#### Question

Four identical rods AB, CD, CF and DE are joined as shown in following figure . The length, cross-sectional area and thermal conductivity of each rod are l, A and K respectively. The ends A, E and F are maintained at temperature T_{1}, T_{2} and T_{3} respectively. Assuming no loss of heat to the atmosphere, find the temperature at B.

#### Solution

Let the temperature at junction B be T.

Let *q*_{1}, *q*_{2} and *q*_{3} be the heat currents, i.e. rate of flow of heat per unit time in AB, BCE and BDF, respectively.

From the diagram, we can see that*q*_{1} = *q*_{2} + *q*_{3}

The rate of flow of heat is given by

`q = (KA DeltaT)/l`

Using this tn the above equation, we get

`(A (T_1 - T))/l = (KA ( T + T_3))/(31/2) + (KA (T-T_2))/(31/2)`

`⇒ T_1 - T = (2(T - T_3))/3+(2(T- T_2))/3`

⇒ 3 (T_{1 }- T) = 2T - 2T_{3} + 2T - 2T_{2}

`⇒ T = (-3T_1 + 2( T_2 +T_3))/7`