Everything about Multipole Expansion totally explained
A
multipole expansion is a
mathematical series representing a
function that depends on angles — usually
the two angles on a sphere. These series are useful because they can often be
truncated, meaning that only the first few terms need to be retained for a good approximation to the original function. The function being expanded may be
complex in general. Multipole expansions are very frequently used in the study of
electromagnetic, and
gravitational fields, where the fields at distant points are given in terms of sources in a small region. The multipole expansion with angles is often combined with an expansion in radius. Such a combination gives an expansion describing a function throughout three-dimensional space.
The multipole expansion is expressed as a sum of terms with progressively finer angular features. For example, the initial term — called the zero-th, or monopole,
moment — is a constant, independent of angle. The following term — the first, or
dipole, moment — varies once from positive to negative around the sphere. Higher-order terms (like the
quadrupole and
octupole) vary more quickly with angles.
Most commonly, the series is written as a sum of
spherical harmonics. Thus, we might write a function
as the sum
» potential is the
electric potential of an infinite line charge.
General mathematical properties
Mathematically, multipole expansions are related to the underlying rotational symmetry of the physical laws and their associated differential equations. Even though the source terms (such as the masses, charges, or currents) may not be symmetrical, one can expand them in terms of
irreducible representations of the rotational
symmetry group, which leads to spherical harmonics and related sets of
orthogonal functions. One uses the technique of
separation of variables to extract the corresponding solutions for the radial dependencies.
Further Information
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