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When a positive charge q is placed in the center of a spherical cavity of a spherical neutral conductor, it induces a negative charge $q$ to line the inner wall of the conductor, and a positive charge $q$ to line the outer wall. I've seen this described in Griffith's and Halliday, Resnick, and Walker.

At the center the point charge should not feel any force. My question is if the point charge is moved slightly off-center, will it feel a force? I've read in Griffiths and Purcell that charges cannot be arranged in stable equilibrium. Does this apply to induced charges as well? I am curious because I have not seen the implications of Laplace's equation and Earnshaw's theorem pointed out for this system.

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Yes the charge will be attracted to the point of the cavity closest to the charge. Earnshaw's theorem applies to the single charge, but not to the electrons in the conductor. This is because you need quantum mechanics to describe the metal. Earnshaw's theorem is only true for classical systems.

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