Question about applying Maxwell's Equations to analyze a copper shaft and magnetic bearing

Question

fellow physics people I have a question regarding the mathematics behind how a magnet will slow down as it falls through a copper pipe. I am aware of the mechanism in which this works, the falling magnet results in a changing magnetic flux which induces eddy currents in the copper pipe which creates an opposing magnetic field. However, I am not exactly sure how Maxwell's equations can be used to show the relationship between the speed of the falling magnet and the magnitude of the opposing magnetic field.

Similarly, I was wondering if it is possible to set the experiment up slightly differently and get the same result. Imagine if instead of a copper pipe we have a copper shaft and a donut-shaped magnet. If we insert the shaft through the hole in the magnet and drop the magnet will the magnet's motions be slowed down? if so can we write down equations to describe this system?

I really would appreciate any help with setting up the equations or even a setting up of how the free body diagrams might differ in these two situations!

Answer 1

However, I am not exactly sure how Maxwell's equations can be used to show the relationship between the speed of the falling magnet and the magnitude of the opposing magnetic field

Well, basic principle must be simple. Lorentz force on electrons due to moving copper pipe is : $$ \mathbf {F} = q\, \left(\mathbf {v} + \mathbf {V}\right) \times \mathbf {B} $$, Where $\mathbf {v}$ is electron relative speed in metal sheet and $\mathbf {V}$ - copper pipe movement speed relative to magnet. So the more you tug magnet in a metal pipe,- the greater Lorentz force you induce and, consequently opposing magnetic field. The rest details I leave to you to find out.