Electric field of a spherical cap

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Gavroy
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Hi

I am looking for the electric field caused by a uniformly charged spherical cap.
Actually, I need only the potential inside the sphere. Is there anybody who knows how to do this. Frankly, I do not have a clue. Or could somebody at least give me integral, that I have to solve?
 

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You can find the solution here: http://faculty.uml.edu/cbaird/all_homework_solutions/Jackson_3_2_Homework_Solution.pdf . Apparently one needs Legendre polynomials to represent the solution.
 

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