$A$ 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 ....... $^{\circ}C$.

Two plates of equal area are placed in contact. Their thicknesses are $2$ and $3$ units respectively. The temperature of the outer surface of the first plate is $-25^{\circ}C$ and the temperature of the outer surface of the second plate is $25^{\circ}C$. Find the temperature of the contact surface if: $(a)$ they are made of the same material,$(b)$ their thermal conductivities are in the ratio $2:3$.

Find the equivalent thermal conductivity of the system.

$A$ metal rod of length $2\, m$ has cross-sectional areas $2A$ and $A$ as shown in the figure. The two ends are maintained at temperatures $100\,^{\circ}C$ and $70\,^{\circ}C$. The temperature of the junction point $C$ is ........ $^{\circ}C$.

The ratio of the diameters of two metallic rods of the same material is $2 : 1$ and their lengths are in the ratio $1 : 4$. If the temperature difference between their ends is equal,the rate of flow of heat in them will be in the ratio: (in $:1$)

$A$ rectangular slab consists of two cubes of copper and brass of equal sides having thermal conductivities in the ratio $4: 1$. If the free face of brass is at $0^{\circ} C$ and that of copper is at $100^{\circ} C$,then the temperature of their interface is (in $^{\circ} C$)