A black body has maximum wavelength $\lambda_m$ at $2000\,K$. Its corresponding wavelength at $3000\,K$ will be

If the temperature of the sun were to increase from $T$ to $2T$ and its radius from $R$ to $2R$, then the ratio of the radiant energy received on earth to what it was previously will be

Solar radiation emitted by the sun resembles that emitted by a black body at a temperature of $6000\,K$. Maximum intensity is emitted at wavelength of about $4800{\text{ }}\mathop {\text{A}}\limits^o $. If the sun was cooled down from $6000\,K$ to $3000\,K$, then the peak intensity would occur at a wavelength of …….. $\overset{o}{\mathop{A}}$

Three metal rods of the same material an identical in all respects are joined as shown in the figure. The temperatures at the ends are maintained as indicated. Assuming no loss of heat from the curved surfaces of the rods, the temperature at the junction $X$ would be ……. $^oC$

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 material of thermal conductivity $K_2.$ The two ends of the combined system are maintained at two different temperatures. There is no loss of heat across the cylindrical surface and the system is in steady state. The effective thermal conductivity of the system is