Multipole Definition and 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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  1. ergospherical

    I Linearised gravity approach to Lense Thirring metric

    Doing some revision and getting confused. It's under GR but may as well be under electromagnetism or calculus because that is where the problem is. Taking a shell of mass ##\rho = M\delta(r-R)/(4\pi R^2)## and four velocity corresponding to rotation about ##z## axis i.e. ##U = (1, -\omega y...
  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. F

    A Calculate variance on the ratio of 2 angular power spectra

    In the context of Survey of Dark energy stage IV, I need to evaluate the error on a new observable called "O" which is equal to : \begin{equation} O=\left(\frac{C_{\ell, \mathrm{gal}, \mathrm{sp}}^{\prime}}{C_{\ell, \mathrm{gal}, \mathrm{ph}}^{\prime}}\right)=\left(\frac{b_{s p}}{b_{p...
  5. Athenian

    Finding the Monopole and Multipole Moments of the Electric Potential

    My first attempt revolved mostly around the solution method shown in this "site" or PowerPoint: http://physics.gmu.edu/~joe/PHYS685/Topic4.pdf . However, after studying the content and writing down my answer for the monopole moment as equal to ##\sqrt{\frac{1}{4 \pi}} \rho##, I found out the...
  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. I

    Is the Quadrupole Moment Non-Zero?

    I have been throwing everything I can at this. I believe that both the monopole and dipole are zero, but I have no clue as to how to evaluate the quadrupole moment.
  8. 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...
  9. 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}...
  10. DoobleD

    I Getting the wrong multipole for 1st acoustic peak

    I'm trying to do a simple calculation, but there must be something wrong. The wavelength ##\lambda_1## corresponding to first acoustic peak of the CMB is related to the sound horizon at last scattering, ##d_{hs}##, by : ## \lambda_1 = 2d_{hs} ## (see for instance slide 14 on Wayne Hu PDF...
  11. 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...
  12. 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 =...
  13. Jules Winnfield

    I How is the first multipole calculated from the Plank Study?

    Reading through the Plank 2013 Results we can see that the angular scale is ##0.0104147## or ##0.60^\circ##. However, the Power Spectrum chart clearly shows the first multipole at ##220## ##l##. Using the relation $$\theta = \frac {180^\circ}{l},$$I calculate the first multipole to peak at...
  14. sitkican

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

    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) + · ·...
  15. 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...
  16. 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...
  17. 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 =...
  18. 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...
  19. 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...
  20. F

    Electric dipole moment for a uniformly charged ring

    Homework Statement Text description: Let V(z) be the potential of a ring of charge on the axis of symmetry at distance z from the center. Obtain the first two non-vanishing terms of the multipole expansion for V(z) with z>>a where a is the radius of the ring. Can you see by symmetry that the...
  21. K

    Potential from a simple Quadrupole expansion

    Hi everyone! I'm currently working on this problem for which I am getting inconsistencies depending on how I do it. I'm trying to find the potential due to the quadrupole moment of the following distribution: +q at (0,0,d), -2q at (0,0,0), and +q at (0,0,-2d) I am doing this using two...
  22. 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...
  23. P

    Multipole Map with Healpix routines

    Hello, I have to create the CMB multipole map from a Planck Data Map with Healpix routines on IDL, and I just don't got a clue of how it must be done! Can anybody help me?Thanks! physfed
  24. U

    Multipole approximation outside conducting sphere

    Homework Statement A dipole is placed next to a sphere (see image), at a large distance what is E proportional to? 3. The Attempt at a Solution or lack thereof I'm having trouble figuring out what's happening in any variations of these. How does the dipole affect the sphere's charge...
  25. 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...
  26. Q

    Poles Arrangement in Multipole Generator

    In multipole generators, why do the poles facing each other have the same polarity? Wouldn't this give rise to a zero net current? Consider this link: http://electriciantraining.tpub.com/14177/css/14177_40.htm Initially, it gives the example of a simple generator with opposite poles...
  27. 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...
  28. 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^{'}...
  29. 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...
  30. 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...
  31. 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...
  32. 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...
  33. 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...
  34. 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...
  35. L

    Fast multipole method question. Point charges vs surface charges

    In many articles authors describe FMM for particles. e.g. http://www.csci.psu.edu/seminars/fallnotes/fmm.pdf http://www.umiacs.umd.edu/labs/cvl/pirl/vikas/publications/FMM_tutorial.pdf In other articles authors describe FMM for charged surface. How does these formulations relates? Main...
  36. D

    Fast Multipole Method: Explained by Derivator

    hi, barnes hut method approximates the interaction by treating a bunch of far away particles as one big particle located in the center of mass of the bunch of particles. My lecture notes say, that the fast multipole methode not only does the above 'barnes hut' approximation, but also...
  37. B

    How to get relation for multipole radiation?

    At the moment i am reading Davydov A.S. Quantum mechanics book. And i need help to derive relation formula for multipole radiation [PLAIN]http://img827.imageshack.us/img827/8974/formulal.png Thank you in advance :)
  38. 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...
  39. 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...
  40. 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...
  41. M

    Statistical moments and multipole moments

    Hello, in statistics, one can derive the moments of a distribution by using a generating function <x^n> = \int dx x^n f(x) = \left( \frac {d}{dt} \int dx \exp(tx) f(x) \right)_{t=0} = \left( \frac d {dt} M(t) \right)_{t=0} Is there a similar method to derive the multipole moments in...
  42. V

    How Is the Multipole Expansion Derived for 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...
  43. 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...
  44. 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} +...
  45. 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...
  46. 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...
  47. 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
  48. 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...
  49. P

    Multipole moments using spherical harmonics

    Hello, My question is fairly simple. My instructor solved in class today Laplace's equation in spherical coordinates which resulted in spherical harmonics. I have not taken any quantum mechanics yet so this is my first exposure to spherical harmonics. What do the "l" and "m" terms in the...
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