Four rods of identical cross-sectional area and made from the same metal form the sides of a square. The temperatures of two diagonally opposite points are $\theta$ and $\sqrt{2}\theta$ respectively in the steady state. Assuming that only heat conduction takes place,what will be the temperature difference between the other two points?

Find the effective thermal resistance between $A$ and $B$ of a cube made up of $12$ rods of the same dimensions,with the given thermal conductivities as shown in the figure. [ $l =$ length of rod,$a =$ cross-sectional area of rod ]

Two plates of equal area are placed in contact with each other. Their thicknesses are $2.0 \, cm$ and $5.0 \, cm$. The temperature of the outer surface of the first plate is $-20^{\circ}C$ and the temperature of the outer surface of the second plate is $20^{\circ}C$. If the plates are made of the same material,find the temperature of the contact surface in $^{\circ}C$.

In the following figure,two insulating sheets with thermal resistances $R$ and $3R$ are shown. The temperature $\theta$ at the interface is ...... $^oC$.

$A$ hollow sphere of inner radius $R$ and outer radius $2R$ is made of a material of thermal conductivity $K$. It is surrounded by another hollow sphere of inner radius $2R$ and outer radius $3R$ made of the same material of thermal conductivity $K$. The inside of the smaller sphere is maintained at $0^oC$ and the outside of the bigger sphere at $100^oC$. The system is in steady state. The temperature of the interface will be ........ $^oC$.

$A$ composite metal bar of uniform cross-section is made up of a length of $25 \ cm$ of copper,$10 \ cm$ of nickel,and $15 \ cm$ of aluminium. Each part is in perfect thermal contact with the adjoining part. The copper end of the composite rod is maintained at $100^{\circ}C$ and the aluminium end at $0^{\circ}C$. The whole rod is covered with an insulating belt so that no heat loss occurs at the sides. If $K_{\text{Cu}} = 2K_{\text{Al}}$ and $K_{\text{Al}} = 3K_{\text{Ni}}$,what will be the temperatures of the $\text{Cu-Ni}$ and $\text{Ni-Al}$ junctions,respectively?