Magnetic Vector Potential


James Clerk Maxwell’s 175th anniversary was celebrated in December 2006.


His most important contribution to science were his equations relating electricity and magnetism,  which ultimately showed that visible light was only a small part of a very broad electromagnetic spectrum.


Faraday’s "Electrotonic" state is Maxwell’s Vector Potential

Michael Faraday invented the transformer. On August 29, 1831, Faraday wound a thick iron ring on one side with insulated wire that was connected to a battery. He then wound the opposite side with wire connected to a galvanometer. What he expected was that a "wave" would be produced when the battery circuit was closed and that the wave would show up as a deflection of the galvanometer in the second circuit. He closed the primary circuit and, to his delight and satisfaction, saw the galvanometer needle jump.

A current had been induced in the secondary coil by one in the primary. When he opened the circuit, however, he was astonished to see the galvanometer jump in the opposite direction. Somehow, turning off the current also created an induced current in the secondary circuit, equal and opposite to the original current.

This phenomenon led Faraday to propose what he called the "electrotonic" state of particles in the wire, which he considered a state of tension. A current thus appeared to be the setting up of such a state of tension or the collapse of such a state. Although he could not find experimental evidence for the electrotonic state, he never entirely abandoned the concept, and it shaped most of his later work.

James Clerk Maxwell, in his Treatise on Electricity and Magnetism , noted of Faraday, regarding his transformer experiment, "He therefore recognized in the second circuit, when in the electromagnetic field, a 'peculiar electrical condition of matter', to which he gave the name of the electrotonic state. He afterwards found that he could dispense with this idea by means of considerations founded on the lines of magnetic force, but even in his latest Researches he says 'again and again the idea of an electrotonic state has been forced on my mind.'

“The whole history of this idea in the mind of Faraday, as shewn in his publicationResearches is well worthy of study. By a course of experiments, guided by intense application of thought, but without the aid of mathamatical calculations, he was led to recognize the existance of something which we now know to be a mathematical quantity, and which may even be called the fundamental quantity in the theory of electromagnetism. “

Mathematicians such as the French Simeon-Denis Poisson, Andre-Marie Ampere, and the German Wilhelm Weber, had derived their electro and magnetic equations on the assumption that electrical charges and magnetic poles act on each other at a distance, while Faraday believed that electrical charges and magnets interact through lines of force which fill all space. To Maxwell, Faraday’s  idea of lines of force  rang true, but only needed a mathematical  expression. Maxwell discovered that incompressible fluid flow was an accurate analogy to the behavior of electrical charges and magnets, providing the mathematical framework for the lines of force concept.  Thus, the equations for describing incompressible fluid flow are analogs for electricity and magnetism, with pressure difference corresponding to potential difference, or voltage, and flow velocity corresponding to strength of electric or magnetic field. The new equations even  accounted for some electrical and magnetic effects which occurred at the boundary between different materials, but which could not be explained by action at a distance. Maxwell thus vindicated Faraday  and turned his ‘vague an varying ‘ lines of force into a new and mathematically impeccable concept, the field.[1]

Maxwell next set out to find a way to express the electrotonic state mathematically. With the aid of work by George Green, William Thompson, and George Gabriel Stokes, Maxwell used differential vectors (at a single point) and found that one of the vectors matched exactly Faraday’s concept of the electrotonic state. Initially he could not think of how to interpret the symbols physically. [2]

The fluid flow analog only worked for static electric and magnetic fields and constant current. As soon as anything changed, the fields behavior was unlike any physical process known at that time. [3] He modeled the dynamic behavior as spinning cells. He developed a mechanical analog for Faraday’s electrotonic state: it was the effect at any point in the field of the angular momentum Of the spinning cells. Like a flywheel, the cells would act as a store of energy, reacting with a counterforce to resist any change in their rotation. This takes the form of an electromotive force which would drive a current. [4]

The next iteration in his theory came when he added elasticity to the cells/spheres. The softer the spring, the greater the electrical displacement. For a given potential difference. Electrostatic energy was potential energy; like a spring; magnetic energy was rotational, like a flywheel, and both could exist in empty space. A change in one always resulted in a change in the other. This new “elastic” model predicted two new phenomenon: 1) a new type of current, which would arise whenever the electric field is changes (displacement current). 2) all materials that have elasticity transmit waves. The EM waves were clearly transverse because the changing electric and magnetic fields were both at right angles to the direction of the wave. He compared his calculated EM wave velocity and compared it to the experimentally  measured speed of light. They  were very close. The physicists of the day believed a medium, an “aether” of some sort,  pervading all space,  was necessary to transmit light waves, so Maxwell’s theory fit right in. [5]

In his next iteration he suspected that the ultimate mechanisms of nature might be beyond our comprehension, so he set his model aside and chose to apply Lagrange’s method of treating the system like a black box: If you know the inputs and the systems general characteristics, you can calculate the outputs without knowledge of the internal mechanism. His first assumption is that  EM fields hold energy, both kinetic and potential. Electromotive and magnetomotive forces not forces in mechanical sense, but act in an analogous way.

Most of the quantities were vectors: The five main vectors were electric and magnetic field intensities, which resembled forces, electric and magnetic flux densities, which resembled strains, and electric current density, which was a kind of flow. Electric charge density  is a scalar. These were the fundamental quantities.  Everything came together beautifully. He showed that all aspects of the behavior of EM systems, including the propagation of light, could be derived from the laws of dynamics.

The theory is embodied in four equations in the 6 major unknowns.

For a point in empty space, Gaussian units:

Div E=0 implies no electric charge is present

Div H=0 implies that no single magnetic poles are present; they always come in north south pairs.

Curl E= -(1/c) dH/dt when the magnetic force changes, it wraps a circular electric force around itself

Curl H= (1/c) dE/dt when the electric force changes, it wraps s circular magnetic force around itself.

For a point in empty space, EM units:

Div E=0 implies no electric charge is present

Div H=0 implies that no single magnetic poles are present; they always come in north south pairs.

Curl E= - dH/dt when the magnetic force changes, it wraps a circular electric force around itself

Curl H= (1/c squared) dE/dt when the electric force changes, it wraps s circular magnetic force around itself.

E is the electric force and H the magnetic force at the arbitrary  point. C is the speed of propagation of the waves. It is also the ratio of the electro of the electromagnetic and electrostatic units of charge. One crucial assumption: that  electric currents exist in empty space. It is these “displacement currents” that give the equations their symmetry and make the waves possible. Without them, dE/dt =0 and the equations collapse. The keystone of his theory, the displacement current, had its origin in the idea that the spinning cells in his construction model could be springy. Heinrich Hertz produced and detected the EM waves predicted by Maxwell’s theory 20 years later.

Div and Curl are ways to describe how these forces vary in the space around the point. + Div is a measure of the tendency to be more outwards than inwards; Curl is a measure of  the force to curl or loop around the point.  [6]

Book endnote:

For compactness,  modern vector notation is used. The substitution is legitimate because Maxwell himself later began the process of modernization. It is usual to include extra vectors B and D. The equations then become:


Div D=0 implies no electric charge is present

Div B=0 implies that no single magnetic poles are present; they always come in north south pairs.

Curl E= -(1/c) dB/dt when the magnetic force changes, it wraps a circular electric force around itself

Curl H= (1/c) dH/dt when the electric force changes, it wraps s circular magnetic force around itself.

D is the density of the electric flux produced by the electric field intensity E.

B is the density of the magnetic flux produced by the magnetic field intensity H.


But in Gaussian units, in empty space, D=E and B=H, allowing the equations to be written using E and H only.


E and H are not forces, strictly speaking, but rather the intensities of the E and H fields. They may be thought of as forces waiting to act, or that would be exerted upon a unit charge or unit magnetic pole if it were placed at that point.

Professor Richard Wolfson provides the following interpretation: [7]

Maxwell said: If  a change in magnetic field results in  an electric field, then does a change in electric field cause a magnetic field?

Div dot E = Rho/EpsilonZero: Shows that an electric charge Rho produces an electric field E.

Div dot  B = 0 Shows there are no magnetic monopoles

Div cross E = -d/dt B Shows a changing magnetic field produces electric field: Faraday’s Law

Div cross B = MuZero J + MuZero  * EpsilonZero* d/dt E Shows a magnetic field arises from either electric current or a change in electric field: Ampere’s Law.

The last two equations show that the electric and magnetic field mutually propagate an electromagnetic wave. The speed of the wave is 1/root(MuZero * EpsilonZero), which is identical to the speed of light, so Maxwell showed that light is an electromagnetic wave.


In the Dynamical Theory Paper,  Maxwell expressed his results in more expansive form. He gave 8 equations for the electric and magnetic fields, remarking that “to eliminate a quantity which expresses a useful idea would be a loss rather than a gain at this stage of our inquiry.” One of the ideas thus expressed was Faraday’s electrotonic state, which in Maxwell’s scheme, became the momentum of the field. Oliver Heaviside  condensed the results to the abbreviated modern form.

Inconsistency in algebraic sign: More detail in Thomas K. Simpson ‘s guided study Maxwell on the Electromagnetic Field. Maxwell generally favored primacy of the field, as Farady had done, but some preferred to choose charge as the fundemental entity, and their case was strengthened when the electron was discovered in 1897. See Daniel M. Siegel Innovation in Maxwell’s Electromagnetic Theory.

There is an interesting short note in the Dynamical Theory paper about gravitation. It was natural for Maxwell to see whether his idea that energy existed in empty space could somehow explain gravity. He soon found that it could not.

End End Note Reference  [8]

“James had so far written each relation involving vectors as a triple set; one for each x, y, and z direction.”

Quaternions were invented by Sir William Rowan Hamilton and have four components: a scalar part, and a vector part in each of the x, y, z directions. Says quaternions were precursor to modern vector system, which dispenses with the scalar. Maxwell found that in some of his equations, 9 ordinary symbols could be replaced by 2 quaternion terms. He also found they made the physical meaning of the equations clearer. He included the shorthand quaternion notation in his Treatise on Electricity and Magnetism, along the conventional longhand notation. To help with the physical interpretation, he coined the terms “curl”,  convergence”, and “gradient” in his quaternion representation. Today “curl”,  convergence”, and “gradient” are used.  Gradient (grad)  represents the direction and rate of change of a scaler quantity in space. “James had started the process that gives us today the system of modern vector analysis. The job was completed around 20 years later by American Josiah Willard Gibbs and the Englishman Oliver Heaviside. [9]

The rate of spatial variation at a point in space of the vector potential, or it’s “curl” gives the magnetic flux density at the point and any change in it with time ( vector potential or the curl of vector potential?) gives rise to an EMF. [10]

Maxwell’s Treatise is probably, after Newton’s Principia, the most renown book in the history of physics. The Treatise takes you along to a certain point, then, when you expect to go on, takes you back to the startand along a new path, which leads to areas inaccessible from the first. The book is not an atlas but an explorers report. In the Treatise, Maxwell made an important new prediction: that EM waves exert a radiation pressure.  He calculated that bright sunlight  presses on the earth’s surface with a force of four pounds per square mile.  In 1900, the Russian physicist, Pyotr Lebsdev experimentally confirmed Maxwell’s prediction. [11]

Hermann Helmholtz took Maxwell’s EM theory seriously. He was professor of physics at Berlin, and on the same level as Lord Kelvin in Britain. In 1879 he persuaded the Berlin Academy of Sciences to offer a prize for a conclusive experimental test of Maxwell’s theory. His star student, Heinrich Hertz took the challenge. He had to produce and detect either displacement currents,  or electromagnetic waves from an electrical circuit. Hertz finally succeded by making use of  a well known property of traveling waves: if they are reflected back upon the source, they form standing waves. Using a spark gap circuit he measured a wavelength of about 30 cm. Since  Hertz first detected the waves, they became known as “hertzian” waves.[12]

Chen Ning Yang, in a paper Vector Potential, Gauge Field and Connection on a Fiber Bundle
confirms that Maxwell identified his vector potential with Faraday's electro-tonic state: "Borrowing from W. Thomson’s (1824-1907) earlier papers on magnetism where the vector potential was introduced (called three functions F, G and H), Maxwell in paper 1) identified this vector potential with Faraday’s intuitive idea of an 'electro-tonic state'”.

The Hertzian/Heaviside lobotomization of Maxwell's Equations

The version of Maxwell’s equations appearing in the standard  technical literature are a set of four vector equations. These four equations show how a changing electric field can produce magnetism, and a changing magnetic field can produce voltage.[13](Called Faraday’s Law) From these equations, radio transmission was predicted. [14]


These four equations, however,  are a modification of Maxwell’s original work. Physicists found Maxwell’s papers confusing, so simplifications were made. [15]

According to Marc Seifer, Heinrich Hertz created an elegant mathematical interpretation of Maxwell’s equations, but at the expense of some aspects of Maxwell’s theory, most notably vector and scalar potentials. Duplicating Hertz’ s work,  Nikola Tesla postulated that these components  should not have been eliminated. [16] Hertz’ s decision to eliminate scalar potentials was also a puzzlement to Oliver Heaviside, who corresponded frequently with the German scientist during the same period. “ I am quite sure you have gone further on than Maxwell,” Heaviside wrote in 1889, “but electrostatical  (scalar) potential and magnatical (scalar) potential ought to remain I think.” Heaviside, however,  like Hertz, was in agreement with the idea of dispensing with vector potentials.[17]

Chen Ning Yang, of the Institute for Theoretical Physics, State University of New York, notes: Heaviside, a brilliant engineer, was very happy with this simplification of Maxwell’s equations. He wrote that it brought “to light interesting relations which were formerly hidden from view by the intervention of the vector potential A, and its parasites J and Ψ. E. Whittaker agreed enthusiastically with Heaviside: The great service which Heaviside now rendered to science was to clear away this accumulation of rubbish...However, the vector and scalar potentials do have measurable meaning in quantum mechanics, and should not be completely eliminated..” [18] Aharonov and Bohm showed that the vector and scalar potentials do have measurable consequences in quantum mechanics. The generations-old dogma, which had started with the work of Hertz and of Heaviside mentioned above, that the vector and scalar potentials were not physically meaningful, had created great resistance toward assigning any physically meaningful role to these potentials.  The final definitive magnetic Aharonov-Bohm experiment, utilizing the new technology of electron holography, was done by Tonomura and collaborators in their beautiful experiments of 1982 and 1986. [19]


Akira Tonomura was responsible for much of the technological development of advanced electron microscopes, which can record not only electron intensity, but phase as well. According to Tonomura, this information can be used to directly observe microscopic magnetic lines of force, or the particle and wave natures of electrons when looking at the fundamentals of quantum mechanics. In 1959, Yakir Aharonov and David Bohm proposed that a moving electron can have its phase altered by the vector potential of the electromagnetic field of a nearby object, without actually encountering the object or its magnetic field. Tonomura's enhanced electron microscope technique was used to test for the AharonovBohm (AB) effect. Using the electron microscope, in several sets of tests, Tonomura demonstrated conclusively  that the AB effect was real. "The AB effect is very subtle," he says, "and there are still many interpretations and implications." Physicists may differ in their interpretation of the AB effect, but no one doubts its existence. [20]

Frontier Perspectives, of Temple University, notes: “The Vector Potential is more fundamental than the electric and magnetic field. That is to say, a changing vector potential induces an electric field, which in turn induces a magnetic field.” “In a quantum system, the vector potential appears directly in the wave equation, analogous to chemical potential. It is capable of producing effects through this quantum field pathway even if its electric and magnetic field components are shielded.” “A British Professor of Bio-electrical Engineering, Cyril Smith, has developed vector potential sensitive instrumentation whose measurement of these frequencies matches that of the dowser estimates. It was found that an alternating vector potential carries the frequency information which may be imprinted on water. However, it was also found that a magnetic field is also required to "format" the water to enable it to accept this vector potential frequency information. [21]

A Hungarian study has confirmed the biological effects of the vector potential via its effects on water. [22]

Dr. William Tiller associates the magnetic vector potential to the “subtle domain” [23] Basically Tiller says that human consciousness can generate a physical space "conditioned" by subtle energy, that this subtle energy is related to the magnetic vector potential of classical electrodynamics, and that this subtle energy may result in psi phenomena.

Tiller asserts that human consciousness in the form of specific intention can be imprinted onto a simple low tech electronic devise (called an IIED: an intention imprinted electrical devise) from a deep human meditative state.  When the IIED is placed in a room it creates a "conditioned space". In a conditioned space oscillations of air and water temp. pH, electrical conductivity of water are global throughout the room. all exhibit same fourier spectral components and are in freq range of 10-2 10-3 hz. Such an effect on the pH of water is thought to require the accessing of magnetic momopoles, a property usually associated with a higher EM gauge symmetry state than normal [24]


The connection between vector potential and subtle energy has been made frequently on the internet. [25]


Maxwells’s Equations and Gravitation
It has been said that Maxwell's original equations provided the groundwork for gravitational propulsion and psychoactive devices. Mahon’s biography notes that Maxwell himself, in his Dynamical Theory paper,  reported that he had tried to explain gravitation in terms of his theory. “He soon found out that it could not…”[26]


Developments within the scientific community have now linked Maxwell’s equations, and specifically vector potential, with inertial forces, gravity, and even anti-gravity [27] Alexandre A. Martins and  Mario J. Pinheiro of the University of Lisbon (Portugal) have written a paper Inertial Forces and the Vector Potential, which suggests the possibility of  manipulating  inertial mass for space propulsion systems. [28] This article was included in the Smithsonian/ NASA ADS Physics Abstract Sevice. [29]


Are such concepts compatible with the Zero Point Field understanding of Newton’s  Laws, developed by Hal Puthoff, Bernie Haisch, and Alfonso Rueda? Paul Davies and William Unrah had found that if you move at constant speed through the vacuum, it all looks the same, but if you accelerate, the vacuum begins to appear like a warm sea of radiation from your perspective. Rueda found that an oscillator, forced to accelerate through the Zero Point Field, will experience resistance proportional to acceleration. A paper was published in the prestigious  mainstream physics journal, Physical Review. The paper demonstrated that the property of inertia is simply resistance to being accelerated through the ZPF. The paper shows that  inertia is a Lorentz force; a force that slows  particles moving through a magnetic field. [30]


In the early 1970’s, William Hooper had been showing that not all electrical fields are the same.  What he called a “motional” electrical field results in a force that can pass through lead, ie. It is unshieldable. [HAARP 14] This is in contrast to the well know electrostatic fields. He holds a US patent on the "All Electric Motional Field Generator." The motional electric field  is not convergent, but uniformly vertical. [31] 




[1] The Man Who Changed Everything: The Life of James Clerk Maxwell, by Basil Mahon. Wiley 2004.  p.56 f.


[2] ibid p. 63 f.

[3] ibid p. 95 f.

[4] ibid p. 103.

[5] Ipid p. 105 f.

[6] Ibid p. 120 f.

[7] Professor Richard Wolfson Physics in Your Life  Lecture 15 , The Teaching Company

[8] Ibid p. 200 f.

[9] ibid p. 141 f.

[10] ibid p. 193. Notes

[11] ibid p. 162. f.

[12] ibid p. 178 f.

[13] Routes of Science: Electricity: Blackbirch Press: 2004. p.18.

[16] Seifer p. 96

[17] Seifer p. 498 note 56



[20] Proceedings of the National Academy of Sciences: Profile of Akira Tonomura

by Philip Downey, Freelance Science Writer



[21] Is a Living System a Macroscopic Quantum System? Cyril W. Smith, Department of Electronic and Electrical Engineering, University of Salford, Salford M5 4WT, England. (Frontier Perspectives Fall/Winter 1998 pp 9-15 )



Effect of Curl-Free Potentials on Water 

Budapest, Hungary


Living objects are complex systems with various harmonized chemical, thermodynamical, and quantum-mechanical processes in aqueous electrolyte environment. We had studied the effect of curl-free magnetic vector-potential on the matrix of the living matter, on the water. The discussed theoretical considerations are in harmony with the presented simple experiments. It is shown that the vector-potential is actually an effective electro-dynamical parameter which could modify the processes in living systems.



[23] Some Science Adventures With Real Magic, William Tiller PhD Walter Dibble, Jr., PhD. p. 21, p.265.


[24] Some Science Adventures With Real Magic, William Tiller Ph.D Walter Dibble, Jr.,Ph.D. p. 265.


[25] Examples:

Subtle Energy Actions and Physical Domain Correlations

The existence of the magnetic vector potential yields the successful quantitative connection in the physical domain of the activity of the subtle energy ... - 42k - Cached - Similar pages

Institute of HeartMath Subtle Energy Research

Subtle energy is used in this context as Einstein used it, that is, ... (A) The magnetic vector potential fields include both an electrodynamic part and ... - 7k - Cached - Similar pages

( Research Notes: Potential Fields

There are three main potential fields: magnetic vector potential `bb A`, ... potential fields have subtle effects on reality at the quantum level. ... - 17k - Cached - Similar pages


Published in: Frontier Perspectives Vol. 7: 16-23, 1998

File Format: PDF/Adobe Acrobat - View as HTML
of “subtle energy”. Even Einstein himself used the term subtle to refer to ..... The vector potential (A) field in the solenoid coil is also not strong ... - Similar pages

Neue Seite 1

The known magnetic vector potential appears to have the role of a ... As a part of consciousness, subtle thought energy is able to control the cosmic ... - 40k - Cached - Similar pages

Thought Energy Synhcronization and Doctor Gregory Nicosia

Subtle energy wave flow along the etheric meridians causes transduced ..... And all are based on magnetic vector potential and natural energy fields. ... - 82k - Cached - Similar pages



[26] The Man Who Changed Everything: The Life of James Clerk Maxwell, by Basil Mahon. Wiley 2004.  p. 202.


[27] Note: 18 October 1979   Abstract  We develop here a new unified theory of the electromagnetic and gravitational field, based on a six-dimensional generalization of Maxwell's equations; additional space-time coordinates are interpreted only as mathematical tools in order to obtain a linear realization of the four-dimensional conformal group.  see also: Einstein's vision I: Classical unified field equations for gravity and electromagnetism using Riemannian quaternions


·  Inertial Forces and the Vector Potential
Manipulating mass for space exploration University of Lisbon Portugal

·  The Connection between Inertial Forces and the Vector Potential
Smithsonisn NASA: Martins article: Abstract:
The inertia property of matter is discussed in terms of a type of induction law related to the extended charged particle's own vector potential. Our approach is based on the Lagrangian formalism of canonical momentum writing Newton's second law in terms of the vector potential and a development in terms of obtaining retarded potentials, that allow an intuitive physical interpretation of its main terms. This framework provides a clear physical insight on the physics of inertia. It is shown that the electron mass has a complete electromagnetic origin and the covariant equation obtained solves the “4/3 mass paradox”. This provides a deeper insight into the significance of the main terms of the equation of motion. In particular a force term is obtained from the approach based on the continuity equation for momentum that represents a drag force the charged particle feels when in motion relatively to its own vector potential field lines. Thus, the time derivative of the particle's vector potential leads to the acceleration inertia reaction force and is equivalent to the Schott term responsible for the source of the radiation field. We also show that the velocity dependent term of the particle's vector potential is connected with the relativistic increase of mass with velocity and generates a stress force that is the source of electric field lines deformation. This understanding broadens the possibility to manipulate inertial mass and potentially suggests some mechanisms for possible applications to electromagnetic propulsion and the development of advanced space propulsion physics. ©2007 American Institute of Physics




[30] The Field Lynn McTaggartHarper Collins 2001 p. 32 f.


Bernie Haisch had read some of Hal's papers and got intereste in the ZPF. He was inspired by the work of Paul Davies and William Unruh at the U of BC. They had found that if you move at constant speed through a vacuum, it all looks the same. but as you accelerate, the vacuum begins to appear like a lukewarm sea of heat radiation from your perspective. Haisch talked Alfonso Rueda into doing a mathamatical analysis of an idealized oscilator moving thru the ZPF. He found that an oscilator forced to move thru the ZPF will experience a resistance proportional to the acceleration. Thus Newtons law f=ma can be derived from electrodynamics. Puthoff, Rueda, and haisch collaborated on a paper that Haisch wrote, and it was published, unchanged, in Physical Review. In the paper they show that inertia is a "lorentz force" - a force that slows particles moving thru a magnetic field.

The Connection between Inertial Forces and the Zero Point Field




[31] []