# Lorentz Force

1. Oct 3, 2012

### PLuz

Hi everyone,

I was wondering, does the Lorentz force can be used to get the self force of a particle over it self?

I'm in doubt because the fields that one uses to compute the force, in the lorentz force expression, are the external fields but one can compute the Liénard-Wiechert potetials and get the fields created by the particle itself. Can I then use those fields to compute de self-force using the Lorentz force expression?

Thank you

Last edited: Oct 3, 2012
2. Oct 3, 2012

### Jolb

If I understand what you're asking, then the answer is no, you can't use the electron's field to find the "self force."

Strictly speaking the Lorentz force law and Maxwell's equations are incompatible with one another. If m is the mass of the electron and q is its charge, then ma=q(E+v×B) would imply that the electron accelerates without losing any energy to radiation. However, according to the Larmor formula, this is incorrect since any accelerating charge must radiate power according to

$$P = \frac{2}{3}\frac{q^2|\vec{a}|^2}{c^3}$$

If we solve for the force on the electron due to its Larmor radiation, we get
$$\mathbf{F}_\mathrm{rad} = \frac{\mu_0 q^2}{6 \pi c} \mathbf{\dot{a}} = \frac{ q^2}{6 \pi \epsilon_0 c^3} \mathbf{\dot{a}}$$
This is called the Abraham-Lorentz force.

This represents the first-order correction to the Lorentz force law. If we go back and put this correction into the Lorentz force law, we again find that we still need to correct for the Larmor radiation due to the new acceleration! There is an infinite regress of corrections needed for the Lorentz force law.

In addition, looking at the formula for the Abraham-Lorentz force, you can solve it and you would find that it admits solutions where the electron spontaneously accelerates off to infinity (infinite velocity). Clearly this is nonsense.

Both of the issues I mentioned above are examples of the electromagnetic theory of fields getting muddled by the introduction of masses. Whenever there's a mass involved in an electromagnetism problem, you run into lots of paradoxical situations like the one above, and the problems usually have to do with self interactions and infinities lying around. One basic question is: what is the energy of an electron? If it is a point particle, then its energy is infinite, and it could use some of this infinite energy to do things like, say, spontaneously accelerating off to infinity. If you ask a professor how these issues get resolved, the usual answer is "Things are better in quantum field theory." But, again, quantum field theory has its own problems.

3. Oct 4, 2012

### ShayanJ

Thanks Jolb,you mentioned interesting things
But I think even in classical theory of electromagnetism,there is a solution(if you take into account SR)
One may ask that how much energy is needed to assemble a globe of charge with radius R and charge q.Then he says q=e and the energy,is the rest energy of electron and from there,you can find a radius for electron which is called classical electron radius.
And in QFT,the answer is renormalization.You may say that's just erasing the problem but I read somewhere there is a physical justification for it for which Kenneth Wilson has won the nobel prize in 1982
Although I should say that self-energy and self-interaction is still a little mysterious to me

4. Oct 4, 2012

### vanhees71

The problem of the self-energy of point particles is not yet really solved within classical electrodynamics. You find a nice elementary introduction (as far as one can speak of "elementary" with regard to an unsolved problem at all ;-)) in The Feynman Lectures, vol. II. A much more detailed exposition is, of course, given in J. D. Jackson, Classical Electrodynamics.

The state of the art is given in

F. Rohrlich, Rohrlich, F. Classical Charged Particles, World Scientific (2007).

5. Oct 4, 2012

### dextercioby

Great post by Jolb.

6. Oct 4, 2012

### PLuz

Thanks Jolb that was exactly what I was looking for.

I've read Griffiths, Jackson and the very nice review by Eric Poisson... I find the explanation of the preacceleration problem being solved within the quantum mechanics point of view just a way to hide the problem, acausal interactions are still there... As Griffiths sais: "it is (to my mind) philosophically repugnant that the theory should countenance it at all"...

Thank you everyone.