A Brief Introduction to Arc Discharges / Plasma Physics 
Depending on how
sophisticated of a design is needed, this material can be relatively
simple or full blown rocket science. So as not to scare anyone away,
I'll start with the simple and easy to understand material and work on
from there.
First,
let's define some terms.
 Neutral Atom 
This is an atom which is electrically neutral, and has the same number
of protons (+1 charge) and electrons (1 charge.) At room temperature
most gases are neutral, or have bonded with other atoms to form
neutral molecules.
 Ionized Atom  An
atom which receives a certain amount of energy may have one or more
electrons knocked out of its orbit thus ionizing it. This results in
a net positive charge on the atom. An atom that looses one electron is
singly ionized, two electrons is doubly ionized, and so on.
 Ion  An ion
refers to an atom or molecule which has either lost or gained an
electron, and as a result is electrically charged. For the purposes of
this discussion, negative ions (an extra electron) are typically not a
significant contributor to the processes described here. Ions can be
formed by collisions with other particles, strong electric fields, or
radiation.
 Ionized Gas  A
gas in which a significant fraction of its atoms are ionized.
 Plasma 
Plasma is a gas which is ionized to the point that it exhibits
significant electrical conductivity and is characterized by high
temperatures.
 Arc
Discharge  This refers to an electrical current that flows through an
ionized gas / plasma. The current must be large enough to selfsustain
the ionized gas through which it flows. It also commonly referred to as
an electrical spark.
Let's start with
something simple like an electric spark.
If you take two wires and put a high enough voltage across them, a
spark will jump between the two.
Fig. 1: Electric Spark
But
how high of a voltage is required to get a spark across a certain gap?
It turns out that this isn't exactly the question we should be asking,
instead we need to know the electric field strength across the gap. An
electric spark will occur if the electric field strength exceeds a
certain value. The electric field is measured in volts per meter, and
is computed by dividing the voltage across the gap by the distance of
the gap. So, for a gap of 1cm with a voltage of 100V, the electric
field strength is 10,000 V/m. An electric field of approximately
3,000,000 V/m is required to cause a spark, so this 100V across 1cm
would not result in a spark  but 30,000 V across 1cm or 3,000V across
0.1cm would.
Now we know how to get a
spark, but why is it that a spark only forms
if the electric field strength is very high? Well, let's look at how an
electric field effects charged particles first. It turns out that an
electrically charged particle in an electric field will feel a force
equal to
Eq 1: Force on a
Charged
Particle in an Electric Field
where
F is the force in Newtons, q is the charge in Coulombs, and E is the
electric field in volts per meter. Also, from physics we know that a
force on a mass will cause an acceleration as related by
Eq. 2:
Relationship
Between Force, Mass, and Acceleration
where
F is the force in Newtons, m is the mass in kilograms, and a is the
acceleration in meters per second per second. Accordingly, placing an
electric
charge in an electric field will cause a force on the charge resulting
in an acceleration in the direction of the electric field. Let's
imagine first an electron is placed
in an electric field of varying intensity.
Fig. 2: Electron
in a
Varying Electric Field
Let's
start with a voltage of 0 Volts across the gap. This results in an
electric field of 0 V/m, and substituting this value into equation 1
results in no force applied to the electron, and it therefor will not
move. Applying a small voltage of 10mV will result in an electric
field of 1 V/m. If we were to follow the substitution through equation
1 and 2 we would find that the electron accelerates at some rate
towards the left. Increasing the electric field by a factor of 10
results in the electron accelerating 10 times more quickly, and so on.
Keep in mind that an electron moving at a certain velocity has some
kinetic energy, as given by
Eq. 3: Kinetic
Energy of
a Moving Mass
Overall,
applying a stronger electric field results in any electrically charged
particles in the field accelerating more quickly, and therefore
possessing more kinetic energy.
Now, let's get back to
how this relates to creating a spark in a gas
across some gap. If we were to look at a gas on an atomic level we
would see mostly neutral molecules and some few electrons. If no
electric field is applied there will be no net movement of the
electrons, but at very weak electric fields the electrons will begin to
migrate as shown. [1]
Fig. 3: Electrons
Begin
to Migrate Across Gap
In
weak electric fields (i.e. low applied voltages) only a very few
electrons will make it all the way across the gap as most will collide
with neutral gas molecules. Increasing the field strength will result
in more free electrons making it across the gap, until at a certain
voltage all of the free electrons will cross the gap. At this point,
increasing the voltage further will not result in more current because
all available free electrons are already crossing the gap. This sort of
discharge is called the First Townsend Discharge region, and can carry
only minuscule currents up to a certain saturation current value. The
current characteristic for this type of discharge is shown below.
Fig. 4: Current
Flow Vs.
Applied E Field for 1st Townsend Discharge
As
the applied electric field is increased, the electrons will increase in
speed and energy. At some point they will acquire enough energy to
knock
additional electrons off of neutral molecules that they collide with as
they cross the gap. These additional electrons will also be accelerated
across the gap and the current will begin to increase again. This sort
of discharge is called the Second Townsend Discharge region, and is
also limited to very small currents. [2]
Fig. 5: Electrons
Ionize Molecules Resulting in Additional Current
Through
the Second Townsend Discharge region the positive ions created by
electron collisions will move slowly in the opposite direction of the
electrons toward the cathode, their lower velocity due to their higher
mass. In the Second Townsend Discharge region these ions do not
contribute significantly to the processes described so far. Up to this
point the current that the gap can carry is a strong function of the
number of naturally occurring free electrons in the gap. Since the
presences of these free electrons is in no way related to the electric
field, first and second Townsend discharges are essentially
externally fueled and are therefore not selfsustaining.
At some higher electric
field strength the positive ions flowing toward
the cathode being to play a critical role. This occurs when the
electric field is strong enough to give the positive ions sufficient
energy that when the collide with the cathode they deposit enough
energy to free electrons from the cathode surface. These free electrons
are then swept towards the anode, creating electron and positive ion
pairs which then repeat this process. [3]
Fig. 6:
SelfSustaining
Discharge Creates its Own Electrons
The
critical process here is the creation of electrons at the cathode's
surface. At some field strength this process becomes efficient enough
to
supply all of the electrons crossing the gap, removing any dependency
on outside sources of current carriers. This is the critical point at
which the discharge quickly transitions to an arc discharge, or a
spark.
By observing the voltage
across an arc discharge we can identify three
key regions. Much of the voltage drop occurs across two invisibly thin
regions directly adjoining the anode and cathode, which are referred to
as
the anode drop region and cathode drop region respectively. The
positive column occupies the remainder of the gap. [4]
Fig. 7: Voltage
Distribution as Function of Distance Into Spark
For
a given current through an arc discharge, the power dissipated in the
cathode drop region (arc current times cathode drop voltage) serves to
heat the cathode and cause it to emit electrons, the power dissipated
in the anode drop region (arc current times anode drop voltage) goes
to heating the anode by electron bombardment (but serves no useful
purpose), and the power dissipated in the positive column maintains the
electron and ion concentrations that support the arc current.
The arc current also
causes magnetic constriction of the arc column,
which confines the current to a narrow crosssection. The magnetic
field created by the arc current interacts with the flowing charges
which creates a force that pushes inward on the arc and counteracts the
tendancy of the heated gas to expand. First, let's look into how
magnetic fields and moving charges interact. A simple example is a
charge moving through a magnetic field which is perpendicular to the
direction of motion.
Fig. 8: Force on a
Charge Moving Through a Magnetic Field
Here,
the charge experiences a force perpendicular to the magnetic field, and
perpendicular to the direction of motion of the charge (the right
hand rule
is any easy way to remember this.) Next we need to establish what the
magnetic field induced by the arc current looks like. In general, a
straight conductor carrying a current will generate a magnetic field
with lines circling the around the conductor.
Fig. 9: Magnetic Field
Around a Cylinderic Conductor
This
is also true when the conductor is a plasma, so if we take into account
the interaction between the magnetic field and the flowing charges we
can see the source of the magnetic constriction.
Fig. 10: Magnetic
Constriction Force
By looking at only a
small portion of the surface of the discharge we
can see that it closely resembles Fig. 8. The interaction of the
magnetic field and motion of the charge results in a force on the
charge towards the center of the conductor. The force is given by:
Eq. 4: Force on a Charge Moving in a
Magnetic Field
where F is the force in
Newtons, q is the charge, v is the velocity of
the charge, and B is the magnetic field. Note that the magnetic field B
is the result of the entire current flowing through the conductor, so
as the current increases so does the magnetic field strength. The
stronger magnetic field in turn results in a greater compressive force
which will tend to reduce the cross section through which the current
flows. The smaller cross section also results in a stronger magnetic
field and greater compressive force.
If these forces were the
only ones at work in an arc discharge, the
magnetic constriction would crush the discharge into an infinitesimally
small cross section, but of course this is not the case since these
compressive forces are opposed by thermal expansion from the very hot
gasses in the arc discharge. In fact, increased current in an arc
discharge will result in increased heat generation which will oppose
the increased compressive forces. The exact dimensions of an arc
discharge are therefore the result of balancing the thermal expansion
due to heating, heat loss due to conduction and radiation, and the
magnetic constriction resulting from the moving charges and magnetic
field.
References
[1] J. D. Cobine, Gaseous Conductors,
New York:
McGrawHill Book Company,
Inc., pp.
143
[2] J. D. Cobine, Gaseous Conductors,
New York:
McGrawHill Book Company,
Inc., pp.
144
[3] J.
D.
Cobine, Gaseous Conductors, New York: McGrawHill
Book Company, Inc., pp.
213
[4] J. D. Cobine, Gaseous Conductors,
New York:
McGrawHill Book Company,
Inc., pp.
299
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