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: State the temperature dependence of resistance.
The resistance of a conductor at temperature t °C is related to its resistance at 0°C by:
\(R_t=R_0(1+\alpha t)\)
where \(\alpha\) is the temperature coefficient of resistance.
Step 2: Write down the given values.
From the problem: \(R_0=20\,\Omega\), \(\alpha=5\times10^{-3}\), and we want \(R_t=2R_0=40\,\Omega\).
Step 3: Substitute into the formula.
\(2R_0=R_0(1+\alpha t)\)
Dividing both sides by \(R_0\) (which is non-zero):
\(2=1+\alpha t\)
Step 4: Solve for \(t\).
\(\alpha t=2-1=1\)
\(t=\frac{1}{\alpha}=\frac{1}{5\times10^{-3}}=200\)
Therefore, \(t=200^\circ C\).
Step 5: Physical interpretation.
As temperature increases, the atoms in the conductor vibrate more vigorously, causing more frequent collisions with conduction electrons. This increases the resistance. At 200°C, the resistance has exactly doubled.
Step 6: State the final answer.
The temperature at which the resistance doubles is:
\(t=200^\circ C\).