Axial multipole moments explained

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

(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.

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$$\backslash Phi(\backslash mathbf)\; =\; \backslash frac\; \backslash frac\; =\backslash frac\; \backslash frac.$$

If the radius r of the observation point is greater than a, we may factor out $\backslash frac$ and expand the square root in powers of

(a/r)< 1 using Legendre polynomials$$\backslash Phi(\backslash mathbf)\; =\; \backslash frac\; \backslash sum\_^\backslash left(\backslash frac\; \backslash right)^\; P\_(\backslash cos\; \backslash theta)\; \backslash equiv\backslash frac\; \backslash sum\_^\; M\_\backslash left(\backslash frac\; \backslash right)\; P\_(\backslash cos\; \backslash theta)$$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} .^[1] 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

(1)/(a) and expand in powers of (r/a)< 1 , once again using Legendre polynomials$$\backslash Phi(\backslash mathbf)\; =\; \backslash frac\; \backslash sum\_^\backslash left(\backslash frac\; \backslash right)^\; P\_(\backslash cos\; \backslash theta)\; \backslash equiv\; \backslash frac\; \backslash sum\_^\; I\_r^\; P\_(\backslash cos\; \backslash theta)$$where the interior axial multipole moments $I\_\; \backslash equiv\; \backslash frac$ 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

λ (\zeta) d\zeta , where λ (\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 λ (\zeta) is significant (denoted \zeta _{max} ), the electric potential may be written$$\backslash Phi(\backslash mathbf)\; =\; \backslash frac\; \backslash sum\_^\; M\_\; \backslash left(\backslash frac\; \backslash right)\; P\_(\backslash cos\; \backslash theta)$$where the axial multipole moments M _{k} are defined$$M\_\; \backslash equiv\; \backslash int\; d\backslash zeta\; \backslash \; \backslash lambda(\backslash zeta)\; \backslash zeta^$$

Special cases include the axial monopole moment (=total charge)$$M\_\; \backslash equiv\; \backslash int\; d\backslash zeta\; \backslash \; \backslash lambda(\backslash zeta),$$the axial dipole moment $M\_\; \backslash equiv\; \backslash int\; d\backslash zeta\; \backslash \; \backslash lambda(\backslash zeta)\; \backslash \; \backslash zeta$, and the axial quadrupole moment $M\_\; \backslash equiv\; \backslash int\; d\backslash zeta\; \backslash \; \backslash lambda(\backslash zeta)\; \backslash \; \backslash zeta^$. Each successive term in the expansion varies inversely with a greater power of

r , e.g., the monopole potential varies as $\backslash frac$, the dipole potential varies as $\backslash frac$, the quadrupole potential varies as $\backslash frac$, etc. Thus, at large distances ($\backslash frac\; \backslash ll\; 1$), 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} would be$$M\_^\; \backslash equiv\; \backslash int\; d\backslash zeta\; \backslash \; \backslash lambda(\backslash zeta)\; \backslash \; \backslash left(\backslash zeta\; +\; b\; \backslash right)^$$

Expanding the polynomial under the integral$$\backslash left(\backslash zeta\; +\; b\; \backslash right)^\; =\; \backslash zeta^\; +\; l\; b\; \backslash zeta^\; +\; \backslash dots\; +\; l\; \backslash zeta\; b^\; +\; b^$$leads to the equation$$M\_^\; =\; M\_\; +\; l\; b\; M\_\; +\; \backslash dots\; +\; l\; b^\; M\_\; +\; b^\; M\_$$If the lower moments

M _{k-1} , M _{k-2} ,\ldots , M _{1} , M _{0} are zero, then M ^{\prime} _{k} = 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 λ (\zeta) is significant (denoted \zeta _{min} ), the electric potential may be written$$\backslash Phi(\backslash mathbf)\; =\; \backslash frac\; \backslash sum\_^\; I\_\; r^\; P\_(\backslash cos\; \backslash theta)$$where the interior axial multipole moments I _{k} are defined$$I\_\; \backslash equiv\; \backslash int\; d\backslash zeta\; \backslash \; \backslash frac$$

Special cases include the interior axial monopole moment (

≠ the total charge)$$M\_\; \backslash equiv\; \backslash int\; d\backslash zeta\; \backslash \; \backslash frac,$$the interior axial dipole moment $M\_\; \backslash equiv\; \backslash int\; d\backslash zeta\; \backslash \; \backslash frac$, 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 ($\backslash frac\; \backslash ll\; 1$), the potential is well-approximated by the leading nonzero interior multipole term.

See also

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

References

1. Book: Eyges, Leonard . The Classical Electromagnetic Field . 2012-06-11 . Courier Corporation . 978-0-486-15235-6 . 22 . en.