Image charge at a dielectric interface ($q^{\prime }=q({ϵ}_1-{ϵ}_2)/({ϵ}_1+{ϵ}_2)$)

For a point electric charge $q$ in a homogeneous medium of permittivity ${ϵ}_1$ beside a plane interface with permittivity ${ϵ}_2$, its potential on the source side is reproduced by a mirrored image charge $q^{\prime }=q({ϵ}_1-{ϵ}_2)/({ϵ}_1+{ϵ}_2)$. The interface has no free surface charge; continuity of tangential electric field and normal electric displacement field fixes this coefficient.

For a real charge $Q$ in medium 1, the potential in medium 2 can be represented by the source $Q_t=2{ϵ}_{2}Q/({ϵ}_1+{ϵ}_2)$ using medium 2's Coulomb prefactor $1/(4\pi {ϵ}_2)$. The same potential uses $2{ϵ}_{1}Q/({ϵ}_1+{ϵ}_2)$ if medium 1's prefactor is retained instead. These different effective source labels do not change the physical field. A decomposition $Q_t=Q+Q^{\prime }$ must state which prefactor it uses.