Question

A non-linear spherical charge distribution carries a density = ar^2 in the region r<a. If a concentric metal shell with radii b and c surrounds the inner charge, calculate the electric field in the four regions starting with region 1, inside the radius a and ending with region 4, outside the concentric spheres.

Answer 1

for r<a:

charge enclosed =integration of a*r^2*4*pi*r^2*dr

=4*pi*a*(r^5/5)

=0.8*pi*a*r^5

if electric field is E,

using Gauss’ law:

E*4*pi*r^2=charge enclosed/epsilon

==>E*4*pi*r^2=0.8*pi*a*r^5/epsilon

==>E=0.2*a*r^3/(epsilon)

for a<r<b:

total charge enclosed=0.8*pi*a*a^5

=0.8*pi*a^6

if electric field is E,

using Gauss law:

E*4*pi*r^2=charge enclosed/epsilon

==>E*4*pi*r^2=0.8*pi*a^6/epsilon

==>E=0.2*a^6/(epsilon*r^2)

for b<r<c:

as it is a metallic hollow cylinder, electric field inside a conductor =0

for r>c:

charge enclosed =

=0.8*pi*a*r^5 with r=a

=0.8*pi*a*a^5=0.8*pi*a^6

if electric field is E,

using Gauss' law,

E*4*pi*r^2=charge enclosed/epsilon=0.8*pi*a^6/epsilon

==>E=0.2*a^6/(epsilon*r^2)