CHAPTER -1
ELECTRIC CHARGES AND FIELD
When a glass rod is rubbed with silk it is able to attract light bodies such as small
pieces of paper, straw, light feathers etc. similarly a plastic comb passed through dry
hair can attract such light objects.
The agency which gives this attracting power is called “electricity”. The name
electricity was taken from the Greek word “elektron” which means amber.
The bodies which acquire this power are said to be electrified or charged. The
electricity produced buy friction is called frictional electricity. If the charges in a
body do not more, then the frictional electricity is also known as „static electricity‟ or
electrostatics.
“The branch of a physics which deal with the study of electric forces, electric
fields and electric potentials due to charges at rest is known as electrostatics”.
Electric charge:
“Electric Charge is an intrinsic property of elementary particles of matter
which gives rise to electric force between various objects “
“The additional property of protons and electrons, which gives rise to electric
force between them, is called electric charge”
The Additional property of protons and electrons, which gives rise to electric
force between them, is called electric charge”.
Electric charge is a scalar quantity. Its SI unit is Coulomb (C). A proton has a
positive charge (+e) and an electron has a negative charge (-e)
Where e=1.6X 10-19C.
Two types of electric charges on different objects
The following experiment reveals this fact;
Experiment 1:
1. Rub a glass rod with silk and suspend it from a rigid support by means of silk
thread. Bring another similarly charges rod near to it. The two rods repel each other.
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2. Now rub a plastic rod with wool and suspend it from a rigid support. Bring
another similarly charged rod near to it. There will be repulsion between the two
rods.
3. Bring a plastic rod rubbed with wool near the charged glass rod. The two rods
attract each other.
Experiment 2:
If a glass rod rubbed with silk, is made to two small pith balls (polystyrene balls)
which are suspended by silk threads, then two balls repel each other, but it as seen
that a pith ball touched with glass rod attracts another pith ball touched with a plastic
rod.
From the above experiments, we note that the charge produced on a glass rod is
different from the charge produced on a plastic rod. Also the charge produced on a
pith ball touched with a glass rod is different from the charge produced on pith ball
touched with a plastic rod. We can conclude that
1. There are only two kinds of electric charges positive & negative.
2. Like charges repel & unlike charges attract each other.
The statement 2 is known as the fundamental law of electro statistics.
The above experiments also demonstrate that the charges are transferred from the
rods to the pith balls on contact. We say that the pith balls have been electrified or
charged by contact.
The property which differentiate two types of charges is called polarity of charge
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Gold leaf electroscope:
Gold leaf electroscope is used to detect charge on the body.
Construction:
It constructs of a box with glass window a metal rod with a small metal sphere is
inserted into the box through a rubber cork. A folded gold leaf is fixed tothe another
end of the metal rod.
Working:
Charged Body is kept in contact with the metal sphere, charge flows from the
charged body into the metal rod and then into the gold leaf, When leaves are charged,
they repel each other. Hence leaves are diverged. The extent of divergence indicates
the amount of charge in the body.
Conductors and Insulators:
Conductors:
“The substances through which electric charges can flow easily are called
conductors”.
They contain a large number of free electrons which make them good conductor of
electricity metals, human and animal bodies, graphite‟s acids, alkalis, etc are
conductors.
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Insulators:
“The substances through which electric charges cannot flow are called
insulators”.
In the atoms of such substances electrons of the outer shell are tightly bound to the
nucleus. Due to the absence of free charge carriers, these substance offer high
resistance to the flow of electricity through them. Most of the non-metals like glass,
diamond, porcelain, plastic, nylon, wood, mica, etc are insulators.
An important difference between conductors and insulators is that when some charge
is transferred to a conductor it readily gets distributed over its entire surface. On the
other hand, if some charge is put on an insulator, it stays at the same place.
Methods of charging:
(a) Charging by friction:
When two bodies are rubbed against each other, at the points of contact,
electrons are transferred from one body to another the body which loses electrons
becomes positively charged and the body which gains electrons becomes negatively
charged. Objects acquiring two kinds of charges on rubbing.
Sl.No
Positive Charge
Negative Charge
1
Glass rod
Silk cloth
2
Fur or woolen cloth
Ebonite, Amber, Rubber rod
3
Woolen coat
Plastic seat
4
Woolen carpet
Rubber shoes
5
Nylon or Acetate
Cloth
6
Dry hair
Comb
(b) Charging by conduction:
If a body is charged with the contact of another body then the process of
charging in known as charging by conduction.
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The nature of charge acquired by the conductor is same as that on the charging
body.
Charging by induction:
Charging of two spheres by induction:
The figure shows the various steps involved in inducing opposite charges on
two metal spheres.
(a) Hold the two metal spheres on insulating stands and place them in contact, as
shown in figure (a).
(b) Bring a positively charged glass rod near the left sphere. The free electrons of the
spheres get attracted towards the glass rod. The left surface of the left spheres
develops an excess of negative charge while the right side of the right sphere
develops an excess of positive charge. However, all of the electrons of the
spheres do not collect at the left face. As the negative charge begins to build up at
the left face, it starts repelling the new incoming electrons. Soon an equilibrium is
established under the action of force of attraction of the rod and the force of
repulsion due to the accumulated electrons. The equilibrium situation is shown in
figure (b).
(c) Holding the glass rod near the left sphere, separate the two spheres by a small
distance, as shown in figure (c). The two spheres now have opposite charges.
(d) Remove the glass rod. The charges on the spheres get redistributed. Their positive
and negative charges face each other, as shown in figure (d). The two spheres
attract each other.
Thus, the two metal spheres get charged by process called charging by induction.
In contrast to the process of charging by contact, here the glass rod not lose any
of its charges.
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Charging of a sphere by induction:
Figure shown the various steps involved in inducing a positive charged on a metal
sphere.
(a) Hold the metal sphere on an insulating stand. Bring a negatively charged plastic
rod near it. The free electrons of the sphere are repelled to the farther end. The
near end becomes positively charged due to deficit of electrons.
(b) When the far end of the sphere is connected to the ground by a connecting wire,
its free electrons flow to the ground.
(c) When the sphere is disconnected from the ground, its positive charge at the near
end remains held there due to the attractive force of the external charge.
(d) When the plastic rod is removed, the positive charged spreads uniformly on the
sphere.
Similarly, the metal sphere can be negatively charged by bringing a positively
charged glass rod near it.
Basic properties of electric charges:
(1) There are two types of electric charges, namely positive and negative. Their
effects tend to cancel each other.
(2) Like charges repel each other and unlike charges attract each other.
(3) The electric charges is additive in nature. In implies that total charge on an object
is algebraic sum of the charges located at different points in the object.
(4) Charges is quantized charges always exists in nature as an integral multiple of the
magnitude of the electronic charge.
Total charge q= ±ne.
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Where, n = 1, 2, 3 is the number of charges on a body.
e is known as fundamental or elementary charge (e =1.6×10-19 c)
(5) The electronics charge of a system is always conserved. This means charge can
neither be created nor be destroyed. But it can be transferred from one body to
another.
(6) Unlike mass, the electric charge on an object is not affected by the motion of the
object.
(7) Charges always reside on the outer surface of a conductor.
(8) Distribution of charges over a conductor depends on its shape.
Comparison of electric charge and mass:
Sl.No. Electric charge
Mass
1
Electric charge on a body may be
positive, negative or zero.
Mass of a body is always a
positive quantity.
2
Electric charge is always quantized.
Quantization of mass is not yet
established.
Q = ± ne
3
Charge is strictly conserved
Mass is not conserved as mass
can be changed into energy and
vice–versa according to the
relation. E =mC2
4
Electrostatic forces between two
electric charges may be attractive or
repulsive depending upon the kind of
charges.
Gravitational force between
two masses is always attractive.
5
Charge may not exist without mass
Mass exists without net charge
also
6
Electrostatic forces between different Gravitational forces between
charges may cancel out
different bodies never cancel
out.
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Coulomb‟s Law:
Force between two point electric charges:
“Coulomb‟s law states that the electrostatic force of attraction or repulsion
between two stationary point charges is directly proportional to the product of the
magnitude of the two charges and inversely proportional to the square of the
distance between them”.
This force acts along the line joining the two charges.
r
q1
q2
If two point charges q1 and q2 are separated by distance r, then the force f
between them is given by,
F ∝ q1 q2 and
This implies,
Fα
Fα
1
𝑟2
𝒒𝟏 𝒒𝟐
𝒓𝟐
Or
F=K
𝒒𝟏 𝒒𝟐
𝒓𝟐
where K is a constant of proportionality called electrostatic force constant.
The value of K depends on the nature of the medium between two charges and the
system of units chosen to measure F, q1, q2 and r.
For the two charges located in free space and in SI units,
we have,
K=
1
4𝜋𝜀0
= 9 X 109Nm2C-2
where 𝜀0 is called permittivity of free space. So we can express coulomb‟s law in SI
units as
F=
𝟏
𝒒𝟏 𝒒𝟐
𝟒𝝅𝜺𝟎
𝒓𝟐
The value of ε0in SI Unit is ε0 = 8.854 X 10-12C2N-1m-2or Fm-1
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Coulomb‟s Law in Vector form:
Colombian forces between like charges (q1 q2> 0) are repulsive consider two
like charges q1 and q2 separated by distance r in vacuum. Magnitude of force between
q1 and q2 in vacuum is given by
F=
1
𝑞1 𝑞2
4𝜋𝜀 0
𝑟2
In Vector form force on charge q1 due to q2 can be written as
F12 =
1
q1 q2
4πε 0
r2
r21
r 21
where r21 =
is a unit vector in the direction from q2 to q1.
r
Similarly force on charge q2 due to q1 can be written as,
F21 =
where r12 =
r 12
r
1
q1 q2
4πε 0
r2
r12
is unit vector in the direction from q1 to q2.
The Coulomb forces between unlike charges (q1q2< 0) are attractive,
proceeding as above, it can be obtained that.
F12 =
and F21 =
1
q1 q2
4πε 0
r2
1
q1 q2
4πε 0
r2
r12
r21
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Importance of vector form:
The vector form of Coulomb‟s law gives following additional information
1. As 𝑟21 = −𝑟12, therefore, 𝐹 21 = - 𝐹 12
This means that the two charges exert equal and opposite force on each other. So
Coulombian forces obeys Newton‟s third law of motion.
2. As the Coulombian forces act along 𝐹 21 = - 𝐹 12 That is, along the line joining the
centers of two charges, so they are central forces.
Unit of charge:
S.I unit of charge is coulomb (C). We can define the unit of charge from coulomb‟s
laws as follows. If q1 =q2= 1C. and r = 1m.
We get,
F=
1
𝑞1 𝑞2
4𝜋𝜀 0
𝑟2
= ( 9 × 109× 1 × 1 )/1= 9 × 109N
One coulomb of charge is that charge which when placed at a distance of one
meter from identical charge in air or vacuum repels the other by a force equal to
9 × 109newton.
Relative permittivity or Dielectric constant:
When two charges are placed in any medium other than air, the force between
them is greatly affected.
“Permittivity is a property of the medium which determines the electric force
between two charges situated in that medium”.
According to Coulomb‟s law, the force between two point charges q1 and q2, placed
in vacuum at distance r form each other is given by,
Fvac =
1
𝑞1 𝑞2
4𝜋𝜀 0
𝑟2
………………….. (1)
where εo is the absolute permittivity of free space.
For the same two charged separated by the same distance in any medium other than
vacuum, the force between them is,
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Fmed =
1
𝑞1 𝑞2
4𝜋𝜀
𝑟2
……….…………. (2)
where ε is the absolute permittivity of the intervening medium.
Dividing equation (1) by (2), we get,
F vac
F med
=
ε
ε0
The ratio of absolute permittivity (ε) of the medium to the absolute permittivity (ε o)
of the vacuum (or air) is called relative permittivity (εr) or dielectric constant (K) of
the given medium. Thus,
εr or K =
ε
ε0
=
F vac
F med
…………… (3)
So we can define dielectric constant in terms of forces between charges as follows,
“The dielectric constant or relative permittivity of medium may be defined as
the ratio of the force between two charges separated by a certain distance in free
space to the force between the same two charges separated by the same distance in
the given medium”.
NOTE:
i. For wax K=2, for glass it is between 2 and 3 (depending in nature of glass). For
mica 4 to 7 for water, K = 80. i.e., when two charges are immersed in water force
between them falls to 1/80 times, the force between same two charges, placed
same distance apart in air.
ii. There is no medium with K < 1. Its minimum value is one for vacuum.
iii. Dielectric constant of a medium with large number of free electrons (like metals)
approaches infinity. For metal K= ∞ i.e, if two charges are separated by metal as
a medium then force between them becomes zero.
Forces between multiple charges:
Principle of superposition:
Coulomb‟s law gives force between two point charges.The principle of
superposition enables us to find the force on a point charge due to a group of point
charges. This principle is based on the property that the forces with which two
charges attract or repel each other are not affected by the presence of other charges.
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“The principle of superposition states that when a number of charges are
interacting, the total force on a given charge is the vector sum of the forces exerted
on it due to all other charges. The forces due to individual charges are not affected
by the presence or absence of other charges”.
Consider that n point charges q1, q2, q3 ……qn are distributed in space in a discrete
manner. The charges are interacting with each other. Let us find the total force on the
charge say q1 due to all other remaining charges.
If the charges q2,q3 ……….qn exert forces F12, F13……..F1n on the charge q1 as
shown in figure, then according to principle of superposition, the total force on
charge q1 is given by,
F1 = F12 +F13 + ……………+ F1n
……..……… (1)
If the distance between the charges q1 and q2 is denoted as r12 and 𝑟12 is unit
vector from charge q2 to q1 then,
F12 =
1
q1 q2
4πε 0
r2 12
r12
Similarly, the force on charge q1 due to other charge q3 is given by,
F13 =
1
q1 q3
4πε 0
r2 13
1
q1 qn
4πε 0
r2 1n
r13
And so on,
F1n =
r1n
Hence in the equation (1) substituting for F12, F13, ………..F1n, the total force
on the charge q1 due to all other charges is given by,
F1 =
1
4πε 0
q1 q2
r2
12
r12 +
q1 q3
r2
13
r13 + ⋯ … … … … .
q1 qn
r 2 1n
r1n
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F1 =
𝒒𝟏
𝟒𝝅𝜺𝟎
𝒏
𝒊=𝟐
𝒒𝒊
𝒓𝟐 𝟏𝒊
𝒓𝟏𝒊
Electric Field:
“The region of space around a charge or a system of charges within which
other charged particles experience electrostatic force is called an electric field”.
Thus, a particle is said to be in an electric field if the particle experiences an
electrostatic force.
Source charge (q):
“The point charge producing electric field is known as a source charge”.
Test charge (qo):
“The infinitesimally small charge experiencing an electrostatic force in the
electric field of source charge is known as a test charge”.
Test charge acts as a detector of the electric field.
Electric Field intensity(E):
“The electric field intensity at point due to a source charge may be defined as
the force experience per unit positive test charge placed at that point”.
The electric field intensity is also called strength of electric field or simply electric
field.
Consider that a positive test charge qo experiences a force F, when placed at the
observation point. Then, electric field at the observation point is given by,
E=
F
q0
It is vector quantity.The direction of the electric intensity is same as the direction of
the force acting on the test charge.
The S.I unit of electric intensity is NC-1or Vm-1 .The dimensional formula for electric
field E is (MLT-3A-1).
Electric field intensity due to point charge:
Consider a point charge +q placed at point O. To find the electric field at P, place
a small positive test charge +qo at P. Then force acting on +qo is,
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F=
1
q q0
4πε 0
r2
r
where r is a unit vector directed from +q to +qo.
E=
F
q0
=
1
q
4πε 0
r2
r
The magnitude of electric field E at P is,
E=
𝟏
𝐪
𝟒𝛑𝛆𝟎
𝐫𝟐
4𝐴 = 𝑟 2
Electric Line of Forces or Electric Field Lines:
The Concept of electric lines of force was introduced by Michael Faraday as an
aid to visualize electric field around electric charges.
“An electric line of force in an electric field is an imaginary straight or curved path
along which a unit positive charge moves or tends to move”.
Lines of force are straight when they represent the electric field due to an isolated
charge and they are curved when they represent the field due to two or more charges
placed nearby.
The electric lines of force are also called Electric flux lines.The bunch or tube of
electric field lines of a charge is called electric flux of that charge.
Properties of Electric Line of Forces:
1. A tangent drawn to a line of force at a point on it gives the direction of the
resultant field at that point
2. Electric lines of force originate from positive charge and terminate on a negative
charge.
3. Electric lines of force are normal to the surface of a charged conductor.
4. Two electric lines of force never intersect one another. If they intersect, then there
will be two tangents. It means that there are two values of the electric field at that
point, which is not possible.
5. The electric lines of force are crowded in the regions of stronger field and are
spread out in the regions of weaker field.
6. Electric lines of force do not pass through a conductor.
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7. A uniform electric field is represented by straight, equal spaced parallel field
lines.
8. Electric lines of force contract lengthwise to represent attraction between two
unlike charges.
9. Electric lines of force exert lateral (sideways) pressure on each other and hence
repel each other to represent repulsion between like charges.
10. Electric lines of force are purely a geometric conceptual representation. It has no
physical existence.
11. The lines of forces reveal the nature of electric field.
Electric flux (ɸ):
“Electric flux through a surface is defined as the number of electric lines of
force passing normally through that surface”.
Electric flux per unit area at any point in the electric field is a measure of electric
intensity at that point.
Electricflux
ɸ
Electric intensity =
=
Area
𝐴
Electric flux is a scalar quantity. The unit of electric flux in NC-1 m2 or Jmc-1. Its
dimensional formula is (M L3 T-3 A-1)
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Suppose that a surface having area S is placed inside electric field of intensity E as
shown in fig. In Order to find the electric flux through the surface of area S, consider
small area element dS of the surface S. The elementary area dS can represent by a
vector dS, which is directed along normal to the area element dS. Suppose that
electric field E makes an angle ɵ with the area vector dS. Then, component of electric
field along normal to the area element dS i.e., along area vector dS is given by
En = E cos ɵ
Hence,electric flux crossing the area element dS in a direction along normal to it
given by,
dɸ =En dS = (E cosɵ) ds
dɸ = E .dS
The electric flux through the whole surface S can be found by integrating the above
over the whole surface S. Therefore, total electric flux through the surface S is given
by
ɸ= ∫E.ds = ∫En dS
Thus, electric flux linked with a surface in an electric field may be defined as the
surface integral of the electric field over that surface.
Electric Dipole:
“An electric dipole is a pair of equal and opposite charges separated by a very
small distance”.
A molecule made up of a positive and a negative ion is an examples of electric dipole
in nature.
Example: H2O, CO2, HCl, etc.
The distance 2a between two charges -q and +q is known as dipole length.
“The straight line joining the negative charge and positive charge and
directed from the negative to the positive charge is called the axis of the dipole”.
“A line passing through the positive and negative charges of the electric
dipole is called the axial line”.
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“A line perpendicular to the axial line and passing through the midpoint of
the line joining the positive and negative charges of the electric dipole is called
equatorial line”.
Electric Dipole Moment (P):
“The electric dipole moment of an electric dipole is defined as the product of
the magnitude of either charge or the dipole length”
Thus,
P = q × 2a
Dipole moment is a measure of strength of electric dipole. The direction of dipole
moment is from negative charge to the positive charge along the line joining the
charges. Dipole moment is a vector quantity. S.I unit of dipole moment is coulomb
meter.
Electric field intensity at a point on the axial line of an electric dipole:
Consider an electric dipole consisting of charges –q and +q separated by a
small distance 2a in free space. Let P be a point on the axial line of the dipole at a
distance r from the center of the dipole on the side of the charge +q.
If E-q is the electric intensity at P due to charge –q at a A,
then,
E-q = -
1
4πε 0
q
r+a 2
P
Suppose E2 is the electric intensity at P due to charge +q at B, then
E+q =
1
4πε 0
q
r−a 2
P
where,P is the unit vector along the dipole axis.
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Hence the resultant electric field at point P is,
Eaxial = E+q + E-q
Eaxial =
Eaxial =
Eaxial =
q
1
r−a 2
4πε 0
−
q
4ar
4πε 0
r2 − a2 2
1
2Pr
4πε 0
r2 − a2 2
1
r+a 2
P
P
P
Here P = q × 2a = dipole moment. If the dipole is short r >> a, a2can be neglected
compared to r2
𝐄axial =
𝟏
𝟐𝐏
𝟒𝛑𝛆𝟎
𝐫𝟑
𝐏
Clearly, electric field at any axial point of the dipole acts along the dipole axis from
negative to positive charge. i.e., in the direction of dipole moment P.
Electric field intensity at a point on the equatorial line of an electric
dipole:
Consider an electric dipole consisting of charges –q and +q separated by
distance 2a and placed in vacuum. Let P be a point on the equatorial line of the dipole
at distance „r‟ from the centre of the dipole.
The distance from each charge to the point P is (r2 + q2)½.
The dipole moment is,
𝑃 = q × 2a 𝑃 .
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where 𝑃 is the unit vector along the dipole axis from –q to +q.
The magnitude of the electric field at p due to the charge –q at A is then,
E-q =
1
𝑞
4𝜋𝜀 0
𝐴𝑃2
=
1
𝑞
4𝜋𝜀 0
𝑟2 + 𝑎2
along PA
The magnitude of the electric field at p due to the +q at B, then
E+q =
1
𝑞
4𝜋𝜀 0
𝐵𝑃2
=
1
𝑞
4𝜋𝜀 0
𝑟2 + 𝑎2
along BP.
Thus the magnitude of E-q and E+q are equal
E-q = E+q =
1
𝑞
4𝜋𝜀 0
𝑟2 + 𝑎2
Let PAB + PBA = Ɵ
Clearly, the components of E1 and E2 normal to the dipole axis will cancel out, the
components parallel to the dipole axis add up. The total electric field E equa is opposite
to P.
𝐸 equa = - (E-q cos Ɵ + E+q cos Ɵ) 𝑃
= - 2E-qcosƟ𝑃
𝐸 equa = - 2 X
since cos Ɵ =
( E-q = E+q)
1
𝑞
4𝜋𝜀 0
𝑟2 + 𝑎2
𝑎
𝑟2 +𝑎2
𝑃
𝑎
𝑟 2 +𝑎 2
𝐸 equa = -
1
𝑃
2
4𝜋𝜀0 𝑟 + 𝑎 2
3/2
𝑃
Where p = q × 2a is the electric dipole moment.
If the dipole is short 2a << r, then a2 may be neglected as compared to r2, then
𝐸 equa = -
1 𝑃
4𝜋𝜀0 𝑟 3
𝑃
Clearly, the direction of electric field at any point on the equatorial line of the dipole
will be anti-parallel to the dipole moment p.
NOTE:
1) At large distances, electric field due to a dipole decreases
1
𝑟3
whereas for a single
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charge electric field decreases as
2)
𝐸𝑎𝑥𝑖𝑎𝑙
1
𝑟2
.
= 2 𝐸𝑒𝑞𝑢𝑎𝑡𝑜𝑟𝑖𝑎𝑙
Expression for the Torque on a Dipole:
Consider electric dipole consisting of two charges –q and +q separated by a
distance 2a, with a dipole moment P in a uniform external electric field E.
Let ɵ be the angle between the dipole axis and the direction of electric field as
shown in fig (a). The force +qE acts on the charge +q along the direction of the field
and a force –qE acts on the charge –q opposite to the direction of field. Therefore the
net force acting on the dipole is +qE –qE = 0.
However, the two forces acting at different points result in a torque on the
dipole.
The magnitude of torque on the dipole is,
τ = either force × perpendicular distance between the force
η = QE × AC
= qE × AB sinƟ
= qE × 2a sin Ɵ
= (q × 2a) E sinƟ
τ = PE sinƟ
The torque on the dipole tends to align it, along the field direction of the
electric field.
In vector notation
τ=P×E
The torque is zero when dipole moment P is along the direction of E.
In a non-uniform electric field, the net force on the dipole will not be zero.
There will be a torque as well as net force on the dipole.
Page | 20
NOTE:
1. It may be pointed out that when dipole is placed in uniform electric filed, it
experiences only a torque. Net force on the dipole is zero. But when the electric
field is non-uniform, it experiences torque as well as a net force.
2. Torque on the dipole becomes zero, when it aligns itself parallel to the esthetic
field (Ɵ = 00)
3. Torque on the dipole is maximum, when dipole is placed as right angles to the
direction of the electric filed (Ɵ = 900)Tmax = P E sin Ɵ = PE.
Continuous distribution of charge:
As said earlier, the electric charge is quantized and the charge on a charged body
is always an integral multiple of the charge on a single electron or a proton. In most
of the practical situation, charge on a charged body is so large as compared to the
magnitude of charge on an electron or a proton that the quantization of charge may be
ignored. In other words, we can assume that on a charged body of reasonable size,
charge has continuous distribution.
The continuous distribution of a charge may be one dimensional, two
dimensional or three dimensional. Accordingly, the distribution of charge is called
linear charge distribution, surface charge distribution and volume charge
distribution respectively.
1. Linear charge density:
When charge is distributed along a line
(straight or curved), the charge distribution is called
linear. Figure shows the uniform distribution of
charge q over the length L of a straight rod. Then,
linear charge density is defined as,
λ =
𝑞
𝐿
Its unit is coulomb meter -1 (C m-1)
For example:
If charge q is uniformly distributed over a circular ring of radius R, then its linear
charge density is,
𝑞
2πR
Page | 21
2. Surface charge density:
When charge is distributed over a surface (plane or
curved), the charge distribution is called surface charge
distribution. Figure shows the uniform distribution of
charge q over a plane surface of area A. Then, surface
charge distribution is defined as,
ζ=
𝑞
𝐴
Its unit is coulomb meter-2 (Cm-2)
For example:
If charge q is uniformly distributed over the surface of a spherical conductor of
radius R, then its surface charge density is
𝑞
ζ=
4π𝑅 2
.
3. Volume charge density:
When charge is distributed over the volume of an object, it is called volume
charge distribution. Figure shows the uniform distribution of charge q over the
volume V of a spherical object. Then, volume charge density is defined as,
ρ=
𝑞
𝑉
Its unit is coulomb m-3 (Cm-3)
For example:
If charge q is uniformly distributed over the entire
volume of a sphere, then its volume charge density is,
𝑞
4/3π𝑅 3
Gauss Theorem:
Gauss theorem deals with net electric flux through a closed surface and net
charge enclosed by the surface.
Gauss theorem states that “the total electric flux through any closed surface
in free space is equal to
𝟏
𝜺𝟎
times the total charge enclosed by the surface.”
Page | 22
Consider a closed surface S enclosing charges +q1, -q2, +q3, -q4+ ………… in free space.
According to Gauss theorem, the total electric flux through the surface is given by
ɸ=
ɸ=
1
(total charge enclosed by the surface)
𝜀0
1
𝜀0
(+q1,
1
ɸ =
𝜀0
-q2, +q3, -q4 + ……………………..)
∑q
This is the mathematical form of Gauss theorem.
NOTE:
Gaussian surface:
The, expression form electric field intensity can be obtained by applying
Coulomb‟s law only in simple cases. In the situation, where Coulomb‟s law or
principle of superposition cannot be applied to calculate the electric field, the same is
achieved by using Gauss‟ law. For this, one has to evaluate the surface integral. So
that surface integral can evaluate easily, a closed surface is chosen cleverly around
the charge distribution. The surface so chosen is called the Gaussian Surface.
Thus, Gaussian surface around a charge distribution (may be a point charge, a line
charge, a surface charge or a volume charge) is a closed surface, such that electric
field intensity at all the points on the surface is same and the electric flux through the
surface is along the normal to the surface.
Gauss's theorem:
Gauss theorem states that “the total flux through a closed surface is 1/Є0 times
the net charge enclosed by the closed surface”.
Mathematically, it can be expressed as
фЄ
=
∫ E.ds =
𝟏
𝜺𝟎
Proof:
Let us consider an isolated positive point charge q, which produces an electric field
surface S is a sphere of radius r centered on q. Then surface S is a Gaussian surface.
Page | 23
Electric field at any point on S is
E=
1
𝑞
4𝜋𝜀 0
𝑟2
This field points radially outward at all points on S. Also, any area element points
radially outwards, so it is parallel to E, i.e., θ = 0°.
Flux through area dS is,
dф Є = E dS = E dS cos 0° = E dS
Total flux through surface S is,
ф Є = ∫ dф Є =∫ E dS = E ∫dS
= E ×Total area of sphere
=
фЄ =
1
𝑞
4𝜋𝜀 0
𝑟2
𝒒
4πr2
𝜺𝟎
This proves Gauss's theorem.
Applications of gauss‟s theorem:
1. Electric field due to an infinity long charged wire by using Gauss's theorem:
Let us consider a thin infinitely long straight wire having a uniform linear
charge density, λ Cm-1. By symmetry, the field E of the line charge is directed
radially outwards and its magnitude is same
at all points equidistant from the line charge.
To determine the field at a distance r from the
line charge, let us consider a cylindrical
Gaussian surface of radius r , length l and
with its axis along the line charge. It has
curved surface S1 and flat circular ends S2
and S3.
Obviously, ds1‖E , dS2 E, dS3 E. So only the
curved surface contributing the total flux.
ф Є = ∫ E . ds
Page | 24
= ∫ E . ds1 + ∫ E . ds2 + ∫ E . ds3
= ∫ E . ds1 + 0 + 0
= E ×area of the curved surface
ф Є =E×2 πrl
Charge enclosed by the Gaussian surface,
q = λl
Using Gauss's theorem, ф Є =
E 2πrl =
E=
𝑞
𝜀0
we get
λl
𝜀0
𝛌
𝟐 𝛑𝐫 𝜺𝟎
Thus the electric field of a line charge is inversely proportional to the distance from
the line charge.
2. Electric field due to a uniformly charged infinite plane sheet by using gauss's
theorem:
Fig: Gaussian surface for a uniformly charged infinite plane sheet
Let us consider a thin, infinite plane sheet of charge with uniform surface charge
density ζ. We wish to calculate its electric field at a point P which is very close to the
conductor at distance r from it.
By symmetry, electric field E points outwards normal to the sheet. Also, it must have
same magnitude and opposite direction at two points P and P' equidistant from the
sheet and on opposite sides. Let us consider a cylindrical Gaussian surface of crosssectional area A and length 2r with its axis perpendicular to the sheet.
Page | 25
As the lines of force are parallel to the curved surface of the cylinder, the flux
through the curved surface is zero. The flux through the plane-end faces of the
cylinder is,
ф Є = EA + EA = 2 EA
Charge enclosed by the Gaussian surface, q = ζ A
According to Gauss's theorem,
фЄ =
2EA =
Or
E=
𝑞
𝜀0
σA
𝜀0
σ
2𝜀 0
E is independent of r, the distance from the plane sheet.
(i) If the sheet is positively charged (ζ > 0), the field is directed away from it.
(ii) If the sheet is negatively charged (ζ < 0), the field is directed towards it.
For a finite larger planar sheet, the above formula will be approximately valid in the
middle regions of the sheet, away from its edges.
3. Electric field due to a uniformly charged thin spherical shell by using gauss's
theorem:
Let us consider a thin charged spherical shell of radius R with uniform surface
charge density ζ . From symmetry, we see that the electric field E at any point is
radial and has same magnitude at points equidistant from the centre of the shell i.e.,
the field is spherically symmetric.
Fig: Gaussian surface for outside points of a thin spherical shell of charge
To determine electric field at any point P at a distance r from O, we choose a
concentric sphere of radius r as the Gaussian surface.
Page | 26
a. When point P lies outside the spherical shell:
The total charge q inside the Gaussian surface is the charge on the shell of radius R
and area 4πR2
q =4πR2 ζ
Flux through the Gaussian surface,
фЄ = E×4 πr2
By Gauss‟s theorem,
фЄ =
𝑞
𝜀0
E × 4 πr2 =
Or
E=
𝑞
𝜀0
1
𝑞
4𝜋𝜀 0
𝑟2
[For r ˃R ]
This field is same as that produced by a charge q placed at the center O. Hence for
points outside the shell, the field due to a uniformly charged shell is as if the entire
charge of the shell is concentrated at its centre.
b. When point P lies on the spherical shell:
The Gaussian surface just encloses the charged spherical shell.
Applying Gauss‟s theorem, E ×4 πr2 =
E=
𝑞
𝜀0
[For r = R ]
𝑞
4 π𝜀 0 𝑅 2
c. When point P lies inside the spherical shell:
Fig: Gaussian surface for inside points of a thin spherical shell of charge.
As it is clear from Fig. The charge enclosed by the Gaussian surface is zero,
i.e, q = 0
Page | 27
Flux through the Gaussian surface,
ф Є = E×4 πr2
Applying Gauss‟s theorem,
фЄ =
𝑞
𝜀0
E×4 πr2 =0
Or
E=0
Hence, electric field due to a uniformly charged spherical shell is zero at all points
inside the shell.
Fig. Shows how E varies with distance r from the centre of the shell of radius r. E is
zero from r=0 to r=R we have
E∝
𝑞
𝑟2
Fig: variation of E with r for a spherical shell of charge.
*****************
Page | 28
CHAPTER-2
ELECTROSTATIC POTENTIAL & CAPACITANCE
Potential difference:
The difference of electrical potential between two points. The electric field around
a charge can be described in two ways
1. By electric field
2. By electrostatic potential or electric potential (V)
The electric field E is a vector quantity, while electric potential is a scalar
quantity
Let us know about potential difference,
Let us consider a point charge +q
located at a point O. let A & B be two points
in its electric field. When a test charge q0 is
moved from A to B, a work WAB has to be
done is moving against the repulsive force
exerted by the charge +q we then calculate the potential difference between points A
& B by the equation
V = VB-VA =
𝑊𝐴𝐵
𝑞0
i.e., “the potential difference between two points in an electric field may be
defined as the amount of work done in moving a unit positive charge from one
point to the other against the electrostatic forces”.
Know it:
In the above definition, we have assumed that the test charge is so small that it
does not disturb the distribution of the source charge. Secondly, we just apply to
much external force on the test charge that it just balances the eruptive electric force
on it and hence does not produce any acceleration in it
SI unit of potential difference is Volt (V) it has been named after the Italian scientist
Alessandro Volta.
1 Volt =
1 𝐽𝑜𝑢𝑙𝑒
1 𝑐𝑜𝑢𝑙𝑜𝑚𝑏
Page | 29
1V = 1 Nm C-1 = 1 JC-1
Hence “the potential difference between two points in an electric field is said
to be 1 volt if 1 joule of work has to be done in moving a positive charge of 1
coulomb from one point to the other against the electrostatic forces.”
Electric potential :
The electric potential at a point located far away from a charge is taken to be zero
in fig. if the point A lies at infinity then
VA = 0, so that
V= VB =
𝑊
𝑞0
Where W is the amount of work done in moving the test charge q0 from
infinity to the point B and VB refers to the potential at point B.
The electric potential at a point in an electric field is the amount of work done in
moving a unit positive charge from infinity to that point against the electrostatic
forces.
Electric potential =
work done
Charge
SI unit of electric potential is volt (v)
The electric potential at a point in an electric field is said to be 1 volt if one joule
of work has to be done in moving a positive charge of 1 coulomb from infinity to that
point against the electrostatic forces.
Electric potential due to a point charge:
Let us consider a positive charge q placed at
the origin O Let P be a point at distance r from
the charge q as shown in Fig. By definition, the
electric potential at point P will be equal to the
amount of work done in bringing a unit positive charge from infinity to the point P.
Suppose a test charge q0 is placed at for A at distance x from O. By coulomb‟s
electrostatic force acting on charge q0 is,
Page | 30
F=
1
𝑞 𝑞0
𝑥2
4𝜋𝜀 0
The force F acts away from the charge q. The small work done is moving the
test charge q0 from A to B through small displacement dx against the electrostatic
force is ,
dw = F˖dx = F dx cos180˚ = - Fdx
The total work done in moving the charge q0 from infinity to the point P will be
=−
Fdx
r 1
∞ 4πε 0
q q0
x2
r −2
x
4πε 0 ∞
q q0
=-
==W =
r
∞
dw = -
W=
q q0
dx
dx
−1 r
4πε 0
x ∞
q q0
1
4πε 0
r
−
1
q q0
4πε 0
r
1
∞
Hence the work done in moving a unit test charge from infinity to the point P, or the
electric potential at point P is,V =
Clearly V∝
1
𝑟
W
q0
=
1
q
4πε 0
r
Thus the electric potential due to a point charge is spherically
symmetric as it depends only on the distance of the observation point from the charge
and not on the direction for that point with respect to the point charge. Moreover, we
note that the potential at infinity is zero.
Page | 31
Fig: shows the variation of electrostatic potential (V∝ 1/r ) and the electrostatic field
(E ∝
1
𝑟
) With distance r from a charge q.
Electric potential due to a dipole:
Electric potential at an axial point of a dipole
Consider an electric dipole consisting of two point charges –q, + q and separated
by distance 2a. Let P be a point on the axis of the dipole at a distance r from its centre
O.
Electric potential at point p due to the dipole is,
V = V1+V2
1
−𝑞
4𝜋𝜀 0
𝐴𝑃
1
−𝑞
4𝜋𝜀 0
𝑟+𝑎
V=
V=
V=
V=
V=
+
𝑞
1
4𝜋𝜀 0
𝑟−𝑎
1
𝑞
4𝜋𝜀 0
𝐵𝑃
+
−
𝑞
2𝑎
4𝜋𝜀 0
𝑟2− 𝑎2
1
𝑃
4𝜋𝜀 0
𝑟2− 𝑎2
1
𝑞
4𝜋𝜀 0
𝑟−𝑎
1
𝑟+𝑎
where P = q x 2a is the dipole moment.
For a short dipole, a2 <<r2
So,
V=
1
𝑃
4𝜋𝜀 0 𝑟 2
Page | 32
Electric potential at an equatorial point of a dipole:
Consider an electric dipole consisting of charges –q & +q and separated by
distance 2a Let P be a point on the perpendicular bisector of the dipole r from its
centre O electric potential at point P due to the
dipole is,
V = V1+V2
V=
V=
1
−𝑞
4𝜋𝜀 0
𝐴𝑃
+
1
−𝑞
4𝜋𝜀 0
𝑟2+ 𝑎2
1
𝑞
4𝜋𝜀 0
𝐵𝑃
+
1
𝑞
4𝜋𝜀 0
𝑟2− 𝑎2
V = 0.
Electric potential at any general point due to dipole:
Consider an electric dipole consisting of two point charges – q & +q and
separated by distance 2a, as shown in fig we wish to determine the potential at a
point p at a distance r from the centre O, the direction OP making an angel 𝜃 with
dipole moment P
Let AP = r1 and BP = r2
Net potential at point p due to the dipole is,
V = V1+V2
V=
1
−𝑞
4𝜋𝜀 0
𝐴𝑃
+
1
𝑞
4𝜋𝜀 0
𝐵𝑃
Page | 33
V=
V=
1
−𝑞
4𝜋𝜀 0
𝑟1
𝑞
1
4𝜋𝜀 0
𝑟2
+
−
1
𝑞
4𝜋𝜀 0
𝑟2
1
𝑟1
To find r1,
From ∆ACO,
cos𝜃 =
cos𝜃 =
𝑎𝑑𝑑
𝑕𝑦𝑝
𝑦
𝑎
where, y = a cos𝜃
Then, r1= r + y
i.e., r1 = r+ a cos𝜃
To find r2,
From ∆BDO,
cos𝜃 =
𝑥
𝑎
where, x = a cos𝜃
Then, r2= r – x
i.e., r2= r- a cos𝜃
w.k.t, V= V1 + V2
V=
V=
V=
q
1
4πε 0
r+a cos θ
q
1
r−a cos θ
− r+ a cosθ+ r+ a cosθ
r2 − a2 cos2 θ
4πε 0
q
−
2a cosθ
4πε 0 r2 − a2 cos2 θ
∵ a2 Cos2θ˂˂ r
V=
V=
q
4πε 0
1
4πε 0
2a cosθ
r2
P cosθ
r2
where P=q×2a is the dipole moment.
Page | 34
Special cases:
When the point p lies on the axial line of the dipole, 𝜃=0˚or 180˚ and
V=
±1
𝑃
.
4𝜋𝜀 0 𝑟 2
i.e., the potential has greatest positive or the greatest negative value.
When the point p lies on the equatorial line of the dipole, 𝜃=90◦ & V=0 i.e., the
potential at any point on the equatorial line of the dipole is zero. However, the
electric field at such points is non-zero.
Differences b/w electric potentials of a dipole and a single charge:
The potential due to a dipole depends not only on distance r but also on the
angle between the position vector r of the observation point and the dipole moment
vector P. The potential due to a single charge depends only on r.
The potential due to a dipole is cylindrically symmetric about the dipole axis. If
we rotate the observation point P about the dipole axis (keeping r and 𝜃 fixed). The
potential V does not change. The potential due to a single charge is spherically
symmetric.
As large distance, the dipole potential falls off as
1
1
r2
while the potential due to a single
charge falls off as .
r
Electric potential due to a system of charges:
Electric potential due to a group of point charges:
Suppose N point charges q1, q2 q3 …… qN lie at distances r1p, r2p, r3p…..rNp from
a point P electric potential at point P due to charge q1 is,
V1=
1
𝑞1
4𝜋𝜀 0
𝑟1𝑝
Similarly, electric potentials at point P due to
other charges will be
Page | 35
V2 =
V3 =
1
𝑞2
4𝜋𝜀 0
𝑟2𝑝
1
𝑞3
4𝜋𝜀 0
𝑟3𝑝
1
𝑞𝑁
4𝜋𝜀 0
𝑟 𝑁𝑝
So on,
VN =
As electric potential is a scalar quantity, so the total potential at point P will be
equal to the algebraic sum of all the individual potentials i.e.,
V = V1+V2+V3+………+VN
V=
1
𝑞1
4𝜋𝜀 0
𝑟1𝑝
+
1
𝑞2
4𝜋𝜀 0
𝑟2𝑝
+
1
𝑞3
4𝜋𝜀 0
𝑟3𝑝
+ ……
1
𝑞𝑁
4𝜋𝜀 0
𝑟 𝑁𝑝
Or,
𝑁
V=
𝑖=0
1
4𝜋𝜀0
𝑞𝑖
𝑟𝑖𝑝
If r1, r2, r3…..rN are the position vectors of the N point charges, the electric
potential at a point whose position vector is r, would be
𝑵
V=
𝟏
𝒊=𝟎 𝟒𝝅𝜺𝟎
𝒒𝒊
𝒓 − 𝒓𝒊
Electric potential due to a uniform by charged thin spherical shell:
Consider a uniformly charged spherical shell of radius R and carrying charge
q. We wish to calculate its potential at point P at distance r from its centre O, as
shown in Fig.
Page | 36
a) When the point P lies outside the shell:
We know that for a uniformly charged spherical shell, the electric field outside
the shell is as if the entire charge is concentrated at the centre. Hence electric
potential at an outside point is equal to that of a point charge located at the centre,
which is given by
V=
1
𝑞
4𝜋𝜀 0
𝑟
[For r>R]
b) When point P lies on the surface of the shell:
Here r = R. Hence the potential on the surface of the shell is,
V=
1
𝑞
4𝜋𝜀 0
𝑅
[For r>R]
c) When point p lies inside the shell:
The electric field at any point inside the shell is zero. Hence electric potential due
to a uniformly charged spherical shell is constant everywhere inside the shell and
its value is equal to that on the surface thus,
V=
1
𝑞
4𝜋𝜀 0
𝑅
[For r>R]
Equipotential surfaces:
An equipotential surface is a surface with a constant value of potential at all
points on the surface.
Page | 37
Since electrostatic potential is constant on an equipotential surface, the potential
difference between any two points on that surface is zero.
Therefore, no work is done is moving a charge from one point to another point on the
equipotential surface. For a single point charge q, the potential is given by
V=
1
𝑞
4𝜋𝜀 0
𝑟
This shows that V is constant if r is constant. Thus, equipotential surfaces of a single
point charge are concentric spherical surfaces with the charge at the centre.
The electric field lines for a single point charge are radial lines starting from a
positive charge or ending on a negative charge. Therefore, electric field line at every
point is normal to the equipotential surface passing through that point.
The drawings of equipotential surfaces given us an alternative visual picture of
electric field in addition to the picture of electric field lines around a charge
configuration.
For uniform electric field E, say along the x-axis, the equipotential surfaces are
planes parallel to the y-z plane.
Page | 38
Relation between electric field and potential :
The quantity
dv
dr
is the rate of change of potential with distance and is called potential.
Potential energy of a system of charges:
Consider the simple case of two point charges q1 & q2 separated by a distance r.
The electric potential due to the charge q1 at a distance r is
V1 =
1
𝑞1
4𝜋𝜀 0
𝑟12
By definition, V1 is the work done is bring a unit positive charge from infinity to
a distance r and q1 against the electric field due to q1. It is also equal to the potential
energy of a unit positive charge at a distance r from q1.
Then potential energy of the charge q2 placed at a distance r from q1 will be
U= V1q2 =
1
𝑞1 𝑞2
4𝜋𝜀 0
𝑟12
Page | 39
This expression also gives potential energy of a system of two charges, equation
is true for any sign of q1 and q2 for like charges q1q2>0, the potential energy is +ve &
+ve work is needed to bring the charges from infinity to a finite distance apart against
the force of repulsion. For unlike charges q1q2<0, a negative work is needed to bring
the charges from infinity to a finite distance apart or a positive work is needed to take
the charges from a finite distance apart to an infinite distance against the force of
attraction .
Eq. (1) can be generalized for a system of any number of charges. Consider a
system of three charges q1, q2, & q3 placed at A, B, and C respectively
AB= r1 , BC= r2 , AC= r3
First the charge q1 is brought from infinity to the position A and no work is
done for this, Next the charge q2 is brought from infinity to B at a distance r1 and q1
therefore Work is done.
And work done is,
U1 =
1
𝑞1 𝑞2
4𝜋𝜀 0
𝑟1
The charges q1 & q2 produce electric field all around. The electric potential at a
point C which is at a distance r3 from q1 & r2 from q2 is,
V=
1
𝑞1
4𝜋𝜀 0
𝑟3
+
1
𝑞2
4𝜋𝜀 0
𝑟2
The work done in bringing q3 from infinity to C is,
U2 =
U2 =
𝑞1
𝑟3
1
4𝜋𝜀0
+
1
𝑞1 𝑞3
4𝜋𝜀 0
𝑟3
1
4𝜋𝜀0
+
𝑞2
𝑟2
𝑞3
1
𝑞2 𝑞3
4𝜋𝜀 0
𝑟2
Hence total work done is assembling the three charges at the locations A, B & C is ,
U = U1+U2=
1
𝑞1 𝑞2
4𝜋𝜀 0
𝑟1
+
1
𝑞1 𝑞3
4𝜋𝜀 0
𝑟3
+
1
𝑞2 𝑞3
4𝜋𝜀 0
𝑟2
Page | 40
U=
𝑞1 𝑞2
𝑟1
+
𝑞1 𝑞3
𝑟3
+
𝑞2 𝑞3
𝑟2
This expression gives the potential energy of a system of the three charges.
Potential energy is an external field:
1. Potential energy of a single charge:
Let us consider an external electric field the charges q is brought from infinity
to the point P in the external electric field then the work done is qV. Where V is
the potential due to external field at P this work is stored in the form of potential
energy of q. If the point P has position vector „r‟ relative to some potential energy
of q at r in an external field = q V(r)
Where V(r) is the external potential at the point P
Thus, if an electron with charge
q = e = 1.6 X 10-19C is accelerated by a potential difference of V=1 volt it would
gain energy of qV – 1.6 x-19 J
The unit of energy is defined as 1electron volt or lev.
i.e., lev = 1.6 x 10-19J the unit based on eV are most commonly used in atomic,
nuclear and particle physics.
2. Potential energy of a system of two charges in an external field:
Let V1 & V2 be the potentials due to external electric field at two points
separate by a distance r. Let a charge q1 be brought from infinity and placed at the
point where external potential is V1.The work done against the external field is
q1V1.
Then a charge q2 is brought from infinity and placed at the other point where the
external potential is V2. Now the work done against the external field is q2V2 and
the work done on q2 against the field due to q1 is,
Therefore the total work done on q2 against two fields (external field & charge q1)
is
U1 =
1
𝑞1 𝑞2
4𝜋𝜀 0
𝑟
+ q2V2
Page | 41
The potential energy of the system of two charges in the external electric field is
equal to the total work done in assembling the configuration of two charges is the
external electric field. It is given by,
U = q1V1 + q2V2+
1
𝑞1 𝑞2
4𝜋𝜀 0
𝑟
Potential energy of a dipole in an external field :
Consider a dipole with charge +q and –q separate by a small distance 2a. The
magnitude of its dipole moment is P=qx2a. Let the dipole be placed in a uniform
electric field E with its dipole moment P making an angel 𝜃 with the direction of
electric field E.
In a uniform electric field, the net force experienced by the dipole is zero But it
experience a torque given by
η = P×E
Whose magnitude is η = PE sin𝜃. This torque will tend to rotate the dipole to
align it with electric field E Let an external torque of the same magnitude be applied
in the opposite direction such that it neutralizes this torque and rotates the dipole
through a small angle d𝜃 without angular acceleration the small amount of work
done during this process is,
dw = η d𝜃 = PE sin𝜃 d𝜃
Therefore the total work done is rotating the dipole from orientation 𝜃1 to 𝜃2 is,
W=
𝜃2
𝑃𝐸
𝜃1
𝑠𝑖𝑛𝜃 𝑑𝜃
W= PE [- cos𝜃]
Page | 42
W = - PE [cos 𝜃2 – cos 𝜃1]
The potential energy of a dipole is an electric field is defined at the amount of the
work done against the electric field in bringing the dipole from infinity and placing it
in the desired infinity and placing it in the desired orientation in the field. The dipole
is perpendicular to the direction of the field
The work done in bringing the charge +q and the charge –q from infinity to the
same potential are equal in magnitude and opposite in sign. Therefore the total work
done is zero.
Work done is zero and potential energy is zero. Work is done only in rotating the
dipole from the position perpendicular to the field to any other position.
This work done is stored as the potential energy of the system.
𝜋
Taking 𝜃1= and 𝜃2=𝜃
2
W=
𝜃
𝜋
2
𝑃𝐸 𝑠𝑖𝑛𝜃 𝑑𝜃
W = PE [- cos 𝜃]
𝜋
W = - PE [ cos 𝜃 – cos ]
2
W = - PE cos 𝜃
W=-P·E
Thus, potential energy of a dipole of dipole moment P when placed at angle 𝜃 to
the direction of electric field E is,
U = - P E 𝒔𝒊𝒏𝜽 = - P · E
Electrostatics of conductors:
Conductor is a substance that can be used to carry electric charges from one place
to the other. They contain mobile charge carriers. All metals are good conductors of
electricity. In metals, the outer electrons (valence) are free to move within the metal.
Following are some important results regarding electrostatics of conductors.
1. Inside a conductor, electric field in zero:
When conductors is placed in an electric field Eext then –ve charges are
induced on the left end and +ve charges are induced on the right end of the
conductor. The process continues till the electric field set up by the induced
Page | 43
charges becomes equal and opposite to the field Eext The net field E= (Eext - Eint )
inside the conductor will be zero.
2. At the surface of a charged conductor, electrostatic field must be normal to
the surface at every point:
If there was a component of electric field directed parallel to the surface,
then the excess charge on the surface would be forced into accelerated motion by
this component. It is charge is set into motion, then the object upon which it is on
is not in a state of electrostatic equilibrium.Therefore, the electric field must be
entirely perpendicular to the conducting surface for objects that are at
electrostatic.
3. The interior of a conductor can have no excess charge in the static situation:
 A natural conductor has equal amounts of positive and negative charges in
every small volume or surface element.
 When the conductor is charged, the excess charge can reside only on the
surface in the static situation, this follows from the Gauss‟s law.
 Consider any arbitrary volume element V inside a conductor. On the closed
surface S bounding the volume element V. electrostatic field is zero. Thus the
total electric flux through S is zero. Hence, by Gauss‟s law, there is no net
charge enclosed by S. This means there is no net charge at any point inside
the conductor and any excess charge must reside at the surface.
4. Electrostatic potential is constant throughout the volume of the conductor
and has the same value (as inside) on its surface:
 Since E=0 inside the conductor and has no tangential component on the
surface no work is done in moving a small test charge within the conductor
and on its surface.
 That is there is no potential difference between any two points inside or on the
surface of the conductor.
 E=
𝛿𝑉
𝛿𝑙
if E is zero then mathematically V should be a constant differentiation
of a constant is zero.
5. Electric field at the surface of a charged conductor:
 We know that E=
𝜍
𝜀0
𝑛 where ζ is the surface charge density and 𝑛 is a unit
vector normal to the surface in the outward direction.
 Let us consider a pill box (a short cylinder) as the Gaussian surface a/bout any
point P on the surface, as shown in Fig.
Page | 44
 The pill box is partly inside and partly outside the surface of the conductor. It
has a small area of cross section 𝛿𝑠 and negligible height.
 Inside the surface, the electrostatic field is zero but outside, the field is normal
to the surface with magnitude E.
 The total flux through the pill box therefore comes only from the outside
(circular) cross section of the pill box
 For small area 𝛿𝑠, E may be considered constant and E and 𝛿𝑠 are parallel or
anti-parallel. The charge enclosed by the pill box is 𝜍𝛿𝑠.
By Gauss‟s law,
E 𝛿𝑠 =
i.e., E =
σ 𝛿𝑠
𝜀0
σ
𝜀0
 Electrified is normal to the surface which is true for both signs of ζ for ζ >0
electric field is normal to the surface outward, for ζ < 0 electric field is
normal to the surface inward.
Electrostatic shielding:
 Consider a conductor with a cavity, with no charges inside the cavity. Therefore
the electrified inside the cavity is zero, whatever be the size and shape of the
cavity and whatever be the charge on the conductor and the external fields in
which it might be placed.
 The electric field inside a charged spherical shell is zero. But if the conductor is
charged or charges are induced on a neutral conductor by an external field, all
charges reside only on the outer surface of a conductor with cavity.
 Whatever may be the charge and field configuration outside any cavity is a
conductor remains shielded from outside electric influence. The field inside the
cavity is always zero. This is known as electrostatic shielding.
Page | 45
Polar and non-polar molecules:
 In a non-polar molecule, the centres of positive and negative charges coincide.
The molecule then has no permanent (intrinsic) dipole moment
Example: O2 and H2 molecules because of their symmetry, have no dipole
moment.
 In a polar molecule the centres of positive and negative charges are separated.
Such molecules have permanent dipole moment. Example : of molecules such as
HCl or Water (H2O)
Dielectrics and Polarization :
 Dielectrics are non-conducting substances and hence they have no charge
carriers.
 In a dielectric free movement of charges is not possible. The external field
induces dipole moment by stretching or re-orienting molecules of the dielectric.
The collective effect of all the molecular dipole moments is net charges on the
surface of the dielectric which produce a field that of a conductor and a
dielectric opposes the external field.
 The opposing field so induced does not exactly cancel the external field but only
reduces it. The extent of the effect depends on the nature of the dielectric.
 The molecules of a substances may be polar or non-polar.
 In an external electric field, the positive and negative charges of a non-polar
molecule are displaced in opposite directions. The displacements stops when the
external force on the constituent charges of the molecule is balanced by the
restoring force (due to internal fields in the molecule)
 The non-polar molecule thus develop an induced dipole moment. The dielectric
is said to be polarized by the external field.
Page | 46
 The induced dipole moments of different molecules add up giving a net dipole
moment of the dielectric in the presence of the external field.
 In a polar molecule dielectric, polar molecular are oriented randomly due to
thermal agitation; so the total dipole moment is zero. When an external field is
applied, the individual dipole moments tend to align with the filed. When
summed over all the molecules, there is then a net dipole moment in the
direction of the external field, i.e., the dielectric is polarized.
 The polarization depends on the external field tending to align the dipoles with
the field and thermal energy tending to disrupt the alignment. The alignment
effect is more important for polar molecular.
 Therefore in both the case, whether polar or non-polar, a dielectric develops a
net dipole moment in the presence of an external field. The dipole moment per
unit volume is called polarization and is denoted by P.
 If the induced dipole moment is in the direction of the field strength then
substances are called linear isotropic dielectrics.
 For linear isotopic dielectrics, P= χeE where χe is a constant called electric
susceptibility of the dielectric medium
 Electric field due to polarization opposes the electric field and ability to decease
the electric intensity is determined by its dielectric constant K.
 The magnitude of electric field applied for which dielectric breakdown takes
place in a dielectric material is called dielectric strength of the material.
Page | 47
Effect of dielectric is an external field
 Let us consider a rectangular dielectric slab placed in a uniform external field E0
parallel to two of its faces.
 The field causes a uniform polarization P of the dielectric. Thus every volume
element ∆V of the slab has a dipole moment P ∆V in the direction of the field.
Anywhere inside the dielectric, the ∆V volume element has no net charge
(through it has net dipole moment) this is because, the positive charge of one
dipole sits close to the negative charge of the adjacent dipole.
 However, at the surfaces of the dielectric normal to the electric field there is a net
charge density.
 As shown in Fig., the positive ends of the dipoles remain unneaturalised at the
right surface and the negative ends at the left surface.
 The unbalanced charges are the induced charges due to the external field thus the
polarized dielectric is equivalent to two charged surfaces with induced surface
charge densities say ζ p and -ζ p
 Clearly, the field produced by these surface charges ζ p opposes the external field.
The total field in the dielectric is thereby, reduced from the case when no
dielectric is present.
Capacitors and Capacitance :
 A capacitor is a system of two conductors separated by an insulator. The
conductors have charges, say Q1 & Q2 and potentials V1 & V2, with potential
difference
V= V1=V2
 The conductors may be charged by connecting them to the two terminals of a
battery Q is called the charge of the capacitor, though this, in fact, is the charge
on one of the conductor the total charge of the capacitor is zero.
Page | 48
 The electric field in the region between the conductors is proportional to the
charge Q
 That is, if the charge on the capacitor is double, the electric field will also be
doubled at every point, now, potential difference V is the work done per unit
positive charge in taking a small test charge from the conductor 2 to 1 against the
field. Therefore, V is also proportional to Q, and the ratio
𝑄
𝑉
is a constant. C =
𝑄
𝑉
the constant C is called the capacitance of the capacitor.
 C is independent of Q or V the Capacitors C depends only on the geometrical
configuration (shape, size, separation) of the system of two conductors.
 The SI unit of capacitance is farad (1 farad = a coulomb volt) or
1 F = 1 C/V
 A capacitor is symbolically shown as while the one with variable capacitance is
shown as
 For large C V is small for a given Q. This means a capacitor with large
capacitance can hold large amount of charge Q at a relatively small V. High
potential difference implies strong electric field around the conductors. A strong
electric field can ionize the surrounding air and accelerate the charges so
produced to be oppositely charged plates, thereby neutralizing the charge on the
capacitor plates, at least partly.
 In other words, the charge of the capacitor leaks away due to the reduction in
insulating power of the intervening medium.
 The maximum electric field that a dielectric medium can withstand without break
down (of its insulating property) is called its dielectric strength for air it is about 3
x 106 Vm-1.
The Parallel Plate Capacitor:
A Parallel plate capacitor consists of two large plane parallel conducting plates
separated by a small distance.
The intervening medium between the plates to be vacuum
Let A be the area of each plate and d be the separation between them
The two plates have charges Q and – Q
Since d is much smaller than the linear dimension of the plates (d2 << A). Plate 1 has
surface charge density ζ =Q/A and plate 2 has a surface charge density -ζ
Page | 49
We know that E=
𝜍
2𝜀 0
𝑛 (refer electric charges and fields chapter for the eqn)_
Applying for outer region I (region above the plate I)
E=
𝜍
2𝜀 0
−
𝜍
=0
2𝜀 0
For outer region II (region below the plate
E=
𝜍
2𝜀 0
−
𝜍
2𝜀 0
=0
In the inner region between the plates 1 and 2, the electric fields due to the two
charged plates add up
E=
𝜍
2𝜀 0
+
𝜍
2𝜀 0
=
𝜍
𝜀0
The direction of electric field is from the positive plate to the negative plate. The
electric field will be uniform in the region between the two plates. The electric field E
is equal to the potential gradient, for uniform electric field we have,
E=
But we know that,
E=
𝑉
or V= Ed
𝑑
𝜍
𝜀0
substituting for ζ
E=
𝑄
𝜀0 𝐴
V = Ed
V=
𝑄𝑑
𝜀0 𝐴
Page | 50
The capacitance C of parallel plate capacitor is,
C=
𝑄
𝑉
Substituting for V in the equation we get,
C=
𝜀0 𝐴
𝑑
Therefore, the capacitance of a parallel plate capacitor with air between the plates is,


Directly proportional to the area of each plate.
Inversely proportional to the distance between the plates.
Effect of Dielectric on the Capacitance:
A Parallel plate capacitor have two large plates, each of area A, separated by a
distance d. The charge on the plates is ± Q, corresponding to the charge density ± ζ
(with ζ = Q/A). When there is vacuum between the plates.
E0 =
𝜍
𝜀0
and potential difference V0 = E0d. The capacitance is C0 =
𝜀0 𝐴
𝑑
Consider a dielectric inserted between the plates fully occupying the
intervening region. The dielectric is polarised by the filed and the effect is equivalent
to two charged sheets with surface charge densities ζp and – ζp. The electric filed in
the dielectric then corresponds to the case when the net surface charge density on the
plates is ± (ζ – ζp).
E=
𝜍 − 𝜍𝑝
𝜀0
So that the potential difference across the plates as V = Ed
𝜍 − 𝜍𝑝
V=
𝜀0
d
For linear dielectrics, to be proportional ζp to E0, thus (ζ-ζp) is proportional to ζ
ζ - ζp =
𝜍
𝐾
where K is a constant characteristic of the dielectric. Clearly K>1
Therefore,
V=
𝜍𝑑
𝜀0 𝐾
=
𝑄𝑑
𝐴𝜀 0 𝐾
The capacitance C, with dielectric between the plates, is then
Page | 51
𝑄
C=
C=
𝑉
𝜀 0 𝐴𝐾
𝑑
……………………. (2)
The products ε0K is called the permittivity of the medium and is denoted by,
ε = ε0K
For vacuum K=1 and ε = ε0 is called the permittivity of the vacuum. The
dimensionless ratio.
K=
𝜀
𝜀0
is called the dielectric constant of the substance
It is clear that K is greater than 1,
We have eqn (1) and eqn(2)
C0 =
𝜀0 𝐴
𝑑
and
C=
𝜀0 𝐴
𝑑
K
Substituting C0 in C, i.e., C0K= C
K=
𝐶
𝐶0
Thus, the dielectric constant of a substance is the factor (>1) by which the
capacitance increases from it vacuum value, when the dielectric is inserted fully
between the plates of a capacitor.
Combination of Capacitors:
We can combine several capacitors of capacitance C1C2 …Cn to obtain a system
with some effective capacitance C Two simple possibilities are
1. Series Combination
2. Parallel Combination
Series Combination:
Page | 52
 Let Capacitors C1 and C2 be connected in series as shown in the figure.
 The left plate of C1 and the right plat of C2 are connected to two terminals of a
battery and have charges Q and – Q respectively.
 It then follows that the right plate of C1 has charge – Q and the left plate of C2 has
charge Q
 This would result in an electric field in the conductor connecting C1 and C2
Charge would flow until the net charge of both C1 and C2 is zero and there is no
electric field in the conductor connecting C1 and C2.
 Thus, in the series combination, charges on the two plates (±Q) are the same on
each capacitor.
 The total potential drop V across the combination is the sum of the potential
drops V1 and V2 across C1 and C2 respectively.
V = V1 + V2
V=
𝑄
𝐶1
𝑄
+
𝐶2
1
V=Q
+
𝐶1
1
𝐶2
Let C be effective capacitance of two capacitors with charge Q and potential
difference V. The effective capacitance of the combination is
C=
𝑄
and V=
𝑉
𝑄
𝐶
Comparing equation 1 and equation 2 we get,
𝑄
𝐶
=Q
1
1
=
𝐶 𝐶
1
1
𝐶1
+
+
1
𝐶2
1
𝐶2
For „n‟ number of capacitors arranged in a similar series way. Equation 3 can be
generalized as,
1
1
=
𝐶 𝐶
1
+
1
𝐶2
+ ……………………….
1
𝐶𝑛
Therefore the reciprocal of effective capacitance in series combination is the sum of
reciprocals of the individual capacitances.
Page | 53
Capacitors in Parallel :
The figure shows two capacitors arranged in parallel
In the case, the same potential difference is applied across both the capacitors. But
the plate charges (±Q1) on capacitor 1 and the plate charges (±Q2) on the capacitor 2
are not necessarily the same Q1 = C1V, Q2 = C2V
The equivalent capacitor is one with charge Q=Q1+Q2 and potential difference V
where, Q = CV
Substituting for Q1 and Q2 we have,
Q = Q1+Q2
Q = C1V + C2V
Q = V (C1 + C2)
we know that, C=
𝑄
𝑉
and Q= CV
Comparing equation 1 and equation 2 we get,
CV = V (C1 + C2)
C = C1 + C2
For „n‟ number of capacitors arranged in a similar series way, Equation 3 can be
generalized as,
C= C1 + C2+ ………………… +Cn
Therefore the effective capacitance in parallel combination is the sum of the
individual capacitances
Energy Stored In a Capacitor:
A capacitor is a device used to store the energy. The process of charging up of a
capacitor involves the transferring of electric charges from its one plate to another.
The work done in charging the capacitor is stored as its electrical potential energy.
Page | 54
Expression for the energy stored in a capacitor:
Let us consider a capacitor of capacitance C. initially its two plates are
uncharged. Suppose the positive charge is transferred from plate 2 to plate1 bit by bit.
In this process, external work has to be done. Because at any stage plate 1 is at higher
potential that the plate 2.
Suppose at any instant the plate 1 and plate 2
have charges Qʹ and -Qʹ respectively as in fig.
then the potential difference between the two
plates will be Vʹ=
𝑄ʹ
𝐶
Suppose now a small additional charge dQʹ be transferred from plate 2 to plate 1. The
work done will be,
dw = Vʹ dQʹ
dw =
𝑄ʹ
𝐶
dQʹ
Therefore the total work done in transferring a charge Q from plate 2 to plate1 will
be, W = 𝑑𝑤
𝑄 𝑄ʹ
0 𝐶
W=
𝑄2
W=
W=
𝑑𝑄ʹ
2𝐶
1
𝑄2
2
𝐶
This work done is stored as electrical potential energy U of the capacitor.
U=
U=
U=
U=
1
𝑄2
2
𝐶
1
C2 V2
2
C
1
2
1
2
CV2
QV
Energy stored in a series combination of capacitors:
For a series combination, Q is constant
Page | 55
Total energy,
U=
U=
𝑄2
1
2
𝐶
𝑄2
1
2
𝐶1
𝑄2
U=
2 𝐶1
1
+
𝐶2
+ ……………………….
𝑄2
𝑄2
1
𝐶𝑛
𝑄2
+ 2𝐶 + 2𝐶 + ……………… + 2𝐶
2
3
𝑛
U = U1 + U2 + U3 + …………………………. Cn
Energy stored in a parallel combination of capacitor:
For a parallel combination, V is constant
Total energy,
U=
𝑄2
1
2
𝐶
U=
1
𝐶2𝑉2
2
𝐶
1
U = CV2
2
1
U = ( C1+C2+C3+………)V2
2
1
1
1
2
2
2
U = C1V2+ C2V2+ C3V2+…….
U = U1+U2+U3+……..
Hence total energy is additive both in series and parallel combination of capacitors.
VAN DE GRAAFT GENERATOR :
Page | 56
 The principle of the machine is as follows
 Consider a large spherical conducting shell of radius R, having a charge Q on it
which spreads uniformly on it. Potential on it is
1
𝑄
4𝜋𝜀 0 𝑅
 Let a small sphere of radius r, carrying charge +q, be placed close to the big shell
of is
1
𝑞
4𝜋𝜀 0 𝑟
on the surface of the smaller shell and
1
𝑞
4𝜋𝜀 0 𝑅
on the bigger
shall due to q.
 The total potential difference due to both charges q and Q V(r) – V(R) is positive.
V(R) =
V(r) =
1
𝑄
4𝜋𝜀 0
𝑅
1
𝑄
4𝜋𝜀 0
𝑅
V(r) - V(R) =
𝑞
1
4𝜋𝜀 0
𝑟
+
+
𝑞
𝑅
𝑞
𝑟
−
1
𝑅
 Let us now connect the smaller and large sphere by a wire.
 The charge q on the smaller will immediately flow on to the bigger shell even
through the charge Q may be quite large. This is due to the reason that potential
on the bigger shell is smaller than smaller shell and charges move from higher to
lower potential.
 Thus q of the small charged sphere is transmitted into the large one, in this way
we can keep adding up large and larger amount of charge on the bigger shell.
 The potential at the outer sphere would also keep rising, at least until we reach
the breakdown field of air. This is the principle of the Van de Graaft generator.
 Van de Graaft generator machine is capable of building up potential difference of
a few million volts, and fields close to the breakdown field of air which is about 3
x 106 V/m.
 A schematic diagram of the van de Graft generator is given in Fig.
 A large spherical conducting shell (of few meters radius) is supported at a height
several meters above the ground on an insulting column.
 A long narrow endless belt insulating material, like rubber or silk, is would
around two pulleys – one at ground level, one at the centre of the shell. This belt
is kept continuously moving by a motor driving the lower pulley.
 It continuously carries positive charge, sprayed on to it by a brush at ground level,
to the top. There is transfers its positive charge to another conducting brush
connected to the large shell. Thus positive charge is transferred to the shell, where
it spreads out uniformly on the outer surface.
Page | 57
 In this way, voltage differences of as much as 6 or 8 million volts (with respect to
ground) can be built up.
 The resulting large electric field are used to accelerate charged particles
(electronic protons, ions) to high energies as in the case of electron microscope.
*******************
Page | 58
CHAPTER-3:
CURRENT ELECTRICITY
Introduction:
Electric charge is a property of a matter by virtue of which it shows tendency
of attraction or repulsion with a charged body.
Current electricity is a branch of physics which deals with the study of
charges in motion.
Charges in motion constitute an electric current. Electric current occurs in
natural phenomena like lightning in which charges flow from the clouds to the earth
through rain. When we apply electric field to a conductor flow of current will takes
place in the direction of applied electric field. Like a river current is the flow of water
molecules, electrical current is the flow of charged particles under the action of
electric field.
Electric current:
Electric current is the measure of amount of charge flow through any cross
section of a conductor. It gives the amount of charge flowing through any cross
section of a substance in unit time (i.e. in one second).
Definition of electric current: Electric current is defined as the time rate of
flow of charges through any cross section of a substance in one second.
Conventional current: Conventional current is the current whose direction
is along the direction of the motion of positive charges under the action of electric
field.
Conventional current is directed along the direction of applied electric field
Electronic current direction is the direction in which electrons moves under
the action of electric field.
The conventional current direction is opposite to that of motion of electrons
under the action of electric field. That is opposite to the direction of applied electric
field. When an electric field is applied to a conductor, electrons will move from
Page | 59
negative terminal to positive terminal through conductor since they are attracted in
the direction opposite to the applied electric field.
Units of electric current: The SI unit of current is ampere (A) Since ampere is a
large unit of current, so commonly used smaller units are
 miliampere (mA)
 microampere (µA)
 nanoampere (nA)
Definition of one ampere: Current through a conductor is said to be one ampere if
one coulomb of charge flows through any cross section of a conductor in one
second.
Why current is a scalar quantity?
Even current has both magnitude and direction it is a scalar quantity but not
vector quantity, because it does not obey vector laws and we can find the net current
by applying simple law of algebra.
Different types of current:
(1) Steady current: A current is said to be steady current if its magnitude and
direction do not changes with time.
Definition of steady current: let Q be the total amount of charge flowing through
any cross section of a substance in a time interval t then steady current is defined
as
I = Q/t.
Varying or variable direct current: current which changes its magnitude and
polarity (or direction) remains same with time is called varying or variable direct
current.
Definition of varying or variable direct current:
Let ∆Q be the net charge flowing across a cross section of a conductor during
the time interval ∆t [i.e., between times t and (t+∆t)].Then, the current at time t across
the cross-section of the conductor is defined as the value of the ratio of ∆Q to ∆t in
the limit of ∆t tending to zero.
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That is,
I = lim𝑡→∞
Δ𝑄
Δ𝑡
Alternating current:
Alternating current is a current whose magnitude varies continuously and
direction changes periodically.
Carriers of current:
The charged particles which are flowing in a definite direction set up electric
current are called carriers of current.
Charge carriers in different phases of the matter:
1. In solid conductors:
Types of conductor in solid phase
Carriers of current
Metals
Free electrons
Intrinsic semiconductor
Electrons and holes
P-type semiconductor
Holes are the majority charge carriers
and electrons are the minority charge
carriers
n-type semiconductor
Electrons are the majority charge
carriers and holes are the minority
charge carriers
2. In liquids:
In electrolyte liquids, the current carriers are positively and negatively charged
ions.
Ex: In the solution of CuSO4 , the current carriers are Cu2+ and SO42- ions.
3. In gases:
In ionized gases, positive and negative ions and electrons are the current
carries.
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In vacuum tubes like radio valves cathode ray oscilloscope, picture tube etc; free
electrons emitted by the heated cathode act as charge carriers.
Microscopic view of electric current in a conductor (OR Mechanism of
the flow of electric charges in a conductor):
Conductors or the metals are the material which has high conductivity or low
resistivity. In conductors charge carriers are negatively charged free electrons. So,
current in conductor is carried by the free electrons in the back ground of fixed
positive ions.
Case1: In the absence of electric field,
In the absence of electric field the electrons are moving randomly due to
thermal motion. The thermal velocity is of the order of 10 5 to 106 ms-1. We can
represent the random motion of free electrons in a conductor by a zigzag path as
shown below. The electrons which are moving due to thermal motion collide with the
fixed ions as well as with the other free electrons. An electron colliding with an ion or
an electron emerges with the speed as before the collision. The direction of its
velocity after the collision is completely random. Because of the randomness of
velocities of electrons, at a given time t the average of these thermal velocities of the
all electrons is zero. So the number of electron passing through any cross section of
the conductor is equal to the number electrons passing in opposite direction. Hence
there will be no net current through the conductor.
Case2: When the electric field is applied:
When an electric field is applied the free electrons which are moving randomly
get accelerated in the direction opposite to the applied electric field. The motion of
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free electrons is still random but they get drifted in the direction opposite to the
applied electric field with a velocity known as drift velocity.
Drift velocity:
Drift velocity is the velocity with which free electrons get drifted in a direction
opposite to the applied electric field.
Derivation for drift velocity:
If we consider a conductor in the absence of electric field the free electrons
which are present in the conductor moves randomly due to its thermal motion. If
there are N number of electrons having velocity v1,v2,…………vN respectively at a
given time t. Then their average velocity will be zero since their direction are
random. If vi be the velocity of ith electron then at a given time t,
(vi) average = ( v1 +v2+………..+VN) / N = 0 ……………(1)
(When the electric field is applied the free electrons get accelerated in the direction opposite to the
applied electric field due to their negative charge.)
The acceleration (a) of an electron in the presence of electric field E is,
a=
−eE
.……………(2)
m
Here e and m are mass and charge of the electron.
(When a ith electron having acceleration a collides with a ion or other free electron its velocity
gained by the electric field is destroyed, and the electrons emerges with a velocity equal to its
velocity due to thermal motion.)
Let vi be the velocity of the electron soon after the collision and Vi be its velocity
soon before the next collision. And let ti be the time elapsed between two collisions.
Then by laws of motion,
Vi = vi + a ti
Drift velocity,
Vd = (Vi)average
(Vi)average = (vi)average + a (ti)average
Here,
(vi)average = 0 (since from equation(1)) and
(ti)average = η
(called as relaxation time)
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Vd= aη
Hence,
From equation (2),
Vd= (
−eE
m
)η
Relaxation time:
It is the average time elapsed between two successive collisions of the free
electrons with the ions or other free electron.
Ohm‟s law :
Statement:
Ohm‟s law states that the study current through a conductor is directly
proportional to the potential difference between two ends of the conductor.
Provided the temperature and other physical conditions remains unchanged.
Explanation:
If I is the steady current through a conductor and V is the potential
difference between its ends then then by ohm‟s law,
I α V.
This implies,
I = KV ……….. (1)
Where K is constant of proportionality and is called conductance. And
conductance is the reciprocal of resistance(R). That is,
K=
1
R
……………. (2)
So from equation (1) & (2) this implies that,
I=
V
R
V = IR
Resistance:
Resistance is the opposition offered by the conductor to the flow of current
through it.
Page | 64
The SI unit of resistance is ohm (Ω)
Definition of one ohm: The resistance of a conductor is said to be one ohm, when a
potential of one volt applied across it produces a current of one ampere.
At a given temperature the resistance of a material is,
 Directly proportional to the length of the conductor(L)
 And inversely proportional to the cross sectional area (A).
That is,
1
R α L and R α .
𝐴
Combining,
1
Rα .
𝐴
And hence for a given conductor
𝐿
R = ρ( ).
𝐴
where ρ is the resistivity of the given material of the conductor.
Resistivity:
Resistivity of a material is defined as it is the resistance of a conductor of unit
length and unit area of cross section.
Ohm‟s law in vector form:
In general if I is the steady current through a conductor and V is the potential
difference between its ends then then by ohm‟s law,
……………… (1)
V = IR
Here R is called resistance.
𝐿
R = ρ ( ).
𝐴
So equation (1) becomes,
𝐿
1
𝐴
𝐴
V = ρ I ( ) = ρ L( ) ………………(2)
𝐼
Current per unit area ( ) is called current density and is denoted by j. the SI unit of
𝐴
current density is A/m2 .
Page | 65
V=jρL
So,
If E is the magnitude of uniform electric field in the conductor whose length is L then
potential difference V= EL
This implies that,
EL = j ρ L
Or
E=jρ
Since the current density is also directed along E it is also a vector. So we can
write the above equation as,
𝐸=𝐽ρ
Or
𝐽 = 𝜍𝐸
Where ζ is the conductivity of the material.
Expression for current (I), current density (𝒋) and conductivity (σ)in
terms of relaxation time:
Consider a conductor of length ∆x and consider a planar area A located
inside the conductor such that the normal to the area is parallel to applied electric
field E.
In the presence of 𝐸 electrons are get drifted opposite to E. due to drift, in an
infinitesimal amount of time ∆t, all electrons to the left of the area at a distance up to
∆x = vd∆t would have cross the area.
If n is the number of free electrons per unit volume of the conductor then
ther will be nA 𝑉𝑑 ∆t electrons in the given conductor.
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Since each electron carries a charge –e, the total charge transported across
this area A to the right in time ∆t is,
-neA 𝑉𝑑 ∆t
Let ∆Q be the amount of charge passing through area A in time ∆t then by
the definition of steady current ∆Q = I ∆t .
The total charge transported in the direction of electric field is the negative
of charge transported from left to right.
That is,
∆Q = I ∆t = - (-neA 𝑉𝑑 ∆t).
This implies that
I = neA 𝑉𝑑 ∆t ………….. (1)
This is the expression for current in terms of drift velocity.
The drift velocity in terms of relaxation time (η) is given by
Vd = −e 𝐸 /m η ……………(2)
Hence expression for current in terms of drift relaxation time is given by
I=
Or
𝑛𝑒𝐴𝑒 𝐸 τ
𝑚
I=
𝑛 𝑒 2𝐴 𝐸 τ
𝑚
……………..(3)
By definition of current density the current I interms of magnitude of current density
𝑗
I = 𝑗A
is given by,
This implies that,
Or
𝑗 A=
𝑗 =
𝑗
𝑛 𝑒 2𝐴 𝐸 τ
𝑚
𝑛𝑒 2 𝐸 τ
=
𝑚
𝑛𝑒 2 τ
𝑚
𝐸
…………… (4)
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The vectors 𝑗 is parallel to 𝐸 hence we can write the above equation in vector form
𝑗 =
as,
𝑛𝑒 2 τ
𝑚
𝐸 …………….(5)
This is the expression for current density in terms of relaxation time. Ohm‟s law in
𝑗 =ζ𝐸
vector form is given by,
…………….(6)
Comparing equation (5) with equation (6) we get,
ζ=
𝑛𝑒 2 τ
𝑚
…………… (7)
Equation (7) is the expression for conductivity in terms of relaxation time.
Resistivity is the reciprocal of conductivity that is ρ =
1
1
𝜍
𝑛𝑒2 τ
𝑚
So, ρ = =
=
𝑚
𝑛𝑒2 τ
1
𝜍
……………. (8)
Equation (8) gives the expression for resistivity in terms of relaxation time.
Limitations of ohm‟s law:
1. Ohm‟s law is applicable for conductors at constant temperature.
2. Ohm‟s law is not applicable for conductors at very low and very high
temperatures.
3. It is not applicable for semiconductors.
4. It holds good only for steady current.
Resistors:
A device which is used to introduce resistance in an electric circuit is called
resistor.
Different types of resistors are used in electronic system some of them are
1. Wire bound resistors and
2. Carbon resistors.
1. Wire bound resistors:
Wire bound resistors are made by winding the wires of an alloy, namely
manganin, constantan, nichrome. These materials are used to make resistors
because there resistivity is highly insensitive to the temperature.
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These resistors have the value of resistance in the range of a fraction of an ohm to a
few hundred ohms.
2. Carbon resistors:
A carbon resistor consists of fine graphite powder mixed with suitable binding
agent moulded into a cylinder with wire leads for electrical connections.
Colour coding of carbon resistors:
Carbon resistors are small in size and hence their values are given using a
colour code. The resistors have a set of co- axial coloured rings on them whose
significance are listed in the table.
Colours
First and second band
Third band
Black
0
100
Brown
1
101
Red
2
102
Orange
3
103
Yellow
4
104
Green
5
105
Blue
6
106
Violet
7
107
Grey
8
108
White
9
109
Gold
-
10-1
Silver
-
10-2
No
colour or
band
-
-
Tolerance band
± 5%
± 10%
± 20%
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 The first two bands from the end indicates the first two digits of value of resistance
in ohm(Ω)
 The third band indicates the decimal multiplier.
 The forth band stands for tolerance or possible variation in percentage about the
indicated values.
Example 1:
If the resistor has following colour bands,
First band – red (2)
Second band – red(2)
Third band – red (102)
Fourth band – silver (± 10%)
Then the value of the resistance of the resistor is (2.2 ×102 Ω) ± 10%
Example 2:
Find the colour code of the resistor having value of resistance 36×104 Ω ± 5%




The first digit in the value of the resistance is 3 so first band is orange.
The second digit in the value of resistance is 6 so the second band is blue.
The decimal multiplier in the value of resistance is 104 so the third band is yellow.
Tolerance or possible variation in percentage about the indicated values is ± 5%.
So the last band is gold.
Hint to remember value of the colour bands:
B. B. R O Y of Great Britain had Very Good Wife wearing Gold Silver Necklace
Temperature dependence of resistivity:
Resistivity of a material is found to be dependent on the temperature. Over a
limited range of temperature the resistivity of a conductor is approximately given by,
ρT = ρ0 [ 1 + α(T – T0 )]
where, ρT – resistivity of the conductor at temperature T
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ρ0 – the resistivity at temperature T0 (i.e, reference temperature)
α – temperature co-efficient of resistivity and its dimension is(temperature)-1
α=
1 𝑑𝜌
𝜌
𝑑𝑡
α is positive for conductors and hence resistivity increases with temperature. And
for semiconductor α is negative and hence resistivity decreases with temperature.
Electrical Energy, Power:
Consider a conductor of resistance R with end points A and B, in which a
current I is flowing from A to B.
A
I
R
B
Let V(A) and V(B) be the potential at the end points A and B respectively.
Then the potential difference between A and B is given by
V = V(A) – V(B) ˃ 0
…………… (1)
The potential energy of charge Q at A and B are QV(A) and QV(B)
respectively.
Suppose in a time interval ∆t, an amount an amount of charge ∆Q = I∆t travels from
A to B then the change in potential energy
∆Upot = Final potential energy – Initial potential energy
This implies that
∆Upot
= ∆Q [V(B) – V(A )] ………….. (2)
From equation (1) & (2)
∆Upot
= - ∆Q V
And
∆Q = I ∆t
So,
∆Upot = - I V ∆t
..…………. (3)
If charges moved without collisions through the conductor then change in kinetic
energy ∆K according to conservation of charges is given by
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∆K = - ∆Upot
That is,
∆K = I V ∆t ˃ 0
…………….. (4)
Thus in case of charges moving through the conductor without collision then
their kinetic energy would increase.
But in practise the charge carriers (free electrons) moves with a steady drift
velocity because of the collisions with the ions. During collision with the ions the
energy gain by the electron due to the electric field is dissipated in the form of heat
energy.
For a time interval ∆t the amount of energy dissipated as heat in the
conductor is given by,
∆W = IV∆t
…………… (5)
Power dissipated(P) across the conductor is the energy dissipated per unit time.
That is,
P=
∆𝑊
………….. (6)
∆𝑡
From equation (5) ∆W = IV∆t so,
P=
And hence
IV ∆t
∆t
P = IV
Using ohm‟s law V = IR, we get
P = I(IR) = I2R
Or
P=
𝑉2
𝑅
This is the equation for power dissipated in the conductor of resistance R
caring a current I.
Transmission voltage:
Consider a device with resistance R to which power P is to be delivered via
transmission cables having a resistance Rc. If V is the voltage across the device and I
current through it then,
P = VI
Page | 72
This implies,
I=
𝑃
………………(1)
𝑉
The power dissipated in the transmission cables Pc is given by,
Pc = I2Rc
………………….(2)
From equation (1) and (2)
Pc =
𝑃2
𝑉2
𝑅𝑐
…………………..(3)
This implies that Pc is inversely proportional to V2 .So in order to reduce
power dissipation transmission cables it must carry current at high value of V.
Combination of resistors – series and parallel:
Resistors can be connected in two ways
1. Series combination and
2. Parallel combination
1. Series combination:
Two or more resistances are said to be in series if they are connected end to
end so that current is same in all the resistance for the applied potential difference.
Let three resistors of resistance R1, R2 & R3 are connected in series. A
potential difference V is applied across the combination. Let V1, V2, V3 be the
potential difference across R1, R2 & R3 respectively. Let I be the current entering the
combination.
Then total potential across the combination is equal to the sum of voltages
across each resistor.
That is,
V = V1 + V2 + V3
……………… (1)
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And from ohm‟s law, V1 = IR1, V2 = IR2, V3 = IR3
This implies that ,
V = IR1 + IR2 + IR3 = I(R1 + R2 + R3)
……………(2)
If we replace the combination by effective resistance Rs then
V = IRs
……………. (3)
Comparing equation (2) and (3)
Rs = R1 + R2 + R3 …………… (4)
Equation (4) gives the expression for equivalent resistance of series
combination of three resistors.
If we have N number of resistors having resistance R1, R2, R3, ……….. RN
connected in series then effective resistance of the combination,
Rs = R1 + R2 + R3 +…………. RN
Thus the effective resistance of the series combination of resistance is equal
to the sum of individual resistance.
2. Parallel combination:
Two or more resistors are said to be in parallel if one end of all the resistors
is joined together and similarly other ends joined together.
Page | 74
Let R1, R2 & R3 be the resistances connected in parallel. Let I1, I2 and I3 be
current flowing through R1, R2 & R3 respectively and let V be the potential across the
combination.
The current (I) entering the combination is given by
I = I1 + I2 + I3
And by Ohm‟s law we can write
I=
𝑉
𝑅1
+
𝑉
𝑅2
+
𝑉
1
=V
𝑅3
𝑅1
+
1
+
𝑅2
1
𝑅3
…………. (1)
If Rp is the effective resistance of the combination then we have,
𝑉
I=
…………….. (2)
𝑅𝑝
From equation (1) and (2) we have
𝑉
𝑅𝑝
1
That is,
𝑅𝑝
1
=V
=
1
𝑅1
+
𝑅1
+
1
𝑅2
1
𝑅2
+
+
1
𝑅3
1
𝑅3
If we have parallel combination of N number of resistors of resistance R1, R2, R3,
……….. RN
Then ,
𝟏
𝑹𝒑
=
𝟏
𝑹𝟏
+
𝟏
𝑹𝟐
+
𝟏
𝑹𝟑
+……………….
𝟏
𝑹𝑵
Thus reciprocal of effective resistance of a parallel combination is equal to the sum of
reciprocals of individual resistances.
Problem:
1. Find the effective resistance of the
combination of resistors in the circuit. If the
voltage between A and C is V, find the
current in the circuit.
Here the resistor R1 is in series with the
Page | 75
23
parallel combination of R2and R3. Replacing R2an R3 by equivalent resistance 𝑅𝑒𝑞
,
1
23
𝑅𝑒𝑞
=
23
Or 𝑅𝑒𝑞
=
1
𝑅2
+
1
𝑅3
𝑅2 𝑅3
𝑅2 +𝑅3
23
The circuit now has 𝑅1 and 𝑅𝑒𝑞
in series and hence their combination can be
123
replaced by equivalent resistance 𝑅𝑒𝑞
123
23
𝑅𝑒𝑞
= 𝑅𝑒𝑞
+ 𝑅1
If the voltage between A and C is V, then by Ohm‟s law I is given by
I=
𝑉
123
𝑅𝑒𝑞
=
𝑉
𝑅 𝑅
𝑅1 + 𝑅 2+𝑅3
2 3
Cells, Emf, Internal Resistance:
Cell is a device which maintains a steady current in an electric circuit.
Basically a cell has two electrodes called the positive (P) and the negative (N) as
shown in the figure. These electrodes are immersed in an electrolytic solution.
Let V+ be the potential difference between positive electrode and the solution
immediately adjacent to it marked A in the figure.
Similarly, the negative electrode develops a negative potential –(V-) relative to the
electrolyte adjacent to it, marked as B in the figure.
Circuit symbol
Page | 76
When there is no current through the cell the potential difference between P and N is
𝛆 = V+ - (V-) = V+ + V………………. (1)
And this is called as electromotive force (emf).
(Note: emf is actually not a force it is potential difference. The name electromotive force however
is used because of historical reasons and was given at a time when the phenomena was not
understood properly).
Definition of electromotive force (emf):
emf is the potential difference between the positive and negative electrodes of
a cell in an open circuit.
If we connect a resistor of finite value of resistance R in between the two
electrodes a current I flows through R and the cell through the electrolytic solution.
The electrolytic solution through wich a current I flows has a finite resistance r called
the internal resistance of the cell.
The potential difference between the electrodes when a current I flows
through the cell is given by,
V = (potential difference between P and A) + (potential difference between A and B) +
(potential difference between B and N)
That is,
V = V+ + V- - Ir = 𝛆 – Ir
…………………… (2)
In the above expression the potential difference between A and B takes negative sign
because the current I flows from B to A.
Here V is also the potential difference across R, we have from Ohm‟s law
V = IR
…………………. (3)
Combining equation (2) and (3)
IR = 𝛆 – Ir
Or
I=
ε
𝑅+𝑟
The maximum current that can be drawn from a cell is for R = 0 and it is Imax =
ε
𝑟
Page | 77
Cells in series and in parallel:
1. Series combination of cells:
Let two cells of emf 𝛆1 and 𝛆2, and internal resistance r1 and r2 be connected
in series as shown in the figure
Here the cells are connected end to end (i.e, one end of the two cells is joined
together leaving the other terminal in either cell free).
Let V(A), V(B) and V(C) be the potentials at points A,B and C as shown in the
figure.
Here V(A) - V(B) is the potential difference between the positive and negative
terminals of the cell so the potential difference between A & B is
VAB = V(A) - V(B) = 𝛆1 - Ir1 …………………. (1)
Similarly
VBC = V(B) - V(C) = 𝛆2 – Ir2 …………………. (2)
The potential difference between the points A and C of the combination is
VAC = V(A) - V(C) = [V(A) - V(B)] – [V(B) - V(C)]
VAC = [𝛆1 - Ir1] – [𝛆2 – Ir2]
VAC = [𝛆1 + 𝛆2 ] –I [r1 + r2 ]
…………….….. (3)
If we replace the combination of the cells by equivalent cell of emf 𝛆eq and
internal resistance req then the potential difference between A & C is given by
VAC = 𝛆eq - I req
………………………… (4)
Comparing equation (3) & (4) we can write
𝛆eq = 𝛆1 + 𝛆2
………………. (5)
req = r1 + r2
…………………(6)
Equation (5) and (6) gives equivalent emf and internal resistance of the
combination of the cells.
Page | 78
Here we connected negative terminal of the first cell with positive terminal of the
second cell. If we connect negative terminal of the first cell with negative terminal of
the second cell then equivalent emf is given by
𝛆eq = 𝛆1 - 𝛆2
……………………. (7)
If there are n number of cells of emf 𝛆1, 𝛆2, 𝛆3………….. 𝛆n and internal
resistance r1, r2, r3 …………rn respectively connected in series. Then equivalent emf
and equivalent resistance of the combination is given by formulae,
𝛆eq = 𝛆1 + 𝛆2 + 𝛆3………….. 𝛆n.
…………….. (8)
req = r1 + r2 + r3 …………….rn.
………………..(9)
2. Parallel combination of cells:
Let two cells of emf 𝛆1 and 𝛆2, and internal resistance r1 and r2 be connected
in parallel as shown in the figure.
Here the currents I1& I2 are the currents leaving the positive electrodes of the
cells. At the point B1, I1& I2 flows in whereas the current I flow out.
Since as much charge flows in as out, we have
I = I1+ I2 …………………… (1)
Let V(B1) and V(B2) be the potentials at B1 and B2 respectively.
Then the potential difference across first cell is,
V = V(B1) - V(B2) = 𝛆1 – I1r1
This implies that, I1 =
ε 1 −𝑉
𝑟1 .
……………………….. (2)
Similarly the potential difference across the second cell is,
V = V(B1) - V(B2) = 𝛆2 – I2r2
Page | 79
This implies that,
I2 =
ε 2 −𝑉
……………….. (3)
𝑟2 .
Substituting equation (2) and (3) in equation (1) we get
I = I1+ I2 =
ε 1.
I=
ε 1 −𝑉
𝑟1 .
ε 2.
+
𝑟1 .
+
ε 2 −𝑉
𝑟2 .
1
-V
𝑟2 .
𝑟1 .
+
1
…………… (4)
𝑟2 .
Rearranging equation (4) we can get the equation for V as
𝜀 1 𝑟2 +𝜀 2 𝑟1
V=
𝑟1 𝑟2
–I
𝑟1 +𝑟2
………………. (5)
𝑟1 +𝑟2
If we replace the combination of the cells by equivalent cell of emf 𝛆eq and
internal resistance req then the potential difference V between B1& B2 is given by
V = εeq - I req
……………………. (6)
Comparing equation (5) with (6) we get
𝜀 1 𝑟2 +𝜀 2 𝑟1
𝛆eq =
req =
𝑟1 +𝑟2
𝑟1 𝑟2
𝑟1 +𝑟2
We can put these equations in a simpler way as
1
𝑟𝑒𝑞
𝜀 𝑒𝑞
𝑟𝑒𝑞
=
=
1
𝑟1 .
𝜀1
𝑟1 .
+
+
1
𝑟2 .
𝜀2
𝑟2 .
If there are n number of cells of emf 𝛆1, 𝛆2, 𝛆3………….. 𝛆n and internal
resistance r1, r2, r3 …………rn respectively connected in parallel. Then equivalent emf
and equivalent resistance of the combination is given by formulae,
1
𝑟𝑒𝑞
𝜀 𝑒𝑞
𝑟𝑒𝑞
=
=
1
𝑟1 .
𝜀1
𝑟1 .
+
+
1
𝑟2
+
.
1
𝑟3
+⋯
.
1
𝑟𝑛 .
𝜀2
𝜀3
𝜀𝑛
𝑟2
𝑟3
𝑟𝑛 .
+
.
+⋯
.
Page | 80
Kirchhoff‟s laws:
Node or Junction: It is a point in a electrical network where more than two
conductors meet.
Electrical network: It is a combination of various circuit elements sources of
emf connected in a complicated manner.
Loop or mesh: A closed path of current in a network is called loop or mesh.
There are two Kirchhoff‟s laws for electrical network they are:


Kirchhoff‟s current law and
Kirchhoff‟s voltage law
Kirchhoff‟s current law (KCL):
Statement:
At any electrical node the algebraic sum of current entering the node is equal
to the algebraic sum of current leaving the node.
Kirchhoff‟s current law also called as Kirchhoff‟s first law of electrical network.
Illustration:
Let different current is flowing in different branches of node as shown in
figure.
Let I1,I4,I5 be the current entering the node and I2, I3 be the current leaving the node.
Applying KCL to the node O we have,
I1+ I5 + I4 = I2+ I3
Or
I1+ I5 + I4 - I2- I3 = 0
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𝐼 =0
That is at any electrical node or junction; the algebraic sum of current is equal to
zero.
KCL follows the law of conservation of charges.
Kirchhoff‟s voltage law (KVL):
Statement:
In any electrical closed circuit the algebraic sum of emf entering is equal to
algebraic sum of products of current and resistance in different branch of network.
That is,
𝐸=
𝐼𝑅
This law is a consequence of conservation of energy. In applying Kirchhoff‟s
laws to electrical networks, the direction of current flow may be assumed either
clockwise or anticlockwise. If the assumed direction of current is not the actual
direction, then on solving the problems, the current will be found to have negative
sign. If the result is positive, then the assumed direction is the same as actual
direction. In the application of Kirchhoff‟s second law, we follow that the current in
clockwise direction is taken as positive and the current in anticlockwise direction is
taken as negative
Kirchhoff‟s current law also called as Kirchhoff‟s second law of electrical
network.
Let us consider the electric circuit given in Fig. Considering the closed loop
ABCDEFA,
I1R2 + I3R3 +I3r3 + I3R5 + I4R6 + I1r1 + I1R1 = E1 + E3
Both cells E1 and E3 send currents in clockwise direction. For the closed loop ABEFA
I1R2 + I2R3 +I2r2 + I4R6 + I1r1 + I1R1 = E1 – E2
Negative sign in E2 indicates that it sends current in the anticlockwise direction.
Page | 82
Sign convention while applying KVL:
(1) For cell
(2) For resistor
Wheatstone bridge:
The Wheatstone bridge has four resistors R1, R2, R3, R4 connected each
other to form a quadrilateral.
Across one pair of diagonally opposite points (A and C in the figure) a
source of emf is connected. This is called the battery arm.
Between the other two vertices, B & D, a galvanometer of resistance G is
connected. This is called galvanometer arm.
Let I1, I2, I3, I4 be the currents through the resistors R1, R2, R3, R4
respectively. And let Ig be the current through the galvanometer of resistance G.
Applying Kirchhoff‟s voltage law to the mesh ABDA we have
-I1R1 + I2R2 + IgG = 0 ………….(1)
Applying Kirchhoff‟s voltage law to the mesh ABDA we have
I4R4 + IgG + I3R3 = 0 ………….(2)
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When the bridge is balanced then Ig = 0, this implies that I1 = I3 and I2 = I4
So equation(1) becomes
-I1R1 + 0 + I2R2 = 0
From this we obtain
𝐼1
𝐼2
=
𝑅2
𝑅1
………………….. (3)
And equation (2) becomes
I4R4 + IgG + I3R3 = 0 = I2R4 + 0 + I1R3
From this we obtain
𝐼1
𝐼2
=
𝑅4
𝑅3
………………. (4)
From equation (3) & (4) we obtain the condition
𝑅2
𝑅1
=
𝑅4
𝑅3
…………………. (5)
Equation (5) relating the four resistors is called the balance condition for the
galvanometer to give zero or null deflection.
Meter bridge:
Meter Bridge uses the concept of Wheatstone‟s network to calculate the
magnitude of unknown resistance. It is a practical form of Wheatstone bridge.
Construction:
 It consists of a wire of length 1m and of uniform cross-sectional area stretched
taut and clamped between two thick L-shaped metal stripes.
 A metal strip is placed in between L-shaped metal stripes with two gaps between
the ends of the stripes.
 One end of the galvanometer is connected to the metallic strip midway between
the two gaps.
 The other end of the galvanometer is connected to a jockey which can slide over
the wire to make electrical connection.
Page | 84
Theory:
An unknown resistance R whose magnitude is to be calculated is connected
across the left gap. Across the right gap, a standard known resistance S is connected.
The jockey is slides over the resistance wire to get zero deflection in the
galvanometer.
Let the distance of the jockey from the end point at zero deflection be l1.
Then the four resistances of the bridge at the balancing point are R, S, Rcml1 and Rcm
(100-l1).
where Rcm is the resistance of the wire per unit centimeter.
Then by balancing condition of the bridge
𝑅
𝑆
So,
=
𝑅𝑐𝑚 𝑙 1
𝑅𝑐𝑚 100−𝑙 1
R=S
=
𝑙1
100−𝑙 1
𝑙1
100−𝑙 1
By choosing various values of S we would get various values of 𝑙1 and
calculate R each time.
Potentiometer:

The instrument consists of a long piece of wire, few meters in length across which
a standard cell is connected.
 The wire is sometimes cut into several pieces placed side by side and connected
at the ends by thick metal strip.
 In the figure, the wires run from A to C. the small vertical portions are the thick
metal strips connecting the various sections of the wire.
Page | 85

A current I flows through the wire which can be varied by a variable resistance
(rheostat, R) in the circuit. Since the wire is uniform, the potential difference
between A and any point at a distance l from A is (l) = ϕl
 Where ϕ is the potential drop per unit length.
Applications of potentiometer:
1. To compare the emf of two cells:





Figure shows an application of the potentiometer to compare the emf of two cells
of emf 𝛆1 and 𝛆2.
The points marked 1, 2, 3 form a two way key. Consider first a position of the key
where 1 and 3 are connected so that the galvanometer is connected to 𝛆1.
The jockey is moved along the wire till at a point N1, at a distance l1 from A,
there is no deflection in the galvanometer
We can apply Kirchhoff‟s loop rule to the closed loop AN1 G31A and get,
Φl1 + 0 - 𝛆1 = 0 ………………… (1)
Similarly if another emf 𝛆2 is balanced against l2 (AN2)
Φl2 + 0 – 𝛆2 = 0 ………………….. (2)
From the two equations
𝜀1
𝜀2
=
𝑙1
𝑙2
This simple mechanism thus allows one to compare the emf‟s of any two sources.
In practice one of the cell is chosen as a standard cell whose emf is known to high
degree of accuracy. The emf of the other cell is then easily calculated from the
above equation.
Page | 86
2. To measure the internal resistance of a cell:
 We can also use potentiometer to measure internal resistance of a cell. For this
the cell whose internal resistance is to be determined is connected across a
resistance box through a key K2, as shown in figure.
 With key K2 open, balance is obtained at length l1 (AN1). Then 𝛆 = ϕl1
 When key K2 is closed, the cell sends a current (I) through the resistance box (R).
if V is the terminal potential difference of the cell and balance is obtained at
length l2(AN2)
So, we have
ε
V
=
𝑙1
𝑙2
……………….. (1)
But, 𝛆 = I (r + R) and V = IR.
ε
(r + R)
V
R
This gives, =
……………… (2)
From equation (1) and (2) we have
(r + R)
R
=
𝑙1
𝑙2
And from this
r=R
𝑙1
𝑙2
− 1 ……………….(3)
Using equation (3) we can find the internal resistance of the given cell.
*********************
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CHAPTER - 4
MOVING CHARGES AND MAGNETISM
Introduction:
Both Electricity and Magnetism have been known for more than 2000 years. In
1820 it was examined that both electricity and magnetism were intimately related.
The Danish physicist Hans Christian Oersted noticed that a current in a straight wire
caused a deflection in the magnetic needle which was kept close to it. He found that
alignment of needle is tangential to an imaginary circle which has the straight wire as
its centre and has its plane perpendicular to the wire.
Oersted‟s Experiment:
Let us consider a magnetic needle NS which can be rotated freely about a vertical
axis in a horizontal plane. Place a wire AB over the needle such that it is parallel to
the needle NS and connect it to a cell and a plug key.
It was observed that:
1. When the wire was held over the needle and the current flows from the south to
the north i.e. from A to B, then the north pole of the magnetic needle gets
deflected towards the west, as shown in the fig(1).
2. If the direction of the current is reversed, then the current now flows from the
north to the south i.e. from B to A, then the North Pole of the magnetic needle
gets deflected towards the east, as shown in the fig (2).
The deflection increases on increasing the current in the wire or bringing the
magnetic needle closer to the wire. If the current stops then the deflection also get
stopped and the magnetic needle comes into its initial position.
From the above experiment Oersted concluded that moving charges or a current
carrying conductor produces a magnetic field in the space around it. In other words
flow of electric charges is the source of magnetic field.
Page | 88
Direction of the deflection of magnetic needle due to electric current can be found
by applying Ampere‟s Swimming Rule: “Imagine a man swimming along the
conductor in the direction of the flow of current facing the needle such that current
enters the feet, then the north pole of the magnetic needle will be deflected towards
his left hand”.
Magnetic Force:
Sources and Fields:
In the chapter 1 we have studied about electric field 𝑬. We have seen that
the interaction between two charges can be considered in two stages.
The charge Q, the source of the field, produces an electric field 𝑬, where
𝑬=
𝐐𝐫
𝟒𝛑𝛆 𝐫 𝟐
Where 𝐫 the unit vector along r and the field is 𝑬 is a vector field. A charge q
interacts with this field and experiences a force 𝑭 given by,
𝑭=q𝑬=
𝐪𝐐𝐫
𝟒𝛑𝛆 𝐫
The field 𝑬 can convey both electric and momentum; we assume that fields are
independent of time. The field at a particular point can be due to one or more
charges. If there are more charges the fields add vectorially.
A static charge produces an electric field 𝑬 at every point in the space around it.
Similarly the current carrying conductor or moving charges produces magnetic
field in the space around .The magnetic field disappears as soon as the current is
switch off or charges stop moving. It means moving charge is a source of both
electric and magnetic field.
Experimentally it is founds to obey principle of superposition: The magnetic
field of several sources is the vector addition of individual magnetic field due to
individual source. i.e. The effective magnetic field 𝑩 at a point due to several
sources is the vector addition of magnetic field of each individual source at that
point. If B1, B2, B3 are the magnetic field at a point due to individual source of the
magnetic field then ,
𝑩 = 𝑩1 + 𝑩2 + 𝑩3 + ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯
The direction of the magnetic field at any point is along the tangent to the
magnetic field line at that point. The direction of the magnetic field lines and thus
Page | 89
magnetic field around a conductor carrying current can be determined by any of
the following rules:
1. Right handed screw rule (Maxwell‟s Cork Screw Rule):
If the forward motion of an imaginary right handed screw is in the
direction of the current through a linear conductor, then the direction of rotation
of the screw gives the direction of the magnetic lines of force around the conductor.
2. Right hand thumb Rule:
If the conductor carrying current is imagined to be held in the right hand
such that the thumb points in the direction of the current, then the tips of the
curled fingers encircling the conductor will give the direction of the magnetic field
lines.
Magnetic Field, Lorentz Force:
“A space or region around a current carrying conductor where the magnetic
effects can be experienced is called as Magnetic field.”
It is produced by moving charges and it is a vector field because it has both
magnitude as well as direction, and it is denoted by 𝑩. Consider a point charge q
moving with a velocity 𝒗 in the presence of both electric field 𝑬 and magnetic
field 𝑩. Let 𝛉 be the angle between 𝒗 and 𝑩.The charge experience a due to the
interaction between the magnetic field produced due to moving charge and magnetic
field applied. The magnitude of the force experienced by the charge is given by,
𝐅 = 𝐁 𝐪 𝐯𝐬𝐢𝐧𝛉
...................... (1)
In vector notation, the force is represented as,
𝑭= 𝐪 𝒗×𝑩
....................... (2)
Page | 90
When we look at the interaction with the magnetic field we find the following
features:
 It depends upon charge of the particle q, the velocity v and the magnetic field B.
 Force on a negative charge is opposite to that on a positive charge .
 The magnetic force q( v ×B) includes a vector product of velocity and magnetic
field.
 The magnetic force is zero if charge is not moving. Only a moving charge feels
the magnetic force.
NOTE:
1. Charge q moves parallel or anti-parallel to the direction of magnetic field 𝑩 i.e.,
𝛉 = 0o or 𝛉 = 180o
Then F = 0
Thus, no force is experienced by a charged particle moving parallel or antiparallel to the direction of the magnetic field.
2. Charge q moves at right angles to the magnetic field 𝑩
i.e., 𝛉 = 90o
Then F = B q v
Thus, the charge experience maximum force.
Definition of Magnetic field:
From the equation 𝐅 = 𝐁 𝐪 𝐯𝐬𝐢𝐧𝛉 .
If q = +1C, v = 1 ms-1, and θ = 90o then, 𝑩 = 𝑭
The magnetic field at a point is defined as
“The magnetic force experienced by a unit charge moving with unit velocity at
right angle to the magnetic field”.
From the equation F = B q vsinθ
⟹ B=
F
q vsin θ
If q = +1C, v = 1 ms , and θ = 90 and 𝑭 = 1N then, 𝑩 = 1tesla.
-1
o
Lorentz Force:
If the charge particle moves in an magnetic field B and electric field E, then the
charge particle experiences a force due to both electric field and magnetic field and it
is given by,
F = Felectric + Fmagnetic
....................................
(1)
F = q E + q ( v× B)
Page | 91
F = q [ E + ( v× 𝐁) ]
…………………(2)
Therefore Lorentz force can be defined as
“The total force experienced by a charged particle in a region where both
electric and magnetic fields are present is called as Lorentz force.”
Magnetic force on a current carrying conductor:
Consider a conducting rod of a uniform cross-sectional area A and length l.
Here we shall assume one kind of charge carriers‟ i.e say electrons in a conducting
rod.
Let „n‟ be the number of density of charge carriers(free electrons).
Total number of charge carriers in the conducting rod is = nAl .
Then the total charge flowing through the conductor = nAlq . ..…(1)
If a steady current I flows through the conducting rod, each charge
carriers has an average drift velocity Vd. In the presence of an external
magnetic field 𝑩, the force on these charge carriers is given as,
F = nAlq ( Vd× 𝑩 ) ............................(2)
where q is the value of the charge on a carrier.
The current density is given as: j = nqVd
∴ F = jAl × 𝑩 ............. (3)
The magnitude of the current density is j =
Hence
F=Il× 𝑩
I
A
, therefore I = j A.
................ (4)
where l is a vector of magnitude l, which is the length of the conducting rod and
with a direction identical to the current I. In this equation , B is the external magnetic
field.It is not the field produced by the current carrying conductor.
If the conducting rod has an arbitrary shape , it can be considered to be made
up of large number of linear strips of length dl.Then the resultant force is
F = 𝚺 𝐈 𝒅𝒍 × 𝐁
................... (6)
The summation can be converted to an integral in most cases.
Page | 92
Motion in a Magnetic Field:
Consider the motion of a charge moving in a magnetic field. In the case of
motion of a charge in a magnetic field, the magnetic force is ⊥er to the velocity of the
particle.The motion of the charged particle is decided by the velocity of the charged
particle.
Here we shall consider a particle of mass „m‟, and charge „q‟ moving with velocity
v in the uniform magnetic field B. The velocity of the particle is resolved into two
components i.e., v and v .The perpendicular force q( v × B), acts as a centripetal
force and produces a circular motion ⊥er to the magnetic field as shown in the fig (6).
“The particle will describe a circle if v and B are ⊥er to each other.”
If velocity has a component along B, this parallel component is unaffected as the
motion along the magnetic field i.e., it produces a linear motion. The motion in a
plane ⊥er to B is as before a circular one, thereby the resultant force is a „Helical
motion‟ as shown in the fig (7).
Hence the charged particle will describe a circular path of radius „r‟ is given by
F=
𝐦𝐯
𝐫
...................... (1)
The magnitude of the magnetic force on a charged particle is,
F = q B vsinθ , if θ= 90° then, F = q v B .........................(2)
∴ From equating equation (1) and (2)
mv
r
∴
=qvB
r=
𝐦𝐯
𝐪𝐁
.....................(3)
Page | 93
The larger the momentum, larger is the radius and bigger is the circular path.The
stronger the magnetic field smaller is the radius and smaller is the circular path.
v
If ω is the angular frequency, then v = r 𝛚 ⟹
=ω
r
we know that, ω = 2πν and
∴
v
=
r
𝛎=
qB
m
𝐪𝐁
𝟐𝛑𝐦
......................(4)
Here 𝛎 is the frequency of rotation. The time taken for one revolution by the
particle in the magnetic field is, T =
T=
𝟏
𝛎
𝟐𝛑𝐦
...................(5)
𝐪𝐁
From equation (4) and (5) it is clear that 𝛎 and T are independent of velocity v.
“The linear distance moved along the magnetic field in one rotation is called Pitch
P.”
We have, P = v T
𝟐𝛑𝐦
...................(6)
P=
V
𝐪𝐁
∴ These four will describe the particular Helix.
Motion in Combined Electric and Magnetic fields:
Velocity Selector:
Let E and B be electric and magnetic fields acting ⊥er to each other and also⊥er
to the velocity v of the charged particle „q‟ moving in it. This particle will experience
both electric and magnetic force which acts in opposite direction as shown in the
fig(8) and the force is given by,
F = Felectric + Fmagnetic
F = q [ E + (v× 𝐁) ]
....................................
(1)
Page | 94
If the velocity v is along x-axis, the electric field E is along y-axis and magnetic
field B is along z-axis,
then v = v i , E = E j and B = B k
The electric force on the charged particle is:
F = q E = q E j along y-axis .
The magnetic force on the charged particle is:
𝐅 = 𝐪 𝐯 × 𝐁 = 𝐪 𝐯 i × 𝐁 k = 𝐪 𝐯𝐁(−j) along negative y-axis.
F = q ( E – v B) j
..................(2)
Hence the forces Felectric and Fmagnetic are in opposite directions. If the values of E and
B are adjusted such a way that the magnitude of these two forces are equal, then the
total force on the charge is zero. Now the particle moves straight along x-axis without
any change in direction. This happens when,
q E = q v B or v =
E
B
This condition can be used to select charged particles of a particular velocity out
of a beam containing charges moving with different speeds.The crossed E and B
fields,therefore serve as a velocity selector.
Only particles with speed
E
B
pass undeflected through the region of crossed fields.
NOTE :
1. J.J Thomson used this method to measure Specific charge of an electron
2. And this principle is also used in Mass Spectrometer to separate ions according to
their specific charge.
Cyclotron:
“Cyclotron is a device used to accelerate charged particles like protons,
deuterons, 𝜶-particles etc., to very high kinetic energy by applying electric and
magnetic fields.” It was invented by E.O.Lawrence and M.S.Livingston in 1934 to
investigate nuclear structure.
Principle:
A charged particle can be accelerated to very high energies by passing it through
the electric field several times.This can be done with the help of a perpendicular
magnetic field which makes the charged particles to move along a circular path.The
Page | 95
frequency does not depend on the speed of the particle and the radius of the circular
orbit.
Construction:
It consists of two D-shaped hallow semicircular metal chambers D1 and D2, called
dees. A source of positive ion is located at the centre of the gap between the dees.
The dees are insulated from each other and enclosed in a highly evacuated chamber.
The dees are connected to a powerful high frequency oscillator. As a result a varying
electric field is produced in the gap between the dees. Since the metallic dees are
electrically shielded inside the dees the electric field is zero.The magnetic field is
perpendicular to the plane of the dees.
Working:
Suppose a positive ion, say a proton, enters the gap between the two dees and
finds dee D2 to be negative(lower potential) and dee D1 to be positive(higher
potential) and the proton gets accelerated towards dee D2.As it enters the dee D2,it
does not experience any electric field due to shielding effect of metallic dee.The
perpendicular magnetic field makes the charged particle proton to describe an
semicircular path within the dee. At the instant the proton comes out of the dee D2.
Due to the high frequency oscillator the polarity of the dees get reversed after every
semicircle. Therefore now dee D2 will be positive and dee D1 will be negative. Proton
now gets accelerated towards dee D1. It moves with the greater speed through D1
describing a larger semicircle than before. Each time the acceleration increases the
energy of the particle. As energy increases, the radius of the circular path increases.
So the path is a spiral one. Ultimately the particle gains high energy while they reach
the periphery of the dees. Finally the accelerated proton is ejected through the exit
port.
Page | 96
Theory:
The ions entering the space inside the dees with a velocity „v‟ is given by q v.
The kinetic energy is given by
1
2
1
Therefore,
mv
2
2
2
mv = q v
Due to the presence of the magnetic field inside the dees the ions travel in a
semicircular path with radius „r‟and is given as,
Bqv=
i.e.,
m v2
⟹
r
v
r
=
Bq
m
Time taken by an proton to describe a semicircular path within the dees is given
by,
t=
π
ω
, where ω is the angular frequency ω =
∴
t=
v
r
πm
Bq
To maintain the circular motion the polarities of the dees has to be reversed in a
time interval of
𝑇
2
, where T is the
T=
period of revolution and is given by, t =
𝑇
2
𝟐𝛑𝐦
𝐪𝐁
Hence the frequency of revolution of the particle will be,
𝛎=
𝐪𝐁
𝟐𝛑𝐦
where 𝛎c is the resonance frequency or cyclotron frequency.
The maximum kinetic energy of the proton will be,
K.E =
K.E =
K.E =
1
2
1
2
mv
m
2
(Bqr
)
m
𝟏 𝐁𝐪 𝐫 𝟐
𝟐
𝐦
Page | 97
The high energy particles produced in a cyclotron are used to bombard nuclei and
study the resulting nuclear reaction and hence to investigate nuclear structure.
Limitations of Cyclotron:
i. Cyclotron cannot accelerate uncharged particles like neutron,
ii. Cyclotron cannot accelerate electrons because they have very small mass.
Uses of Cyclotron:
i. It is used to produce radioactive material for medical purpose e.g., for the
purpose of diagnostics and treatment of chronic diseases.
ii. It is used to synthesise fresh substances.
iii. It is used to improve the quality of solids by adding ions.
iv. It is used to bombard the atomic nuclei with highly accelerated particles to study
the nuclear reactions.
Magnetic field due to a current element:
Biot – Savart Law:
In this law we will study the relation between
current I and magnetic field B. Let us consider a
current element dl of a finite conductor XY
carrying current I. The magnetic field dB due to
current element dl is to be determined at a point P
which is at a distance r. Let 𝜽 be the angle between
dl and the displacement vector r.
According to Biot – Savart Law, the magnitude of the field dB is,
1. Directly proportional to the current I through the conductor.
i.e., dB ∝ I
2. Directly proportional to the length dl of the current element.
i.e., dB ∝ dl
3. Inversely proportional to the square of the distance r of the point from the
current element.
i.e., dB ∝
1
𝑟2
dB ∝
𝐈 𝐝𝐥
𝒓𝟐
Page | 98
The dB direction is perpendicular to the plane containing dl and r.
Thus in vector notation,
dB ∝
dB =
Where
μo
4π
I dl ×r
𝑟3
μ o I dl ×r
4π
𝑟3
is a constant of proportionality and μo is the permeability of free space
and is given by μo = 4𝜋 × 10-7 TmA-1. The magnitude of this field is ,
𝐝𝐁 =
𝛍𝐨 𝐈 𝐝𝐥 𝐬𝐢𝐧𝛉
𝟒𝛑
𝒓𝟐
This law describes the magnetic field produced by a current element.
Special cases:
1. If 𝛉 = 0o , 𝐬𝐢𝐧𝛉 = 0, then dB = 0.
i.e., the magnetic field is zero at points on the axis of the current element.
2. If 𝛉 = 90o , 𝐬𝐢𝐧𝛉 = 1 , then dB is maximum
i.e., the magnetic field due to a current element is maximum in a plane passing
through the element and perpendicular to the axis.
Magnetic field on the axis of a circular current loop:
Let us consider a circular loop carrying a steady current I . we have to find the
magnetic field B at an axial point P at a distance r from the centre O of the loop
whose radius is R as shown in the fig.
Consider a current element dl of the loop. The magnitude dB of the magnetic
field due to current element dl is given by Biot-Savart law,
dB =
since θ = 90o , dB =
μ o I dl sin θ
4π
𝑟2
μo I dl
4π 𝑟2
The field dB is perpendicular to r. The dB
is resolved into two rectangular component,
Page | 99
1. dB 𝐜𝐨𝐬𝛉, which is perpendicular to the axis of the coil.
2. dB 𝐬𝐢𝐧𝛉 , which is along the axis of the coil and away from the centre of the coil.
The vertical components dB 𝐜𝐨𝐬𝛉 will be equal in magnitude and opposite in
direction and hence when summed up they cancel with each other. The horizontal
component dB 𝐬𝐢𝐧𝛉 will lie along the same x-direction and summed up over the loop.
B=
dB sinθ
B=
μo Idl sinθ
4π 𝑟2
B=
μ o I sinθ
4π
𝑟2
dl
The integration of element dl over the entire loop gives 2𝛑𝐑, the circumference of
the loop.Thus the magnetic field at point P due to entire circular loop is ,
B = Bx 𝑖 =
μ o I sin θ
4π 𝑟 2
× 2πR
But 𝑟 2 = x 2 + R2
Bx 𝒊 =
𝛍𝐨 𝐈 𝐑𝟐
𝟐 𝐱 𝟐 + 𝐑𝟐
𝟑/𝟐
𝒊
Special case:
If we find the magnetic field at the centre of the loop, here x=0
μo I
Hence, Bo 𝑖 =
𝑖
2R
Ampere‟s circuital Law:
Ampere‟s circuital law considers an open surface
with a boundary.The boundary is made up of a
number of small line elements and current I is
made to pass through the surface. It gives the
general relation between a current in a wire of
any shape and magnetic field produced around it.
It states that “Line integral of magnetic field B around any closed path in a free
space is equal to 𝝁 times the total current passing through the surface”.
Page | 100
𝐁 𝐝𝐥 = 𝛍𝐨 𝐈
where I is the total current through the surface. The integral is taken over the closed
loop coinciding with the boundary C of the surface.In a simplified form, Ampere‟s
circuital law states that is field B is directed along the tangent to every point on the
perimeter L of a closed curve and its magnitude is constant along the curve, then
BL = μo I
The closed curve is called as Amperian loop.
Magnetic field due to straight conductor of infinite length carrying
current:
Consider a straight conductor of infinite length
carrying a current I from X to Y. Let P be a point
at a perpendicular distance „r‟ from the straight wire
and B be the magnetic field at P. Consider a circle
radius „r‟ around the wire passing through „P‟ as an
amperian loop. The magnetic field B at „P‟ will be
tangential to the circumference of the circular loop.
Hence the magnetic field B is parallel to the line element dl at P. Applying Ampere‟s
circuital law to the circular loop of radius „r‟.
𝐁 𝐝𝐥 = μo 𝐈
B dlcosθ =
B dlcos00
Hence the line integral of the magnetic field along the circular path is
Bdl= B
∴
dl = B (2πr)
B (2πr) = μo I
𝛍 𝐈
B= 𝐨
𝟐𝛑𝐫
From the above results we have,
1. The magnitude of the magnetic field at every point on a circle of radius r is same.
It has cylindrical symmentry.
Page | 101
2. The direction of the magnetic field at every point on the circle is tangential to it.
The lines of constant magnitude of magnetic field form concentric lines. These
circular lines are called magnetic field lines.
3. The magnetic field B decreases as radius r increases.
∵ dB ∝
1
𝑟2
.
The Solenoid and the Toroid:
The Solenoid and the Toroid are two pieces of equipment which generate magnetic
fields.
The Solenoid:
It consists of a long wire wound in the form of a helix having number of turns of
insulated wire such that the plane of a turn is parallel to the plane of the next turn.
Each turn represents a circular loop. A current carrying solenoid is equivalent to a
number of loops placed together.
Fig (a) represents the magnetic field lines for a finite solenoid. It is clear from the
circular loops that the field between two neighbouring turns vanishes.
Fig(b) represents that the field at the interior mid-point P is uniform, strong and
along the axis of the solenoid. The field at the exterior mid-point Q is weak and
moreover is along the axis of the solenoid with no perpendicular or normal
component.
Since the solenoid is made longer it appears like a long cylindrical metal sheet.The
field outside the field approaches zero where the field inside becomes everywhere
parallel to the axis.
Page | 102
Consider a rectangular Amperian loop abcd. Along cd the field is zero and also
along transverse bc and ad, the field component is zero.Thus these two section make
no contribution .Let the field along ab be B, and the relevant length of the Amperian
loop is, L = h.
Let n be the number of turns per unit length, then the total number of turns is nh.
∴ The enclosed current is,
Ie = I (nh)
where I is the current in the solenoid. From the Ampere‟s circuital law
BL =𝛍𝐨 𝐈e
Bh = μo I(nh)
B =𝛍𝐨 𝐧 𝐈
The direction of the field is given by the right-hand rule.The solenoid is used to
obtain a uniform magnetic field.
The Toroid:
“The toroid is a hallow circular ring on which a large number of insulated
turns of a metallic wire are closely wound”.
A solenoid bent into the form of a closed ring is called as a toroid.
Let „I‟ be the current flowing through the toroid.The magnetic field in the open space
inside i.e., at point P and exterior to the toroid i.e., at point Q is Zero. The magnetic
lines of force inside the toroid are circular and concentric with the centre of it.
Page | 103
Consider three circular Amperian loops 1,2 and 3 (dashed lines) having radii r1, r2 and
r3 respectively and point P,S and Q lie on them.The circular area bounded by the
loops 2 and 3, both cut the toroid, hence each turn of current carrying wire is cut once
by the loop 2 and twice by the loop 3.
Let the magnetic field along loop 1 be B1 in magnitude.
According to Ampere‟s circuital law, L= 2πr1.
However loop 1 does not enclose any current Ie= 0.
Thus,
B1 (2𝛑𝐫1)= 𝛍𝐨 (0)
⟹
B1 = 0
∴ The magnetic field at any point P in the open space inside the toroid is zero.
Let the magnetic field along loop3 be B3 in magnitude.
According to Ampere‟s circuital law, L= 2πr3.
However net current enclosed by loop 3 is zero i.e., Ie= 0.
Thus,
B3 (2𝛑𝐫3)= 𝛍𝐨 (0)
⟹
B3 = 0
∴ The magnetic field at any point Q in the open space outside the toroid is zero.
Let the magnetic field along the loop 2 be B in magnitude.
According to Ampere‟s circuital law, L= 2πr.
If „N‟ is the total number of turns in the toroid and I is the current then the net current
enclosed by loop 2 is Ie= NI.
Thus,
B (2𝛑𝐫)= 𝛍𝐨 (NI)
B=
𝛍𝐨 N𝐈
𝟐𝛑𝐫
Page | 104
If „r‟ is the average radius of the toroid and „n‟ be the number of turns per unit length.
Then N= 2πr × n
B=
μ o 2πr n I
2πr
B = 𝛍𝐨 𝐧𝐈
This result is similar to that of solenoid. The magnetic field inside the toriod is
constant and is always tangential.
Force between Two Parallel Currents, the Ampere:
Two straight conductor carrying current placed parallel and close to each other
experience mutual forces due to the interaction
between the magnetic fields produced by the current
in them.
Consider two infinitely long parallel straight
conductors „a‟ and „b‟ separated by a distance „r‟. Let
Ia and Ib be the currents in „a‟ and „b‟ respectively in
the same direction. The magnetic field at any point on
the conductor „b‟ due to current Ia in conductor „a‟is given by,
Ba =
μ 0 Ia
2πr
The conductor „b‟ carrying a current Ib will experience a sideways force due to the
field Ba. The force on the conductor „b‟ of length L due to „a‟ is Fba. The magnitude
of the force is given by,
Fba = Ib L Ba
Fba =
μ 0 Ia Ib L
2πr
......................... (1)
Similarly,the force on the conductor „a‟ due to „b‟is Fab,
Fab =
μ 0 Ia Ib L
2πr
.......................(2)
Page | 105
Since Fab and Fba are equal and opposite, so these forces pull the two conductors
towards each other. Thus, two parallel conductors carrying currents in the same
direction attract each other.
∴ Parallel currents attract, and antiparallel currents repel.
Thus the force between two parallel current carrying conductors is,
F=
μ 0 Ia Ib L
2πr
Definition of Ampere:
The force per unit length of the conductor is,
μ 0 Ia Ib
F=
If Ia = Ib = 1A , r = 1m, then F = 2 × 10-7 Nm-1
2πr
“ One ampere is defined as the steady current which when flows through each of
two infinitely long straight conductors of negligible cross section separated by a
distance of 1meter causes a force of 2 × 10-7 Newton per meter length of each
conductor”.
Torque on Current Loop, Magnetic Dipole:
Torque on a rectangular current loop in a uniform magnetic field:
Consider a rectangular current loop PQRS carrying a current I and suspended
in a uniform horizontal magnetic field of the strength B. The loop is free to rotate
about the vertical axis.
Let PQ= RS = l = length of the loop, SP = QR = b = breadth of the lo
Page | 106
Therefore, PQ × QR = l × b = A = Area of the current loop.
Let „θ‟ be the angle between the plane of the loop and the direction of magnetic field
B.
The Force acting on SP : F2 = I SP × B .
The magnitude of this force is F2 = I b Bsin ∝
.................(1)
By Fleming left hand rule, the force F2 on SP acts vertically upward in the plane
of the coil.
Force acting on QR : F1 = I QR × B .
The magnitude of this force is F1 = I b Bsin∝
..................(2)
By Fleming left hand rule, the force F1 on QR acts vertically downward in the
plane of the coil.
Since the forces F1 and F2 are equal in magnitude and acting in opposite direction
along the same straight line, they cancel each other and will not produce net force or
torque on the loop.
Vertical sides PQ and RS of the loop are always perpendicular to the direction of the
magnetic field irrespective of the position of the loop.
Force acting on the arm PQ : F3 = I PQ × B .
The magnitude of this force is F3 = I b Bsin ∝
F3 = I b Bsin 90o
F3 = I b B
……….......( 3)
By Fleming left hand rule, the direction of this force is perpendicular to the plane
of the loop and is directed outwards.
Force acting on the arm RS : F4 = I RS × B .
The magnitude of this force is F4 = I b Bsin ∝
F4= I b Bsin 90o
F4 = I b B
…………….(4)
By Fleming left hand rule, the direction of this force is perpendicular to the plane
of the loop and is directed inwards.
The forces F3 and F4 are equal in magnitude and opposite in direction.These forces
constitute a couple and tend to rotate the loop in the anti-clock wise direction about
the vertical axis.
Page | 107
Hence,when a current carrying loop is placed in a uniform magnetic field, torque acts
on the loop which tends to rotate the loop.
Torque on the loop 𝝉 = Either force × Arm of the couple.
But arm of the couple = TS = PScos ∝ = bcos ∝
∴
𝝉 = BI l × bcos ∝
𝝉 = BI Acos ∝
When the plane of the loop is parallel to the direction of the magnetic field B,
then ∝ = 0o ;
𝝉 = BI Acos 0o = B I A
which is a maximum value.
When the plane of the loop is perpendicular to the direction of the magnetic field B,
then ∝ = 90o ;
𝝉 = BI Acos 90o = 0
which is a minimum value.
If 𝜃 is the angle between the direction of the magnetic field and the normal (N )
drawn to the plane of the loop, then 𝜃 + ∝ = 90o or ∝ = 90o – 𝜃
∴ cos ∝ = cos (90o – 𝜃) = sin 𝜃
The torque on the loop is , 𝝉 = BIA sin 𝜽
For an rectangular coil of N turns,
𝝉 = N BIA sin 𝜽
Magnetic dipole moment of a current loop:
A current loop acts like a dipole and has a magnetic moment.
“The magnetic moment of a current loop is defined as the product of current I and
the area vector A of the loop”.
It is denoted by m.
m = I A = n IA
where n is the unit vector normal to the loop.
Page | 108
Circular current loop as a magnetic dipole:
The magnetic field on the axis of a circular coil of radius „r‟ carrying current I
is given by,
B=
𝛍𝐨 𝐈 𝐑𝟐
𝟐 𝐱 𝟐 + 𝐑𝟐
𝟑/𝟐
Here x is the distance along the axis fron the centre of the loop. If x >> R, we can
neglect R2 term in the denominator ,
Thus,
μo I R2
B=
2x 3
The area of the loop A = πR2 ,
therefore,
B=
μo I A
2πx3
w.k.t In the magnetic dipole moment m = I A ,
Hence,
𝛍𝐨 𝟐𝐦
B=
𝟒 𝛑𝐱 𝟑
The magnetic dipole moment of a revolving electron:
According to Bohr‟s atomic model in an atom the negatively charged electron (-e)
revolves round the positively charged nucleus (+Ze) in circular orbit of radius „r‟.
Let „V‟ be the orbital speed of the electron. Then the current due to the motion of the
electron is I =
e
T
, where T is the time period of
revolution.
But,
T=
2πr
⟹
V
eV
I=
2πr
The magnetic dipole moment associated with the electron
circulating round the nucleus is,
eVr
eV
μL = I A = I × πr 2 = 2πr πr 2 =
2
This direction of magnetic moment is into the plane of the paper.
μL =
eVr
2
×
me
me
=
e
2m e
× Vr me
Page | 109
μL =
eL
................(1)
2m e
Where L = Vr me is the magnitude of the angular momentum of the electron around
the nucleus .
In vector notation, 𝛍𝐋
=
−𝐞𝐋
𝟐𝐦𝐞
The negative sign indicates the direction of μL and L opposite to each other in the
case of electron. If a positive charged particle is considered then μL and L are in
same direction .
μL
L
=
e
2m e
is called as the gyromagnetic ratio and is a constant.
According to Bohr‟s postualates, L =
nh
2π
Where n = 1,2,3,...........and „h‟ is Planck‟s constant. Then equation (1) becomes
μL =
e
nh
2m e
2π
For the first orbit in Hydrogen atom,where n = 1 ,
μB =
eh
4πm e
= 9.27 × 10-24 Am2
It will have a minimum value and it is called as Bohr magneton.The magnetic dipole
moment of electron is usually expressed in terms of Bohr magneton.
The magnetic dipole moment of an electron due to orbital motion is μL and due to
spin of electron is μs . It is found that μs = μB .
The Moving Coil Galvanometer:
It is a device used to detect and measure accurately very small current in the
order of microamperes. It works on the principle that a current carrying conductor
experiences a mechanical force in a magnetic field construction.
It consists of a rectangular coil ABCD having a large number of turns of insulated
copper wire wound on a light non-magnetic rectangular frame. The coil is suspended
between concave pole pieces of a powerful permanent cylindrical magnet using a
Page | 110
phosphorous- bronze strip. The upper end is connected to a torsion head TH, which in
turn is connected to the terminal T2 fitted on the base. The lower end of the coil is
connected to a spring S which is connected to the terminal T1.
A soft iron cylinder is mounted inside the coil without touching it.This cylinder
increases the strength of the magnetic field. The concave pole pieces provides radial
magnetic field in the gap between the pole pieces and the cylinder. The plane or
concave mirror M is attached to the phosphor bronze strip to note the deflection of
the coil using lamp and scale arrangement. The entire arrangement is enclose in nonmagnetic box with a glass window.
Theory:
Consider a rectangular coil ABCD of „n‟ turns of insulated copper wire. Let B be
the strength of the magnetic field and I be the current flowing through the coil.
Let AB=CD=l=length of the coil, AD=BC= b =breadth of the coil.
A = l × b = area of the coil.
The horizontal sides BC and AD do not experience any force. Since the magnetic
field is radial, the plane of the coil remains parallel to the field for all orientations of
the coil. The vertical sides AB and CD experience a maximum force F= n B I l. The
forces acting on AB and CD are equal in magnitude but acts parallel in opposite
direction according to Flemings left hand rule. These two equal and unlike parallel
forces constitutes a couple.This couple deflects the coil.
Moment of deflecting couple = Force × arm of the couple
= n B I l × b ⟹ nB I A ..................(1)
Page | 111
When the coil deflects, the phosphor bronze wire gets twisted and constitute restoring
couple or twisting couple.
Moment of restoring couple = C θ
....................(2)
where, C is the restoring couple per unit twist or torsional constant of the spring and
θ is the deflection of the coil.
At equilibrium,
Moment of deflecting couple = Moment of restoring couple
nBIA=Cθ
Therefore,
C
I=
where ,
C
nBA
nBA
θ ⟹
I=Kθ
is a constant and is called as reduction factor of the galvanometer.
Therefore, I 𝛼 θ or θ 𝛼 I .
In a moving galvanometer, the deflection is proportional to the current passing
through it. The angle of rotation is measured by lamp and scale arrangement.
Current sensitivity:
If the galvanometer produces a large deflection for a small current, then it said to
be sensitive.
“It is defined as the deflection produced in the galvanometer per unit current
flowing through it”.
Current sensitivity, Is =
θ
I
=
nBA
C
The unit of current sensitivity is rad A-1 or div A-1.
Voltage sensitivity:
“It is defined as the deflection produced in the galvanometer when a unit
voltage is applied across two terminals of the galvanometer".
If R= Resistance of the galvanometer, I = current through it ,then I = VR
Voltage sensitivity Vs =
θ
v
θ
= IR =
nBA
C
×
1
R
I
= Rs
Page | 112
The unit of voltage sensitivity is rad V-1 or div V-1.
Let n turns in the coil of a moving coil galvanometer increased by 2n turns.
Then
θ
I
increases to
2θ
I
Thus the current sensitivity doubles. When the number of turns is doubled ,the length
of the wire is also doubled and the resistance is also doubled i.e., R becomes 2R
Then , Vs =
2nBA
C
×
1
2R
=
nBA
×
C
1
R
=
Is
R
The voltage sensitivity remains the same.
Conversion of galvanometer into an ammeter:
Ammeter is a modified form of a table galvanometer designed to read current
directly in ammeter. It is always connected in series with the circuit so that whole
current which is to be measured will pass through it.
“A galvanometer can be converted into ammeter by connecting a low
resistance called shunt parallel to the galvanometer”.
Consider a pointer galvanometer of resistance G.
Let I be the maximum current to be measured and
Ig be the current required for full scale deflection
in the galvanometer. The remaining current ( I- Ig)
is to flow through the shunt. The shunt resistance S
to be connected in parallel with galvanometer is
calculated as follows,
Potential difference across shunt = Potential difference across the galvanometer.
( I- Ig) S = Ig G
Therefore,
S=
and
Ig =
Ig G
( I− Ig)
S
S+G
I
................(1)
................(2)
By knowing Ig, I and G, S can be calculated.
Page | 113
Since S and G are constants for a given ammeter, from equation (2), Ig 𝛼 I. Hence
scale of the galvanometer can be graduated to give the main current in ampere‟s
directly. An ideal ammeter has Zero resistance.
Conversion of a galvanometer into an voltmeter:
Voltmeter is modified form of a galvanometer designed to measure potential
difference between any two points in a circuit directly in volt.
It is always connected in parallel between the two points across which the potential
difference has to be measured.
“A galvanometer can be converted into a voltmeter by connecting a high
resistance in series to the galvanometer”.
Consider a pointer galvanometer of resistance G.
Let Ig be the current required for full scale deflection
and V be the maximum potential difference to be
measured. Then high resistance R to be connected
in series with galvanometer is calculated as follows,
By ohm‟s law :
V
Ig
= (R +G)
V = Ig ( R +G)
⟹ R=
V
Ig
................(1)
- G .................(2)
By knowing Ig, G and V, R can be calculated. Since R and G are constant for a given
voltmeter, from equation (1) , Ig 𝛼 V. Hence the scale of galvanometer can be
graduated to read the potential difference directly. In ideal voltmeter has infinite
resistance
**************************
Page | 114
CHAPTER 5:
MAGNETISM AND MATTER
Introduction:
In early 600 BC people found a kind of iron ore in a place called Magnesia in
Greece which attracted small bits of iron. The ore was named as magnetite. This
phenomenon of attraction of small bits of iron towards the ore was called magnetism.
The iron ore showing this property was called as a magnet.
When a piece of magnetite was suspended freely, it always points roughly in the
north-south direction. Thus, the natural magnet has attractive and directive
properties. Based on this property, magnetic needles were used for navigation of
ships. Hence magnetite was also called as loadstone or lodestone which means
leading stone.
Natural magnets are weak and have irregular shape, where as artificial magnets
are made using a piece of iron or steel and it has desired shape and strength. Some of
the artificial magnets in use are bar magnet, Horse-shoe magnet, Compass needle and
Magnetic needle.
The bar magnet:
The bar magnet is the most commonly used form of an artificial magnet. It may
be rectangular or cylindrical in shape. The bar magnet has two poles named as the
north-pole and the south pole. When a bar magnet is suspended freely, the pole which
points towards geographic north is called North pole and the pole which points
towards geographic south is called south pole.
Each pole is assigned a polestrength which is also known as magnetic charge. The
polestrength of north pole is +qm and that of south pole is –qm. The distance between
the two poles in a bar magnet is called as magnetic length 2l. The product of
polestrength qm and magnetic length 2l is equal to magnetic moment or moment of
the magnet m.
i.e., 𝐦 = qm × 2𝒍
N
S
Some of the commonly known ideas of magnetism:
1. Like poles repel each other and unlike poles attract each other.
2. Every magnet attracts small pieces of materials like iron, nickel, cobalt and steel
towards it. The attraction is maximum near the ends of the magnet where the
poles are located.
Page | 115
3. Magnetic poles always exist in pairs.The poles of the magnet can never be
separated. If a magnet is cut into half, each piece is a magnet with two poles.
That means, magnetic monopoles do not exist.
4. Repulsion is the sure test of magnetism because, a magnet will attract an iron
piece as well as an unlike pole of another magnet. But it always repels like pole
of another magnet.
The force of attraction or repulsion between two magnetic poles is:
(a) directly proportional to the product of the pole strength or magnetic charge
qm1 and , qm2 and
(b) inversely proportional to the square of the distance between them.
i.e., F ∝
qm 1qm 2
r2
or F =
μο qm 1qm 2
4π
r2
where μο is called absolute permeability of free space or vacuum.
The Magnetic Field Lines:
Magnetic field is the space around a magnet or the space around a
conductor carrying current. To visualize a magnetic field graphically, the concept of
magnetic field lines was introduced.
Properties of magnetic field lines:
1. The magnetic field lines of a magnet or a solenoid form a continuous closed
loops.
2. The tangent to the magnetic field line at a point gives the direction of the net
magnetic field B at that point.
3. No two magnetic lines of force intersect each other. If two magnetic lines of force
intersect then there would be two directions of magnetic field at a point which is
not possible.
4. The magnetic field is strong where the magnetic lines of force are close together
and the magnetic field is weak where the magnetic lines of force are spread out or
spaced out.
Bar magnet as an equivalent solenoid:
The magnetic field lines for a bar magnet and a solenoid carrying current are
similar. A bar magnet can be thought of as a large number of circulating currents like
that in a solenoid. Cutting a magnet into half is similar to cutting a solenoid. In both
cases we get two small parts with weaker magnetic properties. In both bar magnet
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and solenoid, the magnetic field lines are continuous. They emerge from one face and
enter into another face.
The similarity between the bar magnet and the solenoid can be demonstrated by
calculating the magnetic field along the axis of a finite solenoid carrying current.
Consider a solenoid consists of n turns per unit length. Let „a‟ be the radius and
„2l‟ be the length of the solenoid. Let „P‟be a point on the axis of the solenoid at
a distance „r‟ from the centre„o‟of the solenoid.Consider a circular element of
thickness „dx‟ of the solenoid at a distance„x‟ from the centre. Let I be the current
flowing through the solenoid.
∴ The number of turns in the circular element = n dx
∴ The magnitude of the magnetic field at a point P due to the circular element is,
dB =
μo
2πI a 2 ndx
4π
(r−x)2 + a 2
3
2
If the point P is far away from the centre of the solenoid, such that
r>>>>a and r >>>> l then ,by approximation ,
(r − x)2 + a2
3
2
= r3
∴ dB =
μ o 2πI a 2 ndx
r3
4π
The magnitude of the total field is obtained by summing over all the elements or
integrating from the range –l to +l .
dB =
dB =
μo 2πI a2 n
4π r3
μ o 2πI a 2 n
4π
r3
dx
dx
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B=
B=
μ o 2πI a 2 n
4π
r3
𝑙+𝑙
μ o 2πI a 2 n 2𝑙
4π
r3
But the magnitude of the magnetic moment of the solenoid is,
m = total number of turns × current × Area of cross section
m = (n × 2l) × I × 𝜋 𝑎2
∴
B=
𝛍𝐨 𝟐𝐦
𝟒𝛑 𝐫 𝟑
This is similar to the magnetic field on the axial line of a bar magnet. The
magnetic moment of a bar magnet is thus equal to the magnetic moment of an
equivalent solenoid that produces the same magnetic field.
Torque on a magnetic dipole in a uniform magnetic field:
Consider a bar magnet of length 2l and polestrength qm held at a angle 𝛉 to
the direction of a uniform magnetic field 𝐁.
The dipole moment of the bar magnet is,
m = qm × 2𝑙
Force on the north pole = Bqm ; along the direction of field 𝐁
Force on the south pole = Bqm ; opposite to the direction of field 𝐁
Since these two forces are equal in magnitude and opposite in direction and act
along parallel lines, they constitute a couple. This couple tends to rotate the magnet.
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∴ Torque or moment of the couple = force × perpendicular distance
i.e.,
In ∆le NAS, sin θ =
NA
NS
τ = Bqm × NA
NA
=
2𝑙
NA = 2l sin θ
∴
τ = Bqm × 2l sin θ
τ = B m sin θ
In the vector form
𝛕=𝐦 ×𝐁
The magnetic dipole moment is defined as the torque acting on a magnetic dipole
placed normal to a uniform magnetic field of unit strength.
Time period of oscillation of magnetic needle placed in a uniform
magnetic field:
Consider a small magnetic compass needle of magnetic dipole moment m and
moment of inertia I placed in a uniform magnetic field B such that m make an angle
with B . This needle will be under a torque which makes it to oscillate.
Torque acting on the needle is given by,
τ = B m sinθ
This torque is known as restoring torque.
∴ In equilibrium
I
where
𝑑2θ
𝑑𝑡 2
𝑑2θ
𝑑𝑡 2
= - B m sinθ
= angular acceleration.
The negative sign shows that restoring couple opposes the deflecting couple. If is θ
small , by approximation sinθ ≈ θ.
∴
I
𝑑2θ
𝑑𝑡 2
𝑑2θ
𝑑𝑡 2
= -Bmθ
=-
mB θ
I
This represents simple harmonic motion. The square of the angular frequency
is,
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ω2 =
mB
I
⟹
ω=
mB
I
If T is the time period of oscillation, then ω =
T=
2π
2π
T
ω
T = 2π
I
mB
or B =
4 π2 I
m T2
An expression for magnetic potential energy can also be obtained on lines similar to
electrostatic potential energy.
The magnetic potential energy,
Um=
τ (θ) dθ
Um=
mB sinθ dθ
U m = mB
sinθ dθ
U m = - mB cos θ
Um=-𝐦∙𝐁
Analogy between Magnetism and Electrostatics:
1. The electric dipole moment is p = qm × 2𝑎 , where „q‟ is the electric charge and
2a is the distance between the two charges.
The magnetic dipole moment is m = qm × 2𝑙 , where „q‟ is the magnetic charge
and 2l is the distance between the two poles.
2. The magnetic field at large distances due to a bar magnet of magnetic moment
„m‟ can be obtained from the equation of electric field due to electric dipole of
dipole moment „p‟, by replacing electric field E by magnetic field B , Electric
dipole moment p by magnetic dipole moment m and
1
4𝜋𝜀 0
by
μo
4π
.
3. The torque on an electric dipole in an external electric field is τ = p × E and The
torque on an magnetic dipole in an external magnetic field is τ = m × B .
4. The potential energy of electric dipole in aan electric field is U = - p ∙ E , and the
potential energy of a magnetic dipole in a magnetic field is U m = - m ∙ B.
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5. Electric field dipole on a axial line E =
4𝜋𝜀 0 r 3
μ o 2m
Magnetic field dipole on a axial line B =
6. Electric field dipole on a line E =
2p
1
1
4π r 3
p
.
,
4𝜋𝜀 0 r 3
μo m
Magnetic field dipole on a axial line B =
,
4π r 3
.
Gauss‟s law in Magnetism:
It states that,”the net magnetic flux through any closed surface is always
Zero.”
This implies that the number of magnetic field lines leaving the surface is equal to
the number of magnetic field lines entering it.
Consider a closed surface S in a uniform magnetic field 𝐁. Let ∆s be a small
area element of this surface with n along its normal. The magnetic flux through this
area element is
∆ ϕB = B ∙ ∆ s
The total magnetic flux through the closed surface
is,
ϕB =
∆ ϕB =
B ∙ ∆s
According to Gauss‟s law in magnetism
𝛟𝐁 =
𝐁 ∙ ∆𝐬 = 0
This implies that isolated magnetic poles or
monopoles do not exists.
The Earth‟s Magnetism:
The branch of physics which deals with the study of magnetism of the earth is
called terrestrial magnetism or geomagnetism. It is found that the strength of the
earth‟s magnetic field varies from place to place on the earth‟s surface and its value is
of the order of 10-5 tesla.
Earth behave as though there is a bar magnet inside it. Dynamo theory was one of
the theory which gave a valid logical reason for the cause of the earth‟s magnetism.
Dynamo theory says that Earth‟s magnetism may be due to molten charged metallic
fluids (mainly iron and nickel) in the outer core of the earth.The charged fluids
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rotates as the earth rotates.This gives rise to electric currents which are responsible
for the magnetic field.This magnetic field of the earth causes Earth‟s magnetism and
this is known as dynamo effect.The magnetic field lines of the earth resemble that of
a gaint magnetic dipole located at the centre of the earth. The axis of the dipole does
not coincide with axis of rotation of the earth but it is presently inclined by
approximately 11.3o with respect to the axis of rotation of the earth.
Experimental evidences in support of earth‟s magnetism:
1. A freely suspended magnetic needle comes to rest roughly in north-south
direction. This suggests that the earth behaves has a huge magnet with its south
pole lying somewhere near the geographic north pole and its north pole lying
somewhere near the geographic south pole.
2. Existence of neutral points near bar magnet indicates the presence of earth‟s
magnetic field. At these points, the magnetic field of the magnet is cancelled by
the earth‟s magnetic field.
Some definitions in connection with Earth‟s magnetism:
1. Geographic axis:
“The straight line passing through the geographical north and geographical
south poles of the earth is called geographic axis. It is the axis of the rotation of
the earth”.
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2. Magnetic axis:
“The straight line passing through the magnetic north and magnetic south
poles of the earth is called magnetic axis”.
3. Geographic equator:
“It is the great circle on the earth perpendicular to the geographic axis”.
4. Magnetic equator:
“It is the great circle on the earth perpendicular to the magnetic axis”.
5. Geographic meridian:
“The vertical plane passing through the geographical north and geographical
south poles of the earth is called geographic meridian”.
6. Magnetic meridian:
“The vertical plane passing through the imaginary line joining the magnetic
north and magnetic south poles of the earth is called magnetic meridian”.
Magnetic declination and dip:
The earth‟s magnetic field at a place can be completely described by three
parameters which are called elements of earth‟s magnetic field. They are declination,
dip and horizontal component of earth‟s magnetic field.
Magnetic declination (𝛅):
“The angle between the geographical meridian and magnetic meridian at a
place is called magnetic declination at the place”.
Magnetic declination arises because the magnetic axis of the earth does not coincide
with its geographic axis.
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Angle of dip or Magnetic inclination:
“The angle made by the earth‟s total magnetic field 𝐁 with the horizontal
direction in the magnetic meridian is called angle of dip at any
place”.
At the magnetic equator, the dip needle rests horizontally so that
the angle of dip is zero at the magnetic equator. The dip needle
rests vertically at the magnetic poles so that the angle of dip is
90o at the magnetic poles. At all other places, the dip angle lies
between 0o and 90o.
Horizontal component of earth‟s magnetic field (BH):
The total magnetic field of the earth BE can be resolved into rectangular
components one along the horizontal and the other along the vertical.
“The component of the earth‟s total magnetic field BE at a place along the
horizontal direction in the magnetic meridian is called the horizontal component of
the earth‟s magnetic field BH”.
“The component of the earth‟s total magnetic field BE at a place along the vertical
direction in the magnetic meridian is called the vertical component of the earth‟s
magnetic field Bv”.
Horizontal component, BH = BE cos θ
Vertical component, Bv = BE sin θ
Bv
BH
=
B sin θ
B cos θ
=
sin θ
cos θ
= tan θ
BE = BH 2 + BV and θ = tan -1
Bv
BH
Magnetisation and magnetic intensity:
To understand the magnetic properties of elements and compounds we define
and explain few terms
Magnetization ( M):
We know that an electron revolving round the nucleus of an atom has an
magnetic dipole moment. The dipole moments of all the electrons in an atom add
vectorially to give a magnetic moment for the atom.
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“The magnetization M of the sample is defined as the net magnetic moment per
unit volume”.
It is also known as the intensity of magnetization
M=
𝐦𝐧𝐞𝐭
𝐕
.
Its unit is ampere per metre (Am-1).
Magnetic permeability (𝛍):
“The magnetic permeability μ of a material is the ability of the matrial
to allow the passage of magnetic field lines through it”.
It represents the extent to which the magnetic field can penetrate.
The relative magnetic permeability μr of a material medium is defined as the ratio of
magnetic permeability of the material medium (μ) to the magnetic permeability of
vacuum (μr).
μ
i.e., μr =
μ𝑜
The magnetic permeability of vacuum is
𝛍𝒐 = 4𝛑 × 10-7 weber/ ampere-meter (WbA-1m-1) or henry/ metre(Hm-1).
Magnetic induction or magnetic flux density 𝐁 :
The magnetic field B at any point in a medium represents the number of
magnetic field lines per unit area around that point, along the normal to the area. It is
also known as magnetic flux density or magnetic induction.
Magnetic intensity 𝐇:
The degree to which a magnetic field can magnetize a material is represented
in terms of magnetic intensity 𝑯.
It is also known as magnetizing force. It depends only on the source of the magnetic
field.
Consider a long solenoid or toroid of n turns per unit length and carrying a current I.
The magnetic field in the interior of the solenoid is given by,
B = μ nI.
The magnetic intensity H is defined as,
H=
B
μ
= nI
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The magnitude of the magnetic intensity is equal to the number of ampere turns per
unit length of the solenoid or toroid to produce the magnetic field B.
Its unit is ampere / meter (Am-1) .
If the medium inside the solenoid is vacuum then,
H=
𝐵𝑜
or
μ𝑜
where μ𝑜 is permeability of vacuum.
B o = μ𝑜 H
Magnetic susceptibility (𝛘𝐦 ) :
Magnetic susceptibility measures the ability of a substance to take up
magnetization when placed in a magnetic field.
“It is defined as the ratio of the intensity of magnetization M to the
magnetizing field intensity H”.
It is denoted by 𝛘𝐦
Thus, 𝛘𝐦 =
𝐌
𝐇
As magnetic susceptibility is the ratio of two quantities having the same units
(Am-1), so it has no units.
Relation between magnetic permeability and susceptibility:
Consider a long solenoid or toroid of n turns per unit length and carrying a
current I. The magnetic field in the interior of the solenoid when it is filled with
vacuum is given by,
Bo = μo nI = μo H
The net magnetic field B in the interior of the solenoid is expressed as,
B = Bo + Bm
where Bm is the contribution to the magnetic field due to magnetization of the
material.
If M is the magnetization of the material, the Bm = μo M
∴
B = Bo + Bm
B = μo H + μo M
B = μo (H + M )
In vector form,
𝐁 = 𝛍𝐨 (𝐇 +𝐌)
If μ is the permeability of the material inside the solenoid, we have
B = μH
∴ μH = μo (H + M )
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M
i.e., μ = μo 1 + H
since magnetic susceptibility of the material is given by , 𝛘 =
M
H
we have, μ = μo 1 + 𝛘
This is the relation between magnetic permeability μ and susceptibility 𝛘.
Magnetic properties of materials:
There are large variety of substances (elements,compounds and alloys) on the
earth. On the basis of their magnetic properties, Faraday classified these substances
into three categories:
1. Diamagnetic substances:
Daimagnetic substances are those which develop feeble(weak) magnetization
in the opposite direction of the magnetizing field. Such substances are feebly repelled
by magnets and tends to move from stronger to weaker parts of a magnetic field.
Example: copper, Bismuth, lead, zinc, tin, gold, silicon, nitrogen(at STP),water,
silver, mercury, diamond, hydrogen, sodium chloride, etc.,
Properties of Diamagnetic substances:
1. A diamagnetic substance is feebly repelled by a strong magnet. It is because a
diamagnetic substance develops weak magnetization in the opposite direction of
the appield magnetic field.
2. When a diamagnetic substance is placed in a magnetic field, the magnetic field
lines prefer to pass through the surrounding air rather than through the substances.
It is because the induced magnetic field in the diamagnetic substance opposes the
external field.
3. When a rod of diamagnetic substance is suspended in a uniform magnetic field, the
rod comes to rest with its longest axis perpendicular to the field. This is because
rod is feebly repelled by the field.
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4. When placed in a non-uniform magnetic field, a diamagnetic substance moves
from stronger to the weaker parts of the field.
5. The relative permeability(μr ) of a diamagnetic substance is always less than 1.
i.e μr =
B
B0
and B < Bo
6. As a diamagnetic substance develops a weak magnetization in the opposite
direction of the magnetizing field, the magnetic susceptibility (χm ) of diamagnetic
materials is small and negative. It is because μr = 1+ χm and μr < 1
7. The magnetization of the daimagnetic substance lasts so long as the magnetizing
field is applied.
NOTE:
When a metal is cooled to a temperature below its critical temperature in a
magnetic field, it attains both superconductivity and perfect diamagnetism.The
magnetic lines of force get completely repelled from it.
∴ μr =
B
B0
=
0
B0
=0
Since μr = 1+ χm ⟹ χm = -1
This phenomenon of diamagnetism in superconductors is called Meissner effect.
Cause of diamagnetism:
The orbiting electrons in an atom are equivalent to a current loop and hence
possess orbital magnetic moment. In diamagnetic substances, the resultant magnetic
moment in an atom is zero. When an external magnetic field is applied,electrons
having orbital magnetic moment in the same direction are slowed down and those in
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the opposite direction speed up.Thus the substances develops a net magnetic moment
in a direction opposite to that of the applied field.This causes the repulsion.
2. Paramagnetic substances:
Paramagnetic substances are those which develop feeble magnetization in the
direction of the magnetizing field. Such substances are feebly attracted by magnets
and tend to move from weaker to stronger parts of magnetic field.
Example: Manganese, aluminium, chromium, platinum, sodium , copper chloride,
oxygen (at STP) , antimony, etc.,
Properties of Paramagnetic substances:
1. A paramagnetic substance is feebly attracted by a strong magnet. It is because a
paramagnetic substance develops weak magnetisation in the direction of the
applied external magnetic field.
2. When a paramagnetic substance is placed in a magnetic field, the magnetic field
lines prefer to pass through the substance rather than through surrounding
air.Therefore the resultant field B inside the substance is more than the external
field Bo.
3. When a rod of paramagnetic substance is suspended in a uniform magnetic field,
the rod comes to rest with its longest axis along the direction of the external
magnetic field. This is because paramagnetic substance is feebly attracted by the
field.
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4. When placed in a non-uniform magnetic field, a paramagnetic substance moves
from weaker to the stronger parts of the field.
5. The relative permeability(μr ) of a paramagnetic substance has value slightly
greater than 1. i.e μr =
B
B0
and B > Bo
6. As a paramagnetic material develops a small magnetization in the direction of the
magnetizing field, the magnetic susceptibility (χm ) of paramagnetic materials is
small but positive.
It is because μr = 1+ χm and μr > 1
7. The magnetic susceptibility (χm ) of paramagnetic substance varies inversely as the
absolute temperature.
i.e., χm 𝛼
1
T
8. A paramagnetic substance loses its magnetism as soon as the magnetizing field is
removed.
Curie law:
According to Curie law, intensity of magnetization(I) of a paramagnetic substance is
1. Directely proportional to the external magnetic field B. i.e I 𝛼 B
2. Inversely proportional to the absolute temperature (T) of the substance. i.e.,I 𝛼
∴I𝛼
B
T
or I = C
NOTE:
B 𝛼 H so that I 𝛼
χm 𝛼
1
T
or χm =
B
T
1
T
called Curie law and C is called Curie constant.
H
T
C
T
............Curie law.
Cause of paramagnetism:
The atoms or ions or molecules in a paramagnetic material possess a permanent
magnetic dipole moment. In the absence of external magnetic field dipoles of
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paramagnetic substances are randomly oriented and therefore net magnetic moment
of the substance is very small or Zero. Hence the substance does not exhibit
paramagnetism. When suitable external magnetic field is applied, the dipoles are
partially aligned in the direction of the applied field. Therefore the substance is feebly
magnetized in the direction of the applied field.
3.Ferromagnetic substances:
Ferromagnetic substances are those which develop strong magnetization in the
direction of the magnitizing field. They are strongly attracted by magnets and tend to
move from weaker to stronger parts of a magnetic field.
Example: Iron, cobalt, nickel, gadolinium and alloys .
The ferromagnetic material show all the properties of paramagnetic substances, but
to a much greater degree.
Certain ferromagnetic substance retain the magnetism even after the removal of the
applied external magnetic field ,such materials are called Hard ferromagnets. This is
not possible for Diamagnetic or paramagnetic materials. Ferromagnetic substances
which loses its magnetism after the removal of the applied external magnetic field are
called Soft ferromagnets.
Properties of the Ferromagnetic substances:
The ferromagnetic material show all the properties of paramagnetic substances,
but to a much greater degree.
1. A ferromagnetic substance is strongly attracted by a magnet. It is because a
ferromagnetic substance develops strong magnetisation in the direction of the
applied magnetic field.
2. When a ferromagnetic substance is placed in a magnetic field, the magnetic field
lines tend to crowd into the substance. Therefore the resultant field B becomes
much more than the magnetizing field Bo.
3. When a rod of ferromagnetic substance is suspended in a uniform magnetic field,
it quickly aligns itself in the direction of the magnetic field.
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4. When placed in a non-uniform magnetic field, a ferromagnetic substance moves
from weaker to the stronger parts of the field.
5. The relative permeability(μr ) of a ferromagnetic substance is very large.
6. The magnetic susceptibility (χm ) of ferromagnetic materials is positive having very
high value. It is because μr = 1+ χm and μr > >1. For this reason, ferromagnetic
substances can be magnetized easily and strongly.
7. A ferromagnetic substance retains magnetism even after the magnetizing field is
removed.
Curie temperature:
Ferromagnetism decreases with the increase in temperature. When a
ferromagnetic substance is heated, magnetization decreases because random thermal
motions tends to destroy the alignment of domains. At sufficiently high temperature,
the ferromagnetic property of the substance suddenly disappears and substance
becomes paramagnetic.
„The temperature at which of the ferromagnetic substance becomes
paramagnetic is called Curie temperature or Curie point of the substance.‟
Above the Curie point i.e., in the paramagnetic phase, the susceptibility varies
with temperature as
χm =
C
T−T
(T > Tc)
Where C is the constant. It is also known as Curie- Weiss law. This law states
that the susceptibility of a ferromagnetic substance above its Curie temperature is
inversely proportional to the excess of temperature above the Curie temperature.
Cause of Ferromagnetism:
In a ferromagnetic substance, the individual atoms or ions or molecules possess
permanent magnetic dipole moment. The magnetic moments of neighbouring atoms
interact with each other and align themselves spontaneously in a common direction
over a region called domains. Each domain has a typical size of about 1mm and
contains about 1011 atoms. Each domain behaves as a magnet having some dipole
moment. In the absence of any external magnetic field, these domains are on
randomly distributed so that the net magnetic moment is zero. On applying the
magnetic field, the magnetic moment of domains align in the direction of the applied
magnetic field and strongly magnetize the substance.
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Hysteresis:
Consider a sample of ferromagnetic material which is unmagnetized initially
( i.e., M= 0). The magnetic field B in the material is zero. When it is placed in a
solenoid, as the current through the solenoid is increased, the magnetic intensity H
increases. Then the magnetic field B in the material (and its magnetization M)
increases along Oa till magnetization reaches a saturation value as shown in the
figure. Beyond this point, magnetization M remains constant, but B increases linearly
with increase in H. Further H is decreased to zero by reducing the current in the
solenoid to zero. When H =0 it is found that B ≠0 as represented by the curve ab.
Thus, some magnetism is left in the sample.The value of the magnetic field Br in the
sample at this point when magnetic intensity H =0 is retentivity or remanence. Next
the current in the solenoid is reversed and slowly increased. At a particular value of
H at c, the magnetic field inside becomes zero. The value of magnetic intensity Hc at
the point c where the magnetic field inside the sample becomes zero is called
coercivity. As the current is reversed and increased in the magnitude, again
magnetization saturates. This is given by the curve cd. Next the current is reduced to
zero and increased in the reverse direction.Then we get the curves de and efa. The
cycle repeats itself. But , the curve Oa does not retrace itself when H is reduced. It is
found that the magnetic field B in the sample or its magnetization M lag behind the
magnetic intensity H when a sample of ferromagnetic material is taken through a
cycle of magnetization. This phenomenon is called as Hysteresis. The loop abcdefa
obtained for the variation of B and H during a cycle of magnetization is called
hysteresis loop or B-H loop.
The loss of energy per unit volume per cycle of magnetization is called as
hysteresis loss and it is equal to the area of the B –H loop.
NOTE:
“The property of a material to retain the magnetic field even after the removal of
the magnetizing field is called retentivity”.
“The property of a material to retain the magnetization to an extent inspite of the
application of a demagnetizing field is called coercivity”.
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Permanent Magnet and electromagnets:
Substance which at room temperature retain their ferromagnetic property for a
long period of time are called permanent magnets. Permanent magnets are used in
measuring instruments.
The properties of permanent magnets are:
1. The material should have high retentivity so that magnet is strong.
2. The material should have high saturation magnetization.
3. The material should have high coercivity so that it is not easily demagnetized.
4. The material should have high permeability so that it can be easily magnetized.
The materials used for making permanent magnets are steel, alnico, cobalt steel
and ticonal. By placing a ferromagnetic rod in a solenoid and passing current through
it we can make an efficient permanent magnet. The magnetic field of the solenoid
magnetises the rod.
Electromagnets can be made by using soft iron.When placed inside a solenoid
the soft iron rod is magnetized. When the current in the solenoid id switched off,the
magnetism is effectively switched off because soft iron has low retentivity.
Electromagnets are used in electric bells, loudspeakers and telephone diaphragms.
They are used in cranes to lift machinery and bulk quantities of iron and steel.
************************
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CHAPTER -6
ELECTROMAGNETIC INDUCTION
Introduction:
For a long time electricity and magnetism were considered separate and
unrelated phenomenon. In the year 1820 Oersted discovered that a magnetic field
exists around a current carrying conductor. In 1831 Michael Faraday discovered the
induced emf in a varying magnetic field.
“The phenomena in which electric current induced due to varying magnetic
field in a conductor is called electromagnetic induction”.
Experiment of Faraday‟s and Henry:
1. Current induced by a magnet:
In the figure(1) coil is connected to a galvanometer
G. When the north pole of a bar magnet pushed towards
the coil, The pointer in the galvanometer deflects
indicating the presence of electric current in the coil.
When the magnet is pulled away, the pointer deflects in
the opposite direction indicating the reversal of current
direction. The same effects are observed when the bar
magnet is stationary and coil is moved. The defection
lasts as long as the magnet is in motion. Thus, the relative
motion between the coil and magnet is responsible for inducing electric current in the
coil.
NOTE: The induced current is found to be larger if,
1) A strong magnet is used
2) Magnet or coil is moved at fast rate
3) A soft iron rod is used inside the coil.
2. Current induced by current:
In the figure(2) The bar magnet is replaced by a second coil C2 connected to a
battery. The steady current in the coil C2 produces a steady magnetic field. As coil C2
moved towards the coil C1, the galvanometer shows a deflection.
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This indicates the presence of induced current in
the coil C1.When coil C2 is moved away,The
galvanometer shows a deflection in the opposite
direction. The deflection lasts as long as coil C2 is
in motion.
The same effects are observed when the coil C2
held fixed and coil C1 Is moved. Thus the relative
motion between the coils induces the electric current.
3. Current induced by changing current:
In the figure(3) the two coils C1 and C2 are held fixed. Coil C1 is connected to the
galvanometer while the coil C2 is connected to a battery through a tapping key K.
when the tapping key is pressed the galvanometer shows deflection. When the key is
released the galvanometer shows deflection in the opposite direction. If the key is
held pressed continuously, there is no deflection in the galvanometer. Thus the make
and break of one circuit can induce electric current in the other circuit.
Magnetic flux (υ):
“The magnetic flux φ through any surface held in a magnetic field B is
defined as the total number of magnetic lines of force passing the surface
normally. It is measured as the product of surface area and the component of
magnetic field B normal to the surface area”.
Consider a surface area A held in a magnetic field B, such that the component
of magnetic field B normal to the surface is B cosƟ.
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Magnetic flux through the plane surface area A is
Φ = A B cosƟ
θ = B A cosƟ = B.A
where Ɵ ‒ Angle between magnetic field and normal to the surface area.
Magnetic flux is a scalar quantity and its SI unit is weber (wb).
The magnetic flux passing normally through a surface of 1 square meter when
magnetic field of one tesla is applied is called 1 Weber.
Cases:
1. When Ɵ = 90 i.e., The surface is parallel to the magnetic field B.
In this case θ = BA cos 90 = 0
2. When Ɵ = 0 i.e., the surface is perpendicular to the magnetic field B.
In this case θ = BA cos 0 = BA
when a coil of n turns and area A Is placed in a magnetic field, then the
magnetic flux linked with the coil is given by θ = n AB cos Ɵ
Faraday‟s Law of electromagnetic induction:
First law:
“Whenever there is a change in the magnetic flux linked with the circuit an emf
is induced in it”.
This emf lasts as long as there is a change in magnetic flux.
Second law:
“The magnitude of induced emf is directly proportional to the rate of change of
magnetic flux”.
i.e., E α dθ/dt
where
E- Induced emf
θ- Magnetic flux linked with the coil.
Page | 137
Lenz‟s Law:
“The direction of induced emf or induced current in a coil is always opposes
the cause which is responsible for its production”.
Explanation:
When the north pole of the magnet is made to move through a coil then the face „B‟
of the coil behaves as an north pole due to the induced current flowing in
anticlockwise direction hence it opposes the motion of north pole of magnet towards
the coil.
When the north pole of the magnet made to move away from the coil then the face
„B‟ of the coil behave as an south pole due to the induced current flowing in
clockwise direction hence it opposes the motion of north pole of the magnet away
from the coil.
Lenz‟s Law and energy Conservation:
When the north pole of a bar magnet is moved towards a coil, the direction of
induced current is such that the nearer face of the coil acquires north polarity.
Therefore work must be done against the force of repulsion, to bring the magnet
closer to the coil. Similarly when the north pole of a bar magnet is moved away from
the coil, the direction of induced current is such that the nearer face of the coil
acquires south polarity. Therefore work must be done against the force of attraction,
to move the magnet away from the coil. Here the mechanical energy is converted into
electrical energy.
NOTE:
E = - dθ/dt is called Neumann‟s relation. Negative sign indicates the opposing emf.
Page | 138
Motional emf induced in a rod moving in a uniform magnetic field:
“The emf induced in a conductor due to its motion in a magnetic field is called
motional emf”.
Let us consider rectangular conductor PQRS
such that only the side PQ is movable without
any friction. The rod PQ is moved towards the
left with a constant velocity „v‟. rectangular
conductor placed in a magnetic field which is
perpendicular to the plane of the system as
shown in the figure.
If the length RQ= x and RS=l,
the magnetic flux enclosed by the conductor θ = BA cos Ɵ
θ = Blx
( Ɵ =0)
WKT induced emf is given by,
E=-
E=-
dφ
dt
d(Blx )
E= -
dt
Bl dx
dt
E = Blv
where –
dx
dt
=v
Energy consideration in motional emf:
When a conductor of length „l‟ is moved with a velocity „v‟ perpendicular to the
magnetic field „B‟ the Motional emf is given by
e = Blv
Let „r‟ be the resistance of the movable arm PQ, assuming negligible resistance for
the remaining arms, then the current through the conductor is,
Page | 139
I=
e
r
=
Blv
………………… (1)
r
The conductor PQ of length „l‟ and carrying current „I‟ experiences force F in the
perpendicular magnetic field B. The force is given by,
F = Ilb sin 900
lB =
B 2 l2 v
r
The power required to push the conductor against this force is,
P = F× V =
B 2 l2 v
r
V=
B 2 l2 v 2
r
As the conductor is pushed mechanically, The mechanical energy dissipated per
second is,
2
Pj = I r =
B 2 l2 v 2
r
……………….. (2)
Here Pj= P. Thus Mechanical energy is required to move the conductor is converted
into electrical energy and then to thermal energy.
Eddy Current:
Whenever the magnetic flux linked with a metal sheet or blocks changes, an emf
is induced in it. The induced current flow in closed paths perpendicular to the
magnetic flux throughout the metal. These current looks like eddies or whirl-pool in
water and called as Eddy currents. These currents were first discovered by Focault,
So Eddy Currents are also called as Focault current.
“The current induced in a metal sheet when the metal sheet is placed in a
magnetic field are called Eddy current”.
Eddy current also oppose the change in magnetic flux so is governed by Lenz‟s Law.
Page | 140
Experiments to demonstrate eddy‟s current:
Experiment 1:
Take a pendulum in the form of a flat copper
plate as shown in figure.it is free to oscillate
between the pole pieces of an electromagnet. In the
absence of magnetic field, the pendulum swings
freely. As the electromagnet is switched on, the
oscillation of the pendulum get highly damped and
soon it comes to rest. This is because as the copper
plate moves in between the pole pieces of the
magnet, magnetic flux acts through it changes. So eddy currents are set up in it which
according to Lenz‟s law, oppose the motion of copper plate in the magnetic field.
Eddy currents flow anticlockwise as the plate swings into the field and clockwise as
the plate swings out of the field.
Experiment 2:
Now take the pendulum of a flat copper plate
with narrow slot cut across it, as shown in fig.
As the electromagnet is switched on, eddy
currents are set up in the plate but this plate swings
for longer duration compared with the plate without
slots.
This is because the loop has much larger paths
for the electron to travel. Larger paths offers more resistance to electron so eddy
currents are reduced. As a result, the opposition to the oscillation becomes very
small. Eddy currents are undesirable it cause unnecessary heating and wastage of
power. The heat produced by eddy current may damage the insulation of coil. Eddy
current cannot be eliminated but it can be minimized by using laminated plates.
Application of eddy current:
1. Used in braking system of modern train.
2. Large heat produced by eddy current is used to melt the metal in induction
furnace.
3. Used in working of Speedometer.
Page | 141
4. Used in electric meters for measuring electric energy.
5. Used to heat localized tissues of the human body. This branch is called
inductothermy.
Self-Induction:
Self-induction is the phenomena in which an emf is induced in a coil due to
change in the current through it. The property of the coil to oppose any change of
current through it is known as self-induction. Consider a coil connected to a battery B
through a tap key K. when the tap key is pressed, current in the coil begins to grow
from zero to maximum. As the current flows through the coil, it set up a magnetic
field in it.
When the current in the coil increases, the magnetic flux linking the coil also
increases. This increasing magnetic flux set up an induced emf in the coil. But the
induced emf opposes the growth of current in the coil. Thus the growth of current in
the coil is delayed and it takes longer time to attain the maximum value. This is called
Back emf. When the key is released, the current begins to decay from maximum to
zero value. The magnetic flux linking the coil also begin to decrease.
The decreasing magnetic flux sets up an induced emf in the coil. But the induced
emf opposes the decay of current. So, the current does not become zero
instantaneously, but takes some time.
Co-efficient of self-inductance:
The magnetic flux is directly proportional to the current „I‟ in a coil at that
instant.
i.e.
θ α I (or) θ = LI
where, L = Self inductance or co-efficient of self inductance.
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The emf „e‟ due to change in the current in the coil is given by,
dφ
e=-
e=-
dt
d (LI)
e = -L
In this relation L = e when
dI
dt
dt
dI
dt
= 1.
Negative sign indicates the direction of emf. The SI unit of self- inductance is
HENRY. Therefore the self-inductance is said to be one henry if 1 volt of emf is
induced in a coil when the rate of change of current is 1 ampere.
Self-inductance of solenoid:
Consider a long solenoid of cross-sectional area „A‟ and length „l‟ having „n‟
turns per unit length. The magnetic field due to a current I flowing in the solenoid is,
B = µ0nI
The total magnetic flux linked with the solenoid is θ= NBA
where N = nl is the total no of turns.
i.e., θ = nl (µ0nI) A = µ0n2A l I
but we have, θ=LI
Therefore, LI = µ0n2Al (or) L = µ0n2Al
If the solenoid has a some magnetic material of relative permeability µr, then
L = µ0µrn2Al
Energy stored in an inductor:
An inductor is a device having self-inductance (just as a capacitor is a device
having capacitance). The emf of the self- inductance always opposes any change in
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current. Hence it is called back emf. Hence work is done against back emf in
establishing the current. This work is stored as magnetic potential energy.
Let dw be the work done in establishing a current I in the coil in a time dt.
Then, dw = - eI dt
where e is the induced emf.
Since, e = -L
dI
dt
dw = L
dI
dt
I dt
dw = LI dI
The total work done in establishing the current is,
w = ∫dw =1∫ LIdI = L[I2/2] = ½ LI2
(or)
Potential energy, U = ½ LI2
𝐿
If I = 1, w = or L = 2w.
2
Thus, self inductance of a coil is equal to twice the work done in establishing unit
current in the coil.
Mutual Induction:
“Mutual induction is the phenomenon in which an emf is induced in one coil due
to the variation of current in the neighbouring coil”.
Consider two coils primary P and secondary S placed close to each other. The
coil P is connected to a battery B through a tap key K. The coil S connected to a
galvanometer.
Page | 144
When the tap key K is pressed the current in the primary coil increases from zero
to maximum. The current in the primary coil produces a magnetic field which threads
the primary coil as well as secondary coil. As the current in the primary coil
increases, the magnetic flux linked with the secondary coil increases. Therefore, an
emf is induced in the secondary coil which opposes the growth of the current in the
primary coil. The induced emf produced an induced current and a deflection is
observed in the galvanometer. But this induced emf exist only for short duration, till
current in the primary coil reaches a maximum steady value.
When the key is released, the current in the primary coil decreases from maximum
to zero. So the magnetic flux linked with the primary coil as well as secondary coil
decreases.
The decrease in the magnetic flux in the secondary coil induses an emf which
oposes the decay of current in the primary coil. The induced current again causes a
deflection in the galvanometer in the direction opposite to that when the key was
pressed.
Co-efficient of Mutual inductance:
The magnetic flux in the secondary coil is directly proportional to the current
„I‟ in the primary coil.
i.e., θ α I (or) θ = MI
where M =Mutual inductance (or) co-efficient of mutual inductance.
Then the emf „e‟ due to change in current is given by
e=Therefore,
e=-
dφ
but θ = MI
dt
d (MI )
e=-M
In the above relation M = e When
dt
dI
dt
dI
dt
= 1.
Negative sign indicates the direction of emf. The SI unit of mutual inductance is
„HENRY‟. Therefore mutual inductance is said to be 1henry if 1volt of emf is
induced in a coil when rate of current is 1A/sec.
Page | 145
Mutual inductance of two long co-axial solenoid:
Consider two coaxial solenoids S1 and S2 each of length „l‟. let n1 be the number
of turn per unit length and r1 be the radius of the inner solenoid S1 while n2 be the
number of turns per unit length and r2 be the radius of the outer solenoid S2. Imagine a
time varying current I2 through S2 which set up a time varying magnetic flux θ1
through S1.
Then,
θ1 = M12 I2
where M12 = the mutual inductance of solenoid S1 with respect to solenoid S2.
The magnetic field due to current I2 in S2 is, B2 = µ0n2I2
Magnetic flux through S1 is, θ1 = N1B2A1
where N1 = n1l is total number of turns in S1 and A1 = πr12 is the cross sectional area
of S1.
Therefore
θ1 = (n1l) (µ0n2I2) ( πr12)
θ1 = µ0n1n2 πr12l I2
Therefore
M12I2 = µ0n1n2πr12lI2
M12 = µ0n1n2πr12l
Now, consider the reverse case We have, M21 = µ0n1n2πr12l
thus, we get,
M12= M21 = M= µ0n1n2πr12l
Page | 146
where the mutual inductance of two coil depends on their geometry, their separation
and relative orientation.
If the magnetic material of relative permeability µr is present inside the solenoid, then
M= µr µ0n1n2πr12l.
Generator:
A generator or a dynamo is a device which converts mechanical energy into
electrical energy. When the current produced by the dynamo alternates i.e changes its
direction periodically with time the dynamo is called alternator or AC generator.
When the current produced by a dynamo always flows in the same direction, the
dynamo is called DC generator. Generator was designed by NIKOLA TESLA.
Principle:
It works on the principle of electromagnetic induction. Whenever the magnetic
flux linked with a coil changes, an emf induced in the coil. The direction of induced
current is given by Fleming‟s Right hand rule.
Fleming‟s right hand rule or generator rule:
It states that, “if the 1st 3 finger of the right hand is stretched mutually at right
angle to each other then the forefinger indicates the direction of magnetic field,
thumb indicates the direction of motion of the conductor and middle finger
indicates the direction of induced current”.
Construction:
AC generator consists of following parts,
a) Armature: It is a rectangular coil ABCD
consisting of large number of turns of
copper wire wound over a soft iron core.
b) Field magnet: It is a strong magnet having
pole N and S. The armature rotates b/w the
poles about an axis perpendicular to the
field.
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c) Slip rings: The leads of the armature are connected to the slip rings R 1 and R2. Slip
rings rotate with armature.
d) Brushes: Brushes B1 and B2 remains in contact with rings. The brushes pass on the
current from the armature to the external load.
Working:
When the armature rotates, the magnetic flux linked with it changes and current
is induced in the coil which flows through the external load. Consider the rotation of
the coil from the vertical position in the clockwise direction. As CD moves
downwards and AB moves upwards. According to Fleming‟s right hand rule, induced
current flows from A to B and then C to D. During the first half of the rotation
current flows in the coil in the direction ABCD.During the second half of rotation the
current flows in the direction DCBA. The magnitude of induced emf and current
varies, being zero in the vertical position and maximum in the horizontal position. In
this way the generator produces alternating current.
Expression for alternating emf and current:
Consider a rectangular coil of area A consisting of „n‟ turns rotating about an
axis perpendicular to the direction of magnetic field B.
At t = 0. Let the plane of the coil be vertical. Let the coil be rotates through an angle
Ɵ in a time „t‟ with a uniform angular velocity ω. Then the angular velocity
ω=
Ɵ
t
Ɵ = ωt
…………………. (1)
component of B perpendicular to the plane of the coil is BcosƟ
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Magnetic flux linked with the coil at time „t‟ is given by,
θ = nABcosƟ
Substituting for Ɵ,
θ = nABcosωt
…………….(2)
By Lenz‟s law, the induced emf in the coil at time „t‟ is given by,
E=-
dφ
dt
Substituting for θ,
E=-
d (nABcos ωt)
dt
E = - nAB (- ωsinωt )
E = nAB ωsinωt
……………. (3)
Since E = E0 peak value of induced emf
When ωt = 900
E0 = nAB ωsin 900
E0 = nABω
…………… (4)
substituting in equation (3)
E = E0 sinωt
………………(5)
If „R‟ is the resistance of the coil, at any instant time „t‟ the induced current in the coil
is given by,
I=
I=
E
R
E 0 sinωt
R
I = I0 sinωt
where I0 =
E0
R
…………….(6)
, the maximum value of induced current.
Thus, the equation (5) and (6) shows that, a rectangular coil rotating in a magnetic
field produces alternate emf and alternate current i.e., change direction alternatively.
************************
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CHAPTER- 7
ALTERNATING CURRENT
Alternating emf (voltage):
The emf, which changes its direction alternatively with time is called alternating
emf. The sinusoidal alternating emf at any instant of time „t‟ is given by
E = E0 sin ωt ………………. (1)
where, E = Instantaneous emf
E0 = Peak or maximum value of alternating emf
ωt = Phase of alternating emf
Alternating current:
The current, which changes its direction alternatively with time is called
alternating current. The sinusoidal alternating current at any instant of time „t‟ is
given by
I = I0 sin ωt ……………….. (2)
Where
I = Instantaneous alternating current
E0 = Peak or maximum value of alternating current
ωt = Phase of alternating current
Basic Definitions of AC :
Wave form:
“It is the shape of the curve obtained by plotting the instantaneous values of
voltage or current versus tome”.
Instantaneous value:
“It is value of alternating voltage or current at any instant is called
instantaneous value”.
Page | 150
Amplitude:
“The maximum value attained by an alternating current in either direction is
called its amplitude or peak value and is denoted by 𝑰𝟎 ".
Time period:
“The time taken by an alternating current to complete one cycle of its variation
is called its Time period and is denoted by T”.
This time is equal to the time taken by the coil to complete one rotation in the
magnetic field. As the angular velocity of the coil is ω and its angular displacement
in one complete cycle is 2π, so
Time period =
(or)
T=
Angular displacement in a complete cycle
Angular velocity
𝟐𝝅
𝝎
Frequency:
“The number cycles completed per second by an alternating current is called
its frequency” and is denoted by f. The frequency of an alternating current is same
as the frequency of the coil in the magnetic field. Thus
1
𝜔
𝑇
2𝜋
f= =
The alternating current varies with time. It rises from 0 to maximum in one
direction, then falls to 0 and then rises from 0 to maximum in the opposite direction
and again falls to 0, thus completing one full cycle.
The alternating current supplied to our houses has a frequency of 50 cps or 50 Hz.
Phase of AC:
Phase of AC represents its state of variation. It is the fraction of the time period
that has elapsed after the voltage or current has passed through zero value in the
positive direction. During a time interval equal to the period T, the phase changes by
2π radian.
Page | 151
E
0
(a)
(b)
T
π
T
4
2
2
The phases at 0, A, B, C and D are 0, or ,
or π ,
3T
4
or
3π
2
and T or 2π
respectively.
Mean value of alternating current and alternating emf :
The mean value of the alternating current or voltage during the first and second
half of a cycle will be equal in magnitude and opposite in signs. Thus mean value
over one complete cycle will be zero.
Mean or Average value of AC:
“The average value of instantaneous current taken over half of the cycle (half
of the period) of AC is called mean value of alternating current”.
2
It is found to be equal to the times the peak value of AC.
𝜋
i.e.,
Im=
2
I0 = 0.637 I0
𝜋
Similarly, the mean value of alternating emf,
i.e.,
Em =
2
𝜋
E0 = 0.637 E0
Effective value of ac or root mean square value of ac (irms):
The emf and current being sinusoidal, their mean values over a complete cycle is
zero. But their squares are always positive.
Page | 152
“The root mean square value of AC is defined as the square root of mean of
squares of instantaneous currents taken over a complete cycle (period) of
alternating current”.
It is found to be equal to
i.e
Irms =
1
2
1
2
times the peak value of alternating current.
I0 = 0.707 I0
Similarly, the rms value of alternating emf,
i.e
Erms=
1
2
E0 = 0.707 E0
Phasor representation of alternating voltage or current:
1. A line of definite length rotating in the anticlockwise direction with a
constant angular velocity (ω) is called a phasor.
The length of the phasor line is taken equal to the maximum value of the
alternate quantity. Such a rotating line generates a sinusoidal wave.
2. A diagram in which voltage and current are represented as phasors with the
phase angle between them is called a phasor diagram.
3. Circuit elements: The components used in the construction of circuit are
called circuit elements.
The circuit element of AC circuits are resistor, inductance and capacitor.
AC voltage applied to a Resistor:
Consider a circuit containing a pure resistance R applied with an alternating voltage
V = V0 sin ωt
………….. (1)
where, V0 – peak voltage, ω – angular frequency.
Page | 153
Let „I‟ be the current through the circuit at any instant of time„t‟ is given by
I =
I=
V
R
V0 sin ωt
R
I = I0 sin ωt
where I0 =
V
R
………….. (2)
is the maximum or peak value of the current in the circuit. From
equation (1) and (2) we note that both V and I are function of sin ωt. Hence the
voltage and current are in same phase in a purely resistive circuit. This means that
both V and I attain their zero, minimum and maximum values at the same respective
times. This phase relationship s shown graphically in above figure.
AC voltage applied to an inductor:
“A coil having only self-inductance and zero ohmic resistance is called a
pure inductance ”.
Consider a circuit containing a pure inductance „L‟ applied with an alternating
voltage,
V = V0 sin ωt
It is shown in the figure.
………….. (1)
Page | 154
As the alternating voltage flows through the inductor. Its induces an emf in the
inductor. This induced voltage opposes the applied voltage.
𝛆= -L
dI
dt
According Kirchhoff‟s 1st law,
V+𝛆=0
dI
V–L
From equation (1),
dt
( inductor has negligible resistance i.e., R = 0)
=0
V0 sin ωt - L
V0 sin ωt = L
dI
dt
dI
=0
dt
V
dI = sin ωtdt
𝐿
Integrate,
dI =
I=
V
L
sin ωt dt
V − cos ωt
L
ω
+ constant
The constant of integration can be shown to equal to zero.
i.e.
I=
I=
V
𝜔𝐿
V
𝜔𝐿
( - cos ωt)
𝜋
sin (ωt - )
2
𝜋
I = I0 sin (ωt - )
2
𝜋
[∵ - cosωt = sin ( ωt - ) ]
2
…………. (2)
This is the expression for the current through the inductance.
Page | 155
Where I0 =
V
𝜔𝐿
=
𝑉
𝑋
peak value of the current through the inductance and
XL = ω L = 2πf L called inductive reactance.
From equation (1) and (2) it is found that the current lags behind the applied voltage
by π/2 (or) 900 (or voltage leads the current by π/2). Hence the current and voltage are
said to be out of phase. The variation of „V‟ and „I‟ with„t‟ are shown in above figure.
Inductive reactance:
Inductance reactance is the opposition offered by the inductance coil to the flow
of AC through it. If ω is the angular frequency and f is the frequency of the
alternating voltage applied across the inductor, then inductive reactance is,
XL = ωL = 2πfL
The SI unit of inductive reactance is ohm (Ω). It depends only on the self
inductance of the inductor and the frequency of the alternating voltage.
Inductive reactance is directly proportional to the frequency i.e., XL α f. For
direct current f = 0 therefore XL = 0. Thus a pure inductor offers no opposition to
direct current.
AC voltage applied on a capacitor:
Consider a circuit containing pure capacitor „C‟ applied with an alternating
voltage,
V = V0 sin ωt ………… (1)
It is shown in figure.The magnitude of the charge on the plate of the capacitor at any
instant of time „t‟ is given by, q = CV
q = C V0 sin ωt
Page | 156
The current in the circuit at any instant of time „t‟ is given by,
I=
dq
dt
=
d
dt
( C V0 sin ωt)
I = C V0 ω cosωt
𝜋
I = V0 (ωC) sin (ωt + )
I=
(or)
where, I0 =
v
1
ωC
The quantity
1
ωC
v
1
ωC
𝜋
𝜋
[ ∵ cosωt = sin (ωt + ) ]
2
2
sin (ωt + )
2
𝜋
I = I0 sin (ωt + )
………… (2)
2
is the peak value of current.
is analogues to resistance and it is called capacitive reactance
denoted by XC.
Thus capacitive reactance XC =
1
𝜔𝐶
=
1
2𝜋𝑓𝐶
where „f‟ is the frequency of the alternating voltage.
From equation (1) & (2) it is found that the current leads the applied voltage by
𝜋
0
𝜋
2
0
(or) 90 ( voltage lags behind the current by or 90 ) as shown in the above figure.
2
Hence the current and voltage are said to be out of phase.
Capacitive reactance:
Capacitive reactance is the effective opposition offered by capacitance to
alternating current in the circuit. If „ω‟ is the angular frequency and „f‟ is the
frequency of the alternating voltage applied across the capacitor, then capacitive
reactance is,
XC =
1
ωC
=
1
2πfC
Page | 157
The SI unit of capacitive reactance is ohm (Ω). It depends only on the capacitance of
the capacitor and frequency of the alternating voltage. It is inversely proportional to
the frequency i.e., XC =
1
f
as frequency increases capacitive reactance decreases. For
direct current f = 0. Therefore XC = ∞ ( infinity), so capacitor offers infinite
opposition to the direct current through it.
NOTE:
It will not dissipate power therefore, the AC flowing in pure inductance and pure
capacitor is called Wattles or Idle current.
Series LCR circuit:
An ohmic resistance, an inductance and a capacitor connected in series with an
alternating power supply constitute a series LCR circuit.
AC voltage applied to a series LCR circuit:
Consider a AC circuit containing a inductor of pure inductor L, a capacitor of
pure capacitor C and resistor of resistance R be connected in series with an
alternating voltage V.
V = V0 sin ωt ………… (1)
It is shown in figure.
Let „I‟ be the current through the circuit at time„t‟. VL. VC and VR be the potential
difference across L,C and R respectively at that instant.
Then,
VL = XL ,
VC = XC
and VR = IR.
The sum of these voltages is equal to the applied voltage V at time „t‟.
Phasor diagram is shown in the above fig ( here current considered along the positive
x – axis).
Page | 158
In the phasor diagram,
VR and I are in the same phase and is represented by OA.
VL leads the I by 900 and is represented by OB
VC lags behind the I by 900 and is represented by OC
Since VL and VC are in opposite phase, so their resultant voltage is (VL – VC) and is
represented by OD. The diagonal OP of the rectangle OAPD gives the resultant
voltage V. From the triangle OAP,
OP2 = OA2 + AP2
Ve2 = VR2 + (VL – VC)2
Ve2 = Ie2 R2 + ( Ie XL – IeXC)2
Ve2 = Ie2 [R2 +( XL – XC)2]
Ve = Ie 𝑅2 + XL – XC
Ie =
2
Ve
R2+ XL – XC
2
The quantity 𝑅2 + XL – XC 2 represents the effective opposition for the flow of AC
and is called impedance of the circuit. It is denoted by Z.
i.e.,
Z = 𝑅2 + XL – XC
The unit of impedance is ohm. Hence I =
2
𝑉
𝑍
Phase angle:
The angle between the effective voltage and effective current in a AC circuit is
called the phase angle.
From the angle,
tan ϕ =
tan ϕ =
tan ϕ =
V −V
V
IX −IX
IR
X −X
R
ɷ𝐿 −1/ɷ𝐶
ϕ = tan-1 [
𝑅
]
Page | 159
NOTE:
Instantaneous current through the circuit is given by,
I = I0 sin ( ωt - 𝜙 ) for XL ˃ XC
I = I0 sin ( ωt + ϕ) for XL < XC
Electrical resonance:
The phenomenon of current in a series LCR circuit becoming maximum (or
impedance minimum) at a particular frequency (f0) of the applied alternating voltage
is known as electrical resonance. As the frequency of the applied voltage increases,
the inductive reactance XL increases and the capacitive reactance XC decreases.
At particular frequency f = f0, XL = XC and hence Zmin = R.
Hence, XL = XC is the condition for the resonance of LCR circuit.
Consider,
XL = XC
2πf0L =
f0 2 =
f0 =
1
2πf 0 C
1
4𝜋 2
LC
1
2𝜋 𝐿𝐶
Hz
This is the expression for the resonance frequency f0.
NOTE:
Resonance frequency is independent of resistance.
1) XL = 2πfL, at higher frequency (f ˃ f0) ,XL ˃ XC , the circuit is inductive
i.e., ϕ is positive.
2) XC =
1
2πfC
, at lower frequency (f < f0), XC ˃XL , the circuit is capacitive
i.e., ϕ is negative.
Salient features of series resonance LCR circuit:
1.The impedance is minimum.
2. The current is maximum.
Page | 160
3. The LCR circuit behaves as pure resistive circuit.
4. The phase angle is zero.
5. The power factor is unity and the power dissipated will be maximum.
Sharpness of resonance ( q – factor):
The graph of current and frequency of the applied AC voltage for a series LCR
circuit is called the resonance curve. For LCR circuit V = IZ. The peak value of
current is given by,
I0 =
I0 =
𝑉
𝑍
V
𝑅2+ XL – XC
2
For maximum current XL = XC and hence Z = R. Thus I0 =
𝑉
𝑅
This implies that the maximum current varies inversely to the ohmic resistance in the
circuit. For small value of „R‟, the resonance is sharp. At f = f0 the current will be
maximum. The current decreases sharply on either sides of resonance. For larger
value of „R‟ the resonance is flat or less sharp. In such a case the current decreases
slowly on either sides of resonance.
In LCR circuit at resonance, the power loss is maximum. When frequency is
changed from resonant frequency the power loss decreases.
Let I0 be the maximum current in a resonant circuit at the resonant frequency f0.
Let f1 and f2 be two frequencies on either sides of f0 at which the current is
the maximum current i.e.,
I=
1
2
1
2
times
I0
The frequencies f1 and f2 are called half power frequencies. The difference (f2- f1)
is called band width.
The quantity
f
(f2− f1)
is a measure of the sharpness of resonance and it is called
quality factor or Q- factor.
“ The quality factor of a LCR circuit is defined as the ratio of resonant
frequency to band width of resonance circuit”
Page | 161
i.e.,
Q=
f
(f2− f1)
=
resonant frequency
band width
Alternatively the quality factor of a LCR series circuit is defined as the ration of
inductive reactance at resonance (capacitive reactance at resonance) to the resistance
of LCR circuit.
i.e. Q =
𝜔0𝐿
𝑅
=
2𝜋𝑓 𝐿
𝑅
Power In AC Circuits:
The rate at which electrical energy is consumed in an electric circuit is called
its Power. In DC circuit, the power is given by the product of the applied voltage and
current. But in AC circuit, the current and voltage may or may not be in phase. Power
in an AC circuit is given by,
PAv = VrmsIrmscos ϕ
…………(1)
PAv is called the true power or average power and VrmsIrms is called apparent power or
virtual power, cos ϕ is known as power factor of the circuit.
Thus, TRUE POWER = APPARENT POWER × POWER FACTOR
Hence the true power of the AC circuit not only depends on rms values of the voltage
and current but also depends on phase difference between them.
Page | 162
Power Factor:
Power factor is the cosine of the phase difference between the current and
voltage in an AC circuit. It is always positive and not more than 1. The power factor
of a series LCR circuit is given by
𝑅
cos ϕ = = [
𝑍
𝑅
𝑅2
1 2
)
ɷ𝐶
+ (ɷ𝐿 −
]
For a purely inductive or capacitive circuit, ϕ = 900.
Therefore, Power factor = cos 900 = 0
Thus, the power factor becomes minimum value for a purely inductive or capacitive
circuit.
For purely resistive circuit ϕ = 00.
Therefore Power factor = cos 00 = 1
Thus the power factor becomes maximum value for a purely resistive circuit.
“Since PAv = 0 current flowing through the circuit containing only an inductor or a
capacitor is called wattles current” (or) the current which does not dissipates any
energy is called wattles current.
LC Oscillation:
When a charged capacitor is connected to an inductor, the charge oscillates
from one plate of the capacitor to another through the inductor. This result in the
production of electric oscillation called electromagnetic oscillations. “when a charged
capacitor is allowed to discharge through a non – ohmic resistive inductor, electrical
oscillations of constant amplitude and frequency are produced. These oscillation are
called LC – oscillation.
Page | 163
Consider a capacitor of capacitance C with initial charge q0 at t= 0 be connected to
an ideal inductor of self-inductance L. the electrical energy stored in the electric field
of capacitor is given by,
VE =
1 𝑞 02
2 𝐶
As there is no current in the circuit, the energy stored in the magnetic field of the
inductor is zero.
As the circuit is closed, the capacitor begins to discharge through the inductor
and hence current starts flowing through the inductor. As a result magnetic field is set
up around the inductor.
When the capacitor is discharged completely, potential differences across its plates
become zero. The current reaches its maximum value I0. The energy stored in the
magnetic field is given by,
VB =
1
2
LI02
Thus the entire electrostatic energy of the capacitor has been converted into
magnetic field energy of the inductor.
When the capacitor is fully discharged, there is no more current to maintain the
magnetic field around the inductor and hence magnetic field collapses and produces
an emf. According to Lenz‟s law, drives the current in the same direction, to charge
the capacitor in the reverse direction. Now the energy flows from inductor back to the
capacitor. As a result the capacitor get all the energy and charge back to the original
voltage but in opposite direction. Again the capacitor starts discharging in the
opposite direction. Eventually the circuit returns to its original situation.
The whole process repeats, giving continuous oscillation both charge and energy
oscillates back and forth (forward).If the resistance of the LC circuit is zero, there is
no loss of energy and the oscillation will be constant amplitude. Such oscillation are
called undamped oscillation.
If there are resistive losses in the inductor and dielectric losses in the capacitor, a
small part of the energy is lost in every oscillation. As a result the amplitude of the
oscillations decreases. Such oscillations are called damped oscillations.
Page | 164
Frequency Of LC Oscillation:
Let a capacitor C be charged and be connected to an inductor of inductance L.
As the capacitor starts discharging an emf is induced in the inductor .At any instant of
time, the potential difference across the capacitor and the inductor must be equal. If
„q‟ is the charge on the plate of capacitor at any instant of time„t‟.
𝑞
= -L
𝐶
L
𝑑𝐼
𝑑𝑡
𝑑𝐼
𝑑𝑡
𝑞
…………. (1)
+ =0
𝐶
𝑑𝑞
Let „I‟ be the current due to the flow of charge from the capacitor, I =
dI
Hence,
dt
From equation (1)
L
=
d2q
dt 2
𝑑2𝑞
𝑑𝑡 2
𝑑2𝑞
𝑑𝑡 2
This equation has the form
𝑑 2𝑥
𝑑𝑡 2
𝑞
+ =0
𝐶
+{
1
𝐿𝐶
}q=0
+ ω02 x = 0 for a simple harmonic oscillator.
1
The charge oscillates with a natural frequency ω0 =
2πf =
𝑑𝑡
1
𝐿𝐶
=˃
f=
𝐿𝐶
(or)
1
2𝜋 𝐿𝐶
Total energy in lC circuit (or) conservation of energy in lC circuit:
At any instant of time „t‟ the charge left on the capacitor is „q‟ and the current
in the inductor is „I‟.
Energy stored in the capacitor is, VE =
1 𝑞0 2
2 𝐶
The magnetic energy stored in the inductor is, VB =
1
2
LI02
If there is no loss of energy, then the total energy of the LC – circuit at any instant
will be,
V = VE + VB
V=
1 𝑞0 2
2 𝐶
+
1
2
LI02
Page | 165
But,
q = q0cos ω0 t and I =
Therefore, V =
V=
V=
V=
1
2𝐶
1
2𝐶
q02 cos2 ω0 t +
q02 cos2 ω0 t +
1 𝑞0 2
2
𝑐
𝑑𝑞
1
2
1
2
𝑑𝑡
= ω0 q0 sin ω0 t
L ω02 q02 sin2 ω0 t
L
1
𝐿𝐶
q02 sin2 ω0 t
[cos2 ω0 t + sin2 ω0 t ]
𝑞0 2
2𝑐
The initial energy stored in the capacitor is also equal to
𝑞0 2
2𝑐
.
Therefore the total energy is always constant and hence the energy is conserved.
Transformer:
“Transformer is a device which is used to increases or decrease AC voltages”.
A transformer is an electrical device for converting an alternating current at lower
voltage into that at high voltage vice versa. Based on this the transformer are of two
types.
1) Step up Transformer:
The transformer which increases the AC voltage by decreasing the current is
called step up transformer.
2) Step down Transformer:
The transformer which decreases the AC voltage by increasing the current
is called step down transformer.
Principle:
It works on the principle of mutual induction.
Construction:
It consists of two coils (primary and secondary). The primary and the secondary
coils are consists of insulated copper wire. The two coils are wound over opposite
arms of the same soft iron core and insulated from one another. The number of turns
and thickness of wire in a coil depends on nature of transformer.
Page | 166
Working and Theory:
When an alternating voltage is applied across the primary coil of a transformer,
the magnetic flux linked with it continuously changes. The changing magnetic flux
gets linked with secondary coil through the laminated core. As a result alternating
voltage is induced in the secondary coil.
The magnitude of the voltage induced in the secondary coil depend on the voltage
across the primary coil and number of turns in the primary and secondary coil.
If VP and VS are the voltages in primary and secondary and NP and NS are the
number of turns in the primary and secondary respectively.
Then
𝑉𝑠
𝑉𝑝
=
𝑁𝑠
𝑁𝑝
=T
Where T is called turn ratio.
If T ˃ 1, then NS ˃ NP it means VS ˃ VP i.e. the output voltage is greater than the
input voltage such a transformer is called step up transformer.
If T < 1, then NS < NP it means VS < VP i.e. the output voltage is lesser than the input
voltage such a transformer is called step down transformer.
For ideal transformer with no power loss,
Output power = Input power
VS IS = VP IP
Where IP and IS are the values of current in primary and secondary coils.
Page | 167
In practice, always the output power will be less than the input power due to
following losses:
Loss due to heating:
Since the coil have a resistance, heat is produced to joule‟s heating. This can be
minimized by choosing wires of good conductor and proper thickness.
Loss due to flux leakage:
The entire magnetic flux generated by the primary coil. It may not get linked with
secondary coil. This can be minimized by winding the coils inside the coil.
Loss due to eddy current:
Eddy current produced unnecessary heat in the core. This can be minimized by
using laminated plates.
Loss due to hysteresis:
As the core of the transformer is taken through several cycles of magnetization, a
certain amount of energy is lost as heat. This can be minimized by choosing good
materials with narrow hysteresis loop.
************************
Page | 168
CHAPTER-8
ELECTROMAGNETIC WAVES
Introduction:
According to Faraday‟s law of electromagnetic induction, time varying
magnetic field acts as a source of time varying electric field. Maxwell showed that
magnetic field can be produced by electric current as well as time varying electric
field. He investigated Ampere‟s circuital law and found some inconsistency in the
law. He introduced the concept of displacement current to modify the Ampere‟s
circuital law. The modified Ampere‟s circuital law is known as Ampere-Maxwell
circuital law.
According to Maxwell, when either of the field changes with time it gives rise
to other field. This leads to the production of electromagnetic disturbance. This
concept gave birth to electromagnetic waves which propagates in free space with a
speed of light i.e., 3× 108 ms-1.
“Electromagnetic waves are coupled time varying electric and magnetic fields
which propagates in free space”
Displacement current:
According to Ampere‟s circuital law, the line integral of the magnetic field 𝐁
around any closed path is equal to 𝛍𝟎 times the total current passing through the
closed path. i.e.
𝐁 𝐝𝐥 = 𝛍𝐨 𝐈
In 1864 Maxwell found an inconsistency in this relation. The inconsistency can
be understood with the help of the following observations using a parallel plate
capacitor C as follow:
Let I be the time dependent current in the circuit during the charging of the
capacitor. Consider a plane circular loop C1 of radius r whose centre lies on the wire
carrying current and its plane is perpendicular to the direction of the current in the
wire as shown in the [fig a].The magnitude of the magnetic field due to the current is
same at all points on the loop and it is acting tangentially along the circumference of
the loop. If 𝐁 is the magnitude of the magnetic field at any point P, then using
Ampere‟s circuital law for loop C1, we have
Page | 169
B dl = B
dl = B (2πr)
B (2πr) = μo I
∴
𝐁=
𝛍𝐨 𝐈
𝟐𝛑𝐫
................... (1)
Now consider another surface S, shaped like a Tiffin box without a lid. This
surface does not touch the current and has its bottom between the capacitor plates as
shown in [fig b].
Using Ampere‟s circuital law, the magnetic field at P is,
B (2πr) = μo (0)
∴
𝐁 = 0 .................. (2)
From these results we find that there is a magnetic field at P if calculated
through one way (eqn 1) and there is no magnetic field at P if calculated through
another way (eqn 2). Since this contradiction arises from the use of Ampere‟s circuital
law, we can say that Ampere‟s circuital law is logically inconsistency.
According to Maxwell, the convectional understanding is electric current is
essentially the flow of charge (electrons). But current is to be taken as that which
produces a magnetic field. Maxwell resolved the problem of no conduction current
between the plates of the capacitor by stating that during charging and discharging, a
changing electric field is set up between the plates of the capacitor. The changing
electric field is equivalent to a electric current and produces a magnetic field.
Maxwell named the equivalent current as displacement current.
Page | 170
Expression for displacement current:
Let A be the area of the plates and d be the distance between the plates of the
capacitor. The capacitance of the parallel plate capacitor is
C=
𝛆𝟎 𝐀
𝐝
If q is the charge on the capacitor and V is the potential difference between its plates
at any instant of time, then
q = CV or V =
𝐪
𝐜
But electric field between the plates of the capacitor is,
E=
E=
v
d
=
q
cd
d
ε0 A
=
q
d
𝐪
𝛆𝟎 𝐀
The electric flux through the surface S in fig(b) is,
ϕE = E ×A=
𝛟𝐄 =
If
dq
dt
q
ε0 A
×A
𝐪
𝛆𝟎
is the rate of change of charge with time on the plate of the capacitor, then
dϕ E
dt
1 dq
=ε
𝐝𝐪
0
dt
𝐝𝛟𝐄
= 𝛆𝟎
𝐝𝐭
𝐝𝐭
This represents the current through the surface S due to changing electric field and it
is called Maxwell‟s displacement current represented as ID. Thus displacement
current is given by,
I D = 𝛆𝟎
𝐝𝛟𝐄
𝐝𝐭
The displacement current is the missing term in Ampere‟s circuital law.
Page | 171
“Displacement current is that current which appears in the region in which the
electric field and hence the electric flux is changing with time.”
Ampere- maxwell‟s law or generalised ampere‟s law:
The generalised Ampere‟s law is given by,
B dl = μo (IC + ID )
𝐁 𝐝𝐥 = 𝛍𝐨 𝐈𝐂 + 𝛆𝟎
𝐝𝛟𝐄
𝐝𝐭
This law states that “the line integral of magnetic field around any closed path is
equal to 𝝁𝒐 times the sum of conduction current and the displacement current
through that closed path.”
Electromagnetic Waves:
Source of electromagnetic waves:
The important result of Maxwell‟s theory is that accelerated charges radiate
electromagnetic waves. Consider a charge oscillating with some frequency. An
oscillating charge is an example of accelerating charge. It produces an oscillating
electric field in space, which produces an oscillating magnetic field, which in turn is a
source of oscillating electric field, and so on. The oscillating electric field and
magnetic field thus regenerate each other as the wave propagates through space. The
energy associated with the propagating wave comes at the expense of the energy of
the source, namely , the accelerated charge.
According to Maxwell, “Electromagnetic waves are those waves in which there are
sinusoidal variation of electric and magnetic field vectors at right angles to each
other as well as at right angles to the direction of wave propagation”.
Heinrich Hertz produced electromagnetic waves of longer wavelength (6m) using a
spark oscillator and detected them waves successfully. Later on, Jagadish Chandra
Bose succeeded in producing and observing electromagnetic waves of much shorter
wavelength (5mm to 25mm).Guglielmo Marconi succeeded in transmitting
electromagnetic waves over distance of several kilometres. Marconi‟s experiment
was the beginning of the field of communication using electromagnetic waves.
Page | 172
Nature of electromagnetic waves:
The time varying electric field and magnetic field are mutually perpendicular
to each other and also perpendicular to the direction of propagation of the wave. In
the figure electromagnetic wave propagates along the Z–direction. The electric field
Ex is along X -axis and varies sinusoidal with Z -axis at a given time. The magnetic
field By is along the Y-axis and varies sinusoidal with Z -axis at a given time. The
electric field Ex and magnetic field By are perpendicular to each other, and also
perpendicular to the direction of propagation which is in Z-axis.
The equations for the sinusoidal variation of electric field Ex and magnetic field By
with time t are as follows,
Ex = E0 sin(ωt − kZ )
.................(1)
By = B0 sin(ωt − kZ ) ..................(2)
where E0 and B0 are the maximum amplitudes of electric and magnetic fields
respectively. ω is the angular frequency, k is called propagation constant or angular
wave number and it is related to wavelength λ as,
k=
n
2π
λ
n
using eq (1) and eq (2) and the Maxwell‟s eqn ,we can find that,
ω =C k
where C =
If
1
μ 0 ε0
ν is the frequency of the electromagnetic wave, we have
2 πν = C
2π
λ
C =νλ
𝛎=
𝐂
𝛌
Page | 173
Important characteristics and properties of electromagnetic waves:
1. The electromagnetic waves are produced by accelerated electric charge.
2. Electromagnetic waves do not require any material medium for propagation.
3. The speed of electromagnetic waves in free space or vacuum is 3× 108 ms-1 ,
which is given by the relation C =
1
μ 0 ε0
, where μ0 is the permeability and ε0
is the permittivity of free space.
4. The speed of electromagnetic waves in a material medium is given by the
relation v =
1
με
, where μ is the permeability and ε is the permittivity of the
medium.
5. Electromagnetic waves are transverse in nature.
6. The electric field vector E is responsible for the optical effects of an
electromagnetic wave and it is called “light vector”.
7. Electromagnetic waves transfer momentum and energy.
8. Intensity of an electromagnetic wave depends upon the amplitude of electric
1
field/ magnetic field i.e., I = ε0 c E0‟
2
9. Electromagnetic waves undergo reflection, refraction, they exhibit interference,
diffraction and polarisation.
Electromagnetic spectrum:
An orderly distribution of electromagnetic radiations according to their
wavelength or frequency is called the electromagnetic spectrum
Electromagnetic waves consists of,
1. Radio waves
2. Microwaves
3. Infrared –rays
4. Visible light
5. Ultraviolet rays
6. X-rays
7.Gamma rays
Page | 174
Page | 175
1. Radio waves :
Radio waves were discovered by Marconi. The frequency range of radio waves
is from few Hz to 109 Hz and wavelength ranges from 0.1m to several kilometer.
Radio waves are generated by electronic devices called LC oscillators .They exhibit
all properties of electromagnetic waves.
They are subdivide into;
i. SHF (super high frequency)
UHF (ultra high frequency)
VHF (very high frequency)
ii. HF (High frequency)
LF (Low frequency)
MF(Medium frequency)
VLF(Very low frequency) waves
USES:
1. LF radio waves [3kHz – 30kHz] also called long waves ,are used for medium
range communication.
2. MF radio waves [300 kHz - 3MHz] also called medium waves, are used for local
sound broadcast.
3. HF radio waves [3 MHz-30MHz] also called short waves are used for distant
broadcast & communication.
4. VHF radio waves [88MHz -108MHz ] are used for frequency modulated sound
broadcast i.e., FM radio.
5. UHF radio waves [300MHz -3000MHz] are used for TV broadcast and also in
cellular phones.
2. Microwaves :
They were discovered by the Hertz. The frequency ranges from 109Hz to
3×1011Hz and wavelength ranges from 1mm to 0.1m.
They are also called as short-wavelength radio waves. They are generated by special
electronic devices (vacuum tubes) such as Klystrons, Magnetrons and Gunn diodes.
They exhibit all the properties of electromagnetic waves. They can be detected by
crystal detectors.
Page | 176
USES:
1. In radar system electromagnetic waves used in aircraft navigation.
2. In the study of atomic and molecular properties of matter.
3. In domestic appliances like microwaves ovens.
4. In satellite communication.
3. Infrared rays :
They were discovered by William Herschel. The frequency range of IR rays is
from 3×1011 Hz to 4×1014 Hz and the wavelength range is of 750nm to1nm.Sun is
the main natural source of IR rays. These rays are also called as heat waves because
they are produced by hot bodies.
USES:
1. In long distance photography because of their high penetrating power.
2. In the diagnosis of superficial tumours, to improve blood circulation, to cure
sprains etc.
3. To detect erasure and forgery.
4. In physiotherapy to treat paralysis, dislocations and fracture of bones.
5. In solar cookers, solar lights, water heater etc.
6. In the analysis of molecular structure.
7. It is useful in the examiner of old paintings, documents etc.
8. By earth resource satellites to detects healthy crops.
4. Visible light :
It is the most familiar form of electromagnetic waves. It forms a narrow part of
electromagnetic spectrum, which is detected by human eye.
It extends over a very small range of frequencies from 4×1014 Hz to 8×1015Hz and
wavelength range is of about 400nm to 750nm.
The different frequencies of visible light in the decreasing order are associated with
colors ranging from violet to red.
USES:
1. Visible light is useful in photography.
Page | 177
2. It stimulates the sense of sight in human beings, so the beautiful world around
us can be seen.
3. It is useful in optical microscopy.
4. It is useful in astronomy.
5. It is a great source of energy for human life.
5. Ultraviolet rays :
It was discovered by Ritter in 1801. The frequency range of UV rays is about
8×10 Hz to 3×1017Hz and wavelength ranges from 1nm to 400nm.
They are produced by very hot bodies and electronic arc lamps.The sun is an
important natural source of UV radiation .
UV rays are detected by photography, photoelectric effect,fluorescence and
phosphorescence methods.
14
USES:
1. In high resolving power microscopes called ultra microscopes.
2. As activators in various chemical reaction.
3. In the sterilization of air & water in hospitals as they kill bacteria (in operation
theatre, in laboratories preparing blood plasma)
4. To distinguish between fresh eggs from rotten eggs, natural teeth from dentures,
real gems from artificial gems etc.
5. It is helpful in the detection of adulteration.
6. It is helpful in the treatment of certain skin diseases and bone diseases.
7. It is helpful in curing vitamin D deficiency etc.
8. To detect finger prints on a surface.
6. X-rays:
They were discovered by W.Roentgen in 1894 . The wavelength is about
0.001nm to 1nm and frequency range is 3×1017 Hz to 3×1020 Hz .They are
produced in X-ray tubes when high energy electrons are stopped suddenly on a metal
of high atomic number. X-ray have high penetrating power.
USES:
1. To detect foreign bodies like coin, pin, bullets etc inside the human body and
fracture of bones.
2. To destroy malignant tumours and to cure skin diseases.
Page | 178
3. To detect faults ,cracks, air pockets etc in metal castings.
4. In the investigation of structure of crystals.
5. In the treatment of certain forms of cancer.
7. Gamma rays :
They were discovered by Willard .The frequency range of γ-rays is 3×108 Hz to
more than 3× 𝟏𝟎22Hz and wavelength is very smaller than 0.001nm.
Gamma rays are emitted by radioactive nuclei and are produced in nuclear reaction .
They have very high penetrating power.
USES :
1. These are used in medicine to destroy cancer cells.
2. These are used to examine the thick materials for structural flaws.
3. These are used for food preservation.
4. These are used to get valuable information about the structure of atomic nuclei.
***********************
Page | 179
CHAPTER-9
RAY OPTICS AND OPTICAL INSTRUMENTS
“Optics is the branch of physics which deals with the study of nature, production
and rectilinear propagation of light”.
Light is a form of energy, we can see the world around us because of light. Light
is an electromagnetic radiation of wavelength range from 400 nm to 750 nm. Light
travels along a straight line and it is known as rectilinear propagation.
The straight line along which light travel is called a Ray. It is indicated by a
straight line with an arrow. It is the path of the light. The collection or bundles of
light rays is called Beam of light.
 If a number of rays in a beam are parallel, it is called parallel beam.
 If the rays in a beam converge at a point, it is called converging beam.
 If the rays in a beam appears to diverge from a point, it is called diverging
beam.
Optics can be divided into two main branches,
1) Ray optics
2) Wave optics
Ray optics (Geometrical optics):
It concerns itself with particle nature of light and it is based on rectilinear
propagation of light and the laws of reflection and refraction of light. It explain the
formation of images, the aberrations of optical images, working and designing of
optical instruments.
Wave Optics:
It concerns itself with the wave nature of light and is based on the phenomenon
like interference, diffraction and polarization of light.
When the light travelling in one medium falls on the surface of a second medium, the
following three effects may occurs.
Page | 180
 A part of the incident light is turned back into the first medium, this phenomena is
called refection of light.
 A part of the incident light is transmitted into the second medium along a changed
direction, this phenomena is called refraction of light.
 A part of the light energy is observed by the second medium, this phenomena is
called absorption of light.
Reflection of light:
The phenomena in which the incident light is turned back into the same medium
when the light falls on the surface of the another medium is known as reflection of
light.
Laws of reflection:
1) The angle of incidence is equal to the angle of reflection
i.e. < 𝑖 = < 𝑟
2) The incident ray, reflected ray and the normal drawn to the surface
at the point of incidence lie in the same plane.
Laws of reflection are same whether the reflecting surface is plane or curved.
A plane mirror always produces a virtual, erect and same size image at the same
distance behind the mirror as the object in front of the mirror. Also image in a plane
mirror is always laterally inverted. If the ray is incident normally on the reflecting
surface, the ray retrace its path after reflection.
Page | 181
Regular reflection:
When a parallel beam of light falls on a mirror (plane or curved), each ray
of light is reflected from the mirror recording to the laws of reflection. It is called
regular reflection.
Diffuse reflection:
When a parallel beam of light falls on an object, such as wall, floor, paper
etc.., The rays of light are reflected in all the possible directions, such a reflection is
called diffuse reflection.
NOTE:
1. A plane mirror can from a real image only when converged beam of light falls
on the plane mirror.
2. To view full image, a person needs a plane mirror of length equal to half the
height of person.
3. On keeping the incident ray fixed, If the mirror is turned through an angle ɵ then
the reflected ray turns by an angle 2ɵ from its initial path.
The number of images formed when two plane mirrors are inclined by an
angle ɵ and is placed between them is,
n=
If
360
ɵ
360
ɵ
-1
ɵ
is an even integer
n=
If
360
360
ɵ
-1
is an odd integer and the object lies on the bisector of the angle ɵ
(Symmetrically)
n=
If n =
360
ɵ
360
ɵ
is an odd integer and the object does not lies on the bisector of the angle ɵ
(Unusually)
n = ∞, If ɵ = 0 (two parallel mirror) because
360
ɵ
=∞
Page | 182
Reflection at spherical mirror:
A spherical mirror is a part of a hollow sphere whose one side is reflecting and
the other side is reflecting and the other side is opaque. There are two types of
spherical mirrors namely concave mirror and convex mirror.
(1) Concave mirror:
“A spherical mirror whose reflecting surface is towards the centre of the sphere
of which the mirror forms a part is called concave mirror”.
(2) Convex mirror:
“ A spherical mirror whose reflecting is away from the centre of the sphere of
which the mirror forms a part called convex mirror”.
Uses of Spherical Mirror:
Concave Mirror:
As a saving or make up mirror, inspection mirror by dentist, as ophthalmoscope
to observe retina, reflecting type telescope, solar heaters, as a reflector in headlights
of cars, railway engine, search light etc.
Convex Mirror:
As a rear view mirror, reflector in street lamp to provide wide field of view etc.
Some basics definitions:
Pole: “The geometrical centre of the spherical mirror is called pole (p)”.
Aperture: “The diameter of the spherical mirror is called the aperture (AB)”.
Centre of curvature: “The centre of the sphere of which the mirror forms a part is
called the centre of curvature (C)”.
Radius of curvature: “The radius of the sphere of which mirror forms a part is
called the radius of curvature(R)”.
Principle axis: “The straight line joining the pole and centre of curvature of the
spherical mirror extended on both sides is called principle axis mirror”.
Page | 183
Principle focus(F) : When a parallel beam of light close to the principle axis are
incident on a spherical mirror then reflected light rays actually converge at a point
on the principle axis in case of concave mirror or appears to diverge from a point on
its principle axis in case of convex mirror. This fixed point is called principle focus
(F) of the mirror.
Focal length (f): “The distance between the principle focus and the pole of the
mirror is called the focal length of the mirror”.
Object space: “The medium or space in which the incident rays travel is called the
object space”.
Image space: “The medium or space in which the refracted rays travels and forms
a real image is called the image space”.
The concave mirror is also called converging mirror and convex mirror is also called
diverging mirror.
New Cartesian sign conversion:
1. All the distance are measured from the pole of the spherical mirror.
2. The distance measured in the direction of the incident light
are taken as positive, whereas the distance measured in the direction
opposite to the direction of incident light are taken as –ve.
3. Height measured upwards and perpendicular to the principle axis
is taken as +ve and height measure downwards and perpendicular to
the principle axis are taken as –ve.
Page | 184
NOTE:
1. Focal length(f) and radius of curvature(R) are –ve for concave mirror and
positive for a convex mirror.
2. Object is considered to be placed at the left side of the mirror.
Relation between focal length and radius of curvature:
Consider a concave mirror of a small aperture with pole „P‟ centre of curvature
„C‟ and principal focus „F‟. Let a ray of light AB parallel to the principal axis be
incident on the mirror at „B‟ and pass through the principal focus „F‟ after reflection.
Here,
PF – Focal length
CP – Radius of curvature
BC – Normal to mirror at point B
According to Law of reflection <i = <r
As AB is parallel to PC <α = <i
Therefore, an ∆ BFC <r = <α.
Triangle CBF is isosceles So that CF = FB
For a mirror of small aperture FB ≈ FP
Therefore,
CF = FP
Hence,
CP = CF + FP
FP + FP = 2FP
CP = 2FP
( CP = R and FP = f)
R = 2f (or)
f=
𝑅
2
Page | 185
1
Therefore,
Focal length = × Radius of curvature.
2
( This is true for convex mirror also)
Mirror Formula:
Consider a concave mirror of focal length „f‟ and radius of curvature „R‟. Let an
Object AB is placed on the principal axis in front of the concave mirror beyond
centre of curvature „C‟. Its real and inverted image A1 B1 is formed due to reflection.
The light ray AM parallel to the principal axis incident on the mirror, it passes
through the principal focus „F‟ after reflection. The light ray AP incident at some
angle at the pole, its reflected rays obeys laws of reflection. Another ray from A
travelling through centre of curvature „C‟ retraces its path after reflection from the
mirror.
Here all the reflected rays meets at A1 and hence A1 B1 is the real image of the object
AB.
NOTE: For a paraxial light rays, MP can be considered straight and perpendicular to
CP.
From the similar triangles ABC and A1 B1 C
AB
A1 B1
CB
=
.................. (1)
CB 1
From the similar triangles ABP and A1 B1 P
AB
A1 B1
PB
=
PB 1
.................... (2)
From equation (1) & (2)
CB
CB 1
=
PB
PB 1
⟹
PB −PC
PC − PB 1
=
PB
PB 1
Using new Cartesian convention we have,
PB = - U
⟹
PC = -R = - 2f
PB = -V
⟹
Object distance
⟹ Radius of curvature
Image distance
−U – (−R)
−R –(−V)
=
−U
−V
Page | 186
−U + R
U
=
–R + V
V
( -U + R ) V = U (- R + V)
-UV + RV = -UR + UV
UR + RV = 2 UV
Divide throughout by UVR
𝟏
𝐕
𝟏
𝟐
𝐔
𝐑
+ =
𝟏
𝐕
𝟏
+
𝐔
=
(or)
𝟏
𝐟
This is the Mirror equation.
Linear magnification:
Linear magnification of spherical mirror is defined as the ration of the size of
the image to the size of the object. It is denoted by m.
i.e.,
Size of the image
m=
size of the object
⟹
m=
𝑕𝑖
𝑕𝑜
From the similar triangles A1 B1 P and ABP
𝐴1 𝐵 1
𝐴𝐵
=
𝑃𝐵1
𝑃𝐵
By new Cartesian sign conversion,
A1 B1 = - hi, AB = ho, PB1 = -V and PB = - U
-
𝑕𝑖
=
𝑕𝑜
𝑕𝑖
𝑕𝑜
=
m=
−𝑉
−𝑈
=
𝑉
𝑈
−𝑉
𝑈
−𝑽
𝑼
This is the Expression for linear magnification.
(Same for concave and convex mirrors)
Page | 187
NOTE:
1) The magnification (m) is positive for real images and negative for virtual
images.
2) When „m‟ is positive, images must be erect ( i.e virtual)
3) When „m‟ is negative, images must be inverted ( i.e real )
4) As an object held in front of a spherical mirror the distance of the object „u‟ is
always negative.
Refraction of light:
“The phenomenon of change in direction of a ray of light when it travels
obliquely from one transparent medium to another is called refraction of light”
Change in the velocity of light while passing from one
medium to another is responsible for refraction of light.
AB – Refracting surface
PQ – incident Ray
QR – Refracted Ray
MQN – Normal Drawn to AB at Q
d = i ̴ r – Angle of deviation
< 𝑃𝑄𝑀 = i – angle of incidence
< 𝑁𝑄𝑅 = r – angle of refraction
1) When a ray of light travels from rarer medium to denser medium, the refracted
ray bends towards the normal. i.e. i > r
2) When a ray of light travels from denser medium to rarer medium, the refracted
ray bends away from the normal. i.e. i < r. In general d = i ̴ r is called angle of
deviation.
3) The angle between the incident ray and the normal drawn at the point of
incidence is called the angle of incidence.
Page | 188
4) The angle between the refracted ray and the normal drawn at the point of
incidence is called the angle of refraction.
Laws of Refraction:
First law:
“The incident ray, the refracted ray and the normal at the point of incidence, all
lie in the same point”.
Second law (Snell‟s Law):
“The ratio of the sine of the angle of incidence to the sine of angle of refraction
is always constant for a given pair of medium and for light of a given wavelength”.
sin 𝑖
i.e.,
sin 𝑟
= Constant
The constant in the above relation is called refractive index of the second medium
with respect to first medium. It can be written as 1n2 .
When a ray of light travels from medium 1 to medium 2. we have,
𝐬𝐢𝐧 𝒊
𝐬𝐢𝐧 𝒓
= 1n2
Refractive index is a scalar quantity. It neither has a unit nor dimension. It is just a
number.
Absolute refractive index:
“The refractive index of any optical medium with respect to air or
vacuum is called Absolute refractive index”.
It is defined as “the ratio of velocity of light in free space (air) to the velocity of
light in the given optical medium”.
It is represented by n.
𝐶
𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑜𝑓 𝑙𝑖𝑔 𝑕𝑡 𝑖𝑛 𝑎𝑖𝑟
𝑉
𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑜𝑓 𝑙𝑖𝑔 𝑕𝑡 𝑖𝑛 𝑚𝑒𝑑𝑖𝑢𝑚
i.e., n = =
Page | 189
Absolute refractive index is always greater than 1. As C> V. It depends on
Temperature of the medium & Wave length of light.
NOTE:
1. Absolute refractive index of glass ng =
2. Absolute refractive index of water nw =
𝐶
𝑉𝑔
𝐶
𝑉𝑤
Relative refractive index:
“The refractive index of any optical medium with respect to another optical
medium except vacuum or air is called Relative Refractive index”.
It is defined as “the ratio of velocity of light in medium 1 to the velocity of light
in medium 2. It is represented as 1n2”,
i.e.,
1n2 =
𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑜𝑓 𝑙𝑖𝑔 𝑕𝑡 𝑖𝑛 𝑚𝑒𝑑𝑖𝑢𝑚 1
𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑜𝑓 𝑙𝑖𝑔 𝑕𝑡 𝑖𝑛 𝑚𝑒𝑑𝑖𝑢𝑚 2
NOTE:
1) Relative RI of Glass with respect to water can be written as wng.
2) The Relative RI of a denser Medium With respect to rarer medium is >1.
3) The Relative RI of a Rarer medium with respect to denser medium is <1.
4) Relative RI depends on
a) RI of the medium
b) Temperature of the medium
c) wavelength of light.
The Absolute RI of medium 1 is n1 =
The Absolute RI of medium 2 is n2 =
1n2
Hence
1n2
=
=
𝑉1
𝑉2
=
sin 𝑖
sin 𝑟
𝐶
𝑉1
𝐶
𝑉2
𝑛2
𝑛1
=
𝑛2
𝑛1
=
𝑉1
𝑉2
Page | 190
sin 𝑖
Also,
sin 𝑟
=
𝑛2
𝑛1
n1 sin i = n2 sin r
This equation is called general law of refraction.
Principle of reversibility of light:
“It states that if ray of light after suffering any number of reflection and
refraction and its final path reversed, it travels back along the same path in the
opposite direction”.
Explanation:
When light travels from medium „a‟ to medium „b‟ then the relative RI of medium
„b‟ w.r.t „a‟ is given by,
anb
=
sin 𝑖
........................ (1)
sin 𝑟
On reversing path, the light travels from medium „b‟ to medium „a‟ , the
Relative RI of medium „a‟ w.r.t „b‟ is given by,
bna
=
sin 𝑟
....................... (2)
sin 𝑖
×ly above equation (1) & (2) we have,
anb × bna
anb
=
=
sin 𝑖
sin 𝑟
×
sin 𝑟
sin 𝑖
=1
1
n
It follows that RI of medium „b‟ w.r.t „a‟ is equal to the reciprocal of the RI of
medium „a‟ w.r.t „b‟. Accordingly If RI of glass w.r.t air is 3/2. RI of air w.r.t glass
will be 2/3.
NOTE:
Absolute RI of an optical medium is inversely proportional to the wavelength of
light
i.e.,
1
nα .
𝜆
Page | 191
Lateral shift:
The perpendicular distance between the incident ray and the emergent ray,
when ray of light is incident on a parallel sides of the refracting slab is called Lateral
Shift.
From the figure XYX1Y1 ⟹ Parallel sided glass slab,
PQ ⟹ Incident Ray,
QR ⟹ Refracted Ray,
RS ⟹ Emergent Ray,
RT ⟹ Lateral shift.
In Right Angled ∆ QTR,< TQR = i - r
sin (i - r) =
𝑅𝑇
𝑄𝑅
=
𝑑
𝑄𝑅
(or) d = QR sin (i - r)
From the right angled ∆ QN1R we have
cos r =
𝑄𝑁
𝑄𝑅
=
𝑡
𝑄𝑅
(or) QR =
𝑡
cos 𝑟
substituting for QR, we have
d=
𝒕
𝐜𝐨𝐬 𝒓
sin (i – r)
Lateral shift produced by parallel sided glass slab depends on,
a) Thickness of slab
b) Angle of incident
Page | 192
c) RI of glass slab
d) Wavelength of incident light (𝛌)
NOTE:
 Lateral shift is zero for normal incidence because i = r = 0.
 Lateral shift is maximum for gazing incidence i.e. i = 900
𝑡
i.e. Ls (d) =
cos 𝑟
𝑡
=
cos 𝑟
sin (900 – r)
× cos r
LS (d) = t
Thus maximum value of lateral shift will be equal to the thickness of the slab.
Normal Shift:
Suppose that < OBN1 = i and NBC = r
The RI of medium„a‟ with respect to „b‟ is given by bna =
since anb =
1
𝑛
we have,
a
nb =
sin 𝑟
sin 𝑖
sin 𝑖
sin 𝑟
.................... (1)
Now, From the right angled ∆ OAB we have, sin i=
𝐴𝐵
𝑂𝐵
Also, < AIB = < NBC = r
From the right angled ∆ IAB we have, sin r =
𝐴𝐵
𝐼𝐵
Substituting for sin i and sin r in equation (1) we have,
Page | 193
a
nb =
𝐴𝐵/𝐼𝐵
𝐴𝐵/𝑂𝐵
OB ≈ OA, Real depth of the object,
IB ≈ IA, Apparent depth of the object.
Therefore,
a
nb =
𝑅𝑒𝑎𝑙 𝑑𝑒𝑝𝑡 𝑕
𝐴𝑝𝑝𝑒𝑟𝑎𝑛𝑡 𝑑𝑒𝑝𝑡 𝑕
=
𝑂𝐴
𝐼𝐴
The object appears to be raised from its real position „o‟ to its apparent position I.
The distance „OI‟ through which the position of the object appears to be raised is
called normal shift and it is denoted by d.
Therefore,
normal shift (d) = OA – IA
= OA [1 = OA [1 d=t[1-
1
𝑎
𝑏𝑛
𝐼𝐴
𝑂𝐴
1
𝑂𝐴
𝐼𝐴
]
]
]
It may be pointed out that the normal shift in the position of the object depends up on
1. Real depth of the object
2. Thickness of the refracting medium
3. RI of the refracting medium
Refraction effects at sunrise and sunsets:
The sun is visible to an observer before actual sunrise and after actual sunsets.
This is because of atmospheric refraction of light.
The density of air is higher near the surface of earth. The RI of air w.r.t vacuum is
1.0003. When the rays from sun at position S enter at the top of the earth‟s
atmosphere and travel from rarer to denser medium. As a result, they continuously
bend slightly towards the normal. Therefore to an observer on the earth, The sun
which is actually in position S bellow the horizon appears in position S1 above the
horizontal as shown in the figure.
Page | 194
Thus the sun appears to rise early by above two minutes before the actual rise.
For same reason, It appears to set late by above two minutes before it as actually set.
Hence the day becomes longer by about four minutes due to atmospheric refraction.
Total internal reflection:
When a ray of light passes from denser medium to rarer medium the refracted
rays bends away from the normal. As the angle of incidents increases, The angle
refraction also increases. At a particular angle of incidents, The angle of refraction
becomes 900 and refracted ray grazes the surface of separation of two media. This
angle of incidence called critical angle.
“The angle of incidence in the denser medium, for which the angle refraction
is 900 in the rarer medium, is called the critical angle incidence”.
Critical angle is a constant for given pair of media and for a given wavelength. If
the angle of incidence increases more than critical angle, the ray is not refracted but it
is completely reflected back into the denser medium this phenomenon is called Total
internal reflection.
“When a ray of light travelling from denser medium into rarer medium is
incident on the interface of the two media at an angle greater than the critical
angle the light ray is totally reflected back in to the denser medium this
phenomenon is called total internal reflection of light”.
Page | 195
Condition for total internal reflection:
1. The ray of light must travel from a denser medium in to a rarer medium
2. The angle incidence in the denser medium must be greater than critical angle
for the given pair of media and for the given color of light
Application of total internal reflection:
1. Brilliance of diamond:
The brilliance of diamond is due to total internal reflection of light diamond
as high RI (2.4) and low critical angle (240). Therefore, When light enters a
properly cut diamond will undergo multiple total internal reflection and finally
emerges at from its anyone face. Hence diamond is sparkling.
2. Mirage:
“Mirage is an optical phenomenon due to total internal reflection in the
atmospheric layers of different optical density”
It is observed on hot surface such as tar roads and deserts. And summer days
the air near the ground becomes hotter than air. The upper air layers are denser
and lower layers are rarer. When ray of light from tall tree passes from denser to
rarer medium, It bends away from the normal at lower air layer, When the angle
incidence is greater than critical angle, The light rays undergoes total internal
reflection.
For the observer the reflected ray appears to be coming from inside the
ground. The inverted image of tree creates an illusion of reflection of light from a
pound of water. The mirage formed in the hot regions is called inferior mirage.
The mirage formed in the cold region is called superior mirage.
3. Totally Reflecting Glass Prism :
A total reflecting prism is a right angled isosceles prism made of glass. It is
used to turn the incident beam of light through 900 and 1800 and also to invert an
image. They are based on the phenomenon of total internal reflection of light.
Page | 196
The RI of glass is 1.5 and critical angle for glass(air interface) is 420. When light
ray is incident at an angle 450 (>C) on reflecting prisms. It undergoes total internal
reflection.
Totally reflecting prisms find a wind range of application in binoculars, image
inverters etc.
Total reflecting prisms are advantages than plane mirror because,
1. In a plane mirror, only about 90% of incident light is reflected but in reflecting
prisms 100% of incident light is reflects.
2. In mirror more than one image is formed due to multiple reflection but in totally
reflecting prism only one image is formed.
3. Mirrors requires silver coating but prisms does not requires silver coating.
Relation between refractive index and critical angle:
Consider a light ray travel from denser medium to rarer medium. If the ray is
incident at angle „C‟ then the angle of refraction is 900 with usual notation.
we have,
n1 sin i = n2 sin r
From the figure, i = C and r = 900
n1 sin C = n2 sin 900
sin C =
In General,
sin C =
n2
n1
n Rarer
n Denser
Page | 197
If the light ray is travels from denser medium to rarer medium of RI “n” to air,
then
sin C =
𝟏
𝐧
(or) n =
𝟏
𝐬𝐢𝐧 𝐂
This is the relation between RI and critical angle. For a given optical medium and
air, critical angle is high for red and low for violet in the visible range of light.
Optical fibres:
“An optical fibre is a device which effectively conducts lights along any
desired path without loss of energy”.
It works on the principle of total internal reflection. It consists of a thin
transparent fibre made of glass or plastic of high refractive index (about 1.7) known
as core. The core is surrounded by a coating of glass or plastic material of lower RI
(about 1.5) called cladding. The coating prevents absorption of light by dust and
scratches on the surface of the fibre. The core and cladding is further housed in a
jacket known as buffer jacket. A bundle of optical fibres are called a light pipe or
optical cable. For proper working of an optical cable, the optical fibre must be thin,
insulated from each other and parallel to each other. A ray of light entering an optical
fibre meets the interface between the core ad the cladding at an angle greater than
critical angle. Hence it undergoes total internal reflection. This happens repeatedly
and finally the ray emerges at the angle other end with practically no loss of intensity.
Application of optical fibre:
1. Optical fibre are used in communication to transfer information from one point
to another. The information ( data, text, picture, voice etc) are converted into
electrical signals. These signals are converted into lights signals or pulses (Laser
beam) using light converter. These light pulses are then transmitted through
Page | 198
optical fibre to the receiving end. At the receiving end, these light signals are
converted into electrical signals using photo diode. The electrical signals then
converted into the original information by the receiver.
2. Optical fibres are used in endoscopy by doctor to visually examine the inner
parts of a human body.
3. A single optical fibre can replace a large no of copper wire.
4. Optical fibres can transport 3D images.
5. Optical fibres are used to measure temperature, pressure and rate of flow of
liquid.
6. They are used in decorative lamps.
Refraction at a spherical refracting surface:
Consider a spherical refracting surface
separating a rarer medium of RI n1 from a denser
medium of RI n2.
The surface is convex towards rarer medium and
concave towards denser medium. Consider a
point object „O‟ placed on the principle axis at a
distance OM = u from the spherical surface. A ray
OM incident normally on the surface passes
straight. Another ray ON incident on the surface at an angle „i‟ is refracting along NI
with the angle of refraction „r‟. The two refracted rays meet at „I‟. Hence „I‟ is the
real image of the object „O‟ formed at a distance „v‟ from the surface.
Since the aperture is small, by approximately NM = NP
tan NOP ≈ NOP =
tan NCP ≈ NCP =
tan NIP ≈ NIP =
𝑁𝑃
𝑃𝑂
𝑁𝑃
𝑃𝐶
𝑁𝑃
𝑃𝐼
Since the exterior angle of a ∆le is equal to the sum of internal opposite angles.
Page | 199
In the triangle NOC, „i‟ is the exterior angle.
Thus,
i = NOP + NCP
i=
𝑁𝑃
+
𝑃𝑂
𝑁𝑃
................... (1)
𝑃𝐶
In the triangle NCI, NCP is the exterior angle
Thus
NCP = CNI + NIC
NCP = r + NIC
r = NCP – NIP
𝑁𝑃
r=
𝑃𝐶
-
𝑁𝑃
𝑃𝐼
....................... (2)
Using Snell‟s Law n1 sin i = n2 sin r
For small angles, n1 i = n2 r
Using equation (1) & (2) we get,
n1 (
⟹
𝑛1
𝑃𝑂
𝑁𝑃
𝑃𝑂
+
𝑁𝑃
+
𝑛1
𝑃𝐶
𝑃𝐶
=
𝑛2
𝑃𝐶
) = n2 (
-
𝑛2
𝑃𝐼
𝑁𝑃
𝑃𝐶
(or)
-
𝑁𝑃
𝑃𝐼
)
𝑛1
𝑃𝑂
+
𝑛2
𝑃𝐼
=
𝑛2 − 𝑛1
𝑃𝐶
by using Cartesian sign conversion PO = -u PI = +v & PC = + R,
𝑛1
we get,
−𝑢
⟹
𝑛2
𝑣
+
-
𝑛1
+𝑣
𝑛1
𝑢
=
=
𝑛2 − 𝑛1
𝑅
𝑛2 − 𝑛1
𝑅
This is the relation between object distance and image distance in terms of RI and
radius of curvature of the spherical surface.
Lenses:
“A Lens is an optical medium bounded by two refracting surfaces of which at
least one is spherical or cylindrical”.
Lenses commonly used have either two spherical surfaces or one spherical
surface and one plane surface. These lenses are called spherical lenses. (cylindrical
lenses are bounded by cylindrical surfaces) there are different types of lenses namely,
biconvex lens, Plano-convex lens, biconcave lens, Plano- concave lens and concavo –
convex lens etc. As shown in the below fig.
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A convex lens is thicker at the middle than at the edges while a concave lens is
thicker at the edges than the middle.
 A lens which converges a beam of light through it is called converging lens.
 A lens which diverges a beam of light through it is called a diverging lens.
Optical Centre:
“Optical centre of a lens is a point on the principle axis inside the lens such
that all the rays passing through this point will have the emergent ray parallel to
the corresponding incident ray”.
Lens maker‟s Formula:
Consider a thin convex lens of focal length „f‟ and radius of curvature R1 and R2
of its surfaces. Let n2 is the RI of the lens surrounded by a rarer medium of RI n1.
Consider a point object „O‟ placed on the principle axis at a distance „u‟ from the
lens. The ray from „O‟ along the principle axis proceeds undeviated. Another ray OP
incident on the lens refracted along PQ and emerges along QI.
The refracted rays are meet at „I‟ and hence „I‟ is the real image of the object „O‟
formed at a distance „v‟ from the lens.
The formation of image „I‟ is explained in two steps:
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Step 1: Refraction at surface ABC:
In the absence of the second surface ADC. Let I1 be the real image formed
by ABC at a distance v1 from the lens.
𝑛2
⟹
We have,
𝑣
Here,
Therefore,
𝑛1
-
=
𝑢
𝑛2 − 𝑛1
𝑅
v = v1 , u = u and R = R 1
𝑛2
⟹
𝑛1
-
𝑣1
𝑛2 − 𝑛1
𝑅1
=
𝑢
……………… (1)
Step 2 : Refraction at surface ADC:
For the refraction at the surface ADC, the image I1 acts as a virtual object and its
real image „I‟ is formed at a distance „v‟ from the lens in the medium of RI n1.
𝑛2
We have,
𝑣
Here,
𝑛1
-
=
𝑢
𝑛2 − 𝑛1
𝑅
n2 = n1 , v = v , n1 = n2, u = v1, and R = - R2
𝑛1
⟹
Therefore,
𝑣
𝑛2
-
=
𝑣1
𝑛2 − 𝑛1
−𝑅2
......................... (2)
Adding equation (1) and (2) we get,
⟹
𝑛2
𝑣1
n1 (
(
(
1
𝑣
1
𝑣
𝑛1
-
𝑢
1
𝑣
-
1
𝑢
1
𝑢
+
1
𝑢
𝑛1
𝑣
-
𝑛2
𝑣1
=
𝑛2 − 𝑛1
1
) = ( n2 - n1 ) [
)= (
) =(
n2 − n1
𝑛1
𝑛2
𝑛1
)[
− 1) [
-
𝑅1
𝑅1
1
𝑅1
1
𝑅1
𝑅2
𝑅2
-
−𝑅2
1
1
-
𝑛2 − 𝑛1
1
𝑅2
]
]
]
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When the object is at infinity, the parallel rays from the object will converge at
the principle focus.
i.e., when u = infinity we have, v = f.
1
𝑓
= (
𝑛2
𝑛1
1
− 1) [
1
-
𝑅1
𝑅2
]
If the lens is surrounded by air or vacuum, then n1 = 1 and n2 = n
𝟏
𝒇
= ( n- 1 ) [
𝟏
𝑹𝟏
-
𝟏
𝑹𝟐
]
This is Known As Lens Maker‟s Formula.
Power of Lens:
“The ability of a lens to converge (or) diverge the ray of light incident on it is
called the Power of the lens”.
The Power of the lens is defined as “The tangent of the angle by which it converges
(or) diverges a ray of light parallel to the principle axis incident at unit distance
from the optical centre”.
tan ө =
𝑕
𝑓
If, h = 1m
tan ө =
1
𝑓
For small value of ө, tan ө ≈ ө,
Therefore,
ө=
P=
1
𝑓
1
𝑓
Thus, the power of a lens is defined as the reciprocal of its focal length
expressed in metre.
The SI unit of power of a lens is Dioptre (D).
if f = 1m then p =
1
1𝑚
= 1 m-1 = 1 dioptre.
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The power of lens is said to be 1 dioptre, if its focal length is one metre. Power of
convex (or) converging lens is positive and power of concave (or) diverging lens is
negative. Power of the lens immersed in water is ¼ times its power placed in air.
NOTE:
1. A converging lens surrounded by an optically denser medium behaves as
diverging lens.
2. A diverging lens surrounded by an optically denser medium behaves as
converging lens.
Images formation in case of thin lens:
To find the position of image of an object produced by a lens, we take two light
rays coming from the same point of object. After refraction, the point where the
refracted rays are meet gives the position of real image or the point where the
refracted rays are appears to diverge gives the position of virtual image. The two light
rays chosen are,
 A light ray originated from a point of the object parallel to the principle axis of
the lens, after refraction passes through the second principle focus ‟F‟ in case of
convex lens or appears to diverges from the first principle focus „F‟ in case of
concave lens.
 A ray of light passing through the optical centre „C‟ of the lens passes straight
without any deviation.
 A ray of light passing through the first principle focus „F‟ in case of convex lens
or appears to meet at it in case of concave lens, emerge parallel to the principle
axis after refraction.
Linear magnification (m):
Linear magnification produced by a lens is defined as “the ration of size of the
image to the size of the object”.
i.e., linear magnification (m) =
Size of the image
Sie of the object
m=
=
hi
ho
v
u
Linear magnification is positive for virtual image and negative for real image.
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Combination of thin lenses in contact:
Consider two thin lenses L1 and L2 of focal lengths f1 and f2 placed coaxially in
contact. Let „O‟ be an object placed at a distance „u‟ from the combination of the
lenses on their principle axis. „I‟ is the real image of the object „O‟ produced at a
distance „v‟ from the combination of the lenses. The formation of the image can be
explained in two steps.
Step-1: Refraction through the first lens:
In the absence of the second lens L2, the refracted ray through the first lens meet
the principle axis at „I1‟. Hence I1 is the real image of the object „O‟ produced at a
distance v1.
From the lens Formula,
1
𝑣
1
-
𝑢
=
1
𝑓
Here, v = v1, u = u and f = f1
1
𝑣1
-
1
𝑢
=
1
𝑓1
……………… (1)
Step-2: Refraction through the second lens:
In the absence of the first lens L1, the image „I1‟ acts as virtual object and its real
image „I‟ is formed at a distance „v‟.
Here, u = v1 , v = v and f = f2
1
𝑣
-
1
1
=
𝑣1
……………… (2)
𝑓2
Adding equation (1) and (2)
1
𝑣1
1
𝑣
-
1
𝑢
1
𝑢
+
=
1
𝑣
1
𝑓1
-
1
𝑣1
+
1
𝑓2
=
1
𝑓1
+
1
𝑓2
……………….. (3)
𝑇𝑕𝑒 equivalent lens of a combination of lenses is that, single lens which
produces the same effect as that produced by the combination of the lenses.
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If „f‟ is the focal length of the equivalent lens, then the lens formula
1
𝑣
-
1
1
=
𝑢
𝑓
………………….. (4)
Comparing equation (3) and (4)
1
𝑓
=
1
𝑓1
+
1
𝑓2
……………….. (5)
Thus, the reciprocal of the focal length of the equivalent lens is equal to the sum of
the reciprocals of the focal lengths of the lenses in contact.
Power of equivalence lens:
we know, power of lens is given by p =
P1 =
P2 =
1
𝑓1
1
𝑓2
1
𝑓
is the power of the first lens
is the power of the second lens
P = P1 + P2
Thus power of equivalent lens is equal to the algebraic sum of the powers of
individual lenses in contact. For „n‟ numbers of thin lenses in contact
1
𝑓
=
1
𝑓1
+
1
𝑓2
+
1
𝑓3
+ ………..
1
𝑓𝑛
P = P1 + P2 + ………………….. Pn
Magnification of an equivalent lens:
Consider two lenses in contact have linear magnification m1 and m2. Then net
the linear magnification of an equivalent lens formed by two lenses in contact is
given by, m = m1 × m2.
If equivalent lens formed by more than two lenses, then the net linear
magnification is given by, m = m1 × m2 × m3 × m4 × mn
Thus the magnification of combination of lenses is equal to the product of
magnification of individual lenses.
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Prism:
“Prism is a three dimensional transparent optical medium bounded by three
rectangular and two parallel triangular faces”.
Two of the three rectangular faces highly polished and are called
refracting surfaces. The third rectangular face is unpolished and is called base. A line
along which the two refracting surfaces meet is called the refracting edge. The angle
between the refracting surfaces is called the refracting angle or the angle of the prism.
Any section of the prism perpendicular to the refracting edge is called the
principle section of the prism.
Refraction through a Prism:
Consider a principle section ABC of a prism of refractive index „n‟ placed in air.
A ray of monochromatic light PQ incident on the
refracting face AB, refracting along QR and emerges
along RS as shown in the figure.
Let i1 and r1 be the angle of incident and angle
refraction respectively at AB. Let r2 and i2 be the
angles of incidence and emergence respectively at
AC. Normal drawn at Q and R meet at N. Let „A‟ be
the angle of the prism.
From the cyclic quadrilateral AQNR;
A + QNR = 1800
A = 1800 – QNR
…………….. (1)
From the triangle QNR,
r1 + r2 + QNR = 1800
r1 + r2 = 1800 – QNR
……………... (2)
From equation (1) and (2) we get,
A = r1 + r2
………………….(3)
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The angle between the emergent ray and the incident ray extended is called angle of
deviation(d).
Deviation produced by the first refraction at AB = d1 = i1 – r1
Deviation produced by the second refraction at AC = d2 = i2 – r2
Total deviation produced by the ray, D = d1 + d2
D = i 1 – r1 + i 2 – r 2
D = i1 + i2 - r1 – r2
D = i1 + i2 - ( r1 + r2 )
D = i 1 + i2 – A
…………….. (4)
[from eqn (3)]
A graph of angle of incidence versus deviation is shown in below figure. As the
angle of incidence increases from a small value, the angle of deviation decreases
gradually and reaches the minimum value and it is called angle of minimum
deviation (D) and then increases.
When the deviation is minimum the ray passes symmetrically through the prism
i.e. the refracted ray QR passes parallel to the base of the prism. From the graph we
see that for any value of deviation d, there are two values of the angle of incidence
namely i1 & i2.
At minimum deviation, that is when d = D,
i1 = i2 = I and r1 = r2 = r
then, the equation (3) become,
A = r + r = 2r = r =
𝐴
2
………….. (5)
The equation (4) become,
A + D = i + i = 2i = i =
𝐴+𝐷
2
…………… (6)
From Snell‟s Law, the R.I of the prism is,
n=
sin 𝑖
sin 𝑟
substituting i and r we get,
n=
𝐬𝐢𝐧⁡
(
𝐬𝐢𝐧
𝑨+𝑫
)
𝟐
𝑨
(𝟐 )
Page | 208
Dispersion:
Newton in 1666 discovered that when white light was allowed to pass through a
prism, the rays split into various colours.
“The phenomenon in which composite light splits up into its constituent colours on
passing through a suitable optical medium is known as dispersion.”
The band colours obtained after dispersing a composite light is called spectrum.
“A spectrum in which the constituent colours do not overlap and are seen distinctly
is called pure spectrum”.
Eg: Line spectrum
“A spectrum in which the constituent colours overlap and cannot be seen distinctly
is called impure spectrum”.
Eg: Rainbow
The medium which produces dispersion is called a dispersion medium.
Eg: glass, water, diamond etc. Vacuum and air are non-dispersive media.
Reason for dispersion:
Dispersion occurs when light is refracted at an interface between two media. In
vacuum or air, all colours of light have equal velocities. But in a dispersive medium,
light rays of different colours travel with different velocities. Hence refractive index
of the dispersive medium is different for different colours.
In a dispersive medium like glass, speed of violet is minimum, while that of red is
maximum. Therefore refractive index of glass is maximum for violet and least for red
(R.I of the medium is inversely proportional to the velocity). Hence, when a ray of
composite light passes through a dispersive medium (prism) different colours are
deviated to different extents. The deviation is maximum for violet and maximum for
red. This causes the separation of the constituent and hence dispersion occurs.
Page | 209
Rainbow:
“The coloured concentric arcs or the spectrum of sunlight in the form of bows
when sunlight is incident on the rain drops in atmosphere during rain or after the
rain is called rainbow”.
The condition for observing a rainbow is that the sun should be shining in one
part of the sky while it is raining in the opposite part of the sky. Sunlight must be
from the back of the observer and observer should be facing the rain bearing
clouds.Rainbow is formed due to combined effect of dispersion, refraction and
reflection of sunlight by spherical water droplets of rain.
When the parallel beam of sunlight fall on the rain drops, the rays are first
refracted as they enter the raindrops. The deviation will be different wavelength of
light and hence dispersion takes place. Violet colour deviates more than red colour.
When these light rays strike the inner surface of water drop, they undergo total
internal reflection if the angle of incidence greater than critical angle (48°). The
reflected ray is refracted as it comes out. It is found that violet light emerges out at an
angle of (40°) and red light emerged out an angle of 42°. Sometimes two rainbows
are seen. The inner rainbow is called the primary rainbow and the outer rainbow is
called the secondary rainbow.
Primary rainbow:
It is formed due to two refractions and one internal reflection of the light incident
on the raindrops. The primary rainbow has violet colour on the inner edge and red
colour on the outer edge of the rainbow. Other colours of light are seen at
intermediate angles.
Secondary rainbow:
It is formed due to two refractions and two internal reflections of the light
incident on the rain drops. The secondary rainbow has coloured on the inner edge and
violet colour on the outer edge of the rainbow.
Page | 210
More light is absorbed in the drop and hence the secondary rainbow is fainter than the
primary rainbow. The secondary rainbow is larger than the primary rainbow.
Scattering of light:
“The phenomenon of change in the direction of light by the particles of the
medium is called Scattering of light”.
OR
“The phenomenon where the energy of the incident light is first absorbed and
re-emitted by atoms or molecules of the medium in all possible directions is called
scattering of light”.
“If the wavelength of the scattered light is same as that of the incident light, then it
is called coherent scattering”.
Eg: Rayleigh scattering.
“If the wavelength of the scattered light is different from that of the incident light,
then it is called coherent scattering”.
Eg: Raman scattering.
When the dimension of the scattering particles is small compared with the
wavelength of the incident radiation it is called Rayleigh scattering.
Scattering of light by molecules of air (Eg: N2, O2, molecules) is an example of
Rayleigh scattering.
According to the Rayleigh the intensity of scattered light is inversely proportional to
the forth power of the incident light.
That is, I α
1
𝜆4
Hence violet colour scattered most and red colour scattered least.
Page | 211
Blue colour of the sky:
The blue colour of sky is due to scattering by the large number of air molecules
in the atmosphere. According to Rayleigh the intensity of scattered light is inversely
proportional to fourth power of wavelength of incident light.
The wavelength of blue colour is much smaller than that of red colour. Hence
blue colour is scattered much more than red colour (16 times more than that of red
light). Due to this blue colour predominates and the sky appears blue. The blue colour
of the sea is due to the reflection of light from the sky due to the scattering of light by
water molecules. In the absence of atmosphere, there is no scattered light coming
from the sky and hence sky appears black.
White colour of clouds:
A cloud is composed of large sized dust particles and water molecules. These
large sized particles do not obey Rayleigh law of scattering. Therefore, all
wavelengths are scattered nearly equally and there is no change in the intensity of
scattered light. All colours scattered equally merge together to give us the sensation
of white. Hence clouds appear white. (MIE scattering)
Reddish colour of sun during sunrise and sunset:
During the times of sunrise and sunset, the light ray travels larger distance in the
lower and densest part of the earth‟s atmosphere before reaching the earth‟s surface.
1
According to Rayleigh scattering (I ∝ ) larger wavelength orange and red colour is
𝜆
less scattered and other colours are more scattered. Hence least scattered orange and
red colours reach our eye. For this reason, the sun looks reddish at the sunset and
sunrise.
Danger signals are red:
Because wavelength of red colour is large and intensity of scattered light
1
varies inversely as the fourth power of wavelength I ∝ 4 therefore red colour is at
𝜆
least scattered. It can be seen from maximum distance. That is way danger signals
are red.
NOTE:
When the dimension of the scattering particles is comparable with the wavelength
of the incident radiation, scattering takes places in all directions. This phenomena of
scattering in called Tyndall scattering. In Tyndall scattering, the intensity of the
scattered light is independent of wavelength of the incident light, the light of different
wavelengths are equally scattered. Scattering of light by smoke is example of Tyndall
scattering. Scattering observed in a medium of tiny particles, molecular clusters, tiny
water droplets etc is called Mie scattering.
Page | 212
Optical instruments:
“The devices which work on the principle of reflection, refraction and
rectilinear propagation of light are called optical instruments”.
Eg: Microscopes, Telescope, Periscope, Binoculars, Cameras, Etc.,
Human eye:
Human eye is an optical device and acts as a natural camera. The eye is nearly a
spherical ball with a diameter of about 2.5cm.
Parts of the eye and their functions:
1. Cornea: It is the transparent membrane on the front portion of the eyeball
through which light enters the eye.
2. Iris: It is an opaque circular diaphragm having a small central hole called the
pupil. The function of iris is to control the amount of light entering the eye
through the pupil. The pupil becomes small in bright and it becomes wide in dim
light.
3. Eye lens: The eye lens is double convex lens and is made of transparent and
flexible tissues. It is behind the pupil and held in position by ciliary muscles. The
lens focuses the image of objects on the retina of the eye.
4. Ciliary muscles: These muscles hold the eye lens in position. The curvature and
focal length of the lens is changes by ciliary muscles. When these muscles
contracts, the focal length of the lens decreases because curvature of the lens
increases. However when ciliary muscles are relaxed, the focal length of the eye
lens increases.
Page | 213
5. Aqueous humour and vitreous humour: Aqueous humour is a watery fluid
(n=1.337) that fills the space between the cornea and the eyes lens. Vitreous
humour is a transparent jelly like fluid (n=1.437) that fills the space between the
retina and eye lens.
6. Retina: It is a light sensitive membrane on the back interior wall of the eye ball.
A real and inverted image of the object is formed on the retina. It contains light
sensitive cells called rods and cones. Rods are sensitive to intensity of light while
cones are sensitive to colours. These cells change light energy into electrical
signals which send message to brain through optical nerves.
7. Optic nerves: It is the conveyer of light signals from retina to the brain for
interpretation.
Action of the eyes:
The transparent structures like cornea, aqueous humour, eye lens and vitreous
humours together constitute a single converging lens. As the rays from an object
enter the eye, they suffer refractions on passing successively through those structures
and get converged. A real and inverted image is formed on the retina. The light
sensitive cells of retina get activated and generate electrical signals that are sent to the
brain through the optic nerves. Our brain translates the inverted image into an erect
image.
Basic Definitions:
1. Accommodation: The process by which the focal length of the lens is adjusted
automatically by the action of ciliary muscles so that sharp images of the objects
at different positions are formed on the retina is called accommodation of the
eye.
2. Near point: The nearest point from an eye at which an object can be placed so
that its sharp image is formed on the retina is called least distance of distinct
vision or the distance of near point of the eye.
For a normal adult eye, the least distance if distinct vision D=25cm.
3. This distance increases with age, because of decreasing effectiveness of the
ciliary muscles and loss of flexibility of the eye lens.
For point: The farthest point from an eye at which an object can be placed so
that its sharp image is formed on the retina is called as the for point of the eye.
For a normal adult eye, the distance of the far point is infinite from the eye.
4. Range of vision: The distance between the near point and far point of an eye is
known as range of vision. For a normal adult eye, it is about 25cm to infinity.
Page | 214
Defects of vision:
The common defects of vision are:
1.
2.
3.
4.
Short sightedness or myopia.
Long sightedness or Hyper metropia.
Presbyopia.
Astingmatism.
1. Short sightedness or myopia:
The defect of human eye in which light from a distant object focused at a point
in front of the retina is called myopia or short sightedness. Hence the eye can see
close objects clearly and unable to see far off objects clearly.
This occurs due to the increase in the size of the eye ball or decrease in the
focal length of the eye lens. This is corrected by prescribing a concave lens if suitable
focal length to be placed in front of the eye.
2. Long sightedness or Hypermetropia:
The defect of the human eye in which light from near object focused at a
point behind the retina is called hypermetropia or long sightedness. For this effect
eye, far objects are seen clearly and near objects cannot be seen clearly.
For hyper metropic eyes, the near point shifts away from the eye. It is more
than 25cm. This occurs due to the decrease in the size of the ball or increase in the
focal length of the eye lens.
This is corrected by prescribing a convex lens of suitable focal length to be placed in
front of the eye.
Page | 215
Presbyopia:
Far sightedness or long sight defect of vision occurring with age is called
presbyopia.
This occurs due to the weakening of the ciliary muscles, flexibility of the eye with
age and hence decreases in the power of accommodation of the eye. This is corrected
by prescribing a convex lens of suitable focal length to be placed in front of the eye.
Astigmatism:
It is a defect of vision in which a person cannot simultaneously see both
horizontal and vertical lines with the same clarity. This defect occurs when the cornea
is not perfectly spherical so that it has different curvature in different directions.
Astigmatism can be corrected by using a cylindrical lens of suitable radius of
curvature to be placed in front of the eye with an appropriate axis.
Visual angle:
When an object is brought from a large distance towards the eye, the object
appears bigger and bigger. While the actual size of the object does not change, its
apparent size is greater when brought closer to the eye. It is because the apparent size
of an object depends upon the angle it subtends at the eye. It is called visual angle.
“The angle subtended by an object at the eye is called visual angle”.
The greater the visual angle, the larger is the apparent size of the object. This
principle is used in microscopes and telescopes.
Angular magnification or the magnifying power of an optical instrument is defined as
the ratio of the visual angle subtended by the image (formed by the instrument)at the
eye (𝛽) to the visual angle subtended by the unaided (or naked) eye (𝛼)
m=
𝛽
𝛼
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Microscope:
“Microscope is an optical instrument which forms a magnified image of tiny
objects held close to the eye”.
Simple microscope or Magnifying lens:
A simple microscope consists of a biconvex lens of small focal length and is used
to see magnified images of the small objects placed close to the eye. A simple
microscope is based on the fact that when an object is placed between the optic centre
and the focus of a convex lens, a magnified, virtual and erect image of the object is
formed on the same side of the lens as the object.
The lens is held close to the eye and the distance of the object is adjusted, till the
image is formed at least distance of the vision of the eye (D=23sm). For this reason
simple microscope is also called a magnifying glass.
The path of light rays in a simple microscope is a shown in the figure.
The object AB is placed between C and F at a distance „u‟ and its virtual erect
image A1B1 is formed at a distances D equal to least distances of distinct vision.
The magnifying power or angular magnification of a simple microscope is defined
as the ratio of the angle subtended by the image (𝛽) at the eye to the angle subtended
by the object (𝛼) at the eye when both are placed at the least distance of distinct
vision from the eye.
Let < A1 C B1 = β Imagine the object AB to be displaced to A1P at distance D from
the lens.
Let < P C B1 = α
Then magnifying power, m =
𝛽
𝛼
From the figure, since the angles are small by approximations,
tan β = β =
𝐴1 𝐵1
Therefore, m =
𝐷
×
𝐴1 𝐵1
and tan α = α =
𝐷
𝐷
𝐴𝐵
=
𝐴1 𝐵1
𝐴𝐵
=
𝐵1 𝑃
𝐷
=
𝐴𝐵
𝐷
𝑆𝑖𝑧𝑒 𝑜𝑓 𝑖𝑚𝑎𝑔𝑒
𝑆𝑖𝑧𝑒 𝑜𝑓 𝑐𝑏𝑗𝑒𝑐𝑡
This indicated that angular magnification is equal to the linear magnification.
Therefore, m =
𝑣
𝑢
………………….. (2)
From the lens formula,
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1
𝑣
1
1
𝑢
𝑓
- =
Multiplying both sides by v,
𝑣
𝑣
𝑣
𝑣
𝑢
𝑓
- =
𝑣
1–m =
𝑓
𝑣
m=1-
𝑓
Since v = -D
𝑫
m=1+
𝒇
This is the expression for magnifying power of a simple microscope. For Smaller
the focal length, the greater is the magnifying power.
When the image is formed infinity:
In this case the object is placed at the focus of the convex lens. Magnifying
power,
m=
𝛽
…………….. (1)
𝛼
Where α = angle subtended at the eye by the
object when it is placed at least distance of
distinct vision (near point) and β = angle
subtended at the eye by the image at
infinity.Since angles are small by approximation,
tan β ≈ β =
𝐴 𝐵
m =
Therefore,
m=
𝑓
𝐴 𝐵
and tan α = α =
𝑓
×
𝐷
𝐴𝐵
=
𝐴1 𝐵1
𝐶𝐵1
=
𝐴𝐵
𝐷
𝐷
𝑓
𝐷
𝑓
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In this case magnifying power is one less than the magnification when the image is
at the near point. But viewing is quite comfortable because the eye is focused at
infinity.
Uses of magnifying lens:
1. Jewellers and watch makers use the magnifying glass to obtain a magnified view
of tiny parts of jewellery and watch parts.
2. In science laboratories a magnifying glass is used for reading verifier scale etc.
Compound Microscope:
“A compound microscope is an optical instrument used for observing highly
magnified image of tiny objects”.
It consists of two converging lenses.
A lens of short aperture and short focal length (f0) facing the object is called the
objective lens. The other lens of larger focal length (fe) and larger aperture then the
objective is called eyepiece. It is positioned near the eye for viewing the final image.
The two lenses are placed co- axially at the free ends of the tube at suitable from each
other. The distance between the two lenses can be varied by using rack and pinion
arrangement.
The paths of light rays through a compound microscope is shown in the figure.
Let AB is a tiny object placed in front of the objective lens beyond the principal
focus Fo.
The objective forms an inverted, real and enlarged image A1 B1. Now A1 B1 acts
as object for the eyepiece and it is within Fe. Then the eyepiece acts as simple
magnifying lens and it forms erect, virtual enlarged image A2B2. This final image
formed at least distance of distinct vision D and it seen by the eye held closed to the
eyepiece. This is the normal adjustment of the compound microscope.
Angular magnification or magnifying power of a compound microscope is
defined as the ratio of angle (β) subtended at the eye by the final image to the angle
(α) subtended at the eye by the object, when both are at the least distance of distinct
vision from the eye.
Let A2 B2 is the final image formed at a distance D. The object AB is shifted to A2 B3
and it is also at D.
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Then < A3 C2 B2 = α and < A2 C2 B2 = β
Angular Magnification or magnifying power =
𝛽
𝛼
Since the angles are small,
From ∆le A2 C2 B2 ,
tan α α =
𝐴3 𝐴2
𝐷
𝐴 𝐵
=
𝐷
and
From ∆le A2 C2 B2,
tan β β =
Therefore,
m=
m=
𝛽
𝛼
=
𝐴2 𝐵2
𝐷
𝐴2 𝐵2
𝐴2 𝐵2
𝐴1 𝐵1
𝐷
×
×
𝐷
𝐴𝐵
mo =
𝐴2 𝐵2
𝐴1 𝐵1
𝐴1 𝐵1
𝐴𝐵
𝐴2 𝐵2
𝐴𝐵
𝐴1 𝐵1
𝐴𝐵
……………… (1)
m = me × mo
where, me =
=
is magnification produced by the eye piece.
is magnification produced by the objective lens.
Thus, the magnification produced by a compound microscope is given by the
product of magnification produced by the eye piece (me) and objective (mo).
Since the action of eyepiece is like a simple microscope producing the image at
least distance of distinct vision (D).
me = 1 +
𝐷
𝑓𝑒
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For objective lens,
𝐼𝑚𝑎𝑔 𝑒 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒
m0 =
=
𝑂𝑏𝑗𝑒𝑐𝑡 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒
𝑣𝑜
−𝑢 𝑜
The equation (1) becomes,
m=
𝑣𝑜
−𝑢 𝑜
( 1+
𝐷
𝑓𝑒
)
………….. (2)
In this case length of the microscope , L = vo + ue
In practise, the focal length of the objective lens is very small and the object AB
is placed very close to the principal focus of the objective of focal length f0 , by
approximation the object distance AB = uo = fo. The focal length of the eye piece is
very small and the image A1 B1 is formed very close to the eye piece.
Therefore, vo L = length of the microscope tube the distance between the lenses.
Thus,
m=
−𝐿
𝑓𝑜
(1 +
𝐷
𝑓𝑒
)
Hence, smaller the focal lengths of objectives and eyepiece, larger is the
magnification. Further the negative value of the „m‟ tells that final image formed is
inverted with respect to the object.
When final image formed at infinity:
When the final image formed at infinity, magnification power of the eyepiece
me =
m=
m=
𝐷
𝑓𝑒
−𝐿
𝑓𝑜
(
𝐷
𝑓𝑒
)
− 𝐿𝐷
𝑓𝑜 𝑓𝑒
Hence for large magnifying power, f0 & fe must be small.
Telescope:
“A Telescope is an optical instrument which is used to see the distinct objects
clearly and with suitable magnification”.
In general telescopes are divided into two types namely
(1) Refracting telescope (2) Reflecting telescope.
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Refracting telescope:
These telescope make use of converging lenses to see distant objects clearly.
Since lenses form images of objects by refraction of light, these are called refracting
telescope. There are two types namely,
a. Astronomical telescope
b. Terrestrial telescope
a. Astronomical Telescope:
This telescope is used to observe heavenly objects like moon, distant stars and
planet etc.The image formed by this telescope is virtual and inverted.
b. Terrestrial telescope:
This telescope is used to observe the objects on the earth like trees, houses etc.
the image formed by this telescope is virtual and erect.
Reflecting telescope:
These telescope make use of converging mirrors to see distant objects clearly.
Since mirrors form images of objects by reflection of light these are called reflecting
telescope.
Astronomical Refraction Telescope:
An astronomical telescope is an optical instrument which is used to see heavenly
bodied like moon, planet, stars etc. It consists of two achromatic convex lenses
mounted co-axially at the outer ends of two sliding metal tubes.
The lens facing the object is called objective lens And has large focal length and
large aperture. Large aperture helps collecting sufficient light from the distant objects
and provides brighter image. The other lens through which the image of the distant
object is observed is called eyepiece and has small focal length and small aperture.
The aperture is small so that the entire light taken up by the objective enters the eye.
The tube holding the eyepiece can slide into the tube holding the objective with the
help of a rack and pinion adjustment.
The telescope is said to be in normal adjustment when the final image is formed at
infinity. A parallel beam of light from the object is made to fall on the objective lens
of the telescope. It forms a real, inverted and diminished image A1 B1 of the object.
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The eyepiece is adjusted that A1 B1 lies at the focus of the eyepiece (Fe coincide with
Fo) . Hence a magnified image is formed at infinity.
The magnifying power of an astronomical telescope in normal adjustment is
defined as the ratio of the angle subtended at the eye by the final image to the angle
subtended at the eye by the object seen directly, when both the object and the image
are at infinity.
Let < A1 O B1 = α and < A1 E B1 = β
Angular magnification or magnifying power
m=
𝛽
……………..(1)
𝛼
As angles α & β are small,
therefore, α ≈ tan α and β ≈ tan β
m=
𝑡𝑎𝑛 𝛽
𝑡𝑎𝑛 𝛼
………………… (2)
From ∆le A1 O B1, tanα =
From ∆le A1 E B1, tan β =
𝐴1 𝐵1
𝑓𝑜
𝐴1 𝐵1
−𝑓𝑒
Equation (2) become,
m=
m=
𝐴1 𝐵1
−𝑓𝑒
×
𝑓𝑜
𝐴1 𝐵1
𝑓𝑜
−𝑓𝑒
For greater magnification, f0 should be large and fe must be small. The negative sign
indicates that the final image is real and inverted.
In normal adjustment, length of the telescope is equal to ( fo + fe)
NOTE:
When final image is at the near point then,
m=
𝑓𝑜
−𝑓𝑒
(1-
𝑓𝑒
𝐷
)
Note that magnifying power is increased when the image is formed at the near point.
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Reflecting telescope:
An astronomical telescope used to see the distant stars should possess large light
gathering power and high resolving power. Both the light gathering power and the
resolving power for a telescope will be large, if the objective of the telescope is of
large aperture. But the objective lenses of very large aperture are very difficult to
manufacture.
To overcome these limitations of the refracting telescope, the objective lens can
be replaced by a concave parabolic mirrors of large aperture which is free from
spherical and chromatic aberrations. such a telescope is known as Reflecting type
telescope and can form much brighter image of the distinct object.
Cassegrain reflecting telescope:
Telescopes using mirrors as objective are called reflecting telescope. In
reflecting telescope, the objective lens is replaced by parabolic mirror of large
aperture which if free from chromatic and spherical aberrations. The image formed is
very bright and telescope has high resolving power. Such a telescope is known as
cassegrain telescope.
In cassegrain type telescope, there is a parabolic concave mirror of a large
aperture with a hole of suitable diameter at its centre. A parallel beam of light,
parallel to the axis of telescope from a distant star falls on the parabolic concave
reflector . which converges it towards its principal focus F0. the reflected beam
intercepted by convex mirror .the convex mirror forms an inverted image at F.
This inverted image is seen through the eye piece. For normal adjustment, the
magnifying power of the reflecting type telescope is given by,
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m=
Therefore,
m=
𝑓𝑜
𝑓𝑒
=
𝑅
2
𝑓𝑒
𝑅
2𝑓𝑒
where f0 = foal length of the objective
fe = focal length of the eyepiece
R = Radius of curvature of concave mirror
For the final image formed at the least distance of distinct vision
m=
𝑓𝑜
−𝑓𝑒
( 1-
𝑓𝑒
𝐷
)
Advantages of reflecting type telescope:
1. Mirror are free from spherical aberration (formation of non- point, blurred
image of a point object) and chromatic aberration (formation of coloured image
of a white object).
2. A mirror requires grinding and polishing of one surface only. So mirror are
easier to manufacture and cheaper than lenses.
3. Mirrors gather light better than lenses and the final image is brighter.
4. Reflecting type telescope have high light gathering power, large magnification
and large resolving power.
5. A lens of aperture tends to be heavy and therefore difficult to make and support
by its edges. On the other hand a mirror equivalent optical quality weights less
and can be supported over its entire back surface.
Disadvantages of reflecting telescope:
1. They cannot be used for general purpose.
2. It is inconvenient to use because of frequent adjustment.
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Difference between Refracting and reflecting telescope:
Si
No
Reflecting
Refracting Telescope
Telescope
1.
The object is a converging lens
The objective is a concave spherical
mirror
2.
It suffers spherical and chromatic It is free from spherical and chromatic
aberration. Hence the final image aberration. Hence the image formed is
is coloured and blurred.
sharp and bright
3.
It has small light Gathering
power, so a faint image of the
distance star is observed.
It has large light gathering power. So a
bright image of the distant star is
observed.
4.
It is used for general purpose and
is handy
It is used in astronomy and is not
handly.
****************************
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CHAPTER–10
WAVE OPTICS
Nature of light:
Light exhibits both particle and wave nature. That means light exhibits dual
nature. To explain nature of light, different theories are proposed.
1. Newton‟s corpuscular model:
According to this model, light is emitted in the form of a stream of minute
particles called corpuscles. These corpuscles carry energy in the form of kinetic
energy. Also this model predict that speed of light is greater in denser medium
than rarer medium. But experiments shows that speed of light is more in rarer
medium than in denser medium.
This theory explain reflection and refraction but fail to explain interference,
diffraction and polarization.
2. Huygen‟s wave theory:
Wave theory of light was proposed by Christian Huygens in 1678.
According to this theory, light propagates as a mechanical wave in the form of
disturbance through a hypothetical medium called „ether‟.
This theory explain reflection, refraction, interference and diffraction of
light. But fails to explain polarization , photoelectric effect, Raman effect and
Compton effect.
3. Maxwell‟s electromagnetic wave theory:
Electromagnetic wave theory was proposed by Maxwell. According to this
theory, light consists of fluctuating electric and magnetic fields propagated in the
form of electromagnetic waves. These wave can propagate through the vacuum.
This theory explain reflection, refraction, interference and diffraction and
polarization. But fails to explain photoelectric effect, Raman effect and Compton
effect etc.
4. Plank‟s Quantum theory:
This theory was proposed by Max plank in 1900. According to this theory
radiation is in the form of tiny packet of energy called „ quanta‟. The energy of
each quanta or photon is given by E = hυ
where h = plank‟s constant, υ = frequency of radiation.
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This theory explain photoelectric effect, Raman effect and Compton effect
but fails to explain interference, diffraction and polarization.
Huygen‟s principle:
We use huygen‟s principle to derive the laws of reflection and refraction. The
principle of superposition of waves is used to explain interference of light.
Wavefront:
When a small stone is dropped into still water, waves are formed on the surface of
the water in form spreading circles. All the points on the circle, oscillates with the
same phase because they are at the same distance from the source.
“ The locus of all the particles vibrates in the same phase is called Wavefront”
The speed with, which the wavefront moves outwards is called speed of wave. The
energy of the wave travels in the direction perpendicular to the wavefront.
Depending upon the shape of the source of light, wavefront can be classified as 3
category.
1. Spherical wavefront: A spherical wavefront is produced by a point source of
light. It is because, The locus of all the particles vibrates in the same phase are
spheres. Such a wavefront is called spherical wavefront.
2. Cylindrical Wavefront: When the source of light is linear in shape (such as a
slit) cylindrical wavefront produced. It is because The locus of all the particles
vibrates in the same phase is a cylindrical surface and it is called Cylindrical
wavefront.
3. Plane wavefront: If the wavefront originating from the distant source it appear
plane and hence it is called plane wavefront.
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Huygen‟s principle:
Huygen‟s principle is the geometrical method of finding the shape and position of
a wavefront. Huygen‟s principle is stated as follows:
1. Each point on a wavefront is a source of secondary disturbance and the
secondary wavelets originating from these point spread out in all direction
with the speed of the wave.
2. A tangent drawn to secondary wavelets in the forward direction gives the
position of new wavefront.
Let „s‟ is a point source which emits light in all possible direction. Then AB is
the locus of all points which vibrates in same phase it is called initial wavefront. The
secondary wavelets originating from each point on the wavefront travel in the
direction with the same speed „v‟. after a time „t‟ they developes a sphere of radius vt.
The tangential surface CD drawn to all the sphere in the forward direction give the
shape and position of the new wavefront after a time „t‟.
Refraction of a plane wave using huygen‟s principle:
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Let „XY‟ represents the surface separating medium 1 and 2. Let v1 & v2 be the speed
of light in medium 1& 2 respectively (v1 ˃ v2).
Consider a plane wavefront „AB‟ incident in a medium 1 at an angle i on the
refracting surface „XY‟. According to huygen‟s principle every point on the
wavefront AB is the source of secondary wavelets. Let the secondary wavelets from
„B‟ strikes the surface at „C‟ in a time „t‟ then BC = v1t.
The secondary wavelet from „A‟ will travel a distance in a medium 2 in the same
time. Draw an arc with „A‟ as a centre. The tangent from „C‟ touches the arc at „D‟ .
Then AD = v2t and CD represents the “ refracted wavefront”
Let,
< BAC = i = angle of incidence
< DCA = r = angle of refraction
BC = v1t and AD = v2t
From ∆ BAC,
sin i =
𝐵𝐶
𝐴𝐶
From ∆ DCA,
sin r =
Therefore,
sin 𝑖
sin 𝑟
sin 𝑖
sin 𝑟
sin 𝑖
sin 𝑟
=
=
=
𝐴𝐷
𝐴𝐶
𝐵𝐶
𝐴𝐶
𝐵𝐶
𝐴𝐶
𝑣
𝑣
×
=
𝐴𝐶
𝐴𝐷
𝑣1 𝑡
𝑣2 𝑡
sin 𝑖
⟹
sin 𝑟
= Constant
This Represents snell‟s law of refraction.
When light travels from rarer medium to denser medium. It bends towards the
normal. Hence AD is shorter then BC (v1 ˃ v2 )
Also,
BC = λ1 = Wavelength of the light in medium 1
AD = λ2 = Wavelength of the light in medium 2
Therefore,
Therefore,
sin 𝑖
sin 𝑟
𝑣1
𝜆1
=
=
𝑣1
𝑣2
=
𝜆1
𝜆2
𝑣2
𝜆2
Thus when light travel from rarer medium to denser medium(v1 ˃ v2), then
wavelength and speed of propagation decreases but the frequency remains same.
Page | 230
Refraction of a plane wave at a rarer medium:
Consider a plane wavefront AB incident on rarer medium for which v2 ˃ v1. Then
the plan wave bends away from the normal. Hence the angle of refraction is greater
than the angle of incidence. The refracted wavefront is EC.
In this case also snell‟s law of refraction is same and it is given by,
n1sin i = n2sin r
If i = c then r = 900
Therefore,
n1sinc = n2sin900
if i > c there is no refracted wavefront and incident plane wave AB undergoes
total reflection.
𝑛
Then,
sin c = 2
𝑛1
Reflection of a plane wave using huygen‟s principle:
Page | 231
Consider a plane wave front AB incident at an angle i on a reflecting surface
„XY‟. Let „V‟ be the speed of light wave in the medium.
According to huygen‟s principle every point on the wavefront „AB‟ is a source of
secondary wavelets.
Let the secondary wavelets from „B‟ strikes the surface „XY‟ at „C‟ in a time „t‟
then BC = Vt.
The secondary wavelet from „A‟ will travel the same distance AD = Vt in a same
time „t‟ draw an arc with „A‟ as a centre. Then the tangent „CD‟ is the reflected
wavefront.
Let
< BAC = i = angle of incidence
< DCA = r = angle of reflection
Then angle <ABC = <CDA = 900 and BC = AD
Therefore, triangles ABC and DCA are congruent
Hence, <BAC = < DCA
i.e.,
i = r Thus angle of incidence is equal to angle.
Behaviour of a prism, lens and spherical mirror towards a plane
wavefront:
a. Refraction through a prism:
Consider a plane wave passing through a
glass prism. The speed of light is less in
glass than in air. The lower portion of the
incident wavefront travels through a larger
thickness of glass and the upper portions
travel less in glass. Thus the lower portion
of the wavefront gets delayed. This causes a tilt in the emerging wavefront.
b. Behaviour of a lens:
Consider a plane wave incident on a
convex lens. The central part of the incident
plane wave travels the thickest position of
the lens and hence delayed. Hence the
emergent wavefront has a depression at the
centre of the wavefront becomes spherical
with radius „f‟.
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c. Behaviour of a concave mirror:
Consider a plane wave incident on
the concave mirror. Then centre of the
wavefront travel the largest distance to
the pole „P‟ and then reflected. Then the
reflected wavefront become spherical.
Doppler effect in light:
When the source of light or observer is moving, the later wavefront have to
travel for longer or shorter time. As a result there is change in frequency of the light
received by the observer. This is known as Doppler effect. (or)
“The apparent change in the frequency of a light received by the observer due to
the relative motion between a source of light and observer is called Doppler
Effect”.
The apparent frequency of the light increases when the distance between source of
light and observer decreases.
The apparent frequency of the light decreases when the distance between source of
light and observer increases.
When the source of light moves away from the observer. The later wavefront
have to travel a greater distance to reach the observer and hence take longer time.
Therefore frequency measured by the observer decreases. This decreases in frequency
or increases in wavelength during Doppler effect is called “Red shift”.
When the source of light moves towards the observer. There is an increases in
frequency or decreases in wavelength due to Doppler effect. This is called “ Blue
shift”.
Principle of superposition:
According to principle of superposition, when two or more waves travelling
through the medium, superpose one another, a new wave is formed in which resultant
displacement (Y) at any instant is equal to the vector sum of the displacements of
individual waves.
i.e., Y = Y1 + Y2 +………………… YN
1. Constructive superposition: When crest of one wave falls on the crest of the
other wave or when the trough of one wave falls on the trough of the other
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wave, then the resultant amplitude increases and hence resultant intensity
increases. Then the superposition is constructive.
i.e., Y = Y1 + Y2
2. Destructive superposition: When the crest of one wave falls on the trough of
the other wave, then the resultant amplitude decreases and hence resultant
intensity decreases. Then the superposition is destructive.
i.e., Y = Y1 – Y2
Coherent source:
The source of light, which emits light waves of the same frequency, same
wavelength and in same phase or constant phase difference are called “ Coherent
Source”. Two independent source of light cannot be coherent because even though
they may emit light waves of equal wavelength and equal amplitude, they may not
have a constant phase difference.
Interference of light:
“ The modification in the distribution of light energy due to the superposition of
two or more light waves of same nature is called interference of light”.
Due to interference of light, alternating bright and dark bands are observed and they
are called interference fringes or bands.
Eg: Colour pattern on the soap bubbles and colour pattern on oil spills on wet road.
Constructive and destructive interference:
Consider two light waves of the same frequency travelling in a medium in the
same direction. Let a1 and a2 be the amplitude of the two waves. The displacement of
any particle in the medium due to these waves at any instant of time „t‟ are,
y1 = a1 sin ωt and y2 = a2 sin (ωt + δ)
where, δ – phase difference between the waves
ω = 2πf - angular frequency of the waves
According to principle of superposition of waves, the resultant displacement of the
particle is
y = y1 + y2 = a1 sin ωt + a2 sin (ωt + δ)
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= a1 sin ωt + a2 ( sin ωt cos δ + cos ωt sin δ)
= a1 sin ωt + a2 sin ωt cos δ + a2 cos ωt sin δ
y = (a1 + a2 cos δ) sin ωt + a2 cos ωt sin δ
Let
Then,
Therefore,
R cos ө = (a1 + a2 cos δ)
R sin ө = a2 sin δ
………………… (1)
…………………(2)
y = R cos ө sin ωt + R sin ө cos ωt
= R [sin ωt cos ө + cos ωt sin ө]
y = R sin (ωt + ө)
…………….. (3)
Equation (3) represents a simple harmonic vibration of amplitude r and angular
frequency ω. Squaring and adding eq (1) & (2)
R2 cos2 ө + R2 sin2 ө = (a1 + a2 cos δ)2 + (a2 sin δ)2
R2 (cos2 ө + sin2 ө) = a12 + a22 cos2 δ + 2 a1 a2 cos δ + a22 sin2 δ
R2 = a12 + a22 + 2 a1 a2 cos δ (or)
R = a + a + 2 a a cos δ
……………. (4)
Thus the intensity of light is directly proportional to the square of the amplitude of
the wave, for sake of simplicity we assume equal to square of amplitude.
Condition for constructive interference:
When there is constructive interference, the amplitude is maximum and hence the
intensity is also maximum.
From the equation R = a + a + 2 a a cos δ
The resultant amplitude „R‟ is maximum when cos δ = +1
i.e., δ = 2nπ where n = 0,1,2,3 …………. n
Thus „R‟ will be maximum when the phase difference between the two wave is even
multiple of π.
i.e., δ = o, 2π,4π,6π………
Therefore For constructive interference, the phase difference between two light wave
must be equal to even multiple of π (or) the path difference between the waves must
be equal to even multiple of
𝜆
2
where λ is the wavelength of the light.
Page | 235
The maximum amplitude is Rmax = a + a + 2 a a = (a + a ) = a1 + a2
The maximum intensity is Imax α (a1 + a2)2
If a1 = a2 = a,
Rmax = 2a & Imax = 4a2
In such a case the wave are said to constructive interference, since resultant
amplitude is maximum it leads brightness in light wave(bright fringes).
Condition for destructive interference:
When there is destructive interference, the amplitude is minimum and hence the
intensity is also minimum.
From the equation R = a + a + 2 a a cos δ
The resultant amplitude „R‟ is minimum when cos δ = -1
i.e., δ = (2n+1)π where n = 0,1,2,3 …………. n
Thus „R‟ will be minimum when the phase difference between the two wave is odd
multiple of π.
i.e., δ = π,3π,5π.....
Therefore for destructive interference, the phase difference between two light wave
must be equal to odd multiple of π (or) the path difference between the waves must
be equal to odd multiple of
𝜆
2
where λ is the wavelength of the light.
The maximum amplitude is Rmax = a + a − 2 a a =
(a − a ) = a1 - a2
The maximum intensity is Imax α (a1 - a2)2
If a1 = a2 = a, Rmax = 0 & Imax = 0
In such a case the wave are said to destructive interference, since resultant
amplitude is minimum it leads darkness in light wave(dark fringes).
Young‟s double slit experiment:
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The phenomenon of interference of light was first experimentally demonstrated by
Thomas young in 1801. The experimental arrangement is as shown in the above
figure.The light coming from the monochromatic source is made to fall on the narrow
slit „S‟. the light emerged from the narrow slit „S‟ is made to fall on the two narrow
and parallel slits S1 & S2 placed close to each other. The two slits are parallel to S and
equidistant from S. then two slits S1 & S2 acts as two coherent source of light of same
intensity. The cylindrical waves emerging from S1 & S2 interference each other and
produce interference pattern on the screen. The interference pattern consists of
alternating bright and dark bands of equal width. They are called interference bands
or fringes.
The width of interference fringe is given by β =
𝜆𝐷
𝑑
where, λ – wavelength of light, d- distance between the slits and
D – distance between slits and screen.
Intensity distribution curve for interference:
The variation of intensity with phase difference is shown in below figure.
The intensity of light is maximum at the point of constructive interference or the
phase difference between two wave is equal to even multiple of π.
The intensity of light is minimum at the point of destructive interference or the
phase difference between two wave is equal to odd multiple of π. All bright fringes
have maximum intensity and equal width. All dark fringes have zero intensity and
equal width. However energy is neither be created at the point of maximum intensity
nor be destroyed at the point of minimum intensity.
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The energy is only transferred from a dark band to a bright band. Thus the law of
conservation energy holds good during the phenomenon of interference.
Theory of interfernce fringes (or) expression for fringe width:
Let A and B represents two coherent
sources separated by a distance d. let these
two coherent sources emit light waves of
same wavelength λ. Let the screen be placed
at a distance D from the source. Let „C‟ be
the midpoint of AB. Let AE, BF & CO be the
perpendiculars to the screen from A, B & C
respectively. The point „O‟ on the screen is
equidistant from A & B. therefore the path
difference between the two light wave from
A & B reaching „O‟ is zero. Thus the point „O‟ has maximum intensity.
Consider a point „P‟ on the screen at a distance „x‟ from „O‟. let AN be a
normal drawn to the line BP from A such that AP = NP. At the point „P‟ the intensity
is dark or bright depends on the path difference between the waves reaching the point
„P‟ from the two source A & B.
Then Path difference = BP- AP
By applying Pythagoras theorem to the ∆BFP,
𝑑
BP2 = BF2 + FP2 = D2 + [x + ]2
2
Similarly from ∆AEP,
𝑑
AP2 = AE2 + EP2 = D2 + [x - ]2
𝑑 2
]
2
BP2 – AP2 = D2 + [x +
2
2
BP – AP = [x +
𝑑 2
]
2
2
𝑑
- D2 - [x - ]2
- [x
𝑑
- ]2
2
𝑑
(BP + AP) (BP – AP) = [x +
2
2
𝑑
𝑑
2
2
+ x - ] [x +
𝑑
- x+ ]
2
(BP + AP) (BP – AP) = 2xd
(BP – AP) =
2𝑥𝑑
(BP + AP )
Since D>>d, by approximation BP = D & AP = D.
(BP – AP) =
2𝑥𝑑
(D + D)
=
2𝑥𝑑
2D
=
𝑥𝑑
D
Page | 238
Therefore path difference =
𝑥𝑑
………………….. (1)
D
Equation (1) represents the path difference between light wave from A & B at the
point „P‟.
1. For Maximum intensity at „P‟ the path difference must be an even multiple
𝛌
of :
𝟐
i.e.,
𝑥𝑑
D
= 2n
𝜆
(or) x=
2
𝑛𝜆𝐷
𝑑
“ The distance between two consecutive bright or dark fringes in an
interference pattern is called fringe width it is denoted by β”.
β = distance of (n+1)th bright fringe – Distance of n th bright fringe
β = xn+1 - xn
= (n+1)
=
𝑛𝜆𝐷
𝑑
β=
+
𝜆𝐷
𝑛𝜆𝐷
-
𝑑
𝜆𝐷
𝑑
𝑑
-
𝜆𝐷
𝑛𝜆𝐷
𝑑
………………….. (2)
𝑑
2. For Minimum intensity at „P‟ the path difference must be an odd multiple
of
𝛌
𝟐
:
i.e.,
𝑥𝑑
D
= (2n+1)
𝜆
(or) x = (2n+1)
2
𝜆𝐷
2𝑑
β = distance of nth bright fringe – Distance of (n-1) th bright fringe
β = xn – xn-1
β = (2n+1)
β = (2n+1)
β=
β=
𝜆𝐷
2𝑑
𝜆𝐷
𝑑
𝜆𝐷
2𝑑
𝜆𝐷
2𝑑
− (2(n-1)+1)
- [2n – 2 +1]
𝜆𝐷
2𝑑
𝜆𝐷
2𝑑
[ 2n +1 – 2n +1]
…………………. (3)
From equation (2) and (3) it is clear that fringe widths of bright and dark fringes
are equal. Hence they are spaced equally. It is also found that,
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a. β increases if λ increases (β α λ)
b. β α λ larger is the distance between screen and slits, larger will be the fringe
width.
c. β α
1
𝑑
smaller is the distance between coherent sources, the larger will be the
fringe width.
Diffraction of light:
“The Phenomenon of bending of light around the edges of small obstacles
and entering into the region of geometrical shadow is called diffraction of light”.
The diffraction phenomenon is exhibited by the all the types of waves such as
sound waves, light waves, matter waves etc. The amount of bending depends on the
size of the obstacles and wavelength of the wave. The wavelength of light ( ≈ 10-6 m)
is very small compared to the size of ordinary obstacles and apertures. Therefore
diffraction of light is not easily notice in daily life.
The wave length of sound wave is very large compared to that of light wave. Hence
sound waves bend round the corners of buildings and other objects having larger
dimension.
Diffraction of light at a single slit:
Consider a parallel beam of light or a plane
wavefront of wavelength „  ' incident on a
narrow slit „AB‟ of width „d‟. The light is
diffracted by an angle  and focusing on a
screen with the help of convex lens „L‟ The
diffraction pattern obtained on a screen
decreasing intensity on either side.
According to Huygen‟s principle, every point
on the plane wave front „AB‟ incident on the
slit sends out secondary wavelets. These secondary wavelets superpose to produce
diffraction pattern on the screen.
The intensity of light at different points on the screen depends on the path difference
between the secondary wavelets reaching the points. The path difference between the
secondary wavelets from A and B reaching the point „P‟ equal to „BN‟
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From the  ANB,
sin  
BN
BN
 sin  
 BN  d sin 
AB
d
where„d‟ is the width of the slit
1. Intensity at the centre „o‟ of the screen :
All the secondary wavelets reaching the centre „O‟ on the screen are in same
phase. Because the angle of diffraction is zero. Therefore the secondary wavelets
superpose constrictively with each other at „O‟ and produce maximum intensity at
„O‟. The bright band at „O‟ is called principal maximum (or) central maximum.
2. Intesity at a point „p‟ :
If d sin   
The path difference between secondary wavelets from A and B is equal to in
reaching the point „P‟. Let the wavefront „AB‟ is divided into two equal parts AC
and CB. Then the path difference between the secondary wavelets originating
from corresponding points of two parts of the slits is  .
2
As a result destructive interference takes place and the point „P‟ is of minimum
intensity. It is called first minimum. Similarly, if the path difference, a sin   2 ,
then again the intensity on the screen is minimum. It is called secondary
minimum. In general, condition for minima.
d sin   n
where, n=1, 2, 3, …………………3.
3. Intensity at point „q‟:
d sin  
3
2
If,
i.e., The path difference between the secondary wavelets emerging from „A‟ and
„B‟ is 3
2
Then the wave front AB is divided into three equal parts. The secondary waves
from the corresponding points of first two parts is and hence they annul each
other.
But the wavelets emerging from the third part will produce some intensity at
„Q‟. This intensity is very much less than at point „Q‟. The point Q is called first
secondary maximum.
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Similarly, the secondary maximum is located on the screen when the path
difference is 5
2
In general, the condition for secondary maxima is
d sin   (2n  1)

2
where, n=1, 2, 3, ……………………
For small ' ' sin   
Therefore, d
 (2n  1)


2
(2n  1)
2a
Thus the diffraction pattern consists of central maxima and alternate minima and
secondary maxima. The intensity distribution curve is shown in the figure.
Width of central maxima:
“The width of the central maximum is the distance between the first minimum
on either side of the central maximum”.
Let „d‟ be the width of the slit.
d sin   n
but n=1 for I minima Therefore d sin   
sin  
Therefore, 
Also ,



d

d
................................(1)
x
................................(2)
D
Page | 242
Where, x=distance of the I minimum from the central maxima D=the distance of the
screen from the slit.
From equation (1) and (2)
x 

D d
x
D
d
The width of the central maxima = 2 x
2x 
2D
d
If the lens L is placed very close to the slit, the f=D
Linear width of the central maxima =
where,
f
2x 
2f
d
is the focal length of lens.
Difference between interference and diffraction:
SL.NO
INTERFERNCE
DIFFRACTION
1
Interference is due to
superposition of two or
more waves coming from
two coherent sources
Diffraction is due to superposition of
secondary wavelets coming from
different points of the same wavefront
2
Bright and dark fringes have
equal width
Diffraction bands are having unequal
width.
3
Intensity of all bright
fringes is same
The intensity of secondary maxima
decreases with increase in order. The
central maxima has highest intensity.
4
There is a good contrast
between maxima and
minima
There is a poor contrast between bright
and dark bands.
5
The interference of fringes
are larger in number.
The diffraction bands are few in
number.
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Validity of ray optics (fresnel distance):
It is the least distance between slit and screen up to which ray optics is valid.
OR
Fresnel distance is that distance from the slit or aperture to the screen at which the
spreading of light due to diffraction becomes equal of the slit.
It is denoted by Z F
According to ray optics, light travels along a straight line in diffraction at a single slit,
the half angular width of a central maximum gives the spreading of light.
If half angular width is equal to slit width, there is no spreading. Then the ray
optics is valid. We have the condition for minima
d sin   n
for I minima d sin   
sin  
n=1
 


d
------------------- (1)
d
This is the half angular width of central maxima.
The linear spread of central maximum is
sin  
x
D
x  D sin   x  D             (2)
when, x  d
2

d  D   d  D  D  a
d

2
This is the least distance through which light travels before it appreciable
bends from its straight path. This distance is Fresnel distance.
i.e.,
D  Z F , then
Z
F
a
2

This is the expression for Fresnel‟s distance.
If D  Z F , the diffraction effects can be neglected and ray optics valid.
If D  Z F , the spreading due to diffraction is more and ray optics is not valid. Hence
ray optics if the limiting case of wave optics.
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Resolving power of optical instruments:
The image of a point object produced by optical instrument is not a point image
but it is a circular patch with bright central disc surrounded by concentric dark and
bright rings. This is due to diffraction of light.
The nearby point objects are said to be resolved, if they are seen as two separate
distinct objects either with naked eye or though optical instrument.
„The resolving power of an optical instrument is the ability of the instrument
to produce separate image of two nearby objects‟.
Consider two closely lying objects.
 If the images of the two objects appear to be overlapping on one another, they are
said to be unresolved.
 If the images appear to be just separated, they are said to be just resolved.
 If the images are well separated, they are said to be resolved.
Limit of resolution:
“The minimum distance between two point objects at which their images are
just resolved is called limit of resolution”.
Resolving power =
1
limit of resolution
The limit of resolution of human eye is minute or
1
60
degree. Hence eye can
see the two objects separately if the angle subtended by them at the eye is more than
1 minute of arc.
Resolving power of microscope:
The minimum distance between two point objects at which their images are
just resolved is called limit of resolution of microscope.
Page | 245
Limit of resolution of a microscope is given by,
d
min


2n sin 
where, „  ’ is wavelength of light
„n‟ is the R.I. of medium between object and objective lens. Is the semi vertical angle
of cone of light rays coming from the object.
The product nsin  is called the numerical aperture.
The resolving power of a microscope is defined as, “The reciprocal of the minimum
distance between two point objects at which thei images are just resolved”.
Therefore, R.P 
R.P. 
1
d
min
2n sin 

1. The resolving power of a microscope can be increased by using high R.I. oil
(cedal wood oil) between object and objective. The microscope is then called an
oil immersion objective microscope.
2. The R.P. of a microscope can be increased by using a shorter wavelength u.v.
light. Then the microscope is called “ultra-microscope”.
Resolving power of telescope:
The limit of resolution of telescope is defined as “the smallest angular
separation between two distinct objects at which their images are just resolved”.
The limit of resolution of telescope is given by,
 
1.22
D
OR
 
0.61
a
Where, „  ’ is the wave length of light
„D‟ is the diameter of the objective lens,
„a‟ is the radius of objective lens.
“The resolving power of a telescope is defined as the reciprocal of smallest
angular separation between two distant objects at which their images are just
resolved.”
Page | 246
R.P. 
1


R.P. 
D
1.22
Thus the resolving power of a telescope can be increased by increasing the
diameter of the objective lens.
Polarisation of light:
Light waves are electromagnetic waves. That means light consists of an
oscillating electric and magnetic fields vibrating perpendicular to each other and also
perpendicular to the direction of propagation. Hence light waves are transverse
waves. Light is generally represented by the electric field E called light vector.
Ordinary light contains symmetrical vibrations of electric fields in all possible
directions. By some methods it is possible to restrict the vibrations to a single
direction.
“The light in which vibrations are restricted to a single direction or plane is
called plane polarized light”.
“The phenomenon of restricting the vibrations of a light vector in a particular
direction or particular plane is called polarization of light”.
Representation of unpolarised light and plane polarized light:
Unpolarised light:
Unpolarised light contains large number of vibrations in all possible directions
in a plane at right angles to the direction of propagation as shown in figure (a).
Figure (b) represents the unpolarised light in which vibrations are resolved into two
components along two mutually perpendicular directions.
Page | 247
Plane polarised light:
Plane polarized light with vibrations parallel to the plane of the paper is shown
in figure (c).
Plane polarized light with vibrations are perpendicular to the plane of the paper is
shown in figure (d).
 In a plane polarized light, the plane in which the vibrations occur is called the
plane of vibration.
 In a plane polarized light, the plane in which no vibrations occur is called the
plane of polarized.
Polaroids:
Polaroids is a device used to produce plane polarized light. A Polaroid consists of
long chain molecules aligned in a particular direction. When an unpolarised light is
incident on a Polaroid, the electric vectors along the direction of the aligned
molecules of get absorbed and electric vectors perpendicular to the direction of
alignment of molecules passes through it. Hence the transmitted light is called
linearly or plane polarized light.
Page | 248
The direction perpendicular to the direction of the alignment of molecules of the
Polaroid is called pass axis of the Polaroid. The lines drawn on the surface of the
Polaroid shows the pass axis.
Experiment to demonstrate the transverse nature of light using
polaroids:
When unpolarised light passes through the Polaroid „P 1‟ then the intensity of
transmitted light is reduced by half. When the poloroid „P 1‟ is rotated, then there is no
change in the intensity of transmitted light. Another identical poloroid „P 2‟ is placed
in front of „P1‟ so that their pass axis are parallel to each other. Now keeping p1 is
fixed, P2 is alone rotated; the intensity gradually decreases and becomes zero when P 2
is rotated through 900. If the rotation of P2 continued, the intensity increases gradually
and becomes maximum when rotated through 1800. For one complete rotation of „P2‟,
the intensity of light becomes zero twice and minimum twice. The variation in
intensity of light shows that transverse nature of light.
This experiment shows that when unpolarised light passes through Polaroid P1.
The vibrations parallel to pass axis of the Polaroid are transmitted and other
vibrations are blocked. Thus the transmitted light from Polaroid „P 1‟ is plane
polarized and it is called polarizer.
When the pass axis of Polaroid P2 is parallel to pass axis of P1 then light
transmitted through „P2‟ is polarized and intensity is maximum. When the pass axis
of P2 is perpendicular to the pass axis of P1, the vibrations are blocked and no light
emerges through „p2‟. The Polaroid „p2‟ detects the plane polarized light and it is
called analyser.
The following figure shows two sheets of polaroids in the different positions.
Page | 249
In figure (1) the polaroids are at 00 and hence intensity is maximum.
In figure (2), the polaroids P2 makes an angle with P1 and hence shaded portion
shows the decreased intensity of light.
In figure (3), polaroids „P2‟ is rotated through 900, the dark shaded portion shows the
light is completely blocked.
Malus law:
It states that “the intensity of transmitted light through the analyser varies as
the square of the cosine of the angle between the transmission axes of the polarizer
and analyser”.
i.e.,
I ∝ cos2 
Therefore, I = I0cos2 
Thus is Malus Law
Where, I is the intensity of transmitted light through analyser
I0 is the intensity of transmitted polarized light through polarizer
 is the angle between passes axes of polarizer and analyser
Uses fo polaroids:
Polaroids are used
1.
2.
3.
4.
In photographic cameras and 3D movie cameras.
In window glasses of trains and aeroplanes to control the intensity of light.
In sun glasses to cut off the glare produced by light
To eliminate the head glare of automobiles.
Page | 250
Polarisation by scattering:
Plane polarized light is produced due to the scattering light by the molecules of
the earth‟s atmosphere.
When sun light passes through earth‟s
atmosphere it is absorbed by the atoms of
particles or molecules and re-emitted in
different direction. This is called scattering
of light. The scattered light in a direction
perpendicular to the direction of sunlight is
planed polarized. An observer at put angles
to the direction of sun light can see the
polarized light. This scattered polarized light
contains vibrations perpendicular to the
plane of the paper. When sun light is
looking at other angles, then the scattered
light is partially polarized. This can be verified by using polaroids. The intensity of
blue portion for sky varies when viewed through a rotating Polaroid.
Polarisation by reflection:
When ordinary light is reflected from the surface of transparent medium like
glass of water, the reflected light becomes partly polarized or completely polarized.
The degree of polarization depends on the angle of incidence. For a particular angle
of incidence the reflected ray and refracted ray Eire perpendicular to each other.
“The particular angle of incidence for which the reflected light is completely
polarized is called Brewster‟s angle or polarizing angle (iB)”.
It depends on the refractive index of the denser medium.
Page | 251
The reflected light contains vibrations (electric vectors) perpendicular to the plane of
incidence.
Brewster‟s law:
It states that “the tangent of the polarizing angle is numerically equal to the
refractive index of the reflecting medium,
i.e., n=tan iB
where, n is the R.I. of the reflecting medium, iB is the polarizing angle
Proof:
When unpolarised light is incident on reflecting surface at Brewster‟s angle, then
reflected ray is perpendicular to the refracted ray.
i.e., BOC  90
0
 MOB  NOC  90
0
i

B
r 
r

2

2
 iB
According to Snell‟s law,
n
n
sin i B
sin r
sin i B


sin   i B 
2


Page | 252
n
sin i B
cos i B
n = tan
i
B
This is known as Brewster‟s law
****************************
Page | 253
CHAPTER-11
DUAL NATURE OF RADIATION AND MATTER
Introduction:
In metals, “the electrons in the outer most shells of the atoms are loosely bound and
they move freely, such loosely bound electrons are called free electrons”.
These free electrons cannot leave the surface of the metal at ordinary temperature and
under moderate electric fields. The free electron has to overcome the barrier in order
to just escape from the metal surface. “The minimum amount of energy required by
an electron to just escape from the metal surface is called work function”.
The work function is denoted by ɸ0 and is measured in electron volt (eV).
1eV = 1.602 X 10-19J
One electron volt is the kinetic energy gained by an electron when it is accelerated
through potential differences of 1volt.
Electron - Emission:
“The phenomenon of emission of electron from the surface of metal is called
electron emission”.
Types of electron emission (Physical process)
1. Thermionic emission: By suitably heating, sufficient thermal energy can be
imparted to the free electrons to come out of the metal.
2. Field emission: By applying a very strong electric field (of the order 108Vm-1) to
a metal, electrons can be pulled out of the metal.
3. Photo-Electric effect: When light of suitable frequency made to fall on the metal
surface, electrons are emitted from the metal surface. These photo (light) generated
electrons are called photoelectrons.
Photoelectric Effect:
The phenomenon of photoelectric effect was discovered by Heinrich Hertz in
1887. In his experiments on electromagnetic waves, Hertz observed that electric
Page | 254
discharge (high voltage spark) in a cathode ray tube was facilitated when the cathode
was illuminated by ultraviolet light. He did not investigate this observation in detail.
In 1888 , Hallwachs followed up the discovery of Hertz , he observed using a leaf
electroscope that when exposed to ultraviolet radiations,
 A negatively charged zinc plate lost its charge and
 A positively charged zinc plate becomes more positively charged and
 A neutral zinc plate acquired a positive charge. From these observations he
concluded that negatively charged particles, identified to be electrons, are emitted
by the zinc plate during the process.
Now, it is known that when electromagnetic radiations of suitable frequency is
incident on metals, electrons are emitted.
“The phenomenon of emission of electrons from the surface of metals when
irradiated with light of suitable frequency is known as photoelectric effect”.
The electrons liberated in a photoelectric phenomenon are photoelectrons.
The current due to photoelectrons is called photoelectric current or photocurrent.
Photoelectric effect experiment:
 Figure shows a schematic view of the arrangement used for the experimental
study of the photoelectric effect
 It consists of an evacuated glass / quartz tube having a photosensitive plate C
and another metal plate.
 A monochromatic light from the source S of sufficiently short wavelength
passes through the window W and falls on the photosensitive plate C (emitter)
 A transparent quartz window is sealed on the glass tube, which permits
ultraviolet radiations to pass through it and irradiated the photosensitive plate C
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 The electrons are emitted by the plate C and are collected by the plate A
(collector) , by the electric field created by the battery
 The battery maintains the potential differences between the plates C and A, that
can be varied. The polarity of the plates C and A can be reversed by a
commutator. Thus the plate A can be maintained at a desired positive or
negative potential with respect to emitter C
 When the collector plate A is positive with respect to the emitter plate C, the
electrons are attracted to it. The emission of electron cause flow of electric
current in the circuit.
 The potential difference between the emitter and collector plates is measured by
a voltmeter (V) whereas the resulting photo current flowing in the circuit is
measured by a micro ammeter.
 As the intensity and frequency of the incident light is varied, the potential
difference V between the emitter C and the collector A gets varied.
The experiment is conducted to study the dependence of photo current on,
1. Intensity of incident radiation
2. Effect of potential
3. Effect of frequency of incident radiation on stopping potential.
Experimental study of photoelectric effect:
1. Effect of intensity of light on photocurrent :
 The collector A is maintained at a positive potential with respect to emitter C so
that electrons ejected from C are attracted toward collector A . Keeping the
frequency of incident radiation and the accelerating potential fixed, the intensity
of light is varied and the resulting photoelectric light is varied and the resulting
photoelectric current is measured each time .
 It is found that the photocurrent increases linearly with intensity of incident light
as shown below.
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The photocurrent is directly proportional to the number photoelectrons emitted per
second .
This implies that the number of photoelectrons emitted per second is directly
proportional to the intensity of incident radiation.
2. Effect of potential on photoelectric current:
 Keep the plate A at positive accelerating potential with respect to the plate C and
illuminate the Plate C with light of fixed frequency ʋ and foxed intensity I. vary
the positive potential plate A gradually photocurrent increases with the increase
in accelerating potential till a stage is reached when the photocurrent become
maximum and does not increase further with the increase in the accelerating
potential, This maximum value of the photoelectric current is called saturation
current. At this point all the electrons emitted by C are collected by the A
 Keep the plate A at negative with respect to C and gradually increase the
magnitude , it is seen that photocurrent decreases rapidly until it become zero for
a certain value of negative potential V0 on plate A .
“The minimum negative (retarding) potential V0 for which the photocurrent
stops or completely becomes zero is called stopping potential or cutoff”.
 At the stopping potential V0 when no electrons are emitted , the work done by
stopping potential on the fastest electron must be equal to its kinetic energy
1
Kmax = mV2Max = eVo
2
Where M , e, v max are the mass , charge and maximum velocity of the electrons.
We can repeat the experiment with higher intensity radiation (I3> I2>I1) and same
frequencies.
 Thus for the given frequency of the incident radiation, the stopping potential is
independent of its intensity.
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3. Effect of frequency of incident radiation on stopping potential:
 When the intensity of light radiation is varied, for different frequencies the
variation of photocurrent with collector plate potential is observed.
 The resulting variation if as shown in fig. We obtain different values of stopping
potential but the same value of saturation current for incident radiation.
 The energy of the emitted electrons depends on the frequency of the incident
radiations.
 The stopping potential is more negative for higher frequencies of incident
radiation.
 It implies that greater the frequency of incident light , greater is the maximum
kinetic energy of the photoelectrons.
Below graph shows that ,
1. The stopping potential Vo varies linearly with the frequency of incident radiation
for a given photosensitive material.
2. There exists a certain minimum cut-off frequency Vo for which the stopping
potential is Zero.
This implies,
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 The maximum kinetic energy of the photoelectrons varies linearly with the
frequency of incident radiations, but is independent of its intensity.
 For a frequency ʋ of incident radiation , lower than the cut-off frequency ʋo no
photoelectric emission is possible even if the intensity is large. This minimum cutoff frequency ʋo is called threshold frequency. It is different for different
material.
Experimental observations (Law of photoelectric effect):
1. For a given photosensitive material and frequency of incident radiation (above
the threshold frequency ), the photoelectric current is directly proportional to the
intensity of incident light.
2. For a given photosensitive material and frequency of incident radiation, saturation
current is found to be proportional to the intensity of incident radiation whereas
the stopping potential is independent of its intensity.
3. Foe a given photosensitive material, there exists a certain minimum cut-off
frequency of the incident radiation , called the threshold frequency , below which
no emission photoelectrons takes place. Above the threshold frequency ,the
stopping potential or equivalently the maximum kinetic energy of the emitted
photoelectrons increases linearly with the frequency of the incident radiation , but
is independent of its intensity.
4. The photoelectric emission is an instantaneous process without any apparent time
lag
(10-9 or less) even when the incident radiation is made exceedingly dim.
Photoelectric effect and wave theory of light:
The phenomena of interference, diffraction and polarization where explained on
the basis of wave nature of light whereas photoelectric effect explained on the basis
of particle nature of light.
Failure of wave theory to explain photoelectric effect:
According to wave theory light is an electromagnetic wave consisting of electric
and magnetic fields with continuous distribution of energy. This wave picture of light
couldn‟t explain the basic features of light as explained below;
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1. According to wave theory when a wavefronts of light strikes a metal surface the
free electron at the surface absorb the radiant energy continuously. Greater the
intensity of the radiation, greater is the amplitudes of electric and magnetic fields,
and greater is the density of the wave. Hence higher intensity should liberate
photoelectrons with greater kinetic energy. But this is contrary to the
experimental result that the maximum kinetic energy of the photoelectrons does
not depend on the intensity of incident radiation.
2. No matter what the frequency of incident radiation is, a light wave of sufficient
intensity (over a sufficient time) should be able to impart enough energy required
to eject the electrons from the metal surface. Thus the wave theory fails to explain
the existence of threshold frequency.
3. The energy of the light wave is smoothly and evenly distributed across its
advancing wavefront. Each electron intercepts an insignificantly small amount of
this energy and so it should require a finite time to escape from the metal surface.
But actually, the emission is almost instantaneous.
Einstein‟s photoelectric equation: Energy quantum of radiation
In 1905 Albert Einstein used Plank‟s Quantum of radiation and gave famous
equation for photoelectric effect. According to Plank‟s Quantum theory of radiation
the emission or absorption of radiant energy takes place in the form of discrete packet
of energy know as quanta of energy
Quanta of energy is given by E=hυ
Where, υ is the frequency of the radiation ,
h is Plank‟s constant.
Photon shows particle nature .photon has mass m =
E
C2
According to Einstein Photoelectric effect is the result of elastic collision (energy and
momentum is conserved ). If the quantum of energy absorbed exceeds theminimum
energyneeded for the electron to escape from the metal surface the electron is emitted
with maximum kinetic energy.
Kmax = E- ɸ0 = hυ - ɸ0
…………………(1)
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More tightly bound electrons will emerge with kinetic energy less the maximum
value. The intensity of light of a given frequency is determined by the number of
photoelectrons incident per second. The maximum kinetic energy of the emitted
photoelectrons is determined by the energy of each photon.
i.e.,
1
2
mV2Max = hυ - ɸ0
Eqn (1) is called Einstein‟s Photoelectric Equation.
Features of photoelectric effect on the basis of Einstein photoelectric emission :
1. photoelectric emission is the result of an elastic collision between a photon and an
electron. Since the collision lasts for a very short interval of time , there is no time
lag between incident of photon and the emission of photoelectron.
2. If υ > υ0 above the threshold frequency the maximum kinetic energy of the
electrons increases linearly with the frequency of the incident radiation.
K max =
1
2
mV 2max α υ
3. If υ < υ0i.e the frequency of incident radiation is less than the threshold
frequency , the kinetic energy of photoelectrons become negative . so
photoelectric emission does not occur below the threshold frequency.
4. The increase in the intensity means the increase in the number of photons
striking the metal surface per unit time. As each photon ejects only one electron ,
so the number of ejected photoelectrons increases with the increase in intensity of
incident radiation.
If V0be the stopping potential for frequency υ then kinetic energy is given by,
eV0 =hυ - ɸ0
where ɸ0 = hυ0 then,
eV0= hυ -hυ0
V0=
V0 =
hυ
𝑒
𝐡
𝒆
hυ 0
𝑒
υ -
ɸ𝟎
𝒆
This shows that V0 versus υ is a straight line and its independent of the nature of
the material.
The successful explanation of photoelectric effect using the hypothesis of light quanta
and the experimental determination of values h and ɸ0, led to the acceptance of
Einstein‟s picture of photo electric effect.
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Particle nature of light: The Photon
A definite value of energy as well as the momentum is a strong sign that the light
quantum can be associated with the particle. This particle was later named photon.
Characteristics of photon:
1. Photon travel with the speed of light
2. Rest mass of photon is zero.
3. Interaction of radiation with matter, radiations behaves as if it is made up of
particles called photons.
4. Each photon had energy E = hυ and momentum P =
hυ
𝐶
where C is speed of light.
5. Photons are electrically neutral and are not deflected by electric and magnetic
fields.
6. In a photon – particle collision the total energy and momentum are conserved.
7. All photons of light of a particular frequency υ or wavelength λ have same
energy
E = hυ =
hC
λ
and momentum P =
hυ
𝐶
=
h
λ
Wave nature of matter:
Concept of Matter Waves:
Wave theory of electromagnetic radiation explained the phenomenon of
interference , diffraction and polarization . On the other hand, quantum theory of
electromagnetic radiations explained photoelectric effect , Compton effect and black
body radiation.. Thus radiations have both dual nature i.e. Wave and Particle
(quantum nature)
A French Physicist LOUIS DE BROGLIE suggested that the particles like electrons,
protons, neutrons have also dual nature.
The waves associated with the moving material particles are known as de Broglie
waves or matter waves.
Expression for de Broglie wavelength:
de Broglie put forward the hypothesis that moving particles of matter should
display wave like properties under suitable conditions.
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Case1: For photon
According to quantum theory , the energy of a photon is given by,
E = hυ ……………..(1)
Energy of the photon is,
E = mc2 …………….(2)
From (1) and (2)
mc 2 = hυ
where c = υλ and υ =
C
λ
h
mc =
λ=
λ
h
mc
λ=
𝐡
𝑷
where, h = Planck‟s constant
p = mc = momentum of photon
Case 2: For Particle
If instead of a photon, we have a material particle of mass m moving with velocity
v then
λ=
𝐡
𝒎𝑽
which is de-broglie wavelength.
Case 3: For an Electron
Consider an electron of mass m,charge e accelerated from rest though a potential V
the kinetic K of the electron is equal to work done (eV).
K = eV
K=
𝟏
𝟐
mv2 =
P = 2𝑚𝐾 =
p2
2𝑚
2meV
The de Broglie wavelength λ of the electron is then,
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λ=
h
λ=
𝑝
𝒉
2m eV
Substituting the numerical values h, m ,e we get,
λ=
𝟏.𝟐𝟐𝟕
V
nm
NOTE:
In 1929, de Broglie was awarded the Nobel Prize in Physics for his discovery of
the wave nature of electronics.
Heisenberg‟s uncertainty principle:
 The matter-wave picture elegantly incorporated the Heisenberg‟s uncertainty
principle.
 According to the principle, it is not possible to measure both the position and
momentum of an electron at the same time exactly . There is always some
uncertainty (ΔX)
 In the specification of position position and some uncertainity (Δp) in the
specification of momentum . The product of ΔX and Δp is of the order of ħ i.e.
ΔX .Δp = ħ
 if an electron has definite momentum (Δp=0) , by the de Broglie relation , it has a
definite wavelength λ
 A wave of definite (single) wavelengths extends all over space
 By de Broglie relation , the momentum of the electron will also have a spread –an
uncertainity Δp
 By Bohr‟s probability interpretation this means that the electron is not localized
in any finite region of space . that is , its position uncertainity is infinite(ΔX∞)
which is consistent with uncertainity principle.
 It can be shown that the wave packet description together with de Broglie relation
and Bohr‟s probability interpretation reproduces the Heisenberg‟s uncertainity
principle exactly.
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Davisson and Germer Experiment:
The wave nature of electrons was first experimentally verified by C.J Davisson and
L.H Germer in 1927 and independently by J.P Thomson, in 1928 who observed
diffraction effects with beams of electrons scaterred by electrons.
Thomson and Davisson shared Nobel Prize in 1937 for their experimental dicovery
of diffraction of electrons by crystals.
Construction:
 It consists of an electron gun which comprises of a tungsten filament F coated
with the barium oxide and heated by a low voltage power supply (L.T or battery)
 Electrons are emitted by the filament are accelerated to a desired velocity by
applying suitable potential / voltage from a high voltage power supplu (H.T or
battery)
 They are made to pass through a cylinder with fine holes along its axis ,
producing a fine collimated beam T
 The beam is made to fall on the surface of a nickel craytal . The electrons are
scattered in all directions by the atoms of the craystal .
 The intensity of the electron beam , scattered in a given direction , is measured by
the electron detector(collector) . The detector can be moved on a circular scale
and is connected to a sensitive galvanometer , which records the current.
Working:
1. The deflection of the galvanometer is proportional to the intensity of the electron
beam entering the collector.
2. The apparatus is enclosed in an evacuated chamber.
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3. By moving the detector on the circular scale at different positions, the intensity
of the scattered electron beam is measured for different values of angle scattering
θ which is the angle between the incident and the scattered electron beams.
4. The variation of the intensity I of the scattered electrons with the angle of
scattering θ is obtained for different accelerating voltages.
5. The experiment was performed by varying the accelerating voltage from 44 V to
68V
6. It was observed that the strong peak appeared in the intensity I of the electrons
scattered for an accelerating voltage of 54V at a scattering angle θ= 500
7. The appearance of the peak in a particular direction is due to the constructive
interference of electrons scattered from different layers of the regularly spaced
atoms of the crystal .
8. From the electron diffraction method , the wavelength of matter waves was found
to be 0.165nm.
9. The de Broglie wavelength λ associated with electrons , for V = 54V is given by
λ=
λ =
λ =
h
𝑝
𝟏.𝟐𝟐𝟕
V
𝟏.𝟐𝟐𝟕
54
= 0.167 nm
This strikingly confirms the wave nature of electrons and the de Broglie relation.
*********************
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CHAPTER - 12
ATOMS
Introduction:
An atom is the smallest particle of an element. Dalton was the first person to
postulate that matter is made of atoms which are invisible. A theoretical explanation
for the structure of atom is called an atom model. In order to explain the structure of
atom, many models were proposed. The first atom model was proposed by J.J.
Thomson in 1898.
J.J.Thomson Model:
According to this model, the positive charge of the atom is uniformly distributed
throughout the volume of the atom and the negatively charged electrons are
embedded in it like seeds in a watermelon. This model was also called as “plum
pudding” model of the atom. The total positive charge in the atom was evenly and
symmetrically balanced by the negative charges of the electrons.
The Thomson‟s atom model failed to satisfy the results
of the experiments conducted later by Rutherford and others.
Drawbacks of Thomson‟s atom model:
1. According to Thomson‟s atom model, hydrogen should
produce only one spectral line of about 1300 Å, but
experiments reveal that hydrogen spectrum consists of
five different series with several lines in each series.
2. This model could not explain the large angle scattering
of α-particles by heavy atoms in Rutherford‟s α-scattering
experiment.
3. This model could not explain the presence of discrete spectral lines emitted by
hydrogen and other atoms.
Alpha-particle scattering and Rutherford‟s Nuclear model of atom:
In 1911,at the suggestion of Ernst Rutherford, H. Geiger and E. Marsden
performed the 𝛼-scattering experiment.
The experimental arrangement is as shown in the figure. The radioactive source
is kept in a lead cavity which emits α-particles. Alpha particles emitted by the
83Bi
source are collimated into a narrow beam with the help of a lead slit. The collimated
beam of a alpha particles are allowed to fall on a thin gold foil of thickness 2.1× 10-7
m. The α-particles scattered in different directions are observed through a rotatable
detector consisting of a Zinc sulphide ( ZnS ) screen and a microscope. When α214
Page | 267
particles from the gold foil strikes the ZnS screen it produce bright flashes on the
screen.These are observed through a microscope and number of scattered particles for
different angle of scattering θ are noted. “The angle of deviation 𝜽 of the 𝜶-particle
from its initial direction is called scattering angle”.
Observations:
The graph between number of α-particles
scattered and scattering angle is as shown
in the figure.The dots in the graph are due to
data obtained in experiment and solid curve
is the theoretical prediction based on the
assumption that atom has a small positively
charged nucleus at its centre.
It is observed from the graph that ,
1. Most of the α-particles pass straight
through the gold foil. That means they
suffer only small deflections.
2. Only about 0.14% of the α-particles scatter
by more than 1o.
3. Only about 1in 8000 of the incident α-particles are deflected by more than 90o.
Some of them have even retraced their path i.e., angle of deflection will be 180o (
large scale scattering) .
Conclusion:
Large scale scattering indicates that the α-particles are being repelled by a
massive positive charge concentrated in a small space. Hence the atom has a tiny
positively charged core called nucleus containing most of the mass of the atom.
Page | 268
Rutherford‟s atom model:
Some of the essential features of the Rutherford‟s nuclear atom model.
1. The atom has a small, positively charged nucleus. All the positive charge of the
atom and most of the mass of the atom are concentrated in the nucleus.
2. The dimensions of the nucleus are negligibly small as compared to the overall
size of the atom, so that most of the volume occupied by an atom is actually an
empty space. It is suggested that the size of the atom is about 10-10m and size of
the nucleus is about 10-15 m to 10-14 m.
3. The atomic nucleus is surrounded by certain number of electrons. As a whole
atom is electrically neutral, hence the total negative charge of the electrons
surrounding the nucleus is equal to the total positive charge on the nucleus.
4. The electrons in the atom revolve round the nucleus in various circular orbits. The
centripetal force required for the revolution of the electron is provided by the
electrostatic force of attraction between the electron and nucleus.
5. When α-particles comes near to a nucleus, the intense electric field there scatters
it through a large angle.
6. The mass of the nucleus is much larger than the mass of the α-particle. Hence it
can be assumed that the nucleus remains practically at rest during the scattering
process.
Alpha particle trajectory:
Alpha particles are a helium nucleus with charge +2e. When an α-particle is at a
distance r from the nucleus of charge +Ze, the electrostatic repulsion due to the
nucleus is
F=
1
Ze (2e)
4πε 0
r2
Consider the α-particle 1 having initial kinetic energy, K =
1
2
mv2 moving directly
towards a nucleus. As the α-particle approaches the nucleus, its kinetic energy
decreases and potential energy increases. When the α-particle is at a particular
distance D from the nucleus, its kinetic energy becomes zero.This distance D is the
distance of closest approach.
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Distance of closest approach (D):
It is the distance from the nucleus at which the kinetic energy of the approaching
particles becomes zero. At this distance the initial kinetic energy of the α-particle is
completely converted into potential energy. Further, because of electrostatic force of
repulsion the α-particle retraces its path (i.e. θ =180o).
The potential energy of the α-particle at a distance D from the nucleus is
U=
1
Ze (2e)
4πε 0
D
=
1
2Ze 2
4πε 0
D
Since loss in kinetic energy is equal to gain in potential energy,
K=
∴
D=
1
2Ze 2
4πε 0
D
𝟏
𝟐𝐙𝐞𝟐
𝟒𝛑𝛆𝟎
𝐊
D estimates the size of the nucleus.
Consider the α-particle 2 and 3
approaching the nucleus with a small impact parameter b. Because of electrostatic
force of repulsion, the α-particle is scattered through a large angle i.e., θ > 90o
Consider the α-particle 4 , 5, 6, …… approaching the nucleus with a large impact
parameter b. It is scattered through a smaller angle i.e., θ < 90o
The trajectory of a α-particle 3 and 4 will be hyperbola. The path adopted by αparticle in the electrostatic field of target nucleus depends upon the impact parameter.
Impact parameter (b):
It is the perpendicular distance of the initial velocity vector of the α-particle from
the centre of the nucleus.
Electron orbits of rutherford model:
The centripetal force Fc required to keep the electrons in their stable orbits is
provided by the electrostatic force of attraction Fe between electrons and the nucleus.
Consider a hydrogen atom nucleus having charge „+e‟ and electron of charge „-e‟
revolving around the nucleus in orbit of radius „r‟ with velocity „v‟.
We have,
Fc =
m v2
r
................... (1)
And the electrostatic force of attraction between the nucleus and electron is given by,
Fe =
1
(e×e)
4πε 0
r2
................. (2)
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For the stability of the atom both electrostatic force and centripetal force must be
balanced with each other i.e., Fc = Fe
mv 2
r
= 4πε
2
mv (r) =
∴
r=
e2
1
0
r2
e2
4πε 0
𝐞𝟐
𝟏
................. (3)
𝟒𝛑𝛆𝟎 𝐦𝐯 𝟐
This is the relation between the orbit radius and electron velocity.
The kinetic energy of the electron in hydrogen atom is, K =
mv 2 =
But,
K=
e2 1
1
2
mv2
4πε 0 r
𝐞𝟐 𝟏
𝟖𝛑𝛆𝟎 𝐫
This is the expression for kinetic energy of the electron in hydrogen atom.
The potential energy (U) of the electron is,
U = work done = F × displacement
U=
1
(e×e)
4πε 0
r2
U=
(-r)
− 𝐞𝟐 𝟏
𝟒𝛑𝛆𝟎 𝐫
The negative sign indicates that the electrostatic force is in the –r direction.
(i.e., displacement and force are oppositely directed)
The total energy of the electron is,
E=K+U
E=
e2 1
8πε 0 r
e2 1
- 4πε
0
r
Page | 271
E=
− 𝐞𝟐 𝟏
𝟖𝛑𝛆𝟎 𝐫
The total energy of the electron is negative. This implies that the electron is bound to
the nucleus and it will move in a closed, stable orbit.
Limitations of rutherford‟s atomic model:
It failed to account for the stability of atom. The electron revolving round the
nucleus has acceleration along the radius towards the centre. According to the
classical theory of electrodynamics, an accelerated charge radiates electromagnetic
energy. This implies that the electron revolving with acceleration with acceleration in
its orbit loses its energy continuously.
As a result, the electron will „spiral in‟ and finally collapses into the nucleus. With
this the very existence and stability of the atom becomes questionable.
Further if the electron loses its energy continuously, it should give rise to continuous
spectra. This is not in agreement with experimental observation because atomic
spectrum is a discrete line spectrum.
Atomic Spectrum:
When a gas or vapour in atomic state is excited at low pressure by passing
electric current through it, emits radiation. The spectrum of the radiation consists of
bright lines on a dark background with specific wavelengths. This kind of spectrum is
known as line emission spectrum.
It depends on the characteristic of the elements. The study of line emission
spectrum of a material can serve as a type “fingerprint” for identification of the
element of the gas. The line spectrum is due to the excited condition of atoms.
For examples, Hydrogen gives 4 distinct lines.
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When a white light is passed through some gases or vapours, the emergent
spectrum is found to be consists of a bright back ground crossed with one or more
dark lines. This type of spectrum is known as line absorption spectrum. These dark
lines correspond to those wavelengths which were found in the line emission
spectrum.
Spectral series:
Experiments show that, the frequencies of the light emitted by a particular
element exhibit some regular pattern. The spacing between lines within certain sets
decreases in a regular way. Each of these sets of lines is called a spectral series.
Hydrogen is the simplest atom and therefore, has the simplest spectrum. The first
spectral series in the visible region of the hydrogen spectrum was observed by
Balmer. This spectral series is called Balmer series.
In Balmer series, the line with largest wavelength 656.3 nm in the red region is
called 𝐇𝛂 line; the next line with the wavelength 486.1 nm in the blue-green is called
𝐇𝛃 line; the third line with wavelength 434.1 nm in the blue is called 𝐇𝛄 line; the
fourth line with wavelength 410.2 nm in the violet is called 𝐇𝛅 line; and so on. As the
wavelength decreases, the lines appear closer together and are weaker in intensity.
The empirical formula for observed wavelengths in Balmer series is,
𝟏
=R
𝛌
𝟏
𝟐𝟐
−
𝟏
𝐧𝟐
and this is called Balmer formula
Where 𝛌 is the wavelength, R is called Rydberg constant and n= 3,4,5,6 etc.
Page | 273
For Hα line, n=3. For Hβ line, n=4. For Hγ line, n=5. For Hδ line, n=6.
The value of Rydberg constant obtained is R = 1.097 × 107 m-1.
Other series of spectral lines for hydrogen were discovered later by Lyman, Paschen,
Brackett and Pfund series.
The empirical formulas for these series are as following;
Lyman series:
Balmer series:
Paschen series:
Brackett series:
Pfund series:
1
λ
=R
1
λ
1
λ
1
λ
1
λ
=R
=R
=R
=R
1
12
1
22
1
32
1
42
1
52
−
−
−
−
−
1
n2
1
n2
1
n2
1
n2
1
n2
where n= 2, 3, 4…
where n= 3, 4, 5, ...
where n= 4, 5, 6...
where n= 5, 6, 7...
where n= 6, 7, 8...
Bohr model of the hydrogen atom :
Neils Henrik David Bohr applied Planck‟s quantum theory of radiation to
Rutherford‟s atomic model. He combined both classical as well as quantum concepts
to form Bohr‟s theory. He made few assumptions known as Postulates of Bohr‟s
atomic model while proposing a new model for the atom.
Postulates of Bohr‟s Atomic Model:
First postulate:
“An electron in an atom could revolve round the nucleus in a certain stable
orbit without the emission of radiate energy”.
According to this postulate, each atom can exist in certain definite stable states
and each possible state has definite total energy. These states are called the stationary
states and the orbits of electrons in these states are called stationary orbits.
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Second postulate:
“The electron revolves round the nucleus only in those orbits for which the
angular momentum is an integral multiple of
𝒉
𝟐𝝅
, where „h‟ is the Planck‟s
constant”.
The angular momentum L of the electrons revolving round the nucleus in
stationary states is quantised.
i.e., L = n
h
2π
, where n = 1, 2, 3... is an integer called principal quantum number
Third postulate:
“Whenever an electron jumps from a higher non-radiating orbit to a orbit of
lower energy, it emits a radiation (photon). The energy of the photon liberated will
be equal to the difference between the energy of the electron in the two orbits”.
If E1 and E2 are the energy states in two orbits, then the energy of the photon
liberated will be ,
h𝜈 = E2 – E1
Radii of stationary orbits:
Consider an electron of mass m and charge (-e) revolving round a nucleus of
charge +Ze in a circular orbit of radius r with an orbital velocity v. The electrostatic
force of attraction between nucleus and the electron is,
1 Ze (e)
Fe =
4πεo r2
=
1
Ze 2
4πε 0 r2
The centripetal force acting on the electron, Fc =
.............. (1)
m v2
r
.............. (2)
For stability of the atom both electrostatic force and centripetal force must be
balanced with each other i.e. Fc = Fe
mv 2
r
2
1
= 4πε
mv (r) =
Ze 2
2
0 r
Ze 2
4πε 0
............... (3)
From Bohr‟s postulate, the angular momentum of the electron for the nth orbit is
L = mv r =
nh
2π
, where n = 1, 2, 3...
Page | 275
𝐧𝐡
∴ v = 𝟐𝛑𝐦𝐫
Substituting v in eqn 3 we get,
2
Ze 2
n2 h
m
r=
4π2 m2 r2
4πε 0
∴
𝟐
r= 𝐧
𝟐
𝐡 𝛆𝐨
𝛑𝐦𝐙𝐞𝟐
................. (4)
This equation gives the radius of the nth stationary orbit. The radii of the orbits are
proportional to square of the natural numbers.
For hydrogen atom Z = 1
∴
𝟐
r= 𝐧
𝟐
𝐡 𝛆𝐨
𝛑𝐦𝐞𝟐
Bohr radius:
“The radius of the inner most orbit (first orbit n=1) of the hydrogen is called
Bohr radius”.
2
r=
h εo
= 5.29 × 10-11 m = 0.529 Å.
πme2
Orbital speed of electron in a stationary orbit:
Consider an electron of mass m and charge (-e) revolving round a nucleus of
charge +Ze in a circular orbit of radius r with an orbital velocity v. The electrostatic
force of attraction between nucleus and electron is,
1 Ze (e)
Fe =
4πεo r2
=
1
Ze 2
.............. (1)
4πε 0 r2
m v2
The centripetal force acting on the electron, Fc =
r
.............. (2)
For stability of the atom both electrostatic force and centripetal force must be
balanced with each other i.e. Fc = Fe
Page | 276
m v2
r
Ze 2
1
= 4πε
2
mv (r) =
2
0 r
Ze 2
............... (3)
4πε 0
From Bohr‟s postulate, the angular momentum of the electron in the n th stationary
orbit is
L=mvr=
nh
........... (4)
2π
where n = 1,2,3,.....
Dividing eqn 3 by eqn 4 we get,
V=
𝟏
𝐙𝐞𝟐
𝟐𝛆𝟎 𝐡
𝐧
................ (5)
Eqn 5 gives the orbital velocity of an electron in the nth stationary orbit of an atom of
atomic number Z.
For hydrogen Z =1, the orbital velocity is
V=
𝟏
𝐧
𝐞𝟐
𝟐𝛆𝟎 𝐡
............ (6)
The orbital velocity of electron is inversely proportional to the orbit number. Thus
electron has higher velocity in the inner orbits.
Total energy of electron in a stationary orbit:
Consider an electron of mass m and charge (-e) revolving round a nucleus of
charge +Ze in a circular orbit of radius r with an orbital velocity v. By virtue of its
motion electron has kinetic energy (K). Here electron is in the electric field of the
nucleus of charge +Ze, therefore it has a potential energy (U). Thus the total energy
of the electron will be the sum of the potential and kinetic energies.
The potential energy U of the electron at a distance r from the nucleus is,
U=
− 𝟏 𝐙𝐞𝟐
𝟒𝛑𝛆𝟎
𝐫
............ (1)
Page | 277
1
The kinetic energy of the electron is K =
2
mv2 .............. (2)
m v2
The centripetal force acting on the electron, Fc =
r
The electrostatic force of attraction between nucleus and the electron is,
1 Ze (e)
Fe =
4πεo r2
=
1
Ze 2
4πε 0 r2
For stability of the atom both electrostatic force and centripetal force must be
balanced with each other i.e. Fc = Fe
mv 2
r
i.e.,
2
Ze 2
1
= 4πε
mv =
2
0 r
Ze 2 1
............... (3)
4πε 0 r
Substituting eqn 3 in eqn 4, we get
K=
1
Ze 2 1
2
4πε 0 r
K=
𝐙𝐞𝟐
𝟏
............... (4)
𝟒𝛑𝛆𝟎 𝟐𝐫
From eqn 1 and 4 , the total energy of an electron in a stationary orbit of radius r is
given by,
En = K + U
En =
En =
1 Ze 2
4πε0 2r
1 Ze 2
4πε0 r
En =
1
2
Ze 2
4πε0
r
−1
−𝟏 𝐙𝐞𝟐
𝟒𝛑𝛆𝟎 𝟐𝐫
1
........... (5)
w.k.t, the expression for radius of the orbit is, r = n2
2
h εo
π m Ze2
Page | 278
On substituting the value of r in eqn 5
En =
−1 Ze 2
π m Ze2
4πε 0 2
En = −
n2h
2
εo
𝟏
𝐦𝐙𝟐 𝐞𝟒
𝐧𝟐
𝟖𝐡 𝛆𝐨𝟐
𝟐
............. (6)
This expression gives the total energy of electron in the nth stationary orbit of
hydrogen- like atom.
For hydrogen atom Z=1,
∴
𝐦 𝐞𝟒
𝟏
En = −
𝟐
𝐧𝟐
𝟖 𝐡 𝛆𝐨𝟐
𝟏
En = −
𝐧𝟐
(13.6 eV)
Frequency and wave number of emitted radiation in hydrogen:
According to Bohr‟s postulate, when electron in an atom jumps from higher
energy state to lower energy state, the energy difference is given out in the form of
radiation (photon). Let ν is the frequency of the quantum of energy emitted when
electron makes a transition from a higher orbit n2 to a lower orbit n1. If E1 and E2 are
the energies of the electron in the orbits n1 and n2 respectively of hydrogen atom.
“The wave number of a spectral line is defined as the reciprocal of its wavelength”.
i.e., 𝛎 =
𝟏
𝛌
..........(1)
It represents the number of waves present in one metre length. The unit of wave
number is m-1.
mZ2e4
mZ2e4
w.k.t
h ν = E2 – E1 = −
- −
2
2
8εo 2 h n 2
8εo 2 h n 1
hν=
mZ 2 e 4
1
8ε o 2 h 2
n1
−
1
n2
Page | 279
If λ is the wavelength of the emitted radiation and C is the velocity of light,
ν=
Then,
∴
h
C
λ
1
λ
C
λ
=
=
mZ 2 e 4
1
8ε o 2 h 2
n1
mZ 2 e 4
1
8ε o 2 h 3 C
n1
−
−
1
n2
1
........... (2)
n2
Eqn (1) and (2), wave number of a spectral line is,
ν=
Substitute R =
me 4
8ε o 2 h 3 C
1
λ
=
mZ 2 e 4
8ε o 2 h 3
1
C
n1
−
1
n2
= 1.097 ×107 , is Rydberg‟s constant.
ν = R Z2
1
n1
−
1
n2
...........(3)
For hydrogen spectrum, Z=1. Therefore, wave number of spectral lines in hydrogen
spectrum.
𝛎=R
𝟏
𝐧𝟏
−
𝟏
𝐧𝟐
......... (4)
This is called Balmer‟s formula.
Spectral series of hydrogen:
1. Lyman series: It consists of all spectral lines emitted due to the transition of the
electron from higher orbits to the first orbit. i.e., from n2 = 2,3,4,5,..... to n1 = 1.
These spectral lines lie in the ultraviolet region.
2. Balmer series: It consists of all spectral lines emitted due to the transition of the
electron from higher orbits to the second orbit. i.e., from n2 = 3,4,5,..... to n1 = 2.
These spectral lines lie in the visible region.The different visible lines in Balmer
series are represented as Hα , Hβ , Hγ , and Hδ .
Page | 280
3. Paschen series: Here spectral lines are obtained because of the transition of the
electron from the higher orbits to the third orbit. i.e., from n2 = 4,5,6,..... to n1 = 3.
These spectral lines lie in the infrared region.
4. Brackett series: Here spectral lines are obtained because of the transition of the
electron from the higher orbits to the fourth orbit. i.e., from n2 = 5,6,7..... to n1 = 4.
These spectral lines lie in the infrared region.
5. Pfund series: Here spectral lines are obtained because of the transition of the
electron from the higher orbits to the fifth orbit. i.e., from n2 = 6,7,8,..... to n1 = 5.
These spectral lines lie in the infrared region.
Energy Level Diagram:
For hydrogen atom Z=1, The energy of electron in the nth stationary orbit of
hydrogen atom.
En = −
𝟏
𝐧𝟐
𝐦𝐞𝟒
𝟐
𝟖𝐡 𝛆𝐨 𝟐
=−
𝟏
𝐧𝟐
(13.6 eV)
Page | 281
Using this relation the energy of electron in all orbits of hydrogen atom can be
found.The energy levels are represented as shown in the diagram
Here discrete energy levels are represented by horizontal lines. The electron
transitions are represented by vertical lines. Lowest energy level corresponds to n1 = 1
and as the orbit number increases the energy of the electron increases but the energy
difference between orbits decreases.
Ionisation energy:
“The minimum energy required to remove the electron from the ground state of the
atom to infinite distance from the nucleus is called ionisation energy”.
The minimum energy required, E= E∞ - E1
Thus, the ionisation energy of hydrogen atom is 13.6 eV.
Excitation energy:
“The energy required to raise an atom from normal state into an excited state is
called excitation energy”.
First excitation energy of hydrogen atom= E2 – E1 = 10.2 eV
Limitations of Bohr‟s atomic model:
Bohr‟s theory was successful in explaining the line spectrum of hydrogen atom. It
also explains the spectrum due to hydrogen- like atoms.
Some of the limitations are,
1. It cannot explain the spectrum due to complex atoms with more than one electron
revolving round the nucleus.
2. It cannot explain the observed fine structure of spectral lines.
3. It cannot give any information about the intensities of each spectral line.
4. It cannot explain the filling up of electrons in various shells and subshells.
5. It cannot explain the phenomenon of splitting up of spectral lines in the presence
of a magnetic field and in the presence of electric field.
6. This model could not account for the wave nature of electrons.
De-Broglies explanation of Bohr‟s postulate:
According to the postulate Bohr‟s atom model, the angular momentum of
electron revolving around the nucleus is quantised and it has value equal to integral
multiple of
h
2π
Page | 282
i.e.,
L=
nh
2π
where n = 1,2,3,.......
According to de –Broglie, a particle of mass „m‟ moving with speed „v‟ is associated
with a wave of wavelength λ given by, λ =
h
mv
According to de –Broglie, orbiting electron around the nucleus is associated with a
standing wave. Because of the circular path, the electron wave must be a circular
standing wave closing upon itself. If the electron wave does not close upon itself,
destructive interference takes place and the amplitude decreases to zero.
For an electron moving in the nth circular orbit of radius r with speed v,
circumference of the orbit = n λ
2π r = n λ
For de- Broglie wave, the wavelength is,
λ=
∴ 2π r = n
h
mv
h
mv
or
mv r = n
h
2π
i.e., angular momentum is,
L=mvr=n
h
2π
This is the quantum condition proposed by Bohr for angular momentum of electron.
*************************
Page | 283
CHAPTER –13
NUCLEI
Atomic masses and composition of nucleus:
The mass of an atom is very small, compared to a kilogram. Kilogram is not a
very convenient unit to measure such small quantities. Therefore, a different mass
unit is used for expressing atomic masses.
1. This unit is the atomic mass unit (u), defined as 1/12th of the mass of the carbon
(12C)atom. According to this definition
1u =
mass of one 12.𝐶 atom
12
=
= 1.992647 X 10 −26 kg
12
1u = 1.660539 ×10−27 kg
2. Nucleus : The atomic nucleus was discovered in 1911 by Rutherford. According
to Rutherford the central part of the atom which contains entire positive charge .
3. Nucleons: Nucleus consist of Protons and Neutrons .These two constituents of
nucleus is called Nucleons.
NOTE:
In 1935 James Chadwick awarded Nobel Prize for discovery of Neutron
A proton is a positively charged particle and has charge equal to that of an electron.
Mass of proton (mP) = 1.6726 X 10-27 kg = 1.007825a.m.u
A neutron is a neutral particle.
Mass of neutron (mN) = 1.6750 x 10 -27kg = 1.008665a.m.u
4. Nuclides: The nucleus consisting of single Proton is called nuclides. Ex: The
hydrogen nucleus consist of a single proton alone.
The composition of a nucleus can be described using the following
terms and symbols:
1. The number of protons present inside the nucleus of an atom gives the Atomic
Number of an element and is represented is Z .
2. The number of neutrons N . Given by (N =A-Z)
Page | 284
3. The Mass Number of an element is the total number of protons and neutrons
present inside the nucleus. Represented by A. ( A = Z+N).
𝑨
4. Nucleus is represented by 𝒁𝑿 or
element.
Ex : 6 C 10 , 2 He 4
z
X A Where X is chemical symbol of the
5. Isotopes: Isotopes of an element are the atoms of the elements which have the
same atomic number but different mass number.
Ex: Isotopes of carbon ⟹ 6 C 10 , 6 C 11 , 6 C 12 , 6 C 13,6 C 14
Isotopes of oxygen ⟹ 8 O 16 , 8O 17 , 8O 18
1
2
3
Isotopes of hydrogen ⟹
1 H , 1H , 1H
6. Isobars: The atoms of different elements having same mass number but different
atomic numbers are called isobars.
Ex: 1. 11Na 23 and 12Mg23
2. 20Ca 40 and 18Ar 40
7. Isotones: are the nuclei which have the same number of neutrons.
Ex: 1. 8 O16 and 6 C14
2. 1H 3 and 2He 4
Properties Of Nucleus:
1. Nuclear mass: The total mass of nucleus ( i.e. protons and neutrons) present in a
nucleus is called nuclear mass.
Nuclear mass = mass of protons + mass of neutron
M = Z m P + (A-Z) mN
2. Nuclear size: The nucleus is believed to be in spherical .Therefore ,its size is
usually given in terms of radius .It has been found experimentally that the volume
of a nucleus is directly proportional to its mass number A.
R = R0 A1/3.
Where R0 is a constant and its value is 1.2 X 10-15m
3. Nuclear density: The ratio of the mass of the nucleus and its volume is called
nuclear density.
mass of nucleus
mA
Nuclear density =
=
3
Volume of nucleus
Nuclear density (ρ ) =
(4/3)𝜋𝑅0 𝐴
3m
4𝜋𝑅03
Page | 285
Mass-Energy and Nuclear binding energy:
Mass- Energy relation:
According to Einstein‟s theory of relativity Mass and Energy are inter –
convertible.
Mass- Energy equivalence relation is,
E= m C2
where c is the velocity of light in vacuum.
Experimental verification of the Einstein‟s mass –energy relation has been achieved
in the study of nuclear reactions and so in a reaction the conservation law of energy
stares that the initial energy and the final energy are equal provided the energy
associated with the mass is also included.
Mass Defect:
The difference between the sum of masses of nucleons (protons and neutrons)
in the nucleus and the actual mass of the nucleus is called Mass Defect. It is
Represented by ΔM.
Mass defect ΔM = [ Z m P + (A-Z) mN ] – M nuc
Nuclear Binding Energy:
“The total energy required to liberate all the nucleons from the nucleus is
called Nuclear binding energy” and it is represented by Eb.
Binding energy can also be defined as the energy with which the nucleons are
held together within the nucleus.
The mass defect ΔM during the formation of the nucleus is converted into
binding energy of the nucleus according to the relation E=ΔmC2.
E =[ ( Z m P + (A-Z) mN ) – M nuc] C2
The binding energy per nucleon (B.E / Nucleon) is the average energy required
to extract one nucleon from the nucleus to infinite distance.
𝐵.𝐸
𝑛𝑢𝑐𝑙𝑒𝑜𝑛
=
𝑇𝑜𝑡𝑎𝑙 𝑏𝑖𝑛𝑑𝑖𝑛𝑔 𝑒𝑛𝑒𝑟𝑔𝑦
𝑚𝑎𝑠𝑠 𝑛𝑢𝑚𝑏𝑒𝑟
=
𝐸𝑏
𝐴
The greater the binding energy per nucleon, the more stable is the nucleus and vice
versa.
Page | 286
Binding energy curve:
“The curve between binding energy per nucleon and mass number A for
varies nuclei is called binding energy curve”.
The Binding energy curve reveals the following:
1. The Binding energy per nucleon is almost constant for nuclei of middle mass
number (A in between 30 and 170)
2. The Binding energy per nucleon is small for both light nuclei (A<30) and heavy
nuclie (A >170)
3. The nuclei with A = 56 has large binding energy nucleon about 8.75Mev.
We can draw some conclusion from these two observation:
1. The binding energy curve indicates the stability of the nucleus. The greater the
binding energy per nucleon of a nucleus , the more stable the nucleus and vice
versa.
2. When a heavy nucleus split up into lighter nuclei ,then binding energy per
nucleon of lighter nuclei is more than that of the heavy nucleus. The large
amount of energy is liberated and this process is called “ nuclear fission”
When two light nuclei combine to form a heavier nucleus,the binding energy per
nucleon will increase . As a result, energy will be released .This process is called
“nuclear fusion”
Nuclear Forces:
The binding per nucleon for all most elements is about 8Mev. Therefore to
bind the nucleons together, there must be strong attractive forces. These forces of
attraction must be strong enough to overcome the repulsion between the positively
Page | 287
charged protons and to bind both protons and neutrons into a tiny nuclear volume.
These forces are called nuclear forces.
“The strong forces of attraction which firmly hold the nucleons inside the nucleus
are called nuclear forces”.
The following are the some of the characteristics properties of nuclear forces.
1. Nuclear forces are strong attractive forces:
It is greater than coulomb forces between charges.
2. Nuclear forces are charge independent:
The forces between P and N or N and N or P and P is approximately the same
3. Nuclear forces are short range forces i.e they act only over a short range of
distance:
It has been found that forces are quite strong when the distance between the
nucleons is of the order of 10-15m but becomes zero at an inter –nucleon distance
of 10-14 m. When the distance between two nucleons is more than 10 -15 m, the
nuclear forces is negligible.
As the inter nucleon distance decreases, the nuclear forces increases rapidly.
4. Nuclear forces have saturation character:
a nucleon can interact only with limited number of nucleons in its neighborhoods.
5. Nuclear forces are non -central forces:
The force between two nucleons does not act along the line joining their centers.
This shows that the distribution of nucleons in the nucleus is not spherically
symmetric.
6. Nuclear forces are spin dependent:
Nuclear forces depend on the spin of the nuclei.
Page | 288
RADIOACTIVITY:
Radioactivity was discovered by Henry Becquerel in 1896. “The process of
spontaneous disintegration of the nuclei of heavy elements with the emission of
certain types of radiation is called radioactivity”.
There are three types of radioactive decay occur in nature,
1. α – decay
2. β –decay
3. ϒ – decay
1. α – decay (alpha decay):
In which a helium nucleus 2He 4(α particle) is emitted.
Ex:
92
U 238
234
90Th
+ 2He 4 .
In α -decay A decreases by 4 and Z decreases by 2 and it can be represented as,
z
XA
Z-2
Y A-4 + 2He 4
Where X is parent nucleus and Y is daughter nucleus.
Disintegration energy or Q value:
It is the difference between the initial mass energy and the total mass energy of
the decay products.
Q = (Mx – My – MHe) C2
Where Mx , My ,MHe are the masses of the parent nucleus ,daughter nucleus and α
particle. The energy Q is also the net kinetic energy gained in the process.
Properties of α – decay:
1. They are positively charged particle.
2. They produce heating effect.
3. They have very small penetrating power.
2. β –decay (Beta decay):
In which electrons and positrons (particle with the same mass as electrons but
with the opposite charge ) are emitted.
Page | 289
In a beta decay ,a nucleus spontaneously emits an electron (β- decay) or a positron (β+
decay) .
Ex: β- decay
Ex: β+ decay
15
11
-
P 32
32
16S
+e- +ʋ
Na 22
10Ne
22
+e+ +ʋ
The symbols ν- and ν represent antineutrino and neutrino, respectively.
Both are neutral particles, with very little or no mass. These particles are emitted
from the nucleus along with the electron or positron during the decay process.
Neutrinos interact only very weakly with matter. They can even penetrate the earth
without being absorbed.
In β- decay, a neutron transforms into a proton within the nucleus according to
n → p + e– + ν –
In β+ decay a proton transforms into neutron (inside the nucleus)
p → n + e+ + ν
In both β- decay and β+ decay mass no A remains unchanged. In β- decay, the
atomic no Z of the nucleus goes up by 1. In β+decay , the atomic no Z of the nucleus
goes down by 1.
Properties of β – decay:
1. They are negatively charged particle.
2. They have 100 times more penetrating power than alpha –rays.
3. ϒ- decay (gamma decay):
In which high energy photons are emitted. There are energy levels in a nucleus,
just like there are energy levels in atoms. When a nucleus is in an excited state, it can
make a transition to a lower energy state by the emission of electromagnetic
radiation. As the energy differences between levels in a nucleus are of the order of
MeV, the photons emitted by the nuclei have MeV energies and are called gamma
rays.
We represent ϒ- decay as,
(ZXA)*
(ZXA) + ϒ
* indicates a nucleus in an excited state.
Properties of ϒ- decay:
1. They are unchanged.
2. They have very high penetrating power.
Page | 290
3. They travel with velocity of light.
4. They produce photoelectric effect when it incident on certain substances.
Law of Radioactivity Decay:
“The rate of disintegration of atoms at any time is directly proportional to the
number of radioactive atoms present at that instant is called radioactive decay
law”.
Let No be the number of atoms present initially in the radioactive sample.
After time t, the number of radioactive atoms left undecayed in the sample is N.
According to decay law , the rate of disintegration id directly proportional to
the number of radioactive atoms present at that instant of disintegration i.e
𝑑𝑁
𝑑𝑡
αN
𝑑𝑁
= -λ N
𝑑𝑡
………………(1)
Where λ is a constant of proportionality called Decay constant or disintegration
constant .
The negative sign shows that N is decreasing with time.
𝑑𝑁
𝑁
= -λ dt
Integrating both sides , we get,
loge N = - λ t + C ……………(2)
where C is a constant of integration whose value can be found from the initial
conditions .
At t= 0 , N= No .
Substitute in eqn(2), we get C= loge No
Substituting the value of C in eqn (2)
loge N = - λ t + loge No
loge
𝑁
No
𝑁
No
=-λt
= e- λ t
N = No e- λ t …………….(3)
Page | 291
N=0 only at t= ∞
Therefore, a radioactive substances will never disintegrate completely .It is
because the radioactive decay obeys exponential law as shown in the figure,
Half-Life of radioactive substances:
“The half- life of radioactive substances is the time required for one half of
the radioactive substances to disintegrate”.
It is denoted by T1/2
𝟎.𝟔𝟗𝟑
T1/2 =
𝛌
This shows that half-life of a radioactive substances is inversely proportional to
its decay constant.
Mean life or average life of radioactive substances:
All the atoms in a radioactive substances do not disintegrate at the same time. Some
atoms disintegrate earlier whereas others disintegrate after a long time.
“The mean life of a radioactive substances is the sum of the lives of all the
atoms of the substances divided by the total number of atoms present initially in the
substance” .
Let No be the total number of atoms present initially (at t=0) in a radioactive
substances . After time t , the number radioactive atoms present in the sample is N .
Total life of dN atoms = number of atoms X life of each atom
= dN t
Page | 292
Total life of all atoms =
No
0
𝑡𝑑𝑁
As even in a small mass, No is extremely high, it may be taken as infinity .
Total life of all atoms =
∞
0
𝑡𝑑𝑁
Average life of radioactive substances
Ʈ=
We know that,
𝑑𝑁
𝑑𝑡
Total life of all atoms
𝑁0
=
∞
0
𝑡𝑑𝑁
𝑁0
= -λ N ( dN = - λ Ndt )
Ʈ=
Ʈ=
∞
0
𝑡 − λ Ndt
𝑁0
∞
0
𝑡 𝑁0 𝑒 −𝜆𝑡
𝑁0
dt
w.k.t N = No e- λ t
= -λ
Ʈ= λ
∞
0
𝑡 e- λ t dt
∞
𝑡 e- λ t
0
dt
Negative sign is omitted which indicates the decrease in the number of atoms
with time.
Integrating by parts,
𝟏
Ʈ=
𝛌
Thus mean life of radioactive substances is the reciprocal of decay constant.
NOTE:
Relationship between half- life and mean life
𝟎.𝟔𝟗𝟑
Half-life T1/2 =
𝛌
Mean life Ʈ =
𝟏
𝛌
T1/2 = 0.693 Ʈ
Page | 293
Activity:
“The rate of disintegration in a radioactive substances is known as activity".
𝑑𝑁
Activity of radioactive substance A = (minus sign shows that the activity
𝑑𝑡
decreases with time)
𝑑𝑁
Activity (A) = = λN
𝑑𝑡
where A = λ No
Activity A = A0 e- λ t
The radioactivity of substances is expressed in the following units:
1. becquerel (Bq) : One becquerel is one disintegration per second.
2. curie (ci): 1curie =3.7X1010Bq
3. rutherford (Rd): 1rutherford =106Bq
Nuclear Fission:
“The process of splitting of a heavy nucleus into two nuclei of intermediate
masses with the release of energy is called as nuclear fission”.
NOTE:
The huge amount of energy is released because the original mass of the nucleus is
greater than the sum of the masses of the products produced after fission .This
difference in the mass is converted into energy according to Einstein‟s equation
E = ΔmC2.
When Uranium nucleus ( 92U 235) is bombarded by a slow neutron (called
thermal neutron) the 92U 235 nucleus splits into two nuclei with the release of huge
amount of energy .
Page | 294
Ex:
92U
92U
235
235
92U
+ 0n1
+ 0n1
235
+ 0n1
(92U 236)
54Xe
51Sb
140
133
56Ba
141
+36Kr92 + 30n1 +200Mev
+38Sr94 + 20n1 +Q
+41Nb99 + 40n1 +Q
In this process, apart from the energy,on an average 2 and more neutrons are
released. The release of neutron can cause further nuclear fission reaction.
For fission to be self –sustaining, the number of emitted fission neutron should be
more than the incident neutrons .Under such conditions, the fission neutrons keep
increasing , thus maintain the chain reaction . This possibility is determined by a
quantity called neutron multiplication factor, denoted by k.
Chain Reaction:
“A chain reaction is a self -propagating process. During fusion process 2 or 3
neutrons released which can be further used to cause fission process”.
There are two types of chain reactions.
1. Controlled chain reaction
2. Uncontrolled chain reaction
Nuclear Reactor based on thermal neutron fission:
Page | 295
Nuclear reactor is a device in which controlled fission reaction takes place. The
essential parts of a nuclear reactor are,
1. Nuclear fuel
2. Moderator
3. Control rods
4. Coolant and
5. Safety system
 Nuclear fuel is a fissionable fuel embedded in graphite block. The fuel used are
U235 (Uranium) , Th 232(Thorium ) and Plutonium etc..
 Graphite acts as moderator which slowdowns the neutrons .Heavy water and
Beryllium can also He used as moderator .
 Fission reaction is controlled by absorbing neutrons . This is achieved by using
neutron absorbing controls rods . Cadmium , boron , carbon , cobalt , silver etc
rods are used for this purpose .
 The entire system is enclosed in a thick concrete wall to prevent escape of
harmful radiations . The leakage of neutrons can be reduced by using reflectors
inside the surface of the reactor .
 The energy will be mainly in the form of heat which is taken away by the
coolant circulating in the reactor . This heat is used to convert water into high
pressure steam which runs the turbine. The entire reaction is initiated by injection
slow neutrons.
Uses.
1 .Nuclear reactors are used to generate electric power.
2. Specialized nuclear reactors can produce radioactive isotopes for their use in
medical science, agriculture & industry.
3. Nuclear reactors are also used to produce high-velocity beams of neutrons for
their use in nuclear physics.
Nuclear Fusion (Energy Generation in stars):
Page | 296
“The process of combining two light nuclei to form a heavy nucleus with the
release of large amount of energy is known as nuclear fusion”.
When 2 light nuclei are combined to form a heavy nucleus, the mass of the
product nucleus is slightly less than the sum of the masses of the light nuclei fusing
together. As a result, there occurs mass difference in nuclear fusion. This mass
difference results in the release of a very large amount of energy according to the
relation E=(Δm)c2.
Example:
1. When 2 nuclei of heavy hydrogen or deuterium (1H2) are combined the following
reaction is possible.
2
2
3
1
1H +1H
1H +1H +4.0 MeV
2. The nucleus of tritium (1H3) so formed can again fuse with a deuterium nucleus
(1H2) to give the following reaction:
3
2
1H +1H
+0n1+17.6MeV
Energy released per fusion is less than the energy released per fission.
2He
4
Stellar Energy:
The nuclear fusion achieved by raising the temperature of the substance so that
the nuclei have enough kinetic energy to fuse is called thermonuclear fusion.
The energy obtained from sun and other stars is called Stellar energy. The energy
generation in the sun and stars take place by thermonuclear fusion.
The fusion reaction in the sun is a multi-steps process in which hydrogen is
burned into helium, hydrogen being the „fuel‟ and helium the „ashes‟.
There are 2possible cycles namely,
1. Carbon-nitrogen cycle and
2. proton-proton cycle.
1. Carbon nitrogen cycle:
In this cycle carbon atoms acts as a catalyst and 4hydrogen nuclei fuse to form
a helium nucleus with the release of an enormous amount of energy.
2. Proton-proton cycle:
The proton-proton cycle by the thermonuclear reaction occurs is represented by
the following sets of reaction:
Page | 297
1
1
1 H +1 H
2
0
1H +1e +0.42MeV
2[1H1+1H1]
2[1H2+1e0]+E
2[1H2+1H1]
2[2He3]+E
Summing up the above reaction;
41H1
2He
4
+ 21e0+E
Controlled nuclear thermonuclear fusion:
Nuclear fusion is an uncontrolled process. The basic problem in achieving
controlled nuclear fusion is that so solid container can withstand high temperature
involved in fusion.
Radiation hazards:
Everyone knows that radioactivity can be dangerous to health. The cells of the
body may undergo dangerous physical and chemical changes as a results of exposure
to radioactivity.
India‟s atomic energy programme:
The atomic energy programme in India was launched around 1950 under the
leadership of Homi J.Bhabha (1909-1966). The following achievements have been
made in this field:
1. First nuclear reactor named Apsara in India went critical on August 4, 1956. It
used enriched uranium as fuel and water as moderator.
2. The second nuclear reactor named CIRUS (Canada India Research U.S.) was
commissioned in 1960.
******************
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Chapter 14:
SEMICONDUCTOR ELECTRONICS
Any device whose action is based on the controlled flow of electrons through it is
called an electronic device. The branch of physics which deals with these electronic
devices is called electronics.
“The branch of electronics deals with semiconductor devices is called
semiconductor electronics”.
Vacuum tubes:
Before the invention of semiconductor devices vacuum tubes where used in place
of semiconductor devices. Nowadays they are replaced by semiconductor devices.
Some important features of vacuum tubes are as follows:
1. In vacuum tubes, electrons are obtained from a heated cathode and the flow of
electrons controlled by varying voltage between the different electrodes.
2. A vacuum is necessary in the interelectrode region so that electron may not lose
their energy on colliding with the air molecules in their path.
3. As the electron can flow in only direction. i.e, from cathode to anode, so vacuum
tubes are also called as vacuum valves.
4. The vacuum tubes are bulky, consume high power and operate generally at high
voltage (about 100V).
5. They have limited life and reliability.
Semiconductor electronic devices:
In 1930 it was first realised that some solid state semiconductors and their
junctions offer the possibility of controlling the number and direction of flow of
charge carriers through them. Semiconductors are the basic materials used in present
electronic devices like diode, transistor, and integrated circuits.
The following are the important features of semiconductor devices:
1. In a semiconductor device, simple excitations like light, heat or small applied
voltage can change the number of charge carriers.
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2. The charge carriers flow in the solid itself, no vacuum has to be created for the
flow of charges as required in vacuum tubes.
3. It does not require any cathode heating for the production of charge carriers. So it
starts on as soon as it switched on.
4. Semiconductor devices are small in size, consume low power and operate at low
voltages.
5. They have long life and high reliability.
Classification of solids on the basis of their electrical conductivity or
resistivity:
“Resistivity of a material is defined as the resistance offered by a conductor having
unit length and unit area of cross section”.
The conductivity of a material is the reciprocal resistivity.
On the basis of conductivity (ζ) or resistivity (ρ=1/ζ), we can broadly classify solid
into three categories:
1. Metals: They have low resistivity or high conductivity.
a. ρ ≈ 10-2 – 10-8 Ω m
b. ζ ≈ 102 – 108 Ʊ m-1
2. Semiconductors: They have resistivity or conductivity intermediate to metals
and conductor.
a. ρ ≈ 10-5 – 106 Ω m
b.
ζ ≈ 105 – 10-6 Ʊ m-1
3. Insulators: They have high resistivity (or low conductivity).
a. ρ ≈ 1011– 1019 Ω m
b.
ζ ≈ 10-11 – 10-6 Ʊ m-1
Band theory of solids:
Consider a single isolated atom of a solid,
1. In an isolated atom there are single energy levels.
2. An electron revolving in a particular orbit possesses a specified energy.
3. The larger the orbit greater is its energy. Thus an electron in an outer orbit
possesses more energy than in an electron in an inner orbit.
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4. Since discrete energy levels are present in in an atom there is a forbidden energy
region in between two consecutive energy levels.
When the number of atoms brought together to form solid,
1. The electrons which are present in the atom, interacts with the electrons of other
atoms.
2. Due to the interaction the energy levels of each atom are disturbed slightly, and
each energy level splits into a number of levels corresponding to the number of
atoms which are present in the solid.
3. The split levels being around the original level.
4. Thus energy level of single atom is broadened forming a group of very close split
energy levels, which can be regarded as a continuous group, such energy group is
called an energy band.
Energy band: The range of energies possessed an electron in a solid is called energy
band.
There are two important bands,
1. Valence band:
The range of energy possessed by valence electrons in a solid is called valence
band.
2. Conduction band:
The range of energy possessed by conduction electron is called conduction
band.
Once the electrons enters into the conduction band it becomes free to conduct electric
current. These electrons which are in the conduction band are called conduction
electrons.
Conduction electrons are the electrons which responsible for the conduction of
electric current in a conductor.
Conduction Band
Ec
Energy Gap (Eg)
Ev
Valence Band
Page | 301
The valence band and conduction in certain materials overlaps and in some
materials these bands are separated by forbidden energy gap.
Forbidden energy gap (energy gap):
The gap between top of the valence band and bottom of the conduction band on
the energy level diagram is called forbidden energy gap.
Some features of forbidden energy gap:
1. No electrons of solid can stay in a forbidden energy gap as there is no allowed
energy state in this region.
2. The width of the energy gap is a measure of the bondage of valence electrons to
the atom.
3. The greater the energy gap, more tightly the valence electrons are bound to the
nucleus.
Classification of solids on the basis of energy bands
Conductors (metals): One can have metal either when the
 Both the conduction and valence band are partially field,
 Or when the conduction and valence band overlaps.
When the valence band is partially empty, electrons from its lower level can
move to higher level making conduction possible.
When there is an overlap of conduction band and valence band electrons from
valence band can move easily into conduction band. This situation makes large
number of electrons available for electrical conduction.
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Insulators:
In insulators there is a large energy gap exists between valence band and
conduction band. The energy gap is greater
than 3eV. Because of the large energy gap
the electrons in the valence band are not able
to jump into the conduction band by thermal
excitation. So there were no electrons in the
conduction band in the case of insulators. So
insulators are bad conductors of electricity.
Semiconductors:
In semiconductor energy gap is less than
3eV. Due to the small energy gap, thermal
excitation is able to move some of the electrons
from valence band to conduction band. And
these electrons can move in the application of
electric field. In semiconductor only a few
number of electrons available in the conduction
band at room temperature so its conductivity
lies between conductors and insulators.
Semiconductors:
“Semiconductors are the materials whose conductivity lies between conductors
and insulators”.
Semiconductors are classified on two basis,
1. On their chemical composition and
2. Source and the nature of charge carriers.
Classification of semiconductors on their chemical composition:
 Elemental semiconductors:
Example: silicon (Si) and germanium (Ge)
 Compound semiconductors:
Examples are,
Page | 303
Inorganic: CdS, GaAs, CdSe, InP etc.
Organic polymers: polypyrole, polyaniline, polythiophene.
Classification of semiconductor on the basis of the source and the
nature of the charge carriers
Intrinsic semiconductor: The pure semiconductors are called intrinsic
semiconductors. In intrinsic semiconductor number of conduction electrons is equal
to number of holes.
Extrinsic semiconductor: A semiconductor doped with suitable impurity atoms so
as to increase its number of charge carriers is called an extrinsic semiconductor.
Intrinsic semiconductor:
Valence bond model for an intrinsic semiconductor:
If we consider a crystal of semiconductor Ge or Si, each Ge atom has four
valence electrons which it shares with the four neighbouring atoms to form four
covalent bonds.Thus each Ge atom is tetrahedraly bonded to four neighbouring Ge
atoms as shown in the below figure.
Such crystal structure is called diamond like structure. The shared pair of
electrons oscillates back and forth between the two associated atoms. At zero Kelvin
temperature all valence bond in the crystal are perfect. As the temperature increases
the thermal energy of the electrons in the valence bond increases. An electron may
break away from the covalent bond and becomes free to conduct electricity as shown
in the above fig. This electron leaves a vacancy in the covalent bond, and this
vacancy is called hole.
Page | 304
Hole:
“Vacancy of an electron with an effective positive electronic charge is called hole”.
As each electron creates one hole, in intrinsic semiconductor the number density of
free electrons (ne) is equal to the number density of holes (nh) and each is equal to the
intrinsic charge concentration (nhi).
That is,
ne= nh = nhi
Holes as positive charge carriers:
Vacancy of an electron with an effective positive electronic charge is called
hole. The above figure describes hole
as a positive charge carrier. Let hole
be at site 1 as shown in the figure. An
electron at site 2 of the neighbouring
covalent bond may jump to the site 1.
After such jump, a hole is created at
site 2 and site 1 gets occupied by an
electron. Hence the hole is moved
from site 1 to site 2. Under the action
of electric field the moment of holes
is in the electric field direction due to
jumping of bound electrons in the reverse direction from one atom to another.
Current in intrinsic semiconductor:
In intrinsic semiconductor holes and electrons are the charge carriers. And
these charge carriers gives independent electron current Ih and Ie . there for the total
current in intrinsic semiconductor,
I = electron current + hole current
Or
I = Ie + Ih
Limitations of intrinsic semiconductor:
1. Intrinsic semiconductors have low intrinsic charge carrier concentration.
2. As intrinsic charge carriers are always thermally generated, so flexibility is not
available to control their number.
Page | 305
3. For intrinsic semiconductors, ne= nh. they cannot have predominant hole or
electron conduction.
Extrinsic semiconductor:
To overcome the difficulties of intrinsic semiconductor we have to add some
impurity atoms. When a small amount say a few parts per million (ppm), of a suitable
impurity is added to the pure semiconductor, the conductivity of the semiconductor is
increased manifold.
“A semiconductor doped with some suitable impurity atoms so as to increase
its number of charge carriers is called an extrinsic semiconductor”.
Doping: “The process of deliberate addition of a desirable impurity to a pure
semiconductor is called doping”.
Dopants: “The impurity atoms which are added to pure semiconductor in order to
increase its conductivity are called dopants”.
There are two types of dopants used in doping the tetravalent Si or Ge.
1. Pentavalent dopants: they have 5 valence electron.
For example; arsenic (As), antimony (Sb), and Phosporous (p).
2. Trivalent dopants: They have 3 valence electrons.
For example: Indium (In), Boron (B), and Aluminium (Al).
Types of extrinsic semiconductor:
There are two types of extrinsic semiconductor:
1. n-type semiconductor
2. p-type semiconductor
n-type semiconductor:
This semiconductor is obtained by doping the tetravalent semiconductor by Si
or Ge with pentavalent impurities such as arsenic (As), Antimony (Sb), and
phosphorus of group 5 of the periodic table.
Valence bond model of n-type semiconductor
Page | 306
Suppose we dope Si or Ge with a pentavalent element as shown in above fig.
 An atom of pentavalent element occupies the position of an atom in the crystal
lattice of Si or Ge.
 The four electrons of pentavalent atom bond with the four silicon neighbours
while the fifth remains very weakly bound to its parent atom.
 As a result the ionisation energy required to set this electron free is very small
and even at room temperature it will be free to move in the lattice of the
semiconductor.
(for example, the energy required is 0.01 eV for germanium and 0.05 eV for
silicon, to separate this electron from atom. )
 Thus, the pentavalent dopant is donating one extra electron for conduction and
hence is known as donor impurity.
P - type semiconductor:
p-type semiconductor is obtained when Si or Ge is doped with trivalent impurity
like Al, B, In etc.
 Trivalent impurity like Al, B, In, has one valence electron less than that of Si or
Ge.
 When the trivalent impurity atom is added to the Si or Ge crystal the atom can
form covalent bonds with neighbouring three Si or Ge atoms but does not have
any electron to offer to the fourth Si or Ge atom.
Page | 307
 Hence the bond between the fourth neighbour and the trivalent atom has vacancy
or hole as shown in fig.
 Since the neighbouring Si atom in the lattice wants an electron in the place of
hole, an electron in the neighbourhood atom may jump to fill this vacancy leaving
a vacancy or hole at its site.
 Thus the hole is available for conduction of electricity.
 In p-type semiconductor number of holes nh is greater than number of conduction
electrons ne. In this type holes are the majority charge carriers and electrons are
minority charge carriers.
Valance bond model for p-type semiconductor
p-n junction:
The junction between p-type and n-type is known as p-n junction.
A p-n junction is the basic building block of many semiconductor devices like diodes,
transistor, etc.
p-n junction formation:
 A p-n junction is formed by adding precisely a small quantity of pentavalent
impurity to the p-type Si wafer. While doing so part of the p-type Si wafer is
converted into p-type Si
 Two important process occurs during the formation of p-n junction. Those
process are
a. Diffusion
b. Drift
Page | 308
 Due to the concentration gradient of holes and electrons across p and n sides
holes get diffused from p side to n side and electrons get diffused from n side to
p-side.
 As the holes continue to diffuse from p side to n side a layer of negative space
charge on the p side is get developed. Similarly diffusion of electrons forms a
layer of positive space charge on the n side of the junction.
 The space charge region on either side of the junction together is known as
depletion region. (The thickness of the depletion region is of the order of 0.1
micrometre).
 Due to the space charge on either side of the junction an electric field directed
from positive charge towards negative charge develops.
 Due to this electric field electrons on the p side of the junction get drifted to n
side and holes on the n side of the junction get drifted to p-side. Thus drift
current, which is opposite to diffusion current starts.
 Initially, diffusion current is large and drift current is small. As the diffusion
process continues the space charge on either side of the junction increases which
increases the strength of the electric field and hence drift current increases.
 This process continues until the diffusion current equals the drift current so the
net current becomes zero and hence equilibrium is reached. Thus a p-n junction is
formed.
 The space charge across the junction produces a difference of potential across the
junction of the two regions and is called barrier potential.
 This potential prevents the movement of electrons from n-region to p- region.
Semiconductor diode:
“A semiconductor diode is basically a p-n junction with metallic contacts
provided at the ends for the application of an external electric voltage”.
Page | 309
P
-
n
Metallic contact
contact
Depletion Region
Metallic
A p-n junction diode is symbolically represented as shown in the figure. The
direction of arrow indicates the conventional current direction.
+
P
n
-
p-n junction under forward bias:
When an external voltage is applied across a semiconductor diode such that pside is connected to the positive terminal of the battery and n-side to the negative
terminal of the battery it is said to be forward biased.
Page | 310
When the p-n diode is forward biased the direction of applied voltage is
opposite to the barrier potential. As a result,
 The depletion layer width decreases and the barrier height is reduced.
 The holes from p-side and electrons from n-side begin to flow towards the
junction.
 As the applied voltage (V) increases the depletion layer width decreases and the
barrier height is reduced further.
 When V exceeds barrier potential (V0), the majority charge carriers start flowing
easily across the junction and set up a large current in the order of mA called
forward current in the circuit.
 If we increase the voltage above V0 the current increases with increase in applied
voltage.
p-n junction diode under reverse bias:
When an external voltage (V) is applied across the diode such that n-side is positive
and p-side is negative, it is said to be reverse biased.
When a p-n junction is reverse biased the applied voltage V and the barrier
potential V0 are in the same direction as a result of this
 The barrier potential increases to (V0 + V) and hence energy barrier across the
junction increases.
 The majority carriers move away from the junction increasing the width of the
depletion layer.
 Due to this the resistance of the p-n junction becomes very large and no current
flows across the junction by majority charge carriers.
Page | 311
 However at room temperature there are always present some minority charge
carriers, and these charge carriers constitutes a small current in the order of
microampere called as reverse or leakage current.
 The current under reverse bias remains the same for increasing the reverse
voltage until a critical voltage called breakdown voltage (Vbr).
 When reverse voltage reaches breakdown voltage the diode gets breakdown and
the reverse current increases sharply with increase in voltage.
V-I characteristics of a p-n junction diode:
1. Forward bias characteristics: The circuit arrangement to study the V-I
characteristics of a p-n junction diode is shown in the fig.
Fig. (a)
Fig. (b)
 The battery is connected to the diode through a rheostat so that the applied
voltage to the diode can be changed.
 A voltmeter is connected to measure the voltage across the diode. And a
milliammeter is connected in series with the diode to measure the current through
the diode.
 By the help of rheostat the voltage is adjusted for suitable value V from 0V in
steps and corresponding mlliammeter readings (I) are noted.
 A graph between V and I is obtained as shown in the fig.(b)
 From the graph we can see that in forward bias the current first increases very
slowly, almost negligibly till the voltage across the diode crosses a certain value
called threshold voltage or knee voltage. After the threshold voltage (0.3V for
Page | 312
Ge and 0.7V for Si) the diode current increases significantly even for a small
increase in the diode bias voltage
2.
Reverse bias:
The figure shows experimental arrangement for studying characteristic curve of a
p-n junction diode when it is reverse biased.
Fig. (a)
Fig. (b)
 The battery is connected to the diode through a rheostat so that the applied
voltage to the diode can be changed.
 A voltmeter is connected to measure the voltage across the diode. And a
microammeter is connected in series with the diode to measure the reverse current
through the diode.
 By the help of rheostat the reverse voltage is adjusted for suitable value V from
0V in steps and corresponding microammeter readings (I) are noted.
 A graph between V and I is obtained as shown in the fig.(b)
 From the graph we can see that when the diode is reversed biased, the reverse
bias voltage produces a very small current, about a few microamperes which
almost remains constant with bias. This current is called as reverse saturation
current or leakage current.
 When the reverse voltage across the diode reaches a sufficiently large value, the
reverse current suddenly increases to a large value and the voltage is known as
breakdown voltage.
Breakdown voltage or peak inverse voltage:
“The voltage at which breakdown of the junction diode occurs is called breakdown
voltage or peak inverse voltage”.
Page | 313
Dynamic Resistance:
“Dynamic resistance is the ratio of small change in voltage ∆V to a small change in
current ∆I”.
rd =
𝜟𝑽
𝜟𝑰
.
Application of junction diode as a rectifier:
Introduction:
From the V-I characteristics of the diode we see that it allows current to pass only
when it is forward biased. So if an alternating voltage is applied across a diode the
current flows only in that part of the cycle when the diode is forward biased. This
property is used to rectify alternating voltages and the circuit is called rectifier.
Rectifier:
“Rectifier is an electronic circuit which converts AC voltage into DC voltage”.
There are two types of rectifiers they are
1. Half wave rectifier
2. Full wave rectifier
Half wave rectifier:
“A rectifier in which a pulsating voltage will appears across the load only
during the half cycles of the ac input during which the diode is forward biased is
called half wave rectifier”.
The circuit diagram for half wave rectifier is shown in figure.
Page | 314
 The secondary of the transformer supplies the desired ac voltage across terminals
A and B.
 During the positive half of the cycle ac in put voltage at A is positive with respect
point B so the diode is forward biased and it conducts. A current will flow
through RL and hence pulsating input voltage will appears across RL.
 During the negative half of the cycle ac in put voltage at A is negative with
respect point B so the diode is reverse biased and it does not conducts. And hence
no current will flow through RL which results zero voltage drop across RL.
 And the process repeats for every cycle of the ac input voltage.
Full – Wave rectifier:
Circuit Diagram:
Waveforms:
 It consists of two diodes with a centre tapped step down transformer and a load
resistance RL.
 During positive half of the AC input P1 is positive and P2 is negative with respect
to centre point of the secondary coil. Hence diode D1 is forward biased so it
conducts and diode D2 is reverse biased so it does not conduct. So there is a
current through D1 and RL and hence an output voltage across RL.
 During negative half of the AC input P1 is negative and P2 is positive with respect
to centre point of the secondary coil. Hence diode D1 is reverse biased so it so it
does not conduct and diode D2 is forward biased so it conducts. So there is a
current through D2 and RL and hence an output voltage across RL.
 Thus there exists the rectified output voltage across RL during both positive and
negative half cycles of input AC.
Page | 315
Special purpose p-n junction diodes:
Zener Diode:
On the basis of doping level of the p-n junction diode we can classify it into two
types,
1. Avalanche diode:
Lightly doped p-n junction diode which has large depletion layer.
2. Zener diode:
Heavily doped p-n junction diode has thin depletion layer.
“Zener diode is a heavily doped diode which is designed to operate under reverse
bias in the breakdown region with a specific reverse breakdown voltage”.
Zener diode was invented by C. Zener.
I-V characteristics of Zener diode:
A forward characteristics of a Zener diode is same as that of the ordinary diode.
The difference is when it is reverse biased.
Since doping level is very high in Zener diode when compared with ordinary
diode, the width of the depletion region becomes very thin. And it under goes Zener
breakdown when it is reverse biased.
Zener breakdown:
 When the reverse biased voltage is increased, even a small applied voltage creates
a very high electric field in the depletion region.
 Because of the high electric field at the junction, a sufficiently strong force may
exerted on bound electron to tear it out of its covalent bound in the p-region.
 The emission of electron from the host atom due to high electric field is known as
internal field emission or field emission. Due to this electron hole pairs are
generated which increases the reverse current which results Zener breakdown.
 The process by which the covalent bonds in the depletion region are directly
broken by a strong electric field is called Zener breakdown. And the reverse
voltage at which the Zener breakdown takes place is called the Zener breakdown
voltage.
Page | 316
 The I-V characteristics of Zener diode is shown in fig
 It is found that the Zener diode conducts when it is reverse biased, once the
applied voltage exceeds the Zener breakdown voltage Vz.
 For small increase in reverse voltage the reverse current increases but the voltage
across the junction remains almost constant.
The circuit symbol of a Zener diode is shown below.
Zener diode as a voltage regulator:
A DC power supply which maintains the output voltage constant irrespective
of AC mains fluctuations or load variations is known as regulated DC power supply.
A Zener diode is used to obtain a constant DC voltage from DC unregulated
output of a rectifier. The unregulated DC voltage is connected to the Zener diode
through a series resistance RS such that the Zener diode is reverse biased. When the
input voltage increases, the current through RS and that through the Zener diode will
increase. This increases the voltage drop across RS but the voltage across the Zener
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diode remains same. This is because in the breakdown region, the Zener voltage
remains constant even though the Zener current varies (between IZmin to Izmax).
Similarly if the input voltage decreases, the currents through R and Zener will
decrease but Zener voltage VZ will remain constant. Thus the Zener diode acts as a
voltage regulator.
Optoelectronic junction devices:
Optoelectronic junction devices are the semiconductors diodes in which
charge carriers are generated by photons (photo excitation).
Following are the Optoelectronic junction devices:
1. Photodiodes: These are the optoelectronic junction devices which are used for
detecting optical signals.
2. Light emitting diode (LED): It is a semiconductor diode which converts
electrical energy into light energy.
3. Photo voltaic cell: These are the optoelectronic junction devices which converts
light energy into electrical energy.
Photodiodes:
A photodiode is again a special purpose diode fabricated with a transparent
window to allow light to fall on the diode.
It is operated under reverse bias.
Circuit symbol:
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Working:
 When the photodiode is reverse biased the depletion region width increases.
 When it is illuminated with light (photons) with energy (hν) greater than the
energy gap (Eg) of the semiconductor, then electron-hole pairs are generated due
to the absorption of photons.
 Due to electric field of the junction, generated electrons and holes are separated
before they recombine. The direction of electric field is such that electrons reach
n-side and holes reach p-side.
 Electrons are collected on n-side and holes are collected on p-side giving rise to
an emf. When an external load is connected, current flows.
 The magnitude of the photocurrent depends on the intensity of incident light and
it is directly proportional to incident light intensity.
I-V characteristics of photodiode for different illumination intensity are shown in
fig.
Light emitting diode (LED):
“Light emitting diode (LED) is a p-n junction which under forward bias
emits spontaneous radiation”.
Circuit diagram:
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Working principle:
 When the diode is forward biased, electrons are sent from n-side to p-side (where
they are minority carriers) and holes are sent from p-side to n-side (where they
are minority carriers).
 At the boundary the concentration of minority carriers increases compared to the
equilibrium concentration.
 Excess minority carriers on either side of the junction boundary recombine with
majority carriers near the junction.
 On recombination, the energy is released in the form of light energy (photons).
The energy of the photons is equal to or slightly less than the band gap are
emitted.
 When the forward current of the diode is small, the intensity of light emitted is
small. As the forward current increases, intensity of light increases. Further
increase in forward current results in decrease of light intensity.
 LEDs are biased such that the light emitting efficiency is maximum.
Solar cell:
“ A solar cell is basically a p-n junction which generates emf when solar
radiation falls on p-n junction”.
It works on the same principle as of the photodiode, except that no external bias is
applied to the junction area is kept much larger for solar radiation to get maximum
power.
Construction:
 A p-type Si layer of about 300 µm is taken over which a thin layer about 0.3 µm
of n-type Si is grown on one side by diffusion process. The other side of the ptype Si layer is coated with a metal (back contact).
 On the top of the n-type Si layer, metal finger electrode (or metallic grid) is
deposited. This acts as a front contact.
 The metallic grid occupies only a very small fraction of area (< 15%) so that light
can be incident on the cell from top.
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Working:
 When light of suitable frequency is incident on the top surface of the solar cell it
penetrates n-type Si-layer and enters the p-n junction. Due to light close to the
junction electron-hole pairs are generated.
 Due to the electric field of the depletion region electrons are swept to n-side and
holes to p-side.
 The electrons reaching the n-side are collected by the front contact and holes
reaching p-side are collected by the back contact. Thus p-side becomes positive
and n-side becomes negative giving rise to photo voltage.
 When an external load is connected a photocurrent IL flows through the load. A
typical I-V characteristics of a cell is shown in figure.
Junction transistor:
Junction transistor consists of two back to back p-n junctions. It is also called
bipolar junction transistor (BJT).
“A transistor is a three terminal, two semiconductor-junction device whose basic
action is amplification”.
It has three doped regions namely emitter, base and collector.
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The arrowhead on the emitter shows the direction of the conventional current in the
transistor.
Emitter:
It is moderate in size and heavily doped. It supplies large number of majority charge
carriers.
Collector:
It is larger in size compared to emitter and is moderately doped.
Base:
It is the Central region which is very thin and lightly doped .
There are two types of transistors namely n-p-n and p-n-p transistors.
 In n-p-n transistor two segments of n-type semiconductor are separated by a ptype semiconductor.
 In p-n-p transistor two segments of p-type semiconductor are separated by a ntype semiconductor.
The circuit symbol of n-p-n and p-n-p transistors are shown below.
Action of n-p-n transistor:
For the effective working of a transistor its junctions have to be properly
biased. When the transistor is used as an amplifier the emitter-base junction is
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forward biased while the collector-base junction is reverse biased. The transistor is
then said to be in active state.
Basic action of a transistor (n-p-n):
Consider an n-p-n transistor with its emitter- base junction forward biased and
collector-base junction reverse biased. The heavily doped n-type emitter has a high
concentration of electrons (majority carriers). The electrons cross the emitter base
junction and enter the base region in large number as it offers least resistance due to
its forward biased condition. This gives rise to emitter current IE As the p-type base is
very thin and lightly doped only few holes are present in base. As such very few
electrons from the emitter combine with the holes of base, giving rise to base current
IB. The remaining large number of electrons in the base region are minority carriers
there and hence can easily cross the reverse-biased collector- base junction to enter
the collector. This gives rise to collector current IC. The base current is only a small
fraction of the emitter current. It is seen that the emitter current is the sum of base
current and collector current.
IE = IB + IC
IE and IC are in the order of mA such that IE > IC. IB is of the order of µA.
Common emitter (CE) transistor characteristics:
In the CE mode the input is between base and emitter terminals while the
output is between collector and emitter terminals.
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Input characteristic:
The graphical representation of the variation of base current IB with the baseemitter voltage VBE for a fixed value of collector- emitter voltage VCE is called input
characteristic.
VCE is kept fixed, VBE is varied and the variation in IB noted in regular intervals. For
small values of VBE the base current IB is negligible. When VBE exceeds barrier
voltage IB increases sharply even with small increase in VBE.
Increase in VCE appears as increase in VCB hence its effect on IB is negligible.
Hence different values of VCE give almost identical curves.
The input resistance ri of the transistor in CE mode is defined as the ratio of
change in base-emitter voltage to the corresponding change in base current at
constant collector-emitter voltage.
𝚫𝑽
ri = 𝑩𝑬
𝚫𝑰𝑩
𝑉𝐶𝐸 = constant .
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ri ranges from few hundred ohm to few thousand ohm.
Output characteristics:
The graphical representation of the variation of collector current IC collectoremitter voltage VCE for a fixed value of base current IB is called output
characteristic.
Initially for very small values of VCE, IC increases almost linearly – this is since
collector- base junction is not reverse biased and the transistor is not in active state.
The transistor is in the saturation state and current is controlled by VCC (equal
to VCE) in this region. When VCE is increased further, IC increases marginally (very
small change) with VCE for a given base current IB. The output resistance r0 of the
transistor is very high – of the order of 100kΩ or more. It is seen that larger the value
of IB larger is the value of I for a given VCE.
Output resistance (r0)
It is the ratio of the change in collector-emitter voltage to the corresponding
change in collector current at a constant base current IB.
𝜟𝑽
r0 = 𝑪𝑬
𝜟𝑰𝑪
𝐼𝐵 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 .
Transistor as a switch
The transistor acts as a switch when it is used in the cut-off state (region) and
in the saturation state (region).
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Consider the transistor in CE configuration which is base-biased. Applying
Kirchhoff‟s voltage law (KVL) to the input side.
VBB = IBRB+ VBE
Taking VBB as DC input voltage Vi,
Vi = IBRB+ VBE ……………………… (1)
Applying Kirchhoff‟s voltage law (KVL) to the input side
VCE = VCC - ICRC
Taking VCC as DC input voltage V0,
V0 = VCC - ICRC ……………………… (2)
In case of Si transistor, as long as input voltage Vi is less than 0.6V, the transistor
will be in cut-off state and the current IC will be zero.
Thus VO = VCC, from equation (2).
Hence when low input is given the transistor is switched off (it goes to cut-off
state) while output voltage Vo is maximum at VCC. When a high input Vi is given such
that it is enough to drive the transistor into saturation the transistor will be switched
ON – while the output voltage V0 will be very low.
V0 = VCC - ICRC
In saturation region IC will be large hence V0 is very small. LOW input Vi
switches the transistor OFF. While HIGH input Vi switches the transistor ON – Thus
a transistor acts as a switch.
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How transistor works as an amplifier in linear or active region?
The slope of the linear part of the active region of the Vi versus V0 curve gives
the rate of change of output with the input. If ∆V0 and ∆Vi are the changes in the
𝚫𝑽
output and input voltages, then 𝟎 will be the small
𝚫𝑽𝒊
signal voltage gain AV of the amplifier. If the
forward VBB
Is fixed at the midpoint of the active region, then the
base biased CE transistor will behave as an amplifier
𝚫𝑽
with voltage gain 𝟎 . Thus the linear portion of the
𝚫𝑽𝒊
active region of the transistor can be used for the
purpose of amplification.
Transistor amplifier in CE configuration:
The circuit of CE amplifier employing n-p-n transistor is as shown. Here C1
and C2 are Coupling capacitors which block DC and allow only AC. The transistor
operating point is fixed in the middle of the active region. This fixes DC base current
IB and the corresponding collector Current IC while DC voltage VCE would remain
constant. The operating values of VCE and IB determine the operating point of the
amplifier.
Action of transistor as an amplifier CE-configuration:
When Vi = 0 the output voltage V0 = VCE = VCC – ICRC ………………… (1)
During positive half of the ac input signal, emitter-base junction is more forward
biased and the base current IB increases.
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As IB increases, IC increases β times (IC = βIB), the product ICRC increases and hence
from equation (1) V0 decreases during the positive half of the ac input signal.
During negative half of the ac input signal, emitter-base junction is less forward
biased and the base current IB decreases. As IB decreases, IC decreases β times
(IC = βIB), the product ICRC decreases and hence from equation (1) V0 increases
during the negative half of the ac input signal.
Expression for voltage gain of transistor CE – amplifier:
The below figure shows a simple circuit for transistor CE – amplifier.
In the absence of input AC signal vi, applying Kirchhoff‟s voltage law to the
input loop,
VBB = IBRB+ VBE ………………… (1)
When the signal Vi is not zero,
VBB + Vi = (VBE + ∆ VBE) + (IB + ∆ IB ) RB
= (VBE + ∆ IB ri) + (IB + ∆ IB ) RB = VBE + IBRB + ∆ IB (RB + ri ).
This implies that,
VBB + Vi = VBE + IBRB + ∆ IB (RB + ri ) ……………… (2).
Where
ri =
𝚫𝑽𝑩𝑬
𝚫𝑰𝑩
𝑉𝐶𝐸 = constant .
Is the input resistance of the transistor.
From equation (1) & (2)
Vi = ∆ IB (RB + ri ) = ∆ IB r. ………….. (3)
Where,
r = RB + ri is the input resistance of the amplifier.
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Applying Kirchhoff‟s voltage law to the output part,
VCC = VCE + ICRC
The change in IB causes change in IC which in turn causes change in VCE. Voltage
drop cross R also is changed – since VCC is fixed.
∆ VCC = 0 = ∆VCE + ∆ICRC
Or ac output voltage V0 = ∆VCE = - ∆ICRC ………….. (4)
The voltage gain of the transistor amplifier is defined as it is the ratio of ac
output voltage to ac input voltage.
That is,
𝑉
−∆𝐼 𝑅
−∆𝛽 𝑎𝑐 𝑅𝐶
AV = 0 = 𝐶 𝐶 =
𝑉𝑖
Where 𝛽𝑎𝑐 =
∆𝐼𝐶
∆𝐼𝐵
𝑟∆𝐼𝐵
𝑟
is the current gain of the transistor amplifier. The negative sign
represents that output voltage is opposite in phase with the input voltage.
Feedback amplifier and transistor oscillator:
“A feed amplifier is an amplifier in which fraction of output of the amplifier
is fed back in phase or out of phase”.
Two types of feedback are
1. Positive feedback:
A fraction of output is fed back to the input in phase.
2. Negative feedback:
A fraction of output is fed back to the input is out of phase with the input.
Oscillator:
An oscillator is an electronic device which produces sustained electrical
oscillations (ac signal) of constant frequency and amplitude without any external
input.
Principle of an oscillator:
In an oscillator a portion of the output power is returned (feedback part) back
to the input in phase with the input or starting power. An oscillator may be regarded
as a self-sustained transistor amplifier with a positive feedback. (in-phase feedback).
A tank circuit produces oscillations. These oscillations may get damped due to energy
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loss. The transistor amplifier amplifies the
oscillations. The feedback circuit returns or
feeds back a fraction of the output
power of the amplified output from the
amplifier to the tank circuit in phase with the
input signal to that energy loss due to
damping is compensated. This produces
undamped, self-sustained oscillations.
Transistor as an oscillator:
Construction:
The circuit diagram of transistor as an oscillator is shown in figure. Here L-C
circuit is inserted in emitter-base circuit of transistor which is forward biased with
battery VBB. The collector-emitter circuit is reverse biased with battery VCC. A coil L3
inserted in a collector-emitter circuit. It coupled with a coil L1 in such a way that if
increasing magnetic flux is linked with L3, it will support the forward bias of emitter
base circuit and if decreasing magnetic flux is linked with L3, it will oppose the
forward bias of emitter base circuit.
Working:
When key K is closed, IC starts growing through L3. Since L3 is inductively
coupled to L1, increasing IC through L3, induce voltage (feedback) across L1 in such a
way that base-emitter circuit becomes forward biased. This causes increase in IC at a
faster rate and induced voltage increases further across L1. Due to this capacitor C1
gets charged. When the transistor reaches saturation state, the IC increases at lesser
rate and thus induced voltage across L1 decreases. Now the capacitor starts
discharging making the base of the transistor negative. Discharging of capacitor
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drives the transistor in cut off state so that IC becomes zero. This process repeats
again and again and thus sustained electrical oscillations in the output are produced.
Digital electronics and logic gates:
There are two types of electronic signals
1. Analogue signal:
It is an electronic signal which is in the form of continues time-varying
voltage or current.
2. Digital signal:
It is an electronic signal in which only two discrete values of voltage are
possible.
Logic Gates:
“A logic gate is a digital circuit that follows certain logical relationship
between the input and output signals and works on the principles of Boolean
algebra”.
The five common logic gates are NOT, AND, OR, NAND, NOR. Each logic gate
is indicated by a symbol and its function is defined by a truth table that shows all the
possible input logic level combinations with their respective output logic levels.
NOT gate:
It is a single input single output logic gate whose output is the logical inversion
of its input.
Logic symbol:
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Truth table:
Input
Output
A
Y=𝑨
0
1
1
0
AND gate:
An AND gate has two or more inputs with one output.
 The output Y is 1 when both „input A‟ and „input B‟ are 1.
 The out Y is zero otherwise.
That is if all the input is high, the output is high.
Logic symbol:
Truth table:
Inputs
Output Y = A.B
A
B
0
0
0
0
1
0
1
0
0
1
1
1
OR gate:
An OR gate has two or more inputs with one output.
 The output Y is 1 when either input A or B or both are 1.
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 The out Y is 0 when both the inputs are 0.
Logic symbol:
Truth table:
Inputs
A
Output Y = A+B
B
0
0
0
0
1
1
1
0
1
1
1
1
NAND gate:
A NAND gate has two or more inputs with one output. This is an AND gate
followed by NOT gate.
 The output Y is 1 when either input A or B or both are 0.
 The out Y is 0 when both the inputs are 1.
Logic symbol:
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Truth table:
Output Y = 𝑨. 𝑩
Inputs
A
B
0
0
1
0
1
1
1
0
1
1
1
0
NOR gate:
A NOR gate has two or more inputs with one output. This is an OR gate followed
by NOT gate.
 The output Y is 0 when either input A or B or both are 1.
 The out Y is 1 when both the inputs are 0.
Logic symbol:
Truth table:
Output Y = 𝑨 + 𝑩
Inputs
A
B
0
0
1
0
1
0
1
0
0
1
1
0
************************
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CHAPTER-15
COMMUNICATION SYSTEMS
Introduction:
Communication is the act of transmission of information. Every living creature
in the world experiences the need to impart or receive information almost
continuously with others in the surrounding world. For communication to be
successful, it is essential that the sender and the receiver understand a common
language. Man has constantly made endeavors to improve the quality of
communication with other human beings. Languages and methods used in
communication have kept evolving from prehistoric to modern times, to meet the
growing demands in terms of speed and complexity of information. It would be
worthwhile to look at the major milestones in events that promoted developments in
communications, as presented in Table,
YEAR
EVENTS
REMARK
Around
The reporting of the delivery
of a child by queen using
drum beats from a distant
place to King Akbar.
It is believed that minister Birbal
experimented with the
arrangement to decide the number
of drummers posted between the
place where the queen stayed and
the place where the king stayed.
1835
Invention of telegraph by
Samuel F.B. Morse and Sir
Charles Wheatstone
It resulted in tremendous growth
of messages through post offices
and reduced physical travel of
messengers considerably.
1876
Telephone invented by
Alexander Graham Bell and
Antonio Meucci
Perhaps the most widely used
means of communication in the
history of mankind.
1895
Jagadis Chandra Bose and
Guglielmo Marconi
demonstrated wireless
telegraphy.
It meant a giant leap – from an era
of communication using wires to
communicating without using
wires. (wireless)
1936
Television broadcast(John
Logi Baird)
First television broadcast by BBC
1955
First radio FAX transmitted
across continent.(Alexander
Bain)
The idea of FAX transmission
was patented by Alexander Bain
in 1843.
1565 A.D.
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1968
ARPANET- the first internet
came into existence(J.C.R.
Licklider) Fiber optics
developed at
ARPANET was a project
undertaken by the U.S. defence
department. It allowed file
transfer from one computer to
another connected to the network.
1975
Fiber optics developed at Bell
Laboratories
Fiber optical systems are superior
and more economical compared to
traditional communication
systems.
1989-91
Tim Berners-Lee invented the
World Wide Web.
WWW may be regarded as the
mammoth encyclopedia of
knowledge accessible to everyone
round the clock throughout the
year.
Modern communication has its roots in the 19th and 20 century in the work of
scientists like J.C. Bose, F.B. Morse, G. Marconi and Alexander Graham Bell. The
pace of development seems to have increased dramatically after the first half of the
20th century. We can hope to see many more accomplishments in the coming decades.
The aim of this chapter is to introduce th e concepts of communication, namely the
mode of communication, the need for modulation, production and deduction of
amplitude modulation.
Elements of communication system:
Every communication system has three essential elements
 Transmitter
 Communication channel
 Receiver
The block diagram of general communication system is shown in figure.
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The figure depicts the general form of a communication system. In a
communication system the transmitter is located at one place, the receiver is located
at some other place separate from the transmitter and the channel is the physical
medium that connects them.
The information source provides the message to be transmitted. A transducer
converts this message signal into electrical signal before giving it as input to the
transmitter. The transmitter converts this message signal into a form suitable for
transmission through the channel. The massage signals are superimposed on a high
frequency carrier wave. The resultant wave so obtained is called modulated wave.
This modulated wave is sent to the communication channel.
The communication channel caries the modulated wave from the transmitter to
the receiver. When the transmitted signal propagates along the channel it may get
distorted due to channel imperfection and noise.
The receiver receives a distorted version of transmitted signal. At the receiver
end, a demodulator separates the low frequency audio signal from the modulated
signal. The low frequency signal so separated is amplified and fed to the transducer.
The transducer converts the low frequency electrical signal into audio or video signal
and delivers it to the user of information.
Types of basic modes of communication
There are two basic modes of communication:
1. Point to point:
In point to point communication mode, the message is transmitted over a link
between a single transmitter and a receiver. For example telephone represents
point to point communication.
2. Broadcast:
In broad cast mode, there are a large number of receivers corresponding to a
single transmitter. For example, radio and television are broadcast mode of
communication.
Basic terminology used in electronic communication systems
Information source: The function of information source is to produce required
message signal which has to be transmitted.
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Transducer: A transducer is a device which converts message produced by the
information source into time varying electrical signal.
Signal: information converted into electrical signal and suitable for transmission is
called a signal.
Noise: The unwanted signal is called a noise in a communication system.
Transmitter: A transmitter is an arrangement that converts the message signal into a
form suitable for transmission and then transmits it through some communication
channel.
Receiver: A receiver is an arrangement that picks up transmitted signal and processes
it to reproduce the message signal in the suitable form.
Attenuation: It refers to the loss of strength of a signal during its propagation
through the communication channel.
Amplification: It is the process of increasing the strength of the transmitted signal
using some suitable electronic circuit called amplifier.
Range: It is the largest distance between the transmitter and receiver where the signal
is received with sufficient strength.
Band width: Band width is the frequency range over which the communication
system works.
Modulation: Modulation is the process of superimposing the low frequency message
signal (called modulating signal) on a high frequency wave (called carrier wave). The
resulting wave is called modulated wave, which is transmitted.
Demodulation: Demodulation is the process of retrieval of information from the
modulated wave at the receiver.
Bandwidth of siganls
The message signals to be transmitted are of two types,
1. Analog signals and
2. Digital Signals.
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1. Band width of Analog signals: An analog signal is that in which the voltage or
the current varies continuously with time. The message signal to be transmitted is
generally in continuous waveform.
The range of frequencies which are necessary for satisfactory transmission of
information or message contained in the analog signals is called bandwidth of the
analog signal.
a. Band width of speech signals: Speech signals contain frequencies between
300Hz to 3100 Hz. Therefore, speech signals require bandwidth = 3100Hz –
300Hz =2800Hz for commercial telephonic communication.
b. Band width of music signals: The audio range of frequencies produced by
musical instruments is from 20Hz to 20KHz. Therefore, music signals require a
bandwidth of about 20kHz.
c. Bandwidth of video signals: for transmission of pictures, the video signals
required a bandwidth of about 4.2 MHz. Since a T.V. signal contains both
audio and video signals, it is usually allocated a bandwidth of 6MHz for the
transmission of TV signals.
2. Band width of digital signals: A digital signal will have only two discrete
values. The wave will be in form of a square or a rectangular wave or pulse. The
levels of voltage or current will be 0 and 1. Theoretically an infinite band width is
required for digital signals.
A rectangular wave can be considered as the superposition of a large number of
sinusoidal waves of frequencies, f 0 , 2 f 0 , 3 f 0 , 4 f 0 ,.............. nf 0 , n is an integer
extending from zero to infinity. f 0 is called the fundamental frequency and the other
higher frequencies are harmonics. In practice, the contribution from the higher
harmonics to the shape of the wave form is very small and hence they are neglected.
Hence providing a sufficiently wide band width, the data in digital signal is not lost
and the signal is recovered,
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Band width of transmission medium:
The transmission medium delivers the message signals from the transmitter to the
receiver. The commonly used mediums are coaxial cables, optical fibres, free space
etc.
Co-axial Cables :
These are used in signals below 18 GHz. Band width is 750 MHz (approx).
Optical fibres:
These are suitable for microwave and UV waves.Band width -10 Hz.
Free space:
In radio communication; free space is used as the transmission medium. The
frequency range of free space is 105 Hz to 109 Hz. This range is divided further and is
allocated to different service.
Propagation of electromagnetic waves:
In communication with radio waves, the electromagnetic waves transmitted by
the transmitting antenna travel through space and reach the receiver antenna. During
its travel, the strength of the wave keeps on decreasing. Several factors like the
composition of the earth. Influence the path and propagation of the wave.
Ground Wave Propagation:
“Ground wave is that part of the radio waves that travels along the surface of the
earth”
“The propagation of the radio waves along the surface of the earth from the
transmitting antenna to the receiver antenna is called ground wave propagation”.
Ground wave propagation of radio waves is possible only when the antennas are
close to the surface of the earth. For high efficiency, the size of the antenna should at
least be
1 𝑡𝑕
4
of the wave length of the waves. In “Surface wave” propagation, it
induces current in the earth giving rise to resistance losses and dielectric losses. The
wave is attenuated due to the absorption of energy by the ground. The attenuation
increases rapidly with the increase of frequency of the wave. Thus the maximum
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range of coverage is decided by the transmission frequency and power (less than few
MHz).
Sky wave propagation:
“When the radio waves from the transmitting antenna, reach the receiver
antenna after reflection from the ionosphere, it is called sky propagation or
ionospheric propagation”.
Different layers of ionosphere act as “radio mirror” for certain radio
frequencies. Radio waves in the range 2MHz to 30MHz directed towards the
ionosphere are reflected back to earth. Frequencies more than 30MHz penetrate into
the ionosphere and escape to space.
Space wave propagation:
“When the radio waves from the transmitting antenna, travelling in straight
line, directly reaches the receiver antenna, it is called space wave propagation or
line of sight propagation”.
At frequencies above 30 MHz, radio transmission cannot carried out by ground
waves or sky waves. The antennas in this mode of propagation are relatively smaller
and can be placed at heights of several wavelengths from the ground level. But these
may blocked due to the curvature and topography of the earth in the passage part.
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If the height of the transmitting antenna in hT the radio horizon distance of the
transmitting antenna. dT is given by d T  2RhT where R is the radius of the earth
(6400km approx).
The maximum line of sight distance dM between the two antennas with heights hT and
hR above the surface of the earth is given by d M  2RhT  2RhR .
Modulation and its necessity:
Message signals are also called base band signals. This encloses the band of
frequencies representing the original signal from the source. Usually signals are not
single frequency sinusoidal. They will have a range of signals called signal band
width. If the signals belong to audio frequency range (up to 20 KHz) over a long
distance, there will be many factors tending to affect the transmission.
1. Size of the antenna (aerial):
The antenna should have a size at least ¼ the wavelength of the signal. This is
essential for the antenna to properly sense the time variation of the signal. For
example for a signal of frequency of 20 KHz, the wavelength is 15Km and ¼ of
the value is 3.75km. Constructing such a huge antenna is not possible. Hence
direct transmission of signals is not possible, (increasing the frequency of
transmission, the size of the antenna can be reduced). Hence the original low
frequency signal should be translated to higher frequencies or radio frequency
becomes essential.
2. Effective power radiated by an antenna:
In a linear antenna of length L transmitting a signal of  , the power radiated is
2
L
proportional to   . The power radiated increases as the wavelength decreases or
 
frequency increases. For good transmission of the base signal, high powers are
required and for this the frequency should be high.
3. Mixing up of signals from different transmitters:
If base band signals are transmitted by several transmitters, they overlap and
distinguishing them becomes almost impossible. This can solved by using high
frequencies allotting specific bands of frequencies to every transmitter. The signal
can be superposed on a high frequency wave and transmitted. The high frequency
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wave transmitted. The high frequency wave carrying the base band signal is
called the carrier wave.
A sinusoidal carrier wave can be represented as C = A0 sin C t   ............(1)
Where C is the signal strength (Voltage or current), Ac is the amplitude, C is the
angular frequency and  is the initial phase of the carrier wave.
During the process of modulating any of Ac or C or  can be controlled or
modulated, resulting in as amplitude modulation (AM), frequency modulation
(FM) and phase modulation (PM). A digital wave can also be modulated on similar
concepts.
Amplitude modulation:
“When the amplitude of high frequency carrier wave is changed in
accordance with the intensity (amplitude) of the information signal, it is called
amplitude modulation”.
In amplitude modulation, only the amplitude of the carrier wave is changed in
accordance with the intensity of the signal. The modulated wave will have the same
frequency as that of the carrier wave.
„The ratio of change in the amplitude of the carrier wave to the amplitude of
the original currier wave is called modulation factor or modulation index‟.
Modulation factor ,  
A
changeinam plitudeofc arrierwave
 m 
Amplitudeo foriginalc arrierwave(un mod ulated ) A
C
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Let c (t) = Ac sin C t represent carrier wave and m (t) = Am sin C t represent
the sinusoidal message or modulating signal. When audio frequency wave is mounted
over the carrier wave, we get amplitude modulated wave
The amplitude modulated wave is represented by,
cm  Ac  m(t )Sinct.
cm   Ac  Am sin mt Sinct.
 A

cm  Ac 1  m sin mt  sin ct  Ac (1   sin mt ) sin ct.
Ac


The modulated signal contains the message signal.
cm  Ac sin ct  Ac sin mt sin ct
Using sinA sinB =
1
2
[cos (A-B) – cos (A+B)]
1

cm  Ac sin c t  Ac  cos(c  m )t  cos(c  m )t 
2

cm  Ac sin ct 
Ac
2
cos(c  m )t 
Ac
2
cos(c  m )t...............(1)
Equation (1) indicates that
1. The AM wave is equivalent to summation of three sinusoidal waves, one having
amplitude Ac and other two having amplitude
 0 Ac
2
each.
2. The AM wave contains three frequencies namely, unmodulated carrier frequency
c , and two new frequencies (c  m ) and (c  m ) are called lower side and
upper side frequencies.
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If the carrier waves from different broadcasting transmitters are spaced out so
that the side bands do not overlap, different stations can operate without any
overlap on one another.
Production of amplitude modulated wave:
Amplitude modulation can be produced by a variety of methods. A simple
method is as shown in the block diagram.
Let, m (t) = Am sin ect represent the message signal and
C (t) = Ac sin ct represent carrier wave.
Let the two waves be added to produce the signal
This signal x (t) is passed through a square law device which produces an output
represented by,
y (t) = Bx(t) + C x2(t),
where B and C are constants.
y(t ))  BAm sin mt  BAc sin ct  C ( Am sin mt  Ac sin ct ) 2
y(t ))  BAm sin mt  BAc sin C t  C ( Am2 sin m2 t  Ac2 sin c2t  2 Ac Am sin mt sin ct )
Using trigonometric relations,
Sin 2 
1
1  cos 2
and sinA sinB = [cos(A-B) – cos(A+B)]
2
2
we get,
y(t )  BAm sin mt  BAc sin ct 
CAm2
CA2
(1  cos 2mt )  c (1  cos 2ct )
2
2
 CAc Am [cos(c  m )t  cos(c  m )t ]
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y(t )  BAm sin mt  BAc sin ct  {
CAm2 CAc2
CA2
CA2

}  m cos 2mt  c cos 2ct
2
2
2
2
 CAc Am cos(c  m )t  CAc Am cos((c  m )t
The term
CAm2 CAc2
represents a dc term. When the signal y(t) is passed through

2
2
a band pass filter, it rejects the dc part and the frequencies, m , 2m , and 2c , and
retains c , (c  m ) and (c  m )
The output of the band pass filter circuit is,
y(t )  BAc sin ct  CAc Am cos(c  m )t  CAc Am cos(c  m )t
Thus the output from the band pass filter is of the same form as the AM wave (as in
equation (1). The output is taken through an amplifier and then transmitted through a
suitable antenna.
The block diagram of transmitter is shown in figure.
Detection of amplitude modulated wave:
„Detection is the process of recovering the modulating signal from the modulated
carrier wave‟.
The modulated waves transmitted through the transmitting antenna into the
space are received by the receiving antenna. From the received modulated high
frequency carrier wave, audio signal has to be separated.
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Transmitted message gets attenuated in propagating through the intervening
medium. The received signal is therefore amplified. To facilitate the processing of the
signal, the carrier frequency is usually changed to a lower frequency by a device
called intermediate frequency (IF) stage. Then the signal is fed to detector. To
energies the detected signal again it is subjected to further amplification.
Modulated carrier wave contains the frequencies c , (c  m ) and (c  m )
To obtain the message signal, the following procedure is followed.
The AM signal is fed to a rectifier to produce an envelope signal containing the
message. Then the envelop signal is passed through an envelope detector (usually RC
circuit). The output will be the message signal.
A block diagram of detection is as follows.
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