 Relativistic motion in parallel electric and magnetic fields

When uniform $\mathbf{E}$ and $\mathbf{B}$ are parallel, the relativistic Lorentz force splits into independent hyperbolic motion along the fields and circular motion in the perpendicular plane. The circle is periodic in proper time, while electric acceleration makes the helix's longitudinal pitch grow exponentially from one proper-time cyclotron period to the next.