A relativistic particle of charge $q$ and mass $m$ moves in a background electromagnetic field. The four-velocity $u^{\mu}(\tau)$ of the particle at proper time $\tau$ is determined by the equation of motion,

$$m \frac{d u^{\mu}}{d \tau}=q F_{\nu}^{\mu} u^{\nu} .$$

Here $F_{\nu}^{\mu}=\eta_{\nu \rho} F^{\mu \rho}$, where $F_{\mu \nu}$ is the electromagnetic field strength tensor and Lorentz indices are raised and lowered with the metric tensor $\eta=\operatorname{diag}\{-1,+1,+1,+1\}$. In the case of a constant, homogeneous field, write down the solution of this equation giving $u^{\mu}(\tau)$ in terms of its initial value $u^{\mu}(0)$.

[In the following you may use the relation, given below, between the components of the field strength tensor $F_{\mu \nu}$, for $\mu, \nu=0,1,2,3$, and those of the electric and magnetic fields $\mathbf{E}=\left(E_{1}, E_{2}, E_{3}\right)$ and $\mathbf{B}=\left(B_{1}, B_{2}, B_{3}\right)$,

$$F_{i 0}=-F_{0 i}=\frac{1}{c} E_{i}, \quad F_{i j}=\varepsilon_{i j k} B_{k}$$

for $i, j=1,2,3 .]$

Suppose that, in some inertial frame with spacetime coordinates $\mathbf{x}=(x, y, z)$ and $t$, the electric and magnetic fields are parallel to the $x$-axis with magnitudes $E$ and $B$ respectively. At time $t=\tau=0$ the 3 -velocity $\mathbf{v}=d \mathbf{x} / d t$ of the particle has initial value $\mathbf{v}(0)=\left(0, v_{0}, 0\right)$. Find the subsequent trajectory of the particle in this frame, giving coordinates $x, y, z$ and $t$ as functions of the proper time $\tau$.

Find the motion in the $x$-direction explicitly, giving $x$ as a function of coordinate time $t$. Comment on the form of the solution at early and late times. Show that, when projected onto the $y-z$ plane, the particle undergoes circular motion which is periodic in proper time. Find the radius $R$ of the circle and proper time period of the motion $\Delta \tau$ in terms of $q, m, E, B$ and $v_{0}$. The resulting trajectory therefore has the form of a helix with varying pitch $P_{n}:=\Delta x_{n} / R$ where $\Delta x_{n}$ is the distance in the $x$-direction travelled by the particle during the $n$ 'th period of its motion in the $y-z$ plane. Show that, for $n \gg 1$,

$$P_{n} \sim A \exp \left(\frac{2 \pi E n}{c B}\right),$$

where $A$ is a constant which you should determine.