Relativistic Reaction Forces for a Spherical Shell of Charge
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.
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 not sum to zero when a, da/dt, etc. are nonzero. If the shell is tiny (radius r<<1 meter) and its motion is along the x-axis, then the net internal force is found to be
where mem, the so-called electromagnetic mass, is defined to be
The counteracting agent force is thus
All known charged particles are believed also to have mechanical mass. Their total mass can be expressed as
The general equation of motion for a spherical shell of charge, with nonzero mechanical mass, is therefore
Eq. 2-5 reduces to Newton’s second law when mem=0 (uncharged particle).
Abraham and Lorentz aptly dubbed the dax/dt term in Eq. 2-1 the "radiation reaction force" because of its correlation with radiated energy. For example, if the spherical shell oscillates at angular frequencyw, then the net work expended per cycle by the agent’s counteraction equals the field energy flux per cycle through an enclosing surface:
where S=eoc2E X B is the Poynting vector. (The work per cycle expended to counteract –max is zero.) If the tiny shell’s E and B fields at points on the enclosing surface are assumed to be the same as those of a point charge at the shell’s center, then Eq. 2-6 can readily be corroborated by a computer program. Table 2-1 lists the computed left and right sides of Eq. 2-6 for several values of wA<<c. The percent difference, defined to be (Flux-Work) / Work, is also given. The small discrepancies can be attributed to numerical error and to the fact that the non-relativistic formula for the radiation reaction force was used.
Work per Cycle and Energy Flux per Cycle Compared
In view of Newton’s second law, the ax term in Eq. 2-1 might be dubbed the inertial reaction force. As mentioned, the work per cycle, expended by the agent while counteracting this force, is zero:
Or more generally,
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.
"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 allwA. 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,
In Eq. 3.1-1 mem(o) is the rest mass. For a spherical shell of charge it is defined by Eq. 2-2. Einstein concluded that this speed dependence must extend to mechanical mass as well.
It turns out that Newton’s second law for uncharged particles is relativistically correct when expressed in the form
Eq. 3.1-2 becomes
This is the force that a driving agent must exert in order to counteract the relativistically correct inertial reaction force:
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):
where mem will henceforth be understood to be the electromagnetic rest mass. In view of Eq. 3.1-3, Eq. 3.2-1 expands to:
The power expended by the counteracting agent is thus
As in the non-relativistic case (Eq. 2-6), when x = A sin(wt) this power does not integrate to zero over one cycle time. A modified program computes the agent work per cycle over a range of relativistic wA. Table 3.2-1 repeats Table 2-1, but with .85c<wA<.95c. Note the excellent agreement between the relativistically correct work per cycle and the energy flux per cycle.
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:
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:
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:
Or, since the g3moax (inertial) term integrates to zero,
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 in theory be solved to find the motion. In practice this may not always be analytically feasible. Since the radiation reaction force is often much smaller than the inertial reaction force, the abbreviated equation "Fx = g3moax" is often solved in order to obtain an approximate motion. However, this relativistically correct form for Newton’s second law obeys the Work Energy Theorem … any work expended by the driving agent is manifest as changes in kinetic energy and is completely recoverable. Rigorously speaking, Eq. 4-1 more accurately describes the relationship between applied force and particle motion when there is excess charge of one sign or another. The work expended by a driving agent is not fully recoverable; some of that work bleeds away into infinite space as radiant energy.