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 formula_1.
For clarity, we first illustrate the expansion for a single point charge, 
then generalize to an arbitrary charge density formula_2
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 formula_3 (Fig. 1) equals 
If the radius "r" of the observation point is greater than "a", 
we may factor out formula_5 and expand the square root
in powers of formula_6 using Legendre polynomials
where the axial multipole moments 
formula_8 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 formula_9, the axial dipole
moment formula_10 and the axial quadrupole
moment formula_11. 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 formula_12 and expand
in powers of formula_13, once again using Legendre polynomials
where the interior axial multipole moments 
formula_15 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 formula_16, where 
formula_17 represents the charge density at
position formula_18 on the "z"-axis. If the radius "r"
of the observation point P is greater than the largest 
formula_19 for which formula_17
is significant (denoted formula_21), the electric potential 
may be written
where the axial multipole moments formula_23 are defined 
Special cases include the axial monopole moment (=total charge)
the axial dipole moment formula_26, and
the axial quadrupole moment formula_27.
Each successive term in the expansion varies inversely with 
a greater power of formula_28, e.g., the monopole potential 
varies as formula_5, the dipole potential varies 
as formula_30, the quadrupole potential varies 
as formula_31, etc. Thus, at large distances
(formula_32), 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
formula_33 would be
Expanding the polynomial under the integral
leads to the equation
If the lower moments formula_37
are zero, then formula_38. 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
formula_19 for which formula_17
is significant (denoted formula_41), the electric potential may be written
where the interior axial multipole moments formula_43 are defined
Special cases include the interior axial monopole moment (formula_45 the total charge)
the interior axial dipole moment formula_47,
etc. Each successive term in the expansion varies with a greater power of formula_28, e.g., 
the interior monopole potential varies as formula_28, the dipole potential varies 
as formula_50, etc. At short distances (formula_51), 
the potential is well-approximated by the leading nonzero interior multipole term.