Consider a uniformly magnetized sphere of radius $R$, i.e., a sphere that has the same non-zero magnetization $\vec{M}$ throughout.
(a) What are the bound current densities $\vec{J}_b$ and $\vec{K}_b$ inside the sphere and on the surface of the sphere?
Hint: Define a convenient coordinate system and use $\vec{J}_b = \nabla \times \vec{M}$, $\vec{K}_b = \vec{M} \times \hat{n}$.
(b) Consider a hollow rotating spherical shell of radius $R$ with a uniform charge density $\sigma$. Assuming that it is spinning with a constant angular frequency $\omega$ around the z-axis what is the current density inside the sphere and on its surface? Confirm that your answer has the correct units.
(c) The magnetic field inside the spinning sphere from part (b) is given by
$\vec{B}_{in} = \frac{2}{3}\mu_0 \sigma R \omega \hat{z}$.
The magnetic field around the spinning sphere from part (b) is the field of a perfect dipole with a magnetic moment of
$\vec{m} = \frac{4}{3}\pi R^4 \sigma \omega \hat{z}$.
Does this allow you to determine the field inside and outside the uniformly magnetized sphere? If not, explain why not. If yes, determine the magnetic field around the uniformly magnetized sphere.
