Questions tagged [multipole-expansion]

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Symmetrical, spherical charge distribution [closed]

I have this problem from my Electromagnetism class, but I don´t have a lot of idea in how to solve it. Show that for a symmetrical, spherical charge distribution $\rho(\vec{r})=\rho(r)$ (with $r$ ...
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Shifting polynomials in Fast Multipole Method

There is one thing I don't get about the FMM algorithm (of coulombic potential in 2D - https://cims.nyu.edu/~donev/Teaching/WrittenOral/Projects/JasonKaye-WrittenAndOral.pdf). Suppose we have ...
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Spherical harmonics expansion: from scalars to tensors

It is well known that a scalar field on the unit sphere can be expanded in spherical harmonics, see e.g. this. I am wondering if there is a related concept for vector fields and, in general, for any ...
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Building a gravitational potential out of multipole expansions of line segments

I would like to build a gravitational potential out of line segments (numerical). Imagine breaking a wire of mass with arbitrary linear density and path into a collection of small straight wires of ...
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Confusion about exact electrostatic potential of a pure dipole [duplicate]

I asked a version of this question before but the phrasing was unclear, so don't mind me please. Suppose you have a physical dipole with equal and opposite charges Q and -Q, with a specified dipole ...
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Electrostatic potential due to a pure dipole

For a pure dipole, is the electrostatic potential given by the dipole term in the multipole expansion exact for all $r$ (position vector from the origin)? Or is it exact only far away from the dipole? ...
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The multipole expansion of electrostatic potential and large distances

I'm reading Griffiths electrodynamics book and I'm currently studying the multipole expansion of electrostatic potential, and I have two questions if you don't mind: Can I use the multipole expansion ...
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Properties of Vector Spherical Harmonics

In section 5.3.2 of the book Advanced Classical Electromagnetism by Robert Wald, in deriving the multipole expansion for the retarded solution of electromagnetic field in presence of charge-current ...
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Working with decomposition of fields

I'm trying to follow a text I found online. The author decomposes EM fields such $$ \mathbf{E} = \sum_{lm}\left(f_l(r) \mathbf{Y}_{lm} - i \frac{l(l+1)}{r} g_l(r) \mathbf{\Psi}_{lm} - i\left(\frac{d ...
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Can electric fields of multipoles just be described in terms of monopoles, if so then why do we have to do multipole expansion instead of monopoles?

Can we describe the most complex electric field just using monopoles rather than going into multipoles expansion, because using superposition principle we can tell the potential of entire system?
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How important are polarization and magnetization to far-field antenna properties?

Magnetization sources a magnetic field and polarization sources an electric field. Presumably, as an RF current is run through an antenna, it develops locally some magnetization and polarization. In ...
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Is gravitoelectromagnetism monopole or dipole?

Thorns said gravitoelectrism is monopole whereas gravitomagnetism is dipole. Others are saying the latter is monopole: https://doi.org/10.1140/epjc/s10052-021-09696-3 Which on is most likely right? ...
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Convergence of the multipole expansion

Do higher order terms in the multipole expansion for molecules always contribute less than lower, non-vanishing terms? Edit: Although my understanding is that the series is convergent, does that mean ...
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Gravitational waves of objects with the same angular momentum

So I read that gravitational waves are produced when the quadrupole moment of a system is not symmetric. What does it actually mean for the quadrupole moment to be asymmetric? If there are two black ...
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Is a perfectly-absorbing sphere ill-defined?

Suppose we have a spherical object that absorbs without reflection scalar waves (e.g. sound waves) incident on it. We should expect that the incident wave will get a shadow in the corresponding region ...
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Why does a neutral atom have no external electric field?

I've been trying to find an answer to this for some time now. In advance, I imagine the orbital model, nevertheless I cannot imagine that if I am for example closer to a side of the atom, the dipole ...
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How to derive the Debye series expansion of the Mie scattered field?

The usual Mie scattering theory is just a series expansion of the scattered field and the field inside the scattering sphere, done using a cleverly chosen basis. This lets us calculate the full ...
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Is electron diamagnetic resonance (invoking Larmor precession) possible?

Diamagnetism is observed in materials where electrons are paired. These paired electrons have opposite magnetic dipole moments. Therefore, the total magnetic torque on the pair's magnetic dipole ...
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Expressing point charge density in terms of the spherical harmonics

Lets suppose we have two point charges at for $q(0,0,+a)$ and $-q(0,0,-a)$. We can express this as the charge density, by using $$\rho(\vec{r})=q\delta(r-a)\delta(\phi)[\delta(\theta)-\delta(\theta-\...
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Does a symmetric current distribution $\vec{J}(\vec r)$ simplify multipole moments of vector potential?

For a spherically symmetric charge distribution, $\rho(\vec r)=\rho(r)$, all higher-order multipole moments except the monopole term vanish. Does a similar thing happen for the multipoles of a ...
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Why are non-central nuclear forces also called tensor forces?

Experiments suggest that nuclear forces are non-central. Sometimes this is called tensor forces. Why?
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Deriving discrete Quadropole moment from the continuous one

For the dipole moment I can derive the expression for a discrete charge distribution from the continuous one: $p_i=\int\rho(\vec r')x_i'dV'$ And for the electric dipole $\vec p$: $\vec p=\int\rho(\vec ...
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Dipole moment of 2 charges with opposite value

If I have 2 charges $-q$ in origin and $+q$ in position $(0,0,d)^T$ and I want to find the dipole moment of this charge distribution. I easily can use the formula for dipole $\vec p = q\vec r$ or $p_i=...
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Multipoles as a superposition of dipoles?

While navigating on Stack Exchange, I was suddenly wondering about a possibility that I never saw described in my textbooks on electrodynamics. Suppose you have a distribution of electric charges in ...
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About applicability of Gauss’ law in electrostatics

While reading about Gauss’ law in electrostatics, I saw that we do not apply this law to evaluate the fields generated by dipoles, quadrupoles etc. It is only applicable to cases where the fields fall ...
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Multipole Expansion Intuition

So my intuition of the multipole expansion is that we have a certain charge distribution, and the ME tells us that we can approximate that distribution as a superposition of monopoles, dipoles, etc. ...
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Griffiths and Multipole Expansion, what is each variable in the equation?

Griffiths defines the formula for the multipole expansion as $$V(\mathbf r)=\frac{1}{4\pi\epsilon_0}\sum_{n=0}^\infty\frac{1}{r^{(n+1)}}\int(r')^nP_n(\cos\alpha)\rho(\mathbf r')\,d\tau'.$$ But I am ...
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Is an expression of a quadrupole as an expansion of dipoles possible?

Would it be possible to express a quadrupole as an expansion of dipoles? Because a possible definition of a quadrupole seems to be: an electric field equivalent to that produced by two electric ...
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Units in the multipole expansion

In the Wikipedia page (https://en.wikipedia.org/wiki/Multipole_expansion#Expansion_in_Cartesian_coordinates) the expression for the multipole expansion gives the dipole term as $$V(r)=\frac{1}{4\pi\...
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Gauss law apparent violation

If we assume to place a dipole or a quadrupole inside a Gaussian surface, then the Gauss law will give zero electric field outside Gaussian surface at any point since enclosed change is zero. However,...
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Electric dipole moment and charge density

I came across the following expression about the electric dipole moment: $\vec p=\int_V \rho(\vec {r}')\vec r'$ But I don't understand it. What is $\vec r'$ supposed to represent and the density? Are ...
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Why should a dipole have zero net charge?

Why can a dipole not have two unequal charges separated by a distance? Is there any significance for the dipole being defined as electrically neutral?
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Question from Weinberg Lectures on quantum mechanics

In page number 37 of Weinberg's lectures of quantum mechanics book (2nd edition), After Eq.2.1.17, he states the following: The Schrödinger equation (2.1.3) then takes the form $E \psi(x) = -\frac{\...
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Gravitatinal Dipole moments?

So, I was reading about electric dipoles, which is a system of positive and negative charges kept at a distance $d$. But I got an ideas, as to why there seems to be a lot of example of electric ...
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Temporal Variation of Cosmic Microwave Background Radiation

For some time now I am wondering how fast the Cosmic Microwave Background (CMB) Radiation varies with time, in particular, how fast the primary CMB anisotropies are varying. These anisotropies were ...
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Do paired electrons have no higher order magnetic multipoles?

An electron pair consists of an up and down electron in an energetically favorable condition where their magnetic dipole moments cancel each other. However, does that also mean they have no quadrupole ...
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Introduction of a factor $\Delta\ell$ when summing equal distants $C_\ell$

In the context of Legendre expansion with $C_\ell$ quantities, below the following formula which is the error on a $C_\ell$ : $$\sigma_(C_{\ell})=\sqrt{\frac{2}{(2 \ell+1)\Delta\ell}}\,C_{\ell}\quad(1)...
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How do you transform an octopole moment to depend on the origin as the center of mass?

The equation for the origin dependence of a quadrupole moment $\mathbf{Q}$ under translation is: $$\mathbf{Q}'=\mathbf{Q}-[\frac{3}{2}(\mathbf{\mu}\mathbf{a}+\mathbf{a}\mathbf{\mu})-(\mathbf{\mu}\cdot ...
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Expressions for multipole moments

I am trying to understand the multipole expansion, though I don't have a background in EM. Wikipedia gives the expression for the Coulomb potential for dipole and quadrupole moments in terms of $1/r^{...
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Tidal Love numbers and the Schwarzschild metric

I'm puzzled by the statement below: Consider the Einstein equation expanded to the linear order around the Schwarzschild background. This describes perturbation of black hole solution $$g_{\mu\nu}=g_{\...
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Do the magnetic fields of paired electrons vanish everywhere, or do they form a magnetic circuit within the orbital?

As far as I can tell, that paired electrons produce opposing magnetic dipole moments does not rule out the possibility that they produce a toroidal moment. Is it possible in quantum mechanics that ...
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Constellation of charges in a quadrupole

I’d like to know if any constelation of 4 charges is a quadrupole. I have a task where there are 2 positive and 2 negative charged particles in the corners of a rectangle. And this is a quadrupole. Is ...
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5 votes
2 answers
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How can someone mathematically prove that a spherical wave equation solution cannot be a electromagnetic wave? [duplicate]

It's well documented that spherical electromagnetic waves doesn't actually exists. However, most textbooks just brings this concept without any formal demonstration. So, for a spherical wave equation ...
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Potential of a continuous charge distribution and it's dipole term

I've a charge distribution, whose potential will be given as $$\begin{aligned} V(\mathbf{r})=& \frac{1}{4 \pi \epsilon_{0}}\left[\frac{1}{r} \int \rho\left(\mathbf{r}^{\prime}\right) d \tau^{\...
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What is the charge distribution of an Electric Quadrupole?

I'm trying to compute the charge distribution corresponding to the (point) quadrupole moment to gain some intuition (I wanted to know if there can be a quadrupole surface layer like there can be ...
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Taylor expansion of a charge density (multipole radiation)

I'm a little confused about this expansion (maybe because its to late). Could someone write the taylor series down in a more general way (as a sum for example)? It would help me to understand where ...
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Derivation of gravitational wave as quadrupole moment

In Carroll's Spacetime and geometry section 7.5, it is deriving metric $\bar h_{\mu \nu}$ with Lorenz gauge in terms of quadrupole moment of energy density. In Eq.[7.135], \begin{equation} \int d^3y \...
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What is the electrical potential of a quadrupole ion trap?

I learned from Wikipedia that the quadrupolar potential $\phi$ of a quadrupole ion trap is \begin{equation} \phi = \frac{\phi_0}{r_0^2}(\lambda x^2 + \sigma y^2 + \gamma z^2) \end{equation} where $...
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Amplitude decay of electromagnetic and gravitational waves

I recently got confused by an article that claimed that gravitational waves were interesting because that their amplitude of a gravitational wave falls as $1/r$ in the far field limit. Suggesting this ...
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Gravitational Waves, Dipole Radiations and Momentum

I am trying to understand Gravitational Waves and and going through MTW's book. They state that there can be no dipole radiation because of the conservation of momentum but gravitational waves also ...
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