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. $\frac{4 \pi Kr _{1} r _{2}\left(\theta_{2}-\theta_{1}\right)}{ r _{2}- r _{1}}$
• 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. $\frac{ K \left(\theta_{2}-\theta_{1}\right)\left( r _{2}- r _{1}\right)}{4 \pi r _{1} r _{2}}$

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 =\int\limits_{ r _{1}}^{ r _{2}} \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) $=\frac{\theta_{2}-\theta_{1}}{ R }$
$i =\frac{4 \pi Kr _{1} r _{2}}{ r _{2}- r _{1}}\left(\theta_{2}-\theta_{1}\right)$