1. Lecture_02: Outline
Thermal Emission
Blackbody radiation, temperature dependence, Stefan’s law, Wien’s law
Statistical mechanics, Boltzmann distribution, equipartition law
Cavity radiation, Rayleigh-Jeans classical formula

2. Matter under irradiation:
Blackbody radiation
Matter under irradiation:

3. Colors of matter: This is not an emission! Absorbs everything: black
Blackbody radiation
Colors of matter:
Absorbs everything: black
Reflects everything: white
Transparent for everything: no color
Absorbs everything, but yellow: yellow
Absorbs everything, but red: red
This is not an emission!

4. Emission of heated iron:
Blackbody radiation
Emission of heated iron:

5. Emission of heated iron:
Blackbody radiation
Emission of heated iron:

6. Emission of heated iron:
Blackbody radiation
Emission of heated iron:

7. Stefan’s Law: Blackbody radiation
Radiancy, RT, is the total energy emitted per unit time per unit area from a blackbody with temperature T
Spectral radiancy, RT(ν), is the radiancy between frequencies ν and ν+dν
RT(λ)
Stefan-Boltzmann constant:

8. Wien’s displacement law:
Blackbody radiation
Wien’s displacement law:

9. Blackbody radiation
Temperature of stars:
Sun: λmax = 5.1ּ10-7m; North Star: λmax = 3.5ּ10-7m

10. Blackbody radiation
Temperature of stars:
Sun: λmax = 5.1ּ10-7m; North Star: λmax = 3.5ּ10-7m
Sun: T = 5700K; North Star: T = 8300K

11. Temperature of stars: Radiancy of stars:
Blackbody radiation
Temperature of stars:
Sun: λmax = 5.1ּ10-7m; North Star: λmax = 3.5ּ10-7m
Sun: T = 5700K; North Star: T = 8300K
Radiancy of stars:

12. Temperature of stars: Radiancy of stars:
Blackbody radiation
Temperature of stars:
Sun: λmax = 5.1ּ10-7m; North Star: λmax = 3.5ּ10-7m
Sun: T = 5700K; North Star: T = 8300K
Radiancy of stars:
Sun: RT = 5.9ּ107 W/m2; North Star: RT = 2.71ּ108 W/m2

13. Ideal gas: Statistical mechanics
Temperature is related to averaged kinetic energy
For each of projections
(degrees of freedom):

14. Theorem of Equipartition of Energy:
Statistical mechanics
Theorem of Equipartition of Energy:
Each degree of freedom contributes ½kT to the energy of a system, where possible degrees of freedom are those associated with translation, rotation and vibration of molecules

15. Boltzmann Distribution:
Statistical mechanics
Boltzmann Distribution:
Distribution function (number density) nV(E):
It is defined so that nV(E) dE is the number of molecules per unit volume with energy between E and E + dE
Probability P(E):
It is defined so that P(E) dE is the probability to find a particular molecule between E and E + dE

16. Statistical mechanics
Free particle:
Harmonic oscillator:

17. Averaged values: Averaged energy: Averaged velocity: Usually:
Statistical mechanics
Averaged values:
Averaged energy:
Averaged velocity:
Usually:

18. Rayleigh-Jeans formula
Cavity radiation:
A good approximation of a black body is a small hole leading to the inside of a hollow object
The hole acts as a perfect absorber
Energy density, ρT(ν), is the energy contained in a unit volume of the cavity at temperature T between frequencies ν and ν+dν

19. Electromagnetic waves in cavity:
Rayleigh-Jeans formula
Electromagnetic waves in cavity:
E = Emax sin(2πx/λ) sin(2πνt)
With metallic walls,
E = 0 at the wall (x = 0,a)! a

20. Electromagnetic waves in cavity:
Rayleigh-Jeans formula
Electromagnetic waves in cavity:
E = Emax sin(2πx/λ) sin(2πνt)
With metallic walls,
E = 0 at the wall (x = 0,a)! a
Only standing waves
are possible!
n = 1
n = 2
2a/λ = n
n = 3
2aν/c = n
n = 4

21. Allowed frequencies: ν = cn/2a Polarization Rayleigh-Jeans formula1 2 3 4 5 n
Number of allowed waves, N(ν)dν, between frequencies ν and ν+dν
Polarization

22. Two independent waves for each ν, λ
Rayleigh-Jeans formula
Polarization:
E = Emax sin(2πx/λ) sin(2πνt)
Two independent waves for each ν, λ

23. One-dimensional case:
Rayleigh-Jeans formula
One-dimensional case:
Three-dimensional case:
Energy spectrum, ρT(ν)dν, is equal to the number of allowed wave per unit volume times the energy of each wave
Each wave can be considered as a harmonic oscillator with

24. Rayleigh-Jeans formula: