Two rods having thermal conductivity in the ratio of $5 : 3$ having equal lengths and equal cross-sectional area are joined by face to face. If the temperature of the free end of the first rod is $100°C$ and free end of the second rod is $20°C$ . Then temperature of the junction is...... $^oC$

The coefficients of thermal conductivity of copper, mercury and glass are respectively $Kc, Km$ and $Kg$ such that $Kc > Km > Kg$ . If the same quantity of heat is to flow per second per unit area of each and corresponding temperature gradients are $Xc, Xm$ and $Xg$ , then

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......... $^oC$

A metallic prong consists of $4$ rods made of the same material, cross-sections and same lengths as shown below. The three forked ends are kept at $100^{\circ} C$ and the handle end is at $0^{\circ} C$. The temperature of the junction is ............. $^{\circ} C$

A hollow sphere of inner radius $R$ and outer radius $2R$ is made of a material of thermal conductivity $K$. It is surrounded by another hollow sphere of inner radius $2R$ and outer radius $3R$ made of same material of thermal conductivity $K$. The inside of smaller sphere is maintained at $0^o C$ and the outside of bigger sphere at $100^o C$. The system is in steady state. The temperature of the interface will be ........ $^oC$

For the figure shown, when arc $ACD$ and $ADB$ are made of same material, the heat carried between $A$ and $B$ is $H$ . If $ADB$ is replaced with another material, the heat carried becomes $2H$ . If the temperatures at $A$ and $B$ are fixed at $T_1$ and $T_2$ , what is the ratio of the new conductivity to the old one of $ADB$