Three rods AB, BC and BD having thermal conductivities in the ratio 1:2:3 and lengths in the ratio 2:1:1 are joined as shown in Fig. The ends A, C and D are at temperature `T_1`,`T_2` and `T_3` respectively Find the temperature of the junction B. Assume steady state.

Explanation

Let the thermal conductivity of the rod AB, BC and BD ber K, 2K, and 3K respectively. Also let the length be 2L, L and L.
If T be the required temperature of the junction B and assuming `T_1gtTgtT_2` and `TgtT_3`, we have
`(DeltaQ)/(Deltat)]_(AB)=(DeltaQ)/(Deltat)]_(AB)+(DeltaQ)/(Deltat)]_(BD)` (Point rule)
`(KA(T_1-T))/(2L)=(2KA(T-T_2))/(L)+(3KA(T-T_3))/(L)`
`(T_1-T)/(2)=2(T-T_2)+3(T-T_3)`
`T=(1)/(11)(T_1+4T_2 + 6T_3)`