UIUC Physics 436 EM Fields & Sources II Fall Semester, 2015 Lect. Notes 14 Prof. Steven Errede LECTURE NOTES 14

Transcription

1 UIUC Physis 46 EM Fields & Sues II Fall Semese, 05 Le. Nes 4 Pf. Seven Eede LECTURE NOTES 4 EM RADIATION FROM AN ARBITRARY SOURCE: We nw apply he fmalism/mehdlgy ha we have develped in he pevius leues n lw-de muliple EM adiain {E(), M(), E(), M()} an abiay nfiguain f elei hages and uens, nly esiing hese be lalized hage and uen disibuins, nained wihin a finie vlume v nea he igin: F abiay, lalized {al} elei hage and uen densiy disibuins, and J,, he eaded sala and ve penials, espeively ae: V, d d 4 v 4,, v J J A, d d 4 v 4,, v wih and F EM adiain, we assume ha he bsevain / field pin is fa away fm he lalized sue hage / uen disibuin, suh ha: max : max. Then keeping nly up ems linea in : Bu: f And: using: f: Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved

2 UIUC Physis 46 EM Fields & Sues II Fall Semese, 05 Le. Nes 4 Pf. Seven Eede,,,, Nw: Expand, as a Tayl seies in he pesen ime abu he eaded ime, a he igin { 0 }: Defining he eaded ime a he igin: {valid in he fa-zne limi},,,,,...!! Then: Whee:, d, e. d We an dp / negle all highe-de ems beynd he em, pvided ha: max,,,... is saisfied F a hamnially sillaing sysem (i.e. ne wih angula fequeny ω), eah f hese ais e.g., e. is = and hus we have: max if max d, hen d, equivalenly{hee}: max. The w appximains max and max max, me geneally: e amun keeping nly he fis-de {he lwes-de, nn-negligible} ems in. The eaded sala penial V, hen bemes:, V d d,, 4 v 4 v,,... d 4 v, d, 0,... 4 d d v v v Q p p d V,, d, d, d... 4 v v v d Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved

3 UIUC Physis 46 EM Fields & Sues II Fall Semese, 05 Le. Nes 4 Pf. Seven Eede O: V Q p p,... 4 In he sai limi: mnple diple vanishes in he em em sai limi The eaded ve penial, fis de in,, {wih } hen bemes: J J A, d d J, d 4 v 4 v 4 v Giffihs Pblem 5.7 (p. 4) shwed ha f lalized elei hage / uen disibuins nained in he sue vlume v, ha: v Thus: A dp, J, d p, d, Ne ha p p 4 is aleady fis de in any addiinal efinemens ae heefe send de in ; hus, he highe-de ems an be negleed/igned (hee). Nex, we alulae he eaded E and B fields. Sine we ae nly ineesed in he EM 4 adiain fields (in he fa-zne limi), we dp / negle,,, e. ems, and keep nly he adiain-field ems. Ne ha he adiain ems me eniely fm hse ems in he Tayl seies expansins f, and J,,, J,. Sine eaded ime: hen: Thus:, And: in whih we diffeeniae he agumen f bu: p p p V A, p 4 Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved

4 UIUC Physis 46 EM Fields & Sues II Fall Semese, 05 Le. Nes 4 Pf. Seven Eede The eaded elei field f EM adiain in he fa-zne limi is: A, p,, p E V bu: 4 4 E p p p 4 4 {using he BAC-CAB ule}, whee he send ime-deivaive f he al elei diple mmen p eaded ime and mpued fm he igin, { 0 The eaded magnei field f EM adiain in he fa-zne limi is: p B, A, p 4 4 p p 4 4 is evaluaed a he. }: p p 0, Whee in fis sep we have used he elain va {see em () P46 Le. Nes p. and/ Giffihs Equain 0.55, p. 46} and in he las sep n he RHS we have {again} used he elain. 4 B, p whee he send ime-deivaive f he al elei diple mmen p eaded ime and mpued fm he igin, { 0 If we use spheial-pla dinaes, wih he ẑ -axis p T is evaluaed a he p p 0,. }:, hen ning ha: p p z bu: z s sin 0 = p s sin p = sin 4 Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved

5 UIUC Physis 46 EM Fields & Sues II Fall Semese, 05 Le. Nes 4 Pf. Seven Eede psin Thus: E,, And: B,, 4 p sin 4 and we als see ha {again} B,, E,,, B E k The insananeus eaded EM adiain enegy densiy u,, ad u E B 0 Thus: u,,,,,,,, p p sin p in he fa-zne limi is: p sin p sin 6 6 bu: 0 sin sin ad p (Jules/m ) 6 The insananeus eaded Pyning s ve in he fa-zne limi is: S E B,,,,,, ad Was m ad p sin sin,, p ad S u,, 6 6 wih k The insananeus eaded EM pwe adiaed pe uni slid angle in he fa-zne limi is: dp,, p ad ad S,, sin d 6 Was seadian The insananeus eaded al EM pwe adiaed in 4 seadians, wih ve aea elemen da sin dd d in he fa-zne limi is: ad ad p p P S,, da sin sindd S (Was) 6 Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved 5

6 UIUC Physis 46 EM Fields & Sues II Fall Semese, 05 Le. Nes 4 Pf. Seven Eede The insananeus eaded EM adiain linea mmenum densiy in he fa-zne limi is: ad ad p 6 sin,, S,, kg m se The insananeus eaded EM adiain angula mmenum densiy in he fa-zne limi is: ad ad,,,, 0 The sala EM wave haaeisi adiain impedane f he anenna assiaed wih his lwes-de EM adiain is: E E Z Z 0 77 ad H B The sala EM wave adiain esisane f he anenna assiaed wih his lwes-de EM adiain is: R ad I p ad 6 I Ohms Ne ha in he abve, we delibeaely/nsiusly negleed he elei mnple {E(0)} em in he eaded sala penial f fa-zne limi, max : E(0) Q V,, d 4 v 4 As menined peviusly (P46 Le. Nes, p. 4), ha beause f elei hage nsevain, a spheially-symmei elei mnple mmen ann adiae ansveselyplaized EM waves spheial symmey f he mnple mmen esis sillains nly he adial diein hus ne uld ge adiain f ne plaizain fm a eain dω slid angle elemen, bu hen adiain fm he dω s n he sphee als nibue, suh ha he ne EM adiain fm he enie sphee = 0 al desuive inefeene. (Gauss Law - enl Eda Q S independen f he size f he spheially symmei hage disibuin enlsed by he sufae S. Ne als ha f fee-spae EM adiain, B mus be E, and wih bh E and B k, he ppagain diein. Hw d yu d his f a spheially-symmei sue, whee k? Ne als ha if elei hage wee n nseved, hen we wuld ge a eaded elei E(0) Q mnple field ppinal : E, n.b. his says nhing 4 abu he physial size f he spheially-symmei hage disibuin. 0 6 Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved

7 UIUC Physis 46 EM Fields & Sues II Fall Semese, 05 Le. Nes 4 Pf. Seven Eede Cnas he behavi f ansvese waves assiaed wih EM adiain fm a spheiallysymmei sue (an sillaing elei mnple mmen) ( n ne EM adiain) ha f lngiudinal sund waves / ausi waves adiaed fm a spheially symmei sillaing ausi mnple sund sue e.g. a adially inwad / uwad sillaing sphee (a beahing bubble) he lae f whih vey definiely an ppagae / eae sund peisely beause sund waves ae lngiudinal, n ansvese waves!! Nw hink abu he elen f EM adiain fields, elei diple / quaduple / e. highe EM mmens beak he ainal invaiane / ainal symmey assiaed wih he spheial mnple elei hage disibuin f he sue hus ansvese EM waves (EM adiain) an uple suh elei mnple {E(0)} sues and als nes ha lak ainal invaiane!!! In he abve Tayl seies expansins f, and J,, we nly kep ems fis-de in in hese expansins and hen demnsaed ha he fis-de fa-zne limi adiain ems wee assiaed wih he elei diple mmen {E()}. F E() elei diple EM adiain fis-de in f max he insananeus eaded sala and ve penials, elei and magnei fields ae: () p V, 4 () p A, 4 () p E, 4 () p B, 4 n.b. ppinal p 0 deivaive f p (fis ime - veliy ) n.b. ppinal p 0 deivaive f p (send ime - aeleain ) Suppse he (lalized) hage / uen disibuins ae suh ha hee is n (ime-vaying) p, 0 p, 0 p, 0. E() elei diple mmen, and/:, Then he Tayl seies expansin f, and J, fis de in wuld give nhing f penials and fields assiaed wih fa-zne EM adiain. Hweve, highede ems in hese expansins migh give ise nn-vanishing penials and fields. The send de ems in espnd M() magnei diple and E() elei quaduple EM adiain ems in de see/veify his, he send-de nibuin needs be / an be sepaaed u in M() and E() ems. Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved 7

8 UIUC Physis 46 EM Fields & Sues II Fall Semese, 05 Le. Nes 4 Pf. Seven Eede Indeed, if we mpae e.g. he ai f EM pwe adiaed f M() magnei diple vs. E() elei quaduple adiain (in he fa-zne limi): 4 ad M () ad () m whee: e 6 Qzz 60 m b I b q I q e Qzz qdd b q d b Thus: ad M () ad () 4 6 b 4 6 q d 60 5 q 4 5 b 5 4 d Similaly, he hid de ems in in he Tayl seies expansin f, and J, espnd M() magnei quaduple and E() elei uple adiain ems i.e. he hid-de nibuin needs be / an be sepaaed u in M() and E() ems!, J Similaly, he fuh de ems in in he Tayl seies expansin f espnd M() magnei uple and E(4) elei sexuple adiain ems i.e. he fuh-de nibuin an be sepaaed u in M() and E(4) ems! and, And s n, f eah suessive highe-de em in he Tayl seies expansin f, and/ J,!!! Giffihs Example.: a.) An sillaing (i.e. hamnially vaying) elei diple has ime-dependen diple mmen: s whee: s dp sin p p p p d dp d p p p s d d p p z p z Then: z s sin wih: p p z p () V, sin s sin Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved

9 UIUC Physis 46 EM Fields & Sues II Fall Semese, 05 Le. Nes 4 Pf. Seven Eede And: () p p A, sin z z s sin 4 4 () p z, p s 4 4 () p z B, ps 4 4 Bu: z s sin sin sin And: z sin sin sin () Thus: p s V, sin 4 wih: () p A, sin z 4 whee: z s sin () p sin E, s 4 () p sin B, s 4 Cmpae hese esuls f he E() elei diple EM adiain fa-zne limi ase wih hse we bained P46 Leue Nes {see pages 8-}, and/ P46 Leue Nes.5 {he E()/M() summay / mpaisn page } hey ae (f use) idenial! b.) A single, pin elei hage q an have (by definiin) an elei diple mmen p qd whee d is he psiin ve f he pin elei hage q a he eaded ime wih espe he (lal) igin. (n.b. subje all he aveas.e. hie f igin f an EDM having a ne hage see P45 Leue Nes... ) dp dd Hweve: p q qv d d n.b. hese w quaniies d n dp depend n he hie f igin!!! dv And: p q qa d d v a = veliy ve f pin elei hage q a he eaded ime = aeleain ve f pin elei hage q a he eaded ime Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved 9

10 UIUC Physis 46 EM Fields & Sues II Fall Semese, 05 Le. Nes 4 Pf. Seven Eede Eveyhing ges hugh as befe ge he same eaded sala and ve penials, same eaded E and B fields, same u, S, P, e. In paiula, he adiaed EM E() pwe assiaed wih a mving pin hage q is: P q p (Was) Bu: p qa 6 qa Pq Famus Lam fmula (EM pwe adiaed fm a pin hage q) 6 Ne ha he E() EM pwe adiaed by a pin hage q is ppinal he squae f he aeleain a and als is ppinal he squae f he elei hage q. This is he igin f saemen: Wheneve ne aeleaes an elei hage q, i adiaes away EM enegy in he fm f (eal) phns. I is he E() elei diple em whih dminaes his adiain pess. n.b. This is als ue f deeleaing haged pailes he ime-evesed siuain!!! Pq ~ a desn ae abu sign f a {The EM ineain is ime-evesal invaian}!!! Radiain fm aeleaed / deeleaed +q vs. q hages is he same if q q. (Pq desn ae abu he sign f q!) Bu: Pq ~ q s if duble q hen Pq ineases by fa f 4! F he same aeleain/deeleain, high-z nulei adiae EM enegy {in he fm f phns} muh me han e.g. a pn (= hydgen nuleus) pess is knwn as bemssahlung {= baking adiain, auf Deush}. e.g. A fully-sipped uanium nuleus (Zu = 9) gives 9 = 8464 me EM adiain han a pn f he same aeleain, a. EM Pwe Radiaed by a Mving Pin Elei Chage: The eaded elei field f an elei hage q in abiay min is: q E, v u ua 4 whee: u v u w The assiaed eaded magnei field is: B E,, : As menined befe, he fis em in E,, 4 is knwn as he genealized Culmb field, veliy field. q u. v u 0 Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved

11 UIUC Physis 46 EM Fields & Sues II Fall Semese, 05 Le. Nes 4 Pf. Seven Eede The send em in E (a.k.a. he adiain field).,, q 4 u The eaded Pyning s ve is: S, E, B, ule: Use he ABC BA CCA B ua is knwn as he aeleain field S,, E E, E, E E, whee: B, E, Hweve, ne ha n all f his EM enegy flux nsiues EM adiain (eal phns), vi ad S S, S, sme f i is sill in he fm f viual phns, The meaph Giffihs uses, ha f flies aahed a mving gabage uk, is a easnable piue imagine hee. n.b. In de dee he al EM pwe adiaed by a mving pin hage q, we daw a huge sphee f adius eneed n he psiin f he haged paile w a he eaded ime and wai he apppiae ime ineval f he EM adiain adiaed a he eaded ime aive a he sufae f he imaginay sphee. Ne ha in he fa-zne limi, he eaded ime is he e eaded ime f all pins n he sufae f he sphee S. Again, sine he aea f he sphee as Asphee ( ~ ), hen any em in S, will yield a finie answe f adiaed EM pwe, Pad S, da S. 4 5 S, ha vay as,, Hweve, ne ha ems in ha vaies e. will nibue nhing P ad in he limi. F his easn, nly he aeleain fields epesen ue EM adiain (eal phns) hene hei he name, ha f adiain fields: q Ead, ua 4 u Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved

12 UIUC Physis 46 EM Fields & Sues II Fall Semese, 05 Le. Nes 4 Pf. Seven Eede The EM veliy fields d indeed ay EM enegy as he haged paile mves hugh spae-ime, his EM enegy is dagged alng wih i bu i is n in he fm f EM adiain. Ne ha Ead, is (due he ua em). S, vanishes: The send em in ad Sad Ead Ead,,, Ead Ead Nw if he pin hage q happened be {insananeusly} a es ( v ime, hen: u v 0 Then {hee} in his ase { v Bu: {hee}. Then in his ase:,, 0 q u q a a q Ead, ua a 4 4 q 4 4 q a sine 4 q q a a aa q Sad, Ead, a a 4 }: a a s whee θ = pening angle beween and aeleain a. S ad qa sin, s qa 4 4 ) a he eaded Hee again, we see ha n pwe is adiaed in he fwad/bakwad dieins (θ = 0 and θ = π) adiaed pwe is maximum when 90, i.e. when a - ge a dnu-shaped inensiy paen abu he insananeus aeleain ve : a Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved

13 UIUC Physis 46 EM Fields & Sues II Fall Semese, 05 Le. Nes 4 Pf. Seven Eede The pwe adiaed by his pin hage (insananeusly a es a ime ) is: 8 qa sin Pad Sad, da S 6 qa sin d qa P ad qa 6 This fmula was deived assuming v 4 sindd 4 qa 6 Lam pwe fmula {again}!!! v (i.e. nn-elaivisi min). An exa eamen f v 0 0, bu in fa, we ge he same fmula as lng as is (muh) me diffiul / edius. Ne ha in speial elaiviy {ineial (nn-aeleaed) efeene fames}, he hie v meely epesens a judiius hie f an (ineial) efeene fame, wih n 0 lss f genealiy. If we an deemine hw Pad ansfms fm ne efeene fame anhe, hen we an dedue he me geneal v 0 esul (Liénad) fm he (Lam) v 0 esul. (See e.g. Giffihs pblem.69, p. 545). F he v 0 ase, Ead, is me mpliaed (han he v 0 ase). F he v 0 ase, Sad, = he ae f enegy passing hugh he (imaginay) lage-adius sufae S f he sphee, Sad, is NOT he same as he ae f enegy when i lef he haged paile a he eaded ime. Cnside he example f a pesn fiing a seam f bulles (phns) u he windw f a mving a, paallel he diein f min f he a: The ae a whih he bulles sike a age, Rg (#/se) is n he same as he ae f bulles leaving he gun, Rgun (#/se) beause f he elaive min f he a wih espe he age. This is again analgus he Dpple effe. I is puely due a minal gemeial fa (i.e. i is n due speial elaiviy). F bulles mving paallel he a s veliy ve: R R : gun g R g Rgun whee: v Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved

14 UIUC Physis 46 EM Fields & Sues II Fall Semese, 05 Le. Nes 4 Pf. Seven Eede Wheeas f bulles mving ani-paallel he a s veliy ve: R R : gun g R g Rgun whee: F abiay dieins, wih uni ve fm a age: R R : gun g R g R gun whee: v v S if dw = ae f enegy passing hugh sphee f adius hen he ae a whih enegy leaves he d dw dw d dw u dw hage q is: sine: wih u v. d d d d d u u (see P46 Le. Nes, p. 4-5, and/ Giffihs pblem 0.7, p. 44) u v Bu: v = eadain fa dw u dw dw dw v Then: whee: d d d d Thus, he pwe adiaed in a pah f aea da sindd d d sindd = slid angle in whih he EM pwe is adiaed in aea elemen da n he q sufae f he sphee S, wih Ead, ua 4 is given by: u dp ad u u Sad, Ead d q 6 ua 6 u 4 q u 6 6 a u q 6 u u ua 5 u n he sphee S, whee Thus: ad ua 5 6 u dp q d 4 Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved

15 UIUC Physis 46 EM Fields & Sues II Fall Semese, 05 Le. Nes 4 Pf. Seven Eede dp Inegaing d Hweve, he esul f his inegain {again!} yields he famus Liénad fmula: ad d ve he sphee S (i.e. ve θ and φ angles) is a pain. S q Pad a 6 6 a a 6 v a q 6 v Whee: and: 0 = Lenz fa. Ne ha he Liénad fmula edues he Lam fmula f Pad when v. Ne als ha when v, he γ 6 fa in he Liénad fmula ges besek as he haged paile avels lse and lse he speed f ligh, he me ne ies aeleae i (in de make i avel even lse he speed f ligh, ), i adiaes away me and me f he (absbed) enegy as v!!! vey high enegy elen aeleas ae pblemai in his egad, beause he elen is s ligh, mass-wise, e.g. elaive he pn: m 0.5MeV wheeas m 98.8 MeV. e p Giffihs Example.: Suppse v and a disibuin f adiaed pwe ae insananeusly llinea (i.e. paallel eah he). Find he angula d when v a v a (i.e. when v a dpad Then in his ase: u v beause v a 0 Thus: u a v a a v a a Then: ua dp ad q q d 6 u 6 u a 5 5 v ) u v v Wk n denmina em: Wk n numea em: Thus: a a a a a a a a Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved 5

16 UIUC Physis 46 EM Fields & Sues II Fall Semese, 05 Le. Nes 4 Pf. Seven Eede Then: 5 a a dp a a ad q q d 6 6 If we le he ẑ -axis pin alng Then: Thus: 5 5 v v {and hene als alng alng a a s and: v v s : s whee θ = pening angle beween and aeleain a, as shwn n page abve. s 6 s dpad q a d 5 bu: a }: dpad q a d sin 6 s 5 wih: v When β 0: dp q a sin, d 6 ad v 0 Sad : When β : The dnu f EM adiain inensiy is flded fwad by he fa s 5 Ne ha hee is sill n adiain peisely in he fwad diein, ahe i is in a ne whih bemes ineasingly naw as β, f half-angle: max {see Giffihs pblem.5, p. 465} The al EM pwe adiaed in 4 seadians by he pin hage f v a is: dp q a Pad d sindd 5 6 s ad sin d sin sin d 0 5 qa 8 s Le: u = sθ θ = 0 u = + du = sinθdθ θ = π u = 6 Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved

17 UIUC Physis 46 EM Fields & Sues II Fall Semese, 05 Le. Nes 4 Pf. Seven Eede Then: u qa 8 Inegae by pas: vdu uv udv u Pad du 5 qa 4 qa Pad 8 6 Bu: v wih: qa 6 Pad 6 This is he same/idenial esul as bained diely fm he Liénad fmula when v a. I is als knwn as he lassial fmula f bemssahlung ( baking adiain in geman). Again, ne ha beause ad ~ f a P a, he EM pwe adiaed desn depend n he sign i.e. whehe he haged paile is aeleaing deeleaing. Nw i an als be shwn ha he Lenz fa Em, whee E p m v. = al elaivisi enegy assiaed wih a haged paile mving wih 6 Thus, when v, f a given {high} al enegy E, hen ~ m and hus: P ~ m. Cmpaing EM bemssahlung adiain fm an aeleaed elen { m vs. ha f e.g. an aeleaed mun { m MeV 6 6 m m e elen will adiae ad e 0.5MeV } }, f he same al enegy E, an imes me EM enegy han a mun. This explains why muns have suh high peneaing pwe in avesing mae hey lse elaively lile enegy via bemssahlung, wheeas high-enegy elens adiae EM enegy like azy in mae. Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved 7

18 UIUC Physis 46 EM Fields & Sues II Fall Semese, 05 Le. Nes 4 Pf. Seven Eede The Radiain Reain n a Radiaing Chaged Paile Ading he laws f lassial eledynamis, an aeleaing elei hage adiaes elemagnei enegy in he fm f eal phns ( = quana f he EM adiain field). By nsevain f enegy, he EM adiain aies ff / aies away enegy whih mus me a he expense f he haged paile s kinei enegy {sine is es mass ann hange}. In he wds, ne pus in enegy aeleae he haged paile, bu he haged paile winds up being aeleaed less han e.g. an eleially neual paile {f he same es mass f he haged paile}, f he same amun f inpu enegy! The devil is in he mispi deails f peisely hw his is amplished in bh ases. A he mispi level, an eleially haged paile f mass m is aeleaed/ineases is {kinei} enegy T m by absbing EM enegy (eihe in he fm f viual eal phns) fm a sue f EM field(s). In de aeleae/inease he {kinei} enegy T m f an eleially neual paile, i mus inea, a he mispi level, via ne f he fu fundamenal fes f naue, wih a sue {f fields} assiaed wih ha fundamenal fe. {ime} Feynman/Spae-Time Diagam f Elen-Psin Saeing {QED} e e e e In he elemagnei ase, if an eleially haged paile is deeleaed and adiaes EM enegy away in he fm f {eal} phns, by enegy nsevain, he hange in he kinei enegy f he haged paile mus equal he sum f he enegies assiaed wih eah f he n individual {eal} phns adiaed by he haged paile: q n KE E hf n i i i i x {spae} 8 Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved

19 UIUC Physis 46 EM Fields & Sues II Fall Semese, 05 Le. Nes 4 Pf. Seven Eede This implies ha he adiain mus {smehw!} exe a fe, F ad bak n he eleially haged paile i.e. a eil fe, analgus ha assiaed wih fiing a bulle fm a gun. Thus, linea mmenum p mus als nseved in his pess. In he emissin f EM adiain {eal phns}, linea mmenum p h hf i i is als aied away by eah f he {eal} i phns. This mes a he expense f he haged paile s mmenum p q and (nn-elaivisially, f vq ): KEq pq m n n n eil h p p k hf k q i i i i i i i i k = wave ve f he i h phn i B i Thus, if a similaly aeleaed/deeleaed neual paile desn adiae fe quana {f sme kind} beause i is aeleaed/deeleaed, hen beause he eleially-haged paile des adiae EM quana {eal phns} in he aeleain/deeleain pess, hen we an see ha he final-sae KEq KE f similaly aeleaed/deeleaed neual paile f he same mass m and iniial/iginal kinei enegy as ha f he eleially-haged paile. F a nn-elaivisi paile ( v q EM adiaed pwe is: The Radiain Reain Fe n a Chaged Paile E i ) he Lam fmula f he al insananeus P ad qa 6 (Was) Cnsevain f enegy wuld hen imply ha his adiaed EM pwe = he insananeus ae a whih he haged paile lses enegy, due he effe f he EM adiain bakeain / eil fe F : ad dw q a Pq Fad v (Was) d 6 This elain / equain is aually wng. Why??? The easn is, ha we alulaed he adiaed EM pwe by inegaing Pyning s ve Sad, f he EM adiain assiaed wih he aeleaing pin haged paile ve an infinie sphee f adius ; in his alulain he EM veliy fields played n le, sine hey fall ff apidly as a funin f make any nibuin Pad. Hweve, he EM veliy fields d ay enegy beause he al eaded elei field assiaed wih he eleially haged paile is he sum f w ems he EM veliy field and he EM aeleain field ems: E, E v, E a, k i Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved 9

20 UIUC Physis 46 EM Fields & Sues II Fall Semese, 05 Le. Nes 4 Pf. Seven Eede The al eaded EM enegy densiy assiaed wih he al eaded elei field is: v a u, E, E, E, v v a a E, E, E, E, Enegy sed in veliy Css em!!! Enegy sed Enegy sed in field nly (viual phns) in mixue f veliy and aeleain field nly aeleain field (bh (eal phns) Genealized Culmb viual & eal phns!!) fields nly Cnvesin field Radiain fields nly viual eal phns {and vie vesa!} Ne ha: The Genealized Culmb fields vay as The Cnvesin fields vay as The Radiain fields vay as 4 ~ Neihe he Genealized Culmb field ~ n he Cnvesin field nibue ~ EM adiain in he fa-zne limi Clealy, he fis w ems in he EM enegy densiy fmula assiaed wih he elei field have enegy assiaed wih hem. Hweve, his enegy says wih he haged paile i is n adiaed away. As he haged paile aeleaes / deeleaes, enegy is exhanged beween he haged u, abve), his enegy is ieievably aied away (by eal phns) u =. paile and he veliy and aeleain fields. F he lae em (he las/ d em in Thus, a E dw q a Pq Fad v nly auns f he las / d em d 6 u, abve. ( ) in, If we wan knw he al eil fe exeed by he EM veliy and he EM aeleain fields n he pin hage, hen we need knw he al insananeus pwe ls, n jus he adiain-nly nibuin. Thus, in his sense, he em adiain (bak)-eain is a misnme beause i shuld me apppiaely be alled an EM field (bak)-eain. Ne fuhe ha his EM field (bak)- eain is als inimaely nneed wih he issue f he s-alled hidden EM mmenum. Shly, we ll see ha Fad is deemined by he ime deivaive f he aeleain and an be nn-ze even when he aeleain a is insananeusly ze! (The haged paile is n adiaing a ha eaded insan in ime!) a, 0 Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved

21 UIUC Physis 46 EM Fields & Sues II Fall Semese, 05 Le. Nes 4 Pf. Seven Eede By enegy nsevain, he enegy ls by he eleially haged paile in a given mus equal he enegy aied away by he EM eaded ime ineval adiain, plus whaeve exa enegy has been pumped in he EM veliy/genealized Culmb field. If we nside ime inevals suh ha he sysem (nsising f he pinhaged paile q and he EM veliy field see dawing n fllwing page) euns is iniial sae, hen (assuming ha he enegy in he EM veliy fields is he same a ime as a ime, hen he nly ne enegy lss is in he fm f EM adiain (due he emissin f n eal phns). dw q a Thus, while insananeusly Pq Fad v is ine, d 6 by suiably aveaging his elain ve a finie ime ineval, i is valid, wih he esiin ha sae f he sysem is idenial a he eaded imes and : q Fad v d a d 6 F he ase f peidi/hamni min, his means ha he abve inegals mus be aied u ve a leas ne ( me) mplee / full yles, n, n =,,,... F nn-peidi min, he ndiin ha he sysem be idenial a imes and is me diffiul ahieve i is n enugh ha he insananeus veliies and aeleains be equal a n v and, sine he (eaded) fields fahe u (a he pesen ime ) depend and a he ealie eaded ime!!! a Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved

22 UIUC Physis 46 EM Fields & Sues II Fall Semese, 05 Le. Nes 4 Pf. Seven Eede F nn-peidi min, he ndiin ha he sysem be idenial a imes and v v a a, bu all highe deivaives f ehnially equies ha n nly and v mus als likewise be equal a imes and!!! Hweve, in paie, f nn-peidi min, sine he EM veliy fields fall ff apidly wih v v a a, f a bief ime ineval,., i is suffiien ha and The RHS f he abve equain an be inegaed by pas: dv dv dv d v a d d v v d d d d d a Beause f he esiin n v v and a a a he ime endpins and, The em: v v a dv d 0 Thus: q 6 ad F v d a v d O: q ad 0 F a v d 6 Mahemaially, hee ae ls f ways his inegal equain an be saisfied, bu i will eainly be saisfied if: q F ad a 6 Abaham-Lenz fmula This elain is knwn as he Abaham-Lenz fmula f he EM adiain eain fe. q F ad a 6 is he simples pssible fm he EM adiain eain fe an ake. Physially, ne ha his fmula ells us nly abu he ime-aveaged fe {albei} ve a vey bief ime ineval, f he fe mpnen paallel v - beause f he F v F v. iginal em. As suh, i ells us nhing abu ad ad n.b. These aveages ae als esied ime inevals suh ha v v a a. ensue ha and is hsen Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved

23 UIUC Physis 46 EM Fields & Sues II Fall Semese, 05 Le. Nes 4 Pf. Seven Eede q als has disubing, ad The Abaham-Lenz adiain eain fe F a 6 seemingly unphysial impliains ha ae sill n fully undesd day, despie he passage f nealy a enuy! Suppse a haged paile is subje NO exenal fes. Then Newn s nd law says ha: q ad F a ma 6 whee m = (eal) es mass f he haged paile. q ad Then: F ma m a ma : q a a a 6 m 6 m The sluin his linea, fis-de hmgeneus diffeenial equain is: whee a = aeleain a he eaded ze f ime, = 0, and elen is a ime nsan f : e 4 60 se. If a 0, he aeleain expnenially ineases (+ve, if a > 0, ve, if a < 0) as ime pgesses! This is a unaway sluin, whih is CRAZY!!! This an nly be avided if a 0. 6 q m a a e, whih f he Hweve, if he unaway sluins ae exluded n physial gunds, hen he haged paile develps an aausal behavi e.g. if an exenal fe is applied, he haged paile espnds befe he fe as!! This aausal pe-aeleain jumps he gun by nly a sh ime 4 e 60 se, and sine we knw ha quanum mehanis and uneainy piniple ae peaive n sh disane/sh imesales, pehaps his lassial behavi shuldn be unseling us. Neveheless, many i is. (see Giffihs Pblem.9, p. 469 f me aspes/amifiains f he Abaham-Lenz fmula ) Suh diffiulies als pesis in he fully-elaivisi vesin f he Abaham-Lenz equain. Giffihs Example.4 EM Radiain Damping: Calulae he EM adiain damping f an eleially haged paile aahed a sping f naual angula fequeny ω wih diving fequeny = ω The -dimensinal equain f min is: sping ad diving m x m x F mx F F F diving Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved

24 UIUC Physis 46 EM Fields & Sues II Fall Semese, 05 Le. Nes 4 Pf. Seven Eede Wih he sysem sillaing a he diving fequeny ω: Insananeus psiin: x x s Insananeus veliy: x x sin Insananeus aeleain: x x s Insananeus jek: x x sin x sin Thus: x x Thus: mx m x m x F diving Define he damping nsan: (SI unis: se) Then: mx mx m x F diving nd -de linea inhmgeneus diff. eqn. n.b. In his siuain, he EM adiain damping is ppinal v. Cmpae his e.g. nmal mehanial damping, whih is ppinal v (e.g. fiin / dissipain). The Physial Basis f he Radiain Reain We deived he Abaham-Lenz EM adiain eain fe F a x q fm ad 6 nsideain f nsevain f enegy in he EM adiain pess, fm wha was bsevable in he fa-field egin,. Classially, if ne ies deemine his adiain eain fe a he adiaing pin hage, we un in mahemaial diffiulies due he mahemaial pin-behavi f he elei hage (e.g. a is igin) whee he (sai) elei field and espnding sala penial beme singula, his pblem espndingly has infinie enegy densiy a he pin hage. This singula naue is als he pesen f he eaded EM fields assiaed wih a mving pin hage: q E, v u ua 4 u B E u v,, wih: Tday, we knw ha quanum mehanis is peaive, e.g. fm he Heisenbeg uneainy piniple n {e.g. -dimensinal} disane sales f: x p x whee: h = Plank s Cnsan / π, h J-se Then: x p bu: x px me {f elens} 4 Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved

25 UIUC Physis 46 EM Fields & Sues II Fall Semese, 05 Le. Nes 4 Pf. Seven Eede Ne ha: h = 40 ev-nm and: nm = 0 9 m 40 ev-nm / x m 86 fm ( fm = 0 5 m) m 0.5 mev e e The quaniy e 86 fm = edued Cmpn wavelengh f he elen m e h and: e 47 fm = Cmpn wavelengh f elen m e 5 Thus: x e 86 fm 860 m m e F sh disane sales f de xe me 86 fm {and less} he behavi f an elen will be manifesly quanum mehanial in naue. Thus, we shuld n be supised ha when exaplaing lassial EM hey in his sh-disane egime, we bain eneus answes we have n easn expe lassial hey {ninue } hld in he quanum dmain!!! V e q 4 Similaly, we have n business exaplaing quanum mehanis disane sales less han: BH GNme m kg s kg e m = 0 m/s Shwazshild adius f elen (even hizn) Whee GN = Newn s gaviainal nsan. The elen is a blak hle a his disane sale he Shwazshild adius/even hizn f an elen is whee spae & ime inehange les! Hweve, lng befe his egime is eahed, a disane sales espnding he Plank enegy/plank mass m p GN.0 kg.0 GeV.0 ev { GeV = 0 9 ev}, is he egime f quanum gaviy, whee spae-ime iself bemes Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved 5

26 UIUC Physis 46 EM Fields & Sues II Fall Semese, 05 Le. Nes 4 Pf. Seven Eede fam-like (i.e. n ninuus) quanized/diseized {smehw }. The disane sale 5 whee quanum gaviy is peaive is knwn as he Plank lengh: LP GN.60 m. The Plank lengh espnds a ime-sale {knwn as he Plank ime} f P LP GN se Neveheless, bak in he ealy 900 s, ignane f quanum gaviy and quanum mehanis did n sp Abaham, Lenz, Pinaé {and many hes} fm applying lassial EM hey - eledynamis alulae he self-fe / adiain bak-eain n a pin elei hage. These effs by-and-lage mdeled he pin elen as {sme kind f} spaially-exended elei hage disibuin (f finie, bu vey small size), alulains uld hen be aied u and hen (a he end f he alulain) he limi f he size f he hage disibuin 0. In geneal (as we have aleady enuneed his befe in eledynamis), he eaded lassial/maspi EM fe f ne pa (A) aing n anhe pa (B) is n equal and ppsie he fe f B aing n A, Newn s d Law is seemingly vilaed: F F AB BA,, B Adding up he imbalanes f suh fe pais, we bain he ne fe (imbalane) f a hage n iself he self-fe aing n he hage. A H.A. Lenz iginally alulaed he lassial self-fe using a spheial hage disibuin edius see J.D. Jaksn s Classial Eledynamis, d ed., se. 6. and beynd if ineesed in hese deails. A less ealisi mdel f a hage is use a igid dumbbell in whih he al hage q is divided in halves sepaaed by a fixed disane d (simples pssible hage aangemen eluidae he self-fe mehanism): Assume ha he dumbbell mves in x -diein and (f simpliiy) assume ha he dumbbell is insananeusly a es a he eaded ime. Then he eaded elei field a () due () is: q q E, au ua au ua 4 u 8 u 6 Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved

27 UIUC Physis 46 EM Fields & Sues II Fall Semese, 05 Le. Nes 4 Pf. Seven Eede Hee: u {beause v 0 } Ne ha: xdy and hus: d. Ne als ha: fna and: a a x. u xdy and: a xdy ax a We ae in fa nly ineesed in he x -mpnen f E E,, and E,, sine he ŷ -mpnens f will anel when we add fes n he w ends f he dumbbell. Ne fuhe ha sine he w hages n he dumbbell ae bh mving in he same diein / paallel eah he, he magnei fes assiaed ne hage aing n he he will als anel, hus Newn s d Law is manifesly beyed {hee}, in his paiula siuain / nfiguain. If u, hen: u x u x x and sine: xdy hen: u x lxdy x Thus:, And: Then: u q E a u u a x x x 8 u and: a x dy a x a sine: q, 8 E a a x q 8 q 8 Thus: E, a a a a a a a a x a x x q 8 q a bu: d : d 8 q 8 d d a d a q x 8 d E, E, By symmey: x x sine: d Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved 7

28 UIUC Physis 46 EM Fields & Sues II Fall Semese, 05 Le. Nes 4 Pf. Seven Eede The ne eaded fe n he igid dumbbell is: self q d a,,,,, 8 F F F qe qe x Exa d self We nw expand F, in pwes f d. Then when he size d f he eleially-haged dumbbell is aken is limi f d 0, all psiive pwes will disappea....!! Tayl s Theem: x x x x x Reall ha: x v0 and ha: fna 6 Then: x x a a... whee: Bu: d d O: d a a... 6 a a a a a We wan Bu we an slve f in ems f d. Fm abve, i an be seen ha we an slve f d in ems f in ems f d using he evesin f seies ehnique, whih is a fmal mehd ha an be used bain an appximae value f pwes f. T fis de in d, we have:. by igning all highe d d use his as an appximain f baining a ubi ein em: a d d 8 Keep ging... a d d 5 8 a 4 d d 5 d Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved

29 UIUC Physis 46 EM Fields & Sues II Fall Semese, 05 Le. Nes 4 Pf. Seven Eede a a 4 x x a a... d d d Thus: self Then: q d a q a a F, x d... x d d {Ne ha a and a ae evaluaed a he eaded ime.} Using he Tayl seies expansin f self Then: a, we an ewie his esul in ems f he pesen ime: d aa a... a a... a a... q a a F, d... x 4 4d The fis em inside he bakes n he RHS is ppinal aeleain f he hage q. If we pu i n LHS, hen by Newn s nd Law F ma, we see ha i adds he mass m f he dumbbell hee is ineia assiaed wih aeleaing an eleially-haged paile. The al ineial mass f he dumbbell is heefe: q q m mdumbbell m dumbbell 4 4d 4 d O: dumbbell 4 m m q d es mass enegy, E m Ne ha he {epulsive} elesai penial enegy assiaed wih his dumbbell is: q q UE d q V d 4 d 4 d (Jules) The fa ha his wks u pefely is simply due he fa ha he iniial hie f he dumbbell s ienain was delibeaely/nsiusly hsen be ansvese he diein f min. F a lngiudinally iened dumbbell, he EM mass ein is half his amun. F a spheial hage disibuin, he EM mass ein is a fa f ¾!! Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved 9

30 UIUC Physis 46 EM Fields & Sues II Fall Semese, 05 Le. Nes 4 Pf. Seven Eede self The send em inside he bakes n he RHS f he F eain em: in Ne ha F, ad, in, qa qa F ad x x diffes fm Abaham-Lenz esul by a fa f : AL - q ad, elain is he EM adiain F a 6 in The easn f he fa f diffeene is ha physially, Fad, is fe f ne end f he dumbbell aing n he n he i.e. an EM ineain beween he w ends f he dumbbell. Thee is als a fe f eah end f he dumbbell aing n iself an EM self-ineain F, f eah end. When he EM self-ineains f eah end ae inluded (see Giffihs self ad Pblem.0, p. 47), he al EM adiain-eain is: in,, self, qa qa Fad Fad Fad x x whih agees pefely wih Abaham-Lenz adiain-eain fe fmula. Thus, physially we see ha he EM adiain eain is due he fe f he hage aing n iself an {appaen} self-fe! F, F, is valid/well-behaved in limi f Ne als ha ad he size f he dumbbell, d 0). des NOT depend n d ( ad Hweve, ne ha: q dumbbell 4 d m m when d 0!!! The ineial mass f he lassial elen bemes infinie when when d 0, beause: E q q 4 d 4 d U d q V d when d 0!!! {Bu we aleady knew his, as we leaned lng ag, in P45/las semese } Ne ha his unpleasan/awkwad pblem als pesiss in he fully-elaivisi, quanum eledynamial hey {QED}. Infiniies/singulaiies hee ae deal wih/side-sepped by a pess knwn as mass enmalizain, s as avid suh infiniies lk nly a mass diffeenes / enegy diffeenes 0 Pfess Seven Eede, Depamen f Physis, Univesiy f Illinis a Ubana-Champaign, Illinis All Righs Reseved