The temperatures of the two outer surfaces of a composite slab,consisting of two materials having coefficients of thermal conductivity $K$ and $2K$ and thickness $x$ and $4x$,respectively,are $T_2$ and $T_1$ $(T_2 > T_1)$. The rate of heat transfer through the slab in a steady state is $\left( \frac{A(T_2 - T_1)K}{x} \right)f$,where $f$ is equal to:

Two slabs $A$ and $B$ of different materials but of the same thickness are joined end to end to form a composite slab. The thermal conductivities of $A$ and $B$ are $k_1$ and $k_2$ respectively. $A$ steady temperature difference of $12^{\circ} C$ is maintained across the composite slab. If $k_1=\frac{k_2}{2}$, the temperature difference across slab $A$ is (in $^{\circ} C$)

Three rods of the same material and same cross-sectional area,but different lengths $10 \, cm$,$20 \, cm$,and $30 \, cm$,are connected at a junction $O$ as shown in the figure. What is the temperature of the junction $O$ in $^{\circ} C$?

$A$ window used to thermally insulate a room from outside consists of two parallel glass sheets each of area $2.6 \,m^2$ and thickness $1 \,cm$ separated by $5 \,cm$ thick stagnant air. In the steady state, the room-glass interface is at $18^{\circ} C$ and the glass-outdoor interface is at $-2^{\circ} C$. If the thermal conductivities of glass and air are respectively $0.8 \,Wm^{-1} K^{-1}$ and $0.08 \,Wm^{-1} K^{-1}$, the rate of flow of heat through the window is (in $\,W$)

An iron bar $(L_{1}=0.1 \; m, A_{1}=0.02 \; m^{2}, K_{1}=79 \; W m^{-1} K^{-1})$ and a brass bar $(L_{2}=0.1 \; m, A_{2}=0.02 \; m^{2}, K_{2}=109 \; W m^{-1} K^{-1})$ are soldered end to end as shown in the figure. The free ends of the iron bar and brass bar are maintained at $373 \; K$ and $273 \; K$ respectively. Obtain expressions for and hence compute:
$(i)$ the temperature of the junction of the two bars,
$(ii)$ the equivalent thermal conductivity of the compound bar,and
$(iii)$ the heat current through the compound bar.

Two rods of same length and material transfer a given amount of heat in $12 \, s$ when they are joined end to end. But when they are joined lengthwise (parallel),they will transfer the same amount of heat under the same conditions in .......... $s$.