# Alternative way of deriving the Abraham Lorentz force equation?

In Griffiths ("Introduction to Electrodynamics"), he motivates the Abraham Lorentz force (i.e. the radiation reaction force) by considering the energy emitted by a particle due to radiation during one cycle of periodic motion. In particular, he assumes that the energy is emitted from the particle due to the expense of the kinetic energy of the particle. But this assumption can't be true in general. In particular, if a particle is instintaneously at rest, but with a non-zero acceleration, then the work done on the particle (and hence the change in KE) is 0 (since the particle doesn't move), but yet it radiates EM energy. I was thinking that a better way of deriving the radiation reaction force would be to consider the total momentum carried away from the particle due to EM radiation. Due to conservation of momentum, the change in the particle's momentum must be equal and opposite to the momentum carried by the EM wave, and using Newton's second law F=dp/dt, you can calculate the force. Does anybody know a derivation of the Abraham Lorentz force using this method? Thanks in advance!

You are correct that the energy put into radiation per unit time (as predicted by Larmor's formula) is not equal to loss of kinetic energy during that time. However, the standard argument is not assuming this strict balance, it is rather using energy balance over a whole period of oscillation.

Your idea about using EM momentum to find the force on charged sphere should work, however there are difficulties with this.

Very similar idea was analyzed after electron was discovered by Lorentz and Abraham for a sphere, and later by Dirac for point particle.

Lorentz and Abraham worked independently but they both assumed a charged sphere and used standard EM theory to calculate force on the charged sphere due to one part acting on another part. Lorentz found it is difficult to get exact expression for the self-force, but using approximations, one can get approximate expression for the force in terms of series involving derivatives of position: first, there is an inertial term proportional to acceleration $$\mathbf a$$ that depends on distribution of charge in the sphere, and then there is a damping term proportional to $$\dot{\mathbf a}$$ that does not depend on the distribution, only on total charge. The latter term is the Lorentz-Abraham force on charged sphere. Other terms are there too but are usually ignored.

Later Dirac tried to find effect of radiated EM field on the motion of a point charged particle with no structure in the way you suggest, and arrived at the same Lorentz-Abraham force expression (but in addition found its relativistic generalization) [1].

However, Dirac's analysis for point particle is based on a very important and in my opinion incorrect assumption, that he explicitly states in a footnote:

The usual derivation of the stress-tensor is valid only for continuous charge distributions and we are here using it for point charges. This involves adopting as a fundamental assumption the point of view that energy and momentum are localized in the field in accordance with Maxwell's and Poynting's ideas.

This assumption is bad because it introduces infinite EM energy and undefined work term into the theory. This has caused many confused people to write many papers and books with little to no progress.

It is ironic that Dirac himself earlier in the paper quotes Frenkel's paper from 1925 where alternative infinity-free stress-tensor for EM field of point particles is proposed and results in consistent theory. There is no self-force, radiation reaction and related puzzles regarding point particles in Frenkel's variant of point-particle theory. From that point of view, radiation reaction observed in macroscopic theory as antenna radiation resistance is just result of many multi-particle interactions.

[1] Dirac P. Classical theory of radiating electrons (1938), Proc. Roy. Soc A, 167(929), 148-169. https://doi.org/10.1098/rspa.1938.0124

[2] Frenkel J., Zur Elektrodynamik punktfoermiger Elektronen (1925), Zeits. f. Phys., 32, p. 518-534. http://dx.doi.org/10.1007/BF01331692

So if you want to go forward and analyze momentum of EM field to find force on a charged particle, you need to 1) postulate some stress-tensor and 2) work back consequences to the equation of motion, similarly to what Dirac did. However, for point particle and standard Maxwell-Poynting tensor this leads to contradictions due to infinities, so one has to make a change somewhere to make this worthwhile: either work with extended charged distributions, or change the stress-energy tensor to something that does not result in infinities.