Voltage due to electrostatic induction

[mcconohy]

Hello All,
I have an arcane curiosity that cannot be answered anywhere I look (including Griffiths' intro electrodynamics book and the internet). This question seems so simple but I may just be over-thinking it.

If there are two square, parallel, finite, conducting plates and a point change q placed at an arbitrary location in space, what will be the voltage difference between the two plates (i.e. hook a voltmeter up to the two plates, what will the reading be).

Intuitively I know that if the point charge is halfway between the plates, the voltage should be zero because the induced charge should be equal on each plate. Also, the voltage should be zero if the charge is placed far away from the plates because the electric field strength is practically the same on the two plates.
Any help would be great. Thanks in advance!

 

[mfb]

In general, there will be some potential difference. I doubt that there is an analytic solution for arbitrary charge positions. Simulate it?

 

[mcconohy]

So would it be appropriate to use Poisson's equation with two boundary conditions that the voltage is constant for the positions of the conducting plates (or that the electric field is perpendicular to the conductor)?

 

[mfb]

For the numerical simulation? Right.