A solid conducting sphere of radius R is cut through its midplane into two hemispheres. The two
Question:
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?