On the description of electromagnetic fields in slow moving media Abstract. Key words 1. Introduction

Transcription

1 O the desriptio of eletromageti fields i slow movig media Rozov Adrey Leoidovih St. Petersburg State Polytehi Uiversity Pargolovskaya st., 0-40, St. Petersburg, Russia, rozov20@mail.ru\ Abstrat. At preset, to desribe eletromageti fields i movig media Mikowski equatios obtaied o the basis of theory of relativity are used. But importat eletromageti proesses ru uder o relativisti oditios of slow movig media. Therefore, oe should arry out its desriptio i terms of lassial mehais. Aalysis of Mikowski equatios, preseted i the paper, revealed a disrepay betwee a physial model, whih is the base of the equatios, ad kow lassial mehais iformatio: Faraday s experimetal fidigs ad Maxwell theory. As a result, Mikowski equatios are ot a optimal istrumet to arry out aalysis of eletromageti proesses uder o relativisti oditios of slow movig media. The paper proposes a way of desriptio of eletromageti fields i slow movig media o the basis of Maxwell theory withi the frame of lassial mehais. Reeived Galilea ivariat Maxwell equatios lak asymmetry i the desriptio of the reiproal eletrodyami atio of a maget ad a odutor ad oform to kow experimetal data. Key words: eletromageti fields, Maxwell's ad Mikowski equatios, slowly movig media PACS umbers: De, q. Itrodutio I the preset paper the term "eletrodyamis of movig media" is used istead of term eletrodyamis of movig bodies, as oe of the importat appliatios of eletrodyamis is a iteratio betwee eletromageti fields ad plasma. The study of the iteratio proesses betwee eletromageti fields ad movig media beomes ireasigly importat due to the problems of plasma ofiemet i thermoulear fusio reatios. At preset, these proesses are desribed usig Mikowski equatios, derived o the basis of relativity priiple (Mikowski 908) (similar equatios a be obtaied from Loretz eletro theory (see, e.g. Pauli 958)). I istallatios, whih implemet thermoulear fusio reatios, veloities of plasma motio, as a rule, are muh lower tha the relativisti oes. So, it is possible to desribe eletromageti fields uder these oditios i terms of lassial mehais. Le Bella ad L evy-leblod (973) offered a method for aalysis of eletromageti proesses based o Galilea limits of Mikowski equatios. Now this method is widely used (see, e.g. Brow ad Hollad 2003; Motigy ad Rousseaux 2007; Heras 200; Steimetz et al 20). But there are shortomigs of Galilea limits of Mikowski equatios whih have bee formulated by the authors. Amog them there is a importat oe - the lak of proper relatioship betwee movemet of the medium ad eletromageti fields (so the authors had to orret the limits) (Le Bella ad L evy-leblod 973). Mikowski equatios have bee used for the simulatio of plasma istability uder the effet of eletromageti fields i the proesses of thermoulear fusio durig more tha fifty years. But util

2 2 ow a effetive plasma ofiemet by mageti field i these proesses has ot bee solved. So, there are reasos to aalyze the appliability of Mikowski equatios uder o relativisti oditios of slow movig media. 2. Aalysis of Mikowski equatios O derivig eletrodyamis equatios for movig bodies Mikowski followed Poiare-Eistei s mathematial iterpretatio of the priiple of relativity, delarig all iertial systems to be equivalet with respet to physial laws. He hose the oordiate system i whih the body was statioary ad preseted Maxwell equatios for statioary body i it. The he trasformed these equatios i the laboratory oordiate system, relatig to whih the body moved, aordig to the priiple of relativity (Mikowski 908; Pauli 958; Sommerfeld 949). I the result Mikowski reeived equatios osisted of Maxwell equatios for statioary body ad ostitutive equatios whih take ito aout movemet of the body. The essee of Mikowski s method was formulated by Sommerfeld (949): the body kows othig of its motio. But this statemet otradits lassial mehais evidee of the eletromageti idutio pheomea, first of all Faraday s experimetal fidigs (Faraday 994) ad Maxwell theory expressed ito equatios () (2). They treat movemet of the body (medium) i these pheomea as relevat to the soure of mageti field ad, osequetly, idepedet of oordiate systems. More exatly, Faraday ad Maxwell used other terms relevat to the mageti lies of fore they have absolutely the same sese. Therefore, oe should say withi the frame of lassial mehais that the body (medium) does kow of its motio uder these pheomea. To uderstad the ause of this otraditio we have to step aside from Poiare-Eistei s mathematial iterpretatio of the priiple of relativity ad move over to its iitial Galileo s ow physial oe (Seeger 966). Galileo poited to the fat that some mehaial proesses o a movig large ship proeeded differetly depedig o whether or ot they ourred i a idoor abi below dek or o the dek i the ope air. I fat, Galileo formulated the relativity priiple of mehaial proesses oly for iertial systems that arry alog with themselves a laboratory with all its otets (Seeger 966). Thus formulated priiple of relativity together with Faraday s experimetal fidigs lead to the physially proved partiular oordiate system fixed with the soure of mageti field, relatig to whih motio of the medium have to be osidered. Let us follow the mehaism of geeratio of Mikowski equatios shortomigs uder o relativisti oditios of slow movig media. Cosider, for example, Mikowski ostitutive equatio whih has the followig form with the auray withi the terms of order v : D E v H (see, e.g. Ladau et al. 2004; Pauli 958; Sommerfeld 949). Substitutio of this equatio i other Mikowski equatios (Maxwell field equatios for statioary medium) trasforms derivatives of E ito oes of E ad v. As a result, Maxwell idutio equatio for statioary media (7) trasforms ito idutio equatio for movig media (6). But i aother Maxwell field equatio the term v 2 с t appears whih does t have a physial ature. This a have a effet ot oly o the auray of alulatios, but also

3 3 o the possibility of adequate ivestigatio of plasma istability i the pheomea i questio by meas of these equatios. So, we see that it is expediet to use alterative models of eletrodyamis uder o relativisti slow movig media oditios. Aordig to Krotkov et al. (999) eletrodyamis of slow movig media does ot eessitate speial relativity. Let us try to obtai a eletrodyami model for slow movig media withi the frame of lassial mehais. That s why we have to ome bak to the origial Maxwell theory worked out withi the limits of lassial mehais. 3. Derivatio of Maxwell equatios The basis of Maxwell eletrodyamis is omposed of two laws derived experimetally both for statioary ad movig media (see, e.g. Sommerfeld 949): Faraday idutio law s s ds d dt Ampere s law s 4 sds d, d ; () where E, H, B ad I are vetors of eletri ad mageti field stregths, mageti idutio ad urret (iludig displaemet urret) desity, respetively; E s ad H s are tagetial ompoets of vetors E ad H to a losed loop s; B ad I are ormal ompoets of vetors B ad I to a arbitrary surfae δ ofied by the loop s; t time ad the light speed i vauum. The right-had side () geerally desribes the emergey of eletromotive fore i the loop both as a result of variatio of mageti field passig through the loop area i time, ad also due to the motio of the loop i mageti field relative to the soure of mageti field. Without loss of geerality, we will assume that the soure of mageti field is statioary relative to the laboratory oordiate system. I that ase, equatios () (2) desribe the proesses takig plae i the movig loop i the laboratory oordiate system relatig to whih the medium motio is osidered. All further trasformatios of these equatios are arried out also i the laboratory oordiate system. O itegratig (), we apply the Stokes theorem to the left-had side of the equatio, ad i the right-had side we write the total time derivative i terms of its ompoets (i view of the fat that mageti field is a soleoidal oe): s ds d ; d dt s d vd, (4) t where v is the loop veloity vetor. By substitutig (3) ad (4) ito (), we obtai the equatio (2) (3)

4 4 vd 0. (5) t Takig ito aout that the area δ is arbitrary, from (5) oe obtais ( v). (6) t Equatio (6), like the iitial equatio (), desribes both possible variats of eletromageti idutio geeratio. I Mikowski equatios Maxwell idutio equatio uder oditios of statioary media is used as the equatio of eletromageti idutio for the proesses i movig media (Mikowski 908). (7) t Now whe itegratig equatio () for the ase of movig media, equatio (7) is also obtaied as the result of use the relativisti relatio (see, e.g. Beker 933; Paofsky ad Phillips 955). I fat, withi the frame of lassial mehais it meas that the right-had side of () is trasformed similarly to (4), whih orrespods to its represetatio i the laboratory oordiate system, ad the left-had side of () is treated as a expressio preseted i the oordiate system movig together with the medium. As a result of iorret treatmet of equatio () (the left-had ad right-had sides are osidered i differet oordiate systems), the resultig equatio (7) loses a portio of iformatio otaied i equatio (), ad the eed to use a additioal relatio whih takes ito aout the idutio due to motio (Loretz fore relatio) arises. By itroduig vetor potetial A, B= A ad salar potetial, we fid from (6) v. (8) t Let us write (8) i the followig form; mot mov, (9) where E mot is the stregth of eletri field idued by the variatio of mageti field i time mot, (0) t ad E mov the stregth of eletri field idued by the loop motio mov v. () Faraday ivestigated the idutio pheomeo usig odutors (Faraday 994). Aordig to Wilso ad Wilso (93), i ase of dieletris the stregth of eletri field idued by the loop motio is equal to E=E mov, where, (2) here is a dieletri ostat ad μ mageti permeability. So, for the geeral ase of odutors ad dieletris (9) is writte i the followig form:

5 5 E E mot E mov (3) (i ase of odutors =, beause μ ). By meas of oeffiiet it is possible, i partiular, to take a approximate aout of probable presee of impurities i plasma. The differetial form of (3), beig a geeralizatio of (6), looks like ( v). (4) с t с We itegrate equatio (2) takig ito aout all possible types of urrets. Usig the kow desriptios of urrets (see, e.g. Beker 933; Paofsky ad Phillips 955; Pauli 958), as a result of itegratio of (2) we have i a geeral ase D 4 4 j v ( D v), (5) с t с с с where D is vetor of eletri idutio; j odutio urret desity vetor ad ρ desity of free-movig harges. The last two terms of the right-had side of (5), that are abset i Maxwell equatios i statioary media, osider the mageti field idutio by a ovetio urret of harges movig together with the odutor (Rowlad urret) ad a urret that appears at the motio of a dieletri i the eletri field (Roetge urret). have the form Withi Maxwell theory (Maxwell 89) equatios (4) (5) should be appeded with D 4, 0, (6) as well as the equatios of state (ostitutive equatios), whih i media with liear parameters D,, j, (7) where σ odutivity. Ulike the orrespodig Mikowski equatios, the right-had sides of (7) have o members iludig the veloity. I preseted equatios the idutio due to motio is desribed i the field equatios (4) (5). Equatios (4) (7) ostitute Maxwell equatios of eletrodyamis uder oditios of slow movig media. There are 7 salar equatios for 6 ukows. Thus, oe of the equatios ( B=0) represets a additioal limitig oditio. 4. Boudary oditios for Maxwell equatios O the basis of equatios (4) (7) we obtai for a geeral ase of a movig disotiuity surfae the boudary oditios whih we will write as follows

6 6 α v v Rot 2 2, (8) Rot 4 ( 2 ) (js svs ) (v D) 2 (v D), (9) Div 4 D D 2 D s, (20) Div 2 0, (2) where Rot ad Div are symbols of surfae url ad surfae divergee, respetively; idies ad 2 belog to differet (i relatio to ormal ) sides of the boudary; τ is the taget diretio of the boudary; j s ad ρ s boudary surfae desities of the urret vetor ad harge, respetively, ad v s the vetor of boudary veloity. Let us ompare boudary oditios (8) (9) with the boudary oditios resultig from Mikowski equatios for slowly movig media (Ladau et al. 2004; Tamm 979) Rot Rot v 2 2, (22) 4 v j D D 2 s 2, (23) Div 4 D D 2 D s, (24) Div 2 0, (25) where v is the projetio of the boudary veloity vetor o the ormal. The differees of the boudary oditios orrespod to distitios i the equatios (boudary oditios (20) (2) ad (24) (25) oiide together with the orrespodig equatios). 5. Aalysis of Maxwell equatios i slow movig media For oveiee of the further aalysis we will trasform the reeived equatios. By substitutig (4) ad (5) with (6) ad (7), after obvious trasformatios basi field equatios will look as follows: E v, (26) с t B (27) с t с с v j v Oe a see that i ase of medium movemet absee the obtaied equatios will trasform to the kow Maxwell equatios i statioary media. Let us ote eessity of presee of fator i terms of the equatios otaiig medium veloity. At medium depressio the value of these terms will derease, thaks to redutio i, up to zero i emptiess (beause μ ) that orrespods to physis of the pheomeo. The member desribig Rowlad urret ad otaiig medium veloity will be disappearig i these oditios due to medium harge desity redutio.

7 7 Also, it should be oted that the derivatio of the Maxwell equatios does ot require the use of additioal relatio to aout for the idutio due to motio the left part of equatio (26) is obtaied without use of the Loretz fore equatio. Eistei (905) wrote that Maxwell eletrodyamis whe applied to movig bodies, leads to asymmetry i the reiproal eletrodyami atio of a maget ad a odutor whih is t iheret i the pheomeo. It is true oerig Maxwell equatios i the statioary medium (osidered by Eistei) whih do ot desribe eletrodyami pheomea due to motio. The problem was solved withi the theory of relativity. The offered Maxwell equatios whih are reeived withi the frame of lassial mehais desribe eletrodyami pheomea due to motio ad oe a see that these equatios lak asymmetry metioed by Eistei. Let us show Galilea ivariae of the obtaied Maxwell equatios. We show the ivariae of the equatios relatig to oordiate system, movig retiliearly ad i regular itervals relatig to iitial system with ertai veloity w: x x w t, i 2, 3, (28) i i i, where xi ad xi are oordiates of the iitial oordiate system ad ew oordiate system movig relative to it (axes of both systems are parallel), respetively; wi are ompoets of vetor w ad t is time i both systems. respetively. I the vetor form equatio (28) looks like v v w, (29) where v ad v are veloities of medium relatig to iitial ad ew oordiate system, Oe has withi the limits of the lassial mehais E E, B B. Let us write, for example, equatio (26) i ew oordiate system E v w, (30) с t where the rotor is determied o oordiates of ew system xi. Oe a see that (30) ad (26) have the same form. Similarly, equatio (27) keeps the form at Galilea trasformatios. 6. Compariso of the theory with experimetal data It is well-kow that Maxwell equatios i statioary media agree with orrespodig experimetal data. Let us aalyze a agreemet betwee experimetal data ad proposed Maxwell equatios i slow movig media. Let us osider the followig importat ases.. A uipolar idutio: geeratio of a eletromotive fore i a radial elemet of ylidrial permaet maget evely rotatig roud its axis. For these oditios we have formula () orrespodig to kow experimetal data (see, e.g. Ladau et al. 2004; Tamm 979); 2. Propagatio of a plae eletromageti wave i a omageti slow movig medium.

8 8 For the speified oditios i the absee of harges ad urrets, from (4) (7) we have B, t ( B v) D t 2 ( B v) 3 0, 0 4 D. D,, (3) Beause we are iterested oly i terms that are liear by the veloity, the for plae waves propagatig i the 0 diretio alog the ξ oordiate it is possible to assume ( 0 0 ) v v v, (32) u t 0 where u0 is the light speed i the statioary media, u0= m ad m the refratio idex of the medium. Takig ito aout (3)(3), we have ( B v) ( v ) B, ( D v) ( v ) D. (33) Substitutig (32) ad (33) ito (3) () (3) (2) ad takig ito aout (3) (4), we fid B с 0 v u, 0 t с 0 v u. (34) 0 t Calulatig a url from the first equatio (34) ad usig the seod equatio (34), we obtai 2 u 0 u v 2 2 t u 0 0 u0 This wave equatio orrespods to the propagatio veloity 2 v. (35) 2 t 2 2 u u0 0 v u0 0 v, (36) u0 whih to the auray withi the terms of order v oiides with kow experimetal data (Paofsky ad Phillips 955; Tamm 979; Toelat 966). 7. Colusio The Galilea ivariat equatios of eletrodyamis i slow movig media ad the orrespodig boudary oditios are derived o the basis of Maxwell theory withi the frame of lassial mehais. I ase of medium movemet absee the obtaied equatios trasform to the kow Maxwell equatios for statioary media. The derivatio of the Maxwell equatios i slow movig media does ot require the use of additioal relatio to aout for the idutio due to motio Loretz fore relatio. The reeived Maxwell equatios lak asymmetry i the desriptio of the reiproal eletrodyami atio of a maget ad a odutor.

9 A disrepay is revealed betwee a physial model, whih is the base of Mikowski equatios, ad kow lassial mehais iformatio: Faraday s experimetal fidigs ad Maxwell theory. 9 Mikowski equatios desribe pheomea of eletromageti idutio for slow movig media withi the frame of theory of relativity orretly, but at the same time i Mikowski equatios the term v 2 с t appears, whih does ot have a physial ature. As a result, Mikowski equatios are ot a optimal istrumet to arry out aalysis of eletromageti proesses uder o relativisti oditios of slow movig media. Both the aalysis ad the ompariso of the reeived Maxwell equatios with experimetal data prove the appliability of the offered equatios i pratie. Referees [] Beker R., Theorie Der Elektrizitat: Bad II, Eletroetheorie (Verlag ud Druk vo B.G.Teuber, Leipzig 933.) [2] Le Bella M. ad L evy-leblod J. M., Nuovo Cimeto B, (973). [3] Brow H. R. ad Hollad P. R., Studies i the History ad Philosophy of Moder Physis (2003). [4] Eistei A., Aale der Phys., 7, (905). [5] Faraday M., Experimetal Researhes i Eletriity. (Great Books of the Wester World, vol.42. Eylopedia Britaia. Uiversity of Chiago 994.) [6] Jose A. Heras, The Galilea limits of Maxwell's equatios, Am. J. Phys. 78, (200). [7] Krotkov R. V., Pellegrii G. N., Ford N. C. ad Swift A. R., Am. J. Phys. 67, (999). [8] Ladau L. D., Pitaevskii L. P. ad Lifshitz E. M., Eletrodyamis of otiuous media, 2d ed. (Butterworth Heiema, Oxfordm 2004.) [9] Maxwell J. C., A Treatise o Eletriity ad Magetism, 3rd ed, vol. (Claredo, Oxford 89) (reprited (Dover, New York 954.) [0] Mikowski H., Die Grudleihuge fur die elektromagetishe Vorgage i bewegte Koerper (Nahr. Ges. Wiss. Gottige 908.) [] Motigy M. ad Rousseaux G., Am. J. Phys., (2007). [2] Paofsky W. ad Phillips M., Classial Eletriity ad Magetism (Addiso Wesley, Cambridge 42, Mass. 955.) [3] Pauli W., Theory of Relativity (Pergamo, Lodo 958.) [4] Purell E.M., Eletriity ad magetism. Berkeley physis ourse. 2d ed. (Mgraw hill book 984.) [4] Sommerfeld A., Eletrodyamik (Akademishe Verlagsgesellhaft, Leipzig 949.) [5] Steimetz T., Kurz S., Clemes M., "Domais of Validity of Quasistati ad Quasistatioary Field Approximatios," COMPEL, vol. 30, o. 4, , 20. [6] Tamm I. E., Fudametals of the Theory of Eletriity (Mir, Mosow 979.)

10 0 [7] Toelat M.-A., The Priiples of Eletromageti Theory ad of Relativity (Reidel, Dordreht-Hollad 966.) [8] Wilso M. ad Wilso H. A., Phil. Tras. Roy. So., (93).