Classical Electrodynamics · Problem 3.2

Multipoles, Electrostatics of Macroscopic Media, Dielectrics — John David Jackson

Physics — E&M · boundary-value-problems-spherical-coordinates

Problem

A spherical surface of radius RR has charge uniformly distributed over its surface with a density Q/4πR2Q/4\pi R^2, except for a spherical cap at the north pole, defined by the cone θ=α\theta = \alpha. (a) Show that the potential inside the spherical surface can be expressed as Φ=Q8πϵ0∑l=0∞12l+1[Pl+1(cos⁡α)−Pl−1(cos⁡α)]rlRl+1Pl(cos⁡θ)\Phi = \frac{Q}{8\pi\epsilon_0} \sum_{l=0}^{\infty} \frac{1}{2l+1} [P_{l+1}(\cos\alpha) - P_{l-1}(\cos\alpha)] \frac{r^l}{R^{l+1}} P_l(\cos\theta) where, for l=0l = 0, P−1(cos⁡α)=−1P_{-1}(\cos\alpha) = -1. What is the potential outside? (b) Find the magnitude and the direction of the electric field at the origin. (c) Discuss the limiting forms of the potential (part a) and electric field (part b) as the spherical cap becomes (1) very small, and (2) so large that the area with charge on it becomes a very small cap at the south pole.

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