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 calculated 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}=pi (*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
= 36 µA. The dimensions are taken from my Holtz
machine and this current is really what it produces. (It
should produce somewhat more, because there is some polarity
reversal at the charge collectors, but my machine reverts
polarity continuously, what may be reducing the maximum output.)

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 hability 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. 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: 20 March 2002

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