Chapter 30 Problem 13

Problem

A toroid has a rectangular cross section. Show that the self-inductance is

$L={\frac {\mu _{0}N^{2}h}{2\pi }}\ln {\frac {r_{2}}{r_{1}}}$

where $N$ is the total number of turns and $r_{1}$ , $r_{2}$ , and $h$ are shown above. [Hint: Use Ampère’s law to get $B$ as a function of $r$ inside the toroid, and integrate.]

Solution

$\oint {\vec {B}}.d{\vec {l}}=\mu _{0}I_{\textrm {encl}}$