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The figure shows a system of two concentric spheres of radii $r_{1}$ and $r_{2}$ and kept at temperatures $T_{1}$ and $T_{2}$ , respectively. The radial rate of flow of heat in a substance between the two concentric spheres is proportional to

A

\frac{r_{1} r_{2}}{(r_{1} - r_{2})}

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B

(r_{2} - r_{1})

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C

(r_{2} - r_{1}) (r_{1} r_{2})

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D

I n (\frac{r_{2}}{r_{1}})

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Solution

The correct option is A $\frac{r_{1} r_{2}}{(r_{1} - r_{2})}$
Consider a concentric spherical shell of radius r and thickness dr as shown in fig.

The radial rate of flow of heat through this shell in steady state will be $H = \frac{d Q}{d t} = - K A \frac{d T}{d r} = - K (4 π r^{2}) \frac{d T}{d r}$
$\Rightarrow \int_{r 1}^{r 2} \frac{d r}{r^{2}} = - \frac{4 π K}{H} \int_{T_{1}}^{T_{1}} d t$
Which on integration and simplification gives
$H = \frac{d Q}{d t} = \frac{4 π K r_{1} r_{2} (T_{1} - T_{2})}{r_{2} - r_{1}} \Rightarrow \frac{d Q}{d t} \propto \frac{r_{1} r_{2}}{(r_{2} - r_{1})}$

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