On the Larmor and Abraham-Lorentz Formulas for Radiated Power 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 1. Non-Relativistic P 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 1. Non-Relativistic P P . (1-1) Energy conservation requires that E . (1-2) The field vectors There are two distinct formulas for P . (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
Larmor and Abraham-Lorentz Compared Both the Larmor and Abraham-Lorentz formulas for E 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 P . (1-6a) And according to Abraham-Lorentz, . (1-6b) (Of course sin 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 Figure 1-1a Energy Flux vs. Time,
A=.01c
Figure 1-1b Energy Flux per Cycle vs. Azimuth,
A=.01c
2. Relativistic P The relativistically correct version of Larmor’s P . (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
Larmor and Abraham-Lorentz, Relativistic
A
It is an interesting mathematical fact that, whereas the Larmor and Abraham-Lorentz P Figure 2-2a P
wA=.9999c
A=.9999c
Figure 2-2b P
wA=.9999c
A=.9999c
The spiked form of P Figure 2-3a Energy Flux vs. Time,
A=.9999c
Figure 2-3b Energy Flux per Cycle vs. Azimuth,
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,
Figure 2-4b Energy Flux per Cycle vs. Azimuth,
A=c
3. Interpreting Figs. 2-4a and 2-4b. 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 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 P 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. |