# Surface Charges on a Coaxial Cable

Gold Member
Homework Statement:
A long coaxial cable consists of a conducting inner cylinder of radius ##a## and a thick outer conducting cylinder of inner radius ##b## and outer radius ##c## (Note: ##a<b<c##). The surface charge density on the inner cylinder is ##\sigma##. Find the surface charge densities ##\sigma_b## & ##\sigma_c##.
Relevant Equations:
##E_{inside conductor}=0##
Gauss' Law - ##\oint \vec E \cdot d \vec A = \frac {Q_{enclosed}} {\epsilon_0}##
To find ##\sigma_b## I can use a Gaussian surface of a cylinder of length ##L## and radius ##c>r>b##. Since that is inside of the outer conductor, I know the electric field is zero, so I have from Gauss' Law, $$0=2 \pi L\left(b\sigma_b+a\sigma\right)$$ and easily solve for ##\sigma_b##. For ##\sigma_c## however, I am unsure how to proceed. The problem does not give any information about the total charge on the outer cylinder or the coaxial cable as a whole, and it would seem to me that without knowing one of these ##\sigma_c## is arbitrary. Is there something I am missing or is there an issue with the statement of the problem?