See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/277202013
Fluidic Electrodynamics: a new approach to EM
Propulsion
Article · February 2008
CITATIONS
READS
0
65
2 authors:
Alexandre A. Martins
Mario J. Pinheiro
University of Brasília
University of Lisbon
20 PUBLICATIONS 149 CITATIONS
100 PUBLICATIONS 815 CITATIONS
SEE PROFILE
SEE PROFILE
Some of the authors of this publication are also working on these related projects:
Spacecraft flyby anomaly View project
Theoretical Physics View project
All content following this page was uploaded by Mario J. Pinheiro on 14 November 2015.
The user has requested enhancement of the downloaded file.
PHYSICS OF FLUIDS 21, 097103 共2009兲
Fluidic electrodynamics: Approach to electromagnetic propulsion
Alexandre A. Martins1,a兲 and Mario J. Pinheiro2,b兲
1
Institute for Plasmas and Nuclear Fusion and Instituto Superior Técnico,
Av. Rovisco Pais, 1049-001 Lisboa, Portugal
2
Department of Physics and Institute for Plasmas and Nuclear Fusion, Instituto Superior Técnico,
Av. Rovisco Pais, 1049-001 Lisboa, Portugal
共Received 2 December 2008; accepted 11 July 2009; published online 24 September 2009兲
We report on a new methodological approach to electrodynamics based on a fluidic viewpoint. We
develop a systematic approach establishing analogies between physical magnitudes and
isomorphism 共structure-preserving mappings兲 between systems of equations. This methodological
approach allows us to give a general expression for the hydromotive force, thus reobtaining the
Navier–Stokes equations by using the appropriate electromotive force. From this ground we offer a
fluidic approach to different kinds of issues with interest in propulsion, e.g., the force exerted by a
charged particle on a body carrying current; the magnetic force between two parallel currents; the
Magnus force. It is shown how the intermingling between the fluid vector fields and electromagnetic
fields leads to new insights on their dynamics. The new concepts introduced in this work suggest
possible applications to electromagnetic propulsion devices and the mastery of the principles of
producing electric fields of required configuration in plasma medium. © 2009 American Institute of
Physics. 关doi:10.1063/1.3236802兴
I. INTRODUCTION
The analogy between the vector fields of electromagnetism and the hydrodynamic fields provides far-reaching insights into electromagnetic 共EM兲 fields1–4 and it has been
used to solve complex problems, such as the turbulent fluid
flow by Marmanis.5
Several steps have been made in this direction. Holland6
deduced the set of Maxwell’s equations from continuum
mechanics by generalizing the spin-0 theory to a general
Riemannian manifold. On a different ground, the conceptual
similarities between condensed matter and the quantum
vacuum allows one to simulate phenomena in high-energy
physics and cosmology using quantum liquids, Bose–
Einstein condensates, and all scenarios appearing in condensed matter systems.7 The failure to develop a quantum
field theory of gravitation leads to a relativistic description of
rotating space-time that reveals striking similarities between
the vacuum state with quantum condensate.8–10 Meholic and
Froning11 proposed the concept of a superluminal space inside which a vessel would be propelled through the “superlight” continuum, possessing fluid like properties,12 by a field
that interacts with static electromagnetic forces.
A fluid-dynamic approach to the relativistic electromagnetic field has been done by Kaufman,13,14 treating the electromagnetic field as a “fluid” with a “definite and calculable
velocity relative to an observer”14 and taking into account an
additional flow-work term, he obtained a Lorentz-invariant
electron mass. We have obtained similar conclusions through
a different procedure, using the convective derivative of the
EM “fluid.”15–17
These similarities between vast areas of physics, all ina兲
Electronic mail: aam@ist.utl.pt.
Electronic mail: mpinheiro@ist.utl.pt.
b兲
1070-6631/2009/21共9兲/097103/7/$25.00
terconnected with one another, undoubtedly constitute a successful example of the unity of physics and illuminates the
physical reality hidden inside the physical vacuum.
In this paper we introduce a “fluidic electrodynamic”
approach to the electromagnetic fields in terms of the potential functions 共A, ␾兲 and their material derivative, as they
emerge in quantum mechanics as more fundamental quantities than the 共E, B兲 fields, predicting certain quantum interference effects, such as the Aharonov–Bohm 共AB兲 effect and
the single-leg electron interferometer effect known as the
Josephson effect.
The fluidic electrodynamics viewpoint brings farreaching results as analogies are concerned. In addition, the
simplified methodology offered in this article helps solving
problems for either fluid dynamics or electromagnetic theory
by anyone with reasonable knowledge of electrodynamics
and to apply these concepts to situations of experimental
interest with sufficient accuracy, before undergoing more
complicated procedures.
II. FLUIDIC APPROACH TO ELECTROMAGNETISM
The founders of electromagnetism envisaged the
“aether” as an “elastic solid” 共e.g., Faraday’s “electrotonic
state”—a mechanical medium subject to certain states of tension and motion兲, and Maxwell later used the representation
of Faraday’s line of magnetic force as the velocity of an
incompressible fluid,18 introducing in a quantitative form
Faraday’s law of induction under the form E = −⳵A / ⳵t.
Bernhard Riemann attributed the cause of gravitation
and light “in the form of motion of a substance that is continuously spread through all infinite space.”19 Riemann conceived this substance as a physical space whose points move
in geometrical space. Regarding the term aether introduced
hereby, we could argue that it might be better to use it, in-
21, 097103-1
© 2009 American Institute of Physics
Downloaded 06 Apr 2011 to 193.136.137.244. Redistribution subject to AIP license or copyright; see http://pof.aip.org/about/rights_and_permissions
097103-2
Phys. Fluids 21, 097103 共2009兲
A. A. Martins and M. J. Pinheiro
TABLE I. Correspondence of field variables in electromagnetism and hydrodynamics.
Electromagnetism
Hydrodynamics
␳共r , t兲
Hydrodynamic charge n共r , t兲
␧0
Permeability of the vacuum ␮0
Electric potential ␾共r , t兲
Scalar potential ␹
Vector potential A共r , t兲
Electric field E共r , t兲
Magnetic field B共r , t兲
Mass density ␳
Massic enthalpy p / ␳共r , t兲
Velocity potential ⌽
Velocity 共or hydrodynamic momentum兲 u共r , t兲
Lamb vector l共r , t兲
Vorticity ␻共r , t兲
Voltage 共or electric tension兲 U共r , t兲
Electric current I
⳵A
Electromotive force E = − − ⵜ␾ − ⵜ共v · A兲 + 关v ⫻ B兴
⳵t
stead of physical vacuum, or just vacuum, since vacuum usually means “nothing whatsoever,” while physical space confuses the word either with vacuum or mixes up physics with
geometry.20
The special theory of relativity attributed to the aether
the quality of a “superfluous” artifact to think about natural
phenomena, but currently there is a strong need to rethink
this medium as a true physical vacuum. In fact, Dirac postulated the existence of an aether needing an “elaborate mathematics for its description”21 and later he proposed a new
classical theory of electrons formulated in a Hamiltonian
form and not based on gauge transformations.22 According to
him, the 4-vector potential should verify the condition
A␮A␮ = m2 / e2 and the 4-velocity field should be given by
V␮ = eA␮ / m. The velocity V appears in the vector potential
with the physical significance of the velocity with which an
electron’s charge must flow in the ether. Dirac interpreted
this velocity field as something real, even in the absence of
electric charges. By the contrary, Poincaré advocated that the
electromagnetic energy was a fictitious fluid transported in
space according to Poynting’s laws—in fact, according to his
own words, a useful “mathematical fiction.”23
Remarkably, the experimental findings by Graham and
Lahoz24 imply that the vacuum is the seat of “something in
motion,” in the way how Maxwell envisaged the aether.
More recently, it was shown that a medium in uniform motion with velocity v plays the role of the vector potential,
while the charge is proportional to Fresnel’s dragging coefficient for light in moving media.25 In the framework of general relativity, Keech and Corum26 have shown that an electric null current is accompanied by a neutral fluid current,
and it is this null fluid that transports the energy instead of
the electromagnetic field.
Reasoning along this line of thought we thus attribute to
the vector potential the property of the velocity of a fluid
embedded in the physical vacuum. We recall that the physicality of the vector potential is now well proven
experimentally.27 We do not intend here, however, to describe the inner nature of a so pervasive and evasive medium, be it a mechanical medium whose deformations correspond to the electromagnetic fields or a locally preferred
state of rest. However, it is particularly relevant that cur-
1
p共r , t兲 = ␳共r , t兲⌽共r , t兲 + 2 ␳共r , t兲u共r , t兲2
Circulation ⌫
⳵u
p u2
Hydromotive force EH = − − ⵜ +
+ 关u ⫻ ␻兴
⳵t
␳ 2
共 兲
rently there have been advanced some explanations about the
origin of “dark matter,” inventing new particles, as the
weekly interactive massive particle or the neutralino 共e.g.,
Ref. 28兲. Observations of our galaxy and other major galaxies help to discover vast coronas extending far beyond the
visible stellar systems. Although they do not emit visible
light, their mass may exceed the total mass of the stars they
surround. This question might be related to the Graham and
Lahoz experimental findings and quoted above, that something in motion is not taken into account. As we know that
rotating gravitational fields can generate electromagnetic
fields,29 it was recently shown that within the approximation
of the linearized Einstein–Maxwell theory on flat space-time,
an oscillating electric dipole is the source of a spin-2 field,
that is, electromagnetic waves harbor gravitational waves,30
possibly interbreeding each other.
A vector potential vector field is created whenever an
electric charge moves or an electric current is produced by an
electromagnetic system. We do not address here the specific
problem of the motion of a given particle in this fluid, a
problem that is addressed, e.g., in Ref. 31. We are here interested in describing the effects produced by inducing a
flow of the vector potential around material bodies.
To describe the electromagnetic field it is necessary to
define the electric field E共r , t兲, the magnetic field B共r , t兲;
charge density ␳共r , t兲; and the charge velocity v共r , t兲. If we
have a charged particle with position vector ri共r , t兲, the
charge density is ␳共r , t兲 = e␦关r − ri共t兲兴. However, it is more
advantageous to associate the set of Maxwell’s equations
with the electromagnetic potentials A共r , t兲 and ␾共r , t兲
through the relationships15,16,32
E共r,t兲 = −
1 dA
− ⵜ␾
c dt
共1兲
and
B = 关ⵜ ⫻ A兴.
共2兲
Note that we introduced into Eq. 共1兲 the convective derivative d / dt = ⳵ / ⳵t + v · ⵜ, instead of the Maxwell–Einstein operator ⳵ / ⳵t 共see, e.g., Refs. 15–17兲.
We define in Table I the correspondence of field vari-
Downloaded 06 Apr 2011 to 193.136.137.244. Redistribution subject to AIP license or copyright; see http://pof.aip.org/about/rights_and_permissions
097103-3
Phys. Fluids 21, 097103 共2009兲
Fluidic electrodynamics: Approach to electromagnetic propulsion
TABLE II. Equivalent equations in electrodynamics and hydrodynamics.
ⵜ·B⫽0
⳵B
= −关ⵜ ⫻ E兴
⳵t
Faraday’s equation
ⵜ · E = ␳ / ␧0
⳵E 2
= c 关ⵜ ⫻ B兴 − J
⳵t
⌫EM =
ⵜ·␻⫽0
⳵␻
= −关ⵜ ⫻ l兴
⳵t
Thomson’s equation
=
ⵜ · l = −ⵜ2⌽ = n共r , t兲
⳵l 2
= c 关ⵜ ⫻ ␻兴 − I
⳵t
Gauss’s equation
Ampère’s equation
ables in electromagnetism and hydrodynamics. In accordance with Refs. 1 and 33–35, we make the analogy of the
magnetic induction field with vorticity ␻. We also may notice that the electromagnetic analog of one of the three inertial forces that is observed in a frame rotating about a point
with angular velocity ⍀, F3 = −m关⍀̇ ⫻ r兴 is Faraday’s magnetic induction law.36,37
Table II shows the equivalent equations as they are
known in electrodynamics and hydrodynamics. n共r , t兲 denotes the “hydrodynamic” charge density.5,38
We recall here that the vorticity field is defined by
␻ = ⵜ ⫻ u,
共3兲
and it obeys to the causal relationships
ⵜ · ␻ = 0,
共4兲
⳵␻
= − ⵜ ⫻ l.
⳵t
共5兲
and
Here, l defines the Lamb vector, l共r , t兲 = 关␻ ⫻ u兴. This mathematical relationship suggests that the physical entity we call
magnetic field is created by “something” in rotational motion. This idea is embedded in Maxwell’s vortex mechanical
model of the electromagnetic ether that depicts spinning vortices representing a magnetic field, while the lateral motion
of the smaller idle wheel particles would represent the electric current.39 The concept of Lamb vector and hydrodynamic charge have been shown to be useful in locating and
characterizing vortex structures and turbulence.38 The term I
inside the Lamb vector governing equation is the turbulent
current vector.40
We may notice here that the vector potential A in magnetostatics is quite analogous to the potential ␾ 共and it has
the same mathematical solutions兲 in electrostatics 共e.g.,
Ref. 41兲.
Recall that the hydrodynamic circulation is given by
⌫H =
=
冕
冕冕
␥
共u · ds兲
S
共关ⵜ ⫻ u兴 · n兲dS =
冕
冕冕
␥
共A · ds兲
共关ⵜ ⫻ A兴 · n兲dS =
S
共␻ · n兲dS,
S
while in electromagnetism circulation is given by
共6兲
共B · n兲dS = ⌽B . 共7兲
S
Here, ⌫ is the line integral 共circulation兲 around a closed
curve ␥ of the fluid velocity and ds is a unit vector along ␥.
Hence, ⌫ is the analog of the electric current, but it does not
presuppose a “closed circuit.” It is just the circulation around
the curve ␥ that delimits the surface S 共with generality, with
ambiguous shape, e.g., open or closed surface兲 crossed by
the flux of a given “kind” of substance. This is an outcome of
Stoke’s theorem. This quantity ⌫EM is, in fact, the magnetic
flux. The A field is the vector potential and B ⫽ 关ⵜ ⫻ A兴.
Both Eqs. 共6兲 and 共7兲 are applications of the Kelvin–Stokes
theorem relating the surface integral of the curl of a vector
field over a surface S in the Euclidean three-dimensional
space to the line integral of the vector field along ␥ with a
positive orientation, such that ds points counterclockwise
when the surface normal n points toward the viewer.
We have shown in previous publications15–17 that the
electric field is given with generality in the form
E = − ⵜ␾ −
⳵A
+ 关v ⫻ B兴 − ⵜ共v · A兲.
⳵t
共8兲
Equation 共8兲 contains the standard Lorentz force, but also a
new extra term, which come out naturally from the analysis
of the set of Maxwell’s equation in moving electromagnetic
systems,15,17,42 which is ⵜ共v · A兲. We recall that Eq. 共8兲 is an
outcome of the use of the convective derivative d / dt inside
the Maxwell set of equations. It can be readily shown that
d / dt is Galilean invariant, whereas the Maxwell–Einstein operator ⳵ / ⳵t is not.43 Multiplying Eq. 共8兲 by the elementary
charge dq it gives the electrodynamic force law. However, it
appears now a longitudinal force acting on the elementary
charge, −dq ⵜ 共v · A兲. According to Boyer44 this force could
possibly introduce an EM-lag effect on a beam of electrons
accounting for the AB effect. In this case, the AB effect
might be a local effect, making the quantum topological interpretation untenable. Recent experiments,45 however, observe no lag effect for electrons passing a macroscopic solenoid. According to Boyer46 this experiment does not yet rule
out the controversy, since the classical lag effect depends
strongly on the details of the interaction between electrons
and the current of the solenoid.47
From the above considerations we can set up an isomorphism 共structure-preserving mapping兲 between EM-field and
fluid dynamics—what we call fluidic electrodynamics—
obtaining the following expression for an element of hydrodynamic force acting on a given physical system:
EH = − ⵜ⌽ −
冕冕
冕冕
⳵u
+ 关u ⫻ ␻兴 − ⵜu2 .
⳵t
共9兲
From the purely theoretical point of view, Eqs. 共8兲 and 共9兲
are completely analogous mathematically, while describing
different kinds of fluids, all symmetry being recovered.
Equation 共9兲 is equivalent to
Downloaded 06 Apr 2011 to 193.136.137.244. Redistribution subject to AIP license or copyright; see http://pof.aip.org/about/rights_and_permissions
097103-4
Phys. Fluids 21, 097103 共2009兲
A. A. Martins and M. J. Pinheiro
EH = − ⵜ
冉冊
p
du
,
−
␳
dt
共10兲
determining in an unambiguous manner that the scalar potential ␾ has as the hydrodynamic counterpart the massic
enthalpy p / ␳. Using the mathematical identity
1
2
grad共u2兲 ⬅ 关u ⫻ rot u兴 + 共u · ⵜ兲u,
共11兲
into the above Eq. 共9兲, since ␳⌽ = p − ␳u2 / 2, we obtain after a
short calculation the hydrodynamic motive force under the
form
EH = − ⵜ
冉
冊
p 1 2
⳵u
+ u −
+ 关u ⫻ ␻兴,
␳ 2
⳵t
共12兲
obtained after replacing the velocity potential ⌽ by its appropriate expression, as shown on Table I. Multiplying Eq.
共12兲 by the elementary mass dm we obtain a dynamic equation of motion for the elementary particle of mass dm,
dF = dmEH = − dm ⵜ
冉
冊
⳵u
p 1 2
+ u − dm + dm关u ⫻ ␻兴.
⳵t
␳ 2
共13兲
From Eq. 共8兲, the force acting on a moving charge dq
with velocity v relatively to a frame of reference at rest, must
be given by
F = dqE = − dq ⵜ 关␾ + 共v · A兲兴 − dq
⳵A
+ dq关v ⫻ B兴,
⳵t
共14兲
while the corresponding dynamical equation of hydrodynamics for an ideal fluid, the Euler–Gromeko equation, is given
by
−f=−␳
冉
冊
⳵u
␳
− ⵜ p + u2 + ␳关u ⫻ ␻兴,
⳵t
2
共15兲
with f representing an external force per unit volume 关in
international system of units 共SI兲 units N / m3兴 that acts globally on the fluid—a source term. If we consider that this
external force on the fluid is imparted by a “charge,” then the
reaction force on this charge by the surrounding fluid, fext,
will be equal and of opposite direction: fext = −f. Thus, we
have recovered the hydrodynamic equivalent to the electromagnetic force equation, as will be defined in Sec. II A.
Equation 共15兲 has the advantage to appear with the angular
velocity, and as it happens with Euler equation obeys to the
same kind of initial conditions and boundary conditions. It
is worth to recall the contributions of two different kinds
of forces to the fluid dynamics, the gradient of enthalpy
h = p + ␳u2 / 2 and the last term on the right-hand side, which
describes strain condition of the medium. Equation 共15兲 reproduces the dynamics of an incompressible fluid, but it has
been shown by Murad48 that classical incompressible steadystate subsonic flow solutions can be transformed into viscous
solutions by changing the definition of the potential.
The term −dq ⵜ 共v · A兲 in Eq. 共14兲 represents a new term
and it might be important where there is chaotic motion, like
in arcs and plasmas,42 or in special geometries. Other authors
argued the need of another term correcting Lorentz force,
among them, Phipps,49 Graneau,50 Cavalleri et al.,51 and
Monstein,52 but they have not given a consistent formulation
as we do here.
A. Euler and Bernoulli’s integrals of the fluid
dynamic equations
Both the Euler equation and the Euler–Gromeko equation can be integrated in special cases provided certain conditions are given. Here, we follow the presentation given in
standard textbooks 共e.g., Ref. 53兲. Let us consider again Eq.
共14兲. Assuming that our physical system at study is submitted to an external force of the type fext = ␳cE, under the assumption of constant charge density, with ␳c = e共ne − Zni兲 denoting the charge density for a plasma fluid, we can
rearrange Eq. 共14兲 in the following form:
fext = − ␳c
⳵A
− ⵜ关␳c共v · A兲 + ␳c␾兴 + ␳c关v ⫻ B兴.
⳵t
共16兲
In the Coulomb gauge we have ⵜ · A ⫽ 0 and furthermore
let us suppose that we have a potential flow with A ⫽ ⵜ␹,
altogether with ␹ a scalar potential. In hydrodynamics the
analog to ␹ is the potential of velocity ⌽. Hence, let us
suppose instead that our physical system is submitted to an
external massic force deriving from a potential u, fext = −ⵜu.
Of course, this supposition is similar to the supposition taken
with Eq. 共16兲. Note that we make use of the Ampère equation
ⵜ ⫻ H = J = ␳cv 共omitting the displacement current term
⳵⑀E / ⳵t兲 in order to put the term J ⫻ B under the form
关J ⫻ B兴 = 共B · ⵜ兲
冉 冊
B2
B 1
− ⵜ
.
␮ 2
␮
共17兲
It can be shown using standard methods 共e.g., Ref. 54兲 that
by taking the inner product of both members of Eq. 共16兲 by
the elementary displacement vector dr directed along the
streamline the result is the Lagrange integral,
␳c
⳵␹ B2
+
+ ␳c共v · A兲 + ␳c␾ + u = C共t兲.
⳵t 2␮
共18兲
Here, the function C共t兲 depends only on time. Although the
first term on the right-hand side of Eq. 共17兲 plays an important role whenever there is bending and parallel compression
of the magnetic field lines we can assume here approximately straight and parallel field lines, in which case it
vanishes.
Using now Eq. 共17兲 and considering that u = U / ␳m = p
+ 共1 / 2兲␳mv2 with ␳m = mn denoting the mass density, we can
rewrite Eq. 共18兲 under the form
␳c
1
⳵␹
B2
+ 共J · A兲 +
+ ␳c␾ + p + ␳mv2 = C共t兲.
2
⳵t
2␮
共19兲
This is fundamentally the law of conservation of energy, and
it states that the energy of matter plus the energy of this
“fictitious” fluid 共carrying electromagnetic fields兲 is constant
along a streamline. Hence, the nature of the fluid can enter
through this u function, which means here the internal energy per unit mass. The Bernoulli integral can also be obtained in the presence of a B-field for a particle of fluid
Downloaded 06 Apr 2011 to 193.136.137.244. Redistribution subject to AIP license or copyright; see http://pof.aip.org/about/rights_and_permissions
097103-5
flowing along the line of current, since then ⳵A / ⳵t = 0. Envisaging the permanent flux of a given fluid, we aim now to
integrate along a given streamline. If we do the inner product
of Eq. 共16兲 by dr, the differential element of a line of current
it is easily obtained,
p + ␳ c␾ +
B2 1
+ ␳mv2 + 共J · A兲 = C.
2␮ 2
共20兲
C 共in SI units J / m3兲 remains constant as long as we stand
over the line of current, changing value when we change to
another current of line. Besides the contribution of the electric potential and the magnetic pressure terms, Eq. 共20兲
shows the contribution of something new–a pressure term
associated with the product of the electric current density by
the vector potential. This new term represents the onset of a
new kind of interaction through the agency of the vector
potential, that is, by the intermediary of fields with a bigger
range of action 共decaying like 1 / r兲. Also, it shows that the
effective total pressure of the fluid has the contribution of a
another term of energy, 共J · A兲 per unit volume.
Onoochin et al.55 arrived to a similar conclusion, that an
additional magnetic force to the Lorentz force should exist
similar to a magnetic pressure. It should also be related to the
longitudinal forces claimed to exist56–60 and could introduce
an electromagnetic lag in the AB effect.47
The result from Eq. 共20兲 can be applied to magnetocumulative generators, devices that generate electromotive
forces through magnetic compression. It is worthwhile to
remark that, although the value of the vector potential A at a
given point of the physical space is not gauge invariant, the
coupling term J␮A␮ is gauge invariant.61
We may also notice that the nature of the fluid on which
the electromotive force is acting in Eq. 共16兲 does not play
any role. If the fluid is a good conductor of electricity the
resulting equations are those of the magnetohydrodynamics.
Otherwise, the above results have a quite general nature.
Equation 共20兲 helps to master the principle of producing
electric fields of required configuration in the plasma,62 in
particular, understanding the anomalous diffusion of charged
particles in magnetized plasmas.42 Altogether, new insights
for development of high-current accelerators, plasma-optical
systems, and tokamak reactors can be obtained.
B. Magnus’s force and Joukowski’s theorem
The Magnus force is the lift force on a cylinder moving
through a fluid when there is a net circulation of fluid around
the cylinder. Equation 共13兲 allows us to obtain in a different
way the Magnus force or Joukowski’s formula. In addition,
remark that the second term of Eq. 共13兲 is similar to the
Coriolis 共gyroscopic兲 force. To calculate the total 共resultant兲
force acting on a given body with surface S we need to
integrate all over the surface,
F=−
Phys. Fluids 21, 097103 共2009兲
Fluidic electrodynamics: Approach to electromagnetic propulsion
冕冕 冕
␳关␻ ⫻ v兴dSdl.
共21兲
V
Here, the elementary volume is supposed to have cylindrical
symmetry, such as dV = dS · dl. We consider the fluid
incompressible, assumptions usually used to demonstrate
Joukowski’s theorem. If instead we are interested in the force
per unit of transversal length, we integrate to obtain
兩dF兩
=−␳
dl
冕冕
兩关␻ ⫻ v兴 · ndS兩
S
= − ␳兩关⌫H ⫻ v0兴兩 = − ␳⌫Hv0 ,
共22兲
where v0 is the fluid velocity at infinity and where we have
taken the unit vector n along the outer normal to the circulation around the contour ␥, such as ⌫ ⫽ ⌫n.
C. Force exerted on a charged particle by a body
carrying current
We intend to apply the previous methodology to calculate the angular velocity that an ideal solenoid carrying flux
⌽B would communicate to an electrically charged particle of
mass m and charge q, placed at a distance r from the solenoidal axis, in an azimuthal trajectory with velocity V␪. From
Eq. 共6兲, we obtain v␪ = ⌫ / 2␲r, while from Eq. 共7兲, we have
63
v␪ = ⌽B / 2␲r. The canonical momentum can be written immediately, containing the particle mechanical momentum
plus the “field” 共or fluid兲 momentum,
P = mV␪ +
q ⌽B
u␪ = const.
c 2␲r
共23兲
Therefore, the fluidic electrodynamics treatment is rewarding, since considering the second term of Eq. 共13兲, we realize
that the non-null term is ⳵u / ⳵t and, accordingly, we obtain
the force on which is submitted the charged particle 共e.g.,
Refs. 63 and 64兲,
Fp = − q
⳵A
q
=−
⌽̇B .
⳵t
2␲rc
共24兲
D. Force between two parallel currents
of fluid
The calculation of the force exerted between two parallel
currents is a standard application of electromagnetic theory,
hereby applied with the help of our formalism. When considering our Eq. 共13兲, we realize that only the term of
Coriolis has any relevance to the problem. Hence, let us consider two filamentary vortex distant a part of r. The elementary force that filament 共1兲 would exert on filament 共2兲 is
given by
dF2共1兲 = − dm2关␻ ⫻ v1兴,
共25兲
where dm2 denotes the element of mass of a given “kind of
flux” over which the given force is exerted. Considering the
element of mass in the elementary volume dV = dS · dl around
the filamentary axis 共2兲, we can easily obtain
d兩F2共1兲兩
dl2
=−␳
冏冕
S2
冏
关␻2 ⫻ v1兴 · dS2 ,
共26兲
since dm2 = ␳dS2 · dl2. We recall that the velocity induced
around an infinite filamentary vortex 共2兲 ␻2 is given by
v2 = −⌫2 / 2␲r 共e.g., Ref. 65兲. It leads us to
Downloaded 06 Apr 2011 to 193.136.137.244. Redistribution subject to AIP license or copyright; see http://pof.aip.org/about/rights_and_permissions
097103-6
d兩F2共1兲兩
dl2
Phys. Fluids 21, 097103 共2009兲
A. A. Martins and M. J. Pinheiro
=−␳
⌫1
2␲r
冕
共␻2 · dS2兲 = − ␳
S2
⌫ 1⌫ 2
,
2␲r
共27兲
that we can rewrite in the form
dF2共1兲
dl2
=−␳
⌫ 1⌫ 2
ux .
2␲r
共28兲
Considering the orientation of the vorticity, we have an attractive or repulsive force between the two filamentary vortex. Using the analogies shown in Table I it is possible to
establish the electromagnetic force between two currentcarrying wires,
dF2共1兲
dl2
= − ␮0
i 1i 2
ux .
2␲r
共29兲
We can use Eq. 共20兲 to obtain the pressure field around the
filaments. When considering electrical wires acting over the
“electric fluid” in a vacuum 共with no matter motion兲, we
obtain the scalar pressure at a given point ␥ of the streamline,
p共␥兲 = p0 − 共J2 · A1兲.
共30兲
Equation 共30兲 contains an interaction term of pressure, where
J2 is the current density in the second wire 共or, in the analog
filamentary vortex兲 and A1 is the vector potential in the first
wire. When both currents are parallel we have a pressure
decrease of the electric fluid in the region between the two
parallel wires, resulting thus into attraction, while when the
currents are anti-parallel repulsion will prevail. To translate
to the isomorphic relationship in the framework of hydrodynamics, we need to notice that Eq. 共30兲 involves the current inside the conductor, that is, the vortex core, and therefore we need to use instead the expression of the speed
v = ⌫r / 共2␲R2兲, where R is the vortex 共tube兲 radius. Hence,
we need to use Eq. 共25兲 integrating through the surface S to
obtain the pressure at the center r = 0 of the vortex of radius
R 共e.g., Ref. 65兲,
⌫ 1⌫ 2
p = p0 − ␳ 2 2 .
4␲ R
共31兲
Clearly and systematically, we see that this program of fluidic electrodynamics leads us to a new electrodynamic force,
also discussed in Onoochin et al.55 through a different
method.
III. CONCLUSION
The new methodological approach provided by the fluidic electrodynamics approach allows a fast transposition
from electromagnetism to fluid dynamics and vice versa.
From the practical viewpoint, the application of analogies
allowed us to arrive, with respect to the main objective of
our research, at conclusions which are more difficult to obtain by other methods. In addition, we remark that the proposed fluidic electrodynamics framework described in this
paper allows one to study the dynamics of any kind of fluid
共e.g., electrotonic state—EM phenomena; ordinary fluids—
hydrodynamic phenomena兲. In such a framework, a new
electrodynamic force equation was obtained with a new interaction term which represents a longitudinal force exerted
among different elements of a circuit. The isomorphism between EM fields and fluids becomes entirely symmetrical.
The idea of longitudinal forces in electrodynamics was introduced by Ampère and verified by himself and de la Rive with
an experiment done in 1882, where a hairpin was propelled
along two troughs of liquid mercury due to the longitudinal
repulsion 共e.g., Ref. 66兲. Other experiments seem to point
toward the reality of this force, among them, Nasilowski’s
wire fragmentation experiment.57 As it is still going on an
electrodynamic force law controversy,58–60 we believe that
the theoretical frame offered in this paper reinforces the necessity to consider this kind of force as a real one.
However, this electrodynamic force equation 共with a
new term beyond the standard Lorentz equation兲 becomes
Galilean invariant. It was shown, however, that classical
electrodynamics can be formulated consistently with Galilean transformations,67 in particular, through an appropriated
redefinition of the fields constitutive equations.68
Longitudinal forces could introduce an electromagnetic
lag in the AB effect, and this could possibly mean that the
AB effect is local and maybe a classical effect.47
The theoretical framework offered in this paper can be
applied to magnetocumulative generators and may help to
master the principle of producing electric fields of required
configuration in plasmas, which facilitates the development
of high-current accelerators, plasma-optical systems, and
thermonuclear devices.
ACKNOWLEDGMENTS
The authors gratefully acknowledge partial financial
support by the Fundação para a Ciência e a Tecnologia
共FCT兲. We would also like to thank important financial support to one of us 共A.A.M兲 in the form of a Ph.D. Scholarship
from FCT.
1
G. Rousseaux and É. Guyon, “A propos d’ une analogie entre la mécanique des fluides et l’ électromagnetisme,” Bulletin de l’Union des Physiciens 96, 107 共2002兲.
2
M. Liu, “Hydrodynamic theory of electromagnetic fields in continuous
media,” Phys. Rev. Lett. 70, 3580 共1993兲.
3
S. Sridhar, “Turbulent transport of a tracer: An electromagnetic formulation,” Phys. Rev. E 58, 522 共1998兲.
4
A. A. Martins and M. J. Pinheiro, “Inertia, electromagnetism and fluid
dynamics,” AIP Conf. Proc. 969, 1154 共2008兲.
5
H. Marmanis, “Analogy between the Navier–Stokes equations and Maxwell’s equations: Application to turbulence,” Phys. Fluids 10, 1428
共1998兲.
6
P. Holland, “Hydrodynamic construction of the electromagnetic field,”
Proc. R. Soc. London, Ser. A 461, 3659 共2005兲.
7
G. E. Volovik, The Universe in a Helium Droplet 共Clarendon, Oxford,
2003兲.
8
G. Chapline and P. O. Mazur, “Tommy Gold revisited: Why doesn’t the
universe rotate?” AIP Conf. Proc. 822, 160 共2006兲.
9
W. F. Vinem and J. J. Niemela, “Quantum turbulence,” J. Low Temp.
Phys. 128, 167 共2002兲.
10
G. E. Volovik, “Simulation of quantum field theory and gravity in superfluid 3He,” Low Temp. Phys. 24, 127 共1998兲.
11
H. D. Froning, Jr. and G. V. Meholic, “Unlabored transitions between
subluminal and superluminal speeds in a higher dimensional tri-space,”
AIP Conf. Proc. 969, 1129 共2008兲.
12
G. V. Meholic, “Fluidic spacetime and representation of fields in the trispace model of the universe,” AIP Conf. Proc. 1103, 326 共2009兲.
Downloaded 06 Apr 2011 to 193.136.137.244. Redistribution subject to AIP license or copyright; see http://pof.aip.org/about/rights_and_permissions
097103-7
13
Fluidic electrodynamics: Approach to electromagnetic propulsion
H. R. Kaufman, “Fluid dynamic approach to mass of classical electron,”
Phys. Lett. 33A, 9 共1970兲.
14
H. R. Kaufman, “Fluid dynamic approach to the relativistic electromagnetic field,” Lett. Nuovo Cimento 4, 1139 共1970兲.
15
M. J. Pinheiro, “Do Maxwell’s equations need revision?—A methodological note,” Phys. Essays 20, 267 共2007兲 关http://arxiv.org/abs/physics/
0511103兴.
16
A. A. Martins and M. J. Pinheiro, “The connection between inertial forces
and the vector potential,” AIP Conf. Proc. 880, 1189 共2007兲.
17
A. A. Martins and M. J. Pinheiro, “On the electromagnetic origin of inertia
and inertial mass,” Int. J. Theor. Phys. 47共10兲, 2706 共2008兲.
18
A. C. T. Wu and C. N. Yang, “Evolution of the concept of the vector
potential in the description of fundamental interactions,” Int. J. Mod. Phys.
A 21, 3235 共2006兲.
19
B. Riemann, Mathematische Werke, 2nd ed., edited by H. Weber 共B. J.
Teubner, Leipzig, 1897兲.
20
H. L. Armstrong, “Comment on moving transparent media,” Am. J. Phys.
37, 336 共1969兲.
21
P. A. M. Dirac, “Is there an aether?” Nature 共London兲 168, 906 共1951兲.
22
P. A. M. Dirac, “An extensible model of the electron,” Proc. R. Soc.
London, Ser. A 268, 57 共1962兲.
23
H. Poincaré, “La theorie de Lorentz et le principe de reaction,” Archives
Néerlandaises des Sciences Exactes et Naturelles 5, 252 共1900兲.
24
G. M. Graham and D. G. Lahoz, “Observation of static electromagnetic
angular momentum in vacua,” Nature 共London兲 285, 154 共1980兲.
25
U. Leonhardt and P. Piwnicki, “Optics of nonuniformly moving media,”
Phys. Rev. A 60, 4301 共1999兲.
26
T. D. Keech and J. F. Corum, “A new derivation for the field of a timevarying charge in Einstein’s theory,” Int. J. Theor. Phys. 20, 63 共1981兲.
27
A. Tonomura, “Direct observation of thitherto unobservable quantum phenomena by using electrons,” Proc. Natl. Acad. Sci. U.S.A. 102, 14952
共2005兲.
28
D. Clowe, M. Bradac, A. H. Gonzalez, M. Markevitch, S. W. Randall, C.
Jones, and D. Zaritsky, “A direct empirical proof of the existence of dark
matter,” Astrophys. J. 648, L109 共2006兲.
29
E. Teller, “Electromagnetism and gravitation,” Proc. Natl. Acad. Sci.
U.S.A. 74, 2664 共1977兲.
30
B. Bramson, “Do electromagnetic waves harbour gravitational waves?”
Proc. R. Soc. London, Ser. A 462, 1987 共2006兲.
31
V. P. Dmitriyev, “Mechanical model of the Lorentz force and Coulomb
interaction,” Cent. Eur. J. Phys. 6, 711 共2008兲.
32
J. D. Jackson and L. B. Okun, “Historical roots of gauge invariance,” Rev.
Mod. Phys. 73, 663 共2001兲.
33
E. Guyon, J.-P. Hulin, and L. Petit, Hydrodynamique Physique 共CNRS,
Paris, 1991兲.
34
M. D. Semon and J. R. Taylor, “Thoughts on the magnetic vector potential,” Am. J. Phys. 64, 1361 共1996兲.
35
J. Sivardiere, “On the analogy between inertial and electromagnetic
forces,” Eur. J. Phys. 4, 162 共1983兲.
36
M. D. Semon and G. M. Schmieg, “Note on the analogy between inertial
and electromagnetic forces,” Am. J. Phys. 49, 689 共1981兲.
37
D. F. Dempsey, “The rotational analog for Faraday’s magnetic induction
law: Experiments,” Am. J. Phys. 59, 1008 共1991兲.
38
G. Rousseuax, S. Seifer, V. Steinberg, and A. Weibel, “On the Lamb
vector and the hydrodynamic charge,” Exp. Fluids 42, 291 共2007兲.
39
B. J. Hunt, The Maxwellians 共Cornell University Press, Ithaca, 1991兲.
40
H. Marmanis, “Analogy between the electrodynamic and hydrodynamic
equations: Application to turbulence,” Ph.D. thesis, Brown University,
1999.
41
M. Schwartz, Principles of Electrodynamics 共Dover, New York, 1972兲,
p. 141.
Phys. Fluids 21, 097103 共2009兲
42
M. J. Pinheiro, “Anomalous diffusion at edge and core of a magnetized
cold plasma,” J. Phys.: Conf. Ser. 71, 012002 共2007兲.
43
T. Phipps, Jr., “On Hertz’s invariant form of Maxwell’s equations,” Phys.
Essays 6, 249 共1993兲.
44
T. H. Boyer, “Proposed experimental test for the paradoxical forces associated with the Aharonov–Bohm phase shift,” Found. Phys. Lett. 19, 491
共2006兲.
45
A. Caprez, B. Barwick, and H. Batelaan, “Macroscopic test of the
Aharonov–Bohm effect,” Phys. Rev. Lett. 99, 210401 共2007兲.
46
T. H. Boyer, “Comment on experiments related to the Aharonov–Bohm
phase shift,” Found. Phys. 38, 498 共2008兲.
47
T. H. Boyer, “Classical electromagnetic deflections and lag effects associated with quantum interference pattern shifts: Considerations related to the
Aharonov–Bohm effect,” Phys. Rev. D 8, 1679 共1973兲.
48
P. A. Murad, “Closed-form solutions to the transient/steady-state Navier–
Stokes fluid dynamic equations,” AIP Conf. Proc. 813, 1264 共2006兲.
49
T. Phipps, Jr., “Physical significance of the vector potential,” Elec. Spacecraft 26, 20 共1997兲.
50
P. Graneau, Newtonian Electrodynamics 共World scientific, Singapore,
1996兲.
51
R. Angulo, O. Rodrigues, and G. Spavieri, “Does the expression for the
Lorentz force need to be modified?” Hadronic J. 20, 621 共1998兲.
52
C. Monstein, “Electromagnetic induction without magnetic field,” Electric
Spacecraft Journal 24, 29 共1998兲.
53
C. Fédiaevski, I. Voïtkounski, and Y. Faddéev, Mécanique des Fluides
共Mir, Moscow, 1974兲 共French edition兲.
54
L. Landau and E. M. Lifschitz, Mecanique des Fluides 共Mir, Moscow,
1971兲 共French edition兲.
55
V. Onoochin and T. E. Phipps, Jr., “On an additional magnetic force
present in a system of coaxial solenoids,” Electromagn. Phenom. 3, 256
共2003兲.
56
P. Graneau and N. Graneau, “Electrodynamic force law controversy,”
Phys. Rev. E 63, 058601 共2001兲.
57
J. Nasilowksi, in Unduloids and Striated Desintegration of Exploding
Wires, edited by W. G. Chase and H. K. Moore 共Plenum, New York,
1964兲, Vol. 3, pp. 295–313.
58
N. Graneau, T. Phipps, and D. Roscoe, “An experimental confirmation of
longitudinal electrodynamic forces,” Eur. Phys. J. D 15, 87 共2001兲.
59
G. Cavalleri, E. Casaroni, E. Tonni, and G. Spavieri, “Interpretation of the
longitudinal forces detected in a recent experiment of electrodynamics,”
Eur. Phys. J. D 26, 221 共2003兲.
60
P. Graneau, “Electromagnetic jet-propulsion in the direction of current
flow,” Nature 共London兲 295, 311 共1982兲.
61
A. V. Balatsky and B. L. Altshuler, “Persistent spin and mass currents and
Aharonov–Casher effect,” Phys. Rev. Lett. 70, 1678 共1993兲.
62
M. Rickard, D. Dunn-Rankin, F. Winberg, and F. Carlton, “Maximizing
ion-driven gas flows,” J. Electrost. 64, 368 共2006兲.
63
G. T. Trammel, “Aharonov–Bohm paradox,” Phys. Rev. 134, B1183
共1964兲.
64
M. G. Calkin, “Linear momentum of quasistatic electromagnetic fields,”
Am. J. Phys. 34, 921 共1966兲.
65
L. Prandtl and O. G. Tietjens, Fundamentals of Hydro- and Aeromechanics 共Dover, New York, 1934兲.
66
L. Johansson, M.S. thesis, Lund Institute of Technology, 1996.
67
L. Gomberoff, J. Krause, and C. A. López, “Formulation of special relativity by means of Galilean transformations,” Am. J. Phys. 37, 1040
共1969兲.
68
M. A. Miller, Yu. M. Sorokin, and N. S. Stepanov, “Covariance of Maxwell equations and comparison of electrodynamic systems,” Sov. Phys.
Usp. 20, 264 共1977兲.
Downloaded 06 Apr 2011 to 193.136.137.244. Redistribution subject to AIP license or copyright; see http://pof.aip.org/about/rights_and_permissions
View publication stats