Question

The rate of heat conduction in the given two metal rods having the same length is found to be the same when the temperature difference between the ends is kept \(30^\circ C\). If the area of cross section of the first rod is \(8 \times 10^{-2}\,\text{m}^2\), then what will be area of cross section of the second rod? [Given that the ratio of the thermal conductivity of the first rod to that of the second rod is \(1:4\)]

• A. \(2 \times 10^{-2}\,\text{m}^2\)
• B. \(4 \times 10^{-4}\,\text{m}^2\)
• C. \(2 \times 10^{-4}\,\text{m}^2\)
• D. \(4 \times 10^{-2}\,\text{m}^2\)

Hint:

For same heat flow rate, same length, and same temperature difference, \(kA\) remains constant.

Answer 1

The Correct Option is A

Step 1: Use rate of heat conduction formula.
\[ \frac{Q}{t} = \frac{kA\Delta T}{L} \]

Step 2: Apply given condition.
For both rods, rate of heat conduction is same, length is same, and temperature difference is same.
Therefore,
\[ k_1A_1 = k_2A_2 \]

Step 3: Use thermal conductivity ratio.
\[ k_1:k_2 = 1:4 \]
So,
\[ \frac{k_1}{k_2} = \frac{1}{4} \]

Step 4: Substitute in relation.
\[ A_2 = \frac{k_1A_1}{k_2} \]
\[ A_2 = \frac{k_1}{k_2}A_1 \]

Step 5: Put value of \(A_1\).
\[ A_1 = 8 \times 10^{-2}\,\text{m}^2 \]
\[ A_2 = \frac{1}{4} \times 8 \times 10^{-2} \]

Step 6: Simplify.
\[ A_2 = 2 \times 10^{-2}\,\text{m}^2 \]

Step 7: Final answer.
\[ \boxed{2 \times 10^{-2}\,\text{m}^2} \]