26 May, 2008

a basis for K. Meyl's ''New and Dual Field Approach'' of electrodynamics?

In his book
[1, p.130]
R.W. Pohl considers em waves (**E**,**H**)
that travel at constant velocity **u**
coupled by the eqs. (notation slightly modified)

(1.1)
**H** = ε **u** × **E**
and
**E** = − μ **u** × **H** ,

where the matter quantities ε and μ are assumed to be constant.

These ''Pohl waves'' which are selected by K. Meyl as the basis of his
''New and Dual Field Approach'' [2, p.257]
shall be discussed here. Meyl replaced **u** with −**v** (hence
|**v**| = |**u**|) which is rather *inconvenient* since then
the fields move at velocity −**v** with a *annoying* minus sign.
*Physically* that minus sign makes no sense.

First conclusions from (1.1) are the *transversality relations* to be obtained
from the elementary rules of vector algebra

(1.2)
**u** ^ **E** ^
**H** ^ **u** ,

together with the *velocity condition*

(1.3)
εμ **u**² = 1 ,
thus
|**u**| = c = 1/(εμ)^{½}

to be obtained by inserting one of the eqs. (1.1) into the other one using the well-known vector identity

**u** ×(**u** × **a**) = (**u · a**) **u** −
(**u · u**) **a**
where
**a** = εμ **E**
or
**a** = εμ **H** .

The results (1.2) and (1.3) are explicitely mentioned by R.W. Pohl at [1, p.130].

Fields moving at constant velocity **u** means that the fields have a special
dependency in spacetime:

(1.4)
**E**(**r**,t) = **E**(**r** − **u** t)
and
**H**(**r**,t) = **H**(**r** − **u** t) .

This coupled dependency of **H** and **E** on the combination
**x** = **r** − **u** t implies a relation between the time derivation
and a certain directional spatial derivation:

(1.5)
^{∂E}/_{∂t} =
^{∂E(r − u t)}/_{∂t} =
− **u ·** grad **E**
and
^{∂H}/_{∂t} =
^{∂H(r − u t)}/_{∂t} =
− **u ·** grad **H** .

In addition for constant vector **u** we have the identities

(1.6)
curl (**u** × **E**) =
− **u ·** grad **E** + **u** div **E**
and
curl (**u** × **H**) =
− **u ·** grad **H** + **u** div **H**

to obtain

(1.7)
^{∂E}/_{∂t} =
curl (**u** × **E**) − **u** div **E**.
and
^{∂H}/_{∂t} =
curl (**u** × **H**) − **u** div **H** ,

and, using here eqs. (1.1) yields

(1.8)
ε ^{∂E}/_{∂t} =
curl **H** − ε **u** div **E**
and
μ ^{∂H}/_{∂t} =
− curl **E** − μ **u** div **H** .

Therefore, by using the matter relations

(1.9)
**D** = ε **E**
and
**B** = μ **H**

we obtain

(1.10)
^{∂D}/_{∂t} =
curl **H** − **u** div **D**
and
^{∂B}/_{∂t} =
− curl **E** − **u** div **B** .

The terms **u** div **D** and **u** div **B**
represent electrical charges that travel at velocity |**u**| = c, i.e.
at speed of light.

However, in physics up to now no charged particles are known,
that move at speed of light. Therefore, we have to assume
for *realistic* Pohl waves

(1.11)
**u** div **D** = **0**
and
**u** div **B** = **0**

to obtain from eqs. (1.10) the final result, the homogeneous Maxwell equations

(1.12)
^{∂D}/_{∂t} =
curl **H**
and
^{∂B}/_{∂t} =
− curl **E** .

− are transversal,

− move at speed of light with velocity

− satisfy the homogeneous Maxwell equations.

− do not fulfil the superposition principle of waves in case of different propagation vectors u.

Meyl's ''New Approach'' is a mathematically wrong version of the above consideration
on Pohl waves yielding the Maxwell equations (1.10). One could continue with Meyl from (1.10)
by introducing *hypothetical* electrical and magnetical charge densities respectively
[2, eqs.(3.11) and [2, eqs.(3.10), , p.259]

(2.1)
ρ_{el} = div **D**
and
ρ_{mg} = div **B**

Then Meyl [2, eq. (1.5), p.261] and [2, eq. (3.15), p.262]
respectively assumes the validity of ''Ohm's laws" by introducing current densities
and conductivity numbers σ_{el} and σ_{mg}

(2.2)
**j** = **u**ρ_{el} = σ_{el} **E**
and
**b** = **u**ρ_{mg} = σ_{mg} **B**
(remember **B** | | **H** and **u** = −**v**)

However, here at the latest, if not already at the hypothesis of charges
moving at speed of light |**u**| = c, Meyl contradicts himself:

By introducing the eqs. (2.2) Meyl assumes **E** | | **u** and **H** | | **u**
which contradicts the basic equations (1.1) and the transversality properties (1.2)
of the Pohl waves.

Especially K. Meyl should have recognized from the very beginning
the *transversality* of the Pohl waves,
while he asserts the existence of *longitudinal* waves to be contained in
his ''new approach''.

In Sect.4 [2, p.263 ff.] Meyl attempts to derive a ''general'' wave equation [2, eq.
(4.9), p.267]. However, that attempt fails since eq. [2,(3.10+3.15)] is *invalid* as
contradicting the basic equ. [2, (2.3), p.257], i.e. the orthogonality condition
**v** ^ **B**.
In addition Meyl's derivation is wrong due to a vector algebra error.

Meyl does not care for contradictions *within* his ''theory''. At
[2, Figure 9. p. 271]
he displays an ''electric scalar wave (longitudinal)''
where evidenly **v** | | **E**, while a trivial
conclusion of his basic equations (1.1) is **v** ^ **E**.
And at
[2, Figure 10. p. 271] the same for a ''magnetic scalar wave (longitudinal)'' No contradiction!?

[1] R.W. Pohl, Elektrizitätslehre, 21. Auflage, Springer

[2] K. Meyl, Scalar Wave Effects according to Tesla ...,

ANNUAL 2006 OF THE CROATIAN ACADEMY OF ENGINEERING

Almost identical papers to [2]:

[3] K. Meyl, Wireless Tesla Transponder Field-physical basis
. . . according to the invention of Nikola Tesla,

SoftCOM 2006, 14th intern. Conference, 29.09.2006, IEEE and Univ. Split,

Faculty of Electrical Engineering, ISBN 953-6114-89-5, page 67-78

[4] K. Meyl, Far Range Transponder, Field-physical basis for electrically coupled
bidirectional far range transponders,

Proceedings of the 1st RFID Eurasia Conference Istanbul 2007, ISBN 978-975-01566-0-1, page 78-89