Planck’s Law: From Infrared Radiation to Temperature

All objects emit infrared radiation with a temperature above absolute zero. As an object’s temperature rises, the intensity of its emitted energy increases, enabling temperature measurement through the analysis of infrared radiation. Planck’s radiation law establishes the fundamental principle of non-contact temperature measurement, demonstrating that all objects warmer than absolute zero (-273.15°C or 0 K) emit electromagnetic radiation. In the case of a black body—an ideal object that absorbs and emits radiation across all wavelengths—the intensity of infrared emission increases with temperature, allowing for temperature determination based on emitted infrared energy. Planck’s radiation law also indicates that the higher the temperature of a body, the more radiation it emits exponentially across all wavelengths. The spectral radiant exitance per unit area, per unit solid angle, and per unit frequency for radiation wavelength is described in Planck’s law, where c is the speed of light, h is Planck’s constant, and k is the Boltzmann constant [1,2,3].

[math]B_{\lambda }(\lambda ,T)=\frac{2hc^2}{\lambda ^5}\frac{1}{e^{\frac{h\ \bullet \ c}{\lambda \ \bullet \ k_B\ \bullet \ T}}-1}[/math]

Planck’s law can be encountered in several forms and can also be written in terms of the spectral energy density [math]u_{\lambda }(\lambda ,T)[/math]:

[math]u_{\lambda }(\lambda ,T)\ \ =\frac{4\pi }{c}B_{\lambda }(\lambda ,T)[/math]

In the above variants of Planck’s law, use the terms  [math]2h[/math][math]c^{2}[/math] and hc/[math]k_{B}[/math], which comprise physical constants only. Consequently, these terms can be considered as physical constants themselves and are therefore referred to as the first radiation constant [math]c_1=\ 3.74\ \bullet 10^{-16}\ W/m^2[/math] and the second radiation constant [math]c_2=\ 1.44\ \bullet 10^{-2}\ K/m[/math]. In Planck’s law, [math]B_{\lambda }(\lambda ,T)[/math] and [math]M°_{\lambda}\left(\lambda,T\right)[/math], both describe different aspects of blackbody radiation. While [math]B_{\lambda }(\lambda ,T)[/math], is known as spectral radiance, represents the intensity of radiation emitted per unit wavelength and unit solid angle. It is measured in [math]Wm^{-2}\mu[/math][math]m^{-1}sr^{-1}[/math], indicating that it quantifies radiation in a specific direction. In contrast, [Equation] , the spectral emissive power describes the total emitted power per unit area and per unit wavelength, integrating over all directions. Its unit is [math]Wm^{-2}\mu[/math][math]m^{-1}[/math]. The following equation relates the two quantities. 

[math]M°_{\lambda}\left(\lambda,T\right)=\pi.B_{\lambda}\left(\lambda,T\right)[/math]

Therefore, Planck’s law can be reformulated to the spectral radiant exitance [Equation] of a black body into half-space depending on its temperature and wavelength [1,2,3]. 

[math]M°_{\lambda}\left(\lambda,T\right)=\frac{c_{1}}{\lambda^{5}}.\frac{1}{e^{\frac{c_{2}}{\lambda T}}-1}[/math]

These formulas are very abstract, so they cannot be used for many practical applications. But many physical observations derive from it. Emitted IR energy increases with object temperature, and the peak wavelength shifts to shorter wavelengths. With rising temperatures, the maximum of the spectral-specific radiation shifts to shorter wavelengths. At higher temperatures, the amount of infrared radiation increases and can be felt as heat, and more visible radiation is emitted, so the body glows visibly red.  The body is bright yellow or blue-white at higher temperatures and emits significant amounts of short-wavelength radiation. 

At room temperature (~300 K), a body emits thermal radiation that is infrared mainly and invisible. Its peak is around 10 to 14 µm, so most thermal cameras work at this spectral range for the temperature measurement range. Nevertheless, Planck radiation says that the higher the temperature of a body, the more radiation it emits at every wavelength.  

The power emitted from the emitting surface, per unit projected emitting surface area, per unit solid angle, per wavelength unit. Since the radiance is isotropic and independent of direction, the power emitted at an angle to the normal is proportional to the projected area and, therefore, to the cosine of that angle as per Lambert’s cosine law and is unpolarized. 

The following illustration, Figure 1, shows the graphic description of the formula depending on the wavelength with different temperatures.

The inversion of Equation 1 provides the black body temperature for a given spectral intensity at a narrowband wavelength is the following: 

[math]T=\frac{hc}{\lambda \ k_B\ \bullet \ln \left(\frac{2hc^2}{\lambda ^5\ B_{\lambda }}+1\right)}[/math]