Two walls of thicknesses $d_1$ and $d_2$ and thermal conductivities $k_1$ and $k_2$ are in contact. In the steady state,if the temperatures at the outer surfaces are $T_1$ and $T_2$,the temperature at the common wall is

Two thin metallic spherical shells of radii $20 \text{ cm}$ and $30 \text{ cm}$, respectively, are placed with their centers coinciding. $A$ material of thermal conductivity $\alpha$ is filled in the space between the shells. The inner shell is maintained at $300 \text{ K}$ and the outer shell at $310 \text{ K}$. If the rate at which heat flows radially through the material is $40 \text{ W}$, find the value of $\alpha$ (in units of $\text{J s}^{-1} \text{ m}^{-1} \text{ K}^{-1}$).

Copper and iron rods of identical dimensions are coated with wax. The ratio of thermal conductivity of copper to iron is $10:9$. If one end of each rod is placed in hot water,the wax melts. Find the ratio of the lengths of the rods for which the wax melts at the same rate.

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

Four identical conducting rods are joined to form a square $ABCD$. The temperatures of junctions $A, B,$ and $C$ are maintained at $100^{\circ}C, 40^{\circ}C,$ and $0^{\circ}C$ respectively. Choose the $\text{INCORRECT}$ statement for this arrangement in a steady state. (The value of $\frac{KA}{L} = 1 \text{ J}/(s \cdot ^{\circ}C)$).

Two metallic blocks $M_{1}$ and $M_{2}$ of the same cross-sectional area are connected to each other as shown in the figure. If the thermal conductivity of $M_{2}$ is $K$,then the thermal conductivity of $M_{1}$ will be $xK$. Find the value of $x$. [Assume steady-state heat conduction]