130_notes.dvi

Lets sety=hν/kTand solve the equation

y= 5(1−e−y)

I solved this iteratively starting at y=5 and goty= 4.97 implying

hν= 4. 97 kT

If we takeλ= 500 nm, themν= 6× 1014.

T=

hν

5 k

=

(6. 6 × 10 −^34 )(6× 1014 )

(5)(1. 4 × 10 −^23 )

= 6× 103 = 5700

That’s agrees well.

2.6.2 Black Body Radiation from the Early Universe*.

Find the frequencyνat which the the EmissiveE(ν,T) is a maximum for the 2.7 degree cosmic
background radiation. Find the wavelengthλfor whichE(λ,T) is a maximum.

The cosmic background radiation was produced when the universe was much hotter than it is
now. Most of the atoms in the universe were ionized and photons interacted often with the ions
or free electrons. As the universe cooled so that neutral atoms formed, the photons decoupled
from matter and just propagated through space. We see these photons today as the background
radiation. Because the universe is expanding, the radiation has been red shifted down to a much
lower temperature. We observe about 2.7 degrees. The background radiation is very uniform but
we are beginning to observe non-uniformities at the 10−^5 level.

From the previous problem, we can say that the peakλoccurs when

hν = 5kT

ν = 5kT/h

λ = ch/(5kT) =

(3× 108 )(6. 6 × 10 −^34 )

(5)(1. 4 × 10 −^23 )(2.7)

= 1mm

Similarly the peak inνoccurs when

ν= 2. 8 kT/h=

(1. 4 × 10 −^23 )(2.7)

(6. 6 × 10 −^34 )

= 6× 1010 Hz

2.6.3 Compton Scattering*

Compton scattered high energy photons from (essentially) free electronsin 1923. He
measured the wavelength of the scattered photons as a functionof the scattering angle. The figure
below shows both the initial state (a) and the final state, with the photon scattered by an angleθ
and the electron recoiling at an angleφ. The photons were from nuclear decay and so they were of
high enough energy that it didn’t matter that the electrons were actually bound in atoms. We wish
to derive the formula for thewavelength of the scattered photon as a function of angle.