$A$ copper rod of length $10 \ cm$ and cross-sectional area $100 \ cm^2$ is to conduct $4000 \ J/s$ of heat. The thermal conductivity of copper is $400 \ W/m \cdot ^\circ C$. The temperature difference between the two ends of the rod must be maintained at ............. $^\circ C$.

Two different rods $A$ and $B$ are kept as shown in the figure. The ratio of thermal conductivities of $A$ and $B$ is

Three rods of the same dimensions have thermal conductivities $3K, 2K$ and $K$. They are arranged as shown in the figure with their ends at $100\,^{\circ}C, 50\,^{\circ}C$ and $20\,^{\circ}C$. The temperature of their junction is....... $^{\circ}C$

Two containers have the same shape and wall thickness but are made of different materials. They are filled with the same amount of ice at $0^{\circ}C$. If the ice melts completely in $10$ minutes and $25$ minutes respectively,what is the ratio of the thermal conductivities of the materials of the containers?

The temperature difference across two cylindrical rods $A$ and $B$ of the same material and same mass are $40^{\circ} C$ and $60^{\circ} C$ respectively. In steady state,if the rates of flow of heat through the rods $A$ and $B$ are in the ratio $3: 8$,the ratio of the lengths of the rods $A$ and $B$ is (in $: 3$)

The coefficient of thermal conductivity of copper is $9$ times that of steel. In the composite cylindrical bar shown in the figure,what will be the temperature at the junction of copper and steel (in $^{\circ} C$)?