$A$ rod of length $l$ with thermally insulated lateral surface is made of a material whose thermal conductivity $K$ varies as $K = C/T$,where $C$ is a constant. The ends are at temperatures $T_1$ and $T_2$. The heat current density is

The thickness of a uniform rectangular metal plate is $5 \ mm$ and the area of each surface is $5 \ cm^2$. In steady state, the temperature difference between the two surfaces of the plate is $14^{\circ} C$. If the heat flowing through the plate in one second from one surface to the other surface is $42 \ J$, then the thermal conductivity of the metal is

$A$ cylindrical copper rod of length $2 \,m$ and cross-sectional area $2 \,cm^2$ is insulated at its curved surface. One end of the rod is maintained in a steam chamber at $100^{\circ} C$ and the other is maintained in ice at $0^{\circ} C$. The thermal conductivity of copper is $386 \,Js^{-1} \,m^{-1} {}^{\circ} C^{-1}$. Find the temperature at a point which is at a distance of $120 \,cm$ from the colder end. (in $^{\circ} C$)

Two metal cubes $A$ and $B$ of the same size are arranged as shown in the figure. The ends of the combination are maintained at the indicated temperatures. The arrangement is thermally insulated. If the thermal conductivities of $A$ and $B$ are $300 \, W/m \, ^\circ C$ and $200 \, W/m \, ^\circ C$ respectively,find the temperature $t$ at the junction when the steady state is reached.

In which case does the thermal conductivity increase from left to right?

$A$ wall has two layers $A$ and $B$,each made of a different material. Both the layers have the same thickness. The thermal conductivity of the material of $A$ is twice that of $B$. Under thermal equilibrium,the temperature difference across the wall is $36\,^{\circ}C$. The temperature difference across the layer $A$ is ......... $^{\circ}C$