26 May, 2008

## R.W. Pohl's constantly moving em waves − a basis for K. Meyl's ''New and Dual Field Approach'' of electrodynamics?

### 1. R.W. Pohl's waves

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 u² = 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(ru t)                 and                 H(r,t) = H(ru t) .

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

(1.5)                 E/∂t = E(ru t)/∂t = − u · grad E                 and                 H/∂t = H(ru 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 Hu div D                 and                 B/∂t = − curl Eu 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 .

### 2. Remark on Meyl's ''New Approach'' [2]

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)                ρel = div D                 and                 ρmg = 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 σel and σmg

(2.2)                 j = uρel = σel E                 and                 b = uρmg = σmg 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.

### Therefore Meyl's ''New Approach'' breaks down with this contradiction.

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''.

### 3. Further comments on [2]:

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!?

### References

[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