Question

The resistance of a wire at 0 °C is 4 \(\Omega\). If the temperature coefficient of resistance of the material of the wire is \(5 \times 10^{-3} / ^\circ C\), then the resistance of a wire at 50 °C is:

• A. 20 \(\Omega\)
• B. 10 \(\Omega\)
• C. 6 \(\Omega\)
• D. 8 \(\Omega\)
• E. 5 \(\Omega\)

Hint:

To calculate the resistance at a different temperature, use the formula \( R_t = R_0 \left( 1 + \alpha t \right) \), where \(\alpha\) is the temperature coefficient of resistance.

Answer 1

Answer: (E)

The resistance of a material at a different temperature can be calculated using the formula:

\( R_t = R_0 \left( 1 + \alpha t \right) \)

where:

• \( R_t \) is the resistance at temperature \( t \),
• \( R_0 \) is the resistance at \( 0^\circ C \),
• \( \alpha \) is the temperature coefficient of resistance,
• \( t \) is the change in temperature.

Given:

• \( R_0 = 4 \, \Omega \),
• \( \alpha = 5 \times 10^{-3} / ^\circ C \),
• \( t = 50^\circ C \).

Substituting these values into the formula:

\( R_{50} = 4 \, \Omega \left( 1 + 5 \times 10^{-3} \times 50 \right) \)

\( R_{50} = 4 \, \Omega \left( 1 + 0.25 \right) \)

\( R_{50} = 4 \, \Omega \times 1.25 \)

\( R_{50} = 5 \, \Omega \)

Thus, the resistance at \( 50^\circ C \) is \( 5 \Omega \).

Therefore, the correct answer is option (E), \( 5 \Omega \).