Question

Stephan's and Provost law in optics

Answer 1

You probably mean the Stefan–Boltzmann law and Prevost’s law (often spelled “Prevost”). Briefly:

1) Stefan–Boltzmann law
- What it states: The total radiant power emitted per unit surface area of a perfect blackbody (integrated over all wavelengths and directions) is proportional to the fourth power of its absolute temperature:
j* = σ T^4
where j* is the blackbody radiant emittance (W·m⁻²), T is temperature (K), and σ is the Stefan–Boltzmann constant.
- Constant: σ = 5.670374419×10⁻⁸ W·m⁻²·K⁻⁴.
- For real (non‑ideal) surfaces use emissivity ε (0 ≤ ε ≤ 1):
j = ε σ T^4.
- Origin/derivation: Empirically found by Josef Stefan (1879) and derived from thermodynamics and later from Planck’s law (integrating Planck’s spectral radiance over all wavelengths and solid angles). In terms of fundamental constants:
σ = (2π^5 k_B^4) / (15 c^2 h^3).
- Use-cases: estimating radiative heat loss/gain of hot objects, stellar luminosity (L = 4πR^2 σ T^4 for an approximate blackbody star), thermal engineering calculations.
- Limits/assumptions: assumes thermal equilibrium radiation and applies as total (wavelength-integrated) emission; wavelength-dependent behavior requires Planck’s law.

2) Prevost’s law (theory of exchanges)
- Statement: Every body at a finite temperature continuously emits and absorbs thermal radiation. In thermal equilibrium, every two bodies exchange radiation at equal rates so that there is no net exchange.
- Consequences:
- Net radiative heat flow between two bodies is from the hotter to the colder body (because emission rates depend on temperature).
- Forms a conceptual basis for Kirchhoff’s law of thermal radiation: in thermal equilibrium, a body’s emissivity equals its absorptivity at each wavelength and direction.
- Practical expression (net exchange): for two gray surfaces facing each other with emissivities ε1, ε2 and temperatures T1, T2 (and view factor 1), a commonly used approximate net radiative heat transfer is
Q̇_net = A (ε_eff) σ (T1^4 − T2^4),
where ε_eff = [1/ε1 + 1/ε2 − 1]⁻¹ (for diffuse gray surfaces).
- Historical note: Prevost (1791) introduced the exchange idea; it preceded the full development of blackbody theory and Planck’s law.

Relation between the two:
- Prevost’s conceptual law explains that bodies both emit and absorb radiation; Stefan–Boltzmann gives the quantitative total emission from a blackbody (and, with emissivity, from real surfaces). Kirchhoff’s law connects emissivity and absorptivity so that the Stefan–Boltzmann result applies correctly in radiative equilibrium calculations.

If you want, I can:
- show a short derivation of σ from Planck’s law,
- give example calculations (e.g., radiative loss from a 1000 K surface),
- or write the general radiative exchange formula between two surfaces including view factors and surface properties. Which would be most useful?