# Thermal radiation

Jump to: navigation, search

Corresponding Wikipedia article: Black body

According to Kirchhoff's law of radiation, the photonic thermal radiation spectrum depends on the body temperature (average energy of the body particles), and it does not depend on its physic-chemical properties. Different bodies absorb different fraction of the total received radiation, which affects the thermodynamic equilibrium condition. At a constant temperature, the absorbed heat is equal to the released heat.

The black body is an abstract particle system, which absorbs the entire received radiation, and therefore it radiates the maximum possible emission spectrum at a given temperature, if other types of the heat transfer do not exist.

The black body photons represent a system of the 3-dimensional standing waves, the nodes of which contain the material particles. Since each particle emits one photon, the number of photons is the number of standing waves ("Waves", 7) multiplied by 2, because an electromagnetic wave may consist of two orthogonal waves (circular polarization). The spectral density of energy per volume unit of the body is expressed by the average photon energy ("Quantum mechanics", 6): $u(\omega,T)=2\frac{\mathrm{d}n}{\mathrm{d}\omega}\epsilon(\omega)=2\frac{\omega^2}{2\pi^2c^3}\frac{\hbar\omega}{\exp(\hbar\omega/kT)-1}\tag{1}$ $u(\lambda,T)=2\frac{\mathrm{d}n}{\mathrm{d}\lambda}\epsilon(\lambda)=2\frac{4\pi}{\lambda^4}\frac{hc/\lambda}{\exp(hc/\lambda kT)-1}\tag{2}$ The luminous intensity $$I$$ is a radiation power per steradian: $I=\frac{c}{4\pi}u$ The radiation power per area unit of the body surface, according to Lambert's cosine, law is: $f=\pi I=\frac{c}{4}u$ As a result, the spectral density of power per area unit of the black body surface is expressed by well-known Planck’s law: $u(\omega,T)=\frac{\hbar\omega^3}{4\pi^2c^2}\frac{1}{\exp(\hbar\omega/kT)-1}\tag{3}$ $u(\lambda,T)=\frac{2\pi\hbar c^2}{\lambda^5}\frac{1}{\exp(\hbar c/\lambda kT)-1}\tag{4}$ Wien's displacement law is produced by finding the extremum of (4). According to it, as the bodies temperature grows, their own glow color becomes red, orange, yellow and white. The thermal radiation is mainly focused into the infrared and visible ranges. $\lambda_{max}=\frac{b}{T}\;\;\;\;\;b\approx 0,003\;m\cdot K\tag{5}$ Stefan–Boltzmann law is a power, which is radiated by the area unit: $P=\int_0^{\infty}{f(\omega,T)\mathrm{d}\omega}=\frac{\pi^2k^4}{60c^2\hbar^3}=\sigma T^4\tag{6}$