   Chapter 11.11, Problem 32E

Chapter
Section
Textbook Problem

# The resistivity ρ 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 equationρ(t) = ρ20 eα(t−20)where t is the temperature in °C. There are tables that list the values of α (called the temperature coefficient) and ρ20 (the resistivity at 20°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 ρ(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/°C and ρ20 = 1.7 × 10−8 Ω-m. Graph the resistivity of copper and the linear and quadratic approximations for −250°C ≤ t ≤ 1000°C. (c) For what values of t does the linear approximation agree with the exponential expression to within one percent?

(a)

To determine

To find: The expression for the linear and quadratic approximation.

Explanation

Given:

Resistivity is the reciprocal of the conductivity and is denoted by ρ.

The unit of resistivity is ohm-meter.

The resistivity of a given metal is, ρ(t)=ρ20eα(t20).

Calculation:

The given equation is ρ(t)=ρ20eα(t20).

The first derivative of ρ(t) with respect to t is found as,

ddt(ρ(t))=ddt(ρ20eα(t20))=ρ20ddt(eα(t20))=ρ20(eα(t20))ddt(α(t20))=αρ20eα(t20)

That is, ρ(t)=ρ20αeα(t20).

The second derivative will be,

ddt(ρ(t))=ddt(ρ20αeα(t20))=ρ20αddt(eα(t20))=αρ20(eα(t20))ddt(α(t20))=α2ρ20eα(t20)

That is, ρ(t)=ρ20α2eα(t20).

The Taylor series expansion is f(x)f(a)+f(a)1!(xa)+f(a)2!(xa)2+.

Substitute t=20 in ρ(t)=ρ20eα(t20)

(b)

To determine

To graph: The resistivity of copper and the linear and quadratic approximations for 250°Ct1000°C, if α=0.0039/°C and ρ20=1.7×108Ωm.

(c)

To determine

To find: The values of t, where the linear approximation agree with the exponential expression to within one percent.

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