We follow the following procedure for finding capacitance. First put some charges on the two plates uniformly. Then use Gauss’s law to find the electric field between the plates. Use the electric field to find the potential difference. The ratio of charge to potential difference is capacitance by definition.
Let there be \(+\lambda\) be the charge per unit length on the inner cylinder and \(-\lambda\) be the charge per unit length on the outer cylinder. This will give radially outward electric field between the two cylinders. Then Gauss’s law tells us that the electric field between the cylinders has the magnitude
\begin{equation*}
E = \dfrac{\lambda}{2\pi\epsilon_0}\:\dfrac{1}{s},
\end{equation*}
where \(s\) is the distance from the axis. Now we integrate it from the outer cylinder to the inner cylinder, going in the direction of the electric field, to obtain the potential difference \(V\)
\begin{equation*}
V = -\int_{r_2}^{r_1}\:\vec E \cdot d\vec r = \dfrac{\lambda}{2\pi\epsilon_0}\:\ln\left( \dfrac{r_2}{r_1} \right)
\end{equation*}
Writing \(\lambda = Q/L\text{,}\) we can read off the capacitance per unit length \(C/L\) from this formula.
\begin{equation*}
\dfrac{C}{L} = \left[\dfrac{1}{2\pi\epsilon_0}\:\ln\left( \dfrac{r_2}{r_1}\right) \right]^{-1}.
\end{equation*}
