Integral from an electric field calculation

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SUMMARY

The integral discussed is crucial for calculating the electric field of a uniformly charged spherical shell. The specific integral presented is $$ \int_{0}^{\pi} \frac{(z-R\cos{\theta})\sin{\theta}d\theta}{R^2+z^2-2ZR\cos{\theta}}$$, where the variable 'r' in the denominator should be replaced with 'R', representing the radius of the shell. When substituting z=R, the result aligns with the expected electric field of an infinite plane, given by $$ E_z=\sigma/2\epsilon_0 $$.

PREREQUISITES
  • Understanding of calculus, specifically integral calculus.
  • Familiarity with electric field concepts and Gauss's law.
  • Knowledge of spherical coordinates and their applications in physics.
  • Basic understanding of electrostatics and charge distributions.
NEXT STEPS
  • Study the derivation of electric fields from charge distributions using Gauss's law.
  • Explore advanced integral techniques, particularly in spherical coordinates.
  • Learn about the properties of electric fields generated by spherical shells.
  • Investigate the implications of uniform charge distributions in electrostatics.
USEFUL FOR

Physicists, electrical engineers, and students studying electromagnetism who are involved in calculating electric fields from charge distributions.

klawlor419
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Hi all,

Any ideas how to solve this integral?

$$ \int_{0}^{\pi} \frac{(z-R\cos{\theta})\sin{\theta}d\theta}{R^2+z^2-2zr\cos{\theta}}$$

It crops up in a calculation of the electric field for a spherical charged shell. It has a uniform charge smear

Thanks ahead of time
 
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The lower case r in the denominator should be a large R for the radius of the shell.
 
Well, I figured out that when z=R you recover the expected field of an infinite plane.

$$ E_z=\sigma/2\epsilon_0 $$
 

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