\({r}_{1}\) અને \({r}_{2}\) \(\left({r}_{1}<{r}_{2}\right)\) ત્રિજયા ધરાવતા બે ધાતુના પાતળા કવચના કેન્દ્ર એક બીજા પર સંપાત થાય છે. બંને કવચની વચ્ચેની જગયા \({K}\) જેટલી ઉષ્માવાહકતા ધરાવતા દ્રવ્યથી ભરેલી છે. અંદરની કવચ \(\theta_{1}\) તાપમાને અને બહારની કવચ \(\theta_{2}\left(\theta_{1}<\theta_{2}\right)\) તાપમાને રાખેલ છે. આ દ્રવ્યમાં ત્રિજ્યાવર્તી દિશામાં ઉષ્મા વહનનો દર કેટલો હશે?

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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