*Attention:* The IES website will be temporarily unavailable from May 9th to May 13th for site updates. We apologize for any inconvenience and will provide an update closer to the time. Thank you for your continued support of the Illuminating Engineering Society.
DonateJoinMy IES

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}

where:
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 (λ1 – λ2). « Back to Definitions Index