  One of the puzzles to have emerged from the Kepler and TESS missions is the
existence of unexplained dips in the lightcurves of a small fraction of
rapidly-rotating M dwarfs in young open clusters and star-forming regions. We
present a theoretical investigation of one possible explanation - that these
are caused by dust clouds trapped in the stellar magnetic fields. The depth and
duration of the observed dips allow us to estimate directly the linear extent
of the dust clouds and their distances from the rotation axis. The dips are
found to be between 0.4-4.8%. We find that their distance is close to the
co-rotation radius: the typical location for stable points where charged
particles can be trapped in a stellar magnetosphere. We estimate the charge
acquired by a dust particle due to collisions with the coronal gas and hence
determine the maximum grain size that can be magnetically supported, the
stopping distance due to gas drag and the timescale on which dust particles can
diffuse out of a stable point. Using the observationally-derived magnetic field
of the active M dwarf V374 Peg, we model the distribution of these dust clouds
and produce synthetic light curves. We find that for 1 micron dust grains, the
light curves have dips of 1% - 3% and can support masses of order of
$10^{12}$kg. We conclude that magnetically-trapped dust clouds (potentially
from residual disc accretion or tidally-disrupted planetesimal or cometary
bodies) are capable of explaining the periodic dips in the Kepler and TESS
data.
