Electrostatics Notes and MCQs

Notes 51 Pages

Joint Entrance Examination (Main)

Physics

Contributed by

Yash Kuruvilla

1
UNIT - 11
ELECTROSTATICS
Page 1
2
SUMMARY
1. Electric Charge : Just as masses of two particles are responsible for the gravitational force,
charges are responsible for the electric force. Electric charge is an intrinsic property of a particle.
Charges are of two types : (1) Positive charges (2) Nagative charges.
The force acting between two like charges is repulsive and two unlike charges it is attractive
between .
2. Quantization of Electric Charge : The magnitude of all charges found in nature are an integral
multiple of a fundamental charge.
,neQ 
where e is the fundamental unit of charge.
3. Conservation of Electric Charge : Irrespective of any process taking place, the algebraic sum
of electric charges in an electrically isolated system always remains constant.
4. Coulomb's Law : The electric force between two stationary point charges is directly propor-
tional to the product of their charges and inversely proportional to the square of the distance
between them.
2
21
0
2
21
r
qq
4
1
r
qq
kF


If
0qq
21

then there is a repulsion between the two charges and for
0qq
22

, there is a attrac-
tion between the charges.
5. Equation for Force using Columb’s Law, when two charges are placed in a medium having
dielectric constant k.
(1) The electric force


F
experienced by a test charge (q
0
) due to a source charge (q) when
both are placed in a medium having dielectric constant k and separated by a dis-
tance r, is given by :
r
ˆ
kr
qq
4
1
F
2
21
0



F
r
P
O
(q)
(q
o
)
Here
r
ˆ
is the unit vector directed from q to q
0
.
(2) The equation of coulomb's force may be written as follows :
 
r
ˆ
rk
qq
4
1
F
2
21
0


(3) If the source charge and test charge are separated by a number of medium of thickness
1 2 3
d , d , d ........ having dielectric constants ........k,k,k
321
respectively, then the
electric force on charge q
0
due to a charge q is given by
 
0
2 2 2
0
1 1 2 2 3 3
qq1
ˆ
F r
4
k d k d k d


 

0
2
0
i i
qq
1
ˆ
F r
4
k d


 

 
OR

In this equation k
i
is dielectric constant of medium which spreads through the distance d
i
along the line joining q and q
0
.
Page 2
3
For example, see the figure below :
Here, the space between the charges q and q
0
is filled with medium (1, 2, 3). The thickness of
medium 1 is d
1
and its dielectric consant is k
1
Similarly the thickness of medium 2 and 3 is d
2
and
d
3
of medium 3 and their dielectric constants are k
2
and k
3
respectively.
6. Conditions for Equilibrium in Various Cases :
Suppose three charges q
1
, q
2
and q are situated on a straight line as shown below :
If q
1
and q
2
are like charges and q is of unlike charge then,
(1) Force on q
1

 










2
1
2
21
1
0
1
1
r
q
rr
q
4
q
F
(2) Force on q
2
 










2
2
2
21
1
0
2
2
r
q
rr
q
4
q
F
(3) Force on q =









2
2
2
2
1
1
0
r
q
r
q
4
1
F
Now, from above equations, it is clear that various equilibrium conditions can be as follows :
(a) Condition for
1
F
to be zero is,
 
 
2
21
2
1
221
2
2
1
rr
r
q
q
rr
q
r
q




(b) Condition for
2
F
to be zero is,
   
2
21
2
2
1
2
21
1
2
2
rr
r
q
q
rr
q
r
q




(c) Condition for F to be zero is,
2
2
2
2
1
1
r
q
r
q


2
2
2
1
2
1
r
r
r
q

If 21
q,q
and
q
are of same type charges in nature, then,
(1) Charge q will be in equilibrium, if
0
r
q
r
q
4
q
F
2
2
2
2
1
1
0











2
1 2 1 1
2 2 2
1 2 2 2
q q q r
r r q r
   
(2) Charges q
1
and q
2
will not be in equilibrium.
Page 3
4
7. Electric Field Intensity : The electric force acting on a unit positive charges at a given point in
an electric field of a system of charges is called the electric field or the intensity of electric field


E
at that point.
q
F
E 
The SI unit of
E
is
C
N
or
1
Vm

.
If
n21
r...,.........r,r are the position vectors of the charges
n21
q.,.........q,q
respectively, then the
resultant electric field at a point of position vector
r
is,
 
j
n
1j
3
j
j
rr
rr
q
kE 




8. Electric Dipole : A system of two equal and opposite charge, separated by a finite distance is
called electric dipole.
Electric dipole moment


a2qp 
The direction of
p
is from the negative electric charge to the positive electric charge.
9. Electric field of a dipole on the axis of the dipole at point z = z
   
^
3
2kp
E z p for z a
z
  

Electric field of a dipole on the equator of the dipole at point y = y
   
^
3
kp
E y p for y a
y
  

10. The torque acting on the dipole place in an uniform the electric field at an angle  ,
 sinEp||,Ep
11. Electric Flux : Electric flux associated with surface of area
A
, placed in the uniform electric
field.
 cosEAAE
where,

is the angle between
AandE
,
Its SI unit is
2
Nm
C
or V.m.
12. Gauss's Law : The total electric flux associated with the closed surface,
0
S
q
E d a



  

 
where,
q
is the net charge enclosed by the surface.
Page 4
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13. Electric field due to an infinitely long straight charged wire,
,r
ˆ
r
1
2
E
0



where, r is the perpendicular distance from the charged wire.
14. Electric field due to bending of charged rod,
15. Electric field due to uniformly charged thin spherical shell,
(1) Electric field inside the shell
0E 
(2) Electric field at a distance r from the centre outside the shell,
2
2
0
2
r
R
r
q
kE



where, R = radius of spherical shell.
16. Electric field due to a uniformly charged density sphere of radius R,
(1) Electric field inside the region of the sphere,
3
0 0
Q r r
E
4 R 3

 

Page 5
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(2) Electric field outside the sphere,
 
0
2
3
2
0
r3
R
r
r
4
Q
rE





where, Q is the total charge inside the sphere.
17. The information about the work done to take an electric charge from one point to the other in a
given electric field, obtained from the quantities called electric potential and electric potential
energy.
18.


B
A
drE
is the line-integral of electric field between point A and B and it shows the work done by
the electric field in taking a unit positive charge from A and B. Moreover, it does not depend on
the path and
0drE 

.
19. "The work required to be done against the electric field to bring a unit positive charge from
infinite distance to the given point in the electric field, is called the electric potential (V) at that
point".
Electric potential at point P is



P
p
drEV
It unit is
.volt
coulomb
Joule

Symbolically
C
J
V 
Its dimensional formula is
1321
ATLM

Absolute value of electric potential has no importance but only the change in it is important.
20. "The work required to be done against the electric field to bring a given change (q) from infinite
distance to the given point in the electric field is called the electric potential energy of that
electric charge at that point."
p
P
p
qVdrEqU 


The absolute value of electric potential energy has no importance, only the change in it is impor-
tant.
21. Electric potential at point P, lying at a distance r from a point charge q is
r
kq
V
p

22. The electric potential at a point at distance r from an electric dipole is

 
,
r
p
4
1
rv
2
0


( For r > > 2a)
Potential on its axis is
,
r
p
4
1
V
2
0


Potential on its equator is
0V 
23. Electric potential at a point
r
due to a system of point charge
n21
q,.........q,q
situated at position
at position
n21
r,.........r,r
is




n
1i
i
i
rr
kq
V
Page 6
7
The electric potential at point
r
, due to a continuous charge distribution is
 
0
4
1
rV








volume
i
rr
'dr
The electric potential due to a spherical shell is
 
0
1 q
V For r R
4 r

 
and
 
0
1 q
V For r R
4 R

 
24. A surface on which electric potential is equal at all points is called an equipotential surface. The
direction of electric field is normal to the equipotential surface.
25.
dV
E
dl


gives the magnitude of electric field in the direction of
dl

.
To find E from V, in general, we can use the equation
V V V
ˆ ˆ ˆ
E i j k
x y z
 
  
   
 
  
 

The direction of electric field is that in which the rate of decrease of electric potential with
distance
dV
dl

is maximum, and this direction is always normal to the equipotential surface.
26. The electrostatic potential energy of a system of point charges
n21
q.,.........q,q
situated at positions
n21
r,.......r,r
is
n
i j
i 1
ij
i j
kq q
U
r




where
ijij
rrr 
27. The electrostatic potential energy of an electric dipole in an external electric field E, is
U E p Ep cos


   
 
28. When a metallic conductor is placed in an external electric field,
(i) A stationary charge distribution is induced on the surface of the conductor.
(ii) The resultant electric field inside the conductor is zero.
(ii) The net electric charge inside the conductor is zero.
(iv) The electric field at every point on the outer surface of conductor is locally normal to the
surface.
(v) The electric potential inside the region of conductor is the same every where.
(vi) If there is a cavity in the conductor then, even when the conductor is placed in an external
electric field, the resultant electric field inside the conductor and also inside the cavity is
always zero.
This fact is called the electrostatic shielding.
Page 7
8
When electric charge is placed on the metallic conductor :
(i) The electric field inside the conductor is zero everywhere.
(ii) The charge is distributed only on the outer surface of the conductor.
(iii) The electric field on the surface is locally normal, and is equal to n
ˆ
E
0










 .
(iv) If a charge is placed inside the cavity in the conductor, the electric field in the conductor
remains zero.
29. "A device formed by two conductors seprated from each other is called a capacitor." Its capaci-
tance is

V
Q
C
constant. The unit of C is
volt
coulomb
which is also called farad.
F10pF.F10F1
126 

30. The effective capacitance in series connection is C then,
..........
C
1
C
1
C
1
C
1
321

The effective capacitance in parallel connection is C then,
.........CCCC
321

31. The capacitance of the parallel plate capacitor is
d
A
C
0


.
32. The energy stored in the capacitor is
2
VQ
2
CV
C2
Q
U
22

and the energy density = energy
stored per unit volume
,E
2
1
2
0

where
E
electric field.
33. When a dielectric is placed in an external electric field
0
E , polarisation of dielectric occurs due
to electrical induction. The electric field produced by these induced charges is in the opposite
direction to the direction of external electric field. Hence the resultant electric field E, inside the
dielectric is less than the external electric field
0
E .
The dipole moment produced per uint volume is called the intensity of polarisation or in short
polarisation
.n
ˆ
P
b

Since
EP 
, P E
0 e e
x x

  is called the electric susceptibility of the dielectric medium.


1


0 e
x
is called the permittivitty

of the dielectric medium.


0
is called the relative
permittivity of that medium and it is also called the dielectric constant K.
Page 8
9
i.e.
K



 
r
0
E
K 1 , E .
K
  
0
e
x
Thus in the dielectric the electric field reduces to the
th
K
part.
PED
0
 is called the electric displacement. Gauss Law in the presence of dielectric is
written as
qdsD 

, where q is only the net free charge.
34. when there is air (or vacuum) between the plates of a parallel plate capacitor, the capacitance is
d
A
C
0


. On placing a medium of dielectric constant K, the capacitance is
.CK'C 
Thus the
capacitance becomes K times, due to the presence of the dielectric.
35. With the help of Van-De-Graf generator a potential differance of a few nillion volt can be
established.
Page 9
10
CONCEPT MAP
Page 10

Electrostatics Notes and MCQs

Electrostatics Notes and MCQs

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