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If I have a charge $q$ above a grounded conducting plane, I know I can calculate the potential of the charge using image method. Since the plane is grounded, would the potential below the plane be $V=0$, assuming there is no charge below the grounded conducting plane?

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You are right, an infinite grounded conducting plane will shield the lower $1/2$ plane from charges above the plane. If the plane has $0$ resistivity, it will also shield changing magnetic field sources, such as currents and magnets.

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