Preface to the lecture, 1

76

 

fundamental field equation

 

 

Fig. 5.1:    Derivation of the fundamental field equation. 

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Derivation and interpretation

 

77

 

5.1 Fundamental field equation 

We'll start from Ampere's law which provides a value for the current density at any point 

pace and this value corresponds to the vortex density of the magnetic field strength

 

 

The new electric field vortices demand the introduction of a corresponding time constant 

tau

2  

t h a t  should describe the decay of the potential vortices, as an extension. The extended

 

 Faraday law of induction now provides a potential density, that at any point of space 

corresponds to the vortex density of the electric field strength:

 

 

which according to the rules of vector analysis can still be further simplified: 

= rot rot E = 

 - grad div E , where we should remember that the divergence has to 

vanish (div E   =  O.    fig. 3.2, equation 3.7 ), should the corresponding field vortex be 

inserted!

 

Furthermore the following well-known abbreviation can be inserted: 

 = 1/c

2

     (5.6)

 

With that the relation with the speed of light c simplifies to the sought-for field equation:

 

 

This equation describes the spatial (a) and temporal (b, c, d) distribution of a field vector. 

It describes the electromagnetic wave (a, b) with the influences that act damping. As 

dumping terms the well-known eddy current (c) and in addition the newly introduced 

potential vortex (d) appear.

 

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78

 

mathematical interpretation!

 

Field vector:  =  E, H, j, B or D

 

1. elliptic potential equation: 

(stationary:

 

(5.8)

 

2. hyperbolic equation: 

(undamped wave equation) 

(5.9)

 

3. parabolic equation: 

(vortex equation) 

(5.10)

 

decay time of the eddy currents = 

relaxation time:

 

decay time of the potential vortices = 

relaxation time:

 

(5.3)

 

(5.11)

 

Fig.   5.2:     mathematically divisible individual cases. 

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Derivation and interpretation_____________________________________________ 79

 

5.2 Mathematical interpretation of the fundamental field equation

 

Every specialist will be surprised to find the Poisson equation (a, e) again as a term in the 

wave equation. This circumstance forces a completely new interpretation of stationary 

fields upon us. The new damping term, that is formed by the potential vortices (d), is

 

standing in between.

 

Let us start with a mathematical analysis. We have applied the curl to equation 5.4*, then

 

inserted equation 5.1* and obtained a determining equation for the electric field strength

 

E. Of course we could as well have applied the curl to equation 5.1* and inserted equation

 

5.4*. This would have resulted in the determining equation for the magnetic field strength

 

H.

 

If we insert Ohm's law (5.2) and cancel down the specific conductivity, or we put in the 

relations of material (3.5) or (3.6) and cancel down by u respectively    then the field 

equation can likewise be written down for the current density j, for the induction B or for 

the dielectric displacement D.

 

It just is phenomenal that at all events equation 5.7 doesn't change its form at all. The field 

vector is thus arbitrarily interchangeable! This circumstance is the foundation for the claim 

of this field equation to be called fundamental.

 

It does make sense to introduce a neutral descriptive vector    as a substitute for the 

possible field factors E, H, j, B or D.

 

The fundamental field equation 5.7 consists of three different types of partial differential 

equations: a hyperbolic (b), a parabolic (c and d) and an elliptic (e) type. On the left-hand 

side each time the Laplace operator (a) is found which describes the spatial distribution of 

the

 field factor. 

The potential equation of the elliptic type (e) is known as Poisson equation. It describes 

the stationary borderline case of a worn off temporal process (

 resp. 

= O).

 

With this equation potentials and voltages can be calculated exactly like stationary electric 

currents (5.8).

 

The hyperbolic equation (b). known as wave equation, shows a second derivative to time. 

which expresses an invariance with regard to time reversal; or stated otherwise: the direc- 

tion of the time axis can be reversed by a change of sign of t, without this having an influ- 

ence on the course of frequency. Wave processes hence are reversible. Equation 5.7 makes 

clear that a wave without damping by no means can exist in nature. For that both time 

constants 

 would have to have an infinite value, which is not realizable in

 

practice. Seen purely theoretical, undamped waves could withdraw themselves from our

 

measuring technique (5.9).

 

Both vortex equations of the parabolic type (c and d) only show a first derivative to time.

 

With that they are no longer invariant with regard to time reversal. The processes of the

 

formation and the decay of vortices, the so-called diffusion, are as a consequence irre-

 

versible. Seen this way it is understandable that the process of falling apart of the vortex,

 

where the vortex releases its stored energy as heat e.g. in form of eddy losses, can not take

 

place in reverse. This irreversible process of diffusion in the strict fhermodynamic sense

 

increases the entropy of the system (5.10).

 

Because it poses an useful simplification for mathematical calculations, often the different 

types of equations are treated isolated from each other. But the physical reality looks 

different.

 

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