Temperature Variation of Resistance
The resistivity of all materials depends on temperature. Some even become superconductors (zero resistivity) at very low temperatures. (See Figure 3.14.) 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 100 ºC or less), resistivity ρ varies with temperature change ΔT as expressed in the following equation
3.23 ρ=ρ0(1+αΔT),
where ρ0 is the original resistivity and α is the temperature coefficient of resistivity. (See the values of α in Table 3.2 below.) For larger temperature changes, α may vary or a nonlinear equation may be needed to find ρ. Note that 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 close to zero (to three digits on the scale in Table 3.2), and so its resistivity varies only slightly with temperature. This can be useful for making a temperature-independent resistance standard.
Material |
Coefficient α(1/°C) |
Conductors |
|
Silver |
3.8×10−3
|
Copper |
3.9×10−3
|
Gold |
3.4×10−3
|
Aluminum |
3.9×10−3
|
Tungsten |
4.5×10−3
|
Iron |
5.0×10−3
|
Platinum |
3.93×10−3
|
Lead |
4.3×10−3
|
Cu, Mn, Ni alloy |
0.000×10−3
|
Cu, Ni alloy |
0.002×10−3
|
Mercury |
0.89×10−3
|
Nichrome (Ni, Fe, Cr alloy) |
0.4×10−3
|
Semiconductors |
|
Carbon (pure) |
−0.5×10−3
|
Germanium (pure) |
−50×10−3
|
Silicon (pure) |
−70×10−3
|
Table 3.2 Temperature Coefficients of Resistivity α
Note also that α is negative for the semiconductors listed in Table 3.2, 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 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 is directly proportional to ρ. For a cylinder we know R=ρL/A, and so, if L and A do not change greatly with temperature, R will have the same temperature dependence as 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 and A is about two orders of magnitude less than on ρ. Thus,
3.24 R=R0(1+αΔT)
is the temperature dependence of the resistance of an object, where R0 is the original resistance and R is the resistance after a temperature change ΔT. Numerous thermometers are based on the effect of temperature on resistance. (See Figure 3.15.) 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 3.6 Calculating Resistance: Hot-Filament Resistance
Caution must be be used in applying ρ=ρ0(1+αΔT) and R=R0(1+αΔT) for temperature changes greater than 100 ºC. However, 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 (20 ºC) to a typical operating temperature of 2,850 ºC?
Strategy
This is a straightforward application of R=R0(1+αΔT), since the original resistance of the filament was given to be R0=0.350 Ω, and the temperature change is ΔT=2,830 ºC.
Solution
The hot resistance R is obtained by entering known values into the above equation:
3.25 R=R0(1+αΔT)=(0.350 Ω)[1+(4.5×10–3/ºC)(2,830 ºC)]=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.