Jeans formula

The error introduced by use of the Wien equation is less than 1 percent when XT < 3000 pm K. The Wien equation has significant practical value in optical pyrometry for T < 4600 K when a red filter (X = 0.65 pm) is employed. The long-wavelength asymptotic approximation for Eq. (5-102) is known as the Rayleigh-Jeans formula, which is accurate to within 1 percent for XT > 778,000 pm-K. The Raleigh-Jeans formula is of limited engineering utility since a blackbody emits over 99.9 percent of its total energy below the value of XT = 53,000 pm-K. 

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... 

Jeans formula and is applicable only for very long wavelengths. 

At this stage Einstein makes use of the general consideration mentioned above, that the quantum laws must reduce to the classical ones as a limiting case. Clearly in the present problem the limiting case is that of high temperatures, where Jiv is small compared to AT. In this case our formula (2) must become the Rayleigh-Jeans formula (3) of 1, required by the classical theory (and verified by experiment for high temperatures), namely,... 

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... 

We now have experimental values for both the constants a and b occurring in (6). It only remains to compare the limiting form of (6) for long wave lengths and high temperatures with the dassical Rayleigh-Jeans formula. From (6) we have... 

In molecular mechanics, the calculation of electrostatic interactions most often is based on the Coulomb potential energy function (Eq. [15]) using atom-centered point charges q. A few force fields use bond dipoles (jl and the electrostatic interactions are then calculated using the Jeans formula (Eq. [16]).i... 

Combining the two expressions of the number of mode of vibration with the average energy of vibration and reported to the volume of the enclosure of blackbody, results the expression of the density of spectral energy, as Rayleigh-Jeans formula (RJ)... 

For free space, the density p w) of modes at the frequency = 2Ttv is defined by the Rayleigh-Jeans formula (London 1973) ... 

The Rayleigh-Jeans formula for the blackbody radiation curve failed disastrously for short wavelengths and high frequencies but Planck was perhaps motivated mostiy by his interest in Wien s empirical law. At that time it was believed that there would be many tiny discrete oscillators in the cube of metal shown in Figure 10.1 and these oscillators emitted light as a function of temperature so we will follow the classical Rayleigh-Jeans derivation up to a point. Assume the... 

Theory of radiation of an ideal black body from the point of view of wave theory Rayleigh-Jeans formula... 

Planck s formula well describes limiting transitions. So, at hco < kT the exhibitor in the denominator of function (6.6.14) can be decomposed in a series and be limited by two terms. This leads to Rayleigh-Jeans formula which describes very well experiment in this area of frequencies. In contrast, at hm kT the unit in the denominator can be neglected and functions r and/fall according to the exponent that is found by experiment. 

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