On the Larmor and Abraham-Lorentz Formulas for Radiated Power

G.R.Dixon, 6/21/2004

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Software note: The data for all of the figures and tables in this article were generated by Visual Basic programs, which can be viewed by clicking on this link.

1. Non-Relativistic Prad’s.

G.R.Dixon, 6/21/2004

Software note: The data for all of the figures and tables in this article were generated by Visual Basic programs, which can be viewed by clicking on this link.

1. Non-Relativistic Prad’s.

Prad is defined to be that part of the power, expended by an agent that drives an accelerating charge, which equates to the rate at which radiant energy is generated. If the resulting motion is periodic, then Erad is defined to be the expended work per cycle:

. (1-1)

Energy conservation requires that Erad equate to the field energy flux per cycle through some enclosing surface. Specifically, if S=eooc2EXB is the Poynting vector at points on such a surface, then energy conservation requires that

. (1-2)

The field vectors E and B can be computed at points on the enclosing surface by using the general point charge field solutions, which are relativistically rigorous.

There are two distinct formulas for Prad, one advanced by Larmor and the other derived from the radiation reaction force suggested by Abraham and Lorentz. According to Larmor,

. (1-3)

And, assuming the driving agent must counteract the radiation reaction force of Abraham-Lorentz in order to sustain the periodic motion, according to A-L,

. (1-4)

Given a motion of

, (1-5)

a simple program demonstrates that Eqs. 1-3 and 1-4 both satisfy Eq. 1-2 when wA is non-relativistic. Table 1-1 lists the results for a few cases, some non-relativistic and others relativistic. A charge of q=1 coul and an amplitude of 1 meter was modeled. The surrounding surface was a sphere, centered on the origin and with radius R=10l=20pc/w.

Table 1-1

wA (units of c)

Erad (Larmor) (joules)

Erad(Ab-Lorentz) (joules)

Energy Flux per Cycle (joules)

.01

18832

18832

18834

.05

2354049

2354049

2358474

.1

18832392

18832392

18975062

.9

1.37288E10

1.37288E10

6.50642E10

.9999

1.88267E10

1.88267E10

8.96493E11

Larmor and Abraham-Lorentz Compared

Both the Larmor and Abraham-Lorentz formulas for Erad produce results approximately equal to the computed energy flux per cycle through the surrounding surface when wA<<c. However, there are clear discrepancies in the relativistic cases, and these discrepancies cannot be attributed to numerical errors. Since the energy fluxes, which are derived from the relativistically rigorous point charge solutions, are correct in every case, relativistic adjustments must be made to both the Larmor and Abraham-Lorentz Prad formulas.

It is noteworthy that, given the oscillatory motion specified by Eq. 1-5, the maxima of the Larmor and Abraham-Lorentz non-relativistic formulas for Prad are p/2 out of phase. According to Larmor,

. (1-6a)

And according to Abraham-Lorentz,

. (1-6b)

(Of course sin2(wt) and cos2(wt) integrate to the same value over one cycle time.)

It will prove instructive to compute and plot (a) the energy flux through the surrounding surface vs. time, and (b) the net energy flux per cycle vs. azimuth angle q. Figs. 1-1a and 1-1b plot the results for wA=.01c. In this non-relativistic case the energy is radiated continuously in time and space, with most of the energy flux occurring perpendicular to the x-axis. The period of oscillation is t=2.1E-6 seconds. Since the sphere’s radius is an integral number of wavelengths (10l), maximum flux through the surrounding sphere evidently occurs when the charge is at +A … a result that seems to favor the Larmor formula.

Figure 1-1a

Energy Flux vs. Time,

Energy Flux vs. Time, wA=.01c

A=.01c

Figure 1-1b

Energy Flux per Cycle vs. Azimuth,

Energy Flux per Cycle vs. Azimuth, wA=.01c

A=.01c

2. Relativistic Prad’s.

The relativistically correct version of Larmor’s Prad is (for the one-dimensional motion specified by Eq. 1-5):

. (2-1)

In the case of Abraham-Lorentz,

. (2-2)

As discussed in another article, Eq. 2-2 has the interesting consequence that a charge, subject to a constant force, accelerates but does not radiate … a result at odds with Larmor (Eq. 2-1). More importantly for present purposes, Eq. 2-2 satisfies conservation of energy when wA is relativistic, as does Eq. 2-1. Table 2-1 repeats Table 1-1, but for a highly relativistic value of wA. Note the close agreement with the computed energy flux per cycle through a surrounding surface. Only one value of wA is shown because of the large number of iterations that must be made in order to reduce the magnitude of numerical errors.

Table 2-1

wA (units of c)

Erad (Larmor)

Erad(Ab-Lorentz)

Energy Flux per Cycle

.99

1.72447E12

1.72447E12

1.69569E12

Larmor and Abraham-Lorentz, Relativistic

Larmor and Abraham-Lorentz, Relativistic wA

A

It is an interesting mathematical fact that, whereas the Larmor and Abraham-Lorentz Prad maxima are p/2 out of phase in non-relativistic cases, they are in phase in highly relativistic cases. Figs. 2-1a and 2-1b illustrate. A value of wA=.9999c was used to calculate the two Prad’s.

Figure 2-2a

Prad(Larmor) vs. t,

Prad(Larmor) vs. t, wA=.9999c

A=.9999c

Figure 2-2b

Prad(Abraham-Lorentz) vs. t,

Prad(Abraham-Lorentz) vs. t, wA=.9999c

A=.9999c

The spiked form of Prad in Figs. 2-2a and b imply that radiant energy is not emitted as continuously in space and time as is the case when wA<<c. The same program that produced Figs. 1-1a and b can be employed to (a) compute the energy flux through the enclosing sphere as a function of time, and (b) compute the net energy flux per cycle as a function of azimuth angle q when wA=.9999c. Figs. 2-3a and 2-3b plot the results. As expected from the spiked Prad’s in Figs. 2-1a and b, practically all of the radiant energy fluxes through the enclosing sphere at two instants in time. And as Fig. 2-3b illustrates, the energy is compressed along the positive and negative x-axis when wA~c.

Figure 2-3a

Energy Flux vs. Time,

Energy Flux vs. Time, wA=.9999c

A=.9999c

Figure 2-3b

Energy Flux per Cycle vs. Azimuth,

Energy Flux per Cycle vs. Azimuth, wA=.9999c

A=.9999c

It is feasible (if controversial) to compute the functions plotted in Figs. 2-3a and 2-3b, with wA=c. An admittedly ad hoc justification for doing so is that, for the oscillatory motion defined by Eq. 1-5, the particle would have a speed of c only for an instant. In any case, the results are interesting enough to include. Figs. 2-4a and 2-4b repeat Figs. 2-3a and b, but with wA=c.

Figure 2-4a

Energy Flux vs. Time,

Energy Flux vs. Time, wA=c

Figure 2-4b

Energy Flux per Cycle vs. Azimuth,

Energy Flux per Cycle vs. Azimuth, wA=c

3. Interpreting Figs. 2-4a and 2-4b.

A=c

3. Interpreting Figs. 2-4a and 2-4b.

It is clear in Figs. 2-4a and 2-4b that, when wA equals the speed of light, then the radiant energy is quasi-corpuscular. In Fig. 2-4a the energy is radiated at nearly discrete instants in time. And in Fig. 2-4b the energy is compressed almost entirely to the x-axis. (The energy is of course radiated in very small solid angles around the x-axis; q is proportional to the magnitude of those angles.) These results are direct consequences of the point charge field solutions (which are presumably correct).

In this case the period of the motion is t=2.1E-8 seconds. In Figs. 2-2a and/or b, the spikes at approximately 0, t/2 … indicate that the pulses are emitted each time the charge passes through the origin (quite as is the case for the Abraham-Lorentz Prad when wA<<c). The spikes in Fig. 2-4a indicate that the pulses also penetrate the surrounding spherical surface each time the charge passes through the origin. Since the sphere’s radius is an integral number of wavelengths (10l), the pulses evidently propagate away from the emission site at speed c (as expected).

The result that highly relativistic oscillating charges emit negligible energy perpendicular to their paths of oscillation is equally interesting and "unclassical." An excellent thesis project might be to demonstrate that such relativistic re-directions of radiated energy lie at the root of phenomena like synchrotron radiation. The matter will not be pursued in this article.

Suffice it to say that the quasi-corpuscular nature of radiation, when wA~c, might have surprised Maxwell himself. We can only speculate what conclusions he might have drawn, had he known the relativistically correct point charge field solutions and been equipped with a modern PC.