Question 3.1: Show that the potential is constant inside an enclosure comp...

Introduction to Electrodynamics 4th. Edition [EXP-2861]

Show that the potential is constant inside an enclosure completely surrounded by conducting material, provided there is no charge within the enclosure.

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The potential on the cavity wall is some constant, $V_{0}$ (that’s item (iv), in Sect. 2.5.1), so the potential inside is a function that satisfies Laplace’s equation and has the constant value $V_{0}$ at the boundary. It doesn’t take a genius to think of one solution to this problem: $V=V_{0}$ everywhere. The uniqueness theorem guarantees that this is the only solution. (It follows that the field inside an empty cavity is zero—the same result we found in Sect. 2.5.2 on rather different grounds.)

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