About Longitudinal Waves of an Electromagnetic Field

Transcription

1 bou Longudnal Waves of an Eleromagne eld Yur. Sprhev The Sae om Energ Corporaon ROSTOM, "Researh and Desgn Insue of Rado-Eleron Engneerng" - branh of ederal Senf-Produon Cener "Produon ssoaon "Sar" named afer Mhael V.Prosenko", Zarehn, Pena regon, Russa E-mal: ur.sprhev@mal.ru Daed: Sepember, 7 bsra Ths arle presens he soluon o he problem of salar longudnal waves whn he framework of eleromagne poenal whou nrodung an addonal members no he anonal feld Lagrangan or he Mawell equaons ha ensure he esene of a longudnal wave omponen. The Lame equaon for he eleromagne feld s obaned, whh desrbes ransverse and longudnal waves, as well as he ok-podolsk wave equaon for longudnal elerosalar waves ha do no have a magne omponen. I s shown ha he erms of he Loren gauge ondon desrbe a four-dmensonal volume deformaon of he eleromagne feld. Ke words: eleromagne poenal, asmmer ensor, smmer ensor, ansmmer ensor, Mawell equaons, longudnal waves. The Conens:. Inroduon. nsmmer ensor of he eleromagne feld and Mawell's equaons 3. smmer and smmer ensors and Lagrangans of he eleromagne feld 4. The equaons of he eleromagne feld and longudnal waves 5. Referenes. Inroduon There s a problem of longudnal waves n elerodnams. rom he perspeve of quanum elerodnams, he ransmsson of he Coulomb fore s produed b longudnal phoons. However, aordng o Mawell s heor of he eleromagne feld EM, eleromagne waves are ransverse, and a plane eleromagne wave n a vauum does no have a longudnal omponen. Therefore, n quanum elerodnams he ransmsson of he Coulomb fore s performed b ero-spn ransverse phoons, whh are onsdered «unphsal». However, he longudnal neraon self s phsall observable, and real longudnal salar waves are

2 needed o eplan s propagaon n spae wh fne velo. Hene, numerous aemps o eend he sandard elerodnams have been made sne Omura 956, aronov and Bohm 963, and ohers unl a reen revew and analss of hs problem b Modanes 7. Reen ears arles [-9 are devoed o he problem of he esene of salar longudnal waves. The resuls of epermens on he deeon of salar longudnal waves of arfal and naural orgn are desrbed n [-3. ll aemps o nrodue a salar omponen no elerodnams an be dvded no wo he pah. On he frs he pah Omura, aronov and Bohm, ok and Podolsk, and ohers new erms are nrodued no he Lagrangan of he eleromagne feld, whh ensure he appearane of a longudnal omponen n he equaons of feld moon. On he seond he pah new salar erms ha provde a desrpon of he longudnal waves are nrodued no Mawell s equaons, on he bass of her graden nvarane. or hs purpose, eher he salar poenal of a new phsal feld [8, 9 s nrodued no Mawell s equaons or he alread known eleromagne poenal [-6 s used, Thus, all known aemps o solve he problem are redued o he nroduon of new addonal erms no Lagrangan or no he EM equaons. The esene of longudnal epanson/onraon waves mples he esene of an elas onnuum eleromagne vauum. However, n Mawell's heor he eleromagne vauum s nompressble. Ths proper s refleed n he EM desrpon n he form of an ansmmer ensor he rae of whh equals ero. Hene, aemps have been made o nrodue an elas onnuum b analog wh a onnuous elas medum. n eample would be elerodnams based on oka-podolsk Lagrangan [7, n whh an eleromagne analogue of he Lame equaon or he dnam Naver-Sokes equaon s onsrued. Ths represenaon of an elas onnuum orresponds o he deas aeped n quanum elerodnams n regard o eleromagne vauum as plasma, onssng of vrual elerons and posrons. In suh plasma, ransverse and longudnal waves an propagae. In elerodnams, here s also a problem of volaon of Newon's hrd law relaed o he longudnal neraon of urrens. Ths led o he appearane of a hpohess abou he esene of a «salar poenal magne feld» [5, nroduon of whh no Mawell's elerodnams makes possble o preserve he fulfllmen of Newon's hrd law and desrbe he longudnal neraons of urrens. The esene of he "salar magne feld" s onfrmed b dfferen auhors who have ondued epermens on he longudnal neraon of urrens [4-6. In he arles [8, 9 he problem of longudnal neraon s solved b nrodung of he four-dmensonal salar poenal of a new phsal feld no he elerodnams Ths salar poenal s ndependen and s no relaed o Mawell's eleromagne poenal b dfferenal

3 relaons. The onneon beween hs poenal and he lassal heor s realed a he level of soures of he eleromagne feld - harges and urrens. Thus, hs s an addve heor n regard o he Mawell s heor. However, he nroduon of new phsal enes makes sense onl when he soluon o he problem s mpossble n oher was. The am of hs arle s o solve he problem of salar longudnal waves whn he framework of he anonal eleromagne poenal whou nrodung an addonal erms no Lagrangan or no he EM equaons ha ensure he esene of a longudnal wave omponen. EM and eler harges are onsdered n a vauum. The spae-me geomer s aken n he form of pseudo-euldean Mnkowsk spae,,,. The four-dmensonal eleromagne poenal s defned as /,, where φ and are he salar and veor poenals of he EM. The four-dmensonal urren dens s defned as J, J, where ρ and J are he eler harges dens and urren dens. seond rank:. nsmmer ensor of he eleromagne feld and Mawell's equaons EM n four-dmensonal form s desrbed b he anonal ansmmer ensor of he [ Ths ansmmer ensor s a four-dmensonal ovaran roor he omponens of he ensor [ are he dervaves of he salar φ and veor of EM poenals, whh are defned as he srengh omponens of he eler feld E and he magne feld nduon B: E B The Mawell s equaons are obaned from he ensor n he form of s four-dmensonal dvergene under one of he ndes o whh he feld soure s equaed [7 [ J. Le us wre hese equaons n veor form: E / or / B E J or J 3 rom he EM ensor, n he form of he well-known ensor den for an ansmmer ensor, follow wo more of Mawell s equaons: B or 4 3

4 4 B E or 5 3. smmer and smmer ensors and Lagrangans of he eleromagne feld or he esene of longudnal waves of epanson/onraon of EM, here mus be an elas onnuum eleromagne vauum. In he heor of onnuous meda, an elas medum s desrbed b a smmer ensor. Le us onsder he possbl of desrbng suh an envronmen for EM The defnon of a anonal ansmmer ensor [ nludes an asmmer ensor of he seond rank, whh s a four-dmensonal dervave of he eleromagne poenal. Le us denoe as and wre hs ensor n he mar form: 6 Ths asmmer ensor an be unquel deomposed no smmer and ansmmer ensors / / [. Then he anonal ansmmer EM ensor an be wren n he form [. I s lear from hs deomposon ha, n addon o he ansmmer ensor [, one an jusfabl nlude a smmer ensor no he EM desrpon, more presel, hs ensor s mpll onaned n he anonal desrpon of EM [. In he heor of onnuous meda, he ansmmer dsplaemen ensor of a medum s assoaed wh s roaon as a whole, and he smmer ensor s onneed b longudnal and shear deformaons of he medum. Usng hs analog, he ansmmer ensor [, whh s a four-dmensonal ovaran roor, an be assoaed wh he four-dmensonal roaon of he EM, and he smmer ensor an be assoaed wh four-dmensonal deformaons of he EM. Le us wre he smmer EM ensor n he mar form: 7

5 The dagonal omponens of hs ensor desrbe a four-dmensonal volumer deformaon of he EM and represen a four-dmensonal dvergene of he eleromagne poenal. The Loren gauge ondon s wdel used n elerodnams. Thus, he phsal essene of he Loren gauge ondon s he elmnaon of four-dmensonal volume deformaon from he EM equaons. Naurall, when he ondon, s mposed, longudnal waves and neraons of longudnal urrens are eluded from elerodnams. The dvergene of he eleromagne poenal s a salar and s used n mos researh papers o modf he Mawell s equaons. Ths s done b nrodung s hree-dmensonal erms, n he form of addonal erms, no Mawell's equaons. Sne hs dvergene s alread onaned n he EM equaons, s renroduon s ponless. [ The energ-momenum ensor T S [ orresponds o he ansmmer ensor [ obaned n [8. rom hs energ-momenum ensor, n he form of s lnear nvaran, follows he anonal Lagrangan of free EM: L S The smmer ensor [ 4 / E / B desrbes he four-dmensonal deformaon of he EM. Ths ensor orresponds o he energ-momenum ensor desrbng hs deformaon energ. The energ-momenum ensor s obaned from he ensor b he mehod desrbed n [8 T S. rom hs energ-momenum ensor, n he form of s lnear nvaran, follows he EM Lagrangan assoaed wh he four-dmensonal deformaon of EM: L S k k [ 4 The desrpon of he oal energ of he EM an be obaned n he form of an energ-momenum ensor followng from an asmmer ensor. lnear nvaran of he energ-momenum ensor T NS s he oal Lagrangan of EM assoaed wh he four-dmensonal deformaon and he EM roaon: L NS k k I s obvous ha L NS =L S +L S. The omponens of he Lagrangans L S, L S and L NS presen he omponens of he EM energ. The omponens of he L S represen he energ of he fourdmensonal roaon of he EM. The omponens L S represen he energ of four-dmensonal 5

6 deformaon of EM. The omponens of he L S are omponens of he oal energ of he EM. I s neresng o noe ha n he omplee Lagrangan L NS here are no onsuen pars of he oal energ and. Ths s due o he fa ha hese pes of energ are nluded no Lagrangans L S and L S wh dfferen sgns hene, hese pes of energ are fous. Lagrangans L S, L S and L NS an be appled n quanum elerodnams. 4. The equaons of he eleromagne feld and longudnal waves The oal four-dmensonal dvergene of he smmer ensor an be nonero, so we equae wh he four-dmensonal soure of EM J Ths equaon s / equvalen o he equaons J and J. Le us wre down hs fourdmensonal dvergene of he smmer ensor n hree-dmensonal form: / J Eq. 8 s salar and analogous o he Mawell s Eq.. Eq. 9 s a veor equaon and analogous o he Mawell s Eq. 3. Eq. 9 an be wren n he form: J In hs form Eq., represens a omplee eleromagne analogue of he Lame equaon or he Naver-Sokes dnam equaon desrbng he wave moon of a onnuous medum n he lnear heor of elas [9: U U U G where U s he dsplaemen veor of he medum, υ s he velo of longudnal waves, υ s he velo of ransverse waves, and G s he eernal fores. omparson of he Lame equaon wh Eq. shows ha he propagaon velo of longudnal EM waves s 8 9. B he mehods adoped n he heor of elas [9, he waves desrbed b Eq. an be deomposed no longudnal and ransverse waves. In he arles [8-9, he Lame equaon was aken as he bass for onsrung elerodnams wh a longudnal omponen. or hs purpose a new ndependen salar poenal s nrodued n elerodnams. In our ase, he eleromagne analogue of he Lame equaon for EM auomaall follows from he EM ensor whou resorng o an addonal phsal enes and hpoheses, Thus, Eq. shows 6

7 ha he eleromagne analogue of he Lame equaon alread ess n elerodnams and s nroduon wh he help of feld Lagrangan s no requred. or he sa ase, Eqs. 8 and an be wren n he form: / J Eq. for he sa ase ondes wh he Mawell s Eq. and desrbes he Gaussan law for a onsan poenal eler feld. The Eq. dffers from he Mawell s Eq. 3 for he saonar ase b he presene of he frs erm wh he dvergene of he veor poenal. Ths erm represens he graden of he hpoheal «salar poenal magne feld» nrodued b Nkolaev [5 no Mawell's Eq. 3 n order o eplan he longudnal neraon beween sead urrens and he observane of Newon's hrd law n elerodnams. Eq. ondes wh he equaon gven n [6. In hs paper, Eq. was onsrued emprall on he bass of epermenal resuls b supplemenng he Mawell s equaon. In hs arle Eq. s obaned srl mahemaall, as a onsequene of he dvergene of he smmer EM ensor. Thus, Eq. 9 elmnaes he problem of volang of Newon's hrd law n elerodnams. Eqs. 8 and 9 an be wren n he form: / and J pplng he Loren gauge / o hem, we wll oban Mawell's anonal equaons n he Loren gauge [7: / J Le us ake he roor from boh sdes of Eq. 9 and oban he anonal wave equaon for he magne nduon B: J or B B J Le us ake he dvergene from boh sdes of Eq. 8 and he me dervave of Eq. 9. In he resul of summaon of hese equaons and applng smple he ransformaons we wll oban he anonal wave equaon for he eler feld nens E: / J or E E J / 4 Le us ake he dvergene of boh sdes of Eq. and we wll oban he wave equaon: 3 J / 5 7

8 Ths equaon desrbes he longudnal waves of he dvergene of he veor poenal or he wave of he hpoheal «longudnal salar magne feld» of Nkolaev. dsn feaure of hese waves s he absene of he omponen of magne nduon B n hem. Therefore, hese waves orrespond o he name of longudnal elerosalar waves. Takng he me dervaves of boh sdes of Eq. 8 and he dvergene from boh sdes of Eq. 9 and addng he resuls, we wll oban he equaon: J или Ths equaon s he equaon of longudnal elerosalar waves of he dvergene of he eleromagne poenal known n he elerodnams of oka-podolsk and ohers [7-9. The lef-hand sde of Eq. 6 an be ero for an eleromagne poenal ha s no equal ero. Then Eq. 6 an be wren n he form of wo free-sandng equaons: J or J and J rom hs equaon follows ha he wave equaon of elerosalar waves / 6 s jus as fundamenal as he urren dens onservaon equaon J. Ths means ha he moon of harges n aordane wh he equaon J alwas ees he longudnal waves /. Thus, Eqs. 8 and 9 desrbe longudnal neraons of harges and urrens. ll he bas anonal wave equaons of elerodnams follow from hem. Eq., whh s an eleromagne analogue of Lame or Naver-Sokes equaon of he sorop elas medum moon, shows he general laws of moon of all knds of maer. Ths analog allows us o onsder he feld of he four-dmensonal eleromagne poenal as a phsal medum n whh eleromagne waves propagae due o he dnam deformaon of hs medum. Consequenl, he feld of eleromagne poenal an be denfed as a «phsal vauum» or «eleromagne vauum», whh s he soure of vrual phoons and oher elemenar parles. 5. Conluson The problem of longudnal neraon n elerodnams an be solved on he bass of he anonal eleromagne poenal whou nvolvng addonal phsal enes and hpoheses. To solve hs problem, s proposed o appl he epanson of he four-dmensonal dervave of he eleromagne poenal no smmer and ansmmer ensors. rom he 8

9 ansmmer ensor follow Mawell's anonal equaons. rom he smmer ensor follow he Lame equaon for he eleromagne feld, whh desrbes he ransverse and longudnal neraons from he smmer ensor follow all he bas anonal EM equaons, as well as he ok-podolsk wave equaon for longudnal elerosalar waves ha do no have a magne omponen. I follows from he smmer ensor ha he erms of he Loren gauge ondon desrbe a four-dmensonal volume deformaon of he EM. Referenes. G. Modanes, Covaran formulaon of haronov-bohm elerodnams and s applaon o oheren unnelng, arxv: L.M. Hvel and G.C. Gakos, Toward a more omplee elerodnam heor, In. J. Sgnals & Imagng Ss. Engr. 5, I. rbab and Z.. Sa, On he generaled Mawell equaons and her predon of elerosalar wave, Prog. Phs., Dale. Woodsde, Three-veor and salar feld denes and unqueness heorems n Euldean and Mnkowsk spaes, m. J. Phs., Vol. 77, No. 5, pp , Ma 9 5. Van Vlaenderen K.J., Generalsaon of Classal Elerodnams for he Predon of Salar eld Effes, arxv: Van Vlaenderen, K.J. and Waser, Generalsaon of lassal elerodnams o adm a salar feld and longudnal waves, Hadron J., 4, V. ok and B. Podolsk, On he quanaon of eleromagne waves and he nerang of harges on Dra s heor, Sow. Phs., p. 8 93; B. Podolsk and V. ok, Dervaon of Mollers formula from Dra s heor. Sow. Phs., p D. V. Podgan, O.. Zamdoroga, Relavs Dnams of a Charged Parle n an Elerosalar eld, arxv: D. V. Podgan, O.. Zamdoroga, Nonrelavs heor of elerosalar feld and Mawell elerodnams, arxv: G. C. Gakos and T. K. Ish, Deeon of longudnal eleromagne felds n ar, Mrowave Opal Tehn. Le. 6, O.. Zamdoroga n Elerosalar Energ of he Sun: Observaon and Researh, Journal of Modern Phss, 6, 7,

10 . C. Monsen and J. P. Wesle, Observaon of salar longudnal elerodnam waves, Europhs. Le. 59, Ignaev, G.. and Leus, V.., On a superlumnal ransmsson a he phase veloes, n Chubkalo,.E., Pope, V. and Smrnov-Rueda, R. Eds.: Insananeous on a a Dsane n Modern Phss: Pro and Conra, Nova Sene Publ., NY, pp L. Johansson, 999 Longudnal Elerodnam ores and Ther Possble Tehnologal pplaons. Maser of Sene Thess, CODEN: LUTEDX/TET-57/- 55/ G.V. Nkolaev, Nononrador Elerodnams. Theores, epermens, paradoes, Tomsk, NTL Publshng, K. Tomln, Generaled elerodnams, Eas Kaakhsan Sae Tehnal Unvers, Mnsr of Eduaon and Sene, Republ of Kaakhsan 9 [n Russan. 7. L. D. Landau, E. M. Lfshs, The Classal Theor of elds, Oford: Pergamon Press, Yu.. Sprhev, new form of he energ-momenum ensor of he neraon of an eleromagne feld wh a non-ondung medum. The wave equaons. The eleromagne fores, arxv: L. D. Landau, E. M. Lfshs, The Theor of elas, Oford: Pergamon Press, 983.