A Demonstration that When the Poynting Vector Points Away from an Oscillating Point Charge in Motional Rest Frame K, it May Point Toward the Charge in Frame K’

G.R.Dixon, 7/14/2006

In theory the electric and magnetic fields at any instant, and at any point other than that occupied by a point charge, can be solved provided the charge’s past motion is known. In this article we shall consider the fields __arbitrarily close__ to an oscillating point charge.

Let the point charge’s motion in its motional rest frame (inertial frame K) be

. (1)

At time t=0, and at point P=(0,dy,0) the retarded time, t_{R}, practically equals the present time, t. Denoting the retarded velocity as __u___{R}, we then have

, (2a)

(2b)

The general point charge field solutions then indicate^{(1)} that, at space-time point (0,dy,0,0), E_{x} is negative, E_{y}>>-E_{x} is positive, B_{z} is positive, and E_{z}=B_{x}=B_{y}=0. The y-component of the Poynting vector is thus positive:

>0. (3)

__S__ has a component that points away from the charge at point (0,dy,0,0), and presumably this is true at other points on a spherical surface that is centered on the charge and that has an infinitesimal radius equal to dy. This outward power flow constitutes __radiated__ energy. (__S__ also has an x-component associated with "bound" field energy that moves back and forth with the charge.)

We can apply the general field transformations in order to determine E_{x}’, E_{y}’ and B_{z}’ in frame K’ and at time t’=t=0 (where K’ moves in the positive x-direction of frame K at speed v). In general, E_{x}’=E_{x}, and thus E_{x}’ also points in the negative x-direction. However,
Appendix A indicates that

, (4),

where wA is the charge’s speed (at t=t_{R}=0). The transformation of B_{z} thus produces a B_{z}’ that is negative, zero, or positive, depending upon whether wA is less than v, equal to v, or greater than v. This in turn indicates that

; (4a)

, (4b)

. (4c)

In a previous article (where wA=.0001c and v=.95c) plots of the Abraham-Lorentz formula for P_{Rad}(t) and P’_{Rad}(t’) did indeed indicate that P_{Rad}(t=0)>0 whereas P’_{Rad}(t’=0)<0. In
another article it was shown that, in the case of the distinctly different Larmor-Lienard formula for P_{Rad}, P_{Rad}(t=0) and P’_{Rad}(t’=0) are __both greater than zero__ (and are in fact equal). The field transformations and Eq. 4c suggest that the Abraham-Lorentz formula for P_{Rad} is the correct choice.

(1) See Appendix A for the field solutions. For an excellent derivation of the general field solutions, see **INTRODUCTION TO ELECTRODYNAMICS, **2^{nd} Edition, David J. Griffiths, PRENTICE HALL, Chapter 9, Section 9.2.2.