lab

Contents

1 Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Free Fall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 The Earth’s Gravitational Field . . . . . . . . . . . . . . . . . . . . . . . . 165 Newton’s Second Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 Air Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 Vector Addition of Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 Vector Addition of Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 Acceleration In Two Dimensions . . . . . . . . . . . . . . . . . . . . . . . . 2610 Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2811 Conservation of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 3212 Conservation of Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . 3613 Conservation of Momentum in Two Dimensions. . . . . . . . . . . . . . . . . . . 3814 Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4015 The Moment of Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4216 Absolute Zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4617 The Clement-Desormes Experiment . . . . . . . . . . . . . . . . . . . . . . . 4818 The Pendulum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5019 Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5420 Resonance (short version for physics 222) . . . . . . . . . . . . . . . . . . . . . 5821 Standing Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6222 Resonances of Sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6423 Static Electricity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6824 The Oscilloscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7025 The Speed of Sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7626 Electrical Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7827 The Loop and Junction Rules . . . . . . . . . . . . . . . . . . . . . . . . . 8228 Electric Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8429 Magnetism (Physics 206/211) . . . . . . . . . . . . . . . . . . . . . . . . . 8830 The Dipole Field (Physics 222) . . . . . . . . . . . . . . . . . . . . . . . . . 9231 The Earth’s Magnetic Field (Physics 222) . . . . . . . . . . . . . . . . . . . . . 9632 Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10033 The Charge to Mass Ratio of the Electron. . . . . . . . . . . . . . . . . . . . . 10234 Energy in Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10635 RC Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10836 LRC Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11237 Faraday’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11638 Electromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12039 Impedance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12440 Refraction and Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12641 Geometric Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13042 Two-Source Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . 13443 Wave Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13644 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14045 The Photoelectric Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 14446 Electron Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14847 The Hydrogen Atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15448 The Michelson Interferometer . . . . . . . . . . . . . . . . . . . . . . . . . 160

Contents 3

Appendix 1: Format of Lab Writeups . . . . . . . . . . . . . . . . . . . . . . 162Appendix 2: Basic Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . 164Appendix 3: Propagation of Errors . . . . . . . . . . . . . . . . . . . . . . . 170Appendix 4: Graphing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172Appendix 5: Finding Power Laws from Data . . . . . . . . . . . . . . . . . . . . 174Appendix 6: Using the Photogate . . . . . . . . . . . . . . . . . . . . . . . . 176Appendix 7: Using a Multimeter . . . . . . . . . . . . . . . . . . . . . . . . 178Appendix 8: High Voltage Safety Checklist . . . . . . . . . . . . . . . . . . . . 180Appendix 9: Laser Safety Checklist . . . . . . . . . . . . . . . . . . . . . . . 182Appendix 10: The Open Publication License . . . . . . . . . . . . . . . . . . . . 184

4 Contents

Contents 5

1 Interactions

Apparatus

single neodymium magnet . . . . . . . . . . . . . . . . 1/grouptriple neodymium magnet . . . . . . . . . . . . . . . . 1/groupcompasstriple-arm balance . . . . . . . . . . . . . . . . . . . . . . . . 2/groupclamp and 50-cm vertical rod for holding balance upstringtapescissorsheavy-duty spring scalesrubber stoppers

Goal

Form hypotheses about interactions and test them.

Introduction

Why does a rock fall if you drop it? The ancientGreek philosopher Aristotle theorized that it was be-cause the rock was trying to get to its natural place,in contact with the earth. Why does a ball roll if youpush it? Aristotle would say that only living thingshave the ability to move of their own volition, so theball can only move if you give motion to it. Aristo-tle’s explanations were accepted by Arabs and Euro-peans for two thousand years, but beginning in theRenaissance, his ideas began to be modified drasti-cally. Today, Aristotelian physics is discussed mainlyby physics teachers, who often find that their stu-dents intuitively believe the Aristotelian world-viewand strongly resist the completely different versionof physics that is now considered correct. It is notuncommon for a student to begin a physics examand then pause to ask the instructor, ‘Do you wantus to answer these questions the way you told us wastrue, or the way we really think it works?’ The ideaof this lab is to make observations of objects, mostlymagnets, pushing and pulling on each other, and tofigure out some of the corrections that need to bemade to Aristotelian physics.

Some people might say that it’s just a matter ofdefinitions or semantics whether Aristotle is corrector not. Is Aristotle’s theory even testable? Onetestable feature of the theory is its asymmetry. TheAristotelian description of the rock falling and the

ball being pushed outlines two relationships involv-ing four objects:

According to Aristotle, there are asymmetries in-volved in both situations.

(1) The earth’s role is not interchangeable with thatof the rock. The earth functions only as a placewhere the rock tends to go, while the rock is anobject that moves from one place to another.

(2) The hand’s role is not analogous to the ball’s.The hand is capable of motion all by itself, but theball can’t move without receiving the ability to movefrom the hand.

If we do an experiment that shows these types ofasymmetries, then Aristotle’s theory is supported.If we find a more symmetric situation, then there’ssomething wrong with Aristotle’s theory.

Observations

The following important rules serve to keep factsseparate from opinions and reduce the chances ofgetting a garbled copy of the data:

(1) Take your raw data in pen, directly into your labnotebook. This is what real scientists do. The pointis to make sure that what you’re writing down isa first-hand record, without mistakes introduced byrecopying it. (If you don’t have your two lab note-books yet, staple today’s raw data into your note-book when you get it.)

(2) Everybody should record their own copy of theraw data. Do not depend on a ‘group secretary.’

(3) If you do calculations during lab, keep them ona separate page or draw a line down the page andkeep calculations on one side of the line and rawdata on the other. This is to distinguish facts frominferences.

Because this is the first meeting of the lab class,there is no prelab writeup due at the beginning ofthe class. Instead, you will discuss your results with

6 Lab 1 Interactions

your instructor at various points.

A Comparing magnets’ strengths

To make an interesting hypothesis about what willhappen in part C, the main event of the lab, you’llneed to know how the top (single) and bottom (triple)magnets’ strengths compare. It would seem logicalthat the triple magnet would be three times strongerthan the single, but in this part of the lab you’re go-ing to find out for sure.

Orient your magnet this way, as if it’s rolling toward the

compass from the north. With no magnet nearby, the

compass points to magnetic north (dashed arrow). The

magnet deflects the compass to a new direction.

One way of measuring the strength of a magnet isto place the magnet to the north or south of thecompass and see how much it deflects (twists) theneedle of a compass. You need to test the magnetsat equal distances from the compass, which will pro-duce two different angles.1 It’s also important to geteverything oriented properly, as in the figure.2

Make sure to take your data with the magnets farenough from the compass that the deflection angleis fairly small (say 5 to 30 ). If the magnet is closeenough to the compass to deflect it by a large an-gle, then the ratio of the angles does not accuratelyrepresent the ratio of the magnets’ strengths. Afterall, just about any magnet is capable of deflectingthe compass in any direction if you bring it closeenough, but that doesn’t mean that all magnets areequally strong.

1There are two reasons why it wouldn’t make sense to finddifferent distances that produced the same angle. First, youdon’t know how the strengths of the effect falls off with dis-tance; it’s not necessarily true, for instance, that the magneticfield is half as strong at twice the distance. Second, the pointof this is to help you interpret part C, and in part C, the triplemagnet’s distance from the single magnet is the same as thesingle magnet’s distance from the first magnet.

2Laying the magnet flat on the table causes the compassneedle to try to tilt out of the horizontal plane, which it’s notdesigned to do. Turning it so that it faces the compass alsodoesn’t work, because it makes the magnet’s magnetic fieldlie along the same north-south line as the Earth’s, rather thanperpendicular to it.

B Qualitative observations of the interaction oftwo magnets

Play around with the two magnets and see how theyinteract with each other. Can one attract the other?Can one repel the other? Can they act on each othersimultaneously? Do they need to be touching in or-der to do anything to each other? Can A act on Bwhile at the same time B does not act on A at all?Can A pull B toward itself at the same time thatB pushes A away? When holding one of the heaviermagnets, it may be difficult to feel when there is anypush or pull on it; you may wish to have one personhold the magnet with her eyes closed while the otherperson moves the other magnet closer and farther.

C Measurement of interactions between two mag-nets

Once you have your data from parts A and B, youare ready to form a hypothesis about the followingsituation. Suppose we set up two balances as shownin the figure. The magnets are not touching. Thetop magnet is hanging from a hook underneath thepan, giving the same result as if it was on top of thepan. Make sure it is hanging under the center of thepan. You will want to make sure the magnets arepulling on each other, not pushing each other away,so that the top magnet will stay in one place.

The balances will not show the magnets’ true masses,because the magnets are exerting forces on each other.The top balance will read a higher number than itwould without any magnetic forces, and the bot-tom balance will have a lower than normal reading.The difference between each magnet’s true mass andthe reading on the balance gives a measure of how

7

strongly the magnet is being pushed or pulled by theother magnet.

How do you think the amount of pushing or pullingexperienced by the two magnets will compare? Inother words, which reading will change more, or willthey change by the same amount? Write down a hy-pothesis; you’ll test this hypothesis in part C of thelab. If you think the forces will be unequal predicttheir ratio.

Discuss with your instructor your results from partsA and B, and your hypothesis about what will hap-pen with the two balances.

Now set up the experiment described above with twobalances. Since we are interested in the changes inthe scale readings caused by the magnetic forces, youwill need to take a total of four scale readings: onepair with the balances separated and one pair withthe magnets close together as shown in the figureabove.

When the balances are together and the magneticforces are acting, it is not possible to get both bal-ances to reach equilibrium at the same time, becausesliding the weights on one balance can cause its mag-net to move up or down, tipping the other balance.Therefore, while you take a reading from one bal-ance, you need to immobilize the other in the hori-zontal position by taping its tip so it points exactlyat the zero mark.

You will also probably find that as you slide theweights, the pointer swings suddenly to the oppo-site side, but you can never get it to be stable inthe middle (zero) position. Try bringing the pointermanually to the zero position and then releasing it.If it swings up, you’re too low, and if it swings down,you’re too high. Search for the dividing line betweenthe too-low region and the too-high region.

If the changes in the scale readings are very small(say a few grams or less), you need to get the mag-nets closer together. It should be possible to get thescale readings to change by large amounts (up to 10or 20 g).

Part C is the only part of the experiment where youwill be required to analyze random errors using thetechniques outlined in Appendices 2 and 3 at theback of the lab manual. Think about how you canget an estimate of the random errors in your mea-surements. Do you need to do multiple measure-ments? Discuss this with your instructor if you’reuncertain.

Don’t take apart your setup until lab is over, and

you’re completely done with your analysis — it’s no

fun to have to rebuild it from scratch because you

made a mistake!

D Measurement of interactions involving ob-jects in contact

You’ll recall that Aristotle gave completely differentinterpretations for situations where one object wasin contact with another, like the hand pushing theball, and situations involving objects not in contactwith each other, such as the rock falling down tothe earth. Your magnets were not in contact witheach other. Now suppose we try the situation shownbelow, with one person’s hand exerting a force on theother’s. All the forces involved are forces betweenobjects in contact, although the two people’s handscannot be in direct contact because the spring scaleshave to be inserted to measure how strongly eachperson is pulling. Suppose the two people do notmake any special arrangement in advance about howhard to pull. How do you think the readings on thetwo scales will compare? Write down a hypothesis,and discuss it with your instructor before continuing.

Now carry out the measurement shown in the figure.

Self-Check

Do all your analysis in lab, including error analysisfor part C. Error analysis is discussed in appendices2 and 3; get help from your instructor if necessary.

Analysis

In your writeup, present your results from all fourparts of the experiment, including error analysis forpart C.

Analysis

The most common mistake is to fail to address thepoint of the lab. If you feel like you don’t understandwhy you were doing any of this, then you were miss-ing out on your educational experience! See the backof the lab manual for the format of lab writeups.

8 Lab 1 Interactions

Notes For Next Week

(1) Next week, when you turn in your writeup forthis lab, you also need to turn in a prelab writeup forthe next lab in the same notebook. The prelab ques-tions are listed at the end of the description of thatlab in the lab manual. Never start a lab without un-derstanding the answers to all the prelab questions;if you turn in partial answers or answers you’re un-sure of, discuss the questions with your instructoror with other students to make sure you understandwhat’s going on.

(2) You should exchange phone numbers with yourlab partners for general convenience throughout thesemester. You can also get each other’s e-mail ad-dresses by logging in to Spotter and clicking on ‘e-mail.’

9

2 Kinematics

Apparatus

computer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/grouptrack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/groupdynamics cart . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/groupfan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/groupAA batteries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4/groupaluminum slugs . . . . . . . . . . . . . . . . . . . . . . . . . . 2/groupmotion detector . . . . . . . . . . . . . . . . . . . . . . . . . .1/groupprotractor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/groupWD-40

Goal

Learn how to relate the motion of an object to itsposition-versus-time graph.

Introduction

Analyzing motion is the most fundamental thing wedo in physics. The most versatile way of representingmotion is with a graph that has the object’s positionon the upright axis and time on the horizontal axis.It takes some practice to be able to sketch and in-terpret these graphs, but once you get used to them,they become very intuitive.

Apparatus

The object whose motion you’ll study is a cart thatrolls on a track. You can either push the cart byhand, start it moving with a shove, or clamp a fan ontop of it to make it speed up or slow down steadily.To measure the cart’s motion, you’ll use a little sonargun that sends out clicks. When it hears the echofrom the cart, it figures out how far away the cartwas based on the time delay and the known speed ofsound. The sonar gun is connected to a computer,which produces a position-versus-time graph.

Setup

Check that all four wheels on the cart will spin forabout 20-30 seconds if you flick them hard. If theyonly spin for a few seconds, see if you can fix theproblem by spraying WD-40 on the bearings.

Set the cart on the track without the fan. Propthe motion detector (sonar gun) at one end of thetrack so that it is aimed slightly upward. This angleis critical — measure 86 above horizontal with theprotractor, and tape it to the backrest.

With the computer turned off, plug the motion de-tector into the PORT2 plug on the interface box.

Start up the computer. For compactness, I’ll use no-tation like this to describe the computer commands:

Start>Programs>Vernier Software>Logger Pro 2

This is the command to start the computer soft-ware running. ‘Start’ means to click on the startmenu at the bottom left corner of the screen, ‘Pro-grams’ means to select that from the menu, and soon. There are two different versions of the softwareinstalled; use version 2. (Logger Pro 3 doesn’t workwith the interface boxes we have.)

Make sure that the interface box is plugged intoCOM1 (the first COM port) at the back of the com-puter, not COM2. If the computer presents you witha dialog box saying ‘Set Up Interface,’ choose COM1.

Once the program is running, do File>Open, thengo into Probes and Sensors and then into MotionDetector, and open the file of the same name. Atthis point, you may get the following error message,which you can ignore: ‘This file cannot run properlywith this hardware interface.’

You’ll get three graphs on the screen, but you onlywant one, the x− t graph. Click on the x− t graph,and then do View>Graph Layout>One Pane, andthe other two graphs will go away.

If you now click the button to tell it to collect data,the motion detector should start clicking rapidly,and it you move the cart back and forth you shouldsee a graph of its motion. Make sure it is able tosense the cart’s motion correctly for distances from50 cm to the full length of the track. If it doesn’twork when the cart is at the far end of the track,play with the angle of motion detector a little. Ifyou’re having other problems, you may find some

10 Lab 2 Kinematics

relevant debugging information in appendix 6, whichdescribes how to use the computer interface with adifferent sensor.

Observations

In parts Athrough E,you don’t need to take detailednumerical data — just sketch the graphs in your labnotebook. All of your graphs will have garbage dataat the beginning and the end, and you need to makesure you understand what’s what.

A Fast and slow motion

Moving the cart by hand, make a graph for slowmotion and another for fast motion. Make sure themotion is steady, and don’t get confused by the partsof the graph that come before and after your periodof steady pushing. Sketch the graphs and make sureyou understand them.

B Motion in two different directions

Now try comparing the graphs you get for the twodifferent directions of motion. Again, record whatthey look like and figure out what you’re seeing.

C Reproducing a graph

Now see if you can produce a graph that looks likethis:

D Accelerating away from the sensor

Suppose the fan is mounted on the cart as shownin the figure, so that if the cart is released from aposition close to the motion detector, it will beginmoving away from it. Predict what you think thecart’s position-time graph will look like, and showyour prediction to your instructor before getting afan.

Before putting the batteries in the fan, make sure thefan’s switch is off (to the right). Put the batteries inand clamp the fan on the cart.

Set up the situation described above, and comparethe results with what you predicted.

E Slow or Rapid Acceleration

The aluminum slugs can be used to replace two ofthe batteries so that the fan will exert about half asmuch force. Discuss with your partners what youthink will happen if you repeat your previous runwith a weakened fan. Now try it.

F Changing the direction of motion

Change the fan back to full strength.

Now suppose instead of releasing the cart from restclose to the motion detector, you started it movingwith a push toward the motion sensor, from the farend of the track. It will of course slow down andeventually come back. Discuss with your partnerswhat the position-time graph would look like. Nowtry it.

G Rate of changing speed

The goal of this part of the lab is to determinewhether the speed of the cart in part F was changingat a constant rate, i.e., by the same amount everysecond.

Zoom in on the relevant part of your graph from partF. To zoom in, either (a) draw a box with the mouseand click on the magnifying glass icon, or (b) doView>Graph Options>Axis Options to select rangesof time and position values that you want. Print outa big copy; choose landscape mode in the print dialogbox. (Note that if you take different data later, youmay need to fiddle with this again because you’ll bezoomed in on the wrong part of the new graph.)

If that printer isn’t working, here’s what you need todo instead. Do File>Export Data, and select ‘.txt’for the type of the file. Use a text editor such asWordPad to delete the header from the file. Save itin your FC student directory, and also on a floppydisk if you intend to work on it at home. Get intoOpenOffice or Excel, and open the file. Appendix 4describes how to use OpenOffice. Whatever methodyou use, make sure the whole group will end up withcopies.

Rather than trying to read distances from your graph’svertical axis in units of meters, and times from itshorizontal axis in units of seconds, the simplest thingto do is simply to use a ruler to measure verticaland horizontal distances on the graph, and deter-mine the slopes from these; although the resultingslopes won’t be in any standard units, that won’taffect your conclusion.

11

Prelab

The point of the prelab questions is to make sureyou understand what you’re doing, why you’re do-ing it, and how to avoid some common mistakes. Ifyou don’t know the answers, make sure to come tomy office hours before lab and get help! Otherwiseyou’re just setting yourself up for failure in lab.

P1 Make a prediction of the four graphs you’ll ob-tain in parts A and B.

Self-Check

Do the analysis in lab.

Analysis

At one-second intervals, draw nice long tangent lineson the curve from part G and determine their slope.Some slopes will be negative, and some positive.Summarize this series of changing speeds in a table.Did the velocity increase by about the same amountwith every second?

12 Lab 2 Kinematics

3 Free Fall

Apparatus

two stations:Behr free-fall column and weightplumb bobspark generator (CENCO)paper tapeswitch for electromagnet

Goal

Find out whether it is ∆v/∆x or ∆v/∆t that is con-stant for an object accelerating under the influenceof gravity.

Introduction

A fundamental and difficult problem in pre-Newton-ian physics was the motion of falling bodies. Aristo-tle had various incorrect but influential ideas on thesubject, including the assertions that heavier objectsfell faster than lighter ones and that the object onlysped up for a short while after it was dropped andthen continued on at a constant speed. Even amongRenaissance scientists who disagreed with Aristotle’sclaim that the object no longer sped up after a while,there was a great deal of confusion about whether itwas ∆v/∆x or ∆v/∆t that was constant. It seemsobvious to modern physicists that they could notboth be constant, but it was not at all obvious toauthorities such as Domingo de Soto and Albert ofSaxony. Galileo started out thinking they were bothconstant, then realized this was mathematically im-possible, and finally determined from experimentsthat it was ∆v/∆t, now called acceleration, that wasconstant.

The main reason why the confusion persisted for twothousand years was that the methods for measuringtime were inaccurate, and the time required for anobject to fall was very short. Galileo was able tomake settle the issue because he figured out how touse a pendulum to measure time accurately, and alsocame up with the idea of effectively slowing down themotion by studying objects rolling down an inclinedplane, rather than objects falling vertically. He thenfound how to extrapolate from the case of an objectrolling down an inclined plane at an angle θ to the

ideal case of θ=90 , which would be the same as freefall. Galileo’s task would have been a lot simplerif he’d had accurate enough devices for measuringtime, because then he could have simply carried outmeasurements for objects falling vertically. That’swhat you’ll do today.

A Setup

The apparatus consists of a 2-meter tall column witha paper tape running down it. A weight is held at thetop with an electromagnet and then released, fallingright next to the paper tape. (An electromagnetis an artificial magnet that works when you put anelectric current through it, unlike a permanent mag-net, which does not require power.) A spark gener-ator is hooked up to the two vertical wires, and asthe weight falls, sparks cross the gap from the firstwire to the metal flange on the weight, then fromthe flange to the other wire. Sparks are producedonly briefly, at regular intervals of 1/60 of a sec-

14 Lab 3 Free Fall

ond. On their way, the sparks go through the papertape, making dots on it that show the location of theweight at 1/60-s intervals.

First, unplug the spark generator so you don’t getshocked while you’re getting things ready. Use theswitch made from a regular light switch to turn onthe magnet at the top of the column, which operateson 7 volts from the lab’s DC power circuits. Insertthe plumb bob, hanging from the magnet. Use thethree screws on the feet of the column to level theapparatus so the plumb bob’s string is parallel tothe wire.

Replace the plumb bob with the weight. Pull freshtape up from the roll at the bottom, and get thetape straight and centered on the wire.

Plug in the spark generator, and put the functionknob on ‘line,’ which means it will base its cycleof sparks on the AC power from the wall, whichswitches directions once every sixtieth of a second.The red LED should light up. From now on, do notpress the thumb switch to activate the sparks unlessyou are sure nobody is near the vertical wires. Tryit out, and see if you get a spot at the top of thetape, where the weight currently is.

B Observations

Hold down the thumb switch to make the sparksstart, flip the switch to release the weight, and waituntil the weight has fallen in the cup at the bottombefore releasing the thumb switch. You want a nicestraight line of dots on the tape, going all the wayfrom the top to the bottom — you may have to makeadjustments and try a few times before getting agood tape. Take your tape off, and measure thelocations of the dots accurately with a two-meterstick.

Analysis

Since the sparks start before you release the electro-magnet, the first dot at the very top of the tape willgive the starting position of the weight.

If you consider any adjacent pair of dots (avoidingthe top and bottom ones), then measuring the dis-tance between them allows you to calculate an ap-proximation to the speed of the weight, which youcan think of as being its speed at the point half-waybetween the two dots.

Make one plot of speed versus time and another ofspeed versus distance, preferably using a computer,since you will have about thirty data points, and it

would be tedious to plot them all by hand.

Determine whether your data are consistent withconstant ∆v/∆x or ∆v/∆t or neither. Whicheverone it is that is constant, use that as your definitionof the acceleration of gravity, g, in part B.

Self-Check

Appendix 4 discusses graphing. The graphing forthis lab is time-consuming without a computer; sincewe have a limited number of computers in lab, youmay want to go to one of the other campus computerlabs for this. Determine which quantity is constant.

Prelab


P1 How will you tell from your graphs whether itis ∆v/∆x or ∆v/∆t that is constant, or neither ofthem?

15

4 The Earth’s Gravitational Field

Apparatus

(two stations):vertical plank with electromagnetssteel balls (2/station)Linux computers with Audacity installed (in 416 and416P)

Goal

Make a high-precision measurement of the strengthof the Earth’s gravitational field, g, in Fullerton.

Introduction

When objects fall, and all forces other than grav-ity are negligible, we observe that the accelerationis the same, regardless of the object’s mass, shape,density, or other properties. However, the acceler-ation does depend a little bit where on the earthwe do the experiment, and even bigger variations inacceleration can be observed by, e.g., going to themoon. Thus, this acceleration can be considered asa property of space itself, and we can refer to it asthe gravitational field in that region of space. Justas you would use a magnetic compass to find outabout the magnetic field in the classroom, you canuse dropping masses to find out about the gravita-tional field. In this experiment, you’ll measure thegravitational field, g, in the classroom to sufficientlyhigh precision that, if everybody does a good job andwe pool and average everyone’s data to reduce ran-dom errors, we should be able to get a value that ismeasurably different from the generic world-averagevalue you would find in a textbook.

A Measuring g precisely

You will measure g, the acceleration of an object infree fall, using electronic timing techniques. The ideaof the method is that you’ll have two steel balls hang-ing underneath electromagnets at different heights.You’ll simultaneously turn off the two magnets us-ing the same switch, causing the balls to drop atthe same moment. The ball dropped from the lowerheight (h1) takes a smaller time (∆t1) to reach thefloor, and the ball released from the greater height(h2) takes a longer time (∆t2). The time intervalsinvolved are short enough that due to the limita-

tions of your reflexes it is impossible to make goodenough measurements with stopwatch. Instead, youwill record the sounds of the two balls’ impacts onthe floor using the computer. The computer showsa graph in which the x axis is time and the y axisshows the vibration of the sound wave hitting themicrophone. You can measure the time between thetwo visible ‘blips’ on the screen. You will measurethree things: h1, h2, and the time interval ∆t2−∆t1between the impact of the second ball and the first.From these data, with a little algebra, you can findg.

The experiment would have been easier to analyze ifwe could simply drop a single ball and measure thetime from when it was released to when it hit thefloor. But since our timing technique is based onsound, and no sound is produced when the balls arereleased, we need to have two balls. If h1, the heightof the lower ball, could be made very small, then itwould hit the floor at essentially the same momentthe two balls were released (∆t1 would equal 0), and∆t2−∆t1 would be essentially the same as ∆t2. Butwe can’t make h1 too small or the sound would notbe loud enough to detect on the computer.

B Using the computer software

There are three Linux computers that have the rightsoftware and hardware: one in 416P, one on thesouth wall in 416, and one on the east wall. Firstlet’s see how to record yourself on the computer say-ing ‘hello.’ Use the xmix or xmixer program to setthe record and mic levels all the way up. Start upthe sound recording program, called Audacity. Setthe record level on high, using the control marked− . . .+ next to the microphone icon. Record yoursound.

Before you get down to serious science, you may en-joy listening to your own voice reversed in time. Afun diversion is to write a sentence down backwards,read it out loud, and then electronically reverse itso it’s forward again. It sounds sort of like someonewith a thick Hungarian accent.

To find out how long a sound is, you can use thecurson and click to find the time corresponding toa particular point in the graph. Sometimes you arenot sure which wiggles in the visual representation ofthe sound correspond to which parts of the recordedsound. To find out, you can select part of the sound

16 Lab 4 The Earth’s Gravitational Field

Two thumps, as recorded on the computer through the

microphone.

and listen to only that part.

C Finding the interval between two sounds

When you record the sound of the two consecutiveimpacts of the balls, they will look like vertical spikeson the screen. You can practice using hand claps.

To accurately find the time when one of the soundsstarted, first zoom in on it until it’s like you’re see-ing it under a powerful microscope. Click on theonset of the sound, and read the sound off of thebottom of the screen, where it’s displayed to highprecision. (It may be hidden behind the menu barat the bottome of the screen. To fix that, click onthe ‘maximize’ button at the upper right corner ofthe Audacity window.) You should make a seriesof measurements, and make sure they agree at thelevel of about 10−4 s; if they don’t, there’s some-thing wrong with your technique. Also, you shouldcheck that your result for g makes sense.

Audacity will let you keep on making new record-ings, stacking the graphs vertically. However, if youdo this you will introduce significant timing errors.The reason is that Audacity is probably that de-signed for use in multitrack recording of music, soit tries to play back the previously recorded trackswhile recording the new one, and on low-end soundhardware this causes little timing glitches.

Here are some common problems that cause incon-sistent or wrong results: (1) the balls are brushingagainst the electrical wires as they fall; (2) you’remisidentifying the thumps; (3) the surface the ballsare dropping onto has dents in it; (4) you’re not posi-tioning the balls in on the same spot on the magnetsevery time.

D Observations

Measure h1, h2, and the time interval ∆t2 − ∆t1.

Analysis

Extract a value of g from your data.1 Derive errorbars on your result, using the techniques in appen-dices 2 and 3.

Self-Check

Extract the value of g, with error bars. Read Ap-pendix 3 for information on how to do error analysiswith propagation of errors; get help from your in-structor if necessary.

Prelab


P1 If your instructor has assigned homework prob-

lem 27 from ch. 3 of Newtonian Physics, don’t bother

turning in another copy of your work for this prelab

question. Derive an equation for g in terms of thequantities you’ll measure, which are h1, h2, and thetime interval ∆t2 − ∆t1. The point of the lab is tomeasure g, so don’t just say ‘well of course g is 9.8m/s2.’ (You should check your equation by usingthe answer checker for the homework problem.)

1If you feel like it, you can add on a correction to g toaccount for air resistance, which is hr2/3m, where h is theheight in meters from which the ball was dropped, r is theradius of the ball in meters, and m is the ball’s mass in kg.This correction is about at the limit of the accuracy of theexperiment.

17

5 Newton’s Second Law

Apparatus

pulleyspirit levelstringweight holders, not tied to stringtwo-meter stickslotted weightsstopwatchfoam rubber cushions

Goal

Find the acceleration of unequal weights hangingfrom a pulley.

Observations

Set up unequal masses on the two sides of the pulley,and determine the resulting acceleration by measur-ing how long it takes for the masses to move a cer-tain distance. Use the spirit level to make the pulleyvertical; otherwise you get extra friction. Use rela-tively large masses (typically half a kg or a kg eachside) so that friction is not such a big force in com-parison to the other forces, and the inertia of thepulley is negligible compared to the inertia of thehanging masses. Do several different combinations

of masses, but keep the total amount of mass con-

stant and just divide it differently between the twoholders. Remember to take the masses of the holdersthemselves into account. Make sure to perform yourmeasurements with the longest possible distance oftravel, because you cannot use a stopwatch to get anaccurate measurement of very short time intervals.The best results are obtained with combinations ofweights that give times of about 2 to 10 seconds.Also, make sure that the masses are at least a fewhundred grams or so on each side.

Self-Check

Compare theoretical and experimental values of ac-celeration for one of your mass combinations. Checkwhether they come out fairly consistent.

Analysis

Use your measured times and distances to find theactual acceleration, and make a graph of this versusM − m. Show these experimentally determined ac-celerations as small circles. Overlaid on the samegraph, show the theoretical equation as a line orcurve.

Prelab


P1 Criticize the following reasoning: The weightfell 1.0 m in 1 s, so v = 1 m/s, and a = v/t = 1 m/s2.

P2 Since that won’t work, plan how you really willdetermine your experimental accelerations based onyour measured distance and times.

18 Lab 5 Newton’s Second Law

6 Air Friction

Apparatus

coffee filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10/groupstopwatch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/groupcomputer with sonar sensorwire cages to protect sensors

Goal

Determine how the force of air friction depends onthe velocity of a moving object.

As an alternative, you may create your own tech-nique for doing the same sort of experiment for fric-tion between two surfaces wetted with a liquid suchas water, vegetable oil, or machine oil — the resultmight be more interesting, since it is not to be foundin textbooks. If you are interested in doing this, dis-cuss it in advance with your instructor.

Introduction

Friction between solid objects occurs all the time inour daily lives. The frictional force exerted by the airon a solid object is not as often evident, but it is re-sponsible for the wind blowing our hair, for the slowdropping of a feather, and for our cars’ poorer gasmileage at freeway speeds compared to more mod-erate speeds.

The latter effect suggests that air friction might in-crease with speed, unlike solid-solid friction, which isnearly independent of speed. By Newton’s first law,a car or a jet plane cruising at constant speed musthave zero total force on it, so if the air friction forcegets stronger with speed, that would explain whya greater forward-pushing force would be needed totravel at high speeds. For instance, a car travelingat low speed might have a -10 kN air friction forcepushing backward on it, so in order to have zero to-tal force on it the road must be making a forwardforce of +10 kN. At a higher speed, air friction mightincrease to -30 kN, so the road would need to make aforward force of +30 kN. The car convinces the roadto make the stronger force by pushing backward onthe road more strongly: by Newton’s third law, thecar’s force on the road and the road’s force on thecar must be equal in magnitude and opposite in di-rection. The car burns more gas because it must

push harder against the road.

Your goal in this lab is to find a proportionality re-lating the force of air friction to the velocity at whichthe air rushes over the object. For instance, you mayfind the rule

F ∝ v ,

which is a shorthand for

F = (some number)(v) .

The numerical value of ‘some number’ is not veryinteresting, because we would expect it to be dif-ferent for different objects, which is why you wouldwrite your result as F ∝ v. This proportionalitywould tell you for instance that anytime the speedwas doubled, the result would be twice as much airfriction.

Suppose instead you find that doubling the speedmakes the force eight times greater, multiplying thespeed by 10 makes the force 1000 times greater, andso on. In each case, the force is being multipliedby the third power of the increase in the speed, i.e.,F ∝ v3.

Observations

There are two possible methods for measuring theterminal velocity of the filter.

The first method is shown in the figure below. Weuse coffee filters because they don’t tumble or swayvery much as they fall, and because they allow us toeasily change the mass of our falling object by nest-ing more coffee filters inside the bottom one, with-out changing its aerodynamic properties. The filterswill start speeding up when you release them nearthe ceiling, but as they speed up, the upward forceof air friction on them increases, until they reach aspeed at which the total force on them is zero. Onceat this speed, they obey Newton’s first law and con-tinue at constant speed. If the number of coffee fil-ters is small, they will have reached their maximumspeed within the first half a meter or so. By thetime they are even with the edge of the lab bench,they are moving at essentially their full speed. Youcan then use the stopwatch to determine how long ittakes them to cover the distance to the floor, whichwill allow you to find their speed. During this finalpart of the fall, you know the upward force of air

20 Lab 6 Air Friction

friction must be as great as the downward force ofgravity, so you can determine what it was.

A different technique is to drop the filters onto asonar sensor of the type used in lab 2. You can putthe sensor on the floor facing up, and put the wirecage over it to keep it from getting damaged by beingstepped on inadvertently.

For a long time, I had my students do the lab usingonly the first method, but now I’m experimentingwith the second method. A couple of advantages ofthe second method are that (1) it doesn’t depend onhuman reflexes, and (2) it gives you a real-time pic-ture of the motion, so it’s easier to tell whether thefilters are actually reaching terminal velocity. Thelatter is an important issue, because it gives you abetter chance of being able to take data over a widerange of values for F and v, but without runninginto problems with cases where the filters don’t re-ally reach terminal veloicity. On the other hand, thecomputer method has some practical problems, suchas the tendency of the filters to drift sideways insteadof heading straight down onto the sensor. This is anopportunity for you to do something like what realscientists do: use your ingenuity and try differentthings to see what works best!

Note that if the coffee filters get too flattened out,they’ll flutter, giving lousy results.

Take data with stacks of various numbers of coffeefilters. You will get the most clearcut determinationof the power law relationship if your data cover thelargest possible range of values. It’s a good idea totake some data with a large number of filters, drop-ping them from the balcony outside so they have

time to get up to their final speed. This is also theonly way you can tell for sure whether you’re tak-ing data at terminal velocity: the results at the twodifferent heights (inside and outside) should be con-sistent.

Prelab


P1 Suppose you tried to do this lab with stacks ofcoins instead of coffee filters. Assuming you had asufficiently accurate timing device, would it work?

P2 Criticize the following statement:

‘We found that bigger velocities gave bigger air dragforces, which demonstrates the proportionality F ∝v.’

P3 Criticize the following statement:

‘We found F ∝ v7, which shows that you need moreforce to make things go faster.’

Analysis

Use your raw data to compile a list of F and v values.Use the methods explained in Appendix 5 to see ifyou can find a power-law relationship between F andv. This will require fitting a line to a set of data, asexplained in appendix 4. Both fitting a line to dataand finding power laws are techniques you will useseveral more times in this course, so it is worth yourwhile to get help now if necessary in order to getconfident with them.

21

7 Vector Addition of Forces

Apparatus

force table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/groupspirit level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/groupweightsstring

Goal

Test whether the vector sum of the forces acting onan object at rest is equal to zero.

Introduction

Modern physics claims that when a bridge, an earth-quake fault, or an oak tree doesn’t move, it is be-cause the forces acting on it, which combine accord-ing to vector addition, add up to zero. Althoughthis may seem like a reasonable statement, it wasfar from obvious to premodern scientists. Aristotle,for instance, said that it was the nature of each ofthe four elements, earth, fire, water and air, to re-turn to its natural location. Rain would fall fromthe sky because it was trying to return to its natu-ral location in the lakes and oceans, and once it gotto its natural location it would stop moving becausethat was its nature.

When a modern scientist considers a book resting ona table, she says that it holds still because the forceof gravity pulling the book down is exactly canceledby the normal force of the table pushing up on thebook. Aristotle would have denied that this was pos-sible, because he believed that at any one moment anobject could have only one of two mutually exclusivetypes of motion: natural motion (the tendency of thebook to fall to the ground, and resume its naturalplace), and forced motion (the ability of another ob-ject, such as the table, to move the book). Accordingto his theory, there could be nothing like the addi-tion of forces, because the object being acted on wasonly capable of ‘following orders’ from one source ata time. The incorrect Aristotelian point of view hasgreat intuitive appeal, and beginning physics stu-dents tend to make Aristotelian statements such as,‘The table’s force overcomes the force of gravity,’ asif the forces were having a contest, in which the vic-tor annihilated the loser.

Observations

The apparatus consists of a small circular table, witha small metal ring held in the middle by the tensionin four strings. Each string goes over a pulley at theedge of the table, so that a weight can be hung on itto control the tension. The angles can be recordedeither graphically, by sliding a piece of paper un-derneath, or by reading angles numerically off of anangular scale around the circumference of the table.

Use the spirit level to level the table completely us-ing the screws on the feet. Set up four strings withweights, using the small pin to hold the ring in place.Adjust the angles or the amounts of weight or both,until the ring is in equilibrium without the pin, andis positioned right over the center of the table. Avoida symmetric arrangement of the strings (e.g., don’tspace them all 90 degrees apart). The ring is an ex-tended object, so in order to treat it mathematicallyas a pointlike object you should make sure that allthe strings are lined up with the center of the ring,as shown in the figure.

Because of friction, it is possible to change any oneof the weights slightly without causing the ring tomove. This is a potential source of systematic er-rors, but you can eliminate the error completely bythe following method. Find out how much you canincrease or decrease each weight without causing thering to move. Within the range of values that don’tcause slipping, use the center of the range as your

22 Lab 7 Vector Addition of Forces

best value; with this amount of weight, there is nofriction at all in the pulley. The point here is not toredo the entire experiment with a completely differ-ent combination of weights — that would not tellyou anything about friction as a source of error,since even if there was no friction at all, it would bepossible for example to double all the weights andget an equilibrium. Once you’ve set each weight toits friction-free value, leave it that way ; by the timeyou’re done, you will have eliminated friction fromall four pulleys.

Prelab


P1 The weights go on weight holders that hangfrom the string, and the weight holders are each 50g. Criticize the following reasoning: ‘We don’t needto count the mass of the weight holders, because itsthe same on all four strings, so it cancels out.”

P2 Describe a typical scale that you might use fordrawing force vectors on a piece of paper, e.g., howlong might you choose to make a 1-N force? Assumeyour masses are from 500 to 1500 grams.

P3 Graphically calculate the vector sums of thetwo pairs of vectors shown below. As a check onyour results, you should find that the magnitudes ofthe two sums are equal.

Self-Check

Do both a graphical calculation and an analytic cal-culation in lab, without error analysis. Make surethey give the same result. Do a rough check thatthe magnitude of the sum of the forces is small com-

pared to the magnitudes of the individual forces.

Analysis

Calculate the magnitude of vector sum of the forceson the ring, first graphically and then analytically.Make sure the two methods give the same result. Ifthey do not, try measuring the x and y componentsoff of your drawing and comparing them with the xand y components you calculated analytically.

Estimate the possible random error in your finalsum.

Are your results consistent with theory, taking intoaccount the random errors involved?

23

8 Vector Addition of Forces

Based on a lab created by Fream Minton.

Apparatus

unknown weight hung from threepulleys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/groupmeter sticksprotractorsdigital balance

Goal

Use vector addition of forces in three dimensionsto determine the mass of an unknown suspendedweight.

Introduction

Modern physics claims that when a bridge, an earth-quake fault, or an oak tree doesn’t move, it is be-cause the forces acting on it, which combine accord-ing to vector addition, add up to zero. Althoughthis may seem like a reasonable statement, it wasfar from obvious to premodern scientists. Aristotle,for instance, said that it was the nature of each ofthe four elements, earth, fire, water and air, to re-turn to its natural location. Rain would fall fromthe sky because it was trying to return to its natu-ral location in the lakes and oceans, and once it gotto its natural location it would stop moving becausethat was its nature.

When a modern scientist considers a book resting ona table, she says that it holds still because the forceof gravity pulling the book down is exactly canceledby the normal force of the table pushing up on thebook. Aristotle would have denied that this was pos-sible, because he believed that at any one moment anobject could have only one of two mutually exclusivetypes of motion: natural motion (the tendency of thebook to fall to the ground, and resume its naturalplace), and forced motion (the ability of another ob-ject, such as the table, to move the book). Accordingto his theory, there could be nothing like the addi-tion of forces, because the object being acted on wasonly capable of ‘following orders’ from one source ata time. The incorrect Aristotelian point of view hasgreat intuitive appeal, and beginning physics stu-

dents tend to make Aristotelian statements such as,‘The table’s force overcomes the force of gravity,’ asif the forces were in a contest, in which the victorannihilated the loser.

Observations

The setup is shown above. The tension in the stringis very nearly the same on both sides of a good-quality pulley, i.e. one with low friction. Your taskis to use geometrical measurements and measure-ments of the three hanging weights to determinethe unknown mass of the ball hanging in the mid-dle. This will require vector addition in three di-mensions. Once you have determined the weight ofthe unknown, show it to your instructor. Once yourinstructor checks your work for mistakes, you canweigh the ball for comparison with your prediction.

24 Lab 8 Vector Addition of Forces

Prelab


P1 The weights go on weight holders that hangfrom the string, and the weight holders are each 50g. Criticize the following reasoning: ‘We don’t needto count the mass of the weight holders, because itsthe same on all three strings, so it cancels out.”

P2 Suppose the pulleys have nonnegligible friction.Discuss the effect on the results. Would this be arandom error or a systematic error?

Analysis

Carry out a propagation of errors for your predictedmass (see Appendix 3), and discuss whether it isconsistent with your direct measurement.

25

9 Acceleration In Two Dimensions

Apparatus

air track (small)cartphotogate (PASCO) (under lab benches in rm. 418)computerair blowerspower strips for switching CENCO blowers on andoffvernier caliperswood blocks

Goal

Test whether the acceleration of gravity acts like avector.

Introduction

As noted in lab 2, one of the tricky techniques Galileohad to come up with to study acceleration was touse objects rolling down an inclined plane ratherthan falling straight down. That slowed things downenough so that he could measure the time intervalsusing a pendulum clock. Even though you were able,in lab 4, to use modern electronic timing techniquesto measure the short times involved in a vertical fall,there is still some intrinsic interest in the idea ofmotion on an inclined plane. The reason it’s worthstudying is that it reveals the vector nature of accel-eration.

Vectors rule the universe. Entomologists say thatGod must have had an inordinate fondness for bee-tles, because there are so many species of them.Well, God must also have had a special place in herheart for vectors, because practically every naturalphenomenon she invented is a vector: gravitationalacceleration, electric fields, nuclear forces, magneticfields, all the things that tie our universe togetherare vectors.

Setup

The idea of the lab is that if acceleration really actslike a vector, then the cart’s acceleration should equalthe component of the earth’s gravitational accelera-tion vector that is parallel to the track, because the

cart is only free to accelerate in the direction alongthe track. There is almost no friction, since the cartrides on a cushion of air coming through holes in thetrack.

The speed of the cart at any given point can be mea-sured as follows. The photogate consists of a lightand a sensor on opposite sides of the track. Whenthe cart passes by, the cardboard vane on top blocksthe light momentarily, keeping light from getting tothe sensor. The computer detects the electrical sig-nal from the sensor, and records the amount of time,tb, for which the photogate was blocked. Given tb,you can determine the approximate speed that carthad when it passed through the photogate. The useof the computer software is explained in Appendix 6;of the three modes described there, you want to usethe software in the mode in which it measures thetime interval over which the photogate was blocked.Plug the photogate into the DG1 plug on the inter-face box.

Observations

The basic idea is to release the cart at a distance xaway from the photogate. The cart accelerates, andyou can determine its approximate speed, v, when itpasses through the photogate. (See prelab questionP1. Make sure to use vernier calipers to measure thewidth of the vane, w.) From v and x, you can findthe acceleration. You will take data with the tracktilted at several different angles, to see whether thecart’s acceleration always equals the component of gparallel to the track.

26 Lab 9 Acceleration In Two Dimensions

You can level the track to start with by adjustingthe screws until the cart will sit on the track withoutaccelerating in either direction.

The distance x can be measured from the startingposition of the cart to half-way between the pointwhere it first blocks the photogate and the pointwhere it unblocks the photogate. You can determinewhere these positions are by sliding the cart into thephotogate and watching the red LED on the top ofthe photogate, which lights up when it is blocked.

Hints:

Keep in mind that if the cart rebounds at thebottom of the track and comes back up throughthe gate, you will get a second, bogus timereading.

Note that you have no way to measure accu-rately to the total amount of time over whichthe cart picked up speed (which would be sev-eral seconds) — what you measure is the veryshort time required for the cart to pass throughthe photogate.

If you’re using one of the gray air pumps, whichhas a knob to adjust the flow, make sure it’s onthe highest speed, or the cart will drag on thetrack, giving bogus data. It’s easy to mess upthis adjustment, so get the knob set correctlyfor once and for all, and then never touch itagain. To turn the pump on and off, plug theblower in to its own power strip, and use theswitch on the power strip.

Release the cart by hand after starting up theair pump. If you leave the cart on the trackand then turn on the pump, there will be aperiod of time when the pump is first startingup, and the cart will drag.

The variable x actually changes a little whenyou change θ, so don’t just assume it’s alwaysthe same.

You’ll use the photogates again in lab 10, so makesure you understand the technique thoroughly, andtake notes on it so you’ll remember how it’s done.

Self-Check

Find the theoretical and experimental accelerationsfor one of your angles, and see if they are roughlyconsistent.

Prelab


P1 Skip this question if the corresponding home-

work problem from Newtonian Physics has already

been assigned. (a) If w is the width of the vane, andtb is defined as suggested above, what is the speed ofthe cart when it passes through the photogate? (b)Based on v and x, how can you find a?

P2 Should x be measured horizontally, or alongthe slope of the track?

P3 It is not possible to measure θ accurately witha protractor. How can θ be determined based onthe distance between the feet of the air track andthe height of the wood block?

P4 Explain why the following method for findingthe cart’s acceleration is incorrect. ‘The time I gotoff the computer was 0.0237 s. My vane was 2.2 cmwide, so v = 2.2 cm/.0237 s = 93 cm/s. That meansthe acceleration was a = ∆v/∆t = (vf − vi)/∆t =(vf − 0)/∆t = vf/∆t, or 93 cm/s divided by .0237s, which gives 3900 cm/s2.’

Analysis

Extract the acceleration for each angle at which youtook data. Make a graph with θ on the x axis andacceleration on the y axis. Show your measured ac-celerations as points, and the theoretically expecteddependence of a on θ as a smooth curve.

Error analysis is not required for this lab, becausethe random errors are small compared to systematicerrors such as the imperfect leveling of the track,friction, and warping of the track, and the measure-ment of w.

27

10 Conservation Laws

Apparatus

Part A: vacuum pump (Lapine) . . . . . . . . . . . . . . . . . 1electronic balance (large capacity) . . . . . . . . . . . . . . . 1plastic-coated flask . . . . . . . . . . . . . . . . . . . . . . . 1/groupPart B: beaker . . . . . . . . . . . . . . . . . . . . . . . . . . 1/grouppropyl alcohol 200 mL/groupcanola oil 200 mL/groupfunnels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2/group100-mL volumetric flask . . . . . . . . . . . . . . . . . . 1/grouprubber stopper, fitting involumetric flask . . . . . . . . . . . . . . . . . . . . . . . . . . 1/group1-ml pipette and bulb . . . . . . . . . . . . . . . . . . . . 1/groupmagnetic stirrer . . . . . . . . . . . . . . . . . . . . . . . . . . 1/grouptriple-beam balance . . . . . . . . . . . . . . . . . . . . . . .1/group

Goal

People believe that objects cannot be made to disap-pear or appear. If you start with a certain amountof matter, there is no way to increase or decreasethat amount. This type of rule is called a conser-vation law in physics, and this specific law statesthat the amount of matter is conserved, i.e., muststay the same. In order to make this law scientifi-cally useful, we must define more carefully how the‘amount’ of a substance is to be defined and mea-sured numerically. Specifically, there are two issuesthat scientifically untrained people would probablynot agree on:

Should air count as matter? If it has weight,then it probably should count. In this lab, youwill find out if air has weight, and, if so, mea-sure its density.

Should the amount of a substance be defined interms of volume, or is mass more appropriate?In this lab, you will determine whether massand/or volume is conserved when water andalcohol are mixed.

Introduction

Styles in physics come and go, and once-hallowedprinciples get modified as more accurate data comealong, but some of the most durable features of thescience are its conservation laws. A conservation law

is a statement that something always remains con-stant when you add it all up. Most people have ageneral intuitive idea that the amount of a substanceis conserved. That objects do not simply appearor disappear is a conceptual achievement of babiesaround the age of 9-12 months. Beginning at thisage, they will for instance try to retrieve a toy thatthey have seen being placed under a blanket, ratherthan just assuming that it no longer exists. Con-servation laws in physics have the following generalfeatures:

Physicists trying to find new conservation lawswill try to find a measurable, numerical quan-tity, so that they can check quantitatively whetherit is conserved. One needs an operational def-inition of the quantity, meaning a definitionthat spells out the operations required to mea-sure it.

Conservation laws are only true for closed sys-tems. For instance, the amount of water in abottle will remain constant as long as no wa-ter is poured in or out. But if water can get inor out, we say that the bottle is not a closedsystem, and conservation of matter cannot beapplied to it.

The quantity should be additive. For instance,the amount of energy contained in two gallonsof gasoline is twice as much as the amount ofenergy contained in one gallon; energy is addi-tive. An example of a non-additive quantity istemperature. Two cups of coffee do not havetwice as high a temperature as one cup.

Conservation laws always refer to the total amountof the quantity when you add it all up. If youadd it all up at one point in time, and thencome back at a later point in time and add itall up, it will be the same.

How can we pin down more accurately the conceptof the ‘amount of a substance’? Should a gallonof shaving cream be considered ‘more substantial’than a brick? At least two possible quantities cometo mind: mass and volume. Is either conserved?Both? Neither? To find out, we will have to makemeasurements.

We can measure mass by the ‘see-saw method’ —when two children are sitting on the opposite sides

28 Lab 10 Conservation Laws

of a see-saw, the less massive one has to move far-ther out from the fulcrum to make it balance. If weenslave some particular child as our permanent massstandard, then any other child’s mass can be mea-sured by balancing her on the other side and mea-suring her distance from the fulcrum. A more prac-tical version of the same basic principle that doesnot involve human rights violations is the familiarpan balance with sliding weights.

Volume is not necessarily so easy to measure. Forinstance, shaving cream is mostly air, so should wefind a way to measure just the volume of the bub-bly film itself? Precise measurements of volume canmost easily be done with liquids and gases, whichconform to a vessel in which they are placed.

Should a gas, such as air, be counted as having anysubstance at all? Empedocles of Acragas (born ca.492 BC) was the originator of the doctrine that allmaterial substances are composed of mixtures of fourelements: earth, fire, water and air. The idea seemsamusingly naive now that we know about the chem-ical elements and the periodic table, but it was ac-cepted in Europe for two thousand years, and theinclusion of air as a material substance was actu-ally a nontrivial concept. Air, after all, was invis-ible, seemed weightless, and had no definite shape.Empedocles decided air was a form of matter basedon experimental evidence: air could be trapped un-der water in an inverted cup, and bubbles would bereleased if the cup was tilted. It is interesting tonote that in China around 300 BC, Zou Yan cameup with a similar theory, and his five elements didnot include air.

Does air have weight? Most people would probablysay no, since they do not feel any physical sensationof the atmosphere pushing down on them. A delicatehouse of cards remains standing, and is not crushedto the floor by the weight of the atmosphere.

Compare that to the experience of a dolphin, though.A dolphin might contemplate a tasty herring sus-pended in front of it and conjecture that water hadno weight, because the herring did not involuntarilyshoot down to the sea floor because of the weight ofthe water overhead. Water does have weight, how-ever, which a sufficiently skeptical dolphin physicistmight be able to prove with a simple experiment.One could weigh a 1-liter metal box full of water andthen replace the water with air and weigh it again.The difference in weight would be the difference inweight between 1 liter of water of and 1 liter of air.Since air is much less dense than water, this wouldapproximately equal the weight of 1 liter of water.

Our situation is similar to the dolphin’s, as was firstappreciated by Torricelli, whose experiments led himto conclude that ‘we live immersed at the bottomof a sea of...air.’ A human physicist, living her lifeimmersed in air, could do a similar experiment tofind out whether air has weight. She could weigh acontainer full of air, then pump all the air out andweigh it again. When all the matter in a containerhas been removed, including the air, we say thatthere is a vacuum in the container. In reality, aperfect vacuum is very difficult to create. A smallfraction of the air is likely to remain in the containereven after it has been pumped on with a vacuumpump. The amount of remaining air will dependon how good the pump is and on the rate at whichair leaks back in to the container through holes orcracks.

Cautions

Please do not break the glassware! The vacuumflasks and volumetric flasks are expensive.

The alcohol you will be using in this lab is chemicallydifferent from the alcohol in alcoholic beverages. Itis poisonous, and can cause blindness or death if youdrink it. It is not hazardous as long as you do notdrink it.

Observations

A Density of air

You can remove the air from the flask by attach-ing the vacuum pump to the vacuum flask with therubber and glass tubing, then turning on the pump.You can use the scale to determine how much masswas lost when the air was evacuated.

Make any other observations you need in order tofind out the density of air and to estimate error barsfor your result.

B Is volume and/or mass conserved when twofluids are mixed?

The idea here is to find out whether volume and/ormass is conserved when water and alcohol are mixed.The obvious way to attempt this would be to mea-sure the volume and mass of a sample of water, thevolume and mass of a sample of alcohol, and theirvolume and mass when mixed. There are two prob-lems with the obvious method: (1) when you pourone of the liquids into the other, droplets of liquidwill be left inside the original vessel; and (2) the

29

most accurate way to measure the volume of a liq-uid is with a volumetric flask, which only allows onespecific, calibrated volume to be measured.

Here’s a way to get around those problems. Put themagnetic stirrer inside the flask. Pour water througha funnel into a volumetric flask, filling it less thanhalf-way. (Do not use the pipette to transfer thewater.) A common mistake is to fill the flask morethan half-way. Now pour a thin layer of cookingoil on top. Cooking oil does not mix with water,so it forms a layer on top of the water. (Set asideone funnel that you will use only for the oil, sincethe oil tends to form a film on the sides.) Finally,gently pour the alcohol on top. Alcohol does not mixwith cooking oil either, so it forms a third layer. Bymaking the alcohol come exactly up to the mark onthe calibrated flask, you can make the total volumevery accurately equal to 100 mL. In practice, it ishard to avoid putting in too much alcohol throughthe funnel, so if necessary you can take some backout with the pipette.

If you put the whole thing on the balance now, youknow both the volume (100 mL) and the mass ofthe whole thing when the alcohol and water havebeen kept separate. Now, mix everything up withthe magnetic stirrer. The water and alcohol form amixture. You can now test whether the volume ormass has changed.

If the mixture does not turn out to have a volumethat looks like exactly 100 mL, you can use the fol-lowing tricks to measure accurately the excess ordeficit with respect to 100 mL. If it is less than 100

mL, weigh the flask, pipette in enough water to bringit up to 100 mL, weigh it again, and then figure outwhat mass and volume of water you added based onthe change in mass. If it is more than 100 mL, weighthe flask, pipette out enough of the mixture to bringthe volume down to 100 mL, weigh it again, andmake a similar calculation using the change in massand the density of the oil. If you need to pipette outsome oil, make sure to wash and rinse the pipettethoroughly afterwards.

Prelab


P1 Give an example of two things having the samemass and different densities.

P2 Give an example of two things having the samedensity and different masses.

P3 Why can the density of water be given in abook as a standard value under conditions of stan-dard temperature and pressure, while the mass ofwater cannot?

P4 What would your raw data in part A be like ifair had no weight? What would they be like if airdid have weight?

P5 Referring to the section of the lab manual onerror analysis, plan how you will estimate your ran-dom errors.

P6 In part B, pick either mass or volume, and de-scribe what your observations would be like if thatquantity was not conserved.

Self-Check

Do a quick analysis of both parts without error anal-ysis. Plan how you will do your error analysis.

Analysis

A. If your results show that air has weight, determinethe (nonzero) density of air, with an estimate of yourrandom errors.

B. Decide whether volume and/or mass is conservedwhen alcohol and water are mixed, taking into ac-

30 Lab 10 Conservation Laws

count your estimates of random errors.

31

11 Conservation of Energy

Apparatus

air trackcartsprings (steel, 1.5 cm diameter)photogate (PASCO)computerstopwatchesair blowersalligator clipsspring scalesvernier caliperspower strips for switching CENCO blowers on andoffstring

Goal

Test conservation of energy for an object oscillatingaround an equilibrium position.

This could be a vibration of the sun, a water balloon, or

a nucleus.

Introduction

One of the most impressive aspects of the physicalworld is the apparent permanence of so many of itsparts. Objects such as the sun or rocks on earthhave remained unchanged for billions of years, so itmight seem that they are in perfect equilibrium, withzero net force on each part of the whole. In reality,the atoms in a rock do not sit perfectly still at anequilibrium point — they are constantly in vibrationabout their equilibrium positions. The unchangingoblate shape of the sun is also an illusion. The sunis continually vibrating like a bell or a jiggling waterballoon, as shown in the (exaggerated) figure. Thenuclei of atoms also jiggle spontaneously like littlewater balloons. The fact that these types of motion

continue indefinitely without dying out or buildingup relates to conservation of energy, which forbidsthem to get bigger or smaller without transferringenergy in or out.

Our model of this type of oscillation about equilib-rium will be the motion of a cart on an air track be-tween two springs. The sum of the forces exerted bythe two springs should at least approximately obeyHooke’s law,

F = −kx ,

where the equilibrium point is at x = 0. The nega-tive sign means that if the object is displaced in thepositive direction, the force tends to bring it backin the negative direction, towards equilibrium, andvice versa. Of course, there are no actual springsinvolved in the sun or between a rock’s atoms, butwe can still learn about this type of situation in alab experiment with a mass attached to a spring. Inthis lab, you will study how the changing velocity ofthe object, in this case a cart on an air track, canbe understood using conservation of energy. Recallthat for a constant force, the potential energy is sim-ply −Fx, but for a force that is different at differentlocations, the potential energy is minus the area un-der the curve on a graph of F vs. x. In the presentcase, the area formed is a triangle with base = x,height = kx, and

area =1

2base · height

= −1

2kx2

(counted as negative area because it lies below thex axis), so the potential energy is

PE =1

2kx2 .

Conservation of energy, PE +KE = constant, gives

1

2kx2 +

1

2mv2 = constant .

32 Lab 11 Conservation of Energy

Preliminary Observations

You should do both of the following methods of de-termining the spring constant.

Determining the spring constant: method 1

Pull the cart to the side with a spring scale, andmake a graph of F versus x, like the one on page32. To avoid pulling at the wrong angle, it helps ifyou connect the spring scale to the cart with a pieceof string. Find the combined spring constant of thetwo springs, k, from the slope of the graph.

Determining the spring constant: method 2

The second technique for determining k is to pull thecart to one side, release it, and measure the period

of its side-to-side motion, i.e., the time required foreach complete repetition of its vibration. As we’lldiscuss later in the course, the period is nearly in-dependent of the amount of travel, and the springconstant is related to the period and the mass of thecart by the equation k = m(2π/T )2. A small pe-riod indicates a large spring constant, since a pow-erful spring would be required to whip the cart backand forth rapidly. The period, T , can be found veryaccurately by using a stopwatch to time many os-cillations in a row without stopping. This methodtherefore gives a very accurate value for k, whichyou should use in your analysis of the conservationof energy. Your k value from method 1 is still usefulas a check, however.

Observations

The technique is essentially the same as in lab 9,which you may want to review. Instructions for useof the Vernier Timer software are given in Appendix6; you want the mode for measuring how long thephotogate was blocked. The two springs are at-

tached to the cart by sticking them directly throughthe holes in the cart (not through the bumper, whichwould cause the springs to drag on the track). Atthe ends of the track, the springs can be attached us-ing alligator clips, again taking care to attach themhigh enough so they don’t drag.

Throughout the lab, you should only leave the airblower turned on when you are actually using theair track. In the past, we have burned out motorsor even melted hoses by leaving the air blowers oncontinuously.

Before you start taking actual data, check whetheryou have excessive friction by letting the computerrecord data while the cart vibrates back and fortha few times through the photogate. If the air trackis working right, all the time measurements shouldbe nearly the same, but if the data show the cartslowing down a lot from one vibration to the next,then you have a problem with friction. The mostcommon causes of excessive friction are springs thatare dragging on the track or springs that are nothorizontal, and thus tipping the cart and causingone of its edges to drag.

Measure the velocity of the cart for many differentvalues of x by moving the photogate to various po-sitions. Make sure you always release the cart fromrest at the same point, and when you are initiallychoosing this release point, make sure that it is notso far from the center that the springs are completelybunched up or dragging on the track. Don’t forgetthat the x you use in the potential energy shouldbe the distance from the equilibrium position to theposition where the vane is centered on the photogate— if you don’t think about it carefully, it’s easy tomake a mistake in x equal to half the width of thevane.

Prelab


P1 What measurements besides those mentionedabove will you need to do in lab in order to checkconservation of energy?

P2 Find the value of x from the figure below. (I’vemade the centimeter scale unrealistic for readability— the real track is more than a meter long, not 14

33

centimeters.)

Self-Check

Calculate the energies at the extremes, where PE =0 and KE = 0, and see whether the energy is stayingroughly constant. You should do this self-check asearly as possible in the lab, so that you can makesure you’re not spending lots of time collecting datathat turn out to be bogus.

Analysis

Graph PE, KE, and the total energy as functionsof x, with error bars (see appendices 1, 2, and 3),all overlaid on the same plot. Make sure to includethe points with KE = 0 and PE = 0. As a shortcutin your error analysis, it’s okay if you do the erroranalysis for your most typical data-point, in whichthe energy is split roughly 50-50 between PE andKE, and then assume that the same error bars onPE, KE, and total energy apply to all the otherpoints on the graph as well.

Discuss whether you think conservation of energyhas been verified.

34 Lab 11 Conservation of Energy

12 Conservation of Momentum

Apparatus

computer with Logger Pro softwaretrack2 dynamics carts and 2 carts with magnets1-kg weight500 g slotted weightmasking tape2 force sensors with rubber corks

Qualitative Observations

First you’re going to observe some collisions betweentwo carts and see how conservation of momentumplays out. If you really wanted to take numericaldata, it would be a hassle, because momentum de-pends on mass and velocity, and there would be fourdifferent velocity numbers you’d have to measure:cart 1 before the collision, cart 1 after the collision,cart 2 before, and cart 2 after. To avoid all this com-plication, the first part of the lab will use only visualobservations.

Try gently pressing the two carts together on thetrack. As they come close to each other, you’ll feelthem repelling each other! That’s because they havemagnets built into the ends. The magnets act likeperfect springs. For instance, if you hold one cartfirmly in place and let the other one roll at it, theincoming cart will bounce back at almost exactly thesame speed. It’s like a perfect superball.

A Equal masses, target at rest, elastic collision

Roll one cart toward the other. The target cart isinitially at rest. Conservation of momentum readslike this,

M × + M ×=? M × + M × ,

where the two blanks on the left stand for the twocarts’ velocities before the collision, and the twoblanks on the right are for their velocities after thecollision. All conservation laws work like this: thetotal amount of something remains the same. Youdon’t have any real numbers, but just from eye-balling the collision, what seems to have happened?Let’s just arbitrarily say that the mass of a cart isone unit, so that wherever it says ‘M x’ in the equa-tion, you’re just multiplying by one. You also don’t

have any numerical values for the velocities, but sup-pose we say that the initial velocity of the incomingcart is one unit. Does it look like conservation ofmomentum was satisfied?

B Mirror symmetry

Now reenact the collision from part A, but do every-thing as a mirror image. The roles of the target cartand incoming cart are reversed, and the direction ofmotion is also reversed.

M × + M ×=? M × + M × ,

What happens now? Note that mathematically, weuse positive and negative signs to indicate the direc-tion of a velocity in one dimension.

C An explosion

Now start with the carts held together, with theirmagnets repelling. As soon as you release them,they’ll break contact and fly apart due to the re-pulsion of the magnets.

M × + M ×=? M × + M × ,

Does momentum appear to have been conserved?

D Head-on collision

Now try a collision in which the two carts head to-wards each other at equal speeds (meaning that onecart’s initial velocity is positive, while the other’s isnegative).

M × + M ×=? M × + M × ,

E Sticking

Arrange a collision in which the carts will stick to-gether rather than rebounding. You can do this byletting the velcro ends hit each other instead of themagnet ends. Make a collision in which the target isinitially stationary.

M × + M ×=? M × + M × ,

The collision is no longer perfectly springy. Did itseem to matter, or was conservation of momentumstill valid?

36 Lab 12 Conservation of Momentum

F Hitting the end of the track

One end of the track has magnets in it. Take onecart off the track entirely, and let the other cart rollall the way to the end of the track, where it willexperience a repulsion from the fixed magnets builtinto the track. Was momentum conserved? Discussthis with your instructor.

G Unequal masses

Now put a one kilogram mass on one of the carts,but leave the other cart the way it was. Attach themass to it securely using masking tape. A bare carthas a mass of half a kilogram, so you’ve now tripledthe mass of one cart. In terms of our silly (but con-venient) mass units, we now have masses of one unitand three units for the two carts. Make the triple-mass cart hit the initially stationary one-mass-unitcart.

3M × + M ×=? 3M × + M × ,

These velocities are harder to estimate by eye, but ifyou estimate numbers roughly, does it seem possiblethat momentum was conserved?

Quantitative Observations

Now we’re going to explore the reasons why momen-tum always seems to be conserved. Parts H and Iwill be demonstrated by the instructor for the wholeclass at once.

Attach the force sensors to the carts, and put on therubber stoppers. Make sure that the rubber stoppersare positioned sufficiently far out from the body ofthe cart so that they will not rub against the edgeof the cart. Put the switch on the sensor in the+10 N position. Plug the sensors into the DIN1and DIN2 ports on the interface box. Start up theLogger Pro software, and do File>Open>Probes andSensors>Force Sensors>Dual Range Forrce>2-10 NDual Range. Tell the computer to zero the sensors.Try collecting data and pushing and pulling on therubber stopper. You should get a graph showing howthe force went up and down over time. The sensoruses negative numbers (bottom half of the graph) forforces that squish the sensor, and positive numbers(top half) for forces that stretch it. Try both sensors,and make sure you understand what the red and bluetraces on the graph are showing you.

H. Put the extra 1-kilogram weight on one of thecarts. Put it on the track by itself, without the othercart. Try accelerating it from rest with a gentle,

steady force from your finger. You’ll want to set thecollection time to a longer period than the default.Position the track so that you can walk all the wayalong its length (not diagonally across the bench).Even after you hit the Collect button in Logger Pro,the software won’t actually start collecting data untilit’s triggered by a sufficiently strong force; squeezeon one of the sensors to trigger the computer, andthen go ahead and do the real experiment with thesteady, gently force.

What does the graph on the computer look like?

I. Now repeat H, but use a more rapid accelerationto bring the cart up to the same momentum. Sketcha comparison of the graphs from parts H and I.

Discuss with your instructor how this relates to mo-mentum.

J. You are now going to reenact collision A, but don’tdo it yet! You’ll let the carts’ rubber corks bump intoeach other, and record the forces on the sensors. Thecarts will have equal mass, and both forces will berecorded simultaneously. Before you do it, predictwhat you think the graphs will look like, and showyour sketch to your instructor.

Switch both sensors to the +50 N position, and openthe corresponding file on the computer.

Zero the sensors, then check the calibration by bal-ancing a 500 g slotted weight on top, taking data,zooming in, and putting the mouse cursor on thegraph. You will probably find that the absolute cal-ibration of the sensor is very poor when it’s used onthe 50 N scale; keep this in mind when interpretingyour results from the collision.

Now try it. To zoom in on the relevant part of thegraph, use the mouse to draw a box, and then clickon the magnifying glass icon. You will notice byeye that the motion after the collision is a tiny bitdifferent than it was with the magnets, but it’s stillpretty similar. Looking at the graphs, how do youexplain the fact that one cart lost exactly as muchmomentum as the other one gained? Discuss thiswith your instructor before going on.

K. Now imagine – but don’t do it yet – that youare going to reenact part G, where you used unequalmasses. Sketch your prediction for the two graphs,and show your sketch to your instructor before yougo on.

Now try it, and discuss the results with your instruc-tor.

37

13 Conservation of Momentum in Two Di-

mensions

Apparatus

photogate (PASCO) . . . . . . . . . . . . . . . . . . . . . . 1/groupcomputer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/groupsmall steel and plastic balls of various masses 2/groupplastic rulersprotractorscotch tape

Goal

Test whether momentum is conserved in a collisionof two balls.

Introduction

Pool players have an intuitive feeling for conserva-tion of momentum: they can visualize the results ofa collision of two pool balls in advance. They alsoknow that certain shots are impossible. For instance,there is no way to make the cue ball bounce back di-rectly from a collision with another ball (except byputting spin on it, which creates an external frictionforce with the felt). They understand that the anglesare important, so without knowing it, they are doingmental estimates involving momentum as a vector:a quantity that has both magnitude and direction.

In this lab, you will be studying collisions similar tothe collision of the cue ball with an initially station-ary ball. One of the basic principles involved is theconservation of momentum:

The Principle of Conservation of Momentum

No matter how a set of objects interact with eachother, as long as no external force is present, thevector sum of their momenta is conserved. That is,p1i+p2i+. . . = p1f+p2f+. . ., where the ‘. . . ’ meansthat if there are more than two objects involved, alltheir momenta should be added like vectors.

The technique

The idea is to set up an off-center collision, as shownbelow, and measure the initial and final speeds of theballs using the photogate and the computer. The

use of the photogate and the computer software thatworks with it is explained in Appendix 6. Since onany given trial you can only use the photogate tomeasure the speed of a single ball, you will have toreproduce the collision at least three times to mea-sure the three speeds involved. Actually, you willwant to measure each of the three speeds severaltimes in order to get a good estimate of your ran-dom errors.

To reproduce the same initial speed for the projectile(ball 1), you can build a little ramp out of two plasticrulers taped together at a 90-degree angle. A blockof wood can be taped in the ramp at the top to keepthem braced. The block of wood also serves as a

38 Lab 13 Conservation of Momentum in Two Dimensions

convenient reference point: you can release the ballfrom the point where it touches the block.

You should choose a completely asymmetrical setup:two balls of different masses, and a collision in whichthe projectile does not hit the target head-on.

It is critical that you position the target ball at ex-actly the same place every time. Marking the tableand placing the ball on the mark is not good enough.The best technique is to put a piece of scotch tapeon the table and use a ball-point pen to make a tinyimpression in it for the target ball to sit in.

Tips

You want to avoid conditions for which any of thespeeds involved are too slow, because then the ballstend to be accelerated, decelerated, or deflected bytiny bumps in the tabletop. If you notice the ballswandering and wavering as they roll, they are go-ing too slow. Generally speaking, sufficiently highspeeds are achieved if the ramp is at least 7 cm high.Using the heavier ball as the projectile helps to keepthe final speeds high.

A good way to test whether your speeds are sufficientis to measure the angles at which the balls emergefrom the collision, and see if they are the same everytime, to within a tolerance of 5-10 degrees. If theangles are not reproducible to this level of variation,then the balls are not going fast enough.

You will want to use vernier calipers to measure thediameters of the balls. Ask your instructor for helpif you don’t know how to read a vernier scale.

Note that at the instant of collision, the balls aretouching, but their centers are not at the same point.This means you have to be careful about how youmeasure the angles.

If you did not position the photogate at the height ofthe center of the ball whose speed you wanted to de-termine, then the computation of the ball’s speed be-comes complicated — don’t just divide the diameterof the ball by the time from the computer. Discussthis with your instructor once you have a workingsetup.

You should have opposite signs for the componentsof the balls’ final momenta in the direction perpen-dicular to the projectile’s original direction of mo-tion.

You will be putting the photogate in three differentpositions to measure the three velocities. How farfrom the collision should you place it? It should

be as close as possible to the collision, because theballs do gradually slow down as they roll, and youwant to know the speeds immediately before andafter the collision. However, the balls bounce a littleimmediately after the collision, so don’t put the itso close to the collision that they are still bouncingwhen they go through it.

Prelab


P1 Draw an example of a collision, showing theballs before and after it happens, in which |p1i| =0.020 kg ·m/s, |p1f | = 0.010 kg ·m/s, and |p2f | =0.010 kg ·m/s, but momentum was not conserved.(As in the actual lab, the target ball starts at rest.)Explain.

P2 If the magnitude of the initial momentum is thesame as the magnitude of the total final momentum,does that mean momentum was conserved?

Self-Check

Analyze your data without error analysis, and makesure your graphical and analytical results are thesame. Check whether momentum appeared to be atleast approximately conserved.

Analysis

Test whether momentum was conserved, doing yourvector addition once using the analytic method andonce using the graphical method. Take into accountthe random errors in your measurements.

39

14 Torque

Apparatus

meter stick with holes drilled in it . . . . . . . . 1/groupspring scales, calibrated in newtonsweightsstringprotractorshooks

Goal

Test whether the total force and torque on an objectat rest both equal zero.

Introduction

It is not enough for a boat not to sink. It also mustnot capsize. This is an example of a general factabout physics, which is also well known to peoplewho overindulge in alcohol: if an object is to be ina stable equilibrium at rest, it must not only havezero net force on it, to keep from picking up momen-tum, but also zero net torque, to keep from acquiringangular momentum.

Observations

Weigh your meter stick before you do anything else;they don’t all weigh the same amount.

For each spring scale, hang a known weight from it,and adjust the calibration tab so that the scale givesthe correct result.

Construct a setup like the one shown above. Avoidany symmetry in your arrangement. There are fourforces acting on the meter stick:

FH = the weight hanging underneath

FM = Earth’s gravity on the meter stick itself

FL = tension in the string on the left

FR = tension in the string on the right

Each of these forces also produces a torque.

In order to determine whether the total force is zero,you will need enough raw data so that for each torqueyou can extract (1) the magnitude of the force vec-tor, and (2) the direction of the force vector. In orderto add up all the torques, you will have to choose anaxis of rotation, and collect enough raw data to beable to determine for each force (3) the distance fromthe axis to the point at which the force is applied tothe ruler, and (4) the angle between the force vectorand the line connecting the axis with the point wherethe force is applied. Note that the meter stick’s ownweight can be though of as being applied at its centerof mass.

You have a selection of spring scales, so use the rightone for the job — don’t use a 20 N scale to measure0.8 newtons, because it will not be possible to readit accurately. If you need to swap in a new springscale, don’t forget to calibrate it.

Since the analysis requires you to compute the to-tal torque a second time using a different choice ofaxis, you cannot neglect to measure any of the anglesinvolved.

Prelab


P1 You have complete freedom in defining whatpoint is to be considered the axis of rotation — ifone choice of axis causes the total torque to be zero,then any other choice of axis will also cause the to-tal torque to be zero. It is possible to simplify theanalysis by choosing the axis so that one of the fourtorques is zero. Plan how you will do this.

40 Lab 14 Torque

P2 All the torques will be tending to cause rota-tion in the same plane. You can therefore use plusand minus signs to represent clockwise and counter-clockwise torques. Choose which one you’ll call pos-itive. Using your choice of axis, which of the fourtorques, τH , τM , τL, and τR, will be negative, whichwill be positive, and which will be zero?

P3 Suppose that in the figure above, the angle be-tween the meter stick and the hanging weight is 80 ,the mass of the hanging weight is 1 kg, and the massof the meter stick is 0.1 kg. If a student is then try-ing to calculate the x components of the forces FM

and FH , why is it incorrect to say

FM ,x = (0.1 kg)(9.8 m/s2)

and

FH,x = (1 kg)(9.8 m/s2)(cos 80 )?

Analysis

Determine the total force and total torque on themeter stick. For the forces, I think a graphical cal-culation will be easier than a numerical one.

Finally, repeat your calculation of the total torqueusing a different point as your axis. Although you’renormally expected to do your analysis completely in-dependently, for this lab it’s okay if you find the totaltorque for one choice of axis, and your lab partnersdo the calculation their own choices.

Error analysis is not required. For extra credit, youcan do error analysis for one of your total torques.

41

15 The Moment of Inertia

Apparatus

meter stick with hole in center . . . . . . . . . . . . 1/groupnail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1/groupfulcrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1/groupslottedmass set . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/groupduct tapesliding bracket to go onmeter stick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/groupU-shaped hook for hangingweights from bracket . . . . . . . . . . . . . . . . . . . . . 1/groupcomputer Vernier software1/group photogate and adapter box . . . . . . .1/grouptriple-beam balance . . . . . . . . . . . . . . . . . . . . . . .1/group

Goal

Test the equation τtotal = Iα, which relates an ob-ject’s angular acceleration to its own moment of in-ertia and to the total torque applied to it.

Introduction

Newton’s first law, which states that motion in astraight line goes on forever in the absence of a force,was especially difficult for scientists to work out be-cause long-lasting circular motion seemed much moreprevalent in the universe than long-lasting linear mo-tion. The sun, moon and stars appeared to movein never-ending circular paths around the sky. Aspinning top could continue its motion for a muchlonger time than a book sliding across a table. RenDescartes (b. 1596) came close to stating a law of in-ertia like Newton’s, but he thought that matter wasmade out of tiny spinning vortices, like whirlwindsof dust. Galileo, who among Newton’s predecessorscame closest to stating a law of inertia, was also con-fused by the issue of circular versus linear inertia. Anadvocate of the Copernican system, in which the ap-parent rotation of the sun, stars and moon was dueto the Earth’s rotation, he knew that the apparentlymotionless ground, trees, and mountains around himmust be moving in circles as the Earth turned. Wasthis because inertia naturally caused things to movein circles?

Newton, like other giants of science, saw how to focuson the simple rather than the complex. His law of

inertia was completely linear. In his view, all thecommon examples of circular motion really involveda force, which kept things from going straight. Inthe case of a spinning top, for instance, Newton (aconfirmed atomist) would have visualized an atomsin the top as being acted on by some kind of stickyforce from the other atoms, which would keep it fromflying off straight. Linear motion was the simplesttype, needing no forces to keep it going. Circularmotion was more complex, requiring a force to bendthe atoms’ trajectories into circles.

Even though circular motion is inherently more com-plicated than linear motion, some very close analo-gies can be made between the two in the case wherean object is spinning rigidly. (An examples of non-rigid rotation would be a hurricane, in which theinner parts complete a rotation more rapidly thanthe outer parts.) In analogy to Newton’s first law,Ftotal = ma, we have

τtotal = Iα ,

where the angular acceleration α replaces the linearacceleration a, the total torque plays the role givento the total force, and the moment of inertia I isused instead of the mass. In this lab, you are goingto release an unbalanced rotating system — a meterstick on an axle with weights attached to it — andmeasure its angular acceleration in response to thenonzero gravitational torque on it.

Every equation you learned for constant accelerationcan also be adapted to the case of rotation with con-stant angular acceleration, simply by translating allthe variables. For instance, the equation v2

f = 2axfor an object accelerating from rest can be translatedinto the valid rotational formula ω2

f = 2aθ.

The moment of inertia is defined as I =∑

mr2,where m can be thought of as the mass of an indi-vidual atom comprising the rotating body, and r isthe distance of that atom from the axis of rotation.

The word ‘moment’ in ‘moment of inertia’ does notrefer to a moment in time, but is used instead in amore old-fashioned sense of ‘importance’ or ‘weight,’as in ‘matters of great moment.’ The idea is that thefactor of r2 gives more importance to the an atomthat is far from the axis of rotation. Because thesymbol I is used, there is a tendency for students torefer to it as ‘inertia,’ but inertia is a different and

42 Lab 15 The Moment of Inertia

nonquantitative concept, referring to the tendencyof objects to stay at rest or stay in motion.

In practice, it is not practical to carry out a sum overall the atoms. The object whose rotation you willstudy in this lab will consist of a meter stick pivot-ing at its center, with various weights hanging fromit in various places. Both the hanging weights andthe meter stick itself will contribute to the momentof inertia. To a good approximation, each hangingweight can be treated as if all its atoms were con-centrated at its center. Calculus can also be usedto derive formulae for the moments of inertia of ob-jects of various shapes, such as a sphere, a cylinderrotating along its axis, etc. One such formula isI = (1/12)ML2 for the moment of inertia of a rigidrod rotating about an axis passing perpendicularlythrough its center. You can use this formula as agood approximation for the meter stick’s contribu-tion to the moment of inertia, with L = 1 m.

Preliminaries

The meter stick is supported on the fulcrum via anail through the hole in its center. You want to startby producing a balanced arrangement of weights at-tached to the meter stick, as in figure (a) below. Theidea is that if you first balance this configurationcarefully, then you know that the net gravitationaltorque on it is zero. If you then hang another weightfrom the previously empty hanger, as in (b), thenyou know that the total torque simply equals thetorque produced by the earth’s gravitational forceon the added weight.

For ease of adjustment, you can use duct tape, wrappedsticky-side-out, to attach the slotted weights to themeter stick. You can then balance your initial config-uration simply by sliding the weights around. Do notchoose a symmetric setup, i.e., use unequal weights.

The masses need to be slid to the left and right in or-der to achieve equilibrium, but it is less obvious thatit also makes a difference how high the weights are

placed. That is, the center of mass of the whole bal-anced setup must coincide both vertically and hori-zontally with the nail. The concept is shown in thefigure above using a rectangle in place of the actualapparatus. In (d), there will always be a clockwisetorque on the rectangle, because the center of massis to the right of the nail.

In (e), there is zero torque if the rectangle is ini-tially released from this horizontal position, but theequilibrium is unstable, because its center of mass isabove the axis of rotation. Our experiment dependson the cancellation of the gravitational torques oneverything but the extra weight, but in a case like(e), this assumption would only be valid when theapparatus was initially released from horizontal. Laterin the motion, there would be an undesired and un-known extra torque. Although it is visually obviousin this figure that the rectangle’s center of mass istoo high, you can’t tell visually with the actual ap-paratus. The way to tell if the center of mass is toohigh is that if you tilt the meter stick a little bitto the right, it immediately accelerates clockwise,whereas if you tilt it a little to the left, it acceleratescounterclockwise.

In (f), we have a stable equilibrium. Again, there isan unknown, undesired torque unless the rectanglejust happens to be horizontal. You can tell if youhave this situation because the apparatus can swingback and forth about its stable equilibrium position.

You want a neutral equilibrium, i.e., no matter whatangle you release it from, the meter stick just staysthere.

43

Observations

Now add the extra weight so that the meter stickis slightly unbalanced. The idea of this lab is torelease the meter stick and use the photogate to findhow quickly it is moving once it has rotated throughsome angle, using the photogate to find the amountof time required for the tip of the meter stick to passthrough the photogate. From your measurement of∆t using the photogate, you can find ω = ∆θ/∆t,which is an approximation to the meter stick’s finalangular velocity. Instructions for using the computersoftware are given in appendix 6; you want the modefor measuring how long the photogate was blocked.

Once you know the meter stick’s final value of ω,you can extract the angular acceleration. This canthen be compared with the theoretical value of theangular acceleration from τtotal = Iα.

Tips:

You may want to put something under the ful-crum base to raise everything up higher.

Although the balanced configuration, with τtotal =0, still has τtotal = 0 no matter what angle itis at, the torque exerted by the extra weightdoes depend a little on what angle the meterstick is at. This is because of the factor ofsin θ in the definition τ = rF sin θ. Since thetorque is not constant, the angular accelera-tion is not constant, leading to complications.You can avoid this problem by confining allyour measurements to a fairly small range ofpositions near horizontal. As long as θ is fairlyclose to 90 , sin θ is extremely close to 1, andit is a good enough approximation to assume aconstant torque rF producing a constant an-gular acceleration. For instance, as long as θis within 20 above or below horizontal, sin θchanges by no more than 0.06.

Although you want to work only with nearlyhorizontal positions of the meter stick so thatthe torque is approximately constant, you alsoneed to make sure that the total angle tra-versed by the meter stick is still reasonablylarge compared to the angle traversed while themeter stick is blocking the photogate. Other-wise your measurement of ω = ∆θ/∆t will notbe a good approximation to the final instanta-neous angular velocity.

As you will find in your prelab, the angular ac-celeration depends on the square of the angle

∆θ. Measuring this angle accurately is there-fore vital in order to get a good result. A

protractor cannot measure an angle this small

with sufficient accuracy. Use trigonometry todetermine this angle.

It’s easiest if you use radian measure through-out. The equation τtotal = Iα is only true if ais measured in radians/s2.

The sliding bracket and hook contribute bothto the total torque and the moment of inertia,so you’ll have to weigh them.

Prelab


P1 Derive an equation for the experimental valueof the angular acceleration, expressed in terms ofquantities you will actually measure directly, includ-ing the quantities θ and ∆θ defined in the figure be-low. Note that this lab is exactly analogous to theprevious lab where you found a linear accelerationusing a similar setup.

P2 Why would it not be meaningful to try to dealwith the meter stick’s velocity, rather than its angu-lar velocity?

Self-Check

Do all your analysis in lab.

Analysis

Extract theoretical and experimental values of theangular acceleration from your data, and compare

44 Lab 15 The Moment of Inertia

them.

No analysis of random errors is required, because themain source of error is the systematic errors arisingfrom friction and the various approximations, suchas the assumption that sin θ is approximately equalto 1.

45

16 Absolute Zero

Note to the lab technician: The dessicant needs tobe dry before the experiment. If it’s blue, it’s dry. Ifit’s pink, it needs to be pumped on for a few hourswith a vacuum pump while heating it with a hairdryer.

Apparatus

gas capillary tubelarge test tubemercury thermometerglass syringeelectric heating padoven mittslatex tubingicestringfunnels

Introduction

If heat is a form of random molecular motion, thenit makes sense that there is some minimum temper-ature at which the molecules aren’t moving at all.With fancy equipment, physicists have gotten sam-ples of matter to within a fraction of a degree aboveabsolute zero, but they have never actually reachedabsolute zero (and the laws of thermodynamics ac-tually imply that they never can). Nevertheless, wecan determine how cold absolute zero is without evengetting very close to it. Kinetic theory tells us thatthe volume of an ideal gas is proportional to howhigh it is above absolute zero. In this lab, you’llmeasure the volume of a sample of air at tempera-tures between 0 and 100 degrees C, and determinewhere absolute zero lies by extrapolating to the tem-perature at which it would have had zero volume.

Observations

Tie a short piece of string to the thermometer sothat you’ll be able to pull it back out of the beakerwhen you want to without dipping your hands inhot water. Start heating the water up to the boilingpoint. (If you leave the thermometer in the waterwhile it’s heating, you’ll be able to observe later theinteresting fact that the water stops heating up once

it reaches the boiling point.) If the water starts boil-ing before you’re ready, just turn off the heat andreheat it later – it doesn’t cool off very fast.

The capillary tube is sealed at the bottom and openat the top, with a large bulb full of dessicant justbelow the top to keep the air inside dry. There is asmall amount of mercury inside the tube. Right now,the mercury is probably ‘floured,’ i.e., broken up intosmall pieces sticking on the sides of the tube. Theidea is to collect the mercury into a single drop, witha sample of air trapped in the capillary tube underit. The mercury simply acts as a seal. As the airis heated and cooled, it expands and contracts, andyou can measure its volume by watching the mercuryseal rise and drop. By the way, don’t be scared ofthe mercury; mercury vapor is a deadly poison, butliquid mercury is entirely harmless unless you ingestit or get it in an open cut. There is a small filtermade of glass wool at the top end of the bulb, whichwill keep the mercury from getting out.

Remove the gas syringe from the box, being care-ful not to let the glass plunger drop out and break.Connect it to the capillary tube with a piece of tub-ing.

First you need to get the mercury into a single blobin the cavity at the top of the capillary, where itwidens out just below the bulb. If it’s already form-ing a seal across the capillary tube, you won’t beable to get it to move, because it’s trapped betweenthe pressures of the inside air and the outside air.

46 Lab 16 Absolute Zero

You can break the seal by opening the stopcock anddrawing some air out with the syringe. (Note thatthe stopcock has three holes; two are lined up withthe knob, and the third one is on the side markedwith a dot on the knob.) If this doesn’t break theseal, you can very gently tap the capillary tube withyour little finger; a student recently broke a tube bytapping it too hard, although he thought he was be-ing fairly gentle. Now disconnect the tube from thesyringe, and, if necessary, shake it extremely gen-tly upside-down to get all the mercury droplets tocollect in the cavity.

At this point, if you put the tube upright again, themercury drop will sit at the very top of the capil-lary, with a sample of air trapped below it filling theentire tube. This is no good, because most of thetemperatures you’ll be using in this lab are hotterthan room temperature, so you need room for theair sample to expand without forcing the mercuryout into the cavity. Here’s how to get a smaller vol-ume of air trapped under the mercury. Push theplunger all the way into the syringe, open the stop-cock, and connect the syringe to the tube, leavingthe tube horizontal with the mercury in the cavity.Now pull the plunger out until you’ve created a 40%vacuum. If you have the stopcock in the correct po-sition, it should take quite a strong force to pull theplunger out this far. Now bring the tube uprightagain, and gently allow the plunger to slide back in.At this point, the mercury should be about 40% be-low the top of the capillary, and you can disconnectthe syringe.

Detach the syringe and tubing, so from now on, ev-erything is always at constant pressure! We wanttemperature and volume to be the only variablesthat change in this experiment. By leaving every-thing open to the air in the room, we guarantee thatthe pressure will equal the air pressure in the room.

If necessary, bring the water back to a boil, and thenturn off the gas again. Move the Bunsen burneraside, and, being careful not to burn yourself, lowerthe clamp so the test tube is almost touching thetabletop; this way, if it slips out of the clamp, itwon’t fall far enough to break. (I broke one of thetest tubes myself by letting it slip this way.) Insertthe thermometer and the capillary tube, and givethem a minute or so to come to equilibrium withthe water.

You can now start taking a series of temperature andvolume measurements as the water in the test tubegradually cools down towards room temperature.

The cooling process is rapid at first. If you get im-

patient, you can gently pour a small amount of coolwater in the top, making sure to let it equilibrate fora few minutes afterward before taking data. Don’ttry to swirl the test tube around in order to speedup the equilibration – that’s what I was trying to dothe time the test tube slipped out of the clamp andbroke.

When the water gets close to room temperature, thecooling process slows down. At some point, you maywish to fill a beaker with lukewarm water and im-merse the end of the flask in it in order to speed upthe cooling.

Once you have data at temperatures down to nearroom temperature, pour some water off of the icewater, and use it to replace the water in the flask.Make sure you don’t get ice in the flask, which makesit impossible to insert the capillary tube and ther-mometer.

Analysis

Graph the temperature and volume against each other.Does the graph appear to be linear? If so, extrap-olate to find the temperature at which the volumewould be zero.

If your data are nice and linear, then your mainsource of error will be random errors, and you shouldthen determine error bars for your value of absolutezero using the techniques discussed in Appendix 4.

Prelab


P1 Should you measure the volume from the top,the middle, or the bottom of the mercury? Explain.

47

17 The Clement-Desormes Experiment

Apparatus

large flaskglass syringewater manometerhelium (medium size cylinder, $40 from Party City)difluoroethane (sold in cans as gas duster at Fry’s)stopwatchhose clampsgrabber clampsstands

Introduction

Although the theory that matter was made of atomsstarted to be talked about seriously by scientists asearly as Galileo’s time, scientists generally didn’tthink of it as something that was literally true. Theyconsidered the atomic theory to be a useful model,but they thought that any fundamental explanationof real-world phenomena should avoid talking abouthypothetical things like atoms. This feeling was sostrong that the physicist Ludwig Boltzmann, whocame up with an atomic explanation of entropy, wasdriven to suicide by the harsh criticism to which hisideas were subjected. Even more suspect than theexistence of atoms was any attempt to discuss thingslike the shapes of molecules that could be formedby putting them together like tinkertoys; such ideasseemed much too far removed from the possibility ofany experimental testing.

Surprisingly, then, a simple experiment, due to Clem-ent and Desormes, is capable of distinguishing twosamples of gas that differ only by the shape of theirmolecules, even if the gases have the same densityand are composed of molecules having the same mass.

Use the glass syringe to apply a slight overpressureto the air inside the flask, causing the difference inheight between the water in the two sides of themanometer to be about 30 cm. Wait one minuteto make sure the air is in thermal equilibrium withthe room, and then take a pressure reading, p1. Re-lease the pressure by popping the cork for preciselyone second, timed on a stopwatch. The air coolsslightly due to its expansion, because it does me-chanical work as it exits throught the valve. How-ever, because the expansion is rapid, and heat con-

duction is a slow process, we can treat this as insu-lated expansion, as discussed in Appendix 2 of Sim-ple Nature. If the gas is a monoatomic one, suchas helium, then the amount of cooling of the gas, asproved in the book, is given by the relation T ∝ P b,where b = 2/5. If the gas is not monoatomic, how-ever, then its molecules can rotate,1 and at any giventime some of its energy is in the form of kinetic en-ergy along the x, y, and z axes, but some is in theform of rotational kinetic energy. Extracting a givenamount of energy from a diatomic or polyatomic gas,therefore, doesn’t cool it as much as it would cool amonoatomic gas, and it turns out that b = 2/7 for adiatomic gas, and 1/4 for a polyatomic gas.2

Wait one minute for the air to warm back up to roomtemperature. The pressure comes back up somewhatas the air warms back up, and although you shouldwait a full minute to make sure it’s back in thermalequilibrium, most of the rewarming occurs duringthe first few seconds after you finish venting the ini-tial pressure. The pressure will recover to a valuep2 which is less than p1. The ratio p2/p1 gives thevalue of b for the gas.3

I’m still working on improving this lab. The ba-sic idea I have in mind is to have you do the labonce with helium (monoatomic), air (diatomic), and

1An individual atom in a monoatomic gas has essentiallyall its mass concentrated in the nucleus exactly at its center,so it takes an effectively infinite amount of energy to make itrotate with a certain amount of angular momentum.

2You’ll often see this stated in terms of the variable γ =1/(1 − b), which takes on the values 5/3, 7/5, and 4/3.

3In terms of the variable γ, we have γ = p1/(p1 − p2).

48 Lab 17 The Clement-Desormes Experiment

difluoroethane (polyatomic), and observe the differ-ences in the results due to the different shapes ofthe molecules. There are various systematic errorsin the experiment, so my own absolute results forthe b of air haven’t been of extremely high preci-sion; however, in a comparative experiment, I thinkit will be easy to see a difference in b between thegases. One possible problem with the air is that itcontains water vapor, which messes up the thermo-dynamic properties of the air, because water dropletscan condense out of the air when the pressure isdropped suddenly, as when you open a can of beer.The helium and difluoroethane shouldn’t have thisproblem. In the spring semester of 2008, we triedall three gases, and found that it was fairly easy todetect a clear systematic difference between a higherb for air (.20, .29, .33, .29, and .31 for the five labgroups) and a lower one for difluoroethane (.18, .20,.33, .25, and .24), but the results for helium weremuch lower than theory, and barely distinguishablefrom air (.29, .35, .35, .31, and .31, versus .40 ac-cording to theory). This may be because we’re notactually getting the flasks as full of pure helium aswe think we are.

Some of the flasks have holes at both the top and thebottom. With these flasks, it’s a good idea to intro-duce the helium through the bottom hole, since it’slighter than air, and will rise. The difluoroethane,on the other hand, should be put in through the tophole, because it’s heavier than air. I don’t know ifit will be practical to use the helium with the flasksthat only have holes at the top.

Both the helium and the difluoroethane can displacethe beathable air in the classroom, and the amountof helium in the large canister is particularly big. Forthis reason, I’ve been dispensing the helium outsidethe classroom.

The difluoroethane is a liquid when it’s pressurizedinside the can. When you vent some of the pressurethrough the nozzle, the pressure drops, and some ofit vaporizes and comes out. The vaporization con-sumes energy, so the can becomes cold. If you holdthe can upside down and spray it, liquid is emittedrather than gas; this liquid is extremely cold, andcan cause frostbite if it gets on your skin. The gas isnot flammable, and does not harm the ozone layer.Some teenagers have intentionally inhaled it to gethigh, so the manufacturers have added a bitterant.

49

18 The Pendulum

Apparatus

stringcylindrical pendulum bobshooked massesprotractorstopwatchcomputer with photogate and Vernier Timer soft-wareclamps (not hooks) for holding the stringtape measuresmeter sticks

Goal

Find out how the period of a pendulum depends onits length and mass, and on the amplitude of itsswing.

Introduction

Until the industrial revolution, the interest of theworld’s cultures in the measurement of time was al-most entirely concentrated on the construction ofcalendars, so that agricultural cycles could be an-ticipated. Although the Egyptians were the first todivide the day and night into 12 hours, there was notechnology for measuring time units smaller than aday with great accuracy until four thousand yearslater.

Galileo was the first to realize that a pendulum couldbe used to measure time accurately — previously, hehad been using his own pulse to measure the time re-quired for objects to roll down inclined planes. Thelegend is that the idea came to him while he watcheda chandelier swinging during a church service. Sen-tenced to house arrest for suspicion of heresy, hespent the last years of his life trying to build a morepractical pendulum clock that would run for longperiods of time without tending. This technical featwas only achieved later by Christian Huygens. Alongwith the Chinese invention of the compass, accurateclocks were vital for European exploration by sea,because longitude can only be determined by astro-nomical observations combined with accurate mea-surements of time.

Notation and Terminology

When a moving thing, such as a wave, an orbit-ing planet, a wheel, or a pendulum, goes througha repetitive cycle of motion, the time required forone complete cycle is called the period, T . Notethat a pendulum visits any given point once whiletraveling in one direction and once while travelingin the opposite direction. The period is defined ashow long it takes to come back to the same point,traveling in the same direction.

From a to g is one full period of the pendulum. From a

to e is not a full period. Even though the pendulum has

returned at e to its original position in a, it is moving in the

opposite direction, and has not performed every type of

motion it will ever perform.

The amplitude of a repetitive motion is a way ofdescribing the amount of motion. We can definethe amplitude, A, of the pendulum’s motion as themaximum angle to which it rises, i.e., half the totalangle swept out. Let us denote the mass of the bob,or weight at the end of the pendulum, by m, andthe length of the pendulum, from the pivot to themiddle of the weight, as L.

Observations

Make observations to determine how the period, T ,depends on A,L, and m. You will want to use the

50 Lab 18 The Pendulum

technique of isolation of variables. That means thatrather than trying many random combinations ofA,L, and m, you should keep two of them constantwhile measuring T for various values of the thirdvariable. Then you should shift your attention tothe next variable, changing it while keeping the othertwo constant, and so on. Be sure to try quite a fewvalues of the variable you are changing, so you cansee in detail how T depends on each variable.

The period can be measured using the photogate.See appendix 6 for how to use the computer soft-ware; you want the mode that’s meant specificallyfor measuring the period of a pendulum. Note thatthe bob is what is blocking the photogate, so if yourbob is irregularly shaped, your measurements couldbe messed up if it changed orientation between onepass through the photogate and the next. The eas-iest way to make sure this problem doesn’t occur isto use a bob with a circular cross-section, so it hasthe same width no matter which way the photogatecuts through it.

One of the notable differences between the way stu-dents and professional scientists approach experi-ments is that students tend to be timid about explor-ing extreme conditions. In this experiment, there isa big advantage to taking measurements over wideranges of each of the three parameters, because itmay be impossible to ascertain how the period de-pends on a parameter if you only explore a smallrange. When changing L, you can go up to fourmeters if you hang the pendulum from the balcony;however, you should avoid lengths so short that theyare comparable to the size of the bob itself, sincesuch short lengths would have anomalous behavior.

For large values of L, it’s not practical to use thecomputer, so use a stopwatch instead. Don’t justtime one oscillation, because then the precision ofyour timing will be horrible. Measure the time re-quired for some large number of oscillations.

Warning: Since L is measured to the middle of theweight, you must change the length of the stringif you want to vary m while keeping L constant,compensating for the different physical size of thenew weight.

Prelab

The point of the prelab questions is to make sureyou understand what you’re doing, why you’re do-ing it, and how to avoid some common mistakes. Ifyou don’t know the answers, make sure to come tomy office hours before lab and get help! Otherwise

you’re just setting yourself up for failure in lab.

P1 What is the maximum possible amplitude fora pendulum of the type you’ll use, whose bob hangsfrom a string? If you were using a pendulum with astiff rod instead of a string, you could release it fromstraight up. What would its period be if you couldrelease it from exactly straight up?

P2 Referring to appendix 5, how will you tell fromyour log-log plot whether the data follow a powerlaw, i.e., whether it is even appropriate to try toextract p? (If you’ve already done lab 6, it’s exactlythe same technique.)

Self-Check

Figure out which variable T depends on most strongly,and extract p (see below).

Analysis

Graph your data and state your conclusions aboutwhether T depends on A, L and m. Rememberthat on a graph of experimental data, the horizontalaxis should always be the quantity you controlled di-rectly, and the vertical axis should be the quantityyou measured but did not directly select. The pho-togate is so accurate that there is not much pointin putting error bars on your graph — they wouldbe too small to see. Remember, however, that thereare some fairly significant systematic errors, e.g., it ishard to accurately keep L the same when switchingmasses.

It may happen that when you change one of the vari-ables, there are only small, insignificant changes inthe period, but depending on how you graph thedata, it may look like these are real changes in theperiod. Most computer graphing software has a de-fault which is to make the y axis stretch only acrossthe range of actual y data. e.g., if your periodswere all between 0.567 and 0.574 s, then the soft-ware makes an extremely magnified graph, with they axis running only over the short range from 0.567to 0.574 s. On such a scale, it may seem at firstglance that there are some major changes in the pe-riod. To help yourself interpret your graphs, youshould make them all with the same y scale, goingfrom zero all the way up to the highest period youever measured. Then you’ll be comparing all threegraphs on the same footing.

Of the three variables, find the one on which theperiod depends the most strongly, and use the tech-

51

niques outlined in appendix 5 to see if you can findan equation describing the relationship between theperiod and that variable. Assume that the equationis of the form

T = cxp ,

where x would actually be A,L or m, and c and pare constants. The constant p is important, and isexpected to be the same for all pendula. For in-stance, if you find that the mass is the variable thathas the greatest effect on the period, and that therelationship is of the form T = cm3, then you havediscovered something that is probably generally truefor all pendula: that the period is proportional to thecube of the mass. The constant c is just some bor-ing number that’s not worth extracting from yourgraphs; it’s the exponent p that’s interesting anduniversally valid.

52 Lab 18 The Pendulum

19 Resonance

Apparatus

vibrator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/groupstopwatch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/groupmultimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/groupbanana plug cablesThornton power supplies (in lab benches in 416)

Goals

Observe the phenomenon of resonance.

Investigate how the width of a resonance de-pends on the amount of damping.

Introduction

To break a wine glass, an opera singer has to singthe right note. To hear a radio signal, you have tobe tuned to the right frequency. These are examplesof the phenomenon of resonance: a vibrating systemwill respond most strongly to a force that varies witha particular frequency.

Simplified mechanical drawing of the vibrator, front view.

Apparatus

In this lab you will investigate the phenomenon ofresonance using the apparatus shown in the figure.If the motor is stopped so that the arms are locked inplace, the metal disk can still swing clockwise andcounterclockwise because it is attached to the up-right rod with a flexible spiral spring. A push on the

Electrical setup, top view.

disk will result in vibrations that persist for quitea while before the internal friction in the spring re-duces their amplitude to an imperceptible level. Thiswould be an example of a free vibration, in whichenergy is steadily lost in the form of heat, but noexternal force pumps in energy to replace it.Suppose instead that you initially stop the disk, butthen turn on the electric motor. There is no rigidmechanical link to the disk, since the motor and diskare only connected through the very flexible spiralspring. But the motor will gently tighten and loosenthe spring, resulting in the gradual building up of avibration in the disk.

Observations

A Period of Free Vibrations

Start without any of the electrical stuff hooked up.Twist the disk to one side, release it, and determineits period of vibration. (Both here and at pointslater in the lab, you can improve your accuracy bytiming ten periods and dividing the result by ten.)This is the natural period of the vibrations, i.e., theperiod with which they occur in the absence of anydriving force.

B Damping

Note the coils of wire at the bottom of the disk.These are electromagnets. Their purpose is not toattract the disk magnetically (in fact the disk ismade of a nonmagnetic metal) but rather to increasethe amount of damping in the system. Whenever a

54 Lab 19 Resonance

metal is moved through a magnetic field, the elec-trons in the metal are made to swirl around. Asthey eddy like this, they undergo random collisionswith atoms, causing the atoms to vibrate. Vibrationof atoms is heat, so where did this heat energy comefrom ultimately? In our system, the only source ofenergy is the energy of the vibrating disk. The neteffect is thus to suck energy out of the vibration andconvert it into heat. Although this magnetic andelectrical effect is entirely different from mechanicalfriction, the result is the same. Creating damping inthis manner has the advantage that it can be madestronger or weaker simply by increasing or decreas-ing the strength of the magnetic field.

Turn off all the electrical equipment and leave it un-plugged. Connect the circuit shown in the top leftof the electrical diagram, consisting of a power sup-ply to run the electromagnet plus a meter . You donot yet need the power supply for driving the motor.The meter will tell you how much electrical currentis flowing through the electromagnet, which will giveyou a numerical measure of how strong your damp-ing is. It reads out in units of amperes (A), themetric unit of electrical current. Although this doesnot directly tell you the amount of damping force inunits of newtons (the force depends on velocity), theforce is proportional to the current.

Once you have everything hooked up, check withyour instructor before plugging things in and turn-ing them on. If you do the setup wrong, you couldblow a fuse, which is no big deal, but a more seri-ous goof would be to put too much current throughthe electromagnet, which could burn it up, perma-nently ruining it. Once your instructor has checkedthis part of the electrical setup she/he will show youhow to monitor the current on the meter to makesure that you never have too much.

The Q of an oscillator is defined as the number ofoscillations required for damping to reduce the en-ergy of the vibrations by a factor of 535 (a definitionoriginating from the quantity e2π). As planned inyour prelab, measure the Q of the system with theelectromagnet turned off, then with a current of 300mA through the electromagnet, and then 600 mA.You will be using these two current values through-out the lab.

C Frequency of Driven Vibration

Now connect the DC power supply (circular blackand red plugs) on your lab bench to the terminalson the motor labeled ‘motoranschluβ.’ The coarseand fine adjustments to the speed of the motor aremarked ‘groβ’ (gross) and ‘fein’ (fine).

Three of the vibrators have broken ‘motoranschluβ’connections; they are marked. If you have one ofthese, you need to connect the power supply to theother plugs, and control the motor’s frequency fromthe power supply knob. Since this makes it difficultto control the frequency accurately, you should dothe low-Q setup in part F.

Set the damping current to the higher of the twovalues. Turn on the motor and drive the system at afrequency very different from its natural frequency.You will notice that it takes a certain amount oftime, perhaps a minute or two, for the system tosettle into a steady pattern of vibration. This iscalled the steady-state response to the driving forceof the motor.

Does the system respond by vibrating at its naturalfrequency, at the same frequency as the motor, or atsome frequency in between?

D Resonance

With your damping current still set to the highervalue, try different motor frequencies, and observehow strong the steady-state response is. At whatmotor frequency do you obtain the strongest response?

You can save yourself some time if you think of thispart and part F as one unit, and plan ahead so thatthe data you take now are also the data you need forpart F.

E Resonance Strength

Set the motor to the resonant frequency, i.e., thefrequency at which you have found you obtain thestrongest response. Now measure the amplitude ofthe vibrations you obtain with each of the two damp-ing currents. How does the strength of the resonancedepend on damping?

With low amounts of damping, I have sometimes en-countered a problem where the system, when drivennear resonance, never really settles down into a steadystate. The amplitude varies dramatically from oneminute to the next, perhaps because the power sup-ply is not stable enough to control the driving fre-quency consistently enough. If this happens to you,check with your instructor.

F Width of the Resonance

Now measure the response of the system for a largenumber of driving frequencies, so that you can graphthe resonance curve and determine the width of theresonance. Concentrate on the area near the topand sides of the peak, which is what’s important forfinding the FWHM.

55

To make this part less time-consuming, your instruc-tor will assign your group to do only one of the twographs, low-Q or high-Q. Each group will have theirown data for one Q and another group’s data foranother Q.

Prelab


P1 Plan how you will determine the Q of your os-cillator in part B. [Hint: Note that the energy of avibration is proportional to the square of the ampli-tude.]

Self-Check

Make your graphs for part F (see below), and see ifthey make sense. Make sure to make the frequencyaxis expanded enough to get an accurate FWHMfrom the graph,

Analysis

Compare your observations in parts C, D, and Ewith theory.

For part F, construct graphs with the square of theamplitude on the y axis and the frequency on thex axis. The reason for using the square of the am-plitude is that the standard way of specifying thewidth of a resonance peak is to give its full width athalf resonance (FWHM), which is measured betweenthe two points where the energy of the steady-statevibration equals half its maximum value. Energy isproportional to the square of the amplitude. Deter-mine the FWHM of the resonance for each value ofthe damping current, and find whether the expectedrelationship exists between Q and FWHM; make anumerical test, not just a qualitative one. Obviouslythere is no way you can get an accurate FWHM ifthe peak is only as wide as a pencil on the graph —make an appropriate choice of the range of frequen-cies on the x axis.

56 Lab 19 Resonance

20 Resonance (short version for physics

222)

This is a simplified version of lab 19, meant to in-troduce some concepts related to mechanical reso-nance, without any detailed data-taking. The ideais to reinforce the relevant concepts from physics 221so that they can be used as a metaphor for electricalresonances in 222.

Apparatus

vibrator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/groupThornton power supply . . . . . . . . . . . . . . . . . . . 1/groupstopwatch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/groupmultimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/groupbanana plug cables

Goals

Observe the phenomenon of resonance.

Learn how to visualize phases and amplitudesin a plane.

Introduction

To break a wine glass, an opera singer has to singthe right note. To hear a radio signal, you have tobe tuned to the right frequency. These are examplesof the phenomenon of resonance: a vibrating systemwill respond most strongly to a force that varies witha particular frequency.

Apparatus

In this lab you will investigate the phenomenon ofresonance using the apparatus shown in the figure.If the motor is stopped so that the arms are locked inplace, the metal disk can still swing clockwise andcounterclockwise because it is attached to the up-right rod with a flexible spiral spring. A push on thedisk will result in vibrations that persist for quitea while before the internal friction in the spring re-duces their amplitude to an imperceptible level. Thiswould be an example of a free vibration, in whichenergy is steadily lost in the form of heat, but noexternal force pumps in energy to replace it.

Simplified mechanical drawing of the vibrator, front view.

Electrical setup, top view.

Suppose instead that you initially stop the disk, butthen turn on the electric motor. There is no rigidmechanical link to the disk, since the motor and diskare only connected through the very flexible spiralspring. But the motor will gently tighten and loosenthe spring, resulting in the gradual building up of avibration in the disk.

Observations

A Period of Free Vibrations

Start without any of the electrical stuff hooked up.Twist the disk to one side, release it, and use thestopwatch to determine its natural period of vibra-

58 Lab 20 Resonance (short version for physics 222)

tion. (Both here and at points later in the lab, youcan improve your accuracy by timing ten periodsand dividing the result by ten.)

B Damping

Note the coils of wire at the bottom of the disk.These are electromagnets. Their purpose is not toattract the disk magnetically (in fact the disk ismade of a nonmagnetic metal) but rather to increasethe amount of damping in the system. Whenever ametal is moved through a magnetic field, the elec-trons in the metal are made to swirl around. Asthey eddy like this, they undergo random collisionswith atoms, causing the atoms to vibrate. Vibrationof atoms is heat, so where did this heat energy comefrom ultimately? In our system, the only source ofenergy is the energy of the vibrating disk. The neteffect is thus to suck energy out of the vibration andconvert it into heat. Although this magnetic andelectrical effect is entirely different from mechanicalfriction, the result is the same. Creating damping inthis manner has the advantage that it can be madestronger or weaker simply by increasing or decreas-ing the strength of the magnetic field.

Turn off all the electrical equipment and leave it un-plugged. Connect the circuit shown in the top leftof the electrical diagram, consisting of a power sup-ply to run the electromagnet plus a meter . You donot yet need the power supply for driving the motor.The meter will tell you how much electrical currentis flowing through the electromagnet, which will giveyou a numerical measure of how strong your damp-ing is. It reads out in units of amperes (A), themetric unit of electrical current. Although this doesnot directly tell you the amount of damping force inunits of newtons (the force depends on velocity), theforce is proportional to the current.

Once you have everything hooked up, check withyour instructor before plugging things in and turn-ing them on. If you do the setup wrong, you couldblow a fuse, which is no big deal, but a more seri-ous goof would be to put too much current throughthe electromagnet, which could burn it up, perma-nently ruining it. Once your instructor has checkedthis part of the electrical setup she/he will show youhow to monitor the current on the meter to makesure that you never have too much.

The Q of an oscillator is defined as the number ofoscillations required for damping to reduce the en-ergy of the vibrations by a factor of 535 (a definitionoriginating from the quantity e2π). As planned inyour prelab, measure the Q of the system with theelectromagnet turned off, then with a current of 0.25

A through the electromagnet, and then 0.50 A. Youwill be using these two current values throughoutthe lab.

C Frequency of Driven Vibration

Now connect the lab’s DC power supply to the ter-minals on the motor labeled ‘motorpanschluβ.’ Thecoarse and fine adjustments to the speed of the mo-tor are marked ‘groβ’ (gross) and ‘fein’ (fine).

Set the damping current to the higher of the twovalues. Turn on the motor and drive the system at afrequency very different from its natural frequency.You will notice that it takes a certain amount oftime, perhaps a minute or two, for the system tosettle into a steady pattern of vibration. This iscalled the steady-state response to the driving forceof the motor.

Does the system respond by vibrating at its naturalfrequency, at the same frequency as the motor, or atsome frequency in between?

D Resonance

With your damping current still set to the highervalue, try different motor frequencies, and observehow strong the steady-state response is. At whatmotor frequency do you obtain the strongest response?

E Resonance Strength

Set the motor to the resonant frequency, i.e., thefrequency at which you have found you obtain thestrongest response. Now measure the amplitude ofthe vibrations you obtain with each of the two damp-ing currents. How does the strength of the resonancedepend on damping?

F Phase Response

If the disk and the vertical arm were connected rigidly,rather than through a spring, then they would al-ways be in phase. For instance, the disk would reachits most extreme clockwise angle at the same mo-ment when the vertical arm was also all the wayclockwise. But since the connection is not rigid,this need not be the case. Find a frequency sig-nificantly below the resonant frequency, at whichthe amplitude of the steady-state response is per-haps one tenth of the value it would have at res-onance. What do you observe about the relativephase of the disk and the vertical arm? Are they inphase or out of phase? You can describe the phaseby assigning positive phase angles to oscillations inwhich the disk is ahead of the arm, and negativephases when the disk is behind. These phase anglescan range from -180 to 180 . Actually +180 and

59

-180 would represent the same thing: the oscilla-tions have phases that are exactly the opposite. Tryto estimate roughly what the phase angle is. Youdon’t have any way to measure it accurately, but youshould be able to estimate it to the nearest multipleof 45 . Measure the amplitude of the steady-stateresponse as well.

Now measure the phase and amplitude of the re-sponse when the driving force is at the resonant fre-quency.

Finally, do the same measurements when the drivingforce is significantly above resonance.

Analysis

The point of this is to connect the mechanical analogto what you know about the phase response of aresonant LRC circuit. You’re measuring the phasebetween F and x, which is analogous to the phasebetween V and q in electrical terms. However, mostpeople think of AC circuits in terms of V and I, notV and q. The phase relationships you’re expecting,therefore, are those that would hold between F andv = dx/dt, which differ by 90 degrees from the F −xphases you actually measured as raw data.

To complete the electrical analogy, we would reallyprefer to discuss the mechanical analog of impedance.The (constant) driving force from the motor playsthe role of the voltage, while the frequency-dependentamplitude of the vibration plays the role of the cur-rent. Dividing these two quantities gives us some-thing analogous to impedance, and since the drivingforce is always the same, we can say that the in-verse of the amplitude is essentially a measure ofthe impedance.

To summarize, you have a complex impedance whoseamplitude and phase angle you can determine fromyour data. Plot the impedances at the various fre-quencies in the complex plane.

Prelab


P1 Plan how you will determine the Q of your os-cillator in part B. [Hint: Note that the energy of avibration is proportional to the square of the ampli-

tude.]

60 Lab 20 Resonance (short version for physics 222)

21 Standing Waves

Apparatus

stringweights, including 1-gram weightspulleyvibratorpaperclipsmetersticksbutcher paperscissorsweight holders

Goals

Observe the resonant modes of vibration of astring.

Find how the speed of waves on a string de-pends on the tension in the string.

Introduction

The Greek philosopher Pythagoras is said to havebeen the first to observe that two plucked stringssounded good together when their lengths were inthe proportion of two small integers. (This is assum-ing the strings are of the same material and underthe same tension.) For instance, he thought a pleas-ant combination of notes was produced when onestring was twice the length of the other, but that thecombination was unpleasant when the ratio was, say,1.4 to 1 (like the notes B and F). Although differentcombinations of notes are used in different culturesand different styles of music, there is at least somescientific justification for Pythagoras’ statement. Wenow know that a plucked string does not just vibrateat a single frequency but simultaneously at a wholeseries of frequencies f1, 2f1, 3f1,... These frequen-cies are called the harmonics. If one string is twicethe length of the other, then its lowest harmonic is athalf the frequency of the other string’s, and its har-monics coincide with the odd-numbered harmonicsof the other string. If the ratio is 1.4 to 1, however,then there is essentially no regular relationship be-tween the two sets of frequencies, and many of theharmonics lie close enough in frequency to produceunpleasant beats.

Setup

The apparatus allows you to excite vibrations at afixed frequency of 120 Hz (twice the frequency ofthe alternating current that runs the vibrator). Thetension in the string can be controlled by varying theweight.

You may find it helpful to put a strip of white butcherpaper behind the black string for better visual con-trast.

It’s important to get the vibrator set up properlyalong the same line as the string, not at an angle.

Observations

Observe as many modes of vibration as you can. Youwill probably not be able to observe the fundamen-tal (one hump) because it would require too muchweight. In each case, you will want to fine-tune theweight to get as close as possible to the middle ofthe resonance, where the amplitude of vibration isat a maximum. When you’re close to the peak ofa resonance, an easy way to tell whether to add orremove weight is by gently pressing down or liftingup on the weights with your finger to see whetherthe amplitude increases or decreases.

For large values of N , you may find that you need touse a paperclip instead of the weight holder, in orderto make the mass sufficiently small. Keep in mind,however, that you won’t really improve the qualityof your data very much by taking data for very highvalues of N , since the 1-gram precision with whichyou can locate these resonances results in a poorrelative precision compared to a small weight.

Prelab

The point of the prelab questions is to make sureyou understand what you’re doing, why you’re do-ing it, and how to avoid some common mistakes. Ifyou don’t know the answers, make sure to come to

62 Lab 21 Standing Waves

my office hours before lab and get help! Otherwiseyou’re just setting yourself up for failure in lab.

P1 Should the whole length of the string be countedin L, or just the part from the vibrator to the pulley?

P2 How is the tension in the string, T , related tothe mass of the hanging weight?

P3 How can the velocity of the waves be deter-mined if you know the frequency, f , the length ofthe string, L, and the number of humps, N?

Self-Check

Do your analysis in lab.

Analysis

Use the techniques given in appendix 5 to see if youcan find a power-law relationship between the veloc-ity of the waves in the string and the tension in thestring. (Do not just try to find the correct powerlaw in the textbook, because besides observing thephenomenon of resonance, the point of the lab is toprove experimentally what the power-law relation-ship is.)

63

22 Resonances of Sound

Apparatus

wave generator (PASCO PI-9587C) . . . . . . . 1/groupspeaker (Thornton) . . . . . . . . . . . . . . . . . . . . . . . 1/group100 mL graduated cylinder . . . . . . . . . . . . . . . 1/groupLinux computers with FFT Explorer installed (in416 and 416P)flexible whistling tube . . . . . . . . . . . . . . . . . . . . . . . . . . . 1tuning fork marked with frequency, mounted on awooden box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1aluminum rod, 3/4-inch dia, about 1 m long2

Goals

Find the resonant frequencies of the air insidea cylinder by two methods.

Measure the speeds of sound in air and in alu-minum.

Introduction

In the womb, your first sensory experiences were ofyour mother’s voice, and soon after birth you learnedto distinguish the particular sounds of your parents’voices from those of strangers. The human ear-brainsystem is amazingly sophisticated in its ability toclassify vowels and consonants, recognize people’svoices, and analyze musical sound. Until the 19th-century investigations of Helmholtz, the whole pro-cess was completely mysterious. How could we soeasily tell a cello from a violin playing the same note?A radio station in Chicago has a weekly contest inwhich jazz fanatics are asked to identify instrumen-talists simply by their distinctly individual timbres— how is this possible?

Helmholtz found (using incredibly primitive nonelec-tronic equipment) that part of the answer lay in therelative strengths of the overtones. The psycholog-ical sensation of pitch is related to frequency, e.g.,440 Hz is the note ‘A.’ But a saxophonist playingthe note ‘A’ is actually producing a rich spectrum offrequencies, including 440 Hz, 880 Hz, 1320 Hz, andmany other multiples of the lowest frequency, knownas the fundamental. The ear-brain system perceivesall these overtones as a single sound because they areall multiples of the fundamental frequency. (The Ja-vanese orchestra called the gamelan sounds strange

to westerners partly because the various gongs andcymbals have overtones that are not integer multi-ples of the fundamental.)

One of the things that would make ‘A’ on a clarinetsound different from ‘A’ on a saxophone is that the880 Hz overtone would be quite strong for the sax-ophone, but almost entirely missing for the clarinet.Although Helmholtz thought the relative strengthsof the overtones was the whole story when it cameto musical timbre, actually it is more complex thanthat, which is why electronic synthesizers still do notsound as good as acoustic instruments. The timbredepends not just on the general strength of the over-tones but on the details of how they first build up(the attack) and how the various overtones fade inand out slightly as the note continues.

Why do different instruments have different soundspectra, and why, for instance, does a saxophonehave an overtone that the clarinet lacks? Many mu-sical instruments can be analyzed physically as tubesthat have either two open ends, two closed ends, orone open end and one closed end. The overtonescorrespond to specific resonances of the air columninside the tube. A complete treatment of the subjectis given in your textbook, but the basic principle isthat the resonant standing waves in the tube musthave an antinode (point of maximum vibration) atany closed end of the tube, and a node (point of zerovibration) at any open end.

Using the Wave Generator

The wave generator works like the amplifier of yourstereo, but instead of playing a CD, it produces asine wave whose frequency and amplitude you cancontrol. By connecting it to a speaker, you can con-vert its electrical currents to sound waves, makinga pure tone. The frequency of the sine wave cor-responds to musical pitch, and the amplitude cor-responds to loudness. Use the output labeled LOΩ. The wave generator can create enough voltage togive a mildly unpleasant tingling sensation in yourhand if you touch the leads. None of the electri-cal apparatus used in this lab, however, is any moredangerous than a home TV or stereo.

64 Lab 22 Resonances of Sound

Setup

Unplug the wave generator. Check the fuse in theback of the wave generator to make sure it is notblown, then put it back in. Plug in the wave gener-ator and turn off the on/off switch at the top right.Turn the ‘amplitude’ knob of the wave generator tozero, and then turn on the on/off switch.

Plug the speaker into the wave generator. The ba-nana plugs go in the two holes on the right. Set thefrequency to something audible. Wait 30 secondsfor the wave generator to warm up, then turn theamplitude knob up until you hear a sound.

The wave generator and the speaker are not reallydesigned to work together, so if you leave the volumeup very high for a long time, it is possible to blowthe speaker or damage the wave generator. Also, thesine waves are annoying when played continuously atloud volumes!


Observations

This lab has three parts, A, B, and C. It is not reallypossible for more than one group to do part A in thesame room, both because their sounds interfere withone another and because the noise becomes annoy-ing for everyone. Your instructor will probably havethree groups working on part A at one time, onegroup in the main room, one in the small side room,and one in the physics stockroom. Meanwhile, theother groups will be doing parts B and C.

A Direct Measurement of Resonances by Lis-tening

Set up the graduated cylinder so its mouth is cov-ering the center of the speaker. Find as many fre-quencies as possible at which the cylinder resonates.When you sweep through those frequencies, the soundbecomes louder. To make sure you’re really hearinga resonance of the cylinder, make sure to repeat eachobservation with the cylinder removed, and makesure the resonance goes away. For each resonance,take several measurements of its frequency — if youare careful, you can pin it down to within ±10 Hzor so. You can probably speed up your search sig-nificantly by calculating approximately where youexpect the resonances to be, then looking for them.

B Electronic Measurement of Resonances ofan Air Column

The resonances of the air column in a cylinder canalso be excited by a stream of air flowing over anopening, as with a flute. In this part of the lab, youwill excite resonances of a long, flexible plastic tubeby grabbing it at one end and swinging it in a cir-cle. The frequency of the sound will be determinedelectronically. Note that your analysis for these res-onances will be somewhat different, since the tubeis open at both ends, and it therefore has differentpatterns of resonances from the graduated cylinder,which was only open at one end.

To measure the frequency, you will use a computer toanalyze the sound. There are two Linux computersthat have the right software and hardware. As awarmup before attempting the actual measurementswith the whistling tube, try the following. First,start up the program if nobody else has already doneso. It is called FFT Explorer, and you can run it bydouble-clicking on its icon on the desktop. In realtime, the program will monitor the sound cominginto the microphone, and display a graph of loudnessversus frequency. Try whistling. The frequency atwhich you whistled should show up as a prominentpeak. You may need to play with the frequency andloudness scales, using the two menus on the lowerright. If you’re not careful, it’s easy to get confusedby setting a frequency range that’s too narrow, sothat the peak you want isn’t even on the graph. It’sa good idea to try it first on a very wide frequencyscale, and then narrow the scale to the narrowest onethat allows you to see the peak. When you get thegraph you want to see, you can freeze it by clickingon the stop button. Although the software doesn’tgive any convenient way to read off the frequencyof the peak with high precision, you can accomplishthat by measuring on the screen with a ruler, andinterpolating.

Debugging software problems:

If sound input doesn’t work, or mysteriouslystops working, it’s typically because Linux’ssound system (called ALSA) is upset; this canbe fixed by logging out, and then logging backin again.

Now try the whole procedure with the tuning forkinstead of whistling, and make sure you can use thecomputer to obtain the frequency inscribed on thefork. You can put the mic inside the wooden boxthat the tuning fork is mounted on. Although thesoftware doesn’t let yu zoom in on the peak, you

65

can lay a ruler on the screen, and interpolate fairlyaccurately.

Once you have done these warmups, you are ready toanalyze the sound from the whistling tube. You onlyneed to analyze data from one frequency, althoughif you’re not sure which mode you produced, it maybe helpful to observe the pattern of the frequencies.(If you guess wrong about which mode it was, you’llfind out, because the value you extract for the speedof sound will be way off.)

C The Speed of Sound in Aluminum

The speed of sound in dense solid is much fasterthan its speed in air. In this part of the lab, youwill extract the speed of sound in aluminum froma measurement of the lowest resonant frequency ofa solid aluminum rod. You will use the computerfor an electronic measurement of the frequency, asin part B.

Grab the rod with two fingers exactly in the middle,hold it vertically, and tap it on the floor. You willhear two different notes sounding simultaneously. Aquick look at their frequencies shows that they arenot in a 2:1 ratio as we would expect based on ourexperiences with symmetric wave patterns. This isbecause these two frequencies in the rod are actu-ally two different types of waves. The higher note isproduced by longitudinal compression waves, whichmeans that an individual atom of aluminum is mov-ing up and down the length of the rod. This typeof wave is analogous to sound waves in air, whichare also longitudinal compression waves. The lowernote comes from transverse vibrations, like a vibrat-ing guitar string. In the transverse vibrations, atomsare moving from side to side, and the rod as a wholeis bending.

If you listen carefully, you can tell that the trans-verse vibration (the lower note) dies out quickly, butthe longitudinal mode keeps going for a long time.That gives you an easy way to isolate the longitudi-nal mode, which is the one we’re interested in; justwait for the transverse wave to die out before youfreeze the graph on the computer.

The rod is symmetric, so we expect its longitudi-nal wave patterns to be symmetric, like those of thewhistling tube. The rod is different, however, be-cause whereas we can excite a variety of wave pat-terns in the tube by spinning it at different speeds,we find we only ever get one frequency from the rodby tapping it at its end: it appears that there isonly one logitudinal wave pattern that can be ex-cited strongly in the rod by this method. The prob-

lem is that we then need to infer what the pattern is.Since you hold the rod at its center, friction shouldvery rapidly damp out any mode of vibration thathas any motion at the center. Therefore there mustbe a node at the center. We also know that at theends, the rod has nothing to interact with but theair, and therefore there is essentially no way for anysignificant amount of wave energy to leak out; wetherefore expect that waves reaching the ends have100% of their energy reflected. Since energy is pro-portional to the square of amplitude, this meansthat a wave with unit amplitude can be reflectedfrom the ends with an amplitude of either R = +1(100% uninverted reflection) or −1 (100% inverted).In the R = −1 case, the reflected wave would can-cel out the incident wave at the end of the rod, andwe would have a node at the end, as in lab 21. Inthe R = +1 case, there would be an antinode. Butwhen you tap the end of the rod on the floor, you areevidently exciting wave motion by moving the end,and it would not be possible to excite vibrations bythis method if the vibrations had no motion at theend. We therefore conclude that the rod’s patternof vibration must have a node at the center, andantinode at the ends. There is an infinite number ofpossible wave patterns of this kind, but we will as-sume that the pattern that is excited strongly is theone with the longest wavelength, i.e., the only nodeis at the center, and the only antinodes are those atthe ends. If you feel like it, there are a couple ofpossible tests you can try do to check whether thisis the right interpretation. One is to see if you candetect any other frequencies of longitudinal vibra-tion that are excited weakly. Another is to predictwhere the other nodes would be, if there were morethan one, and then see if the vibration is killed bytouching the rod there with your other hand; if thereis a node there, touching it should have no effect.

Prelab


P1 Find an equation to predict the frequencies ofthe resonances in parts A and B. Note that theywill not be the same equations, since one tube issymmetric and the other is asymmetric.

66 Lab 22 Resonances of Sound

Self-Check

Extract the speed of sound from either part A orpart B, without error analysis, and make sure youget something close to the accepted value.

Analysis

Make a graph of wavelength versus period for theresonances of the graduated cylinder, check whetherit looks like it theoretically should, and if so, findthe speed of sound from its slope, with error bars,as discussed in appendix 4.

Use the data from part B to find a second value ofthe speed of sound, also with error bars.

The effective length of the cylinder in part A shouldbe increased by 0.4 times its diameter to account forthe small amount of air beyond the end that also vi-brates. For part B, where the whistling tube is openat both ends, you should add 0.8 times its diameter.

When estimating error bars from part B, you maybe tempted to say that it must be perfectly accurate,since its being done by a computer. Not so! You willsee that the peak is a little ragged, and that meansyou cannot find the frequency with perfect accuracy.

Extract the speed of sound in aluminum from yourdata in part C, including error bars.

67

23 Static Electricity

Apparatus

scotch taperubber rodheat lampfurbits of paperrods and strips of various materials30-50 cm rods, and angle brackets, for hanging chargedrods

Goal

Determine the qualitative rules governing electricalcharge and forces.

Introduction

Newton’s law of gravity gave a mathematical for-mula for the gravitational force, but his theory alsomade several important non-mathematical statementsabout gravity:

Every mass in the universe attracts every othermass in the universe.

Gravity works the same for earthly objects asfor heavenly bodies.

The force acts at a distance, without any needfor physical contact.

Mass is always positive, and gravity is alwaysattractive, not repulsive.

The last statement is interesting, especially becauseit would be fun and useful to have access to somenegative mass, which would fall up instead of down(like the ‘upsydaisium’ of Rocky and Bullwinkle fame).

Although it has never been found, there is no theo-retical reason why a second, negative type of masscan’t exist. Indeed, it is believed that the nuclearforce, which holds quarks together to form protonsand neutrons, involves three qualities analogous tomass. These are facetiously referred to as ‘red,’‘green,’ and ‘blue,’ although they have nothing todo with the actual colors. The force between two ofthe same ‘colors’ is repulsive: red repels red, green

repels green, and blue repels blue. The force betweentwo different ‘colors’ is attractive: red and green at-tract each other, as do green and blue, and red andblue.

When your freshly laundered socks cling together,that is an example of an electrical force. If the grav-itational force involves one type of mass, and thenuclear force involves three colors, how many typesof electrical ‘stuff’ are there? In the days of Ben-jamin Franklin, some scientists thought there weretwo types of electrical ‘charge’ or ‘fluid,’ while othersthought there was only a single type. In this lab, youwill try to find out experimentally how many typesof electrical charge there are.

Observations

Stick a piece of scotch tape on a table, and then layanother piece on top of it. Pull both pieces off thetable, and then separate them. If you now bringthem close together, you will observe them exertinga force on each other. Electrical effects can also becreated by rubbing the fur against the rubber rod.

Your job in this lab is to use these techniques totest various hypotheses about electric charge. Themost common difficulty students encounter is thatthe charge tends to leak off, especially if the weatheris humid. If you have charged an object up, youshould not wait any longer than necessary beforemaking your measurements. It helps if you keep yourhands dry.

A Repulsion and/or attraction

Test the following hypotheses. Note that they aremutually exclusive, i.e., only one of them can be true.

A1) Electrical forces are always attractive.

A2) Electrical forces are always repulsive.

A3) Electrical forces are sometimes attractive andsometimes repulsive.

Interpretation: Once you think you have tested thesehypotheses fairly well, discuss with your instructorwhat this implies about how many different types ofcharge there might be.

68 Lab 23 Static Electricity

B Are there forces on objects that have not beenspecially prepared?

So far, special preparations have been necessary inorder to get objects to exhibit electrical forces. Thesepreparations involved either rubbing objects againsteach other (against resistance from friction) or pullingobjects apart (e.g. overcoming the sticky force thatholds the tape together). In everyday life, we do notseem to notice electrical forces in objects that havenot been prepared this way.

Now try to test the following hypotheses. Bits of pa-per are a good thing to use as unprepared objects,since they are light and therefore would be easilymoved by any force. Do not use tape as an un-charged object, since it can become charged a littlebit just by pulling off the roll.

B1) Objects that have not been specially preparedare immune to electrical forces.

B2) Unprepared objects can participate in electricalforces with prepared objects, and the forces involvedare always attractive.

B3) Unprepared objects can participate in electricalforces with prepared objects, and the forces involvedare always repulsive.

B4) Unprepared objects can participate in electricalforces with prepared objects, and the forces involvedcan be either repulsive of attractive.

Hypotheses B1 through B4 are mutually exclusive.

C Rules of repulsion and/or attraction and thenumber of types of charge

Test the following mutually exclusive hypotheses:

C1) There is only one type of electric charge, andthe force is always attractive.

C2) There is only one type of electric charge, andthe force is always repulsive.

C3) There are two types of electric charge, call themX and Y. Like charges repel (X repels X and Y repelsY) and opposite charges attract (X and Y attracteach other).

C4) There are two types of electric charge. Likecharges attract and opposite charges repel.

C5) There are three types of electric charge, X, Yand Z. Like charges repel and unlike charges attract.

The only way to keep all your observations straightis to make a square table, in which the rows andcolumns correspond to the different objects you’retesting against each other for attraction and repul-

sion. To test C3 versus C5, you’ll need to see if youcan successfully explain your whole table by labelingthe objects with only two labels, X and Y.

Some of the equipment may look identical, but notbe identical. In particular, some of the clear rodshave higher density than others, which may be be-cause they’re made different types of plastic, or glass.This could affect your conclusions, so you may wantto check, for example, whether two rods with thesame diameter, that you think are made of the samematerial, actually weigh the same.

In general, you will find that some materials, andsome combinations of materials, are more easily chargedthan others. For example, if you find that the ma-hogony rod rubbed with the weasel fur doesn’t chargewell, then don’t keep using use it! The white plasticstrips tend to work well, so don’t neglect them.

Discuss your conclusions with your instructor.

Self-Check

The following are examples of incorrect reasoningabout this lab. As a self-check, it would be a verygood idea to figure out for yourself in each case whythe reasoning is logically incorrect or inconsistentwith Newton’s laws. You do not need to do this inwriting — it is just to help you understand what’sgoing on. If you can’t figure some of them out, askyour instructor before leaving lab.

(1) ‘The first piece of tape exerted a force on thesecond, but the second didn’t exert one on the first.’

(2) ‘The first piece of tape repelled the second, andthe second attracted the first.’

(3) ‘We observed three types of charge: two thatexert forces, and a third, neutral type.’

(4) ‘The piece of tape that came from the top waspositive, and the piece from the bottom was nega-tive.’

(5) ‘One piece of tape had electrons on it, and theother had protons on it.’

(6) ‘We know there were two types of charge, notthree, because we observed two types of interactions,attraction and repulsion.’

Writeup

Explain what you have concluded about electricalcharge and forces. Base your conclusions on yourdata!

69

24 The Oscilloscope

Apparatus

oscilloscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/groupmicrophone (RS 33-1067) . . . . . . . . . . . . . . . . . 1/groupmicrophone (Shure C606) . . . . . . . . . . . . . . . . . . . . . . . .1PI-9587C sine wave generator . . . . . . . . . . . . .1/groupamplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/groupvarious tuning forks

If there’s an equipment conflict with respect to thesine wave generators, the HP200CD sine wave gen-erators can be used instead.

Goals

Learn to use an oscilloscope.

Observe sound waves on an oscilloscope.

Introduction

One of the main differences you will notice betweenyour second semester of physics and the first is thatmany of the phenomena you will learn about arenot directly accessible to your senses. For example,electric fields, the flow of electrons in wires, and theinner workings of the atom are all invisible. Theoscilloscope is a versatile laboratory instrument thatcan indirectly help you to see what’s going on.

The Oscilloscope

An oscilloscope graphs an electrical signal that variesas a function of time. The graph is drawn from left toright across the screen, being painted in real time asthe input signal varies. In this lab, you will be usingthe signal from a microphone as an input, allowingyou to see sound waves.

The input signal is supplied in the form of a voltage.You are already familiar with the term ‘voltage’ fromcommon speech, but you may not have learned theformal definition yet in the lecture course. Voltage,measured in metric units of volts (V), is defined asthe electrical potential energy per unit charge. Forinstance if 2 nC of charge flows from one terminal ofa 9-volt battery to the other terminal, the potentialenergy consumed equals 18 nJ. To use a mechanicalanalogy, when you blow air out between your lips,

the flowing air is like an electrical current, and thedifference in pressure between your mouth and theroom is like the difference in voltage. For the pur-poses of this lab, it is not really necessary for you towork with the fundamental definition of voltage.

The input connector on the front of the oscilloscopeaccepts a type of cable known as a BNC cable. ABNC cable is a specific example of coaxial cable(‘coax’), which is also used in cable TV, radio, andcomputer networks. The electric current flows inone direction through the central conductor, and re-turns in the opposite direction through the outsideconductor, completing the circuit. The outside con-ductor is normally kept at ground, and also serves asshielding against radio interference. The advantageof coaxial cable is that it is capable of transmittingrapidly varying signals without distortion.

Most of the voltages we wish to measure are not bigenough to use directly for the vertical deflection volt-age, so the oscilloscope actually amplifies the inputvoltage, i.e., the small input voltage is used to con-trol a much large voltage generated internally. Theamount of amplification is controlled with a knob onthe front of the scope. For instance, setting the knobon 1 mV selects an amplification such that 1 mV atthe input deflects the electron beam by one squareof the 1-cm grid. Each 1-cm division is referred toas a ‘division.’

The Time Base and Triggering

Since the X axis represents time, there also has tobe a way to control the time scale, i.e., how fastthe imaginary ”penpoint” sweeps across the screen.For instance, setting the knob on 10 ms causes it tosweep across one square in 10 ms. This is known asthe time base.

In the figure, suppose the time base is 10 ms. Thescope has 10 divisions, so the total time required for

70 Lab 24 The Oscilloscope

the beam to sweep from left to right would be 100ms. This is far too short a time to allow the userto examine the graph. The oscilloscope has a built-in method of overcoming this problem, which workswell for periodic (repeating) signals. The amountof time required for a periodic signal to perform itspattern once is called the period. With a periodicsignal, all you really care about seeing what one pe-riod or a few periods in a row look like — once you’veseen one, you’ve seen them all. The scope displaysone screenful of the signal, and then keeps on over-laying more and more copies of the wave on top ofthe original one. Each trace is erased when the nextone starts, but is being overwritten continually bylater, identical copies of the wave form. You simplysee one persistent trace.

How does the scope know when to start a new trace?If the time for one sweep across the screen just hap-pened to be exactly equal to, say, four periods of thesignal, there would be no problem. But this is un-likely to happen in real life — normally the secondtrace would start from a different point in the wave-form, producing an offset copy of the wave. Thou-sands of traces per second would be superimposedon the screen, each shifted horizontally by a differ-ent amount, and you would only see a blurry bandof light.

To make sure that each trace starts from the samepoint in the waveform, the scope has a triggering cir-cuit. You use a knob to set a certain voltage level,the trigger level, at which you want to start eachtrace. The scope waits for the input to move acrossthe trigger level, and then begins a trace. Once thattrace is complete, it pauses until the input crossesthe trigger level again. To make extra sure that it isreally starting over again from the same point in thewaveform, you can also specify whether you want tostart on an increasing voltage or a decreasing volt-age — otherwise there would always be at least two

points in a period where the voltage crossed yourtrigger level.

Setup

To start with, we’ll use a sine wave generator, whichmakes a voltage that varies sinusoidally with time.This gives you a convenient signal to work with whileyou get the scope working. Use the black and whiteoutputs on the PI-9587C.

The figure on the last page is a simplified drawingof the front panel of a digital oscilloscope, showingonly the most important controls you’ll need for thislab. When you turn on the oscilloscope, it will takea while to start up.

Preliminaries:

Press DEFAULT SETUP.

Use the SEC/DIV knob to put the time baseon something reasonable compared to the pe-riod of the signal you’re looking at. The timebase is displayed on the screen, e.g., 10 ms/div,or 1 s/div.

Use the VOLTS/DIV knob to put the voltagescale (Y axis) on a reasonable scale comparedto the amplitude of the signal you’re lookingat.

The scope has two channels, i.e., it can ac-cept input through two BNC connectors anddisplay both or either. You’ll only be usingchannel 1, which is the only one represented inthe simplified drawing. By default, the oscil-loscope draws graphs of both channels’ inputs;to get rid of ch. 2, hold down the CH 2 MENUbutton (not shown in the diagram) for a coupleof seconds. You also want to make sure thatthe scope is triggering on CH 1, rather thanCH 2. To do that, press the TRIG MENUbutton, and use an option button to select CH1 as the source. Set the triggering mode tonormal, which is the mode in which the trig-gering works as I’ve described above. If thetrigger level is set to a level that the signalnever actually reaches, you can play with theknob that sets the trigger level until you getsomething. A quick and easy way to do thiswithout trial and error is to use the SET TO50automatically sets the trigger level to mid-way between the top and bottom peaks of thesignal.

71

You want to select AC, not DC or GND, onthe channel you’re using. You are looking ata voltage that is alternating, creating an al-ternating current, ‘AC.’ The ‘DC’ setting isonly necessary when dealing with constant orvery slowly varying voltages. The ‘GND’ sim-ply draws a graph using y = 0, which is onlyuseful in certain situations, such as when youcan’t find the trace. To select AC, press theCH 1 MENU button, and select AC coupling.

Observe the effect of changing the voltage scale andtime base on the scope. Try changing the frequencyand amplitude on the sine wave generator.

You can freeze the display by pressing RUN/STOP,and then unfreeze it by pressing the button again.


Now try observing signals from the microphone. Byfeeding the mic’s signal through the amplifier andthen to the scope, you can make the signals easierto see.

As of fall 2008, we’re in the process of testing a bet-ter mic (Shure brand) to replace the Radio Shackones. We have one of the Shure ones. If your groupis the one that gets it, please relay the informationabout how it worked through your instructor andback to Ben Crowell. Some notes about this mic: Aswith the Radio Shack mics, polarity matters. Thetip of the phono plug connector is the live connec-tion, and the part farther back from the tip is thegrounded part. You can connect on to the phonoplug with alligator clips. You don’t need the am-plifier. Notes for instructors: This mic was $30 atFry’s. It has an unusually high gain, -52 dBV/Pa at1 kHz, which helps to make the signals clean enoughto see well on a scope without preamplification. Itsoutput impedance is 600 ohms. The main reason theRS 33-1067 mics have a poorer S/N ratio is that thecables are not coax, so they pick up a lot of noisein differential mode. The RS 33-3013 mics are notreally any better for this application; although theydo have coax cables, they have a very low gain. Weshould buy phono-to-BNC connectors for the Shuremics.

Once you have your setup working, try measuringthe period and frequency of the sound from a tuningfork, and make sure your result for the frequency isthe same as what’s written on the tuning fork.

Don’t crank the gain on the amplifier all the wayup. If you do, the amplifier will put out a distorted

waveform. Use the highest gain you can use withoutcausing distortion.

Observations

A Periodic and nonperiodic speech sounds

Try making various speech sounds that you can sus-tain continuously: vowels or certain consonants suchas ‘sh,’ ‘r,’ ‘f’ and so on. Which are periodic andwhich are not?

Note that the names we give to the letters of thealphabet in English are not the same as the speechsounds represented by the letter. For instance, theEnglish name for ‘f’ is ‘ef,’ which contains a vowel,‘e,’ and a consonant, ‘f.’ We are interested in the ba-sic speech sounds, not the names of the letters. Also,a single letter is often used in the English writing sys-tem to represent two sounds. For example, the word‘I’ really has two vowels in it, ‘aaah’ plus ‘eee.’

B Loud and soft

What differentiates a loud ‘aaah’ sound from a softone?

C High and low pitch

Try singing a vowel, and then singing a higher notewith the same vowel. What changes?

D Differences among vowel sounds

What differentiates the different vowel sounds?

E Lowest and highest notes you can sing

What is the lowest frequency you can sing, and whatis the highest?

Prelab


P1 In the sample oscilloscope trace shown on page70, what is the period of the waveform? What is itsfrequency? The time base is 10 ms.

P2 In the same example, again assume the timebase is 10 ms/division. The voltage scale is 2 mV/division.Assume the zero voltage level is at the middle of thevertical scale. (The whole graph can actually beshifted up and down using a knob called ‘position.’)


What is the trigger level currently set to? If the trig-ger level was changed to 2 mV, what would happento the trace?

P3 Referring to the chapter of your textbook onsound, which of the following would be a reasonabletime base to use for an audio-frequency signal? 10ns, 1µ s, 1 ms, 1 s

P4 Does the oscilloscope show you the period ofthe signal, or the wavelength? Explain. (If you’re inPhysics 222, skip this one, because you don’t knowabout the definition of wavelength yet.)

Analysis

The format of the lab writeup can be informal. Justdescribe clearly what you observed and concluded.

73

A simplified diagram of the controls on a digital oscilloscope.


25 The Speed of Sound

Based on a lab by Hans Rau.

Apparatus

oscilloscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/groupoptical bench . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/groupHP function generator . . . . . . . . . . . . . . . . . . . . 1/grouptransducers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2/group

Goal

Measure the speed of sound.

Introduction

There are several simple methods for getting a roughestimate of the speed of sound, for instance tim-ing an echo, or watching the kettledrum player ata symphony and seeing how long the sound takes toarrive after you see the mallet strike the drumhead.The latter method, comparing vision against hear-ing, assumes that the speed of light is much greaterthan the speed of sound, the same assumption thatis used when estimating the distance to a lightningstrike based on the interval between the flash andthe thunder. The assumption is a good one, sincelight travels about a million times faster than sound.Military jets routinely exceed the speed of sound,but no human has ever traveled at speeds even re-motely comparable to that of light. (The electronsin your television set are moving at a few percent ofthe speed of light, and velocities of 0.999999999999times the speed of light can be attained in particleaccelerators. According to Einstein’s theory of rela-tivity, motion faster than light is impossible.)

In this lab, you will make an accurate measurementof the speed of sound by measuring the wavelengthand frequency of a pure tone (sine wave) and com-puting

v = λf .

We will be using sound with a frequency of about35-40 kHz, which is too high to be audible. This hasthe advantage of eliminating the annoying din of sixlab groups producing sine waves at once. Such high-frequency, inaudible sound is known as ultrasound.Ultrasound at even higher frequencies, in the MHz

range, is used for imaging fetuses in the womb.

Setup

The setup is shown below. A transducer is a de-vice capable of acting as either a speaker or a micro-phone. The function generator is used to create avoltage that varies sinusoidally over time. This volt-age is connected through two coax cables, to the os-cilloscope and the first transducer, used as a speaker.The sound waves travel from the first transducer tothe second transducer, used as a microphone. Youwill be using both channels of the scope to displaygraphs of two waveforms at the same time on theoscilloscope. As you slide one transducer along theoptical bench, changing the distance between them,you will change the phase of one wave relative to theother. Thus, you can determine the distance corre-sponding to a given number of wavelengths and ex-tract the wavelength of the sound waves accurately.The wavelength of the sound will be roughly a fewcm. The frequency can be read from the knob onthe function generator. (The time scale of an os-cilloscope typically has a systematic error of about2-5%, so you should not use a measurement of theperiod from the scope for this purpose.)

When setting up the scope, you will need to selectone channel or the other to trigger on. You canselect the voltage scales for the two channels inde-pendently, but they always have the same time base.

The most common problem in this lab is that someelectrical current gets through the metal optical bench,causing the receiving transducer to pick up the orig-inal input signal directly, rather than by receivingthe sound waves. A precaution that usually worksis to connect the optical bench to the ground con-tact of the scope (use an alligator clip to attach tothe body of the bench). It is easy to check whether

76 Lab 25 The Speed of Sound

the problem exists: put your hand between the twotransducers to absorb the sound, and you should seethe amplitude of the signal from the receiver becomemuch smaller. The receiving transducer will receivesound best at frequencies in the range of 35-40 kHz,so keep the frequency in that range.

When you connect the function generator to boththe scope and the transmitting transducer, you’llprobably end up connecting a BNC-to-banana con-nector to the function generator, and then putting asecond banana connector into the back of the first.It’s important to make sure that the little tabs marked‘GND’ are on the same side of both connectors.

If you are still having problems after taking the abovesteps, try replacing one of the transducers — someof the transducers are unreliable.

Observations

Determine the wavelength and frequency of the soundwaves using the oscilloscope. Find out the tempera-ture in the lab.

Do a quick analysis, without error analysis, duringlab, to see if your result is reasonable.

Prelab


P1 The drawings show two different configurationsof the transducers on the optical bench.

The scales are in cm. By sliding the right-hand

transducer from the position shown in the first draw-ing to the position shown in the second drawing, thestudent swept one trace past five complete cyclesof the other trace. (The actual optical benches areabout a meter long, not 8 or 9 cm as shown.) Whatis the wavelength of the ultrasound? [Self-check: youshould get 0.6 cm]

P2 Does it matter which transducer you move?

P3 You can choose through how many wavelengthsyou will move the transducer. What effect will thishave on the accuracy of your determination of thespeed of sound?

P4 What is a reasonable value for the speed ofsound?

Self-Check

Do an analysis without error bars before leaving lab,and check that your speed of sound is reasonable.

Analysis

Determine the speed of sound from your data, anduse the techniques discussed in appendix 3 to deriveerror bars.

Compare your result with the previously determinedvalue of

v = (20.1)√

T ,

where v is in m/s and T is the absolute temperature,found by adding 273 to the Celsius temperature. Isit statistically consistent with your value?

77

26 Electrical Resistance

Apparatus

DC power supply (Thornton) . . . . . . . . . . . . . 1/groupdigital multimeters (Fluke and HP) . . . . . . . 2/groupresistors, various valuesunknown electrical componentselectrode pastealligator clipsspare fuses for multimeters — Let students replacefuses themselves.

Goals

Measure curves of voltage versus current forthree objects: your body and two unknownelectrical components.

Determine whether they are ohmic, and if so,determine their resistances.

Introduction

Your nervous system depends on electrical currents,and every day you use many devices based on elec-trical currents without even thinking about it. De-spite its ordinariness, the phenomenon of electriccurrents passing through liquids (e.g., cellular flu-ids) and solids (e.g., copper wires) is a subtle one.For example, we now know that atoms are composedof smaller, subatomic particles called electrons andnuclei, and that the electrons and nuclei are elec-trically charged, i.e., matter is electrical. Thus, wenow have a picture of these electrically charged par-ticles sitting around in matter, ready to create anelectric current by moving in response to an exter-nally applied voltage. Electricity had been used forpractical purposes for a hundred years, however, be-fore the electrical nature of matter was proven at theturn of the 20th century.

Another subtle issue involves Ohm’s law,

I =∆V

R,

where ∆V is the voltage difference applied across anobject (e.g., a wire), and I is the current that flowsin response. A piece of copper wire, for instance,has a constant value of R over a wide range of volt-ages. Such materials are called ohmic. Materials

with non-constant are called non-ohmic. The inter-esting question is why so many materials are ohmic.Since we know that electrons and nuclei are boundtogether to form atoms, it would be more reasonableto expect that small voltages, creating small electricfields, would be unable to break the electrons andnuclei away from each other, and no current wouldflow at all — only with fairly large voltages shouldthe atoms be split up, allowing current to flow. Thuswe would expect R to be infinite for small voltages,and small for large voltages, which would not beohmic behavior. It is only within the last 50 yearsthat a good explanation has been achieved for thestrange observation that nearly all solids and liquidsare ohmic.

Terminology, Schematics, and Re-

sistor Color Codes

The word ‘resistor’ usually implies a specific typeof electrical component, which is a piece of ohmicmaterial with its shape and composition chosen togive a desired value of R. Any piece of an ohmicsubstance, however, has a constant value of R, andtherefore in some sense constitutes a ‘resistor.’ Thewires in a circuit have electrical resistance, but theresistance is usually negligible (a small fraction of anOhm for several centimeters of wire).

The usual symbol for a resistor in an electrical schematicis this , but some recent schematics use

this . The symbol represents a fixed

source of voltage such as a battery, while repre-sents an adjustable voltage source, such as the powersupply you will use in this lab.

In a schematic, the lengths and shapes of the linesrepresenting wires are completely irrelevant, and areusually unrelated to the physical lengths and shapesof the wires. The physical behavior of the circuitdoes not depend on the lengths of the wires (un-less the length is so great that the resistance of thewire becomes non-negligible), and the schematic isnot meant to give any information other than thatneeded to understand the circuit’s behavior. All thatreally matters is what is connected to what.

For instance, the schematics (a) and (b) above are

78 Lab 26 Electrical Resistance

completely equivalent, but (c) is different. In thefirst two circuits, current heading out from the bat-tery can ‘choose’ which resistor to enter. Later on,the two currents join back up. Such an arrangementis called a parallel circuit. In the bottom circuit, aseries circuit, the current has no ‘choice’ — it mustfirst flow through one resistor and then the other.

Resistors are usually too small to make it convenientto print numerical resistance values on them, so theyare labeled with a color code, as shown in the tableand example below.

Setup

Obtain your two unknowns from your instructor.Group 1 will use unknowns 1A and 1B, group 2 willuse 2A and 2B, and so on.

Here is a simplified version of the basic circuit youwill use for your measurements of I as a function of∆V . Although I’ve used the symbol for a resistor,

the objects you are using are not necessarily resis-tors, or even ohmic.

Here is the actual circuit, with the meters included.In addition to the unknown resistance RU , a knownresistor RK (∼ 1kΩ is fine) is included to limit thepossible current that will flow and keep from blow-ing fuses or burning out the unknown resistance withtoo much current. This type of current-limiting ap-plication is one of the main uses of resistors.

Observations

A Unknown component A

Set up the circuit shown above with unknown com-ponent A. Most of your equipment accepts the ba-nana plugs that your cables have on each end, butto connect to RU and RK you need to stick alligatorclips on the banana plugs. See Appendix 7 for in-formation about how to set up and use the two mul-timeters. Do not use the pointy probes that comewith the multimeters, because there is no convenientway to attach them to the circuit — just use the ba-nana plug cables. Note when you need three wires to

79

come together at one point, you can plug a bananaplug into the back of another banana plug.

Measure I as a function of ∆V . Make sure to takemeasurements for both positive and negative volt-ages.

Often when we do this lab, it’s the first time in sev-eral months that the meters have been used. Thesmall hand-held meters have a battery, which maybe dead. Check the battery icon on the LCD screen.

B Unknown component B

Repeat for unknown component B.

C The human body

Now do the same with the body of one member ofyour group. This is not dangerous — the maxi-mum voltage available from your power supply isnot enough to hurt you. (Children usually figureout at some point that touching the terminals of a 9V battery to their tongue gives an interesting sensa-tion. The currents you will use in this lab are ten toa hundred times smaller.) You may wish to keep thevoltage below about 5 V or so. At voltages muchhigher than that (10 to 12 V), a few subjects getirritated skin.

You will not want to use the alligator clips. Withthe power supply turned off, put small dabs of theelectrode paste on the subject’s left wrist and justbelow the elbow, and simply lay the banana plugconnectors in the paste. The subject should avoidmoving. The paste is necessary because without it,most of the resistance would come from the connec-tion through the dry epidermal skin layer, and theresistance would change erratically. The paste is arelatively good conductor, and makes a better elec-trical connection.

Prelab


P1 Check that you understand the interpretationsof the following color-coded resistor labels:

blue gray orange silver = 68 kΩ ± 10%blue gray orange gold = 68 kΩ ± 5%blue gray red silver = 6.8 kΩ ± 10%black brown blue silver = 1 MΩ ± 10%

Now interpret the following color code:

green orange yellow silver = ?

P2 Fit a line to the following sample data and usethe slope to extract the resistance (see Appendix 4).

Your result should be consistent with a resistor colorcode of green-violet-yellow.

P3 Plan how you will measure I versus ∆V forboth positive and negative values of ∆V , since thepower supply only supplies positive voltages.

P4 Would data like these indicate a negative resis-tance, or did the experimenter just hook somethingup wrong? If the latter, explain how to fix it.

P5 Explain why the following statements aboutthe resistor RK are incorrect:

a) ‘You have to make RK small compared to RU , soit won’t affect things too much.’

b) ‘RK doesn’t affect the measurement of RU , be-cause the meters just measure the total amount thepower supply is putting out.’

c) ‘RK doesn’t affect the measurement of RU , be-cause the current and voltage only go through RK

after theyve already gone through RU .”

80 Lab 26 Electrical Resistance

Analysis

Graph I versus ∆V for all three unknowns. Decidewhich ones are ohmic and which are non-ohmic. Forthe ones that are ohmic, extract a value for the resis-tance (see appendix 4). Don’t bother with analysisof random errors, because the main source of error inthis lab is the systematic error in the calibration ofthe multimeters (and in part C the systematic errorfrom the subject’s fidgeting).

Programmed Introduction to Prac-

tical Electrical Circuits

Physics courses in general are compromises betweenthe fundamental and the practical, between explor-ing the basic principles of the physical universe anddeveloping certain useful technical skills. Althoughthe electricity and magnetism labs in this manualare structured around the sequence of abstract the-oretical concepts that make up the backbone of thelecture course, it’s important that you develop cer-tain practical skills as you go along. Not only willthey come in handy in real life, but the later partsof this lab manual are written with the assumptionthat you will have developed them.

As you progress in the lab course, you will find thatthe instructions on how to construct and use circuitsbecome less and less explicit. The goal is not tomake you into an electronics technician, but neithershould you emerge from this course able only to flipthe switches and push the buttons on prepackagedconsumer electronics. To use a mechanical analogy,the level of electrical sophistication you’re intendedto reach is not like the ability to rebuild a car enginebut more like being able to check your own oil.

In addition to the physics-based goals stated at thebeginning of this section, you should also be devel-oping the following skills in lab this week:

(1) Be able to translate back and forth between schemat-ics and actual circuits.

(2) Use a multimeter (discussed in Appendix 7),given an explicit schematic showing how to connectit to a circuit.

Further practical skills will be developed in the fol-lowing lab.

81

27 The Loop and Junction Rules

Apparatus

DC power supply (Thornton) . . . . . . . . . . . . . 1/groupmultimeter (Fluke) . . . . . . . . . . . . . . . . . . . . . . . 1/groupresistors

Goal

Test the loop and junction rules in two electricalcircuits.

Introduction

If you ask physicists what are the most fundamen-tally important principles of their science, almost allof them will start talking to you about conserva-tion laws. A conservation law is a statement that acertain measurable quantity cannot be changed. Aconservation law that is easy to understand is theconservation of mass. No matter what you do, youcannot create or destroy mass.

The two conservation laws with which we will beconcerned in this lab are conservation of energy andconservation of charge. Energy is related to voltage,because voltage is defined as V = PE/q. Chargeis related to current, because current is defined asI = ∆q/∆t.

Conservation of charge has an important consequencefor electrical circuits:

When two or more wires come together at a point ina DC circuit, the total current entering that pointequals the total current leaving it.

Such a coming-together of wires in a circuit is calleda junction. If the current leaving a junction was,say, greater than the current entering, then the junc-tion would have to be creating electric charge outof nowhere. (Of course, charge could have beenstored up at that point and released later, but thenit wouldn’t be a DC circuit — the flow of currentwould change over time as the stored charge wasused up.)

Conservation of energy can also be applied to anelectrical circuit. The charge carriers are typicallyelectrons in copper wires, and an electron has a po-tential energy equal to −eV . Suppose the electronsets off on a journey through a circuit made of re-

sistors. Passing through the first resistor, our sub-atomic protagonist passes through a voltage differ-ence of ∆V1, so its potential energy changes by −e∆V1.To use a human analogy, this would be like going upa hill of a certain height and gaining some gravi-tational potential energy. Continuing on, it passesthrough more voltage differences, −e∆V2, −e∆V3,and so on. Finally, in a moment of religious tran-scendence, the electron realizes that life is one bigcircuit — you always end up coming back where youstarted from. If it passed through N resistors be-fore getting back to its starting point, then the totalchange in its potential energy was

−e (∆V1 + . . . + ∆VN ) .

But just as there is no such thing as a round-triphike that is all downhill, it is not possible for theelectron to have any net change in potential energyafter passing through this loop — if so, we wouldhave created some energy out of nothing. Since thetotal change in the electron’s potential energy mustbe zero, it must be true that ∆V1 + . . . + ∆VN = 0.This is the loop rule:

The sum of the voltage differences around any closedloop in a circuit must equal zero.

When you are hiking, there is an important distinc-tion between uphill and downhill, which depends en-tirely on which direction you happen to be travelingon the trail. Similarly, it is important when apply-ing the loop rule to be consistent about the signsyou give to the voltage differences, say positive ifthe electron sees an increase in voltage and negativeif it sees a decrease along its direction of motion.

Observations

A The junction rule

Construct a circuit like the one in the figure, usingthe Thornton power supply as your voltage source.To make things more interesting, don’t use equalresistors. Use nice big resistors (say 100 kΩ to 1MΩ) — this will ensure that you don’t burn up theresistors, and that the multimeter’s small internalresistance when used as an ammeter is negligible incomparison. Insert your multimeter in the circuit tomeasure all three currents that you need in order totest the junction rule.

82 Lab 27 The Loop and Junction Rules

B The loop rule

Now come up with a circuit to test the loop rule.Since the loop rule is always supposed to be true, it’shard to go wrong here! Make sure that (1) you haveat least three resistors in a loop, (2) the whole cir-cuit is not just a single loop, and (3) you hook in thepower supply in a way that creates non-zero voltagedifferences across all the resistors. Measure the volt-age differences you need to measure to test the looprule. Here it is best to use fairly small resistances, sothat the multimeter’s large internal resistance whenused in parallel as a voltmeter will not significantlyreduce the resistance of the circuit. Do not use re-sistances of less than about 100 Ω, however, or youmay blow a fuse or burn up a resistor.

Prelab


P1 Draw a schematic showing where you will in-sert the multimeter in the circuit to measure thecurrents in part A.

P2 Invent a circuit for part B, and draw a schematic.You need not indicate actual resistor values, sinceyou will have to choose from among the values actu-ally available in lab.

P3 Pick a loop from your circuit, and draw a schematicshowing how you will attach the multimeter in thecircuit to measure the voltage differences in part B.

P4 Explain why the following statement is incor-rect: ‘We found that the loop rule was not quite true,but the small error could have been because the re-sistor’s value was off by a few percent compared tothe color-code value.’

Self-Check

Do the analysis in lab.

Analysis

Discuss whether you think your observations agreewith the loop and junction rules, taking into accountsystematic and random errors. If this is your firsttime doing error analysis, read appendices 2 and 3.

Programmed Introduction to Prac-

tical Electrical Circuits

The following practical skills are developed in thislab:

(1) Use a multimeter without being given an explicitschematic showing how to connect it to your circuit.This means connecting it in parallel in order to mea-sure voltages and in series in order to measure cur-rents.

(2) Use your understanding of the loop and junc-tion rules to simplify electrical measurements. Theserules often guarantee that you can get the same cur-rent or voltage reading by measuring in more thanone place in a circuit. In real life, it is often mucheasier to connect a meter to one place than another,and you can therefore save yourself a lot of troubleusing the rules rules.

83

28 Electric Fields

Apparatus

board and U-shaped probe rulerDC power supply (Thornton)multimeterscissorsstencils for drawing electrode shapes on paper

Goals

To be better able to visualize electric fields andunderstand their meaning.

To examine the electric fields around certaincharge distributions.

Introduction

By definition, the electric field, E, at a particularpoint equals the force on a test charge at that pointdivided by the amount of charge, E = F/q. We canplot the electric field around any charge distributionby placing a test charge at different locations andmaking note of the direction and magnitude of theforce on it. The direction of the electric field atany point P is the same as the direction of the forceon a positive test charge at P. The result would bea page covered with arrows of various lengths anddirections, known as a ‘sea of arrows’ diagram..

In practice, Radio Shack does not sell equipment forpreparing a known test charge and measuring theforce on it, so there is no easy way to measure elec-tric fields. What really is practical to measure at anygiven point is the voltage, V , defined as the elec-trical energy (potential energy) that a test chargewould have at that point, divided by the amountof charge (E/Q). This quantity would have unitsof J/C (Joules per Coulomb), but for conveniencewe normally abbreviate this combination of units asvolts. Just as many mechanical phenomena can bedescribed using either the language of force or thelanguage of energy, it may be equally useful to de-scribe electrical phenomena either by their electricfields or by the voltages involved.

Since it is only ever the difference in potential en-ergy (interaction energy) between two points thatcan be defined unambiguously, the same is true for

voltages. Every voltmeter has two probes, and themeter tells you the difference in voltage between thetwo places at which you connect them. Two pointshave a nonzero voltage difference between them ifit takes work (either positive or negative) to movea charge from one place to another. If there is avoltage difference between two points in a conduct-ing substance, charges will move between them justlike water will flow if there is a difference in levels.The charge will always flow in the direction of lowerpotential energy (just like water flows downhill).

All of this can be visualized most easily in termsof maps of constant-voltage curves (also known asequipotentials); you may be familiar with topograph-ical maps, which are very similar. On a topographi-cal map, curves are drawn to connect points havingthe same height above sea level. For instance, a cone-shaped volcano would be represented by concentriccircles. The outermost circle might connect all thepoints at an altitude of 500 m, and inside it youmight have concentric circles showing higher levelssuch as 600, 700, 800, and 900 m. Now imagine asimilar representation of the voltage surrounding anisolated point charge. There is no ‘sea level’ here, sowe might just imagine connecting one probe of thevoltmeter to a point within the region to be mapped,and the other probe to a fixed reference point veryfar away. The outermost circle on your map mightconnect all the points having a voltage of 0.3 Vrelative to the distant reference point, and withinthat would lie a 0.4-V circle, a 0.5-V circle, and soon. These curves are referred to as constant-voltagecurves, because they connect points of equal volt-age. In this lab, you are going to map out constant-voltage curves, but not just for an isolated pointcharge, which is just a simple example like the ide-alized example of a conical volcano.

You could move a charge along a constant-voltagecurve in either direction without doing any work,because you are not moving it to a place of higherpotential energy. If you do not do any work whenmoving along a constant-voltage curve, there mustnot be a component of electric force along the surface(or you would be doing work). A metal wire is aconstant-voltage curve. We know that electrons in ametal are free to move. If there were a force alongthe wire, electrons would move because of it. In factthe electrons would move until they were distributedin such a way that there is no longer any force on

84 Lab 28 Electric Fields

them. At that point they would all stay put andthen there would be no force along the wire and itwould be a constant-voltage curve. (More generally,any flat piece of conductor or any three-dimensionalvolume consisting of conducting material will be aconstant-voltage region.)

There are geometrical and numerical relationshipsbetween the electric field and the voltage, so eventhough the voltage is what you’ll measure directlyin this lab, you can also relate your data to electricfields. Since there is not any component of elec-tric force parallel to a constant-voltage curve, elec-tric field lines always pass through constant-voltagecurves at right angles. (Analogously, a stream flow-ing straight downhill will cross the lines on a topo-graphical map at right angles.) Also, if you dividethe work equation (∆energy) = Fd by q, you get(∆energy)/q = (F/q)d, which translates into ∆V =−Ed. (The minus sign is because V goes down whensome other form of energy is released.) This meansthat you can find the electric field strength at a pointP by dividing the voltage difference between the twoconstant-voltage curves on either side of P by thedistance between them. You can see that units ofV/m can be used for the E field as an alternative tothe units of N/C suggested by its definition — theunits are completely equivalent.

A simplified schematic of the apparatus, being used with

pattern 1 on page 86.

A photo of the apparatus, being used with pattern 3 on

page 86.

Method

The first figure shows a simplified schematic of theapparatus. The power supply provides an 8 V volt-age difference between the two metal electrodes, drawnin black. A voltmeter measures the voltage differ-ence between an arbitrary reference voltage and apoint of interest in the gray area around the elec-trodes. The result will be somewhere between 0 and8 V. A voltmeter won’t actually work if it’s not partof a complete circuit, but the gray area is intention-ally made from a material that isn’t a very goodinsulator, so enough current flows to allow the volt-meter to operate.

The photo shows the actual apparatus. The elec-trodes are painted with silver paint on a detachableboard, which goes underneath the big board. Whatyou actually see on top is just a piece of paper onwhich you’ll trace the equipotentials with a pen. Thevoltmeter is connected to a U-shaped probe with ametal contact that slides underneath the board, anda hole in the top piece for your pen.

Turn your large board upside down. Find the smalldetachable board with the parallel-plate capacitorpattern (pattern 1 on page 86) on it, and screw it tothe underside of the equipotential board, with thesilver-painted side facing down toward the tabletop.Use the washers to protect the silver paint so that itdoesn’t get scraped off when you tighten the screws.Now connect the voltage source (using the providedwires) to the two large screws on either side of theboard. Referring to Appendix 7 on how to use a

85

multimeter, connect the multimeter so that you canmeasure the voltage difference across the terminalsof the voltage source. Adjust the voltage source togive 8 volts.

If you press down on the board, you can slip the pa-per between the board and the four buttons you seeat the corners of the board. Tape the paper to yourboard, because the buttons aren’t very dependable.There are plastic stencils in some of the envelopes,and you can use these to draw the electrodes accu-rately onto your paper so you know where they are.The photo, for example, shows pattern 3 traced ontothe paper.

Now put the U-probe in place so that the top isabove the equipotential board and the bottom of itis below the board. You will first be looking forplaces on the pattern board where the voltage is onevolt — look for places where the meter reads 1.0 andmark them through the hole on the top of your U-probe with a pencil or pen. You should find a wholebunch of places there the voltage equals one volt,so that you can draw a nice constant-voltage curveconnecting them. (If the line goes very far or curvesstrangely, you may have to do more.) You can thenrepeat the procedure for 2 V, 3 V, and so on. Labeleach constant-voltage curve. Once you’ve finishedtracing the equipotentials, everyone in your groupwill need one copy of each of the two patterns youdo, so you will need to photocopy them or simplytrace them by hand.

Repeat this procedure with another pattern. Groups1 and 4 should do patterns 1 and 2; groups 2 and 5patterns 1 and 3; groups 3, 6, and 7 patterns 1 and4.

Prelab


P1 Looking at a plot of constant-voltage curves,how could you tell where the strongest electric fieldswould be? (Don’t just say that the field is strongestwhen you’re close to ‘the charge,’ because you mayhave a complex charge distribution, and we don’thave any way to see or measure the charge distribu-tion.)

P2 What would the constant-voltage curves looklike in a region of uniform electric field (i.e., one inwhich the E vectors are all the same strength, andall in the same direction)?

Self-Check

Calculate at least one numerical electric field valueto make sure you understand how to do it.

You have probably found some constant-voltage curvesthat form closed loops. Do the electric field patternsever seem to close back on themselves? Make sureyou understand why or why not.

Make sure the people in your group all have a copyof each pattern.

Analysis

A. After you have completed the plots for two pat-terns, you should try to draw in electric field vectors.You will then have two different representations ofthe field superimposed on one another. Rememberthat electric field vectors are always perpendicularto constant-voltage curves. The electric field linespoint from high voltage to low voltage, just as theforce on a rolling ball points downhill.

B. Select at least five places on each plot and deter-mine the electric field strength (E) at each of them.Make sure to include the two points that appear tohave the strongest and weakest fields.

C. For the parallel-plate capacitor, in what regionwas the electric field relatively uniform?

86 Lab 28 Electric Fields

29 Magnetism (Physics 206/211)

Apparatus

bar magnet (stack of 6 Nd)compassgraph paper, with 1 cm squaresHall effect magnetic field probesLabPro interfaces, DC power supplies, and USB ca-bles

Goal

Find how the magnetic field of a bar magnet changeswith distance along one of the magnet’s lines of sym-metry.

Introduction

A Qualitative Mapping of the Magnet’s Field

You can use a compass to map out part of the mag-netic field of a bar magnet. The compass is affectedby both the earth’s field and the bar magnet’s field,and points in the direction of their vector sum, but ifyou put the compass within a few cm of the bar mag-net, you’re seeing mostly its field, not the earth’s.Investigate the bar magnet’s field, and sketch it inyour lab notebook.

B Variation of Field With Distance: Deflectionof a Magnetic Compass

You can infer the strength of the bar magnet’s fieldat a given point by putting the compass there andseeing how much it is deflected.

The task can be simplified quite a bit if you restrictyourself to measuring the magnetic field at pointsalong one of the magnet’s two lines of symmetry,shown in the figure.

If the magnet is flipped across the vertical axis, thenorth and south poles remain just where they were,and the field is unchanged. That means the entiremagnetic field is also unchanged, and the field at apoint such as point b, along the line of symmetry,must therefore point straight up.

If the magnet is flipped across the horizontal axis,then the north and south poles are swapped, and thefield everywhere has to reverse its direction. Thus,the field at points along this axis, e.g., point a, must

point straight up or down.

Line up your magnet so it is pointing east-west.Choose one of the two symmetry axes of your mag-net, and measure the deflection of the compass attwo points along that axis, as shown in the secondfigure, at the end of the lab. As part of your prelab,you will use vector addition to find an equation forBm/Be, the magnet’s field in units of the Earth’s, interms of the deflection angle θ. For your first point,find the distance r at which the deflection is 70 de-grees; this angle is chosen because it’s about as bigas it can be without giving very poor relative preci-sion in the determination of the magnetic field. Foryour second data-point, use twice that distance. Bywhat factor does the field decrease when you doubler?

Note that the measurements are very sensitive to therelative position and orientation of the bar magnetand compass. You can position them accurately bylaying them both on top of a piece of graph paper,but before you set all that up, get a preliminaryestimate of the distances you’ll be using, becauseotherwise you can end up wasting your time.

Based on your two data-points, form a hypothesisabout the variation of the magnet’s field with dis-tance according to a power law B ∝ rp.

C Variation of Field With Distance: Hall EffectMagnetometer

In this part of the lab, you will test your hypothesisabout the power law relationship B ∝ rp; you willfind out whether the field really does obey such alaw, and if it does, you will determine p accurately.

88 Lab 29 Magnetism (Physics 206/211)

This part of the lab uses a device called a Hall ef-fect magnetometer for measuring magnetic fields. Itworks by sending an electric current through a sub-stance, and measuring the force exerted on thosemoving charges by the surrounding magnetic field.The probe only measures the component of the mag-netic field vector that is parallel to its own axis. Plugthe probe into the LabPro interface, connect the in-terface to the computer’s USB port, and plug the in-terface’s DC power supply in to it. Start up version3 of Logger Pro, and it will automatically recognizethe probe and start displaying magnetic fields on thescreen, in units of mT (millitesla). The probe hastwo ranges, one that can read fields up to 0.3 mT,and one that goes up to 6.4 mT. You can select ei-ther one using the switch on the probe. To test yourhypothesis with good precision, you need to obtaindata over the widest possible range of fields. Alwaysuse the more sensitive 0.3 mT scale whenever possi-ble, because it will give better precision for low fields.Be careful, however, because if you expose the probeto a field that’s beyond its maximum range, it willgive incorrect readings. Although you have an ex-pectation about the direction of the field (based bothon symmetry arguments and on your qualitative re-sults from part A), it’s a good idea to try orientingthe probe in different ways to see what happens.

Two extra complications are that the Earth’s fieldis adding on to the magnet’s field, and the abso-lute calibration of the probe is very poor by de-fault. You can make the computer take care of bothof these issues automatically, by zeroing the sensor(Experiment>Zero) when it is exposed only to theEarth’s field. This causes the computer to impose acalibration such that the Earth’s field is consideredto be exactly zero. You may need to redo the calibra-tion each time you switch scales. If you then carryout the whole measurement with the probe and themagnet’s field both aligned east-west, the Earth’sfield has no effect.

Prelab


P1 Suppose that when the compass is 11.0 cm fromthe magnet, it is 45 degrees away from north. Whatis the strength of the bar magnet’s field at this loca-tion in space, in units of the Earth’s field?

P2 Find Bm/Be in terms of the deflection angle θ.As a special case, you should be able to recover youranswer to P1.

Analysis

Determine the magnetic field of the bar magnet asa function of distance. No error analysis is required.Look for a power-law relationship using the log-loggraphing technique described in appendix 5. Doesthe power law hold for all the distances you investi-gated, or only at large distances?

89

Measuring the variation of the bar magnet’s field with respect to distance

90 Lab 29 Magnetism (Physics 206/211)

30 The Dipole Field (Physics 222)

Apparatus

bar magnetcompassgraph paper, with 1 cm squaresHall effect magnetic field probesLabPro interfaces, DC power supplies, and USB ca-bles

Goal

Find how the magnetic field of a bar magnet changeswith distance along one of the magnet’s lines of sym-metry.

Introduction

This lab is designed to be used along with section10.3 of Simple Nature, which is about the superpo-sition (i.e., addition) of fields. That section is aboutelectric fields, and the basic principle is that if wehave two sets of sources (charges) that would indi-vidually create fields E1 and E2, then their combinedfield is the vector sum E1 +E2. Static electric fields,however, are difficult to control and measure. Mag-netic fields are much easier to work with, and thesame vector addition principle applies to them. Inthis lab, you’ll expose a magnetic compass to thesuperposed magnetic fields of the earth and a barmagnet.

A Qualitative Mapping of the Dipole’s Field

You can use a compass to map out part of the mag-netic field of a bar magnet. It turns out that thebar magnet is the magnetic equivalent of an electricdipole. The compass is affected by both the earth’sfield and the bar magnet’s field, and points in thedirection of their vector sum, but if you put the com-pass within a few cm of the bar magnet, you’re seeingmostly its field, not the earth’s. Investigate the barmagnet’s field, and sketch it in your lab notebook.You should see that it looks like the field a dipole.

B Variation of Field With Distance: Deflectionof a Magnetic Compass

Magnetic fields are actually measured in units ofTesla (T), but for the purposes of this part of the lab,we’ll just measure the fields in units of the earth’s

magnetic field. That is, we define the earth’s mag-netic field to have a strength of exactly 1.0 in Fuller-ton.1 You can infer the strength of the bar magnet’sfield at a given point by putting the compass thereand seeing how much it is deflected. The standardnotation for magnetic field is B, so we can notate thefields of the earth and the magnet as Be and Bm.

The task can be simplified quite a bit if you restrictyourself to measuring the magnetic field at pointsalong one of the magnet’s two lines of symmetry,shown in the figure.

If the magnet is flipped across the vertical axis, thenorth and south poles remain just where they were,and the field is unchanged. That means the entiremagnetic field is also unchanged, and the field at apoint such as point b, along the line of symmetry,must therefore point straight up.

If the magnet is flipped across the horizontal axis,then the north and south poles are swapped, and thefield everywhere has to reverse its direction. Thus,the field at points along this axis, e.g., point a, mustpoint straight up or down.

Line up your magnet so it is pointing east-west.Choose one of the two symmetry axes of your mag-net, and measure the deflection of the compass attwo points along that axis, as shown in the secondfigure, at the end of the lab. As part of your prelab,you will use vector addition to find an equation forBm/Be, the magnet’s field in units of the Earth’s, interms of the deflection angle θ. For your first point,

1Actually we’re defining its horizontal component to beone unit — the compass can’t respond to vertical fields. Thedip angle of the magnetic field in Fullerton is fairly steep.

92 Lab 30 The Dipole Field (Physics 222)

find the distance r at which the deflection is 70 de-grees; this angle is choses because it’s about as big asit can be without giving very poor relative precisionin the determination of the magnetic field. For yoursecond data-point, use twice that distance. By whatfactor does the field decrease when you double r?

Note that the measurements are very sensitive to therelative position and orientation of the bar magnetand compass. You can position them accurately bylaying them both on top of a piece of graph paper,but before you set all that up, get a preliminaryestimate of the distances you’ll be using, becauseotherwise you can end up wasting your time.

Based on your two data-points, form a hypothesisabout the variation of the dipole’s field with dis-tance according to a power law B ∝ rp. (If you’vedone homework problems 11 and 16 in chapter 10 ofSimple Nature, then you know what p should be foran electric dipole, based on vector addition of theelectric fields of two charges.)

C Variation of Field With Distance: Hall EffectMagnetometer

In this part of the lab, you will test your hypothesisabout the power law relationship B ∝ rp; you willfind out whether the field really does obey such alaw, and if it does, you will determine p accurately.

This part of the lab uses a device called a Hall effectmagnetometer for measuring magnetic fields. Youdon’t know enough about magnetism yet to under-stand the theory behind the operation of the de-vice, so you can just think of it as a mysterious littleprobe, like a wand, that you can place at some pointin space and measure the magnetic field. The probeonly measures the component of the magnetic fieldvector that is parallel to its own axis. Plug the probeinto the LabPro interface, connect the interface tothe computer’s USB port, and plug the interface’sDC power supply in to it. Start up version 3 ofLogger Pro, and it will automatically recognize theprobe and start displaying magnetic fields on thescreen, in units of mT (millitesla). The probe hastwo ranges, one that can read fields up to 0.3 mT,and one that goes up to 6.4 mT. You can select ei-ther one using the switch on the probe. To test yourhypothesis with good precision, you need to obtaindata over the widest possible range of fields. Al-ways use the more sensitive 0.3 mT scale wheneverpossible, because it will give better precision for lowfields. Be careful, however, because if you expose theprobe to a field that’s beyond its maximum range, itwill give incorrect readings. Although you have anexpectation about the direction of the field (based

both on symmetry arguments and on your qualita-tive results from part A), it’s a good idea to tryorienting the probe along different axes to see whathappens. In general, if you want to use the probe tomeasure a field whose direction and magnitude areboth unknown, you need to orient the probe alongtwo different axes, and determine the two compo-nents separately.

Two extra complications are that the Earth’s fieldis adding on to the magnet’s field, and the abso-lute calibration of the probe is very poor by de-fault. You can make the computer take care of bothof these issues automatically, by zeroing the sensor(Experiment>Zero) when it is exposed only to theEarth’s field, and aligned perpendicular to it. Thiscauses the computer to impose a calibration suchthat the Earth’s field is considered to be exactly zero.You may need to redo the calibration each time youswitch scales. If you then carry out the whole mea-surement with the probe and the magnet’s field bothaligned east-west, the Earth’s field has no effect.

D Variation of Field With Angle: Hall Effect Mag-netometer

Homework problems 11 and 16 in chapter 10 of Sim-ple Nature, predict that for an electric dipole, thefield in the midplane is exactly half as strong as theon-axis field, at the same distance. Test this predic-tion.

Also, find the magnitude of the field at an angleof 45 degrees between the midplane and the axis.Since you don’t know the direction of the field atthis location based on symmetry arguments (and youonly know it very roughly based on mapping with acompass in part A), you’ll need to measure both ofthe field’s components at this location.

As you plan your observations in this part, you’llneed to think about what is the best distance atwhich to place the probe. If the distance is too large,you may find that the field is too weak to measurewith good precision. If the distance is too small, thenthe physical size of the probe becomes an issue, sincethe exact location at which the probe measures thefield is ill-defined. (The probe measures a voltagecreated by the field in a sample of some material,and that sample has a finite size.)

In part C, all the fields were along a single line, andthere were no angles involved. That made it simpleto get rid of the effect the Earth’s field. That doesn’twork in this part, however. One way of handling thedifficulty is to flip the magnet by 180 degrees, andfind the difference between the readings for the two

93

opposite orientations of the magnet, which shouldequal twice the magnet’s field. The Earth’s field can-cels out. This means that you need a total of fourdifferent measurements at each point in space, cover-ing all four possible combinations of the orientationof the probe along x or y with both orientations ofthe magnet.

Prelab


P1 Suppose that when the compass is 11.0 cm fromthe magnet, it is 45 degrees away from north. Whatis the strength of the bar magnet’s field at this loca-tion in space, in units of the Earth’s field?

P2 Find Bm/Be in terms of the deflection angle θ.As a special case, you should be able to recover youranswer to P1.

Analysis

Determine the magnetic field of the bar magnet asa function of distance. No error analysis is required.Look for a power-law relationship using the log-loggraphing technique described in appendix 5. Doesthe power law hold for all the distances you inves-tigated, or only at large distances? Compare thispower law result with the result for the variation ofan electric dipole’s field with distance.

94 Lab 30 The Dipole Field (Physics 222)

Measuring the variation of the bar magnet’s field with respect to distance

95

31 The Earth’s Magnetic Field (Physics 222)

Apparatus

digital multimeterneodymium magnet (6 discs stuck together)magnetic compassresistorsdecade resistor boxesrulersthread1-m aluminum rodstopwatchphotogatelaseraluminum rods, and clampsD cell batteries and holders . . . . . . . . . . . . . . . 2/groupHelmholtz coils (e/m apparatus)high-precision Helmholtz coil (one set)Hall effect magnetic field probesLabPro interfaces, DC power supplies, and USB ca-bles

Goal

Determine the horizontal component of the Earth’smagnetic field in Fullerton, to high precision.

Observations

Since you’ve already used the Hall effect magneticfield probes in lab 30, you might think that it wouldbe relatively trivial to measure the Earth’s magneticfield precisely. However, the calibration of thoseprobes is quite poor, so it’s not possible to get resultswith error bars smaller than about 10-20%.

The geometry of a Helmholtz coil.

The basic idea of the more precise technique usedin this lab is to hang a permanent magnet from a

thread, and observe the period of its oscillations inthe Earth’s magnetic field. The idea is that if theEarth’s field is stronger, there is a stronger torquetrying to align the magnet north-south, and the fre-quency of the oscillations will therefore be higher.By measuring the frequency of the oscillations, wecan work backward and infer the strength of the hor-izontal component of the Earth’s field.

A contour map of the field of a Helmholtz coil (top view

of the horizontal plane cutting through the center).

One reason the technique isn’t quite that simple isthat the frequency of the oscillations also dependson other quantities, including the magnet’s dipolemoment and moment of inertia, that are very diffi-cult to measure with better than about 10% preci-sion. A trick for getting around this problem is tosuperimpose a known southward magnetic field onthe Earth’s northward one, and adjust the knownfield so as to cancel the Earth’s. Reducing the fieldincreases the period of the oscillations, and if wecould exactly cancel the horizontal component of theEarth’s field, then the period would be infinite. Theknown field is supplied by a type of electromagnetcalled a Helmholtz coil, shown in the first figure. Itconsists of two circular coils of wire, with their axescoinciding. In the classic design (which is what’sreally properly called a Helmholtz coil), the separa-tion h between the planes of the two coils is equal totheir radius, b. Having h = b turns out to produce

96 Lab 31 The Earth’s Magnetic Field (Physics 222)

the most uniform possible field near the center of thewhole arrangement, in the sense that all the field’sderivatives up to the fourth derivative equal zero.The second figure (from the Wikipedia article, copy-left licensed by Wikipedia) is a contour map showinghow little the field actually varies over a fairly largevolume in the center. The ‘octopus’ in the middleis the region in which the field is between 99% and101% of its value at the center.

Even this version of the experiment turns out to needsome further tweaking. It is difficult to align the axisof the coils with the Earth’s field, so we typically endup with a misalignment, φ, which is a few degrees.Therefore, the fields do not really cancel, and as thecurrent through the coils is tuned through the op-timal value, the horizontal field becomes small, butnot zero, and swings around gradually from northto south. It becomes difficult to pick off the currentthat produces the maximum period, partly becausethe period of the oscillations is not quite indepen-dent of amplitude, and it becomes difficult to con-trol the amplitude of the oscillations properly whenthe equilibrium orientation is constantly changing.Even if we could precisely recognize the current thatgave the maximum period, that would be the currentthat canceled out the component of the Earth’s fieldalong the coils’ axis, i.e., we would be taking thevector (Bx,By), and changing it to (Bx, 0), wherey is the axis of the coils. Thus we would really bemeasuring, By = B cos φ, rather than |B|. To getaround this problem, you can use the following itera-tive method: (1) Align the coils’ axis approximatelywith the earth’s field by eyeballing the alignmentagainst a magnetic compass. (2) Tune the currentin the coil to the point where the magnet’s equilib-rium orientation is perpendicular to the earth’s field.This is pretty close to the current that would havecanceled the earth’s field, if the alignment had beenperfect. In this state, the magnet will point either tothe east or to the west, depending on the direction ofthe error in alignment. (3) Carefully, slowly rotatethe apparatus until the magnet’s equilibrium orien-tation shifts to the north-south line. This is a statein which the coil’s field is exaclty on the same lineas the Earth’s, but their magnitudes are slightly dif-ferent. (4) Tune the current again to maximize theperiod. In this final step, it becomes important tocontrol the amplitude of the oscillations. As shownin the figure, the error in the period is less than 0.1%for amplitudes of less than about 10 degrees.

The problem now boils down to the accurate deter-mination of the field at the center of the Helmholtzcoils for a given amount of current, i.e., the ratio

The dependence of the period on amplitude. For an-

gles less than 20 degrees, the motion is nearly simple

harmonic, and the period is independent of amplitude to

within about 1%. Higher amplitudes can be used, but it

becomes much more important to control the initial am-

plitude.

B/I. You’ll derive the relevant expression as one ofyour prelab questions. It depends on the accuratemeasurement of the dimensions b and h. In generalit’s fairly difficult to construct magnet coils so thattheir dimensions are accurately determinable, andthe coils you’ll use are no exception. They consistof somewhat irregular bundles of wire tied togetherwith cable ties, and they aren’t even circular; theirvertical diameter is significantly different from theirhorizontal diameter. As closely as I’ve been able todetermine, they have h = 14.7 ± 0.3 cm, and an av-erage b of about 15.1 ± 0.3 cm, but these error barsare uncomfortably large. They have N = 130 turnsof wire on each coil, i.e., 260 turns on each completeset of Helmholtz coils.

Because of these problems, I’ve constructed a Helm-holtz coil that has a much more precisely measur-able geometry. You can calculate B/I for the pre-cise coils, whose dimensions are carefully constructedand easy to measure: h = b = 11.15± .05 cm. Theyhave N = 5 turns of wire in each coil. Althoughthere is only one copy of the precise Helmholtz coil,and it wouldn’t be convenient to use for this labanyway (they produce weak fields, and their interioris not very accessible), we can calibrate your coilsagainst them. I’m planning to do this as a studentlab for the first time in spring 2009, and we’ll usethe data from that semester as a calibration for thecoils, by comparing Bearth/I for them with Bearth/Ifor the precise coils.

The Helmholtz coils we’re using are actually meant

97

for lab 33, and they have a big, extremely expensivevacuum tube stuck inside them for that purpose.With your instructor’s help, very carefully detachthe base from the tube. Then unscrew the yokesthat hold the tube in place, and put the tube out ofthe way in the stockroom, with cusioning to makesure it doesn’t get broken.

You need precisely controlled, steady currents forthis lab, and DC power supplies aren’t stable enough,so you’ll use batteries instead. To control the cur-rent precisely, you’ll use the decade resistor boxes,which are variable resistors that let you dial up anydecimal number of ohms that you want.

We want to keep all magnetic materials far awayfrom the magnet. Clamp the 1-m aluminum rod tothe vertical steel post, and hang the magnet from it,far from the post.

There are several possible methods for measuringthe period of the oscillations, and one of my goalsfor spring 2009 is to have my students test drivethem. One is to use a stopwatch to time, say, 20oscillations. A second method would be to use themagnetic field probe and graph the field as a functionof time. A third method would be to use a photo-gate, in pendulum mode as described in appendix6. The photogates have steel screws in them, so youcan’t use them in the ordinary way, with the mag-net swinging through the infrared beam that goesacross the center of the gate. Instead, you can openthe shutter on the inside of the photogate to changeit into a mode where it senses light from the beam ofan external laser. The photogate can then be phys-ically far away from the magnet so that the screwsdon’t affect the measurement. A possible problemwith the photogate method is that it requires theamplitude of the oscillations to be big enough sothat the magnet blocks and unblocks the photogate,but with oscillations that big, the dependence of theperiod on amplitude could be a significant sourceof error unless the amplitude was very accuratelycontrolled. This problem could possibly be solvedby attaching a cardboard vane to the magnet, andthat would also get rid of the safety problem causedby reflecting the laser beam from the shiny magnet.Of these methods, it’s possible that one might bethe most convenient for rough initial measurements,while another would work best for the final, accuratemeasurement.

When you’re done with all this, what you’ve actuallymeasured is the magnetic field inside the building.Many buildings have magnetic building materials, sothe fields inside them are different from the Earth’s

field. To correct for this, measure the period of themagnet’s oscillation inside and outside. If they aresignificantly different, correct according to B1/B2 =(T2/T1)

2; this follows from adapting the equationω =

√

k/m for simple harmonic motion to the caseof rotation, with the torque τ = m × B playing therole of the restoring force.

(Note to myself: Some of my own further notes aboutthe lab are embedded in comments in the LaTeXsource code for the lab manual.)

Prelab


P1 For an electromagnet consisting of a single cir-cular loop of wire of radius b, the field at a point onits axis, at a distance z from the plane of the loop,is given by

B =2πkIb2

c2(b2 + z2)3/2.

Starting from this equation, derive an equation forthe magnetic field at the center of a pair of Helmholtzcoils, in terms of h, b, and N . Find B/I for boththe high-precision coils and the low-precision ones,based on the given values of h, b, and N . (The B/Ifor the low-precision ones is useful as a check, but haspoor precision, which is why you’ll calibrate againstthe high-precision ones.)

P2 Estimate the current that will be required inthe low-precision coils in order to cancel the Earth’sfield, about 2 × 10−5 T.

Analysis

Find the earth’s magnetic field, with error bars.

98 Lab 31 The Earth’s Magnetic Field (Physics 222)

32 Relativity

Apparatus

magnetic balance . . . . . . . . . . . . . . . . . . . . . . . . . 1/groupmeter stick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/groupmultimeter (BK, not HP) . . . . . . . . . . . . . . . . . 1/grouplaser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1/groupvernier calipers . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/groupdecade resistor box (General Radio) . . . . . . 1/groupstapleslab’s DC power supply (30 A)

Goal

Measure the speed of light.

Introduction

Oersted discovered that magnetism is an interac-tion of moving charges with moving charges, butit wasn’t until almost a hundred years later thatEinstein showed why such an interaction must exist:magnetism occurs as a direct result of his theory ofrelativity. Since magnetism is a purely relativisticeffect, and relativistic effects depend on the speed oflight, any measurement of a magnetic effect can beused to determine the speed of light.

Setup

The idea is to set up opposite currents in two wires,A and B, one under the other, and use the repulsionbetween the currents to create an upward force onthe top wire, A. The top wire is on the arm of a bal-ance, which has a stable equilibrium because of theweight C hanging below it. You initially set up thebalance with no current through the wires, adjustingthe counterweight D so that the distance between thewires is as small as possible. What we care about isreally the center-to-center distance (which we’ll callR), so even if the wires are almost touching, there’sstill a millimeter or two worth of distance betweenthem.) By shining a laser at the mirror, E, and ob-serving the spot it makes on the wall, you can veryaccurately determine this particular position of thebalance, and tell later on when you’ve reproduced it.

If you put a current through the wires, it will raisewire A. The torque made by the magnetic repulsion

is now canceling the torque made by gravity directlyon all the hardware, such as the masses C and D.This gravitational torque was zero before, but nowyou don’t know what it is. The trick is to put a tinyweight (a staple) on top of wire A, and adjust thecurrent so that the balance returns to the position itoriginally had, as determined by the laser dot on thewall. You now know that the gravitational torqueacting on the original apparatus (everything exceptfor the staple) is back to zero, so the only torquesacting are the torque of gravity on the staple andthe magnetic torque. Since both these torques areapplied at the same distance from the axis, the forcescreating these torques must be equal as well. Byweighing a block of staples, you can determine theweight of one staple, and infer the magnetic forcethat was acting.

It’s very important to get the wires A and B per-fectly parallel. You also need to minimize the resis-tance of the apparatus, or else you won’t be able toget enough current through it to cancel the weightof the staple. Most of the resistance is at the pol-ished metal knife-edges that the moving part of thebalance rests on. It may be necessary to clean thesurfaces, or even to freshen them a little with a fileto remove any layer of oxidation. Since everyoneis sharing the same power supply, you can’t turn aknob to control the voltage being applied to yoursetup. Instead, you need to put the decade resistorbox in series in your circuit, and use it to control thecurrent that flows.

Analysis

The mass of an aluminum atom is 4.48 × 10−26 kg.Let’s assume that each aluminum atom contributes

100 Lab 32 Relativity

one conduction electron, and that the wires havemasses per unit length of 2 g/m — these two as-sumptions are only roughly right, but you’ll see laterthat they end up not mattering.

You can now calculate the number of coulombs permeter of conduction electrons, −λ, in your wires. Bycombining this with your measured levitation cur-rent, you can find the average velocity, v, at whichthe electrons were drifting through the wire. This ve-locity is quite small compared to the speed of light,so the relativistic effect is slight. However, as youfound when you did the prelab, the amount of chargein a piece of ordinary matter is huge, so even a slighteffect is enough to produce a measurable result.

Now imagine yourself as one of the moving electronsin the top wire. In your frame of reference, the elec-trons in the other strip are moving at velocity −2v,and for each such electron there is a correspondingproton moving at velocity −v relative to you. (Youdon’t care about the protons and electrons that arepaired off in atoms, because they cancel each other.)Both the electrons and the protons are squashed to-gether by the relativistic contraction of space, so wehave

λp = λ1

√

1 − v2/c2

λe = −λ1

√

1 − (2v)2/c2.

In the frame of reference fixed to the tabletop, thesewould have canceled each other out, but in yourframe of reference, we have

λtotal = λp + λe

= λ

[

1√

1 − v2/c2− 1

√

1 − (2v)2/c2

]

You may want to try calculating this directly justfor fun, but unless your calculator has unusuallyhigh precision, it will round off to zero, since thegamma factors are both very close to one. To geta useful result, we need to use the approximation(1 − ǫ)−1/2 ≈ 1 + ǫ/2, which results in

λtotal ≈ −λ3v2

4c2.

In your frame of reference, the electric field of thischarge is what is responsible for repelling you andcausing the upward electric force on the wire. Theelectric force can be calculated by applying Gauss’

law to a cylinder of radius R and length ℓ:

ΦE = 4πkqin

(E)(2πRℓ) = 4πkλtotalℓ

E =2kλtotal

R

The electrical force Eq = Eλℓ cancels out the grav-itational force mg acting on the staple, so ignoringplus and minus signs, we have

Eλℓ = mg

6kλ2v2

4Rc2=

mg

ℓ

But λv is just the current, so

6kI2

4Rc2=

mg

ℓ

Solving for c, we have

c = I

√

6kℓ

4Rgm

Note that although I asked you to calculate v andλ for physical insight, it turns out that all you re-ally need to know is their product, which equals thecurrent you read on your meter.

Your final result is the speed of light, with error bars.

Prelab


Do the laser safety checklist, Appendix 9, tear it out,and turn it in at the beginning of lab. If you don’tunderstand something, don’t initial that point, andask your instructor for clarification before you startthe lab.

P1 Calculate −λ, the number of coulombs per me-ter in the tabletop’s frame of reference, using theassumptions given above. Answer: −7 × 103 C/m

P2 This is a huge amount of charge! Why doesn’tit produce any measurable electrical forces when thewire is just lying there without being connected toany electrical circuit?

101

33 The Charge to Mass Ratio of the Electron

Apparatus

vacuum tube with Helmholtzcoils (Leybold ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Cenco 33034 HV supply . . . . . . . . . . . . . . . . . . . . . . . . . 112-V DC power supplies (Thornton) . . . . . . . . . . . . . 1multimeters (Fluke or HP) . . . . . . . . . . . . . . . . . . . . . . 2compass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1ruler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1banana-plug cables

Goal

Measure the charge-to-mass ratio of the electron.

Introduction

Why should you believe electrons exist? By the turnof the twentieth century, not all scientists believedin the literal reality of atoms, and few could imag-ine smaller objects from which the atoms themselveswere constructed. Over two thousand years hadelapsed since the Greeks first speculated that atomsexisted based on philosophical arguments withoutexperimental evidence. During the Middle Ages inEurope, ‘atomism’ had been considered highly sus-pect, and possibly heretical. Finally by the Vic-torian era, enough evidence had accumulated fromchemical experiments to make a persuasive case foratoms, but subatomic particles were not even dis-cussed.

If it had taken two millennia to settle the questionof atoms, it is remarkable that another, subatomiclevel of structure was brought to light over a periodof only about five years, from 1895 to 1900. Mostof the crucial work was carried out in a series ofexperiments by J.J. Thomson, who is therefore oftenconsidered the discoverer of the electron.

In this lab, you will carry out a variation on a crucialexperiment by Thomson, in which he measured theratio of the charge of the electron to its mass, q/m.The basic idea is to observe a beam of electrons ina region of space where there is an approximatelyuniform magnetic field, B. The electrons are emittedperpendicular to the field, and, it turns out, travelin a circle in a plane perpendicular to it. The force

of the magnetic field on the electrons is

F = qvB , (1)

directed towards the center of the circle. Their ac-celeration is

a =v2

r, (2)

so using F = ma, we can write

qvB =mv2

r. (3)

If the initial velocity of the electrons is provided byaccelerating them through a voltage difference V ,they have a kinetic energy equal to qV , so

1

2mv2 = qV . (4)

From equations 3 and 4, you can determine q/m.Note that since the force of a magnetic field on amoving charged particle is always perpendicular tothe direction of the particle’s motion, the magneticfield can never do any work on it, and the particle’sKE and speed are therefore constant.

You will be able to see where the electrons are going,because the vacuum tube is filled with a hydrogengas at a low pressure. Most electrons travel large dis-tances through the gas without ever colliding with ahydrogen atom, but a few do collide, and the atomsthen give off blue light, which you can see. AlthoughI will loosely refer to ‘seeing the beam,’ you are re-ally seeing the light from the collisions, not the beam

102 Lab 33 The Charge to Mass Ratio of the Electron

of electrons itself. The manufacturer of the tube hasput in just enough gas to make the beam visible;more gas would make a brighter beam, but wouldcause it to spread out and become too broad to mea-sure it precisely.

The field is supplied by an electromagnet consistingof two circular coils, each with 130 turns of wire(the same on all the tubes we have). The coils areplaced on the same axis, with the vacuum tube atthe center. A pair of coils arranged in this type ofgeometry are called Helmholtz coils. Such a setupprovides a nearly uniform field in a large volumeof space between the coils, and that space is moreaccessible than the inside of a solenoid.

Safety

You will use the Cenco high-voltage supply to makea DC voltage of about 300 V . Two things automat-ically keep this from being very dangerous:

Several hundred DC volts are far less danger-ous than a similar AC voltage. The householdAC voltages of 110 and 220 V are more dan-gerous because AC is more readily conductedby body tissues.

The HV supply will blow a fuse if too muchcurrent flows.

Do the high voltage safety checklist, Appendix 8,tear it out, and turn it in at the beginning of lab. Ifyou don’t understand something, don’t initial thatpoint, and ask your instructor for clarification beforeyou start the lab.

Setup

Before beginning, make sure you do not have anycomputer disks near the apparatus, because the mag-netic field could erase them.

Heater circuit: As with all vacuum tubes, the cath-ode is heated to make it release electrons more easily.There is a separate low-voltage power supply builtinto the high-voltage supply. It has a set of plugsthat, in different combinations, allow you to get var-ious low voltage values. Use it to supply 6 V to theterminals marked ‘heater’ on the vacuum tube. Thetube should start to glow.

Electromagnet circuit: Connect the other Thorntonpower supply, in series with an ammeter, to the ter-minals marked ‘coil.’ The current from this power

supply goes through both coils to make the magneticfield. Verify that the magnet is working by using itto deflect a nearby compass.

High-voltage circuit: Leave the Cenco HV supplyunplugged. It is really three HV circuits in one box.You’ll be using the circuit that goes up to 500 V.Connect it to the terminals marked ‘anode.’ Askyour instructor to check your circuit. Now plug inthe HV supply and turn up the voltage to 300 V .You should see the electron beam. If you don’t seeanything, try it with the lights dimmed.

Observations

Make the necessary observations in order to findq/m, carrying out your plan to deal with the effectsof the Earth’s field. The high voltage is supposedto be 300 V, but to get an accurate measurementof what it really is you’ll need to use a multimeterrather than the poorly calibrated meter on the frontof the high voltage supply.

The beam can be measured accurately by using theglass rod inside the tube, which has a centimeterscale marked on it.

Be sure to compute q/m before you leave the lab.That way you’ll know you didn’t forget to measuresomething important, and that your result is reason-able compared to the currently accepted value.

There is a glass rod inside the vacuum tube with acentimeter scale on it, so you can measure the diam-eter d of the beam circle simply by looking at theplace where the glowing beam hits the scale. This ismuch more accurate than holding a ruler up to thetube, because it eliminates the parallax error thatwould be caused by viewing the beam and the ruleralong a line that wasn’t perpendicular to the plane ofthe beam. However, the manufacturing process usedin making these tubes (they’re probably hand-blownby a glass blower) isn’t very precise, and on many ofthe tubes you can easily tell by comparison with thea ruler that, e.g., the 10.0 cm point on the glass rodis not really 10.0 cm away from the hole from whichthe beam emerges. Past students have painstakinglydetermined the appropriate corrections, k, to add tothe observed diameters by the following electricalmethod. If you look at your answer to prelab ques-tion P1, you’ll see that the product Br is always afixed quantity in this experiment. It therefore fol-lows that Id is also supposed to be constant. Theymeasured I and d at two different values of I, anddetermined the correction k that had to be added totheir d values in order to make the two values of Id

103

equal. The results are as follows:

serial number k (cm)98-16 0.099-08 -0.699-10 -0.299-17 +0.299-56 +0.3

If your apparatus is one that hasn’t already had its kdetermined, then you should do the necessary mea-surements to calibrate it.

Prelab


The week before you are to do the lab, briefly famil-iarize yourself visually with the apparatus.

Do the high voltage safety checklist, Appendix 8,tear it out, and turn it in at the beginning of lab. Ifyou don’t understand something, don’t initial thatpoint, and ask your instructor for clarification beforeyou start the lab.

P1 Derive an equation for q/m in terms of V , rand B.

P2 For an electromagnet consisting of a single cir-cular loop of wire of radius b, the field at a point onits axis, at a distance z from the plane of the loop,is given by

B =2πkIb2

c2(b2 + z2)3/2.

Starting from this equation, derive an equation forthe magnetic field at the center of a pair of Helmholtzcoils. Let the number of turns in each coil be N (inour case, N = 130), let their radius be b, and let thedistance between them be h. (In the actual experi-ment, the electrons are never exactly on the axis ofthe Helmholtz coils. In practice, the equation youwill derive is sufficiently accurate as an approxima-tion to the actual field experienced by the electrons.)If you have trouble with this derivation, see your in-structor in his/her office hours.

P3 Find the currently accepted value of q/m forthe electron.

P4 The electrons will be affected by the Earth’smagnetic field, as well as the (larger) field of the

coils. Devise a plan to eliminate, correct for, or atleast estimate the effect of the Earth’s magnetic fieldon your final q/m value.

P5 Of the three circuits involved in this experi-ment, which ones need to be hooked up with theright polarity, and for which ones is the polarity ir-relevant?

P6 What would you infer if you found the beamof electrons formed a helix rather than a circle?

Analysis

Determine q/m, with error bars.

Answer the following questions:

Q1. Thomson started to become convinced dur-ing his experiments that the ‘cathode rays’ observedcoming from the cathodes of vacuum tubes werebuilding blocks of atoms — what we now call elec-trons. He then carried out observations with cath-odes made of a variety of metals, and found thatq/m was the same in every case. How would thatobservation serve to test his hypothesis?

Q2. Why is it not possible to determine q and mthemselves, rather than just their ratio, by observingelectrons’ motion in electric or magnetic fields?

Q3. Thomson found that the q/m of an electronwas thousands of times larger than that of ions inelectrolysis. Would this imply that the electrons hadmore charge? Less mass? Would there be no way totell? Explain.

104 Lab 33 The Charge to Mass Ratio of the Electron

34 Energy in Fields

Apparatus

Heath coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/group0.01 µF capacitors . . . . . . . . . . . . . . . . . . . . . . . . 1/groupDaedalon function generator . . . . . . . . . . . . . . 1/groupPASCO PI-9587C sine-wave generator . . . . 1/grouposcilloscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/group

Goal

Observe how the energy content of a field relates tothe field strength.

Introduction

A simplified version of the circuit.

The basic idea of this lab is to observe a circuit likethe one shown in the figure above, consisting of a ca-pacitor and a coil of wire (inductor). Imagine thatwe first deposit positive and negative charges on theplates of the capacitor. If we imagined that the uni-verse was purely mechanical, obeying Newton’s lawsof motion, we would expect that the attractive forcebetween these charges would cause them to comeback together and reestablish a stable equilibriumin which there was zero net charge everywhere inthe circuit.

However, the capacitor in its initial, charged, statehas an electric field between its plates, and this fieldpossesses energy. This energy can’t just go away,because energy is conserved. What really happensis that as charge starts to flow off of the capacitorplates, a current is established in the coil. This cur-rent creates a magnetic field in the space inside andaround the coil. The electric energy doesn’t justevaporate; it turns into magnetic energy. We endup with an oscillation in which the capacitor andthe coil trade energy back and forth. Your goal isto monitor this energy exchange, and to use it todeduce a power-law relationship between each fieldand its energy.

The actual circuit.

The practical realization of the circuit involves somefurther complications, as shown in the second figure.

The wires are not superconductors, so the circuit hassome nonzero resistance, and the oscillations wouldtherefore gradually die out, as the electric and mag-netic energies were converted to heat. The sine wavegenerator serves both to initiate the oscillations andto maintain them, replacing, in each cycle, the en-ergy that was lost to heat.

Furthermore, the circuit has a resonant frequencyat it prefers to oscillate, and when the resistance isvery small, the width of the resonance is very nar-row. To make the resonance wider and less finicky,we intentionally insert a 10 kΩ resistor. The induc-tance of the coil is about 1 H, which gives a resonantfrequency of about 1.5 kHz.

The actual circuit consists of the 1 H Heath coil, a0.01 µF capacitance supplied by the decade capaci-tor box, a 10 kΩ resistor, and the PASCO sine wavegenerator (using the GND and LO Ω terminals).

Observations

Let E be the magnitude of the electric field betweenthe capacitor plates, and let E be the maximumvalue of this quantity. It is then convenient to definex = E/E, a unitless quantity ranging from −1 to 1.Similarly, let y = B/B for the corresponding mag-netic quantities. The electric field is proportionalto the voltage difference across the capacitor plates,which is something we can measure directly usingthe oscilloscope:

x =E

E=

VC

VC

Magnetic fields are created by moving charges, i.e.,by currents. Unfortunately, an oscilloscope doesn’tmeasure current, so there’s no equally direct way toget a handle on the magnetic field. However, allthe current that goes through the coil must also gothrough the resistor, and Ohm’s law relates the cur-rent through the resistor to the voltage drop across

106 Lab 34 Energy in Fields

it. This voltage drop is something we can measurewith the oscilloscope, so we have

y =B

B=

I

I=

VR

VR

To measure x and y, you need to connect channels1 and 2 of the oscilloscope across the resistor andthe capacitor. Since both channels of the scope aregrounded on one side (the side with the ground tabon the banana-to-bnc connector), you need to makesure that their grounded sides both go to the piece ofwire between the resistor and the capacitor. Further-more, one output of the sine wave generator is nor-mally grounded, which would mess everything up:two different points in the circuit would be grounded,which would mean that there would be a short acrosssome of the circuit elements. To avoid this, loosenthe banana plug connectors on the sine wave genera-tor, and swing away the piece of metal that normallyconnects one of the output plugs to the ground.

Tune the sine wave generator’s frequency to reso-nance, and take the data you’ll need in order to de-termine x and y at a whole bunch of different placesover one cycle.

Some of the features of the digital oscilloscopes canmake the measurements a lot easier. Doing Acquire>Averagetells the scope to average together a series of up to128 measurements in order to reduce the amountof noise. Doing CH 1 MENU>Volts/Div>Fine al-lows you to scale the display arbitrarily. Rather thanreading voltages by eye from the scope’s x-y grid, youcan make the scope give you a measuring cursor. DoCursor>Type>Time. Use the top left knob to movethe cursor to different times. Doing Source>CH 1and Source>CH 2 gives you the voltage measure-ment for each channel. (Always use Cursor 1, neverCursor 2.)

The quality of the results can depend a lot on thequality of the connections. If the display on thescope changes noticeably when you wiggle the wires,you have a problem.

Analysis

Plot y versus x on a piece of graph paper. Let’sassume that the energy in a field depends on thefield’s strength raised to some power p. Conservationof energy then gives

|x|p + |y|p = 1 .

Use your graph to determine p, and interpret yourresult.

Prelab


P1 Sketch what your graph would look like forp = 0.1, p = 1, p = 2, and p = 10. (You shouldbe able to do p = 1 and p = 2 without any compu-tations. For p = 0.1 and p = 10, you can either runsome numbers on your calculator or use your math-ematical knowledge to sketch what they would turnout like.)

107

35 RC Circuits

Apparatus

oscilloscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/groupPasco PI-9587C function generator1/group unknown capacitor . . . . . . . . . . . . . . .1/groupknown capacitors, 0.05 µF . . . . . . . . . . . . . . . 1/groupresistors of various values

Goals

Observe the exponential curve of a dischargingcapacitor.

Determine the capacitance of an unknown ca-pacitor.

Introduction

God bless the struggling high school math teacher,but some of them seem to have a talent for mak-ing interesting and useful ideas seem dull and use-less. On certain topics such as the exponential func-tion, ex, the percentage of students who figure outfrom their teacher’s explanation what it really meansand why they should care approaches zero. That’sa shame, because there are so many cases where it’suseful. The graphs show just a few of the importantsituations in which this function shows up.

The credit card example is of the form

y = aet/k ,

while the Chernobyl graph is like

y = ae−t/k ,

In both cases, e is the constant 2.718 . . ., and k isa positive constant with units of time, referred toas the time constant. The first type of equation isreferred to as exponential growth, and the secondas exponential decay. The significance of k is thatit tells you how long it takes for y to change by afactor of e. For instance, an 18% interest rate onyour credit card converts to k = 6.0 years. Thatmeans that if your credit card balance is $1000 in1996, by 2002 it will be $2718, assuming you neverreally start paying down the principal.

An important fact about the exponential function isthat it never actually becomes zero — it only gets

closer and closer to zero. For instance, the radioac-tivity near Chernobyl will never ever become exactlyzero. After a while it will just get too small to poseany health risk, and at some later time it will get toosmall to measure with practical measuring devices.

Why is the exponential function so ubiquitous? Be-cause it occurs whenever a variable’s rate of change

108 Lab 35 RC Circuits

is proportional to the variable itself. In the creditcard and Chernobyl examples,

(rate of increase of credit card debt)

∝ (current credit card debt)

(rate of decrease of the number of radioactive atoms)

∝ (current number of radioactive atoms)

For the credit card, the proportionality occurs be-cause your interest payment is proportional to howmuch you currently owe. In the case of radioactivedecay, there is a proportionality because fewer re-maining atoms means fewer atoms available to de-cay and release radioactive particles. This line ofthought leads to an explanation of what’s so specialabout the constant e. If the rate of increase of a vari-able y is proportional to y, then the time constantk equals one over the proportionality constant, andthis is true only if the base of the exponential is e,not 10 or some other number.

Exponential growth or decay can occur in circuitscontaining resistors and capacitors. Resistors andcapacitors are the most common, inexpensive, andsimple electrical components. If you open up a cellphone or a stereo, the vast majority of the parts yousee inside are resistors and capacitors. Indeed, manyuseful circuits, known as RC circuits, can be builtout of nothing but resistors and capacitors. In thislab, you will study the exponential decay of the sim-plest possible RC circuit, shown below, consisting ofone resistor and one capacitor in series.

Suppose we initially charge up the capacitor, mak-ing an excess of positive charge on one plate and anexcess of negative on the other. Since a capacitorbehaves like V = Q/C, this creates a voltage dif-ference across the capacitor, and by Kirchoff’s looprule there must be a voltage drop of equal magni-tude across the resistor. By Ohm’s law, a currentI = V/R = Q/RC will flow through the resistor,and we have therefore established a proportionality,

(rate of decrease of charge on capacitor)

∝ (current charge on capacitor) .

It follows that the charge on the capacitor will decayexponentially. Furthermore, since the proportional-ity constant is 1/RC, we find that the time constant

of the decay equals the product of R and C. (It maynot be immediately obvious that Ohms times Faradsequals seconds, but it does.)

Note that even if we put the charge on the capac-itor very suddenly, the discharging process still oc-curs at the same rate, characterized by RC. ThusRC circuits can be used to filter out rapidly varyingelectrical signals while accepting more slowly varyingones. A classic example occurs in stereo speakers. Ifyou pull the front panel off of the wooden box thatwe refer to as ‘a speaker,’ you will find that thereare actually two speakers inside, a small one for re-producing high frequencies and a large one for thelow notes. The small one, called the tweeter, notonly cannot produce low frequencies but would ac-tually be damaged by attempting to accept them.It therefore has a capacitor wired in series with itsown resistance, forming an RC circuit that filtersout the low frequencies while permitting the highsto go through. This is known as a high-pass filter.A slightly different arrangement of resistors and in-ductors is used to make a low-pass filter to protectthe other speaker, the woofer, from high frequencies.

Observations

In typical filtering applications, the RC time con-stant is of the same order of magnitude as the pe-riod of a sound vibration, say ∼ 1 ms. It is thereforenecessary to observe the changing voltages with anoscilloscope rather than a multimeter. The oscillo-scope needs a repetitive signal, and it is not possi-ble for you to insert and remove a battery in thecircuit hundreds of times a second, so you will usea function generator to produce a voltage that be-comes positive and negative in a repetitive pattern.Such a wave pattern is known as a square wave. Themathematical discussion above referred to the expo-nential decay of the charge on the capacitor, but anoscilloscope actually measures voltage, not charge.As shown in the graphs below, the resulting volt-age patterns simply look like a chain of exponentialcurves strung together.

Make sure that the yellow or red ‘VAR’ knob, onthe front of the knob that selects the time scale, is

109

clicked into place, not in the range where it movesfreely — otherwise the times on the scope are notcalibrated.

A Preliminary observations

Pick a resistor and capacitor with a combined RCtime constant of ∼ 1 ms. Make sure the resistor isat least ∼ 10kΩ, so that the internal resistance ofthe function generator is negligible compared to theresistance you supply.

Note that the capacitance values printed on the sidesof capacitors often violate the normal SI conventionsabout prefixes. If just a number is given on the ca-pacitor with no units, the implied units are micro-farads, mF. Units of nF are avoided by the manufac-turers in favor of fractional microfarads, e.g., insteadof 1 nF, they would use ‘0.001,’ meaning 0.001 µF.For picofarads, a capital P is used, ‘PF,’ instead ofthe standard SI ‘pF.’

Use the oscilloscope to observe what happens to thevoltages across the resistor and capacitor as the func-tion generator’s voltage flips back and forth. Notethat the oscilloscope is simply a fancy voltmeter, soyou connect it to the circuit the same way you woulda voltmeter, in parallel with the component you’reinterested in. Make sure the scope is set on DC, notAC, by doing CH 1>Coupling>DC. A complicationis added by the fact that the scope and the func-tion generator are fussy about having the groundedsides of their circuits connected to each other. Thebanana-to-BNC converter that goes on the input ofthe scope has a small tab on one side marked ‘GND.’This side of the scope’s circuit must be connected tothe ‘LO’ terminal of the function generator. Thismeans that when you want to switch from measur-ing the capacitor’s voltage to measuring the resis-tor’s, you will need to rearrange the circuit a little.

If the trace on the oscilloscope does not look like theone shown above, it may be because the functiongenerator is flip-flopping too rapidly or too slowly.The function generator’s frequency has no effect onthe RC time constant, which is just a property of

the resistor and the capacitor.

If you think you have a working setup, observe theeffect of temporarily placing a second capacitor inparallel with the first capacitor. If your setup isworking, the exponential decay on the scope shouldbecome more gradual because you have increasedRC. If you don’t see any effect, it probably meansyou’re measuring behavior coming from the internalR and C of the function generator and the scope.

Use the scope to determine the RC time constant,and check that it is correct. Rather than readingtimes and voltages by eye from the scope’s x-y grid,you can make the scope give you a measuring cur-sor. Do Cursor>Type>Time, and Source>CH 1 .Use the top left knob to move the cursor to differenttimes.

B Unknown capacitor

Build a similar circuit using your unknown capacitorplus a known resistor. Use the unknown capacitorwith the same number as your group number. Takethe data you will need in order to determine the RCtime constant, and thus the unknown capacitance.

As a check on your result, obtain a known capacitorwith a value similar to the one you have determinedfor your unknown, and see if you get nearly the samecurve on the scope if you replace the unknown ca-pacitor with the new one.

Prelab


P1 Plan how you will determine the capacitanceand what data you will need to take.

Analysis

Determine the capacitance, with error analysis (ap-pendices 2 and 3).

110 Lab 35 RC Circuits

36 LRC Circuits

Apparatus

Heath coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/group0.05 µF capacitor . . . . . . . . . . . . . . . . . . . . . . . . . 2/groupPasco PI-9587C generator (under lab benches in 416)1/grouposcilloscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/group

Goals

Observe the resonant behavior of an LRC cir-cuit.

Observe how the properties of the resonancecurve change when the L, R, and C values arechanged.

Introduction

Radio, TV, cellular phones — it’s mind-bogglingto imagine the maelstrom of electromagnetic wavesthat are constantly pass through us and our sur-roundings. Perhaps equally surprising is the factthat a radio can pick up a wave with one partic-ular frequency while rejecting all the others nearlyperfectly. No seasoned cocktail-party veteran couldever be so successful at tuning out the signals thatare not of interest. What makes radio technologypossible is the phenomenon of resonance, the prop-erty of an electrical or mechanical system that makesit respond far more strongly to a driving force thatvaries at the same frequency as that at which the de-vice naturally vibrates. Just as an opera singer canonly break a wineglass by singing the right note, aradio can be tuned to respond strongly to electricalforces that oscillate at a particular frequency.

A simplified version of the circuit.

Circuit

As shown in the figure, the circuit consists of theHeath coil, a 0.05 µF capacitor, a 47-ohm resistor,

and the sine wave generator to supply a driving volt-age. You will study the way the circuit resonates,i.e., responds most strongly to a certain frequency.

Some added complications come from the fact thatthe function generator, coil, and oscilloscope do notbehave quite like their idealized versions. The coildoesn’t act like a pure inductor; it also has a certainamount of DC resistance, simply because the wirehas finite resistance. So in addition to the 47-ohmresistor, you will have 62 ohms of resistance comingfrom the resistance of the wire in the coil. There isalso some internal resistance from the function gen-erator itself, amounting to 600 ohms when you usethe outputs marked ‘high Ω.’ The ‘R’ of the circuitis really the sum of these three series resistances.

You will also want to put the oscilloscope in ACcoupling mode, which filters out any DC component(additive constant) on the signal. The scope accom-plishes this filtering by adding in a very small (20pF) capacitor, which appears in parallel in the cir-cuit because an oscilloscope, being a voltmeter, isalways used in parallel. In reality, this tiny parallelcapacitance is so small compared to capacitance ofthe 0.05 µF capacitor that the resulting correctionis negligible (and that’s a good thing, because if itwasn’t negligible, the circuit wouldn’t be a simpleseries LRC circuit, and its behavior would be muchmore complicated).

Observations

A Observation of Resonance

By connecting the oscilloscope to measure the volt-age across the resistor, you can determine the amountof power, P = V 2/R, being taken from the sinewavegenerator by the circuit and then dissipated asheat in the resistor. Make sure that your circuit ishooked up with the resistor connected to the groundedoutput of the amplifier, and hook up the oscilloscopeso its grounded connection is on the grounded sideof the resistor. As you change the frequency of thefunction generator, you should notice a very strongresponse in the circuit centered around one particu-lar frequency, the resonant frequency fo. (You couldmeasure the voltage drop across the capacitor or theinductor instead, but all the pictures of resonancecurves in your textbook are graphs of the behaviorof the resistor. The response curve of a capacitor or

112 Lab 36 LRC Circuits

inductor still has a peak at the resonant frequency,but looks very different off to the sides.)

The inductance of your solenoid is roughly 1 H basedon the approximation that it’s a long, skinny solenoid(which is not a great approximation here). Based onthis, estimate the resonant frequency of your circuit,

ωo =1√LC

.

Locate ωo accurately, and use it to determine theinductance of the Heath coil accurately.

B Effect of Changing C

Change the capacitance value by putting two capac-itors in parallel, and determine the new resonantfrequency. Check whether the resonant frequencychanges as predicted by theory. This is like tuningyour radio to a different frequency. For the rest ofthe lab, go back to your original value of C.

C The Width of the Resonance

The width of a resonance is customarily expressed asthe full width at half maximum, ∆f , defined as thedifference in frequency between the two points wherethe power dissipation is half of its maximum value.Determine the FWHM of your resonance. You aremeasuring voltage directly, not power, so you needto find the points where the amplitude of the voltageacross the resistor drops below its peak value by afactor of

√2.

D Effect of Changing R

Replace the resistor with a 2200-ohm resistor, andremeasure the FWHM. You should find that theFWHM has increased in proportion to the resistance.(Remember that your resistance always includes theresistance of the coil and the output side of the am-plifier.)

E Ringing

An LRC circuit will continue oscillating even whenthere is no oscillating driving force present. This

unforced behavior is known as ‘ringing.’Drive yourcircuit with a square wave. You can think of this asif you are giving the circuit repeated ‘kicks,’ so thatit will ring after each kick.

Choose a frequency many many times lower thanthe resonant frequency, so that the circuit will havetime to oscillate many times in between ‘kicks.’ Youshould observe an exponentially decaying sine wave.

The rapidity of the exponential decay depends onhow much resistance is in the circuit, since the re-sistor is the only component that gets rid of energypermanently. The rapidity of the decay is custom-arily measured with the quantity Q (for ‘quality’),defined as the number of oscillations required for thepotential energy in the circuit to drop by a factor of535 (the obscure numerical factor being e2π). Forour purposes, it will be more convenient to extractQ from the equation

Vpeak,i = Vpeak,0 · exp

[

− πt

QT

]

where T is the period of the sine wave, Vpeak,0 is thevoltage across the resistor at the peak that we useto define t = 0, and Vpeak,i is the voltage of a laterpeak, occurring at time t.

Collect the data you will need in order to determinethe Q of the circuit, and then do the same for theother resistance value.

F The Resonance Curve

Going back to your low-resistance setup, collect volt-age data over a wide range of frequencies, coveringat least a factor of 10 above and below the resonantfrequency. You will want to take closely spaced datanear the resonance peak, where the voltage is chang-ing rapidly, and less closely spaced points elsewhere.Far above and far below the resonance, it will be con-venient just to take data at frequencies that changeby successive factors of two.

(At very high frequencies, above 104 Hz or so, youmay find that rather than continuing to drop off, theresponse curve comes back up again. I believe thatthis effect arises from nonideal behavior of the coil athigh frequencies: there is stray capacitance betweenone loop and the next, and this capacitance acts likeit is in parallel with the coil.)

In engineering work, it is useful to create a graph ofthe resonance curve in which the y axis is in decibels,

113

db = 10 log10

(

P

Pmax

)

= 20 log10

(

V

Vmax

)

,

and the x axis is a logarithmic frequency scale. (Onthis graph, the FWHM is the width of the curve at 3db below the peak.) You will construct such a graphfrom your data.

Analysis

Check whether the resonant frequency changed bythe correct factor when you changed the capacitance.

For both versions of the circuit, compare the FWHMof the resonance and the circuit’s Q to the theoreticalequations

∆ω =R

L

and

Q =ωo

∆ω.

Note that there are a total of three resistances inseries: the 62-ohm resistance of the coil, the 47-ohm resistor, and the ∼ 50-ohm resistance of thesine-wave generator’s output. No error analysis isrequired, since the main errors are systematic onesintroduced by the nonideal behavior of the coil andthe difficulty of determining an exact, fixed value forthe internal resistance of the output of the amplifier.

Graph the resonance curve — you can probably saveyourself a great deal of time by using a computer todo the calculations and graphing. To do the calcula-tions, you can go to my web page, www.lightandmatter.com. Go to the lab manual’s web page, and then click on‘data-analysis tool for the LRC circuits lab’. Onceyour data are ready to graph, I suggest using com-puter software to make your graph (see Appendix4).

On the high-frequency end, the impedance is dom-inated by the impedance of the inductor, which isproportional to frequency. Doubling the frequencydoubles the impedance, thereby cutting the currentby a factor of two and the power dissipated in the re-sistor by a factor of 4, which is 6.02 db. Since a factorof 2 in frequency corresponds in musical terms to oneoctave, this is referred to as a 6 db/octave roll-off.Check this prediction against your data. You should

also find a 6 db/octave slope in the limit of low fre-quencies — here the impedance is dominated by thecapacitor, but the idea is similar. (More complex fil-tering circuits can achieve roll-offs more drastic than6 db/octave.)

Prelab


P1 Using the rough value of L given in the labmanual, compute a preliminary estimate of the an-gular frequency ωo, and find the corresponding fre-quency fo.

P2 Express Q in terms of L, R, and C.

P3 Show that your answer to P2 has the rightunits.

P4 Using the rough value of L given in the labmanual, plug numbers into your answer to P2, andmake a preliminary estimate of the Q that you ex-pect when using the lower of the two resistance val-ues. Your result should come out to be 6 (to one sigfig of precision).

P5 In part D, you could measure t and T usingthe time scale on the scope. However, all we careabout is their ratio t/T ; think of a technique fordetermining t/T that is both more precise and easierto carry out.

114 Lab 36 LRC Circuits

37 Faraday’s Law

Apparatus

function generator . . . . . . . . . . . . . . . . . . . . . . . . 1/groupsolenoid (Heath) 1/group plus a few moreoscilloscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/group10-ohm power resistor . . . . . . . . . . . . . . . . . . . . 1/group4-meter wire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/grouppalm-sized pieces of iron or steelmasking taperulers

Goals

Observe electric fields induced by changing mag-netic fields.

Test Faraday’s law.

Introduction

Physicists hate complication, and when physicist Mi-chael Faraday was first learning physics in the early19th century, an embarrassingly complex aspect ofthe science was the multiplicity of types of forces.Friction, normal forces, gravity, electric forces, mag-netic forces, surface tension — the list went on andon. Today, 200 years later, ask a physicist to enu-merate the fundamental forces of nature and themost likely response will be ‘four: gravity, electro-magnetism, the strong nuclear force and the weaknuclear force.’ Part of the simplification came fromthe study of matter at the atomic level, which showedthat apparently unrelated forces such as friction, nor-mal forces, and surface tension were all manifesta-tions of electrical forces among atoms. The otherbig simplification came from Faraday’s experimentalwork showing that electric and magnetic forces wereintimately related in previously unexpected ways, sointimately related in fact that we now refer to thetwo sets of force-phenomena under a single term,‘electromagnetism.’

Even before Faraday, Oersted had shown that therewas at least some relationship between electric andmagnetic forces. An electrical current creates a mag-netic field, and magnetic fields exert forces on anelectrical current. In other words, electric forcesare forces of charges acting on charges, and mag-netic forces are forces of moving charges on moving

charges. (Even the magnetic field of a bar magnet isdue to currents, the currents created by the orbitingelectrons in its atoms.)

Faraday took Oersted’s work a step further, andshowed that the relationship between electricity andmagnetism was even deeper. He showed that a chang-ing electric field produces a magnetic field, and achanging magnetic field produces an electric field.Faraday’s law,

ΓE = −dΦB/dt

relates the circulation of the electric field around aclosed loop to the rate of change of the magneticflux through the loop. It forms the basis for suchtechnologies as the transformer, the electric guitar,the amplifier, and generator, and the electric motor.

Observations

A Qualitative Observations

To observe Faraday’s law in action you will first needto produce a varying magnetic field. You can do thisby using a function generator to produce a currentin a solenoid that that varies like a sine wave as afunction of time. The solenoid’s magnetic field willthus also vary sinusoidally.

The emf in Faraday’s law can be observed around aloop of wire positioned inside or close to the solenoid.To make the emf larger and easier to see on an os-cilloscope, you will use 5-10 loops, which multipliesthe flux by that number of loops.

The only remaining complication is that the rate ofchange of the magnetic flux, dΦB/dt, is determinedby the rate of change of the magnetic field, whichrelates to the rate of change of the current throughthe solenoid, dI/dt. The oscilloscope, however, mea-sures voltage, not current. You might think thatyou could simply observe the voltage being suppliedto the solenoid and divide by the solenoid’s 62-ohmresistance to find the current through the solenoid.This will not work, however, because Faraday’s lawproduces not only an emf in the loops of wire but alsoan emf in the solenoid that produced the magneticfield in the first place. The current in the solenoid isbeing driven not just by the emf from the functiongenerator but also by this ‘self-induced’ emf. Eventhough the solenoid is just a long piece of wire, it

116 Lab 37 Faraday’s Law

does not obey Ohm’s law under these conditions.To get around this difficulty, you can insert the 10-ohm power resistor in the circuit in series with thefunction generator and the solenoid. (A power re-sistor is simply a resistor that can dissipate a largeamount of power without burning up.) The powerresistor does obey Ohm’s law, so by using the scopeto observe the voltage drop across it you can inferthe current flowing through it, which is the same asthe current flowing through the solenoid.

Create the solenoid circuit, and hook up one channelof the scope to observe the voltage drop across thepower resistor. A sine wave with a frequency on theorder of 1 kHz will work.

Wind the 2-m wire into circular loops small enoughto fit inside the solenoid, and hook it up to the otherchannel of the scope.

As always, you need to watch out for ground loops.The output of the function generator has one of itsterminals grounded, so that ground and the groundedside of the scope’s input have to be at the same placein the circuit.

The signals tend to be fairly noise. You can cleanthem up a little by having the scope average over aseries of traces. To turn on averaging, do Acquire>Average>128.To turn it back off, press Sample.

First try putting the loops at the mouth of the solenoid,and observe the emf induced in them. Observe whathappens when you flip the loops over. You will ob-serve that the two sine waves on the scope are out ofphase with each other. Sketch the phase relationshipin your notebook, and make sure you understand interms of Faraday’s law why it is the way it is, i.e.,why the induced emf has the greatest value at a cer-tain point, why it is zero at a certain point, etc.

Observe the induced emf at with the loops at severalother positions such as those shown in the figure.

Make sure you understand in the resulting variationsof the strength of the emf in terms of Faraday’s law.

B A Metal Detector

Obtain one of the spare solenoids so that you havetwo of them. Substitute it for the loops of wire, sothat you can observe the emf induced in the secondsolenoid by the first solenoid. If you put the twosolenoids close together with their mouths a few cmapart and then insert a piece of iron or steel betweenthem, you should be able to see a small increase inthe induced emf. The iron distorts the magnetic fieldpattern produced by the first solenoid, channelingmore of the field lines through the second solenoid.

C Quantitative Observations

This part of the lab is a quantitative test of Fara-day’s law. Going back to the setup for part A, mea-sure the amplitude (peak-to-peak height) of the volt-age across the power resistor. Choose a positionfor the loops of wire that you think will make itas easy as possible to calculate dΦB/dt accuratelybased on knowledge of the variation of the currentin the solenoid as a function of time. Put the loopsin that position, and measure the amplitude of theinduced emf. Repeat these measurements with a fre-quency that is different by a factor of two.

Self-Check

Before leaving, analyze your results from part C andmake sure you get reasonable agreement with Fara-day’s law.

Analysis

Describe your observations in parts A and B andinterpret them in terms of Faraday’s law.

Compare your observations in part C quantitativelywith Faraday’s law. The solenoid isn’t very long, sothe approximate expression for the interior field of along solenoid isn’t very accurate here. To correct forthat, multiply the expression for the field by (cosβ+cos γ)/2, which you derived in homework problem11-30 in Simple Nature, where β and γ are anglesbetween the axis and the lines connecting the pointof interest to the edges of the solenoid’s mouths.

Prelab

The point of the prelab questions is to make sureyou understand what you’re doing, why you’re do-

117

ing it, and how to avoid some common mistakes. Ifyou don’t know the answers, make sure to come tomy office hours before lab and get help! Otherwiseyou’re just setting yourself up for failure in lab.

P1 Plan what raw data you’ll need to collect forpart C, and figure out the equation you’ll use totest whether your observations are consistent withFaraday’s law.

118 Lab 37 Faraday’s Law

38 Electromagnetism

Apparatus

solenoid (Heath) . . . . . . . . . . . . . . . . . . . . . . . . . . 1/grouposcilloscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/group2-meter wire with banana plugs . . . . . . . . . . .1/groupmagnet (stack of 6 Nd) . . . . . . . . . . . . . . . . . . . 1/groupmasking tapestring

Goals

Observe electric fields induced by changing mag-netic fields.

Build a generator.

Discover Lenz’s law.

Introduction

Physicists hate complication, and when physicist Mich-ael Faraday was first learning physics in the early19th century, an embarrassingly complex aspect ofthe science was the multiplicity of types of forces.Friction, normal forces, gravity, electric forces, mag-netic forces, surface tension — the list went on andon. Today, 200 years later, ask a physicist to enu-merate the fundamental forces of nature and themost likely response will be ‘four: gravity, electro-magnetism, the strong nuclear force and the weaknuclear force.’ Part of the simplification came fromthe study of matter at the atomic level, which showedthat apparently unrelated forces such as friction, nor-mal forces, and surface tension were all manifesta-tions of electrical forces among atoms. The otherbig simplification came from Faraday’s experimentalwork showing that electric and magnetic forces wereintimately related in previously unexpected ways, sointimately related in fact that we now refer to thetwo sets of force-phenomena under a single term,‘electromagnetism.’

Even before Faraday, Oersted had shown that therewas at least some relationship between electric andmagnetic forces. An electrical current creates a mag-netic field, and magnetic fields exert forces on anelectrical current. In other words, electric forcesare forces of charges acting on charges, and mag-netic forces are forces of moving charges on moving

charges. (Even the magnetic field of a bar magnet isdue to currents, the currents created by the orbitingelectrons in its atoms.)

Faraday took Oersted’s work a step further, andshowed that the relationship between electricity andmagnetism was even deeper. He showed that a chang-ing electric field produces a magnetic field, and achanging magnetic field produces an electric field.Faraday’s work forms the basis for such technologiesas the transformer, the electric guitar, the trans-former, and generator, and the electric motor. Italso led to the understanding of light as an electro-magnetic wave.


In this lab you will use a permanent magnet to pro-duce changing magnetic fields. This causes an elec-tric field to be induced, which you will detect usinga solenoid (spool of wire) connected to an oscillo-scope. The electric field drives electrons around thesolenoid, producing a current which is detected bythe oscilloscope. If you haven’t used an oscilloscopebefore, your instructor will help you to get started.It’s simply a device for graphing a measured voltageas a function of time.

A A constant magnetic field

Do you detect any signal on the oscilloscope whenthe magnet is simply placed at rest inside the solenoid?Try the most sensitive voltage scale.

B A changing magnetic field

Do you detect any signal when you move the magnetor wiggle it inside the solenoid or near it? Whathappens if you change the speed at which you movethe magnet?

C Moving the solenoid

What happens if you hold the magnet still and movethe solenoid?

The poles of the magnet are its flat faces. In laterparts of the lab you will need to know which is north.Determine this now by hanging it from a string andseeing how it aligns itself with the Earth’s field. Thepole that points north is called the north pole of themagnet. The field pattern funnels into the body ofthe magnet through its south pole, and reemerges at

120 Lab 38 Electromagnetism

its north pole.

D A generator

Tape the magnet securely to the eraser end of a pen-cil so that its flat face (one of its two poles) is like thehead of a hammer, and mark the north and southpoles of the magnet for later reference. Spin the pen-cil near the solenoid and observe the induced signal.You have built a generator. (I have unfortunatelynot had any luck lighting a lightbulb with the setup,due to the relatively high internal resistance of thesolenoid.)

Trying Out Your Understanding

E Changing the speed of the generator

If you change the speed at which you spin the pencil,you will of course cause the induced signal to have alonger or shorter period. Does it also have any effecton the amplitude of the wave?

F A solenoid with fewer loops

Use the two-meter cable to make a second solenoidwith the same diameter but fewer loops. Comparethe strength of the induced signals.

G Dependence on distance

How does the signal picked up by your generatorchange with distance?

Try to explain what you have observed, and discussyour interpretations with your instructor.

Lenz’s Law

Lenz’s law describes how the clockwise or counter-clockwise direction of the induced electric field’s whirlpoolpattern relates to the changing magnetic field. Themain result of this lab is a determination of howLenz’s law works. To focus your reasoning, here arefour possible forms for Lenz’s law:

1. The electric field forms a pattern that is clockwisewhen viewed along the direction of the B vector ofthe changing magnetic field.

2. The electric field forms a pattern that is counter-clockwise when viewed along the direction of the Bvector of the changing magnetic field.

3. The electric field forms a pattern that is clockwisewhen viewed along the direction of the ∆B vector ofthe changing magnetic field.

4. The electric field forms a pattern that is coun-

terclockwise when viewed along the direction of the∆B vector of the changing magnetic field.

Your job is to figure out which is correct.

The most direct way to figure out Lenz’s law is tochopping motion that ends up with the magnet inthe solenoid, observing whether the pulse inducedis positive or negative. What happens when youreverse the chopping motion, or when you reversethe north and south poles of the magnet? Try allfour possible combinations and record your results.

To set up the scope, press DEFAULT SETUP. Thisshould have the effect of setting the scope on DCcoupling, which is what you want. (If it’s on AC cou-pling, it tries to filter out any DC part of the inputsignals, which distorts the results.) To check thatyou’re on DC coupling, you can do CH 1 MENU,and check that Coupling says DC.

Make sure the scope is on DC coupling, not AC cou-pling, or your pulses will be distorted.

It can be tricky to make the connection between thepolarity of the signal on the screen of the oscilloscopeand the direction of the electric field pattern. Thefigure shows an example of how to interpret a posi-tive pulse: the current must have flowed through thescope from the center conductor of the coax cable toits outer conductor (marked GND on the coax-to-banana converter).

Prelab


P1 The time-scale for all the signals is determined

121

by the fact that you’re wiggling and waving the mag-net by hand, so what’s a reasonable order of magni-tude to choose for the time base on the oscilloscope?

Self-Check

Determine which version of Lenz’s law is correct.

122 Lab 38 Electromagnetism

39 Impedance

Observe how the impedances of capacitors andinductors change with frequency.

Observe how impedances combine according tothe arithmetic of complex numbers.

Setup

We’ll start by observing the impedance of a capaci-tor. Ideally, what we want is this:

However, we want to know not just the amplitudeof the voltage and current sine-waves but the phaserelationship between them as well, which we can’tget from a regular meter. We need to use an oscil-loscope, and oscilloscopes only measure voltage, notcurrent. This leads us to something like the follow-ing setup:

Here ch. 2 tells us the voltage across the resistor,which is related to the current in the resistor accord-ing to Ohm’s law. By the junction rule, the currentin the resistor is the same as the current through thecapacitor.

But even now, we’re not out of the woods. In thissetup, the ground of ch. 2 is connected to the samewire as the active (+) connection to ch. 1, which

would cause ch. 1 to read zero, and would shortacross the capacitor as well. Instead, we need this:

Now both GND connections are going to the samepoint in the circuit. Because we’ve swapped the con-nections to ch. 1, its trace will be upside-down, andinconsistent with ch. 2. There is a special control onthe scope for inverting ch. 2, which makes the twochannels consistent again.

Observations

A Impedance of the capacitor

Hook up the circuit as shown, using a 1 kΩ resistanceand a 0.2 µF capacitance. The HP signal genera-tor has a ground strap connecting one of its outputterminals to ground. Disconnect this ground strap,since grounding either side of the signal generatorwould mean that either the resistor or the capacitorwould be connected to ground on both sides. Try afrequency of 100 Hz.

Observe the phase relationship between VC , on ch.1, and the signal on ch. 2, which essentially tellsus the current IC except for a factor of 1/R. Sketchthis phase relationship in your raw data. BecauseVC = q/C and I = dq/dt, the current through thecapacitor should be proportional to dV/dt. Basedon the phase relationship you observed, does thisseem to be true?

Measure the phase angle numerically from the oscil-loscope. Is it what you expect?

Determine the magnitude of the capacitor’s impedance.

124 Lab 39 Impedance

Suppose you represent the signal that is ahead inphase using a point that is more counterclockwisein the complex plane. Sketch the locations of thevoltage and current in the complex plane. (You canarbitrarily choose one of them to be along the realaxis if you like.) Where would the impedance thenlie in the plane?

Now change the frequency to 1000 Hz, and see whatchanges. Sketch your new impedance in the com-plex plane. Do you find the expected relationshipbetween impedance and frequency?

B Inductance of the Heath coil

Make the measurements you need in order to calcu-late the theoretical inductance of the inductor, usingthe equation derived in the prelab. The approxima-tion may be off by as much as a factor of two, sincethe solenoid isn’t long and skinny, but it’s useful soyou have some idea of what to expect.

C Impedance of the inductor

Now repeat all the above steps using the Heath coilas an inductor.

D Impedances in series

Put the capacitor and inductor in series, and collectthe data you’ll need in order to determine their com-bined impedance at several frequencies ranging from100 to 1000 Hz.

Analysis

Use your data from part C to determine an experi-mental value of the coil’s inductance, and comparewith the theoretical result based on your measure-ments in part B.

Graph the theoretical and experimental impedanceof the series combination in part D, overlaying themon the same graph. Show theory as a curve and ex-periment as discrete data-points. Do the same kindof graph for the parallel combination.

125

40 Refraction and Images

Apparatus

rectangular block of plastic (20x10x5 cm,from blackboard optics kit), or plastic box with wa-ter in itlaserspiral plastic tube and fiber optic cable for demon-strating total internal reflectionrulerprotractorbutcher paper

Goals

Observe the phenomena of refraction and totalinternal reflection.

Locate a virtual image in a plastic block byray tracing, and compare with the theoreticallypredicted position of the image.

Introduction

Without the phenomenon of refraction, the lens ofyour eye could not focus light on your retina, and youwould not be able to see. Refraction is the bending ofrays of light that occurs when they pass through theboundary between two media in which the speed oflight is different. Light entering your eye passes fromair, in which the speed of light is 3.0× 108 m/s, intothe watery tissues of your eye, in which it is about2.2 × 108 m/s. Since it is inconvenient to write orsay the speed of light in a particular medium, weusually speak in terms of the index of refraction, n,defined by

n = c/v,

where c is the speed of light in a vacuum, and v isthe speed of light in the medium in question. Thus,vacuum has n = 1 by definition. Air, which is notvery dense, does not slow light down very much, soit has an index of refraction very close to 1. Waterhas an index of refraction of about 1.3, meaning thatlight moves more slowly in water by a factor of 1/1.3.

Refraction, the bending of light, occurs for the fol-lowing reason. Imagine, for example, a beam of lightentering a swimming pool at an angle. Because ofthe angle, one side of the beam hits the water first,

and is slowed down. The other side of the beam,however, gets to travel in air, at its faster speed, forlonger, because it enters the water later — by thetime it enters the water, the other side of the beamhas been limping along through the water for a littlewhile, and has not gotten as far. The wavefront istherefore twisted around a little, in the same waythat a marching band turns by having the people onone side take smaller steps.

Quantitatively, the amount of bending is given bySnell’s law:

ni sin θi = nt sin θt,

where the index i refers to the incident light and in-cident medium, and t refers to the transmitted lightand the transmitting medium. Note that the an-gles are defined with respect to the normal, i.e., theimaginary line perpendicular to the boundary.

Also, not all of the light is transmitted. Some is re-flected — the amount depends on the angles. In fact,for certain values of ni, nt, and θi, there is no valueof θt that will obey Snell’s law (sin θt would haveto be greater than one). In such a situation, 100%of the light must be reflected. This phenomenon isknown as total internal reflection. The word inter-nal is used because the phenomenon only occurs forni > nt. If one medium is air and the other is plasticor glass, then this can only happen when the incidentlight is in the plastic or glass, i.e., the light is try-ing to escape but can’t. Total internal reflection isused to good advantage in fiber-optic cables used totransmit long-distance phone calls or data on the in-ternet — light traveling down the cable cannot leakout, assuming it is initially aimed at an angle closeenough to the axis of the cable.

Although most of the practical applications of thephenomenon of refraction involve lenses, which havecurved shapes, in this lab you will be dealing almost

126 Lab 40 Refraction and Images

exclusively with flat surfaces.

Preliminaries

Check whether your laser’s beam seems to be roughlyparallel.

Observations

A Index of refraction of plastic

Make the measurements you have planned in orderto determine the index of refraction of the plasticblock (or the water, whichever you have). The laserand the block of plastic can simply be laid flat on thetable. Make sure that the laser is pointing towardsthe wall.

B Total internal reflection

Try shining the laser into one end of the spiral-shaped plastic rod. If you aim it nearly along theaxis of the cable, none will leak out, and if you putyour hand in front of the other end of the rod, youwill see the light coming out the other end. (It willnot be a well-collimated beam any more because thebeam is spread out and distorted when it undergoesthe many reflections on the rough and curved insidethe rod.)

There’s no data to take. The point of having this aspart of the lab is simply that it’s hard to demonstrateto a whole class all at once.

C A virtual image

Pick up the block, and have your partner look side-ways through it at your finger, touching the sur-face of the block. Have your partner hold her ownfinger next to the block, and move it around un-til it appears to be as far away as your own finger.Her brain achieves a perception of depth by subcon-sciously comparing the images it receives from hertwo eyes. Your partner doesn’t actually need to beable to see her own finger, because her brain knowshow to position her arm at a certain point in space.Measure the distance di, which is the depth of theimage of your finger relative to the front of the block.

Now trace the outline of the block on a piece of pa-per, remove the block, mark the location of the im-age, and put the block back on the paper. Shinethe laser at the point where your finger was origi-nally touching the block, observe the refracted beam,and draw it in. Repeat this whole procedure severaltimes, with the laser at a variety of angles. Finally,

extrapolate the rays leaving the block back into theblock. They should all appear to have come from thesame point, where you saw the virtual image. You’llneed to photocopy the tracing so that each personcan turn in a copy with his or her writeup.

Prelab

The point of the prelab questions is to make sureyou understand what you’re doing, why you’re do-ing it, and how to avoid some common mistakes. Ifyou don’t know the answers, make sure to come tomy office hours before lab and get help! Otherwise

127

you’re just setting yourself up for failure in lab.


P1 Laser beams are supposed to be very nearlyparallel (not spreading out or contracting to a focalpoint). Think of a way to test, roughly, whether thisis true for your laser.

P2 Plan how you will determine the index of re-fraction in part A.

P3 You have complete freedom to choose any in-cident angle you like in part A. Discuss what choicewould give the highest possible precision for the mea-surement of the index of refraction.

Analysis

Using your data for part A, extract the index of re-fraction. Estimate the accuracy of your raw data,and determine error bars for your index of refrac-tion.

Using trigonometry and Snell’s law, make a the-oretical calculation of di. You’ll need to use thesmall-angle approximation sin θ ≈ tan θ ≈ θ, for θmeasured in units of radians. (For large angles, i.e.viewing the finger from way off to one side, the rayswill not converge very closely to form a clear virtualimage.)

Explain your results in part C and their meaning.

Compare your three values for di : the experimentalvalue based on depth perception, the experimentalvalue found by ray-tracing with the laser, and thetheoretical value found by trigonometry.

128 Lab 40 Refraction and Images

41 Geometric Optics

Apparatus

optical bench . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/groupconverging lens (unknown focallength to be measured) . . . . . . . . . . . . . . . . . . . 1/groupconverging lens, longest availablefocal length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1/groupconverging lens, f = 50 mm1/group lamp and arrow-shaped mask . . . . 1/groupfrosted glass screen . . . . . . . . . . . . . . . . . . . . . . . 1/group

Goals

Observe a real image formed by a convex lens,and determine its focal length.

Construct a telescope and measure its angularmagnification.

Introduction

The credit for invention of the telescope is disputed,but Galileo was probably the first person to use onefor astronomy. He first heard of the new inventionwhen a foreigner visited the court of his royal pa-trons and attempted to sell it for an exorbitant price.Hearing through second-hand reports that it con-sisted of two lenses, Galileo sent an urgent messageto his benefactors not to buy it, and proceeded toreproduce the device himself. An early advocate ofsimple scientific terminology, he wanted the instru-ment to be called the ‘occhialini,’ Italian for ‘eye-thing,’ rather than the Greek ‘telescope.’

His astronomical observations soon poked some gap-ing holes in the accepted Aristotelian view of theheavens. Contrary to Aristotle’s assertion that theheavenly bodies were perfect and without blemishes,he found that the moon had mountains and the sunhad spots (the marks on the moon visible to thenaked eye had been explained as optical illusions oratmospheric phenomena). This put the heavens onan equal footing with earthly objects, paving theway for physical theories that would apply to thewhole universe, and specifically for Newton’s law ofgravity. He also discovered the four largest moonsof Jupiter, and demonstrated his political savvy bynaming them the ‘Medicean satellites’ after the pow-erful Medici family. The fact that they revolved

around Jupiter rather than the earth helped makemore plausible Copernicus’ theory that the planetsdid not revolve around the earth but around the sun.Galileo’s ideas were considered subversive, and manypeople refused to look through his telescope, eitherbecause they thought it was an illusion or simplybecause it was supposed to show things that werecontrary to Aristotle.

The figure on the next page shows the simplest re-fracting telescope. The object is assumed to be atinfinity, so a real image is formed at a distance fromthe objective lens equal to its focal length, fo. Bysetting up the eyepiece at a distance from the imageequal to its own focal length, fE , light rays that wereparallel are again made parallel.

The point of the whole arrangement is angular mag-nification. The small angle θ1 is converted to a largeθ2. It is the small angular size of distant objects thatmakes them hard to see, not their distance. There isno way to tell visually whether an object is a thirtymeters away or thirty billion. (For objects within afew meters, your brain-eye system gives you a senseof depth based on parallax.) The Pleiades star clus-ter can be seen more easily across many light yearsthan Mick Jagger’s aging lips across a stadium. Peo-ple who say the flying saucer ‘looked as big as anaircraft carrier’ or that the moon ‘looks as big as ahouse’ don’t know what they’re talking about. Thetelescope does not make things ‘seem closer’ — sincethe rays coming at your eye are parallel, the finalvirtual image you see is at infinity. The angularmagnification is given by

MA = θ2/θ1

(to be measured directly in this lab)

MA = fo/fE

(theory)

Observations

A Focal length of a convex lens

Use your unknown convex lens to project a real im-age on the frosted glass screen. For your object, usethe lamp with the arrow-shaped aperture in front ofit. Make sure to lock down the parts on the opti-cal bench, or else they may tip over and break theoptics!

130 Lab 41 Geometric Optics

B The telescope

Use your optical bench and your two known lensesto build a telescope. Since the telescope is a devicefor viewing objects at infinity, you’ll want to take itoutside.

The best method for determining the angular magni-fication is to observe the same object with both eyesopen, with one eye looking through the telescope andone seeing the object without the telescope. Goodprecision can be obtained, for example, by looking ata large object like a coke machine, and determiningthat a small part of it, whose size you can measurewith a ruler, appears, when magnified, to cover somelarger part of it, which you can also measure.

Your brain is not capable of focusing one eye at onedistance, and the other at another distance. There-fore it’s important to get your telescope adjustedprecisely so that the image is at infinity. You can dothis by focusing your naked eye on a distant object,and then moving the objective until the image popsinto focus in the other eye. Theoretically this wouldbe accomplished simply by setting the lenses at thedistance shown in the diagram, but in reality, a smallamount of further adjustment is necessary, perhapsbecause the quality control on the focal lengths ofthe lenses is not perfect.

Prelab



P1 In part A, do you want the object to be closerto the lens than the lens’ focal length, exactly at adistance of one focal length, or farther than the focallength? What about the screen?

P2 Plan what measurements you will make in partA and how you will use them to determine the lens’focal length.

P3 It’s disappointing to construct a telescope witha very small magnification. Given a selection oflenses, plan how you can make a telescope with the

greatest possible magnification.

Analysis

Determine the focal length of the unknown lens, witherror bars.

Find the angular magnification of your telescope fromyour data, with error bars, and compare with the-ory. Do they agree to within the accuracy of themeasurement?

131

A refracting telescope

132 Lab 41 Geometric Optics

42 Two-Source Interference

Apparatus

ripple tank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/groupyellow foam pads . . . . . . . . . . . . . . . . . . . . . . . . . 4/grouplamp and unfrosted straight-filament bulb1/group wave generator . . . . . . . . . . . . . . . . . . .1/groupbig metal L-shaped arms for hangingthe wave generator . . . . . . . . . . . . . . . . . . . . . . . 1/grouplittle metal L-shaped arms with yellowplastic balls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2/grouprubber bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2/groupwhite plastic screen . . . . . . . . . . . . . . . . . . . . . . . 1/groupThornton DC voltage source . . . . . . . . . . . . . . 1/groupsmall rubber stopper . . . . . . . . . . . . . . . . . . . . . 1/grouppower strip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/groupbucket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/groupmop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1flathead screwdriver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1rulers and protractorskimwipes and alcohol for cleaningbutcher paper

Goals

Observe how a 2-source interference pattern ofwater waves depends on the distance betweenthe sources.

Observations

Light is really made of waves, not rays, so when wetreated it as rays, we were making an approximation.You might think that when the time came to treatlight as a wave, things would get very difficult, andit would be hard to predict or understand anythingwithout doing complicated calculations.

Life isn’t that bad. It turns out that all of the most

important ideas about light as a wave can be seenin one simple experiment, shown in the first figure.1

A wave comes up from the bottom of the page, andencounters a wall with two slits chopped out of it.The result is a fan pattern, with strong wave motioncoming out along directions like X and Z, but novibration of the water at all along lines like Y. Thereason for this pattern is shown in the second figure.The two parts of the wave that get through the slitscreate an overlapping pattern of ripples. To get toa point on line X, both waves have to go the samedistance, so they’re in step with each other, and re-inforce. But at a point on line Y, due to the unequaldistances involved, one wave is going up while theother wave is going down, so there is cancellation.The angular spacing of the fan pattern depends onboth the wavelength of the waves, λ, and the dis-tance between the slits, d.

The ripple tank is tank that sits about 30 cm abovethe floor. You put a little water in the tank, andproduce waves. There is a lamp above it that makesa point-like source of light, and the waves cast pat-terns of light on a screen placed on the floor. Thepatterns of light on the screen are easier to see andmeasure than the ripples themselves.

In reality, it’s not very convenient to produce a double-slit diffraction pattern exactly as depicted in the firstfigure, because the waves beyond the slits are soweak that they are difficult to observe clearly. In-stead, you’ll simply produce synchronized circularripples from two sources driven by a motor.

Put the tank on the floor. Plug the hole in the side ofthe tank with the black rubber stopper. If the plasticis dirty, clean it off with alcohol and kimwipes. Wetthe four yellow foam pads, and place them aroundthe sides of the tank. Pour in water to a depth ofabout 5-7 mm. Adjust the metal feet to level thetank, so that the water is of equal depth throughout

1The photo is from the textbook PSSC physics, which hasa blanket permission for free use after 1970.

134 Lab 42 Two-Source Interference

the tank. (Do not rotate the wooden legs them-selves, just the feet.) If too many bubbles form onthe plastic, wipe them off with a ruler.

Make sure the straight-filament bulb in the lightsource is rotated so that when you look in throughthe hole, you are looking along the length of the fil-ament. This way the lamp acts like a point sourceof light above the tank. To test that it’s orientedcorrectly, check that you can cast a perfectly sharpimage of the tip of a pen.

Clamp the light source to the post and turn it on.Put the white plastic screen on the floor under thetank. If you make ripples in the water, you shouldbe able to see the wave pattern on the screen.

The wave generator consists of a piece of wood thathangs by rubber bands from the two L-shaped metalhangers. There is a DC motor attached, which spinsan intentionally unbalanced wheel, resulting in vi-bration of the wood. The wood itself can be usedto make straight waves directly in the water, butin this experiment you’ll be using the two little L-shaped pieces of metal with the yellow balls on theend to make two sources of circular ripples. The DCmotor runs off of the DC voltage source, and themore voltage you supply, the faster the motor runs.

Start just by sticking one little L-shaped arm in thepiece of wood, and observing the circular wave pat-tern it makes. Now try two sources at once, in neigh-boring holes. Pick a speed (frequency) for the motorthat you’ll use throughout the experiment — a fairlylow speed works well. Measure the angular spacingof the resulting diffraction pattern for several valuesof the spacing, d, between the two sources of ripples.

How do you think the angular spacing of the wavepattern seems to depend mathematically on d? Con-struct a graph to test whether this was really true.If you’re not sure what mathematical rule to guess,you can use the methods explained in Appendix 5and look for any kind of a power law relationship.

135

43 Wave Optics

Apparatus

helium-neon laser1/group optical bench with posts & holders 1/grouphigh-precision double slits . . . . . . . . . . . . . . . . 1/grouprulersmeter stickstape measuresbutcher paper

Goals

Observe evidence for the wave nature of light.

Determine the wavelength of the red light emit-ted by your laser, by measuring a double-slitdiffraction pattern. (The part of the spectrumthat appears red to the human eye covers quitea large range of wavelengths. A given type oflaser, e.g., He-Ne or solid-state, will produceone very specific wavelength.)

Determine the approximate diameter of a hu-man hair, using its diffraction pattern.

Introduction

Isaac Newton’s epitaph, written by Alexander Pope,reads:

Nature and Nature’s laws lay hid in night.

God said let Newton be, and all was light.

Notwithstanding Newton’s stature as the greatestphysical scientist who ever lived, it’s a little ironicthat Pope chose light as a metaphor, because it wasin the study of light that Newton made some of hisworst mistakes. Newton was a firm believer in thedogma, then unsupported by observation, that mat-ter was composed of atoms, and it seemed logical tohim that light as well should be composed of tinyparticles, or ‘corpuscles.’ His opinions on the sub-ject were so strong that he influenced generationsof his successors to discount the arguments of Huy-gens and Grimaldi for the wave nature of light. Itwas not until 150 years later that Thomas Youngdemonstrated conclusively that light was a wave.

Young’s experiment was incredibly simple, and couldprobably have been done in ancient times if some

savvy Greek or Chinese philosopher had only thoughtof it. He simply let sunlight through a pinhole in awindow shade, forming what we would now call acoherent beam of light (that is, a beam consistingof plane waves marching in step). Then he held athin card edge-on to the beam, observed a diffrac-tion pattern on a wall, and correctly inferred thewave nature and wavelength of light. Since Roemerhad already measured the speed of light, Young wasalso able to determine the frequency of oscillation ofthe light.

Today, with the advent of the laser, the productionof a bright and coherent beam of light has becomeas simple as flipping a switch, and the wave natureof light can be demonstrated very easily. In this lab,you will carry out observations similar to Young’s,but with the benefit of hindsight and modern equip-ment.

Observations

A Determination of the wavelength of red light

Set up your laser on your optical bench. You willwant as much space as possible between the laserand the wall, in order to let the diffraction patternspread out as much as possible and reveal its finedetails.

Tear off two small scraps of paper with straight edges.Hold them close together so they form a single slit.Hold this improvised single-slit grating in the laserbeam and try to get a single-slit diffraction pattern.You may have to play around with different widthsfor the slit. No quantitative data are required. Thisis just to familiarize you with single-slit diffraction.

Make a diffraction pattern with the double-slit grat-ing. See what happens when you hold it in yourhand and rotate it around the axis of the beam.

The diffraction pattern of the double-slit grating con-sists of a rapidly varying pattern of bright and darkbars, with a more slowly varying pattern superim-posed on top. (See the figure two pages after thispage.) The rapidly varying pattern is the one thatis numerically related to the wavelength, λ, and thedistance between the slits, d, by the equation

∆θ = λ/d,

where θ is measured in radians. To make sure you

136 Lab 43 Wave Optics

can see the fine spacing, put your slits several metersaway from the wall. This will necessitate shining itacross the space between lab tables. To make it lesslikely that someone will walk through the beam andget the beam in their eye, put some of the smalldesks under the beam. The slit patterns we’re usingactually have three sets of slits, with the followingdimensions:

w (mm) d (mm)A .12 .6B .24 .6C .24 1.2

The small value of d is typically better, for two rea-sons: (1) it produces a wider diffraction pattern,which is easier to see; (2) it’s easy to get the beam ofthe laser to cover both slits. If your diffraction pat-tern doesn’t look like the one in the figure on page136, typically the reason is that you’re only cover-ing one slit with the beam (in which case you get asingle-slit diffraction pattern), or you’re not illumi-nating the two slits equally (giving a funny-lookingpattern with little dog-bones and things in it).

Think about the best way to measure the spacing ofthe pattern accurately. Is it best to measure from abright part to another bright part, or from dark todark? Is it best to measure a single spacing, or takeseveral spacings and divide by the number to findwhat one spacing is? Do it.

Determine the wavelength of the light, in units ofnanometers. Make sure it is in the right range forred light.

B Diameter of a human hair

Pull out one of your own hairs, hold it in the laserbeam, and observe a diffraction pattern. It turnsout that the diffraction pattern caused by a narrowobstruction, such as your hair, has the same spac-ing as the pattern that would be created by a sin-gle slit whose width was the same as the diameterof your hair. (This is an example of a general theo-rem called Babinet’s principle.) Measure the spacingof the diffraction pattern. (Since the hair’s diame-ter is the only dimension involved, there is only onediffraction pattern with one spacing, not superim-posed fine and coarse patterns as in part A.) De-termine the diameter of your hair. Make sure thevalue you get is reasonable, and compare with theorder-of-magnitude guess you made in your prelabwriteup.

Prelab


Read the safety checklist.

P1 Roughly what wavelength do you expect redlight to have?

P2 It is not practical to measure ∆θ directly us-ing a protractor. Plan how you will determine ∆θindirectly, via trigonometry.

P3 Make a rough order-of-magnitude guess of thediameter of a human hair.

Analysis

Determine the wavelength of the light and the diam-eter of the hair, with error bars.

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A double-slit diffraction pattern.

138 Lab 43 Wave Optics

44 Polarization

Apparatus

laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1/groupcalcite crystal (flattest available) . . . . . . . . . .1/grouppolarizing films . . . . . . . . . . . . . . . . . . . . . . . . . . . 2/groupNa gas discharge tube . . . . . . . . . . . . . . . . . . . . 1/groupphotovoltaic cell and collimator . . . . . . . . . . . 1/group

Goals

Make qualitative observations about the polar-ization of light.

Test quantitatively the hypothesis that polar-ization relates to the direction of the field vec-tors in an electromagnetic wave.

Introduction

It’s common knowledge that there’s more to lightthan meets the eye: everyone has heard of infraredand ultraviolet light, which are visible to some otheranimals but not to us. Another invisible feature ofthe wave nature of light is far less well known. Elec-tromagnetic waves are transverse, i.e., the electricand magnetic field vectors vibrate in directions per-pendicular to the direction of motion of the wave.Two electromagnetic waves with the same wavelengthcan therefore be physically distinguishable, if theirelectric and magnetic fields are twisted around indifferent directions. Waves that differ in this wayare said to have different polarizations.

An electromagnetic wave has electric and magnetic field

vectors that vibrate in the directions perpendicular to its

direction of motion. The wave’s direction of polarization is

defined as the line along which the electric field lies.

Maybe we polarization-blind humans are missing outon something. Some insects and crustaceans can de-

tect polarization, and a neuroscientist at the Univer-sity of Pennsylvania has recently found evidence thata freshwater fish called the green sunfish can see thepolarization of light (Discover magazine, Oct. 1996).Most sources of visible light (such as the sun or alight bulb) are unpolarized. An unpolarized beamof light contains a random mixture of waves withmany different directions of polarization, all of themchanging from moment to moment, and from pointto point within the beam.


Before doing anything else, turn on your gas dis-charge tube, so it will be warmed up when you areready to do part E.

A Double refraction in calcite

Place a calcite crystal on this page. You will see twoimages of the print through the crystal.

To understand why this happens, try shining thelaser beam on a piece of paper and then insertingthe calcite crystal in the beam. If you rotate thecrystal around in different directions, you should beable to get two distinct spots to show up on thepaper. (This may take a little trial and error, partlybecause the effect depends on the correct orientationof the crystal, but also because the crystals are notperfect, and it can be hard to find a nice smoothspot through which to shine the beam.)

In the refraction lab, you’ve already seen how a beamof light can be bent as it passes through the interfacebetween two media. The present situation is similarbecause the laser beam passes in through one face ofthe crystal and then emerges from a parallel face atthe back. You have already seen that in this type ofsituation, when the beam emerges again, its direc-tion is bent back parallel to its original direction, butthe beam is offset a little bit. What is different hereis that the same laser beam splits up into two parts,which bumped off course by different amounts.

What’s happening is that calcite, unlike most sub-stances, has a different index of refraction dependingon the polarization of the light. Light travels at adifferent speed through calcite depending on how theelectric and magnetic fields are oriented compared tothe crystal. The atoms inside the crystal are packedin a three-dimensional pattern sort of like a stack of

140 Lab 44 Polarization

oranges or cannonballs. This packing arrangementhas a special axis of symmetry, and light polarizedalong that axis moves at one speed, while light polar-ized perpendicular to that axis moves at a differentspeed.

It makes sense that if the original laser beam wasa random mixture of all possible directions of po-larization, then each part would be refracted by adifferent amount. What is a little more surprising isthat two separated beams emerge, with nothing inbetween. The incoming light was composed of lightwith every possible direction of polarization. Youwould therefore expect that the part of the incominglight polarized at, say, 45 compared to the crystal’saxis would be refracted by an intermediate amount,but that doesn’t happen. This surprising observa-tion, and all other polarization phenomena, can beunderstood based on the vector nature of electricand magnetic fields, and the purpose of this lab isto lead you through a series of observations to helpyou understand what’s really going on.

B A polarized beam entering the calcite

A single laser beam entering a calcite crystal breaks up

into two parts, which are refracted by different amounts.

The calcite splits the wave into two parts, polarized in

perpendicular directions compared to each other.

We need not be restricted to speculation about whatwas happening to the part of the light that enteredthe calcite crystal polarized at a 45 angle. You canuse a polarizing film, often referred to informally as a‘Polaroid,’ to change unpolarized light into a beam ofonly one specific polarization. In this part of the lab,you will use a polarizing film to produce a beam oflight polarized at a 45 angle to the crystal’s internal

axis.

If you simply look through the film, it doesn’t looklike anything special — everything just looks dim-mer, like looking through sunglasses. The light reach-ing your eye is polarized, but your eye can’t tell that.If you looked at the film under a microscope, you’dsee a pattern of stripes, which select only one direc-tion of polarization of the light that passes through.

Now try interposing the film between the laser andthe crystal. The beam reaching the crystal is nowpolarized along some specific direction. If you rotatethe film, you change beam’s direction of polariza-tion. If you try various orientations, you will be ableto find one that makes one of the spots disappear,and another orientation of the film, at a 90 anglecompared to the first, that makes the other spot goaway. When you hold the film in one of these direc-tions, you are sending a beam into the crystal thatis either purely polarized along the crystal’s axis orpurely polarized at 90 to the axis.

By now you have already seen what happens if thefilm is at an intermediate angle such as 45 . Twospots appear on the paper in the same places pro-duced by an unpolarized source of light, not just asingle spot at the midpoint. This shows that thecrystal is not just throwing away the parts of thelight that are out of alignment with its axis. Whatis happening instead is that the crystal will accept abeam of light with any polarization whatsoever, andsplit it into two beams polarized at 0 and 90 comparedto the crystal’s axis.

This behavior actually makes sense in terms of thewave theory of light. Light waves are supposed toobey the principle of superposition, which says thatwaves that pass through each other add on to eachother. A light wave is made of electric and magneticfields, which are vectors, so it is vector addition we’re

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talking about in this case. A vector at a 45 anglecan be produced by adding two perpendicular vec-tors of equal length. The crystal therefore cannotrespond any differently to 45-degree polarized lightthan it would to a 50-50 mixture of light with 0-degree and 90-degree polarization.

The principle of superposition implies that if the 0 and

90 polarizations produce two different spots, then the

two waves superimposed must produce those two spots,

not a single spot at an intermediate location.

C Two polarizing films

So far I’ve just described the polarizing film as adevice for producing polarized light. But one canapply to the polarizing film the same logic of super-position and vector addition that worked with thecalcite crystal. It would not make sense for the filmsimply to throw away any waves that were not per-fectly aligned with it, because a field oriented on aslant can be analyzed into two vector components,at 0 and 90 with respect to the film. Even if onecomponent is entirely absorbed, the other compo-nent should still be transmitted.

Based on these considerations, now think about whatwill happen if you look through two polarizing filmsat an angle to each other, as shown in the figureabove. Do not look into the laser beam! Just lookaround the room. What will happen as you changethe angle θ?

D Three polarizing films

Now suppose you start with two films at a 90 angleto each other, and then sandwich a third film be-tween them at a 45 angle, as shown in the two fig-

ures above. Make a prediction about what will hap-pen, and discuss your prediction with your instructorbefore you make the actual observation.

Quantitative Observations

E Intensity of light passing through two polar-izing films

In this part of the lab, you will make numerical mea-surements of the transmission of initially unpolarizedlight transmitted through two polarizing films at anangle θ to each other. To measure the intensity ofthe light that gets through, you will use a photocell,which is a device that converts light energy into anelectric current.

You will use a voltmeter to measure the voltageacross the photocell when light is shining on it. Aphotovoltaic cell is a complicated nonlinear device,but I’ve found empirically that under the conditionswe’re using in this experiment, the voltage is pro-portional to the power of the light striking the cell:twice as much light results in twice the voltage.

This measurement requires a source of light that isunpolarized, constant in intensity, and comes froma specific direction so it can’t get to the photocellwithout going through the polaroids. The ambientlight in the room is nearly unpolarized, but variesrandomly as people walk in front of the light fixtures,etc. The laser beam is constant in intensity, but asI was creating this lab I found to my surprise that itis partially polarized, with a polarization that variesover time. A more suitable source of light is thesodium gas discharge tube, which makes a nearlymonochromatic, unpolarized yellow light. Make sureyou have allowed it to warm up for at least 15-20minutes before using it; before it warms up, it makesa reddish light, and the polaroids do not work verywell on that color.

Make measurements of the relative intensity of lighttransmitted through the two polarizing films, using avariety of angles θ. Don’t assume that the notches onthe plastic housing of the polarizing films are a good

142 Lab 44 Polarization

indication of the orientation of the films themselves.

Prelab


P1 Given the angle θ between the polarizing films,predict the ratio |E′|/|E| of the transmitted electricfield to the incident electric field.

P2 Based on your answer to P1, predict the ra-tio P ′/P of the transmitted power to the incidentpower.

P3 Sketch a graph of your answer to P2. Super-imposed on the same graph, show a qualitative pre-diction of how it would change if the polaroids werenot 100% perfect at filtering out one component ofthe field.

Analysis

Discuss your qualitative results in terms of superpo-sition and vector addition.

Graph your results from part E, and superimpose atheoretical curve for comparison. Discuss how yourresults compare with theory. Since your measure-ments of light intensity are relative, just scale thetheoretical curve so that its maximum matches thatof the experimental data. (You might think of com-paring the intensity transmitted through the two po-laroids with the intensity that you get with no po-laroids in the way at all. This doesn’t really work,however, because in addition to acting as polarizers,the polaroids simply absorb a certain percentage ofthe light, just as any transparent material would.)

143

45 The Photoelectric Effect

Apparatus

Hg gas discharge tubelight aperture assemblylens/grating assemblyphotodiode module, support base, and coupling roddigital multimeter (Fluke)pieces of plywoodgreen and yellow filters

Goals

Observe evidence that light has particle prop-erties as well as wave properties.

Measure Planck’s constant.

Introduction

The photoelectric effect, a phenomenon in whichlight shakes an electron loose from an object, pro-vided the first evidence for wave-particle duality:the idea that the basic building blocks of light andmatter show a strange mixture of particle and wavebehaviors. At the turn of the twentieth century,physicists assumed that particle and wave phenom-ena were completely distinct. Young had shown thatlight could undergo interference effects such as diffrac-tion, so it had to be a wave. Since light was a wavecomposed of oscillating electric and magnetic fields,it made sense that when light encountered matter,it would tend to shake the electrons. It was onlyto be expected that something like the photoelectriceffect could happen, with the light shaking the elec-trons vigorously enough to knock them out of theatom. The best theoretical estimates, however, werethat light of ordinary intensity would take millionsof years to do the trick — it would take that longfor the electron slowly to absorb enough energy toescape.

The actual experimental observation of the photo-electric effect was therefore an embarrassment. Itstarted up immediately, not after a million years.Albert Einstein, better known today for the theoryof relativity, was the first to come up with the rad-ical, and correct, explanation. Einstein simply sug-gested that in the photoelectric effect, light was be-having as a particle, now called a photon. The beam

of light could be visualized as a stream of machine-gun bullets. The electrons would be small targets,but when a ‘light bullet’ did score a hit, it packedenough of an individual wallop to knock the elec-tron out immediately. Based on other experimentsinvolving the spectrum of light emitted by hot, glow-ing objects, Einstein also proposed that each photonhad an energy given by

E = hf ,

where f is the frequency of the light and h is Planck’sconstant.

In this lab, you will perform the classic experimentused to test Einstein’s theory. You should refer tothe description of the experiment in your textbook.Briefly, you will expose the metal cathode of a vac-uum tube to light of various frequencies, and deter-mine the voltage applied between the cathode andanode that just barely suffices to cut off the pho-toelectric current completely. This is known as thestopping voltage, Vs. According to Einstein’s theory,the stopping voltage should obey the equation

eVs = hf − Es,

where Es is the amount of energy required by anelectron to penetrate the surface of the cathode andescape.

Optical setup.

Setup

You can use the Hg gas discharge tube to producemonochromatic light with the following wavelengths:

144 Lab 45 The Photoelectric Effect

color wavelength (nm)ultraviolet 365violet 405blue 436green 546yellow 578

The diffraction grating splits up the light into theselines, so you can make one line at a time enter thephotodiode. Slit 1 slides into the slot in the front ofthe discharge tube. The lens serves to create focusedimages of slit 1 at the photodiode. The lens anddiffraction grating are housed in a single unit, whichis attached to a pair of rods (not shown) projectingfrom slit 1. Do not drop the lens and diffractiongrating — I have already damaged one by droppingit, and they cost $200 to replace. For measurementswith the green and yellow lines, green and yellowfilters are used to help eliminate stray light of othercolors — they stick magnetically on the front of thecollimator tube. Slit 2 and the collimator tube keepstray light from getting in.

The photodiode module is held on top of a post ona rotating arm. The ultraviolet line is invisible, butthe front of slit 2 is coated with a material that flu-oresces in UV light, so you can see where the lineis.

Circuit.

Circuit

The circuit in fig. (a) above is the one shown intextbooks for this type of experiment. Light comesin and knocks electrons out of the curved cathode.If the voltage is turned off, there is no electric field,so the electrons travel in straight lines; some willhit the anode, creating a current referred to as the

photocurrent. If the voltage is turned on, the electricfield repels the electrons from the wire electrode, andthe current is reduced or eliminated. The stoppingvoltage would be measured by increasing the voltageuntil no more current was flowing. We used to usea setup very similar to this in this course, but itwas difficult to get good data because it was hard tojudge accurately when the current had reached zero.

The circuit we now use, shown in fig. (b), uses acute trick to determine the stopping voltage. Thephotocurrent transports electrons from the cathodeto the anode, so a net positive charge builds up onthe cathode, and a negative charge on the anode. Asthe charge and the voltage increase, the photocur-rent is reduced, until finally the voltage reaches thestopping voltage, and no more current can flow. Youthen read the voltage off of the voltmeter. Whenyou have the next color of light shining on the cath-ode, you momentarily close the switch, dischargingthe photodiode, and then take your next measure-ment. The only disadvantage of this setup is thatyou cannot adjust the voltage yourself and see howthe photocurrent varies with voltage.

Setup

Move the housing containing the grating and lensuntil you get a good focus at the front of the photo-diode box. The square side needs to be facing awayfrom the discharge tube.

Diffraction patterns are supposed to be symmetric,i.e., the m = 1 and m = −1 maxima should be iden-tical. In reality, there is something strange aboutthis setup that can cause the shorter wavelengthlines (especially the UV line) to be extremely dimon one side. Check which one is brighter on yourapparatus.

Just because the light gets in through slit 2 doesnot mean it is getting in to the photodiode. Theoriginal design of the apparatus allowed the photo-diode module to twist around on its post, and ithad to be adjusted carefully by trial and error. Be-cause students were getting frustrated with this, Iepoxied the photodiode modules onto their posts inthe right orientation. This makes it impossible todisassemble the apparatus and put it in its storagebox, but should get rid of the hassles with orient-ing it. However, you should still check that it’s ori-ented correctly, because it’s possible that your setupwas a little different from mine, I’m the epoxy canbe cracked by rough handling, and the screw at thebase of the post can also get loose. There are three

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things you should check to make sure the orientationis right: (1) Sighting along the tube like a gun, youshould see that it looks like it’s lined up with thecenter of the grating. (2) The tube can be lifted outon a hinge so that you can see the glass photodi-ode tube inside the box; check that light is actuallyfalling on the opening on the side of the tube. (3)Take data using the UV line. If you don’t get a big-ger voltage for this line than for the others, then thelight is not making it in to the photodiode.

Observations

You can now determine the stopping voltages corre-sponding to the five different colors of light.

Hints:

The biggest possible source of difficulty is straylight. The room should be dark when you doyour measurements.

The shortest wavelengths of light (highest fre-quencies), for which the energy of the pho-tons is the highest, readily produce photoelec-trons. The photocurrent is much weaker forthe longer wavelengths. Start with the short-wavelength line and graduate to the more diffi-cult, lower frequencies. Don’t forget the filtersfor the yellow and green lines!

If the button to zero the voltage doesn’t work,it is because the batteries are dead.

When you hit the button to zero the voltage, itmay actually pop up high and then come backdown. This is normal. (It’s acting like an RCcircuit with a long RC time constant).

Check the batteries in your photodiode modulebefore you start, using the two banana plugsdesigned for this purpose. If your batteries aredead, you need to replace them. I’ve also seencases where the batteries are on the borderlineat the beginning of the lab, and then die com-pletely during the lab; in this situation, you’llnotice that the stopping voltages you’re mea-suring change over the course of the lab, anddon’t make sense. It won’t hurt to check thebatteries at the end of the lab as well as at thebeginning.

Where the lines hit the white front of slit 2,they should be sharp, and should not overlap.You can adjust the focus by moving the lensand grating in or out. If you can’t get a good

focus, check and make sure that the square sideof the unit is away from the Hg tube.

The photodiode module can be rotated on itspost so that the light goes straight down thetube. If you don’t line it up correctly, you’llbe able to tell because the voltage will creepup slowly, rather than shooting up to a certainvalue and stopping. There is a screw that issupposed to allow you to lock the photodiodeinto position at the correct angle. Make sureto loosen the screw before trying to aim thephotodiode, and lock it once it’s aimed cor-rectly. If your photodiode won’t lock in place,you need to tighten the aluminum post thatforms the base of the box.

Prelab



P1 In the equation eVs = hf − Es, verify that allthree terms have the same units.

P2 Plan how you will analyze your data to deter-mine Planck’s constant.

Analysis

Extract Planck’s constant from your data, with errorbars (see appendix 4). Is your value consistent withthe accepted value given in your textbook?

Every electron that absorbs a photon acquires a ki-netic energy equal to hf . Thus it would seem thatif the voltage is less than the stopping voltage, ev-ery electron should have enough energy to reach theother electrode. Give two reasons why many elec-trons do not reach the other electrode even whenthe voltage is less than the stopping voltage.

146 Lab 45 The Photoelectric Effect

46 Electron Diffraction

Apparatus

cathode ray tube (Leybold 555 626)high-voltage power supply (new Leybold)100-kΩ resistor with banana-plug connectorsVernier calipers

Goals

Observe wave interference patterns (diffractionpatterns) of electrons, demonstrating that elec-trons exhibit wave behavior as well as particlebehavior.

Learn what it is that determines the wave-length of an electron.

Introduction

The most momentous discovery of 20th-century physicshas been that light and matter are not simply madeof waves or particles — the basic building blocks oflight and matter are strange entities which displayboth wave and particle properties at the same time.In our course, we have already learned about theexperimental evidence from the photoelectric effectshowing that light is made of units called photons,which are both particles and waves. That proba-bly disturbed you less than it might have, since youmost likely had no preconceived ideas about whetherlight was a particle or a wave. In this lab, however,you will see direct evidence that electrons, which youhad been completely convinced were particles, alsodisplay the wave-like property of interference. Yourschooling had probably ingrained the particle inter-pretation of electrons in you so strongly that youused particle concepts without realizing it. Whenyou wrote symbols for chemical ions such as Cl−

and Ca2+, you understood them to mean a chlorineatom with one excess electron and a calcium atomwith two electrons stripped off. By teaching you tocount electrons, your teachers were luring you intothe assumption that electrons were particles. If thislab’s evidence for the wave properties of electronsdisturbs you, then you are on your way to a deeperunderstanding of what an electron really is — botha particle and a wave.

The electron diffraction tube. The distance labeled as

13.5 cm in the figure actually varies from about 12.8 cm

to 13.8 cm, even for tubes that otherwise appear identical.

Method

What you are working with is basically the samekind of vacuum tube as the picture tube in your tele-vision. As in a TV, electrons are accelerated througha voltage and shot in a beam to the front (big end)of the tube, where they hit a phosphorescent coat-ing and produce a glow. You cannot see the electronbeam itself. There is a very thin carbon foil (it lookslike a tiny piece of soap bubble) near where the neckjoins the spherical part of the tube, and the elec-trons must pass through the foil before crossing overto the phosphorescent screen.

The purpose of the carbon foil is to provide an ultra-fine diffraction grating — the ‘grating’ consists ofthe crystal lattice of the carbon atoms themselves!

148 Lab 46 Electron Diffraction

As you will see in this lab, the wavelengths of theelectrons are very short (a fraction of a nanometer),which makes a conventional ruled diffraction gratinguseless — the closest spacing that can be achieved ona conventional grating is on the order of one microm-eter. The carbon atoms in graphite are arranged insheets, each of which consists of a hexagonal patternof atoms like chicken wire. That means they are notlined up in straight rows, so the diffraction patternis slightly different from the pattern produced by aruled grating.

Also, the carbon foil consists of many tiny graphitecrystals, each with a random orientation of its crys-tal lattice. The net result is that you will see a brightspot surrounded by two faint circles. The two circlesrepresent cones of electrons that intersect the phos-phor. Each cone makes an angle θ with respect tothe central axis of the tube, and just as with a ruledgrating, the angle is given by

sin θ = λ/d

where λ is the wavelength of the wave. For a ruledgrating, d would be the spacing between the lines.In this case, we will have two different cones withtwo different θ’s, θ1 and θ2, corresponding to twodifferent d′s, d1 and d2. Their geometrical meaningis shown below.

The carbon atoms in the graphite crystal are arranged

hexagonally.

Safety

This lab involves the use of voltages of up to 6000 V.Do not be afraid of the equipment, however; thereis a fuse in the high-voltage supply that limits theamount of current that it can produce, so it is notparticularly dangerous. Read the safety checklist on

high voltage in Appendix 8. Before beginning thelab, make sure you understand the safety rules, ini-tial them, and show your safety checklist to yourinstructor. If you don’t understand something, askyour instructor for clarification.

In addition to the high-voltage safety precautions,please observe the following rules to avoid damagingthe apparatus:

The tubes cost $1000. Please treat them withrespect! Don’t drop them! Dropping them wouldalso be a safety hazard, since they’re vacuum tubes,so they’ll implode violently if they break.

Do not turn on anything until your instructorhas checked your circuit.

Don’t operate the tube continuously at thehighest voltage values (5000-6000 V). It producesx-rays when used at these voltages, and the strongbeam also decreases the life of the tube. You canuse the circuit on the right side of the HV supply’spanel, which limits its own voltage to 5000 V. Don’tleave the tube’s heater on when you’re not actuallytaking data, because it will decrease the life of thetube.

Setup

You setup will consist of two circuits, a heater circuitand the high-voltage circuit.

The heater circuit is to heat the cathode, increas-ing the velocity with which the electrons move inthe metal and making it easier for some of themto escape from the cathode. This will produce thefriendly and nostalgia-producing yellow glow whichis characteristic of all vacuum-tube equipment. Theheater is simply a thin piece of wire, which acts asa resistor when a small voltage is placed across it,producing heat. Connect the heater connections, la-beled F1 and F2, to the 6-V AC outlet at the backof the HV supply.

The high-voltage circuit’s job is to accelerate theelectrons up to the desired speed. An electron thathappens to jump out of the cathode will head ‘down-hill’ to the anode. (The anode is at a higher voltagethan the cathode, which would make it seem likeit would be uphill from the cathode to the anode.However, electrons have negative charge, so they’relike negative-mass water that flows uphill.) The highvoltage power supply is actually two different powersupplies in one housing, with a left-hand panel forone and a right-hand panel for the other. Connectthe anode (A) and cathode (C) to the right-hand

149

panel of the HV supply, and switch the switch onthe HV supply to the right, so it knows you’re usingthe right-hand panel.

The following connections are specified in the doc-umentation, although I don’t entirely understandwhat they’re for. First, connect the electrode X tothe same plug as the cathode.1 Also, connect F1 toC with the wire that has the 100-kΩ resistor splicedinto it. The circuit diagram on page 152 summarizesall this.

Check your circuit with your instructor before turn-ing it on!

Observations

You are now ready to see for yourself the evidence ofthe wave nature of electrons, observe the diffractionpattern for various values of the high voltage, andfigure out what determines the wavelength of theelectrons. You will need to do your measurementsin the dark.

You will measure the θ’s, and thus determine thewavelength, λ, for several different voltages. Eachvoltage will produce electrons with a different veloc-ity, momentum, and energy.

Hints:

While measuring the diffraction pattern, don’ttouch the vacuum tube — the static electricfields of one’s body seem to be able to perturbthe pattern.

It is easiest to take measurements at the high-est voltages, where the electrons pack a wallopand make nice bright rings on the phosphor.Start with the highest voltages and take dataat lower and lower voltages until you can’t seethe rings well enough to take precise data. Toget unambiguous results, you’ll need to takedata with the widest possible range of voltages.

In order to reach a definite conclusion aboutwhat λ is proportional to, you will need accu-rate data. Do your best to get good measure-ments. Pay attention to possible problems in-curred by viewing the diffraction patterns fromdifferent angles on different occasions. Try re-peating a measurement more than once, andseeing how big your random errors are.

1If you look inside the tube, you can see that X is an extraelectrode sandwiched in between the anode and the cathode.I think it’s meant to help produce a focused beam.

You need to get data down to about 2 or 3kV in order to get conclusive results from thisexperiment. The tubes are not quite identi-cal, and were not designed to operate at suchlow voltages, so they haven’t been tested un-der those conditions. Experience has shownthat some of the tubes work at lower voltagesthan others. The group that has the tube thatworks the best at low voltages can share theirlow-voltage data with the other groups.

Prelab



Read the safety checklist.

P1 It is not practical to measure θ1 and θ2 directlywith a protractor. Come up with a plan for how toget the angles indirectly using trigonometry.

The figure shows the vacuum tube as having a par-ticular shape, which is a sphere with the foil andphosphor at opposite ends of a diamater. In reality,the tubes we’re using now are not quite that shape.To me, they look like they may have been shapedso that the phosphor surface is a piece of a spherecentered on the foil. If so, then arc lengths acrossthe phosphor can be connected to diffraction anglesvery simply via the definition of radian measure.

P2 If the voltage difference across which the elec-trons are accelerated is V , and the known mass andcharge of the electron are m and e, what are theelectrons’ kinetic energy and momentum, in termsof V ,m, and e? (As a numerical check on your re-sults, you should find that V = 5700 V gives KE =9.1 × 10−16 J and p = 4.1 × 10−23 kg·m/s.)

P3 All you’re trying to do based on your graphs isjudge which one could be a graph of a proportional-ity, i.e., a line passing through the origin. Becauseof this, you can omit any constant factors from theequations you found in P1. When you do this, whatdo your expressions turn out to be?

P4 Why is it not logically possible for the wave-length to be proportional to both p and KE? Toboth 1/p and 1/KE?


P5 I have suggested plotting λ as a function ofp, KE, 1/p and 1/KE to see if λ is directly propor-tional to any of them. Once you have your raw data,how can you immediately rule out two of these fourpossibilities and avoid drawing the graphs?

P6 On each graph, you will have two data-pointsfor each voltage, corresponding to two different mea-surements of the same wavelength. The two wave-lengths will be almost the same, but not exactlythe same because of random errors in measuring therings. Should you get the wavelengths by combiningthe smaller angle with d1 and the larger angle withd2, or vice versa?

Analysis

Once you have your data, you can try plotting λ asa function of, say, the kinetic energy, KE, of theelectrons, and see if it makes something simple likea straight line. Make sure your graph includes theorigin (see below). You could also try plotting λas a function of the electrons’ momentum, p, or asa function of other quantities such as 1/KE, 1/p,etc. You can simplify your analysis by leaving outconstant factors.

What does λ seem to be proportional to? Your datamay cover a small enough range of voltage that morethan one graph may look linear. You can rule oneout by checking whether a line fit through the datapoints would pass near the origin, as it must for aproportionality. This is why it is important to haveyour graph include the origin.

151

The circuit for the new setup.


47 The Hydrogen Atom

Apparatus

H gas discharge tube . . . . . . . . . . . . . . . . . . . . . 1/groupHg gas discharge tube . . . . . . . . . . . . . . . . . . . . 1/groupspectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/groupdiffraction grating, 600 lines/mm . . . . . . . . . 1/groupsmall screwdriver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1black cloth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1piece of plywood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1block of wood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1penlight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/group

Goals

Observe the visible line spectrum of hydrogen.

Determine the mass of the electron.

Introduction

What’s going on inside an atom? The question wouldhave seemed nonsensical to physicists before the 20thcentury — the word ‘atom’ is Greek for ‘unsplit-table,’ and there was no evidence for subatomic par-ticles. Only after Thomson and Rutherford had demon-strated the existence of electrons and the nucleus didthe atom begin to be imagined as a tiny solar system,with the electrons moving in elliptical orbits aroundthe nucleus under the influence of its electric field.The problem was that physicists knew very well thataccelerating charges emit electromagnetic radiation,as for example in a radio antenna, so the acceler-ation of the electrons should have caused them toemit light, steadily lose energy, and spiral into thenucleus, all within a microsecond,.

Luckily for us, atoms do not spontaneously shrinkdown to nothing, but there was indeed evidence thatatoms could emit light. The spectra emitted by veryhot gases were observed to consist of patterns of dis-crete lines, each with a specific wavelength. Theprocess of emitting light always seemed to stop shortof finally annihilating the atom — why? Also, whywere only those specific wavelengths emitted?

In this lab, you will study the spectrum of light emit-ted by the hydrogen atom, the simplest of all atoms,with just one proton and one electron. In 1885, be-fore electrons and protons had even been imagined,

a Swiss schoolteacher named Johann Balmer discov-ered that the wavelengths emitted by hydrogen wererelated by mysterious ratios of small integers. For in-stance, the wavelengths of the red line and the blue-green line form a ratio of exactly 20/27. Balmer evenfound a mathematical rule that gave all the wave-lengths of the hydrogen spectrum (both the visibleones and the invisible ones that lay in the infraredand ultraviolet). The formula was completely empir-ical, with no theoretical basis, but clearly there werepatterns lurking in the seemingly mysterious atomicspectra.

The first step toward understanding Balmer’s nu-merology was Einstein’s theory that light consistedof particles (photons), whose energy was related totheir frequency by the equation Ephoton = hf , orsubstituting f = c/λ, Ephoton = hc/λ .

According to this theory, the discrete wavelengthsthat had been observed came from photons with spe-cific energies. It seemed that the atom could existonly in specific states of specific energies. To getfrom an initial state with energy Ei to a final statewith a lower energy Ef , conservation of energy re-quired the atom to release a photon with an energyof Ephoton = Ei − Ef .

Not only could the discrete line spectra be explained,but if the atom possessed a state of least energy(called a ‘ground state’), then it would always endup in that state, and it could not collapse entirely.Knowing the differences between the energy levels ofthe atom, it was not too difficult to figure out theatomic energy levels themselves. Niels Bohr showedthat they obey a relatively simple equation,

En = −mk2e4

2~2· 1

n2

where n is an integer labeling the level, k is theCoulomb constant, e is the fundamental unit of charge,~ is Planck’s constant over 2π, and me is the mass ofthe electron. All the energies of the photons in theemission spectrum could now be explained as differ-ences in energy between specific states of the atom.For instance the four visible wavelengths observed byBalmer all came from cases where the atom ended upin the n = 2 state, dropping down from the n = 3,4, 5, and 6 states.

Although the equation’s sheer size may appear for-midable, keep in mind that the quantity in paren-theses is just a numerical constant, and the variation

154 Lab 47 The Hydrogen Atom

of energy from one level to the next is of the verysimple mathematical form 1/n2. It was because ofthis basic simplicity that the wavelength ratios like20/27 occurred. The minus sign occurs because theequation includes both the electron’s potential en-ergy and its kinetic energy, and the standard choiceof a reference-level for the potential energy resultsin negative values.

Along with the nice formulas came a whole newset of subversive concepts: that nature is randomin certain ways, that the building blocks of natureare both particles and waves, and that subatomicparticles do not follow well-defined trajectories asthey travel through space. Today these ridiculous-sounding ideas are taken for granted by working physi-cists, and we are so sure of the theory behind Bohr’sequation that it is now used as one of the most accu-rate ways of determining the mass of the electron. Ina previous lab, you measured the charge-to-mass ra-tio of the electron, but like the experiment by Thom-son on which it was based, that technique was un-able to give the charge and mass separately. Mod-ern techniques allow us to measure wavelengths oflight, and therefore energies of photons, with highprecision, so if all the other fundamental constantsin Bohr’s equation are known, we can solve for themass of the electron. This lab is really the only ex-ample of a high-precision experiment that you’ll doin this course — done correctly, it allows the de-termination of the electron’s mass to five significantfigures!

Method

The apparatus you will use to observe the spectrumof hydrogen is shown in the figure. For a given wave-length, the grating produces diffracted light at manydifferent angles: a central zeroth-order line at θ = 0,first-order lines on both the left and right, and so onthrough higher-order lines at larger angles. The lineof order m occurs at an angle satisfying the equationmλ = d sin θ.

To measure a wavelength, you will move the tele-scope until the diffracted first-order image of the slitis lined up with the telescope’s cross-hairs and thenread off the angle. Note that the angular scale onthe table of the spectroscope actually gives the anglelabeled α in the figure, not θ.

Sources of systematic errors

There are three sources of systematic error:

angular scale out of alignment: If the angularscale is out of alignment, then all the angleswill be off by a constant amount.

factory’s calibration of d: The factory thatmade the grating labeled it with a certain spac-ing (in lines per millimeter) which can be con-verted to d (center-to-center distance betweenlines). But their manufacturing process is notall that accurate, so the actual spacing of thelines is a little different from what the labelsays.

orientation of the grating: Errors will be causedif the grating is not perpendicular to the beamfrom the collimator, or if the lines on the grat-ing are not vertical (perpendicular to the planeof the circle).

Eliminating systematic errors

A trick to eliminate the error due to misalignment ofthe angular scale is to observe the same line on boththe right and the left, and take θ to be half the differ-ence between the two angles, i.e., θ = (αR − αL)/2.Because you are subtracting two angles, any sourceof error that adds a constant offset onto the anglesis eliminated. A few of the spectrometers have theirangular scales out of alignment with the collimatorsby as much as a full degree, but that’s of absolutelyno consequence if this technique is used.

Regarding the calibration of d, the first person whoever did this type of experiment simply had to makea diffraction grating whose d was very precisely con-structed. But once someone has accurately mea-sured at least one wavelength of one emission lineof one element, one can simply simply to determinethe spacing, d, of any grating using a line whosewavelength is known.

You might think that these two tricks would be enoughto get rid of any error due to misorientation of thegrating, but they’re not. They will get rid of anyerror of the form θ → θ + c or sin θ → c sin θ, butmisorientation of the grating produces errors of theform sin θ → sin θ + c. The adjustment of the orien-tation of the grating is described later.

Optics

The figure below shows the optics from the side, withthe telescope simply looking down the throat of thecollimator at θ = 0. You are actually using the op-

155

tics to let you see an image of the slit, not the tubeitself. The point of using a telescope is that it pro-vides angular magnification, so that a small changein angle can be seen visually.

A lens is used inside the collimator to make the lightfrom the slit into a parallel beam. This is important,because we are using mλ = d sin θ to determine thewavelength, but this equation was derived under theassumption that the light was coming in as a parallelbeam. To make a parallel beam, the slit must belocated accurately at the focal point of the lens. Thisadjustment should have already been done, but youwill check later and make sure. A further advantageof using a lens in the collimator is that a telescopeonly works for objects far away, not nearby objectsfrom which the reflected light is diverging strongly.The lens in the collimator forms a virtual image atinfinity, on which the telescope can work.

The objective lens of the telescope focuses the light,forming a real image inside the tube. The eyepiecethen acts like a magnifying glass to let you see theimage. In order to see the cross-hairs and the imageof the slit both in focus at the same time, the cross-hairs must be located accurately at the focal pointof the objective, right on top of the image.

Setup

Turn on both gas discharge tubes right away, to letthem get warmed up.

Adjusting the optics at the start of the experiment isvital. You do not want to fail to get the adjustmentsright and then spend several frustrating hours tryingfruitlessly to make your observations.

First you must check that the cross-hairs are at thefocal point of the objective. If they are, then the im-age of the slits formed by the objective will be at thesame point in space as the crosshairs. You’ll be ableto focus your eye on both simultaneously, and therewill be no parallax error depending on the exact po-sition of your eye. The easiest way to check this isto look through the telescope at something far away(& 50 m), and move your head left and right to see ifthe crosshairs move relative to the image. Slide theeyepiece in and out to achieve a comfortable focus.If this adjustment is not correct, you may need tomove the crosshairs in or out; this is done by slidingthe tube that is just outside the eyepiece tube. (Youneed to use the small screwdriver to loosen the screwon the side, which is recessed inside a hole. The holemay have a dime-sized cover over it.)

The white plastic pedestal should have already beenadjusted correctly to get the diffraction grating ori-ented correctly in three dimensions, but you shouldcheck it carefully. There are some clever featuresbuilt into the apparatus to help in accomplishingthis. As shown in the figure, there are three axesabout which the grating could be rotated. Rotationabout axis 1 is like opening a door, and this is ac-complished by rotating the entire pedestal like a lazySusan. Rotation about axes 2 (like folding down atailgate) and 3 are accomplished using the tripod ofscrews underneath the pedestal. The eyepiece of thetelescope is of a type called a Gauss eyepiece, witha diagonal piece of glass in it. When the grating isoriented correctly about axes 1 and 2 and the tele-scope is at θ = 0, a beam of light that enters throughthe side of the eyepiece is partially reflected to thegrating, and then reflected from the grating back tothe eye. If these two axes are correctly adjusted, thereflected image of the crosshairs is superimposed onthe crosshairs.

First get a rough initial adjustment of the pedestalby moving the telescope to 90 degrees and sight-ing along it like a gun to line up the grating. Nowloosen the screw (not shown in the diagram) thatfrees the rotation of the pedestal. Put a desk lampbehind the slits of the collimator, line up the tele-scope with the m = 0 image (which may not beexactly at α = 180 degrees), remove the desk lamp,cover the whole apparatus with the black cloth, andposition a penlight so that it shines in through thehole in the side of the eyepiece. Adjust axes 1 and2. If you’re far out of adjustment, you may see partof a circle of light, which is the reflection of the pen-light; start by bringing the circle of light into yourfield of view. When you’re done, tighten the screwthat keeps the pedestal from rotating. The pedestalis locked down to the tripod screws by the tensionin a spring, which keeps the tips of two of the screwssecure in dimples underneath the platform. Don’tlower the screws too much, or the pedestal will nolonger stay locked; make a habit of gently wigglingthe pedestal after each adjustment to make sure it’snot floating loose. Two of the spectrometers havethe diagonal missing from their eyepieces, so if youhave one of those, you’ll have to borrow an eyepiecefrom another group to do this adjustment.

For the adjustment of axis 3, place a piece of maskingtape so that it covers exactly half of the slits of thecollimator. Put the Hg discharge tube behind theslits. The crosshairs should be near the edge of thetape in the m = 0 image. Move the telescope out to alarge angle where you see one of the high-m Hg lines,


and adjust the tripod screws so that the crosshairsare at the same height relative to the edge of thetape.

Observations

Now put the Hg tube behind the collimator. Makesure the hottest part of the tube is directly in front ofthe slits; you will need to use the piece of plywood toraise the spectrometer to the right height. You wantthe tube as close to the slits as possible, and linedup with the slits as well as possible; you can adjustthis while looking through the telescope at an m = 1line, so as to make the line as bright as possible.

If your optics are adjusted correctly, you should beable to see the microscopic bumps and scratches onthe knife edges of the collimator, and there shouldbe no parallax of the crosshairs relative to the imageof the slits.

Here is a list of the wavelengths of the visible Hglines, in nm, to high precision:

404.656 violet There is a dimmer violetline nearby at 407.781 nm.

435.833 blue491.604 blue-green dim546.074 green There is also a dimmer

blue-green line.yellow This is actually a complex

set of lines, so it’s not use-ful for calibration.

You’ve seen the mercury spectrum before, in thephotoelectric effect lab, but you will notice some dif-ferences here. You will be able to see various dim-mer lines as well as the bright ones, and you wantto avoid mistaking those for the bright ones thatwe’re using for calibration. Also, as noted above,some lines that appear to be single lines in a low-resolution spectrum are actually mutliple lines. Thetable gives the wavelengths in vacuum. Althoughwe’re doing the lab in air, our goal is to find whatthe hydrogen wavelengths would have been in vac-uum; by calibrating using vacuum wavelengths formercury, we end up getting vacuum wavelengths forhydrogen as well.

Start by making sure that you can find all of themercury lines in the correct sequence — if not, thenyou have probably found some first-order lines andsome second-order ones. If you can find some linesbut not others, use your head and search for themin the right area based on where you found the linesyou did see. You may see various dim, fuzzy lights

through the telescope — don’t waste time chasingthese, which could be coming from other tubes orfrom reflections. The real lines will be bright, clearand well-defined. By draping the black cloth overthe discharge tube and the collimator, you can getrid of stray light that could cause problems for youor others. Put a box behind the discharge tube toblock the light coming out through the hole in theback as well. Do a quick and rough check that theangles at which you observe the violet line are closeto the approximate one predicted in prelab questionP1.

Now try swapping in the hydrogen tube in place ofthe mercury tube, and go through a similar processof acquainting yourself with the four lines in its vis-ible spectrum, which are as follows:

violet dimpurpleblue-greenred

Again you’ll again have to make sure the hottestpart of the tube is in front of the collimator; thisrequires putting books and/or blocks of wood underthe discharge tube.

By coincidence, three of the mercury lines lie veryclose to three of the hydrogen lines: violet next toviolet, blue next to purple, and blue-green next toblue-green. We’ll exploit this coincidence to reducesystematic errors. The idea is that if we have a hy-drogen wavelength, λ, that we want to measure, anda nearby known line λc from the mercury spectrumthat we can use for calibration, then we can sidestepthe determination of the grating’s spacing entirelyby using the relation

λ =sin θ

sin θcλc .

Although we’ve tried hard to eliminate systematicerrors through careful adjustment of the optics, someerrors will still remain. But these errors vary smooth-ly with angle, so by calibrating against known linesthat occur at nearly the same angle, we can mini-mize their effects. For each line, you’ll need a totalof four angles: αL and αR for the calibration lineat m = ±1, and a similar pair of angles for the hy-drogen line. Note that the knob that adjusts thewidth of the slit only moves the right-hand knife-edge, which appears to be on the left in the invertedimage. Therefore, adjusting the slit changes the lo-cation of the center of the slit, and such adjustmentsshould not be done between the αL and αR measure-ments of the same line. My experience is that thedimness of the violet hydrogen line makes its wave-

157

length difficult to measure accurately, so includingthe violet-violet pair actually worsens the quality ofthe final result. Therefore you end up just measuringtwo pairs (eight angles).

The angles are measured using a vernier scale, whichis similar to the one on the vernier calipers you havealready used in the first-semester lab course. Yourfinal reading for an angle will consist of degrees plusminutes. (One minute of arc, abbreviated 1’, is 1/60of a degree.) The main scale is marked every 30minutes. Your initial, rough reading is obtained bynoting where the zero of the vernier scale falls onthe main scale, and is of the form ‘xxx 0’ plus alittle more’ or ‘xxx 30’ plus a little more.’ Next, youshould note which line on the vernier scale lines upmost closely with one of the lines on the main scale.The corresponding number on the vernier scale tellsyou how many minutes of arc to add for the ‘plus alittle more.’

As a check on your results, everybody in your groupshould take independent readings of every angle youmeasure in the lab, nudging the telescope to the sideafter each reading. Once you have independent re-sults for a particular angle, compare them. If they’reconsistent to within one or two minutes of arc, aver-age them. If they’re not consistent, figure out whatwent wrong.

Prelab



P1 The nominal (and not very accurate) spacingof the grating is stated as 600 lines per millimeter.From this information, find d, and predict the anglesαL and αR at which you will observe the 404.656 nmviolet mercury line.

P2 Make sure you understand the first three vernierreadings in the figure, and then interpret the fourthone.

P3 In what sequence do you expect to see the Hglines on each side? Make a drawing showing thesequence of the angles as you go out from θ=0.

P4 The visible lines of hydrogen come from the

3 → 2, 4 → 2, 5 → 2, and 6 → 2 transitions. Basedon E = hf , which of these should correspond towhich colors?

P5 Based on the Bohr equation, predict the ratio

λblue−green

λpurple

for hydrogen, expressing your answer as the ratio oftwo integers.

Self-Check

Before leaving lab, make sure that your wavelengthsare consistent with your prediction from prelab ques-tion P5, to a precision of no worse than about onepart per thousand.

Analysis

Throughout your analysis, remember that this isa high-precision experiment, so you don’t want toround off to less than five significant figures.

We assume that the following constants are alreadyknown:

e = 1.6022 × 10−19 C

k = 8.9876 × 109 N·m2/C2

h = 6.6261 × 10−34 J·sc = 2.9979 × 108 m/s

The energies of the four types of visible photonsemitted by a hydrogen atom equal En − E2, wheren = 3, 4, 5, and 6. Using the Bohr equation, we have

Ephoton = A

(

1

4− 1

n2

)

,

where A is the expression from the Bohr equationthat depends on the mass of the electron. From thetwo lines you’ve measured, extract a value for A.If your data passed the self-check above, then youshould find that these values for A agree to no worsethan a few parts per thousand at worst. Computean average value of A, and extract the mass of theelectron, with error bars.

Finally, there is a small correction that should bemade to the result for the mass of the electron be-cause actually the proton isn’t infinitely massive com-pared to the electron; in terms of the quantity mgiven by the equation on page 154, the mass of theelectron, me, would actually be given by me = m/(1−m/mp), where mp is the mass of the proton, 1.6726×10−27 kg.


The spectrometer

Optics.

Orienting the grating.

Prelab question 2.

159

48 The Michelson Interferometer

Apparatus

Michelson interferometer . . . . . . . . . . . . . . . . . .1/groupNa and H gas discharge tubes . . . . . . . . . . . . 1/grouptools inside drawer . . . . . . . . . . . . . . . . . . . . 1 set/group2 × 4 piece of wood . . . . . . . . . . . . . . . . . . . . . . . 3/groupcolored filters (Cambosco and others)

Goals

Determine the wavelength of a line of the emis-sion spectrum of sodium or hydrogen.

The Michelson interferometer is a device for measur-ing the wavelength of light, used most famously inthe Michelson-Morley experiment of 1887, which waslater interpreted as disproving the existence of theluminiferous aether and supporting Einstein’s theoryof special relativity.

As shown in the figure, the idea is to take a beamof light from the source, split it into two perpendic-ular beams, send it to two mirrors, and then recom-bine the beams again. If the two light waves are inphase when recombined, they will reinforce, but ifthey are out of phase, they will cancel. Since thetwo waves originated from the splitting of a singlewave, the only reason they would be out of phasewas if the lengths of the two arms of the apparatuswere unequal. Mirror A is movable, and the distancethrough which it moves can be controlled and mea-sured extremely accurately using a micrometer con-nected to the mirror via a lever. If mirror A is moved

by distance equal to a quarter of a wavelength of thelight, the total round-trip distance traveled by thewave is changed by half a wavelength, which switchesfrom constructive to destructive interference, or viceversa. Thus if the mirror is moved by a distance d,and you see the light go through n complete cyclesof appearance and disappearance, you can concludethat the wavelength of the light was λ = 2d/n.

To make small and accurate adjustments of the mir-ror easier to do, the micrometer is connected to itthrough a level that reduces the amount of move-ment by a factor k, approximately equal to 5.23;the micrometer reads the bigger distance D = kdthat it actually travels itself, so the wavelength isλ = 2D/kn.

Another trick to make the apparatus easier to useis that the mirrors A and B are slightly curved.This means that instead of seeing a field of lightthat varies uniformly between dark and bright asyou turn the knob, instead you see a set of concen-tric rings (called fringes), which expand or contractdepending on which direction you turn the knob.

Turn on the sodium discharge tube, and let it warmup until it’s yellow.

Remove the drawer from the box, and take out thetool kit. Unscrew the screws on the bottom of thebox that lock the interferometer to the floor of thebox, and very carefully take the instrument out ofthe box. Screw the two aluminum legs into the bot-tom of the interferometer, and lay a piece of woodflat under the third leg, which is a threaded rod; thismakes the apparatus level.

Place the discharge tube near the entrance windowof the apparatus. If you look through the viewingwindow, you will see the image of the tube itself,reflected through the mirrors. To make this into auniform circle of light, place the ground glass screen(inside the bag of tools) in the bracket at the en-trance window.

Mirror B needs to be perfectly perpendicular to mir-ror A, and its vertical plane needs to be matched tomirror A’s. This is adjusted using the knobs on mir-ror B, one for vertical adjustment and one for hor-izontal. A rough initial adjustment can be done byaligning the two images of the circular entrance win-dow. You can then hang the metal pointer (from thebag of tools) on the top of the ground glass screen,

160 Lab 48 The Michelson Interferometer

and do a better adjustment so that the two images ofthe pointer’s tip coincide. You should now see a setof very fine concentric circular interference fringes,centered on a point outside of the field of view. Thefinal, fine adjustment is obtained by bringing thecenter of this pattern to the center of the field ofview.

The micrometer has a millimeter scale running from0 to 25 mm, with half-millimeter divisions on thebottom. To take a reading on it, first read thenumber of millimeters and half-millimeters based onwhere the edge of the cylindrical rotating part lies onthis scale. Then add on the reading from the vernierscale that runs around the circumference of the ro-tating part, which runs from 0.000 to 0.500 mm. Youshould be able to estimate to the nearest thousandthof a millimeter (tenth of a vernier division).

While looking at the interference fringes, turn theknob on the micrometer. You will see them eitherexpand like smoke rings, or contract and disappearinto the center, depending on which way you turnthe knob. Rotate the knob while counting about 50to 100 fringes, and record the two micrometer read-ings before and after. The difference between theseis D. It helps if you prop your head on the table,and move the micrometer knob smoothly and contin-uously. Moving your head disturbs the pattern, andhalting the micrometer knob tends to cause backlashthat confuses the count of fringes by plus or minusone.

It has been an ongoing project to get these spec-trometers back in operation and fully calibrated forthe first time in many years. In spring 2006, my stu-dents in physics 223 gave them a thorough test drive.In spring 2007, we started taking data to determinek accurately for each spectrometer, using the knownwavelength of the sodium emission line at 589 nm.Their data are on sheets inside each spectrometer’sbox. That class also experimented with using theapparatus to measure the wavelengths of some linesin the spectrum of hydrogen, which is of some fun-damental interest because it is the simplest of allatoms. Since hydrogen’s spectrum, unlike sodium’s,includes several different visible lines of similar in-tensity, this required using colored filters to selectthe desired line. They found that filter #2 fromthe Cambosco box worked well for the red line, and#8 for the blue-green line. Lines with short wave-lengths were more difficult to do. For the next classthat does the lab, my goal is to accumulate morecalibration data, so we can start to detect whethercertain data points are off because of ±1 errors incounting the number of fringes. I would also like to

make more progress in measuring lines of the hydro-gen spectrum accurately.

161

Appendix 1: Format of Lab Writeups

Lab reports must be three pages or less, not countingyour raw data. The format should be as follows:

Title

Raw data — Keep actual observations separate from

what you later did with them.

These are the results of the measurements you takedown during the lab, hence they come first. Youshould clearly mark the beginning and end of yourraw data, so I don’t have to sort through many pagesto find your actual presentation of your work, below.Write your raw data directly in your lab book; don’twrite them on scratch paper and recopy them later.Don’t use pencil. The point is to separate facts fromopinions, observations from inferences.

Procedure — Did you have to create your own

methods for getting some of the raw data?

Do not copy down the procedure from the manual.In this section, you only need to explain any meth-ods you had to come up with on your own, or caseswhere the methods suggested in the handout didn’twork and you had to do something different. Do notdiscuss how you did your calculations here, just howyou got your raw data.

Abstract — What did you find out? Why is it im-

portant?

The ‘abstract’ of a scientific paper is a short para-graph at the top that summarizes the experiment’sresults in a few sentences. If your results deviatedfrom the ideal equations, don’t be afraid to say so.After all, this is real life, and many of the equa-tions we learn are only approximations, or are onlyvalid in certain circumstances. However, (1) if yousimply mess up, it is your responsibility to realizeit in lab and do it again, right; (2) you will neverget exact agreement with theory, because measure-ments are not perfectly exact — the important issueis whether your results agree with theory to roughlywithin the error bars.

The abstract comes first in your writeup, but you’llwrite it last, so leave a little space for it.

The abstract is not a statement of what you hopedto find out. It’s a statement of what you did findout. It’s like the brief statement at the beginningof a debate: ‘The U.S. should have free trade withChina.’ It’s not this: ‘In this debate, we will discusswhether the U.S. should have free trade with China.’

If this is a lab that has just one important numericalresult (or maybe two or three of them), put themin your abstract, with error bars where appropriate.There should normally be no more than two to fournumbers here. Do not recapitulate your raw datahere — this is for your final results.

If you’re presenting a final result with error bars,make sure that the number of significant figures isconsistent with your error bars. For example, if youwrite a result as 323.54± 6 m/s, that’s wrong. Yourerror bars say that you could be off by 6 in the ones’place, so the 5 in the tenths’ place and the four inthe hundredths’ place are completely meaningless.

If you’re presenting a number in scientific notation,with error bars, don’t do it like this

1.234 × 10−89 m/s ± 3 × 10−92 m/s ,

do it like this

(1.234 ± 0.003) × 10−89 m/s ,

so that we can see easily which digit of the result theerror bars apply to.

Justification and Reasoning — Convince me of

what you claimed in your abstract.

Cconvince me that the statements you made aboutyour results in the abstract follow logically from yourdata. This will typically involve both calculationsand logical arguments. Continuing the debate meta-phor, if your abstract said the U.S. should have freetrade with China, this is the rest of the debate, whereyou convince me, based on data and logic, that weshould have free trade.

In your calculations, the more clearly you show whatyou did, the easier it is for me to give you partialcredit if there is something wrong with your final re-sult. If you have a long series of similar calculations,you may just show one as a sample. If your prelabinvolved deriving equations that you will need, re-peat them here without the derivation. Try to layout complicated calculations in a logical way, go-ing straight down the page and using indentation tomake it easy to understand. When doing algebra,try to keep everything in symbolic form until thevery end, when you will plug in numbers.

162 Lab Appendix 1: Format of Lab Writeups

Model Lab Writeup

Comparison of Heavy and Light Falling Objects- Galileo Galilei

Raw Data

(Galileo’s original, somewhat messy notes go here.)

He does not recopy the raw data to make them looknicer, or mix calculations with raw data.

Procedure

We followed the procedure in the lab manual withthe following additions: (1) To make sure both ob-jects fell at the same time, we put them side by sideon a board and then tipped the board. (2) We waiteduntil there was no wind.

Abstract

We dropped a cannon ball weighing two hundredpounds and a musket ball weighing half a pound si-multaneously from the same height. Both hit theground at nearly the same time. This contradictsAristotle’s theory that heavy objects always fall fasterthan light ones.

height of drop = 200 ± 4 cubits

amount by which cannon

ball was ahead at the bottom < 1 hand’s breadth

Justification and Reasoning

From a point 100 cubits away from the base of thetower, the top was at a 63 angle above horizontal.The height of the tower was therefore

100 cubits × tan 63 = 200 cubits.

We estimated the accuracy of the 100-cubit horizon-tal measurement to be ±2 cubits, with random errorsmainly from the potholes in the street, which madeit difficult to lay the cubit-stick flat. If it was 102cubits instead of 100, our result for the height of thetower would have been 204 cubits, so our error barson the height are ±4 cubits.

It is common knowledge that a feather falls moreslowly than a stone, but our experiment shows thatheavy objects do not always fall much more rapidly.We do not have any data on feathers, but we sug-gest that extremely light objects like feathers arestrongly affected by air resistance, which would benearly negligible for a cannonball. We think we saw

the cannon ball leading at the bottom by a slightmargin (1 hand’s breadth), but we could not be sure.It is possible that the musket ball was just notice-ably affected by air resistance. In any case, the Aris-totelian theory is clearly wrong, since it predicts thatthe cannon ball, which was 400 times heavier, wouldhave taken one 400th the time to hit the ground.

163

Appendix 2: Basic Error Analysis

No measurement is perfectly ex-

act.

One of the most common misconceptions about sci-ence is that science is ‘exact.’ It is always a strug-gle to get beginning science students to believe thatno measurement is perfectly correct. They tend tothink that if a measurement is a little off from the‘true’ result, it must be because of a mistake — ifa pro had done it, it would have been right on themark. Not true!

What scientists can do is to estimate just how faroff they might be. This type of estimate is calledan error bar, and is expressed with the ± symbol,read ‘plus or minus.’ For instance, if I measure mydog’s weight to be 52 ± 2 pounds, I am saying thatmy best estimate of the weight is 52 pounds, and Ithink I could be off by roughly 2 pounds either way.The term ‘error bar’ comes from the conventionalway of representing this range of uncertainty of ameasurement on a graph, but the term is also usedwhen no graph is involved.

Some very good scientific work results in measure-ments that nevertheless have large error bars. Forinstance, the best measurement of the age of the uni-verse is now 15±5 billion years. That may not seemlike wonderful precision, but the people who did themeasurement knew what they were doing. It’s justthat the only available techniques for determiningthe age of the universe are inherently poor.

Even when the techniques for measurement are veryprecise, there are still error bars. For instance, elec-trons act like little magnets, and the strength of avery weak magnet such as an individual electron iscustomarily measured in units called Bohr magne-tons. Even though the magnetic strength of an elec-tron is one of the most precisely measured quantitiesever, the best experimental value still has error bars:1.0011596524 ± 0.0000000002 Bohr magnetons.

There are several reasons why it is important in sci-entific work to come up with a numerical estimateof your error bars. If the point of your experimentis to test whether the result comes out as predictedby a theory, you know there will always be somedisagreement, even if the theory is absolutely right.You need to know whether the measurement is rea-sonably consistent with the theory, or whether thediscrepancy is too great to be explained by the lim-

itations of the measuring devices.

Another important reason for stating results with er-ror bars is that other people may use your measure-ment for purposes you could not have anticipated.If they are to use your result intelligently, they needto have some idea of how accurate it was.

Error bars are not absolute limits.

Error bars are not absolute limits. The true valuemay lie outside the error bars. If I got a better scale Imight find that the dog’s weight is 51.3±0.1 pounds,inside my original error bars, but it’s also possiblethat the better result would be 48.7 ± 0.1 pounds.Since there’s always some chance of being off by asomewhat more than your error bars, or even a lotmore than your error bars, there is no point in be-ing extremely conservative in an effort to make ab-solutely sure the true value lies within your statedrange. When a scientist states a measurement witherror bars, she is not saying ‘If the true value is out-side this range, I deserve to be drummed out of theprofession.’ If that was the case, then every scientistwould give ridiculously inflated error bars to avoidhaving her career ended by one fluke out of hun-dreds of published results. What scientists are com-municating to each other with error bars is a typicalamount by which they might be off, not an upperlimit.

The important thing is therefore to define error barsin a standard way, so that different people’s state-ments can be compared on the same footing. Byconvention, it is usually assumed that people esti-mate their error bars so that about two times out ofthree, their range will include the true value (or theresults of a later, more accurate measurement withan improved technique).

Random and systematic errors.

Suppose you measure the length of a sofa with atape measure as well as you can, reading it off tothe nearest millimeter. If you repeat the measure-ment again, you will get a different answer. (Thisis assuming that you don’t allow yourself to be psy-chologically biased to repeat your previous answer,and that 1 mm is about the limit of how well youcan see.) If you kept on repeating the measurement,

164 Lab Appendix 2: Basic Error Analysis

you might get a list of values that looked like this:

203.1 cm 203.4 202.8 203.3 203.2203.4 203.1 202.9 202.9 203.1

Variations of this type are called random errors, be-cause the result is different every time you do themeasurement.

The effects of random errors can be minimized by av-eraging together many measurements. Some of themeasurements included in the average are too high,and some are too low, so the average tends to bebetter than any individual measurement. The moremeasurements you average in, the more precise theaverage is. The average of the above measurementsis 203.1 cm. Averaging together many measurementscannot completely eliminate the random errors, butit can reduce them.

On the other hand, what if the tape measure was alittle bit stretched out, so that your measurementsalways tended to come out too low by 0.3 cm? Thatwould be an example of a systematic error. Sincethe systematic error is the same every time, aver-aging didn’t help us to get rid of it. You probablyhad no easy way of finding out exactly the amountof stretching, so you just had to suspect that theremight a systematic error due to stretching of thetape measure.

Some scientific writers make a distinction betweenthe terms ‘accuracy’ and ‘precision.’ A precise mea-surement is one with small random errors, while anaccurate measurement is one that is actually closeto the true result, having both small random errorsand small systematic errors. Personally, I find thedistinction is made more clearly with the more mem-orable terms ‘random error’ and ‘systematic error.’

The ± sign used with error bars normally impliesthat random errors are being referred to, since ran-dom errors could be either positive or negative, whereassystematic errors would always be in the same direc-tion.

The goal of error analysis

Very seldom does the final result of an experimentcome directly off of a clock, ruler, gauge or meter.It is much more common to have raw data consist-ing of direct measurements, and then calculationsbased on the raw data that lead to a final result.As an example, if you want to measure your car’sgas mileage, your raw data would be the number ofgallons of gas consumed and the number of milesyou went. You would then do a calculation, dividing

miles by gallons, to get your final result. When youcommunicate your result to someone else, they arecompletely uninterested in how accurately you mea-sured the number of miles and how accurately youmeasured the gallons. They simply want to knowhow accurate your final result was. Was it 22 ± 2mi/gal, or 22.137 ± 0.002 mi/gal?

Of course the accuracy of the final result is ulti-mately based on and limited by the accuracy of yourraw data. If you are off by 0.2 gallons in your mea-surement of the amount of gasoline, then that amountof error will have an effect on your final result. Wesay that the errors in the raw data ‘propagate’ throughthe calculations. When you are requested to do ‘er-ror analysis’ in a lab writeup, that means that you

165

are to use the techniques explained below to deter-mine the error bars on your final result. There aretwo sets of techniques you’ll need to learn:

techniques for finding the accuracy of your rawdata

techniques for using the error bars on your rawdata to infer error bars on your final result

Estimating random errors in raw

data

We now examine three possible techniques for es-timating random errors in your original measure-ments, illustrating them with the measurement ofthe length of the sofa.

Method #1: Guess

If you’re measuring the length of the sofa with ametric tape measure, then you can probably make areasonable guess as to the precision of your measure-ments. Since the smallest division on the tape mea-sure is one millimeter, and one millimeter is also nearthe limit of your ability to see, you know you won’tbe doing better than ± 1 mm, or 0.1 cm. Making al-lowances for errors in getting tape measure straightand so on, we might estimate our random errors tobe a couple of millimeters.

Guessing is fine sometimes, but there are at least twoways that it can get you in trouble. One is that stu-dents sometimes have too much faith in a measuringdevice just because it looks fancy. They think thata digital balance must be perfectly accurate, sinceunlike a low-tech balance with sliding weights on it,it comes up with its result without any involvementby the user. That is incorrect. No measurement isperfectly accurate, and if the digital balance onlydisplays an answer that goes down to tenths of agram, then there is no way the random errors areany smaller than about a tenth of a gram.

Another way to mess up is to try to guess the errorbars on a piece of raw data when you really don’thave enough information to make an intelligent esti-mate. For instance, if you are measuring the rangeof a rifle, you might shoot it and measure how farthe bullet went to the nearest centimeter, conclud-ing that your random errors were only ±1 cm. Inreality, however, its range might vary randomly byfifty meters, depending on all kinds of random fac-tors you don’t know about. In this type of situation,you’re better off using some other method of esti-mating your random errors.

Method #2: Repeated Measurements and the Two-Thirds Rule

If you take repeated measurements of the same thing,then the amount of variation among the numbers cantell you how big the random errors were. This ap-proach has an advantage over guessing your randomerrors, since it automatically takes into account allthe sources of random error, even ones you didn’tknow were present.

Roughly speaking, the measurements of the lengthof the sofa were mostly within a few mm of the aver-age, so that’s about how big the random errors were.But let’s make sure we are stating our error bars ac-cording to the convention that the true result willfall within our range of errors about two times outof three. Of course we don’t know the ‘true’ result,but if we sort out our list of measurements in order,we can get a pretty reasonable estimate of our errorbars by taking half the range covered by the mid-dle two thirds of the list. Sorting out our list of tenmeasurements of the sofa, we have

202.8 cm 202.9 202.9 203.1 203.1203.1 203.2 203.3 203.4 203.4

Two thirds of ten is about 6, and the range coveredby the middle six measurements is 203.3 cm - 202.9cm, or 0.4 cm. Half that is 0.2 cm, so we’d esti-mate our error bars as ±0.2 cm. The average of themeasurements is 203.1 cm, so your result would bestated as 203.1 ± 0.2 cm.

One common mistake when estimating random er-rors by repeated measurements is to round off allyour measurements so that they all come out thesame, and then conclude that the error bars werezero. For instance, if we’d done some overenthu-siastic rounding of our measurements on the sofa,rounding them all off to the nearest cm, every singlenumber on the list would have been 203 cm. Thatwouldn’t mean that our random errors were zero!The same can happen with digital instruments thatautomatically round off for you. A digital balancemight give results rounded off to the nearest tenth ofa gram, and you may find that by putting the sameobject on the balance again and again, you alwaysget the same answer. That doesn’t mean it’s per-fectly precise. Its precision is no better than about±0.1 g.

Method #3: Repeated Measurements and the Stan-dard Deviation

The most widely accepted method for measuring er-ror bars is called the standard deviation. Here’s howthe method works, using the sofa example again.


(1) Take the average of the measurements.

average = 203.1 cm

(2) Find the difference, or ‘deviation,’ of each mea-surement from the average.

−0.3 cm −0.2 −0.2 0.0 0.00.0 0.1 0.1 0.3 0.3

(3) Take the square of each deviation.

0.09 cm2 0.04 0.04 0.00 0.000.00 0.01 0.01 0.09 0.09

(4) Average together all the squared deviations.

average = 0.04 cm2

(5) Take the square root. This is the standard devi-ation.

standard deviation = 0.2 cm

If we’re using the symbol x for the length of thecouch, then the result for the length of the couchwould be stated as x = 203.1± 0.2 cm, or x = 203.1cm and σx = 0.2 cm. Since the Greek letter sigma(σ) is used as a symbol for the standard deviation, astandard deviation is often referred to as ‘a sigma.’

Step (3) may seem somewhat mysterious. Why notjust skip it? Well, if you just went straight fromstep (2) to step (4), taking a plain old average ofthe deviations, you would find that the average iszero! The positive and negative deviations alwayscancel out exactly. Of course, you could just takeabsolute values instead of squaring the deviations.The main advantage of doing it the way I’ve outlinedabove are that it is a standard method, so people willknow how you got the answer. (Another advantageis that the standard deviation as I’ve described ithas certain nice mathematical properties.)

A common mistake when using the standard devi-ation technique is to take too few measurements.For instance, someone might take only two measure-ments of the length of the sofa, and get 203.4 cmand 203.4 cm. They would then infer a standard de-viation of zero, which would be unrealistically smallbecause the two measurements happened to comeout the same.

In the following material, I’ll use the term ‘standarddeviation’ as a synonym for ‘error bar,’ but that doesnot imply that you must always use the standarddeviation method rather than the guessing methodor the 2/3 rule.

There is a utility on the class’s web page for calcu-lating standard deviations.

Probability of deviations

You can see that although 0.2 cm is a good figurefor the typical size of the deviations of the mea-surements of the length of the sofa from the aver-age, some of the deviations are bigger and some aresmaller. Experience has shown that the followingprobability estimates tend to hold true for how fre-quently deviations of various sizes occur:

< 1 standard deviation about 2 times out of 3

1-2 standard deviations about 1 time out of 4

2-3 standard deviations about 1 time out of 20

3-4 standard deviations about 1 in 500

4-5 standard deviations about 1 in 16,000

> 5 standard deviations about 1 in 1,700,000

The probability of various sizes of deviations, shown

graphically. Areas under the bell curve correspond to

probabilities. For example, the probability that the mea-

surement will deviate from the truth by less than one stan-

dard deviation (±1σ) is about 34 × 2 = 68%, or about 2

out of 3. (J. Kemp, P. Strandmark, Wikipedia.)

Example: How significant?

In 1999, astronomers Webb et al. claimed to have

found evidence that the strength of electrical forces in

the ancient universe, soon after the big bang, was slightly

weaker than it is today. If correct, this would be the first

example ever discovered in which the laws of physics

changed over time. The difference was very small, 5.7±1.0 parts per million, but still highly statistically signifi-

cant. Dividing, we get (5.7 − 0)/1.0 = 5.7 for the num-

ber of standard deviations by which their measurement

was different from the expected result of zero. Looking

at the table above, we see that if the true value really

was zero, the chances of this happening would be less

than one in a million. In general, five standard devia-

tions (‘five sigma’) is considered the gold standard for

statistical significance.

However, there is a twist to this story that shows how

statistics always have to be taken with a grain of salt.

167

In 2004, Chand et al. redid the measurement by a

more precise technique, and found that the change was

0.6 ± 0.6 parts per million. This is only one standard

deviation away from the expected value of 0, which

should be interpreted as being statistically consistent

with zero. If you measure something, and you think you

know what the result is supposed to be theoretically,

then one standard deviation is the amount you typically

expect to be off by — that’s why it’s called the ‘standard’

deviation. Moreover, the Chand result is wildly statisti-

cally inconsistent with the Webb result (see the exam-

ple on page 171), which means that one experiment or

the other is a mistake. Most likely Webb at al. under-

estimated their random errors, or perhaps there were

systematic errors in their experiment that they didn’t re-

alize were there.

Precision of an average

We decided that the standard deviation of our mea-surements of the length of the couch was 0.2 cm,i.e., the precision of each individual measurementwas about 0.2 cm. But I told you that the average,203.1 cm, was more precise than any individual mea-surement. How precise is the average? The answeris that the standard deviation of the average equals

standard deviation of one measurement√number of measurements

.

(An example on page 170 gives the reasoning thatleads to the square root.) That means that you cantheoretically measure anything to any desired preci-sion, simply by averaging together enough measure-ments. In reality, no matter how small you makeyour random error, you can’t get rid of systematic er-rors by averaging, so after a while it becomes point-less to take any more measurements.


Appendix 3: Propagation of Errors

Propagation of the error from a

single variable

In the previous appendix we looked at techniquesfor estimating the random errors of raw data, butnow we need to know how to evaluate the effects ofthose random errors on a final result calculated fromthe raw data. For instance, suppose you are given acube made of some unknown material, and you areasked to determine its density. Density is definedas ρ = m/v (ρ is the Greek letter ‘rho’), and thevolume of a cube with edges of length b is v = b3, sothe formula

ρ = m/b3

will give you the density if you measure the cube’smass and the length of its sides. Suppose you mea-sure the mass very accurately as m = 1.658±0.003 g,but you know b = 0.85±0.06 cm with only two digitsof precision. Your best value for ρ is 1.658 g/(0.85 cm)3 =2.7 g/cm3.

How can you figure out how precise this value for ρis? We’ve already made sure not to keep more thantwosignificant figures for ρ, since the less accuratepiece of raw data had only two significant figures.We expect the last significant figure to be somewhatuncertain, but we don’t yet know how uncertain. Asimple method for this type of situation is simply tochange the raw data by one sigma, recalculate theresult, and see how much of a change occurred. Inthis example, we add 0.06 cm to b for comparison.

b = 0.85 cm gave ρ = 2.7 g/cm3

b = 0.91 cm gives ρ = 2.0 g/cm3

The resulting change in the density was 0.7 g/cm3,so that is our estimate for how much it could havebeen off by:

ρ = 2.7 ± 0.7 g/cm3 .

Propagation of the error from sev-

eral variables

What about the more general case in which no onepiece of raw data is clearly the main source of error?For instance, suppose we get a more accurate mea-surement of the edge of the cube, b = 0.851 ± 0.001cm. In percentage terms, the accuracies of m andb are roughly comparable, so both can cause sig-

nificant errors in the density. The following moregeneral method can be applied in such cases:

(1) Change one of the raw measurements, say m, byone standard deviation, and see by how much thefinal result, ρ, changes. Use the symbol Qm for theabsolute value of that change.

m = 1.658 g gave ρ = 2.690 g/cm3

m = 1.661 g gives ρ = 2.695 g/cm3

Qm = change in ρ = 0.005 g/cm3

(2) Repeat step (1) for the other raw measurements.

b = 0.851 cm gave ρ = 2.690 g/cm3

b = 0.852 cm gives ρ = 2.681 g/cm3

Qb = change in ρ = 0.009 g/cm3

(3) The error bars on ρ are given by the formula

σρ =√

Q2m + Q2

b ,

yielding σρ = 0.01 g/cm3. Intuitively, the idea hereis that if our result could be off by an amount Qm

because of an error in m, and by Qb because of b,then if the two errors were in the same direction, wemight by off by roughly |Qm| + |Qb|. However, it’sequally likely that the two errors would be in oppo-site directions, and at least partially cancel. The ex-pression

√

Q2m + Q2

b gives an answer that’s smallerthan Qm +Qb, representing the fact that the cancel-lation might happen.

The final result is ρ = 2.69 ± 0.01 g/cm3.

Example: An average

On page 168 I claimed that averaging a bunch of mea-

surements reduces the error bars by the square root of

the number of measurements. We can now see that

this is a special case of propagation of errors.

For example, suppose Alice measures the circumfer-

ence c of a guinea pig’s waist to be 10 cm, Using the

guess method, she estimates that her error bars are

about ±1 cm (worse than the normal normal ∼ 1 mm

error bars for a tape measure, because the guinea pig

was squirming). Bob then measures the same thing,

and gets 12 cm. The average is computed as

c =A + B

2,

where A is Alice’s measurement, and B is Bob’s, giving

11 cm. If Alice had been off by one standard devia-

tion (1 cm), it would have changed the average by 0.5

170 Lab Appendix 3: Propagation of Errors

cm, so we have QA = 0.5 cm, and likewise QB = 0.5cm. Combining these, we find σc =

p

Q2

A+ Q2

B= 0.7

cm, which is simply (1.0 cm)/√

2. The final result is

c = (11.0 ± 0.7) cm. (This violates the usual rule for

significant figures, which is that the final result should

have no more sig figs than the least precise piece of

data that went into the calculation. That’s okay, be-

cause the sig fig rules are just a quick and dirty way

of doing propagation of errors. We’ve done real propa-

gation of errors in this example, and it turns out that the

error is in the first decimal place, so the 0 in that place

is entitled to hold its head high as a real sig fig, albeit a

relatively imprecise one with an uncertainty of ±7.)

Example: The difference between two measurements

In the example on page 167, we saw that two groups

of scientists measured the same thing, and the results

were W = 5.7 ± 1.0 for Webb et al. and C = 0.6 ± 0.6for Chand et al. It’s of interest to know whether the

difference between their two results is small enough to

be explained by random errors, or so big that it couldn’t

possibly have happened by chance, indicating that some-

one messed up. The figure shows each group’s results,

with error bars, on the number line. We see that the two

sets of error bars don’t overlap with one another, but er-

ror bars are not absolute limits, so it’s perfectly possible

to have non-overlapping error bars by chance, but the

gap between the error bars is very large compared to

the error bars themselves, so it looks implausible that

the results could be statistically consistent with one an-

other. I’ve tried to suggest this visually with the shading

underneath the data-points.

To get a sharper statistical test, we can calculate the

difference d between the two results,

d = W − C ,

which is 5.1. Since the operation is simply the subtrac-

tion of the two numbers, an error in either input simply

causes an error in the output that is of the same size.

Therefore we have QW = 1.0 and QC = 0.6, resulting

in σd =p

Q2

W+ Q2

C= 1.2. We find that the difference

between the two results is d = 5.1 ± 1.2, which differs

from zero by 5.1/1.2 ≈ 4 standard deviations. Looking

at the table on page 167, we see that the chances that

d would be this big by chance are extremely small, less

than about one in ten thousand. We can conclude to a

high level of statistical confidence that the two groups’

measurements are inconsistent with one another, and

that one group is simply wrong.

171

Appendix 4: Graphing

Review of Graphing

Many of your analyses will involve making graphs.A graph can be an efficient way of presenting datavisually, assuming you include all the informationneeded by the reader to interpret it. That meanslabeling the axes and indicating the units in paren-theses, as in the example. A title is also helpful.Make sure that distances along the axes correctlyrepresent the differences in the quantity being plot-ted. In the example, it would not have been correctto space the points evenly in the horizontal direction,because they were not actually measured at equallyspaced points in time.

Graphing on a Computer

Making graphs by hand in your lab notebook is fine,but in some cases you may find it saves you time todo graphs on a computer. For computer graphing,I recommend OpenOffice, which is free, open-sourcesoftware. It’s installed on the computers in rooms416 and 418. Because OpenOffice is free, you candownload it and put it on your own computer athome without paying money. If you already knowExcel, it’s very similar — you almost can’t tell it’sa different program.

Here’s a brief rundown on using OpenOffice:

On Windows, go to the Start menu and choosePrograms, OpenOffice.org, and Calc. On Linux,do Applications, Office, OpenOffice.org, Spread-sheet.

Type in your x values in the first column, andyour y values in the second column. For sci-entific notation, do, e.g., 5.2e-7 to represent5.2 × 10−7.

Select those two columns using the mouse.

From the Insert menu, do Chart.

When it offers you various styles of graphs tochoose from, choose the icon that shows a scat-ter plot, with dots on it (XY Chart).

Adjust the scales so the actual data on the plotis as big as possible, eliminating wasted space.To do this, right-click anywhere on the axis,choose the Scale tab, uncheck Automatic, andput in the lower and upper limits you want.

Fitting a Straight Line to a Graph

by Hand

Often in this course you will end up graphing somedata points, fitting a straight line through them witha ruler, and extracting the slope.

In this example, panel (a) shows the data, with errorbars on each data point. Panel (b) shows a bestfit, drawn by eye with a ruler. The slope of thisbest fit line is 100 cm/s. Note that the slope shouldbe extracted from the line itself, not from two datapoints. The line is more reliable than any pair ofindividual data points.

In panel (c), a ‘worst believable fit’ line has beendrawn, which is as different in slope as possible fromthe best fit, while still pretty much staying consis-tent the data (going through or close to most of theerror bars). Its slope is 60 cm/s. We can thereforeestimate that the precision of our slope is +40 cm/s.

There is a tendency when drawing a ‘worst believablefit’ line to draw instead an ‘unbelievably crazy fit’line, as in panel (d). The line in panel (d), with avery small slope, is just not believable compared tothe data — it is several standard deviations awayfrom most of the data points.

172 Lab Appendix 4: Graphing

Fitting a Straight Line to a Graph

on a Computer

It’s also possible to fit a straight line to a graph usingcomputer software such as OpenOffice. E.g., lab 47(the hydrogen atom) is a high-precision lab, and it’snot possible to get a sufficiently accurate result byhand.

To do this in OpenOffice, double-click on one of your

data points. A dialog box will come up. Selectthe Statistics tab, and under ‘Regression curves,’ se-lect the icon showing a line being fit to some data.This will cause the line to be drawn on your graph.To display the equation of the line, double-click onthe graph so that it’s surrounded by a gray border;then right-click on the line, and do Insert Regres-sion Curve Equation. By default your equation willonly have the slope and y-intercept shown with threesig figs; if you need more precision, double-click onthe graph so it’s outlined in gray, right-click on theequation, do Object Properties, Numbers, Scientific,and add more zeroes after the decimal place underFormat code.

How accurate is your slope? A method for gettingerror bars on the slope is to artificially change oneof your data points to reflect your estimate of howmuch it could have been off, and then redo the fitand find the new slope. The change in the slopetells you the error in the slope that results from theerror in this data-point. You can then repeat thisfor the other points and proceed as in appendix 3.In some cases, such as the absolute zero lab and thephotoelectric effect lab, it’s very hard to tell howaccurate your raw data are a priori ; in these labs,you can use the typical amount of deviation of thepoints from the line as an estimate of their accuracy.

173

Appendix 5: Finding Power Laws from Data

For many people, it is hard to imagine how scientistsoriginally came up with all the equations that cannow be found in textbooks. This appendix explainsone method for finding equations to describe datafrom an experiment.

Linear and nonlinear relationships

When two variables x and y are related by an equa-tion of the form

y = cx ,

where c is a constant (does not depend on x or y),we say that a linear relationship exists between xand y. As an example, a harp has many strings ofdifferent lengths which are all of the same thicknessand made of the same material. If the mass of astring is m and its length is L, then the equation

m = cL

will hold, where c is the mass per unit length, withunits of kg/m. Many quantities in the physical worldare instead related in a nonlinear fashion, i.e., therelationship does not fit the above definition of lin-earity. For instance, the mass of a steel ball bearingis related to its diameter by an equation of the form

m = cd3 ,

where c is the mass per unit volume, or density, ofsteel. Doubling the diameter does not double themass, it increases it by a factor of eight.

Power laws

Both examples above are of the general mathemati-cal form

y = cxp ,

which is known as a power law. In the case of alinear relationship, p = 1. Consider the (made-up)experimental data shown in the table.

h=height of rodentat the shoulder(cm)

f=food eaten perday (g)

shrew 1 3rat 10 300capybara 100 30,000

It’s fairly easy to figure out what’s going on justby staring at the numbers a little. Every time youincrease the height of the animal by a factor of 10, itsfood consumption goes up by a factor of 100. Thisimplies that f must be proportional to the square ofh, or, displaying the proportionality constant k = 3explicitly,

f = 3h2 .

Use of logarithms

Now we have found c = 3 and p = 2 by inspection,but that would be much more difficult to do if theseweren’t all round numbers. A more generally appli-cable method to use when you suspect a power-lawrelationship is to take logarithms of both variables.It doesn’t matter at all what base you use, as long asyou use the same base for both variables. Since thedata above were increasing by powers of 10, we’ll uselogarithms to the base 10, but personally I usuallyjust use natural logs for this kind of thing.

log10 h log10 fshrew 0 0.48rat 1 2.48capybara 2 4.48

This is a big improvement, because differences areso much simpler to work mentally with than ratios.The difference between each successive value of his 1, while f increases by 2 units each time. Thefact that the logs of the f ′s increase twice as quicklyis the same as saying that f is proportional to thesquare of h.

Log-log plots

Even better, the logarithms can be interpreted visu-ally using a graph, as shown on the next page. Theslope of this type of log-log graph gives the powerp. Although it is also possible to extract the pro-portionality constant, c, from such a graph, the pro-portionality constant is usually much less interestingthan p. For instance, we would suspect that if p = 2for rodents, then it might also equal 2 for frogs orants. Also, p would be the same regardless of whatunits we used to measure the variables. The con-stant c, however, would be different if we used dif-ferent units, and would also probably be different forother types of animals.

174 Lab Appendix 5: Finding Power Laws from Data

Appendix 6: Using the Photogate

The photogate

The photogate is a U-shaped thing about 10 cmacross, with an invisible infrared beam going acrossthe gap of the U, like the infrared beam of a TV re-mote control. When something blocks the beam, anelectrical signal is sent through a wire to the com-puter. We will use the photogate by sending movingobjects through it. The computer tells you for howlong the photogate was blocked, allowing you to cal-culate the speed of the object as it passed through.Plug the photogate into the DG1 plug on the inter-face box.

Using the software

If you’re using the ULI interface (beige box), useLogger Pro 2, and make sure the interface box isturned on before you boot up the computer.

If you’re using the USB interface, use Logger Pro 3.

From the Start menu at the lower left corner of thescreen, run Logger Pro (in Programs>Vernier Soft-ware). It asks for permission to scan for the rightport — say OK. (If it complains that it can’t find theport, you may be able to fix the problem if you quitLogger Pro, power the interface off and on again,and then get back in Logger Pro and try again.)

The next step depends on what mode you are usingthe software in.

Using the software in different modes

For various labs, there will be three different modesin which we’ll use the software. From the File menu,do Open, and locate the file you need:

Mode for measuring how long the photogatewas blocked: Probes & Sensors > Photogate> One Gate Timer

Mode for measuring the time between two in-terruptions of the photogate: ...

Mode for measuring the period of a pendulum:Probes & Sensors > Photogate > Pendulum

If there is no button for collecting data, it’s becausethe interface box wasn’t turned on when you bootedup. Reboot.

Using the data

Often you may find that the software rounds off tooseverely. For instance, when you’re in the mode formeasuring how long the photogate was blocked, youwant more than the three decimal places it offers bydefault in the Delta-T column. To fix this, double-click on the title of the Delta-T column, and selecta greater number of significant figures.

176 Lab Appendix 6: Using the Photogate

Appendix 7: Using a Multimeter

The most convenient instrument for measuring cur-rents and voltage differences is called a digital mul-timeter (DMM), or simply a multimeter. ‘Digital’means that it shows the thing being measured ona calculator-style LCD display. ‘Multimeter’ meansthat it can measure current, voltage, or resistance,depending on how you have it set up. Since we havemany different types of multimeters, these instruc-tions only cover the standard rules and methods thatapply to all such meters. You may need to check withyour instructor regarding a few of the particulars forthe meter you have available.

Measuring current

When using a meter to measure current, the me-ter must be in series with the circuit, so that everyelectron going by is forced to go through the me-ter and contribute to a current in the meter. Manymultimeters have more than one scale for measur-ing a given thing. For instance, a meter may havea milliamp scale and an amp scale. One is used formeasuring small currents and the other for large cur-rents. You may not be sure in advance what scaleis appropriate, but that’s not big problem — onceeverything is hooked up, you can try different scalesand see what’s appropriate. Use the switch or but-tons on the front to select one of the current scales.The connections to the meter should be made at the‘common’ socket (‘COM’) and at the socket labeled‘A’ for Amperes.

Measuring voltage

For a voltage measurement, use the switch or but-tons on the front to select one of the voltage scales.(If you forget, and hook up the meter while theswitch is still on a current scale, you may blow afuse.) You always measure voltage differences witha meter. One wire connects the meter to one pointin the circuit, and the other connects the meter toanother point in a circuit. The meter measures thedifference in voltage between those two points. Forexample, to measure the voltage across a resistor,you must put the meter in parallel with the resistor.The connections to the meter should be made at the‘common’ socket (‘COM’) and at the socket labeled‘V’ for Volts.

Blowing a fuse is not a big deal.

If you hook up your multimeter incorrectly, it is pos-sible to blow a fuse inside. This is especially likely tohappen if you set up the meter to measure current(meaning it has a small internal resistance) but hookit up in parallel with a resistor, creating a large volt-age difference across it. Blowing a fuse is not a bigproblem, but it can be frustrating if you don’t real-ize what’s happened. If your meter suddenly stopsworking, you should check the fuse.

178 Lab Appendix 7: Using a Multimeter

Appendix 8: High Voltage Safety Checklist

Name:

Never work with high voltages by yourself.

Do not leave HV wires exposed - make surethere is insulation.

Turn the high-voltage supply off while workingon the circuit.

When the voltage is on, avoid using both handsat once to touch the apparatus. Keep one hand inyour pocket while using the other to touch the ap-paratus. That way, it is unlikely that you will get ashock across your chest.

It is possible for an electric current to causeyour hand to clench involuntarily. If you observe thishappening to your partner, do not try to pry theirhand away, because you could become incapacitatedas well — simply turn off the switch or pull the plugout of the wall.

180 Lab Appendix 8: High Voltage Safety Checklist

Appendix 9: Laser Safety Checklist

Name:

Before beginning a lab using lasers, make sure youunderstand these points, initial them, and show yoursafety checklist to your instructor. If you don’t un-derstand something, ask your instructor for clarifi-cation.

The laser can damage your eyesight perma-nently if the beam goes in your eye.

When you’re not using the laser, turn it off orput something in front of it.

Keep it below eye level and keep the beam hor-izontal. Don’t bend or squat so that your eye is nearthe level of the beam.

Keep the beam confined to your own lab bench.Whenever possible, orient your setup so that thebeam is going toward the wall. If the beam is goingto go off of your lab bench, use a backpack or a boxto block the beam.

Don’t let the beam hit shiny surfaces such asfaucets, because unpredictable reflections can result.

182 Lab Appendix 9: Laser Safety Checklist

Appendix 10: The Open Publication License

Copyright (c) 1999-2001 B. Crowell and V. Roundy.All rights reserved.

These materials are open-content licensed under theOPL 1.0 license. A copy of the license is given below,and the original is available at http://opencontent.org.

LICENSE

Terms and Conditions for Copying, Distributing, andModifying

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1. You may copy and distribute exact replicas of theOpenContent (OC) as you receive it, in any medium,provided that you conspicuously and appropriatelypublish on each copy an appropriate copyright no-tice and disclaimer of warranty; keep intact all thenotices that refer to this License and to the absenceof any warranty; and give any other recipients ofthe OC a copy of this License along with the OC.You may at your option charge a fee for the mediaand/or handling involved in creating a unique copyof the OC for use offline, you may at your optionoffer instructional support for the OC in exchangefor a fee, or you may at your option offer warrantyin exchange for a fee. You may not charge a fee forthe OC itself. You may not charge a fee for the soleservice of providing access to and/or use of the OCvia a network (e.g., the Internet), whether it be viathe world wide web, FTP, or any other method.

2. You may modify your copy or copies of the Open-Content or any portion of it, thus forming worksbased on the Content, and distribute such modifica-tions or work under the terms of Section 1 above,provided that you also meet all of these conditions:

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184 Lab Appendix 10: The Open Publication License

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185

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