A volume V in vacuum is bounded by a surface S consisting of several separate conducting surfaces S_i. One conductor is held at unit potential and all the other conductors at zero potential.

(a) Show that the capacitance of the one conductor is given by

C = є₀∫|ÑФ|² d³x

Where Ф(х) is the solution for the potential.

(b) Show that the true capacitance С is always less than or equal to the quantity

C(Ψ) = є₀∫_v|ÑΨ|²d³x

Where ψ is any trial function satisfying the boundary conditions on the conductors. This is a variational principle for the capacitance that yields an upper bound.