[UPSC CSE 2013] Electromagnetic Theory (Q363, 15M)

(i) Considering an isotropic, linear, non-conducting, non-magnetic and inhomogeneous dielectric medium with \vec{D}=\epsilon\vec{E}=\epsilon_0n^2(x,y,z)\vec{E}, show that the electromagnetic wave equation for the field \vec{E} is given by \nabla^2\vec{E}+\vec{\nabla}\left(\frac{1}{n^2}\vec{\nabla}n^2\cdot\vec{E}\right)-\mu_0\varepsilon_0n^2\frac{\partial^2\vec{E}}{\partial t^2}=0. \setcounter{enumi}{1}

(ii) Write down the scalar equation for E_x from the above equation.

(iii) Interpret physically the situation if we move from homogeneous to an inhomogeneous medium.

(iv) Obtain the similar vector equation for the magnetic field \vec{H} in inhomogeneous medium.