= Poynting flux and terminal power
{c}
{title2=$P=\sum_k\phi_kI_k$}
For static <electric fields> and <magnetic fields>, <Ampère's law> and $\mathbf E=-\nabla\phi$ give $\boldsymbol{\mathcal P}=\mathbf E\times\mathbf H=\phi\mathbf J_f-\nabla\times(\phi\mathbf H)$. <Stokes theorem> on an oriented separating surface then gives
$$
\int_\Sigma\boldsymbol{\mathcal P}\cdot\mathbf n\,dS
=\sum_k\phi_k I_k-\oint_{\partial\Sigma}\phi\mathbf H\cdot d\boldsymbol\ell,
$$
where each <perfect conductor> crossing it has constant <electric potential> $\phi_k$ 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>.
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