# Learn about the proof of Lamb-Chaplygin dipole

• gerar
In summary, the conversation is about the Lamb-Chaplygin dipole solution and the search for its proof. The proof can be found on pages 166-174 of the first link (link1) provided. The green lines represent the stream function and the steps involved in deriving the horizontal momentum equation. By transforming to polar coordinates and using a Bassel's function, the equation can be satisfied.
gerar
I learn about Lamb-Chaplygin dipole and try to find the proof of this topic.

can anyone tell me please on which pages (and of which link) the proof is found?
I think the proof is found only at link1: 166-174 but I'm not sure.

any help appreciated!

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explanation the next steps in other words (dipole)

I am trying to understand the lamb-chaplygin dipole solution.
can someone explain me the green lines in other words please?

this is the proof:

When introducing the stream function, the steps that you usually take are as follows:
1) replace u and v by the stream function.
2) Derive the horizontal momentum equation (for u) with respect to y and the other with respect to x.
3) Eliminate the pressure term, to end up with a single equation in ψ.

so:

v = -∂ψ/∂x, u = ∂ψ/∂y

(∂^2)ψ/∂x^2 + (∂^2)ψ/∂y^2 = f(ψ)

if we put f(ψ) = -(k^2)*ψ. where k is a constant, and transform to polar coordinates r, θ, we get:

(d^2)ψ/ dr^2 + (1/r) * (dψ/dr) + (1/r^2) * (d^2)ψ/ dθ^2 + (k^2)*ψ = 0

which is satisfied by:

where Js is a bassel's function.

any help appreciated! thanks!

## 1. What is the Lamb-Chaplygin dipole?

The Lamb-Chaplygin dipole is a theoretical model used to explain the behavior of electrons in a magnetic field. It describes how the electrons are distributed in a dipolar magnetic field and the resulting magnetic moment.

## 2. Who developed the proof of the Lamb-Chaplygin dipole?

The proof of the Lamb-Chaplygin dipole was developed by physicists Willis Lamb and Sergey Chaplygin in the 1940s.

## 3. How does the Lamb-Chaplygin dipole differ from other dipole models?

The Lamb-Chaplygin dipole differs from other dipole models in that it takes into account the quantum mechanical effects of the electrons' motion, rather than treating them as classical particles. It also includes a correction factor for the magnetic moment that is more accurate at high magnetic fields.

## 4. What is the significance of the Lamb-Chaplygin dipole in physics?

The Lamb-Chaplygin dipole is significant in physics because it provides a more accurate understanding of the behavior of electrons in a magnetic field. It has been used in various experiments and has led to further developments in quantum mechanics and magnetism.

## 5. Are there any real-world applications of the Lamb-Chaplygin dipole?

Yes, the Lamb-Chaplygin dipole has been applied in various fields such as nuclear physics, astrophysics, and material science. It has also been used in the development of magnetic resonance imaging (MRI) technology.

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