The temperature $\theta$ at the junction of two insulating sheets,having thermal resistances $R_{1}$ and $R_{2}$ as well as top and bottom temperatures $\theta_{1}$ and $\theta_{2}$ (as shown in figure) is given by

$A$ composite metal bar of uniform cross-section is made up of a length of $25 \ cm$ of copper,$10 \ cm$ of nickel,and $15 \ cm$ of aluminium. Each part is in perfect thermal contact with the adjoining part. The copper end of the composite rod is maintained at $100^{\circ}C$ and the aluminium end at $0^{\circ}C$. The whole rod is covered with an insulating belt so that no heat loss occurs at the sides. If $K_{\text{Cu}} = 2K_{\text{Al}}$ and $K_{\text{Al}} = 3K_{\text{Ni}}$,what will be the temperatures of the $\text{Cu-Ni}$ and $\text{Ni-Al}$ junctions,respectively?

Six identical rods are arranged as shown in the figure. The temperature of the junction $B$ will be......... $^oC$

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

An insulated container is filled with ice at $0\,^{\circ}\text{C}$,and another container is filled with water that is continuously boiling at $100\,^{\circ}\text{C}$. In a series of experiments,the containers are connected by various thick metal rods that pass through the walls of the container as shown in the figure.
In experiment $I$: a copper rod is used and all ice melts in $20$ minutes.
In experiment $II$: a steel rod of identical dimensions is used and all ice melts in $80$ minutes.
In experiment $III$: both the rods are used in series and all ice melts in $t_{10}$ minutes.
In experiment $IV$: both rods are used in parallel and all ice melts in $t_{20}$ minutes.

Two walls have thicknesses $d_1$ and $d_2$ and thermal conductivities $k_1$ and $k_2$,respectively. In the steady state,the temperatures of the outer surfaces are $T_1$ and $T_2$. What is the temperature at the interface of the two walls?