Cylindrical Capacitor. A cylindrical capacitor consists of two coaxial conducting cylindrical shells separated by a certain dielectric. One cylindrical shell is earthed and other is used to store charge on it.
Expression for capacitance of a cylindrical capacitor
Let P and Q be two conducting cylindrical shells of a cylindrical capacitor of radii a and b as shown in Fig.
Let a charge +q be given to cylinder P, so negative charge will be induced on the inner surface of cylinder Q, whose outer surface is earthed.
Let λ be linear charge density. The electric field intensity E on Gaussian cylinder (as shown by dotted lines) of radius r is given by
E = \(\frac{1}{2\piɛ_o} \frac{\lambda}{r}\)
The potential difference of cylindrical shells P and Q is given by
.......(1)
Total charge on the cylinder, q = λl .................(2)
If C is the capacitance of the cylindrical capacitor, then
C = \(\frac{q}{V}\)
Using eq. (1) & (2), we get
C = \(\frac{\lambda l2 \pi \varepsilon_0}{\lambda \ log_e\ b/a} = \frac{2\pi\varepsilon_0l}{log_e\ b/a}\)
or C = \(\frac{2 \pi \varepsilon_0l}{2.303\ log_{10}\ b/a}\)
