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’
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
At time t=0, and at point P=(0,dy,0) the retarded time, tR, practically equals the present time, t. Denoting the retarded velocity as uR, we then have
The general point charge field solutions then indicate(1) that, at space-time point (0,dy,0,0), Ex is negative, Ey>>-Ex is positive, Bz is positive, and Ez=Bx=By=0. The y-component of the Poynting vector is thus positive:
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 Ex’, Ey’ and Bz’ 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, Ex’=Ex, and thus Ex’ also points in the negative x-direction. However, Appendix A indicates that
where wA is the charge’s speed (at t=tR=0). The transformation of Bz thus produces a Bz’ 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
In a previous article (where wA=.0001c and v=.95c) plots of the Abraham-Lorentz formula for PRad(t) and P’Rad(t’) did indeed indicate that PRad(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 PRad, PRad(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 PRad 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, 2nd Edition, David J. Griffiths, PRENTICE HALL, Chapter 9, Section 9.2.2.