Are I(nu, T) and I(lamda, T) Equivalent in Planck's Law?

In summary, the wikipedia page on Planck's law of black body radiation discusses two different formulas for spectral radiance: I(nu, T) and I(lamda, T). These formulas have different units, J/m^2 and J/s/m^3 respectively. However, they can be seen as equivalent expressions and are commonly used in calculations. It is important to remember the definitions and that I(nu, T) represents density with respect to the frequency scale while I(lamda, T) represents density with respect to the wavelength scale. Using the same notation for both can be confusing, but it is a common practice.
  • #1
nicktacik
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On the wikipeida page http://en.wikipedia.org/wiki/Planck's_law_of_black_body_radiation

two formula's are given for spectral radiance, I(nu, T) and I(lamda,T). However, I(nu, T) seems to have units of J/m^2 and I(lamda, T) seems to have units of J/s/m^3. My homework question is to show that these expression are equivalent, however the fact that the units don't work boggles my mind. Does each expression represent something different? Can someone please shed some light on this?
 
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  • #2
Remember the definitions.
Remember that for example I(nu,T) gives a density with respect to the frequency scale. The units you have indicated could be seen in another light: W/m²/Hz. I like the "/Hz" units! It is convenient to think that: I(nu,T)*d(nu)*Surface is a power.

Observe that using the same notations for I(nu, T) and I(lamda,T) is not really a good idea because these are different functions, but it is common practice.
 
  • #3
Oh, so I(nu, T) is really I per unit frequency and I(lamda, T) is really I per unit wavelength?
 

Related to Are I(nu, T) and I(lamda, T) Equivalent in Planck's Law?

What is Planck's law?

Planck's law, also known as the Planck radiation law, is a fundamental law of physics that describes the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature. It was first proposed by German physicist Max Planck in 1900.

What is the problem with Planck's law?

The problem with Planck's law is that it only accurately describes the spectral density of radiation emitted by a black body at high frequencies (short wavelengths). At low frequencies (long wavelengths), the predictions of Planck's law deviate from experimental observations, a phenomenon known as the "ultraviolet catastrophe."

Why does Planck's law fail at low frequencies?

This failure is due to the assumption made by Planck's law that the energy of radiation is continuous and can take on any value. However, according to the principles of quantum mechanics, energy is quantized and can only take on certain discrete values. This discrepancy leads to an overestimation of the energy at low frequencies, resulting in the ultraviolet catastrophe.

How was the problem with Planck's law solved?

The problem with Planck's law was solved by Albert Einstein in 1905 when he introduced the concept of photons, or packets of energy, to explain the discreteness of energy observed in quantum mechanics. This led to the development of the quantum theory of radiation and a modified version of Planck's law that accurately describes the spectral density of radiation at all frequencies.

What are the implications of the solution to the problem with Planck's law?

The solution to the problem with Planck's law revolutionized our understanding of the behavior of matter and energy at the atomic and subatomic level. It paved the way for the development of quantum mechanics, which has numerous applications in modern technology, including transistors, lasers, and computers.

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