Two thin metallic spherical shells of radii $r_{1} a n d r_{2} (r_{1} < r_{2})$ 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 $θ_{1}$ and the outer shell at temperature $θ_{2} (θ_{1} < θ_{2})$ . The rate at which heat flows radially through the material is

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$\frac{π r_{1} r_{2} (θ_{2} - θ_{1})}{r_{2} - r_{1}}$

$\frac{4 π K r_{1} r_{2} (θ_{2} - θ_{1})}{r_{2} - r_{1}}$

$\frac{K (θ_{2} - θ_{1})}{r_{2} - r_{1}}$

$\frac{K (θ_{2} - θ_{1}) (r_{2} - r_{1})}{4 π r_{1} r_{2}}$

answer is A.

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Detailed Solution

Resistance of spherical shell of thickness dr and radius r is
Question Image
$d R = \frac{d r}{K (4 π r^{2})} [using R = \frac{1}{K} \frac{l}{A}]$
Total resistance
$R = \int_{r_{1}}^{r_{2}} \frac{d r}{K (4 π r^{2})} \Rightarrow R = \frac{1}{4 π K} (\frac{1}{η} - \frac{1}{r_{2}})$

Current $= \frac{θ_{2} - θ_{1}}{R}$ $= \frac{4 π K r_{1} r_{2}}{r_{2} - r_{1}} (θ_{2} - θ_{1})$