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

Transcription

1 1 MOTION OF AN ELECTRON IN CLASSICAL AND RELATIVISTIC ELECTRODYNAMICS AND AN ALTERNATIVE ELECTRODYNAMICS Musa D. Abdullahi 1 Bujumbura Street, Wuse, Abuja, Nigeria musadab@outlook.om Abstrat As the fore exerted by an eletri field is propagated at the eloity of light, an eletri harge moing with eloity is subjet to aberration of eletri field. For an eletron of mass m and harge e moing with aeleration d/ in an eletri field of magnitude E, the aelerating fore, in aordane with Newton s seond law of motion, is put as etor F = ee( )/ = m(d/), where ( ) is the relatie eloity between the eletron and the aelerating fore. With F = 0 at =, the eletron is aelerated to the speed of light as a limit. The eletron an reole in a irle with onstant speed. The relatiisti mass-eloity formula is found to be orret where an eletron moes perpendiular to an eletri field, as in irular reolution around a nuleus. It is argued that relatiisti mass is not a physial quantity but the ratio eer/ of eletrostati fore to aeleration in a irle of radius r, whih beomes infinitely large for retilinear motion in a irle of infinite radius. An alternatie eletrodynamis is deeloped for an eletron aelerated to the speed of light at onstant mass and with emission of radiation. Radiation ours if there is a hange in kineti or potential energy of a moing eletron. Cirular reolution in Rutherford s nulear model of the hydrogen atom is shown to be stable outside quantum mehanis. Keywords: Aberration of eletri field, aeleration, harge, field, fore, mass, radiation, relatiity, speed, eloity. 1. Introdution There are now three systems of eletrodynamis in physis. Classial eletrodynamis is appliable to eletrially harged partiles moing at a speed that is muh slower than that of light. Relatiisti eletrodynamis is for partiles moing at speeds omparable to that of light. Quantum eletrodynamis is for atomi partiles moing at ery high speeds. There should be one system of eletrodynamis appliable to all partiles at all speeds up to that of light, as a limit. This paper presents an alternatie system, named radiatie eletrodynamis. An Eletron Aelerated to the Speed of Light with Constant Mass & Emission of Radiation

2 Classial eletrodynamis is based on the seond law of motion, originated by Galileo Galilei in 1638 [1], but enuniated by Isaa Newton []. The theory of speial relatiity was formulated in 1905 mainly by Albert Einstein [3, 4]. The quantum theory was deised by Max Plank [5], Louis de Broglie [6] and others. Relatiisti eletrodynamis redues to lassial eletrodynamis at low speeds. The relatiity and quantum theories, howeer, are inompatible at high speeds. Both the relatiity and quantum theories annot be orret. One or both theories may be wrong. Indeed, speial relatiity is under attak by physiists: Bekmann [7] and Renshaw [8]. In this paper, quantum theory and general relatiity are not onsidered while Einstein s theory of speial relatiity is hallenged. The paper introdues an alternatie eletrodynamis, appliable to an eletrially harged partile, like an eletron, moing in an eletri field at speeds up to that of light, with emission of radiation and mass of a partile remaining onstant. 1.1 Newton s Seond Law of Motion For a body moing with eloity at time t, Newton s seond law of motion, whih inludes the first and third laws, relates the rate of hange of eloity or aeleration d/ produed on a body of mass m, to the impressed fore F, in the etor equation: d F = m (1) Aording to equation (1), where mass m is a onstant independent of eloity, the aeleration beomes zero, and the body moes in a straight line with onstant speed, if the aelerating fore F redues to zero or if the mass m beomes infinitely large. With the emergene of speial relatiity, where mass m is supposed to inrease with, Newton s seond law of motion was modified. The law now relates the impressed fore F to rate of hange of momentum m, thus: d F = ( m ) () This paper assumes the alidity of Newton s seond law of motion (equation 1), but where mass m remains onstant and aelerating fore F redues to zero at the eloity of light, of magnitude. 1. Coulomb s Law of Eletrostati Fore Coulomb s law, first published in 1785, is the most important priniple in physis. It gies the fore F of attration between an An Eletron Aelerated to the Speed of Light with Constant Mass & Emission of Radiation

3 eletron of harge e and a positie eletri harge Q, separated by a distane r in spae, as etor: eq F = û = ee (3) 4πε o r where û is a unit etor in the diretion of E, the eletrostati field due to harge Q, and ε o is the permittiity of spae. The urrent problem of physis lies in making Coulomb s law independent of eloity of the eletron, of harge e, moing with eloity in the eletrostati field of intensity E. This paper is based on the fore between two eletri harges being dependent on the magnitude and diretion of relatie eloity between the harges. It alls for a modifiation of equation Relatiisti Mass-eloity Formula Aording to Newton s seond law of motion, a fore an aelerate a partile to a speed greater than that of light, with its mass remaining onstant. But experiments with aelerators hae shown that no partile, not een the eletron, the lightest partile known in nature, an be aelerated beyond the speed of light. The theory of speial relatiity explains this limitation by positing that the mass of a partile inreases with its speed, beoming infinitely large at the speed of light. That sine an infinitely large mass annot be aelerated any faster by any finite fore, the speed of light beomes the limit to whih a body an be aelerated. This is a plausible proposition. The well-known relatiisti mass-eloity formula is: mo m = = γ mo (4) 1 where m is the mass of a partile moing with speed relatie to an obserer, m o is the rest mass, is the speed of light in a auum and γ is the Lorentz fator. Equation (4), where m is a physial quantity, beoming infinitely large at the speed of light, is under hallenge in this paper. The diffiulty with infinite mass, at the speed of light, in equation (4), is the Ahilles heel of the theory of speial relatiity. Resoling this diffiulty, by allowing a moing partile, suh as an eletron, to reah the speed of light with its mass remaining onstant, is the purpose of this paper. Suh a resolution, giing the speed of light as the ultimate speed, without infinite mass of a moing partile, should bring great relief to physiists all oer the world. 3 An Eletron Aelerated to the Speed of Light with Constant Mass & Emission of Radiation

4 The proponents of speial relatiity just ignore the problem with equation (4). They say that it is the momentum, not the mass, whih inreases with speed. They aoid the diffiulty altogether by arguing that the speed neer really reahes that of light, or that partiles moing at the speed of light (photons) hae zero rest mass. But eletrons are easily aelerated and hae been aelerated to pratially the speed of light as demonstrated by William Bertozzi in 1964 [9], using a linear aelerator of 15MeV energy. Eletron aelerators, betatrons and eletron synhrotrons of oer 15x10 6 MeV, hae been built and operated with eletrons moing at the speed of light for all pratial purposes. A most remarkable demonstration of the existene of a uniersal limiting speed, equal to the speed of light, was in an experiment by William Bertozzi of the Massahusetts Institute of Tehnology [9]. The experiment (see Table1) showed that eletrons aelerated through energies of 15 MeV or oer, attain, pratially, the speed of light. Bertozzi measured the heat energy J deeloped when a stream of aelerated eletrons hit an aluminium target at the end of their flight path, in a linear aelerator. He found the heat energy released J to be nearly equal to the potential energy P lost, to gie P = J = K, where K was the kineti energy lost. Bertozzi identified J as solely due to the kineti energy K gained by the eletrons, on the assumption that the aelerating fore on an eletron of harge e moing in an eletri field of magnitude E, is ee, independent of the speed of the eletron. Bertozzi might hae made a mistake in equating the potential energy P lost with the kineti energy K gained by the eletrons. The energy equation should hae been P = J + R = K + R, where R was the energy radiated by the aelerated eletrons. Radiation is propagated at the speed of light with maximum in a diretion perpendiular to the aeleration of the eletrons. The transerse radiation had no heating effet, as there was no omponent impinging at the same point or on the same target as the aelerated eletrons. The heating effet should be due to a stream of eletrons hitting the target. The transerse radiation, whih might hae been negligible, should be a result of aberration of eletri field to be desribed here. 1.4 Larmor Formula for Radiation Power Larmor formula of lassial eletrodynamis, desribed by Griffith [10], gies the radiation power R p of an aelerated eletron as proportional to the square of its aeleration. For an eletron reoling 4 An Eletron Aelerated to the Speed of Light with Constant Mass & Emission of Radiation

5 with onstant speed in a irle of radius r with entripetal aeleration of magnitude a = /r, Larmor lassial formula for radiation power gies R p = (e /6πε o 3 )a = (e /6πε o 3 ) 4 /r, where ε o is the permittiity of spae. Speial relatiity adopted this formula [10] and gies radiation power R = γ 4 R p, where the Lorentz fator γ is defined in equation (4). The relatiisti fator γ 4 means that the radiation power inreases explosiely as the speed approahes that of light. Aording to Larmor formula, the hydrogen atom, onsisting of an eletron reoling round a heay positiely harged nuleus, would radiate energy as it aelerates and spirals inward to ollide with the nuleus, leading to the ollapse of the atom. But atoms are the most stable entities known in nature. Use of Larmor formula was unfortunate as it led physis astray early in the 0 th entury. It required the brilliant, but erroneous, hypotheses of Niels Bohr s [11] quantum theory to stabilize and retain Rutherford s [1] nulear model of the hydrogen atom. This pushed physis into the quagmire of quantum mehanis. In the alternatie eletrodynamis, there is no need for Bohr s quantum theory to stabilize the nulear model of the hydrogen atom. In this paper it is shown that irular reolution of an eletron round a nuleus, with onstant speed, is without irradiation. Radiation omes only if there is a hange in the potential energy or kineti energy of a harged partile moing in an eletri field. 1.5 Abraham-Lorentz Formula for Radiation Fore. Abraham-Lorentz formula [10] of lassial eletrodynamis gies the radiation reation fore R f on an aelerated eletron, proportional to the rate of hange of aeleration, as R f = (e/6πε o 3 )da/. The orretness of Lorentz-Abraham formula is doubtful, as the salar produt of R f and eloity does not gie Lamor formula for radiation power R p in lassial eletrodynamis. There is no formula for radiation reation fore in relatiisti eletrodynamis. As suh there should hae been no emission of radiation under relatiisti eletrodynamis. But there must be radiation, the most ommon phenomenon in nature. Speial relatiity found a way out by adopting a modified Lamor formula as R = γ 4 R p. 1.6 Aberration of Eletri Field Figure1 depits an eletron of harge e and mass m, moing at a point P with eloity, in an eletrostati field of intensity E due to a 5 An Eletron Aelerated to the Speed of Light with Constant Mass & Emission of Radiation

6 stationary soure harge +Q at an origin O. For motion at an angle θ to the aelerating fore F, the eletron is subjeted to aberration of eletri field. This is a phenomenon similar to aberration of light disoered by the English astronomer James Bradley in 175 [13]. It is one of the most signifiant disoeries in siene but now seldom mentioned in physis. This relegation may be beause aberration shows the relatiity ( - ) of eloity of light with respet to an obserer or objet moing with eloity, ontrary to the relatiisti ardinal priniple of onstany of speed of light. In aberration of eletri field, as in aberration of light, the diretion of the eletri field, indiated along PN by the eloity etor, as (Figure 1), appears shifted by an aberration angle α, from the instantaneous line PO, suh that: sinα = sinθ (5) where the speeds and are the magnitudes of the eloities and respetiely. The referene diretion is the diretion of the aelerating fore F. Equation (5) was first deried by astronomer James Bradley with respet to light radiation from a star. Aberration of eletri field, whih is missing in lassial and relatiisti eletrodynamis, is used in the formulation of radiatie eletrodynamis presented here as an alternatie eletrodynamis. The speed of light is explained, as an ultimate limit, without infinite mass of an aelerated harged partile. 6 Figure 1 Vetor diagram depits angle of aberration α as a result of an eletron of harge e and mass m moing, at a point P, with eloity, at an angle θ to the aelerating fore F. The unit etor û is in the diretion of the eletrostati field of intensity E due to a stationary soure harge +Q at the origin O. An Eletron Aelerated to the Speed of Light with Constant Mass & Emission of Radiation

7 The result of aberration of eletri field is that the aelerating fore on a moing eletron depends on the eloity of the eletron in an eletri field. If the aelerating fore is redued to zero at the speed of light, that speed beomes an ultimate limit, aording to Newton s first law of motion. The differene F - (-ee) between the aelerating fore F (on an eletron moing with eloity ) and the eletrostati fore -ee (on a stationary eletron) gies the radiation reation fore R f. Radiation power R p is simply deried as salar produt -.R f, in ontrast to Abraham-Lorentz and Larmor formulas of lassial eletrodynamis.. Retilinear Motion in Classial eletrodynamis The aelerating fore F exerted on an eletron of harge e and mass m moing at time t with eloity and aeleration (d/), in an eletrostati field of intensity E, in aordane with Coulomb s law (equation 3) and Newton s seond law of motion (equation 1), is gien by etor equation: d F = ee = m (6) For an eletron aelerated in the opposite diretion of a uniform eletrostati field of onstant intensity E = Eû in the diretion of unit etor û, equation (6) beomes: d F = eeû = m û (7) Equation (7) is a first order differential equation with solution: = at at = (8) where the speed = 0 at time t = 0 and a = ee/m is the onstant of aeleration. The speed may reah an infinitely large alue if the fore is applied for a suffiiently long time t. An eletron moing with eloity in the positie diretion of the eletri field, suffers a deeleration and the equation of retilinear motion beomes: d F = eeû = m û (9) The solution of equation (9) for an eletron deelerated from speed of light by a uniform eletri field of magnitude E, is: 7 An Eletron Aelerated to the Speed of Light with Constant Mass & Emission of Radiation

8 8 = at at = 1 (10) The eletron is deelerated to a stop in time t = /a. 3. Retilinear Motion in Relatiisti Eletrodynamis In relatiisti eletrodynamis, the aelerating fore F exerted on an eletron of harge e and mass m, moing with eloity at time t in an eletrostati field of intensity E, is: d F = eeû = ( m) û where mass m is supposed to inrease with speed in aordane with equation (4), so that d mo F = eeû = û (11) 1 For a onstant field of magnitude E, equation (11) is also a first order differential equation with solution as: at = (1) a t 1+ where speed = 0 at time t = 0 and a = ee/m o is a onstant. Equation (1) makes the speed of light the ultimate limit as time t, in ontrast to equation (8). In relatiisti eletrodynamis, an eletron moing with an infinitely large mass, at the speed of light annot be deelerated and stopped by any finite fore. Suh a moing eletron ontinues to moe at the speed of light, gaining potential energy without losing kineti energy, ontrary to the priniple of onseration of energy. 4. Motion in the Alternatie Eletrodynamis The fore exerted on an eletron, moing with eloity, by an eletrostati field, is propagated at the eloity of light relatie to the stationary soure harge and transmitted with eloity ( ) relatie to the moing eletron. The eletron an be aelerated to the eloity of An Eletron Aelerated to the Speed of Light with Constant Mass & Emission of Radiation

9 light and no faster. In Figure 1 the eletron may be aelerated in the diretion of the fore with θ = 0 or it may be deelerated against the fore with θ = π radians or it may reole in a irle, at a speed perpendiular to the aelerating field, with θ = π/ radians. Reolution in a irle of onstant radius r is with onstant speed and onstant entripetal aeleration /r. The aelerating fore F (Figure 1), on an eletron of harge e and mass m moing at time t with eloity and aeleration (d/), in an eletrostati field of magnitude E, is proposed as gien by the etor equation and Newton s seond law of motion, thus: ee d F = ( ) = m (13) where is the eloity of light, of magnitude, at aberration angle α to the aelerating fore F and ( ) is the relatie eloity of transmission of the fore with respet to the moing eletron. The fore, propagated at eloity of light, annot ath up and impat on an eletron also moing with eloity =. With no fore on the eletron, it ontinues to moe with onstant speed, in aordane with Newton s first law of motion. Equation (13) may be regarded as an extension, amendment or modifiation of Coulomb s law of eletrostati fore between two eletri harges (equation 3), taking into onsideration the relatie eloity between the harges. In equation (13), the eletri field experiened by a moing harged partile may also be regarded as dependent on the magnitude and diretion of eloity of the partile in the field. Equation (5) linking the angle θ with the aberration angle α (Figure 1) and equation (13) are the twin equations of the alternatie eletrodynamis. Equation (13) is the basi expression of radiatie eletrodynamis as the alternatie eletrodynamis. Expanding equation (13) by taking the modulus of the etor ( ), with respet to the angles θ and α (Figure 1), gies: ee ( ) ee { os ( )} m d F = = + θ α û = (14) where (θ α) is the angle between the etors and and û is a unit etor in the diretion of eletri field of magnitude E. 4.1 Equations of Retilinear Motion For an eletron aelerated in a straight line, where θ = 0, equations (5) and (14) gie: 9 An Eletron Aelerated to the Speed of Light with Constant Mass & Emission of Radiation

10 ee 1 F = û = m d û (15) This is a first order differential equation. The solution of equation (15) for an eletron aelerated by a uniform eletri field of onstant magnitude E, from initial speed u, is: at = { u} exp (16) For aeleration from zero initial speed (u = 0), equation (16) beomes: at = 1 exp (17) where a = ee/m is the onstant of aeleration. Figure.C1 is a graph of / against at/ for equation (17). The eletron will be aelerated, by the eletri field, to a maximum or ultimate speed equal to that of light, with emission of radiation. The distane x, oered in time t by an eletron = ( ) aelerated from zero initial speed, is obtained by integrating equation (17), with respet to time t, to gie: at x = t + exp 1 (18) a For an eletron deelerated in a straight line, where θ = π radians, equations (5) and (14) gie the deelerating fore F, in aordane with Newton s seond law of motion, as: 1 F = ee + û = m d û (19) Soling the differential equation (19) for an eletron deelerated from speed u, by a uniform eletri field of magnitude E, gies: at = { + u} exp (0) For an eletron deelerated from the speed of light, equation (0) beomes: at = exp 1 (1) Figure.C is a plot of / against at/ aording to equation (1). The eletron will be deelerated to a stop ( = 0) in time t = (/a)ln = 10 An Eletron Aelerated to the Speed of Light with Constant Mass & Emission of Radiation

11 0.693/a, haing lost kineti energy 0.5m, equal to the potential energy gained plus the energy radiated. Energy is radiated wheneer there is a hange in the kineti energy or potential energy of an eletron moing in an eletri field. Figure is a graph of / against at/ for an eletron aelerated from zero initial speed or an eletron deelerated from speed of light, by a uniform eletri field: the solid lines, A1 and A aording to lassial eletrodynamis (equations 8 and 10), the dashed ure B1 (for equation 1) and line B aording to relatiisti eletrodynamis and the dotted ures C1 and C aording to equations (17) and (1) of the alternatie eletrodynamis. At law speeds the three systems of eletrodynamis oinide for aelerated eletrons but there is a marked departure for deeleration from the speed of light. Figure / (speed in units of ) against at/ (time in units of /a) for an eletron of harge e and mass m = m o aelerated from zero initial speed or deelerated from the speed of light, by a uniform eletrostati field of magnitude E, where a = ee/m; the lines A1 and A aording to lassial eletrodynamis (equations 8 and 10), the dashed ure B1 (equation 1) and line (B) aording to relatiisti eletrodynamis and the dotted ures C1 and C aording to equations (17) and (1) of the alternatie eletrodynamis. 11 An Eletron Aelerated to the Speed of Light with Constant Mass & Emission of Radiation

12 The distane x = ( ), oered in time t by an eletron deelerated from the speed of light, is obtained by integrating equation (1) to gie: at x = 1 exp t () a After a long time t the eletron moes in the opposite diretion with speed and radiation of energy. In equations (1) and () it is seen that an eletron entering a uniform deelerating field at a point with speed, omes to a stop in time t = 0.693/a, at a distane X = /a from the point of entry, haing lost kineti energy equal to 0.5m, gained potential equal to eex = 0.307m and radiated energy equal to 0.193m. The eletron will ome bak to the starting point (x = 0) in time t = 1.594/a, with speed and kineti energy 0.176m, haing lost potential energy equal to eex = 0.307m and radiated energy equal to 0.131m. The eletron will then be aelerated to the speed of light, as the ultimate limit, with emission of radiation. The eletron ontinues to radiate energy een as it moes in an eletri field with onstant speed of light equal to and loss of potential energy. 4. Equations of Cirular Motion For θ = π/ radians we get motion perpendiular to the field and reolution is in a irle of radius r with onstant speed and aeleration ( /r)û. Equations (5) and (14), with mass m = m o (rest mass) and noting that os(π/ α ) = sinα = /, gie the aelerating fore F as etor: F = ee 1 û = m û = m o û (3) r r Equation (3) gies the salar equation: mo ee = = ζ r r 1 eer mo ζ = = = γ m 1 1 An Eletron Aelerated to the Speed of Light with Constant Mass & Emission of Radiation o (4)

13 Equation (1) for relatiisti mass m and equation (4) for ζ (zeta ratio) are idential but obtained from two different points of iew. In equation (1), of relatiisti eletrodynamis, the quantity m inreases with speed, beoming infinitely large at speed. In equation (4), of the alternatie eletrodynamis, mass remains onstant at the rest mass m o, and the quantity ζ = {(ee)/( /r)} is the ratio of magnitude of the radial eletrostati fore ( ee) on a stationary eletron, to the entripetal aeleration ( /r) in irular motion. This quantity ζ, the zeta ratio, may beome infinitely large at the speed of light, without any diffiulty. At the speed of light, the eletron moes with zero aeleration in an ar of a irle of infinite radius, whih is a straight line, to make the ratio ζ also infinite without any problem. Equating ζ with physial mass, whih has olume and weight, is an expensie ase of mistaken identity. The presene of Lorentz fator γ in equation (4) should be noted. It is the result of motion of a harged partile perpendiular to an eletri field and has nothing to do with mass. In lassial eletrodynamis, radius r of irular reolution for an eletron of harge e and mass m, in a radial eletri field of magnitude E due to a positiely harged nuleus, is gien by: m mo r = ro ee = ee = (5) where m = m o is a onstant and r o is the lassial radius. In relatiisti eletrodynamis, where mass m aries with speed in aordane with equation (4), the radius of reolution beomes: m mo r = = = γ ro (6) ee ee 1 In the alternatie eletrodynamis, where m = m o is a onstant, the radius r of reolution, obtained from equation (3), beomes: mo m r = = = γ ro (7) ee 1 ee 1 Relatiisti eletrodynamis and the alternatie eletrodynamis gie the same expression for radius of reolution in irular motion as r = γr o, but for different reasons. The inrease in radius with speed was misonstrued in speial relatiity as due to inrease in mass with speed, rather than the result of aelerating fore dereasing with speed. This 13 An Eletron Aelerated to the Speed of Light with Constant Mass & Emission of Radiation

14 is where laims of experimental obserations aurately onfirming relatiisti eletrodynamis, not radiatie eletrodynamis, went wrong. 5. Radiation Reation Fore and Radiation Power in the Alternatie Eletrodynamis The aelerating fore on a moing eletron is less than the eletrostati fore ee on a stationary eletron. The differene between the aelerating fore F and the eletrostati fore ee is the radiation reation fore R f = F ( ee), that is always present when a harged partile is aelerated by an eletri field. This is analogous to a fritional fore, whih always opposes motion, against whih work done appears as radiation. A simple and useful expression for radiation reation fore R f is missing in lassial and relatiisti eletrodynamis and it makes the differene. The diretion of maximum emission of eletromagneti radiation, from an aelerated harged partile, is perpendiular to the diretion of aeleration. For retilinear motion, with θ = 0 (Figure 1), equation (13) gies the radiation reation fore R f, in the diretion of unit etor û, as: ee ee ee R f = ( ) û + eeû = û = (8) In deeleration, with θ = π radians, R f = (ee/)û = (ee/), as (8). Radiation power is R p =.R f, the salar produt of R f and eloity. The salar produt is obtained, with referene to Fig. 1, as: ee Rp = R f = ( ) + ee Rp = ee osθ os( θ α ) + (9) For retilinear motion with θ = 0 or θ = π radians, equations (5) for θ and α and equation (9) gie radiation power in retilinear motion, as: Rp =. R f = ee (30) Positie radiation power, as gien by equation (30), means that energy is radiated in aelerated and deelerated motions. Radiation makes all the differene between the three systems of eletrodynamis. In irular reolution, where is orthogonal to E and R f, the radiation power R p (salar produt) is zero, as an be asertained from equations (5) and (9) with θ = π/ radians and os(θ α) = sinα = /. 14 An Eletron Aelerated to the Speed of Light with Constant Mass & Emission of Radiation

15 Equation (9) is ery signifiant in the alternatie (radiatie) eletrodynamis. It makes irular reolution of an eletron, around a entral fore of attration, as in Rutherford s nulear model of the hydrogen atom, stable, without reourse to Bohr s quantum theory. Equations (8), (9) and (30) are the radiation formulas of the alternatie (radiation) eletrodynamis. These equations are in ontrast to those of lassial eletrodynamis where radiation fore is proportional to the rate of hange of aeleration (Abraham-Lorentz formula) and the radiation power is proportional to the square of aeleration (Larmor formula). There is no formula for radiation reation fore in relatiisti eletrodynamis. Speial relatiity adopted a modified Larmor formula: R = γ 4 R p, for radiation power to aount for radiation from aelerated harges. We now onsider hange in potential energy and radiation for an aelerated or deelerated eletron in three systems of eletrodynamis: lassial, relatiisti and radiatie. 6. Potential Energy in Classial Eletrodynamis In lassial eletrodynamis, the magnitude of aelerating fore on an eletron of harge e and onstant mass m, moing at time t with speed and aeleration of magnitude d/, in the opposite diretion of an eletrostati field of magnitude E, is gien, in aordane with Coulomb s law of eletrostati fore and Newton s seond law of motion, by equation (6): d ee = m For retilinear motion in the diretion of a displaement x, we obtain the differential equation: d d ee = m m = dx (31) The potential energy P lost by the eletron or work done on the eletron, in being aelerated with onstant mass m = m o, through distane x from an origin (x = 0), to a speed from rest, is gien by the definite integral: x ( ) ( ) (3) P = ee dx = m m d 0 0 Integrating, equation (3) gies: x 1 P = ee dx = m 0 ( ) 15 An Eletron Aelerated to the Speed of Light with Constant Mass & Emission of Radiation

16 This is equal to the kineti energy K gained by the eletron. P 1 = (33) m Here, with no onsideration of radiation, the potential energy lost is equal to the kineti energy gained by an aelerated eletron. A graph of / against P/m is shown as A1 in Figure 3. In lassial eletrodynamis, an eletron, 0 moing at the speed of light, an be deelerated to a stop and may be aelerated in the opposite diretion to reah a speed greater than. The potential energy P gained, equal to the kineti energy K lost, in deelerating an eletron from the speed of light to a speed, within a distane x in a field of magnitude E, without radiation, is: P 1 = 1 m A graph of / against P/m is shown as A in Figure Potential Energy in Relatiisti Eletrodynamis 16 An Eletron Aelerated to the Speed of Light with Constant Mass & Emission of Radiation (34) In relatiisti eletrodynamis, the kineti energy K gained by an eletron or the work done, in being aelerated by an eletri field E, through a distane x, to a speed from rest, is the potential energy P lost. There is no onsideration of energy radiation in this situation. The kineti energy K of a partile of mass m and rest mass m o moing with speed, is gien by the relatiisti equation: 0 x 1 P= ee dx = m d = m x ( ) ( ) ( ) ( ) = = = ee dx K P m m mo P = m o 1 P 1 = 1 (35) m o 1 where m o is the rest mass (at = 0) and the speed of light in a auum. The amount of kineti energy is supposed to be aounted for o

17 by the inrease in mass. Bertozzi s experiment was onduted to erify equation (35) and it did so in a remarkable way. A graph of / against P/m o, for equation (35), is shown as dashed ure B1 in Figure 3. In relatiisti eletrodynamis, an eletron moing at the speed of light (with an infinitely large mass), annot be stopped by any deelerating fore. The eletron ontinues to moe at the same speed, (dashed line B in Figure 3) gaining potential energy without losing kineti energy. This is the point of departure between relatiisti eletrodynamis and the alternatie (radiatie) eletrodynamis. 8. Potential Energy and Radiation in the Alternatie Eletrodynamis In the alternatie eletrodynamis, the fore F (Figure 1), is gien by equations (5) and (14). For an eletron aelerated in a straight line, the equations, with θ = 0, gie: ee 1 F = û = m d û (36) The salar equation is: d d ee 1 = m = m (37) dx Potential energy P lost in aelerating an eletron to speed from 0, is gien by the integral: ( ) x d (38) P = ee dx = m Resoling the right-hand integral into partial frations, we obtain: 1 P = m 1 d 0 (39) 1 P m ln 1 = m (40) p = ln 1 (41) m Energy radiated R in aeleration is obtained by subtrating the kineti energy gained, K = ½ m, from potential energy P lost, thus: 17 An Eletron Aelerated to the Speed of Light with Constant Mass & Emission of Radiation

18 1 R = P K = m ln 1 m m R = m ln (4) In aeleration, (potential energy) equation (41), for the alternatie eletrodynamis, should be ompared with equation (35) for relatiisti eletrodynamis and equation (33) for lassial eletrodynamis. For a deelerated eletron, equations (5) and (14), with θ = π radians, gie: ee 1 F = + û = m d û (43) d d ee 1+ = m = m (44) dx Potential energy P gained in deelerating the eletron through distane x, from speed to, is: x d P = ee( dx) = m 0 (45) 1+ Resoling the integrand into partial frations and integrating, the potential energy gained is: 1 P = m 1 d (46) 1+ 1 P = m ln 1+ + m 1 (47) Graphs of P/m against / are shown as C1 and C, in Figure 3 for equations (41) and (47). Energy radiated R is the kineti energy lost minus the potential energy gained, thus: 1 1 R = m( ) m ln 1+ m 1 R 1 1 = m + + ln 1+ (48) 18 An Eletron Aelerated to the Speed of Light with Constant Mass & Emission of Radiation

19 9. Speed Versus Potential Energy in Three Systems of Eletrodynamis 9.1 Classial Eletrodynamis Potential energy P lost by an eletron in being aelerated from rest to the speed of light, equal to kineti energy K gained, is gien by equation (33). Potential energy gained in deeleration from the speed of light, to rest, equal to kineti energy K lost, is gien by equation (34). 9. Relatiisti Eletrodynamis Potential energy lost or kineti energy gained in aeleration to the speed of light is gien by equation (35). No equation is obtainable for deeleration from the speed of light. 9.3 Alternatie Eletrodynamis Potential energy expressions are equations (41) and (47). Equation (41) indiates a speed limit equal to that of light, with large loss of potential, as demonstrated by Bertozzi s experiment (Table 1). 9.4 Bertozzi s Experiment In the experiment performed by William Bertozzi [9], the speed of high-energy eletrons was determined by measuring the time T that TABLE 1. RESULTS OF BERTOZZI S EXPERIMENTS WITH ELECTRONS ACCELERATED IN TIME T THROUGH 8.4 METRES AND ENERGY P IN A LINEAR ACCELERATOR P MeV P/m o T x 10-8 se. (m o = 0.5 MeV, = 8.4/T m/se) V x 10 8 m/se / Bertozzi s Experiment / Classial Equation 33 / Relatiisti Equation 35 / Alternatie Equation * * * * * * *Inserted in Figure 3. is required for them to traerse a distane of 8.4 metres after haing been aelerated through a potential energy P inside a linear aelerator. Bertozzi s experimental data, reprodued in Table 1, learly 19 An Eletron Aelerated to the Speed of Light with Constant Mass & Emission of Radiation

20 demonstrated that eletrons aelerated through potential energy of 15 MeV attain, for all pratial purposes, the speed of light. A graph with three results of Bertozzi s experiment is shown in Figure 3. An experiment does not tell a lie but human interpretation of the result may but wrong. This is the ase in the expression of the relatiisti mass-eloity formula (equation 4). In a linear or yli aelerator, with eletrons or protons moing at the speed of light in the apparatus or mahine, where is the infinitely large mass, oneied as the mass of the whole unierse? Figure 3. / (speed in units of ) against P/m (potential energy in units of m ) for an eletron of mass m aelerated from zero initial speed or deelerated from the speed of light, the solid ures (A1 and A) aording to lassial eletrodynamis (equation 33 and 34), the dashed ure (B1) aording to relatiisti eletrodynamis (equation 35) and the dotted ures (C1 and C) aording to the alternatie eletrodynamis (equations 41 and 47). The three solid squares are the result of Bertozzi s experiment (Table 1). 0 An Eletron Aelerated to the Speed of Light with Constant Mass & Emission of Radiation

21 10. Speed and Kineti Energy in Three Systems of Eletrodynamis In lassial eletrodynamis, kineti energy gained by an eletron of mass m in being aelerated, to a speed, is K = ½ m, same as equation (33) for potential energy P lost. K = (49) m In lassial eletrodynamis, kineti energy lost in deeleration from the speed of light to a speed, is K = ½ m( ), same as equation (34) for potential energy P gained. K 1 = 1 (50) m Equations (49) and (50) are the same for lassial and relatiisti eletrodynamis. These are respetiely illustrated as ure (A1 & C1) and ure (A & C) in Figure 4. If potential energy lost is equal to kineti energy gained, there should hae been no energy radiation in lassial eletrodynamis. But, as there must be energy radiation, Larmor formula for radiation power is inoked to explain emission of radiation. In relatiisti eletrodynamis, kineti energy gained by an eletron of rest mass m o in being aelerated to a speed, from rest, is gien by: mo K = m mo = m o 1 K 1 = 1 (51) m o 1 The kineti energy K is supposed to be aounted for in the inrease of mass of the eletron. Emission of radiation is explained in a modified form of Lamor formula. Figure 4 shows graphs of / (speed in units of ) against K/m (kineti energy, gained or lost, in units of m ) for an eletron of harge e and mass m aelerated from zero initial speed or deelerated from the speed of light, by a uniform field of magnitude E, where a = 1 An Eletron Aelerated to the Speed of Light with Constant Mass & Emission of Radiation

22 ee/m o is a onstant. The dotted ures (A1 & C1) and (A & C) are in aordane with the lassial eletrodynamis and the alternatie eletrodynamis (equations 49 and 50). The dashed ure B1 is in aordane with relatiisti eletrodynamis (equation 51). No ure is obtainable in relatiisti eletrodynamis for an eletron deelerated from the speed of light. Figure 4 / (speed in units of ) against K/m (kineti energy lost or gained in units of m ) for an eletron aelerated from zero initial speed or deelerated from the speed of light, under 3 systems of eletrodynamis: A1 &A Classial, B1 Relatiisti and C1&C Alternatie. 11. Speed Versus Energy Radiation in the Alternatie Eletrodynamis Figure 5 shows graphs of / (speed in units of ) against R/m energy radiated (in units of m ) for an eletron of mass m and harge An Eletron Aelerated to the Speed of Light with Constant Mass & Emission of Radiation

23 -e aelerated by an eletri field from zero initial speed or deelerated from the speed of light aording to the alternatie eletrodynamis. The ure C1 is for an aelerated eletron (equation 4) and ure C for a deelerated eletron (equation 48). Energy is always radiated, under aeleration or deeleration. There are no suh energy radiation graphs from the points of iews of lassial eletrodynamis and relatiisti eletrodynamis. Radiation, the most ommon phenomenon in nature, makes the all differene between the three systems of eletrodynamis: lassial, relatiisti and the alternatie. The alternatie system of eletrodynamis may rightly be alled radiatie eletrodynamis. Indeed, radiatie eletrodynamis explains the speed of light as a limit and desribes the motion of an eletron up to the speed of light with onstant mass and emission of radiation. Figure 5 Graphs of / (speed in units of ) against R/m energy radiated (in units of m ) for an eletron aelerated from zero initial speed to a speed or deelerate from the speed of light to a speed under the alternatie (radiatie) eletrodynamis. (No suh graphs are obtainable from the point of iew of lassial and relatiisti eletrodynamis). 3 An Eletron Aelerated to the Speed of Light with Constant Mass & Emission of Radiation

24 1. Conluding Remarks 1. Quantum mehanis is not required to desribe the motion of an eletron in an eletri field. Cirular motion of an eletron around a positiely harged nuleus, as in the Rutherford s nulear model of the hydrogen atom, is inherently stable.. Relatiisti eletrodynamis and the alternatie eletrodynamis are in agreement with Bertozzi s experiment for aelerated eletrons in retilinear motion, speed of light being a limit, but for different reasons. 3. The speed of light as a limit is learly demonstrated by relatiisti eletrodynamis on the basis of mass of a moing partile inreasing to beome infinitely large at the speed of light, and the alternatie eletrodynamis on the basis of aelerating fore or field reduing to zero at the speed of light. 4. The relatiisti and alternatie eletrodynamis gie the same expression γr o for radius of irular reolution for an eletron around a entral fore of attration, where γ is Lorentz fator and r o the lassial radius. Asribing inrease in radius to inrease in mass with speed, not to derease in aelerating fore with speed, was a mistake. 5. The fator γ is due to motion of harged partiles perpendiular to an eletri field. There is nothing like mass expansion in atuality. 6. In the alternatie eletrodynamis, the aelerating fore on a reoling harged partile redues to zero, and it moes in an ar of a irle of infinite radius, whih is a straight line. 7. Coulomb s law of eletrostati fore needs to be modified (as in equation 13) to take into onsideration the eloity of an eletri harge in an eletri field or the relatie eloity between two eletri harges moing in spae. Making Coulomb s law independent of eloity of a harged partile in an eletri field is the blight of physis. 8. Eliminating an infinitely large mass of a partile at the speed of light, and identifying relatiisti mass as the ratio of eletrostati fore to aeleration in irular motion, should bring great relief to physiists. 9. The eloity of light, of magnitude as speed, is a onstant relatie to the soure but it an be added to or subtrated from by the motion of an obserer. The eloity of light s from a soure moing with eloity u, relatie to an obserer moing with eloity, is etor s = + (u ), in ontrast to Einstein s eloity addition rule. 10. The lassial Abraham-Lorentz formula for radiation reation fore, being proportional to the rate of hange of aeleration, is not 4 An Eletron Aelerated to the Speed of Light with Constant Mass & Emission of Radiation

25 ompatible with Lamor formula for radiation power being proportional to the square of aeleration. These radiation formulas are defetie. 11. Absene of a formula for radiation reation fore, in relatiisti eletrodynamis, whih atually dismisses radiation, is a serious defet. 1. The answer as to whih eletrodynamis is orret may be found in the motion of eletrons deelerated from the speed of light. In the alternatie (radiatie) eletrodynamis, an eletron moing at the speed of light is easily brought to rest on entering a deelerating field at a point. In an experiment with a narrow burst of eletrons aelerated as near to the speed of light as possible and made to enter a deelerating field at a point, the eletrons being stopped at all inalidates relatiisti eletrodynamis. Suh eletrons being stopped and turned bak on their traks, to return to the point of entry at speed equal to 0.594, with emission of radiation, eliminates relatiity and inaugurates radiatiity. Referenes [1] P. Lenard; Great Man of Siene, G. Bell and Sons Ltd., London (1958), pp [] I. Newton (1687); Mathematial Priniples of Natural Philosophy (Translated by F. Cajori), Uniersity of California Press, Berkeley (1964). [3] A. Einstein; On the Eletrodynamis of Moing Bodies, Ann. Phys., 17 (1905), 891 [4] A. Einstein; & H.A. Lorentz, The Priniples of Relatiity, Matheun, London (193). [5] M. Plank; Ann. Phys,, 4 (1901), 533. [6] L. de Broglie; Researhes in Quantum Mehanis, Dotoral Thesis, Uniersity of Paris, 194. [7] P. Bekmann; Einstein Plus Two, The Golem Press, Boulder (1987). [8] C. Renshaw; Galilean Eletrodynamis, Vol. 7/6, 1996 [9] W. Bertozzi; Speed and Kineti Energy of Relatiisti Eletrons", Am. J. Phys., 3 (1964), Also online at: [10] D.J. Griffith; Introdution to Eletrodynamis, International Edition, Pearson Eduation In., pp (Fourth ed. 013). [11] N. Bohr; Phil. Mag., 6 (1913), 476. [1] E. Rutherford; Phil. Mag., 1 (1911), 669. [13] 5 An Eletron Aelerated to the Speed of Light with Constant Mass & Emission of Radiation