Question

For some applications, it is important that the value of a resistance not change with temperature. For example, suppose you made a 18 kΩkΩ resistor from a carbon resistor and a Nichrome wire-wound resistor connected together so the total resistance is the sum of their separate resistances. The temperature coefficients of resistivity of carbon and Nichrome at 20∘C∘C are -0.0005 (∘C)−1(∘C)−1 and 0.0004 (∘C)−1(∘C)−1, respectively.

1.

What value should the carbon resistor have (at 20∘C∘C) so that the combination is temperature independent?

Express your answer to one significant figure and include the appropriate units.

2.

What value should the Nichrome wire-wound resistor have (at 20 ∘C∘C) so that the combination is temperature independent?

Express your answer to one significant figure and include the appropriate units.

Answer 1

given
total resistance, R = 18 kohm
let R1 is the resistance of carbon and R2 is the resistance of Nichrome at 20 degrees Celsius
1)
R1 + R2 = 18 kohm

alfa1 = -0.0005 C^-1
alfa2 = 0.0004 C^-1

when decrease in resistance of carbon is equal to increase in resistance of Nichrome, total resistance remains constant.

|delta_R1| = |delta_R2|

R1*|alfa1|*delta_T = R2|alfa|*delta_T2

R1/R2 = |alfa2|/|alfa1|

R1/R2 = 0.0004/0.0005

R1/R2 = 4/5

==> R2 = 5*R1/4

from equation 1

R1 + R2 = 18

R1 + 5*R1/4 = 18

4*R1 + 5*R1 = 18*4

9*R1 = 18*4

R1 = 8 k ohms <<<<<<<<------------Answer

2) from equation 1

R1 + R2 = 18

R2 = 18 - R1

R2 = 18 - 8

R2 = 10 k ohms   <<<<<<<<------------Answer