# Axial multipole moments

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.

## 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

$\Phi (\mathbf {r} )={\frac {q}{4\pi \varepsilon }}{\frac {1}{R}}={\frac {q}{4\pi \varepsilon }}{\frac {1}{\sqrt {r^{2}+a^{2}-2ar\cos \theta }}}.$ 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 $M_{k}\equiv qa^{k}$ 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 $M_{1}=qa$ and the axial quadrupole moment $M_{2}\equiv qa^{2}$ . This illustrates the general theorem that the lowest non-zero multipole moment is independent of the origin of the coordinate system, but higher multipole 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 $(r/a)<1$ , once again 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

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

$\left(\zeta +b\right)^{l}=\zeta ^{l}+lb\zeta ^{l-1}+\ldots +l\zeta b^{l-1}+b^{l}$ $M_{k}^{\prime }=M_{k}+lbM_{k-1}+\ldots +lb^{l-1}M_{1}+b^{l}M_{0}$ 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

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.