[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 _λ = dI_λ/dA΄ = c_1L λ^-5 [e(^c₂^/λT) – 1]^-1 ,where:
L_λ    = the spectral radiance
dI_λ   = 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 c_1L and c₂ are constants designated as the first and second radiation constants.
c_1L and c₂ are constants designated as the first and second radiation constants.

 

Note: The designation c_1L 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 c_1L 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 = πc_1L. 

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

Then c_1L 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, c_1L = 1.191062 x 10⁴ W·μm⁴/(cm²·sr), with L_λ being given in W/(cm²·sr·μm). 

The currently recommended value* of c₂ is 1.438786 x 10^‑2 m·K.[1]

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

 

$$
Q = \int_{\lambda_1}^{\lambda_2} Q_\lambda \, d\lambda
$$

$$
Q = A t c_1 \int_{\lambda_1}^{\lambda_2} \lambda^{-5} \left[ e^{\frac{c_2}{\lambda T}} – 1 \right]^{-1} \, d\lambda
$$

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).

 

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[1] NBS Special Publication 398, Fundamental Physical Constants. Washington, DC: U.S. Department of Commerce; Jan 1974.

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