Chapter 30 Problem 19

Problem

For the toroid of Problem 13, determine the energy density in the magnetic field as a function of $r$ ($r_{1}<r<r_{2}$) and integrate this over the volume to obtain the total energy stored in the toroid, which carries a current $I$ in each of its $N$ loops.

Solution

Since the energy density is a function of radius only, we treat the toroid as cylindrical shells each with differential volume $dV=2\pi rhdr$ .

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

$B(2\pi r)=\mu _{0}NI$

$B={\frac {\mu _{0}NI}{(2\pi r)}}$

<math>u_b=\frac{B^2}{2\mu_0}=\frac{1}{2\mu_0}\left( \frac{\mu_0 NI}{(2 \pi r)}\right)^2)=\frac{\mu_0 N^2 I^2 }{8\pi^2 r^2}