Radiation distribution formula Planck

Figure 4-5. Wavelength distributions of the sun s photons incident on the earth s atmosphere and its surface. The curve for the solar irradiation on the atmosphere is an idealized one based on Planck s radiation distribution formula (Eq. 4.3a). The spectral distribution and the amount of solar irradiation reaching the earth s surface depend on clouds, other atmospheric conditions, altitude, and the sun s angle in the sky. The pattern indicatedby the lower curve is appropriate at sea level on a clear day with the sun overhead.

The shape of the curve depicting the wavelength distribution of photons incident upon the earth s atmosphere can be closely predicted using Planck s radiation distribution formula ... 

If we know the surface temperature of a blackbody, we can predict the wavelength for maximal radiation from it. To derive such an expression, we differentiate Planck s radiation distribution formula with respect to wavelength and set the derivative equal to zero.4 The relation obtained is known as Wien s displacement law ... 

To integrate Planck s radiation distribution formula over all wavelengths, x can conveniently be substituted for V( T) and hence dx= —(liT)(lf) )dk, so dk= -X2Tdx = -dx/(Tx2). The total energy radiated is thus ... 

The Planck distribution formula describes the spectral intensity of the radiation field from a black body as... 

The photon energy mode density can be expressed as the product of Equations 3.28 and 3.32 giving the well-known Planck s formula (Planck s black body radiation distribution law) ... 

At low pressure, the only interactions of the ion with its surroundings are through the exchange of photons with the surrounding walls. This is described by the three processes of absorption, induced emission, and spontaneous emission (whose rates are related by the Einstein coefficient equations). In the circumstances of interest here, the radiation illuminating the ions is the blackbody spectrum at the temperature of the surrounding walls, whose intensity and spectral distribution are given by the Planck blackbody formula. At ordinary temperatures, this is almost entirely infrared radiation, and near room temperature the most intense radiation is near 1000 cm". ... 

The total emission of radiant energy from a black body takes place at a rate expressed by the Stefan-Boltzmann (fourth-power) lav/ while its spectral energy distribution is described by Wien slaws, ormore accurately by Planck s equation, as well as by a n umber of oilier empirical laws and formulas, See also Thermal Radiation,... 

The major selling point of standard cosmology is the observed isotropic microwave background radiation, with black-body spectrum. In a closed universe it needs no explanation. Radiation, which accumulates in any closed cavity, tends, by definition, to an equilibrium wavelength distribution according to Planck s formula (Figure 2.5). 

One which absorbs completely any heat or light radiation reaching it and reflects none. It remains in equilibrium with the radiation reaching and leaving it, and at a given steady temperature emits radiation (black body radiation) with a flux density and spectral energy distribution which are characteristic of that temperature and is described by Planck s radiation formula [41],... 

Having developed the Planck distribution function for photons, we will now use it to obtain the Planck formula for the spectrum of black body radiation. 

Planck radiation formula eq. (15.7) gives the distribution of energy per unit volume per unit frequency. The same formula for the distribution of energy per unit volume per wavelength is... 

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