Particles which move in the magnetosphere's plasma gain an electric charge which depends on the density and temperature of the plasma and the sunlight stream. If motion is slow enough, it is possible to consider that the micro-particle's electric charge is in quasi-equilibrium. For certain conditions, the Hamilton function can be written for the problem with a variable electric charge and, hence, the methods of the analysis of systems of Hamilton equations can be applied for research of such micro-particle motion. Although these conditions are strong enough, they correspond to the statement of many real problems. The spatial distribution of plasma in the Earth's plasmasphere is described by a model of a two-component plasma. In the present paper, the capability of propagation of results which were received earlier for a case of the motion of a quasi-equilibrium electric charge in the Earth plasmasphere has been shown. The received result shows that there is an opportunity for long orbital holding (not less than one month) of micro-particles of space dust in the Earth's plasmasphere.