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 . For clarity, we first illustrate the expansion for a single point charge, then generalize to an arbitrary charge density localized to the z-axis.
Axial multipole moments of a point charge[edit]
The electric potential of a point charge q located on the z-axis at (Fig. 1) equals
If the radius r of the observation point is greater than a, we may factor out and expand the square root in powers of using Legendre polynomials
Conversely, if the radius r is less than a, we may factor out and expand in powers of , once again using Legendre polynomials
General axial multipole moments[edit]
To get the general axial multipole moments, we replace the point charge of the previous section with an infinitesimal charge element , where represents the charge density at position on the z-axis. If the radius r of the observation point P is greater than the largest for which is significant (denoted ), the electric potential may be written
Special cases include the axial monopole moment (=total charge)
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 would be
Expanding the polynomial under the integral
Interior axial multipole moments[edit]
Conversely, if the radius r is smaller than the smallest for which is significant (denoted ), the electric potential may be written
Special cases include the interior axial monopole moment ( the total charge)
See also[edit]
- Potential theory
- Multipole expansion
- Spherical multipole moments
- Cylindrical multipole moments
- Solid harmonics
- Laplace expansion
References[edit]
- ^ Eyges, Leonard (2012-06-11). The Classical Electromagnetic Field. Courier Corporation. p. 22. ISBN 978-0-486-15235-6.