Problem 2.16

A long coaxial cable (Fig. 2.26) carries a uniform volume charge density ρ on the inner cylinder (radius a), and a uniform surface charge density on the outer cylindrical shell (radius b). This surface charge is negative and is of just the right magnitude that the cable as a whole is electrically neutral. Find the electric field in each of the three regions: (i) inside the inner cylinder (s b). Plot |E| as a function of s.

To find the electric field in each of the three regions of a long coaxial cable, we can use Gauss’s law.

First, let’s consider the region inside the inner cylinder (s < a). The electric field in this region is solely due to the uniform volume charge density on the inner cylinder. By Gauss’s law, the electric field in this region is given by:

E = (1/ε) ∫ ρ dV

where ε is the permittivity of free space. Since the charge density is uniform, we can simplify this to:

E = (ρ/ε)V

where V is the volume of the inner cylinder. Therefore, the electric field inside the inner cylinder is: