Method of image charges

Remarks

Example (plane): let $\Omega = \set{x_3 > 0}$ and let a charge $q$ sit at $a = (0, 0, d)$ with $d > 0$. The image is the charge $-q$ at $a' = (0, 0, -d)$. Each point $x$ with $x_3 = 0$ is at the same distance from $a$ and $a'$, so the two potentials cancel there. Example (sphere): let $\Omega$ be the exterior of the ball of radius $R$ centred at $0$, and let a charge $q$ sit at distance $d > R$ from $0$. The image is the charge $-qR/d$ at the point on the same ray at distance $R^2/d$ from $0$. Non-example: for the plane, the charge $+q$ at $a'$ is not an image charge, since the potential on $x_3 = 0$ is then twice that of $q$, not zero. With $q = 1$ and $\varepsilon_0$ removed, the potential of the charge and its images is the Green's function of $\Omega$ for the Dirichlet problem, up to a constant factor.