Question 26 of 126

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

A$\dfrac{K\left({\theta }_{2}-{\theta }_{1}\right)}{{r}_{2}-{r}_{1}}$B$\dfrac{K\left({\theta }_{2}-{\theta }_{1}\right)\left({r}_{2}-{r}_{1}\right)}{4\pi {r}_{1}{r}_{2}}$C$\dfrac{\pi K{r}_{1}{r}_{2}\left({\theta }_{2}-{\theta }_{1}\right)}{{r}_{2}-{r}_{1}}$D$\dfrac{4\pi K{r}_{1}{r}_{2}\left({\theta }_{2}-{\theta }_{1}\right)}{{r}_{2}-{r}_{1}}$

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Answer: (D) $\dfrac{4\pi K{r}_{1}{r}_{2}\left({\theta }_{2}-{\theta }_{1}\right)}{{r}_{2}-{r}_{1}}$

Equivalent resistance $(R)=\dfrac{{r}_{2}-{r}_{1}}{K4\pi {r}_{2}{r}_{1}}$
Thus, Heat flow $=\dfrac{4\pi K\left({r}_{1}{r}_{2}\right)\left({\theta }_{2}-{\theta }_{1}\right)}{\left({r}_{2}-{r}_{1}\right)}$