College Physics for AP® Courses

# 20.3Resistance and Resistivity

College Physics for AP® Courses20.3 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.

The information presented in this section supports the following AP® learning objectives and science practices:

• 1.E.2.1 The student is able to choose and justify the selection of data needed to determine resistivity for a given material. (S.P. 4.1)
• 4.E.4.2 The student is able to design a plan for the collection of data to determine the effect of changing the geometry and/or materials on the resistance or capacitance of a circuit element and relate results to the basic properties of resistors and capacitors. (S.P. 4.1, 4.2)
• 4.E.4.3 The student is able to analyze data to determine the effect of changing the geometry and/or materials on the resistance or capacitance of a circuit element and relate results to the basic properties of resistors and capacitors. (S.P. 5.1)

## 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 20.13 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 $AA size 12{A} {}$.

Figure 20.13 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) $600×10−3600×10−3$
Germanium $(1−600)×10−3(1−600)×10−3 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 $10 8 − 10 14 10 8 − 10 14$
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 20.14.) 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.

Figure 20.14 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 $4 . 3 × 10 − 3 4 . 3 × 10 − 3 size 12{3 "." 9 times "10" rSup { size 8{ - 3} } } {}$
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 Temperature 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 20.15.) 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 20.15 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 20.16

## Applying the Science Practices: Examining Resistance

Using the PhET Simulation “Resistance in a Wire”, design an experiment to determine how different variables – resistivity, length, and area – affect the resistance of a resistor. For each variable, you should record your results in a table and then create a graph to determine the relationship.

### Footnotes

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