Magnetic moment of a rotating charged spherical shell ($\mathbf{m}=Q{a}^2\mathbf{\omega }/3$)

For a uniformly charged spherical shell rotating rigidly in the quasistatic nonrelativistic model, a latitude strip carries current $dI=\sigma \omega a^2\sin \theta d\theta$. Its loop area is $\pi a^2{\sin }^2\theta$, so integration uses ${\int }_0^{\pi }{\sin }^3\theta d\theta =4/3$ and gives $\mathbf{m}=Q{a}^2\mathbf{\omega }/3$. For an isolated conducting shell, $Q=4\pi {ϵ}_{0}aV$. Clockwise and counterclockwise descriptions require a viewing direction; the vector formula fixes the sign without that ambiguity.