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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 $${\displaystyle \lambda (z)}$$ 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 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 $${\frac {1}{r}}$$ 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 $$\frac{1}{a}$$ 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 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 $${\displaystyle {\frac {1}{r^{3}}}}$$, 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 ( $$\neq$$ 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 r, e.g., the interior monopole potential varies as r, the dipole potential varies as $$r^{2}$$ , etc. At short distances ( $${\displaystyle {\frac {r}{\zeta _{min}}}\ll 1}$$ ), the potential is well-approximated by the leading nonzero interior multipole term.

Potential theory
Multipole moments
Multipole expansion
Spherical multipole moments
Cylindrical multipole moments
Solid harmonics
Laplace expansion

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