Classical Electrodynamics · Problem 1.20

Boundary-Value Problems in Electrostatics: I — John David Jackson

Physics — E&M · capacitance-comparison-conductors

Problem

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

S_1

and in the other configuration have surface

S_1'

, with

S_1'

totally inside

S_1

. ((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

C

of the conductor with surface

S_1

that encloses

S_1'

. ((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,

C \approx 0.655(4\pi\epsilon_0 a)

. ((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?

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