Electric potential inside a hollow sphere with non-uniform charge

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The discussion revolves around the challenge of determining the charge distribution within a hollow sphere that has a non-uniform charge. The user struggles to derive the correct results from the given electric potential and notes that Gauss's Law is ineffective due to the lack of symmetry, resulting in zero electric flux. A suggestion is made to expand the potential using Legendre polynomials to address the problem. This approach aims to provide a clearer understanding of the electric potential in the context of non-uniform charge distribution. The conversation highlights the complexities of applying classical electrostatic principles in non-standard scenarios.
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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