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Questions tagged [multipole-expansion]

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2
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2answers
207 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 ...
7
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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 ...
1
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2answers
144 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 ...
3
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1answer
99 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 ...
2
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1answer
681 views

Linear, homogenous and isotropic dielectric in electrostatic field. Why do I consider two potentials (inside & outside sphere)?

Presentation of the problem : We have a uniform homogenous isotropic dielectric sphere in an electrostatic field. To solve this problem, we remark that we have an azimuthal symmetry. So the ...
6
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1answer
385 views

Dependence of the multipole moments on the origin

Multipole moments of a system are defined with an explicit refrence to the co-ordinate system, e.g. $$\boldsymbol{d}=\int dV\, \rho\,\boldsymbol{r}\\ \boldsymbol{\mu}=\frac{1}{2c}\int dV\, [\...
-1
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2answers
149 views

Integrals with azimuthal symmetry

The dipole moment is given by $$\textbf{p}=\iiint \textbf{r'}\rho(\textbf{r'}) \;\mathrm{dV}$$ let's say that the charge configuration is a sphere with radius R and that the charge density is $\rho$ ...
1
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1answer
2k views

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 $\rho_0$,...
3
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1answer
755 views

Understanding multipole expansion in classical electrodynamics

I am trying to better understand what the multipole expansion means from a phyiscal point of view. Although mathematically, one may say it is just another form of a series expansion, in this case, the ...
2
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1answer
135 views

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 ...
1
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1answer
641 views

Is octupole moment traceless?

I derived expressions for quadrupole $\Theta_{ij}$ and octupole moment $\Omega_{ijk}$: $$\Theta_{ij} = \frac{1}{2}\int dV \rho(\mathbf{r}')\left[3r'_ir'_j - r'^2\delta_{ij} \right],$$ $$\Omega_{ijk} = ...
0
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0answers
68 views

Are there transitions of mixed multipolarity in charged ions?

In nuclear physics transitions with mixed multipolarity, ex. (M1+E2), play an important role due to d-wave admixture. I was wondering if analogous situation has ever been considered in atomic ...
1
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0answers
45 views

Does a spherical potential mean spherical mass distribution? [closed]

I came across this question is Binney & Tremaine's galactic dynamics. I'm pretty sure that the answer is no, but struggling to come up with an example.
1
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2answers
350 views

Is there a relation between the Legendre generator function and Spherical Harmonics for a Potential?

Recently I had to solve a simple problem in which I had a sphere of radius $R$ with a constant potential (but with different sign), on both of the hemispheres, and I was asked to get the electrostatic ...
0
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1answer
101 views

Multipole expansion of an azimuthally symmetric charge distribution

Given a localized distribution of charge with a charge density: $$\rho(\vec{r})=\rho(r,\theta)=\frac{1}{64\pi}r^2\frac{1}{e^r}\sin^2\theta$$ And I know that: $$ \phi(\vec{r})=\sum_{l=0}^{\infty} \...
4
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3answers
882 views

Are gravitational quadrupole moment, second moment of mass, and moment of inertia the same?

my understanding of moments is that they refer to distributions about an expected value, which allows us to make the multipole expansion. I read that: the zeroth moment of mass refers to the mass of ...
0
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1answer
464 views

How do multipole moments relate to a Taylor expansion, with regards to Newtonian potential?

Given the Newtonian gravitational potential, $$ \phi(\mathbf{x}) = - \int \frac{G \rho(\mathbf{x'})}{|\mathbf{x} - \mathbf{x'}|}$$ One can construct a 'multipole expansion' by using the Taylor ...
0
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0answers
33 views

Are there atoms with compatible electric and magnetic multipole transitions

I know that usually E1-transition is dominant over all the higher multipoles in atomic spectroscopy. My question is: are there any natural/fabricated atomic systems where magnetic transitions are ...
3
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1answer
175 views

Multipole expansion of energy-momentum tensor

In this paper it is stated that the multipole moments of the stress-energy tensor $T^{\mu \nu}$ are given by $$ \int_{x^0 = const} T^{\mu \nu} \delta x^{\alpha_1}\cdots \delta x^{\alpha_n} \sqrt{g} \:...
1
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0answers
106 views

Tight coupling of photons and baryons before recombination

I am trying to understand the equations of CMB anisotropies. In the book I am studying with, the author wants to show that, when solving for the photon perturbations in the tightly coupled limit (...
11
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2answers
3k views

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 ...
1
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1answer
144 views

Why do we need poles?

The electric force is the attraction or repulsion between charges. If we for example had a metal with only positive charges, and another metal with only negative charges. The two metal pieces will ...
0
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2answers
283 views

Quadrapole moment of a real dipole [closed]

What is the quadrapole moment of a real dipole with 2 opposite charges seperated by distance $a$?
4
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1answer
606 views

The dipole radiation pattern and spherical harmonics $Y_{10}$

I am studying the multipole expansion of electromagnetic wave radiation pattern, and it is said that any fields can be decomposed into the spherical harmonics $Y_{lm}$. However, for $l=1$, which ...
1
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0answers
75 views

How to understand the multipole expansion? [closed]

How to understand the multipole expansion? I just can't understand the description given in Wikipedia. It seems on Wikipedia that the formula is about a function only with angle variable, but I think ...
1
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1answer
89 views

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 ...
0
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1answer
136 views

Point charge above infinite plate - with Legendre Polynomials [closed]

From the method of images we know that the potential everywhere above a grounded plate with a point charge above it is equal to $$ V(x,y,z) = (\frac{q}{4 \pi \epsilon_0})[ \frac{1}{\sqrt{x^2 + y^2 + (...
19
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3answers
3k views

Why is dipole the simplest source in electrodynamics?

I see this sort of statement in many materials, for example this: The smallest radiating unit is a dipole, an electromagnetic point source. and this: The simplest infinitesimal radiating ...
3
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1answer
705 views

How does a electric quadrupole oscillate?

I know that in static a electric quadrupole is made of two positive charges and two negative charges, distributed as in the following figure: What I don't understand is when it oscillates and ...
7
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2answers
204 views

How many truly different multipolar charge distributions are there?

Dipolar charge distributions are essentially all the same: regardless of how one adds up a combination of the form $$ \sigma(\theta,\varphi) = \operatorname{Re}\left[\sum_{m=-1}^1 a_m Y_{1m}(\theta,\...
-1
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2answers
1k views

Why does the electric field in a dipole cancel out at distances much larger than the separation of the two charges forming a dipole ($r \gg 2a$)?

The electric field of the electric dipole is not zero. Since the charge $q$ and $–q$ are separated by some distance, the electric fields due to them, when added, do not exactly cancel out. However, ...
-1
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1answer
2k views

What is an electric quadrupole and how to find its net electric field on a point which is on the dipole axis? [closed]

In a question of my book it says that an electric quadrupole is a system of two dipoles of equal magnitude but opposite in sign. If that's the case and since they are vectors, won't they cancel out ...
5
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2answers
522 views

Meaning of lowest multipoles in CMB spectrum?

What is the meaning attributed to the apparent rise of the CMB at very low mutipoles (i.e. at the beginning of the experimental spectrum), if any? Clarification: The data at that region have large ...
1
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1answer
402 views

Multipole expansion of the magnetic energy of a distribution of current density

My professor left me as an exercise the multipole expansion of the energy of a distribution of current density in a magnetic field but I don't manage even to understand how to start. The energy of ...
0
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0answers
729 views

Quadrupole moment tensor

I am currently working on gravitational waves, and as in every lecture on general relativity I derived the symmetric trace-free perturbation of a Minkowski metric with the Lorenz gauge condition, ...
0
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2answers
259 views

Is there a legendre polynomial series expansion for the gravitational potential in General Relativity (GR)?

In Newtonian mechanics we perform a multipole expansion on the gravitational potential $V(r)=-GM/r$ by a series expansion of Legendre polynomials. Then the Hamiltonian is given by \begin{equation} H ...
1
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0answers
156 views

Does it make sense an inertial mass dipole moment for massless particles?

Electric and magnetic dipoles exist in nature even with zero net electric and magnetic charge Electrons have a unit of charge, about $10^{-31}$ kg of inertial mass, zero electric dipole moment, $\...
23
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5answers
3k views

Far away from a charged conductor, the field is like a point charge. Where's the point located?

In the framework of classical electrodynamics, at distances much greater than a conductor's dimension, the field ought to approach that of a point charge located at the conductor. But where at? For ...
0
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2answers
833 views

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=\...
5
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0answers
124 views

What is the minimal number of point charges required to model an electric octupole?

In an answer to this question, I showed that a general electric quadrupole can be modeled by a combination of seven point charges (or possibly less if one or two of the principal quadrupole moments is ...
2
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1answer
523 views

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'...
2
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1answer
35 views

Choosing $A_l=0$ to guarantee bounded potential in infinity

I'm taking a course in Electrodynamics and quite often, when using the spherical approach $$\Phi=\sum\limits_{l~=~0}^{\infty}\left(A_lr^l+B_lr^{-(l+1)}\right)P_l(\cos\gamma),$$ there's the argument ...
1
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1answer
83 views

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 ...
0
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2answers
847 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 ...
1
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1answer
304 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 ...
0
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1answer
511 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,$...
0
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1answer
323 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 ...
1
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1answer
1k 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 $+...
0
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0answers
207 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 ...
6
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2answers
288 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?