Question

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 $K$ 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 :-

• A. $\frac{4\pi K{r}_1{r}_2\left({\theta }_2-{\theta }_1\right)}{r_2-r_1}$
• 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. $\frac{K\left({\theta }_2-{\theta }_1\right)\left(r_2-r_1\right)}{4\pi r_1{r}_2}$

Answer 1

Answer: (A)

Thermal resistance of spherical sheet of thickness $dr$ and radius $r$ is
$dR=\frac{dr}{K\left(4\pi r^2\right)}$
$R=\underset{r_1}{\overset{r_2}{\int }}\frac{dr}{K\left(4\pi r^2\right)}$
$R=\frac{1}{4\pi K}\left(\frac{1}{r_1}-\frac{1}{r_2}\right)=\frac{1}{4\pi K}\left(\frac{r_2-r_1}{r_1{r}_2}\right)$
Thermal current (i) $=\frac{{\theta }_2-{\theta }_1}{R}$
$i=\frac{4\pi K{r}_1{r}_2}{r_2-r_1}\left({\theta }_2-{\theta }_1\right)$