Magnetic dipole moment of a rigidly rotating charge distribution

A circular charge distribution rotating with angular speed $\omega$ can be decomposed into current loops. A hoop of line density $\eta$ and radius $R$ has $m=\pi \eta \omega R^3\hat{n}$; a disc of surface density $\sigma$ has $m=\pi \sigma \omega R^4\hat{n}/4$.

A uniformly charged solid sphere of total charge $Q$, radius $R$ and angular velocity $\mathbf{\omega }$ has magnetic dipole momentFor a uniformly charged thin spherical shell the coefficient is $1/3$ instead of $1/5$. At large distances the magnetic field is a magnetic dipole field, $\mathbf{B}={\mu }_0[3(\mathbf{m}\cdot \hat{\mathbf{r}})\hat{\mathbf{r}}-\mathbf{m}]/(4\pi r^3)$. Its angularly averaged magnitude and root mean square magnitude both scale as $r^{-3}$; its angularly averaged vector is zero. Steady rotation of this axisymmetric charge distribution produces magnetostatics, not radiation.

A uniformly charged rotating sphere has an exactly dipolar exterior magnetic vector potential. For a concentric equatorial loop of radius $b$ outside the sphere, Stokes theorem gives ${\Phi }_b=2\pi b{A}_{\phi }(b,\pi /2)={\mu }_{0}m/(2b)$. If its effective source electric current is defined by $m=\pi a^2{I}_{\mathrm{eff}}$, its mutual inductance is ${\mu }_0\pi a^2/(2b)$. The finite-source vector potential is regular through the spanning disk; an exterior point-dipole expression should not be integrated through its singular origin.