Two rods $A$ and $B$ of different materials but same cross-section are joined as shown in the figure. The free end of $A$ is maintained at $100^{\circ}C$ and the free end of $B$ is maintained at $0^{\circ}C$. If $l_2 = 2l_1$,$K_1 = 2K_2$ and the rods are thermally insulated from the sides to prevent heat losses,then the temperature $\theta$ of the junction of the two rods is ........ $^{\circ}C$.

$A$ 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 ........ $^oC$.

Two plates of the same thickness with coefficients of thermal conductivities $K_1$ and $K_2$ and cross-sectional areas $A_1$ and $A_2$ are connected as shown. Find the effective coefficient of thermal conductivity $K_{eq}$.

Two identical metal wires of thermal conductivities $K_{1}$ and $K_{2}$ respectively are connected in series. The effective thermal conductivity of the combination is

$A$ wall has two layers $A$ and $B$,each made of different materials. Both layers are of the same thickness. The thermal conductivity of material $A$ is twice that of $B$. If in the steady state,the temperature difference across the wall is $24^{\circ} C$,then the temperature difference across the layer $B$ is (in $^{\circ} C$)

$A$ cylinder of radius $R$ is surrounded by a cylindrical shell of inner radius $R$ and outer radius $2R$. The thermal conductivity of the material of the inner cylinder is $K_1$ and that of the outer cylinder is $K_2$. Assuming no loss of heat,the effective thermal conductivity of the system for heat flowing along the length of the cylinder is