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 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.