Friday Feb. 6, 2015
A "hardcopy" of the notes that follow
will be provided in class.
We'll spent the entire class examining ion counters and
conductivity meters. A crude and quickly assembled
conductivity meter will be demonstrated at some point in class.
Ion counters measure the concentration of small ion charge
carriers in the air (as we will see they can only measure one
polarity of charge carrier at a time). A conductivity meter
measures the air's conductivity. And, again, it isn't able
to measure the total conductivity because both positively and
negatively charged small ions make the air conductive. A
conductivity meter is only able to measure the contribution that
small ions of a single polarity make. A measurement of total
conductivity would require operating two conductivity meters at
the same time.
Ions counters and conductivity meters often make use of the
cylindrical capacitor geometry shown below. The outer
electrode is connected to plus or minus voltage (V), the inner
electrode is kept at ground potential.
If we assume that we are
looking a just a small section in the middle of a much longer
cylinder (i.e. staying away from the open ends), the E field
will have just a radial component, Er. The solution
above, obtained using Gauss' Law, is the same as the field
around an infinitely long line of charge with uniform line
charge density (the charge is induced on the inner, grounded,
conductor when the outer cylinder is raised to potential
V). A caution:
λ above represents line charge
density (Coulombs/meter). We'll later be using λ
to represent conductivity.
Next we will try to find an expression for the
electric field in terms of the potential difference between
the two cylinders.
If the outer cylinder is connected to positive potential,
the grounded inner cylinder will have negative charged induced
on it. The E field will point inward as shown in the
figure above.
It is a relatively easy matter to determine the capacitance
Let's look first at how an ion counter works. An ion
counter will measure the concentration of small ions in the
air (the concentration of ions of one polarity).
The electric field will cause a positively charged
small ion entering the cylindrical capacitor at the left to
drift toward the inner conductor with a drift velocity (vd).
In time dt, the small ion will
drift a distance dr
(Point 1). The drift velocity, Point 2, is just the
electrical mobility, Be,
times the electric field. T at Point 3 is the time it will take the
small ion to drift from the outer cylinder to the inner
cylinder.
Now in order to be "counted" the small ion must make it to
the center conductor before it travels a distance L, the
length of the cylindrical capacitor.
This means (Point 4) that T must be less than L/u, where u is the speed at
which the air is traveling along the length of the
cylindrical capacitor (we assume u is uniform, that there
is no dependence on r).
Mobile ions are more likely to make it to the center
electrode. So another way of looking at this is in
terms of a critical mobility, Bc.
Ions with a mobility Be greater than Bc will be collected,
the others won't. The volumetric flow rate is an
easier parameter to measure than the horizontal
speed. So we can rewrite Bc in terms of flow rate.
We want Bc to be
small so that all of the small ions have a mobility
greater than Bc
and can be counted. Clearly the lower the flow rate
and the longer the tube, the more time the small ions will
spend in the capacitor and the more likely they will be
collected. Increasing the potential difference
between the two cylinders will increase the strength of
the electric field and the inward drift velocity of the
small ions.
We'll assume that all of the small ions of one polarity are
collected by the center electrode as they pass through the
cylinder. The current flowing to the center electrode
would then be the product of the small ion concentration,
the charge per small ions, and the volumetric flow
rate. Later in this lecture we will look at
instrumentation that could be used to measure this (small)
current.
When functioning as a conductivity meter, only the small
ions in a portion of the volume of air flowing through the
cylindrical capacitor are collected (the green shaded volume
in the figure below).
On an earlier figure (Point 3) we determined the time, T,
needed to travel from the outer electrode to the inner
electrode (from b/2
to a/2).
We'll write down the expression again but substitute rc for b/2 (see Point 7 below).
Note a small error in the notes handed out in class in
the 2nd and 3rd equation below has been corrected.
The last term at Point 8 is the rate at which the green
shaded volume is flowing through the cylinder times N q. This is the
charge collected at the inner conductor per unit time and is
the signal current. Let's solve the expression for isignal and then try to
relate the signal current to conductivity.
We substitute in for
capacitance and next we recall that N Be q is
conductivity.
There is a linear relationship between isignal and V. The slope
of a plot of isignal vs
V should
provide an estimate of conductivity. And
again because we collect only one polarity of small ion
we aren't measuring the total conductivity. The
total conductivity depends on charge carriers of both
polarities.
The following figure shows a conventional ion
counter/conductivity meter design and an op-amp circuit
that could be used to measure the signal current
(Vout = signal current x R).
Note how the op-amp keeps the center conductor at
ground potential. Because the signal current is
very small, a large feedback resistor is needed in the
op-amp circuit (1013 ohms was used in
the "instrument" demonstrated in class).
When the outer cylinder is connected to +V as shown
above, positive charge carriers will move to the center
conductor and will produce the current signal shown
above. Because this is connected to the -
input of the amplifier we should see a negative polarity
output voltage. Similarly when the outer cylinder
is connected to -V, negative charge carriers will flow
to the center conductor, the signal current will reverse
polarity and the output voltage will be positive.
We did verify that this was happening with the crude
instrument that was operated in class. Here's a
summary of those measurements (the numbers below are
from the 2013 edition of the course, I'll update them
with 2015 numbers after class).
The meter was first operated with 0 volts connected
to the outer cylinder. The amplifier output
voltage was 0.8 v. When the outer cylinder was
connected to +20 v, the output was -2.0. This is a
net change of -2.0 - 0.8 volts = - 2.8 v.
An output voltage of +3.0, a change of 3.0 - 0.8 = +2.2
volts, was observed when the outer cylinder was
connected to -20 volts.
Thus the instrument was operating more or less as
expected.
The photograph below is from a paper describing
measurements of air conductivity under thunderstorms in
Florida ("Ground
Level Measurements of Air Conductivities Under
Florida Thunderstorms," R.J. Blakeslee & E. P.
Krider, J. Geophys. Res., 97, 12947-12951, 1992).
Three conductivity meters were operated
simultaneously. One was connected to positive
voltage, one was grounded, and one was connected to
negative voltage, much as we did in the class
demonstration.
The sensors were kept in a wooden box (uncovered in
the photo) to protect them and the electronics from
rain. Air was drawn in through the three tubes at
right.
In this last figure we get a better idea of how this
instrument can function as either a conductivity meter
or an ion counter and how it transitions from one to the
other.
For a given rate of air flow through the cylindrical
capacitor we monitor the signal current as the potential
difference between the outer and inner conductors is
increased. As V increases small ions in a growing
volume of air are collected and measured. The
signal current increases. Eventually all of the
small ions are collected and the signal current flattens
out (saturates).
The slope of the linearly increasing, early portion
of the plot (shaded green above) provides an estimate of
conductivity. The amplitude of the
signal current, once it has flattened out (blue) can be
related to small ion concentration.
A common household ionization-type smoke detectors is
really just a very basic conductivity meter.
Alpha particles from
a small amount of radioactive Americium-241 ionizes
the air between two metal plates. The plates
are connected to a battery and the voltage
difference causes a weak current to flow between the
plates.
The current flowing through the ionization chamber
drops significantly when smoke enters the
chamber. This is because the charge carriers
quickly stick to any smoke particles that enter the
ionization chamber and suffer a large drop in their
electrical mobility. The drop in current is
sensed and used to sound the alarm.
The smoke detector was opened and the conductivity
chamber was placed near the opening of the class
conductivity meter. My thinking was that
some of the additional charge carriers created by
the Americium-241 source would be drawn into the
conductivity meter and we should be able to see a
larger output signal. That did seem to be the
case.
Here's a good
video explanation embedded in a
larger Wikipedia article.
An to finish off another example of another
conductivity "meter", the so-called Nu-Klear
Detector.
The instructions read "Shake gently until some
beads float. Seek shelter at once if all beads
drop. Remain in shelter until some beads
float." Shaking the device charges the red
beads in the center cylinder (probably another
example of triboelectric charging). If
ionizing radiation is present (such as would be the
case following a nuclear explosion) the air in the
cylinder would become conducting and would
neutralize the charge on the beads. They'd
fall to the bottom of the device. You
can read more about it here.