Question

Potential inside a cubical box having all plates grounded and top maintained at v

Answer 1

This is a rather simple problem. Such problems are best approached by using the ansatz approach - meaning trying to guess the form the solution will take, and if your guess for the solution satisfies both Laplace's equation and the boundary conditions then the uniqueness theorem guarantees that you have found THE solution. Now, regarding your problem, the zero boundary conditions for x and y are reminiscent of standing waves with nodes at both ends. So we can guess the x and y dependencies to be sin

mπx

L

and sin

nπy

L

respectively. Your statement k

2

x

+k

2

y

+k

2

z

=0 doesn't look right since the sum of three positive real nos. cannot be zero. Rather, you would have