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 $=\sigma \omega R$
$\therefore B={\mu }_0\sigma \omega R$
$\therefore u=\frac{1}{2{\mu }_0}{B}^2=\frac{1}{2{\mu }_0}{\mu }_0^2\cdot (\sigma \omega R{)}^2$
$u=\frac{{\mu }_0{\sigma }^2{\omega }^2{R}^2}{2}$