1.2. Spherical Shells of Charge.
Although the concept of a point charge is useful, it turns out that the field momentum of any finite point charge, moving at any velocity other than zero, is infinite! In order to avoid this singularity, it is useful to consider spherical shells of charge. (A finite point charge can then be considered to be the limit of such a distribution as the radius shrinks to zero.)
1.2.1. Modeling the Spherical Shell Using Small Point Charges.
Perhaps ironically, we can approximate a spherical shell of charge q as a distribution of smaller, equal point charges, preferably with each charge at the vertex of a symmetric polyhedron (said polyhedron inscribed in the sphere). Each such point charge will experience a small force in the net field of the other constituent charges. And the entire distribution will experience a net force (which may or may not be zero) in its own field.
Ideally the spherical shell would be modeled as an infinite number of infinitesimal point charges. For computing purposes we must settle for a finite number of small but finite charges.
1.2.2. Implied Agent Forces.
In considering an oscillating charge it was implicitly assumed that some external agent exerted a mechanical (i.e., non-electromagnetic) force on the oscillating charge. In a similar fashion, an external agent must (often implicitly) exert a mechanical force on each small point charge comprising a constant-radius spherical shell … even when the shell is at rest. For when the shell is at rest, each constituent point charge experiences a radially outward-pointing electric force in the net electric field of all the other charges. If this force is not mechanically counteracted in every case, the shell’s radius (or shape) cannot remain constant. In effect a constituent point charge exerts an outward mechanical force on the constraining agent and the agent exerts an equal and oppositely directed mechanical force on the charge. One might even go further and suggest that it is the charge/field that exerts the force on the agent. The charge might be thought of as the "mechanical hook" whereby the agent interacts with the field.
1.2.3. Net Forces when a = 0.
When a spherical shell has zero acceleration (and hence constant velocity) the agent forces that hold the shell together sum to zero. Thus no work is expended by the agent to maintain the spherical distribution of infinitesimal point charges.
1.2.4. Net Forces when da/dt=0 (and when a is Nonzero).
When da/dt=0 and a is nonzero but constant, then owing to time delays the forces between a spherical shell’s diametrically opposed point charges do not consistently sum to zero. Computed solutions for the fields of a finite number of small but finite point charges indicate that the charges (and hence the agent, that holds the shell together) collectively experience a net electric force pointing opposite to a. In order to maintain the distribution’s shape, the driving agent must counteract these acceleration-opposing forces in addition to counteracting the omnipresent repulsive forces. And although the repulsive forces sum to zero, the acceleration-opposing forces do not! Consequently the driving agent, which counteracts this net electric force, must exert a net force in the same direction as a. In effect the spherical shell exhibits an inertia that opposes its acceleration. For future reference we shall refer to the acceleration-opposing net force, that the shell experiences in its own field, as the inertial reaction force. And (as already suggested), the agent’s counteraction to this force is dubbed the inertial force. The net nonzero force that an external agent must exert when a is nonzero of course brings Newton’s 2nd law to mind.
1.2.5. Net Forces when da/dt is Nonzero.
Unlike neutral matter, in the case of a spherical shell of charge there is yet an additional, reactive force component on each charge increment when da/dt is nonzero! And here again the driving agent must counteract these myriad tiny forces if the distribution’s shape is to be maintained. It turns out that this particular component of the total reaction force points in the same direction as da/dt. Thus the driving agent’s net counteraction must point opposite to da/dt. This force, that the charges experience in their own net electric field when da/dt is nonzero, is usually called the radiation reaction force. And the agent’s counteraction constitutes the radiative component of the total agent force.
As will be discussed, a noteworthy difference between the inertial and radiative agent force components is that (a) the magnitude of the inertial component depends on the spherical shell’s radius, whereas (b) the magnitude of the radiative component is independent of the radius (and more generally of the distribution’s shape).