Three rods AB, BC and BD having thermal conductivities in the ratio 1:2:3and length in the ratio 2:1:1 are joined as shown in figure . 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 conductivites of the rods AB, BC be K,2K and 3K respectively . Also , let their lengths be 2L ,L and L .
If T be the requied tempertaureof the junction B and assuming ` T_(1) gt T gt T_(2) gtT_(3)`
We have `(Delta Q)/(DeltaT)]_(AB) = (DeltaQ)/(Deltat)]_(BC) + (DeltaQ)/(Deltat)]_(BD)` i.e., `(KA(T_(i)-T))/(2L) = (2KA(T-T_(2)))/(L) + (3KA(T-T_(3)))/(L)`
(or) `(T_(1)- T)/(2) = 2(T - T_(2)) + 3(T- T_(1)) (or) T = (1)/(11) (T_(1) + 4T_(2) + 6T_(3))`