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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 - EINSTEIN’s 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

K1

rc

,,

rv

nn

,

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.

FARADAY’s 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 „average“ velocity

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.

BROWN’s 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

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[28] MOON Parry, and Domina Eberle SPENCER, “A New Electrodynamics”, Journal Franklin Institute

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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

FcosFx

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

FsinFx

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

FcosFx

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.