[UPSC IFoS 2022] Electromagnetic Theory (Q317, 10M)

Consider a medium with an electric current density \vec{\text{J}}, permittivity \varepsilon and conductivity \sigma, which is described by Ohm's law \vec{\text{E}} = \sigma \vec{\text{J}}. Write down Maxwell's equations in such a medium. Using these equations, derive the wave equation for the electric field \vec{\text{E}}. Show that plane waves of the form \vec{\text{E}} = \vec{\text{E}}_0 \text{ e}^{\text{i}(\omega\text{t} - \text{kx})}, where \vec{\text{E}}_0 is a constant vector, are solutions of the wave equation but with complex k. Determine the explicit expression for k in terms of \varepsilon, permeability \mu_0, \sigma and the angular frequency \omega.