Potential distribution in metal cylinder

[vickcro]

I have an electrode that I'm trying to model. It's got v=0 at z=0, and v=V at z=H. I've been playing around with the poisson equation, and keep getting terrible solutions to PDEs involving Bessel functions. Anyone know if there's a better way to do this (since I can't seem to solve it with the boundary conditions anyhow...and it's not a homework problem, so there's no one to ask)

Thanks!

 

[Prologue]

What kind of symmetry is there?

 

[Meir Achuz]

The answer depends on the boundary condition at r=R.
Bessel function expansion is the easiest way to do it.
The solution is given in most math-physics and EM textbooks.