In the case of a spherical shell, any da/dt-induced radiation reaction force is independent of the shell’s radius. Thus given some shell of radius R, we may increase or decrease R (keeping the charge constant) and the shell will experience the same radiation reaction force to a given da/dt. Given two concentric shells with different radii, we may increase/decrease the radius of either one so that the radii are equal. In this case the combination will experience the same radiation reaction force as a single shell whose charge equals the sum of the two original charges.

In Newtonian mechanics, which deals with "neutral matter," there is no da/dt-related component to the agent force. Newton’s 2^nd law states simply that

. (1.4.2_1)

(Here we have written m_mech to signify "mechanical mass" … the inertial mass of an uncharged particle, to the extent such a concept corresponds to anything in the real world).

Since all "uncharged" atoms consist of equal amounts of positive and negative charge, and since the radiation reaction force is proportional to q² (and not to q), it might be wondered why F=ma works so well in the case of atoms (and more generally in cases of objects composed of atoms). In brief, why did Newton find that Eq. 1.4.2_1 works in cases of "neutral matter," when neutral matter is in fact composed of electric charges?

We can appreciate why this is so if we model a typical atom as a central, spherical shell of positive charge, surrounded by a larger spherical shell of equal negative charge. At first glance, since F_radiative is proportional to q², it might seem that (a) when such a distribution is vibrated as a whole, then (b) there should be double the non-conservative work expended per cycle than is required to vibrate either shell alone.

Here, however, we encounter the Equivalence Principle suggested above. Assuming we have equal magnitude charges, two such superimposed shells amount to no charge at all! By inference no power is expended to oscillate the concentric pair when their radii are unequal. The only requirement for this equivalence is that the two shells be concentric during the oscillation.

Note that the "Equivalence Principle" does not apply to the conservative (inertial) power expended to counteract the inertial reaction force (which is proportional to 1/R). Here two superimposed, equal radii shells are not equivalent to concentric shells with different radii. Assuming the charge magnitudes are equal, the inner shell has a greater electromagnetic mass.

It is also worth noting that two oppositely signed, oscillated shells with a net charge of zero will radiate when their centers do not coincide. As will be discussed later, the agent’s counteraction of the interactive force between these two shells will include a radiative component. It is only when the shells are concentric (as in our simplistic model of an oscillated atom) that they are equivalent to a single shell that has zero charge density.

Of course if we have two concentric spheres of same-sign charge then, so far as power expended per cycle to counteract the radiation reaction force goes, the pair is equivalent to a single shell of arbitrary radius and with a charge equal to the sum of the two original charges. Indeed the radius can be arbitrarily small, the implication being that the same radiated power occurs when a finite radius shell and an equal-magnitude point charge are vibrated. It is only in the case of the conservative (or inertial) power expenditure that the 1/R dependence of m_elecmag results in a point charge singularity.