If the temperature of the sun is doubled,the rate of energy received by the earth will be increased by a factor of:

$A$ star emits radiation with maximum intensity at a wavelength of $289.9 \, nm$. What is the intensity of the radiation emitted by the star? (Given: Stefan's constant $\sigma = 5.67 \times 10^{-8} \, W/m^2K^4$,Wien's constant $b = 2898 \, \mu m \cdot K$)

Suppose the sun expands so that its radius becomes $100$ times its present radius and its surface temperature becomes half of its present value. The total energy emitted by it then will increase by a factor of

$A$ black sphere has radius $R$ whose rate of radiation is $E$ at temperature $T$. If the radius is made $\frac{R}{3}$ and the temperature is made $3T$,the rate of radiation will be:

Two spheres made of the same material have radii in the ratio $1:2$. Both are at the same temperature. The ratio of heat radiation energy emitted per second by them is:

Two bodies $A$ and $B$ at temperatures $T_1 \ K$ and $T_2 \ K$ respectively have the same dimensions. Their emissivities are in the ratio $16:1$. At $T_1 = x T_2$,they radiate the same amount of heat per unit area per unit time. The value of $x$ is