The Dirichlet Green function for the region outside an earthed conducting sphere of radius

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Electricity, Magnetism & Electromagnetism · Conductors, capacitance and boundary-value electrostatics

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

Problem

The Dirichlet Green function for the region outside an earthed conducting sphere of radius aa, centred at the origin, is built from the sphere's image-charge solution. For a unit point charge at r′\mathbf r' (with r′=∣r′∣>ar'=|\mathbf r'|>a), the image charge −a/r′-a/r' sits at (a2/r′2)r′(a^2/r'^2)\mathbf r', so that for any field point r\mathbf r (with r=∣r∣>ar=|\mathbf r|>a) outside the sphere, G(r,r′)=14πε0[1∣r−r′∣−a/r′∣r−(a2/r′2)r′∣].G(\mathbf r,\mathbf r')=\frac{1}{4\pi\varepsilon_0}\left[\frac{1}{|\mathbf r-\mathbf r'|}-\frac{a/r'}{\left|\mathbf r-(a^2/r'^2)\mathbf r'\right|}\right]. Let γ\gamma be the angle between r\mathbf r and r′\mathbf r'. Show, by direct calculation, that G(r,r′)=G(r′,r)G(\mathbf r,\mathbf r')=G(\mathbf r',\mathbf r) for every pair of points r,r′\mathbf r,\mathbf r' outside the sphere.

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