Electrostatic Potential in a Box

[azupol]

Homework Statement

A box shaped rectangular metal cavity of sides a, b and c along the x, y and z axes,
respectively, has one corner at the origin. Of the six sides, all are grounded except the one
at x=a and the one at y=b which are held at potentials of V₁ and V₂ , respectively. Find the
potential V(x,y,z) everywhere inside the cavity.

There are 6 boundary conditions, I think I've got them all correct:
i.) V(a, y, z)=V₁
ii.)V(x, b, z)=V₂
iii.)V(x, y, c)=0
iv.)V(x ,y, 0)=0
v.)V(0, y, z)=0
vi.)V(x, 0, z)=0

Homework Equations

$\Delta V = 0$

The Attempt at a Solution

I know how to approach this, we assume the potential can take the form V(x,y,z)= X(x)Y(y)Z(z), twice differentiate with respect to each spatial coordinate and plug it into the Laplace equation. From there we get three ODEs, which are each equal to a constant. This is where I get stuck. I know the choice of these constants, in particular their signs, is dependent on the boundary conditions. I'm stuck at this point, I don't know which sign to choose for each constant such that A+B+C=0, since the form of each separated solution will change based on these constants. Previous problems I've done involved one of the sides of the box going to infinity, so it was easy to choose constants to satisfy V->0 as a coordinate went to infinity.

 

[Orodruin]

I suggest you work by separating your inhomogeneous problem into two (one where V1=0 and one where V2=0).

The variable separation is mainly efficient for finding eigenfunctions of the Laplace operator with homogeneous boundary conditions. For each of the inhomogeneous problems after the separation, think about which dimensions you have homogeneous boundary conditions in - for these dimensions the possible separation constants will be given by the homogeneity. The possible values of the last constant will then be given by those of the other two. You then have to construct a series solution which you adjust to fit the boundary conditions in the last dimension.