  A classical analysis suggests that an external magnetic field can cause
trajectories of charge carriers on a superconducting domain wall or cosmic
string to bend, thus expelling charge carriers with energy above the mass
threshold into the bulk. We study this process by solving the Dirac equation
for a fermion of mass $m_f$ and charge $e$, in the background of a domain wall
and a magnetic field of strength $B$. We find that the modes of the charge
carriers get shifted into the bulk, in agreement with classical expectations.
However the dispersion relation for the zero modes changes dramatically --
instead of the usual linear dispersion relation, $\omega_k =k$, the new
dispersion relation is well fit by $\omega \approx m_f tanh(k/k_*)$ where
$k_*=m_f$ for a thin wall in the weak field limit, and $k_*=eBw$ for a thick
wall of width $w$. This result shows that the energy of the charge carriers on
the domain wall remains below the threshold for expulsion even in the presence
of an external magnetic field. If charge carriers are expelled due to an
additional perturbation, they are most likely to be ejected at the threshold
energy $\sim m_f$.
