PHYSICS OF LEVITATION

Dimitri S. H. Charrier

Dimitri Charrier / 2019 / http://dscharrier.free.fr

Intelligenti pauca

‘Few words suffice for he who understands’

Anonymous author

‘[relativity theories] are made for believers: a believer will never oppose his guru’

Demetris Christopoulos (researchgate.net)

Edition 2019

PREFACE

When I first got interested to explore models and make experiments of innovative thrusters I expected in the end that it could trigger passionate reactions. It is a healthy reaction. Beginning at my individual level, the secondary effect was the appropriation of this topic by other people. It means that the transfer of the passion, message and idea is a success, in other words the message I wanted to express has been received and well appreciated. I experienced it mainly after the publication of articles.

While writing this essay ‘Physics of Levitation’, I felt it was an interesting exercise. Maybe because this domain is a little child of Physics. Personally when I grow up in my education and instruction at school and university, I wish I could have had in my hands such essay - even if it is really a personal vision - gathering concepts and manners to levitate a solid object above the ground, such easy thinking experiment, still today submitted to technological limits. Indeed, in the literature such essays or books blending common levitating means with still hypothetical ones are scarce, for not saying nonexistent. It has at least to be read once for getting a reminder and a view on how basic effects, laws and principles of physics can be turned into a beautiful scientific tool. A high potential of technological applications and therefore benefits is possible.

However most of all, I consider chapter 3 ‘Thermodynamic approach’ as a revolution. Indeed, it reminds that the coupling of different physics can give birth to new visions and opportunities. It is clearly not an engineering book. Do not hope to be capable to build immediately a machine to levitate after having read this essay. It is more an outsider than a well conventional, standard essay or book, although its frame is strictly using academic and recognized notions of physics.

Dimitri Charrier

FOREWORDS

If you read these lines, it means you are animated by a natural curiosity about levitation, which is the key preliminary to innovative propulsions. This edition gathers notes and concepts which I have written and collected since 2002. Future editions might come with more examples of levitations, including advanced models. So far, this edition contains demonstrations based on rigorous and classical physics. Still, the difficulty was to give an apparent coherence in the construction of this topic, especially because new models are presented. Keeping a good size ratio between the chapters was a challenge.

17 years of reflection led to the publication of models, proposals, experimental results in influencing scientific journals. I rapidly experienced after publishing articles lively exchanges with anonymous and passionate readers. Also comments found on internet demonstrate that the society has an interest on the breakthroughs in propulsion and thruster physics. Particularly with sensitive measurements and their coupling with the Earth magnetic field, this part is discussed in Chapter 3, 3-2. It was time to publish to the largest audience as possible, both impulsion levitation (Chapter 1, part 1-4) and thermodynamic (Chapter 3, part 3-1) models. Graduated students, amateurs, professional physicists or even electrical engineers re-demonstrated the published models and some also tried to replicate the micronewton electromagnetic thruster. The interest of this essay is to re-open a debate on the principle.

Is it possible to levitate an inertial object above the ground?

I will try to give an answer to this question all along this essay. Any technology is attainable only if the models are understood in advance, followed and designed into a device. The levitation domain is at the junction between classical mechanics, classical electrodynamics and thermodynamics. Indeed, if we only restricts to the 1^st and 3^rd laws of Newton, it is not possible to move and change direction of an inertial object launched in one direction in an empty space without changing its momentum. Unless there is a loss of mass. Tsiolkovsky model illustrates a nice use of classical mechanics and is the basics one for rocket engineers. In addition, I led a study showing it is possible to change momentum not only in a short scale of time due to retarded electromagnetic interaction, Lenz’s law and Lorentz’s force, but especially due to asymmetrical electromagnetic interactions and heat transfers leading to an apparent loss of momentum. Therefore, mechanics, electromagnetism and thermodynamics need to be associated in one hybrid model for highlighting new applications such as levitation.

A motivation when exposing such models is to invite teachers at university level to include such topics. Being a student I was interested in overcoming mindset barriers, especially in the side sciences such as levitation. Indeed, physics courses at the university are sometime taught in separated silos. Sometimes footbridges exist, between electromagnetism and thermodynamics in superconductivity for instance. Authors writing a manuscript unless motivated by getting a graduate degree or by sub-contracting with an editor take a personal initiative. Most of the time they want to give a testimony of a discovery, express personal feelings, denunciate an injustice, explain a model, a philosophical concept, etc. Writing lines never falls in the ear of deaf. As soon as the knowledge of one individual is shared and beneficial for a large community then the challenge is won. Clearly the levitation topic deserved that ink and paper are spent. Moreover when touching the boundaries of the common human knowledge, a trait of poetry, epistemology and philosophy is necessary, which is a mean to overcome the visible and material condition of the human.

This topic intends also to reconcile two communities: applied physics community and the theoretical physics community. The first one might believe that the second one misses the instinct of physical laws driving universal law which is essential. While the second one sometimes believes that the first one is composed with people having failed their graduate degrees. This situation can be compared, to some extent, with the opposition between work and capital, they need each other. Those two communities sometimes do not understand each other. Applied physics, likely because the effects are more impressive on the daily economical life, such as solid state science is present everywhere:

Ø Superconductors

o magnetic flux trapping and Meissner effect

o high current densities field in coils for magnetic resonance imaging

Ø Metals

o high magnetic permeability in transformers

o high electrical conductivity in cables

Ø Inorganic semiconductors

o doped p-n junction in diodes

o solar photon conversion in photovoltaic diodes

o switches and amplifiers with transistors

Ø Organic semiconductors

o conjugated π-systems and applications as for inorganic semiconductors

o printed electronics

o zero band gap semiconductor as graphene

Ø Insulators:

o low electrical conductivity in high electric field applications

o low thermal conductivity

Ø Glass:

o wave guides as optical fibers

Ø Nuclear:

o nuclear reactor

o neutrinos

Solid state science and their applications shown above are not exhaustive.

Theoretical physics like general relativity and cosmology are a little bit spared by economical profit models. Funding from states is justified because in average we all realize as taxpayers that this is a common interest for human being to understand where we are coming from and where we are. Moreover, general relativity opens a fantastic vision of our world with curved space, contraction of time with acceleration and gravity. This vision opens doors to time travels (to the past and to the future, see the great book of Paul Davies ‘How to Build a Time Machine’) and more speculatively to space travel with the highly hypothetical worm holes.

Come back to Earth, levitation technologies could find patronage in private institutes because of the clear interest of mobility in our current economic model. Means of mechanical transportation still constitutes a part of the heart of manufacturing industries: cars, trains, planes, rockets, vactrain. Note there, the thermal combustion is nowadays a major and direct way to produce energy for transportation.

The law of physics do not follow law of economics. In our modern civilization, making profit or money is often related to energy recuperated out of a transformation process: services, agriculture or manufacture. Organizing a money based on energy unit because 1 joule will remain 1 joule, is a risky proposal in our organization. The dream to see a currency based on a physical quantity is a dream of physicist. The capability of banks to print bank notes is made possible by debts, based on future and constant hypothetical prosperity and human creativity.

However, society can also approve that public money is spent because levitation is also something unknown with a large potentiality.

If private or public investors do not contribute in putting human and material resources, levitation technologies will remain as a side science, standing as a poor relative.

Coming back to the topic of our interest, this work is dedicated to open minded scientists having a basic background in physics.

For the time being, I hope you will enjoy the approach on how this niche subject is treated. I have a light presentiment that it will be a referent essay.

CONTENTS

PREFACE.. iii

FOREWORDS. v

CONTENTS. x

Chapter 1 PHYSICAL MODEL. 1

1-1 Introduction. 2

1-2 Laws of physics. 4

1-3 State of the art in levitation means. 8

1-4 Innovative model of levitation: impulsion. 19

Chapter 2 MECHANICS IN ELECTROMAGNETISM... 24

2-1 Lorentz’ force. 25

2-2 Magnetic pressure. 26

2-3 Closed loop and momentum conservation. 29

Chapter 3 THERMODYNAMIC APPROACH.. 34

3-1 One case: Coil-Disc device. 35

3-2 Work and force expression. 42

3-3 Heat dissipation. 43

3-4 Epistemology. 44

REFERENCES. 45

PAPER – 2010. 51

PAPER – 2012. 72

CONFERENCE – 2016. 86

INTERNATIONAL SYTEM BASE UNIT.. 99

FIGURES, TABLES AND QUOTES. 102

Chapter 1 PHYSICAL MODEL

‘Concrete is abstract made familiar by use’

Paul Langevin (La Notion de Corpuscules et d'Atomes, Hermann, Paris, 1934, p.45)

1-1 Introduction

‘Levitation’ term coming from Latin levitas "lightness" is often attached with an original reputation coming from esoteric traditions of levitating humans in worldwide cultures, in tales, even magicians entertain us with levitation tricks. Levitation and immortality are classified as myths, sometimes reserved to blessed people or demigods. The levitation discussed in this essay is a real levitation process for an object to sustain in vacuum in a stable position, with the only presence of gravitational force. No other external fields or forces are exerted on the object to counter the gravitation. Only the object aims to produce a counter force. The last chapter focuses on the principle of energy conservation which prevails on the momentum conservation. Indeed, emitting heat coming from Joule heating is responsible for a real breakthrough technology. Heat can be dissipated by infrared radiation, conduction or convection if a fluid surrounds the hot part.

In the following chapters, speculative physics like vacuum energy, gravity shields, lacking of experimental demonstrations are explicitly chosen to be not enumerated and assessed. Only a preference remains of rigorous laws taught in the universities. Also, because the levitation process is not a zero-g exerted on the object still feeling the gravitation.

Chapter 1 introduces the topic with a first part 1-1 containing a reminder of laws of physics used for the demonstrations. Going through all those concepts in detail is not the objective here, it might be the easiest part accessible to read for non-experts. It is rather a reminder of physical laws. A brief review of real and hypothetical ways to sustain a solid object in a fluid (air or liquid) and in vacuum is given in part 1-3. Finally, a model is given in part 1-4, this is novel as far as the author knows.

Chapter 2 aims to entertain the reader with three cases treated in parts 2-1, 2-2 and 2-3. Those three parts are new and never published as far as the author knows. They demonstrate that classical mechanics and classical electromagnetism do not bring satisfaction alone for getting a true levitation. A third element of physics is needed: thermodynamics.

Chapter 3 is the binder between Chapters 1 and 2 by proposing a thermodynamic approach in order to answer our question of levitation. Particularly, a note is given on the ‘Earth magnetic field controversy’ and the micronewton electromagnetic thruster. Finally, a last part opens a discussion on the epistemology face to the presented topic. It was necessary to give a philosophical point of view before closing this subject.

1-2 Laws of physics

A succession of major concepts in classical physics is given. Basic laws of mechanics (Table 1-1), postulates of special relativity (Table 1-2 and Table 1-3), laws of electromagnetism (Table 1-4), thermodynamics (Table 1-5) and thermal radiations (Table 1-6) are the fundamentals for describing levitation mechanisms and motion in general. Regarding the equations and symbols in tables 1-3, 1-4 and 1-6, the curious readers are invited to explore specialized books in the respective areas.

Table 1‑1 Newton's laws of motion [1], [2]

	Newton's laws of motion
1^st law	In an inertial frame of reference, an object either remains at rest or continues to move at a constant velocity, unless acted upon by a force.
2^nd law	In an inertial reference frame, the vector sum of the forces F on an object is equal to the mass m of that object multiplied by the acceleration  of the object: F = ma. (It is assumed here that the mass m is constant)
3^rd law	When one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body.

Table 1‑2 Einstein’s postulates of special relativity [3]

	Postulates of special relativity
1^st postulate	The laws of physics are invariant in all inertial systems
2^nd postulate	The speed of light in a vacuum is the same for all observers, regardless of the motion of the light source

Table 1‑3 Consequences of special relativity

	Consequences of special relativity postulates
Time dilation
Length contraction

Table 1‑4 Laws of electromagnetism [4]

	Laws of electromagnetism
Joule	Electric current through a conductor produces heat
Lenz
	e is the electromotive force (V), F the magnetic flux
Biot-Savart
Coulomb
Lorentz
London
Retardation
Jefimenko

See the considered first modern book on the topic from Sadi Carnot [5] in French.

Table 1‑5 Laws of thermodynamics

	Laws of thermodynamics
0^th law	If two systems are in thermal equilibrium with a third system, they are in thermal equilibrium with each other. This law helps define the concept of temperature
1^st law	When energy passes, as work, as heat, or with matter, into or out from a system, the system's internal energy changes in accord with the law of conservation of energy. Equivalently, perpetual motion machines of the first kind (machines that produce work with no energy input) are impossible
2^nd law	In a natural thermodynamic process, the sum of the entropies of the interacting thermodynamic systems increases. Equivalently, perpetual motion machines of the second kind (machines that spontaneously convert thermal energy into mechanical work) are impossible
3^rd law	The entropy of a system approaches a constant value as the temperature approaches absolute zero. With the exception of non-crystalline solids (glasses) the entropy of a system at absolute zero is typically close to zero, and is equal to the natural logarithm of the product of the quantum ground states.

Table 1‑6 Thermal radiation laws

	Radiation laws
Stefan-Boltzmann
Wien
Planck

1-3 State of the art in levitation means

Gravity is a long range interaction. As in electromagnetism, several attempts demonstrating that this interaction can be shielded failed so far. Electric field cannot escape or enter from a copper Faraday cage, while flux lines of magnetic field do. Magnetic shields are made of high magnetic permeability materials such as iron. In case of gravity, a hypothetical material with a high gravitational permeability could play a role of shield.

Note at this point, the concept of gravity shield is different from a concept of anti-gravity device. The first is constituted with walls capable to trap gravitational waves, but they still have a mass (the shield) submitted to gravitation, while the anti-gravity device is rather a concept flirting with speculative physics where its walls repulses the gravitation waves. Unfortunately, all massive bodies with (positive) masses attract each other unless other repulsive interaction prevails.

Let’s come back to the concept of gravity shield. For instance why not imagine a huge horizontal hollow pipe turning around its central axis. The mass of the pipe submitted to gravitation must transfer information to the attractive and massive body (Earth) of its mass while it is turning on itself. Likely the effect of gravity inside the hollow pipe rotating at high speed could be reduced due to the screening of gravity wave. This could be due to loss of gravitational information exchange. This purely hypothetical exercise has been theorized in the past, but without any convincing experimental setup so far to support it. In the 90’s, a debate emerged with a rotating superconductor shield [6], but as discussed in 3-4, science is not only to demonstrate a fact, a law, but consists in being reproducible at any place, at any time by anybody having proper equipment and skills [7]. A whole area could be funded by states in order to go in that direction. For getting funding a proof of concept must be established at first place.

On the other hand, proof of concepts and even working devices are capable to levitate using the known laws of physics, as this Chapter is dedicated to.

A clear distinction is shown in Table 1‑7 between assisted levitation from ground installation and self-levitating devices. Even in those two families, different environmental supports are possible:

§ In fluids (Navier–Stokes equations)

o Gas flux (see Figure 1‑1 the author performing a ping pong levitation)

o Acoustic trapping (see Figure 1‑2 a potential energy distribution)

o Liquid, water (see Figure 1‑3 a case of granite stone levitating above a water film)

o Ionized gas, plasma (magnetohydrodynamic, Biefeld-Brown effect)

§ Electromagnetic waves

o Electrostatics[*] (see Figure 1‑4 a case of electrostatic levitator in vacuum)

o Magnetostatics^*

o Optics

o Radio frequency

o Superconductivity (see Figure 1‑5 for high temperature superconductors)

o Quantum electrodynamics

§ None

o Rockets

o Coupled or hybrid physics

In fluids, as well as in liquids and in air, the magnetohydrodynamic, also noted MHD, offers the theoretical and sometimes practical opportunities to sustain and move solid devices. As the cross-section between fluid mechanics and electromagnetism, in particular with the Lorentz’ law, the mathematical description of MHD can be found here [8], [9]. Charge discharges and plasma physics are mechanisms already present in Nature: active sun emitting charged particles, solar wind and plasma streams, Earth’s magnetosphere and ionosphere. In the technological world, plasma can be found in lighting with different types of lamps (fluorescent, plasma displays) and in prototype controlled fusion reactors. For both applications, the power is generally supplied from ground installation, i.e. requiring an important amount of energy. I think it is important to mention, here, that for levitating and moving a solid device in a fluid with MHD, it is technically possible. Past works reported some MHD sea-water propulsion for naval applications, as discussed by Tixador [10]. In France, Petit is a famous public author and specialist on MHD [11], he often quoted “the silence barrier” in opposition with the “sound barrier”. Indeed, MHD is a way to suppress the sound barrier due to annihilation of air contact. One might wonder the amount energy necessary for transporting a device in the sea and in air. Surely, due to Archimedes law, less energy is required for transporting a heavy submarine in sea-water than in air over a same distance, the submarine being heavier than air. Ionizing a fluid and using ions to propel a device seems less constraining in a liquid than in a gas. In air, a huge amount of energy is required at least to maintain a steady altitude called levitation. Then an additive energy is needed for moving in any direction. The next part 1-4 deals with the notion of impulsion in such situation.

Theoretically, MHD propulsion is a beautiful way to maintain a device in a steady altitude in air, or to propel in a fluid.

Investments have been made for military applications.

In 2018, an announcement by the state of Russia was made concerning an ultra-fast missile, above Mach 5. Obviously, no clear indications are available, although MHD technology might be involved. As mentioned above, the energy embedded in a device is a key parameter for achieving levitation. In the case of the Russian missile, a nuclear power reactor seems to feed the propulsion. It is publicly said that the nuclear energy is converted into heat for compressing air. It is less sure that a nuclear reactor creates electricity with a turbine delivering a power, a voltage U and a current I. But why not? The dream of a portative nuclear reactor is maybe real, like a pocketreactor.

Space technologies are forwarding a breakthrough physical concepts satisfying again a compromise between cost, energy consumption and spacecraft speed. An interesting book treating those topics was edited and published by the American Institute of Aeronautics and Astronautics in 2009 [12].

In Table 1-7, cases are left intentionally empty because of potential existent phenomena, not reported, not known by the author. Other ways of levitations called trapping or tweezers are not exhaustively presented here because they involve nanometer or micrometer size objects, such as atoms, ions, aggregates, molecules, etc. Famous trapping methods are magnetic trapping using the magnetic moment of atoms, for instance in Bose-Einstein condensates. Other trapping techniques reported are named Penning trap and magneto-optical trap.

Levitation is a particular case of propulsion. Ideal levitation characterizes a floating solid object with no physical contact as being fixed and stable in Cartesian  or spherical  coordinates. We get respectively during a reasonable period of time  or . Indeed in some real levitation cases, instabilities or perturbations particularly in fluids are met and prevent the stability conditions. Since gravity depends on altitude, levitation is no longer an issue when the terrestrial acceleration  becomes close to zero at high altitude. The fundamentals of propulsion physics are known and not sophisticated where laws and principles are recognized. Tsiolkovsky rocket equation is calculated with basic Newtonian mechanics. Solid state physics such as superconductivity is likely the most recent one. Flotation and levitation due to external static field forces are treated in an article [13]. There are cheap and expensive ways to maintain levitation of a solid objects. It depends on their weight mainly.

Table 1‑7 Ways for levitating solid objects met in Nature and technologies

Support	Physics domain	Levitation
		Ground installation		Embedded system, autonomous
		no power supply	power supply	no power supply	power supply
Fluids	Fluid mechanics	Particles in air Brownian motion	Air flux Liquid flux [14]	Aerostat Submarine	Hummingbird
			Hair dryer and ping pong ball		Drone Helicopter
	Mechanics		Acoustic pressure [15]		S.A.S.E.R.[†]
	Plasma physics				M.H.D.[‡] [8], [9]
					Biefeld-Brown [16]
Electro-magnetic waves / Photons	Electro-statics		Electro-static levitator [17]	Charged clouds
	Magneto-statics	Diamagnet [18]	Feedback [19]		Lorentz’s force[§] Magnetic pressure[**]
		Paramagnet
		Ferromagnet [20]
	Electro-magnetism Optics		Optical tweezers		Photon rocket
			Beam-powered propulsion
			Photonic laser
	Electro-magnetism Radio-frequency		Lorentz’s force: coil above a plate
			Spinning magnets: hover-board
	Quantum electro-dynamics	Casimir effect			E.M. drive [21]–[24]
None	Mechanics				Rocket engine
					Closed rocket engine[††]
					Impulsion rocket[‡‡]
	Coupling physics				Hybrid physics [§§]

Figure 1‑1 The author performing the air flux trap with a ping pong ball

Figure 1‑2 Potential energy distribution of ultrasonic standing wave [15]

Figure 1‑3 “Kugel fontain”: a stone levitating with a film of water [14]

Figure 1‑4 Ti Zr-Ni alloy in an electrostatic levitator chamber (Physics Today)

A complete list of magnetic levitation means is shown in Table 1‑8, some comments must be added: i/ magnetic levitation is submitted to intrinsic instability as shown for instance in Figure 1‑6 and Figure 1‑7 with a saddle like magnetic well, ii/ in case of solid objects, they are found sometimes in natural state, or need to be purified and transformed with a cost of energy. The typical physical property is the unit-less magnetic susceptibility c. Diamagnetic materials (the ideal being the superconductors c = -1 also called the Meißner–Ochsenfeld effect) have a negative susceptibility, while paramagnetic and ferromagnetic materials have a positive susceptibility [25].

Susceptibilities of typical materials are shown in Table 1‑9. Carbon atoms organized in graphite structure show a relatively low susceptibility, C atoms are abundant. Since a decade now that the exfoliation of highly ordered pyrolitic graphite (HOPG) layers is a best practice for investigating electronic properties of graphene. For the exercise, the author made the experiment of levitating a B pencil above alternated tiny magnets, as shown in Figure 1‑8, inspired by other works [18].

Figure 1‑5 Are room temperature superconductors possible? [26]

Table 1‑8 Complete map of magnetic levitation and examples

				Levitating object
				Metal	Solid object			Coil /
						/
Ground installation	Metal						[27]		coil above plate
	Solid object					wip	wip	wip
					[18]	[20]	wip
				[27]	wip	wip	wip[***]
	Coil				[28]
				[29]

Table 1‑9 Magnetic susceptibilities of different elements [30]

Elements	(c10^-9)	Elements	(c10^-9)
Ag	-19,5	Al	16,5
Bi	-280,1	Ti	153
C Graphite	-6
Cu	-5,46
In	-107
Pb	-23
Sn	-37

Figure 1‑6 Iron ball floating in air [20]

Figure 1‑7 Instable potential well (saddle like) under static magnetic field

Figure 1‑8 The author performing a diamagnetic levitation of a B pencil lead

1-4 Innovative model of levitation: impulsion

A basic physical model of levitation is given, it could be valid for any type of physical ways as long as an impulsion is given. The model treats a uniform field force, a solid object is capable of producing impulses controlled by itself. The case of gravitational field is treated, i.e. maintaining an object in sustentation is similar with suppressing the field force. The interest is considerable for technological point of view, it reduces the constraints of Earth attraction when taking off and traveling into space. Nowadays technologies of physical thrusters offer nanonewton or micronewton forces are not able to lift an object by itself, i.e. offering enough impulsion to its center-of-mass to counter balance a uniform gravitational force field. The following part gives physical parameters dedicated to the instrumentation controlling the levitation stability. With Newton equation, time, frequency energy and power considerations are given. The model opens routes toward design of levitating devices.

A body with mass m in the vicinity of massive body such our Earth is submitted to attraction force described with Newton equation:

Equation 1.1

where  is the acceleration depending on the massive body mass, gravitational constant and distance between two bodies, on Earth . The free fall is perfectly described using the Equation 1.1. Now consider the body capable of giving an impulse force in the direction opposite to attraction.

Along an axis z with , as shown in Figure 1‑9, in free fall the acceleration is noted

Equation 1.2

Figure 1‑9 Free fall axis

Initial conditions are

§ rest position at height

§ impulse against gravity direction at speed

Both speed and acceleration can be calculated

Equation 1.3

Time when   is then

Equation 1.4

Moreover the principle of energy conservation gives us that

Equation 1.5

Therefore the time when both speed and height are at equal zero

Equation 1.6

The time in Equation 1.6 is nothing else the time when the solid object reaches a new rest position before falling in free fall, as shown in Figure 1‑10. The principle of apparent levitation is to give an impulse at each new rest position and in such way that  is very small. It is a pseudo levitation that still looks like a levitation on small time scales, while on long time scale the object moves up.

From quantitative point of view, the calculation of power necessary for insuring one jump is:

Equation 1.7

where  is the energy coming the pulse of force in one period, ie it is the energy necessary to elevate the object from  to , and  the frequency of energy pulse.

Note the  is equal to kinetic or potential energy  in amplitude range of height h. Therefore Equation 1.7 becomes:

Equation 1.8

Now the power related in 1 second when N pulses are submitted such as , therefore the power per second unit is

Equation 1.9

Figure 1‑10 Apparent and pseudo-stable levitation

It is noticeable to see that the power does not depend on . In this model, the pseudo-levitation is due to the choice of giving an impulse at each new rest position, in the end the object rises. A better levitation case would consist in leaving the object falling down to the position  , i.e. the initial position, as shown in Figure 1‑11. There, a new energy pulse  against gravity is not enough to rise the object again because of the gained kinetic energy  coming from the free fall. Double of energy  is required for i) suppressing the kinetic energy coming from the free fall, ii) giving impulse to go from  to .

In the end the power calculation for one period of impulse is equal to Equation 1.8 because both time and energy doubled. Finally, the power over N impulses created during one second is equal to Equation 1.9.

Since gravity and its acceleration depend on the geophysics, a local acceleration must be measured in real-time in order to adjust a stable levitation.

Ideal thrusters not losing weight are still speculative, however, in the next chapter we will go through original ideas which have been reported in public cultures, in patents, comics, pseudo-scientific blogs, etc.

Figure 1‑11 Stable levitation model

Chapter 2 MECHANICS IN ELECTROMAGNETISM

‘There is no exquisite beauty (...)

without some strangeness in the proportion’

Edgar Poe to Lord Verulam

Sapere Aude

‘Dare to know’

Horace (Epistles, I, II, 40)

2-1 Lorentz’ force

The famous Lorentz equation giving the force F exerted by a magnetic field B on a current I circulating in a conducting wire.

Equation 2.1

The situation capable to naively create a thrust is to place a magnet having a magnetic field B perpendicular to a metallic wire having a current I, as shown in Figure 2‑1. The question now is to ask ourselves if the rigid device magnet-wire can move in one direction.

Figure 2‑1 Permanent magnet creates a force F in a conductor with a current I

In the device magnet-wire the asymmetry stands only in the geometry, the magnetic field B emitted by the magnet creates a force on the wire and inversely the current I circulating in the wire.

Having experimented by myself I can confirm no displacements were visible. Here again if the heat losses were asymmetrical between magnet and wire, likely a full thermodynamic treatment could demonstrate that the device creates a thrust.

2-2 Magnetic pressure

Another idea is the magnetic pressure exerted on a conical coil. The magnetic pressure is an important concept when designing and building solenoids with elevated magnetic field. A long cylindrical solenoid, as shown in Figure 2‑2, having N’ spires per meter is flowed by a current I. Then the current per length unit is N’I.

Figure 2‑2 Cylindrical coil represented with magnetic flux lines

The metallic wound wire constituting the cylindrical coil has diameter a. In the coil, the magnetic field is axial and uniform

Equation 2.2

Having a null magnetic field outside the coil, inside the winding, the average magnetic field B is half of term given in Equation 2.2.

Equation 2.3

However, inside the winding, the magnetic field B is axial and the current density is tangential. Then, the volume force density taken from Lorentz Equation 2.1 is equal at  (you can check that the dimensions are indeed N/m³ i.e. a force in Newton per volume unit) and is oriented radially outside the coil. The volume force density of magnetic field exerted on the winding is like having a gas pressure, like if a gas under pressure was trapped inside the coil. What is the value of this pressure? The volume force density F’ is given by JB where

Equation 2.4

Then the volume force density (N/m³) at the winding is equal at

Equation 2.5

In the end the magnetic pressure (N/m²) exerted on the wire having a diameter a is equal at

Equation 2.6

Note that the pressure depends on the squared of magnetic field. Note also that we have considered a null magnetic field outside the solenoid. Is the magnetic pressure significant? Let’s take B = 1 T. Then the pressure is equal at 

Equation 2.7

Usually designing high magnetic field coils require a care for preventing mechanical deformation with wires as well as made of copper or superconductive materials.

In the case of a conical coil, as shown in Figure 2‑3, we can launch the question, what is the effect of magnetic pressure on the coil. Does the pressure exerted on the winding trigger a thrust toward the direction of the little base of the cone? This would be possible if the magnetic force is indeed normal (perpendicular) to the coil surface. For now I do not have a mathematical demonstration to show that the coil does move or not toward the little base of the cone. What is certain is that the magnetic field amplitude decreases when going from the little base to the large base of the cone.

However, if we can come back to the analogy of gas pressure contained in a conical balloon. Does a conical balloon move toward its small base cone? Surely not.

Note that the conical shape is met in device called ‘RF resonant cavity thruster’ designed by Roger Shawyer. Interesting and less interesting communications are now in the public domain [21-22], [24], [31-36]. Again, nano, micro newton are involved here. Think big, start small.

Figure 2‑3 Conical coil represented with magnetic flux lines

2-3 Closed loop and momentum conservation

In this part, I would like to entertain the reader interested in propulsion techniques which do not expel chemicals or ions. Some people call this technology ‘closed systems’ if they indeed only use internal and closed mechanical motion and if they do not interact with the environment by expelling mass or heat (radiation, conduction or convection). Closed system must mean also adiabatic system, this is the discussion of the last chapter. Closed systems seem to violate the principle of momentum conservation. It seems naive at first place to treat about this subject, however, tenths of honest people developed the wackiest devices sometimes patented. The run for money nowadays makes lose all notions of physics.

Those closed systems are in opposition with open systems having an interaction with the environmental space. I will demonstrate that ideal closed mechanical systems are impossible to use as a propulsion and thruster technology. However, real systems encounter mechanical frictions leading to heat loss and this opens a possible way to explore for applications. The conservation of energy might supplant conservation of momentum as discussed in the last chapter.

To put the context, let’s start to draw a simple picture. A fisherman on a little boat lost his paddles. Fortunately, his bucket water filled is plenty of big alive fish. Then, he has the idea to throw them away from the boat in one opposite direction of his wished target on waterside. By the way, he noticed the boat took some momentum when he thrown very hard.

The first example of propulsion without propellant appearing in patents having sometimes several tenths of pages starts from the following idea: the current chemical rockets use a propulsion of ejected matter into space, so why not recuperating the matter into a closed circuit, for eventually decelerate it and reaccelerate. Gyroscope thruster also starts with similar considerations.

In order to demonstrate that it does not work, I took the example of two masses M that I launch in the back of a ship creating so a thrust in front of the ship. The two masses would be recuperated on two opposite circular circuits or rails guiding the motion in such way that the ship still keeps a rectilinear direction.

Figure 2‑4 Schematic of two masses launched in 2 symmetrical rails

The thrust is the force that accelerated masses exchange with the ship, the kinetic energy is equal to the integral of the force F on the length L on which force is exerted:         

Equation 2.8

According to Figure 2‑4 and Figure 2‑5, the mass M starts its curved path at a point x = r and y = 0 at a speed V, then the mass reaches α after a time .

Equation 2.9

Figure 2‑5 Circle on which mass the M stays confined in closed circuit

where N is the number of position that will be integrated along the axis y, I will calculate the contribution of initial force on the y component, for that I use the quantity of motion .

Equation 2.10

Concretely, it means that the mass M transfers both energy and momentum along axis Oy.  is the time that the mass M travels on the circle the angle . Let’s define d a being the distance between  and  on the circle, then the following can be written:

Equation 2.11

Then the quantity of motion is written:

Equation 2.12

The resultant of momentum quantity  along the y axis is expressed as the sum of initial momentum quantity  plus the reaction of mass M on the curvature which is Equation 2.12 with N going to infinity, plus the contribution of mass M when stopping abruptly at the position .

Equation 2.13

A mathematical numerical software can show that

Equation 2.14

Therefore the resultant becomes null:

Equation 2.15

It can be comforting, intuitive and logical for some, or, crazy, insane and impossible for others, but the fact is it is not possible to use gyroscopes, closed circuit, for moving a full body without losing its mass.

However.

And now it is becoming interesting.

If some losses are generated when our system of two masses follows the circular path, then the Equation 2.14 is no longer valid. The momentum would be not transferred entirely into the device, but would be partly lost in the loss mechanism. The losses can have different origins, they can be mechanical friction in case of mechanical contact or eddy current heating in case of magnetic rails, etc. In any cases, the losses are a conversion of mechanical momentum into heat. In case of adiabatic closed system, the internal temperature increases, dissipation is then recommended by conduction (and convection) and radiation. Therefore, the thermodynamics physics enters here as a trigger for alternative ways for thrust. As a consequence, such system is partially closed. It is mechanically closed but because heat exchange such as infrared radiations must exit for preventing the adiabatic system to increase in temperature.

Chapter 3 THERMODYNAMIC APPROACH

‘Words exist because of meaning.

Once you've gotten the meaning, you can forget the words.

Where can I find a man who has forgotten words so I can talk with him?’

Zhuangzi (369-286 BC)

Labor omnia improbus vincit

‘A hard work defeats everything’

Virgil (Georgics, I, 144-145)

3-1 One case: Coil-Disc device

In Chapter 2 we have seen that the action-reaction principle works perfectly for mechanical systems and that any tricks to circumvent this specific law of Nature is not possible. In the same chapter, we have seen a non-contact interaction based on magnetic interaction, and the same conclusion remains, for every distant or retarded interaction a counter interaction exists. However, a bias exists that is to create an asymmetry in the losses that are frictions in mechanics, joule heating in metals submitted by electromagnetic waves and eddy currents. As much as surprising it might be, this the latter one we want to enhance in order to create a breaking in the action-reaction principle.

In this Chapter the case of an electromagnetic actuator creating a non-contact force on a disc is taken. This choice is not fully arbitrary since it has been theorized, tested and published in peer review journals. The device Coil-Disc is taken as an illustration of the demonstration, but any other device satisfying the breaking by creating asymmetrical losses is satisfying too.

In the following three lines, we can see the basic notions:

1. Electromagnetic actuator and the Lenz’s law: electromagnetic energy

2. Lorentz force: mechanical work

3. Thermal dissipation: Joule and thermal losses

Conservation of mechanical momentum is possible when the term 3. is null and is not possible when losses are present. Therefore conservation of energy supplants the action-reaction law.

As shown in Figure 3‑1, in thermodynamics:

§ the open system stands for open exchange of matter and energy with the environment

§ the closed system stands for open exchange of energy but not for the matter

§ the isolated system stands no exchange of energy and matter to the environment

In vacuum the only way to expel heat is an emissive radiation likely in the infrared spectrum from the black body radiation. This is not in any case comparable to a laser thruster which is directional. Here the radiation of the device is required for preventing an internal heating, and is preferably chosen to be isotropically evacuated for avoiding temperature rise on the surface.

Figure 3‑1 Definitions of (1) open, (2) closed, and (3) isolated systems

The two basic postulates to keep in mind are energy conservation and momentum conservation.

The famous experiment of the jumping ring can be practiced even at home when plugging a solenoid to the electrical network, illustrated in Figure 3‑2. There are plenty of videos on the internet showing that a ring placed in front of a solenoid coil filled with an iron core, the ring jumps away from the coil. Be aware for risks of electrical accident when performing the experiment at home. The fast transient current flowing in the coil is responsible for a fast transient magnetic flux in the ring, itself opposed in direction to the induced one. Now the idea is to attach rigidly both coil and ring together. As discussed in Chapter 3, the only way to reduce the counter force coming from the ring is to increase its losses from joule heating.

Figure 3‑2 Coil-ring device with magnetic field lines

In one hand, when cooling down the ring its resistance becomes much smaller, then the induced electromotive and Lorentz forces becomes much higher and the jump becomes much higher. The trick to use a copper ring at room temperature is to guarantee heat dissipation therefore energy loss. For a given shape of impulse electromagnetic wave coming from the coil, the losses in the copper ring must be enhanced in order to reduce the amplitude of waves coming back from the ring. In the meantime the jump becomes smaller. A full model has been built, simulated and published [37]. Further a test device has been tried and showed success [29]. Lively comments came immediately after the publication of the results. The major criticism is the fact the Earth magnetic field was not properly shielded during the experiment. In fact, in the equation of motion, the device is sensitive on fast changing magnetic field and not to a constant magnetic fields such as the Earth one. Note that the model itself has not been so far contradicted, on the contrary. Another aim of this Chapter is to eventually challenge the model.

In the following we will discuss the device coil-disc instead of coil-ring. In the model we assume a disc as being an assembly of tiny rings stuck the ones against the others, as shown in Figure 3‑3.

Figure 3‑3 Assumption of a disc as being discretized with multi-rings

Let’s write now the equation of energy conservation. First the energy emitted by the coil can be described as the sum of the two sides.

Equation 3.1

The first term of Equation 3.1 is the energy lost in vacuum the empty side of the coil. The second term of Equation 3.1 is the energy coming the coil reaching the disc  while a tiny part  can be lost in vacuum too if the device is not magnetic tight.

Equation 3.2

Now let’s give a description of the first term which is the parameter for driving energy balance between mechanic work , heat dissipation , and counter-force coming reemitted electromagnetic waves from the disc to the coil . Indeed the first term of Equation 3.2 is the sum of different energies:

Equation 3.3

Indeed the electromagnetic waves coming from the coil create eddy currents on the metallic disc, and those currents submitted to magnetic field, the Lorentz force creates a work, the first term of Equation 3.3. In the same time, eddy currents create joule heating (black body radiation), the second term of Equation 3.3. Eddy currents might reemit electromagnetic waves with an amplitude depending on the eddy currents, third term of Equation 3.3, which is responsible of the counter force, and a counter work to the coil .

Equation 3.4

Similar reasoning as stated above can be done.

Equation 3.5

As for the transmission from coil to disc, now let’s describe the energy process from disc to coil.

Equation 3.6

Now having put the basic fundamentals of energy exchanges, the only way to produce a useful work in the coil-disc device is to have the following condition satisfied.

Equation 3.7

We can go even further by establishing another condition .

When looking at equations, from Equation 3.3 to Equation 3.6, we clearly see that  is smaller and is a fraction of . However this does not mean that . The weight of the three terms in the master Equation 3.3 is not known. An exercise is to assess the contribution of each term of heat dissipations  and . When having resistive materials such copper, aluminum at room temperature, we can assume some heat dissipations. In opposite when having low resistive materials at low temperatures (resistive or superconducting materials), the heat dissipations become negligible.

In Table 3-1 different situations of resistive materials are summarized. Starting from normal metals such as copper, aluminum for resistive metals at room temperatures, to low resistive metals generally realized at low temperatures either with normal metals or superconductors. Remind again that the term  is the energy dissipated in the coil coming from the counter electromagnetic wave. The easiest case to realize in a laboratory is the case #1 where no constraining cryogenic technologies are required. The cases #2, #3 and #4 require cryogenic technologies and all the drawbacks and difficulties appear. On the other hand, the chance to observe any total work or displacement is small in the case #1 if not having very sensitive equipment measuring tiny displacements. In summary the most favorable situation is qualitatively the case #2 where the  is expected to be larger than . The case #2 is a favorable situation for creating a thermal asymmetry in the system.

Whatever the cases of the Table 3-1 to experiment, to improve, to challenge, what matters in the end for a final user is to get the Equation 3.7 satisfied not only on short time scales like the time for the electromagnetic wave to travel forward and back but on long time scales, as long as the coil emits transient electromagnetic waves. A cross-talk between newly reemitted wave and previous back waves must be considered.

Table 3‑1 Qualitative assessment of different cases where heat dissipation and work are confronted in the device coil-disc

Case	Disc	Balance sheet Equation 3.3	Coil	Balance sheet Equation 3.6
#1	Resistive		Resistive
#2	Low resistive		Resistive
#3	Resistive		Low resistive
#4	Low resistive		Low resistive

A gedankenexperiment, proposal of experiments, preliminary results and first exchanges via normal communications, erratum and corrections are already available to the public domain [29], [38–41]. The only valid and solid hypothesis is the energy conservation in the coil-disc system between provided electrical energy, work and heat dissipation. The latter coming from joule losses triggers the asymmetrical forces [37], [42]. Route of improvement is to increase the work which in turns balances joule losses. This technology turns out to be a way to produce a brutal acceleration by a short pulse of force also called impulse, necessary to get levitation as stated in Chapter 1.

However, for lifting heavy loads, both an important electrical generator and an optimized design for dissipating heat are required.

3-2 Work and force expression

The force expression on the disc after a first wave travel coming from the coil was given in the references [29], [37], [40], and was confronted with experimental data.

Equation 3.8

where t, r₂, r, B_r1, B_z1, S₂ and L₂ are respectively, the disc thickness, the disc and integration radius, the radial and axial magnetic field components at the extremity of the coil, the disc area and the ring integration inductance. Experiments were performed with an angle between the geological north-south line and the coil axis at 15°. The disc was oriented to the south-east and the coil to the north-west. Measurements were performed at Latitude: 46.6592 and Longitude: 0.3596. A magnetic field component (along B_z1) such as the Earth magnetic field which is not time dependent within the time scale of experiments does not create a force on the disc since only the time dependent fields contribute into the force of Equation 3.8. This explanation can close the ‘Earth magnetic field controversy’ related to the measurements. The issue of surrounding environment of the coil-disc system contributing to the total thrust might be raised when taking into account sources of artifacts, such as mirror charging effects. A care was taken in order to prevent proximity mirror effects by using an immediate environment made of wood.

Since involved forces are very tiny, several techniques are reported to be efficient [36], [43–66] as soon as are satisfied the following conditions: a proper calibration, a noise-free environment. Finally, the confidence of the experimenter for not recording stray signals might be important as well.

3-3 Heat dissipation

As discussed in the previous parts, heat must be released from the system. In vacuum, radiation dominates therefore in this part we remind the physics of radiation.

Heat comes from the induced current and is given by .

Beside the final temperature  can be calculated with  where ,  and  are respectively the mass, mass heat capacity and initial temperature in Kelvin.

Wien law gives a relationship between the wavelength  with the highest emission from a perfect black body and the temperature, their product is constant.

Equation 3.9

For instance, at 300 K, , the emission is in the infrared spectrum.

Stefan-Boltzmann law gives the thermal flux (thermal power) emitted by a surface S of a black body at a temperature T

Equation 3.10

where   is the Stefan-Boltzmann constant. The surface flux emitted by a black body at a temperature is then .

3-4 Epistemology

In this last part, the underlying treated notions are about scientific integrity and repeatability in science. The sad story of innovative concepts in physics particularly advanced ones is that some:

Ø cannot be submitted to experimental validations because of lack of appropriate technical instruments at one moment of history

Ø cannot be repeated by other peers showing negative results or positive results

Ø can be repeated by other peers showing negative results or positive results.

A knowledge in theoretical and experimental sciences is shared and promoted via edition of an academic book, publication of an article with an independent and specialized referee, and this whatever the area of science. Tacit terms of reproducibility in experimental physics are [67]:

1. Initial skepticism on the facts (thanks to Pyrron)

2. Realism on principle

3. Methodological materialism

4. Rationality (and logics).

The physical world Re is the sum of different sets. They are perceptible world P (even distorted), plus a fraction n (n<1) of imaginary I world, plus an imperceptible C world, plus a not imaginable G world. Those sets might have dependence, for instance, imperceptible C world can superpose a part of a not imaginable G world. Also P is accessible with human senses, instruments converting non accessible signals by humans into accessible.

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PAPER – 2010

Loss of linear momentum in an electrodynamics system: from an analytical approach to simulations

D.S.H. Charrier[†††]

Institut P`, SP2MI - Teleport 2, Boulevard Marie et Pierre Curie, BP 30179, 86962 Futuroscope Chasseneuil, France

ABSTRACT

The purpose of this investigation is to discuss the useful force in a coil-ring system by electromagnetomechanical conversion. It is shown that a net force in the inertial, closed and isolated coil-ring system is created. Analytical and practical tools toward new experiments for electromagnetic thrusters are given.

Keywords: retarded fields, momentum, electromagnetic pressure, thruster

I - INTRODUCTION

In this paper, a question arises: is it possible to create a thrust[i] using retarded electromagnetic fields in an inertial, closed and isolated system without exerting an external force? An ultra fast current ramp in an emitter coil induces a repulsive force in a receptor coaxial conductive disc or ring. Using this principle, magnetic actuators can provide high forces reported until 130 000 N.[ii] This force is not anecdotal and motivates this work. The aim is to convert a part or the whole force into a real thrust using the property of retarded fields. A gedankenexperiment is proposed and simulated. Due to the retarded field there is a loss of the momentum not only on short time scales (ns) but on long times (> seconds). The causality of a physical phenomenon is still source of fascinating debates and has theoretical consequences that are controversial in modern gravitation theories.[iii] Here, the coil-ring problem is treated from the retarded waves travel until resultant force calculations.

Into the frame of classical electrodynamics, the explanation of the created force in the coil-ring system may be detailed considering the retarded regime. Thus, the current pulse generates a spectrum of electromagnetic waves that leave the coil, covers the coil-to-ring distance at light speed and reaches the conductive ring surface. Here, it induces eddy currents which become the source of repulsive electromotive force. The newly created electromagnetic spectrum covers back the ring-to-coil distance at light speed to produce an induced current in the coil and so on. The electromagnetic waves in retarded regime for time dependent current distributions are entirely described by the generalized Jefimenko equations[iv] (also called Heaviside-Feynman formula[v]). These equations take into account retardation effects becoming important at extremely high frequencies, by essence. At lower frequencies the retardation can be neglected. It is shown in the paper that only high frequencies (> 3 GHz) are involved, by looking at the spatial scales. The retarded fields were mainly studied in particle physics[vi]^{,[vii],[viii],[ix]} and for moving atomic dipoles.[x]^,[xi] More recently, Jiménez et al.[xii] gave the exact electromagnetic field in retarded and static regime produced by a finite wire. So far, the electromagnetic field produced by a single circular ring in retarded coordinates with a time dependent current has not been reported in the literature. In this study, the dimension of wire sections is very small compared to the diameter of rings.

In the second part, the generalized equations of Jefimenko are given for one emitter ring in transient regime. In the third part, the induced electromotive force acting on a receptor ring and the generalized equations of Jefimenko for the single ring situation are simulated. The simulations show that the net force between an emitter ring and a receptor ring is not equal to zero. In the last part, the case of a realistic thruster is discussed using a magnetic core.

II – EXPRESSIONS OF THE RETARTED FIELD OF A SINGLE RING

The Jefimenko equations give the exact expressions of retarded electric  and magnetic  fields in space and time created by a transient (time dependent) current distribution. They are generally written as follows:[xiii]^,[xiv]

  (1)

   (2)

where and are respectively the electric and magnetic constants, c is the light velocity, a the charge density and  the current density. R () is the distance separating the current sources and observation points while  denotes the (retarded) coordinates of the current source where the electromagnetic waves traveling at light speed are created.

The particular case of a ring with a transient current i is analyzed using the cylindrical coordinates (z,q,r) (see also the inset of Fig. 1):

       (3)

where r is the ring radius, S the ring section and dl the unit length along the ring. As there is no charge density in the ring (a = 0) and injecting Eq. (3) into Eqs. (1) and (2), the analytical electromagnetic field in transient regime created by a ring is written in cylindrical coordinates:

       (4)

(5)

(6)

with

Eqs. (4)-(6) are the generalized equations of Jefimenko for a single ring with transient current. To confirm the consistence of Eqs. (4)-(6), the analytical Biot-Savart expression of the magnetic field along the z axis of the ring with a DC current  should be recovered from Eq. (6) by fixing the boundary condition  and ρ → 0. The numerical calculations of elliptical functions were done with Mathematica 6. In Fig. 1, the continuous curve is the magnetic field according to the analytical equation. The empty circles are the numerical calculated points using Eq. (6) with. The good agreement between the known analytical expression and the calculated one is a validation of Eq. (6). As an additional check, far from the coil the expression of the fields should approach those of a radiating magnetic dipole. For that let’s define the magnetic dipole moment of the ring . The static magnetic dipole assumes a constant current  and a negligible ring radius . With the later conditions, Eqs. (5) and (6) should be identical with those reported in literature and .[xv] Because of cylindrical symmetry, the radial components are equal (). The magnetic field of a dipole moment is shown in Fig. 2. Four dashed lines are indexed with letters A, B, C and D. The rescaled magnetic field  of magnetic field through these dashed lines are shown in Fig. 3 a) and b) in arbitrary units. Using Mathematica, the Fig. 3 a) and b) show the analytical curves (lines) versus the numerical points (symbols). The numerical points are obtained with Eqs. (5) and (6) using the dipole approximations  and . The good agreement between analytical and numerical calculations is a second validation of Eqs. (5) and (6). Further on, a force study gives the estimation of net thrust between two coaxial rings.

III – LOSS OF THE LINEAR MOMENTUM

Assumptions must be given on the dimensionality of the system. Typical length scales are the size of the emitter and receptor rings, their separating distance, and the typical wavelength corresponding to the working frequency. The standard actuators have typically an emitter-to-receptor distances z around the millimeter, which involves that the electromagnetic wavelength participating in the force is at maximum in the order of millimeter. Longer wavelengths cannot excite electrons on the metal participating in the total force. However, other contribution might come from the dimension of the system d which is typically at maximum in the order of decimetre. The frequency  to treat must be higher than the threshold frequency  in vacuum. This relatively high threshold frequency justifies the use of Jefimenko equations. The Eqs. (4)-(6) take into account the fact that in the transient regime the waves emitted by different parts of the emitter reach different parts of the receptor and these in retarded regime, by integrating all the spatial and temporal effects. However, the calculation of eddy currents is far from trivial. A deeper Fourier analysis (not shown here) gives the exact electromagnetic spectrum. The skin depth  of eddy currents in transient regime is written as . and  are the electronic conductivity and relative magnetic permeability, respectively.[xvi] Note in the transient regime, the skin depth  does not depend on the frequency, but on the time t. A Fourier transform is therefore useless. Let’s consider thus the interaction between two coaxial rings, as shown in inset of Fig. 5. The magnetic field B₁ is produced at a time  by means of transient current . It leaves the emitter ring, covers the ring-to-ring distance z₀ at light speed and starts inducing the current  in the receptor ring at a time . A Laplace force is created in the receptor ring. The Laplace force is the particular case of Lorentz force for a current distribution and both are valid in transient regimes.  must be determined from the induced electromotive force  where L₂, R₂ and F_B2 are respectively the self-inductance, resistance and magnetic flux of the receptor ring. In the situation of a transient current pulse , the self-inductance term is dominant and the current is simply . The field B₂, produced by i₂, covers back the ring-to-ring distance and modifies the current inside the emitter ring at a time  (is a minor delay due to slowing light speed into the metallic crystal of the receptor ring). Now the emitter ring is subject to the force  starting at , and so on. Note that both  and  are times dependent. If the emitter and receptor rings are jointly attached, the resultant of the system  can be calculated in ranges of times. The normalized instantaneous resultants  along z axis are shown below for the first three ranges of times  where n indexes the wave front travel. The initiating transient current  starts at t = 0:

(7)

(8)

(9)

where B_r1, B_z1, B_r2 and B_z2 are calculated using Eqs. (5) and (6). The induced currents  and  decrease versus time and finally will not significantly contribute to the final force . The maximum instantaneous resultant is reached in the first receptor-to-emitter electromagnetic wave travel . After this range of time, switches simultaneously opening the emitter and receptor circuits or an adequate signal would reset to zero the currents. Consequently, they annihilate the Laplace forces F₁₂ and F₂₁. When after deriving Eqs. (5) and (6), which account all the dynamics of the electromagnetic field, Eqs. (7)-(9) give the net force versus time and wave front travel which Eqs. (5) and (6) do not give. To circumvent the space scale issues, a magnetic core will be introduced in the discussion part.

Thus, a current pulse into the emitter ring applied during  and a subsequent opening of emitter and receptor circuits create a maximum net force . The fast opening of circuits will create a self-induction that will prolong the waiting time by t_s-i before imposing a new pulse into the emitter ring. In short, to continuously get a net force , a current ramp should be applied during  at a frequency . To get a reasonable thrust  in vacuum, calculations showed a working distance and frequency at respectively z₀ = 0.1 mm and , the latter frequency is out of current possibilities. The resultant is interesting if only it can be used not only on time scales of current pulses (typically ns), but on much longer time scales. The fundamental question is to check the validity of interrupting the current loops to get a zero force without inducing a counter-induction. Before having a second current pulse, a subsequent waiting time must get rid of remaining fields and also of remaining counter-forces. In practice, the ultra fast current switches can be made using field effect transistors (FETs)[xvii], ultra fast actuators running at most at 1 GHz[xviii] or even semiconductor junctions.[xix] All three should run at very high frequencies.

Clearly, the inertial, closed and isolated coil-ring system moves in one direction without applying external forces. The question raised in the introduction is answered: a thrust is created in the coil-ring system using retarded fields. An adapted electronics can force the system to move not only after one current ramp, but also after series of current ramps. A possible explanation of the observed loss of the mechanical linear momentum is the loss of the so-called electromagnetic momentum. The electromagnetic waves exert a pressure, and it is expressed in static regime as follows .^xiii The total pressure  present in the coil-ring system is not symmetric, i.e. the coil exerts an electromagnetic pressure on the ring placed on only one side of the coil. In the other side of the coil, the pressure exerted in vacuum is lost and is likely responsible for the observed loss of the mechanical linear momentum. An additional explanation of the loss of the momentum is the internal heating. In brief the eddy currents created in the receptor ring by high frequency electromagnetic waves will be themselves source of electromagnetic waves. However, the spectrum covered by these induced waves likely contains infrared light since thermal heating accompanies eddy currents in metals. To avoid parasitic electromagnetic crosstalk, a special care must be done on a proper electrical shielding of current leads.

To summarize, the typical working frequencies should not exceed hundreds of GHz due to the mentioned technical limitations of switches. The next section discusses a way to reduce the working frequency by introducing a magnetic core.

IV – DISCUSSION ON A REAL THRUSTER

An option to decrease the frequency  in order to have a workable frequency without losing the amplitude of thrust  is to use a coaxial magnetic guide material with a high magnetic permeability µ_r as shown in Fig. 4. The receptor ring is placed in vacuum, in the end of the core, and therefore its received magnetic field has to be described in vacuum only. The internal field of the magnetic core will create a phase delay , discussed in detail below. At high frequencies the treatment of magnetic field inside a core is simpler using complex permeability equations than using brutal finite element calculations such as the Schwarz-Christoffel mapping.[xx] At high frequencies, the magnetic field can be decomposed as   in a core and  in vacuum. The permeability in transient regime is then deduced as

where  and .[xxi]  is an intrinsic value of magnetic materials and is associated with losses (such as hysteresis or eddy currents), e.g. in static regime . An analytical treatment^xxi gives the loss factor  where  and  are respectively the relaxation time of magnetic domains and constant depending on magnetic susceptibility, under transient magnetic field. While  increases, especially for high frequencies, the loss factor  becomes small and the remaining losses are attributed by eddy currents losses. They can be avoided by reducing the dimension of the core in one dimension (hysteresis losses are also reduced at high frequencies). According to the Fig. 4, the path followed by the magnetic field suggests that the section at the end of cylinder can be attributed to a coil without core where the flux F_B1 remains the same through the cylinder. As discussed above, the losses are tiny  within the range of frequencies involved () and such magnetic materials firstly decrease the light speed by a factor  and secondly act as a spacer of length z_µr between the two rings. The total distance separating the two rings is then  where the receptor ring is placed at z₀ from the end of a cylindrical magnetic core. Then, a current ramp should be applied during  at frequency  . Moreover, for a number  of emitter rings, is proportional to  and a time delay  should be taken into account where  is the spatial elongation of  wires, see Fig. 4. At ,  were numerically calculated using Eqs. (5), (6) and (8), e.g. for an emitter ring radius r₁ = 10 cm, a receptor ring inductance L₂ = 10 mH, and z_µr = 10 cm,  mm,  and  (Fig. 5).  means that the time delay due to spatial elongation  is considered small compared to the delay of coil-to-ring distance. The current i₁ was applied with a ramp of 50 mA/ns.  were calculated versus z₀ and for different receptor radius r₂. A maximum force is shown when the two rings have equal radius. When the receptor ring is near the magnetic core (z₀ = 0.1 mm), Fig. 6 shows a maximum of  for r₁ = r₂ = 10 cm. The inset of Fig. 6 shows a full period of the calculated thrust  versus time. The maximum attainable force is . The abrupt force at  is due to  in Eqs. (5) and (6). The low forces in the present thruster with  are competitive with other existing thrusters.ⁱ For instance, electrostatic propulsion system based on Xenon ions reveals a thrust of 0.2 N. Pulsed plasma using Teflon^® shows a thrust of 0.1 N. The main advantage of the present thruster is the direct conversion of electricity to kinetic energy without using propellant. There is an way to increase the thrust by increasing . For higher , the delay due to the spatial elongation cannot be ignored any more. The force of 130 000 N was reachedⁱⁱ using much higher turns  and running a longer time of current ramp, obtaining then higher magnetic fields. Moreover, the possibility to play with dimensions, transient currents and magnetic cores help the experimental feasibility.

In conclusion, the generalized equations of Jefimenko for the single ring with a transient current show good agreement with the Biot-Savart expression in static regime. A second validation is the good agreement between analytical and numerical calculations in the dipole moment approximations. Due to dimensionality, the frequencies f involved in the resultant force are higher than the threshold frequency  in vacuum. This relatively high threshold frequency justifies the use of Jefimenko equations. To circumvent space and time scale issues, a magnetic core is introduced. The introduction of a magnetic core creates a phase delay  that is shown to be small. The simulations of instantaneous forces between emitter and receptor rings show the existence of a net thrust in one direction for a small time slot. A direct consequence is a possible loss of the linear momentum. The thermal heating and more likely the asymmetrical electromagnetic pressure exerted in the coil-ring system are responsible for the loss. To benefit of a maximum net electromagnetic thrust , the emitter circuit should be opened before receiving the magnetic response of the receptor ring. This latter might be opened at the same time to avoid counter-induction effects. The experimental requirements are high frequency signals switches into the emitter and receptor circuits. A thrust of was obtained for .

ACKNOWLEDGEMENTS

The author would like to thank O.D. Jefimenko for his inspiring original works.

FIGURE CAPTIONS:

Fig. 1: Continuous line: norm of magnetic field along z axis according to the known Biot-Savart equation for a ring radius r = 10 cm and a current i = 1 A. Empty circles: norm of magnetic field using Eq. (6) with the same parameters and with . The inset shows the ring in the (Oxy) plane, the cylindrical coordinates (z,q,r), the ring radius r and the observation point.

Fig. 2: Magnetic dipole moment centered at (x;z) = (0;0) using Eqs. (5) and (6). The four dashed lines are indexed A, B, C and D. See text for more info.

Fig. 3: Rescaled magnetic field calculated (symbols) with Eqs. (5) and (6) under dipole approximation  versus the rescaled analytical expressions (dashed lines and lines). The rescaled magnetic fields are calculated along the x axis a) and z axis b) along two axes separating the centre (5 and 10 cm).

Fig. 4: Cylindrical magnetic core placed inside the emitter coil. The receptor ring is placed outside the magnetic core. See text for more info.

Fig. 5: Numerically calculated instantaneous net force (in N) for a system of emitter-receptor vs. their separating distance z₀ and for different receptor radius r₂. The emitter radius was taken r₁ = 10 cm, a receptor ring inductance L₂ = 10 mH, a magnetic core with z_µr = 10 cm. The current i₁ was applied with a ramp of 50 mA/ns. The inset shows the interaction of two coaxial emitter (left) and receptor (right) rings. F_B1, F_B2 are the magnetic fluxes, ,  the Laplace force for the receptor and emitter rings respectively. The outer dashed box represents the system of jointed rings with an instantaneous resultant . The inset does not show the magnetic core for simplicity.

Fig. 6: Numerical instantaneous resultant  vs. receptors radius r₂, for an emitter-receptor distance z₀ = 0.1 mm and an emitter radius r₁ = 10 cm. The inset of Fig. 3 shows a full period of the calculated thrust  versus time with r₁ = r₂ = 10 cm, L₂ = 10 mH, z_µr = 10 cm and .

CHARRIER FIG. 1

CHARRIER FIG. 2

CHARRIER FIG. 3

CHARRIER FIG. 4

CHARRIER FIG. 5

CHARRIER FIG. 6

REFERENCES

PAPER – 2012

Micronewton electromagnetic thruster

D.S.H. Charrier[‡‡‡]

Institut P` (CNRS – Université de Poitiers – ENSMA), UPR 3346, SP2MI – Téléport 2, Bd Marie et Pierre Curie, BP 30179, 86962 Futuroscope, France

A low cost and light electromagnetic thruster, consisting in a disc rigidly attached with a coaxial coil, shows steady recoil by losing its linear momentum. The signal applied in the device is a square electric potential. A continuous thrust is observed on the center-of-mass in one single direction under electromagnetic excitation for various voltages and nominal high frequencies. At 1 kHz with 20 V amplitude, the recoil force reaches 4 μN (micronewton). The recoil is numerically quantified with induced electromotive and Lorentz forces. The presented device directly converts electric energy into kinetic energy.

Continuous efforts are done for improving performances of thrusters^1–8. Loss of linear momentum between two macroscopic bodies is demonstrated in the presented device. Classical electrodynamics is a powerful tool for calculating self-force in current loops^9–11 and sometimes leads into linear momentum loss phenomena between two bodies as soon as retarded interactions play an important role¹². It offers a physical way for producing a thrust, i.e. a direct conversion of electric potential energy into kinetic energy. This device is hoped to be used in satellites for accurate trajectory correction maneuvers, for example, thanks to a reduced use of propellant. The device is designed firstly for creating electromagnetic waves, secondly for benefiting electrodynamics effects and finally for measuring a displacement resulting from the lost momentum. The design retained is the magnetic actuator-like device consisting of a coil and a ring (or a disc). As for the principle of magnetic actuators^13-15, a fast current ramp in a coil produces a repulsive force in a coaxial metallic ring. Classical magnetic actuators apply a single ultra fast current ramp (a capacitive discharge) in order to create, at first wave front travel, a fast magnetic flux variation in the receptive ring inducing a repulsive Lorentz force. Later, at second wave front travel, the induced ring has time to create itself a counter force in the coil and so on, leading to center-of-mass oscillations with damped response. The counter force, due to the second wave front travel, is supposedly severely damped because the reemitted electromagnetic field amplitude by the metallic ring is reduced compared to the emitted one. The amplitude difference is partly converted into dissipation in the metal. Finally, the total force is due to asymmetrical properties and geometry between coil and ring. In the device, series of ultra fast successive current ramps is applied in the coil for creating a continuous force. The displacement of a rigid coil-disc device is measured for various voltages and frequencies of excitation. Numerical calculations of an analytical model partially predict the displacements.

An analytical model of force, after a first wave front travel, was validated using numerical calculations¹⁶. The model recovered both the Biot-Savard law in the constant uniform current case and the dipole moment in the dipole approximations. B_ρ1 and B_z1 being, respectively, the radial and axial retarded magnetic field components at the extremity of the coil, they are valid for a small section of coil wire compared to coil diameter. They are expressed as follows:

(1)

(2)

with .

H.m^-1 and  m.s^-1 are, respectively, the magnetic permeability and speed of light in vacuum.  is the time-dependent current in the coil with retarded coordinates . r₁, N_t, r₂ and S₂ are, respectively, the radius and number of turn of the coil, the radius and area of the disc. The Lorentz force calculations are chosen instead of those of Poynting energy flux density because they offer a direct comparison with experiments. Since the Lorentz force is valid in transient regime, and using the induced electromotive force, one can deduce the resulting force pushing a conductive ring with a ultrafast current ramp circulating in a coaxial coil. The force at first wave front travel is expressed as:

                    (3)

where [4] is the inductance of an elementary integration ring and r_w the wire radius of the ring.

Here, a disc is used in the experiments instead of a ring because of higher force sensitivity. The Eq. (3) must integrate the whole disc radius . The wire radius r_w of the ring becomes, by definition, the half thickness of the disc. Finally, the pushing force on a rigid and coaxial disc is expressed as follows:

       (4)

where t is the disc thickness. As the coil pushes the disc in one direction, the idea to glue the disc with the coil as a rigid thruster leads to consider the force of Eq. (4) as the total force of thruster, at first wave front travel.

Let’s define, z = z_μr + z₀ the distance separating the coil and disc. z_μr is the distance covered by the magnetic lines in a metallic core before reaching the vacuum. z₀ is the distance covered by the magnetic lines in vacuum before reaching the disc. In total, the numerical code contains four physical values (μ₀, c, t_ramp, i₁), six geometrical parameters (r₁, N_t, r₂, r_w, z₀, z_μr), three coordinates of integration (q, r, r). t_ramp is the ramp duration. In all calculations presented here, the disc was composed of 100 meshing elements because a numerical convergence of the force of Eq. (4) was observed between 50 and 1000 meshing elements.

Micro and nanonewton forces can be measured with rotational^18,19, torsion^1,20 or translational²¹ pendulums. As shown in Fig. 1, the rigid coil-disc device was hung on a translational pendulum by two metallic wires used as current leads at height  of 8.5 cm. The mass pendulum  was 76 g. In order to extract torque contributions, two mirrors for displacement measurements were placed symmetrically with respect to the coil-disc device. The coil (750 turns N_t) had an interior radius r₁ of 8 mm. The core of laminated steel plates was over the coil extremity by a distance  of 1.25 cm. The disc (coaxial with the coil and core) with a radius of 1.35 cm for a thickness t (2r_w) of 0.6 mm was made of copper. The total distance separating the coil from the disc is then  +  as mentioned above. The thickness z₀ of 0.55 mm was measured optically. A square-shape signal generator (Hameg HM8131-2) was used as high frequency power supply. The RL frequency f_coil of the circuit was 476 Hz, while the RL frequency f_coilnocore without the core was 1380 Hz. The ramp duration t_ramp was at nearly 22 ns and the current speed i₁ was 14.4 A.ms^-1.

In Fig. 2 (a), the raw data of displacements versus time measured on the two channels are shown at rest situation (0 V before 0 second) and with electromagnetic excitation (100 Hz with amplitude of 20 V after 0 second). The top and bottom vertical axis are the raw data of measured displacement in micrometer versus time. Both displacements  and  are positive during excitation and show a clear deviation from their origin at rest position. The stronger oscillations appear after the first signal impulses since the most important kinetic energy is delivered, the coil-disc device going from rest position to excited position. The device then oscillates around a continuous displacement until the damping from the pendulum is suppressed. The motion is created by the continuous electromagnetic excitation and demonstrates a first-kind electromagnetic thruster. In Fig. 2 (b), successive cycles of rest/excitation are shown with a periodicity of 200 seconds, the excitation frequency is 15 MHz at 20 V amplitude. Before 0 second, the device was excited explaining the damped signal the first hundred seconds. A test situation was to observe the motion of the coil without copper disc. In the limit of the experimental accuracy, no displacement was observed agreeing the expectation to get zero thrust, since no repulsive force was created.

In Fig. 3 (a), the real experimental displacement  is shown in μm (Symbols) versus the voltage amplitude in V at 15 MHz. Numerical calculations of d (Continuous line) using Eq. (1), (2), (4) and  are shown for comparing the model and the experiments. Although the calculated displacement as a function of voltage seems quadratic-like, the experimental points (without error bars) seem linear. Within the error bars, the theory fails to explain a weaker experimental displacement at 20 V. The model can be however used for low voltages. The discrepancy at 20 V between experiment and theory is attributed to less sharp slope of square excitation signal coming from the power supply, because if less amplitude of field is delivered into the coil, the total driving force will be lower as Eq. (4) suggests.

In order to see the effect of the excitation frequency on the displacement, measurements performed with and without steel plates are shown in Fig. 3 (b) on top the top vertical axis. In both cases the displacement is steady at around 0.5 μm above 1 kHz and the delivered mechanical power in W (bottom vertical axis) increases. In summary, the low energy consumption of the device associated with the weak force is very promising for long term applications for small satellites.

Acknowledgments

The author is very grateful to Fabrice Berneau and Jozef Vesely for stimulating discussions on the acquisition of data. This work was financially supported by the French CPER (Région Poitou-Charentes).

Figure captions:

Fig.1: Top view of the device hung on the pendulum. The origin O₁ of displacement for mirror 1 is located at rest position of pendulum. The origin O₂ of displacement for mirror 2 is located at rest position of pendulum.

Fig. 2: Both graphs are measured with a square-shape signal at 20 V in the coil with core. The top and bottom vertical axis show raw data of displacement vs. time in micrometer, measured on channel 1 for mirror 1 and on channel 2 for mirror 2 at 100 Hz (a) and at 15 MHz (b). (b) The signal is applied with cycled excitations.

Fig. 3: The displacement  is calculated with  where  and  are the displacements measured on channel 1 and channel 2, respectively. (a) Experimental displacements (Symbols) and numerical calculations (Continuous line) vs. the excitation voltage at 15 MHz (coil with core). (b) Experimental displacements and mechanical power vs. the applied frequency at 20 V.

CHARRIER FIG. 1

CHARRIER FIG. 2a

CHARRIER FIG. 2b

CHARRIER FIG. 3a

CHARRIER FIG. 3b

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CONFERENCE – 2016

This article was prepared in response to reactions of the 2012 paper. It was a complementary communication with the conference held in Baltimore in the American Physical Society.

Frequency spectrum analysis on the force contribution in a micronewton electromagnetic thruster

D.S.H. Charrier[§§§]^,†

^*Institut P` (CNRS – Université de Poitiers – ENSMA), UPR 3346, SP2MI – Téléport 2, Bd Marie et Pierre Curie, BP 30179, 86962 Futuroscope, France

^†Nexans Research Center, 29 rue Pré Gaudry, 69007 Lyon, France

A frequency analysis on results published in 2012 is given in this letter, where experimental displacements of the coil-disc device obtained at different generator square signal frequencies (from 100 Hz to 15 MHz) are compared with monochromatic AC currents calculated from RL and RLC_stray circuit impedances. In time domain, a square signal can be decomposed in sum of monochromatic (Dirac-like) contributions in frequency domain. The analysis revealed that i) the RL circuit has a working device onset frequency above 170 kHz ii) the stray capacitance C_stray of the coil must be much below the order of pF for having working coil-disc device frequencies. The onset frequency at 170 kHz is coherent with a previous prediction, the device having decimeter scale d at maximum, the monochromatic working threshold frequency was found at about f_th = c/d » 3 GHz.

In 2012, experimental results were reported on an electromagnetic thruster¹ although a similar original concept was reported before by Goodwin.^2,3 Since then, the order of magnitude of thrust or at least the proof-of-principle on a laboratory scale device were not reported by other works. The thrust was explained by an electromagnetic momentum emitted by the coil transferred into mechanical momentum on the metallic disc. Transient regime, asymmetrical coil-disc device, Joule heating accompanying eddy currents helped in creating a total net force and not only on short time scales. In this letter, different physical and theoretical aspects are given to complement past experiments that could help experimentalists willing to reproduce the proof-of-principle and a frequency signal analysis is reported. The force equation was proposed and confronted to experimental results with a relative success:

Equation 1

where t, r₂, r, B_r1, B_z1, S₂ and L₂ are respectively, the disc thickness, the disc and integration radius, the radial and axial magnetic field components at the extremity of the coil, the disc area and the ring integration inductance. Experiments were performed with an angle between the geological north-sourth line and the coil axis at 15°. The disc was oriented to the south-east and the coil to the north-west. Measurements were performed at Latitude: 46.66 and Longitude: 0.36. A magnetic field component (along B_z1) such as the Earth magnetic field which is not time dependent within the time scale of experiments does not create a force on the disc since only the time dependent fields contribute into the force of Equation 1. The issue of surrounding environment of the coil-disc system contributing to the total thrust might be raised when taking into account sources of artifacts, such as mirror charging effects. A care was taken in order to prevent proximity mirror effects by using an immediate environment made of wood.

            Recently, works were published on either advices⁴ or recommendations⁵ for measuring respectively micronewton or even nanonewton. In Ref¹, optical fibers placed in the vicinity of a metal plate collected reflected light. It was shown that the reflected light is linearly proportional to the fiber to metal plate distance in a certain range of distance.⁶ The link between collected light and displacement was done using a calibration curve: input reflected light voltage versus displacement in micrometer measured with a micrometer screw. Regarding the acquisition of displacements, the following procedure was followed: acquisition at rest position, a pause of acquisition just before switching on the power supply, few seconds of waiting time, acquisition again. The few seconds of waiting time of displacement acquisition was necessary due to the too high displacement provoked by the first square shape ramps up of the signal. This minor note about the transient regime does not change the displacement observed in the steady regime which was used for the analysis. Therefore all the data as shown in ¹ do not show few seconds of displacement after the power supply was switched on. Moreover, complementary trials were performed where the two optical fiber sensors were placed in opposite direction in such way that all artifacts between the coil-disc device and the measurement setup was uncorrelated. Regarding the model and the interpretation of the results, several comments are given here. As proposed by Lafleur⁷, the need to consider the speed of sound is of interest when looking at the force establishment on the whole body. The calculations proposed in here⁸ were to consider the total force at a time t without regarding any mechanical steady state or relativistic effect. Since the total motion of the rigid or elastic body under non-uniform acceleration is also source of intense investigations, as referenced by Franklin⁹ and Dmitriev¹⁰, no further model was built. As shown in Table 1, the measurement setup and the device characteristics can be decomposed in different frequency spectrum. The terms ‘monochromatic’ and ‘polychromatic’ refer respectively to Dirac-like frequency and rich Fourier decomposed frequency spectrum. The thrust measurement setup was composed by pendulum and sampling frequencies. Both are monochromatic. The oscillating frequency of the pendulum  = 1,71 Hz with g the standard gravitational constant (9,81 m s^-2) and h the pendulum height  (8,5 cm) is crucial when working a low generator square wave frequency. To be relevant, the generator frequency must be at least two times higher than the pendulum oscillating frequency. At low frequency the system coil hung on the pendulum might enter into resonant oscillations. The sampling frequency  was about 10 Hz. The data of displacements done with a generator frequency above  = 20 Hz were relevant. As shown in Table 1, the coil was excited with a square wave generator where the periodic wave is monochromatic and the Fourier decomposed square signal is necessary polychromatic. From the disc side, both working device electromagnetic wave frequencies and eddy current and skin depth frequencies are polychromatic.

In order to identify the unknown working frequency of the device, complementary calculations in frequency domain are presented. The calculations presented in Ref ¹ used a linear and resistive ramp up to calculate the current and its derivative as function of time. Here the RL circuit is treated rather than simple resistive R circuit. Then the Equation 2 was used to calculate the current.

             Equation 2

where U, i, R_coil and L_coil are respectively the voltage, current, resistance and inductance of the coil. Taking conditions of i_coil(t=0) = 0 and t = L_circuit/R_circuit then both current and its derivative can be calculated as a function of time as shown in Equation 3, Equation 4 and in Figure 1.

Equation 3

    Equation 4

In frequency domain, the RL circuit is described using the impedance terms as shown in Equation 5:

Equation 5

where w is equal et 2pf with the f the frequency of the signal. Using the norm of impedance, it is possible to use the impedance to deduce the current flowing in the coil in one single frequency,

        Equation 6

Using the frequency domain, the derivative di_coil(w)/dt is equal at zero.

Inductances are known to behave as capacitor at high frequencies because the presence of isolated spires influence themselves. As shown in Figure 2, the stray capacitance coming from the inductance is added in the impedance calculations. The impedance term is shown in Equation 7.

Equation 7

where C_stray is the stray capacitance that is to be defined. Impedance terms contrarily to Equation 3 and Equation 4 allow a frequency-domain study rather than a time-domain study. As shown in Figure 3, according to the stray capacitance values (RLC circuit), the current in the coil has a minimum while in the case of RL circuit the current decreases with the frequency. In Figure 4, the experimental and calculated displacements are shown. The experimental displacement was obtained with the so-called polychromatic square wave generator while the calculated displacements are monochromatic coming from the impedance terms. The aim of this representation is to identify the working monochromatic frequencies. For instance, in the case of stray capacitance of value 1 pF, less than two decades of frequencies are involved in the displacement while in the case of RL or RLC with 1 fF, a large working frequency spectrum is identified above 170 kHz (~1,8 km wavelength). Electromagnetic momentum transfer into mechanical momentum in the disc is possible using a continuum spectrum of Dirac-like frequencies above 170 kHz. Indeed frequencies below 170 kHz are not playing a role in the force. This observation is consistent with what it has been proposed in the past⁸. In term of experimental improvements, the exciting signal must be monitored accurately in order to extract the full Fourier decomposed spectrum. The same spectrum can be inserted in the calculation and confronted to experiments. The force can be increased by decreasing the resistivity of disc, using low temperatures. Stray capacitance can be reduced by replacing N spires into a single large flat metal sheet or using electronic means of compensation.

Figure captions:

Figure 1: Current i and its derivative di/dt in time domain in the RL device circuit.

Figure 2: Equivalent circuit of the operational device under electromagnetic excitation. Stray capacitance between the coil spires changes the device response.

Figure 3: Currents calculations in monochromatic frequency domain using RL and RLC impedances (1 nF and 1 pF stray capacitances).

Figure 4: Real displacement as produced by a polychromatic square wave generator (Fourier decomposition of a square contains infinity a monochromatic frequencies) versus monochromatic calculated displacements.

Tables:

Table 1: Decomposition of frequencies present in the experiments reported in Ref¹.

CHARRIER TABLE1

CHARRIER FIG. 1

CHARRIER FIG. 2

CHARRIER FIG. 3

CHARRIER FIG. 4

References

¹ D.S.H. Charrier, Appl. Phys. Lett. 101, 034104 (2012).

² D.P. Goodwin, AIP Conf. Proc. 552, 976 (2001).

³ D.P. Goodwin, AIP Conf. Proc. 699, 1175 (2004).

⁴ J. Soni and S. Roy, Rev. Sci. Instrum. 84, 095103 (2013).

⁵ J.E. Polk, A. Pancotti, T. Haag, S. King, M. Walker, J. Blakely, and J. Ziemer, in 33rd Int. Electr. Propuls. Conf. (2013), pp. 1–24.

⁶ C. Coupeau, J.C. Girard, and J. Grilhé, J. Vac. Sci. Technol. B Microelectron. Nanom. Struct. 16, 1964 (1998).

⁷ T. Lafleur, Appl. Phys. Lett. 105, 146101 (2014).

⁸ D.S.H. Charrier, Prog. Electromagn. Res. M 13, 69 (2010).

⁹ J. Franklin, Found. Phys. 43, 1489 (2013).

¹⁰ A.L. Dmitriev, AIP Conf. Proc. 969, 1163 (2008).

INTERNATIONAL SYTEM BASE UNIT

	Name	SI unit	Symbol	Symbolic expression
	Mass	kilogram	M	kg
	Length	meter	L	m
	Time	second	T	s
	Current	ampere	I	A
	Temperature	kelvin	T	K
	Volt	volt	U	kg.m^2.s^-3.A^-1
	Resistance	ohm	Ω	kg.m^2.s^-3.A^-2
	Inductance	henry	H	kg.m^2.s^-2.A^-2
	Capacitance	farad	F	kg^−1.m^−2.s^4.A^2
	Electric field			kg⋅m⋅s^−3⋅A^−1
Chap. 1	Force	newton	N	kg.m.s^-2
	Work, Energy, Heat	joule	J	kg.m^2.s^-2
	Power	watt	W	kg.m^2.s^-3
Chap. 2	Momentum			kg.m.s^-1
	Magnetic induction	tesla	T	kg.s^-2.A^-1
	Pressure	pascal	Pa	kg.m^-1.s^-2

Term	Definition extracted from en.oxforddictionaries.com
Truth	That which is true or in accordance with fact or reality.
Proof	Evidence or argument establishing a fact or the truth of a statement.
Effect	A physical phenomenon, typically named after its discoverer.
Phenomenon	A fact or situation that is observed to exist or happen, especially one whose cause or explanation is in question.
Law	A statement of fact, deduced from observation, to the effect that a particular natural or scientific phenomenon always occurs if certain conditions are present.
	A generalization based on a fact or event perceived to be recurrent.
Theorem	A general proposition not self-evident but proved by a chain of reasoning; a truth established by means of accepted truths.
Lemma	A subsidiary or intermediate theorem in an argument or proof.
Principle	A fundamental truth or proposition that serves as the foundation for a system of belief or behavior or for a chain of reasoning.
	A general scientific theorem or law that has numerous special applications across a wide field.
Postulate	A thing suggested or assumed as true as the basis for reasoning, discussion, or belief.
Axiom	A statement or proposition which is regarded as being established, accepted, or self-evidently true.
Hypothesis	A supposition or proposed explanation made on the basis of limited evidence as a starting point for further investigation.
	A proposition made as a basis for reasoning, without any assumption of its truth.
Scepticism	A sceptical attitude; doubt as to the truth of something.
Criticism	The expression of disapproval of someone or something on the basis of perceived faults or mistakes.

Constant in physics		Value	Unit
Speed of light in vacuum	c	299 792 458	m.s^-1
Boltzmann constant	k	1.38064852 × 10^-23	kg.m^2.s^-2.K^-1
Newtonian constant of gravitation	G	6.67408(31) × 10^-11	kg^−^1⋅m^3.s^−^2
Planck constant	h	6.626 070 040(81) × 10^-34	kg.m^2.s^-1
Magnetic constant	µ0	4p × 10^-7	kg.m.s^-2.A^-2
Electric constant	e0	8.854 187 817 × 10^-12	kg^−^1.m^−^3.s^4⋅A^2
Electric charge	C	1.602 × 10^-19	s.A
Stefan-Boltzmann constant	s	5.670 367 × 10^-8	kg.s^3.K^-4

FIGURES, TABLES AND QUOTES

Figure 1‑1 The author performing the air flux trap with a ping pong ball

Figure 1‑2 Potential energy distribution of ultrasonic standing wave [15]

Figure 1‑3 “Kugel fontain”: a stone levitating with a film of water [14]

Figure 1‑4 Ti Zr-Ni alloy in an electrostatic levitator chamber (Physics Today)

Figure 1‑5 Are room temperature superconductors possible? [26]

Figure 1‑6 Iron ball floating in air [20]

Figure 1‑7 Instable potential well (saddle like) under static magnetic field

Figure 1‑8 The author performing a diamagnetic levitation of a B pencil lead

Figure 1‑9 Free fall axis

Figure 1‑10 Apparent and pseudo-stable levitation

Figure 1‑11 Stable levitation model

Figure 2‑1 Permanent magnet creates a force F in a conductor with a current I

Figure 2‑2 Cylindrical coil represented with magnetic flux lines

Figure 2‑3 Conical coil represented with magnetic flux lines

Figure 2‑4 Schematic of two masses

Figure 2‑5 Circle on which mass the M stays confined in closed circuit

Figure 3‑1 Definitions of (1) open, (2) closed, and (3) isolated systems

Figure 3‑2 Coil-ring device with magnetic field lines

Figure 3‑3 Assumption of a disc as being discretized with multi-rings

Table 1‑1 Newton's laws of motion [1], [2]

Table 1‑2 Einstein’s postulates of special relativity [3]

Table 1‑3 Consequences of special relativity

Table 1‑4 Laws of electromagnetism [4]

Table 1‑5 Laws of thermodynamics

Table 1‑6 Thermal radiation laws

Table 1‑7 Ways for levitating solid objects met in Nature and technologies

Table 1‑8 Complete map of magnetic levitation and examples

Table 1‑9 Magnetic susceptibilities of different elements [30]

Table 3‑1 Qualitative assessment of different cases where heat dissipation and work are confronted in the device coil-disc

I give the reasons why I chose the quotes for each chapter.

Page 1:

I first read this quote in 1998 at the library of the Jopeph Fourier University. It summarizes the notion of ‘concrete’ as being the ‘abstract’ becoming familiar by use. Let’s say that life, laws of Nature are irrational. Even if equations can describe the dynamics of motions and interactions, in particular we do not know the origin at physical constants.

Page 24:

First quote

If our eyes could only see perfectly symmetrical systems, or only equilibrium situations, we would not be able to catch unexpected phenomena and consequently not able to anticipate any new experiments. I think creativity must born from asymmetry in Nature, also of course from provoked situations.

Second quote:

In my opinion, any question must be animated by a fast or a long process of search. To give an answer to all questions is not mandatory while the approach of constant questioning is. The process of (fast or long) searching is healthy, a constant questioning too.

Page 34:

First quote

This quote is very important. In other words, the rational point of view must invite anyone to separate the person bringing a message from the content of the message. Personally I prefer to discuss on the content of a message rather on its author. Seriousness of a messenger can sometimes bring seriousness in his message, this is rarely reciprocal, i.e. a message from an author with a poor fame will not have a same impact than a famous one.

Second quote

Indeed a hard work defeats everything. Here, the work is not quantifiable in Joule. Also, a lot of work and energy can defeat gravity.

---

[*] Earnshaw’s theorem and Maxwell equations demonstrate that the extensions term ‘static’ does not prevent unstable levitation, retardation of electromagnetic interaction is one explanation

[†] Sound Amplification by Stimulated Emission of Radiation

[‡] Magnetohydrodynamics

[§] Chapter 2, part 2-1

^[**] Chapter 2, part 2-2

[††] Chapter 2, part 2-3

[‡‡] Chapter 1, part 1-4

[§§] Chapter 3

[***] wip : works in principle, no reference found

[†††] dimitri.charrier@univ-poitiers.fr

[‡‡‡] dimitri.charrier@nexans.com

[§§§] dimitri.charrier@nexans.com

---

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