Resistance and Resistivity

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 [link] 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 size 12{R} {}

is directly proportional to its length L 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, R size 12{R} {}

is inversely proportional to the cylinder’s cross-sectional area A size 12{A} {}

.

A cylindrical conductor of length L and cross section A is shown. The resistivity of the cylindrical section is represented as rho. The resistance of this cross section R is equal to rho L divided by A. The section of length L of cylindrical conductor is shown equivalent to a resistor represented by symbol R.

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 ρ size 12{ρ} {}

of a substance so that the resistance R size 12{R} {}

of an object is directly proportional to ρ size 12{ρ} {}

. Resistivity ρ size 12{ρ} {}

is an intrinsic property of a material, independent of its shape or size. The resistance R size 12{R} {}

of a uniform cylinder of length L size 12{L} {}

, of cross-sectional area A size 12{A} {}

, and made of a material with resistivity ρ size 12{ρ} {}

, is

R=ρLA. size 12{R = { {ρL} over {A} } "."} {}

[link] gives representative values of ρ 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.

Resistivities ρ size 12{ρ} {} of Various materials at 20º C
Material Resistivity ρ size 12{ρ} {} ( Ω ⋅ m size 12{ %OMEGA cdot m} {} )
Conductors
Silver 1 . 59 × 10 − 8 size 12{1 "." "59" times "10" rSup { size 8{ - 8} } } {}
Copper 1 . 72 × 10 − 8 size 12{1 "." "72" times "10" rSup { size 8{ - 8} } } {}
Gold 2 . 44 × 10 − 8 size 12{2 "." "44" times "10" rSup { size 8{ - 8} } } {}
Aluminum 2 . 65 × 10 − 8 size 12{2 "." "65" times "10" rSup { size 8{ - 8} } } {}
Tungsten 5 . 6 × 10 − 8 size 12{5 "." 6 times "10" rSup { size 8{ - 8} } } {}
Iron 9 . 71 × 10 − 8 size 12{9 "." "71" times "10" rSup { size 8{ - 8} } } {}
Platinum 10 . 6 × 10 − 8 size 12{"10" "." 6 times "10" rSup { size 8{ - 8} } } {}
Steel 20 × 10 − 8 size 12{"20" times "10" rSup { size 8{ - 8} } } {}
Lead 22 × 10 − 8 size 12{"22" times "10" rSup { size 8{ - 8} } } {}
Manganin (Cu, Mn, Ni alloy) 44 × 10 − 8 size 12{"44" times "10" rSup { size 8{ - 8} } } {}
Constantan (Cu, Ni alloy) 49 × 10 − 8 size 12{"49" times "10" rSup { size 8{ - 8} } } {}
Mercury 96 × 10 − 8 size 12{"96" times "10" rSup { size 8{ - 8} } } {}
Nichrome (Ni, Fe, Cr alloy) 100 × 10 − 8 size 12{"100" times "10" rSup { size 8{ - 8} } } {}
Semiconductors1
Carbon (pure) 3.5 × 10 –5
Carbon ( 3.5 − 60 ) × 10 –5
Germanium (pure) 600×10−3
Germanium (1−600)×10−3 size 12{ \( 1 - "600" \) times "10" rSup { size 8{ - 3} } } {}
Silicon (pure) 2300
Silicon 0.1–2300
Insulators
Amber 5 × 10 14 size 12{5 times "10" rSup { size 8{"14"} } } {}
Glass 10 9 − 10 14 size 12{"10" rSup { size 8{9} } - "10" rSup { size 8{"14"} } } {}
Lucite >10 13 size 12{>"10" rSup { size 8{"13"} } } {}
Mica 10 11 − 10 15 size 12{"10" rSup { size 8{"11"} } - "10" rSup { size 8{"15"} } } {}
Quartz (fused) 75 × 10 16 size 12{"75" times "10" rSup { size 8{"16"} } } {}
Rubber (hard) 10 13 − 10 16 size 12{"10" rSup { size 8{"13"} } - "10" rSup { size 8{"16"} } } {}
Sulfur 10 15 size 12{"10" rSup { size 8{"15"} } } {}
Teflon >10 13 size 12{>"10" rSup { size 8{"13"} } } {}
Wood 10 8 − 10 11 size 12{"10" rSup { size 8{8} } - "10" rSup { size 8{"11"} } } {}
Calculating Resistor Diameter: A Headlight Filament

A car headlight filament is made of tungsten and has a cold resistance of 0.350Ω 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=ρLA size 12{R = { {ρL} over {A} } } {}

to find the cross-sectional area A 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=ρLA size 12{R = { {ρL} over {A} } } {}

, is

A=ρLR. size 12{A = { {ρL} over {R} } "."} {}

Substituting the given values, and taking ρ size 12{ρ} {}

from [link], yields

A = ( 5.6 × 10 –8 Ω ⋅ m ) ( 4.00 × 10 –2 m ) 0.350 Ω = 6.40 × 10 –9 m 2 .

The area of a circle is related to its diameter D size 12{D} {}

by

A=πD24. size 12{A = { {πD rSup { size 8{2} } } over {4} } "."} {}

Solving for the diameter D size 12{D} {}

, and substituting the value found for A 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 . 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" "." {} } } {}

Discussion

The diameter is just under a tenth of a millimeter. It is quoted to only two digits, because ρ 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 [link].) 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 size 12{"100"°C} {}

or less), resistivity ρ size 12{ρ} {}

varies with temperature change ΔT size 12{DT} {}

as expressed in the following equation

ρ=ρ0(1 +αΔT), size 12{ρ = ρ rSub { size 8{0} } \( "1 "+ αΔT \) ","} {}

where ρ0 size 12{ρ rSub { size 8{0} } } {}

is the original resistivity and α size 12{α} {}

is the temperature coefficient of resistivity. (See the values of α size 12{α} {}

in [link] below.) For larger temperature changes, α size 12{α} {}

may vary or a nonlinear equation may be needed to find ρ size 12{ρ} {}

. Note that α 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 α size 12{α} {}

** close to zero (to three digits on the scale in [link]), and so its resistivity varies only slightly with temperature. This is useful for making a temperature-independent resistance standard, for example.

A graph for variation of resistance R with temperature T for a mercury sample is shown. The temperature T is plotted along the x axis and is measured in Kelvin, and the resistance R is plotted along the y axis and is measured in ohms. The curve starts at x equals zero and y equals zero, and coincides with the X axis until the value of temperature is four point two Kelvin, known as the critical temperature T sub c. At temperature T sub c, the curve shows a vertical rise, represented by a dotted line, until the resistance is about zero point one one ohms. After this temperature the resistance shows a nearly linear increase with temperature T.

Tempature Coefficients of Resistivity α size 12{α} {}
Material Coefficient α(1/°C)2
Conductors
Silver 3 . 8 × 10 − 3 size 12{3 "." 8 times "10" rSup { size 8{ - 3} } } {}
Copper 3 . 9 × 10 − 3 size 12{3 "." 9 times "10" rSup { size 8{ - 3} } } {}
Gold 3 . 4 × 10 − 3 size 12{3 "." 4 times "10" rSup { size 8{ - 3} } } {}
Aluminum 3 . 9 × 10 − 3 size 12{3 "." 9 times "10" rSup { size 8{ - 3} } } {}
Tungsten 4 . 5 × 10 − 3 size 12{4 "." 5 times "10" rSup { size 8{ - 3} } } {}
Iron 5 . 0 × 10 − 3 size 12{5 "." 0 times "10" rSup { size 8{ - 3} } } {}
Platinum 3 . 93 × 10 − 3 size 12{3 "." "93" times "10" rSup { size 8{ - 3} } } {}
Lead 3 . 9 × 10 − 3 size 12{3 "." 9 times "10" rSup { size 8{ - 3} } } {}
Manganin (Cu, Mn, Ni alloy) 0 . 000 × 10 − 3 size 12{0 "." "000" times "10" rSup { size 8{ - 3} } } {}
Constantan (Cu, Ni alloy) 0 . 002 × 10 − 3 size 12{0 "." "002" times "10" rSup { size 8{ - 3} } } {}
Mercury 0 . 89 × 10 − 3 size 12{0 "." "89" times "10" rSup { size 8{ - 3} } } {}
Nichrome (Ni, Fe, Cr alloy) 0 . 4 × 10 − 3 size 12{0 "." 4 times "10" rSup { size 8{ - 3} } } {}
Semiconductors
Carbon (pure) − 0 . 5 × 10 − 3 size 12{ - 0 "." 5 times "10" rSup { size 8{ - 3} } } {}
Germanium (pure) − 50 × 10 − 3 size 12{ - "50" times "10" rSup { size 8{ - 3} } } {}
Silicon (pure) − 70 × 10 − 3 size 12{ - "70" times "10" rSup { size 8{ - 3} } } {}

Note also that α size 12{α} {}

is negative for the semiconductors listed in [link], 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 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 R0 size 12{R rSub { size 8{0} } } {}

is directly proportional to ρ size 12{ρ} {}

. For a cylinder we know R=ρL/A size 12{R=ρL/A} {}

, and so, if L size 12{L} {}

and A size 12{A} {}

do not change greatly with temperature, R size 12{R} {}

will have the same temperature dependence as ρ 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 L size 12{L} {}

and A size 12{A} {}

is about two orders of magnitude less than on ρ size 12{ρ} {}

.) Thus,

R = R 0 ( 1 + αΔT ) size 12{R = R rSub { size 8{0} } \( "1 "+ αΔT \) } {}

is the temperature dependence of the resistance of an object, where R0 size 12{R rSub { size 8{0} } } {}

is the original resistance and R size 12{R} {}

is the resistance after a temperature change ΔT size 12{DT} {}

. Numerous thermometers are based on the effect of temperature on resistance. (See [link].) 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.

A photograph showing two digital thermometers used for measuring body temperature.

Calculating Resistance: Hot-Filament Resistance

Although caution must be used in applying ρ=ρ0(1 +αΔT) size 12{ρ = ρ rSub { size 8{0} } \( "1 "+ αΔT \) } {}

and R=R0(1 +αΔT) size 12{R = R rSub { size 8{0} } \( "1 "+ αΔT \) } {}

for temperature changes greater than 100ºC 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

) to a typical operating temperature of 2850ºC size 12{"2850"°"C"} {}

?

Strategy

This is a straightforward application of 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 Ω size 12{R rSub { size 8{0} } =0 "." "350"` %OMEGA } {}

, and the temperature change is ΔT=2830ºC size 12{ΔT="2830"°"C"} {}

.

Solution

The hot resistance R 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.

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

Conceptual Questions

In which of the three semiconducting materials listed in [link] 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.)

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 [link].)

Part a of the figure shows a voltage V applied along the length of a rectangular bar using a battery. The current is shown to emerge from the positive terminal, pass along the length of the rectangular bar, and enter the negative terminal of the battery. The resistance of the rectangular bar along the length is shown as R and the current is shown as I. Part b of the figure shows a voltage V applied along the width of the same rectangular bar using a battery. The current is shown to emerge from the positive terminal, pass along the width of the rectangular bar, and enter the negative terminal of the battery. The resistance of the rectangular bar along the width is shown as R prime, and the current is shown as I prime.

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

Explain why R=R0(1 +αΔT) size 12{R = R rSub { size 8{0} } \( "1 "+ αΔT \) } {}

for the temperature variation of the resistance R size 12{R} {}

of an object is not as accurate as ρ=ρ0(1 +αΔT) size 12{ρ = ρ rSub { size 8{0} } \( "1 "+ αΔT \) } {}

, which gives the temperature variation of resistivity ρ size 12{ρ} {}

.

Problems & Exercises

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

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.

If the 0.100-mm diameter tungsten filament in a light bulb is to have a resistance of 0.200 Ω

at 20.0ºC size 12{"20" "." 0°"C"} {}

, how long should it be?

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

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).

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.)

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

(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?

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 size 12{"20" "." 0°"C"} {}

. Over what temperature range can it be used?

− 5º C to 45ºC

Of what material is a resistor made if its resistance is 40.0% greater at 100ºC size 12{"100"°"C"} {}

than at 20.0ºC size 12{"20" "." 0°"C"} {}

?

An electronic device designed to operate at any temperature in the range from –10.0ºC to 55.0ºC 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

(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 77.7Ω size 12{"77" "." 7 %OMEGA } {}

at 20.0ºC size 12{"20" "." 0°"C"} {}

? (b) What is its resistance at 150ºC size 12{"150"°"C"} {}

?

Assuming a constant temperature coefficient of resistivity, what is the maximum percent decrease in the resistance of a constantan wire starting at 20.0ºC size 12{"20" "." 0°"C"} {}

?

0.06%

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

A copper wire has a resistance of 0.500 Ω size 12{0 "." "500 " %OMEGA } {}

at 20.0ºC size 12{"20" "." 0°"C"} {}

, and an iron wire has a resistance of 0.525 Ω size 12{0 "." "525 " %OMEGA } {}

at the same temperature. At what temperature are their resistances equal?

− 17º C size 12{-"17"°"C"} {}

(a) Digital medical thermometers determine temperature by measuring the resistance of a semiconductor device called a thermistor (which has α=–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 37.0ºC size 12{"37" "." 0°"C"} {}

(normal body temperature)? (b) The negative value for α 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.)

Integrated Concepts

(a) Redo [link] taking into account the thermal expansion of the tungsten filament. You may assume a thermal expansion coefficient of 12×10−6/ºC size 12{"12"´"10" rSup { size 8{-6} } /°"C"} {}

. (b) By what percentage does your answer differ from that in the example?

(a) 4.7Ω size 12{4 "." 7 %OMEGA } {}

(total)

(b) 3.0% decrease

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

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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