  Inspired by a question of Colin de Verdi\`{e}re and Truc we study the
dynamics of a classical charged particle moving in a bounded planar domain
$\Omega$ under the influence of a magnetic field $\mathbf{B}$ which blows up at
the boundary of the domain. We prove that under appropriate blow-up conditions
the particle will never reach the boundary. As a corollary we obtain
completeness of the magnetic flow. Our blow-up condition is that $\mathbf{B}$
should not be integrable along normal rays to the boundary, while its
tangential derivative should be integrable along those same rays.
