UIUC Physics 436 EM Fields & Sources II Fall Semester, 2011 Lect. Notes 18.5 Prof. Steven Errede LECTURE NOTES 18.5

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1 LCTUR NOTS 8.5 The Lrentz Transfrmatin f and B Fields: We hae seen that ne bserer s -field is anther s B -field (r a mixture f the tw), as iewed frm different inertial referene frames (IRF s). What are the mathematial rules / phsial laws f {speial} relatiit that gern the transfrmatins f B in ging frm ne IRF(S) t anther IRF(S')??? In the immediatel preeding leture ntes, the reader ma hae ntied sme tait / impliit assumptins were made, whih we nw make expliit: ) letri harge q (like, speed f light) is a Lrentz inariant salar quantit. N matter hw fast / slw an eletriall-harged partile is ming, the strength f its 9 eletri harge is alwas the same, iewed frm an/all IRF s: e.6 Culmbs {n.b. eletri harge is als a nsered quantit, alid in an / all IRF s.} Sine the speed f light is a Lrentz inariant quantit, then sine then s is and thus and must be separatel Lrentz inariant quantities, i.e Farads/meter 7 same in an / all IRF s 4π Henrs/meter ) The Lrentz transfrmatin rules fr and B are the same, n matter hw the and B fields are prdued - e.g. frm sures: q (harges) and/r I (urrents); r frm fields: e.g. B t, et. The Relatiisti Parallel-Plate Capaitr: The simplest pssible eletri field: Cnsider a large -plate apaitr at rest in IRF(S ). It arries surfae harges ± σ n the tp/bttm plates and has plate dimensins and w {in IRF(S )!} separated b a small distane d, w. In IRF(S ): letri field as seen in IRF(S ): σ N B -field present in IRF(S ): B n urrents present! n.b. is nn-zer nl in the gap regin between -plates Prfessr Steen rrede, Department f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis 5-. All Rights Resered.

2 Nw nsider examining this same apaitr setup frm a different IRF(S), whih is ming t the right at speed (as iewed frm the rest frame IRF(S ) f the -plate apaitr) i.e. + x is the elit f IRF(S) relatie t IRF(S ). Viewed frm the ming frame IRF(S), the -plate apaitr is ming t the left (i.e. alng the x -axis) with speed. The plates alng the diretin f mtin hae als Lrentzntrated b a fatr f ( ) ( ), i.e. the length f plates in IRF(S) is nw {n.b. the plate separatin d and plate width w are unhanged in IRF(S) sine bth d and w are t diretin f mtin!!} Sine: Q Area σ TOT TOT Q but: Q TOT Lrentz inariant quantit w QTOT QTOT And: σ A w but: w w and d d sine bth t diretin f mtin. Thus: QTOT But: σ w σ w σ σ σ σ ( ) σ Thus: σ σ but sine: > σ > σ σ σ The surfae harge densit n the plates f apaitr in IRF(S) is higher than in IRF(S ) b fatr f ( ) T an bserer in IRF(S), the plates f the -plate apaitr are ming in the Thus the eletri field in IRF(S) is: σ σ σ, x diretin. where: σ σ The supersript is t expliitl remind us that f mtin. Here, + x between IRF s. is fr -fields t the diretin Prfessr Steen rrede, Department f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis 5-. All Rights Resered.

3 Nw nsider what happens when we rtate the {islated} -plate apaitr b 9 in σ IRF(S ), then x in IRF(S ), but in the ming frameirf(s): The eletri field in IRF(S) is: σ x QTOT QTOT QTOT But: σ σ Area w w σ σ x x -field in IRF(S ) d is nw Lrentz-ntrated in IRF(S): d d but has n effet n {in IRF(S)}, beause des nt depend n d! Wh??? Beause here, the -plates were first harged up (e.g. frm a batter) and then disnneted frm the batter! Ptential differene V ( IRF( S) ) V( IRF( S) ) Δ Δ!!! Sine: ΔV ΔV d d ΔV d ΔV d but: d d ΔV ΔV d d Δ V ( IRF( S) ) Δ V( IRF( S) ) The -plate apaitr is deliberatel nt nneted t an external batter (whih wuld keep Δ V nstant, but then we wuld hae σ σ in the ase and σ σ in the ase. Currents wuld then flw (transitriall) in bth situatins. Nte that we als want t hang n t/utilize the Lrentz-inariant nature f Q TOT, whih is anther reasn wh the batter is disnneted Griffiths xample.: The letri Field f a Pint Charge in Unifrm Mtin A pint harge q is at rest in IRF(S ). An bserer is in IRF(S), whih mes t the right (i.e. in the + x diretin) at speed relatie t IRF(S ). What is the -field f the eletri harge q, as iewed frm the ming frame IRF(S)? In the rest frame IRF(S ) f the pint harge q, the eletri field f the pint harge q is: Prfessr Steen rrede, Department f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis 5-. All Rights Resered.

4 q r 4 π r q where: x 4π q x ( x + + z) 4π q ( x + + z) z 4π z ( x + + z) r x + + z But: and: u ( ) Then in IRF(S), whih is ming t right (i.e. in the + x diretin) at speed relatie t IRF(S): x z q x 4π q x ( x + + z) 4π q ( x + + z) z z 4π ( x + + z) n.b. These relatins are urrentl expressed in terms f the IRF(S ) rdinate (x,, z ) f the field pint P. Hweer we want/need the IRF(S) expressed in terms f the IRF(S) rdinate (x,, z) f the field pint P. Use the inerse Lrentz transfrmatin n rdinates: In IRF(S) at time t: Obseratin / field pint in IRF(S) at time t: Inerse Lrentz Transfrmatin: x x+ t R R Rx+ R+ Rz ( ) x R x z ( ) z z Rz 4 Prfessr Steen rrede, Department f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis 5-. All Rights Resered.

5 Then: x + + z r R + R + R x z s + sin R θ R θ Frm abe figure: R Rsθ x R R s θ. x Then sine: ( R Rx + R + Rz ) ( R + Rz ) R sin θ Sine: R R ( s θ sin θ ) In IRF(S): +. x z q R 4π θ θ x ( R s + R sin ) q R 4π θ θ ( R s + R sin ) q R 4π θ θ z ( R s + R sin ) q R 4π θ θ ( R s + R sin ) Piked up Lrentz fatr frm Lrentz transfrmatin f rdinate Piked up Lrentz fatr frm Lrentz transfrmatin f field with: R Rx+ R+ Rz x z Or: q π θ θ 4 R ( s + sin ) R R R where: R R sine: R RR r: R R R q π 4 s θ + sin θ R R with: Thus: Or: q 4π sin θ + sin θ R R q 4π sin θ + sin θ sin θ R R Prfessr Steen rrede, Department f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis 5-. All Rights Resered. 5

6 Thus: q 4π R R sin θ Heaiside expressin fr the retarded eletri field r, t expressed in terms f the present time t!!! ret ( ) The unit etr R pints alng the line frm the present psitin f the harged partile at time t!!! (See Griffiths Ch., quatin.68, page 49) R beause the fatr is present in the numeratr f all f the x, z, mpnents!!! But wait!!! This isn t the entire str fr the M fields in IRF(S)!!! In the first example f the hrizntal -plate apaitr, whih was ming with relatie elit x as seen b an bserer in IRF(S), shwn in the figure belw: The ming surfae-harged plates f the -plate apaitr as iewed b an bserer in IRF(S) nstitute surfae urrents: K σ x ± with: σ σ These tw surfae urrents reate a magneti field in IRF(S) between the plates f the -plate apaitr!!! Side / edge-n iew: B-fields prdued (use right-hand rule): Ampere s Ciruital Law: C B i d I Bw± Kw enl 6 Prfessr Steen rrede, Department f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis 5-. All Rights Resered.

7 K σ B+ > z z > K+ σ B z z + < + + < Add B and B t get the ttal B -field: + ( ) ( ) ( ) ( ) + K σ B ( > d) + z + z K σ B ( d) z z < σ σ BTOT ( > d) z+ z K σ and: σ σ σ σ In IRF(S): BTOT ( d) z z σ z Kz σ σ B ( ) TOT < + z z Thus, in IRF(S) {whih mes with elit + x relatie t IRF(S )} we hae fr the hrizntal -plate apaitr: σ σ nl exists in regin d: where: σ σ and: β B nl exists in regin d : B σ z σ z β In IRF(S): The fat that B exists / is nn-zer nl where exists / is nn-zer is nt an aident / nt a mere inidene! The spae-time prperties assiated with rest frame IRF(S ) are rtated (Lrentz-transfrmed) in ging t IRF(S). in IRF(S ) nl exists between plates in IRF(S ) Gets spae-time rtated (Lrentz-transfrmed) in ging t IRF(S) and B in IRF(S) nl exist between the apaitr plates in IRF(S). and B between plates in IRF(S) mes frm / is assiated with between plates in IRF(S ) Prfessr Steen rrede, Department f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis 5-. All Rights Resered. 7

8 Pint is: M field energ densit u M (x,,z,t) must be nn-zer in a gien IRF in rder t hae M fields present at spae-time pint (x,,z,t)! In IRF(S ): u( r, t) ( r, t) nl M field energ densities u and u nl in regin between plates f -plate apaitr. In IRF(S): u( r, t) (, ) r t + B ( r, t) Spae-time ( rt, ) pint funtins n.b. If u in ne IRF(S ) u in anther IRF(S). Mmentar Aside: Hidden Mmentum Assiated with the Relatiisti Parallel Plate Capaitr. Between plates f -plate apaitr in IRF(S): σ σ where: σ σ and: β B σ z σ z β S B z x x x z σ σ z x z x Pnting s etr pints in the diretin f mtin f the -plate apaitr, as seen b an bserer in IRF(S), and nl exists/is nn-zer in the regin between plates f the -plate apaitr. Between plates f -plate apaitr (nl!): ( ) M field linear mmentum densit in IRF(S): M S σ x. Pints in the diretin f mtin f the -plate apaitr, as seen b an bserer in IRF(S), nl exists/is nn-zer in the regin between plates f the -plate apaitr. M field linear mmentum in IRF(S): p Vlume(IRF( S)) V M M M S Where the lume (in IRF(S)):, where:, w w and: d d {here}. VS wd wd Define the lume (in IRF(S ): V wd. Thus: VS V. wd p σ V x σ x σ wd x σ Vx ( ) M S V The hidden mmentum is: phidden ( m ) Nte that: m B and: m IA z. with: I Kd i Kw where: K σ. The rss-setinal area is: i sine and d d, w w. A d d 8 Prfessr Steen rrede, Department f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis 5-. All Rights Resered.

9 Side iew (in IRF(S)): In IRF(S): σ m IA z Kwd ( σ ) w dz σ ( σ ) w ( σ )( ) dz wd z x σ σ σ ( ) ( ) ( ) σ V p hidden w d z + w d x+ V x using Thus In IRF(S): p ( ) hidden m B +σ V x But: pm ( B) VS σv x Thus, we (again) see that: phidden pm fr the relatiisti -plate apaitr!!! Nte that in IRF(S ): B But nte that: phidden ( m B) S Ε B p., thus phidden pm is als alid in IRF(S ). The numerial alue f hidden mmentum is referene frame dependent, i.e. it is nt a Lrentz-inariant quantit, just as relatiisti mmentum p {in general} is referene frame dependent/is nt a Lrentz-inariant quantit. M Nw let us return t task f determining the Lrentz transfrmatin rules fr and B : Fr the ase f the relatiisti hrizntal -plate apaitr, let us nsider a third IRF(S ) that traels t the right (i.e. in the + x -diretin) with elit + x relatie t IRF(S). In IRF(S ), the M fields are: σ and: B σ z Frm use f instein s elit additin rule: is the elit f IRF(S ) relatie t IRF(S ): + + with: ( ) and: σ σ Prfessr Steen rrede, Department f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis 5-. All Rights Resered. 9

10 We need t express and B {defined in IRF(S )} in terms f and B{defined in IRF(S)}. σ σ where: ( ) σ and: σ and: ( ) σ σ Ε but: B σ z σ z B σ z Nw (after sme algebra): {in IRF(S)} σ but: σ but: σ B σz σz {in IRF(S)} ( ) + + where: ( ) ( ) ( ) But: ( + ) ( + ) and: ( + ) with: ( ) In IRF(S ): σ ( + ) σ B σ z + Cmpare these t the and B fields in IRF(S): In IRF(S): σ σ B σz σ z + ( ) σ + where: σ σ and: z ( ) σ + z β β Using: we an rewrite in IRF(S ) as: σ σ + ( ) σ σ where: x and: Bz in IRF( S ) in IRF( S ) B Bz z B B z x z Bz ( ) Ver Useful Table(s): x z x z z x z x z x x z Prfessr Steen rrede, Department f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis 5-. All Rights Resered.

11 B B ( z) ( z) Or simpl: ( Bz) Likewise, again using: we an rewrite B in IRF(S ) as: σ B ( ) ( ) σ + z σ z σ z σ z z B Bz z in IRF( S ) in IRF( S ) B Bz Bz z B z z z z Or simpl: B B z z Thus, we nw knw hw the and B z fields transfrm. Next, in rder t btain the Lrentz transfrmatin rules fr z and B, we align the apaitr plates parallel t x- plane instead f x-z plane as shwn in the figure belw: In IRF(S) the {nw} rtated fields are: r σ r z z z r r B + σ B Use the right-hand rule t get rret sign!!! The rrespnding and B fields in IRF(S') are (repeating the abe methdlg): z ( z + B) and: B B + z As we hae alread seen (b rienting the plates f apaitr parallel t -z plane): n.b. there was n ampaning B -field in this ase! x x Thus we are nt able t dedue the Lrentz transfrmatin fr B x ( t diretin f mtin) frm the -plate apaitr prblem. Prfessr Steen rrede, Department f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis 5-. All Rights Resered.

12 S alternatiel, let us nsider the lng slenid prblem, with slenid {and urrent flw} riented as shwn in the figure belw: In IRF(S): In IRF(S): B Bx nix x We want t iew this frm IRF(S'), whih is ming with elit + x relatie t IRF(S). In IRF(S): n N L # turns/unit length, N ttal # f turns, L length f slenid in IRF(S). Viewed b an bserer in IRF(S'), the slenid length ntrats: L L in IRF(S') N N In IRF(S'): n n # turns/unit length in IRF(S'), where L L β Hweer, time als dilates in IRF(S') relatie t IRF(S) affets urrents: ( ) I dq in IRF(S) dt dq dt dq dt I I dt dt dt dt in IRF(S'). But: dt dt I I B B x nix n x ( ) I x nix Bxx B B x Bx Lngitudinal / parallel-t-bst diretin B-field des nt hange!!! Thus, we nw hae a mplete set f Lrentz transfrmatin rules fr and B, fr a Lrentz transfrmatin frm IRF(S) t IRF(S'), where IRF(S') is ming with elit + x relatie t IRF(S): x x B x Bx ( Bz) B B + z z ( z B) B B z z β β Just stare at/pnder these relatins fr a while take ur (prper) {spae-}time D u pssibl see a wee bit f Maxwell s equatins aft here??? ;) Prfessr Steen rrede, Department f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis 5-. All Rights Resered.

13 Tw limiting ases warrant speial attentin:.) If B in lab IRF(S), then in IRF(S') we hae: B ( ) ( z z z z ) But: + x B ( ) in IRF(S')!!!.) If in lab IRF(S), then in IRF(S') we hae: ( B ) ( z Bz B z B z) But: + x B in IRF(S') i.e. magneti part f Lrentz fre law!!! Griffiths xample.4: Magneti Field f a Pint Charge in Unifrm Mtin A pint eletri harge q mes with nstant elit + z in the lab frame IRF(S). Find the magneti field B in IRF(S) assiated with this ming pint eletri harge. Nte that in the rest frame f the eletri harge {IRF(S )} that B eerwhere. In IRF(S) the eletri harge q is ming with elit + z But the eletri field f ming pint harge q in IRF(S) is: (See pages -6 abe/griffiths xample.) B ( ) ( ) q sin θ ϕ 4π R ( ) sin θ and: B ( ) ( ) q R 4π R ( ) sin θ where: θ s ( R ) If the pint eletri harge q is heading diretl twards u (i.e. alng the ϕ aims unter-lkwise: (ut f page). R i, R sin θ ϕ. + ẑ diretin) then NOT: In the nn-relatiisti limit ( ): B R q 4π R The Bit-Saart Law fr a ming pint harge!!! NOT ALSO: H.C. Øersted disered the link between eletriit and magnetism in 8. It wasn t until 95, with instein s speial relatiit paper that a handful f humans n this planet finall understd the prfund nature f this relatinship a timespan f 85 ears apprximatel a human lifetime passed! (see / read handut n Øersted) Prfessr Steen rrede, Department f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis 5-. All Rights Resered.

14 4 Prfessr Steen rrede, Department f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis 5-. All Rights Resered.

15 Prfessr Steen rrede, Department f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis 5-. All Rights Resered. 5

16 6 Prfessr Steen rrede, Department f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis 5-. All Rights Resered.

17 The letrmagneti Field Tensr F The relatins fr the Lrentz transfrmatin f and B in the lab frame IRF(S) t and B in IRF(S'), where IRF(S') is ming with elit + x relatie t IRF(S) are: Ε mpnents B mpnents x x B x Bx ( β z) B ( B + β z ) z ( z + β ) Bz ( Bz β ) B B Ε mpnents B mpnents β β It is readil apparent that the and B field ertainl d nt transfrm like the spatial / -D etr parts f e.g. tw separate ntraariant 4-etrs ( and B ), beause the (rthgnal) mpnents f and B t the diretin f the Lrentz transfrmatin are mixed tgether {as seen in the ase fr B in IRF(S) resulting in B ( ) in IRF(S') and the ase fr in IRF(S) resulting in B in IRF(S')}. Prfessr Steen rrede, Department f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis 5-. All Rights Resered. 7

18 Nte als that the frm f the Lrentz transfrmatin fr s. mpnents is swithed {here} fr the fields!!! Reall that true relatiisti 4-etrs transfrm b the rule a Λ a, e.g. fr a Lrentz bst frm IRF(S) IRF(S') alng + x. Speifiall, reall that x Λ x is expliitl: t ( t β x) Parallel mpnents: x ( x βt) Perpendiular Cmpnents: z z Fr a Lrentz transfrmatin x the Λ tensr has the frm: x x B x Bx ( βbz) B ( B + β z ) z ( z + βb) B z ( Bz β ) Λ x frm IRF(S) IRF(S') alng + x, Rw index Clumn index β β Λ Thus, there is n wa that the and B fields an be nstrued as being the spatial / -D etr mpnents f ntraariant 4-etrs and B. (What wuld be their tempral mpnents/ salar unterparts: and B???) It turns ut that the -D spatial etrs and B are the mpnents f a 4 4 rank-tw M field tensr, F!!! A 4 4 rank tw tensr t λσ Lrentz transfrms ia tw Λ -fatrs (ne fr eah index): Where t λσ is: t Λ Λ t λσ λ σ Rw index Clumn index Clumn # Rw # t t t t t t t t t λσ t t t t t t t t 8 Prfessr Steen rrede, Department f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis 5-. All Rights Resered.

19 The 6 mpnents f the 4 4 send rank tensr t λσ need nt all be different (e.g. Λ ) the 6 mpnents f t λσ ma hae smmetr / anti-smmetr prperties: λσ Smmetri 4 4 rank tw tensr: t + t λσ Anti-smmetri 4 4 rank tw tensr: t t λσ σλ Fr the ase f a smmetri 4 4 rank tw tensr t + t, f the 6 ttal mpnents, are unique, but 6 are repeats: t t, t t, t t t t, t t t t t t t t t t t t t t t t t t t t λσ sm. t λσ σλ Fr the ase f an anti-smmetri 4 4 rank tw tensr t t, f the 6 ttal mpnents, nl 6 are unique, 6 are repeats (but with a minus sign!) and the 4 diagnal elements must all be (i.e. t t t t ): t t, t t, t t t t, t t t t t t t t t t λσ anti sm. t t t t t t t S it wuld seem that the spatial / -D and B etrs an be represented b an anti-smmetri 4 4 rank tw ntraariant tensr!!! λσ Let s inestigate hw the Lrentz transfrmatin rule t ΛλΛ σt fr a 4 4 rank tw anti-smmetri tensr t λσ wrks (6 unique nn-zer mpnents). Starting with t Λ Λ t λσ : anti sm. Clumn # Rw # σλ σλ λ σ β β Λλ ( r Λ σ ) t t t t t t λσ anti sm. t t t t t t t Prfessr Steen rrede, Department f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis 5-. All Rights Resered. 9

20 We see that Rw # Λ Λ Λ λ, unless λ r : r β Clumn # Rw # Λ We als see that Λ σ, unless σ r : β r Clumn # λσ There are nl 4 nn-zer terms in the sum: t ΛΛ t ΛΛ t +ΛΛ t +ΛΛ t +ΛΛ t But: t and t, whereas t t. Λ λ σ ( ΛΛ ΛΛ ) ( ( β )( β )) ( β ) ( β ) t t t t t But: β t t One (e.g. u!!!) an wrk thrugh the remaining 5 f the 6 distint ases fr (r fr mpleteness sake, wh nt wrk ut all 6 ases expliitl!!!)... t Λ Λ t λσ λ σ The mplete set f six rules fr the Lrentz transfrmatin f a 4 4 rank tw ntraariant λσ anti-smmetri tensr t Λ Λ t are: t λ σ t x x ( β ) ( βbz) ( β ) z ( z + βb) t t t t t t t t B x Bx ( + β ) B ( B + β z ) ( β ) B z ( Bz β ) t t t t t t As an be seen, the abe Lrentz transfrmatin rules fr this tensr are indeed preisel thse we deried n phsial grunds fr the M fields! Thus, we an nw expliitl nstrut the eletrmagneti field tensr ntraariant anti-smmetri tensr: F a 4 4 rank tw F F F F F x z F F F F F F B B x z F F F F F F Bz Bx F F F F F F z B Bx Prfessr Steen rrede, Department f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis 5-. All Rights Resered.

21 F x z x Bz B Bz Bx z B Bx F -field representatin f (SI units: Vlts/m). Nte the spae-time struture f anti-smmetri F all nn-zer elements hae spae-time attributes there are n time-time r spae-spae mpnents!!! Nte that there are simple alternate / equialent frms f the field tensr If we diide ur F b, we get Griffith s definitin f F : i.e. F ν Griffiths x z x Bz B ν FOurs Bz Bx z B Bx Nte that there exists an equialent, but different wa f embedding anti-smmetri 4 4 rank tw tensr: Instead f mparing:, z ( z + βb), ( ), t ( t + βt ) a) x x, ( βbz) with: a') t t t t βt and als mparing: + Ε, B z ( Bz βε ), ( + ), t ( t βt ) b) B x Bx, B ( B β z) with: b') t t t t βt F. B-field representatin f and B in an We uld instead mpare {a) with b')} and {b) with a')}, and thereb btain the s-alled anti-smmetri rank tw dual tensr G : F (SI units: Tesla) G Bx B Bz Bx z B z x Bz x -field representatin f G (SI units: Vlts/m) Nte that the G elements an be btained diretl frm a dualit transfrmatin (!!!): B, B F b arring ut Nte that the dualit transfrmatin leaes the Lrentz transfrmatin f (i.e. φ dualit 9 ). and B unhanged!!! Again, nte that ur representatin f the dual tensr diiding urs b, i.e.: G is simpl related t Griffiths b Prfessr Steen rrede, Department f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis 5-. All Rights Resered.

22 n.b. Fr the sake f mpatibilit with Griffiths bk, we will use his definitins f the field strength tensrs. Please be aware that these definitins d ar in different bks! G Griffiths Bx B Bz Bx z Gurs B z x Bz x B-field representatin f G (SI units: Tesla) With instein s 95 publiatin f his paper n speial relatiit, phsiists f that era sn realized that the and B -fields were indeed intimatel related t eah ther, beause f the unique struture f spae-time that is assiated with the unierse in whih we lie. Prir t Øersted s 8 diser that there was indeed a relatin between eletri s. magneti phenmena, phsiists thught that eletriit and magnetism were separate / distint entities. Maxwell s ther f eletrmagnetism unified these tw phenmena as ne, but it was nt until instein s 95 paper, that phsiists trul understd wh the were s related! It is indeed amazing that the eletrmagneti field is a 4 4 rank tw anti-smmetri ntraariant tensr, F (r equialentl, )!!! (six mpnents f whih are unique.) G - We hae learned that the 4-etr prdut f an tw (bna-fide) relatiisti 4-etrs, e.g. ab ab Lrentz inariant quantit (i.e. ab ab same alue in all / an IRF) - Likewise, the tensr prdut a Lrentz inariant quantit. A B AB f an tw relatiisti rank- tensrs is als - Thus, the tensr prduts f M field tensrs F G FG F and are all Lrentz inariant quantities. G, namel F F, G G, and - Speifiall, it an be shwn that (see Griffiths Prblem.5, page 57 P46 HW#4): F F G G B ( um u) Lrentz inariant quantit!!! 4 And: F G FG ( i B) Lrentz inariant quantit!!! - Obiusl, F F FF, G G GG and inariants that an be frmed / nstruted using F F G FG and are the nl Lrentz G!!! - Nte that this is b n means as bius / lear, tring t frm Lrentz inariant quantities starting diretl frm the and B fields themseles!!! n.b. Sine M energ densit um + B, Pnting s etr S B, M field linear mmentum densit S M and M field angular mmentum densit M r M annt be frmed frm an linear mbinatins f F F, G G r F G, then u M, S, M and M annt be Lrentz inariant quantities! Prfessr Steen rrede, Department f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis 5-. All Rights Resered.