Questions tagged [multipole-expansion]

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Potential due to a charge distribution and multipole expansion

I'm studying on the Griffiths' book Introduction to Electrodynamics and a doubt came to me reading about the multipole expansion. In chapter 2.3.4 this formula is shown $$ V(\vec{r}) = \frac{1}{4\pi\...
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Can a quadrupole form in a purely organic crystalline material?

I'm a chemist so bear with me here. This is a question about small molecules, such as biphenyls. In the solid-state biaromatic systems without steric hinderance forms planar conjugated structures. By ...
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Transforming from the non-traceless to the traceless quadrupole moment

Today I was working over the multipole expansion for electric potentials, and on my own, I got to the non-traceless definition of the quadrupole moment $ Q_{\alpha \beta} = \int{\rho(\mathbf{r'})\...
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Why is $A \propto 1/r^3, B \propto 1/r^4$, far away from circular loops

Two equal circular current loops are placed coaxially with each other. The loops have equal but opposite currents $I$. $$ A \propto r^{-l} $$ $$ B \propto r^{-k} $$ , where $A$ is the vector ...
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Components of Electric Quadrupole Oscillator Strength

Fermi's Golden Rule states that the rate of a transition of an electron going from the ground state $0$ into some state $n$, is directly proportional to the square of the first order perturbation $\...
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62 views

How can a potential be non-central?

I'm studying nucleon-nucleon interactions and I'm reading that the potential for said interaction has a non-central (or tensor) component. Now, I understand that, when describing a 2-bodies problem, ...
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32 views

Electric octupole in cartesian coordinates

I am new to this: Does anyone know to define different electric octupole moments in cartesian coordinates? I am looking for expressions that look like this (for electric dipoles).
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Understanding the multipole Expansion for Quadrupoles

Potential of a Quadrupole is given as, $$V(r) = \frac{1}{4\pi e_0}\left(\frac{1}{R}q+\frac{1}{R^2}\sum_{i=x,y,z}\hat{R_i}\vec{p_i} + \frac{1}{R^3}\sum_{i,j=x,y,z}\hat{R_i}\hat{R_j}Q_{ij}\right)$$ ...
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22 views

Multipole expansion

consider a charge distribution say q, 2q, 3q, kept on the vertices of an equilateral triangle. There exist monopole, dipole ,qudrupole etc terms in the potential expression. these 3 charges are ...
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58 views

How to calculate the Quadrupole moment via integration?

I have following problem: calculate the quadrupole moment of following arrangement, where $e$ is the charge and $a$ is the distance between the charges. Hint: The quadrupole moment is defined ...
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Question about the Multipole Expansion second term (quadrupole)

I was wondering if the $n=2$ term in the potential of a multipole: $V(r)= \dfrac{1}{4 \pi \epsilon_0} \sum \dfrac{1}{r^{(n+1)}} \int (r')^n P_n(cos \alpha) \rho(r')dv'$ was the same as this ...
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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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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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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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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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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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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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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 ...
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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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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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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
114 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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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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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
126 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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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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69 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
58 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
275 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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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\...
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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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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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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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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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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
105 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 ...
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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
363 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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1k 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
32 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
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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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434 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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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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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
148 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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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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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
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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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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 ...