Approach
45
3.3 Duality
Duality is a fundamental physical principle. Opposite, but one another complementing
phenomena can be assigned
to each other in pairs, like e.g. (see fig. 8.8):
F i r s t of all we find the duality confirmed in the case of the electromagnetic wave
spreading in a homogeneous medium. Here the field pointers of E and H are directed
perpendicular to each other and are in a fixed relation to each other. But if the wave is
damped in the presence of matter, for instance by eddy currents, then by basing on
Maxwell's field theory the duality will vanish.
A good example for perfect duality provides the integral of a field strength vector along
the path from a to b:
Urn if the integration takes place along a closed path then the circulation yields:
According to Ampere's law (3.1) the magnetic field can thus form enclosed currents and
spatially spreading eddy currents. The electric field on the other hand should be
irrotational (3.2).
Let's take the case that the electromagnetic wave is damped by eddy currents and the
magnetic field in this way becomes a vortex field. The electric field itself that, as said, is
in a fixed relation and perpendicular to the vortex field H, will show all the vortex-typical
properties. Hence nothing would be more obvious as to also grant the electric field a
formation of vortices:
Critics of this dual approach, like for instance Jackson
or Lehner
, point out that with
reference to the fourth Maxwell equation the electric field should be understood as a
source field:
46
Flow vortices
Fig. 3.4a: Velocity distribution v(R) for a vortex with
rigid-
body rotation
Fig. 3.4b: Velocity distribution v(R) in a potential vortex
(see Lugt
).
Lugt, Hans J.: Vortex flow in nature and technology. Krieger publishing
company, Florida 1995; page 30 and 31, ISBN 0-89464-916-7
Approach ____________________________________________________________ 47
For a complete duality from the existence of electric monopoles, individual in the space
charge density
contained charge carriers, the claim for magnetic monopoles is derived.
In spite of intensive search such north or south pole particles however until now could not
be found. Herein from the sight of criticism is seen a confirmation for the assumption that
Maxwell's field theory is self-contained and hence in principle may not be extended. The
critics h
ave a problem of causality: They postulate source fields that at the same time
should
be vortex fields. But if one asks how one should imagine such a field that is scalar
and at the same time vectorial, then it looks as if no one has ever made any thoughts about
it.
The from causality derived solution of the problem of lacking duality requires to extend
the Maxwell theory in one point, by introducing the potential vortex of the electric field
here and at the same time make a cut in another place:
div D = O
(3.7)
With this formulation, the assumption of a freedom of sources in principle, the complete
duality already is reached: Now neither magnetic nor electric monopoles exist (Fig. 3.3)!
At first we have to accept the loss of the electron hoping that the calculation in the end
works out: the "exchange" vortices against particles, by which the quanta can be banned
from the field theory, suggests that the elementary particles themselves are nothing else as
spherical vortices that have found to an own physical reality.
3.4 Flow vortices
In fluid engineering convincing and strong indications for the correctness of the chosen
approach can be found. It benefits us that hydrodynamic vortices are visible or can be the
injection of smoke, e.g. in a wind-tunnel.
Already Leonardo da Vinci had observed at liquids that there exist two dual basic types
of plane vortices: "Among the vortices one is slower at
the centre than at the sides, another
is faster at the centre than at the sides."
A vortex of the first type, also called "vortex with rigid-body rotation", is formed for
instance by a liquid in a centrifuge that due to its inertia of mass is pressed to the edge
because there the largest velocity exists. In an analogous way the electromagnetic vortex
in electrically conductive material shows the well-known "skin effect" (Fig. 3.4a).
To explain the other vortex Newton describes the experiment where a rod is dipped into a
liquid as viscous as possible and then is turned. In this potential vortex the velocity of the
particle increases the closer to the rod it is (Fig. 3.4b).
The duality of both vortex phenomena becomes obvious when we make ourselves clear
that in the experiment with the centrifuge the more liquid presses to the edge the less
viscous the medium is. And that on the other hand the potential vortex forms the stronger
the more viscous the medium is. As conclusion we read in text books that the viscosity of
the liquid decides whether a vortex with rigid-body rotation or a potential vortex is
formed.