Two spheres made of different materials have radii in the ratio $2:1$ and wall thicknesses in the ratio $1:4$. They are completely filled with ice. If the time taken to melt the ice completely in the larger sphere is $25 \ min$ and in the smaller sphere is $16 \ min$,then the ratio of the thermal conductivities of the materials of the larger and smaller spheres is ..........

There are two identical vessels filled with equal amounts of ice. The vessels are made of different metals. If the ice melts in the two vessels in $20$ and $35$ minutes respectively,the ratio of the coefficients of thermal conductivity of the two metals is:

$A$ $5 \ cm$ thick ice block is present on the surface of water in a lake. The temperature of the air is $-10^{\circ}C$. How much time will it take to double the thickness of the block? (Given: $L = 80 \ cal/g$,$K_{ice} = 0.004 \ cal/s \cdot cm \cdot ^{\circ}C$,$\rho_{ice} = 0.92 \ g/cm^3$)

Rate of flow of heat through a cylindrical rod is $H_1$. The temperatures at the ends of the rod are $T_1$ and $T_2$. If all the dimensions of the rod become double and the temperature difference remains the same,the rate of flow of heat becomes $H_2$. Then:

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.

$A$ 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 ........... $W\, m^{-2}$.