On Negative Radiated Power

G.R.Dixon, 10/02/2004

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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 Fradreact, Eq. 3 is independent of the shell’s radius. In particular it applies to a point charge (the limit of a spherical shell with zero radius).

If a point charge has the motion

, (4)

then Eq. 3 becomes


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 –Fradreactv:


Using values of q=1 coul, A=1m, and wA=.95c, Fig. 1 plots this Prad over the range 0<t<2p/w.

Figure 1



Prad(t), wA=.95c


As in the non-relativistic formula for Prad, maximum radiated power occurs at times t=0, p/w and 2p/w ... moments when the charge passes through the origin. Also as in the non-relativistic case, no radiated power is generated at times t=p/2w and 3p/2w, when the charge is at +A (where Fv=0). The interesting feature in the relativistic case is the intervals of time during which the radiated power is actually negative. Examination of the figure’s data file indicates that the radiated power is greater than zero only where Prad spikes. For most of the time Prad is negative.

The energy radiated per cycle is

. (7)

For the power plotted in Fig. 1, numerical integration indicates that

Erad = 1.71 x 1011 joules. (8)

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) Prad will be zero whenever

. (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 Prad depends only upon v, a and da/dt at any given instant, we might try changing Eq. 4 in such a way that, during the times when Prad is positive in Fig. 1, v is altered to constantly equal its value when Prad goes negative (so that a and da/dt become zero whenever Prad is positive in the figure). Fig. 2a depicts the altered velocity. Figs. 2b and c plot x(t) and a(t) respectively.

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

Prad(t), Modified Motion

Of particular note is the result that the total "emitted" radiant energy per cycle for the new motion computes to a negative value:

Erad = -9.17 x 1010 joules. (11)

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 Prad (Eq. 9). Notwithstanding such objections, one cannot help but wonder if there is some more realistic motion that produces a net positive amount of work per cycle being done on the driving agent as said agent counteracts the radiation reaction force. In such an event energy conservation suggests that the net energy flux per cycle through a surrounding surface would be negative!

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, energy fluxes in a direction opposite to the wave’s propagation! Eventually the wave would reach the complementary charge (wherever it may be) and would theoretically cause it to emit precisely the amount of radiant energy per cycle as is absorbed by the driving agent in Fig. 3. In effect, when the point charge is driven as depicted in Fig. 2a, the driving agent would hypothetically "borrow energy" from the net field of the driven charge and its opposite sign complement, said complement re-depositing the energy at some later time. It should be noted that engineering a system where a "point" charge can be driven as depicted in Figs. 2a-c (or some variation where da/dt is always finite) may be challenging. Among other things, the assumption is that the charge is driven by a "contact" (non-electromagnetic) force, somewhat like a charged pith ball being driven by an applied force.

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 memv, where mem is the charge’s electromagnetic mass. Similarly, the energy in the magnetic field is demonstrably memv2/2. Let us now suppose that a decelerating force is brought to bear at time t1, causing the charge to come to rest at time t2. At this time the force is withdrawn. And let us suppose that the amount of work done on the decelerating agent, during the time interval t1<t<t2 all but equals the energy in the magnetic field prior to time t1. (Some relatively small amount of positive work may have to be done to produce a small amount of radiation, but this can be minimized by keeping the applied force constant most of the time.) At time t3>t2 we can recognize three zones in all of space: (1) An "inner" zone where there is only an electrostatic field; (2) A "transitional" zone, where the electric and magnetic fields depend upon the nature of the acceleration that occurred in the time frame t1<t<t2; and (3) An "outer" zone, all the way out to infinity, where the fields are still those of a charge moving with constant velocity. This being the case, we have a situation where the driving agent has absorbed practically all of the initial energy in the magnetic field by time t2; yet there is still magnetic field energy in the outer zone all the way out to infinity.

One suggested solution to this difficulty is that the energy in the transition zone might actually be less than the electrostatic field energy that will eventually exist there (once the "wave" has passed on out into infinite space). And in the process of rising to the final electrostatic field energy, energy must presumably flow into the transition zone from the outer zone. In brief, the "wave" propagates outward, but (on average) energy fluxes inward.

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.