A2 PHYSICS - Notes.pdf

1
Table of Contents
FURTHER MECHANICS................................................................................................... 2
ELECTRIC FIELDS ......................................................................................................... 20
MAGNETIC FIELDS........................................................................................................ 60
PARTICLE PHYSICS....................................................................................................... 85
NUCLEAR PHYSICS..................................................................................................... 105
THERMODYNAMICS ................................................................................................... 118
OSCILLATIONS ............................................................................................................ 138
ASTROPHYSICS........................................................................................................... 158

3
Momentum
Momentum is a vector quantity. The magnitude of momentum is equal to the product of
mass and velocity of an object. The direction of momentum is parallel to the direction of
velocity of the object.
𝑝 = 𝑚𝑣
The rate of change of motion is proportional to the unbalanced force, and this change takes
place along the direction of force.
𝐹 ∝
𝑚𝑣 − 𝑚𝑢
𝑡
𝐹 = 𝑘 ∙
𝑚𝑣 − 𝑚𝑢
𝑡
𝐹 = 𝑘 ∙
𝑚(𝑣 − 𝑢)
𝑡
As we know,
𝑣 − 𝑢
𝑡
= 𝑎
Therefore,
𝐹 = 𝑘𝑚𝑎
Here, k=1. Therefore,
𝐹 = 𝑚𝑎
1 unit of force is defined as the magnitude of force which causes an acceleration of 1m/s2
when it acts upon an object of mass 1kg.
Impulse
Impulse is a vector quantity. The magnitude of impulse is equal to the product of force and
its time of action (time of collision).
𝐹𝑡 = 𝑚𝑣 − 𝑚𝑢 = 𝑖𝑚𝑝𝑢𝑙𝑠𝑒

4
Momentum against Time Graphs
Figure 1a Figure 1b
In figure 1a, the gradient is constant which represents unbalanced force is constant. In
figure 1b, the initial gradient of graph is zero, which indicates that initial unbalanced force
on object is zero. The gradient of the graph gradually increases which indicates increasing
force.
𝐺𝑟𝑎𝑑𝑖𝑒𝑛𝑡 =
∆𝑦
∆𝑥
Therefore,
𝐺𝑟𝑎𝑑𝑖𝑒𝑛𝑡 =
∆𝑝
∆𝑡
𝐺𝑟𝑎𝑑𝑖𝑒𝑛𝑡 = 𝐹𝑜𝑟𝑐𝑒
Force-Time Graphs
Figure 1a Figure 1b

5
Figure 2a Figure 2b
In figure 1a, a constant force acts on an object. The total change in momentum of an object
can be determined by calculating the area of the shaded region. Figure 1b represents the
change in momentum between t1 and t2. In figure 2a, a large force acts on an object for a
small time period, and in figure 2b, a small force acts for a long time period. the areas under
both the graphs are almost equal, which represents equal change in momentum.
Conservation Law of Momentum
The total momentum of a system remains conserved during a collision or explosion,
provided that no external force acts on the system.
Conservation Law of Momentum in 2 Dimensions
Before Collision Collision After Collision
𝑝!!
= 𝑝!"
𝑚"𝑣"𝑐𝑜𝑠𝜃" + 𝑚#𝑣#𝑐𝑜𝑠𝜃# = 𝑚"𝑣"𝑐𝑜𝑠𝛼" + 𝑚#𝑣#𝑐𝑜𝑠𝛼#
𝑝$!
= 𝑝$"
𝑚"𝑣"𝑠𝑖𝑛𝜃" − 𝑚#𝑣#𝑠𝑖𝑛𝜃# = 𝑚#𝑣#𝑠𝑖𝑛𝛼# − 𝑚"𝑣"𝑠𝑖𝑛𝛼"

6
𝑝% = 𝑝&
Relationship Between Kinetic Energy and Momentum
𝐸' =
"
#
𝑚𝑣#
𝑝 = 𝑚𝑣
𝑣 =
𝑝
𝑚
Therefore,
𝐸' =
"
#
𝑚 A
𝑝
𝑚
B
#
𝐸' =
"
#
× 𝑚 ×
𝑝#
𝑚#
𝑝 = D2𝑚𝐸'
From DeBroglie’s Equation,
𝜆 =
ℎ
𝑝
𝜆 =
ℎ
D2𝑚𝐸'
A high speed beam of particles is used to determine the internal structure of small particles
like protons and neutrons. Due to very large amount of energy, these particles have very
small wavelengths. If this wavelength is comparable to the target particle, the diffraction
pattern can be used to determine the internal structure.

7
Explosion
In case of explosion, large particles split into two or more smaller particles.
𝑚 = 𝑚" + 𝑚#
Force on A by B = 𝐹
Force on B by A = 𝑅
According to Newton’s second law,
𝐹 =
𝑑𝑝(
𝑑𝑡
𝑅 =
𝑑𝑝)
𝑑𝑡
According to Newton’s third law,
𝐹 = −𝑅
𝑑𝑝(
𝑑𝑡
= −
𝑑𝑝)
𝑑𝑡
𝑑𝑝(
𝑑𝑡
+
𝑑𝑝)
𝑑𝑡
= 0
𝑑
𝑑𝑡
(𝑝( + 𝑝)) = 0
𝑑
𝑑𝑡
(𝑡𝑜𝑡𝑎𝑙 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚) = 0
If an object splits from rest, total momentum before explosion was zero. According to
conservation law, total momentum after collision is also zero. If it splits in two parts, they
will move in opposite directions.
𝑝( = −𝑝)
J2𝑚(𝐸'#
= −J2𝑚)𝐸'$
2𝑚(𝐸'#
= 2𝑚)𝐸'$
𝐸'#
𝐸'$
=
𝑚)
𝑚(
In case of explosions, particles with larger mass gain smaller kinetic energy.

8
Conservation Law of Momentum in 2 Dimensional Explosion
Before Explosion AfterExplosion
𝑚 = 𝑚" + 𝑚# + 𝑚*
𝑝!!
= 𝑝!"
𝑚𝑢 = 𝑚#𝑣#𝑐𝑜𝑠𝜃# + 𝑚"𝑣"𝑐𝑜𝑠𝜃" − 𝑚*𝑣*
𝑝$!
= 𝑝$"
0 = 𝑚#𝑣#𝑠𝑖𝑛𝜃# − 𝑚"𝑣"𝑠𝑖𝑛𝜃"

9
Elastic and Inelastic Collisions
If the total kinetic energy of a system decreases during a collision, it is called an inelastic
collision, and if it remains same, it is called elastic collision. In real life collisions, kinetic
energy is converted to heat, sound and elastic strain energy.
𝐸'!
=
"
#
𝑚"𝑢"
#
+
"
#
𝑚#𝑢#
#
𝐸'"
=
"
#
𝑚"𝑣"
#
+
"
#
𝑚#𝑣#
#
Therefore,
"
#
𝑚"𝑢"
#
+
"
#
𝑚#𝑢#
#
=
"
#
𝑚"𝑣"
#
+
"
#
𝑚#𝑣#
#
𝑚"𝑢"
#
+ 𝑚#𝑢#
#
= 𝑚"𝑣"
#
+ 𝑚#𝑣#
#
The law of conservation of momentum is followed by both collisions, provided that no
external force is applied to it. In the cases below, both the objects in the experiments are
equally massive.
Collision 1:
𝑚𝑢 + 0 = (𝑚 + 𝑚)𝑣
𝑚𝑢 = 2𝑚𝑣
𝑣 =
𝑢
2
𝐸'!
=
"
#
𝑚𝑢#
𝐸'"
=
"
#
× (2𝑚) × A
𝑢
2
B
#
=
"
+
𝑚𝑢#
Therefore,
𝐸'!
= 2𝐸'"

10
Collision 2:
"
#
𝑚𝑢#
+ 0 =
"
#
𝑚𝑣"
#
+
"
#
𝑚𝑣#
#
"
#
𝑚𝑢#
=
"
#
𝑚(𝑣"
#
+ 𝑣#
#)
𝑢#
= 𝑣"
#
+ 𝑣#
#

11
Circular Motion
If a motion of a particle is such that its distance from a fixed point remains constant with
time, this motion is called circular motion.
Properties of circular motion:
• It has constant speed
• Velocity changes
• Constant distance from arc to centre
• Acceleration towards the centre of the circle
• Centripetal force towards the centre of the circle
Angular Displacement
The figure above shows a particle moving in a circular path of radius rm. It moves from point
A to point B along the circular path. Distance travelled by the particle is,
𝑠 = 𝑎𝑟𝑐 𝑜𝑓 𝐴𝐵
The angle produced by the arc at the circle’s centre (centre of the circular path) is called the
angular displacement. The unit of angular displacement is radians.
𝑠 = 𝑟𝜃
𝜃 =
𝑠
𝑟
For complete circle,
𝜃 = 2𝜋
Therefore,
𝑠 = 2𝜋𝑟

12
Angular Velocity
Angular displacement per unit time is called angular velocity.
𝐴𝑛𝑔𝑢𝑙𝑎𝑟 𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦 =
𝐴𝑛𝑔𝑢𝑙𝑎𝑟 𝐷𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡
𝑇𝑖𝑚𝑒
𝜔 =
𝑠
𝑡
For complete rotation,
𝜔 =
2𝜋
𝑡
Therefore,
𝜔 = 2𝜋𝑓
Where, f is the frequency of rotation.
Relationship Between Angular Velocity and Linear Speed
𝑣 =
𝑑
𝑡
𝑣 =
𝑠
𝑡
𝑣 =
𝑟𝜃
𝑡
∴ 𝑣 = 𝜔𝑟

13
Rotation per Minute (RPM)
This is used as a unit of angular velocity. It represents the number of complete rotation
within one minute.
Centripetal Acceleration
If a particle moves in a constant speed in a circular path or constant angular velocity, its
motion is called uniform circular motion. In the figure above, the particle is moving in a
circular path with constant speed. At any moment, the velocity is parallel to the tangent of
the curved path.
𝑣, , 𝑣- and 𝑣. represents velocity at three points. In case of uniform circular motion, all
these vectors have the same length, which indicates same speed. But there is change in
velocity due to the change in direction. Rate of change of velocity is called acceleration. In a
circular path, particles are always accelerating even though the speed remains constant.
This acceleration is called centripetal acceleration.

14
According to this vector triangle, change in velocity takes place towards the centre of the
circular path. Thus, constant acceleration is directed towards the centre. Magnitude of
centripetal acceleration can be found using the equation,
𝑎. =
𝑣#
𝑟
𝑎. =
𝜔#
𝑟#
𝑟
𝑎. = 𝜔#
𝑟
We can also say,
𝑎. = (2𝜋𝑓)#
𝑟
𝑎. = 4𝜋#
𝑓#
𝑟
Centripetal Force
In a circular path, an object always accelerates towards its centre. According to Newton’s
second law, an unbalanced force is needed for the acceleration. This force acts along the
direction of acceleration. Thus, an unbalanced force is needed to keep the object moving in
a circular path. This force is called centripetal force.
The magnitude of the centripetal force can be found using the equation,
𝐹 = 𝑚𝑎
Therefore,
𝐹. =
𝑚𝑣#
𝑟
𝐹. = 𝑚𝜔#
𝑟
Centripetal force is not a particular type of force. At different conditions, it is provided by
different sources. Actually, unbalanced force towards the centre provides the unbalanced
force.
Velocity or displacement in a circular path is parallel to the tangent of the circular path.

15
Change in apparent weight due to Rotational Motion
An object of mass mkg is placed on a point P, where radius of the earth is rm. Two forces act
on the object. They are gravitational force and the normal contact force.
Since the object is moving in a circular path, there must be an unbalanced force on the
object towards the centre, which providesthe necessary centripetal force. According to the
free body force diagram, unbalanced force towards the centre is given by the equation,
𝐹 = 𝑚𝑔 − 𝑅
𝐹. = 𝑚𝜔#
𝑟
Therefore,
𝑚𝑔 − 𝑅 = 𝑚𝜔#
𝑟
𝑅 = 𝑚𝑔 − 𝑚𝜔#
𝑟
𝑅 = 𝑚(𝑔 − 𝜔#
𝑟)
According to this equation, apparent weight, which is equal to the normal reaction force, is
less than the actual weight of the object.
At P, the normal reaction force is,
𝑅 = 𝑚(𝑔 − 𝜔#
𝑟)

16
If the object moves towards the pole, apparent weight of the object increases, due to the
decreasing radius.
𝑅 = 𝑚(𝑔 − 𝜔#
𝑟𝑐𝑜𝑠𝜃)
If θ = 90o
,
𝑅 = 𝑚(𝑔 − 𝜔#
𝑟𝑐𝑜𝑠90)
𝑅 = 𝑚𝑔
Motion in a Vertical Circular Path
At A,
𝑇 − 𝑚𝑔 =
𝑚𝑣#
𝑟
𝑇 =
𝑚𝑣#
𝑟
+ 𝑚𝑔
At B,
𝑇 =
𝑚𝑣#
𝑟
At C,
𝑇 + 𝑚𝑔 =
𝑚𝑣#
𝑟
𝑇 =
𝑚𝑣#
𝑟
− 𝑚𝑔

17
Speed Breaker
A car is moving over a speed breaker at a height of rm. According to its free body force
diagram,
𝑊 − 𝑅 =
𝑚𝑣#
𝑟
𝑅 = 𝑊 −
𝑚𝑣#
𝑟
𝑅 = 𝑚𝑔 −
𝑚𝑣#
𝑟
If the speed of the car is increased, normal reaction force decreases. If the car is at rest, v is
zero. So, the normal reaction force is equal to weight. The magnitude of centripetal force is
large when R is smallest or zero. At this condition,
𝑚𝑔 −
𝑚𝑣#
𝑟
= 0
𝑚𝑔 =
𝑚𝑣#
𝑟
𝑣 = D𝑟𝑔
If the speed of the car exceeds this critical speed, it will take off and move along the tangent
of the curved path.
The car takes off if,
𝑣 > D𝑟𝑔

18
Satellites
Satellites are moving in a circular path around planets. Due to the change in direction of
motion, satellites are always accelerating towards the centre of the circular path. For this
acceleration, centripetal force is needed, which is provided by the gravitational force.
𝐹/ = 𝐺 ∙
𝑚0𝑚1
𝑟#
𝐹. = 𝑚1𝜔#
𝑟
Therefore,
𝐺 ∙
𝑚0𝑚1
𝑟#
= 𝑚1𝜔#
𝑟
𝐺 × 𝑚0 = Y
2𝜋
𝑡
Z
#
× 𝑟*
𝑡#
=
4𝜋#
𝐺𝑚0
∙ 𝑟*
Where,
+2%
/3&
is a constant.
If this time period is equal to the rotational time period of a planet, the satellite remains
stationary with respect to a point on the surface of the planet. Such satellites are called
geostationary satellites.

19
Experiment to Determine the Relationship between
Centripetal Force and Speed of an Object
Apparatus:
A rubber stopper, a few loads (of different masses), metre ruler, stopwatch, marker, glass
tube.
Procedure:
The stopper is attached to one end of the string, which passes through the glass tube.
Another end of the string is attached to a known mass. When the stopper moves in a
circular path of circular radius,
𝐶𝑒𝑛𝑡𝑟𝑖𝑝𝑒𝑡𝑎𝑙 𝐹𝑜𝑟𝑐𝑒 = 𝑊𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑆𝑢𝑠𝑝𝑒𝑛𝑑𝑒𝑑 𝑂𝑏𝑗𝑒𝑐𝑡
𝑇 = 𝐹. =
𝑚𝑣#
𝑟
Also,
𝑇 = 𝑚𝑔
Therefore,
𝑚𝑔 =
𝑚𝑣#
𝑟
𝑣#
= 𝑟𝑔
The speed of the stopper is gradually increased, until it reaches a particular radius. When it
is in equilibrium state, the total time for a particular number of rotations is measured using
the stopwatch. It is used to calculate average time period. The mass of the freely suspended
load is gradually increased. For each load, time period is calculated.

21
Electric Fields
Electric charge is one of the fundamental properties of all particles. A particle can be
positively charged or negatively charged. Some particles can also be neutral.
Electric field is defined as the space where the charged particles experience a force. Electric
field strength of a point inside the field is defined as force per unit charge.
𝐸𝑙𝑒𝑐𝑡𝑟𝑖𝑐 𝐹𝑖𝑒𝑙𝑑 𝑆𝑡𝑟𝑒𝑛𝑔𝑡ℎ =
𝐹𝑜𝑟𝑐𝑒
𝐶ℎ𝑎𝑟𝑔𝑒
𝐸 =
𝐹
𝑄
Unit of electric field strength = Nc-1
Electric field is a vector quantity. The direction of field strength is the direction of a force on
a positively charged particle on an electric field. A negatively charged particle experiences
force on the opposite direction of the electric field.
According to Newton’s Second Law, an unbalanced force causes acceleration.
𝐹 = 𝑚𝑎
𝐹 = 𝑄𝐸
Therefore,
𝑚𝑎 = 𝑄𝐸
𝑎 =
𝑄𝐸
𝑚
𝑎 represents acceleration of a charged particle in an electric field if the field strength is 𝐸.

22
Electric Field Lines
These are imaginary lines used to represent the shape and relative strength of an electric
field. These lines can be straight or curved. These lines represent the direction of force on a
positive charge from an isolated charge. For an isolated positive charge, electric field is
directed outwards, and for a negative charge, it is directed inwards. In case of a combination
of charges, electric field lines are started from positive charge to negative charge. If the field
lines are closer to each other, it represents stronger electric field. Uniform electric field is
defined as the space remains unchanged. In this case, field lines are parallel to each other,
and have constant separation.
The particle “x” is at rest between the parallel plates. Mass of x is 12.6x10-3
g. Charge of x is
62x10-6
c. To balance the downward weight of the object, there must be an upward force
which is provided by the electric field. To balance the downward weight for this particular
object there must be an upward force on x.
𝐹 = 𝑄𝐸
𝐹 = 𝑚𝑔
Therefore,
𝑄𝐸 = 𝑚𝑔
𝐸 =
𝑚𝑔
𝑄

23
Potential Difference
The potential difference between two points is defined as the amount of work done per unit
charge, to move it from one point to another. The work done to move Q charge from A to B
is W J. Thus, the potential difference between these two points is a scalar quantity and its
unit is volts (V).
In an electric field, amount of work done to move a charged particle from one point to
another does not depend on its path of motion. It only depends on the potential difference
of the initial and final path (point).
Work Done, 𝑊 = 𝑄𝑉
Electronvolt is another unit of energy. It is used to express a very small amount of energy or
work done. It is defined as the amount of work done to transfer an electron with a potential
difference of 1V.
1eV = 1.6x10-19
J

24
Relationship between Potential Difference and Electric Field
Strength
Electric potential at A is 𝑉( and at B is 𝑉).
Thus, the potential difference,
𝑉 = 𝑉( − 𝑉)
Distance of AB = 𝑑
Amount of work done for 𝑄 charge to move from A to B is 𝑊 = 𝑄𝑉.
If electric field strength is 𝐸,
𝐹 = 𝑄𝐸
𝑊 = 𝐹𝑑
Therefore,
𝑊 = 𝑄𝐸𝑑
Again,
𝑊 = 𝑄𝑉
Therefore,
𝑄𝑉 = 𝑄𝐸𝑑
𝑉 = 𝐸𝑑

25
Relationship between Potential Difference and Kinetic
Energy
In this figure, two vertical parallel plates are used to produce a horizontal uniform electric
field. This two plates are connected to a DC source, of potential difference V volts, where A
has a higher potential and B has a lower potential. A positively charged particle, x, is placed
close to A. it experiences force along the direction of the electric field lines. According to
Newton’s second law, this force causes acceleration, and its kinetic energy increases.
𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝐾𝑖𝑛𝑒𝑡𝑖𝑐 𝐸𝑛𝑒𝑟𝑔𝑦 = 𝑊𝑜𝑟𝑘 𝐷𝑜𝑛𝑒
"
#
𝑚𝑣#
− 0 = 𝑄𝑉
𝑣#
=
2𝑄𝑉
𝑚
𝑣 = b
2𝑄𝑉
𝑚
From relationship between kinetic energy and momentum,
𝑝 = D2𝑚𝐸'
𝐸' =
𝑝#
2𝑚
𝐸' = 𝑄𝑉
Therefore,
𝑝#
2𝑚
= 𝑄𝑉
𝑝 = D2𝑚𝑄𝑉

26
From de Broglie Equation,
𝜆 =
ℎ
𝑝
𝜆 =
ℎ
D2𝑚𝑄𝑉
Electron Gun
An electron gun is a device which is used to produce a beam of high speed electrons. The
filament is connected across a high voltage source. When current flows through the
filament, electrical energy is converted to thermal energy. By using this energy, bond of
electron is broken. The produced electron has no kinetic energy. To accelerate this electron,
an electric field is produced by using two parallel plates, and accelerating potential
difference is applied across the plates. Due to this voltage, speed of electrons increases, and
they gain higher kinetic energy. Velocity of the produced electron can be found from the
formula,
"
#
𝑚𝑣#
= 𝑄𝑉
𝑣 = b
2𝑄𝑉
𝑚

27
The accelerating potential difference of the electron gun is VA. Thus, the speed of the
electron produced by the gun is ux. A potential difference is produced across the horizontal
plates, x and y. Thus, a vertical electric field is produced by these two plates. As the plate x
has higher potential, the electric field is directed vertically downwards. The beam of
electron enters the vertical electric field along the horizontal direction. At initial moment,
horizontal velocity,
𝑢! = b
2𝑒𝑉(
𝑚
𝑢$ = 0
As the electron enters horizontally, the vertical component of its velocity is zero. If there is
no vertical electric field, electrons move in a horizontal path, which is represented by the
dotted line. In presence of an electric field, electrons deflect in upward direction. Point P
represents the final point of electrons inside the electric field. Beyond this point, electrons
move in a straight path, following Newton’s first law of motion. The deflecting potential
difference between the horizontal plates, x and y, is 𝑉4.
𝑢! = b
2𝑒𝑉(
𝑚
𝑢$ = 0
Thus, the electric field strength,
𝐸 =
𝑉4
𝑑
𝐹 = 𝑄𝐸
Therefore,
𝐹 =
𝑄𝑉4
𝑑

28
According to Newton’s second law of motion,
𝐹 = 𝑚𝑎
Therefore,
𝑚𝑎 =
𝑄𝑉4
𝑑
Hence, in this case,
𝑎$ =
𝑄𝑉4
𝑚𝑑
Since a parabolic motion is taking place,
𝑢! = 𝑣! = b
2𝑒𝑉(
𝑚
𝑢! =
𝑠
𝑡
𝑡 =
𝑠
𝑢!
Also,
𝑠 = 𝑢𝑡 +
"
#
𝑎𝑡#
If the displacement is ℎ, therefore,
ℎ = 𝑢$𝑡 +
"
#
𝑎$𝑡#
ℎ = 0 +
"
#
×
𝑒𝑉4
𝑚𝑑
× 𝑡#
ℎ =
"
#
×
𝑒𝑉4
𝑚𝑑
× Y
𝑠
𝑢!
Z
#
ℎ =
"
#
×
𝑒𝑉4
𝑚𝑑
×
𝑠#
2𝑒𝑉(
𝑚
ℎ =
𝑉4𝑠#
4𝑑𝑉(
At point P, the final vertical velocity,
𝑣 = 𝑢 + 𝑎𝑡
𝑣$ = 𝑢$ + 𝑎$𝑡

29
𝑣$ = 0 +
𝑒𝑉4
𝑚𝑑
×
𝑠
𝑢!
𝑣$ =
𝑒𝑉4𝑠
𝑚𝑑𝑢!
Resistant tangential velocity,
𝑣 = J𝑣!
# + 𝑣$
#
tan 𝜃 =
𝑣$
𝑣!
tan 𝜃 =
𝑒𝑉4𝑠
𝑚𝑑𝑢!
÷ 𝑢!
tan 𝜃 =
𝑒𝑉4𝑠
𝑚𝑑𝑢!
#
Equipotential
These are imaginary lines or surface in an electric field, where all the points have the electric
potential. The metal plates x and y produces uniform horizontal electric field lines. If their
potential difference is V volts, A, B and C represents positions of equipotentials. The amount
of work done to move charged particle from one form to another equipotential does not
depend on the distance. It depends on the charge of the equipotentials.
𝐸 =
𝑉
𝑑
𝑉 = 𝐸𝑑

30
The potential difference between A and B,
𝑉" = 𝐸𝑑"
The potential difference between B and C,
𝑉# = 𝐸𝑑#
Therefore,
𝑉"
𝑉#
=
𝐸𝑑"
𝐸𝑑#
𝑉"
𝑉#
=
𝑑"
𝑑#
In a uniform electric field, potential difference between the equipotentials remains constant
if they have constant separation or distance.

31
Coulomb’s Law
When two charged particles are close to each other, they interact with each other by
electrostatic force. The magnitude of this force is calculated by using Coulomb’s law.
Coulomb’s Law states that the magnitude of electrostatic force between particles is directly
proportional to the product of their charges and inversely proportional to their distance
squared.
Charge of A = 𝑄"
Charge of B = 𝑄#
Therefore,
𝐹5 =
𝑘𝑄"𝑄#
𝑑#
Here,
𝑘 =
1
4𝜋𝜀
Where, ε is the permittivity of the medium.
k is considered as 8.99x109
Nm2
c-2
for our purposes.
Electric Field Strength
Electric field strength is defined as the force acting per unit charge.
𝐹 =
𝑘𝑄𝑄"
𝑑#
𝐸 =
𝐹
𝑄"
Therefore,
𝐸 =
𝑘𝑄𝑄"
𝑑#
×
1
𝑄"
𝐸 =
𝑘𝑄
𝑑#
The 𝑄 charge produces electric field around it. P is a point at 𝑑 distance from 𝑄 charge. A
test charge 𝑄" is placed at point P. The electric field strength at the point P can be calculated
by the equation,

32
𝐸 =
𝑘𝑄
𝑑#
Therefore,
𝐸 ∝
1
𝑑#
Electric field strength against
"
6%
graph is a straight line passing through the origin. It
represents inverse square law between the field strength and distance.
Electric Field Strength of a Hollow Spherical Object
In case of a sphere, or a spherical shaped conductor, all the charges are distributed evenly
over the surface. This electric fields cancel each other inside the sphere. Thus, the resultant
force inside the sphere is zero. Outside the sphere, the electric field follows the inverse
square law, in such a way, that the charge is concentrated inside the sphere (centre of the
sphere).

33
Resultant Electric Field Strength
𝑟" > 𝑟#
Two charged particles are placed dm away from each other. Since they have same polarity,
their electric fields are directed in opposite directions in the same space between them. At a
certain point, these two electric fields have same magnitude. Since their direction is
opposite, resultant field strength at the point is zero. It is called neutral or null point. If P
represents the null point between 𝑄" and 𝑄#, then at P, we can say,
𝐸" = 𝐸#
𝑘𝑄"
𝑟"
#
=
𝑘𝑄#
𝑟#
#
𝑄"
𝑄#
= Y
𝑟"
𝑟#
Z
#
Electric Field Strength for Non-Identical Charges
𝑄" > 𝑄#

34
From 𝑄" to point P, electric field strength is directed towards right, because the electric field
strength 𝐸# is greater than 𝐸". P is not the centre because magnitude of 𝑄" is greater than
𝑄#.
|𝑄"| > |𝑄#|
If two oppositely charged particles are placed, resultant field strength between charged
particles become large, and it starts to diminish as it leaves the charged particle. In this case,
neutral point can be detected at a place outside, and not between the charged particles.
Distance of this neutral point will be greater from the larger charge.
Experiment to Determine Electrostatic Force between Two
Charged Particles
Figure 1 Figure 2
In figure 1, a charged object, A, is placed on an electronic balance, by using a non-
conductive stand. The mass of the object is recorded. In figure 2, another charged object, B,
is placed above A, by a non-conducting support. If they have the same polarity, object A

35
experiences a downward force. Due to this downward force, reading on the electronic
balance increases. If they have opposite polarity, upward force acts on A. Thus, reading on
the electronic balance decreases. Therefore, from the difference between the two readings,
magnitude of electrostatic force can be found, using the equation,
𝐹 = ∆𝑚𝑔
𝑇 sin 𝜃 = 𝐹 𝑇 cos 𝜃 = 𝐹
𝐹 =
𝑘𝑄𝑄
𝑑#
Therefore,
𝑇 sin 𝜃 =
𝑘𝑄#
𝑑#
Now,
𝑇 sin 𝜃
𝑇 cos 𝜃
=
𝑘𝑄#
𝑑#
×
1
𝑚𝑔
tan 𝜃 =
𝑘𝑄#
𝑑#𝑚𝑔
𝑄 = b
𝑑#𝑚𝑔 tan 𝜃
𝑘

36
Capacitor
In the circuit diagram above, x and y are two parallel metal plates connected to a DC source,
of an EMF of VD. The space between the metal plates, x and y, are occupied by non-
conductive di-electric material. When the switch is turned on, current should not flow
through the circuit, due to the broken path at x and y. but in practical, a decrease in current
can be observed for a small period of time.
As the metal plate x is connected to the positive terminal of the cell, electrons move from x
to the positive terminal. Thus the metal plate x becomes positively charged. Metal plate y is
connected to the negative terminal. Due to electrostatic repulsion, electrons move from
negative terminal of the cell to y. Thus, the plate y becomes negatively charged. Due to the
opposite polarity, a potential difference is produced across the parallel plates. If the
potential difference between x and y is VC, and across the resistor is VR, and according to
Kirchoff’s law,
𝑉4 = 𝑉7 + 𝑉8
𝑉8 = 𝑉4 − 𝑉7
We know that,
𝑉 = 𝐼𝑅
Therefore,
𝐼8 =
𝑉4 − 𝑉7
𝑅
When 𝑡 = 0, charge on the parallel plates is zero. Thus, there is no potential difference
across the capacitor, and hence, the potential difference across the resistor is largest, and
maximum current flows through the circuit.
𝐼3,! =
𝑉4
𝑅
As current flows through the circuit, potential difference between the metal plates x and y
gradually increases, and the current through the circuit decreases. When the parallel plates
store sufficient charge, their potential difference becomes equal to the EMF of the cell. The

37
potential difference across the resistor drops to zero. According to Ohm’s law, current
through the circuit becomes zero. At this condition, parallel plates have largest possible
charge.
If the plates are connected across an electric appliance, it can provide energy. Thus the
arrangement can store electric potential energy, by creating an electric field between the
plates. This arrangement is called the capacitor.
To transfer more charge into the capacitor, its potential difference must be increased.
Charge of the capacitor is proportional to the potential difference between x and y.
𝑄 ∝ 𝑉
𝑄 = 𝐶𝑉
The proportionality constant, 𝐶, is called the capacitance. Capacitance is defined as the
amount of charge stored by a capacitor when the potential across its two plates is 1 volt.
Unit = c/V or Farad (F)
In practice, a Farad is a very large unit. For real life appliances, milliFarad and microFarad is
used.

38
Charge against Voltage Graphs
The equation, 𝑄 = 𝐶𝑉, represents linear relationship between potential difference and
charge. Thus, the graph is a straight line through the origin. In this case, the applied
potential difference is considered which is varied using a variable resistor. Thus, the
potential difference is an independent variable, and is plotted across the x-axis. The
dependent variable is charge, and is plotted across the y-axis. The gradient of this graph
gives capacitance. The potential difference across the capacitor depends on the amount of
charge of the parallel plates.
Capacitors can come in many types, for example, the parallel plates can be turned to a
cylinder, to make large surface area, while keeping the capacitor compact. Such capacitors
are known as cylindrical capacitors.
The capacitance of a capacitor depends on:
1. Area of parallel plates
2. Distance between the plates.
3. Permittivity of the di-electric material

39
𝐶 =
𝜀𝐴
𝑑
ε = Permittivity
A = Surface Area
d = Distance
According to Work-Energy Theorem, work done is equal to energy transferred.
𝑊 =
"
#
𝑄𝑉
𝑄 = 𝐶𝑉
Therefore,
𝑊 =
"
#
× 𝐶𝑉 × 𝑉
𝑊 =
"
#
𝐶𝑉#
Again,
𝑉 =
𝑄
𝐶
Therefore,
𝑊 =
"
#
× 𝑄 ×
𝑄
𝐶
𝑊 =
𝑄#
2𝐶

40
Efficiency of a Capacitor
IfΔQ is the amount of charge passing through the circuit, then total work done by the cell,
𝑉
9 = 𝑉8 + 𝑉7
Therefore,
𝑊 = ∆𝑄𝑉
9
Amount of energy stored by the capacitor,
𝐸 = ∆𝑄𝑉
.
Amount of energy lost due to resistance,
𝐸 = ∆𝑄𝑉8
A1 represents the amount of energy stored by the capacitor. A2 represents the amount of
energy lost due to resistance. Total area, (A1+A2), represents the amount of energy provided
by the cell. Thus, efficiency of the charging process of this capacitor is 50%.

41
Series Combination of Capacitors
If a capacitor is connected across a DC source, two parallel plates store equal and opposite
charge. Thus, resultant charge of a capacitor is zero. If this capacitor is connected across an
appliance, charges flow from one plate to another through the circuit. Thus, the charge of
the capacitor refers to the magnitude of charge on one plate.
In the circuit diagram above, the capacitors are connected in series across a DC source.
Metal plate A of the capacitor X is connected to the positive terminal of the cell. Thus, it
becomes positively charged. Similarly, metal plate D of capacitor Y becomes negatively
charged. Due to the broken path, charge cannot transfer between metal plates B and C. But
their plates get polarity due to electrostatic induction.
Because of the series configuration, both capacitors store equal amount of charge, but
resultant charge that can be provided by this arrangement is equal to that of one capacitor.
For series configuration, we know,
𝑉 = 𝑉" + 𝑉#
𝑄
𝐶1
=
𝑄
𝐶"
+
𝑄
𝐶#
1
𝐶1
=
1
𝐶"
+
1
𝐶#
For n number of capacitors,
1
𝐶1
=
1
𝐶"
+
1
𝐶#
+ ⋯ +
1
𝐶:
For n number of identical capacitors,
𝐶1 =
𝐶
𝑛

42
Parallel Combination of Capacitors
In this circuit, if two capacitors are connected in parallel against a DC source, the two
capacitors will have the same potential difference. The amount of charge stored by
capacitor X is Q1 and capacitor Y is Q2. Therefore,
𝑄" = 𝐶"𝑉
𝑄# = 𝐶#𝑉
Total charge stored by this combination,
𝑄0 = 𝑄" + 𝑄#
If resultant capacitance of the capacitor is CP, the total charge will be,
𝑄0 = 𝐶0𝑉
𝑄0 = 𝑄" + 𝑄#
𝐶0𝑉 = 𝐶"𝑉 + 𝐶#𝑉
𝐶0 = 𝐶" + 𝐶#
For n number of capacitors,
𝐶0 = 𝐶" + 𝐶# + ⋯ + 𝐶:
For n number of identical capacitors,
𝐶0 = 𝑛𝐶

43
Energy Stored in Series and Parallel Combination of Capacitors
Figure 1 Figure 2
In figure 1, two identical capacitors, X and Y, are connected in series. So, their total
capacitance,
𝐶1 =
𝐶
2
Work Done,
𝑊 =
"
#
𝐶1𝑉#
𝑊 =
"
#
× Y
𝐶
2
Z × 𝑉#
𝑊 =
"
+
𝐶𝑉#
In figure 2, two identical capacitors, X and Y, are connected in parallel. So, their total
capacitance,
𝐶0 = 2𝐶
Work Done,
𝑊 =
"
#
𝐶0𝑉#
𝑊 =
"
#
× (2𝐶) × 𝑉#
𝑊 = 𝐶𝑉#

44
Charging of Capacitors
In the circuit above, a two way switch is used to charge and discharge a capacitor. Charge
flows through a resistor, R, when the switch is connected to point A. Charge flows from the
cell to the capacitor. Thus, the potential difference of the capacitor gradually increases with
time. By following Kirchoff’s Voltage Rule, the potential difference across the resistor
decreases with time. At any point, it is given by the formula,
𝑉
9 = 𝑉8 + 𝑉7
At initial moment, the potential difference across the capacitor is zero. Thus, VR has the
largest magnitude. When 𝑡 = 0, we know that 𝑉7 = 0. So,
𝑉8 = 𝑉
9
𝐼 =
𝑉8
𝑅
𝐼3,! =
𝑉
9
𝑅
If the current through the circuit after t seconds is I, and the potential difference across the
resistor is VR, then we know,
𝑉8 = 𝐼𝑅
𝑉8 =
𝑑𝑄
𝑑𝑡
∙ 𝑅
If the amount of charge after t seconds is Q coulombs, then the potential difference across
the capacitor is,
𝑉7 =
𝑄
𝐶

45
From Kirchoff’s Voltage Rule, we know,
𝑉
9 = 𝑉8 + 𝑉7
𝑉
9 =
𝑑𝑄
𝑑𝑡
∙ 𝑅 +
𝑄
𝐶
𝑑𝑄
𝑑𝑡
∙ 𝑅 = 𝑉
9 −
𝑄
𝐶
𝑑𝑄
𝑑𝑡
∙ 𝑅 =
𝐶𝑉
9 − 𝑄
𝐶
p
𝐶
𝐶𝑉
9 − 𝑄
𝑑𝑄 = p
1
𝑅
𝑑𝑡
𝐶 p
1
𝐶𝑉
9 − 𝑄
𝑑𝑄 = p
1
𝑅
𝑑𝑡
𝐶 ln|𝐶𝑉
9 − 𝑄| = −
𝑡
𝑅
+ 𝑘
ln|𝐶𝑉
9 − 𝑄| = −
𝑡
𝑅𝐶
+
𝑘
𝐶
𝐶𝑉
9 − 𝑄 = 𝑒;
<
87
=
>
7
𝐶𝑉
9 − 𝑄 = 𝑒;
<
87 × 𝑒
>
7
𝐶𝑉
9 − 𝑄 = 𝐾𝑒;
<
87
𝐶𝑉
9 − 𝑄 = 𝐶𝑉
9𝑒;
<
87
𝑄 = 𝐶𝑉
9 − 𝐶𝑉
9𝑒;
<
87
𝑄 = 𝐶𝑉
9 Y1 − 𝑒;
<
87Z
𝑄 = 𝑄9 Y1 − 𝑒;
<
87Z
Now, from the equation,
𝑄 = 𝑄9 − 𝑄9𝑒;
<
87

46
We can plot a graph.
The charge of a capacitor varies exponentially with time. The time constant, Tau, is found
using the equation,
𝜏 = 𝑅𝐶
𝜏 =
𝑉
𝐼
×
𝑄
𝑉
𝜏 =
𝑄
𝐼
𝜏 = 𝑡
The product of resistance and capacitance of a circuit gives a particular time, which is called
time constant of the circuit. At this time constant, the capacitor stores 63% of total charge.
As we know, at initial moment, charge of the capacitor is zero.
𝑄 = 𝑄9 Y1 − 𝑒;
<
87Z
When 𝑡 = 0,
𝑄 = 𝑄9 Y1 − 𝑒;
?
87Z
𝑄 = 𝑄9(1 − 1)
𝑄 = 0
At 𝜏 time,
𝑄 = 𝑄9 A1 − 𝑒;
@
87B
𝑄 = 𝑄9 Y1 − 𝑒;
87
87Z

47
𝑄 = 𝑄9(1 − 𝑒;")
𝑄 ≅ 0.63𝑄9
Identify the Equation of Current at Time, t seconds
When time, t=0,
𝐼9 =
𝑉
9
𝑅
This current gradually decreases.
At time 𝑡 seconds,
𝐼 =
𝑑𝑄
𝑑𝑡
𝐼 =
𝑑
𝑑𝑡
Y𝑄9 − 𝑄9𝑒;
<
87Z
𝐼 = 0 − 𝑄9𝑒;
<
87 × Y−
1
𝑅𝐶
Z
𝐼 =
𝑄9𝑒;
<
87
𝑅𝐶
𝐼 =
𝑄9
𝑅𝐶
∙ 𝑒;
<
87
𝐼 = 𝐼9𝑒;
<
87
This equation represents the variation of current through the circuit, at a particular time
period. According to this equation, current decreases exponentially with time.
At 𝜏 time,
𝐼 = 𝐼9𝑒;
@
87
𝐼 = 𝐼9𝑒;
87
87
𝐼 = 𝐼9𝑒;"
𝐼 ≈ 0.37𝐼9

48
At 𝜏 time, the current decreases to about 37% of the initial current.
The potential difference across a capacitor is 𝑉7, where,
𝑉7 =
𝑄
𝐶
𝑉7 =
𝑄9 Y1 − 𝑒;
<
87Z
𝐶
𝑉7 =
𝑄9
𝐶
∙ Y1 − 𝑒;
<
87Z
𝑉7 = 𝑉
9 Y1 − 𝑒;
<
87Z
At a certain time, the voltage across the fixed resistor can be found using the equation,
𝑉8 = 𝐼𝑅
𝑉8 = 𝐼9𝑒;
<
87 × 𝑅
𝑉8 = 𝐼9𝑅𝑒;
<
87

49
𝑉8 = 𝑉
9𝑒;
<
87
At time 𝑡 = 0,
𝑉8 = 𝑉
9
At time 𝑡 = ꝏ,
𝑉8 = 𝑉
9𝑒;
ꝏ
87
𝑉8 = 0
Discharging of a Capacitor
When the switch is connected to the point B, the capacitor starts to discharge through the
resistor. At initial moment of the discharge process, the capacitor has the largest amount of
charge. As time passes, charge of the capacitor gradually decreases. According to Kirchoff’s
Voltage rule, we know,
𝑉
9 = 𝑉7 + 𝑉8

50
When the cell is removed, 𝑉? = 0
𝑉7 + 𝑉8 = 0
𝑉7 =
𝑄
𝐶
𝑉8 =
𝑑𝑄
𝑑𝑡
∙ 𝑅
Therefore,
𝑄
𝐶
+
𝑑𝑄
𝑑𝑡
∙ 𝑅 = 0
𝑑𝑄
𝑑𝑡
∙ 𝑅 = −
𝑄
𝐶
p
1
𝑄
𝑑𝑄
A
A'
= − p
1
𝑅𝐶
𝑑𝑡
<
?
[ln 𝑄]A'
A
= − {
𝑡
𝑅𝐶
|
?
<
ln }
𝑄
𝑄9
} = −
𝑡
𝑅𝐶
𝑄
𝑄9
= 𝑒;
<
87
𝑄 = 𝑄9 ∙ 𝑒;
<
87
When 𝑡 = 𝜏,
𝜏 = 𝑅𝐶
𝑄 = 𝑄9 ∙ 𝑒;
87
87
𝑄 = 𝑄9 ∙ 𝑒;"
𝑄 = 0.37𝑄9

51
The potential difference across the capacitor,
𝑉7 =
𝑄
𝐶
𝑉7 =
𝑄9 ∙ 𝑒;
<
87
𝐶
𝑉7 = 𝑉
9 ∙ 𝑒;
<
87
Current,
𝐼 =
𝑑𝑄
𝑑𝑡
𝐼 =
𝑑
𝑑𝑡
Y𝑄9 ∙ 𝑒;
<
87Z
𝐼 = 𝑄9 ∙ 𝑒;
<
87 × Y−
1
𝑅𝐶
Z

52
𝐼 = −
𝑄9
𝑅𝐶
∙ 𝑒;
<
87
𝐼 = −𝐼9 ∙ 𝑒;
<
87
In this equation, the negative sign represents opposite direction of current flow through the
resistor.
Experiment to Determine Capacitance
Graphical Method
A two way switch is connected to point A, to charge the capacitor. When the capacitor is
fully charged, reading of the ammeter drops to zero. The two way switch is connected to B
to discharge the capacitor through a known resistor. The ammeter is used to record the
current through the circuit. By using the timer, time for each current is record. By using this
reading, a current against time graph is plotted.

53
From this graph, time constant can be determined. By substituting the value of t and R, we
can find the capacitance.
𝑅𝐶 = 𝜏
𝐶 =
𝜏
𝑅
Mathematical Method
During this process, current through the circuit decreases, which is represented by,
𝐼 = −𝐼9 ∙ 𝑒;
<
87
ln(𝐼) = ln Y𝐼9 ∙ 𝑒;
<
87Z
ln(𝐼) = ln(𝐼9) + ln Y𝑒;
<
87Z
ln(𝐼) = ln(𝐼9) −
𝑡
𝑅𝐶
ln(𝐼) = −
1
𝑅𝐶
+ ln(𝐼9)

54
𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡 = −
1
𝑅𝐶
𝑅𝐶 × 𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡 = −1
𝐶 = −
1
𝑅 × 𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡
Charge against Time Graphs
If the resistance of the circuit is increased, the initial current, 𝐼9 =
C(
8
, decreases.
Thus, the initial gradient of the graph becomes smaller. Due to the large resistance, the time
constant increases, and the capacitor takes longer time to charge. The maximum charge,
𝑄9 = 𝐶𝑉
9, does not depend on the resistance. Thus, the final charge of the capacitor
remains unchanged.
𝐼9 =
𝑉?
𝑅

55
If the capacitance is increased, maximum charge of the arrangement increases, and time
constant becomes large, but the initial current through the circuit remains same, and the
gradient of the graph remains unchanged.
↑ 𝑄 = ↑ 𝐶 𝑉
↑ 𝜏 = 𝑅 ↑ 𝐶
If the EMF of the cell is increased, the time constant remains unchanged, but initial current
and maximum charge becomes large.

56
Properties of Current against Time Graphs
Area under the graph represents amount of charge transferred. In this case, the shaded area
represents amount of charge transferred into the capacitor between time 𝑡" and 𝑡#. By
measuring area under the graph, we can estimate the amount of charge stored in a
capacitor. If the resistance is increased, initial current through the circuit decreases, but
time constant increases. But maximum of the final charge of the capacitor does not depend
on the resistance. Thus, area under the graph should be equal.
If the capacitance is increased, initial current remains same, but time constant and final
charge becomes large.

57
If EMF of the cell is increased, time constant remains same, but initial current and final
charge becomes large.
Millikan’s Oil Drop Experiment
The atomizer is used to produce oil droplets. Initially, these oil droplets are projected
horizontally, so vertical component of velocity is zero. Due to gravitational pull, downward
velocity of the oil droplets increases. Thus, upward drag forces on the oil droplets increases
with time. When oil droplets move in terminal velocity, total upward force becomes equal
to total downward force.

58
We know that,
𝜌9%D =
𝑚𝑎𝑠𝑠
𝑣𝑜𝑙𝑢𝑚𝑒
𝑚𝑎𝑠𝑠 = 𝜌9%D × 𝑉
𝑉 =
+
*
∙ 𝜋𝑟*
∴ 𝑚 = 𝜌9%D ×
+
*
∙ 𝜋𝑟*
𝑚 =
+
*
∙ 𝜋𝑟*
𝜌9%D
𝑊 = 𝑈 + 𝐹
𝑚𝑔 = 𝜌&𝑣9𝑔 + 𝐹
+
*
∙ 𝜋𝑟*
𝜌9%D =
+
*
∙ 𝜋𝑟*
𝜌,%E𝑔 + 6𝜋𝑟𝜂𝑣F
+
*
∙ 𝜋𝑟*
𝜌9%D −
+
*
∙ 𝜋𝑟*
𝜌,%E𝑔 = 6𝜋𝑟𝜂𝑣F
𝑟 =
+
* ∙ 𝜋𝑟*
𝑔(𝜌9%D − 𝜌,%E)
6𝜋𝜂𝑣F
𝑟#
=
9𝜂𝑣F
2𝑔(𝜌9%D − 𝜌,%E)
𝑟 = b
9𝜂𝑣F
2𝑔(𝜌9%D − 𝜌,%E)
The oil droplets become charged by friction inside the cylinder. Two horizontal plates, A and
B, are used. A is negatively charged, and B is positively charged. So, the electric field is
directed towards the upward direction. The magnitude of this field is,
𝐸 =
𝑉
𝑑

59
When a positively charged oil drop enters the region between the two plates, the drop
experiences a force, which is in the upward direction. This force is provided by the electric
field. By using a suitable potential difference, the oil drop can be brought to rest. At this
condition, frictional force becomes zero, because its speed is zero. Now, the downward
force is balanced by upthrust and the electric force.
𝑊 = 𝑈 + 𝐹5
𝑊 − 𝑈 = 𝐹5
Now,
𝐹5 = 𝐹 = 6𝜋𝑟𝜂𝑣F
𝑄𝑉
𝑑
= 6𝜋𝑟𝜂𝑣F
𝑄 =
6𝜋𝑑𝜂𝑣F
𝑉
× 𝑟
𝑄 =
6𝜋𝑑𝜂𝑣F
𝑉
× b
9𝜂𝑣F
2𝑔(𝜌9%D − 𝜌,%E)
Where, 𝑑 = plate separation

61
Magnetic Fields
A magnetic field is a space where a magnet or a moving charged particle experience a force. Like an
electric field, a magnetic field is a vector quantity. Thus, it has a magnitude and direction.
Properties of Charged Particles
• A static charged particle produces electric field.
• A moving charged particle produces both electric field and magnetic field.
Poles of a Magnet (Magnetic Poles)
Poles represent the point where the strength of the magnet (or magnetic field) is largest. A freely
suspended magnet is directed along north-south direction. Direction of magnetic field is defined as
the direction of force experienced on the individual north pole inside the magnetic field.
Same poles repel each other, and opposite poles attract. This phenomenon is known as magnetic
interaction.
Magnetic Field Strength
Magnetic field is represented by imaginary lines, which are called magnetic field lines. Separation
between these field lines represent relative field strengths. If the field lines are closer to each other,
it represents stronger magnetic field. In the figures above, magnetic field lines are passing through
Area, A. The total number of field lines through a particular area is called magnetic flux or magnetic
field density.
! = #$
% is the magnetic flux. Its unit is Weber (Wb). B is the magnetic field strength, and its unit is Tesla
(T). The area is represented by A, in m2.

62
This equation is applicable if the magnetic field lines is perpendicular to the surface. In figure 3, the
magnetic field lines form an angle, Theta, with the surface, A. The component of this magnetic field
strength perpendicular to the surface is,
! sin %
Thus, magnetic flux is,
& = !( sin %
Therefore, the magnetic flux is maximum when the angle is 90o, and minimum when it is parallel.
Properties of Magnetic Field Lines
• Magnetic field lines are continuous. They follow a complete path or loop.
• Magnetic field does not intersect each other.
• If the magnetic field lines are parallel to each other, and have constant separation, it
represents uniform magnetic field.
Force on a Moving Charged Particle in a Magnetic Field
A moving charged particle produces a magnetic field around it. If a charged particle is projected
through a magnetic field, it experiences a force due to the interactions of the two fields.
Magnitude of this force is,
) = !*+ sin %
Where, Q = charge of particle
v = velocity of the particle
B = magnetic field strength
θ = angle between magnetic field and velocity
If a charged particle moves perpendicularly to the direction of the magnetic field, it experiences
maximum force.
) = !*+ sin 90
) = !*+
If the charged particle moves parallel to the direction of the magnetic field lines, the force
experienced is minimum.
) = !*+ sin 0
) = 0

63
The magnitude of the magnetic field strength is equal to the amount of force that acts on 1c of
charge when it moves at 1m/s, perpendicular to the direction of magnetic field.
The direction of force on a moving charged particle can be determined by Fleming’s left hand rule. If
the index finger is placed along the direction of magnetic field, the middle finger is placed along the
direction of velocity, then the thumb gives the direction of force, on a positive charge inside the
magnetic field. A negative charged particle experiences force in the opposite direction of the thumb.
In the diagram above, an electron and a positron are projected horizontally, through a magnetic
field. This magnetic field is directed inwards. According to Fleming’s left hand rule, an upward force
acts on the positron, and it follows a curved path. Due to the negative charge, the electron
experiences a force to the opposite direction given by Fleming’s left hand rule. Thus, the electron
deflects opposite to the direction of positron.
Denotes magnetic field into the plane
Denotes magnetic field out of the plane
Motion of an Electron in a Uniform Magnetic Field
This figure represents the path of motion of an electron in a region of uniform magnetic field.
Because of its charge, the moving electron experience a force inside the magnetic field. The

64
direction of this force can be determined by Fleming’s left hand rule. In this case, the angle between
velocity and the magnetic field is 90o, so the force will be,
! = #$%
! = #&%
The direction of this force is perpendicular to the direction of velocity. Thus, this force provides
centripetal force. Due to this force, the charged particle follows a circular path in the magnetic field.
Centripetal force can be found using the equation,
!! =
'%"
(
So, we can say,
'%"
(
= #&%
(&# = '%
( =
'%
&#
This figure represents the path of an electron and a positron inside a uniform magnetic field, where
they have different speeds. Due to the opposite charges, they experience force in the opposite
directions. The positron follows an anticlockwise path, and the electron follows a clockwise path.

65
! =
#$
%&
Momentum,
' = #$ = (2#*!
Therefore,
! =
(2#*!
%&
If m, Q and B are constants,
! ∝ (*!
If the kinetic energy remains constant, the charged particle follows a uniform circular path of
constant radius. But in practice, the kinetic energy gradually decreases, due to collision with other
particles. Thus, the radius of the circular path gradually decreases, and it follows an inward spiral
path.
This is the path of motion, as the kinetic energy is decreasing. An accelerating charged particle
produces an electromagnetic wave. When a charged particle moves in a circular path, it accelerates
due to the constant change in velocity. As the accelerating charged particle emits electromagnetic
waves, its kinetic energy decreases by following the law of conservation of energy. Thus, the radius
of the circular path decreases.

66
Magnetic Field Arounda Current Carrying Wire
When current flows through a wire, it produces a magnetic field in the space around the wire.
Direction of the magnetic field can be determined by using right hand grip rule. If the thumb of the
right hand is placed along the direction of current flow, then the curled fingers give the direction of
the magnetic field.
Magnitude of this magnetic field strength depends on:
• The amount of current through the wire.
• Perpendicular distance of the point from the wire.
If the current through the wire is I, then magnetic field strength at P can be written as,
! =
#!$
2&'
Where, µ0is the permittivity of free space.
Force on a Current Carrying Wire
A current carrying wire produces a magnetic field. If it is placed in an external magnetic field, it
experiences force, due to interaction between the two magnetic fields. The magnitude of this force
can be found by using the formula,
( = !$) sin -
B is the magnetic field strength, I is the current through the wire, l is the length of wire inside the
magnetic field, and θ is the angle between the wire (current) and the magnetic field.

67
The direction of the force on the wire can be determined by using Fleming’s left hand rule. In this
diagram, direction of force on the current carrying wire is inward. If the wire is placed perpendicular
to the magnetic field, it will experience maximum force. If it is parallel, the force will be zero.
Current is the flow of electrons through a conductor. When these electrons move through a
magnetic field, force act on each of the electrons. As a result, the wire experiences force.
! = #$% sin )
* = $+
$ =
*
+
Therefore,
! = # ×
*
+
× % × sin )
! = #* ×
%
+
× sin )
! = #*- sin )
Magnetic Field Around a Current Carrying Solenoid
When current flows through a solenoid, it produces a magnetic field, which is similar to a bar
magnet. It has a north and a south pole. The magnetic field lines are directed from north pole to
south pole. North pole of the solenoid can be determined by the right hand grip rule. If the curled
fingers are placed along the direction of current flow, the thumb shows the north pole of the
solenoid. Magnitude of the strength of the magnetic field of a current carrying solenoid can be found
by,
# =
.!/$
%
Where, / = total number of turns

68
Number of turns per unit length, !,
! =
#
$
Therefore,
% = &!!'
Electric Motor
Figure 1a Figure 1b
An electric motor is a device which converts electrical energy to kinetic energy or mechanical energy.
In the figures above, uniform magnetic fields are produced by using two opposite poles. This uniform
magnetic field is directed towards right, from north to south. A rectangular loop of conducting wire
is placed inside the field and is connected to a DC source. In figure 1a, the source provides a
clockwise current through the loop. The direction of current is upwards through the side AB.
According to Fleming’s left hand rule, inward force acts on AB. The magnitude of this force is,
( = %'$ sin ,
As the angle between the magnetic field and current is 90o,
( = %'$
Side BC is parallel to the direction of the magnetic field, and thus, no force is experienced by BC.
Direction of current flow through the wire CD is downwards. According to Fleming’s left hand rule,
outward magnetic force acts on the wire. Due to equal and opposite parallel forces, a moment acts
on the loop, which causes it to rotate. As the loop rotates, magnitude of this force gradually
decreases. The figure 1b represents the condition of the loop after 180o rotation. The direction of
the coil does not change. If a DC source is connected, without using a commutator, then the coil
would vibrate instead of rotating.

69
Rotation of the loop can be increased by:
• Increasing the current through the loop.
• Using stronger magnetic field.
• Increasing the number of turns of the loop.
• Increasing length of AB and CD.
• Introducing soft iron core.
Hall Voltage
ABCD is a rectangular metallic plate. A DC source is connected across the length of the metal plate.
Current flows through the positive terminal of the cell through the metal plate. Thus, electrons flow
in the opposite direction of the current flow. Due to inward magnetic field, force acts on moving
electrons. According to Fleming’s left hand rule, direction of force on this negatively charged
electron is upward. Thus, the side AB of the metal plate becomes negatively charged, and CD
becomes relatively negatively charged. There will be a potential difference across the width of the
metal. This potential difference is called the Hall Voltage.
Due to the Hall Voltage, an electric field is created inside the metal plate. This field is directed
upward. If the width of the metal plate is d, magnitude of this electric field is,
! =
#!
$
#! = Hall Voltage across the width
In this field, negatively charged electron experiences downward force. The magnitude of this force
is,
%" = &#!

70
!! = #" ×
%#
&
The upward magnetic field of the plate is equal to the provided by the electric field. For this reason,
this electron remains undeflected, and the voltmeter will give a constant Hall Voltage.
#"%#
&
= '#"(
)%#
&
= ')(
%# = ('&
Therefore,
%# ∝ '
Faraday’s Experiment on Electromagnetic Induction
A centre-zero galvanometer is connected in series with a conduction loop. A bar magnet is moved
towards and away from the loop, and the deflection of the galvanometer is observed.
Observations:
• When the north pole of the bar magnet is moved towards the loop, the deflection of the
galvanometer’s dial indicates the presence of current through the loop.
• When the north pole of the bar magnet is moved away from the loop, the galvanometer’s
dial deflects in the opposite direction. This indicates that the flow of current through the
loop is reversed.

71
• The direction of current flow alters if the south pole of the magnet is moved towards the
loop.
• If the magnet is moved faster, the deflection of the galvanometer becomes larger, which
indicates larger current flow through the loop.
• If the loop is moved towards or away from the stationary magnet, dial deflects.
• If the loop and the magnet remains stationary, no current is observed. However, if both of
them move towards or away (have a relative motion), current can be observed.
In the figure above, the magnet is moved towards the stationary loop. When the magnet is at A, the
number of field lines, or magnetic flux, in the loop is very small, due to the large distance between
the magnet and the loop.
When the magnet is at B, the number of field lines in the loop increases, due to smaller distance
between the magnet and the loop. So, there is a change in magnetic flux through the loop. When
there is a relative motion between the magnet and the loop, due to this change in magnetic flux, an
EMF is induced across the loop, which causes current through the loop. This is called Induced EMF,
and the process is called electromagnetic induction. If they remain stationary, or both moves with
the same velocity, there is no change in magnetic flux through the loop. Thus, no EMF is induced.
Area of loop = A
Magnetic field strength = B
Magnetic flux,
! = #$ sin (
For N number of turn of coil,
)! = #$) sin (
According to Faraday’s law, the rate of change of magnetic flux produces the induced EMF.
Induced EMF,
* = −
∆)!
∆-
* = −
.()!)
.-

72
! = −
$
$%
('() sin -)
In this equation, the negative sign represents the magnitude of the induced EMF is such that it
opposes the change creating it. According to Faraday’s law of induction, we know,
! = −
$()/)
$%
If ), - and ( are constants,
! = −
$
$%
('() sin -)
! = −() sin - ∙
$'
$%
EMF can be changed by changing the magnetic field strength through the loop, which is possible by
moving the magnet or the coil towards or away from each other. The EMF can also be changed by
changing the area of the loop.
If ), ' and - are constants,
! = −') sin - ∙
$(
$%
EMF can also be changed by changing the angle.
If ), ' and ( are constants,
! = −'() ∙
$
$%
(sin -)
We know that,
- = 1%
Therefore,
! = −'() ∙
$
$%
(sin 1%)
! = −'()1 cos 1%
! = −!! cos 1%

73
Origin of Induced EMF
In the figure, a conductor AB of length ! is moved downward through a magnetic field. The magnetic
field is directed inwards. Due to the nature of metallic bonding, the conductor contains a large
number of delocalized electrons. A moving charged particle experiences a force inside the magnetic
field. Thus, end A becomes negatively charged, and end B becomes positively charged. This can be
defined by using Fleming’s right hand rule. Because of this, a potential difference is produced
between A and B.

74
Due to the potential difference between AB, an electric field is produced in the conductor. Thus, the
electric field is directed towards left. Because of this field, the electrons experience force towards
the right. The magnitude of force on electron,
!! = #$% sin )
% is the speed of electrons in the magnetic field. If the electric field inside the conductor is $,
magnitude of the electric force on *,
!" = $*
As we know, electric field strength,
* =
+
,
Therefore,
!" =
$+
,
This force acts towards the right on the opposite direction of the magnetic force. When two forces
become equal, the potential difference between the two ends of the conductor becomes constant.
At this condition,
!" = !!
$+
,
= #$%
-
,
= %#
When the conductor AB is moved through the magnetic field, an EMF is induced in the conductor,
obeying Faraday’s law.
Induced EMF depends on:
• Magnetic field strength
• Length of the conductor
• Velocity of the conductor
• Angle between the magnetic field lines and velocity
If two ends of the conductor is connected to metal wire, current will flow through the circuit due to
the balanced EMF. Current flows from higher potential B to lower potential A through external
circuit. As the current flows through a complete path, it will flow from B to A. The direction of the
induced current through the conductor can be determined by Fleming’s right hand rule. If the index
finger is placed along the direction of the magnetic field lines, and the thumb is placed along the
direction of velocity, then the middle finger gives the direction of current through the conductor.

75
Lenz’s Law
The direction of induced EMF is such that it opposes the change creating it.
This law helps to explain the conservation of energy, in case of electromagnetic induction.
Figure 1 Figure 2
In figure 1, the north pole of a bar magnet is moved towards a coil. Due to the change in magnetic
flux, an EMF is induced across the coil. This induced EMF produces a current through the loop. When
current flows through the coil, it produces circular magnetic fields. According to Lenz’s law, the
induced EMF is such that the coil produces a magnetic field with north pole at point A. to move
against this repulsive force, energy is needed. This energy is provided by an external source. Thus,
the energy of the external source decreases. By following the law of conservation of energy, an
equal amount of energy is formed across the coil as electrical potential energy.
In figure 2, the north pole of the bar magnet is pulled away from the coil. According to Lenz’s law,
the direction of the induced EMF is such that a south pole is formed at end A of the coil. So, there
must be a magnetic attraction between the poles. Thus, energy is needed to pull magnet away from
the coil. This supplied energy is converted to electrical potential energy by electromagnetic
induction.
Verification of Lenz’s Law

76
In the figure, A and B are two identical magnets. They have the same initial height from the ground.
When these magnets are released, they move downwards due to the gravitational pull of the earth.
Magnet A moves through the loop completely, until it hits the ground, and the magnet B drops
directly to the ground. As magnet A approaches towards the loop, an EMF is induced across the
loop, due to the change in magnetic flux. Direction of the induced current is such that the loop
produces North Pole above, by obeying Lenz’s law. Due to repulsion between two forces, the
resultant downward force becomes less than the actual weight of the magnet A. Thus, the magnet
accelerates at a lower rate, and takes a longer time to reach to the ground than magnet B. As the
magnet B moves due to gravitational acceleration of the earth, the time taken to reach the ground
will be less. This time can be measured accurately by using suitable instruments.
If magnet A takes longer time than magnet B, Lenz’s law is verified.
Figure 1
In figure 1, an oscilloscope is connected to a coil. A magnet is released, which passes through the coil
due to gravitational acceleration. Due to the change in velocity, there is a change in magnetic flux,
and thus an EMF is induced across the coil, which can be measured from the oscilloscope. As the
magnet approaches towards the loop, the magnetic flux linkage increases. Due to its acceleration,
magnetic flux increases at an increasing rate. Variation of change in magnetic flux is due to the
motion of the magnet. According to Faraday’s law, an induced EMF is produced, which can be found
by using the equation,
! = −
$(&')
$)

77
Gradient of the magnetic flux linkage against time graph is the induced EMF, as shown in figure 2.
Initial gradient of the graph is zero. At this instant, rate of change of magnetic flux is zero. According
to Faraday’s law, the induced EMF is also zero. As the gradient increases, the magnitude of the
induced EMF also increases. At point A, the rate of change of magnetic flux is largest, when the north
pole of the magnet is just entering the coil. At point B, magnitude of the magnetic flux linkage has
greatest value, but gradient is zero, which represents that the induced EMF is zero. After time t2, the
magnet is moving away from the coil. Thus, the magnetic flux linkage decreases, and the induced
EMF increases in the opposite direction.
In figure 3, the graph represents the variation of induced EMF across the coil with time. Due to the
gravitational pull, thespeed of the magnet gradually increases. The magnet moves away from the
coil at a higher speed than when the magnet approaches the coil. Thus, the negative peak of the
EMF has the largest amplitude. When the North Pole of the magnet approaches the coil, by
following Lenz’s law, the direction of current is such that the coil produces a North Pole above it.
When the magnet is moving away from the coil, by following Lenz’s law, a North Pole is formed at
the bottom of the coil, which attracts the South Pole of the magnet. Thus, the magnitude of the
induced EMF is slightly decreased.
Expression of Induced EMF
Position 1 Position 2 Position 3
ABCD is a metal loop. The length of each side of the loop is !. The loop is pulled at a constant speed "
m/s, through a uniform magnetic field, along the horizontal direction. As the magnetic field and
direction of velocity is perpendicular, so the magnetic flux is,
# = %&
For position 1, area inside the magnetic field,
& = '!
Due to the motion of the loop, there is a change in magnetic flux, which causes an induced EMF.

78
Therefore,
! = #$%
We know that the magnitude of induced EMF,
& =
'()!)
'+
& =
'
'+
(#$%)
& = #% ∙
'$
'+
& = #%-
At position 1, the side AB cuts the magnetic field due to the motion of the loop. According to
Fleming’s right hand rule, direction of the induced current is from A to B. as the current passes
through the complete path, the induced EMF causes anticlockwise current through the loop.
At position 2, the loop is completely inside the magnetic field. So, there is no change in magnetic flux
through the loop. By following Faraday’s law, the magnitude of induced EMF would be zero.
However, the magnitude of the induced EMF would have not been zero if the loop was accelerating.
At position 3, the side CD cuts the magnetic field lines due to the motion of the loop. According to
Fleming’s right hand rule, the direction of current in CD is from D to C. So, a clockwise current passes
through the loop. The magnitude of the induced current at position 1 and 3 can be found using the
equation,
. =
&
/
. =
#%-
/

79
Eddy Current
If there is a change in magnetic flux through a metal plate, a current is induced in the metal plate,
which follows a complete path through the metal plate, and obeys Lenz’s law, opposing the change
creating it.
There are two types of currents which are induced:
1. Induced useful current
2. Induced wasted current
The induced wasted current is called eddy current. However, this current has many applications
nowadays. For instance, in case of transformers, it contains a metal core. Induced useful current is
produced in the secondary coil. But eddy current is also produced on the surface and inside the
metal. To reduce this eddy current, we should make the metal core of the transformer with thin
sheets of metal, laminated (wrapped) with an insulator. Eddy current is useful in induction cooker,
induction braking system, and metal detectors.
Experiment to Observe the Effect of Eddy Current
Figure 1 Figure 2
In figure 1, a simple pendulum is constructed using a metal plate. If it is released from its maximum
displacement, it will swing for a long time period in absence of a magnetic field. In figure 2, a metal
plate moves inside a magnetic field. Due to the change in magnetic flux, eddy current is produced in

80
the metal plate. By following Lenz’s law, the motion of the metal plate is opposed due to induced
current. As a result, this metal plate comes to rest in a very short time. In figure 3, a splitted metal
plate moves through the magnetic field. Due to this, broken current is produced. That’s why, amount
of eddy current decreases. As a result, the plate experiences small force, and swings for a longer
time period.
Induction Cooker
An induction cooker contains a metal coil. When current flows through the coil, it produced a
magnetic field. These magnetic field lines pass through the conductor. Due to the AC source, the
direction of current through the conductor changes with time. As a result, there is a rate of change
of magnetic flux through the conductor, which in turn produces eddy current. As the eddy current
flows, the temperature of the container increases.
Induction Braking System
In a magnetic braking system, the metal disc of the vehicle’s wheel rotates inside a magnetic field,
which is produced by electromagnets. In normal conditions, current through the electromagnet is

81
zero. Thus, the metal disc moves freely through the electromagnet. When brake is applied, which
means that the switch of the electromagnet is closed, and current flows through the
electromagnets’ coils, the electromagnets produce a magnetic field which passes through the metal
disc. Due to the rotation of the disc, there is a change in magnetic flux, which produces eddy current
in the metal disc. By following Lenz’s law, the direction of the eddy current is such that the motion of
the disc is opposed. Thus, its speed decreases, and the car slows down. In this case, the kinetic
energy of the car is converted to thermal energy.
Metal Detectors
A metal detector contains a primary coil, called transmitter, and a secondary coil, called receiver. An
AC source is connected across the transmitter. Current flows through the primary coil, and it
produces a magnetic field around it. Due to the suitable arrangement, this magnetic field lines
cannot pass through the receiver. As an AC source is connected, the magnetic field lines across the
primary coil changes with time. In presence of a metal, an eddy current is produced in the metal,
due to the change in magnetic flux. Because of this current, the metal produces a magnetic field
around it, which changes continuously with time. These magnetic field lines pass through the
secondary coil, which causes an induced EMF across the receiver. Thus, the presence of metal can be
detected from a voltmeter connected across the receiver.

82
Generator
Figure 1 Figure 2
An electrical generator is used to produce electrical energy from mechanical energy. In figure 1,
ABCD is a metal loop, which is placed inside a uniform magnetic field. When the loop rotates, there
is a change in magnetic flux, which causes EMF. This loop is connected to an external circuit by the
help of a slip ring commutator. It allows rotation of the loop without changing the terminals.
However, the connection of the wires shifts from left to right. In figure 1, AB of the loop is moved
upwards and CD is moved downwards. At this point, the direction of velocity of AB and CD is
perpendicular to the direction of the magnetic field. Due to the interactions of the magnetic field
lines, current is induced on the sides AB and CD. According to Fleming’s right hand rule, current
through AB is from A to B, and the current in CD is from C to D. Since AD and BC are parallel to the
magnetic field lines, there are no interactions with the magnetic field lines. But, there is a current
through this side, as current flows through a complete path. In figure 1, a clockwise current flows
through the loop and this current flow from X to Y. At initial moment, the angle between velocity
and magnetic field lines is 90o. We can calculate the magnitude of induced EMF by the equation,
! = #$% sin )
Figure 2 represents the condition of the loop after 180o rotation. At this moment, AB is moved
downwards and CD is moved outwards. By following Fleming’s right hand rule, a clockwise current is
produced, from D to A, and this current flow through the circuit from Y to X. Thus, continuous
rotation of the loop causes an alternating current. For multiple turns of wires,
*+ = #,* sin )
In this expression, ) represents the angle between the area of loop and the magnetic field lines. If
the loop rotates at a constant angular velocity, then,

83
Figure 3a Figure 3b
Figure 3a represents the change in magnetic flux linkage through the loop with time. According to
Faraday’s law, rate of change of magnetic flux produces an induced EMF. Figure 3b shows the
variation of change of induced EMF with time. Magnitude of induced EMF is largest when,
cos $% = ±1
The magnitude of the induced EMF can be increased:
• By increasing the number of turns of wires in the loop.
• By increasing the area of the loop.
• By using stronger magnets.
• By moving the loop faster.
If the loop rotates faster, the rate of interaction of magnetic field lines is higher. Thus, the induced
EMF increased. At this high speed, the loop takes shorter time to complete one complete rotation.
Thus, the time period decreases and the frequency increases.

84
Transformer
Transformers are used to increase or decrease a supply voltage according to the aim. There are two
types of transformers:
1. Step-up Transformers
2. Step-down Transformers
In a step-up transformer, the voltage is increased, and in a step-down transformer, the voltage is
decreased.
This is done by taking the advantage of magnetic field lines and the number of turns of wires in the
coils. In a step-up transformer, the number of turns in the primary coil is less than the number of
turns in the secondary coil. In a step-down transformer, the number of turns in the primary coil is
greater than the number of turns in the secondary coil.
In case of transformers, the number of turns, voltage, and current follow a ratio:
!!
!"
=
#!
#"
=
$!
$"
!! = Number of loops in secondary coil
!" = Number of loops in primary coil
#! = Voltage across secondary coil
#" = Voltage across primary coil
$! = Current in secondary coil
$" = Current in primary coil

86
Alpha Particle Scattering Experiment
In this experiment, a beam of alpha particles is projected through a gold foil, and the deflection is
observed. A natural source of alpha particles is Radon. It is placed in a metal or lead container with a
small opening. Thus, a narrow beam of alpha particles are produced and the deflection of alpha
particles through the gold foil can be observed. This arrangement took place inside a vacuum
chamber, so that the velocity of the alpha particles is not affected.
Observations:
• Most of the alpha particles move in a straight line or is slightly deflected.
• Some of the alpha particles are deflected at a large angle.
• Very few alpha particles are deflected at or greater than 90o which is called backscattering.
Conclusion:
• Most of the space inside an atom is empty.
• There is a positively charged centre, called nucleus.
• Mass of the positively charged centre is very large compared to that of the negative charged
electron. The nucleus contains most of the mass of the atom.

87
In this experiment, a narrow beam of alpha particles is used, to measure deviation accurately. Gold
foil was used as it is a malleable material and can be penetrated easily. Moreover, in case of other
thick metal plate, the alpha particles will be deflected multiple times, and a random pattern of alpha
particles would have been produced. Vacuum chamber was used to prevent random collision of
alpha particles with air particles. If random collision took place, the alpha particles would lose their
kinetic energy.
NOTE: It is wise to use gold foil of 1 atom thickness, which will make the experiment much more
reliable. If there are multiple layers of atoms, the alpha particles will be deflected several times and
proper deflection cannot be observed.
Wave-Particle Duality
When a beam of electrons passes through a crystal, it diffracts, which indicates wave nature of
electrons. Similarly, photoelectric effect represents the particle nature of photons.
! = ℎ$
! = %&!
Therefore,
ℎ$ = %&!
ℎ&
'
= %&!
%& =
ℎ
'
( =
ℎ
'
We know that momentum,
( = %& = )2%!"
Therefore, from de Broglie’s Equation,
' =
ℎ
(
' =
ℎ
)2%!"

88
Particle Accelerators
According to Einstein, relation between mass and energy can be explained by the equation,
! = #$!
According to Einstein, if any object increases its speed with respect to any object stationary observer,
its mass increases due to inertia. It happens more significantly if the object travels close to the speed
of light. If it reaches the speed of light, its mass increases to infinity, which results in infinite energy,
according to Einstein’s theory, which is proven mathematically, but not experimentally, due to
obvious engineering problems.
In a nuclear reactor, energy is produced from mass. It is also possible to make mass from energy.
When a high-speed particle collides against a target, the kinetic energy of the particle decreases. By
following the law of conservation of mass-energy, the kinetic energy is converted to mass. Thus,
fundamental particles are produced. Accelerators are used to produce high speed beam of particles.
Linear Accelerators (LINAC)
%" > %# > %$ > %! > %%
In a LINAC, charged particles are accelerated in a straight path, through a series of drift tubes. These
tubes are connected across an alternating voltage source. Thus, there is a potential difference
between each consecutive tubes.

89
Figure 2a represents the variation of potential of terminals with time. At time ! = 0, a positively
charged particle, like proton, is at a point between tube 1 and tube2, which is represented by figure
2b. At this instant, tube 1 is positive and tube 2 is negative. Because of this, a horizontal electric field
is produced between the tubes. The positively charged [particle experiences a force along the
direction of the electric field, and it begins to accelerate. There is not electric field inside the tube.
That’s why, the charged particle moves with a constant speed inside the tubes.
At time ! = !!, the positively charged particle is at a position between tubes 2 and 3. At this instant,
the tube 2 is positive and the tube 3 is negative. Due to the electric field, the charged particle
experiences a force along the direction of the electric field lines, which is also the direction of
motion. Thus the particle accelerates and its kinetic energy increases. Due to the synchronized
alternating source, the charged particle experiences force along the direction of its velocity. Thus it
travels through the gap between the two tubes and finally, a high speed beam is produced.
For this arrangement, a source of constant frequency or time period of alternating current source
should be used. For continuous acceleration, the charged particle should remain inside the tube for
half time period. At constant speed, the distance travelled by the charged particle within its half time
period is,
$ =
%!
2
As speed increases, within the same time, the proton travels larger distance. To keep it synchronized,
the length of the tube should be increased. When the speed of the particle becomes comparable to
the speed of light, after a certain point, it cannot increase its speed anymore. As the particle is at a
high speed, its mass increases. This extra mass is known as relativistic mass. The relativistic mass can
be found by the equation,
' =
'"
(1 −
#!
$!
Where, '" = rest mass, m = relativistic mass, % = speed of the particle, and + = speed of light.
NOTE: When the speed of a particle reaches closer and closer to the speed of light, its speed
becomes constant, but its kinetic energy still increases due to increasing mass. When this high speed
particle beam collide against a target, its kinetic energy decreases, which is converted to new mass
(in the form of particles).

90
Targets can be arranged in two ways:
Fixed Target Experiment
In case of fixed target experiment, there is a resultant momentum before collision. Thus, the particle
must have a resultant momentum after collision. By following the conservation law, the particle has
kinetic before collision. The total energy given by the accelerator is not converted into mass.
Collision Beam Experiment
In case of collision beam experiment, the total momentum before collision is zero. If two particles
move with same speed in opposite directions, according to the conservation law, the total
momentum after collision must be zero. Thus it is possible that the particle comes to rest after
collision. The total kinetic energy after the collision can be used to produce mass. This method is
highly efficient in terms of energy to mass conversion, but the probability of collision of particles is
lower.
Cyclotron
Inside a magnetic field, charged particles follow a circular path, because centripetal force is provided
by the magnetic field.
!! = !"
#$% =
&%#
'
' =
&%
#$
' =
(
#$
' =
)2&+$
#$
'# =
2&+$
##$#
Therefore,
+$ ∝ '#

91
In a cyclotron, charged particles are accelerated in a circular path. It is accelerated using the
semicircular Dees, electric field and magnetic field. The metal Dees, X and Y, are connected across an
alternating voltage source.
Figure 2a represents variation of potential of terminal A with time. This arrangement is placed inside
a uniform magnetic field, and it is perpendicular to the surface of the Dees. A positively charged
particle, like proton, is placed in the middle of the gap between the two Dees.

92
Figure 2b represents the position of a proton between two Dees, at time, t=0. It experiences a force
along the direction of the electric field lines, and it accelerates because a resultant force acts on it.
Thus, the kinetic energy of the proton increases in the space between the Dees. Inside the Dees,
there is no accelerating electric field. Thus, the particle moves with constant speed, but it
accelerates by changing the direction of motion due to the magnetic field. That’s why, it follows a
circular path inside the semicircular Dees.
Figure 2c represents the motion of the proton inside the Dees. It experiences force along the
direction of motion, and thus its kinetic energy increases.
According to the equation,
! =
#$
%&
Radius of the circular path increases as the particle moves with greater speed, and it will follow
outward spiral path. Velocity if the particle increases each time it passes throughthe gap between
the Dees. For its continuous acceleration, the particle should spend half time period inside each
Dees.

93
! =
#$
%&
$ =
%&!
#
We know that,
$ = '!
Therefore,
'! =
%&!
#
2)* =
%&
#
* =
%&
2)#
* is called the cyclotron frequency. If an AC source of this frequency is applied, the particles remain
synchronized with the time period of the source.
+ =
1
*
Therefore,
+ =
2)#
%&
+
2
=
)#
%&
!
"
is the time spent by the particle in each Dee.
As speed of the particle in the cyclotron increases, it continues to increase its speed, until it reaches
the speed of light.

94
Synchrotron
BM = Binding Magnet
RFAC = Radio Frequency Accelerating Cavity
FM = Focusing Magnet
In a synchrotron, charged particle accelerates in a circular path of constant radius. Inside the RFAC,
an alternating synchronized electric field is used to accelerate the charged particle. Binding magnets
are used to provide centripetal force, which keeps the charged particle moving in a circular path.
This magnetic field is produced by electromagnets. The strength of the magnets can be modified.
The radius of the circular path increases according to the equation,
! =
#$
%&
To keep the radius constant, magnetic field is modified when the speed of the particle increases.
After passing through the RFAC, the particle accelerates in a circular path, maintaining constant
radius. When a charged particle accelerates, it radiates electromagnetic waves. In a synchrotron,
charged particles move in a circular path, and reach a speed comparable to the speed of light. Due to
its circular motion, its acceleration takes place, even at constant speed. As a result, it radiates
electromagnetic radiation, which is called synchrotron radiation. Because of this radiation, a large
amount of energy is lost to the surroundings. Focusing magnets help to focus all the particles to a
concentric beam. Particle detectors are used to detect the path of motion of these particles.

95
Bubble Chamber
Bubble chambers contain liquid hydrogen. The temperature of hydrogen is higher than its boiling
point, but it remains in liquid phase due to high pressure. If pressure is released, hydrogen changes
its phase from liquid to gas. Thus, bubble is formed inside the liquid hydrogen. This bubble formation
initiates around the impurities, when a particle is produced and pass through the bubble chamber. It
causes ionization around its path of motion. Thus, ions acts as impuritiesand bubbles are produced
around the path of motion of the particles. A magnetic field is used to deflect the charged particles.
From the direction of their deflection charge of the particles can be identified, and their mass-charge
ratio can be calculated from the radius of the path.
Examples:
Electron and positron curl is formed due to the magnetic field.
An electron loses its energy quickly because it radiates electromagnetic radiation. That’s why it is
spiraling inwards.
A particle comes to rest, and leaves a dense track near the end as its ionizing power increases.

96
A neutral particle decays into some other particles. Two of them are charged, and one is neutral.
Particles and Antiparticles
Each particle has an antiparticle. Particles and antiparticles have same properties, except the charge.
Electron and positron are two antiparticles of each other. Electron is the particle, and positron is an
antiparticle. They both have the same mass, but have equal and opposite charge. The charge of an
electron is -1.6x10-19c, and the charge of a positron 1.6x10-19c.
Antiparticles have:
• Same mass as the original particle.
• Opposite charge of the original particle.
• Opposite spin of the original particle.
• Opposite value of baryon number, lepton number, and strangeness.
The first antiparticle discovered was anti-electron, which is named as positron. It is usually notes as
e+. Other antiparticles are denoted as the normal symbol of the particle, but with a bar over it.
Pair Production
A particle and an antiparticle can be produced from a high energy photon, or by collision between
two other particles. The photon must have sufficient energy to produce the rest mass of two
particles. So, its energy must be at least twice the rest energies of the two particles. If it is greater
than this, the surplus energy is converted into the kinetic energy of the particles.
According to the conservation law of mass-energy, the energy of the photon is equivalent to the
energy of the produced particles.

97
Annihilation
When a particle and its antiparticle interacts, they are converted to energy, in the form of photons.
This process of mass to energy conversion is called annihilation. If a particle and its antiparticle
produce two photons, so we can say, according to the conservation law,
2ℎ# = 2%&!
ℎ# = %&!
Electronvolt (eV)
It is another unit of energy. This is the energy required to move 1 electron which is accelerated
through a potential difference of 1V. So, we know, 1eV = 1.6x10-19J.
Rest Mass
The mass of subatomic particles are always described as their rest mass. In other words, mass of
subatomic particle which is not moving. This is because Einstein’s special theory of relativity says
that the mass of anything increases when it is moving, and since the particle can move very fast, this
increment can be considerable.
Rest Energy
This is linked to the rest mass. According to the equation, E=mc2, the rest energy can be converted to
rest mass by dividing with c2. The rest energy is usually measured in electronvolts.
Spin
This is an important property of subatomic particles. This can sometimes be considered as angular
momentum. Spin takes values such as 0, ±
"
!
, ±1, ±
#
!
, ±2, and so on.

98
Particle Classification
All particles can be classified into hadrons and leptons. Hadrons experience strong nuclear force,
however, leptons do not.
Leptons
Properties:
• They have spin ½ or -½.
• They are acted on weak nuclear force.
• They are fundamental particles and cannot be sub-divided further.
• All leptons have lepton number +1, and all anti-leptons have lepton number -1.
• All particles which are not leptons have lepton number 0.
The most familiar example of leptons is electrons. Electrons are stable and they do not decay.
Muons and Taus are also leptons. They decay quite readily into other particles. Electrons, Muons,
and Taus, each have their corresponding neutrinos. They have no charge and mass, and only interact
very weakly with matter. Hence, they are very hard to detect.
All six leptons have antiparticles with opposite spin, charge and lepton number.
Particle Symbol Charge Antiparticle
Electron e- -1 e+
Electron Neutrino ve 0 v̅e
Muon μ- -1 μ+
Muon Neutrino vμ 0 v̅μ
Tau τ- -1 τ+
Tau Neutrino vτ 0 v̅τ
Hadrons
Hadrons are sub-divided into two groups. They are Baryons and Mesons.
Baryons
• They have spin 0.5, -0.5, 1.5, or -1.5.
• They are not fundamental particles.
• They are composed of quarks.
• All baryons have baryon number +1, and all anti-baryons have baryon number -1.
• All other particles other than baryons have baryon number 0.
• Baryons are the heaviest group of particles.

99
Protons and neutrons are baryons. The only stable baryon is proton. It has a half-life of about 1032
years. So, proton decay will be so rare that it is virtually unobservable. All other baryons decay
readily, most with a half-life of about 13 minutes when they are outside the nucleus. A neutron
decays to produce a proton, an electron, and an anti-electron neutrino.
Baryon Chart
Proton p +1 p̅
Neutron n 0 n̅
Lambda λ 0 λ5
Sigma+ Σ+ +1 Σ5+
Sigmao Σo 0 Σ5o
Sigma- Σ- -1 Σ5-
Xi+ +1
Xio 0
Xi- -1
Particles
Leptons Hadrons
Baryons Mesons
Quarks

100
Mesons
• Mesons have mass between leptons and baryons.
• Their spins are whole numbers (0, +1, -1, +2, -2).
• They are not fundamental particles. They consist of quarks.
• All mesons have a very short half-life.
Meson Chart
Pion+ π+ +1 π̅+
Piono πo 0 π̅ o
Pion- π- -1 π̅ -
Kaon+ κ+ +1 κ̅ +
Kaono κo 0 κ̅ o
Kaon- κ- +1 κ̅ -
Eta η 0 η̅
Quarks
• There are 6 quarks altogether. Each has its corresponding anti-quark.
• Quarks experience strong nuclear force.
• Quarks are the constituent particles of hadrons.
• They are considered as fundamental particles.
• Quarks have not been observed in isolation.
• Quarks have baryon number 1/3, and anti-quarks have baryon number -1/3.
• Quarks have spin 0.5 or -0.5.
• Quarks and anti-quarks have lepton number 0.
• Baryons are formed from three quarks.
• Mesons are formed from 1 quark and a non-corresponding anti-quark.
Quark Table
Up u +2/3 u̅
Down d -1/3 d;
Charm s +2/3 s̅
Strange c -1/3 c̅
Top t +2/3 t̅
Bottom b -1/3 b;

101
Strangeness
Strangeness explains why some reactions cannot take place.
Properties:
• Strange quarks have strangeness -1.
• The anti-quark of strange quark has strangeness +1.
• All other quarks and leptons have strangeness 0.
The strangeness of a hadron can be found by adding the strangenesses of its constituent quarks.
Quark Compositions and Strangenesses of some Hadrons:
Particle Quarks Strangeness
Proton uud 0
Neutron udd 0
Pion+ ud- 0
Kaono ds̅ +1
Kaon+ us̅ +1
Sigma+ uus -1
Lambda uds -1
Fundamental Forces
There are four fundamental forces:
• Gravitational force – It acts between masses and it is always attractive in nature. Its range is
infinite.
• Electromagnetic force – This is the force between all charged objects. It can be attractive or
repulsive. Its range is infinite.
• Weak nuclear force – It acts on all particles, that is, on both leptons and quarks. It has a
range less than 10-17m. It is responsible for beta decay and interactions involving quark
change. The electromagnetic and weak nuclear forces are now thought to be different
aspects of the same force, so they are sometimes called electroweak force together.
• Strong nuclear force – It acts on hadrons and quarks. Its range is very short and it acts only
within the nucleus. It is responsible for holding the nucleus together.
The order of strength is:
Strong nuclear > Electromagnetic > Weak Nuclear > Gravitational

102
Particle Exchange Model for Four Interactions
The idea behind this model is that forces are acting because of virtual particles being exchanged
between interacting particles. The virtual particles are considered to form clouds surrounding the
interacting particles.
Large Hadron Collider (LHC)
The Large Hadron Collider is a giant synchrotron, over 8km in diameter and built 100m under the
ground, bordered between Switzerland and France. This machine is designed to collide protons with
each other. Scientists believe that it will produce new particles which were not seen after the Big
Bang.
There are four critical experiments in the LHC. They are:
Compact Muon Solenoid (CMS) – This discovers the Higgs Boson, a new fundamental particle. From
CMS experiments, it is hoped that the LHC will make mini black holes, dark matter, super symmetric
particles, gravitons, etc.
Large Hadron Collider Beauty (LHCB) – This detector looks for the decay of bottom and charm
quarks from mesons. Scientists want to observe why our universe contains mostly matter and very
little antimatter. Theoretically, they should be in equal amounts.
A Toroidal LHC Apparatus (ATLAS) – This is done to verify the new fundamental particle, Higgs
Boson. This also wants to figure out extra dimensions in space.
A Large Ion Collision Experiment (ALICE) – The idea of this experiment is to find quark-gluon plasma
which has been predicted by quantum mechanics theory.
Detectors must be capable of:
• Measuring momentum and signs of charge.
• Measuring energy.
• Identifying the charged particle (if any) like electrons, muons, etc.
• Inferring the presence of the undetectable neutral particle, neutrino.
NOTE: Anti-hydrogen was made by LHC, but it did not last long.

103
Antimatter
This is the matter composing of antiparticles. Antimatter is a matter which has electrical charges
reversed. Anti-electrons (positrons) are like electrons with a positive charge. Antimatter and matter
behave same way towards gravity.
Law of Conservation of Particle Interaction
When particles interact, some of their properties remain conserved. These properties are:
• Momentum
• Mass-Energy
• Charge
• Baryon number
• Lepton number
• Strangeness
Momentum
During particle interaction, the total momentum remains conserved, provided that no external force
is acting on them.
Mass-Energy
During particle interaction, energy can be used to produce mass and mass can be used make energy.
In a reaction, if energy is produced, total mass decreases. On the other hand, if mass of the products
become large, it means that energy is provided during this reaction. If initial mass of two interacting
particles is mi and mf, the change in mass is,
∆" = $"! − ""$
& = "'#
If the final mass of the system is larger than the initial mass, it ensures energy is provided to the
system.
Charge
During particle interaction, the total charge remains conserved.

104
Baryon number
Charge Characteristics of quarks:
Quarks Relative Charge Exact Charge
u, c, t +
!
"
+
!#
"
u̅, c̅, t̅ −
!
"
−
!#
"
d, s, b −
$
"
−
#
"
d*, s̅, b* +
$
"
+
#
"
The total number of baryon before interaction is equal to the total number of baryon after
interaction. As we know, baryons are composed of three quarks. Individual quarks of a baryon have
a baryon number of 1/3, and the individual antiquarks of an antibaryon have a baryon number of
-1/3.
Lepton Number
All leptons have lepton number 1, and all anti-leptons have lepton number -1. All other particles
other than leptons have lepton number 0.
Strangeness
Strangeness is -1 for all strange quarks, and +1 for all anti-strange quarks. All other quarks have
strangeness 0.
Boson Table
Force Exchange Boson Symbol Charge
Electromagnetic Photon γ 0
Weak Nuclear W boson
Z boson
W-
W+
Zo
-1
+1
0
Strong Nuclear Gluon g 0
Gravitational Graviton undetermined undetermined

106
Stability of Nucleus
All atoms contain a nucleus at their centres. Protons, which are positively charged, remains inside
the nucleus, while the electrons, which are negatively charged, revolves (orbits) around the nucleus.
The total charge of an atom is zero, even though the nucleus is positively charged, because there are
equal numbers of protons and electrons in an atom, and due to the fact that the charge of a proton
is equal and opposite to the charge of an electron.
The total number of protons in an atom is called the atomic number, and the total number of
nucleons (sum of protons and neutrons) is called the mass number, or the atomic mass. Each
element has atoms of distinct atomic and mass numbers, characteristic to the particular element.
Inside the nucleus, electrostatic repulsive force acts between protons. Therefore, the protons tend
to move away from each other. However, they are held together in the nucleus due to strong
nuclear force.
The stability of a nucleus depends on the ratio of protons and neutrons in the nucleus. The pattern
of stability can be analyzed from a neutron number against proton number graph.
For small nuclei, whose proton numbers are not high, stability is achieved if they have equal
numbers of protons and neutrons in their nuclei. If the number of protons is more, the electrostatic
repulsive force increases, but the strong nuclear force does not increase at the same proportion, due
to its short range. When a proton is added to the nucleus, it will exert roughly the same force of
repulsion on the other protons inside the nucleus. This is because all protons have nearly the same
separation. However, strong nuclear force is only effective between adjacent neighbors. To make
the nucleus stable, more neutrons should be added. The extra neutrons will provide the strong
nuclear force, which will reduce the effect of the electrostatic repulsion force to the extent that the
nucleons stay together. Thus, larger nuclei achieve stability if the number of neutrons is greater than
the number of protons.
Most stable nuclei have equal numbers of protons and neutrons. This implies that two neutrons and
two protons (an alpha particle) is the most stable composition for a nucleus. Thus, 16O, 28Si, and 56Fe
are also elements with stable nuclei. When more protons and neutrons are added, nuclei move to
higher energy levels, and become unstable again. To achieve stability once again, the nuclei tries to
return to its lower energy levels. Such nuclei undergoes break-down, emitting radiation. The types of
radioactive decay are alpha decay, beta decay, and decay by emitting gamma radiation.

A2 PHYSICS - Notes.pdf

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A2 PHYSICS - Notes.pdf