Finding the electric potential above a charged metal toroid (electrostatic lens)

[Callumnc1]

Homework Statement:

Please see below

Relevant Equations:

Please see below

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

 

[BvU]

 

[Callumnc1]

View attachment 321660

Thank you for your reply @BvU!

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

Many thanks!

 

[haruspex]

Thank you for your reply @BvU!

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.

 

[Callumnc1]

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!