The Color of Heat
While the Stefan-Boltzmann law tells us how much total power a hot object radiates, Wien's Displacement Law tells us exactly what color (wavelength) that radiation will be.
Formulated by Wilhelm Wien in 1893, the law states that the peak wavelength of the electromagnetic radiation emitted by a black body is inversely proportional to its absolute temperature. Simply put: as an object gets hotter, the peak color it glows shifts from invisible Infrared, to Red, to Yellow, to White, and eventually to Blue and Ultraviolet.
Stellar Astrophysics
- Star Colors: If you look at the night sky, you will notice stars have different colors. Betelgeuse is distinctly red, while Sirius is a piercing blue-white. Wien's Law allows astronomers to determine the exact surface temperature of a star simply by looking at its color. Betelgeuse is relatively "cool" (about $3500 , \text{K}$), so its peak wavelength is in the red spectrum. Sirius is blisteringly hot (nearly $10,000 , \text{K}$), pushing its peak wavelength deep into the blue and ultraviolet spectrum.
- Blacksmithing: Before modern laser thermometers, a blacksmith relied entirely on Wien's Law to judge the temperature of the steel in the forge. When the steel is $800^\circ\text{C}$, it glows a dull cherry red. At $1000^\circ\text{C}$, it shifts to a bright orange. At $1200^\circ\text{C}$, it becomes a blinding lemon-yellow.
- Cosmic Microwave Background: The remnants of the Big Bang are still glowing, but the universe has expanded and cooled so much over 13.8 billion years that this "afterglow" is now at a temperature of just $2.725 , \text{K}$. According to Wien's Law, the peak wavelength of a $2.7 , \text{K}$ object is in the microwave spectrum, exactly where radio telescopes detect it.
The Formula
Example Calculation
You want to find the peak wavelength of the radiation emitted by the Sun, which has a surface temperature of $5778 , \text{K}$. Wien's displacement constant ($b$) is $2.89777 \times 10^{-3} , \text{m}\cdot\text{K}$.
- Divide Constant by Temperature ($b / T$): $(2.89777 \times 10^{-3}) / 5778 = 5.015 \times 10^{-7} , \text{meters}$.
The peak wavelength is roughly $501.5 , \text{nanometers}$. If you look at an electromagnetic spectrum chart, $500 , \text{nm}$ is right in the middle of the visible light spectrum—it's a bright blue-green. Our Sun technically emits more green light than any other color, though our eyes perceive the massive mixture of all the visible wavelengths as white light.