Driving Forces and Self Forces, Driving Powers and Field Energy Flux Rates G.R.Dixon
Note: The graphs in this article were generated by Visual Basic programs in Appendix A. 1. Overview. 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<E (2) It computes and plots (a) P, the rate at which the driving force does work, and (b) dF The subject charge has magnitude q = 1 coul and radius R = .005 meters. The motion of the shell’s geometric center is: . (1-1) The driving force is assumed to be a "contact" (non-electromagnetic) force, and hence the The average 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 dF The electromagnetic (rest) mass of a spherical shell of charge is: . (1-2) The relativistic formula for the driving force is (in 1 dimension): . (1-3) (That is, no mechanical mass is assumed.) The rate at which this force does work is: . (1-4)
Fig. 2-1a plots F vs. t (see Eq. 1-3), and Fig. 2-1b plots -q<E . (2-1) Newton’s third law suggests that q<E , (2-2a) . (2-2b) Figure 2-1a
Figure 2-1b
-q<E
Fig. 2-2a plots P vs. t, and Fig. 2-2b plots dF Figure 2-2a
P, wA = .001c Figure 2-2b
dF The practical equality of F and –q<E
It is clear in Eqs. 2-2a and b that F
Figure 3a
P, t ~ 0 Figure 3b
dF 4. Energy Conservation. The fact that the curves for P and dF We may verify that energy conservation is satisfied by computing (a) W, the total work done per cycle by the driving force, and (b) E
, (4-1a) . (4-1b) 5. Conclusions. 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 …" 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 dF 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 dF
1. The Feynman Lectures on Physics, Volume II (Electromagnetism), Fourth printing – July 1966, ADDISON-WESLEY PUBLISHING COMPANY INC., READING, MASSACHUSETTS, pp 28-5. |