Question

Stefan's and provost law in optics

Answer 1

You are probably referring to the Stefan–Boltzmann law (commonly called Stefan’s law) and Prevost’s law (the “law of exchanges”) — two fundamental statements about thermal (radiant) energy. Briefly:

1) Stefan–Boltzmann law
- Statement: The total radiative power per unit area emitted by a blackbody is proportional to the fourth power of its absolute temperature:
E_bb = σ T^4
where σ (Stefan–Boltzmann constant) = 5.670374419×10^−8 W·m^−2·K^−4.
- For a real surface (gray body) with emissivity ε (0 ≤ ε ≤ 1):
E = ε σ T^4.
- Origins: Empirical result (Stefan, 1879) given a thermodynamic derivation by Boltzmann (1884); microscopically explained by Planck’s law and the frequency integration that produces the T^4 dependence.
- Related quantities: the volumetric energy density of blackbody radiation u = a T^4 with a = 4σ/c, and the radiative flux F = (c/4) u = σ T^4.
- Example: At T = 300 K a blackbody emits ≈ 459 W/m^2.
- Limitations/notes: Stefan–Boltzmann gives total (integrated over all wavelengths) power. Spectral distribution is given by Planck’s law; emissivity can vary with wavelength and angle. Near-field (sub-wavelength distances) and non-equilibrium situations can violate the simple surface-law form.

2) Prevost’s law (law of exchanges) and its consequences
- Statement (Prevost, 1790s): Every body continually emits and receives thermal radiation. At thermal equilibrium the exchanges balance, so a body’s net radiative heat gain depends on the difference between what it absorbs and what it emits.
- Practical consequence: Two bodies at different temperatures exchange net radiative power; for simple gray, diffuse surfaces facing each other, net per-unit-area power ≈ ε σ (T1^4 − T2^4) (more generally you must include view/factor geometry and surface properties).
- Connection to Kirchhoff’s law: Prevost’s reasoning helped lead to Kirchhoff’s law of thermal radiation: at thermal equilibrium, a body’s emissivity equals its absorptivity at each wavelength and direction. This is why blackbodies (perfect absorbers) are also perfect emitters.
- Use in optics/thermal engineering: Prevost’s idea underpins radiative heat-transfer calculations, thermal camera calibration, design of blackbody sources, coatings to control emission/absorption, and estimating net radiation between surfaces (with view factors).

Short formula summary
- Blackbody emitted power per unit area: E_bb = σ T^4.
- Real surface: E = ε(λ,θ) integrated over wavelength/angle; often approximated as E ≈ ε σ T^4 for gray, diffuse surfaces.
- Net exchange between two simple gray surfaces: Q_net = A ε_eff σ (T1^4 − T2^4) (with ε_eff and geometrical view factors as required).

If you’d like, I can:
- Derive the Stefan–Boltzmann law from Planck’s law (integration) or Boltzmann’s thermodynamic argument;
- Show how to compute net radiative transfer between two finite surfaces including view factors;
- Give examples for specific temperatures or materials (wavelength-dependent emissivity).