2. Thermal radiation

Light as a particle: Planck's radiation law (2/2)

Radiation as a function of the wavelength

Microwaves are commonly indicated by their frequency whereas visible light is mostly characterised by the use of wavelengths. Planck's radiation law in dependency of the wavelengths indicates:

u_{λ} = \frac{d U_{λ}}{d λ} = \frac{8 π h c}{λ^{5}} \frac{1}{exp {h c / λ k T} - 1}

Question: How to convert frequencies into wavelengths? ↓

The relation of frequency and wavelength is $c = f λ$ . Applied to Planck's radiation law in frequency representation given on the previous page, the result will not be the same as shown above.

This is caused by the fact that we are dealing with a differential equation since the spectral energy density is a differential. That is why not only $f$ is converted into $λ$ , but also the differential $d f$ has to be converted into the differential $d λ$ :

\frac{d f}{d λ} = c \frac{d (\frac{1}{λ})}{d λ} = - c \frac{1}{λ^{2}}

The negative sign indicates that an increasing wavelength corresponds to a decreasing frequency. However, both quantities are always positive. As a result, one would focus on the modulus of the derivative:

| \frac{d f}{d λ} | = c \frac{1}{λ^{2}}

If the frequency differential in the frequency-dependent radiation law is replaced by

| d f | = c \frac{d λ}{λ^{2}}

the wavelength-dependent form shown above is obtained.

200-400 K 400-1000 K 1000-2500 K 2500-4000 K 4000-7000 K

Emission graphs of black bodies as a function of the wavelength at different temperatures in Kelvin. Wavelengths at the abscissa and in the unit of the energy density are in nanometres (nm).

How to realise a cavity radiator, which emits radiation as a black body obeying Planck's radiation law? This significantly depends on the emitters temperature, in other words on the desired wavelength in the spectrum.

Equations ↓

Planck emitters

At room temperature, an emitter which has a maximum radiation rate of about 10μm in thermal infrared is relatively simple to be installed. A box made from black cardboard displaying a little hole is suitable, as Student experiments with black cubes show. The opening appears to be of a darker black than the black cardboard. Of the cavity, infrared radiation passes out of which the spectrum approximately results in the progression of Planck's radiation law for room temperature.

For applications with high precision, where the emitter's temperature shall be variable in a range from room temperature up to 1500°C resp. 1200 K, heater ovens are used which resemble the ones used 120 years ago in the Light-technical laboratory of the Physikalisch-Technische Reichsanstalt in Berlin. High care is put into the outer isolation and even heating of the inner space to obtain a thermic equilibrium on the inside. The temperature measured by the thermometer on the inner walls will then adequately correspond to the radiation temperature of the Planck radiation coming out of the opening.

A cavity radiator for temperatures of about 1500°C. In the front-right, one can see radiation receiver of which the sensitivity is calibrated with the cavity radiator.
Source: Federal Institute for Metrology METAS, Bern, Switzerland

More radiation laws are presented on the pages which follow. They are in parts mathematically ambitious and not required for a first understanding. You may directly proceed to the pages about solar radiation.

Understanding Spectra from the Earth

2. Thermal radiation

Light as a particle: Planck's radiation law (2/2)

Radiation as a function of the wavelength

Planck emitters