Particle motion in electric and magnetic fields (due 2025 Nov 10)

The motion of a particle with charge \(q\) and mass \(m\) under the influence of an electric field \(\vec{E}\) and magnetic field \(\vec{B}\) is given by (RL, Eq. 4.85b, but in SI units), \[ \frac{d\vec{p}}{dt} = q \left(\vec{E} + \vec{v} \times \vec{B}\right), \] where \(\vec{p}=m\vec{u}=m\gamma\vec{v}\) is the spatial part of the 4-momentum (with \(\vec{u}\) the spatial part of the 4-velocity and \(\vec{v}=d\vec{x}/dt\) the 3-velocity), and \(\gamma=\sqrt{1+(u/c)²} = 1 / \sqrt{1-(v/c)²}\) the usual lorentz factor.

1. Derive the motion of a particle initially at rest at the origin in a constant electric field \(\vec{E}=E_{0}\hat{x}\). It is easiest to start by deriving \(u_{\rm x}(t)\). Show that \(\gamma-1\propto{}t^{2}\) at early times and \(\gamma\propto{}t\) at late times. What is the physical interpretation of the timescale at which the behaviour changes? Make plots of \(u_{\rm x}(t)\), \(v_{\rm x}(t)\), \(\gamma(t)\), and \(x(t)\).
2. Derive the motion of a particle with initial velocity \(v_{0}\hat{y}\) in a constant magnetic field \(\vec{B}=B_{0}\hat{z}\). Again, it is easiest to work with \(\vec{u}\), writing out \(d\vec{u}/dt\) and showing that this implies that \(d\gamma/dt=0\). After that, take another derivative with time to derive the solution (or just show that it works by inserting the known solution).

Simulate the motion of a particle. I recommend first reading the description of various integration methods in Ripperda et al., 2018ApJS..235...21R, in particular §2.1.1 for the relativistic version of the Boris method. After that, you may decide to write your own code implementing it, or you can start with my version of the boris pusher written by L. Fuster & G. Bogopolsky (where I extended it to include relativity; it also uses astropy units, is SI-based, and allows simulating multiple particles at the same time).

1. Verify your code by reproducing the analytic results above (include some comparison plots).
2. Try combinations, first with \(\vec{E}\) as a small perturbation on \(\vec{B}\) (see Pad §3.8.5 to guide expectations), then vice versa, and finally with both chosen such that their effects have similar magnitude.
3. Try something more complex of your own choice (for this part in particular, collaboration and cross-verification are greatly encouraged!). Examples might be calculating trajectories in a dipole magnetic field, including mirroring; verifying the Bremsstralung results; including the effects of radiation reaction; calculating E and B from the particles (the latter needs retarded potentials, so more difficult and possibly quite slow; see my accelerating charge example). Do ask if uncertain.