At time t = 0, a charge distribution $$\rho \left( {\overrightarrow {\bf{r}} ,\,0} \right)$$  exists within an ideal homogeneous conductor of permittivity $$\varepsilon $$ and conductivity $$\sigma $$. At a later time $$\rho \left( {\overrightarrow {\bf{r}} ,\,t} \right)$$  is given by

A. $$\rho \left( {\overrightarrow {\bf{r}} ,\,t} \right) = \rho \left( {\overrightarrow {\bf{r}} ,\,0} \right)\exp \left( { - \frac{{\sigma t}}{\varepsilon }} \right)$$

B. $$\rho \left( {\overrightarrow {\bf{r}} ,\,t} \right) = \frac{{\rho \left( {\overrightarrow {\bf{r}} ,\,0} \right)}}{{1 + {{\left( {\frac{{\sigma t}}{\varepsilon }} \right)}^2}}}$$

C. $$\rho \left( {\overrightarrow {\bf{r}} ,\,t} \right) = \rho \left( {\overrightarrow {\bf{r}} ,\,0} \right)\exp \left[ { - {{\left( {\frac{{\sigma t}}{\varepsilon }} \right)}^2}} \right]$$

D. $$\rho \left( {\overrightarrow {\bf{r}} ,\,t} \right) = \rho \left( {\overrightarrow {\bf{r}} ,\,0} \right)\exp \frac{\varepsilon }{{\sigma t}}\sin \left( {\frac{{\sigma t}}{\varepsilon }} \right)$$

Answer: Option A