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Unit-1 Elasto-dynamics
Page 1
Unit-1
Elasto-dynamics
Syllabus:
Simple Harmonic Motion, Electric Flux, displacement vector, Columb law,
Gradient, Divergence, Curl, Gauss Theorem, Stokes theorem, Gauss law in
dielectrics, Maxwell’s equation: Integral & Differential form in free space,
isotropic dielectric medium.
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Periodic motion:
If an object repeats its motion on a definite path after a regular time interval then such type of motion is
called periodic motion.
1) Vibratory motion or oscillatory motion
2) Uniform circular motion
3) Simple harmonic motion
Vibratory motion:
If a body in periodic motion moves to and fro about a definite point on a single path, the motion of the body
is said to be vibratory or oscillatory motion.
Mean or equilibrium position:
The point on either side of which the body vibrates is called the mean position or equilibrium position of the
motion.
Time period:
The definite time after which the object repeats its motion, is called time period and it is denoted by .
Frequency:
The number of complete oscillation in one second is called the frequency of that body, it is represented by
the letter or or � its unit is .
Uniform circular motion:
Figure(1): Uniform circular motion
Let an object is moving on a circular path of radius with uniform angular velocity � = .
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Page 3
In right angle triangle Δ
∠ = � +� �
= cos � +� �
= cos � +� �
= .� cos � +� �
But � =
so = .� cos� � +� �
Similarly
= sin � +� �
= sin � +� �
= .� sin � +� �
= r.� sin� � +� �
Both equation (1) and (2) represents the uniform circular motion.
Simple (armonic Motion S(M :
When a body moves periodically on a straight line on either side of a point, the motion is called the simple
harmonic motion.
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Page 4
Graphical representation of SHM
Figure(2): Graphical representation of SHM
Displacement in SHM:
Let a particle is moving on a circular path with uniform angular velocity " " and the radius of the circular
path is " "; then movement of the point on their axis i.e. and is the SHM about the mean position
Figure(3): SHM
Let at time � =� the particle is on point and after time the position of the particle is then
In Δ
= sin
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= .� sin …………………………………………… (1)
= .� sin�
This equation represents the displacement of foot dropped from the position of particle on � −� .
Velocity in SHM:
Differentiating equation (1) with respect to we get-
= .� sin�
= rω cos� …………….
= √ � −� sin
= √ −� sin
= √ −� Using (1)
(i) In equilibrium condition � =�
So
= √ −�
=
(ii) In the position of maximum displacement i.e. � =�
So
= √ −�
=
Acceleration:
Again differentiating equation (2) we get-
=
�
= rω cos�
= −rω sin�
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Page 6
= − ………………………………………………………… (3)
+� =
This is a second order differential equation which denotes the equation of SHM in the differential form
Again by equation (3)
= −
Multiplying by i.e. the mass of the particle executing SHM then
= −
= −
Here negative sing shows that the direction of displacement and acceleration are opposite to one another
So ∝ −
� =�
Time period and frequency:
=
⇒
= √
�
= √
�
= √
�
= √
�
And = √�
Question: A uniform circular motion is given by the equation � =� � sin � +� ., find
1) Amplitude
2) Angular frequency
3) Time period
4) Phase
Sol: Given: � =� � sin� +� .
Comparing the given equation with the standard equation of uniform circular motion i.e. � =
� sin � +� �
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Page 7
We get
� =�
� =�
�
� = = =� .
� = = =� .
Question: A particle is moving with SHM in a straight line. When the displacement of the particle from
equilibrium position has values and , the corresponding position has valocities and
show that the time period of oscillation is given by
� =� √
−�
−�
Sol: In the SHM the velocity is given by-
= √ −� …………………………………… (1)
At velocity is
So
= √ −�
Squaring both sides
= −� …………………………. (2)
Again at the velocity is
So
= −� …………………………. (3)
By equation (2) and (3)
−� = −� � −� −�
−� = −�
−�
−�
=
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Page 8
=
−�
−�
= √
−�
−�
…………………………. (4)
Now � =
So
= √
−�
−�
Question: If the earth were a homogeneous sphere and a straight hole was bored in it through
the centre, then a body dropped in the hole, execute SHM. Calculate the time period
of its vibration. Radius of the earth is . � ×� 6
and � =� . −
Solution: The time period of oscillation executed by the body dropped in the hole along the
diameter of earth
� =� √=� √
. � ×� 6
.
=� .
Energy of a particle executing SHM:
A particle executing SHM possess potential energy on the account of its position and kinetic energy
on account of motion.
Potential energy:
We know that the acceleration in a simple harmonic motion is directly proportional to the displacement
and its direction is towards the mean position
= −
Let is the mass of particle executing SHM then the force acting on the particle will be-
= .�
= −
If the particle undergoes an infinitesimal displacement against the restoring force, then the small amount of
work done against the restoring force is given by
= − .�
Here negative sign shows that the restoring force is acting the displacement than
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Page 9
=
So the total amount of work done
= ∫�
=
This work done is equal to the potential energy of the particle at displacement
i.e. =
Kinetic energy:
If is the velocity of the particle executing SHM, when the displacement is then kinetic energy
=
But for SHM � =� √ −�
Where is the amplitude of SHM
So
= � √ −�
⇒ = −� ……………………………. (2)
Total energy:
Now the total energy
= � +�
⇒ = + −�
⇒ = + −
⇒ =
Thus we find that the total energy:
1) � ∝�
2) � ∝� of SHM
3) � ∝� of SHM
Graphical representation of total energy of SHM
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Page 10
Figure(4): Total energy of SHM
Position vector:
A position vector expresses the position of a point P in space in terms of a displacement from an arbitrary
reference point O (typically the origin of a coordinate system). Namely, it indicates both the distance and
direction of an imaginary motion along a straight line from the reference position to the actual position of
the point.
Displacement Vector:
A displacement is the shortest distance from the
initial to the final position of a point P. Thus, it is
the length of an imaginary straight path, typically
distinct from the path actually travelled by
particle or object. A displacement vector
represents the length and direction of this
imaginary straight path. Figure(5): Displacement vector
Area Vector:
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In many problems the area is treated as a vector,
an area element is represented by ⃗⃗⃗⃗ , such
that the area representing the area vector ⃗⃗⃗⃗ is
perpendicular to the area element. The length of
the vector ⃗⃗⃗⃗ represents the magnitude of the
area element
Figure(6): Area vector
Coulomb’s Law:
According to it the force of attraction or repulsion
between the two point charges is directly
proportional to the product of the magnitude of the
charges and inversely proportional to the square of
the distance between them.
If two charges and are separated at a distance
form one another then the force between these
charges will be-
Figure(7): Two electric charges separated a distance r
i) Force is proportional to the product of the magnitude of the charges i.e. � ∝� .�
ii) The force is inversely proportional to the distance between the charges i.e. � ∝
So
� ∝
.�
� =�
.�
Where is a proportionality called electrostatic force constant, its value depends on the nature of the
medium in which the two charges are located and also the system of units adopted to measure ,� and .
So
� =� .
.�
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Case 1:(when the medium between the charges is air or vacuum )
As we know that the force between the charges is given as-
� =� .
.�
If we put =� =� and � =� then
� =�
So is the force feels by two charges of placed apart from one another in vacuum or free space.
Its value is � =� � ×�9
� ×� ×�
Case 2:(When the medium between the charges is other than the vacuum)
If the changes are located in any other medium then
� = . =� � ×�9
.
Where is the dielectric constant of relative permittivity.
Putting this value in equation (1) we get
′
= .
.�
Where ′ is the force in the medium
′
= .
.�
Where � =� is called the relative permittivity of the medium.
Vector form of the Coulomb’s Law
Consider two like charges and present at and in vacuum at a distance apart. The two charges
will exert equal repulsive force on each other,
Let be the force on charge due to the charge and be the force on charge due to charge .
According to the Coulo s’ la , the ag itude of fo e o ha ge and is given by
| |. | | =
.
………………………… (1)
Let ̂ and ̂ are the unit vectors in the direction from to and vice versa.
So the force is along the direction of unit vector ̂ , we have
⃗ = .
.�
̂
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Page 13
And
⃗ = .
.�
̂
These two equations show the Coulo s’ la i e to fo .
Electric flux:
Number of electric lines of forces passing normally through the surface, when held in the electric field. It is
denoted by � . There are two types of electric flux-
1. Positive electric flux: When electric lines of forces leave any body through its surface it is considered
as positive electric flux.
2. Negative electric flux: When lines of forces enter through any surface, it is considered as the
negative electric flux.
Measurement: Let us consider a small area ⃗⃗⃗⃗ of a
closed surface . The electric field ⃗ produced
due to the charge will be radially outwards
which will be along ̂. Now the normal to the
surface area is ⃗⃗⃗⃗ as shown in the figure,
hence the angle between ⃗⃗⃗⃗ and ̂ is �
So the electric lines of forces from the surface
area will be given as-
� = ⃗ .� ⃗⃗⃗⃗
� = � cos� �………….
Figure(8): Electric flux
Where � cos� �is the component of electric field ⃗ along ⃗⃗⃗⃗ .
Hence the electric flux through a small elementary surface area is equal to the product of the small area and
normal component of ⃗⃗ along the direction of the elementary area⃗⃗⃗⃗⃗ .
Over the hole surface,
� = ∮ � cos� �
� = ∮ ⃗ .� ⃗⃗⃗⃗ ………………………… (2)
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Gradient of a scalar field:
The gradient of a scalar function � is a vector whose magnitude
is equal to maximum rate of chcnge of scalar function � with
respect to the space variable ∇⃗⃗ and has direction along that
change.
�� =
�
̂
In the scalar field let there be two level surfaces and close
together characterised by the scalar function � and �� +� �
respectively. Consider point and on the level surfaces and
respectively. Let and � +�⃗⃗⃗⃗ be the position vector of and
. Then ⃗⃗⃗⃗⃗ = ⃗⃗⃗⃗ =� ̂ � +� ̂ � +�̂
Now as � is a function of ,� ,� i.e.
�� =� � ,� ,�
Then the total differentiation of this function can be given as
Figure(9): Gradient of a scalar field
� =
�
� +
�
� +
�
� = ( ̂
�
+� ̂
�
+� ̂
�
)� .� ( ̂ � +� ̂ � +�̂ )
� = ∇⃗⃗ � ⃗⃗⃗⃗ …………………………………………………… (1)
Agian if represents the distance along the normal from point to the surface to point , then
=
In the ∆
= cos� �
= cos� �
Now if we consider a unit vector along as ̂
then
= ⃗⃗⃗⃗ .� ̂ …………………………………………………… (2)
If we proceed form to then value of scalar function � increases by an amount �
� =
�
� =
�
�
⃗⃗⃗⃗ .� ̂ [Usi g ……………………………. (3)
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(∇⃗⃗ .� �).� ⃗⃗⃗⃗ =
�
�
⃗⃗⃗⃗ .� ̂
(∇⃗⃗ .� �) =
�
�
̂
� =
�
�
̂
Note: ∇⃗⃗ =� ̂ +� ̂ +� ̂ is called del or Nabla operator.
Note: � = ∇⃗⃗ .� �
� = ( ̂ +� ̂ +� ̂ )� .� �
� = ( ̂
�
+� ̂
�
+� ̂
�
)
Note: The gradient of a scalar field has great significant in physics. The negative gradient of
electric potential of electric field at a point represents the electric field at that point. i.e.
⃗ =� −
Note: The gradient of a scalar field is a vector quantity.
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Divergence of a vector field:
The divergence of a vector field at a
certain point ,� ,� is defined as the
outward flux of the vector field per unit
volume enclosed through an infinitesimal
closed surface surrounding the point " ".
=� lim
�→
.� ⃗⃗⃗⃗
�
=� lim
�→
�
�
Consider a infinitesimal rectangular box with
sides Δ ,� Δ ,� Δand one corner at the point
,� ,� in the region of any vector
function with rectangular faces
perpendicular to co-ordinates axis.
Figure(10): divergence of a vector field
The flux emerging outwards from
surface i.� e.� surface, �
= ∬ ̅ .�
� = ∬ ( ̂ ̅ +� ̂̅ +� ̂ ̅ ).� ̂Δ ,� Δ
Where ̅̅̅
is the average of the vector function over thesurface i.e. surface
� = ∬ ̅̅̅̅̅.� Δ .� Δ…………………………………………. (1)
Similarly
The flux emerging out from the
surface i.e. surface ,� �
= ∬ ̅ .�
� = ∬ ( ̂ ̅ +� ̂̅ +� ̂ ̅ ).� − ̂Δ ,� Δ
� = ∬ − ̅ .� Δ .� Δ………………………………………. (2)
Thus net outwards flux of vector through the two faces perpendicular to � −axis,
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� = � +� �
� = ∬ ̅ −� ̅ Δ .� Δ ……………….. (3)
But ̅̅̅̅̅ −� ̅̅̅̅̅ = � +� Δ ,� ,�−� ,� ,�
̅̅̅̅̅ −� ̅̅̅̅̅ = Δ …………………………………………………… (4)
Where
��
is the variation of with distance along � −axis by equation (2) and (3)
Thus net outward flux of vector function through the two faces perpendicular to � −axis
� =
��
Δ Δ ,� Δ [ Using equation (3)
Similarly perpendicular to � −axis
� = Δ Δ Δ
Similarly perpendicular to � −axis
� = Δ Δ Δ
Therefore whole outward flux through infinitesimal box
� = � +� �+� �
� = + + Δ Δ Δ
Now at any point, which is the flux enclosed per unit infinitesimal volume surrounding that point is
given by-
= lim
Δ Δ Δ →
�
Δ Δ Δ
=
lim
Δ Δ Δ →
( + + )� Δ Δ Δ
Δ Δ Δ
= + +
= ( ̂ +� ̂ +� ̂ ) ( ̂ +� ̂ +� ̂ )
= ∇⃗⃗ .�
Note: Divergence of a vector field is a scalar quantity.
Note: If =� +
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Page 18
it indicates the existence of the source of fluid at that point.
Note: If =� −
It means fluid is flowing towards the point and thus there exist a sink for the fluid.
Note: If =�
It means the fluid is flowing continuously from that point. In other words this means that the flux of
the vector function entering and leaving this region is equal. This condition is called solenoidal
vector.
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Curl of a vector field:
If is any vector field at any point and an
infinitesimal test area at point then
=� lim
→
∮ .� ⃗⃗⃗⃗
̂
Let us consider an infinitesimal rectangular area
with sides Δ and Δ parallel to � −�
plane in the region of vector function ⃗⃗ .
Let the coordinate of be ,� ,� . If
,� ,� are the Cartesian components of
at then
⃗⃗ =� ̂ +� ̂ +� ̂
Figure(11): Curl of a vector field
Now the line integral of vector field
along the path
= ∫� .� ⃗⃗⃗⃗
= (̂̅ +� ̂̅ +� ̂̅ ).� ̂ Δ
= ̅ Δ
Where ̅ is the average value of � −component of the vector function over the path
Similarly for the Path
= ∫� .� ⃗⃗⃗⃗
= (̂̅ +� ̂̅ +� ̂̅ ).� − ̂ Δ
= − ̅ Δ
Where ̅ is the average value of � −component of vector function over the path .
Hence the contribution to line integral ∮ ⃗⃗ . ⃗⃗⃗⃗ form two path and parallel to � −axis is
= −
= − − Δ
As the rectangle is infinitesimal the difference between the average of .� .̅ − ̅ along these two
paths may be approximated to the difference between the values of at and
Thus-
̅ − ̅ = −�
̅ − ̅ = ,� � +� Δ ,�−� ,� ,�
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Page 20
̅ − ̅ = Δ
Hence the contribution to the line integral ∮ .� ⃗⃗⃗⃗ from the path and
= Δ Δ …………………………………………… (2)
Similarly by the path and
= Δ Δ …………………………………………… (3)
Therefore the line integral along the whole rectangular form (2) and (3) is given by-
= ∮� +� .� ⃗⃗⃗⃗
= ∮� .� ⃗⃗⃗⃗
= − Δ Δ ……………………………… (4)
Now = lim
Δ Δ →
=
lim
Δ Δ →
( − )� Δ Δ
Δ Δ
= − ……………………………………. (5)
Similarly
=
( − ) ……………………………………. (6)
and = − ……………………………………. (7)
Summing up the results given in (5), (6) and (7) we get
= ̂ +� ̂ +� ̂
= ̂� ( − )� +� ̂� (− )� +�̂ ( − )
=
[
̂ ̂ ̂
]
= ∇⃗⃗ ×�
Note: The curl of a vector field is sometime called circulation or rotation or simply .
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Note: If =� then vector field is called Lamellar field.
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Page 22
Gauss’ Divergence Theorem:
According to this theorem the volume integral of
divergence of a vector field over a volume is
equal to the surface integral of that vector field
taken over the surface which enclosed that
volume . i.e.
∭( ) � =� ∬�.� ⃗⃗⃗⃗
�
Consider a volume enclosed by a surface this
volume can be divided into small elements of
volumes ,� …� …�� which are enclosed by the
elementary surface ,� …� …� …� …��
respectively. By definition the flux of a vector
field diverging out of the ℎ
element is
Figure(12): Gauss’ Di e ge e tho e
( )�
=
.� �
⃗⃗⃗⃗⃗⃗
�
�
( )�
.� � = ∬� .� ⃗⃗⃗⃗
�
………………………………………………… (1)
On LHS of equation we add the quantity ( )�
.� � for each element ,� …� …��
∑( )�
.� �
�
�=
= ∭( )
�
On RHS of equation (1) if we add the quantity .� ⃗⃗⃗⃗
�
for each ,� …� …� …� …�� we get the terms only on
the outer surface
Sum comes out to be
∑� ∬�.� �
⃗⃗⃗⃗⃗⃗
�
�
�=
= ∬� .� ⃗⃗⃗⃗
So putting these values in equation (1) we get
So ∭( )
�
= ∬� .� ⃗⃗⃗⃗
This is the Gauss’ di e ge e theo e .
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Page 23
Stokes theorem:
According to this theorem, the line integral of a vector field along the boundary of a closed curve is
equal to the surface integral of curl of that vector field when the surface integration is done over a surface
enclosed by the boundary i.e.
∮� .� ⃗⃗⃗ =� ∬� .� ⃗⃗⃗⃗
Figure(13): Stokes theorem
Consider a vector which is a function of position. We are to find the line integral
∮ .� ⃗⃗⃗ along the boundary of a closed curve . If we divide the area enclosed by the curve in two parts by
a line , we get two closed curve and . The line integral of vector along the boundary of will be
equal to the sum of integral of along and
∮� .� ⃗⃗⃗
= ∮� .� ⃗⃗⃗ +� ∮�.� ⃗⃗⃗
Similarly if we divide the area enclosed by the curve in small element of area …� …� …� …by the
curve ,� …� …� …� ….As shown in the figure. Then the sum of line integrals along the boundary of these
curves ,� …� …� …� ..(taken anticlockwise) will be
∮� .� ⃗⃗⃗
= ∑� ∮�.� ⃗⃗⃗
By the definition of curl, we have
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=
∮ .� ⃗⃗⃗
�
.� �
= ∮� .� ⃗⃗⃗
∮� .� ⃗⃗⃗
= ∑� .� �
⃗⃗⃗⃗⃗⃗⃗ =� ∬� .� ⃗⃗⃗⃗
∮� .� ⃗⃗⃗
= ∬� .� ⃗⃗⃗⃗
Gauss Law
According to this law, the net electric flux through any closed surface is times of the total charge
present inside it.
� = ………………………… (1)
But by the definition of electric flux
⇒ � = ∬� ⃗ .� ⃗⃗⃗⃗ …� …� …� …� …� …� …� …� …� …(2)
So by equation (1) and (2)
so ∬� ⃗ .� ⃗⃗⃗⃗
=
This is the i teg al fo of Gauss’ la .
Proof:
Case1:
When the charge lies inside the arbitrary
closed surface.
Let charge lies inside the arbitrary surface at
point
Now let us consider an infinitesimal area ⃗⃗⃗⃗
on this surface which contain the point , the
direction of the area vector ⃗⃗⃗⃗ is
perpendicular to the surface and electric field
Figure(14): Gauss Law
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Page 25
⃗ makes an angle � with ⃗⃗⃗⃗ then electric field
will be given as-
⃗ = ……………………………………. (3)
Now the flux emerging out of the surface area ⃗⃗⃗⃗ will be
� = ⃗ .� ⃗⃗⃗⃗
⇒ � = � cos� �
Where � is the angle between ⃗ and ⃗⃗⃗⃗
So putting the value of ⃗ we get
� = � cos� �
⇒ � = � cos� �
But
� cos� �
=� i.e. solid angle
� =
Now total flux
� = ∫
⇒ � = ∫�
But � =�
� =
⇒ � =
Case 2:
When the charge lies outside the closed surface then the flux entering and leaving the surface
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Page 26
area will be equal and opposite then �� =�
Gauss law in the differential form (Poisson’s equation and Laplace’s equation
If the charge is continuous distributed over the volume and charge density is
then = ∭�
�
Now by Gauss theorem the flux emerging out of this surface which enclosed volume
∬� ⃗ .� ⃗⃗⃗⃗
= ∭�
�
…� …� …� …� …� …� …� …� …� …(1)
By Gauss divergence theorem
∬� ⃗ .� ⃗⃗⃗⃗
= ∭� ⃗
�
…� …� …� …� …� …� …� …� …� …(2)
⇒ ∭� ⃗
�
= ∭�
�
⇒ ∭� ( ⃗ − )
�
=
But as we know that � ≠�
So ⃗ − =
⇒ ⃗ = …� …� …� …� …� …� …� …� …� …(3)
This is the diffe e tial fo of Gauss’ la a d also k o as Poisso ’s e uatio
Now if we consider the charge less volume then � =�
So ⃗ = …� …� …� …� …� …� …� …� …� …(4)
This equation is Laplace equation.
Again by equation (3)
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Page 27
⃗ =
We know that ⃗ =� −
So
− =
⇒
−∇⃗⃗ .� ∇⃗⃗ =
⇒
∇ = −
⇒
+ + = −
Gauss law in Presence of dielectrics :
The Gauss’ la elates the ele t i flu a d ha ge. The theo e states that the et ele t i flu a oss a
arbitrary closed surface drown in an electric field is equal to times the total charge enclosed by the
surface. Now we want to extend this theorem for a region containing free charge embedded in dielectric.
In figure the dotted surface in an imaginary closed surface drown in a dielectric medium. There is certain
amount of free charge in the volume bounded by surface. Let us assume that free charge exists on the
surface of three conductors in amount ,� ,� …� ..In a dielectric there also exits certain amount of
polarisation (bound) charge .
He e Gauss’ theo e
∬� ⃗ .� ⃗⃗⃗⃗
= ( ′
+� ) ……………….
Where � =� +� +� is the total free charge and is the polarisation (bound) charge by
= ∬ ⃗ .� ⃗⃗⃗⃗
+ +
+� ∭ −
�
……………….
Here is the volume of the dielectric enclosed by . As there is no boundary of the dielectric at ,
therefore the surface integral in equation (2) does not contain a contribution from . If we transform
volume integral in (2) into surface integral by means of Gauss divergence theorem, we must include
contribution from all surface bounding , namely ,� ,� and i..e.
∫ −
�
= [ ∬ ⃗ .� ⃗⃗⃗⃗
+ +
+� ∭ −
�
]
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Page 28
Using above equation, we note that
= ∬ ⃗ .� ⃗⃗⃗⃗
+ +
……………….
Substituting this value in (1)
We get
∬� ⃗ .� ⃗⃗⃗⃗
= � − ∬� ⃗ .� ⃗⃗⃗⃗
∬ ⃗ +
⃗
.� ⃗⃗⃗⃗ =
Multiplying through by
∬( ⃗ +� ⃗ ).� ⃗⃗⃗⃗
= ……………….
This equation states that the flux of the vector ⃗ +� ⃗ through a closed surface is equal to the total
free charge enclosed by the surface. This vector quantity is named as electric displacement ⃗⃗ i.e.
⃗⃗ = ⃗ +� ⃗ ……………………..
Evidently electric displacement ⃗⃗ has the same unit as ⃗ . i.e. charge per unit area.
In terms of electric displacement vector ⃗⃗ , equation (4) becomes
∬� ⃗⃗ .� ⃗⃗⃗⃗
= ……………………..
i.e. the flux of electric displacement vector across an arbitrary closed surface is equal to the total free
charge enclosed by the surface.
This esult is usuall efe ed to as Gauss’ theo e i diele t i .
If e o side i to a la ge u e of i fi itesi al olu e ele e ts, the Gauss’ theo e a
expressed as
∬� ⃗⃗ .� ⃗⃗⃗⃗
= ∭�
�
……………………..
Where is the charge density at a point within volume element such that � →�.
∭� ⃗⃗ .�
�
= ∭�
�
∭ ⃗⃗ −� .�
�
=
Volume is arbitrary, therefor we get
⃗⃗ −� =
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Page 29
⃗⃗ =
This result is called differential fo of Gauss’ theo e i a diele t i .
The main advantage of this method is that the total electrostatic field at each point in the dielectric
medium may be expressed as the sum of parts
,� ,� = ⃗⃗ ,� ,� − ,� ,� …………………..….
Where the first term ⃗⃗ is related to free charge density through the divergence and the second
theorem is proportional to the polarisation of the medium. In vacuum � =� so ⃗ =
⃗⃗
Electric Polarization �
When a dielectric is placed in any external electric field then the dielectric gets polarized and
induced electric dipole moment is produced which is proportional to the external applied electric
field. Now if there are number of dipoles induced in per unit volume of dielectric then total
polarization will be-
⃗ = ��
⃗⃗⃗⃗⃗ …� …� …� …� …� …� …� …� …� …(1)
But
��
⃗⃗⃗⃗⃗ ∝ ⃗⃗⃗⃗
So
��
⃗⃗⃗⃗⃗ = ��
⃗⃗⃗⃗
Putting this value in equation (1) we get
⇒ ⃗ = ��
⃗⃗⃗⃗
It is clear from the above equation that the direction of polarization is in the direction of the
applied external electric field. And the unit is /
Electric displacement
We know that the value of electric field depends on the nature of the material, so to study the
dielectric we need such a quantity which does not depends on the nature of the material and this
quantity is known as electric displacement vector ⃗⃗ .
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Page 30
Both ⃗ and ⃗⃗ are same except that we define ⃗ by a force in a charge placed at a point while the
displacement vector is measure by the displacement flux per unit area at that point.
∭� ⃗⃗ .� ⃗⃗⃗⃗ =
Or
=
⇒ = �
Where � is the surface charge density.
Relation between ⃗⃗ and ⃗⃗
We know that the Gauss law is given as-
∬� ⃗ .� ⃗⃗⃗⃗
=
Where is the permittivity of the dielectric medium
⇒ ⃗ = .
But
�
=� so we have ⃗ = ⃗⃗ ⇒� ⃗⃗ =� ⃗
⇒� ⃗⃗ =� ⃗⃗
� � =�
Where is the permittivity of the free space
Current:
Current for study current
� =
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Page 31
If the charge passing per unit time is not constant, then the current at any instant will be given as
� =
Current density:
=
⃗⃗⃗⃗
= .� ⃗⃗⃗⃗
= ∫� .� ⃗⃗⃗⃗ =
From the above equation we can see that the current is the flux of current density as
�� =� ∫�⃗ ⃗⃗⃗⃗
Its SI unit is
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Page 32
Equation of continuity:
The law of conservation of charge is called the equation of the continuity.
� =� ∬�.� ⃗⃗⃗⃗
For steady current charge does not stay at any
place, so the current will be constant for all the
places.
Figure(17): Flux of current
⇒ = ∬� .� ⃗⃗⃗⃗ =�
By divergence theorem
⇒ ∬� .� ⃗⃗⃗⃗
= ∭� .�
�
So ∭� .�
�
=
On differentiating we get
=
This is the equation of continuity for study current.
Now if current is not stationary i.e. if the current is the function of the time and position
then = ∬� .� ⃗⃗⃗⃗ =� −
Here negative sign shows that the charge is reduced with respect to time.
But if is the charge per unit volume then-
= ∭� .�
�
So ∬� .� ⃗⃗⃗⃗
= − ∭� .�
�
⇒ ∭� .�
�
= − ∭� .�
�
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Page 33
⇒ ∭� ( + )�
�
=
⇒ = −
This is the equation of continuity for time varying current.
Maxwell’s equations
James Clerk Maxwell took a set of known experimental laws (Faraday's Law, Ampere's Law) and
unified them into a symmetric coherent set of Equations known as Maxwell's Equations. These
equations are nothing but the relation between electric field and magnetic field in terms of
divergence and curl.
S.N. Name Integral form Differential form
1
Gauss’ La fo
electricity
∬� ⃗ .� ⃗⃗⃗⃗ = ∭�
�
⃗ =
2
Gauss’ la fo
magnetism
∬� ⃗ .� ⃗⃗⃗⃗ =� ⃗ =�
3
Fa ada ’s La of
induction
∮� ⃗ .� ⃗⃗⃗ = ∬� ⃗ .� ⃗⃗⃗⃗
⃗ =� −
⃗⃗
4 A pe e’s la ∮� ⃗ .� ⃗⃗⃗ =� ∬�⃗⃗⃗ .� ⃗⃗⃗⃗ + ∬� ⃗⃗ .� ⃗⃗⃗⃗ ⃗ =� � +�
⃗
Maxwell’s first equation Gauss’ law in electric):
Let us consider a volume which is enclosed in a surface , the Gauss’ la the ele t i flu is
given as
∬� ⃗ .� ⃗⃗⃗⃗
= � …� …� …� …� …� …� …� …� …� …(1)
Where is the totat charge enclosed in the volume
Now if is the volume charge density then
= ∭�
�
…� …� …� …� …� …� …� …� …� …
(2)
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Page 34
⇒ ∬� ⃗ .� ⃗⃗⃗⃗
= ∭�
�
This is the i teg al fo of Ma ell’s e uatio .
B Gauss’ di e ge e theo e
⇒ ∬� ⃗ .� ⃗⃗⃗⃗
= ∭� ⃗
�
So by applying this on above equation we get
⇒ ∭� ⃗
�
= ∭�
�
⇒ ∭� ( ⃗ − )�
�
=
But � ≠� so
⇒ ⃗ − =
⇒ ⃗ =
⇒ ⃗ =
⇒ ⃗⃗ = [ � ⃗⃗ =� ⃗
Maxwell’s second equation Gauss’ law in magnetism :
Since the magnetic lines of forces are closed curves so the magnetic flux entering any orbitri
surface should be equal to leaving it
mathematically
⇒ ∬� ⃗ .� ⃗⃗⃗⃗
= � …� …� …� …� …� …� …� …� …�(1)
This is i teg al fo of Ma ell’s se o d e uatio .
No Gauss’ di e ge e theo e
⇒ ∬� ⃗ .� ⃗⃗⃗⃗
= ∭� ⃗
�
So equation (1) can be written as-
⇒ ∭� ⃗
�
=
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Page 35
As � ≠� so
⇒ ⃗ =
Maxwell’s third equation Faraday’s law :
A o di g to Fa ada ’s la of ele t o ag eti i du tio if the ag eti flu li ked ith a losed
circuit changes with time then a is induced in the close circuit which is known as induced
the direction of the induced will be such as it oppose the change in the magnetic flux.
It is given as
⇒ = −
�
…� …� …� …� …� …� …� …� …� …(1)
But Gauss’ theo e e k o that
⇒ � = ∬� ⃗ . ⃗⃗⃗
So = − ∬� ⃗ ⃗⃗⃗⃗
Now if ⃗ is the electric field produced due to the change in the magnetic flux then the induced
will be equal to the line integral of ⃗ along the circuit. i.e.
⇒ = ∮ ⃗ .� ⃗⃗⃗ …� …� …� …� …� …� …� …� …� …(2)
⇒ ∮� ⃗ .� ⃗⃗⃗
= − ∬� ⃗ ⃗⃗⃗⃗
⇒ ∮� ⃗ .� ⃗⃗⃗
= −� ∬
⃗
⃗⃗⃗⃗ …� …� …� …� …� …� …� …� …� …(3)
No Stokes’ theo e
⇒ ∮� ⃗ .� ⃗⃗⃗
= ∬� ⃗ .� ⃗⃗⃗⃗
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Applying this to the above equation, we get
∬� ⃗ .� ⃗⃗⃗⃗
= −� ∬
⃗
⃗⃗⃗⃗
⇒ ∬� ⃗ +
⃗
� ⃗⃗⃗⃗ =
As ⃗⃗⃗⃗ ≠�
So
⃗ +
⃗
=
⇒ ⃗ = −
⃗
Maxwell’s fourth equation Maxwell’s correction in Ampere’s law
A pe e’s La is gi e as
⇒ ⃗ =
This equation is true only for time independent electric field and to correct this equation for time
varying field a term must be added
⇒ ⃗ = ( +� ) …� …� …� …� …� …� …� …� …� …(1)
Taking of both side and for simplicity writing as
⇒ ( ⃗ ) = ( +� )
But divergence of curl of any quantity is always zero so ( ⃗ )� =�
Then ( +� ) = ………………………………………. (2)
⇒ = − …� …� …� …� …� …� …� …� …� …(3)
But by the equation of continuity
⇒ = − …� …� …� …� …� …� …� …� …� …(4)
A d Ma ell’s fi st e uatio
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⃗ =
⇒ = ⃗ …� …� …� …� …� …� …� …� …� …(5)
By (4) and (5)
⇒ = − ( ⃗ )
⇒ = − ( ⃗⃗ ) …� …� …� …� …� …� …� …� …� …(6)
Again by (3) and (6)
⇒ − = − ( ⃗⃗ )
⇒ = ( ⃗⃗ )
⇒ = � ( ⃗⃗ )
⇒ =
⃗⃗
Putti g this alue i A pe e’s la e get
⃗ =� +
⃗⃗
This is Ma ell’s fou th e uatio .
For vacuum ⃗ =� and � =�
So
⃗⃗ = +�
⃗
⇒ ⃗⃗ = +�
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Unit-II (LASER)
Unit-2
LASER
Syllabus:
Properties of lasers, types of lasers, derivation of Einstein A & B
Coefficients, Working He-Ne and Ruby lasers.
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LASER:
The word LASER is acronym for light amplification by stimulated emission of radiation. Laser source produces
coherent, monochromatic, least divergent, unidirectional and high intense beam. Einstein gave the theoretical
description of stimulated emission in 1917. In 1954 G.H. Towne developed microwave amplifier MASER using
Einstein’s theo y a d put forward to light and first Laser was developed.
Characteristics of Laser beam:
i) Coherent: The Laser light is coherent. A Laser emits the light waves of same wavelength and in
same phase.
ii) Monochromatic: If the light coming from a source has only one frequency or single wavelength is
called monochromatic source and the light is called monochromatic light. In case of Laser beam it
has the wavelength confirmed to very narrow range of a few angstroms.
iii) Divergence: Divergence is the measure of its spread with distance. The angular spread in ordinary
light is very high because of its propagation in the form of a spherical wave front. The divergence
in the Laser beam is negligible. A very small divergence is due to the diffraction of Laser light when
it emerges out from the partially silvered mirror.
iv) Directionality: An ordinary source of light emits light in all directions. In case of Laser the photons
of a particular direction are only allowed to escape. Thus the Laser beam is highly directional.
v) Intensity: The intensity of ordinary light decreases as it travels in the space. This is because of its
spreading. The Laser does not spread with distance. It propagates in the space in the form of
narrow beam and its intensity remains almost constant over long distance.
Three Quantum Process:
1. Induced absorption: When an atom gains some energy by any mean in the ground state, the electrons of
the atoms absorbs some energy and are excited to high energy level.
Let us consider two energy levels � and � of an atom. Suppose this atom is expose to light radiation it can
excite the atom from ground state � to the high energy state � by absorbing a photon of frequency �. The
frequency � is given as
� =
� − �
ℎ
This process is called the induced absorption.
Pictorially it is represented as in figure(1)
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Figure(1):Induced absorption
This may also be shown by the equation
+ ℎ� ⟶∗
[* represents the exited state ]
The probability of absorption transition is given by
∝ �
= �
Where � is the energy state density
And the number of absorption transition in material is equal to the product of number of atoms at � and
absoption transition is given as
=
= �
Where is the number of atoms in ground state �
2. Spontaneous Emission: When an atom at lower energy level is exited to the high energy level, it cannot stay
in the exited state for relatively longer time. In a time of about −8
��, the atom reverts to the lower energy
state by releasing a photon of energy ℎ� = �− �. This emission of photon by an atom without any external
input is called spontaneous emission.
Figure(2): Spontaneous emission
We may write the transition as
∗
⟶ + ℎ�
Probability of spontaneous emission depends only on the properties of energy states and is depends on the
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photon density. It is equal to the life time of �
=
is Ei stei ’s oeffi ie t fo spo te ious e issio .
Number of spontaneous transition depends only on number of atoms in the excited state � . Thus
=
Process of spontaneous emission cannot be controlled from outside and photon are emitted in random order
so light is non-directional, non-monochromatic, incoherent and no amplification of light takes place.
3. Stimulated Emission: In 1916 Einstein predicted the existence of stimulated emission. A photon of
appropriate energy when incidents to an atom which is in the exited state, then it may causes the de-
excitation by the emission of an additional photon of same frequency as that of incident one, now the two
photons of same frequency moves together. This process is called the stimulated or induced emission. The
emitted photon have same direction, phase, energy and state of polarization as that of incident photon we
can rewrite the transition as
∗
+ ℎ� ⟶ + ℎ�
The probability of stimulated emission is given by
� ∝ �
� = �
is Ei stei ’s oeffi ie t of sti ulated e issio .
The number of stimulated transition in a material is given by
� = �
Where is the number of atoms in excited state �
The light produced by this process is essentially directional, monochromatic, and coherent, the outstanding
feature of this process is the multiplication of photons i.e. if there are exited atoms, photons will be
produced.
Figure (3): Multiplication of stimulated photons into an avalanche
Population Inversion:
In the thermal equilibrium number of atoms in higher energy levels is less than population of lower energy
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level . Then if � and � are two energy levels with population and then by Boltzmann relatin.
= × (
� − �
)
Since � > �⟹ < . In this situation the system absorbes electromagnetic radiation incident on it for
laser action to take place, the higher energy level should be more populated as compared to the lower energy
state i.e. > .
Thus the process by which the population of a particular high energy state is made more than that of a
specified lower energy state is called population inversion.
Figure(4): Population inversion
Meta stable States:
An atom in the exited state has very short life time which is of the order of −8
��. Therefore even if
continuous energy is given to the atoms in ground state to transfer them to exited state they immediately
comes back to the ground state. Thus population inversion cannot be achieved. To achieve population
inversion, we
Figure(5):
must have energy states which has a longer lifetime. The life time of meta stable state is −
to −
��
which is time of exited states thus allows accumulation of large number of excited atoms and
result in population inversion.
Components of Laser
the essential components of Laser are-
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Active Medium:
The active medium is the material in which the laser action takes place. It may be solid, liquid, or gas. The
medium determines the wavelength of the laser radiation. Atoms are characterized by the large number of
the energy levels, but all types of atoms are not suitable for Laser operation. Even in a medium consisting of
different species of atoms, only a fraction of atoms of particular type have energy level system suitable for
achieving population inversion. Such atoms can produce more stimulated emission than spontaneous
emission causes amplification of light. These atoms are called active center. The rest of the medium acts as
the host medium and supports the active medium. Thus the active medium is the one which when excites,
reaches the state of population inversion and promotes stimulated emission leading to light amplification.
Figure(6): Component of LASER
Optical Resonator:
It is specially designed cylindrical tube having two opposite optically plane mirrors with active medium filled
between them, one mirror is fully silvered and other is partially silvered and are normal to the light intensity
by multiple reflection. Science active medium is maintained in population inversion state photon produced
through spontaneous emission produces the stimulated emission in every direction. The photons having
parallel to the axis of the resonators are only augmented while other are reflected trough the walls of
resonator. If these unidirectional photons reach fully reflecting mirror they reflects and while transverse
through the medium they produce the stimulated emission in other atoms thus increased stimulated photons
reaches partially silvered mirror. At this end some photons are transmitted and other are reflects back in the
medium. This process repeats itself again and again.
Working of optical resonator:
a) Non-exited medium before pumping.
b) Optical pumping.
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c) Spontaneous/stimulated emission.
d) Optical feedback.
e) Light amplification.
f) Light oscillation and laser output.
Pumping:
The process by which we can achieve the population inversion is called the pumping.
Pumping Schemes:
Figure(7): Pumping scheme
Two level pumping scheme:
A two level pumping scheme is not suitable for obtaining population inversion. The time span ∆ , for which
atom has to stay at upper level � , must be longer for achieving population inversion condition.
As according to the Heise e g’s uncertainty principle
∆�. ∆ ≥
ℏ
Figure (8): Two level pumping scheme.
∆ will be longer if ∆� is smaller i.e. � is narrow. If ∆� is smaller, the pumping efficiency is smaller as a
consequence of which less number of atoms are exited and population inversion is not achieved.
Three level pumping:
Let an atomic system has three energy levels, the state � is the ground state, � is the metastable state and
� is the energy excited state. When light is incident, the atom are rapidly exited to upper most state � . They
Pumping
Scheme
Two Level
Pumping
Scheme
Three Level
Pumping
Scheme
Four Level
Pumping
Scheme
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comes back in the lower energy level.
The atom does not stay at the � level for long time. The probability of spontaneous transition � ⟶ �is
comparable to � ⟶ �, � is the metastable state. Science probability of � ⟶ � transition is extremely
small when the medium is expose to a large number of photons a large number of atoms will be exited to the
higher energy level � . Some of these atoms make spontaneous transition to the � state trough the radiative
transition.
As the spontaneous transition from � to � occurs rarely. The atoms get trapped into the state � . This
process continues when more than half of the ground state atoms accumulate at � , the population inversion
is achieved between � and � .
Figure(9): Three Level Pumping
In this scheme a very useful-pumping process is required because to achieve population inversion more than
half of ground state atoms must be pumped to the upper state.
Four Level Pumping:
In four level pumping process the active medium are pumped from ground state � to the uppermost level �
from where they rapidly fall to intermediate � level i.e. meta stable state, leaving level � empty. Now � is
populated inversely with respect to � .
If a triggering incident beam has frequency � the transition � ⟶ �is the stimulated transition. It could be
the atom from � may go to � sponteniously. This transition � ⟶ �is non radiative transition.
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Figure(10): Four level pumping
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Einstein’s coefficient:
Let there is a lasering medium in which the number of atoms in the ground state are and number of atoms
in the excited state are , � is the energy density of radiation for frequency �.
The rate of self-emission
∝
=
And the rate of stimulated emission
∝
And ∝ �
∴ ∝ �
= �
The rate of absorption
∝
And ∝ �
= �
Here, coefficient and and are respectively called the Ei stei ’s and coefficients. It is clear that
the rate of stimulated emission and rate of absorption determined by the same coefficient . This is why
simulated emission is also called the inverse absorption.
Relation between Einstein’s and coefficient:
Let there be an assembly of atoms in thermal equilibrium at temperature T with radiation frequency �.
Since the rate of absoption of radiation i.e. transition from state � ⟶ �is proportional to the energy desity
of radiation � . The number of transition per unit time per unit volume from � → �is given by
= �
Where is the number of atoms in energy state � and is the probability of the transition from � → �
Similarly the number of transition per unit time per unit volume from state � → �may be given as
= { + � }
Where is the number of atpms in energy state � and is the probability of the transition from � → �
In equilibrium state
=
� = { + � }
� = + �
� − � =
[ − ] � =
� =
�
� −�
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� = [
�
�
�
�
−�
]
� = [�
�
�
�
−
�
�
]
� = [�
�
�
�
−
]
By Boltzmann distribution law
= �− � / �
= �− � / �
Where is the total number of atoms
�
�
= � � −� / �
Then
� = [
{
� −�
�� }
�
�
−
]
But � − �= ℎ�
So
� = [
{
ℎ�
�� }
�
�
−
]
But a o di g to the Pla k’s the e e gy de sity of the adiatio of f e ue y � at temperature T is given by
� =
8�ℎ�
. [
( ℎ�/��)−
]
On comparing
=
�ℎ�
�
and =
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Unit-II (LASER)
Ruby Laser:
Solid state laser is the first laser operated successfully. It was fabricated by Mainman in 1960. Ruby is the
lasing medium consist of the crystal of mixture and the . Here some aluminum atoms are
replaced by the . % cromiume atoms.
Construction:
Chromium atoms doped into the aluminum atoms. The active medium in ruby with which main laser action
takes place is +
ions.
Length of the cylindrical rod lies in between 2 to 20cm and the diameter of the rod is about 0.1 to 2cm. The
end faces of the rod are polished flat and parallel. In this one face is partially silvered and other face is fully
silvered.
Ruby rod is surrounded by the helical Xenon photo flash lamp which provides the pump energy to rise the
chromium atom to higher energy level. The parallel ends rod forms an optical cavity so that the photon
traveling along the axis of the optical cavity gets reflects back and fro the end surfaces.
Working:
The energy level of +
ions on the crystal lattice. Consists of three level systems. Upper energy level is short
lived state.
Figure(11):
Figure(12)
When a flash light falls upon the ruby rod, the Å radiation photon are absorbed by +
ions which are
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pumped to the exited state � . The transition from � to � is the optical pumping transition.
Now the +
ions in the exited state give a part of their energy to the crystal lattice and decay to the meta
stable state � . Hence the transition from � to � is radiation less transition. Metastable state � is long lived
state; hence the number of +
ions goes on increasing, while due to pumping the number in the ground
state � goes on decreasing.
Population inversion is established between the � and � . The spontaneous photon emitted by +
ion at
� level is of the wave length of about Å.
Drawbacks :
1) Efficiency of ruby Laser is very low.
2) The Laser Output is not continuous occurs in the form of pulse of microseconds duration.
3) The Laser requires the high pumping power.
4) The defects due to crystalline impurities are also presents in the laser.
Figure(13): Ruby Laser output
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Gas Laser:
Gas Lasers are most widely used Lasers. The ranges from low power Lasers like Helium-Neon Laser to high
power Laser like laser. These lasers operate with rarified gases as the active medium and are excited by
and electrical discharge.
In gases the energy levels of the atoms involves the lasing process are narrow and as such require sources
with sharp wavelength to excite atoms. Most common method to excite gas molecules is by passing an
electric discharge through the gas electrons present in the discharge through the gas electrons presents in the
discharge transfer energy to atoms of laser gas by collision.
He-Ne Laser:
Helium-Neon Laser was first gas Laser to be invented by Ali-Jawan in 1961. The pumping method employed in
He-Ne Laser is electrical pumping method and is based on four level pumping scheme. Since He-Ne laser is a
gas laser so He-Ne laser have sharp energy levels.
Construction:
It consists of a long discharge tube made up of fused quartz which is − �in the length and
in the diameter. The tube is filled with �� and � gases under the pressure of Hg and
. of Hg respectively. And are filled in the ratio ranging from : :. Neon is the active center and
have energy levels suitable for laser transition. While He atoms help in exiting Neon atom. The electrodes are
provided in the discharge tube to provided discharge in the gas which are connected to a high power supply.
The optical cavity of laser consists plane and highly reflecting mirror at one end of the laser tube and a Plano-
concave output mirror of an approximately % transmission at the other end.
To minimize reflection Laser the discharge tube edges are cut at the angle. This arrangement causes the laser
output to be linearly polarized.
Working:
A high voltage is applied across the gas mixture produces electrical breakdown of the gas into ions and
electrons. Fast moving electrons are collide with Helium and Neon atoms and exit them to high energy level.
�� atom are more easily excitable than Ne atoms as they are lighter.
The life time of the energy levels � and � of He is more therefore these levels of He becomes densely
populated. As the � energy levels � and � are close to the exited levels � and � of He. The probability of
the atoms transferring their energy to Ne atom by inelastic collision is greater than the probability of coming
ground state � by spontaneous emission. Since the pressure of the He is 10 times greater than the pressure
of Neon, the levels � and � of Neon are densely populated than any other energy levels.
Photons with the energy ℎ� stimulate the transition from � to � , � to � and � to � . During these
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transition radiation are emitted with the wavelength of . � , Å and . � respectivly.
Figure(16): He-Ne Laser
Figure(17): Energy level diagram of He-Ne Laser
From the energy levels � spontaneous emission occurs in the energy level � . Since the energy level � is the
lower Metastable state then the possibility of atom in the level � getting de-exited to the level � may occur,
if it happened then number of atoms in ground state will go on diminishing and the laser ceases to function.
This can be protected by reducing the diameter of the tube so that atoms in � follows direct transition to the
level � through collision with the walls of tube.
The He-Ne Laser operate in continuous wave mode.
Application of Laser:
1) The laser beam is used to vaporize unwanted materials during the manufacturing of electronic circuits
on semiconductors chips.
2) Laser is used to detect and destroy the enemy missiles during war.
3) Metallic rod can be melted and joined by means of laser beam.
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4) Low price semiconductor lasers are used in CD players, laser printers.
5) High power lasers are used to leasing thermo nuclear reactions which would become the ultimate
exhaust little power source for human civilization.
6) Laser is also being employed for separating the various isotopes of an element.
7) Laser beam are also been used to the internal confinement of plasma.
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Unit-3 (Fiber Optics)
Unit-3
Fibre Optics
Syllabus:
Fibre Optics: Light guidance through optical fibre, types of fibre, numerical
aperture, V-Number, Fibre dispersion (through ray theory in step index
fibre), block diagram of fibre optic communication system
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Fibre Optics:
Fiber optics is the technology in which signals are converted from electrical into optical signals transmitted
through a thin glass-fiber and re-converted into electrical signals.
Definition:
An optical fiber is a transparent medium as thin as human hair, made of glass or clear plastic designed to guide
light waves along its length.
Total Internal Reflection:
When light waves goes into denser medium through
rare medium then they goes away from the normal. If
the angle of incidence exceeds the critical angle then
the refracted ray comes back in to the same medium,
this phenomenon is called the total internal
reflection.
Figure(1):Total internal reflection
Principle of optical fiber cable:
The propagation of light in the optical fiber from one end to another end is based on the principal of total
internal reflection (TIR). When light enters through one end it suffers successive TIR from side walls and
travels along the fiber length in a zigzag path.
Construction:
An optical fiber is cylindrical in shape and has three co-axial regions. The inner most region is the light guiding
region known as core, whose diameter is of the order of . It is surrounded by a co-axial middle region
known as cladding. The diameter of cladding is of the order of , the refractive index of cladding is
always lower than that of the core. The purpose of the cladding is to make the light to be confined to the core.
Light launched into the core and striking the core cladding interface at an angle greater than critical angle will
be reflected back into the core. The outermost region is called sheath or jacket, which is made up of plastic or
polymer. The sheath protects the cladding and core from abrasion and the harmful contamination of moisture
and also increases the mechanical strength of the fiber. Optical fiber is used to transmit light signal over long
distance. Optical fiber requires a light source for launching light into the fiber at its input and a photo detector
to receive light at its output end
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. As the diameter of the optical fiber is very small,
LEDs and laser diodes are used as light source. At the
receiver end semiconductor photodiodes are used for
detection of light pulses and convert the optical
signals into electrical form.
Figure(2):Optical fiber cable
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Light Propagation in the Fibber:
Let us consider the light propagation in the optical fiber. The end at which the light enters the fiber is called
the launching end. Let the refractive index of the core is and that of cladding is as < . Let the
outside medium from which the light is launched have the refractive index . Let the light ray enters the
fiber at an angle �� with the axis and strikes core-cladding interface at an angle �. If � > �the ray will suffer
total internal reflection and remains within the fiber.
Figure (3): Propagation of light through optical fiber cable.
Fractional Refractive Index:
It is the ratio of the difference of the refractive index of core and cladding to refractive index of core. It is
denoted by ∆ and is expressed as
∆=
−
Where = refractive index of the core
= refractive index of cladding
It has no dimension and its order is . this parameter is always positive because > . In order to guide
light effectively through the fiber ∆≪ typically of the order of 0.01
Acceptance Angle:
Applying Snell’s law at the laun hing end
si ��
si ��
=
sin �� = sin �� …………………………………………………………………... (1)
Now In Δ � + �� + =
⟹ �� = − �
So putting in equation (1)
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sin �� = sin − �
sin �� = cos � …………………………………………………………………... (2)
Now �� = �� when � = �
Applying Snell’s law
sin � = sin �
sin � = ∵ sin =
But cos � = √ − sin�
cos � = √ − ()
cos � = √
−
cos � =
√ −
………………………………………………………… (3)
Therefore, putting the value in equation (2) we get
sin �� = ×
√ −
sin �� =
√ −
Let air be the medium at launching end so =
Then sin �� = √ −
� �� = sin− √ −
The angle � �� is called the acceptance angle of the fiber. Acceptance is the maximum angle that are light
rays can have relative to the axis of the fiber and propagate down the fiber.
In 3D the light rays contained within the cone having a fall angle � �� are accepted and transmitted along
the fiber. Therefore the cone is called the acceptance cone.
Figure(4): Acceptance cone= � ��
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Numerical Aperture:
Numerical aperture determines the light gathering ability of the fiber. This is defined as the � of the angle
of acceptance angle.
� = sin ��
But sin �� = √ − , so
� = √ −
Relation between �� and ��
We know that numerical aperture is given as
� = √ −
� = √ + −
� = √ + − ×
� = √
+ −
But
+
≈ and
−
= ∆
so � = √ ∆
� = √ ∆
Normalized frequency V-
number
Optical fiber is characterized by a parameter caused V-number or normalized frequency. Normalized
frequency is the relation between refractive indices and wavelength, and is given by
� =
��
√ −
Where =radius of core
=free space wavelength
But we know that
√ − = � = √ ∆
so � =
��
�
� =
��
√ ∆
� =
�
√ ∆ Where =
V- number helps in determining the number of modes that can propagates through a fiber from above relation
number of modes that can propagates through a fiber increase with increase in � .
Maximum number of modes in multi-mode step index fiber is given by � =
�
maximum number of mode in
multi-mode graded index fiber is given by
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� =
�
Also
For single mode fiber, � < .
For multi-mode fiber, � > .
The corresponding wavelength is called cut-off wavelength.
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Mode of Propagation:
Figure(5): Mode of Propagation
The total possible number allowed path in an optical fiber is known as modes.
When light propagates at an angle close to the critical angle are high order modes and when modes
propagates with angles longer than critical angle are low order mode. The zero order rays travels along the
axis are known as axial ray. On the basis of modes of light propagation optical fiber are of two types:
1) Single mode fiber: - It supports only one mode of propagation.
2) Multi-mode fiber: - It supports number of modes for propagation.
Refractive Index Profile:
It is a plot of refractive index drawn on one of the axis (say-X) and the distance from axis of the core other axis
(say-Y). On the basis of refractive index profile, there are two types of fibers-
1) Steps index fiber: In this refractive index of the core is constant throughout the core.
2) Graded Index Fiber: In this the refractive index of core varies smoothly over the diameter of the core.
Types of the optical fiber:
Based on the profile and modes of propagation optical fiber are of three types-
1) Single mode step index fiber: The diameter of typical SMSIF is about − which is of the order
of wavelength of light used. SMSIF has a very thin fiber, the refractive index changes abruptly at the
core-cladding interface for which it is called step index fiber. In this fiber light travels along the axis of
the fiber. The � (i.e. numerical aperture) and ∆ (i.e. fractional refractive index) have very small
values for single mode fiber and thus have very low acceptance. Therefore the light occupying in fiber
becomes difficult. Costly laser diodes are used to launch the light into the fiber. A single mode fiber
has very small value of ∆ and allows only one mode to propagates through them therefore intermodal
dispersion does not exists in single mode fiber and thus have high data transfer rate.
2) Multi-mode step index fiber: This fiber is similar to single mode fiber only it has a large diameter of
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the order of − . Large compared to the wavelength of light. In multi mode fiber the light
follows a zigzag path. It allows more than one but finite number of modes to propagate through them.
The NA is larger because of large core diameter the signal having path length along the axis of the
fiber is shorter while the other zigzag path longer resulting in higher intermodal dispersion which
means lower data rate and less efficient transmission. LEDs or laser source can be used for launching
of light in this kind of fiber. This kind of fiber. This kind of fiber is used for short range communication.
3) Multi-mode Graded Index fiber: Multimode fiber have a core having refractive index at the center is
very high and decreases as we move towards the cladding, such profile causes a periodic focusing of
light propagation to the fiber. It allows more than one mode to propagate through them and the core
diameter ranges from − the acceptance angle and � decreases with distance from the
axis. The number of modes in this fiber is half that of multimode step index fiber. Therefore gives
lower dispersion. Since the � of this fiber is less than multimode step index fiber, it makes coupling
fiber to the source more difficult. Hence LEDs or laser light source can be used for launching the light
in them; these are used in medium range communication.
Refractive index profile:
Index profile is the refractive index distribution across the core and cladding of fibre. Some fibre has a step
index profile, in which the core has one uniformly distributed index. Other optical fibre has a graded index
profile, in which refractive index varies gradually as a function of radial distance from the axis of the fibre.
Multimode Step Index
(MMSI OFC)
Multimode Graded Index
(MMGI OFC)
Single mode Step Index
(SMSI OFC)
Fibre cross-
section
� Large Gradually decreases with
distance from axis
Very small
∆ Large Very small
Acceptance
angle �
Large acceptance angle Gradually decreases with
distance from axis
Low acceptance angle
Number of
modes
Allow finite number of
modes � = �/
Number of mode are half of
MMSI OFC i.e.
� = �/
Only single mode is possible
Range Short range
communication
Medium range
communication
Long range communication
Data rate Lower data rate Lower data rate Higher data rate
Efficiency Lower efficient Lower efficient Highly efficient
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Light source LED LED Costly LASER diode
Coupling Comparatively easy Very difficult difficult
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Pulse Dispersion:
High pulse launched into a fiber decrease in amplitudes as it travels along the fiber decrease in amplitude as it
travels along the fiber due to laser. It also spreads during travel so its output pulse become wider than input
pulse these are of three types:
1) Intermodal Dispersion: It is due to difference in propagation time in different modes.
2) Intramodal Dispersion: It results due to difference in wavelength, since fiber light consists of groups of
waves.
3) Wave guided dispersion: It happens due to wave guiding properties of fiber.
Fiber Losses:
The losses in optical fiber may be due to following causes:
1. Rayleigh scattering losses: The glass in optical fiber is an amorphous solid that is formed by allowing
the glass to cool from its molten state at high temperature, until it freezes. During the forming
process, some defects are causes in fiber which allows scattering a small portion of light passing
through the glass, creating losses. It affects each wavelength differently.
2. Absorption Losses: The ultraviolet absorption, infrared absorption and ion resonance absorption
these three mechanisms contribute to absorption losses in glass fiber. The oxygen ions in pure silica
have very tightly bounded and only the ultraviolet light photons have enough energy to be observe.
Infrared absorption takes place because photons of light energy are absorbers by the atoms within the
glass molecules and converted to the random vibration.
3. Micro bend Losses: Due to small irregularities in the cladding, causes light to be reflected at angle
where there is no further reflection.
4. Macro bend Losses: It is a bend in the entire cable which causes certain modes not to be reflect and
therefore causes losses to the cladding.
Figure (6): Macro and micro band losses in optical fiber cable.
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5. Temperature Changes: A temperature change from 0 to ℃ could add as much as to the cable
losses. Stress (Strain and tension) could add another .
6. Attenuation Losses: Attenuation losses of an optical fiber is defined as the ratio of optical output
power � from a fiber of length � to the output power �� . In symbol � is expressed attenuation in
� / .
� =
�
log [
��
�
]
In case a fiber is an ideal when �� = � , therefore � = which means that there will no
attenuation loss. In actual practice, a low loss fiber may have � = /.
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Calculation of dispersion for step indeed fiber:
Figure(7): Propagation of light through the optical fiber cable
is the time taken by ray to travel + by velocity � then
=
+
………………………………………… (1)
If be the refractive index of core and is speed of light in vacuum, then
=
�
�
From the figure in ∆ �
�
sec ��
⟹ � sec ��
and
�
sec ��
⟹ � sec ��
Putting the values in equation (1) we get
=
� sec �� + � sec ��
� + �sec ��
sec� ………….. 2
As the ray in the fiber propagates by a series of total internal reflection at the interface, the time taken by the
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ray in traversing an axial length of the fiber will be
� = .c s ��
………………………………… (3)
Time taken by rays making zero angle with fiber axis will be minimum i.e.
� �
=
. cos
= …………………… (4)
The maximum time is given by
� ��
= .c s ��
………………………………… (5)
Now y Snell’s law
si ��
si ��
=
But �� = for �� = � (i.e. critical angle)
sin �
sin
=
sin � = ……………………………….......... (6)
From the figure is clear that
� + �� + =
� = − ��
So by equation (8)
sin − �� =
cos ��
= ………………………………... (7)
Putting the value of cos �� in equation (5)
� =
.
�
�
� =
.
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Fibre Optics communication System:
The optical fibers are widely used for communication purpose. The fiber optics communication system is almost
similar to ordinary communication system. Simply the systems consist of transmitter, information channel and
recover.
Transmitter:
The transmitter converts electrical signals (Information signal) into optical signals. Mainly transmitter consists of
1) Transducer: If input signal is other than the electrical signals, we use a transducer which consists a non-
electrical message into electrical signal.
2) Modulator: The output of transducer is connected as the input of modulator, with the help of
modulator electrical messages are converted into the desired form. There are two kinds of modulators;
digital and analog.
3) Light source: The function of light source is to generate carrier waves on which the information signal is
impressed and transmitted. The light sources used are light emitting diodes (LEDs) or LASER diodes.
These are known as optical oscillators.
4) Input channel coupler: It transfers the signals to information channel i.e. optical fiber in a proper
manner.
5) Information Channel: It is a link between transmitter and receiver.
Figure(8): The optical fiber communication system
Receiver:
Receiver converts the signals into electrical signals; it consists of-
1) Output channel coupler: its main function is to direct the light emerging from optical fiber into the
photodetector.
2) Photodetector: The photodetector converts the light wave into an electric current. The detector output
includes the message, which is separated from the carrier in next step.
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3) Signal processor: the information from the carrier wave is separated by signal processor includes
amplifiers and filters. The optical signal, if needed, amplified and undesired frequencies are filtered by
the processor.
4) Signal restorer: while traveling through the optical fiber the signal progressively attenuated and
distorted due to various laser and dispersion occurring in the fiber. Thus the signal should be amplified
and restorers are used for this purpose.
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Unit IV- Quantum Mechanics
Page 1
Unit-4
Quantum Mechanics
Syllabus:
Black body radiation, ultraviolet catastrophe, Crompton effect, plates
theory of radiation, phase and group velocity, particle in a box, uncertainty
principle, well-behaved wave equation, Schrodinger equation, application
to particle in a box.
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Page 2
Black body:
A black body is one which absorbs all types of heat radiation
incident on it when radiations are permitted to fall on black body
they are neither reflected nor transmitted.
A black body is known as black body due to the fact that whatever
may the colour of the incident radiation the body appears black by
absorbing all kind of radiations incident on it.
A perfect black does not exists thus a body representing close
proximity to perfect black body so it can be considered as a black
body.
A hollow sphere is taken with fine hole and a point projection in
front of the hole and is coated with lamp black on its inner surface
shows the close proximity to the black body, when the radiation
enter through hole, they suffer multiple reflection and are totally
absorbed.
Figure(1): Black body
Black Body radiation:
A body which completely absorbs radiation of all
radiations of all wavelength/frequencies incident
on it and emits all of them when heated at higher
temperature is called black body. The radiation
emitted by such a body is called black body
radiation. So the radiation emitted form a black
body is a continuous spectrum i.e. it contains
radiation of all the frequencies.
Distributions of the radiant energy over different
wavelength in the black body radiation at a given
temperature are shown in the figure.
Black body radiation is a common synonym for
thermal radiation.
Figure(2): Black body radiation
Radiation:
Radiation is a process which the surface of an object radiates its thermal energy in the form of the
electromagnetic waves.
Radiations are of two types
Radiation
Ionising
radiation
Non-ionising
radiation
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Page 3
Emissivity:
The emissivity of a material is the irradiative power of its surface to emit heat by radiation, usually it is shown
by or . It is the ratio of energy radiated by a material to the energy radiated by the black body.
True black body has maximum emissivity � =� (highly polished silver has an emissivity for about . at
least.)
Plank’s Quantum Hypothesis:
Plank assumes that the atoms of the wall of blackbody behave as an oscillator and each has a characteristic
frequency of oscillation. He made the following assumption-
1) An oscillator can have any arbitrary value of energy but can have only discrete energies as per the
following relation
� =� ℎ�
Where � =� , , , � …� ..and � and ℎ a e k o as f e ue a d Pla k’s o sta t.
2) The oscillator can absorb or emit energy only in the form of packets of energy ℎ� but not
continuously.
� =� ℎ�
Average energy of Plank’s Oscillators:
If be the total number of oscillations and as the total energy of these oscillators, then average energy will
be given by the relation.
̅ = ……………………………………………………………………………. (1)
Now consider ,� ,� …� …� …� …� .� .�as the number of oscillators having the energy values
,� ℎ�,� ℎ�� …� .� .� ℎ�espe ti el . The the Ma ell’s dist i utio fo ula
= +� +� +� ⋯� …� …� …� …� ..
= � +�−
ℎ�
�� +� −
ℎ�
�� +� ⋯� …� …� .�
=
( � −�−
ℎ�
��)
.............................................................. (2)
And the total energy
= ×� + ×� ℎ�+ ×� ℎ�+� ⋯� ….
= ×� +� ( −
ℎ�
�� ×� ℎ�)� +� (−
ℎ�
�� ×� ℎ�)� +� ⋯
= −
ℎ�
�� ×� ℎ�[ � +� −
ℎ�
�� +� −
ℎ�
�� +� ⋯� …� .� .]
=
−
ℎ�
��
ℎ�
( � −�−
ℎ�
��)
………………………………………….. (3)
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Page 4
Putting the value of and from above equations in equation (1) we get-
̅ =
̅ =
ℎ� −
ℎ�
��
( � −�−
ℎ�
��)
̅ =
ℎ�
(
ℎ�
�� −� )
…………………………………………..………. (4)
This is the e p essio fo the a e age e e g i Pla k’s os illato s.
Plank’s radiation formula:
The average density of radiation � in the frequency range � and �� +� �depending upon the average of an
oscillator is given by-
� � =
��
�� ×�̅ …………………………………………..……… (5)
� � =
�� ℎ�
(
ℎ�
�� −� )
�
� � =
�ℎ �
(
ℎ�
�� −� )
� …………………………………………..……… (6)
The a o e elatio is k o as the Pla k’s adiatio fo ula i te s of the f e ue . This la a also e
expressed in terms of wavelength � of the radiation. Since �� =
�
for electromagnetic radiation, �� =
−
�
�. Further we know that the frequency is reciprocal of wavelength or in other words an increase in
frequency corresponds to a decrease in wavelength. therefore
� � = − � �
� � = −
�ℎ �
−
�
�
(
ℎ
�� −� )
� � =
�ℎ
�
(
ℎ
�� −� )
�
…………………………………………..… (7)
The a o e elatio is k o as the Pla k’s la i te s of a ele gth �
Wien’s law and Rayleigh-Jeans law:
With the help of Pla k’s adiatio Wie ’s la a d Ra leigh-Jens law can be derive. When the wavelength �
and temperature � are very small, then
ℎ�
�� ≫� . Therefore, can be neglected in the denominator of
equation (7).
� � =
�ℎ
�
−
ℎ
��
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Page 5
By substituting �ℎ � =� and
ℎ
�
=� , we get
� � =
�
−
�� …………………………………………..………. (8)
This is k o as Wie ’s la , hi h is alid at lo te pe atu e � and small wavelength �.
For higher temperature � and large wavelength �,
ℎ�
�� can be approximated to � +
ℎ
��
. Then we have from
equation (7)
� � =
�ℎ
� � +
ℎ
� �
−�
�
� � =
� �
�
� ………………….…………………………………………..… (9)
This is known as Rayleigh-Jeans law.
Ultraviolet Catastrophe:
One of the nagging questions at the time concerned the spectrum of radiation emitted by a so-called black
body. A perfect black body is an object that absorbs all radiation that is incident on it. Perfect absorbers are
also perfect emitters of radiation, in the sense that heating the black body to a particular temperature causes
the black body to emit radiation with a spectrum that is characteristic of that temperature. Examples of black
bodies include the Sun and other stars, light bulb filaments, and the element in a toaster. The colours of
these objects correspond to the temperature of the object. Examples of the spectra emitted by objects at
particular temperatures are shown in Figure 3
Figure 3: The spectra of electromagnetic radiation emitted by hot objects. Each spectrum corresponds to a
particular temperature. The black curve(dotted line) represents the predicted spectrum of a 5000 K black
body, according to the classical theory of black bodies
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Page 6
At the end of the 19th century, the puzzle regarding blackbody radiation was that the theory regarding how
hot objects radiate energy predicted that an infinite amount of energy is emitted at small wavelengths, which
clearly makes no sense from the perspective of energy conservation. Because small wavelengths correspond
to the ultraviolet end of the spectrum, this puzzle was known as the ultraviolet catastrophe. Figure 27.1
shows the issue, comparing the theoretical predictions to the actual spectrum for an object at a temperature
of 5000 K. There is clearly a substantial disagreement between the curves
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Page 7
Matter wave:
According to Louis de-Broglie every moving matter particle is surrounded by a wave whose wavelength
depends up on the mass of the particle and its velocity. These waves are known as matter wave or de-
Broglie waves.
Wavelength of the de-Broglie wave:
Consider a photon whose energy is given by � =� ℎ�� =
ℎ
�
[ � � =� ��……………………………………… (1)
Where ℎ is Pla k’s o sta t. � ×� −
, � is the frequency and � is the wavelength of photon.
No Ei stei ’s ass e e g elatio
= ……………………………………… (2)
=
ℎ
�
� =
ℎ
� =
ℎ
Where � =�
In place of the photon a material particle of mass is moving with velocity then
� =
ℎ
…………………………….…… (3)
(i)
Now we know that the kinetic energy of the material particle of mass moving with velocity � is given by-
=
= �
= [ � � =� ]
= √
So by equation (3)
� =
ℎ
√
(ii)
According to kinetic theory of gasses the average kinetic energy of the material particle is given by � =
� where � =� . � ×�−
/ i.e. Boltzmann constant
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Page 8
= �
= �
= � � � =
= √ �
So by equation (3)
� =
ℎ
√ �
……………………………… (4)
Group or Envelope of the wave:
When a mass particle moves with some velocity than it emits the matter waves, those waves interacts each
other and where there they interfere constructively they form an envelope around the particle which is
known as wave group or simply envelope.
Figure(2): Formation of the wave packet
Group velocity:
Group velocity of a wave is the velocity with which the overall shape of the a e’s amplitudes (modulation
or envelope) of the wave propagates through space. It is denoted by �.
Phase velocity:
The phase velocity of a wave is the rate at which the phase or the wave propagates in the space. It is
denoted by �.
Expression for Group velocity and phase velocity:
Let us suppose that the wave group arises from the combination of two waves that have some amplitude
but differ by an amount ∆ in angular frequency and an mount ∆ in wave number.
= � cos � −� ………………………………
= � cos[ � +� ∆� − � +� ∆] ………………………………
By the principle of superposition
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Page 9
= +� …………….…………………………… (3)
= [cos � −� +� cos{ � +� ∆� − � +� ∆}]
Using the identity
cos� � +� cos� � =� � cos� (
� +�
)� cos� (
� −�
)
And cos −� =� cos� �
� +� � =
� −� +� { � +� ∆ � −� � −� ∆ }
� +� � =
� −� � +� ∆ � −� ∆
� +� � =
�+∆ �− � −∆�
� +� � =
+∆ �− �+∆�
� −� � =
� −� −� { � +� ∆ � −� � −� ∆
� −� � =
� −� � −� � −� ∆ � +� � +
� −� � =
−∆ � +� ∆
� −� � =� −
∆ � −� ∆
� =� � [ � cos{
� +� ∆� − � +� ∆
} .� cos{
∆ � −� ∆
}]
Let � +� ∆ � =�and � +� ∆ � =�
So we have
= [cos� (
� −�
) .� cos� (
∆ � −� ∆
)]
⟹ = [cos � −� .� cos�
∆
� −
∆�
]…………………… (4)
This is the resultant wave equation of superposition of two waves having the amplitude
� cos�
∆
� −
∆�
and phase cos � −� where and are mean values of angular frequency and
prapogation constant of the wave.
Phase velocity:
Since phase � −� =
Differentiating with respect to we get
� −� =
⟹
�
= �
But � =
�
phase
velocity
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Page 10
⟹
� =
�
= �
...................................................... (5)
Group Velocity:
⟹ ∆
� −
∆� =
⟹ ∆ = ∆�
⟹ ∆
∆�
=
�
So the group velocity
� =
�
=
�
........................................................ (6)
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Page 11
Relation between Group velocity and phase velocity
1. For dispersive and non-dispersive medium:
But by equation (5) i.e. � =
�
⟹� � =��
Putting into equation (6) we get
⟹
� =
( �)
⟹
� = �.� � +�
�
⟹
� = � � +� (
�
�
)
�
�
�
�
⟹
� = � � +� (
�
�
)
�
� �−
⟹
� = � � +� (
�
)
�
−�− .� �
⟹
� = � −� �
�
�
Different cases:
1) If
��
�
=� i.e. if the phase velocity does not depends on the wavelength then � =� �, such a medium is
called the non dispersive medium.
2) If
��
�
≠� i.e. if it has positive values then � <� �, then such a medium is called the dispersive
medium.
2. Relativistic particle:
Let us consider a de-Broglie wave associated with a moving particle of rest mass and velocity , then the
and will be given by
⟹ = ��
⟹ =
�
ℎ
� �� =
ℎ
⟹ =
�
ℎ
.
√ � −
……………………………….. (8)
And
⟹ =
�
�
⟹
=
�
ℎ
�
� �� =
ℎ
�
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Page 12
⟹
=
� �
ℎ√ −
�
�
...................................................(9) � � =
√ −
�
�
Now phase velocity � =
�
So
� = (
��
ℎ
.
√ −
�
� )
(
� �
ℎ√ −
�
� )
� =
�
ℎ
.
√ −
�
�
� ×�
ℎ√ −
�
�
� �
� =
�
………………………………………………………………… (10)
Now group velocity � =
�
The expression can be written as
� = ……………………………………………………………… (11)
In order to find the value of � we have to solve the following terms-
⟹
=
[
�
ℎ
.
√ � − ]
[By equation (8)]
⟹
= �
ℎ
√ � −
−
⟹
= �
ℎ
(− ) � −
−
.� (− )
⟹
= �
ℎ
� −
−
……………………………………… (12)
Again
⟹ =
[
�
ℎ√ � − ]
⟹ =
�
ℎ
[
√ � − ]
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