The Electromagnetic Mass of a Charged Particle

Transcription

1 In mmry f M.I. Kuligina ( ) Th Elctrmagntic Mass f a Chargd Particl V.A. Kuligin, G.A. Kuligina, M.V. Krnva Dpartmnt f Physics, Stat Univrsity Univrsittskaya Sq. 1, Vrnh , Russia A slutin f th f lctrmagntic mass is btaind in th framwrk f Maxwll s quatins. Thr is a mathmatical prf that th lctrmagntic mass psssss th standard prprtis f th inrtial mass. W cnclud that Maxwll s quatins dal with tw kinds f filds. Ths ar lcal filds f chargd particls and th prpagating filds f th lctrmagntic wav. Intrductin Pag 6 APEIRON Vl. 3 Nr. 1 January 1996 Tw aspcts f th lctrmagntic mass (EMM) ar cnsidrd. Th first aspct is th classical prblm f th EMM. Shrtly aftr th discvry f th law f nrgy cnsrvatin by Pynting, it turnd ut that th EMM did nt mt th standard prprtis f an inrtial Mass. Blw w shall cnsidr xampls. Th scnd aspct is th prblm f chargd particl mdls. It is knwn that chargd particl mdls ar nt stabl bcaus f Culmb frcs, which must brak th charg. Th hypthsis was advancd that th inrtial mass f a particl was qual t th sum f th EMM and th nn-lctrmagntic mass (NEM), crating a stabl stat in th particl. Hr a cnstraint was st by th EMM prblm: th bad prprtis f EMM must b balancd by th thr bad prprtis f NEM. Withut a slutin t th prblm f th EMM, th sarch fr a mdl cannt succd (Ivannk 1949; ynman t al. 1964). It smd that a slutin might b prvidd by quantum thry. Hwvr, this did nt ccur. Mrvr, w knw that many difficultis with quantum thry hav classical rts. Th prblm f th EMM is n such prblm. Thus w hav a vicius circl. Is thr a way ut? Epistmlgy rquirs that intrnal cntradictins must nt b allwd in any scintific thry. Thy hav t b rslvd by changing th intrprtatin, transfrmatin f th mdl r mdificatins t th mathmatical frmalism f th thry. Our task is t analy th prblm f th EMM (first aspct). 1. Th Elctrmagntic Mass Prblm W bgin th analysis with xampls whr th prblm can b sn clarly. Th authrs wish t pint ut that th dnsity f nrgy f lctrmagntic wavs is dscribd wll by Pynting s vctr. Hwvr, Pynting s vctr is nt in agrmnt with mchanics. Nwtn s mchanics stats that th cnnctin btwn th mass m and th mmntum P is P=mv In th sam way, th rlatin btwn th dnsity f mass w/c and th dnsity f nrgy flw S is as fllws: S= w v (1.1) Analgusly, w may writ th dnsity f nrgy flw S f th charg. S = wv (1.) whr w = 3 bgradφg th dnsity f lctrmagntic nrgy f a charg. W shall nt cnsidr rlativistic xampls, sinc SRT has pistmlgical rrrs (Kuligin t al. 1989, 1990, 1994). Exampl 1. Lt us assum a charg with unifrm lctrical dnsity. Th charg mvs alng th x-axis with cnstant vlcity v. r cmparisn, w slct tw pints n

2 . W rgard SRT as a qustinabl thry (Kuligin t al. 1989, 1990, 1994). W must thrfr us th mathmatical frmalism f SRT (Lrnt transfrmatin) with xtrm car. igur 1 th surfac f th charg, as shwn in igur 1. With Pynting s vctr w can btain th dnsitis f th lctrmagntic flux at th pint. S =[ E H]= (grad ϕ ) v (pint 1) S =[ E H]=0 (pint ) Th vlcitis and dnsitis f th masss hav qual valus at th tw pints. At pint 1, th flux dnsity S is gratr than xpctd by factr f. At pint, th flux dnsity S is qual t r. What has happnd? If w cnsidr rlativistic vlcitis, thn w hav th prblm f th 4 3 factr, which is discussd in many txtbks (.g. Panfsky and Phillips 196). Exampl. Hr w shall dal with a chargd plan f infinit xtnt. Th plan is plttd in igur If th plan mvs upward with vlcity vycvy << ch, thn th flux dnsity is qual t: S =[ E H]= (grad ϕ ) v (1.3) Hr again w find a vilatin f th classical rul (1.). In any part f th chargd plan, flux dnsity is twic as high as th flux dnsity as in Equatin (1.). W hav th altrnativ rsult if th plan is mvd alng th x-axis: bcaus f th symmtry th magntic fild is absnt. Cnsquntly, th flux dnsity is qual t r. S =[ E H]= (grad ϕ ) v (1.4) Onc again, w find th paradx. In natur, inrtial mass is a scalar quantity. Lgically w must accpt that it has t acquir tnsr prprtis! What prprtis must NEM hav s that th full mass f th particl psssss th standard inrtial prprtis? Mrvr, any EMM f a charg which has th asymmtrical frm (fr xampl llipsidal r tridal frm), must hav tnsr prprtis. Any studnt can chck this. But this is nnsns! It is knwn that th EMM dpnds n intractins (Kuligin t al. 1986). r instanc, if a charg is changd by factr 3 withut any chang in vlum, thn th EMM is changd by a factr 9, nt by a factr 3. rm this pint f viw w shall wrk ut th prblm fr th fr charg, whr thr ar n intractins and th vlcity is cnstant. irst f all, w must ascrtain th cnnctin btwn Nwtn s mchanics and Maxwll s quatins. W writ Maxwll s quatins in Lrnt s gaug and btain th nn-rlativistic quatins, which ar crrct up t scnd rdr f v/c. A =-µ j (.1) φ 1 φ diva + = 0 c whr = + + x y ϕv A = c ρ = (.) (.3) (.4) and j= ρ v (.5) Additinal Equatins (.4) and (.5) ar ncssary fr th analysis. W must shw that Equatins (.1), (.) and (.3) ar cnsistnt with classical mchanics. r this purps, w rplac th vctr ptntial A in Equatin (.1) by th scalar ptntial φ (using Equatins (.4) and (.5)). b A + µ j= 1 rt -gradϕ v c + b-grad ϕg+ vdivb-gradϕg (.6) =0 In th mchanics f cntinuus mdia w hav th prf f th cnditin, whn th vctr a and th intnsity f its fild lins ar cnsrvd (Kchin 1965): a rt[ a v]+ + vdiv a=0 If w rplac th vctr a c by E = gradφ, thn w btain Maxwll s quatin (.1) fr th fr charg. g. Umv s Vctr Nw w shall slv th prblm f EMM in th framwrk f th nn-rlativistic cas nly. Tw cnsidratins lad us t this apprach. 1. Histrically Maxwll s quatins ars du t Culmb s law, Ampr s law and araday s law. W must us xprimntal laws hr. igur APEIRON Vl. 3 Nr. 1 January 1996 Pag 7

3 Similarly, w btain Equatin (.7) frm Equatin (.3). This is th cntinuity quatin f th ptntial in th mchanics f cntinuus mdia. div vϕ + ϕ =0 (.7) In Equatin (.8) th scalar ptntial is gnratd by th surc ρ. ρ φ = (.8) It can radily b sn that quasi-static lctrdynamics and mchanics hav similar quatins. Earlir wrk (Kuligin t al. 1986) dmnstrats this rsult. Nw w bgin th prf f th law f cnsrvatin f nrgy. Prf Lt φ b th ptntial f th surc ρ (Equatin (.8). W writ th intgral I. I = 1 ρ φ 3 d r φ φ d r = 3 (.9) whr d 3 r is a vlum lmnt. With Gauss s frmula w may writ I =- ϕ gradϕ n d σ + grad d 4 b ϕg τ (.10) whr dσ is th surfac lmnt and n is unit surfac nrmal. On th thr hand with Equatin (.6) and Equatin (.7), w may writ Equatin (.9) in th fllwing frm I =- gradϕ v gradϕ n dσ gradϕ dτ 4 Cmparisn f Equatin (.10) with Equatin (.11) yilds Pag 8 APEIRON Vl. 3 Nr. 1 January 1996 (.11) S u n d σ + t w d t =0 (.1) whr S u is th dnsity f lctrmagntic flux r Umv s vctr, R S U V (.13) ϕ Su = - grad ϕ + gradϕ v grad ϕ = vw T W Hr Equatin (.7) was usd; w is th dnsity f lctrmagntic nrgy. (.14) w = gradϕ Equatin (.1) is Umv s law f nrgy cnsrvatin, which was prvd by Umv (1874) fr th mchanics f cntinuus mdia. A scnd prf f Umv s law was givn by us (Kuligin t al. 1986). It is clar that Equatin (.13) and Equatin (.14) crrspnd t th quatins f Nwtn s mchanics (1.1) and (1.). With this rsult, w can calculat th crrct lctrmagntic flux dnsity in xampls discussd prviusly. Nw w calculat th EMM and th mmntum f a charg f arbitrary frm m = w dx dyd ; P S 3. Kintic Enrgy Equilibrium = udx dyd ; P = w v Nw w shall prv anthr imprtant rsult: th kintic nrgy quilibrium quatin. W shall shw that th EMM psssss kintic nrgy. This fact is nt particularity nw. Hwvr, w must hav th full pictur f th phnmnn. irst w cnsidr th physical mdl f th chang f kintic nrgy f th fild. If xtrnal frcs act n th charg, thn th charg is acclratd and its kintic nrgy is changd. Th chang is cnnctd with th currnt dnsity j and th vctr ptntial A. Th acclratd mtin f th charg can b tratd as th jump frm n instantanusly c-mving inrtial fram t th nxt fram. Th instantanusly cmving inrtial fram and th nn-inrtial fram hav qual vlcitis at n instant. Th fild = gradφ is nt tim-dpndnt and th vctr ptntial A is qual t r in th instantanusly c-mving fram. Th acclratd mtin f th charg inducs th additinal lctrical fild E, which is causd by th chang f vctr ptntial A vr tim (s Appndix 1). Th fild E cannt b cnsidrd as a ngligibl quantity. In th instantanusly c-mving fram th fild is qual t A v E =- 1 ϕ =- c (3.1) Th dnsity f th pwr which is gnratd by th charg is qual t pk = ρ = =- µ * v jai Ev 4 K J (3.) Th pwr dnsity ds nt dpnd n th inrtial fram in Nwtn s mchanics. Nw w shall dscrib this mdl mathmatically. T prv th quatin w us Grn s frmula f vctr ptntial E M d τ = dive div M+rtE rtm n dτ whr E and M ar th vctr ptntials f tw arbitrary filds. Lt E = b 1 A t g b th fild which is gnratd by th acclratd charg and M = A µ b th vctr ptntial f th fild dividd by µ. In this cas w btain full kintic nrgy quilibrium quatin, and w can writ th diffrntial frm f this quatin : whr: wk div S k + + pk =0 (3.3) a) pk =- 1 j A =- ja I 4 K J (3.4) is th dnsity f pwr which changs kintic nrgy ; and 1 b) wk = ivba + rt A 4µ (3.5)

4 is th kintic nrgy dnsity. With Equatin (.4) w hav Nwtn s rsult w k * A A I div A + rta v w = c grad = v c = v ϕ µ c) Sk (3.6) i.. th kintic nrgy flux dnsity. W nw illustrat this kintic nrgy quilibrium quatin with a simpl xampl. =- 1 µ 4. Chang f Enrgy f a Currnt Elmnt In quasi-static lctrdynamics th vctr ptntial f a currnt lmnt is qual t : I t dl d A = µ (4.1) 4 π r Substituting Equatin (4.1) int Equatin (3.6) and Equatin (3.8) w hav th fllwing rsults. 1. Th kintic nrgy dnsity is qual t : d µ I( t)dl wk = (4.) 4πr Th distributin f th kintic nrgy dnsity is radially symmtric.. Th kintic nrgy flux dnsity is d Sk = r d wk (4.3) Nw w discuss th pculiar prprtis f th flux dnsity d S k a) Th chang f d ω k is assciatd with d S k. Th flux dnsity d S k dpnds n th chang f squard currnt I in tim. If th currnt incrass, thn th flux dnsity d S k is psitiv and d S k is dirctd tward th radius. This flux incrass th kintic nrgy f th lctrical fild. If th currnt I dcrass, thn th flux cms back tward th currnt withut lss. Th flux tnds t cnsrv th prvius currnt in tim. Th flux dnsity d S k dcrass as 1/ r 3 in spac. b) If th currnt changs, thn th kintic nrgy flux appars simultanusly thrughut spac. c) Cntrary t Umv s vctr, which dals with th transfr f nrgy with vlcity v, th kintic nrgy flux is cnnctd nly with th acclratin f th charg. Th lctrical fild is A E =- 1 W can rgard this as intgral EM (slf-inductin) f a currnt lmnt. This analgy is givn fr illustratin. I Cnclusins 1. W hav invstigatd th prblm f th EMM f a fr charg. Nt that in th prf n hypthss wr usd. Th EMM has Nwtn s mmntum and classical kintic nrgy within th framwrk f Maxwll s quatins.. Th inrtial mass m f th chargd particl is qual t m = m + mn whr: m is th lctrmagntic mass and m n is th nn-lctrmagntic mass. Using th inductin mthd w can prv that th NEM has th standard prprtis f th inrtial mass. Th thsis can b xtndd t th gnral cas. Any inrtial mass must hav standard mchanical prprtis, which d nt dpnd n natur f th mass. This is a vry imprtant rsult. 3. W cnclud that Maxwll s quatins dal with tw kinds f filds : a) th filds f chargs (Culmb s ptntials and Umv s vctrs; th rst EMM f th charg is nt qual t r); b) th filds f th lctrmagntic wavs (rtardd ptntials and Pynting s vctrs; th rst EMM f th lctrmagntic wav is qual t r). If w us nly rtardd ptntials in ur rsarch, thn w cannt giv a full and crrct pictur f natur. It is als pssibl that th quantum prprtis f particls may b xplaind by classical mthds. Th prblm f a classical mdl (r structur) f chargs is nw f prim imprtanc. Acknwldgmnt Th authrs wish t thank Dr. C. Whitny f Tufts Univrsity fr hlpful advic and suggstins in prparing th manuscript. Appndix W writ th intgral variabl f th charg which intract with th ptntial frcs. Th charg dnsity is cnstant and rtatin f th charg is absnt. All pints f th charg mv with th sam vlcity. L * v S= M I O - µ 1- + Λ dτ dtp c (A.1) N * * * * whr: µ = µ + µ n ; µ is th lctrmagntic mass dnsity; µ * n is dnsity f nn-lctrmagntic mass. Th pndrmtiv quatin fllws frm Equatin (A.1). d i d i d i µ * µ * µ * v + v rt v - grad c +grad Λ =0 (A.) a) Supps that xtrnal frcs ar absnt (Λ = 0 ). Th particl is stabl if th fllwing cnditin is mt: grad µ * =gradµ * + grad µ * =0 (A.3) n Q APEIRON Vl. 3 Nr. 1 January 1996 Pag 9

5 b) If xtrnal frcs xist (Λ 0 ), thn w must supps that th structur f th particl is cnsrvd and, hnc, th cnditin (Equatin (A.3)) applis. Nw Equatin (A.) is multiplid by v. With Equatin (A.3) w can writ th prduct. d i d i gradλ 0 (A.4) * * v µ v µ nv + v = Th first trm f Equatin (3.4) is th lctrmagntic pwr f th acclratd charg. I K J (A.5) pk =- v v = 1 j A =- ja µ * 4 Rcall that ρ and φ ar nt tim-dpndnt. Rfrncs ynman, R., Lightn, R.B. and Sands, M., Th inman Lcturs n Physics, vl., Addisn-Wsly. Ivannk, D.D., Sklv, A.A., Klassichskaya Triya Plya Nauka, Mscw,(in Russian). Kchin, N.E., Vctr Calculatins and Elmnts f Tnsr Calculatins, Nauka, Mscw, (in Russian). Kuligin, V.A., Kuligina, G.A. and Krnva, M.V.,1986. Mchanics f quasi-nutral systms f chargs and laws f cnsrvatin f nn-rlativistic lctrdynamics, dpsitd with VINITI, Apr. 9, 1986, # V. 86, Mscw, (in Russian). Kuligin, V.A., Kuligina, G.A. and Krnva, M.V., Lrnt s transfrmatin and pistmlgy, dpsitd with VINITI, Jan. 1, 1989, # V. 89, Mscw, (in Russian). Kuligin, V.A., Kuligina, G.A. and Krnva, M.V., Th paradxs f rlativity mchanics and lctrdynamics, dpsitd with VINITI July 4, 1990, # V. 90. (in Russian). Kuligin, V.A., Kuligina, G.A. and Krnva, M.V., Epistmlgy and spcial rlativity, Apirn 0:1. Panfsky, W. and Phillips, M., 196. Classical Elctricity and Magntism, Addisn-Wsly. Umv (Umff), N.A., Bwg-Glich. d. Enrgi in cntin. Krprn, Zitschriff d. Math. and Phys. V. XIX, Schlmilch. Pag 10 APEIRON Vl. 3 Nr. 1 January 1996