  We extend a recent classical mechanical analog of Bohr's atom consisting of a
scalar field coupled to a massive point-like particle [P. Jamet, A. Drezet, ``A
mechanical analog of Bohr's atom based on de Broglie's double-solution
approach'', Chaos 31, 103120 (2021)] by adding and studying the contribution of
a uniform weak magnetic field on their dynamics. In doing so, we are able to
recover the splitting of the energy levels of the atom called Zeeman's effect
within the constraints of our model and in agreement with the semiclassical
theory of Sommerfeld. This result is obtained using Larmor's theorem for both
the field and the particle, associating magnetic effects with inertial Coriolis
forces in a rotating frame of reference. Our work, based on the old `double
solution' theory of de Broglie, shows that a dualistic model involving a
particle guided by a scalar field can reproduce the normal Zeeman effect.
