Two metallic blocks $M_{1}$ and $M_{2}$ of the same cross-sectional area are connected to each other as shown in the figure. If the thermal conductivity of $M_{2}$ is $K$,then the thermal conductivity of $M_{1}$ will be $xK$. Find the value of $x$. [Assume steady-state heat conduction]

Three rods each of length $l$ and cross-sectional area $A$ are joined in series between two heat reservoirs as shown in the figure. Their thermal conductivities are $2K$,$K$,and $\frac{K}{2}$,respectively. Assuming that the conductors are insulated from the surroundings,the temperatures $T_1$ and $T_2$ of the junctions in the steady-state condition are,respectively:

One end of a thermally insulated rod is kept at a temperature $T_1$ and the other at $T_2$. The rod is composed of two sections of length $l_1$ and $l_2$ and thermal conductivities $K_1$ and $K_2$ respectively. The temperature at the interface of the two sections is

The walls of a closed cubical box of edge $60 \text{ cm}$ are made of material of thickness $1 \text{ mm}$ and thermal conductivity $4 \times 10^{-4} \text{ cal s}^{-1} \text{ cm}^{-1} {}^{\circ}\text{C}^{-1}$. The interior of the box is maintained $1000^{\circ}\text{C}$ above the outside temperature by a heater placed inside the box and connected across a $400 \text{ V}$ $DC$ supply. The resistance of the heater is: (in $\Omega$)

The ratio of the diameters of two metallic rods of the same material is $2 : 1$ and their lengths are in the ratio $1 : 4$. If the temperature difference between their ends is equal,the rate of flow of heat in them will be in the ratio: (in $:1$)

$A$ metal rod of length $10 \text{ cm}$ and area of cross-section $2.8 \times 10^{-4} \text{ m}^2$ is covered with a non-conducting substance. One end of it is maintained at $80^{\circ} \text{C}$,while the other end is put in ice at $0^{\circ} \text{C}$. It is found that $20 \text{ g}$ of ice melts in $5 \text{ min}$. The thermal conductivity of the metal in $\text{J s}^{-1} \text{ m}^{-1} \text{ K}^{-1}$ is (Latent heat of ice is $80 \text{ cal g}^{-1}$.)