# Engineering Physics (kas 101t/201t)

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PHYSICS ENGINEERING PHYSICS (KAS 101T/201T)

Module - 1 Relativistic Mechanics:

[8]

Frame of reference, Inertial & non-inertial frames, Galilean transformations, MichelsonMorley experiment, Postulates of special theory of relativity, Lorentz transformations, Length contraction, Time dilation, Velocity addition theorem, Variation of mass with velocity, Einstein‟s mass energy relation, Relativistic relation between energy and momentum, Massless particle.

Module- 2 Electromagnetic Field Theory:

[8]

Continuity equation for current density, Displacement current, Modifying equation for the curl of magnetic field to satisfy continuity equation, Maxwell‟s equations in vacuum and in non conducting medium, Energy in an electromagnetic field, Poynting vector and Poynting theorem, Plane electromagnetic waves in vacuum and their transverse nature. Relation between electric and magnetic fields of an electromagnetic wave, Energy and momentum carried by electromagnetic waves, Resultant pressure, Skin depth.

Module- 3 Quantum Mechanics:

[8]

Black body radiation, Stefan‟s law, Wien‟s law, Rayleigh-Jeans law and Planck‟s law, Wave particle duality, Matter waves, Time-dependent and time-independent Schrodinger wave equation, Born interpretation of wave function, Solution to stationary state Schrodinger wave equation for one-Dimensional particle in a box, Compton effect.

Module- 4 Wave Optics:

[10]

Coherent sources, Interference in uniform and wedge shaped thin films, Necessity of extended sources, Newton‟s Rings and its applications. Fraunhoffer diffraction at single slit and at double slit, absent spectra, Diffraction grating, Spectra with grating, Dispersive power, Resolving power of grating, Rayleigh‟s criterion of resolution, Resolving power of grating.

Module- 5Fibre Optics & Laser:

[10]

Fibre Optics: Introduction to fibre optics, Acceptance angle, Numerical aperture, Normalized frequency, Classification of fibre, Attenuation and Dispersion in optical fibres. Laser: Absorption of radiation, Spontaneous and stimulated emission of radiation, Einstein‟s coefficients, Population inversion, Various levels of Laser, Ruby Laser, He-Ne Laser, Laser applications.

Course Outcomes: 1. To solve the classical and wave mechanics problems 2. To develop the understanding of laws of thermodynamics and their application in various processes 3. To formulate and solve the engineering problems on Electromagnetism & Electromagnetic Field Theory 4. To aware of limits of classical physics & to apply the ideas in solving the problems in their parent streams

Reference Books: 1. Concepts of Modern Physics - AurthurBeiser (Mc-Graw Hill)

Program Outcomes (PO’s) relevant to the course:

PO1 Engineering knowledge: Apply the knowledge of mathematics, science, engineering fundamentals, and an engineering specialization to the solution of complex engineering problems.

PO2 Problem analysis: Identify, formulate, review research literature, and analyze complex engineering problems reaching substantiated conclusions using first principles of mathematics, natural sciences, and engineering sciences.

PO3 Design/development of solutions: Design solutions for complex engineering problems and design system components or processes that meet the specified needs with appropriate consideration for the public health and safety, and the cultural, societal, and environmental considerations.

PO4 Conduct investigations of complex problems: Use research-based knowledge and research methods including design of experiments, analysis and interpretation of data, and synthesis of the information to provide valid conclusions

PO5 Modern tool usage: Create, select, and apply appropriate techniques, resources, and modern engineering and it tools including prediction and modeling to complex engineering activities with an understanding of the limitations.

PO6 The engineer and society: Apply reasoning informed by the contextual knowledge to assess societal, health, safety, legal and cultural issues and the consequent responsibilities relevant to the professional engineering practice.

PO7 Environment and sustainability: Understand the impact of the professional engineering solutions in societal and environmental contexts, and demonstrate the knowledge of, and need for sustainable development.

PO8 Ethics: Apply ethical principles and commit to professional ethics and responsibilities and norms of the engineering practice..

PO9 Individual and team work: : Function effectively as an individual, and as a member or leader in diverse teams, and in multidisciplinary settings

PO10 Communications: Communicate effectively on complex engineering activities with the engineering community and with society at large, such as, being able to comprehend and write effective reports and design documentation, make effective presentations, and give and receive clear instructions.

PO11 Project management and finance: : Demonstrate knowledge and understanding of the engineering and management principles and apply these to one’s own work, as a member and leader in a team, to manage projects and in multidisciplinary environments.

PO12 Life-long learning: Recognize the need for, and have the preparation and ability to engage in independent and life-long learning in the broadest context of technological change.

COURSE OUTCOMES (COs): Engineering Physics (KAS-101T/201T),

YEAR OF STUDY: 2020-21

Course outcomes CO1 CO2 CO3 CO4 CO5

Statement (On completion of this course, the student will be able to -)

Understand the basics of relativistic mechanics.(K2) Develop EM-wave equations using Maxwell’s equations.(K1) Understand the concepts of quantum mechanics.(K2) Describe the various phenomena of light and its applications in different fields.(K1) Comprehend the concepts and applications of fiber optics and LASER.(K2)

Mapping of Course Outcomes with Program Outcomes (CO-PO Mapping)

1. Slight ( low) 2. Moderate ( medium)

3. Substantial (high)

COs/POs CO1 CO2 CO3 CO4 CO5

PO1 PO2 PO3 PO4 PO5 PO6 PO7 PO8 PO9 PO10 PO11 PO12

3 3

3

3 3

3

3

3 3

3

2

3

3

3

3

KAS-

3 3

3

3

101T/201T

Engineering Physics Module-1-Relativistic Mechanics

KAS 101T/201T CO1

1.1 Introduction of Newtonian Mechanics

The universe, in which we live, is full of dynamic objects. Nothing is static here Starting from giant stars to tiny electrons, everything is dynamic. This dynamicity of universal objects leads to variety of interactions, events and happenings. The curiosity of human being/scientist to know about these events and laws which governs them. Mechanics is the branch of Physics which is mainly concerned with the study of mobile bodies and their interactions.

A real breakthrough in this direction was made my Newton in 1664 by presenting the law of linear motion of bodies. Over two hundred years, these laws were considered to be perfect and capable of explaining everything of nature.

1.2 Frames of Reference

The dynamicity of universal objects leads to variety of interactions between these objects leading to various happenings. These happenings are termed as events. The relevant data about an event is recorded by some person or instrument, which is known as “Observer”.

The motion of material body can only described relative to some other object. As such, to locate the position of a particle or event, we need a coordinate system which is at rest with respect to the observer. Such a coordinate system is referred as frame of reference or observers frame of reference.

“A reference frame is a space or region in which we are making observation and measuring physical(dynamical) quantities such as velocity and acceleration of an object(event).” or

“A frame of reference is a three dimensional coordinate system relative to which described the position and motion(velocity and acceleration) of a body(object)”

Fig.1.1 represents a frame of reference(S), an object be

situated at point P have co-ordinate

i.e.

P(

), measured by an Observer(O). Where be the time

of measurement of the co-ordinate of an object. Observers in

different frame of references may describe same event in

different way.

Fig 1.1: frame of reference

Example- takes a point on the rim of a moving wheel of cycle. for an observer sitting at the center of the wheel, the path of the point will be a circle. However, For an observer standing on the ground the path of the point will appear as cycloid.

1

Engineering Physics

KAS 101T/201T

Module-1-Relativistic Mechanics

CO1

There are two types of frames of reference.

1. Inertial or non-accelerating frames of reference 2. Non-inertial or accelerating frames of reference

The frame of reference in which Newton's laws of motion hold good is treated as inertial frame of reference. However, the frame of reference in which Newton's laws of motion not hold good is treated as non-inertial frame of reference. Earth is non-inertial frame of reference, because it has acceleration due to spin motion about its axis and orbital motion around the sun.

1.3 Galilean Transformations

The Galilean transformations equations are used to transform the coordinates of position and time from one inertial frame to the other. The equations relating the coordinates of a particle in two inertial frames are called as Galilean transformations. Consider the two inertial frames of reference F and F'. Let the frame F' is moving with constant velocity v with reference to frame F. The frames F and F' are shown in Fig.1

Let some event occurs at the point P at any instant of time t. The coordinates of point P with respect to frame F are x, y, z, t and with respect to frame F' are x', y', z', t'. Let at t = t'= 0, the origin O of frame F and O' of frame F' coincides with one another. Also axes x and x' are parallel to v. Let y' and z' are parallel to y and z respectively.

From Fig.

x=x'+ vt……………………………...(1)

x'= x - vt……………………………...(2)

As there is no relative motion along y and zaxes, we can writ

y'=y….…(3),

z'=z….…(4), and t'=t….…(5)

These equations are called as Galilean transformation equations. The inverse Galilean transformation can be written as,

x=x' + vt, y=y', z'=z and t=t'

Hence transformation in position is variant only along the direction of motion of the frame and remaining dimensions( y and z) are unchanged under Galilean Transformation. At that era scientist were assumed time should be absolute.

2

Engineering Physics Module-1-Relativistic Mechanics

KAS 101T/201T CO1

b) Transformation in velocities components:

The conversion of velocity components measured in frame F into their equivalent components in the frame F' can be known by differential Equation (1) with respect to time we get,

u'x= = (

)

hence u'x= ux-V Similarly, from Equation (3) and (4) we can write

u'y= uy and u'z= uz In vector form,

u'=u – v

Hence transformation in velocity is variant only along the direction of motion of the frame and remaining dimensions( along y and z) are unchanged under Galilean Transformation.

c) Transformation in acceleration components:

The acceleration components can be derived by differentiating velocity equations with respect to time,

a'x=

(

)

a'x= ax

In vector form a' = a

This shows that in all inertial reference frames a body will be observed to have the same acceleration. Hence acceleration are invariant under Galilean Transformation.

Failures of Galilean transformation:

According to Galilean transformations the laws of mechanics are invariant. But under Galilean transformations, the fundamental equations of electricity and magnetism have very different forms.

Also if we measure the speed of light c along x-direction in the frame F and then in the frame F' the value comes to be c' = c – vx. But according to special theory of relativity the speed of light c is same in all inertial frames.

1.4 Michelson-Morley Experiment

“The objective of Michelson - Morley experiment was to detect the existence of stationary medium ether (stationary frame of reference i.e. ether frame.)”, which was assumed to be required for the propagation of the light in the space.

3

Engineering Physics

KAS 101T/201T

Module-1-Relativistic Mechanics

CO1

In order to detect the change in velocity of light due to relative motion between earth and hypothetical medium ether, Michelson and Morley performed an experiment which is discussed below. The experimental arrangement is shown in Fig

Light from a monochromatic source S, falls on the semi-silvered glass plate G inclined at an angle 45° to the beam. It is divided into two parts by the semi silvered surface, one ray 1 which travels towards mirror M1 and other is transmitted, ray 2 towards mirror M2. These two rays fall normally on mirrors M1 and M2 respectively and are reflected back along their original paths and meet at point G and enter in telescope. In telescope interference pattern is obtained.

If the apparatus is at rest in ether, the two reflected rays would take equal time to return the glass plate G. But actually the whole apparatus is moving along with the earth with a velocity say v. Due to motion of earth the optical path traversed by both the rays are not the same. Thus the time taken by the two rays to travel to the mirrors and back to G will be different in this case.

Let the mirrors M1 and M2 are at equal distance l from the glass plate G. Further let c and v be the velocities at light and apparatus or earth respectively. It is clear from Fig. that the reflected ray 1 from glass plate G strikes the mirror M1at A' and not at A due to the motion of the earth.

The total path of the ray from G to A' and back will be GA'G'.

∴ From Δ GA'D

(GA')2=(AA')2+ (A'D)2...(1)

As (GD =AA')

If t be the time taken by the ray to move from G to A', then from Equation (1), we have (c t)2=(v t)2 + (l)2

Hence t= √ If t1 be the time taken by the ray to travel the whole path GA'G', then

4

Engineering Physics Module-1-Relativistic Mechanics

KAS 101T/201T CO1

t1= 2t = √ t1= ((

= (

)

)) ……………….(2) Using Binomial Theorem

Now, in case of transmitted ray 2 which is moving longitudinally towards mirror M2. It has a velocity (c – v) relative to the apparatus when it is moving from G to B. During its return journey, its velocity relative to apparatus is (c + v). If t2 be the total time taken by the longitudinal ray to reach G', then

T2= ( ) ( ) after solving

t2= ((

)) ……………….(3)

Thus, the difference in times of travel of longitudinal and transverse journeys is

Δt = t2-t1 = = ((

)) ((

))

Δt = ……………….(4) The optical path difference between two rays is given as,

Optical path difference (Δ) =Velocity × t = c x Δt

= c x =

Δ = ……………….(5)

If λ is the wavelength of light used, then path difference in terms of wavelength is, = λ Michelson-Morley perform the experiment in two steps. First by setting as shown in fig and secondly by turning the apparatus through 900. Now the path difference is in opposite direction i.e. path difference is -λ . Hence total fringe shift ΔN = Michelson and Morley using l=11 m, λ= 5800 x 10-10 m, v= 3 x 104 m/sec and c= 3 x 108 m/sec ∴ Change in fringe shift ΔN = λ substitute all these values

=0.37 fringe

5

Engineering Physics

KAS 101T/201T

Module-1-Relativistic Mechanics

CO1

But the experimental were detecting no fringe shift. So there was some problem in theory calculation and is a negative result. The conclusion drawn from the Michelson-Morley experiment is that, there is no existence of stationary medium ether in space.

Negative results of Michelson - Morley experiment

1. Ether drag hypothesis: In Michelson - Morley experiment it is explained that there is no relative

motion between the ether and earth. Whereas the moving earth drags ether alone with its motion

so the relative velocity of ether and earth will be zero.

2.

Lorentz-Fitzgerald Hypothesis: Lorentz told that the length of the arm (distance between the pale

and the mirror M2) towards the transmitted side should be L(√1 – v2/c2) but not L. If this is taken

then theory and experimental will get matched. But this hypothesis is discarded as there was no

proof for this. 3. Constancy of Velocity of light: In Michelson - Morley experiment the null shift in fringes was

observed. According to Einstein the velocity of light is constant it is independent of frame of

reference, source and observer.

Einstein special theory of Relativity (STR)

Einstein gave his special theory of relativity (STR) on the basis of M-M experiment

Einstein’s First Postulate of theory of relativity: All the laws of physics are same (or have the same form) in all the inertial frames of reference moving with uniform velocity with respect to each other. (This postulate is also called the law of equivalence).

Einstein’s second Postulate of theory of relativity: The speed of light is constant in free space or in vacuum in all the inertial frames of reference moving with uniform velocity with respect to each other. (This postulate is also called the law of constancy).

1.5 Lorentz Transformation Equations

Consider the two observers O and O' at the origin of the inertial frame of reference F and F' respectively as shown in Fig. Let at time t = t' = 0, the two coordinate systems coincide initially. Let a pulse of light is flashed at time t = 0 from the origin which spreads out in the space and at the same time the frame F' starts moving with constant velocity v along positive X-direction relative to the frame F. This pulse of light reaches at point P, whose coordinates of position and time are (x, y, z, t) and (x',y', z', t') measured by the observer O and O' respectively. Therefore the transformation equations of x and x' can be given as,

x'=k (x – v t)………………………....(1)

Where k is the proportionality constant and is independent of x and t.

6

Engineering Physics

KAS 101T/201T

Module-1-Relativistic Mechanics

CO1

The inverse relation can be given as,

x=k (x' + v t') ………………… (2)

As t and t' are not equal, substitute the value of x' from Equation (1) in Equation (2)

x=k [k (x – vt) + vt'] or =(

)

t'=

or

t'=

(

) ………………………....(3)

According to second postulate of special theory of relativity the speed of light c remains constant. Therefore the velocity of pulse of light which spreads out from the common origin observed by observer O and O' should be same.

∴ x=c t and x' = ct'………………………………………...(4)

Substitute the values of x and x' from Equation (4) in Equation (1) and (2) we get

ct'=k (x – v t) = k (ct – v t) or ct'=kt (c – v) ................................(5)

and similarly

ct=k t' (c + v) ................................(6)

Multiplying Equation (5) and (6) we get, c2 t t'=k2t t' (c2 – v2) hence

after solving

k= ................................(7)

√

Hence equation (7) substitute in equation (1), then Lorentz transformation in position will be

x'= ,

√

y=y', z'=z

Calculation of Time: equation (7) substitute in equation (3),

t'=

(

)

From equation ( 7),

or, then above equation becomes

t'=

=

or,

t'= ( )

7

Module - 1 Relativistic Mechanics:

[8]

Frame of reference, Inertial & non-inertial frames, Galilean transformations, MichelsonMorley experiment, Postulates of special theory of relativity, Lorentz transformations, Length contraction, Time dilation, Velocity addition theorem, Variation of mass with velocity, Einstein‟s mass energy relation, Relativistic relation between energy and momentum, Massless particle.

Module- 2 Electromagnetic Field Theory:

[8]

Continuity equation for current density, Displacement current, Modifying equation for the curl of magnetic field to satisfy continuity equation, Maxwell‟s equations in vacuum and in non conducting medium, Energy in an electromagnetic field, Poynting vector and Poynting theorem, Plane electromagnetic waves in vacuum and their transverse nature. Relation between electric and magnetic fields of an electromagnetic wave, Energy and momentum carried by electromagnetic waves, Resultant pressure, Skin depth.

Module- 3 Quantum Mechanics:

[8]

Black body radiation, Stefan‟s law, Wien‟s law, Rayleigh-Jeans law and Planck‟s law, Wave particle duality, Matter waves, Time-dependent and time-independent Schrodinger wave equation, Born interpretation of wave function, Solution to stationary state Schrodinger wave equation for one-Dimensional particle in a box, Compton effect.

Module- 4 Wave Optics:

[10]

Coherent sources, Interference in uniform and wedge shaped thin films, Necessity of extended sources, Newton‟s Rings and its applications. Fraunhoffer diffraction at single slit and at double slit, absent spectra, Diffraction grating, Spectra with grating, Dispersive power, Resolving power of grating, Rayleigh‟s criterion of resolution, Resolving power of grating.

Module- 5Fibre Optics & Laser:

[10]

Fibre Optics: Introduction to fibre optics, Acceptance angle, Numerical aperture, Normalized frequency, Classification of fibre, Attenuation and Dispersion in optical fibres. Laser: Absorption of radiation, Spontaneous and stimulated emission of radiation, Einstein‟s coefficients, Population inversion, Various levels of Laser, Ruby Laser, He-Ne Laser, Laser applications.

Course Outcomes: 1. To solve the classical and wave mechanics problems 2. To develop the understanding of laws of thermodynamics and their application in various processes 3. To formulate and solve the engineering problems on Electromagnetism & Electromagnetic Field Theory 4. To aware of limits of classical physics & to apply the ideas in solving the problems in their parent streams

Reference Books: 1. Concepts of Modern Physics - AurthurBeiser (Mc-Graw Hill)

Program Outcomes (PO’s) relevant to the course:

PO1 Engineering knowledge: Apply the knowledge of mathematics, science, engineering fundamentals, and an engineering specialization to the solution of complex engineering problems.

PO2 Problem analysis: Identify, formulate, review research literature, and analyze complex engineering problems reaching substantiated conclusions using first principles of mathematics, natural sciences, and engineering sciences.

PO3 Design/development of solutions: Design solutions for complex engineering problems and design system components or processes that meet the specified needs with appropriate consideration for the public health and safety, and the cultural, societal, and environmental considerations.

PO4 Conduct investigations of complex problems: Use research-based knowledge and research methods including design of experiments, analysis and interpretation of data, and synthesis of the information to provide valid conclusions

PO5 Modern tool usage: Create, select, and apply appropriate techniques, resources, and modern engineering and it tools including prediction and modeling to complex engineering activities with an understanding of the limitations.

PO6 The engineer and society: Apply reasoning informed by the contextual knowledge to assess societal, health, safety, legal and cultural issues and the consequent responsibilities relevant to the professional engineering practice.

PO7 Environment and sustainability: Understand the impact of the professional engineering solutions in societal and environmental contexts, and demonstrate the knowledge of, and need for sustainable development.

PO8 Ethics: Apply ethical principles and commit to professional ethics and responsibilities and norms of the engineering practice..

PO9 Individual and team work: : Function effectively as an individual, and as a member or leader in diverse teams, and in multidisciplinary settings

PO10 Communications: Communicate effectively on complex engineering activities with the engineering community and with society at large, such as, being able to comprehend and write effective reports and design documentation, make effective presentations, and give and receive clear instructions.

PO11 Project management and finance: : Demonstrate knowledge and understanding of the engineering and management principles and apply these to one’s own work, as a member and leader in a team, to manage projects and in multidisciplinary environments.

PO12 Life-long learning: Recognize the need for, and have the preparation and ability to engage in independent and life-long learning in the broadest context of technological change.

COURSE OUTCOMES (COs): Engineering Physics (KAS-101T/201T),

YEAR OF STUDY: 2020-21

Course outcomes CO1 CO2 CO3 CO4 CO5

Statement (On completion of this course, the student will be able to -)

Understand the basics of relativistic mechanics.(K2) Develop EM-wave equations using Maxwell’s equations.(K1) Understand the concepts of quantum mechanics.(K2) Describe the various phenomena of light and its applications in different fields.(K1) Comprehend the concepts and applications of fiber optics and LASER.(K2)

Mapping of Course Outcomes with Program Outcomes (CO-PO Mapping)

1. Slight ( low) 2. Moderate ( medium)

3. Substantial (high)

COs/POs CO1 CO2 CO3 CO4 CO5

PO1 PO2 PO3 PO4 PO5 PO6 PO7 PO8 PO9 PO10 PO11 PO12

3 3

3

3 3

3

3

3 3

3

2

3

3

3

3

KAS-

3 3

3

3

101T/201T

Engineering Physics Module-1-Relativistic Mechanics

KAS 101T/201T CO1

1.1 Introduction of Newtonian Mechanics

The universe, in which we live, is full of dynamic objects. Nothing is static here Starting from giant stars to tiny electrons, everything is dynamic. This dynamicity of universal objects leads to variety of interactions, events and happenings. The curiosity of human being/scientist to know about these events and laws which governs them. Mechanics is the branch of Physics which is mainly concerned with the study of mobile bodies and their interactions.

A real breakthrough in this direction was made my Newton in 1664 by presenting the law of linear motion of bodies. Over two hundred years, these laws were considered to be perfect and capable of explaining everything of nature.

1.2 Frames of Reference

The dynamicity of universal objects leads to variety of interactions between these objects leading to various happenings. These happenings are termed as events. The relevant data about an event is recorded by some person or instrument, which is known as “Observer”.

The motion of material body can only described relative to some other object. As such, to locate the position of a particle or event, we need a coordinate system which is at rest with respect to the observer. Such a coordinate system is referred as frame of reference or observers frame of reference.

“A reference frame is a space or region in which we are making observation and measuring physical(dynamical) quantities such as velocity and acceleration of an object(event).” or

“A frame of reference is a three dimensional coordinate system relative to which described the position and motion(velocity and acceleration) of a body(object)”

Fig.1.1 represents a frame of reference(S), an object be

situated at point P have co-ordinate

i.e.

P(

), measured by an Observer(O). Where be the time

of measurement of the co-ordinate of an object. Observers in

different frame of references may describe same event in

different way.

Fig 1.1: frame of reference

Example- takes a point on the rim of a moving wheel of cycle. for an observer sitting at the center of the wheel, the path of the point will be a circle. However, For an observer standing on the ground the path of the point will appear as cycloid.

1

Engineering Physics

KAS 101T/201T

Module-1-Relativistic Mechanics

CO1

There are two types of frames of reference.

1. Inertial or non-accelerating frames of reference 2. Non-inertial or accelerating frames of reference

The frame of reference in which Newton's laws of motion hold good is treated as inertial frame of reference. However, the frame of reference in which Newton's laws of motion not hold good is treated as non-inertial frame of reference. Earth is non-inertial frame of reference, because it has acceleration due to spin motion about its axis and orbital motion around the sun.

1.3 Galilean Transformations

The Galilean transformations equations are used to transform the coordinates of position and time from one inertial frame to the other. The equations relating the coordinates of a particle in two inertial frames are called as Galilean transformations. Consider the two inertial frames of reference F and F'. Let the frame F' is moving with constant velocity v with reference to frame F. The frames F and F' are shown in Fig.1

Let some event occurs at the point P at any instant of time t. The coordinates of point P with respect to frame F are x, y, z, t and with respect to frame F' are x', y', z', t'. Let at t = t'= 0, the origin O of frame F and O' of frame F' coincides with one another. Also axes x and x' are parallel to v. Let y' and z' are parallel to y and z respectively.

From Fig.

x=x'+ vt……………………………...(1)

x'= x - vt……………………………...(2)

As there is no relative motion along y and zaxes, we can writ

y'=y….…(3),

z'=z….…(4), and t'=t….…(5)

These equations are called as Galilean transformation equations. The inverse Galilean transformation can be written as,

x=x' + vt, y=y', z'=z and t=t'

Hence transformation in position is variant only along the direction of motion of the frame and remaining dimensions( y and z) are unchanged under Galilean Transformation. At that era scientist were assumed time should be absolute.

2

Engineering Physics Module-1-Relativistic Mechanics

KAS 101T/201T CO1

b) Transformation in velocities components:

The conversion of velocity components measured in frame F into their equivalent components in the frame F' can be known by differential Equation (1) with respect to time we get,

u'x= = (

)

hence u'x= ux-V Similarly, from Equation (3) and (4) we can write

u'y= uy and u'z= uz In vector form,

u'=u – v

Hence transformation in velocity is variant only along the direction of motion of the frame and remaining dimensions( along y and z) are unchanged under Galilean Transformation.

c) Transformation in acceleration components:

The acceleration components can be derived by differentiating velocity equations with respect to time,

a'x=

(

)

a'x= ax

In vector form a' = a

This shows that in all inertial reference frames a body will be observed to have the same acceleration. Hence acceleration are invariant under Galilean Transformation.

Failures of Galilean transformation:

According to Galilean transformations the laws of mechanics are invariant. But under Galilean transformations, the fundamental equations of electricity and magnetism have very different forms.

Also if we measure the speed of light c along x-direction in the frame F and then in the frame F' the value comes to be c' = c – vx. But according to special theory of relativity the speed of light c is same in all inertial frames.

1.4 Michelson-Morley Experiment

“The objective of Michelson - Morley experiment was to detect the existence of stationary medium ether (stationary frame of reference i.e. ether frame.)”, which was assumed to be required for the propagation of the light in the space.

3

Engineering Physics

KAS 101T/201T

Module-1-Relativistic Mechanics

CO1

In order to detect the change in velocity of light due to relative motion between earth and hypothetical medium ether, Michelson and Morley performed an experiment which is discussed below. The experimental arrangement is shown in Fig

Light from a monochromatic source S, falls on the semi-silvered glass plate G inclined at an angle 45° to the beam. It is divided into two parts by the semi silvered surface, one ray 1 which travels towards mirror M1 and other is transmitted, ray 2 towards mirror M2. These two rays fall normally on mirrors M1 and M2 respectively and are reflected back along their original paths and meet at point G and enter in telescope. In telescope interference pattern is obtained.

If the apparatus is at rest in ether, the two reflected rays would take equal time to return the glass plate G. But actually the whole apparatus is moving along with the earth with a velocity say v. Due to motion of earth the optical path traversed by both the rays are not the same. Thus the time taken by the two rays to travel to the mirrors and back to G will be different in this case.

Let the mirrors M1 and M2 are at equal distance l from the glass plate G. Further let c and v be the velocities at light and apparatus or earth respectively. It is clear from Fig. that the reflected ray 1 from glass plate G strikes the mirror M1at A' and not at A due to the motion of the earth.

The total path of the ray from G to A' and back will be GA'G'.

∴ From Δ GA'D

(GA')2=(AA')2+ (A'D)2...(1)

As (GD =AA')

If t be the time taken by the ray to move from G to A', then from Equation (1), we have (c t)2=(v t)2 + (l)2

Hence t= √ If t1 be the time taken by the ray to travel the whole path GA'G', then

4

Engineering Physics Module-1-Relativistic Mechanics

KAS 101T/201T CO1

t1= 2t = √ t1= ((

= (

)

)) ……………….(2) Using Binomial Theorem

Now, in case of transmitted ray 2 which is moving longitudinally towards mirror M2. It has a velocity (c – v) relative to the apparatus when it is moving from G to B. During its return journey, its velocity relative to apparatus is (c + v). If t2 be the total time taken by the longitudinal ray to reach G', then

T2= ( ) ( ) after solving

t2= ((

)) ……………….(3)

Thus, the difference in times of travel of longitudinal and transverse journeys is

Δt = t2-t1 = = ((

)) ((

))

Δt = ……………….(4) The optical path difference between two rays is given as,

Optical path difference (Δ) =Velocity × t = c x Δt

= c x =

Δ = ……………….(5)

If λ is the wavelength of light used, then path difference in terms of wavelength is, = λ Michelson-Morley perform the experiment in two steps. First by setting as shown in fig and secondly by turning the apparatus through 900. Now the path difference is in opposite direction i.e. path difference is -λ . Hence total fringe shift ΔN = Michelson and Morley using l=11 m, λ= 5800 x 10-10 m, v= 3 x 104 m/sec and c= 3 x 108 m/sec ∴ Change in fringe shift ΔN = λ substitute all these values

=0.37 fringe

5

Engineering Physics

KAS 101T/201T

Module-1-Relativistic Mechanics

CO1

But the experimental were detecting no fringe shift. So there was some problem in theory calculation and is a negative result. The conclusion drawn from the Michelson-Morley experiment is that, there is no existence of stationary medium ether in space.

Negative results of Michelson - Morley experiment

1. Ether drag hypothesis: In Michelson - Morley experiment it is explained that there is no relative

motion between the ether and earth. Whereas the moving earth drags ether alone with its motion

so the relative velocity of ether and earth will be zero.

2.

Lorentz-Fitzgerald Hypothesis: Lorentz told that the length of the arm (distance between the pale

and the mirror M2) towards the transmitted side should be L(√1 – v2/c2) but not L. If this is taken

then theory and experimental will get matched. But this hypothesis is discarded as there was no

proof for this. 3. Constancy of Velocity of light: In Michelson - Morley experiment the null shift in fringes was

observed. According to Einstein the velocity of light is constant it is independent of frame of

reference, source and observer.

Einstein special theory of Relativity (STR)

Einstein gave his special theory of relativity (STR) on the basis of M-M experiment

Einstein’s First Postulate of theory of relativity: All the laws of physics are same (or have the same form) in all the inertial frames of reference moving with uniform velocity with respect to each other. (This postulate is also called the law of equivalence).

Einstein’s second Postulate of theory of relativity: The speed of light is constant in free space or in vacuum in all the inertial frames of reference moving with uniform velocity with respect to each other. (This postulate is also called the law of constancy).

1.5 Lorentz Transformation Equations

Consider the two observers O and O' at the origin of the inertial frame of reference F and F' respectively as shown in Fig. Let at time t = t' = 0, the two coordinate systems coincide initially. Let a pulse of light is flashed at time t = 0 from the origin which spreads out in the space and at the same time the frame F' starts moving with constant velocity v along positive X-direction relative to the frame F. This pulse of light reaches at point P, whose coordinates of position and time are (x, y, z, t) and (x',y', z', t') measured by the observer O and O' respectively. Therefore the transformation equations of x and x' can be given as,

x'=k (x – v t)………………………....(1)

Where k is the proportionality constant and is independent of x and t.

6

Engineering Physics

KAS 101T/201T

Module-1-Relativistic Mechanics

CO1

The inverse relation can be given as,

x=k (x' + v t') ………………… (2)

As t and t' are not equal, substitute the value of x' from Equation (1) in Equation (2)

x=k [k (x – vt) + vt'] or =(

)

t'=

or

t'=

(

) ………………………....(3)

According to second postulate of special theory of relativity the speed of light c remains constant. Therefore the velocity of pulse of light which spreads out from the common origin observed by observer O and O' should be same.

∴ x=c t and x' = ct'………………………………………...(4)

Substitute the values of x and x' from Equation (4) in Equation (1) and (2) we get

ct'=k (x – v t) = k (ct – v t) or ct'=kt (c – v) ................................(5)

and similarly

ct=k t' (c + v) ................................(6)

Multiplying Equation (5) and (6) we get, c2 t t'=k2t t' (c2 – v2) hence

after solving

k= ................................(7)

√

Hence equation (7) substitute in equation (1), then Lorentz transformation in position will be

x'= ,

√

y=y', z'=z

Calculation of Time: equation (7) substitute in equation (3),

t'=

(

)

From equation ( 7),

or, then above equation becomes

t'=

=

or,

t'= ( )

7

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