# Potential Inside Conducting Cube

A cubical box with sides of length L consists of six metal plates. Five sides of the box { the plates at $x=0, x=L, y=0, y=L, z=0$ - are grounded. The top of the box (at z = L) is made of a separate sheet of metal, insulated from the others, and held at a constant potential $V_0$. Find the potential inside the box.

I am attempting to set up this problem with Legendre polynomials. Since the top portion is grounded, this affects the "potential" region I can take, and I am unclear on how about to go setting it up.

I have set up the Laplace as follows:

$1\over x$${d^2x\over dx^2} + 1\over y$${d^2y\over dy^2} +$$1\over z$${d^2z\over dz^2} = 0$

In which at this point, must each component equal a constant?

which gives $k_x^2+k_y^2+k_z^2 = 0$

$1\over x$${d^2x\over dx^2} = k_x^2 1\over y$${d^2y\over dy^2} = k_y^2$

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