Electric Potential Inside an Infinite Rectangular Trough

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1. May 2, 2015

Zachreham

1. The problem statement, all variables a
nd given/known data

A rectangular trough extends infinitely along the z direction, and has a cross section as shown in the figure. All the faces are grounded, except for the top one, which is held at a potential V(x) = V_0 sin(7pix/b). Find the potential inside the trough.
The length along the x direction is b, and the length along the y direction is a.

2. Relevant equations

Laplace's equation in rectangular coordinates

3. The attempt at a solution

I know that my solution will be V(x,y,z)=X(x)*Y(y)*Z(z) and have already solved the x-dependent and y-dependent parts of the solution. After applying boundary condition I obtained:
X(x) = Asin(n*pi*x/b) where n= 1, 2, 3...
Y(y) = Csin(m*pi*x/a) where m= 1, 2, 3...

Solving for the Z-dependent part is when i run into some trouble...
I know that in a case where we're dealing with a finite rectangle with 5 faces being grounded and one being held at a certain potential that the Z solution would be comprised of sinh and cosh and the eigenvalue would be gamma^2= -(npi/b)^2 - (mpi/a)^2. However, the potential doesn't necessarily vanish at one end point in this problem, because our trough is not finite.

2. May 3, 2015

TSny

Hello. Welcome to PF!

The trough is infinite in the ±z-directions and none of the boundary conditions depend on z. This suggests trying to find a solution V(x, y, z) that does not depend on z.