Five rods of same dimensions are arranged as shown in the fig.They have thermal conductivities, $k_{1}, k_{2}, k_{5}, k_{4}$ and $k_{3}$ when points A and B are maintained at different temperatures. No heat flows through the central rod if:

k_{1} k_{4} = k_{2} k_{3}

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k_{1} = k_{4}

and

k_{2} = k_{3}

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\frac{k_{1}}{k_{4}} = \frac{k_{2}}{k_{3}}

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k_{1} k_{2} = k_{3} k_{4}

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Solution

The correct option is A $k_{1} k_{4} = k_{2} k_{3}$
No heat will pass if $T_{C} = T_{D}$ and all lengths are equal
$k_{1} (T_{D} - T_{B}) = k_{2} (T_{C} - T_{B})$
$k_{3} (T_{D} - T_{A}) = k_{4} (T_{C} - T_{A})$
Since $T_{C} = T_{D}$
$\frac{k_{1}}{k_{3}} = \frac{k_{2}}{k_{4}}$
Thus, $k_{1} k_{4} = k_{2} k_{3}$

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