A Theoretical Investigation of Magnetic Monopoles

1. A Theoretical Investigation of Magnetic Monopoles Chad A. Middleton Mesa State College October 22, 2009

2. A Brief History of the Magnetic Monopole…. • “On the Magnet”, Pierre de Maricourt, Letter to Siger de Foucaucourt (1269) • Petrus Peregrinus defines magnetic poles and observes that they are never seen in isolation. • “Law of Magnetic Force”, C.A. Coulomb (1788) • Establishes for magnetic poles that force varies inversely as the square and is proportional to the product of the pole strength. • “The Action of Currents on Magnets”, H.C Oersted (1820) • Provides the first sign that electricity and magnetism are connected. • “Electrodynamic Model of Magnetism”, A. M. Ampere (1820) • Asserts that all magnetism is due to moving electric charges, explaining why magnets do not have isolated poles. • Principle of magnetic ambiguity

3. A Brief History of the Magnetic Monopole…. • “On the Possible Existence of Magnetic Conductivity and Free Magnetism”, P. Curie, Seances Soc. Phys. (Paris, 1894) pp. 76-77 • 1stpost-Amperian proposal of isolated poles “Quantized Singularities in the Electromagnetic Field”, P.A.M. Dirac, Proc. R. Soc. London Ser. A 133, 60-72 (1931) • “The Theory of Magnetic Monopoles”, P.A.M. Dirac, Phys. Rev. 74, 817-830 (1948) • Concludes that product of magnitude of an isolated electric charge and magnetic pole must be an integral multiple of a smallest unit. • “First Results from a Superconductive Device for Moving Magnetic Monopoles”, B. Cabrera, Phys. Lett. 48, 1378-1380 (1982) • Reports a signal in an induction detector, which in principle is unique to a monopole.

4. Maxwell’s Equations in Integral form (in vacuum) Gauss’ Law for E-field Gauss’ Law for B-field Faraday’s Law Ampere’s Law with Maxwell’s Correction

5. Using the Divergence Theoremand Stokes’ Theorem… • The Divergence Theorem • Stokes’ Theorem … for a general vector field

6. Maxwell’s Equations in differential form (in vacuum) Gauss’ Law for E-field Gauss’ Law for B-field Faraday’s Law Ampere’s Law with Maxwell’s Correction these plus the Lorentz force completely describe Classical Electromagnetic Theory

7. Taking the divergence of the 4th Maxwell Eqn yields.. Equation of Continuity = Conservation of Electric Charge

8. Taking the curl of the 3rd & 4th eqns (in free space when e = Je = 0) yield.. The waveequations for the E-, B-fields with predicted wave speed Light = EM wave!

9. Back to Maxwell’s Equations… Gauss’ Law for E-field Gauss’ Law for B-field Faraday’s Law Ampere’s Law with Maxwell’s Correction • Maxwell’s equations are almost symmetrical • allow for the existence of a magnetic charge density,m& a magnetic current, Jm

10. Maxwell’s Equations become… Gauss’ Law for E-field Gauss’ Law for B-field Faraday’s Law Ampere’s Law with Maxwell’s Correction the Lorentz force becomes

11. Taking the divergence of the 3rd & 4th eqns yield.. Equation of Continuity  Electric & Magnetic Charge are each conserved separately

12. Does the existence of magnetic charges have observable EM consequences?Not if all particles have the same ratio of qm/qe !

13. Maxwell’s Equations are Invariant under the Duality Transformations • Matter of convention to speak of a particle possessing qe & not qm (so long as qe / qm = constant for all particles)

14. So long as qe / qm = constant for all particles… Set: This sets the Mixing Angle: and yields: Notice: • for this choice of α, our original Maxwell’s Equations are recovered! • existence of monopoles = existence of particles with differentα

15. Dirac Quantization Condition Dirac showed that the existence of even a singleMagnetic Monopole (a.k.a a particle with a differentmixing angle) requires qe , qmbe quantized. where “Quantized Singularities in the Electromagnetic Field”, P.A.M. Dirac, Proc. R. Soc. London Ser. A 133, 60-72 (1931) “The Theory of Magnetic Monopoles”, P.A.M. Dirac, Phys. Rev. 74, 817-830 (1948)