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CptXray

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## Homework Statement

There's a metal cunducting cube with edge length ##a##. Three of its walls: ##x=y=z=0## are grounded and the other three walls: ##x=y=z=a## are held at a constant potential ##\phi_{0}## . Find potential inside the cube.

## Homework Equations

The potential must satisfy Laplace equation $$\Delta \phi = 0$$

## The Attempt at a Solution

First, I postulate that a potential is a product:

$$\phi(x,y,z) = X(x)Y(y)Z(z)$$.

Plugging it into a Laplace equation(further I'm going to write functions of ##X(x)##, ##Y(y)##, ##Z(z)## as ##X##, ##Y##, ##Z## and derivatives as ##X'## and so on):

##\Delta\phi = X''YZ + Y''XZ +Z''XY = XYZ\bigg(\frac{X''}{X} + \frac{Y''}{Y} + \frac{Z''}{Z} \bigg) = 0##

Dividing both sides by ##XYZ## I get:

$$\frac{X''}{X} + \frac{Y''}{Y} + \frac{Z''}{Z} = 0$$

So, every term of the sum above has to be a constant: ##\frac{X''}{X} = -\alpha^2##, ##\frac{Y''}{Y} = -\beta^2##, ##\frac{Z''}{Z} = \gamma^2## it gives relation between ##\alpha##, ##\beta## and ##\gamma##: ##\gamma^2 = \alpha^2 + \beta^2##

I know that general solutions are:

\begin{cases}

X = Ae^{i\alpha x} + Be^{-i\alpha x}

\\

Y = Ce^{i\beta y} + De^{-i\beta y}

\\

Z = Ee^{\gamma z} + Fe^{-\gamma z}

\end{cases}

Applying boundary conditions for grounded walls ##\phi(x=y=z=0) = 0##

##X(0) = A + B = 0⇒X = \tilde{X}\sin{\alpha x}##

##Y(0) = C + D = 0⇒Y = \tilde{Y}\sin{\beta y}##

##X(0) = E + F = 0⇒Z = \tilde{Z}\sinh{\gamma z}##

where ##\tilde{X}##, ##\tilde{Y}##, ##\tilde{Z}## ##∈## ##ℝ## and are constants.

And at this point I have a problem, because when I try to find ##\alpha## and ##\beta## I don't get quantized solutions to build a series solution. Most likely it's my attempted solution that is wrong, but I'd be really happy for some guidlines or hints from you guys.

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