\(r _{1}\) तथा \(r _{2}\) त्रिज्याओं \(\left( r _{1}\,<\, r _{2}\right)\) के दो पतले धातुत्वीय गोलीय कोशों को संपातित केन्द्रों के साथ रखा जाता है। दोनों कोशों के मध्य का भाग उष्मीय चालकता \(K\) के पदार्थ से भरा है। आन्तरिंक कोश को ताप \(\theta_{1}\) तथा बाहय कोश को ताप \(\theta_{2}\) (जहाँ \(\theta_{1}\,<\,\theta_{2}\) ) पर पोषित किया जाता है। पदार्थ से त्रिज्यीय प्रवाहित उष्मा की दर होती है।

A \(\frac{4 \pi {Kr}_{1} {r}_{2}\left(\theta_{2}-\theta_{1}\right)}{{r}_{2}-{r}_{1}}\)
B \(\frac{\pi{r}_{1} {r}_{2}\left(\theta_{2}-\theta_{1}\right)}{{r}_{2}-{r}_{1}}\)
C \(\frac{{K}\left(\theta_{2}-\theta_{1}\right)}{{r}_{2}-{r}_{1}}\)
D \(\frac{{K}\left(\theta_{2}-\theta_{1}\right)\left({r}_{2}-{r}_{1}\right)}{4 \pi {r}_{1} {r}_{2}}\)

Verified Solution

Answer & Solution

Correct Answer

(A) \(\frac{4 \pi {Kr}_{1} {r}_{2}\left(\theta_{2}-\theta_{1}\right)}{{r}_{2}-{r}_{1}}\)

Step-by-step Solution

Detailed explanation

Thermal resistance of spherical sheet of thicleness \(dr\) and radius \(r\) is \({d} {R}=\frac{{dr}}{{K}\left(4 \pi {r}^{2}\right)}\) \({R}=\int_{{r}_{1}}^{{r}_{2}} \frac{{dr}}{{K}\left(4 \pi {r}^{2}\right)}\)…

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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