Magnetic energy balance in resistive magnetohydrodynamics (${\dot{E}}_B=-{\int }_V\mathbf{u}\cdot (\mathbf{J}\times \mathbf{B})dV-\eta {\int }_V{J}^{2}dV$)

For constant electrical resistivity $\eta$ and magnetic permeability ${\mu }_0$, let $E_B={\int }_V{B}^2/(2{\mu }_0)dV$ 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.