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In this question, we solve a paradox of a different type. For a fairly simple system of charges proposed by W. Shockley and R. P. James in 1967, understanding the conservation of linear momentum requires careful relativistic analysis. If a point charge is located near a magnet of changing magnetization, there's an induced electric force on the charge, but no apparent reaction on the magnet. The process may be slow enough that any electromagnetic radiation (and any momentum carried away by it) is negligible. Thus, apparently we get a cannon without recoil. In our analysis of this system, we will demonstrate that in relativistic mechanics, a composite body may hold a nonzero mechanical momentum while remaining stationary.
It is pointed out that Coleman and Van Vleck make a major blunder in their discussion of the Shockly-James paradox by designating relativistic “hidden mechanical momentum” as the basis for resolution of the paradox. This blunder has had a wide influence in the current physics literature, including erroneous work on the Shockley-James paradox, on Mansuripur’s paradox, on the motion of a magnetic moment, on the Aharonov-Bohm phase shift, and on the Aharonov-Casher phase shift. Although hidden mechanical momentum is indeed dominant for non-interacting particles moving in a closed orbit under the influence of an external electric field, the attention directed toward hidden mechanical momentum represents a fundamental misunderstanding of the classical electromagnetic interaction between a multiparticle magnet and an external point charge. 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.[....]Coleman and Van Vleck correctly refer to the requirement of special relativity that the total momentum of a steady-state system must vanish, and therefore they conclude that the magnet must contain some internal momentum; however, they mention only the possibility of relativistic mechanical internal momentum. For a magnet consisting of a single moving charge (or many non-interacting charges), the internal momentum is indeed mechanical. For a magnet consisting of a few interacting particles, the internal momentum will involve both mechanical and electromagnetic momentum. For an interacting-multiparticle magnet or for a few-interacting-particle magnet of very low velocity charges, the internal electromagnetic momentum dominates the internal mechanical momentum of the magnet, and this leads to qualitative changes in the momentum balance of the system consisting of a magnet and a point charge. Third, for a steady-state situation when the induced charge distribution of the magnet is essentially electrostatic, the internal electromagnetic momentum is equal in magnitude and opposite in direction from the familiar external electromagnetic momentum. Thus the total electromagnetic momentum of the system is zero.[25]
An alternate resolution to the “Mansuripur paradox”Francis Redfern∗Texarkana College, Texarkana, TX 75599(Dated: March 19, 2015)AbstractIn 2013 an article in the journal Science online declared that the paradox proposed by Masud Mansuripur was resolved. This paradox concerns a point charge-Amperian magnetic dipole system as seen in a frame of reference where they are at rest and one in which they are moving. In the latter frame an electric dipole appears on the magnetic dipole. A torque is then exerted upon the electric dipole by the point charge, a torque that is not observed in the at-rest frame. Mansuripur points out this violates the relativity principle and suggests the Lorentz force responsible for the torque be replaced by the Einstein-Laub force. The “resolution” of the paradox reported by Science, based on numerous papers in the physics literature, preserves the Lorentz force but depends on the concept of “hidden momentum”. Here I propose a different resolution based on the overlooked fact that the charge-magnetic dipole system contains linear and angular electromagnetic field momentum. The time rate of change of the field angular momentum in the frame through which the system is moving cancels that due to the charge-electric dipole interaction. From this point of view hidden momentum is superfluous.[....]Mansuripur’s argument is that the the Lorentz formalism needs to be corrected by including the angular momentum density of the magnetization of the magnetic dipole that results from Einstein-Laub, r×(ϵoE ×M), where M is the magnetization. This term cancels the supposed hidden angular momentum density to resolve the paradox [20]. As it turns out, this term “fixes” the problem by accounting for the lack of consideration of the role of the field angular momentum when the Lorentz formalism is not properly applied. Hence, there is no paradox in either the Lorentz and Einstein-Laub formalisms if the magnetic dipole is Amperian[.]