Surface Charges on a Coaxial Cable

[cpburris]

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?

 

[kuruman]

Strictly speaking there is the issue that the total charge on the outer conductor is not specified as you correctly pointed out. I think that in this case you may safely assume that the outer conductor bears no net charge.