 Poynting theorem

The Poynting theorem is the local energy balance of classical electromagnetism. For vacuum field laws, with current density $\mathbf{J}$, the electromagnetic energy per unit volume isandThus a decrease in local electromagnetic energy can supply outward energy flux or work on electric charge. The term $\mathbf{J}\cdot \mathbf{E}$ is power transferred to matter per unit volume. Its sign matters: positive values remove field energy, while negative values describe matter supplying the field. The magnetic part of the Lorentz force supplies no work term.

For a fixed volume $V$, the divergence theorem gives the directly measurable balanceFor example, $2.0\mathrm{W}$ entering a region with $0.50\mathrm{W}$ leaving and $1.20\mathrm{W}$ transferred to matter means stored electromagnetic energy increases at $0.30\mathrm{W}$. In source-free vacuum only the boundary flux changes the stored energy; a nonzero local Poynting vector can still have zero net closed-surface flux.

A brief proof takes the dot product of the Ampère-Maxwell equation with $\mathbf{E}/{\mu }_0$ and uses Faraday's law with $\mathbf{B}/{\mu }_0$. The identity $\nabla \cdot (\mathbf{E}\times \mathbf{B})=\mathbf{B}\cdot (\nabla \times \mathbf{E})-\mathbf{E}\cdot (\nabla \times \mathbf{B})$ turns the remaining terms into the divergence of the Poynting vector. ^[ref].