In estimating the capacitance of a given configuration of conductors, comparison with known capacitances is often helpful. Consider two configurations of n conductors in which the (n – 1) conductors held at zero potential are the same, but the one conductor whose capacitance we wish to know is different. In particular, let the conductor in one configuration have a closed surface S1 and in the other configuration have surface S1’, with S1’ totally inside S1.
(a) Use the extremum principle of Section 1.12 and the variational principle of Problem 1.17 to prove that the capacitance C’ of the conductor with surface S’1 is less than or equal to the capacitance С of the conductor with surface S1 that encloses S1’.
(b) Set upper and lower limits for the capacitance of a conducting cube of side a. Compare your limits and also their average with the numerical value, С ( 0.655(4πє0a).
(c) By how much do you estimate the capacitance per unit length of the two-wire system of Problem 1.7 will change (larger? smaller?) if one of the wires is replaced by a wire of square cross section whose side is equal to its diameter?