Question

(a)The magnetic moment of a macroscopic body is defined as $\int_{V} M d \tau .$ Prove the relationship

$$ \int_{V} \vec{M} d \tau=\int_{V} \vec{r} \rho_{M} d \tau+\oint_{S} \vec{r} \sigma_{M} d a $$

where $\mathrm{S}$ is the surface bounding $\mathrm{V}$. [Hint: Refer to the similar problem involving polarization $\vec{P}]$ (b) $\mathrm{A}$ permanent magnet in the shape of a sphere of radius $R$ has uniform magnetization $\vec{M}_{0}$ in the direction of the polar axis. determine teh magnetic moment of the magnet from both the right and left sides of the equation in part (a).