Two bars of thermal conductivities K and 3K and lengths 1cm and $2\,\mathrm{cm}$ respectively have equal cross-sectional area, they are joined lengths wise as shown in the figure. If the temperature at the ends of this composite bar is $0^\circ C$ and $\mathbf{K}^2 / l$ respectively (see figure), then the temperature $\phi$ of the interface is
$(a) 50°C (b) \frac{100}{3} °C (c) 60°C (d) \frac{200}{3} °C$
Text Solution
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Temperature of interface
$\theta = \frac{K_1 \theta_1 l_2 + K_2 \theta_2 l_1}{K_1 l_2 + K_2 l_1} = \frac{K \times 0 \times 2 + 3K \times 100 \times 1}{K \times 2 + 3K \times 1} = \frac{300K}{5K} = 60^\circ C$
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