But there many notes that students should be read in preparation.
The notes below are intended mainly for the Phy262L students who have to do extended reports for this lab.
But Phy112 students can profit from the notes too even though they don't have to do some things like the error analysis and the log-log plot.
The readings below must be read completely the lab period to get the prepartion mark. There are NO part marks for having read parts of the readings.
Yes, it is a lot of preparation readings. But the actual lab manipulations are quite short. You should have plenty of time in the first week to get a good set of data, create the 3 required figures, do the error analysis, and divide up the report writing tasks among the group members. There is only one lab report per group. In the second week of the extended lab, there will be lots of time to retake any data if needed, complete the lab report, proofread the lab report, and hand it in by the following Monday, February 14, 5pm (either to me, under my door, or by email attachment).
Remember this lab is 15 % of the course grade in itself. So it's a big deal in itself. It's also a big deal in learning error analysis and report writing skills---which is the main raison d'être for the lab.
These articles help you to understand how a field mill works. Then read all 3 pages of About Electric Field Mill Operation.
You must do 2 shielding measurements. The manual just says one.
First, try a neutral ungrounded metal plate. The metal plate with the plastic holder can be used for this. Holding the metal plate with your hand grounds it.
Second, use an ungrounded metal plate. Holding the plate with your hand grounds it.
Explain the different results in your lab report.
But they are designed for course laboratories, not professional work, and so are intentionally simplified devices and don't have a lot of options.
One peculiarity is that the read-out is in kilovolts even though field mills measure electric field, not electric potential.
Recall:
Delta V = - integral (vec E)*(dvec s)
where (vec E) is the electric field
and
(dvec s) is a differential displacement vector.
What the read-out means is that if the measured
electric field is constant and extends 100 mm and
points in the direction of the extension, then
the read-out electric potential would be
the change in electric potential over that 100 mm.
So to convert the read-out to electric field, one uses the formula:
V_read-out
E = ---------------- ,
100 mmm
but this just gives E in MKS units of kV/mm.
To convert to V/m (which are the MKS units
of electric field), one uses the formula
V_read-out
E = ---------------- * con1 * con2 = V_read-out * con .
100 mmm
Find conversion factors
(AKA factors of unity) con1, con2, and con.
.
Where is the electric field actually being measured? Well the manufacturer has declined to specify, but about 0.3 +/- 0.1 cm below the rotating shutter of the field mill seems to be the location of the sensor plate. So that is the location of the measured electric field---as far as I can tell.
Electric field is a vector field. So what is the direction of the electric field measured?
The read-out is positive for a downward field (i.e., a field pointing straight down along the axis of the rotating shutter).
The read-out is negaiive for a upward field (i.e., a field pointing straight up along the axis of the rotating shutter).
What if the electric field is not aligned with the axis of the rotating shutter? Some partial measurement of the electric field that would be present at the sensor plate in the absense of the field mill is being done, but it's not clear exactly how to interpret that partial measurement---the manufacturer leaves us guessing.
So for quantitative measurements, it is best to make sure the electric field is aligned with the axis of the rotating shutter.
In this experiment, the graphite-covered ball is aligned with that axis for all quantitative measurements.
Just to give you comparison values for electric field, the natural electrical field of the Earth is about 150 V/m near the surface and points downward and the electrical breakdown field of air is about 3*10**6 V/m.
The former value is pretty small and perhaps negligible compared to the error in the electric field in our experiment. When the latter value is reached, one can get an electrical discharge which can happen a from static electricity or when a circuit broken or in the form of lightning. The electrical breakdown field of air will not be reached in the measurments.
kq
(vec E) = ------- (hat r) ,
r**2
where k=c**2*mu_0/(4*pi) exactly (Wkipedia: Coulomb's law)
approximately 8.987 551 787 * 10**9 N m**2/C**2
approximately 9 * 10**9 N m**2/C**2
approximately 10**10 N m**2/C**2,
q is charge in coulombs,
r is distance from the center of the distribution to the point of evaluation,
(hat r) is a unit vector point from the center of the distribution to the point of evaluation.
In this experiment, we are only interested in the magnitude of the electric field,
and so will be studying the formula
kq
E = ------- .
r**2
In our case, the spherically symmetric charge distribution is charge on the graphite covering of a ball.
Graphite is a conductor and any charge put on the graphite will spread into a spherically symmetric distribution---which we know by symmetry and mutual repulsion of like charge---unless the system is perturbed from spherical symmetry by an external electric field.
There are two procedures of charging the ball.
What is sign of the charge on the ball?
Well click on the triboelectric effect and determine what charge the styrofoam gets and that is the ball's charge.
.
We just hold the styrofoam near the ball and ground the ball by touching it with our hand while the hand also touches a good ground like the top of the field mill.
You remove your hand before removing the styrofoam.
What kind of charging is this?
What is the charge on the ball in this case?
< blink>ANSWER.
So use the second procedure, despite what the manual says.
The electric field is measured using the field mill---only take the absolute value---the sign does NOT matter to the analysis of the formula.
If you charge the ball they I asked for, it should have positive charge and give a positive reading on the field mill.
Convert the electric field read-outs to MKS units using the conversion formula derived above.
The distance measurements can be written down in centimeters, but for plotting convert to meters---or otherwise there is a heck of confusion in interpreting the plots.
The distance measurements are from the CENTER of the charge distribution to the location of the sensor plate of the field mill which is 0.3 +/- 0.1 cm below the top of the field mill---as far as I can tell.
You should use distances (in centimeters): 3, 4, and thereafter in 2 centimeter increments as far as you can go which is about 30 centimeters. NOTE I ask you to go closer than the lab manual recommends.
You need to estimate an error (formally an uncertainty) for the distances.
In this lab, we use them.
I'd guess it is of order 0.3 cm---but you must decide from your own judgment about how well you are determining distance.
You must also estimate an error in the electric field measurements.
The first thing to do is to check if there is a background electric field when all charged objects are remote from the field mill.
There may be a significant one or not.
The actual background field should be of order natural electrical field of the Earth
(i.e., about 150 V/m).
You should try to field down to this level before charging the ball, but with the ball just above the
field mill.
Then discharge the ball to ground by touching the ball
and field mill top simultaneously.
The field may still be much larger than the 150 V/m.
In this case, try wiping the plastic handle may have
charge.
Since the handle is an insulator, grounding it will not remove
charge.
Wiping the handle with a water-dampened paper towel should remove this
charge.
Do NOT drop any alcohol or water into the field mill.
The actual electric field measurements
will also fluctuate, and so give an estimate of error.
You should generate a table something like the following:
All plots will be done with Graphical Analysis
(which is on the lab room computers)
or
a similar package if you know how to use one.
It's pretty straightforward to use.
There is a help feature.
Groups should share their expertise as they acquire it with
Graphical Analysis.
Plots can be printed out on the lab room printer.
Now for the figures:
The electric field (or its
proxy) is on vertical axis for all figures: it is the dependent variable---the one
do not control directly in taking the data.
The distance (or its proxy) is plotted on the horizontal axis: it is the independent
variable---the one you directly control in taking the data.
The data points will just get a curve that rises as r gets small.
The curve should be qualitatively consistent with the
electric field formula
being tested: i.e.,
Graphical Analysis will try
to join the points by straight line segments or give a best fit line.
Turn those features off.
The straight line segments and best fit line are not useful in the analysis of the
data since they are not the theoretically expected behavior.
If the data is linear, your data is very bad.
If
Graphical Analysis will do a curve
fit to the data, then try that.
If not, just draw a smooth free-hand curve that fits the data within
error bars (see below).
Data points and error bars:
For Figure 1 a representative data point with
error bars
suffices.
A representative data point is an artifical data point isolated
from the real data points and labeled representative data point
on the plot or in the figure caption.
The error bars give
the size of the
errors
in the two graphical dimensions.
For example of date points with,
error bars see
TGraphErrors Example.
Lots of other examples can be found by googling Images for
error bar.
What if the
error bars are too small
to plot.
Then you don't plot them, but mention that they are too small to plot in the text
or figure caption.
For example, say that you had a theoretical formula
To the eye, a plot of y versus x is just a curve that is
qualitatively consistent with the formula.
But what if the data were really following a y=ax**2
formula.
How could you tell?
There are various ways.
Plotting y versus x**3 is a standard way.
You have linearized the data.
You should get a straight line with in
error
if your data actually obeys the formula.
The slope is what?
The intercept is what?
For another example, say had the theoretical formula
Now in our case, the theoretical formula is
Theoretically, what should be the slope
and what should be the intercept?
.
Graphical Analysis
will plot a best-fit line for your linearized data and will give the
best fit slope and intercept and a
Pearson product-moment correlation coefficient.
Use only the data points that fit the linear trend.
We know that when the ball gets too close to the sensor plate that deviations from
electric field formula
for a spherically symmetric charge distribution may occur.
Why do we know this? Which way do we expect the deviations to occur: up or down?
You should explain this in the report.
We can also expect deviations when the distance r gets too large.
Why? Explain in your report.
But as far as I can tell
Graphical Analysis will not give
errors for the
slope and intercept.
The correlation coefficient
is a measure of how well the data is fit by a line, but it actually means takes more explanation than
I can go into here.
If the data fits a line perfectly,
the correlation coefficient
gives 1 and if the data are completely uncorrelated it gives 0.
But in between 0 and 1, the
correlation coefficient
is ambiguous actually and requires interpretation.
Is a value of 0.8 good or bad for a linear fit?
It depends actually.
To give an extreme example,
if you have just two data points, they are fit by a line perfectly
and you get a
correlation coefficient of 1.
But the data may not come from linear data population at all.
Usually, if you have a fair number of data points and
the correlation coefficient is
greater than 0.9, one would say the data is fairly linear.
But there is still no quantitative judgment without a bit more work.
So I require worst-fit lines drawn by hand.
What you do is put by hand error bars
on a few of the data points that span the full range of data that best follows the linear trend---if any does at all.
Say the initial variable is x and the linearized variable is z=1/sqrt(x).
The error in the linearized variable
can be calculated in several ways, but a simple way is
The procedure is a bit tedious, but it works well enough.
Remember that errors
usually have only one significant figure.
You should avoid using data points that are obvious outliers---they are probably some extreme error
in measurement: e.g., you wrote down the wrong number.
Then draw two worst-fit lines that lie within the
error bars of the data points that follow the linear trend.
One has the lowest slope possible consistent with
error and the
other the highest.
The slopes and intercepts of the worst-fit lines allow you to estimate
errors for the slope and intercept of
the best-fit line.
If the best-fit line slope and intercept agree within estimated
error
with the theoretical predictions, then the theory has been confirmed.
If not, then not.
There are marginal cases, of course.
You should discuss what you have found.
If you don't get agreement within error, then
you should say so.
You should try to explain this (away).
Using worst-fit lines is a crude error
analysis technique, but it works.
This is plot I require in addition to those required in the manual.
For example, say that you had a theoretical power-law formula
.
The slope is the power of the power-law formula.
The intercept is the logarithm of the coefficient of the formula.
Now in our case, the theoretical formula is
Theoretically, what should be the slope
and what should be the intercept?
.
You must do a best-fit line and worst-fit line analysis for Figure 3
similar to that done for Figure 2.
Use only the data points that fit the linear trend.
We know that when the ball gets too close to the sensor plate that deviations from
electric field formula
for a spherically symmetric charge distribution may occur.
Why do we know this? Which way do we expect the deviations to occur: up or down?
You should explain this in the report.
We can also expect deviations when the distance r gets too large.
Why? Explain in your report.
The manual specifies most of the ingredients.
But I can add a few more points.
To get the background field in this case, first
move all charged objects away.
Actually, before any measurements, clean the plastic rod with alcohol.
The background field value will certainly fluctuate and fluctuations give an estimate of the
error in the background
electric field.
Table of Data
------------------------------------------------------------------------
distance E-field E-field-E-field_background
(m) (V/m) (V/M)
------------------------------------------------------------------------
0.010 +/- 0.003 (10.0 +/- 0.5)*10**4 (9.8 +/- 0.7)*10**4
0.020 9.7*10**4 9.5*10**4
0.030 9.4*10**4 9.2*10**4
and so on.
The numbers given are just representative, not actual measurements.
I've assumed a background field of (0.2 +/- 0.2)*10**4 V/m.
The correction for the background field is
(E +/- dE) - (E_background +/- dE_background) = (E-E_background) +/- (dE+dE_background)
where ``d'' indicates an error value.
Your last significant figures
should NOT be smaller than your
error
Everyone's data will be a bit different because the balls will all
have different charges.
The errors
are only given in the first row if they are all the same in a column---which they
should be in this case.
You will have to write the
errors in by hand
since you will be using
the Graphical Analysis software
(unless you use similar package)
which does not apparently support
errors---but maybe
it does and I've just not figured it out yet.
The middle column doesn't have to be presented, but you should say what your
background E-field is and what its
error in your report.
kq
E = ------- .
r**2
A by-eye analysis cannot say more.
But you should say whatever you can about the figure.
Graphical Analysis will
put on the data points, but unless it has more capabilities than I know of,
it will not put on
error bars.
y=ax**3 .
You took data to test this formula.
y=a/sqrt(x) .
To linearize this formula you plot y against what?
kq
E = ------- .
r**2
So to linearize what should E be plotted against?
But you probably shouldn't use all the data points for the best-fit line.
The best-fit line is determined by a
least squares for a line algorithm.
You have to do a little work to get the error bars
for linearized variables.
Delta z = (1/2)*(|1/sqrt(x-dx)-z|+|1/sqrt(z+dz)-z|) .
y=ax**p .
Take the base 10 logarithm of both sides to get
log(y)=log(a) + p*log(x) .
What is the slope?
What is the intercept?
kq
E = ------- .
r**2
So on a log-log plot what should be ploted against what?
Just as for Figure 2,
you probably shouldn't use all the data points for the best-fit line.
