What is Multipole expansion: Definition and 40 Discussions

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,



{\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



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



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


{\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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  1. K

    A Confused about multipole expansion of vector potential

    Hello! I found an expression in this paper (eq. 1) for the multipole expansion of the vector potential. I am not sure I understand what form do the vector spherical harmonics (VSH) have. Also, for example, the usual hyperfine interaction operator is given by...
  2. P

    I Vector Potential Multipole Expansion

    when you do a multipole expansion of the vector potential you get a monopole, dipole, quadrupole and so on terms. The monopole term for a current loop is μI/4πr*∫dl’ which goes to 0 as the integral is over a closed loop. I am kinda confused on that as evaulating the integral gives the arc length...
  3. Ahmed1029

    I Exact electrostatic potential of a pure dipole using multipole expansion

    If I have a physical dipole with dipole moment p. Now, this formula for potential (V) is a good approximation when r is much larger than both r1 and r2 in the picture below. It's however said that for a pure dipole for which the separation between charges goes to zero and q goes to infinity, the...
  4. patric44

    Exploring Nuclear Quadrupole Moment and Deformation

    hi guys I have read the other day about how the nuclear quadruple moment descries the deformation of the nucleus, however i can't get my head around how is that!, I am familiar with the multiple expansion in which we can describe the potential of an arbitrary charge distribution by the following...
  5. F

    A Photometric Galaxy Clustering Error and Poisson Noise

    The error on photometric galaxy clustering under the form of covariance which is actually a standard deviation expression for a fixed multipole ##\ell## : ## \sigma_{C, i j}^{A B}(\ell)=\Delta C_{i j}^{A B}(\ell)=\sqrt{\frac{2}{(2 \ell+1) f_{\mathrm{sky}} \Delta \ell}}\left[C_{i j}^{A...
  6. Tony Hau

    What is the meaning of r' in the Multipole Expansion?

    The diagram of the problem should look something like this: ,which is just the normal spherical coordinate.To calculate the potential far away, we use the multipole expansion. ##I_o## in the expansion is ok, because ##(r^{'})^{0} = 1##. However, I am wondering how I should calculate ##I_1##...
  7. pallab

    Multipole expansion for the case r'>>r and r>>r'

    for the case, r>>r' the higher-order term like 1/r^2 and above that is negligible. so V(r)=cons.*1/r*P0(cos a) but for the case r'>>r will it be V(r)=cons.*1/r'[ summation Pn(cos a')t'^n] where t'=r/r' now if we neglect higher-order term of r/r' then V(r)=cons.*1/r'*P0(cos a') which is...
  8. WeiShan Ng

    Deriving magnetic dipole moment from multipole expansion

    Homework Statement This is from Griffith's Introduction to Electrodynamics, where the book is deriving the magnetic dipole moment from multipole expansion of the vector potential The vector potential of a current loop can be written as $$\mathbf{A(r)}=\frac{\mu_0 I}{4\pi} \left[ \frac{1}{r}...
  9. S

    A Multipole expansion of linearized field equations

    I read Chris Hirata's paper on gravitational waves (http://www.tapir.caltech.edu/~chirata/ph236/lec10.pdf) where he performs a multipole expansion of the gravitational source. I got most of it, apart from the part where he expands the inverse distance function into a series : More specifically...
  10. grandpa2390

    Multipole Expansion Homework: Potential in Spherical Coordinates

    Homework Statement Four particles are each placed a distance a from the origin 3q at (0,a) -2q at (a,0) -2q at (-a,0) q at (0,-a) find the simple approximate formula for the potential valid at points far from the origin. Express in Spherical coordinates Homework Equations P=qr ##V =...
  11. D

    Electrostatic polarization of an axially symmetric conductor

    Homework Statement A grounded Z-axis symmetric closed conductor has a single point charge at the origin within it, inducing negative charge onto its inner surface. Given the induced charge density from the unit point charge, find the surface charge induced instead by a unit dipole at the...
  12. sitkican

    Multipole Expansion of a Thin rod

    Homework Statement Consider a very thin rod lying on the z axis from z = −L/2 to z = L/2. It carries a uniform charge density λ. Show that away from the rod, at the point r (r >>L), the potential can be written as V (r, θ) = (2Lλ/4πε0)(1/L)[ 1 + 1/3(L/2r)2P2(cos θ) + 1/3(L/2r)4 P4(cos θ) + · ·...
  13. JulienB

    Multipole expansion of a line charge distribution

    Homework Statement Hi everybody! I'm very stuck trying to solve this problem, hopefully some of you can give me a clue about in which direction I should go: Determine the multipole expansion in two dimensions of the potential of a localized charge distribution ##\lambda(\vec{x})## until the...
  14. 1

    Finding the magnetic field of a loop at far distances

    Homework Statement Loop of current ##I## sitting in the xy plane. Current goes in counter clockwise direction as seen from positive z axis. Find: a) the magnetic dipole moment b) the approximate magnetic field at points far from the origin c) show that, for points on the z axis, your answer is...
  15. D

    Testing multipole expansion Application ID: 31901

    like in the manual i have done the following steps 1.built 100nm radius sphere 2.pressed build all 3. added air material 4.added silicon materal withrefractive index 1.5 5. selected all the domains and pressed Build All but when i press both "test application" or "compute" i get an empty...
  16. P

    Multipole expansion of Vector Potential (A)

    Homework Statement So my teacher, as we made the multipole expansion of Vector Potential (\vec A) decided to proof that the monopole term is zero doing something like this: ∫∇'⋅ (J.r'i)dV' = ∮r'iJ ndS' = 0 The first integral, "opening" the nabla: J⋅(∇r'i) + r'i(∇⋅J) this must be equals 0 J =...
  17. phys-student

    Multipole expansion of polarized cylinder

    Homework Statement I need to calculate the electric field on the midplane of a uniformly polarized cylinder at a large distance from the center of the cylinder. The question also says that because the distance is large compared to the radius the dipole dominates the multipole expansion...
  18. L

    Multipole Expansion: Understanding Electric & Magnetic Fields

    Hello, I was hoping someone could help make the concept of electric multipole/ magnetic multipole expansions clearer. I think my most fundamental question is: Are dipole, quadrupole and up fields just a shortcut to using the superposition principle on a charge distribution in space or do they...
  19. E

    Multipole expansion - small problem

    Homework Statement Jackson 4.7 Given a localized charge distribution: \rho(r)=\frac{1}{64\pi}r^{2} e^{-r} sin^{2}\theta make the multipole expansion of the potential due to this charge distribution and determine all nonvanishing moments. Write down the potential at large distances as a...
  20. ShayanJ

    Point charges and multipole expansion

    Consider the following charge distribution:A positive charge of magnitude Q is at the origin and there is a charge -Q on each of the x,y and z axes a distance d from the origin. I want to expand the potential of this charge distribution using spherical coordinates.Here's how I did it...
  21. L

    Multipole expansion. Problems with understanding derivatives

    Hi everyone Homework Statement I want to find the multipole expansion of \Phi(\vec r)= \frac {1}{4\pi \epsilon_0} \int d^3 r' \frac {\rho(\vec r')}{|\vec r -\vec r'|} Homework Equations Taylor series The Attempt at a Solution My attempt at a solution was to use the Taylor series. I...
  22. A

    Dipole term in multipole expansion

    Hi. I'm having some difficult in understanding something about the dipole term in a multipole expansion. Griffiths writes the expansion as a sum of terms in Legendre polynomials, so the dipole term in the potential is writen \frac{1}{4 \pi \epsilon r^{2}}\int r^{'}cos\theta^{'}\rho dv^{'}...
  23. X

    Multipole Expansion Homework: Invariance w/ Orthogonal Rotation

    Homework Statement Given the multipole moment of the mass distribution how would I go about determining that the multipole moment expansion is invariant. I Homework Equations http://cohengroup.ccmr.cornell.edu/courses/phys3327/HW2/hw2.pdf The Attempt at a Solution I need to explicitly show...
  24. J

    Multipole Expansion: Quadrupole Moment Calculation

    Homework Statement Four point charges: q at a^z; q at -a^z; -q at a^y and -q at -a^y where ^z and ^y are the unit vectors along the z and y axes. Homework Equations Find the approximate expression (i.e. calculate the first non-zero term in the multipole expansion) for the...
  25. A

    Multipole expansion of point charge placed inside a cube

    Homework Statement A cube of side a is fi lled with a uniform charge density distribution of total charge Q. A point charge +Q is placed at the center of the cube. Show all odd electrostatic multipole moments vanish. (i.e., 2^l poles with odd l). Show that among the even moments, those...
  26. G

    Force acting between bodies - using multipole expansion

    Homework Statement Actually, this is not truly a homework, I'm just ineterested in how to solve problems, like the one below. So, we have two conductive spheres, at a distance R from each other, the radii are r1 and r2 (r1 and r2 are comperable in size, while R is significantly larger than...
  27. fluidistic

    Finding Multipole Expansion for Azimuthally Symmetric Charge Distribution

    Homework Statement I'm given a charge density rho (\rho (r) = r^2 \sin ^2 \theta e^{-r}) and I'm asked to find the multipole expansion of the potential as well as writing explicitely all the non vanishing terms. Homework Equations Not sure and this is my problem. The Attempt at a...
  28. S

    Multipole Expansion of Dipole on Z-Axis w/ Spherical Harmonics

    given a dipole on z-axis(+q at z=a and -q at z= -a) , find out the non vanishing multipoles using spherical harmonics. can somebody tell me how to do this problem using spherical harmonics..because when we write charge density using dirac delta function in spherical polar coordinates. then we...
  29. D

    Multipole expansion on a sphere

    Homework Statement A sphere of radius R, centered at the origin, carries charge density ρ(r,θ) = (kR/r2)(R - 2r)sinθ, where k is a constant, and r, θ are the usual spherical coordinates. Find the approximate potential for points on the z axis, far from the sphere. Homework Equations...
  30. M

    Taking legendre polynomials outside the integral in a multipole expansion

    Homework Statement A chare +Q is distributed uniformly along the z axis from z=-a to z=+a. Find the multipole expansion. Homework Equations Here rho has been changed to lambda, which is just Q/2a and d^3r to dz. The Attempt at a Solution I have solved the problem correctly...
  31. M

    Usefulness of multipole expansion of skalar potential

    Upper undergraduate here. Lot of time spent in studying, but can't find acceptable answers in what follows. To be more specific, my questions are related on "Classical Electrodynamics", Jackson 2nd edition, Sect. 4.2. (and 4.1 of course), titled "Multipole expansion of the Energy of a Charge...
  32. V

    Multipole Expansion when r<r'

    Homework Statement Find the multipole expansion for the case when r<r', where r is the distance from the origin to the observation point and r' is the distance from the origin to the source point. Homework Equations...
  33. S

    Jackson 6.4 (Multipole Expansion)

    Homework Statement Jackson 6.4b Homework Equations Multipole expansion especially Eq 4.9 in Jackson which is for a Quadrupole The Attempt at a Solution I found the result in 6.4a. The rho over there tells us that there is a charge density inside the sphere. Since the charge density...
  34. H

    Multipole Expansion in Electrodynamics: Simplifying with Taylor Series

    Hi, I'm just working through some electrodynamics notes, and am a bit stuck following a particular Taylor expansion, the author starts with: \frac{1}{R_1}=\frac{1}{r} [1+(\frac{l}{r})^2-2\frac{l}{r}cos(\theta)]^-0.5 Which he then says by assuming l<<r and expanding we get...
  35. B

    Simple Taylor or Multipole Expansion of Potential

    Hullo, Somehow, I couldn't get the TeX to come out right. I have been trying to learn scheme theory (algebraic geometry) and completely forgotten how to do this simple calculus type stuff... Homework Statement Let V be a potential of the form [tex]V = \left(\frac{1}{r} +...
  36. H

    Spherical layer charge distribution and multipole expansion

    Hi, I have a random spherical distribution of N charges between radiuses R1 and R2. N is up to 10^9 or more.I want to calculate the electrostatic potential closed to the origin of the sphere. R1 and R2 are much bigger than the distance of this point to the origin. So I thought about using...
  37. R

    Multipole Expansion Homework: Calculate Approx. Electrostatic Potential

    Homework Statement I have to calculate the approximate electrostatic potential far from the origin for the following arrangement of three charges: +q at (0,0,a), -q at (0,a,0) and (0,-a,0). I have to give the final answer in spherical coordinates and keep the first two non-zero terms in the...
  38. G

    Multipole Expansion - Electrostatic Case

    Im having a little problem with this question Not sure where to start but I believe that a 3D taylor series expansion might be useful. Please could someone urgently help me out as it is due in a few hours! Thanks for your time. GM
  39. L

    Solve Multipole Expansion Problem: Find Exact Potential on Z Axis

    Question: Assume the chrages to be on the z axis with the midway between them. Find the potential exactly for a field point on the z axis. Okay, so I found the potential which is v = k*p/(z^2-0.25*l^2) k is the constant 1/4*pi*epsilon, l stands for the length between the two point...