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.[2]
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,