Multipole moments

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Multipole moments are the coefficients of a series expansion of a potential, usually involving powers (or inverse powers) of the distance to the origin, as well as some angular dependence. In principle, a multipole expansion provides an exact description of the potential and generally converges under two conditions: (1) if the sources (e.g., charges) are localized close to the origin and the point at which the potential is observed is far from the origin; or (2) the reverse, i.e., if the sources (e.g., charges) are located far from the origin and the potential is observed close to the origin. In the first (more common) case, the coefficients of the series expansion are called exterior multipole moments or simply multipole moments whereas, in the second case, they are called interior multipole moments. The zeroth-order term in the expansion is called the monopole moment, the first-order term is denoted as the dipole moment, and the third, fourth, etc. terms are denoted as quadrupole, octupole, etc. moments.

The potential at a position $\mathbf{r}$ within a charge distribution can often be computed by combining interior and exterior multipoles.

[edit] Examples of multipoles

There are many types of multipole moments, since there are many types of potentials and many ways of approximating a potential by a series expansion, depending on the coordinates and the symmetry of the charge distribution. The most common expansions include:

Axial multipole moments of a $\frac{1}{R}$ potential,
Spherical multipole moments of a $\frac{1}{R}$ potential, and
Cylindrical multipole moments of a $\ln \ R^{ }$ potential

Examples of $\frac{1}{R}$ potentials include the electric potential, the magnetic potential and the gravitational potential of point sources. An example of a $\ln \ R^{ }$ potential is the electric potential of an infinite line charge.

[edit] General mathematical properties

Multipole moments in mathematics and mathematical physics form an orthogonal basis for the decomposition of a function, based on the response of a field to point sources that are brought infinitely close to each other. These can be thought of as arranged in various geometrical shapes, or, in the sense of distribution theory, as directional derivatives.

In practice, many fields can be well approximated with a finite number of multipole moments (although an infinite number may be required to reconstruct a field exactly). A typical application is to approximate the field of a localized charge distribution by its monopole and dipole terms. Problems solved once for a given order of multipole moment may be linearly combined to create a final approximate solution for a given source.