# Wien's Law Calculator

Created by Davide Borchia
Last updated: Nov 30, 2022

If you have ever wondered why we talk of "cold" or "warm" light, you are in the right place: with the Wien's law calculator, you will learn the relationship between the light emitted by a body and its temperature.

Here you will learn:

• What is a black-body and why it puzzled physics;
• The features of the spectrum of light emitted by a black body;
• What is the Wien's displacement law; and
• How to calculate the peak wavelength or the temperature using the formula for Wien's displacement law.

We will also give you some examples of Wien's law, that will make the dependence of wavelength on temperature as clear as ever!

Take an opaque body in thermal equilibrium (at a constant temperature) able to absorb all incident radiation. If we poke a small hole in that body, the light coming out of it would have a spectrum only dependent on the temperature of the body itself.

We just described a black-body. Classical physics crashed against this object at the end of the XIX century when the theoretical models forecasted a behavior (indefinite increase in temperature) that was nowhere near the experimental observations. Unknowingly, the physicist who performed those first experiments witnessed the dawn of quantum physics.

## Wien's law of displacement

Wien, together with Lummer, was among the first to analyze the light coming out of a heated oven. A dark oven kept at a constant temperature is a good approximation of a black body. Wien took the spectrogram of the light filtering through a pinhole and found a behavior summarized by the following graph:

The quantities measure the spectral radiance of the emitted light. Each curve corresponds to a temperature (the temperature of the oven), with the radiance sampled over the wavelengths.

As you can see, the curves have a defined peak wavelength: that's where classical physics breaks, leaving space for the quantum interpretation of the natural world. Wien noticed two things:

• There is no ultraviolet catastrophe: the spectra always reach zero after peaking;
• The peak wavelength is inversely proportional to the temperature: for higher temperatures, the spectra peak at shorter wavelengths (more energetic photons). Use Omni's to see how temperature is relevant to energy.

We'll see the consequences of these observations after learning the formula for Wien's displacement law.

## Wien's displacement law formula

Our Wien's law calculator implements the Wien's displacement law formula:

$\lambda_{\text{peak}}= \frac{b}{T}$

Where:

• $\lambda_{\text{peak}}$ is the peak wavelength;
• $b$ is the Wien's constant for displacement; and
• $T$ is the temperature of the black-body.

The Wien's displacement constant has value $b = 2.897\ 771\ 955... \cdot 10^3\ \text{m}\cdot\text{K}$.

## Examples of the Wien's law of displacement

Wien's law of displacement gives colors to our daily lives, from the warm color of a bonfire to the stars shining at night. Let us give you some examples.

Take Antares the brightest star of the Scorpion constellation. Stars are almost ideal black-bodies: knowing the peak wavelength of the emission spectrum of Antares, we use Wien's law to calculate the temperature of the surface of the star. Antares' light peaks at $\lambda_{\text{peak}}=900\ \text{nm}$:

\begin{align*} T_{\text{Antares}} &= \frac{b}{\lambda_{\text{peak}}} \\ \\ &= \frac{2.898 \cdot 10^3\ \text{m}\cdot\text{K}}{900\cdot 10^{-9}\ \text{m}}\\ \\ &=3,\!220\ \text{K} \end{align*}

Antares has a bright, red color. What about Sirius, the brightest star in the Earth's sky? Its color is white, almost cold. The peak wavelength is, in this case, $\lambda_{\text{peak}}=300\ \text{nm}$. Input this value in our Wien's law calculator:

\begin{align*} T_{\text{Sirius}} &= \frac{b}{\lambda_{\text{peak}}} \\ \\ &= \frac{2.898 \cdot 10^3\ \text{m}\cdot\text{K}}{300\cdot 10^{-9}\ \text{m}}\\ \\ &=9,\!600\ \text{K} \end{align*}

Sirius' surface is almost three times as bright as the surface of Antares, and the peak wavelength reflects this difference. The discovery of the relation between color (wavelength) and the temperature was a breakthrough in astrophysics, and we owe it much of the knowledge of our cosmic neighborhood: we even use it to measure the temperature of black holes. Check the black hole collision calculator to learn more about this kind of object.

Davide Borchia
Black body temperature
K
Peak wavelength
nm
Peak frequency
THz
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