Poynting flux and terminal power ($P={\sum }_k{\varphi }_k{I}_k$)

For static electric fields and magnetic fields, Ampère's law and $\mathbf{E}=-\nabla \varphi$ give $\mathcal{P}=\mathbf{E}\times \mathbf{H}=\varphi {\mathbf{J}}_f-\nabla \times (\varphi \mathbf{H})$. Stokes theorem on an oriented separating surface then giveswhere each perfect conductor crossing it has constant electric potential ${\varphi }_k$ and signed electric current $I_k$. The line term is absent for a closed surface and must vanish in the limiting sense at infinity for a noncompact surface. Under that condition the incoming Poynting flux equals terminal power. If the signed electric currents sum to zero, the result is independent of the reference for electric potential.