Problem 2756 An infinite cylinder of radius R has a linear c

Problem 27.56

An infinite cylinder of radius R has a linear charge density . The volume charge density (C/m3) within the cylinder (rR) is (r)=0rR, where 0 is a constant to be determined.

Part B

Use Gauss\'s law to find an expression for the electric field E inside the cylinder, rR. Give your answer as a multiple of /0.

Solution

Here, we have a uniform linear charge density, however, the volume charge density varies from the axis.

We assume a cylinderical shell at radius r and thickness dr and of unit length, coaxial with the infinite cylinder.

The density at the given r would be: rRPo

Hence, the charge inside the shell = rRPo x 2*pi*r dr

Integrating the differential charge for r = 0 to r =R

We get: Q(for unit length) = (2/3)RPo R^3

However, we know that the linear charge density is , hence the charge calculate above should be equal to

Hence, = (2/3)RPo R^3

or, Po = (3 / 2 R^4)

Part B) We assume a Gaussian surfac or radius r, and of unit length and coaxial with the cylinder.

Hence, E(2 r) = Charge included / 0

Charge included =  (3 / 2 R^4)R * (2 r)dr =  (3 / R^3)* (r)dr = (3 / 2R^3)* r^2

That is, E(2 r) = (3 / 2R^3)* r^2 / 0

Electric field = (3 r) / (4R^3 0) = (/0)* (3r / 4R^3 )