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I am in Purcell's E&M book at the section explaining why the field is zero inside a hollow conductor of any shape. The proof given is that the potential function inside the conductor must obey Laplace's equation, and that the boundary of the region (in this case a rectangular metal box) is an equipotential. It calls the potential on the box some constant phi, then states that one solution (i.e. value of the potential inside the box) is that the potential is constant throughout the volume and equal to phi, the potential on the surface. Then it states by the uniqueness theorem that this is the only solution.

I understand all of this except why one couldn't say: one solution is that the potential is constant throughout the volume and *not equal to* phi (the potential on the surface of the box still being equal to phi of course). Therefore this is the only solution. The proof would come out the same, but it is this mathematical step that confuses me. Thanks,

Tomkat