Transcript Cosmology, Inflation, & Compact Extra Dimensions

A Theoretical Investigation of
Magnetic Monopoles
Chad A. Middleton
Mesa State College
October 22, 2009
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
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.
Maxwell’s Equations in Integral form
(in vacuum)
 E  dA     dV
1
e
Gauss’ Law for E-field
0 V
A
 B  dA  0
Gauss’ Law for B-field
A

C
B
Ed 
 dA
A t

E 
 B  d  0  Je  0 t  dA
C
A
Faraday’s Law
Ampere’s Law with
Maxwell’s Correction
Using the Divergence Theorem
and Stokes’ Theorem…
 F  dA 
  FdV
A
  F  dA
 Fd
C

• The Divergence
Theorem
V
• Stokes’ Theorem
A
… for a general vector field
F  F (x,t)
Maxwell’s Equations in differential form
(in vacuum)
 E 
1
0
e
Gauss’ Law for E-field
 B  0
Gauss’ Law for B-field
B
 E  
t
Faraday’s Law
E
 B  0 J e  00
t
these plus

F  qe E  v  B

Ampere’s Law with
Maxwell’s Correction
the Lorentz force completely describe
Classical Electromagnetic Theory
Taking the divergence of the 4th Maxwell Eqn yields..
e
 Je  
t

Equation of Continuity
=
Conservation of Electric Charge
Taking the curl of the 3rd & 4th eqns
(in free space when e = Je = 0) yield..
2
1

E
2
 E 2 2
c t
2
1  B
2
 B 2 2
c t
The wave equations for the
E-, B-fields with
predicted wave speed
c

1
00
 3.0 108 m /s
Light = EM wave!
Back to Maxwell’s Equations…
 E 
1
0
e
 B  0
 E 
Gauss’ Law for E-field
Gauss’ Law for B-field
B

t
E
 B  0 J e  00
t
Faraday’s Law
Ampere’s Law with
Maxwell’s Correction

Maxwell’s equations are almost symmetrical
F  qe E  v  B
 allow for the existence of a
magnetic charge density, m & a magnetic current, Jm

Maxwell’s Equations become…
1
e
Gauss’ Law for E-field
 B  0m
Gauss’ Law for B-field
B
 E  0 J m 
t
Faraday’s Law
 E 
0
E
 B  0 J e  00
t
the Lorentz force becomes
Ampere’s Law with
Maxwell’s Correction

E 
F  qe E  v  B  qm B  v  2 
c 



Taking the divergence of the 3rd & 4th eqns yield..
e
 Je  
t
m
 Jm  
t

Equation of Continuity
Electric & Magnetic Charge are each

conserved separately
Does the existence of
magnetic charges have
observable EM
consequences?
Not if all particles have
the same ratio of qm/qe !
Maxwell’s Equations are Invariant under the
Duality Transformations
E  E 'cos  cB'sin
cB  E 'sin   cB'cos

cq ,cJ  cq ',cJ 'cos  q ',J 'sin
q ,J  cq ',cJ 'sin  q ',J 'cos
e
m
e
m
e
e
m
m
e
e
m
m
 Matter of convention to speak of a particle possessing

qe & not qm
(so long as qe / qm = constant for all particles)
So long as qe / qm = constant for all particles…
Set:
qm  0  cq e 'sin   qm 'cos 
This sets the Mixing Angle:
qm '
tan 
cqe '

and yields:
Jm  0
Notice:
• for this choice of α, our original Maxwell’s
Equations are recovered!

• existence of monopoles = existence of particles with different α
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.
0qeqm
n
2

where
n
“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)
