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

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

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

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

$u_{b}={\frac {B^{2}}{2\mu _{0}}}={\frac {1}{2\mu _{0}}}\left({\frac {\mu _{0}NI}{2\pi r}}\right)^{2}={\frac {\mu _{0}N^{2}I^{2}}{8\pi ^{2}r^{2}}}$

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

$U=\int u_{b}dV=\int _{r_{1}}^{r_{2}}{\frac {\mu _{0}N^{2}I^{2}}{8\pi ^{2}r^{2}}}2\pi rhdr={\frac {\mu _{0}N^{2}I^{2}h}{4\pi }}\int _{r_{1}}^{r_{2}}{\frac {dr}{r}}={\frac {\mu _{0}N^{2}I^{2}h}{4\pi }}\ln \left({\frac {r_{1}}{r_{2}}}\right)$