# Axial multipole moments

**Axial multipole moments** are a series expansion
of the electric potential of a
charge distribution localized close to
the origin along one
Cartesian axis,
denoted here as the *z*-axis. However,
the axial multipole expansion can also be applied to
any potential or field that varies inversely
with the distance to the source, i.e., as .
For clarity, we first illustrate the expansion for a single point charge,
then generalize to an arbitrary charge density
localized to the *z*-axis.

## Axial multipole moments of a point charge

The electric potential of a point charge *q* located on
the *z*-axis at (Fig. 1) equals

If the radius *r* of the observation point is **greater** than *a*,
we may factor out and expand the square root
in powers of using Legendre polynomials

where the **axial multipole moments**
contain everything
specific to a given charge distribution; the other parts
of the electric potential depend only on the coordinates
of the observation point **P**. Special cases include the axial
monopole moment , the axial dipole
moment and the axial quadrupole
moment . This
illustrates the general theorem that the lowest
non-zero multipole moment is independent of the
origin of the coordinate system,
but higher multipole moments are not (in general).

Conversely, if the radius *r* is **less** than *a*,
we may factor out and expand
in powers of , once again using Legendre polynomials

where the **interior axial multipole moments**
contain
everything specific to a given charge distribution;
the other parts depend only on the coordinates of
the observation point **P**.

## General axial multipole moments

To get the general axial multipole moments, we replace the
point charge of the previous section with an infinitesimal
charge element , where
represents the charge density at
position on the *z*-axis. If the radius *r*
of the observation point **P** is greater than the largest
for which
is significant (denoted ), the electric potential
may be written

where the axial multipole moments are defined

Special cases include the axial monopole moment (=total charge)

- ,

the axial dipole moment , and the axial quadrupole moment . Each successive term in the expansion varies inversely with a greater power of , e.g., the monopole potential varies as , the dipole potential varies as , the quadrupole potential varies as , etc. Thus, at large distances (), the potential is well-approximated by the leading nonzero multipole term.

The lowest non-zero axial multipole moment
is invariant under a shift *b* in
origin, but higher moments generally
depend on the choice of origin. The shifted multipole moments
would be

Expanding the polynomial under the integral

leads to the equation

If the lower moments are zero, then . The same equation shows that multipole moments higher than the first non-zero moment do depend on the choice of origin (in general).

## Interior axial multipole moments

Conversely, if the radius *r* is smaller than the smallest
for which
is significant (denoted ), the electric potential may be written

where the interior axial multipole moments are defined

Special cases include the interior axial monopole moment ( the total charge)

- ,

the interior axial dipole moment , etc. Each successive term in the expansion varies with a greater power of , e.g., the interior monopole potential varies as , the dipole potential varies as , etc. At short distances (), the potential is well-approximated by the leading nonzero interior multipole term.