  We study the radial and vertical stability of dust grains launched with all
charge-to-mass ratios at arbitrary distances from rotating planets with complex
magnetic fields. We show that the aligned dipole magnetic field model analyzed
by Jontof-Hutter and Hamilton (2012) is an excellent approximation in most
cases, but that fundamentally new physics arises with the inclusion of
non-axisymmetric magnetic field terms. In particular, large numbers of distant
negatively-charged dust grains, stable in a magnetic dipole, can be driven to
escape by a more complex field. We trace the origin of the instability to
overlapping Lorentz resonances which are extremely powerful when the
gravitational and electromagnetic forces on a dust grain are comparable. These
resonances enable a dust grain to tap the spin energy of the planet to power
its escape. We also explore the relatively minor influence of different launch
speeds and the far more important effects of variable grain charge. Only the
latter are capable of significantly affecting the micron-sized grains that
dominate visible and infrared images of faint dust rings. Finally, we present
full stability maps for Earth, Jupiter, Saturn, Uranus, and Neptune with
magnetic fields modeled out to octupole order. Not surprisingly, dust in the
tortured magnetic fields of Uranus and Neptune show the greatest instability.
