[QUE/EM-05003] --- EM-PROBLEM --- Point charge and a dielectric sphere embedded in another medium

A point charge \(q\) is embedded at the center of a sphere with dielectric  constant \(\kappa_1\). The sphere is itself embedded in an infinite volume with dielectric constant \(\kappa_2\).

1. Find the electric field inside and outside the sphere;
2. Obtain the surface charge density of bound charges on the surface of the sphere;
3. Compute the electric field due to the bound charges outside the sphere;
4. Verify that the electric field outside, as obtained in part (a) is sum of the fields due to the bound charges and the field of free charge \(q\) embedded in dielectric medium 1.