Three rods of same dimensions are arranged as shown in figure. They have thermal conductivities K₁ , K₂and K₃. The points P and Q are maintained at different temperatures for the heat to flow at the same rate along PRQ and PQ. Which of the following options is correct?

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$K_{3} = \frac{1}{2} (K_{1} + K_{2})$

$K_{3} = 2 (K_{1} + K_{2})$

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

$K_{3} = K_{1} + K_{2}$

answer is C.

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Detailed Solution

Rate of flow of heat along PQ

${(\frac{dQ}{dt})}_{PQ} = \frac{K_{3} AΔθ}{l}$ …………(i)

Rate of flow of heat along PRQ

${(\frac{dQ}{dt})}_{PRQ} = \frac{K_{s} AΔθ}{2 l}$

Effective conductivity for series combination of two rods of same length

$K_{s} = \frac{2 K_{1} K_{2}}{K_{1} + K_{2}}$

So ${(\frac{dQ}{dt})}_{PRQ} = \frac{2 K_{1} K_{2}}{K_{1} + K_{2}} \cdot \frac{AΔθ}{2 l} = \frac{K_{1} K_{2}}{K_{1} + K_{2}} \cdot \frac{AΔθ}{l}$ ……(ii)

$Equating Eqs. (i) and (ii) K_{3} = \frac{K_{1} K_{2}}{K_{1} + K_{2}}$

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