- #1
peterpiper
- 14
- 0
Homework Statement
Starting with:
G = [itex]\frac{I}{V}= \frac{-\int_{s}\sigma\mathbf{E}\cdot d\boldsymbol{s}}{-\int_{l}\boldsymbol{E}\cdot d\boldsymbol{l}}[/itex]
Derive the conductance per unit length, G', of a coaxial cable by assuming a line charge of [itex]\rho_{l}[/itex] on the center conductor.
Homework Equations
[itex]\boldsymbol{E}=\boldsymbol{\hat{r}}\frac{\rho _{l}}{2\pi\epsilon_{0}r}[/itex]
Let a be the radius of the inner conductor and let b be the radius of the outer conductor.
The Attempt at a Solution
So far my only attempt has been plugging E straight into the integrals with the region of integration being a to b on both integrals and 0 to 2[itex]\pi[/itex] on the double integral. I get the correct coefficient of [itex]2\pi\sigma[/itex] but I'm supposed to have a ln(b/a) in the denominator. I'm kinda baffled as to what I'm supposed to do. I'm assuming I'm doing something stupid in the double integral but those never were my strong suit in math. Any help would be greatly appreciated as this is the last bit of work I need to do to finish my E-Mag course. Thanks in advance for anything you may be able to offer me.