Electrostatics - Conducting torus and a point charge

In summary, the problem involves a metal conducting torus with a point charge located on the torus' axis and the task is to calculate the charge distribution on the torus and the electrostatic force on the point charge. The method of images is used, substituting the torus with a charged loop. The problem simplifies to finding the electrostatic potential of a charged loop, which can be complicated. The attempt at a solution includes trying to use spherical coordinates and then toroidal coordinates, but getting stuck at finding the charge distribution in toroidal coordinates. The torus is assumed to be grounded.
  • #1
thepero
2
0

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!
 
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  • #2
Is the torus grounded?
 
  • #3
Sorry for that. Yes, the torus is grounded.
 

1. What is a conducting torus?

A conducting torus is a three-dimensional shape that resembles a donut, with a circular cross-section and a hole in the middle. It is typically made of a conducting material, such as metal, and is used to study the behavior of electric fields and charges.

2. How does a conducting torus interact with a point charge?

A conducting torus can interact with a point charge through the creation of electric fields. When a point charge is placed near a conducting torus, the charge will induce an electric field in the torus, causing the charges within the torus to redistribute. This redistribution can result in a net charge on the surface of the torus.

3. What is the relationship between the electric field and charge on a conducting torus?

The electric field on a conducting torus is directly proportional to the charge. This means that as the charge on the torus increases, so does the strength of the electric field. Additionally, the electric field on a conducting torus is strongest at the surface and decreases as you move towards the center of the torus.

4. Can a conducting torus shield a point charge?

Yes, a conducting torus can act as a shield for a point charge. When a point charge is placed inside a conducting torus, the electric field created by the charge is cancelled out by the opposite electric field created by the induced charges on the surface of the torus. This results in a net electric field of zero outside the torus, effectively shielding the point charge from external influences.

5. How can the behavior of a conducting torus and a point charge be studied?

The behavior of a conducting torus and a point charge can be studied through experiments and simulations. By manipulating the position, charge, and other parameters of the point charge and the torus, scientists can observe and measure the resulting electric fields and charges on the surface of the torus. This can provide valuable insights into the principles of electrostatics and help in the development of new technologies.

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