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

The magnetic Lorentz force is perpendicular to velocity, so it does no work and $\gamma$ remains constant over an orbit when radiative losses are neglected. For the transverse momentum, $p_{\perp }=\gamma m_e{v}_{\perp }$ and $|{\dot{\mathbf{p}}}_{\perp }|={\omega }_g{p}_{\perp }=eB{v}_{\perp }$. HenceSince $10\mu \mathrm{G}=10^{-9}\mathrm{T}$,This is the gyromotion frequency, not the much higher characteristic synchrotron radiation frequency.