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University Physics Volume 2

9.3 Resistivity and Resistance

University Physics Volume 29.3 Resistivity and Resistance
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  1. Preface
  2. Unit 1. Thermodynamics
    1. 1 Temperature and Heat
      1. Introduction
      2. 1.1 Temperature and Thermal Equilibrium
      3. 1.2 Thermometers and Temperature Scales
      4. 1.3 Thermal Expansion
      5. 1.4 Heat Transfer, Specific Heat, and Calorimetry
      6. 1.5 Phase Changes
      7. 1.6 Mechanisms of Heat Transfer
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    2. 2 The Kinetic Theory of Gases
      1. Introduction
      2. 2.1 Molecular Model of an Ideal Gas
      3. 2.2 Pressure, Temperature, and RMS Speed
      4. 2.3 Heat Capacity and Equipartition of Energy
      5. 2.4 Distribution of Molecular Speeds
      6. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    3. 3 The First Law of Thermodynamics
      1. Introduction
      2. 3.1 Thermodynamic Systems
      3. 3.2 Work, Heat, and Internal Energy
      4. 3.3 First Law of Thermodynamics
      5. 3.4 Thermodynamic Processes
      6. 3.5 Heat Capacities of an Ideal Gas
      7. 3.6 Adiabatic Processes for an Ideal Gas
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    4. 4 The Second Law of Thermodynamics
      1. Introduction
      2. 4.1 Reversible and Irreversible Processes
      3. 4.2 Heat Engines
      4. 4.3 Refrigerators and Heat Pumps
      5. 4.4 Statements of the Second Law of Thermodynamics
      6. 4.5 The Carnot Cycle
      7. 4.6 Entropy
      8. 4.7 Entropy on a Microscopic Scale
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
  3. Unit 2. Electricity and Magnetism
    1. 5 Electric Charges and Fields
      1. Introduction
      2. 5.1 Electric Charge
      3. 5.2 Conductors, Insulators, and Charging by Induction
      4. 5.3 Coulomb's Law
      5. 5.4 Electric Field
      6. 5.5 Calculating Electric Fields of Charge Distributions
      7. 5.6 Electric Field Lines
      8. 5.7 Electric Dipoles
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
    2. 6 Gauss's Law
      1. Introduction
      2. 6.1 Electric Flux
      3. 6.2 Explaining Gauss’s Law
      4. 6.3 Applying Gauss’s Law
      5. 6.4 Conductors in Electrostatic Equilibrium
      6. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    3. 7 Electric Potential
      1. Introduction
      2. 7.1 Electric Potential Energy
      3. 7.2 Electric Potential and Potential Difference
      4. 7.3 Calculations of Electric Potential
      5. 7.4 Determining Field from Potential
      6. 7.5 Equipotential Surfaces and Conductors
      7. 7.6 Applications of Electrostatics
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    4. 8 Capacitance
      1. Introduction
      2. 8.1 Capacitors and Capacitance
      3. 8.2 Capacitors in Series and in Parallel
      4. 8.3 Energy Stored in a Capacitor
      5. 8.4 Capacitor with a Dielectric
      6. 8.5 Molecular Model of a Dielectric
      7. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    5. 9 Current and Resistance
      1. Introduction
      2. 9.1 Electrical Current
      3. 9.2 Model of Conduction in Metals
      4. 9.3 Resistivity and Resistance
      5. 9.4 Ohm's Law
      6. 9.5 Electrical Energy and Power
      7. 9.6 Superconductors
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    6. 10 Direct-Current Circuits
      1. Introduction
      2. 10.1 Electromotive Force
      3. 10.2 Resistors in Series and Parallel
      4. 10.3 Kirchhoff's Rules
      5. 10.4 Electrical Measuring Instruments
      6. 10.5 RC Circuits
      7. 10.6 Household Wiring and Electrical Safety
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    7. 11 Magnetic Forces and Fields
      1. Introduction
      2. 11.1 Magnetism and Its Historical Discoveries
      3. 11.2 Magnetic Fields and Lines
      4. 11.3 Motion of a Charged Particle in a Magnetic Field
      5. 11.4 Magnetic Force on a Current-Carrying Conductor
      6. 11.5 Force and Torque on a Current Loop
      7. 11.6 The Hall Effect
      8. 11.7 Applications of Magnetic Forces and Fields
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    8. 12 Sources of Magnetic Fields
      1. Introduction
      2. 12.1 The Biot-Savart Law
      3. 12.2 Magnetic Field Due to a Thin Straight Wire
      4. 12.3 Magnetic Force between Two Parallel Currents
      5. 12.4 Magnetic Field of a Current Loop
      6. 12.5 Ampère’s Law
      7. 12.6 Solenoids and Toroids
      8. 12.7 Magnetism in Matter
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    9. 13 Electromagnetic Induction
      1. Introduction
      2. 13.1 Faraday’s Law
      3. 13.2 Lenz's Law
      4. 13.3 Motional Emf
      5. 13.4 Induced Electric Fields
      6. 13.5 Eddy Currents
      7. 13.6 Electric Generators and Back Emf
      8. 13.7 Applications of Electromagnetic Induction
      9. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    10. 14 Inductance
      1. Introduction
      2. 14.1 Mutual Inductance
      3. 14.2 Self-Inductance and Inductors
      4. 14.3 Energy in a Magnetic Field
      5. 14.4 RL Circuits
      6. 14.5 Oscillations in an LC Circuit
      7. 14.6 RLC Series Circuits
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    11. 15 Alternating-Current Circuits
      1. Introduction
      2. 15.1 AC Sources
      3. 15.2 Simple AC Circuits
      4. 15.3 RLC Series Circuits with AC
      5. 15.4 Power in an AC Circuit
      6. 15.5 Resonance in an AC Circuit
      7. 15.6 Transformers
      8. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
    12. 16 Electromagnetic Waves
      1. Introduction
      2. 16.1 Maxwell’s Equations and Electromagnetic Waves
      3. 16.2 Plane Electromagnetic Waves
      4. 16.3 Energy Carried by Electromagnetic Waves
      5. 16.4 Momentum and Radiation Pressure
      6. 16.5 The Electromagnetic Spectrum
      7. Chapter Review
        1. Key Terms
        2. Key Equations
        3. Summary
        4. Conceptual Questions
        5. Problems
        6. Additional Problems
        7. Challenge Problems
  4. A | Units
  5. B | Conversion Factors
  6. C | Fundamental Constants
  7. D | Astronomical Data
  8. E | Mathematical Formulas
  9. F | Chemistry
  10. G | The Greek Alphabet
  11. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
    13. Chapter 13
    14. Chapter 14
    15. Chapter 15
    16. Chapter 16
  12. Index

Learning Objectives

By the end of this section, you will be able to:
  • Differentiate between resistance and resistivity
  • Define the term conductivity
  • Describe the electrical component known as a resistor
  • State the relationship between resistance of a resistor and its length, cross-sectional area, and resistivity
  • State the relationship between resistivity and temperature

What drives current? We can think of various devices—such as batteries, generators, wall outlets, and so on—that are necessary to maintain a current. All such devices create a potential difference and are referred to as voltage sources. When a voltage source is connected to a conductor, it applies a potential difference V that creates an electrical field. The electrical field, in turn, exerts force on free charges, causing current. The amount of current depends not only on the magnitude of the voltage, but also on the characteristics of the material that the current is flowing through. The material can resist the flow of the charges, and the measure of how much a material resists the flow of charges is known as the resistivity. This resistivity is crudely analogous to the friction between two materials that resists motion.

Resistivity

When a voltage is applied to a conductor, an electrical field EE is created, and charges in the conductor feel a force due to the electrical field. The current density JJ that results depends on the electrical field and the properties of the material. This dependence can be very complex. In some materials, including metals at a given temperature, the current density is approximately proportional to the electrical field. In these cases, the current density can be modeled as

J=σE,J=σE,

where σσ is the electrical conductivity. The electrical conductivity is analogous to thermal conductivity and is a measure of a material’s ability to conduct or transmit electricity. Conductors have a higher electrical conductivity than insulators. Since the electrical conductivity is σ=J/Eσ=J/E, the units are

σ=[J][E]=A/m2V/m=AV·m.σ=[J][E]=A/m2V/m=AV·m.

Here, we define a unit named the ohm with the Greek symbol uppercase omega, ΩΩ. The unit is named after Georg Simon Ohm, whom we will discuss later in this chapter. The ΩΩ is used to avoid confusion with the number 0. One ohm equals one volt per amp: 1Ω=1V/A1Ω=1V/A. The units of electrical conductivity are therefore (Ω·m)−1(Ω·m)−1.

Conductivity is an intrinsic property of a material. Another intrinsic property of a material is the resistivity, or electrical resistivity. The resistivity of a material is a measure of how strongly a material opposes the flow of electrical current. The symbol for resistivity is the lowercase Greek letter rho, ρρ, and resistivity is the reciprocal of electrical conductivity:

ρ=1σ.ρ=1σ.

The unit of resistivity in SI units is the ohm-meter (Ω·m)(Ω·m). We can define the resistivity in terms of the electrical field and the current density,

ρ=EJ.ρ=EJ.
(9.6)

The greater the resistivity, the larger the field needed to produce a given current density. The lower the resistivity, the larger the current density produced by a given electrical field. Good conductors have a high conductivity and low resistivity. Good insulators have a low conductivity and a high resistivity. Table 9.1 lists resistivity and conductivity values for various materials.

Material Conductivity, σσ
(Ω·m)−1(Ω·m)−1
Resistivity, ρρ
(Ω·m)(Ω·m)
Temperature
Coefficient, αα
(°C)−1(°C)−1
Conductors
Silver 6.29×1076.29×107 1.59×10−81.59×10−8 0.0038
Copper 5.95×1075.95×107 1.68×10−81.68×10−8 0.0039
Gold 4.10×1074.10×107 2.44×10−82.44×10−8 0.0034
Aluminum 3.77×1073.77×107 2.65×10−82.65×10−8 0.0039
Tungsten 1.79×1071.79×107 5.60×10−85.60×10−8 0.0045
Iron 1.03×1071.03×107 9.71×10−89.71×10−8 0.0065
Platinum 0.94×1070.94×107 10.60×10−810.60×10−8 0.0039
Steel 0.50×1070.50×107 20.00×10−820.00×10−8
Lead 0.45×1070.45×107 22.00×10−822.00×10−8
Manganin (Cu, Mn, Ni alloy) 0.21×1070.21×107 48.20×10−848.20×10−8 0.000002
Constantan (Cu, Ni alloy) 0.20×1070.20×107 49.00×10−849.00×10−8 0.00003
Mercury 0.10×1070.10×107 98.00×10−898.00×10−8 0.0009
Nichrome (Ni, Fe, Cr alloy) 0.10×1070.10×107 100.00×10−8100.00×10−8 0.0004
Semiconductors[1]
Carbon (pure) 2.86×10−62.86×10−6 3.50×10−53.50×10−5 −0.0005
Carbon (2.861.67)×10−6(2.861.67)×10−6 (3.560)×10−5(3.560)×10−5 −0.0005
Germanium (pure) 600×10−3600×10−3 −0.048
Germanium (1600)×10−3(1600)×10−3 −0.050
Silicon (pure) 2300 −0.075
Silicon 0.123000.12300 −0.07
Insulators
Amber 2.00×10−152.00×10−15 5×10145×1014
Glass 10−910−1410−910−14 10910141091014
Lucite <10−13<10−13 >1013>1013
Mica 10−1110−1510−1110−15 1011101510111015
Quartz (fused) 2.00×10−152.00×10−15 75×101675×1016
Rubber (hard) 10−1310−1610−1310−16 1013101610131016
Sulfur 10−1510−15 10151015
TeflonTM <10−13<10−13 >1013>1013
Wood 10−810−1110−810−11 10810111081011
Table 9.1 Resistivities and Conductivities of Various Materials at 20 °C [1] Values depend strongly on amounts and types of impurities.

The materials listed in the table are separated into categories of conductors, semiconductors, and insulators, based on broad groupings of resistivity. Conductors have the smallest resistivity, and insulators have the largest; semiconductors have intermediate resistivity. 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 we will explore in later chapters.

Check Your Understanding 9.5

Copper wires use routinely used for extension cords and house wiring for several reasons. Copper has the highest electrical conductivity rating, and therefore the lowest resistivity rating, of all nonprecious metals. Also important is the tensile strength, where the tensile strength is a measure of the force required to pull an object to the point where it breaks. The tensile strength of a material is the maximum amount of tensile stress it can take before breaking. Copper has a high tensile strength, 2×108Nm22×108Nm2. A third important characteristic is ductility. Ductility is a measure of a material’s ability to be drawn into wires and a measure of the flexibility of the material, and copper has a high ductility. Summarizing, for a conductor to be a suitable candidate for making wire, there are at least three important characteristics: low resistivity, high tensile strength, and high ductility. What other materials are used for wiring and what are the advantages and disadvantages?

Interactive

View this interactive simulation to see what the effects of the cross-sectional area, the length, and the resistivity of a wire are on the resistance of a conductor. Adjust the variables using slide bars and see if the resistance becomes smaller or larger.

Temperature Dependence of Resistivity

Looking back at Table 9.1, you will see a column labeled “Temperature Coefficient.” The resistivity of some materials has a strong temperature dependence. In some materials, such as copper, the resistivity increases with increasing temperature. In fact, in most conducting metals, the resistivity increases with increasing temperature. The increasing temperature causes increased vibrations of the atoms in the lattice structure of the metals, which impede the motion of the electrons. In other materials, such as carbon, the resistivity decreases with increasing temperature. In many materials, the dependence is approximately linear and can be modeled using a linear equation:

ρρ0[1+α(TT0)],ρρ0[1+α(TT0)],
(9.7)

where ρρ is the resistivity of the material at temperature T, αα is the temperature coefficient of the material, and ρ0ρ0 is the resistivity at T0T0, usually taken as T0=20.00°CT0=20.00°C.

Note also that the temperature coefficient αα is negative for the semiconductors listed in Table 9.1, 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 with temperature is also related to the type and amount of impurities present in the semiconductors.

Resistance

We now consider the resistance of a wire or component. The resistance is a measure of how difficult it is to pass current through a wire or component. Resistance depends on the resistivity. The resistivity is a characteristic of the material used to fabricate a wire or other electrical component, whereas the resistance is a characteristic of the wire or component.

To calculate the resistance, consider a section of conducting wire with cross-sectional area A, length L, and resistivity ρ.ρ. A battery is connected across the conductor, providing a potential difference ΔVΔV across it (Figure 9.13). The potential difference produces an electrical field that is proportional to the current density, according to E=ρJE=ρJ.

Picture is a schematic drawing of a battery connected to a conductor with the cross-sectional area A. Current flows from high potential side to the low potential side of the conductor.
Figure 9.13 A potential provided by a battery is applied to a segment of a conductor with a cross-sectional area A and a length L.

The magnitude of the electrical field across the segment of the conductor is equal to the voltage divided by the length, E=V/LE=V/L, and the magnitude of the current density is equal to the current divided by the cross-sectional area, J=I/A.J=I/A. Using this information and recalling that the electrical field is proportional to the resistivity and the current density, we can see that the voltage is proportional to the current:

E=ρJVL=ρIAV=(ρLA)I.E=ρJVL=ρIAV=(ρLA)I.

Resistance

The ratio of the voltage to the current is defined as the resistance R:

RVI.RVI.
(9.8)

The resistance of a cylindrical segment of a conductor is equal to the resistivity of the material times the length divided by the area:

RVI=ρLA.RVI=ρLA.
(9.9)

The unit of resistance is the ohm, ΩΩ. For a given voltage, the higher the resistance, the lower the current.

Resistors

A common component in electronic circuits is the resistor. The resistor can be used to reduce current flow or provide a voltage drop. Figure 9.14 shows the symbols used for a resistor in schematic diagrams of a circuit. Two commonly used standards for circuit diagrams are provided by the American National Standard Institute (ANSI, pronounced “AN-see”) and the International Electrotechnical Commission (IEC). Both systems are commonly used. We use the ANSI standard in this text for its visual recognition, but we note that for larger, more complex circuits, the IEC standard may have a cleaner presentation, making it easier to read.

Figure A shows the ANSI symbol for a resistor. Figure B shows the IEC symbol for a resistor.
Figure 9.14 Symbols for a resistor used in circuit diagrams. (a) The ANSI symbol; (b) the IEC symbol.

Material and shape dependence of resistance

A resistor can be modeled as a cylinder with a cross-sectional area A and a length L, made of a material with a resistivity ρρ (Figure 9.15). The resistance of the resistor is R=ρLAR=ρLA.

Picture is a schematic drawing of a resistor. It is a uniform cylinder of length L and cross-sectional area A.
Figure 9.15 A model of a resistor as a uniform cylinder of length L and cross-sectional area A. Its resistance to the flow of current is analogous to the resistance posed by a pipe to fluid flow. The longer the cylinder, the greater its resistance. The larger its cross-sectional area A, the smaller its resistance.

The most common material used to make a resistor is carbon. A carbon track is wrapped around a ceramic core, and two copper leads are attached. A second type of resistor is the metal film resistor, which also has a ceramic core. The track is made from a metal oxide material, which has semiconductive properties similar to carbon. Again, copper leads are inserted into the ends of the resistor. The resistor is then painted and marked for identification. A resistor has four colored bands, as shown in Figure 9.16.

Picture is a schematic drawing of a resistor. It contains four colored bands: red, black, green, and grey.
Figure 9.16 Many resistors resemble the figure shown above. The four bands are used to identify the resistor. The first two colored bands represent the first two digits of the resistance of the resistor. The third color is the multiplier. The fourth color represents the tolerance of the resistor. The resistor shown has a resistance of 20×105Ω±10%20×105Ω±10%.

Resistances range over many orders of magnitude. Some ceramic insulators, such as those used to support power lines, have resistances of 1012Ω1012Ω or more. A dry person may have a hand-to-foot resistance of 105Ω105Ω, whereas the resistance of the human heart is about 103Ω103Ω. A meter-long piece of large-diameter copper wire may have a resistance of 10−5Ω10−5Ω, and superconductors have no resistance at all at low temperatures. As we have seen, resistance is related to the shape of an object and the material of which it is composed.

Example 9.5

Current Density, Resistance, and Electrical field for a Current-Carrying Wire Calculate the current density, resistance, and electrical field of a 5-m length of copper wire with a diameter of 2.053 mm (12-gauge) carrying a current of I=10mAI=10mA.

Strategy We can calculate the current density by first finding the cross-sectional area of the wire, which is A=3.31mm2,A=3.31mm2, and the definition of current density J=IAJ=IA. The resistance can be found using the length of the wire L=5.00mL=5.00m, the area, and the resistivity of copper ρ=1.68×10−8Ω·mρ=1.68×10−8Ω·m, where R=ρLAR=ρLA. The resistivity and current density can be used to find the electrical field.

Solution First, we calculate the current density:

J=IA=10×10−3A3.31×10−6m2=3.02×103Am2.J=IA=10×10−3A3.31×10−6m2=3.02×103Am2.

The resistance of the wire is

R=ρLA=(1.68×10−8Ω·m)5.00m3.31×10−6m2=0.025Ω.R=ρLA=(1.68×10−8Ω·m)5.00m3.31×10−6m2=0.025Ω.

Finally, we can find the electrical field:

E=ρJ=1.68×10−8Ω·m(3.02×103Am2)=5.07×10−5Vm.E=ρJ=1.68×10−8Ω·m(3.02×103Am2)=5.07×10−5Vm.

Significance From these results, it is not surprising that copper is used for wires for carrying current because the resistance is quite small. Note that the current density and electrical field are independent of the length of the wire, but the voltage depends on the length.

The resistance of an object also depends on temperature, since R0R0 is directly proportional to ρ.ρ. For a cylinder, we know R=ρLAR=ρLA, so if L and A do not change greatly with temperature, R has the same temperature dependence as ρ.ρ. (Examination of the coefficients of linear expansion shows them to be about two orders of magnitude less than typical temperature coefficients of resistivity, so the effect of temperature on L and A is about two orders of magnitude less than on ρ.)ρ.) Thus,

R=R0(1+αΔT)R=R0(1+αΔT)
(9.10)

is the temperature dependence of the resistance of an object, where R0R0 is the original resistance (usually taken to be 20.00°C)20.00°C) and R is the resistance after a temperature change ΔT.ΔT. The color code gives the resistance of the resistor at a temperature of T=20.00°CT=20.00°C.

Numerous thermometers are based on the effect of temperature on resistance (Figure 9.17). One of the most common thermometers is based on 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.

Picture is a photograph of two digital oral thermometers.
Figure 9.17 These familiar thermometers are based on the automated measurement of a thermistor’s temperature-dependent resistance.

Example 9.6

Calculating Resistance Although caution must be used in applying ρ=ρ0(1+αΔT)ρ=ρ0(1+αΔT) and R=R0(1+αΔT)R=R0(1+αΔT) for temperature changes greater than 100°C100°C, for tungsten, the equations work reasonably well for very large temperature changes. A tungsten filament at 20°C20°C has a resistance of 0.350Ω0.350Ω. What would the resistance be if the temperature is increased to 2850°C2850°C?

Strategy This is a straightforward application of R=R0(1+αΔT)R=R0(1+αΔT), since the original resistance of the filament is given as R0=0.350ΩR0=0.350Ω and the temperature change is ΔT=2830°CΔT=2830°C.

Solution The resistance of the hotter filament R is obtained by entering known values into the above equation:

R=R0(1+αΔT)=(0.350Ω)[1+(4.5×10−3°C)(2830°C)]=4.8Ω.R=R0(1+αΔT)=(0.350Ω)[1+(4.5×10−3°C)(2830°C)]=4.8Ω.

Significance Notice that the resistance changes by more than a factor of 10 as the filament warms to the high temperature and the current through the filament depends on the resistance of the filament and the voltage applied. If the filament is used in an incandescent light bulb, the initial current through the filament when the bulb is first energized will be higher than the current after the filament reaches the operating temperature.

Check Your Understanding 9.6

A strain gauge is an electrical device to measure strain, as shown below. It consists of a flexible, insulating backing that supports a conduction foil pattern. The resistance of the foil changes as the backing is stretched. How does the strain gauge resistance change? Is the strain gauge affected by temperature changes?

Picture is a schematic drawing of a strain gauge device that consists of the conducting pattern deposited on the insulated surface. Metal contacts are made to the two large pads at the origin of the conducting pattern.

Example 9.7

The Resistance of Coaxial Cable Long cables can sometimes act like antennas, picking up electronic noise, which are signals from other equipment and appliances. Coaxial cables are used for many applications that require this noise to be eliminated. For example, they can be found in the home in cable TV connections or other audiovisual connections. Coaxial cables consist of an inner conductor of radius riri surrounded by a second, outer concentric conductor with radius roro (Figure 9.18). The space between the two is normally filled with an insulator such as polyethylene plastic. A small amount of radial leakage current occurs between the two conductors. Determine the resistance of a coaxial cable of length L.

Picture is a schematic drawing of a coaxial cable. It consists of a central metal core encapsulated by the dielectric insulator. Metal shield surrounds dielectric insulator. The whole assembly in inserted in the plastic jacket.
Figure 9.18 Coaxial cables consist of two concentric conductors separated by insulation. They are often used in cable TV or other audiovisual connections.

Strategy We cannot use the equation R=ρLAR=ρLA directly. Instead, we look at concentric cylindrical shells, with thickness dr, and integrate.

Solution We first find an expression for dR and then integrate from riri to roro,

dR=ρAdr=ρ2πrLdr,R=rirodR=riroρ2πrLdr=ρ2πLriro1rdr=ρ2πLlnrori.dR=ρAdr=ρ2πrLdr,R=rirodR=riroρ2πrLdr=ρ2πLriro1rdr=ρ2πLlnrori.

Significance The resistance of a coaxial cable depends on its length, the inner and outer radii, and the resistivity of the material separating the two conductors. Since this resistance is not infinite, a small leakage current occurs between the two conductors. This leakage current leads to the attenuation (or weakening) of the signal being sent through the cable.

Check Your Understanding 9.7

The resistance between the two conductors of a coaxial cable depends on the resistivity of the material separating the two conductors, the length of the cable and the inner and outer radius of the two conductor. If you are designing a coaxial cable, how does the resistance between the two conductors depend on these variables?

Interactive

View this simulation to see how the voltage applied and the resistance of the material the current flows through affects the current through the material. You can visualize the collisions of the electrons and the atoms of the material effect the temperature of the material.

Citation/Attribution

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