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20.3 Resistance and resistivity  (Page 2/6)

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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} } } {}
Semiconductors Values depend strongly on amounts and types of impurities
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 14

Calculating resistor diameter: a headlight filament

A car headlight filament is made of tungsten and has a cold resistance of 0 . 350 Ω 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 = ρL A 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 = ρL A size 12{R = { {ρL} over {A} } } {} , is

A = ρL R . 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 = πD 2 4 . 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.

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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 100º C size 12{"100"°C} {} or less), resistivity ρ size 12{ρ} {} varies with temperature change Δ T size 12{DT} {} as expressed in the following equation

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

where ρ 0 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.
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.

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Source:  OpenStax, College physics for ap® courses. OpenStax CNX. Nov 04, 2016 Download for free at https://legacy.cnx.org/content/col11844/1.14
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