Constants Part A For some applications, it is important that the value of a resi

Question

Constants Part A For some applications, it is important that the value of a resistance not change with temperature. For example, suppose you made a 90 k 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 are-0.0005 ("C) 1 and 0.0004 (°C) 1, respectively What value should the carbon resistor have (at 20° C) so that the combination is temperature independent? Express your answer to one significant figure and include the appropriate units Roc- Value Units Submit Request Answer Part B What value should the Nichrome wire-wound resistor have (at 20 °C) so that the combination is temperature independent? Express your answer to one significant figure and include the appropriate units alue Units Submit Request Answen

Explanation / Answer

Suppose resistor of Carbon wire is R1 & Resistance of Nichrome wire is R2

Now Given that

R1 + R2 = 90 k-Ohm = 90000 Ohm

Now Relation between resistance and temperature is given by

Rnew = R0*(1 + alpha*(T - T0))

T0 = 20 C

For Carbon wire

R1' = R1*(1 + alpha1*(T - 20))

For Nichrome wire

R2' = R2*(1 + alpha2*(T - 20))

Now we need value of R1 and R2 such that Req remains constant for any Temperature, So

R1' + R2' = R1 + R2

R1*(1 + alpha1*(T - 20)) + R2*(1 + alpha2*(T - 20)) = R1 + R2

R1 + R1*alpha1*(T - 20)) + R2 + R2*alpha2*(T - 20)) = R1 + R2

R1*alpha1*(T - 20) + R2*alpha2*(T - 20) = 0

alpha1 = temperature coefficient of resistivity of carbon = -0.0005 /C

alpha2 = temperature coefficient of resistivity of Nichrome = 0.0004 /C

R1*(-0.0005)*(T - 20) + R2*(0.0004)*(T - 20) = 0

R1*0.0005*(T - 20) = R2*0.0004*(T - 20)

R1*0.0005 = R2*0.0004

5*R1 = 4*R2

R2 = (5/4)*R1 = 1.25*R1

So now

R1 + R2 = 90000

R1 + 1.25*R1 = 90000

2.25*R1 = 90000

R1 = 90000/2.25

R1 = 40000 ohm

R1 = 4*10^4 Ohm = Resistance of Carbon

R2 = 1.25*R1 = 1.25*40000 = 50000

R2 = 5*10^4 ohm = Resistance of Nichrome

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