Laws of thermal radiation

Thermal radiation is electromagnetic radiation emitted by a body because of its temperature, in the range about $0.1$ to $100\ \mu$m (ultraviolet, visible and infrared).

Planck's law

It gives the spectral emissive power of a black body at wavelength $\lambda$ and absolute temperature $T$:

$$E_{b\lambda} = \frac{C_1\,\lambda^{-5}}{\exp\left(C_2/\lambda T\right)-1}$$

$C_1 = 3.742\times10^8\ \text{W}\,\mu\text{m}^4/\text{m}^2$, $C_2 = 14\,388\ \mu\text{m K}$. At every wavelength $E_{b\lambda}$ rises with $T$; each curve has a peak.

Wien's displacement law

Setting $dE_{b\lambda}/d\lambda = 0$ gives the wavelength at the peak. With $x = C_2/\lambda T$, the condition becomes

$$x\,e^{x} = 5\,(e^{x}-1)\;\Rightarrow\; x = 4.965$$

so

$$\lambda_{max}T = \frac{C_2}{4.965} = 2898\ \mu\text{m K}$$

The peak moves to shorter wavelengths as $T$ rises (a heated iron goes from dull red to white). The sun (5800 K) peaks near $0.5\ \mu$m, a room-temperature body (300 K) near $9.7\ \mu$m.

Stefan-Boltzmann law

Integrating Planck's law over all wavelengths gives the total emissive power of a black body. Put $x = C_2/\lambda T$, so $\lambda = C_2/xT$:

$$E_b = \int_0^\infty E_{b\lambda}\,d\lambda = \frac{C_1T^4}{C_2^4}\int_0^\infty\frac{x^3}{e^x-1}\,dx = \frac{C_1T^4}{C_2^4}\cdot\frac{\pi^4}{15} = \sigma T^4$$ $$\sigma = \frac{\pi^4C_1}{15\,C_2^4} = 5.67\times10^{-8}\ \text{W/m}^2\text{K}^4$$

For a real (grey) surface $E = \varepsilon\sigma T^4$.

Kirchhoff's law

At thermal equilibrium the emissivity of a surface equals its absorptivity, $\varepsilon_\lambda = \alpha_\lambda$. For a grey (diffuse) surface $\varepsilon = \alpha$ at all conditions. A good absorber is a good emitter.

Lambert's cosine law

The radiation intensity $I$ of a diffuse surface is the same in all directions, so emissive power in direction $\theta$ from the normal varies as $\cos\theta$, and total emissive power $E = \pi I$.

Frequency, wavelength and temperature

Wavelength and frequency are related by $\lambda = c/\nu$ ($c = 3\times10^8$ m/s in vacuum). A hotter body emits at higher frequency (shorter wavelength) for its peak, according to Wien's law, and also emits more energy at all frequencies according to the $T^4$ law.