Separation of Variables to Laplace's Equation in Electrostatics

[chaos333]

Homework Statement

A cubical box (side length a) consists of five metal plates, which are welded
together and grounded (Fig. 3.23). The top is made of a separate sheet of metal, insulated
from the others, and held at a constant potential V0. Find the potential inside the box. [What
should the potential at the center (a/2, a/2, a/2) be? Check numerically that your formula is
consistent with this value.

Relevant Equations

d^2v/dx^2+d^2v/dy^2+d^2v/dz^2=0

A bit messy but the bottom is supposed to be the potential function

 

[Orodruin]

First of all, in the future please use the LaTeX feature rather than posting pictures of writing. Apart from being more legible, it is then possible to quote particular passages of your post.

As for the solution itself: First of all, you are not answering the first question: "What should the potential at the center be?" This is a relatively simple question which is easily answerable.

Second, your constant $c_n$ should be $c_{nm}$ and be inside of the sums.

I did not check the calculations explicitly because they are difficult to read, but in general the approach looks fine. You also need to evaluate at the middle of the box and show numerically that it approaches the correct value.