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If a conductor is connected to the earth, the potential of the conductor becomes equals to that of the earth, i.e, zero. (Potential of earth is often taken as zero).
If the conductor was at some other potential, charges will flow from it to the earth to bring it's potential to zero.
Suppose two identical metal plates having charges q1 and a2 are placed facing each other at some seperation, and if we join the right (second) plate with the ground, then the charge appearing on the outer surface of the right plate will go to the earth.
Any metal object connected to earth need not to have zero charge. If a conductiong bodyis isolated to the earth, it will have no charge. If the conducting bodyis connected to the earthhave any charged object near it to the bdy will not have zero charge,.