Conductive contribution to complex permittivity (${\epsilon }_{r,\mathrm{eff}}={\epsilon }_r+i\sigma /(\omega {\epsilon }_0)$)

With phasor convention $e^{-i\omega t}$, the Ampere-Maxwell law in a scalar conductor reads $\nabla \times \mathbf{H}=(\sigma -i\omega {\epsilon }_0{\epsilon }_r)\mathbf{E}$. Factoring $-i\omega {\epsilon }_0$ gives the displayed effective relative complex permittivity. Its positive imaginary conductive term expresses loss in this convention; the sign reverses for $e^{+i\omega t}$. This combines conduction and displacement currents for the same field and must not be confused with the material's loss-free relative permittivity alone.