Question

Solid uniform conducting sphere of mass ${}^{\prime }{m}^{\prime }$ and charge $Q$ rotates about its axis of symmetry with constant angular velocity ${}^{\prime }{\omega }^{\prime },$ then the ratio of magnetic moment to the moment of inertia of the sphere is $\frac{xQ\omega }{6m},$ then $x$ is (Neglect induced charges due to centrifugal force)

Answer 1

Solid sphere is conducting sphere
$\Rightarrow$ mass is distributed in the whole volume, but charge would be distributed on the surface only.

$\Rightarrow$ magnetic moment of follow shell of mass $M$, radius $R$ and charge $Q$

$\frac{\mu }{L}=\frac{Q}{2M}$

$\Rightarrow \mu =\frac{Q}{2M}\times \frac{2}{3}M{R}^2\omega$

$\Rightarrow \mu =\frac{Q{R}^2\omega }{3}$

given solid sphere of mass $M$, and charge $Q$ will have

$I=\frac{2}{5}M{R}^2$

$\frac{\mu }{I}=\frac{\frac{Q}{3}{R}^2\omega }{\frac{2}{5}M{R}^2}=\frac{5Q\omega }{6m}$

$\Rightarrow x=5$