Classical Electron in the Field of a Magnetic Monopole

To date, there is no conclusive experimental evidence for the existence of magnetic monopoles, the magnetic analog of electric charges. All known magnetic effects arise from magnetic dipoles or from electrical currents. Much of the interest in magnetic monopoles started with a proposal by P. A. M. Dirac in 1931 [1] that if even one magnetic monopole exists in the Universe, a necessary consequence is the quantization of electric charge. Several theories beyond the Standard Model, proposed by 't Hooft, Polyakov, and others, also predict the existence of monopoles. Magnetic monopoles added to Maxwell's equations would create a theory of higher symmetry than the present version of electrodynamics [2].
This Demonstration is concerned with the interaction between a classical electric charge and a magnetic monopole. The problem was actually solved a long time ago by Poincaré [3] (see also [4] and [5]). For historical continuity, we use Gaussian electromagnetic units. The field of a point monopole of magnetic charge is given by , in complete analogy with Coulomb's law for an electric charge. The Lorentz force on an electron of mass and charge moving with velocity is then , leading to Newton's equation of motion .
The equations of motion can be completely solved with a few vector operations. Taking the scalar product with , we have ; therefore , so that the speed of the electron is a constant (for ). Taking the scalar product with gives , so that , with the solution , where , the initial separation of the electron and monopole and .
Taking the vector product of with Newton's equation, we find . The orbital angular momentum is evidently not a constant of the motion (although its magnitude is). Note also that . Instead, the appropriate constant of the motion is the vector (somewhat reminiscent of the Runge–Lenz vector for the Coulomb problem). The angular momentum of the electromagnetic field can be calculated from , a result first obtained by J. J. Thomson. Thus the vector , sometimes called the Poincaré vector, represents the total angular momentum: mechanical plus electromagnetic. It is shown by a blue arrow. Since is perpendicular to , . Also, , thus the trajectory of the position vector is evidently confined to the surface of a right circular cone (the Poincaré cone) with constant slant angle with respect to the axis .
The motion of about the origin with an angular velocity determines the -dependence of the trajectory. We have , so that integration gives
The electron spin is, in its lowest energy state, parallel everywhere to the magnetic field and does not contribute to the motion.
In the graphic, all variables are scaled relative to . The monopole is marked with a blue cross, while the electron's trajectory is show in red. The trajectory is actually a geodesic on the surface of the cone, which would follow a straight line if the cone were unrolled. To keep within the scale of the diagram, the values of are limited to be between 1.001 and 1.005 (multiples of ).


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The existence of magnetic monopoles would imply the quantization of electric charge. As a major implication in Dirac's paper [1], this is perhaps the most intriguing aspect of magnetic monopoles.
The electromagnetic vector potential is related to the magnetic induction by . With for a magnetic monopole, two possible forms of the vector potential are and . The first form is singular for along the negative axis, while the second is singular for along the positive axis. For values of or , either form is valid and they must be related by a gauge condition: , where .
Gauge invariance in quantum mechanics requires that alternative representations of the wavefunction differ by a phase factor χ/ℏ ), in the present case . Comparing the wavefunctions for and we should find . This implies that , , so that the electric charge (as well as the magnetic pole strength ) must therefore occur as integral multiples of some elementary magnitude.
[1] P. A. M. Dirac, "Quantized Singularities in the Electromagnetic Field," in Proceedings of the Royal Society (A133), London: The Royal Society, 1931 pp. 60–72.
[2] J. D. Jackson, Classical Electrodynamics, 3rd ed., New York: John Wiley & Sons, 1999 pp. 273–280.
[3] H. Poincaré, Comptes Rendus, 123, 1896 pp. 520–521.
[4] I. R. Lapidus and J. L. Pietenpol, "Classical Interaction of an Electric Charge with a Magnetic Pole," American Journal of Physics, 28(1), 1960 pp. 17–18.
[5] G. Nadeau, "Concerning the Classical Interaction of an Electric Charge with a Magnetic Monopole," American Journal of Physics, 28(6), 1960 p. 566.
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