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Questions tagged [multipole-expansion]

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42 views

Clarification on Taylor expansion in linear quadrupole multipole expansion in cartesian coordinates

In my textbook, given the electric linear quadrupole on the z axis, as in the figure, the author considers the functions $$ \frac{1}{r_i} = \frac{1}{\sqrt{(x-x_i)^2+(y-y_i)^2+(z-z_i)^2}}, $$ with $r = ...
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11 views

What is the nature of the multipoles used to shape/re-direct an e-beam?

Imagine having an electron beam, just like in an electron microscope. Now imagine that we have some multipoles prepared in the beams path to influence the shape/path of the beam. My question is: ...
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1answer
26 views

Reconstructing Charge Distribution from Multipole Expansion

Let $\rho$ be a static, discrete or continuous charge distribution, and $\phi(\mathbf{r})$ the corresponding electric potential. We may expand $\phi$ in a multipole series, $$ \phi(\mathbf{r}) = \frac{...
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1answer
35 views

Does a uniformly moving charge induce electric quadrupole moment densities in surrounding space?

The electric field of a uniformly moving charge is cylindrically symmetric around an axis parallel to its velocity vector. It varies inversely to the square of the distance. The electric field of a ...
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0answers
42 views

Is there a way to know the non-zero coefficient of $\cot(\theta)$ expansion in spherical harmonics?

I'm currently trying to find an analytical solution to the Poisson equation for a given distribution using a multipole expansion. During this task, I found the radial expansion and everything else, ...
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0answers
43 views

Difference between monopole moment and charge or mass itself?

I'm trying to understand the difference between monopole moment and charge or mass itself. What I found is related to magnetic monopole that is irrelevant. I want to know in multiple expansion of ...
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0answers
61 views

Why gravitational waves can only be generated by a time-varying quadrupole moment of the mass distribution?

The (rather old) source I dispose of "Sexl, Urbantke : Gravitation and Cosmology" describes the radiation of gravitational waves only rather sketchy. So why gravitational waves are only generated by ...
2
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2answers
68 views

Electric quadrupole and octupole moments for nuclei

I am getting slightly confused as to which nuclei cab exhibit quadrupole and octupole excitations. In This link it says closed shell nuclei cannot exhibit quadrupole oscillations because if their ...
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22 views

Diffraction by induced quadrupoles

This is a very interesting problem that I've been struggling to solve for a week, so I decided to ask for some orientation, as I think it could also be of interest for the community. Let's consider a ...
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0answers
30 views

The magnetic and electric dipole moment 'popping out' from multipole expansions

I interpret electric dipole moments and magnetic dipole moments as intrinsic properties of certain materials, but in Griffiths it's literally picked out of expressing the potential and magnetic vector ...
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0answers
23 views

How to extract quadrupole moment and its error from $\chi^2 < \chi^2 + 1$ surface?

I have following info: plot of $\chi^2$ minimization of 208-Rn Coulomb excitation data, Surface corresponds to regions $\chi^2 < \chi^2 + 1$ with error bars of 1$\sigma$, mean lifetime is 8 $\pm$ 0....
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1answer
87 views

Quadrupole moment of Kerr spacetime

In this paper this paper, the Kerr black hole is described as having quadrupole moment of $q=J^2/M$ (which means $q=a^2M$ using $J=aM$) whereas in this paper it says in the abstract that the limiting ...
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2answers
118 views

Total charge for Dipole and Quadrupole moment

I have been struggling on what to put for $Q(total)$ in the equations for the dipole and quadrupole moment for the potential: $$ V_{quad}(\mathbf r) \approx \frac{2Qd^2}{4\pi\epsilon_0} \left( \frac{...
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1answer
33 views

Question about multipole expansion of electrostatic potential?

If I take a certain dipole and by dipole I mean that two charges of opposite sign differentiated by very small distance. And if I take the formula of multipole expansion of potential then I see no ...
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1answer
120 views

The fields of Liénard and Wiechert and Poynting vector

EDIT: I know that the electric and magnetic fields depend not only on speed but also on acceleration and can both be expressed as the sum of two contributions: \begin{equation} \overline{E} (\bar{r},...
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1answer
30 views

Relevance of electromagnetic multipole transitions

In what kind of systems higher electromagnetic multipole transitions (like electric quadrupole transitions) become important or at least measurable? Is it for antennas in radiofrequency? Is it in the ...
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1answer
66 views

Applications of multipole expansion in gravitational problems [closed]

I want to know what exactly are practical applications of multipole expansion in some problems concerning gravitation. I have also read in Subtle is the Lord by Abraham Pais that Einstein had ...
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1answer
55 views

Looking for an understanding of Toroidal Moments [duplicate]

The wikipedia page is not so enlightening Apparantly the Neutrino, if it is a Dirac particle, has a Toroidal Moment. Does this mean the Dirac neutrino would interact electromagnetically as well as ...
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3answers
194 views

Electric quadrupole moment and electric octupole moment

From a Chinese electrodynamics textbook, first we take a $1/r$ Taylor expansion into the electric potential equation and simplify it, then we get the final equation, and we call each term different ...
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0answers
18 views

Express polarization moments in terms of temperature quadrupoles

I am trying to compute the polarisation moments in the tight coupling limit which is an exercise from Dodelson's Modern cosmology where the evolution equation is given by $$\dot{\Theta}_p+ik\mu\...
2
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1answer
60 views

Can I replace a multipole expansion by a combination of separate dipoles?

If I want to be able to model a magnetic field flux density $\mathbf{B}$ from a magnetic source located at the origin at a position $\mathbf{r}$, it is my understanding that I can represent $\mathbf{B}...
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3answers
233 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, ...
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17 views

Relative strengths of multipole contributions in atom photo-excitations

I have heard and read many times that dipole contribution to the photo-absorption matrix element is orders of magnitude stronger than quadrupole. To be exact, they are related as: $\frac{\langle ...
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1answer
51 views

Electric quadrupole - tensor identity

In classical electrodynamics, we introduce the electric quadrupole moment $$D^{ij}\equiv\int y^i y^j \rho \mathrm{d}^3y$$ and its reduced (trace-less) version $$\mathcal{D}^{ij}\equiv D^{ij} - \frac{1}...
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1answer
61 views

Griffiths Multipole Expansion and $Q$ going to zero

Griffiths states that given the multipole expansion: $$V(\vec{r})=\frac{1}{4\pi\epsilon_o}\sum_{n=0}^\infty\frac{1}{r^{(n+1)}}\int(\vec{r}')^nP_n(cos(\theta')\rho(\vec{r}')d\tau'$$ for large $r$ the ...
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2answers
54 views

Does the quadrupole moment tensor contracted with itself yield Kronecker delta?

I have trouble understanding the Kronecker delta and how it comes up in tensor equations. I know the metric contracted with itself gives the Kronecker delta which either is 0 or 1 depending on if the ...
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1answer
99 views

The greatest quadruple moment

Let us consider the following problem: it is necessary to find the shape of the body with fixed mass and density which at large distances compared with its characteristic dimensions would give the ...
3
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1answer
65 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 $...
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3answers
249 views

Isn't magnetism governed by the inverse square law? [duplicate]

Why does magnetism appear to decay much faster than gravity with distance? A clear indication of this is the fact that a magnet that in short distance able to overcome gravity and pick up some object,...
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1answer
762 views

Electric field from a quadrupole

In the following problem, I have already solved for the value of the potential, and I would like to tackle the extra exercise, which asks for the electric field of a point quadrupole: At every ...
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1answer
28 views

Expressing interaction between two classical charge distributions in terms of multipole moments

I am interested in expressing the interaction energy between two classical charge distributions in terms of the multipole moments of each of the charge distributions. For example, let's assume each of ...
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1answer
99 views

How did scientists before Coulomb ensured this fundamental property of magnets? [closed]

Firstly, please note that I am talking about the period BEFORE electricity and magnetism were unified. So I am NOT seeking for answers based on Ampere atomic current model of magnets. I have read the ...
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2answers
382 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,...
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2answers
897 views

Does the dipole moment depend on the choice of origin?

Does the dipole moment depend on the choice of origin if the total charge Q is not zero? for a system of charges neutral overall? How can I show that mathematically? Also I need some drawings to ...
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1answer
99 views

Gravitational waves of an oscillating Schwarzschild black hole

Gravitational waves are produced by an accelerated mass, similar to the production of light waves by an accelerated charge. The amount of gravitational energy released from a rotating object can be ...
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1answer
116 views

Cross Product in Spherical Coordinates

I am looking at an example problem from Greiner's Classical Electrodynamics (chapter 21 , page 441) about the Hertzian Dipole where the radiation will require a cross product ($\vec{d} \times \hat{n}$,...
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0answers
81 views

Multipole expansions: test on a function $\zeta=\zeta(t)$

Considering the potential $\psi(r)$ of a sphere of mass $M$ with density $\rho(\mathbf r')$, connected by a small volume positioned in the $P'$ point as shown in the figure: The $\psi(r)$ is: $$\...
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0answers
299 views

Why is the electric field of an axial quadrupole not the same as the electric field of two axial dipoles, at far distance?

An axial electric quadrupole, made of four inline charges $(+q, -q, -q, +q)$ with opposite charges a distance $a$ apart, and the two $-q$ charges adjacent, has an electric field at a remote point $P$ ...
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1answer
64 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 ...
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2answers
80 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 ...
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1answer
198 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 ...
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1answer
40 views

How are these multipole moments related to the ones from electrodynamics?

Let $f : \mathbb{R}^3\to \mathbb{R}$ be a continuous function of compact support. Its Fourier transform is $$\mathfrak{F}[f](k)=\int f(x)e^{ikx}dx=\int f(x)\sum_{n=0}^\infty \dfrac{i^n}{n!}k_{a_1}\...
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1answer
100 views

Does the electric quadrupole vanish if $|\psi|^2$ is spherically symmetric?

In the book Nuclear and Particle Physics by B.R. Martin it is said that the quantum mechanical analogue to the electric quadrupole $$ Q \equiv \frac{1}{e}\sum\limits_i\int\psi^*q_i(3z_i^2-r^2)\psi d^3\...
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43 views

What is a force multipole?

In a recent talk about physics and mechanics inside the cell, I heard such terms as 'force monopole' and 'force dipole'. What do such terms mean? Are they talking about the angular distributions, ...
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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?
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1answer
185 views

Contribution of quadrupole term using Quadrupole moment

I read this derivation in Griffiths that if we obtain dipole moment vector $\vec{p}$ , then dipole term in the potential (due to general distribution far away at $\vec{r}$ ) can be written as $$\frac{...
2
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2answers
198 views

Why does the electric field strength for a dipole go as $1/r^3$?

I've been given the following graphic to help wrap my head around this. If the potential can be shown to represent a $1/r^2$ relation, then I'm more than happy to accept that the electric field is ...
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5answers
2k 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 ...
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2answers
132 views

Why can large objects at greater distance be treated as a point particle?

Why can large objects at greater distance be treated as a point particle? "The bodies of our solar system are so far apart compared with their diameters that they can be treated as particles to an ...
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1answer
94 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 ...