Preface to the lecture, 1

558

 

The field-theoretical approach 

The new and dual field approach consists of 

equations of transformation

 

of the electric 

and      of the magnetic field

 

 (27.1)   and    

             (27.2)

 

unipolar induction 

equation of convection

 

 

•

 

Formulation according to the rules of duality 

•

 

Grimsehl

  speaks of the ,,equation of convection", 

according to which moving charges produce a magnetic field 

and   so-called   convection   currents   (referring   to   Rontgen 

1885, Himstedt, Rowland 1876, Eichenwald and others)

 

•

 

Pohl

 gives examples for the equations of transformation, 

•

 

he writes the equations beneath each other 

 (27.3) and    

         (27.4)

 

•  and points out that for 

 

one equation changes into the other one!

 

The new and dual approach roots in textbook physics! 

Fig. 27.8: The new and dual field approach

 

:     see Part 1, chapter 6.5 

:    Grimsehl: Lehrbuch der Physik, 2.Bd., 17,Aufl. Teubner Verl. 1967, S. 130. 

:   R.W.Pohl:    Einfuhrung   in    die    Physik,   Bd.2    Elektrizitatslehre,    21.Aufl. 

Springer-Verlag 1975, Seite 77 

---

Faraday versus Maxwell

 

559

 

27.8 The field-theoretical approach

 

The duality between E- and H-field and the commutability asks for a corresponding dual 

formulation to the Faraday-law (27.1). Written down according to the rules of duality 

there results an equation (27.2), which occasionally is mentioned in some textbooks. 

While both equations in the books of Pohl

 and of Simonyi

 are written down side by 

side having equal rights and are compared with each other, Grimsehl

 derives the dual 

regularity (27.2) with the help of the example of a thin, positively charged and rotating 

metal ring. He speaks of ,,equation of convection", according to which moving charges 

produce a magnetic field and so-called convection currents. Doing so he refers to 

workings of Rontgen 1885, Himstedt, Rowland 1876, Eichenwald and many others more, 

which today hardly are known.

 

In his textbook also Pohl gives practical examples for both equations of transformation. 

He points out that one equation changes into the other one, if as a relative velocity v the 

speed of light c should occur. This question will also occupy us.

 

We now have found a field-theoretical approach with the equations of transformation, 

which in its dual formulation is clearly distinguished from the Maxwell approach. The 

reassuring conclusion is added: The new field approach roots entirely in textbook 

physics, as are the results from the literature research. We can completely do without 

postulates.

 

Next thing to do is to test the approach strictly mathematical for freedom of 

contradictions. It in particular concerns the question, which known regularities can be 

derived under which conditions. Moreover the conditions and the scopes of the derived 

theories should result correctly, e.g. of what the Maxwell approximation consists and why 

the Maxwell equations describe only a special case.

 

27.9 Derivation of Maxwell's field equations

 

As a starting-point and as approach serve the equations of transformation of the 

electromagnetic field, the Faraday-law of unipolar induction and the according to the rules 

of duality formulated law (eq. 27.1, 2). If we apply the curl to both sides of the equations 

then according to known algorithms of vector analysis the curl of the cross product each 

time delivers the sum of four single terms. Two of these again are zero for a non- 

accelerated relative motion in the x-direction with v = dr/dt.

 

One term concerns the vector gradient (v grad)B, which can be represented as a tensor. 

By writing down and solving the accompanying derivative matrix giving consideration to 

the above determination of the v-vector, the vector gradient becomes the simple time 

derivation of the field vector B(r(t)) (eq. 27.10, according to the rule of eq. 27.11).

 

:     R.W.Pohl:    Einfuhrung   in    die   Physik,    Bd.2    Elektrizitatslehre,    21.Aufl. 

Springer-Verlag 1975, Seite 76 und 130

 

:    K. Simonyi: Theoretische Elektrotechnik, 7.Aufl. VEB Berlin 1979, Seite 924. 

:   Grimsehl: Lehrbuch der Physik, 2,Bd., 17.Aufl. Teubner Verl. 1967, S. 130.

 

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560

 

Derivation of Maxwell's field equations

 

As approach serve the equations of transformation (fig. 27.5) of 

the electric and of the magnetic field: 

 (27.1) and    

          (27.2)

 

If we apply the curl to the respective cross product: 

 (27.5)  and   

    (27.6)

 

then according to the algorithms

 four sum terms are delivered: 

 

(27.5) 

  (27.6)

 

where 2 of them are zero because of:     

  (27.7) 

•

 

the divergence of v(t) disappears: 

div v  =   0   ,  (27.8) 

•

 

and will be zero as well:    

.  (27.9) 

•

 

there remain the vector gradients: 

 

and   

  

, (27.10) 

•  according to the rules

 in general (with eq. 27.7): 

 

(27.11)

 

•  A comparison of the coefficients of both field equations 

(27.12) 

 

(27.13)

 

with the Maxwell equations results in: 

•  for the potential density b  =  - v div B  =  0       ,    (27.14) 

(eq. 27.12    = law of induction, if b = 0  resp.   div B = 0) 

• 

for the current density j  =  - v div D  = 

    , (27.1a 

(eq. 27.13   = Ampere's law, if j = with v moving negative 

charge carriers ( = electric space charge density). 

Fig. 27.9: ____ Derivation of Maxwell's field equations as a 

special case of the equations of transformation 

:     Bronstein u.a.: Taschenbuch der Mathematik, 4.Neuaufl. Thun 1999, S. 652

 

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