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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 Solution The force Fe on the electric charge e due to the toroid causes a time rate of change of the mechanical momentum Pe of the electron according to Fe dPe dt 1 and likewise the force FT on the toroid changes the mechanical momentum PT of the latter according to dPT 2 FT dt The paradox which dates back to Ampe re is that the magnetic interaction of a moving charge and a current as well as the magnetic interaction of two moving charges does not in general obey Newton s third law Fe FT so that the total mechanical momentum of the system Pmech Pe PT is not constant in time in apparent violation of Newton s rst law for an isolated system The resolution of such paradoxes is that electromechanical systems in general possess an additional momentum PEM associated with the interaction of the charges and currents with the electromagnetic eld such that the total momentum of an isolated system Pe PT PEM in the present example is constant in time A further subtlety is that the sum Pmech PEM while constant may appear to have a nonzero value for an isolated system at rest However a hidden mechanical momentum Ph can be identi ed that restores the total momentum of a system at rest to zero 2 1 2 1 1 Analysis in the Lab Frame The Electromagnetic Momentum For systems in which e ects of radiation and of retardation can be ignored the electromagnetic momentum can be calculated in various equivalent ways 7 in Gaussian units PEM A dVol c E B dVol 4 c J dVol c2 3 where is the electric charge density A is the magnetic vector potential in the Coulomb gauge where A 0 E is the electric eld is the electric scalar potential and J is the electric current density The rst form is due to Faraday 8 and Maxwell 9 the second 1 Toroids with a simple helical winding have a net azumithal current that leads to an external magnetic eld The idealized toroidal eld considered here could be better approximated by a double helical winding with one winding in the opposite sense to the other See for example 4 2 This paradox was revived in 5 6 without reference to Cullwick 2 form is due to Poynting 10 and Abraham 11 and the third form was introduced by Furry 12 To calculate the electromagnetic momentum using the rst form of eq 3 we need the vector potential AT of the toroid at the position of the charge e but we do not need the vector potential of the charge since the toroid is assumed to be electrically neutral The vector potential of the toroid obeys AT BT BT 4 where the magnetic eld is BT 2I ac inside the toroid and zero outside and is a unit vector in the azimuthal direction in a cylindrical coordinate system z The toroid is centered on the origin with the z axis as its axis as shown in the gure below with radius b exaggerated for clarity Cullwick notes 3 that the relation 4 has the same form as Maxwell s equation for the magnetic eld due to a conducting wire that forms a solid torus of the same dimensions as the hollow toroidal magnet when the wire carries azimuthal current density J J 4 4 J J 5 c c From the Biot Savart law we know that the magnetic eld along the axis of the current loop is for b a 2 b2J a2 Bloop 0 0 z z 6 c z 2 a2 3 2 Comparing eqs 4 and 5 we see that on replacing 4 J in eq 6 by 2I a we obtain the vector potential on the axis of the toroid when b a Bloop AT 0 0 z b2I a z 2 c z a2 3 2 7 Hence the electromagnetic momentum of the system when charge e is at position z on the axis of the toroid is b2 Ie eAT 0 0 z a z 8 PEM 2 2 c c z a2 3 2 3 which is independent of the velocity of the charge To calculate the electromagnetic momentum using the second form of eq 3 we note that the electric eld at the toroid due to charge e has magnitude Ee e z 2 a2 on average and that the z component of Ee BT which is the only one remaining after the integral over the toroid volume is Ee BT a z 2 a2 Hence 3 PEM e 2I a b2Ie E e BT 2 a b2 a dVol 2 z z 2 2 2 4 c z a ac z 2 a2 4 c c z a2 3 2 9 For completeness we calculate the electromagnetic momentum using the third form of eq 3 We must keep the rst correction to the spatial dependence of the electric potential e of charge e over the toroid Referring to …


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