A spherical shell with radius a has conductivity in the angular range 1 <

A spherical shell with radius a has conductivity σ in the angular range α₁ < θ < π − α₂. Otherwise, the shell is perfectly conducting and a potential difference V is maintained between θ = 0 and θ = π.

(a) Solve Laplace’s equation to find the potential, surface current density, and resistance of the shell between θ = 0 and θ = π.

(b) Divide the shell into many thin rings. Find the resistance of each and combine them to find the resistance and confirm the answer derived in part (a).

The substitution y = ln[tan(θ/2)] will be useful.