 Dipole-induced charge on a grounded plane

A normal electric dipole comprising $+q$ at height $h$ and $-q$ at height $h+d$, with $d\ll h$, induces the displayed leading surface charge density on a grounded plane. The image charge construction first gives ${\sigma }_q(a)=-qh/[2\pi (a^2+h^2{)}^{3/2}]$ for one point charge; subtraction at heights $h$ and $h+d$ gives $-d{\partial }_h{\sigma }_q$ to first order. The central negative region ends at $a=\sqrt{2}h$, outside which the induced surface charge density is positive. Integrating over a disk gives $Q(<a)=-qd{a}^2/(a^2+h^2{)}^{3/2}$ to this order, so the total induced electric charge is zero. At finite $d$, the exact sign-change radius obeys $a^2=h^{2/3}(h+d{)}^{2/3}[h^{2/3}+(h+d{)}^{2/3}]$.