# Questions tagged [multipole-expansion]

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### 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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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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### Calculating multipole expansion outside uniformly charged sphere

When calculating the potential outside a uniformly charged sphere with charge density $\rho$ and radius R using the multipole expansion, it makes sense that only the monopole term survives (since it ...
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### What do moments of inertia do in the potential terms of Lagrangians?

I am struggling to understand the Lagrangian computed in this paper. In particular, a binary spacecraft-debris system is assumed as below. The analysis goes as follows. 1- I am in trouble to ...
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### Why is quadruple moment zero for spherically symmetric charge distribution about centre?

How can we show that for a spherically symmetric charge distribution, the dipole, quadrupole and all higher moments about the centre of the distribution are identically zero. As we already know that ...
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### Multipole moment of an octupole

I have a cube with each side of length a centered on the origin. Each of the corners carries a charge of magnitude q, but the sign of the charge alternates such that no edge connects corners with the ...
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### Does a uniformly moving charge induce electric quadrupole moment densities in surrounding space?

The electric field of a uniformly moving charge is cylindrically symmetric around an axis parallel to its velocity vector. It varies inversely to the square of the distance. The electric field of a ...
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### Is there a way to know the non-zero coefficient of $\cot(\theta)$ expansion in spherical harmonics?

I'm currently trying to find an analytical solution to the Poisson equation for a given distribution using a multipole expansion. During this task, I found the radial expansion and everything else, ...
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### Difference between monopole moment and charge or mass itself?

I'm trying to understand the difference between monopole moment and charge or mass itself. What I found is related to magnetic monopole that is irrelevant. I want to know in multiple expansion of ...
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### Why gravitational waves can only be generated by a time-varying quadrupole moment of the mass distribution?

The (rather old) source I dispose of "Sexl, Urbantke : Gravitation and Cosmology" describes the radiation of gravitational waves only rather sketchy. So why gravitational waves are only generated by ...
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### Electric quadrupole and octupole moments for nuclei

I am getting slightly confused as to which nuclei cab exhibit quadrupole and octupole excitations. In This link it says closed shell nuclei cannot exhibit quadrupole oscillations because if their ...
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### How to extract quadrupole moment and its error from $\chi^2 < \chi^2 + 1$ surface?

I have following info: plot of $\chi^2$ minimization of 208-Rn Coulomb excitation data, Surface corresponds to regions $\chi^2 < \chi^2 + 1$ with error bars of 1$\sigma$, mean lifetime is 8 $\pm$ 0....
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### Quadrupole moment of Kerr spacetime

In this paper this paper, the Kerr black hole is described as having quadrupole moment of $q=J^2/M$ (which means $q=a^2M$ using $J=aM$) whereas in this paper it says in the abstract that the limiting ...
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### Griffiths Multipole Expansion and $Q$ going to zero

Griffiths states that given the multipole expansion: $$V(\vec{r})=\frac{1}{4\pi\epsilon_o}\sum_{n=0}^\infty\frac{1}{r^{(n+1)}}\int(\vec{r}')^nP_n(cos(\theta')\rho(\vec{r}')d\tau'$$ for large $r$ the ...
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### Does the quadrupole moment tensor contracted with itself yield Kronecker delta?

I have trouble understanding the Kronecker delta and how it comes up in tensor equations. I know the metric contracted with itself gives the Kronecker delta which either is 0 or 1 depending on if the ...
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If considering a hollow conducting sphere with a surrounding uniform charge distribution, for example, it will have a constant and uniform potential throughout the inside of the hollow sphere because $... 3answers 2k views ### Isn't magnetism governed by the inverse square law? [duplicate] Why does magnetism appear to decay much faster than gravity with distance? A clear indication of this is the fact that a magnet that in short distance able to overcome gravity and pick up some object,... 1answer 4k views ### Electric field from a quadrupole In the following problem, I have already solved for the value of the potential, and I would like to tackle the extra exercise, which asks for the electric field of a point quadrupole: At every point ... 1answer 52 views ### Expressing interaction between two classical charge distributions in terms of multipole moments I am interested in expressing the interaction energy between two classical charge distributions in terms of the multipole moments of each of the charge distributions. For example, let's assume each of ... 1answer 104 views ### How did scientists before Coulomb ensured this fundamental property of magnets? [closed] Firstly, please note that I am talking about the period BEFORE electricity and magnetism were unified. So I am NOT seeking for answers based on Ampere atomic current model of magnets. I have read the ... 2answers 563 views ### Quadrupole moment tensor definition I'm not sure what the proper definition of the quadrupole moment tensor is. In the book on gravitational waves by Maggiore, the definition is $$M^{ij}=\int d^3x T^{00}x^ix^j. \tag{3.37}$$ ( Maggiore,... 2answers 3k views ### Does the dipole moment depend on the choice of origin? Does the dipole moment depend on the choice of origin if the total charge Q is not zero? for a system of charges neutral overall? How can I show that mathematically? Also I need some drawings to ... 1answer 119 views ### Gravitational waves of an oscillating Schwarzschild black hole Gravitational waves are produced by an accelerated mass, similar to the production of light waves by an accelerated charge. The amount of gravitational energy released from a rotating object can be ... 1answer 449 views ### Cross Product in Spherical Coordinates I am looking at an example problem from Greiner's Classical Electrodynamics (chapter 21 , page 441) about the Hertzian Dipole where the radiation will require a cross product ($\vec{d} \times \hat{n}$,... 0answers 95 views ### Multipole expansions: test on a function$\zeta=\zeta(t)$Considering the potential$\psi(r)$of a sphere of mass$M$with density$\rho(\mathbf r')$, connected by a small volume positioned in the$P'$point as shown in the figure: The$\psi(r)$is:$$\... 0answers 925 views ### Why is the electric field of an axial quadrupole not the same as the electric field of two axial dipoles, at far distance? An axial electric quadrupole, made of four inline charges$(+q, -q, -q, +q)$with opposite charges a distance$a$apart, and the two$-q$charges adjacent, has an electric field at a remote point$P$... 1answer 92 views ### When can we consider dipoles? A dipole is a collection of two oppositely charged particles held at some distance, but if two charges are unequally charged (but oppositely charged) how do we take the dipole between them? Do we only ... 2answers 117 views ### Is electic field is always asymptotic to$r^{\alpha}$for some rational$\alpha$? Suppose you have an electric field in three dimensions created by some finite (but possibly arbitrarily high) number of point charges, each with charge equal to an integer multiple (positive or ... 1answer 562 views ### How does one prove that the lowest-order nonvanishing multipole moment of a charge distribution is independent of the origin, for arbitrary$\ell\$?

The multipole moments of a distribution are independent of origin if all the lower terms are zero. I can explicitly verify this statement by hand up to the quadrupole level, but is there any straight ...