Questions tagged [multipole-expansion]

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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. ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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=\...
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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 ...
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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 ...
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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 $\rho_0$,...
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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 (...
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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'...
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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 ...
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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 ...
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, ...