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I was recently assigned a problem in my modern physics class regarding finding the mean energy of Cosmic Microwave Background (CMB) photons. The problem reads as follows:

The universe is permeated with primordial microwave radiation that has a mean wavelength of about 2.5 mm.

(a) Find the mean energy (in eV) of the CMB photons, given that the energy of a photon is related to its frequency by the relationship E = hf (where f=frequency), where h is Planck's constant.

(b) Consider a cosmic-ray proton that has total relativistic energy E in the CMB reference frame. Find E such that the Doppler shifted CMB photons striking the proton head-on have a mean energy of 100 MeV.

I believe I have solved (a), with the solution as follows:

E = hf = h(c/wavelength)

E(mean) = h(c/mean wavelength)

= (4.136x10^-15 eV*s)[(3.0x10^8 m/s)/2.5x10^-3 m]

= 4.963 x 10^-4 eV

The second part, (b), I have no idea how to find. Any help would be much appreciated!

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# Homework Help: Finding mean energy of cosmic microwave background photons

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