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PHYSICS
ELECTROSTASTICS
ELECTROSTATICS
ELECTRIC CHARGE
Charge is the property associated with matter due to which is produces and experiences
electrical and magnetic effects. The study of electrical effects of charge at rest is called
electrostatics.
The strength of particle’s electric interaction with objects around it depends on its
electric charge, which can be either positive or negative. An object with equal amounts
of two kinds of charge is electrically neutral, whereas one with an imbalance is
electrically charged.
In the table given below, if a body in the first column is rubbed against a body in the
second column, the body in first column will acquire positive charge, while that in the
second column will acquire negative charge.
TABLE I
Sl. No
1.
2.
3.
4.
5.
First Column
Glass Rod
Flannes or cat skin
Woollen cloth
Woollen cloth
Woollen cloth
Second Column
Silk Rod
Ebonite rod
Amber
Rubber shoes
Plastics objects
Electric Charge: Electric charge can be written as ne where n is a positive or negative
integer and e is a constant of nature called the elementary charge (approximately 1.60 x
10-19C). Electric charge is conserved, the (algebraic) net charge of any isolated system
cannot be changed.
Regarding charge following points are worth nothing:
(a)
Like charges repel each other and unlike charges attract each other.
(b)
Charge is a scalar and can be of two types; positive or negative.
(c)
Charge is quantized, i.e., the charge on anybody will be some integral multiple of
e, i.e.,
Q - ± ne. where n- 1, 2, 3……………
(d)
Charge on anybody can never be (1/3e), 1.5e etc.
The electrostatic unit of charge is stat-coulomb and electromagnetic unit is abcoulomb in CGS system. But in SI system the unit of charge is coulomb, I coulomb
=1/10ab-coulomb = 3x109 stat-coulomb.
NOTE: Recently, it has been discovered that elementary particles such as proton or
neutron are composed of quarks having charge (±1/3)e and (±2/3)e. However, as quarks
do not exist in Free State, the quantum of charge is still e.
Example-1: How many electrons are there in one coulomb of negative charge?
Sol: The negative charge is due to presence of excess electrons, since they carry
negative charge. Because an electron has a charge whose magnitude is e = 1.6x10 -19 C,
the number of electrons
N=q/e=1.0/1.6x10-19
N=6.25x1018
(5) Unit and dimensional formula
S.I. unit of charge is Ampere x sec = coulomb ©, smaller S.I. units are mC, µC.
C.G.S. unit of charge is Stat coulomb or e.s.u. Electromagnetic unit of charge is ab
coulomb
1C=3x109 stat coulomb = 1/10 ab coulomb.
Dimensional formula [Q]=[AT]
(6) Charge is
Transferable: It can be transferred from one body to
another
Associated with mass: Charge cannot exist without
mass but reverse is not true.
Conserved: It can neither be created nor be destroyed.
Invariant: Independent of velocity of charged particle.
→
→
(7) Electric charge produced electric field (E), magnetic field (B) and electromagnetic
radiations.
→
→
→
+ v = 0 + v = constant + v ≠ constant
→
E only
→
→
E and B
→→
E, B and radiates energy
(8) Point charge: A finite size body may behave like a point charge if it produces an
inverse square electric field. For example an isolated charged sphere behave like a point
charge at very large distance as well as very small distance close to it’s surface.
(9) Charge on a conductor: Charge given to a conductor always resides on it’s outer
surface. This is way a solid and hollow conducting sphere of same outer radius will hold
maximum equal charge. If surface is uniform the charge distributes uniformly on the
surface and for irregular surface the distribution of charge, i.e., Charge density is not
uniform. It is maximum where the radius of curvature is minimum and vice versa. i.e.,
(1/R). This is why charge leaks from sharp points.
(10) Charge Distribution: It may be of two types
(i)
Discrete distribution of charge: A System consisting of ultimate individual
charges
(ii)
Continuous distribution of charge: An amount of charge distribute uniformly or
non-uniformly on a body. It is of following three types
(a)
Line charge distribution: Charge on a line e.g. charged straight wire,
circular charged ring etc.
Charge = Linear charge density
Length
S.I. unit is C/M
Diamension is [L-1TA]
(b)
Surface charge distribution: Charge distributed on surface e.g. plane sheet
of charge, conducting sphere, conducting cylinder of
=Charge = Surface charge density
Area
S.I. Unit is C/m2
(c)
Volume charge density: Charge distributes through out the volume of the body
e.g. charge on a dielectric sphere
= charge= Volume charge density
Volume
S.I. Unit is C/m3
Method of Charging:
A body can be charged by following methods.
(1) By Friction: By rubbing two bodies together, both positive and negative charges in
equal amounts appear simultaneously due to transfer of electrons from one body to
the other.
(i)
When a glass rod is rubbed with silk, the rod becomes positively charged
while the silk becomes negatively charged. The decrease in the mass of glass
rod is equal to the total mass of electrons lost by it.
(ii)
Ebonite on rubbing with wool becomes negatively charged making the wool
positively charged.
(iii) Clouds also get charged by friction.
(iv) A comb moving through dry hair gets electrically charged. It starts attracting
small bits of paper.
(2)
By electrostatic induction: If a charged body is brought near an uncharged body,
one side of neutral body (closer to charged body) becomes oppositely charged while the
other side becomes similarly charged.
Induced charge can be lesser or equal to inducing charge (but never greater) and its
maximum value is given by Q’=-Q[1-1/K]
Where Q is the inducing charge and K is the dielectric constant of the material of the
uncharged body. It is also known as specific inductive capacity (SIC) of the medium, or
relative permittivity Er of the medium (relative means with respect to free space)
Different dielectric constants
Medium
Vaccum
Air
Paraffin Wax
Rubber
Transformer oil
Glass
K
1
1.0003
2.1
3
4.5
5-10
Medium
Mica
Silicon
Germanium
Glycerin
Water
Metal
K
6
12
16
50
80
(3) Charging by conduction: Take two conductors, one charged and other uncharged.
Bring the conductors in contact with each other. The charge (whether –ve or +ve) under
its own repulsion will spread over both the conductors. Thus the conductors will be
charged with the same sign. This is called as charging by conduction (through contact).
1 Coulomb = 3 X 109 esu of charge =
emu of charge
(1 Faraday = 96500 coulomb, 1 Amp Hr = 3600 coulombs)
The esu of charge is also called Static coloumb (stat. coul.) or frankline (Fr).
= emu of charge = 3X 1010 = C
esu of charge
Frankline (i.e. esu of charge) is the smallest unit of charge, while faraday is largest.
[Remember – Faraday is unit of capacity]
Properties of charge –
1. Like charges repel and opposite charges attract.
Ex. – A (+)ve charge sphere will attract –
Sol. – (i) (-) ve charged
(ii) Neutral
2. Charge is a scalar and can be of two types only viz positive and negative. This is
because it adds algebraically and represents excess or deficiency of electrons.
3. Charge is transferable – If a charged body is put in contact with an uncharged
body, uncharged body becomes charged due to transfer of electrons from one
body to the other. If the charged body is positive and it will withdraw some
electrons from uncharged body and if negative will transfer some of its excess
electrons to the uncharged body.
The process of charge transfer is called ‘Conduction’ and in conduction –
1. The charged body loses some of its charge (which is equal to the charge gained
by uncharged body).
2. The charges on both the books are similar if initially one is charged and other
uncharged.
3. The charge gained by uncharged body is always lesser than initial charge present
on charged body, i.e. whole of the charge cannot be transferred by conduction
from one body to the other. Actually, the flow of charge stops when both acquire
same potential.
Exception:Ex: Can ever the whole charge of a body be transferred to the other? If yes how and if
not why?
Ans: Yes, if the charged body is enclosed by a conducting body and connected to it, the
whole charge will be transferred to the conducting body, as charge resides on the outer
surface of a conductor.
4. Charge is invariant – This means that charge like phase is independent of frame
of reference, i.e. charge on a body does not charge whatever be its speed. While
charge density or mass of a body depends on its speed and increases with
increase in speed.
5. Charge is always associated with Mass I.e. charge cannot exist without mass
though mass can exist without charge, so:
a. The particles such as photon or neutrino which have no (rest) mass can
never have a charge (as charge cannot exist without mass).
b. As charge cannot exist without mass, the presence of charge is a
convincing proof of existence of mass.
c. In charging, the mass of a body charges.
6. Charge is conserved – In isolated system, total charge does not charge with time,
though individual charges may charge i.e. charge can neither be created nor
destroyed. It therefore, follows that simultaneously equal quantities of positive
and negative charge can appear or disappear. This is what actually happens in
pair production and anniweatron.
Conservation of charge is also found to hold good in all types of reactions either
chemical, nuclear or decay.
 In pair production and anniweatron neither mass nor energy is conserved
separately but (mass + energy) is conserved.
 In pair production, presence of nucleus is a must to conserve momentum. In
absence of nucleus, both energy and momentum will not be conserved
simultaneously and the process cannot take place.
7. Accelerated charge radiates energy: Electromagnetic theory has established that
a charged particle at rest produces only electric field in the space surrounding it.
However, if the charged particle is in unaccelerated motion it produces both
electric and magnetic fields but doesnot radiate energy. And if the motion of
charged particle is accelerated it not only produces electric and magnetic fields
but also radiates energy in the space surrounding the charge in the form of
electromagnetic waves.
V
E
V = Constant
E & B
But no radiation
V ≠ Constant
E, B and radiates energy
8. Similar charges repel each other while dissimilar attract.
The True test of electrification is repulsion and not attraction as attraction may
also take place between a charged and an uncharged body and also between two
similarly charged bodies.
Quest: Can two similarly charged bodies ever attract each other?
Ans: Yes, when the charge on one body (Q) is much greater than that on the
other (q) and they are close enough to each other so that force of attraction
between (Q) and induced charge on the other exceeds the force of repulsion
between (Q) and (q).
 If the charges are point, no induction will take place and hence, two similar
point charges can never attract each other.
9. Charge resides on the outer surface of a conductor because like charge repel and
try to get as far as possible from one another and stay at the farthest distance
from each other which is outer surface of the conductor. This is why a solid and a
hollow conductor sphere of same outer radius soap bubble expand on charging.
10. In case of conducting body no doubt charge resides on its outer surface, the
distribution of charge, i.e. charge density is not uniform. It is maximum where the
radius of curvature is minimum and vice-versa, i.e. σ α (1/R). This is why charge
leaks from sharp points.
Proof: a. As conductor is an equipotential surface, i.e. Vs = constant and incase of
spherical conductor
1 q
4rf 0 R
with q = 4rR 2 v
1 4rR 2 v =
So, o s =
Cons tan t
4rf 0
R
1
i.e. vR = Cons tan t or va
R
os =
b. Lighting rods are made up of conductors with one of their ends earthed while
the other sharp and protects a building rom lighting either by neutralizing or
conducting charge of the cloud to the ground.
11. A body can be charged by friction, induction or conduction.
In friction when two bodies are rubbed together, electrons are transferred from
one body to the other. As a result of this one body becomes positively charged
while the other negatively charged. E.g. when a glass rod is rubbed with silk, the
rod becomes positively charged while the silk negatively. However, ebonite on
rubbing with wool becomes negatively charged making the wool positively
charged. Clouds also become charged by friction. In charging by friction in
accordance with conservation of charge, both positive and negative charges in
equal amounts appear simultaneously due to transfer of electrons from one body
to the other.
In case of induction it is worth noting that –
1. Inducing body neither gains nor loses charge.
2. The nature of induced charge is always opposite to that of inducing charge is
always opposite to that of inducing charge.
3. Induced charge can be lesser or equal to inducing charge (but never greater)
and its maximum value is given by –
q’ = -q(1 – 1/K)
Where q is the inducing charge and K is the dielectric constant of the
material of the uncharged body.
4. For metals in electrostatics, k = 3, So, q’ = -q
i.e. in metals induced charge is equal and opposite to inducing charge.
5. Induction takes place only in bodies (either conducting or non-conducting)
and not in particles.
12. If a charged body is brought near a neutral body, the charged body will attract
opposite charge and repel similar charge present in the neutral body. As a result
of this one side of the neutral body becomes positive while the other negative
this process is called “Electrostatic Induction”.
13. Charge can be detected and measured with the help of gold leaf electroscope,
electrometer, voltammeter or ballistic – galvanometer. In case of gold leaf
electroscope –
a. If a charged body is brought near uncharged electroscope, charge on the disc
of electroscope will be opposite to that of body while leaves similar to that of
body and leaves while diverse.
b. If an uncharged electroscope is touched by a charged body, disc and leaves
both acquire charge similar to that of body and leaves will diverse.
c. If electroscope is charged by induction, disc and leaves both will acquire charge
opposite to that of inducting body and leaves will diverse. In fig © electroscope is
charged by induction using a positive charged body.
d. If a charged body is brought near a charged electroscope, the leaves will
further diverse if the charge on the body is similar to that on the electroscope
and will usually converge if opposite. This is how we determine the nature of
charge.
If the induction effect is strong enough leaves offer converging may again
diverse.
Ex: What is the difference between ‘charging by induction and charging by conduction’?
1. In induction the two bodies are ‘close to each other’, while in conduction touch
each other.
2. In induction charge on inducing body remains uncharged while in conduction
charge on charging body charges.
3. In induction induced charge is always opposite in nature to the ‘inducing charge’
while in conduction the charge on the two bodies is always of same nature.
4. In ‘induction’ induced charge can be equal in magnitude to inducing charge but in
conduction ‘charge transferred’ is usually lesser than initial charge present.
COULOMB’s LAW –
Coulomb found that force between two point charges at rest –
1. Varies directly as the magnitude of each charge, i.e. Fα q1 X q2
2. Varies inversely as the square of distance between them, i.e. Fα 1/r 2
3. Depends on the nature of medium between the charges.
4. Is always along the line joining the charges.
5. Is attractive if charges are unlike and repulsive if like.
Fα q1 X q2
α 1/r2
Fair =
1 q1 q2
4rf 0 r 2
Fmed =
q1 q2
1
4rf 0 K r 2
1 =
Km 2
9X109 2 (SI Unit)
4rf0
c
K = Constant = Characterizes the medium between the charges and is called
dielectric constant, specific inductive capacity (S.I.C) or relative permittivity and
for vaccum, free space or air its value is taken to be 1.
F12 =
q1 q2
r|
4rf0 Kr3 12
q1 q2
= 1
r 12
4rf r3
f = f0 K
f
K = = f r (relative permittivity or
f0
dielectric cons tan t)
r
q1 K q2
F12 = Force on q1 due to q2.
S
r12 = Unit vector directed to q1 from q2
F12 !
f0 = 8.85 X 10-12 F/m
 The equilibrium of a charged particle under the action of Colombian forces along
can never be stable. This statement is known as Earnshaw’s Theorem.
Ex: A copper atom consists of copper nucleus surrounded by 29 electrons. The atomic
weight of copper is 63.5g/mol. Let us now take two pieces of copper each weighing 10g.
Let us transfer one electron from one piece to another for every 1000 atoms in a piece.
What will be the Coulomb force between the two pieces after the transfer of electrons if
they are 2.10cm apart?
23
Sol.: No of atoms in 10gm of copper = 6X 10 X 10 = 9.45 X 1022
63.5
Total electron transferred = 1 X 9.45 X 1022 = 9.45 X 1019
1000
q = ne = 9.45 X 1019 X 1.6 X 10-19 = 15.12c
Treating each piece of copper as point charge, electric force between them from
coulomb’s law when they are 10 cm apart
F=
Ex.
Sol.
9 X 109 X (15.2) 2
= 2.08 X 1014 3
(10 X 10-2) 2
(a)
Two similar point charges q1 and q2 are placed at a distance r apart in
air. If a dielectric slab of thickness t and dielectric const. K is put
between the charges, calculate the coulomb force of repulsion.
(b)
If the thickness of the slab covers half the distance between the
charges, the coulombs force repulsive is reduced 4 : 9. Calculate the in
the ratio dielectric constant of the slab.
(a)
1 q1q2
1 q1q2

2
40 r '
40 Kr2
r'  r K
If there exists a slab of thickness t and dielectric constant K, the effective
air separation between the charges will be:
F=
(b)
q1q2
1
40 r  t   t K 2


F
4

F0
9
q1q2
r

 r  
2
1
40
4
1
=
9 40
r

K

2
q1q2
r2
2
K=4
PRINCIPLE OF SUPERPOSITION OF COULOMB'S LAW:
The resultant force on a test charge is a vector sum of forces due to individual
charges.
Fres  F1  F2  F3  F4 


Equilibrium of Charge Particle:
If the net force acting on the charge particle is zero that we say that, the charge
particle is in equilibrium.
Equilibrium of a charge
Case (Q1, Q2)
when Q1 and Q2 both are of similar nature
Let |Q1| < |Q2|
For q to be in equilibrium,
Fq = 0,
KqQ1
KQ2q

2
2
x
r  x
Q1
Q2

x
rx
x
Q1
Q1  Q2
r
nearer to charge of smaller magnitude.
Case 2:
Q1 and Q2
when Q1 and Q2 are of opposite nature.
KQ1q
x2
F1 =
KQ2q
F2 =
r  x
2
KQ1q
KQ2q

2
2
x
r  x
Q1
x2

Q2 r  x 2
Q1
Q2

x
rx
x Q2  r Q1  x Q1
x  r Q1
x=

r Q1
Q1  Q2

From charge of smaller magnitude.
Ex.
Two point charges +q and +4q are placed at a distance and apart. Find the
magnitude, sign and location of a third charge which makes the system in
equilibrium.
Force on charge q due to 4q (repulsive) (in the direction of BA).
In order to make A in equilibrium, a negative charge (let q 1) be placed between
A and B at a distance x from A.
For the equilibrium of A
1 q  4q
1 q  q1

40 l2
40 x2
q4x2 = l2 q1
Considering the equilibrium of C,
1 q1  4q
1 q  q1

2
40 l  x
40 x2
4x2 = (l – x)2
2x = + (l – x)
x=
l
3
 l
4q  
 3
 q1 
l2
Ex.
2

4q
9
A pith ball of mass 9 × 10–5 kg carries a charge of 5 µc. What must be the
magnitude and sign of the charge on a pith ball B held 2 cm directly above the
pith ball A, such that the pith ball A remain's stationary?
FAB = m1g
=
1 q1q2
40 AB2
92 = 7.84 µC.
Ex.
Three identical spheres each having a charge q and radius R, are kept in such
away that each touches the other two. Find the magnitude of the electric
force on any sphere due to other two.
FAB =
1
q2
BA
40 2R 2
FAC
1
q2

CA
40 2R 2
FA  3 FAB
1
3  9
=
 
40 4  R 
Ex.
2
Two equally charged identical metal spheres A and B repel each other with a
force 2 × 10–5 N. Another identical uncharged sphere C is touched to B and
then placed at the mid point between A and B. What is the net electric force
on C? [R 1981]
K  q2
F=
r2
= 2 × 10–5 N
when sphere C touches B, the charge on B, q will distribute equally on B and C
as spheres are identical conductors.
For conductors in contact
V1 = V2
q1 q2

r1
r2
r1  r2
 q1  q2
Also
q 1 + q2 = q
 q1  q2 
q
2
So sphere C will experience a force
FCA =
 q
Kq  
 2
r
 
2
2
 2F
along AB due to charge on A.
FCB =
 q  q
K   
 2  2
r
 
2
2
F
along BA due to charge on B.
So the net force on C due to charges on A and B
FC = FCA – FCB
= 2F – F = F
along AB
Ex.
Three identical spheres each having a charge q and radius R, are kept in such a
way that each touches the other two. Find the magnitude of the electric force
on any sphere due to other two.
Sol.
As for external points a charged sphere behaves as if the whole of its charge
where concentrated at its centre.
Force on A due to B
FAB =
=
1 qq
40 2R 2
1 q2
42 4R 2
along BA
Force on A due to C
FAC =
1 qq
40 2R 2
1 q2
=
40 4R 2
along CA
FAB = FAC = F
FA =
=
F2  F2  2FF cos 60
3F
1
3  q
FA =
 
40 4  R 
Ex.
2
Five point charges, each of value +q are placed on five vertices of a regular
hexagon of side LM. What is the magnitude of the force on a point charge of
value –q coulomb placed at the centre of a hexagon?
If there had been a sixth charge +q at the remaining vertex of hexagon force
due to all the six charges on –q at O will be zero (as the forces due to individual
charges will balance each other), i.e.,
FR  0
Now if f is the force due to sixth charge and F due to remaining five charges,
F F  0
i.e., F   f
or f = F
=
1 qq
40 L2
F=f
1  q
=
 
40  L 
Ex.
2
An α-particle passes rapidly, through the exact centre of a hydrogen molecule
moving on a line perpendicular to the inter-nuclear axis. The distance
-particle experience the
maximum force and what is it?
Sol.
FR = 2F cos Ө
along BA
1
2e2
F
40 x2  a2


x
cos Ө =
x
2
1
2 2
a

with a =
b
2
FR = 2 
1
2e2

40 x2  a2

 x
e2x
i.e., FR =

2
3
2 2
0 x  a
For FR to be max,
dFR
0
dx
x= 
a
2

2
x
1
2 2
a

b
Ans.
2 2
=
8e2
Fmax. =
3 3 0b2
Ex.
A point charge q is situated at a distance d from one end of a thin non
conducting rod of length L having a charge Q (uniformly distributed along its
length). Find the magnitude of electric force between the two.
dF =
1 qdQ
40 x2
dQ =
Q
dx
L
dF =
1 qQ
dx
40 Lx2
1 qQ
F=
40 L
d L

d
dx
x2
d L
1 qQ  1 
=

40 L  x  d
=
1 qQ  1
1 


40 L  d d  L 
F=
Ex.
1
qQ
40 d  d  L 
Two identical charged spheres are suspended by strings of equal length the
strings make an angle of 30° with each other. When suspended in a liquid of
density 0.8 gm/cc, the angle remains the same. What is the dielectric constant
of the liquid? (Density of the material of sphere is 1.6 gm/cc).
T cos Ө = mg
T sin Ө = F …(2)
….(1)
tan Ө =
F
mg
….(3)
When the balls are suspended in a liquid of density σ and dielectric constant K,
 1
times, i.e.,
K 
the electric force will become 

F
while
K 
F' = 

weight mg' = mg – th
= mg – vσ g


mg' = mg 1  


V=
m

tan Ө' =
=
F'
mg'
F


Kmg 1  


'  
K=
Ex.

1.6

2
   1.6  0.8
(a)
Two similar helium filled spherical balloons tied to a 5 gm weight with
strings and each carrying an electric charge q float in equilibrium as
shown in fig. Find the magnitude of q in eqn assuming that the charge
on each balloon acts as it were concentrated at its centre.
(b)
Find the volume of each balloon. Neglect the weight of the unfilled
balloons and assume that the density of air = 0.00129 gm/cc and
density of helium inside the balloon = 0.0002 gm/cc).
Equilibrium of weight
….(1)
Equilibrium of a balloon
F = T sin Ө
Th – mg = T cos Ө
i.e. Vσ g – Vƿ g = T cos Ө …(3)
F=
mg
tan 
2
and     Vg 
F=
q=
mg
…(4)
2
qq
in eqn. units
x2
x2 
and V =
mg
tan 
2
m
2    
= 1665 eqn of charge
V=
5
2 0.00129  0.0002
5  105
=
218
= 2294 cc
TRANSLATORY EQUILIBRIUM
When several forces act on a body simultaneously in such a way that the
resultant force on the body is zero, i.e.,
F0
With F  Fi
The body is said to be in translatory equilibrium.
1.
As if a vector is zero all its components must vanish, i.e. in equilibrium
as–
F  Fi  0
Fx  0,
Fy  0
and Fz  0
So in equilibrium forces along x-axis must balance each other and same
is true for other directions.
2.
As for a body
F0
means ma  0
or
dV
0
dt
V  const or zero
i.e., if a body is in translatory equilibrium it will be either at rest or in
uniform motion.
If it is at rest, the equilibrium is called static, otherwise dynamic.
3.
If the forces are conservative, then as for conservative force
dV 

 F   dr  and for equilibrium (F = 0)
So F = 
i.e.,
dV
0
dr
dV
0
dr
i.e., in conservative fields at equilibrium potential energy is optimum,
i.e., in equilibrium potential energy is maximum or minimum or
constant.
4.
Dynamic equilibrium types:
Types of dynamic equilibrium.
(a)
Stable equilibrium:
If on slight displacement from equilibrium position a body has
tendency to regain its original position, it is said to be in stable
equilibrium. In case of stable equilibrium potential energy is
 d2V

minimum  2   ve and so centre of gravity is lowest.
 dr

(b)
Unstable equilibrium:
If on slight displacement from equilibrium position body moves in
the direction of displacement, the equilibrium is said to be
unstable. In this situation potential energy of the body is
 d2V

maximum  2  negative and so centre of gravity is
 dr

highest.
Examples–
(c)
Neutral equilibrium:
If on slight displacement from equilibrium position a body has no
tendency to come back to original position or to move in the
direction of displacement, it is said to be in neutral equilibrium. In
this situation potential energy of the body is constant
 d2V


0
 dr2
 and so centre of gravity remains at constant height.
5.
In case of stable equilibrium lesser the potential energy or lower the
centre of gravity, i.e., greater the base area more stable is the
equilibrium.
6.
If we plot graphs between
F
U
and , at equilibrium F will be zero
r
r
while U will be optimum (max or min or constant). If
U = min
i.e.,
d2U
dr 2
= (+) ve, equilibrium is stable.
U = max
d2U
i.e.,
dr 2
= negative, equilibrium is unstable.
and U = constt.
d2U
i.e.,
= 0,
dr 2
Equilibrium is neutral.
ELECTRIC FIELD AND POTENTIAL
*
The space surrounding an electric charge q in which another charge q0
experiences a (electrostatic) force of attraction, or repulsion, is called the
electric field of the charge q.
*
q ԑ Source charge
Point charge a group of point charges continuous distribution of charges
q0 ԑ test charge must be vanishingly small so that it does not modify the
Electric field of the source charge.
Intensity (or strength) of Electric field–
E
F
q0
The intensity of electric field at a point in an electric field is the ratio of the
force acting on the test-charge placed at that point to the magnitude of the
test-charge.
If the intensity of electric field E at a point in an electric field be known, then
we can determine the force F acting on a charge q placed at that point by the
following eqn.
F  qE
E  MLT 3A 1 
gl
gl
 gl 
2
2

1
mv2
2
 v  2gl

v
l
=
2gl
l
=
2g
l

2g
Ans.
l
Example-6:
An inclined plane making an angle 30° with the horizontal is placed in a
kg and charge 0.01 C is allowed to slide down from rest from a height of 1 m.
If the coefficient of friction is 0.2, find the time it will take the particle to
reach the bottom.
Sol.
The different forces on the particle are shown in figure.
From,
N = mg cos 30° + qԑ cos 60°
Friction
f = µN
= µ mg cos 30° + µ ԑ cos 60°
Now the total force F acting along the inclined plane is
F = mg sin 30° – µN – qԑ cos 30°
or F = mg sin 30° – mg cos 30° – µqԑ cos 60° – qԑ cos 30°
Thus acceleration is
or a =
F
m
= g sin 30° – µg cos 30°

q
q
cos 60 
cos 30
m
m
or a = 9.8 × 0.5 – 0.2 × 9.8
 3  0.2  0.01  100
0.01  100
3
 

 0.5 


1
1
2
 2 
Now, distance travelled in time t is
s= 0
1 2
at
2
or t =
 2  2


a 
[As s =
1
 2]
sin 30
or =
 4 


2.237 
= 1.345 sec.
Example-7:

In space horizontal Electric field  E 

mg 
exist as shown in figure and a
q 
mass m attached at the end of a light rod. If mass m is released from the
position shown in figure find the angular velocity of the rod when it passes
through the bottom most position
Sol.
(A)
g
l
(B)
2g
l
(C)
3g
l
(D)
5g
l
According to work energy theorem:
w = ∆T
WE + Wg =
1
mv2  0 ….(1)
2
WE = qE l sinӨ ,
Wg = mg (l – l cos Ө)
– l cos Ө)
=
1
mv2 from eqn. (1)
2
mg l sin Ө + mg l – mg l cos Ө
=
1
mv2
2

mg 
QE 

q 

Electric Lines of Force:
Faraday gave a new approach for representation of electric field in the form of electric
lines of force. Electric lines of force are graphical representation of
electric field. “An electric line of force is an imaginary line or curve drawn
through a region of space so that its tangent at any point is in the
direction of the electric field vector at that point.”
This model of electric field has the following characteristics:
(i)
Electric lines of force are originated from positive charge and terminal into
negative charge.
(ii) The number of electric lines of force originates from a point charge q is q/ԑ0.
Electric lines of force may be fraction.
(iii) The number of lines per unit area that pass through a surface perpendicular to
the electric field lines is proportional to the strength of field in that region.
(iv)
No electric lines of force cross each other. If two electric lines of force cross
each other, it means electric field has two directions at the point of
cross. This is not physically possible.
(v)
Electric lines of force for two equal positive point charges are said to
have rotational symmetry about the axis joining the charges.
(vi)
Electric lines of force for point positive charge and a nearby negative
point charge that are equal in magnitude are said to have rotational
symmetry about an axis passing through both charges in the plane of the
page.
(vii)
Electric dipole Electric lines of force due to infinitely large sheet of
positive charge is normal to the sheet.
(viii)
No electrostatic lines of force are present inside a conductor. Also
electric lines of force are perpendicular to the surface of conductor. For
example if a conducting sphere is placed in a region where uniform
electric field is present, then induced charges are developed on the
sphere.
(ix)
If a charged particle is released from rest in region where only uniform
electric field is present, then charged particle move along an electric line
of force. But if charged particle has initial velocity, then the charged
particle may or may not follow the electric lines of force.
(x)
Electric lines of force inside the parallel plate capacitor is uniform. It
shows that field inside the parallel plate capacitor is uniform. But at the
edge of plates, electric lines of force are curved. It shows electric lines of
force at the edge of plates is non-uniform This is known as fringing
effect.
If the size of plates are infinitely large, then fringing effect can be
neglected.
(xi)
If a metallic plate is introduced between plates of a charged capacitor,
then electric lines of force can be discontinuous.
(xii)
If a dielectric plate is introduced between plates of a charged capacitor,
then number of lines of forces in dielectric is lesser than that in case of
vacuum space.
(xiii)
Electrostatics electric lines of force can never be closed loops, as a line
can never start and end on the same charge. Also if a line of force is a
closed curve, work done round a closed path will not be zero and electric
field will not remain conservative.
(xiv)
Lines of force have tendency to contract longitudinally like a stretched
elastic string producing attraction between opposite charges and repel
each other laterally resulting in, repulsion between similar charges and
edge-effect (curving of lines of force near the edges of a charged
conductor).
ELECTRIC POTENTIAL AND ELECTRIC POTENTIAL DIFFERENCE:
Electric Potential:
"Electric potential at any point in a electric field is equal to the ratio of the work
done in bringing a test charge from infinity to that point, to the value of test
charge."
Suppose, W be the work required in bringing a test charge q 0 from infinity to a
point b against the repulsive force F acting on it, then potential at the point b is
Vb 
Wb
q0
Since, W and q0 both are scalar quantities; the potential is also a scalar quantity.
Electric Potential Difference:
The potential difference between two points in an electric field is equal to the
ratio of work done in moving a test charge from one point to the other, to the
value of test charge. Suppose W work be done in bringing a small test charge q 0
from the point a to a point b against the repulsive force acting on it, then
potential difference between the points is
Vb  Va 
Wab
q0
Obviously, potential difference is also a scalar quantity.
IMPORTANT FEATURES
1.
Electric potential due to a point charge q:
From the definition of potential,
V
U
q0
1 qq0
.
40 r
=
q0
or V =
1 q
.
40 r
Here, r is the distance from the point charge q to be point at which the
potential is evaluated.
If q is positive, the potential that it produces is positive at all points; if q
is negative, it produces a potential that is negative everywhere. In either
case, V is equal to zero at r = 3 .
2.
Electric potential due to a system of charges:
Just as the electric field due to a collection of point charges is the vector
sum of the fields produced by each charge, the electric potential due to a
collection of point charges is the scalar sum of the potentials due to each
charge.
V
3.
1
40
qi
r
i
i
In the equation V 
1
40
qi
 r , if
i
the whole charge is at equal
i
distance r0 from the point where V is to be evaluated, then we can write,
V=
1 qnet
.
40 r0
Where qnet is the algebraic sum of all the charges of which the system is
made.
Example-11:
In a regular polygon of n sides, each corner is at a distance r from the centre.
Identical charges are placed at (n – 1) corners. At the centre, the intensity is E
and the potential is V. The ratio
V
has magnitude.
E
(A)
(C)
Sol.
E=
rn
n  1
r
(B)
r(n – 1)
(D)
r n  1
n
q
4 0 r2
and v 
n  1 q
4 0 r
n  1 q

4 0 r
v

q
E
4 0 r2
= (n – r)
TABLE : Electric Potential of Various Systems
S.No.
1.
2.
First Column
Isolated charge
A ring of charge
Second Column
V
q
40r
E
q
40
  2
R  x2  x


20
3.
A disc of charge
E
4.
Infinite sheet of charge
Not defined
5.
Infinitely long line of charge
Not defined
6.
Finite line of charge
V
q
R2  x2

sec   tan 
ln
40
sec   tan 
7.
Charged spherical shell
(a)
Inside 0 < r < R
V
(b)
Outside r > R
V
8.
Solid sphere of charge
(a)
q
40r
Inside 0 < r < R
E
(b)
q
40R
R 2
6 0

r2 
3


R 2 

Outside r > R
V
q
40r