Two concentric spheres have radii a, b (b>a) and each is divided into two hemispheres by the same horizontal plane. The upper hemisphere of the inner sphere and the lower hemisphere of the outer sphere are maintained at potential V. The other hemispheres are at zero potential.
Determine the potential in the region a<=r<=b as a series in Legendre polynomials. Including terms at least up to l=4. Check your solution against known results in the limiting cases b--> infinity, and a--> 0.
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The solution shows in detail how to obtain the Legendre polynomial expansion of two concentric spheres, their opposite halves are held at constant potential and the other half is grounded.