If the thermal conductivity of aluminum is $0.5 \ cal/cm \cdot s \cdot ^\circ C$,then the temperature gradient required to conduct $10 \ cal/s \cdot cm^2$ in the steady state is ...... $^\circ C/cm$.

$A$ 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).

Two bars of thermal conductivities $K$ and $3K$ and lengths $1\, cm$ and $2\, cm$ respectively have equal cross-sectional area. They are joined length-wise as shown in the figure. If the temperatures at the ends of this composite bar are $0\,^{\circ}C$ and $100\,^{\circ}C$ respectively,then the temperature $\varphi$ of the interface is......... $^oC$.

Two rods are connected as shown. The rods are of the same length and same cross-sectional area. In the steady state,the temperature $(\theta)$ of the interface will be........ $^oC$

Steam at $373 K$ is passed through a tube of radius $10 cm$ and length $2 m$. The thickness of the tube is $5 mm$ and the thermal conductivity of its material is $390 W m^{-1} K^{-1}$. Calculate the heat lost per second. The outside temperature is $0^{\circ}C$.

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