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?

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:

#mathcalE = - (d Phi)/(dt)# (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 #Phi# 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:

#mathcalE = - alpha (dx)/(dt) = - alpha v#

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 = mg - alpha v#.

This is a separable DE in form:

#(dv)/(dt) = g - alpha/m v#

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

#v(t) = (mg )/alpha (1 - e^(-alpha/m t))#

And #lim_(t to oo) v(t) = (mg )/alpha#

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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