Understanding the Abraham-Lorentz Formula and its Paradox Explained by Griffiths

[Silviu]

Hello! Griffiths is deriving the Abraham-Lorentz formula, which calculates the force that the radiation puts on a charged, accelerating particle (i.e. the force that makes it harder to accelerate a charged particle than a neutral one). For the non-relativistic case, the formula is: $$F_{rad}=\frac{\mu_0 q^2}{6 \pi c}\dot{a}$$ He then mentions a paradox based on the fact that if the particle is subject to no external forces, based on Newton's second law you obtain: $$F_{rad}=\frac{\mu_0 q^2}{6 \pi c}\dot{a}=ma$$ and if you solve for $a$, you obtain that the acceleration increases spontaneously with time (then he talks about certain attempts to solve this issue). I am a bit confused about what is the actual issue. If I understand the formula well, this $F_{rad}$ appears when you have a change in the acceleration. So it is the change in the acceleration that creates the force, not the force that creates the change in the acceleration. However, he assumes that you have no external forces, i.e. the particle moves at constant speed (or stands still). So with no external forces the acceleration would remain the same (which is zero). So the equation would reduce to $$F_{rad}=ma=0$$ He indeed mentions that if we ASSUME that the acceleration is zero we can avoid this problem (and have another one instead), but I am not sure why would we ASSUME it is zero. Isn't it obvious it is zero, as you have no external force, so nothing to change the acceleration? Thank you!

 

[jedishrfu]

The Wikipedia article talks about some of the oddities of this formula:

https://en.wikipedia.org/wiki/Abraham–Lorentz_force

where they mention some pathological solutions of a particle being affected by the force before it generates it called pre-acceleration solutions.

 

[MisterX]

Hello, the Abraham-Lorentz formula is invalid as an instantaneous expression for force. It can only be applied in a time averaged sense, to get the average drag of a particle that undergoes periodic motion.

There are various attempts to try and reconcile it and other classical expressions for radiation reaction force with reality, such as the assumptions about initial conditions that you described, but I have found none of them satisfactory. My experience is a number of sources too hastily claim they have resolved this issue, and present something flawed, but there may be some worthwhile things to be found by digging deeper into the literature.

This issue of self force or radiation reaction force has been written about and studied in depth, but is not necessarily settled.

If you read Griffith's own papers on this subject or related subjects you may come out only even more confused. They may be worth looking at to appreciate the unresolved aspects of classical E&M. Take this one for example, The Fields of a Charged Particle in Hyperbolic Motion.

I think Feynman and Schwinger's thoughts on the subject to be worth examining. Feynman's work as a graduate student was on a related subject and he continued to tinker with it, and made some interesting comments on it. Schwinger wrote about synchrotron radiation in his Electrodynamics book.

Part of why this issue hasn't been settled is that it is the force is only a significant factor for devices we have built that involve periodic motion.
Until recently, at least, the amount of $F_{rad}$ for a linear accelerator was small enough that its exact details could be ignored.

So I reiterate this is one of the unresolved aspects of classical E&M and is intimately tied with others such as "does a uniformly accelerated particle radiate?" and "Does a particle radiate in gravitational free fall?"

As far as I know the answer is "we don't know."

 

[romsofia]

I think you're better off reading Jackson for this kind of issues. I believe chapter 16 refers to self-force, so you should be able to find a solution there.