Past exam of the physics B course of the University of Cambridge / 2018 / Paper 1 / Section B / B6 / Breakdown voltage / Solution

The largest gap field is at the apex: $|E{|}_{\max }=3V/d$. If the fixed breakdown threshold is $E_{\mathrm{bd}}$, a smooth parallel-plate gap permits $V_{\mathrm{flat}}=E_{\mathrm{bd}}d$, whereas this small hemispherical boss permitsThus the maximum voltage is reduced by a factor of three, within the stated local-field approximation. This comparison holds the material dielectric strength and gap $d$ fixed.