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A Faraday cage is best understood as an approximation to an ideal hollow conductor.
In the case of a hollow conductor (Faraday cage), the equipotential region includes the space inside.
If Coulomb's law is not exactly valid, then that would cause the presence of an electric field inside a hollow conductor when it is subjected to an external electric field.
Faraday cup utilizes a physical principle according which the electrical charges delivered to the inner surface of a hollow conductor are redistributed around its outer surface due to mutual self-repelling of charges of the same sign - phenomenon discovered by Faraday (angl.