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College Physics

20.3 Resistance and Resistivity

College Physics20.3 Resistance and Resistivity
  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

Material and Shape Dependence of Resistance

The resistance of an object depends on its shape and the material of which it is composed. The cylindrical resistor in Figure 20.11 is easy to analyze, and, by so doing, we can gain insight into the resistance of more complicated shapes. As you might expect, the cylinder’s electric resistance RR size 12{R} {} is directly proportional to its length LL size 12{L} {}, similar to the resistance of a pipe to fluid flow. The longer the cylinder, the more collisions charges will make with its atoms. The greater the diameter of the cylinder, the more current it can carry (again similar to the flow of fluid through a pipe). In fact, RR size 12{R} {} is inversely proportional to the cylinder’s cross-sectional area AA size 12{A} {}.

A cylindrical conductor of length L and cross section A is shown. The resistivity of the cylindrical section is represented as rho. The resistance of this cross section R is equal to rho L divided by A. The section of length L of cylindrical conductor is shown equivalent to a resistor represented by symbol R.
Figure 20.11 A uniform cylinder of length LL size 12{L} {} and cross-sectional area AA size 12{A} {}. Its resistance to the flow of current is similar to the resistance posed by a pipe to fluid flow. The longer the cylinder, the greater its resistance. The larger its cross-sectional area AA size 12{A} {}, the smaller its resistance.

For a given shape, the resistance depends on the material of which the object is composed. Different materials offer different resistance to the flow of charge. We define the resistivity ρρ size 12{ρ} {} of a substance so that the resistance RR size 12{R} {} of an object is directly proportional to ρρ size 12{ρ} {}. Resistivity ρρ size 12{ρ} {} is an intrinsic property of a material, independent of its shape or size. The resistance RR size 12{R} {} of a uniform cylinder of length LL size 12{L} {}, of cross-sectional area AA size 12{A} {}, and made of a material with resistivity ρρ size 12{ρ} {}, is

R=ρLA.R=ρLA. size 12{R = { {ρL} over {A} } "."} {}
20.18

Table 20.1 gives representative values of ρρ size 12{ρ} {}. The materials listed in the table are separated into categories of conductors, semiconductors, and insulators, based on broad groupings of resistivities. Conductors have the smallest resistivities, and insulators have the largest; semiconductors have intermediate resistivities. Conductors have varying but large free charge densities, whereas most charges in insulators are bound to atoms and are not free to move. Semiconductors are intermediate, having far fewer free charges than conductors, but having properties that make the number of free charges depend strongly on the type and amount of impurities in the semiconductor. These unique properties of semiconductors are put to use in modern electronics, as will be explored in later chapters.

Material Resistivity ρ ρ size 12{ρ} {} ( Ω m Ω m size 12{ %OMEGA cdot m} {} )
Conductors
Silver 1 . 59 × 10 8 1 . 59 × 10 8 size 12{1 "." "59" times "10" rSup { size 8{ - 8} } } {}
Copper 1 . 72 × 10 8 1 . 72 × 10 8 size 12{1 "." "72" times "10" rSup { size 8{ - 8} } } {}
Gold 2 . 44 × 10 8 2 . 44 × 10 8 size 12{2 "." "44" times "10" rSup { size 8{ - 8} } } {}
Aluminum 2 . 65 × 10 8 2 . 65 × 10 8 size 12{2 "." "65" times "10" rSup { size 8{ - 8} } } {}
Tungsten 5 . 6 × 10 8 5 . 6 × 10 8 size 12{5 "." 6 times "10" rSup { size 8{ - 8} } } {}
Iron 9 . 71 × 10 8 9 . 71 × 10 8 size 12{9 "." "71" times "10" rSup { size 8{ - 8} } } {}
Platinum 10 . 6 × 10 8 10 . 6 × 10 8 size 12{"10" "." 6 times "10" rSup { size 8{ - 8} } } {}
Steel 20 × 10 8 20 × 10 8 size 12{"20" times "10" rSup { size 8{ - 8} } } {}
Lead 22 × 10 8 22 × 10 8 size 12{"22" times "10" rSup { size 8{ - 8} } } {}
Manganin (Cu, Mn, Ni alloy) 44 × 10 8 44 × 10 8 size 12{"44" times "10" rSup { size 8{ - 8} } } {}
Constantan (Cu, Ni alloy) 49 × 10 8 49 × 10 8 size 12{"49" times "10" rSup { size 8{ - 8} } } {}
Mercury 96 × 10 8 96 × 10 8 size 12{"96" times "10" rSup { size 8{ - 8} } } {}
Nichrome (Ni, Fe, Cr alloy) 100 × 10 8 100 × 10 8 size 12{"100" times "10" rSup { size 8{ - 8} } } {}
Semiconductors1
Carbon (pure) 3.5 × 10 –5 3.5 × 10 –5
Carbon ( 3.5 60 ) × 10 –5 ( 3.5 60 ) × 10 –5
Germanium (pure) 600×103600×103
Germanium (1600)×103(1600)×103 size 12{ \( 1 - "600" \) times "10" rSup { size 8{ - 3} } } {}
Silicon (pure) 2300 2300
Silicon 0.1–2300 0.1–2300
Insulators
Amber 5 × 10 14 5 × 10 14 size 12{5 times "10" rSup { size 8{"14"} } } {}
Glass 10 9 10 14 10 9 10 14 size 12{"10" rSup { size 8{9} } - "10" rSup { size 8{"14"} } } {}
Lucite >10 13 >10 13 size 12{>"10" rSup { size 8{"13"} } } {}
Mica 10 11 10 15 10 11 10 15 size 12{"10" rSup { size 8{"11"} } - "10" rSup { size 8{"15"} } } {}
Quartz (fused) 75 × 10 16 75 × 10 16 size 12{"75" times "10" rSup { size 8{"16"} } } {}
Rubber (hard) 10 13 10 16 10 13 10 16 size 12{"10" rSup { size 8{"13"} } - "10" rSup { size 8{"16"} } } {}
Sulfur 10 15 10 15 size 12{"10" rSup { size 8{"15"} } } {}
Teflon >10 13 >10 13 size 12{>"10" rSup { size 8{"13"} } } {}
Wood 10 8 10 11 10 8 10 11 size 12{"10" rSup { size 8{8} } - "10" rSup { size 8{"11"} } } {}
Table 20.1 Resistivities ρ ρ size 12{ρ} {} of Various materials at 20º C 20º C

Example 20.5 Calculating Resistor Diameter: A Headlight Filament

A car headlight filament is made of tungsten and has a cold resistance of 0.350Ω0.350Ω size 12{0 "." "350" %OMEGA } {}. If the filament is a cylinder 4.00 cm long (it may be coiled to save space), what is its diameter?

Strategy

We can rearrange the equation R=ρLAR=ρLA size 12{R = { {ρL} over {A} } } {} to find the cross-sectional area AA size 12{A} {} of the filament from the given information. Then its diameter can be found by assuming it has a circular cross-section.

Solution

The cross-sectional area, found by rearranging the expression for the resistance of a cylinder given in R=ρLAR=ρLA size 12{R = { {ρL} over {A} } } {}, is

A=ρLR.A=ρLR. size 12{A = { {ρL} over {R} } "."} {}
20.19

Substituting the given values, and taking ρρ size 12{ρ} {} from Table 20.1, yields

A = ( 5.6 × 10 –8 Ω m ) ( 4.00 × 10 –2 m ) 0.350 Ω = 6.40 × 10 –9 m 2 . A = ( 5.6 × 10 –8 Ω m ) ( 4.00 × 10 –2 m ) 0.350 Ω = 6.40 × 10 –9 m 2 .
20.20

The area of a circle is related to its diameter DD size 12{D} {} by

A=πD24.A=πD24. size 12{A = { {πD rSup { size 8{2} } } over {4} } "."} {}
20.21

Solving for the diameter DD size 12{D} {}, and substituting the value found for AA size 12{A} {}, gives

D = 2 A p 1 2 = 2 6.40 × 10 –9 m 2 3.14 1 2 = 9.0 × 10 –5 m . D = 2 A p 1 2 = 2 6.40 × 10 –9 m 2 3.14 1 2 = 9.0 × 10 –5 m . alignl { stack { size 12{D =" 2" left ( { {A} over {p} } right ) rSup { size 8{ { {1} over {2} } } } =" 2" left ( { {6 "." "40"´"10" rSup { size 8{ +- 9} } " m" rSup { size 8{2} } } over {3 "." "14"} } right ) rSup { size 8{ { {1} over {2} } } } } {} # =" 9" "." 0´"10" rSup { size 8{ +- 5} } " m" "." {} } } {}
20.22

Discussion

The diameter is just under a tenth of a millimeter. It is quoted to only two digits, because ρρ size 12{ρ} {} is known to only two digits.

Temperature Variation of Resistance

The resistivity of all materials depends on temperature. Some even become superconductors (zero resistivity) at very low temperatures. (See Figure 20.12.) Conversely, the resistivity of conductors increases with increasing temperature. Since the atoms vibrate more rapidly and over larger distances at higher temperatures, the electrons moving through a metal make more collisions, effectively making the resistivity higher. Over relatively small temperature changes (about 100ºC100ºC size 12{"100"°C} {} or less), resistivity ρρ size 12{ρ} {} varies with temperature change ΔTΔT size 12{DT} {} as expressed in the following equation

ρ=ρ0(1 +αΔT),ρ=ρ0(1 +αΔT), size 12{ρ = ρ rSub { size 8{0} } \( "1 "+ αΔT \) ","} {}
20.23

where ρ0ρ0 size 12{ρ rSub { size 8{0} } } {} is the original resistivity and αα size 12{α} {} is the temperature coefficient of resistivity. (See the values of αα size 12{α} {} in Table 20.2 below.) For larger temperature changes, αα size 12{α} {} may vary or a nonlinear equation may be needed to find ρρ size 12{ρ} {}. Note that αα size 12{α} {} is positive for metals, meaning their resistivity increases with temperature. Some alloys have been developed specifically to have a small temperature dependence. Manganin (which is made of copper, manganese and nickel), for example, has αα size 12{α} {} close to zero (to three digits on the scale in Table 20.2), and so its resistivity varies only slightly with temperature. This is useful for making a temperature-independent resistance standard, for example.

A graph for variation of resistance R with temperature T for a mercury sample is shown. The temperature T is plotted along the x axis and is measured in Kelvin, and the resistance R is plotted along the y axis and is measured in ohms. The curve starts at x equals zero and y equals zero, and coincides with the X axis until the value of temperature is four point two Kelvin, known as the critical temperature T sub c. At temperature T sub c, the curve shows a vertical rise, represented by a dotted line, until the resistance is about zero point one one ohms. After this temperature the resistance shows a nearly linear increase with temperature T.
Figure 20.12 The resistance of a sample of mercury is zero at very low temperatures—it is a superconductor up to about 4.2 K. Above that critical temperature, its resistance makes a sudden jump and then increases nearly linearly with temperature.
Material Coefficient αα(1/°C)2
Conductors
Silver 3 . 8 × 10 3 3 . 8 × 10 3 size 12{3 "." 8 times "10" rSup { size 8{ - 3} } } {}
Copper 3 . 9 × 10 3 3 . 9 × 10 3 size 12{3 "." 9 times "10" rSup { size 8{ - 3} } } {}
Gold 3 . 4 × 10 3 3 . 4 × 10 3 size 12{3 "." 4 times "10" rSup { size 8{ - 3} } } {}
Aluminum 3 . 9 × 10 3 3 . 9 × 10 3 size 12{3 "." 9 times "10" rSup { size 8{ - 3} } } {}
Tungsten 4 . 5 × 10 3 4 . 5 × 10 3 size 12{4 "." 5 times "10" rSup { size 8{ - 3} } } {}
Iron 5 . 0 × 10 3 5 . 0 × 10 3 size 12{5 "." 0 times "10" rSup { size 8{ - 3} } } {}
Platinum 3 . 93 × 10 3 3 . 93 × 10 3 size 12{3 "." "93" times "10" rSup { size 8{ - 3} } } {}
Lead 3 . 9 × 10 3 3 . 9 × 10 3 size 12{3 "." 9 times "10" rSup { size 8{ - 3} } } {}
Manganin (Cu, Mn, Ni alloy) 0 . 000 × 10 3 0 . 000 × 10 3 size 12{0 "." "000" times "10" rSup { size 8{ - 3} } } {}
Constantan (Cu, Ni alloy) 0 . 002 × 10 3 0 . 002 × 10 3 size 12{0 "." "002" times "10" rSup { size 8{ - 3} } } {}
Mercury 0 . 89 × 10 3 0 . 89 × 10 3 size 12{0 "." "89" times "10" rSup { size 8{ - 3} } } {}
Nichrome (Ni, Fe, Cr alloy) 0 . 4 × 10 3 0 . 4 × 10 3 size 12{0 "." 4 times "10" rSup { size 8{ - 3} } } {}
Semiconductors
Carbon (pure) 0 . 5 × 10 3 0 . 5 × 10 3 size 12{ - 0 "." 5 times "10" rSup { size 8{ - 3} } } {}
Germanium (pure) 50 × 10 3 50 × 10 3 size 12{ - "50" times "10" rSup { size 8{ - 3} } } {}
Silicon (pure) 70 × 10 3 70 × 10 3 size 12{ - "70" times "10" rSup { size 8{ - 3} } } {}
Table 20.2 Tempature Coefficients of Resistivity α α size 12{α} {}

Note also that αα size 12{α} {} is negative for the semiconductors listed in Table 20.2, meaning that their resistivity decreases with increasing temperature. They become better conductors at higher temperature, because increased thermal agitation increases the number of free charges available to carry current. This property of decreasing ρρ size 12{ρ} {} with temperature is also related to the type and amount of impurities present in the semiconductors.

The resistance of an object also depends on temperature, since R0R0 size 12{R rSub { size 8{0} } } {} is directly proportional to ρρ size 12{ρ} {}. For a cylinder we know R=ρL/AR=ρL/A size 12{R=ρL/A} {}, and so, if LL size 12{L} {} and AA size 12{A} {} do not change greatly with temperature, RR size 12{R} {} will have the same temperature dependence as ρρ size 12{ρ} {}. (Examination of the coefficients of linear expansion shows them to be about two orders of magnitude less than typical temperature coefficients of resistivity, and so the effect of temperature on LL size 12{L} {} and AA size 12{A} {} is about two orders of magnitude less than on ρρ size 12{ρ} {}.) Thus,

R = R 0 ( 1 + αΔT ) R = R 0 ( 1 + αΔT ) size 12{R = R rSub { size 8{0} } \( "1 "+ αΔT \) } {}
20.24

is the temperature dependence of the resistance of an object, where R0R0 size 12{R rSub { size 8{0} } } {} is the original resistance and RR size 12{R} {} is the resistance after a temperature change ΔTΔT size 12{DT} {}. Numerous thermometers are based on the effect of temperature on resistance. (See Figure 20.13.) One of the most common is the thermistor, a semiconductor crystal with a strong temperature dependence, the resistance of which is measured to obtain its temperature. The device is small, so that it quickly comes into thermal equilibrium with the part of a person it touches.

A photograph showing two digital thermometers used for measuring body temperature.
Figure 20.13 These familiar thermometers are based on the automated measurement of a thermistor’s temperature-dependent resistance. (credit: Biol, Wikimedia Commons)

Example 20.6 Calculating Resistance: Hot-Filament Resistance

Although caution must be used in applying ρ=ρ0(1 +αΔT)ρ=ρ0(1 +αΔT) size 12{ρ = ρ rSub { size 8{0} } \( "1 "+ αΔT \) } {} and R=R0(1 +αΔT)R=R0(1 +αΔT) size 12{R = R rSub { size 8{0} } \( "1 "+ αΔT \) } {} for temperature changes greater than 100ºC100ºC size 12{"100"°"C"} {}, for tungsten the equations work reasonably well for very large temperature changes. What, then, is the resistance of the tungsten filament in the previous example if its temperature is increased from room temperature ( 20ºC 20ºC ) to a typical operating temperature of 2850ºC2850ºC size 12{"2850"°"C"} {}?

Strategy

This is a straightforward application of R=R0(1 +αΔT)R=R0(1 +αΔT) size 12{R = R rSub { size 8{0} } \( "1 "+ αΔT \) } {}, since the original resistance of the filament was given to be R0=0.350 ΩR0=0.350 Ω size 12{R rSub { size 8{0} } =0 "." "350"` %OMEGA } {}, and the temperature change is ΔT=2830ºCΔT=2830ºC size 12{ΔT="2830"°"C"} {}.

Solution

The hot resistance RR size 12{R} {} is obtained by entering known values into the above equation:

R = R 0 ( 1 + αΔT ) = ( 0 . 350 Ω ) [ 1 + ( 4.5 × 10 –3 / ºC ) ( 2830º C ) ] = 4.8 Ω. R = R 0 ( 1 + αΔT ) = ( 0 . 350 Ω ) [ 1 + ( 4.5 × 10 –3 / ºC ) ( 2830º C ) ] = 4.8 Ω.
20.25

Discussion

This value is consistent with the headlight resistance example in Ohm’s Law: Resistance and Simple Circuits.

PhET Explorations: Resistance in a Wire

Learn about the physics of resistance in a wire. Change its resistivity, length, and area to see how they affect the wire's resistance. The sizes of the symbols in the equation change along with the diagram of a wire.

Figure 20.14

Footnotes

  • 1 Values depend strongly on amounts and types of impurities
  • 2 Values at 20°C.
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