Questions tagged [multipole-expansion]

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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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34 views

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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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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How to know when quadupole moment is zero or not qualitatively?

Let us take dipole moment of discrete charges system: The equation $$\vec{p}=\sum q_ir'_i= \sum q_{+i}r'_i + \sum q_{-i}r'_i$$ $$Q_{+i}R'_{+i} - Q_{-i}R'_{-i}$$ ie dipole moment is zero when centers ...
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31 views

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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53 views

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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How to calculate the variance on the ratio of 2 angular power spectra?

In the context of Survey of Dark energy stage IV, I need to evaluate the error on a new observable called "O" which is equal to : $$ O=\left(\frac{C_{\ell, \mathrm{gal}, \mathrm{sp}}^{\prime}...
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44 views

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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69 views

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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41 views

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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53 views

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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38 views

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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27 views

Physical interpretation of multipolariy of gamma radiation

So I understand the theory behind the multipolarity of gamma radiation but I was wondering what the physical interpretation behind it is. Gamma rays are in themselves electromagnetic waves but how do ...
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28 views

Correct expression for multipole expansion

Panofsky and Philips state the multipole expansion of a potential due to a volume charge distribution $\rho$ in a finite volume $V$ given by $\phi(\mathbf R)=\frac{1}{4\pi\epsilon_0}\int_V\frac{\rho(\...
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1answer
60 views

Cosmic microwave background dipole anisotropy

I'm trying to wrap my head around dipole anisotropies of the Cosmic Microwave Background (CMB). I think my confusion is rooted in a misunderstanding of how multipole expansion is used in describing ...
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67 views

Electric field of a dipole from superposition and from multipole expansion

This question came in my physics test: Charge density in a one dimensional space is given by $\rho=Q[\delta(x-x_0)-\delta(x+x_0)]$. The electric field due to this charge distribution at point (2$x_0$,...
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233 views

Can you expand a real scalar field $\phi(t,\mathbf{x})$ in terms of spherical harmonics?

A massless real scalar admits the expansion $$ \phi(t,\mathbf{x}) = \int \frac{d^3\mathbf{p}}{(2\pi)^{3/2} \sqrt{2|\mathbf{p}|}} \bigg( e^{ - i |\mathbf{p}| t + i \mathbf{p} \cdot \mathbf{x} } a_{\...
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Do dipolar gravitational waves exist?

There seems to be some controversy (see A, B) on this topic, so I'm posting a new question for discussion and clarification. By definition, one cannot accelerate the center of mass of a closed system (...
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1answer
66 views

Are Field Beams Possible?

I was recently wondering why the field force between two objects is proportional to the square root of the distance (sometime you just take things for granted). This comes from the inverse-square law, ...
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Sources of quadrupole radiation from black hole

During ringdown from BH merger, the event horizon is (obviously) very far from equilibrium. This is a statement about mass distribution in spacetime. Suppose I fly a rocket across the event horizon (...
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1answer
77 views

Radiating quadrupole [closed]

I'm trying to solve the following problem. Two charges $+q$ are located along the $z$ axis at $z=\pm a \sin \omega t$. Determine the lowest non-vanishing multipole moments, the vector potential and ...
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23 views

Why there cannot be source-free spherical symmetrical EM disturbances? [duplicate]

On the paper by John Wheeler that first introduced the concept of "geons", he states that when one tries to construct such geons with spherical symmetry, incoherence of the different EM ...
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36 views

Is there a system that only contains electric dipole moment but not higher electric multipole moments?

Does there exist such thing, if it does, which charge distribution could generate it? If it doesn't exist, can it be proven?
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Multipole expansion to approximate magnetic field of two currents with given distance and the field of a dipol with given momentum in 2D

I tried to approach this problem numerically, and someone pointed out that this was a classical multipole expansion problem. I dont know anything about multipole expansion, So i ask for help here. My ...
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42 views

The factor $3$ in the definition of the quadrupole moment tensor

I can find two different ways of writing the quadrupole moment tensor $$Q = \int \mathrm{d}^3r \rho(r) \left(3 r\otimes r - |r|^2I\right)$$ or $$Q = \int \mathrm{d}^3r \rho(r) \left(r\otimes r - \frac{...
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Why does Jackson's book use different method to calculate the vectorial multiple expansion coefficients in chapter 10.3 and 9.7?

In chapter 9.7: $$Z_0a_E(l,m)f_l(kr)=-\frac{k}{\sqrt{l(l+1)}}\int{Y^*_{lm}\mathbf{r\cdot E}d\Omega}\tag{9.123}$$ In chapter 10.3 $$a_\pm(l,m)j_l(kr)=\int{\mathbf X^*_{lm}\mathbf\cdot\mathbf E(\mathbf ...
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26 views

Why electric field symmetry along $z$-axis implies vanishing of odds multipole moments

In the Jackson exercise 6.4, we have the following electric field inside a rotating sphere, $$ \vec{E} = -\frac{2}{3} \mu_0 M \omega r \hat{r} $$ Where $r$ is the radial coordinate in cylindrical ...
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1answer
32 views

Why does a magnetic transition $\mathrm{M}(\ell)$ have approximately the same probability as an electric $\mathrm{E}(\ell+1)$ transition?

When discussing electromagnetic decays and multipolarity, B. Povh, et al.$^1$ state that the magnetic transition $\mathrm{M}(\ell)$ have approximately the same probability as an electric $\mathrm{E}(\...
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What does the dipole moment really represent?

Wikipedia gives the most general expression for the $n^{\rm th}$ moment $\mu_n$ of a physical quantity $\Lambda$ as: $$ \mu_n = \int {\bf x}^n \space \lambda({\bf x}) \space \rm d^3 x$$ provided that ...
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45 views

Point charges appearance in the far-field regime - electrostatics

What would the electric field and potential of three point charges (for example -q, -q and +2q, equally spaced) look like in the far-field regime as opposed to the near-field regime? We are only being ...
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60 views

Electric quadrupole allowed transitions mathematical proof

Consider the electric quadruple moment operators as follows: $Q_{20} = \frac{e}{2}(x^2+y^2-2z^2) $ $Q_{2 \pm1} = \frac{e\sqrt{6}}{2}z(x\pm iy) $ $Q_{2 \pm2} = - \frac{e\sqrt{6}}{4}(x\pm iy)^2 $ I ...
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47 views

Multipole terms of Electrostatic energy

If we consider the Potential and electric field of a Dipole, could we infer that for a dipole : $V \propto 1/r^2$, $E \propto 1/r^3$ , $da \propto r^2$, then Electrostatic potential energy should ...
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1answer
71 views

Why does this volume integral vanish?

I am stuck on this problem concerning the gravitational potential of a body. The body has a mass density $\rho(\mathbf x)$ and I have to calculate a contribution to the total gravitational potential ...
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32 views

Electrostatic energy of a multipole

In classical electrostatics, we learn that the electrostatic potential of an electric dipole at a distance $r$ is proportional to $1/r^2$. Then putting two dipoles together to form a quadrupole, the ...
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What is the potential $\phi(x,y)$ inside quadrupole/Paul trap?

Suppose we have a configuration like the figure below. I'm asked to compute the potential $\phi (x,y) $ inside the trap. I'm attempting this by solving the Laplace equation $\nabla^2 \phi = 0$ ...
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1answer
77 views

Electric potential for half-grounded sphere [closed]

I have a sphere where the upper half surface has a potential of $V_0$ and the lower half is grounded and I have to find the potential everywhere (using the Laplace solution for spherical coordinates). ...
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Electric Quadrupole Moment of the Deuteron. 1/4 factor

Sometimes in books I use to see this expression $$\widehat{Q}_{0}=\left(3 z^{2}-r^{2}\right)=\sqrt{\frac{16 \pi}{5}} r^{2} Y_{20}(\theta, \phi)$$ for quadrupole electric moment of the deuteron but ...
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Partition function of quadrupole? [closed]

I would like to compute the partition function Z for the quadrupole moment, but it is not as easy as the one for the dipole. Could anyone help please?
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Modelling forces between/on toroidal dipoles/anapoles

I've been trying to find something more concrete about these moments and how they might interact within fields, but haven't found anything clear. Supposedly these moments are "left over" ...
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2answers
545 views

How to understand observed electric quadrupole moment of deuteron?

Deuteron is p-n, so naively should have zero electric quadrupole moment. However, experimentally it turns out quite large: $0.2859\ e\cdot fm^2$ from https://en.wikipedia.org/wiki/Deuterium#...
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93 views

What is the rank of the quadrupole moment tensor in QM and nuclear physics?

For an assignment on Quantum Mechanics, we have been given an expression for the quadrupole moment of the deuteron $$\langle Q\rangle = \langle(2z^2-x^2-y^2)\rangle$$ It is known that the quadrupole ...
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1answer
42 views

Can the ground state of an even-even nucleus have a quadrupole moment?

The question I'm stuck with is the following: The quadrupole moment in a way describes the shape of the electric charge distribution of a nucleus right? Oblate if negative, prolate if positive and 0 ...
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215 views

If all the multipole expansions apart from the monopole is zero, then is the charge density spherically symmetric?

Say I have a charge density $\rho(\vec{x})$ in some finite volume $V$, such that all of the multipole expansions apart from the monopole moment (meaning dipole, quadrupole and so forth) are zero. Does ...
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1answer
78 views

Multipole expansion of $V$ in powers of $\frac{1}{r}$

If the total charge is $0$, why should the dominant term be the dipole? Also, for a dipole consisting of only two opposite charges separated by a distance $d$, i dont understand how can there be ...
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Relating two different expressions for quadrupole moments [closed]

How do we get from the expression $$\frac{1}{2} (3(\vec{r} \cdot \vec{r}')^2-r'^2 r^2)$$ to $$\sum_{i,j=1}^3 r_ir_j \frac{1}{2} (3r'_ir'_j-r'^2\delta_{ij})$$ If someone could go step by step and ...
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1answer
30 views

Multipole Translation Theorem in FMM [closed]

Consider the standard multipole expansion for the coulomb potential often used in the fast multipole method algorithm (Theorem 5.2). A translation theorem is used in the algorithm (Theorem 5.3). Is O(...
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93 views

Multipole expansion sphere

I saw the solution on griffiths, I didnt understand why in the monopole term they integrate from 0 to r, and then in the dipole they integrate on theta from 0 to pi and in the quadrupole the ...
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60 views

Is linear response only the first term in a series expansion?

So in the theory of linear response, the goal is to look at how certain dynamical variables (or operators in QFT) respond to an external source. To be more concrete, suppose that $x$ obeys (in index ...
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56 views

What would the multi-pole expansion of a spherical shell of magnets look like?

A singular magnet can be expanded in terms of the dipole term, while a set of four magnets with alternating sides facing the inside is described in terms of the quadrupole term. What would happen if ...
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1answer
356 views

Quadrupole moment

What information does the quadrupole provide? I've seen many definitions on the internet, but I don't understand the relation between them. What is the relation between knowing that the quadrupole ...

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