= Conducting hemispherical protrusion in a uniform field
{title2=$\Phi=E_0z(1-a^3/r^3)$}
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 $E_0z$ at large distance. The outward normal field gives surface charge $-3\varepsilon_0E_0\cos\theta$ on the hemisphere and $-\varepsilon_0E_0(1-a^3/s^3)$ 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.
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