# Motion between magnet and charged particle

A moving charge is affected by a magnet i.e it experiences a force.

If the magnet is moving and the charge is stationary then here should be no force on the charged particle. However, from the frame of reference of the magnet the charge is moving, and it should experience a force.

But how is it possible that there is a force in one frame and no force in the other? Force is not a relative quantity and each observer should agree on the presence of the force, shouldn't they?

• This is similar to the question that I asked here. – Andrei Geanta Nov 2 '17 at 14:56

If the magnet is moving then its magnetic field is changing in time, and Faraday's law of induction, $$\nabla \times \mathbf E = -\frac{\partial \mathbf B}{\partial t},$$ tells you that this requires an electric field to be present. This electric field then exerts a force on the charge, regardless of whether it is moving or stationary, and causes it to accelerate.

My answer refers to the case, that the external field is homogeneous over some area and you move your magnet inside this area. Otherwise see Emilio’s answer.

If the magnet is moving and the charge is stationary then here should be no force on the charged particle...

... is a right statement. Furthermore there is a second surprising case. A homopolar generator (Faraday disc) shows the next behavior:

If the magnetic field is provided by a permanent magnet, the generator works regardless of whether the magnet is fixed to the stator or rotates with the disc.

If you understand how the electron in an external magnetic field gets deflected Lorentz force then you are able to explain both phenomena.

An electron obey a magnetic dipole moment. This intrinsic magnet gets aligned in an external magnetic field. Now, if the electron moves the alignment is accompanied by a deflection due to the gyroscopic effect. Since the deflection is an acceleration the electron radiates. This emission of a photon disalign the tiny magnet again and this cycle repeats until the electron comes to rest in the center of its spiral path.

For your case, the electrons magnetic dipole moment gets aligned, but since it is not moving, no deflection occurs. For the case the magnet rotates together with the disc, the electron moves in circles and is accelerated, emits photons, gets aligned by the magnetic field again and the electron moves outwards or inwards (depends from the direction of rotation and the direction of the magnetic field) despite the magnetic field is in rest to the disc.

• This is incorrect. For one, the magnetic dipole moment is irrelevant for these dynamics (you would get the same apparent paradox for a hypothetical charged particle with no magnetic dipole moment). More importantly, the homogeneity (or not) of the magnetic field is irrelevant: if there is a force in the magnet's frame, then there is a force in the particle's frame, and therefore there is an electric field in the latter, regardless of whether the magnetic field is homogeneous or not. A homogeneous $B$ means that $E$ will be locally conservative, not that it will be zero. – Emilio Pisanty Nov 2 '17 at 15:57
• @EmilioPisanty For one, a charged subatomic particle without magnetic dipole does not exist, but an atom without magnetic dipole moment in sum gets not deflected. For second, we had a discussion here about would the neutron - with its magnetic moment, but without charge - be deflected or not. In my understanding, it will. For third, how you explane the homopolar generator? And for fourthly, what are the inconsistencies in my answer (if not to use the argument that this is not teached at the moment)? – HolgerFiedler Nov 2 '17 at 18:38
• In this context, the existence or not of charged particles with zero magnetic dipole moment is irrelevant; EM theory needs to be consistent for that case whether they exist or not. (As a matter of fact, they do exist, though - alpha particles have zero angular momentum, so by the Wigner-Eckart theorem they're forbidden from having a nonzero dipole moment or indeed any directional quantity.) – Emilio Pisanty Nov 2 '17 at 19:00
• And as for your answer, it is incorrect to assume that the behaviour in the middle of a region with homogeneous $B$ is independent of the behaviour at the edges. (More importantly, your answer is inconsistent with special relativity, but I guess that's what happens to answers that willfully ignore correct material that's already been provided.) I don't understand why you think the homopolar motor is relevant, either, but you have a long history of posting irrelevant material and at some point I stopped asking why. – Emilio Pisanty Nov 2 '17 at 19:09
• I do have a (personal) request, though: If you don't understand the material, ask, don't post incorrect answers whose only effect is creating misconceptions that harm your readers and that other people will need to fix later on. Posts like this one only create more work for people that could be doing more useful stuff, and they provide negative value to this site. – Emilio Pisanty Nov 2 '17 at 19:11