Radiation-force density without Ohm's law (${\overline{f}}_x=-{\partial }_x[{\mu }_0\overline{H_z^2}/2+{ϵ}_0\overline{E_y^2}/2]$)

For a monochromatic transverse field depending on $x,t$, with vacuum permittivity and permeability and arbitrary induced transverse current, Ampère's law gives $J_y=-{\partial }_x{H}_z-{ϵ}_0{\partial }_t{E}_y$. Multiplying by ${\mu }_0{H}_z$ and averaging Faraday's law gives the displayed gradient. No constitutive assumption $J=\sigma E$ is required. A steady average removes the field-momentum time derivative.