Is the Abraham-Lorents force formula in wikipedia correct?

[atyy]

Abraham-Lorentz formula

Maybe the formula is problematic? Parrott suggests there's a problem with the relativistic form of the equation (Appendix 1, http://arxiv.org/abs/gr-qc/9303025). Maybe also try a related discussion by Rohrlich (http://arxiv.org/abs/0804.4614). I've read Parrott's discussion, and it seems reasonable to me, I'm still trying to figure out Rohrlich's.

 

[snoopies622]

Thanks for the references, atyy. I take it that classical EM has no answer to this question?

 

[Bob_for_short]

Thanks for the references, atyy. I take it that classical EM has no answer to this question?

No, at least what is written in textbooks. The AL formula is bad. If at some moment the acceleration is constant, the third derivative makes such a solution unstable. You are bound to obtain a self-accelerating solution.

Mathematically you need three initial constants.

Try to make numerical 1D calculations. You will be obliged to specify the initial acceleration.
If you consider the third derivative by perturbation theory, it means singularly perturbed equation. There is a whole science of it.

Bob.

 

[snoopies622]

So if I ask, "I have a particle of rest mass m₀ and charge q and I want to accelerate it at rate a, what force must I apply to it?", classical electromagnetism cannot answer this question? To me this seems a little like asking about the self-inductance of a long straight wire.

If

<br /> <br /> V = L \frac {di}{dt}<br /> <br />

and

<br /> <br /> \frac {di}{dt} = \rho \frac {dv}{dt}<br /> <br />

where \rho is charge per unit length, then there is an electric potential and therefore a reactive force even if the acceleration is constant, though I don't know if this approach is valid when dealing with a single charged particle.

 

[Bob_for_short]

The problem with one charge is different. We have mechanical equations and wave equations. We know from experience how external fields act on the electron. Then we make two suppositions:

1) The radiated field should be incerted into the particle equation,

2) The electron is pointlike.

The latter supposition (electron model) gives the exact "friction" force expressed only via particle variables. Now the question is if this modelling is good physically and mathematically.

The analysis shows that it is bad. This model is bad.

The model without friction is even better since the losses are small, so most people use just external forces.

The ideas of self-action and pointlikenes of the electron are bad. Please read my articles on these subjects (arxiv:0811.4416 and arxiv:0806.2635).

Vladimir Kalitvianski.

 

[snoopies622]

So if I ask, "I have a particle of rest mass m₀ and charge q and I want to accelerate it at rate a, what force must I apply to it?", classical electromagnetism cannot answer this question?

Perhaps I should ask a more modest question: what are the experimental values? Even if the data can't be explained using classical physics, I would like to know what the relationship looks like. Has an electromagnetic reaction force ever been measured for a charged particle undergoing constant acceleration?

 

[Bob_for_short]

When the electron is non relativistic, its radiative losses are calculated with Lorentz-Abraham formula considered as a small perturbation. In that case the "friction" term is a known function of time (third derivative of non perturbed trajectory) and no mathematical problem arise. This approach is sufficiently accurate since the electron radiated many photons (classical radiation). In QED it corresponds to the inclusive picture (sum over different final photon states) which coincides with the classical results. This is what is done in linear accelerators.

I know that in case of ultra relativistic electron in a magnetic field the electron radiates very energetic photons that spread the electron orbit in an arbitrary way (radiation happens not continuously but by chance). So the beam radial width is determined with quantum rather than classical radiation mechanism.

Bob.

 

[snoopies622]

I'm afraid I don't know what you mean by, "considered as a small perturbation". For constant acceleration, the Abraham-Lorentz formula gives no reaction force at all. Are you saying that the force is so small that it can be ignored? I should note that I am not asking specifically about the case of an elementary particle like an electron, just any object with charge q accelerating at rate a.

 

[Bob_for_short]

I'm afraid I don't know what you mean by, "considered as a small perturbation". For constant acceleration, the Abraham-Lorentz formula gives no reaction force at all. Are you saying that the force is so small that it can be ignored? I should note that I am not asking specifically about the case of an elementary particle like an electron, just any object with charge q accelerating at rate a.

Yes, for non relativistic case is it really small, so zero is as good as non zero but very small correction. For a more massive body (with a smaller ratio charge/mass) the radiative losses are even smaller. That is why the Newtonian mechanics (without radiative "friction") works fine for macroscopic bodies.

Bob.

 

[snoopies622]

Thanks, Bob. I didn't realize we were talking about such small quantities. I just did a quick calculation using the Larmor formula to find how much power would be emitted by one Coulomb of charge accelerating at one meter per second squared. Very small indeed!