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

1 Axial multipole moments of a point charge
2 General axial multipole moments
3 Interior axial multipole moments
4 See also
5 External links

[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 $M 0 = q$ , 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

$\Phi(\mathbf{r}) = \frac{q}{4\pi\varepsilon a} \sum_{k=0}^{\infty} \left( \frac{r}{a} \right)^{k} P_{k}(\cos \theta ) \equiv \frac{q}{4\pi\varepsilon} \sum_{k=0}^{\infty} I_{k} r^{k} P_{k}(\cos \theta )$

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 $ζ m a x$ ), 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 ( $\frac{\zeta_{max}}{r} \ll 1$ ), 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 $ζ m i n$ ), 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.