Download Introduction The Wimshurst machine is an electrostatic generator

Introduction
The Wimshurst machine is an electrostatic generator that builds a
potential across two electrodes by the action of a hand crank. It was invented
by the British engineer James Wimshurst and first described in 18833.
Earlier
machines, such as those by Voss and Holtz, had been of a similar form, but
lacked some of the Wimshurst’s elegant
innovations.
Description
The Wimshurst machine consists
of two circular dielectric plates mounted
on parallel on an axle with conducting
strips spaced evenly along the edges on
the outer surface, as shown in the diagram. A hand crank is connected to a
pulley system that causes the plates to counter-rotate when the crank is turned.
On two ends of the system, a U-shaped wire is situated around the outer
surface of the two plates, and has a number of small points facing toward the
plates, but not quite touching them. These U-shaped wires are connected to the
output electrodes where sparks can be generated. The final essential component
is a charge transfer rod for each plate, which has conducting brushes that slide
along the surface of the plate and actually make contact with the sectors as
they rotate by. These rods have an opposite inclination to the horizontal for
each plate.
Electrostatic Induction
The machine’s functioning is based on the
principle of electrostatic induction. Electrostatic
induction is a process by which the charge density
of a conductor can be increased. It works by
bringing a charged conductor near to one side of the
original conductor. This causes the free charges in
the original conductor to separate into positive and
negative on opposing sides due to the Coulomb repulsion from the introduced
conductor. Then a third conductor with neutral charge is brought into contact
with the other side of the original conductor, causing a flow of charge onto the
third conductor to equalize the potential. Since only one polarity of charge is on
each side, this has the effect of draining one type of charge, while leaving the
other, thus increasing the net charge density. If the third conductor is then
grounded, the process can be repeated, yielding a further increase in charge
density. The Wimshurst machine serves the purpose of mechanically repeating
this process of electrostatic induction.
Explanation
The first issue in the functioning of the Wimshurst machine is that the
induction process needs a charged object to start with in order to carry out the
charge displacement part of the process. However, the machine does not require
any preparation. This charge actually comes from the small amount of imbalance
in the charges that is always present in a real-world system4. There is no way
of knowing which polarity the system starts in, so some machines use a small
piece of fur to provide a bias, which allows more predictable behavior4.
The mechanism of charge buildup is carried out by the charge transfer
rods. The charge in the sectors is always
distributed such that sectors on one side of
the charge transfer rod have one polarity and the other side has the other
polarity. The charge transfer rod has brushes on both ends so that when it
makes contact with one sector, it also makes contact with opposite sector,
which has opposite polarity. If there were just one plate, then the charge
transfer rod would simply move all charges in both sectors to cancel each
other, leaving zero net charge. However, the other plate has regions behind the
brushes that have a net charge (gray regions in diagram). These cause the
charge transfer rod to overcompensate for the charge imbalance between the
sectors that it connects. Having just one of these gray regions would cause an
overcompensation to the point of negating the original imbalance. However, there
is a region on each end, so it actually transfers more charge than would be
needed to negate the charge on each sector, resulting in a greater charge on
each sector than initially, but with swapped polarities. This means the charge
transfer rod transfers more than twice the amount of charge that it would
normally transfer to simply neutralize the two sectors it connects. It takes a lot
of force to build up a high charge density against Coulomb repulsion, and here
the key is that the region on the other plate is larger than the sector that is
being charged, so it exerts more force without the need for larger charge
density.
Notice the elegance in the design here. For induction, you need to
ground the conductors in between cycles. This normally means that a connection
must be made to some large capacitor such as the earth. But the Wimshurst
machine uses a clever trick—instead of dumping small amounts of charge into a
capacitor, it uses a modified version of induction where charge just gets moved
back and forth within the machine. The important distinction is that normal
induction is concerned with keeping the same polarity of charge on a given
conductor, which requires a place to dump the opposite charge. But the
Wimshurst machine allows the sectors to oscillate polarity so that they can be
used as a dumping location and a buildup location simultaneously.
In order for the charges to build up to large levels, it is required that
the other plate is increasing its charge in the gray regions, otherwise the
buildup will be quickly limited by the increasing Coulomb repulsion due to larger
charge densities. Here we find another elegant idea in the design of the
Wimshurst machine: the same arrangement is used for the other plate, but with
the neutralizing bar in the opposite inclination to the horizontal. Since each plate
is half one polarity and half the other polarity, with the division being the
charge transfer rod, this creates 4 quadrants in the combined system. The top
and bottom quadrants have opposite polarity on each plate, and the left and
right quadrants have the same polarity on each plate. This guarantees that the
gray regions will always be oppositely charged, and charged opposite to the top
plate.
The final aspect of the mechanism of the Wimshurst machine’s functioning
is the charge collection. The U-shaped wires with points are situated around the
left and right ends of the device. These locations have the same polarity of
charge on both plates. Since the charge on the sector passing under is
increasing on each cycle, there will be a potential difference between the
sectors and the points on the wire. The points cause electrical breakdown in
the air at low voltages because V = Q/(4πε 0)r = (σ/ ε 0)r = Er which means
that a smaller radius of curvature creates a larger electric field for a given
potential2. This breakdown is not visible because it creates a constant stream
rather than a powerful spark because it doesn’t need as much voltage as a
spark2. This creates an actual transfer of charges, which may be collected
simply with the capacitance of the electrode, or with a Leyden jar. A Leyden jar
provides additional capacitance, which allows for more current in the sparks. The
voltage of the sparks will remain the same since that is only determined by the
breakdown of the air in the spark gap.
Properties
One interesting observation is that sparks are generated with no charged
objects to begin with and no connection to ground. Where do these charges
come from? It turns out that the charges just come from the free charges in
the conducting sectors. But are there enough electrons in the metal to do this?
Suppose the sectors were made of a total of 1 gram of Aluminum. The density
of Al is 27g/mol so there are 6.02*1023/27=2.23*1022 atoms of Al in the whole
apparatus. A good Leyden jar with a capacitance of 1*10-6 Farads at 100,000
Volts would store Q=CV=0.1 Coulombs = 6.24*1017 electrons. This is only one
in every 35,700 electrons that is taken off the conducting sectors. After the
spark, the electrodes are neutralized since the spark starts and ends on the
machine, and the charges are effectively recycled for the next spark.
There are limits to the length of spark that can be generated by the
Wimshurst machine. The spark length is determined by the voltage difference
between the two electrodes. There are design limits to the amount of voltage
that the machine can tolerate. If the potential between a sector right after the
brush and right before the brush is large enough to cause breakdown, the
sectors will be electrically neutralized, which would force the process to start
over if many sectors had this breakdown occur.
There are also limits to the amount of current that can be generated by
the machine (this limit can be offset by providing a large Leyden jar and
waiting longer for the sparks). The limit is again due to the electrical breakdown
in air.
The maximum electric field allowable before ionization of air begins is E =
3*106
= σ/ ε 0, so σ = 2.655*10-5 C/m2. Therefore, even if all of the excess charge is
transferred to the charge collectors at the terminals, the current will be I=σA,
where A is the area of sectors that passes under the collectors in one second3.
However, the collector will be also be charged, so it is unlikely that all the
excess charge will be transferred. The area passing under the collectors is a
function of the size of the sectors and the rotational speed of the plates, so
cranking the plates faster will allow for increased current. Faster cranking will
also minimize the effects of charge leakage throughout the system, providing
higher efficiency.
There is a resistance to cranking due to the coulomb repulsion of the
collectors with incoming sectors. The collector is always charged to nearly the
same potential as the sectors because of the constant transfer occurring. If we
model the incoming sector as a point charge Q and let the width of a sector
be w, the distance from the plates to the collectors be b, and the spacing
between sectors be d, then the Coulomb force is
where x is the distance of the center of the incoming sector from the collector.
To determine the average force over one repetition, we integrate from (d-w/2) to
0 and divide by the distance (d-w/2), resulting in
Doubling this quantity since there are two disks with the same effect, and
doubling again since there is another sector of opposite charge that is being
pulled away, and multiplying by the radius of the disk yields
which is the amount of torque that the user must supply to the crank.
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Image Credit: Electrostatic Induction from Ohanian, Physics Volume II
Prof. Douglas Cline
http://www.coe.ufrj.br/~acmq/whyhow.html
http://madsci.org/posts/archives/may98/895429674.Ph.r.html