= Induced charge as harmonic measure
{title2=$Q_j=-q\,w_j(\mathbf r_0)$}
In a grounded conducting enclosure, let the <harmonic function> $w_j$ equal one on boundary component $j$ and zero on the others. A <point charge> $q$ at $\mathbf r_0$ induces total <electric charge> $Q_j=-q w_j(\mathbf r_0)$ on that component. The weighting <harmonic function> is its <harmonic measure>.
To prove the identity, let $\phi$ be the zero-boundary <electric potential>, so $\nabla^2\phi=-q\delta^{(3)}(\mathbf r-\mathbf r_0)/\epsilon_0$. <Green's second identity> gives $-qw_j(\mathbf r_0)/\epsilon_0=\int_{\partial D}w_j\partial_n\phi\,dS$, where $\mathbf n$ points out of the vacuum domain into the metal. The <electrostatic boundary conditions at a conductor> give $\sigma=\epsilon_0\partial_n\phi$, proving the result. In an unbounded domain require the additional boundary term at infinity to vanish. Between concentric spheres, $w_a(r)=a(b-r)/(r(b-a))$; between parallel planes $z=0,d$, $w_0=1-z/d$ and $w_d=z/d$.
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