$A$ brass rod of length $2 \ m$ and radius $1 \ cm$ has one end maintained at $250 \ ^oC$. When steady state is reached, the rate of heat flow through any cross-section is $0.5 \ cal \ s^{-1}$. Find the temperature of the other end in $^oC$. (Thermal conductivity of brass $K = 0.26 \text{ cal s}^{-1} \text{ cm}^{-1} \text{ } ^\circ\text{C}^{-1}$)

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:

Two concentric spheres of radii $r_1$ and $r_2$ are maintained at temperatures $T_1$ and $T_2$ respectively. The rate of radial heat flow between the two concentric spheres is proportional to:

An ice box used for keeping eatables cold has a total wall area of $1\;m^2$ and a wall thickness of $5.0\;cm$. The thermal conductivity of the ice box material is $K = 0.01\;J/(m\cdot s\cdot ^\circ C)$. It is filled with ice at $0^\circ C$ along with eatables on a day when the temperature is $30^\circ C$. The latent heat of fusion of ice is $334 \times 10^3\;J/kg$. The amount of ice melted in one day is ........ $g$ $(1\;day = 86,400\;s)$.

$A$ brass boiler has a base area of $0.15\; m^{2}$ and thickness $1.0\; cm$. It boils water at the rate of $6.0\; kg/min$ when placed on a gas stove. Estimate the temperature (in $^oC$) of the part of the flame in contact with the boiler. Thermal conductivity of brass $= 109\; J s^{-1} m^{-1} K^{-1}$; Heat of vaporisation of water $= 2256 \times 10^{3}\; J kg^{-1}$.

Find the equivalent thermal conductivity of the system.