Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 2 / 16D / d / Solution

For $\mathbf E=(0,0,E)$, $\mathbf B=0$, and initial momentum $(p_0,0,0)$,
$$
p_x=p_0,\qquad p_z=qEt,\qquad
\mathcal E(t)=\sqrt{\mathcal E_0^2+c^2q^2E^2t^2}.
$$
With the initial <position> at the origin,
$$
z(t)=\frac{\mathcal E(t)-\mathcal E_0}{qE},
\qquad
x(t)=\frac{cp_0}{qE}
\operatorname{arsinh}\frac{cqEt}{\mathcal E_0}.
$$

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 Incoming links: Relativistic motion in a constant electric field with transverse momentum