The electric field E(r, t) at a point r at time t in a metal due to the passage of electrons can be described by the equation $${\nabla ^2}\overrightarrow {\bf{E}} \left( {\overrightarrow {\bf{r}} ,\,t} \right) = \frac{1}{{{c^2}}}\left[ {\frac{{{\partial ^2}\overrightarrow {\bf{E}} \left( {\overrightarrow {\bf{r}} ,\,t} \right)}}{{\partial {t^2}}} + \omega {'^2}\overrightarrow {\bf{E}} \left( {\overrightarrow {\bf{r}} ,\,t} \right)} \right]$$
where, $$\omega '$$ is a characteristic associated with the metal and c is the speed of light in vacuum. The dispersion relation corresponding to the plane wave solutions of the form exp $$\left[ {i\left( {\overrightarrow {\bf{i}} .\overrightarrow {\bf{r}} - \omega t} \right)} \right]$$   is given by

A. $${\omega ^2} = {c^2}{k^2} - \omega {'^2}$$

B. $${\omega ^2} = {c^2}{k^2} + \omega {'^2}$$

C. $$\omega = ck - \omega '$$

D. $$\omega = ck + \omega '$$

Answer: Option A