= Dipole-induced charge on a grounded plane
{title2=$\sigma(a)=\dfrac{qd(a^2-2h^2)}{2\pi(a^2+h^2)^{5/2}}$}
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 $a=\sqrt2h$, 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 $a^2=h^{2/3}(h+d)^{2/3}[h^{2/3}+(h+d)^{2/3}]$.
Back to article page
