KCET · Physics · Thermal Properties of Matter
Two identical rods \(A C\) and \(C B\) made of two different metals having thermal conductivities in the ratio \(2: 3\) are kept in contact with each other at the end \(C\) as shown in the figure. \(A\) is at \(100^{\circ} \mathrm{C}\) and \(B\) is at \(25^{\circ} \mathrm{C}\). Then the junction \(C\) is at
- A \(55^{\circ} \mathrm{C}\)
- B \(60^{\circ} \mathrm{C}\)
- C \(75^{\circ} \mathrm{C}\)
- D \(50^{\circ} \mathrm{C}\)
Answer & Solution
Correct Answer
(A) \(55^{\circ} \mathrm{C}\)
Step-by-step Solution
Detailed explanation
Let the temperature of the junction be \(\theta\). Heat flow through the rod is given by
\(Q =\frac{K A\left(\theta_{1}-\theta_{2}\right)}{d} \)
\( \text {Here, } K_{1}(100-\theta) =K_{2}(\theta-25) \)
\( \Rightarrow \frac{K_{1}}{K_{2}} =\frac{\theta-25}{100-\theta} \)
\( \text {But } \frac{K_{1}}{K} =\frac{2}{3} (given) \)
\( \therefore \frac{2}{3} =\frac{\theta-25}{100-\theta} \)
\( \Rightarrow 3 \theta-75 =200-2 \theta \)
\( \therefore 5 \theta =275 \)
\( \therefore \theta =55^{\circ} \mathrm{C}\)
\(Q =\frac{K A\left(\theta_{1}-\theta_{2}\right)}{d} \)
\( \text {Here, } K_{1}(100-\theta) =K_{2}(\theta-25) \)
\( \Rightarrow \frac{K_{1}}{K_{2}} =\frac{\theta-25}{100-\theta} \)
\( \text {But } \frac{K_{1}}{K} =\frac{2}{3} (given) \)
\( \therefore \frac{2}{3} =\frac{\theta-25}{100-\theta} \)
\( \Rightarrow 3 \theta-75 =200-2 \theta \)
\( \therefore 5 \theta =275 \)
\( \therefore \theta =55^{\circ} \mathrm{C}\)
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