1. Lecture 4a. Blackbody Radiation Energy Spectrum of Blackbody Radiation - Rayleigh-Jeans Law - Rayleigh-Jeans Law - Wien’s Law - Wien’s Law - Stefan-Boltzmann Law - Stefan-Boltzmann Law

2. Energy Spectrum of Blackbody Radiation The average energy of photons with frequency between and +d (per unit volume): u( ,T) - the energy density per unit photon energy for a photon gas in equilibrium with a blackbody at temperature T. - the spectral density of the black-body radiation (the Plank’s radiation law) u as a function of the energy:  = average number of photons within this freq. range photon energy

3. Classical Limit (small, large ), Rayleigh-Jeans Law This equation predicts the so- called ultraviolet catastrophe – an infinite amount of energy being radiated by a black body at high frequencies or short wavelengths. Rayleigh-Jeans Law At low frequencies or high temperatures: - purely classical result (no h), can be obtained directly from the equipartition theorem

4. Rayleigh-Jeans Law (cont’d) In the classical limit of large : u as a function of the wavelength:

5. High Limit, Wien’s Displacement Law The maximum of u( ) shifts toward higher frequencies with increasing temperature. The position of maximum: Wien’s displacement law - discovered experimentally by Wilhelm Wien Numerous applications (e.g., non-contact radiation thermometry) - the “most likely” frequency of a photon in a blackbody radiation with temperature T u(,T) Nobel 1911 At high frequencies/low temperatures:

6. Solar Radiation The surface temperature of the Sun - 5,800K. As a function of energy, the spectrum of sunlight peaks at a photon energy of Spectral sensitivity of human eye: - close to the energy gap in Si, ~1.1 eV, which has been so far the best material for solar cells

7. Stefan-Boltzmann Law of Radiation The (average) photon density: The total energy of photons per unit volume : (the energy density of a photon gas) the Stefan- Boltzmann Law the Stefan-Boltzmann constant - increases as T 3 The average energy per photon: (just slightly less than the “most” probable energy)

8. Power Emitted by a Black Body For the “uni-directional” motion, the flux of energy per unit area c  1s energy density u 1m 2 T Integration over all angles provides a factor of ¼: Thus, the power emitted by a unit-area surface at temperature T in all directions: The total power emitted by a black-body sphere of radius R: (the hole size must be >> the wavelength) Consider a black body at 310K. The power emitted by the body: While the emissivity of skin is considerably less than 1, it still emits a considerable power in the infrared range. For example, this radiation is easily detectable by modern techniques (night vision). Some numbers:

9. Sun’s Mass Loss Beiser 9.22. The Sun’s mass is 2 ·10 30 kg, its radius is 7·10 8 m, and its surface temperature is 5,800K. Find the mass loss for the Sun in one second. How many years are needed for the Sun to lose 1% of its mass by radiation? This result is consistent with the flux of the solar radiation energy received by the Earth (1370 W/m 2 ) being multiplied by the area of a sphere with radius 1.5·10 11 m (Sun-Earth distance). the mass loss per one second 1% of Sun’s mass will be lost in

10. The Greenhouse Effect Transmittance of the Earth atmosphere Absorption: Emission: the flux of the solar radiation energy received by the Earth ~ 1370 W/m 2  = 1 – T Earth = 280K R orbit = 1.5·10 11 mR Sun = 7·10 8 m In reality  = 0.7 – T Earth = 256K To maintain a comfortable temperature on the Earth, we need the Greenhouse Effect ! However, too much of the greenhouse effect leads to global warming:

11. Problem The cosmic microwave background radiation (CMBR) has a temperature of approximately 2.7 K. (a) What wavelength λ max (in m) corresponds to the maximum spectral density u(λ,T) of the cosmic background radiation? (a) (b) What is approximately the number of CMBR photons hitting the earth per second per square meter [i.e. photons/(s·m 2 )]? (b) The average energy per photon:

12. Problem The frequency peak in the spectral density of radiation for a certain distant star is at 1.7 x 10 14 Hz. The star is at a distance of 1.9 x 10 17 m away from earth and the energy flux of its radiation as measured on earth is 3.5x10 -5 W/m 2. a) What is the surface temperature of the star? b) What is the total power emitted by 1 m 2 of the surface of the star? c) What is the total electromagnetic power emitted by the star? d) What is the radius of the star? (a) (b) (c) (d)

13. Problem a)What is the energy flux of the Sun’s radiation at Mercury's orbit? b)What is the total power absorbed by Mercury? [Hint: Consider that it appears as a flat disk to the Sun and it absorbs all of the incident radiation.] c) If Mercury is in thermodynamic equilibrium, it will emit the same total power as it receives from the Sun. Assuming that the temperature of the "hot“ side of Mercury is uniform, find this temperature. d) What is the peak frequency of the radiation absorbed by Mercury? e) What is the peak frequency of the radiation emitted by Mercury? (a) (b) (c)- hemi-sphere Planet Mercury revolves and rotates at the same rate, so one side of the planet is always facing the Sun. Mercury is a distance of 5.8 x 10 10 m from the Sun, and has a radius of 2.44 x 10 6 m. The radius of the Sun is 7·10 8 m and its total power output is 4 x 10 26 W. In this problem treat the planet as if it were a black body

14. Problem (cont’d) (d) (e)