Question

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?

Answer 1

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

() .