Answers to Chapter 22 Problems

Chapter 1: Understanding the physical universe
- 1.1 The programme of physics
- 1.2 The building blocks of matter
- 1.3 Matter in bulk
- 1.4 The fundamental interactions
- 1.5 Exploring the physical universe: the scientific method
- 1.6 The role of physics: its scope and applications
Chapter 2 Using mathematical tools in physics
- 2.1 Applying the scientific method
- 2.2 The use of variables to represent displacement and time
- 2.3 Representation of data
- 2.4 The use of differentiation in analysis: velocity and acceleration in linear motion
- 2.5 The use of integration in analysis
- 2.6 Maximum and minimum values of physical quantities: general linear motion
- 2.7 Angular motion: the radian
- 2.8 The role of mathematics in physics
- Chapter 2 Problems
  - Answers to Chapter 2 problems
Chapter 3 The causes of motion: dynamics
- 3.1 The concept of force
- 3.2 The first law of dynamics (Newton's first law)
- 3.3 The fundamental dynamical principle (Newton's second law)
- 3.4 Systems of units: SI
- 3.5 Time dependent forces: oscillatory motion
- 3.6 Simple harmonic motion
- 3.7 Mechanical work and energy
- 3.8 Plots of potential energy functions
- 3.9 Power
- 3.10 Energy in simple harmonic motion
- 3.11 Dissipative forces: damped harmonic motion
  - 3.11.1 Trial solution technique for solving the damped harmonic motion equation
- 3.12 Forced oscillations
- 3.13 Non-linear dynamics: chaos
- 3.14 Phase space representation of dynamical systems
- Chapter 3 Problems
  - Answers to Chapter 3 problems
Chapter 4 Motion in two and three dimensions
- 4.1 Vector physical quantities
- 4.2 Vector algebra
- 4.3 Velocity and acceleration vectors
- 4.4 Force as a vector quantity: vector form of the laws of dynamics
- 4.5 Constraint forces
- 4.6 Friction
- 4.7 Motion in a circle: centripetal force
- 4.8 Motion in a circle at constant speed
- 4.9 Tangential and radial components of acceleration
- 4.10 Hybrid motion: the simple pendulum
  - 4.10.1 Large angle corrections for the simple pendulum
- 4.11 Angular quantities as vectors: the cross product
- Chapter 4 Problems
  - Answers to Chapter 4 Problems
Chapter 5 Force fields
- 5.1 Newton's law of universal gravitation
- 5.2 Force fields
- 5.3 The concept of flux
- 5.4 Gauss' lew for gravitation
- 5.5 Applications of Gauss’ law
- 5.6 Motion in a constant uniform field: projectiles
- 5.7 Mechanical work and energy
- 5.8 Power
- 5.9 Energy in a constant uniform field
- 5.10 Energy in an inverse square law field
- 5.11 Moment of a force: angular momentum
- 5.12 Planetary motion: circular orbits
- 5.13 Planetary motion: elliptical orbits and Kepler's laws
  - 5.13.1 Conservation of the Runge-Lens vector
- Chapter 5 Problems
  - Answers to Chapter 5 problems
Chapter 6 Many-body interactions
- 6.1 Newton's third law
- 6.2 The principle of conservation of momentum
- 6.3 Mechanical energy of systems of particles
- 6.4 Particle decay
- 6.5 Particle collisions
- 6.6 The centre of mass of a system
- 6.7 The two-body problem: reduced mass
- 6.8 Angular momentum of systems of particles
- 6.9 Conservation principles in physics
- Chapter 6 Problems
  - Answers to Chapter 6 problems
Chapter 7 Rigid-body dynamics
- 7.1 Rigid bodies
- 7.2 Rigid bodies in equilibrium: statics
- 7.3 Torque
- 7.4 Dynamics of rigid bodies
- 7.5 Measurement of torque: the torsion balance
- 7.6 Rotation of a rigid body about a fixed axis: moment of inertia
- 7.7 Calculation of moments of inertia: the parallel axis theorem
- 7.8 Conservation of angular momentum of rigid bodies
- 7.9 Conservation of mechanical energy in rigid body systems
- 7.10 Work done by a torque: torsional oscillations: rotational power
- 7.11 Gyroscopic motion
  - 7.11.1 Precessional angular velocity of a top
- 7.12 Summary - connection between rotational and translational motions
- Chapter 7 problems
  - Answers to Chapter 7 problems
Chapter 8 Relative motion
- 8.1 Applicability of Newton's laws of motion: inertial reference frames
- 8.2 The Galilean transformation
- 8.3 The CM (centre-of-mass) reference frame
- 8.4 Example of a non-inertial frame: centrifugal force
- 8.5 Motion in a rotating frame: the Coriolis force
- 8.6 The Foucault pendulum
  - 8.6.1 Precession of a Foucault pendulum
- 8.7 Practical criteria for inertial frames: the local view
- Chapter 8 problems
  - Answers to Chapter 8 Problems
Chapter 9 Special relativity
- 9.1 The velocity of light
  - 9.1.1 The Michelson-Morley experiment
- 9.2 The Principle of Relativity
- 9.3 Consequences of the Principle of Relativity
- 9.4 The Lorentz transformation
- 9.5 The Fitzgerald-Lorentz contraction
- 9.6 Time dilation
- 9.7 Paradoxes in special relativity
  - 9.7.1 Simultaneity: quantitative analysis of the twin paradox
- 9.8 Relativistic transformation of velocity
- 9.9 Momentum in relativistic mechanics
- 9.10 Four-vectors: the energy-momentum 4-vector
- 9.11 Energy-momentum transformations: relativistic energy conservation
  - 9.11.1 The force transformations
- 9.12 Relativistic energy: mass-energy equivalence
- 9.13 Units in relativistic mechanics
- 9.14 Mass-energy equivalence in practice
- 9.15 General relativity
- Chapter 9 problems
  - Answers to Chapter 9 Problems
Chapter 10 Continuum mechanics: mechanical properties of materials: microscopic models of matter
- 10.1 Dynamics of continuous media
- 10.2 Elastic properties of solids
- 10.3 Fluids at rest
- 10.4 Elastic properties of fluids
- 10.5 Pressure in gases
- 10.6 Archimedes' principle
- 10.7 Fluid dynamics; the Bernoulli equation
- 10.8 Viscosity
- 10.9 Surface properties of liquids
- 10.10 Boyle's law (Mariotte's law)
- 10.11 A microscopic theory of gases
- 10.12 The SI unit of amount of matter; the mole
- 10.13 Interatomic forces: modifications to the kinetic theory of gases
- 10.14 Microscopic models of condensed matter systems
- Chapter 10 problems
  - Answers to Chapter 10 Problems
Chapter 11 Thermal physics
- 11.1 Friction and heating
- 11.2 The SI unit of thermodynamic temperature; the kelvin
- 11.3 Heat capacities of thermal systems
- 11.4 Comparison of specific heat capacities: calorimetry
- 11.5 Thermal conductivity
- 11.6 Convection
- 11.7 Thermal radiation
- 11.8 Thermal expansion
- 11.9 The first law of thermodynamics
- 11.10 Change of phase: latent heat
- 11.11 The equation of state of an ideal gas
- 11.12 Isothermal, isobaric and adiabatic processes: free expansion
- 11.13 The Carnot cycle
- 11.14 Entropy and the second law of thermodynamics
- 11.15 The Helmholtz and Gibbs functions
- Chapter 11 problems
  - Answers to Chapter 11 Problems
Chapter 12 Microscopic models of thermal systems: kinetic theory of matter
- 12.1 Microscopic interpretation of temperature
- 12.2 Polyatomic molecules: principle of equipartition of energy
- 12.3 Ideal gas in a gravitational field: the ‘law of atmospheres’
- 12.4 Ensemble averages and distribution functions
- 12.5 The distribution of molecular velocities in an ideal gas
- 12.6 Distribution of molecular speeds
- 12.7 Distribution of molecular energies; Maxwell-Boltzmann statistics
- 12.8 Microscopic interpretation of temperature and heat capacity in solids
- Chapter 12 problems
  - Answers to Chapter 12 Problems
Chapter 13 Wave motion
- 13.1 Characteristics of wave motion
- 13.2 Representation of a wave which is travelling in one dimension
- 13.3 Energy and power in a wave motion
- 13.4 Plane and spherical waves
- 13.5 Huygens’ principle: the laws of reflection and refraction
- 13.6 Interference between waves:
- 13.7 Interference of waves passing through openings: diffraction
- 13.8 Standing waves
  - 13.8.1 Standing waves in three dimensional cavity
- 13.9 The Doppler effect
- 13.10 The wave equation
- 13.11 Waves along a string
- 13.12 Waves in elastic media: longitudinal waves in a solid rod
- 13.13 Waves in elastic media: sound waves in gases
- 13.14 Superposition of two waves of slightly different frequencies: wave and group velocities
- 13.15 Other waveforms: Fourier analysis
- Chapter 13 problems
  - Answers to Chapter 13 Problems
Chapter 14 Introduction to quantum mechanics
- 14.1 Physics at the beginning of the twentieth century
- 14.2 The blackbody radiation problem; Planck’s quantum hypothesis
- 14.3 The specific heat capacity of gases
- 14.4 The specific heat capacity of solids
- 14.5 The photoelectric effect
  - 14.5.1 Example of an experiment to study the photoelectric effect
- 14.6 The X-ray continuum
- 14.7 The Compton effect: the photon model
- 14.8 The de Broglie hypothesis: wave-particle duality
- 14.9 Interpretation of wave-particle duality
- 14.10 The Heisenberg uncertainty principle
- 14.11 The wavefunction: expectation values
- 14.12 The Schrödinger (wave mechanical) method
  - 14.12.1 Expectation value of momentum
- 14.13 The free particle
- 14.14 The time-independent Schrödinger equation: eigenfunctions and eigenvalues
  - 14.14.1 Derivation of the Ehrenfest theorem
- 14.15 The infinite square potential well
- 14.16 Potential steps
- 14.17 Other potential wells and barriers
- 14.18 The simple harmonic oscillator
  - 14.18.1 Ground state of the simple harmonic oscillator
- 14.19 Further implications of quantum mechanics
- Chapter 14 problems
  - Answers to Chapter 14 Problems
Chapter 15 Electric currents
- 15.1 Electric currents
- 15.2 The electric current model; electric charge
- 15.3 The unit of electric current; the ampere
- 15.4 Heating effect revisited: electrical resistance
- 15.5 Strength of a power supply: emf
- 15.6 Resistance of a circuit
- 15.7 Potential difference
- 15.8 Effect of internal resistance
- 15.9 Comparison of emfs: the potentiometer
- 15.10 Multiloop circuits
- 15.11 Kirchhoff's rules
- 15.12 Comparison of resistances: the Wheatstone bridge
- 15.13 Power supplies connected in parallel
- 15.14 Resistivity and conductivity
- 15.15 Variation of resistance with temperature
- Chapter 15 problems
  - Answers to Chapter 15 Problems
Chapter 16 Electric fields
- 16.1 Electric charge at rest
- 16.2 Electric fields: electric field strength
- 16.3 Force between point charges: Coulomb's law
- 16.4 Electric flux and electric flux density
- 16.5 Electric fields due to systems of charges
- 16.6 The electric dipole
- 16.7 Gauss’ law for electrostatics
- 16.8 Applications of Gauss’s law
- 16.9 Potential difference in electric fields
- 16.10 Electric potential
- 16.11 Equipotential surfaces
- 16.12 Determination of electric field strength from electric potential
- 16.13 Acceleration of charged particles
- 16.14 The laws of electrostatics in differential form
- Chapter 16 problems
  - Answers to Chapter 16 Problems
Chapter 17 Electric fields in materials
- 17.1 Conductors in electric fields
- 17.2 Insulators in electric fields; polarisation
- 17.3 Electric susceptibility
- 17.4 Boundaries between dielectric media
- 17.5 Ferroelectricity and paraelectricity; permanently polarised materials
- 17.6 Uniformly polarised rod; the ‘bar electret’
- 17.7 Microscopic models of electric polarisation
- 17.8 Capacitors
- 17.9 Examples of capacitors with simple geometry
- 17.10 Energy stored in an electric field
- 17.11 Capacitors in series and in parallel
- 17.12 Charge and discharge of a capacitor through a resistance
- 17.13 Measurement of permittivity
- Chapter 17 problems
  - Answers to Chapter 17 Problems
Chapter 18 Magnetic fields
- 18.1 Magnetism
- 18.2 The work of Ampère, Biot and Savart
- 18.3 Magnetic pole strength
- 18.4 Magnetic field strength
- 18.5 Ampère's law
- 18.6 The Biot-Savart law
- 18.7 Applications of the Biot-Savart law
- 18.8 Magnetic flux and magnetic flux density
- 18.9 Magnetic fields of permanent magnets; magnetic dipoles
- 18.10 Forces between magnets; Gauss’ law for magnetism
- 18.11 The laws of magnetostatics in differential form
- Chapter 18 problems
  - Answers to Chapter 18 Problems
Chapter 19 Electric currents and moving charges in magnetic fields
- 19.1 Forces between currents magnets
- 19.2 The force between two long parallel wires
- 19.3 Current loop in a magnetic field
- 19.4 Magnetic fields due to moving charges
- 19.5 Force on a moving electric charge in a magnetic field
- 19.6 Applications of moving charges in uniform magnetic fields; the classical Hall effect
- 19.7 Charge in a combined electric and magnetic field; the Lorentz force
- 19.8 Magnetic dipole moments of charged particles in closed orbits
- 19.9 Polarisation of magnetic materials; magnetisation, magnetic susceptibility
- 19.10 Paramagnetism and diamagnetism
- 19.11 Boundaries between magnetic media
- 19.12 Ferromagnetism; the magnetic needle revisited
- 19.13 Moving coil meters and electric motors
- 19.14 Electric and magnetic fields in moving reference frames
- Chapter 19 problems
  - Answers to Chapter 19 Problems
Chapter 20 Electromagnetic induction: time-varying emfs
- 20.1 The principle of electromagnetic induction
- 20.2 Simple applications of electromagnetic induction
- 20.3 Self-inductance
- 20.4 The series L-R circuit
- 20.5 Discharge of a capacitor through an inductor and resistor
- 20.6 Time-varying emfs: mutual inductance: transformers
- 20.7 Alternating current (a.c.)
- 20.8 Alternating current transformers
- 20.9 Resistance, capacitance and inductance in a.c. circuits
- 20.10 The series L-C-R circuit: phasor diagrams
- 20.11 Power in an a.c. circuit
- Chapter 20 problems
  - Answers to Chapter 20 Problems
Chapter 21 Maxwell’s equations; electromagnetic radiation
- 21.1 Reconsideration of the laws of electromagnetism: Maxwell's equations
- 21.2 Plane electromagnetic waves
- 21.3 Experimental observation of electromagnetic radiation
- 21.4 The electromagnetic spectrum
- 21.5 Polarization of electromagnetic waves
- 21.6 Energy, momentum and angular momentum in electromagnetic waves
- 21.7 The photon model revisited
- 21.8 Reflection of electromagnetic waves at an interface between non-conducting media
- 21.9 Electromagnetic waves in a conducting medium
- 21.10 Invariance of electromagnetism under the Lorentz transformation
- 21.11 Maxwell’s equations in differential form
- Chapter 21 problems
  - Answer to Chapter 21 problems
Chapter 22 Wave optics
- 22.1 Electromagnetic nature of light
- 22.2 Coherence: the laser
- 22.3 Diffraction at a single slit
- 22.4 Two slit interference and diffraction: Young's double slit experiment
- 22.5 Multiple slit interference: the diffraction grating
- 22.6 Diffraction of X-rays: Bragg scattering
- 22.7 The SI unit of luminous intensity, the candela
- Chapter 22 problems
  - Answers to Chapter 22 Problems
Chapter 23 Geometrical optics
- 23.1 The ray model: geometric optics
- 23.2 Reflection of light
- 23.3 Image formation by spherical mirrors
- 23.4 Refraction of light
- 23.5 Refraction at successive plane interfaces
- 23.6 Image formation by spherical lenses
- 23.7 Image formation of extended objects: magnification
- 23.8 Dispersion of light
- Chapter 23 problems
  - Answers to Chapter 23 problems
Chapter 24 Atomic Physics
- 24.1 Atomic models
- 24.2 The spectrum of hydrogen: the Rydberg formula
- 24.3 The Bohr postulates
- 24.4 The Bohr theory of the hydrogen atom
- 24.5 The quantum mechanical (Schrödinger) solution of the one-electron atom
  - 24.5.1 The angular and radial equations for a one-electron atom
  - 24.5.2 The radial solutions of the lowest energy state of hydrogen
- 24.6 Interpretation of the one-electron atom eigenfunctions
- 24.7 Intensities of spectral lines: selection rules
  - 24.7.1 Radiation from an accelerated charge
  - 24.7.2 Expectation value of the electric dipole moment
- 24.8 Quantisation of angular momentum
  - 24.8.1 The angular momentum quantisation equations
- 24.9 Magnetic effects in one-electron atoms: the Zeeman effect
- 24.10 The Stern-Gerlach experiment: electron spin
  - 24.10.1 The Zeeman effect
- 24.11 The spin-orbit interaction
  - 24.11.1 The Thomas precession
- 24.12 Identical particles in quantum mechanics: the Pauli exclusion principle
- 24.13 The periodic table: multielectron atoms
- 24.14 The theory of multielectron atoms
- 24.15 Further uses of the solutions of the one-electron atom
- Chapter 24 problems
  - Answers to Chapter 24 problems
Chapter 25 Electrons in solids: quantum statistics
- 25.1 Bonding in molecules and solids
- 25.2 The classical free electron model of solids
- 25.3 The quantum mechanical free electron model of solids: Fermi energy
- 25.4 The electron energy distribution at zero K
- 25.5 Electron energy distributions at T > zero K
  - 25.5.1 The quantum distribution functions
- 25.6 Specific heat capacity and conductivity in the quantum free electron model
- 25.7 Quantum statistics: systems of bosons
- 25.8 Superconductivity
- Chapter 25 problems
  - Answers to Chapter 25 problems
Chapter 26 Semiconductors
- 26.1 The band theory of solids
- 26.2 Conductors, insulators and semiconductors
- 26.3 Intrinsic and extrinsic (doped) semiconductors
- 26.4 Junctions in conductors
- 26.5 Junction in semiconductors; the p-n junction
- 26.6 Biased p-n junctions; the semiconductor diode
- 26.7 Photodiodes, particle detectors and solar cells
- 26.8 Light emitting diodes; semiconductor lasers
- 26. 9 The tunnel diode
- 26.10 Transistors
- Chapter 26 problems
  - Answers to Chapter 26 problems
Chapter 27 Nuclear and particle physics
- 27.1 Properties of atomic nuclei
- 27.2 Nuclear binding energies
- 27.3 Nuclear models
- 27.4 Radioactivity
- 27.5 alpha-, beta- and gamma-decay
- 27.6 Detection of radiation: units of radioactivity
- 27.7 Nuclear reactions
- 27.8 Nuclear fission and nuclear fusion
- 27.9 Fission reactors
- 27.10 Thermonuclear fusion
- 27.11 Sub-nuclear particles
- 27.12 The quark model
- Chapter 27 problems
  - Answers to Chapter 27 problems
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Chapter 22 Wave optics
Chapter 22 problems
Answers to Chapter 22 Problems

22.1.1 114 K

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22.1.2 the intensity is reduced by a factor of ten

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22.1.3 (a) 0° (b) 45° (c) 63.4°

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22.1.4 (a) 0 (b) I₀/8

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22.1.5 (a) 722 nm (b) 726 nm

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22.2.1 7.96 x 10¹⁴ W m^–2

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22.2.2 (a) 0.40 W m^–2 (b) 10¹⁶ W

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22.3.1 74.5 m

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22.3.2 0.17 mm; 7 mm

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22.4.1 2.95 mm

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22.5.1 (a) 4.01 × 10^–6 m (b) 1.00 × 10^–6 m (c) n = 0, 1, 2, 3, 5 and 6

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22.5.2 8.4 × 10⁵ lines m^–1

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22.5.3 10⁻⁶ m

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22.6.1 1.3 × 10⁻¹⁰ m

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22.6.2 (a) 2.9 × 10⁻¹⁰ m (b) 4

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22.6.3 14.9°

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Answers to Chapter 22 Problems

Understanding Physics

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