Electrostatic shielding by an open hemispherical conductor: an exactly solvable boundary-element case study for undergraduates

Abstract A grounded, zero-thickness hemispherical conductor of radius \(R\) with a point charge \(q\) at the centre of its parent sphere is superficially close to the familiar closed conducting shell, yet the open rim changes both the electrostatics and the numerical analysis: shielding becomes partial, and the induced surface charge is singular at the rim. The problem also admits a classical closed-form solution through Kelvin's treatment of the charged spherical bowl, which we rederive in a form suited to an upper-undergraduate course: \(Q_{\mathrm{ind}}/q=-(\tfrac12+1/\pi)\), the pole density is \(\sigma(0)/\sigma_0=-(\tfrac12+2/\pi)\) with \(\sigma_0=q/(4\pi R^2)\), the rim divergence is \(\sigma/\sigma_0\sim-(2/\pi)/\sqrt{\cos\theta}\), and the residual far-field monopole is exactly \((\tfrac12-1/\pi)q\). We then use the exact solution as the verification target of an axisymmetric boundary-element calculation in which the inverse-square-root edge singularity is extracted analytically by the substitution \(u=\sqrt{\cos\theta}\). The comparison exposes a cautionary result of direct pedagogical value: a mesh sequence whose successive values change at the \(10^{-7}\) level can appear converged while its true error, dominated by the slowly decaying quadrature error of the fixed panel rule, remains several times larger---a discrepancy revealed only by comparison with a refined quadrature rule and the exact solution. We quantify the full error budget---discretisation, panel quadrature, and an independent boundary-potential check---and organise the material as a predict--compute--compare activity in which students must coordinate physical and algorithmic reasoning. Starter and reference codes are provided as supplementary material.