# Cylindrical Capacitor & Spherical Capacitor

### Cylindrical Capacitor :

The capacitance of a system of two co-axial conducting, hollow cylinders of radius R1 and R2and length l , (l >> R1 , R2) can be found out as follows.

Applying Gauss law on a cylinder of radius r.

$\displaystyle E(2\pi r l ) = \frac{\lambda l}{\epsilon_0}$

$\displaystyle E = \frac{\lambda }{2 \pi \epsilon_0 r}$

P. D across the cylinder $\displaystyle V = \int \vec{E}.\vec{dr}$

$\displaystyle V = \frac{\lambda }{2 \pi \epsilon_0} \; ln \frac{R_2}{R_1}$

$\displaystyle C = \frac{Q}{V} = \frac{\lambda l}{V}$

$\displaystyle C = \frac{2\pi \epsilon_0 l}{ln \frac{R_2}{R_1}}$

### Spherical Capacitor :

The capacitance of a system of two concentric conducting spherical shells of radii R1 and R2 can be found out as follows.

Applying Gauss’s law on a concentric sphere of radius r

$\displaystyle E(4 \pi r^2 ) = \frac{Q}{\epsilon_0 }$

$\displaystyle E = \frac{Q}{4 \pi\epsilon_0 r^2}$

P.D across the shell is

$\displaystyle V = \frac{Q}{4 \pi\epsilon_0 } (\frac{1}{R_1} – \frac{1}{R_2})$

Capacitance C = Q/V

$\displaystyle C = \frac{4 \pi \epsilon_0 R_1 R_2 }{R_2 – R_1}$

Exercise : Three identical metallic plates are kept parallel to one another at separations a & b as shown in figure. The outer plates are connected by a thin conducting wire and a charge Q is placed on the central plate. Find the charge on all the six surfaces.