## 23.1 Black Body Radiation When a piece of metal is heated sufficiently, it glows and changes color from red to orange to white. This change occurs because the color of light emitted by an object depends on its temperature. Scientists use the concept of a **black body**—a perfect absorber and emitter of radiation—to understand this phenomenon. **Wien's Displacement Law** allows us to determine the temperature of distant stars by analyzing the wavelength of light they emit. ### Black Body A black body is an ideal object that absorbs all radiation falling on it. It is perfectly opaque and non-reflecting. Additionally, a black body is a good emitter of radiation. Although purely theoretical, stars serve as excellent practical examples of black bodies. The radiation emitted by a black body has a characteristic spectrum determined entirely by its temperature. ### Wien's Displacement Law Wien's Displacement Law describes the relationship between thermal radiation, temperature, and emission spectra. The law states that hotter objects emit shorter wavelengths (such as blue light), while cooler objects emit longer wavelengths (such as red light). This law relates the observed peak wavelength of light emitted by a star to its surface temperature. **Principle:** The black body radiation curve for different temperatures peaks at a wavelength ($\lambda_{\max}$), which is inversely proportional to the absolute temperature ($T$). **Mathematical Relation:** $$\lambda_{\max } \propto \frac{1}{T}$$ or $$\lambda_{\max } = \frac{\text{constant}}{T}$$ The constant has a value of $2.9 \times 10^{-3}\ \mathrm{m \cdot K}$. **Wien's Displacement Law Equation:** $$\lambda_{\max }T = 2.9 \times 10^{-3}\ \mathrm{m \cdot K}$$ **Implications:** - As the temperature of a body increases, the wavelength at peak intensity decreases. This explains why hotter stars appear white or blue, while cooler stars appear red or yellow. - As the temperature of a body increases, the intensity of radiation at each wavelength increases significantly. ### Stefan-Boltzmann Law While Wien's law deals with the peak wavelength, the total energy radiated is governed by the Stefan-Boltzmann Law. It states that the total energy radiated per unit surface area of a black body across all wavelengths per unit time (also known as the black-body radiant exitance or spectral intensity) is directly proportional to the fourth power of the black body's thermodynamic temperature $T$. $$E = \sigma T^4$$ Where $\sigma$ is the Stefan-Boltzmann constant ($5.67 \times 10^{-8}\ \mathrm{W \cdot m^{-2} \cdot K^{-4}}$). ### Planck's Quantum Theory Classical physics failed to explain the black body radiation curves, particularly at short wavelengths (the "Ultraviolet Catastrophe"). In 1900, Max Planck explained the distribution by assuming that energy is radiated or absorbed in discrete packets called "quanta". $$E = hf$$ Where $h$ is Planck's constant ($6.63 \times 10^{-34}\ \mathrm{J \cdot s}$). | Law/Constant | Formula | Value/Description | | :----------- | :------ | :----------------- | | Wien's Displacement Law | $\lambda_{\max }T = b$ | $b = 2.9 \times 10^{-3}\ \mathrm{m \cdot K}$ | | Stefan-Boltzmann Law | $E = \sigma T^4$ | $\sigma = 5.67 \times 10^{-8}\ \mathrm{W \cdot m^{-2} \cdot K^{-4}}$ | | Planck's Equation | $E = hf$ | $h = 6.63 \times 10^{-34}\ \mathrm{J \cdot s}$ | **Significance:** Wien's Law is crucial for determining the surface temperatures of stars and other celestial objects based on their emitted light, connecting observational astronomy to fundamental physics.