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Why must the electrostatic potential inside a hollow charged conductor be the same at every point?

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Last updated date: 28th Feb 2024
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Hint Electric field lines originate from a positive charge and terminate on a negative charge. For a potential inside a surface to be zero, the work done in moving a unit test charge in the region must also be zero.

Complete step by step answer
For a charged hollow conductor, since the entire surface of the conductor has the same charge on it, no electric field line can originate from one point on the surface and end on another point on the surface as electric field lines must originate from a positive charge and end on a negative charge. But since we only have one kind of charge on the surface, electric field lines cannot pass through the inside of the conductor and hence we can say that there is no electric field inside the conductor.
Since the electric field is zero, from the relation $ E = - dV/dr $ , we can conclude the potential has to be constant since only then can the derivative of the potential with respect to the position can be zero. So we can conclude that the electrostatic potential inside the hollow charged conductor must be the same at every point and only then can the electric field inside the conductor be zero which we proved earlier.

Note
Even if there is a charged particle outside of the conductor, the conductor will induce a charge that is opposite and distributed such that the net charge on it will be equal in magnitude but opposite in polarity of the charged particle outside it. But there will be no electric field induced inside the conductor again since the surface of the conductor will have the same charge distributed all over its surface and hence the potential will be the same at every point inside the conductor.
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