Electrokinetic devices in air

Content:

The physics of electrokinetic devices applying and
adapting the Child-Langmuir Law derivation for
vacuum diodes
Part 2: electrokinetic devices in air
by Leon Tribe (leon.tribe@gmail.com)
19th September 2007

Recommended Reading
‘Introduction to Electrodynamics’ by David J. Griffiths. This and other related physics
textbooks can be purchased here: Amazon Electrokinetic Physics Books Links

Introduction
The Child-Langmuir Law describes the characteristics of a parallel plate vacuum
diode. By using this approach, we can derive a one dimensional expression for the
characteristic properties of Lifters and related electrokinetic devices and by drawing
an analogy to the vacuum diode, gain insights into the Lifter’s subtle properties. A
possible explanation for the Biefeld-Brown effect is given for both the air and vacuum.

Derivation of the characteristic equations for a electrokinetic
device in air
Let us approximate our electrokinetic device as 2 plates, one plate grounded and the
other at V  V0 . Let them be a distance ‘d’ apart and let us assume they are very large
so that we can neglect end effects (Area = A >> d2). Therefore, all properties will only
depend on ‘x’ and we can make a 1-D approximation.

d

A

x=0
V=0

x=d
V = V0

We know:
d 2V
 ( x)
(Poisson’s equation) (1)

2
0
dx

Copyright © 2007 Leon Tribe. All rights reserved.

E

dV
(Definition of potential) (2)
dx

v( x)  E ( x) (Blanc’s Law for the mobility of ions in a medium) (3)
NB: We have now departed from our vacuum derivation in that the motion of each
electron is defined by a new equation due to the influence of air on the motion of the
ions formed.
Also
I   jA   vA (Definition of current) (4)
Combining (1) and (4) we get
d 2V
I
(5)

2
v( x) A 0
dx
Combining this with (3) yields
d 2V
I
(6)

2
E ( x) A 0
dx
Combining this with (2) yields
d 2V
I
(7)

2
dV
dx

A 0
dx
Rearranging yields
2

dV d 2V 1 d  dV 
I
(8)


 
2
dx dx
2 dx  dx 
A 0
Rearranging and integrating yields
2

2I
 dV 
x  C (9)

 
A 0
 dx 
Assuming at x = 0, E = 0, we get
dV  2 I
 
dx  A 0


x 

1/ 2

 2I
  
 A 0




1/ 2

x1 / 2 (10)

Integrating by x yields
 2I
V ( x)   
 A 0




1/ 2

2 3/ 2
x  C (11)
3

At x = 0, V = 0 so
 2I
V ( x)   
 A 0




1/ 2

2 3/ 2 
8
x   
3
 9A 0




1/ 2

I 1 / 2 x 3 / 2 (12)

Copyright © 2007 Leon Tribe. All rights reserved.

At x = d, V  V0 , therefore taking (12) and rearranging for I, we get
1

2

V
8 
  03
I  V0 d  
d
 9A 0 
3

2

 9A 0 

 (13)
8 

This is the air equivalent of the Child-Langmuir law. The current goes with the square
of the potential and inversely with the cube of the gap length.
We also obtain
2

V0  9A 0
2
V ( x)   

3
8
 A 0 d 

 



1/ 2

2 3/ 2  9 
x  3
3
 4d 

1/ 2

V0

2 3/ 2
x  V0 d 3 / 2 x 3 / 2 (14)
3

Combining (2) and (14) yields
3
E ( x)   V0 d 3 / 2 x1 / 2 (15)
2
Combining (3) and (15) yields
3
v( x)   V0d 3 / 2 x1 / 2 (16)
2
Combining (1), (2) and (15) yields
dE
3
1
3
 ( x)   0
  0 V0 d 3 / 2 x 1/ 2   0 V0 d 3 / 2 x 1/ 2 (17)
dx
2
2
4
The force on the electrons in the gap is defined by the Lorentz force law for
electrostatic charges
F ( x)  A ( x) E ( x) (18)
Combining this with (15) and (17) yields
3
3
9 2
F ( x)  A ( x) E ( x)  A 0 V0 d 3 / 2 x 1 / 2 V0 d 3 / 2 x1 / 2  A 0 V0 d 3 (18)
4
2
8
What is interesting about this result is that it is independent of x. Integrating over the
gap we get
d
9
2
FT   F ( x)dx  AV0  0 d 2 (19)
0
8
Combining (13) and (19) we get

9
 9A 0 
FT  A 0 d 2 Id 3  

8
8 

1



Id

(20)

Given the force on the device will be equal and opposite to the force on the ions in the
gap, we obtain our familiar form of
Id
FEK 
(21)

Copyright © 2007 Leon Tribe. All rights reserved.

For completeness, we can also calculate the opposing force due to the change in
momentum.
F (d ) 

d (mv)
dv
3
3
1
9 2
 mv
 m V0d 3 / 2 x1 / 2 V0d 3 / 2 x 1 / 2  m V0  2 d 3 (22)
dt
dx
2
2
2
8

Therefore our total force on the device is really the difference of (21) and (22), that is
2

9 2 2 2
9 2 9  V0   m 2
2
FFinal  m V0  d  AV0  0 d    
 A 0  (23)
8
8
8 d   d

Consequences for electrokinetic devices
While ions are formed in the medium surrounding the electrokinetic device, the
Biefeld-Brown effect can be explained in terms of a loss of momentum to the medium
through collisions between the air and the ions. Simply put, while in a vacuum the
pull on the device from the charge-driven force exactly cancels the mass-driven
movement when the electron is collected, for air, some of the ion’s momentum is
already diminished through collisions with the air so its impact with the collector is
correspondingly reduced. The net effect will be an observed force towards the
emitting electrode with no directly observable mechanism, what is often referred to
erroneously as the ‘unbalanced force’.

Comparison of experimental observation to the model
While a literature search of the major peer-reviewed journals resulted in little
evidence of experimentation with electrokinetic devices, a number of sites exist on the
internet where experimental results have been published. While not peer-reviewed, it
is still instructive to determine where reality and the model coincide and where they
do not. Any reported experiment which is inconsistent with this model can be
replicated and gives direction for refining the model.
Experimental
Observation

Source

Consistent
with
proposed
model?
Yes

Details

Electrokinetic
devices show
movement in air

Multiple.
http://www.blazelabs.com/e-exp04.asp
http://jlnlabs.imars.com/lifters/logbook/index.htm

Increasing the
temperature of
the emitter,
increases
emitter current
and thrust
Making the
electrodes out of
different
materials affects
the observed
force
Measured force
is dependant on
the medium the

http://www.blazelabs.com/e-exp07.asp

Yes

http://www.blazelabs.com/e-exp08.asp

No

http://www.blazelabs.com/e-exp12.asp

Yes

For a fixed gap, the
force is proportional to
the current. Therefore
an increase in current
will lead to an increase
in observed force
The model makes no
comment on the
materials used for the
electrodes and therefore
they should not affect
the observed force
The ion mobility
constant,  , is
dependant on the

The model predicts a
force in air

Copyright © 2007 Leon Tribe. All rights reserved.

device is
operating in
Electrokinetic
devices stop
working when
put in a vacuum

http://www.blazelabs.com/l-vacuum.asp
http://www.blazelabs.com/nasatest.pdf

Yes

Wire polarity
affects the force

http://www.blazelabs.com/l-doe.asp

Yes

There is an
asymmetric
force. The force
on the emitter is
different to the
collector

http://jlnlabs.imars.com/lifters/asymmetric/index.htm

Yes

medium the ion is
travelling through so
this is consistent
In Part one, we predict
that if no current is lost
to the vacuum
container, there will be
no net force observed
The ion mobility
constant is dependant
on the type of ions
formed which depends
on the polarity
As the field at the
emitter is zero, the
induced charge on the
emitter is also zero.
However, as there is a
field on the collector,
there is also an induced
charge on the collector.
As F=qE, this means
the collector
experiences a force but
the emitter does not.
This logic also applies
to the vacuum i.e. if
there is current loss in
the vacuum, the force
will act on the collector,
not the emitter

Areas for further research
For a given working device, it should be relatively simple to adjust the key parameters
and measure the effect on the other measurable parameters i.e. plot the IV curve,
determine the relationship between gap, voltage, current and observed force. This can
then be compared to the relationships predicted by this model and the model’s
effectiveness can be assessed.
While vacuum tests appear to confirm an electrokinetic device will not work in a
vacuum, quantitative tests of electrokinetic device performance at different air
pressures will yield information on how performance is affected by, for instance,
atmospheric and weather changes.
In one experiment it was suggested that the electrode material can affect performance.
This is a remarkable result and if the specific quality of the material which affects
electrokinetic performance can be identified, this could be exploited for better
performance.
Finally, while experiments have been performed in different gases, the model also
allows for a measurable force in dielectric liquids. Experiments could be performed
on different liquids, including water to see how the force is affected. While the force
in the air is quite weak, it may be discovered that the electrokinetic force could be
employed for other purposes such as liquid transport or underwater propulsion.

Copyright © 2007 Leon Tribe. All rights reserved.

Acknowledgements
My thanks go to Evgenij Barsukov for the inspiration to fully develop this derivation,
to Daniel Boyd for helping the document be ‘bullet-proof’ and to Steven Dufresne for
the web space.

Copyright © 2007 Leon Tribe. All rights reserved.

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