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Compatibility of Larmor's formula with radiation reaction for a radiating charge

It is shown that the well-known disparity in classical electrodynamics between the power losses calculated from the radiation reaction and that from Larmor's formula, is succinctly understood when a proper distinction is made between quantities expressed in terms of a "real time" and those expressed in terms of a retarded time. It is explicitly shown that an accelerated charge, taken to be a sphere of vanishingly small radius $r_{\rm o} $, experiences at any time a self-force proportional to the acceleration it had at a time $r_{\rm o} /c$ earlier, while the rate of work done on the charge is obtained by a scalar product of the self-force with the instantaneous (present) value of its velocity. Now if the retarded value of acceleration is expressed in terms of the present values of acceleration, then we get the rate of work done according to the radiation reaction equation, however if we instead express the present value of velocity in terms of its time-retarded value, then we get back the familiar Larmor's radiation formula. From this simple relation between the two we show that they differ because Larmor's formula, in contrast with the radiation reaction, is written not in terms of the real-time values of quantities specifying the charge motion but is instead expressed in terms of the time-retarded values. Moreover, it is explicitly shown that the difference in the two formulas for radiative power loss exactly matches the difference in the temporal rate of the change of energy in the self-fields between the retarded and real times. From this it becomes obvious that the ad hoc introduction of an acceleration-dependent energy term, usually referred to in the prevalent literature as Schott-term, in order to make the two formulas comply with each other, is redundant.