= 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 $E\gg m_ec^2$, $\gamma\simeq5000/0.5=10^4$. Since $\nu_{\rm orb}\simeq c/(2\pi R)$, the supplied characteristic-frequency scaling gives
$$
\boxed{\lambda_{\rm char}\sim\frac{2\pi R}{\gamma^3}\simeq6.3\times10^{-10}\,\mathrm m.}
$$
This is subnanometer X-ray radiation, with a beaming angle of order $10^{-4}\,\mathrm{rad}$. The conventional critical-angular-frequency normalization $\omega_c=3\gamma^3c/(2R)$ gives $4.2\times10^{-10}\,\mathrm m$ instead; the question's order-of-magnitude <frequency> prescription does not distinguish these order-one factors.
\Image[/past-exam-of-the-physics-course-of-the-year-4-of-the-natural-sciences-course-of-the-university-of-cambridge/2013/paper-1-synchrotron-beaming.png]
{title=Instantaneous synchrotron beaming along the tangent; opening exaggerated for visibility}
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