# Black Body Radiation

In a blackbody, Absorptance α equals Emissivity ε, and α = ε = 1.

Practical existing object is not like this, but it is described with transmittance τ and refrectance ρ as follows,

α + ρ + τ = 1

Energy radiated from the blackbody is described as follows and called “Planck’s law”.

In order to obtain whole radiant emittance of the blackbody, integrate the equation (1) through the full the full wavelengths (0 to infinity). The result is as follows and called “Stefan-Bolzmann’s equation”.

The temperature of blackbody can be obtained directly from the radiant energy of blackbody by this equation. In order to find out the wavelength on the maximum spectral radiant emittance, differentiate Planck’s law and make the value to 0.

This equation is called “Wien’s displacement law”.

Where in above (1) to (3),

Wλ : Spectral radiant emittance per unit wavelength and unit area

[W/ cm2 · μm]

λm : Wavelengthofmaximum spectral radiant emittance [ μ m]

λ : Wavelength [μm]

h : Plank’s constant =6. 6 2 6 1 x 1 0 – 3 4 [W· s 2 ]

T : Absolutetemperature [K]

c : Lightvelocity=2.9 9 7 9 x 1 0 1 0 [ cm/ s ]

k : Bolzmann constant = 1 . 3 8 0 7 x 1 0 – 2 3 [W· s /K]

σ : Stefan-Bolzmann constant = 5 . 6 7 0 5 x 1 0 – 1 2 [W/ cm2 ·K4 ]

c1: Fist radiationconstant=3.7 4 1 8 x 1 0 4 [ / cm2 · μm4 ]

c2: Second radiation constant = 1 . 4 3 8 8 x 1 0 4 [ μm·K]

In radiation of normal object, as the emissivity is ε (<1) times of the blackbody, multiply above equation by ε. Following Fig. 6.2 is spectral radiant emittance of a blackbody.

(a) is shown by logarithmic scale and (b) is shown by linear scale.

The graphs in Fig. 6.2 shows that wavelength and spectral radiant emittance vary with the temperature.

Fig. 6.2 shows, as the temperature rises, the peak of spectral radiant emittance is shifting to shorter wavelength side. This phenomenon is some in the visible light region, an object at low temperature appears red, and as the temperature increases, it changes to yellowish and then whitish color.

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