Electric potential inside a hollow sphere with non-uniform charge

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SUMMARY

The discussion centers on calculating the electric potential inside a hollow sphere with a non-uniform charge distribution. The user encountered difficulties in determining the charge distribution from the given potential and noted that Gauss's Law is ineffective due to the lack of symmetry, resulting in zero electric flux. The recommended approach to resolve this issue is to expand the potential using Legendre polynomials, which can effectively handle non-uniform charge distributions.

PREREQUISITES
  • Understanding of electric potential and charge distribution
  • Familiarity with Gauss's Law and its applications
  • Knowledge of Legendre polynomials and their use in electrostatics
  • Basic principles of electrostatics and field theory
NEXT STEPS
  • Study the application of Legendre polynomials in electrostatics
  • Review advanced topics in electric potential and charge distributions
  • Explore the limitations of Gauss's Law in non-symmetric charge configurations
  • Investigate numerical methods for solving electric potential problems
USEFUL FOR

Physicists, electrical engineers, and students studying electrostatics, particularly those interested in complex charge distributions and potential calculations.

RodolfoM
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Homework Statement
The electric potential on the surface of a hollow spherical shell of radius 𝑅 is 𝑉0 𝑐𝑜𝑠𝜃, where 𝑉0 is a constant. In this problem we use spherical coordinates with origin at the center of the shell. What is the potential inside the shell?

Answer: 𝑉(𝑟,𝜃) = 𝑟/𝑅 𝑉0 𝑐𝑜𝑠𝜃
Relevant Equations
Gauss's Law, Point charge potential.
I tried to find the charge distribution using the given potential but couldn't produce the correct result. Also, Gauss's Law doesn't help, as the electric flux is 0 but we don't have any symmetry. Can someone please shine a light on this? Thanks in advance..
 
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