Question

At what temperature the resistance of a conductor becomes 20% more than its resistance at \(27^\circ\) C? (The value of the temperature coefficient of resistance of the conductor is \(2.0 \times 10^{-4}\)/K.)

• A. 900 K
• B. 1100 K
• C. 1300 K
• D. 1450 K

Hint:

Use \(R = R_0[1+\alpha \Delta T]\) with \(R = 1.2R_0\).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Resistance of a conductor changes almost linearly with temperature. The temperature coefficient \(\alpha\) tells the fractional change per kelvin.

Step 2: Key Formula:
\[ R_T = R_{27}\,[1 + \alpha (T - T_0)] \] where \(T_0 = 27 ^\circ C = 300\) K.

Step 3: Put the values:
We need \(R_T = 1.2\,R_{27}\). So \[ 1.2 = 1 + 2.0\times10^{-4}\,\Delta T \] \[ 2.0\times10^{-4}\,\Delta T = 0.2 \] \[ \Delta T = \frac{0.2}{2.0\times10^{-4}} = 1000 \text{ K} \]

Step 4: Find the final temperature:
\(T = 27 ^\circ C + 1000 = 1027 ^\circ C\). In kelvin, \(T = 300 + 1000 = 1300\) K.

Step 5: Check the options:
900 K and 1100 K give a rise of only 600 K and 800 K, so the resistance would rise only 12% and 16%. 1450 K gives a rise of 1150 K, that is 23%. Only 1300 K gives exactly 20%.

Final Answer:
The conductor reaches 20% more resistance at 1300 K. \[ \boxed{1300\ \text{K}} \]