Is There a Logical Difficulty with the Larmor Formula?

[GRDixon]

The non-relativistic Larmor formula for the power radiated by a point charge is proportional to the charge’s acceleration squared. When the charge’s velocity and acceleration are collinear, the radiated power is proportional to gamma^6 times a^2. If the charge has simple harmonic motion, say x = A sin(wt), then the acceleration and radiated power are maximum when the charge is at rest (at x = A and x = -A). But assuming the charge is driven by some force, F, the radiated power should be Fv, which equals zero at the turning points. Does this constitute a logical difficulty with the Larmor formula?

 

[torquil]

An interesting problem. I found some of your writing on the subject, using google. Could the power output at the maximum positions be accounted for by the energy that is released by the disappearing magnetic field around the charge?

However, the magnetic field is proportional to the velocity (at least when moving at a constant velocity), and the energy density therefore prop. to the square of v. The rate of change of this is 0 at the points where this Larmor formula problem occurs, so that's kinda bad. On the other hand, the magnetic field is not really the same as for a charge of constant velocity, so maybe it could work?

Maybe you have already considered this?

Torquil

 

[GRDixon]

Maybe you have already considered this?

Torquil

I haven't, but will now. My thought was that a charge distribution becomes totally unlength-contracted at the turning points, and possibly length-contraction entails stresses, al la Poincare stresses, in the moving charge. This much I have been able to demonstrate: at distances large compared to a distribution's radius, the flux of power through a surrounding surface does suggest that maximum power output occurs at the turning points. It's all rather confusing. Surely some external force must drive the simple harmonic motion! Thanks for your thought-provoking response.

 

[torquil]

Surely some external force must drive the simple harmonic motion!

I would expect this to be an incident electromagnetic field? The electrons movements would subsequently produce this secondary Larmor radiation. My hunch is that the effect is not related to a finite charge distribution, simply because that was not assumed in the derivation of the Larmor formula.

Is the power released by the decompression of the charge distribution dependent on its overall size, for a given total charge Q? And if so, what happens to it when the overall size of the distribution approaches zero?

Torquil

 

[GRDixon]

I would expect this to be an incident electromagnetic field? The electrons movements would subsequently produce this secondary Larmor radiation. My hunch is that the effect is not related to a finite charge distribution, simply because that was not assumed in the derivation of the Larmor formula.

Is the power released by the decompression of the charge distribution dependent on its overall size, for a given total charge Q? And if so, what happens to it when the overall size of the distribution approaches zero?

Torquil

The energy in the compressed charge (or Poincare stresses, or whatever) is an ad hoc attempt to account for an inequality between (a) the energy in the electrostatic field of a spherical shell of charge, and (b) the shell's electromagnetic mass times c^2. Both of these energies depend upon q^2/R. If U is the field energy and m is the electromagnetic mass, then the inequality is U=(3/4)mc^2. As Feynman observes, "This formula was discovered before relativity, and when Einstein and others began to realize that it must always be that U=mc^2, there was great confusion." (from "The FL on Physics", V2, Chap 28.) As you can see, when the overall size approaches zero, U and mc^2 both approach infinity (assuming q is fixed). As far as I know, the idea that the stress energy increases with speed was my own ad hoc contribution. But I'm not convinced that this explains the "Larmor/Fv Disconnect."