= 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>, $p_\perp=\gamma m_ev_\perp$ and $|\dot{\mathbf p}_\perp|=\omega_gp_\perp=eBv_\perp$. Hence
$$
\boxed{\nu_g=\frac{\omega_g}{2\pi}=\frac{eB}{2\pi\gamma m_e}.}
$$
Since $10\ \mu\mathrm G=10^{-9}\ \mathrm T$,
$$
\boxed{\nu_g=2.80\times10^{-5}\ \mathrm{Hz},\qquad
\omega_g=1.76\times10^{-4}\ \mathrm{s^{-1}},\qquad T_g=3.57\times10^4\ \mathrm s\simeq9.9\ \mathrm h.}
$$
This is the <gyromotion> <frequency>, not the much higher characteristic <synchrotron radiation> <frequency>.
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