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 temperatureindependent resistance standard, for example.
Table 2: Tempature Coefficients of Resistivity α α size 12{α} {}
Material 
Coefficient αα(1/°C) 
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 
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.
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.
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.