A numerical integration program was developed to follow the paths of dust particles in the Jovian ring, including the acceleration due to gravity and the Lorentz and drag accelerations arising from the motions of the charged dust through the Jovian plasma. Particles were assumed to start from circular, noninclined Keplerian orbits. The orbit of a 2.5-&mgr;-radius spherical dust particle of density 2 g/cm3-10V will become significantly perturbed. The ring will tend to warp northwards near 130--160¿ longitude (where the longitudinal component of the Jovian magnetic field is strongest); the maximum excursion of these Jupiter ring grains is about 0.1¿, consistent with a distance of 220 km above the equatorial plane. This distance is more than an order of magnitude larger than the observed upper limit to the half-thickness of the ring. Either the particles are larger or the voltages on them less than what has been inferred by previous workers. The plasma near the ring may be considerably cooler than was estimated. Particles of 0.3 &mgr; with -10 V potentials, starting as before in circular orbits near the outer edge of the ring (1.78 RJ), are spread from 1.68 to 198 RJ and inclined up to 7¿ out of the equatorial plane. Their paths do not follow Keplerian orbits, and particle positions are not symmetric about the equatorial plane. Particles of 0.4-&mgr; radius show much less asymmetry in their orbits than the 0.3-&mgr; particles, while particles of 0.2 &mgr; or less are perturbed into the Jupiter cloudtops within a few tens of hours.