Chapter 24, Problem 58EAP

An infinite cylinder of radius R has a linear charge density $\lambda$ .

The volume charge density (C/m³) within the cylinder (r $\le$ R) is

$\rho (r)=r{\rho }_0/R$ , where ${\rho }_0$ is a constant to be determined.

a. Draw a graph of $\rho$ versus x for an x-axis that crosses the cylinder

perpendicular to the cylinder axis. Let x range from -2R to 2R.

b. The charge within a small volume dV is $dq=\rho dV$ . The integral

of $\rho dV$ over a cylinder of length L is the total charge $Q=\lambda L$

within the cylinder. Use this fact to show that ${\rho }_0=3\lambda /2\pi R^2$

Hint: Let dV be a cylindrical shell of length L, radius r, and thick-

ness dr. What is the volume of such a shell?

c. Use Gauss's law to find an expression for the electric field

strength E inside the cylinder, r $\le$ R, in terms of $\lambda$ and R.

d. Does your expression have the expected value at the surface,

r = R? Explain.