## Quick Reference

A law stated by Max Planck giving the distribution of energy radiated by a black body. It introduced into physics the novel concept of energy as a quantity that is radiated by a body in small discrete packets rather than as a continuous emission. These small packets became known as quanta and the law formed the basis of quantum theory. The **Planck formula** gives the energy radiated per unit time at frequency ν per unit frequency interval per unit solid angle into an infinitesimal cone from an element of the black-body surface that is of unit area in projection perpendicular to the cone's axis. The expression for this **monochromatic specific intensity***I*_{ν} is:

*I*_{ν} = 2*hc*^{−2}ν^{3}/[exp(*h*ν/*kT*− 1)],

where *h* is the Planck constant, *c* is the speed of light, *k* is the Boltzmann constant, and *T* is the thermodynamic temperature of the black body. *I*_{ν} has units of watts per square metre per steradian per hertz (W m^{−2} sr^{−1} Hz^{−1}). The monochromatic specific intensity can also be expressed in terms of the energy radiated at wavelength λ per unit wavelength interval; it is then written as *I*_{λ}, and the Planck formula is:

*I*_{λ} = 2*hc*^{2}λ^{−5}/[exp(*hc*/λ*kT*) − 1].

There are two important limiting cases of the Planck formula. For low frequencies ν << *kT*/*h* (equivalently, long wavelengths λ >> *hc*/*kT*) the **Rayleigh–Jeans formula** is valid:

*I*_{ν} = 2*c*^{−2}ν^{2}*kT*,

or

*I*_{λ} = 2*c*λ^{−4}*kT*,

Note that these expressions do not involve the Planck constant. They can be derived classically and do not apply at high frequencies, i.e. high energies, when the quantum nature of photons must be taken into account. The second limiting case is the **Wien formula**, which applies at high frequencies ν >> *kT*/*h* (equivalently, short wavelengths λ << *hc*/*kT*):

*I*_{ν} = 2*hc*^{−2}ν^{3} exp(−*h*ν/*kT*),

or

*I*_{λ} = 2*hc*^{2}λ^{−5} exp(−*hc*/λ*kT*).

See also Wien's displacement law.

*Subjects:*
Physics.

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