= Magnetic energy balance in resistive magnetohydrodynamics
{title2=$\dot E_B=-\int_V\mathbf u\cdot(\mathbf J\times\mathbf B)\,dV-\eta\int_VJ^2\,dV$}
For constant <electrical resistivity> $\eta$ and <magnetic permeability> $\mu_0$, let $E_B=\int_VB^2/(2\mu_0)\,dV$ and $\mathbf J=\nabla\times\mathbf B/\mu_0$. In a fixed volume with zero boundary energy flux, multiply the <resistive induction equation> by $\mathbf B/\mu_0$ and integrate by parts. This gives the displayed balance. The first term transfers <energy> between the <magnetic field> and the fluid; the second is nonpositive and represents <Joule heating>. The latter does not imply that the total <magnetic energy> must decrease if fluid motion continually supplies it.
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