Laplace equation for parallel plate condersers

[gulsen]

I've recently started studying Laplace's equation and it's solution under various simple circumstances in electrostatics. I tried to solve the equation for a parallel plate condenser system, but I couldn't meet the boundary conditions. I had two plates, one placed on xz plane at y=0 (with potential = 0), second parallel to it, at y=d (with potential V_0). I placed them such that they're symmetrical in x and z, i.e., y-axis crosses midpoints of plates; therefore the potential should be an even function of x and z. Noting that V(0,0,0) = 0 I wrote the solution:

A\cosh(kx) \cosh(lz) \sinh(my)
with k^2 + l^2 + m^2 = 0 and let A be any complex number.

I assumed that potential should drop to zero when x,z \to \pm \infty, and this's the boundary condition that doesn't meet with my "solution".

Can anyone help me working out the solution, or forward me to some resource on it?
Thanks!

 

[lpfr]

Your formula cannot fit the potential. Of course, k and l are zero (because of symmetry the potential does not depends on x or z).
But the potential in this problem is a linear function of y. There is no way to fit a sinh into a straight line.
Then your assumption about the mathematical form of the solution is wrong.
Let's start with Laplace's equation:

{\partial^2 \varphi\over \partial x^2 } +<br /> {\partial^2 \varphi\over \partial y^2 } +<br /> {\partial^2 \varphi\over \partial z^2 } = 0.

The first and third terms are zero. Then:
{\partial^2 \varphi\over \partial y^2 }= 0.

Then: \varphi = ay+b

 

[gulsen]

This can't be true because plates are not infinite, and field lines are no longer straight lines when we approach to the edges:

http://www.regentsprep.org/Regents/physics/phys03/aparplate/plate2.gif

And how do we say potential itself does not depend on x either z? Apparently they do --even though the assumption that field lines were straight, they should vanish in the outside region between plates, which is defined by x and y.

I remember this problem is not easily solved (possibly from Feynman lectures)

 

[lpfr]

Well, if you had said that the plates where finite. I wouldn't have bothered to answer.
This problem, as Feynman said is not soluble analytically.