Questions tagged [multipole-expansion]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
28
votes
3answers
5k views

Why there is no dipole gravitational wave?

I have read that "thanks to conservation of momentum" there is no dipole gravitational radiation. I am confused about this, since I cannot see the difference with e.m. radiation. Is this due to the ...
23
votes
5answers
3k views

Far away from a charged conductor, the field is like a point charge. Where's the point located?

In the framework of classical electrodynamics, at distances much greater than a conductor's dimension, the field ought to approach that of a point charge located at the conductor. But where at? For ...
19
votes
3answers
3k views

Why is dipole the simplest source in electrodynamics?

I see this sort of statement in many materials, for example this: The smallest radiating unit is a dipole, an electromagnetic point source. and this: The simplest infinitesimal radiating ...
19
votes
2answers
2k views

Can one force the electric quadrupole moments of a neutral charge distribution to vanish using a suitable translation?

For a system of electric charges $q_i$, at positions $\mathbf{r}_i$, with a nonzero net charge $Q=\sum_i q_i$, one can define a "centre of charge" in the obvious way as $$ \mathbf{r}_c=\frac{1}{Q}\...
17
votes
1answer
1k views

Do neutrons exhibit momentary small charges due to the movement of its quarks?

So in a similar way to electrons moving in atoms, causing induced dipole-dipole interactions, can neutrons momentarily attract or repel?
12
votes
2answers
2k views

Hexadecapole potential using point particles?

We can get monopole $1/r$, dipole $1/r^2$, quadrupole $1/r^3$ and octupole $1/r^4$ potential falloff by placing opposite point charges at the corners of a point, line, square and cube, respectively. ...
12
votes
1answer
911 views

Does a pendulum necessarily emit gravitational waves?

A question about the behaviour of a pendulum in a frictionless vacuum recently made it back to the front page, and a few comments below John Rennie's excellent answer set me thinking about one ...
11
votes
2answers
3k views

What does the monopole/quadrupole moment of the Earth signify?

I'm currently reading about orbits of near-Earth satellites and some terminology is getting thrown around that I'm not sure I understand what they actually mean: The Earth's monopole moment and the ...
11
votes
2answers
342 views

What's the $\ell$ in the Bicep2 paper mean?

The BICEP experiment's recent announcement included the preprint of their paper, BICEP2 I: Detection of $B$-mode polarization at degree angular scales. BICEP2 Collaboration. To be submitted. BICEP-...
10
votes
2answers
797 views

What are the advantages of multipole expansion of potentials?

When I see the equations of multipolar expansions they "look" to me harder than the original expressions. For example: Multipole expansion - spherical form, in Wikipedia I bet that this is not the ...
9
votes
3answers
6k views

What is the physical meaning of the terms in the multipole expansion?

I have a few questions on multipole expansions and I have read about the topic in many places but could not find an answer to my questions, so please be patient with me. The electrostatic potential ...
7
votes
5answers
3k views

Electric field falls off faster than $\frac{1}{r^2}$ for large distances

An excerpt from a book; The electric field due to a charge configuration with total charge zero, is not zero; but for distances large compared to the size of the configuration, its field falls off ...
7
votes
2answers
211 views

How many truly different multipolar charge distributions are there?

Dipolar charge distributions are essentially all the same: regardless of how one adds up a combination of the form $$ \sigma(\theta,\varphi) = \operatorname{Re}\left[\sum_{m=-1}^1 a_m Y_{1m}(\theta,\...
6
votes
2answers
317 views

Multipole Expansion: Electrostatics

Why in multipole expansion (or the terms therein) goes as mono-, di-, quadru-, octu-, or more specifically why are they in powers of 2? Why can't we have hexapole for instance?
6
votes
1answer
408 views

How do aspherical gravitational monopoles look like?

I was recently pointed by laboussoleestmonpays to a beautiful paper from some time ago, Aspherical gravitational monopoles. Alain Connes, Thibault Damour and Pierre Fayet. Nucl. Phys. B 490 no. 1-2 ...
6
votes
1answer
432 views

Dependence of the multipole moments on the origin

Multipole moments of a system are defined with an explicit refrence to the co-ordinate system, e.g. $$\boldsymbol{d}=\int dV\, \rho\,\boldsymbol{r}\\ \boldsymbol{\mu}=\frac{1}{2c}\int dV\, [\...
5
votes
3answers
1k views

One question about sextupole

In multipole expansion, we use monopole, dipole, quadrupole or octupole to describe an electromagnetic field. But I saw someone use sextupole to describe transition states. If we expand an ...
5
votes
3answers
1k views

Why does a monopole not radiate energy in electodynamics?

Why there is no monopole radiation in Electromagnetic field? I read somewhere that it is impossible because it violates charge conservation. I don't understand how? How charge conservation gets ...
5
votes
2answers
571 views

Meaning of lowest multipoles in CMB spectrum?

What is the meaning attributed to the apparent rise of the CMB at very low mutipoles (i.e. at the beginning of the experimental spectrum), if any? Clarification: The data at that region have large ...
5
votes
2answers
266 views

Computing a “best-fit” of discrete points from a multipole expansion, i.e. invert the multipole moments

Take a field $\phi(\bf{x})$ created from a charge distribution contained within a radius $R$. The multipole expansion in spherical harmonics $Y_{\ell,m}$ outside of $R$ is approximated by: $$ \phi({\...
5
votes
2answers
308 views

Potential generated by a hollow sphere with a hole

The sphere has radius $R$ and is missing its "pole" - meaning that in the area $\theta\leq\alpha$ there is nothing. The object has a homogenous charge density $\sigma=\frac{Q}{\pi R^2}$ I'm trying to ...
5
votes
0answers
135 views

What is the minimal number of point charges required to model an electric octupole?

In an answer to this question, I showed that a general electric quadrupole can be modeled by a combination of seven point charges (or possibly less if one or two of the principal quadrupole moments is ...
5
votes
1answer
203 views

Toroid moments tensor decomposition

I am currently working on my bachelor's thesis on the anapole / toroidal moment and it seems that I am stuck with a tensor decomposition problem. I have actually never had a course about tensors, so ...
4
votes
1answer
4k views

Existence of Tripoles?

With multipole expansions, we speak only of monopoles, dipoles, and $2^n$-poles. Why is there nothing like a tripole? So how would something like $rsin(3 \theta)$ be expressed with a multipole ...
4
votes
1answer
1k views

Traceless multipole moments vs non-traceless moments

There are two different possibilities to define the electric quadrupole tensor: On the one hand, one can define \begin{align}Q_{kl} = \int \rho(\mathbf r') \cdot r'_k \, r'_l d^3r',\end{align} while ...
4
votes
2answers
90 views

Is electic field is always asymptotic to $r^{\alpha}$ for some rational $\alpha$?

Suppose you have an electric field in three dimensions created by some finite (but possibly arbitrarily high) number of point charges, each with charge equal to an integer multiple (positive or ...
4
votes
3answers
1k views

Are gravitational quadrupole moment, second moment of mass, and moment of inertia the same?

my understanding of moments is that they refer to distributions about an expected value, which allows us to make the multipole expansion. I read that: the zeroth moment of mass refers to the mass of ...
4
votes
1answer
699 views

The dipole radiation pattern and spherical harmonics $Y_{10}$

I am studying the multipole expansion of electromagnetic wave radiation pattern, and it is said that any fields can be decomposed into the spherical harmonics $Y_{lm}$. However, for $l=1$, which ...
4
votes
1answer
72 views

all multipolar terms of nuclear fields must be confined?

When usually speaking about QCD confinement, one refers to the fact that color charges over lengths above the hadron size are fully neutral, which implies that monopolar charges over an extended ...
3
votes
3answers
249 views

Systematic expansion of $e^{i\vec{k}\cdot\vec{r}}$ in atomic physics in terms of Legendre polynomials and identifying different $l$ terms

In the context of light-matter interaction one often makes the approximation $e^{i\vec{k}\cdot\vec{r}}\approx 1$. Keeping higher order terms in $e^{i\vec{k}\cdot\vec{r}}$ give magnetic dipole, ...
3
votes
3answers
1k views

Why is the gravitational multipole expansion taken with the origin at the centre of mass instead of the centre of gravity?

Wikipedia's article on the multipole expansion of the gravitational potential expands it as $$ V(\mathbf{x}) = - \frac{GM}{|\mathbf{x}|} - \frac{G}{|\mathbf{x}|} \int \left(\frac{r}{|\mathbf{x}|}\...
3
votes
1answer
1k views

Physical meaning of multipole moment

Is there a physical interpretation for multipole moments? For a quantity governed by the Laplace equation ($\nabla^2 \omega = 0$), I understand that the general solution is given by the multipole ...
3
votes
2answers
922 views

What is the first non-vanishing multipole moment of this configuration?

Imagine that you have a triangle where each side has the length $a$ and a charge $q$ sitting at every vertex. Additionally, we have a charge $-3q$ sitting in the center of the triangle. What is the ...
3
votes
3answers
2k views

How does the gravity of a massive non-spherical object act on things around it?

Firstly, not sure if this question ought to be in the space SE site. Please let me know if it should. (Posted in both for now) Secondly, I don't know a whole lot about physics (I'm just inquisitive). ...
3
votes
1answer
544 views

What does a hexadecapole look like? [duplicate]

Two dipoles can form a quadrupole, two quadropoles an octopole. The textbook by Griffith then says ' and so on'. So how would a hexadecapole really look like? My impression was that the construction ...
3
votes
1answer
1k views

Meaning of terms and interpretation in the electric multipole expansion

In section 3.4.1 of Griffiths' Introduction to Electrodynamics, he discusses electric multipole expansion. He derives the formula or the electric potential of a dipole, which I follow, but right ...
3
votes
1answer
320 views

Intuitive explanation of difference in $r$-dependence between dipole and monopole

For an electric monopole, its potential scales with $\frac{1}{r}$, where $r$ is the distance from the point of interest to the charge. However, for a dipole, its potential scales with $\frac{1}{r^2}$. ...
3
votes
2answers
406 views

Why 3 dipole terms in a multipole expansion?

As can be seen on this page http://en.wikipedia.org/wiki/Multipole_expansion when we take a multipole expansion without assuming azimuthal symmetry we end up with $2l+1$ coefficients for the $l^{th}$ ...
3
votes
2answers
301 views

Objective measure of anisotropy

If we know a function $f(\phi, \theta)$ in $\mathbb R^3$ only over a convex surface (which for simplicity let's assume a sphere of radius $r$), is there any measure for the degree of anisotropy over ...
3
votes
2answers
275 views

Any quadrupole approximation? Any example?

In atomic and molecular physics we quite often encounter with electric dipole approximation. The dipole approximation we do when the wave-length of the type of electromagnetic radiation which induces, ...
3
votes
2answers
468 views

Quadrupole moment tensor definition

I'm not sure what the proper definition of the quadrupole moment tensor is. In the book on gravitational waves by Maggiore, the definition is $$M^{ij}=\int d^3x T^{00}x^ix^j. \tag{3.37}$$ ( Maggiore,...
3
votes
1answer
77 views

When can we consider dipoles?

A dipole is a collection of two oppositely charged particles held at some distance, but if two charges are unequally charged (but oppositely charged) how do we take the dipole between them? Do we only ...
3
votes
1answer
113 views

Why don't higher order multipole moments “stack” like the monopole and dipole moments do?

Electric charge is notorious for needing only a relatively small number of electrons or protons to move to produce macroscopically visible effects. Similarly, electric and magnetic dipoles produce big ...
3
votes
1answer
201 views

Multipole expansion of energy-momentum tensor

In this paper it is stated that the multipole moments of the stress-energy tensor $T^{\mu \nu}$ are given by $$ \int_{x^0 = const} T^{\mu \nu} \delta x^{\alpha_1}\cdots \delta x^{\alpha_n} \sqrt{g} \:...
3
votes
1answer
2k views

Method of image charges (semisphere on a metal)

I'm currently trying to study ahead for the upcoming semester since I'm on break and I'm stuck on the method of image charges. I've tried watching some youtube videos on that topic and I thought I ...
3
votes
1answer
73 views

What is the potential inside a hollow conducting sphere with multipoles uniformly surrounding it?

If considering a hollow conducting sphere with a surrounding uniform charge distribution, for example, it will have a constant and uniform potential throughout the inside of the hollow sphere because $...
3
votes
1answer
927 views

Understanding multipole expansion in classical electrodynamics

I am trying to better understand what the multipole expansion means from a phyiscal point of view. Although mathematically, one may say it is just another form of a series expansion, in this case, the ...
3
votes
1answer
859 views

How does a electric quadrupole oscillate?

I know that in static a electric quadrupole is made of two positive charges and two negative charges, distributed as in the following figure: What I don't understand is when it oscillates and ...
3
votes
0answers
79 views

Can gravitational waves observed far from a black hole tell us anything about the multipole moments of a dynamical horizon?

In a paper by Ashtekar et al in 2013 on the approach to the final state to a stationary black hole they study the evolution of the multipole moments of dynamical horizons, which relax away (except for ...
3
votes
0answers
129 views

Higher order multipolar second harmonic generation in centrosymmetric materials

As is pointed in this question, second harmonic generation is forbidden in the bulk of the materials possessing centrosymmetry. In some papers it is said that in the dipolar approximation the SHG ...