multipolexpansies

Multipole expansions are a mathematical technique used to approximate a physical quantity, such as an electric or magnetic potential, at points far from its source. This approximation is achieved by representing the quantity as a sum of terms, each corresponding to a "multipole" of increasing order. The lowest order term is the monopole, representing the total charge or dipole moment of the source. Subsequent terms, such as the dipole, quadrupole, and octupole, capture more detailed features of the source distribution.

The utility of multipole expansions lies in their ability to simplify complex problems. When the observation point is sufficiently far from the source, the higher-order multipole terms become progressively smaller and can often be neglected, yielding a manageable approximation. This is particularly useful in electromagnetism, where the potential created by a distribution of charges or currents can be complex to calculate precisely. By using a multipole expansion, one can often determine the dominant behavior of the field at large distances with just the first few terms.

The mathematical formulation involves expressing the potential as a series expansion in terms of spherical harmonics and radial functions. The coefficients of this expansion are determined by integrals over the source distribution. The monopole term is simply proportional to the total charge. The dipole term depends on the dipole moment, which is a vector quantity related to the first moment of the charge distribution. Higher-order terms involve more complex moments of the source distribution. The convergence of the expansion is guaranteed for distances greater than the extent of the source.