One end of a thermally insulated rod is kept at a temperature $T_1$ and the other at $T_2$ . The rod is composed of two sections of length $l_1$ and $l_2$ and thermal conductivities $K_1$ and $K_2$ respectively. The temperature at the interface of the two section is

Bottom of a lake is at $0^{\circ} C$ and atmospheric temperature is $-20^{\circ} C$. If $1 cm$ ice is formed on the surface in $24 \,h$, then time taken to form next $1 \,cm$ of ice is ......... $h$

Two bars of thermal conductivities $K$ and $3K$ and lengths $1\,\, cm$ and $2\,\, cm$ respectively have equal cross-sectional area, they are joined lengths wise as shown in the figure. If the temperature at the ends of this composite bar is $0\,^oC$ and $100\,^oC$ respectively (see figure), then the temperature $\varphi $ of the interface is......... $^oC$

Surface of the lake is at $2^{\circ} C$. The temperature of the bottom of the lake is ....... $^{\circ} C$

A metallic rod of cross-sectional area $9.0\,\,cm^2$ and length $0.54 \,\,m$, with the surface insulated to prevent heat loss, has one end immersed in boiling water and the other in ice-water mixture. The heat conducted through the rod melts the ice at the rate of $1 \,\,gm$ for every $33 \,\,sec$. The thermal conductivity of the rod is ....... $ Wm^{-1} K^{-1}$

In the Ingen Hauz’s experiment the wax melts up to lengths $10$ and $25 cm$ on two identical rods of different materials. The ratio of thermal conductivities of the two materials is