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Cullwick’s Paradox



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Cullwick s Paradox Charged Particle on the Axis of a Toroidal Magnet Kirk T McDonald Joseph Henry Laboratories Princeton University Princeton NJ 08544 June 4 2006 updated June 13 2013 1 Problem In an induction linac 1 a toroidal solenoid magnet carries a time dependent current I t such that the induced electric eld can transfer energy from the magnet to charged particles that move along the axis of the toroid Discuss the force and momentum balance in an idealized induction linac consisting of a single magnet whose form is a torus of major radius a and minor radius b a and a single electron of charge e that moves along the symmetry axis of the toroid The current I is the total current crossing any major circle on the surface of the torus While actual induction linacs contain high permeability ferrites inside the toroid whose windings are made from shielded or unshielded conductors it su ces here to consider a nonconducting toroid without ferrites whose currents are due to electric charges xed on the rims of rotating disks Neighboring disks have opposite charges and rotate in opposite senses so that the net electric charge and the net mechanical angular momentum of the toroid is zero This con guration of a nonconducting toroid has no azimuthal current in contrast to a single layer helical winding on the toroid which includes in e ect a single azimuthal current loop You may assume that unlike the case of an induction linac the velocity v of the moving charge e of rest mass m is small compared to c the speed of light and that the time variation of the current in the toroid is slow enough that radiation and retarded e ects can be ignored Provide an analysis in the rest frame of the moving charge as well as in the lab frame i e the rest frame of the toroid 1 Cullwick 2 3 has noted that this example is paradoxical because no force is exerted on the moving charge when the current is constant in the toroid 1 but the moving charge exerts a nonzero force on the toroid 2 2



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