Is V = xy the correct potential function for this geometry?

[satchmo05]

Homework Statement

Determine whether or not V = xy is the correct potential function for the geometry shown in the figure: http://img268.imageshack.us/img268/446/48401266.jpg for the region 0 < x< 1 and 0 < y < 1. Why or why not? Assume that the region between the conducting plates has a relative permittivity of ε_r = 3.0.

Image for problem shown at this URL: http://img268.imageshack.us/img268/446/48401266.jpg

Homework Equations

There are multiple equations for the potential function V. I do not know which one to use/how to use it.

The Attempt at a Solution

Don't really know to get started on this one. I would appreciate any help that you can provide. I just need to know how to get started. Thanks for the help!

 

[Mindscrape]

You're testing whether or not the trial function obeys laplace's equation with the correct boundary conditions.

 

[satchmo05]

Ah, that makes sense Mindscape. Thank you. Ok, so I have determined that del^2(V) does in fact equal 0. What does this tell me? Is it the correct potential function? By the uniqueness principle, can I conclude that V=xy is the only potential function for this design, given the boundary conditions?

 

[Mindscrape]

You also need to check the boundary conditions for V(x,y):

V(0,y)=0
V(x,0)=0
V(1,y)=1
V(x,1)=1

 

[satchmo05]

I thought there were 2 criteria though for determining whether or not a given potential function is the solution the DiffEQ. I have already determined that the Laplacian holds true, what is the other?

 

[Mindscrape]

Well, if you're going to guess a solution, you'd better make sure that it satisfies the PDE to start with, and secondly that it satisfies the actual physical conditions when all is said and done.

V(x,y)=5x+7y

is that a solution? Why or why not?

Do you know how to find the solution from scratch if you had to? Maybe that would help you understand a little better.