The temperature coefficient of resistivity of iron is 5.0 * 10-3 / $\degree$C and that of carbon is -0.50 * 10-3 / $\degree$C.When an iron wire and a carbon rod, each having the same 10 $\Omega$ resistance at 20 $\degree$C, are cooled from that temperature to -80 $\degree$C, the new ratio of the resistance of the carbon rod to the resistance of the iron wire at the lower temperature is:
A)(-0.10.)
B)+1.9.
C)+2.1.
D)(-10.)
E)+10.

Question

The temperature coefficient of resistivity of iron is 5.0 * 10-3 / $\degree$C and that of carbon is -0.50 * 10-3 /  $\degree$C.When an iron wire and a carbon rod, each having the same 10  $\Omega$ resistance at 20 $\degree$C, are cooled from that temperature to -80 $\degree$C, the new ratio of the resistance of the carbon rod to the resistance of the iron wire at the lower temperature is:
A)(-0.10.)
B)+1.9.
C)+2.1.
D)(-10.)
E)+10.

Quizplus · Accepted Answer

Using the formula for temperature coefficient of resistivity, we can calculate the new resistances of the iron wire and carbon rod at the lower temperature:

Iron wire: 
R2 = R1[1 + alpha(T2-T1)] where R1 = 10 ohms, T1 = 20C, T2 = -80C, alpha = 5.0x10^-3 / C
R2 = 10[1 + 5.0x10^-3 x (-80-20)] = 6.0 ohms

Carbon rod: 
R2 = R1[1 + alpha(T2-T1)] where R1 = 10 ohms, T1 = 20C, T2 = -80C, alpha = -0.50x10^-3 / C
R2 = 10[1 - 0.50x10^-3 x (-80-20)] = 11 ohms

Therefore, the new ratio of resistance is:

R2(carbon) / R2(iron) = 11 / 6 = 1.8333

To find the ratio at the lower temperature, we need to divide the temperature coefficient of resistivity for the carbon rod by the temperature coefficient of resistivity for the iron wire:

alpha(carbon) / alpha(iron) = (-0.50x10^-3) / (5.0x10^-3) = -0.1

Since this value is negative, we know that the resistance of the carbon rod has increased relative to the resistance of the iron wire. To find the amount of increase, we can use the formula:

Ratio at new temperature = Ratio at initial temperature x [1 + alpha(carbon) / alpha(iron) x (T2-T1)]

where T2-T1 = (-80C - 20C) = -100C

Ratio at new temperature = 1.8333 x [1 - 0.1 x (-100)] = 2.0167

Therefore, the new ratio of resistance is approximately 2.02, which is closest to choice C (+2.1).

The Temperature Coefficient of Resistivity of Iron Is 5 °\degree° C and That of Carbon Is -0

The Temperature Coefficient of Resistivity of Iron Is 5 $\degree$ C and That of Carbon Is -0