CHAPTER 11

ELECTROSTATICS
CHARGE
It is the inherent property of certain fundamental particles. It
accompanies them wherever they exist. Commonly known
charged particles are proton and electron. The charge of a
proton is taken as positive and that of electron is taken as
negative. It is represented by symbol e.
e = 1.6 × 10–19 coulomb
Positive and negative sign were arbitrarily assigned by
Benzamin Franklin. This does not mean that charge of proton is
greater than charge of electron.

Quantization of Charge
Electric charges appear only in discrete amounts, it is said to be
quantized.

Conservation of Charge
For an isolated system, the total charge remains constant, charge
is neither created nor destroyed, and it is transferred from one
body to the other.

COULOMB’S LAW

1
The force of interaction of two stationary point charges in
vacuum is directly proportional to the product of these charges
and inversely proportional to the square of their separation
kq1 q 2
F=
r2

Where, k is a constant which depends on the system of units. Its

value in SI unit is

k = 9  109 Nm2C-2

The constant is often written in the form

1
k= 40

Where, 0 is called the permittivity constant which is

numerically equal to

o = 8.85  1012 C2 /Nm2

ELECTRIC FIELD
.

The electric field strength ( E ) at a point is defined as the force
per unit charge experienced by a test charge qt, placed at that
point.

 F
E =
qt

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LINES OF FORCE
The electric fieldlines or lines of force are helpful in
visualizing field patterns. They provide the following basic
information:
(a) The direction of the field is along the tangent to a line of
force.
(b) The strength or magnitude of the field is proportional to
the number of lines that cross a unit area perpendicular to
the line.

GAUSS’ LAW
 1
The net flux of E through a closed surface equals 0
times

the net charge enclosed by the surface.

  q
 .dS  
E
0

Table 1 Electric field E due to Various Charge Distributions

1. Isolated point charge  1 q
E = r̂
40 r 2

3
 +q r P E

2. A Ring of Charge E|| = 0

+ + q 1 qx
E =
+ + E|| = 0 40 ( R 2  x 2 ) 3 / 2
+ R +
+ +
+ x P E  0
+
+ +
+ +
+ +

3. A Disc of Charge E|| = 0

  x 
E = 1  
2 o  2 
x R 
2
+
+ + 
+  E|| = 0
++ R +
Where  is the surface
++
++
++ +
+ + ++ + x P E  0
++ + + charge density
+ + +
+ +
+

4. Infinite Sheet of Charge 

E =
2 o

+
where  is the surface
E|| = 0
+ + + charge density
+

+ + +
+
+ + +
x P E  0
+
+ +
+

5. Infinitely Long Line of Charge E|| = 0


E=
2o r

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 E|| = 0 where  is the linear
+
+
+
+ charge density
+
+
x P E  0
+
+

6. Finite Line of Charge 

E= [sin  + sin ]
4o x
+
+
+
 
+

P
E E|| = [ cos cos]
+
+
x
4o x
+
+ E||
where  is the linear
charge density

7. Uniformly charged sphere Inside 0 rR


++ +  r
+ +
+ ++ R + ++ E =
+  + + + 3 o
+ + + + + ++
+ ++ + + +
Outside r R
+ + + +
++ + +
+ +

 r  R  3
E =   r̂
3 o  r 

where  is the volume

charge density.

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POTENTIAL

Electric potential, V is defined as the change in electrostatic

potential energy per unit charge.
U
V = q

The SI unit of electric potential is the volt (V).

Relationship between E and V

We know that

V =
Wext
q


Now Wext=  Fext .ds

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Table 2 Electric Potential V due to Various Charge
Distribution

1. Isolated Charge 1 q
V= 4o r
+q r P

2. A Ring of Charge 1 q
V = 4
o R2  x2
+ + q
+ +
+ R +
+ +
+ x P
+
+ +
+ +
+ +

3. A Disc of Charge
+
+ +
+ 
++ R +
++
V=

2 o
R 2
 x2  x 
++
++ +
+ + ++ + x P
++ + +
+ + +
+ +
+

4. A sphere of Charge Inside 0 rR

+
++ + R 2  r2 
3  2 
+
+ ++ R + ++ V=
+  + + + + 6 o  R 
+ ++ ++ +
+ ++ + + + r P
+ + + +
++ + +
+ + Outside

rR

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 R 3 1
V=  
3 o r

where  is the volume

charge density.

1. Electric field intensity due to Dipole

  
(i) Along the axis E p E

 x x
 2kp 
E ||  
x3 The E is parallel to p .

The direction of electric

field along the axis is in
the same direction as
that of the dipole
moment.
(ii) Along the bisector
 
kp
E 
y3

The direction of electric field along the bisector is opposite

to that of the dipole moment.

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2. Electric Potential Due to a Dipole Moment

(i) Along the axis

E
2kp
V|| = y
x2
p

(ii) Along the bisector y

V = 0 E

 
The E  is anti-parallel to p

3. Dipole in an External Uniform Field

(i) Torque

If a dipole is oriented at E

+ E
q
an angle  to an uniform d
 
qE -
electric field as shown in

p
the figure, the charges  
E
experience equal and An electric dipole experiences a torque in
an electric field.
opposite forces. So there
is no net force on the
dipole. However, there is
a net torque on the dipole.
  
τ  pE (20)

The magnitude of the torque is

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  = pEsin 

(ii) Potential Energy

The potential energy of a dipole in an external field is given by
 
U =  p .E

CAPACITORS
A capacitor is a device that stores electrical energy. The
capacitance of the capacitor is defined as the magnitude of
the charge on one plate divided by the magnitude of the
potential difference V between them
q
C=
V

Capacitance depends on the size and shape of the plates

and the material between them. It does not depend on q or
V individually. The SI unit of capacitance is the farad (F).
1 farad = 1 coulomb/volt

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1. Parallel Plate Capacitor A
+ + + + + +q

= q
A E
d
 q
E= 
o o A
     q
qd Parallel plate capacitor
V = Ed = o A

q o A
C= 
V d

2. Spherical Capacitor 


 + q
q  1 1 b
V=  a +
4o  a b 
+
+ + 

E
q 4o 4o +
C =  
V 1 1 1 1
  
a b a b
Spherical capacitor
4 o ab
or C=
ba

3. Cylindrical Capacitor

E= 2 o r
ar b -
+

V = VaVb=   E dr
a

b
l
Cylindrical capacitor

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  b
or V= ln
2o a

q  l 2o l
C=   (26)
V V b
ln
a

Energy stored in a Capacitor

The energy stored in a capacitor is equal to the work done
to charge it.
q
dW= Vdq=   dq
C 

The charge moves through the wires, not across the gap
between the plates.
The total work done to transfer charge Q is
Q
q Q 2 QV 1
W=  C dq  2C  2  2 CV
2

Since the charge on each plate is unaffected the capacitance

in the presence of the dielectric is
q 0 kqo
C=   kCo
A Vo

The capacitance of the capacitor increases by a factor k.

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