Three rods `AB,BC`and `BD` having thremal conductivities in the ratio `1:2:3` and lengths in the ratio `2:1:1` are joined as shown in The ends `A,C` and `D` are at temperature `theta_(1),theta_(2)` and `theta_(3)` respectively Find the temperature of the junction `B`
(Assume steady state and `theta_(1)gt theta gt theta_(2)gttheta_(3))`
.

Explanation

Let the thermal conductivities of the rods `AB,BC` and `BD` be `K,2K` and `3K` respectively and their lenths be `2L,L` and `L` we have
`[(DeltaQ)/(Deltat)]_(AB)=[(DeltaQ)/(Deltat)]_(BC)+[(DeltaQ)/(Deltat)]_(BD)`
`i.e(KA(theta_(1)-theta))/(2L) =(2KA(theta-theta_(2)))/(L)+(3KA(theta-theta_(3)))/(L)`
`:.theta=(1)/(11) (theta_(1)+4theta_(2)+6theta_(3))` .