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College Physics

23.5 Electric Generators

College Physics23.5 Electric Generators
  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

Electric generators induce an emf by rotating a coil in a magnetic field, as briefly discussed in Induced Emf and Magnetic Flux. We will now explore generators in more detail. Consider the following example.

Example 23.3 Calculating the Emf Induced in a Generator Coil

The generator coil shown in Figure 23.20 is rotated through one-fourth of a revolution (from θ = θ = to θ = 90º θ = 90º ) in 15.0 ms. The 200-turn circular coil has a 5.00 cm radius and is in a uniform 1.25 T magnetic field. What is the average emf induced?

The figure shows a schematic diagram of an electric generator. It consists of a rotating rectangular coil placed between the two poles of a permanent magnet shown as two rectangular blocks curved on side facing the coil. The magnetic field B is shown pointing from the North to the South Pole. The two ends of this coil are connected to the two small rings. The two conducting carbon brushes are kept pressed separately on both the rings. The coil is attached to an axle with a handle at the other end. Outer ends of the two brushes are connected to the galvanometer. The axle is mechanically rotated from outside by an angle of ninety degree that is a one fourth revolution, to rotate the coil inside the magnetic field. A current is shown to flow in the coil in clockwise direction and the galvanometer shows a deflection to left.
Figure 23.20 When this generator coil is rotated through one-fourth of a revolution, the magnetic flux Φ Φ changes from its maximum to zero, inducing an emf.

Strategy

We use Faraday’s law of induction to find the average emf induced over a time ΔtΔt size 12{Δt} {}:

emf=NΔΦΔt.emf=NΔΦΔt. size 12{"emf"= - N { {ΔΦ} over {Δt} } } {}
23.11

We know that N=200N=200 size 12{N="200"} {} and Δt=15.0msΔt=15.0ms size 12{Δt="15" "." 0`"ms"} {}, and so we must determine the change in flux ΔΦΔΦ size 12{ΔΦ} {} to find emf.

Solution

Since the area of the loop and the magnetic field strength are constant, we see that

ΔΦ=Δ(BAcosθ)=ABΔ(cosθ).ΔΦ=Δ(BAcosθ)=ABΔ(cosθ). size 12{ΔΦ=Δ \( ital "BA""cos"θ \) = ital "AB"Δ \( "cos"θ \) } {}
23.12

Now, Δ(cosθ)=1.0Δ(cosθ)=1.0 size 12{Δ \( "cos"θ \) = - 1 "." 0} {}, since it was given that θ θ goes from to 90º 90º . Thus ΔΦ=ABΔΦ=AB size 12{ΔΦ= - ital "AB"} {}, and

emf=NABΔt.emf=NABΔt. size 12{"emf"=N { { ital "AB"} over {Δt} } } {}
23.13

The area of the loop is A=πr2=(3.14...)(0.0500m)2=7.85×103m2A=πr2=(3.14...)(0.0500m)2=7.85×103m2 size 12{A=πr rSup { size 8{2} } = \( 3 "." "14" "." "." "." \) \( 0 "." "0500"`m \) rSup { size 8{2} } =7 "." "85" times "10" rSup { size 8{ - 3} } `m rSup { size 8{2} } } {}. Entering this value gives

emf=200(7.85×103m2)(1.25 T)15.0×103 s=131V.emf=200(7.85×103m2)(1.25 T)15.0×103 s=131V. size 12{"emf"="200" { { \( 7 "." "85" times "10" rSup { size 8{ - 3} } " m" rSup { size 8{2} } \) \( 1 "." "25"" T" \) } over {"15" "." 0 times "10" rSup { size 8{ - 3} } " s"} } ="131"" V"} {}
23.14

Discussion

This is a practical average value, similar to the 120 V used in household power.

The emf calculated in Example 23.3 is the average over one-fourth of a revolution. What is the emf at any given instant? It varies with the angle between the magnetic field and a perpendicular to the coil. We can get an expression for emf as a function of time by considering the motional emf on a rotating rectangular coil of width ww size 12{w} {} and height size 12{l} {} in a uniform magnetic field, as illustrated in Figure 23.21.

The figure shows a schematic diagram of an electric generator with a single rectangular coil. The rotating rectangular coil is placed between the two poles of a permanent magnet shown as two rectangular blocks curved on side facing the coil. The magnetic field B is shown pointing from the North to the South Pole. The North Pole is on the left and the South Pole is to the right and hence the direction of field is from left to right. The angular velocity of the coil is given as omega. The velocity vector v of the coil makes an angle theta with the direction of field.
Figure 23.21 A generator with a single rectangular coil rotated at constant angular velocity in a uniform magnetic field produces an emf that varies sinusoidally in time. Note the generator is similar to a motor, except the shaft is rotated to produce a current rather than the other way around.

Charges in the wires of the loop experience the magnetic force, because they are moving in a magnetic field. Charges in the vertical wires experience forces parallel to the wire, causing currents. But those in the top and bottom segments feel a force perpendicular to the wire, which does not cause a current. We can thus find the induced emf by considering only the side wires. Motional emf is given to be emf=Bℓvemf=Bℓv size 12{"emf"=Bℓv} {}, where the velocity v is perpendicular to the magnetic field BB size 12{B} {}. Here the velocity is at an angle θθ size 12{θ} {} with BB size 12{B} {}, so that its component perpendicular to BB size 12{B} {} is vsin θvsin θ size 12{v"sin"θ} {} (see Figure 23.21). Thus in this case the emf induced on each side is emf=Bℓvsinθemf=Bℓvsinθ size 12{"emf"=Bℓv"sin"θ} {}, and they are in the same direction. The total emf around the loop is then

emf=2Bℓvsinθ.emf=2Bℓvsinθ. size 12{"emf"=2Bℓv"sin"θ} {}
23.15

This expression is valid, but it does not give emf as a function of time. To find the time dependence of emf, we assume the coil rotates at a constant angular velocity ωω size 12{ω} {}. The angle θθ size 12{θ} {} is related to angular velocity by θ=ωtθ=ωt size 12{θ=ωt} {}, so that

emf=2Bℓvsinωt.emf=2Bℓvsinωt. size 12{"emf"=Bℓv"sin"ωt} {}
23.16

Now, linear velocity v v is related to angular velocity ω ω by v=v= size 12{v=rω} {}. Here r=w/2r=w/2 size 12{r=w/2} {}, so that v=(w/2)ωv=(w/2)ω size 12{v= \( w/2 \) ω} {}, and

emf=2Bℓw2ωsinωt=(w)sinωt.emf=2Bℓw2ωsinωt=(w)sinωt. size 12{"emf"=2Bℓ { {w} over {2} } ω"sin"ωt= \( ℓw \) Bω"sin"ωt} {}
23.17

Noting that the area of the loop is A=wA=w size 12{A=ℓw} {}, and allowing for NN size 12{N} {} loops, we find that

emf = NAB ω sin ωt emf = NAB ω sin ωt size 12{"emf"= ital "NAB"ω"sin"ωt} {}
23.18

is the emf induced in a generator coil of NN size 12{N} {} turns and area AA size 12{A} {} rotating at a constant angular velocity ω ω in a uniform magnetic field BB size 12{B} {}. This can also be expressed as

emf=emf0sinωt,emf=emf0sinωt, size 12{"emf"="emf" rSub { size 8{0} } "sin"ωt} {}
23.19

where

emf 0 = NAB ω emf 0 = NAB ω size 12{"emf" rSub { size 8{0} } = ital "NAB"ω} {}
23.20

is the maximum (peak) emf. Note that the frequency of the oscillation is f=ω/f=ω/ size 12{f=ω/2π} {}, and the period is T=1/f=/ωT=1/f=/ω size 12{T=1/f=2π/ω} {}. Figure 23.22 shows a graph of emf as a function of time, and it now seems reasonable that AC voltage is sinusoidal.

The first part of the figure shows a schematic diagram of a single coil electric generator. It consists of a rotating rectangular loop placed between the two poles of a permanent magnet shown as two rectangular blocks curved on side facing the loop. The magnetic field B is shown pointing from the North to the South Pole. The two ends of this loop are connected to the two small rings. The two conducting carbon brushes are kept pressed separately on both the rings. The loop is rotated in the field with an angular velocity omega. Outer ends of the two brushes are connected to an electric bulb which is shown to glow brightly. The second part of the figure shows the graph for e m f generated E as a function of time t. The e m f is along the Y axis and the time t is along the X axis. The graph is a progressive sine wave with a time period T. The crest maxima are at E zero and trough minima are at negative E zero.
Figure 23.22 The emf of a generator is sent to a light bulb with the system of rings and brushes shown. The graph gives the emf of the generator as a function of time. emf0emf0 size 12{"emf" rSub { size 8{0} } } {} is the peak emf. The period is T=1/f=/ωT=1/f=/ω size 12{T=1/f=2π/ω} {}, where ff size 12{f} {} is the frequency. Note that the script E stands for emf.

The fact that the peak emf, emf0=NABωemf0=NABω size 12{"emf" rSub { size 8{0} } = ital "NAB"ω} {}, makes good sense. The greater the number of coils, the larger their area, and the stronger the field, the greater the output voltage. It is interesting that the faster the generator is spun (greater ωω size 12{ω} {}), the greater the emf. This is noticeable on bicycle generators—at least the cheaper varieties. One of the authors as a juvenile found it amusing to ride his bicycle fast enough to burn out his lights, until he had to ride home lightless one dark night.

Figure 23.23 shows a scheme by which a generator can be made to produce pulsed DC. More elaborate arrangements of multiple coils and split rings can produce smoother DC, although electronic rather than mechanical means are usually used to make ripple-free DC.

The first part of the figure shows a schematic diagram of a single coil D C electric generator. It consists of a rotating rectangular loop placed between the two poles of a permanent magnet shown as two rectangular blocks curved on side facing the loop. The magnetic field B is shown pointing from the North to the South Pole. The two ends of this loop are connected to the two sides of a split ring. The two conducting carbon brushes are kept pressed separately on both sides of the split rings. The loop is rotated in the field with an angular velocity w. Outer ends of the two brushes are connected to an electric bulb which is shown to glow brightly. The second part of the figure shows the graph for e m f generated as a function of time. The e m f is along the Y axis and the time t is along the X axis. The graph is a progressive and rectified sine wave with a time period T. The sine wave has only positive pulses. The crest maxima are at E zero.
Figure 23.23 Split rings, called commutators, produce a pulsed DC emf output in this configuration.

Example 23.4 Calculating the Maximum Emf of a Generator

Calculate the maximum emf, emf0emf0 size 12{"emf" rSub { size 8{0} } } {}, of the generator that was the subject of Example 23.3.

Strategy

Once ωω size 12{ω} {}, the angular velocity, is determined, emf0=NABωemf0=NABω size 12{"emf" rSub { size 8{0} } = ital "NAB"ω} {} can be used to find emf0emf0 size 12{"emf" rSub { size 8{0} } } {}. All other quantities are known.

Solution

Angular velocity is defined to be the change in angle per unit time:

ω=ΔθΔt.ω=ΔθΔt. size 12{ω= { {Δθ} over {Δt} } } {}
23.21

One-fourth of a revolution is π/2π/2 size 12{l} {} radians, and the time is 0.0150 s; thus,

ω = π/2 rad0.0150 s = 104.7 rad/s. ω = π/2 rad0.0150 s = 104.7 rad/s.
23.22

104.7 rad/s is exactly 1000 rpm. We substitute this value for ωω size 12{ω} {} and the information from the previous example into emf0=NABωemf0=NABω size 12{"emf" rSub { size 8{0} } = ital "NAB"ω} {}, yielding

emf0 = NABω = 200(7.85×103m2)(1.25T)(104.7rad/s) = 206V. emf0 = NABω = 200(7.85×103m2)(1.25T)(104.7rad/s) = 206V.alignl { stack { size 12{"emf" rSub { size 8{0} } = ital "NAB"ω} {} # " "="200" \( 7 "." "85" times "10" rSup { size 8{ - 3} } " m" rSup { size 8{2} } \) \( 1 "." "25"" T" \) \( "104" "." 7" rad/s" \) {} # " "="206"" V" {} } } {}
23.23

Discussion

The maximum emf is greater than the average emf of 131 V found in the previous example, as it should be.

In real life, electric generators look a lot different than the figures in this section, but the principles are the same. The source of mechanical energy that turns the coil can be falling water (hydropower), steam produced by the burning of fossil fuels, or the kinetic energy of wind. Figure 23.24 shows a cutaway view of a steam turbine; steam moves over the blades connected to the shaft, which rotates the coil within the generator.

Photograph of a steam turbine connected to a generator.
Figure 23.24 Steam turbine/generator. The steam produced by burning coal impacts the turbine blades, turning the shaft which is connected to the generator. (credit: Nabonaco, Wikimedia Commons)

Generators illustrated in this section look very much like the motors illustrated previously. This is not coincidental. In fact, a motor becomes a generator when its shaft rotates. Certain early automobiles used their starter motor as a generator. In Back Emf, we shall further explore the action of a motor as a generator.

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