Four rods of identical cross-sectional area and made from the same metal form the sides of a square. The temperatures of two diagonally opposite points are $\theta$ and $\sqrt{2}\theta$ respectively in the steady state. Assuming that only heat conduction takes place,what will be the temperature difference between the other two points?

$A$ wall consists of two layers $A$ and $B$. Both layers have the same thickness but are made of different materials. The thermal conductivity of $A$ is twice that of $B$. In the steady state,the temperature difference across the two ends of the wall is $36^{\circ}C$. What is the temperature difference across the two ends of layer $A$ in $^{\circ}C$?

In a composite rod,two rods of different lengths and the same cross-sectional area are joined end to end. If $K$ is the equivalent coefficient of thermal conductivity of the composite rod,then $\left( \frac{\ell_1 + \ell_2}{K} \right)$ is equal to:

$A$ 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.

For the shown figure,calculate the equivalent thermal resistance if the bricks are made of the same material of thermal conductivity $K$.