Planck's radiation law

A law of physics which gives the spectral energy distribution of the heat radiation emitted from a so-called blackbody at any temperature. Discovered by Max Planck, this law laid the foundation for the advent of the quantum theory because it was the first physical law to postulate that electromagnetic energy exists in discrete bundles, or quanta. See Heat radiation, Quantum mechanics

Planck's radiation law

See black body.

Planck’s Radiation Law

the law of energy distribution in the spectrum of equilibrium radiation (electromagnetic radiation in thermodynamic equilibrium with matter) at a certain temperature. It was first derived by M. Planck in 1900 on the basis of the hypothesis of energy quanta. The law gives the spectral dependence of the energy density p (the energy of the radiation per unit volume) on the frequency v or wavelength λ = c/v, where c is the speed of light. Since the blackbody emissive power u = (c/4)ρ (the radiant energy emitted from the surface per unit area and time) is proportional to the energy density, it is also given by the law. The functions ρ_{v, T} and u_{v, T} or ρ_{λ, T} and u_{λ, T} for a unit interval of the frequencies or wavelengths are universal functions of v or X and of T; they are independent of the nature of the substance with which the radiation is in equilibrium.

Planck’s radiation law is expressed by the formula

where h is Planck’s constant and k is the Boltzmann constant. Function (2) is illustrated for various temperatures in Figure 1. The maximum of the function shifts toward the short-wavelength end of the spectrum with increasing T.

Figure 1

Other laws of equilibrium radiation result from Planck’s radiation law. Integration with respect to v (or λ) from 0 to ∞ gives the values of the total energy density of radiation for all frequencies—the Stefan-Boltzmann radiation law:

We also obtain the total emissive power of a black body:

In the high-frequency region the energy of the photon is much greater than the thermal energy (hv ≫ kT), and Planck’s radiation law becomes Wien’s radiation law:

ρ_{v, T} = (8πhv³/c³)e^–hv/kT

In the low-frequency region, where kT ≫ hv, it becomes the Rayleigh-Jeans radiation law:

ρ_{v, T} = (8πv²/c³)kT

These laws are thus the limiting cases of Planck’s radiation law. Wien’s displacement law also follows from Planck’s radiation law, which may be represented in the form p_{v, T} = v³f(v/t), where f(v/T) is a function only of the ratio of v and T.

Planck’s radiation law is in agreement with experimental data and can be used to calculate the values of h and k. The temperatures of heated bodies—for example, the surfaces of stars—can be determined on the basis of the law by means of pyrometers. For temperatures greater than 2000°K, the only reliable method of temperature determination makes use of the blackbody radiation laws and Kirchhoffs radiation law. Planck’s radiation law is also used in calculations of light sources.

The law was derived by A. Einstein in 1916 through the examination of the quantum transitions for atoms in equilibrium with radiation. Planck’s radiation law can also be obtained as a consequence of the Bose-Einstein statistics.

M. A. EL’IASHEVICH

Planck's radiation law

Planck's radiation law