Electrostatic potential
I'm not enough of a mathematician to see or visualize why this
is true (other than demonstrating that the curl of a gradient is
zero). We'll just have to accept that on faith.
Actually figuring out what the scalar function needs to be is
another problem.
The curl of the radius vector r over r3
(magnitude of r cubed) term in the expression for electric field
is zero.
The scalar function in this case is - (1/r)
We'll insert this into the
expression for electric field.
It is often much simpler to determine the electrostatic
potential because it is a scalar quantity. The
electric field can then be determined by taking the gradient
of the potential.
We'll
probably only look at the animation below in class.
When the curl of a vector field is zero, the vector
field is irrotational. There's a pretty good
online
animation that shows both irrotational and
rotational vortices (spinning fluids).
The pattern of electric field vectors around a
positive charge would look something like the figure
below. We can imagine that the arrows represent
fluid motion and can picture what would happen if a
small object were placed in the moving fluid (as was
done in the animation above). It would move
outward without any rotation.
An irrotational field has another important
property. A line integral from from one point to
another will be independent of the integration
path.
Because the curl is zero we can use Stokes' Theorem to
say the line integral of the vector around a closed loop
is also zero. The rest of the argument follows
fairly simply from that.
Sometimes rather than starting with Φ and
then determining the E field, we might know the E field
and want to determine Φ.
We can determine the
potential as shown above (there's no reason rref
needs to be the upper limit, it could just as easily be
the lower limit)
Let's assume a point charge and substitute in an
expression for E into the left integral above.
We'll set this equal to the earlier expression
We get our earlier result (provided we assume that Φ(r = ∞)
is zero)
We can write the
electric field as the gradient of the electrostatic
potential and then substitute that into Gauss' Law.
We obtain Poisson's Equation. Laplace's equation
applies in situations where the volume space charge
density is zero. We'll be using Laplace's equation
in our next lecture. Here is a handout
with vector differential operators (Laplacian,
curl, gradient and divergence) in cartesian,
cylindrical, and spherical coordinate systems.
"Fast" and "slow" E field antenna systems
Another example of a field mill record, one hour of actual
thunderstorm and lightning fields recorded at the Kennedy Space
Center, is shown below (from: Livingston and Krider
(1978)). The abrupt transitions are caused by
lightning and are superimposed on a static field of about 3
kV/m (negative potential gradient corresponds to a positive
E field pointing upward toward negative charge in the bottom
of a thunderstorm).
A field mill can be used to determine
when a thunderstorm becomes electrified and monitor
electrical activity in a thunderstorm. Note
that a fairly large dynamic range (-12 kV/m to +12 kV/m) is
needed to insure that the E field remains on scale.
Later in the semester we will see that measurements of the
lightning field changes like these made at multiple
locations can be used to determine the magnitude and
location of the charge neutralized by the lightning
discharge.
A cloud-to-ground (CG) lightning discharge lasts about 1 second
or so and appears as just an abrupt field change on the record
above. We're going to find that a lightning discharge
consists of a series of different processes that occur on
millisecond, microsecond and even sub microsecond time
scales. The figure below illustrates this (don't worry
about all the details and names at this point, we'll come back
to this later in the semester). An electric field mill can
measure static and slowly varying fields but wouldn't faithfully
resolve all the field changes and variations that occur on these
faster time scales. We need a different kind
of measuring system.
One way of measuring these faster time varying electric
fields is to use a flat plate antenna (aka flush plate dipole
antenna). It basically consists of a large flat grounded
plate that would be positioned on the ground (ideally flush with
the surrounding ground). A smaller circular insulated
sensor plate is found inside a center hole as shown in the
photograph below (the antenna is on the classroom floor in this
photograph). The center plate "senses" the
electric field and is insulated from ground.
We look under the top plate of the antenna in the next
picture.
The center sensor plate is supported on 3 insulating nylon or
Teflon spacers (frosty white color). The top end of the
supports are covered with "rain hats" (empty, silver colored
catfood cans) to try to keep the insulators dry during rainy
weather. A wire connection to the center plate connects to
a coaxial cable to carry the signal to processing and recording
equipment.
In some ways the operation of this antenna is similar to the
field mill. In this case a time varying E field causes
current to flow to and from the center sensor plate (you don't
need to repeatedly cover and uncover the sensor plate).
This current is proportional to the time derivative of the
electric field (σ in the
equation below is the surface charge density on the sensor
plate).
Integrating the current gives an output signal that is
proportional to E.
In the circuit above the antenna is connected to a capacitor,
this is a passive integrator. Some kind of measuring
device would then be connected across the capacitor.
The value of the RC decay time constant determines whether the
antenna works as a "slow" or a "fast" antenna system.
A Slow E field system has a long, 1 to 10 second
decay time and would be appropriate if you wanted to study
an entire lightning discharge with faster time resolution
than a field mill would provide. A couple of actual
Slow E field records are shown below (also from the
Livingston and Krider (1978) paper cited earlier).
Note first the much faster time scale,
0.4 seconds full scale in this case. What
would appear as a single abrupt field change on a
field mill recording has been spread out in time.
The step changes in the E field are
lightning strokes to the ground. The top example shows
a 3 stroke cloud-to-ground discharge. The second
discharge has 4 strikes to ground.
Because the Slow E antenna system does not have DC response
the static E field (which can be several kV/m) is
effectively filtered out (like switching from DC to AC
coupling on an oscilloscope). Lightning field changes
can be examined with higher gain. Because the signals
of interest last from 0.5 to 1 second, a decay time constant
of 10 seconds would be appropriate here. The
slow E field would decay back to zero during the interval
between lightning discharges.
To give you some appreciation for how recording methods
have changed, the signals above (recorded in the 1970s) were
(I believe) displayed on a storage oscilloscope and
photographed with a Polaroid film camera!
Because of the long decay time constant, charged
precipitation falling on a Slow E field antenna can drive
the signal off scale. Inverted antennas are sometimes
used to avoid that problem.
This antenna is mounted on the roof of the Penthouse
atop the PAS Building. Because of its exposed position
and the chance that it could be struck by lightning, signals
are brought into the Penthouse on a fiber optic cable.
The E field variations that occur during a
individual return stroke could be examined by increasing the
vertical gain and displaying the Slow E field signals on an
even faster time scale. However, as sketched below,
the long time decay would mean the signal might not decay
back to zero in the interval between return strokes.
A slow E field record (a
few seconds long decay time constant) displayed on a much
faster time scale.
The solution is to shorten the RC decay time constant.
This turns the slow antenna into a fast antenna system.
Same antenna but with a much shorter (milliseconds long) decay
time constant.
A decay time of about 1 to 10 milliseconds would be
long enough to accurately record the Fast E field variations
for the first 100 μs or so but would allow the signal to
decay back to zero in the interval between strokes. A short
decay time constant would also mean charged precipitation
would be less likely to drive the Fast E signal off scale.
The operation of the flat plate
antenna was demonstrated in the last few minutes of class.
The setup is shown below
We first need to create a time varying
electric field of known amplitude. We do this by
placing a flat metal screen on insulators a known distance
(0.125 m) above the top of the antenna. A square wave
signal from a function generator is connected to the test
plate.
The electric field created by the test screen is just the
voltage on the screen (10 volts) divided by the distance
between screen and the antenna (0.125 m); 80 V/m.
The test signal and the antenna output signal are shown on
the oscilloscope displays above. The test signal was a 0
to 10 volt square wave and is displayed on a 5 volt/div scale
above. The noisier antenna signal is shown on the second
(lower) channel. Note the exponential decay of the signal
that occurs because the 1 M Ω input impedance of the
oscilloscope is connected in parallel with the 1000 pF
integrating capacitor. The signals are being displayed on
a 1 ms/div time scale. A closeup view of the oscilloscope
display is shown at right.
Let's first make sure we understand why the antenna signal is
positive.
+ 10 volts on the plate above the antenna produces a downward
pointing E field of 80 V/m. This causes negative charge to
be induced on the antenna sensor plate, meaning meaning the
direction of the antenna current is from the antenna to
ground. This produces a positive going output voltage on
the oscilloscope display.
We'll do a quick calculation of the voltage that we would expect
from this setup and see how well it agrees with the actual
measurements.
We would expect a peak signal of about 80
mV, we seem to be getting about 60 mV. That's not too
bad. I suspect the reason for the difference is that
there may be some stray capacitance between the antenna and
ground that we would need to add to the 1000 pF in the
integrating capacitor.
As a final test, the antenna was connected to a small (50
pF) integrating capacitor. That should increase the
amplitude of the antenna signal and shorten the decay time
constant.
The oscilloscope display above confirms
this. Though again the signal isn't as large and the
time decay as short as calculations would predict.
Stray capacitance is probably again the reason for the
discrepancy. The stray capacitance may well be larger
than the 50 pF integrating capacitor that we are using in
this test.
References:
J.M.
Livingston and E.P. Krider, "Electric Fields Produced by
Florida Thunderstorms," J. Geophys. Res., 83, 385-401, 1978.
