# Homework Help: Capacitance of Parallel cylinders

1. Sep 4, 2011

### jfy4

1. The problem statement, all variables and given/known data
Two long, cylindrical conductors of radii $a_1$ and $a_2$ are parallel and separated by a distance $d$, which is large compared with either radius. Show that the capacitance per unit length is given approximately by
$$C=\pi\epsilon_0 \left(\ln\frac{d}{a}\right)^{-1}$$
where $a$ is the geometric mean of the two radii.
Approximately what gauge wire (state diameter in millimeters) would be necessary to make a two-wire transmission line with a capacitance of $1.2\times10^{-11}$ F/m if the separation of the wires was 0.5 cm? 1.5 cm? 5.0 cm?

2. Relevant equations
$$C=\frac{Q}{\phi}$$
$$\phi=-\int \vec{E}\cdot d\vec{l}$$

3. The attempt at a solution
Honestly, I'm pretty lost on this... I'm not sure how to construct this from scratch, my only work is from backwards, and I feel shady about it...

Here it is:
We know that $C=Q/\phi$ and we are given the capacitance per unit length $C'=C/l$, so I did this
$$\phi=\frac{q}{C' l}=\frac{\lambda}{\pi\epsilon_0}[\ln(d)-\ln(\sqrt{a_1 a_2})]$$
with $\lambda$ the charge per unit length, which can be written as
$$\phi=-\int_{d}^{\sqrt{a_1 a_2}} \frac{\lambda}{\pi\epsilon_0 }\frac{1}{r}dr$$
Then the electric field would be
$$\vec{E}=\frac{\lambda}{\pi\epsilon_0 r}\hat{r}$$
Now if I were to continue with this (and I'm not committed to this :) ), how could I manage to write this as a superposition of two cylindrical electric fields from two different conductors with equal and opposite charges...?

Can someone give me a hand here?