Resistance and Resistivity
Learning Objectives
By the end of this section, you will be able to:- Explain the concept of resistivity.
- Use resistivity to calculate the resistance of specified configurations of material.
- Use the thermal coefficient of resistivity to calculate the change of resistance with temperature.
Material and Shape Dependence of Resistance
$R=\frac{\rho L}{A}\\$
.Material | Resistivity ρ ( Ω ⋅ m ) |
---|---|
Conductors | |
Silver | 1. 59 × 10^{−8 } |
Copper | 1. 72 × 10^{−8 } |
Gold | 2. 44 × 10^{−8 } |
Aluminum | 2. 65 × 10^{−8 } |
Tungsten | 5. 6 × 10^{−8 } |
Iron | 9. 71 × 10^{−8 } |
Platinum | 10. 6 × 10^{−8 } |
Steel | 20 × 10^{−8 } |
Lead | 22 × 10^{−8 } |
Manganin (Cu, Mn, Ni alloy) | 44 × 10^{−8 } |
Constantan (Cu, Ni alloy) | 49 × 10^{−8 } |
Mercury | 96 × 10^{−8 } |
Nichrome (Ni, Fe, Cr alloy) | 100 × 10^{−8 } |
Semiconductors^{[3]} | |
Carbon (pure) | 3.5 × 10^{5 } |
Carbon | (3.5 − 60) × 10^{5 } |
Germanium (pure) | 600 × 10^{−3} |
Germanium | (1−600) × 10^{−3} |
Silicon (pure) | 2300 |
Silicon | 0.1–2300 |
Insulators | |
Amber | 5 × 10^{14 } |
Glass | 10^{9 } − 10^{14 } |
Lucite | >10^{13 } |
Mica | 10^{11 } − 10^{15 } |
Quartz (fused) | 75 × 10^{16 } |
Rubber (hard) | 10^{13 } − 10^{16 } |
Sulfur | 10^{15 } |
Teflon | >10^{13 } |
Wood | 10^{8 } − 10^{11 } |
Example 1. Calculating Resistor Diameter: A Headlight Filament
A car headlight filament is made of tungsten and has a cold resistance of 0.350 Ω. 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$R=\frac{\rho L}{A}\\$
to find the cross-sectional area 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$R=\frac{\rho L}{A}\\$
, is$A=\frac{\rho L}{R}\\$
.
Substituting the given values, and taking ρ from Table 1, yields$\begin{array}{lll}A& =& \frac{\left(5.6\times {\text{10}}^{-8}\Omega \cdot \text{m}\right)\left(4.00\times {\text{10}}^{-2}\text{m}\right)}{\text{0.350}\Omega }\\ & =& \text{6.40}\times {\text{10}}^{-9}{\text{m}}^{2}\end{array}\\$
.
The area of a circle is related to its diameter D by$A=\frac{{\pi D}^{2}}{4}\\$
.
Solving for the diameter D, and substituting the value found for A, gives$\begin{array}{lll}D& =& \text{2}{\left(\frac{A}{p}\right)}^{\frac{1}{2}}=\text{2}{\left(\frac{6.40\times {\text{10}}^{-9}{\text{m}}^{2}}{3.14}\right)}^{\frac{1}{2}}\\ & =& 9.0\times {\text{10}}^{-5}\text{m}\end{array}\\$
.
Discussion
The diameter is just under a tenth of a millimeter. It is quoted to only two digits, because ρ is known to only two digits.Temperature Variation of Resistance
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 100ºC or less), resistivity ρ varies with temperature change ΔT as expressed in the following equation
ρ = ρ_{0 }(1 +αΔT),
where ρ_{0} is the original resistivity and α is the temperature coefficient of resistivity. (See the values of α in Table 2 below.) For larger temperature changes, α may vary or a nonlinear equation may be needed to find ρ. Note that 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 α close to zero (to three digits on the scale in Table 2), and so its resistivity varies only slightly with temperature. This is useful for making a temperature-independent resistance standard, for example.Material | Coefficient (1/°C)^{[4]} |
---|---|
Conductors | |
Silver | 3.8 × 10^{−3 } |
Copper | 3.9 × 10^{−3 } |
Gold | 3.4 × 10^{−3 } |
Aluminum | 3.9 × 10^{−3 } |
Tungsten | 4.5 × 10^{−3 } |
Iron | 5.0 × 10^{−3 } |
Platinum | 3.93 × 10^{−3 } |
Lead | 3.9 × 10^{−3 } |
Manganin (Cu, Mn, Ni alloy) | 0.000 × 10^{−3 } |
Constantan (Cu, Ni alloy) | 0.002 × 10^{−3 } |
Mercury | 0.89 × 10^{−3 } |
Nichrome (Ni, Fe, Cr alloy) | 0.4 × 10^{−3 } |
Semiconductors | |
Carbon (pure) | −0.5 × 10^{−3 } |
Germanium (pure) | −50 × 10^{−3 } |
Silicon (pure) | −70 × 10^{−3 } |
R = R_{ 0 } ( 1 + αΔT )
is the temperature dependence of the resistance of an object, where R_{0} is the original resistance and R is the resistance after a temperature change ΔT. Numerous thermometers are based on the effect of temperature on resistance. (See Figure 3.) 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.Example 2. Calculating Resistance: Hot-Filament Resistance
Although caution must be used in applying ρ = ρ_{0}(1 +αΔT) and R = R_{0}(1 +αΔT) for temperature changes greater than 100ºC, 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 ) to a typical operating temperature of 2850ºC?
Strategy
This is a straightforward application of R = R_{0}(1 +αΔT), since the original resistance of the filament was given to be R_{0 }= 0.350 Ω, and the temperature change is ΔT = 2830ºC.Solution
The hot resistance R is obtained by entering known values into the above equation:$\begin{array}{lll}R & =& {R}_{0}\left(1+\alpha\Delta T\right)\\ & =& \left(0.350\Omega\right)\left[1+\left(4.5\times{10}^{-3}/º\text{C}\right)\left(2830º\text{C}\right)\right]\\ & =& {4.8\Omega}\end{array}\\$
.
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.
Section Summary
- The resistance R of a cylinder of length L and cross-sectional area A is $R=\frac{\rho L}{A}\\$, where ρ is the resistivity of the material.
- Values of ρ in Table 1 show that materials fall into three groups—conductors, semiconductors, and insulators.
- Temperature affects resistivity; for relatively small temperature changes ΔT, resistivity is $\rho ={\rho }_{0}\left(\text{1}+\alpha \Delta T\right)\\$, where ρ_{0} is the original resistivity and$\alpha$is the temperature coefficient of resistivity.
- Table 2 gives values for α, the temperature coefficient of resistivity.
- The resistance R of an object also varies with temperature: $R={R}_{0}\left(\text{1}+\alpha \Delta T\right)\\$, where R_{0} is the original resistance, and R is the resistance after the temperature change.
Conceptual Questions
1. In which of the three semiconducting materials listed in Table 1 do impurities supply free charges? (Hint: Examine the range of resistivity for each and determine whether the pure semiconductor has the higher or lower conductivity.)
2. Does the resistance of an object depend on the path current takes through it? Consider, for example, a rectangular bar—is its resistance the same along its length as across its width? (See Figure 5.)
3. If aluminum and copper wires of the same length have the same resistance, which has the larger diameter? Why?
4. Explain why
$R={R}_{0}\left(1+\alpha\Delta T\right)\\$
for the temperature variation of the resistance R of an object is not as accurate as $\rho ={\rho }_{0}\left({1}+\alpha \Delta T\right)\\$
, which gives the temperature variation of resistivity ρ.Problems & Exercises
1. What is the resistance of a 20.0-m-long piece of 12-gauge copper wire having a 2.053-mm diameter?
2. The diameter of 0-gauge copper wire is 8.252 mm. Find the resistance of a 1.00-km length of such wire used for power transmission.
3. If the 0.100-mm diameter tungsten filament in a light bulb is to have a resistance of 0.200 Ω at 20ºC, how long should it be?
4. Find the ratio of the diameter of aluminum to copper wire, if they have the same resistance per unit length (as they might in household wiring).
5. What current flows through a 2.54-cm-diameter rod of pure silicon that is 20.0 cm long, when 1.00 × 10^{3}V is applied to it? (Such a rod may be used to make nuclear-particle detectors, for example.)
6. (a) To what temperature must you raise a copper wire, originally at 20.0ºC, to double its resistance, neglecting any changes in dimensions? (b) Does this happen in household wiring under ordinary circumstances?
7. A resistor made of Nichrome wire is used in an application where its resistance cannot change more than 1.00% from its value at 20.0ºC. Over what temperature range can it be used?
8. Of what material is a resistor made if its resistance is 40.0% greater at 100ºC than at 20.0ºC?
9. An electronic device designed to operate at any temperature in the range from –10.0ºC to 55.0ºC contains pure carbon resistors. By what factor does their resistance increase over this range?
10. (a) Of what material is a wire made, if it is 25.0 m long with a 0.100 mm diameter and has a resistance of 77.7 Ω at 20.0ºC? (b) What is its resistance at 150ºC?
11. Assuming a constant temperature coefficient of resistivity, what is the maximum percent decrease in the resistance of a constantan wire starting at 20.0ºC?
12. A wire is drawn through a die, stretching it to four times its original length. By what factor does its resistance increase?
13. A copper wire has a resistance of 0.500 Ω at 20.0ºC, and an iron wire has a resistance of 0.525 Ω at the same temperature. At what temperature are their resistances equal?
14. (a) Digital medical thermometers determine temperature by measuring the resistance of a semiconductor device called a thermistor (which has α = –0.0600/ºC) when it is at the same temperature as the patient. What is a patient’s temperature if the thermistor’s resistance at that temperature is 82.0% of its value at 37.0ºC (normal body temperature)? (b) The negative value for α may not be maintained for very low temperatures. Discuss why and whether this is the case here. (Hint: Resistance can’t become negative.)
15. Integrated Concepts (a) Redo Exercise 2 taking into account the thermal expansion of the tungsten filament. You may assume a thermal expansion coefficient of 12 × 10^{−6}/ºC. (b) By what percentage does your answer differ from that in the example?
16. Unreasonable Results (a) To what temperature must you raise a resistor made of constantan to double its resistance, assuming a constant temperature coefficient of resistivity? (b) To cut it in half? (c) What is unreasonable about these results? (d) Which assumptions are unreasonable, or which premises are inconsistent?
Glossary
- resistivity:
- an intrinsic property of a material, independent of its shape or size, directly proportional to the resistance, denoted by ρ
- temperature coefficient of resistivity:
- an empirical quantity, denoted by α, which describes the change in resistance or resistivity of a material with temperature
Selected Solutions to Problems & Exercises
1. 0.104 Ω3. 2.8 × 10^{−2}m
5. 1.10 × 10^{−3}A 7. −5ºC to 45ºC
9. 1.03
11. 0.06%
13.−17ºC
15. (a) 4.7 Ω (total) (b) 3.0% decrease
Licenses and Attributions
More Study Resources for You
Show More