# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...