Past exam of the physics course of the year 4 of the natural sciences course of the University of Cambridge / 2013 / Paper 1 / Section A / A7 / Solution

Relativistic synchrotron radiation is concentrated within angle of order $1/\gamma$ around the instantaneous velocity, tangent to the circular orbit. The narrow instantaneous beam sweeps around with the particle; over a whole orbit the emission forms a thin fan about the orbital plane. With $E\gg m_e{c}^2$, $\gamma \simeq 5000/0.5=10^4$. Since ${\nu }_{\mathrm{orb}}\simeq c/(2\pi R)$, the supplied characteristic-frequency scaling givesThis is subnanometer X-ray radiation, with a beaming angle of order $10^{-4}\mathrm{rad}$. The conventional critical-angular-frequency normalization ${\omega }_c=3{\gamma }^{3}c/(2R)$ gives $4.2\times 10^{-10}\mathrm{m}$ instead; the question's order-of-magnitude frequency prescription does not distinguish these order-one factors.