$A$ heat flux of $4000 \, J/s$ passes through a rod of length $10 \, cm$ and cross-sectional area $100 \, cm^2$. The thermal conductivity of copper is $400 \, W/m^{\circ}C$. The ends of the rod must be maintained at a temperature difference of ....... $^{\circ}C$.

The two ends of a rod of length $L$ and a uniform cross-sectional area $A$ are kept at two temperatures $T_1$ and $T_2$ $(T_1 > T_2)$. The rate of heat transfer,$\frac{dQ}{dt}$,through the rod in a steady state is given by:

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 plates $A$ and $B$ have thermal conductivities $84 \, W m^{-1} K^{-1}$ and $126 \, W m^{-1} K^{-1}$ respectively. They have the same surface area and same thickness. They are placed in contact along their surfaces. If the temperatures of the outer surfaces of $A$ and $B$ are kept at $100^{\circ} C$ and $0^{\circ} C$ respectively,then the temperature of the surface of contact in steady state is $.......... \, ^{\circ} C$.

Find the equivalent thermal conductivity of the system.

Three rods of identical cross-sectional area and made from the same metal form the sides of an isosceles triangle $ABC$,right-angled at $B$. The points $A$ and $B$ are maintained at temperatures $T$ and $\sqrt{2}T$ respectively. In the steady state,the temperature of point $C$ is $T_C$. Assuming that only heat conduction takes place,$\frac{T_C}{T}$ is equal to