. (7_1)

The point charge field solutions are used to compute E_Rad. Since these solutions are based on Maxwell’s equations, they are presumably rigorous and the article accordingly tests the formulas used to compute W. Only non-relativistic cases (where the maximum speed is <<c) are considered. The relativistic formulas for W will be considered in one or more later sections.

Readers who wish to run the Basic programs (or their equivalent) on their own computers are advised that the computation of retarded time, and values derived from it, seems to work well only when the field evaluation point is at least a half wavelength from the source charge. Very small differences between the current time and the retarded time often do not produce accurate results, given the limited number of significant digits in most floating point processors. (Floating point processors compute products of very large and very small values accurately, but information is lost when two such numbers are added.)

In order for any charge to move periodically, a driving agent must theoretically counteract the charge’s inertial and radiation reaction forces. Furthermore, if two or more charges move periodically and are close enough to influence one another, then theoretically the driving agent must also counteract the interactive forces.

(7_2)

The power expended by the inertial reaction force (and by the agent’s counteraction to it) integrates to zero over a cycle time. Hence the inertial reaction force plays no role in the energy flux per cycle through an enclosing surface. The electromagnetic mass is accordingly of no consequence, and we may stipulate that the charges are point charges. (The radiation reaction force of a point charge is the same as that of a spherical shell of charge, etc.) The non-relativistic formula for the radiation reaction force is:

. (7_3)

At the other charge’s position the point charge field solutions can be used to compute q_i’s E_x field at time t, and hence the interactive force on that other charge.

Denoting the net interactive force acting on charge q_i as F_i(Interact), the power expended by the driving agent on charge q_i is

. (7_4)

Thus W_i, the net work per cycle expended on q_i, is

. (7_5)

And of course W, the work per cycle expended on both charges, is

. (7_6)

An important rule is that, whereas the E and B field vectors of the individual charges are additive, the Poynting vectors are not. In brief, the electric and magnetic fields of the individual charges must be summed before computing the Poynting vector at any point on the enclosing surface.

Two Runs are made, one for a lone charge and one for a pair of charges. In the lone charge case the agent theoretically need only counteract that charge’s radiation reaction force. In the two-charge case the agent must counteract the radiation reaction force, plus the interactive force experienced in the other charge’s electric field.

The values for the oscillation amplitude(s), etc., are specified in Tables 7_1 and 7_2 (for Runs 1 and 2 respectively), along with the computed values of E_Rad and W. The 1-charge program can easily be altered, and the 2_charge program can readily be modified to deal with more than two charges. Note how W = E_Rad in every case.