   Chapter 17, Problem 66AP

Chapter
Section
Textbook Problem

When a straight wire is heated, its resistance changes according to the equationR = R0 [1 + α(T − T0)](Eq. 17.7), where α is the temperature coefficient of resistivity. (a) Show that a more precise result, which includes the length and area of a wire change when it is heated, is R = R 0 [ 1   +   α ( T − T 0 ) ] [ 1   +   α ′ ( T − T 0 ) ] [ 1   +   2 α ′ ( T − T 0 ) ] where α′ is the coefficient of linear expansion. (See Topic 10.) (b) Compare the two results for a 2.00-m-long copper wire of radius 0.100 mm, starting at 20.0°C and heated to 100.0°C.

(a)

To determine
The formula to calculate the variation of resistance with change in temperature on taking into account the variation of length and area of cross section of resistor with respect to temperature.

Explanation

Given Info: Variation of resistance with respect to temperature is given by R=R0(1+α(TT0)) .

Explanation:

Formula to calculate the resistance at temperature T0 is,

R0=ρ0L0A0 (I)

• R0 is the resistance at temperature T0
• ρ0 is the resistivity of the material of resistor at temperature T0
• L0 is the length of the resistor at temperature T0
• A0 is the area of cross section of the resistor at temperature T0

Formula to calculate the resistance at temperature T is,

R=ρLA (II)

• R is the resistance at temperature T
• ρ is the resistivity of the material of resistor at temperature T
• L is the length of the resistor at temperature T
• A is the area of cross section of the resistor at temperature T

Formula to calculate the variation of resistivity with respect to temperature is,

ρ=ρ0(1+α(TT0)) (III)

• ρ is the resistivity at temperature T ,
• ρ0 is the resistivity at temperature T0 ,
• α is the temperature coefficient of resistivity,

Formula to calculate the variation of L with respect to temperature is,

L=L0(1+α(TT0)) (IV)

• L0 is the length of the resistor at temperature T0 ,
• α is the coefficient of linear expansion,

Formula to calculate the variation of L with respect to temperature is,

A=A0(1+2α(TT0)) (V)

• A0 is the area of cross section of the resistor at temperature T0

Substitute equation (II), equation (III) and equation (IV) in equation (I) and rewrite R

(b)

To determine
The resistance of copper wire at 100.0°C using formulas with and without considering the temperature dependence of length and area. Compare the results.

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