Question

A metal with temperature coefficient of resistance has a value 200, its initial resistance is given by 40 \( \Omega \). For an increase in 30 \( ^{\circ} \)C to 35 \( ^{\circ} \)C what will be the final resistance value?

• A. \( \text{40 K}\Omega \)
• B. \( \text{4 K}\Omega \)
• C. \( \text{40 }\Omega \)
• D. \( \text{400 }\Omega \)

Hint:

The temperature coefficient of resistance (\( \alpha \)) for most metals is a small positive value, typically in the order of \( 10^{-3} \) to \( 10^{-5} \) per degree Celsius. A positive \( \alpha \) means resistance increases with temperature. If an unusually large value for \( \alpha \) is given, it often implies a unit conversion or a specific context (like a thermistor, but even then, 200 is extremely high for \( \alpha \)). In such ambiguous cases, checking if a slight modification to the given value (e.g., by powers of 10) leads to one of the options is a useful strategy. Here, assuming \( 200 \times 10^{-6} \) or similar makes sense.

Answer 1

The Correct Option is C

Understanding the Problem:

• A metal's resistance changes with temperature due to its temperature coefficient of resistance (α).
• Given: Initial resistance \( R_0 = 40 \, \Omega \), temperature coefficient \( \alpha = 200 \), initial temperature \( T_0 = 30 \, \degree \text{C} \), final temperature \( T = 35 \, \degree \text{C} \).
• We need to find the final resistance \( R \) after the temperature change.

Formulae:

• The change in resistance with temperature is given by: \[ R = R_0 \left(1 + \alpha \Delta T \right) \] where \( \Delta T = T - T_0 \).

Calculating the Change in Temperature:

\[ \Delta T = T - T_{0} = 35 \, \degree \text{C} - 30 \, \degree \text{C} = 5 \, \degree \text{C} \]

Calculating the Final Resistance:

\[ R = R_0 \left(1 + \alpha \Delta T \right) \]

\[ R = 40 \left(1 + 200 \times 5 \right) \]

\[ R = 40 \left(1 + 1000 \right) \]

\[ R = 40 \times 1001 = 40040 \, \Omega \]

Verification:

• The given options seem inconsistent with the calculation (40 Ω, 400 Ω, 4 kΩ, 40 kΩ).
• However, the correct calculation yields \( 40040 \, \Omega \), which is closest to \( 40 \, \text{k}\Omega \).

Final Answer:

The final resistance is \( 40 \, \text{k}\Omega \).