Questions tagged [multipole-expansion]

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Electrostatic polarization of an axially symmetric conductor

A single point charge at the origin induces charge onto a grounded Z-axis symmetric conductor. The induced charge is given in cylindrical coordinates as $\sigma(r, z)$ since the Z symmetry means there ...
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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 ...
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1answer
412 views

Gravitational Multipole Moments

I'm trying to write the gravitational potential of two masses $m_1$ and $m_2$ using the multipolar expansion of the potential: $$V = G\sum\frac{1}{r^{n+1}}\int r'^n P_n(\cos \theta) \rho(r') dV'$$ I ...
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937 views

Why isn't the electric quadrupole moment always zero?

We know that the electric quadrupole moment ($Q$) describes whether the nucleus of an atom is prolate ($Q<0$), oblate ($Q>0$) or spherical ($Q=0$). We also know that $$Q=\frac{2j-1}{2(j+1)}Q_0,$...
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1answer
441 views

Multipole expansion in electrostatics [closed]

I saw that in potential due to charge distribution at some far points is composed of monopole,dipole,quadrupole etc terms.Does that mean the infinite charges in that distribution distributes as ...
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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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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 ...
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2answers
424 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?
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2answers
3k views

Electric octupole moment in cartesian coordinates

I'm trying to calculate the symmetric traceless tensor for the octupole moment in cartesian coordinates... I have to deal with the electrostatic potential of the form: $\Phi^{(4)}(\textbf{r})=\int d^{...
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1answer
2k views

Quadrupole moment of point charges?

We know that due to expansion, the Quadrupole potential equals $$1/4\pi \epsilon r^3 . \int (r^\prime)^2(\frac32. \cos^2\theta^\prime-\frac12)\rho(r^\prime)d\tau$$ but what is the equation for point ...
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2answers
875 views

Multipole expansion of the electromagnetic field

In Jackson's Classical Electrodynamics, section 9.7, he develops the multipole expansion of the electromagnetic fields in terms of the vector spherical harmonics and the spherical Bessel and Hankel ...
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1answer
207 views

Do gravitational waves radiate isotropically?

Do gravitational waves radiate equally in all directions? If so, is this an inherent property of all gravitational waves or is it just due to how they are normally produced? In other words, in a ...
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1answer
3k views

Dipole moment of a single point charge

Kindly refer to the Multipole Expansion section (chapter 3) of David Griffith's Introduction to Electrodynamics. Let us first discuss about why multipole expansion is needed. As far as I understand, ...
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80 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 ...
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58 views

multipole moments of dipole with finite spacing

Can a dipole with finite spacing between poles be represented by pure multipoles centered at the origin? Say for example that I have a dipole with finite spacing $2\epsilon$ between the poles. I ...
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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 ...
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1answer
149 views

What is the affection of stress tensor of spacetime by the energy/mass density moment of a photon? [closed]

First of all what kind of moment exhibits the photon under its propagation to spacetime continium -quadrupole,dipole or monopole! Please, explain me- why. Do Give some arguements! When it propagetes ...
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2answers
3k views

Quadrupole moment traceless?

I've been doing some exercises in my workbook and every time I came across a problem involving a quadrupole moment it was always assumed that it was traceless without providing any proof (at least ...
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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 ...
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1answer
71 views

What is the location of point charge with respect to the original charged body? [duplicate]

we say if distance between two charged bodies is large as compared to its size then we take them as point charges and assume all its charge content to be concentrated at one point in space. What is ...
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0answers
195 views

Quadrupole and Multipoles in Physics

I am a little confused over the notion of quadrupole and higher moments in physics in general. The first time I saw it was in electromagnetism, when we did multipole expansion to analyze higher ...
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0answers
110 views

External and internal multipole expansion for axisymmetric potential - the region of convergence

Say, we have a system of electrodes exhibiting symmetry around a certain axis. We know the explicit expression for the potential on the axis $\phi (z)$. We want to find the potential for any point in ...
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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 ...
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0answers
47 views

strength of each dipole charge

If I have a two dimensional dipole whose line charges are located on the y axis, I know that the electric flux through a gaussian pill box containing both the charges will be zero, that is \begin{...
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0answers
83 views

Series expansion of potential due to a dipole displaced from the origin

I learn that we can expand the electric potential in an infinite series of $\rho$ and $\cos(n \phi)$ when solving the Laplace equation in polar coordinates. The problem I want to consider is the ...
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1answer
271 views

Is the transition electric quadrupole or magnetic dipole?

If a nucleus makes a transition from 0$^+$ ground state to 2$^+$ excited state, then will the transtion have E2 character, or M1? Or partly, both? Should the matrix elements of both E2 and M1 be ...
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3answers
2k 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}|}\...
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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 ...
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1answer
38 views

Numerical computation of electric multipoles

I'm trying to do a plot of the first multipole terms in Mathematica. My plot isn't what I expected, so maybe my problem is in the mathematics. Following the Griffiths' book on Electrodynamics, the ...
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1answer
92 views

Where does the net charge on a hemisphere appear to act from?

I was wondering if, like centre of mass, there is anything called "centre of charge".
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67 views

find cylindrical multipole coefficients

How does one find the coefficients to the cylindrical multipole expansion? I have the harmonic function $\omega = \dfrac{f(\theta)}{\sqrt{r}}$ where $f(\theta)$ is function with a period of $2\pi$ ...
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2answers
924 views

Multipole expansion in cylindrical coordinates

I am seeking the general solution for the Laplace equation in cylindrical coordinates or $$\nabla^2 \omega = 0. $$ In several texts, the general solution can be found via separation of variables ...
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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 ...
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1answer
150 views

Introductory literature on the multipole expansions including toroidal moments

I wish to understand the general idea of the multipole expansion in the context of classical electrodynamics and, especially, the concept of the toroidal moments. All the papers on the subject I've ...
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82 views

Multipole expansion of Woods-Saxon potential?

When can a distribution be expanded in multipoles? What is the basic requirement? Can it be done for a potential like the Woods-Saxon form?
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1answer
880 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 ...
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1answer
513 views

Physical intuition for quadrupole source

In his Theory of Vortex Sound M. S. Howe defines sources "mathematically" (i.e. dipole is a source that could be described as a vector and than there is proved that it's equivalent to a two point ...
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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 ...
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2answers
300 views

What determines the factors of the multipole expansion?

The multiple expansion of a potential V has contributing terms proportional to $\frac{1}{r^{n+1}}$ where $n=0,1,2...$. First, why are we interested only in integer powers of r? Second, why are we ...
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1answer
3k views

Find out gradient of electric potential at ${\bf r}$ created by eletric dipole of moment ${\bf p}$ [closed]

Suposing an electric dipole of moment ${\bf p}$ located at the origin which creates an electric potential at ${\bf r}$ given by $$\psi(\textbf{r})=\frac{\textbf{p}\centerdot\textbf{r}}{4\pi\epsilon_0r^...
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1answer
487 views

The Electric Quadrupole

I've read the following sentence: "Every electric circuit with two pairs of accessible terminals is called a quadrupole." I was wondering why does it happen that the multipole expansion gives us a ...
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133 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 ...
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0answers
106 views

Physical significance of toroidal moments

The electric field of a toroidal dipole moment $\vec t$ is the same as the electric field of a electric dipole moment $\vec p$, except that it is scaled with the factor $ik$. The $i$ makes the ...
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2answers
66 views

Is it inevitable to compute the quadruople tensor in components? Why? [closed]

I was trying to determine the quadrupole tensor for a given charge distribution in one go from this equation: $$\overleftrightarrow{D}=\int d^3r \varrho(\vec{r})\left(3\vec{r} \circ \vec{r}-r^2\hat{I}...
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0answers
47 views

Approximating electrostatic grids as a multipole expansion

Is there a known good summary, or a succinct algorithm to compute the far-field approximations of an arbitrary set of electrostatic surfaces set at different potentials? I'm looking to model a ...
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2answers
241 views

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 \sqrt{\frac{4\pi}{2l+1}...
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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 ...
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3answers
2k 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 ...
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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 ...
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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 ...