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.
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The resistance of an object depends on its shape and the material of which it is composed. The cylindrical resistor in Figure 1 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 R is directly proportional to its length 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, R is inversely proportional to the cylinder’s cross-sectional area A.
Figure 1. A uniform cylinder of length L and cross-sectional area 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 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 resistivityρ of a substance so that the resistance R of an object is directly proportional to ρ. Resistivity ρ is an intrinsic property of a material, independent of its shape or size. The resistance R of a uniform cylinder of length L, of cross-sectional area A, and made of a material with resistivity ρ, is
ho L}{A}\
Table 1 gives representative values of 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.
Table 1. Resistivities ρ of Various materials at 20º C MaterialResistivity 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<1> | |
Carbon (pure) | 3.5 × 105 |
Carbon | (3.5 − 60) × 105 |
Germanium (pure) | 600 × 10−3 |
Germanium | (1−600) × 10−3 |
Silicon (pure) | 2300 |
Silicon | 0.1–2300 |
Insulators | |
Amber | 5 × 1014 |
Glass | 109 − 1014 |
Lucite | >1013 |
Mica | 1011 − 1015 |
Quartz (fused) | 75 × 1016 |
Rubber (hard) | 1013 − 1016 |
Sulfur | 1015 |
Teflon | >1013 |
Wood | 108 − 1011 |
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
ho L}{A}\
Solution
The cross-sectional area, found by rearranging the expression for the resistance of a cylinder given in
ho L}{A}\
ho L}{R}\
Substituting the given values, and taking ρ from Table 1, yields
ight)left(4.00 imes { ext{10}}^{-2} ext{m}
ight)}{ ext{0.350}Omega }\ & =& ext{6.40} imes { ext{10}}^{-9}{ ext{m}}^{2}end{array}\
The area of a circle is related to its diameter D by
Solving for the diameter D, and substituting the value found for A, gives
ight)}^{frac{1}{2}}= ext{2}{left(frac{6.40 imes { ext{10}}^{-9}{ ext{m}}^{2}}{3.14}
ight)}^{frac{1}{2}}\ & =& 9.0 imes { ext{10}}^{-5} ext{m}end{array}\
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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.
Figure 2. 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.
where ρ0 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.
Table 2. Tempature Coefficients of Resistivity αMaterialCoefficient (1/°C)<2>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 |
Note also that α is negative for the semiconductors listed in Table 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 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 R0 is directly proportional to ρ. For a cylinder we know R = ρL/A, and so, if L and A do not change greatly with temperature, R will have the same temperature dependence as 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 L and A is about two orders of magnitude less than on ρ.) Thus,
R = R 0 ( 1 + αΔT )
is the temperature dependence of the resistance of an object, where R0 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.
Figure 3. These familiar thermometers are based on the automated measurement of a thermistor’s temperature-dependent resistance. (credit: Biol, Wikimedia Commons)
Although caution must be used in applying ρ = ρ0(1 +αΔT) and R = R0(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 = R0(1 +αΔT), since the original resistance of the filament was given to be R0 = 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:
ight)\ & =& left(0.350Omega
ight)left<1+left(4.5 imes{10}^{-3}/º ext{C}
ight)left(2830º ext{C}
ight)
ight>\ & =& {4.8Omega}end{array}\
Discussion
This value is consistent with the headlight resistance example in Ohm’s Law: Resistance and Simple Circuits.
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Section Summary
The resistance R of a cylinder of length L and cross-sectional area A is
ho L}{A}\
ho ={
ho }_{0}left( ext{1}+alpha Delta T
ight)\
ight)\
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.)