Five rods of 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 if

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Solution

The correct option is C
For no current flow between C and D
${(\frac{Q}{t})}_{A C} = {(\frac{Q}{t})}_{C B} \Rightarrow \frac{K_{1} A (θ_{A} - θ_{C})}{l} = \frac{K_{2} A (θ_{C} - θ_{B})}{l}$
$\Rightarrow \frac{θ_{A} - θ_{C}}{θ_{C} - θ_{B}} = \frac{K_{2}}{K_{1}}$ ....(i)
Also ${(\frac{Q}{t})}_{A D} = {(\frac{Q}{t})}_{D B} \Rightarrow \frac{K_{3} A (θ_{A} - θ_{D})}{l} = \frac{K_{4} A (θ_{D} - θ_{B})}{l}$
$\Rightarrow \frac{θ_{A} - θ_{D}}{θ_{D} - θ_{B}} = \frac{K_{4}}{K_{3}}$ ....(ii)
It is given that $θ_{C} = θ_{D}$ , hence from equation (i) and (ii) we get $\frac{K_{2}}{K_{1}} = \frac{K_{4}}{K_{3}} \Rightarrow K_{1} K_{4} = K_{2} K_{3}$

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