The figure shows the cross-section of the outer wall of a house buit in a hill-resort to keep the house insulated from the freezing temperature of outside. The wall consists of teak wood of thickness `L_(1)` and brick of thickness `(L_(2) = 5L_(1))`, sandwitching two layers of an unknown material with identical thermal conductivites and thickness. The thermal conductivity of teak wood is `K_(1)` and that of brick is `(K_(2) = 5K)`. Heat conducion through the wall has reached a steady state with the temperature of three surfaces being known. `(T_(1) = 25^(@)C, T_(2) = 20^(@)C` and `T_(5) = - 20^(@)C)`. Find the interface temperature `T_(4)` and `T_(3)`.

Explanation

Let interface area be A. the thermal resistance of wood.
`R_1=(L_1)/(K_1 A)`
And that of brick wall
`R_2=(L_2)/(K_2 A)=(5L)/(5K_1 A)=R_1`
Let thermal resistance of the each sand witch layer = R. Then the above wall can be visualized as a circuit

thermal current through each wall is same.
Hence `(25-20)/(R_1)=(20-T_3)/(R)=(T_3-T_4)/(R)=(T_4+20)/(R_1)`
`rArr 25-20 = T_4+20`
`rArr T_4=-15^@ C`
also, `20-T_3=T_3-T_4`
`rArr T_3=(20+T_4)/(2)=2.5^@ C`