Floating potential of a conductor near another conductor with known potential with respect to the earth

Question

Electrostatics textbooks delight at computing the electric field or the potential created by a known distribution of charge, ignoring the fact that this is rather a non practical question. In practicality, you have a voltage generator and conductors. You know the output voltage of the generator with respect to the earth, and hence the electrical potential of the connected conductor. It turns out that I don't see, in my old books of physics, how to handle electrostatic questions related to the floating potential of conductors, even though they should be solvable somehow. To put flesh on bones, I propose the following general question:

Assume x,y,z is an axes system, and that the y-z plane is occupied by a conductive plate at a known potential V with respect to the earth. Now, a conductive material M of neutral global electrical charge is placed at some distance of the plate, and is not connected to anything else. Furthermore, the distance of M to the plate is much smaller than the distance of M to anything connected to the earth (so that, the direct influence of the earth on M can be neglected). What is the potential of M with respect to the earth ?

For the sake of simplicity, it can be assumed that M is a copper cylindrical rod of length 1 along the x axis, located from x = 1 to x = 2. Regarding the unspecified parameters, any assumption can be done. Also, if the "infinite plate" is problematic, it can be assumed it's a large disk or a large rectangular plate.

Answer 1

In empty space, you should have $\nabla^2V=0$. For conductors of known potential, you have a boundary condition $V=V_i$ where $V_i$ is the voltage on the $i$th conductor. For conductors of unknown voltage, you know that:

1) $\nabla V$ is perpendicular to the conductor's surface.

2) $\oint_{\text{conductor surface}} (\nabla V) \cdot d\vec{A} = 0$

(Since the conductor has no net charge.)

Together with the boundary conditions at infinity, this should be enough constraints to solve the problem. Of course, that leaves the question of how one should mathematically handle this.

In terms of mathematical advice, I don't have much. It seems like it would be a good idea to pretend that the floating conductors actually each have their own fixed potential. Then solve the equation for the set of potentials that satisfies (2) for all the floating conductors.

Answer 2

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

Answer 3

First solve the problem with the plate at 0 V and the conductor M at 1 V. Calculate the charge $Q_1$ on M (this can be done by integrating $E$ over the surface of M and using Gauss' law). Call this Case A.

Next, solve the problem with the plate at its actual voltage $V$ and the conductor at 0 V. Again, find the charge $Q_2$ on the conductor. Call this case B.

You can see the the actual situation, with the plate at $V$ and the conductor at (unknown) $V_M$ can be written schematically as

Actual situation = $V_M$*(Case A) + Case B.

Thus the actual charge $Q$ on the conductor M (can be zero) must obey $$ Q = V_M*Q_1 + Q_2, $$ so that $$ V_M = (Q - Q_2)/Q_1. $$