Laws of Radiation

This law gives the spectral distribution of radiant energy inside a blackbody.

Where:

T = Absolute temperature of the blackbody

h = Planck’s constant (6.626 x 10-34 Js)

c = Speed of light (2.998 x 108 m s-1)

k = Boltzmann's constant (1.381 x 10-23 JK-1)

λ = Wavelength in m

The spectral radiant exitance from a non perturbing aperture in the blackbody cavity, M_eλ (λ,T), is given by:

L_eλ (λ,T), the spectral radiance at the aperture is given by:

The curves in Figure 3 show M_Bλ plotted for blackbodies at various temperatures. The output increases and the peak shifts to shorter wavelengths as the temperature, T, increases.

Integrating the spectral radiant exitance over all wavelengths gives:

σ is called the Stefan-Boltzmann constant

This is the Stefan-Boltzmann law relating the total output to temperature.

If M_e(T) is in W m-2, and T in kelvins, then σ is 5.67 x 10-8 Wm-2 K-4.

At room temperature a 1 mm2 blackbody emits about 0.5 mW into a hemisphere. At 3200 K, the temperature of the hottest tungsten filaments, the 1 mm2, emits 6 W.

This law relates the wavelength of peak exitance, λ_m, and blackbody temperature, T:

λ_mT = 2898 where T is in kelvins and λ_m is in micrometers.

The peak of the spectral distribution curve is at 9.8 µm for a blackbody at room temperature. As the source temperature gets higher, the wavelength of peak exitance moves towards shorter wavelengths. The temperature of the sun’s surface is around 5800K. The peak of a 6000 blackbody curve is at 0.48 µm, as shown in Fig. 3.

The radiation from real sources is always less than that from a blackbody. Emissivity (ε) is a measure of how a real source compares with a blackbody. It is defined as the ratio of the radiant power emitted per area to the radiant power emitted by a blackbody per area. (A more rigorous definition defines directional spectral emissivity ε(θ,φ,λ,T). Emissivity can be wavelength and temperature dependent (See Figure 2).

As the emissivity of tungsten is less than 0.4 where a 3200 K blackbody curve peaks, the 1 mm2 tungsten surface at 3200 K will only emit 2.5 W into the hemisphere. If the emissivity does not vary with wavelength then the source is a “graybody”.

Kirchoff’s Law states that the emissivity of a surface is equal to its absorptance, where the absorptance (α) of a surface is the ratio of the radiant power absorbed to the radiant power incident on the surface.

Lambert’s Cosine Law holds that the radiation per unit solid angle (the radiant intensity) from a flat surface varies with the cosine of the angle to the surface normal (Figure 4). Some Oriel^® Light Sources, such as arcs, are basically spherical. These appear like a uniform flat disk as a result of the cosine law. Another consequence of this law is that flat sources, such as some of our low power quartz tungsten halogen filaments, must be properly oriented for maximum irradiance of a target. Flat diffusing surfaces are said to be ideal diffusers or Lambertian if the geometrical distribution of radiation from the surfaces obeys Lambert’s Law. Lambert’s Law has important consequences in the measurement of light. Cosine receptors on detectors are needed to make meaningful measurements of radiation with large or uncertain angular distribution.