Limits on weak magnetic confinement of neutral atoms

It is shown that when a magnetic field is used to support neutral atoms against the gravitational force mg, the total curvature of the field magnitude B must be larger than m^2 g^2/(2 \mu^2 B), where mu is the magnetic moment of the atoms. This limits the minimum confinement strength obtainable for a trapped atomic gas. It is also conjectured that the curvature must be larger than twice this value for a magnetic potential that varies in only one or two dimensions, such as an atomic waveguide.