 Solution

For a $5$ GeV electron, ${\gamma }_e=E/(m_e{c}^2)\simeq 9785$. With $\rho =10$ m, the stated synchrotron critical frequency formula givesAt the same total beam energy and bend radius, a proton has ${\gamma }_p=5/0.9383\simeq 5.329$, and the same supplied formula givesThe ratio is $({\gamma }_p/{\gamma }_e{)}^3\simeq (m_e/m_p{)}^3$, explaining the strong suppression for protons. The critical-energy formula is an ultrarelativistic approximation, excellent for the electron and only approximate for this moderately relativistic proton. If “5 GeV proton beam” instead denotes kinetic energy, then ${\gamma }_p=1+5/0.9383$ and the estimate is about $7.50\times 10^{-6}$ eV; the two beam-energy conventions should not be mixed.