A particle of mass $m$ and charge $q$ moves relativistically under the influence of a constant electric field $E$ in the positive $z$-direction, and a constant magnetic field $B$ also in the positive $z$-direction.

In some inertial observer's coordinate system, the particle starts at

$$x=R, \quad y=0, \quad z=0, \quad t=0$$

with velocity given by

$$\dot{x}=0, \quad \dot{y}=u, \quad \dot{z}=0,$$

where the dot indicates differentiation with respect to the proper time of the particle. Show that the subsequent motion of the particle, as seen by the inertial observer, is a helix.

a) What is the radius of the helix as seen by the inertial observer?

b) What are the $x$ and $y$ coordinates of the axis of the helix?

c) What is the $z$ coordinate of the particle after a proper time $\tau$ has elapsed, as measured by the particle?