Rayleigh–Jeans law

In 1900, the British physicist Lord Rayleigh derived the λ−4 dependence of the Rayleigh–Jeans law based on classical physical arguments, relying upon the equipartition theorem. This law predicted an energy output that diverges towards infinity as wavelength approaches zero (as frequency tends to infinity). Measurements of the spectral emission of actual black bodies revealed that the emission agreed with Rayleigh's calculation at low frequencies but diverged at high frequencies, reaching a maximum and then falling with frequency, so the total energy emitted is finite. Rayleigh recognized the unphysical behavior of his formula at high frequencies and introduced an ad hoc cutoff to correct it, but experimentalists found that his cutoff did not agree with data. Hendrik Lorentz also presented a derivation of the wavelength dependence in 1903. More complete derivations, which included the proportionality constant, were presented in 1905 by Rayleigh and Sir James Jeans and independently by Albert Einstein. == Comparison to Planck's law ==

In 1900 Max Planck empirically obtained an expression for black-body radiation expressed in terms of wavelength (Planck's law): B_\lambda(T) = \frac{2hc^2}{\lambda^5} \frac{1}{e^\frac{hc}{\lambda k_\text{B} T} - 1}, where h is the Planck constant, and is the Boltzmann constant. Planck's law does not suffer from an ultraviolet catastrophe and agrees well with the experimental data, but its full significance (which ultimately led to quantum theory) was only appreciated several years later. Since e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots, then in the limit of high temperatures or long wavelengths, the term in the exponential becomes small, and the exponential is well approximated with the Taylor polynomial's first-order term: e^{\frac{hc}{\lambda k_\text{B} T}} \approx 1 + \frac{hc}{\lambda k_\text{B} T}. So \frac{1}{e^\frac{hc}{\lambda k_\text{B} T} - 1} \approx \frac{1}{\frac{hc}{\lambda k_\text{B} T}} = \frac{\lambda k_\text{B} T}{hc}. This results in Planck's blackbody formula reducing to B_{\lambda}(T) = \frac{2 ck_\text{B} T}{\lambda^4}, which is identical to the classically derived Rayleigh–Jeans expression. The same argument can be applied to the blackbody radiation expressed in terms of frequency . In the limit of small frequencies, that is h \nu \ll k_\text{B} T , B_\nu(T) = \frac{2 h\nu^3}{c^2} \frac{1}{e^\frac{h\nu}{k_\text{B} T} - 1} \approx \frac{2 h\nu^3}{c^2} \cdot \frac{k_\text{B} T}{h\nu} = \frac{2 \nu^2 k_\mathrm{B} T}{c^2}. This last expression is the Rayleigh–Jeans law in the limit of small frequencies. == Consistency of frequency- and wavelength-dependent expressions ==

When comparing the frequency- and wavelength-dependent expressions of the Rayleigh–Jeans law, it is important to remember that \frac{dP}{d\lambda} = B_{\lambda}(T) and \frac{dP}{d\nu} = B_{\nu}(T). Note that these two expressions then have different units, as a step d\lambda in wavelength is not equivalent to a step d\nu in frequency. Therefore, B_{\lambda}(T) \neq B_{\nu}(T), even after substituting the value \lambda = c/\nu, because B_\lambda(T) has units of energy emitted per unit time per unit area of emitting surface, per unit solid angle, per unit wavelength, whereas B_\nu(T) has units of energy emitted per unit time per unit area of emitting surface, per unit solid angle, per unit frequency. To be consistent, we must use the equality B_\lambda \, d\lambda = dP = B_\nu \, d\nu, where both sides now have units of power (energy emitted per unit time) per unit area of emitting surface, per unit solid angle. Starting with the Rayleigh–Jeans law in terms of wavelength, we get B_\lambda(T) = B_\nu(T) \frac{d\nu}{d\lambda}, where \frac{d\nu}{d\lambda} = \frac{d}{d\lambda} \left(\frac{c}{\lambda}\right) = -\frac{c}{\lambda^2}. This leads to B_\lambda(T) = \frac{2k_\text{B} T\left(\frac{c}{\lambda}\right)^2}{c^2} \times \frac{c}{\lambda^2} = \frac{2ck_\text{B} T}{\lambda^4}. == Other forms of Rayleigh–Jeans law ==

Depending on the application, the Planck function can be expressed in 3 different forms. The first involves energy emitted per unit time per unit area of emitting surface, per unit solid angle, per spectral unit. In this form, the Planck function and associated Rayleigh–Jeans limits are given by B_\lambda(T) = \frac{2 hc^2}{\lambda^5} \frac{1}{e^\frac{hc}{\lambda k_\text{B} T} - 1} \approx \frac{2ck_\text{B} T}{\lambda^4} or B_\nu(T) = \frac{2h\nu^3}{c^2} \frac{1}{e^\frac{h\nu}{k_\text{B} T} - 1} \approx \frac{2k_\text{B} T\nu^2}{c^2}. Alternatively, Planck's law can be written as an expression I(\nu, T) = \pi B_\nu(T) for emitted power integrated over all solid angles. In this form, the Planck function and associated Rayleigh–Jeans limits are given by I(\lambda, T) = \frac{2\pi hc^2}{\lambda^5} \frac{1}{e^\frac{hc}{\lambda k_\text{B} T} - 1} \approx \frac{2\pi ck_\text{B} T}{\lambda^4} or I(\nu, T) = \frac{2\pi h\nu^3}{c^2} \frac{1}{e^\frac{h\nu}{k_\text{B} T} - 1} \approx \frac{2 \pi k_\text{B} T\nu^2}{c^2}. In other cases, Planck's law is written as u(\nu, T) = \frac{4\pi}{c} B_\nu(T) for energy per unit volume (energy density). In this form, the Planck function and associated Rayleigh–Jeans limits are given by u(\lambda, T) = \frac{8 \pi hc}{\lambda^5} \frac{1}{e^\frac{hc}{\lambda k_\text{B} T} - 1} \approx \frac{8\pi k_\text{B} T}{\lambda^4} or u(\nu, T) = \frac{8\pi h\nu^3}{c^3} \frac{1}{e^\frac{h\nu}{k_\text{B} T} - 1} \approx \frac{8 \pi k_\text{B} T\nu^2}{c^3}. == See also ==