Question

Show that Planck’s distribution function reduces to Rayleigh-Jeans law and Wien’s law

under appropriate limiting conditions​

Answer 1

Answer:

The Rayleigh-jeans law and Wien's law with experimental results at large wavelengths (low frequencies) but strongly disagrees at short wavelengths (high frequencies). This inconsistency between observations and the predictions of classical physics is commonly known as the ultraviolet catastrophe.

Explanation:

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Answer 2

Answer:

Explanation:

Planck's law states that the spectral radiance, B(λ, T), of a blackbody at a given wavelength, λ, and temperature, T, is given by:

B(λ, T) = (2hc^2)/λ^5 * 1/(e^(hc/λkT) - 1)

Under certain conditions, this equation can reduce to Wien's law and Rayleigh-Jeans law.

Wien's law states that the wavelength at which a blackbody emits the most radiation, λmax, is inversely proportional to the temperature of the blackbody: λmax T = constant ( usually denoted as b)

To get Wien's law from Planck's law, we can take the derivative of B(λ, T) with respect to λ and set it equal to zero, which will give us the maximum wavelength.

λmax = b/T

Rayleigh-Jeans law states that the intensity of radiation emitted by a blackbody is directly proportional to the temperature of the blackbody: I(λ) ∝ T

To get Rayleigh-Jeans law from Planck's law, we can take the limit of Planck's law as λ tends to infinity, where the intensity becomes directly proportional to the temperature:

B(λ, T) = (2hc^2)/λ^4 * T

So, in summary, Planck's law reduces to Wien's law and Rayleigh-Jeans law under certain conditions, Wien's law holds true at high temperatures where the maximum wavelength is inversely proportional to temperature, and Rayleigh-Jeans law holds true at the long wavelength where the intensity is directly proportional to temperature.