The radiation emitted by a star $A$ is $10000$ times that of the sun. If the surface temperatures of the sun and the star $A$ are $6000 \ K$ and $2000 \ K$ respectively, the ratio of the radii of the star $A$ and the sun is: (in $: 1$)

$A$ black body at a high temperature $T \ K$ radiates energy at the rate $E \ W/m^2$. When the temperature falls to $(T/2) \ K$,the radiated energy will be:

Calculate the surface temperature of the planet,if the energy radiated by unit area in unit time is $5.67 \times 10^4 \, W$. (Assume the planet to be a black body).

$A$ wire of length $10 \ cm$ and diameter $0.5 \ mm$ is used in a bulb. The temperature of the wire is $1727^{\circ} C$ and power radiated by the wire is $94.2 \ W$. Its emissivity is $\frac{x}{8}$ where $x=$ . . . . . . (Given $\sigma=6.0 \times 10^{-8} \ W \ m^{-2} \ K^{-4}$,$\pi=3.14$ and assume that the emissivity of wire material is same at all wavelengths.)

Write the value and dimensional formula of the Stefan-Boltzmann constant.

$A$ black body,at a temperature of $227\,^{\circ}C$,radiates heat at a rate of $7\, cal\, cm^{-2} \,s^{-1}$. At a temperature of $727\,^{\circ}C$,the rate of heat radiated in the same units will be ..... units.