# What is Multipole: Definition + 51 Threads

A multipole expansion is a mathematical series representing a function that depends on angles—usually the two angles used in the spherical coordinate system (the polar and azimuthal angles) for three-dimensional Euclidean space,

R

3

{\displaystyle \mathbb {R} ^{3}}
. Similarly to Taylor series, multipole expansions are useful because oftentimes only the first few terms are needed to provide a good approximation of the original function. The function being expanded may be real or complex-valued and is defined either on

R

3

{\displaystyle \mathbb {R} ^{3}}
, or less often on

R

n

{\displaystyle \mathbb {R} ^{n}}
for some other

n

{\displaystyle n}
.
Multipole expansions are used frequently in the study of electromagnetic and gravitational fields, where the fields at distant points are given in terms of sources in a small region. The multipole expansion with angles is often combined with an expansion in radius. Such a combination gives an expansion describing a function throughout three-dimensional space.The multipole expansion is expressed as a sum of terms with progressively finer angular features (moments). The first (the zeroth-order) term is called the monopole moment, the second (the first-order) term is called the dipole moment, the third (the second-order) the quadrupole moment, the fourth (third-order) term is called the octupole moment, and so on. Given the limitation of Greek numeral prefixes, terms of higher order are conventionally named by adding "-pole" to the number of poles—e.g., 32-pole (rarely dotriacontapole or triacontadipole) and 64-pole (rarely tetrahexacontapole or hexacontatetrapole). A multipole moment usually involves powers (or inverse powers) of the distance to the origin, as well as some angular dependence.
In principle, a multipole expansion provides an exact description of the potential, and generally converges under two conditions: (1) if the sources (e.g. charges) are localized close to the origin and the point at which the potential is observed is far from the origin; or (2) the reverse, i.e., if the sources are located far from the origin and the potential is observed close to the origin. In the first (more common) case, the coefficients of the series expansion are called exterior multipole moments or simply multipole moments whereas, in the second case, they are called interior multipole moments.

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2. ### Multipole expansions, calculating the various moments of point charges

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3. ### I Vector Potential Multipole Expansion

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4. ### I Exact electrostatic potential of a pure dipole using multipole expansion

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6. ### Finding the Monopole and Multipole Moments of the Electric Potential

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7. ### What is the meaning of r' in the Multipole Expansion?

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8. ### Is the Quadrupole Moment Non-Zero?

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9. ### Multipole expansion for the case r'>>r and r>>r'

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15. ### Multipole Expansion of a Thin Rod: How to Derive the Potential?

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16. ### Multipole expansion of a line charge distribution

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18. ### Multipole expansion of Vector Potential (A)

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19. ### Multipole expansion of polarized cylinder

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20. ### Multipole Expansion: Understanding Electric & Magnetic Fields

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21. ### Electric dipole moment for a uniformly charged ring

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22. ### Potential from a simple Quadrupole expansion

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23. ### Multipole expansion - small problem

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26. ### Point charges and multipole expansion

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27. ### Poles Arrangement in Multipole Generator

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29. ### Dipole term in multipole expansion

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30. ### Multipole Expansion Homework: Invariance w/ Orthogonal Rotation

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38. ### How to get relation for multipole radiation?

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39. ### Multipole expansion on a sphere

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40. ### Taking legendre polynomials outside the integral in a multipole expansion

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41. ### Usefulness of multipole expansion of skalar potential

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42. ### Statistical moments and multipole moments

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43. ### How Is the Multipole Expansion Derived for r<r'?

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44. ### Multipole Expansion in Electrodynamics: Simplifying with Taylor Series

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45. ### Simple Taylor or Multipole Expansion of Potential

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46. ### Spherical layer charge distribution and multipole expansion

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47. ### Multipole Expansion Homework: Calculate Approx. Electrostatic Potential

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48. ### Multipole Expansion - Electrostatic Case

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49. ### Solve Multipole Expansion Problem: Find Exact Potential on Z Axis

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50. ### Multipole moments using spherical harmonics

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