Question

Consider a toroid of circular cross-section of radius $b$ , major radius $R$ much greater than minor radius $b$ , (see diagram) find the total energy stored in magnetic field of toroid -

• A. $\frac{B^2{\pi }^2{b}^{2}R}{2{\mu }_0}$
• B. $\frac{B^2{\pi }^2{b}^{2}R}{4{\mu }_0}$
• C. $\frac{B^2{\pi }^2{b}^{2}R}{8{\mu }_0}$
• D. $\frac{B^2{\pi }^2{b}^{2}R}{{\mu }_0}$

Answer 1

Answer: (D)

$B=\frac{{\mu }_{0}Ni}{2R}$
$\varphi =\pi b^2\times B\times N$
$\varphi =Li$
$L=\frac{\varphi }{i}=\frac{{\mu }_0{N}^2{b}^2}{2R}$ , with $b<<<R$
$E=\frac{1}{2}L{i}^2=\frac{{\mu }_0{N}^2{i}^2}{4R}{b}^2$