  We examine the dynamics of electron beams that, in free space, are
self-accelerating, in the presence of an additional magnetic field. We focus
our attention in the case of Airy beams that follow parabolic trajectories and
in generalized classes of beams associated with power-law trajectories. We
study the interplay between beam self-acceleration and the circular motion
caused by the magnetic field. In the case of Airy beams, using an integral
representation, we find closed-form solutions for the electron wavefunction. We
also derive asymptotic formulas for the beam trajectories both for Airy beams
and for self-accelerating power-law beams. A ray optics description is rather
useful for the interpretation of the beam dynamics. Our results are in
excellent comparison with direct numerical simulations.
