Black Body Radiation

Black body radiation is also known as thermal electromagnetic radiation within the body or surrounding the body that is in thermodynamic equilibrium with the environment that is emitted by the black body which is considered to be an idealized opaque, non-reflective body. The intensity of the radiation depends upon the temperature and is inversely proportional to the specific spectrum of wavelengths. This has been assumed for the sake of uniformity and a constant approach to the theory and the calculations.

Many ordinary bodies emit thermal electromagnetic radiation spontaneously which can be considered as black body radiation. A body that is in a perfect insulated enclosure and is in thermal equilibrium containing the black body radiation, will only emit it if a hole is made on the wall of the body such that the hole is small enough to not disturb the equilibrium or have a negligible effect on the equilibrium. 

In a dark room, most of the bodies that emit black body radiation appear black at room temperature because it radiates in the infrared spectrum that is not perceived by the human eye at all. As because the eyes do not have the ability to perceive the radiations below the visible range wavelength, even if the objective physical spectrum of infrared radiation is at its peak, due to its lowest just faintly visible temperature it subjectively appeared grey. Also, the human eye does not have the capacity to perceive colours in low light. It appears dull red when the object becomes a little hotter. It becomes bright red, orange, yellow, white, and ultimately blue-white with the increase in the temperature. 

Though planets and stars are not in thermal equilibrium with their surroundings and are not perfect black bodies, black body radiation is employed as a first approximation to determine the energy they emit. As the Black holes absorb all the radiation that is falling on them, thus they are near to perfect black bodies. It also has been proposed that with a temperature that is dependent on the mass of the black holes, they emit black body radiation. The black body radiation formula is as follows:-

\[B_{v} (v, T) = (\frac{2hv^{3}}{ c^{2}}) (\frac{1}{e^{hv/kt}-1})\]

Where, 

Bv (v, T) = Spectral radiance

h = Planck constant

v = Frequency of electromagnetic radiation

c = Speed of light in vacuum

k = Boltzmann constant

T = Absolute temperature of the body

(Image will be Uploaded soon)

Black Body Radiation Theory

The black body radiation theory is as divided into two segments of explanation that is as follows:-

1. Spectrum: The continuous frequency spectrum which is a characteristic of black body radiation, depends only on the body of the temperature which is known as Planck’s spectrum or Planck’s law. At a definite frequency, the spectrum of the radiation reaches its peak which then increases with the increase in the temperature and at room temperature it falls in the infrared region of the electromagnetic spectrum. When the temperature reaches 5000C steadily, black bodies start to emit significant amounts of visible light. Thus when viewed by the humans in the dark it appears ghosty-grey in appearance. The glow becomes visible with the rise in the temperature in a manner that it appears dull-red initially, which then turns to yellow, and eventually a "dazzling bluish-white" as the temperature rises further. When it emits a substantial fraction of its energy as ultraviolet radiation, it starts to appear white in colour. The black body radiation gives a sneak peek into the thermodynamic equilibrium state of cavity radiation.

(Image will be Updated soon)

2. Black Body: A perfect black body is a physical body that absorbs all the incident electromagnetic radiations irrespective of its angle of radiance or frequency. The perfect black body is an idealization as a perfect black body does not exist in nature. Graphite and lampblack are the perfect examples of approximations to a black material because their emissivities are greater than 0.95. The radiative intensity of the blackbody radiation has a unique absolutely stable distribution that can persist in thermodynamics equilibrium in a cavity. In equilibrium, the total intensity of radiation which is admitted and reflected by a body for each frequency is solely determined by the equilibrium temperature and it does not depend upon the shape, structure of the body or the material. A perfectly black body which is also known as a perfect absorber of spectral radiation is solely due to the emission as there is no reflected radiation. The blackbody is also considered a diffusion emitter in addition. Therefore all the radiation that happens from a black body is considered as radiation from the black body at thermal equilibrium.

There is a visible glow of light from the black body radiation if the temperature of the object is high enough. The temperature at which all the solids glow at a dim brand is about 798 the Caribbean which is also known as a Draper point. A small opening on a large uniformly heated OPEC wall cavity at 1000K very look red from the outside and it looks white from the outside at 6000 K. It does not matter how the oven is constructed for what material has been used for the construction as long as the light entering into the oven is almost absorbed by its wall it is considered to be a good approximation of the black body radiation. Therefore the light and the colour that comes out of the opening of the wall is considered complete as the function of the cavity temperature alone. Therefore the curve that is obtained by plotting a graph between the amount of energy inside the oven per unit volume and per unit frequency interval versus normal frequency is called the black body curve. Such different curves are obtained at different temperatures.

Equations of Black Body Radiation

The equations of the Black body radiation which are also black body radiation formula is as follows:-

1. Planck’s Equation: 

\[B_{v} (v, T) = (\frac{2hv^{3}}{ c^{2}}) (\frac{1}{e^{hv/kt}-1})\]

Where, 

 Bv (v, T) = Spectral radiance

h = Planck constant

v = Frequency of electromagnetic radiation

 c = Speed of light in vacuum

 k = Boltzmann constant

T = Absolute temperature of the body

2. Wien's Displacement Law: Wien's displacement law establishes a relationship between the black body radiation at any temperature with the spectrum at any other temperature. If a shape of a spectrum at one temperature is known then it gets very easy to calculate the shape at any other temperature with the help of this law. Thus the intensity of the spectra is calculated on the basis of a function of its wavelength or of the frequency. Wien's displacement law states that the spectral radiance of black-body radiation per unit wavelength, peaks at the wavelength λ_peak  is given by:-

λ_peak = b/T

Where, 

T = absolute temperature of the body

b = Wien's displacement constant which is also known as constant of proportionality that is equal to 2.897771955...×10⁻³ m⋅K, or b ≈ 2898 μm⋅K.

3. Stefan–Boltzmann Law: By integrating BvT cos (θ) with frequency the radiance L whose unit will be equal to power (area * solid angle), the equation will be as follows:-

L = 2π5/15 (k4T4/c2h3)(1/π) = σT4 (cos θ /π)

By using, \[\int_{0}^{\infty}\] dx (x2/(ex-1)) = π4/15 

Where, x = hv/kT and σ = (2π5/15)((k4/c2h3) = 5. 670373 x 10-8 W/m2K4 is the Stefan–Boltzmann constant.

On a side note, at a distance d, the intensity dI per area dA of radiating surface is the useful expression 

dI = σT4 (cos θ /πd2)dA

Where the radiation is perpendicular to the receiving surface. 

By subsequently integrating L over the solid angle Ω for all azimuthal angle (0 to π) and polar angle θ from 0 to π /2, we arrive at the Stefan–Boltzmann law: the power j* emitted per unit area of the surface of a black body is directly proportional to the fourth power of its absolute temperature:

j* = σT4

As we used,

\[\int cos \theta d\Omega = \int_{0}^{2\pi} \int_{0}^{\pi/2} cos \theta sin \theta d\theta d\Phi = \pi\]

Kirchhoff Law of Black Body Radiation

Gustav Robert Kirchhoff (1824–1887) stated in 1860 that “at thermal equilibrium, the power radiated by an object must be equal to the power absorbed.” Therefore from this law, the observation that has been concluded is that if an object absorbs the radiation incident upon it is equal to 100% then it should re-radiate a hundred percent of it as well. This has already been stated as the black body radiation. But in reality most of such sources are not a complete black body radiator. Therefore it is taken as the energy that is incident upon the surface is either reflected or transmitted. Therefore the ratio of radiant emittance W’ and the radiant emittance W of emissivity of a black body is as follows:- 

Therefore, \[\varepsilon = \frac{W^1}{W}\]

The below figure, therefore, classifies the different types of radiation sources with this relationship where the curve for the black body is given with ε =1. It is the Planck’s curve. Therefore the curve of the black body is proportional to all the planck's curve for all the wavelengths. For a selective radiator with temperature and wavelength, the spectral radiant emittance varies a lot.

(Image will be uploaded soon)