A solid conducting sphere of radius R is cut through its midplane into two hemispheres. The two halves are then brought back together so that the two halves are electrically isolated by a narrow gap of thickness S << R. A charge q is then placed on the upper hemisphere, while the tower hemisphere remains uncharged. In this problem, you may ignore edge effects which occur at distances of order S from the equator where the two spheres meet.

a. What is the charge per unit area σ ₁ (r) on the planar surface of the upper hemisphere, as a function of distance r from the center of the sphere?

b. What is the charge per unit area σ ₂ (r) on the planar surface of the lower hemisphere as a function of the distance r from the center of the sphere?

c. What is the charge per unit area σ ₃ (θ) on the outer surface of the upper hemisphere, where θ is the polar angle of a point on the surface, with the north pole at θ = 0 and the equator at θ = π/2?

d. What is the charge per unit area σ ₄ (θ) on the outer surface of the lower hemisphere. where θ on the lower hemisphere ranges from π/2 (equator) to π at the south pole?

e. What is the electric field E(r) in the gap between the hemispheres?

f. What is the electric potential difference V between the hemispheres?