# Questions tagged [multipole-expansion]

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### Symmetrical, spherical charge distribution [closed]

I have this problem from my Electromagnetism class, but I don´t have a lot of idea in how to solve it. Show that for a symmetrical, spherical charge distribution $\rho(\vec{r})=\rho(r)$ (with $r$ ...
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### Shifting polynomials in Fast Multipole Method

There is one thing I don't get about the FMM algorithm (of coulombic potential in 2D - https://cims.nyu.edu/~donev/Teaching/WrittenOral/Projects/JasonKaye-WrittenAndOral.pdf). Suppose we have ...
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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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### Building a gravitational potential out of multipole expansions of line segments

I would like to build a gravitational potential out of line segments (numerical). Imagine breaking a wire of mass with arbitrary linear density and path into a collection of small straight wires of ...
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### Confusion about exact electrostatic potential of a pure dipole [duplicate]

I asked a version of this question before but the phrasing was unclear, so don't mind me please. Suppose you have a physical dipole with equal and opposite charges Q and -Q, with a specified dipole ...
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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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### 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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### Multipoles as a superposition of dipoles?

While navigating on Stack Exchange, I was suddenly wondering about a possibility that I never saw described in my textbooks on electrodynamics. Suppose you have a distribution of electric charges in ...
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### About applicability of Gauss’ law in electrostatics

While reading about Gauss’ law in electrostatics, I saw that we do not apply this law to evaluate the fields generated by dipoles, quadrupoles etc. It is only applicable to cases where the fields fall ...
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### Multipole Expansion Intuition

So my intuition of the multipole expansion is that we have a certain charge distribution, and the ME tells us that we can approximate that distribution as a superposition of monopoles, dipoles, etc. ...
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### Griffiths and Multipole Expansion, what is each variable in the equation?

Griffiths defines the formula for the multipole expansion as $$V(\mathbf r)=\frac{1}{4\pi\epsilon_0}\sum_{n=0}^\infty\frac{1}{r^{(n+1)}}\int(r')^nP_n(\cos\alpha)\rho(\mathbf r')\,d\tau'.$$ But I am ...
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### Is an expression of a quadrupole as an expansion of dipoles possible?

Would it be possible to express a quadrupole as an expansion of dipoles? Because a possible definition of a quadrupole seems to be: an electric field equivalent to that produced by two electric ...
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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 ...
1 vote
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### 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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### 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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### 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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### 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 ...
1 vote
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### 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], \int d^3y \...
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