Assuming the sun to be a spherical body of radius $R$ at a temperature of $T$ $K$, evaluate the total radiant power, incident on earth, at a distance $r$ from the sun- (when radius of earth is $r_0$)

The temperature of a black body reduces to half of its actual value. By what fraction will the amount of radiations given out reduce?

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

A cylindrical metallic rod, in thermal contact with two reservoirs of heat at its two ends, conducts an amount of heat $Q$ in time $t.$ The metallic rod is melted and the  material is formed into a rod of half the radius of the original rod. Amount of heat  conducted by the new rod, when placed in thermal contact with same two reservoirs in  time $t$ , is

A cylinder of radius $R$ made of a material of thermal conductivity$ k_1$ is surrounded by a cylindrical shell of inner radius $R$ and outer radius $2R$ made of a material of thermal conductivity $k_2$. The two ends of the combined system are maintained at different temperatures. There is no loss of heat from the cylindrical surface and the system is in steady state. The effective thermal conductivity of the system is