physics 163 lab manual
TRANSCRIPT
Physics 163
Lab Manual
Revised Fall 2021
This publication was developed with the support of a Hispanic Serving Institution STEM &
Articulation grant from the U.S. Department of Education.
Rule Why?
Plan on staying the entire 3 hours every time. It happens every time; if you leave I count you
absent.
Read the manual before class. You’ll go home sooner.
Get to lab on time. Lateness = absence, 2nd absence reduces course
grade.
Don’t interrupt during lecture. This saves time and keeps people focused.
Don’t use your cell. It distracts and slows down lab.
Don’t go to the bathroom and text/call people. You leave your lab group hanging and irritate me.
Rotate responsibilities (measuring, using Excel, etc). Everyone will need all basic lab skills in future
classes and on lab exams.
Use Excel formulas to compute your data. Mistakes are quick to correct; less time in the long
run; good job skill.
Don’t tear down apparatus until the end of class. I will check your data for errors and ask you to show
me how you did the experiment.
Check print preview before printing. Saves wasting paper.
Print out a single data sheet for me to look over. Ensures credit and prevents mistakes.
Make sure you understand the conclusion questions. So it is easier to write up later.
Make sure you were checked in an out before
leaving.
Leaving early is the same as an absence!
Write up the lab the same day you acquire data.
Turn in the lab at the appropriate time.
It is fresh in your mind, you get it over with, & that
is the standard practice for higher level labs.
Checklist for data tables:
• Times New Roman, 10 or 11 pt, variables are italicized, units are not italicized
• the δ symbol, the ∆ symbol, and parentheses are not italicized
• numeric subscripts are not italicized while letters in subscripts are italicized (m1 or mexp)
• the %precision and %difference do not have units (everything else does have units)!
• the cells which will contain data are centered with borders (Dom has more strict guidelines here)
• constants (the same for all experiments) are written at the top in a separate table
• prefixes, subscripts, and superscripts are appropriately used
• the number of sig figs is appropriate for all cells (based on measuring device, calculations, etc)
• error calculations only have one sig fig (I use two for numbers over 10%)
Checklist for graphs:
• gridlines removed
• graph has title using variables (in italics) but no
units in title
• axis labels have variables in italics with units
not italicized
• graph fills the entire field (graph size should be
about 1/3 to 1/2 a page)
• no legend for single set of data (a legend is only
used for multiple data sets on a single graph)
• if trendline is shown there is an equation with R2
value on the chart
• show the data as points only (no connecting
smooth line)
• for graphs with both theoretical & experimental data: theory is a smooth line (no points) while experiment is a
only points (no line)
• the graph and data table are sized such that it fits on a single sheet of paper (not always possible)
• everything except the legend has appropriate subscripts and superscripts
• title is y-axis label versus x-axis label
1
Table of Contents
First Day Activity (writing & plotting in Excel) 3
Coulomb’s Law 7
Measuring with DMM’s 9
Mapping the Electric Field With MATLAB (screenshots included) 13
Ohm’s Law and Resistivity 17
Series & Parallel Circuits 21
Characteristic of a Diode 31
B-Fields 33
Magnetic Fields with Oral Presentations 35
Motors 59
Oscilloscopes and Function Generators 65
RC Lab 75
Series LRC Circuit 79
RLC Filter Circuit 83
Radio Lab 85
2
3
PHYS 163 First Day of Lab Activity
The purpose of this activity is to help you learn required skills for writing up lab reports. For this assignment each
person must do their own work (no collaboration is allowed for computer work). Answer all numbered questions in
order on a separate sheet of paper. Answer in complete sentences; answer in way that makes rewriting of the
question unnecessary.
Checklist to turn in:
Two sentences describing the intro eqt’ns
A paragraph for the procedure
A data sheet with data & graph for the example of Sample Graph Type 2 (Exp & Th on same graph)
A copy of the graph from the example of Sample Graph type 1
Answers to the numbered questions in complete sentences.
Practice writing equations in your introduction In some lab you are asked to use the following equations:
∆ = − 12
(1)
= −−2∆ (2)
Write two sentences that properly explain the variables used in these two equations. Read the examples and
guidelines in the lab manual to see examples of how this can be done. A few points to consider:
• Explain the variables either immediately before or immediately after their first use.
• Do not re-define variables (e.g., once you properly define g and ∆y for the first equation, don’t do it again).
• Think: is ∆y distance, displacement, or something else in this case? Also, equation 2, written as it is,
implies that v is not speed but rather …
• Typically, you do not include units for the equations, that info will be obvious in the data tables!
• Try to make the equations in your introduction formatted just like a textbook.
• For the intro section, one can use present or past tense but you must use third person.
Practice writing the procedure section
Pretend you were given the following list of instructions in your lab manual. Use this information to write part of
your procedure. I am not interested in what units you used (that info is in the data tables already). I am interested in
your use of the 3rd person and past tense. Look in your lab manual for both guidelines and a sample report. This
should help you figure out what to do. Try to keep your sentences clear and concise. In general, it is better to use
two or three short sentences rather than one long sentence.
• Charge a rubber rod by rubbing it with fur.
• Touch the rod to the metal ring.
• Use the electric field sensor to measure the electric field at the center of the ring.
• Move the electric field sensor along the axis of the ring in 1.0 cm increments. Record measurements of the
electric field at each position.
• Plot the electric field versus position along the axis of the ring.
4
Graphing with both experimental and theoretical values (Sample Graph Type 2)
For this task, focus on using the equation given to you, not understanding the equation.
We will discuss in lecture this very problem and I will help you understand the equation
more at that time.
You are told you have a uniform ring of charge with radius R containing a uniform
distribution of charge q. You are given the fact that the magnitude of the electric field a
distance x from the center of the ring (along the axis of the ring) is given by
= /
The units of charge are given by Coulombs (C). Suppose you are given a ring of charge with a radius of 10.0 cm
and a charge of 5.00×10-12 C. The constant is called the Coulomb constant given by k=8.99×109 N·m2/C2.
In some hypothetical lab you obtain the following experimental data (put in the correct units for Eexp in your work).
All measurements of x were done with a meter stick precise to 5 mm. The field measurements were precise to 0.05
units. Use the formula above to generate the theoretical values in Excel (some are provided to check your work).
x (m) Eth (?) Eexp (?)
0.000 0.00 0.62
0.020 ? 0.96
0.040 1.4 1.40
0.060 ? 1.71
0.080 1.7 1.75
0.100 ? 1.69
0.120 1.42 1.55
0.140 ? 1.40
0.160 1.07 1.24
0.180 0.927 1.11
0.200 ? 0.99
Create a graph of like that in Sample Graph Type 2 in
your lab manual (see the table of contents for help). It
should look like the one shown with some small changes:
• put the appropriate labels on each axis (variable
italicized, unit not italicized)
• put a title (variables given as y-variable vs x-
variable with no units in the title)
Use the checklist in the front pages of your lab manual to
be sure you have everything looking good.
Tips: I had the x-axis scale run 0.000 to 0.210 to avoid
cutting off the last data point. To do this I right clicked
on the x-axis and clicked “Format Axis”. The error bars
can be added in using the Chart Layout Tab on the far
right side. I used fixed values of 0.05 for the vertical error bars and 0.005 for the horizontal error bars. Finally, I
right clicked on one of the experimental points, clicked “Format Data series”, and reduced the marker size down to 2
points. You should add in appropriate axis labels and a title. Double check the checklist before you are done!
x
R
5
Example of plotting data with only experimental values (Sample Graph Type 1)
Before starting this part of the assignment, look once more at the Excel sheet provided to you. Go to the bottom left
corner of the Excel file and click the tab for Sample Graph Type 1. This part of the assignment refers to the data and
graph shown on there.
A block is held above the ground and released from rest. In this case, the distance traveled (d) is given by
= 12
(3)
where g is the magnitude of the acceleration due to gravity and t is elapsed time.
Look at the graph shown in Sheet 2 of the provided Excel file. I purposely left the axes and title off of the graph to
make you think. What is plotted (what versus what) for you? Hint: look at the values on each axes and compare
them to the data.
The easiest types of graphs to understand are linear graphs. Because of this, one often presents any graphical data in
linear form if possible. Rather than plotting d vs t one could plot d vs t2.
Create such a linear graph. The graph should look like Sample Graph Type 1 from the manual. Be sure to include a
trendline. Think: should a legend be used on this type of graph? Correctly label the axes and give the graph a title.
Print a copy of this graph only (I don’t need the data or the graph I created).
Answer the following questions in complete sentences. Your answer to the question should make clear what
question was asked.
SAMPLE GRAPH TYPE 2 QUESTIONS
1) Why does it make sense to record the units for R in meters rather than cm?
2) Why do the sig figs of the Eth change for the bottom part of the table? Hint: consider all the variables (and
the commensurate sig figs) that went into the calculation of Eth.
3) Usually it is inappropriate to use a legend. In what cases should one use a legend?
4) By comparing the proximity of the dots to the smoothed line one can easily see regions where the
experimental data fit well to the theory. What location on the axis of the ring appears to have the worst?
Note: using a least squares algorithm one could quantify the error.
5) Notice that the experimental data points always lie slightly above the theoretical line. Does this indicate
random errors or systematic errors? For help on this you might consider googling “random versus
systematic error” and look for examples. I found the link from www.physics.umd.edu pretty easy to read.
SAMPLE GRAPH TYPE 1 QUESTIONS
6) Determine the value of g from one of the coefficients of the d vs t trendline. Do not obtain a value of g for
each point of the graph and average them. Show work rather than writing an answer in a complete sentence
for this question.
7) Determine the value of g from the slope of the graph. Do not obtain a value of g for each point of the graph
and average them. Show work rather than writing an answer in a complete sentence for this question.
Hint: compare equation (3) to y = mx + b to determine how g relates to the slope m.
6
7
Coulomb’s Law
Materials: bases, rods (large rods that fit the bases), right angle clamps, small rods, thread, latex gloves, scissors,
electrostatics experiment (assorted glass, plastic, and rubber rods with pieces of silk, wool, and fur), pith balls
(preferably large), electroscopes
Objective: 1) Determine the electrostatic charge on two small conductors by means of Coulomb’s Law. 2) Estimate
the error associated with your calculation of the electric charge. 3) Learn about charging by induction.
Theory: Coulomb’s Law states that the force F between charged particles is along the
line joining the particles and is proportional to the charge on each one and inversely
proportional to the square of the distance between them.
= |!|||
For today’s lab ke is Coulomb’s constant, q1 and q2 are the respective charges, and R is
the center-to-center separation distance between them.
In the apparatus shown in the diagram at the right, two small conducting bodies of
equal mass m and diameter D are suspended by thread from horizontal support rods.
Notice that L is the distance from the center of mass of the sphere to the point of
contact on the rod above it (not the length of string used). Notice that the spheres are
at least 20 cm from any surfaces or rods.
Introducing charge to the conducting spheres (while they are in contact with each
other) will cause them to assume equal charges (provided they have the same shape
and size). At some angle the bodies will be in equilibrium under the action of the
thread tension T, gravitational force mg, and the electrostatic repulsive force F given
by Coulomb’s Law. The equilibrium condition allows F (and thus q) to be found in
terms of mg and tanθ. Note: if θ is small, the small angle approximation may be used:
θθ tansin ≈ .
The final goal of today’s lab is to determine the charge on each sphere. To find a
theoretical result for today’s lab you will need to determine tanθ in terms of L and R.
This result can then be used in Coulomb’s law with the assumption that the charge on
each sphere is equal (q1=q2=q) and is uniformly distributed on each sphere. Be sure to
state this assumption in your lab notes! Before setting up the apparatus use all this
information and determine an algebraic expression for q in terms of the variables
ke, m, g, L, and R.
For the theory on charging by induction see your textbook or do a Google search. Do
this after the experiment on charging the spheres.
Procedure for Determining Charge on Spheres and Estimating Error:
Measure the masses of the two metalized pith spheres to be electrified, using an
analytical balance.
Wearing latex gloves, suspend the spheres from the rod using more than 1m of thread.
The oils from your skin can get on the thread and allow charge to escape from the spheres! Avoid handling the silk
insulating thread with bare hands in order to prevent the silk from becoming a conducting path. Make sure
that the charged bodies are 20 cm away from any surface/edge of the table (ruler is the exception).
q=
θ
mg
F
T
R
θ
Side view before introducing charge to the spheres
Side view after introducing charge to the spheres
L
20cm
>1m
Typical separation after charging is very small!
8
Mount a horizontal meter stick so that the separation of the bodies can be viewed against the meter stick. The meter
stick should be placed far enough from the bodies so that they are not disturbed. At the same time, get the
instruments as close as possible to reduce errors. Some trial and error is required to figure out how close you can get
before you are affecting the experiment (try out the next paragraph).
Charge a plastic rod using fur with as much charge as you can get (you should hear the crackling of static before you
stop). Charge the bodies by bringing them into contact with the charged rod. Watch the spheres fly apart and
bounce off each other a few times. Eventually they should settle down and remain slightly apart. Try adding a little
more charge. If you can get a center-to-center separation of about 5 cm you are doing about as good as you can.
Determine the separation by viewing the bodies against the scale from some distance away. Think: what points on
the spheres bodies (edges or centers) should define separation for Coulomb’s law?
Determine the amount of charge on each charged body. Determine the number of electrons on each body. Repeat
steps 5 through 7 for a second quantity of charge. Try to get as much charge as possible for this final calculation.
Using the error appendices in this manual, perform an error estimate similar to the first “Example of a Typical
Experiment.” Use this to get a result like qexp=3.4±0.7 nC. Do this only for your largest value of qexp.
• You’ll need to estimate your error in measuring L and R (they probably are worse than 1mm but better than
10 cm). They won’t necessarily have the same errors!
• Choose the best prefix to tabulate your data (don’t write 4.2×10-5C when you could write 42 µC).
• SHOW ALL STEPS OF THE ERROR CALCULATION EXPLICITLY (follow template in the appendix).
• Final result for an error calculation should only have ONE SIG FIG!
Procedure for Learning about Charging by Induction
Lastly, learn how to charge an electroscope by induction then draw a
series of 3-5 pictures describing the induction process. Be sure to
include 1) words explaining what is going on in each picture and 2)a
bunch of +’s and –‘s throughout the electroscope in each picture
including on the leaves of the electroscope. Remember that the
electrons redistribute (not the protons…think why) and an object is left
with EXCESS + or – charge as a result. An example of how to start
this series of figures is shown at left.
Conclusions:
1) The assumption that charge is evenly distributed on the spheres NOT a good one. Sketch (1/4 page in size)
two negatively charged spheres after reaching equilibrium (spaced slightly apart). Show the excess
negative charges on the spheres as minus signs in your picture. Where will most of them reside?
2) Which measurement had the largest error in today’s experiment (the leading contributor to error)? Estimate
the percent error in this measurement. Again, if you measured a length as 12±3cm you would get
3/12=25%error.
3) Regarding the error for the charge on the spheres: why is it important to get as much charge as possible?
Hint: No matter how much charge you get on the spheres you always have large reading errors for R. What
happens to the percent error in R if the spheres don’t have much charge?
Rod w/ excess negative charge,
other protons and electrons are
present but not drawn
Electroscope neutral w/ charges
evenly distributed when
charged rod is far away
9
Measuring w/ DMM’s
Apparatus: Each group should have ten 10Ω resistor and one 56Ω resistor banana cables, alligator clips, 1.5 V D-
cell batteries, single battery holders, 1 handheld digital multimeter per station, 1 benchtop multimeter per station,
DMM red leads, DMM black leads, galvanometers (coil with compass…might need both the large 40 turn and
smaller 15 turn coils)
Objectives: Learn how to use a DMM to measure resistance, voltage, and current.
Comment on DMM data. Sometimes the DMM will output 4 sig figs and sometimes only 3.
Always record all sig figs output by the DMM. In Excel, look for these buttons somewhere in
the top/middle that allow you to change the sig figs in each cell. Also, when measuring with a
DMM start with the dial at the highest possible setting then work the dial down to the lowest
setting which does not overload. This gives the maximum number of sig figs for each
measurement. Lastly, turn off the DMM and open the switch on any circuit when not in use to
save battery life. This is important with two 3-hour labs back to back!
PART I: The nominal value of a resistance is the value stated by the manufacturer (e.g. 10Ω or 4.7kΩ). Obtain 10
resistors with a nominal value of 10Ω. Before you start measuring them, verify they all have the same color bands
in exactly the same order. Google “resistor color code” to verify each resistor’s color bands match the nominal
value stated on the little plastic bag. If you have trouble discerning the colors (for instance orange and red often
look similar) get a second opinion. Give your instructor any resistors which are not in the correct bag. If you do not
have 10 resistors after checking all the resistor color codes, try a different nominal value of resistance and start over.
As a group measure the resistance of all 10 resistors with a DMM and tabulate the data. Note the tolerance band of
each resistor in your table. They will probably each be 20%, 10%, or 5% but check it just in case. For each resistor,
compare the resistance measured with a DMM to the nominal value and state if each resistor is within the tolerance
stated by the manufacturer. Lastly, obtain the average and standard deviation of your measured values using the
AVERAGE and STDEV functions in Excel. If you divide the standard deviation by the average this roughly gives
you a “% spread” of your measured resistors. Compare this to the tolerance band stated by the manufacturer…
PART II: Obtain one additional resistor with nominal resistance 56Ω. If 56Ω resistors are
unavailable, ensure your two resistors have different nominal values. Prepare a simple circuit that
employs a 1.5 V battery, your resistor, and a switch. Note: if no switch is available, you may consider
the act of connecting and disconnecting one of your wires as your switch. Also, ensure you have a
benchtop DMM and a handheld DMM at your station for this part.
You will be measuring voltage, resistance, and current for this circuit. The measurement of current is
often done improperly by students just starting out. This can cause blown fuses, frustration, and
confusion. Use the break and replace method. Put the DMM in series with the resistor, not in
parallel. See the lower figure at right.
Note: if you don’t know how large the current will be, first use the 10 A setting and the 10 A cable
positions. If your measured current is less than 0.20 A using this mode, switch both the cables and the
dials to the 200 mA or less positions on the DMM. If you don’t know what I’m talking about, ask the
instructor for help in person BEFORE you attempt current measurements.
0.23 A
A
10
Measure the following quantities with an OPEN SWITCH (open circuit):
• The resistance of the load resistor with the benchtop DMM
• The voltage + across the battery with the benchtop DMM
To notice:
1) Measuring the resistance of a resistor is best done before the resistor is placed in the circuit. Other
elements in the circuit will mess with the reading of your resistor. If you tried to measure the resistance
while the circuit was connected, you perhaps found an over limit error message on the DMM screen.
2) The open circuit voltage and closed voltage are nearly, but not exactly the same. For small values of load
resistance the difference can actually be quite dramatic!
Measure the following quantities with the DMM with a CLOSED SWITCH (closed circuit):
• The voltage +, across the battery with the benchtop DMM
• The current - through the load resistor with the handheld DMM.
Notice the measured values can differ depending on the status of the circuit (open versus closed). We will learn
later that no battery is ideal; all real batteries have internal resistance. The internal resistance of the battery
effectively lowers the voltage of the battery while in operation.
In addition, putting a DMM in series to measure current will also add some hidden resistance. Further complicating
matters, the internal resistance of DMM while measuring current changes as the dial is rotated (as sensitivity setting
is changed). If we restrict ourselves to resistors between 10Ω and 56Ω we should be able to set the dial to the
200mA setting and leave it there. We can then lump this internal resistance of the ammeter in with the internal
resistance of the battery. Note: the DMM in parallel measuring voltage is does not significantly affect the internal
resistance for reasons we can discuss when we get to circuits.
The amount the voltage is lowered is a function of both the load resistance and the total internal resistance
(including both battery and DMM used as an ammeter). In our circuit internal resistance (.) is given by
. = // −
where // is given by
// = +-
Do four total trials (use both the 10Ω & the 56Ω with two different batteries).
Have each different student do one set of measurements so each student gets the practice.
Use δ// = //1234545 6 237
7 6 where δ. = δ// δ. Make Excel formulas do the calculations for
you but check one by hand to verify you typed in the formulas correctly. Remember to use your unrounded result
for δ+ in your calculation of δ. when checking by hand…that’s what Excel does! Also, think about your units!
11
Be sure to check the following items:
• Use the DMM value, not nominal value, of resistance; measure 8 before it is connected to a circuit!
• Get the max sig figs possible for each measurement (should almost always be 3…4 if first digit is 1)
• Adjust sig figs of each cell in Excel to include all sig figs from DMM (important if last digit is zero)
• The delta symbol (δ) can be created using lower case “d” in the symbol font.
• The capital omega symbol (Ω) can be created using upper case “W” in the symbol font.
• Variables like , +, and 9 are italicized (with the exception of numerical subscripts).
• Units like V and A are NOT italicized.
• The lower case delta symbol (δ) indicates uncertainty or error. Estimate the errors based on your readings
from the DMM or by searching for the uncertainty in an online manual for the DMM.
• δ. is different from the other uncertainties. Since you calculated . you must also calculate δ..
PART III: A galvanometer is a device used to measure current. A simple galvanometer can
be made by placing a compass at the center of a coil of wire. If current flows through the
wire the compass needle deflects. Try building the circuit shown at right. Start with a large
resistor. When the switch is closed, observe the compass.
Now open the switch and replace the large resistor with a significantly smaller resistor. Upon
closing the switch, once again observe the deflection of the compass needle. Does it deflect
more, less, or the same amount as the larger resistor.
Finally, take the coil out of the circuit. Measure the resistance of the coil with a DMM.
Make a mental note: is the coil resistance negligible compared to in your opinion?
Don’t re-write the questions. Answer in full sentences which make obvious what question was asked.
Conclusions:
1) When measuring the resistance of a resistor with the DMM one often gets 3-4 sig figs. For example,
suppose a 1.0kΩ resistor with 10% tolerance band is measured with the DMM. The student determines
resistance is = 1023 < 1Ω. Even with a 10% tolerance band, the measuring error is not <100Ω.
What is the point of the tolerance band if it doesn’t really tell you the error in the resistors value?
What does the tolerance band tell you?
2) The manufacturer of the resistor uses the tolerance band to alert users of the possible range of resistance
values. A 1.0kΩ resistor with 10% tolerance band implies measured resistance should be a value between
900&1100Ω (even though it is nominally 1000Ω). When you use a DMM, you typically get a much
smaller measurement error (?). That said, if no DMM measurement is made, the tolerance band can be
used as a worst case scenario number for ?. No question here; no answer required. I just want to stress
you should measure resistors with a DMM in this class (as opposed to using the nominal value).
3) If internal resistance (.) is less than 1% of load resistance (), . has negligible effect on measurements.
What minimum load resistance is required to make internal resistance negligible for your circuits? Support
your answer by stating numerical values of internal resistance from your circuits in the answer to this
question. Random Note: we will learn later power is delivered to the load resistor at maximum rate if load
resistance equals internal resistance.
12
4) Why does running current through the coil of wire cause the compass needle to deflect? You can probably
web search for this answer if needs be. In particular, when current is running through the coil, what type of
field is created?
5) Consider the circuit shown at right. Assume is a large resistor. Assume the coil is
essentially a wire with MUCH less resistance. It is very possible the coil in this circuit will get
hot and melt (and no longer be functional). Explain why the coil will get very hot and possibly
be destroyed. Hint: in a circuit you may have heard that most of the current takes the path of
least resistance.
6) Do web searches for “dead battery voltage” and “internal resistance over time”. From your
research, answer the following (listing websites you used to determine the answers):
a) When a battery dies does battery voltage drop to zero?
b) As a battery dies, does internal resistance increase, decrease, or remain relatively
constant?
13
Electric Field Mapping Version 3.0 Instructions
Apparatus: field mapping apparatus, we only require 1 plate per group…we have 6 plates so use one of each then
repeat as necessary, Power Supplies, one DMM per group, 1 cm cross grid (use the internet), laptops (or PCs) with
MATLAB installed
What to submit is listed on last page!
Goals:
1) For a single charge configuration, tabulate + versus , data for every spot on a 1cm cross grid. Please
use a single sheet in Excel with no column or row headings. The data starts in cell A1.
2) Save your raw data as a csv file (comma delimited file). Select “Save As” and look for csv…
3) Input the data to MATLAB in two ways (cut and paste and using the csvread function).
4) Create a contour map showing equipotentials in 0.5V increments.
5) Compute additional tables of data for C ≈ − E4EC and F ≈ − E4
EF.
6) Generate a quiver plot in MATLAB showing the electric field arrows.
7) Display the quiver plot overlaid on top of the contour plot.
8) Save your final image on a flash drive or to the web. As a group you will stand in front of the class and
explain the interesting features of your equipotentials and associated electric field map.
9) Write answers to the conclusion questions (see last page). Submit a single document for your group (not
one document for each member). Handwritten responses only. Answer questions in full sentences.
Answers should make clear what question was asked.
Acquiring Data
Select one person in your group to become the MATLAB expert.
*The MATLAB expert should skip to the next page while the others complete this section on data acquisition.
Connect the experimental apparatus as demonstrated by your instructor.
*Please ensure there is a washer between the charge pattern and the screw.
*Please ensure the screw is tight after you ensure the washer is in place to protect the metal paint on the pattern.
Check for any loose connections anywhere a cable connects; check the wand as well!
Get some 1 cm cross gird paper off the internet with a web search or by reading the next line.
You can probably find a link to a 1 cm cross grid at http://www.robjorstad.com/Phys163/163Lab.htm.
Attach your 1 cm cross grid paper on top of the apparatus.
Use 0.0V for the low potential and some convenient voltage like 5.0V or 10.0V for the high potential.
Move the wand around in an orderly fashion (i.e. do one row at a time, always left to right).
*At each cross on the grid, slightly squeeze the wand until you can get a voltage measurement.
Don’t overthink it! Type in a number with three sig figs and move on.
If the third sig fig is unstable, WHO CARES! Take your best guess and move on.
*If you can’t get a number when the wand hits something underneath the board, make an estimate and move on.
Record every voltage for the entire grid. Do not write them on paper; immediately type them into a spreadsheet.
Save the data as a comma delimited file (CSV file) using the save as function.
Once you have gotten this done, you should have all the data required to complete the above numbered list.
Meet up with your MATLAB trainer on page 4…
14
MATLAB Expert Instructions
While the others acquire data, your goal is to practice getting data into MATLAB and making the plots using fake
data I have created for you. Hopefully you will be done practicing with the fake data before they give you real data
for your charge configuration.
Find the fake data sets online (probably at http://www.robjorstad.com/Phys163/163Lab.htm). Try to get the datasets
into MATLAB in two different ways: cut and paste and using the csvread function.
You might be wondering why you should bother with the cumbersome csvread function?
Answer: For large datasets in real science applications it is often impractical to cut and paste.
1) Cut and paste directly from Excel into MATLAB.
a. Highlight the values in Excel.
b. Hit CTRL-C to copy.
c. Go to MATALB and look for a button near the top saying “+ New Variable”
d. Click in the top right cell of the new variable (called unnamed at this point).
e. Hit CTRL-V to paste the data into MATLAB.
f. Rename the dataset in MATLAB by clicking on the name in one of the right windows.
2) Save each Excel file as a CSV file using the save as function.
a. Verify your data has no text or extra spaces. The data should start in cell A1.
b. I recommend saving them to the desktop so they are easy to find.
3) Determine the filepath of your CSV file.
a. Right click on the desktop icon and select
“properties”. A pop-up window such as the one at
right should appear.
b. Usually the 3rd or 4th line down tells you the start of
the filepath name. For the screen shot image at
right the filepath name is
C:\Users\Podium\Desktop\163VData.csv
c. On a piece of scratch paper, write down your
filepath name.
4) Use the csvread function to input the data to MATALB
a. Search the help file in MATLAB (or do a web search) for “csvread”.
b. Please give your matrix a descriptive name…something other than M.
c. Based on my choice of filepath I might type
Vdata = csvread(‘C:\Users\Podium\Desktop\163VData.csv’)
5) Make a contour plot
a. Search the help file in MATLAB for “contour” or do a web search for “MATLAB contour”.
b. Read carefully and learn how to adjust the number of contours.
c. Think: if your data ranges 0 to 5.0 V, what number of contours is needed to get 1 V increments?
How about 0.5 V increments? How do your answers change if 0 to 4.0 V is used instead?
d. Search the web or the help file to figure out how to add axis labels, titles, etc.
e. Figure out how to add the scale bar on the side of the figure.
f. Look carefully at the data set versus the figure. Is it flipped vertically, horizontally, both, or
neither?
g. When you print, nothing is in color. I need you to include It will help to add labels to the contour
levels. Do a web search to see different ways of getting this done. For my choice of filename,
something like this might work:
contour(Vdata, ‘ShowText’, ‘on’)
More on next page…
15
6) Use your + versus , data to generate C and F data.
a. When you download the fake data for C and F, you will
notice half of the data is missing! This download is
supposed to help you check your work. Forget about that
file for now and read the following bullets to learn how to
make the data yourself!
b. The upper screen shot at right shows + versus , data.
This is the only full data sete of the fake data.
c. The lower screen shot at right shows F data.
d. To compute F ≈ − E4EF in cell S2 of lower screenshot I
used the following equation in Excel:
=-(B3-B1)/2
i. Why the 2? Our grid spacing was 1 cm per cross.
I went from the cross above to the cross below.
The total spacing is 2 cm.
ii. Why above to below? This is an attempt to
average out errors from calculations using two
adjacent points. We will still have large errors for
a few points but hopefully this will be a decent
estimate.
iii. Notice all HI′K on the edge are zero. I set them
to zero as our Excel formula can’t work! For
instance, there is no cell B0 to compute F for
cell S1.
iv. Now use the test files (fake data files) to check your results. You’ll notice I left about
half of the numbers blank to force you to figure out the equation to make it work.
CHALLENGE: You could use MATLAB to create the C & F data using operations on matrix elements. It would
take a bit of thought and effort but a clever student could surely figure this out. NOTE: that same clever student
should be wise enough to wait until after the lab is done to make an attempt…
7) Use your C and F data to generate a quiver plot. Search for quiver plot. Figure it out.
8) Try to overlay the quiver plot on top of the contour plot. Hint: try a web search for “Combine Contour Plot
and Quiver Plot”. Wilson Phillips would be proud…
At this point, you should be able to take your actual + versus , data set, generate C & F data, and create a
contour plot with quiver plot overlaid on top. In addition, you should be able to train your partners to do the same.
Go on to the next page.
+ versus , data
F versus , data
Use these
two values
of + to
compute
Δ+ in the
vertical
direction
Use this
Δ+ to get
F in
between
the two
points
used
16
Putting it all together
Those who acquired data: give your final data set to the MATLAB trainer. Go to page 2 and learn how to make
contour and quiver plots using the instructions. Try to figure it out yourself and leave the MATALAB trainer alone
until she or he finishes your final figure.
MATLAB Trainer: Make the final plot showing a quiver plot overlaid upon the contour plot. This is showing you
the electric field vector on top of the equipotentials. Make sure your final plot has axis labels and contour labels. In
addition, somehow figure out a number for your largest electric field arrow in M
NO and write it on the plot. You could
do this by hand if desired by looking at the largest arrow and checking the numbers in the Excel file. Alternatively,
you could create a table in Excel of the magnitude of the field given by = C F then look for the largest
number using the MAX function in Excel.
LAST 30 MINUTES each group will stand up and show their plot to the rest of the class. Each group has a
different plot and we can hopefully learn about each unique charge configuration that way. Expect to show your
plot, let people look at it for a minute. Ideally, you will have already completed the questions below. Each lab
group member could prepare an answer to give to the class in case I call on you. Each group member should
prepare a different question in case I have time to ask all each of you to say something.
TO TURN IN:
Print out a single copy of your final plot. Put all names on it in a corner somewhere.
On a second sheet of paper write all names neatly in upper right hand corner.
Write the class section and class time.
Write answers to the conclusion questions below.
Submit a single document for your group (not one document for each group member).
Handwritten responses only.
Be neat and organized.
Answer questions in full sentences.
Answers should make clear what question was asked.
Basically, re-phrase my question in the answer you give to me.
1. Which way do your electric field vectors point: from high-to-low or low-to-high voltage?
2. Which way should your electric field vectors point: parallel or perpendicular to the equipotentials?
Optional: can you think of any reasons why your calculated values might not be perfectly parallel or
perpendicular?
3. Where is the field strongest/weakest on your plots: close to the electrodes or far from the electrodes? Hrlp
me understand your answer by referring to regions on your plot using the , coordinates. For example,
you might say:
“The field is strongest along the vertical line = 5.0cm and weakest near the point (2.0, 6.0).”
4. Is there anywhere on your plot where the electric field is roughly constant in magnitude? Hint: if the field
has constant magnitude, what must be true about the equipotential spacing? Again, reference coordinates.
5. For our lab today, there is always a constant potential difference between adjacent equipotentials. If the
lines are spaced closely together, does this imply a large or small field? Does your data support this?
6. Imagine someone released a proton from rest in the middle of your map. Which way would it accelerate at
the instant it is released?
7. If an electron was instead placed in the middle of the map, which way would it move AND would it
accelerate faster, slower, or at the same rate? To be clear, we comparing the accelerations of the electron
and proton at the instant they are each released from rest in the middle of your map. EXPLAIN WHY FOR
CREDIT.
8. If a charge was placed on your equipotential map and released from rest, would it be appropriate to model
the motion using the constant acceleration kinematics equations?
17
Ohm’s Law and Resistivity
Apparatus: resistivity apparatus, PASCO Science Workshop 750 Interface & Power Supplies, 2 digital multimeters
per group (handheld & bench top if necessary), DMM red leads, DMM black leads, resistors (100 Ω),
micrometers or calipers
Instructor Note: The Pasco power supplies have a 300 mA limit. If you are using them you’ll need to have some
resistor in series with the long skinny wires or you’ll get data affected by the current limit! Also, the
bench top DMMs will give good data as long as the appropriate dial setting is used (max out the sig figs
without going over limit).
Goals:
1) Build a simple circuit
2) Measure Δ+, 9 and in that simple circuit
3) Experimentally determine the relationship between voltage and current
4) Experimentally determine the relationship between length and resistance
For this experiment you will need a long wire with a small cross-section. This skinny wire will act as a resistor.
The resistance of the wire is given by
= PQR = 4P
S Q
where ρ is the resistivity of the wire, d is the diameter of the wire, and L is the length of the wire. For a cylindrical
wire R = TUVW is the cross-sectional area of the wire.
Important comments:
Resistance is a property of an object. Example: The resistor (the object) has resistance 101.4Ω.
Resistivity is a property of a material. Example: Copper (the material) has a resistivity of 17nΩ ∙ m.
The units of resistivity are “nano-ohm meters”, NOT “nano-ohms per meter”.
Part I: Measure the resistance of 1.00 m of skinny wire with the DMM.
Part II: Now, for the same 1.00 m of wire, do a four-point measurement of the resistance while sweeping the DC
voltage. See this explained with figures on the next page.
Set the power supply at Δ+Z[, = 1.00V and record both Δ+\7[ and 9\7[ . Change the power supply to 2.00 V
and record both Δ+\7[ and 9\7[ again. Repeat this procedure for all source voltages between -5.00 V and 5.00V.
Hint: to get negative voltages you may need to switch the red and black leads at the power supply!
Be sure to record the ]^_`ab across the wire (not the ]^Kcdaeb of the power supply).
Plot Δ+\7[ on the -axis and 9\7[ on the -axis. This is the standard way because technically the voltage is the
independent variable and current is the dependent variable. Get the slope of the graph and use it to determine the
resistance of the wire. CAREFUL: the slope is not the resistance but rather… (check the units)!
18
Part III: Repeat the previous experiment with the following changes. Select a convenient voltage (say 1.00 V).
Measure both current and voltage for 1.00 m of wire. Use this information to determine the resistance of 1.00 m of
wire. Then repeat this for 2.00 m, 3.00m, etc on up to 10.0 m of wire. A table of versus Q can then be recorded.
From this information a plot of vs. Q can be made. The slope of this plot
should be given by slope =f? . This shows that an experimental value of the
resistivity is given by PCh =?∙ ijklm. Determine your diameter using
micrometers so you can calculate PCh.
Be careful: If you squeeze too hard with the calipers (or micrometer)
the wire compresses and gives an incorrect measurement! Use the flat
portion of the caliper…not the knife edge. Note: this measurement has
only two sig figs AND gets squared in the computation. Squaring implies
the error in this measurement is doubled in the final result! Close the
caliper until you first begin to feel opposition to your motion in the calipers
and this is your measurement.
Compare this to the accepted value of some common materials in the table
shown at the end of the lab.
Four-probe measurements: To make a four-probe measurement, connect
the circuit as shown in the figure. Notice that there are four “probes” – the
two leads from the DMM measuring voltage and the two leads from the
DMM measuring current.
The DMM on the right measures the voltage across only the segment of
wire. I recommend using the bench top DMM for this measurement. The
DMM on bottom measures the current through the circuit.
Note: To change the length of the wire one can connect at the different
“taps”.
These values of voltage and current can be used to find the resistance for
each measured I and ∆V using by Ohm’s Law (∆V=IR).
The four point measurement technique is useful for measuring very small resistances that typically are hard to
measure with DMM’s. Here is a very simple web resource that discusses four-point measurements:
http://www.sciencebuddies.org/science-fair-projects/project_ideas/Elec_p025.shtml
0.23 A
A
2.07 V
V
DC
pwr
Approximately what it
looks like in real life
Schematic
\7[
A
100Ω
Δ+Z[, V
100Ω resistor to
reduce heating
10.0m
wire
DMM measuring
current in series
DMM
measuring
voltage in
parallel
19
Conclusions:
1) Suppose you have a connected circuit with some resistors in it. Which method, using a DMM to measure
resistance directly (Part I) or using the four-probe measurements (part II) is able to accurately determine the
resistance of a single resistor in the circuit while connected to power?
2) Suppose that, instead of 1.00 m of wire you had only 1.00 cm of wire. Which measurement technique (part
I or Part II) should be used in this situation and why? Hint: is the resistance of the wiring in the rest of the
circuit still negligible for such a small length of wire?
3) A device is called “ohmic” if it has a linear IV plot. Was your long skinny wire ohmic? Note: it is
important to plot both positive and negative values for your IV plot because some devices could look ohmic
for positive voltages but not negative voltages.
4) Suppose you made an IV plot of a small length of wire. As the voltage is increased, the current should
increase (as well as the temperature of the wire). What should happen to the resistance in the wire? What
should happen to the slope of the graph? Sketch a plausible IV plot for a small length of wire that gets very
hot as 5V is applied. Is this device ohmic? An example of such a device is an incandescent light bulb.
5) Suppose you compared two wires of the same material. Wire 1 is longer than wire 2 while the cross-
sectional area is the same. Which wire (if any) has a larger resistivity? Explain.
6) Suppose you compared two wires of the same material. Wire 1 is a smaller gauge than wire 2 while the
length is the same. Use a web search to determine how gauge relates to diameter. Which wire (if any) has
a larger resistance? Explain.
7) The most common error I have seen in this lab is squeezing the calipers too hard when measuring the
diameter of the wire. If you did squeeze too hard, how would this affect your final result for PCh? More
specifically, if you squeezed to hard on the calipers do you make your experimental result have a more
positive or more negative % difference? Assume you choose the appropriate value for P/n.
Table of Resistivities (near room temperature)
material ρth (nΩ.m)
aluminum 26.5
tungsten 52.8
nickel 69.3
steel, plain 180
steel, stainless 720
nichrome (80/20) 1090
nichrome (70/30) 1180
nichrome (max) 1500
phos. Bronze, grade A 35.92
phos. Bronze, grade C 132.6
phos. Bronze, grade D 156.7
phos. Bronze, grade E 95.79
20
21
Series & Parallel Circuits
Prior to lab: The pre-lab for this lab starts just after the conclusion questions. This assignment is actually really
good practice for both exams and the real world. If you fully understand the pre-lab algebra and make the Excel
sheet in advance you can expect the lab to take 2-2.5 hours instead of 3!
DMM & Breadboard Playlist:
https://www.youtube.com/playlist?list=PLBQTyyPKj9WbW41JW83crz62DJ4wM2wC6
Apparatus: batteries, battery holders, resistors (one 22 Ω, 33 Ω, and 47 Ω for each group), breadboards, banana
cables, alligator clips, DMMs (handheld and benchtop), DMM red leads, DMM black leads, jumper wires
Goals:
1) Know how to build a series circuit, parallel circuit, and the two different mixed circuits
2) Know how to measure ∆+, 9, and in those simple circuits (and compute o)
3) Verify relationships between ∆+’s and 9’s amongst the branches of various circuits
Theory: Four circuits are drawn on the pages following the conclusion questions. The circuits show the various
stages of combining series and parallel circuits as well as two mixed circuits. This basic process of combining
resistors to draw equivalent circuits can be done using the following rules:
For Series Resistors: p = ! ⋯
For Parallel Resistors: !
rst = !ru !
rV !rv ⋯
Shortcut for Two Parallel Resistors: p = rurV
ruwrV In today’s lab you will first calculate the equivalent resistance for each circuit. We will then assume that the voltage
of the battery Δ+x is equal to the voltage across the equivalent resistance. This ignores internal resistance and
implies that terminal voltage is the same as the ideal maximum or open-circuit voltage of the battery (also called the
electro-motive force or EMF). Read the first two pages of the chapter on Direct Current Circuits in your textbook to
help you answer conclusion questions on this.
Once you know p and Δ+pyou can determine the current in the equivalent resistor 9p and the power delivered to
it op . Since the equivalent circuit is a single resistor and a single battery this current must match that of the battery.
Also, assuming the internal resistance is negligible, the power delivered to the resistor should match the power
output of the battery. To determine the voltage across, the current through, and the power delivered to each resistor
one can then use two simple rules.
1) Resistors in series have the same current.
2) Resistors in parallel have the same voltage.
For the lab assignment, you are to do the following:
• Measure the actual values of each with a DMM (prior to connecting the in the circuit).
• Connect the circuit.
• Measure the voltage across the battery and each resistor while the circuit is operating.
• Measure the current through the battery and each resistor while the circuit is operating.
• Use your measured values of 9 and Δ+ to calculate the actual power delivered to each resistor (and the
power delivered by the battery). Note: to get the power delivered to an item simply use oC = 9CΔ+C .
• Make one sketch of each circuit showing all components with the measured values of , Δ+, and 9 and the
calculated value of o for each component.
• Answer all conclusion questions.
• Verify with your instructor what work, if any, you are required to show (I usually say no work is ok today).
22
Building Circuit 1:
Build a series circuit (Circuit 1) with two resistors in series. Connect the battery to the breadboard using banana
cables and alligator clips (see Figure 1). Connect the wires and the resistors in the breadboard as shown in Figure 2.
Do not go on until every member of the group can build the circuit AND measure all currents and voltages.
Compute each power. Don’t forget to record the value of each resistance! When measuring resistance, be sure to
isolate each resistor prior to making the measurement. If you try to measure resistance of a resistor while it is in a
circuit you may unintentionally record the measurement of some equivalent resistance.
Figure 1: A top view of the circuit in its entirety.
Figure 2: Close-up of the breadboard. Notice that the yellow wire is securely fastened to the red terminal by
tightening the knob and compressing the metal part of the wire to the metal part under the red knob. Row 50
connects the yellow wire to one end of the resistor. Row 55 connects the two the resistors. Row 60 connects the
green wire to the other end of the 2nd resistor. Lastly the green wire connects securely to the black terminal. Going
back to figure 1 you can see that the banana cables and battery complete the circuit by connecting the red terminal to
the black terminal.
23
Continue with Circuits 2-4: Build Circuits 2, 3, and 4. For each circuit measure all currents and voltages.
Compute each power. Don’t forget to record the value of each resistance. When measuring resistance, be sure to
isolate each resistor prior to making the measurement. If you try to measure resistance of a resistor while it is in a
circuit you may unintentionally record the measurement of some equivalent resistance.
There is a tip for measuring current below these figures.
Figure 3: Notice both resistors now connect row 50 to 55. This means any current that comes into row 50 has two
channels to flow to row 55.
It gets tricky when you try to measure currents in a parallel circuit. A technique I call “break & replace” seems to
help some students make fewer errors. This technique is shown in the Figures 4a and 4b below.
Figure 4a: Just before the break and replace method.
Notice that the DMM is using the 200 mA setting &
the cables are in the proper place for measuring
current with the 200 mA setting. The banana cables
from the DMM both have alligator clips. One of the
alligator clips has a small wire gripped in its teeth.
Figure 4b: To measure the current through the left
resistor I “break” the circuit by removing one leg of
the resistor and connecting it to one side of the DMM
measuring current. I then “replace” the removed
resistor wire with the alligator clip wire from the
other terminal in the DMM. This causes current
through the DMM which can be measured.
24
Conclusions:
Correctly use the phrases current through, power delivered, and voltage across (or potential difference).
1) For the simple series circuit (Circuit 1):
a. Compare the VOLTAGES ACROSS each resistor. Which has more potential difference: the
larger resistor or the smaller resistor…or do they have the same potential difference?
b. Determine the % difference between the theoretical and experimental voltages for each resistor.
2) For the simple parallel circuit (Circuit 2):
a. Compare the CURRENTS THROUGH each resistor. Which carries more current: the larger
resistor or the smaller resistor…or do they carry the same current?
b. Determine the % difference between the theoretical and experimental currents for each resistor.
3) For Circuit 3:
a. Consider currents 9y// , 9!, 9, and 9. Which two should be equal?
b. In theory, we expect 9y// = 9! 9. Determine an experimental summed value using the equation
9y//Ch = 9!Ch 9Ch. Compare the experimental summed current 9y//Ch to the theoretically
predicted value with a % difference.
c. Consider potential differences Δ+y// , Δ+!, Δ+, and Δ+. Which two should be equal?
d. In theory, we expect Δ+y// = Δ+ Δ+. Determine an experimental summed value using the
equation Δ+y//Ch = Δ+Ch Δ+Ch. Compare the experimental summed value to the
theoretical value with a % difference.
4) For Circuit 4:
a. Consider currents 9y// , 9!, 9, and 9. Which two should be equal?
b. In theory, we expect 9y// = 9! 9. Determine an experimental summed value using the equation
9y//Ch = 9!Ch 9Ch. Compare the experimental summed current to the theoretical value
with a % difference.
c. Consider potential differences Δ+y// , Δ+!, Δ+, and Δ+. Which two should be equal?
d. In theory, we expect Δ+y// = Δ+! Δ+. Determine an experimental summed value using the
equation Δ+y//Ch = Δ+!Ch Δ+Ch. Compare the experimental summed value to the
theoretical value with a % difference.
5) For each circuit compare the sum of power delivered to all resistors to battery power with a % difference.
6) Consider two 12Ω’s in parallel with an ideal 6V battery versus five 12Ω’s in parallel with an ideal 6V
battery. Assume wire resistance is negligible. Draw each circuit and determine the equivalent resistance,
current through each resistor, the voltage across each resistor, the power delivered to each resistor, the total
current and the total power delivered by the battery in each case.
a. Which circuit has more equivalent resistance in total?
b. Which battery drives more total current?
c. Which battery will run out faster?
7) A household circuit is analogous to the scenario described in the previous problem. As one turns on more
lights/appliances, one is effectively adding additional resistance in parallel to the household circuit.
a. What happens to current drawn from the power company as one turns on additional appliances?
b. Does turning on a 2nd appliance decrease the potential difference across an already operating
appliance?
8) The utility company charges you by the kW ∙ hr. Is this a unit of voltage, current, resistance, power, or
something else? Support your work by converting 1.00kW ∙ hr to an SI unit. Side note: SI stands for
“systeme internationale”.
9) Batteries often include a rating in units of A ∙ hr. Is this a unit of voltage, current, resistance, power, or
something else? Support your work by converting 1.00A ∙ hr to an SI unit. Side note: Amps are
technically an SI unit while Coulombs are not.
25
26
Version A Version B Version C Version D Version E Version F
! 33 Ω 47 Ω 22 Ω 33 Ω 47 Ω 22 Ω
22 Ω 33 Ω 33 Ω 47 Ω 22 Ω 47 Ω
47 Ω 22 Ω 47 Ω 22 Ω 33 Ω 33 Ω
∆+xy// 1.5 V 1.5 V 1.5 V 1.5 V 1.5 V 1.5 V
As a pre-lab assignment, determine algebraic theoretical values for each potential difference, current, and power
listed in the first four circuit diagrams shown below (two on the next page). I have already started the first one for
you to give you an idea of how the answers should look. Circuit 4 gets pretty ugly. I recommend using a single
page for Circuits 1 & 2, Circuit 3 on its own on the next page, then Circuit 4 on its own page. A non-trivial circuit
(Circuit 5) is discussed as an optional extra credit assignment as well. It requires creating a simulation.
In addition, each student should create an Excel spreadsheet for computing the theoretical values for any of the
above versions prior to coming to lab. I made a screen shot of the one I made on the page after Circuit 5. When you
arrive in class, your instructor will give each group a different version. You should be able to input in the DMM
measured values of resistance and DMM measured open circuit battery voltage for any version in your spreadsheet
to have it predict theoretical currents, powers, etc. You will not be submitting this, but this is the real job skill so
each student really needs to be doing this…
o! = ~!!
o =
∆+! = ~ !!
9! = ~!
9 =
∆+xy// = ~
9xy// = ~p
= ~!
oxy// = ~p
= ~!
∆+ =
Circuit 1
o! =
o =
∆+! =
9! = 9 =
∆+xy// = ~
9xy// =
oxy// =
∆+ =
Circuit 2
!
!
27
o =
o =
∆+ =
9 =
9 =
∆+xy// = ~
9xy// =
oxy// =
∆+ =
Circuit 3
o! = o =
∆+! =
9! = 9 =
∆+xy// = ~ ∆+ =
Circuit 4
o! =
∆+! =
9! =
9xy// =
oxy// =
o =
9 =
∆+ =
!
!
28
∆+ =
9 =
9 =
∆+xy// = ~
9xy// =
∆+ =
Circuit 5: This circuit (shown below) is optional unless your instructor suggests otherwise. I might give
up to 2 extra lab points for completion of this activity (discuss with instructor prior to attempting).
Let ~ = 1.2V, ! = = 10Ω, = W = 22Ω & = 33Ω. Use Tinkercad to create a simulation (see
image at bottom of page). Include DMMs in the sim so you can display voltages and currents for each
resistor. TIP: after clicking “Create New Circuit” in Tinkercad, use the search function on the right side
of the screen to search for “multimeter” or “power supply”.
Build the actual circuit and compare measured voltages and currents to the simulated values. When
submitting your work, include a screen shot (printout) of your simulation. Also submit and a data sheet
showing the measured values the % difference between simulated and experimental values.
Challenge: Compute theoretical currents using KVL/KCL for these nominal values.
∆+W =
9W =
∆+! =
9! =
9 = ∆+ =
!
W
29
Below is a screen shot of an Excel Workbook I made. I change the resistances in cells b1, b2, and b3 and all of the
voltages, currents and powers will auto update. If you make such a table, you can check your work against this
screen shot by using the same values. If you want, you can check your algebraic work against this screen shot as
well. Simply plug in the resistance values shown in the screen shot into your algebraic formulas and verify you get
the same results as my screen shot!
Watch out: notice I used mA for currents (and mW for power). By doing this the current values are much easier to
read and match the units the DMM will produce when you do the experiment in class. You’ll want to think
carefully about either multiplying or dividing by 1000 as appropriate.
30
31
Characteristic of a Diode
Apparatus: Diodes (not LEDs), resistors, ceramic capacitors, PASCO Science Workshop 750 Interface & Power
Supplies, 2 digital multimeters per group (handheld & bench top if necessary), DMM red leads, DMM black leads,
breadboards, jumper wires, oscilloscopes, oscilloscope probes, very small flathead screwdriver.
Purpose: To explore the IV-plot of a diode and determine if the diode is an ohmic device.
Theory: An ohmic device is a device which obeys Ohm’s Law (Vacross the device=Ithrough the deviceR). This must be true
for all I and V the device experiences. It is easiest to verify if a device is ohmic by looking at its IV plot. Only
devices with linear IV plots are ohmic (see figure below). Notice an annoying feature of IV plots: voltage (the
independent variable) is on the x-axis while the current (the dependent variable) is on the y-axis. This is standard
practice. Unfortunately, this means that the resistance of a device at any voltage is not given by the slope of the IV
plot but rather the inverse of the slope of the IV plot.
Procedure:
1) Measure the resistance R of a resistor (about 1 kΩ) and record the result. Use a breadboard to put the
resistor in series with the diode. Be sure that the resistor connects to the unbanded end of the diode. For
kicks try measuring the resistance of a diode with the DMM. Then switch the leads and try it again!
2) Connect the free end of the resistor to the positive terminal of the power amplifier and the negative and the
ground to the banded end of the diode. See figure.
3) Set the power amplifier to DC and the voltage setting to 0.0 volts. Record both the voltage across the
resistor VR and the voltage of the power supply Vpwr. From these you should be able to determine the
voltage across the diode Vd. From Ohm’s Law the current in the circuit can be calculated from I = VR /R.
This one should be easy.
4) Now increase the voltage by 0.1 volt increments from 0 to 2 volts, and record VR and Vpwr. Use this
information to generate the table of values for Vd and Id.
5) Now return to 0.0 volts. Reverse the polarity of the circuit by switching the cables in the DC power supply.
This time going by 0.5 volt increments, repeat the previous step. Keep in mind that when you tabulate
these values in your IV plot the values will correspond to negative voltages.
DC Power +
Supply -
V
Non-linear = non-ohmic Non-linear = non-ohmic Linear = ohmic
I (A) I (A) I (A)
V (V) V (V) V (V)
32
6) Plot Id versus Vd (remember which variable goes on which axis for an IV plot). Is the diode an ohmic
device? There should be a voltage where the slope suddenly changes. This is called the knee. Locate the
knee on your graph and label it.
7) Plot I vs Vpwr and locate the knee on the graph. From Ohm’s law for both diode & resistor, I = Vpwr / RT,
where RT is the equivalent resistance of the circuit. From your graph calculate the RT before and after the
“knee” and compare it to RT that you measured in Part 1. Compare it to R, the resistance of the resistor.
Note: I believe the power supply has an internal resistance of about 50 Ω.
8) Now use the triangle wave on the power supply. Set the frequency to 1 kHz and the amplitude to 2.0 V.
Ask your instructor to show you how to set-up an oscilloscope. Notice this set-up makes the same graph
you did in 1 millisecond (if the frequency of your function generator is 1 kHz)! Very powerful…Make a
sketch of the graph on the oscilloscope. The sketch should be ½ page in size with correctly labeled the
axes with units. Indicate a scale on each axis for credit. Also include a title above the sketch.
9) Make a “half-wave” rectifier by switching the power supply to AC sine wave generator. You should see a
sine wave with half it curves flattened out. This is a method of partially converting an AC current to DC
current. Make a sketch of the graph on the oscilloscope. Think: ½ page, labels, units, scale, & title.
10) Try connecting a capacitor across the resistor in parallel. What happens? Try varying the frequency. The
capacitor is more effective at smoothing out the curve at higher frequencies. Why? What is the
relationship between τ (RC) and the generator frequency for effective smoothing? Make a sketch of the
graph on the oscilloscope at 1 kHz. Think: ½ page, labels, units, scale, & title.
11) A more effective method of converting AC to DC is to construct a full wave rectifier.
The above circuit shows a possible arrangement for a full wave rectifier. The rectified output voltage
appears across the resistor. Try to understand this circuit by tracing the path of the current for each half of
the generator cycle. Make a sketch of the graph on the oscilloscope at 1 kHz. Think: ½ page, labels, units,
scale, & title. Specifically state whether or not you are still using the capacitor in parallel with the resistor.
For each numbered step above be sure to answer any questions asked. In your notes clearly organize and label
which graphs and tables go with which number step. It may help to print them all out then cut them up with scissors
and label them by number (and include good titles). Don’t forget to do quality sketches on parts 8 through 11. By
quality I mean they should look just like a graph: ½ page in size with axis labels (with units and scale) and titles.
sine wave
33
B-Fields
In this lab you will measure the relative B-field at various locations relative to the center of a toroid of wire (the big
loop of wire). Download the XL worksheet as specified by your instructor and print necessary parts. Connect
the 200 loop coil of wire in series with an AC power supply. Do not use the resistor in series with the 200-turn coil.
Use the outer terminals on the big loop and set the driving voltage on the Pasco signal output to 5 VAC at a
frequency of 20 kHz. Connect the 2000 turn detector coil to the DMM’s leftmost inputs. Set the DMM to measure
200 mV AC. Measure the relative magnitude of the B-field by centering the hole of the probe coil over the position
you wish to measure. The hole of the detector coil and the hole of the big coil should both be facing the same
direction.
Part I: Plot the relative strength of the B-field versus position in the plane of the loop. Using the grid found in the
XL worksheet named “grid” take values at every intersection on the grid (approx every 2 cm). Use team work and
trade off to get it done quickly. See the image below for grid concept. Tabulate your data in the XL template
provided by your instructor in the sheet named “part 1”. After inputting the data make a “surface plot” in XL.
Choose the option for a 2D surface plot (“contour plot”).
Part II: Plot the relative strength of the B-field versus horizontal position on axis. Take values of the B-field every
cm for 15 cm on both sides of the loop and one value at the center of the loop. See the image on the next page for an
idea about the experiment. When plotting this data, follow the instructions provided on the XL worksheet. Calculus
people only: show experimental data as points only (no line) and the theory calculation as no points with a smooth
line. Data points which are close to the smooth line are good agreement with the theory. Closeness of fit can be
quantified by performing some sort of RMS value on each data point with the theory value. Hint: since you
normalized the B-field to 1 at the center use that value to determine the current for your theoretical model.
~2.7 mV
VAC
The B-field measurement of coordinate (5, 3)
(5,3) x (cm) V(mV)
5 3 2.7
y (cm)
34
Conclusions:
1) What do you notice about the size of the B-field in Part I as you move radially away from the center?
2) Is the trend isotropic (the same in all directions) or anisotropic (not isotropic)?
3) Does the radial trend change as you move outside the loop?
4) Is there any region where the field is “flat”? Here flat means relatively constant. Where is the field most
flat?
5) You are asked to do an experiment and exhibit a constant magnetic field on a large sample…for example
your thumb. What place in the coil will all parts of the thumb feel approximately the same B-field?
6) Why do you think MRI magnets huge coils?
7) Is the B-field equal to zero outside the coil?
8) Which direction shows the B-field dropping faster: moving out from the center radially (part I) or on axis
(part II)? Does this agree with theory? (Consider only the region of space within a few cm of the center.)
9) Calculus people only: MRI’s typically use B-fields on the order of 1 Tesla (at the center of the MRI magnet
coil) and have radii of about 0.5 m. Estimate the size of the B-field 2m from the center (hint: see text for
formula). Compare this to the earth’s magnetic field of size 0.5 Gauss. Note: 104G = 1T.
10) Should you walk into a MRI room (while someone is being scanned) with something magnetic or easily
magnetized in your pocket? Which is more dangerous a small magnetizable object or a large one?
Plotting B-Field along the
horizontal axis perpendicular to
the coil ( to the big loop of wire) x (cm)
V (mV)
-4.0
1.7
35
Magnetic Fields with Oral Presentations (10-12 min presentation)
General instructions to all groups
This lab will take place over three weeks.
• The first week you acquire all possible data you can think of. Some of it may end up being useless, but
better to get extra stuff and not use it than to re-build your apparatus to take data in week 2.
• The second week you get any data you missed and prepare your presentation.
• The third week you present to the class.
• Missing any of these three weeks negatively significantly impacts other students in your lab group.
As such, extra penalties may be imposed for any tardiness or absence during these three weeks.
Take photos AND videos of yourself doing the experiment. While videos are often much more useful for
explaining things, they can add a level of headache in terms building your presentation. The videos might help you
write up your talk even if you end up using only photos later on. Include a ruler in every photo for scale!
If you can do an experiment, you should be able to plot SOMETHING.
A single data point is pretty darn meaningless to other scientists. To create a
plot, typically you try to design an experiment wherein you vary one
parameter (say mass) and only one other parameter is affected (say current).
Ideally, you should have a theoretical prediction of how changing the mass
causes the current to vary. You then create a table of values that looks like
the one at right.
Choosing the correct prefix is crucial to looking good. Looking good is almost as important as being good in
terms of getting a job. It’s all marketing these days. When you present info to others you are selling yourself as a
product. The labels on your data are like the packaging. Compare the tables below. I think you will agree the third
version looks better AND is easier for a reader to understand at a glance.
Scientific
Engr w/o
prefix
Engr w/
prefix
x (m) x (m) x (km)
2.00E+03 2.00E+03 2.00
5.00E+03 5.00E+03 5.00
1.000E+04 10.00E+03 10.00
2.000E+04 20.00E+03 20.00
5.000E+04 50.00E+03 50.00
1.0000E+05 100.00E+03 100.00
Tip: while doing your work/calculations in a spreadsheet, it is easiest to use scientific notation.
After doing all the work, make copies of each data column using engineering notation with appropriate prefix.
Any time you take data, always think about how many sig figs you should show. Always write down an
estimate for the error/uncertainty for every measurement. Every once in a while this can make the difference
between understanding your experiment and being completely clueless.
Summary:
1. Always think about sig figs on every measurement/calculation.
2. Always record some kind of error estimate for every measurement/calculation.
3. Take photos and videos of your work as you go (including a ruler for scale in each shot).
4. Use scientific notation for calculations; use engineering notation (with prefix) for final presentation.
g 9/n mA 9Ch mA
10.0 51.5 62.3
20.0 103.0 115.2
50.0 257.5 291.8
36
Scientists, and everyone else, hate reading data tables. It is
much better to present your data graphically. General trends are
more easily internalized by the audience. Plots show the
audience you tested a particular scenario under a wide range of
conditions.
Note: A wise choice of prefix keeps the numbers on your axes
easy to read. If you chose prefixes wisely for your final data
table, the plot should look great! As an example, I created a plot
for gravitational potential energy versus position for a set of
masses. Think how messy the axes would be had I not used giga-
meters and giga-Joules…
Your plots will likely look like the one shown in your lab manual appendices entitled Sample Graph Type II.
This type of graph is useful because it shows both the theoretically predicted values and the experimentally
determined values. The audience can instantly recognize how well theory matches experiment. If theory is in good
qualitative agreement with experiment, the experimental dots take on the same shape as the theoretical smooth line.
If theory is in good quantitative agreement with experiment, for this class we assume 70% (or more) of the
experimental error bars are in contact with the theoretical smooth line. This topic relates to confidence intervals…
Each group must produce at least one plot (not a table) of some experimental data compared to theoretical data.
Most groups will have several plots. Usually, but not always, having more plots makes for an easier talk. Each plot
will take several minutes to explain and rapidly chews up the required time.
Questions to ask yourself while preparing your presentation slides with a plot:
• Is the plot as big as possible? Use an entire slide (no border) for each plot to make them easy to read.
• Can we read the numbers on your graph? Is every font size at least 18 pnt?
• Are you using the same font as the rest of the presentation? Or are you doing your best to use at most three
fonts throughout your talk? Excessive font changes distract from content.
• Are the sig figs used in the axis labels comparable to the sig figs of the underlying data?
• Units are NOT italicized.
• Variables ARE italicized.
• I prefer you to use variables (instead of words) in your axis labels. Other instructors differ on this.
• Note: I sometimes make axes in a thicker black line and gridlines, if useful, in a thinner dotted line.
• Only include a legend if two or more data sets are shown on a single plot. You will probably have two sets
(experimental & theoretical) on the same plot. You should probably have a legend.
• Typically, experimental data is shown as points (or just error bars) while a theoretical data set is shown as a
smoothed line. Exception: a large experimental data set (>20 points?) is often shown as a solid line.
-200
-150
-100
-50
0
-200 -100 0 100 200
x (Gm)
U (GJ)
37
Once you know the plot(s) you plan to explain, form a 10-12 min presentation around it.
Your presentation should include slides for the following:
• Title slide with awesome picture, title of experiment, and your full names
• Goal slide
• Theory slides
• Procedure slides
• Plot(s)
• Conclusions
For a 40 minute video on slide design, try a web search for “giving an effective presentation Stanford”.
Otherwise, continue reading below.
Title slide: Maybe you took a picture that really exemplifies your topic. Or consider a funny cartoon that relates to
your topic (cite the website you get it from). Try to keep the joke funny but inoffensive.
Goal slide: Tell us what you plan to cover, no more than three items if possible.
Theory and procedure slides often get blended together. You might show a picture of what you hope to do, then
show an FBD or other calculation supporting what you just talked about. Then show another picture of what
happens next, then explain some more math, etc. No matter how you organize it, make sure it flows logically.
Usually it helps a lot to have photos AND a simplified drawing. Sometimes photos get too busy and the drawing
can help the audience see what is really important. Word or ppt graphics should work just fine; I can help you as
well. You could also sketch a simplified drawing and scan it in (or use your phone to snap a photo).
Regarding cut and paste of my work:
• Don’t do it.
• One goal of making you do this is to improve your ability to make figures and use the equation editor
Regarding videos:
• If you are showing a video you made, have multiple ways to show it (embedded, online & flash drive).
• Be careful when embedding it in the talk. Test the embedding to verify it works.
• Have it saved on a flash drive and load it up ready to go BEFORE you start talking as a back-up.
• Know the best time to start/stop the clip so you do not waste time.
Regarding calculations/derivations/equations:
• Don’t show every little step of algebra. Show the starting point and explain how it relates to your pictures,
your drawings, or both if it helps. Skip to the final result. Perhaps, if one step is particularly counter-
intuitive/tricky/important, show that sub-step.
• Have the full calculation worked out in your notes with all the gory details just in case I ask. Note: usually
I can see if it is wrong right away so make sure you double check with me before presenting to make sure
you get it right!
• USE THE EQUATION EDITOR for all equations AND any time you put a variable in a figure. It should
automatically get all your italics and fonts correct. It will make your equations look professional and make
the audience think you pay attention to detail!
38
Regarding plots:
• Correctly use engineering notation to clean up your plots
• Include theory and experiment on the same plot whenever possible
• Don’t put the plot slide up and say, “Here is our plot. Moving on…”. Work up some things to say about it!
• Compare the plot to the theoretical equations whenever possible. You may end up flipping back and forth
between the final result of your theory and the plots. Perhaps you say stuff like “Here the mass is small so
we expect the current to be large. As the mass increases we expect the current to decrease. Our equations
predict current is inversely proportional to the square root of mass. This smooth line shows what the theory
predicts while the dots indicate our experimental parameters. We see the data does not agree well with the
theory for small masses but seems to match up quite well (within about ___%) for most of the large
masses.”
• Point out interesting spots where things don’t match your predictions.
• If your data doesn’t match theory well, explain why the data is noisy or has large errors.
• Have the data table on your flash drive just in case (but don’t include it in the presentation).
Conclusions:
• Remind us of what you wanted to do in your original goal slide
• Summarize super-briefly how
• Propose future experiments that could elucidate related topics or further clarify your own experiment
Regarding the scoring of your presentation:
• 40 points total (10 points for data day, 10 points for prep day, 20 points for presentation day)
• Show up to all meetings for the next three weeks.
• If you miss a day you will lose AT LEAST 10 points per day missed.
• If you miss more than one day additional penalties to your overall course grade are incurred.
• If you do not participate fully on any day, your score is reduced.
• Each group member must do an equal portion of the talking or all group members lose points.
• Stay within the time limits or you will lose points.
• Practice your talk and time it at least twice as a group prior to presenting.
• Follow the slide design guidelines. I reduce points for using too many words, formatting/style,
inappropriate use of italics (or lack thereof), subscripts, superscripts, font size, color schemes, excessive
clutter, slides too sparse, etc. Basically,if something distracts the audience (me) from focusing on the
content, you lose points.
• If the content is wrong, you also lose points.
• Being a decent speaker (eye contact, loud enough, keeping us awake, etc) is worth a few points as well.
• Work hard, show me several rough drafts, verify your equations are correct (and formatted correctly).
• Test drive your talk on the big screen in advance to make sure it shows up well.
39
General tips on Measurement Settings for Mag Field Sensor
TIP: Expect only 1-2 sig figs for your mag field measurements. On 100x precision is 0.1 G; on 10x it’s 1 G.
TIP: For fields larger than 10 G use 10x; for fields less than 10 G use 100x.
MEASUREMENT SETTINGS FOR SOLENOIDS:
Use 1.5A current as measured by the power supply display. Set magnetic force sensor on 100x sensitivity and leave
it. Open DataStudio and install a “Magnetic Field Sensor”. Look on the upper left side of the screen. Drag the 100x
setting to “DIGITS” (on lower let side of screen). If for some reason you reach 9-10 G, switch to 10x on the probe
and in Data Studio. Watch out! If the sensor locks up you might need to re-boot DataStudio and restart the probe.
MEASUREMENT SETTINGS FOR COIL:
Use 1.5A current as measured by the power supply display. Set magnetic force sensor on 10x sensitivity and leave
it. Open DataStudio and install a “Magnetic Field Sensor”. Look on the upper left side of the screen. Drag the 10x
setting to “DIGITS” (on lower let side of screen). When your magnetic field drops below 10 G you can switch to
100x sensitivity. Watch out! If the sensor locks up you might need to re-boot DataStudio and restart the probe.
MEASUREMENT SETTINGS FOR LONG STRIGHT WIRE:
Use 1.7A current as measured by the power supply display. Remember your experimental current is effectively 44(1.7A) = 75A. Open DataStudio and install a “Magnetic Field Sensor”. You will need to use 10x sensitivity
close to the wire and 100x sensitivity a few cm away. You must ensure the sensor switches are visible from above
while you take data. At each grid point you must take both radial and axial measurements to get both components of
the magnetic field. Historically the sensor has not stayed well-zeroed (tared) over all measurements. At the end we
can subtract off any unusual offsets and add that to error.
MEASUREMENT SETTINGS FOR NEODYMIUM MAGNET: Set magnetic force sensor on 1x sensitivity and leave it. Open DataStudio and install a “Magnetic Field Sensor”.
Look on the upper left side of the screen. Drag the 1x setting to “DIGITS” (on lower let side of screen).
WATCH OUT! The most common error is using 10x on the probes and 100x in DataStudio (or vice versa).
Before each set of measurements, ensure the setting on the probe and the screen match.
Just prior to each set of measurements, move the sensor far away from the coil/solenoid without changing the probe
orientation. Then zero the sensor using the TARE button.
After taking measurements on the 10x setting, you might try for more sig figs on any measurement less than 10 G.
WATCH OUT! In many instances it appears changing the sensitivity settings will cause the mag field sensors to
malfunction. If you change the sensitivity setting, move the probe around a bit to verify it is working properly
(verify the probe reading is large close to the wire and small far from the wire). If a malfunction occurs, try closing
Data Studio and setting up the sensor from scratch. Several times this has fixed this problem for me.
WATCH OUT! Earth’s mag field is about 0.5 Gauss where 104 Gauss = 1 Tesla. For magnetic field measurements
smaller than 5 G the earth’s field is potentially a 10% error! For on axis data, one way to reduce this error is to
take a first set of data (yZ[U!). Next reverse the current direction and retake the data (yZ[U). By
subtracting your two measurements and dividing by two the contribution from the earth’s field should drop out and
you should get higher precision measurements. No one ever has time to do this… yZ[U! = ,7+ + y[/n yZ[U = ,7− + y[/n yZ[U! − yZ[U2 = ,7+
40
Option 1: Compare solenoids to coils.
Determine vs along the axis of a
coil. See Sample Graph Type II in lab
manual appendices.
Determine vs along three different
solenoids. Compare each solenoid to
both the coil and to other solenoids.
The magnetic field along the central
axis of a solenoid is given by
Z = 92 + Q212 + Q26 + .
− − Q212 − Q26 + .
where = is the number of turns per unit length, 9 is current, Q is total length of the solenoid, . is the radius of the
solenoid, and is the distance (on-axis) from the center of the solenoid. You should be able to derive the above
result for Z (in terms of ) by finding the appropriate problem in the workbook. The trick is to take your long
solenoid and slice it into coils…
We already derived the mag field along the axis of a coil
,7 = 9.2(. + )/
where . is the radius of the coil, is the number of turns, 9 is current, and is distance from the center of the coil.
To compare the coil to the solenoid, modify the current such that the product 9 for each solenoid equals 9 for the
coil. I think you need to cut the current in half for the coil, if I remember correctly, because the solenoids each have
100 turns while the coil has 200 turns.
Be careful: carefully distinguish between and .
Plot vs showing both experimental and theory values on the same plot.
Perhaps show an additional version with all data sets on the same plot for easy comparison?
Discuss which solenoids, large vs small , have the worst problems with fringing fields. Support your conclusion
by showing the fringing fields on the plots. Compare experiment to theory with a percent difference.
Now measure vs (, ) in the plane of the coil and in the plane at the end of your largest diameter solenoid. Use
Matlab to make a contour plot. I’m basically curious to see how uniform the field at the end of the big radius
solenoid is compared to the coil. Include points both inside and outside the coil/solenoid! See option 2 in-plane
measuring procedure for more detailed instructions on this. Think: the solenoid mag field should or shouldn’t
change much as you vary the location in plane?
Continues next page…
For longest solenoid measure every 2 cm.
For other solenoids (and coil) measure every 1 cm.
Always measure for about 15 cm past each end.
Assume center of each solenoid is = 0 (or = 0).
41
To address:
• Two of your solenoids have the same radius & number of turns but one is twice as long. Should they have
the same magnetic field strength or no? Does your data support your reasoning?
• Two of your solenoids have the same length & number of turns but one has twice the radius. Should they
have the same magnetic field strength or no? Does your data support your reasoning?
• Magnetic energy density in your air core solenoids is given by = xV20. Rank the solenoids according to
magnetic energy density clearly indicating any approximate ties.
• Consider the two solenoids with same length & number of turns but differing radii. The magnetic field
inside a solenoid is not supposed to depend on the length if each solenoid uses the same current. That
means for the same input current they should also have the same magnetic field energy density. Since the
larger radius solenoid encloses more volume that implies you have more total magnetic field energy for the
larger radius solenoid for the same current input. Is this a contradiction? Let’s discuss…
42
Option 2: Circular vs triangular coil of equal perimeter.
Before you do anything, take a guess: if circular coil and triangular coil have same
number of turns and equal perimeter, which should have the larger mag field at the
center?
Use the workbook to determine on the axis of the coil and the triangle distance
from the center. Relate side length i of the triangle to coil radius ..
Hint: they have equal perimeter.
= √398S ii12 + 1i3 +
ON-AXIS: Plot vs along axis of both the circular and triangular coils. See Sample Graph Type II in lab manual
appendices.
IN-PLANE: Plot vs (, ) of both coils and show contour plots using Matlab.
Plan on explaining your derivation in detail to the class. You might have a few more intermediate steps on your
theory slides than other people.
Note: when before you start explaining your derivation, ask the class to take a vote and ask them your initial
question: if circular coil and triangular coil have same number of turns and equal perimeter, which should have the
larger mag field at the center?
At the end of your derivation, state which one should win. If possible, try to think of a heuristic argument (one that
requires no calculation) for which one should win. I might be able to help.
i .
i
On-axis measuring procedure:
Support a ruler along the axis of the coil.
Ensure the field sensor is centered on the axis (not the ruler).
Measure every 1 cm along the axis.
Measure 15 cm from the center in either direction.
Assume center is = 0 (or = 0).
In-plane measuring procedure: Lay the coil flat on the 1 cm by 1 cm grid. Measure at every intersection for a quadrant of the coil.
Include the sign of each measurement. Tabulate your data in Excel without any column headings.
Save the file as a csv (comma delimited) file.
Import the csv file to Matlab and make a contour plot.
Your grid should cover half the triangle but only a quarter of
the circle or square.
Include points both inside and outside each coil!
43
Option 3: Circular vs square coil of equal perimeter. Similar to Option 2 but use a square coil.
Before you do anything, take a guess: if circular coil and square coil have same number of turns and equal
perimeter, which should have the larger mag field at the center?
Use the workbook to determine on the axis of the coil and the square distance from the center.
You will want to follow a derivation similar to the one shown for the triangle in option 2.
The derivation is nearly identical with the following exceptions:
1. Make a new figure.
2. Use ℎ = . 3. Use = 4Z/! instead of = 3Z/!.
4. Relate side length i of the square to coil radius .. Hint: they have equal perimeter.
From several web sources I found
Zpy[ = 92S ii4 + 1i2 +
Do all the same things described in Option 2.
44
Option 4: Helmholtz coils
ON-AXIS: Plot vs for two coils (current both running same direction) with each of five spacings: 0, 0.5, , 1.5, and 2. Perhaps it is best to first show vs for each data set separately then show all curves on a single
plot of vs . Color code the experimental dots to match each smooth theoretical line to make your presentation
more clear. Assuming the current in each coil is running down on the front side (and to the right is ) one finds: //y = ! +
//y = 9.2(. + )/ + 9.2(. + )/
//y = 9. 2 1(. + )/ + 1(. + )/¡
//y = 9. 2 ¢£¤£¥ 1¦. + 22 + 6§
/ + 1¦. + 22 − 6§
/£©£ª
IN-PLANE: Plot « vs (, ) and do a contour plot in Matlab. Use the plane halfway between your Helmholtz
coils. You do not need to do this for other coil spacings…just the Helmholtz coils. This plot shows the uniformity
of the field (or lack thereof) as you get away from the central axis. It gives an idea of the practical usable space that
has constant in between the coils. This will contrast nicely with the talk from Option 1.
More on next page…
On-axis measuring procedure:
Support a ruler along the axis of the coil.
Ensure the field sensor is centered on the axis (not the ruler).
Measure every 1 cm along the axis.
Measure 15 cm from the center in either direction.
Assume center is = 0 (or = 0).
Do the experiment a total of five times: = 0, r , , 1.5¬2.
In-plane measuring procedure (only for = 8 case): Suspend the 1 cm by 1 cm grid halfway between the two
coils, parallel to the plane of the coils, covering 1 full
quadrant of the coils. Measure « at every grid intersection for a 15 by 15 grid.
Tabulate your data in Excel without any column headings.
Save the file as a csv (comma delimited) file.
Import the csv file to Matlab and make a contour plot.
= 0
= ®k-jil¬®-
!
45
Note: be sure to look up Helmholtz coils. Only when the hoop-to-hoop spacing is is this arrangement of two coils
called Helmholtz coils. In your theory section, spend a little extra time explaining your theoretical equation for vs . Focus your energy on the Helmholtz equation case. Show the class UxU« and
UVxU«V are both equal to zero midway
between the two coils for Helmholtz coils. Also, think about why this might be a useful thing for an experiment.
Or, consider teaching the class about a Maxwell Coil…you could build one if you are really crazy.
As an easy variation, you could instead reverse the current direction on one of the coils for = .
This would make for a fun final question to the audience to challenge their understanding. You could say “What
should happen if I use spacing but reverse the current in the left coil?” Then show a final type two graph for two
coils with one current reversed.
46
Option 5: The magnetic field of a single straight wire is very small. To make the field larger, a single leg of a very
large square coil can be considered as an infinite straight wire. This allows us to use the number of turns to safely
multiply the current to a measureable level. Our massive square is about 1.0 m per side with 44 turns.
PLOT 1: Plot vs . from the wire and compare to theory = ¯°±T[ (use radii of 1-12 cm in 1 cm increments). Read
about Sample Graph Type 2 in the lab manual appendices.
Watch out! Remember your experimental current is 449²³³ because of the 44 turn multiplier previously discussed.
PLOT 2: Measure vs (, ) for points on a 1-cm grid surrounding the wire. At each grid intersection, record both
the axial and radial modes. These should be the values of C & F.determine the components of the magnetic field
in the - and -dierctions (along the two axes of your grid). Use these components to determine the magnitude
AND direction of the field at all points in space similar to what was done in the electric field mapping lab.
Create a graphic where the field at each location is drawn as an arrow to scale. Show an arrowhead indicating the
field direction at each point. I think the best way to do this is to read the helpfile in Matlab on the “quiver” function.
Create a second plot with perfect theoretical values for comparison. One way to do this is to create a comma
delimited file in Excel (called csv file). Another method might be to use the “meshgrid” function in Matlab. Better
yet, use the technique described in Option 7. If another group is doing option 7, collaborate with them and farm out
your theory work to them in exchange for giving them some experimental data.
Note: far from the wire the field components should be zero. Close to the wire the field components should be large.
If you notice your mag field changing from positive to negative as you move away from the wire, it is probably not
zeroed well. It is probably not your fault either.
Before taking data do the best you can to zero the sensor about 20-30 cm away from the wire, in the plane of the
grid, in the orientation you expect to use with the grid. If you still end up recording both positive and negative
values, determine the largest negative value for the -component of the field. Add this amount to each C. Do a
similar process for each F. By doing this all fields should be positive and the fields farthest from the wire should
be zero.
Is it reasonable to treat our system as an infinitely long wire? Determine the theoretical magnitude and direction
of the field produced by each of the big square wire’s four segments at a point 10.0 cm from the midpoint of the
right side of the square. Should any of the components cancel or no? Do the other three segments increase or
decrease the theoretical magnetic field close to the wire? Also, compare the size of the field produced by the right
1.00 m segment to one produced by an infinite straight wire. What % error is introduced by treating the wire as
infinite as opposed to only being 1.00 m long?
TOP VIEW
PLOT 1: Measure 12 points
going outwards along solid line
using 1 cm increment. Be certain
to note the distance between the
wire’s center and your first data
PLOT 2: Measure at every grid intersection for a 12 cm by 12 cm grid
(1cm intervals). Be certain to note the distance between the wire’s center
and your first grid point. Note: you may need to adjust the signs of your C
and F data to make the directions in MATLAB match the right hand rule.
SIDE VIEW
• For both plots, be aware of axial/radial mode switch on sensor
• Ensure switches are visible from above while taking data
This ensures you measure C and F instead of «and F
1x
10
x
100
x
wire
Rec
ord
th
e cu
rren
t d
irec
tion j
ust
in
cas
e…
47
Option 6: Transformer response versus frequency and core type (use new transformer equipment)
One year I had some students build some air core transformers with turn ratios of 5:80, 10:80, 20:80, 40:80, 80:80.
Note: there should be four wires sticking out of each tube; two for connecting to the first solenoid and two for the
second solenoid. You will connect one of the solenoids to an AC power source (function generator) operating with
the output cranked to max for a sine wave at 1.00 kHz.
I floated an oscilloscope to measure the max voltage across the primary coil while in operation. At the same time
you can measure the max voltage on the secondary coil. Since the turns ratio is known, we can compare the voltage
ratio to the turns ratio and use a percent difference to quantify any discrepancy.
The turns ratio (´µ) is given the symbol ¶ = ´µ. The transformer equation predicts +·+ = ¶
Tip: Your instructor can show you how to use the measure button on our oscilloscopes to speed up data collection.
Hopefully you can fill out the table below (or something comparable). Notice you will use most of the coils twice.
Sometimes the smaller number of turns will be the primary (called a step-up transformer). Sometimes the larger
number of turns will be the primary (step-down transformer). ¹ (Hz) · ¸ ¶ +· (+) + (+) +·/+ % diff % err
5 80 = +·/+ − ¶¶ × 100% Discuss with
instructor…
10 80
…
80 80
80 40
…
80 5
Finally, try repeating the experiment using an iron core. Repeat again using an aluminum core.
Now repeat with all three cores variations at 10.0 kHz, 100 Hz, and 10.0 Hz.
Compress this data table down using a plot of voltage ratio versus turns ratio. Show the theory as a smooth line and
the experimental data as dots only (no line). Make a second version of this plot using logarithmic horizontal axis.
Note: the voltage across the primary will probably not be exactly the same for each case…I’m not sure so measure it
(and the secondary voltage) for every trial just in case…
Look at real transformers, get some pictures and discuss practical design features/issues. Maybe just talk about the
different types of transformers that exist in the real world. Perhaps read a discussion online about air core vs soft
iron core vs steel. Make sure you cite any web resources. If at all possible, narrow the scope of what you discuss
and try and give the class 1-3 simple takeaways about real life transformers at the end of your talk as a way to tie
your data set into the real world. I’m thinking out loud here. Figure out why aluminum decreases the effectiveness
of the transformer.
Describe how one might convert AC voltage from a standard outlet to 9 VDC for charging a phone. Basically,
explain how a transformer is used in conjunction with diodes and capacitors to create an AC to DC converter. A sub
step is learning about rectification.
More next page…
48
The information below comes from my workbook. Perhaps it will give you some ideas of things to discuss.
Transformers: A transformer is a device consisting of two coils of wire in close proximity. Many transformers
have an iron core. This essentially means you have two coils of wire wrapped on a chunk of iron. One coil can be
thought of as the input. The input coil is call the primary. The output coil is called the secondary. If the magnetic
field created by the primary changes an induced emf is created in the secondary. In this manner electricity is
transferred from the primary to the secondary without and electrical contact!
The equation relating coil voltages (for a lossless transformer) is ∆+∆+! = ! = ¶
where ∆+! is primary voltage, ∆+ is secondary voltage,! is number of turns in the primary, is number of turns
in the secondary, and ¶ is called the “turns ratio”.
Q1: Why do transformers require an AC signal (think function generator or wall socket) connected to the input
instead of a DC signal (think battery or DC power supply)?
Q2: A step-up transformer has a secondary voltage greater than the primary voltage. Which coil, primary or
secondary has more turns? Is the turns ratio greater than or less than 1?
Q3: Based on the previous question, what is the logical name for a transformer with turns ratio less than 1?
Q4: Assume energy losses in a transformer are minimal. We know power relates to energy via o = 9Δ+. Use this
information to determine a relationship between the currents in the primary & secondary.
Q5: Explain why the use of transformers with AC power helps power companies reduce energy waste along
transmission lines.
Q6: In real life, our air core transformers do not transfer all of the energy from the input source to the output source.
Explain why?
Challenge/Going Further: We will soon learn that a coil of wire (or solenoid) has inductance. For an oscillating
source signal operating at frequency ¹ the impedance (») of a circuit with resistance and inductance Q is given by » = + (2S¹Q)
Impedance is measured in ohms.
For an AC source, this impedance can be used to determine the current in the circuit (similar to Ohm’s law) using +yC = 9yC»
Here +yC is the amplitude (or maximum value) of voltage. Similar for 9yC .
The power delivered to such a circuit is given by oy¼ = 9[Z
where 9[Z = !√ 9yC ≈ 0.70719yC.
Rearrange this formula to determine the average power in terms of +yC, , ¹, and Q.
Look online to find a formula determine for the inductance of an air core solenoid (probably tightly wound for all
but the low turns solenoids).
For each data point, use this rearranged formula to determine the power input to the solenoid.
For each data point, use the rearranged formula to determine the power output of the solenoid.
See if the ratio of power more closely matches the turns ratio.
49
Option 7: Attach a magnet to an oscillating glider. Allow the magnet to oscillate near a pick-up coil.
1) Plot pick up coil peak voltage versus angle comparing theory to experiment with a plot similar to Sample
Graph Type II in lab manual appendix.
2) Create a theoretical model for the signal (voltage versus time) and compare theory to experiment with a
plot similar to Sample Graph Type II in lab manual appendix.
First do the procedure outlined in the numbered bullets. Then do the math model described on the following four
pages. Extremely confusing at first but, in my opinion, pretty amazing and cool.
1. Start with the coil area vector perpendicular to motion of the magnet (magnet goes towards edge of
coil…not towards hole in the center). This orientation will be called 90 degrees. The above picture
shows 0 degrees…not 90 degrees. Figure out the max distance you can stretch the glider on the air track
without having the magnet smash into the edge of the coil.
2. Ensure the coil can rotate with its center always equidistant from the magnets closest approach.
3. Now set the coil to zero degrees. Oscillate the magnet. Record a video. From this video you can
determine the period of the oscillation. Also record the amplitude of this oscillation. Use this same
oscillation amplitude for all measurements.
4. Using a Data Studio Voltage Sensor the peak voltage in the pick up coil from the 1st oscillation. Call this
peak voltage ℰyC. Be sure to include the sign of the first peak.
5. Sweep the angle in 10 degree increments from 0 deg to 180 deg. For each angle, predict peak voltage
(relative to peak voltage from 90 deg). We predict the measured voltage should be ±ℰyC cos ¿ depending
on how the electrical cables are attached.
6. Record experimental peak voltage for each angle. For each measurement
a. Keep the center of the coil position unchanged.
b. Keep the oscillation identical.
c. Use the height of the first peak.
d. Be sure to include the sign of each measurement (first peak could be negative).
7. You should be able to plot peak voltage in the pick-up coil vs angle. You should have a theoretical and
number and experimental number for each. In the lab manual appendices see Sample Graph type II for
formatting tips.
8. Repeat experiment with a larger magnet, a longer period (add mass to glider), or a longer amplitude.
9. Measure the magnetic field strength versus distance in 1 cm increments for up to 15 cm from each magnet
face. This data might be useful later.
50
Magnetic flux is Φx = ∙ R = R cos ¿.
For a coil of wire, remember that R points perpendicular to the plane of the coil.
Notice there is no flux when lies in the plane of the coil. Flux is maximum when is perpendicular to the coil.
A changing magnetic flux through a coil induces a voltage called induced emf (ℰ).
The induced emf creates a current that opposes the change in flux. The equation is thus
ℰ = − (R cos ¿) = − (S. cos ¿) where, for a circular loop of wire, R = S..
If you use a coil of wire instead of a single loop the equation becomes
ℰ = − (S. cos ¿) where is the number of turns. The coil of wire used to obtain the induced voltage is sometimes called a “pick-up”
coil.
In our experiments today we will try various trials at various fixed angles. We will not be changing the angles
during any given trial. The number of turns and radii of our coils will remain constant between all trials. Therefore,
for our experiments today
ℰ = −(S. cos ¿)
Using the chain rule, we can convert the time derivative to a spatial derivative
ℰ = −(S. cos ¿)
ℰ = −(S. cos ¿)
In theory you should be able to plot vs for the permanent magnet using the mag field sensor. If possible, use
small increments (perhaps 1 cm…perhaps 2 mm…no clue here). In this special case, do an exponential trendline.
This trendline is an approximate function for (). Note: to use an exponential curve fit in Excel, only use non-zero
values of . In my fake data below I assumed the sensor stopped giving non-zero values for > 6cm.
z (m) B (T)
0.01 0.0074
0.02 0.0041
0.03 0.0025
0.04 0.0006
0.05 0.0001
0.06 0.0001
UxU« is the derivative of your trendline. In
my fake data case I find  = −. ÃbÄÃÅ.Æ.
B = 0.03e-97.39z
R² = 0.94
0
0.004
0.008
0.012
0.00 0.02 0.04 0.06
B (T)
z (m)
B vs z
51
You should be able to estimate the size of by either making a video and using Tracker or by using the equations
discussed in the “oscillations crash course” discussed next.
To get () the speed of the magnet at any distance from the coil:
First get the spring constant of each spring. Simply
hang a mass on it and let it come to equilibrium. See
figure near right. We see = «st. A lot of our springs
have ≈ 3 ÇO.
When you place springs in parallel (by adding springs to
either end of the glider) the effective spring constant is
simply the sum of all springs. ÈÈ = ! + +⋯
See figure far right showing springs in parallel. Expect ÈÈ ≈ 6 ÇO
Oscillations terms: R = Amplitude = maximum distance from equilibrium in meters É = Period = time to complete one cycle in seconds ¹ = Frequency = the # of cycles per second in Hertz Ê = Angular frequency = 2S¹ in rad/sec Ê = 2S¹ = 2SÉ
For a horizontal mass-spring system where the mass is released
from rest on the right side of the figure the following equations
are valid:
Strictly speaking, ZFZ/ = !Zh[7Z +7U[ .
Fortunately for us, with our super light springs Zh[7Z ≪ 7U[ . Therefore ZFZ/ ≈ 7U[ .
I made some plots on the next page showing what we might expect for an oscillation. Look at those now.
= R cos(Ê) = −ÊR sin(Ê) Ê = Í ÈÈZFZ/
7 = R yC = −ÊR 7 = 0
p
p =
p
R = yC
R = yC
52
keff (N/m) msystem (kg) ω (rad/sec) A (m) Period (s)
6 0.600 3.16 0.100 1.99
t (s) z (m) v (m/s) 0.00 0.100 0.000 0.05 0.099 -0.050 … … …
2.00 0.100 -0.013
In our magnet and coil experiment, we want to know
the velocity of the magnet when it is distance from the
coil…not distance from the equilibrium position. If
is the distance from the equilibrium position to the
center of the coil we want to use = + . Notice:
when is negative (left of equilibrium), the calculation
for gives a number smaller than .
-0.100
-0.050
0.000
0.050
0.100
0.00 0.50 1.00 1.50 2.00
z (m)
t (s)
z vs t
p
coil
53
We can now put this together with UxU« and try to predict the shape of the electrical signal!
ℰ = −(S. cos ¿) ℰ = −(S. cos ¿)(−2.92mÄÎÏ.W«)(−ÊR sinÊ
Note: at each value of in the UxU« equation, we can sub in
= = R cosÊ
This gives
~ = −S. cos ¿−2.92mÄÎÏ.W(UwÐ NÑÒ(Ó/))(−ÊR sinÊ
I will assume the following constants for my fake oscillation data.
keff (N/m) msystem (kg) ω (rad/sec) A (m) Period (s)
6 0.600 3.16 0.100 1.99
If we have an oscillation with amplitude R = 10cm = 0.10m, the distance between the coil and the equilibrium
position must be slightly larger. Let us assume = 12cm = 0.12m.
For now I will assume S. cos ¿ = 1m for simplicity. In theory you could determine this number using
measurements. I find ℰ = −2.92mÄÎÏ.W(.!w.! NÑÒ(.!Ô/))(0.316 sin(3.16)) I plugged all that crap into Excel and found this:
Ok…what is the point of all this? I thought, if you were really crazy, you could follow the same procedure I did and
predict the size and shape of your electrical signal. Then you could export the data from the scope (or use a data
studio voltage sensor) to get experimental voltage versus time data.
Ultimately, put the experimental curve from the scope on the same plot as your theoretical plot and see how close
the dots are to the curve. I think this is a lot of work but might be a fun capstone project to your three semester
physics sequence. I personally did it just because I thought it was possible and if we could actually predict the shape
of this insane looking electrical signal!
-0.030
-0.020
-0.010
0.000
0.010
0.020
0.030
0.00 0.50 1.00 1.50 2.00
EMF (V)
t (s)
EMF vs t
54
Option 8: MATLAB MODELLING
My idea was to have you figure out how to download, use, and modify the bits of code on the following link:
https://d-arora.github.io/Doing-Physics-With-Matlab/mpDocs/cemB03.pdf
Related mscript here:
https://github.com/D-Arora/Doing-Physics-With-Matlab/blob/master/mpScripts/cemB03.m
The presentation would essentially explain equation 1 on page 2 of the first link, what a saturation level is and why
it is needed, the idiosyncrasies of the code starting at page 5 "The number of calculation..." and continuing until
halfway through page 7 (or all the way to page 9 if needed).
Start by making your own versions of the plots shown in Figure 7 (page 11), Figure 13 (page 16), Figure 15 (page
18), Figure 17 (page 21).
As proof of your understanding, consider the arrangement of currents (all out of the
page with equal magnitude) shown at right. The wires are arranged on the corners
of an equilateral triangle such that one wire is on the -axis and two are on the -
axis. Create a theoretical equation for the magnetic field strength along the -axis.
Next, modify the code so you can generate the contour plots, quiver plots, and (). Try to modify the call of the quiver function in the code to show not just the
direction but also the magnitude.
Collaborate with members of the group doing Option 5 to get some experimental
data. They use a wire carrying the equivalent of 75 A. Put the wire in the bottom
left corner of a square 12 cm on a side. Modify the call of the quiver function in
the code to show not just the direction but also the magnitude.
As further proof of your understanding, simulate the magnetic field surrounding a slab of current and/or a coax.
One could make a coax with a single wire at the center surrounded by a ring of currents. Ensure the sum of the
currents in the ring is equal in magnitude and opposite in direction to the central wire. One could build the slab out
of a larger number of adjacent wires each carrying a tiny fraction of the slab’s total current. Compare your results to
that predicted by an infinite slab with uniform current density. Compare to a single wire carrying the same total
current. Adjust the saturation level to see some detail as you see fit. Modify the code to reduce computation time to
a manageable level.
More ideas for things to talk about:
• What is a saturation level and why is it useful/necessary for Biot-Savart computations? Include images of
the same data with two different saturation levels to help emphasize this point.
• Does this simulation handle locations inside the wires?
• What are the units of the parameters?
• How is computation time affected by partition size? Use the case of a single wire with partition sizes of
1000, 2000, 3000, 4000, 5000, and 8000. Tabulate the simulation run time for each case. Create a plot
showing run time versus partition size. Show quiver plots the of 2k and 4k partition size to demonstrate
what the extra computation time gets you in terms of resolution.
• How is run time affected by the number of wires? Using a partition size of 2k make a plot of run time
versus number of wire for 1, 2, 3, 4, 5, and 8 wires.
• Challenge: How might one parallelize this code?
55
OPTIONS 9 & 10: Python modelling (put two groups on it, one focuses on motion, other field calculations?)
First focus on creating a simulation to determine the magnetic field at all points in space near a vertical current
carrying wire. The key to success is to make simple sub goals.
1) First draw a vertical wire Q = 20m tall centered on the origin. Imagine this is the wire carrying current 9 = 1.00A directed upwards (+Õ). Define a single point of interest (POI). Draw an arrow at this POI
representing the size of the magnetic field. To do this, split the wire up into about = 40 segments. Be
sure your code allows us to vary both the length of the wire & the number of segments…this is
crucial. The contribution to the field at the POI is given by the BS-law: = 94S ∙ i × .. = 94S ∙ i × ..
In this equation i = ‖i‖ is the length of a wire segment and . is displacement from that segment (the
source) to the POI. Tip: you will probably need to scale the arrow to make it visible! Recall i is defined
to point in the direction of current but has units of length. The total field would be the integral (in our case
a sum) over all segments. Tip: remember to offset source position by ds/2!!!
Compare your computational result to the exact result of Cy,/ = 92S¬ ¿ with a percent difference. Here ¬ is the distance to the wire and is tangent to the circle of radius ¬.
TIP: Be careful…I think ¬ = kmk¹ℎmim Ø + + √ + You have to figure out which one.
TIP: You won’t be able to see the arrow unless you choose an appropriate scale factor. THINK: arrows
between 1 and 10 will fit on the screen fairly well.
TIP: Include a saturation value Zy/ . If your code computes a value of > Zy/ you want the code to
display an arrow corresponding to Zy/ instead of .
TIP: Use an if statement to prevents the code from computing if you input a POI with . = 0 or ¬ = 0.
2) Make a copy of your previous code for safe keeping.
3) Create an alternative display style for the magnetic field arrow. Make your arrow a fixed size (say 1 or 2 in
length) and then align the arrow with the field at whatever POI you choose. This time, scale the arrow’s
color vector or opacity (or both) to correspond to . For example, if > Zy/ you could make the color
red (vector (1, 0, 0)). If < Zy/ us white (vector (1, 1, 1)) with kl¬®- = xxÚÛÜ. Read a help file…
4) We want our simulation to produce results within 5% of Cy,/ for a wide range of POI’s. We also want
the code to run in a timely manner. It is worth testing this cheesy little code in a few ways to get a feel for
how to optimize performance version precision.
a. First set the POI to (1, 0, 0) with Q = 20 and = 200. Think: the POI is 1 m from the wire and
we are splitting the wire up into i = 10cm long segments. Record the % error in your sim.
b. Keeping the POI at (1, 0, 0) & Q = 20, change until the error is approximately 5% and record
this info. If it cannot be done, record the smallest % error you can get.
c. Next set the POI to (5, 0, 0) with Q = 20 and = 200. Change until the error is approximately
5% and record this info. If it cannot be done, record the smallest % error you can get.
d. Now reset to POI to (1, 0, 0) with Q = 20 and = 200. While keeping K fixed at 0.1,
determine what length of the wire gives 5% errors and record this info. If it cannot be done,
record the smallest % error you can get.
e. Fix the segment length at i = 0.01. What value of Q gives 5% error for the POI at (1, 0, 0)?
f. Hopefully now you have some idea of how small i needs to be to make your simulation within
5% of the exact value for various positions near the wire.
56
5) Create a circular loop of wire of radius = 5 in the -plane using 36 segments. Create a sensor arrow
and check the field using an exact result from class along the axis of the loop. Note: for points off-axis, we
can only check our results with conceptual reasoning about approximate field size and direction.
6) Now go back to the straight wire code. Work on getting arrows at several points along the -axis. Try
points spaced out every 2 units…. with 5 positive and 5 negative. Hopefully you remembered to put in the
if statement that skips drawing the arrow if = ∞. Save this code and make a copy.
7) Extend your work to make arrows at points on a gird in the -plane.
8) Extend your work to make arrows at all points on a 3D lattice.
For the first half of your presentation:
• What was your goal? What purpose does your code serve in terms of physics?
• Show a quick screen shot of the key parts of the code (BS law computation)…do not drag us through the
code line by line. Tip: create some figures to help you explain the computation to the class. Emphasize
that an integral and a sum are essentially the same thing if you slice up i small enough.
• Talk about the limitations of your code. Summarize your findings regarding how close or far you can be to
the wire and how it affects your choice of i. Try to make quantitative statements by include percent
errors when discussing. For example, don’t say things like “it looked good” or “it was really close”….try
saying things to get errors under 5% we required blah blah blah”. Note: I understand it is complex as there
are many cases. Think of your job as finding a good balance of giving people a feel for your code’s
precision while not boring us to death. I’m estimating you spend no more than 2 minutes on this part.
• TIP: Show screen shots of the code with the various cases. Draw Cy,/ and yhh[C at the same time on
the screen in different colors. While you drone on about %’s to make me happy and get a good grade,
students can see the arrows and get a gut feeling for your code’s precision without getting bored.
• Show us a screen shots of the full 3D lattice of arrows. Include current up vs current down. TIP: show
both on same slide and ask class which one has current up. Explain to the class what a saturation value
is and why it was needed. Explain super briefly how you implemented your arrows. If people ask more
questions, tell them to talk to you afterwards because I said so. Move on.
• Now make two screen shots where the current runs horizontally. For one case have current running to the
right. For the other case have current running to the left. Tip: don’t change your code…just run it then
rotate the image before each screen shot. For each screen shot, first make the current up or down in your
original code then rotate the image before taking your screen shot! On each screen shot include a 2D lattice
of mag-field arrows in the -plane. Ask the class which current is running to the right to make them
practice the right hand rule. Wait thirty seconds for them to figure it out and answer. It will seem like a
long time but wait them out. TIP: put both screen shots on same slide so it is easy to compare.
Now I expect you to do something extra & cool!
Go to the next page to see how I expect you to fulfill the other half of your requirements.
57
Try some or all of these:
A) Make simulations of the coils used by every other option. These include straight wire (already done),
triangular, square, Helmholtz (two circular coils of equal radius separated by distance ), and solenoids
(many circular coils right next to each other). You will need to coordinate with each other group to figure
out the best way to add in your sims to their presentations. Discuss successful strategies with your
instructor and other group members on the presentation prep days. It would neat to see a doubly wound
solenoid vs a singly wound solenoid. Also, some solenoids of different sizes corresponding to the ones we
have in the lab.
B) Also, it would be neat to simulate the field at the midpoint of a side of a giant square compared to that of a
straight wire. Get some % errors for the comparison. This might actually help me redesign some lab
equipment to save space and make Angus happy! Goal: determine the smallest size square loop I can use
such that the field near the midpoint of one side has negligible contributions from the other sides? Think:
once fields drop below about 0.5 G, the earth’s field dominates contributions from the wire regardless of %
errors from theory.
C) MOST FUN? Use magnetic force to update the momentum of a moving charge on one of the following:
1) Start simple with a charged particle with velocity in the -plane while points out of the plane.
Calculate the speed required for circular motion and verify your sim works correctly. Also, estimate
the maximum appropriate time step size by calculating the period of the orbit. You’ll probably need a
time step at least 100 times smaller than the period for the motion to be smooth. Try time step sizes of Þ!, Þ, Þ, Þ! & ÞÔ. To check each case, draw a circle on the screen of the correct radius then run your
charge (with a trail). Make screen shots of each case for your presentation. Save your code.
2) What should happen if you modify the speed to 15% above your previous result? Have some
fun…take a guess before you run the code! Repeat with the speed 15% below. During your
presentation, hopefully you can show these sims or make vids?
3) Simulate charged particle released from rest from the -axis with constant electric field upwards and
magnetic field out of the page.
4) Simulate a charged particle entering a region of constant magnetic field at some small angle and have
it travel in a corkscrew path.
D) Coolest combo of everything: Try a web search for magnetic mirror. A magnetic bottle is made from two
magnetic mirrors. One could think of the earth’s magnetic field as similar to the field of a giant bar
magnet. A giant bar magnet could be modeled by a solenoid (a bunch of coils next to each other).
Simulate a solenoid with some charges moving around it with a wide range of velocities (vary both
magnitude & direction). See if any get trapped by the fringing fields around the solenoid. Perhaps a
simpler way to test this (I have no clue if it will work) is to try something like the picture below. I’m
guessing such an orientation of coils (all currents running same way) would cause mag field lines
somewhat like the blue lines shown. I do not know this for certain. If those fields lines are correct, charges
with appropriate velocities should bounce back and forth without ever touching the coils! This is called a
magnetic bottle. If I recall correctly, this effect can be seen at the very ends of fluorescent light tubes. This
effect occurs in the van Allen belts surrounding the earth. From Wikipedia: Ions from the solar wind are
trapped in the van Allen belts preventing the destruction of atmosphere but also endangering low earth orbit
satellites.
58
59
Motors
Apparatus (up to 14 groups of 2 for this lab): Scissors, Template (provided by instructor), tape, 2 large paper
clips per group, 5m motor wire per group (28-30 gauge has worked in the past, higher gauge means less weight and
easier to rotate but also easier to break when sanding off enamel), Two 8” segments of stiff, non-insulated wire
(have available entire box of wire), Sandpaper, dissection needles, DMMs, DMM red leads, DMM black leads,
PASCO Science Workshop 750 Interface & Power Supplies, 1-Ohm power resistors
A current-carrying wire running perpendicular to a magnetic field experiences a force. A current-carrying loop of
wire exposed to the same field may feel multiple forces on different parts of the loop. Using the right hand rule
determine the direction of the force experienced by each segment of wire in the loop pictured below.
Assume a battery is connected to points a and h so that the current shown is produced. Assume that the loop of wire
is free to rotate about the vertical axis shown running between segments ab and gh. What will happen to the loop of
wire at the instant shown? Explain below.
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
Again the wire is free to rotate about the vertical axis. Suppose the wire was able to rotate 180º? Assume the wires
twist up a little bit at the base causing current to run in the opposite direction. What happens to torque on the loop?
______________________________________________________________________________
______________________________________________________________________________
ab
bc
cd
de
ef
fg
a
c
e
f g
h
b
d
Note: which forces are equal in magnitude and
which cause a significant torque?
B-field
points to
the LEFT
Direction of Force on Wire Segment:
60
In order to make a motor useful one typically wants to reverse the direction of the current flow every time the loop
rotates 180º. To do this one cannot have the wires being permanently connected at a and g. Wires or pieces of
metal are used to lightly brush against the wire segments ab and gh. These wire segments are called, ingeniously,
brushes. The wire segments ab and gh are called commutators.
Visit the following website to get a feel for how the motor works. The animation is useful.
http://www.walter-fendt.de/ph14e/electricmotor.htm
On the following page is a top view of what your constructed motor will look like. Note that upon a half-revolution
the commutators will be connecting to opposite brushes. This causes the current to still continue clockwise through
the loop and thus causes the loop to experience a torque which keeps the loop rotating.
Construct the motor using the template on thick paper. Tips on
construction:
• Be extremely cautious about properly sanding wires for
making electrical contact. Once you put the motor
together it is difficult to re-sand a wire without
mutilating your commutators.
• Be careful cutting out the box. Small errors propagate
and cause shoddy construction. Shoddy construction
creates unnecessary friction which makes the motor
unable to run.
• Every loop counts so don’t get sloppy. Be sure you
always wrap the loops in one direction.
• When you get ready to run the motor be careful to
TURN OFF THE POWER SUPPLY IF THE MOTOR
ISN’T RUNNING. If the motor doesn’t turn and the
power is on you might ignite it.
Stage 1
1) Cut out both the box portion and armature portion of the
motor template on solid lines.
2) Fold both parts on dotted lines.
3) Punch holes in black dots.
4) Tape the box portion into a shoe-box shape and tape it securely (see instructor’s motor).
5) Fold the armature over on itself (see instructor’s motor).
6) Pierce a straightened paper clip thru the black dots on the armature. This paper clip will be the axle of your
motor so try to make it as straight as possible.
7) A tiny piece of tape can be used to keep the armature from sliding around on the axle.
Stage 2
1) Take the 5m portion of wire and sand about 3” on each end.
2) Check if the ends are adequately sanded with a DMM. Use the diode setting and listen for the beep. A
solid beeping tone indicates all the insulation has been properly stripped. Check both sides of both ends for
solid tone. Repeat the sanding until this is the case.
3) Wrap the 5m wire around the armature over and over in the same direction until all 5m has been used up.
The small portions of stripped wire should remain along the axle.
4) At this point your armature should look approximately like the picture on the next page. The stripped
portions of the 5m wire are going to become your commutators. Try to bend them into the shape shown in
the picture. Try to make the commutators symmetrical.
Magnet
Brush
Commutator
61
Stage 3
1) Now run one of the “brushes” (the non-insulated wire) over the axle and commutators and thru the black
dots. Run the second “brush” under both the axle and the commutators.
2) Align the brushes so they make contact with the commutators when the commutators and armature are
vertically oriented.
3) Your commutators and brushes must make good electrical contact when the wire is vertically oriented
regardless of which commutator is touching the top or bottom brush.
4) Check the electrical connection using the diode mode of your DMM. See stage 2 step 2 for instructions.
Be sure to rotate the armature 360 degrees. You should hear a beep every time the armature is vertically
oriented.
Stage 4 (READ ALL INSTRUCTIONS FIRST OR YOUR MOTOR MAY BURST INTO FLAME!)
1) Connect one lead of a power supply to a small resistor (say 1 Ohm).
2) If the power supply has a current limiter, set it to maximum. Set the voltage to 5 V. Verify your power
supply is giving you 5V with the DMM.
3) Connect the resistor to one of the brushes.
4) Connect the other lead of the power supply to the other brush.
5) “Kick start” the motor by flicking it lightly with your finger. You will probably see a few sparks. If the
motor stops you should IMMEDIATELY turn off the voltage or you will burn out your motor and cry.
6) If you have trouble try increasing the voltage to get the motor started.
7) Optimize your motor by tweaking the commutators and brushes to reduce the minimum operating voltage.
To improve the motor you want the brushes in contact with the armatures for as much time as possible
without causing too much friction.
8) Check for and reduce friction on the axle.
Challenge: Get your motor running with one 1.5V battery (without the resistor) and estimate the horsepower of
your motor. Hint: to estimate power you need ∆E/∆t. Try to have a lifting contest with your rivals/peers.
axle armature
Commutators:
insulation stripped
Commutators lined up
with black dots
Taped but
stripped wires
not touching
axle
Taped but
stripped
wires don’t
touch axle
One brush over
the top, other
beneath
Securely fasten uninsulated brush wires to box with tape.
Make sure end of wire accessible to connect to battery.
62
If you get you motor to twitch but not turn over you get 7 points out of 10.
1 extra point for the first page of the lab handout correctly completed.
1 extra point for running the motor on a power supply and completing the data table below.
1 extra point for completing the conclusion questions (on next page).
Optional: 1 extra point for completing computation of duty cycle and RPMs (discussed after conclusion questions).
Scores over 100% on this lab are possible.
CONCLUSION QUESTIONS ON THE NEXT PAGE.
Use appropriate sig figs &
include units
Power supply voltage used to lift paper clip
Average current at that voltage
Power input to motor (Pin=Iavg∆V)
Mass of paper clips lifted
Distance paper clips lifted
Energy expended lifting the paper clips
Time to lift the clips
Power Output of Motor (Pout=E/t)
Efficiency (η=Pout /Pin)
63
Conclusions:
1) For typical student motors with no resistors in series, the power supply usually shows an applied voltage of 3.0 V
with average current of 0.30A. This implies the resistance coil should be about 10 Ω…right? Let’s learn more…
a) Determine the actual resistance of the coil using ,7 = fÐ . Assume you used 28 gauge wire (0.321 mm in
diameter) that is 5.0 meters long to make your coil. Copper has a resistivity of 17 nΩ⋅m.
b) The resistance found in the previous part is the true resistance in the coil. Assume your duty cycle is about
20% (current is actually flowing in the motor only 20% of the time). What is the expected average current
if a 3.0 V potential difference was applied to this resistance? Hint: use Ohm’s law and multiply by the duty
cycle to get the average expected current.
c) In practice, the coil spinning in the presence of an external
magnetic field creates back EMF. If we ignore the internal
resistance of the power supply, we might model this scenario using
the circuit shown at right. Notice the back EMF of the motor
effectively cancels out part of the applied voltage from the power
supply. In equation form we could write Δ+h\[ZhhF − ày,á = -y¼7y,/yCh[7/,7 Use this equation to estimate the size of the back EMF for the your
motor.
More on Back EMF and motor limitations: A coil moving in the presence of a magnetic field can experience an
induced EMF. In this case, the motor coil has an induced EMF that is called a back EMF. This back EMF acts
somewhat like a battery that OPPOSES current flow in the motor. So in your motor the power supply tries to spin
the coil one direction while the back EMF tries to oppose that spinning motion. This is why it is important to
unplug the motor if it stops spinning – when the coil is motionless no back EMF produced and much more current
flows (which burns out the motor). The back EMF effectively reduces the voltage applied to the circuit. Read the
section in your book on back EMF.
There will also be drag on the motor as it spins. It turns out that both the drag and the back EMF increase with
rotation rate. Friction between moving parts will also tend to inhibit the motion of the motors. As the current flows
the wires get hot and reduce the current from an initial maximum. All these factors (some more than others) will
contribute to setting an upper limit on the speed of the motor. Furthermore, one cannot simply apply more and more
voltage to speed it up or the wires will melt!
ày,á ∆+h\[ZhhF
,7
64
The duty cycle and RPM measurements are optional for 1 extra point
(scores over 100% possible). First finish all other parts.
To measure duty cycle, run the motor with a 1-ohm power resistor in series off
the power supply. Use an oscilloscope to measure the voltage across the 1-ohm
power resistor just like you would with a DMM. See the figure at right for what
the circuit might look like. Finally get your motor running.
Once the motor is running, the scope will display voltage versus time data for the
power resistor. The signal will be pretty messy looking. Try adjusting the
horizontal divisions knob on the far right side of the scope until you see what
looks like 4 to 8 messy blobs on your screen. Hit the RUN STOP button to
capture a cycle.
Look closely at the messy patterns. Hopefully you see a messy blob, then a
second messy blob with different shape. Hopefully you notice a third messy
blob which looks nearly identical to the first. The fourth messy blob should look like the second blob. The idea is
the odd blobs correspond to your commutators in one orientation while the even blobs correspond to the
commutators rotated 180 degrees.
If the voltage is zero across the resistor (in series with your motor) we know the motor is not receiving power.
If the voltage is non-zero, the motor is getting power.
Duty cycle is the percentage of time the motor receives power.
To measure RPMs, determine the period of your motor’s waveform.
This is the time for one full revolution.
Remember: the time for a revolution should correspond to the time between the start of the first messy blob to the
start of the third messy blob!!!
Do a conversion to get RPMs.
Duty Cycle
(% of time current runs in motor)
RPMs
Motor
R
+ Pwr Supply
-
O-scope
Ch. 1
65
Oscilloscopes and Function Generators
Apparatus: oscilloscope (and power cable) with one probe, function generator (and power cable) with BNC to
alligator clip adaptor, grey 2-to-3 prong adapter for wall sockets, one 100 Ω resistor, one 1kΩ resistor, and one 10
kΩ resistor
Today’s lab is designed to get you familiarized with oscilloscopes and function generators. Oscilloscopes are used
to measure time-varying electrical signals (often called waveforms). Function generators are used to produce time-
varying electrical signals (waveforms). By the time you finish today’s lab you will hopefully be able to do the
following:
• Produce a desired waveform of given type, amplitude, and frequency
• Properly connect a circuit involving an oscilloscope and function generator
• Accurately evaluate various waveform parameters using the oscilloscope
THE BASICS OF THE FUNCTION GENERATOR
The function generator is used to create waveforms with various amplitudes, shapes, and frequencies. A brief
overview of the controls is as follows.
• At bottom right are several metal connectors called BNC connectors. The rightmost of these is labeled
OUTPUT. That is where you connect the BNC to alligator clip cable so the output can be connected to
other electronic equipment. Do this now.
• The dial adjacent to the OUTPUT connection labeled OUTPUT LEVEL controls the amplitude of the wave
produced. Turn this up to about the 12 o’clock position now (this is roughly halfway to max).
• At the top right are three buttons indicating square, sine, or triangle wave. One of these buttons must be
depressed or no wave will be produced. Press the sine wave button now if it is not already depressed.
• All along the top you see a row of buttons indicating the range of the function generator. Frequencies
between 50 and UP TO 500 Hz can be produced while the 500 button is depressed. Again, no wave will
be produced unless one of these buttons is pressed so press the 5 kHz button now (allowing waves between
500 and 5000 Hz to be produced).
• In the bottom left are the frequency controls COARSE and FINE. Again, turn each of these knobs to about
the 12 o’clock position. In general, before each adjustment of the frequency always reset the FINE adjust
to the 12 o’clock position. This allows you to make fine adjustments to the frequency either up or down
which is handy.
• For our function generators there is a -20dB button. Press this button now if it is not already depressed.
This button is used to decrease the output level. For most of the waves we’ll be using today the small
output level is perfect. If you needed a larger output you would ensure this button is not depressed.
• Look for a button marked DC offset. We will use this button later but for now make sure it is NOT
depressed. The DC offset simply adds a set voltage to any wave produced. For example, adding a 3V DC
offset shifts any waves produced by the function generator up by 3V.
THE BASICS OF THE SCOPE
Turn on the scope and allow it a minute to power up. While it is powering up locate the following groups of
controls: triggering, horizontal, vertical.
• The horizontal controls adjust the scale/location of the graph on the screen along the horizontal (time) axis.
• The vertical controls adjust the scale/location of the graph on the screen along the vertical (voltage) axis.
• The Ch1 – Ch4 inputs all measure voltage. Notice that each channel has its own dial so each channel can
have a different vertical scale. The inputs are color coded so that the yellow channel uses a yellow line on
the screen.
• The probe is the small black device with a metal tip (positive lead) and tiny alligator clip (negative lead).
The positive lead is just like the red cable coming off a DMM measuring voltage while the alligator clip is
just like the cable coming out of the black (or COM) terminal of the DMM.
66
AUTORANGE
The scope’s AUTORANGE button will cause the scope to continually adjust the horizontal and vertical axes of the
scope to get the waveform visible on the screen. If you get lost at any point today, try hitting AUTORANGE and
usually the scope does a good job of resetting itself and getting a waveform on the screen.
RUN STOP
The RUN STOP button will take a snapshot of the waveform (in case it is fluttering too much too read). You should
not be using the RUN STOP button at all today. We did use it when characterizing your motors as those signals
were volatile and hard to read without the RUN STOP button.
PROBES & PROBE CALIBRATION
Check if the probe is on 10x or 1x mode; there is a switch on the handle of the probe that says 10x or 1x. Hit Ch 1
menu. On the right side of the scope screen, verify the probe mode on screen (1x, 10x, etc) matches the switch on
the probe itself. Before using your probe check the calibration of the probe. To do this, connect a probe to CH1.
The tiny hook lead should then be connected to the top piece of metal next to the screen labeled PROBE ADJUST.
The ground lead (alligator clip) should be connected to the bottom piece of metal next to the screen. If all goes well
you should see a square wave appear (you may have to press the AUTORANGE button once or twice). If the wave
does not appear square some screwdriver adjustments will need to be made with your instructor. This should be
checked at the beginning of each lab for each probe. Be sure to check the probe reads a very square wave on the
scope screen on both the 1x and 10x settings before proceeding.
DIVISIONS
The distance between the dotted lines on the scope screen are called divisions (div). The small tick marks are used
to help you read to the 10th of a division. Somewhere on the scope screen you should see indications of the sec/div
used for the horizontal axis and V/div for the vertical axis. These tell you the units for the graph as well!
MEASURE
Try pressing the MEASURE button near the top of the scope. This activates the keys adjacent to the scope screen.
Try pressing the buttons next to the screen and you’ll see various things you can measure including the period,
frequency, peak-to-peak voltage, max voltage, etc of the waveform on the screen. These are very useful tools but
for this class you are also required to determine the period, frequency, and peak-to-peak voltage using the V/div and
sec/div listed on screen. On a lab practical I will often give a photo of the screen so you are not able to use the
MEASURE button to do the work for you.
TRIGGERING
Triggering sets the level and slope (positive or negative) of the graph at a particular spot on the scope screen.
COUPLING
The three most common choices for coupling are DC, AC and ground. Using the ground coupling mode removes
any signal and displays a flat line at 0V. This trick helps you identify what height on the screen corresponds to 0V!
Remember, since one can use the vertical position dial to translate the graph up or down 0V will not always be at the
center of the screen. If a waveform has only positive voltages, it is convenient to set 0V to the bottom of the screen.
DC coupling shows both AC and Dc components of a waveform. AC coupling subtracts off any DC offset.
GROUNDING ISSUES BETWEEN THE SCOPE & THE FUNCTION GENERATOR
Ground is considered to be the electric potential of the earth. By plugging in a three-prong device (say the scope) to
the wall socket it is tying its ground to the electric potential of the earth. When the scope and the function generator
are both plugged into wall sockets they both have a ground connection. If one then connects the ground of the scope
(alligator clip on the probe) and the ground of the function generator (black cable) to different points in a circuit a
short circuit problem is introduced. The circuit will not produce results as it was designed.
Some function generators produce outputs that are not grounded to the earth even though they are plugged in using
three-prong adapters. One nice thing about the floating ground output is that it will not cause a short circuit (as
described above) when the oscilloscope is connected. HOWEVER, until you know more, do not in general float the
ground of electronic devices. Also, for now, only connect one probe or channel of the scope to a circuit at a time. If
you wish to read further: http://www.aspowertechnologies.com/resources/pdf/Floating%20Output.pdf
67
Worksheets to turn in start here. 1) Which mode of the probe (10x or 1x) is affected by the adjustment screw? ________________________
2) What is graphed on the vertical axis? Voltage or time? _________________________
3) What is graphed on the horizontal axis? Voltage or time? _______________________
4) What happens when you turn the dial labeled Horizontal Position (the little Horizontal dial)? Does the
scale of the graph change or is the graph translated?
_____________________________________________________
5) What happens when you turn the dial labeled Vertical Position (the little Ch 1 dial)? Does the scale of the
graph change or is the graph translated?
_____________________________________________________
6) What happens when you turn the Horizontal sec/div dial? Scale change or translation?
_____________________________________________________
7) What happens when you turn the Ch 1 V/div dial? Scale change or translation?
_____________________________________________________
8) Effectively the _______________________________dial can be used to zoom in or out on the voltage axis
of the graph while the ______________________________ dial can be used to zoom in or out on the time
axis. Meanwhile the ______________________________ dial is used to align the voltage axis with a
convenient point on the screen while the _____________________________ dial is used to align the time
axis with a convenient point on the screen.
68
• Connect the 1x probe to the PROBE COMP (small brass loops near bottom right of screen) on the scope.
• Hit AUTORANGE and wait for the scope to react.
• Now adjust the V/div and sec/div until you get the largest possible graph on the screen that clearly shows at
least two full wave forms. If the edges of the waveform are right at the edge of the screen, decrease wither
the V/div or sec/div one increment even if the graph becomes a little smaller (this clearly shows the
periodicity of the wave).
• Now use the Horizontal Position dial to line up the graph with the axes on the screen.
• Verify 0 Volts aligns with the horizontal axis at the middle of the screen. You might need to hit Ch1 Menu
and fiddle with the “coupling”. If not aligned, use the Vertical Position dial to make the adjustment.
9) Make a sketch of the screen below.
On the t-axis, include both numbers and units (get this from the sec/div on screen).
On the V-axis, include both numbers and units (get this from the sec/div on screen).
On the sketch, clearly indicate the period by drawing a horizontal line (with arrowheads) the length of the
period.
Label this line with T = # with units (use the number obtained from your sketch…not the measure button).
Clearly indicate the peak-to peak voltage by drawing a vertical line (with arrowheads).
Label this line with Vpp = # with units (use the number obtained from your sketch…not the measure button).
Compute ¹ = !·[7U and R = ¬lj-m = ! Δ+hyá/hyá. Think about how many sig figs are reasonable from reading the plot. Don’t forget units!!!
f=____________________________ A=____________________________
69
• On the function generator, make sure the -20db button is depressed.
• On the function generator, set the FINE dial to the 12 o’clock position.
• Now press the 5 kHz button on the function generator (allowing frequencies UP TO 5 kHz).
• Near the far right of the function generator are three buttons for sine, square, and triangle wave. Ensure the
sine wave button is depressed.
• Set the function generator to roughly 1 kHz frequency using the COARSE dial. Make it exactly 1 kHz
using the FINE dial.
• Turn up the OUTPUT dial all the way (approximately the 5 o’clock position).
• Connect the BNC-to-alligator clip cable to the output BNC connector on the front panel of the function
generator. Notice there is a red and a black output on this cable. Black is tied to ground.
• First connect the tip of the scope probe to the Red output of the function generator. Hit the AUTORANGE
button and wait for the scope to respond. You should see a graph appear even though the ground connector
of the scope is not connected to the Black ground of the function generator. This is because the connection
is essentially already made.
10) In the last step, it appears that a complete circuit is not made but the two devices are actually connected in
two places. One is where the positive or red leads touch. How are the scope and the function generator
grounds connected? Hint: consider the power cables plugged into the wall for each piece of equipment.
__________________________________________________________________________________________
__________________________________________________________________________________________
Now incorrectly connect the Black cable from the function generator output to the tip of the probe. Try connecting
the ground of the probe (little black alligator clip) to the Red output from the function generator. No graph will
appear on the screen because the potential difference across the probe is zero (both tip and ground are connected to
ground). That means the voltage produced by the function generator is going somewhere else. In general this is not
a good idea as you could fry out some electronics this way. In this special case no damage is done.
• Leaving the cables incorrectly attached (Black to tip and Red to ground), turn off the power to the function
generator.
• Unplug the power cable. Then connect the grey, 2-to-3 prong converter on the end of the function
generator power cable and plug it back in. Be sure the metal ground connector (small metal loop on the
adapter plug) does not touch any metal on the outlet.
• Turn the function generator back on and, if necessary, hit AUTORANGE on the scope. You should now
see the graph on the scope even though the grounds appear to be incorrectly connected. Since the function
generator is not connected to any ground at the electric socket, it is said to be floating.
In general, never float the ground of anything unless you are a competent electrician or have further training.
If you have ground issues, ask your instructor for help rather than risk electric shock. The 2-to-3 prong
converter is designed to be safely used only when the grounding loop is securely fastened to the screw at the
center of the outlet. Failure to do this can be lethal.
Finally, reconnect the probe in the proper fashion and remove the 2-to-3 prong converter. One student in a lab
called these “death plugs”. Try a web search for “cheater plugs” if you want to learn more.
70
Adjust the wave until the output is 2 Volts peak-to-peak (which is 1V amplitude). These questions assume you are
adjusting the triggering just after hitting the Autorange button. Now go the section of controls on the scope
labeled TRIGGERING. Try adjusting the triggering level and note what happens to the waveform. Look at the very
edge of the screen (right or left) and you might see the little arrow indicating the triggering level. Look at the top or
bottom edge of the screen to see the triggering position (where the waveform starts). Try pressing the slope button.
In the sentences below, circle the best answers to fill in the blanks.
11) The trigger _(level or slope)_ is the Voltage of the waveform at the _(left side, right side, or the origin)__.
12) The trigger _(level or slope)_ determines the slope of the waveform at the _(left end, right end, or origin)_.
Adjust the triggering level until the arrow is above the top of the waveform. Notice the graph gets blurred out. If
the waveform never reaches the triggering level, the scope does not have a clear starting point for making the graph.
• Set up a random waveform (each student must use a unique, randomized waveform).
• Try using the 500 Hz, 5 kHz, or 50 kHz range on the function generator.
• Then use the course adjust dial of the function generator to select some oddball frequency.
• Lastly, adjust the output level dial of the function generator so the amplitude is randomized as well.
• Use the V/div & sec/div dials to make the graph as large as possible while still having at least one full
waveform appear on the screen.
• Use the horizontal and vertical position dials to line up the waveform in some convenient location. You
may also want to check the ground level by hitting Ch1 Menu and looking at the “Coupling” mode.
13) Sketch the graph below including numbers and units on both the V-axis and t-axis. Tip: you could snap a
photo of your waveform so you can sketch it while the other students set-up their waveforms…
Sketch the period and peak-to-peak voltages on the graph as done in the previous sketch.
Obtain estimates for the frequency and amplitude of the waveform (think about sig figs & units).
fscreen=____________________________ Ascreen=____________________________
Verify these values using the MEASURE button from the scope (include units).
fmeasurebutton=_______________________ AfromVppmeasurebutton=_____________________
71
Now use the function generator to set up a 0.5V amplitude sine wave at a frequency of 10 kHz. Remember: you’ll
need to ensure that the -20dB button is depressed. Hit AUTORANGE on the scope. Then hit the Channel 1 Menu
button. Find a button just to the right of the screen associated with coupling. Press the button a few times to see the
various options of coupling. Notice that Ground shows you where 0V is located. The most commonly used modes
are “DC Coupling” and “AC Coupling”.
Turn on the DC offset button of your function generator by pressing it; it should remain depressed after pressing.
Adjust the dial for the DC offset on the function generator.
Try the same thing while in AC coupling on the scope.
14) While the scope is in DC coupling mode, how does rotating the DC offset dial (of the function generator)
clockwise affect the graph on the screen?
_______________________________________________________________________________________
15) While the scope is in AC coupling mode, how does rotating the DC offset dial (of the function generator)
clockwise affect the graph on the screen?
______________________________________________________________________________________
The DC offset on the function generator adds or subtracts a constant voltage to the waveform. Note that this
differs from simply turning the vertical position dial on the scope. If you add too much DC offset to a waveform
it will eventually distort because of the limitations of the function generator.
For example, by adding a 2V DC offset to a 2V amplitude wave implies the wave will oscillate between 0V and 4V
(not -2V and +2V). This property could potentially be useful (or harmful) to your electronics.
On the contrary, suppose you had the same 2V amplitude wave without any DC offset. You can use the position
dials to cause the waveform on the screen to appear as an oscillation between 0V and 4V. However, the actual
output of the function generator is still oscillating between -2V and 2V. If, for example, your electronics cannot
handle a reverse bias (negative voltage) you would be in trouble.
72
Measuring in a real circuit:
For the last portion of the assignment you will connect a circuit like the one
shown. First we will derive some theoretical results about this circuit. Then you
will make measurements with the scope (just like a DMM) and compare the
experimental values to the theoretical values with a percent difference.
The symbol at the very bottom of the figure with three horizontal lines in a
triangular shape indicates the ground of the circuit. R is your load resistor (the
resistor you are measuring). The scope is in parallel with the load resistor and is
measuring voltage (just like you did with a DMM). The function generator has its internal resistance explicitly
shown in figure as Rfg. Note: for our function generators Rfg=50 Ω. Rfg is in series with the load resistor R. Since
the scope has a very large resistance (10 MΩ or more) in parallel with the load resistor R, it has negligible effect on
the circuit.
16) Verify scope resistance is negligible by finding the equivalent resistance of the 10
MΩ scope in parallel with the 10 kΩ resistor. Determine the percent change from
10 kΩ when the scope is placed in parallel with it. Show work and put your
answers in the boxes.
17) We can model function generator as an ideal source with potential differnce
Vfg in series with a small internal resistance of Rfg. Determine the theoretical
current 9/n in this circuit in terms of Vfg, R, and Rfg. Show all work for credit.
Determine the theoretical voltage Δ+r/n across the resistor R in terms of Vfg,
R, and Rfg.
R (in kΩ)
when 10 MΩ
in parallel
with 10 kΩ
percent
change from
10 kΩ
9/n
Δ+r/n
scope R
Vfg
10MΩ 10kΩ
R
Rfg
Vfg
73
• Now actually set up the circuit as shown.
• Use the controls on the function generator to create a square wave
that oscillates between 0.0 V and 1.0 V operating at 1.0 kHz. If you
adjust the dials while the scope is connected (and use the
MEASURE button) you should be able to set this up fairly quickly.
You’ll need 1.0 V peak to peak plus 0.5 V DC offset. Check it with
the scope before connecting it to any resistor.
o If you are having trouble, you might hit the Ch1 Menu
button and verify the Probe Voltage on the scope screen is set to the same setting (1x vs 10x) as
the switch on the probe itself.
o You may also need to change the coupling mode (also on screen after hitting Ch1 Menu). To use
the DC offset you probably want to have the scope set in DC coupling.
• Once this is done, disconnect the scope probe from the function generator leads.
• Obtain three resistors with resistances of about 0.1 kΩ, 1 kΩ, and 10 kΩ.
• Measure the values of these resistors with the DMM and record the measured values (not the approximate
values) in the table below.
• Predict the theoretical voltage across the load resistor using your previous results.Now select one of the
resistors and connect it to the function generator leads.
• Now connect the probe leads from the scope so as not to cause a short circuit ground error. Remember the
alligator clip is ground on the scope so it must go to ground of the function generator.
• Measure the voltage across the resistor. By this I mean look at the scope and read off the max voltage (or
use the MEASURE button). It will not be exactly 1.0 V but should be somewhat close to it. Fill in this
value in the Δ+rCh column of the table below.
• Determine a %difference between the theoretical and experimental values.
• Repeat this process for the other two resistors.
²³³ (kΩ) Δ+r/n (V) Δ+rCh (V) %diff
18
19
20
21) Suppose you have a function generator operating with a 1.00V amplitude with no
load. The internal resistance of this function generator is 50.0 Ω. What load resistor
R should be used if we want 99% of the function generator’s 1.00 V amplitude across
the load? Again, assume the internal resistance of the function generator is in series
with the load resistor. Include units and use three sig figs.
21)
scope R
Vfg
74
75
RC Lab
Part I: RC Circuit
Below you see a picture of two circuits (and a giant hand holding a stopwatch…or is it tiny?): one with a battery,
resistor and a capacitor & one with only a resistor and a capacitor. In this experiment you first charge the
capacitor(s). Higher capacitance implies higher amounts of stored energy. The energy will dissipate according to an
exponential decay with time constant τ =RC (look up discharging a cap in your text).
Get the materials. You will be running the first part of this experiment with the following RC combinations:
C=0.1 F with R=4.7 kΩ (measure the actual value of resistance with a DMM)
C=0.1 F with R=1 kΩ (measure the actual value of resistance with a DMM)
C=0.025 F with R=4.7 kΩ (measure the actual value of resistance with a DMM)
Calculate the time constants and predict which combination should decay the fastest/slowest.
First charge the cap. Connect a resistor and a capacitor in parallel (start with the slowest RC). Connect the cap
(and thus also the resistor) in parallel with 3 batteries provided by your instructor (or use a 4.5V current limited DC
source). Measure the voltage across the cap to ensure it has reached 4.5 V (or the max voltage put out by the
batteries).
Now disconnect the batteries/power. Immediately after disconnecting the battery portion of the circuit you should
start a stopwatch and simultaneously record the DC voltage across the resistor with a DMM. With your stopwatch
tabulate the voltage versus time ON THE RESISTOR ONLY. Take values for a length of time equal to 5τ at
roughly 10-30 second intervals. If you want to you could try to get this going using Data Studio but I expect your
final results to done using Excel.
Repeat the same experiment with different RC combinations. Use a 0.1 F cap with a resistor of about 4.7 kΩ, a
0.1 F cap with a resistor of about 1 kΩ, and a 0.025 F cap with a resistor of about 4.7 kΩ. Measure all resistors to
get accurate values for the resistance.
Check! Keep track of your data and label it well so you don’t get confused. By now you should have three data
sets showing one column of t and the other column being V. Be sure each data set is label with both the R and the C
used for that data set.
3
Batteries
Cap
R
Cap
Resistor
DMM
?
?
Charging the capacitor Discharging the capacitor
76
Part 1: Graphing Mayhem!
Plot the voltage on the resistor versus time (VR vs. t) for the three sets of data. Be sure to show the appropriate
curve fit on the plot with regression coefficient and equation on a linear-linear plot. Determine τ from the trendline
equation.
Plot lnVR vs. t for the three sets of data. Perform the appropriate curve fit to the data and notice what happens to
the curve defined by your data. Determine τ from the trendline equation. Which graph gives a more accurate value
of τ? See checklist for what to include for your report.
Part II: Another RC Circuit
Select a capacitor and resistor set such that τ is about 0.1 msec (try R=1kΩ and
C=0.1µF). Connect a function generator with a square wave function (if none is
available use the PASCO interface box) to R and C as shown to the right. Set
the function generator to produce a 1 kHz square wave with 2.0 V amplitude and
DC offset of 1.0 V (you may need to hit the 20dB button). To do this first
connect the function generator to the scope directly (without the R and C) and
adjust the dials until it is reading appropriately. Use DC coupling (press the Ch1
menu button to get to this).
Connect the leads of the 10x probe from Channel 1 as seen in the diagram. Use
the coupling switch set to DC mode (not AC mode like last week). The ground
should be connected between the resistor and capacitor (not between the resistor
and the function generator). The function generator will need to be floated
from ground using the grounding plug adapters.
Now connect a second 10x probe to channel 2 of the scope. You need not use
the ground connection because the ground connection is already between the
resistor and capacitor from channel 1. Connect the lead of the probe to the
opposite side of C in order to read the voltage across the capacitor. Display both
channel 1 and channel 2 of the scope at the same time. What do you notice
about the voltages on the cap and resistor? Do they add up to the voltage across
the function generator? Try using the INVERT button (press the Ch2 menu button to get to this). What happens?
Should you be using the INVERT button? Hint: read conclusion questions.
SKETCH the waveforms seen on channel 1 of the scope. You may choose to use a flash drive to export the waveform to your computer and print it in lieu of
sketching.
Be sure to modify the time scale and voltage scale so that the sketch scope displays the largest possible
waveform that includes one decay.
Label your sketch with the scale of each axis.
Specifically note the time required for the voltage to decrease to half of the maximum value (t½).
Determine the time constant from t½.
C
R
F.Gen
O-scope
Ch. 1 Ch. 2
77
Conclusions:
Part I:
1) How do the time constants relate to the fitting parameters of your graph? Specifically, for each RC combination
list the theoretical and experimentally determined time constants (τ) with percent differences. Explain both
predicted trends and discrepancies.
2) Could we have plotted the voltage on a capacitor instead of the resistor; what would differ and what would
remain the same? Explain and defend your answer.
3) What sources of error dominate the experiment? How could the experiment be improved?
Part II:
4) In Part II you noted the time to drop to half of the maximum voltage. Use this information to determine the
capacitance of the cap you used in the experiment (and if available compare to the known value with a percent
difference). Hint: consider τ=RC and also include the input impedance of the function generator if appropriate. 5)
Why is it reasonable to compare an inverted channel 2 signal to a non-inverted channel 1 signal (hint: how should
VR + VC compare to Vfg)?
6) When comparing the two signals, what happens to the voltage (absolute value) on the resistor over time? What
happens to the voltage (absolute value) on the capacitor over time? What is true about the sum of these two voltages
over time?
Checklist:
Raw data (with borders) with variables(not words) as column headings (in italics) and units (not italics)
Three graphs (use graph which determines τ most accurately: VR vs. t or lnVR vs. t)
On each graph show trendline, R2, and calculation of τ Sketch/print-out of waveforms for part II showing approximately 1-2 periods of the waveform
Annotate your sketch/print-out with the time for the VR to drop by 50%
Determine the time constant from t½
Answer all conclusion questions using full sentences
78
79
Series ã8ä Circuit
Apparatus: function generator, BNC to alligator clip cable, O-scopes, TWO
probes per table, LCR meter, breadboards (but no jumper cables), DMMs and
probe leads
Each lab group requires one each of the following circuit components
• 10.0mH inductor
• 3.3mH inductor
• 0.1mF ceramic capacitor
• 1.0kΩ resistor
Each lab group is assigned one of these (lab assistant’s choice)
• 4.7kΩ resistor
• 4.3kΩ resistor
• 240Ω resistor
• 220Ω resistor
Goal: Pair up with another lab group. One lab group must use the large inductor, the other group the small inductor.
Assemble the series Qç circuit shown. Simultaneously measure the function generator (+Z[,háÄhá) and the
voltage across the resistor (+r) for a particular source frequency. While keeping source voltage fixed, sweep source
frequency. For each frequency, tabulate peak current (-yC) and phase angle (è). Make plots showing theoretical
and experimental values (use Sample Graph Type II formatting). Make the horizontal axis logarithmic. Repeat all
this for a second Qç circuit using a different value of . At the end, share your two plots with the other group.
This section describes how to get experimental data from a scope.
A typical scope screen might look a bit like the plot at right. Assume each horizontal division is 0.2 msec.
Assume each vertical division is 2.0 V.
Source (pk-to-pk) voltage is 10 vertical divisions (+Z[,háÄhá = 20.0V).
Peak-to-peak voltage across the resistor is about half that (+r = 10.0V). The period is about 14 horizontal divisions (Þ = 2.8msec). The frequency is ¹ = !Þ = !.ê×!ëvs = 357Hz.
The resistor peaks just before the source.
Recall current is given by 4ír .
Therefore, -() leads +Z[,() if resistor voltage leads source voltage!
Here current leads the source voltage (or source voltage lags current).
This is a capacitively dominated circuit (ELI the ICE man).
We expect the phase angle should be negative.
The time between the two peaks is about 1.5 horizontal divisions ( = 0.30msec). The phase can be determined using a ratio
î/Þ = ïÔ° This gives
è = 360° ?Þ = 360° (0.30msec)(2.8msec) = 360° (1.5boxes)(14boxes) = −39° If -() leads +Z[,() we assume è is negative; the circuit is capacitively dominated (ELI the ICE man).
Note: for our circuits all phase angles should be between −90°&90°. If you get a phase angle outside these limits,
reconsider the direction you traveled when determining the time between peaks.
C
L
R
Red
Black
~
O-scope
Ch. 2 Ch. 1
V_R
V_source
Þ = Æóôõ = . ö÷øùú
ûü = . ýóôõ = þ.þ÷øùú
80
To determine theoretical values, we will use the following set of equations:
Relating peak-to-peak source voltage (+Z[,háÄhá)
to max source voltage (+Z[,yC) +Z[,yC = +Z[,háÄhá2
Relating peak source voltage to peak current -yC = +Z[,yC»
Total Impedance (note: has units of Ω) » = + ( − )
Phase angle (typically given in degrees… even though one must almost
immediately convert to radians to do anything useful) è = tanÄ! −
Capacitive Reactance (note: has units of Ω) = 1Êç = 12S¹ç
Inductive Reactance (note: has units of Ω) = ÊQ = 2S¹Q
WATCH OUT!!! In the previous two formulas, the symbol ¹ stands for source frequency (the frequency of the
function generator). It IS NOT resonance frequency ¹ = !T√.
The equations listed below are not necessary to complete this lab.
I am including them as it may be helpful to your understanding of the series LRC circuit.
Source voltage as function of time +Z[,() = +Z[,yC sin(Ê) Current as a function of time
Notice a positive phase angle -() = -yC sin(Ê − è)
WATCH OUT!!! In many resources you will see -() = -yC sin(Ê) & +Z[,() = +Z[,yC sin(Ê + è). Shifting -() to the left by è (relative to +Z[,()) is the same as shifting +Z[,() to the right by è.
WATCH OUT!!! In the previous two formulas, the symbol Ê = 2S¹ stands for source frequency (the
frequency of the function generator).
WATCH OUT!!! People often say source frequency for Ê even though it is actually source angular frequency.
Peak voltage across the capacitor +yC = -yC
Voltage across the capacitor as a function of time
Hint: ELI the ICE man…+() must lag current…thus the extra − T +() = +yC sin(Ê − è − S2)
Peak voltage across the inductor +yC = -yC
Voltage across the inductor as a function of time
Hint: ELI the ICE man…+() must lead current…thus the extra + T +() = +yC sin(Ê − è + S2)
81
Reminder of what you are supposed to get done:
• Simultaneously measure the function generator (+Z[,háÄhá) and the
voltage across the resistor (+r) for a particular source frequency.
• While keeping source voltage fixed, sweep source frequency.
• For each frequency, tabulate peak current (-yC) and phase angle (è).
• Repeat all this for a second Qç circuit using a different value of .
• At the end, swap plots with another group so each group has 2 plots plots
for 4 distinct cases of Q, , and ç.
Now making plots:
• Make the horizontal axis logarithmic for each case.
• Make a plot of -yC vs Ê including both theoretical and experimental
values (use Sample Graph Type II formatting). You might consider trying
to put all 4 experimental and all 4 theoretical curves in one plot. If too
hard to read, perhaps plot 1 or 2 data sets per plot.
• Make a plot of è vs Ê including both theoretical and experimental values (use Sample Graph Type II
formatting). You might consider trying to put all 4 experimental and all 4 theoretical curves in one plot. If
too hard to read, perhaps plot 1 or 2 data sets per plot.
Conclusion Questions:
1) Suppose you wanted to increase the resonance frequency. You have only one resistor and one inductor but you
do have a second identical capacitor. Would you want to put the capacitor in series or parallel with the first
capacitor to increase the resonant frequency? Explain why.
2) As resistance in the circuit increases, will the resonant frequency increase, decrease, or stay the same? Explain.
3) Consider the plot of -yC vs Ê. Generally speaking, it looks a bit like a hill or mountain. What happens to the
shape of this hill/mountain as resistance increases? Discuss both hill/mountain height and width.
4) The function generator has 50 Ω of internal impedance that was not accounted for in our theory. Does this
omission affect our calculations of each Ith? Approximately what % error is introduced by ignoring the 50 Ω
internal impedance when R = 1.0 kΩ? Compare this result to your graph…does it make sense?
Note: ignoring the 50 Ω of internal impedance has no effect on our measurement of Iexp. This is because we used
Iexp=VR/R. Here VR was the voltage across only the resistor of in the circuit and R is just that resistance so this ratio
should be correct.
This is a good trick to remember. Whenever you need to know the current in a particular part of a circuit, you can
always measure the voltage across a resistor in that part of the circuit and divide by R to get the current.
C
L
R
Red
Black
~
O-scope
Ch. 2 Ch. 1
82
83
RLC Filter Circuit
To the right is a picture of an RLC filter circuit. Fill in the table below and
estimate the resonant frequency of the circuit. Select an L and a C that provides a
resonant frequency of about fres ~ 10 kHz (try L=3.3mH and C=0.1µF).
Select a 1 kΩ resistor. Measure the actual values of L, R, and C with the LCR
meter or a DMM. List your values below with units.
Version A: Use the same L and C with a second, smaller R (say approx 250 Ω).
Version B: Use the same R and C with a second L.
L=____________ C=_____________ R1=_____________
R2=___________ ωres = ___________ fres = ____________
When completing the table below be sure to watch out for Hz vs. rad/sec!
Component Eqt’n for
Reactance (χ)
χ at low ω ?
(big or small)
χ at high ω ?
(big or small)
χ w/ small L or C ?
(big or small)
χ w/ large L or C ?
(big or small)
Inductor (L) χL=
Capacitor (C) χC=
Equation for total impedance in this circuit: » = Í + ¦
2ÓÄ u
6§
Z at lo ω’s:____________ Z at ω0:_______________ Z at hi ω’s:______________
Note that Ith=Vsource/Z while Iexp=VR/R. Notice that as you change the value of ω, Ith should change! In today’s lab
we want to plot how Ith changes as ω changes and compare that to experimental data.
To get data:
Remember that the function generator uses f while the above eqt’ns use ω.
Determine your target frequencies. These are the approximate values of the frequencies you will use on the
function generator.
0.05f0 0.1f0 0.2f0 0.5f0 0.7f0 0.8f0 0.9f0 f0 1.1f0 1.2f0 1.3f0 1.5f0 2f0 5f0 10f0 20f0
Turn on the function generator at a frequency near the theoretical resonance f0 on the function generator.
Be sure the output cable is connected to the output BNC connector!
Set the amplitude to a convenient size (say 10.0 V amplitude). To verify this, first connect the function
generator directly to the scope without any resistors, etc. Then use the MEASURE and AutoRange buttons
on the scope.
Remember that VRmax on the resistor is half of the peak to peak value of the sine wave on your scope. You
may be able to set the scope to read out this information for you. It is best to determine Vmax by dividing
Vpk-pk by two using the scopes measure button. This avoids having to think about any spurious DC offset
caused by the function generator.
Use the scope’s MEASURE button to record the frequency and amplitude for a frequency near each target
frequency. Don’t worry about hitting the exact target frequency.
L
R ~
C
O-scope
Ch. 1
Red
Black
84
Tabulate I vs. ωωωω for both sets of data. Hint: make separate columns in XL to convert f to ω and VR to Iexp for your
plots.
Create a column of theoretical data. Use your equation for Z and the function generator voltage (Vsource) to
determine an equation for Ith in terms of R, L, C, Vsource, and ω.
Plot I vs. ωωωω for both theory and experiment on the same chart. Repeat for the second set of data. This graph
should look like Sample Graph Type II in the appendices. Use a logarithmic scale for the ω-axis by right-clicking
that axis and selecting “format axis”. Look around for the Logarithmic check-box. Show the theory as a smooth
line with no points and show the experiment as points only with no smooth line. Since you have more than one set
of data on the same graph you’ll need a legend to explain the various curves.
For conclusion:
1) As frequency increases, the current in the impedance of the circuit asymptotically approaches what value?
2) Suppose one replaced the function generator in the circuit with an ideal battery with emf E & no internal
resistance. Sketch a picture of such a circuit and clearly label the current through each element of the circuit and
voltage across each element of the circuit. Each current will be either E /R or 0 and the voltages will be either E or 0.
3) In theory we said that the LC in parallel had infinite impedance at resonance. This is not strictly true. One can
determine an estimate of the impedance at resonance by comparing Ith at resonance to the theoretical maximum of
Vsource/R. Taking the ratio gives
9/ny/[Z9/nyC = +Z[,»y/[Z +Z[, = »y/[Z Use your experimental data to come up with an estimate for the experimental value of Z at resonance.
4) Compare the LC being in series (the previous lab) to the LC being in parallel (this lab). Sketch plots of Z vs ω
and I vs ω for each type of circuit. Clearly label your work so it is easy to follow.
Notice that ignoring the 50Ω of internal impedance has no effect on our measurement of Iexp. This is because we
used Iexp=VR/R. Here VR was the voltage across only the resistor of in the circuit and R is just that resistance so this
ratio should be correct. This is a good trick to remember. Whenever you need to know the current in a particular
part of a circuit, you can always measure the voltage across a resistor in that part of the circuit and divide by R to get
the current.
5) Notice, however, that ignoring the 50Ω of internal impedance causes us to miscalculate our theoretical values.
For each resistor used, will you be over or underestimating? By what % for each resistor?
85
Radio Lab
An AM radio can be constructed out of very simple materials. The heart of the radio circuit lies in its ability to
select one station (one frequency) at a time. Analogous to the resonant frequency of a swinging mass, some electric
circuits have resonant frequencies. These circuits will be excited by electromagnetic waves very easily at this
frequency (or station) while other frequencies (other stations) cannot excite currents in the circuit.
How does a radio pick up a signal in the first place?
A loop of wire (an antenna) sits on your radio. Electromagnetic waves pass through the loop. The magnetic field of
the wave is changing from pointing one way to pointing the other. In other words, the antenna (the loop of wire) sits
in a region where the B-field is changing. A changing B-field creates a changing magnetic flux through the loop of
wire. A changing flux through the wire creates an EMF and thus a current in the antenna (the loop of wire).
Basic Alternating Current (AC) circuits
An AC circuit is a circuit with a sinusoidal current running through it. This means the current first flows clockwise
then counter-clockwise then clockwise then…it keeps alternating. Typically we say
tVV ωsinmax= or tII ωsinmax=
Where V is voltage, I is current, ω is angular frequency (recall fπω 2= ) and t is time. Remember that ω
essentially means how fast the current alternates.
Resistors and batteries are not the only important components in AC circuits. Typically both inductors (coils of
wire) and capacitors (remember the parallel plates?) appear in AC circuits. Inductors are labeled with the letter L
and henry (H) is the unit of inductance. Capacitors are labeled with C and are measured in Farads (F).
Though we cannot strictly speak of the “resistance” of a capacitor or inductor, we can speak of the reactance of
these. Reactance is represented by the letter χ. Reactance, used only in AC circuits, has units of Ohms and is thus
analogous to resistance. The reactance of inductors and capacitors are given by eqt’ns in your text. Find the eqt’ns
and fill in the following table.
Component Eqt’n for
Reactance (χ)
χ at low ω ?
(big or small)
χ at high ω ?
(big or small)
χ w/ small L or C ?
(big or small)
χ w/ large L or C ?
(big or small)
Inductor (L) χL=
Capacitor (C) χC=
Using the capacitance of the capacitor in today’s lab find the reactance for ω=3000 Hz.
______________________________________________________________________________
Using L=40mH find the reactance at 3000 Hz.
______________________________________________________________________________
86
These questions refer to the LRC Series Circuit seen to the right.
1) At very low frequencies (ω ~ 0) how much impedance does the circuit have?
___________________________________________________
2) How much current can flow?
___________________________________________________
At very high frequencies (ω ~ ∞) how much impedance does the circuit have?
How much current can flow?
__________________________________
Do you think there will be a frequency that allows a maximum of current to flow? If so what is the name of this
frequency?
__________________________________
Sketch a plot of current flow in the circuit as a function of ω.
Indicate the resonant frequency ω res on the graph. I am
interested in the shape of the graph; don’t worry about getting
proportions correct as that is impossible with the amount of info
I’ve given you.
The resonance frequency ωres is determined (in an LRC-series
circuit like the one pictured above) by setting the inductive
reactance (χL) equal to the capacitive reactance (χC). Using this
information find the resonance frequency as a function of L and C
(in terms of L and C). List it below and show your work.
The AM frequency band runs from 530-1600 kHz(note that this is f and not ω). Determine roughly the value of
inductance (in H) you will need in order to set the resonant frequency to the following stations. Fill in the table to
the right. Hint: first determine what size capacitor we are using in this lab!
Build your radio per the instructions given. Look up what stations are available in the Santa Maria area. Try and
listen in to some of them. You can tune your radio by adjusting the inductance. The chart you just made may or
may not help in tuning.
f (not ω) in kHz Required L in H
530
700
900
1100
1350
1600
Current
ω
Imax
C
L
R~0Ω
AC
Voltage
~
87
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
More resistance in a circuit broadens the resonance peak of your graph. Do you want a broad peak or a narrow one?
Does this jive with the amount of resistance in your radio circuit? Explain.
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
Good Job!