Question

A conducting sphere of radius a, at potential Vo, is surrounded by a thin concentric spherical shell of radius b, over which someone has glued a surface charge density σ(8)-k cos θ, where k is a constant and θ is the polar spherical coordinate. (a) Find the potential in each region: (i)r > b, and () a<r<b. [5 points] [Hint: start from the general solution of Laplace's equation in spherical coordinates, but allow for different coefficients in the radial part in the two different regions; find the coefficients using the continuity of the potential at ra and r b and the discontinuity of the electric field at r- b]. (b) Find the induced surface charge density σίθ) on the conductor. [2.5 points] (c) What is the total charge of this system? Check that your answer is consistent with the behavior of V at large r. [2.5 points] Answer: V(r, 6)-

Answer 1

As the configuration is azimuthally symmetric Electric potential in spherical polar co-ordinates is v(cos theta) = sigma^infinity_l = 1 (A_l r^l + B_l/r^l + 1) P_l (cos theta) equation (1) a) For r > b In equation (1) A_l = 0 for all l because v rightarrow 0 at infinity therefore v(r, theta) = sigma^infinity_l = 1 B_l/r^l + 1 P_l (cos theta) For a < r < b v(r, theta) = Sigma [C_r r^l + D_l/r^l + 1] P_l(cos theta) For r < a, v(r, theta) = v_0 using following boundary conditions. \r\n V is continuous at a . . . (a) V is continuous at b - - - (b) At b Delta [partial differential V/partial differential r] = -1/epsilon_0 sigma(theta) equation (c) Here sigma(theta) is the surface charge density substitute k cos theta for sigma (theta) Delta [partial differential v/partial differential r] = -k cos theta/epsilon_0 From boundary condition (b) v is continuous at