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.
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...
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...
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...
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...
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...
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##...
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.
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...
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
$$\mathbf{A(r)}=\frac{\mu_0 I}{4\pi} \left[ \frac{1}{r}...
I'm trying to do a simple calculation, but there must be something wrong.
The wavelength ##\lambda_1## corresponding to first acoustic peak of the CMB is related to the sound horizon at last scattering, ##d_{hs}##, by :
## \lambda_1 = 2d_{hs} ## (see for instance slide 14 on Wayne Hu PDF...
I read Chris Hirata's paper on gravitational waves (http://www.tapir.caltech.edu/~chirata/ph236/lec10.pdf) where he performs a multipole expansion of the gravitational source.
I got most of it, apart from the part where he expands the inverse distance function into a series :
More specifically...
Homework Statement
Four particles are each placed a distance a from the origin
3q at (0,a)
-2q at (a,0)
-2q at (-a,0)
q at (0,-a)
find the simple approximate formula for the potential valid at points far from the origin. Express in Spherical coordinates
Homework Equations
P=qr
##V =...
Reading through the Plank 2013 Results we can see that the angular scale is ##0.0104147## or ##0.60^\circ##. However, the Power Spectrum chart clearly shows the first multipole at ##220## ##l##. Using the relation $$\theta = \frac {180^\circ}{l},$$I calculate the first multipole to peak at...
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) + · ·...
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...
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...
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 =...
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...
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...
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...
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...
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...
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
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...
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...
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...
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...
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^{'}...
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...
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...
Homework Statement
A cube of side a is filled 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...
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...
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...
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...
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...
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...
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 :)
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...
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...
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...
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...
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...
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...
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} +...
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...
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...
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
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...
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...