= Electric dipole above a grounded conducting plane
{title2=$U=-p^2(1+\cos^2\theta)/(64\pi\epsilon_0a^3)$}
An ideal <electric dipole> at height $a$ above a grounded plane has an image <electric dipole moment> with reversed tangential component and unchanged normal component. If $\theta$ is its angle to the normal, its conductor-response <potential energy> is the displayed expression. The factor one half in $U=-\mathbf p\cdot\mathbf E_{\rm im}/2$ accounts for induced <surface charge>. Therefore its downward <force> magnitude is $3p^2(1+\cos^2\theta)/(64\pi\epsilon_0a^4)$, and its orienting <torque> follows from $-\partial_\theta U$. The two normal orientations are stable against tilting; the parallel orientation is unstable. The net induced plane <electric charge> is zero for a neutral source, although its local <surface charge density> is nonzero.
Back to article page
