There are actually two parts to this "self" force. The one associated with a nonzero da/dt was first suggested by Abraham and Lorentz and dubbed the radiation reaction force. The second part of the "self" force has been called the inertial reaction force because (in non-relativistic cases) it equates to –m_ElecMaga, where m_ElecMag is the electromagnetic mass discussed in a previous section. The total "self" force is the sum of the radiation reaction force and the inertial reaction force. Provided some driving agent counteracts this total self force, periodic motion is possible.

Abraham and Lorentz’s non-relativistic formula for the radiation reaction force is

. (6_1)

Note that, unlike the inertial reaction force (or more rigorously unlike the electromagnetic mass), the radiation reaction force is independent of the charge distribution specifics. Any charge distribution, q, has the radiation reaction force specified by Eq. 6_1.

The total reaction force … the sum of the radiation reaction force and the inertial reaction force … is

. (6_2)

This is the force that a driving agent must somehow counteract if the charge is to maintain a periodic motion.

A charge distribution frequently discussed in the current book is the spherical shell of charge, of radius R and magnitude q. As discussed in a previous section, the electromagnetic mass of this particular distribution is

. (6_3)

Such a distribution can be modeled as an infinite number of infinitesimal point charges. If the shell is at rest, then each point charge experiences an outward, interactive, electrostatic force in the net electric field of all the other point charges. And as previously discussed, some extra-Maxwellian agent must counteract each such interactive force if the shell’s radius is to remain constant. The interactive forces in the resting case (and more generally when the distribution’s velocity is constant) sum to zero. And, since a and da/dt are zero in this case, there are presumably no reaction forces.

If the shell oscillates non-relativistically (v_max<<c) then its shape can usually be assumed to be maintained at all times. Each infinitesimal constituent charge now experiences an "infinitesimal infinitesimal" electric "self" force in its own, infinitesimal, acceleration-induced electric field (always present when a and/or da/dt are nonzero). And each constituent charge also experiences an infinitesimal interactive force in the net electric field of all the other constituent point charges. Furthermore, owing to time delays inherent in the field equations, the infinite number of infinitesimal interactive forces do not now sum to zero (although the infinite number of "infinitesimal infinitesimal" self forces do). In order for the motion to be periodic, an agent must counteract the finite sum of all the interactive forces.

In theory the following algorithm could be used to compute the total reaction force, and hence to check on the validity of Eq. 6_2. (1) At the location of each point charge, at every time epoch, compute the net electric field and thence the net interactive force experienced by that charge and attributable to the other charges. (2) Sum all the interactive forces experienced by the individual charges, to obtain the net reaction force exhibited by the entire spherical shell.

In practice we cannot of course divide a spherical shell up into an infinite number of infinitesimal point charges (at least not for purposes of computation). However, it is perfectly feasible to divide the shell up into a finite number of quasi-point charges. We say "quasi-point charges" because the electromagnetic mass (and hence the inertial reaction force) for a true, finite point charge is theoretically infinite. What we aim for is a compromise of sorts, where each quasi-point charge has a finite electromagnetic mass, but where the point charge field solutions can be used to compute that charge’s electric field at the other quasi-point charge positions. An obvious value for the quasi-point charge’s electromagnetic mass is m_ElecMag/N, where m_ElecMag is the electromagnetic mass of the spherical shell of charge (see Eq. 6_3), and N is the number of quasi-point charges used to approximate it.

Let us model a spherical shell as six quasi-point charges which, at time t=0, have the following coordinates: (R,0,0), (0,0,-R), (-R,0,0), (0,0,R), (0,R,0), and (0,-R,0). Using the point charge field solutions, it is a simple matter to approximate the total reaction force for the entire spherical shell.

Fig. 6_1 plots the total theoretical reaction force for the continuous distribution of charge (Eq. 6_2). Fig. 6_2 plots the computed total interactive forces among the 6 quasi-point charges. Note how the curves closely approximate one another in shape and phase.

It is worth noting that any continuous distribution of charge can be approximated using an array of quasi-point charges. In non-relativistic cases it can be assumed that the underlying distribution maintains its shape as its speed varies. If nothing else, Fig. 6_2 suggests that Abraham and Lorentz were right. According to Maxwell’s equations (and more pointedly the point charge field solutions and the Lorentz force law), a spherical shell of charge does indeed experience the theoretical force specified by Eq. 6_2. The radiation reaction part of this total force, which is proportional to 1/c³, is small in maximum magnitude compared to the inertial reaction part (which is proportional to 1/c²). Nonetheless its presence is clearly indicated in Fig. 6_2, where the total force is slightly negative at time t=0, when the spherical shell’s center passes through the origin where the inertial reaction force is zero.

The power, expended by the driving agent to counteract the inertial reaction force, integrates to zero over any given cycle time. But the power expended to counteract the radiation reaction force integrates to a positive value in any given cycle time. And as will be shown in a later section, this net expended agent work per cycle equates to the net energy flux per cycle through a surface that encloses the oscillation site.