JEE Advanced Physics Modern Physics PDF

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 About Pearson Pearson is the world’s learning company, with presence across 70 countries worldwide. Our unique insights and world-class expertise comes from a long history of working closely with renowned teachers, authors and thought leaders, as a result of which, we have emerged as the preferred choice for millions of teachers and learners across the world. We believe learning opens up opportunities, creates fulfilling careers and hence better lives. We hence collaborate with the best of minds to deliver you class-leading products, spread across the Higher Education and K12 spectrum. Superior learning experience and improved outcomes are at the heart of everything we do. This product is the result of one such effort. Your feedback plays a critical role in the evolution of our products and you can contact us - [email protected]. We look forward to it. F01_Physics for JEE Mains and Advanced_Modern Physics_Prelims.indd 1 11/16/2019 2:21:07 PM  This page is intentionally left blank F01_Physics for JEE Mains and Advanced_Modern Physics_Prelims.indd 2 11/16/2019 2:21:07 PM 3 THIRD EDITION JEE ADVANCED PHYSICS Modern Physics Rahul Sardana  Copyright © 2020 Pearson India Education Services Pvt. Ltd Published by Pearson India Education Services Pvt. Ltd, CIN: U72200TN2005PTC057128. No part of this eBook may be used or reproduced in any manner whatsoever without the publisher’s prior written consent. This eBook may or may not include all assets that were part of the print version. The publisher reserves the right to remove any material in this eBook at any time. ISBN 978-93-539-4169-7 eISBN: 978-93-539-4424-7 Head Office: 15th Floor, Tower-B, World Trade Tower, Plot No. 1, Block-C, Sector 16, Noida 201 301, Uttar Pradesh, India. Registered Office: The HIVE, 3rd Floor, Pillaiyar Koil Street, Jawaharlal Nehru Road, Anna Nagar, Chennai 600 040, Tamil Nadu, India. Phone: 044-66540100 website: in.pearson.com, Email: [email protected] F01_Physics for JEE Mains and Advanced_Modern Physics_Prelims.indd 4 11/16/2019 2:21:07 PM  Contents Chapter Insight xiii Preface xix About the Author xx CHAPTER Dual Nature of Radiation and Matter 1.1 1 Matter Waves  1.1 de-Broglie’s Postulate  1.1 Characteristics of Matter Waves 1.2 de-Broglie Wavelength Associated with the Charged Particles  1.2 de-Broglie Wavelength Associated with Uncharged Particles: Thermal Neutrons 1.3 Heisenberg’s Uncertainty Principle 1.3 Quantum Nature of Light and Plank’s Quantum Theory  1.4 Properties of Photons  1.5 Number of Photons Emitted Per Second by a Source 1.6 Intensity of Light Due to a Light Source 1.7 Photon Flux in a Light Beam 1.7 Photon Density in a Light Beam  1.8 Momentum of a Photon  1.11 Force and Radiation Pressure Due to a Photon Beam Incident Normally on a Surface 1.11 Force and Radiation Pressure Due to a Photon Beam Incident on a Perfectly Absorbing Surface at an Angle of Incidence  1.14 Force and Radiation Pressure Due to a Photon Beam Incident on a Perfectly Reflecting Surface at an Angle of Incidence  1.15 Force and Radiation Pressure Due to a Photon Beam Incident on a Partially Reflecting Surface at an Angle of Incidence  1.16 Force Exerted on any Object in the Path of a Light Beam  1.18 Emission of Electrons  1.21 Photoelectric Effect 1.21 Some Important Terms 1.22 Plank’s Quantum Theory 1.22 Experimental Setup for Photoelectric Effect  1.22 F01_Physics for JEE Mains and Advanced_Modern Physics_Prelims.indd 5 11/16/2019 2:21:07 PM  vi Contents Laws of Photoelectric Emission 1.23 Einstein’s Explanation of Photo-electric Effect 1.23 Photoelectric Effect cannot be Explained Using Classical Wave Theory of Light  1.24 Characteristics of Photo Electric Effect  1.29 Graph Between Kmax and ν  1.31 Determination of Photoelectric Current  1.31 Solved Problems  1.35 Practice Exercises 1.45 Single Correct Choice Type Questions 1.45 Multiple Correct Choice Type Questions 1.58 Reasoning Based Questions  1.61 Linked Comprehension Type Questions 1.62 Matrix Match/Column Match Type Questions 1.65 Integer/Numerical Answer Type Questions  1.68 Archive: JEE Main  1.69 Archive: JEE Advanced 1.74 Answer Keys–Test Your Concepts and Practice Exercises  1.78 CHAPTER Atomic Physics 2.1 2 Structure of an Atom: An Introduction 2.1 Thomson Empirical Model/Thomson Plum Pudding Model 2.1 Rutherford’s Atomic Model 2.2 Experimental Arrangement  2.2 Observations  2.2 Conclusions 2.2 Rutherford’s Atom-model Postulates 2.3 Distance of Closest Approach  2.3 Trajectory of an Alpha Particle and Impact Parameter 2.4 Failure of Rutherford Model 2.5 Bohr’s Atomic Model  2.6 Bohr’s Theory of the Hydrogen like Atoms 2.7 Frequency of Emitted Radiation  2.10 Hydrogen Spectrum  2.11 Hypothetical Bohr’s Model  2.17 Recoil of an Atom Due to Electron Transition  2.19 Atomic Collisions 2.21 Effect of Mass of Nucleus on Bohr Model  2.24 About Rydberg Constant  2.25 Ritz Combination Principle  2.27 F01_Physics for JEE Mains and Advanced_Modern Physics_Prelims.indd 6 11/16/2019 2:21:07 PM  Contents vii Merits and Demerits of Bohr’s Theory  2.28 Critical Potential 2.29 Resonance Potential 2.29 Excitation Potential  2.29 Ionisation Potential  2.29 X-rays  2.32 X-ray Spectra Classification  2.33 Moseley’s Law  2.35 Properties of X-rays 2.37 Bragg’s Law  2.39 Intensity of Transmitted X-ray  2.39 Solved Problems  2.41 Practice Exercises 2.50 Single Correct Choice Type Questions 2.50 Multiple Correct Choice Type Questions 2.62 Reasoning Based Questions  2.65 Linked Comprehension Type Questions 2.67 Matrix Match/Column Match Type Questions 2.72 Integer/Numerical Answer Type Questions  2.76 Archive: JEE Main  2.77 Archive: JEE Advanced 2.80 Answer Keys–Test Your Concepts and Practice Exercises  2.86 CHAPTER Nuclear Physics  3.1 3 Nucleus and Nuclear Structure 3.1 Atomic Mass Unit (u or amu) 3.1 Isotopes 3.2 Isobars 3.2 Isotones 3.2 Nuclear Radius  3.2 Nuclear Density 3.2 The Nuclear Force  3.2 Yukawa Theory or Meson Theory of Nuclear Forces  3.3 Mass Energy Equivalence 3.3 Mass Defect and Binding Energy 3.4 Binding Energy (BE): Revisited 3.4 Variation of Binding Energy Per Nucleon with Mass Number A 3.6 Packing Fraction 3.6 Nuclear Stability 3.7 F01_Physics for JEE Mains and Advanced_Modern Physics_Prelims.indd 7 11/16/2019 2:21:08 PM  viii Contents Radioactivity 3.8 Laws of Radioactive Disintegration  3.8 Half Life (T1/2)  3.9 Mean Life (t ) 3.10 Parallel Radioactive Disintegration 3.11 Activity of Radioactive Substance (A) 3.12 Specific Activity 3.12 Units of Radioactivity 3.12 Nuclear Radiations  3.17 Radioactive Series  3.19 Successive Disintegration and Radioactive Equilibrium  3.20 Equilibrium  3.20 Artificial Transmutation – Nuclear Reactions 3.23 Laws Governing Nuclear Reactions 3.23 Q-value of a Nuclear Reaction  3.24 Case-I: Exoergic Reaction 3.24 Case-II: Endoergic Reaction 3.24 Energetics of Nuclear Reactions 3.27 Alpha Decay 3.29 Beta Decay 3.30 Types of Beta Decay 3.31 Pauli’s Neutrino Hypothesis and Neutrino Properties 3.33 Continuous Energy Spectrum for Beta Rays 3.33 Q-value for Beta Decay Process 3.34 Gamma Decay  3.35 Classification of Nuclear Reactions  3.37 Nuclear Fission  3.42 Chain Reaction  3.43 Nuclear Fusion  3.44 Nuclear Holocaust  3.46 Use of Radioisotopes  3.46 Solved Problems  3.49 Practice Exercises 3.58 Single Correct Choice Type Questions 3.58 Multiple Correct Choice Type Questions 3.72 Reasoning Based Questions  3.75 Linked Comprehension Type Questions 3.76 Matrix Match/Column Match Type Questions 3.80 F01_Physics for JEE Mains and Advanced_Modern Physics_Prelims.indd 8 11/16/2019 2:21:08 PM  Contents ix Integer/Numerical Answer Type Questions  3.83 Archive: JEE Main  3.85 Archive: JEE Advanced 3.87 Answer Keys–Test Your Concepts and Practice Exercises  3.96 CHAPTER Semiconductor Devices and Applications  4.1 4 Introduction  4.1 Functioning of Vacuum Tube(s) 4.1 Disadvantages of Using Vacuum Tube(s) 4.1 Semiconductor Electronics 4.1 Classification of Metals, Conductors and Semiconductors on the Basis of Conductivity 4.2 Energy Bands  4.2 Forbidden Energy Gap (DEg) 4.3 Band Theory of Classification of Metals, Insulators and Semiconductors  4.3 Holes in Semiconductors  4.4 Current Carriers in Semiconductors  4.4 Increasing the Conductivity of a Semiconductor: Doping Process  4.5 p-type (Extrinsic) Semiconductor  4.5 n-type (Extrinsic) Semiconductor  4.6 Relation Between the Number Density of Intrinsic and Extrinsic Charge Carriers 4.6 Electrical Resistivity of Semiconductors 4.7 p-n Junction  4.9 Semiconductor Diode  4.11 p-n Junction Diode Under Forward Bias 4.12 p-n Junction Diode Under Reverse Bias 4.13 V-I Characteristics of a p-n Junction Diode  4.16 Dynamic Resistance of a Diode 4.16 Zener and Avalanche Breakdown in Reverse Bias of a Diode 4.17 Avalanche Breakdown 4.18 Application of Junction Diode as a Rectifier  4.18 Filter Circuits  4.21 Zener Diode: Special Purpose Junction Diode 4.22 Zener Diode as a Voltage Regulator  4.22 Opto Electronic Junction Devices  4.24 Junction Transistor  4.27 Transistor Construction 4.28 Working of Transistor  4.28 F01_Physics for JEE Mains and Advanced_Modern Physics_Prelims.indd 9 11/16/2019 2:21:08 PM  x Contents Action of NPN Transistor 4.29 Action of PNP Transistor  4.29 Three Configurations of a Transistor 4.30 Current Gain for a Transistor  4.31 Relation Between a and β 4.31 Common Emitter Transistor Characteristics  4.32 Input Characteristic Curve of a Transistor  4.32 Output Characteristic Curve of a Transistor  4.33 Transfer Characteristic Curve of a Transistor  4.34 Current Amplification Factor (b ) 4.35 Advantages of Transistors  4.35 Drawbacks of Transistors 4.35 Transistor as a Switch  4.35 Amplifying Action of a Transistor  4.37 Transconductance 4.37 Concept of an Amplifier 4.37 AC Voltage Gain (Av) 4.38 NPN Transistor as Common Base Amplifier  4.38 Current, Voltage and Power Gain for a Common Base Amplifier 4.39 PNP Transistor as Common Base Amplifier  4.39 NPN Transistor as Common Emitter Amplifier  4.40 Current, Voltage and Power Gain for a Common Emitter Amplifier 4.41 PNP Transistor as Common Emitter Amplifier  4.41 Transistor as an Oscillator 4.44 Working of an Oscillator  4.45 Barkhausen’s Criterion for Sustained Oscillations 4.46 Digital Electronics  4.46 Logic Gates 4.47 Boolean Algebra Basics  4.47 The ‘OR Gate’ 4.48 The AND Gate  4.49 The ‘NOT Gate’  4.50 Combination of Gates 4.51 The ‘XOR Gate’ 4.52 Logic Gates Using NAND Gates  4.52 Logic Gates Using NOR Gates  4.53 F01_Physics for JEE Mains and Advanced_Modern Physics_Prelims.indd 10 11/16/2019 2:21:08 PM  Contents xi Solved Problems  4.56 Practice Exercises 4.61 Single Correct Choice Type Questions 4.61 Archive: JEE Main  4.74 Answer Keys–Practice Exercises  4.81 CHAPTER Communication Systems  5.1 5 Introduction  5.1 Roots of Modern Communication  5.2 Elements of a Communication System  5.2 Modes of Communication  5.3 About Jagadis Chandra Bose (1858–1937)  5.3 Basic Terminologies Used in Electronic Communication Systems 5.3 Bandwidth of Analog Signals 5.5 Digital Signals in the Form of Sine Waves 5.5 Bandwidths of Communication Channels  5.6 Propagation of Electromagnetic Waves  5.7 Ground Wave Propagation  5.7 Sky Wave Propagation 5.8 Space Wave Propagation 5.9 Determination of Range  5.10 Modulation and Its Necessity  5.11 Amplitude Modulation  5.12 Modulation Index in terms of Amax and Amin 5.15 Power and Current Relation in AM Wave 5.16 Production of Amplitude Modulated Wave 5.16 Production of Amplitude Modulated Carrier Wave Using a Common Emitter Amplifier 5.17 Antenna 5.17 Receiver 5.18 Receiver for Detection of an Amplitude Modulated Wave 5.18 Detection/Demodulation of an Amplitude Modulated Wave 5.19 Junction Diode as a Detector/Demodulator 5.19 The Internet 5.21 World Wide Web (WWW) 5.21 Chat  5.21 Facsimile (FAX) 5.21 Mobile Telephony 5.22 F01_Physics for JEE Mains and Advanced_Modern Physics_Prelims.indd 11 11/16/2019 2:21:08 PM  xii Contents Practice Exercises 5.23 Single Correct Choice Type Questions 5.23 Archive: JEE Main  5.27 Answer Keys–Practice Exercises  5.31 Hints and Explanations Chapter 1: Dual Nature of Radiation and Matter  H.3 Chapter 2: Atomic Physics H.38 Chapter 3: Nuclear Physics  H.87 Chapter 4: Semiconductor Devices and Applications  H.133 Chapter 5: Communication Systems  H.146 F01_Physics for JEE Mains and Advanced_Modern Physics_Prelims.indd 12 11/16/2019 2:21:08 PM  ChaPter InsIGhtDual Nature of 1 CHAPTER Radiation and Matter 2 CHAPTER Learning Objectives Atomic Physics Help the students 3 Learning Objectives CHAPTER set an aim to After reading this chapter, you will be able to: After reading this chapter, you will be able to understand concepts and problems based on: achieve the major take-aways from a (b) de-Broglie’s Hypothesis (c) Photon properties Nuclear Physics (a) Understand the concept of mater waves (d) Radiation pressure (e) Photoelectric emission (f) Photoelectric effect and properties. particular chapter All this is followed by a variety of Exercise Sets (fully solved) which contain questions as per the AQ1 Learning AfterJEE latest reading Objectives this pattern. Atchapter, the endyou will be able of Exercise Sets,to: a collection of problems asked previously in JEE (Main Chapter 4: Semiconductor Devices and Applications 4.15 and Advanced) are also given. After reading this chapter, you will be able to understand concepts and problems based on: (a) Understand the structure of atom (e) The Bohr’s Theory is applied on Hydrogen : Modern Physics D AQ1 1 4Ω (b) Based on various models like Thompson like atoms for understanding the concepts Model of energy and various transitions of an atom A B D2 Number of photons emitted per second by a one (c) Rutherford’s Model including X-Rays and applications..2 eV 8Ω Learning After reading Objectives this chapter, you will be able to: milliwatt source is MATTER(d)WAVES Bohr’s Model along with their postulates and where, h is the Planck’s constant whose value is given SOLUTION After reading this chapter, you will be able to understand drawbacks by h concepts = 6.63 × 10 and−34problems based on: Js and Psource 10 −3 −1 n= = ≈ 2.3 × 1015 sAQ2 19 in the form of AQ1 Light possesses (a) this All Understand isdual followed theby nature basics ai.e. variety of(g)Exercise it behaves Nuclear Sets both Radiations (fully solved) which (n) contain Beta Decay Spectrum questions Chapter as per the 3: Nuclear Physics 3.5 12 V The given Esingle photon circuit 4.35 × 10 −is a wheatstone bridge. ofJEENucleus and p = momentum ofpreviously the particle as a wave latest and aspattern. AtNuclear a particle. theIn endsome (h) of Exercise Successive Disintegration Sets, a collection phenomena of problems (o) asked Gamma Decay in JEE (Main 2 ( 1.6 × 10 −19 ) ( 2 ) structure and Advanced) are also and (i) Nuclear given.polarisation, it Reactions (p) Classifi cation of Nuclear Number of photons emitted for a quantum 10 Ω 10 Ω e.g., interference, efficiency IlluSTRATION diff4raction de-BroglieBindingassumed this expression energy per nucleon in isanalogy with −31 behaves as(b)waves Properties because of Nuclear they are Forces ( j) Energetics explained on the of Nuclear Reactions photon because momentum of photon is 9.1 × 10diode en circuit, D1 is forwardof 0.5% and biased is A B A nucleus (c) Mass has Defect binding basis of Wave theory while in some other phenomena energy of 100 ReactionsMeV. It fur- (q) Nuclear BE 92 Fission s −1 biased. Hence, D1 will conduct but D(2 ther (d) releases MeV energy. h (r) Chain=Reaction = 7.66 MeV rse N = 2.3 × 1015 ) × 0.5 A e.g. Bphotoelectric Binding effect, 10Compton Energy effect,Find the new it(k)behaves Alpha as binding p = Decay A 12 and Nuclear 100 C D (e) energy particles (photons). Nuclear of the Stability nucleus. (l) Beta Decay l Fusion. 10 Ω 10 Ω STRUCTURE (f) Radioactivity OFdemands AN ATOM: (m) Pauli’s Neutrino is Hypothesis attracted towards the7centre of uniformly charged hIlluSTRATION efore, current through 8 Ω resistance ⇒ N will = 1.15 × 1013 s −1 Since nature symmetry, therefore de- ⇒sphere l =while they exert a force of repulsion upon each SOluTION hereas of hotons theintensity Wm −2 4 Ω resistance current 2through AN INTRODUCTION Broglie All this that thought is followed matter by a variety must have of dual Exercise nature. Sets (fully solved) pwhich contain Calculate the questions as per the binding energy per nucleon for 10 20 Ne, Afterlatest JEE pattern. releasing AtMeVthe end of Exercise Sets, other.ofThe a collection electrons problems asked get themselves previously in JEE arranged (Main in such ce of area 1 × 10 −4 m 2 and Thus photo-electric Sincecurrent we observe that AQ2The particle nature of 10matter is, itwell will become known more and stable If m is the56mass of particle Fe and 238 U. and Given v thethatvelocity, mass then of neutron is All matter and and is madetheupbinding Advanced) hence ofalso are tinygiven. particlesofknown energy the as nucleus awill way that 26 the force of92attraction and that of repulsion q Ne hence de-Broglie thought that material particles must momentum ofother. particle = mv. of proton is , p mass 53% 12 of the incident pho- P( R 13 ) ( −9 ) atoms. increase. There areSo, about 105 different kinds of atoms, balance each 1. 008665 amu When disturbed, electronsisvibrate1. 007825 amu , =. Calculate = 3 A the number of I = = = ne = 1 =. 15 × 10 1. 6 × 10 possess wave-nature. new binding energy of the nucleus is 4 t t Q S and they combine with each other in different ways to and fro mass within of 10 20 the NeatomTheory is 19and l =.h992440 with cause amu , massofof 56 emission 26 Fe is er second and their mini- ⇒ I = 1.84 × 10 −6 Hence A =no 1.84 μ A will from through current to form the diode groups ( BE and ) called = 100 + 10 = 110 new molecules. All matter has been MeV So, de-Broglie visible, infra-red wavelength 55.93492and amu ultra-violet and mv light. mass of 92238 …(2) U is 238.050783 amu. rgies13 ( in eV ). Given that ION hence it will not offer any resistance.found NuClEuS de-BROGLIE’S So, thetonet AND be resist- composed NuClEAR POSTuLATE of atoms STRuCTuRE or molecules, and the A = Z+N Illustrations If Ek is kinetic energy of Positively particle,charged then basic Nucleusknowledgewas of discovered atoms and their by Rutherford. constitution gives No electronsSOluTION are present inside matter the nucleus. figure has 23ance between A and B will be According used in the circuit shown in theILLuSTRATION IlluSTRATION to de-Broglie 5 a material particleThe nucleus in motion Elaborative ⎧ and p 2 ⎫its us of valuable information about the behaviour ofprotons matter. voltage drop of 0.5 V at all current and a ( 10 function )( 10 ) ( 10 ) must anhave atom Find consists a wave likeofcharacter the binding two types energy andofofparticles, the56 Fe. Atomic 26 wavelength massIf of Xp is= the2mE Bindingchemicalenergy symbol for an element of nucleus ⎨∵ XEKA =then is given⎬ by the The surface of a metal of work ϕ0)(is10illumi- nucleus equation K is represented as simple Z X or as theory A ZX ⎩ Z A 2 m⎭ power rating of 200 mW. What should be RAB = + 20andΩ neutrons = associated 56 withtogether Fe is it55is.9349 given called u byand as Nucleons. that of 1 H is 1. 00783 u. Mass emitted per second nated by light whose electric field 10 + 10 component 10 + 10 varies A protonEMPIRICAL has a positiveMODEL/ charge equal to +e = 1.6 × ⇒ h THOMSON l= …(3) 2mE(KBE )Ne = ⎡helps ( A - Z ) mthe students of the resistor R , connected in series with Area ) for obtaining 0.53maximum current? with time as E = E0 [ 1 + ccos os ( ω t ) ] si sinn ( ω 0 t ). Calculate10-19lC=ofand THOMSON hneutron = 1. a PLUM mass equal 1.00867 to m PUDDING u. = 16726231 × 10-27 kg. p MODEL …(1) ATOMIC MASS uNIT ⎣ (u OR amu) ( n + Zmp - M Z X ⎦ c A )⎤ 2 the maximum kinetic energy of photoelectrons emit-It was discovered by Goldstein. p h photon ) 100 × ILLUSTRATION 15 to understand ( ) SOluTION S OluluTION TION R ted from the surface. Two ammeters A1 and A2 are connected J.J. Thomson A neutron gave across has the, noficharge rst ideai.e. regarding is a neutral structure parti- Atomic and ⇒ (nuclear BE )Ne =masses( Δm ) c2are ⎡ generally 10 mn + 10expressed =Electron mp - M 10 ( 20 ) Ne ⎤ c 2 in terms of atomic massthe illustrations 56 -27 = 26 ⎣ ⎦ 0.53 a diode and resistor respectively of as andThe cleatom. shown The its number model mass in the is m ofknown isn protons = 1.6749286 in him after × Feas 2610 kgand Thomson’s. Thus,theanumber unit (a.m.u.). It is the nearest × I = 6.25 × 1011 s −1 SOLuTION atom model. ofisneutrons According is ( Ato Z ) a=proton. - this, 56 - 26positive entire = 30. charge where,Thomson’s integer value atom model of mass represented in a.m.u. (atomic model supporting the of 9) Figure. neutron slightly heavier than 100 Thomson’s The given electric field component is is distributed However, 01_Modern Physics_Part The 1.indd 1 for binding uniformly problems energy inofthe (unless Feform 56 and isuntil of specifi a sphere. ed), mass unit). atom satisfied the requirements Δm = ( 10 ( 1.008665 ) + 10 ( 1.007825 ) ) - 19 11/9/2019.992440 9:56:32 AM of photoelectrons 2isV E = E0 si n ( ω 0 t ) + E0 ssin in ( ω 0 t ) co A(2 ) coss ω t 20 Ω Negatively we take charged electrons 26are arranged within = ( here the atom and the demands ) 2 ⎛ Mass 1 =model, theory of 12 ofhydrogen Camu atom at Please electro-magnetic ⎞ note theory. sin this sphere BE lying 26 × 1and.00783 u-+2730 there. The× 1model.00867isu popu- - 55.9349 uAccordingc 1 amu ⇒to=this Δm 0.17246 can give rise to AQ3 mp ≈ mn = 1.6726231 × 10 kg 12 ⎜⎝line. that theory ⎟ and is 5V larly known as plum-pudding model. Every electron a single spectral rest in ground Experimentally, state ⎠ hydrogen E0 ⇒ number BE = (of u ) in c 2 the nucleus is called ⇒ ( BE )Ne = ( 0.17246 )( 9315 ) = 160.64 MeV rgy is, diode (or circuit) is given by⇒ E = E 0 sin si n ( ω 0 t ) + 2 ⎡ ⎣ sin s in ( ω 0 + ω ) t + sin si n ( ω 0 − ω ) t ⎤ The ⎦ total 0.52878 protons found ⇒ 1