6.0 Capacitance of a cylindrical capacitor

6.1 Working method to find capacitance of any capacitor

When a metallic cylinder of radius $a$ is placed coaxially inside an earthed hollow metallic cylinder of radius $b$ $(>a)$, we get cylindrical capacitor. If a charge $q$ is given to the inner cylinder, induced charge $-q$ will reach to the inner surface of the outer cylinder.

Assume that the capacitor is of very large length $(l>>b)$ so that the lines of forces are radial. Using Gauss's law we can prove that $$E(r) = \frac{\lambda }{{2\pi {\varepsilon _ \circ }r}}$$

Here, $\lambda$=charge per unit length, $a \leqslant r \leqslant b$

Therefore, the potential difference between the cylinders $$V = - \int\limits_a^b {\overrightarrow {E.} } \overrightarrow {dr} = - \int\limits_a^b {\frac{\lambda }{{2\pi {\varepsilon _ \circ }r}} = \frac{\lambda }{{2\pi {\varepsilon _ \circ }}}} \ln \left( {\frac{b}{a}} \right)$$$$\therefore \frac{\lambda }{V} = \frac{{charge /length}}{{potential\ difference}} = \frac{{capacitance}}{{length}} = \frac{{2\pi {\varepsilon _ \circ }}}{{\ln (b/a)}}$$

Hence, capacitance per unit length is $$\frac{{2\pi {\varepsilon _ \circ }}}{{\ln (b/a)}}$$