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 $\Omega$)

$A$ heat source at $T_1 = 10^3\, K$ is connected to another heat reservoir at $T_2 = 10^2\, K$ by a copper slab which is $1\, m$ thick. Given that the thermal conductivity of copper is $0.1\, W\, K^{-1}\, m^{-1}$,the energy flux through it in the steady state is ........... $W\, m^{-2}$.

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 lengths $l_1$ and $l_2$ and thermal conductivities $K_1$ and $K_2$ respectively. The temperature at the interface of the two sections is

Two identical containers are filled with the same amount of ice. The containers are made of different metals. If the ice in both containers melts in $20$ minutes and $35$ minutes respectively,find the ratio of their thermal conductivities.

Three rods of identical cross-section and lengths are made of three different materials of thermal conductivity $K_{1}, K_{2},$ and $K_{3}$,respectively. They are joined together at their ends to make a long rod. One end of the long rod is maintained at $100^{\circ} C$ and the other at $0^{\circ} C$. If the joints of the rod are at $70^{\circ} C$ and $20^{\circ} C$ in steady state and there is no loss of energy from the surface of the rod,the correct relationship between $K_{1}, K_{2}$ and $K_{3}$ is:

Three metal rods made of copper, brass, and steel, each with a cross-sectional area of $4 \,cm^2$, are joined as shown in the figure. Their lengths are $46 \,cm, 13 \,cm$, and $12 \,cm$ respectively. Their coefficients of thermal conductivity are $0.92, 0.26$, and $0.12$ respectively, all in $CGS$ units. The rods are thermally insulated from the surroundings except at the ends. The rate of flow of heat through the copper rod, in $cal \,s^{-1}$, is: