Two identical square metal rods are welded end-to-end as shown in figure $(a)$. $20 \text{ calories}$ of heat flows through them in $4 \text{ minutes}$. If they are joined as shown in figure $(b)$,the same amount of heat will flow in ...... minutes.

Two metal plates of equal thickness have thermal conductivities $K_1$ and $K_2$ respectively. They are joined together to form a single plate. The equivalent thermal conductivity of this composite plate is ..........

$A$ cylinder of radius $R$ is made of a material with thermal conductivity $K_1$. It is surrounded by a cylindrical shell of a material with thermal conductivity $K_2$,having an inner radius $R$ and an outer radius $2R$. The two ends of this composite system are maintained at two different temperatures. There is no heat loss from the surface of the cylinder in the steady state. Find the equivalent thermal conductivity of the system.

Two metal slabs of same cross-sectional area have thicknesses $d_1$ and $d_2$,and thermal conductivities $K_1$ and $K_2$ respectively,are connected in series. The free ends of the two slabs are kept at temperatures $T_1$ and $T_2$ $(T_1 > T_2)$. The temperature $T$ of their common junction is

Two identical conducting rods are first connected independently to two vessels,one containing water at $100^{\circ}C$ and the other containing ice at $0^{\circ}C$. In the second case,the rods are joined end to end and connected to the same vessels. Let $q_1$ and $q_2$ $g/s$ be the rate of melting of ice in the two cases respectively. The ratio $q_2/q_1$ is

Three rods $A, B,$ and $C$ of thermal conductivities $K, 2K$ and $4K$,cross-sectional areas $A, 2A$ and $2A$ and lengths $2l, l$ and $l$ respectively are connected as shown in the figure. If the ends of the rods are maintained at temperatures $100\,^oC$,$50\,^oC$,and $0\,^oC$ respectively,then the temperature $\theta$ of the junction is