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Two thin metallic spherical shells of radii r_{1}$ and $r_{2}\left(r_{1}<r_{2}\right) are placed with their centres coinciding. A material of thermal conductivity \mathrm{K} is filled in the space between the shells. The inner shell is maintained at temperature \theta_{1} and the outer shell at temperature \theta_{2}\left(\theta_{1}<\theta_{2}\right). The rate at which heat flows radially through the material is :
 
Option: 1 \frac{\mathrm{K}\left(\theta_{2}-\theta_{1}\right)}{\mathrm{r}_{2}-\mathrm{r}_{1}}
Option: 2 \frac{\mathrm{K}\left(\theta_{2}-\theta_{1}\right)\left(\mathrm{r}_{2}-\mathrm{r}_{1}\right)}{4 \pi \mathrm{r}_{1} \mathrm{r}_{2}}
Option: 3 \frac{4 \pi \mathrm{Kr}_{1} \mathrm{r}_{2}\left(\theta_{1}-\theta_{2}\right)}{\mathrm{r}_{2}-\mathrm{r}_{1}}
Option: 4 \frac{\pi r_{1} r_{2}\left(\theta_{2}-\theta_{1}\right)}{r_{2}-r_{1}}

Answers (1)

best_answer

H=\frac{K\left ( 4\pi x^{2} \right )}{dx}\left ( d\theta \right )\\

\frac{H}{4\pi K}\int_{r_{1}}^{r_{2}}\frac{dx}{x^{2}}=\int_{\theta_{2}}^{\theta_{1}}d\theta\\

\frac{H}{4\pi K}\times \left [ -\frac{1}{x} \right ]_{r_{1}}^{r_{2}}=\left [ \theta \right ]_{\theta_{2}}^{\theta_{1}}\\

\frac{H}{4\pi K}\left [ \frac{-1}{r_{2}}+\frac{1}{r_{1}} \right ]=\left ( \theta_{2}-\theta_{1} \right )\\

\frac{H\left ( r_{2}-r_{1} \right )}{4\pi K\left ( r_{1}r_{2} \right )}=\theta_{1}-\theta_{2}\\

H=\frac{4\pi K\left ( r_{1}r_{2} \right )\left ( \theta_{1}-\theta_{2} \right )}{\left ( r_{2}-r_{1} \right )}

The correct option is (3)

Posted by

vishal kumar

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