Theoretical Physics

Theoretical Physics Third Edition......Page 1
Half-Title......Page 2
Title-Page......Page 4
Copyright Page......Page 5
Preface To The Second Edition......Page 6
Preface ToThe Third Edition......Page 7
Contents......Page 8
Introduction......Page 26
Part 1: Mathematical Introduction......Page 30
1. The Concept of a Vector......Page 32
2. Addition and Subtraction of Vectors; Multiplication of a Vector by a Scalar......Page 33
3. The Scalar Product of two Vectors......Page 35
4. The Vector Product of two Vectors. The Directed Plane Area as a Vector......Page 37
5. Multiple Products......Page 40
6. Differentiation of a Vector with respect to a Scalar; Application to the Theory of Space Curves......Page 42
7. Space Derivatives of a Scalar Quantity......Page 45
8. The Concept of Divergence and Gauss’s Theorem......Page 48
9. The Curl of a Vector, and Stokes’s Theorem......Page 50
10. The Operator ∇......Page 55
11. Calculation of the Gradient in a Vector Field; Fundamental Principles of Tensor Analysis......Page 57
12. Calculation of more complicated Vector Differential Expressions with the help of the Nabla Operator......Page 62
13. Differential Vector Operations in Curvilinear Orthogonal Co-ordinates......Page 65
14. Degeneration of the Vector Differential Operations at Surfaces of Discontinuity in the Field......Page 68
15. Fundamentals of the Matrix Calculus......Page 71
1. Simple Harmonic Vibrations......Page 75
2. Representation of more complicated Periodic Phenomena by Series of Harmonic Terms. Fourier Series. The Fourier Integral......Page 79
3. Modulated Vibrations and Beats......Page 82
4. Combination of Vibrations along different Axes. Lissajous' Figures......Page 83
5. The Propagation of Periodic Disturbances in the form of Waves......Page 86
6. Combination of several Waves having the same Direction of Propagation; Linearly and Elliptically Polarized Waves; Group Velocity......Page 89
7. Combination of Waves having the same Frequency but different Directions of Propagation. Standing Waves......Page 92
1. Conformal Mapping of one Plane on another......Page 94
2. The Cauchy-Riemann Conditions and the Differential Equation of Laplace......Page 96
3. Line Integrals in the Gauss Plane; the Cauchy Integral Theorem......Page 97
2. Derivation of the Euler-Lagrange Differential Equation......Page 100
Part 2: Mechanics......Page 104
1. The Fundamental Concepts of Kinematics......Page 106
2. Newton’s Second Law of Motion......Page 108
3. Time Integral and Path Integral of the Force. Work and Energy......Page 110
4. Conservative Forces; Potential......Page 112
5. Central Forces; the Law of Areas......Page 113
6. Gravitational Forces; Planetary Motion......Page 115
7. Quasi-elastic Forces and Harmonic Vibrations......Page 118
8. Harmonic Vibrations with Frictional Resistance......Page 119
9. Forced Vibrations; Resonance......Page 121
10. Non-harmonic Vibrations; Sudden Changes of Amplitude......Page 125
11. Mechanics of a Constrained Particle. The Simple Pendulum......Page 128
1. Theorem concerning the Motion of the Centre of Mass......Page 133
2. Angular Momentum of a System of Particles......Page 135
3. Total Energy of a System of Particles......Page 136
4. The Principle of Virtual Displacements, D'Alembert’s Principle and the Lagrangian Equations of the First Kind......Page 138
5. The Lagrangian Equations of the Second Kind for Arbitrary Coordinates (Generalized Co-ordinates)......Page 143
6. Generalized Momentum Co-ordinates. Hamilton’s Equations......Page 146
7. Hamilton’s Principle......Page 148
8. Canonical Transformations......Page 149
9. Cyclic Variables. The Hamilton-Jacobi Differential Equation......Page 151
10. Periodic and Multiply Periodic Systems. Angle Variables; the Angle Variables of the Keplerian Motion......Page 153
1. Selected Topics in the Kinematics of Rigid Bodies......Page 163
2. General Statics and Dynamics of Rigid Bodies. Equivalence of Systems of Forces acting upon Rigid Bodies......Page 168
3. Rotation of a Rigid Body about a Fixed Axis. Moment of Inertia and its Calculation......Page 171
4. Motion of a Rigid Body about a Fixed Point. Elements of the Theory of the Top......Page 176
1. The Geometry of Small Displacements......Page 186
2. State of Stress of a Body under Strain......Page 192
3. The Conditions of Equilibrium of an Elastic Body......Page 194
4. Relations between the Strain Tensor and the Stress Tensor......Page 196
5. Energy of Elastically Deformed Bodies; Elastic Potential......Page 200
6. Elementary Treatment of the Bending of a Cantilever Beam......Page 201
7. Waves in Unbounded Elastic Media (Seismic Waves.) Longitudinal Waves in Bars......Page 205
8. Transverse Vibration of Stretched Strings and Membranes......Page 208
1. Equilibrium of Fluid Bodies (Hydrostatics)......Page 216
2. The Fundamental Hydrodynamical Equations......Page 219
3. Irrotational Flow......Page 222
4. General Theorems concerning Vortex and Circulation......Page 228
5. Plane Circulatory Motion......Page 232
6. Undulatory Propagation of Disturbances in Fluids (Sound Waves)......Page 237
7. Hydrodynamics of Viscous Fluids......Page 239
8. Surface Tension of Liquids......Page 247
1. Space and Time in Newtonian Mechanics......Page 253
2. Inertial Frames. The Galilean Transformation......Page 254
3. Accelerated Frames of Reference. Free Fall on the Rotating Earth......Page 256
4. Moving Frames of Reference in Acoustics. The Doppler Effect......Page 260
5. Moving Frames of Reference in Optics. The Michelson-Morley Experiment......Page 262
6. The Relativistic Conception of Space and Time. The Lorentz Transformation......Page 265
7. Immediate Consequences of the Lorentz Transformation......Page 269
8. Geometric Representation of the Lorentz Transformation. The Fourdimensional World. Calculation with World-vectors......Page 274
9. Newton’s Second Law in the Theory of Relativity. The Variability of Mass and the Inertia of Energy......Page 279
10. Fundamental Concepts of the Generalized Theory of Relativity......Page 285
Part 3: Field Theory of Electromagnetic and Optical Phenomena......Page 288
1. Definitions......Page 290
2. Electrical Charge (Quantity of Electricity) as the Source of Flux......Page 291
3. The Electrostatic Potential......Page 294
4. Simple Examples of the Electrostatic Field in a Vacuum (or in Air)......Page 296
1. Formal Introduction of the Concepts “ Dielectric Displacement ” and “ Free Charge ”. Boundary Conditions at the Surface of Separation of Two Dielectrics......Page 304
2. Polarization of Dielectrics......Page 306
3. Simple Examples of the Electrostatic Field in Dielectrics......Page 309
1. Potential Energy of Systems of Charges in a given Field......Page 312
2. Total Energy of the Electrostatic Field.......Page 313
3. Forces and Equilibrium in the Electrostatic Field. Theory of the Manometer Method for the Electrical Susceptibility of a Liquid......Page 317
1. Ohm’s Law......Page 320
2. Generation of Heat in a steady Electric Field......Page 322
1. Comparison of Electrostatic and Magnetostatic Fields......Page 324
2. Calculation of the Magnetostatic Field accompanying a given Distribution of Electric Currents in a Vacuum......Page 327
3. Calculation of the Magnetic Field accompanying Electric Currents when Ferromagnetic Materials are present......Page 332
4. Ponderomotive Forces on Conductors in a Magnetic Field......Page 335
1. The Law of Induction. Maxwell’s Equations......Page 338
2. Self and Mutual Induction......Page 340
3. Steady Alternating-current Circuits......Page 343
4. Non-stationary States (Transient Phenomena) in Alternating-current Circuits......Page 346
5. Resistance and Inductance of Wires for Alternating Currents. Skin Effect......Page 351
1. The Electrical Analogue of the Law of Induction......Page 354
2. The Wave Equation for the Propagation of Fields in Dielectrics......Page 355
3. The Poynting Vector of Energy Flow.......Page 358
4. Propagation of Electromagnetic Waves in Conduction Media......Page 360
5. Hertz’s Solution of the Field Equations. The Hertzian Oscillator......Page 362
1. Unified Rigorous Derivation of the Field Equations and of the Boundary Conditions......Page 370
2. Waves in the Boundary Layer. The Ground Wave......Page 372
3. Consequence of the Boundary Conditions for Insulating Media. The Optical Laws of Reflection and Refraction......Page 376
4. Polarization and Intensity Relationships for Reflection and Refraction. Fresnel’s Formulae......Page 377
5. Total Reflection......Page 381
6. Absorbing Media. The Optics of Metals......Page 384
1. The Field Equations for Anisotropic Bodies......Page 387
2. Plane Electromagnetic Waves in Anisotropic Media......Page 390
3. Normal Surface and Wave Surface. The Optical Axes......Page 395
4. Refraction of Plane Waves at the Surface of an Anisotropic Medium......Page 398
1. The General Diffraction Problem and Attempts to solve it. Kirchoff’s Formula......Page 403
2. Reciprocal Theorems of the Theory of Diffraction. Classification of Diffraction Phenomena......Page 406
3. Fraunhofer Diffraction by a Slit and by One-, Two- and Three-dimensional Gratings......Page 409
4. Fresnel Diffraction Phenomena at a Slit and at a Circular Aperture. Zone Plates......Page 417
1. The Fundamentals of Geometrical Optics. Laws of Fermat and of Malus......Page 422
2. The Properties of Collinear Projection......Page 424
3. The Practical Problem of Image Formation. Abbe’s Sine Law. General Path of an Elementary Pencil from a Point Source......Page 430
4. The Resolving Power of Optical Systems......Page 436
5. The Fundamentals of Interference Optics. Interference Fringes......Page 437
Part 4: The Theory of Electricity. II. The Atomistic Nature of Electrical Phenomena......Page 442
1. The Fundamental Phenomena of Electrolytic Conduction and their Interpretation......Page 444
2. Dependence of Electrolytic Conductivity upon Concentration. The Theory of Debye-Hückel and Onsager......Page 448
1. Direct Determination of the Elementary Electrical Charge by the Millikan Oil Drop Method......Page 452
2. The Nature of the Cathode Ray Particles. The Electron......Page 453
3. Survey of the Possible Methods of Generating Carriers of Electricity in Gases......Page 458
4. The Separately-sustained Electrical Discharge. Spark Discharge......Page 460
5. Self-maintaining Discharge; Glow and Arc Discharge......Page 464
6. The Origin of Cathode, Canal, and Positive Rays. The Mass Spectrograph......Page 467
1. Electrons as Carriers of Current in Metals......Page 471
2. Derivation of Ohm’s Law for Metals......Page 473
3. Conduction of Heat in Metals; the Law of Wiedemann and Franz......Page 474
4. Objections to the Theory developed above. The Electron Theory of Pauli and Sommerfeld......Page 476
1. The Origin of Electrical and Magnetic Polarization......Page 477
2. Theory of Dielectric Polarization, Optical Index of Refraction and Dispersion......Page 478
3. Parelectric Susceptibility......Page 482
4. Paramagnetic, Ferromagnetic and Antiferromagnetic Susceptibility......Page 485
5. Magnetism Induced by Revolving Electrons. The Magnetomechanical Parallelism. Theory of Diamagnetic Susceptibility......Page 490
1. The Fundamental Equations......Page 493
2. Steady Fields......Page 495
3. Optical Behaviour of Superconductors......Page 496
1. Electromagnetic Induction in Moving Bodies from the Standpoint of the Electron Theory......Page 498
2. Magnetic Effects of Moving Charges......Page 501
3. Propagation of Electromagnetic Waves in Moving Media......Page 504
4. Relativistic-invariant Form of the Electromagnetic Equations......Page 507
Part 5: The Theory of Heat. Phenomenological Part......Page 512
1. Definition of Temperature, Quantity of Heat, Thermal Capacity and Specific Heat......Page 514
2. The Differential Equation of Heat Conduction; Initial and Boundary Conditions......Page 516
3. A Simple Example of the Integration of the Equation of Heat Conduction: Penetration of the Daily and Yearly Temperature Variations into the Interior of the Earth......Page 517
1. Definition of the Thermodynamic Variables and the Relationships between them......Page 519
2. The Equation of State of an Ideal Gas......Page 521
3. The Equation of State of a Real Gas......Page 524
1. Formulation of the First Law......Page 528
2. Specific Heat at Constant Volume and at Constant Pressure. The Energy Function of a Gas......Page 530
3. Adiabatic Change......Page 533
4. Application of the First Law to Thermochemistry......Page 537
1. The Carnot Cycle and the Ideal Heat Engine......Page 541
2. Formulation and Interpretation of the Second Law......Page 545
3. Conditions of Equilibrium for Systems under Various Conditions. Thermodynamic Potentials......Page 551
4. Connexion between the Internal Energy and the Equation of State......Page 554
5. Electrocaloric and Magnetocaloric Phenomena......Page 555
1. Gibbs' Phase Rule: a General Theorem on the Maximum Number of Possible Phases......Page 558
2. The Vapour Pressure Curve and the Melting-point Curve......Page 561
3. Chemical Equilibrium in a Mixture of Ideal Gases. The Law of Mass Action......Page 566
4. Chemical Equilibrium in a System consisting of Dilute Solutions and Ideal Gases......Page 570
5. Thermodynamic Equilibrium of Dilute Solutions of Strong Electrolytes......Page 574
1. Free Energy as a Measure of Chemical Affinity; Determination of this Quantity for the Galvanic Cell......Page 580
2. Formulation of the Nernst Heat Theorem......Page 582
3. Consequences for the Specific Heats and Temperature Coefficients. The Chemical Constant of a Gas......Page 584
4. Unattainability of the Absolute Zero......Page 586
Part 6: The Theory of Heat. Statistical Part......Page 588
1. Bernoulli’s Formula; Boyle’s Law......Page 590
2. Number of Collisions and Mean Free Path for Real Gases......Page 592
3. Viscosity and Heat Conduction in Gases. Determination of Avogadro’s Number and of the Size of the Molecule......Page 595
4. Derivation of some Properties of Crystals on the Molecular Theory......Page 600
1. Entropy and Probability......Page 606
2. Calculation of the most Probable Distribution of Density in an Ideal Gas......Page 607
3. Representation of the Distribution of Position and Velocity by means of Phase Space. Liouville’s Theorem......Page 611
4. The Maxwell-Boltzmann Energy Distribution......Page 613
5. Applications of the Maxwell-Boltzmann Energy Distribution......Page 616
6. The Law of Equipartition and its Application to the Specific Heats......Page 622
7. Fluctuations. Transition from Microscopic to Macroscopic Motion......Page 625
1. The Quantum Condition for the Subdivision of the Phase Space of an Oscillator......Page 630
2. Temperature Variation of the Vibrational Contribution to the Specific Heats......Page 632
3. The Debye Theory of the Specific Heats of Solids......Page 635
1. Analogy between a Gas and Radiation filling an Enclosure. Radiation Pressure......Page 640
2. The Connexion between Energy Density and Surface Brightness......Page 643
3. The Connexion between Emission and Absorption in Thermodynamic Equilibrium. Kirchhoff’s Law......Page 646
4. Planck’s Law of Radiation......Page 648
1. Preliminary Remarks concerning the Duality of Waves and Corpuscles......Page 652
2. The Bose-Einstein Statistics......Page 653
3. The Fermi-Dirac Statistics and its Application to Electrical Conductivity......Page 658
4. Entropy Constant and Chemical Constant of an Ideal Monatomic Gas......Page 663
Part 7: The Structure of Atoms and Molecules And the Theory of Spectra......Page 666
1. Investigation of the Structure of the Atom; the Scattering of x......Page 668
2. Conclusions from the Experiments on the Scattering of α-Particles......Page 672
3. The Bohr Model of the Hydrogen Atom......Page 674
4. Consideration of the Motion of the Nucleus. The Spectrum of Hydrogen and the Spectrum of Ionized Helium......Page 678
5. X-ray Spectra. Moseley’s Law......Page 680
6. The Correspondence Principle......Page 682
7. The Atomic Spectra of the Alkalis, the Alkaline Earths and Similar Systems......Page 684
8. Perturbation of Electron Orbits by External Forces. Stark Effect and Zeeman Effect......Page 689
9. Difficulties arising in the Atomic Theory of Magnetism. Explanation in Terms of the Spinning Electron......Page 694
10. The Theory of Multiplets and of their Zeeman Effects. Quantum Theory of Paramagnetic Susceptibility......Page 696
11. The Structure of the Periodic System of the Elements. Pauli’s Principle......Page 701
12. Elements of the Theory of Band Spectra......Page 707
1. The Wave and Particle Aspects of Light. Compton Effect and Raman Effect......Page 714
2. The Inexactness of Atomic Observations......Page 717
3. Matter Waves......Page 720
4. The Proper Values of the Wave Equation......Page 724
5. The Hydrogen Atom......Page 725
6. Rotation Spectra of Diatomic Molecules......Page 729
7. The Physical Meaning of the Ψ Function. Intensity Relations for Spectral Lines......Page 730
1. The Theory of Perturbations in Wave Mechanics......Page 734
2. The Chemical Bond. Formation of the Hydrogen Molecule......Page 739
3. The Theory of Dispersion and of the Raman Effect in Wave Mechanics......Page 743
4. The Surmounting of Potential Barriers in the Wave Mechanics......Page 748
5. Energy Bands of Electrons in Metals......Page 752
6. The Role of Lattice Defects in Dielectric Crystals......Page 762
1. General Remarks on the Physics of the Atomic Nucleus......Page 766
2. The Hyperfine Structure of Spectral Lines as a Connecting Link between the Physics of the Outer Shells and that of the Nucleus......Page 770
3. Radioactivity......Page 772
4. Artificial Transformation and Excitation of the Nucleus......Page 774
5. The Neutron......Page 781
6. The Positron......Page 783
7. Artificial Radioactivity......Page 787
8. The Role of Protons and Neutrons in Nuclear Structure......Page 788
9. The Liquid Drop Model of the Nucleus......Page 794
10. The Fission of Uranium......Page 798
11. Liquid Drop Treated as a Fermi Gas......Page 800
12. Independent Particle and Shell Models of the Nucleus; Nuclear Moments......Page 803
13. Beta Transformations and the Neutrino Hypothesis......Page 808
14. Mesons......Page 809
15. Brief Survey of our Knowledge concerning Cosmic Rays......Page 811
Part 8: Selected Topics From Several Fields of Technical Importance......Page 814
1. Condition for Absence of Distortion and its Relation to the Sine Condition......Page 816
2. Focal Length of an Electron Lens......Page 818
1. The Phenomenon of Piezoelectricity......Page 823
2. Application of Piezoelectricity to the Stabilization of Oscillating Circuits......Page 825
1. Characteristic Curve of a Thermionic Tube......Page 830
2. Oscillations of a Plasma......Page 832
1. Behaviour of High-polymer Chain Molecules......Page 834
2. A Model of Muscular Action......Page 836
1. Generalization of x......Page 838
2. Bessel Functions of the First Kind......Page 841
3. Bessel Functions of the Second and Third Kinds. Asymptotic Values of Bessel Functions......Page 847
4. Spherical Harmonics......Page 850
Solutions of The Exercises......Page 856
Tables......Page 896
References for Further Study......Page 899
Index......Page 906
Back Cover......Page 917