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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 Blackbody radiation Matter under irradiation: 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! Blackbody radiation Emission of heated iron: Blackbody radiation Emission of heated iron: Blackbody radiation Emission of heated iron: c Blackbody radiation Stefan’s Law: 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 0 0 RT RT ( )d RT ( )d RT T 4 Stefan-Boltzmann constant: 5.67 108 W m2 K 4 RT(λ) Blackbody radiation Wien’s displacement law: max T max cT 2.898 10 3 Blackbody radiation Temperature of stars: Sun: λmax = 5.1ּ10-7m; North Star: λmax = 3.5ּ10-7m c max T 2.898 10 m K 3 Blackbody radiation Temperature of stars: Sun: λmax = 5.1ּ10-7m; North Star: λmax = 3.5ּ10-7m c max T 2.898 10 m K 3 Sun: T = 5700K; North Star: T = 8300K Blackbody radiation Temperature of stars: Sun: λmax = 5.1ּ10-7m; North Star: λmax = 3.5ּ10-7m c max T 2.898 10 m K 3 Sun: T = 5700K; North Star: T = 8300K Radiancy of stars: RT T 4 5.67 108 W m2 K 4 Blackbody radiation Temperature of stars: Sun: λmax = 5.1ּ10-7m; North Star: λmax = 3.5ּ10-7m c max T 2.898 10 m K 3 Sun: T = 5700K; North Star: T = 8300K Radiancy of stars: RT T 4 5.67 108 W m2 K 4 Sun: RT = 5.9ּ107 W/m2; North Star: RT = 2.71ּ108 W/m2 Statistical mechanics Ideal gas: Temperature is related to averaged kinetic energy 1 3 2 KE m v kT 2 2 For each of projections (degrees of freedom): 1 1 1 1 2 2 2 m v x m v y m vz kT 2 2 2 2 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 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 nV (E ) n0 e E kT e E kT P(E ) kT Statistical mechanics Free particle: E KE mv 2 2 P (v ) e mv 2 2 kT kT Harmonic oscillator: 1 1 2 KE m vx kT 2 2 1 PE kT 2 E E KE PE kT Statistical mechanics Averaged values: Averaged energy: E E EP(E )dE 0 P(E )dE 0 Averaged velocity: v v vP(v)dv 0 P(v)dv 0 0 0 Usually: P(E )dE P(v)dv 1 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 T ( ) RT ( ) 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 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)! Only standing waves are possible! 2a/λ = n 2aν/c = n a n=1 n=2 n=3 n=4 Rayleigh-Jeans formula Allowed frequencies: ν = cn/2a 2a ( d ) c 2a c 1 2 3 4 5 n Number of allowed waves, N(ν)dν, between frequencies ν and ν+dν 2a N ( )d 2 d c Polarization Rayleigh-Jeans formula Polarization: E = Emax sin(2πx/λ) sin(2πνt) Two independent waves for each ν, λ Rayleigh-Jeans formula One-dimensional case: Three-dimensional case: 4a N ( )d d c 4a 3 N ( )d 3 2 2 d c Energy spectrum, ρT(ν)dν, is equal to the number of allowed wave per unit volume times the energy of each wave 1 T ( )d 3 N ( )d E a Each wave can be considered as a harmonic oscillator with E kT Rayleigh-Jeans formula Rayleigh-Jeans formula: 8 2 kT T ( )d d 3 c