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As I understand it, the potential of a charged conductor is the work done in bringing a unit positive charge to its surface from infinity. Is this correct?

When a conductor is grounded, then any excess electrons on it go into the ground. Is this correct? If so, why?

Assuming that they do go into the ground, the conductor now has no charge and therefore produces no field. So the integral of -E dr from infinity must be zero. Is this why the potential of a grounded conductor is zero?