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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.
Figure 1: Point charge on the z axis; Definitions for axial multipole expansion
Axial multipole moments of a point charge[]
The electric potential of a point charge q located on
the z-axis at
(Fig. 1) equals
![{\displaystyle \Phi (\mathbf {r} )={\frac {q}{4\pi \varepsilon }}{\frac {1}{R}}={\frac {q}{4\pi \varepsilon }}{\frac {1}{\sqrt {r^{2}+a^{2}-2ar\cos \theta }}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ad0a2c41accf41bbb20529c0853e1edb82f934e)
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
![{\displaystyle \Phi (\mathbf {r} )={\frac {q}{4\pi \varepsilon r}}\sum _{k=0}^{\infty }\left({\frac {a}{r}}\right)^{k}P_{k}(\cos \theta )\equiv {\frac {1}{4\pi \varepsilon }}\sum _{k=0}^{\infty }M_{k}\left({\frac {1}{r^{k+1}}}\right)P_{k}(\cos \theta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fb447c90ffb3fb8dd26be19fb899af61c1b3775)
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
![{\displaystyle \Phi (\mathbf {r} )={\frac {q}{4\pi \varepsilon a}}\sum _{k=0}^{\infty }\left({\frac {r}{a}}\right)^{k}P_{k}(\cos \theta )\equiv {\frac {1}{4\pi \varepsilon }}\sum _{k=0}^{\infty }I_{k}r^{k}P_{k}(\cos \theta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e108b868bb02152ff0f6c040cfbccd384775c70d)
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
![{\displaystyle \Phi (\mathbf {r} )={\frac {1}{4\pi \varepsilon }}\sum _{k=0}^{\infty }M_{k}\left({\frac {1}{r^{k+1}}}\right)P_{k}(\cos \theta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f976beff590d8b1f79e5604af1bd79bc18b2ec0d)
where the axial multipole moments
are defined
![{\displaystyle M_{k}\equiv \int d\zeta \ \lambda (\zeta )\zeta ^{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53376061fa21c6b50e43e7171a1f1bfa9f5fc62a)
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
![{\displaystyle M_{k}^{\prime }\equiv \int d\zeta \ \lambda (\zeta )\ \left(\zeta +b\right)^{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c79bed483c51c3f7287e7c590b79a958c35c2748)
Expanding the polynomial under the integral
![{\displaystyle \left(\zeta +b\right)^{l}=\zeta ^{l}+lb\zeta ^{l-1}+\ldots +l\zeta b^{l-1}+b^{l}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6335213c945ae17706fa92b23a50c3802bccbbfc)
leads to the equation
![{\displaystyle M_{k}^{\prime }=M_{k}+lbM_{k-1}+\ldots +lb^{l-1}M_{1}+b^{l}M_{0}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be7192975d701b4b033d1805c002daf45a5caded)
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
![{\displaystyle \Phi (\mathbf {r} )={\frac {1}{4\pi \varepsilon }}\sum _{k=0}^{\infty }I_{k}r^{k}P_{k}(\cos \theta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd81e8b3d07d0569ac0ff996a9dcde81986463db)
where the interior axial multipole moments
are defined
![{\displaystyle I_{k}\equiv \int d\zeta \ {\frac {\lambda (\zeta )}{\zeta ^{k+1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/563b521415e6155ccaf74fa64e33d4fab20c1ca5)
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.
See also[]
References[]