A long nonconducting solid cylinder of radius R has a nonuni

A long, nonconducting, solid cylinder of radius R has a nonuniform volume charge density rho that is a function of radial distance r from the cylinder axis: rho = Ar^2. Where A is a constant with the appropriate units. What is the magnitude of the electric field at radius r where r R. State your answers in terms of the given variables, using epsilon_0 when needed.

Solution

We have a long cylinder with charge density as a function of r from the central axis as: Ar^2

That is at any distance r and cylindrical shell of thickness dr, the charge contained = 2*pi*r*Ar^2*dr

We integrate upto r to get net charge contained as a function of r as 2/3*pi*r^3

Assume a gaussian surface of unit length and cylindrical in shape with the axis coinciding with that of the cylinder and radius r < R

That is E* 2*pi*r = 2*pi*r^3 /  3₀  

Hence E * r = r^2 / 3₀

Therefore, E = r/ 3₀

b.) For r > R, we cosider a gaussian surface with r > R

Hence, E*2*pi*r = 2pi*R^3 / 3₀

That is, E = R^3 / 3r₀