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## Homework Statement

A surface at z = 0 is held at potential V (x, y) = V0 cos(qx) sin(qy). Find the

potential in the region z > 0.

## Homework Equations

Laplace's equation in Cartesian coordinates

## The Attempt at a Solution

I wrote at least a page of my past 2 attempts at a solution. For some reason I was logged out during writing it and when I logged back in, everything was gone....

Anyway to summarize: (note I need only boundaries, I can do the math)

Attempt 1: Make a geometry and assume boundary conditions. That is, find the potential inside this geometry that extends only to points above the origin and with it carries the solution to z > 0

Attempt 2: Solve it directly from laplace's equation.

I think I knw V(x) and V(y) at the surface potential from the problem set up, so all I have to do is find V(z) from laplace's equation.

X(x) = Cos(qx)

Y(y) = sin(qy)

Z(z) = ???

Plugging these in I get:

Z(z) = e^(-qz) or e^(qz)

But these must have some coefficients and I suppose I can work backwards and find the boundary conditions that satisfy general solutions to each 2nd order Diff Eq with these potentials. Then find the coefficients from Fourier's trick.

I have been working on this for days....Any help would be greatly greatly and I mean greatly appreciated!

Thank you!

Attempt 2: