Classical Electrodynamics · Problem 2.16

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

Physics — E&M · poisson-equation-unit-square-uniform-charge

Problem

A two-dimensional potential exists on a unit square area (

0 \leq x \leq 1

0 \leq y \leq 1

) bounded by ``surfaces'' held at zero potential. Over the entire square there is a uniform charge density of unit strength (per unit length in

z

). Using the Green function of Problem 2.15, show that the solution can be written as

\Phi(x, y) = \frac{4}{\pi^3\epsilon_0}\sum_{m=0}^{\infty}\frac{\sin[(2m+1)\pi x]}{(2m+1)^3}\left\{1 - \frac{\cosh[(2m+1)\pi(y - \frac{1}{2})]}{\cosh[(2m+1)\pi/2]}\right\}