Three rods of same dimensions have thermal conductivities $3 K$, $2 K$ and $K$. They are arranged as shown…

Three rods of same dimensions have thermal conductivities $3 K$, $2 K$ and $K$. They are arranged as shown in the figure below. Then in the steady state the temperature of the junction $P$ is

1. $\frac{200}{3}{ }^{\circ} \mathrm{C}$
2. $\frac{100^{\circ}}{3}{ }^{\circ} \mathrm{C}$
3. $75^{\circ} \mathrm{C}$
4. $\frac{50}{3}{ }^{\circ} \mathrm{C}$

Solution

Amount of heat transmitted from one point to another is $ Q=\frac{k A t\left(T_1-T_2\right)}{d} $ where, $k$ is thermal conductivity, $t$ is time and $T_1-T_2$ is temperature difference. At junction, $ \begin{aligned} Q & =Q_1+Q_2 \\ \frac{3 k A(100-T) t}{d} & =\frac{2 k A(T-50) t}{d}+\frac{k A(T-0) t}{d} \\ 300 k A t-3 k A t T & =2 k A t T-100 k A t+k A T t \\ 400 k A t & =6 k A t T \\ T & =\frac{400}{6} \\ T & =\frac{200^{\circ}}{3} C \end{aligned} $

Asked in: AP EAMCET 2018 (23 Apr Shift 1)

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