While the electromagnetic interaction between the photons and the charged electrons and positrons kept them in equilibrium with each other during the early universe, show that the direct electromagnetic Coulomb interaction energy between the electrons and positrons was small compared with the relativistic kinetic energy of those species. Show that the ratio between the Coulomb and kinetic energies is of the order of the fine structure constant:

\[\frac{u_{\text {coulomb }}}{u_{e}} \approx \alpha=\frac{e^{2}}{4 \pi \varepsilon_{0} \hbar c}=\frac{1}{137.036}\]