Understanding Physics

MICHAEL MANSFIELD AND COLM O'SULLIVAN

Physics Department

University College Cork

Ireland

JOHN WILEY & SONS CHICHESTER . NEWYORK . WEINHEIM . BRISBANE . SINGAPORE . TORONTO

Published in association with — J

PRAXIS PUBLISHING P R A X I S CHICHESTER X - — ^

Contents

Preface xv

1 Understanding the physical universe 1

1.1 The programme of physics 1 1.2 Building blocks of matter 2 1.3 Matter in bulk 6

1.4 The fundamental interactions 6 1.5 Exploring the physical universe: the scientific method 7 1.6 The role of physics: its scope and applications 10

2 Using mathematical tools in physics 12

2.1 Applying the scientific method 12 2.2 Use of variables to represent displacement and time 12 2.3 Representation of data 14 2.4 Use of differentiation in analysis: velocity and acceleration in

linear motion 17 2.5 Use of integration in analysis 21 2.6 Maximum and minimum values of physical variables: general

linear motion 26 2.7 Angular motion: the radian 27 2.8 Using mathematics in physics 30

Worked Examples 31 Problems 34

3 Causes of motion: dynamics 36

3.1 The concept of force 36 3.2 The first law of dynamics (Galileo's principle/Newton's first law) 37 3.3 The fundamental dynamical principle (Newton's second law) 38 3.4 Systems of units: SI 42 3.5 Time-dependent forces: oscillatory motion 45

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3.6 Simple harmonic mot ion 48

Worked Examples 52

Problems 53

4 Motion in two and three dimensions 55

4.1 Vector physical quantit ies 55

4.2 Velocity and acceleration vectors 59

4.3 Mot ion w i th constant acceleration: projecti le mot ion 60

4.4 Force as a vector quanti ty: vector fo rm of the laws of dynamics 62

4.5 Constraint forces 64

4.6 Friction 66

4.7 Mot ion in a circle: centr ipetal force 68

4.8 Mot ion in a circle at constant speed 69

4.9 Tangential and radial components of acceleration 71

4.10 Hybrid mot ion : the simple pendulum 72

4.11 Angular quantit ies as vectors: the cross product 73

Worked Examples 75

Problems 78

5 Fields and energy 81

5.1 Newton's law of gravi tat ion 81

5.2 General force fields 83

5.3 Mechanical work 85

5.4 Potential energy: conservative fields 87

5.5 Mechanical energy 90

5.6 Power 92

5.7 Plots of potent ia l energy funct ions 92

5.8 Energy in a constant un i form f ie ld 94

5.9 Energy in an inverse square law f ield 95

5.10 Energy in simple harmonic mot ion 97

5.11 Damped harmonic mot ion 99

5.12 Moment of a force: angular momentum 102

5.13 Planetary mot ion : circular orbits 104

*5.14 Planetary mot ion : ell iptical orbits and Kepler's laws 105

*5.15 Gravitational forces exerted by shells and spheres 109

Worked Examples 110

Problems 116

6 Many-body interactions 119

6.1 Newton's th i rd law 119

6.2 The principle of conservation of momentum 122

6.3 Mechanical energy of systems of particles 124

6.4 Decays and collisions 124

6.5 The centre of mass of a system 128

* indicates a more advanced section

Contents vii

6.6 The two-body problem: reduced mass 130

6.7 Collisions in LAB and CM coordinate systems 133

6.8 Angular momentum of systems of particles 139

6.9 Conservation principles in physics 141

Worked Examples 142

Problems 145

7 Rigid-body dynamics 148

7.1 Rigid bodies 148

7.2 Rigid bodies in equi l ibr ium: statics 149

7.3 Torque 151

7.4 Dynamics of r igid bodies 152

7.5 Measurement of to rque: the torsion balance 153

7.6 Rotation of a rigid body about a f ixed axis: moment of inertia 154

7.7 Calculation of moments of inert ia: the parallel axes theorem 155

7.8 Conservation of angular momentum of r igid bodies 158

7.9 Conservation of mechanical energy in r igid-body systems 159

7.10 Work done by a torque: torsional oscillations; rotat ional power 162

7.11 Gyroscopic mot ion 164

7.12 Summary: the connection between rotat ional and

translat ional motions 166

Worked Examples 167

Problems 169

8 Relativemotion 172

8.1 Appl icabi l i ty of Newton's laws of mot ion : inertial reference

frames 172

8.2 The Galilean transformations 173

8.3 The centre of mass (CM) f rame 176

8.4 Example of a non-inert ial f rame: centr i fugal force 178

8.5 The Earth as a rotat ing f rame: effective g 179

*8.6 The Coriolis force 180

*8.7 The Foucault pendulum 183

8.8 Practical criteria fo r inertial frames: the local view 185

Worked Examples 187

Problems 191

9 Special relativity 193

9.1 The velocity of l ight 193

9.2 The Principle of Relativity 194

9.3 Consequences of the Principle of Relativity 195

9.4 The Lorentz transformations 198

9.5 The Fitzgerald-Lorentz contract ion 201

9.6 Time di lat ion 202

* indicates a more advanced section

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9.7 Paradoxes in special relativity 204 *9.8 Simultaneity: quantitative analysis of the twin paradox 205 9.9 Relativistic transformation of velocity 208 9.10 Momentum in relativistic mechanics 210 9.11 4-vectors: the energy-momentum 4-vector 212

9.12 Energy-momentum transformations: relativistic energy conservation 214

9.13 Relativistic energy: mass-energy equivalence 215 9.14 Units in relativistic mechanics 219 9.15 Mass-energy equivalence in practice 220 9.16 General relativity 221

Worked Examples 222 Problems 226

10 Continuum mechanics: mechanical properties of materials 228

10.1

10.2

10.3

10.4

10.5

10.6

10.7

10.8

10.9

10.10

10.11

10.12

10.13

10.14

Dynamics of cont inuous media

Elastic properties of solids

Fluids at rest

Elastic properties of f luids

Pressure in gases

Archimedes' Principle

Fluid dynamics

Viscosity

Surface properties of liquids

Pressure in gases

A microscopic theory of gases

The mole

Interatomic forces: modifications to the kinetic theory of gases

Microscopic models of condensed matter systems

Worked Examples

Problems

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229

233

234

235

236

238

241

243

245

247

250

252

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11 Thermal physics 260

11.1 Friction and heating 260 11.2 Temperature scales 262 11.3 Heat capacities of thermal systems 265 11.4 Comparison of specific heat capacities: calorimetry 267 11.5 Thermal conductivity 268

11.6 Convection 269 11.7 Thermal radiation 270 11.8 Thermal expansion 272 11.9 The first law of thermodynamics 274 11.10 Change of phase: latent heat 276

* indicates a more advanced section

1.11 The equation of State of an ideal gas 277

1.12 Isothermal, isobaric and adiabatic processes 278

1.13 The Carnot cycle 281 1.14 Entropy and the second law of thermodynamics 283 1.15 Helmholz and Gibbs functions 286 1.16 Microscopic interpretation of temperature 287 1.17 Polyatomic molecules: principle of equipartition of energy 291

1.18 Ideal gas in a gravitational field: the' law of atmospheres' 293

1.19 Ensemble averages and distribution functions 294 1.20 Distribution of molecular velocities in an ideal gas 296 1.21 Distribution of molecular speeds, momenta and energies 298 1.22 Microscopic interpretation of temperature and heat capacity

in solids 300 Worked Examples 301 Problems 304

Wave motion 307

12.1 Characteristics of wave motion 307 12.2 Representation of a wave travelling in one dimension 310

12.3 Energy and power in a wave motion 312 12.4 Plane and spherical waves: the concept of flux 313 12.5 Huygens'principle: the laws of reflection and refraction 315 12.6 Geometrical optics 318 12.7 Interference between waves 322 12.8 Diffraction: resolving power 325 12.9 Two-slit interference and diffraction: Young's double slits 329

12.10 Multiple slit interference: the diffraction grating 331 12.11 X-ray diffraction: Bragg scattering 335 12.12 Standing waves 336 12.13 The Doppler effect 338 12.14 Coherence: the laser 341 12.15 The wave equation 343 12.16 Waves along a string 344 12.17 Waves in elastic media: longitudinal waves in a solid rod 345

12.18 Waves in elastic media: sound waves 346 12.19 Superposition of two waves of slightly different frequencies:

wave and group velocities 347

12.20 Other waveforms: Fourier analysis 349 Worked Examples 352 Problems 355

Introduction to quantum mechanics 358

13.1 Physics at the beginning of the twentieth century 358

13.2 Blackbody radiation 359 13.3 The photoelectric effect 362

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13.4 The X-ray cont inuum 365

13.5 The Compton effect: the photon model 367

13.6 The de Broglie hypothesis: electron di f f ract ion 370

13.7 Interpretat ion of wave-particle dual i ty 371

13.8 The Heisenberg uncertainty principle 373

13.9 The wavefunct ion: expectation values 376

13.10 The Schrödinger (wave mechanical) method 378

13.11 The free particle 379

13.12 The t ime- independent Shrödinger equat ion: eigenfunct ions

and eigenvalues 383

13.13 The inf in i te square potent ial wel l 385

13.14 The potent ia l step 388

13.15 Other potent ia l wells and barriers 394

*13.16 The simple harmonic oscillator 398

13.17 Further implications of quantum mechanics 401

Worked Examples 402

Problems 404

14 Electric currents 407

14.1 Electric currents 407

14.2 Force between currents 409

14.3 The uni t o f electric current 411

14.4 Heating effect revisited: electrical resistance 411

14.5 Strength of a power supply: emf 414

14.6 Resistance of a circuit 415

14.7 Potential dif ference 416

14.8 Effect of internal resistance 417

14.9 Comparison of emfs: the potent iometer 419

14.10 Resistivity 420

14.11 Var iat ion of resistance w i t h temperature 422

14.12 Mul t i loop circuits 423

14.13 Ki rchhof fs rules 424

14.14 Comparison of resistances: Wheatstone bridge 426

14.15 Power supplies connected in parallel 426

Worked Examples 427

Problems 430

15 Electric fields 432

15.1 The electric charge model 432

15.2 Interpretat ion of electric current in terms of charge 434

15.3 Electric f ie ld strength and electric f lux density 436

15.4 Electric fields due to po in t charges 440

15.5 Forces between point charges: Coulomb's law 443

* indicates a more advanced section

15.6 Potential difference in electric fields 446 15.7 Electric potential 450 15.8 Capacitors 455 15.9 Capacitors in series and in parallel 459 15.10 Charge and discharge of a capacitor through a resistance 460 15.11 Dielectric materials 463

Worked Examples 464 Problems 468

16 Magnetic fields 470

16.1 Magnetism 470 16.2 Work of Ampere, Biot and Savart 473 16.3 Magnetic pole strength 474 16.4 Magnetic field strength 475

16.5 Ampere's law 476 16.6 The Biot-Savart law 479 16.7 Applications of the Biot-Savart law 481 16.8 Magnetic flux and magnetic flux density 482 16.9 Magnetic fields due to systems of poles 484

16.10 Forces between magnets 485 16.11 Forces between currents and magnets 486 16.12 The permeability of vacuum 487 16.13 Current loop in a magnetic field 488 16.14 Magnetic dipoles and magnetic materials 490 16.15 Moving coil meters and electric motors 495

16.16 Magnetic fields due to moving charges 496 16.17 Force on a charge in a magnetic field 497 16.18 Magnetic dipole moments of charged particles in closed orbits 500

*16.19 Electric and magnetic fields in moving reference frames 501 Worked Examples 505 Problems 508

17 Electromagnetic induction: time-varying emfs 510

17.1 The principle of electromagnetic induction 510 17.2 Simple applications of electromagnetic induction 514 17.3 Self-inductance 516 17.4 Series L-R circuit 518 17.5 Discharge of a capacitor through an inductance and a

resistance 520

17.6 Time-varying emfs: mutual inductance, transformers 522 17.7 Alternating current (a.c.) 524 17.8 Alternating current transformers 527

* indicates a more advanced section

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17.9 Resistance, capacitance and inductance in a.c. circuits 529

17.10 Series L-C-R circuit: phasor diagrams 531 17.11 Power in an a.c. circuit 535 17.12 Faraday's law revisited 536

Worked Examples 536 Problems 539

18 Maxwell's equations: electromagnetic radiation 541

18.1 Reconsideration of the laws of electromagnetism: Maxwell's equations 541

18.2 Plane electromagnetic waves 544 18.3 Experimental observation of electromagnetic radiation 548 18.4 The electromagnetic spectrum 549 18.5 Energy in electromagnetic waves 551 18.6 Momentum in electromagnetic waves 553 18.7 Polarisation of electromagnetic waves 555 18.8 The photon model revisited 561

*18.9 Invariance of electromagnetism under the Lorentz transformation 563 Worked Examples 564 Problems 566

19 Atomic physics 569

19.1 Atomic models 569

19.2 The hydrogen spectrum: the Rydberg formula 572 19.3 The Bohr postulates 573

19.4 The Bohr theory of the hydrogen atom 574 19.5 The quantum mechanical (Schrödinger) solution of the

one-electron atom 579 *19.6 The radial solutions of the lower energy states of hydrogen 584

19.7 Interpretation of the one-electron atom eigenfunctions 587 19.8 Intensities of spectral lines: selection rules 591

19.9 Space quantisation: quantisation of angular momentum 593 19.10 Magnetic effects in one-electron atoms: the Zeeman effect 594 19.11 The Stern-Gerlach experiment: electron spin 596 19.12 The spin-orbit interaction 599 19.13 Identical particles in quantum mechanics: the Pauli exclusion

principle 600

19.14 The periodic table: multielectron atoms 603 19.15 The theory of multielectron atoms 608 19.16 Further uses of the solutions of the one-electron atom 609

Worked Examples 610 Problems 612

* indicates a more advanced section

Electrons in solids: quantum statistics 615

20.1 Bonding in molecules and solids 615 20.2 The classical free electron model of solids 620 20.3 The quantum mechanical free electron model: the Fermi

energy 623 20.4 The electron energy distribution at OK 626 20.5 Electron energy distributions at T > 0 К 629 20.6 Specific heat capacity and conductivity in the quantum free

electron model 630 20.7 The band model of solids 632 20.8 Semiconductors 634 20.9 Junctions in conductors and semiconductors: the p-n

junctions 637 20.10 Transistors 643 20.11 The Hall effect 646 20.12 Quantum statistics: systems of bosons 647 20.13 Superconductivity 649

Worked Examples 651 Problems 653

Nuclear physics, particle physics and astrophysics 656

656 659 660 666 667 672 674 676 678 680 684 690 695 704 708 710

Answers to problems 712 Appendix A Mathematical rules and formulas 717 Appendix В Some fundamental physical constants 738 Appendix С Some astrophysical and geophysical data 740 Appendix D The periodic table of the elements 741 Bibliography 743 Index 744

21.1 21.2 21.3 21.4 21.5 21.6 21.7 21.8 21.9 21.10 21.11 21.12 21.13 21.14

Properties of atomic nuclei Nuclear binding energies Nuclear models Radioactivity a-, ß- and y-decay Detection of radiation: units of radioactivity Nuclear reactions Nuclear fission and nuclear fusion Fission reactors Thermonuclear fusion

Sub-nuclear particles The quark model Physics of stars The origin of the Universe Worked Examples Problems