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A long rod has one end at $0 ° C$ and other end at a high temperature. The coefficient of thermal conductivity varies with distance from the low temperature end as $K = K_{0} (1 + a x)$ , where $K_{0} = 10^{2}$ SI unit and $a = 1 m^{- 1}$ . At what distance from the first end the temperature will be $100 ° C$ ? The area of cross-section is $1 c m^{2}$ and rate of heat conduction is $1 W$ .

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$3 m$

$2.7 m$

$1.5 m$

$1.7 m$

answer is B.

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Detailed Solution

$H = \frac{T D}{R}$

$∴ R = \frac{T D}{H} = \frac{100 - 0}{1}$

$= 100 k W^{- 1}$

Now, $R = \int_{0}^{x} d R = \int_{0}^{x} \frac{d x}{K_{0} (1 + a x) A}$

$100 = \int_{0}^{x} \frac{d x}{10^{2} (1 + x) (10^{- 4})}$

Solving this equation we get,

$x = 1.7 m$

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