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The problem of a charged particle moving in an arbitrary prescribed trajectory near a perfectly conducting sphere is formulated in terms of the magnetic-field integral equation on the sphere. The integral equation is solved with the aid of the solutions of the associated eigenvalue problem. A detailed study of the canonical problem, where the charged particle moves around a sphere in a circular orbit, reveals that a sufficiently accurate solution can be obtained by solving two independent quasi-static problems. Validity criteria are established for the quasi-static solutions in terms of the particle's speed and distance from the sphere. These quasi-static problems can be easily generalized and solved for particles of arbitrary motion and conductors of arbitrary shape. Induced surface currents and charges on the sphere as well as the rate of energy loss of the charged particles are calculated.