Question

The resistance of a wire at room temperature ${30^\circ }C$ is found to be $10\Omega$ . Now to increase the resistance by $10\%$ , the temperature of the wire must be: [The temperature coefficient of resistance of the material of the wire is 0.002 per Celsius]
(A) ${36^\circ }C$
(B) ${83^\circ }C$
(C) ${63^\circ }C$
(D) ${33^\circ }C$

Answer 1

Hint In this question, we are given the initial conditions of the wire. We know that when a wire is heated, it experiences gain in resistance. This gain in resistance should be equal to $10\%$ . We can also infer that if the original resistance was R, the new resistance should be 1.1R.
Use the following relation
${R_T} = {R_0}(1 + \alpha T)$

Complete step by step solution
According to the question
Temperature coefficient of resistance of the wire,
$\alpha = 0.002{/^\circ }C$
Resistance at temperature,
${30^\circ }C = {R_{30}} = 10\Omega$
We know,
${R_T} = {R_0}(1 + \alpha T)$
Putting the value of ${R_{30}}$ , $\alpha$ and T in above equation we have
$10 = {R_0}(1 + 30\alpha )$ … (i)
After $10\%$ increase in resistance the new resistance becomes
$10 + 10 \times \dfrac{{10}}{{100}} = 11\Omega$
Again, for the new resistance we have
$11 = {R_0}(1 + \alpha T)$ … (ii)
Dividing equation (ii) by (i), we get
$\dfrac{{11}}{{10}} = \dfrac{{{R_0}(1 + \alpha T)}}{{{R_0}(1 + 30\alpha )}}$
${R_0}$ gets cancelled out as it’s the same material and same reference temperature.
Or, $\dfrac{{11}}{{10}} = \dfrac{{(1 + \alpha T)}}{{(1 + 30\alpha )}}$
Or, $11(1 + 30\alpha ) = 10(1 + \alpha T)$
Or, $11 + 330\alpha = 10 + 10\alpha T$
Or, $1 + 330\alpha = 10\alpha T$
Or, $\dfrac{{1 + 330\alpha }}{{10\alpha }} = T$
Putting the value of $\alpha$ , we get
$T = \dfrac{{1 + 330 \times 0.002}}{{10 \times 0.002}}$
Or, $T = {83^ \circ }C$ (Option B)
Option B is correct answer

Note The above calculation is made with the consideration that the reference temperature is the same. Although in case of different reference temperature the above formula becomes,
${R_T} = {R_0}(1 + \alpha (T - {T_0}))$
Where, ${R_T}$ = Conductor resistance at temperature “T”.
${R_0}$ = Conductor resistance at reference temperature.
$\alpha$ = Temperature coefficient of resistance of the material of the conductor.
T = Conductor temperature in degree Celsius.
${R_0}$ = Reference temperature in degree Celsius.
Resistance of a conductor increases with the increasing temperature because the chance of electrons colliding with protons increases. Resistance decreases with decreasing temperature. However, in some cases when the temperature is decreased beyond a critical temperature its resistance becomes zero, such materials are called superconductors. The critical temperature for mercury is 4.15 K.