In addition to length and cross-section, temperature also affects the resistance value of a material. Both the thermal velocity of the electrons and their collisions with the lattice atoms increase at higher temperatures, which effectively leads to an increase in the specific resistance. For not too large temperature changes, the change in resistance is approximately proportional to the temperature. Around room temperature, the change in resistivity can be written as
$$
\Delta \varrho = \varrho_0 \alpha \Delta T
$$
where the temperature coefficient $\alpha$ is introduced as the proportionality constant. Just like $\varrho_0$, this coefficient depends on the material. Typically, $\varrho_0$ is measured and specified at $20^\circ$C. The total resistivity is then calculated according to
$$
\varrho = \varrho_0 + \Delta \varrho
$$
Substituting the earlier relation yields
$$
\varrho = \varrho_0 (1 + \alpha \Delta T)
$$
This relationship is used in resistance-based thermometers, where temperature is inferred from the change in resistance value.
Example: A copper wire heats up due to current flow and reaches a temperature of $40^\circ$C. By how much does the specific resistivity increase? Answer: 0.42%
Table: Specific resistivities and temperature coefficients of some materials
| Material | Resistivity [$\Omega \cdot \mathrm{mm}^2/\mathrm{m}$] | Temp. Coefficient [1/K] |
|---|---|---|
| Gold | $2.214 \cdot 10^{-2}$ | $3.9 \cdot 10^{-3}$ |
| Copper (pure) | $1.721 \cdot 10^{-2$ | $3.9 \cdot 10^{-3}$ |
| Constantan | $0.5$ | $5 \cdot 10^{-5}$ |
| Carbon | $350$ | $2 \cdot 10^{-4}$ |
| Muscle tissue | $2 \cdot 10^6$ | — |