Question

The cosmic microwave background radiation is light left over from the Big Bang that has been Doppler-shifted to microwave frequencies by the expansion of the universe. It now fills the universe with 450 photons/cm³ at an average frequency of 160 GHz. How much energy from the cosmic microwave background, in MeV, fills a small apartment that has 95 m² of floor space and 2.5-m-high ceilings?

Short answer

Energy is equal to 70.72 MeV

Step-by-step solution

Step 1. Given information is :Universe is filled as 450 photons/cm3Average frequency = 160 GHz

We need to find out the energy from the cosmic microwave background, in MeV, that fills a small apartment that has 95 m² of floor space and 2.5-m-high ceilings.

Step 2. Using the density of photons.

$$\begin{gathered} n=\frac{N}{V} \\ nisthedensity. \\ V=Ah \\ Aisthearea \\ histheheight \\ N=nV \\ {f}_{net}=N{f}_{avg.} \\ {f}_{net}isthenetfrequencyand{f}_{avg.}istheaveragefrequency. \\ {f}_{net}=nAh{f}_{avg.} \\ {E}_{net}=h{f}_{net} \\ {E}_{net}=hnAh{f}_{avg.} \\ {E}_{net}=\left(6.63\times {10}^{-34}\right)\left(450\times {10}^6\right)\left(Ah\right)\left(160\times {10}^{9}Hz\right) \\ {E}_{net}=1.133\times {10}^{-11}J=70.72MeV \end{gathered}$$