Chapter 13.7, Problem 13P

Find the electrostatic potential outside a conducting sphere of radius $a$ placed in an originally uniform electric field, and maintained at zero potential. Hint: Let the original field $E$ be in the negative $z$ direction so that $E=-E_{0}k.$ Then since $E=-\nabla \Phi ,$ where $\Phi$ is the potential, we have $\Phi =E_{0}z=E_{0}r\cos \theta$ (Verify this!)for the original potential. You then want a solution of Laplace’s equation ${\nabla }^{2}u=0$ which is zero at $r=a$ and becomes $u~\Phi$ for large $r$ (that is, far away from the sphere). Select the solutions of Laplace’s equation in spherical coordinates which have the right $\theta$ and $\varphi$ dependence (there are just two such solutions) and find the combination which reduces to zero for $r=a$.