Relativistic Reaction Forces for a Spherical Shell of Charge G.R.Dixon, 1/15/04
1. Overview.
In this article the relativistic forms of the inertial and radiation reaction forces are discussed. The "classical" forms were discussed in a previous article. As in that article, a tiny spherical shell of charge (radius r<<1 meter) oscillates along the x-axis. The relativistic version of the radiation reaction force is tested by computing the energy flux per cycle through an enclosing spherical surface and comparing that result with the work per cycle done by the driving agent (who counteracts the reaction forces). The programs used to compute these quantities are included in an appendix. 2. Review.
In electromagnetic theory a spherical shell of charge cannot maintain its dimensions without the non-electromagnetic intervention of a constraining agent. Each increment of charge in the shell experiences an infinitesimal radial force outward in the electric field of the other increments. And an external agent must counteract each of these tiny forces if r, the shell radius, is to remain constant. If the shell is permanently at rest in some inertial frame, then the internal electric forces sum to zero. In this the case the net counteracting agent force also sums to zero, and the agent is usually not explicitly mentioned. Lorentz and others found, however, that the net internal forces (and hence the counteracting agent forces) do (2-1) where m . (2-2) The counteracting agent force is thus . (2-3) All known charged particles are believed also to have . (2-4) The general equation of motion for a spherical shell of charge, with nonzero mechanical mass, is therefore . (2-5) Eq. 2-5 reduces to Newton’s second law when m Abraham and Lorentz aptly dubbed the da (2-6) where Table 2-1
Work per Cycle and Energy Flux per Cycle Compared In view of Newton’s second law, the a . (2-7) Or more generally, . (2-8) If the same programs are run for relativistic wA … particularly if the program that computes the agent work per cycle is run for such relativistic oscillators … then it is clear that the reaction forces require relativistic adjustment. For example, Table 2-2 lists the computed non-relativistic agent work per cycle and the energy flux per cycle for a few relativistic values of wA. Table 2-2
"Classical" Work per Cycle and Energy Flux per Cycle Compared 3. Relativistic Adjustments.
The point charge field solutions are accurate at all source charge speeds. Thus they compute to the correct energy flux per cycle through a surrounding surface for all wA. However, the left half of Eq. 2-6 does not produce approximately equal works per cycle when wA is relativistic. Both the inertial and radiation reaction forces of Eq. 2-1 require relativistic adjustment. These adjustments are discussed individually below. 3.1 The Relativistic Inertial Reaction Force.
Before Special Relativity was introduced, Lorentz and others realized that electromagnetic mass cannot be a constant attribute of a charged particle. Rather it must be a function of the particle’s speed. In one dimension, . (3.1-1) In Eq. 3.1-1 m It turns out that Newton’s second law for uncharged particles is relativistically correct when expressed in the form (3.1-2) . Or, since , (3.1-3) Eq. 3.1-2 becomes (3.1-4) . This is the force that a driving agent must exert in order to counteract the relativistically correct . (3.1-5) As in the non-relativistic case, the work per cycle expended to counteract this force is zero. 3.2 The Relativistic Radiation Reaction Force.
The relativistically correct expression for the radiation reaction force is (in the case of a spherical shell of charge): , (3.2-1) where m . (3.2-2) The power expended by the counteracting agent is thus . (3.2-3) As in the non-relativistic case (Eq. 2-6), when x = A sin(wt) this power does Table 3.2-1
Work per Cycle and Energy Flux per Cycle Compared 4. Relativistic Electrodynamics. Given a spherical shell of charge, the one-dimensional relativistically correct equation of motion is: . (4-1) This is the force that a driving agent must apply in order to counteract the inertial and radiation reaction forces. The agent power expenditure is: . (4-2) If the motion is sinusoidal (say x=A sin(wt)), then the expended agent work per cycle equals the field energy flux per cycle through an enclosing surface: . (4-3) Or, since the g (4-4) . By Fourier analysis this result extends to all periodic motions. The program used to generate Table 3.2-1 can easily be modified to show that this is the case. It is noteworthy that the preceding discussion is predicated upon a shift from the customary paradigm for dynamics. Usually it is the force experienced by a particle that is presumed to be given, and the particle’s motion that is then calculated. In the present discussion it is the motion that is given (e.g. x=A sin(wt)) and the force required to maintain that motion that is computed. Of course given a force, Eq. 4-1 can |