Multipole Moment

Multipole Moment

In mathematics, especially as applied to physics, multipole moments are the coefficients of a series expansion of a potential due to continuous or discrete sources (e.g., an electric charge distribution). A multipole moment usually involves 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(the second-order), fourth(the third-order), etc. terms are denoted as quadrupole, octupole, etc. moments.

The potential at a position within a charge distribution can often be computed by combining interior and exterior multipoles.