Rivnya nnya Ma ksvella ce osnovni rivnyannya klasichnoyi elektrodinamiki yaki opisuyut elektrichne ta magnitne pole stvorene zaryadami j strumami Zmist 1 Rivnyannya elektrodinamiki v diferencialnij formi 1 1 SGSG 1 1 1 U vakuumi 1 1 2 U seredovishi 1 2 Mizhnarodna sistema velichin 2 Poyasnennya 3 Istorichna dovidka 4 Neinvariantnist vidnosno peretvoren Galileya 5 Tablicya rivnyan u ISQ 6 Otrimannya rivnyan Maksvella u vakuumi iz vikoristannyam STV principu superpoziciyi ta zakonu Kulona 7 Nezalezhnist rivnyan Maksvella 8 Primitki 9 Div takozhRivnyannya elektrodinamiki v diferencialnij formi RedaguvatiForma zapisu rivnyan Maksvella zalezhit vid sistemi odinic Zdebilshogo fiziki koristuyutsya formoyu zapisu v sistemi SGSG U Mizhnarodnij sistemi velichin ISQ na bazi yakoyi pobudovana Mizhnarodna sistema odinic SI vibrana forma zapisu v yakij ne figuruyut mnozhnik 4 p displaystyle 4 pi nbsp ta shvidkist svitla s Ideya polyagala v tomu shob zapisati rivnyannya Maksvella yak najfundamentalnishi rivnyannya v najprostishij formi Odnak ce prizvelo do poyavi zajvih mnozhnikiv v inshih osnovnih rivnyannyah napriklad zakoni Kulona Krim togo napruzhenosti elektrichnih ta magnitnogo poliv otrimali rizni rozmirnosti sho z tochki zoru fiziki ye velikim nedolikom Oskilki rivnyannya Maksvella opisuyut rozpovsyudzhennya elektromagnitnih hvil to bazhano takozh shob yihnya shvidkist shvidkist svitla vhodila v rivnyannya SGSG Redaguvati U vakuumi Redaguvati U diferencijnij formi rivnyannya Maksvella dlya vakuumu mayut takij viglyad rot B 1 c E t 4 p c j displaystyle text rot mathbf B frac 1 c frac partial mathbf E partial t frac 4 pi c mathbf j nbsp rot E 1 c B t displaystyle text rot mathbf E frac 1 c frac partial mathbf B partial t nbsp div B 0 displaystyle text div mathbf B 0 nbsp div E 4 p r displaystyle text div mathbf E 4 pi rho nbsp Rivnyannya zapisani v sistemi SGS Tut E displaystyle mathbf E nbsp napruzhenist elektrichnogo polya B displaystyle mathbf B nbsp vektor magnitnoyi indukciyi r displaystyle rho nbsp gustina elektrichnogo zaryadu j displaystyle mathbf j nbsp gustina elektrichnogo strumu c displaystyle c nbsp shvidkist svitla U seredovishi Redaguvati U rechovini elektrichne ta magnitni polya harakterizuyutsya dodatkovimi vektorami elektrichnoyu indukciyeyu ta napruzhenistyu magnitnogo polya zv yazanih z vidpovidno napruzhenistyu elektrichnogo polya j magnitnoyu indukciyeyu spivvidnoshennya yaki nazivayut materialnimi U zagalnomu viglyadi materialni spivvidnoshennya mayut skladnu nelokalnu formu tomu pri zapisu osnovnih rivnyan elektrodinamiki yih ne navodyat Rivnyannya nabirayut viglyadu rot H 1 c D t 4 p c j J D displaystyle text rot mathbf H frac 1 c frac partial mathbf D partial t frac 4 pi c mathbf j J dot D nbsp rot E 1 c B t B displaystyle text rot mathbf E frac 1 c frac partial mathbf B partial t dot B nbsp div B 0 displaystyle text div mathbf B 0 nbsp div D 4 p r f displaystyle text div mathbf D 4 pi rho f nbsp Tut r f displaystyle rho f nbsp gustina vilnih zaryadiv Vnesok zv yazanih zaryadiv vrahovuyetsya pri viznachenni vektora elektrichnoyi indukci D displaystyle mathbf D nbsp Mizhnarodna sistema velichin Redaguvati U Mizhnarodnij sistemi velichin ISQ navit dlya vakuumu vvodyatsya dvi dodatkovi harakteristiki elektromagnitnogo polya vektor elektrichnoyi indukciyi ta napruzhenist magnitnogo polya U vakuumi voni pov yazani z napruzhenistyu elektrichnogo polya ta magnitnoyu indukciyeyu za dopomogoyu stalih mnozhnikiv B m 0 H displaystyle mathbf B mu 0 mathbf H nbsp H B m 0 M displaystyle textbf H textbf B mu 0 textbf M nbsp D e 0 E P displaystyle mathbf D varepsilon 0 mathbf E textbf P nbsp de e 0 displaystyle varepsilon 0 nbsp elektrichna stala m 0 displaystyle mu 0 nbsp magnitna stala P M displaystyle textbf P textbf M nbsp polyarizaciya ta namagnichenist sumarni dipolni momenti d V displaystyle dV nbsp tomu sistema diferencijnih rivnyan Maksvella maye takij viglyad rot H D t j displaystyle text rot mathbf H frac partial mathbf D partial t mathbf j nbsp rot E B t displaystyle text rot mathbf E frac partial mathbf B partial t nbsp div B 0 displaystyle text div mathbf B 0 nbsp div D r displaystyle text div mathbf D rho nbsp U rechovini rivnyannya zberigayut svij viglyad za vinyatkom togo sho materialni spivvidnoshennya tobto zv yazkok mizh D displaystyle mathbf D nbsp ta E displaystyle mathbf E nbsp B displaystyle mathbf B nbsp ta H displaystyle mathbf H nbsp mayut skladnishu formu i zamist gustini usih elektrichnih zaryadiv r displaystyle rho nbsp vrahovuyutsya tilki vilni elektrichni zaryadi Poyasnennya RedaguvatiPershe rivnyannya Maksvella zakon Ampera viznachaye magnitne pole stvorene strumom iz gustinoyu j displaystyle mathbf j nbsp abo zh navedene zminnim elektrichnim polem Druge rivnyannya Maksvella zakon Faradeya viznachaye elektrichne pole yake vinikaye pri zmini napruzhenosti magnitnogo polya Tretye rivnyannya Maksvella teorema Gausa stverdzhuye sho ne isnuye monopolnih magnitnih zaryadiv Chetverte rivnyannya Maksvella rivnyannya Puassona stverdzhuye sho navkolo elektrichnih zaryadiv isnuye elektrichne pole Ce rivnyannya analogichne zakonu Kulona Istorichna dovidka RedaguvatiZgidno z legendoyu pristupayuchi do roboti nad stvorennyam zagalnoyi teoriyi elektromagnitnih yavish Dzhejms Klerk Maksvell virishiv sho chitatime tilki eksperimentalni roboti Pri vivedenni svoyih rivnyan vin opiravsya na zakon Kulona yakij viznachav silu vzayemodiyi mizh zaryadami zakon Ampera sho viznachav silu vzayemodiyi mizh strumami zakon elektromagnitnoyi indukciyi Faradeya vidsutnist eksperimentalnih danih sho vkazuvali b na isnuvannya magnitnogo monopolya ta matematichnij aparat rozvinutij pri vivchenni yavish v oblasti mehaniki j gidrodinamiki Elektrichne ta magnitni polya Maksvell uyavlyav sobi yak mehanichni zburennya pevnogo seredovisha efiru V 1820 roci Gans Kristian Ersted viyaviv 1 sho prohodyachi cherez drit galvanichnij strum zmushuye vidhilyatisya magnitnu strilku kompasa Take vidkrittya prityaglo shiroku uvagi vchenih togo chasu V tomu zh 1820 roci Zhan Batist Bio ta Feliks Savar eksperimentalno znajshli viraz 2 dlya magnitnoyi indukciyi yaka vinikaye zakon Bio Savara Laplasa i Andre Mari Amper viyaviv sho vzayemozv yazok na vidstani z yavlyayetsya takozh mizh dvoma drotami cherez yaki prohodit strum Amper uviv termin elektrodinamichnij i visunuv gipotezu sho prirodnij magnetizm pov yazanij z isnuvannyam v magniti kilcevih strumiv 3 Vpliv strumu na magnit viyavlenij Erstedom prizvelo Majkla Faradeya do ideyi pro te sho povinen isnuvati zvorotnij vpliv magnitu na strumi Pislya trivalih eksperimentiv v 1831 roci Faradej vidkriv sho magnit yakij peremishayetsya bilya providnika porodzhuye v providniku elektrichnij strum Ce yavishe bulo nazvano elektromagnitnoyu indukciyeyu Faradej vviv ponyattya polya sil deyakogo seredovisha sho znahoditsya mizh zaryadami i strumami Jogo mirkuvannya mali yakisnij harakter odnak voni zrobili velicheznij vpliv na doslidzhennya Maksvella Pislya vidkrittiv Faradeya stalo yasno sho stari modeli elektromagnetizmu Amper Puasson ta inshi nepovni Nezabarom z yavilasya teoriya Vebera zasnovana na dalekodiyi Prote vidtodi vsya fizika krim teoriyi tyazhinnya mala spravu lishe z blizkodiyeyu optika termodinamika mehanika sucilnih seredovish tosho Gaus Riman i ryad inshih vchenih vislovlyuvali pripushennya sho svitlo maye elektromagnitnu prirodu tak sho teoriya elektromagnitnih yavish tezh povinna buti blizkodiyevoyu Cej princip stav suttyevoyu osoblivistyu teoriyi Maksvella Maksvell vpershe opublikuvav svoyi rivnyannya v 1861 roci V 1864 pobachila svit insha jogo pracya v yakij rivnyan bulo visim oskilki voni vklyuchali inshi zakoni yaki zaraz ne zavedeno vklyuchati v chislo rivnyan Maksvella V 1884 Gevisajd za dopomogi Gibbsa vibrali pershu sistemu 4 h rivnyan i perepisali yiyi u vektornij formi blizkij do suchasnoyi Neinvariantnist vidnosno peretvoren Galileya RedaguvatiRivnyannya Maksvella zminyuyut svij viglyad pri perehodi vid odnoyi inercijnoyi sistemi koordinat do inshoyi yaksho pravila cogo perehodu zadavati klasichnimi peretvorennyami Galileya Cya obstavina malo hvilyuvala Maksvella j inshih vchenih XIX storichchya oskilki vvazhalosya sho rivnyannya spravedlivi lishe v odnij sistemi koordinat tij sho zv yazana z neporushnim efirom U 1887 roci Larmor znajshov peretvorennya pri yakih rivnyannya Maksvella ne zminyuyut viglyadu pri perehodi vid odnoyi neinercijnoyi sistemi koordinat do inshoyi Ci peretvorennya buli nazvani peretvorennyami Lorenca Lorenc otrimav yih u nablizhenomu viglyadi troshki ranishe Same ci peretvorennya Ejnshtejn poklav v osnovu specialnoyi teoriyi vidnosnosti yaka vidmovilasya vid ideyi pro isnuvannya efiru Pislya cogo rivnyannya Maksvella nabuli statusu universalnogo zakonu prirodi spravedlivogo v bud yakij sistemi koordinat Prote yihnya interpretaciya dokorinno vidriznyayetsya vid idej na osnovi yakih Maksvell yih viviv Tablicya rivnyan u ISQ RedaguvatiU ISQ rivnyannya elektrodinamiki mayut nastupnij viglyad Lp Diferencialne rivnyannya Integralne rivnyannya Nazva Yavishe kotre opisuye rivnyannya1 E B t displaystyle nabla times vec E frac partial vec B partial t nbsp L E d l d F B d t displaystyle oint limits L vec E cdot mbox d vec l frac mbox d Phi B mbox d t nbsp zakon Faradeya Zminne u chasi magnitne pole viklikaye vihrove elektrichne pole 2 H j D t displaystyle nabla times vec H vec j frac partial vec D partial t nbsp L H d l I d F D d t displaystyle oint limits L vec H cdot mbox d vec l I frac mbox d Phi D mbox d t nbsp Zakon Ampera rozshirenij Maksvellom Elektrichnij strum i zminne elektrichne pole stvoryuyut magnitne pole 3 D r displaystyle nabla cdot vec D rho nbsp S D d s V r d V displaystyle oint limits S vec D cdot mbox d vec s int limits V rho cdot mbox d V nbsp zakon Gausa dlya elektriki Dzherelo elektrichnogo polya zaryadi4 B 0 displaystyle nabla cdot vec B 0 nbsp S B d s 0 displaystyle oint limits S vec B cdot mbox d vec s 0 nbsp Zakon Gausa dlya magnitnogo polya Ne isnuye zaryadu magnitnogo polya silovi liniyi magnitnogo polya zamkneni de D elektrichna indukciya Kl m B magnitna indukciya T E napruzhenist elektrichnogo polya V m H napruzhenist magnitnogo polya A m FD potik elektrichnoyi indukciyi Kl A s FB magnitnij potik Vb j gustina strumu A m r gustina zaryadu Kl m3 displaystyle nabla cdot nbsp operator divergenciyi 1 m displaystyle nabla times nbsp operator rotora 1 m Otrimannya rivnyan Maksvella u vakuumi iz vikoristannyam STV principu superpoziciyi ta zakonu Kulona RedaguvatiIz otrimannya virazu dlya sili Lorenca napruzhenistyu elektrichnogo polya zaryadu sho ruhayetsya ye viraz E Q g r r 2 g 2 c 2 r u 2 3 2 1 displaystyle mathbf E frac Q gamma mathbf r left mathbf r 2 frac gamma 2 c 2 mathbf r cdot mathbf u 2 right frac 3 2 qquad 1 nbsp a indukciyeyu magnitnogo polya B 1 c u E displaystyle mathbf B frac 1 c mathbf u times mathbf E nbsp Yaksho u 1 displaystyle 1 nbsp pidstaviti r 0 displaystyle mathbf r 0 nbsp to znachennya vidpovidno napruzhenosti ta indukciyi bude neskinchenno velikim Dlya uniknennya cogo mozhna shtuchno vvesti konstantu a 2 displaystyle a 2 nbsp yak dodanok u znamennik 1 displaystyle 1 nbsp regulyarizaciya Todi modifikovanij viraz nabude viglyadu E Q g r r 2 g 2 r u 2 c 2 a 2 3 2 2 displaystyle mathbf E frac Q gamma mathbf r left mathbf r 2 gamma 2 frac mathbf r cdot mathbf u 2 c 2 a 2 right frac 3 2 qquad 2 nbsp Dlya togo shob pokazati u yakij miri tochki prostoru ye dzherelami ta stokami elektrichnogo ta magnitnogo poliv treba vzyati divergenciyu vid napruzhenosti elektrichnogo polya ta vid indukciyi magnitnogo polya Z urahuvannyam poperednih peretvoren Poperedni peretvorennya r d i v x i y j z k x x y y z z 3 displaystyle nabla mathbf r div x mathbf i y mathbf j z mathbf k frac partial x partial x frac partial y partial y frac partial z partial z 3 nbsp g r a d r r x i r y j r z k r r displaystyle grad r frac partial r partial x mathbf i frac partial r partial y mathbf j frac partial r partial z mathbf k frac mathbf r r nbsp f r f r x x f r y y f r z z f r x x f r y y f r z z r x f x r y f y r z f z f r r g r a d f displaystyle nabla varphi mathbf r frac partial varphi r x partial x frac partial varphi r y partial y frac partial varphi r z partial z varphi frac partial r x partial x varphi frac partial r y partial y varphi frac partial r z partial z r x frac partial varphi partial x r y frac partial varphi partial y r z frac partial varphi partial z varphi nabla mathbf r mathbf r cdot grad varphi nbsp g r a d r u 2 2 g r a d r u r u 2 g r a d r x u x r y u y r z u z r u 2 u x i u y j u z k r u 2 u r u displaystyle grad mathbf r cdot mathbf u 2 2grad mathbf r cdot mathbf u mathbf r cdot mathbf u 2grad r x u x r y u y r z u z mathbf r cdot mathbf u 2 u x mathbf i u y mathbf j u z mathbf k mathbf r cdot mathbf u 2 mathbf u mathbf r cdot mathbf u nbsp z virazu 2 displaystyle 2 nbsp mozhna otrimati E 4 p Q d a r displaystyle nabla mathbf E 4 pi Q delta a mathbf r nbsp vivedennya E Q g r r 2 g 2 r u 2 c 2 a 2 3 2 3 Q g r 2 g 2 r u 2 c 2 a 2 3 2 3 2 k Q g r r 2 g 2 r u 2 c 2 a 2 5 2 r r 2 r 2 g 2 u r u c 2 displaystyle nabla mathbf E nabla frac Q gamma mathbf r r 2 gamma 2 frac mathbf r cdot mathbf u 2 c 2 a 2 frac 3 2 frac 3Q gamma r 2 gamma 2 frac mathbf r cdot mathbf u 2 c 2 a 2 frac 3 2 frac 3 2 frac kQ gamma mathbf r r 2 gamma 2 frac mathbf r cdot mathbf u 2 c 2 a 2 frac 5 2 left frac mathbf r r 2r 2 gamma 2 mathbf u frac mathbf r cdot mathbf u c 2 right nbsp 3 Q g r 2 g 2 r u 2 c 2 a 2 3 2 3 k Q g r 2 g 2 r u 2 c 2 r 2 g 2 r u 2 c 2 a 2 5 2 3 Q g a 2 r 2 g 2 r u 2 c 2 a 2 5 2 4 p 3 Q g a 2 4 p r 2 g 2 r u 2 c 2 a 2 5 2 displaystyle frac 3Q gamma r 2 gamma 2 frac mathbf r cdot mathbf u 2 c 2 a 2 frac 3 2 frac 3kQ gamma r 2 gamma 2 frac mathbf r cdot mathbf u 2 c 2 r 2 gamma 2 frac mathbf r cdot mathbf u 2 c 2 a 2 frac 5 2 frac 3Q gamma a 2 r 2 gamma 2 frac mathbf r cdot mathbf u 2 c 2 a 2 frac 5 2 frac 4 pi 3Q gamma a 2 4 pi r 2 gamma 2 frac mathbf r cdot mathbf u 2 c 2 a 2 frac 5 2 nbsp de d a r 3 g a 2 4 p r 2 g 2 r u 2 c 2 a 2 5 2 displaystyle delta a mathbf r frac 3 gamma a 2 4 pi r 2 gamma 2 frac mathbf r cdot mathbf u 2 c 2 a 2 frac 5 2 nbsp trivimirna delta funkciya Diraka yaka dozvolyaye zapisati prostorovu gustinu zaryadu zoseredzhenogo v odnij tochci Z neyi vidno sho E 0 displaystyle nabla mathbf E 0 nbsp u kozhnij tochci krim yak pri r 0 a gt 0 displaystyle r 0 a gt 0 nbsp u yakij E displaystyle nabla mathbf E infty nbsp Zvidsi mozhna stverdzhuvati bazuyuchis na viznachenni divergenciyi sho elektrichnij zaryad tochka u danomu vipadku yaka ye dzherelom elektrichnoyi indukciyi Perejshovshi do neperervnogo rozpodilennya zaryadiv u ob yemi ta vikoristavshi aksiomu principa superpoziciyi poliv sumu trivimirnih delta funkcij Diraka mozhna zaminiti ob yemnoyu gustinoyu E 4 p r r i Q i d a r r i 3 displaystyle nabla mathbf E 4 pi rho quad mathbf rho sum i Q i delta alpha mathbf r mathbf r i qquad 3 nbsp Rivnyannya 3 displaystyle 3 nbsp ye pershim rivnyannyam Maksvella Iz nogo mozhna otrimati bagato fizichnih naslidkiv Odin z cih naslidkiv polyagaye u tomu sho silovi liniyi polya pochinayutsya na dodatnomu zaryadi i mozhut zamikatisya lishe na vid yemnomu oskilki dlya dodatnogo zaryadu E displaystyle nabla mathbf E nbsp vidpovidaye vitoku polya a dlya vid yemnogo jogo stoku Analogichno mozhna otrimati velichinu divergenciyi magnitnoyi indukciyi Dlya cogo treba urahuvati nastupni poperedni vivedennya Poperedni peretvorennya f a f a a g r a d f displaystyle nabla times varphi mathbf a varphi nabla times mathbf a mathbf a times grad varphi nbsp r i j k x y z r x r y r z z y i x z j y x k x y k y z j z x i 0 displaystyle nabla times mathbf r begin vmatrix mathbf mathbf i amp mathbf mathbf j amp mathbf mathbf k frac partial partial x amp frac partial partial y amp frac partial partial z r x amp r y amp r z end vmatrix frac partial z partial y mathbf i frac partial x partial z mathbf j frac partial y partial x mathbf k frac partial x partial y mathbf k frac partial y partial z mathbf j frac partial z partial x mathbf i 0 nbsp Todi koristuyuchis tim sho odrazu u 0 displaystyle nabla mathbf u 0 nbsp mozhna otrimati sho B 0 4 displaystyle nabla mathbf B 0 qquad 4 nbsp Vivedennya B 1 c u E 1 c u E 1 c u r Q g r 2 g 2 r u 2 c 2 a 2 3 2 1 c u r g r a d k Q g r 2 g 2 r u 2 c 2 a 2 3 2 displaystyle nabla mathbf B nabla cdot frac 1 c mathbf u times mathbf E frac 1 c mathbf u cdot nabla times mathbf E frac 1 c left mathbf u cdot nabla times mathbf r frac Q gamma r 2 gamma 2 frac mathbf r cdot mathbf u 2 c 2 a 2 frac 3 2 right frac 1 c left mathbf u cdot left mathbf r times grad left frac kQ gamma r 2 gamma 2 frac mathbf r cdot mathbf u 2 c 2 a 2 frac 3 2 right right right nbsp r 0 1 c u r 3 2 Q g r 2 g 2 r u 2 c 2 a 2 5 2 r r 2 r 2 g 2 u r u c 2 displaystyle nabla times mathbf r 0 frac 1 c left mathbf u cdot left mathbf r times frac 3 2 frac Q gamma r 2 gamma 2 frac mathbf r cdot mathbf u 2 c 2 a 2 frac 5 2 left frac mathbf r r 2r 2 gamma 2 mathbf u frac mathbf r cdot mathbf u c 2 right right right nbsp 1 c u r 3 r Q g r 2 g 2 r u 2 c 2 a 2 5 2 1 c u r 3 u r u Q g 3 c 2 r 2 g 2 r u 2 c 2 a 2 5 2 1 c u 3 r u r u Q g 3 c 2 r 2 g 2 r u 2 c 2 a 2 5 2 displaystyle frac 1 c left mathbf u cdot left mathbf r times frac 3 mathbf r Q gamma r 2 gamma 2 frac mathbf r cdot mathbf u 2 c 2 a 2 frac 5 2 right right frac 1 c left mathbf u cdot left mathbf r times frac 3 mathbf u mathbf r cdot mathbf u Q gamma 3 c 2 r 2 gamma 2 frac mathbf r cdot mathbf u 2 c 2 a 2 frac 5 2 right right frac 1 c left mathbf u cdot frac 3 mathbf r times mathbf u mathbf r cdot mathbf u Q gamma 3 c 2 r 2 gamma 2 frac mathbf r cdot mathbf u 2 c 2 a 2 frac 5 2 right nbsp 1 c r u u 3 r u Q g 3 c 2 r 2 g 2 r u 2 c 2 a 2 5 2 0 displaystyle frac 1 c mathbf r cdot mathbf u times mathbf u frac 3 mathbf r cdot mathbf u Q gamma 3 c 2 r 2 gamma 2 frac mathbf r cdot mathbf u 2 c 2 a 2 frac 5 2 0 nbsp Zvidsi ochevidno vihodyachi z ponyattya divergenciyi sho zhodna z tochok prostoru u poli zaryadu sho ruhayetsya vklyuchayuchi tochku polozhennya samogo zaryadu ne ye dzherelom magnitnogo polya Rivnyannya 4 displaystyle 4 nbsp ye drugim rivnyannyam Maksvella Teper dlya viznachenosti zakruchenosti polya v tochkah mozhna vzyati rotor vid E B displaystyle mathbf E mathbf B nbsp Z urahuvannyam zhe togo sho shvidkist ruhu ISV postijna mozhna zapisati yavnij viraz dlya rotora magnitnogo polya B 1 c u E 1 c u E 1 c E u 5 displaystyle nabla times mathbf B nabla times frac 1 c mathbf u times mathbf E frac 1 c mathbf u mathbf E cdot nabla frac 1 c mathbf E mathbf u cdot nabla qquad 5 nbsp Dovedennya B 1 c i j k x x x u y E z u z B y u z E x u x B z u x E y u y E x 1 c u x E y E z i u y E x E z j u z E x E y k displaystyle nabla times mathbf B frac 1 c begin vmatrix mathbf i amp mathbf j amp mathbf k frac partial partial x amp frac partial partial x amp frac partial partial x u y E z u z B y amp u z E x u x B z amp u x E y u y E x end vmatrix frac 1 c u x E y E z mathbf i u y E x E z mathbf j u z E x E y mathbf k nbsp 1 c u x E y E y E z i u y E y E y E z j u z E y E y E z k 1 c E x u x i E y u y j E z u z k 1 c u E 1 c E u displaystyle frac 1 c u x E y E y E z mathbf i u y E y E y E z mathbf j u z E y E y E z mathbf k frac 1 c E x u x mathbf i E y u y mathbf j E z u z mathbf k frac 1 c mathbf u mathbf E cdot nabla frac 1 c mathbf E mathbf u cdot nabla nbsp Cej viraz mozhna vidozminiti za dopomogoyu nastupnih mirkuvan Pri analizi ruhu ISV vidnosno zaryadu treba viraziti radius vektor r displaystyle mathbf r nbsp u yavnomu viglyadi r r 0 u t E k Q g r 0 u t r 0 u t 2 g 2 r 0 u t u 2 c 2 a 2 3 2 displaystyle mathbf r mathbf r 0 mathbf u t Rightarrow mathbf E frac kQ gamma mathbf r 0 mathbf u t mathbf r 0 mathbf u t 2 gamma 2 frac mathbf r 0 mathbf u t cdot mathbf u 2 c 2 a 2 frac 3 2 nbsp Todi chastinna pohidna po chasu napruzhenosti elektrichnogo polya bude rivna E t i 1 3 E r 0 i u i t r 0 i u i t t i 1 3 E r i u i u E 6 displaystyle frac partial mathbf E partial t sum i 1 3 frac partial mathbf E partial r 0 i u i t frac partial r 0 i u i t partial t sum i 1 3 frac partial mathbf E partial r i u i mathbf u cdot nabla mathbf E qquad 6 nbsp Pidstavivshi 6 displaystyle 6 nbsp u 5 displaystyle 5 nbsp mozhna otrimati B 1 c u E 1 c E u 1 c 4 p Q d a r u 1 c E t 1 c 4 p j 1 c E t 7 displaystyle nabla times mathbf B frac 1 c mathbf u mathbf E cdot nabla frac 1 c mathbf E mathbf u cdot nabla frac 1 c 4 pi Q delta a mathbf r mathbf u frac 1 c frac partial mathbf E partial t frac 1 c 4 pi mathbf j frac 1 c frac partial mathbf E partial t qquad 7 nbsp de j i Q i d a r i u displaystyle mathbf j sum i Q i delta a mathbf r i mathbf u nbsp gustina strumu Rivnyannya 7 displaystyle 7 nbsp ye tretim rivnyannyam Maksvella Z nogo vidno sho pri elektrichnij strum abo zmina jogo u chasi porodzhuyut vihrove magnitne pole Rotor zhe vid napruzhenosti elektrichnogo polya bude riven E Q g r r 2 g 2 r u 2 c 2 a 2 3 2 r Q g r 2 g 2 r u 2 c 2 a 2 3 2 3 Q g r g 2 c 2 u u r r r 2 g 2 c 2 u r 2 5 2 3 u r r u Q g 3 c 2 r 2 g 2 r u 2 c 2 a 2 5 2 8 displaystyle nabla times mathbf E nabla times frac Q gamma mathbf r r 2 gamma 2 frac mathbf r cdot mathbf u 2 c 2 a 2 frac 3 2 nabla times mathbf r frac Q gamma r 2 gamma 2 frac mathbf r cdot mathbf u 2 c 2 a 2 frac 3 2 3Q gamma frac left left mathbf r frac gamma 2 c 2 mathbf u mathbf u cdot mathbf r right times mathbf r right left r 2 frac gamma 2 c 2 mathbf u cdot mathbf r 2 right frac 5 2 frac 3 mathbf u times mathbf r mathbf r cdot mathbf u Q gamma 3 c 2 r 2 gamma 2 frac mathbf r cdot mathbf u 2 c 2 a 2 frac 5 2 qquad 8 nbsp Viraz 8 displaystyle 8 nbsp analogichno do 6 displaystyle 6 nbsp mozhna peretvoriti Todi E 1 c B t 9 displaystyle nabla times mathbf E frac 1 c frac partial mathbf B partial t qquad 9 nbsp Dovedennya Analogichno do pohidnoyi po chasu napruzhenosti E displaystyle mathbf E nbsp mozhna obchisliti pohidnu vid indukciyi magnitnogo polya B displaystyle mathbf B nbsp B t i 1 3 B r 0 i u i t r 0 i u i t t u B Q g c u u r r 2 g 2 c 2 u r 2 3 2 displaystyle frac partial mathbf B partial t sum i 1 3 frac partial mathbf B partial r 0 i u i t frac partial r 0 i u i t partial t mathbf u nabla mathbf B frac Q gamma c mathbf u nabla left frac mathbf u times mathbf r left r 2 frac gamma 2 c 2 mathbf u cdot mathbf r 2 right frac 3 2 right nbsp Oskilki u u r u u r u u 0 displaystyle mathbf u nabla mathbf u times mathbf r mathbf u times mathbf u nabla mathbf r mathbf u times mathbf u 0 nbsp to vektornij dobutok mozhna vinesti za znak operatora pohidnoyi Todi 1 c u r Q g u r 2 g 2 r u 2 c 2 a 2 3 2 1 c 3 Q g 1 g 2 u 2 u r u r r 2 g 2 r u 2 c 2 a 2 5 2 1 c 3 Q g 3 u r u r r 2 g 2 r u 2 c 2 a 2 5 2 c E displaystyle frac 1 c mathbf u times mathbf r frac Q gamma mathbf u nabla r 2 gamma 2 frac mathbf r cdot mathbf u 2 c 2 a 2 frac 3 2 frac 1 c frac 3Q gamma 1 gamma 2 u 2 mathbf u times mathbf r mathbf u cdot mathbf r r 2 gamma 2 frac mathbf r cdot mathbf u 2 c 2 a 2 frac 5 2 frac 1 c frac 3Q gamma 3 mathbf u times mathbf r mathbf u cdot mathbf r r 2 gamma 2 frac mathbf r cdot mathbf u 2 c 2 a 2 frac 5 2 c nabla times mathbf E Rightarrow nbsp E 1 c B t displaystyle Rightarrow nabla times mathbf E frac 1 c frac partial mathbf B partial t nbsp Rivnyannya 9 displaystyle 9 nbsp ye chetvertim rivnyannyam Maksvella Z nogo vidno sho rotor napruzhenosti elektrichnogo polya zminyuyetsya tilki todi koli ye nestacionarne magnitne pole i vidpovidno napruzhenist elektrichnogo polya ne sferichno simetrichna cherez relyativistski efekti ye vidilenij napryam ruhu zaryadu U vipadku iz zaryadom yakij pokoyitsya pole sferichno simetrichne tomu dlya nogo rotor riven nulyu Na ostachu zalishilos napisati pro dvi aksiomi kozhna z yakih maye dosit vagomij vnesok u mozhlivist zastosuvannya otrimanih rivnyan dlya elektrodinamiki Persha aksioma polyagaye u postulyuvanni vektornoyi prirodi elektromagnitnogo polya Yaksho b priroda elektromagnitnogo polya bula tenzornoyu to dlya jogo opisannya znadobilisya b rivnyannya na kshtalt rivnyan ZTV Napriklad yaksho formalno zastosuvati tu zh metodiku sho prodemonstrovana u comu rozdili do zakonu Vsesvitnogo tyazhinnya to mozhna otrimati rivnyannya shozhi do rivnyan Maksvella yak i viraz dlya sili podibnij do virazu sili Lorenca Prote yih virnist ne pidtverdzhuyetsya eksperimentalno hoch yakisno voni i virno opisuyut dinamiku til u gravitacijnomu poli za umovi spravedlivosti principu superpoziciyi Druga zh aksioma pov yazana z postulyuvannyam nezalezhnosti rivnyan Maksvella vid priskorennya zaryadu sho stvoryuye pole Tobto voni spravedlivi dlya bud yakih mozhlivih vipadkiv ruhu zaryadu Okrim cogo varto napisati pro princip superpoziciyi Vin mozhe buti zastosovanij do tih pir poki polya sho stvoryuyutsya zaryadami ne stanut nastilki silnimi sho budut vplivati na prostir chas unemozhlivlyuyuchi predstavlennya vektoriv harakteristik polya sistemi cherez linijnu kombinaciyu vektoriv zaryadiv ciyeyi sistemi Nezalezhnist rivnyan Maksvella RedaguvatiKoristuyuchis rivnyannyam neperervnosti mozhna pereviriti sistemu rivnyan Maksvella na nevirodzhenist Vzyavshi divergenciyu vid rotornogo rivnyannya dlya indukciyi magnitnogo polya bez pidstanovki E displaystyle nabla mathbf E nbsp i virazivshi z rivnyannya neperervnosti j r t displaystyle nabla mathbf j frac partial rho partial t nbsp mozhna otrimati 1 c j 1 c E t 1 c t E 4 p r 0 E 4 p r f x y z displaystyle frac 1 c nabla mathbf j frac 1 c frac partial nabla mathbf E partial t frac 1 c frac partial partial t left nabla mathbf E 4 pi rho right 0 Rightarrow nabla mathbf E 4 pi rho f x y z nbsp Analogichno mozhna vzyati divergenciyu vid chetvertogo rivnyannya Maksvella E B t 0 B g x y z displaystyle nabla cdot nabla times mathbf E frac partial nabla mathbf B partial t 0 Rightarrow nabla mathbf B g x y z nbsp Takim chinom iz drugoyi pari rivnyan Maksvella mozhna otrimati pershu tilki z tochnistyu do funkcij vid koordinat yaki ne zalezhat vid chasu Strogo dovesti zhe koristuyuchis lishe cimi dvoma rivnyannyami sho funkciyi rivni nulyu nemozhlivo Tomu u comu sensi rivnyannya Maksvella usogo yih visim dvi pari po tri rivnyannya oskilki rotorni rivnyannya rozpadayutsya na tri komponentnih rivnyannya ye nezalezhnimi Primitki Redaguvati Ersted G K Doslidi yaki stosuyutsya dij elektrichnogo nezvyazku na magnitnu strilu v kn Amper A M Elektrodinamika storinka 433 439 J B Biot and F Savart Note sur le Magnetisme de la pile de Volta Annales Chim Phys vol 15 pp 222 223 1820 Kniga Mario Locci Istoriya fiziki storinki 253 257Div takozh RedaguvatiSila Lorenca Peretvorennya LorencaCya stattya potrebuye dodatkovih posilan na dzherela dlya polipshennya yiyi perevirnosti Bud laska dopomozhit udoskonaliti cyu stattyu dodavshi posilannya na nadijni avtoritetni dzherela Zvernitsya na storinku obgovorennya za poyasnennyami ta dopomozhit vipraviti nedoliki Material bez dzherel mozhe buti piddano sumnivu ta vilucheno sichen 2020 Otrimano z https uk wikipedia org w index php title Rivnyannya Maksvella amp oldid 38211189
