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 \(Q?\)
\(\left(\sigma~ \text{is Stefan-Boltzmann constant}\right)\)
1. \(\dfrac{Q}{4\pi R^2\sigma}\)

2. \(\left(\dfrac{Q}{4\pi R^2\sigma}\right )^{\dfrac{-1}{2}}\)

3. \(\left(\dfrac{4\pi R^2 Q}{\sigma}\right )^{\dfrac{1}{4}}\)

4. \(\left(\dfrac{Q}{4\pi R^2 \sigma}\right)^{\dfrac{1}{4}}\)

A spherical black body with a radius of \(12\) cm radiates \(450\)-watt power at \(500\) K. If the radius were halved and the temperature doubled, the power radiated in watts would be:
1. \(225\)
2. \(450\)
3. \(1000\)
4. \(1800\)

If the sun’s surface radiates heat at \(6.3\times 10^{7}~\text{Wm}^{-2}\) ${\mathrm{}}^{}$then the temperature of the sun, assuming it to be a black body, will be:
\(\left(\sigma = 5.7\times 10^{-8}~\text{Wm}^{-2}\text{K}^{-4}\right)\)
1. \(5.8\times 10^{3}~\text{K}\)
2. \(8.5\times 10^{3}~\text{K}\)
3. \(3.5\times 10^{8}~\text{K}\)
4. \(5.3\times 10^{8}~\text{K}\)