Classical Electrodynamics · Problem 2.15

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

Physics — E&M · green-function-square-region

Problem

(a) Show that the Green function

G(x, y; x', y')

appropriate for Dirichlet boundary conditions for a square two-dimensional region,

0 \leq x \leq 1

0 \leq y \leq 1

, has an expansion

G(x, y; x', y') = 2\sum_{n=1}^{\infty}g_n(y, y')\sin(n\pi x)\sin(n\pi x')

where

g_n(y, y')

satisfies

\left(\frac{\partial^2}{\partial y'^2} - n^2\pi^2\right)g_n(y, y') = -4\pi\delta(y' - y) \quad \text{and} \quad g_n(y, 0) = g_n(y, 1) = 0

(b) Taking for

g_n(y, y')

appropriate linear combinations of

\sinh(n\pi y')

and

\cosh(n\pi y')

in the two regions,

y' < y

and

y' > y

, in accord with the boundary conditions and the discontinuity in slope required by the source delta function, show that the explicit form of

G

G(x, y; x', y') = 8\sum_{n=1}^{\infty}\frac{1}{n\sinh(n\pi)}\sin(n\pi x)\sin(n\pi x')\sinh(n\pi y_<)\sinh[n\pi(1 - y_>)]

where

y_<(y_>)

is the smaller (larger) of

y

and

y'