Axial multipole moments

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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 \frac{1}{R}. For clarity, we first illustrate the expansion for a single point charge, then generalize to an arbitrary charge density \lambda(z) 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[edit]

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


\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}}.

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


\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 )

where the axial multipole moments M_{k} \equiv q a^{k} 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 M_{0}=q, the axial dipole moment M_{1}=q a and the axial quadrupole moment M_{2} \equiv q a^{2}. This illustrates the general theorem that the lowest non-zero multipole moment is independent of the 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 \frac{1}{a} and expand in powers of (r/a)<1 using Legendre polynomials


\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 )

where the interior axial multipole moments I_{k} \equiv \frac{q}{a^{k+1}} 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[edit]

To get the general axial multipole moments, we replace the point charge of the previous section with an infinitesimal charge element \lambda(\zeta)\ d\zeta, where \lambda(\zeta) represents the charge density at position z=\zeta on the z-axis. If the radius r of the observation point P is greater than the largest \left| \zeta \right| for which \lambda(\zeta) is significant (denoted \zeta_\text{max}), the electric potential may be written


\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 )

where the axial multipole moments M_{k} are defined


M_{k} \equiv \int d\zeta \ \lambda(\zeta) \zeta^{k}

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


M_{0} \equiv \int d\zeta \ \lambda(\zeta)
,

the axial dipole moment M_{1} \equiv \int d\zeta \ \lambda(\zeta) \ \zeta, and the axial quadrupole moment M_{2} \equiv \int d\zeta \ \lambda(\zeta) \ \zeta^{2}. Each successive term in the expansion varies inversely with a greater power of r, e.g., the monopole potential varies as \frac{1}{r}, the dipole potential varies as \frac{1}{r^{2}}, the quadrupole potential varies as \frac{1}{r^{3}}, etc. Thus, at large distances (\frac{\zeta_\text{max}}{r} \ll  1), 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 M_{k}^{\prime} would be


M_{k}^{\prime} \equiv \int d\zeta \ \lambda(\zeta) \ 
\left(\zeta + b \right)^{k}

Expanding the polynomial under the integral


\left( \zeta + b \right)^{l} = \zeta^{l} + l b \zeta^{l-1} + \ldots + l \zeta b^{l-1} + b^{l}

leads to the equation


M_{k}^{\prime} = M_{k} + l b M_{k-1} + \ldots + l b^{l-1} M_{1} + b^{l} M_{0}

If the lower moments M_{k-1}, M_{k-2},\ldots , M_{1}, M_{0} are zero, then M_{k}^{\prime} = M_{k}. 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[edit]

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


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

where the interior axial multipole moments I_{k} are defined


I_{k} \equiv \int d\zeta \ \frac{\lambda(\zeta)}{\zeta^{k+1}}

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


M_{0} \equiv \int d\zeta \ \frac{\lambda(\zeta)}{\zeta}
,

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

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