Two rods $A$ and $B$ of different materials are welded together as shown in figure. Their thermal conductivities are $K_{1}$ and $K_{2}$ . The thermal conductivity of the composite rod will be :

$2 (K_{1} + K_{2})$
$\frac{K_{1} + K_{2}}{2}$
$\frac{3 (K_{1} + K_{2})}{2}$
$K_{1} + K_{2}$

K_{1} + K_{2}

\frac{K_{1} + K_{2}}{2}

\frac{3 (K_{1} + K_{2})}{2}

2 (K_{1} + K_{2})

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Solution

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$R_{1} = \frac{d}{k_{1} A}$
$R_{2} = d K_{2} A$
$R e q = \frac{R_{1} R_{2}}{R_{1} + R_{2}}$
$\frac{d}{k e q (2 A)} = \frac{(\frac{d}{k_{1} A}) (\frac{d}{k_{2} A})}{\frac{d}{A} (\frac{1}{k_{1}} + \frac{1}{k_{2}})}$
$\frac{1}{2 k e q} = \frac{\frac{1}{k_{1} k_{2}}}{\frac{k_{1} + k_{2}}{k_{1} k_{2}}} = \frac{1}{k_{1} + k_{2}}$
$k e q = \frac{k_{1} + k_{2}}{2}$

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