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| '''Axial multipole moments''' are a [[series expansion]]
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| of the [[electric potential]] of a
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| charge distribution localized close to
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| the [[origin (mathematics)|origin]] along one
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| [[Cartesian coordinates|Cartesian axis]],
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| denoted here as the ''z''-axis. However,
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| the axial multipole expansion can also be applied to
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| any potential or field that varies inversely
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| with the distance to the source, i.e., as <math>\frac{1}{R}</math>.
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| For clarity, we first illustrate the expansion for a single point charge,
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| then generalize to an arbitrary charge density <math>\lambda(z)</math>
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| localized to the ''z''-axis.
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| [[Image:Point axial multipole.svg|frame|right|Figure 1: Point charge on the z axis; Definitions for axial multipole expansion]]
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| ==Axial multipole moments of a [[point charge]]==
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| The [[electric potential]] of a [[point charge]] ''q'' located on
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| the ''z''-axis at <math>z=a</math> (Fig. 1) equals
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| :<math>
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| \Phi(\mathbf{r}) =
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| \frac{q}{4\pi\varepsilon} \frac{1}{R} =
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| \frac{q}{4\pi\varepsilon} \frac{1}{\sqrt{r^{2} + a^{2} - 2 a r \cos \theta}}.
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| </math>
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| If the radius ''r'' of the observation point is '''greater''' than ''a'',
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| we may factor out <math>\frac{1}{r}</math> and expand the square root
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| in powers of <math>(a/r)<1</math> using [[Legendre polynomials]]
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| :<math>
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| \Phi(\mathbf{r}) =
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| \frac{q}{4\pi\varepsilon r} \sum_{k=0}^{\infty}
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| \left( \frac{a}{r} \right)^{k} P_{k}(\cos \theta ) \equiv
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| \frac{1}{4\pi\varepsilon} \sum_{k=0}^{\infty} M_{k}
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| \left( \frac{1}{r^{k+1}} \right) P_{k}(\cos \theta )
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| </math>
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| where the '''axial multipole moments'''
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| <math>M_{k} \equiv q a^{k}</math> contain everything
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| specific to a given charge distribution; the other parts
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| of the [[electric potential]] depend only on the coordinates
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| of the observation point '''P'''. Special cases include the axial
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| [[Monopole (mathematics)|monopole]] moment <math>M_{0}=q</math>, the axial [[dipole]]
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| moment <math>M_{1}=q a</math> and the axial [[quadrupole]]
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| moment <math>M_{2} \equiv q a^{2}</math>. This
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| illustrates the general theorem that the lowest
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| non-zero multipole moment is independent of the
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| [[origin (mathematics)|origin]] of the [[coordinate system]],
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| but higher multipole multipole moments are not (in general). | |
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| Conversely, if the radius ''r'' is '''less''' than ''a'',
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| we may factor out <math>\frac{1}{a}</math> and expand
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| in powers of <math>(r/a)<1</math> using [[Legendre polynomials]]
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| :<math>
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| \Phi(\mathbf{r}) =
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| \frac{q}{4\pi\varepsilon a} \sum_{k=0}^{\infty}
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| \left( \frac{r}{a} \right)^{k} P_{k}(\cos \theta ) \equiv
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| \frac{1}{4\pi\varepsilon} \sum_{k=0}^{\infty} I_{k}
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| r^{k} P_{k}(\cos \theta )
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| </math>
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| where the '''interior axial multipole moments'''
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| <math>I_{k} \equiv \frac{q}{a^{k+1}}</math> contain
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| everything specific to a given charge distribution;
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| the other parts depend only on the coordinates of
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| the observation point '''P'''.
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| ==General axial multipole moments==
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| To get the general axial multipole moments, we replace the
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| point charge of the previous section with an infinitesimal
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| charge element <math>\lambda(\zeta)\ d\zeta</math>, where
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| <math>\lambda(\zeta)</math> represents the charge density at
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| position <math>z=\zeta</math> on the ''z''-axis. If the radius ''r''
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| of the observation point '''P''' is greater than the largest
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| <math>\left| \zeta \right|</math> for which <math>\lambda(\zeta)</math>
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| is significant (denoted <math>\zeta_\text{max}</math>), the [[electric potential]]
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| may be written
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| :<math> | |
| \Phi(\mathbf{r}) =
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| \frac{1}{4\pi\varepsilon} \sum_{k=0}^{\infty} M_{k}
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| \left( \frac{1}{r^{k+1}} \right) P_{k}(\cos \theta )
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| </math>
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| where the axial multipole moments <math>M_{k}</math> are defined
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| :<math>
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| M_{k} \equiv \int d\zeta \ \lambda(\zeta) \zeta^{k}
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| </math>
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| Special cases include the axial [[Monopole (mathematics)|monopole]] moment (=total [[electric charge|charge]])
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| :<math>
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| M_{0} \equiv \int d\zeta \ \lambda(\zeta)
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| </math>,
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| the axial [[dipole]] moment <math>M_{1} \equiv \int d\zeta \ \lambda(\zeta) \ \zeta</math>, and
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| the axial [[quadrupole]] moment <math>M_{2} \equiv \int d\zeta \ \lambda(\zeta) \ \zeta^{2}</math>.
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| Each successive term in the expansion varies inversely with
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| a greater power of <math>r</math>, e.g., the monopole potential
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| varies as <math>\frac{1}{r}</math>, the dipole potential varies
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| as <math>\frac{1}{r^{2}}</math>, the quadrupole potential varies
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| as <math>\frac{1}{r^{3}}</math>, etc. Thus, at large distances
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| (<math>\frac{\zeta_\text{max}}{r} \ll 1</math>), the potential is well-approximated
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| by the leading nonzero multipole term.
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| The lowest non-zero axial multipole moment
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| is invariant under a shift ''b'' in
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| [[origin (mathematics)|origin]], but higher moments generally | |
| depend on the choice of origin. The shifted multipole moments
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| <math>M_{k}^{\prime}</math> would be
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| :<math>
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| M_{k}^{\prime} \equiv \int d\zeta \ \lambda(\zeta) \
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| \left(\zeta + b \right)^{k}
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| </math>
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| Expanding the polynomial under the integral
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| :<math>
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| \left( \zeta + b \right)^{l} = \zeta^{l} + l b \zeta^{l-1} + \ldots + l \zeta b^{l-1} + b^{l}
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| </math>
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| leads to the equation
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| :<math> | |
| M_{k}^{\prime} = M_{k} + l b M_{k-1} + \ldots + l b^{l-1} M_{1} + b^{l} M_{0}
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| </math>
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| If the lower moments <math>M_{k-1}, M_{k-2},\ldots , M_{1}, M_{0}</math>
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| 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).
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| ==Interior axial multipole moments== | |
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| Conversely, if the radius ''r'' is smaller than the smallest
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| <math>\left| \zeta \right|</math> for which <math>\lambda(\zeta)</math>
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| is significant (denoted <math>\zeta_{min}</math>), the [[electric potential]] may be written
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| :<math> | |
| \Phi(\mathbf{r}) =
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| \frac{1}{4\pi\varepsilon} \sum_{k=0}^{\infty} I_{k}
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| r^{k} P_{k}(\cos \theta )
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| </math>
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| where the interior axial multipole moments <math>I_{k}</math> are defined
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| :<math>
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| I_{k} \equiv \int d\zeta \ \frac{\lambda(\zeta)}{\zeta^{k+1}}
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| </math>
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| Special cases include the interior axial [[Monopole (mathematics)|monopole]] moment (<math>\neq</math> the total charge)
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| :<math>
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| M_{0} \equiv \int d\zeta \ \frac{\lambda(\zeta)}{\zeta}
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| </math>,
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| the interior axial [[dipole]] moment <math>M_{1} \equiv \int d\zeta \ \frac{\lambda(\zeta)}{\zeta^{2}}</math>,
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| etc. Each successive term in the expansion varies with a greater power of <math>r</math>, e.g.,
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| the interior monopole potential varies as <math>r</math>, the dipole potential varies
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| as <math>r^{2}</math>, etc. At short distances (<math>\frac{r}{\zeta_{min}} \ll 1</math>),
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| the potential is well-approximated by the leading nonzero interior multipole term.
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| ==See also==
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| *[[Potential theory]]
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| *[[Multipole moments]]
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| *[[Multipole expansion]]
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| *[[Spherical multipole moments]]
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| *[[Cylindrical multipole moments]]
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| * [[Solid harmonics]]
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| * [[Laplace expansion (potential)|Laplace expansion]]
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| ==References==
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| <references/>
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| {{Unreferenced|date=April 2008}}
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| ==External links==
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| [[Category:Electromagnetism]]
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| [[Category:Potential theory]]
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