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1) When a charge 'q' is given to an isolated conductor, its potential will change. 2) The change in potential depends on the size and shape of the conductor.

I could understand the point no. 1. Since the potential depends on the charge and distance, I could understand the first pint.

Where from the size and shape of the conductor comes here in potential?

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I guess that by "giving a charge to a conductor" you mean a charged capacitor plate (which is a conductor). The potential of such a capacitor plate is meaningless if there is only one plate. The system needs at least one other conductor, so that you can ask "What is the potential difference between those two conductors?"

To answer this question, you would evenly distribute the charge over the surface of one of the capacitor plates. The other capacitor plate is uncharged. Then, you have valid boundary conditions to solve the Poisson equation, which gives you the potential difference between the two conductors as its solution. The solution depends on the exact shape of both conductors; if the distance is large compared to the size of the conductors, the potential difference will be equal to the potential of a point charge.

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