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I thought I had a decent understanding of Electrostatics, but this problem is not making sense to me.

If I applied a voltage to a conducting plate +V and had a ground in some arbitrary location, how could one calculate the electric field around the plate? If a charge is placed near this plate (in a vacuum for simplicity), how would the charge behave?

I understand this would be easily solved if we had some idea of the charge distribution on the plate, but I don't know how to make the connection between this fixed potential, and a charge distribution.

I was thinking maybe Poisson's equation could be used, but I don't know how to apply it.

The answer I'm leaning to is that a charge near the plate would experience negligible electric field because the voltage of the plate is only meaningful with respect to the ground. Since the ground and plate are at an arbitrary distance away, the field lines from one to the other would be fairly weak, and thus negligible.

Is this the correct understanding for this problem?

Thanks in advance.

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You could calculate the field around the metal plate by using Laplace equation assuming as boundary conditions constant potential on the plate and that the ground with zero potential is far away. The solution depends on the shape of you plate. If your plate was a sphere you could do it analytically.The electric field will be highest near the plate, especially near the edges, and a charge will experience the strongest electrostatic force there.

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