Physics-Class 12 Textbook PDF

Principles of PHYSICS- II Grade XII Rajendra Pd. Koirala Prajjwal Khanal Assistant Professor Lecturer of Physics Central Department of Physics Tribhuvan University, Kathmandu PRINCIPLES OF PH YSI C...

Principles of PHYSICS- II Grade XII Rajendra Pd. Koirala Prajjwal Khanal Assistant Professor Lecturer of Physics Central Department of Physics Tribhuvan University, Kathmandu PRINCIPLES OF PH YSI CS - II Edition: First 2018 Reprint 2019 Reprint 2020 ©Authors Publishers: Asmita Books Publishers and Distributors (P) Ltd. Kathmandu, Nepal Tel. 01-4168216/4168274 website: www.asmitapublication.com.np facebook: www.facebook.com/asmitapublication email: [email protected] Distributors: Kasthamandap Pustak Pasal Bhotahity, Kathmandu Tel. 01-4224048 Price: Rs. 825/- ISBN: 978-9937-615-29-7 Printed in: Kathmandu, Nepal  Preface There are a number of textbooks of +2 levels in the market. So, obviously, a natural question arises, why another one again? The major reason is that most of the textbooks tend to be dry and formal and hence often difficult or complex for the students. It was thus essential to develop a textbook that could touch the pulse of students hence, this textbook is developed with a new approach. Our approach is to recognize that physics is a description of reality starting each topic with concrete observations and experiences to enable students directly related to it. Not only does this book make the material more interesting and easier to understand but also it is closer to the way that physics is actually practiced worldwide. This book entitled "Principles of Physics- II" covers the latest syllabus of class XII. The main objectives of this book are two folds: to provide the student with a clear and logical presentation of the basic concepts and principles of physics, and to strengthen an understanding of the concepts and principles through a broad range of interesting applications to the real world. This book is an end product of our uninterrupted two decade long teaching experience. We have tried to solve all the difficulties of the students through this book. The basic parts presented in this book are explanation of theory, mathematical formulae, related figures, answers to the short questions, worked out examples and adequate self practice questions. In some observational facts, reliable reference books are mentioned to avoid the confusion for the reader. In the numerical portions, 'ALP' refers to Advanced Level Physics and 'UP' refers to University Physics. SI system of unit is used throughout the book. We wish to acknowledge our indebtedness to the many international books which have been consummated. We would like to express our profound and sincere gratitude to our family, colleagues, students, readers, etc. from different part of the country who have adopted this book and sent us their compliments and valuable suggestions through available means. In this regard, special mention goes to Mr. Prakash Pantha, Mr. Akash Pokhrel, Mr. Shesh Nath Chaudhary, Sanjaya K. Sharma, Diwash Dahal, Bipin Bhattarai, Roshan Shrestha, Laxman Aryal and all of our students. Last but not least, Mr. Manoj Kumar Sharma, managing director of Asmita Books Publishers & Distributors (P) Ltd. deserves our acclaim for meticulous efforts and suggestions to present this collective effort to carry out these matters in this form. We are also very much thankful to Mr. Bipin Kumar Acharya for his valuable advice and suggestions in preparing the book. Mr. Niraj Bhattarai deserves thanks and appreciation for his outstanding type settings and layout for this book. Humbly, we would like to request our esteemed readers to kindly send us the valuable suggestions for the improvement of the book and to notify of any errors they might come across while going through it. By which both will be thankfully acknowledged and incorporated in the next edition. Finally, we would like to thank almighty for this endless blessings and kindness. June 2018 Authors Teaching hours: 150T +50P Syllabus Full marks: 100 (75T + 25 P) Nature of course: Theory +Practical Pass Marks: 27T + 8P Course Contents Unit-1 Waves and Optics TH 40 Waves TH 23 1. Wave Motion- Wave motion; Longitudinal and transverse waves; Progressive and stationary waves; Mathematical description of a wave. LH4 2. Mechanical Waves- Speed of wave motion; Velocity of sound in solid and liquid; Velocity of sound in gas; Laplace’s correction; Effect of temperature, pressure, humidity on velocity of sound. LH5 3. Wave in Pipes and Strings- Stationery waves in closed and open pipes; Harmonics and overtones in closed and open organ pipes; End correction in pipes; Resonance Tube experiment; Velocity of transverse waves along a stretched string; Vibration of string and overtones; Laws of vibration of fixed string. LH6 4. Acoustic Phenomena- Sound waves: Pressure amplitude; Characteristics of sound: Intensity; loudness, quality and pitch; Beats; Doppler’s effect; Infrasonic and ultrasonic waves; Noise pollution: Sources, health hazard and control. LH8 Physical Optics TH 17 1. Nature and Propagation of Light- Nature and sources of light; Electromagnetic spectrum; Huygen’s principle, Reflection and Refraction according to wave theory; Velocity of light: Foucault’s method; Michelson’s method. LH6 2. Interference- Phenomenon of Interferences; Coherent sources; Young’s two slit experiment; Newton’s ring LH4 3. Diffraction- Diffraction from a single slit; Diffraction pattern of image; Diffraction grating; Resolving power of optical instruments LH4 4. Polarization- Phenomenon of polarization; Brewster’s law; transverse nature of light; Polaroid LH3 Unit-2 Electricity and Magnetism TH 55 Current Electricity TH 20 1. D.C. Circuit- Electric Currents; Drift velocity and its relation with current; Ohm’s law; Electrical Resistance; Resistivity; Conductivity; Super conductors; Perfect Conductors; Current-voltage relations; Ohmic and Non-Ohmic resistance; Resistances in series and parallel, Potential Divider, Conversion of galvanometer into voltmeter and ammeter, Ohmmeter; Electromotive force: Emf of a source, internal resistance; Work and power in electrical circuits; Joule’s law and its verification. LH9 2. Electrical Circuits- Kirchhoffs laws; Wheatstone bridge circuit; P.O.Box, Meter Bridge; Potentiometer; Comparison of e.m.f’s., measurement of internal resistance of a cell. LH7 3. Thermoelectric Effect- Seebeck Effect; Thermocouples, Peltier effect: Variation of thermoelectric emf with temperature, Thermopile, Thomson effects. LH2 4. Chemical Effect of Current- Faraday’s laws of electrolysis; Faraday’s constant, Verification of Faraday laws of electrolysis. LH2 Magnetic Field of Current TH 35 1. Magnetic Field- Magnetic field lines and magnetic flux; Oersted’s experiment; Force on moving charge, Force on Conductor; Force and Torque on rectangular coil, Moving coil galvanometer; Hall effect; Magnetic field of a moving charge; Biot and Savart law and its application to (i) a circular coil (ii)a long straight conductor (iii) a long solenoid; Ampere’s law and its application to (i)a long straight conductor (ii) a straight solenoid (iii) a toroidal solenoid; Forces between two parallel conductors carrying current- definition of ampere. LH14 2. Magnetic Properties of Materials- Elements of earth magnetism and their variation; Dip and Dip circle; Flux density in magnetic material; Relative permeability; Susceptibility; Hysteresis, Dia,-Para- and Ferro-magnetic materials. LH5 3. Electromagnetic Induction- Faraday’s laws; Induced electric fields; Lenz’s law, Motional electromotive force; AC generators; eddy currents; Self inductance and Mutual inductance; Energy stored in an inductor; Transformer. LH8 4. Alternating Currents- Peak and RMS Value of AC current and Voltages, AC through resistor, capacitor and inductor; Phasor diagram, Series circuits containing combination of resistor, capacitor and inductor; Series Resonance, Quality factor; Power in AC circuits: Power factor; choke coil. LH 8 Unit-3 Modern Physics TH 55 1. Electrons and Photons- Electrons: Milikan’s oil drop experiment, Gaseous discharge at various pressure; Cathode rays, Motion of electron beam in electric and magnetic fields; Thomson’s experiment to determine specific charge of electrons. Photons: Quantum nature of radiation; Einstein’s photoelectric equation; Stopping potential; Measurement of Plank’s constant, Milikan’s experiment LH 10 2. Solids and Semiconductor Devices- Structure of solids; Energy bands in solids (qualitative ideas only); Difference between metals, insulators and semi-conductors using band theory; Intrinsic and extrinsic semi-conductors; P-N Junction; Semiconductor diode: Characteristics in forward and reverse bias; Full wave rectification; Filter circuit; Zener diode; Transistor: Common emitter characteristics, Logic gates; NOT, OR, AND, NAND and NOR., Nanotechnology (introductory idea) LH 11 3. Quantization of Energy- Bohr’s theory of hydrogen atom; Spectral series; Excitation and ionization potentials; Energy level; Emission and absorption spectra, De Broglie Theory; Duality; Uncertainly principle. Lasers: He- Ne laser, Nature and production, properties and uses. X-rays: Nature and production; uses: X-rays, X-rays diffraction, Bragg’s law. LH9 4. Nuclear Physics- Nucleus: Discovery of nucleus; Nuclear density; Mass number; Atomic number; Atomic mass; Isotopes; Einstein’s mass-energy relation, Mass Defect; Binding energy; Fission and fusion. LH6 5. Radioactivity- Alpha-particles; Beta-particles, Gamma rays; Laws of radioactive disintegration; Half-life and decay constant; Geiger-Muller Tube; Radio carbon dating; Medical use of nuclear radiation; Health hazards and safety precautions. LH7 6. Nuclear Energy and Other Sources of Energy- Sources of energy; Conservation and degradation of energy; Transformation of energy. Nuclear energy: Energy released from fission and fusion; Thermal and Hydroelectric power; Wind energy; Biofuels; Solar energy; Solar constant; Solar devices; Global energy consumption pattern and demands; Energy use in Nepal. Fuels and pollution: Global Warming; Acid rain. LH9 7. Particle Physics and Cosmology- Particles and antiparticles, Quarks and Leptons, baryons, mesons. Universe- Hubble law; Big Bang; Critical density; Dark matter LH3  Contents Unit-1: Waves and Optics 1. Wave Motion 1.1 Introduction 1 1.2 Wave Motion 2 1.3 Graphical Representation of Waves 7 1.4 Basic Terminologies of Wave 8 1.5 Progressive Wave 10 1.6 Differential Form of Wave Equation 12 1.7 Interference of Sound 13 1.8 Stationary Wave 15 1.9 Stationary Waves in Boundary 17  Tips for MCQs 20  Worked Out Problems 21  Challenging Problems 23  Conceptual Questions with Answers 24  Exercises 26  Multiple Choice Questions 28  Hints to Challenging Problems 29 2. Mechanical Waves 2.1 Introduction 33 2.2 Speed of Mechanical Wave 36 2.3 Speed of Sound in Gaseous Medium 38 2.4 Factors Affecting the Speed of Sound in a Gas 40  Tips for MCQs 42  Worked Out Problems 43  Challenging Problems 46  Conceptual Questions with Answers 47  Exercises 50  Multiple Choice Questions 52  Hints to Challenging Problems 53 3. Waves in Pipes and Strings 3.1 Tone, Note, Harmonics and Overtones 57 3.2 Organ Pipes 58 3.3 Open Organ Pipe 60 3.4 End Correction of Organ Pipe 63 3.5 Forced and Damped Oscillation 64 3.6 Resonance 64 3.7 Resonance Tube Apparatus 66 3.8 Waves in String 68 3.9 Modes of Vibration of a Stretched String 70 3.10 Verification of Laws of Vibrating Strings 74  Tips for MCQs 76   Worked Out Problems 78  Challenging Problems 81  Conceptual Questions with Answers 83  Exercises 87  Multiple Choice Questions 90  Hints to Challenging Problems 91 4. Acoustic Phenomena 4.1 Introduction 95 4.2 Pressure Amplitude 95 4.3 Characteristics of Sound 97 4.4 Relations of Intensity and Amplitude of Wave 99 4.5 Intensity Level 102 4.6 Infrasonics, Audible, Ultrasonics and Supersonics 104 4.7 Beats 105 4.8 Doppler’s Effect 110 4.9 Noise, Noise Pollution and its Control 119  Tips for MCQs 120  Worked Out Problems 121  Challenging Problems 125  Conceptual Questions with Answers 126  Exercises 130  Multiple Choice Questions 133  Hints to Challenging Problems 133 5. Speed of Light 5.1 Introduction 137 5.2 Foucault's Method 137 5.3 Importance of measuring speed of light 141  Tips for MCQs 142  Worked Out Problems 142  Challenging Problems 143  Conceptual Questions with Answers 144  Exercises 145  Multiple Choice Questions 146  Hints to Challenging Problems 147 6. Physical Optics 6.1 Introduction 149 6.2 Electromagnetic Waves 150 6.3 Wavefronts and Wavelets 152 6.4 Wave Theory of Light 154 6.5 Laws of Reflection of Light from Wave Theory 155 6.6 Laws of Refraction of Light from Wave Theory 156  Tips for MCQs 158  Conceptual Questions with Answers 158  Exercises 160  Multiple Choice Questions 160 7. Interference of Light 7.1 Introduction 163 7.2 Coherent Sources 163 7.3 Analytical Treatment of Interference of Light 166 7.4 Young’s Double Slit Experiment 171 7.5 Theory of interference 172 7.6 Interference in a thin film 176 7.7 Newton's Ring 177  Tips for MCQs 181  Worked Out Problems 181  Challenging Problems 184  Conceptual Questions with Answers 185  Exercises 188  Multiple Choice Questions 190  Hints to Challenging Problems 192 8. Diffraction of Light 8.1 Introduction 195 8.2 Classification of Diffraction 196 8.3 Fraunhofer Diffraction at a Single Slit 197 8.4 Diffraction Grating 202 8.5 Resolving Power of Optical Instruments 204  Tips for MCQs 205  Worked Out Problems 206  Challenging Problems 208  Conceptual Questions with Answers 209  Exercises 211  Multiple Choice Questions 213  Hints to Challenging Problems 214 9. Polarization of Light 9.1 Introduction 217 9.2 Polarization of waves 217 9.3 Polarization Methods 219 9.4 Poloroids 219 9.5 Experimental Demonstration of Transverse Nature of Light 220 9.6 Malus' law 221 9.7 Polarization by Reflection 222 9.8 Brewster's law 223  Tips for MCQs 223  Worked Out Problems 224  Challenging Problems 224  Conceptual Questions with Answers 225  Exercises 226  Multiple Choice Questions 227  Hints to Challenging Problems 228  Unit-2: Electricity and Magnetism 10. Direct Current Circuit 10.1 Introduction 229 10.2 Electric Circuit 229 10.3 Electric Current 230 10.4 Metallic Conduction 233 10.5 Ohm's Law 234 10.6 Resistance and Resistivity 236 10.7 Colour Code for Resistors 240 10.8 Combinations of Resistors 241 10.9 Voltage Divider Circuit 244 10.10 Current Divider 244 10.11 Superconductivity 245 10.12 Electrical Devices 247  Tips for MCQs 250  Worked Out Problems 252  Challenging Problems 258  Conceptual Questions with Answers 259  Exercises 264  Multiple Choice Questions 268  Hints to Challenging Problems 270 11. Heating Effect of Current 11.1 Introduction 273 11.2 Joules Law of Heating 273 11.3 Electric Energy and Power 275 11.4 Electromotive Force 276 11.5 Terminal Potential Difference 277 11.6 Internal Resistance of a Cell 277 11.7 Relation of emf, Terminal Potential Difference and Internal Resistance of a Cell 278 11.8 Combination of Cells 279  Tips for MCQs 281  Worked Out Problems 282  Challenging Problems 286  Conceptual Questions with Answers 288  Exercises 291  Multiple Choice Questions 293  Hints to Challenging Problems 294 12. Electric Circuit 12.1 Introduction 299 12.2 Kirchhoff's Laws 299 12.3 Wheat Stone Bridge 302 12.4 Meter Bridge 303 12.5 Post office box (P.O. Box) 305  12.6 Potentiometer 306 12.7 Comparison of Emfs of two cell 307 12.8 Measurement of internal Resistance of the cell 308  Tips for MCQs 309  Worked Out Problems 310  Challenging Problems 312  Conceptual Questions with Answers 314  Exercises 315  Multiple Choice Questions 317  Hints to Challenging Problems 318 13. Thermoelectricity 13.1 Introduction 321 13.2 Thermoelectric Effect 321 13.3 Seebeck Effect 322 13.4 Variation of Thermo-emf (E) with Temperature () 322 13.5 Peltier Effect 324 13.6 Thomson's Effect 325 13.7 Thermopile 326  Tips for MCQs 327  Worked Out Problems 327  Conceptual Questions with Answers 328  Exercises 330  Multiple Choice Questions 331 14. Chemical Effect of Current 14.1 Introduction 333 14.2 Electrolysis 334 14.3 Faraday's Law of Electrolysis 335 14.4 Faraday's Constant 337  Tips for MCQs 338  Worked Out Problems 339  Challenging Problems 340  Conceptual Questions with Answers 341  Exercises 342  Multiple Choice Questions 343  Hints to Challenging Problems 344 15. Magnetic Effect of Current 15.1 Introduction 345 15.2 Oersted Discovery 345 15.3 Rules of Finding the Direction of Magnetic Field 345 15.4 Lorentz Force 347 15.5 Magnetic Force on a Current Carrying Conductor 347 15.6 Torque on Rectangular Current Loop and Magnetic Moment 349 15.7 Magnetic Moment 350  15.8 Moving Coil Galvanometer 351 15.9 Biot-Savart Law 353 15.10 Applications of Biot-Savart's Law 354 15.11 Statement of Ampere's Circuital Law 363 15.12 Application of Ampere's circuital law 363 15.13 Force between two conductors carrying current 366 15.14 Magnetic force between two parallel conductors 366 15.15 Hall Effect 369  Tips for MCQs 370  Worked Out Problems 372  Challenging Problems 377  Conceptual Questions with Answers 378  Exercises 382  Multiple Choice Questions 385  Hints to Challenging Problems 387 16. Magnetism 16.1 Introduction 391 16.2 Geographical Meridian and Magnetic Meridian 392 16.3 Magnetic Elements of the Earth 392 16.4 Apparent Dip 394 16.5 Domain Theory of Ferromagnetism 396 16.6 Magnetic Properties of Materials 396 16.7 Magnetic Substances 399 16.8 Magnetic Hysteresis 401  Tips for MCQs 403  Worked Out Problems 404  Challenging Problems 405  Conceptual Questions with Answers 405  Exercises 408  Multiple Choice Questions 409  Hints to Challenging Problems 409 17. Electromagnetic Induction 17.1 Introduction 411 17.2 Electromagnetic Induction 411 17.3 Magnetic Flux and Induction Explained 412 17.4 Faraday's Laws of Electromagnetic Induction 413 17.5 Lenz law and direction of induced emf 415 17.6 Motional emf 417 17.7 Emf induced in a rotating coil in uniform magnetic field 418 17.8 Inductor and Inductance 420 17.9 Self inductance 420 17.10 Energy stored in an inductor 421 17.11 Mutual Induction 423 17.12 A.C. Generator 426 17.13 Transformer 427  Tips for MCQs 430   Worked Out Problems 431  Challenging Problems 434  Conceptual Questions with Answers 436  Exercises 439  Multiple Choice Questions 441  Hints to Challenging Problems 442 Unit-3: Modern Physics 18. Alternating Currents 18.1 Introduction 445 18.2 Alternating Current 445 18.3 RMS Value of A.C. 448 18.4 Phasors 449 18.5 A.C. Through Resistor 450 18.6 A.C. Through Inductor 451 18.7 A.C. Through Capacitor 453 18.8 A.C. Through R–L Series Circuit 454 18.9 A.C. Through R-C Circuit 456 18.10 L-C-R Series Circuit in A.C. 457 18.11 Power in LCR Circuit 459 18.12 Q Factor in LCR Circuit 461 18.13 Wattless Current 461 18.14 Choke Coil 462  Tips for MCQs 462  Worked Out Problems 464  Challenging Problems 468  Conceptual Questions with Answers 470  Exercises 472  Multiple Choice Questions 475  Hints to Challenging Problems 476 19. Electrons 19.1 Introduction 481 19.2 Particle Nature of Electricity 481 19.3 Millikan's Oil Drop Experiment 482 19.4 Motion of Electron in Uniform Electric Field 484 19.5 Motion of Electron in Uniform Magnetic Field 486 19.6 Specific Charge of Electron 489 19.7 Determination of Specific Charge (e/m) of an Electron by J.J. Thomson's Experiment 489 19.8 Conduction Through Gases 491 19.9 Discharging Mechanism 493 19.10 Cathode Rays and Their Production 494  Tips for MCQs 496  Worked Out Problems 497  Challenging Problems 501  Conceptual Questions with Answers 502   Exercises 505  Multiple Choice Questions 508  Hints to Challenging Problems 508 20. Photons 20.1 Introduction 511 20.2 Quantum Nature of Light 511 20.3 Photoelectric Effect 512 20.4 Einstein's Equation of Photoelectric Effect 513 20.5 Laws of Photoelectric Emission 515 20.6 Millikan's Verification of Einstein's Equation of Photoelectric Effect 517 20.7 Photocell 518  Tips for MCQs 519  Worked Out Problems 520  Challenging Problems 522  Conceptual Questions with Answers 523  Exercises 527  Multiple Choice Questions 529  Hints to Challenging Problems 531 21. Semiconductor 21.1 Introduction 533 21.2 Band Theory of Solids 533 21.3 Semiconductors 535 21.4 Charge Carriers in Semiconductor 536 21.5 Types of Semiconductor 536 21.6 P-type semiconductor 537 21.7 P-N Junction Diode (Semiconductor diode) 539 21.8 Working of a P-N Junction Diode 540 21.9 Diode Characteristics and Its Study 541 21.10 Semiconductor Diode as Rectifier 542 21.11 Filter Circuits 544 21.12 Zener Diode 544 21.13 Transistors 545 21.14 Working of a Transistor 546 21.15 Transistor Configuration 547 21.16 Transistor as an Amplifier 549 21.17 Logic Gate 551 21.18 Nanotechnology 553  Tips for MCQs 554  Conceptual Questions with Answers 556  Exercises 561  Multiple Choice Questions 562 22. Atomic Models 22.1 Introduction 565 22.2 Rutherford's Atomic Model 565 22.3 Bohr's Atomic Model 566 22.4 Energy of Electron 569 22.5 Spectral Series of Hydrogen Atom 573 22.6 Heinsenberg Uncertainty Principle 577 22.7 Excitation Energy and Excitation Potential 580 22.8 Ionization Energy and Ionization Potential 580 22.9 Emission and Absorption Spectra 581 22.10 Laser 583  Tips for MCQs 587  Worked Out Problems 589  Challenging Problems 592  Conceptual Questions with Answers 593  Exercises 598  Multiple Choice Questions 601  Hints to Challenging Problems 603 23. X-rays 23.1 Introduction 607 23.2 Production of X-Rays 607 23.3 X-ray Spectra 610 23.4 X-rays Diffraction 612 23.5 Bragg's Law 613  Tips for MCQs 615  Challenging Problems 618  Conceptual Questions with Answers 618  Exercises 621  Hints to Challenging Problems 622 24. Nuclear Physics 24.1 Introduction 625 24.2 Nucleus of an Atom 625 24.3 Constituents of a Nucleus 626 24.4 Nuclear Density 626 24.5 Atomic Number and Atomic Mass 627 24.6 Representation of a Nucleus of an Atom 627 24.7 Isotopes, Isobars, Isotones 627 24.8 Einstein's Mass-Energy Relation 628 24.9 Units of Energy 629 24.10 Atomic Mass Unit 629 24.11 Mass Defect 630 24.12 Packing Fraction 631  24.13 Binding Energy 631 24.14 Nuclear Reaction 632 24.15 Nuclear Fusion Reaction 636  Tips for MCQs 637  Worked Out Problems 639  Challenging Problems 642  Conceptual Questions with Answers 642  Exercises 646  Multiple Choice Questions 648  Hints to Challenging Problems 649 25. Radioactivity 25.1 Introduction 651 25.2 Radioactive Decay 652 25.3 Stability of Nucleus and Radioactive Isotopes 652 25.4 Nature of Radioactivity 653 25.5 Alpha Rays (-rays) 653 25.6 Beta Rays (-rays) 654 25.7 Gamma rays (-rays) 655 25.8 Laws of Radioactive Transformation 656 25.9 Radioactive Decay Law 657 25.10 Number of Atoms Left After nth Half Lives 659 25.11 Kinetic Energy of Emitted -particle from nucleus 660 25.12 Uses of Radioactive Nuclei 661 25.13 Geiger Muller Counter: A Radiation Detector 662 25.14 Radiation Hazard 664  Tips for MCQs 666  Worked Out Problems 666  Challenging Problems 669  Conceptual Questions with Answers 670  Exercises 673  Multiple Choice Questions 675  Hints to Challenging Problems 676 26. Nuclear Energy and Other Sources of Energy 26.1 Introduction 679 26.2 Energy and Energy Sources 679 26.3 Conservation of Energy and Degradation of Energy 680 26.4 Global Energy Consumption Pattern 681 26.5 Energy use in Nepal 682 26.6 Nuclear Energy 682 26.7 Renewable Energy and Nonrenewable Energy 684 26.8 Pollution 687 26.9 Air Pollution 688  26.10 Water Pollution 689 26.11 Ozone Layer 690 26.12 Green House Effect 691 26.13 Acid Rain 692  Conceptual Questions with Answers 693  Exercises 695 27. Particle Physics 27.1 Introduction 697 27.2 History of Elementary Particles 697 27.3 Particles and antiparticles 698 27.4 Annihilation 699 27.5 Pair Production 699 27.6 Concept of Spin 699 27.7 Classification of Elementary Particles 700 27.8 Fermions 700 27.9 Leptons 700 27.10 Quarks 701 27.11 Bosons 702 27.12 Hadrons 702 27.13 Mesons 703 27.14 Baryons 704 27.15 Three Generations of Quarks and Leptons 705  Tips for MCQs 706  Conceptual Question Answer 706  Exercises 708  Multiple Choice Questions 708 28. Cosmology 28.1 Introduction 709 28.2 The Universe 709 28.3 Evolution of Star 712 28.4 Big Bang 715 28.5 Expanding of Universe 716 28.6 Hubble's Law 717 28.7 Critical Density 718 28.8 Dark Matter and Dark Energy 719  Tips for MCQs 720  Worked Out Problems 720  Conceptual Questions with Answers 721  Exercises 722  Multiple Choice Questions 723  Bibliography 724  Appendix (Including Model Questions) 725  WAVE MOTION 1.1 Introduction When we throw a stone in a quiet pond, nice circular ripples emerge on the surface of water that move in a concentric pattern outward from the point of disturbance as shown in Fig. 1.1. These ripples are the waves or more precisely surface waves. Though the spreading pattern of these surface waves seem nice and simple, the physics behind it is quite complex. Actually, the stone displaces the water molecules at the point of impact from their equilibrium positions. These molecules execute a back and forth vibration and in doing so, all other neighbouring molecules through out the surface are forced to do the same about their mean Fig. 1.1: Ripples in a pond positions. So, a kind of disturbance seems to propagate from the point of impact in radially outward direction. This disturbance travelling from one point to another point is called a wave in motion. During wave motion, the particles though displaced from their mean position, do not actually travel from one point to another. Rather, they transfer their energies to neighbouring molecules during the vibration and it is the energy that is being transported from one point to another. Thus, we can say that wave motion is a mode of energy transfer from one point to another point. As an analogy, following example is relevant to understand the wave motion. When a person standing at last of a very long queue pushes another person in front of him, he loses his balance and all other person ahead in the queue receive a gentle push and hence lose balance to some extent as shown in Fig. 1.2. However, all of them in the line manage to return back to their initial position. Therefore, the disturbance in the form of push Fig. 1.2: Queue in front of temple 2 Principles of Physics - II created at the end of a queue travelled throughout the queue to the front. But, there is no actual displacement of the person in the queue from end to front. This is the real way of wave motion in a material medium. On contrary, a running stream of water carries energy with itself as it moves along. This is not the way of energy transfer in discussion for our present situation. 1.2 Wave Motion We can define a wave as an activity that transmits energy from one point to another point without actual transfer of matter. The most common waves that we come through in our daily life are sound waves, water ripples, light waves, etc. It might appear in water ripples that, the water has moved along the wave from its initial position; however it is not the case. In fact, at the onset of water ripples the water molecules vibrate up and down and transfer its energy to the neighbouring water molecules and thus, a chain of energy transfer is created without transfer of molecules from its mean position. Characteristics of Wave Motion i. Wave motion is a disturbance propagating in a medium. ii. It transfers energy as well as momentum from one point to another. iii. It has finite and fixed speed depending on the nature of the medium and is given by v = f. iv. When it travels in a medium, there is a continuous phase difference among the successive medium particles. v. The vibrating particles of the medium posses a kinetic as well as potential energy. vi. The phenomena such as reflection, refraction, interference and diffraction are shown by all types of waves but polarization is shown only by transverse wave. Types of Wave Motion There are three ways of energy transfer by waves and hence there are three types of wave motion. i. Electromagnetic wave ii. Mechanical wave iii. Matter wave i. Electromagnetic wave: The wave which does not require medium for its propagation is called electromagnetic wave. For example, light, heat, radio waves. The magnitude of electromagnetic field varies during propagation of electromagnetic wave. All electromagnetic waves such as - rays, X-rays, microwaves etc. are non-mechanical waves. ii. Mechanical wave: The wave which requires medium for its propagation is called mechanical wave. For example, waves on springs and strings, water waves, sound waves, seismic waves etc. are mechanical waves. For the propagation of a mechanical wave, the medium should have two properties: elasticity, and inertia. Due to elastic property of a medium, the mechanical wave is also called an elastic wave. The medium must be continuous to propagate such wave. iii. Matter wave: The waves associated with the microscopic particles such as electrons, protons, neutrons, atoms and molecules, when they are in motion are called matter waves. The concept of matter wave was first introduced by de Broglie, so it is also called de Broglie waves. Although, these waves can be generalized to the large mass objects, they are not detectable. Matter waves are very important for the quantum mechanical description of matters. Electronic waves (i.e. matter waves) are used to visualize the very small particles in electron microscope.  Wave Motion  Chapter 1  3 Types of Mechanical Waves There are two types of a mechanical wave based on the direction of vibration of medium particles or the fields. i. Transverse wave ii. Longitudinal wave i. Transverse wave If the particles of a medium vibrate perpendicularly to the propagation of the wave, then the wave is called transverse wave. These waves travel in the form of crests and troughs as shown in Fig. 1.3. For example, waves on strings, water ripples etc. Displacement→ Particle C C Time→ O C = Crest T = Trough T Fig. 1.3: Transverse wave Propagation of Transverse Wave Wave transports energy from one point to another. In transverse wave, the direction of oscillation of particles in a medium and direction of propagation of wave are perpendicular to each other. To study the propagation of transverse wave, consider nine points on a medium in which every point T lies in phase difference of 8 , where T is the time period of oscillation of particles. It means, the T disturbance of every preceding point transfers it to succeeding point after the time period 8. The process of wave propagation is described below. i. When t = 0, the particle 1 remains at rest as shown in Fig 1.4 (i), the displacement of the particle is determined by, y(t = 0) = a sin t = a sin 0 = 0 T ii. When t = 8 , the particle 1 executes simple harmonic motion (SHM) with displacement, T T 2 T a yt = 8  = a sin  8 = a sin T  8 =   2 At the same instant, the particle 2 just starts SHM as shown in Fig 1.4 (ii) 2T iii. When t = 8 , the particle 1 executes SHM with displacement 2T 2T 2 2T y t = 8  = a sin  8 = a sin T  8 = a   In this condition, the particle displaces with maximum amplitude in positive direction. a At that instant, particle 2 is displaced by and particle 3 just starts SHM as shown in 2 Fig 1.4 (iii). 4 Principles of Physics - II 3T iv. When t = 8 , the particle 1 executes SHM with displacement 3T 3T 2 3T a y t = 8  = a sin  8 = a sin T  8 =   2 a At the same instant, particle 2 has maximum displacement, particle 3 is displaced by and 2 particle 4 starts executing SHM as shown in Fig 1.4 (iv). 4T v. When t = 8 , the particle 1 executes SHM with displacement 4T 4T 2 4T y t = 8  = a sin  8 = a sin T  8 = 0   a a Particle 1 returns to the mean position, particles 2, 3, and 4 have displacements , a, 2 2 respectively. Particle 5 starts executing SHM as shown in Fig 1.4 (v) Similarly, the displacement of particle 1 executes SHM in next half cycle making the displacement as shown in Fig. 1.4 (vi), (vii), (viii), (ix). 5T 5T 2 5T a vi. When t = 8 , y t = 8  = a sin T  8 = –   2 6T 6T 2 6T vii. When t = 8 , y t = 8  = a sin T  8 = – a   7T 7T 2 7T a viii. When t = 8 , y t = 8  = a sin T  8 = –   2 8T 2 ix. When t = 8 = T, y (t = T) = a sin T  T = 0 Thus, the transverse wave propagates in a medium. The process of formation of a complete transverse wave is shown in Fig 1.4. 1 2 3 4 5 6 7 8 9 t=0 (i) T t=8 (ii) 1 2 3 4 5 6 7 8 9 2T t= 8 (iii) 1 2 3 4 5 6 7 8 9 3T t= 8 (iv) 1 2 3 4 5 6 7 8 9  Wave Motion  Chapter 1  5 4T t= 8 (v) 1 2 3 4 5 6 7 8 9 5T 2 t= 8 (vi) 1 3 4 5 6 7 8 9 6T 3 t= 8 (vii) 1 2 4 5 6 7 8 9 7T 4 t= 8 (viii) 1 2 3 5 6 7 8 9 8T 2 5 t= 8 (ix) 1 3 4 6 7 8 9 Fig. 1.4: Propagation of transverse wave ii. Longitudinal wave If the particles of a medium vibrate along the direction of propagation of the wave, the wave is called longitudinal wave. These waves travel in the form of compressions and rarefactions as shown in Fig. 1.5. During compressions and rarefactions, the pressure of the medium changes. That is why, they are C R C R also called pressure or compression waves. For example, waves on springs Fig 1.5: Propagation of along length, sound waves in air etc. Whenever, a wave propagates in a longitudinal wave medium, there is transfer of energy from one point to the another but, the net displacement of the particle is zero. So, when a particle in an elastic medium is disturbed from its mean position, a restoring force (property of elastic medium) acts in it; as a result it executes SHM. But, the disturbance in the form of energy is transferred to the surrounding particles and this disturbance forms a pattern of propagation known as wave propagation. The graphical representation of longitudinal wave is shown in Fig. 1.6. Propagation of Longitudinal Wave Consider an elastic medium in which the particles of the medium have to and fro motion (i.e. simple harmonic motion). Consider nine particles 1, 2, 3, 4, 5, 6, 7, 8 and 9 arranged linearly. At t = 0, all the particles occupy their mean position. When the particle at 1 is disturbed, then the disturbance is transferred to all other particle continuously. The transfer of disturbance can be explained below. 6 Principles of Physics - II i. When t = 0, particle 1 is set into vibration. Density→ T C C ii. When t = 8 , particle 1 sends the disturbance to particle 2. So, particle 2 is set into vibration. Time→ O 2T iii. When t = 8 , particle 1 reaches to C = Compression R = Rarefaction R extreme position, particle 2 sends the disturbance to particle 3 and hence, Fig. 1.6: Wave form of longitudinal wave particle 3 is set into vibration. 3T iv. When t = 8 , particle 1 starts returning back to the left. Particle 2 reaches to extreme position. Particle 3 sends the disturbance to the particle 4 and is set into vibration. 4T v. When t = 8 , particle 1 returns backs to its mean position. Particle 2 starts returning back to the left. Particle 3 reaches to extreme position. Particle 4 sends disturbance to particle 5. Similarly, the disturbance travels to particles 6, 7, 8 and 9. Then, particle 1 starts oscillating in 8T the opposite direction. Hence, the one cycle of oscillation is completed at time t = T = 8. Thus, the longitudinal wave transfers energy (disturbance) in an elastic medium. The formation of complete longitudinal wave is shown in Fig. 1.7. 0 1 2 3 4 5 6 7 8 9 T 8 2T 8 3T 8 4T 8 5T 8 6T 8 7T 8 T C R C Fig. 1.7: Propagation of longitudinal wave Difference between Longitudinal Wave and Transverse Wave Longitudinal Wave Transverse Wave 1. The particles of the medium vibrate along 1. The particles of the medium vibrate at the direction of propagation of waves. right angle to the direction of propagation of wave. 2. In this type of wave motion, a series of 2. In this type of wave motion, compressions compressions and rarefactions are formed. and rarefactions are not formed. One crest One compression and one rarefaction and one trough constitute one wave. constitute one wave.  Wave Motion  Chapter 1  7 3. Sound waves in air and water medium 3. Light waves are transverse in nature. travel as longitudinal wave. 4. It can travel in all types of media, solid, 4. It can travel in solid and in liquid at lower liquid and gas. depth from the surface but not in gases. 5. The pressure and density vary and are 5. The pressure and density remain the same maximum at the compression region and through out any region. minimum at rarefaction region. 6. If a wave is longitudinal it is mechanical, 6. If a wave is non-mechanical, it is but if a wave is mechanical it may or may transverse, but if a wave is transverse it not be longitudinal. may or may not be non-mechanical. 1.3 Graphical Representation of Waves When a disturbance is created at a point of a medium, the particles in the medium get displaced from their mean position. This displacement of the particles imparts disturbances to the neighbouring particles. Thus, the disturbance travels to the surroundings forming a regular pattern of vibration of particles in the medium, which is called wave. The direction of displacement of the particles may be parallel or perpendicular to the direction of propagation of wave. The net displacement of particle is zero, although the disturbance travels long distance away. In this process, the particles execute simple harmonic motion (SHM). Therefore, the displacement of particles in a medium can be written in terms of equation of SHM, when wave travels, 2  y = a sin t – x... (1.1)    Physically, this equation implies that, the particle displacement (y) depends on two variables; the distance of wave propagation (x) from the mean position, and instantaneous time of oscillation (t) of a particular particle. They are explained as follows: i. Displacement versus distance graph: If the displacement of a particle is taken along y-axis and distance of wave propagation along x-axis, the graph so drawn is called displacement versus distance graph as shown in Fig. 1.8. In transverse wave, the displacement of particles and distance of wave propagation are perpendicular to each other. So, it is easier to visualize the graph. But, in case of longitudinal wave, the direction of particle displacement (y) is parallel to the direction of wave propagation. Nevertheless, the graphical representation can be visualized by taking the displacement of particles (y) perpendicular to the direction of propagation of wave. Therefore, the nature of graph is shown in the Fig. 1.8. Y Displacement amplitude O X Distance Wavelength Fig. 1.8: Displacement-distance graph for a wave 8 Principles of Physics - II Y Displacement a O Time Time period Fig. 1.9: Displacement-time graph for a wave ii. Displacement versus time graph: If the displacement of particle is taken along y-axis and time of oscillation of a particle is taken along x-axis, the graph so drawn is known as displacement versus time graph. The nature of the graph is shown in Fig. 1.9. 1.4 Basic Terminologies of Wave Compression (C): The region at which the particles in the medium come  closer is known as compression region. In this region, the particles in a medium come closer and hence density and pressure increase as shown in Fig. 1.10. Rarefaction (R): The region at which the particles in the medium C R C R move away from each other is called rarefaction region. In this region, Fig 1.10: Longitudinal wave the particles in a medium move away from each other and hence density and pressure decrease as shown in Fig. 1.10. Crest: The position of maximum positive displacement i.e. the upper-most point of the transverse wave is called crest. In Fig. 1.11, C symbolizes the crest. Trough: The position of maximum negative displacement i.e. the lowest point of the transverse wave is called trough. In Fig. 1.11, T symbolizes the trough. Wavelength: The distance travelled by a wave in one complete cycle is called wavelength. It is denoted by . It is the distance between, either any two nearest crests or troughs in case of transverse waves (or any two rarefactions or compressions in case of longitudinal waves). Y C  a O Time T  Fig 1.11: Wave motion The wavelength can also be defined as the separation between any two nearest points which are in the same phase. Amplitude: The maximum displacement of the particles in a medium about their mean position is known as amplitude. It is denoted by 'a' or 'A'. Time period: The time in which a particle of medium completes one vibration about its mean position is known as time period of wave. It is denoted by 'T'.  Wave Motion  Chapter 1  9 Frequency: The number of oscillations per second is called frequency. It is denoted by 'f'. It can also be defined as the number of waves passing through a point per unit time. For N number of complete waves, the frequency is, N N 1  f = t = NT = T It is measured in cycle/s which is also called hertz.  1 Hz = 1 cycle/s. Wave speed: The linear distance covered by a wave per unit time in its direction of propagation is called its wave speed. Distance along propagation of wave Wave speed (v) = Time taken As we know, the wave travels distance '' in time period T. So, Wave length () v = Time period (T) 1 v =T v = f... (1.2) i.e., wave speed = frequency × wave length Equation (1.2) is an important relation between the speed of a wave, its frequency and wave length. This relation is valid for all kinds of waves including mechanical and electromagnetic waves. Particle Speed The longitudinal wave propagates due to the oscillation of molecules of an elastic medium. The speed of particle when it oscillates to transfer the energy from one particle to another is known as particle speed. The displacement of a particle from its mean position is written as, y = a sin (t – ) dy dt = a cos (t – ) dy  Speed of particles, v = dt = a cos (t – ) The velocity of oscillating particles depends on its phase, varying from zero to maximum. The maximum value of speed of particle is, vmax = a … (1.3) Phase of a Wave The position of an oscillating particle during time 't' can be described in terms of angular displacement from its mean position. This angular displacement of the oscillating particle in a medium which describes its location is known as phase or phase angle of a wave. The wave equation for simple harmonic motion is, y = a sin t The angular term 't' gives the phase of a wave. When one wave is ahead of another by some angle, the difference of angle between them is represented by phase difference (). Then, the phase of oscillation is represented by (t – ) or (t + ). 10 Principles of Physics - II Therefore, the wave equation of SHM is, y = a sin (t – ) For a wave, moving opposite to the above condition, y = a sin (t + ) Relation between Phase Difference and Path Difference The path refers the linear displacement and phase refers the angular displacement of two points in wave propagating medium. Therefore, the linear displacement of two points in a medium is called path difference of these points of the wave. It is denoted by x. Similarly, the angular displacement of two points in a wave is called their phase difference. It is denoted by . Consider a OA wave travelling along x-direction as shown in Fig. 1.12. Let us take a point P in the x- axis at distance x from the origin O. The relation between phase difference () and path difference (x) of two points O and P are as follows: As we know, for path difference , the phase difference is 2. For path difference , the corresponding phase difference is 2.  2 Y For path difference 1, the corresponding phase difference is.  A O 2 x P For, path difference x, the corresponding phase difference is x. X   Therefore, Fig. 1.12: A complete cycle of wave 2 Phase difference () = x... (1.4)  2 i.e. phase difference () = × path difference (x) wavelength () This is the relation between path difference and phase difference. The waves are said to be in the same phase if they have a phase difference of even integral multiples of . Similarly, the waves are said to be out of phase (opposite phase), if they have a phase difference of odd integral multiples of . 1.5 Progressive Wave A wave which travels forward in a medium with maximum transfer of energy from one particle to another particle is called progressive wave. Progressive wave is also called travelling wave. For example, water wave, light wave, sound wave, etc. are progressive waves. Progressive Wave Equation Consider a progressive wave travelling along x-direction from a reference origin O with speed v. The displacement time graph for the progressive wave is shown in Fig. 1.13. The particles in the medium execute simple harmonic motion, while the progressive wave travels from one point to another. The wave equation for such condition is written as, y = a sin (t – )... (1.5) Where, y = displacement of a particle in a medium a = amplitude  = angular velocity  = phase difference  Wave Motion  Chapter 1  11 Fig. 1.13 : Progressive wave Also, the phase difference is related to the path difference. So, 2 = x  Putting this value of  in equation (1.5), we get, 2  y = a sint – x... (1.6)    2 Now, taking  = T in equation (1.6), we get, 2 2  y = a sin  T t – x    t x y = a sin 2  T – ... (1.7)   v Also,  = 2f = 2 , v is the wave velocity. So, equation (1.6) can also be written as,  2vt 2  y = a sin  – x     2  y = a sin (vt – x)... (1.8)  2 2 The term is called propagation constant or wave vector denoted by k i.e. k =. So, equation(1.6)   can be written as, y = a sin (t – kx)... (1.9) Equations (1.6), (1.7), (1.8) and (1.9) are the progressive wave equations written in several alternative forms. If the wave travels in opposite direction i.e. along negative X-axis, the equation (1.9) becomes, y = a sin (t + kx)... (1.10) Hence, the general progressive wave equation is given by, y = a sin (t ± kx)... (1.11) 12 Principles of Physics - II Characteristics of a Progressive Wave i. Every particle of a medium executes periodic motion. ii. The amplitude of each particle of the medium is same, but there exists phase difference between them. iii. The distance between two successive crests of a transverse wave and distance between a compression and rarefaction is wavelength. iv. The change in pressure and density of a medium is similar in case of progressive waves. v. A progressive wave travels forward, undamped and unobstructed. vi. No particle remains permanently at rest. vii. Energy is transferred across every plane along the direction of propagation. viii. The progressive wave may be longitudinal or transverse. 1.6 Differential Form of Wave Equation The general wave equation is y = a sin (t – kx)... (1.12) Differentiating equation (1.12) with respect to 't', dy dt = a cos (t – kx) Again, differentiating, d2 y dt2 = –a sin (t – kx) 2 = –2y 1 d2y  y = – 2 dt2... (1.13)  Now, differentiating (1.12) with respect to 'x', dy dx = –ak cos (t – kx) Again, differentiating, d2 y dx2 = –ak sin (t – kx) 2 = –k2y 1 d2y  y = –k2 dx2... (1.14) Equating (1.13) with (1.14), we get, 1 d2y 1 d2y – 2 dt2 = –k2 dx2  k2 d2y d2y = 2 dt2 dx2 1 d2y d2y ... k = 4 / 1 1 =  2 2 2 v2 dt2 = dx2  2 42f2 = f22 v2 d2y 1 d2y  dx2 – v2 dt2 = 0... (1.15) Equation (1.15) is the differential form of wave equation.  Wave Motion  Chapter 1  13 Principle of Superposition of Waves When two or more waves meet simultaneously at a point of a medium, the particles in the medium oscillate with new displacement so that a new wave pattern is formed. This phenomenon of formation of new wave by mixing of two or more waves is known as superposition of wave. The principle of superposition of waves states that the resultant displacement of the particle is equal to the vector sum of individual displacements due to different waves. If y be the resultant displacement of a particle and y1, y2,... are displacements due to individual waves, then according to the principle of superposition of waves, we have y = y1 ± y2 ±...... (1.16) 1.7 Interference of Sound The phenomenon of redistribution of energy in the resultant sound wave formed by the superposition of two sound waves having same frequency (or wavelength) and constant phase difference when travelling in same direction is called interference of sound wave. There are two types of interference of a wave. i. Constructive interference ii. Destructive interference The amplitude becomes maximum in the constructive interference and hence intensity of sound becomes maximum. In the destructive interference, amplitude and intensity becomes minimum. In the process of interference, there is only transference of energy from one part to another. The energy missing at one point re-appears at another point. There is only redistribution of energy without any destruction or creation of energy, and so the law of conservation of energy is fully obeyed. Interference occurs in both transverse and longitudinal waves. Expression of Interference of Two Waves Let y1 and y2 be the displacements of particles in a medium due to waves of same angular frequency . Let a1 and a2 be the arbitrary amplitudes of these waves when travelling in the same direction with phase difference . The wave equations for these waves are, y1 = a1 sin (t – kx)... (1.17) y2 = a2 sin (t – kx + )... (1.18) 2 Where, k = , called wave vector  Applying superposition principle, y = y1 + y2... (1.19) Using (1.17) and (1.18) in (1.19), we get, y = a1 sin (t – kx) + a2 sin (t – kx + ) = a1 sin (t – kx) + a2 sin (t – kx) cos  + a2 cos (t – kx) sin  = (a1 + a2 cos ) sin (t – kx) + (a2 sin ) cos (t – kx) Putting, a1 + a2 cos  = A cos ... (1.20) a2 sin  = A sin ... (1.21) Where, A is the amplitude of resultant wave and  is the phase angle. 14 Principles of Physics - II Therefore, y = A cos  sin (t – kx) + A sin  cos (t – kx) y = A sin (t – kx + )... (1.22) The equation (1.22) gives the wave equation of a harmonic wave. To find the amplitude of resultant wave, the equations (1.20) and (1.21) can be squared and added, A2 cos2 + A2 sin2  = (a1 + a2 cos )2 + (a2 sin )2 A2 (cos2 + sin2) = a21 + 2a1a2 cos  + a22 cos2  + a22 sin2  A2 = a21 + 2a1a2 cos  + a22 A = a21 + a22 + 2a1 a2 cos ... (1.23) To find the phase angle, dividing (1.21) by (1.20), we get, A sin  a2 sin  = A cos  a1 + a2 cos  a2 sin  tan  = a1 + a2 cos  a2 sin     = tan–1 ... (1.24) a1 + a2 cos  Cases: i. When original waves overlap in phase, i.e.  = 0. A = a1 + a 2 For, a1 = a2 = a A = 2a, (maximum amplitude) This interference is called constructive interference. In Fig.1.14, two waves (represented by dotted lines) moving along the same positive x direction with same frequency and phase superimpose to form a single wave (represented by a solid line) which has same frequency but, has maximum amplitude (constructive interference). Y a2 Amax a1 X  Fig.1.14: Constructive interference of two waves ii. When original waves overlap out of phase,  = 180º A = a1 – a2 For a1 = a2 A = 0 (minimum amplitude) This interference is called destructive interference. In Fig. 1.15, two waves (represented by dotted lines) moving along the same +ve x-direction with same frequency but opposite phase superimpose to form a single wave (represented by solid line) which has same frequency but has minimum amplitude (destructive interference).  Wave Motion  Chapter 1  15 Y a1 Amin X a2  Fig 1.15: Destructive interference of two waves (a1 > a2) 1.8 Stationary Wave When two progressive waves of same amplitude and frequency travel in a medium in exactly the opposite direction, a resultant wave is formed. This resultant wave is called stationary wave. It is also called standing wave. No energy is transferred from a particle to surrounding particles while stationary wave is formed in a medium. Each particle has its own characteristics of vibration. Hence, the amplitude of vibration of the different particles are different, ranging from zero to some maximum value. The position of particle at the zero displacement is called node (N) and the position of particle at which the maximum displacement takes place is called antinode (AN). The formation of stationary wave in a string is explained below: Consider a rope tied to a rigid support of pole at one end, the next end being held by our hand. Now let us give a gentle up and down jerk to the free end so that a pulse travels along the length of string which reaches the next end until it is reflected Fig. 1.16: Formation of stationary wave back by the rigid pole. After reflection, the pulse again travels along the length of string but changes its direction. Thus, the reflected pulse (wave) travel back to our hand and hence we have now two waves which are traveling in the opposite directions and they combine to produce a resultant wave which appears to be stationary wave Fig. 1.16. We cannot see the original wave and reflected wave separately, only a stationary wave is visible. The frequency of the progressive wave and the stationary wave is the same. When stationary waves are formed, the amplitude becomes maximum and strain becomes minimum at certain points. At other certain points, the amplitude becomes minimum and strain becomes maximum. The points, where the amplitude is maximum and strain is minimum, are called antinodes (A.N.) and the points, where the amplitude is minimum and strain is maximum are called nodes (N). Antinodes and nodes are formed alternately in the standing wave. Thus, the wave in which antinodes and nodes are formed alternately is called a stationary wave. Stationary Wave Equation Let y1 and y2 be the displacements of two progressive waves of same amplitude a and wave length  travelling in opposite direction simultaneously with the same velocity v. The equations of these waves may be expressed as follows, 16 Principles of Physics - II y1 = a sin (t – kx)... (1.25) y2 = a sin (t + kx)... (1.26) Thus, the resultant displacement of the particle of medium due to both the waves will be determined from the principle of superposition, = y1 + y2 = a sin (t – kx) + a sin (t + kx) = a [sin (t – kx) + a sin (t + kx)]  y = 2a cos kx. sin t y = A sin t... (1.27) Equation (1.27) represents a simple harmonic wave whose amplitude is A = 2a cos kx. It is evident that, for different values of x, the amplitude will have different values. Obviously, the frequency of stationary wave is equal to the interfering waves i.e. there is no change in frequency. Condition for maximum amplitude The amplitude A = 2 a cos kx will be maximum if, A = ± 2a So, 2a cos kx = ± 2a cos kx =  1 2x cos =1  2x cos = cos n , where n = 0, 1, 2, 3,...  2x or, = n  n or, x= 2... (1.28) The equation (1.28) is the condition for antinode formation,  For n = 0, x0 = 0, For n = 1, x1 = 2 2 3 For n = 2, x2 = 2 For n = 3, x3 = 2 Hence, the antinodes occurs at the positions where, Phase difference () = 0, , 2, 3,... n, and  2 3 n Path difference (x) = 0, 2  2  2..., 2 Therefore, the condition of antinode is,  3 n x = 0, 2 , , 2 , …, 2 n (n  1)  The distance between two consecutive antinodes = 2 – 2 = 2. Condition for minimum amplitude The amplitude A = 2a cos kx will be minimum if, A = 0 So, 2a cos kx = 0  Wave Motion  Chapter 1  17 cos kx = 0 2x cos =0  2x  cos = cos (2n + 1) 2 , where n = 0, 1, 2, 3,...  2x  or, = (2n + 1) 2   or, x = (2n + 1) 4... (1.29) The equation (1.29) is the condition for node formation.  3 For n = 0, x0 = 4 For n = 1, x1 = 4 5 7 For n = 2, x2 = 4 For n = 3, x3 = 4 9  3  For n = 4, x4 = 4 Similarly, x = 4  4... , (2n + 1)4. Hence, the nodes occur at the positions where,  3  Phase difference () = 2  2 ,... (2n + 1) 2  3  Path difference (x) = 4  4 ,... , (2n + 1) 4    The distance between two consecutive nodes = (2n + 1) 4 – {2(n  1) + 1} 4 = 2 , which is equal to the distance between two consecutive antinodes.  n  The distance between any consecutive node and antinode = (2n + 1) 4 – 2 = 4. 1.9 Stationary Waves in Boundary During the propagation of sound wave in air, the nodes are formed both at rarefactions and compressions but the antinodes are formed in between these rarefactions and compressions. At compression, a cross section is found in which the neighbouring molecules exert pushing force of equal and opposite direction so that this cross section becomes stationary (rest). Similarly at rarefaction, a cross section is found in which the neighbouring molecules exert pulling force equal and opposite direction so that this cross section is also at rest. The wave pattern for stationary wave depends on the type of boundary (the position from which the wave reflects). There are two types of boundaries viz., open boundary and closed boundary. i. Stationary wave in open boundary: Open boundary is that boundary from which the waves are reflected but the particles are not reflected rather they move along in forward direction. Such boundaries do not have rigid surfaces to reflect the wave. From such boundaries, compressions are reflected as rarefactions and vice-versa. The equation of stationary wave in equation (1.27) is determined considering the open boundary. The vibration pattern of particles and wave form in this boundary is as shown in Fig. 1.17. 18 Principles of Physics - II Particles pile up Particles pull apart y AN (i) N x (ii) Fig 1.17: (i) Vibration pattern of particles, (ii) Stationary wave in open boundary. At the boundary, for x = 0 A = 2a cos kx = 2a = Amax i.e. antinode is formed. ii. Stationary wave in closed boundary: Closed boundary is such boundary from which both waves and particles reflect back. Such boundaries have rigid surface to reflect the wave. From these boundaries rarefactions are reflected as rarefactions and compressions are reflected as compressions. However, the incident particles reflect with phase reversal of . So, the displacement equation of particles which are incident on the boundary and reflected from the boundary can be respectively written in the following forms.  y1 = a sin (t – kx)... (1.30) y2 = a sin (t + kx + )... (1.31)  From superposition principle, y = y1 + y2... (1.32) Using (1.30) and (1.31) in (1.32) y = a sin (t – kx) + a sin (t + kx + ) = a sin (t – kx) + a sin ( + (t + kx)) = a sin (t – kx) – a sin (t + kx) = a [sin (t – kx) – sin (t + kx)] t – kx – t – kx t – kx + t + kx = a 2 sin cos  2 2  = –2a sin kx cos t = A cos t... (1.33) where, amplitude of resultant wave is A = –2a sin kx At the boundary x = 0, A = 0 The vibration pattern of particles and wave form at the closed boundary is shown in Fig 1.18.  Wave Motion  Chapter 1  19 Particles pile up Particles pull apart y AN (i) N x (ii) Fig 1.18: (i) Vibration pattern of particles, (ii) Stationary wave in closed boundary. Characteristics of a Stationary Wave i. Only the particles other than those at the nodes executes periodic motion. ii. The phase difference between particles of the medium is same, but amplitude is different. iii. In case of stationary wave each particle attains its stationary position twice during one complete vibration. iv. In this wave, nodes and antinodes are formed alternately and the separation between any two consecutive nodes or antinodes is /2. v. The amplitude is minimum at nodes and maximum at antinodes. vi. In the stationary wave, the change in pressure and density of the medium is not uniform. It is maxim