UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 20 Prof. Steven Errede LECTURE NOTES 20

Transcription

1 UIUC Phsis 436 M Fields & Sres II Fall Semester, 05 Let. Ntes 0 Prf. Steen rrede LCTUR NOTS 0 RLATIVISTIC LCTROYNAMICS f MOVING MACROSCOPIC MIA We hae disssed the relatiisti aspets f eletrdnamis assiated with ming eletri J, J. harges and eletri rrents: Fr marspi media, restriting rseles here (fr simpliit/larit s sake) t linear, hmgenes, istrpi materials, sh as lass A dieletris and magneti materials that are desribed in lassial eletrdnamis b the ailiar 3- marspi fields: B = salar qantit = salar qantit = marspi eletri permittiit = marspi magneti permeabilit f the material. f the material. Ke = salar qantit Km = salar qantit = relatie eletri permittiit f the = relatie magneti permeabilit f the material. material (a.k.a. dieletri nstant ). e Ke = salar qantit m Km = salar qantit = eletri sseptibilit f the material. = magneti sseptibilit f the material. e r: e m r: Nte that:,, K, K,, are all defined in the rest/prper frame f the linear material. e m e m Nte als that:, and hene: Z are all Lrent inariant qantities: 8 = speed f light/m waes in free spae/am = ms. marspi eletri permittiit f free spae/am = marspi magneti permeabilit f free spae/am = m Fm m Z = marspi impedane f free spae/am = Ths, fr linear marspi materials, the fllwing 3- etr qantities are defined in the rest/prper frame f the linear material: r t, r, t rt, ert, n prt, rt, rt, nmrt, m e and: B rt, rt, n.b.,, K, K,, are nt m = eletri plariatin = eletri diple mment per nit lme. = magnetiatin = magneti diple mment per nit lme. e m e m Lrent inariant qantities!!!,, are Lrent inariant!!! Prfessr Steen rrede, epartment f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis All Rights Resered.

2 UIUC Phsis 436 M Fields & Sres II Fall Semester, 05 Let. Ntes 0 Prf. Steen rrede Ths, we btain the {sal} nstittie/ailliar relatins fr linear media defined in the rest/prper frame f the linear material: r, t r, tr, t and: r, t Br, tr, t In rder t g er t a relatiisti frmlatin f these relatins, we mst be arefl/preise in their phsial meaning. In partilar, the eletri permittiit, and magneti permeabilit, are defined in the rest frame f the material i.e. is the prper eletri permittiit and is the prper magneti permeabilit. Ths, the marspi fields,, and,, B are all defined in the rest/prper frame f the linear material i.e. the are prper marspi eletrmagneti fields. Bease we alread knw/nderstand the Lrent transfrmatin prperties f and B, we an write the ailiar/nstittie relatins as: r t r t r t,,, and: Brt, rt, rt, The relatiisti M field tensr F is: F 0 0 B B B 0 B B B 0 and its relatiisti dal tensr G is: G 0 B B B B 0 B 0 B 0 We nstrt a rank- anti-smmetri tensr (analgs t F ) fr the and fields. It desribes the (relatiisti si-mpnent) eletrmagneti displaement field. {n.b. again, mparing frms in different tetbks, ariatins fr will be fnd that depend n the hie/definitin f the metri g sed and nentins r.e. erall nstants, et.!!! }. Taken tgether, the ailiar/nstittie relatins sggest replaing B B weer, it is a little leaner if we als diide thrgh b, i.e. replae. B F Prfessr Steen rrede, epartment f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis All Rights Resered.

3 UIUC Phsis 436 M Fields & Sres II Fall Semester, 05 Let. Ntes 0 Prf. Steen rrede Then bemes: SI nits hek: Cl Amps, m m Cl m Amp m se m We als nstrt the relatiisti dal f, whih is (the analg f G ), defined as: The relatiisti dal tensr ma als be btained diretl frm b arring t an apprpriate dalit transfrmatin n the and -fields, analgs t that fr the and B -fields: s Bsin 90 B B s sin s sin 90 s sin r: r: s sin B sins B s sin sins Then fr 90, we replae: and: in : Likewise, we ma analgsl define/nstrt a rank- anti-smmetri tensr fr the and fields desribing the (relatiisti si-mpnent) eletrmagneti plariatin field. Sine Clmbs has the same SI nits as and Amps has the same SI nits as meter meter we see that: 0 Nte the mins sign reersal here fr 0,, relatie t,, de t 0 s. BM!!! 0 Prfessr Steen rrede, epartment f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis All Rights Resered. 3

4 UIUC Phsis 436 M Fields & Sres II Fall Semester, 05 Let. Ntes 0 Prf. Steen rrede G Again, we an nstrt a relatiisti dal tensr f, fr F and/r, fr }, defined as: whih we all {the analg f Again, nte the mins sign reersal here fr,, relatie t,, de t s. BM!!! Again, the relatiisti dal tensr an be btained diretl frm b arring t the same dalit transfrmatin as abe, bt n the and fields. Sine has the same SI nits as, and has the same SI nits as we see that: s sin 90 s sin r: s sin sins Then fr 90, we replae: and: in : Frm the lassial eletrdnami ailiar/nstittie relatins: r: and: B in terms f the M field tensrs and their dals are: F and: and: B we see that their (mbined) relatiisti eqialents, G n.b. mst be + here!!! r: F and: G n.b. If we define F then we mst hae the + sign in bease the dal M field tensr G F nnets t: G F. G M 4 Prfessr Steen rrede, epartment f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis All Rights Resered.

5 UIUC Phsis 436 M Fields & Sres II Fall Semester, 05 Let. Ntes 0 Prf. Steen rrede Ths the marspi relatiisti M field tensrs and their dals are: F 0 0 B B B 0 B B B 0 G 0 B B B B 0 B 0 B 0 F Let s epliitl hek the rretness/alidit f these relatins assming things are internall rret within eah f the M field tensrs T, then we nl need t hek tw nn-er elements: Fr the 0 T mpnents: Fr the T mpnents: F r: G r: B B F r: BM B G r: Ths, we btain the relatiisti ailiar/nstittie relatins and their dals: F and: G r: F and: G Prfessr Steen rrede, epartment f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis All Rights Resered. 5

6 UIUC Phsis 436 M Fields & Sres II Fall Semester, 05 Let. Ntes 0 Prf. Steen rrede Lrent Transfrmatins f the Relatiisti Marspi letrdnamial Fields, F, G,, and Their als We alread knw/hae shwn that in the relatiisti M fields in frame IRF(S') are related t thse in IRF(S) ia: F F and G G, where the Lrent transfrmatin tensr. Fr linear marspi media, we hae linear relatins between,, and als between, B,, whih hld relatiistiall between F and als G. We epliitl see sing the Lrent transfrmatins f indiidal field qantities defined in IRF(S) t the frame IRF(S'):, F F and: That: F F F F hlds in IRF( S ). Likewise, fr the dal tensrs we epliitl see sing the Lrent transfrmatins f indiidal field qantities defined in IRF(S) t the frame IRF(S'):, G G and: That: G G G G hlds in IRF( S ). qialentl, we an alternatiel epress these as a simple Lrent transfrmatin f the marspi,, and B,, fields, e.g. fr a Lrent transfrmatin frm the rest/prper frame IRF(S) where the marspi,, and B,, fields are defined, t anther IRF( S ) ming with elit t ne f the ˆ, ˆ, r ˆ aes ( eg.. ˆ). 6 Prfessr Steen rrede, epartment f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis All Rights Resered.

7 UIUC Phsis 436 M Fields & Sres II Fall Semester, 05 Let. Ntes 0 Prf. Steen rrede We hae alread shwn {in P436 Letre Ntes 9 p. -3} that the mpnents f transfrm as: that are and t the bst diretin IRF( S ): IRF(S): IRF( S ): IRF(S): and B B B Ths: B B B B B where: and: IRF( S ): IRF(S): IRF( S ): IRF(S): And similarl: IRF( S ): IRF(S): IRF( S ): IRF(S): n.b. nte the sign reersals here relatie t the abe relatins!!! And we als hae: In IRF( S ): In IRF(S): P P BM BM n.b. We see frm these tw relatins that in the rest/prper frame IRF(S 0 ) f a plaried dieletri with eletri plariatin 0 (r a magnetied material with magnetiatin 0 ) if the linear material is ming e.g. with elit ˆ in the lab frame IRF(S), an bserer in the lab frame will see a mbinatin f eletri plariatin and magnetiatin in bth ases! n.b.,, are all Lrent inariant qantities. Prfessr Steen rrede, epartment f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis All Rights Resered. 7

8 UIUC Phsis 436 M Fields & Sres II Fall Semester, 05 Let. Ntes 0 Prf. Steen rrede Relatiisti Inariants Assiated with Marspi Relatiisti M Fields We hae alread shwn/disssed that fr the and B -field tensrs and F G, there are tw and nl tw relatiisti Lrent inariant qantities: ) F F F F G G G G B M field energ densit differene U U M 4 ) F G FG G F GF B Transersalit f and B Sine the (inner) prdt f a ntraariant tensr f rank-n with an ther ariant tensr f the same rank-n is a Lrent inariant qantit (i.e. same ale in an inertial referene frame, we see that fr, F, and their dals, G, we an literall hae a field da / g wild {!!!} and frm man additinal Lrent inariant qantities, sh as: 3) 4) 4 5) 6) 4 We an als frm the rssed Lrent inariants: See these letre ntes, p.0 fr details. 7) 8) 9) F F F F G G G G F F F F G G G G As well as: 0) ) ) F F F F G G G G G G G G F F F F Ths, we an frm a ttal f niqe bi-linear Lrent inariant qantities sing the relatiisti marspi M field tensrs F,, P and their dals G,, M. 8 Prfessr Steen rrede, epartment f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis All Rights Resered.

9 UIUC Phsis 436 M Fields & Sres II Fall Semester, 05 Let. Ntes 0 Prf. Steen rrede We epliitl wrk t the first fr f these new Lrent inariant qantities: 3) ) ) Prfessr Steen rrede, epartment f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis All Rights Resered. 9

10 UIUC Phsis 436 M Fields & Sres II Fall Semester, 05 Let. Ntes 0 Prf. Steen rrede 6) M We an als frm higher-rder tri-linear and qadri-linear inariants frm mbinatins f F,, P andg,, M, e.g.: F (and all allwed permtatins): F G (and all allwed permtatins): Shrthand: Shrthand: FFF GGG FFG et. FFFF et. FF GG FF FFF FFP GGM FFM FFFP F G FFFG F GM FFF FPP GMM FFFM PPP MMM et. et. e.g. F P G and all allwed permtatins e.g. F P G M and all allwed permtatins Nte: Obisl, nt all f these mbinatins f fields will be trl niqe! e.g. and: F F F F F F F F B F F F F B 4 B 0 Prfessr Steen rrede, epartment f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis All Rights Resered.

11 UIUC Phsis 436 M Fields & Sres II Fall Semester, 05 Let. Ntes 0 Prf. Steen rrede Mawell s qatins in Tensr Ntatin fr Marspi Linear Media The lassial eletrdnamis ersin f Mawell s eqatins fr marspi linear media are: where: tt free bnd.) Gass Law: tt free bnd The ailiar/nstittie relatin gies: tt free and P bnd, e.) N magneti mnples: B 0 3.) Farada s Law: B t 4.) Ampere s Law: B J t The ailiar/nstittie relatin tt e where: B J J J J P M P tt free bnd bnd gies: B Mawell s displaement rrent term J, plariatin rrent: t M J free and M J bnd where: P J J t t t t p J bnd p bnd t B, m m Relatiistiall, we hae seen that Mawell s fr eqatins fr free spae / the am {i.e. n matter present} are ntained within the tw tensr relatins: F F Jtt G and: G 0 Gien that: F and: G Prfessr Steen rrede, epartment f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis All Rights Resered.

12 UIUC Phsis 436 M Fields & Sres II Fall Semester, 05 Let. Ntes 0 Prf. Steen rrede We see that the inhmgens relatiisti Mawell relatins are: F F Jtt free J free free, J free J t P bnd Jbnd bnd, J bnd P M P Jbnd Jbnd Jbnd t Whereas the hmgens relatiisti Mawell relatins are: B 0 M G G 0 B M t t t The relatiisti nstittie relatins fr linear, hmgenes, nifrm, istrpi media {i.e. the relatiisti analgs f and B } were prpsed b. Minkwski as: free n.b. and: F G The relatiisti M field tensrs are ntrated with the prper 4-elit. The relatiisti M field tensrs are defined in the rest/prper frame IRF(S) f the linear material. is the prper eletri permittiit and is the prper magneti permeabilit. Bth qantities are defined in the rest/prper frame IRF(S) f the linear material. is the (ntraariant) prper 4-elit f the material. Prfessr Steen rrede, epartment f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis All Rights Resered.

13 UIUC Phsis 436 M Fields & Sres II Fall Semester, 05 Let. Ntes 0 Prf. Steen rrede We an easil shw that these frmlae gie the rret nstittie relatins when the linear. ere, the sm er llapses t a single term: material is at rest, i.e. when,0,0,0 In the rest/prper frame f the linear material: F G F G B B If the linear medim is ming e.g. in the lab frame IRF(S) with rdinar elit then:,,. Then fr 0 : F and: G Fr 0 : F F 0 F F F F r: And: G G G G G3 B B B B G B r: B Similarl fr : F F F F F B B B And: 0 3 F B ( B 0 3 G G 0 G G G 3 B B 0 G B B Prfessr Steen rrede, epartment f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis All Rights Resered. 3

14 UIUC Phsis 436 M Fields & Sres II Fall Semester, 05 Let. Ntes 0 Prf. Steen rrede Ths fr: F and: G We btain: B and: B ** B Insert: B Int: Sle fr : B B 4 Use the BAC CAB rle n the first triple rss-prdt: B B 4 Bt: Ths: 4 B B 4 B efine: and: = speed f prpagatin f M waes in the linear medim. B Then: 4 Frm eqatin ** (RS p abe): B B Insert: Sle fr : 4 Prfessr Steen rrede, epartment f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis All Rights Resered.

15 UIUC Phsis 436 M Fields & Sres II Fall Semester, 05 Let. Ntes 0 Prf. Steen rrede B B B B Bt: B B B B B B B Bt: and: B B Ths, the marspi and -fields in a linear medim ming with rdinar elit in IRF(S) in terms f the and B fields present in IRF(S) are: 4 B with: B B and: Nte that and are the prper eletri permittiit and the prper magneti permeabilit f the linear medim i.e. the are defined in the rest/prper frame f the linear medim. When, then {i.e. the nn-relatiisti limit}. Keeping nl terms linear in : B B B and: Usall er small Usall er small Prfessr Steen rrede, epartment f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis All Rights Resered. 5