 Solution

For constant $r=R>0$, the radial equation requires $\dot{\theta }(\dot{\theta }+2e\mu /m)=0$. Select the stipulated nonzero angular velocity and define the positive cyclotron frequency ${\Omega }_c=2e\mu /m$. ThenThis is a circle in the transverse plane combined with an independent axial oscillation, as in uniform magnetic field with axial harmonic confinement. Viewed from positive $z$, positive charge rotates clockwise. Its three-dimensional path lies on the cylinder $r=R$ and reverses its vertical motion; it is generally not a constant-pitch helix. For $Z=0$ it is a horizontal circle. This orbit has $p_{\theta }=-e\mu R^2$.