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)

__F___{RadReact} is thus the same for all charge distributions (including point charges) of magnitude q.

Regarding d__a__/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, __F___{RadReact} 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 F_{RadReact} is (in one dimension)

. (2)

Thus more generally __F___{RadReact} is presumably zero only when both __a__ and d__a__/dt are zero.

The hypothesis that d__a__/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 __F___{RadReact}:

. (3)

(If there are multiple, interacting charges, then (-F_{RadReact}-F_{Interact}) should be integrated for each charge.) Clearly this is true if P_{Rad}, the instantaneous rate at which radiant energy is emitted, equates to the rate at which –__F___{RadReact} 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 –F_{RadReact}v is alternately negative and positive. This result is particularly intriguing if there are selected oscillations where –F_{RadReact}v 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 (-F_{RadReact}-F_{Interact}) , and (c) sum the results to get the energy radiated per cycle.