Driving Agent Power Expenditure, Oscillating Charge

G.R.Dixon, 2/27/04

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As discussed in numerous other articles on this website, a charged particle that is forced to oscillate exerts two reaction forces on the driving agent: (1) an inertial reaction force, and (2) a radiation reaction force. The driving agent must counteract these two forces in order to maintain the particle’s motion. For example, suppose that a charge’s motion is:

. (1)

The relativistically correct expression for the radiation reaction force is:

. (2)

The part of the total agent force that counteracts this radiation reaction force is:

. (3)

And the rate at which this force does work is:

. (4)

In this article it is shown that, under appropriate circumstances, Prad is spiked in time. By energy conservation the radiant energy is accordingly emitted as pulses (at points in time), rather than over extended intervals of time.

Let us begin by expressing Fx_rad in terms of the charge’s motion. From Eq. 1:

, (5a)

, (5b)

, (5c)

(5d)

(5e)

Thus:

(6)

.

Let:

, (8a)

. (8b)

Fig. 1, whose data points were computed by a simple program, plots Prad vs. time when wA=.001c. Prad varies smoothly in time, with two maxima when the charge is at the origin.

Figure 1

Prad(t),

Prad(t),

Prad(t), wA=.001c

A=.001c

Prad(t),

Prad(t), wA=.001c

A=.001c

A=.001c

A=.001c

Fig. 2 plots Prad(t) when wA=.9c. Note how the agent power expenditure is more compressed around the moments when the particle passes through the origin. Also of interest is the fact that the agent actually does negative work before and after each Prad maximum.

Figure 2

Prad(t),

Prad(t),

Prad(t),

Prad(t), wA=.9c

A=.9c

A=.9c

A=.9c

Figs. 3 and 4 plot Prad(t) for wA=.999999c and .999999999999c respectively. In the most extreme relativistic case the spiking of Prad is virtually complete. The driving agent does virtually all of its work when the particle is at the origin!

Figure 3

Prad(t),

Prad(t),

Prad(t),

Prad(t), wA=.999999c

A=.999999c

A=.999999c

A=.999999c

Figure 4

Prad(t),

Prad(t),

Prad(t),

Prad(t), wA=.999999999999c

A=.999999999999c

The fact that the driving agent does virtually all of its work (in the extremely relativistic case wA~c) when the particle passes through the origin is consistent with the compression of energy flux through a surface surrounding the oscillating charge (as shown in a previous paper). It was also seen in that paper that the energy flux was virtually all along the positive and negative x-axis when wA~c. These two results together indicate that an extremely relativistic, oscillating charge emits its radiation in particulate form. This phenomenon, first suggested by Einstein to explain the photoelectric effect, is not as exotic as first believed; it is a natural consequence of Maxwell’s equations when an oscillating charge’s maximum speed is extremely relativistic.