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11

Mechanical Force on Unit Area of a Charged Cylinder

Consider a small element AB of area dS. Total charge = Q Surface charge density = s Charge on element AB = s dS Electric field intensity at a point P near surface AB E = At point P, E = E_{1} + E_{2} (in magnitude) where E_{1} is due to charge on AB E_{2} is due to the remaining charge Q - s dS At point S (inside the conductor), E=0 Since E = 0, E_{1} = E_{2} (in magnitude) \ force experienced by the charged element AB, F_{AB} = E_{2} (s dS) = E^{2} kÎ_{0} dS \ Force per unit area= F_{AB}/dS = E^{2}kÎ_{0}

\

F

A

= E^{2} kÎ_{0}

Energy density (Energy per unit volume) in an electric field:

Consider a charged conductor which carries a charge Q. Consider a small element AB of area dS on the charged conductor. If s is surface charge density then the charge carried by small element AB =s dS. The charge (s dS) is located in the field due to the charge (Q-sdS) on the surface of the remaining part of the conductor. Therefore, the force on the element AB is given by F_{AB} = Electric field xs dS ( F = qE) = E/2s dS = 1/2 (s/ke_{0})s dS ( E = s/ke_{0}) F_{AB} = 1/2(s^{2}/ke_{0}) dS (1)

This force is directed normally outwardS. Suppose, under the action of this force, the element AB moves through a distance dx. Therefore, the work done dW = F_{AB} dx = 1/2(s^{2}/ke_{0}) dS dx (using 1) But dS.dx = Volume ( dV) of the region swept out by the element AB dW = 1/2(s^{2}/ke_{0}) dV This work is stored as the electrostatic potential energy in volume dV.

Energy stored in volume ( dV) = 1/2(s^{2}/ke_{0}) dV Energy stored per unit volume (Energy density) is given by U = dW/ dV

=

1

2

s^{2}

kÎ_{0}

Since E = s/ke_{0},

U =

(E^{2}kÎ_{0})

12.

Capacity of a Conductor

It is defined as the quantity of charge that must be deposited on the conductor to raise its potential by unity.

Capacity C =

Q

V

where Q :

charge on conductor

V :

potential of the conductor

Capacity is measured in terms of farad. One Farad: If the potential of a conductor is raised by 1 volt, when a charge of 1 coulomb is deposited on it, then its capacity is said to be one farad.

Other smaller units are: 1 milli farad = 1mf = 10^{-}^{3} f 1 micro farad = 1 mf = 10^{-}^{6} f 1 pico farad = 1 pf = 10^{-}^{12} f

13.

Concept of a Condenser

An arrangement consisting of two parallel conductors separated from each other by air or some other insulating medium (dielectric) is called a Condenser or a Capacitor. Principle: The presence of an earthed conductor near a charged body reduces the potential of the charged body and increases its capacity. The capacity of a condenser is defined as the ratio of the charge to the potential difference between the two plates.

(a)

C=

V_{A}

(b)

C¢=

V_{A} - V_{B}

C¢ > C

14.

A Plane Parallel Plate Condenser with a Dielectric

A plane parallel plate condenser consists of two metal plates separated by a distance ' d' of an insulating medium (dielectric).

The capacitance of a parallel plate capacitor is given by,

C =

Î A

d

A Î_{0} k

where A:

surface area of its plate

d:

distance between the plates

Î:

permittivity of the dielectric

For air k = 1 \ C_{a}=

A Î_{0}

C

C_{a}

= k