On Negative Radiated Power G.R.Dixon, 10/02/2004
Note: The Visual Basic program code that produced the results in this article is provided in Appendix A. G.R.Dixon, 10/02/2004 Note: The Visual Basic program code that produced the results in this article is provided in Appendix A. Theoretically the relativistic formula for the radiation reaction force experienced by a spherical shell of charge (in one dimension) is . (1) Or, since , (2) the radiation reaction force is . (3) Like the non-relativistic formula for F If a point charge has the motion , (4) then Eq. 3 becomes (5) Presumably the driving agent counteracts this reaction force (as well as the inertial reaction force). This being the case, Prad, the rate at which the driving agent does work in counteracting the radiation reaction force, is –F (6) Using values of q=1 coul, A=1m, and wA=.95c, Fig. 1 plots this P Figure 1
P
A=.95c
As in the non-relativistic formula for P The energy radiated per cycle is . (7) For the power plotted in Fig. 1, numerical integration indicates that E Multiplication of the negative of Eq. 3 by the velocity yields the more general formula for radiated power in terms of the velocity and acceleration: . (9) An interesting feature of Eq. 9 is that the radiated power at any moment depends only upon the kinematics at that moment. And theoretically the equation applies to any one-dimensional motion. In any case, it is clear in Eq. 6 that (for the oscillatory motion specified by Eq. 4) P . (10) Thus as Fig. 1 illustrates, when an oscillating charge is sufficiently relativistic, zero power will be radiated not only at the turning points but also at points close to and on either side of the origin. Since P Figure 2a
Altered Velocity Figure 2b
Altered x(t) Figure 2c
Altered a(t) The results of such a modification are interesting. Fig. 3 depicts the computed radiated power for this new motion. Figure 3
P Of particular note is the result that the total "emitted" radiant energy per cycle for the new motion computes to a negative value: E For such a motion the driving agent hypothetically absorbs energy from the charge’s electromagnetic field every cycle! The motion depicted in Figs. 2a-c is objectionable on two grounds. First, in any real world case we might expect the acceleration to vary smoothly in time. Secondly, the instantaneous changes in the acceleration imply infinite singularities in da/dt and accordingly in P Such a negative power flux would be puzzling, since it must be wondered where the inexhaustible supply of energy would come from. The answer perhaps lies in an aspect of Maxwellian theory often glossed over, namely that no charge exists without an equal, opposite amount of charge somewhere else. As the point charge is driven with the motion depicted in Figs. 2a-c, a "waveform" theoretically propagates away into infinite space. The novel feature of this waveform would be that, on average, A plausibility argument can be suggested for such temporary field energy deficits by considering the case of a charge moving with a constant velocity of magnitude much less than c. The momentum in such a charge’s electromagnetic field is readily shown to equal m One suggested solution to this difficulty is that the energy in the transition zone might actually be The feasibility of such "waves" with net energy back-flow may seem novel at first. Yet the example discussed above practically begs their existence. The whole matter warrants further consideration. |