Question

A small conducting sphere of radius r carrying a charge q is surrounded by a large concentric shell of radius R on which charge Q is placed. using gauss theorem derive the expression for the electric field at a point X. 1) b/w the sphere and a shell ( r < x < R). 2) outside the spherical shell ​

Answer 1

according to Gauss theorem, electric flux is the ratio of charge enclosed inside the gaussian surface to the permittivity of medium.

e.g., electric flux = Q/$\epsilon_0$

or, E.A = Q/$\epsilon_0$

(1) between the sphere and shell, r < x < R

charge enclosed inside the gaussian surface (spherical shell of radius x ) is q

electric flux = E.A = q/$\epsilon_0$

or, E × 4πx² = q/$\epsilon_0$

or, E = q/4π$\epsilon_0$x²

hence, electric field between the sphere and shell is E = q/4π$\epsilon_0$x²

(2)outside the spherical shell, x > R

charge enclosed inside the gaussian surface (spherical shell of radius x > R) is (q + Q)

so, electric flux = E' .A' = (q + Q)/$\epsilon_0$

or, E' × 4πx² = (q + Q)/$\epsilon_0$

or, E = (q + Q)/4π$\epsilon_0$x²

hence, Electric field outside the spherical shell is E = (q + Q)/4π$\epsilon_0$x²