Wien's Displacement Law

Astrophysics | AQA A-Level Physics

Key Definition

Wien's displacement law: the black-body radiation curve for different temperatures peaks at a wavelength that is inversely proportional to the temperature. Expressed as $\lambda_{\text{max}} T = 2.9 \times 10^{-3}$ m K.

Black-body radiation

All bodies emit a spectrum of thermal radiation in the form of electromagnetic waves.
An ideal black-body radiatorA theoretical object that absorbs and emits all wavelengths of electromagnetic radiation. Stars are the best real-world approximation to a perfect black body. is one that absorbs and emits all wavelengths. Stars are the best approximation to this.
The radiation emitted has a characteristic spectrum determined by the temperature alone. This can be represented on a graph of intensity against wavelength.
As the temperature increases, the peak of the curve moves to a shorter wavelength and a higher intensity.

The equation

Wien's displacement law relates the peak emission wavelength of a body to its surface temperature:

$$\lambda_{\text{max}} T = 2.9 \times 10^{-3} \text{ m K}$$

Where $\lambda_{\text{max}}$ = the wavelength at which the intensity is maximum (m), and $T$ = the surface temperature of the body (K).
The key part is the inverse relationship: the higher the temperature, the shorter the peak wavelength.

Colour and temperature

Hotter objects tend to appear white or blue (short peak wavelength), while cooler objects tend to appear red or yellow (long peak wavelength).
This gives a direct link between a star's colour and its surface temperature:

Star Colour	Temperature / K
Blue	> 33 000
Blue-white	10 000 – 30 000
White	7 500 – 10 000
Yellow-white	6 000 – 7 500
Yellow	5 000 – 6 000
Orange	3 500 – 5 000
Red	< 3 500

Common Mistake

The temperature in Wien's law must be in kelvin (K). If the question gives temperature in degrees Celsius, convert first. When drawing a black-body curve, remember that the left side rises steeply, the right side falls gradually (and never reaches zero), and the curve must not cross the intensity axis (there are no negative wavelengths).