## Capacitance Of Cylindrical Capacitor

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# Capacitance of a Cylindrical Capacitor

Consider a cylindrical capacitor formed by two coaxial cylinders A and B of radii a and b respectively and each of length l. Air is the medium between A and B. The outer cylinder B is earthed.If a charge + q is given to the inner cylinder, then an equal charge – q is induced on the inner surface of the outer cylinder and a charge +q on the outer surface of the outer cylinder. The charge +q induced on the outer surface of the outer cylinder flows to the earth.

The electric field at a point P in the space between the two cylinders at a distance r from the axis is

E = ___

__1______ q. …. (1)

2πε

_{0}l r

The potential difference V between the cylinders A and B is

V = - ∫a

**E .**dl ... (2)

Here, d

**l**is the vector displacement along a path from B to A.

Now,

**E**is radially outward and d

**l**is inward. Therefore

**E .**d

**l**= E dl cos 180 = - E dl.

As we move a distance dl from B to A, we move in the direction of decreasing r. So dl = - dr. Thus

**E .**dl = E dr.

Eq (2) becomes, V = - ∫a E dr

= - ___

__q_____ ∫a

__dr__

2πε

_{0}l __ r

= - __

___q_____ {log

_{e}r] = - ____

__q______ {log

_{e}a – log

_{e}b}

__ 2πε

_{0}l ______ 2πε

_{0}l

= __

___q_____ log

_{e}

__b__

2πε

_{0}l __ a

Hence the capacitance of the cylindrical capacitor is

C =

__q__=

__2πε__

_{0}lV log

_{e}(b/a)

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