A bar magnet falling inside a vertical metal tube reaches a terminal velocity even if the tube is evacuated so that there is no air resistance. Explain?

1 Answer
May 9, 2017

As the magnet moves through a (non-magnetic) conducting metal tube, it will induce an EMF in the tube according to Faraday's Law. That creates an opposite-acting magnetic force that is recognised in Lenz's law .

If we take Faraday's Law:

E=dΦdt (the negative sign recognises that the EMF acts to oppose the motion creating it, ie Lenz's law)

...then, we might suggest that the change in flux Φ is entirely due to the change in position v of the magnet with time, as the strength of the magnet, and the geometry of the magnet and metal tube, are fixed.

On that basis:

E=αdxdt=αv

It also seems reasonable to argue that the back magnetic force is therefore a function of v. From there Newton's Law tells us that:

ma=mgαv.

This is a separable DE in form:

dvdt=gαmv

With v(0)=0, this has solution:

v(t)=mgα(1eαmt)

And limtv(t)=mgα

That's very much a first stab but it suggest that there will be a trade off and that a terminal velocity exists.

Here's a really cool YouTube Vid on it

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