Question

The temperature coefficient of resistance of a wire is 0.00125 per degree celsius. At 300K its resistance is 1 ohm. The resistance of the wire will be 2 ohm at following temperature:
A. 1154K.
B. 400K.
C. 600K.
D. 1400K.

Answer 1

To solve this question, we have to remember that the resistance of a metallic conductor increases with increase in temperature. The dependence of resistance with temperature is given by, ${R_t} = {R_0}\left( {1 + \alpha t} \right)$ where ${R_0}$ is resistance at ${0^0}C$ and ${R_t}$ is the resistance at ${t^0}C$ and $\alpha$ is the temperature coefficient of resistance.

Complete answer:
Given that,
Temperature coefficient, $\alpha$ = 0.00125.
When t = 300 K, resistance, ${R_1}$ = 1 ohm.
We can write this as:
${R_1} = 1 = {R_0}\left( {1 + 0.00125 \times 300} \right)$ ……. (i)
Similarly,
When resistance, ${R_2}$ = 2 ohm, temperature = t K.
We can write this as:
${R_2} = 2 = {R_0}\left( {1 + 0.00125 \times t} \right)$ ……… (ii)
Dividing equation (ii) by (i), we will get
$\Rightarrow \dfrac{{{R_2}}}{{{R_1}}} = \dfrac{2}{1} = \dfrac{{{R_0}\left( {1 + 0.00125 \times t} \right)}}{{{R_0}\left( {1 + 0.00125 \times 300} \right)}}$
Solving this,
$\Rightarrow 2\left( {1 + 0.00125 \times 300} \right) = \left( {1 + 0.00125 \times t} \right)$
$\Rightarrow 2.75 = \left( {1 + 0.00125 \times t} \right)$
$\Rightarrow \dfrac{{2.75 - 1}}{{0.00125}} = t$
$\Rightarrow 1400K = t$
Hence, the temperature of the wire will be 1400 K when resistance is 2 ohm.

So, the correct answer is “Option D”.

Note:
This type of questions can also get solved by using the formula, $\alpha = \dfrac{{{R_2} - {R_1}}}{{{R_1}\left( {{t_2} - {t_1}} \right)}}$ per $^0C$ or per K. We can also solve in terms of resistivity, ${\alpha _r} = \dfrac{{{\rho _2} - {\rho _1}}}{{{\rho _1}\left( {{t_2} - {t_1}} \right)}}$ per $^0C$ or per K.