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 flow at the same rate alig PRQ and PQ which of the following option is correct?

Options

A. `K_3=(1)/(2)(K_1+K_2)`
B. `K_3=K_1+K_2`
C. `K_3=(K_1K_2)/(K_1+K_2)`
D. `K_3=2(K_1+K_2)`

Answer

C

Explanation

Rate of flow of heat along PQ
`((dQ)/(dt))_(PQ)=(K_3ADeltatheta)/(l)` .(i)
Rate of flow of heat along PRQ
`((dQ)/(dt))_(ORQ)=(K_SaDeltatheta)/(2l)`
Effective conductivity for series combination of two rods of same length
`K_S=(2K_1K_2)/(K_1+K_2)`
So `((dQ)/(dt))_(PRQ)=(2K_1K_2)/(K_1+K_2)(ADeltatheta)/(2l)=(K_1K_2)/(K_1+K_2)(ADeltatheta)/(l)` .(ii)
Equating Eq. (i) and (ii) `K_3=(K_1K_2)/(K_1+K_2)`