(Solved): Problem 13.3. Gauss's law for dielectrics a ) = A charged conducting sphere of radius a is enclose ...

Problem 13.3. Gauss's law for dielectrics a ) = A charged conducting sphere of radius a is enclosed in a concentric spherical shell of dielectric with dielectric constant k. The dielectric shell has inner radius a and outer radius b, so that it fits tightly around the conducting sphere. The conducting sphere has a total electric charge of Q. In addition, the dielec- tric shell has a free charge density p(r) = pola)?, which exists only for a b. Make sure you draw a new picture showing the dielectric shell, the Gaussian surface, and multiple electric fields and normal vectors with tails on the Gaussian surface. All radii must be clearly marked. d) What is the polarization in the dielectric, and what are the polarization charge densities in the bulk and on the surfaces of the dielectric? Explain clearly how normal vectors are defined. e) State the two boundary conditions that must be satisfied at the dielectric interface that forms the outer surface of the dielectric shell at r = b. Show that the electric fields that you found in parts (a) and (b) explicitly satisfy both of these boundary conditions.