Floating potential of a conductor in an electrical field

[coquelicot]

I'm interested in 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.

 

[Henryk]

Because M is conductive, the potential will be constant. Assuming that M is absolutely neutral (wrt. charge) and much closer to the plate at the potential V than ground, it will be at the potential just a bit lower than V. The exact potential would depend on the ratio of effective distances of the M to the plate and ground.

 

[coquelicot]

Thank you. Yes I know that but the question is: Neglecting the effect of the ground on M, do you have a mathematical mean to handle this problem. For example, assuming M is a rod of cylindrical shape, extending from x = 1 to 2.