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On the Independence of the Radiation Reaction Force from a Charge Distribution’s Specifics.

G.R.Dixon, 4/25/2005

The non-relativistic formula for the radiation reaction force of a charge, q, is

. (1)

FRadReact is thus the same for all charge distributions (including point charges) of magnitude q.

Regarding da/dt, we shall assume that it refers to the motion of a distribution’s "center of charge" (or ‘cq’ for short). Even if a given distribution’s shape changes in time, FRadReact is theoretically zero so long as the cq remains at rest in some inertial frame (i.e. is "inertial"). Note that the relativistically rigorous formula for FRadReact is (in one dimension)

. (2)

Thus more generally FRadReact is presumably zero only when both a and da/dt are zero.

The hypothesis that da/dt and a refer to the motion of a distribution’s cq implies that a spinning distribution of charge will not radiate so long as its cq is inertial. Spinning rings of charge (and distributions built up from them, e.g. spinning spheres), with their magnetostatic fields, of course do not radiate. But more generally any spinning distribution will theoretically not radiate so long as its cq is inertial. This despite the fact that, for asymmetric distributions, E and B may vary in time.

It is not difficult to show that the radiant energy flux per cycle time, through a fixed surface that surrounds a lone oscillating charge, equals the work done per cycle by the negative of FRadReact:

. (3)

(If there are multiple, interacting charges, then (-FRadReact-FInteract) should be integrated for each charge.) Clearly this is true if PRad, the instantaneous rate at which radiant energy is emitted, equates to the rate at which –FRadReact does work:

. (4)

Although the fields of a given distribution, at points external to the distribution, may depend upon the distribution’s specifics, the radiant flux per cycle through a surrounding surface evidently depends only upon the motion of the cq. Indeed for purposes of computing the radiant energy flux per cycle, a point charge (of size q) located at the cq can theoretically always be substituted for the extended distribution … a useful fact since the fields of a point charge are readily computed.

An intriguing feature of a point charge that oscillates through a resting spherical distribution (of opposite sign) is that –FRadReactv is alternately negative and positive. This result is particularly intriguing if there are selected oscillations where –FRadReactv does zero net work per cycle. In light of the above, this possibility can be extended to negative "clouds" of charge, oscillating back and forth over an embedded, much more compact positive nucleus. So far as radiation is concerned, the effect would be the same as that of a negative point charge at the negative cloud’s center. From the perspective of the relatively huge cloud of negative charge, the oscillation may seem slight. However, only a small amplitude of oscillation would suffice to result in the cq’s movement entirely through the tiny positive nucleus, with room to spare. And, if the negative cloud is distributed in a spherical shell, then the negative charge itself never need actually make contact with the central nucleus!

The hypothesis that the motion of a distribution’s cq is the contributing factor in the emission of radiant energy is rich in possibilities. The software for multiple like-signed point charges (oscillating with a common frequency) can easily be modified to deal with opposite-signed point charges, and hence to opposite-signed extended distributions of charge. One need only (a) substitute point charges for the various distributions, (b) compute the work per cycle done by each charge’s (-FRadReact-FInteract) , and (c) sum the results to get the energy radiated per cycle.