= Steady toroidal induction by spherical differential rotation
{title2=$B_\phi=-\frac{B_0\sin\theta\cos\theta}{\eta r^3}\int_0^rx^4\Omega(x)\,dx$}
An axial uniform <magnetic field> and localized radial <differential rotation> generate a steady <toroidal magnetic field>. The angular factor $\sin\theta\cos\theta$ is an eigenfunction of the toroidal angular diffusion operator with eigenvalue $-6$. Its radial amplitude satisfies $\eta(f''+2f'/r-6f/r^2)=-B_0r\Omega'$. Differentiating the integral in the title proves the formula. It is regular at the origin and decays at infinity when $\Omega$ is finite at zero and decays sufficiently rapidly.
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