Lorentz law with moving magnetic field

Question

Trying to understand Lorentz law. If we look at the charge that is moving in stationary magnetic field - the Lorentz force is applied to it, but when we switch to observer that is moving with charge that force still apply as electric field (if I am not mistaken). Does it work with following example (see picture)

Magnet stuck between two ferromagnetic discs. This system rotates around axis shown on the picture. In one spot between discs there is an electric charge that is stationary to observer. Point on disc surface is not inertial frame of reference because acceleration applied. But what if we look for very small rotation angle and huge disc radius when displacement close to linear. Does Lorentz force applied to this charge in this scenario?

Answer 1

For the observer who sees a stationary test charge, there will of course be no Lorentz force (other than to the extent that the test charge is attracted to any polarization that it, itself, induces in the discs; I will neglect this effect for the remainder of the answer).

Suppose the linear velocity of the disc near the position of the test charge is $v$, and we now consider frame that is moving at velocity $v$ with respect to the original frame, so that the linear velocity of the disc near the position of the test charge is zero in this new frame, and the test charge is moving at velocity $-v$. In that case, the test charge will experience both electric and magnetic forces because the Lorentz transformation of the magnetic field from the original frame results in an electromagnetic field that has both electric and magnetic components. The electric and magnetic forces that act on the test charge will cancel each other, and the test charge will retain the constant velocity $-v$.

You might be wondering, in the new frame, how can it be that there is an electric field from the discs? What is producing that electric field, from the point of view of physics in the new frame?

Recall that when a current-carrying wire is viewed from a different inertial frame, an electric field appears because Lorentz contraction gives the wire a net charge density in the new frame. Now think of the magnetic field from a permanent magnet as being produced by tiny current loops. If such a loop moves with a velocity that is not perpendicular to the plane of the loop, Lorentz contraction will result in an electric polarization (i.e. positive charge density in one half of the loop and negative in the other), which results in an electric field. In the end, when modelling the field of an electron in classical electromagnetism, we usually take the limit as the size of the loop goes to zero while keeping the magnetic dipole moment constant. When this limit is taken, the electric dipole moment remains.

(When doing realistic calculations involving multiple reference frames, we don't actually need to think about tiny current loops. We just use the polarization-magnetization tensor. You can write it down in one frame, where it's purely magnetic, and see what the electric components are in another frame.)