# Electrokinetic devices in air

### Content:

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

9A 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

9A 0

3

2

9A 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 9A 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) V0d 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

9A 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 V0d 3 / 2 x1 / 2 V0d 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.