Poission equation, spherical harmonics, looking for reference

[Derivator]

Hi folks,

I'm looking for a derivation of the following statement (formula 76)
http://img845.imageshack.us/img845/1550/screenshot4op.png

Do you know any reference, where I can find a bit more detailed description? I reckon, you can find it in Jackson's electrodynamic book, but I couldn't find it.derivator

 

[Polyrhythmic]

Well, if you take a look at the original definition of the electrostatic potential, which is basically given by something proportional to q/|r-r'|. In order to get q, you have to integrate over the charge density rho which is given in your text. The factor of 1/|r-r'| is also given there, so you just have to combine those two and calculate.

 

[Meir Achuz]

Try Sec. 4.2.5 in Franklin "Classical Electromagnetism".

 

[Studiot]

Weinberger "A first Course in Partial Differential Equations" treats this in his chapter on Legendre and associated functions p 192 ff

But any good book on partial diffs should have a similar treatment.

 

[PJC01]

,

The statement you have shared is known as the Poisson equation, and it is a fundamental equation in electrostatics and other fields of physics. It describes the relationship between the electric potential and the charge distribution in a given system. The formula you have shared is a specific form of the Poisson equation in spherical coordinates, which is often used to solve problems involving spherical symmetry.

To understand the derivation of this equation, I would recommend consulting textbooks on electromagnetism and mathematical methods in physics. As you mentioned, Jackson's "Classical Electrodynamics" is a well-known reference for this topic. Other resources that may be helpful include Griffiths' "Introduction to Electrodynamics" and Arfken and Weber's "Mathematical Methods for Physicists."

I would also suggest looking for online lectures or notes from university courses on electromagnetism, as these often provide detailed derivations and explanations of the Poisson equation. Additionally, there are many research papers and articles available that discuss the use and derivation of the Poisson equation in various physical systems.

I hope this helps in your search for a more detailed description of the Poisson equation in spherical coordinates. Keep exploring and learning, and don't hesitate to reach out to experts in the field for further guidance.