Heat flows radially outward through a spherical shell of outside radius $R_{2}$ and inner radius $R_{1} .$ The temperature of inner surface of shell is $θ_{1}$ and that of outer is $θ_{2} .$ At what radial distance from centre of shell the temperature is just half way between $θ_{1}$ and $θ_{2} .$

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$\frac{3 R_{1} + R_{2}}{2}$

$\frac{4 R_{2} + 3 R_{1}}{3}$

$\frac{8 R_{2} R_{1}}{4 R_{1} + 3 R_{2}}$

$\frac{2 R_{1} R_{2}}{R_{1} + R_{2}}$

answer is D.

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Detailed Solution

Resistance of spherical shell

$\begin{array}{l} R_{0} = \frac{1}{4 π k} [\frac{1}{R_{1}} - \frac{1}{R_{2}}]; R^{1} = \frac{1}{4 π K} [\frac{1}{R_{1}} - \frac{1}{r}] \\ (θ_{1} - θ_{2}) \to R θ_{1} - (\frac{θ_{1} + θ_{2}}{2}) = \frac{R_{1}}{R} (θ_{1} - θ_{2}) \\ \Rightarrow θ_{1} - (\frac{θ_{1} + θ_{2}}{2}) = \frac{R^{1}}{R} (θ_{1} - θ_{2}) \\ θ_{1} - \frac{θ_{1} + θ_{2}}{2} = \frac{r - R_{1}}{R_{1} r} = \frac{R_{1} R_{2}}{R_{2} - R_{1}} (θ_{1} - θ_{2}) \\ \Rightarrow \frac{2 R_{1} R_{2}}{R_{1} + R_{2}} \end{array}$