Question

A long thin non-conducting cylindrical pipe of radius $R$ carrying a uniform surface charge density $\sigma $ is rotating about its axis with a constant angular speed $\omega $ . The energy density of the magnetic field inside the tube is

• A. $\frac{\mu _{0} \sigma ^{2} \omega ^{2} R^{4}}{2}$
• B. $\frac{\mu _{0} \sigma ^{2} \omega ^{2} R^{2}}{2}$
• C. $\frac{\mu _{0} \sigma ^{2} \omega ^{2} R^{2}}{4}$
• D. $\frac{\mu _{0} \sigma ^{2} \omega ^{2} R^{4}}{4}$

Answer 1

Answer: (B)

The rotating cylinder is equivalent to a solenoid with current per unit length $=\sigma \omega R$
$\therefore B=\mu_{0} \sigma \omega R$
$\therefore u=\frac{1}{2 \mu_{0}} B^{2}=\frac{1}{2 \mu_{0}} \mu_{0}^{2} \cdot(\sigma \omega R)^{2}$
$u=\frac{\mu_{0} \sigma^{2} \omega^{2} R^{2}}{2}$