Static Multipole Field

The manner in which static multipole moments of a source give rise to a multipole field is illustrated by the following

Problem (Multipole field of an asymmetric static source).

Given:

1. The potential inside and outside a sphere of radius $r=r_0$ satisfies the Laplace equation
   $$\nabla^2\psi =0\,.$$

2. The value of the potential on the sphere is
   $$\psi (r_0,\theta ,\varphi )=\frac{g(\theta ,\varphi )}{r_0}\,.$$

Find the potential $\psi (r,\theta ,\varphi )$ inside $(r<r_0)$ and outside $(r_0<r)$ the sphere.

The potential may be an electrical potential, in which case its value on the sphere is determined by the charge distribution on sphere. By contrast, if the potential is a gravitational potential, its value on the sphere is determined by the mass distribution.

In either case, the governing equation would be

$$\nabla^2\psi =-\frac{\delta (r-r_0)}{r^2} g(\theta ,\varphi )\,,$$

as one can verify after we have found the solution to the given problem.

There are two boundary conditions implicit in the given problem, namely

$$\psi (r=0,\theta ,\varphi )=\textrm{finite}$$

and

$$\psi (r=\infty ,\theta ,\varphi )=0\,.$$

The second boundary condition expresses the fact that there are no masses (or charges) distributed at very large $r$ . The two boundary conditions demand that the radial part of the potential be

$$R_\ell (r)\propto \left\{\begin{array}{cl} \left(\frac{r}{r_0}\right)^\ell &r<r_0\\ \left(\frac{r_0}{r}\right)^{\ell +1} &r_0<r\end{array}\right.$$

The total solution is, therefore,

$$\psi(r,\theta,\phi) = \sum^\infty_{\ell =0}\left\{ \begin{a... ...ay}{l} \textrm{INSIDE}\\ \\ \textrm{OUTSIDE}~, \end{array}$$

where

$$\langle Y^m_\ell ,g\rangle = \int^\pi_0 \int^{2\pi}_0 \overline{Y}^m_\ell (\theta ,\varphi )g(\theta ,\varphi )\sin\theta d\theta d\varphi \,.$$

Let us exhibit explicitly the exterior potential. It is a superposition of various ``multipole'' potential fields,

$$\psi = Y^0_0 (\theta ,\varphi )\frac{\langle Y^0_0 ,g\rangle}{r} + \sum^1_{m=-1} Y^m_1 (\theta ,\varphi )\frac{\langle Y^m_1,g\rangle r_0} {r^2}$$

$$+\sum^2_{m=-2} Y^m_2(\theta ,\varphi )\frac{\langle Y^m_2,g\rangle r^2_0} {r^3} +\cdots$$

They are called the monopole, dipole, quadrupole, ... and $2^\ell$ -pole fields respectively. The constant numerators express the source strengths of the fields. These numerators are called the monopole moment, dipole moment (which has three components), quadrupole moment (which has five components), $\dots$ and $2^\ell$ -pole moment (which has $2\ell +1$ components). Each one of them is an example of a multipole moment.

Analogous descriptive names hold for the interior field.