Teorema Pojntinga angl Poynting s theorem teorema sho opisuye zakon zberezhennya energiyi elektromagnitnogo polya Teorema bula dovedena u 1884 roci Dzhonom Genri Pojntingom Vse zvoditsya do nastupnoyi formuli u t S J E displaystyle frac partial u partial t nabla cdot mathbf S mathbf J cdot mathbf E De S vektor Pojntinga J gustina strumu i E elektrichne pole Gustina energiyi u displaystyle u ϵ 0 displaystyle epsilon 0 elektrichna stala m 0 displaystyle mu 0 magnitna stala u 1 2 e 0 E 2 B 2 m 0 displaystyle u frac 1 2 left varepsilon 0 mathbf E 2 frac mathbf B 2 mu 0 right Teorema Pojntinga v integralnij formi t V u d V V S d A V J E d V displaystyle frac partial partial t int V u dV oint partial V mathbf S d mathbf A int V mathbf J cdot mathbf E dV de V displaystyle partial V poverhnya sho obmezhuyuye ob yem V displaystyle V U tehnichnij literaturi teorema zazvichaj zapisuyetsya nastupnim chinom u displaystyle u gustina energiyi S e 0 E E t B m 0 B t J E 0 displaystyle nabla cdot mathbf S varepsilon 0 mathbf E cdot frac partial mathbf E partial t frac mathbf B mu 0 cdot frac partial mathbf B partial t mathbf J cdot mathbf E 0 de e 0 E E t displaystyle varepsilon 0 mathbf E cdot frac partial mathbf E partial t gustina energiyi elektrichnogo polya B m 0 B t displaystyle frac mathbf B mu 0 cdot frac partial mathbf B partial t gustina energiyi magnitnogo polya i J E displaystyle mathbf J cdot mathbf E potuzhnist vtrat Dzhoulya na odinicyu ob yemu Zmist 1 Dovedennya 2 Uzagalnennya 3 Alternativni formi 4 DzherelaDovedennya RedaguvatiTeorema mozhe buti dovedena z dopomogoyu dvoh rivnyan Maksvella dlya prostoti vvazhayemo sho seredovishe ce vakuum m 1 e 1 dlya zagalnogo vipadku z dovilnim seredovishem potribno u formuli do kozhnogo e0 i m0 pripisati e i m E B t displaystyle nabla times mathbf E frac partial mathbf B partial t nbsp Domnozhivshi dvi chastini rivnyannya na B displaystyle mathbf B nbsp otrimayemo B E B B t displaystyle mathbf B cdot nabla times mathbf E mathbf B cdot frac partial mathbf B partial t nbsp Rozglyanemo spochatku rivnyannya Maksvella Ampera B m 0 J ϵ 0 m 0 E t displaystyle nabla times mathbf B mu 0 mathbf J epsilon 0 mu 0 frac partial mathbf E partial t nbsp Domnozhivshi dvi chastini rivnyannya na E displaystyle mathbf E nbsp otrimayemo E B E m 0 J E ϵ 0 m 0 E t displaystyle mathbf E cdot nabla times mathbf B mathbf E cdot mu 0 mathbf J mathbf E cdot epsilon 0 mu 0 frac partial mathbf E partial t nbsp Vidnyavshi pershe rivnyannya z drugogo otrimayemo E B B E m 0 E J ϵ 0 m 0 E E t B B t displaystyle mathbf E cdot nabla times mathbf B mathbf B cdot nabla times mathbf E mu 0 mathbf E cdot mathbf J epsilon 0 mu 0 mathbf E cdot frac partial mathbf E partial t mathbf B cdot frac partial mathbf B partial t nbsp Nareshti E B m 0 E J ϵ 0 m 0 E E t B B t displaystyle nabla cdot mathbf E times mathbf B mu 0 mathbf E cdot mathbf J epsilon 0 mu 0 mathbf E cdot frac partial mathbf E partial t mathbf B cdot frac partial mathbf B partial t nbsp Oskilki vektor Pojntinga S displaystyle mathbf S nbsp viznachayetsya kak S 1 m 0 E B displaystyle mathbf S frac 1 mu 0 mathbf E times mathbf B nbsp ce rivnoznachno S ϵ 0 E E t B m 0 B t J E 0 displaystyle nabla cdot mathbf S epsilon 0 mathbf E cdot frac partial mathbf E partial t frac mathbf B mu 0 cdot frac partial mathbf B partial t mathbf J cdot mathbf E 0 nbsp Uzagalnennya RedaguvatiMehanichna energiya u teoremi viznachayetsya yak t u m r t S m r t J r t E r t displaystyle frac partial partial t u m mathbf r t nabla cdot mathbf S m mathbf r t mathbf J mathbf r t cdot mathbf E mathbf r t nbsp de u m kinetichna energiya gustini u sistemi Vona mozhe buti opisana yak suma kinetichnoyi energiyi chastinok a u m r t a m a 2 r a 2 d r r a t displaystyle u m mathbf r t sum alpha frac m alpha 2 dot r alpha 2 delta mathbf r mathbf r alpha t nbsp S m displaystyle mathbf S m nbsp potik energiyi abo mehanichnij vektor Pojntinga S m r t a m a 2 r a 2 r a d r r a t displaystyle mathbf S m mathbf r t sum alpha frac m alpha 2 dot r alpha 2 dot mathbf r alpha delta mathbf r mathbf r alpha t nbsp Rivnyannya neperervnosti energiyi abo zakon zberezhennya energiyi t u e u m S e S m 0 displaystyle frac partial partial t left u e u m right nabla cdot left mathbf S e mathbf S m right 0 nbsp Alternativni formi RedaguvatiMozhna otrimati j inshi formi teoremi Pojntinga Zamist togo shob vikoristovuvati vektor potoku S E B displaystyle mathbf S propto mathbf E times mathbf B nbsp mozhna vibrati formu Avraama E H displaystyle mathbf E times mathbf H nbsp formu Minkovskogo D B displaystyle mathbf D times mathbf B nbsp abo yakus inshu Dzherela RedaguvatiEric W Weisstein Poynting Theorem From ScienceWorld A Wolfram Web Resource 5 1 TEOREMA UMOVA POJNTINGA c 91 Osnovi elektrodinamiki Klubis Ya D Shkatulyak N M Otrimano z https uk wikipedia org w index php title Teorema Pojntinga amp oldid 38024814
