Relation between spectral intensity and spectral energy density

In summary, while deriving the Planck radiation formula in Principles of Lasers by Svelto, equation 2.2.3 states that the spectral intensity at a hole in the cavity wall is equal to the speed of light in vacuum divided by four times the refractive index of the medium and multiplied by the spectral energy density inside the cavity. This is in the case of a monochromatic wave propagating in one direction, where the energy per unit time passing through a given area must come from the volume filled with the energy density. The factor of 1/4 in the equation is justified by considering the energy leaving through a circular hole of radius ##\delta r## per unit time, which must be equal to the energy crossing
  • #1
IcedCoffee
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TL;DR Summary
How is the following relation between spectral intensity and spectral energy density derived?
In Principles of Lasers by Svelto, while deriving the Planck radiation formula, equation 2.2.3 says $$I_{\nu} = \frac {c_0} {4n} \rho_\nu$$
where ##I_\nu## is the spectral intensity at some hole in the cavity wall (energy per time per area per frequency),
##c_0## is the speed of light in vacuum,
##n## is the refractive index of the medium inside the cavity,
and ##\rho_\nu## is the spectral energy density inside the cavity (energy per volume per frequency).

I understand that in the case of monochromatic wave propagating in one direction, ##I = \frac {c_0} {n} \rho##
since multiplying both sides by ##dt## would give the amount of energy passing through a given area (perpendicular to the direction of propagation),
which must originate from the volume filled with the energy density ##\rho##.

However, in this case, where does the factor of ## \frac 1 4## come from?
 
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  • #2
Here's a picture:
1616499347556.png


Since the entire cavity is in a steady-state, the energy leaving through the circular hole of radius ##\delta r## per unit time must be equal to the energy crossing the hemisphere ##S_2## per unit time. (I guess we are also saying that the medium does not exist between ##S_2## and the hole?) Now, in time ##dt## such energy must come from the shaded area. Since radiation inside the cavity is not directional like plane wave, only a fraction of energy inside the shaded area will cross the surface ##S_2##.

In such integration, how do I justify the factor 1/4 ?
 
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Related to Relation between spectral intensity and spectral energy density

1. What is spectral intensity?

Spectral intensity refers to the amount of energy per unit time per unit area that is emitted or absorbed by a material at a specific wavelength or frequency.

2. How is spectral intensity measured?

Spectral intensity is typically measured using a spectrometer, which measures the intensity of light at different wavelengths. The resulting data can be plotted on a graph to show the spectral intensity distribution.

3. What is spectral energy density?

Spectral energy density refers to the amount of energy per unit volume that is contained within a specific wavelength or frequency range. It is a measure of how much energy is present in a given spectral region.

4. How are spectral intensity and spectral energy density related?

The spectral intensity at a specific wavelength is directly proportional to the spectral energy density at that wavelength. This means that as the spectral intensity increases, the spectral energy density also increases.

5. Why is the relationship between spectral intensity and spectral energy density important?

Understanding the relationship between spectral intensity and spectral energy density is important in many fields of science, including astronomy, spectroscopy, and materials science. It allows us to analyze the properties of different materials and to study the behavior of light in different environments.

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