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Conduction Law

A system has ...

Question

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

A

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

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B

l o g (\frac{r_{2}}{r_{1}})

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C

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

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D

(r_{2} - r_{1})

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Solution

The correct option is A $\frac{r_{1} r_{2}}{r_{2} - r_{1}}$
Consider an elemental spherical shell of thickness dx at radius x.
Thermal resistance of this shell is given by
$d R = \frac{d x}{K (4 π x^{2})} {F r o m R = \frac{L}{K A}} \int d R = R = \frac{1}{K} \int_{r_{1}}^{r_{2}} \frac{d x}{4 π x^{2}} = \frac{1}{4 π K} [\frac{1}{r_{1}} - \frac{1}{r_{2}}] = \frac{(r_{2} - r_{1})}{4 π K (r_{1} r_{2})}$
Therefore, rate of heat flow is:
$H = \frac{T_{1} - T_{2}}{R} = (T_{1} - T_{2}) (\frac{4 π K (r_{1} r_{2})}{r_{2} - r_{1}}) H \propto \frac{(r_{1} r_{2})}{r_{2} - r_{1}}$

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