Driving Forces and Self Forces,
Driving Powers and Field Energy Flux Rates
Note: The graphs in this article were generated by Visual Basic programs in Appendix A.
This article performs the following two exercises:
(1) It computes and plots (a) F, the force needed to make a tiny spherical shell of charge oscillate, and (b) –q<Ex>, thedriving agent's reaction to the "self" force experienced by the charge in its own electric field (where <Ex> is the average x-component of the field over the spherical shell's surface);
(2) It computes and plots (a) P, the rate at which the driving force does work, and (b) dFE/dt, the rate at which field energy fluxes through the charge’s surface.
The subject charge has magnitude q = 1 coul and radius R = .005 meters. The motion of the shell’s geometric center is:
The driving force is assumed to be a "contact" (non-electromagnetic) force, and hence the E and B fields are solely those of the driven charge.
The average E and B fields in the charge’s surface are assumed to be the same as those of a point charge at the spherical shell’s center. The legitimacy of this assumption will be assessed (a) by comparing F with –q<Ex>, and (b) by comparing P with dFE/dt. The point charge field solutions are relativistically rigorous, and thus the relativistically correct formula for the driving force is used, even though wA is non-relativistic.
Regarding comparisons (a) and (b) cited above, it may prove instructive to distinguish bound field energy from radiant field energy. Bound field energy will be loosely defined to be energy that stays with the charge. Radiant field energy escapes permanently into infinite space. Since bound field energy is presumably distributed throughout space, the amount at any moment within a spherical volume (with the charge at its center) must increase as the sphere’s radius increases. Consequently the average positive or negative dFE/dt through any spherical surface must decrease as the surface’s radius is increased. dFE/dt through any such surface can therefore be expected to be somewhat less than P. However, it might be expected that the two plots vs. time will have similar shapes.
The electromagnetic (rest) mass of a spherical shell of charge is:
The relativistic formula for the driving force is (in 1 dimension):
(That is, no mechanical mass is assumed.) The rate at which this force does work is:
2. F and –q<Ex>; and P and dFE/dt Compared.
Fig. 2-1a plots F vs. t (see Eq. 1-3), and Fig. 2-1b plots -q<Ex>. Note that, for most practical purposes,
Newton’s third law suggests that q<Ex> is a reaction force. Indeed Eq. 1-3 suggests that this reaction force consists of 2 parts, an inertial reaction force (a la newtonian mechanics) and a radiation reaction force (first suggested by Abraham and Lorentz):
F,wA = .001c
-q<Ex>,wA = .001c
Fig. 2-2a plots P vs. t, and Fig. 2-2b plots dFE/dt through a sequence of (stationary) spherical surfaces, each instantaneously coincident with the spherical shell of charge. Note that dFE/dt is less than P, although the two curves have similar shapes.
P, wA = .001c
dFE/dt, wA = .001c
The practical equality of F and –q<Ex> in Figs. 2-1a and b, and the similar shapes of the curves for P and dFE/dt in Figs. 2-2a and b, lend support to the assumption that the average E field of the spherical shell is practically the same as the average E field of a point charge at the shell’s center.
3. Abraham and Lorentz.
It is clear in Eqs. 2-2a and b that F(InertReact) nominally dominates F(RadReact). The exception occurs when |t| ~ 0. At these times the acceleration is practically zero and |da/dt| and |v| have their maximum values. Thus P is positive at and close to t = 0. This positive power expenditure is not evident in the figures of Sect. 2 because the plotting software scales things so that everything fits. It is instructive to plot P and dFE/dt in the immediate vicinity of t = 0. Figs. 3-1a and b do so. dFE/dt and P are both positive, as indicated by the Abraham-Lorentz formula for the radiation reaction force (and the driving agent’s counteraction to it). Interestingly enough, dFE/dt is actually greater than P.
P, t ~ 0
dFE/dt, t ~ 0
4. Energy Conservation.
The fact that the curves for P and dFE/dt in Figs. 2-2a and b have different maxima and minima may at first seem to violate energy conservation. However, as already pointed out, some of the driving force’s power expenditure is used to change bound field energy when the charge’s center is at any x other than –A, 0, and A.
We may verify that energy conservation is satisfied by computing (a) W, the total work done per cycle by the driving force, and (b) Erad, the total amount of energy that fluxes through an enclosing spherical surface in a cycle time. (In the case of periodic motion, the net bound energy flux per cycle time is zero.) In order to lessen the effects of the relatively large fluxes associated with bound field energy, a larger spherical surface can be utilized, say a single, fixed one, centered on the Origin and with a radius of 5 meters. W and Erad compute to:
In discussing the phenomenon of electromagnetic self-forces, the late R. Feynman wrote, "If we have a charged particle and we push on it for awhile, there will be some momentum in the electromagnetic field. Momentum must have been poured into the field somehow. Therefore there must have been a force pushing on the electron in order to get it going --- a force in addition to that required by its mechanical inertia, a force due to its electromagnetic interaction. And there must be a corresponding force back on the "pusher." But where does that force come from? The picture is something like this …"(1)
We may of course paraphrase Feynman, and make a similar statement about a force acting over a distance, and the accompanying flux of field energy out into the electromagnetic field.
Indeed given the results obtained in this article, we can be a good deal more explicit about the mechanisms involved. In non-relativistic cases the bound field energy within the spherical shell’s surface is evidently only a small fraction of the total field energy, and the power expended by the driving force on the one hand, and dFE/dt on the other, do not differ by much. (This might be contrasted with the infinite power needed to drive a point charge, where dFE/dt through any spherical surface of finite radius is nonetheless finite.)
The essential correctness of the formula for the driving force is indicated when the work per cycle, done by said force, and the energy flux per cycle through an enclosing surface, are computed and found to be equal. (At points on an adequately large, stationary, enclosing surface, the tiny spherical shell’s fields (and Poynting vector) are presumably indistinguishable from the fields of a point charge at the spherical shell of charge’s center.)
An important result in the present monograph is the demonstration of the power expended by the driving agent’s counteraction to the radiation reaction force of Abraham and Lorentz. Although greatly overshadowed by the inertial forces most of the time, the radiation reaction force becomes dominant in a brief window of time close to t = 0 (in the case of sinusoidal motion). At these times the charge’s acceleration is nearly zero, and both the radiation reaction force and the particle’s velocity have their maximum magnitudes. The fact that the computed dFE/dt during such times is positive (as P is) testifies to the essential reality of the radiation reaction force. That is, the attendant positive dFE/dt is strictly a consequence of Maxwell’s equations (since the point charge field solutions are grounded in those equations). It is not based upon the power theoretically expended to counteract the radiation reaction force.
1. The Feynman Lectures on Physics, Volume II (Electromagnetism), Fourth printing – July 1966, ADDISON-WESLEY PUBLISHING COMPANY INC., READING, MASSACHUSETTS, pp 28-5.