To solve this problem, we need to apply two important principles of thermal radiation: Wien's Displacement Law and Stefan-Boltzmann Law.
Wien's Displacement Law states that the wavelength (\(\lambda_{\text{max}}\)) at which the emission of a black body spectrum is maximum is inversely proportional to the absolute temperature (T) of the black body:
\[\lambda_{\text{max}} \times T = b\]where \(b\) is Wien's constant, given as \(2.9 \times 10^{-3} \text{ mK}\).
Given \(\lambda_{\text{max}} = 2900 \, \text{Å} = 2900 \times 10^{-10} \text{ m}\).
Using Wien's Law:
\(T = \frac{b}{\lambda_{\text{max}}}\)
Substitute the values:
\(T = \frac{2.9 \times 10^{-3}}{2900 \times 10^{-10}} \, \text{K}\) \(T = \frac{2.9 \times 10^{-3}}{2.9 \times 10^{-7}} \, \text{K}\) \(T = 10,000 \, \text{K}\)
Now that we have the temperature, we can use the Stefan-Boltzmann Law, which relates the total energy radiated per unit surface area of a black body in unit time (intensity, \(I\)) to the fourth power of its temperature \(T\):
\(I = \sigma T^4\)
where \(\sigma\) is Stefan-Boltzmann's constant, \(5.67 \times 10^{-8} \, \text{Wm}^{-2}\text{K}^{-4}\).
Substitute the values:
\(I = 5.67 \times 10^{-8} \times (10,000)^4\) \(I = 5.67 \times 10^{-8} \times 10^{16}\) \(I = 5.67 \times 10^{8} \, \text{Wm}^{-2}\)
The intensity of radiation emitted by the perfect radiator is \(5.67 \times 10^{8} \, \text{Wm}^{-2}\). Thus, the correct option is:
\(5.67 \times 10^8\) Wm\(^{-2}\)
A particle is moving in a straight line. The variation of position $ x $ as a function of time $ t $ is given as:
$ x = t^3 - 6t^2 + 20t + 15 $.
The velocity of the body when its acceleration becomes zero is: