Flux coupling of a rotating charged sphere to a circular loop (${\Phi }_b={\mu }_{0}m/(2b)$)

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.