KCET · Physics · Thermal Properties of Matter

Two slabs are of the thicknesses \(d_{1}\) and \(d_{2}\). Their thermal conductivities are \(\mathrm{K}_{1}\) and \(\mathrm{K}_{2}\) respectively. They are in series. The free ends of the combination of these two slabs are kept at temperatures \(\theta_{1}\) and \(\theta_{2}\). Assume \(\theta_{1}>\theta_{2}\). The temperature \(\theta\) of their common junction is

A \(\frac{\mathrm{K}_{1} \theta_{1}+\mathrm{K}_{2} \theta_{2}}{\theta_{1}+\theta_{2}}\)
B \(\frac{\mathrm{K}_{1} \theta_{1} \mathrm{~d}_{1}+\mathrm{K}_{2} \theta_{2} \mathrm{~d}_{2}}{\mathrm{~K}_{1} \mathrm{~d}_{2}+\mathrm{K}_{2} \mathrm{~d}_{1}}\)
C \(\frac{\mathrm{K}_{1} \theta_{1} \mathrm{~d}_{2}+\mathrm{K}_{2} \theta_{2} \mathrm{~d}_{1}}{\mathrm{~K}_{1} \mathrm{~d}_{2}+\mathrm{K}_{2} \mathrm{~d}_{1}}\)
D \(\frac{\mathrm{K}_{1} \theta_{1}+\mathrm{K}_{2} \theta_{2}}{\mathrm{~K}_{1}+\mathrm{K}_{2}}\)

Verified Solution

Answer & Solution

Correct Answer

(C) \(\frac{\mathrm{K}_{1} \theta_{1} \mathrm{~d}_{2}+\mathrm{K}_{2} \theta_{2} \mathrm{~d}_{1}}{\mathrm{~K}_{1} \mathrm{~d}_{2}+\mathrm{K}_{2} \mathrm{~d}_{1}}\)

Step-by-step Solution

Detailed explanation

For first slab

Heat current, \(\mathrm{H}_{1}=\frac{\mathrm{K}_{1}\left(\theta_{1}-\theta\right) \mathrm{A}}{\mathrm{d}_{1}}\)
For second slab,
Heat current, \(\mathrm{H}_{2}=\frac{\mathrm{K}_{2}\left(\theta-\theta_{2}\right) \mathrm{A}}{\mathrm{d}_{2}}\)
As slabs are in series
\(\mathrm{H}_{1} =\mathrm{H}_{2} \)
\( \therefore \frac{\mathrm{K}_{1}\left(\theta_{1}-\theta\right) \mathrm{A}}{\mathrm{d}_{1}} =\frac{\mathrm{K}_{2}\left(\theta-\theta_{2}\right) \mathrm{A}}{\mathrm{d}_{2}} \)
\( \Rightarrow \theta =\frac{\mathrm{K}_{1} \theta_{1} \mathrm{~d}_{2}+\mathrm{K}_{2} \theta_{2} \mathrm{~d}_{1}}{\mathrm{~K}_{2} \mathrm{~d}_{1}+\mathrm{K}_{1} \mathrm{~d}_{2}}\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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