Two metal cubes $A$ and $B$ of the same size are arranged as shown in the figure. The extreme ends of the combination are maintained at the indicated temperatures. The arrangement is thermally insulated. The coefficients of thermal conductivity of $A$ and $B$ are $300 \; W/m^{\circ}C$ and $200 \; W/m^{\circ}C$,respectively. After steady state is reached,the temperature of the interface will be ...... $^{\circ}C$.

$A$ 'thermacole' icebox is a cheap and an efficient method for storing small quantities of cooked food in summer in particular. $A$ cubical icebox of side $30 \,cm$ has a thickness of $5.0 \,cm$. If $4.0 \,kg$ of ice is put in the box,estimate the amount of ice (in $kg$) remaining after $6 \,h$. The outside temperature is $45 \,^{\circ}C$,and the coefficient of thermal conductivity of thermacole is $0.01 \,J \,s^{-1} \,m^{-1} \,K^{-1}$. Heat of fusion of water $= 335 \times 10^{3} \,J \,kg^{-1}$.

The two ends of a rod of length $L$ and a uniform cross-sectional area $A$ are kept at two temperatures $T_{1}$ and $T_{2}$ $(T_{1} > T_{2})$. The rate of heat transfer,$\frac{dQ}{dt}$ through the rod in a steady state is given by:

$1.56 \times 10^5 \ J$ of heat is conducted through a $2 \ m^2$ wall of $12 \ cm$ thickness in one hour. The temperature difference between the two sides of the wall is $20^{\circ} C$. The thermal conductivity of the material of the wall is (in $W \ m^{-1} \ K^{-1}$):

Heat is conducted across a composite block of two slabs of thickness $d$ and $2d$. Their thermal conductivities are $2k$ and $k$ respectively. All the heat entering the face $AB$ leaves from the face $CD$. The temperature in $^oC$ of the junction $EF$ of the two slabs is:

The ends $Q$ and $R$ of two thin wires,$PQ$ and $RS$,are soldered (joined) together. Initially,each of the wires has a length of $1 \,m$ at $10^{\circ} C$. Now,the end $P$ is maintained at $10^{\circ} C$,while the end $S$ is heated and maintained at $400^{\circ} C$. The system is thermally insulated from its surroundings. If the thermal conductivity of wire $PQ$ is twice that of the wire $RS$ and the coefficient of linear thermal expansion of $PQ$ is $1.2 \times 10^{-5} \,K^{-1}$,the change in length of the wire $PQ$ is (in $\,mm$)