“Spherical multipole moments”的意思、由来-开放百科全书

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Spherical multipole moments

释义

Spherical multipole moments of a point charge
General spherical multipole moments
Note
Interior spherical multipole moments
Interaction energies of spherical multipoles
Special case of axial symmetry
See also
External links

{{Unreferenced|date=October 2010}}Spherical multipole moments are the coefficients in a series expansion

of a potential that varies inversely with the distance R to a source, i.e., as 1/R. Examples of such potentials are the electric potential, the magnetic potential and the gravitational potential.

For clarity, we illustrate the expansion for a point charge, then generalize to an arbitrary charge density . Through this article, the primed coordinates such as

refer to the position of charge(s), whereas the unprimed coordinates such as refer to the point at which the potential is being observed. We also use spherical coordinates throughout, e.g., the vector has coordinates where is the radius, is the colatitude and is the azimuthal angle.

Spherical multipole moments of a point charge

The electric potential due to a point charge located at is given by

where

is the distance between the charge position and the observation point

and is the angle between the vectors and .

If the radius of the observation point is greater than the radius of the charge,

we may factor out 1/r and expand the square root in powers of using Legendre polynomials

This is exactly analogous to the axialmultipole expansion.

We may express in terms of the coordinates

of the observation point and charge position using the

spherical law of cosines (Fig. 2)

Substituting this equation for into

the Legendre polynomials and factoring the primed and unprimed

coordinates yields the important formula known as the spherical harmonic addition theorem

where the functions are the spherical harmonics.

Substitution of this formula into the potential yields

which can be written as

where the multipole moments are defined

.

As with axial multipole moments, we may also consider

the case when

the radius of the observation point is less

than the radius of the charge.

In that case, we may write

which can be written as

where the interior spherical multipole moments are defined as the complex conjugate of irregular solid harmonics

The two cases can be subsumed in a single expression if

and are defined

to be the lesser and greater, respectively, of the two

radii and ; the

potential of a point charge then takes the form, which is sometimes referred to as Laplace expansion

General spherical multipole moments

It is straightforward to generalize these formulae by replacing the point charge

with an infinitesimal charge element

and integrating. The functional form of the expansion is the same

where the general multipole moments are defined

Note

The potential Φ(r) is real, so that the complex conjugate of the expansion is equally valid. Taking of the complex conjugate leads to a definition of the multipole moment which is proportional to Y_lm, not to its complex conjugate. This is a common convention, see molecular multipoles for more on this.

The free Python package [https://github.com/maroba/multipoles multipoles] is available for computing spherical multipole moments and multipole expansions.