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Conduction

Five rods of ...

Question

Five rods of the same dimensions are arranged as shown in the figure. They have thermal conductivities $K_{1}, K_{2}, K_{3}, K_{4} a n d K_{5}$ .When points A and B are maintained at different temperatures no heat flows through the central rod CD if:

K_{1} = K_{4} a n d K_{2} = K_{3}

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K_{1} K_{4} = K_{2} K_{3}

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K_{1} K_{2} = K_{3} K_{4}

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K_{1} K_{3} = K_{2} K_{4}

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Solution

The correct option is A $K_{1} = K_{4} a n d K_{2} = K_{3}$
Given that $θ_{C} = θ_{D}$
Let temperature of $A = θ_{1}$
Temperature of $B = θ_{2}$
Therefore, $θ_{C} = \frac{K_{1} θ_{1} + K_{2} θ_{2}}{K_{1} + K_{2}}$
and, $θ_{D} = \frac{K_{3} θ_{1} + K_{4} θ_{2}}{K_{3} + K_{4}}$
As the temperature of both the node are same,on equating both of them we find,
$\frac{K_{1} θ_{1} + K_{2} θ_{2}}{K_{1} + K_{2}} = \frac{K_{3} θ_{1} + K_{4} θ_{2}}{K_{3} + K_{4}}$
$K_{1} θ_{1} K_{3} + K_{1} θ_{1} K_{4} + K_{2} θ_{2} K_{4} = K_{1} θ_{1} K_{3} + K_{1} θ_{2} K_{4} + K_{2} θ_{1} K_{3} + K_{2} θ_{2} K_{4}$
$k_{2} k_{3} (θ_{1} - θ_{2}) = k_{1} k_{4} (θ_{1} - θ_{2})$
$K_{1} K_{4} = K_{2} K_{3}$

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