Questions tagged [multipole-expansion]

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14
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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. ...
5
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3answers
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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 ...
24
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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}\...
21
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3answers
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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 ...
31
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3answers
7k 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 ...
8
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2answers
229 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,\...
10
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3answers
7k 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 ...
13
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1answer
1k 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 ...
5
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1answer
2k 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 ...
1
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2answers
2k views

Why does magnetic force vary in proportion to the cube of the distance instead of the square?

Magnetic forces vary from gravity and electromagnetic radiation (such as light) in that gravity and radiation diminish by the square of the distance from the source, but magnetism diminishes by the ...
6
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1answer
422 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 ...
2
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1answer
324 views

Can one force the octupole moments of a charge distribution (neutral and with vanishing dipole moment) to vanish using a suitable translation?

In a previous question, I noted that if you have a charge distribution with nonzero charge, then it is possible to choose an origin (at the centre of charge) which makes its dipole moment vanish, and ...
12
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4answers
1k views

What does the dipole moment really represent?

Wikipedia gives the most general expression for the $n^{\rm th}$ moment $\mu_n$ of a physical quantity $\Lambda$ as: $$ \mu_n = \int {\bf x}^n \space \lambda({\bf x}) \space \rm d^3 x$$ provided that ...
11
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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
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2answers
1k 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 ...
6
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1answer
2k 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 ...
5
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1answer
224 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 ...
1
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1answer
625 views

Why three families of multipole moments?

There are three families of multipole moments: The electric multipole moments, the magnetic multipole moments and the toroidal multipole moments. Is there any reason why there are this three families ...
1
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1answer
565 views

How does one prove that the lowest-order nonvanishing multipole moment of a charge distribution is independent of the origin, for arbitrary $\ell$?

The multipole moments of a distribution are independent of origin if all the lower terms are zero. I can explicitly verify this statement by hand up to the quadrupole level, but is there any straight ...
5
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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 ...
1
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0answers
201 views

How does the Wigner-Eckart theorem rule Multipole Expansion?

I am wondering why a spin-S particle have only the term up to $k=2S$ in his multipole expansion ? It seems that the Wigner-Eckart theorem shows the relation between spin and multipole expansion but I ...
0
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2answers
1k views

How is the potential in multipole expansion independent from the origin chosen?

Consider a charge distribution $\rho(x',y',z')$ and a point $P=(x,y,z)$ where we want to calculate the potential with multipole expansion. Suppose also that the total charge is zero, that is $$Q=\...
7
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5answers
5k 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 ...
2
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0answers
324 views

Positive/negative octupole moment of nuclei?

Does octupole moment of nuclear charge distribution show any positive/negative character, like the quadrupole moment does? Quadrupole moment has prolate and oblate types, but what about ...
2
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1answer
4k views

Understanding the integral for the electric dipole moment of a charge distribution

In problem 3.35 of Griffiths' Introduction to electrodynamics, he states: A solid sphere, radius $R$, is centered at the origin. The “northern” hemisphere carries a uniform charge density $\rho_0$,...
2
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1answer
89 views

Do dipolar gravitational waves exist?

There seems to be some controversy (see A, B) on this topic, so I'm posting a new question for discussion and clarification. By definition, one cannot accelerate the center of mass of a closed system (...
2
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1answer
827 views

Approximate point quadrupole as point charges

Given a quadrupole moment, $Q$, how can one approximate the resulting potential as a potential due to a set of point charges? What extra degrees of freedom arise by doing so? As an analogy to what I'...
1
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2answers
704 views

Why is quadruple moment zero for spherically symmetric charge distribution about centre?

How can we show that for a spherically symmetric charge distribution, the dipole, quadrupole and all higher moments about the centre of the distribution are identically zero. As we already know that ...
1
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1answer
94 views

Multipole expansion relative to point $z=-d$

I'm attempting to solve a problem where the solution involves the multipole expansion of a the relative vector between a point at $z=-d, x=y=0$, i.e., $r = -d, \theta = \pi$ and a point at r. We ...
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1answer
459 views

Potential of a dipole with actual physical extension?

I think everybody here knows the equation that gives the potential of a point like dipole, but how does the field look like if you have e.g. a metal sphere with radius $R$ and a certain dipol moment, ...
1
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1answer
2k views

Choice of Origin in Dipole moment calculation

I understand (Choice of Origin in Multipole Expansion in Electrostatics) that in multipole expansion I need not choose any particular origin during calculation. But in case of, say 2 point charges $+...
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2answers
498 views

Is there a relation between the Legendre generator function and Spherical Harmonics for a Potential?

Recently I had to solve a simple problem in which I had a sphere of radius $R$ with a constant potential (but with different sign), on both of the hemispheres, and I was asked to get the electrostatic ...
0
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0answers
253 views

Choice of Origin in Multipole Expansion in Electrostatics

In a multipole expansion in electrostatics, how do we select the origin? For instance, if there is a linear continuous distribution of charge $+Q$ (say from $x = -a$ to $x = +a$). Is there any rule ...