A rectangular metal box is earthed (held at ) on five of its six faces; the sixth face

PHYSICS · Subjective · 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

A rectangular metal box is earthed (held at V=0V=0) on five of its six faces; the sixth face, the top, is left open. The interior of the box is vacuum, with no free charge anywhere inside. Some external apparatus maintains a known field entering through the open top, so that on that face the outward normal derivative of the potential is fixed at ∂V/∂n=−E0(x,y)\partial V/\partial n = -E_0(x,y), with E0(x,y)E_0(x,y) a given function and n^\hat{\mathbf n} the outward normal of the box. Suppose V1(r)V_1(\mathbf r) and V2(r)V_2(\mathbf r) are two functions, each satisfying ∇2V=0\nabla^2 V=0 throughout the interior, each equal to zero on the five earthed faces, and each satisfying ∂V/∂n=−E0(x,y)\partial V/\partial n=-E_0(x,y) on the open face. Using Green's first identity, show that V1=V2V_1=V_2 everywhere inside the box.

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