Electric dipole above a grounded conducting plane ($U=-p^2(1+{\cos }^2\theta )/(64\pi {ϵ}_0{a}^3)$)

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 $U=-\mathbf{p}\cdot {\mathbf{E}}_{\mathrm{im}}/2$ accounts for induced surface charge. Therefore its downward force magnitude is $3p^2(1+{\cos }^2\theta )/(64\pi {ϵ}_0{a}^4)$, 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.