Black Body Radiation: quick revision, 15 bites

What is a Black Body?

Perfect absorber, perfect emitter.

A **black body** is an idealized object that **absorbs all incident electromagnetic radiation**, regardless of frequency or angle of incidence. It does not reflect or transmit any radiation. Because it absorbs everything, it appears perfectly black at thermal equilibrium.

Takeaway. A black body is a perfect absorber and emitter of radiation.

Emission Spectrum

Hot black bodies glow with a continuous spectrum.

A black body at a given temperature emits radiation across a **continuous spectrum**—all wavelengths are present, but the intensity varies with wavelength. The spectrum depends only on the body's temperature, not on its material or shape.

Takeaway. The emission spectrum of a black body is continuous and temperature-dependent.

Temperature Dependence

Hotter means brighter and bluer.

The total energy emitted per unit area per unit time (radiant exitance) increases dramatically with temperature. This is quantified by the **Stefan-Boltzmann law**: $E = \sigma T^4$, where $\sigma$ is the Stefan-Boltzmann constant ($5.67 \times 10^{-8} \, \text{W m}^{-2} \text{K}^{-4}$).

Takeaway. Radiant power scales with the fourth power of absolute temperature.

Stefan-Boltzmann Law

Total power emitted: T to the fourth.

Takeaway. The Stefan-Boltzmann law gives the total power radiated by a black body.

Wien's Displacement Law

Peak wavelength shifts with temperature.

The wavelength at which the emission is maximum ($\lambda_{\text{max}}$) is inversely proportional to temperature. This is **Wien's displacement law**: $\lambda_{\text{max}} = \frac{b}{T}$, where $b \approx 2.898 \times 10^{-3} \, \text{m K}$.

Takeaway. Hotter objects emit peak radiation at shorter wavelengths.

Wien's Law Formula

Calculate peak wavelength easily.

Takeaway. Wien's law lets you find the peak emission wavelength from temperature.

Planck's Radiation Law

The exact spectrum formula.

**Planck's law** describes the spectral radiance of a black body at a given temperature. It was derived by Max Planck in 1900 and introduced the concept of energy quantization: $E = h\nu$, where $h$ is Planck's constant ($6.626 \times 10^{-34} \, \text{J s}$).

Takeaway. Planck's law gives the exact intensity at each wavelength and introduced quantization.

Planck's Formula

The quantum formula for black body radiation.

Takeaway. Planck's formula accurately predicts the black body spectrum.

Ultraviolet Catastrophe

Classical physics predicted infinite energy.

Before Planck, classical physics (Rayleigh-Jeans law) predicted that the intensity of radiation would increase indefinitely at shorter wavelengths, leading to an absurd infinite energy emission—the **ultraviolet catastrophe**.

Takeaway. The ultraviolet catastrophe showed the need for quantum theory.

Stars as Black Bodies

Stars approximate black bodies.

Stars, including the Sun, emit spectra that closely match black body radiation. By analyzing a star's spectrum, astronomers can determine its surface temperature using Wien's law.

Takeaway. Black body radiation explains the colors of stars.

Thermal Imaging and Incandescence

From night vision to light bulbs.

**Thermal imaging** cameras detect infrared radiation emitted by warm objects, which follows black body principles. **Incandescent light bulbs** produce light by heating a filament to high temperatures, emitting a continuous spectrum.

Takeaway. Black body radiation underpins thermal imaging and incandescent lighting.

Confusing Black Body with Black Color

Black body ≠ black paint.

A common misconception is that a black body must be black in color. In reality, a black body at high temperatures emits visible light (e.g., a glowing piece of metal). The term 'black' refers to its perfect absorption at all wavelengths, not its appearance.

Takeaway. A black body can glow brightly when hot; 'black' refers to absorption, not color.

Remember the Laws

Stefan for power, Wien for peak.

Use the mnemonic: **'Stefan says power, Wien says peak.'** Stefan-Boltzmann law gives total power ($T^4$), Wien's law gives the wavelength of maximum emission ($1/T$).

Takeaway. Stefan for total power, Wien for peak wavelength.

Exam Tip: Units and Constants

Know your constants.

Takeaway. Memorize the constants and their units for quick problem solving.

Cosmic Microwave Background

The universe's afterglow is a black body.

The **cosmic microwave background (CMB)** radiation, left over from the Big Bang, is a near-perfect black body spectrum at a temperature of about 2.725 K. Its measurement provides strong evidence for the Big Bang theory.

Takeaway. The CMB is a black body remnant of the early universe.