Electrostatics - Conducting torus and a point charge

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SUMMARY

The discussion focuses on calculating the charge distribution on a grounded metal conducting torus influenced by a point charge located on its axis. The method of images is employed to simplify the problem by substituting the torus with a charged loop. The participant encounters difficulties in deriving the charge distribution using toroidal coordinates and expresses uncertainty about the correct formulation of the electrostatic potential. The challenge lies in transitioning from spherical to toroidal coordinates while managing the complexities of the infinite value in the coordinate system.

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  • Understanding of electrostatic potential and charge distribution
  • Familiarity with the method of images in electrostatics
  • Knowledge of toroidal coordinates and their applications
  • Experience with delta functions in charge distribution modeling
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Homework Statement


We have a metal conducting torus and a point charge that is located on the torus' axis (location on the axis is arbitrary). Calculate the (influenced) charge distribution on the torus and the electrostatic force on the point charge.


Homework Equations


Equation for electrostatic potential (volume integral over charge distribution etc.)


The Attempt at a Solution


OK, so the problem is pretty straightforward. I'm trying to solve this problem with method of images where I substituted the torus with a charged loop. There are lots of variables, so I'm not going to write everything. The problem simplifies to an electrostatic potential of a charged loop and this is where it gets complicated (for me :)). At first, I tried spherical coordinates, but I got an elliptical integral at the end. So, I tried toroidal coordinates, which seemed a bit complicated. I got stuck at getting the charge distribution of a charged loop in toroidal coordinates. I'm used to write it with delta functions and don't know how to do that in toroidal coordinates. I'm using (u,v,phi) for notation and the values for coordinates on the loop are u=0 rad, v=inf, phi=[o,2pi]. I don't know if this is right, but the problem is the infinite value of v.
So, I have two questions: Am I even on the right track with the toroidal coordinates and how would one write a charge distribution?

Regards!
 
Physics news on Phys.org
Is the torus grounded?
 
Sorry for that. Yes, the torus is grounded.
 

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