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:

mathcalE = - (d Phi)/(dt)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 PhiΦ is entirely due to the change in position vv 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 vE=αdxdt=αv

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

ma = mg - alpha vma=mgαv.

This is a separable DE in form:

(dv)/(dt) = g - alpha/m vdvdt=gαmv

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

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

And lim_(t to oo) v(t) = (mg )/alphalimtv(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

() .