Question

The figure shows a spherical shell with uniform volume charge density ρ = 2.18 nC/m3, inner radius a = 9.30 cm, and outer radius b = 2.6a. What is the magnitude of the electric field at radial distances (a) r = 0; (b) r = a/2.00, (c) r = a, (d) r = 1.50a, (e) r = b, and (f) r = 3.00b?

Answer 1

inner radius of the spherical shell a = 9.30 cm = 9.30 x 10^-2 m

outer radius of the spherical shell b = 2.6a = 24.18 x 10^-2 m

volume charge density ρ = 2.18 nC/m^3 = 2.18 x 10^-9 C/m^3

let the radius of the Gaussian surface is r.

at a ≤ r ≤ b, the enclosed charge q = (ρ)(4/3)(π)[(r^3-a^3)]

therefore electric field is given by

E = k[q/r^2] where, k = 1/4πε_0

E = [ρ/3ε_0][(r^3-a^3)/r^2]

a)

given r =0

so, the charge q lie on the surface of the shell is equal to zero.

then the electric field E = 0 N/C

b)

given r = a/2

at this point the charge is also zero.

therefore electric field E = 0 N/C

c)

similarly at r = a , the charge q = 0 C

hence E = 0 N/C

d)

given r = 1.5a

permitivity of the free space ε_0 = 8.85 x 10^-12 C^2/N.m^2

electric field E = [ρ/3ε_0][(r^3-a^3)/r^2]

E = [(2.18 x 10^-9/3 x 8.85 x 10^-12)]*[0.1395^3 - 0.093^3] / 0.1395^2

E = (82.11 x 0.00191) / 0.1395^2

E = 8.060 N/C

e)

given r = b = 24.18 x 10^-2 m

electric field

E = [ρ/3ε_0][(b^3-a^3)/r^2]

E = [(2.18 x 10^-9/3 x 8.85 x 10^-12)]*[0.2418^3 - 0.093^3] / 0.2418^2

E = (82.11 x 0.01333) / 0.2418^2

E = 18.725 N/C

f)

given r = 3b = 0.7254 m

electric field E =[ρ/3ε_0][b^3-a^3)/(3b)^2]

E = [(2.18 x 10^-9/3 x 8.85 x 10^-12)]*[0.2418^3 - 0.093^3] / 0.7254^2

E = (82.11 x 0.01333) / 0.7254^2

E = 2.08 N/C