KCET · Physics · Thermal Properties of Matter
A, B and C are the three identical conductors but made from different materials. They are kept in contact as shown.
Their thermal conductivities are \(\mathrm{K}, 2 \mathrm{~K}\) and \(\frac{\mathrm{K}}{2}\). The free end of \(\mathrm{A}\) is at \(100^{\circ} \mathrm{C}\) and the free end of \(\mathrm{C}\) is at \(0^{\circ} \mathrm{C}\). During steady state, the temperature of the junction of \(\mathrm{A}\) and \(\mathrm{B}\) is nearly
- A \(37^{\circ} \mathrm{C}\)
- B \(71^{\circ} \mathrm{C}\)
- C \(29^{\circ} \mathrm{C}\)
- D \(63^{\circ} \mathrm{C}\)
Answer & Solution
Correct Answer
(B) \(71^{\circ} \mathrm{C}\)
Step-by-step Solution
Detailed explanation
Let \(\mathrm{R}\) be the thermal conductivity of conductor, then thermal conductivity of conductor \(B=\frac{R}{2}\) and thermal conductivity of conductor \(C=2 R\).
\(\therefore\) Heat current, \(\mathrm{H}=\frac{100^{\circ}-0^{\circ}}{\mathrm{R}+\frac{\mathrm{R}}{2}+2 \mathrm{R}}=\frac{200}{7 \mathrm{R}}\)
If \(\mathrm{T}^{\prime}\) be the temperature of the junction of \(\mathrm{A}\) and B, then
\(\mathrm{H}=\frac{100-\mathrm{T}^{\prime}}{\mathrm{R}} \text { or } \frac{200}{7 \mathrm{R}}=\frac{100-\mathrm{T}^{\prime}}{\mathrm{R}}\)
or
\(\mathrm{T}^{\prime}=\frac{500}{7}=71^{\circ} \mathrm{C}\)
\(\therefore\) Heat current, \(\mathrm{H}=\frac{100^{\circ}-0^{\circ}}{\mathrm{R}+\frac{\mathrm{R}}{2}+2 \mathrm{R}}=\frac{200}{7 \mathrm{R}}\)
If \(\mathrm{T}^{\prime}\) be the temperature of the junction of \(\mathrm{A}\) and B, then
\(\mathrm{H}=\frac{100-\mathrm{T}^{\prime}}{\mathrm{R}} \text { or } \frac{200}{7 \mathrm{R}}=\frac{100-\mathrm{T}^{\prime}}{\mathrm{R}}\)
or
\(\mathrm{T}^{\prime}=\frac{500}{7}=71^{\circ} \mathrm{C}\)
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