Distance between 3d voltage probes
The conductivity of the probes is not going to be trivial, you'll need to know your field strength from the dipole and assume that it is indeed pointed in the direction that you have indicated. If the dipole is not pointed in that direction then it will make the calculations ugly because symmetry will be thrown out the window as a gradient will exist across the disk and cylinder.

Secondly dipoles aren't point charges, I'm not sure what the field equation is for yours, but it will need to be understood to a reasonable degree of accuracy. Maybe you've worked out the equations already, if you have great, if you haven't I'll include them.  

Gauss's law will be your best friend here: 

Integrate the fields from the dipole across the surface of the disk of the probe. Then the cylinder. I would imagine that after the cylinder is worked out it will be found that the cylinders surface will 'collect' a small percentage  of your total charge and can be neglected. 

This will work if you assume that the disk\probe has no charge (everything in the real world has a charge that will have to be minimized). It also neglects edge effects from the corner and these may or may not contribute substantial error.

The second thing you will want to add in is effects from materials, like electric permeability and whatever your solution does to the electric field

After the paper calculations are done, if you have some time left, I'd throw the whole thing into a 2d static electric field solver and then compare that to the hand calculations and see if the models are close if they are, then you can use the simpler assumptions.