglebSi.

kristaluri meseri samganzomilebiania da bevrad rTulia erTganzomil-

ebian difraqciul meserTan SedarebiT. Sesabamisad, rentgenis sxivebis dif-

raqciis Sedegad miRebuli suraTis gaTvla sakmaod rTulia. amitom praq-

tikaSi xSirad sargebloben rusi mecnieris vulfisa da ingliseli mecni-

erebis mama-Svili bregebis mier SemuSavebuli martivi da mosaxerxebeli meTo-

diT, romlis arsi mdgomareobs imaSi, rom rentgenis sxivebis difraqcia

SeiZleba ganvixiloT rogorc monokristalis∗ paraleluri kristalogra-

fiuli (atomuri) sibrtyeebidan maTi arekvlis Sedegi. magram Cveulebrivi

arekvlisagan gansxvavebiT aseT arekvlas adgili aqvs mxolod im SemTxvevaSi,

∗ erTi mTliani kristali, romlis moculoba warmoadgens erTian kristalur mesers.

246

roca mezobeli sibrtyeebidan arekvlili talRebisaTvis sruldeba

interferenciuli maqsimumis piroba∗ (sur. 70.1).

( )...,3,2,1==Δ kkλ (70.1)

rentgenis sxivebisaTvis nebismieri

nivTierebis gardatexis maCvenebeli 1=n ,

amitom, rogorc (70.1) suraTidan Cans, '1 da

'2 sxivebs Soris optikuri svlaTa sxvaoba

Θ=+=Δ sin2dDFED ,

sadac Θ aris kuTxe dacemul sxivebsa da

kristalografiul sibrtyes Soris (`sria-

lis kuTxe~), xolo d _ manZili sibrtyeebs Soris. Δ -s miRebuli mniSvnelo-

bis CasmiT (70.1)-Si miiReba vulf-bregebis cnobili formula:

λkd =Θsin2 . (70.2)

(70.2) formula gamoiyeneba ori mniSvnelovani amocanis gadasawyvetad:

1. cnobili Θ,λ da k -s mixedviT kristalografiul sibrtyeebs Soris d

manZilis gansazRvra anu kristalis struqturis dadgena. es meTodi udevs

safuZvlad rentgenostruqturul analizs.

2. meore amocana pirvelis Sebrunebulia da mdgomareobs cnobili Θ,d da k -

s mixedviT λ -s gansazRvraSi. es meTodi udevs safuZvlad rentgenul

speqtroskopias.

radgan Θsin ar SeiZleba iyos erTze meti, (70.2) formulidan gamomdina-

reobs, rom kristalis sivrcul meserze difraqcia SesaZlebelia mxolod

maSin, roca 2λ

>d ; Tu 2λ

<d , maSin difraqcias adgili ara aqvs da brtyeli

talRa gadis kristalSi mimarTulebis Seucvlelad, anu ise, rogorc

erTgvarovan garemoSi. amis gamo utolobas d2>λ uwodeben kristalis

optikuri erTgvarovnobis pirobas λ talRis sigrZis sxivebisaTvis. aqedan ga-

momdinare, sinaTlis sxivebisaTvis ( 7105 −⋅≈λ m) yvela kristali ( 109 1010 −− −≈d m)

unda iyos optikurad erTgvarovani, rasac sinamdvileSi ara aqvs adgili.

∗ sibrtyeze ganlagebuli nawilakebis (ionebi, atomebi, molekulebi) qmnian im araerTgvarovne-bebs, romlebzec xdeba rentgenis sxivebis gabneva yvela mimarTulebiT, magram gabneuli sxive-bidan interferenciis Sedegad erTmaneTs aZliereben mxolod im mimarTulebis sxivebi, rom-lebic akmayofileben (70.1) pirobas.

sur. 70.1

247

aixsneba es imiT, rom Cveni msjeloba samarTliania mxolod idealuri

kristalisaTvis, romlis nawilakebs Soris manZili zustad erTnairia. sinam-

dvileSi, realur kristalSi, es manZilebi siTburi moZraobis gamo swrafad

da qaosurad icvleba. garda amisa, realuri kristali yovelTvis Seicavs mina-

revebs da sxvadasxva defeqtebs. yovelive amis gamo, realuri kristali ar

SeiZleba iyos sinaTlis talRebisaTvis mTlianad optikurad erTgvarovani,

ramac unda gamoiwvios sinaTlis gabneva.

rentgenis sxivebis farTo praqtikuli gamoyeneba dafuZnebulia mis did

gamWolisunarianobaze da imaze, rom es unari damokidebulia nivTierebis

simkvriveze _ simkvrivis SemcirebiT igi izrdeba.

amazea dafuZnebuli rentgenodefeqtoskopia _ sxeulebsa da liTonis

nakeTobebSi sxvadasxva defeqtebis (bzarebis, fuWvilebis anu carieli adgi-

lebis) aRmoCena.

rentgenis sxivebis am Tvisebazea agreTve dafuZnebuli rentgenodiagnos-

tika _ adamianis Sinagan organoebSi sxvadasxva arabunebrivi kerebis (wylule-

bis, simsivnuri wamonaqmnebisa da sxv.) aRmoCena, mkvrivi qsovilebis (Zvlebis)

motexilobis an bzarebis aRmoCena da sxv.

Tavi IX

sinaTlis dispersia, STanTqma da gabneva

$71. dispersiis movlena. niutonis cdebi. normaluri da anomaluri

dispersia.

viciT, ($54) rom sinaTlis sxivi (kona) samwaxnaga prizmaSi gavlisas

ixreba fuZisken. magram Tu sxivi TeTri sinaTlisaa (mag. mzis), igi ara marto

ixreba fuZisken, aramed iSleba ferad-ferad sxivebad. am movlenas sinaTlis

dispersia∗ ewodeba, xolo miRebul suraTs _ speqtri. speqtri 7 feris

zolisgan Sedgeba, romlebic TandaTan, uwyvetad gadadian erTimeoreSi. es

ferebia: wiTeli, narinjisferi, yviTeli, mwvane, cisferi, lurji da iisferi.

yvelaze naklebad gadaixreba wiTeli feris sxivi, yvelaze metad _iisferi

∗ warmoiqmneba laTinuri sityvisagan dispersio _ gabneva, mimofantva.

248

(sur. 71.1). vinaidan garemos gardatexis maCvenebeli damokidebulia am garemoSi

sinaTlis siCqareze ⎟⎠⎞

⎜⎝⎛ =

υcn , gamodis, rom wiTeli sxivis gavrcelebis siCqare

udidesia, iisferis ki umciresi.

sinaTlis dispersiiT aixsneba

cisartyelas warmoSoba. am SemTxve-

vaSi prizmebis rols TamaSobs

wvimis wylis wveTebi.

sinaTlis dispersiis pirveli

eqsperimentuli gamokvleva Caatara

niutonma (1672 w.). niutonis cda

genialurad martivi iyo _ sinaTlis wyaros warmoadgenda mzis sxivebiT

ganaTebuli (fanjris) darabis viwro xvreli, xolo ekrans _kedeli. rodesac

niutonma diafragmis saSualebiT speqtridan gamoyo erT-erTi feris sxivi

(vTqvaT, wiTeli) da gaatara prizmaSi, igi fuZisken ki daixara, magram ferebad

ar daSlila, rac imas niSnavda, rom prizma ki ar aferadebda TeTr sinaTles,

ki ar cvlida mas, aramed mxolod Slida nawilebad. speqtris Svidive feris

SekrebiT niutonma kvlav TeTri sinaTle miiRo.

am cdebis safuZvelze niutonma gaakeTa Semdegi daskvnebi:

1. TeTri sinaTle rTuli sinaTlea, igi Sedgeba Svidi sxvadasxva ferisagan.

2. sxvadasxva feris sxivisaTvis mocemuli nivTierebis gardatexis maCvenebeli

sxvadasxvaa, ris gamoc prizmaSi gavlisas TeTri sinaTle iSleba ferebad.

3. speqtris Svidive feris Sekrebisas kvlav miiReba TeTri sinaTle.

sinaTlis talRuri Teoriis Tanaxmad, sinaTlis fers gansazRvravs tal-

Ris sigrZe anu rxevis sixSire. maSasadame, sinaTlis dispersia imiTaa gamo-

wveuli, rom gardatexis maCvenebeli damokidebulia talRis sigrZeze.

( )λfn = . (71.1) (71 . 1) formulas ewodeba dispersiis zogadi formula, xolo

sazogadod, dispersia ganimarteba rogorc gardatexis maCveneblis

damokidebuleba talRis sigrZeze (sixSireze) . sinaTlis dispersia

dispersiis kerZo gamovlenaa.

Cvens mier ganxilul magaliTSi gardatexis maCvenebeli talRis sigrZis

zrdasTan erTad mcirdeba. aseT dispersias normaluri ewodeba. normalur

sur. 71.1