Induced charge as harmonic measure ($Q_j=-q{w}_j({\mathbf{r}}_0)$)

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 $\varphi$ be the zero-boundary electric potential, so ${\nabla }^2\varphi =-q{\delta }^{(3)}(\mathbf{r}-{\mathbf{r}}_0)/{ϵ}_0$. Green's second identity gives $-q{w}_j({\mathbf{r}}_0)/{ϵ}_0={\int }_{\partial D}{w}_j{\partial }_n\varphi dS$, where $\mathbf{n}$ points out of the vacuum domain into the metal. The electrostatic boundary conditions at a conductor give $\sigma ={ϵ}_0{\partial }_n\varphi$, 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 $w_d=z/d$.