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College Physics

23.11 Reactance, Inductive and Capacitive

College Physics23.11 Reactance, Inductive and Capacitive
  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

Many circuits also contain capacitors and inductors, in addition to resistors and an AC voltage source. We have seen how capacitors and inductors respond to DC voltage when it is switched on and off. We will now explore how inductors and capacitors react to sinusoidal AC voltage.

Inductors and Inductive Reactance

Suppose an inductor is connected directly to an AC voltage source, as shown in Figure 23.45. It is reasonable to assume negligible resistance, since in practice we can make the resistance of an inductor so small that it has a negligible effect on the circuit. Also shown is a graph of voltage and current as functions of time.

Part a of the figure describes an A C voltage source V connected across an inductor L. The voltage across the inductance is shown as V L. Part b of the figure describes a graph showing the variation of current and voltage across the inductance as a function of time. The voltage V L and current I L is plotted along the Y axis and the time t is along the X axis. The graph for current is a progressive sine wave from the origin. The graph for voltage V is a cosine wave and an amplitude slightly less than the current wave.
Figure 23.45 (a) An AC voltage source in series with an inductor having negligible resistance. (b) Graph of current and voltage across the inductor as functions of time.

The graph in Figure 23.45(b) starts with voltage at a maximum. Note that the current starts at zero and rises to its peak after the voltage that drives it, just as was the case when DC voltage was switched on in the preceding section. When the voltage becomes negative at point a, the current begins to decrease; it becomes zero at point b, where voltage is its most negative. The current then becomes negative, again following the voltage. The voltage becomes positive at point c and begins to make the current less negative. At point d, the current goes through zero just as the voltage reaches its positive peak to start another cycle. This behavior is summarized as follows:

AC Voltage in an Inductor

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.

Current lags behind voltage, since inductors oppose change in current. Changing current induces a back emf V=L(ΔI/Δt)V=L(ΔI/Δt) size 12{V= - L \( ΔI/Δt \) } {}. This is considered to be an effective resistance of the inductor to AC. The rms current II size 12{I} {} through an inductor LL size 12{L} {} is given by a version of Ohm’s law:

I=VXL,I=VXL, size 12{I= { {V} over {X rSub { size 8{L} } } } } {}
23.51

where V V is the rms voltage across the inductor and XLXL size 12{X rSub { size 8{L} } } {} is defined to be

XL=fL,XL=fL, size 12{X rSub { size 8{L} } =2π ital "fL"} {}
23.52

with ff size 12{f} {} the frequency of the AC voltage source in hertz (An analysis of the circuit using Kirchhoff’s loop rule and calculus actually produces this expression). XLXL size 12{X rSub { size 8{L} } } {} is called the inductive reactance, because the inductor reacts to impede the current. XLXL size 12{X rSub { size 8{L} } } {} has units of ohms (1 H=1 Ωs1 H=1 Ωs, so that frequency times inductance has units of (cycles/s)(Ωs)=Ω(cycles/s)(Ωs)=Ω size 12{ \( "cycles/s" \) \( ` %OMEGA cdot s \) = %OMEGA } {}), consistent with its role as an effective resistance. It makes sense that XLXL size 12{X rSub { size 8{L} } } {} is proportional to LL size 12{L} {}, since the greater the induction the greater its resistance to change. It is also reasonable that XLXL size 12{X rSub { size 8{L} } } {} is proportional to frequency ff size 12{f} {}, since greater frequency means greater change in current. That is, ΔItΔIt size 12{ΔI} {} is large for large frequencies (large ff size 12{f} {}, small ΔtΔt size 12{Δt} {}). The greater the change, the greater the opposition of an inductor.

Example 23.10 Calculating Inductive Reactance and then Current

(a) Calculate the inductive reactance of a 3.00 mH inductor when 60.0 Hz and 10.0 kHz AC voltages are applied. (b) What is the rms current at each frequency if the applied rms voltage is 120 V?

Strategy

The inductive reactance is found directly from the expression XL=fLXL=fL size 12{X rSub { size 8{L} } =2π ital "fL"} {}. Once XLXL size 12{X rSub { size 8{L} } } {} has been found at each frequency, Ohm’s law as stated in the Equation I=V/XLI=V/XL size 12{I=V/X rSub { size 8{L} } } {} can be used to find the current at each frequency.

Solution for (a)

Entering the frequency and inductance into Equation XL=fLXL=fL size 12{X rSub { size 8{L} } =2π ital "fL"} {} gives

X L = fL = 6.28 ( 60.0 / s ) ( 3.00 mH ) = 1.13 Ω at 60 Hz . X L = fL = 6.28 ( 60.0 / s ) ( 3.00 mH ) = 1.13 Ω at 60 Hz .
23.53

Similarly, at 10 kHz,

XL=fL=6.28( 1.00 × 10 4 /s )(3.00 mH)=188 Ω at 10 kHz.XL=fL=6.28( 1.00 × 10 4 /s )(3.00 mH)=188 Ω at 10 kHz. size 12{X rSub { size 8{L} } =2π ital "fL"=6 "." "28" \( 3 "." "00"" mH" \) ="188" %OMEGA } {}
23.54

Solution for (b)

The rms current is now found using the version of Ohm’s law in Equation I=V/XLI=V/XL size 12{I=V/X rSub { size 8{L} } } {}, given the applied rms voltage is 120 V. For the first frequency, this yields

I=VXL=120 V1.13 Ω=106 A at 60 Hz.I=VXL=120 V1.13 Ω=106 A at 60 Hz.
23.55

Similarly, at 10 kHz,

I=VXL=120 V188 Ω=0.637 A at 10 kHz.I=VXL=120 V188 Ω=0.637 A at 10 kHz. size 12{I= { {V} over {X rSub { size 8{L} } } } = { {"120"" V"} over {"188 " %OMEGA } } =0 "." "637"" A"} {}
23.56

Discussion

The inductor reacts very differently at the two different frequencies. At the higher frequency, its reactance is large and the current is small, consistent with how an inductor impedes rapid change. Thus high frequencies are impeded the most. Inductors can be used to filter out high frequencies; for example, a large inductor can be put in series with a sound reproduction system or in series with your home computer to reduce high-frequency sound output from your speakers or high-frequency power spikes into your computer.

Note that although the resistance in the circuit considered is negligible, the AC current is not extremely large because inductive reactance impedes its flow. With AC, there is no time for the current to become extremely large.

Capacitors and Capacitive Reactance

Consider the capacitor connected directly to an AC voltage source as shown in Figure 23.46. The resistance of a circuit like this can be made so small that it has a negligible effect compared with the capacitor, and so we can assume negligible resistance. Voltage across the capacitor and current are graphed as functions of time in the figure.

Part a of the figure shows a capacitor C connected across an A C voltage source V. The voltage across the capacitor is given by V C. Part b of the diagram shows a graph for the variation of current and voltage across the capacitor as functions of time. The voltage V C and current I C is plotted along the Y axis and the time t is along the X axis. The graph for current is a progressive sine wave from the origin starting with a wave along the negative Y axis. The graph for voltage is a cosine wave and amplitude slightly less than the current wave.
Figure 23.46 (a) An AC voltage source in series with a capacitor C having negligible resistance. (b) Graph of current and voltage across the capacitor as functions of time.

The graph in Figure 23.46 starts with voltage across the capacitor at a maximum. The current is zero at this point, because the capacitor is fully charged and halts the flow. Then voltage drops and the current becomes negative as the capacitor discharges. At point a, the capacitor has fully discharged (Q=0Q=0 size 12{Q=0} {} on it) and the voltage across it is zero. The current remains negative between points a and b, causing the voltage on the capacitor to reverse. This is complete at point b, where the current is zero and the voltage has its most negative value. The current becomes positive after point b, neutralizing the charge on the capacitor and bringing the voltage to zero at point c, which allows the current to reach its maximum. Between points c and d, the current drops to zero as the voltage rises to its peak, and the process starts to repeat. Throughout the cycle, the voltage follows what the current is doing by one-fourth of a cycle:

AC Voltage in a Capacitor

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.

The capacitor is affecting the current, having the ability to stop it altogether when fully charged. Since an AC voltage is applied, there is an rms current, but it is limited by the capacitor. This is considered to be an effective resistance of the capacitor to AC, and so the rms current II size 12{I} {} in the circuit containing only a capacitor CC size 12{C} {} is given by another version of Ohm’s law to be

I=VXC,I=VXC, size 12{I= { {V} over {X rSub { size 8{C} } } } } {}
23.57

where VV size 12{V} {} is the rms voltage and XCXC size 12{X rSub { size 8{C} } } {} is defined (As with XLXL size 12{X rSub { size 8{L} } } {}, this expression for XCXC size 12{X rSub { size 8{C} } } {} results from an analysis of the circuit using Kirchhoff’s rules and calculus) to be

XC=1fC,XC=1fC, size 12{X rSub { size 8{C} } = { {1} over {2π ital "fC"} } } {}
23.58

where XCXC size 12{X rSub { size 8{C} } } {} is called the capacitive reactance, because the capacitor reacts to impede the current. XCXC size 12{X rSub { size 8{C} } } {} has units of ohms (verification left as an exercise for the reader). XCXC size 12{X rSub { size 8{C} } } {} is inversely proportional to the capacitance CC size 12{C} {}; the larger the capacitor, the greater the charge it can store and the greater the current that can flow. It is also inversely proportional to the frequency ff size 12{f} {}; the greater the frequency, the less time there is to fully charge the capacitor, and so it impedes current less.

Example 23.11 Calculating Capacitive Reactance and then Current

(a) Calculate the capacitive reactance of a 5.00 mF capacitor when 60.0 Hz and 10.0 kHz AC voltages are applied. (b) What is the rms current if the applied rms voltage is 120 V?

Strategy

The capacitive reactance is found directly from the expression in XC=1fCXC=1fC size 12{X rSub { size 8{C} } = { {1} over {2π ital "fC"} } } {}. Once X C X C has been found at each frequency, Ohm’s law stated as I=V/XCI=V/XC size 12{I=V/X rSub { size 8{C} } } {} can be used to find the current at each frequency.

Solution for (a)

Entering the frequency and capacitance into XC=1fCXC=1fC size 12{X rSub { size 8{C} } = { {1} over {2π ital "fC"} } } {} gives

XC = 1fC = 16.28(60.0/s)(5.00 μF)=531 Ω at 60 Hz. XC = 1fC = 16.28(60.0/s)(5.00 μF)=531 Ω at 60 Hz.alignl { stack { size 12{X rSub { size 8{C} } = { {1} over {2π ital "fC"} } } {} # " "= { {1} over {6 "." "28" \( "60" "." 0/s \) \( 5 "." "00" μF \) } } ="531 " %OMEGA " at 60 Hz" {} } } {}
23.59

Similarly, at 10 kHz,

XC = 1fC=16.28(1.00×104/s)(5.00 μF) = 3.18 Ω at 10 kHz.XC = 1fC=16.28(1.00×104/s)(5.00 μF) = 3.18 Ω at 10 kHz.alignl { stack { size 12{X rSub { size 8{C} } = { {1} over {2π ital "fC"} } = { {1} over {6 "." "28" \( 1 "." "00" times "10" rSup { size 8{4} } /s \) \( 5 "." "00" μF \) } } } {} # " "=3 "." "18" %OMEGA " at 10 kHz" {} } } {}
23.60

Solution for (b)

The rms current is now found using the version of Ohm’s law in I=V/XCI=V/XC size 12{I=V/X rSub { size 8{C} } } {}, given the applied rms voltage is 120 V. For the first frequency, this yields

I=VXC=120 V531 Ω=0.226 A at 60 Hz.I=VXC=120 V531 Ω=0.226 A at 60 Hz. size 12{I= { {V} over {X rSub { size 8{C} } } } = { {"120"" V"} over {"531 " %OMEGA } } =0 "." "226"" A"} {}
23.61

Similarly, at 10 kHz,

I=VXC=120 V3.18 Ω=37.7 A at 10 kHz.I=VXC=120 V3.18 Ω=37.7 A at 10 kHz. size 12{I= { {V} over {X rSub { size 8{C} } } } = { {"120"" V"} over {3 "." "18 " %OMEGA } } ="37" "." 7" A"} {}
23.62

Discussion

The capacitor reacts very differently at the two different frequencies, and in exactly the opposite way an inductor reacts. At the higher frequency, its reactance is small and the current is large. Capacitors favor change, whereas inductors oppose change. Capacitors impede low frequencies the most, since low frequency allows them time to become charged and stop the current. Capacitors can be used to filter out low frequencies. For example, a capacitor in series with a sound reproduction system rids it of the 60 Hz hum.

Although a capacitor is basically an open circuit, there is an rms current in a circuit with an AC voltage applied to a capacitor. This is because the voltage is continually reversing, charging and discharging the capacitor. If the frequency goes to zero (DC), XCXC size 12{X rSub { size 8{C} } } {} tends to infinity, and the current is zero once the capacitor is charged. At very high frequencies, the capacitor’s reactance tends to zero—it has a negligible reactance and does not impede the current (it acts like a simple wire). Capacitors have the opposite effect on AC circuits that inductors have.

Resistors in an AC Circuit

Just as a reminder, consider Figure 23.47, which shows an AC voltage applied to a resistor and a graph of voltage and current versus time. The voltage and current are exactly in phase in a resistor. There is no frequency dependence to the behavior of plain resistance in a circuit:

Part a of the diagram shows a resistor R connected across an A C voltage source V. The voltage drop across the resistor R is given by V R.Part b of the diagram shows a graph showing the variation of voltage V R and current I R with time t. the V R and current I R are plotted along Y axis and time t is along the X axis. Both I and V are progressive cosine waves. The amplitude of I wave is more than V wave.
Figure 23.47 (a) An AC voltage source in series with a resistor. (b) Graph of current and voltage across the resistor as functions of time, showing them to be exactly in phase.

AC Voltage in a Resistor

When a sinusoidal voltage is applied to a resistor, the voltage is exactly in phase with the current—they have a phase angle.

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