  We study plasmons in a rectangular two-dimensional (2D) electron system in
the vicinity of a planar metal electrode (gate) and in the presence of a
perpendicular uniform magnetic field, using Maxwell's equations and neglecting
retardation effects. The conductivity of the 2D system is characterized by the
dynamical Drude model without taking collisional relaxation into account, which
well describes both high mobility graphene and other field effect transistor
structures, including quantum wells like Ga(Al)As, in the terahertz and in some
cases sub-terahertz frequency ranges. Without a magnetic field, we analytically
find the current distribution and frequency of plasma eigenmodes when the
plasmon wavelength is much larger than the distance to the gate, i.e. in the
fully screened limit. To find an approximate solution in a magnetic field, we
expand current in the complete set of eigenmodes without magnetic fields.
Analytical asymptotic expressions in weak and strong magnetic fields were
obtained for the lower modes. Unlike the disk and stripe, the frequencies of
these modes tend to zero as the magnetic field tends to infinity. We also
discuss a direct analogy to rotational gravity shallow water waves, where
size-quantized Poincare waves correspond to size-quantized magnetoplasmons,
while Kelvin waves correspond to edge magnetoplasmons.
