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= Magnetic monopole =
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Introduction
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In particle physics, a magnetic monopole is a hypothetical elementary
particle that is an isolated magnet with only one magnetic pole (a
north pole without a south pole or vice versa).
A magnetic monopole would have a net "magnetic charge". Modern
interest in the concept stems from particle theories, notably the
grand unified and superstring theories, which predict their existence.
Magnetism in bar magnets and electromagnets is not caused by magnetic
monopoles, and indeed, there is no known experimental or observational
evidence that magnetic monopoles exist.
Some condensed matter systems contain effective (non-isolated)
magnetic monopole quasi-particles, or contain phenomena that are
mathematically analogous to magnetic monopoles.
Pre-twentieth century
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Many early scientists attributed the magnetism of lodestones to two
different "magnetic fluids" ("effluvia"), a north-pole fluid at one
end and a south-pole fluid at the other, which attracted and repelled
each other in analogy to positive and negative electric charge.
However, an improved understanding of electromagnetism in the
nineteenth century showed that the magnetism of lodestones was
properly explained not by magnetic monopole fluids, but rather by a
combination of electric currents, the electron magnetic moment, and
the magnetic moments of other particles. Gauss's law for magnetism,
one of Maxwell's equations, is the mathematical statement that
magnetic monopoles do not exist. Nevertheless, it was pointed out by
Pierre Curie in 1894 that magnetic monopoles 'could' conceivably
exist, despite not having been seen so far.
Twentieth century
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The 'quantum' theory of magnetic charge started with a paper by the
physicist Paul Dirac in 1931. In this paper, Dirac showed that if
'any' magnetic monopoles exist in the universe, then all electric
charge in the universe must be quantized (Dirac quantization
condition). The electric charge 'is', in fact, quantized, which is
consistent with (but does not prove) the existence of monopoles.
Since Dirac's paper, several systematic monopole searches have been
performed. Experiments in 1975 and 1982 produced candidate events that
were initially interpreted as monopoles, but are now regarded as
inconclusive. Therefore, it remains an open question whether monopoles
exist.
Further advances in theoretical particle physics, particularly
developments in grand unified theories and quantum gravity, have led
to more compelling arguments (detailed below) that monopoles do exist.
Joseph Polchinski, a string-theorist, described the existence of
monopoles as "one of the safest bets that one can make about physics
not yet seen". These theories are not necessarily inconsistent with
the experimental evidence. In some theoretical models, magnetic
monopoles are unlikely to be observed, because they are too massive to
create in particle accelerators (see below), and also too rare in the
Universe to enter a particle detector with much probability.
Some condensed matter systems propose a structure superficially
similar to a magnetic monopole, known as a flux tube. The ends of a
flux tube form a magnetic dipole, but since they move independently,
they can be treated for many purposes as independent magnetic monopole
quasiparticles. Since 2009, numerous news reports from the popular
media have incorrectly described these systems as the long-awaited
discovery of the magnetic monopoles, but the two phenomena are only
superficially related to one another. These condensed-matter systems
remain an area of active research. (See below.)
Poles and magnetism in ordinary matter
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All matter ever isolated to date, including every atom on the periodic
table and every particle in the standard model, has zero magnetic
monopole charge. Therefore, the ordinary phenomena of magnetism and
magnets have nothing to do with magnetic monopoles.
Instead, magnetism in ordinary matter comes from two sources. First,
electric currents create magnetic fields according to Ampère's law.
Second, many elementary particles have an 'intrinsic' magnetic moment,
the most important of which is the electron magnetic dipole moment.
(This magnetism is related to quantum-mechanical "spin".)
Mathematically, the magnetic field of an object is often described in
terms of a multipole expansion. This is an expression of the field as
the sum of component fields with specific mathematical forms. The
first term in the expansion is called the 'monopole' term, the second
is called 'dipole', then 'quadrupole', then 'octupole', and so on. Any
of these terms can be present in the multipole expansion of an
electric field, for example. However, in the multipole expansion of a
'magnetic' field, the "monopole" term is always exactly zero (for
ordinary matter). A magnetic monopole, if it exists, would have the
defining property of producing a magnetic field whose 'monopole' term
is non-zero.
A magnetic dipole is something whose magnetic field is predominantly
or exactly described by the magnetic dipole term of the multipole
expansion. The term 'dipole' means 'two poles', corresponding to the
fact that a dipole magnet typically contains a 'north pole' on one
side and a 'south pole' on the other side. This is analogous to an
electric dipole, which has positive charge on one side and negative
charge on the other. However, an electric dipole and magnetic dipole
are fundamentally quite different. In an electric dipole made of
ordinary matter, the positive charge is made of protons and the
negative charge is made of electrons, but a magnetic dipole does 'not'
have different types of matter creating the north pole and south pole.
Instead, the two magnetic poles arise simultaneously from the
aggregate effect of all the currents and intrinsic moments throughout
the magnet. Because of this, the two poles of a magnetic dipole must
always have equal and opposite strength, and the two poles cannot be
separated from each other.
Maxwell's equations
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Maxwell's equations of electromagnetism relate the electric and
magnetic fields to each other and to the motions of electric charges.
The standard equations provide for electric charges, but they posit no
magnetic charges. Except for this difference, the equations are
symmetric under the interchange of the electric and magnetic fields.
In fact, symmetric Maxwell's equations can be written when all charges
(and hence electric currents) are zero, and this is how the
electromagnetic wave equation is derived.
Fully symmetric Maxwell's equations can also be written if one allows
for the possibility of "magnetic charges" analogous to electric
charges. With the inclusion of a variable for the density of these
magnetic charges, say , there is also a "magnetic current density"
variable in the equations, .
If magnetic charges do not exist - or if they do exist but are not
present in a region of space - then the new terms in Maxwell's
equations are all zero, and the extended equations reduce to the
conventional equations of electromagnetism such as (where is
divergence and is the magnetic field).
In Gaussian cgs units
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The extended Maxwell's equations are as follows, in Gaussian cgs
units:
Maxwell's equations and Lorentz force equation with magnetic
monopoles: Gaussian cgs units
Name Without magnetic monopoles With magnetic monopoles
Gauss's law |colspan="2"| \nabla \cdot \mathbf{E} = 4 \pi
\rho_{\mathrm e}
Gauss's law for magnetism \nabla \cdot \mathbf{B} = 0 \nabla \cdot
\mathbf{B} = 4 \pi \rho_{\mathrm m}
Faraday's law of induction -\nabla \times \mathbf{E} =
\frac{1}{c}\frac{\partial \mathbf{B}} {\partial t} -\nabla \times
\mathbf{E} = \frac{1}{c}\frac{\partial \mathbf{B}} {\partial t} +
\frac{4 \pi}{c}\mathbf{j}_{\mathrm m}
Ampère's law (with Maxwell's extension) |colspan="2"| \nabla \times
\mathbf{B} = \frac{1}{c}\frac{\partial \mathbf{E}}{\partial t} +
\frac{4 \pi}{c} \mathbf{j}_{\mathrm e}
!Lorentz force law |\mathbf{F}=q_{\mathrm
e}\left(\mathbf{E}+\frac{\mathbf{v}}{c}\times\mathbf{B}\right)
|\mathbf{F}=q_{\mathrm
e}\left(\mathbf{E}+\frac{\mathbf{v}}{c}\times\mathbf{B}\right) +
q_{\mathrm
m}\left(\mathbf{B}-\frac{\mathbf{v}}{c}\times\mathbf{E}\right)
In these equations is the 'magnetic charge density', is the
'magnetic current density', and is the 'magnetic charge' of a test
particle, all defined analogously to the related quantities of
electric charge and current; is the particle's velocity and is the
speed of light. For all other definitions and details, see Maxwell's
equations. For the equations in nondimensionalized form, remove the
factors of .
In SI units
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In SI units, there are two conflicting units in use for magnetic
charge : webers (Wb) and ampere·meters (A·m). The conversion between
them is , since the units are by dimensional analysis (H is the henry
- the SI unit of inductance).
Maxwell's equations then take the following forms (using the same
notation above):
Maxwell's equations and Lorentz force equation with magnetic
monopoles: SI units
Name rowspan=2 | Without magnetic monopoles colspan=2 | With
magnetic monopoles
Weber convention Ampere·meter convention
Gauss's Law colspan="3" | \nabla \cdot \mathbf{E} =
\frac{\rho_{\mathrm e}}{\varepsilon_0}
Gauss's Law for magnetism \nabla \cdot \mathbf{B} = 0 \nabla \cdot
\mathbf{B} = \rho_{\mathrm m} \nabla \cdot \mathbf{B} =
\mu_0\rho_{\mathrm m}
Faraday's Law of induction -\nabla \times \mathbf{E} = \frac{\partial
\mathbf{B}} {\partial t} -\nabla \times \mathbf{E} = \frac{\partial
\mathbf{B}} {\partial t} + \mathbf{j}_{\mathrm m} -\nabla \times
\mathbf{E} = \frac{\partial \mathbf{B}} {\partial t} +
\mu_0\mathbf{j}_{\mathrm m}
Ampère's Law (with Maxwell's extension) colspan="3" | \nabla \times
\mathbf{B} = \frac{1}{c^2} \frac{\partial \mathbf{E}} {\partial t} +
\mu_0 \mathbf{j}_{\mathrm e}
Lorentz force equation \mathbf{F}=q_{\mathrm
e}\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}\right) \begin{align}
\mathbf{F} ={} &q_{\mathrm
e}\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}\right) +\\
&\frac{q_{\mathrm m}}{\mu_0}\left(\mathbf{B}-\mathbf{v}\times
\frac{\mathbf{E}}{c^2}\right) \end{align} \begin{align} \mathbf{F}
={} &q_{\mathrm
e}\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}\right) +\\
&q_{\mathrm
m}\left(\mathbf{B}-\mathbf{v}\times\frac{\mathbf{E}}{c^2}\right)
\end{align}
Tensor formulation
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Maxwell's equations in the language of tensors makes Lorentz
covariance clear. The generalized equations are: