Image charge

The method of images is a trick for solving electrostatics problems involving conductors. Instead of integrating over the induced charge distribution on the conductor's surface — which is usually unknown in advance — you replace the conductor with a cleverly placed fictitious "image charge" that, together with the real charge, produces exactly the right boundary conditions.

The canonical example is a point charge +q above an infinite grounded conducting plane. The conductor must be an equipotential at V = 0. Remove the conductor entirely, and instead place a fictitious −q the same distance below where the plane used to be. The two charges together produce a potential that is zero on the plane by symmetry. In the half-space above the plane, the field is identical to the original problem's — but now you have a simple two-charge system you can compute by inspection. The image charge is not real; no actual charge exists below the plane. It is a calculational device that reproduces the effect of the induced surface charges on the real conductor.

The method works because Laplace's equation has unique solutions once boundary conditions are specified. If you find any field that satisfies the equation in the region of interest and matches the boundaries, you have found the field. Image configurations are known for a point charge near a grounded plane, near a grounded sphere, between two intersecting planes, and a handful of other high-symmetry cases. Beyond those, numerical methods take over — but within their domain, images turn problems that would require surface integrals into arithmetic.

Siméon Denis Poisson introduced the method of images in his 1813 work on electrostatics and potential theory, building on the mathematical framework Laplace had developed for gravitation. William Thomson (Lord Kelvin) extended it dramatically in 1848, applying it to the grounded sphere and giving the method its modern, fully geometrical form.