Three rods of same dimensions are arranged as shown in Fig. They have thermal conductivities `K_1`,`K_2` and `K_3`. The points P and Q are maintained at different temperature for the heat to flow at the same rate along PRQ and PQ. Which of the following options correct?

Options

A. `K_(3) = (1)/(2) (K_(1) + K_(2))`
B. `K_(3) = K_(1) + K_(2)`
C. `K_(3) = (K_(1) K_(2))/(K_(1) + K_(2))`
D. `K_(3) = 2 (K_(1) + K_(2))`

Answer

C

Explanation

(3) `R_(1) = (lambda)/(K_(1)), R_(2) - (lambda)/(K_(2)), R_(3) = (lambda)/(K_(3))`
`R_(PRQ) = R_(1) + R_(2)`
`R_(PQ) = R_(3)`
`theta_(P) - theta_(Q) = i R_(PRQ) = i R_(PQ)`
`R_(1) + R_(2) = R_(3)`
`(lambda)/(K_(1)) + (lambda)/(K_(2)) = (lambda)/(K_(3))`
`K_(3) = (K_(1) K_(2))/(K_(1) + k_(2))`