# I Forces acting on a magnet which is deflecting a charged particle

#### S_Wildjocks

Summary
Considering a charged particle moving through a magnetic field, what forces does the particle exert on the magnet that is causing it to deflect?
Summary: Considering a charged particle moving through a magnetic field, what forces does the particle exert on the magnet that is causing it to deflect?

Hi all,

probably a dumb question, but what force(s) does a charged particle exert on a magnet as it passes through it's magnetic field and deflects off?

For example, if I have a charged particle, moving at some velocity, and it is deflected through 90deg due to the presence of a magnetic field, what forces does the particle exert on the magnet?

Probably the best way of imagining what I'm asking is to imagine a charged particle travelling through space, and which passes by a bar magnet (which is aligned so that the particle deflects around it) which is also floating in space. What force is exerted on each?

I'm guess that the particle exerts an equal but opposite force to the force due to the magnet and which is exerted on it (F=qv×B=qvBsinθ), however, surely the mass of the particle matters too right? So if I were to look at it in a purely classical sense it would be F=mv^2/r... the radius of deflection will obviously be dependent on the field strength, particle velocity & mass, etc. But what equation should I use?

Which is it? Is it both?... I suspect that neither will give me the correct answer. I'm sure I'm missing lots here, some version on the Lorentz equation?

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#### meopemuk

I'm guess that the particle exerts an equal but opposite force to the force due to the magnet and which is exerted on it (F=qv×B=qvBsinθ),
I think you asked a very good question.

Surely, in Maxwell's theory we can calculate the magnetic field created by the magnet and use Lorentz formula to find out the force acting on the particle. Likewise, we can calculate electric and magnetic fields created by the particle and find out the total force applied to the atoms from which the magnet is made.

However, as far as I can tell, these formulas do not guarantee that the two forces will be equal but opposite. In general, Maxwell's electrodynamics has difficulties in ensuring the conservation of the total momentum or the total angular momentum in systems of interacting charges. Some of these problems were discussed in the literature under the names of "Cullwick paradox" and "Trouton-Noble paradox".

Eugene.

#### S_Wildjocks

Hi Eugene,

thanks for that - that's really interesting, I had a brief look at the two paradoxes (they look like some serious rabbit holes one could lose themselves in!).

I was thinking alright that the charged particle itself will also have both an associated electric and magnetic field. So, am I right in saying that there will be a force acting between the particle and magnet (charged particle moving in magnetic field etc..) but there will also be another force, one created between the interaction of the particle's magnetic field and that of the magnet?

You pointed out that Maxwell's equations have difficulty conserving momentum... so what happens experimentally? Momentum is conserved? So basically if I fire a charged particle into a magnetic field it will deflect without losing (or gaining) velocity, albeit the angular velocity will change?

s.

#### meopemuk

The law of conservation of the total momentum is the basic law of physics, which cannot be violated.

When you fire a charged particle into a magnetic field, then the particle will deflect, the magnet will gain some speed, and some radiation will be created. These things should work in concert so that the total momentum before the collision (particle+magnet) is equal to the total momentum after the collision (particle+magnet+radiation). AFAIK, in Maxwell's theory you cannot prove that this conservation law works in all circumstances. I've seen people introducing so-called "hidden momentum" when they have a trouble with the momentum conservation law in electrodynamics.

Eugene.

#### Ibix

It's easy to prove that Maxwell's equations respect energy and momentum conservation. You write the Lagrangian and show that there are time and space translation symmetries and Noether's theorem does the rest. But it can be very tricky to do the book keeping for a practical situation. So it's not that the equations have trouble with momentum and energy conservation - but the user may well have trouble.

Without actually doing any maths myself: In this case you've got an accelerating charge, so it's very likely that the field is carrying away momentum. Your charge is also moving, and hence has a magnetic field, which means that I'd expect a force on the magnet from that. Actually solving that (which includes worrying about the equations of motion of particle and magnet, since this affects the field) is, I suspect, messy.

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