Poynting flux in the neighbourhood of a point charge in arbitrary motion and radiative power losses

Abstract

We examine the electromagnetic fields in the neighbourhood of a ‘point charge’ in arbitrary motion and thereby determine the Poynting flux across a spherical surface of vanishingly small radius surrounding the charge. We show that the radiative power losses from a point charge turn out to be proportional to the scalar product of the instantaneous velocity and the first time-derivative of the acceleration of the charge. This may seem to be discordant with the familiar Larmor formula where the instantaneous power radiated from a charge is proportional to the square of acceleration. However, it seems that the root cause of the discrepancy actually lies in Larmor’s formula, which is derived using the acceleration fields but without due consideration for the Poynting flux associated with the velocity-dependent self-fields ‘co-moving’ with the charge. Further, while deriving Larmor’s formula, one equates the Poynting flux through a surface at some later time to the radiation loss by the enclosed charge at the retarded time. Poynting’s theorem, on the other hand, relates the outgoing radiation flux from a closed surface to the rate of energy decrease within the enclosed volume, all calculated for the same given instant only. Here we explicitly show the absence of any Poynting flux in the neighbourhood of an instantly stationary point charge, implying no radiative losses from such a charge, which is in complete conformity with energy conservation. We further show how Larmor’s formula is still able to serve our purpose in the vast majority of cases. It is further shown that Larmor’s formula in general violates momentum conservation and, in the case of synchrotron radiation, leads to a potentially incorrect conclusion about the pitch angle changes of the radiating charges, and that only the radiation reaction formula yields a correct result, consistent with special relativity.