 Conducting hemispherical protrusion in a uniform field

A grounded hemispherical boss of radius $a$ on the grounded plane $z=0$ in an otherwise uniform downward electric field has the displayed exterior electric potential, with $r^2=x^2+y^2+z^2$. It vanishes both on the hemisphere $r=a$ and on the surrounding plane, and tends to $E_{0}z$ at large distance. The outward normal field gives surface charge $-3{\epsilon }_0{E}_0\cos \theta$ on the hemisphere and $-{\epsilon }_0{E}_0(1-a^3/s^3)$ on the plane at cylindrical radius $s>a$. The apex field is $3E_0$, so a field-limited breakdown voltage is reduced by a factor three. For a capacitor with finite spacing $d\gg a$, this is a leading local approximation; it is not an exact solution on the distant second plate.