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Koch, M. & Wiest, J. (2007). Discussion of magnetic field saturation phenomenons by interpretation of Maxwell's second equation. PHILICA.COM Article number 92.

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Discussion of magnetic field saturation phenomenons by interpretation of Maxwell’s second equation

Martin Kochunconfirmed user (Med. Elektronik, Technische Universitat Munchen)
Joachim Wiestconfirmed user (R&D, cellasys GmbH)

Published in physic.philica.com

It can be dangerous for the personnel, if magnetic flux densities across electric material, e.g. the iron cores of electric machines become saturated. Saturation is a non-linear field effect, which might be impossible to be described by standard classic electrodynamics. An experiment is described where a moving dynamic field is applied to a magnetic washer. Its result, namely the repulsion of the washer away from an active electromagnet cannot be explained by state of the art knowledge. An extension of the constitutive second Maxwell equation for the description of saturated magnetic flux densities is discussed.

Article body


In industry, where heavy duty electric machines are used, field-problems occur occasionally, which endanger personnel. These field-problems are caused by magnetic flux densities, which are forced into saturation.

A magnetic flux density can be called saturated, when the relation between field H and flux density B is non-linear. In practice, flux density saturation means, that the magnetic flux B is not only crossing the soft iron, but is also using other paths to close itself. It may happen, as the senior author witnessed, that the flux crosses the wedge area of a generator (Fig. 1b).


Figure 1 — a) Normal distribution of a magnetic flux density. The diamagnetic wedge area is bypassed by the magnetic flux. b) Non-classic magnetic saturated flux density. The diamagnetic wedge area is crossed by the magnetic flux. The residual moisture in the resin is boiled by the flux energy (via eddy currents) and the winding disintegrate.

This wedge part as indicated in figure 1 was composed of a two-component material (resin and fibre), which contained approximately 10% residual moisture. In figure 1b the magnetic flux left the soft iron, crossed the wedge and heated up the mixture of resin, fibre and water. First the water boiled, than the resin melted and consequently the remaining soft fibre frame alone was unable to retain the massive copper windings of the rotor. Because the rotor was spinning at 3000 rpm, the generator was destroyed with the consequent risk to the operators. The reason for the damage was the poor design of the generator, which allowed magnetic flux densities about 2,2 Tesla to occur, where the saturation limit in soft iron is 2,16 [1]. The magnetic flux density phenomenon is also theme of the experiment described in this paper. Here, a travelling magnetic field at a constant speed affects a soft iron washer.

All engineering students are taught that electromagnetic events are described by the Maxwell equations [2] and the relation between electric field and the magnetic flux density is based on Faradays induction law, stated in the second Maxwell equation (1).


A magnetic field is described by the first Maxwell equation [2].



Materials and Methods

Figure 2 shows the experiment. A transparent plastic box was placed on a field generator (Dynamic Marker [3]) and covered with a soft iron field return path. A 10 mm thick spacer was placed between the box and the field generator, to bypass the stray fields. Two metal washers where laid in the bottom of the box. The left one of nonmagnetic copper and the right washer was made out of soft iron (magnetite). Round shaped washers were used in the experiment in order to obtain easily field saturation spots (contact spots washer/field return path). Both washers were mass produced standard parts, with an outer diameter of 14.6 mm, an inner diameter of 5.5 mm and a thickness of 1mm. The weight of the magnetite washer was 1.2 g and that of the copper washer 1.4 g. A painted line on the washers to indicate the rotation can be seen in figure 4b, 4c, 4d.

Figure 2 — Experimental setup. A dynamic magnetic field is applied along the air gap with the help of a field generator, the field is closed by a field return path on top of the plastic box. The box maintains the air gap. A copper- (left) and a soft iron-washer (right) are placed in the air gap. To shortcut the stray field, a non-magnetic spacer is placed on top of the generator under the box.


Figure 3 — The magnetic field in a two-dimensional drawing. The field is spread and evenly distributed between field generator and field return path and moves in s direction. The velocity v in horizontal direction is constant. Heff = 6,4E3 A/m, f = 15Hz.



The results of the experiment are shown in figure 4. A video of the “live” experiment can be viewed at http://www.feldkraft.de/Filme/Spring.wmv [3].

Figure 4 — Effects observed on a copper washer (left) and a magnetite washer (right), due to an application of the magnetic field. a) Rest position, no movement, no field; b) field applied, only the magnetite washer first jitters and moves and then jumps around in a random manner; c) field applied, the random behaviour stops and the magnetite washer is suspended beneath the field return path; d) field applied, the magnetite washer rotates, while hovering.


In figure 4a no field is generated and both washers lay motionless on the bottom of the plastic box.

In figure 4b a slowly wandering low frequent (15Hz) magnetic field is built up by a field generator. The field lines are evenly distributed and cross the air gap (plastic box) twice. The soft iron washer stands up and moves around in a random manner still upright. In figure 4c the bouncing motion of the iron washer stops below the field return path and becomes almost stationary. The turning motion of the washer can be observed by the line marked on the washer.

In figure 4d the soft iron washer is rotating gently, as seen again by the line, hovering below the field return path. The rotation of the washer depends on the frequency of the applied magnetic field. The rotation can be made either clockwise or anticlockwise.

During the experiment the copper washer remained stationary (no movement at all).

If the system (experimental set) is turned upside down, no lifting movement occurs.



The movement of the soft iron washer can be separated into three main events:

1) The stand up.

2) The jumping and hovering of the washer.

3) The rotation.

The stand up of the washer is explainable by the well known classic electric Lorentz force. The rotation of the washer can be explained by the Einstein-de-Haas-effect. Einstein and de Haas showed that a ferromagnetic bar can be moved due to magnetisation [4]. In the experiment made in 1915 the alternating half turn movement (clockwise and anticlockwise) of the bar resulted from an AC magnetic field. The complete revolution of the soft iron washer, as shown in the experiment, results from the travelling AC magnetic field. In both cases a microscopic alternation of the angular momentum produces a macroscopic turning-motion.

In the opinion of the authors, aspect two, the jumping of the soft iron washer and its hovering under the field return path cannot be explained by the state of the art technology. Here the magnetic washer moves, non-classically, away from the active electromagnet, instead of being classically attracted to it. Electron eddy currents are unlikely to cause the jumping, because the experiment was run at a low frequency (15Hz) and furthermore there was no noticeable heating effect on the copper washer, so the eddy currents were far below 2,5 A/mm². The displacement current density (Equation 3) can also be ignored due to the low frequency.


The source of a magnetic field is always an electric current. This current cannot be noticed by the washer. But the soft iron washer is able to detect the moving magnetic field, and this is what activates the motion of the iron washer in the air gap (plastic box). It is therefore a magnetic effect, which moves the soft iron washer.


There are two non-linear aspects in the experiment, namely ferromagnetism and saturation. Firstly, Heisenberg showed in 1928 that ferromagnetism is non-classic [5]. Not knowing this, Einstein and de Haas misinterpreted their experiment quantitatively (Landé-factor). Secondly, there is a saturation of the magnetic flux density, at the point where the washer touches the magnetic field return path. The damped oscillating currents of the coil systems in the field generator are superposed, to generate a wandering field. The permanent turning with constant speed of the hovering washer shows that the washer registers a constantly moving magnetic field, which is not damped and might therefore be compared with a harmonic oscillator.

The standard form of equation 1 can be deduced, if the surface area A is a closed bowl. Then, the integral along boundary line s (figure 5b) becomes zero and the integral across the area A spherical (figure 5a) (5) [2].

Figure 5 — a) Closed surface area A where boundary line s is zero (Equation 5). b) A surface area A exists and here the electric field can be integrated along. Area A has a diameter of 2r.


This shows the classical and well-known source free (divergence free) nature of the classical magnetic field.

In figure 5b area A is the open spot of the bowl, with the boundary line s=2π
. The boundary s is also the circle-line of the circle with the diameter 2r. In the experiment this circular area corresponds to the small spot, where the washer touches the magnetic field return path. If the bowl area is not totally closed (case 5b), we gain this second area, where the left side of equation 1 may be integrated along (Stokes!). Vector Bs and vector A (the vector of the open spot area with an “s-boundary”) points in the same direction.

Supposing that non-classic, but electrically charged particles [6], which are moving in a two dimensional area, produce a time dependent flux, this flux per area corresponds to an induction, which is oppositely directed to an incoming, time dependent, saturated magnetic flux beam.

Upper non-classical phenomenon occurs only, when a ferromagnetic crystalline area is magnetized into saturation. This magnetic effect occurs, if a non-linear relation exists between H and B. Then a non-classic (Remember: no eddy currents possible, because of low frequency) magnetic counter flux density is produced across the area, where the original and saturated magnetic field beam is entering (washer/field return path). This counter flux across an area may be seen as a magnetic charge [6]. Considering this, the first equation (1) must be rewritten.


The flux field lines of the magnetic charge are penetrating a closed two-dimensional crystalline area. The area vector and the incoming field vector are aligned (7). After the integration procedure, differentiation is executed.


In equation 7 the vector points into an opposite direction than vector , therefore the scalar product receives a minus sign. This aspect inserted into Eq.6 delivers Eq.8


The right side of Eq.8 has to be differentiated according to the product rule.


And if:


Equation 10 is valid, saying area A is kept constant and K is a constant without dimension. Area A is encircled by boundary s.




In equation 13 a non-linear, time dependent magnetic flux penetrating across area A, produces a counter flux and is thus weakened. This occurs in the spot-area (contact of the washer with the magnetic field return path) where the magnetic flux is crossing. The driving power for the counter flux is a vortex field E (Eq.13), which produces a voltage in the washer, which might move non-classic particles with an electric charge.


It could not be recognized, that a force was acting on the non-magnetic copper washer in the experiment (Figures 4 a) to 4 d)). Since only the magnetic washer was activated and moved, the force character of the effect must be a magnetic one.

To produce a pure magnetic force (aside of a magnetic field) a magnetic charge is needed. The magnetic charge is a magnetic flux. A magnetic flux can only be adequate to a magnetic charge, if it is – in total – penetrating a closed area. This closed area is the spot like field saturated contact point “washer/field return path”. This aspect motivates us to split the integral area, which consequently leads to equation (6) and the following. But let us emphasize that for this case no longer standard classical electrodynamics are appropriate, what counts are Paul Dirac’s ingenious ideas about magnetic charges [8, 9, 10]. Dirac derived 1930 in his equations about the quantization of electric charges [11] - uncontradicted until now – the existence of a magnetic charge. Dirac’s tools had been quantum mechanics, ours had been based on an extension of the second Maxwell Equation. If we are right, we may postulate that the standard Maxwell Equations (div B = 0), which are relativistic correct, do not fulfill quantum mechanics. To do so, they have to be extended!

In order to outline once more the authors’ line of thinking: An active magnetic field generator (producing a wandering magnetic field) should normally attract a magnetic iron washer in a classical way. But instead of being attracted, the magnetic washer is repelled from the field generator near by and hovers at a constant speed below the field return path.

This uncommon behavior might be seen as a non-classical effect, which is not explainable with the “classical” Maxwell equation set. The extension of Maxwell’s second equation in the shown way allows an explanation of the observed phenomenon. Considering this, however, interesting new aspects in basic research and technology are imaginable:

It is possible, to generate a very high magnetic field density with tiny and cost effective devices. Magnetic AC fields possess no permittivity constants and it does appear, as if non-classic, but electrically charged particles are technically exploitable.

It is possible to explain experiments as described by Kamau et al. [7].

It is possible to set up new models for fusion of biological membranes [12].

A control aspect of the author’s line of thinking would be a superconductivity experiment where the fluxes in equation 13 compensate. The voltage in the washer could not be produced by an electric vortex field, in a situation, where the washer is in a state of superconductivity. This second experiment would produce a necessary final proof of our investigation. We hope that technically and scientifically the author’s aspects are sufficiently interesting to deserve further study.


The authors wish to thank Prof. Bernhard Wolf and the staff of the Heinz Nixdorf – Department for Medical Electronics of the Technical University of Munich for their support during the ongoing project.




[1] Tipler, P. (2000). Physik, Heidelberg/Berlin, GER: Spectrum Akademischer Verlag.

[2] Bosse, G. (1967). Grundlagen der Elektrotechnik, BI-Hochschultaschenbuch.

[3] http://www.feldkraft.de.

[4] Gerthsen, C. (1969). Physik, Springer Verlag.

[5] Pais, A. (1982). Subtle is the Lord, New York.

[6] Bihler, P. (2004). Geheimnisse der Quanteninformatik, Univ. Karlsruhe.

[7] Kamau, S.W., Hassa, P.O., Seitz, B., Petri-Fink, A., Hofmann, H., Hofmann-Amtenbrink, M., von Rechenberg, B. & Hottinger, M.O. (2006). Enhancement of the efficiency of non-viral gene delivery by applicatioin of pulsed magetic field. Nucleic Acids Research, 34(5).

[8] Dirac, P.A.M. (1930). Quantised Singularities in the Electromagnetic Field. The Royal Society, A, 126.

[9] Jackson, J.D. (1999). Classical Electrodynamics, Library of congress, 3rd edition.

[10] Song, J.S. (1996). Theory of Magnetic Monopoles and Electric-Magnetic Duality: A Prelude to S-Duality. Journal of Undergraduate Sciences, 3.

[11] Booss, B. (1976). Mathematisierung der Einzelwissenschaften, Birkhaeuser Verlag.

[12] Koch, M., Otto, A., Wiest, J. & Wolf, B. (2007). Can single electrons initiate fusion of biological membranes? Biophysical Reviews and Letters, 2(1).



Appendix A: Symbols


Electric field




Magnetic flux density, saturated




Magnetic Field


Current density


Electric flux density


Magnetic flux density, virtual




Period length






Angular frequency






Magnetic flux, virtual

Vector symbol



Information about this Article
Peer-review ratings (from 1 review, where a score of 100 represents the ‘average’ level):
Originality = 50.00, importance = 25.00, overall quality = 75.00
This Article was published on 1st May, 2007 at 17:18:09 and has been viewed 27009 times.

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The full citation for this Article is:
Koch, M. & Wiest, J. (2007). Discussion of magnetic field saturation phenomenons by interpretation of Maxwell’s second equation. PHILICA.COM Article number 92.

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1 Peer review [reviewer #47336unconfirmed user] added 30th September, 2011 at 16:30:57

The article states:
“Saturation is a non-linear field effect, *which might be impossible to be described by standard classic electrodynamics*”, (with the added emphasis via *…*). The last part of the sentence is stated in the abstract as some kind of conjecture for which there is no supporting evidence. Moreover, the proposed magnetic monopoles are suspect as one has never observed what the authors’ proposed in this article—the physical isolation of magnetic (mono) poles, called “magnetic charges” in this articles. The suggested “magnetic charges” are definitely inconsistent with both electrodynamics, either classical or quantum (QED), and relativity theories. The invocation of such ‘magnetic charges’ is quite unnecessary for explaining the nonlinear field effects such as saturation discussed in this article. Saturation effects do occur in the presence of magnetic dipoles, and in the case of ferromagnetic domains, that are quite well understood without the need for postulating any unobserved and unobservable “magnetic charges”. The presence of saturation is explained in QED via multiple-magnon dispersion, which is well-known to be nonlinear. (*References to such explanations and detailed QED calculations in peer-reviewed articles published several decades ago in standard scientific journals are available upon request.)

Originality: 2, Importance: 1, Overall quality: 3

2 Author comment added 12th October, 2011 at 16:32:19

Dear reviewer 47336,
as already reciprocated (27.19.11) to your earlier comments (27.10.11) on our Philica article no.99, we recommend you to read the Science (which is a high profile “peer review” journal) article from 2009: “Dirac Strings and Magnetic Monopoles in Spin Ice”. You should reconsider your statement ´… the proposed magnetic monopoles are suspect as one has never observed what the authors proposed in this article…,…the (never seen) physical isolation of magnetic monopoles, called “magnetic charges” in this articles … ´ when reflecting the work of D.J.P. Morrison et al. and others.

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