A Theoretical Investigation of Magnetic Monopoles

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A Theoretical Investigation of Magnetic Monopoles

• Chad A. Middleton
• Mesa State College
• October 22, 2009

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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

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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
• 1st post-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.

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Maxwells Equations in Integral form (in vacuum)

Gauss Law for E-field Gauss Law for
B-field Faradays Law Amperes Law with
Maxwells Correction
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Using the Divergence Theorem and Stokes Theorem

• The Divergence Theorem
• Stokes Theorem

for a general vector field
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Maxwells Equations in differential form (in
vacuum)

Gauss Law for E-field Gauss Law for
B-field Faradays Law Amperes Law with
Maxwells Correction
these plus
the Lorentz force completely describe Classical
Electromagnetic Theory
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Taking the divergence of the 4th Maxwell Eqn
yields..

Equation of Continuity Conservation of Electric
Charge
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Taking the curl of the 3rd 4th eqns (in
free space when ?e Je 0) yield..

The wave equations for the E-, B-fields with
predicted wave speed
Light EM wave!
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Back to Maxwells Equations

Gauss Law for E-field Gauss Law for
B-field Faradays Law Amperes Law with
Maxwells Correction

• Maxwells equations are almost symmetrical
• allow for the existence of a
• magnetic charge density, ?m a magnetic
  current, Jm

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Maxwells Equations become

Gauss Law for E-field Gauss Law for
B-field Faradays Law Amperes Law with
Maxwells Correction
the Lorentz force becomes
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Taking the divergence of the 3rd 4th eqns
yield..

Equation of Continuity ? Electric Magnetic
Charge are each conserved
separately
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Does the existence of magnetic charges have
observable EM consequences?Not if all
particles have the same ratio of qm/qe !

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Maxwells 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)

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So long as qe / qm constant for all particles

Set
This sets the Mixing Angle
and yields

• Notice
• for this choice of a, our original Maxwells
  Equations are recovered!
• existence of monopoles existence of particles
  with different a

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Dirac Quantization Condition

Dirac showed that the existence of even a single
Magnetic Monopole (a.k.a a particle with a
different mixing angle) requires qe , qm be
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)