Question

In the radiation emitted by a black body, the ratio of the spectral densities at frequencies \( 2\nu \) and \( \nu \) will vary with \( \nu \) as:

• A. \( \left[ e^{h\nu / k_B T} - 1 \right]^{-1} \)
• B. \( \left[ e^{h\nu / k_B T} + 1 \right]^{-1} \)
• C. \( \left[ e^{h\nu / k_B T} - 1 \right] \)
• D. \( \left[ e^{h\nu / k_B T} + 1 \right] \)

Hint:

For black body radiation, the ratio of the spectral densities at different frequencies can be found by applying Planck's law and using the exponential dependence of the intensity.

Answer 1

The Correct Option is A

Step 1: Understanding black body radiation.
The spectral density of radiation emitted by a black body is governed by Planck's law. The ratio of the spectral densities at frequencies \( 2\nu \) and \( \nu \) follows the form \( \frac{B(2\nu)}{B(\nu)} \), where \( B(\nu) \) is the Planck radiation formula. The ratio depends on the exponential terms in the denominator, leading to the form \( \left[ e^{h\nu / k_B T} - 1 \right]^{-1} \).
Step 2: Conclusion.
Thus, the correct answer is option (A).