The thickness of a metallic plate is $0.4 \ cm$. The temperature difference between its two surfaces is $20^{\circ}C$. The quantity of heat flowing per second is $50 \ \text{calories}$ through an area of $5 \ cm^2$. In the $CGS$ system,the coefficient of thermal conductivity is:

The ends $Q$ and $R$ of two thin wires,$PQ$ and $RS$,are soldered (joined) together. Initially,each of the wires has a length of $1 \,m$ at $10^{\circ} C$. Now,the end $P$ is maintained at $10^{\circ} C$,while the end $S$ is heated and maintained at $400^{\circ} C$. The system is thermally insulated from its surroundings. If the thermal conductivity of wire $PQ$ is twice that of the wire $RS$ and the coefficient of linear thermal expansion of $PQ$ is $1.2 \times 10^{-5} \,K^{-1}$,the change in length of the wire $PQ$ is (in $\,mm$)

One end of a copper rod of uniform cross-section and of length $3.1 \ m$ is kept in contact with ice at $0^{\circ}C$ and the other end with water at $100^{\circ}C$. At what point along its length should a temperature of $200^{\circ}C$ be maintained so that in steady state,the mass of ice melting is equal to the mass of steam produced in the same interval of time? Assume that the whole system is insulated from the surroundings. Latent heat of fusion of ice and vaporisation of water are $80 \ cal/g$ and $540 \ cal/g$ respectively.

The length of a metal rod is $20 \ cm$ and its area of cross-section is $4 \ cm^2$. If one end of the rod is kept at a temperature of $100^{\circ} C$ and the other end is kept in ice at $0^{\circ} C$, then the mass of the ice melted in $7 \ minutes$ is (Thermal conductivity of the metal $= 90 \ W \ m^{-1} \ K^{-1}$ and latent heat of fusion of ice $= 336 \times 10^3 \ J \ kg^{-1}$) (in $g$)

$A$ 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).

An ice box is used to keep food cool,having a surface area of $1 \, m^2$ and a thickness of $5.0 \, cm$. The thermal conductivity of the ice box material is $K = 0.01 \, J/(m \cdot s \cdot ^\circ C)$. The inside temperature is maintained at $0 \, ^\circ C$ and the outside temperature is $30 \, ^\circ C$. If the latent heat of fusion of ice is $334 \times 10^3 \, J/kg$,calculate the mass of ice that melts in one day.