2. Relativistic Electrodynamics.
In this section, cases are considered where maximum particle speeds are close to c.
Two relativistic precepts in the dynamics of uncharged particles are that (1) a particle’s mechanical mass is a function of its speed, and (2) Newton’s 2nd law, as he originally presented it, is relativistically valid:
(2_1)
. (2_2)
The dependence of electromagnetic mass has the same form as Eq. 2_1.
In the case of a charged particle, the relativistically correct formula for the radiation reaction force is:
. (2_3)
2.1. The Relativistically Rigorous Radiation Reaction Force.
If one compares the energy flux per cycle time, out through a surface enclosing an oscillating point charge, with the agent work per cycle expended to counteract ada/dt (where a=q2/6peoc3), then as vmax (or wA) approaches c, significant inequalities are encountered. Since the point charge field solutions (used to compute S at points on the enclosing surface) are relativistically correct, it must be that the non-relativistic expression for the radiation reaction force requires adjustment. The relativistically correct formula in one dimension is
. (2.1_1)
When this formula is used, then it is found that, for all wA<c,
. (2.1_2)
In the case of a charge going in a circle at constant speed, the vector equation 2_3 simplifies to
. (2.1_3)
Here g is a constant and da/dt points opposite to v at all times. Thus a driving agent, in counteracting FRadReact, must exert a constant force in the same direction as v. In order to keep the orbit from decaying, the particle must be driven around the circle by a constant-magnitude, tangential mechanical force. Among other things, the rate at which the driving agent does work is constant in time.
As in the case of oscillating motion, the above results can be compared with the energy flux per cycle time through an enclosing surface. Here again Eq. 2.1_2 is found to be satisfied. Furthermore, it is found that the rate of field energy flux through the enclosing surface is constant in time, quite as the driving agent’s power expenditure is.