It is pointed out that the interaction of a magnet and a point charge has not been properly understood because the mutual interactions of the magnet's current carriers have been neglected. The magnet-point-charge interaction is important for understanding some theoretical paradoxes, such as the Shockley-James paradox, and for interpreting some experimentally observed effects, such as the Aharonov-Bohm and Aharonov-Casher phase shifts. Coleman and Van Vleck provide a discussion of the Shockley-James paradox where they note that internal relativistic mechanical momentum (hidden momentum) can be carried by the current carriers of the magnet. Although internal mechanical momentum is indeed dominant for noninteracting particles moving in a closed orbit under the influence of an external electric field, the presence of interactions among the magnet's current carriers leads to an internal electromagnetic momentum, which does not seem to be recognized in the physics literature. In the interacting multiparticle situation, the external charge induces an electrostatic polarization of the magnet, which leads to an internal electromagnetic momentum in the magnet where both the electric and magnetic fields for the momentum are contributed by the magnet particles. This internal electromagnetic momentum for the interacting multiparticle situation is equal in magnitude and opposite in direction compared to the familiar external electromagnetic momentum where the electric field is contributed by the external charged particle and the magnetic field is that due to the magnet. In the present article, the momentum balance of the Shockley-James situation for a system of a magnet and a point charge is calculated in detail for a magnet model consisting of two interacting point charges, which are constrained to move in a circular orbit on a frictionless ring with a compensating negative charge at the center.