Electrostatic Potential of cylindrical surface

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SUMMARY

The discussion focuses on calculating the electrostatic potential of a cylindrical surface with a height \( h \) and radius \( R \), where the charge density is defined as \( \sigma(z) = \sigma_0 z \). The participant expresses difficulty in applying Laplace's equation in cylindrical coordinates due to the lack of symmetry and the complexity introduced by the varying charge density. They consider integrating the electric field to find the potential but seek guidance on handling the boundary conditions and the integration of the charge density over the area.

PREREQUISITES
  • Understanding of electrostatics, specifically electric potential and charge density.
  • Familiarity with Laplace's equation and Poisson's equation in cylindrical coordinates.
  • Knowledge of integration techniques for varying charge densities.
  • Basic concepts of electric fields and their relationship to potential.
NEXT STEPS
  • Study the application of Laplace's equation in cylindrical coordinates for electrostatic problems.
  • Learn about integrating charge densities to find electric fields and potentials.
  • Research boundary conditions relevant to cylindrical geometries in electrostatics.
  • Explore examples of varying charge densities and their effects on electric fields and potentials.
USEFUL FOR

This discussion is beneficial for physics students, particularly those studying electrostatics, as well as educators and researchers looking to deepen their understanding of cylindrical charge distributions and their impact on electric potential calculations.

greygasher
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Homework Statement


The figure shows a section of a cylindrical surface, height h and radius R. The curved surface extends from the z-axis to the y-axis only and has a charge density given by σ(z)= σ0z where σ0is some constant. ind the electrostatic potental at a. (a is at the origin)

I'm sorry I'm not sure how to get the picture up here. It's not that complicated, just a cylindrical shell but it's cut in a way that nothing cancels out (as I hope is obvious from the problem description)

Homework Equations


(1) Laplace/Poisson

or

(2)
690ab6aada2735237ee2fba30dc1698b.png


The Attempt at a Solution


I'm really kind of at a loss. I could solve the Laplace for cylindrical coordinates (as that is where our lectures have been heading) but I feel like there's absolutely no symmetry to exploit and it would be way too much work. I would almost just want to find the electric field and integrate to get the potential, but we've never really done (nor can I find much help online with) a varying charge density. So I get stuck with finding the enclosed charge.
If I end up needing to use Laplace, I'd like some help with my boundary conditions and probably making sense of the whole mess. I can follow the derivation alright but the solution is a hair away from being over my head.
 
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what about integrating the charge density over the area.
[tex] V(r)= \int_{a}^{b}\frac{\sigma}{4\pi\epsilon_0r}da[/tex]
 

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