Method of image charges

The name originates from the replacement of certain elements in the original layout with fictitious 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.

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

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.

above an infinite grounded conducting plane in the xy-plane is a dipole moment at

with equal magnitude and direction rotated azimuthally by π.

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, Similar to the conducting plane, the case of a planar interface between two different dielectric media can be considered.

) 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.

[3] Unlike the case of the metal, the image charge

It may not 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).

Referring to the figure, we wish to find the potential inside a grounded sphere of radius R, centered at the origin, due to a point charge inside the sphere at position

(For the opposite case, the potential outside a sphere due to a charge outside the sphere, the method is applied in a similar way).

The image of this charge with respect to the grounded sphere is shown in red.

due to both charges alone is given by the sum of the potentials: Multiplying through on the rightmost expression yields: and it can be seen that on the surface of the sphere (i.e. when r = R), the potential vanishes.

If we assume for simplicity (without loss of generality) that the inner charge lies on the z-axis, then the induced charge density will be simply a function of the polar angle θ and is given by: The total charge on the sphere may be found by integrating over all angles: Note that the reciprocal problem is also solved by this method.

The image of an electric point dipole is a bit more complicated.

If the dipole is pictured as two large charges separated by a small distance, then the image of the dipole will not only have the charges modified by the above procedure, but the distance between them will be modified as well.

lying inside the sphere of radius R will have an image located at vector position

(i.e. the same as for the simple charge) and will have a simple charge of: and a dipole moment of: The method of images for a sphere leads directly to the method of inversion.

are the spherical coordinates of the position, then the image of this harmonic function in a sphere of radius R about the origin will be If the potential

, then the image potential will be the result of a series of charges of magnitude

, then the image potential will be the result of a charge density

The field of a positive charge above a flat conducting surface, found by the method of images.
Method of images for an electric dipole moment in a conducting plane
Diagram illustrating the image method for Laplace's equation for a sphere of radius R . The green point is a charge q lying inside the sphere at a distance p from the origin, the red point is the image of that point, having charge − qR / p , lying outside the sphere at a distance of R 2 / p from the origin. The potential produced by the two charges is zero on the surface of the sphere.
Field lines outside a grounded sphere for a charge placed outside the sphere.
Several surfaces require an infinite series of point image charges.