Latchman PhysicsGRE Solutions

7/30/2019 Latchman PhysicsGRE Solutions

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The Physics GRE Solution Guide

GR8677 Test

http://groups.yahoo.com/group/physicsgre_v2

November 3, 2009

Author:

David S. Latchman


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David S. Latchman 2009


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Preface

This solution guide initially started out on the Yahoo Groups web site and was pretty

successful at the time. Unfortunately, the group was lost and with it, much of the thehard work that was put into it. This is my attempt to recreate the solution guide andmake it more widely avaialble to everyone. If you see any errors, think certain thingscould be expressed more clearly, or would like to make suggestions, please feel free todo so.

David Latchman

Document Changes

05-11-2009 1. Added diagrams to GR0177 test questions 1-25

2. Revised solutions to GR0177 questions 1-25

04-15-2009 First Version


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Contents

Preface i

Classical Mechanics xv

0.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

0.2 Newtons Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi

0.3 Work & Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

0.4 Oscillatory Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii

0.5 Rotational Motion about a Fixed Axis . . . . . . . . . . . . . . . . . . . . xxii

0.6 Dynamics of Systems of Particles . . . . . . . . . . . . . . . . . . . . . . . xxiv

0.7 Central Forces and Celestial Mechanics . . . . . . . . . . . . . . . . . . . xxiv

0.8 Three Dimensional Particle Dynamics . . . . . . . . . . . . . . . . . . . . xxvi

0.9 Fluid Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvi

0.10 Non-inertial Reference Frames . . . . . . . . . . . . . . . . . . . . . . . . xxvii

0.11 Hamiltonian and Lagrangian Formalism . . . . . . . . . . . . . . . . . . . xxvii

Electromagnetism xxix

0.12 Electrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxix

0.13 Currents and DC Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiv

0.14 Magnetic Fields in Free Space . . . . . . . . . . . . . . . . . . . . . . . . . xxxiv

0.15 Lorentz Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiv

0.16 Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiv

0.17 Maxwells Equations and their Applications . . . . . . . . . . . . . . . . . xxxiv

0.18 Electromagnetic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiv


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iv Contents0.19 AC Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiv

0.20 Magnetic and Electric Fields in Matter . . . . . . . . . . . . . . . . . . . . xxxiv

0.21 Capacitance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxv

0.22 Energy in a Capacitor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxv

0.23 Energy in an Electric Field . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxv

0.24 Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxv

0.25 Current Destiny . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxv

0.26 Current Density of Moving Charges . . . . . . . . . . . . . . . . . . . . . xxxv

0.27 Resistance and Ohms Law . . . . . . . . . . . . . . . . . . . . . . . . . . xxxv

0.28 Resistivity and Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . xxxvi

0.29 Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxvi

0.30 Kirchoffs Loop Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxvi

0.31 Kirchoffs Junction Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxvi

0.32 RC Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxvi

0.33 Maxwells Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxvi

0.34 Speed of Propagation of a Light Wave . . . . . . . . . . . . . . . . . . . . xxxvii

0.35 Relationship between E and B Fields . . . . . . . . . . . . . . . . . . . . . xxxvii

0.36 Energy Density of an EM wave . . . . . . . . . . . . . . . . . . . . . . . . xxxvii i

0.37 Poyntings Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxvii i

Optics & Wave Phonomena xxxix

0.38 Wave Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxix

0.39 Superposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxix

0.40 Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxix

0.41 Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxix0.42 Geometrical Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxix

0.43 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxix

0.44 Doppler Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xl

0.45 Snells Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xl

Thermodynamics & Statistical Mechanics xli

0.46 Laws of Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . xli

0.47 Thermodynamic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . xli



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Contents v0.48 Equations of State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xli

0.49 Ideal Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xli

0.50 Kinetic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xli

0.51 Ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xli

0.52 Statistical Concepts and Calculation of Thermodynamic Properties . . . xlii

0.53 Thermal Expansion & Heat Transfer . . . . . . . . . . . . . . . . . . . . . xlii

0.54 Heat Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xlii

0.55 Specific Heat Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xlii

0.56 Heat and Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xlii

0.57 First Law of Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . xlii

0.58 Work done by Ideal Gas at Constant Temperature . . . . . . . . . . . . . xli i i

0.59 Heat Conduction Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . xliii

0.60 Ideal Gas Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xliv

0.61 Stefan-Boltzmanns FormulaStefan-Boltzmanns Equation . . . . . . . . xliv

0.62 RMS Speed of an Ideal Gas . . . . . . . . . . . . . . . . . . . . . . . . . . xliv

0.63 Translational Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . xliv

0.64 Internal Energy of a Monatomic gas . . . . . . . . . . . . . . . . . . . . . xliv

0.65 Molar Specific Heat at Constant Volume . . . . . . . . . . . . . . . . . . . xlv

0.66 Molar Specific Heat at Constant Pressure . . . . . . . . . . . . . . . . . . xlv

0.67 Equipartition of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . xlv

0.68 Adiabatic Expansion of an Ideal Gas . . . . . . . . . . . . . . . . . . . . . xlvi i

0.69 Second Law of Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . xlvi i

Quantum Mechanics xlix

0.70 Fundamental Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xlix0.71 Schrodinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xlix

0.72 Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . liv

0.73 Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . liv

0.74 Wave Funtion Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . liv

0.75 Elementary Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . liv

Atomic Physics lv

0.76 Properties of Electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lv

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vi Contents0.77 Bohr Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lv

0.78 Energy Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lvi

0.79 Atomic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lvi

0.80 Atomic Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lvi

0.81 Selection Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lvii

0.82 Black Body Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lvii

0.83 X-Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lviii

0.84 Atoms in Electric and Magnetic Fields . . . . . . . . . . . . . . . . . . . . lix

Special Relativity lxiii

0.85 Introductory Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lxiii0.86 Time Dilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lxiii

0.87 Length Contraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lxiii

0.88 Simultaneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lxiii

0.89 Energy and Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . lxiv

0.90 Four-Vectors and Lorentz Transformation . . . . . . . . . . . . . . . . . . lxv

0.91 Velocity Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lxvi

0.92 Relativistic Doppler Formula . . . . . . . . . . . . . . . . . . . . . . . . . lxvi0.93 Lorentz Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . lxvi

0.94 Space-Time Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lxvii

Laboratory Methods lxix

0.95 Data and Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . lxix

0.96 Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lxxi

0.97 Radiation Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lxxi0.98 Counting Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lxxi

0.99 Interaction of Charged Particles with Matter . . . . . . . . . . . . . . . . lxxi i

0.100Lasers and Optical Interferometers . . . . . . . . . . . . . . . . . . . . . . lxxi i

0.101Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lxxii

0.102Fundamental Applications of Probability and Statistics . . . . . . . . . . lxxii

GR8677 Exam Solutions lxxiii

0.103Motion of Rock under Drag Force . . . . . . . . . . . . . . . . . . . . . . . lxxi i i



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Contents vii0.104Satellite Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lxxiv

0.105Speed of Light in a Dielectric Medium . . . . . . . . . . . . . . . . . . . . lxxiv

0.106Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lxxiv

0.107Inelastic Collision and Putty Spheres . . . . . . . . . . . . . . . . . . . . . lxxv

0.108Motion of a Particle along a Track . . . . . . . . . . . . . . . . . . . . . . . lxxvi

0.109Resolving Force Components . . . . . . . . . . . . . . . . . . . . . . . . . lxxvi

0.110Nail being driven into a block of wood . . . . . . . . . . . . . . . . . . . . lxxvii

0.111Current Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lxxvii

0.112Charge inside an Isolated Sphere . . . . . . . . . . . . . . . . . . . . . . . lxxvii i

0.113Vector Identities and Maxwells Laws . . . . . . . . . . . . . . . . . . . . lxxix

0.114Doppler Equation (Non-Relativistic) . . . . . . . . . . . . . . . . . . . . . lxxix

0.115Vibrating Interference Pattern . . . . . . . . . . . . . . . . . . . . . . . . . lxxix

0.116Specific Heat at Constant Pressure and Volume . . . . . . . . . . . . . . . lxxix

0.117Helium atoms in a box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lxxx

0.118The Muon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lxxxi

0.119Radioactive Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lxxxi

0.120Schrodingers Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . lxxxi i

0.121Energy Levels of Bohrs Hydrogen Atom . . . . . . . . . . . . . . . . . . lxxxi i

0.122Relativistic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lxxxi i i

0.123Space-Time Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lxxxi i i

0.124Lorentz Transformation of the EM field . . . . . . . . . . . . . . . . . . . lxxxiv

0.125Conductivity of a Metal and Semi-Conductor . . . . . . . . . . . . . . . . lxxxiv

0.126Charging a Battery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lxxxv

0.127Lorentz Force on a Charged Particle . . . . . . . . . . . . . . . . . . . . . lxxxv

0.128K-Series X-Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lxxxv

0.129Electrons and Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lxxxvi

0.130Normalizing a wavefunction . . . . . . . . . . . . . . . . . . . . . . . . . lxxxvii

0.131Right Hand Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lxxxvii i

0.132Electron Configuration of a Potassium atom . . . . . . . . . . . . . . . . . lxxxviii

0.133Photoelectric Effect I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lxxxix

0.134Photoelectric Effect II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lxxxix

0.135Photoelectric Effect III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . lxxxix



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viii Contents0.136Potential Energy of a Body . . . . . . . . . . . . . . . . . . . . . . . . . . . lxxxix

0.137Hamiltonian of a Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xc

0.138Principle of Least Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . xc

0.139Tension in a Conical Pendulum . . . . . . . . . . . . . . . . . . . . . . . . xc

0.140Diode OR-gate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xci

0.141Gain of an Amplifier vs. Angular Frequency . . . . . . . . . . . . . . . . xci

0.142Counting Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xci

0.143Binding Energy per Nucleon . . . . . . . . . . . . . . . . . . . . . . . . . . xcii

0.144Scattering Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . xcii

0.145Coupled Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xcii

0.146Collision with a Rod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xciv

0.147Compton Wavelength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xciv

0.148Stefan-Boltzmanns Equation . . . . . . . . . . . . . . . . . . . . . . . . . xciv

0.149Franck-Hertz Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . xcv

0.150Selection Rules for Electronic Transitions . . . . . . . . . . . . . . . . . . xcv

0.151The Hamilton Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . xcv

0.152Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xcvi

0.153Debye and Einstein Theories to Specific Heat . . . . . . . . . . . . . . . . xcvi i

0.154Potential inside a Hollow Cube . . . . . . . . . . . . . . . . . . . . . . . . xcvi i

0.155EM Radiation from Oscillating Charges . . . . . . . . . . . . . . . . . . . xcvi i i

0.156Polarization Charge Density . . . . . . . . . . . . . . . . . . . . . . . . . . xcvi i i

0.157Kinetic Energy of Electrons in Metals . . . . . . . . . . . . . . . . . . . . . xcvi i i

0.158Expectation or Mean Value . . . . . . . . . . . . . . . . . . . . . . . . . . . xcix

0.159Eigenfuction of Wavefunction . . . . . . . . . . . . . . . . . . . . . . . . . xcix

0.160Holograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c

0.161Group Velocity of a Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . ci

0.162Potential Energy and Simple Harmonic Motion . . . . . . . . . . . . . . . ci

0.163Rocket Equation I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cii

0.164Rocket Equation II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cii

0.165Surface Charge Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cii

0.166Maximum Power Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . ciii

0.167Magnetic Field far away from a Current carrying Loop . . . . . . . . . . ciii



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Contents ix0.168Maxwells Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . civ

0.169Partition Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cv

0.170Particle moving at Light Speed . . . . . . . . . . . . . . . . . . . . . . . . cv

0.171Car and Garage I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cv

0.172Car and Garage II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cvi

0.173Car and Garage III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cvi

0.174Refrective Index of Rock Salt and X-rays . . . . . . . . . . . . . . . . . . . cvi

0.175Thin Flim Non-Reflective Coatings . . . . . . . . . . . . . . . . . . . . . . cvii

0.176Law of Malus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cvii

0.177Geosynchronous Satellite Orbit . . . . . . . . . . . . . . . . . . . . . . . . cvi i i

0.178Hoop Rolling down and Inclined Plane . . . . . . . . . . . . . . . . . . . cvi i i

0.179Simple Harmonic Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . cix

0.180Total Energy between Two Charges . . . . . . . . . . . . . . . . . . . . . . cx

0.181Maxwells Equations and Magnetic Monopoles . . . . . . . . . . . . . . . cx

0.182Gauss Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cxi

0.183Biot-Savart Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cxii

0.184Zeeman Effect and the emission spectrum of atomic gases . . . . . . . . cxii

0.185Spectral Lines in High Density and Low Density Gases . . . . . . . . . . cxiii

0.186Term Symbols & Spectroscopic Notation . . . . . . . . . . . . . . . . . . . cxi i i

0.187Photon Interaction Cross Sections for Pb . . . . . . . . . . . . . . . . . . . cxiv

0.188The Ice Pail Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . cxiv

0.189Equipartition of Energy and Diatomic Molecules . . . . . . . . . . . . . . cxiv

0.190Fermion and Boson Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . cxv

0.191Wavefunction of Two Identical Particles . . . . . . . . . . . . . . . . . . . cxv

0.192Energy Eigenstates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cxvi

0.193Braggs Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cxvii

0.194Selection Rules for Electronic Transitions . . . . . . . . . . . . . . . . . . cxvi i

0.195Moving Belt Sander on a Rough Plane . . . . . . . . . . . . . . . . . . . . cxvi i i

0.196RL Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cxviii

0.197Carnot Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cxx

0.198First Order Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . cxxi i

0.199Colliding Discs and the Conservation of Angular Momentum . . . . . . cxxii



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x Contents0.200Electrical Potential of a Long Thin Rod . . . . . . . . . . . . . . . . . . . . cxxi i i

0.201Ground State of a Positronium Atom . . . . . . . . . . . . . . . . . . . . . cxxiv

0.202The Pinhole Camera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cxxiv

Constants & Important Equations cxxvii

.1 Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cxxvii

.2 Vector Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cxxvii

.3 Commutators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cxxvii i

.4 Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cxxix



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List of Tables

0.67.1Table of Molar Specific Heats . . . . . . . . . . . . . . . . . . . . . . . . . xlvi

0.140.1Truth Table for OR-gate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xci

0.189.1Specific Heat, cv for a diatomic molecule . . . . . . . . . . . . . . . . . . . cxiv

.1.1 Something . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cxxvii


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xii List of Tables



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List of Figures


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xiv List of Figures



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Classical Mechanics

0.1 Kinematics

0.1.1 Linear Motion

Average Velocity

v =x

t=

x2 x1t2 t1 (0.1.1)

Instantaneous Velocity

v = limt0

xt

=dxdt

= v(t) (0.1.2)

Kinematic Equations of Motion

The basic kinematic equations of motion under constant acceleration, a, are

v = v0 + at (0.1.3)

v2 = v2

0+ 2a (x

x0) (0.1.4)

x x0 = v0t + 12at2 (0.1.5)

x x0 = 12 (v + v0) t (0.1.6)

0.1.2 Circular Motion

In the case ofUniform Circular Motion, for a particle to move in a circular path, a

radial acceleration must be applied. This acceleration is known as the CentripetalAcceleration


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xvi Classical MechanicsCentripetal Acceleration

a =v2

r(0.1.7)

Angular Velocity

=v

r(0.1.8)

We can write eq. (0.1.7) in terms of

a = 2r (0.1.9)

Rotational Equations of Motion

The equations of motion under a constant angular acceleration, , are

= 0 + t (0.1.10)

= + 0

2t (0.1.11)

= 0t +12t2 (0.1.12)

2 = 20 + 2 (0.1.13)

0.2 Newtons Laws

0.2.1 Newtons Laws of Motion

First Law A body continues in its state of rest or of uniform motion unless acted uponby an external unbalanced force.

Second Law Thenetforceonabodyisproportionaltoitsrateofchangeofmomentum.

F = dpdt

= ma (0.2.1)

Third Law When a particle A exerts a force on another particle B, B simultaneouslyexerts a force on A with the same magnitude in the opposite direction.

FAB = FBA (0.2.2)

0.2.2 Momentum

p = mv (0.2.3)



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Work & Energy xvii0.2.3 Impulse

p = J =

Fdt = Favgdt (0.2.4)

0.3 Work & Energy

0.3.1 Kinetic Energy

K 12

mv2 (0.3.1)

0.3.2 The Work-Energy Theorem

The net Work done is given byWnet = Kf Ki (0.3.2)

0.3.3 Work done under a constant Force

The work done by a force can be expressed as

W = Fx (0.3.3)

In three dimensions, this becomes

W = F r = Fr cos (0.3.4)

For a non-constant force, we have

W =

xf

xi

F(x)dx (0.3.5)

0.3.4 Potential Energy

The Potential Energy is

F(x) = dU(x)dx

(0.3.6)

for conservative forces, the potential energy is

U(x) = U0

x

x0

F(x)dx (0.3.7)



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xviii Classical Mechanics0.3.5 Hookes Law

F = kx (0.3.8)

where kis the spring constant.

0.3.6 Potential Energy of a Spring

U(x) =12

kx2 (0.3.9)

0.4 Oscillatory Motion

0.4.1 Equation for Simple Harmonic Motion

x(t) = A sin (t + ) (0.4.1)

where the Amplitude, A, measures the displacement from equilibrium, the phase, , isthe angle by which the motion is shifted from equilibrium at t = 0.

0.4.2 Period of Simple Harmonic Motion

T =2

(0.4.2)

0.4.3 Total Energy of an Oscillating System

Given thatx = A sin (t + ) (0.4.3)

and that the Total Energy of a System is

E = KE + PE (0.4.4)

The Kinetic Energy is

KE =12

mv2

=12

mdx

dt

=12mA22 cos2 (t + ) (0.4.5)



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Oscillatory Motion xixThe Potential Energy is

U=12

kx2

= 12kA2 sin2 (t + ) (0.4.6)

Adding eq. (0.4.5) and eq. (0.4.6) gives

E =12

kA2 (0.4.7)

0.4.4 Damped Harmonic Motion

Fd =

bv =

b

dx

dt

(0.4.8)

where b is the damping coefficient. The equation of motion for a damped oscillatingsystem becomes

kx bdxdt

= md2x

dt2(0.4.9)

Solving eq. (0.4.9) govesx = Aet sin (t + ) (0.4.10)

We find that

=b

2m(0.4.11)

=

k

m b

2

4m2

=

20

b2

4m2

=

20 2 (0.4.12)

0.4.5 Small Oscillations

The Energy of a system is

E = K+ V(x) =12

mv(x)2 + V(x) (0.4.13)

We can solve for v(x),

v(x) =

2m

(E V(x)) (0.4.14)where E V(x) Let the particle move in the potential valley, x1 x x2, the potentialcan be approximated by the Taylor Expansion

V(x) = V(xe) + (x xe)dV(x)

dx

x=xe+

12(x xe)2

d2V(x)dx2

x=xe

+ (0.4.15)



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xx Classical MechanicsAt the points of inflection, the derivative dV/dx is zero and d2V/dx2 is positive. Thismeans that the potential energy for small oscillations becomes

V(x)

V(xe) +

1

2k(x xe)2

(0.4.16)where

k

d2V(x)dx2

x=xe

0 (0.4.17)

As V(xe) is constant, it has no consequences to physical motion and can be dropped.We see that eq. (0.4.16) is that of simple harmonic motion.

0.4.6 Coupled Harmonic Oscillators

Consider the case of a simple pendulum of length, , and the mass of the bob is m1.For small displacements, the equation of motion is

+ 0 = 0 (0.4.18)

We can express this in cartesian coordinates, x and y, where

x = cos (0.4.19)y = sin (0.4.20)

eq. (0.4.18) becomesy + 0y = 0 (0.4.21)

This is the equivalent to the mass-spring system where the spring constant is

k= m20 =mg

(0.4.22)

This allows us to to create an equivalent three spring system to our coupled pendulumsystem. The equations of motion can be derived from the Lagrangian, where

L = T V=

12

m y21 +12

m y22 12

ky21 +12y2 y1

2+

12

ky22

=12

m

y12 + y22 1

2

ky21 + y

22

+

y2 y1

2 (0.4.23)We can find the equations of motion of our system

d

dt

L

yn

=

L

yn(0.4.24)

1Add figure with coupled pendulum-spring system



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Oscillatory Motion xxiThe equations of motion are

m y1 = ky1 + y2 y1

(0.4.25)

m y2 = ky2 + y2 y1 (0.4.26)We assume solutions for the equations of motion to be of the form

y1 = cos(t + 1) y2 = B cos(t + 2)y1 = y1 y2 = y2 (0.4.27)

Substituting the values for y1 and y2 into the equations of motion yieldsk+ m2

y1 y2 = 0 (0.4.28)

y1 +k+ m2

y2 = 0 (0.4.29)

We can get solutions from solving the determinant of the matrix

k+ m2 k+ m2

= 0 (0.4.30)Solving the determinant gives

m22 2m2 (k+ ) + k2 + 2k = 0 (0.4.31)

This yields

2 =

k

m=

g

k+ 2

m=

g

+

2m

(0.4.32)

We can now determine exactly how the masses move with each mode by substituting2 into the equations of motion. Where

2 =k

mWe see that

k+ m2 = (0.4.33)Substituting this into the equation of motion yields

y1 = y2 (0.4.34)

We see that the masses move in phase with each other. You will also noticethe absense of the spring constant term, , for the connecting spring. As themasses are moving in step, the spring isnt stretching or compressing and henceits absence in our result.

2 =k+

mWe see that

k+ m2 = (0.4.35)Substituting this into the equation of motion yields

y1 = y2 (0.4.36)Here the masses move out of phase with each other. In this case we see thepresence of the spring constant, , which is expected as the spring playes a role.It is being stretched and compressed as our masses oscillate.



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xxii Classical Mechanics0.4.7 Doppler Effect

The Doppler Effect is the shift in frequency and wavelength of waves that results froma source moving with respect to the medium, a receiver moving with respect to themedium or a moving medium.

Moving Source If a source is moving towards an observer, then in one period, 0, itmoves a distance ofvs0 = vs/f0. The wavelength is decreased by

= vsf0

v vsf0

(0.4.37)

The frequency change is

f =v

= f0

v

v

vs

(0.4.38)

Moving Observer As the observer moves, he will measure the same wavelength, , asif at rest but will see the wave crests pass by more quickly. The observer measuresa modified wave speed.

v = v + |vr| (0.4.39)The modified frequency becomes

f =v

= f0

1 +

vrv

(0.4.40)

Moving Source and Moving Observer We can combine the above two equations

=v vs

f0(0.4.41)

v = v vr (0.4.42)To give a modified frequency of

f =v

=

v vrv vs

f0 (0.4.43)

0.5 Rotational Motion about a Fixed Axis

0.5.1 Moment of Inertia

I =

R2dm (0.5.1)

0.5.2 Rotational Kinetic Energy

K=12I2 (0.5.2)



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Rotational Motion about a Fixed Axis xxiii0.5.3 Parallel Axis Theorem

I = Icm +Md2 (0.5.3)

0.5.4 Torque

= r F (0.5.4) = I (0.5.5)

where is the angular acceleration.

0.5.5 Angular Momentum

L = I (0.5.6)

we can find the Torque

=dL

dt(0.5.7)

0.5.6 Kinetic Energy in Rolling

With respect to the point of contact, the motion of the wheel is a rotation about thepoint of contact. Thus

K= Krot =12

Icontact2 (0.5.8)

Icontact can be found from the Parallel Axis Theorem.

Icontact = Icm +MR2 (0.5.9)

Substitute eq. (0.5.8) and we have

K=12

Icm +MR

22

=12

Icm2 +

12

mv2 (0.5.10)

The kinetic energy of an object rolling without slipping is the sum of hte kinetic energy

of rotation about its center of mass and the kinetic energy of the linear motion of theobject.



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xxiv Classical Mechanics0.6 Dynamics of Systems of Particles

0.6.1 Center of Mass of a System of Particles

Position Vector of a System of Particles

R =m1r1 + m2r2 + m3r3 + + mNrN

M(0.6.1)

Velocity Vector of a System of Particles

V =dR

dt=

m1v1 + m2v2 + m3v3 + + mNvNM

(0.6.2)

Acceleration Vector of a System of Particles

A =dV

dt

=m1a1 + m2a2 + m3a3 + + mNaN

M(0.6.3)

0.7 Central Forces and Celestial Mechanics

0.7.1 Newtons Law of Universal Gravitation

F =

GMm

r2

r (0.7.1)

0.7.2 Potential Energy of a Gravitational Force

U(r) = GMmr

(0.7.2)

0.7.3 Escape Speed and Orbits

The energy of an orbiting body is

E = T+ U

=12mv2

GMm

r (0.7.3)



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Central Forces and Celestial Mechanics xxvThe escape speed becomes

E =12

mv2esc GMm

RE= 0 (0.7.4)

Solving for vesc we findvesc =

2GM

Re(0.7.5)

0.7.4 Keplers Laws

First Law The orbit of every planet is an ellipse with the sun at a focus.

Second Law A line joining a planet and the sun sweeps out equal areas during equal

intervals of time.Third Law The square of the orbital period of a planet is directly proportional to the

cube of the semi-major axis of its orbit.

T2

R3= C (0.7.6)

where C is a constant whose value is the same for all planets.

0.7.5 Types of Orbits

The Energy of an Orbiting Body is defined in eq. (0.7.3), we can classify orbits by theireccentricities.

Circular Orbit A circular orbit occurs when there is an eccentricity of 0 and the orbitalenergy is less than 0. Thus

12

v2 GMr

= E < 0 (0.7.7)

The Orbital Velocity is

v =

GM

r(0.7.8)

Elliptic Orbit An elliptic orbit occurs when the eccentricity is between 0 and 1 but thespecific energy is negative, so the object remains bound.

v =

GM

2r

1a

(0.7.9)

where a is the semi-major axis



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xxvi Classical MechanicsParabolic Orbit A Parabolic Orbit occurs when the eccentricity is equal to 1 and the

orbital velocity is the escape velocity. This orbit is not bounded. Thus

1

2v2

GM

r = E = 0 (0.7.10)

The Orbital Velocity is

v = vesc =

2GM

r(0.7.11)

Hyperbolic Orbit In the Hyperbolic Orbit, the eccentricity is greater than 1 with anorbital velocity in excess of the escape velocity. This orbit is also not bounded.

v= GM

a (0.7.12)

0.7.6 Derivation of Vis-viva Equation

The total energy of a satellite is

E =12

mv2 GMmr

(0.7.13)

For an elliptical or circular orbit, the specific energy is

E = GMm2a

(0.7.14)

Equating we get

v2 = GM2

r 1

a

(0.7.15)

0.8 Three Dimensional Particle Dynamics

0.9 Fluid Dynamics

When an object is fully or partially immersed, the buoyant force is equal to the weightof fluid displaced.

0.9.1 Equation of Continuity

1v1A1 = 2v2A2 (0.9.1)



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Non-inertial Reference Frames xxvii0.9.2 Bernoullis Equation

P +12v2 + gh = a constant (0.9.2)

0.10 Non-inertial Reference Frames

0.11 Hamiltonian and Lagrangian Formalism

0.11.1 Lagranges Function (L)

L = T

V (0.11.1)

where T is the Kinetic Energy and V is the Potential Energy in terms of GeneralizedCoordinates.

0.11.2 Equations of Motion(Euler-Lagrange Equation)

L

q=

d

dt

L

q

(0.11.2)

0.11.3 Hamiltonian

H= T+ V

= pq L(q, q) (0.11.3)

where

H

p= q (0.11.4)

Hq

= Lx

= p (0.11.5)



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xxviii Classical Mechanics



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Electromagnetism

0.12 Electrostatics

0.12.1 Coulombs Law

The force between two charged particles, q1 and q2 is defined by Coulombs Law.

F12 =1

40

q1q2

r212

r12 (0.12.1)

where 0 is the permitivitty of free space, where

0 = 8.85 1012C2N.m2 (0.12.2)

0.12.2 Electric Field of a point charge

The electric field is defined by mesuring the magnitide and direction of an electricforce, F, acting on a test charge, q0.

E Fq0

(0.12.3)

The Electric Field of a point charge, q is

E =1

40

q

r2r (0.12.4)

In the case of multiple point charges, qi, the electric field becomes

E(r) =1

40

ni=1

qi

r2i

ri (0.12.5)

Electric Fields and Continuous Charge Distributions

If a source is distributed continuously along a region of space, eq. (0.12.5) becomes

E(r) =1

40 1

r2rdq (0.12.6)


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xxx ElectromagnetismIf the charge was distributed along a line with linear charge density, ,

=dq

dx(0.12.7)

The Electric Field of a line charge becomes

E(r) =1

40

line

r2rdx (0.12.8)

In the case where the charge is distributed along a surface, the surface charge densityis,

=Q

A=

dq

dA(0.12.9)

The electric field along the surface becomes

E(r) =1

40

Surface

r2rdA (0.12.10)

In the case where the charge is distributed throughout a volume, V, the volume chargedensity is

=

Q

V =

dq

dV (0.12.11)The Electric Field is

E(r) =1

40

Volume

r2rdV (0.12.12)

0.12.3 Gauss Law

The electric field through a surface is

=

surface S

d =

surface S

E dA (0.12.13)

The electric flux through a closed surface encloses a net charge.

E dA = Q

0(0.12.14)

where Q is the charge enclosed by our surface.



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Electrostatics xxxi0.12.4 Equivalence of Coulombs Law and Gauss Law

The total flux through a sphere is

E dA = E(4r2) = q

0(0.12.15)

From the above, we see that the electric field is

E =q

40r2(0.12.16)

0.12.5 Electric Field due to a line of charge

Consider an infinite rod of constant charge density, . The flux through a Gaussiancylinder enclosing the line of charge is

=

top surface

E dA +

bottom surface

E dA +

side surface

E dA (0.12.17)

At the top and bottom surfaces, the electric field is perpendicular to the area vector, sofor the top and bottom surfaces,

E dA = 0 (0.12.18)At the side, the electric field is parallel to the area vector, thus

E dA = EdA (0.12.19)Thus the flux becomes,

=

side sirface

E dA = E

dA (0.12.20)

The area in this case is the surface area of the side of the cylinder, 2rh.

= 2rhE (0.12.21)

Applying Gauss Law, we see that = q/0. The electric field becomes

E =

20r(0.12.22)

0.12.6 Electric Field in a Solid Non-Conducting Sphere

Within our non-conducting sphere or radius, R, we will assume that the total charge,Q is evenly distributed throughout the spheres volume. So the charge density of oursphere is

=Q

V =Q

43R

3 (0.12.23)



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xxxii ElectromagnetismThe Electric Field due to a charge Q is

E =Q

40r2(0.12.24)

As the charge is evenly distributed throughout the spheres volume we can say thatthe charge density is

dq = dV (0.12.25)

where dV = 4r2dr. We can use this to determine the field inside the sphere bysumming the effect of infinitesimally thin spherical shells

E =

E0

dE =

r0

dq

4r2

=

0

r0

dr

=Qr

430R

3(0.12.26)

0.12.7 Electric Potential Energy

U(r) =1

40qq0r (0.12.27)

0.12.8 Electric Potential of a Point Charge

The electrical potential is the potential energy per unit charge that is associated with astatic electrical field. It can be expressed thus

U(r) = qV(r) (0.12.28)

And we can see that

V(r) =1

40

q

r(0.12.29)

A more proper definition that includes the electric field, E would be

V(r) = C

E d (0.12.30)

where C is any path, starting at a chosen point of zero potential to our desired point.

The difference between two potentials can be expressed such

V(b) V(a) = b

E d+a

E d

= b

a E d (0.12.31)



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Electrostatics xxxiiiThis can be further expressed

V(b) V(a) =b

a

(V) d (0.12.32)

And we can show thatE = V (0.12.33)

0.12.9 Electric Potential due to a line charge along axis

Let us consider a rod of length, , with linear charge density, . The Electrical Potentialdue to a continuous distribution is

V =

dV = 140 dq

r (0.12.34)

The charge density isdq = dx (0.12.35)

Substitutingthisintotheaboveequation,wegettheelectricalpotentialatsomedistancex along the rods axis, with the origin at the start of the rod.

dV =1

40

dq

x

= 140dx

x (0.12.36)

This becomesV =

40ln

x2x1

(0.12.37)

where x1 and x2 are the distances from O, the end of the rod.

Now consider that we are some distance, y, from the axis of the rod of length, . Weagain look at eq. (0.12.34), where r is the distance of the point P from the rods axis.

V =1

40 dq

r

=1

40

0

dxx2 + y2

12

=

40ln

x +

x2 + y2

12

0

=

40ln

+

2 + y2

12 ln y

=

40 ln+

2 + y212

d

(0.12.38)



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xxxiv Electromagnetism0.13 Currents and DC Circuits

2

0.14 Magnetic Fields in Free Space

3

0.15 Lorentz Force

4

0.16 Induction

5

0.17 Maxwells Equations and their Applications

6

0.18 Electromagnetic Waves

7

0.19 AC Circuits

8

0.20 Magnetic and Electric Fields in Matter

9



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Capacitance xxxv0.21 Capacitance

Q = CV (0.21.1)

0.22 Energy in a Capacitor

U=Q2

2C

=CV2

2

=

QV

2 (0.22.1)

0.23 Energy in an Electric Field

u Uvolume

=0E2

2(0.23.1)

0.24 Current

I dQdt

(0.24.1)

0.25 Current Destiny

I =

A

J dA (0.25.1)

0.26 Current Density of Moving Charges

J =I

A= neqvd (0.26.1)

0.27 Resistance and Ohms Law

R V

I (0.27.1)



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xxxvi Electromagnetism0.28 Resistivity and Conductivity

R = L

A(0.28.1)

E = J (0.28.2)

J = E (0.28.3)

0.29 Power

P = VI (0.29.1)

0.30 Kirchoffs Loop Rules

Write Here

0.31 Kirchoffs Junction Rule

Write Here

0.32 RC Circuits

E

IR

Q

C

= 0 (0.32.1)

0.33 Maxwells Equations

0.33.1 Integral Form

Gauss Law for Electric Fields

closed surface

E

dA =Q

0(0.33.1)



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Speed of Propagation of a Light Wave xxxviiGauss Law for Magnetic Fields

closed surface

B dA = 0 (0.33.2)

Amperes Law B ds = 0I+ 00 d

dt

surface

E dA (0.33.3)

Faradays Law E ds = d

dt

surface

B dA (0.33.4)

0.33.2 Differential Form

Gauss Law for Electric Fields

E = 0

(0.33.5)

Gauss Law for Magnetism B = 0 (0.33.6)

Amperes Law

B = 0J + 00Et

(0.33.7)

Faradays Law

E = Bt

(0.33.8)

0.34 Speed of Propagation of a Light Wave

c =100

(0.34.1)

In a material with dielectric constant, ,

c =

c

n(0.34.2)

where n is the refractive index.

0.35 Relationship between E and B Fields

E = cB (0.35.1)

E B = 0 (0.35.2)



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xxxviii Electromagnetism0.36 Energy Density of an EM wave

u =1

2B

2

0+ 0E

2 (0.36.1)

0.37 Poyntings Vector

S =10

E B (0.37.1)



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Optics & Wave Phonomena

0.38 Wave Properties

1

0.39 Superposition

2

0.40 Interference

3

0.41 Diffraction

4

0.42 Geometrical Optics

5

0.43 Polarization

6


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xl Optics & Wave Phonomena0.44 Doppler Effect

7

0.45 Snells Law

0.45.1 Snells Law

n1 sin1 = n2 sin2 (0.45.1)

0.45.2 Critical Angle and Snells Law

The critical angle, c, for the boundary seperating two optical media is the smallestangle of incidence, in the medium of greater index, for which light is totally refelected.

From eq. (0.45.1), 1 = 90 and 2 = c and n2 > n1.

n1 sin90 = n2sinc

sinc =n1n2

(0.45.2)



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Thermodynamics & Statistical Mechanics

0.46 Laws of Thermodynamics

1

0.47 Thermodynamic Processes

2

0.48 Equations of State

3

0.49 Ideal Gases

4

0.50 Kinetic Theory

5

0.51 Ensembles

6


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xlii Thermodynamics & Statistical Mechanics0.52 Statistical Concepts and Calculation of Thermody-

namic Properties

7

0.53 Thermal Expansion & Heat Transfer

8

0.54 Heat Capacity

Q = CTf Ti

(0.54.1)

where C is the Heat Capacity and Tf and Ti are the final and initial temperaturesrespectively.

0.55 Specific Heat Capacity

Q = cmTf ti

(0.55.1)

where c is the specific heat capacity and m is the mass.

0.56 Heat and Work

W =

VfVi

PdV (0.56.1)

0.57 First Law of Thermodynamics

dEint = dQ dW (0.57.1)

where dEint is the internal energy of the system, dQ is the Energy added to the systemand dWis the work done by the system.



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Work done by Ideal Gas at Constant Temperature xliii0.57.1 Special Cases to the First Law of Thermodynamics

Adiabatic Process During an adiabatic process, the system is insulated such that thereis no heat transfer between the system and its environment. Thus dQ = 0, so

Eint = W (0.57.2)

If work is done on the system, negative W, then there is an increase in its internalenergy. Conversely, if work is done by the system, positive W, there is a decreasein the internal energy of the system.

Constant Volume (Isochoric) Process If the volume is held constant, then the systemcan do no work, W = 0, thus

Eint = Q (0.57.3)

If heat is added to the system, the temperature increases. Conversely, if heat isremoved from the system the temperature decreases.

Closed Cycle In this situation, after certain interchanges of heat and work, the systemcomes back to its initial state. So Eint remains the same, thus

Q = W (0.57.4)

The work done by the system is equal to the heat or energy put into it.

Free Expansion In this process, no work is done on or by the system. Thus Q =W = 0,

Eint = 0 (0.57.5)

0.58 Work done by Ideal Gas at Constant Temperature

Starting with eq. (0.56.1), we substitute the Ideal gas Law, eq. (0.60.1), to get

W = nRT

VfVi

dV

V

= nRTln Vf

Vi(0.58.1)

0.59 Heat Conduction Equation

The rate of heat transferred, H, is given by

H =Q

t= kA

TH TCL

(0.59.1)

where kis the thermal conductivity.



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xliv Thermodynamics & Statistical Mechanics0.60 Ideal Gas Law

PV = nRT (0.60.1)

wheren = Number of molesP = PressureV = VolumeT = Temperature

and R is the Universal Gas Constant, such that

R 8.314 J/mol. K

We can rewrite the Ideal gas Law to sayPV = NkT (0.60.2)

where kis the Boltzmanns Constant, such that

k=R

NA 1.381 1023J/K

0.61 Stefan-Boltzmanns FormulaStefan-Boltzmanns Equa-

tionP(T) = T4 (0.61.1)

0.62 RMS Speed of an Ideal Gas

vrms =

3RTM

(0.62.1)

0.63 Translational Kinetic Energy

K=32

kT (0.63.1)

0.64 Internal Energy of a Monatomic gas

Eint =32nRT (0.64.1)



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Molar Specific Heat at Constant Volume xlv0.65 Molar Specific Heat at Constant Volume

Let us define, CV such that

Q = nCVT (0.65.1)

Substituting into the First Law of Thermodynamics, we have

Eint + W = nCVT (0.65.2)

At constant volume, W = 0, and we get

CV =

1n

EintT (0.65.3)

Substituting eq. (0.64.1), we get

CV =32

R = 12.5 J/mol.K (0.65.4)

0.66 Molar Specific Heat at Constant Pressure

Starting with

Q = nCpT (0.66.1)

and

Eint = Q W nCVT = nCpT+ nRT

CV = Cp R (0.66.2)

0.67 Equipartition of Energy

CV =

f

2

R = 4.16f J/mol.K (0.67.1)

where f is the number of degrees of freedom.



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xlvi Thermodynamics & Statistical Mechanics

DegreesofFreedom

PredictedMolarSp

ecificHeats

Mo

lecule

Tran

slational

Rotational

Vibrational

Total(f)

CV

CP=

CV+

R

Mo

natomic

3

0

0

3

32

R

52R

Dia

tomic

3

2

2

5

52

R

72R

Pol

yatomic(Linear)

3

3

3n

5

6

3R

4R

Pol

yatomic(Non-Linear)

3

3

3n

6

6

3R

4R

Table0.67.1:TableofMolarSpecificHeats



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Adiabatic Expansion of an Ideal Gas xlvii0.68 Adiabatic Expansion of an Ideal Gas

PV = a constant (0.68.1)

where = CPCV.We can also write

TV1 = a constant (0.68.2)

0.69 Second Law of Thermodynamics

Something.



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xlviii Thermodynamics & Statistical Mechanics



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Quantum Mechanics

0.70 Fundamental Concepts

1

0.71 Schr odinger Equation

Let us define to be = Aei(t

xv ) (0.71.1)

Simplifying in terms of Energy, E, and momentum, p, we get

= Aei(Etpx)

(0.71.2)

We obtain Schrodingers Equation from the Hamiltonian

H = T+ V (0.71.3)

To determine E and p,

2

x2= p

2

2 (0.71.4)

t=

iE

(0.71.5)

and

H =p2

2m+ V (0.71.6)

This becomesE = H (0.71.7)

E = i

tp2 = 2

2

x2

The Time Dependent Schrodingers Equation is

i

t=

2

2m2

x2+ V(x) (0.71.8)

The Time Independent Schrodingers Equation is

E = 2

2m2

x2 + V(x) (0.71.9)


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l Quantum Mechanics0.71.1 Infinite Square Wells

Let us consider a particle trapped in an infinite potential well of size a, such that

V(x) =

0 for 0 < x < a for |x| > a,so that a nonvanishing force acts only at a/2. An energy, E, is assigned to the systemsuch that the kinetic energy of the particle is E. Classically, any motion is forbiddenoutside of the well because the infinite value ofVexceeds any possible choice ofE.

Recalling the Schrodinger Time Independent Equation, eq. (0.71.9), we substitute V(x)and in the region (a/2, a/2), we get

2

2m

d2

dx2 = E (0.71.10)

This differential is of the formd2

dx2+ k2 = 0 (0.71.11)

where

k=

2mE2

(0.71.12)

We recognize that possible solutions will be of the form

cos kx and sin kx

As the particle is confined in the region 0 < x < a, we say

(x) =

A cos kx + B sin kx for 0 < x < a0 for |x| > a

We have known boundary conditions for our square well.

(0) = (a) = 0 (0.71.13)

It shows that

A cos0 + B sin0 = 0

A = 0 (0.71.14)

We are now left with

B sin ka = 0ka = 0;; 2; 3;

(0.71.15)

Whilemathematically,n can bezero, thatwould meanthere would be nowavefunction,so we ignore this result and say

kn = na

for n = 1, 2, 3,



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Schrodinger Equation liSubstituting this result into eq. (0.71.12) gives

kn =n

a=

2mEn

(0.71.16)

Solving for En gives

En =n222

2ma2(0.71.17)

We cna now solve for B by normalizing the functiona0

|B|2 sin2 kxdx = |A|2 a2

= 1

So |A|2 = 2a

(0.71.18)

So we can write the wave function asn(x) =

2a

sin

nx

a

(0.71.19)

0.71.2 Harmonic Oscillators

Classically, the harmonic oscillator has a potential energy of

V(x) =12

kx2 (0.71.20)

So the force experienced by this particle is

F = dVdx

= kx (0.71.21)

where kis the spring constant. The equation of motion can be summed us as

md2x

dt2= kx (0.71.22)

And the solution of this equation is

x(t) = A cos0t +

(0.71.23)

where the angular frequency, 0 is

0 =

k

m(0.71.24)

The Quantum Mechanical description on the harmonic oscillator is based on the eigen-function solutions of the time-independent Schrodingers equation. By taking V(x)from eq. (0.71.20) we substitute into eq. (0.71.9) to get

d2

dx2 = 2m2

k2x2 E = mk

2x2 2Ek



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lii Quantum MechanicsWith some manipulation, we get

mk

d2

dx2=

mk

x2 2E

m

k

This step allows us to to keep some of constants out of the way, thus giving us

2 =

mk

x2 (0.71.25)

and =2E

m

k=

2E0

(0.71.26)

This leads to the more compact

d2

d2

=2 (0.71.27)

where the eigenfunction will be a function of. assumes an eigenvalue anaglaousto E.

From eq. (0.71.25), we see that the maximum value can be determined to be

2max =

mk

A2 (0.71.28)

Using the classical connection between A and E, allows us to say

2max =

mk

2Ek

= (0.71.29)

From eq. (0.71.27), we see that in a quantum mechanical oscillator, there are non-vanishing solutions in the forbidden regions, unlike in our classical case.

A solution to eq. (0.71.27) is() = e

2/2 (0.71.30)

where

dd

= e2/2

andd

d2= 2exi

2/2 e2/2 =2 1

e

2/2

This gives is a special solution for where

0 = 1 (0.71.31)

Thus eq. (0.71.26) gives the energy eigenvalue to be

E0 = 020 = 02 (0.71.32)



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Schrodinger Equation liiiThe eigenfunction e2/2 corresponds to a normalized stationary-state wave function

0(x, t) =

mk

22

18

e

mk x2/2eiE0t/ (0.71.33)

This solution of eq. (0.71.27) produces the smallest possibel result of and E. Hence,0 and E0 represents the ground state of the oscillator. and the quantity 0/2 is thezero-point energy of the system.

0.71.3 Finite Square Well

For the Finite Square Well, we have a potential region where

V(x) = V0 for a x a0 for |x| > a

We have three regions

Region I: x < a In this region, The potential, V = 0, so Schrodingers Equation be-comes

2

2md2

dx2= E

d2

dx2= 2

where = 2mE

This gives us solutions that are

(x) = A exp(x) + B exp(x)As x , the exp(x) term goes to ; it blows up and is not a physicallyrealizable function. So we can drop it to get

(x) = Bex for x < a (0.71.34)

Region II:

a < x < a In this region, our potential is V(x) = V0. Substitutin this into

the Schrodingers Equation, eq. (0.71.9), gives

2

2md2

dx2 V0 = E

ord2

dx2= l2

where l

2m (E + V0)

(0.71.35)

We notice that E > V0, making l real and positive. Thus our general solutionbecomes

(x) = C sin(lx) + D cos(lx) for a < x < a (0.71.36)



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liv Quantum MechanicsRegion III: x > a Again this Region is similar to Region III, where the potential, V = 0.

This leaves us with the general solution

(x) = F exp(x) + G exp(x)

As x , the second term goes to infinity and we get

(x) = Fex for x > a (0.71.37)

This gives us

(x) =

Bex for x < aD cos(lx) for 0 < x < aFex for x > a

(0.71.38)

0.71.4 Hydrogenic Atoms

c

0.72 Spin

3

0.73 Angular Momentum

4

0.74 Wave Funtion Symmetry

5

0.75 Elementary Perturbation Theory

6



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Atomic Physics

0.76 Properties of Electrons

1

0.77 Bohr Model

To understand the Bohr Model of the Hydrogen atom, we will take advantage of ourknowlegde of the wavelike properties of matter. As we are building on a classicalmodel of the atom with a modern concept of matter, our derivation is considered to besemi-classical. In this model we have an electron of mass, me, and charge, e, orbitinga proton. The cetripetal force is equal to the Coulomb Force. Thus

140

e2r2

= mev2r

(0.77.1)

The Total Energy is the sum of the potential and kinetic energies, so

E = K+ U=p2

2me |f race240r (0.77.2)

We can further reduce this equation by subsituting the value of momentum, which wefind to be

p2

2me

=12

mev2 =

e2

80r

(0.77.3)

Substituting this into eq. (0.77.2), we get

E =e2

80r e

2

40r= e

2

80r(0.77.4)

At this point our classical description must end. An accelerated charged particle, likeone moving in circular motion, radiates energy. So our atome here will radiate energyand our electron will spiral into the nucleus and disappear. To solve this conundrum,Bohr made two assumptions.

1. The classical circular orbits are replaced by stationary states. These stationarystates take discreet values.


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lvi Atomic Physics2. The energy of these stationary states are determined by their angular momentum

which must take on quantized values of.

L = n (0.77.5)

We can find the angular momentum of a circular orbit.

L = m3vr (0.77.6)

From eq. (0.77.1) we find v and by substitution, we find L.

L = e

m3r

40(0.77.7)

Solving for r, gives

r =L2

mee2/40 (0.77.8)We apply the condition from eq. (0.77.5)

rn =n22

mee2/40= n2a0 (0.77.9)

where a0 is the Bohr radius.a0 = 0.53 1010 m (0.77.10)

Having discreet values for the allowed radii means that we will also have discreetvalues for energy. Replacing our value ofrn into eq. (0.77.4), we get

En = me2n2

e2

40

= 13.6

n2eV (0.77.11)

0.78 Energy Quantization

3

0.79 Atomic Structure

4

0.80 Atomic Spectra

0.80.1 Rydbergs Equation

1= RH

1n2

1n2

(0.80.1)



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Selection Rules lviiwhere RH is the Rydberg constant.

For the Balmer Series, n = 2, which determines the optical wavelengths. For n = 3,weget the infrared or Paschen series. The fundamental n = 1 series falls in the ultraviolet

region and is known as the Lyman series.

0.81 Selection Rules

6

0.82 Black Body Radiation

0.82.1 Plank Formula

u(f,T) =8

c3f3

eh f/kT 1 (0.82.1)

0.82.2 Stefan-Boltzmann Formula

P(T) = T4 (0.82.2)

0.82.3 Weins Displacement Law

maxT= 2.9 103 m.K (0.82.3)

0.82.4 Classical and Quantum Aspects of the Plank Equation

Rayleighs Equation

u(f,T) = 8f2

c3kT (0.82.4)

We can get this equation from Planks Equation, eq. (0.82.1). This equation is a classicalone and does not contain Planks constant in it. For this case we will look at thesituation where h f < kT. In this case, we make the approximation

ex 1 + x (0.82.5)

Thus the demonimator in eq. (0.82.1) becomes

eh f/kT 1 1 + h fkT

1 = h fkT

(0.82.6)



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lviii Atomic PhysicsThus eq. (0.82.1) takes the approximate form

u(f,T) 8hc3

f3kT

h f=

8f2

c3kT (0.82.7)

As we can see this equation is devoid of Planks constant and thus independent ofquantum effects.

Quantum

At large frequencies, where h f > kT, quantum effects become apparent. We canestimate that

eh f/kT 1 eh f/kT (0.82.8)

Thus eq. (0.82.1) becomesu(f,T) 8h

c3f3eh f/kT (0.82.9)

0.83 X-Rays

0.83.1 Bragg Condition

2d sin = m (0.83.1)

for constructive interference offparallel planes of a crystal with lattics spacing, d.

0.83.2 The Compton Effect

The Compton Effect deals with the scattering of monochromatic X-Rays by atomictargets and the observation that the wavelength of the scattered X-ray is greater thanthe incident radiation. The photon energy is given by

E= h =hc

(0.83.2)

The photon has an associated momentum

E = pc (0.83.3)

p = Ec

=h

c=

h

(0.83.4)

The Relativistic Energy for the electron is

E2 = p2c2 + m2e c4 (0.83.5)

where p p = P (0.83.6)



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Atoms in Electric and Magnetic Fields lixSquaring eq. (0.83.6) gives

p2 2p p +p2 = P2 (0.83.7)Recall that E= pc and E = cp, we have

c2p2 2c2p p + c2p2 = c2P2E

2 2E E cos + E2 = E2 m2e c4 (0.83.8)Conservation of Energy leads to

E+ mec2 = E + E (0.83.9)

Solving

E E = E mec2E

2

2E E + E = E2

2Emec2 + m2e c

4 (0.83.10)2E E 2E E cos = 2Emec2 2m2e c4 (0.83.11)

Solving leads to

= = hmec

(1 cos) (0.83.12)

where c = hmec is the Compton Wavelength.

c =h

mec= 2.427 1012m (0.83.13)

0.84 Atoms in Electric and Magnetic Fields

0.84.1 The Cyclotron Frequency

A test charge, q, with velocity v enters a uniform magnetic field, B. The force acting onthe charge will be perpendicular to v such that

FB = qv B (0.84.1)

or more simply FB = qvB. As this traces a circular path, from Newtons Second Law,we see that

FB =mv2

R= qvB (0.84.2)

Solving for R, we getR =

mv

qB(0.84.3)

We also see thatf =

qB

2m(0.84.4)

The frequency is depends on the charge, q, the magnetic field strength, B and the massof the charged particle, m.



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lx Atomic Physics0.84.2 Zeeman Effect

The Zeeman effect was the splitting of spectral lines in a static magnetic field. This issimilar to the Stark Effect which was the splitting in the presence in a magnetic field.

In the Zeeman experiment, a sodium flame was placed in a magnetic field and itsspectrum observed. In the presence of the field, a spectral line of frequency, 0 wassplit into three components, 0 , 0 and 0 + . A classical analysis of this effectallows for the identification of the basic parameters of the interacting system.

The application of a constant magnetic field, B, allows for a direction in space in whichthe electron motion can be referred. The motion of an electron can be attributed to asimple harmonic motion under a binding force kr, where the frequency is

0 =1

2

k

me(0.84.5)

The magnetic field subjects the electron to an additional Lorentz Force, ev B. Thisproduces two different values for the angular velocity.

v = 2r

The cetripetal force becomesmev

2

r= 422rme

Thus the certipetal force is

422rme

= 2reB + kr for clockwise motion

422rme = 2reB + kr for counterclockwise motionWe use eq. (0.84.5), to emiminate k, to get

2 eB2me

0 = 0 (Clockwise)

2 +eB

2me 0 = 0 (Counterclockwise)

As we have assumed a small Lorentz force, we can say that the linear terms in aresmall comapred to 0. Solving the above quadratic equations leads to

= 0 +eB

4mefor clockwise motion (0.84.6)

= 0 eB4me for counterclockwise motion (0.84.7)

We note that the frequency shift is of the form

=eB

4me(0.84.8)

If we view the source along the direction of B, we will observe the light to have two

polarizations, a closckwise circular polarization of0+ and a counterclosckwisecircular polarization of0 .



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Atoms in Electric and Magnetic Fields lxi0.84.3 Franck-Hertz Experiment

The Franck-Hertz experiment, performed in 1914 by J. Franck and G. L. Hertz, mea-sured the colisional excitation of atoms. Their experiement studied the current ofelectrons in a tub of mercury vapour which revealed an abrupt change in the currentat certain critical values of the applied voltage.2 They interpreted this observation asevidence of a threshold for inelastic scattering in the colissions of electrons in mer-cury atoms.The bahavior of the current was an indication that electrons could losea discreet amount of energy and excite mercury atoms in their passage through themercury vapour. These observations constituted a direct and decisive confirmation ofthe existence os quantized energy levels in atoms.

2Put drawing of Franck-Hertz Setup



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lxii Atomic Physics



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Special Relativity

0.85 Introductory Concepts

0.85.1 Postulates of Special Relativity

1. The laws of Physics are the same in all inertial frames.

2. The speed of light is the same in all inertial frames.

We can define =

11 u2

c2

(0.85.1)

0.86 Time Dilation

t = t (0.86.1)

where t is the time measured at rest relative to the observer, t is the time measuredin motion relative to the observer.

0.87 Length Contraction

L =L

(0.87.1)

where L is the length of an object observed at rest relative to the observer and L is thelength of the object moving at a speed u relative to the observer.

0.88 Simultaneity

4


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lxiv Special Relativity0.89 Energy and Momentum

0.89.1 Relativistic Momentum & Energy

In relativistic mechanics, to be conserved, momentum and energy are defined as

Relativistic Momentum

p = mv (0.89.1)

Relativistic Energy

E = mc2 (0.89.2)

0.89.2 Lorentz Transformations (Momentum & Energy)

px = px E

c

(0.89.3)

py = py (0.89.4)

pz = pz (0.89.5)E

c=

E

cpx

(0.89.6)

0.89.3 Relativistic Kinetic Energy

K= E mc2 (0.89.7)

= mc2

1

1 v2c2

1

(0.89.8)= mc2

1 (0.89.9)

0.89.4 Relativistic Dynamics (Collisions)

Px =

Px Ec

(0.89.10)

Py = Py (0.89.11)

Pz = Pz (0.89.12)E

c = E

c Px

(0.89.13)



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Four-Vectors and Lorentz Transformation lxv0.90 Four-Vectors and Lorentz Transformation

We can represent an event in S with the column matrix, s,

s =

xyzict

(0.90.1)

A different Lorents frame, S, corresponds to another set of space time axes so that

s =

x

y

z

ict

(0.90.2)

The Lorentz Transformation is related by the matrix

x

y

z

ict

=

0 0 i0 1 0 00 0 1 0

i 0 0

xyzict

(0.90.3)

We can express the equation in the form

s = Ls (0.90.4)

The matrix L contains all the information needed to relate position fourvectors forany given event as observed in the two Lorentz frames S and S. If we evaluate

sTs =

x y z ict

xyzict

= x2 + y2 +z2 c2t2 (0.90.5)

Similarly we can show that

sT

s = x2

+ y2

+z2

c2

t2

(0.90.6)We can take any collection of four physical quantities to be four vector provided thatthey transform to another Lorentz frame. Thus we have

b =

bxbybzibt

(0.90.7)

this can be transformed into a set of quantities of b in another frame S such that it

satisfies the transformation b = Lb (0.90.8)



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lxvi Special RelativityLooking at the momentum-Energy four vector, we have

p =

pxpy

pziE/c

(0.90.9)

Applying the same transformation rule, we have

p = Lp (0.90.10)

We can also get a Lorentz-invariation relation between momentum and energy suchthat

pTp = pTp (0.90.11)

The resulting equality gives

p2x +p2y +p

2z E

2c2

= p2x +p2y +p

2z E2c2 (0.90.12)

0.91 Velocity Addition

v =v u1 uv

c2(0.91.1)

0.92 Relativistic Doppler Formula

= 0

c + u

c u let r =

c uc + u

(0.92.1)

We have

receding = r0 red-shift (Source Receding) (0.92.2)

approaching =0r

blue-shift (Source Approaching) (0.92.3)

0.93 Lorentz Transformations

Given two reference frames S(x,y,z, t)and S(x,y,z, t), where the S-frame is movingin the x-direction, we have,

x = (x ut) x = (x ut) (0.93.1)y = y y = y (0.93.2)z = y y = y (0.93.3)

t = t

u

c2 x

t = t +

u

c2 x

(0.93.4)



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Space-Time Interval lxvii0.94 Space-Time Interval

(S)2 = (x)2 + y2

+ (z)2 c2 (t)2 (0.94.1)Space-Time Intervals may be categorized into three types depending on their separa-tion. They are

Time-like Interval

c2t2 > r2 (0.94.2)

S2 > 0 (0.94.3)

When two events are separated by a time-like interval, there is a cause-effectrelationship between the two events.

Light-like Interval

c2t2 = r2 (0.94.4)

S2 = 0 (0.94.5)

Space-like Intervals

c2t2 < r2 (0.94.6)S < 0 (0.94.7)



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lxviii Special Relativity



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Laboratory Methods

0.95 Data and Error Analysis

0.95.1 Addition and Subtraction

x = a + b c (0.95.1)The Error in x is

(x)2 = (a)2 + (b)2 + (c)2 (0.95.2)

0.95.2 Multiplication and Division

x = a bc

(0.95.3)

The error in x is x

x

2=

a

a

2+

b

b

2+

c

c

2(0.95.4)

0.95.3 Exponent - (No Error in b)

x = ab

(0.95.5)The Error in x is

x

x= b

a

a

(0.95.6)

0.95.4 Logarithms

Base e

x = ln a (0.95.7)


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lxx Laboratory MethodsWe find the error in x by taking the derivative on both sides, so

x =d ln a

da a

= 1a

a

=a

a(0.95.8)

Base 10

x = log10 a (0.95.9)

The Error in x can be derived as such

x =d(log a)

daa

=

ln aln10

daa

=1

ln10a

a

= 0.434a

a(0.95.10)

0.95.5 Antilogs

Base e

x = ea (0.95.11)

We take the natural log on both sides.

ln x = a ln e = a (0.95.12)

Applaying the same general method, we see

d ln xdx

x = a

xx

= a (0.95.13)

Base 10

x = 10a (0.95.14)



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Instrumentation lxxiWe follow the same general procedure as above to get

log x = a log10log x

dx x=a

1ln10

d ln adx

x = a

x

x= ln10a (0.95.15)

0.96 Instrumentation

2

0.97 Radiation Detection

3

0.98 Counting Statistics

Lets assume that for a particular experiment, we are making countung measurementsfor a radioactive source. In this experiment, we recored N counts in time T. Thecounting rate for this trial is R = N/T. This rate should be close to the average rate, R.The standard deviation or the uncertainty of our count is a simply called the

N rule.

So =

N (0.98.1)

Thus we can report our results as

Number of counts = N

N (0.98.2)

We can find the count rate by dividing by T, so

R =N

T N

T(0.98.3)

The fractional uncertainty of our count is NN

. We can relate this in terms of the countrate.

R

R=

NT

NT

=N

N

=

N

N

=1N (0.98.4)



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lxxii Laboratory MethodsWe see that our uncertainty decreases as we take more counts, as to be expected.

0.99 Interaction of Charged Particles with Matter5

0.100 Lasers and Optical Interferometers

6

0.101 Dimensional Analysis

Dimensional Analysis is used to understand physical situations involving a mis ofdifferent types of physical quantities. The dimensions of a physical quantity areassociated with combinations of mass, length, time, electric charge, and temperature,represented by symbols M, L, T, Q, and , respectively, each raised to rational powers.

0.102 Fundamental Applications of Probability and Statis-

tics

8



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GR8677 Exam Solutions

0.103 Motion of Rock under Drag Force

From the information provided we can come up with an equation of motion for the

rock.mx = mg kv (0.103.1)

If you have seen this type of equation, and solved it, youd know that the rocks speedwill asymtotically increase to some max speed. At that point the drag force and theforceduetogravitywillbethesame. Wecanbestanswerthisquestionthroughanalysisand elimination.

A Dividing eq. (0.103.1) by m gives

x = g k

m x (0.103.2)

We see that this only occurs when x = 0, which only happens at the top of theflight. So FALSE.

B At the top of the flight, v = 0. From eq. (0.103.2)

x = g (0.103.3)

we see that this is TRUE.

C Again from eq. (0.103.2) we see that the acceleration is dependent on whether therock is moving up or down. Ifx > 0 then x < g and ifx < 0 then x > g. So thisis also FALSE.

D If there was no drag (fictional) force, then energy would be conserved and the rockwill return at the speed it started with but there is a drag force so energy is lost.The speed the rock returns is v < v0. Hence FALSE.

E Again FALSE. Once the drag force and the gravitational force acting on the rock isbalanced the rock wont accelerate.

Answer: (B)


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lxxiv GR8677 Exam Solutions0.104 Satellite Orbits

The question states that the astronaut fires the rockets jets towards Earths center. We

infer that no extra energy is given to the system by this process. Section 0.7.5, showsthat the only other orbit where the specific energy is also negative is an elliptical one.

Answer: (A)

0.105 Speed of Light in a Dielectric Medium

Solutions to the Electromagnetic wave equation gives us the speed of light in terms ofthe electromagnetic permeability, 0 and permitivitty, 0.

c = 100 (0.105.1)where c is the speed of light. The speed through a dielectric medium becomes

v =10

=1

2.100

=c2.1

(0.105.2)

Answer: (D)

0.106 Wave Equation

We are given the equation

y = A sin

t

T x

(0.106.1)

We can analyze and eliminate from what we know about this equation

A The Amplitude, A in the equation is the displacement from equilibrium. So thischoice is incorrect.

B As the wave moves, we seek to keep the

tT

x

term constant. So as t increases, weexpect x to increase as well as there is a negative sign in front of it. This meansthat the wave moves in the positive x-direction. This choice is also incorrect.

C The phase of the wave is given by

tT

x, we can do some manipulation to show

2

t

T x

= 2f t kx = 0= t kx (0.106.2)



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Inelastic Collision and Putty Spheres lxxvOr rather

kx = t (0.106.3)

Differentiating eq. (0.106.3) gives us the phase speed, which is

v = T

(0.106.4)

This is also incorrect

E From eq. (0.106.4) the above we see that is the answer.

Answer: (E)

0.107 Inelastic Collision and Putty Spheres

We are told the two masses coalesce so we know that the collision is inelastic andhence, energy is not conserved. As mass A falls, it looses Potential Energy and gainsKinetic Energy.

Mgh0 =12

Mv20 (0.107.1)

Thusv20 = 2gh0 (0.107.2)

Upon collision, momentum is conserved, thus

Mv0 = (3M +M) v1= 4Mv1

v1 = v04 (0.107.3)

The fused putty mass rises, kinetic energy is converted to potential energy and we findour final height.

12

(4M) v21 = 4Mgh1 (0.107.4)

This becomesv21 = 2gh1 (0.107.5)

Substituting eq. (0.107.3), we get

v04

2= 2gh1 (0.107.6)

and substituting, eq. (0.107.2),2gh016

= 2gh1 (0.107.7)

Solving for h1,

h1 =h0

16

(0.107.8)

Answer: (A)



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lxxvi GR8677 Exam Solutions0.108 Motion of a Particle along a Track

As the particle moves from the top of the track and runs down the frictionless track,

its Gravitational Potential Energy is converted to Kinetic Energy. Lets assume that theparticle is at a height, y0 when x = 0. Since energy is conserved, we get3

mgy0 = mg(y0 y) + 12mv2

12

v2 = gy (0.108.1)

So we have a relationship between v and the particles position on the track.

The tangential ac

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