Understanding Physics Book

Preface to Second Edition xv

1 Understanding the physical universe 1

1.1 The programme of physics 1
1.2 The building blocks of matter 2
1.3 Matter in bulk 4
1.4 The fundamental interactions 5
1.5 Exploring the physical universe: the scientific method 5
1.6 The role of physics: its scope and applications 7

2 Using mathematical tools in physics 9

2.1 Applying the scientific method 9
2.2 The use of variables to represent displacement and time 9
2.3 Representation of data 10
2.4 The use of differentiation in analysis: velocity and acceleration in linear motion 12
2.5 The use of integration in analysis 16
2.6 Maximum and minimum values of physical variables: general linear motion 21
2.7 Angular motion: the radian 23
2.8 The role of mathematics in physics 25

3 The causes of motion: dynamics 31

3.1 The concept of force 31
3.2 The first law of dynamics (Newton’s first law) 32
3.3 The fundamental dynamical principle (Newton’s second law) 33
3.4 Systems of units: SI 36
3.5 Time dependent forces: oscillatory motion 38
3.6 Simple harmonic motion 40
3.7 Mechanical work and energy: power 44
3.8 Energy in simple harmonic motion 48
3.9 Dissipative forces: damped harmonic motion 50
3.10 Forced oscillations 54
3.11 Nonlinear dynamics: chaos 56

4 Motion in two and three dimensions 63

4.1 Vector physical quantities 63
4.2 Vector algebra 64
4.3 Velocity and acceleration vectors 67
4.4 Force as a vector quantity: vector form of the laws of dynamics 69
4.5 Constraint forces 70
4.6 Friction 72
4.7 Motion in a circle: centripetal force 74
4.8 Motion in a circle at constant speed 75
4.9 Tangential and radial components of acceleration 77
4.10 Hybrid motion: the simple pendulum 78
4.11 Angular quantities as vectors: the cross product 79

5 Force fields 87

5.1 Newton’s law of universal gravitation 87
5.2 Force fields 88
5.3 The concept of flux 89
5.4 Gauss’ law for gravitation 90
5.5 Motion in a constant uniform field: projectiles 94
5.6 Mechanical work and energy 96
5.7 Energy in a constant uniform field 102
5.8 Energy in an inverse square law field 103
5.9 Moment of a force: angular momentum 105
5.10 Planetary motion: circular orbits 107
5.11 Planetary motion: elliptical orbits and Kepler's laws 108

6 Many-body interactions 117

6.1 Newton’s third law 117
6.2 The principle of conservation of momentum 120
6.3 Mechanical energy of systems of particles 121
6.4 Particle decay 122
6.5 Particle collisions 123
6.6 The centre of mass of a system of particles 127
6.7 The two-body problem: reduced mass 128
6.8 Angular momentum of a system of particles 131
6.9 Conservation principles in physics 132

7 Rigid body dynamics 141

7.1 Rigid bodies 141
7.2 Rigid bodies in equilibrium: statics 142
7.3 Torque 143
7.4 Dynamics of rigid bodies 144
7.5 Measurement of torque: the torsion balance 145
7.6 Rotation of a rigid body about a fixed axis: moment of inertia 146
7.7 Calculation of moments of inertia: the parallel axis theorem 147
7.8 Conservation of angular momentum of rigid bodies 149
7.9 Conservation of mechanical energy in rigid body systems 149
7.10 Work done by a torque: torsional oscillations: rotational power 152
7.11 Gyroscopic motion 154
7.12 Summary: connection between rotational and translational motions 155

8 Relative motion 161

8.1 Applicability of Newton's laws of motion: inertial reference frames 161
8.2 The Galilean transformation 162
8.3 The CM (centre-of-mass) reference frame 165
8.4 Example of a noninertial frame: centrifugal force 170
8.5 Motion in a rotating frame: the Coriolis force 171
8.6 The Foucault pendulum 175
8.7 Practical criteria for inertial frames: the local view 176

9 Special relativity 183

9.1 The velocity of light 183
9.2 The principle of relativity 184
9.3 Consequences of the principle of relativity 184
9.4 The Lorentz transformation 187
9.5 The Fitzgerald-Lorentz contraction 190
9.6 Time dilation 191
9.7 Paradoxes in special relativity 192
9.8 Relativistic transformation of velocity 193
9.9 Momentum in relativistic mechanics 194
9.10 Four vectors: the energy-momentum 4-vector 196
9.11 Energy-momentum transformations: relativistic energy conservation 198
9.12 Relativistic energy: mass-energy equivalence 199
9.13 Units in relativistic mechanics 202
9.14 Mass-energy equivalence in practice 202
9.15 General relativity 203
9.16 Simultaneity: quantitative analysis of the twin paradox 204

10 Continuum mechanics: mechanical properties of materials 211

10.1 Dynamics of continuous media 211
10.2 Elastic properties of solids 212
10.3 Fluids at rest 215
10.4 Elastic properties of fluids 217
10.5 Pressure in gases 217
10.6 Archimedes’ principle 218
10.7 Fluid dynamics 220
10.8 Viscosity 223
10.9 Surface properties of liquids 224
10.10 Boyle’s law (Mariotte’s law) 226
10.11 A microscopic theory of gases 227
10.12 The mole 230
10.13 Interatomic forces: modifications to the kinetic theory of gases 230
10.14 Microscopic models of condensed matter systems 232

11 Thermal physics 239

11.1 Friction and heating 239
11.2 Temperature scales 240
11.3 Heat capacities of thermal systems 242
11.4 Comparison of specific heat capacities: calorimetry 243
11.5 Thermal conductivity 244
11.6 Convection 245
11.7 Thermal radiation 246
11.8 Thermal expansion 248
11.9 The first law of thermodynamics 249
11.10 Change of phase: latent heat 251
11.11 The equation of state of an ideal gas 252
11.12 Isothermal, isobaric and adiabatic processes: free expansion 252
11.13 The Carnot cycle 256
11.14 Entropy and the second law of thermodynamics 258
11.15 The Helmholtz and Gibbs functions 260
11.16 Microscopic interpretation of temperature 261
11.17 Polyatomic molecules: principle of equipartition of energy 263
11.18 Ideal gas in a gravitational field: the ‘law of atmospheres’ 265
11.19 Ensemble averages and distribution functions 266
11.20 The distribution of molecular velocities in an ideal gas 267
11.21 Distribution of molecular speeds, momenta and energies 269
11.22 Microscopic interpretation of temperature and heat capacity in solids 271

12 Wave motion 277

12.1 Characteristics of wave motion 277
12.2 Representation of a wave which is travelling in one dimension 279
12.3 Energy and power in a wave motion 281
12.4 Plane and spherical waves 282
12.5 Huygens’ principle: the laws of reflection and refraction 282
12.6 Interference between waves 284
12.7 Interference of waves passing through openings: diffraction 288
12.8 Standing waves 290
12.9 The Doppler effect 293
12.10 The wave equation 294
12.11 Waves along a string 295
12.12 Waves in elastic media: longitudinal waves in a solid rod 296
12.13 Waves in elastic media: sound waves in gases 297
12.14 Superposition of two waves of slightly different frequencies: wave and group velocities 298
12.15 Other wave forms: Fourier analysis 300

13 Introduction to quantum mechanics 307

13.1 Physics at the beginning of the twentieth century 307
13.2 The blackbody radiation problem 308
13.3 The photoelectric effect 311
13.4 The X-ray continuum 313
13.5 The Compton effect: the photon model 314
13.6 The de Broglie hypothesis: electron waves 316
13.7 Interpretation of wave-particle duality 318
13.8 The Heisenberg uncertainty principle 319
13.9 The wavefunction: expectation values 322
13.10 The Schr€odinger (wave mechanical) method 323
13.11 The free particle 324
13.12 The time-independent Shr€odinger equation: eigenfunctions and eigenvalues 327
13.13 The infinite square potential well 328
13.14 The potential step 331
13.15 Other potential wells and barriers 336

14 Electric currents 347

14.1 Electric currents 347
14.2 Force between currents 349
14.3 The unit of electric current 350
14.4 Heating effect revisited: electrical resistance 351
14.5 Strength of a power supply: emf 353
14.6 Resistance of a circuit 354
14.7 Potential difference 354
14.8 Effect of internal resistance 356
14.9 Comparison of emfs: the potentiometer 358
14.10 Multiloop circuits 359
14.11 Kirchhoff’s rules 360
14.12 Comparison of resistances: the Wheatstone bridge 361
14.13 Power supplies connected in parallel 362
14.14 Resistivity 363
14.15 Variation of resistance with temperature 365

15 Electric fields 371

15.1 The electric charge model 371
15.2 Interpretation of electric current in terms of charge 373
15.3 Electric fields: electric field strength 374
15.4 Forces between point charges: Coulomb’s law 376
15.5 Electric flux and electric flux density 376
15.6 Electric fields due to systems of point charges 378
15.7 Gauss’ law for electrostatics 381
15.8 Potential difference in electric fields: electric potential 383
15.9 Acceleration of charged particles 388
15.10 Dielectric materials 389
15.11 Capacitors 391
15.12 Capacitors in series and in parallel 395
15.13 Charge and discharge of a capacitor through a resistor 396

16 Magnetic fields 403

16.1 Magnetism 403
16.2 The work of Ampere, Biot and Savart 405
16.3 Magnetic pole strength 406
16.4 Magnetic field strength 407
16.5 Ampere’s law 408
16.6 The Biot-Savart law 410
16.7 Applications of the Biot-Savart law 411
16.8 Magnetic flux and magnetic flux density 413
16.9 Magnetic fields due to systems of poles 413
16.10 Forces between magnets 414
16.11 Forces between currents and magnets 415
16.12 The permeability of vacuum 416
16.13 Current loop in a magnetic field 417
16.14 Magnetic dipoles and magnetic materials 419
16.15 Moving coil meters and electric motors 423
16.16 Magnetic fields due to moving charges 425
16.17 Force on an electric charge in a magnetic field 425
16.18 Magnetic dipole moments of charged particles in closed orbits 427
16.19 Electric and magnetic fields in moving reference frames 428

17 Electromagnetic induction: time-varying emfs 437

17.1 The principle of electromagnetic induction 437
17.2 Simple applications of electromagnetic induction 440
17.3 Self-inductance 441
17.4 The series L-R circuit 444
17.5 Discharge of a capacitor through an inductor and a resistor 446
17.6 Time-varying emfs: mutual inductance: transformers 447
17.7 Alternating current (a.c.) 449
17.8 Alternating current transformers 453
17.9 Resistance, capacitance and inductance in a.c. circuits 454
17.10 The series L-C-R circuit: phasor diagrams 456
17.11 Power in an a.c. circuit 459

18 Maxwell's equations: electromagnetic radiation 465

18.1 Reconsideration of the laws of electromagnetism: Maxwell’s equations 465
18.2 Plane electromagnetic waves 468
18.3 Experimental observation of electromagnetic radiation 470
18.4 The electromagnetic spectrum 471
18.5 Polarisation of electromagnetic waves 473
18.6 Energy, momentum and angular momentum in electromagnetic waves 476
18.7 Reflection of electromagnetic waves at an interface between nonconducting media 479
18.8 Electromagnetic waves in a conducting medium 480
18.9 The photon model revisited 483
18.10 Invariance of electromagnetism under the Lorentz transformation 484

19 Optics 489

19.1 Electromagnetic nature of light 489
19.2 Coherence: the laser 492
19.3 Diffraction at a single slit 493
19.4 Two slit interference and diffraction: Young’s double slit experiment 496
19.5 Multiple slit interference: the diffraction grating 498
19.6 Diffraction of X-rays: Bragg scattering 501
19.7 The ray model: geometrical optics 504
19.8 Reflection of light 505
19.9 Image formation by spherical mirrors 506
19.10 Refraction of light 508
19.11 Refraction at successive plane interfaces 512
19.12 Image formation by spherical lenses 513
19.13 Image formation of extended objects: magnification 517
19.14 Dispersion of light 520

20 Atomic physics 527

20.1 Atomic models 527
20.2 The spectrum of hydrogen: the Rydberg formula 529
20.3 The Bohr postulates 530
20.4 The Bohr theory of the hydrogen atom 531
20.5 The quantum mechanical (Schr€odinger) solution of the one-electron atom 534
20.6 The radial solutions of the lowest energy state of hydrogen 538
20.7 Interpretation of the one-electron atom eigenfunctions 539
20.8 Intensities of spectral lines: selection rules 543
20.9 Quantisation of angular momentum 544
20.10 Magnetic effects in one-electron atoms: the Zeeman effect 545
20.11 The Stern–Gerlach experiment: electron spin 547
20.12 The spin–orbit interaction 549
20.13 Identical particles in quantum mechanics: the Pauli exclusion principle 550
20.14 The periodic table: multielectron atoms 552
20.15 The theory of multielectron atoms 554
20.16 Further uses of the solutions of the one-electron atom 555

21 Electrons in solids: quantum statistics 559

21.1 Bonding in molecules and solids 559
21.2 The classical free electron model of solids 563
21.3 The quantum mechanical free electron model: the Fermi energy 565
21.4 The electron energy distribution at 0 K 568
21.5 Electron energy distributions at T > 0 K 570
21.6 Specific heat capacity and conductivity in the quantum free electron model 571
21.7 The band theory of solids 573
21.8 Semiconductors 574
21.9 Junctions in conductors and semiconductors: p-n junctions 576
21.10 Transistors 581
21.11 The Hall effect 583
21.12 Quantum statistics: systems of bosons 584
21.13 Superconductivity 585

22 Nuclear physics, particle physics and astrophysics 589

22.1 Properties of atomic nuclei 589
22.2 Nuclear binding energies 591
22.3 Nuclear models 592
22.4 Radioactivity 595
22.5 a-, b- and g-decay 597
22.6 Detection of radiation: units of radioactivity 600
22.7 Nuclear reactions 602
22.8 Nuclear fission and nuclear fusion 603
22.9 Fission reactors 604
22.10 Thermonuclear fusion 606
22.11 Subnuclear particles 609
22.12 The quark model 613
22.13 The physics of stars 617
22.14 The origin of the universe 622

Worked examples 625

Problems 627

Answers to problems 629

Appendix A: Mathematical rules and formulas 639
Appendix B: Some fundamental physical constants 659
Appendix C: Some astrophysical and geophysical data 661

Bibliography 663

Index 665

Inside front cover: Summary of notations used in text
Inside back cover: The periodic table (Appendix D)