Short on time? This page is the recap. For explanations, Figure 1 and the worked examples, use the full study guide.
Recap
- All matter spontaneously turns some internal (thermal) energy into electromagnetic radiation.
- A blackbody absorbs all radiation falling on it. At constant temperature it must emit at the same rate as it absorbs.
- A blackbody’s spectrum is continuous and depends only on temperature. It is plotted as intensity per unit wavelength against wavelength.
- Curve features: near zero at short λ, steep rise, one peak, long tail. A hotter curve is higher at every wavelength and peaks at a shorter wavelength. The area under it is the power per unit area.
- Classical physics predicted ever-rising emission at short wavelengths (the “ultraviolet catastrophe”). Planck’s law fits the data by assuming light energy comes in quanta E = hf.
- Hot objects change colour from dull red to white as T rises; stars that look bluish are hotter than stars that look reddish.
Key relationships
| Relationship | Symbols and units | Use it to |
|---|---|---|
| λ_max = b/T | b = 2.898 × 10⁻³ m·K; T in K; λ in m | link peak wavelength and temperature |
| P = σAT⁴ | σ = 5.67 × 10⁻⁸ W/(m²·K⁴); A in m² | find total power emitted |
| P/A = σT⁴ | W/m² | compare surfaces at different T |
| A = 4πr² | sphere | surface area of a star or ball |
| P₂/P₁ = (A₂/A₁)(T₂/T₁)⁴ | ratios | compare two blackbodies without σ |
Assumptions behind the numbers
- The object is modelled as an ideal blackbody; real surfaces emit somewhat less.
- T is the absolute temperature, uniform over the surface.
- P = σAT⁴ is the power emitted, not the net power after absorbing from the surroundings.
Mistakes to avoid
- Using °C. Convert to kelvin first.
- Doubling T and doubling P. P scales as T⁴.
- λ_max ∝ T. It is inversely proportional.
- Crossing curves. A hotter curve lies above a cooler one everywhere.
- Peak only. A blackbody emits at all wavelengths, not just at λ_max.
- Forgetting the area. Two stars at the same T can emit very different total powers if their radii differ.
Quick self-check
- Where is the peak for a blackbody at 2900 K? (About 1.0 × 10⁻⁶ m, 1.0 µm, infrared)
- A blackbody’s temperature is tripled. By what factor does its power change? (3⁴ = 81)
- A star’s peak wavelength is half that of another. Compare their temperatures. (The first is twice as hot)
- What did Planck assume that classical physics did not? (Light energy is emitted in quanta, E = hf)
Next: practice questions.