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In physics, the Rayleigh–Jeans Law, first proposed in the early 20th century, attempts to describe the spectral radiance of electromagnetic radiation at all wavelengths from a black body at a given temperature through classical arguments. For wavelength $\lambda$, it is;

$B_\lambda(T) = \frac{2 c k T}{\lambda^4}$

where c is the speed of light, k is Boltzmann's constant and T is the temperature in kelvins. For frequency $\nu$, it is;

$B_\nu(T) = \frac{2 \nu^2 k T}{c^2}$.

The Rayleigh–Jeans expression agrees with experimental results at large wavelengths (or, equivalently, low frequencies) but strongly disagrees at short wavelengths (or high frequencies). This inconsistency is commonly known as the ultraviolet catastrophe.[1] [2]

Historical development Edit

In 1900, the British physicist Lord Rayleigh derived the $\lambda^{-4}$ dependence of the Rayleigh–Jeans law based on classical physical arguments.[3] A more complete derivation, which included the proportionality constant, was presented by Rayleigh and Sir James Jeans in 1905. The derived expression had two problems: It predicted an energy output that diverges towards infinity as wavelength approaches zero (as frequency tends to infinity) and measurements of energy output at short wavelengths disagreed with the result from the Rayleigh–Jeans expression.

Comparison to Planck's law Edit

In 1900 Max Planck empirically derived an expression for Blackbody Radiation expressed in terms of wavelength λ = c /ν:

$B_\lambda(T) = \frac{2 c^2}{\lambda^5}~\frac{h}{e^\frac{hc}{\lambda kT}-1}$

where h is Planck's constant. The Planck 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. In the limit of a very very high temperatures or long wavelengths, the term in the exponential becomes small, and so the denominator becomes approximately hc /λkT by power series expansion. Specifically,

$\frac{1}{e^\frac{hc}{\lambda kT}-1} \approx \frac{1}{\frac{hc}{\lambda kT}} = \frac{\lambda kT}{hc}$

which results in the full Planck Function reducing to

$B_{\lambda}(T) = \frac{2ckT}{\lambda^4}$

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 $\nu = \frac{c}{\lambda}$ in the limit of small frequency:

$B_{\nu}(T) = \frac{2h\nu^3/c^2}{e^\frac{h\nu}{kT} - 1} \approx \frac{2h\nu^3}{c^2}* \frac{kT}{h\nu} = \frac{2kT\nu^2}{c^2}$

For both the frequency and wavelength dependent expressions, the Planck Function reduces to the Rayleigh–Jeans law in the limit of large wavelength or low frequency.

Consistency of frequency and wavelength dependent expressions Edit

When comparing the frequency and wavelength dependent expressions of the Rayleigh–Jeans law it is important to remember that

$B_{\lambda}(T) \neq B_{\nu}(T)$

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

$B_{\lambda} \, d\lambda = B_{\nu} \, d\nu$

where both sides now have units of 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

$B_{\lambda}(T) = B_{\nu}(T) \times \frac{d\nu}{d\lambda}$

where

$\frac{d\nu}{d\lambda} = \frac{d}{d\lambda}\left(\frac{c}{\lambda}\right) = -\frac{c}{\lambda^2}$.

$B_{\lambda}(T) = \frac{2kT\left( \frac{c}{\lambda}\right)^2}{c^2} \times \frac{c}{\lambda^2} = \frac{2ckT}{\lambda^4}$.

Other forms of Rayleigh–Jeans law Edit

Depending on the application, the Plank 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 unit frequency. In this form, the Plank Function and associated Rayleigh–Jeans limits are given by

$B_\lambda(T) = \frac{2 c^2}{\lambda^5}~\frac{h}{e^\frac{hc}{\lambda kT}-1} \approx \frac{2c kT}{\lambda^4}$

or

$B_{\nu}(T) = \frac{2h\nu^2/c^2}{e^\frac{h\nu}{kT} - 1} \approx \frac{2kT\nu^2}{c^2}$

Alternatively, Planck's law can be written as an expression $u(\nu,T) = \pi I(\nu,T)$ for emitted power integrated over all solid angles. In this form, the Plank Function and associated Rayleigh–Jeans limits are given by

$u(\lambda,T) = \frac{2\pi c^2}{\lambda^5}~\frac{h}{e^\frac{hc}{\lambda^4 kT}-1} \approx \frac{2\pi ckT}{\lambda^4}$

or

$u(\nu,T) = \frac{2\pi h\nu^2/c^2}{e^\frac{h\nu}{kT} - 1} \approx \frac{2 \pi kT\nu^2}{c^2}$

In other cases, Plank's Law is written as $\rho(\nu,T) = \frac{4\pi}{c} I(\nu,T)$ for energy per unit volume (energy density). In this form, the Plank Function and associated Rayleigh–Jeans limits are given by

$\rho(\lambda,T) = \frac{8 \pi c}{\lambda^5}~\frac{h}{e^\frac{hc}{\lambda kT}-1} \approx \frac{8\pi kT}{\lambda^4}$

or

$\rho(\nu,T) = \frac{8\pi h\nu^3/c^3}{e^\frac{h\nu}{kT} - 1} \approx \frac{8 \pi kT\nu^2}{c^3}$