Three rods AB, BC and BD havingthermal conductivitiesin the ratio 1:2:3and lengthin the ratio 2:1:1 are joined as shown in figure . The ends A , C and Dare at temperature T_(1), T_(2) and T_(3) respectively . Find the temperature of the junctionB . Assume steady state .

Three rods AB, BC and BD havingthermal conductivitiesin the ratio 1:2:

1.	Three rods AB, BC and BD havingthermal conductivitiesin the ratio 1:2:3and lengthin the ratio 2:1:1 are joined as shown in figure . The ends A , C and Dare at temperature T_(1), T_(2) and T_(3) respectively . Find the temperature of the junctionB . Assume steady state .
Answer» SOLUTION :Let the thermal conductivites of the rods AB, BC be K,2K and 3K respectively . Also, lettheirlengths be 2L ,L and L . IfT be the requied tempertaureof the junctionB 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))`