Skip to main content

Questions tagged [multipole-expansion]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
1 vote
1 answer
44 views

From where does this equation of quadrupole potential of a quadrupole mass filter derive from?

Quoting from thesis "Chip-scale quadrupole mass filters for a Micro-Gas Analyzer" (page 23): As I am solving quadrupole mass filter equation, I am stuck from where quadrupole potential ...
khushbu's user avatar
  • 11
3 votes
2 answers
154 views

The puzzling interaction between an anapole moment and external fields

Consider an electrical current distribution with only an [anapole, or toroidal moment] but no electrical or magnetic multipole moments, like this current on a torus: Its magnetic field is completely ...
Jos Bergervoet's user avatar
30 votes
5 answers
14k views

How can a point source emit spherical EM waves when they are forbidden by Maxwell's equations?

I know that there exist plane wave solutions to the Maxwell equations in free space, and I tried solving them for a spherical wave emanating from a point but could find no solution consistent with the ...
Thanos's user avatar
  • 419
1 vote
1 answer
183 views

Why does a group of charges with spherical symmetry, where each oscillates radially while retaining spherical symmetry, not radiate? [duplicate]

I am thinking about the problem, "Why does a group of charges with spherical symmetry, where each oscillates radially while retaining spherical symmetry, not radiate?" I figured out that ...
KingWangZZang's user avatar
0 votes
1 answer
67 views

Can an oscillatory field produced by a time-dependent charge distribution fall off faster than $1/r$?

An electric charge has a field that decays at large distances like $1/r^2$. A charge distribution with total charge zero, but with a nonvanishing dipole moment, has a field that decays at large ...
user196574's user avatar
  • 2,282
-1 votes
1 answer
32 views

Electric Field of a Linear Electric Quadrupole Along x Axis [closed]

A charge $+q$ is at $x=-d$. Another $+q$ is at $x=+d$ and $-2q$ is at the origin. a) Derive expressions for the electric field along the x-axis as a function of x. b) show that $E∝\frac{1}{r^4}$ for $...
TEGNO's user avatar
  • 1
2 votes
0 answers
57 views

Do neutral atoms have an electric field? [duplicate]

The charge of an atom is the sum of its nuclear charges (protons and electrons). If a atom is neutral, does it mean it does not have an net electric field? I thought about this a lot, here is some of ...
Kryptic Coconut's user avatar
0 votes
0 answers
20 views

Difference between Mutlipole moments calculated with normal integration and with Wigner-$D$ matrices

Im learning about Wigner-$D$ matrices and the applications to spherical harmonics. Now I wanted to test my knowledge, but i failed miserably :( (worked on this the whole day). So here is my problem: I ...
Stefan283's user avatar
0 votes
2 answers
29 views

Couple acting on quadrupole due to a point charge

I wish to find the couple acting on the quadrupole due to q', assuming r>>a. Here is my working: Force acting on -2q: $E_{at \ -2q} = \frac{q'}{4\pi\epsilon r^2} \\ F_1 = (-2q) E = \frac{-2qq'}{...
psychgiraffe's user avatar
0 votes
1 answer
70 views

Can a point charge be asymmetric?

The derivation of Coulombs law from Maxwell's first equation for a point charge assumes that the field is symmetric along a sphere. What happens if this assumption is removed? Could there be other ...
Kids Free's user avatar
1 vote
1 answer
31 views

Is quadrupole contribution to gravitational potential the sum of the contribution of all $m$ values?

Many of the sources I find on multipole expansions seem to be about electric potential and involve matrices. However, in my classical mechanics class we have not used matrices for multipole expansions ...
toomanyfeet's user avatar
0 votes
0 answers
28 views

Finding quadrupole moment

The potential of a $2^n$ pole can be found by, $$ \varphi_n =\frac{p^{(n)}}{4\pi\epsilon_0 n!}\frac{\partial^{'n}}{\partial l_n \partial l_{n-1} ... \partial l_1} \bigg(\frac{1}{r} \bigg) $$ where $$ ...
Refrigerator's user avatar
0 votes
1 answer
65 views

Electromagnetic Dipole Radiation Derivation

I was reading this wikepedia page, about the dipole radiation, and I was wondering how to derive the $\mathbf E$ and $\mathbf B$ fields in this situation. I've started using the retarded potentials: $$...
Álvaro Rodrigo's user avatar
1 vote
1 answer
63 views

Why is the quadrupole tensor for a hollow sphere a diagonal matrix?

I have determined a quadrupole tensor for a hollow sphere with radius $R$ and charge density $\rho(\theta)$. The tensor has the following form:$$Q=\left(\begin{array}{ccc}Q_{11} & 0 & 0 \\ 0 &...
CherryBlossom1878's user avatar
0 votes
0 answers
67 views

Electric Potential due to an ellipsoid

Recently I came across the following problem: Suppose $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ is an ellipse with surface charge density $\sigma=\sigma_0\sin(\theta)\cos(\phi)$ where $\...
Amit Kumar Basistha's user avatar
10 votes
2 answers
1k views

Multipole expansion of gravitational field

Just as in electrostatics we can expand in a Taylor series the electrostatic potential as the infinite sum of the contributions of a monopole, a dipole, a quadrupole, etc., could we apply the same ...
Lagrangiano's user avatar
  • 1,631
0 votes
0 answers
35 views

Expansion coefficients in multipole expansion as spherical tensors

This is a relatively simple question, but I cannot find a clear answer: Given the multipole expansion of a real scalar function, $$ f(r,\theta,\phi) = \sum_{\lambda\mu} f_{\lambda\mu}(r) Y_{\lambda\mu}...
kc9jud's user avatar
  • 211
3 votes
0 answers
57 views

Function composition and multipole expansion

Assume that I have some function $g(r,\theta,\phi)$ which I have expanded in a multipole series: $$ g(r,\theta,\phi) = \sum_{\ell=0}^\infty \sum_{m=-\ell}^{+\ell} g_{\ell m}(r)\, Y_{\ell m}(\theta,\...
kc9jud's user avatar
  • 211
4 votes
1 answer
162 views

Are there any forces that are inversely proportional to the fourth power of the distance involved?

As I humbly asked on Astronomy SE, I was wondering if there any forces or similar phenomena that scale inversely proportionally to the fourth power of the distance. There are plenty of things that ...
user267545's user avatar
2 votes
2 answers
452 views

Parity of Photons

In nuclear physics, while studying gamma decay (Nuclear physics, Roy and Nigam, 1st ed, pp 450) I have read that the parity of photons depends on the type of multipole radiation they represent. Means ...
Sagar K. Biswal's user avatar
2 votes
1 answer
112 views

Optimal position of negative charge to "neutralise" a positive distribution

This question stems from trying to understand the notion of center of charge and if the analytical definition of this center depends on what exactly is minimized (the dipole moment or the total ...
Quillo's user avatar
  • 5,068
0 votes
1 answer
69 views

Potential - metal sphere in a uniform electric field

I ran into trouble while reading example 3.8 from Griffiths' Introduction to Electrodynamics (4th ed.). The example is as follows: "An uncharged metal sphere of radius $R$ is placed in an ...
driver99's user avatar
0 votes
0 answers
79 views

Multipole expansion of a scalar quantity which is a function of a unit vector in 3D

I have a known scalar quantity $A(\mathbf{\hat{e}})$ which is a function of a unit vector in 3D \begin{equation} \mathbf{\hat{e}}=(e^1,e^2,e^3)=(\sin\theta\cos\phi, \sin\theta\sin\phi,\cos\theta). \...
user1712948's user avatar
20 votes
5 answers
3k views

Purpose of Using Taylor Series and Multipole Expansion to Approximate Potential

I'm currently taking a third-year electromagnetism course (we use Griffiths), and we have begun covering approximations of our potential function, $\text{V} =\int\frac{k \text{dQ}}{\textbf{||r||}}$, ...
opaque_dragon's user avatar
0 votes
0 answers
40 views

Multipole Expansion of the Electromagnetic Field of A Rotating Particle

In the case of a particle $q$ orbiting in a circle with $\omega_0$ and $r_0$, the current density $\mathbf{j}(\mathbf{r},t)$ contains frequency at $\omega_0$ (dipole) and $2\omega_0$ (quadrupole), $3\...
Firestar-Reimu's user avatar
0 votes
1 answer
48 views

Radial and angular parts of matrix elements for alkali atoms - calculating Quadrupole matrix elements

I am trying to calculate the quadrupole matrix element for an alkali ($^{87}$Rb specifically) and have come to the expression for the matrix element being as follows $\left<f|Q_q^k|i\right>=\int ...
D. Brown's user avatar
  • 101
-1 votes
1 answer
60 views

Why the charge is zero in this problem? [closed]

Say we have a bar centered on the origin, orientated with the z-axis, such that $a$ is the measure of the bar and it has a linear density that varies within $z$ with the following expression $\lambda(...
Ulshy's user avatar
  • 69
0 votes
1 answer
36 views

Meaning of electric dipole moment

what is the meaning of electric dipole moment? Or why do we need to define electric dipole moment?
Physics Ed's user avatar
0 votes
1 answer
77 views

Multipole expansion, same charges

I know dipole is defined with 2 opposite charges. That's why in EM dipoles exist, while in gravity they do not. However, I view multipole expansion as a way to describe how the distribution of charges ...
Matteo's user avatar
  • 77
1 vote
1 answer
106 views

Multipole expansion and spectral decomposition

We can always write a Hermitian operator in the form of its spectral decomposition: $\hat{A}=\sum_i \lambda_i | \chi_i(\boldsymbol{r})\rangle\langle\chi_i(\boldsymbol{r}')|$ where $\lambda_i$ are the ...
Guiste's user avatar
  • 464
1 vote
1 answer
89 views

Conservation of angular momentum in electric quadrupole transition

I know that We can classify radiative transitions of electrons between different energy levels as electric multipole transitions or magnetic multipole transitions. My question is: In electric ...
Dinesh Katoch's user avatar
1 vote
1 answer
95 views

Fourier transform of electric quadrupole potential

I'm struggling to find the scattering amplitude by the first Born approximation (Fourier transform) given by $$ f(\vec{k}_f,\vec{k}_i)= -\frac{1}{2\pi}\langle \vec{k}_f | {V}| \vec{k}_i\rangle = -\...
Alan Falkowski's user avatar
0 votes
1 answer
137 views

Why is the potential field inside a quadrupole field given by $V=k(\alpha x^{2} +\beta y^{2} +\gamma z^{2})$ for constants $k,\alpha,\beta,\gamma$?

I am seeking an explanation regarding the origin of the potential inside a quadrupole field. The common understanding is that the potential can be described by the equation: $$V=k(\alpha x^{2} +\beta ...
Adrien Amour's user avatar
1 vote
2 answers
149 views

Can ideal dipoles be associated to a covariant four-current?

I am trying to check if the classical electromagnetic sources from a point electric/magnetic dipole do form a true four-current. In this SE post, it is shown that a point electric charge do transform ...
Woe's user avatar
  • 388
1 vote
2 answers
139 views

Star with quadrupole in binary system violates Newton's 3th law?

Suppose that, in a binary system of two stars, the star A (and only the star A) has a non-zero quadrupole moment $Q_A$. Then, the star B feels the usual gravity force plus an additional force, ...
gravitone123's user avatar
0 votes
1 answer
64 views

Using spherical harmonics for the charged interaction of particles

I am currently inversitgating a system consists of positive and negative point particles satisfying charge neutrality condition. They have a unit charge +1 and -1. In most case, we deal with coulomb ...
user avatar
2 votes
1 answer
117 views

How are multipolar expansion of earth magnetic field computed?

In the study of the geomagnetic field, an expansion in spherical harmonics is used to represent the scalar magnetic potential: the first terms give the dipole approximation, then the quadrupole, etc. ...
Weier's user avatar
  • 206
1 vote
1 answer
76 views

Is there a generalization of Gauss's Law for enclosed dipoles, quadrupoles, etc

Given the electrostatic field $\mathbf E$, its integral over a closed surface $\mathcal A$ is the total charge enclosed by it: $$\epsilon_0\oint_{\mathcal A} \mathbf E \cdot d \mathbf A = Q_{\mathcal ...
hyportnex's user avatar
  • 19.7k
13 votes
1 answer
318 views

Is there a Lorentz invariant electromagnetic quadrupole moment tensor?

I'm familiar with the electric and magnetic quadrupole moment tensors. However, I'm bothered that these objects are tensors only in the sense of spatial rotations. After all, Maxwell's equations and ...
Aiden's user avatar
  • 1,888
1 vote
0 answers
58 views

Multipole expansion - different derivation leads to the dipole decreasing in $r^{-2}$ and $r^{-3}$

To prepare myself for the electrodynamics course i used Griffith, in which his derivation gives $$V(r)=\frac{1}{4\pi \epsilon_0}(\frac{1}{r}\int \rho(r')d \tau ' + \frac{1}{r^2}\int r'cos\alpha \rho(r'...
Tomy's user avatar
  • 287
-2 votes
1 answer
83 views

Multipole Expansion of Point Charge at Origin

The multipole expansion for the potential of some collection of point charges can be written as $$ V=\frac{1}{4\pi\epsilon_0r}\sum_{i}q_i\sum_{n=0}^{\infty}(\frac{r_i}{r})^nP_n(\cos\alpha)\,, $$ where ...
Bruno Nowak's user avatar
1 vote
3 answers
346 views

Why is the quadrupole moment of a spherical object equal to zero (when looking at the formula in cartesian coordinates)?

Is there a simple way to understand why $$\int \rho(x,y,z) (2z^2-x^2-y^2) dxdydz$$ is equal to zero if the density has spherical symmetry?
Yannenou's user avatar
0 votes
1 answer
138 views

Is it true that an electric monopole does not radiate? [duplicate]

In the third edition of Griffith's Introduction to Electrodynamics, in section 11.1.4, he says that an electric monopole does not radiate because the electric charge is conserved. But we do know that ...
Solidification's user avatar
0 votes
0 answers
50 views

Meaning of reference radius in multipole expansion

I have a given electrostatic potential $\Phi(x,y)$ in 2D and want to see how different it is from a some multipole potential (eg. quadropole). The expansion in the multipole terms is $\Phi(x,y)= \...
Agnieszka's user avatar
  • 195
2 votes
0 answers
164 views

Why there can't be an isotropic radiator?

Why there can't be an isotropic radiator? I know you can prove it using Maxwell's equations but I don't see how. Can someone please help?
Yusaku Fujiki's user avatar
2 votes
0 answers
121 views

Multipole Expansion: Why is my Taylor Series Wrong?

Let's start by a formula. The scalar electrostatic Potential is given by: $$\phi(\mathbf{r}) = \dfrac{1}{4\,\pi\varepsilon_0}\,\int \dfrac{\rho(\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|}\,\mathrm{d^3 r'}$...
Leon's user avatar
  • 462
5 votes
1 answer
432 views

How to prove that spherical and Cartesian $l$th multipole moments have the same number of independent components?

This is Problem 4.3 from Jackson Classical Eletrodynamics (3rd edition). I have searched online about this problem but have not found any satisfactory solutions. In the problem, Jackson says that $q_{...
Ulysses Zhan's user avatar
2 votes
4 answers
827 views

EM radiation from monopole v.s. dipole

I'm confused about two conflicting ideas: the first is that monopoles cannot emit EM radiation (see here), and the second is that an accelerating point charge does produce EM radiation (see here). I ...
Chris's user avatar
  • 163
0 votes
1 answer
329 views

Quadrupole moment being tensor in electromagnetism

I was reading "Lectures on Electromagnetism" by Ashok Das and he says that because the moment of a monopole is a scalar, the moment of a dipole is a vector then the quadrupole moment is a ...
Pacur's user avatar
  • 1
1 vote
1 answer
75 views

Would a gravitational wave accelerate a single ball?

Suppose I have two balls floating in space. If a gravitational wave with the correct polarization passed by, it would create an oscillating strain causing the balls to accelerate together, then apart, ...
Steve Andrews's user avatar

1
2 3 4 5
7