The Spectrum of Blackbody Radiation

All objects emit electromagnetic radiation depending on their temperature. A perfect emitter known as a blackbody radiates energy across a continuous spectrum, with intensity that varies smoothly with wavelength. Wien’s displacement law describes where this spectrum reaches its maximum. Specifically, it states that the peak wavelength ${\lambda }_{\text{max}}$ is inversely proportional to absolute temperature: ${\lambda }_{\text{max}}=\frac{b}{T}$, where $b$ is Wien’s displacement constant.

Derivation and Constant

The constant $b$ has a value of 2.897×10⁻³ m·K. Wien deduced this relation in the 1890s while studying thermal spectra from heated bodies. Later, Max Planck showed that Wien’s law emerges from his more general blackbody formula by differentiating it with respect to wavelength and setting the derivative to zero. The result elegantly links temperature and color—hotter objects emit radiation that peaks at shorter wavelengths.

Practical Significance

A classic example of Wien’s law occurs in astronomy. Stars of different surface temperatures appear distinct colors because their blackbody peaks fall at visible or near-visible wavelengths. The Sun, with a surface temperature around 5,800 K, has its peak near 500 nm, in the green portion of the spectrum. Cooler stars emit more strongly in the red or infrared, while very hot stars peak in the ultraviolet. By measuring a star’s color, astronomers estimate its temperature and glean insights into its size, age, and composition.

Using the Calculator

To find the peak wavelength for any temperature, simply enter the temperature in kelvins into the form above. The script computes ${\lambda }_{\text{max}}=\frac{b}{T}$ and displays the result in nanometers and micrometers. The tool is useful for students exploring radiation laws, engineers designing thermal cameras, and anyone curious about the color of glowing objects.

Interpreting the Numbers

As temperature increases, the peak shifts to shorter wavelengths. Doubling the temperature halves the peak wavelength. For example, a metal heated to 1,500 K glows a dull red with a peak around 1,900 nm, while at 3,000 K it becomes bright orange with a peak near 970 nm. This relation explains why incandescent bulbs produce more visible light as the filament gets hotter. It also shows why cooler objects emit mainly infrared radiation beyond human sight.

Relationship to Other Laws

Wien’s displacement law is one piece of the broader theory of blackbody radiation that includes Planck’s law and the Stefan-Boltzmann law. Planck’s law gives the spectral intensity at every wavelength, while the Stefan-Boltzmann law relates the total power radiated to the fourth power of temperature. The peak wavelength from Wien’s law complements these by indicating where the emission is most intense. Together, they provide a comprehensive picture of thermal radiation.

Limitations

The law applies strictly to ideal blackbodies. Real materials may deviate due to emissivity that varies with wavelength. For instance, the peak of sunlight is slightly offset from the theoretical value because the Sun’s atmosphere is not a perfect blackbody. Nonetheless, Wien’s law remains an excellent approximation for many materials and serves as an essential benchmark in spectroscopy.

Exploring Temperature Ranges

This calculator covers everything from cryogenic temperatures, where peak emission lies in the far infrared, to stellar temperatures with peaks in the ultraviolet. By entering different values, you can explore how the color of an object changes as it heats up. Candle flames at about 1,800 K emit a warm glow around 1,600 nm, while molten steel at 2,700 K peaks around 1,070 nm. The progression continues until objects become so hot that their emission shifts beyond the visible into ultraviolet.

Broader Applications

Temperature sensing, astronomy, and even forensic science make use of Wien’s law. Infrared cameras detect the peak emission from objects to gauge their temperatures. Astronomers analyze stellar spectra to infer surface temperatures and classify stars. In industrial settings, pyrometers rely on the relationship between color and temperature to estimate the heat of furnaces or molten metal without direct contact. The simple inverse proportionality between wavelength and temperature gives rapid insight in all these scenarios.

Historical Context

Wilhelm Wien’s research came during a period of intense study into thermal radiation. His displacement law and related work paved the way for Planck’s quantum hypothesis, which revolutionized physics. The connection between temperature and peak wavelength helped solve the “ultraviolet catastrophe” predicted by classical theories. Recognizing that the spectrum has a maximum at a finite wavelength, rather than diverging as frequency increases, was a crucial step toward modern quantum mechanics.

Conclusion

Wien’s displacement law distills a complex phenomenon into a simple formula linking temperature with color. By entering a single value into this calculator, you immediately see the wavelength at which a blackbody radiates most intensely. This knowledge illuminates everyday experiences, from the glow of heated metal to the hues of distant stars, revealing the deep interconnection between temperature and light.