Q: Three rods of same dimension s are arranged as shown in figure. They have thermal conductivities K_{1}, K_{2} and K_{3}. 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?

(a) $\displaystyle K_3 = \frac{1}{2}(K_1 + K_2) $

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

(c) $ \displaystyle K_3 = \frac{K_1 K_2}{K_1 + K_2} $

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

Ans:(c)

$ \displaystyle Thermal \, Resistance \propto \frac{1}{Thermal \, conductivity} $

$ \displaystyle R = \frac{\lambda}{K} $

$ \displaystyle R_1 = \frac{\lambda}{K_1} \, R_2 = \frac{\lambda}{K_2} \, R_3 = \frac{\lambda}{K_3} $

R_{PRQ} = R_{1} + R_{2}

R_{PQ} = R_{3}

θ_{P} – θ_{Q} = i R_{PRQ} = i R_{PQ}

R_{1} + R_{2} = R_{3}

$ \displaystyle \frac{\lambda}{K_1} + \frac{\lambda}{K_2} = \frac{\lambda}{K_3} $

$ \displaystyle \frac{1}{K_1} + \frac{1}{K_2} = \frac{1}{K_3} $

$ \displaystyle K_3 = \frac{K_1 . K_2}{K_1 + K_2 } $