Electromagnetic energy, momentum and forces in a dielectric medium with losses

Transcription

1 leroane ener, oenu and fores n a deler edu wh losses Yur A. Srhev he Sae Ao ner Cororaon ROSAO, "Researh and esn Insue of Rado-leron nneern" - branh of Federal Senf-Produon Cener "Produon Assoaon "Sar" naed afer hael.proseno", Zarehn, Pena reon, Russa -al: ur.srhev@al.ru aed: a, 7 Absra Fro he ener-oenu ensors of he eleroane feld and he ehanal eneroenu, he equaons of ener onservaon and balane of eleroane and ehanal fores are obaned. he equaon for he Abraha fore n a deler edu wh losses s obaned Kewords: he ener-oenu ensor, he eleroane fores, eleroane oenu, he Abraha fore. he onens. Inroduon. ner-oenu ensors 3. quaons of onservaon of ener and oenu 4. leroane fores 5. Conluson Referenes. Inroduon he roble of eleroane oenu and eleroane fores arsn as a resul of he neraon of eleroane radaon wh a deler edu s he subje of nuerous sudes. Soe reen aers on he hee are lsed n Refs. [ 5] e al. In hese arles, he fors of he eleroane oenu n he edu, he shaes of he eleroane ener-oenu ensors, he Abraha fore and oher eleroane fores are onsdered. In arle [6] fro ensors of eleroane feld and eleroane nduon he of neraon of an eleroane feld wh a deler edu and he equaons of onservaon of ener and oenu n a deler edu whou losses are obaned. quaons of onservaon of he eleroane oenu are sulaneousl he equaons of he balane of eleroane fores of neraon of eleroane radaon wh he edu. ere wo ases are ossble.

2 In he frs ase, when lh asses hrouh a ransaren edu, here are no losses,.e. absoron and saern of lh. hen he bea eleroane ener reans unhaned before enern he edu and afer en, and he dverenes of he n he edu and ousde he edu equal ero, sne here s no eleroane ener sn. In hs ase, he equaons of onservaon of onl eleroane ener and oenu follow fro. If here are no losses n he edu, hen here s onl an ehane of reave ener beween he wave and he edu. In hs ase, he fores arsn n he edu are reave fores and he do no erfor wor n. hs ase s onsdered n [6]. In he seond ase, when lh asses hrouh he edu, here are losses. hen ar of he eleroane radaon ener reans n he edu. In hs ase, he dverene of he eleroane ener-oenu ensor s no ero, sne here s a sn of eleroane ener. In hs ase, ehanal fores arse ha erfor wor on deforaon, hean, or oon of he edu. hs wor onsues eleroane ener of radaon. In hs ase, he equaons of onservaon of eleroane ener and oenu us be suleened b he ehanal ener and oenu of he edu, whh are desrbed b he ehanal ener-oenu ensor. In hs ase, he dverene of he ehanal ener ensor s he soure, and he dverene of he eleroane ener-oenu ensor s a sn. hen he roess of ransferrn of eleroane ener o he edu an be wren n he for of equaons and, where s he eleroane ener-oenu ensor, s he ehanal ener-oenu ensor dverenes for dfferen ndes have dfferenes, sne he ensor s aser. In hs ase, he neraon of lh wh he edu s reave and ave or ole. he urose of hs aer s o onsder eleroane ener, oenu, and eleroane fores n a deler edu wh losses.. ner-oenu ensors We an wre he anonal n s eneral for [6] где W S µ, ν =,,, 3;, =,, 3 W he ner dens; S he ner flu dens Ponn veor; he oenu dens; he ensor of he oenu flu dens he sress ensor he oonens of he obaned n [6] have he for: W S 3.

3 3 Afer her subsuon no he anonal ensor, he new has he for: 3 I dffers fro nows's b daonal ers.: Is lnear nvaran I s he he Larane dens funon for eleroane feld n he edu, and s lnear nvaran for he rofeld or vauu I s a quadra nvaran of he anser eleroane feld ensor: I I he soro deler edu s desrbed b he anonal aeral equaons and, where ε and µ, resevel, he relave deler and ane ereabl s of he edu. 3 For vauu = =. hen he for vauu an be wren n he for: 4 he ehanal ener-oenu ensor s obaned as he ensor rodu of he veor of he four-densonal velo of he edu oon, o he four-densonal dens veor of he ehanal oenu, P, where s he ass dens, s he dens of he ehanal edu oon. Sne he salar rodus of he ed velo oonens are ero, we oban a ser ehanal ener-oenu ensor n he for: W S P 5 Is oonens have he for: W S v Is lnear nvaran has he for: v I Afer he ranson fro ass dens o ass hs nvaran orresonds o he anonal eresson nown n ehans as a anonal eresson of he relavs Larane funon [6] for a free arle aen wh he oose sn. 3. quaons of onservaon of ener and oenu n a edu wh losses he equaons for he onservaon of ener and oenu n a edu wh losses follow fro he and 5 n he for of her four-densonal dverenes and

4 . In he eneral ase, he s aser and for eah of s ndes one an wre down wo rous of equaons: q. 6 and 8 are equaons of he balane of eleroane and ehanal ener. q. 7 and 9 are equaons of he balane of eleroane and ehanal oenu. In q. 6 and 8, 7 and 9, he rh-hand sdes of he equaons are equal. hs s due o he ser of he ehanal ener-oenu ensor. Fro he equaons of onservaon of he ener dens 6 and 8 follows he equaon: or he lef sde of hs equaon s he dverene of he oenu dens n he nows s for, and he rh-hand sde s he dverene of he oenu dens n he Abraha s for. Fro q. 7 and 9, an no aoun he equal of her rh-hand sdes follows he equaon: leroane fores n a deler edu wh losses leroane fores, ore resel, he dens of eleroane fores n a onnuous deler edu wh losses are deerned n he for of dervaves of he dens of he eleroane oenu wh rese o e. hen q. 7 and 9 an be rearded as equaons of he balane of eleroane and ehanal fores n he edu. q. 7 for he oenu dens n he Abraha s for an be wren n he for of a balane of eleroane and ehanal fores: A 3 q. 9 for he oenu dens n he nows s for an be wren n he for of a balane of eleroane and ehanal fores: 3 3 alane fore equaons are dsnushed b he seond ers. Le us fnd hs dfferene n he for of he dfferene beween q. 3 and : 4

5 5 3 3 A he lef sde of hs balane fore equaon s he Abraha fore. Fnal equaon for he Abraha fore n a deler edu wh losses has he for: F A 4 hs equaon does no dffer fro he equaon of he Abraha fore for a lossless edu [6]. Fro q. and 3 follows ha n he resene of losses, eleroane fores beause ehanal fores n he edu ha erfor wor on oon, deforaon and hean of he edu. q. 4 for he Abraha fore shows ha s a vore reave fore ha does no deend on he losses n he edu. If he edu s desrbed b anonal aeral equaons of he for and and ε and µ are onsans or salar funons, hen he veors and, and are ollnear and he rh-hand sde of he q. 4 wh veor rodus of hese veors s ero. hen he Abraha fore s ero. If he edu s desrbed b he anonal aeral equaons, hen he non-daonal oonens of he sress ensor are ero, and he eleroane fores nfluenn he edu are deerned onl b s daonal oonens. hen A he q. 8 and of he fores balane n he edu an be wren n he for of a snle equaon: 5 Aln veor denes o hs equaon, we oban eleroane and ehanal fores n he eanded for: hs equaon desrbes he oal balane of eleroane and ehanal fores n a deler edu wh losses. In s lef sde here are eleroane fores, and on he rh sde here are ehanal fores. For onsan ε and μ hs equaon s slfed: Coarn ndvdual eleroane and ehanal fores n hs equaon one an see her olee orresondene o eah oher. 7 Conluson he desron of ener, eleroane oenu and eleroane fores n a deler edu wh losses are obaned. I s shown ha he Abraha fore s a reave fore and

6 does no deend on losses n he deler edu. A olee dealed balane equaon for all eleroane and ehanal fores n a deler edu wh losses s obaned. Referenes. I. rev, nows oenu resuln fro a vauu edu an roedure, and a bref revew of nows oenu eerens, Annals of Phss Rodro edna and J. Sehan, he ener-oenu ensor of eleroane felds n aer, arxv: aro G. Slvernha, Revsn he Abraha-nows lea, arxv: Joseh J. sonano, leroane oenu n a eler: a a o ass Analss of he nows-abraha ebae, arxv: Yur A. Srhev, quaon for he Abraha fore n non-ondun edu and ehods for s easureen, arxv: Yur A. Srhev, A new for of he ener-oenu ensor of he neraon of an eleroane feld wh a non-ondun edu. he wave equaons. he eleroane fores, arxv: hael. Crenshaw, he Role of Conservaon Prnles n he Abraha--nows Conrovers, arxv: C.J. Sheard,.A. Ke, Phs. Rev. A Nesereno, A.. Nesereno, J. ah. Phs C. Wan, Is he Abraha eleroane fore hsal? O, 6 7, Pablo L. Saldanha, J. S. Olvera Flho, dden oenu and he Abraha-nows debae, arxv: asso esa, A Coarson beween Abraha and nows oena, Journal of odern Phss, 6, 7, L. Zhan, W. She, N. Pen, U. Leonhard, New J. Phs G. era and K. P. Snh, Phs. Rev. Le. 5, Cho, Par, llo S, Oh K, Ooehanal easureen of he Abraha Fore n an Adaba Lqud Core Oal Fber Waveude, arxv: Landau L, Lfshs he Classal heor of Felds Oford: Peraon Press,