Electric Potential Inside an Infinite Rectangular Trough

[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.

Homework Equations

Laplace's equation in rectangular coordinates

The Attempt at a Solution

[/B]
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.

 

[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.