Question

The resistance of a wire at \(20^\circ C\) is \(20\Omega\) and at \(500^\circ C\) is \(60\Omega\). At which temperature its resistance will be \(25\Omega\)?

• A. \(50^\circ C\)
• B. \(60^\circ C\)
• C. \(70^\circ C\)
• D. \(80^\circ C\)

Hint:

Use ratio method for linear variation: \[ \frac{\Delta R}{\text{total change}} = \frac{\Delta T}{\text{total temperature range}} \]

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Resistance increases linearly with temperature for metallic conductors. This relationship allows us to use ratios to determine unknown temperatures.
Step 2: Key Formula or Approach:
The change in resistance \(\Delta R\) is proportional to the change in temperature \(\Delta T\).
\[ \frac{R_{2} - R_{1}}{T_{2} - T_{1}} = \frac{R_{3} - R_{1}}{T_{3} - T_{1}} \]
Step 3: Detailed Explanation:
Let:
\(R_{1} = 20\text{ }\Omega, T_{1} = 20^{\circ}\text{C}\).
\(R_{2} = 60\text{ }\Omega, T_{2} = 500^{\circ}\text{C}\).
We want to find \(T_{3}\) for \(R_{3} = 25\text{ }\Omega\).
Calculate the rate of change of resistance with temperature:
\[ \frac{\Delta R}{\Delta T} = \frac{60 - 20}{500 - 20} = \frac{40}{480} = \frac{1}{12}\text{ }\Omega/^{\circ}\text{C} \]
Now set up the equation for the unknown temperature:
\[ \frac{25 - 20}{T_{3} - 20} = \frac{1}{12} \]
\[ \frac{5}{T_{3} - 20} = \frac{1}{12} \]
Cross-multiplying:
\[ T_{3} - 20 = 60 \implies T_{3} = 80^{\circ}\text{C} \]
Step 4: Final Answer:
The resistance will be \(25\text{ }\Omega\) at \(80^{\circ}\text{C}\).