   Chapter 17, Problem 32P

Chapter
Section
Textbook Problem

An engineer needs a resistor with a zero overall temperature coefficient of resistance at 20.0°C. She designs a pair of circular cylinders, one of carbon and one of Nichrome as shown in Figure P1 7.32. The device must have an overall resistance of R1 + R2 = 10.0 Ω independent of temperature and a uniform radius of r = 1.50 mm. Ignore thermal expansion of the cylinders and assume both are always at the same temperature. (a) Can she meet the design goal with this method? (b) If so, state what you can determine about the lengths L1 and L2 of each segment. If not, explain. Figure P17.32

(a)

To determine
Whether it is possible to construct a pair of circular cylinders made of carbon and nichrome with overall zero temperature coefficient of resistance.

Explanation

Given Info: The pair of circular cylinders is made up of carbon and nichrome. The device has overall resistance 10Ω . The radius of device is 1.50mm . The device has overall zero temperature coefficient of resistance at 20.0°C .

Explanation:

The formula to calculate the resistance of carbon at temperature T is,

R1=R01(1+α1(TT0))                                 (I)

• R01 is the resistance of carbon at some reference temperature T0 ,
• α1 is the temperature coefficient of resistivity of carbon,
• T0 is the reference temperature and its value is 20.0°C

The formula to calculate the resistance of nichrome at temperature T is ,

R2=R02(1+α2(TT0))                                 (II)

• R02 is the resistance of nichrome at some reference temperature T0 ,
• α2 is the temperature coefficient of resistivity of nichrome,

Reff=R1+R2                                                      (III)

• Reff is the overall resistance of the device.

Use equation (I) and (II) in (III) to rewrite Reff .

Reff=R01(1+α1(TT0))+R02(1+α2(TT0))

Substitute 10Ω for Reff to rewrite the above equation.

R01(1+α1(TT0))+R02(1+α2(TT0))=10ΩR01+R02+R01α1(TT0)+R02α2(TT0)=10Ω

The value R01α1(TT0)+R02α2(TT0) must be zero since overall resistance must not vary with the temperature

(b)

To determine
The length of carbon and nichrome resistors

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