Question

Consider a toroid, having a circular cross-section of radius $b$ , major radius $R \, \left(R > > b\right)$ , having $N$ turns and carrying current $I$ . Find the total energy stored in the toroid.

• A. $\frac{\mu _{0} N^{2} I^{2} b^{2}}{2 R}$
• B. $\frac{\mu _{0} N^{2} I^{2} b^{2}}{3 R}$
• C. $\frac{\mu _{0} N^{2} I^{2} b^{2}}{6 R}$
• D. $\frac{\mu _{0} N^{2} I^{2} b^{2}}{4 R}$

Answer 1

Answer: (D)

The magnetic field inside a toroid is,
$B=\frac{\mu _{0} N i}{2 \pi R}$
Flux linked with the toroid,
$\phi=\pi b^{2}\times B\times N$
Also, flux is related to self-inductance as,
$\phi=Li$
$L=\frac{\phi}{i}=\frac{\mu _{0} N^{2} b^{2}}{2 R}$ with $b < < < R$
Energy $=\frac{1}{2}Li^{2}=\frac{\mu _{0} N^{2} I^{2} b^{2}}{4 R}$