question_answer

The figure shows a system of two concentric spheres of radii r1 and r2 and kept at temperatures T1 and T2, respectively. The radial rate of flow of heat in a substance between the two concentric spheres is proportional to [AIEEE 2005]

A)            \[\frac{{{r}_{1}}\,{{r}_{2}}}{({{r}_{1}}-{{r}_{2}})}\]

B)            \[({{r}_{2}}-{{r}_{1}})\]

C)            \[({{r}_{2}}-{{r}_{1}})({{r}_{1}}\,{{r}_{2}})\]

D)            In \[\left( \frac{{{r}_{2}}}{{{r}_{1}}} \right)\]

Correct Answer: A

Solution :

                   Consider a concentric spherical shell of radius r and thickness dr as shown in fig. The radial rate of flow of heat through this shell in steady state will be \[H=\frac{dQ}{dt}=-KA\frac{dT}{dr}=-K\,(4\pi {{r}^{2}})\frac{dT}{dr}\] Þ \[\int_{\,{{r}_{1}}}^{\,{{r}_{2}}}{\frac{dr}{{{r}^{2}}}=-\frac{4\pi K}{H}\int_{\,{{T}_{1}}}^{{{T}_{1}}}{dT}}\] Which on integration and simplification gives \[H=\frac{dQ}{dt}=\frac{4\pi K{{r}_{1}}{{r}_{2}}({{T}_{1}}-{{T}_{2}})}{{{r}_{2}}-{{r}_{1}}}\]Þ \[\frac{dQ}{dt}\propto \frac{{{r}_{1}}{{r}_{2}}}{({{r}_{2}}-{{r}_{1}})}\]