# Three rods of same dimension s are arranged as shown in figure. They have thermal conductivities K1, K2 and K3.

Q: Three rods of same dimension s 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. Which of the following options is correct? (a) $\displaystyle K_3 = \frac{1}{2}(K_1 + K_2)$

(b) K3 = K1 + K2

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

(d) K3 = 2( K1 + K2)

Ans:(c)

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

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

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

RPRQ = R1 + R2

RPQ = R3

θP – θQ = i RPRQ = i RPQ

R1 + R2 = R3

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

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

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