Past exam of the astrophysics course of the University of Oxford / 2022 / 4 / a / Solution

An ultrarelativistic electron has kinetic energy much larger than its rest energy: $(\gamma -1)m_e{c}^2\gg m_e{c}^2$, equivalently $\gamma \gg 1$. Its speed obeys $v/c=\sqrt{1-{\gamma }^{-2}}\simeq 1-1/(2{\gamma }^2)$ and its total energy is approximately $pc$. At $\gamma =10^6$, $E=\gamma m_e{c}^2\simeq 511\mathrm{GeV}$.