2. Energy Fluxes, Near and Far. (S/W reference: Appendix
A.2)
Let point charge q=1 coul have the motion
. (2_1)
We shall apply the algorithm from Sect. 1 to compute Ex, Ey, Bz and Sy (Poynting Vector) at two points on the y-axis, one "near" and one "far" from the origin. The wavelength is defined to be
. (2_2)
Let us say that the near point is at y = l/10 = .063 meters, and the far point is at y = 5l = 3.14 meters.
Fig. 2_1 plots Ex(y = l/10, t) (red trace) and cBz(y = l/10, t) (blue trace) over the time interval 0<t<2p/w. Fig. 2_2 plots the same functions at y = 5l. Note how Ex and Bz appear to be out of phase at y = l/10, and in phase at y = 5l.
Figure 2_1
Ex(t) (Red Trace) and cBz(t) (Blue Trace) at y =
l/10
Figure 2_2
Ex(t) (Red Trace) and cBz(t) (Blue Trace) at y = 5
l
Not surprisingly, Ex at y = 5l is a scaled down version of Ex at y = l/10, and similarly for Bz. But because of the phase differences the Poynting vector component, Sy, is quite different at these two distances from the origin. Figs. 2_3 and 2_4 plot Sy(t) at y = l/10 and at y = 5l respectively.
Figure 2_3
Sy(t), y =
l/10
Figure 2_4
Sy(t), y = 5
l
At y =
l/10 field energy appears to flux in and out, with somewhat more fluxing outward than inward each cycle. At y = 5l the energy appears to flux almost exclusively outward in pulsed fashion.
The alternating outward and inward fluxing of field energy in the "near fields" more or less parallels the rise and fall of particle kinetic energy. For reasons that will be elaborated upon in Sect. 3 this is characterized as predominantly "inertial" field energy flux, and it is evidently a short range phenomenon.
Mixed in with the near-fields inertial energy flux is a weaker "radiant" energy flux. As evidenced in Fig. 2_4, this persists over a much longer (theoretically infinite) range. Worth noting now (and shown in later sections) is the result that If S is integrated over an enclosing surface, over a cycle time, then it is found that a small amount of radiant energy permanently escapes into infinite space every cycle. Energy Conservation requires that an equal amount of work be done each cycle by whatever agent causes the charge to oscillate. A position-dependent (or "conservative") force, such as that exerted by a spring, cannot therefore maintain the periodic motion. Such a force must be augmented in such a way that the net expended work per cycle equals the field energy flux per cycle through a fixed, enclosing surface. Field energy fluxes through such surfaces will be computed and discussed in later Sections.