At sea level and normal temperatures, the molecules of the gases that compose common air get ionized in the presence of an electric field of about 30 kV/cm. This imposes a fundamental limit to all electrostatic phenomena, and affect fundamentally the behavior of electrostatic machines operating in open air.

**Maximum output current in electrostatic machines:**

For electrostatic machines, the limit affects the maximum charge density
that a charge-transport surface can carry. In all machines, the collection
of the generated charge occurs when it's deposited over a flat surface,
and the electric field produced by the charge is perpendicular to that
surface, directed only away from the surface. Using Gauss' theorem, we
find that the charge density *p* required to generate an electric
field *E* perpendicular to a flat surface is:

*p = **e*_{0 }*E*

where *e*_{0} is the
permittivity of vacuum, practically the same of the air, *e*_{0
}= 8.85x10^{-12}. Considering *E *= 3x10^{6}
V/m, the result is:

*p*_{max }= 26.55 µC/m^{2}.

The maximum output current of any machine can then be estimated as:

*i*_{max }= *p*_{max }*A*

where *A* is the area of charge carrier surface that passes under
the charge collectors in one second. For an idealized rotating disk
machine, of any kind that transports charge at just *one side* of
the disk with the electric field pointing only away from the surface, and
removes all the charge from the surface at the charge collectors, the
maximum output current *i*_{max} would be calculated as:

*i*_{max }= π (*r*_{max}^{2 }- *r*_{min}^{2})
*f* *p*_{max}

where the active area of the machine is considered as a ring with maximum
radius *r*_{max }and minimum radius *r*_{min},
meters, and *f* is the rotating speed in turns per second.

Ex: A Holtz machine turning at 40 turns/second, and with *r*_{max
}= 13 cm and *r*_{min }= 9 cm, assuming discharge of
the disk at the charge collectors, produces *i*_{max }=
3.14 x (0.13^{2 }- 0.09^{2}) x 40 x 26.55 = 29.35 µA. The
dimensions are taken from my Holtz machine and
this current is close to what it really produces. (It produces somewhat
more, about 36 µA, because there is some polarity reversal at the charge
collectors, but my machine reverts polarity continuously, turning the
measurement difficult.)

Symmetrical Toepler machines work considering
just one disk, and follow the relation closely, as I can verify in my
machines. The structure of the machine effectively removes the charges
from the disk surfaces, without polarity reversal.

My double Voss machine, produces about 4 x *i*_{max}.
The mechanism in this case is probably a polarity reversal in the disks at
the charge collectors caused by the intense electric field from the
charged inductors in the inner disks. Essentially the same phenomenon that
happens at the neutralizers when there is little current drain at the
charge collectors. The same machine, with just one side operating,
produces about 2 x *i*_{max}. A Holtz machine, if freed from
the polarity reversals, could probably reach the same limit, or somewhat
less because a Holtz machine transports some charge at the back of the
rotating disk, with polarity opposite to the polarity of the charges in
the frontal area.

Wimshurst machines act as if only one disk
were used for charge transport (fact observed by researchers in the late
1800's [26], that noticed that a machine that
collects charge from just one disk produces practically the same current).
A reason for this is that the capacitance between the disks couples a
voltage decrease at one disk, as it discharges to a charge collector, to
the other disk, preventing its discharge. The second disk acts only as an
inductor plate, causing a polarity reversal at the disk that is
discharged. The partial use of the disk surfaces by the sectors cause some
current decrease, and these machines produce a current that is at most 2 x
*i*_{max}, depending on how much of the active area is used
by the sectors, and in the ability of the sectors in capturing charges
from their surroundings. A current or about *i*_{max} is
the most common case. It's useless to use brushes at the charge
collectors, in an attempt to discharge both disks simultaneously, because
this prevents the polarity reversal that would occur, and the drained
charge remains the same.

Sectorless Wimshurst machines, or Bonetti machines,
can
easily reach 2 x *i*_{max}, due to the more effective use of
the disk surfaces. They produce a little more of current than a Voss
machine of the same size. My best machine
behaves in this way.

Some multiple machines can exceed the limit by significant amounts.
Apparently the close proximity of charge transport surfaces cause some
shielding of the inner disks by the outer disks, allowing denser charge
density (about two times greater) at the inner disks. This is what I can
measure in my triplex Wimshurst machine, that
produces two times more current than the consideration of just two
machines predicts, or about 4 x *i*_{max}. With just one
section operating it reaches only *i*_{max}. A similar
effect was observed by early experimenters with this type of machine. The
exact mechanism of the current increase, however, is not clear yet.

Friction machines are affected in the same way, if good enough to reach
the limit, ideally no more than 4 x *i*_{max}. in a Ramsden
type machine, that has four friction pads. But these machines turn slowly
and the reports in old texts indicate that they operate well below the
limit, with strong dependency on how the rubbing pads are made. The Van
de Graaff generator also follows a similar rule, as mentioned in
the first papers about it [p4]. Its current
would be the product of the maximum charge density and the area per second
moved by the belt. A single belt or disk can also allow some of the
electric field to point across the material, if there is no charged
surface of the same polarity at the other side, and this can also increase
the limit, up to two times. To enclose all the charged surfaces in solid
dielectrics, as done in Wommelsdorf and Wehrsen
machines, reduces losses, but doesn't necessarily eliminate the limit, as
breakdown continues to limit the electric field at the surfaces of the
insulators, what limits the density of the available charges. Wommelsdorf
machines with inductors at both sides of each rotating disk act as Voss
machines, with intensified charge generation at the neutralizers and
polarity reversal at the charge collectors, but measurements show that
they don't reach more than 2 x *i*_{max} for each rotating
disk. A Wehrsen machine with two rotating disks is a double Holtz machine,
and shall produce about 4 x *i*_{max }(see this table).

In most influence machines, the output current grows exponentially until the losses equalize the current generation, or the limit calculated above is reached. For an usual machine that has only to charge its Leyden jars and is reasonably well insulated, the limit is reached almost immediately after the machine starts to operate, and the machine works practically as a current source after this. The output current only declines significantly when the output voltage is high enough to divert most of the generated charges to internal corona and sparks in the machine.

When the limit is exceeded, visible sparks and corona appear at the charge transport surfaces, removing the excess of charge. This problem can be solved by changing the insulating material around the charge transport surfaces to something that supports more intense electric field without ionization. This is more effectively done by running the machine in a pressurized gas, as hydrogen, or even in specially treated liquid dielectrics, as done in the relatively recent (1950's-1960's) machines developed by Felici.

**A curious observation about the actual number of surface atoms
ionized:**

This is a calculation that explains why electrostatic forces are so weak
and everything is so dependent on surface details and humidity. Imagine a
copper plate charged to the limit of surface charge density:

The number of atoms in 1 m^{3} of pure copper can be calculated as
*n* = 8.60x10^{28} atoms.

The number of atoms in 1 m^{2} of copper surface is then *n*^{2/3}
= 1.95x10^{19 }atoms.

Considering the charge or one electron, the maximum charge in 1 m^{2},
26.6 uC, is equivalent to 26.6x10^{-6}/1.60x10^{-19} =
1.66x10^{14} electrons.

Dividing the number of surface atoms by the number of electrons we obtain
that only one in 117000 surface atoms is ionized!

Developed and Maintained by Antonio Carlos M. de
Queiroz

Created: 23 June 2000

Last update: 8 November 2017

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