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The resistivity $\rho$ of a conducting wire is the reciprocal of the conductivity and is measured in units of ohm-meters $(\Omega-m)$ . The resistivity of a given...

Question

The resistivity $\rho$ of a conducting wire is the reciprocal of the conductivity and is measured in units of ohm-meters $(\Omega-m)$ . The resistivity of a given metal depends on the temperature according to the equation $$ \rho(t)=\rho_{20} e^{\alpha i t-20 )} $$ where $t$ is the temperature in $^{\circ} \mathrm{C}$ . There are tables that list the values of $\alpha$ (called the temperature coefficient) and $\rho_{20}$ (the resistivity at $20^{\circ} \mathrm{C} )$ for various metals.

The resistivity $\rho$ of a conducting wire is the reciprocal of the conductivity and is measured in units of ohm-meters $(\Omega-m)$ . The resistivity of a given metal depends on the temperature according to the equation $$ \rho(t)=\rho_{20} e^{\alpha i t-20 )} $$ where $t$ is the temperature in $^{\circ} \mathrm{C}$ . There are tables that list the values of $\alpha$ (called the temperature coefficient) and $\rho_{20}$ (the resistivity at $20^{\circ} \mathrm{C} )$ 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 $\rho(t)$ 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 $\alpha=0.0039 /^{\circ} \mathrm{C}$ and $\rho_{20}=1.7 \times 10^{-8} \Omega-\mathrm{m} .$ Graph the resistivity of copper and the linear and quadratic approximations for $-250^{\circ} \mathrm{C} \leqslant t \leqslant 1000^{\circ} \mathrm{C}$ (c) For what values of $t$ does the linear approximation agree with the exponential expression to within one percent?



Answers

The resistivity $ \rho $ of a conducting wire is the reciprocal of the conductivity and is measured in units of ohm-meters $ (\Omega\text{-m}). $ The resistivity of a given metal depends on the temperature according to the equation

$ \rho(t) = \rho_{20}e^{\alpha^{t-20}} $

where $ t $ is the temperature in $ ^oC. $ There are tables that list the values of (called the temperature coefficient) and $ \rho_{20} $ (the resistivity at $ 20^oC $ ) 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 $ \rho(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 $ \alpha = 0.0039/^oC $ and $ \rho_{20} = 1.7 x 10^{-8} \Omega\text{-m.} $. Graph the resistivity of copper and the linear and quadratic approximations for $ -250^oC \le t \le 1000^oC. $

(c) For what values of $ t $ does the linear approximation agree with the exponential expression to within one percent?


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