A conducting shell of radius R is cut in half. The two hemispherical pieces are electrically separated from each other but left close together, so that the distance separating the two halves can be neglected. The upper half is maintained at a constant potential V-Vo, and the lower half is grounded (V0).

a) Determine the boundary condition at r R as a function of θ and expand it in a series of Legendre polynomials, retaining the terms up to l = 3 only.
b) Calculate the electrostatic potential V at all points outside of the surface of the conductors. Neglect terms falling faster than 1/r

The boundary condition and the voltage for the entire sphere are in the explanation content.

Explanation: Conducting sphere of radius r is made up of two hemispherical shells

separated by a thin insulating ring. They are kept at different potentials, +V and –V

Please find the attached file for the solution

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