Separation of Variables to Laplace's Equation in Electrostatics

AI Thread Summary
The discussion focuses on the application of separation of variables to solve Laplace's equation in electrostatics, emphasizing the importance of clarity in mathematical notation. Participants highlight the need for using LaTeX for better readability and quoting capabilities. A specific question regarding the potential at the center is noted as unanswered, indicating a gap in the solution provided. Additionally, there is a correction regarding the notation of constants, suggesting that ##c_n## should be ##c_{nm}## within the sums. Overall, the approach is deemed generally acceptable, but further evaluation and numerical verification are necessary.
chaos333
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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
1721178908349.png
1721178923350.png

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