Engineering Physics (2nd Edition) PDF

Engineering Physics Second Edition About the Authors Hitendra K Malik is currently Professor of Physics at the Indian Institute of Technology Delhi, from where he received his PhD degree in the field of Plasma Physics, in 1995 at the age of 24. He...

Engineering Physics Second Edition About the Authors Hitendra K Malik is currently Professor of Physics at the Indian Institute of Technology Delhi, from where he received his PhD degree in the field of Plasma Physics, in 1995 at the age of 24. He has been a merit scholarship holder throughout his academic career. He is the recipient of Career Award from AICTE, Government of India, for his teaching and research, Outstanding Scientist Award from VIF, India, for his contributions to Science, and 2017 Albert Nelson Marquis Lifetime Achievement Award from USA. In addition, he received the prestigious Erasmus Mundus Visiting Fellowship from European Union (Germany and France), JSPS Fellowship (two times) from Japan, FRD Fellowship from South Africa and DAAD Fellowship from Germany. Owing to his worldwide recognition, his name has been included in ‘Marquis Who’s Who’ in 2011, published from USA. Based on the survey conducted by ResearchGate (RG), his scientific score has been found within top 5% of the scientists and researchers all over the world. Professor Malik is highly cited in India and abroad for his research work and books with h-index of 24 and i10-index of 70. Governments of India, Germany and France, through DST, CSIR, DRDO, AICTE, DAAD, CEFIPRA, etc., have provided him funding to accomplish 12 sponsored research projects. He is on the editorial board of 5 reputed research journals (including Springer). In recognition of his outstanding research and teaching contributions, he has been asked to deliver more than 50 keynote and invited talks in India, Japan, South Korea, USA, France, Germany, South Africa, and Turkey. Also, he has been chief guest in various universities, mentor of faculty colleagues of engineering institutions, and member of organizing and advisory committees of national and international conferences held in India and abroad. He has guided 80 PhD, postgraduate and undergraduate theses, including 22 PhD theses in the area of laser/ microwave plasma interactions, particle acceleration, solitons, Terahertz radiation, Hall thrusters, plasma material interaction, and nanotechnology. He has published more than 330 scientific papers in high impact factor journals and conferences, including 19 independent articles. He has been reviewer for 72 Journals of international repute, several sponsored research projects (Indian and Foreign agencies), and 18 PhD theses. He is an expert member of academic and administrative bodies of 14 different universities and institutions from 8 states of India including UGC. Apart from this book, he has also authored another textbook on Laser-Matter Interaction, CRC Press, 3 Chapters in the Books Wave Propagation, InTechOpen Science, Croatia (featured as highly downloaded chapter), Society, Sustainability and Environment, Shivalik Prakashan, New Delhi, and Plasma Science and Nanotechnology, Apple Academic Press, exclusive worldwide distribution by CRC Press, a Taylor & Francis Group. Ajay Kumar Singh has almost two decades of teaching experience in several engineering institutions across North India. Currently, he is Professor of Physics at the Department of Applied Sciences, Maharaja Surajmal Institute of Technology (MSIT), Janakpuri, New Delhi. He has also served as the Head of Department at MSIT. Earlier, he was associated as Professor (2003–2012) at the Department of Applied Science and Humanities, Dronacharya College of Engineering, Haryana. Dr. Singh completed his PhD from Aligarh Muslim University in the year 1999. During his PhD, his work specifically focused on Uranium concentration in rock samples, soil samples and fly ash samples. He also investigated radon levels in low and high background areas. He has published more than 20 research papers and several articles in national and international journals and conferences. He has edited and co-authored several books on Environment, Water Resources, Nuclear Physics, and Engineering Physics. His book on Engineering Physics Practical and Tutorials has been highly appreciated by students. He is a life member of Plasma Science Society of India (PSSI). Dr. Singh was the ‘B.Tech First Year Syllabus Revision Committee’ coordinator representing all the affiliated engineering colleges of Guru Gobind Singh Indraprastha University (GGSIPU). He is academic coordinator of PhD scholars enrolled at MSIT under University School of Information, Communication and Technology, GGSIPU, Dwarka, New Delhi. He is also supervisor of PhD students under USICT, which is a premier constituent institute of GGSIPU. Dr. Singh has also been teaching a special course on Nanotechnology for the USICT PhD students. He is also the teacher representative in the governing board of Maharaja Surajmal Institute of Technology.  Engineering Physics Second Edition Hitendra K MaliK Professor, Department of Physics Indian Institute of Technology Delhi ajay KuMar SingH Professor, Department of Applied Sciences Maharaja Surajmal Institute of Technology (MSIT) New Delhi McGraw Hill Education (India) Private Limited CHENNAI McGraw Hill Education Offices Chennai New York St Louis San Francisco Auckland Bogotá Caracas Kuala Lumpur Lisbon London Madrid Mexico City Milan Montreal San Juan Santiago Singapore Sydney Tokyo Toronto  McGraw Hill Education (India) Private Limited Published by McGraw Hill Education (India) Private Limited 444/1, Sri Ekambara Naicker Industrial Estate, Alapakkam, Porur, Chennai - 600 116 Engineering Physics, 2e Copyright © 2018, 2010, by McGraw Hill Education (India) Private Limited. 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However, neither McGraw Hill Education (India) nor its authors guarantee the accuracy or completeness of any information published herein, and neither McGraw Hill Education (India) nor its authors shall be responsible for any errors, omissions, or damages arising out of use of this information. This work is published with the understanding that McGraw Hill Education (India) and its authors are supplying information but are not attempting to render engineering or other professional services. If such services are required, the assistance of an appropriate professional should be sought. Typeset at The Composers, 260, C.A. Apt., Paschim Vihar, New Delhi 110 063 and printed at Cover Printer: Visit us at: www.mheducation.co.in  Dedicated to OMENDRA Bhaiya and all those moments that remain with me as a source of inspiration and help me to move ahead with great success, satisfaction and optimistic approach Brief Contents Foreword xix Preface to the Second Edition xxi Preface to the First Edition xxiii 1. Interference 1 2. Diffraction 63 3. Polarisation 121 4. Lasers and Holography 155 5. Fibre Optics 186 6. Electron Optics 208 7. Waves and Oscillations 233 8. Simple Harmonic Motion and Sound Waves 259 9. Sound Waves and Acoustics of Buildings 284 10. Dielectrics 313 11. Electromagnetism 328 12. Theory of Relativity 395 13. Applied Nuclear Physics 451 14. Crystal Structure 517 15. Development of Quantum Mechanics 551 16. Quantum Mechanics 595 17. Free Electron Theory 634 18. Band Theory of Solids and Photoconductivity 654 19. Magnetic Properties of Solids 685 20. Superconductivity 716 21. X-Rays 735 viii Brief Contents 22. Nanoscience and Nanotechnology 751 Appendices Appendix 1 Measurements and Errors 779 Appendix 2 Optics 790 Appendix 3 Mechanical Properties of Materials 794 Chapter-wise Answers to Objective Type Questions 805 Index 809  Contents ix Contents Foreword xix Preface to the Second Edition xxi Preface to the First Edition xxiii 1. Interference 1 Learning Objectives 1 1.1 Young’s Double Slit Experiment 2 1.2 Concept of Waves and Huygens’ Principle 2 1.3 Phase Difference and Path Difference 3 1.4 Coherence 4 1.5 Coherent Sources 5 1.6 Analytical Treatment of Interference 6 1.7 Conditions for Sustained Interference 8 1.8 Multiple Beam Superposition 9 1.9 Interference by Division of Wavefront 10 1.10 Interference by Division of Amplitude 16 1.11 Applications of Interference in the Field of Engineering 32 1.12 Scientific Applications of Interference 35 1.13 Homodyne and Heterodyne Detection 35 Summary 37 Solved Examples 38 Objective Type Questions 57 Short-Answer Questions 58 Practice Problems 58 Unsolved Questions 60 2. Diffraction 63 Learning Objectives 63 2.1 Young’s Double Slit Experiment: Diffraction or Interference? 64 2.2 Difference between Diffraction and Interference 64 2.3 Types of Diffraction 64 2.4 Fresnel’s Half-period Zones 66 2.5 Zone Plate 68 2.6 Fresnel’s Diffraction by a Circular Aperture 72 x Contents 2.7 Fraunhofer Diffraction by a Single Slit 76 2.8 Fraunhofer Diffraction by Double Slits 80 2.9 Fraunhofer Diffraction by N Slits: Diffraction Grating 84 2.10 Application of Diffraction Grating 90 2.11 Resolving Power of an Optical Instrument 91 2.12 Resolving Power of a Telescope 93 2.13 Resolving Power of a Microscope 94 2.14 Resolving Power of a Plane Diffraction Grating 96 2.15 Dispersive Power of a Plane Diffraction Grating 97 Summary 98 Solved Examples 99 Objective Type Questions 116 Short-Answer Questions 118 Practice Problems 118 Unsolved Questions 119 3. Polarisation 121 Learning Objectives 121 3.1 Mechanical Experiment Showing Polarisation of Transverse Wave 122 3.2 Difference between Unpolarised Light and Polarised Light 123 3.3 Means of Production of Plane–Polarised Light 123 3.4 Theory of Production of Plane, Circularly and Elliptically Polarised Light 135 3.5 Optical Activity 138 3.6 Specific Rotation 138 3.7 Laurent’s Half-shade Polarimeter 138 3.8 Biquartz Polarimeter 139 3.9 Saccharimeter 140 3.10 Photoelasticity 141 Summary 142 Solved Examples 143 Objective Type Questions 150 Short-Answer Questions 151 Practice Problems 152 Unsolved Questions 153 4. Lasers and Holography 155 Learning Objectives 155 4.1 Absorption and Emission of Radiation 156 4.2 Population Inversion 159 4.3 Characteristic of Laser Light 161 4.4 Main Components of Laser 163 4.5 Types of Laser 163 4.6 Applications of Lasers 169 4.7 Laser Cooling 170 4.8 Holography 170  Contents xi 4.9 Holography Versus Conventional Photography 171 4.10 Recording and Reconstruction of Image on Holograph 172 4.11 Types of Holograms 174 4.12 Applications of Holography 177 Summary 178 Solved Examples 179 Objective Type Questions 183 Practice Problems 184 5. Fibre Optics 186 Learning Objectives 186 5.1 Fundamental Ideas about Optical Fibre 187 5.2 Optical Fibres as a Dielectric Waveguide 188 5.3 Types of Optical Fibres 188 5.4 Acceptance Angle and Numerical Aperture 190 5.5 Fibre Optics Communication 193 5.6 Optical Fibre Sensors 196 5.7 Optical Fibre Connector 197 5.8 Optical Fibre Couplers 197 5.9 Applications of Optical Fibre Couplers 198 Summary 198 Solved Examples 200 Objective Type Questions 206 Short-Answer Questions 206 Practice Problems 207 6. Electron Optics 208 Learning Objectives 208 6.1 Specific Charge of an Electron 208 6.2 Determination of Specific Charge of an Electron: Thomson’s Method 209 6.3 Motion of an Electron in Uniform Electric and Magnetic Fields 210 6.4 Electrostatic and Magnetostatic Focusing 219 6.5 Scanning Electron Microscope (SEM) 220 6.6 Scanning Tunneling Microscope (STM) 225 Summary 226 Solved Examples 227 Objective Type Questions 231 Practice Problems 232 7. Waves and Oscillations 233 Learning Objectives 233 7.1 Translational Motion 233 7.2 Vibrational or Oscillatory Motion 233 7.3 Simple Harmonic Motion (SHM) 234 7.4 Differential Equation of SHM and Its Solution 234 7.5 Simple Pendulum 239 xii Contents 7.6 Mass-string System 241 7.7 Damped Harmonic Oscillator 243 7.8 Attenuation Coefficients of a Vibrating System 245 7.9 Forced Vibrations 247 7.10 Resonance 250 Summary 251 Solved Examples 252 Objective Type Questions 257 Short-Answer Questions 257 Practice Problems 258 8. Simple Harmonic Motion and Sound Waves 259 Learning Objectives 259 8.1 Superposition of Two SHMs 260 8.2 Sound Wave and its Velocity 260 8.3 Sound Displacement 261 8.4 Standing Waves 261 8.5 Standing Waves in Air Columns 262 8.6 Doppler Effect 264 8.7 Supersonic and Shock Waves 266 8.8 Derivation of Sound Speed 267 8.9 Intensity of Sound and Its Derivation 268 8.10 Sound-Intensity Level 269 8.11 Interference of Sound Waves in Time: Beats 270 8.12 Relation between Displacement and Pressure Amplitude 271 8.13 Lissajous Figures 272 8.14 Endoscopy 274 Summary 275 Solved Examples 276 Objective Type Questions 280 Practice Problems 282 Unsolved Questions 283 9. Sound Waves and Acoustics of Buildings 284 Learning Objectives 284 9.1 Audible, Ultrasonic and Infrasonic Waves 284 9.2 Production of Ultrasonic Waves 285 9.3 Absorption and Dispersion of Ultrasonic Waves 290 9.4 Detection of Ultrasonic Waves 291 9.5 Applications of Ultrasonic Waves 292 9.6 Types of Acoustics 295 9.7 Acoustics of Buildings 296 9.8 Factors Affecting the Architectural Acoustics 300 Summary 301 Solved Examples 302  Contents xiii Objective Type Questions 308 True or False 311 Practice Problems 312 10. Dielectrics 313 Learning Objectives 313 10.1 Dielectric Constant 313 10.2 Types of Dielectrics 314 10.3 Polarisation of Dielectrics 315 10.4 Types of Polarisation 317 10.5 Gauss’s Law in Dielectrics 318 10.6 Dielectric Loss 319 10.7 Clausius-Mosotti Equation 321 Summary 322 Solved Examples 323 Objective Type Questions 325 Short-Answer Questions 326 Practice Problems 326 11. Electromagnetism 328 Learning Objectives 328 11.1 Charge Density 329 11.2 Del Operator 330 11.3 Gradient 331 11.4 Divergence 331 11.5 Curl 332 11.6 Fundamental Theorem of Calculus 333 11.7 Fundamental Theorem for Gradient 333 11.8 Gauss’s or Green’s Theorem 334 11.9 Stokes’ Theorem 335 11.10 Electric Field and Electric Potential 335 11.11 Poisson’s and Laplace’s Equations 336 11.12 Capacitor 337 Æ 11.13 Magnetic Flux Density (B ) 338 Æ 11.14 Magnetic Field Strength (H ) 338 11.15 Ampere’s Circuital Law 338 11.16 Electrostatic Boundary Conditions 339 11.17 Scalar and Vector Potentials 341 11.18 Continuity Equation 342 11.19 Maxwell’s Equations: Differential Form 343 11.20 Maxwell’s Equations: Integral Form 348 11.21 Significance of Maxwell’s Equations 350 11.22 Maxwell’s Displacement Current and Correction in Ampere’s Law 351 11.23 Eletromagnetic (EM) Wave Propagation in Free Space 352 11.24 Transverse Nature of Electromagnetic Waves 355 xiv Contents 11.25 Maxwell’s Equations in Isotropic Dielectric Medium: EM Wave Propogation 355 11.26 Maxwell’s Equations in Conducting Medium: EM Wave Propagation and Skin Depth 357 11.27 Electromagnetic Energy Density 360 11.28 Poynting Vector and Poynting Theorem 361 11.29 Wave Propagation in Bounded System: Waveguide 363 11.30 Coaxial Cable 364 Summary 365 Solved Examples 369 Objective Type Questions 388 Short-Answer Questions 390 Practice Problems 391 12. Theory of Relativity 395 Learning Objectives 395 12.1 Frame of Reference 395 12.2 Galilean Transformation 396 12.3 Michelson-Morley Experiment 398 12.4 Postulates of Special Theory of Relativity 401 12.5 Lorentz Transformation 401 12.6 Length Contraction 403 12.7 Time Dilation 405 12.8 Addition of Velocities 406 12.9 Variation of Mass with Velocity 407 12.10 Einstein’s Mass Energy Relation 409 Summary 411 Solved Examples 413 Objective Type Questions 447 Short-Answer Questions 448 Practice Problems 449 Unsolved Questions 450 13. Applied Nuclear Physics 451 Learning Objectives 451 13.1 Basic Properties of Nucleus 452 13.2 Nuclear Forces 454 13.3 Binding Energy of Nucleus 455 13.4 Nuclear Stability 457 13.5 Nuclear Models 457 13.6 Nuclear Shell Model 457 13.7 Nuclear Liquid Drop Model 459 13.8 Radioactivity 460 13.9 Discovery of Neutron 469 13.10 Nuclear Reactions: Conservation Laws 471 13.11 Nuclear Fission 473  Contents xv 13.12 Nuclear Fusion 476 13.13 Controlled Fusion 477 13.14 Particle Accelerators 481 Summary 487 Solved Examples 492 Objective Type Questions 508 Short-Answer Questions 512 Practice Problems 513 Unsolved Questions 515 14. Crystal Structure 517 Learning Objectives 517 14.1 Types of Solids 518 14.2 Unit Cell 518 14.3 Types of Crystals 519 14.4 Translation Vectors 520 14.5 Lattice Planes 521 14.6 Miller Indices 521 14.7 Simple Crystal Structures 523 14.8 Coordination Number 523 14.9 Interplanar Spacing 524 14.10 Nearest Neighbour Distance and Atomic Radius 525 14.11 Packing Fraction 526 14.12 Potential Energy Curve and Nature of Interatomic Forces 528 14.13 Different Types of Bonding Forces 529 14.14 Crystal Structure Analysis 531 14.15 Point Defects in Solids 534 Summary 537 Solved Examples 539 Objective Type Questions 546 Short-Answer Questions 548 Practice Problems 549 Unsolved Questions 550 15. Development of Quantum Mechanics 551 Learning Objectives 551 15.1 Blackbody Radiation: Spectral Distribution 552 15.2 Planck’s Quantum Hypothesis 554 15.3 Simple Concept of Quantum Theory 556 15.4 Wave Particle Duality 557 15.5 Photoelectric Effect 557 15.6 de Broglie Waves: Matter Waves 559 15.7 Compton Effect: Compton Scattering 561 15.8 Phase and Group Velocities: de Broglie Waves 565 Summary 569 xvi Contents Solved Examples 570 Objective Type Questions 591 Short-Answer Questions 592 Practice Problems 593 Unsolved Questions 594 16. Quantum Mechanics 595 Learning Objectives 595 16.1 Heisenberg Uncertainty Principle 596 16.2 Wave Function and Its Physical Significance 601 16.3 Time Independent Schrödinger Equation 602 16.4 Time Dependent Schrödinger Equation 603 16.5 Operators 604 16.6 Applications of Schrödinger Equation 605 16.7 Quantum Statistics 614 Summary 617 Solved Examples 618 Objective Type Questions 630 Short-Answer Questions 632 Practice Problems 632 Unsolved Questions 633 17. Free Electron Theory 634 Learning Objectives 634 17.1 Lorentz–Drude Theory: Classical Free Electron Theory of Metals 635 17.2 Applications of Lorentz–Drude Theory 636 17.3 Limitations of Lorentz–Drude or Free Electron Theory 638 17.4 Quantum Theory of Free Electrons 638 17.5 Thermionic Emission 644 Summary 645 Solved Examples 646 Objective Type Questions 651 Short-Answer Questions 652 Practice Problems 653 18. Band Theory of Solids and Photoconductivity 654 Learning Objectives 654 18.1 Kronig-Penney Model 655 18.2 One- and Two-Dimensional Brillouin Zones 659 18.3 Effective Mass of an Electron 660 18.4 Distinction between Insulators, Semiconductors and Conductors (Metals) 661 18.5 Intrinsic Semiconductor 664 18.6 Extrinsic Semiconductor 667 18.7 Hall Effect 669 18.8 Photoconductivity 671 18.9 Simple Model of Photoconductor 671  Contents xvii 18.10 Effect of Traps 674 18.11 Applications of Photoconductivity 675 Summary 676 Solved Examples 678 Objective Type Questions 680 Practice Problems 683 19. Magnetic Properties of Solids 685 Learning Objectives 685 19.1 Magnetic Moment of an Electron 687 19.2 Classification of Magnetic Materials 688 19.3 Comparison of Properties of Paramagnetic, Diamagnetic and Ferromagnetic Materials 689 19.4 Classical Theory of Diamagnetism (Langevin’s Theory) 689 19.5 Classical Theory of Paramagnetism (Langevin’s Theory) 693 19.6 Classical Theory of Ferromagnetism 696 Æ Æ 19.7 Hysteresis: Nonlinear Relationship between B and H 697 19.8 Energy Loss Due to Hysteresis 698 19.9 Importance of Hysteresis Curve 699 19.10 Magnetic Circuits 700 19.11 Forces on Magnetic Materials 700 19.12 Magnetic Materials and Their Applications 700 Summary 702 Solved Examples 705 Objective Type Questions 712 Short-Answer Questions 714 Practice Problems 714 20. Superconductivity 716 Learning Objectives 716 20.1 Electrical Resistivity of Solids and Phonons 717 20.2 Properties of Superconductors 717 20.3 Classification of Superconductors 719 20.4 Effect of Magnetic Field 719 20.5 Isotope Effect 720 20.6 London Equations 720 20.7 Penetration Depth 722 20.8 Cooper Pairs 723 20.9 Bose–Einstein Condensation 724 20.10 BCS Theory: Qualitative Explanation 724 20.11 Coherence Length 725 20.12 High Temperature (Hi-Tc) Superconductivity 726 20.13 Application of Superconductivity 726 Summary 727 Solved Examples 729 xviii Contents Objective Type Questions 732 Short-Answer Questions 734 Practice Problems 734 21. X-Rays 735 Learning Objectives 735 21.1 Origin of X-rays 736 21.2 Properties of X-rays 736 21.3 X-ray Spectra 737 21.4 Moseley’s Law 738 21.5 Practical Applications of X-rays 740 Summary 740 Solved Examples 741 Objective Type Questions 747 Short-Answer Questions 748 Practice Problems 749 Unsolved Questions 749 22. Nanoscience and Nanotechnology 751 Learning Objectives 751 22.1 How Nanomaterials are Differenet from Bulk Materials? 752 22.2 Difference between Nanoscience and Nanotechnology 752 22.3 Quantum Confinement and Classification of Nanostructures 752 22.4 Nanoscale in 1-D 753 22.5 Nanoscale in 2-D 760 22.6 Nanoscale in 3-D 760 22.7 Applications of Nanotechnology 768 22.8 Limitations of Nanotechnology 770 22.9 Disadvantages of Nanotechnology 770 Summary 771 Solved Examples 773 Objective Type Questions 774 Practice Problems 776 Appendices Appendix 1 Measurements and Errors 779 Appendix 2 Optics 790 Appendix 3 Mechanical Properties of Materials 794 Chapter-wise Answers to Objective Type Questions 805 Index 809  Foreword xix Foreword It gives me immense pleasure to see the present textbook on “Engineering Physics” which covers almost the entire syllabus taught at undergraduate level at different engineering colleges and institutions throughout India. I complement the authors and appreciate their efforts in bringing out this book written in a very simple language. The text is comprehensive and the explanation of topics is commendable. I understand that this book carries all the elements required for a good presentation. I have been a student of IIT Kharagpur and later on taught at IIT Delhi. Being a part of the IIT system, I recognise that the rigorous and enriching teaching experience at IITs originating from the interaction with the best engineering students and their strong feedback results in continuous evolution and refinement of the teachers. This spirit is reflected in the comprehensive and in-depth handling of important topics in a very simple manner in this book. I am happy to note that this textbook has been penned down by IITian and hope that it would serve to be a good textbook on the subject. Since this book also covers advanced topics, it will be an important learning resource for the teachers, and those students who wish to develop research skills and pursue higher studies. I hope that the book is well received in the academic world. Professor Prem Vrat Vice-Chancellor, U.P. Technical University, Lucknow Founder Director, IIT Roorkee  Preface to the Second Edition xxi Preface to the Second Edition The first edition of the textbook was appreciated by the teachers and students of many universities, engineering colleges and institutes, including IIT’s throughout India. Words of appreciation were also received from faculty colleagues from Japan, China, Taiwan, Russia, Canada, South Korea, Pakistan, Bangladesh, Turkey, Iran, South Africa, Germany, France, United Kingdom, and United States of America. Students preparing for GATE/CSIR competitive examinations also suggested for more examples in the book and inclusion of topics of postgraduate level. The students very enthusiastically informed us about the utility of the book for the preparation of interviews for admission in PhD programmes at IITs and other universities (including foreign universities) or to get government jobs in India. In view of all the above points, we have come up with the second edition of the book, where we have used simple language for explaining each and every topic. We have included more physical insight, wherever required. Some chapters are thoroughly revised in terms of new topics and solved problems. We have also updated advanced topics keeping in mind the research going on in these fields. The solutions to the Objective- Type Questions are also provided at the end of the book. In particular, Chapter 4 includes details of the topic Population Inversion which covers various schemes for the same, i.e., two-level, three-level and four-level systems. In Chapter 5, a topic on Optical Fibres as a Dielectric Waveguide is included. After Chapter 7 on Waves and Oscillations, a new Chapter 8 on Simple Harmonic Motion and Sound Waves has been included that discusses standing waves, supersonic and shock waves, in addition to sound waves, Doppler effect and Lissajous figures. Chapter 9 on Sound Waves and Acoustics of Buildings has been thoroughly revised. In this chapter, Recording and Reproduction of Sound has been withdrawn and other topics are revisited. New topics on ultrasonics have been included which talk about production of ultrasonic waves and their absorption, dispersion, detection and applications. In Chapter 10 on Dielectrics, a topic Energy Stored in an Electrostatic Field is withdrawn as its concept is discussed in Chapter 11 on Electromagnetism. Moreover, details of Clausius-Mosotti equation are revised with the inclusion of physical insight of this equation. The chapter on Electromagnetism has been thoroughly revised. For example, Section 11.21 has been rewritten in order to make the readers understand which form of the Maxwell’s equations is appropriate for free space, dielectric medium and conducting medium and how are these equations modified in these media. Bound charges and bound currents are also discussed. The solution to wave equation in conducting medium is included as Section 11.28.1, where dispersion relation, skin depth and phase relationship of the electric and magnetic field vectors are discussed. New solved problems, objective- type questions and other practice problems are also included in order to provide an indepth knowledge on the electromagnetic fields and their propagation in different media. In Chapter 12 on Theory of Relativity, physical insight to two interesting topics, viz. Length Contraction and Time Dilation is provided. Several new solved problems on various topics are also provided for the readers. Chapter 13 on Applied Nuclear Physics has been thoroughly revised and new topics are included on xxii Preface to the Second Edition basic properties of nucleus, nuclear forces, binding energy of nucleus, nuclear stability and various nuclear models, in addition to more equations and problems, both solved and unsolved. Introduction part of Chapter 16 on Quantum Mechanics has been revised. The topic on Thermionic Emission (Section 17.7) has been shortened but significance of Richardson’s equation is included. The earlier Chapter 21 on Photoconductivity and Photovoltaics has been withdrawn but its important topics, viz. photoconductivity, simple model of photoconductor and effect of traps, are included in Chapter 18 on Bond Theory of Solids and Photoconductivity. The much important Chapter 22 on Nanophysics has been rewritten in view of recent advances in the field. Now, it is renamed as Nanoscience and Nanotechnology. Certain new topics are included to clarify how nanomaterials are different from bulk materials and to know the differences between nanoscience and nanotechnology. The chapter very systematically discusses the nanoscales in 1D, 2D, 3D and OD. Particu- larly, nanowires, carbon nanotubes, inorganic nanotubes, biopolymers, nanoparticles, buckyballs/fullerenes and quantum dots are discussed in detail along with the methods of their synthesis, properties and their applications. Finally, the applications, limitations and disadvantages of nanotechnology are also discussed. The exhaustive OLC supplements of the book can be accessed at http://www.mhhe.com/malik/ep and contain the following: For Instructors Solution Manual Chapter-wise Power Point slides with diagrams and notes for effective lecture presentations For Students A sample chapter A Solved Question Paper An e-guide to aid last minute revision need We believe the readers shall find the second edition of the book more beneficial in terms of syllabus covered, quality of topics, large number of solved problems aimed at providing physical insight to various topics, and teaching various methods of solving difficult problems. The systematic approach adopted in the present book shall certainly help the teachers and students providing for crystal clear understanding of the topics and carrying out research in the related fields. This edition will be vital in enhancing the self confidence of our UG and PG students which will help them in advancing their careers. Finally, we look forward to receive feedback from the teachers and students on the recent edition of the book. H K Malik Ajay K Singh Publisher’s Note: McGraw Hill Education (India) invites suggestions and comments, all of which can be sent to [email protected] (kindly mention the title and author name in the subject line). Piracy-related issues may also be reported. Preface to the First Edition Physics is a mandatory subject for all engineering students, where almost all the important elements of the subject are covered. Finally, these evolve as different branches of the engineering course. The book entitled Engineering Physics has been written keeping in mind the need of undergraduate students from various engineering and science colleges of all Indian universities. It caters to the complete syllabus for both–Physics-I and Physics-II papers in the first year Engineering Physics course. The aim of writing this book has been to present the material in a concise and very simple way so that even weak students can grasp the fundamentals. In view of this, every chapter starts with a simple introduction and then related topics are covered with a detailed description along with the help of figures. Particularly the solved problems (compiled from University Question Papers) are at the end of each chapter. These problems are not merely numerical; many of them focus on reasoning and require thoughtful analysis. Finally, the chap- ters carry unsolved questions based on which the students would be able to test their knowledge as to what they have acquired after going through various chapters. A chapter-end summary and list of important formu- lae will be helpful to students for a quick review during examinations. The rich pedagogy consists of solved examples (450), objective-type questions (230), short-answer questions (224) and practice problems (617). The manuscript has been formulated in such a way that students shall grasp the subject easily and save their time as well. Since the complete syllabus is covered in a single book, it would be highly convenient to both. The manuscript contains 22 chapters which have been prepared as per the syllabus taught in various colleges and institutions. In particular, the manuscript discusses optics, lasers, holography, fibre optics, waves, acoustics of buildings, electromagnetism, theory of relativity, nuclear physics, solid state physics, quantum physics, magnetic properties of solids, superconductivity, photoconductivity and photovoltaic, X-rays and nanophysics in a systematic manner. We have discussed advanced topics such as laser cooling, Bose-Einstein condensation, scanning electron microscope (SEM), scanning tunnelling microscope (STM), controlled fusion including plasma, Lawson criterion, inertial confinement fusion (ICF), plasma based accelerators, namely, plasma wake field accelerator, plasma beat wave accelerator, laser wake field accelerator and self- modulated laser wake field accelerator, and nanophysics with special emphasis on properties of nanoparticles, carbon nanotubes, synthesis of nanoparticles and applications of nanotechnology. These will be of interest to the teachers who are involved in teaching postgraduate courses at the universities and the students who opt for higher studies and research as their career. Moreover, a series of review questions and problems at the end of each chapter together with the solved questions would serve as a question bank for the students preparing for various competitive examinations. They will get an opportunity to learn the subject and test their knowledge on the same platform. The structuring of the book provides in-depth coverage of all topics. Chapter 1 discusses Interference. Chapter 2 is on Diffraction. Chapter 3 is devoted to Polarization. Coherence and Lasers are described in xxiv Preface to the First Edition Chapter 4. Chapter 5 discusses Fibre Optics and its Applications, while Electron Optics is dealt with in Chapter 6. Chapter 7 describes Waves and Oscillations. Chapter 8 is on Sound Waves and Acoustics. Chapter 9 is on Dielectrics. Electromagnetic Wave Propagation is described in Chapter 10. Chapter 11 discusses the Theory of Relativity. Chapter 12 is devoted to Nuclear Physics. Crystal Structure is described in Chapter 13. Chapter 14 deals with the Development of Quantum Physics, while Chapter 15 is on Quantum Mechanics. Chapter 16 discusses Free Electron Theory. Band Theory of Solids is explained in Chapter 17. Chapter 18 describes the Magnetic Properties of Solids. Chapter 19 is on Superconductivity. Chapter 20 explains X-rays in detail while Chapter 21 is on Photoconductivity and Photovoltaics. Finally, Chapter 22 discusses Nanophysics in great detail. The manuscript has been organised such that it provides a link between different topics of a chapter. In order to make it simpler, all the necessary mathematical steps have been given and the physical feature of the mathematical expressions is discussed as and when required. The exhaustive OLC supplements of the book can be accessed at http://www.mhhe.com/malik/ep and contain the following: For Instructors Solution Manual Chapter-wise Power Point slides with diagrams and notes for effective lecture presentations For Students A sample chapter Link to reference material Solved Model Question Paper Answers to objective type questions given in the book. We would like to thank the entire team of Tata McGrawHill Education specifically Vibha Mahajan, Shalini Jha, Tina Jajoriya, Dipika Dey, Sohini Mukherji, Priyanka Negi and Baldev Raj for bringing out this book in a very short time span. The reviewers of the book also deserve a special mention for taking out time to review the book. Their names are given below. A K Jain IIT Roorkee Dhirendra Kumar Meerut Institute of Engineering and Technology, Uttar Pradesh Vinay Kumar SRMS CET, Bareilly Prerna Garg Meerut Institute of Technology, Uttar Pradesh Amit Kumar Srivastava Aryavrat Institute of Technology and Management, Lucknow Shyam Singh Aryavart Institute of Technology and Management, Lucknow R S Tiwari Apollo Institute of Engineering, Kanpur Kamlesh Pathak SVNIT, Surat, Gujarat Kanti Jotania M S University, Baroda, Gujarat Vijayalakshmi Sanyal Bharathiyar College of Engineering and Technology, Karaikal, Tamil Nadu A K Meikap NIT, Durgapur, West Bengal K Sivakumar Anna University, Chennai H K Malik Ajay K Singh Interference 1 Learning Objectives After reading this chapter you will be able to LO5 Discuss analytical treatment of interference and conditions for LO1 Explain interference through Young’s sustained interference double slit experiment LO6 Examine multiple beam superposition LO2 Describe the concept of wave and and interference by division of Huygen’s principle wavefront and amplitude LO3 Illustrate phase and path difference LO7 Review engineering/scientific LO4 Explain coherence, its various types applications of interferences including and coherent sources homodyne and heterodyne detection Introduction You would have seen beautiful colours in soap films or patch of oil floating on the surface of water. Moreover, the colour gets changed when you watch it from different angles. Did you ever try to find out the reason? In scientific language, this takes place due to the phenomenon of interference. The phenomenon of interference of light tells us about the wave nature of the light. In optics, the interference means the superposition of two or more waves which results in a new wave pattern. Here, we are talking about the interaction of waves emerging from the same source or when the frequencies of these waves are the same. In the context of light, which is an electromagnetic wave, we say that when the light from two different sources moves in the same direction, then these light wave trains superimpose upon each other. This results in the modification of distribution of intensity of light. According to the principle of superposition, this is called the interference of light. More precisely the interference can be defined as the interaction between two or more waves of the same or very close frequencies emitted from coherent sources (defined later), where the wavefronts are combined according to the principle of superposition. The resulting variation in the disturbances produced by the waves is called the interference pattern. Thomas Young, in 1802, explained the interference successfully in his double slit experiment. 2 Engineering Physics 1.1 YOung’s DOubLe sLit experiment LO1 The phenomenon of interference may be better understood by taking two point light sources S1 and S2 which produce similar waves (Fig. 1.1). Let the sources S1 and S2 be at equal distances from the main source S while being close to each other. Since the sources emit waves in all the directions, the spherical waves first pass through S and then S1 and S2. Finally these waves expand into the space. The crests of the waves are represented by complete arcs and the troughs by dotted arcs. It is seen that constructive interference takes place at the points where the crests due to one source meet the crests due to another source or where their troughs meet each other. In this case, the resultant S1 amplitude will be the sum of the amplitudes of the separate waves and hence the intensity of the light will be maximum at these points. Similarly, S at those points where crests due to one source meet the troughs due to S2 another source or vice-versa, the resultant amplitude will be the difference of the amplitudes of the separate waves. At these points the intensity of the waves (or light) will be minimum. Therefore, due to the intersection of these lines, an alternate bright and dark fringes are observed on the Screen screen placed at the right side of the sources S1 and S2. These fringes are Figure 1.1 obtained due to the phenomenon of interference of light. 1.2 COnCept Of Waves anD HuYgens’ prinCipLe LO2 A wave is a disturbance that propagates through space and time, usually with the transference of energy from one point to another without any particle of the medium being permanently displaced. Under this situation, the particles only oscillate about their equilibrium positions. If the oscillations of the particles are in the direction of wave propagation, then the wave is called longitudinal wave. However, if these oscillations take place in perpendicular direction with the direction of wave propagation, the wave is said to be transverse in nature. In electromagnetic waves, such as light waves, it is the changes in electric and magnetic fields which represent the wave disturbance. The progress of the wave propagation is described by the passage of a waveform through the medium with a certain velocity called the phase velocity or wave velocity. However, the energy is transferred at the group velocity of the waves making the waveform. The wave theory of the light was proposed in 1678 by Huygens, a Dutch scientist. On the basis of his wave theory, he explained satisfactorily the phenomena of reflections, refraction etc. In the beginning, Huygens’ supposed that these waves are longitudinal waves but later he came to know that these waves are transverse in nature. Huygens’ gave a hypothesis for geometrical construction of the position of a common wavefront at any instant when the propagation of waves takes place in a medium. The wavefront is an imaginary surface joining the points of constant phase in a wave propagated through the medium. The way in which the wavefront is propagated further in the medium is given by Huygens’ principle. This principle is based on the following assumptions: (i) Each point on the given wavefront acts as a source of secondary wavelets. (ii) The secondary wavelets from each point travel through space in all the directions with velocity of light. (iii) A surface touching the secondary wavelets tangentially in the forward direction at any given time constructs the new wavefront at that instant. This is known as secondary wavefront.  Interference 3 In order to demonstrate the Huygens’ principle, we consider P1 P2 P P1 the propagation of a spherical wavefront (Fig. 1.2a) or plane P wavefront (Fig. 1.2b) in an isotropic (uniform) medium P2 1 1 (for example, ether) emerging from a source of light S. At any time, suppose PQ is a section of the primary wavefront 2 2 drawn from the source S. To find the position of the wavefront after an interval t, we take points 1, 2, 3,... on the primary 3 3 wavefront PQ. As per Huygens’ principle, these points act S 4 4 as the source of secondary wavelets. Taking each point as the centre, we draw spheres of radii ct, where c is the speed 5 5 of light. These spherical surfaces represent the position of 6 6 secondary wavelets at time t. Further, we draw a surface P1Q1 that touches tangentially all these secondary wavelets Q2 in the forward direction. This surface P1Q1 is the secondary Q Q2 Q Q1 Q1 wavefront. Another surface P2Q2 in the backward direction is not called the secondary wavefront as there is no backward (a) (b) flow of the energy during the propagation of the light waves. Figure 1.2 1.3 pHase DifferenCe anD patH DifferenCe LO3 As mentioned, the interference pattern is obtained when the two or more waves superimpose each other. In order to understand this pattern it is very important to know about the path and phase differences between the interfering waves. 1.3.1 phase Difference Two waves that have the same frequencies and different phases are known to have a phase difference and are said to be out of phase, with each other. If the phase difference is 180°, then the two waves are said to be in antiphase and if it is 0°, then they are in phase as shown in Fig. 1.3(a and b). If the two interfering waves meet at a point where they are in antiphase, then the destructive interference occurs. However, if these two waves meet at a point where they are in the same phase, then the constructive interference takes place. (a) (b) Figure 1.3 1.3.2 path Difference In Fig. 1.4, while the two wave crests are traveling a different distance from their sources, they meet at a point P in such a way that a crest meets a crest. For this particular location on the pattern, the difference in distance traveled is known as path difference. 4 Engineering Physics 1.3.3 Relation between Path Difference and phase Difference It is clear from the positions of crests or troughs of the waves r1 that if the path difference between the two waves is equal to S1 P the wavelength l, the corresponding phase difference is 2p (360°). Suppose for a path difference of d the corresponding phase difference is f. Then it is clear that r2 S2 2p f= d d = r2 − r1 l 2p (Path difference) Phase difference = ¥ Path difference (i) l Figure 1.4 This can be made clearer with the help of Fig. 1.4, where two sources of waves S1 and S2 are shown. The wavelength of these sources is l and the sources are in phase at S1 and S2. The frequencies of both the waves are taken to be the same as f. Therefore, the angular frequency w = 2pf. They travel at the same speed and the 2p propagation constant for them is k =. We can write the wave equations for both the waves at point P as l y1 = a cos(wt – kr1) for the wave emerging from source S1 and y2 = a cos(wt – kr2) for the wave emerging from source S2 Here (wt – kr1) is the phase f1 and (wt – kr2) is the phase f2. Therefore, the phase difference between them is f1 – f2, given by f1 – f2 = wt – kr1 – wt + kr2 = k(r2 – r1). 2p Using Eq. (i) and k = , the path difference is obtained as l Path difference d = r2 – r1. 1.4 COHerenCe LO4 Coherence is a property of waves that helps in getting stationary interference, i.e., the interference which is temporally and spatially constant. During interference the waves add constructively or subtract destructively, depending on their relative phases. Two waves are said to be coherent if they have a constant relative phase. This also means that they have the same frequency. Actually the coherence is a measure of the correlation that exists between the phases of the wave measured at different points. The coherence of a wave depends on the characteristics of its source. 1.4.1 Temporal Coherence Temporal coherence is a measure of the correlation between the phases of a wave (light) at different points along the direction of wave propagation. If the phase difference of the wave crossing the two points lying along the direction of wave propagation is independent of time, then the wave is said to have temporal coherence. Temporal coherence is also known as longitudinal coherence. This tells us how monochromatic a source is. In Fig. 1.5A, a wave traveling along the positive x-direction is shown, where two points A and B are lying on the x-axis. Let the phases of the wave at these points at any instant t be fA and fB, respectively, and at a later time t¢ they be f¢A and f¢B. Under this situation, if the phase difference fB – fA = f B¢ - f A¢ , then the wave is said to have temporal coherence.  Interference 5 x axis A B Figure 1.5A 1.4.2 Spatial Coherence Spatial coherence is a measure of the correlation between the phases of a wave (light) at different points transverse to the direction of propagation. If the phase difference of the waves crossing the two points lying on a plane perpendicular to the direction of wave propagation is independent of time, then the wave is said to have spatial coherence. This tells us how uniform the phase of the wavefront is. In Fig. 1.5B, a wave traveling along the positive x-direction is shown, where PQRS is a transverse plane and A and B are the two points situated on this plane within the waveforms. Let the waves crossing these points at any time t have the same phase f and at a later time t¢ the phases of the waves are again the same but equal to f¢. Under this situation, the waves are said to have spatial coherence. Q P A x axis B x axis R S Figure 1.5B 1.4.3 Coherence Time and Coherence Length A monochromatic source of light emits radiation of a single frequency (or wavelength). In practice, however, even the best source of light emits radiations with a finite range of wavelengths. For a single frequency wave, the time interval over which the phase remains constant is called the coherence time. The coherence time is generally represented by Dt. In a monochromatic sinusoidal wave the coherence time is infinity because the phase remains constant throughout. However, practically the coherence time exists and the distance traveled by the light pulses during this coherence time is known as coherence length DL. The coherence length is also called the spatial interval, which is the length over which the phase of the wave remains constant. The coherence length and coherence time are related to each other according to the following formula DL = cDt 1.5 COHerent sOurCes LO4 Two sources of light are said to be coherent, if they emit waves of the same frequency (or wavelength), nearly the same amplitude and maintain a constant phase difference between them. Laser is a good example of coherent source. In actual practice, it is not possible to have two independent sources which are coherent. This can be explained as follows. A source of light consists of large number of atoms. According to the atomic 6 Engineering Physics theory, each atom consists of a central nucleus and the electrons revolve around the nucleus in different orbits. When an atom gets sufficient energy by any means, its electrons jump from lower energy level to higher energy level. This state of an atom is called an excited state. The electron lives in this state only for about 10–8 seconds. After this interval of time the electrons fall back to the inner orbits. During this process, the atoms radiate energy in the form of light. Out of the large number of atoms some of them emit light at any instant of time and at the next instant other atoms do so and so on. This results in the emission of light waves with different phases. So, it is obvious that it is difficult to get coherent Many Source Points Many Wavelengths light from different parts of the same source (Fig. 1.6). Therefore, two independent sources of light Figure 1.6 can never act as coherent sources. 1.5.1 Production of Coherent Light from Incoherent Sources An ordinary light bulb is an example of an incoherent source. We can produce coherent light from such an incoherent source, though we will have to a lot of the light. If we use spatially filter the light coming from such source, we can increase the spatial coherence (Fig. 1.7). Further, spectrally filtering of the light increases the temporal coherence. This way we can produce the coherent light from the incoherent source. Spatial Filter Spectral Filter Coherent Light Incoherent Source Pinhole Wavelength Filter Figure 1.7 1.6 anaLYtiCaL treatment Of interferenCe LO5 Let us consider the superposition of two waves of same frequency w and a constant phase difference f traveling in the same direction. Their amplitudes are taken as a1 and a2, respectively. The displacement due to one wave at any instant is given by y1 = a1 sin wt (i) and the displacement due to another wave at the same instant is given by y2 = a2 sin (wt + f) (ii)  Interference 7 According to the principle of superposition, the resultant displacement (yR) is given by yR = y1 + y2 (iii) = a1 sin wt + a2 sin (wt + f) = a1 sin wt + a2 sin wt cos f + a2 cos wt sin f = (a1 + a2 cos f) sin wt + a2 sin f cos wt (iv) Assuming a1 + a2 cos f = A cos q (v) a2 sin f = A sin q (vi) We obtain using Eq. (iv) – (vi) yR = A sin (wt + q) (vii) On squaring and adding Eqs. (v) and (vi), we have 2 2 A2 (sin2q+ cos2q) = a2 sin2f + a12 + 2a1a2 cos f + a2 cos2f A2 = a12 + a22 (sin2f + cos2f) + 2a1a2 cos f (viii) The resultant intensity is therefore given by 2 2 I = A2 = a1 + a2 + 2a1a2 cos f (ix) The angle q can be calculated from Eqs. (v) and (vi) as a2 sin f tan q = (x) a1 + a2 cos f 1.6.1 Condition for Constructive Interference It is clear from Eq. (ix) that the intensity, I will be maximum at points where the values of cos f are +1, i.e, phase difference f be 2np, with n = 0, 1, 2, 3,.... Then the maximum intensity is obtained from Eq. (ix) as Imax = (a1 + a2)2 (xi) In other words, the intensity will be maximum when the phase difference is an integral multiple of 2p. In this case, I max > a12 + a22 Thus, the resultant intensity will be greater than the sum of the individual intensities of the waves. If a1 = a2 = a, then Imax = 4a2 1.6.2 Condition for Destructive Interference It is clear from Eq. (ix) that the intensity I will be minimum at points where cos f = –1. i.e., where phase difference f = (2n + 1)p, with n = 0, 1, 2, 3,.... Then Eq. (ix) gives Imin = (a1 – a2)2 (xii) Therefore, it is clear that in destructive interference the intensity will be minimum when the phase difference f is an odd multiple of p. If a1 = a2, then Imin = 0 8 Engineering Physics If a1 π a2, then Imin π 0 I min < a12 + a22 Thus, in the case of destructive interference the resultant intensity will be less than the sum of the individual intensities of the waves. Figure 1.8 represents the intensity variation with phase differences f graphically (for a1 = a2 = a). I 4a2 –7p –6p –5p –4p –3p –2p –p 0 p 2p 3p 4p 5p 6p 7p f Figure 1.8 1.6.3 Conservation of Energy The resultant intensity due to the interference of two waves a1 sin wt and a2 sin (wt + f) is given by Eq. (ix), reproduced below 2 2 I = a1 + a2 + 2a1a2 cos f 2 2 \ Imax = a1 + a2 + 2a1a2 = (a1+ a2)2 and Imin = a12 + a22 – 2a1a2 = (a1 – a2)2 If a1 = a2 = a then Imax = 4a2 and Imin = 0 Therefore, average intensity (Iav) will be obtained as Iav = 2a2 2 2 For unequal amplitudes a1 and a2 the average intensity would be (a1 + a2 ). Thus, in interference only some part of energy is transferred from the position of minima to the position of maxima, and the average intensity or energy remains constant. This shows that the phenomenon of interference is in accordance with the law of conservation of energy. 1.7 COnDitiOns fOr sustaineD interferenCe LO5 Sustained interference means a constant interference of light waves. In order to obtain such interference, the following conditions must be satisfied (i) The two sources should emit waves of the same frequency (wavelength). If it is not so, then the positions of maxima and minima will change with time.  Interference 9 (ii) The waves from the two sources should propagate along the same direction with equal speeds. (iii) The phase difference between the two interfering waves should be zero or it should remain constant. It means the sources emitting these waves must be coherent. (iv) The two coherent sources should be very close to each other, otherwise the interference fringes will be very close to each other due to the large path difference between the interfering waves. For the large separation of the sources, the fringes may even overlap and the maxima and minima will not appear distinctly. (v) A reasonable distance between the sources and screen should be kept, as the maxima and minima appear quite close if this distance is smaller. On the other hand, the large distance of the screen reduces the intensity. (vi) In order to obtain distinct and clear maxima and minima, the amplitudes of the two interfering waves must be equal or nearly equal. (vii) If the source is not narrow, it may act as a multi source. This will lead to a number of interference patterns. Therefore, the coherent sources must be narrow. (viii) In order to obtain the pattern with constant fringe width and good intensity fringes, the sources should be monochromatic and the background should be dark. 1.7.1 Condition of Relative Phase Shift This is regarding the introduction of additional phase change between the interfering waves when they emerge after reflecting from two different surfaces. In most of the situations, the reflection takes place when the beam propagates from the medium of lower refractive index to the medium of higher refractive index or vice-versa. When the reflection occurs with light going from a lower index toward a higher index, the condition is called internal reflection. However, when the reflection occurs for light going from a higher index toward a lower index, the condition is referred to as external reflection. A relative phase shift of p takes place between the externally and internally reflected beams so that an additional path difference of l/2 is introduced between the two beams. If both the interfering beams get either internally or externally reflected, no phase shift takes place between them. 1.8 muLtipLe beam superpOsitiOn LO6 In Section 1.6, we have given theoretical analysis of the interference due to the superposition of two waves of the same frequency and the constant phase difference. The intensity of the interference pattern showed its dependence on the amplitudes of the interfering waves. However, now we consider a large number of waves of the same frequency and amplitude, which propagate in the same direction. The amount by which each wave train is ahead or lags behind the other is a matter of chance. Based on the amplitude and intensity of the resultant wave, we can examine the interference. We assume n number of wave trains whose individual amplitudes are equal (= a, say). The amplitude of the resultant wave can be understood as the amplitude of motion of a particle undergoing n simple harmonic motions (each of amplitude a) at once. In this case, if all these motions are in the same phase, the resultant wave will have an amplitude equal to na and the intensity would be n2a2, i.e., n2 times that of one wave. However, in our case, the phases are distributed purely at random, as shown in Fig. 1.9 as per graphical method of compounding amplitudes. Here, the phases f1, f2, f3,... take arbitrary values between 0 and 2p. The intensity due to the superposition of such waves can be calculated by the square of the resultant amplitude A. In order to find A2, we should square the sum of the 10 Engineering Physics projections of all vectors a along the x-direction and add it to the square of the corresponding sum along the y-direction. The summation of projections along x-direction are given by the following expression a(cos f1 + cos f2 + cos f3 +... + cos fn) y a f3 f2 A a a f1 O X Figure 1.9 The square of quantity in the parentheses gives the terms of the form cos2 f1, 2 cos f1 cos f2, etc. It is seen that the sum of these cross product terms increases approximately in proportion to number n. So we do not obtain a definite result with one given array of arbitrarily distributed waves. For a large number of such arrays, we find their average effect in computing the intensity in any physical problem. Under this situation, it is safe to conclude that these cross product terms will average to zero. So we consider only the cos2 f terms. Similarly, for the y projections of the vectors we obtain sin2 f terms. With this we have I ª A2 = a2(cos2 f1 + cos2 f2 + cos2 f3 +... + cos2 fn) + a2(sin2 f1 + sin2 f2 + sin2 f3 +... + sin2 fn). Using the identity sin2 fp + cos2 fp = 1, the above expression reduces to I ª a2 ¥ n. Since a2 is the intensity due to a single wave, the above relation shows that the average intensity resulting from the superposition of n waves with arbitrary phases is n times of a single wave. It means the resultant amplitude A increases in proportion with in length as n gets increased. 1.9 interferenCe bY DivisiOn Of WavefrOnt LO6 This method uses multiple slits, lenses, prisms or mirrors for dividing a single wavefront laterally to form two smaller segments that can interfere with each other. In the division of a wavefront, the interfering beams of radiation that left the source in different directions and some optical means is used to bring the beams back together. This method is useful with small sources. Double slit experiment is an excellent example of interference by division of wavefront. Fresnel’s biprism is also used for getting interference pattern based on this method. 1.9.1 Fresnel’s Biprism Fresnel’s Biprism is a device by which we can obtain two virtual coherent sources of light to produce sustained interference. It is the combination of two acute angled prisms which are joined with their bases in such a way that one angle becomes obtuse angle q¢ of about 179° and remaining two angles are acute angles each of about 1/2°, as shown in Fig. 1.10.  Interference 11 X A S1 d d B q¢ 2d S C q S2 D A¢ Y Figure 1.10 Let monochromatic light from slit S fall on the biprism, placed at a small distance from S. When the light falls on upper part of the biprism, it bends downward and appears to come from source S1. Similarly, the other part of the light when falls on the lower part of the biprism, bends upward and appears to come from source S2. Here, the images S1 and S2 act as two virtual coherent sources of light (Fig. 1.10). Coherent sources are the one that have a constant or zero phase difference throughout. In the situation, on placing the screen XY on right side of the biprism, we obtain an alternate bright and dark fringes in the overlapping region BC. 1.9.1.1 Theory of Fringes X Let A and B be two virtual coherent sources of light separated by a distance P 2d. The screen XY, on which the fringes are obtained, is separated by a distance D from the two coherent sources, as shown in Fig. 1.11. The point C on the screen is equidistant from A and B. Therefore, the path xn difference between the two waves from sources A and B at point C is zero. Thus the point C will be the centre of a bright fringe. On both sides A N of C, alternately bright and dark fringes are produced. d 2d S C Draw perpendiculars AN and BM from A and B on the screen. Let the d B M distance of a point P on the screen from the central bright fringe at C be D xn. Y From geometry, we have Figure 1.11 NP = xn – d; MP = xn + d In right angled DANP, AP2 = AN2 + NP2 (i) 2 2 = D + (xn – d) 12 Engineering Physics È ( x - d )2 ˘ D 2 Í1 + n ˙ Î D2 ˚ 1/2 È ( x - d )2 ˘ AP = D Í1 + n ˙ Î D2 ˚ È 1 ( xn - d ) 2 ˘ AP = D Í1 + ˙ , [as (xn - d ) 1) reflected along BC and partly refracted along BF at r r r r an angle r. At point F the wave BF is again partly I G reflected from the second surface along FD and partly F t i Air emerges out along FK and so on. In this situation, the interference occurs between reflected waves BC and J L DE and also between the transmitted waves FK and K GL (Fig. 1.16). Figure 1.16  Interference 17 The path difference between the reflected rays D = (BF + FD)in film – (BM)in air D = m (BF + FD) – BM Q BF = FD \ D = 2 mBF – BM (i) In the right angled DBFH, t t cos r = or BF = (ii) BF cos r BH and tan r = or BH = t tan r t BD = 2 ¥ BH \ BD = 2t tan r (iii) In the DBMD, BM sin i = or BM = BD sin i BD \ BM = 2t tan r sin i (iv) From Eqs. (i), (ii) and (iv), we get t D = 2m - 2t tan r sin i (v) cos r sin i Q m= or sin i = m sin r (vi) sin r 2 mt sin r 2 mt \ D= - 2t m sin r = [1 - sin 2 r ] cos r cos r cos r D = 2mt cos r (vii) Equation (vii) represents only the apparent path difference and does not represent the effective total path difference. When the light is reflected from the surface of an optically denser medium in case of rad BC, a phase change of pa equivalent to path difference of l/2 is introduced. Therefore, the total path difference between BC and DE will be D = 2mt cos r + l/2 (viii) Condition for Maxima: To have a maximum at a particular point, the two rays should arrive there in phase. So the path difference must contain a whole number of wavelength, i.e., D = nl, n = 0, 1, 2..., (ix) From Eq. (viii) and (ix), we get 2mt cos r + l/2 = nl 2mt cos r = nl – l/2 2mt cos r = (2n – 1)l/2 (x) 18 Engineering Physics Condition for Minima: To have a minimum at a particular point, the two rays should arrive there in out of phase (odd multiple of p) for which the path difference must contain a half odd integral number of wavelength, i.e, Ê 1ˆ D = Án + ˜ l (xi) Ë 2¯ Using Eq. (viii), we obtain 2mt cos r = nl where, n = 0, 1, 2, 3, … (xii) It should be noted that the interference pattern will not be perfect because the intensities of the rays BC and DE are not the same and their amplitudes are different. In order to obtain the interference between the transmitted waves, we calculate the path difference between the waves, FK and GL as under D = (FD + DG)in film – (FJ)in air D = m[FD + DG] – FJ Q FD = DG \ D = 2mFD – FJ (xiii) DI t t In DFDI, cos r = = or FD = (xiv) FD FD cos r FI FI and tan r = = or FI = t tan r DI t FG = 2t tan r (xv) In right angled DFJG, FJ sin i = or FJ = FG sin i FG \ FJ = 2t tan r sin i (xvi) From Eq. (xiii), (xiv) and (xvi), we get 2mt D= - 2t tan r sin i cos r 2mt sin r È sin i ˘ = cos r - 2t cos r m sin r Í m = sin r ˙ Î ˚ 2mt = [1 - sin 2 r ] = 2 mt cos r cos r Since these two waves are emerging from the same medium, the additional phase difference (or path difference) will not be introduced. Therefore, the total path difference D = 2mt cos r (xvii) Condition for Maxima: As discussed, it is possible when D = nl (xviii)  Interference 19 From Eqs. (xvii) and (xviii), we get 2mt cos r = nl where, n = 0, 1, 2, 3, … (xix) Condition for Minima: For obtaining minimum intensity, we should have Ê 1ˆ D = Án + ˜ l Ë 2¯ Ê 1ˆ which gives 2mt cos r = Á n + ˜ l where, n = 0, 1, 2, 3, …(xx) Ë 2¯ Thus, the conditions for interference with transmitted light are obviously opposite to those obtained with reflected light. Hence, if the film appears dark in the reflected light, it will appear bright in the transmitted light and vice-versa. This shows that the interference pattern in the reflected and transmitted lights are complimentary to each other. (i) Necessity of an Extended Source of Light for Interference in Thin Films When a thin transparent film is exposed to white light and seen in the reflected light, different colours are seen in the film. These colours arise due to the interference of the light waves reflected from the top and bottom surfaces of the film. The path difference between the reflected rays depends upon the thickness t, refractive index m of the film and the angle q of inclination of the incident rays. The light which comes from any point from the surface of the film will include the colour whose wavelength satisfies the equation 2mt cos r = (2n – 1) l/2 and only this colour will be present with the maximum intensity in the reflected light. When the transparent film of a large thickness as compared to the wavelength of the light, is illuminated by white light, the path difference at any point of the film will be zero. In the case of such a thick film, at a given point, the condition of constructive interference is satisfied by a large number of wavelengths, as l2) which are very close to each other (as Sodium D lines). The two wavelengths form their separate fringe patterns but as l1 and l2 are very close to each other and thickness of air film is small, the two patterns practically coincide with each other. As the mirror M1 is moved slowly, the two patterns separate slowly and when the thickness of air film is such that the dark fringe of l1 falls on bright fringe of l2, the result is maximum indistinctness. Now the mirror M1 is further moved, say through a distance x, so that the next indistinct position is reached. In this position, if n fringes of l1 appear at the centre, then (n +1) fringes of l2 should appear at the centre of the field of view. Hence l1 l x=n and x = (n + 1) 2 2 2 2x or n= (i) l1 2x and (n + 1) = (ii) l2 32 Engineering Physics On subtracting Eq. (i) from Eq. (ii), we get 2x 2x (n + 1) - n = - l2 l1 2 x(l1 - l2 ) or 1= l1l2 2 l1l2 lav 2 or (l1 - l2 ) = = where l1 l2 = lav is the square of mean of l1 and l2. 2x 2x Thus measuring the distance x moved by mirror M1 between the two consecutive positions of maximum indistinctness, the difference between two wavelengths of the source can be determined, if lav is known. (iii) Determination of Thickness and Refractive Index of a Thin Transparent Sheet The Michelson’s interferometer is adjusted for producing straight white light fringes and cross-wire is set up on the central bright fringe. Now insert thin transparent plate in the path of one of the interfering waves. On the inclusion of a plate of thickness t and refractive index m, the path difference is increased by a factor of 2(m – 1)t. The fringes are therefore shifted. The mirror M1 is now moved till the central fringe is again brought back to its initial position. The distance x traveled by the mirror M1 is measured by micrometer. Therefore x 2 x = 2( m - 1)t or t = (iii) ( m - 1) From Eq. (iii), we can write x m= +1 (iv) t Thus, by knowing the thickness of the transparent sheet and the distance x, we can calculate the refractive index of the sheet with the help of a Michelson’s interferometer. 1.11 appLiCatiOns Of interferenCe in tHe fieLD Of engineering LO7 The phenomenon of interference arises in many situations and the scientists and engineers have taken advantage of interference in designing and developing various instruments. 1.11.1 Testing of Optical Flatness of Surfaces An example of the application of interference method is the testing of optical components for surface quality. The most important example is that of optical flats. However, the methods used for fl at surfaces can be adapted simply to test spherical surfaces. 1.11.1.1 Flatness Interferometers With these interferometers we can compare the flatness of two surfaces by placing them in contact with slight wedge of air between them. This gives a tilt and thus the fringes start originating like that of Newton’s ring between the two surfaces. To get half wavelength contours of the space between the surfaces, they should be viewed from infiity. Further, to avoid the risk of scratching, a desirable distance should be there between the two surfaces. Most common examples of flatness interferometers are Fizeau and Twyman interferometers.  Interference 33 (i) Fizeau Interferometer S In this type of interferometer, the sources and viewing point are kept at infinity (Fig. 1.28). This interferometer generates interference between the surface of a test sample and a reference surface that is brought close to the test sample. The interference images are recorded and analysed by an imaging optic system. However, the contrast and the shape of the interference signals depend on the reflectivity of the test samples. (ii) Twyman-Green Interferometer This is an important instrument used to measure defects in optical components such as lenses, prisms, plane parallel windows, laser rods and plane mirrors. Twyman-Green interferometer, shown in Fig. 1.29 resembles Michelson interferometer in the beam splitter and mirror arrangement. However, the difference lies in the way of their illumination. In the case of Twyman-Green interferometer, we use a monochromatic point source Figure 1.28 which is located at the principal focus of a well-corrected lens whereas in Michelson interferometer an extended source is used. If the mirrors M1 and M2 are perpendicular to each other and the beam-splitter BS makes an angle of 45° with the normal of each mirror, then the interference is exactly analogous to thin film interference at normal incidence. Therefore, completely constructive interference is obtained when d = ml/2, where d is the path difference between the two arms adjusted by translating the mirror M1. The complete destructive interference is obtained when d = (m + 1/2)l/2. With the help of rotation of mirror M2 we can see fringes of equal thickness on the screen, as the angle of incidence is constant. This situation is analogous to interference pattern observed with collimated light and a thin film with varying thickness. In order to test the optical components, one of the mirrors is intentionally tilted to create fringes. Then the quality of the component can be determined from the change in the fringe pattern when the component is placed in the interferometer. Lens testing is specifically important for quantifying aberrations and measuring the focal length. Rotate M1 BS M2 S Lens Translate Screen Figure 1.29 34 Engineering Physics 1.11.2 Nonreflecting or Antireflecting (AR) Coatings Interference-based coatings were invented in November 1935 by Alexander Smakula, who was working for the Carl Zeiss optics company. Antireflecting coatings are a type of optical coatings. These are applied to the surface of lenses and other optical devices for reducing reflection. This way the efficiency of the system gets improved since less light is lost. For example, in a telescope the reduction in reflections improves the contrast of the image by elimination of stray light. In another applications a coating on eyeglass lenses makes the eyes of the wearer more visible. The anti-reflecting coatings can be mainly divided into three groups. 1.11.2.1 Single-layer Interference Coatings The simplest interference non-reflecting coating consists of a single quarter-wave layer of transparent material. The refractive index of this material is taken to be equal to the square root of the substrate’s refractive index. This theoretically gives zero reflectance at the center wavelength and decreased reflectance for wavelengths in a broad band around the center. The use of an intermediate layer to form an antireflection coating can be thought of as analogous to the technique of impedance matching of electrical signals. A similar method is used in fibre optic research where an index matching oil is sometimes used to temporarily defeat total internal reflection so that light may be coupled into or out of a fiber. The antireflection coatings rely on an intermediate layer not only for its direct reduction of reflection coefficient, but also use the interference effect of a thin layer. If the layer thickness is controlled precisely and it is made exactly one quarter of the light’s wavelength (l/4), then it is called a quarter-wave coating (Fig. 1.30). In this case, the incident beam I, when reflected from the second interface, will travel exactly half its own wavelength further than the beam reflected from the first surface. The two reflected beams R1 and R2 will destructively interfere as they are exactly out of phase and cancel each other if their intensities are equal. Therefore, the reflection from the surface is suppressed and all the energy of the beam is propagated through the transmitted beam T. In the calculation of the reflection from a stack of layers, the transfer-matrix method can be used. I λ 4 n0 n1 ns l R1 T R2 Figure 1.30  Interference 35 1.11.2.2 Multilayer Coatings or Multicoating Multiple coating layers can also be used for reflection reduction. It is possible if we design them such that the reflections from the surfaces undergo maximum destructive interference. This can be done if we add a second quarter-wave thick higher-index layer between the low-index layer (for example, silica) and the substrate. Under this situation, the reflection from all three interfaces produces destructive interference and antireflection. Optical coatings can also be made with near-zero reflectance at multiple wavelengths or optimum performance at angles of incidence other than 0°. 1.11.2.3 Absorbing Antireflecting Coatings Absorbing antireflecting coatings are an additional category of antireflection coatings. These coatings are useful in situations where low reflectance is required and high transmission through a surface is unimportant or undesirable. They can produce very low reflectance with few layers. They can often be produced more cheaply or at greater scale than standard non-absorbing anti-reflecting coatings. In sputter deposition system for such films, titanium nitride and niobium nitride are frequently used. 1.11.2.4 Practical Problems with AR Coatings Real coatings do not reach perfect performance, though they are capable of reducing a surface’s reflection coefficient to less than 0.1%. Practical details include correct calculation of the layer thickness. This is because the wavelength of the light is reduced inside a medium and this thickness will be l0/4n1, where l0 is the vacuum wavelength and n1 is the refractive index of the film. Finding suitable materials for use on ordinary glass is also another difficulty, since few useful substances have the required refractive index (n ª 1.23) which will make both reflected rays exactly eq