# Wien's Displacement Law ## Black Body Radiator - An **ideal black body radiator** absorbs and emits all wavelengths of electromagnetic radiation - Stars represent the closest real-world approximation to ideal black body radiators - Perfect black bodies are theoretical constructs rather than physical objects - The radiation spectrum from a black body is uniquely determined by its temperature alone - No other factors influence the characteristic emission pattern - Intensity-wavelength graphs reveal how thermodynamic temperature correlates with peak emission wavelength - Different temperatures produce distinctive spectral curves - Each curve peaks at a specific wavelength determined by the object's temperature

A black body is an idealized object that absorbs all incident electromagnetic radiation and emits radiation with a spectrum determined solely by its temperature

### Black Body Radiation Curves The graph below shows how radiation intensity varies with wavelength for black bodies at different temperatures:

Key observations from the curves: - **Hotter objects emit more radiation** at all wavelengths (higher overall intensity) - **Peak wavelength shifts to shorter values** as temperature increases (Wien's displacement law) - The 10,000 K curve (blue-white star) peaks around 290 nm in the ultraviolet region - The 5,800 K curve (yellow star, similar to our Sun) peaks around 500 nm in the visible green-yellow region - The 3,500 K curve (red star) peaks around 830 nm in the near-infrared region ## Wien's Displacement Law - Wien's displacement law connects the observed wavelength of light from an object to its surface temperature - The fundamental principle states: **The black body radiation curve for different temperatures peaks at a wavelength that is inversely proportional to the temperature** - This relationship can be expressed mathematically as: $$\lambda_{max} \propto \frac{1}{T}$$ - Where: - $\lambda_{max}$ = the peak wavelength at maximum intensity (m) - $T$ = the surface temperature of the object (K) ### Temperature and Emission Characteristics - The equation reveals two key consequences of increasing temperature: - **Shorter peak wavelength**: hotter objects emit predominantly at shorter wavelengths, appearing white or blue; cooler objects emit at longer wavelengths, appearing red or yellow - **Greater overall intensity**: the radiation intensity increases at all wavelengths as temperature rises ## Star Classification by Temperature and Colour | Star Colour | Temperature / K | |------------|----------------| | blue | >33 000 | | blue-white | 10 000 − 30 000 | | white | 7500 − 10 000 | | yellow-white | 6000 − 7500 | | yellow | 5000 − 6000 | | orange | 3500 − 5000 | | red | < 3500 |

Temperature in Wien's Law must be in Kelvin (K). Convert from °C by adding 273 if necessary.

## Worked Example **Question:** The star Sirius has a spectrum that peaks at a wavelength of 290 nm, while the star Antares peaks at a wavelength of 750 nm. Determine which of these two stars, Antares or Sirius, has a cooler surface temperature. **Answer:** **Step 1: List the known quantities** - Peak emission wavelength of Sirius, $\lambda_S = 290$ nm - Peak emission wavelength of Antares, $\lambda_A = 750$ nm **Step 2: Recall Wien's displacement law** $$\lambda_{max} \propto \frac{1}{T}$$ **Step 3: Compare the relative surface temperatures** Since $\lambda_{max}T = \text{constant}$ for all black bodies: $$\frac{\lambda_A}{\lambda_S} = \frac{T_S}{T_A}$$ Rearranging for the temperature ratio: $$\frac{T_S}{T_A} = \frac{750}{290} \approx 2.6$$ **Step 4: Write a concluding statement** - The surface temperature of Sirius is approximately 2.6 times hotter than Antares - Therefore, **Antares is significantly cooler than Sirius**

Objects with longer peak wavelengths have lower temperatures—the relationship is inverse

## Observational Application: The Orion Constellation - Different stars in the same constellation exhibit different colours due to temperature variations - Consider two prominent stars in Orion: - **Rigel** (blue-white appearance) has a peak wavelength around 263 nm → hotter surface temperature - **Betelgeuse** (red appearance) has a peak wavelength around 828 nm → cooler surface temperature - Visual colour directly indicates surface temperature - Blue/white stars are extremely hot - Red/orange stars are relatively cool

Wien's Law allows astronomers to measure stellar temperatures from Earth without physical contact—simply analyze the spectrum