What Is the Electric Potential on a Charged Cone?

[SummerPhysStudent]

Thanks for taking the time to look at this. I'm getting ready to go to grad school, and I'm realizing that although I did ok in my classes, there are large gaps in my knowledge of physics. That said, I'm currently trying to work my way through an E&M book, and now I'm stuck.

Here's the problem

Pretend that you have an open ended cone with the vertex at the origin and the fat end a distance located at a height R. Coincidently, the Radius of the cone at this height is also R. This cone also carries a uniform surface charge sigma. Find the potential difference between the vertex of the cone and the point at the center of the its base.

So obviously, V = 1 / (4 * pi * e0) * 2pi * int(sqrt(2)*r'dr / r) * sigma

(right)

so it's pretty easy to solve for the vertex, but I can't figure out how to solve the integral for the other point.
Thanks for the help
James

 

[maverick280857]

Try setting up the integrand using cylindrical coordinates $$(r, \theta, z)$$ since that is an obvious choice for cylindrical symmetry (note: you can also use spherical coordinates...they might seem more logical at first sight).

 

[M98Ranger]

Hi James,

I understand your frustration with trying to understand a difficult concept in physics. It's great that you are taking the initiative to work through an E&M book to improve your understanding.

To solve this problem, you can use the concept of electric potential on a charged ring. The cone can be thought of as a series of charged rings stacked on top of each other. The potential at any point on the cone can be found by summing the potential contributions from each ring.

To find the potential at the center of the base, you can integrate the potential contributions from each ring at a distance R from the center. This integral can be simplified using the substitution u = r/R, which will result in a simpler form to solve.

I recommend looking up examples of finding electric potential on a charged ring to get a better understanding of the concept and how it can be applied to this problem. Also, don't hesitate to reach out to your professors or classmates for help in understanding the material. Good luck with your studies!