Three rods made of the same material and having the same cross-sectional area but different lengths $10 \ cm$,$20 \ cm$,and $30 \ cm$ are joined as shown. The temperature of the joint is ....... $^oC$.

Copper and iron rods of identical dimensions are coated with wax. The ratio of thermal conductivity of copper to iron is $10:9$. If one end of each rod is placed in hot water,the wax melts. Find the ratio of the lengths of the rods for which the wax melts at the same rate.

Two rods $A$ and $B$ of the same cross-sectional area $A$ and length $l$ are connected in series between a source $(T_1 = 100^{\circ}C)$ and a sink $(T_2 = 0^{\circ}C)$ as shown in the figure. The rods are laterally insulated. If $G_A$ and $G_B$ are the temperature gradients across rod $A$ (thermal conductivity $3K$) and rod $B$ (thermal conductivity $K$),then:

Two slabs have thicknesses $d_{1}$ and $d_{2}$. Their thermal conductivities are $K_{1}$ and $K_{2}$ respectively. They are in series. The free ends of the combination of these two slabs are kept at temperatures $\theta_{1}$ and $\theta_{2}$. Assume $\theta_{1} > \theta_{2}$. The temperature $\theta$ of their common junction is

Two identical rods of copper and iron are coated with wax uniformly. When one end of each is kept at the temperature of boiling water,the lengths up to which wax melts are $8.4 \ cm$ and $4.2 \ cm$ respectively. If the thermal conductivity of copper is $0.92$,then the thermal conductivity of iron is:

Two thin metallic spherical shells of radii $r_{1}$ and $r_{2}$ $(r_{1} < r_{2})$ are placed with their centres coinciding. $A$ material of thermal conductivity $K$ is filled in the space between the shells. The inner shell is maintained at temperature $\theta_{1}$ and the outer shell at temperature $\theta_{2}$ $(\theta_{1} < \theta_{2})$. The rate at which heat flows radially through the material is: