# Calculating Electric Potential for a Non-Negligible Thickness Toroid

• member 731016
In summary, the conversation discusses a problem involving calculating the electric potential in a toroid with a non-negligible thickness. The solution is given, but there is a doubt about the linear charge density and its relation to the toroid's dimensions. The conversation ends with a question about how to calculate the electric potential with a non-neglected thickness.
member 731016
Homework Statement
Relevant Equations
For A.1 of this problem,

The solution is

However, I have a doubt about the linear charge density ##\lambda##.

I don't understand how ##\lambda = \frac {q}{2\pi R} ## since this is not a thin ring, but has a non-negligible width of ##2a##

I think that the toroid has a larger area than thin circle with a circumference ##2\pi R## so linear charge density should be less than that expression.

EDIT: How would we calculate the electric potential if the thickness was not neglected?

Many thanks !

Problem from:
https://www.ipho2021.lt/uplfiles/Th2.pdf
https://www.ipho2021.lt/uplfiles/Th2-Solution.pdf

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member 731016
BvU said:

However, how would we calculate the electric potential if the thickness was not neglected?

Many thanks!

Callumnc1 said:

However, how would we calculate the electric potential if the thickness was not neglected?

Many thanks!
Since it is metallic, your first challenge would be to figure out the charge distribution. Good luck with that.

member 731016, nasu and BvU
haruspex said:
Since it is metallic, your first challenge would be to figure out the charge distribution. Good luck with that.
Thank you for your reply @haruspex! Yeah that seems hard!

## What is the electric potential of a toroid with non-negligible thickness?

The electric potential of a toroid with non-negligible thickness is determined by integrating the contributions of each infinitesimal charge element over the entire volume of the toroid. This involves solving complex integrals that take into account the geometry and charge distribution of the toroid.

## How does the thickness of the toroid affect the calculation of electric potential?

The thickness of the toroid introduces additional complexity to the calculation of electric potential because it requires considering the three-dimensional distribution of charge rather than just a surface or line charge. This affects the distance from each charge element to the point where the potential is being calculated, making the integrals more complex.

## What mathematical techniques are commonly used to calculate the electric potential of a toroid with non-negligible thickness?

Common mathematical techniques include using cylindrical coordinates to simplify the geometry, applying Gauss's law for symmetry considerations, and performing volume integrals. Numerical methods and computational tools are often employed to handle the complexity of these integrals.

## Can symmetry be used to simplify the calculation of electric potential for a toroid?

Yes, symmetry can be used to simplify the calculation. For a uniformly charged toroid, the problem exhibits cylindrical symmetry, which can reduce the complexity of the integrals. However, the non-negligible thickness still requires careful consideration of the three-dimensional charge distribution.

## Are there any approximations that can simplify the calculation of electric potential for a toroid with non-negligible thickness?

Approximations can be made under certain conditions, such as assuming the thickness is small compared to the major radius of the toroid. In such cases, the toroid can be approximated as a series of concentric rings or thin shells, simplifying the integrals. However, these approximations may introduce errors and are not always applicable.

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