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eep

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**E&M: Using Laplace's Equation to solve for a conducting "slit"**

## Homework Statement

The set up is as follows: You have a conductor at potential 0 along the y-axis at x=0. You have another conductor at potential V=Vo running along the x-axis at y=0. You have a third conductor at potential V=Vo running along the x-axis at y=a. The "touching" corners of the conductors are all insulated from one another. Determine what the potential is by using Laplace's equation.

## Homework Equations

[tex]\nabla^2V = 0[/tex]

[tex]V(x,y) = X(x)Y(y)[/tex]

[tex]\frac{1}{X}\frac{{d^2}X}{d^2x} = k^2[/tex]

[tex]\frac{1}{Y}\frac{{d^2}X}{d^2y} = -k^2[/tex]

## The Attempt at a Solution

So we're basically just trying to solve this by separation of variables. I know Y(y) must oscillate as the potential is Vo at two points. After applying boundry conditions in the Y-direction, and taking the simplest form, I get

[tex]Y(y) = {V_o}cos(\frac{n{\pi}y}{a})[/tex]

However in the X-direction I only having one boundry condition, giving me

[tex]

X(x) = 2sinh(\frac{n{\pi}x}{a})

[/tex]

The general solution is a sum over N of X*Y, however as x goes to infinity, X goes to infinity which makes me doubt my solution for X. If the conductor at x=0 was at potential Vo, and the conductors at y=0,a were at 0, I could apply the boundry condition that V goes to 0 as x goes to infinity, but I don't think I can apply that in this case. Suggestions?