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Understanding Physics MICHAEL MANSFIELD AND COLM O'SULLIVAN Physics Department University College Cork Ireland JOHN WILEY & SONS CHICHESTER . NEW YORK . WEINHEIM . BRISBANE . SINGAPORE . TORONTO Published in association with PRAXIS PUBLISHING CHICHESTER PR Contents Preface 1 Understanding the physical universe 1.1 1.2 1.3 1.4 1.5 1.6 The programme of physics Building blocks of matter Matter in bulk The fundamental interactions Exploring the physical universe: the scientific method The role of physics: its scope and applications 2 Using mathematical tools in physics 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 Applying the scientific method Use of variables to represent displacement and time Representation of data Use of differentiation in analysis: velocity and acceleration in linear motion Use of integration in analysis Maximum and minimum values of physical variables: general linear motion Angular motion: the radian Using mathematics in physics Worked Examples Problems 3 Causes of motion: dynamics 3.1 3.2 3.3 3.4 3.5 The concept of force The first law of dynamics (Galileo's principle/Newton's first law) The fundamental dynamical principle (Newton's second law) Systems of units: SI Time-dependent forces: oscillatory motion xv 1 1 2 6 6 7 10 12 12 12 14 17 21 26 27 30 31 34 36 36 37 38 42 45 vi Contents 3.6 4 5 48 52 53 Motion in two and three dimensions 55 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 55 59 60 62 64 66 68 69 71 72 73 75 78 Vector physical quantities Velocity and acceleration vectors Motion with constant acceleration: projectile motion Force as a vector quantity: vector form of the laws of dynamics Constraint forces Friction Motion in a circle: centripetal force Motion in a circle at constant speed Tangential and radial components of acceleration Hybrid motion: the simple pendulum Angular quantities as vectors: the cross product Worked Examples Problems Fields and energy 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 *5.14 *5.15 6 Simple harmonic motion Worked Examples Problems Newton's law of gravitation General force fields Mechanical work Potential energy: conservative fields Mechanical energy Power Plots of potential energy functions Energy in a constant uniform field Energy in an inverse square law field Energy in simple harmonic motion Damped harmonic motion Moment of a force: angular momentum Planetary motion: circular orbits Planetary motion: elliptical orbits and Kepler's laws Gravitational forces exerted by shells and spheres Worked Examples Problems 81 81 83 85 87 90 92 92 94 95 97 99 102 104 105 109 110 116 Many-body interactions 119 6.1 6.2 6.3 6.4 6.5 119 122 124 124 128 Newton's third law The principle of conservation of momentum Mechanical energy of systems of particles Decays and collisions The centre of mass of a system * indicates a more advanced section Contents vii 6.6 6.7 6.8 The two-body problem: reduced mass Collisions in LAB and CM coordinate systems Angular momentum of systems of particles 130 133 139 6.9 Conservation principles in physics Worked Examples Problems 141 142 145 7 Rigid-body dynamics 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 7.12 Rigid bodies Rigid bodies in equilibrium: statics Torque Dynamics of rigid bodies Measurement of torque: the torsion balance Rotation of a rigid body about a fixed axis: moment of inertia Calculation of moments of inertia: the parallel axes theorem Conservation of angular momentum of rigid bodies Conservation of mechanical energy in rigid-body systems Work done by a torque: torsional oscillations; rotational power Gyroscopic motion Summary: the connection between rotational and translational motions Worked Examples Problems 8 Relative motion 8.1 8.2 8.3 8.4 8.5 *8.6 *8.7 8.8 9 Applicability of Newton's laws of motion: inertial reference frames The Galilean transformations The centre of mass (CM) frame Example of a non-inertial frame: centrifugal force The Earth as a rotating frame: effective g The Coriolis force The Foucault pendulum Practical criteria for inertial frames: the local view Worked Examples Problems 148 148 149 151 152 153 154 155 158 159 162 164 166 167 169 172 172 173 176 178 179 180 183 185 187 191 Special relativity 193 9.1 9.2 9.3 9.4 9.5 9.6 193 194 195 198 201 202 The velocity of light The Principle of Relativity Consequences of the Principle of Relativity The Lorentz transformations The Fitzgerald-Lorentz contraction Time dilation * indicates a more advanced section viii Contents 9.7 *9.8 9.9 9.10 9.11 9.12 9.13 9.14 9.15 9.16 10 11 Paradoxes in special relativity Simultaneity: quantitative analysis of the twin paradox Relativistic transformation of velocity Momentum in relativistic mechanics 4-vectors: the energy-momentum 4-vector Energy-momentum transformations: relativistic energy conservation Relativistic energy: mass-energy equivalence Units in relativistic mechanics Mass-energy equivalence in practice General relativity Worked Examples Problems 204 205 208 210 212 214 215 219 220 221 222 226 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 10.14 228 229 233 234 235 236 238 241 243 245 247 250 252 254 256 258 Dynamics of continuous media Elastic properties of solids Fluids at rest Elastic properties of fluids 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 Thermal physics 260 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9 11.10 260 262 265 267 268 269 270 272 274 276 Friction and heating Temperature scales Heat capacities of thermal systems Comparison of specific heat capacities: calorimetry Thermal conductivity Convection Thermal radiation Thermal expansion The first law of thermodynamics Change of phase: latent heat * indicates a more advanced section Contents 11.11 11.12 11.13 11.14 11.15 11.16 11.17 11.18 11.19 11.20 11.21 11.22 12 The equation of state of an ideal gas Isothermal, isobaric and adiabatic processes The Carnot cycle Entropy and the second law of thermodynamics Helmholz and Gibbs functions Microscopic interpretation of temperature Polyatomic molecules: principle of equipartition of energy Ideal gas in a gravitational field: the'law of atmospheres' Ensemble averages and distribution functions Distribution of molecular velocities in an ideal gas Distribution of molecular speeds, momenta and energies Microscopic interpretation of temperature and heat capacity in solids Worked Examples Problems 277 278 281 283 286 287 291 293 294 296 298 300 301 304 Wave motion 307 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 12.10 12.11 12.12 12.13 12.14 12.15 12.16 12.17 12.18 12.19 307 310 312 313 315 318 322 325 329 331 335 336 338 341 343 344 345 346 Characteristics of wave motion Representation of a wave travelling in one dimension Energy and power in a wave motion Plane and spherical waves: the concept of flux Huygens'principle: the laws of reflection and refraction Geometrical optics Interference between waves Diffraction: resolving power Two-slit interference and diffraction: Young's double slits Multiple slit interference: the diffraction grating X-ray diffraction: Bragg scattering Standing waves The Doppler effect Coherence: the laser The wave equation Waves along a string Waves in elastic media: longitudinal waves in a solid rod Waves in elastic media: sound waves Superposition of two waves of slightly different frequencies: wave and group velocities 12.20 Other waveforms: Fourier analysis Worked Examples Problems 13 Introduction to quantum mechanics 13.1 13.2 13.3 Physics at the beginning of the twentieth century Blackbody radiation The photoelectric effect " 347 349 352 355 358 358 359 362 ix x Contents 13.4 13.5 13.6 13.7 13.8 13.9 13.10 13.11 13.12 13.13 13.14 13.15 *13.16 13.17 14 15 The X-ray continuum The Compton effect: the photon model The de Broglie hypothesis: electron diffraction Interpretation of wave-particle duality The Heisenberg uncertainty principle The wavefunction: expectation values The Schrodinger (wave mechanical) method The free particle The time-independent Shrodinger equation: eigenfunctions and eigenvalues The infinite square potential well The potential step Other potential wells and barriers The simple harmonic oscillator Further implications of quantum mechanics Worked Examples Problems 365 367 370 371 373 376 378 379 383 385 388 394 398 401 402 404 Electric currents 407 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 14.9 14.10 14.11 14.12 14.13 14.14 14.15 407 409 411 411 414 415 416 417 419 420 422 423 424 426 426 427 430 Electric currents Force between currents The unit of electric current Heating effect revisited: electrical resistance Strength of a power supply: emf Resistance of a circuit Potential difference Effect of internal resistance Comparison of emfs: the potentiometer Resistivity Variation of resistance with temperature Multiloop circuits Kirchhoff's rules Comparison of resistances: Wheatstone bridge Power supplies connected in parallel Worked Examples Problems j Electric fields 432 15.1 15.2 15.3 15.4 15.5 432 434 436 440 443 The electric charge model Interpretation of electric current in terms of charge Electric field strength and electric flux density Electric fields due to point charges Forces between point charges: Coulomb's law * indicates a more advanced section Contents 15.6 15.7 15.8 15.9 15.10 15.11 Potential difference in electric fields Electric potential Capacitors Capacitors in series and in parallel Charge and discharge of a capacitor through a resistance Dielectric materials Worked Examples Problems 16 Magnetic fields 16.1 16.2 16.3 16.4 16.5 16.6 16.7 16.8 16.9 16.10 16.11 16.12 16.13 16.14 16.15 16.16 16.17 16.18 *16.19 17.6 17.7 17.8 470 Magnetism Work of Ampere, Biot and Savart Magnetic pole strength Magnetic field strength Ampere's law The Biot-Savart law Applications of the Biot-Savart law Magnetic flux and magnetic flux density Magnetic fields due to systems of poles Forces between magnets Forces between currents and magnets The permeability of vacuum Current loop in a magnetic field Magnetic dipoles and magnetic materials Moving coil meters and electric motors Magnetic fields due to moving charges Force on a charge in a magnetic field Magnetic dipole moments of charged particles in closed orbits Electric and magnetic fields in moving reference frames 470 473 474 475 476 479 481 482 484 485 486 487 488 490 495 496 497 500 501 Worked Examples Problems 505 508 17 Electromagnetic induction: time-varying emfs 17.1 17.2 17.3 17.4 17.5 446 450 455 459 460 463 464 468 The principle of electromagnetic induction Simple applications of electromagnetic induction Self-inductance Series L-R circuit Discharge of a capacitor through an inductance and a resistance Time-varying emfs: mutual inductance, transformers Alternating current (a.c.) Alternating current transformers * indicates a more advanced section 510 510 514 516 518 520 522 524 527 xi xii Contents 18 17.9 Resistance, capacitance and inductance in a.c. circuits 17.10 Series L-C-R circuit: phasor diagrams 17.11 Power in an a.c. circuit 17.12 Faraday's law revisited Worked Examples Problems 529 531 535 536 536 539 Maxwell's equations: electromagnetic radiation 541 18.1 Reconsideration of the laws of electromagnetism: Maxwell's equations ^ 18.2 Plane electromagnetic waves 18.3 . Experimental observation of electromagnetic radiation 18.4 The electromagnetic spectrum 18.5 Energy in electromagnetic waves 18.6 Momentum in electromagnetic waves 18.7 Polarisation of electromagnetic waves 18.8 The photon model revisited *18.9 Invariance of electromagnetism under the Lorentz transformation Worked Examples Problems 19 Atomic physics 19.1 19.2 19.3 19.4 19.5 *19.6 19.7 19.8 19.9 19.10 19.11 19.12 19.13 19.14 19.15 19.16 Atomic models The hydrogen spectrum: the Rydberg formula " The Bohr postulates The Bohr theory of the hydrogen atom The quantum mechanical (Schrodinger) solution of the one-electron atom The radial solutions of the lower energy states of hydrogen Interpretation of the one-electron atom eigenfunctions Intensities of spectral lines: selection rules Space quantisation: quantisation of angular momentum Magnetic effects in one-electron atoms: the Zeeman effect The Stern-Gerlach experiment: electron spin The spin-orbit interaction Identical particles in quantum mechanics: the Pauli exclusion principle The periodic table: multielectron atoms The theory of multielectron atoms Further uses of the solutions of the one-electron atom Worked Examples Problems * indicates a more advanced section 541 544 548 549 551 553 555 561 563 564 566 569 569 572 573 574 579 584 587 591 593 594 596 599 600 603 608 609 610 612 Contents 20 Electrons in solids: quantum statistics 615 20.1 20.2 20.3 615 620 20.4 20.5 20.6 20.7 20.8 20.9 20.10 20.11 20.12 20.13 Bonding in molecules and solids The classical free electron model of solids The quantum mechanical free electron model: the Fermi energy The electron energy distribution at OK Electron energy distributions at T > 0 K Specific heat capacity and conductivity in the quantum free electron model The band model of solids Semiconductors Junctions in conductors and semiconductors: the p-n junctions Transistors The Hall effect Quantum statistics: systems of bosons Superconductivity Worked Examples Problems 21 Nuclear physics, particle physics and astrophysics 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- 0- 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 orig i n of the U n iverse Worked Examples Problems Answers to problems Appendix A Mathematical rules and formulas Appendix B Some fundamental physical constants Appendix C Some astrophysical and geophysical data Appendix D The periodic table of the elements Bibliography Index 623 626 629 630 632 634 637 643 646 647 649 651 653 656 656 659 660 666 667 672 674 676 678 680 684 690 695 704 708 710 712 717 738 740 741 743 744 xiii