Explain the ide of resistivity.Use resistivity to calculate the resistance of specified configurations the material.Use the heat coefficient of resistivity to calculate the readjust of resistance v temperature.

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The resistance of an object depends on its shape and the product of which that is composed. The cylindrical resistor in figure 1 is straightforward to analyze, and, by for this reason doing, us can gain insight into the resistance the more complicated shapes. As you can expect, the cylinder’s electric resistance R is straight proportional come its size L, comparable to the resistance that a pipe to fluid flow. The longer the cylinder, the more collisions charges will certainly make with its atoms. The higher the diameter the the cylinder, the much more current that can lug (again comparable to the circulation 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 size L and also cross-sectional area A. That resistance to the circulation of present is similar to the resistance make by a pipeline to liquid flow. The longer the cylinder, the better its resistance. The bigger its cross-sectional area A, the smaller its resistance.


For a offered shape, the resistance relies on the product of which the thing is composed. Different materials offer various resistance to the circulation of charge. We define the resistivityρ of a problem so that the resistance R of an item is directly proportional come ρ. Resistivity ρ is an intrinsic building of a material, elevation of its shape or size. The resistance R that a uniform cylinder of length L, the cross-sectional area A, and also made that a material with resistivity ρ, is


R=\frac\rho LA\\.

Table 1 gives representative worths of ρ. The materials noted in the table are separated into categories that conductors, semiconductors, and insulators, based on wide groupings that resistivities. Conductors have the the smallest resistivities, and insulators have the largest; semiconductors have intermediate resistivities. Conductors have varying however large complimentary charge densities, whereas most charges in insulators are bound come atoms and are not totally free to move. Semiconductors space intermediate, having much fewer complimentary charges than conductors, but having properties the make the number of cost-free charges rely strongly ~ above the type and quantity of impurities in the semiconductor. These distinctive properties the semiconductors are placed to use in contemporary electronics, as will certainly be discover in later on chapters.


Table 1. Resistivities ρ of Various products at 20º C MaterialResistivity ρ ( Ω ⋅ m )
Conductors
Silver1. 59 × 10−8
Copper1. 72 × 10−8
Gold2. 44 × 10−8
Aluminum2. 65 × 10−8
Tungsten5. 6 × 10−8
Iron9. 71 × 10−8
Platinum10. 6 × 10−8
Steel20 × 10−8
Lead22 × 10−8
Manganin (Cu, Mn, Ni alloy)44 × 10−8
Constantan (Cu, Ni alloy)49 × 10−8
Mercury96 × 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
Silicon0.1–2300
Insulators
Amber5 × 1014
Glass109 − 1014
Lucite>1013
Mica1011 − 1015
Quartz (fused)75 × 1016
Rubber (hard)1013 − 1016
Sulfur1015
Teflon>1013
Wood108 − 1011

Example 1. Calculating Resistor Diameter: A Headlight Filament


A auto headlight filament is make of tungsten and also has a cold resistance the 0.350 Ω. If the filament is a cylinder 4.00 cm long (it might be coiled to conserve space), what is its diameter?

Strategy

We can rearrange the equation R=\frac\rho LA\\ to discover the cross-sectional area A of the filament from the given information. Then its diameter can be found by assuming it has a one cross-section.

Solution

The cross-sectional area, uncovered by rearranging the expression for the resistance the a cylinder offered in R=\frac\rho LA\\, is

A=\frac\rho LR\\.

Substituting the offered values, and taking ρ from Table 1, yields

\beginarraylllA& =& \frac\left(5.6\times \text10^-8\Omega \cdot \textm\right)\left(4.00\times \text10^-2\textm\right)\text0.350\Omega \\ & =& \text6.40\times \text10^-9\textm^2\endarray\\.

The area the a one is pertained to its diameter D by

A=\frac\pi D^24\\.

Solving because that the diameter D, and also substituting the value discovered for A, gives

\beginarraylllD& =& \text2\left(\fracAp\right)^\frac12=\text2\left(\frac6.40\times \text10^-9\textm^23.14\right)^\frac12\\ & =& 9.0\times \text10^-5\textm\endarray\\.

Discussion

The diameter is simply under a tenth that a millimeter. The is quoted to only two digits, because ρ is recognized to only two digits.


Figure 2. The resistance the a sample of mercury is zero at an extremely low temperatures—it is a superconductor up to about 4.2 K. Above that critical temperature, that is resistance makes a sudden jump and also then increases almost linearly through temperature.


where ρ0 is the initial resistivity and also α is the temperature coefficient the resistivity. (See the values of α in Table 2 below.) For bigger temperature changes, α might vary or a nonlinear equation may be required to find ρ. Keep in mind that α is confident for metals, definition their resistivity rises with temperature. Part alloys have actually been developed specifically to have actually a tiny temperature dependence. Manganin (which is do of copper, manganese and nickel), because that example, has actually α close to zero (to three digits ~ above the scale in Table 2), and so that resistivity varies only slightly v temperature. This is helpful for making a temperature-independent resistance standard, for example.


Table 2. Tempature Coefficients of Resistivity αMaterialCoefficient (1/°C)<2>
Conductors
Silver3.8 × 10−3
Copper3.9 × 10−3
Gold3.4 × 10−3
Aluminum3.9 × 10−3
Tungsten4.5 × 10−3
Iron5.0 × 10−3
Platinum3.93 × 10−3
Lead3.9 × 10−3
Manganin (Cu, Mn, Ni alloy)0.000 × 10−3
Constantan (Cu, Ni alloy)0.002 × 10−3
Mercury0.89 × 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 additionally that α is negative for the semiconductors detailed in Table 2, an interpretation that your resistivity to reduce with enhancing temperature. Castle become much better conductors at greater temperature, due to the fact that increased thermal agitation rises the number of free charges obtainable to bring current. This property of diminish ρ through temperature is also related to the form and lot of impurities existing in the semiconductors. The resistance of one object likewise depends top top temperature, since R0 is directly proportional come ρ. Because that a cylinder we know ρL/A, and also so, if L and A perform not change greatly with temperature, R will have actually the same temperature dependence together ρ. (Examination the the coefficients of linear growth shows them come be about two orders of magnitude less than typical temperature coefficients the resistivity, and so the impact of temperature top top L and also A is around two assignment of magnitude much less than top top ρ.) Thus,


R = R 0 ( 1 + αΔT )

is the temperature dependence of the resistance of one object, where R0 is the original resistance and R is the resistance after ~ a temperature adjust ΔT. Plenty of thermometers are based upon the effect of temperature top top resistance. (See figure 3.) among the most usual is the thermistor, a semiconductor crystal v a solid temperature dependence, the resistance of i m sorry is measure up to acquire its temperature. The an equipment is small, so the it quickly comes into thermal equilibrium v the part of a person it touches.


Figure 3. These familiar thermometers are based on the automatic measurement the a thermistor’s temperature-dependent resistance. (credit: Biol, Wikimedia Commons)


Although caution have to be supplied in using ρ ρ0(1 +αΔT) and also R0(1 +αΔT) because that temperature transforms greater than 100ºC, for tungsten the equations work-related reasonably well for very huge temperature changes. What, then, is the resistance the the tungsten filament in the previous instance if that temperature is raised from room temperature ( 20ºC ) to a typical operating temperature that 2850ºC?

Strategy

This is a straightforward applications of R0(1 +αΔT), since the original resistance of the filament was offered to it is in R0 = 0.350 Ω, and the temperature readjust is Δ= 2830ºC.

Solution

The hot resistance R is obtained by entering well-known values into the over equation:

\beginarraylllR & =& R_0\left(1+\alpha\Delta T\right)\\ & =& \left(0.350\Omega\right)\left<1+\left(4.5\times10^-3/º\textC\right)\left(2830º\textC\right)\right>\\ & =& 4.8\Omega\endarray\\.

Discussion

This value is constant with the headlight resistance instance 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 R=\frac\rho LA\\, where ρ is the resistivity of the material.Values of ρ in Table 1 show that materials loss into three groups—conductors, semiconductors, and insulators.Temperature affect resistivity; for reasonably small temperature changes ΔT, resistivity is \rho =\rho _0\left(\text1+\alpha \Delta T\right)\\ , where ρ0 is the original resistivity and also \text\alpha is the temperature coefficient the resistivity.Table 2 provides values for α, the temperature coefficient that resistivity.The resistance R of one object additionally varies through temperature: R=R_0\left(\text1+\alpha \Delta T\right)\\, where R0 is the original resistance, and R is the resistance ~ the temperature change.

Conceptual Questions


1. In i m sorry of the three semiconducting materials listed in Table 1 do impurities supply free charges? (Hint: study the selection of resistivity because that each and also determine whether the pure semiconductor has actually the greater or lower conductivity.)