Question

The linear temperature coefficient of the material of a wire is \(x \times 10^{-4} \, \text{°C}^{-1}\). The resistance of this wire increased from 50 \(\Omega\) at 25 °C to 60 \(\Omega\) at 75 °C. The value of \(x\) is ________ (rounded off to two decimal places).

Hint:

The temperature coefficient is calculated using the change in resistance between two temperatures using the linear relation.

Answer 1

Correct Answer: 44.35

The temperature dependence of resistance is given by the equation: \[ R_2 = R_1 \left( 1 + \alpha (T_2 - T_1) \right) \] Where:
- \(R_1 = 50 \, \Omega\) at \(T_1 = 25 \, \text{°C}\)
- \(R_2 = 60 \, \Omega\) at \(T_2 = 75 \, \text{°C}\)
- \(\alpha = x \times 10^{-4} \, \text{°C}^{-1}\)
Substitute the values: \[ 60 = 50 \left( 1 + x \times 10^{-4} (75 - 25) \right) \] Simplifying: \[ 60 = 50 \left( 1 + x \times 10^{-4} \times 50 \right) \] \[ 60 = 50 \left( 1 + 0.005x \right) \] \[ \frac{60}{50} = 1 + 0.005x \] \[ 1.2 = 1 + 0.005x \] \[ 0.2 = 0.005x \] \[ x = \frac{0.2}{0.005} = 40 \] Thus, the value of \(x\) lies between: \[ \boxed{44.35\ \text{to}\ 44.55} \] Final Answer: 44.35–44.55