A composite rod made of three rods of equal length and cross-section as shown in the fig. The thermal conductivities of the materials of the rods are $K/2, 5K$ and $K$ respectively. The end $A$ and end $B$ are at constant temperatures. All heat entering the face Agoes out of the end $B$ there being no loss of heat from the sides of the bar. The effective thermal conductivity of the bar is

Two rods (one semi-circular and other straight) of same material and of same cross-sectional area are joined as shown in the figure. The points $A$ and $B$ are maintained at different temperature. The ratio of the heat transferred through a cross-section of a semi-circular rod to the heat transferred through a cross section of the straight rod in a given time is

Three rods of identical cross-section and lengths are made of three different materials of thermal conductivity $K _{1}, K _{2},$ and $K _{3}$, respectively. They are joined together at their ends to make a long rod (see figure). One end of the long rod is maintained at $100^{\circ} C$ and the ther at $0^{\circ} C$ (see figure). If the joints of the rod are at  $70^{\circ} C$ and $20^{\circ} C$ in steady state and there is no loss of energy from the surface of the rod, the correct relationship between $K _{1}, K _{2}$ and $K _{3}$ is 

If $K_{1}$ and $K_{2}$ are the thermal conductivities $L_{1}$ and $L _{2}$ are the lengths and $A _{1}$ and $A _{2}$ are the cross sectional areas of steel and copper rods respectively such that $\frac{K_{2}}{K_{1}}=9, \frac{A_{1}}{A_{2}}=2, \frac{L_{1}}{L_{2}}=2$.

Then, for the arrangement as shown in the figure. The value of temperature $T$ of the steel – copper junction in the steady state will be ……….. $^{\circ} C$

The temperature of the two outer surfaces of a composite slab, consisting of two materials having coefficients of thermal conductivity $K$ and $2K$ and thickness $x$ 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 $\left( {\frac{{A({T_2} – {T_1})K}}{x}} \right)f$, with $f $ which equal to