Question

5. A hollow sphere of radius R has a potential on the surface of V(θ, d) Vo cos θ. There is no a) Find the potential everywhere inside and outside the sphere. b) Find the electric field everywhere inside the sphere. (You will find it easier to convert the potential to Cartesian coordinates and then find the field.) c) Find the charge density σ(0) on the surface of the sphere using Gauss' law. charge inside or outside the sphere.

Answer 1

Given that v(theta, phi) = v_0 cos theta there is no charge inside or outside the sphere rightarrow Poisson's equation is given by nabla^2 v = -e/epsilon_0 the change is zero both for r lessthanorequalto R and r greaterthanorequalto R then Poisson's equation in Laplace's equation is nabla^2 v = 0 rightarrow the general solution for the equation is v(r, theta) = sigma^infinity + l = 0 Alr^l pl(cos theta) (r lessthanorequalto R) and v(r, theta) = sigma^infinity_l = 0 Bl r^-(l + 1) pl(cos theta) (r greaterthanorequalto R) now, comparing the given potential with Legendre's polynomial v(r, theta) = k cos theta v(r, theta) = kP_1 (cos theta) A_1 = k/R & B_1 = kR^2 therefore v(r, theta) = kR^2 /r^2 cos theta (r greaterthanorequalto R) & v(r, theta) = kr/R cos theta (r lessthanorequalto R) hence the charge density sigma(theta) sigma(theta) = -epsilon_0 (partial differential v_out /partial differential r - partial differential v_in /partial differential r) sigma(theta) = -epsilon_0 /R (3k cos theta) rightarrow therefore the charge density sigma(theta) on the surface of