Classical Electrodynamics · Problem 1.17

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

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

Problem

A volume VV in vacuum is bounded by a surface SS consisting of several separate conducting surfaces SiS_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=ϵ0∫V∣∇Φ∣2 d3xC = \epsilon_0 \int_V |\nabla\Phi|^2\, d^3x where Φ(x)\Phi(\mathbf{x}) is the solution for the potential. ((b)) Show that the true capacitance CC is always less than or equal to the quantity C[Ψ]=ϵ0∫V∣∇Ψ∣2 d3xC[\Psi] = \epsilon_0 \int_V |\nabla\Psi|^2\, d^3x where Ψ\Psi is any trial function satisfying the boundary conditions on the conductors. This is a variational principle for the capacitance that yields an upper bound.

Browse the problem library