Classical Electrodynamics · Problem 1.18

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

Physics — E&M · capacitance-variational-lower-bound

Problem

Consider the configuration of conductors of Problem 1.17, with all conductors except

S_1

held at zero potential. ((a)) Show that the potential

\Phi(\mathbf{x})

anywhere in the volume

V

and on any of the surfaces

S_i

can be written

\Phi(\mathbf{x}) = \frac{1}{4\pi\epsilon_0} \oint_{S_1} \sigma_1(\mathbf{x}')G(\mathbf{x}, \mathbf{x}')\, da'

where

\sigma_1(\mathbf{x}')

is the surface charge density on

S_1

and

G(\mathbf{x}, \mathbf{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

S_1

absent). Show also that the electrostatic energy is

W = \frac{1}{8\pi\epsilon_0} \oint_{S_1} da \oint_{S_1} da'\, \sigma_1(\mathbf{x})G(\mathbf{x}, \mathbf{x}')\sigma_1(\mathbf{x}')

where the integrals are only over the surface

S_1

. ((b)) Show that the variational expression

C^{-1}[\sigma] = \frac{\displaystyle\oint_{S_1} da \oint_{S_1} da'\, \sigma(\mathbf{x})G(\mathbf{x}, \mathbf{x}')\sigma(\mathbf{x}')}{4\pi\epsilon_0\left[\displaystyle\oint_{S_1} \sigma(\mathbf{x})\, da\right]^2}

with an arbitrary integrable function

\sigma(\mathbf{x})

defined on

S_1

, is stationary for small variations of

\sigma

away from

\sigma_1

. Use Thomson's theorem to prove that the reciprocal of

C^{-1}[\sigma]

gives a lower bound to the true capacitance of the conductor

S_1