Generally,Poisson's equation can be solved with appropriate boundary conditions to get potential from which the surface charge densities can be obtained. Most textbooks on Classical electrodynamics treat disc as a special case of the more general ellipsoid,and derive the charge distribution from it(e.g.Zangwill,Jackson,Smythe etc),so does other Physics StackExchange answers like- Charge distribution on a conducting solid disc . Are there other ways to get the surface charge density without using the "squashed ellipsoid"? Is there any general(analytic or numerical)method to solve for the charge density of an arbitrary shape,(like the conducting disc)simply from its geometry,noting that the potential or field is not known?


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