Poynting's theorem

Remarks

The proof is a direct computation. By the product rule $\nabla \cdot (E \times B) = B \cdot (\nabla \times E) - E \cdot (\nabla \times B)$, one of the vector calculus identities. Faraday's law replaces $\nabla \times E$ by $-\partial_t B$, and the Ampère–Maxwell law replaces $\nabla \times B$ by $\mu_0 J + \mu_0 \varepsilon_0 \partial_t E$. Dividing by $\mu_0$ gives $\nabla \cdot S = -\partial_t u - J \cdot E$. The integral form follows from the divergence theorem.

The term $J \cdot E$ is the power per unit volume that the field does on the charges, by the Lorentz force. In a resistor with $J = \sigma E$ it is the Joule heating $\sigma \abs{E}^2$. In a linear, non-dispersive medium with symmetric permittivity and permeability, the same computation gives $\partial_t u + \nabla \cdot (E \times H) = -J_f \cdot E$ with $u = \frac{1}{2} (E \cdot D + B \cdot H)$.

Example: a long straight resistive wire carrying a steady current has an axial $E$ and a circular $B$, so $S$ points radially into the wire. The flux of $S$ into a segment of the wire equals the Joule heat produced in it, $I^2 R$ for a segment of resistance $R$.

Going deeper

For a student

The theorem is the local conservation law of energy for Maxwell's equations, a continuity equation with a source term. For a solution in vacuum with $J = 0$ and enough decay at infinity, it gives conservation of the total energy $\int u \, dV$. This is a basic a priori estimate in the analysis of the Maxwell system. In the covariant formulation it is the time component of the conservation law of the electromagnetic stress–energy tensor, whose space components give the momentum balance with the Maxwell stress tensor.