Dimensional formula for thermal conductivity is (here $K$ denotes the temperature)

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 $\Omega$)

$A$ large cylindrical rod of length $L$ is made by joining two identical rods of copper and steel of length $(\frac{L}{2})$ each. The rods are completely insulated from the surroundings. If the free end of the copper rod is maintained at $100\,^oC$ and that of the steel rod at $0\,^oC$,then the temperature of the junction is........$^oC$ (Thermal conductivity of copper is $9$ times that of steel).

$A$ wall is made up of two layers $A$ and $B$. The thickness of the two layers is the same,but materials are different. The thermal conductivity of $A$ is double that of $B$. In thermal equilibrium,the temperature difference between the two ends is $36^{\circ}C$. Then the difference of temperature at the two surfaces of $A$ will be ....... $^{\circ}C$

Two thin metallic spherical shells of radii $r_{1}$ and $r_{2}$ $(r_{1} < r_{2})$ are placed with their centres coinciding. $A$ material of thermal conductivity $K$ is filled in the space between the shells. The inner shell is maintained at temperature $\theta_{1}$ and the outer shell at temperature $\theta_{2}$ $(\theta_{1} < \theta_{2})$. The rate at which heat flows radially through the material is:

$A$ slab of stone of area $0.36\, m^2$ and thickness $0.1\, m$ is exposed on the lower surface to steam at $100^{\circ} C$. $A$ block of ice at $0^{\circ} C$ rests on the upper surface of the slab. In one hour $4.8\, kg$ of ice is melted. The thermal conductivity of the slab is .......... $J/m/s/^{\circ} C$ (Given latent heat of fusion of ice $= 3.36 \times 10^5\, J/kg$)