Method of image charges

The method of image charges (also known as the method of images and method of mirror charges) is a basic problem-solving tool in electrostatics. The name originates from the replacement of certain elements in the original layout with imaginary charges, which replicates the boundary conditions of the problem (see Dirichlet boundary conditions or Neumann boundary conditions). The validity of the method of image charges rests upon a corollary of the uniqueness theorem, which states that the electric potential in a volume V is uniquely determined if both the charge density throughout the region and the value of the electric potential on all boundaries are specified. Alternatively, application of this corollary to the differential form of Gauss' Law shows that in a volume V surrounded by conductors and containing a specified charge density ρ, the electric field is uniquely determined if the total charge on each conductor is given. Possessing knowledge of either the electric potential or the electric field and the corresponding boundary conditions we can swap the charge distribution we are considering for one with a configuration that is easier to analyze, so long as it satisfies Poisson's equation in the region of interest and assumes the correct values at the boundaries. The simplest example of method of image charges is that of a point charge, with charge q, located at above an infinite grounded (i.e.: ) conducting plate in the xy-plane. To simplify this problem, we may replace the plate of equipotential with a charge −q, located at . This arrangement will produce the same electric field at any point for which (i.e., above the conducting plate), and satisfies the boundary condition that the potential along the plate must be zero. This situation is equivalent to the original setup, and so the force on the real charge can now be calculated with Coulomb's law between two point charges.