Planck’s radiation distribution

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 energy density in a radiation field in equilibrium with a black body at a temperature T may be calculated by the methods of quantum statistical mechanics (Chapter XV) and, as mentioned in Chapter I, is given by Planck s radiation distribution law as... 

FIGURE 4.7 Experimental test of Planck s distribution for blackbody radiation. The dots represent experimental data acquired at T = 1646 K. The continuous curve represents Planck s predicted distribution, with the parameter h = 6.63 X 10 J s. Agreement between experiment and theory is spectacular, demonstrating the validity of Planck s theory and also determining the value of the previously unknown parameter/ . 

The majority of pulse calorimetric measurements use pyrometry, which is non-contact (optical) measurement of the thermal radiation emitted fi om any heated body or substance according to Planck s radiation law for black body radiation. Planck s law describes the spectral distribution of black body radiance which provides the basis for the International Temperature Scale (ITS-90) [76], especially above the freezing point of silver [77]. Because Planck s law is only... 

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],... 

Planck s radiation law gives the spectral distribution of blackbody radiation. It may be expressed as... 

Contrary to the impression that one might have from a traditional course in introductory calculus, well-behaved functions that cannot be integrated in closed form are not rare mathematical curiosities. Examples are the Gaussian or standard error function and the related function that gives the distribution of molecular or atomic speeds in spherical polar coordinates. The famous blackbody radiation cuiwe, which inspired Planck s quantum hypothesis, is not integrable in closed form over an arbitiar y inteiwal. 

The distribution of energy under idealized conditions is given by Planck s equation in which the energy radiated into a hemisphere in W/cm in... 

Pyrometers Planck s distribution law gives the radiated energy flux qb(X, T)dX in the wavelength range X to X -1- dX from a black surface ... 

Total heat transfer consists of radiation at different frequencies. The distribution of radiation energy in a spectrum and its dependency on temperature is determined from Planck s law of radiation. M ,and are the spectral radiation intensities for a blackbody ... 

If the emissive power E of a radiation source-that is the energy emitted per unit area per unit time-is expressed in terms of the radiation of a single wavelength X, then this is known as the monochromatic or spectral emissive power E, defined as that rate at which radiation of a particular wavelength X is emitted per unit surface area, per unit wavelength in all directions. For a black body at temperature T, the spectral emissive power of a wavelength X is given by Planck s Distribution Law ... 

The Stefan-Boltzmann Law and Wien s Law for black body radiation have been unified into Planck s Law for black body radiation, from which Planck s constant was first introduced. Planck s analysis of the spectral distribution of black body radiation led him to an understanding of the quantisation of energy and radiation and the role of the photon in the theory of radiation. The precise law relates the intensity of the radiation at all wavelengths with the temperature and has the form ... 

This is Planck s famous radiation law, which predicts a spectral energy density, p , of the thermal radiation that is fully consistent with the experiments. Figure 2.1 shows the spectral distribution of the energy density p for two different temperatures. As deduced from Equation (2.2), the thermal radiation (also called blackbody radiation) from different bodies at a given temperature shows the same spectral shape. In expression (2.2), represents the energy per unit time per unit area per frequency interval emitted from a blackbody at temperature T. Upon integration over all frequencies, the total energy flux (in units of W m ) - that is, Atot = /o°° Pv Av - yields... 

A space entirely surrounded by material walls of sufficient thickness to be impenetrable to radiation is traversed in all directions by waves of every possible frequency. Unit volume contains a definite amount of radiant energy —the radiation density—determined only by the temperature of the walls, and distributed among the different frequencies in accordance with Planck s law. 

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,... 

Blackbody radiation sources are accurate radiant energy standards of known flux and spectral distribulion. They are used for calibrating other infrared sources, detectors, and optical systems. The radiating properties of a blackbody source are described by Planck s law. Energy distribution... 

Planck s law is universally accepted today, and blackbody radiation is a tremendously important concept in physics, chemistry, and biology. The blackbody distribution is graphed on a log scale for a variety of temperatures in Figure 5.2. 

Planck s constant was discovered as part of the solution to a nineteenth century conundrum in physics, known as the black-body problem. The challenge was to model the wavelength distribution of radiation emitted through the aperture in a closed cavity at various temperatures6. The standard equations of statistical thermodynamics failed to produce the observed spectrum, unless it was assumed that the energy of radiation with frequency v was an integral multiple of an elementary energy quantum hv. 

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