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The relative permittivity and hence absolute permittivity of a conductor is infinity. Now coulomb force between two point charges is inversely proportional to the permittivity of the medium between the two charges.

  1. Then does this mean if two point charges are separated by a conductor then coulomb force between two point charges is zero?
  2. When charge is given to a conductor at a point inside the conductor, the charges get distributed on the surface to experience minimum repulsion.According to the first case if the force is zero, then the charges should not experience any repulsion and should remain at the point where they are given. So at what point am i going wrong. Please Explain.
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(1) Then does this mean if two point charges are separated by a conductor then coulomb force between two point charges is zero?

No, it just means that EM fields don't propagate through a conductor. They do however hit the surface and the free electrons in the conductor then will create a charge separation and hence a potential difference will be induced.

The opposite end of the conductor will now lack the charges that left to to go join the EM field on the other side, and hence a charge on this end will "see" this and hence feel a force. See method of images for more on this.

(2) When charge is given to a conductor at a point inside the conductor, the charges get distributed on the surface to experience minimum repulsion.According to the first case if the force is zero, then the charges should not experience any repulsion and should remain at the point where they are given. So at what point am i going wrong.

Your reasoning here is more or less alright. Though since point (1) does not hold, given a detailed answer to this question is moot.

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