Resources on non-spherical conductor surface charges

[albertrichardf]

Hi all,
I know qualitatively that charges tend to concentrate on sharp edges of conducting surfaces. I have tried searching online for a mathematical treatment of such a phenomenon, but I cannot find anything that's quite rigorous. I'd appreciate it if someone could guide me towards such resources.
Thank you.

 

[ZapperZ]

Hi all,
I know qualitatively that charges tend to concentrate on sharp edges of conducting surfaces. I have tried searching online for a mathematical treatment of such a phenomenon, but I cannot find anything that's quite rigorous. I'd appreciate it if someone could guide me towards such resources.
Thank you.

Define "mathematical treatment".

If you want a plug-and-chug equation, there is no such thing. This is really a complex boundary-value problem involving finding the solution to Poisson's equation. Only the simplest and highly-symmetric cases will you find an analytical solution. Otherwise, you have to solve it numerically, typically using finite-element analysis.

Zz.

 

[albertrichardf]

Thanks for answering. I mean, are there any analytical methods to solve, or approximate a solution to Poisson's equation in these cases? Or at least show mathematically, that the charges tend to concentrate at the tip of a pointed surface?

 

[robphy]

The textbook proof goes this way.
Consider two conducting spheres, with radii $R_1$ and $R_2$, separated by a large distance $r \gg R_1,R_2$ but connected by a wire. The surfaces have the same electric potential.
If the total charge is $Q_1+Q_2$, what are $Q_1$ and $Q_2$ (how must that total charge be distributed)?
See, e.g., https://www.feynmanlectures.caltech.edu/II_06.html#Ch6-S11