I Relation between spectral intensity and spectral energy density

AI Thread Summary
The discussion centers on the derivation of the Planck radiation formula, specifically the relationship between spectral intensity (Iν) and spectral energy density (ρν) in laser physics. It clarifies that Iν is defined as energy per time per area per frequency, while ρν represents energy per volume per frequency. The factor of 1/4 in the equation arises from the integration of energy crossing a surface in a steady-state cavity, accounting for the isotropic nature of radiation, which means only a fraction of the energy contributes to the intensity measured. The explanation emphasizes that the energy leaving through a hole must equal the energy crossing a hemisphere surrounding the hole, justifying the inclusion of the 1/4 factor. The discussion concludes by confirming the understanding of these relationships in the context of laser physics.
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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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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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