If the radius of a star is \(R\) and it acts as a black body, what would be the temperature of the star at which the rate of energy production is \(Q\)?
1. \(\frac{Q}{4\pi R^2\sigma}\)
2. \(\left(\frac{Q}{4\pi R^2\sigma}\right )^{\frac{-1}{2}}\)
3. \(\left(\frac{4\pi R^2 Q}{\sigma}\right )^{\frac{1}{4}}\)
4. \(\left(\frac{Q}{4\pi R^2 \sigma}\right)^{\frac{1}{4}}\)

The total radiant energy per unit area, normal to the direction of incidence, received at a distance \(R\) from the centre of a star of radius \(r,\) whose outer surface radiates as a black body at a temperature \(T\) K is given by: (Where \(\sigma\) is Stefan’s constant):
1. \(\frac{\sigma r^{2}T^{4}}{R^{2}}\)

2. \(\frac{\sigma r^{2}T^{4}}{4 \pi R^{2}}\)

3. \(\frac{\sigma r^{2}T^{4}}{R^{4}}\)

4. \(\frac{4\pi\sigma r^{2}T^{4}}{R^{2}}\)

A black body at \(227^{\circ}~\mathrm{C}\) radiates heat at the rate of \(7~ \mathrm{cal-cm^{-2}s^{-1}}\).  At a temperature of \(727^{\circ}~\mathrm{C}\), the rate of heat radiated in the same units will be:
1. \(60\)
2. \(50\)
3. \(112\)
4. \(80\)

For a black body at a temperature of 727ºC, its radiating power is 60 watts and the temperature of the surroundings is 227ºC. If the temperature of the black body is changed to 1227ºC then its radiating power will be:

1. 304 W

2. 320 W

3. 240 W

4. 120 W

Unit of Stefan's constant is:

1. Watt-m²-K⁴

2. Watt-m²/K⁴

3. Watt/m²–K

4. Watt/m² K⁴

A sphere maintained at a temperature of 600 K, has a cooling rate R in an external environment of 200 K temperature. If its temperature falls to 400 K, then its cooling rate will be:

1. $\frac{3}{16}R$

2. $\frac{16}{3}R$

3. $\frac{9}{27}R$

4. None 

Radiation energy corresponding to the temperature T of the sun is E. If its temperature is doubled, then its radiation energy will be:
1. 32 E
2. 16 E
3. 8 E
4. 4 E