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I've read that electric field inside a conductor is zero, and that potential of a two points say A and B on the surface of the conductor is same (constant). The latter puts me in a state of confusion. If $V$ is constant then there is no electric field. But there is an accumulation of charges on the surface of a conductor. Then why is $V$ constant? Shouldn't it vary since the electric field on the surface of the conductor is NOT zero?

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The potential difference between two points in an electric field can be zero in two cases. The first one, as you point out, is when $\vec E = 0$.

The second case however, is when $\vec E.d \vec s$ is always $0$, where $d\vec s$ is an infinitesimal displacement used in path integrals. This can happen when vectors $\vec E$ and $d\vec s$ are perpendicular.

On the surface of a conductor, the electric field $\vec E$, due to charges on it's surface are always perpendicular to the surface at any point (Otherwise, there would be a component of electric field along the surface, and the charges would move. The conductor would then no longer be in electrostatic equilibrium.) Hence, for any two points $A$ and $B$ on the surface of the conductor: $$\Delta V = -\int_{A}^B \vec E.d \vec s = 0 => V_A = V_B$$

Hope this helps.

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