Origin of "Hidden Momentum Forces" on Magnets

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Abstract
Shockley and James have noted that an electrically neutral magnet whose moment is changing with time exerts a force on an electric charge of negligible velocity at a large distance from it, but that there is no obvious corresponding back-action of the charge on the magnet, although required by general considerations of the conservation of momentum. In the present paper, it is shown that the back-force is a consequence of the relativity corrections to the motion of the particles composing the magnet. The proof is given generally in terms of the relativistic theorem on the motion of the "center of energy" and explicitly in terms of a Lagrangian for a system of particles obtained many years ago by Darwin. The effect of spin is examined and found not to affect the action-reaction balance. In the Appendix, the properties of the center of energy are utilized to show how the Darwin Lagrangian should be modified when there are nonelectric classical forces acting between the particles of the magnet.

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