Classical Electrodynamics · Problem 1.21

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

Physics — E&M · variational-poisson-unit-square

Problem

A two-dimensional potential problem consists of a unit square area (0≤x≤10 \leq x \leq 1, 0≤y≤10 \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 zz). ((a)) Apply the variational principle (1.63) for the Poisson equation with the ``variational'' trial function Ψ(x,y)=A⋅x(1−x)⋅y(1−y)\Psi(x, y) = A \cdot x(1 - x) \cdot y(1 - y) to determine the best value of the constant AA. [I use quotation marks around ``variational'' because there are no parameters to vary except the overall scale.] ((b)) The exact (albeit series) solution for this problem is [see Problems 2.15 and 2.16] 4πϵ0Φ(x,y)=16π2∑m=0∞sin⁡[(2m+1)πx](2m+1)3{1−cosh⁡[(2m+1)π(y−12)]cosh⁡[(2m+1)π/2]}4\pi\epsilon_0\Phi(x,y) = \frac{16}{\pi^2}\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\} For y=0.25y = 0.25 and y=0.5y = 0.5, plot and compare the simple variational solution of part a with the exact solution as functions of xx.

Related theory

Browse the problem library