# Questions tagged [multipole-expansion]

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### Multipole expansion

consider a charge distribution say q, 2q, 3q, kept on the vertices of an equilateral triangle. There exist monopole, dipole ,qudrupole etc terms in the potential expression. these 3 charges are ...
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### How to calculate the Quadrupole moment via integration?

I have following problem: calculate the quadrupole moment of following arrangement, where $e$ is the charge and $a$ is the distance between the charges. Hint: The quadrupole moment is defined ...
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I was wondering if the $n=2$ term in the potential of a multipole: $V(r)= \dfrac{1}{4 \pi \epsilon_0} \sum \dfrac{1}{r^{(n+1)}} \int (r')^n P_n(cos \alpha) \rho(r')dv'$ was the same as this ...
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### Are gravitational quadrupole moment, second moment of mass, and moment of inertia the same?

my understanding of moments is that they refer to distributions about an expected value, which allows us to make the multipole expansion. I read that: the zeroth moment of mass refers to the mass of ...
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### How does a electric quadrupole oscillate?

I know that in static a electric quadrupole is made of two positive charges and two negative charges, distributed as in the following figure: What I don't understand is when it oscillates and ...
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### Multipole expansion of the electromagnetic field

In Jackson's Classical Electrodynamics, section 9.7, he develops the multipole expansion of the electromagnetic fields in terms of the vector spherical harmonics and the spherical Bessel and Hankel ...
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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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In multipole expansion, we use monopole, dipole, quadrupole or octupole to describe an electromagnetic field. But I saw someone use sextupole to describe transition states. If we expand an ...
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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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### The magnetic and electric dipole moment 'popping out' from multipole expansions

I interpret electric dipole moments and magnetic dipole moments as intrinsic properties of certain materials, but in Griffiths it's literally picked out of expressing the potential and magnetic vector ...
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This is a very interesting problem that I've been struggling to solve for a week, so I decided to ask for some orientation, as I think it could also be of interest for the community. Let's consider a ...
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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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### Applications of multipole expansion in gravitational problems [closed]

I want to know what exactly are practical applications of multipole expansion in some problems concerning gravitation. I have also read in Subtle is the Lord by Abraham Pais that Einstein had ...
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### Systematic expansion of $e^{i\vec{k}\cdot\vec{r}}$ in atomic physics in terms of Legendre polynomials and identifying different $l$ terms

In the context of light-matter interaction one often makes the approximation $e^{i\vec{k}\cdot\vec{r}}\approx 1$. Keeping higher order terms in $e^{i\vec{k}\cdot\vec{r}}$ give magnetic dipole, ...
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In classical electrodynamics, we introduce the electric quadrupole moment $$D^{ij}\equiv\int y^i y^j \rho \mathrm{d}^3y$$ and its reduced (trace-less) version $$\mathcal{D}^{ij}\equiv D^{ij} - \frac{1}... 1answer 69 views ### Griffiths Multipole Expansion and Q going to zero Griffiths states that given the multipole expansion:$$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 ... 2answers 56 views ### 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 ... 1answer 101 views ### The greatest quadruple moment Let us consider the following problem: it is necessary to find the shape of the body with fixed mass and density which at large distances compared with its characteristic dimensions would give the ... 1answer 65 views ### What is the potential inside a hollow conducting sphere with multipoles uniformly surrounding it? 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 275 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,... 2answers 845 views ### Why does magnetic force vary in proportion to the cube of the distance instead of the square? Magnetic forces vary from gravity and electromagnetic radiation (such as light) in that gravity and radiation diminish by the square of the distance from the source, but magnetism diminishes by the ... 1answer 100 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 839 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 ... 1answer 459 views ### How do multipole moments relate to a Taylor expansion, with regards to Newtonian potential? Given the Newtonian gravitational potential,$$ \phi(\mathbf{x}) = - \int \frac{G \rho(\mathbf{x'})}{|\mathbf{x} - \mathbf{x'}|}$$One can construct a 'multipole expansion' by using the Taylor ... 1answer 29 views ### Expressing interaction between two classical charge distributions in terms of multipole moments I am interested in expressing the interaction energy between two classical charge distributions in terms of the multipole moments of each of the charge distributions. For example, let's assume each of ... 1answer 99 views ### How did scientists before Coulomb ensured this fundamental property of magnets? [closed] Firstly, please note that I am talking about the period BEFORE electricity and magnetism were unified. So I am NOT seeking for answers based on Ampere atomic current model of magnets. I have read the ... 2answers 406 views ### Quadrupole moment tensor definition I'm not sure what the proper definition of the quadrupole moment tensor is. In the book on gravitational waves by Maggiore, the definition is$$M^{ij}=\int d^3x T^{00}x^ix^j. \tag{3.37}$$( Maggiore,... 2answers 976 views ### Does the dipole moment depend on the choice of origin? Does the dipole moment depend on the choice of origin if the total charge Q is not zero? for a system of charges neutral overall? How can I show that mathematically? Also I need some drawings to ... 0answers 81 views ### Multipole expansions: test on a function \zeta=\zeta(t) Considering the potential \psi(r) of a sphere of mass M with density \rho(\mathbf r'), connected by a small volume positioned in the P' point as shown in the figure: The \psi(r) is:$$\...
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 ($\vec{d} \times \hat{n}$,...