Static Poynting flow

Time-independent crossed electric and magnetic fields can carry a nonzero energy flux $\mathbf{S}=\mathbf{E}\times \mathbf{H}$. A steady coaxial circuit is an example: radial electric and azimuthal magnetic fields carry energy axially from source to load. Static energy density does not imply zero flux.

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.

For a resistive upper strip at $z=d$ with $V(x)=V_0-IRx/l$ and an ideal lower strip at zero potential, the narrow-gap approximation gives $\varphi =zV(x)/d$ and $\mathbf{H}=(0,I/a,0)$. The Poynting vector has both an axial flux and an upward flux into the lossy conductor. Axial power decreases by $I^{2}R$ along the line, exactly equal to the integrated upper-surface flux. The ideal lower conductor receives no steady dissipation.