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. $\frac{{\mu }_0{N}^2{I}^2{b}^2}{2R}$
• B. $\frac{{\mu }_0{N}^2{I}^2{b}^2}{3R}$
• C. $\frac{{\mu }_0{N}^2{I}^2{b}^2}{6R}$
• D. $\frac{{\mu }_0{N}^2{I}^2{b}^2}{4R}$

Answer 1

Answer: (D)

The magnetic field inside a toroid is,
$B=\frac{{\mu }_{0}Ni}{2\pi R}$
Flux linked with the toroid,
$\varphi =\pi b^2\times B\times N$
Also, flux is related to self-inductance as,
$\varphi =Li$
$L=\frac{\varphi }{i}=\frac{{\mu }_0{N}^2{b}^2}{2R}$ with $b<<<R$
Energy $=\frac{1}{2}L{i}^2=\frac{{\mu }_0{N}^2{I}^2{b}^2}{4R}$