Question

The ratio of the resistance of conductor at temperature 15\(^\circ\)C to its resistance at temperature 37.5\(^\circ\)C is 4:5. The temperature coefficient of resistance of the conductor is

• A. \(\dfrac{1}{25}\) \(^\circ\)C\(^{-1}\)
• B. \(\dfrac{1}{50}\) \(^\circ\)C\(^{-1}\)
• C. \(\dfrac{1}{80}\) \(^\circ\)C\(^{-1}\)
• D. \(\dfrac{1}{75}\) \(^\circ\)C\(^{-1}\)

Hint:

\(R \propto (1 + \alpha T)\). Set up a ratio equation and solve for \(\alpha\) algebraically. Cross-multiply carefully.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
\(R_T = R_0(1 + \alpha T)\). Ratio: \(\dfrac{R_1}{R_2} = \dfrac{1 + \alpha T_1}{1 + \alpha T_2}\).
Step 2: Detailed Explanation:
\[ \frac{4}{5} = \frac{1 + 15\alpha}{1 + 37.5\alpha} \] \[ 4(1 + 37.5\alpha) = 5(1 + 15\alpha) \] \[ 4 + 150\alpha = 5 + 75\alpha \] \[ 75\alpha = 1 \Rightarrow \alpha = \frac{1}{75}\ ^\circ\text{C}^{-1} \]
Step 3: Final Answer:
Temperature coefficient \(\alpha = \mathbf{\dfrac{1}{75}}\) \(^\circ\)C\(^{-1}\).