Questions tagged [multipole-expansion]
The multipole-expansion tag has no usage guidance.
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Exploring Quantum Wiggles with AI: Gravitational Waves and the Double-Slit Mystery [closed]
I would like to preface this by stating that I am by no means a physicist, but I am an engineer who can't stop thinking about how similar the macrocosmic universe is to the quantum realm.
I asked ...
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Meaning of electric dipole moment
what is the meaning of electric dipole moment?
Or why do we need to define electric dipole moment?
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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 ...
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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 ...
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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 ...
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Fourier transform of electric quadrupole potential
I'm struggling to find the scattering amplitude by the first Born approximation (Fourier transform) given by
$$
f(\vec{k}_f,\vec{k}_i)= -\frac{1}{2\pi}\langle \vec{k}_f | {V}| \vec{k}_i\rangle = -\...
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Generalization of Thorne integral [duplicate]
How can we generalize the integral expressions given by Thorne in [his paper in equation (2.5)? The formula given in his paper cannot be directly translated to terms that has similar structure but ...
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Why is the potential field inside a quadrupole field given by $V=k(\alpha x^{2} +\beta y^{2} +\gamma z^{2})$ for constants $k,\alpha,\beta,\gamma$?
I am seeking an explanation regarding the origin of the potential inside a quadrupole field. The common understanding is that the potential can be described by the equation: $$V=k(\alpha x^{2} +\beta ...
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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 ...
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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, ...
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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 ...
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Product of rank-2 spherical tensors
I have been reading up a lot of material on spherical tensors to try and help me with a calculation of the electric quadrupole energy shift of an atom, and have run into a block due to my lack of ...
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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.
...
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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: $$\epsilon_0\oint_{\mathcal A} \mathbf E \cdot d \mathbf A = Q_{\mathcal ...
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Is there a Lorentz invariant electromagnetic quadrupole moment tensor?
I'm familiar with the electric and magnetic quadrupole moment tensors. However, I'm bothered that these objects are tensors only in the sense of spatial rotations. After all, Maxwell's equations and ...
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Multipole expansion - different derivation leads to the dipole decreasing in $r^{-2}$ and $r^{-3}$
To prepare myself for the electrodynamics course i used Griffith, in which his derivation gives
$$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'...
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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 ...
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Quadrupole magnetic trap principle
I read that, the principle of the magnetic quadrupole trap confining is that an atom, with a magnetic dipole moment, subjected to a inhomogeneous magnetic field, is driven towards the point where the ...
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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?
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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 ...
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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)= \...
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Are multipoles the coefficients of the charge distribution for a basis of functions?
If I calculate the poles (monopole, dipole, quadrupol ...) for a charge distribution, then I will get something of the form:
\begin{align}
Q \text{ or } \vec{Q} \text{ or } Q_{ij} \text{ or ....} = \...
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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?
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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'}$...
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Weinberg eq 9.1.53, Post-Newtonian Expansion Question
In Weinberg's Gravitation and Cosmology, there is a derivation of the Post-Newtonian expansion. In this post $(i)$ means the tensor is of order $(v/c)^i$ The derivation goes along the lines of the ...
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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_{...
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Stationary points of the interaction energy between two magnetic dipoles
I'm currently trying to find the stationary points of the interaction energy between two magnetic dipoles and I have the interaction energy as a function of polar angles of the two magnetic dipoles $U=...
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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 ...
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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 ...
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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, ...
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Applicablity of multipole-expansion (ME)
I have three questions about applicability of ME from nerdy physicists:
Is Multipole-Expansion a general mathematical decomposition tool? or it is only applicable in physics? (physics = gravity, ...
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Inverse of equation with Legendre polynomials [closed]
I have to find the inverse of the equation of the multipole moment of a field
\begin{equation}
\Theta_l(k)=\int_{-1}^1 \frac{d\mu}{2}\mathcal{P}_l(\mu)\Theta(k,\mu),
\end{equation}
where $\mathcal{P}...
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Multipole moments of line charge
I am supposed to calculate the multipole moments of a line charge with total charge Q spread from $z=-a$ to $z=a$ on the z-axis. I know that each multipole moment is given by:
$$
q_{l,m}=\int_{\mathbb{...
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How can I derive the quadrupole momentum tensor in terms of dyadic product of wave vector and electric field?
In page 3 of PhysRevB.78.121101 they used an expression of electric quadrupole tensor as
$Q_{ij} = i \alpha_Q(k_i E_j + k_j E_i)$ or $\overleftrightarrow{Q}=i\alpha_Q(\vec{k}\vec{E}+\vec{E}\vec{k})$
, ...
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I can't seem to figure out a way to compute a gradient without reference coordinates
I'm not sure if this question is better asked here or in Mathematics but here it goes:
I'm studying electric dipoles, and this exercise I'm working on asks for the energy between 2 dipoles, given by $$...
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Spherical harmonics expansion: from scalars to tensors
It is well known that a scalar field on the unit sphere can be expanded in spherical harmonics, see e.g. this. I am wondering if there is a related concept for vector fields and, in general, for any ...
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Electrostatic potential due to a pure dipole
For a pure dipole, is the electrostatic potential given by the dipole term in the multipole expansion exact for all $r$ (position vector from the origin)? Or is it exact only far away from the dipole? ...
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The multipole expansion of electrostatic potential and large distances
I'm reading Griffiths electrodynamics book and I'm currently studying the multipole expansion of electrostatic potential, and I have two questions if you don't mind:
Can I use the multipole expansion ...
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Properties of Vector Spherical Harmonics
In section 5.3.2 of the book Advanced Classical Electromagnetism by Robert Wald, in deriving the multipole expansion for the retarded solution of electromagnetic field in presence of charge-current ...
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Working with decomposition of fields
I'm trying to follow a text I found online.
The author decomposes EM fields such
$$
\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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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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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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