The surface of a hollow conducting sphere of inner radius a is divided into an even number of equal segments by a set of planes; their common line of intersection is the z axis and they are distributed uniformly in the angle ϕ. (The segments are like the skin on wedges of an apple, or the earth's surface between successive meridians of longitude.) The segments are kept at fixed potentials ±V, alternately.
(a) Set up a series representation for the potential inside the sphere for the general case of 2n segments, and carry the calculation of the coefficients in the series far enough to determine exactly which coefficients are different from zero. For the nonvanishing terms, exhibit the coefficients as an integral over cosθ.
(b) For the special case of n=1 (two hemispheres) determine explicitly the potential up to and including all terms with l=3. By a coordinate transformation verify that this reduces to result (3.36) of Section 3.3.