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# Resistance and Resistivity

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Summary:

• 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

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 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} {}.

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} } "."} {}
(1)

Table 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.

Table 1: Resistivities ρ ρ size 12{ρ} {} of Various materials at 20º C 20º C

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} } } {}

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)
600×103600×103

Germanium
(1600)×103(1600)×103 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 11 10 8 10 11 size 12{"10" rSup { size 8{8} } - "10" rSup { size 8{"11"} } } {}

### Example 1: 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} } "."} {}
(2)

Substituting the given values, and taking ρρ size 12{ρ} {} from Table 1, yields

A = ( 5.6 × 10 –8 Ω m ) ( 4.00 × 10 –2 m ) 1.350 Ω = 6.40 × 10 –9 m 2 . A = ( 5.6 × 10 –8 Ω m ) ( 4.00 × 10 –2 m ) 1.350 Ω = 6.40 × 10 –9 m 2 .
(3)

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} } "."} {}
(4)

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" "." {} } } {}
(5)

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 2.) 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} } $$"1 "+ αΔT$$ ","} {}
(6)

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 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 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 α α size 12{α} {}

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} } } {}

3 . 9 × 10 3 3 . 9 × 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} } } {}

Note also that αα size 12{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 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} } $$"1 "+ αΔT$$ } {}
(7)

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 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=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.
(8)

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 RR size 12{R} {} of a cylinder of length LL size 12{L} {} and cross-sectional area AA size 12{A} {} is R=ρLAR=ρLA size 12{R = { {ρL} over {A} } } {}, where ρρ size 12{ρ} {} is the resistivity of the material.
• Values of ρρ size 12{ρ} {} in Table 1 show that materials fall into three groups—conductors, semiconductors, and insulators.
• Temperature affects resistivity; for relatively small temperature changes ΔTΔT size 12{DT} {}, resistivity is ρ=ρ0(1 +αΔT)ρ=ρ0(1 +αΔT) size 12{ρ = ρ rSub { size 8{0} } $$"1 "+ αΔT$$ } {}, where ρ0ρ0 size 12{ρ rSub { size 8{0} } } {} is the original resistivity and α α is the temperature coefficient of resistivity.
• Table 2 gives values for αα size 12{α} {}, the temperature coefficient of resistivity.
• The resistance RR size 12{R} {} of an object also varies with temperature: R=R0(1 +αΔT)R=R0(1 +αΔT) size 12{R = R rSub { size 8{0} } $$"1 "+ ΔαT$$ } {}, where R0R0 size 12{R rSub { size 8{0} } } {} is the original resistance, and R R is the resistance after the temperature change.

## Conceptual Questions

### Exercise 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.)

### Exercise 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.)

### Exercise 3

If aluminum and copper wires of the same length have the same resistance, which has the larger diameter? Why?

### Exercise 4

Explain why R=R0(1 +αΔT)R=R0(1 +αΔT) size 12{R = R rSub { size 8{0} } $$"1 "+ αΔT$$ } {} for the temperature variation of the resistance RR size 12{R} {} of an object is not as accurate as ρ=ρ0(1 +αΔT)ρ=ρ0(1 +αΔT) size 12{ρ = ρ rSub { size 8{0} } $$"1 "+ αΔT$$ } {}, which gives the temperature variation of resistivity ρρ size 12{ρ} {}.

## Problems & Exercises

### Exercise 1

What is the resistance of a 20.0-m-long piece of 12-gauge copper wire having a 2.053-mm diameter?

0.104 Ω 0.104 Ω

### Exercise 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.

### Exercise 3

If the 0.100-mm diameter tungsten filament in a light bulb is to have a resistance of 0.200 Ω 0.200 Ω at 20.C20.C size 12{"20" "." 0°"C"} {}, how long should it be?

#### Solution

2 . 8 × 10 2 m 2 . 8 × 10 2 m size 12{2 "." "82"´"10" rSup { size 8{-2} } " m"} {}

### Exercise 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).

### Exercise 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 1.00 × 10 3 V is applied to it? (Such a rod may be used to make nuclear-particle detectors, for example.)

#### Solution

1 . 10 × 10 3 A 1 . 10 × 10 3 A size 12{1 "." "10"´"10" rSup { size 8{-3} } " A"} {}

### Exercise 6

(a) To what temperature must you raise a copper wire, originally at 20.0ºC 20.0ºC , to double its resistance, neglecting any changes in dimensions? (b) Does this happen in household wiring under ordinary circumstances?

### Exercise 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.C20.C size 12{"20" "." 0°"C"} {}. Over what temperature range can it be used?

#### Solution

C to 45ºC C to 45ºC

### Exercise 8

Of what material is a resistor made if its resistance is 40.0% greater at 100ºC100ºC size 12{"100"°"C"} {} than at 20.C20.C size 12{"20" "." 0°"C"} {}?

### Exercise 9

An electronic device designed to operate at any temperature in the range from –10.C to 55.C–10.C to 55.C size 12{"10" "." 0°"C to 55" "." 0°"C"} {} contains pure carbon resistors. By what factor does their resistance increase over this range?

1.03

### Exercise 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 77.7Ω77.7Ω size 12{"77" "." 7 %OMEGA } {} at 20.C20.C size 12{"20" "." 0°"C"} {}? (b) What is its resistance at 150ºC150ºC size 12{"150"°"C"} {}?

### Exercise 11

Assuming a constant temperature coefficient of resistivity, what is the maximum percent decrease in the resistance of a constantan wire starting at 20.C20.C size 12{"20" "." 0°"C"} {}?

0.06%

### Exercise 12

A wire is drawn through a die, stretching it to four times its original length. By what factor does its resistance increase?

### Exercise 13

A copper wire has a resistance of 0.500 Ω0.500 Ω size 12{0 "." "500 " %OMEGA } {} at 20.C20.C size 12{"20" "." 0°"C"} {}, and an iron wire has a resistance of 0.525 Ω0.525 Ω size 12{0 "." "525 " %OMEGA } {} at the same temperature. At what temperature are their resistances equal?

#### Solution

17º C 17º C size 12{-"17"°"C"} {}

### Exercise 14

(a) Digital medical thermometers determine temperature by measuring the resistance of a semiconductor device called a thermistor (which has α=0.0600/ºCα=0.0600/ºC size 12{α"=-"0 "." "0600"/°"C"} {}) 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.C37.C size 12{"37" "." 0°"C"} {} (normal body temperature)? (b) The negative value for αα size 12{α} {} may not be maintained for very low temperatures. Discuss why and whether this is the case here. (Hint: Resistance can’t become negative.)

### Exercise 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 12×106/ºC12×106/ºC size 12{"12"´"10" rSup { size 8{-6} } /°"C"} {}. (b) By what percentage does your answer differ from that in the example?

#### Solution

(a) 4.7Ω4.7Ω size 12{4 "." 7 %OMEGA } {} (total)

(b) 3.0% decrease

### Exercise 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?

## Footnotes

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

## 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

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