Q168 / 180 · KCET 2007 · Physics

Two identical rods $A C$ and $C B$ made of two different metals having thermal conductivities in the ratio $2: 3$ are kept in contact with each other at the end $C$ as shown in the figure. $A$ is at $100^{\circ} \mathrm{C}$ and $B$ is at $25^{\circ} \mathrm{C}$. Then the junction $C$ is at

  • Correct answer
    A
    $55^{\circ} \mathrm{C}$
  • B
    $60^{\circ} \mathrm{C}$
  • C
    $75^{\circ} \mathrm{C}$
  • D
    $50^{\circ} \mathrm{C}$

Solution

Let the temperature of the junction be $\theta$. Heat flow through the rod is given by $$ \begin{aligned} & Q &=\frac{K A\left(\theta_{1}-\theta_{2}\right)}{d} \\ & \text { Here, } K_{1}(100-\theta) &=K_{2}(\theta-25) \\ \Rightarrow & \frac{K_{1}}{K_{2}} &=\frac{\theta-25}{100-\theta} \\ & \text { But } & \frac{K_{1}}{K} &=\frac{2}{3} (given) \\ \therefore & \frac{2}{3} &=\frac{\theta-25}{100-\theta} \\ \Rightarrow & 3 \theta-75 &=200-2 \theta \\ \therefore & 5 \theta &=275 \\ \therefore & \theta &=55^{\circ} \mathrm{C} \end{aligned} $$

Hence, the correct answer is option: 'A'.

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