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On Electrodynamics of Uniform Moving Charges

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The theory of electrodynamics exists since more than hundred years and is used for almost every electromagnetic application. But there still exist debates for example about the existence of a motional electric field outside current carrying wires. This essay examines the force between uniform moving charges with some applications and experiments and shows a request for an additional γ-factor on the formula for the electric field of a uniform moving charge. Two possibilities to explain this additional factor are given.
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copyright © (2000, 2015) by André Waser, CH-8840 Einsiedeln Page 1
On Electrodynamics of Uniform
Moving Charges
André Waser
*
Issued: July 21, 2000
Revised: Dec. 30, 2015
The theory of electrodynamics exists since more than hundred years and is
used for almost every electromagnetic application. But there still exist debates
for example about the existence of a motional electric field outside current car-
rying wires. This essay examines the force between uniform moving charges
with several applications and experiments and shows a request for an addition-
al -factor on the Lienard-Wiechert formula for the electric field of a uniform
moving charge.
Keywords: moving charges; induction; special relativity
Introduction
The electrodynamics of moving bodies has motivated EINSTEIN [3] to formulate the theory of
special relativity. He recognizes the all electrodynamic processes underlying principle of rela-
tivity. Not the movement against an aether has to be understood as the cause for electrodynam-
ic effects but the relative motion between two inertial systems. With his second more funda-
mental postulate of the absolute constancy of the velocity of light independent of the velocity
of the source - EINSTEINs theory was able to describe effects with relativistic velocities much
better than previous theories based on aether concepts.
EINSTEIN was the first who recognized that the electric and magnetic forces depends on the
movement of the associated reference frame and that the question about the seat of the electro-
motive force in unipolar induction is therefore meaningless [3]. This can be traced back to
forces between charges only. Generally, it must be possible to describe the electromagnetic
theory only as forces between charges only. Some time ago MOON & SPENCER presented a
new electrodynamics without using the magnetic field concept [28]-[30]. This paper is another
attempt to use a formulation without the magnetic field concept for forces between uniform
moving charges.
A special case, where this forces can be studied, is the motional electric field, first reported
by William HOOPER [21] and later also established by EDWARDS [12] and EDWARDS et. al.
[13]. About a year later BARTLETT and WARD [5] denied the existence of this effect. Frequent-
ly some papers were published about this effect [3] until EDWARDS et. al. [24] changed their
measurement setup and then also claimed, that this motional electric field does not exist. By
examining the experiments cited above and by the existing theoretical foundation the author
believes, that the motional electric field really exists, but the measurement setup greatly influ-
ences the result due to the inductive nature of the motional electric field.
*
Gerbestrasse 1, CH-8840 Einsiedeln, andre.waser@bluewin.ch
Page 2 copyright © 2000, 2015 by André Waser, CH-8840 Einsiedeln
Forces Between Moving Charges
For the force between charges the following geometry shall be valid:
Figure 1. Geometry of electrodynamics between charges.
In 1846 Wilhelm Eduard WEBER published for the force between moving charges (with
adapted notation) [40]:
2
20
12 2 2
20
q1 d r 1 dr
1r
q 2 dt
4 r c dt




 



 




Fr
(2.1)
what can be written as [3]:
220
12 2 2
20
qv3
11
q2
4 r c c


 


 

F r a rcos
(2.2)
In 1954 Parry MOON and Domina Eberle SPENCER have presented the equation reprinted
below [28], which can be applied to many electrodynamic applications.
2
0 2 0
11
2 2 2
200
01
0
qq
v 3 d r
1 v t
q 2 dt c
4 r c 4 c r
1 1 r
qt
4 r r c
 
 


 
 
 




  

Fra
r
cos
(2.3)
with:
F: Force on charge q2 [N]
qi: Electric charge q1 and q2 [As]
v: Relative velocity of charge q1 with respect to q2 [m / s]
a0: Acceleration unit vector of q1 with respect to q2 []
r0: Unit vector of distance from q1 to q2 []
r: Distance from q1 to q2 [m]
: Angle between the vectors a0 and v [rad]
The first term describes the AMPERE law and is addressed to moving charges. The second term
corresponds to the acceleration between charges as it is the case for example with alternating
currents and the third term MOON and SPENCER introduced for time varying charges. If ele-
mentary charges are used for calculation only, the third term reduces to the COULOMB law,
because elementary charges are looked upon as constant in time.
The equation (2.3) has some contradictions, because the force between moving charges is
derived with the principle of “action at a distance”, whereas the force between accelerating
charges deals with a finite signal propagation velocity c. Because of this MOON & SPENCER
introduced the force of a time dependent charge (they called it MAXWELL force). This time
dependent part was needed to describe effects of radiated waves.
Alfred LIENARD [25] and Emil WIECHERT [42] have deduced the retarded potentials of
charges, from which the general COULOMB-FARADAY law can be derived [18]-a:
copyright © (2000, 2015) by André Waser, CH-8840 Einsiedeln Page 3
 
2
0
q 1 1
t4 c t Kr c t Kr
Kr
,

 

  

nn
Er
(2.4)
with
,
from which the electric LIENARD-WIECHERT field of a charge is
     
 
20 0 0
13 2 2
0
0
q11
r c r
41

 
 



  
E r r r a
r
. (2.5)
These equations splits neatly into two parts. The first term depends on the velocity v but not on
the acceleration a of charge q1 and have vector components parallel to r and v, whereas the
second term is proportional to a and the vector direction is always perpendicular to r. This
equation applies also to relativistic velocities.
Forces Between Uniformly Moving Charges
The acceleration terms should now be suppressed. For uniform motion MOON & SPENCER
published a reduced equation [27], which can be applied to induction problems
022
2
0
1
22
3
11
r4
q
qr
F

cos
(2.6)
This equation, which is also included in WEBER’s equation (2.2), describes all processes using
direct current or uniformly moving charges. MOON and SPENCER have demonstrated [26] that
equation (2.6) is the only possible relation between charges to describe the original equations
of AMPERE [1] correctly. In opposite a huge number of possible equations between current
elements are known (for example GAUSS [16], GRASSMANN [17], NEUMANN [31],
HELMHOLTZ [19], RIEMANN [35], ASPTEN [2]). Because only one equation of forces between
charges describes the phenomenon of induction in opposite to many formulas, which uses
current elements, the formulation with charges is now considered more fundamental than the
others. Because equation (2.6) cannot be used for relativistic velocities [27], an adaptation is
needed.
The retarded velocity depending part of the LIENARD-WIECHERT field can also be de-
scribed as a function of the present position rp at time t. This corresponds to an “action at a
distance” formulation as in equation (2.6). For the present position rp the force between two
charges is [18]-b
 
2
p0
1p
23
22
202
q1
q4r
1sin

  
Fr
(2.7)
This equation can also be deduced from the electromagnetic field tensor [18]-c. Nevertheless,
equation (2.7) cannot be transformed into (2.6) without an adaptation. For small velocities
(v « c) equation (2.7) can be decomposed with a Taylor series to
 
222
11
3
22
00
2
qq
1 1 3
1
4 4 2 2
1cos
sin
 

 


  

 
(2.8)
Actually the first application presented later in this essay with a linear unipolar generator shows
that (2.7) leads to a wrong result. Equation (2.7) can only be used for the force calculation
between moving charges, if the following correction is made:
Page 4 copyright © 2000, 2015 by André Waser, CH-8840 Einsiedeln
2 2 2 2
2
Correction
1 3 1 3
1 1 1
2 2 2
1

 
 
   
 
 
 
 

cos cos
(2.9)
The corrected equation for the action at a distance force is
 
2
p0 2 2
v11
p
2 3 2
22
22 00
2
1
qq
3
11
q q 2
4 r 4 r
1cos
sin
 

 


 

 
F
Fr
(2.10)
Force Field Plots of Uniformly Moving Charges with Low Velocities
Despite the fact that with standard induction applications the relative motion between charges
is about one billion times smaller than the speed of light c, the equations (2.7) and (2.10) deliv-
ers fundamental discrepancy when used for calculation of induction processes. For further
analyzing we plot the radial velocity depending field of equations (2.7) and (2.6), where the
COULOMB field is subtracted. Then the radial field for the low speeds of v0=c/106 and
v1=c/2106 results in the plots below:
0
30
60
90
120
150
180
210
240
270
300
330
gn 0
gn 1
n
0
30
60
90
120
150
180
210
240
270
300
330
hn 0
hn 1
n
Figure 2. Radial velocity depending field according to equation (2.7).
0
30
60
90
120
150
180
210
240
270
300
330
10 12
0
gn 0
gn 1
n
Figure 3. Radial field part according to the WEBER-MOON-SPENCER equation (2.6).
The traditional equation (2.7) shows the same radial field for a slow moving charge as the
WEBER-MOON-SPENCER equation (2.6), with the exception, that it is rotated for exactly /2.
v v
v
copyright © (2000, 2015) by André Waser, CH-8840 Einsiedeln Page 5
Applications
Unipolar Induction Generator Linear Type
This example has been presented by MOON & SPENCER [29]. It should again be used to verify
the traditional equation (2.7) against the new equation (2.10). In this experiment a metal plate is
moving along an endless (for simplification) current carrying wire. Then in this metal plate a
voltage V is induced in inverse y-direction according to figure 4.
Figure 4. Unipolar induction on uniform moving plate along a current carrying wire.
FARADAYs equation E = v B delivers the value:
aa
0
22
22
aa
Iv Iv al
V d d
ya
2 c 2 c
ll
ln


    
  

y
v B s s
(3.1)
Now figure 5 applies for an observer resting with the charge q1+.
Figure 5. Force between moving charges in an atomic “cell” of two conductors.
MOON & SPENCER have shown [27] that equation (2.6) leads to the result
 
2
1 2 1 2
00
2 3 5 2
2 2 2 2
2 0 0
22
I v I v
y 3 y 1
d
2y
q 2 c 2 c
yy

 
 
   
Fyy
(3.2)
Page 6 copyright © 2000, 2015 by André Waser, CH-8840 Einsiedeln
The direction of the resulting electric field E is opposite to the force acting on the negative
charge q2- and points therefore in the negative direction of y0. The finally detected voltage V on
a voltmeter follows from integration of all field parts along the y direction:
a
1 2 1 2
22
00
a
I v I v
dy a l
Vya
2 c 2 c
lln

    
  
(3.3)
This is identical to (3.1). Beneath the force on q2- there acts also an equal but opposite force on
the fixed ions q2+ in point P. This means that on the moving plate a force acts toward the cur-
rent carrying wire. When a current is flowing though the moving plate along the y direction, the
forces on the negative and positive charges in the plate are not balanced so that it is expected to
find a weak force on the moving plate. With the setup of normally done measurements where
the moving plate has to be fixed somehow against the current wire - this force may be to week
to catch the attention of the experimenters.
Now we check the traditional equation (2.7) for application to this problem. The velocity
terms with higher order than 2 are suppressed for simplicity. Then it is:
   
0 2 0 22
11
2 3 2
22
00
22
qq
11
1 1 3
4 4 2
q r r
1cos
sin
 
 

  
 
F r r
(3.4)
Again figure 5 applies. Then the electric field calculates:
 
20
2
1 2 1 2 2
0
q q v v
1
1 1 3
4 2 c r
cos

 

 


 


r
F
(3.5)a
 
20
2
1 2 2 2
0
q q v
1
1 1 3
4 2 c r
cos

 

 


 


r
F
(3.5)b
 
2
2
20
2
1 i 2
22
q0
v 2v v
e13
42c r
cos
 
  

r
F F F
(3.6)
With v1 « v2 it follows for the force acting on q2-:
 
2
0
q2
12
22
20
ev v 1 3
q 4 c r
cos
 

Fr
(3.7)
 
012 2
22
20
Iv
d1 3 d
q r 4 c cos
 

Fr
(3.8)
 
2
12
02 3 5
2 2 2 2
022
Iv y 3y d0
2c yy

 
    
Ey
(3.9)
The negative charge q2- experienced no force so the electric field is interpreted to be zero. The
traditional equation (2.7) leads to the wrong result.
copyright © (2000, 2015) by André Waser, CH-8840 Einsiedeln Page 7
AMPERE’s Force Law
This example describes a setup with the electron drift velocity v1 only according to figure 6.
Figure 6. Geometry of force between two parallel wires
The force between two parallel conductors of the length l is known as AMPERE’s force law:
y2 I2
0
0
y
F
l
(3.10)
Applying (2.6) the force between the charges is:
Like Currents: Opposite Currents:
2
0
0
2
qr
4e
2
r
F
2
0
0
2
qr
4e
2
r
F
2
2
20
2
1
22
q0
v
e3
11
42
cr
cos
 

 


 

r
F
2
2
20
2
1
22
q0
v
e3
11
42
cr
cos
 

 


 

r
F
2
2
20
2
1
22
q0
v
e3
11
42
cr
cos
 

 


 

r
F
2
2
20
2
1
22
q0
v
e3
11
42
cr
cos
 

 


 

r
F
2
0
0
2
qr
4e
2
r
F
 
2
2
20
12
22
q0
2v
e3
11
42
cr
cos
 

 


 


r
F
The force on conductor 2 is given with the sum of all forces acting on the charges q2+ and q2-:
2
20
2
1
222
0
v
e3
1
22
cr
cos

 

 
r
F
2
20
2
1
222
0
v
e3
1
22
cr
cos

 

 
r
F
The expansion from the conductor cell with the cross section A1 and the electron density N1
leads to the force on a current element of length d:
00
2 2 2 2 2
1 1 1 2 2 2 1 2
22 2 2 2
00
N A ev N A ev I I
33
d 1 d 1 d
22
2 c r 2 c r
cos cos
   
   
   
 
   
rr
F
(3.11)
With integration over the wire length 1 the force on the current element Id2 can be derived as
0 2 0
2 1 2 1 2
2 2 2
200
d I I I I
31
1d
d 2 y
2 c r 2 c
sin
cos


 

 

Fyy
(3.12)
whereas follows immediately the AMPÈRE force law (3.10). If instead of (2.6) the equation
(2.8) would be applied, again the result would be a zero force.
Page 8 copyright © 2000, 2015 by André Waser, CH-8840 Einsiedeln
Motional Electric Field
A special case, where the radial force field of two current carrying wires with opposite currents
can be studied is an arrangement according to figure 7. It refers to the motional electric field,
first reported by William HOOPER [21]-[23] in 1969.
For two very close arranged conductors, which carries the same current but in opposite di-
rection, it is I1 = -I2 and approximately y1 = y2, so that on first glance one supposes, there exists
no magnetic field B and also no vector potential field A. Therefore, an external charge q3
should not experience a force due to the currents. ASSIS et. al. [4] have shown, that this is not
correct. And besides of the motional electric field there exists also a force proportional to the
current, which should not be considered here in more detail.
Figure 7. Geometry of Motional Electric Field force
With (2.6) and with the same calculation method as before the force on q3 becomes
2
00
3 1 1 2 2
22
3 0 0
N A v 1 Iv 1
q 4 c y 4 c y
   
 
FE y y
(3.13)
The same equation was published by ASSIS [3] and WESLEY [41]. The result fits nicely to the
experiment of HOOPER. The motional electric field E is oriented centripetal to the two wires. It
is very important to note here, that under any circumstances this force cannot be shielded.
HOOPER concludes therefore, that this force corresponds to the gravitational force and deliv-
ered a rough calculation of the attracting force between two hydrogen atoms [20].
This impossibility to shield the radial electrical force field of every current carrying wire is
valid for almost all known arrangements, that means for ordinary coils as well as for caduceus
coils. In comparison with the other forces of an ordinary coil on an external charge, this force
of the motional electric field is extremely small, what could be the reason that it is not well
known. For example, a current of 5000 A gives at a distance of 10cm to the wires according to
figure 9 and with a drift velocity of about 1cm/s an electric field strength of about 50V/m.
Such small fields usually are not measured with ordinary induction measurement equipment.
It might be of special interest, that living organisms can react to fields of an even smaller
amplitude as it has been shown by the experiments of Glen REIN with bacteria exposed in a
field of a caduceus coil [33], [34]. The discussion about the influence of electric fields to living
organisms is illuminated in another way. So even in a region, where ordinary measurements
show, that no electric or magnetic fields are active, this very small and even not detected re-
maining fields still are able to influence biological processes.
copyright © (2000, 2015) by André Waser, CH-8840 Einsiedeln Page 9
The SANSBURY Experiment
The force direction of the motional electric field depends not on the direction of the current
but on the polarity of the test charge q3. SANSBURY [36] has confirmed this behaviour in an
experiment. In this experiment a charged torque bar was placed close to a current carrying
conductor as shown in figure 8.
Figure 8. Geometry of the SANSBURY experiment.
The silver foil was charged with an adjustable 3kV source against the current loop. To ini-
tialise the experiment, the silver foil was charged against the wire carrying no current, so that
the torque pendulum stabilised its initial position shown in figure 8. Then the current was set to
900A what forces the torque pendulum to move. Now the SANSBURY experiment shows the
opposite sign than equation (3.13), that is, the negative charged foil was attracted to the cur-
rent-carrying conductor instead of repelled.
A suggestion, what probably can be the cause for this, is the movement of the midpoint of
the torque bar shown in figure 8. According to the calculation given in Appendix A there exists
the following force distribution along the y-axis:
Figure 9. Force Fx(y) of current loop on torque bar
It shows, that at the position, where the silver foil is located (y~0), the force on a positive test
charge acts in the negative x-direction toward the current loop, but at the position y = 7cm of
the midpoint of the torque pendulum, the force is in the positive x-direction. This positive force
Page 10 copyright © 2000, 2015 by André Waser, CH-8840 Einsiedeln
at y = 7 cm is even higher than shown in figure 9, because the current loop acts on the charged
72 cm long suspension wire, too.
The charge densities along the copper suspension and the silver foil is not known, so it is
difficult to calculate the total momentum on the pendulum. SANSBURY has noticed a high in-
stability of the measurement when the current is set on, what doesn’t enable to make precise
readings (it was, for example, not possible to measure the angular deflection of the torque bar
as a function of the current intensity). This instability can be explained with the force character-
istic on the torque bar shown in figure 9 as well as on the forces on the suspension wire.
The EDWARDS Experiments
In 1974 EDWARDS [12] reported an electric field due to conduction current in a supercon-
ducting coil. Two years later Edwards et al. [12] measured a motional electric field proportion-
al to I2 with high accuracy. About a year later BARTLETT et al. [5] denied the EDWARDS effect
partly based on the measurements with a spinning coil. Finally, in 1991 LEMON et al. [24]
changed their measurement set-up previously used to demonstrate the EDWARDS effect and
reported then a negative result also. So what happened?
It is important that the motional electric field is not an electrostatic field but merely an in-
duced field. The circulation along a closed path in the vicinity of a uniform moving charge is
not always zero (PURCELL [32]), as can be seen for example from figure 2 and 3. For an infi-
nite long, straight current carrying wire the following scheme applies:
Figure 10. Different cases for an induced Motional Electric Field
The HOOPER experiment is exactly case 1b) of figure 10. The experimental set-up of
EDWARDS et al. is close to case 1a). The signal lead and the support tube to the electrometer
corresponds definitively to case 1a). Because the superconducting wire is not straight but
winded along a ring, different supply wire positions relative to the superconducting coil experi-
ence a different motional electric field. And this is the case inside the brass shield of the coil
assembly. So the outcome of the experiment depends on the supply wire and shield positions
relative to the superconducting coil. And exactly in this region EDWARDS et al. have changed
the set-up between their two publications [13] and [24]. This could be an explanation why in
earlier runs they measured a signal proportional to I2 and at later runs not.
copyright © (2000, 2015) by André Waser, CH-8840 Einsiedeln Page 11
The BARTLETT Experiment
The experimental set-up of BARTLETT and WARD [5] corresponds at first glance to case
4b) of figure 10. But again the moving current wire is not straight and therefore the motional
electric field depends on the position of a test charge around the spinning coil. Generally, rot E
is not zero around every closed path between the two shells so that for example not every part
of a sphere will contain the same charge density. In this radial direction, in which the motional
electric field induction is a maximum, also the local charge density on a sphere is a maximum.
In the radial direction the motional electric field decreases with 1/r, so that a charge density on
the inner sphere at a given radial direction is reduced by 1/r on the outer sphere. That means,
for a radial direction a charge density difference should be detectable. But in the signal line
from the inner sphere to the lock-in amplifier an e.m.f. will be induced too, which corresponds
to the local radial potential between the inner and outer spheres and therefore cancels the
measuring value out. The measured value is about zero, as reported by BARTLETT and WARD.
BARTLETT and WARD gave some other tests about the charge’s dependency of its velocity
which uses accelerated charges inside atoms. This seems problematic because it is well known
that an electron bound to an atomic nucleus does not “move” around an orbit but merely has is
state defined by quantum mechanics.
About the Correction Factor
As shown with equation (2.9) the correction factor must be applied to the LIENARD-
WIECHERT field to make it usable for induction phenomena. The question arises, what does this
additional factor mean.
Obviously, the force between (uniform) moving charges is transformed with the factor .
This transformation of force between uniform moving coordinate systems is described with the
“Four-Vector-Force” in the framework of Special Relativity, which in turn is derived from the
local time of the moving charges (for example FEYNMAN[15]).
It’s worth to note, that the whole derivation of LIENARD-WIECHERT potentials and fields
has been made without explicit use of LORENTZ transformation. To apply these potentials and
fields correctly to induction experiments solely by relative motions between charges (thus
without any magnetic fields), it’s absolutely necessary to use the relativistic factor .
Page 12 copyright © 2000, 2015 by André Waser, CH-8840 Einsiedeln
Further Experiments
The Biefeld-Brown Effect (Hypothesis)
Around the year 1920 Thomas Townsend BROWN and his mentor Dr. P.A. BIEFELD have
exercised with free hanging capacitors. With experiments with electron ray tubes BROWN has
discovered, that each time he deflected the electron beam between two conducting plates with a
strong electric field, a small but detectable force appeared. For further investigations Brown
constructed several types of capacitor arrangements. He discovered that a capacitor stressed
with a high voltage tended to accelerate against the direction of the electric field lines. BROWN
tested his experiments in air, oil and even in vacuum, but the capacitor always shows the same
behavior (but with different magnitude) independent of the surrounding medium. Finally, his
work led BROWN to the application of several patents [6]-[10]. The force on the capacitor
depends on the following points:
1) proportional to the applied DC voltage
2) proportional to the current between the electrodes
3) reverse proportional to the square of the distance between the electrodes
4) proportional to the product of the electrode masses
5) Week seasonal dependency to day and month cycles (Sun- and Moon position)
The items 1-3 points on an electrodynamic cause between relatively moving charges, the
items 4 and 5 are not covered with the presented theory herein and its treatment should be put
back for the moment. Extremely important is BROWN‘s reported behavior 2, which is probably
not widely known, but which has been confirmed by the author with some simple experiments.
The have a force on the capacitor arrangement, it is obviously necessary, that the current does
not drop to zero. For a further analysis a refer to figure 11.
Figure 11: Biefeld-Brown Effect shown with two oppositely charged balls
With the knowledge of the preceding examples it is easily understandable, that the moving
charges in the supply wires as well as in the free flight path between the charged balls Q+ and
Q- examine forces to the remaining positive ions in the balls. Because of the high voltages (the
voltages are in the range of 30kV...300kV) it is expected that the velocity va is substantial
higher than the velocity vw in the conductors, so that vw can be neglected. In addition, va is not
constant. Because of the electric field an electron q- will be continuously accelerated away from
Q- towards Q+.
In the previous examples the cause for the movement of the charges in a wire (i.e. a voltage
source or an external electric field) was not taken into consideration to calculate the forces. For
the first time the causing charge is now also this charge, on which the force must be calculated
in this experiment. Because of this the COULOMB field will again not be taken into calculation
but only the force depending on the relative movement of the charges.
copyright © (2000, 2015) by André Waser, CH-8840 Einsiedeln Page 13
A moving electron q- with an „averagevelocity
a
v
(the influence of the acceleration
should now be neglected) causes at the mean distance r = r+ = r- the velocity depending force
F+ on the positive charge Q+:
2
0
2
a
0r
c
v
2
1
4q
Q
rF

(5.1)
A force with the same magnitude but in opposite direction effects also to the electron q-. On the
charged ball Q- an analogue force is applied:
2
0
2
a
0r
c
v
2
1
4q
Q
rF

(5.2)
There is again a reaction force in opposite direction to the electron q-. The total force on the
capacitor construction with the balls Q+ and Q- is the sum of both above forces:
2
0
2
a
0r
c
v
2qQ r
F

(5.3)
As a result, the whole composition moves in the direction from the negative ball to the positive
charged ball. This is confirmed by the experiment.
The reaction forces to the moving electron reduces its acceleration, which will become zero
for v = c. For v = c the reaction force is exactly equal to the causing force originating from the
relative movement between the ‘free’ electron and the charged balls Q.
BROWNs first three statements can be justified qualitatively:
1) proportional to the applied DC voltage: The higher the voltage U, the higher is the
stored charge in Q+ and Q-, and the higher is the resulting force.
2) proportional to the supply current (Leakage current): The more electrons are in-
volved, the higher is the resulting force.
3) inverse proportional to the square of the distance between the balls: This relation can
be found in the equation (5.3).
Usually this experiment is explained with the movement of the surrounding, ionized air. But
this argument cannot explain why the composition always shows a distinct higher force when
the voltage is switched on (i.e. when the current has its maximum). In addition, this argument
does not explain why the experiment works in vacuum also.
The asymmetries Brown used in his apparatus for the shape of the anode and cathode can
be explained, when the acceleration depending forces are also taken into consideration (which
are not subject to this paper). With a view to some pictures in Brown’s patents (for example
[7]: figures1,5,6) it is evident that the resulting force can be optimized when the cross-section
of the anode Q+ is made very small. For this reason, Brown mostly used simply wires instead
of other forms for the anode. Then the angle between the velocity vector a and r is always
close to zero and the force between the charges is not reduced by the acceleration depending
force part.
With the still used assumption that with a normal experimental setup for the Biefeld-
Brown effect no relativistic speeds are involved, the explanation for the first three points can be
regarded as completed. But totally open is the reported counteraction with the gravitational
force (points 4 and 5). The speculation should be allowed here, that gravity is finally also a
force between charges only, so that an interaction between gravity and electric fields seems
possible. Another indication that an interaction between inertia and electric fields exists is
given by Erwin SAXL[37],[38] with his very high precision measurements with a torsion pen-
dulum inside an electric charged FARADAY cage. This is very interesting for further investiga-
tions.
Page 14 copyright © 2000, 2015 by André Waser, CH-8840 Einsiedeln
From this hypothetical explanation to the BIEFELD-BROWN-Effect two new experimental
proposals can be formulated. Usually high voltage sources do not allow to have high currents
also, so it is difficult to have both properties within one source. Therefore, two sources should
be used. The high voltage circuit U1 is used as usually done in the Brown experiments. A sec-
ond current source U2 is designed in such a way, that it is able to deliver a high current between
the two charged poles Q+ and Q-. For safety reason and one end of each source are electrically
connected, the other poles not, of course:
Figure 12: (left) and 13 (right): Two experiment proposals to increase the BIEFELD-
BROWN Effect
The conductor “Leiterstrecke” should be able to transport a huge number of charges (electrons)
with a maximum high velocity. A superconductor or an electron tube would be excellent. With
a normal conductor there are many free charges available but the mean drift velocity is very
small. So when using a normal conductor, the current must be increased which needs a higher
conductor cross section and minimizes the possibility to observe the effect due to the higher
weight of the total arrangement.
More experiments are analyzed in der German original paper.
Conclusions
The presented examples of electrodynamic applications with uniform moving charges have
shown, that a second-order electric force field around conductors exists. Because this field does
not behave like a static field but more like an induced field, it is somewhat difficult to measure.
The electric LIENARD-WIECHERT field needs a correction factor when it is applied to in-
duction.
copyright © (2000, 2015) by André Waser, CH-8840 Einsiedeln Page 15
References
[1] AMPERE André-Marie, “Mémoire sur la théorie mathématique des Phénomènes électrodyna-
miques, uniquement déduite de l’expérience”, Collection de Mémoires relatifs à la physique, Paris
3 (1887) 27
[2] ASPTEN Harold, “The Law of Electrodynamics”, Journal of the Franklin Institute 287 No.2 (Feb-
ruary 1969) 179-183
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No.1 (1991) 109-113
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Stationary Resistive Wire Carrying a Constant Current”, Foundations of Physics 29 (February
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[5] BARTLETT D. F. and B. F. L. WARD, „Is an electron’s charge independent of its velocity?“, Physi-
cal Review D 16 No.12 (15 December 1977) 3453-3458
[6] BROWN Thomas Townsend, “Electrostatic Motor”, US Patent 1'974'483 (25 Sep. 1934)
[7] BROWN Thomas Townsend, “Electrokinetic Apparatus”, US Patent 2'949'550 (16 Aug. 1960)
[8] BROWN Thomas Townsend, “Electrokinetic Transducer”, US Patent 3'018‘394 (23 Jan. 1962)
[9] BROWN Thomas Townsend, “Electrokinetic Generator”, US Patent 3'022'430 (20 Feb. 1962)
[10] BROWN Thomas Townsend, “Electrokinetic Apparatus”, US Patent 3'187'206 (1 June 1965)
[11] EINSTEIN Albert, “Zur Elektrodynamik bewegter Körper”, Annalen der Physik und Chemie 17, IV.
Folge (30 June 1905) 891-921
[12] EDWARDS W. Farrell, „Measurement of an Electric Field Due to Conduction Currents“, Utah State
University Press, Monograph Series 21 ISBN 0-87421-071-2 (1974) 1-10
[13] EDWARDS W. Farrell, C.S. KENYON, and D.K. LEMON, “Continuing Investigation into Possible
Electric Fields Arising from Steady Conduction Currents”, Physical Review D 14 (15 August
1976) 922-938
[14] FARADAY Michael, Experimental Researches in Electricity”, 1-3 (1831) 3090-3115
[15] FEYNMAN Richard P, Robert B. LEIGHTON and Matthew SANDS, “The Feynman Lectures on Phys-
ics”, Addison-Wesley (1963) Vol. II, Section 216
[16] GAUSS Carl Friedrich, “Zur mathematischen Theorie der elektrodynamischen Wirkung”, Werke
Göttingen 5 (1867)
[17] GRASSMANN Hermann, “Neue Theorie der Elektrodynamik”, Annalen der Physik und Chemie 64
No 1 (1845) 1-18
[18] HEALD Mark A., and Jerry B. Marion, “Classical electromagnetic radiation, Saunders College
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gewandte Mathematik 75 (1873) 35-66
[20] HOOPER William J., “Equivalence of the Gravitational Field and a Motional Electric Field”, Pro-
ceedings of the Boulder Conference on High Energy Physics (18-22 August 1969)
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October 1971)
[23] HOOPER William J., “Apparatus for Generating Motional Electric Field”, US Patent 3'656’013 (11
April 1972)
[24] LEMON D. K., W. Farrell EDWARDS and C. S. KENYON, „Electric potentials associated with steady
conduction currents in superconducting coils”, Physics Letters A 162 (1992) 105-114
[25] LIENARD Alfred, „Champ électrique et magnétique produit par une charge électrique“, L’Eclairage
Electrique 16 (July 1898) 5-14, 53-59, 106-112
[26] MOON Parry, and Domina Eberle SPENCER, “Interpretation of the AMPÈRE Experiments, Journal
Franklin Institute 257 (March 1954) 203-220
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nal Franklin Institute 257 (April 1954) 305-315
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[28] MOON Parry, and Domina Eberle SPENCER, “A New Electrodynamics, Journal Franklin Institute
257 (May 1954) 369-382
[29] MOON Parry, and Domina Eberle SPENCER, “On Electromagnetic Induction, Journal Franklin
Institute 260 (September 1955) 213-225
[30] MOON Parry, and Domina Eberle SPENCER, “Some Electromagnetic Paradoxes, Journal Franklin
Institute 260 (November 1955) 375-395
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Akademie der Wissenschaften, Abhandlungen (1845)
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New York (1965) 161
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[35] RIEMANN Berhard, “Schwere, Elektrizität und Magnetismus”, C. Rümpler Verlag, Hannover
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[36] SANSBURY Ralph, “Detection of a force between a charged metal foil and a current-carrying con-
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[38] SAXL Erwin J., “Device and Method for Measuring Gravitational and Other Forces”, US Patent
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549-573
copyright © (2000, 2015) by André Waser, CH-8840 Einsiedeln Page 17
Appendix A
The force in x-direction on the torque bar of the SANSBURY experiment [36] is calculated with
the geometry of figure 12.
Figure 12: Geometry of the SANSBURY Experiment
For the wire element AB the force on the torque bar is:
 
2
2
AB
ryl  
,
 
2
2
AB
ry
cos l
 
 
 
,
x0
FcosFx
 
   
   
C
D
0
2
AB 1 AB
22
20 AB
x
x3
0
AB 1
2 3 2 5 2
22
22
x
20
22
0C
1D
2 3 2 3 2
22
22
0CD
d Iv d 3
1
q2
4 c r
y Iv 3d
q2
4c yy
x
Iv x
3
2
8c x y x y
cos
ll
ll

 


 
    
 
    






   

   



 


 
Fr
Fx
x
For the wire element BC the force on the torque bar is:
22
BC C
rx  
,
22
BC C
rx
cos  
 

,
CC
22
BC C
xx
rx
sin 
 

,
x0
FsinFx
 
 
 
   
 
0
2
BC BC
1
22
20 BC
x2
y
0
BC C C
1
2 3 2 5 2
2 2 2 2
y
20CC
33
22
0CC
1
2 3 2 3 2
22
22
0C C C C
dIv d 3
1
q2
4 c r
y x x
Iv 3d
q2
4c xx
2x y y 2x y y
Iv
8c x x y x x y
l
l
cos
l l l l
ll


 

 
    
   
 
    






   

   



 


 
Fr
Fx
x
And finally for the wire element CD the force on the torque bar is:
Page 18 copyright © 2000, 2015 by André Waser, CH-8840 Einsiedeln
 
2
2
CD
ryl  
,
 
2
2
CD
ry
cos l

 
 
,
x0
FcosFx
 
   
   
D
C
0
2
CD CD
1
22
20 CD
x
x3
0
CD 1
2 3 2 5 2
22
22
x
20
22
0C
1D
2 3 2 3 2
22
22
0CD
dIv d 3
1
q2
4 c r
yIv 3d
q2
4c yy
x
Iv x
3
2
8c x y x y
cos
ll
ll

 


 
    
 
    






   

   



 


 
Fr
Fx
x
The total force Fx(y) in x-direction on the torque bar is then the sum of the three wire element
forces. This total force is shown in figure 10.
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We present the results for the electric and magnetic fields due to linear and circular current distributions according to Weber's theory. We show how the electric field predicted by Weber's law is compatible with the anomalous diffusion in plasmas. Finally, we discuss some modern experiments related to this topic and compare the results of these experiments with a prediction based on Weber's law.
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From attempts to observe electric fields arising from charge-neutral, current-carrying superconducting coils (NbTi, Nb and Pb) we conclude that effects previously reported are not fundamental departures from conventional electromagnetic theory. The null results predicted by Maxwell's theory are confirmed to within two parts in one thousand.