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College Physics

Section Summary

College PhysicsSection Summary
  1. Preface
  2. 1 Introduction: The Nature of Science and Physics
    1. Introduction to Science and the Realm of Physics, Physical Quantities, and Units
    2. 1.1 Physics: An Introduction
    3. 1.2 Physical Quantities and Units
    4. 1.3 Accuracy, Precision, and Significant Figures
    5. 1.4 Approximation
    6. Glossary
    7. Section Summary
    8. Conceptual Questions
    9. Problems & Exercises
  3. 2 Kinematics
    1. Introduction to One-Dimensional Kinematics
    2. 2.1 Displacement
    3. 2.2 Vectors, Scalars, and Coordinate Systems
    4. 2.3 Time, Velocity, and Speed
    5. 2.4 Acceleration
    6. 2.5 Motion Equations for Constant Acceleration in One Dimension
    7. 2.6 Problem-Solving Basics for One-Dimensional Kinematics
    8. 2.7 Falling Objects
    9. 2.8 Graphical Analysis of One-Dimensional Motion
    10. Glossary
    11. Section Summary
    12. Conceptual Questions
    13. Problems & Exercises
  4. 3 Two-Dimensional Kinematics
    1. Introduction to Two-Dimensional Kinematics
    2. 3.1 Kinematics in Two Dimensions: An Introduction
    3. 3.2 Vector Addition and Subtraction: Graphical Methods
    4. 3.3 Vector Addition and Subtraction: Analytical Methods
    5. 3.4 Projectile Motion
    6. 3.5 Addition of Velocities
    7. Glossary
    8. Section Summary
    9. Conceptual Questions
    10. Problems & Exercises
  5. 4 Dynamics: Force and Newton's Laws of Motion
    1. Introduction to Dynamics: Newton’s Laws of Motion
    2. 4.1 Development of Force Concept
    3. 4.2 Newton’s First Law of Motion: Inertia
    4. 4.3 Newton’s Second Law of Motion: Concept of a System
    5. 4.4 Newton’s Third Law of Motion: Symmetry in Forces
    6. 4.5 Normal, Tension, and Other Examples of Forces
    7. 4.6 Problem-Solving Strategies
    8. 4.7 Further Applications of Newton’s Laws of Motion
    9. 4.8 Extended Topic: The Four Basic Forces—An Introduction
    10. Glossary
    11. Section Summary
    12. Conceptual Questions
    13. Problems & Exercises
  6. 5 Further Applications of Newton's Laws: Friction, Drag, and Elasticity
    1. Introduction: Further Applications of Newton’s Laws
    2. 5.1 Friction
    3. 5.2 Drag Forces
    4. 5.3 Elasticity: Stress and Strain
    5. Glossary
    6. Section Summary
    7. Conceptual Questions
    8. Problems & Exercises
  7. 6 Uniform Circular Motion and Gravitation
    1. Introduction to Uniform Circular Motion and Gravitation
    2. 6.1 Rotation Angle and Angular Velocity
    3. 6.2 Centripetal Acceleration
    4. 6.3 Centripetal Force
    5. 6.4 Fictitious Forces and Non-inertial Frames: The Coriolis Force
    6. 6.5 Newton’s Universal Law of Gravitation
    7. 6.6 Satellites and Kepler’s Laws: An Argument for Simplicity
    8. Glossary
    9. Section Summary
    10. Conceptual Questions
    11. Problems & Exercises
  8. 7 Work, Energy, and Energy Resources
    1. Introduction to Work, Energy, and Energy Resources
    2. 7.1 Work: The Scientific Definition
    3. 7.2 Kinetic Energy and the Work-Energy Theorem
    4. 7.3 Gravitational Potential Energy
    5. 7.4 Conservative Forces and Potential Energy
    6. 7.5 Nonconservative Forces
    7. 7.6 Conservation of Energy
    8. 7.7 Power
    9. 7.8 Work, Energy, and Power in Humans
    10. 7.9 World Energy Use
    11. Glossary
    12. Section Summary
    13. Conceptual Questions
    14. Problems & Exercises
  9. 8 Linear Momentum and Collisions
    1. Introduction to Linear Momentum and Collisions
    2. 8.1 Linear Momentum and Force
    3. 8.2 Impulse
    4. 8.3 Conservation of Momentum
    5. 8.4 Elastic Collisions in One Dimension
    6. 8.5 Inelastic Collisions in One Dimension
    7. 8.6 Collisions of Point Masses in Two Dimensions
    8. 8.7 Introduction to Rocket Propulsion
    9. Glossary
    10. Section Summary
    11. Conceptual Questions
    12. Problems & Exercises
  10. 9 Statics and Torque
    1. Introduction to Statics and Torque
    2. 9.1 The First Condition for Equilibrium
    3. 9.2 The Second Condition for Equilibrium
    4. 9.3 Stability
    5. 9.4 Applications of Statics, Including Problem-Solving Strategies
    6. 9.5 Simple Machines
    7. 9.6 Forces and Torques in Muscles and Joints
    8. Glossary
    9. Section Summary
    10. Conceptual Questions
    11. Problems & Exercises
  11. 10 Rotational Motion and Angular Momentum
    1. Introduction to Rotational Motion and Angular Momentum
    2. 10.1 Angular Acceleration
    3. 10.2 Kinematics of Rotational Motion
    4. 10.3 Dynamics of Rotational Motion: Rotational Inertia
    5. 10.4 Rotational Kinetic Energy: Work and Energy Revisited
    6. 10.5 Angular Momentum and Its Conservation
    7. 10.6 Collisions of Extended Bodies in Two Dimensions
    8. 10.7 Gyroscopic Effects: Vector Aspects of Angular Momentum
    9. Glossary
    10. Section Summary
    11. Conceptual Questions
    12. Problems & Exercises
  12. 11 Fluid Statics
    1. Introduction to Fluid Statics
    2. 11.1 What Is a Fluid?
    3. 11.2 Density
    4. 11.3 Pressure
    5. 11.4 Variation of Pressure with Depth in a Fluid
    6. 11.5 Pascal’s Principle
    7. 11.6 Gauge Pressure, Absolute Pressure, and Pressure Measurement
    8. 11.7 Archimedes’ Principle
    9. 11.8 Cohesion and Adhesion in Liquids: Surface Tension and Capillary Action
    10. 11.9 Pressures in the Body
    11. Glossary
    12. Section Summary
    13. Conceptual Questions
    14. Problems & Exercises
  13. 12 Fluid Dynamics and Its Biological and Medical Applications
    1. Introduction to Fluid Dynamics and Its Biological and Medical Applications
    2. 12.1 Flow Rate and Its Relation to Velocity
    3. 12.2 Bernoulli’s Equation
    4. 12.3 The Most General Applications of Bernoulli’s Equation
    5. 12.4 Viscosity and Laminar Flow; Poiseuille’s Law
    6. 12.5 The Onset of Turbulence
    7. 12.6 Motion of an Object in a Viscous Fluid
    8. 12.7 Molecular Transport Phenomena: Diffusion, Osmosis, and Related Processes
    9. Glossary
    10. Section Summary
    11. Conceptual Questions
    12. Problems & Exercises
  14. 13 Temperature, Kinetic Theory, and the Gas Laws
    1. Introduction to Temperature, Kinetic Theory, and the Gas Laws
    2. 13.1 Temperature
    3. 13.2 Thermal Expansion of Solids and Liquids
    4. 13.3 The Ideal Gas Law
    5. 13.4 Kinetic Theory: Atomic and Molecular Explanation of Pressure and Temperature
    6. 13.5 Phase Changes
    7. 13.6 Humidity, Evaporation, and Boiling
    8. Glossary
    9. Section Summary
    10. Conceptual Questions
    11. Problems & Exercises
  15. 14 Heat and Heat Transfer Methods
    1. Introduction to Heat and Heat Transfer Methods
    2. 14.1 Heat
    3. 14.2 Temperature Change and Heat Capacity
    4. 14.3 Phase Change and Latent Heat
    5. 14.4 Heat Transfer Methods
    6. 14.5 Conduction
    7. 14.6 Convection
    8. 14.7 Radiation
    9. Glossary
    10. Section Summary
    11. Conceptual Questions
    12. Problems & Exercises
  16. 15 Thermodynamics
    1. Introduction to Thermodynamics
    2. 15.1 The First Law of Thermodynamics
    3. 15.2 The First Law of Thermodynamics and Some Simple Processes
    4. 15.3 Introduction to the Second Law of Thermodynamics: Heat Engines and Their Efficiency
    5. 15.4 Carnot’s Perfect Heat Engine: The Second Law of Thermodynamics Restated
    6. 15.5 Applications of Thermodynamics: Heat Pumps and Refrigerators
    7. 15.6 Entropy and the Second Law of Thermodynamics: Disorder and the Unavailability of Energy
    8. 15.7 Statistical Interpretation of Entropy and the Second Law of Thermodynamics: The Underlying Explanation
    9. Glossary
    10. Section Summary
    11. Conceptual Questions
    12. Problems & Exercises
  17. 16 Oscillatory Motion and Waves
    1. Introduction to Oscillatory Motion and Waves
    2. 16.1 Hooke’s Law: Stress and Strain Revisited
    3. 16.2 Period and Frequency in Oscillations
    4. 16.3 Simple Harmonic Motion: A Special Periodic Motion
    5. 16.4 The Simple Pendulum
    6. 16.5 Energy and the Simple Harmonic Oscillator
    7. 16.6 Uniform Circular Motion and Simple Harmonic Motion
    8. 16.7 Damped Harmonic Motion
    9. 16.8 Forced Oscillations and Resonance
    10. 16.9 Waves
    11. 16.10 Superposition and Interference
    12. 16.11 Energy in Waves: Intensity
    13. Glossary
    14. Section Summary
    15. Conceptual Questions
    16. Problems & Exercises
  18. 17 Physics of Hearing
    1. Introduction to the Physics of Hearing
    2. 17.1 Sound
    3. 17.2 Speed of Sound, Frequency, and Wavelength
    4. 17.3 Sound Intensity and Sound Level
    5. 17.4 Doppler Effect and Sonic Booms
    6. 17.5 Sound Interference and Resonance: Standing Waves in Air Columns
    7. 17.6 Hearing
    8. 17.7 Ultrasound
    9. Glossary
    10. Section Summary
    11. Conceptual Questions
    12. Problems & Exercises
  19. 18 Electric Charge and Electric Field
    1. Introduction to Electric Charge and Electric Field
    2. 18.1 Static Electricity and Charge: Conservation of Charge
    3. 18.2 Conductors and Insulators
    4. 18.3 Coulomb’s Law
    5. 18.4 Electric Field: Concept of a Field Revisited
    6. 18.5 Electric Field Lines: Multiple Charges
    7. 18.6 Electric Forces in Biology
    8. 18.7 Conductors and Electric Fields in Static Equilibrium
    9. 18.8 Applications of Electrostatics
    10. Glossary
    11. Section Summary
    12. Conceptual Questions
    13. Problems & Exercises
  20. 19 Electric Potential and Electric Field
    1. Introduction to Electric Potential and Electric Energy
    2. 19.1 Electric Potential Energy: Potential Difference
    3. 19.2 Electric Potential in a Uniform Electric Field
    4. 19.3 Electrical Potential Due to a Point Charge
    5. 19.4 Equipotential Lines
    6. 19.5 Capacitors and Dielectrics
    7. 19.6 Capacitors in Series and Parallel
    8. 19.7 Energy Stored in Capacitors
    9. Glossary
    10. Section Summary
    11. Conceptual Questions
    12. Problems & Exercises
  21. 20 Electric Current, Resistance, and Ohm's Law
    1. Introduction to Electric Current, Resistance, and Ohm's Law
    2. 20.1 Current
    3. 20.2 Ohm’s Law: Resistance and Simple Circuits
    4. 20.3 Resistance and Resistivity
    5. 20.4 Electric Power and Energy
    6. 20.5 Alternating Current versus Direct Current
    7. 20.6 Electric Hazards and the Human Body
    8. 20.7 Nerve Conduction–Electrocardiograms
    9. Glossary
    10. Section Summary
    11. Conceptual Questions
    12. Problems & Exercises
  22. 21 Circuits and DC Instruments
    1. Introduction to Circuits and DC Instruments
    2. 21.1 Resistors in Series and Parallel
    3. 21.2 Electromotive Force: Terminal Voltage
    4. 21.3 Kirchhoff’s Rules
    5. 21.4 DC Voltmeters and Ammeters
    6. 21.5 Null Measurements
    7. 21.6 DC Circuits Containing Resistors and Capacitors
    8. Glossary
    9. Section Summary
    10. Conceptual Questions
    11. Problems & Exercises
  23. 22 Magnetism
    1. Introduction to Magnetism
    2. 22.1 Magnets
    3. 22.2 Ferromagnets and Electromagnets
    4. 22.3 Magnetic Fields and Magnetic Field Lines
    5. 22.4 Magnetic Field Strength: Force on a Moving Charge in a Magnetic Field
    6. 22.5 Force on a Moving Charge in a Magnetic Field: Examples and Applications
    7. 22.6 The Hall Effect
    8. 22.7 Magnetic Force on a Current-Carrying Conductor
    9. 22.8 Torque on a Current Loop: Motors and Meters
    10. 22.9 Magnetic Fields Produced by Currents: Ampere’s Law
    11. 22.10 Magnetic Force between Two Parallel Conductors
    12. 22.11 More Applications of Magnetism
    13. Glossary
    14. Section Summary
    15. Conceptual Questions
    16. Problems & Exercises
  24. 23 Electromagnetic Induction, AC Circuits, and Electrical Technologies
    1. Introduction to Electromagnetic Induction, AC Circuits and Electrical Technologies
    2. 23.1 Induced Emf and Magnetic Flux
    3. 23.2 Faraday’s Law of Induction: Lenz’s Law
    4. 23.3 Motional Emf
    5. 23.4 Eddy Currents and Magnetic Damping
    6. 23.5 Electric Generators
    7. 23.6 Back Emf
    8. 23.7 Transformers
    9. 23.8 Electrical Safety: Systems and Devices
    10. 23.9 Inductance
    11. 23.10 RL Circuits
    12. 23.11 Reactance, Inductive and Capacitive
    13. 23.12 RLC Series AC Circuits
    14. Glossary
    15. Section Summary
    16. Conceptual Questions
    17. Problems & Exercises
  25. 24 Electromagnetic Waves
    1. Introduction to Electromagnetic Waves
    2. 24.1 Maxwell’s Equations: Electromagnetic Waves Predicted and Observed
    3. 24.2 Production of Electromagnetic Waves
    4. 24.3 The Electromagnetic Spectrum
    5. 24.4 Energy in Electromagnetic Waves
    6. Glossary
    7. Section Summary
    8. Conceptual Questions
    9. Problems & Exercises
  26. 25 Geometric Optics
    1. Introduction to Geometric Optics
    2. 25.1 The Ray Aspect of Light
    3. 25.2 The Law of Reflection
    4. 25.3 The Law of Refraction
    5. 25.4 Total Internal Reflection
    6. 25.5 Dispersion: The Rainbow and Prisms
    7. 25.6 Image Formation by Lenses
    8. 25.7 Image Formation by Mirrors
    9. Glossary
    10. Section Summary
    11. Conceptual Questions
    12. Problems & Exercises
  27. 26 Vision and Optical Instruments
    1. Introduction to Vision and Optical Instruments
    2. 26.1 Physics of the Eye
    3. 26.2 Vision Correction
    4. 26.3 Color and Color Vision
    5. 26.4 Microscopes
    6. 26.5 Telescopes
    7. 26.6 Aberrations
    8. Glossary
    9. Section Summary
    10. Conceptual Questions
    11. Problems & Exercises
  28. 27 Wave Optics
    1. Introduction to Wave Optics
    2. 27.1 The Wave Aspect of Light: Interference
    3. 27.2 Huygens's Principle: Diffraction
    4. 27.3 Young’s Double Slit Experiment
    5. 27.4 Multiple Slit Diffraction
    6. 27.5 Single Slit Diffraction
    7. 27.6 Limits of Resolution: The Rayleigh Criterion
    8. 27.7 Thin Film Interference
    9. 27.8 Polarization
    10. 27.9 *Extended Topic* Microscopy Enhanced by the Wave Characteristics of Light
    11. Glossary
    12. Section Summary
    13. Conceptual Questions
    14. Problems & Exercises
  29. 28 Special Relativity
    1. Introduction to Special Relativity
    2. 28.1 Einstein’s Postulates
    3. 28.2 Simultaneity And Time Dilation
    4. 28.3 Length Contraction
    5. 28.4 Relativistic Addition of Velocities
    6. 28.5 Relativistic Momentum
    7. 28.6 Relativistic Energy
    8. Glossary
    9. Section Summary
    10. Conceptual Questions
    11. Problems & Exercises
  30. 29 Introduction to Quantum Physics
    1. Introduction to Quantum Physics
    2. 29.1 Quantization of Energy
    3. 29.2 The Photoelectric Effect
    4. 29.3 Photon Energies and the Electromagnetic Spectrum
    5. 29.4 Photon Momentum
    6. 29.5 The Particle-Wave Duality
    7. 29.6 The Wave Nature of Matter
    8. 29.7 Probability: The Heisenberg Uncertainty Principle
    9. 29.8 The Particle-Wave Duality Reviewed
    10. Glossary
    11. Section Summary
    12. Conceptual Questions
    13. Problems & Exercises
  31. 30 Atomic Physics
    1. Introduction to Atomic Physics
    2. 30.1 Discovery of the Atom
    3. 30.2 Discovery of the Parts of the Atom: Electrons and Nuclei
    4. 30.3 Bohr’s Theory of the Hydrogen Atom
    5. 30.4 X Rays: Atomic Origins and Applications
    6. 30.5 Applications of Atomic Excitations and De-Excitations
    7. 30.6 The Wave Nature of Matter Causes Quantization
    8. 30.7 Patterns in Spectra Reveal More Quantization
    9. 30.8 Quantum Numbers and Rules
    10. 30.9 The Pauli Exclusion Principle
    11. Glossary
    12. Section Summary
    13. Conceptual Questions
    14. Problems & Exercises
  32. 31 Radioactivity and Nuclear Physics
    1. Introduction to Radioactivity and Nuclear Physics
    2. 31.1 Nuclear Radioactivity
    3. 31.2 Radiation Detection and Detectors
    4. 31.3 Substructure of the Nucleus
    5. 31.4 Nuclear Decay and Conservation Laws
    6. 31.5 Half-Life and Activity
    7. 31.6 Binding Energy
    8. 31.7 Tunneling
    9. Glossary
    10. Section Summary
    11. Conceptual Questions
    12. Problems & Exercises
  33. 32 Medical Applications of Nuclear Physics
    1. Introduction to Applications of Nuclear Physics
    2. 32.1 Medical Imaging and Diagnostics
    3. 32.2 Biological Effects of Ionizing Radiation
    4. 32.3 Therapeutic Uses of Ionizing Radiation
    5. 32.4 Food Irradiation
    6. 32.5 Fusion
    7. 32.6 Fission
    8. 32.7 Nuclear Weapons
    9. Glossary
    10. Section Summary
    11. Conceptual Questions
    12. Problems & Exercises
  34. 33 Particle Physics
    1. Introduction to Particle Physics
    2. 33.1 The Yukawa Particle and the Heisenberg Uncertainty Principle Revisited
    3. 33.2 The Four Basic Forces
    4. 33.3 Accelerators Create Matter from Energy
    5. 33.4 Particles, Patterns, and Conservation Laws
    6. 33.5 Quarks: Is That All There Is?
    7. 33.6 GUTs: The Unification of Forces
    8. Glossary
    9. Section Summary
    10. Conceptual Questions
    11. Problems & Exercises
  35. 34 Frontiers of Physics
    1. Introduction to Frontiers of Physics
    2. 34.1 Cosmology and Particle Physics
    3. 34.2 General Relativity and Quantum Gravity
    4. 34.3 Superstrings
    5. 34.4 Dark Matter and Closure
    6. 34.5 Complexity and Chaos
    7. 34.6 High-temperature Superconductors
    8. 34.7 Some Questions We Know to Ask
    9. Glossary
    10. Section Summary
    11. Conceptual Questions
    12. Problems & Exercises
  36. A | Atomic Masses
  37. B | Selected Radioactive Isotopes
  38. C | Useful Information
  39. D | Glossary of Key Symbols and Notation
  40. Index

23.1 Induced Emf and Magnetic Flux

  • The crucial quantity in induction is magnetic flux ΦΦ size 12{Φ} {}, defined to be Φ=BAcosθΦ=BAcosθ size 12{Φ= ital "BA""cos"θ} {}, where BB size 12{B} {} is the magnetic field strength over an area AA size 12{A} {} at an angle θθ size 12{θ} {} with the perpendicular to the area.
  • Units of magnetic flux ΦΦ size 12{Φ} {} are Tm2Tm2 size 12{T cdot m rSup { size 8{2} } } {}.
  • Any change in magnetic flux ΦΦ size 12{Φ} {} induces an emf—the process is defined to be electromagnetic induction.

23.2 Faraday’s Law of Induction: Lenz’s Law

  • Faraday’s law of induction states that the emfinduced by a change in magnetic flux is
    emf = N Δ Φ Δt emf = N Δ Φ Δt size 12{"emf"= - N { {ΔΦ} over {Δt} } } {}

    when flux changes by ΔΦΔΦ size 12{ΔΦ} {} in a time ΔtΔt size 12{Δt} {}.

  • If emf is induced in a coil, N N is its number of turns.
  • The minus sign means that the emf creates a current II size 12{I} {} and magnetic field BB size 12{B} {} that oppose the change in flux ΔΦΔΦ size 12{ΔΦ} {} —this opposition is known as Lenz’s law.

23.3 Motional Emf

  • An emf induced by motion relative to a magnetic field B B is called a motional emf and is given by
    emf=Bℓv(B, , andv perpendicular),emf=Bℓv(B, , andv perpendicular), size 12{"emf"=Bℓv} {}
    where size 12{ℓ} {} is the length of the object moving at speed vv size 12{v} {} relative to the field.

23.4 Eddy Currents and Magnetic Damping

  • Current loops induced in moving conductors are called eddy currents.
  • They can create significant drag, called magnetic damping.

23.5 Electric Generators

  • An electric generator rotates a coil in a magnetic field, inducing an emfgiven as a function of time by
    emf=NABωsinωt,emf=NABωsinωt, size 12{"emf"= ital "NAB"ω"sin"ωt} {}
    where AA size 12{A} {} is the area of an NN size 12{N} {}-turn coil rotated at a constant angular velocity ωω size 12{ω} {} in a uniform magnetic field BB size 12{B} {}.
  • The peak emf emf0emf0 size 12{"emf" rSub { size 8{0} } } {} of a generator is
    emf0=NABω.emf0=NABω. size 12{"emf" rSub { size 8{0} } = ital "NAB"ω} {}

23.6 Back Emf

  • Any rotating coil will have an induced emf—in motors, this is called back emf, since it opposes the emf input to the motor.

23.7 Transformers

  • Transformers use induction to transform voltages from one value to another.
  • For a transformer, the voltages across the primary and secondary coils are related by
    VsVp=NsNp,VsVp=NsNp, size 12{ { {V rSub { size 8{s} } } over {V rSub { size 8{p} } } } = { {N rSub { size 8{s} } } over {N rSub { size 8{p} } } } } {}
    where VpVp size 12{V rSub { size 8{p} } } {} and VsVs size 12{V rSub { size 8{s} } } {} are the voltages across primary and secondary coils having NpNp size 12{N rSub { size 8{p} } } {} and NsNs size 12{N rSub { size 8{s} } } {} turns.
  • The currents IpIp size 12{I rSub { size 8{p} } } {} and IsIs size 12{I rSub { size 8{s} } } {} in the primary and secondary coils are related by IsIp=NpNsIsIp=NpNs size 12{ { {I rSub { size 8{s} } } over {I rSub { size 8{p} } } } = { {N rSub { size 8{p} } } over {N rSub { size 8{s} } } } } {}.
  • A step-up transformer increases voltage and decreases current, whereas a step-down transformer decreases voltage and increases current.

23.8 Electrical Safety: Systems and Devices

  • Electrical safety systems and devices are employed to prevent thermal and shock hazards.
  • Circuit breakers and fuses interrupt excessive currents to prevent thermal hazards.
  • The three-wire system guards against thermal and shock hazards, utilizing live/hot, neutral, and earth/ground wires, and grounding the neutral wire and case of the appliance.
  • A ground fault interrupter (GFI) prevents shock by detecting the loss of current to unintentional paths.
  • An isolation transformer insulates the device being powered from the original source, also to prevent shock.
  • Many of these devices use induction to perform their basic function.

23.9 Inductance

  • Inductance is the property of a device that tells how effectively it induces an emf in another device.
  • Mutual inductance is the effect of two devices in inducing emfs in each other.
  • A change in current ΔI1/ΔtΔI1/Δt size 12{ΔI rSub { size 8{1} } /Δt} {} in one induces an emf emf2emf2 size 12{"emf" rSub { size 8{2} } } {} in the second:
    emf2=MΔI1Δt,emf2=MΔI1Δt, size 12{"emf" rSub { size 8{2} } = - M { {ΔI rSub { size 8{1} } } over {Δt} } } {}
    where M M is defined to be the mutual inductance between the two devices, and the minus sign is due to Lenz’s law.
  • Symmetrically, a change in current ΔI2/ΔtΔI2/Δt size 12{ΔI rSub { size 8{2} } /Δt} {} through the second device induces an emf emf1emf1 size 12{"emf" rSub { size 8{1} } } {} in the first:
    emf1=MΔI2Δt,emf1=MΔI2Δt, size 12{"emf" rSub { size 8{1} } = - M { {ΔI rSub { size 8{2} } } over {Δt} } } {}
    where M M is the same mutual inductance as in the reverse process.
  • Current changes in a device induce an emf in the device itself.
  • Self-inductance is the effect of the device inducing emf in itself.
  • The device is called an inductor, and the emf induced in it by a change in current through it is
    emf=LΔIΔt,emf=LΔIΔt, size 12{"emf"= - L { {ΔI} over {Δt} } } {}
    where LL size 12{L} {} is the self-inductance of the inductor, and ΔI/ΔtΔI/Δt size 12{ΔI/Δt} {} is the rate of change of current through it. The minus sign indicates that emf opposes the change in current, as required by Lenz’s law.
  • The unit of self- and mutual inductance is the henry (H), where 1 H=1 Ωs1 H=1 Ωs size 12{1`H=1` %OMEGA cdot s} {}.
  • The self-inductance LL size 12{L} {} of an inductor is proportional to how much flux changes with current. For an NN size 12{N} {}-turn inductor,
    L=NΔΦΔI .L=NΔΦΔI . size 12{L=N { {ΔΦ} over {ΔI} } } {}
  • The self-inductance of a solenoid is
    L=μ0N2A(solenoid),L=μ0N2A(solenoid), size 12{L= { {μ rSub { size 8{0} } N rSup { size 8{2} } A} over {ℓ} } } {}
    where NN size 12{N} {} is its number of turns in the solenoid, AA size 12{A} {} is its cross-sectional area, size 12{ℓ} {} is its length, and μ0=×10−7Tm/Aμ0=×10−7Tm/A size 12{μ rSub { size 8{0} } =4π times "10" rSup { size 8{"-7"} } `T cdot "m/A"} {} is the permeability of free space.
  • The energy stored in an inductor EindEind size 12{E rSub { size 8{"ind"} } } {} is
    Eind=12LI2.Eind=12LI2. size 12{E rSub { size 8{"ind"} } = { {1} over {2} } ital "LI" rSup { size 8{2} } } {}

23.10 RL Circuits

  • When a series connection of a resistor and an inductor—an RL circuit—is connected to a voltage source, the time variation of the current is
    I=I0(1et/τ)    (turning on).I=I0(1et/τ)    (turning on). size 12{I=I rSub { size 8{0} } \( 1 - e rSup { size 8{ - t/τ} } \) } {}
    where I0=V/RI0=V/R size 12{I rSub { size 8{0} } =V/R} {} is the final current.
  • The characteristic time constant ττ size 12{τ} {} is τ=LRτ=LR size 12{τ= { {L} over {R} } } {} , where L L is the inductance and R R is the resistance.
  • In the first time constant ττ size 12{τ} {}, the current rises from zero to 0.632I00.632I0 size 12{0 "." "632"I rSub { size 8{0} } } {}, and 0.632 of the remainder in every subsequent time interval ττ size 12{τ} {}.
  • When the inductor is shorted through a resistor, current decreases as
    I=I0et/τ    (turning off).I=I0et/τ    (turning off). size 12{I=I rSub { size 8{0} } e rSup { size 8{ - t/τ} } } {}
    Here I0I0 size 12{I rSub { size 8{0} } } {} is the initial current.
  • Current falls to 0.368I00.368I0 size 12{0 "." "368"I rSub { size 8{0} } } {} in the first time interval ττ size 12{τ} {}, and 0.368 of the remainder toward zero in each subsequent time ττ size 12{τ} {}.

23.11 Reactance, Inductive and Capacitive

  • For inductors in AC circuits, we find that when a sinusoidal voltage is applied to an inductor, the voltage leads the current by one-fourth of a cycle, or by a 90º 90º phase angle.
  • The opposition of an inductor to a change in current is expressed as a type of AC resistance.
  • Ohm’s law for an inductor is
    I=VXL,I=VXL, size 12{I= { {V} over {X rSub { size 8{L} } } } } {}
    where VV size 12{V} {} is the rms voltage across the inductor.
  • XLXL size 12{X rSub { size 8{L} } } {} is defined to be the inductive reactance, given by
    XL=fL,XL=fL, size 12{X rSub { size 8{L} } =2π ital "fL"} {}
    with ff size 12{f} {} the frequency of the AC voltage source in hertz.
  • Inductive reactance XLXL size 12{X rSub { size 8{L} } } {} has units of ohms and is greatest at high frequencies.
  • For capacitors, we find that when a sinusoidal voltage is applied to a capacitor, the voltage follows the current by one-fourth of a cycle, or by a 90º 90º phase angle.
  • Since a capacitor can stop current when fully charged, it limits current and offers another form of AC resistance; Ohm’s law for a capacitor is
    I=VXC,I=VXC, size 12{I= { {V} over {X rSub { size 8{C} } } } } {}
    where VV size 12{V} {} is the rms voltage across the capacitor.
  • XCXC size 12{X rSub { size 8{C} } } {} is defined to be the capacitive reactance, given by
    XC=1fC.XC=1fC. size 12{X rSub { size 8{C} } = { {1} over {2π ital "fC"} } } {}
  • XCXC size 12{X rSub { size 8{C} } } {} has units of ohms and is greatest at low frequencies.

23.12 RLC Series AC Circuits

  • The AC analogy to resistance is impedance Z Z , the combined effect of resistors, inductors, and capacitors, defined by the AC version of Ohm’s law:
    I 0 = V 0 Z or I rms = V rms Z , I 0 = V 0 Z or I rms = V rms Z , size 12{I rSub { size 8{0} } = { {V rSub { size 8{0} } } over {Z} } " or "I rSub { size 8{ ital "rms"} } = { {V rSub { size 8{ ital "rms"} } } over {Z} } ,} {}
    where I0I0 size 12{I rSub { size 8{0} } } {} is the peak current and V0V0 size 12{V rSub { size 8{0} } } {} is the peak source voltage.
  • Impedance has units of ohms and is given by Z=R2+(XLXC)2Z=R2+(XLXC)2 size 12{Z= sqrt {R rSup { size 8{2} } + \( X rSub { size 8{L} } - X rSub { size 8{C} } \) rSup { size 8{2} } } } {}.
  • The resonant frequency f0f0 size 12{f rSub { size 8{0} } } {}, at which XL=XCXL=XC size 12{X rSub { size 8{L} } =X rSub { size 8{C} } } {}, is
    f0=1LC.f0=1LC. size 12{f rSub { size 8{0} } = { {1} over {2π sqrt { ital "LC"} } } } {}
  • In an AC circuit, there is a phase angle ϕϕ size 12{ϕ} {} between source voltage VV size 12{V} {} and the current II size 12{I} {}, which can be found from
    cosϕ=RZ,cosϕ=RZ, size 12{"cos"ϕ= { {R} over {Z} } } {}
  • ϕ=ϕ= size 12{ϕ=0 rSup { size 8{ circ } } } {} for a purely resistive circuit or an RLC circuit at resonance.
  • The average power delivered to an RLC circuit is affected by the phase angle and is given by
    Pave=IrmsVrmscosϕ,Pave=IrmsVrmscosϕ, size 12{P rSub { size 8{"ave"} } =I rSub { size 8{"rms"} } V rSub { size 8{"rms"} } "cos"ϕ} {}
    cosϕcosϕ size 12{"cos"ϕ} {} is called the power factor, which ranges from 0 to 1.
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