Two materials having coefficients of thermal conductivity $3K$ and $K$ and thickness $d$ and $3d$,respectively,are joined to form a slab as shown in the figure. The temperatures of the outer surfaces are $\theta_2$ and $\theta_1$ respectively $(\theta_2 > \theta_1)$. The temperature at the interface is

What is the temperature (in $^oC$) of the steel-copper junction in the steady state of the system shown in the figure? Length of the steel rod $= 15.0 \; cm,$ length of the copper rod $= 10.0 \; cm,$ temperature of the furnace $= 300^{\circ} C,$ temperature of the other end $= 0^{\circ} C.$ The area of cross-section of the steel rod is twice that of the copper rod. (Thermal conductivity of steel $= 50.2 \; J s^{-1} m^{-1} K^{-1};$ and of copper $= 385 \; J s^{-1} m^{-1} K^{-1}$)

Mud houses are cooler in summer and warmer in winter because

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

$A$ body of length $1 \; m$ having a cross-sectional area of $0.75 \; m^2$ has heat flow through it at the rate of $6000 \; J/s$. Find the temperature difference if the thermal conductivity $K = 200 \; J \cdot m^{-1} \cdot s^{-1} \cdot K^{-1}$.

Two containers have the same shape and wall thickness but are made of different materials. They are filled with the same amount of ice at $0^{\circ}C$. If the ice melts completely in $10$ minutes and $25$ minutes respectively,what is the ratio of the thermal conductivities of the materials of the containers?