The resistivity p of a conducting wire is the reciprocal of the conductivity and is measured in units of ohm-meters (Ω-m). The resistivity of a given metal depends on the temperature according to the equation p(t) = p20ea(t–20) where t is the temperature in oC. There are tables that list the values of α (called the temperature coefficient) and p20 (the resistivity at 20oC) for various metals.
Except at very low temperatures, the resistivity varies almost linearly with temperature and so it is common to approximate the expression for p(t) by its first- or second-degree Taylor polynomial at t = 20.
(a) Find expressions for these linear and quadratic approximations.
(b) For copper, the tables give α = 0.0039/oC and p20 = 1.7 X 10–8 Ω-m. Graph the resistivity of copper and the linear and quadratic approximations for – 250oC < t < 100020oC.
(c) For what values of does the linear approximation agree with the exponential expression to within one percent?