Question

A small conducting sphere of radius r is lying concentrically inside a bigger hollow conducting sphere of radius R. The bigger and smaller sphere are charged with Q and q(Q > q) and are insulated from each other. The potential difference between the spheres will be 

(A) [1 / (4π∈₀)] [(q / r) – (Q / R)] 

(B) [1 / (4π∈₀)] [(q / r) – (q / R)] 

(C) [1 / (4π∈₀)] [(Q / R) + (q / r)] 

(D) [1 / (4π∈₀)] [(q / R) – (Q / r)]

Answer 1

The correct option (B) [1/(4π∈₀)] [(q/r) – (q/R)]

Explanation:

The potential difference between two sphere of radius r & R with r < R & charge of each q, Q respectively is

[1/(4π∈₀)][(q/r) – (q/R)]

The potential on outer sphere due to charge Q = [1/(4π∈₀)] ∙ (Q/R) 

on the inner sphere potential due to q = [1/(4π∈₀)] (q/r)

∴ Total potential on inner sphere = [1/(4π∈₀)](q/r) + [1/(4π∈₀)]  ∙ (Q/R)    (1)

on the outer sphere, potential due to q is [1/(4π∈₀)] ∙ (q/R)

∴ Total potential on outer sphere

= [1/(4π∈₀)](q/R) + [1/(4π∈₀)](Q/R)  (2)

∴ from (1) & (2) V_(r) – V_(R) = [1/(4π∈₀)][(q/r) – (q/R)] as r < R hence

(1/r) – (1/R) is positive quantity