Nearly a similar problem I have just encountered, which I have found the solution with FEMM simulation.

Let there be 100 V charged cylinder of 0.5 cm radius (planar problem) having 10 cm depth. Let there be a plate having 100 Sq cm area placed near the charged sphere as shown in the following planar geometry.

After implementing and solving above physical system in FEMM, found out the flux D.n through the plate using the plate as contour.

Put the charge equivalent to "Flux through plate/20.0 on plate by using conductor properties. Again obtaining the solution. Follwing voltage observed every where on the plate. Also equipotential lines suggests correct solution.

I don't know why I need to divide by 20. Somebody from acadamics may explain this in a better way

Nearly a similar problem I have just encountered, which I have found the solution with FEMM simulation.

Let there be 100 V charged cylinder of 0.5 cm radius (planar problem) having 10 cm depth. Let there be a plate having 100 Sq cm area placed near the charged sphere as shown in the following planar geometry.

Put the zero charge on the plate by using conductor properties. Again obtaining the solution. Following voltage observed everywhere on the plate. Also, equipotential lines suggest the correct solution.

I don't know the theory behind it. Somebody from academics may explain this in a better way