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Concept of Square Root

Five rods of ...

Question

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

K_{1}

K_{4}

and

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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None of these

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Solution

The correct option is B $K_{1}$ $K_{4}$ = $K_{2}$ $K_{3}$
For no heat flow between $A$ and $B$ , heat flow throght $DA$ will be equal to the heat flow through $AC$ ,

${(\frac{Q}{t})}_{D A} = {(\frac{Q}{t})}_{A C}$

Substituting the values from figure

$\Rightarrow \frac{K_{3} A (θ_{D} - θ_{A})}{l} = \frac{K_{1} A (θ_{A} - θ_{C})}{l}$

where, $θ_{A}, θ_{B}, θ_{C},$ and $θ_{D}$ are the tempratute at the respective points, $l$ is the length and $A$ is the cross-section area for each rod.

$\Rightarrow \frac{(θ_{D} - θ_{A})}{(θ_{A} - θ_{C})} = \frac{K_{1}}{K_{3}} . . . (1)$

Similarly,
${(\frac{Q}{t})}_{D B} = {(\frac{Q}{t})}_{B C}$

$\Rightarrow \frac{K_{4} A (θ_{D} - θ_{B})}{l} = \frac{K_{2} A (θ_{B} - θ_{C})}{l}$

$\Rightarrow \frac{(θ_{D} - θ_{B})}{(θ_{B} - θ_{C})} = \frac{K_{2}}{K_{4}} . . . (2)$

It is given that $θ_{A} = θ_{B}$ , hence from equation $(1)$ and $(2)$ , we get

$\frac{K_{1}}{K_{3}}$ = $\frac{K_{2}}{K_{4}}$

$∴ K_{1}$ $K_{4}$ = $K_{2}$ $K_{3}$

Hence, option (b) is correct answer.

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Concept of Square Root

Standard VII Mathematics

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