Planck radiation law

[2.13.4] An expression representing the spectral radiance of a blackbody as a function of the wavelength and temperature. This law is commonly expressed by the formula:

L_{lambda} = dI_{lambda }/dA'=c_{1L}lambda^{-5}left [e^{(c_{2}/lambda T)}-1 right ]^{-1}

Lλ    = the spectral radiance
dIλ   = the spectral radiant intensity
dA’  = the projected area (dA cosθ ) of the aperture of the blackbody
e      = the base of natural logarithms ( approx. 2.71828)
Τ     = absolute temperature c1L and c2 are constants designated as the first and second radiation constants.

Note: The designation c1L is used to indicate that the equation in the form given here refers to the radiance L, or to the intensity I per unit projected area A’, of the source. Numerical values are commonly given not for c1L but for c1, which applies to the total flux radiated from a blackbody aperture (that is, in a hemisphere [2π steradians]) so that, with the Lambert cosine law taken into account, c1 = πc1L.

The currently recommended value of c1 is 3.741832 x 10-16 W·m2, or 3.741832 x 10-12 W·cm2.

Then c1L is 1.191062 x 10-16 W·m2/sr, or 1.191062 x 10-12 W·cm2/sr.

If, as is more convenient, wavelengths are expressed in micrometers and area in square centimeters, c1L = 1.191062 x 104 W·μm4/(cm2·sr), with Lλ being given in W/(cm2·sr·μm).

The currently recommended value of c2 is 1.438786 x 10-2 m·K.[note]Values of radiation constants from NBS Special Publication 398.[/note]

The Planck law in the following form gives the energy radiated from the blackbody in a given wavelength interval (λ1 – λ2):

Q=int _{lambda 1}^{lambda 2}Qlambda dlambda

Q=Atc_{1}int _{lambda 1}^{lambda 2}lambda^{-5} left [ e^{(c_{2}/lambda t)} -1 right ]^{-1 }dlambda
If A is the area of the radiation aperture or surface in square centimeters, t is time in seconds, λ is wavelength in micrometers, and c1 = 3.741832 x 104 W·μm4/cm2, then Q is the total energy in watt seconds (joules) emitted from this area (that is, in the solid angle 2π) in time t, within the wavelength interval (λ1λ2). « Back to Definitions Index