(la) A grounded conducting sphere of radius a is covered with a shell of dielectric material, extending to an outer radius b, and of dielectric constant e. There is a vacuum outside the dielectric. The coated sphere is placed in an asymptotically constant electric field, approaching E = (0, 0, Eo) (Cartesian components) at large distances. Using the fact that the general azimuthally symmetric solution of Laplace's equation has the form o(r, 0) = (Aer' + Ber-(-1) P (cos 0) (1) in spherical polar coordinates, solve for the potential everywhere outside the conduct- ing sphere of radius a. [Recall Po(x) = 1, Pi(x) = x, P(x) = }(3x2 -1), etc.] Note: You will need to consider two expansions for the potential: $1(r, 0), for a Erb, $2 (r, # ) , for r 2 b, and then solve for the coefficients by imposing the boundary condition at r = a, the junction conditions at r = b, and the asymptotic condition at infinity. Remember that, as always, you can simplify the calculations by first identifying which subset of the modes in the expansions will actually arise, and including only these.b) Starting from the expression U = 1/(84) fy(E . D)d , show, by performing an integration by parts, that the electrostatic energy inside the shell of dielectric material can be expressed as E U dielectric = (2) Or ds)- where S, is the spherical surface at r = b. Calculate Udielectric explicitly for the configuration in Qu. (4a), using this formula. c) Show that if a, b and En are kept fixed, the energy Udielectric is maximised when the dielectric constant is given by 2(63 - a3) E = Emax 2a3+ 63 (3) Calculate the corresponding maximum value Umax of Udielectric