Black Body Radiation: Wien's Displacement Law

The question asks for the peak wavelength ($\lambda_{max}$) of radiation emitted by a black body at a specific temperature. This can be determined using Wien's displacement law.

Wien's Displacement Law Formula

Wien's displacement law relates the peak wavelength of emitted radiation to the temperature of the black body:

$ \lambda_{max} = \frac{b}{T} $

Where:

• $ \lambda_{max} $ is the peak wavelength in meters (m).
• $ b $ is Wien's displacement constant, approximately $ 2.898 \times 10^{-3} $ m·K.
• $ T $ is the absolute temperature in Kelvin (K).

Calculating Peak Wavelength

Given the temperature $ T = 4000 $ K:

1. Substitute values into the formula:

   $ \lambda_{max} = \frac{2.898 \times 10^{-3} \text{ m·K}}{4000 \text{ K}} $

2. Perform the calculation:

   $ \lambda_{max} = \frac{2.898 \times 10^{-3}}{4 \times 10^3} \text{ m} $

   $ \lambda_{max} = \frac{2.898}{4} \times 10^{-6} \text{ m} $

   $ \lambda_{max} = 0.7245 \times 10^{-6} \text{ m} $

3. Convert to micrometers ($\mu$m):

   Since $ 1 \text{ μm} = 10^{-6} $ m, the result is:

   $ \lambda_{max} = 0.7245 \text{ μm} $

Conclusion

Comparing the calculated value $ 0.7245 $ $\mu$m to the options provided, it is closest to $ 0.72 $ $\mu$m.