(Solution) - Consider the configuration of conductors of Problem 1 17 with a -(2025 Original AI-Free Solution)

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Consider the configuration of conductors of Problem 1.17, with all conductors except S1 held at zero potential.
(a) Show that the potential ?(?) anywhere in the volume V and on any of the surfaces Si can be written

Where ?1 (?') is the surface charge density on S1 and G(x, x') is the Green function potential for a point charge in the presence of all the surfaces that are held at zero potential (but with S1 absent). Show also that the electrostatic energy is



Where the integrals are only over the surface S1
(b) Show that the variational expression



With an arbitrary integrable function ?(x) defined on S1, is stationary for small variations of ? away from ?1. Use Thomson's theorem to prove that the reciprocal of ??1[?] gives a lower bound to the true capacitance of the conductorS1.