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Q.

A rod of length $l$ and cross section area A has a variable thermal conductivity given by $k = α T$ , where $α$ is a positive constant and T is temperature in kelvin. Two ends of the rod are maintained at temperatures $T_{1}$ and $T_{2} (T_{1} > T_{2})$ . Heat current flowing through the rod will be

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a

$\frac{A α (T_{1}^{2} - T_{2}^{2})}{2 l}$

b

$\frac{A α (T_{1}^{2} - T_{2}^{2})}{4 l}$

c

$\frac{A α (T_{1}^{2} - T_{2}^{2})}{3 l}$

d

$\frac{A α (T_{1}^{2} - T_{2}^{2})}{l}$

answer is D.

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Detailed Solution

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Heat current : $i = - k A \frac{d T}{d x}$
   $i d x = - k A d T$
   $i \int_{0}^{l} d x = - A α \int_{T_{1}}^{T_{2}} T d T$
   $\Rightarrow i l = - A α \frac{(T_{2}^{2} - T_{1}^{2})}{2 a}$
   $\Rightarrow i = \frac{A α (T_{1}^{2} - T_{2}^{2})}{2 l}$

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