Rayleigh-Jeans distribution

Fluctuations near Tc behave as classical fields in sense of Rayleigh-Jeans 3-momentum distribution is n(p) T/E(p) in the vicinity of... 

This is the celebrated Einstein derivation of Planck s law to complete it one takes into account that for large T the distribution must become identical with the Rayleigh-Jeans law. )... 

Show that Equation 5.9, the Rayleigh-Jeans law, is identical to the Planck black-body distribution in the limit as h —> 0. 

This implies that the higher-energy modes are less populated than what is implied by the equipartion principle. Substituting this value, rather than kT, into the Rayleigh-Jeans formula (1.21), we obtain the Planck distribution law... 

Q.7.4 Show that (a) the Rayleigh-Jeans law is a special case of Planck distribution law for the blackbody spectrum. Show also that (b) the Wein displacement law can be derived from Planck s distribution law. 

If a mode of oscillation can possess any arbitrary amount of energy from zero to infinity, there is no reason for the Rayleigh-Jeans formula to be incorrect. Let us suppose, for the sake of argument, that an oscillator cannot have any arbitrary energy but may have energy only in integral multiples of a certain unit of energy e. Then the distribution of a collection of these oscillators is discrete and we can represent it by... 

Measurements at wavelengths longer than the peak of emission yield the form of the dust emissivity law there, but more importantly they also provide a unique handle to the dust density distribution. For grain temperatures occurring even at large radius in these sources, the Rayleigh-Jeans approximation is satisfactory at X > 300y. Then since the calculated dust temperature varies slowly with radius a any major variation of... 

Rayleigh found the v2 dependence, Jeans later supplied the rest). Their distribution function g(v) increases as v2, with no provision for a fall-off to zero as the frequency and the energy go to infinity ("ultraviolet catastrophe"). 

Planck wanted to understand black body radiation. The black body may be modeled by a box, with a small hole (shown in Fig. 1.1). We heat the box up, wait for the system to reach a stationary state (at a fixed temperature), and see what kind of electromagnetic radiation (intensity as a function of frequency) comes out of the hole. In 1900, Rayleigh and Jeans tried to apply classical mechanics to this problem, and they calculated correctly that the black body would emit the electromagnetic radiation with a distribution of frequencies. However,... 

At a given temperature T the intensity distribution (at a given frequency v. Fig. l.l.b) has a single maximum. As the temperature increases, the maximum should shift towards higher frequencies (a piece of iron appears red at 500 °C, but bluish at 1000 °C). Just like Rayleigh and Jeans, Max Planck was unable to derive... 

Following the work of Kirchhoff on the connection between emission and absorption coefficients, it had been proved that the radiation inside a totally enclosed cavity maintained at a uniform temperature was a function of the temperature alone and was identical with the radiation which would be emitted by a perfectly black body at the same temperature. The spectral distribution of the radiation had been investigated experimentally and it was found that the intensity increased slowly with decreasing frequency until it reached a peak, after which it decreased very rapidly (Fig.1.1). However, all attempts to derive an equation giving the intensity as a function of frequency had failed. The most convincing approach made by Rayleigh and Jeans on the basis of classical thermodynamics gave the result... 

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