Two conducting cylinders of equal length but different radii are connected in series between two heat baths kept at temperatures $T_1 = 300 \ K$ and $T_2 = 100 \ K$,as shown in the figure. The radius of the bigger cylinder is twice that of the smaller one and the thermal conductivities of the materials of the smaller and the larger cylinders are $K_1$ and $K_2$ respectively. If the temperature at the junction of the two cylinders in the steady state is $200 \ K$,then $K_1 / K_2 = \dots$

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 $\,mm$)

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:

Two sheets of thickness $d$ and $2d$ and the same area are touching each other on their faces. The temperatures $T_A$,$T_B$,and $T_C$ shown are in a geometric progression with a common ratio $r = 2$. Then,the ratio of the thermal conductivity of the thinner sheet to the thicker sheet is:

Three rods of the same dimensions have thermal conductivities $3K, 2K$,and $K$. They are arranged as shown in the figure. The temperature of the junction in the steady state is:

The coefficient of thermal conductivity of a rod depends on its