Two isolated conducting plates, A and B

PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Conductors, capacitance and boundary-value electrostatics · Conductors, capacitance and boundary-value electrostatics

The Boundary-Value Problem, Ch Uniqueness, images and the Green function of a region

Problem

Two isolated conducting plates, A and B, are held apart in vacuum inside a large earthed conducting enclosure. Plate A carries a fixed total charge +Q+Q and plate B a fixed total charge −Q-Q (charged once, then left isolated), and there is no other free charge in the enclosure. Neither plate's potential is specified. Suppose V1(r)V_1(\mathbf r) and V2(r)V_2(\mathbf r) are two potentials, each satisfying ∇2V=0\nabla^2V=0 in the vacuum region, each equal to zero on the earthed enclosure, and each consistent with the given charges, that is −ε0∮SA∂V∂n dS=Q,−ε0∮SB∂V∂n dS=−Q,-\varepsilon_0\oint_{S_A}\frac{\partial V}{\partial n}\,dS=Q,\qquad -\varepsilon_0\oint_{S_B}\frac{\partial V}{\partial n}\,dS=-Q, with n^\hat{\mathbf n} the outward normal of each plate. ((a)) Using Green's first identity applied to W=V1−V2W=V_1-V_2, show that V1=V2V_1=V_2 everywhere in the enclosure, so that both the field and the potential of each plate are uniquely determined by QQ. ((b)) Identify which piece of the data given above plays the role of the reference that pins down the plates' actual potential values (and not merely the field), and say what would be left undetermined if that piece of data were absent.

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