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If a thin metal plate is placed between two charges $+q$ and $+q$, will this cause a change in the electrostatic force acting on one charge due to another? What is the concept behind this? What will happen if the metal plate is thick?

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The metal plate, being a good conductor, will have its electrons rearrange in such a way as to neutralise the electric field inside the plate.

The electrons would tend to bunch up in the plate at the point(s) closest to each of the two charges, alterring the electric field that they're exposed to and changing the electrostatic force on them.

Strictly speaking however, the force applied by one charge to the other remains the same. But the force that each particle experiences will be reduced by the partial cancellation of the positive field by the proximal electrons bunched up in the plate.

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Will this cause a change in the electrostatic force acting on one charge due to the other?

The answer is no.

The two charges will induce some charges on the metal plate. This will of course change the electric field, but according to the principle of superposition, the force on one charge due to another charge is not affected by the presence of other charges (i.e., the charges on the metal plate) near it. So the force due to one charge $+q$ on the other charge $+q$ will remain the same.

But the total force on the charged particle may be different because of the presence of induced charges on the metal plate.

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Force on any charge increases when metal plate is placed between them as the effective distance between two charges decreases but in case of glass or other dielectrics it decreases.

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I think that the force exerted by each charge on each other will be equal to $0$, as the electric field of charges will not pass through the metal plate. But the net force on each charges will remain the same and the force will be exerted by the charge induced on the metal plate by both charges.

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