Contents

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 Figure 20.11 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 $R R size 12{R} {}$ is directly proportional to its length $L L size 12{L} {}$ , 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, $R R size 12{R} {}$ is inversely proportional to the cylinder’s cross-sectional area $A A size 12{A} {}$ .

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.

Figure 20.11 A uniform cylinder of length

L L size 12{L} {}

and cross-sectional area

A A 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

A A 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 $ρ ρ size 12{ρ} {}$ of a substance so that the resistance $R R size 12{R} {}$ of an object is directly proportional to $ρ ρ size 12{ρ} {}$ . Resistivity $ρ ρ size 12{ρ} {}$ is an intrinsic property of a material, independent of its shape or size. The resistance $R R size 12{R} {}$ of a uniform cylinder of length $L L size 12{L} {}$ , of cross-sectional area $A A size 12{A} {}$ , and made of a material with resistivity $ρ ρ size 12{ρ} {}$ , is

R = \frac{ρL}{A} . R = \frac{ρL}{A} . size 12{R =  {  {ρL}  over  {A} } "."} {}

20.18

Table 20.1 gives representative values of $ρ ρ size 12{ρ} {}$ . 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.

Material	Resistivity $ρ ρ size 12{ρ} {}$ ( $Ω \cdot m Ω \cdot m size 12{ %OMEGA cdot m} {}$ )
Conductors
Silver	$1 . 59 \times 10^{- 8} 1 . 59 \times 10^{- 8} size 12{1 "." "59" times "10" rSup { size 8{ - 8} } } {}$
Copper	$1 . 72 \times 10^{- 8} 1 . 72 \times 10^{- 8} size 12{1 "." "72" times "10" rSup { size 8{ - 8} } } {}$
Gold	$2 . 44 \times 10^{- 8} 2 . 44 \times 10^{- 8} size 12{2 "." "44" times "10" rSup { size 8{ - 8} } } {}$
Aluminum	$2 . 65 \times 10^{- 8} 2 . 65 \times 10^{- 8} size 12{2 "." "65" times "10" rSup { size 8{ - 8} } } {}$
Tungsten	$5 . 6 \times 10^{- 8} 5 . 6 \times 10^{- 8} size 12{5 "." 6 times "10" rSup { size 8{ - 8} } } {}$
Iron	$9 . 71 \times 10^{- 8} 9 . 71 \times 10^{- 8} size 12{9 "." "71" times "10" rSup { size 8{ - 8} } } {}$
Platinum	$10 . 6 \times 10^{- 8} 10 . 6 \times 10^{- 8} size 12{"10" "." 6 times "10" rSup { size 8{ - 8} } } {}$
Steel	$20 \times 10^{- 8} 20 \times 10^{- 8} size 12{"20" times "10" rSup { size 8{ - 8} } } {}$
Lead	$22 \times 10^{- 8} 22 \times 10^{- 8} size 12{"22" times "10" rSup { size 8{ - 8} } } {}$
Manganin (Cu, Mn, Ni alloy)	$44 \times 10^{- 8} 44 \times 10^{- 8} size 12{"44" times "10" rSup { size 8{ - 8} } } {}$
Constantan (Cu, Ni alloy)	$49 \times 10^{- 8} 49 \times 10^{- 8} size 12{"49" times "10" rSup { size 8{ - 8} } } {}$
Mercury	$96 \times 10^{- 8} 96 \times 10^{- 8} size 12{"96" times "10" rSup { size 8{ - 8} } } {}$
Nichrome (Ni, Fe, Cr alloy)	$100 \times 10^{- 8} 100 \times 10^{- 8} size 12{"100" times "10" rSup { size 8{ - 8} } } {}$
Semiconductors1 1 Values depend strongly on amounts and types of impurities
Carbon (pure)	$3.5 \times 10^{–5} 3.5 \times 10^{–5}$
Carbon	$(3.5 - 60) \times 10^{–5} (3.5 - 60) \times 10^{–5}$
Germanium (pure)	$600 \times 10^{- 3} 600 \times 10^{- 3}$
Germanium	$(1 - 600) \times 10^{- 3} (1 - 600) \times 10^{- 3} size 12{ $ 1 - "600" $ times "10" rSup { size 8{ - 3} } } {}$
Silicon (pure)	$23002300$
Silicon	$0.1–2300 0.1–2300$
Insulators
Amber	$5 \times 10^{14} 5 \times 10^{14} size 12{5 times "10" rSup { size 8{"14"} } } {}$
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)	$75 \times 10^{16} 75 \times 10^{16} size 12{"75" times "10" rSup { size 8{"16"} } } {}$
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	$10^{8} - 10^{11} 10^{8} - 10^{11} size 12{"10" rSup { size 8{8} } - "10" rSup { size 8{"11"} } } {}$

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 $0.350 Ω 0.350 size 12{0 "." "350" %OMEGA } {}$ . 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 $R = \frac{ρL}{A} R = \frac{ρL}{A} size 12{R = { {ρL} over {A} } } {}$ to find the cross-sectional area $A A size 12{A} {}$ 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 $R = \frac{ρL}{A} R = \frac{ρL}{A} size 12{R = { {ρL} over {A} } } {}$ , is

A = \frac{ρL}{R} . A = \frac{ρL}{R} . size 12{A =  {  {ρL}  over  {R} } "."} {}

20.19

Substituting the given values, and taking $ρ ρ size 12{ρ} {}$ from Table 20.1, yields

\begin{array}{l} A & = & \frac{(5.6 \times 10^{–8} Ω \cdot m) (4.00 \times 10^{–2} m)}{0.350 Ω} \\ = & 6.40 \times 10^{–9} m^{2} . \end{array} \begin{array}{l} A & = & \frac{(5.6 \times 10^{–8}) (4.00 \times 10^{–2}}{0.350} \\ = 6.40 \times 10^{–9} \end{array}

20.20

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

A = \frac{{πD}^{2}}{4} . A = \frac{{πD}^{2}}{4} . size 12{A =  {  {πD rSup { size 8{2} } }  over  {4} } "."} {}

20.21

Solving for the diameter $D D size 12{D} {}$ , and substituting the value found for $A A size 12{A} {}$ , gives

\begin{array}{l} D & = & 2 {(\frac{A}{p})}^{\frac{1}{2}} = 2 {(\frac{6.40 \times 10^{–9} m^{2}}{3.14})}^{\frac{1}{2}} \\ = & 9.0 \times 10^{–5} m . \end{array} \begin{array}{l} D & = & 2 {(\frac{A}{p})}^{\frac{1}{2}} = 2 {(\frac{6.40 \times 10^{–9}}{3.14})}^{\frac{1}{2}} \\ = 9.0 \times 10^{–5} \end{array} 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 $ρ ρ size 12{ρ} {}$ 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 Figure 20.12.) 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 $100º C 100º C size 12{"100"°C} {}$ or less), resistivity $ρ ρ size 12{ρ} {}$ varies with temperature change $Δ T Δ T size 12{DT} {}$ as expressed in the following equation

ρ = ρ_{0} (1 + α Δ T), ρ = ρ_{0} (1 + α Δ T), size 12{ρ = ρ rSub { size 8{0} }  \( "1 "+ αΔT \) ","} {}

20.23

where $ρ_{0} ρ_{0} size 12{ρ rSub { size 8{0} } } {}$ is the original resistivity and $α α size 12{α} {}$ is the temperature coefficient of resistivity. (See the values of $α α size 12{α} {}$ in Table 20.2 below.) For larger temperature changes, $α α size 12{α} {}$ may vary or a nonlinear equation may be needed to find $ρ ρ size 12{ρ} {}$ . Note that $α α size 12{α} {}$ 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 $α α size 12{α} {}$ 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.

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.

Figure 20.12 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 $α α$ (1/°C)2 2 Values at 20°C.
Conductors
Silver	$3 . 8 \times 10^{- 3} 3 . 8 \times 10^{- 3} size 12{3 "." 8 times "10" rSup { size 8{ - 3} } } {}$
Copper	$3 . 9 \times 10^{- 3} 3 . 9 \times 10^{- 3} size 12{3 "." 9 times "10" rSup { size 8{ - 3} } } {}$
Gold	$3 . 4 \times 10^{- 3} 3 . 4 \times 10^{- 3} size 12{3 "." 4 times "10" rSup { size 8{ - 3} } } {}$
Aluminum	$3 . 9 \times 10^{- 3} 3 . 9 \times 10^{- 3} size 12{3 "." 9 times "10" rSup { size 8{ - 3} } } {}$
Tungsten	$4 . 5 \times 10^{- 3} 4 . 5 \times 10^{- 3} size 12{4 "." 5 times "10" rSup { size 8{ - 3} } } {}$
Iron	$5 . 0 \times 10^{- 3} 5 . 0 \times 10^{- 3} size 12{5 "." 0 times "10" rSup { size 8{ - 3} } } {}$
Platinum	$3 . 93 \times 10^{- 3} 3 . 93 \times 10^{- 3} size 12{3 "." "93" times "10" rSup { size 8{ - 3} } } {}$
Lead	$3 . 9 \times 10^{- 3} 3 . 9 \times 10^{- 3} size 12{3 "." 9 times "10" rSup { size 8{ - 3} } } {}$
Manganin (Cu, Mn, Ni alloy)	$0 . 000 \times 10^{- 3} 0 . 000 \times 10^{- 3} size 12{0 "." "000" times "10" rSup { size 8{ - 3} } } {}$
Constantan (Cu, Ni alloy)	$0 . 002 \times 10^{- 3} 0 . 002 \times 10^{- 3} size 12{0 "." "002" times "10" rSup { size 8{ - 3} } } {}$
Mercury	$0 . 89 \times 10^{- 3} 0 . 89 \times 10^{- 3} size 12{0 "." "89" times "10" rSup { size 8{ - 3} } } {}$
Nichrome (Ni, Fe, Cr alloy)	$0 . 4 \times 10^{- 3} 0 . 4 \times 10^{- 3} size 12{0 "." 4 times "10" rSup { size 8{ - 3} } } {}$
Semiconductors
Carbon (pure)	$- 0 . 5 \times 10^{- 3} - 0 . 5 \times 10^{- 3} size 12{ - 0 "." 5 times "10" rSup { size 8{ - 3} } } {}$
Germanium (pure)	$- 50 \times 10^{- 3} - 50 \times 10^{- 3} size 12{ - "50" times "10" rSup { size 8{ - 3} } } {}$
Silicon (pure)	$- 70 \times 10^{- 3} - 70 \times 10^{- 3} size 12{ - "70" times "10" rSup { size 8{ - 3} } } {}$

Table 20.2 Tempature Coefficients of Resistivity

α α size 12{α} {}

Note also that $α α size 12{α} {}$ 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 $ρ ρ size 12{ρ} {}$ 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 $R_{0} R_{0} size 12{R rSub { size 8{0} } } {}$ is directly proportional to $ρ ρ size 12{ρ} {}$ . For a cylinder we know $R = ρL / A R = ρL / A size 12{R=ρL/A} {}$ , and so, if $L L size 12{L} {}$ and $A A size 12{A} {}$ do not change greatly with temperature, $R R size 12{R} {}$ will have the same temperature dependence as $ρ ρ size 12{ρ} {}$ . (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 $L L size 12{L} {}$ and $A A size 12{A} {}$ is about two orders of magnitude less than on $ρ ρ size 12{ρ} {}$ .) Thus,

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

20.24

is the temperature dependence of the resistance of an object, where $R_{0} R_{0} size 12{R rSub { size 8{0} } } {}$ is the original resistance and $R R size 12{R} {}$ is the resistance after a temperature change $Δ T Δ T size 12{DT} {}$ . Numerous thermometers are based on the effect of temperature on resistance. (See Figure 20.13.) 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.

Figure 20.13 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 = ρ_{0} (1 + α Δ T) ρ = ρ_{0} (1 + α Δ T) size 12{ρ = ρ rSub { size 8{0} } $ "1 "+ αΔT $ } {}$ and $R = R_{0} (1 + α Δ T) R = R_{0} (1 + α Δ T) size 12{R = R rSub { size 8{0} } $ "1 "+ αΔT $ } {}$ for temperature changes greater than $100º C 100º C size 12{"100"°"C"} {}$ , 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 $2850º C 2850º C size 12{"2850"°"C"} {}$ ?

Strategy

This is a straightforward application of $R = R_{0} (1 + α Δ T) R = R_{0} (1 + α Δ T) size 12{R = R rSub { size 8{0} } $ "1 "+ αΔT $ } {}$ , since the original resistance of the filament was given to be $R_{0} = 0 . 350 Ω R_{0} = 0 . 350 Ω size 12{R rSub { size 8{0} } =0 "." "350"` %OMEGA } {}$ , and the temperature change is $Δ T = 2830º C Δ T = 2830º C size 12{ΔT="2830"°"C"} {}$ .

Solution

The hot resistance $R R size 12{R} {}$ is obtained by entering known values into the above equation:

\begin{array}{l} R & = & R_{0} (1 + α Δ T) \\ = & (0 . 350 Ω) [1 + (4.5 \times 10^{–3} / ºC) (2830º C)] \\ = & 4.8 Ω. \end{array} \begin{array}{l} R & = & R_{0} (1 + α Δ T) \\ = (0 . 350 Ω) [1 + (4.5 \times 10^{–3} / ºC) (2830º C)] \\ = 4.8 Ω. \end{array}

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.

Access multimedia content Open the interactiveOpen the interactive

Figure 20.14