1. Non-Relativistic Electrodynamics.
In this section it is assumed that particle maximum speeds are much less than c, the speed of light. Among other things, particle mechanical and electromagnetic masses are assumed to be constant attributes of a particle. In general, the relativistic term (1-v2/c2)1/2 does not appear.
1.1. The Fields of a Point Charge.
Several equivalent formulas for the electric (E) and magnetic (B) fields of a point charge have been derived over the years. A useful form for computing purposes is provided by Griffiths:
, (1.1_1)
, (1.1_2)
. (1.1_3)
In these equations r is the vector from the point charge’s retarded position to the field evaluation point; v is the retarded velocity; and a is the retarded acceleration. In brief, r, v, and a correspond to the retarded time, tr.
In order to compute the fields at an arbitrary field evaluation point (which can be any point other than that occupied by the charge itself), the charge’s past motion must be known. Provided with this information, a computer can readily approximate the retarded quantities by means of an iterating routine. And having done this, it can compute E and B directly.
An excellent subject for such an exercise consists of an oscillating point charge whose motion is
. (1.1_4)
It is useful to define the wavelength of this oscillation to be
. (1.1_5)
Let us consider 2 points on the y-axis, say one at y=.05l and a second at y=l. (The first point is sometimes said to be in the "near fields" region, and the second in the "far fields.")
If we are concerned with the field energy fluxes along the y-axis, we can compute Ex and Bz in each case and then the y-component of the Poynting vector,
. (1.1_6)
Fig. 1.1_1 plots Sy vs. time at y=.05l. Fig. 1.1_2 does the same at y=l.
Figure 1.1_1
Sy, Near Fields Region
Figure 1.1_2
Sy, Far Fields Region
It turns out that, in the near fields region, field energy primarily fluxes away from and back toward the charge. And, like the charge’s motion, this energy flux is nearly sinusoidal. Energy considerations suggest that there is a correspondence between this flux and the power expended by the external agent that mechanically drives the charge. For reasons to be made clear shortly, we shall refer to this part of the agent-provided driving force as the "inertial" force. Note that the energy pouring into the fields in one part of a cycle flows back to the driving agent in a succeeding fraction of a cycle.
In the far fields region the energy flux is practically all outward and pulsed. Presumably part of the positive power, expended by the driving agent in one part of a cycle, cannot be recouped in a succeeding part of a cycle; it is permanently lost to infinite space. We shall refer to this part of the total agent force as the "radiative" force. In general a driving force may have both inertial and radiative parts:
. (1.1_7)
Note the figures’ implication that the inertial part is "conservative" in the sense that the net work per cycle expended by it is apparently zero. The radiative part is "non-conservative." The net work per cycle expended by it is greater than zero.