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.

Also Read :

Capacitance of a Parallel Plate Capacitor
Series & Parallel Combination of Capacitors
Redistribution of Charge, Common Potential & Loss of Energy
Parallel Plate Capacitor with different Charges
Effect of Dielectrics on Charged Capacitor
Force b/w the Plates of a Parallel Plate Capacitor

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