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1 new form of he ener-momenum ensor of he neraon of an eleromane feld wh a non-ondun medum. The wave equaons. The eleromane fores Yur. Sprhev Researh and esn Insue of Rado-leron nneern - branh of ederal Sae Unar nerprse of ederal Senf-Produon Cener "Produon ssoaon "Sar" named afer hael V.Proseno" -mal: ur.sprhev@mal.ru aed: prl, 7 bsra The arle desrbes a new approah o obann he ener-momenum ensor of eleromane feld n medum whou he use of awell's equaons and Ponn heorem. The ener-momenum ensor has new quales and onsequenes. Is lnear nvaran s Larane dens of he eleromane feld. rom he ensor follows he equaon of onservaon of ener dens, he equaon of flu ener dens and wave equaons for hese ener values. Wave equaon for momenum dens desrbes smulaneous ransfer of momenum and anular momenum reardless of radaon polaraon. rom he ensor follow he balane equaons of he eleromane fores for he momenum dens n he forms of he nows and braham, whh proves her equal and muual supplemenaon. quaon for braham fore s obaned as well. I s shown ha s dverene s equal o ero. Tensor and he balane equaons of eleromane fores n a onnuous medum are derved. Kewords: he ener-momenum ensor, braham-nows onrovers, he eleromane fores, braham fore, wave equaons, momenum dens, anular momenum. The onens. Inroduon.. The ensors of he eleromane feld and eleromane nduon.. The ener-momenum ensors n elerodnams. 4. ervaon of ener-momenum ensor from feld and nduon ensors. 5. The equaons for he onservaon and wave equaons for ener and momenum of he eleromane feld. 6. leromane fores n onnuous meda and her ensor. 7. Conluson bloraph

2 . Inroduon The problem of neraon of eleromane felds wh medum s ben dsussed for ears, bu a unque soluon has no been found e. In reen ears aemps o reae maerals wh unque eleromane values are ben underaen. Therefore, he ssue of neraon of wh medum has one o he fronburner. leromane fores n onnuous medum are usuall epeed o be found n he form of four-dmensonal dverene of ener-momenum ensor of T [], plan a e role n hs as. The problem of fndn eleromane fores n a onnuous medum an be dvded no wo pars. The frs one s o hoose he form of T. The seond one s o hoose maeral equaons desrbn he eleromane properes of he medum. The am of hs arle s o show he soluon o he frs problem, whh sas ha here s no defne answer o he queson wheher one of he nown forms of T s orre. The mos frequenl dsussed forms of T are nows s ensor and braham s ensor. or eample, hs s done, n he arles [] - [], [4] []. In he arles [] - [], [5], [8], [] he auhors are ondun a omparave analss of he resuls follown from T n he forms of nows and braham and ve her preferene o braham s ensor. In he arles [4], [6], [7], [9] advanaes of T n he form of nows and dsadvanaes of T n he form of braham are shown. ordn o he auhors opnon n he arles [4] and [6] braham s ensor s onsdered o be non-relavsall ovaran, and herefore he preferene s ven o nows s varan of ensor. In he arle [] s noed ha "n mos suaons, he resuls obaned on he bass of he of braham s and nows s ensors, are absoluel denal". ordn o he auhors opnon n he arle [], n he framewor of purel marosop approah, s mpossble o mae an unambuous hoe of T form. n mporan aspe of hs debae s he queson of he esene of he braham fore. Ths fore appears when omparn ensors of nows and braham as a neessar addon o he braham s ensor. One of he reasons for esene of dfferen pons of vew on he T forms s he la of sron mahemaal mehod of dervaon of he T. In he well-nown researh wors s obaned no b mahemaal dervaon, bu b he mehod of onsruon from he dfferen pars. rom awell equaons and he epresson for he Loren fore wh he use of Ponn s heorem he e he equaons whh are nerpreed as equaons of onservaon of ener and momenum. urher, members of hese equaons are nerpreed as dervaves of he T omponens. These "buldn" pars of he T are he ener dens and he momenum dens of he, ener flu dens he Ponn veor, a hree-dmensonal ensor of momenum flu dens or hreedmensonal sress ensor. Ths mehod has a eran freedom n hoosn omponen pars of T and leads o he fa ha he, somemes, are hosen b he auhors on he bass of eneral onsderaon and undersood n dfferen was. Ths provoes a debae. Ths mehod was used o

3 buld T n he forms of nows, er eavsde, braham, elmhol - braham, braham rlloun Paevs, Polevo Rov, e. These forms of T orrespond o a eran represenaon forms of eleromane fores. Ths mehod of dervn T has a snfan drawba. The man drawba s ha relave deler and mane permv of he medum n he Ponn s heorem are onsdered o be onsan [], s a resul, he obaned resuls are no unversal. In relaon o eleromane fores n he arle [] s noed ha he are obaned b "onsequenal mehod. eanwhle, s desrable o oban fore and oher quanes ener dens, ener flu, momenum dens wh he help of a unfed mehod based on he feld equaons". The feld equaons ssem of awell's equaons are derved from he ansmmer ensor of he, so s mehodall reasonable o derve T drel from eleromane feld ensor, omn he sep of obann awell's equaons, Ponn s heorem and he hoe of he omponen pars of T. Wh hs mehod n hand, he form of he T and follown from he equaons for ener and momenum are ompleel and unquel deermned b he ansmmer ensor of. Thus, no reasons for a debae arse. Ths arle presens sron onssen mahemaal approah for dervn T, he equaons of onservaon of dens of eleromane ener and momenum dens of eleromane fores, he wave equaons for he dens of ener and momenum. The feaure of hs mehod s ha does no use awell's equaons and he Ponn s heorem bu uses fundamenal elemen - ansmmer ensor of he, whh onans all neessar nformaon. The ansmmer ensor of he self follows from he four-dmensonal dervave of he eleromane poenal ansmer ensor of - hrouh s deomposon no smmer and ansmmer ensors. Thus, he onl sarn pon of he proposed mehod s he fourdmensonal eleromane poenal μ. In he mehod presened below all he equaons of onservaon and balane of he eleromane fores resul from T n he form of s fourdmensonal dverene. Ths approah s smple, desrpve, and wh proper nal posulae and he absene of mahemaal errors ves he orre resul. T and follown equaons, derved wh he help of hs mehod, are suable for onsderaon of he eneral ases of neraon of wh he medum, sne no resrons on he forms of maeral equaons are mposed.. The ensors of he eleromane feld and eleromane nduon The eomer of spae-me s aen as he pseudo-uldean nows spae,,, []. our-dmensonal eleromane poenal μ, s respevel, defned as φ/,, where φ and are he salar and veor poenals of. smmer 4-ensor of he seond ran νμ s derved as four-dmensonal dervaon of he eleromane poenal μ : /, / r, r

4 4 where r - oordnaes n uldean spae. The asmmer ensor of he eleromane feld νμ n mar represenaon has he form: rom hs ensor, n he form of s four-dmensonal dverene for eah of he ndes and, follow he equaons of n he poenals s assumed ha he feld soures are mssn: The frs wo equaons are he awell s equaons n he Loren albraon whou he soures, and he oher wo are dervaves of he albraon of he Loren ondon. ansmmeraon and smmeraon, ensor an be unquel deomposed no smmer and ansmmer ensors: ] [ In hs form ansmmer ensor of he n he mar represenaon has he form: ] [ where - eler feld nens; - mane feld nduon. rom hs ensor follow awell's equaons for vauum or mrofeld. Zommerfeld. [] dvded he eleromane values no fore values and quan values. ore values: eler feld nens and he mane feld nduon. Quanave values: he nduon of eler feld and he mane feld nens. Pars of values and, and an be ombne respevel no he ansmmer ensors of he ] [ and eleromane nduon I ] [ f []. We wre he ensor I ] [ f b analo o he ensor of n he form: ] [ f 4

5 rom hs ensor awell's equaons for onnuous meda follow. The relaonshp beween and, and s defned b maeral equaons. or vauum or mrofeld, and /, where and are, respevel, eler and mane onsans. or wea n sorop non-ferromane deler medum whou dsperson s usuall ae he maeral equaon n he form: и / 5 Where and are he relave deler permv and mane permeabl of he medum. where. The ensors of ener-momenum n elerodnams The anonal T n he eneral form s: W T S W ener dens; ν, µ=,,, ;, =,, 6 S he ener flu dens Umov-Ponn veor; he dens of momenum; dens momenum flu ensor he enson ensor. The reeved nows s and braham s ensors, whh were obaned on he bass of Ponn s heorem have reeved he reaes prevalene n elerodnams. Componens T 6 n he form of nows have he form: W / S /. Componens T 6 n he form of braham have he forms: W / S / /. fer subsuon of he omponens of braham s form n T 6, beomes smmer. 4. The dervaon of he ensor of ener-momenum ensor of felds and nduon We wll derve he T 6 drel from he ensors of he and I 4 whou he nvolvemen of awell's equaons and Ponn s heorem. ner values are he produs of he fore values and he quan values [, p. ]. The ener of neraon of wh he medum s a quadra form from he enson and nduon of eler and mane felds. Sne he nens and nduon of eler and mane felds are he omponens of he ensors and 4, he quadra form of her omponens are omponens of he T. Therefore, we wll e T n he form of a salar produ of ensors of he and I 4. 5

6 The salar produ of wo ensors of he seond ran are ompued usn he formula [ p. 8]: P a b ν, µ=,,, man he follown replaemen: a [ ] and b f [ ] we wll oban: T [ f ν, η, µ=,,, ] [ ] where he members wh he same ndes are summed. Ths formula wll le us fnd he omponens of T 6: T T T T T T T T / / / T T T T T T T T These omponens of T an be wren as: where W S / T 7 W, S,,, =,, Comparson of he omponens of T wh nows s and braham s ensors shows ha s lose o nows s T and dffers from b daonal omponens,.e., he ener dens W and he daonal omponens of he hree-dmensonal ensor of flu dens of momenum. The man op of dsusson n arles []-[] s he nd of momenum dens n T. In he form of nows momenum dens s represened as =, bu n he form of braham s represened as =/. The dens of he momenum n T 7 has he form denal o nows s. In he eneral ase T 7 s asmmer as well as nows s T. or he medum desrbed b he maeral equaons 5, T 7 has a smmer form: T / / or vauum and mrofeld T 7 also has a smmer form: 8 T / / / 9 Lnear nvarans T of 7 9 are 6

7 I I I / These nvarans represen he Larane dens of he, and a lnear nvaran I for he mrofeld s also a quadra nvaran of ansmmer ensor of he. Suh posve quales are absen n nown forms of T. 5. The equaons for he onservaon and wave equaons for ener and momenum of he eleromane feld The equaons of onservaon of eleromane ener and momenum follow from T 7 n he form of s four-dmensonal dverene. In eneral ase T 7 s asmmer and for eah of s ndes wo roups of equaons an be ven an no onsderaon he form of equaon of T 7, s possble no o mae an dsnons beween ovaran and onravaran ndes: а T and б T or а W S and б W S In he frs roup we wll reeve he equaons of ener dens onservaon of and ener flow S: In he seond roup we wll reeve he equaons of ener dens onservaon and momenum dens n meda : / rom he equaons of he ener dens onservaon and follows he equaon: / or / or.e. he dverene momenum dens n he forms of he nows and braham are equal. Tan dervaves wh me from boh pars of he las equaon, we wll oban: or The epresson n braes represens he braham fore. Consequenl, follows from T 7 ha he dverene of he braham fore s equal o ero. Ths onluson follows from he nows s ensor. There are no resrons on onsuve equaons n equaons. Therefore, equaons are unversal and desrbe he onservaon laws of ener dens, eleromane ener flu dens and momenum dens n all pes of maeral equaons for 7

8 and I. The epresson n braes represens he power of braham. Consequenl, T 7 follows ha he dverene of he braham fore s equal o ero. Ths onluson follows from he nows ensor. In equaons, here are no resrons on onsuve equaons. Therefore, equaons are ener and desrbe he onservaon laws of ener dens, flu dens eleromane ener and dens of he pulse n all pes of maeral equaons for and I. The resuln equaons were obaned for he saonar medum, bu due o relavs ovarane of and I ensors, hese equaons are also ovaran, and, when usn he nown formulas of ranson, are vald for a movn medum. or vauum or mrofeld equaons and are redued o one equaon 4 Tan no onsderaon ha he salar produ of he med omponens of he veor s equal o ero, equaons and are smplfed and also redued o one equaon: 5 pandn he seond erm of he equaon 4 an no aoun awell equaons, redue equaon o form: Subsun hs epresson no equaon 4, we wll oban /, or wh aura o onsans we wll oban equaon for he eleromane wave equaon, equaon 5 an be wren n he form: /. Tan no aoun hs 6 espe he fa ha he eleromane ener ravels n waves, here are no wave equaons for ener, ener flow and eleromane momenum n elerodnams. Le us oban hese equaons. Consdern he equaons 4 and 6 as a ssem, and dvdn he unnown quanes n a sandard wa, we wll derve he wave equaon for he ener dens of he eler feld: and he wave equaon for flu dens S of he eleromane ener S S 7 8 Tan no aoun he equaon mane feld ener: / equaon 7 ves us he wave equaon for he 9 8

9 vdn equaon 9 on, we derve he wave equaon for he dens of eleromane momenum: Thus, equaons 7 - desrbe he sruure of ener and momenum of eleromane radaon. Le us onsder hs queson n deal. The wave equaon an be wren n he form: I s nown ha eleromane radaon smulaneousl ransfers momenum and anular momenum. quaon desrbes boh of hese haraerss of radaon. If we elmnae he erm from he equaon, we wll oban he lassal 'alember wave equaon, whh desrbes he ransfer of momenum dens. The seond erm of he equaon desrbes he double rulaon of he momenum dens n a losed loop,.e. desrbes he ransfer of anular momenum dens. In elerodnams here are dfferen pons of vew on anular momenum of he, dsussed n he arles [4] - [6]. In he arle [5] s noed ha n lassal elerodnams he parado of "null-hel" of a plane eleromane wave ess, when he equaons of he do no desrbe he ransfer proess of anular momenum of a plane wave, whh onrads he deas of quanum elerodnams abou he nernal anular momenum spn of a phoon, ndependen of radaon polaraon. Ths problem s solved b he equaon, whh shows ha he erm, whh desrbes double roaon of he veor of momenum dens, desrbes he ransfer of he anular momenum dens. Ths orresponds o quanum elerodnams undersandn of he nernal anular momenum spn of he phoon. Sne he veor of momenum dens here has dual roaon, mples ha he eleromane wave has a orodal momen [7] of he momenum dens. ene, he nernal anular momenum of he phoon s also he orodal anular momenum of he eleromane feld. Thus, equaon elmnaes he parado of "ero hel" of eleromane wave, shown he spral of moon of eleromane ener n, reardless of radaon polaraon and brns he lassal elerodnams and quanum elerodnams loser o eah oher. 6. leromane fores n onnuous meda and her ensor leromane fores, more aurael he dens of eleromane fores n a onnuous non-onduve medum, are defned as dervaves of he eleromane momenum dens a he me. In he absene of eernal fores, hares and urrens, equaons and follown from T 7, an be onsdered as he balane equaons of eleromane fores n he meda. quaon for he momenum dens n he form of braham an be wren as: 9

10 / quaon for he momenum dens n he form of nows an be wren n he form: rom he nows s ensor wo smlar equaons for he momenum dens n he forms of braham and nows follow, bu he are no equal o equaons and : / / / rom he braham s ensor follows an equaon jus for he momenum dens n he form of braham: / / / The eleromane fores n a non-onduve medum are defned b wo quanes nduon of he eler feld and mane feld nens, whh respevel depend on he eleral and mane haraerss of he meda. Then equaon wh he momenum dens n he form of braham, whh nludes he mane feld nens, desrbes he eleromane fores assoaed wh mane haraerss of he medum, and he equaon wh he momenum dens n he form of nows, whh nludes he flu dens of eler feld, desrbes he eleromane fores assoaed wh he eleral haraerss of meda. or brev sae, we wll all hese he denses of eleromane fores respevel, he mane and eler fores. ased on hs, we an onlude ha from T 7 and he nows ensor, follow he desrpon of boh - eler and mane fores n he meda,.e. he eleromane fores are desrbed n full, and from T n he form of braham follows onl he desrpon of he mane fores. Ths suess ha he braham s ensor s nomplee. In eneral ase eler and mane fores have dfferen values, and he dfferene n hese eleromane fores s he braham fore. Sne he braham s ensor does no nlude hs fore, n order o oban all he fores we have o add hs fore. In eneral erms, braham fore s wren as he dfferene of he equaons for he hanes of he momenum n nows form and n he form of braham []: / rom equaons and follows ha he braham fore an be wren also n he form of a dfferene of he dverene of he sress ensor : or nall, he equaon braham fore an be wren as: / 4

11 Ths equaon also follows from nows s T. quaon 4 onfrms he onluson made n haper 5 ha he dverene of he braham fore s equal o ero. s n equaons and, here are no resrons on onsuve equaons, and he are unversal for an meda, hs apples o he equaons of eleromane fores - 4. rom T 7 and nows follow an equaon of fores for he momenum dens n he forms of nows and braham, hene he subje of dsusson abou whh of hese forms of momenum dens s he "rh" one, has no meann, as boh forms are orre and omplemen eah oher. rom equaon 4 follows an mporan onluson. If he meda s desrbed b he anonal maeral equaons of he form and /, and are onsans or salar funons, hen he veors and, and are ollnear and he rh sde of he equaon 4 veor produ of hese veors s equal o ero. Then he braham fore s equal o ero, and T 7 and nows s ensor are smmer. Thus, braham fore whh s ommonl wren as equae o ero. or hs ase, he nondaonal omponens of he sress ensor s equal o ero, and he eleromane fores an on meda, are deermned onl b s daonal members. Then he equaons of eleromane fores and an be wren n he form of one equaon: f / rom equaon 5 we an onlude ha dependn on manude relaon of he values of relave deler permv and mane permeabl of he medum he eleromane fore an hane s sn or beome ero. or he ase of ollnear veors and, and, when he braham fore s equal o ero, from nows s and braham s ensors follow smlar equaon for he eleromane fores: 5 f / / 6 Comparson of hs equaon wh he equaon 5 shows her prnpal dfferene. our-dmensonal eleromane fores are defned as four-dmensonal dervaves of T. The fore balane equaons and are obaned n he form of a dverene of T. u he eleromane fore an be obaned n a more eneral wa as he omponens of a ensor of eleromane fores T. To oban hs ensor we wll ae he four-dmensonal dervave of T 7. Sne dfferenaon nreases ensor ran, he T s a hrd-ran ensor: W S / T 7 or a beer undersandn of eleromane fores he omponens of T an be wren n he form of her four-dmensonal dervaves:

12 W S / Obaned equaons for eleromane fores are unversal for an meda, desrbed b he ensor of eleromane nduon. equaons 5: Le us derve equaons for eleromane fores n he meda desrbed b he maeral W S / / Le us fnd four-dmensonal dverene n eah of he pars of ndes of T 7 and we wll oban balane equaons of eleromane fores n a onnuous medum n he absene of eernal fores: ll equaons, obaned from T 7, are rue for a saonar medum, bu beause of s relavs ovarane, when usn he nown formulas of ranson, he are also vald for a unforml movn medum. 7. Conluson rom he ensors of he and I whou he nvolvemen of awell's equaons and Ponn s heorem we ve srl mahemaall derved T 7 from whh follow he equaons of onservaon of eleromane ener dens, ener flu dens and momenum dens. T 7 has an mporan feaure ha s no observed n oher forms of T, namel, ha s lnear nvaran presens self as a quadra nvaran of ensor and Laranan dens of a he same me, ombnn hese fundamenal ener values of he. New wave equaons for eleromane ener dens, ener flu dens and momenum dens, ha ddn es n elerodnams before, were obaned from T 7. The wave equaon for momenum dens desrbes ener sruure of eleromane radaon and smulaneous ransfer of he eleromane momenum and anular momenum, reardless of he polaraon of he radaon. Ths elmnaes he problem of "null-hel" of plane

13 eleromane wave and draws lassal elerodnams and quanum elerodnams loser o eah oher. Obaned balane equaons of eleromane fores n a onnuous medum lead o he onluson abou he equal and muual omplemenaon of nows s and braham s forms of momenum dens. Ths allows o end he dsusson on hs ssue. The equaon braham fore prevousl absen n he elerodnams s obaned from T 7. I s shown ha braham fore ess onl n he medum where he veors and, and are no ollnear. Tensor of eleromane fores s obaned from T 7. Possbl o derve equaons for eleromane ener dens, ener flu dens and momenum dens from T 7 proves s orreness. Obann hese equaons from he nown forms of Ts s mpossble. bloraph. Sobel sn V Sov. Phs. Usp Gnbur V L Sov. Phs. Usp Gnbur V L, Uarov V Sov. Phs. Usp ] 4. Veselao V G Phs. Usp aarov V P, Ruhade Phs. Usp Veselao V G, Shhavlev V V Phs. Usp avdovh V Phs. Usp aarov V P, Ruhade Phs. Usp Veselao V G Phs. Usp Topn I N, Levna K Phs. Usp Tonnela, undamenals of eleromanes and he relav heor, 96. Zommerfeld., lerodnam,.; 958. Kohn N.., Veor alulus and fundamenals of ensor alulus,.; Rodro edna, J Sephan, The ener-momenum ensor of eleromane felds n maer, arxv: arne S., Resoluon of he braham-nows dlemma, Phs. Rev. Le paes. 6. Saldanha P.L., vson of he momenum of elero- mane waves n lnear meda no eleromane and maeral pars, Ops press ansurpur, Resoluon of he braham-nows onrovers, Op. Comm Pablo L. Saldanha, J. S. Olvera lho, dden momenum and he braham- nows debae, arxv: v

14 9. assmo Tesa, Comparson beween braham and nows omena, Journal of odern Phss, 6, 7, -8. Cho, Par, llo S, Oh K, Opomehanal easuremen of he braham ore n an daba Lqud Core Opal ber Waveude, arxv: Crenshaw and T.. ahder, ner-momenum ensor of he eleromane feld n a deler, Op. Comm Iver rev, planaon for he ransverse radaon fore observed on a verall hann fber, arxv: ansurpur, Phs. Rev. Le., I. rev, Phs. Rev. Le., Joseph J. sonano, leromane omenum n a eler: a a o ass nalss of he nows-braham ebae, arxv: Pablo L. Saldanha, vson of he ner and of he omenum of leromane Waves n Lnear eda no leromane and aeral Pars, arxv: G.. Waler,. G. Laho and G. Waler, Can. J. Phs. 5, G.. Waler and. G. Laho, Naure London 5, I. rev, permens n phenomenoloal elerodnams and he eleromane ener-momenum enso, Phs. Rep. 5, 979, Se.... I. rev and S. Е. llnsen, eeon of he braham fore wh a suesson of shor opal pulses, Phs. Rev. 86, 58.. I. rev, Commen on Observaon of a push fore on he end fae of a nanomeer sla flamen eered b ouon lh, Phs. Rev. Le., I. rev and S. Е. llnsen, Transverse radaon fore n a alored opal fber, Phs. Rev. R 8, 86. Tomґas Ramos, Gullermo. Rublar and Yur N. Obuhov, rs prnples approah o he braham-nows onrovers for he momenum of lh n eneral lnear non-dspersve meda, arxv:.58v 4. Vul fslon K S Phs. Usp Soolov I V Phs. Usp arabanov L Phs. Usp ubov V, Tosunan L lemenar phss of parle and aom nule,