Blackbody Radiation Thomas Wedgwood 1792 Objects in Kilns All Become Red at Same Temperature Independent of Composition

Blackbody Radiation Thomas Wedgwood 1792 objects in kilns all become red at same temperature independent of composition

Blackbody Radiation various observers in mid-1800s spectrum is continuous in contrast to discrete spectrum of heated gases

infrared: melting ice cube

visible: spectrum of Hg vapor

Blackbody Radiation Gustav Kirchhoff 1859 for an object in thermal equilibrium with radiation its emitted radiation is equal to the radiation it absorbs • a good absorber is a good emitter P universal function = Rtot = eJ (T ) for all objects: as A yet unknown power per area

• emissivity e = 1 for perfect absorber (blackbody) Blackbody Radiation

Josef Stefan 1879 discovers universal function • Stefan’s Law – empirically determined P = R = eσT 4 A tot W σ = 5.67 × 10−8 m2K 4 • Ludwig Boltzmann derives this later

Blackbody Radiation blackbody radiation distribution • continuous • well-deﬁned peak wavelength • long-wavelength tail, asymmetric • area under curve proportional to power P R(λ) power per area per wavelength interval, such that: R = R λ dλ tot ∫ ( )

Blackbody Radiation Wilhelm Wien 1893 • relates peak wavelength to object temperature • Wien’s Displacement Law – empirically determined

−3 λpeakT = 2.898 × 10 mK but... what physical mechanism is responsible for R ( λ ) ? Blackbody Radiation classical Rayleigh-Jeans Equation • model: radiation is composed of standing waves in a cavity 1 R(λ) = cEaven(λ) 4 number density of modes: how many ways per unit 8π volume can radiation of a n(λ) = 4 given wavelength oscillate λ in cavity

2πc but... what is the average R(λ) = 4 Eave energy per mode of λ oscillation?

Blackbody Radiation Maxwell-Boltzmann energy distribution statistical thermodynamics: given a ﬁxed amount of energy and a ﬁxed number of particles, how are particles arranged in available energy states? Boltzmann factor: probability E − k T of occupying state of energy P(E) ∝ e B E what happens as E gets large compared to kT?

Blackbody Radiation ... so the average energy per mode of oscillation is...

∞ EP E dE average = value*probability ∫ ( ) N.B. energy is a 0 continuous Eave = ∞ quantity in ∫ P(E)dE must normalize classical physics 0 ∞ − E ∫ Ee kBT dE 0 = ∞ = ... = kBT − E ∫ e kBT dE 0 Blackbody Radiation Rayleigh-Jeans Equation

2πc 2πckBT R(λ) = Eave = (whew!) λ 4 λ 4

The Ultraviolet R(λ) Catastrophe (oh no!)

Blackbody Radiation Max Planck 1895 • model: cavity resonators (oscillating electric charges) emit and absorb only discrete energies n is an integer En = nhf f is frequency of oscillation h is a constant of proportionality

The Quantum Hypothesis

Boltzmann factor: probability E nhf − n − of occupying state of energy kBT kBT Pn (En ) ∝ e = e E=nhf N.B. energy is quantized Blackbody Radiation ... so the average energy per mode of oscillation is...

∞ average = value*probability ∑EnPn (En ) n=0 Eave = ∞ ∑Pn (En ) must normalize n=0 ∞ nhf − ∑nhfe kBT n=0 hc λ = ∞ nhf = hc − λk T ∑e kBT e B −1 n=0

Blackbody Radiation Planck Equation

2 5 2πc 2πhc λ where h = 6.63×10−34 J ⋅s R(λ) = 4 Eave = hc λ e λkBT −1 at long wavelengths: hc hc e λkBT −1 ≈ 1+ −1 so λk T B agrees with 2 5 2 5 2πhc λ 2πhc λ 2πckBT Rayleigh-Jeans R(λ) = hc ≈ = 4 λkBT hc k T equation at e −1 λ B λ long wavelengths

Blackbody Radiation Planck Equation

2 5 2πc 2πhc λ h = 6.63×10−34 J ⋅s R(λ) = 4 Eave = hc where λ e λkBT −1 at short wavelengths: hc hc 2πhc2 λkBT λkBT e −1 ≈ e so R(λ) = hc λ 5e λkBT