Black body radiation curves: Planck's law

Planck's law

The concept of a black body

In physics, a black body is an ideal object that perfectly absorbs all electromagnetic energy it receives, without reflecting or transmitting any of it. This complete absorption of light results in thermal agitation (random movements of molecules or atoms) that causes the emission of electromagnetic radiation, known as black body radiation.
These movements are caused by the temperature of the black body. The higher the temperature, the faster the movements and the greater the thermal agitation.
Such an object does not exist in reality, but the concept of a black body is used as an important theoretical model in thermodynamics and quantum physics.

Black body radiation

The radiation emitted by a black body depends only on its temperature. This emission, described by the Planck law, has a continuous spectrum whose maximum shifts toward shorter wavelengths as the temperature increases. This behavior is at the heart of many applications in astrophysics and quantum physics.

Planck's equation

The Planck law describes the distribution of electromagnetic energy (or the distribution of photon density) radiated by a black body at a given temperature, as a function of wavelength.

Planck's equation is as follows:

\[ I(\nu,T)=\frac{8\pi h\nu^{3}}{c^{2}}\,\frac{1}{e^{\frac{h\nu}{k_B T}}-1} \]

where:
\(I(\nu,T)\) is the radiation intensity per unit area, solid angle, and frequency
\(\nu\) is the frequency of the radiation
\(T\) is the temperature of the black body
\(h\) is the Planck constant
\(k_B\) is the Boltzmann constant
\(c\) is the speed of light

A major discovery in physics

The Planck law was a major discovery in physics, because it provided a quantitative explanation for the experimental observation that the electromagnetic spectrum of a black body is a bell-shaped curve. This law also had a profound impact on the development of quantum mechanics, notably through the work of Max Planck (1858-1947), who introduced the notion of energy quanta to explain black body radiation. The amount of radiation emitted by a black body depends only on its temperature and wavelength.

Applications of Planck's law

Planck's law is applicable to a wide variety of electromagnetic radiation sources, including stars, nebulae, ovens, incandescent lamps, lasers, diodes, and also semiconductors.
Planck's law is a fundamental law of physics that has many practical applications, both in astrophysics and in engineering.

Wien's law

The maximum intensity of the spectrum

Planck's equation predicts that the electromagnetic spectrum curve of a black body is a bell-shaped curve, with a maximum intensity at a specific wavelength. The wavelength of maximum intensity, \(\lambda_{\text{max}}\), is given by the Wien law:

\[ \lambda_{\text{max}} = \frac{b}{T} \]

where \(b\) is the Wien displacement constant, \(b \approx 2.8978 \times 10^{-3}\ \text{m}\cdot\text{K}\). This relation can also be written as \(\lambda_{\text{max}} T = b\).

Why is it called the ultraviolet catastrophe?

At the end of the 19th century, classical physics predicted that the intensity radiated by a black body should increase without limit as the wavelength decreased. A black body should therefore have emitted infinite energy in the ultraviolet and beyond. Yet observations showed, on the contrary, a maximum followed by a rapid decrease. This contradiction between theory and experiment was called the ultraviolet catastrophe. Planck resolved it by assuming that energy is exchanged in discrete quanta, opening the way to quantum physics.

Key takeaways

A black body perfectly absorbs all electromagnetic energy it receives and re-emits it according to its temperature alone. The Planck law describes the distribution of this radiation, while the Wien law indicates the wavelength of maximum emission. The higher the temperature, the more this maximum shifts toward shorter wavelengths. These laws founded quantum physics and make it possible, in particular, to measure the temperature of stars.