Question 100 of 126

The temperature $\theta$ at the junction of two insulating sheets, having thermal resistances ${R}_{1}$ and ${R}_{2}$ as well as top and bottom temperatures ${\theta }_{1}$ and ${\theta }_{2}$ (as shown in figure) is given by :

A$\dfrac{{\theta }_{1}{R}_{2}-{\theta }_{2}{R}_{1}}{{R}_{2}-{R}_{1}}$B$\dfrac{{\theta }_{2}{R}_{2}-{\theta }_{1}{R}_{1}}{{R}_{2}-{R}_{1}}$C$\dfrac{{\theta }_{1}{R}_{2}+{\theta }_{2}{R}_{1}}{{R}_{1}+{R}_{2}}$D$\dfrac{{\theta }_{1}{R}_{1}+{\theta }_{2}{R}_{2}}{{R}_{1}+{R}_{2}}$

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Answer: (C) $\dfrac{{\theta }_{1}{R}_{2}+{\theta }_{2}{R}_{1}}{{R}_{1}+{R}_{2}}$

Heat flow rate will be same through both
$∴\dfrac{{\theta }_{1}-\theta }{{R}_{1}}=\dfrac{\theta -{\theta }_{2}}{{R}_{2}}$
${R}_{2}{\theta }_{1}-{R}_{2}\theta ={R}_{1}\theta -{R}_{1}{\theta }_{2}$
$\theta =\dfrac{{R}_{2}{\theta }_{1}+{R}_{1}{\theta }_{2}}{{R}_{1}+{R}_{2}}$