Discrepancy between power radiated and the power loss due to radiation reaction for an accelerated charge
aa r X i v : . [ phy s i c s . c l a ss - ph ] O c t Article
Discrepancy between Power Radiated and the PowerLoss Due to Radiation Reaction for an AcceleratedCharge
Ashok K. Singal 0000-0002-8479-7656
Astronomy and Astrophysics Division, Physical Research Laboratory, Navrangpura, Ahmedabad - 380 009,India; [email protected]: 13 September 2020; Accepted: date; Published: date
Abstract:
We examine here the discrepancy between the radiated power, calculated from the Poyntingflux at infinity, and the power loss due to radiation reaction for an accelerated charge. It is emphasizedthat one needs to maintain a clear distinction between the electromagnetic power received by distantobservers and the mechanical power loss undergone by the charge. In the literature, both quantitiesare treated as almost synonymous; the two in general could, however, be quite different. It is shownthat in the case of a periodic motion, the two formulations do yield the power loss in a time averagedsense to be the same, even though, the instantaneous rates are quite different. It is demonstrated thatthe discordance between the two power formulas merely reflects the difference in the power going inself-fields of the charge between the retarded and present times. In particular, in the case of a uniformlyaccelerated charge, power going into the self-fields at the present time is equal to the power that wasgoing into the self-fields at the retarded time plus the power going in acceleration fields, usually calledradiation. From a study of the fields in regions far off from the time retarded positions of the uniformlyaccelerated charge, it is shown that effectively the fields, including the acceleration fields, remain aroundthe ‘present’ position of the charge which itself is moving toward infinity due to its continuous constantacceleration, with no other Poynting flow that could be termed as ‘radiation emitted’ by the charge.
Keywords: classical electromagnetism; applied classical electromagnetism; radiation by movingcharges; radiation or classical fields
1. Introduction
In electromagnetic radiation by a point charge, the radiated power is proportional to the squareof the acceleration, known as Larmor’s formula [1–3]. On the other hand, the consequent radiationreaction on the charge is directly proportional to the rate of change of the acceleration of the charge[4–8]. The two formulations do not seem to be conversant with each other. This apparent discordancebetween the two formulations has survived without a proper, universally acceptable, solution for longerthan a century. Larmor’s formulation is believed to be more rigorous than the radiation-reactionformulation, though there are a large number of arguments, based on energy-momentum conservationlaws, that suggest that there is something amiss in Larmor’s radiation formula [9,10]. For instance, theradiation pattern for the slowly moving charge ( v ≪ c ) has a sin θ dependence with respect to theacceleration vector, [1–3], consequently the net momentum carried away by the radiation, averaged overthe solid angle, is nil. Therefore, from the momentum conservation law, such a radiating charge cannotsuffer any momentum losses. However, due to a finite amount of power going into electromagneticradiation, as per Larmor’s formula, the kinetic energy of the charge must be reducing with time. Howcould a radiating charge lose kinetic energy without a loss of momentum? Further, an acceleratedcharge, that may be instantly stationary, has zero kinetic energy at that instant. However, according of 16 to Larmor’s formula, the charge would be undergoing kinetic energy losses proportional to the square ofacceleration, even though its kinetic energy may be zero. In order to be still compatible with Larmor’sformula, these energy-momentum conservation problems have been circumvented by proposing anacceleration-dependent term, called Schott energy, within electromagnetic fields, that may be lyingsomewhere in the vicinity of the charge [6,11–16]. However, recently, from a critical examination of theelectromagnetic fields of a uniformly accelerated charge [17], no Schott energy was found anywhere inthe near vicinity of the charge, or for that matter, even in the far-off regions.Here, we critically examine the relation between the two formulations and demonstrate that amathematical subtlety in the application of Poynting’s theorem is being missed when we try to use theenergy-momentum conservation laws to compare the two formulas.We shall, unless otherwise specified, confine ourselves only to non-relativistic motion, as the sameset of disparities get carried over to the relativistic case [10]. Further, we shall assume a one-dimensionalmotion with acceleration parallel to the velocity and also throughout use the cgs system of units.
2. Two Discrepant Formulations for Radiation Losses from an Accelerated Charge
The electromagnetic field ( E , B ) at a time t , of an arbitrarily moving charge e , is written as [1–3,18], E = (cid:20) e ( n − v / c ) γ r ( − n · v / c ) + e n × { ( n − v / c ) × ˙ v } rc ( − n · v / c ) (cid:21) t ′ , B = n × E , (1)where all quantities in square brackets are to be evaluated at the retarded time t ′ = t − r / c .As the acceleration contributes only to the transverse fields, we shall, unless otherwise specified,leave the radial fields aside and consider, henceforth, only the transverse fields. It is to be emphasizedthat not only the acceleration fields, even the velocity fields have a transverse field component, normal tothe radial direction along n .With the help of the vector identity v = n ( v . n ) − n × { n × v } , transverse components of theelectromagnetic field of a charge, having a non-relativistic motion and therefore comprising only linearterms in velocity ( v ) and acceleration ( ˙ v ), can be written from Eq. (1) as E = (cid:20) e n × ( n × v ) cr + e n × ( n × ˙ v ) c r (cid:21) t ′ = (cid:20) e n × ( n × ( v + ˙ v r / c )) cr (cid:21) t ′ , B = (cid:20) − e n × v cr − e n × ˙ v c r (cid:21) t ′ = − (cid:20) e n × ( v + ˙ v r / c ) cr (cid:21) t ′ . (2)To calculate the radiated electromagnetic power, we make use of the radial component of thePoynting vector [1–3], n · S = c π n · ( E × B ) = c π ( n × E ) · B = c π ( B ) . (3)Accordingly, one gets for the the radial component of the Poynting vector n · S = e (cid:2) ( v + ˙ v r / c ) (cid:3) t ′ π r c sin θ . (4)The sin θ pattern implies that the rate of momentum being carried in the electromagnetic radiation iszero. ˙ p em = of 16 However, the net Poynting flow through a spherical surface, Σ of radius r , around the charge, for a large r , is P em = Z r → ∞ d Σ ( n · S ) = e c Z π d θ sin θ (cid:2) ( v + ˙ v r / c ) (cid:3) t ′ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r → ∞ = e c h ˙ v i t ′ . (6)This is Larmor’s famous result for the electromagnetic power radiated from an accelerated chargedparticle [1–3]. Since the contribution of velocity fields ( ∝ r ), for a large enough r , seems negligible, withthe Poynting flow due to the acceleration fields being independent of r (Eq. (6)), the common perceptionis that in all cases, the acceleration fields ( ∝ r ) alone represent radiation from a charge.Presumably, using the energy-momentum conservation laws [19,20], we can compute the mechanicalenergy-momentum losses of the radiating charged particle. For instance, the momentum of the chargewould not change due to radiation damping, F = − ˙ p em = T , of the charged particle should change due to radiation losses at a rated T d t = −P em . (8)Now, Eqs. (7) and (8) do not seem mutually consistent since the charged particle cannot lose kinetic energywithout losing momentum. In fact, some problem is inherently present in Eq. (8) itself, as in the rest frameof the charge, the energy loss rate is finite ( ∝ ˙ v ) even when the charged particle has no kinetic energy( v = ) to lose. It may be pointed out here that such a power loss into radiation can happen, withoutany change in the kinetic energy of the emitting charge, only if there were a loss of internal (rest mass!)energy, without an accompanying loss of momentum [21]. However, we do not contemplate a radiatingcharged particle to be converting its rest mass energy into electromagnetic radiation; after all, a radiatingelectron still remains an electron at the end of the emission of radiation.Somewhere something is amiss! Actually, in the above formulation, which is the standard text-book approach, one is equating thePoynting flux at time t to the mechanical power loss of the charge at a retarded time t − r / c , purportedlyusing Poynting’s theorem of energy conservation. However, there is a fallacy in this particular step asPoynting’s theorem does not directly relate the Poynting flux through a closed surface at a time t to powerlosses by the enclosed charge at a retarded time t − r / c . Since the electromagnetic fields at r at time t doget determined by the charge motion at the retarded time t ′ = t − r / c , one may intuitively be tempted toequate the electromagnetic power represented by the Poynting flux at r at time t to the mechanical powerloss of the charge at the retarded time t ′ = t − r / c . Our common-sense notion of causality may, however,be in conflict with the strict mathematical definition of Poynting’s theorem and the ensuing applicationof energy-momentum conservation laws to electromechanical systems could sometimes lead us astray.It is such an oversight in this case that has mostly been the cause of confusion in this long drawn outcontroversy. In Poynting’s theorem, all quantities are to be calculated, strictly for the same instant of time[1–3].Applying the Poynting’s theorem in an appropriate manner, one obtains the instantaneousmechanical power loss of the charge, in terms of the real time values of the charge motion [22], as P me = − e c ¨ v · v , (9) of 16 with all quantities evaluated for the same common instant, say, t .Now, one needs to maintain a clear distinction between the electromagnetic power received bydistant observers (Eq. (6)) and the instantaneous mechanical power loss of the charge (Eq. (9)). In theliterature both power rates are treated as almost synonymous. However, as we can see, the two are notnecessarily the same (cf. Eqs. (6) and (9)).The difference in the two power formulas is P me − P em = − e c ¨ v · v − e c [ ˙ v · ˙ v ] t ′ = − e c d ( ˙ v · v ) d t . (10)The rightmost term in Eq. (10) is the total time derivative of a term, known as Schott energy, believed to bean acceleration-dependent energy, − e ( ˙ v · v ) /3 c , in the electromagnetic fields, lying in the near vicinityof the charge [6,11–16]. This elusive, century-old term does not seem to have been encountered elsewherein physics. As we shall demonstrate in Section 2.3, the Schott term is not any real electromagnetic energyin the fields and makes an appearance in the above equation merely because the power going in theself-field of an accelerated charge between“real” and “retarded” times is different.In the same way, exploiting Maxwell’s stress tensor, from the momentum conservation theorem wecan arrive at the expression for the rate of change of momentum of the charge [8], written as˙ p me = e c ¨ v . (11)The result in Eq. (11), is known as the Abraham–Lorentz radiation reaction formula, and was earlierderived from a detailed computation of the self-force due to a rate of change of acceleration ( ¨ v ) of theinstantly stationary ( v =
0) charge, in a quite involved manner [1,2,4–7].
Does the discrepancy in two formulations imply that Larmor’s formula cannot be applied forcomputing mechanical power losses for a radiating charge?In the case of a periodic motion of period T , there is no difference in the radiated power integratedor averaged between t to t + T and t ′ to t ′ + T , therefore Larmor’s formula, does yield a correct averagepower loss by the charge for a periodic case.Let us write the motion of a harmonically oscillating charge (like in a radio antenna) as x = x o sin ( ω t + ψ ) . (12)Then v = ˙ x = ω x o cos ( ω t + ψ ) , (13)˙ v = ¨ x = − ω x o sin ( ω t + ψ ) = − ω x , (14)¨ v = − ω x o cos ( ω t + ψ ) = − ω v . (15)For such a motion, one gets from Larmor’s formula the radiative power ∝ ˙ v = ω x sin ( ω t + ψ ) while the radiation reaction formula yields a power loss ∝ − ¨ v · v = ω x cos ( ω t + ψ ) . Both expressionsyield the same result when averaged over a full cycle, however, the instantaneous rates are very different.It means the power spectrum, which gives average power at each frequency, would be the same in mostcases, where the actual motion of the charge could be Fourier analysed. Of course in a non-periodic caselike that of a uniformly accelerated charge, where a Fourier analysis is not possible, the two formulascould yield discordant answers. of 16 Figure 1.
The self-force f on a charged spherical shell of a small radius ǫ , moving non-relativisticallywith an acceleration a = ˙ v . The net self-force on the charged shell at any instant is proportional tothe acceleration it had at a time interval ǫ / c earlier [23]. This implies that for a uniform acceleration a , the self-force on the charge would only be an ‘inertial’ force − m el a , where m el = e /3 ǫ c is theelectromagnetic mass of the charge [24], without any radiation reaction, whatsoever, consistent with thefact that there is no radiation emitted in this case. In order to understand the genesis of the difference between Eqs. (6) and (9), which respectively areat the retarded and real times, we compute the rate of work done by the self-force of an accelerated charge.We consider the charge to be in the shape of a spherical shell of a sufficiently small radius ǫ , though, aswe shall see, the final results sought by us will be independent of the value of ǫ . We compute the force oneach infinitesimal element of the charged shell due to the time-retarded fields from the remainder parts ofthe shell and perform a double integration over the shell to get the total self-force on the charge [1]. Thenet self-force at a time t on the accelerated charged spherical shell of radius ǫ turns out to be proportionalto the acceleration it had at a retarded time t ′ = t − ǫ / c (Fig. 1) [23]. f t = − e ǫ c ˙ v t ′ , (16)where ˙ v t ′ is the acceleration of the charge at the retarded time t ′ . Then, for the charge moving withvelocity v t at time t , the work being done against self-force of the charge is P t = − f t · v t = e ǫ c ˙ v t ′ · v t . (17)Because work is done against the self-force, this is the rate at which energy is being put into the fields ofthe charge.By expressing the acceleration at time t ′ in terms of its real-time value at t , to a first order, we have˙ v t ′ = ˙ v t − ¨ v ǫ / c . (18) of 16 Then from Eq. (16) for the self-force, in terms of real-time values, we can write f t = − e ǫ c ˙ v t + e c ¨ v t , (19)where the last term is the well-known Abraham–Lorentz radiation reaction (Eq. (11)).From this we get the formula for power going into the fields (Eq. (17)), but now expressed in termsof the real time values, as P t = e ǫ c ( ˙ v · v ) t − e c ( ¨ v · v ) t . (20)The first term on the right hand side is the rate of change of self-field energy of the accelerated charge atthe real time t , and the second term is power loss due to radiation reaction (Eq. (9)), again evaluated at t .On the other hand, if we express the velocity itself in terms of its value at the retarded time t ′ , to afirst order in ǫ / c , we have v t = v t ′ + ˙ v t ′ ǫ / c . (21)Substituting in (Eq. (17)), we get P t = e ǫ c [ ˙ v · v ] t ′ + e c [ ˙ v · ˙ v ] t ′ . (22)The first term on the right hand side shows the rate of change of self-field energy of the accelerated charge at the retarded time t ′ , and the second term is Larmor’s formula (Eq. (6)), again evaluated at t ′ .From Eqs. (20) and (22), we get − e c ( ¨ v · v ) t − e c [ ˙ v · ˙ v ] t ′ = e ǫ c [ ˙ v · v ] t ′ − e ǫ c ( ˙ v · v ) t (23)It shows that the difference in the two power formulas, Eqs. (9) and (6), which respectively are at real andretarded times, is merely the difference in the power going in self-fields of the charge between retardedand present times.Now, we can write the right hand side of Eq. (23) as2 e ǫ c [ ˙ v · v ] t ′ − e ǫ c ( ˙ v · v ) t = − e ǫ c d ( ˙ v · v ) d t ǫ c = − e c d ( ˙ v · v ) d t , (24)a result independent of ǫ . This demonstrates that the elusive Schott term is not some actual hiddenenergy in the near fields but shows up in Eq. (10) merely due to the different rates of energy change inthe self-fields between retarded and present times of an accelerated charge. This is consistent with thefindings from a critical examination of the electromagnetic fields of a uniformly accelerated charge [17],where, contrary to the suggestions in the literature [11–16], no Schott energy term was found anywherein the near vicinity of the charge, or for that matter, even in far-off regions.
3. A Uniformly Accelerated Charge
In the derivation of Larmor’s formula (Eq. (5)), which is a standard text-book material [1–3], it isassumed that any contribution of velocity fields could be neglected. This assumption holds true in almostall cases, except in that of a uniformly accelerated charge, where the velocity may change monotonicallywith time [9]. of 16
Actually, in the case of a uniform acceleration, in the expressions for the fields (Eq. (2)), the retardedvelocity of the charge would be related to the present value of velocity, v o = [ v + ˙ v r / c ] t ′ . Then thetransverse components of the electromagnetic fields become E = e n × ( n × v o ) cr , B = − e n × v o cr . (25)Thus we see that what all the acceleration fields do in this case is to make the instantaneous transversefields everywhere directly proportional to the instantaneous present velocity v o of the accelerated charge. Since the self-field energy of a charge moving with a uniform velocity depends upon the magnitudeof the velocity (see, e.g., [9]), when a charge is accelerated, its self-field energy must change too,depending upon the change in velocity. It is therefore imperative that the acceleration fields providefor the changes taking place in the energy in self-fields. As the acceleration, ˙ v , changes the velocityof the charge to say, v o = v + ˙ v r / c , the acceleration fields ( ∝ ˙ v / r ) ensure that the transverse fieldsaccordingly remain ‘updated’ ( ∝ v o / r ), to remain synchronized with the ‘present’ value of the velocityof the charge, and the energy in self-fields is always equal to that required because of the ‘present’velocity of the accelerated charge. The conventional wisdom, on the other hand, is that the accelerationfields, exclusively and wholly, represent power irreversibly lost as radiation, given by Larmor’s formula.Thus there may be something amiss in the standard picture where one does not even consider that thePoynting flux from the acceleration fields might be contributing toward the changing self-field energy ofthe accelerating charge. After all a stationary charge has zero self-field energy in transverse fields, andthe growth in the self-field energy as the charge picks up speed due to acceleration, could have come onlyfrom the acceleration fields. The radiation actually would only be that part of the Poynting flux which isover and above the value determined by the change occurring in the instantaneous velocity of the charge.Employing the formula for the electromagnetic field energy E = Z V E + B π d V , (26)it is possible to compute the electromagnetic field energy, not only for a charge moving with a uniformvelocity, but even in the case of a charge moving with a uniform acceleration [9]. For instance, thetransverse field energy of the uniformly accelerated charge, in a shell of volume 4 π r d r , enclosed betweenspheres Σ and Σ of radii r and r + d r , isd E = e (cid:18) v c (cid:19) d rr . (27)We can integrate over r to get the total energy in the transverse fields outside a sphere of radius ǫ as, E = e c ǫ v . (28)Since the integral diverges for r →
0, we restricted the lower limit of r to a small ǫ , which may representthe radius of the charged particle.One can also calculate the energy in fields of a charge moving with a uniform velocity v o and exactlythe same amount of field energy is found around the charge. Thus it is clear that the acceleration fields of 16 in the case of a uniformly accelerated charge add just sufficient energy in the self-fields so as to make thetotal field energy equal to that required because of the ‘present’ velocity of the accelerating charge. Thisis true even in the case of the charges moving with relativistic velocities [9].That the Poynting flux in the acceleration fields feeds the self-field energy in the case of a uniformlyaccelerated charge, is further seen from a comparison of the self energy changes between the real and theretarded times. Since in the case of a uniformly accelerated charge ¨ v =
0, then from Eq. (23), we get2 e ǫ c ( ˙ v · v ) t = e ǫ c [ ˙ v · v ] t ′ + e c [ ˙ v · ˙ v ] t ′ . (29)From Eq. (29), it is obvious that in the case of a uniformly accelerated charge, power going into theself-fields at the present time t is equal to the power that was going into the self-fields at the retardedtime t ′ plus the power going in acceleration fields, usually called Larmor’s formula for radiative losses.Instead of any losses being suffered by the charge, the energy in its self-fields is actually being constantlyaugmented by the acceleration fields. There is no other power term in the formulation that could becalled radiation emitted by the uniformly accelerated charge.We can compute the net momentum as well, in the self-fields of a uniformly accelerated charge, fromthe volume integral p = Z V E × B π c d V . (30)Due to the azimuth symmetry about the direction of motion, the transverse component of the electric field(Eq. (25)) makes a nil contribution to the momentum, when integrated over the solid angle. However, theradial component, e n / r , does make a net finite contribution, which would be along the direction ofmotion. Accordingly, we get p = e v o ǫ c Z π d θ sin θ = e ǫ c v o = m el v o , (31)where m el = e /3 ǫ c is the electromagnetic mass of the charge [24]. Thus we see that as the chargevelocity changes to v o due to the acceleration, the acceleration fields contribute to the self-fields of thecharge, so that the field momentum becomes m el v o , in accordance with the instantaneous velocity v o .Thus both the energy and momentum in the self-fields of the uniformly accelerated charge aregetting constantly updated by its acceleration fields in accordance with its ‘present’ velocity at any instant. In the derivation of Larmor’s formula (Eq. (6)), one assumed that the velocity fields would alwaysmake a negligible contribution to the Poynting flow, for large r . However, in the case of a uniformlyaccelerated charge, the contribution of velocity fields could match that of the acceleration fields, for all r .From Eq. (25), we find the Poynting flux to be P = e v r c . (32)The power passing through the spherical surface in the case of a uniformly accelerated charge is ∝ v / r .A similar transverse component of the electromagnetic field (Eq. (25)) is also seen in the case ofa charge moving with a uniform velocity v o , equal to the “present” velocity of the accelerated charge.Therefore, a Poynting flux exactly similar to Eq. (32) is also present in the case of a uniformly moving of 16 charge, where we know there are no radiation losses and the Poynting flow through a surface aroundtime-retarded position of the charge is merely due to the “convective” flow of fields, along with themoving charge. However, with respect to the ‘present’ position of a charge, there is no radial Poyntingflux in this case. Taking a cue from this, even for a uniformly accelerated charge, one should examinethe Poynting flux vis-à-vis the ‘present’ position of the accelerated charge, to find out if there indeedis some radiation taking place. As the energy in the self-fields must be "co-moving" with the charge,(otherwise the self-fields would lag behind, and no longer remain about the charge to qualify as itsself-fields), and there should accordingly be a Poynting flow. Therefore not all of the Poynting flowmay constitute radiation. The radiated power would be the part of the Poynting flow that is detachedfrom the charge [3], i.e., it should be over and above the energy changes in the self-fields of the charge,as determined from the changing velocity of the charge. As we saw from the energy-momentum in thefields in Section 3.1, there is no such excess energy in fields to be termed as radiation in the case of auniformly accelerated charge.It is evident from Eq. (25) that the transverse component of electromagnetic field, at least in theinstantaneous rest frame ( v o =
0) of a uniformly accelerated charge, is nil. This happens due toa systematic cancellation of acceleration fields by the transverse component of velocity fields, in theinstantaneous rest frame, both for the electric and magnetic fields, at all distances. That the magneticfield is zero everywhere in this case was first pointed out by Pauli [25], using Born’s solutions [26], whoinferred from it that no wave zone would be formed and hence there is no radiation from a uniformlyaccelerated charge.3.2.1. A definition of radiation at infinity incompatible with Green’s theoremIt has been claimed that Pauli’s statement, that contradicts Larmor’s formula, is invalid on thegrounds that a limit to large r at a fixed time, say, t =
0, is implied therein [27,28]. It has been assertedthat the radiation should instead be defined by the total rate of energy emitted by the charge at theretarded time t ′ , and is to be calculated by integrating over the surface of the light sphere in the limit ofinfinite r = c ( t − t ′ ) for a fixed emission time t ′ , with both t → ∞ and r → ∞ [27,28]. The two limitingprocedures, one with t fixed and the other with t ′ fixed, do not yield the same result and from that it hasbeen concluded that Pauli’s observation that B = t is a mere curiositythat may be of some interest but does not imply an absence of radiation [27,28].If we carefully examine the reason why a fixed emission time t ′ is being chosen for defining‘radiation’ [27,28], we can see that this choice makes the contribution to the Poynting flow, from thevelocity fields at t ′ , for a large enough r , negligible. However, for a uniformly accelerated charge, onecannot ignore the contribution of the velocity fields to the Poynting flow, as v ( t ′ ) ∝ − ˙ vr / c . Moreover, inthis case, there is something unusual happening about the fields at large r vis-à-vis the charge location atlarge t , which we shall discuss in Section 3.3.Actually in Green’s retarded solution, the scalar potential φ at a field point x , for instance, isdetermined at time t from the volume integral [1] φ ( x , t ) = Z [ ρ ( x ′ )] t ′ r d x ′ . (33)Here the charge density ρ ( x ′ ) at x ′ , enclosed within square brackets, and at a distance r = | x − x ′ | fromthe field point, is to be determined at the retarded time t ′ = t − r / c . A similar expression is there for thevector potential as well.Thus here x and t are specified first and the volume integral of ρ / r at the corresponding retardedtimes is then computed. Pauli’s argument is consistent with this procedure. In fact, the radiation defined Figure 2.
Angular distribution of the electric field strength with respect to the time-retarded position z r of the uniformly accelerated charge, moving along the z -axis with velocity v → c and the correspondingLorentz factor γ ≫
1. Due to the relativistic beaming, the field strength is mostly appreciable only withina cone of angle θ ∼ γ about the direction of motion. When at time t , the fields from the retarded position z r are at the spherical light-front of radius r = ct , the charge meanwhile has moved to z o , quite close to thespherical light-front. The circle represented by points P on the spherical light-front r = ct where the fieldstrength is maximum as a function of θ , lies almost vertically above z o , the ‘present’ position of the charge,and thus are not very far from it, implying that the field at large r is still around the ‘present’ location ofthe charge. by first fixing the emission time, t ′ [27,28], strictly speaking, may not be in tune with Green’s retardedtime solution, and could sometime lead to wrong conclusions, especially in the limit r → ∞ .It may be pointed out that for a “point” charge e moving with velocity v , first fixing the point chargeposition x ′ at the retarded-time t ′ , to determine the potential this way, yields φ = e / r , while the morecorrect approach of first fixing the field point x at time t , leads to φ = e / [ r ( − n · v / c )] , the correctexpression for the potential [24]. Conclusions about radiation from a uniformly accelerated charge, contrary to ours, seem to havebeen drawn previously. This was because, firstly only the acceleration fields were being considered, anapproach which though might be valid in vast majority of cases of radiation from an accelerated charge,but is not valid in the case of a uniformly accelerated charge. The reason being that in the latter case thevelocity at the retarded time being v ∝ ˙ vt = ˙ vr / c , the velocity fields, v / r ∝ ˙ vr / cr become comparableto the acceleration fields, ∝ ˙ v / rc , for all r . Secondly, almost no attention has generally been paid to the‘present’ location of the charge vis-à-vis the fields that move to r = ct . As we will show, during theintervening time interval t = r / c , the charge is almost keeping in step with the fields, being only a finitedistance ∝ γ behind for all t , with γ ever increasing due to the uniform acceleration. As such, the fieldsremain appreciable along the direction of motion only in a small, finite region ∝ γ about the ‘present’position of the charge, very similar to the uniform velocity case where electric field is ever appreciableonly near the ‘present’ position of the charge, in a region whose extent falls as 1/ γ and where the fieldstrength is mostly along the direction normal to the direction of motion (see, e.g., [29]). In the literature,almost no attention has been paid to the charge position relative to the light-front of the supposed to beradiation fields or vice-versa, in the case of a uniformly accelerated charge.Since we want to examine far fields at large r , this would also imply large values of t = r / c . Now, auniform acceleration for a long duration could make the motion of the charge relativistic, accordingly, inthis Section, we shall no longer assume the motion to be non-relativistic.Let the charge moving with a uniform acceleration, a ≡ γ ˙ v along + z axis, was momentarilystationary at time t = z = α , chosen, without any loss of generality, so that α = c / a . The position and velocity of the charge, before or after, at any other time t are then given by [27,29,30] z o = ( α + c t ) , v o = c t / z o and γ o = z o / α , which implies ct = αγ o v o / c .In a typical radiation scenario, the radiated energy moves away ( r → ∞ ), with the charge responsibleremaining behind, perhaps not very far from its location at the corresponding retarded time, e.g., inlocalized charge or current distributions in a radiating antenna. This of course necessarily implies that notonly the motion of the charge is bound, its velocity and acceleration are having, some sort of oscillatorybehaviour, even if not completely regular. However, in the case of a uniform acceleration, such is notthe case. Due to a constant acceleration, the charge picks up speed, and after a long time its motionwill become relativistic, with v → c and the corresponding Lorentz factor becoming very large ( γ ≫ θ ∼ γ [1–3], about thedirection of motion.One comes across such instances of relativistic beaming in the synchrotron radiation, where due toan extremely relativistic motion ( v ≈ c ) of the gyrating charge, the radiation is confined to a narrow angle ∼ γ about the instantaneous direction of motion [1,31]. Furthermore, in extragalactic radio sources,due to highly relativistic motion of a radio source component with respect to the observer’s frame ofreference, the radio emission appears confined to a narrow cone of emission with a cone-opening angle ∼ γ [32].In our present case, the charge, moving with a velocity v → c , is not very far behind the sphericallight-front of radius r = ct . The charge, with v ≈ c ( − γ ) , moves a distance ∼ ct ( − γ ) alongthe z -axis, while the circle of maxima of the field, represented by P at r = ct , has moved along the z -axisa distance, r cos θ ≈ ct ( − θ /2 ) ≈ ct ( − γ ) , thus the field maxima lies in a plane normal to the z -axis that passes nearly through the ‘present’ position of the charge on the z -axis (Fig. 2), and the fieldsare all around the charge. The electric field, in fact, very much resembles that of a charge moving witha uniform velocity equal to the ‘present’ velocity of the uniformly accelerated charge, with the field ina plane normal to the direction of motion. Thus, as the fields move toward infinity, so does the chargeand the fields are confined along the direction of motion in a small, finite region ct /2 γ ∼ α /2 γ aboutthe ‘present’ position of the charge, very similar to the uniform velocity case where electric field is everappreciable only near the ‘present’ position of the charge, in a region whose extent falls as 1/ γ . As wasshown in Section 3.1, the fields actually are the self-fields of the charge that due to the acceleration fields,increase in strength, as the charge picks up speed, to a value expected from that of the charge movingwith a uniform velocity equal to the ‘present’ velocity of uniformly accelerated charge, and accordingly,there is no radiation being ‘emitted’ by the charge.We can verify the above statements explicitly by a comparison of the fields of a uniformly acceleratedcharge, which may have a relativistic ‘present’ velocity v o → c and a corresponding Lorentz factor γ o ≫ v o and thus the sameLorentz factor γ o .The electromagnetic fields of the charge moving with a uniform acceleration, is given in cylindricalcoordinates ( z , ρ , φ ), as [27,29,30] E z = − e α ( α + c t + ρ − z ) / ξ E ρ = e α ρ z / ξ B φ = e α ρ ct / ξ , (34)where ξ = [( α + c t − ρ − z ) + α ρ ] . The above solution is restricted to a region z > − ct with a discontinuity in the fields at z = − ct [27,29,30]. These field expressions are equivalent to the field expressions in terms of retarded-time quantities,and can be derived in the case of a uniformly accelerated charge starting from Eq. (1), using algebraictransformations [29].On the other hand, the electromagnetic field of the charge moving with a uniform velocity v o , canbe written in a spherical coordinates ( R , Θ ), or in cylindrical coordinates ( ρ , ∆ z ), centered at the “present”charge position [1–3], as E = e ˆ R R γ [ − ( v / c ) sin Θ ] = e γ R [ ρ + γ ∆ z ] , (35)The magnetic field in both cases is given by B = v o × E . Equation (35) can be derived in the case of auniformly moving charge from velocity fields (the first term in the square brackets in Eq. (1)) [1–3].As is well known, for a charge moving relativistically with a uniform velocity, the electric fieldcomponent perpendicular to the direction of motion is stronger by a factor γ relative to the componentalong the direction of motion, with the field lines becoming oriented perpendicular to the direction ofmotion [1–3]. Moreover, for a large γ , the field becomes negligible, except in a narrow zone along thedirection of motion, with the field lines confined mostly within a small angle, ∆ z / ρ ∼ γ , with respectto a plane normal to the direction of motion and passing through the ‘present’ charge position [29].Now if we plot the electric field (Eq. (34)) of the uniformly accelerated charge, for a large r = ct ,which also implies v o → c and γ o ≫
1, and compare it with the field (Eq. (35)) of a charge moving withthe same, but a uniform, velocity v o and thus having the same γ o , we find that the fields are quite similarin both cases. Figure 3 shows a comparison of the electric fields in both cases for γ o = v o = c . In both cases fields are very similar and extend, from the “present” charge position, indirection normal to the direction of motion.Figure 4 shows the corresponding Poynting flow, almost indistinguishable in both cases, with theoverall Poynting flow in each case being along the direction of motion of the charge, confirming that thePoynting flow for a uniformly accelerated charge merely represents the “convective” flow of self-fields,along with the moving charge, like in the case of a charge moving with a uniform velocity. Of course, inthe case of a uniformly accelerated charge, the self-field strength continuously keeps getting ‘updated’due to acceleration fields, in tune with the changing charge velocity due to its uniform acceleration.Naturally, there is no radiation reaction in the case of a uniformly accelerated charge since no fieldenergy is being ‘radiated away’ from such a charge. This, of course, also makes the case of a uniformlyaccelerated charge fully conversant with the strong principle of equivalence.In order to avoid a contradiction with Larmor’s radiation formula, it has been suggested thatthe radiation emitted from the uniformly accelerated charge goes beyond the horizon, in regions ofspace-time inaccessible to an observer co-accelerating with charge [30,33]. Actually, it is a misconceptionas from Eq. (34), E ρ = z = t , implying that there is no component of Poynting fluxthrough the z = t = z -component of Poynting vector due to δ -fields is present at z =
0, causallyrelated to the charge during its uniform velocity before an acceleration was imposed at an infinite past.The δ -field, is, in fact, not causally related to the charge during its uniform acceleration, whose influenceat that time lies only in the z > z > t = z =
0. In fact, it has been shown that becauseof a rate of change of acceleration at the time when the acceleration was first imposed on the charge, anevent with which the δ -field has a causal relation, the charge underwent radiation losses [22], owing to Figure 3.
The electric field distribution (a) of a uniformly accelerated charge, with a ‘present’ velocity v o = c , corresponding to γ o =
100 (b) of a charge moving with a uniform velocity v o = c ,corresponding to γ o = ∼ γ with respect to the electric field lines that begin from the charge position z o , in plane perpendicularto the direction of motion. the Abraham-Lorentz radiation reaction [4,5,7,8], thereby neatly explaining the total energy lost by thecharge into δ -field during a transition from a uniform velocity phase to the uniform acceleration phaseat infinite past [29]. In fact, as has been demonstrated here, all fields, including the acceleration fields,having a genesis from the uniform accelerated charge, remain around the moving charge and are notradiated away or dissociated from the charge as long as it continues moving with a uniform acceleration.
4. Conclusions
We showed that a discrepancy between two formulations of the power going into electromagneticradiation and the mechanical power loss of the radiating charge, merely reflects the difference in thepower going in self-fields of the charge between the retarded and present times. It was shown thatequating the Poynting flux at time t , given by Larmor’s formula, to the mechanical power loss of thecharge at a retarded time t ′ , is not in accordance with Poynting’s theorem, where all quantities need tobe calculated, strictly for the same instant of time. It was further shown that in the case of a periodic Figure 4.
The Poynting vector for a charge (a) moving with a uniform proper acceleration, and is presentlyat z o moving with a ‘present’ velocity v o = c , corresponding to γ o =
100 (b) moving with a uniformvelocity v o = c , corresponding to γ o = r = ct is shown in the caseof uniformly accelerated charge, which looks like a plane on this scale. The overall Poynting flow inboth cases is along the direction of motion of the charge. Arrows show Poynting vector directions atdifferent distances from the charge. The length of an arrow is not a direct indicator of the magnitude ofthe corresponding Poynting vector, the plot shows the trend only qualitatively. In fact, the magnitude ofthe Poynting vector, represented by larger arrows, is maximum at the plane normal to the direction ofmotion, passing through the charge at z o , and drops rapidly off the plane. At the positions of smallerarrows, shown in the figure, the magnitude of the Poynting vector falls as much as by a factor of ∼ . motion, where there is no difference in the radiated power averaged over the period starting at t or t ′ ,Larmor’s formula does yield a correct, average power loss by the charge, an argument which, however,cannot be applied in the case of a uniformly accelerated charge. It was shown that for a charge uniformlyaccelerated, all its fields, including the acceleration fields, originating from the time retarded positionsof the charge, are not radiated away from it and remain around the ‘present position of the charge as itsself-fields. Funding:
This research received no external funding.
Conflicts of Interest:
The author declares no conflict of interest.
References
1. Jackson, J. D. Classical electrodynamics 2nd ed.; Wiley: New York, USA, 1975.2. Panofsky, W. K. H; Phillips, M. Classical electricity and magnetism, 2nd ed.; Addison-Wesley: MA, USA, 1962.
3. Griffiths, D. J. Introduction to electrodynamics, 3rd ed.; Prentice: New Jersey, USA, 1999.4. Abraham, M.
Theorie der elektrizitat, Vol II: Elektromagnetische theorie der strahlung
Teubner: Leipzig, Germany,1905.5. Lorentz, H. A. The theory of electron, Teubner: Leipzig, Germany, 1909. Reprinted 2nd ed, Dover: New York,USA, 1952.6. Schott, G. A. Electromagnetic radiation, Cambridge University Press: Cambridge, UK, 1912.7. Yaghjian, A. D. Relativistic Dynamics of a charged sphere, 2nd ed.; Springer: New York, USA, 2006.8. Singal, A. K. Radiation reaction from electromagnetic fields in the neighborhood of a point charge. Am. J. Phys. , 85, 202–206.9. Singal, A. K. The equivalence principle and an electric charge in a gravitational field II. A uniformly acceleratedcharge does not radiate. Gen. Rel. Grav. , 29, 1371–1390.10. Singal, A. K. Disparities of Larmor’s/Liénard’s formulations with special relativity and energy-momentumconservation. J. Phys. Comm. , 2, 031002.11. Teitelboim, C. Splitting of the Maxwell tensor: radiation reaction without advanced fields. Phys. Rev. D ,1, 1572–1582.12. Eriksen E.; Grøn, Ø. Electrodynamics of hyperbolically accelerated charges V. The field of a charge in the Rindlerspace and the Milne space. Ann. Phys. , 313, 147–196.13. Heras, j. A; O’Connell, R. F. Generalization of the Schott energy in electrodynamic radiation theory. Am. J. Phys. , 74, 150–153.14. Hammond, R. T. Relativistic particle motion and radiation reaction in electrodynamics. El. J. Theor. Phys. ,23, 221–258.15. Rowland, D. R. Physical interpretation of the Schott energy of an accelerating point charge and the question ofwhether a uniformly accelerated charge radiates. Eur. J. Phys. , 31, 1037–1051.16. Grøn, Ø. The significance of the Schott energy for energy-momentum conservation of a radiating chargeobeying the Lorentz–Abraham–Dirac equation. Am. J. Phys. , 79, 115–122.17. Singal, A. K. The fallacy of Schott energy-momentum. Phys. Ed. (IAPT) , 36, No. 1, 4.18. Singal, A. K. A first principles derivation of the electromagnetic fields of a point charge in arbitrary motion, Am.J. Phys. , 79, 1036-1041.19. Hartemann, F. V; Luhmann Jr, N. C. Classical electrodynamical derivation of the radiation damping force. Phy.Rev. Lett. , 74, 1107–1110.20. Rohrlich, F. The dynamics of a charged sphere and the electron. Am. J. Phys. , 65, 1051–1056.21. Mould, R. A. Basic Relativity, Springer: New York, USA, 1994.22. Singal, A. K. Poynting flux in the neighbourhood of a point charge in arbitrary motion and radiative powerlosses. Eur. J. Phys. , 37, 045210.23. Singal, A. K. Compatibility of Larmor’s formula with radiation reaction for an accelerated charge. Found. Phys. , 46, 554–574.24. Feynman, R.; Leighton, R. B.; Sands, M. The Feynman lectures on physics Vol. II; Addison-Wesley: Mass, USA,1964.25. Pauli, W. Relativitätstheorie in Encyklopadie der Matematischen Wissenschaften, V , Teubner: Leipzig,Germany, 1921. Translated as Theory of relativity Pergamon: London, UK, 1958.26. Born, M. Die Theorie des starren Elektrons in der Kinematik des Relativitätsprinzips. Ann. Phys.
30 1–56.27. Fulton, T; Rohrlich, F. Classical radiation from a uniformly accelerated charge. Ann. Phys. , 9, 499–517.28. Rohrlich, F.
Classical charged particles . World Scientific: Singapore, 2007.29. Singal, A. K. A discontinuity in the electromagnetic field of a uniformly accelerated charge. J. Phys. Commun. , 4, 095023.30. Boulware, D. G. Radiation from a uniformly accelerated charge. Ann. Phys. , 124, 169-188.31. Rybicki, G. B.; Lightman, A. P.
Radiative processes in astrophysics . Wiley: New York, USA, 1979.32. Rees, M. J. Appearance of relativistically expanding radio sources. Nature , 211 468–470
33. Almeida, C. de; Saa, A. The radiation of a uniformly accelerated charge is beyond the horizon: A simplederivation. Am. J. Phys.2006