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Magnetic Multipole Tensor

When the electric scalar potential is expanded into spherical coordinates, one gets \begin{align} \phi (\vec r) = \frac{1}{4\pi\varepsilon_0} \sum_{l=0}^{\infty} \sum_{m=-l}^l ...
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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 monopole does 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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Potential due to a Uniform Ring [closed]

Consider a uniform ring of mass $M$ and radius $a$ (?). I would like to prove that for $r > a$: $$\Phi(r) = \frac{GM}{r} \sum_0^\infty[P_n(0)]^2 \left(\frac{a}{r}\right)^n $$ Where $\Phi(r)$ is ...
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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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475 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 ...
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Magnetic multipoles in spherical harmonic

Does someone know explain me how to identify the multipoles magnetic terms of the multipolar expansion (Dipole, quadrupole, etc) in spherical harmonics?
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319 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 ...
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Electrodynamic multipole expansion

I am reading Jackson, Classical Electrodynamics, and I have a question regarding the electrodynamic multipole expansion (with page numbers I refer to the 3rd edition). So on page 409, he gives in ...
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196 views

Rotation of Taylor expansion of a scalar

I have a scalar magnetic field in a volume expressed by the formula $$B(x,y)=B_0 + \frac{\partial B}{\partial x}(x-x_0) + \frac{\partial B}{\partial y}(y-y_0)$$ which approximates the ...
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39 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 ...
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87 views

The anapole moment, derivation from Dirac current density

Basically I am looking for a way to expand the electromagnetic interaction energy $W = A_{\mu}j^{\mu}$ (both $A$ and $j$ obtained from the Dirac equation) similar to the classical expansion in ...
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162 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 ...
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1answer
76 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 ...
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1answer
128 views

quadrupole moment and higher for simple current loop

I'm working on a field configuration that needs to die off rapidly and I got to a $1/r^5$ dependence with canceling the dipole moment of the system cleverly, but to go get better arrangements I need ...
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1answer
122 views

Clarification of multipole expansion for a point charge

In Griffith's electrodynamic: 3.4.2 He pointed out that the monopole term is the exact potential for a single point charge. However I was under the impression that different configuration of a ...
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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. ...
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Proton as superposition of hadrons: $\vert p\rangle = c_0\vert p_0\rangle+c_1\vert h\rangle+\cdots$

I have a question regarding hadron fluctuations. For instance on page 85 in Feynman's "Photon-Hadron Interactions" equation 15.2 reads: $$\tag1\vert \omega\rangle = \vert ...
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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 ...
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1answer
79 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 $\frac{1}{r^2}$. ...
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1answer
134 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. ...
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285 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 ...
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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, ...
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1answer
248 views

Density matrix and irreducible tensor operators

I'm reading those lecture notes on atomic physics. Yesterday I posed a question on reducible tensors, and today I have a question on their relation to the density matrix. If there's any information ...
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2answers
194 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 $l^{th}$ ...
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Question about “quadrupole radiation” vector potential formula derivation

I tried to get an expression for $\mathbf A (\mathbf x )$ in quadrupole approximation. After some transformations of Liénard–Wiechert vector potential I got, as in many books, $$ \mathbf A \approx ...
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176 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, ...
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1answer
661 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 ...
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1answer
172 views

Are multipole fields, multipole expansion, and multipole radiation the same thing?

Interaction between electromagnetic radiation and nuclei can be written in terms of multipole radiation. Are multipole fields, multipole expansion and multipole radiation the same thing? I have found ...
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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. ...
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0answers
354 views

What is the magnetic quadrupole moment of a nucleus in cylindrical coordinates?

What is the magnetic quadruple moment of a nuclei in cylindrical coordinates? The quadrupole moment of a nucleus is zero in spherical coordinates but in the cylindrical coordinates it can't be ...
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429 views

Forcing quadrupole moments to vanish for a neutral system

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 $$ ...
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1answer
634 views

Simple quadrupole moment

I have a very simple problem: There is a charge $-q$ at $(0, 0, d)$ and $(0, 0, -d)$ as well a charge $2q$ at $(0, 0, 0)$. I have to calculate the quadrupole moment using spherical coordinates. I use ...
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1answer
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Center of charge in quadrupol tensor

In theoretical classical electrodynamics we defined the quadrupol tensor of $n$ charges $q_k$ at positions (from origin or center of charge, see below) $\vec r_k$ like so: $$Q_{ij} = \sum_{k=1}^n q_k ...
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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: $$ ...
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1answer
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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 ...
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quadripolar moment in curved space

So, i'm going over the Thorne's derivation of the quadrupolar radiation term, and they write the core term as: $$ \frac{3 r_i r_j - 2 r^2 \delta_{ij}}{4 r^5} $$ But if i try to obtain this term by ...
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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 ...