Question

If the radiation emitted by a perfect radiator has maximum intensity at a wavelength of 2900 Å, the intensity of radiation emitted by it is (Stefan-Boltzmann's constant = \(5.67 \times 10^{-8}\) Wm\(^{-2}\)K\(^{-4}\) and Wein's constant = \(2.9 \times 10^{-3}\) mK)

• A. \(5.67 \times 10^8\) Wm\(^{-2}\)
• B. \(5.67\) Wm\(^{-2}\)
• C. 5670 Wm\(^{-2}\)
• D. 2.9 Wm\(^{-2}\)

Hint:

This is a two-step problem common in thermal radiation. 1. Use Wien's Law (\(\lambda_{max} T = b\)) to find the temperature from the peak wavelength. 2. Use the Stefan-Boltzmann Law (\(I = \sigma T^4\)) to find the total intensity from the temperature. Make sure all units are in the SI system before calculating.

Answer 1

The Correct Option is A

Step 1: Find the temperature of the perfect radiator using Wien's Displacement Law.
Wien's Law states that \( \lambda_{max} T = b \), where b is Wien's constant.
We are given the wavelength of maximum intensity, \( \lambda_{max} = 2900 \) Å.
Convert this wavelength to meters: \( \lambda_{max} = 2900 \times 10^{-10} \text{ m} = 2.9 \times 10^{-7} \text{ m} \).
Wien's constant is \( b = 2.9 \times 10^{-3} \) mK.
Now, solve for the temperature T.
\( T = \frac{b}{\lambda_{max}} = \frac{2.9 \times 10^{-3}}{2.9 \times 10^{-7}} = 1 \times 10^{(-3 - (-7))} = 10^4 \) K.
Step 2: Find the intensity of radiation using the Stefan-Boltzmann Law.
The Stefan-Boltzmann Law states that the total intensity (or emissive power) I of a perfect radiator (black body) is given by \( I = \sigma T^4 \), where \(\sigma\) is the Stefan-Boltzmann constant.
We are given \( \sigma = 5.67 \times 10^{-8} \) Wm\(^{-2}\)K\(^{-4}\).
We found \( T = 10^4 \) K.
Substitute these values into the formula.
\( I = (5.67 \times 10^{-8}) \times (10^4)^4 \).
\( I = 5.67 \times 10^{-8} \times 10^{16} = 5.67 \times 10^{16-8} = 5.67 \times 10^8 \) Wm\(^{-2}\).