Blackbody radiation and Plank s law

Transcription

1 lakbody radiation and Plank s law blakbody problem: alulating the intensity o radiation at a given wavelength emitted by a body at a speii temperature Max Plank, 900 quantization o energy o radiation-emitting osillators: only ertain energies or the radiation-emitting osillators in the avity wall are allowed Albert Einstein, 905 extended quantization o energy o radiation-emitting osillators to quantization o light photons Niels ohr, 93 quantum model o the atom explained the photoeletri eet

2 lakbody radiation and Plank s law blakbody is an objet that absorbs all eletromagneti radiation alling on it an onsequently appears blak the opening to the avity is a good approximation o a blakbody: ater many reletions all o the inident energy is absorbed radiation is in thermal equilibrium with the avity (e.g. oven avity) beause radiation has exhanged energy with the walls many times similar to the thermal equilibrium o a luid with a ontainer spetral energy density u(, T) is energy per unit volume per unit requeny o the radiation within the blakbody avity u(, T) depends only on temperature and light requeny and not on the physial and hemial makeup o the blakbody all the objets in the oven, regardless o their hemial nature, size, or shape, emit light o the same olor blakbody is a peret absorber and ideal radiator

3 Max Plank, 900 spetral energy density o blakbody radiation u(, T) h k 3 h 3 h k T e J s is Plank's onstant J/K is oltzmann's onstant radiated d intensity towards ultraviolet atastrophe lassial Rayleigh-Jeans law quantum Plank law limiting behaviors at high requenies h k T >> at low requenies h k T << h kt e h kt e 8 8 u(, T) e 3 3 π h π h h kt e 3 h k T 3 kt h kt e + h k T +... h h h k T u(, T) e h h k T 3 3 k T Wien s exponential law, 893 lassial Rayleigh-Jeans law Rayleigh-Jeans law is the lassial limit obtained when h 0

4 Max Plank: blakbody radiation is produed by vibrating submirosopi eletri harges, whih he alled resonators the walls o a avity are omposed o resonators vibrating at dierent requeny Classial Maxwell theory: An osillator o requeny ould have any value o energy and ould hange its amplitude ontinuously by radiating any ration o its energy Plank: the total energy o a resonator with requeny ould only be an integer multiple o h. (During emission or absorption o light) resonator an hange its energy only by the quantum o energy Eh E nh where n,,3,... E h Ε h 3h h Plank: allowed energy levels o a resonator h 0 photon absorption photon emission

5 all systems vibrating with requeny are quantized and lose or gain energy in disrete pakets or quanta Eh Consider a pendulum m0. kg, l m, displaed by θ0 0 ( )( )( )( ) E mgl θ 0 ( os ) 0. kg 9.8 m/s m os0.5 0 J g 9.8 m/s 0.5 Hz π l π m 3 3 E h J s 0.5 s J ( )( ) 3 E J. 0 E.5 0 J E is large is large E is small m is small 3 quantum o energy lassial physis: quantization is unobservable bl and energy loss or gain looks ontinuum quantum physis energy hange o atomi osillator sending out green light ( J s )( m/s ) 9 h E h J.3 ev 9 λ 50 0 m a more appropriate unit o energy or desribing atomi proesses 9 ev.60 0 J

6 Max Plank: u(, T ) d EN( ) d average energy emitted per osillator number o osillators with requeny between and +d N( ) d d 3 E h e h kt h u(, T) d d 3 h kt e h u( λ, T) dλ dλ 5 h kt λ e λ λ Plank distribution untion intensity radiated towards ultraviolet atastrophe lassial Rayleigh-Jeans law quantum Plank a law in lassial Rayleigh-Jeans theory E kt u (, T ) d k 3 Td or lassial theory predits unlimited energy emission in the ultraviolet region ultraviolet atastrophe in quantum theory ultraviolet atastrophe is avoided: E tents to zero at high beause the irst allowed energy level Eh is so large or large ompared to the average thermal energy available k T that oupation o the irst exited state is negligibly small

7 spetral energy density u(, T) is energy per unit volume per unit requeny o the radiation within the blakbody avity u( λ, T) h 5 h λ e λ k T J (, T) u(, T ) power density e J(, T) is power emitted per unit area per unit requeny Wien s displaement law, 893: the wavelength marking the maximum power emission o a blakbody, λ max, shits towards shorter wavelengths with inreasing temperature λmax ~ T 3 λ T m K max Stean-oltzmann law, 879: The total power per unit area emitted at all requenies by a blakbody, e total,, is proportional to the orth power o its temperature e e d σt σ total W m K The Stean-oltzmann onstant 3 5 πh πk T x π k etotal u λ T dλ dλ dx T σt 3 x h 0 e 5 h (, ) 5 h kt λ ( e λ ) x h λk T π 5

8 or perpendiular radiated energy derivation o J(, T) u(, T) δxδt radiated power area A beause hal the power will be going in the x diretion radiated power [ energy density] volume time A δx A δx u(, T) u(, T) u(, T) A δ t δ x or any angle θ volume perpendiular radiated power [ energy density] osθ time A δx A δx u(, T) os θ u(, T) os θ u(, T) Aos θ δt δx os θ averaging over θ and dividing by A to get power density radiated power u (, T ) A os θ (, ) J T u(, T ) A A os θ

9 Estimate the surae temperature o the Sun and ind λ max or the Sun emission. The Sun radius R S 7.0x0 8 m The Earth-to-Sun distane R.5x0 m The total power rom the Sun at the Earth e total 00 W/m etotal ( RS ) σt Stean-oltzmann law e ( R ) π R e ( R ) π R onservation o energy ttl total S S total ttl etotal R R σ RS ( ) T etotal ( RS ) σ ( 00 W/m ) (.5 0 m) 8 ( ) T 5800 K W/m K m m K m K 9 λmax m 500 nm T 5800 K eye s sensitivity peak Wien s displaement law

10 Plank, 900: osillators in the walls o the blakbody are quantized Einstein, 905: light itsel is omposed o quanta o energy photon a quantum o eletromagneti radiation (a quantum o light) E x E x photon Eh photons Eh 3 photons E3h Ephoton h the photoeletri eet

11 Einstein, 906: in addition to arrying energy Eh a photon arries a momentum pe/h/ direted along its line o motion Peter Debye, Arthur Holly Compton, 93: sattering o x-rays photons rom eletrons ould be explained by treating photons as partiles with energy h and momentum h/ and by onserving energy and momentum o the photon-eletron pair in a ollision partile properties p o light x-rays are eletromagneti waves with short wavelengths Frequeny () photon ht energy h h h kv kev 0 λ 0 m x-rays were disovered by Wilhelm Roentgen in 895: X-rays are generated tdwhen high-speed eletrons strike a metal target and give up some o their energy when they interat with the orbital eletrons o an atom