Preface to the lecture, 1

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:

 

 

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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

 

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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.

 

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