  We study the dynamics of a trapped, charged Brownian particle in presence of
a time dependent magnetic field. We calculate work distributions for different
time dependent protocols. In our problem thermodynamic work is related to
variation of vector potential with time as opposed to the earlier studies where
the work is related to time variation of the potentials which depends only on
the coordinates of the particle. Using Jarzynski identity and Crook's equality
we show that the free energy of the particle is independent of the magnetic
field, thus complementing the Bohr-van Leeuwen theorem. We also show that our
system exhibits a parametric resonance in certain parameter space.
