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 λ(z) localized to the z-axis.

Figure 1: Point charge on the z axis; Definitions for axial multipole expansion
Figure 1: Point charge on the z axis; Definitions for axial multipole expansion


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[edit] Axial multipole moments of a point charge

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 M0 = q, the axial dipole moment M1 = qa 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{q}{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.


[edit] 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 \lambda(\zeta)\ d\zeta, where λ(ζ) represents the charge density at position z = ζ on the z-axis. If the radius r of the observation point P is greater than the largest \left| \zeta \right| for which λ(ζ) is significant (denoted ζmax), 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 Mk 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_{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).


[edit] Interior axial multipole moments

Conversely, if the radius r is smaller than the smallest \left| \zeta \right| for which λ(ζ) is significant (denoted ζmin), 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 Ik 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 r2, etc. At short distances (\frac{r}{\zeta_{min}} \ll 1), the potential is well-approximated by the leading nonzero interior multipole term.

[edit] See also

[edit] References

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