MHT-CET : Physics Entrance Exam

### MHT - CET : Physics - Electrostatics Page 2

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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 = E1 + E2 (in magnitude)
where E1 is due to charge on AB
E2 is due to the remaining charge Q -
s dS

At point S (inside the conductor), E=0
Since E = 0, E1 = E2 (in magnitude)
\ force experienced by the charged element AB,

FAB = E2 (
s dS) = E2 kÎ0 dS
\ Force per unit area= FAB/dS = E2kÎ0

\

 F A

= E2 kÎ0

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
FAB = Electric field x
s dS ( F = qE)
= E/2
s dS
= 1/2 (
s/ke0)s dS ( E = s/ke0)
FAB = 1/2(
s2/ke0) 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 = FAB
dx
= 1/2(
s2/ke0) dS dx    (using 1)

But
dS.dx = Volume ( dV) of the region swept out by the element AB
dW = 1/2(
s2/ke0) dV
This work is stored as the electrostatic potential energy in volume
dV.

Energy stored in volume ( dV) = 1/2(s2/ke0) dV
Energy stored per unit volume (Energy density) is given by
U =
dW/ dV

=

 1 2

 s2 kÎ0

Since E = s/ke0,

U =

 1 2

(E2kÎ0)

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
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=

 Q VA

(b)

C¢=

 Q VA - VB

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 Î0 k d

 where A: surface area of its plate d: distance between the plates Î: permittivity of the dielectric

For air k = 1 \ Ca=

 A Î0 d

\

 C Ca

= k

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