Variation of Resistance with Temperature

Resistivity is a property of a material that VARIES with temperature. Fig. 11.11 shows how the resistivity of copper varies as a function of absolute temperature (in kelvin); over a fairly wide range, this variation is very nearly LINEAR, and can be written as

where $T_0$ is a chosen reference temperature (commonly $0^\circ$C) and $\rho_0$ is the resistivity of the material at that reference temperature. Rearranging Eq. (11.35) for $\alpha$,

Here $\alpha$ is called the TEMPERATURE COEFFICIENT OF RESISTIVITY. Since resistance $R$ of a fixed sample scales the same way as $\rho$ does (the sample's length and area do not change appreciably), the same relation holds directly for resistance:

The temperature coefficient of resistance $\alpha$ is therefore defined as the fractional (per unit original resistance) increase in resistance per degree rise in temperature, measured at the chosen reference temperature; its SI unit is $^\circ$C$^{-1}$ or K$^{-1}$ (per degree Celsius or per degree kelvin -- numerically identical, since the Celsius and Kelvin scales share the same size of degree, so only a TEMPERATURE DIFFERENCE matters here, not the absolute temperature value, and either scale can be used interchangeably). For small temperature differences, this can equivalently be written in differential form:

$\alpha=\frac{1}{R_0}\frac{dR}{dT}\qquad\text{--- (11.38)}$ …