Two cylinders $P$ and $Q$ have the same length and diameter and are made of different materials having thermal conductivities in the ratio $2 : 3$. These two cylinders are combined to make a single cylinder. One end of $P$ is kept at $100^{\circ}C$ and the other end of $Q$ is kept at $0^{\circ}C$. The temperature at the interface of $P$ and $Q$ is ...... $^{\circ}C$.

$A$,$B$ and $C$ are three identical conductors but made from different materials. They are kept in contact as shown. Their thermal conductivities are $K$,$2K$ and $K/2$. The free end of $A$ is at $100^{\circ} C$ and the free end of $C$ is at $0^{\circ} C$. During steady state,the temperature of the junction of $A$ and $B$ is nearly (in $^{\circ} C$)

Two rectangular blocks,having identical dimensions,can be arranged either in configuration $I$ or in configuration $II$ as shown in the figure. One of the blocks has thermal conductivity $k$ and the other $2k$. The temperature difference between the ends along the $x$-axis is the same in both the configurations. It takes $9 \ s$ to transport a certain amount of heat from the hot end to the cold end in the configuration $I$. The time to transport the same amount of heat in the configuration $II$ is: (in $s$)

Three conducting rods of the same material and cross-section are shown in the figure. The temperatures at $A$, $D$, and $C$ are maintained at $20 ^oC$, $90 ^oC$, and $0 ^oC$ respectively. If there is no heat flow in $AB$, find the ratio of the lengths of $BD$ and $BC$.

The temperatures of the two outer surfaces of a composite slab,consisting of two materials having coefficients of thermal conductivity $K$ and $2K$ and thicknesses $x$ and $4x$ respectively,are $T_2$ and $T_1$ $(T_2 > T_1)$. The rate of heat transfer through the slab in a steady state is $\left( \frac{A(T_2 - T_1)K}{x} \right)f$,where $f$ is equal to:

$A$ 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