Chapter 21, Problem 15Q

To determine

To Explain: A bar magnet achieved terminal velocity while falling inside metal tube even though it has been evacuated.

Explanation of Solution

Introduction:According to Faraday's Law, when the magnet moves through a conducting metal tube (or non-magnetic), it will induce an EMF in tube. Also, it creates an opposite-acting magnetic force which is recognised in Lenz's law.

By taking Faraday’s law, $EMF=-N\frac{\Delta {\varphi }_B}{\Delta t}$ . In this equation the minus sign represents direction the induced EMF acts.

The change in flux $\varphi$ is entirely due to the change in positionof the magnet with time, as the strength of the magnet, and the geometry of the magnet and metal tube, are fixed. So,

  $EMF=-\alpha \frac{dx}{dt}=-\alpha v$

It seems practical to argue that back magnetic force is function ofposition $v$ .

From the Newton's Law: $ma=mg-\alpha v$ , which is separable differential equation in form: $\frac{dv}{dt}=g-\frac{\alpha }{m}v$

With $v\left(0\right)=0$ , this has solutions:

  $v\left(t\right)=\frac{mg}{\alpha }\left(1-e^{-\frac{\alpha }{m}t}\right)$ and $\underset{t\to \infty }{\lim }v\left(t\right)=\frac{mg}{\alpha }$