Classical Electrodynamics · Problem 2.14

Boundary-Value Problems in Electrostatics: II — John David Jackson

Physics — E&M · hollow-cylinder-four-quarters

Problem

A variant of the preceding two-dimensional problem is a long hollow conducting cylinder of radius

b

that is divided into equal quarters, alternate segments being held at potential

+V

and

-V

. (a) Solve by means of the series solution (2.71) and show that the potential inside the cylinder is

\Phi(\rho, \phi) = \frac{4V}{\pi}\sum_{n=0}^{\infty}\left(\frac{\rho}{b}\right)^{4n+2}\frac{\sin[(4n+2)\phi]}{2n+1}

(b) Sum the series and show that

\Phi(\rho, \phi) = \frac{2V}{\pi}\tan^{-1}\!\left(\frac{2\rho^2 b^2 \sin 2\phi}{b^4 - \rho^4}\right)