MOTION OF AN ELECTRON IN CLASSICAL AND RELATIVISTIC ELECTRODYNAMICS AND AN ALTERNATIVE ELECTRODYNAMICS

Transcription

1 MOTION OF AN ELECTRON IN CLASSICAL AND RELATIVISTIC ELECTRODYNAMICS AND AN ALTERNATIVE ELECTRODYNAMICS Musa D. Abdullahi, U.M.Y. Uniersity P.M.B. 18, Katsina, Katsina State, Nigeria Abstrat Fr an eletrn f mass m and harge e ming with elity and aeleratin d/ in an eletri field f magnitude E, the aelerating fre is put as etr F = ee( )/ = m(d/), where ( ) is the relatie elity between the eletrial fre prpagated with elity f light and the eletrn. The eletrn is aelerated t the speed f light r it reles in a irle at a nstant speed. The relatiisti mass-elity frmula is rret fr irular relutin and mass in that frmula is the rati f eletrstati fre ee t aeleratin /r in a irle f radius r, whih is infinitely large fr retilinear mtin. An eletrdynamis is deelped fr an eletrn aelerated t the speed f light at nstant mass and with emissin f radiatin, ntrary t lassial and relatiisti eletrdynamis. Radiatin urs if there is a hange in kineti energy. Keywrds: Aberratin, aeleratin, harge, field, fre, mass, radiatin, relatiity, elity 1. Intrdutin There are nw three systems f eletrdynamis in physis. Classial eletrdynamis is appliable t eletrially harged partiles ming at a speed that is muh slwer than that f light. Relatiisti eletrdynamis is fr partiles ming at a speed mparable t that f light. Quantum eletrdynamis is fr atmi partiles ming at ery high speeds. There shuld be ne system f eletrdynamis appliable t all partiles at speeds up t that f light. Classial eletrdynamis is based n the send law f mtin, riginated by Galile Galilei in 1638 [1], but enuniated by Isaa Newtn []. The thery f speial relatiity was frmulated in 1905 mainly by Albert Einstein [3, 4]. The quantum thery was deised by Max Plank [5], Luis de Brglie [6] and thers. Relatiisti eletrdynamis redues t lassial eletrdynamis at lw speeds as nrmally enuntered. The relatiity and quantum theries are inmpatible at high speeds. Bth the relatiity and quantum theries, therefre, annt be rret. One f the theries r bth theries may be wrng. Indeed, speial relatiity is under attak by physiists: Bekmann [7] and Renshaw [8]. Relatiity is the bne f ntentin in this paper. The paper intrdues an alternatie eletrdynamis, appliable t an eletrially harged partile, like an eletrn, ming in an eletri field at speeds up t that f light, with mass f a ming partile remaining nstant and with emissin f radiatin. 1.1 Newtn s Send Law f Mtin Fr a bdy ming with elity at time t, Newtn s send law f mtin, whih inludes the first and third laws, relates the rate f hange f elity r aeleratin d/ prdued n a bdy f mass m, t the impressed fre F, in the etr equatin: d F = m (1) Arding t equatin (1), where mass m is a nstant independent f elity, the aeleratin bemes zer, and the bdy mes in a straight line with nstant speed, if the aelerating fre F redues t zer r if the mass m bemes infinitely large. With the adent f the thery f speial relatiity, where m inreases with, Newtn s send law f mtin was mdified. The law nw relates fre F t the rate f hange f mmentum m, thus: 1 An Eletrdynamis f a Partile Aelerated t the Speed f Light with Cnstant Mass. Musa D. Abdullahi 013

2 d F= ( m ) () This paper assumes the alidity f Newtn s send law f mtin, but where mass m remains nstant and aelerating fre F redues t zer at the elity f light, f magnitude. 1. Culmb s Law f Eletrstati Fre Culmb s law, first published in 1785, is the mst imprtant priniple in physis. It gies the fre F f attratin between an eletrn f harge e and a psitie eletri harge Q, separated by a distane r in spae, as etr: eq F= = ee (3) 4πε r where E is the eletrstati field intensity due t harge Q and ε is the permittiity f spae. The urrent prblem f physis lies in making Culmb s law independent f elity f the eletrn, f harge e, ming with elity in the eletrstati field f intensity E. 1.3 Relatiisti Mass-elity Frmula Arding t Newtn s send law f mtin, a fre an aelerate partile t a speed greater than that f light, with its mass remaining nstant. But experiments with aeleratrs hae shwn that n partile, nt een the eletrn, the lightest partile knwn in nature, an be aelerated beynd the speed f light. The thery f speial relatiity explains this limitatin by psiting that the mass f a partile inreases with its speed, beming infinitely large at the speed f light. That sine an infinite mass annt be aelerated any faster by any finite fre, the speed f light bemes the limit t whih a bdy an be aelerated. This is a plausible prpsitin. The relatiisti mass-elity frmula is: m m= =γ m (4) where m is the mass f a partile ming with speed relatie t an bserer, m is the rest mass, is the speed f light in a auum and γ is the Lrentz fatr. Equatin (4), where m is a physial quantity, beming infinitely large at the speed f light, is the bne f ntentin in this paper. The diffiulty with infinite mass, at the speed f light, in equatin (4), is the Ahilles heel f the thery f speial relatiity. Resling this diffiulty, by allwing a ming partile t reah the speed f light with its mass remaining nstant, is the purpse f this paper. Suh a reslutin, giing the ultimate speed withut infinite mass, wuld bring great relief t physiists all er the wrld. The prpnents f speial relatiity just ignre the prblem with equatin (4). They say that it is the mmentum, nt the mass, whih inreases with speed. They aid the diffiulty altgether by arguing that the speed neer really reahes that f light, r that partiles ming at the speed f light (phtns) hae zer rest mass. But eletrns are easily aelerated and hae been aelerated t pratially the speed f light as demnstrated by William Bertzzi in 1964 [9], using a linear aeleratr f 15 MeV energy. Eletrn aeleratrs, betatrns and eletrn synhrtrns f er 10 6 MeV, hae been built and perated with eletrns ming at the speed f light fr all pratial purpses. A mst remarkable demnstratin f the existene f a uniersal limiting speed, equal t the speed f light, was in an experiment by William Bertzzi f the Massahusetts Institute f Tehnlgy [9]. The experiment (see Table1) shwed that eletrns aelerated thrugh energies f 15 MeV r er, attain, pratially, the speed f light. Bertzzi measured the heat energy J deelped when a stream f aelerated eletrns hit an aluminium target at the end f their flight path, in a linear aeleratr. He fund the heat energy released J t be nearly An Eletrdynamis f a Partile Aelerated t the Speed f Light with Cnstant Mass. Musa D. Abdullahi 013

3 equal t the ptential energy P lst, t gie P = J = K, where K was the kineti energy lst. Bertzzi identified J as slely due t the kineti energy K lst by the eletrns, n the assumptin that the aelerating fre n an eletrn f harge e ming in an eletri field f magnitude E, is ee, independent f the speed f the eletrn. Bertzzi might hae made a mistake in equating the ptential energy P with the kineti energy K f the eletrns. The energy equatin shuld hae been P = J + R = K + R, where R was the energy radiated. Radiatin is prpagated at the speed f light with maximum in a diretin perpendiular t the aeleratin f the eletrns. The transerse radiatin had n heating effet, as there was n mpnent impinging at the same pint r n the same target as the aelerated eletrns. This radiatin is a result f aberratin f eletri field. 1.4 Larmr Frmula fr Radiatin Pwer Larmr frmula f lassial eletrdynamis, desribed by Griffith [10], gies the radiatin pwer R p f an aelerated eletrn as prprtinal t the square f its aeleratin. Fr an eletrn reling with nstant speed in a irle f radius r with entripetal aeleratin f magnitude /r, Larmr lassial frmula gies R p = (e /6πε r ) 4 / 3, where ε is the permittiity f spae. Speial relatiity adpted this frmula [10] and gies radiatin pwer R = γ 4 R p, where the Lrentz fatr γ is defined in equatin (4). The relatiisti fatr γ 4 means that the radiatin pwer inreases explsiely as the speed apprahes that f light. Arding t Larmr frmula, the hydrgen atm, nsisting f an eletrn reling rund a heay psitiely harged nuleus, wuld radiate energy as it aelerates and spirals inward t llide with the nuleus, leading t the llapse f the atm. But atms are the mst stable entities knwn in nature. Use f Larmr frmula was unfrtunate as it led physis astray early in the 0 th entury. It required the brilliant hyptheses f Niels Bhr s [11] quantum mehanis t stabilize and retain the Rutherfrd s [1] nulear mdel f the hydrgen atm. In the alternatie eletrdynamis, there is n need fr Bhr s quantum thery t stabilize the nulear mdel f the hydrgen atm. In this paper it is shwn that irular relutin f an eletrn rund a nuleus, with nstant speed, is withut irradiatin. Radiatin mes nly if there is a hange in the ptential energy r kineti energy f a harged partile ming in an eletri field. 1.5 Aberratin f Eletri Field Figure1 depits an eletrn f harge e and mass m, ming at a pint P with elity, in an eletrstati field f intensity E due t a statinary sure harge +Q at an rigin O. Fr mtin at an angle θ t the aelerating fre F, the eletrn is subjeted t aberratin f eletri field. This is a phenmenn similar t aberratin f light disered by the English astrnmer James Bradley in 175 [13]. In aberratin f eletri field, as in aberratin f light, the diretin f the eletri field, indiated alng PN by the elity etr, as shwn in Figure 1, appears shifted by an aberratin angle α, frm the instantaneus line PO, suh that: sinα = sinθ (5) where the speeds and are the magnitudes f the elities and respetiely. The referene diretin is the diretin f the aelerating fre F. Equatin (5) was first deried by astrnmer James Bradley with respet t light radiatin frm a star. Aberratin f eletri field, whih is missing in lassial and relatiisti eletrdynamis, is used in the frmulatin f the alternatie eletrdynamis. The result f aberratin f eletri field is that the aelerating fre n a ming eletrn depends n the elity f the eletrn in an eletri field. If the aelerating fre is redued t zer at the speed f light, that speed bemes the ultimate limit, in ardane with 3 An Eletrdynamis f a Partile Aelerated t the Speed f Light with Cnstant Mass. Musa D. Abdullahi 013

4 Figure 1. Vetr diagram depits angle f aberratin α as a result f an eletrn f harge e and mass m ming, at a pint P, with elity, at an angle θ t the aelerating fre F. The unit etr û is in the diretin f the eletrstati field f intensity E due t a statinary sure harge +Q at the rigin O. Newtn s first law f mtin. Als, the differene between the aelerating fre F (n a ming eletrn) and the eletrstati fre ee (n a statinary eletrn) gies the radiatin reatin fre, frm whih the radiatin pwer is deried, in ntrast t Abraham-Lrentz frmula and Larmr frmula f lassial eletrdynamis.. Equatins f Retilinear Mtin in Classial eletrdynamis The aelerating fre F exerted n an eletrn f harge e and mass m ming at time t with elity and aeleratin (d/), in an eletrstati field f intensity E, in ardane with Culmb s law (equatin 3) and Newtn s nd law f mtin (equatin 1), is: d F= ee = m (6) Fr an eletrn aelerated in the ppsite diretin f a unifrm eletrstati field f nstant intensity E = Eû in the diretin f unit etr û, equatin (6) bemes: d F= eeû= m û (7) Equatin (7) is a first rder differential equatin with slutin: = at at = (8) where speed = 0 at time t = 0 and a = ee/m is a nstant. The speed may reah any alue. An eletrn ming with elity in the psitie diretin f the field, suffers a deeleratin and the equatin f mtin bemes: d F = eeû = m û (9) The slutin f equatin (9) fr an eletrn deelerated frm speed f light by a unifrm field f magnitude E, is: 4 An Eletrdynamis f a Partile Aelerated t the Speed f Light with Cnstant Mass. Musa D. Abdullahi 013

5 = at at = (10) The eletrn is deelerated t a stp in time t = /a. 3. Equatins f Retilinear Mtin in Relatiisti Eletrdynamis In relatiisti eletrdynamis, the aelerating fre F exerted n an eletrn f harge e and mass m, ming with elity at time t in an eletrstati field f intensity E, is: d F = eeû = ( m) û where mass m inreases with speed in ardane with equatin (4), s that d m F= eeû= û (11) Fr a nstant field f magnitude E, equatin (11) is als a first rder differential equatin with slutin as: at = (1) at 1+ where speed = 0 at time t = 0 and a = ee/m is a nstant. Equatin (1) makes the speed f light the ultimate limit as time t, in ntrast t equatin (8). In relatiisti eletrdynamis, an eletrn ming at the speed f light annt be deelerated and stpped by any finite fre. Suh a ming eletrn ntinues t me at the speed f light, gaining ptential energy withut lsing kineti energy. 4. Equatins f Mtin in the Alternatie Eletrdynamis The fre exerted n an eletrn, ming with elity, by an eletrstati field, is prpagated at the elity f light relatie t the sure harge and transmitted with elity ( ) relatie t the ming eletrn. The eletrn an be aelerated t the elity f light and n faster. In Figure 1 the eletrn may be aelerated in the diretin f the fre with θ = 0 r it may be deelerated against the fre with θ = π radians r it an rele in a irle, at nstant speed, perpendiular t the aelerating field, with θ = π/ radians. The aelerating fre F (Figure 1), n an eletrn f harge e and mass m ming at time t with elity and aeleratin (d/), in an eletrstati field f magnitude E, is prpsed as gien by the etr equatin and Newtn s send law f mtin, thus: ee d F = ( ) = m (13) where is the elity f light, f magnitude, at aberratin angle α t the aelerating fre F and ( ) is the relatie elity f transmissin f the fre with respet t the ming eletrn. The fre, prpagated at elity f light, annt ath up and impat n an eletrn als ming with elity =. With n fre n the eletrn, it ntinues t me with nstant speed, in ardane with Newtn s first law f mtin. Equatin (13) may be regarded as an extensin, amendment r mdifiatin f Culmb s law f eletrstati fre between tw eletri harges (equatin 3), taking int nsideratin the relatie elity between the harges. In equatin (13), the eletri field experiened by a ming harged partile may als be regarded as dependent n elity f the partile in the field. Equatin (5) linking the angle θ with the aberratin angle α (Figure1) and equatin (13) are the twin equatins f the alternatie eletrdynamis. Equatin (13) is the basi expressin 5 An Eletrdynamis f a Partile Aelerated t the Speed f Light with Cnstant Mass. Musa D. Abdullahi 013

6 f the alternatie eletrdynamis. Expanding equatin (13) by taking the mdulus f the etr ( ), with respet t the angles θ and α (Figure 1), gies: ee ( ) ee { s( )} m d F= = + θ α û = (14) where (θ α) is the angle between the etrs and. 4.1 Equatins f Retilinear Mtin Fr an eletrn aelerated in a straight line, where θ = 0, equatins (5) and (14) gie: ee 1 F = û = m d û (15) This is a first rder differential equatin. The slutin f equatin (15) fr an eletrn aelerated by a unifrm eletri field f nstant magnitude E, frm initial speed u, is: at = { u} exp (16) Fr aeleratin frm zer initial speed (u = 0), equatin (16) bemes: at = exp (17) where a = ee/m is a nstant. Fig..C1 is a graph f / against at/ fr equatin (17). The eletrn will be aelerated, by the eletri field, t an ultimate speed equal t that f light. The distane x= ( ), ered in time t by an eletrn aelerated frm zer initial speed, is btained by integrating equatin (17), t gie: at x = t+ exp 1 (18) a Fr an eletrn deelerated in a straight line, where θ = π radians, equatins (5) and (14) gie the deelerating fre F as: 1 F = ee + û = m d û (19) Sling the differential equatin (19) fr an eletrn deelerated frm speed u, by a unifrm eletri field f magnitude E, gies: at = { + u} exp (0) Fr an eletrn deelerated frm speed f light, equatin (0) bemes: at = exp 1 (1) Figure.C is a plt f / against at/ arding t equatin (1). The eletrn will be deelerated t a stp ( = 0) in time t = (/a)ln = 0.693/a, haing lst kineti energy 0.5m, equal t the ptential energy gained plus the energy radiated. Energy is radiated wheneer there is a hange in the kineti energy r ptential energy f a ming eletrn. Figure shws a graph f / against at/ fr an eletrn aelerated frm zer initial speed r an eletrn deelerated frm speed f light, by a unifrm eletri field: the slid lines, A1 and A arding t lassial eletrdynamis (equatins 8 and 10), the dashed ure B1 (fr equatin 1) and line B arding t relatiisti eletrdynamis and the dtted ures C1 and C arding t equatins (17) and (1) f the alternatie eletrdynamis. At law speeds the three systems f eletrdynamis inide fr aelerated eletrns but there is a marked departure fr eletrns deelerated frm the speed f light. 6 An Eletrdynamis f a Partile Aelerated t the Speed f Light with Cnstant Mass. Musa D. Abdullahi 013

7 Figure. / (speed in units f ) against at/ (time in units f /a) fr an eletrn f harge e and mass m = m aelerated frm zer initial speed r deelerated frm the speed f light, by a unifrm eletrstati field f magnitude E, where a = ee/m; the lines A1 and A arding t lassial eletrdynamis (equatins 8 and 10), the dashed ure B1 (equatin 1) and line (B) arding t relatiisti eletrdynamis and the dtted ures C1 and C arding t equatins (17) and (1) f the alternatie eletrdynamis. ( ) The distane x=, ered in time t by an eletrn deelerated frm the speed f light, is btained by integrating equatin (1) t gie: at x= exp t () a In equatins (1) and () it is seen that an eletrn entering a unifrm deelerating field at a pint with speed, mes t a stp in time t = 0.693/a, at a distane X = /a frm the pint f entry, haing lst kineti energy equal t 0.5m, gained ptential equal t eex = 0.307m and radiated energy equal t 0.193m. The eletrn will me bak t the starting pint (x = 0) in time t = 1.594/a, with speed and kineti energy 0.176m, haing lst ptential energy equal t eex = 0.307m and radiated energy equal t 0.131m. The eletrn will then be aelerated t the speed f light, as the ultimate limit, with emissin f radiatin. These results are nt btainable frm lassial r relatiisti eletrdynamis. 4. Equatins f Cirular Mtin Fr θ = π/ radians we get relutin is in a irle f radius r with nstant speed and entripetal aeleratin ( /r)û. Equatins (5) and (14), with mass m = m (rest mass) and nting that s(π/ α ) = sinα = /, gie the aelerating fre F as: F= ee û= m û= m û r r 7 An Eletrdynamis f a Partile Aelerated t the Speed f Light with Cnstant Mass. Musa D. Abdullahi 013

8 m ee = =ζ (3) r r eer m ζ = = (4) Equatin (1) fr relatiisti mass m and equatin (4) fr ζ (zeta rati) are idential but btained frm tw different pints f iew. In equatin (1), f relatiisti eletrdynamis, the quantity m inreases with speed, beming infinitely large at speed. In equatin (4), f the alternatie eletrdynamis, mass m remains nstant at the rest mass m, and the quantity ζ = {(ee)/( /r)} is the rati f magnitude f the radial eletrstati fre ( ee) n a statinary eletrn, t the entripetal aeleratin ( /r) in irular mtin. This quantity ζ, the zeta rati, may beme infinitely large at the speed f light, withut any diffiulty. At the speed f light, the eletrn mes with zer aeleratin in an ar f a irle f infinite radius, whih is a straight line, t make the rati ζ als infinite withut any prblem. Equating ζ with physial mass m, whih has a weight, is an expensie ase f mistaken identity. In lassial eletrdynamis, radius r f irular relutin fr an eletrn f harge e and mass m, in a radial eletri field f magnitude E due t a psitiely harged nuleus, is: m m r = r ee = ee = (5) where m = m is a nstant and r is the lassial radius. In relatiisti eletrdynamis, where mass m aries with speed in ardane with equatin (4), the radius f relutin bemes: m m r = = =γ r (6) ee ee In the alternatie eletrdynamis, where m = m is a nstant, the radius r f relutin, btained frm equatin (4), bemes: m m r = = =γ r (7) ee ee Relatiisti eletrdynamis and the alternatie eletrdynamis gie the same expressin fr radius f relutin in irular mtin as r = γr, but fr different reasns. This inrease in radius with speed was misnstrued in speial relatiity as inrease in mass with speed. 5. Radiatin Reatin Fre and Radiatin Pwer The aelerating fre n a ming eletrn is less than the eletrstati fre ee n a statinary eletrn. The differene between the aelerating fre F and the eletrstati fre ee is the radiatin reatin fre R f = F ( ee), that is always present when a harged partile is aelerated by an eletri field. This is analgus t a fritinal fre, whih always ppses mtin. A simple and useful expressin fr radiatin reatin fre R f is missing in lassial and relatiisti eletrdynamis and it makes all the differene. The diretin f maximum emissin f eletrmagneti radiatin, frm an aelerated harged partile, is perpendiular t the diretin f aeleratin. Fr retilinear mtin, with θ = 0 (Figure1), equatin (13) gies the radiatin reatin fre R f, in the diretin f unit etr û, as: ee ee ee R f = ( ) û+ eeû= û= (8) 8 An Eletrdynamis f a Partile Aelerated t the Speed f Light with Cnstant Mass. Musa D. Abdullahi 013

9 In deelerated retilinear mtin, with θ = π radians, R f = (ee/)û = (ee/), same as (8). Radiatin pwer is R p =.R f, the salar prdut f R f and elity. The salar prdut is btained, with referene t Figure 1, as: ee Rp = R f = ( ) + ee Rp = ee sθ s( θ α) + (9) Fr retilinear mtin with θ = 0 r θ = π radians, equatins (5) fr θ and α and equatin (9) gie radiatin pwer as: Rp = R. f = ee (30) Psitie radiatin pwer, as gien by equatin (30), means that energy is radiated in aelerated and deelerated mtins. In irular relutin, where is rthgnal t E and R f, the radiatin pwer R p (salar prdut f and R f ) is zer, as an be asertained frm equatins (5) and (9) with θ = π/ radians and s(θ α) = sinα = /. Equatin (9) is signifiant in the alternatie eletrdynamis. It makes irular relutin f an eletrn, rund a entral fre f attratin, as in Rutherfrd s nulear mdel f the hydrgen atm, stable, utside Bhr s quantum thery. Equatins (8), (9) and (30) are the radiatin frmulas f the alternatie eletrdynamis. These equatins are in ntrast t thse f lassial eletrdynamis where radiatin fre is prprtinal t the rate f hange f aeleratin (Abraham-Lrentz frmula) and the radiatin pwer is prprtinal t the square f aeleratin (Larmr frmula). There is n frmula fr radiatin reatin fre in relatiisti eletrdynamis. Speial relatiity adpted a mdified Larmr frmula: R = γ 4 R p. We nw nsider hange in ptential energy and radiatin fr an aelerated r deelerated eletrn in three systems f eletrdynamis 6. Ptential Energy in Classial Eletrdynamis In lassial eletrdynamis, the magnitude f aelerating fre n an eletrn f harge e and nstant mass m, ming at time t with speed and aeleratin f magnitude d/, in the ppsite diretin f an eletrstati field f magnitude E, is gien, in ardane with Culmb s law f eletrstati fre and Newtn s send law f mtin, by equatin (6): d ee = m Fr retilinear mtin in the diretin f a displaement x, we btain the differential equatin: d d ee = m m = dx (31) The ptential energy P lst by the eletrn r wrk dne n the eletrn, in being aelerated with nstant mass m = m, thrugh distane x frm an rigin (x = 0), t a speed frm rest, is: ( ) ( ) x 0 (3) 0 P= ee dx = m m d Integrating, equatin (3) gies: x 1 P= ee( dx) = m 0 This is equal t the kineti energy K gained by the eletrn. P m 1 = (33) 9 An Eletrdynamis f a Partile Aelerated t the Speed f Light with Cnstant Mass. Musa D. Abdullahi 013

10 Here, with n nsideratin f radiatin, the ptential energy lst is equal t the kineti energy gained by an aelerated eletrn. A graph f / against P/m is shwn as A1 in Figure 3. In lassial eletrdynamis, an eletrn, ming at the speed f light, an be deelerated t a stp and may be aelerated in the ppsite diretin t reah a speed greater than. The ptential energy P gained, equal t the kineti energy K lst, in deelerating an eletrn frm the speed f light t a speed, within a distane x in a field f magnitude E, withut radiatin, is: x 1 P= ee( dx) = m ( d) m( ) 0 = P 1 = 1 (34) m A graph f / against P/m is shwn as A in Figure Ptential Energy in Relatiisti Eletrdynamis In relatiisti eletrdynamis, the kineti energy K gained by an eletrn r the wrk dne, in being aelerated by an eletri field E, thrugh a distane x, t a speed frm rest, is the ptential energy P lst. There is n nsideratin f energy radiatin in this situatin. The kineti energy K f a partile f mass m and rest mass m ming with speed, is gien by the relatiisti equatin: x 0 0 x ( ) = = = ee dx K P m m ( ) = = = ee dx K P m m m P= m P 1 = 1 (35) m where m is the rest mass (at = 0) and the speed f light in a auum. The amunt f kineti energy is suppsed t be aunted fr by the inrease in mass. Bertzzi s experiment was nduted t erify equatin (35) and it did s in a remarkable way. A graph f / against P/m, fr equatin (35), is shwn as B1 in Figure 3. In relatiisti eletrdynamis, an eletrn ming at the speed f light (with infinite mass), annt be stpped by any deelerating fre. The eletrn ntinues t me at the same speed, (line B in Figure 3) gaining ptential energy withut lsing kineti energy. This is the pint f departure between relatiisti and the alternatie eletrdynamis 8. Ptential Energy and Radiatin in the Alternatie Eletrdynamis In the alternatie eletrdynamis, the fre F (Figure 1), is gien by equatins (5) and (14). Fr an eletrn aelerated in a straight line, the equatins with θ = 0, gie: ee 1 F= û= m d û (36) The salar equatin is: d d ee = m = m (37) dx 10 An Eletrdynamis f a Partile Aelerated t the Speed f Light with Cnstant Mass. Musa D. Abdullahi 013

11 Ptential energy P lst in aelerating an eletrn thrugh distane x, t a speed frm rest, is: P= ee dx = m 0 0 Resling the right-hand integral int partial fratins, we btain: 1 P = m 1 d 0 (39) P m ln 1 = m (40) p = ln 1 (41) m The energy radiated R in aeleratin is btained by subtrating the kineti energy gained, K = ½ m, frm ptential energy P lst, thus: 1 R= P K = m ln m m R= m ln + + (4) In aeleratin, equatin (41), fr the alternatie eletrdynamis, shuld be mpared with equatin (35) fr relatiisti eletrdynamis and equatin (33) fr lassial eletrdynamis. Fr a deelerated eletrn, equatins (5) and (14), with θ = π radians, gie: ee 1 F= + û= m d û (43) d d ee 1+ = m = m (44) dx Ptential energy P gained in deelerating the eletrn thrugh distane x, frm speed t, is: x d P = ee( dx) = m (45) 0 1+ Resling the integrand int partial fratins and integrating, the ptential energy gained is: 1 P = m d (46) 1+ 1 P= m ln 1+ + m (47) Graphs f P/m against / are shwn as C1 and C, in Figure 3 fr equatins (41) and (47). Energy radiated R is kineti energy lst minus ptential energy gained, thus: 1 1 R= m( ) m ln 1+ m R ( ) x d (38) 1 1 = m + + ln 1+ (48) 11 An Eletrdynamis f a Partile Aelerated t the Speed f Light with Cnstant Mass. Musa D. Abdullahi 013

12 9. Speed Versus Ptential Energy in Bertzzi s Experiment In the experiment perfrmed by William Bertzzi [9], the speed f high-energy eletrns was determined by measuring the time T required fr them t traerse a distane f 8.4 metres after haing been aelerated thrugh a ptential energy P inside a linear aeleratr. Bertzzi s experimental data, reprdued in Table 1, learly demnstrate that eletrns aelerated thrugh ptential energy f 15 MeV attain, pratially, the speed f light. TABLE 1. RESULTS OF BERTOZZI S EXPERIMENTS WITH ELECTRONS ACCELERATED IN TIME T THROUGH 8.4 M AND ENERGY P IN AN ACCELERATOR (m = 0.5 MeV, = 8.4/T m/se) P T x 10-8 MeV P/m se. x 10 8 m/se / Experiment / Classial Equatin 33 / Relatiisti Equatin 35 / Alternatie Equatin * * * * * * Figure 3. / (speed in units f ) against P/m (ptential energy in units f m ) fr an eletrn f mass m aelerated frm zer initial speed r deelerated frm the speed f light, the slid lines (A1 and A) arding t lassial eletrdynamis (equatin 33 and 34), the dashed ure (B1) arding t relatiisti eletrdynamis (equatin 35) and the dtted ures (C1 and C) arding t the alternatie eletrdynamis (equatins 41 and 47). The slid squares are the result f Bertzzi s experiment (Table 1). 1 An Eletrdynamis f a Partile Aelerated t the Speed f Light with Cnstant Mass. Musa D. Abdullahi 013

13 10. Speed and Kineti Energy in 3 Systems f Eletrdynamis In lassial eletrdynamis, kineti energy gained by an eletrn f mass m in being aelerated, t a speed, is K = ½ m, same as equatin (33) fr ptential energy P lst. K = (49) m In lassial eletrdynamis, kineti energy lst in deeleratin frm the speed f light t a speed, is K = ½ m( ), same as equatin (34) fr ptential energy P gained. K 1 = 1 (50) m Equatins (49) and (50) are the same fr lassial and relatiisti eletrdynamis. These are respetiely illustrated as ure (A1 & C1) and ure (A & C) in Figure 4. Figure 4. / (speed in units f ) against K/m (kineti energy lst r gained in units f m ) fr an eletrn aelerated frm zer initial speed r deelerated frm the speed f light, under 3 systems f eletrdynamis In relatiisti eletrdynamis, kineti energy gained by an eletrn f rest mass m in being aelerated t a speed, frm rest, is: m K = m m = m K 1 = 1 (51) m In relatiisti eletrdynamis, an eletrn ming at the speed f light =, with infinite mmentum, annt be stpped by any fre. It is suppsed t ntinue ming at speed. 13 An Eletrdynamis f a Partile Aelerated t the Speed f Light with Cnstant Mass. Musa D. Abdullahi 013

14 Figure 4 shws graphs f / (speed in units f ) against K/m (kineti energy, gained r lst, in units f m ) fr an eletrn f harge e and mass m aelerated frm zer initial speed r deelerated frm the speed f light, by a unifrm field f magnitude E, where a = ee/m is a nstant. The dtted ures (A1 & C1) and (A & C) are in ardane with the lassial eletrdynamis and the alternatie eletrdynamis (equatins 49 and 50). The dashed ure B1 is in ardane with relatiisti eletrdynamis (equatin 51). N ure is btainable in relatiisti eletrdynamis fr an eletrn deelerated frm the speed f light. 11. Speed Versus Energy Radiatin Figure 5 shws graphs f / (speed in units f ) against R/m energy radiated (in units f m ) fr an eletrn f mass m and harge e aelerated by an eletri field frm zer initial speed r deelerated frm the speed f light arding t the alternatie eletrdynamis. The ure C1 is fr an aelerated eletrn (equatin 4) and ure C fr a deelerated eletrn (equatin 48). Energy is always radiated, under aeleratin r deeleratin. There are n suh energy radiatin graphs frm the pints f iews f lassial eletrdynamis and relatiisti eletrdynamis. Radiatin, the mst mmn phenmenn in nature, makes the differene between the three systems f eletrdynamis. Figure 5. Graphs f / (speed in units f ) against R/m energy radiated (in units f m ) fr an eletrn aelerated frm zer initial speed r deelerated frm the speed f light, under the alternatie eletrdynamis. 14 An Eletrdynamis f a Partile Aelerated t the Speed f Light with Cnstant Mass. Musa D. Abdullahi 013

15 1. Cnluding Remarks Quantum eletrdynamis is nt required in desribing the mtin f an eletrn in an eletri field. Relatiisti eletrdynamis and the alternatie eletrdynamis appear t be in agreement fr an aelerated eletrn in retilinear mtin (equatins 1 and 17), as depited in Fig.. Bertzzi s experimental results appear t be in agreement with relatiisti eletrdynamis (equatin 35) and the alternatie eletrdynamis (equatin 41) fr an aelerated eletrn in retilinear mtin, as depited in Figure 3. Relatiisti and the alternatie systems f eletrdynamis demnstrate learly the speed f light as a limit; relatiisti eletrdynamis n the basis f mass f a ming partile inreasing t beme infinitely large at the speed f light and the alternatie eletrdynamis n the basis f aelerating fre r aelerating field reduing t zer at the speed f light. Atually, it is the aelerating field E experiened, as well as the aelerating fre ee, whih depends n the speed f a harged partile in the field, as harge e is a nstant. Relatiisti eletrdynamis and the alternatie eletrdynamis gie the same expressin fr radius f irular relutin as γr fr an eletrn rund a entre f fre f attratin, where Lrentz fatr γ is defined in equatin (1) and r is the lassial radius as expressed in equatin (7). In irular relutin f a harged partile, derease in aelerating fre with speed, in ardane with the alternatie eletrdynamis, has the same effet (inrease in radius) as apparent inrease f mass with speed in ardane with relatiisti eletrdynamis. This may explain the apparent agreement between relatiisti eletrdynamis and relutin f harged partiles (eletrns and prtns) in yli aeleratrs. At the speed f light the aelerating fre n a reling harged partile redues t zer, in ardane with the alternatie eletrdynamis, and it mes in a irle f infinite radius, whih is a straight line. This is in ntrast t relatiisti eletrdynamis where inrease in radius is misinterpreted as being the result f mass inreasing with speed. In the alternatie eletrdynamis, the fre exerted n an eletri harge ming in an eletri field, depends n the elity f the harge. This is tantamunt t mdifying Culmb s law f eletrstati fre, taking int nsideratin the relatie elity between tw eletri harges ming in spae. This is nt the ase in lassial and relatiisti eletrdynamis where the Culmb fre is independent f the relatie elities. The questin nw is: Whih ne f the eletrdynamis is rret? The answer may be fund in the mtin f eletrns deelerated frm the speed f light. Arding t lassial eletrdynamis, an eletrn f mass m entering a retarding field at a pint (x = 0), with speed f light, is brught t rest after lsing kineti energy 0.500m, equal t the ptential energy gained, withut energy radiatin. The eletrn may then be aelerated bakwards t reah the pint f entry with speed and may reah a speed greater than withut radiatin after a lng time. Arding t relatiisti eletrdynamis, an eletrn ming at the speed f light (with infinitely large mass, mmentum and energy), annt be stpped by any finite fre. The eletrn is suppsed t me at the speed f light gaining ptential energy withut lsing kineti energy, ntrary t the priniple f nseratin f energy. In the alternatie eletrdynamis an eletrn ming at the speed f light (with kineti energy 0.500m ) n entering a retarding field at a pint (x = 0), is easily brught t rest after gaining ptential energy equal t 0.307m and radiating energy equal t 0.193m (Figures and 5). The kineti energy K lst minus the ptential energy P gained is equal t the energy R radiated (K - P = R). The eletrn is then aelerated bakwards t return t the pint f entry (x = 0) at speed = 0.594, lsing ptential energy equal t 0.307m, gaining kineti energy ½ m = 0.176m and radiating energy equal t 0.131m (Figure 3). The ptential energy P lst minus the kineti energy K gained is equal t the energy R radiated (P - K = R). Energy radiated in the rund trip, entering with speed and returning bak with speed 0.594, is 15 An Eletrdynamis f a Partile Aelerated t the Speed f Light with Cnstant Mass. Musa D. Abdullahi 013

16 0.34m. The eletrn may then be aelerated bakwards t reah an ultimate speed with radiatin f energy. Fr eletrns aelerated by an eletri field, lassial eletrdynamis is a failure but relatiisti eletrdynamis, the alternatie eletrdynamis and Bertzzi s experiment appear t be in agreement (Figures and 3). The piture is mpletely different fr deelerated eletrns. It is energy radiatin that makes all the differene. In the alternatie eletrdynamis, an eletrn ming at the speed f light is easily brught t rest n entering a deelerating field at a pint. In an experiment with a narrw burst f eletrns aelerated as near t the speed f light as pssible and made t enter a deelerating field at a pint, the eletrns being stpped inalidates relatiisti eletrdynamis. Suh eletrns being stpped and turned bak n their traks, t return t the pint f entry with speed equal t 0.594, inalidates relatiisti eletrdynamis and inaugurates the alternatie eletrdynamis. 1. Referenes [1] P. Lenard; Great Man f Siene, G. Bell and Sns Ltd., Lndn (1958), pp [] I. Newtn (1687); Mathematial Priniples f Natural Philsphy (Translated by F. Cajri), Uniersity f Califrnia Press, Berkeley (1964). [3] A. Einstein; On the Eletrdynamis f Ming Bdies, Ann. Phys., 17 (1905), 891 [4] A. Einstein; & H.A. Lrentz, The Priniples f Relatiity, Matheun, Lndn (193). [5] M. Plank; Ann. Phys,, 4 (1901), 533. [6] L. de Brglie; Researhes in Quantum Mehanis, Dtral Thesis, Uniersity f Paris, 194. [7] P. Bekmann; Einstein Plus Tw, The Glem Press, Bulder (1987). [8] C. Renshaw; Galilean Eletrdynamis, Vl. 7/6, 1996 [9] W. Bertzzi; Speed and Kineti Energy f Relatiisti Eletrns", Am. J. Phys., 3 (1964), Als nline at: [10] D.J. Griffith; Intrdutin t Eletrdynamis, Prentie-Hall, Englewd Cliff, New Jersey (1981), pp (Third ed. 1998), [11] N. Bhr; Phil. Mag., 6 (1913), 476. [1] E. Rutherfrd; Phil. Mag., 1 (1911), 669. [13] 16 An Eletrdynamis f a Partile Aelerated t the Speed f Light with Cnstant Mass. Musa D. Abdullahi 013