= Magnetic-energy injection by differential boundary rotation
{title2=$\dot E_B=\frac{2\pi}{\mu_0}\int f(\psi)(\Omega_{\rm out}-\Omega_{\rm in})\,d\psi$}
The general ideal magnetic-energy balance has outward-surface contribution $\mu_0^{-1}\oint[(u\cdot B)B-B^2u]\cdot dS$ and volume contribution $-\int u\cdot(J\times B)dV$. The latter vanishes for a <force-free magnetic field>. For an axisymmetric boundary rotating tangentially, $u\cdot B=\Omega f(\psi)$ and the directed flux of a thin tube is $2\pi d\psi$. Pair its exit and entry endpoints to obtain the displayed energy injection, counting each boundary endpoint once. Equal endpoint angular <velocities> inject no <magnetic energy>.
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