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.
Reflection in a conducting plane
Point charges
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.
The potential at any point in space, due to these two point charges of charge +q at +a and −q at −a on the z-axis, is given in cylindrical coordinates as
The surface charge density on the grounded plane is therefore given by
In addition, the total charge induced on the conducting plane will be the integral of the charge density over the entire plane, so:
The total charge induced on the plane turns out to be simply −q. This can also be seen from the Gauss's law, considering that the dipole field decreases at the cube of the distance at large distances, and the therefore total flux of the field though an infinitely large sphere vanishes.
Because electric fields satisfy the superposition principle, a conducting plane below multiple point charges can be replaced by the mirror images of each of the charges individually, with no other modifications necessary.
Electric dipole moments
The image of an electric dipole moment p at above an infinite grounded conducting plane in the xy-plane is a dipole moment at with equal magnitude and direction rotated azimuthally by π. That is, a dipole moment with Cartesian components will have in image dipole moment . The dipole experiences a force in the z direction, given by
and a torque in the plane perpendicular to the dipole and the conducting plane,
Reflection in a dielectric planar interface
Similar to the conducting plane, the case of a planar interface between two different dielectric media can be considered. If a point charge is placed in the dielectric that has the dielectric constant , then the interface (with the dielectric that has the dielectric constant ) will develop a bound polarization charge. It can be shown that the resulting electric field inside the dielectric containing the particle is modified in a way that can be described by an image charge inside the other dielectric. Inside the other dielectric, however, the image charge is not present.
Unlike the case of the metal, the image charge is not exactly opposite to the real charge: . It may even have the same sign, if the charge is placed inside the stronger dielectric material (charges are repelled away from regions of lower dielectric constant). This can be seen from the formula.