Separation of Variables to Calculate Potential Inside Box

[Seraphim755]

My friends and I are in our first senior-level physics course at the University of Alabama in Huntsville, Introductory E&M. At the moment, we're working on using separation of variables to calculate electric potentials inside different objects given certain boundary conditions. One, however, is giving us problems.

1. Homework Statement

A cubical box of side length a consists of four metal plates that are welded together and grounded. The top and bottom faces of the cube are made of separate metal sheets and insulated from the others. These faces are held at a constant potential V₀. Find the potential inside the box.

Homework Equations

Standard form for separation of variables for this problem leads to the form:
X(x) = Asin(kx) + Bcos(kx)
Y(y) = Csin(ly) + Dcos(ly)
Z(z) = Ee^√(k2+l2)z + Fe^-√(k2+l2)z

With variables being changed based on the boundary conditions in the problems, which can be found by considering which plates are grounded. For this particular problem, we believe the boundary conditions to yield:

V = 0 @ x = 0 , x = a
V = 0 @ y = 0 , y = a
V = V₀ @ z = 0 , z = a

The Attempt at a Solution

[/B]
We worked a similar problem where only one plate had potential, the top plate. It was fairly straightforward, with the X and Y portions turning into sine functions and the Z portion becoming a hyperbolic sine function thanks to the exponentials. However, we are not sure what to do with this given the bottom plate having potential as well. Our teacher mentioned a method that he referred to as "Fourier's trick" that is a bit much to try to type out. Not sure if this is well-known terminology, but I thought I'd mention it.

Thanks in advance for the help, we deeply appreciate it!

 

[Chestermiller]

My friends and I are in our first senior-level physics course at the University of Alabama in Huntsville, Introductory E&M. At the moment, we're working on using separation of variables to calculate electric potentials inside different objects given certain boundary conditions. One, however, is giving us problems.

1. Homework Statement

A cubical box of side length a consists of four metal plates that are welded together and grounded. The top and bottom faces of the cube are made of separate metal sheets and insulated from the others. These faces are held at a constant potential V₀. Find the potential inside the box.

Homework Equations

Standard form for separation of variables for this problem leads to the form:
X(x) = Asin(kx) + Bcos(kx)
Y(y) = Csin(ly) + Dcos(ly)
Z(z) = Ee^√(k2+l2)z + Fe^-√(k2+l2)z

With variables being changed based on the boundary conditions in the problems, which can be found by considering which plates are grounded. For this particular problem, we believe the boundary conditions to yield:

V = 0 @ x = 0 , x = a
V = 0 @ y = 0 , y = a
V = V₀ @ z = 0 , z = a

The Attempt at a Solution

[/B]
We worked a similar problem where only one plate had potential, the top plate. It was fairly straightforward, with the X and Y portions turning into sine functions and the Z portion becoming a hyperbolic sine function thanks to the exponentials. However, we are not sure what to do with this given the bottom plate having potential as well. Our teacher mentioned a method that he referred to as "Fourier's trick" that is a bit much to try to type out. Not sure if this is well-known terminology, but I thought I'd mention it.

Thanks in advance for the help, we deeply appreciate it!

Consider the symmetry of Z with respect to z, assuming z = 0 is at the center of the box.

Chet

 

[vela]

What exactly is the difficulty you're running into?