Poynting theorem

For electromagnetic energy densityMaxwell's equations implyThe field energy lost from a volume becomes outward electromagnetic flux or work on charges.

In vacuum the electromagnetic energy density isand the field energy in a region is $U_{\mathrm{EM}}=\int udV$.

In vacuum-permeability magnetohydrodynamics, the magnetic energy in a volume is $E_B=\int B^2/(2{\mu }_0)dV$. Its volume density equals the magnetic pressure.

For a vacuum plane wave, $B=\hat{k}\times E/c$, soThe electric and magnetic energy densities are equal, and the time averages satisfy $⟨S⟩=c⟨u⟩\hat{k}$.

For symmetric time-independent constitutive tensors,and the macroscopic Maxwell equations give ${\partial }_{t}u+\nabla \cdot S=-E\cdot J$.