Temperature Variation of Resistance

The resistivity of all materials depends on temperature. Some even become superconductors (zero resistivity) at very low temperatures. (See this figure.) 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 \(\text{100º}\text{C}\) or less), resistivity \(\rho \) varies with temperature change \(\Delta T\) as expressed in the following equation

\(\rho ={\rho }_{0}(\text{1}+\alpha \Delta T)\text{,}\)

where \({\rho }_{0}\) is the original resistivity and \(\alpha \) is the temperature coefficient of resistivity. (See the values of \(\alpha \) in this table below.) For larger temperature changes, \(\alpha \) may vary or a nonlinear equation may be needed to find \(\rho \). Note that \(\alpha \) 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 \(\alpha \) close to zero (to three digits on the scale in this table), 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.

Tempature Coefficients of Resistivity \(\alpha \)

Material	Coefficient \(\alpha \)(1/°C)
Conductors
Silver	\(3\text{.}8×{\text{10}}^{-3}\)
Copper	\(3\text{.}9×{\text{10}}^{-3}\)
Gold	\(3\text{.}4×{\text{10}}^{-3}\)
Aluminum	\(3\text{.}9×{\text{10}}^{-3}\)
Tungsten	\(4\text{.}5×{\text{10}}^{-3}\)
Iron	\(5\text{.}0×{\text{10}}^{-3}\)
Platinum	\(3\text{.}\text{93}×{\text{10}}^{-3}\)
Lead	\(3\text{.}9×{\text{10}}^{-3}\)
Manganin (Cu, Mn, Ni alloy)	\(0\text{.}\text{000}×{\text{10}}^{-3}\)
Constantan (Cu, Ni alloy)	\(0\text{.}\text{002}×{\text{10}}^{-3}\)
Mercury	\(0\text{.}\text{89}×{\text{10}}^{-3}\)
Nichrome (Ni, Fe, Cr alloy)	\(0\text{.}4×{\text{10}}^{-3}\)
Semiconductors
Carbon (pure)	\(-0\text{.}5×{\text{10}}^{-3}\)
Germanium (pure)	\(-\text{50}×{\text{10}}^{-3}\)
Silicon (pure)	\(-\text{70}×{\text{10}}^{-3}\)

Note also that \(\alpha \) is negative for the semiconductors listed in this table, 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 \(\rho \) 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 \({R}_{0}\) is directly proportional to \(\rho \). For a cylinder we know \(R=\mathrm{\rho L}/A\), and so, if \(L\) and \(A\) do not change greatly with temperature, \(R\) will have the same temperature dependence as \(\rho \). (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 \(\rho \).) Thus,

\(R={R}_{0}(\text{1}+\alpha \Delta T)\)

is the temperature dependence of the resistance of an object, where \({R}_{0}\) is the original resistance and \(R\) is the resistance after a temperature change \(\Delta T\). Numerous thermometers are based on the effect of temperature on resistance. (See this figure.) 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.

These familiar thermometers are based on the automated measurement of a thermistor’s temperature-dependent resistance. (credit: Biol, Wikimedia Commons)

Example: Calculating Resistance: Hot-Filament Resistance

Although caution must be used in applying \(\rho ={\rho }_{0}(\text{1}+\alpha \Delta T)\) and \(R={R}_{0}(\text{1}+\alpha \Delta T)\) for temperature changes greater than \(\text{100º}\text{C}\), 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 (C\(\text{20ºC}\)) to a typical operating temperature of \(\text{2850º}\text{C}\)?

Strategy

This is a straightforward application of \(R={R}_{0}(\text{1}+\alpha \Delta T)\), since the original resistance of the filament was given to be \({R}_{0}=0\text{.}\text{350 Ω}\), and the temperature change is \(\Delta T=\text{2830º}\text{C}\).

Solution

The hot resistance \(R\) is obtained by entering known values into the above equation:

\(\begin{array}{lll}R& =& {R}_{0}(1+\alpha \Delta T)\\ & =& (0\text{.}\text{350 Ω})[\text{1}+(4.5×{\text{10}}^{–3}/\text{ºC})(\text{2830º}\text{C})]\\ & =& \text{4.8 Ω.}\end{array}\)

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.

Temperature Variation of Resistance