# Why there are three equations governing blackbody emission?

I learnt that the spectrum of a black body can be explained using

1. Planck distribution
2. Rayleigh -Jeans equation and
3. Weins displacement law

Can anyone tell me why there are three different equations that explain the same physical phenomena and when should I consider which equation?

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## 1 Answer

Historically, two people (or groups of people) independently came up with different equations to model the blackbody equations in different parts of the spectrum. Rayleigh-Jeans law (classically derived) is valid for longer wavelengths and Wien's law (not Wien's displacement law) is valid for shorter wavelengths.

The Planck Distribution approaches the two laws in its limits, as in, for shorter wavelengths it is approximately equivalent to Wien's law and for longer wavelengths, it is approximately equivalent to Rayleigh-Jeans law. However, the quantum mechanically formulated Planck's law is accurate at all wavelengths, so it is the one that should always be used. One can use the other laws, for example, for brightness temperature definitions in radio astronomy, they use the RJ law for convenience, but that's because the wavelengths are long enough, and the approximation of the Planck's law gives the same result. (https://en.wikipedia.org/wiki/Planck%27s_law#Approximations)

Wien's displacement law only relates the peak wavelength to the temperature, which is a different law completely, though Planck's law (differentiating the function to find the maximum) also gives the same result. Though in this case, the displacement law is not approximate. It is accurate and can be used whenever you want.

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Thanks that makes things clear. But still I have a doubt. If Wein's law is accurate then why is it called wein approximation? (see here: en.wikipedia.org/wiki/Wien_approximation) –  Vivek V K Jul 5 '14 at 14:17
Also why do you say that the displacement law is a totally different law when It is shown in the same page I mentioned above as a special case of planck's law? –  Vivek V K Jul 5 '14 at 14:19
Wien's law is not accurate! It's only approximately close to the Planck's law for shorter wavelengths! Hence the term approximation. Also, Wien's displacement law is different from the other three, as in it only describes the relation between the peak wavelength and the temperature of the blackbody. $\lambda_{max} T = const$ (en.wikipedia.org/wiki/Wien's_displacement_law) The other 3 laws talk about the spectrum of a black body, i.e. the intensity as a function of wavelength/frequency, and Wien's law is one of them. (en.wikipedia.org/wiki/Wien_approximation) –  Takku Jul 5 '14 at 15:01
yeah that clears it. thanks :) –  Vivek V K Jul 5 '14 at 15:07