A metal spherical shell of radius R with its center at x = y=z=0 is cut in half along its intersection with the plane z=0. The two halves are separated by an infinitesimal gap and the upper and lower hemispheres are brought to voltages + V and - V respectively. Show that the potential inside the sphere is 1 ()! 41+ 3 r, 0) V (!)? 21 2 o (-1)(*)** 120 2+ P2+1 (cos ), = I=0 where is the polar angle measured relative to the positive z-axis. Hint: V20 =0 inside and on a spherical shell. is clearly independent of $. Following the usual procedure of separation of variables, we find that the solution is of the form r, o) --- [Air! + Bir-i-'][C; Pl(cos 6) + DIQ. (coso)]. I=0 Immediately set B, = 0 (all 1) so 4(0, 0) is finite and set Di = 0 (all I) so 4 (r, 0) and lr, ) are finite. The problem is now one of mathematics-determining the constants AC = ai such that the boundary conditions are satisfied: for O