Question

A thick insulating spherical shell has inner radius a and outer radius b. The shell

carries a uniform volume charge density ρ0.

(a) Consider a spherical Gaussian surface of radius r concentric with the shell. How

much charge is enclosed in the Gaussian surface for r < a, a < r < b, and r > b?

(b) What does symmetry dictate about the magnitude and direction of the electric

field?

(c) Determine the electric field everywhere (i.e., what is the electric field for r < a,

a < r < b, and r > b).

Answer 1

(a) For the Gaussian surface at r<a there will be no charge enclosed, hence the total electric field will be 0, inside the insulating shell, For the Gaussian surface at r>b, there will be an enclosed charge of, (charge density x volume)

Now for the region a<r<b, we get the enclosed charge as,

(b) From the spherical symmetry condition the electric field will be pointed away from the center or towards the center, depending on the charge on the sphere.

(c) For the Gaussian surface at r<a there will be no charge enclosed, hence the total electric field will be 0, inside the insulating shell, For the Gaussian surface at r>b, there will be an enclosed charge of,

Hence the electric field will be, (from Gauss's law),

Now for the region a<r<b, we get the enclosed charge as,

Hence the electric field is,