Question

Three rods of same dimensions are arranged as shown in figure they have thermal conductivities K₁, K₂ and K₃. The points P and Q are maintained at different temperatures for the heat to flow at the same rate along PRQ and PQ then which of the following option is correct

• A. $K_{3} = \frac{1}{2}(K_{1} + K_{2})$
• B. $K_{3} = K_{1} + K_{2}$
• C. $K_{3} = \frac{K_{1}K_{2}}{K_{1} + K_{2}}$
• D. $K_{3} = 2(K_{1} + K_{2})$

Answer 1

Answer: (C)

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

Answer 2

Rate of flow of heat along PQ $\left( \frac{dQ}{dt} \right)_{PQ} = \frac{K_{3}A\Delta\theta}{l}$ ….(i)

Rate of flow of heat along PRQ $\left( \frac{dQ}{dt} \right)_{PRQ} = \frac{K_{s}A\Delta\theta}{2l}$

Effective conductivity for series combination of two rods of

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

So $\left( \frac{dQ}{dt} \right)_{PRQ} = \frac{2K_{1}K_{2}}{K_{1} + K_{2}}.\frac{A\Delta\theta}{2l} = \frac{K_{1}K_{2}}{K_{1} + K_{2}}.\frac{A\Delta\theta}{l}$

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