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'''Axial multipole moments''' are a [[series expansion]]
of the [[electric potential]] of a
charge distribution localized close to 
the [[origin (mathematics)|origin]] along one 
[[Cartesian coordinates|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 <math>\frac{1}{R}</math>.
For clarity, we first illustrate the expansion for a single point charge, 
then generalize to an arbitrary charge density <math>\lambda(z)</math>
localized to the ''z''-axis.

[[Image:Point axial multipole.svg|frame|right|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 <math>z=a</math> (Fig. 1) equals 

:<math>
\Phi(\mathbf{r}) = 
\frac{q}{4\pi\varepsilon} \frac{1}{R} =
\frac{q}{4\pi\varepsilon} \frac{1}{\sqrt{r^{2} + a^{2} - 2 a r \cos \theta}}. 
</math>

If the radius ''r'' of the observation point is '''greater''' than ''a'', 
we may factor out <math>\frac{1}{r}</math> and expand the square root
in powers of <math>(a/r)<1</math> using [[Legendre polynomials]]

:<math>
\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 )
</math>

where the '''axial multipole moments''' 
<math>M_{k} \equiv q a^{k}</math> 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 (mathematics)|monopole]] moment <math>M_{0}=q</math>, the axial [[dipole]]
moment <math>M_{1}=q a</math> and the axial [[quadrupole]]
moment <math>M_{2} \equiv q a^{2}</math>.  This 
illustrates the general theorem that the lowest
non-zero multipole moment is independent of the 
[[origin (mathematics)|origin]] of the [[coordinate system]], 
but higher multipole multipole moments are not (in general).

Conversely, if the radius ''r'' is '''less''' than ''a'', 
we may factor out <math>\frac{1}{a}</math> and expand
in powers of <math>(r/a)<1</math> using [[Legendre polynomials]]

:<math>
\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 )
</math>
where the '''interior axial multipole moments''' 
<math>I_{k} \equiv \frac{q}{a^{k+1}}</math> 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 <math>\lambda(\zeta)\ d\zeta</math>, where 
<math>\lambda(\zeta)</math> represents the charge density at
position <math>z=\zeta</math> on the ''z''-axis.   If the radius ''r''
of the observation point '''P''' is greater than the largest 
<math>\left| \zeta \right|</math> for which <math>\lambda(\zeta)</math>
is significant (denoted <math>\zeta_\text{max}</math>), the [[electric potential]] 
may be written

:<math>
\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 )
</math>

where the axial multipole moments <math>M_{k}</math> are defined 

:<math>
M_{k} \equiv \int d\zeta \ \lambda(\zeta) \zeta^{k}
</math>

Special cases include the axial [[Monopole (mathematics)|monopole]] moment (=total [[electric charge|charge]])

:<math>
M_{0} \equiv \int d\zeta \ \lambda(\zeta)
</math>,

the axial [[dipole]] moment <math>M_{1} \equiv \int d\zeta \ \lambda(\zeta) \ \zeta</math>, and
the axial [[quadrupole]] moment <math>M_{2} \equiv \int d\zeta \ \lambda(\zeta) \ \zeta^{2}</math>.
Each successive term in the expansion varies inversely with 
a greater power of <math>r</math>, e.g., the monopole potential 
varies as <math>\frac{1}{r}</math>, the dipole potential varies 
as <math>\frac{1}{r^{2}}</math>, the quadrupole potential varies 
as <math>\frac{1}{r^{3}}</math>, etc.  Thus, at large distances
(<math>\frac{\zeta_\text{max}}{r} \ll  1</math>), 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 (mathematics)|origin]], but higher moments generally
depend on the choice of origin.  The shifted multipole moments
<math>M_{k}^{\prime}</math> would be

:<math>
M_{k}^{\prime} \equiv \int d\zeta \ \lambda(\zeta) \ 
\left(\zeta + b \right)^{k}
</math>

Expanding the polynomial under the integral
:<math>
\left( \zeta + b \right)^{l} = \zeta^{l} + l b \zeta^{l-1} + \ldots + l \zeta b^{l-1} + b^{l}
</math>
leads to the equation
:<math>
M_{k}^{\prime} = M_{k} + l b M_{k-1} + \ldots + l b^{l-1} M_{1} + b^{l} M_{0}
</math>
If the lower moments <math>M_{k-1}, M_{k-2},\ldots , M_{1}, M_{0}</math>
are zero, then <math>M_{k}^{\prime} = M_{k}</math>.  The same equation shows that multipole moments higher than the first non-zero moment do depend on the choice of [[origin (mathematics)|origin]] (in general).

==Interior axial multipole moments==

Conversely, if the radius ''r'' is smaller than the smallest
<math>\left| \zeta \right|</math> for which <math>\lambda(\zeta)</math>
is significant (denoted <math>\zeta_{min}</math>), the [[electric potential]] may be written

:<math>
\Phi(\mathbf{r}) = 
\frac{1}{4\pi\varepsilon} \sum_{k=0}^{\infty} I_{k}
r^{k} P_{k}(\cos \theta )
</math> 

where the interior axial multipole moments <math>I_{k}</math> are defined

:<math>
I_{k} \equiv \int d\zeta \ \frac{\lambda(\zeta)}{\zeta^{k+1}}
</math>

Special cases include the interior axial [[Monopole (mathematics)|monopole]] moment (<math>\neq</math> the total charge)

:<math>
M_{0} \equiv \int d\zeta \ \frac{\lambda(\zeta)}{\zeta}
</math>,

the interior axial [[dipole]] moment <math>M_{1} \equiv \int d\zeta \ \frac{\lambda(\zeta)}{\zeta^{2}}</math>,
etc.  Each successive term in the expansion varies with a greater power of <math>r</math>, e.g., 
the interior monopole potential varies as <math>r</math>, the dipole potential varies 
as <math>r^{2}</math>, etc.  At short distances (<math>\frac{r}{\zeta_{min}} \ll  1</math>), 
the potential is well-approximated by the leading nonzero interior multipole term.

==See also==

*[[Potential theory]]
*[[Multipole moments]]
*[[Multipole expansion]]
*[[Spherical multipole moments]]
*[[Cylindrical multipole moments]]
* [[Solid harmonics]]
* [[Laplace expansion (potential)|Laplace expansion]]

==References==
<references/>
{{Unreferenced|date=April 2008}}

==External links==

[[Category:Electromagnetism]]
[[Category:Potential theory]]