= Flux coupling of a rotating charged sphere to a circular loop
{title2=$\Phi_b=\mu_0m/(2b)$}
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 $\Phi_b=2\pi b A_\varphi(b,\pi/2)=\mu_0m/(2b)$. If its effective source <electric current> is defined by $m=\pi a^2I_{\rm eff}$, 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.
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