Question

The resistance of a wire at 0°C is 20 Ω. If the temperature coefficient of resistance is \(5\times10^{-3}\), the temperature at which the resistance will be double of that at 0°C is:

• A. \(10^\circ\text{C}\)
• B. \(200^\circ\text{C}\)
• C. \(250^\circ\text{C}\)
• D. \(300^\circ\text{C}\)

Hint:

For temperature dependence of resistance, \[ R_t=R_0(1+\alpha t) \] If resistance becomes double, then \[ 1+\alpha t=2 \] which gives \[ t=\frac{1}{\alpha} \] directly.

Answer 1

The Correct Option is B

Step 1: Use the relation between resistance and temperature.
The resistance at temperature \(t^\circ\text{C}\) is given by \[ R_t=R_0(1+\alpha t) \] where, \[ R_0=20\,\Omega \] is the resistance at \[ 0^\circ\text{C} \] and \[ \alpha=5\times10^{-3}\,^\circ\text{C}^{-1} \]

Step 2: Apply the condition for doubled resistance.
The resistance becomes double of its initial value. Therefore, \[ R_t=2R_0 \] Using the formula, \[ 2R_0=R_0(1+\alpha t) \] Cancelling \(R_0\), \[ 2=1+\alpha t \] \[ \alpha t=1 \] \[ t=\frac{1}{\alpha} \]

Step 3: Substitute the value of \(\alpha\).
\[ t=\frac{1}{5\times10^{-3}} \] \[ t=\frac{1}{0.005} \] \[ t=200^\circ\text{C} \]

Step 4: Final conclusion.
Hence, the required temperature is \[ \boxed{200^\circ\text{C}} \]