Axial Multipole Moments
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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 ${\displaystyle {\frac {1}{R}}}$. For clarity, we first illustrate the expansion for a single point charge, then generalize to an arbitrary charge density ${\displaystyle \lambda (z)}$ 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

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

${\displaystyle \Phi (\mathbf {r} )={\frac {q}{4\pi \varepsilon }}{\frac {1}{R}}={\frac {q}{4\pi \varepsilon }}{\frac {1}{\sqrt {r^{2}+a^{2}-2ar\cos \theta }}}.}$

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

${\displaystyle \Phi (\mathbf {r} )={\frac {q}{4\pi \varepsilon r}}\sum _{k=0}^{\infty }\left({\frac {a}{r}}\right)^{k}P_{k}(\cos \theta )\equiv {\frac {1}{4\pi \varepsilon }}\sum _{k=0}^{\infty }M_{k}\left({\frac {1}{r^{k+1}}}\right)P_{k}(\cos \theta )}$

where the axial multipole moments ${\displaystyle M_{k}\equiv qa^{k}}$ 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 ${\displaystyle M_{0}=q}$, the axial dipole moment ${\displaystyle M_{1}=qa}$ and the axial quadrupole moment ${\displaystyle M_{2}\equiv qa^{2}}$. 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 ${\displaystyle {\frac {1}{a}}}$ and expand in powers of ${\displaystyle (r/a)<1}$, once again using Legendre polynomials

${\displaystyle \Phi (\mathbf {r} )={\frac {q}{4\pi \varepsilon a}}\sum _{k=0}^{\infty }\left({\frac {r}{a}}\right)^{k}P_{k}(\cos \theta )\equiv {\frac {1}{4\pi \varepsilon }}\sum _{k=0}^{\infty }I_{k}r^{k}P_{k}(\cos \theta )}$

where the interior axial multipole moments ${\displaystyle I_{k}\equiv {\frac {q}{a^{k+1}}}}$ 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 ${\displaystyle \lambda (\zeta )\ d\zeta }$, where ${\displaystyle \lambda (\zeta )}$ represents the charge density at position ${\displaystyle z=\zeta }$ on the z-axis. If the radius r of the observation point P is greater than the largest ${\displaystyle \left|\zeta \right|}$ for which ${\displaystyle \lambda (\zeta )}$ is significant (denoted ${\displaystyle \zeta _{\text{max}}}$), the electric potential may be written

${\displaystyle \Phi (\mathbf {r} )={\frac {1}{4\pi \varepsilon }}\sum _{k=0}^{\infty }M_{k}\left({\frac {1}{r^{k+1}}}\right)P_{k}(\cos \theta )}$

where the axial multipole moments ${\displaystyle M_{k}}$ are defined

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

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

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

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

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

Expanding the polynomial under the integral

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

${\displaystyle M_{k}^{\prime }=M_{k}+lbM_{k-1}+\ldots +lb^{l-1}M_{1}+b^{l}M_{0}}$

If the lower moments ${\displaystyle M_{k-1},M_{k-2},\ldots ,M_{1},M_{0}}$ are zero, then ${\displaystyle M_{k}^{\prime }=M_{k}}$. 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 ${\displaystyle \left|\zeta \right|}$ for which ${\displaystyle \lambda (\zeta )}$ is significant (denoted ${\displaystyle \zeta _{min}}$), the electric potential may be written

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

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

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

Special cases include the interior axial monopole moment (${\displaystyle \neq }$ the total charge)

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

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