Download the lab manual for Physics 206/211. - Light and Matter

Lab Manual for Physics 206/211

Benjamin CrowellFullerton College

www.lightandmatter.com

Copyright (c) 1999-2011 by B. Crowell. This lab manual is subject to the Creative Commons CC-BY-SAlicense.

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Contents

1 Electricity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Resistivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 The Loop and Junction Rules . . . . . . . . . . . . . . . . . . . . . . . . . 144 Electric Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 Magnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 Electromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 The Charge to Mass Ratio of the Electron. . . . . . . . . . . . . . . . . . . . . 349 Radioactivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3810 Two-Source Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . 4211 Refraction and Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4412 Geometric Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4813 Wave Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5214 The Photoelectric Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 5615 Electron Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6016a Setup of the Spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . 6416b The Mass of the Electron . . . . . . . . . . . . . . . . . . . . . . . . . . . 6816c The Nitrogen Molecule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

Appendix 1: Format of Lab Writeups . . . . . . . . . . . . . . . . . . . . . . 80Appendix 2: Basic Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . 82Appendix 3: Propagation of Errors . . . . . . . . . . . . . . . . . . . . . . . 88Appendix 4: Graphing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90Appendix 5: Finding Power Laws from Data . . . . . . . . . . . . . . . . . . . . 92Appendix 6: High Voltage Safety Checklist . . . . . . . . . . . . . . . . . . . . 94Appendix 7: Laser Safety Checklist . . . . . . . . . . . . . . . . . . . . . . . 96

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1 Electricity

Apparatusscotch taperubber rodheat lampfurbits of paperrods and strips of various materials30-50 cm rods, and angle brackets, for hanging chargedrodspower supply (Thornton), in lab benches . .1/groupmultimeter (PRO-100), in lab benches . . . . 1/groupalligator clipsflashlight bulbsspare fuses for multimeters — Let students replacefuses themselves.

GoalsDetermine the qualitative rules governing elec-trical charge and forces.

Light up a lightbulb, and measure the currentthrough it and the voltage difference across it.

IntroductionNewton’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 some

negative mass, which would fall up instead of down(like the “upsydaisium” of Rocky and Bullwinklefame).

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, greenrepels green, and blue repels blue. The force be-tween two different “colors” is attractive: red andgreen attract each other, as do green and blue, andred and blue.

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 oth-ers thought there was only a single type. In the firstpart of this lab, you will try to find out experimen-tally how many types of electrical charge there are.

The unit of charge is the coulomb, C; one coulombis defined as the amount of charge such that if twoobjects, each with a charge of one coulomb, are onemeter apart, the magnitude of the electrical forcebetween them is 9 × 109 N. Practical applicationsof electricity usually involve an electric circuit, inwhich charge is sent around and around in a cir-cle and recycled. Electric current, I, measures howmany coulombs per second flow past a given point; ashorthand for units of C/s is the ampere, A. Voltage,V , measures the electrical potential energy per unitcharge; its units of J/C can be abbreviated as volts,V. Making the analogy between electrical interac-tions and gravitational ones, voltage is like height.Just as water loses gravitational potential energy bygoing over a waterfall, electrically charged particleslose electrical potential energy as they flow througha circuit. The second part of this lab involves build-ing an electric circuit to light up a lightbulb, andmeasuring both the current that flows through thebulb and the voltage difference across it.

6 Lab 1 Electricity

ObservationsThe 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. (I will deduct 25% from your grade if youmix calculations and raw data.)

(4) Never write numbers without units. Withoutunits, a number is meaningless. There is a big dif-ference bewteen “Johnny is six” and “Johnny is sixfeet.” (I will deduct 25% from your grade if youwrite numbers without units.)

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 withyour instructor at various points.

A Inferring the rules of electrical repulsion andattraction

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.

To keep this lab from being too long, the class willpool its data for part A. Your instructor will organizethe results on the whiteboard.

i. Repulsion and/or attraction

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

A) Electrical forces are always attractive.

R) Electrical forces are always repulsive.

AR) Electrical forces are sometimes attractive andsometimes repulsive.

Interpretation: Once the class has tested these hy-potheses thoroughly, we will discuss what this im-plies about how many different types of charge theremight be.

ii. 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 it off the roll.

U0) Objects that have not been specially preparedare immune to electrical forces.

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

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

UAR) Unprepared objects can participate in elec-trical forces with prepared objects, and the forcesinvolved can be either repulsive or attractive.

These four hypotheses are mutually exclusive.

Once the class has tested these hypotheses thor-oughly, we will discuss what practical implicationsthis has for planning the observations for part iii.

iii. Rules of repulsion and/or attraction and thenumber of types of charge

Test the following mutually exclusive hypotheses:

1A) There is only one type of electric charge, andthe force is always attractive.

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1R) There is only one type of electric charge, andthe force is always repulsive.

2LR) There are two types of electric charge, callthem X and Y. Like charges repel (X repels X andY repels Y) and opposite charges attract (X and Yattract each other).

2LA) There are two types of electric charge. Likecharges attract and opposite charges repel.

3LR) There are three types of electric charge, X, Yand Z. Like charges repel and unlike charges attract.

On the whiteboard, we will make a square table,in which the rows and columns correspond to thedifferent objects you’re testing against each otherfor attraction and repulsion. To test hypotheses 1Athrough 3LR, you’ll need to see if you can success-fully explain your whole table by labeling the objectswith only one label, X, or whether you need two orthree.

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 of different types of plastic, orglass. This could affect your conclusions, so you maywant to check, for example, whether two rods withthe same diameter, that you think are made of thesame material, actually weigh the same.

In general, you will find that some materials, andsome combinations of materials, are more easily charg-ed than others. For example, if you find that themahogony rod rubbed with the weasel fur doesn’tcharge well, then don’t keep using it! The white plas-tic strips tend to work well, so don’t neglect them.

Once we have enough data in the table to reach adefinite conclusion, we will summarize the resultsfrom part A and then discuss the following examplesof incorrect reasoning about this 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 bottom was negative.”(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.”

B Measuring current and voltage

As shown in the figure, measuring current and volt-age requires hooking the meter into the circuit intwo completely different ways.

The arrangement for the ammeter is called a seriescircuit, because every charged particle that travelsthe circuit has to go through each component in arow, one after another. The series circuit is arrangedlike beads on a necklace.

The setup for the voltmeter is an example of a paral-lel circuit. A charged particle flowing, say, clockwisearound the circuit passes through the power supplyand then reaches a fork in the road, where it has achoice of which way to go. Some particles will passthrough the bulb, others (not as many) through themeter; all of them are reunited when they reach thejunction on the right.

Students tend to have a mental block against set-ting up the ammeter correctly in series, because itinvolves breaking the circuit apart in order to in-sert the meter. To drive home this point, we willact out the process using students to represent thecircuit components. If you hook up the ammeter in-correctly, in parallel rather than in series, the meterprovides an easy path for the flow of current, so alarge amount of current will flow. To protect themeter from this surge, there is a fuse inside, whichwill blow, and the meter will stop working. This isnot a huge tragedy; just ask your instructor for areplacement fuse and open up the meter to replace

8 Lab 1 Electricity

it.

Unscrew your lightbulb from its holder and lookclosely at it. Note that it has two separate elec-trical contacts: one at its tip and one at the metalscrew threads.

Turn the power supply’s off-on switch to the off po-sition, and turn its (uncalibrated) knob to zero. Setup the basic lightbulb circuit without any meter init. There is a rack of cables in the back of the roomwith banana-plug connectors on the end, and mostof your equipment accepts these plugs. To connectto the two brass screws on the lightbulb’s base, you’llneed to stick alligator clips on the banana plugs.

Check your basic circuit with your instructor, thenturn on the power switch and slowly turn up theknob until the bulb lights. The knob is uncalibratedand highly nonlinear; as you turn it up, the voltage itproduces goes zerozerozerozerozerosix! To light thebulb without burning it out, you will need to find aposition for the knob in the narrow range where itrapidly ramps up from 0 to 6 V.

Once you have your bulb lit, do not mess with theknob on the power supply anymore. You do not evenneed to switch the power supply off while rearrang-ing the circuit for the two measurements with themeter; the voltage that lights the bulb is only abouta volt or a volt and a half (similar to a battery), soit can’t hurt you.

We have a single meter that plays both the role ofthe voltmeter and the role of the ammeter in this lab.Because it can do both these things, it is referred toas a multimeter. Multimeters are highly standard-ized, and the following instructions are generic onesthat will work with whatever meters you happen tobe using in this lab.

Voltage difference

Two wires connect the meter to the circuit. At theplaces where three wires come together at one point,you can plug a banana plug into the back of anotherbanana plug. At the meter, make one connectionat the “common” socket (“COM”) and the other atthe socket labeled “V” for volts. The common plug iscalled that because it is used for every measurement,not just for voltage.

Many multimeters have more than one scale for mea-suring a given thing. For instance, a meter mayhave a millivolt scale and a volt scale. One is usedfor measuring small voltage differences and the otherfor large ones. You may not be sure in advance whatscale is appropriate, but that’s not a big problem —

once everything is hooked up, you can try differentscales and see what’s appropriate. Use the switchor buttons on the front to select one of the voltagescales. By trial and error, find the most precise scalethat doesn’t cause the meter to display an error mes-sage about being overloaded.

Write down your measurement, with the units ofvolts, and stop for a moment to think about whatit is that you’ve measured. Imagine holding yourbreath and trying to make your eyeballs pop outwith the pressure. Intuitively, the voltage differenceis like the pressure difference between the inside andoutside of your body.

What do you think will happen if you unscrew thebulb, leaving an air gap, while the power supplyand the voltmeter are still going? Try it. Inter-pret your observation in terms of the breath-holdingmetaphor.

Current

The procedure for measuring the current differs onlybecause you have to hook the meter up in series andbecause you have to use the “A” (amps) plug on themeter and select a current scale.

In the breath-holding metaphor, the number you’remeasuring now is like the rate at which air flowsthrough your lips as you let it hiss out. Based onthis metaphor, what do you think will happen tothe reading when you unscrew the bulb? Try it.

Discuss with your group and check with your in-structor:(1) What goes through the wires? Current? Volt-age? Both?(2) Using the breath-holding metaphor, explain whythe voltmeter needs two connections to the circuit,not just one. What about the ammeter?While waiting for your instructor to come aroundand discuss these questions with you, you can go onto the next part of the lab.

Resistance

The ratio of voltage difference to current is calledthe resistance of the bulb, R = ∆V/I. Its units ofvolts per amp can be abbreviated as ohms, Ω (capitalGreek letter omega).

Calculate the resistance of your lightbulb. Resis-tance is the electrical equivalent of kinetic friction.Just as rubbing your hands together heats them up,objects that have electrical resistance produce heatwhen a current is passed through them. This is whythe bulb’s filament gets hot enough to heat up.

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When you unscrew the bulb, leaving an air gap, whatis the resistance of the air?

Ohm’s law is a generalization about the electricalproperties of a variety of materials. It states that theresistance is constant, i.e., that when you increasethe voltage difference, the flow of current increasesexactly in proportion. If you have time, test whetherOhm’s law holds for your lightbulb, by cutting thevoltage to half of what you had before and checkingwhether the current drops by the same factor. (Inthis condition, the bulb’s filament doesn’t get hotenough to create enough visible light for your eye tosee, but it does emit infrared light.)

List of materials for static electricity

You don’t have to know anything about what thevarious materials are in order to do this lab, but hereis a list for use by instructors and the lab technician:

• scotch tape (used as two different objects, topand bottom)

• teflon fabric (brown, coarse)

• teflon rods (white, rigid, slippery, skinny)

• PVC pipe

• polyurethane rods (brown, flexible)

• nylon (?) fabric (blue)

• fur

Notes For Next Week(1) Next week, when you turn in your writeup forthis lab, you also need to turn in a prelab writeupfor the next lab. The prelab questions are listedat the end of the description of that lab in the labmanual. Never start a lab without understandingthe answers to all the prelab questions; if you turnin partial answers or answers you’re unsure of, dis-cuss the questions with your instructor or with otherstudents to make sure you understand what’s goingon.

(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.”

Rules and OrganizationCollection of raw data is work you share with yourlab partners. Once you’re done collecting data, youneed to do your own analysis. E.g., it is not okay fortwo people to turn in the same calculations, or on alab requiring a graph for the whole group to makeone graph and turn in copies.

You’ll do some labs as formal writeups, others asinformal “check-off” labs. As described in the syl-labus, they’re worth different numbers of points, andyou have to do a certain number of each type by theend of the semester.

The format of formal lab writeups is given in ap-pendix 1 on page 80. The raw data section mustbe contained in your bound lab notebook. Typicallypeople word-process the abstract section, and anyother sections that don’t include much math, andstick the printout in the notebook to turn it in. Thecalculations and reasoning section will usually justconsist of hand-written calculations you do in yourlab notebook. You need two lab notebooks, becauseon days when you turn one in, you need your otherone to take raw data in for the next lab. You mayfind it convenient to leave one or both of your note-books in the cupboard at your lab bench wheneveryou don’t need to have them at home to work on;this eliminates the problem of forgetting to bringyour notebook to school.

For a check-off lab, the main thing I’ll pay attentionto is your abstract. The rest of your work for acheck-off lab can be informal, and I may not ask tosee it unless I think there’s a problem after readingyour abstract.

10 Lab 1 Electricity

11

2 Resistivity

Apparatuspower supply (Thornton), in lab benches . .1/groupmultimeter (PRO-100), in lab benches . . . . 1/groupdigital multimeters (FlukeHP) . . . . . . . . . . . . 1/groupresistors of various valuesalligator clipsspare fuses for multimeters — Let students replacefuses themselves.Play-Doh (Crayloa super soft dough, $10/3 lb atschool supply store on Commonwealth)1 g aluminum slotted weights for use as electricalcontacts

GoalsMeasure how the electrical resistance of a cylin-der depends on its dimensions.

Invent a constant of proportionality incorpo-rated into this relationship, giving a measureof the intrinsic electrical properties of a mate-rial called its resistivity.

IntroductionYour 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 proved at theturn of the 20th century.

We observe that some materials, such as metals,conduct electricity more easily than others, such aswood. This suggests that measurements of the re-sistance of an object made out of a certain sub-stance can be used to learn things about the atomic-

level structure of the substance (e.g., how many freecharge carriers it has per unit volume), or as a testto identify an unknown substance.

ObservationsA A known resistor

In lab 1, you found the resistance of a lightbulb.To keep things simple, that setup used only a sin-

12 Lab 2 Resistivity

gle multimeter, but that required rearranging howeverything was hooked up every time you wantedto switch between measuring voltage and measuringcurrent. In today’s lab, you will use a similar setup,but with two meters simultaneously connected, onefor each purpose. Build your setup, and test it bymeasuring data on a known resistor. To avoid creat-ing large currents or making the resistor so hot thatit will burn your fingers, use one with a fairly highvalue, at least in the kiloohm range.

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 above.

As a test of whether your resistor is actually ohmic,take data at two different voltages and determinewhether the current varies in proportion.

B Dependence of resistance on dimensions

For a fixed substance such as sand or plywood, theresistance is not a fixed number of ohms. The resis-tance of a sample depends on its size, its shape, andthe way in which the electrical contacts are made toit. In this part of the lab you will take measurementswith cylindrical samples of play-doh, with the elec-trical contacts made using the thin disk-shaped 1 galuminum weights.

Two different numbers are required in order to de-fine the dimensions of a cylinder. (There are variousways to define these; you can make your own choice.)Measure the resistance for different values of thesetwo variables. Practice control of variables by keep-ing one constant while varying the other. You willlearn the most by taking extreme values of the vari-ables.

Leaving the power supply on for a long time causescorrosion to the electrical contacts and chemical changesin the play-doh, so don’t do that.

To get good results:

Quite a bit of the resistance comes from thesurface of contact between the aluminum diskand the play-dough. What seems to matter themost is that you need a very smooth surface,because on a rough surface, most of the metalisn’t even making contact.

It seems to help if you cut into the play-doughand then press the play-dough against the con-tact with your fingers to make better contact.

I got better results by subtracting the resis-tance found by placing the two contacts very

close, where the resistance was almost entirelydue to the contacts.

AnalysisFor each variable in part B, use the graphing tech-nique given in appendix 5 to see if you can find apower law that relates the resistance to that vari-able. If the results are at least approximately con-sistent with power laws, then you can combine themto create a relationship of the formR = ρ(. . .)p(. . .)q,where (. . .) represents the two variables, p and q areconstants found from your graph, and the constantof proportionality ρ (Greek letter rho) is a propertyof the substance, called its resistivity. If you findthat p and q have values that are close to nice sim-ple numbers, carry out your analysis under the as-sumption that those simple numbers are right. Whatunits does ρ have? Find the resistivity of play-doh.

PrelabThe 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 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 = ?

13

3 The Loop and Junction Rules

ApparatusDC power supply (Thornton) . . . . . . . . . . . . . 1/groupmultimeter (PRO-100, in lab benches) . . . . 1/groupresistors

GoalTest the loop and junction rules in two electricalcircuits.

IntroductionIf 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.

ObservationsA 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 equal re-sistors. Use resistors with values in the range of (say1 kΩ to 10 MΩ). If they’re much higher than that,the currents will be too low for the PRO-100 me-ters to measure accurately. If they’re much smallerthan that, you could burn up the resistors, and themultimeter’s internal resistance when used as an am-meter might not be negligible in comparison. Insertyour multimeter in the circuit to measure all three

14 Lab 3 The Loop and Junction Rules

currents that you need in order to test the junctionrule.

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.


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 quitetrue, but the small error could have been becausethe resistor’s value was off by a few percent com-pared to the color-code value.”

Self-CheckDo the analysis in lab.

AnalysisDiscuss whether you think your observations agreewith the loop and junction rules, taking into accountsystematic and random errors.

Programmed Introduction to Prac-tical Electrical CircuitsThe 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.

15

4 Electric Fields

Apparatusboard and U-shaped proberulerDC power supply (Thornton)multimeterscissorsstencils for drawing electrode shapes on paper

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

To examine the electric fields around certaincharge distributions.

IntroductionBy 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 that

can be defined unambiguously, the same is true forvoltages. 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 topograph-ical map, curves are drawn to connect points hav-ing the same height above sea level. For instance, acone-shaped volcano would be represented by con-centric circles. The outermost circle might connectall the points at an altitude of 500 m, and inside ityou might have concentric circles showing higher lev-els such as 600, 700, 800, and 900 m. Now imaginea similar representation of the voltage surroundingan isolated point charge. There is no “sea level”here, so we might just imagine connecting one probeof the voltmeter to a point within the region tobe mapped, and the other probe to a fixed refer-ence point very far away. The outermost circle onyour map might connect all the points having a volt-age of 0.3 V relative to the distant reference point,and within that would lie a 0.4-V circle, a 0.5-Vcircle, and so on. These curves are referred to asconstant-voltage curves, because they connect pointsof equal voltage. In this lab, you are going to mapout constant-voltage curves, but not just for an iso-lated point charge, which is just a simple examplelike the idealized 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 distributed

16 Lab 4 Electric Fields

in such a way that there is no longer any force onthem. 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 withpattern 1 on page 18.

A photo of the apparatus, being used with pattern 3 onpage 18.

MethodThe 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 18) 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. Connect the multimeter so that you can mea-

17

sure the voltage difference across the terminals of thevoltage source. Adjust the voltage source to give 8volts.

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.

If you’re using the PRO-100 meters, they will tryto outsmart you by automatically choosing a range.Most people find this annoying. To defeat this mis-feature, press the RANGE button, and you’ll see theAUTO indicator on the screen turn off.

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.


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-CheckCalculate 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.

AnalysisOn each plot, find the strongest and weakest electricfields, and calculate them.

On top of your plots, draw in electric field vectors.You will then have two different representations ofthe field superimposed on one another.

As always when drawing vectors, the lengths of thearrows should represent the magntitudes of the vec-tors, although you don’t need to calculate them allnumerically or use an actual scale. Remember thatelectric field vectors are always perpendicular to constant-voltage curves. The electric field lines point fromhigh voltage to low voltage, just as the force on arolling ball points downhill.

18 Lab 4 Electric Fields

19

5 Relativity

Apparatusmagnetic balance . . . . . . . . . . . . . . . . . . . . . . . . . 1/groupmeter stick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/groupmultimeter (BK or PRO-100, not HP) . . . . 1/grouplaser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1/groupvernier calipers . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/groupphotocopy paper, for use as a weightDC power supply (Mastech, 30 A)box of special cablesscissors

GoalMeasure the speed of light.

IntroductionOersted 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.

SetupThe 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 raise

wire A. The torque made by the magnetic repulsionis 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 on top of wire A, and adjust the current sothat the balance returns to the position it originallyhad, as determined by the laser dot on the wall. Younow know that the gravitational torque acting on theoriginal apparatus (everything except for the weight)is back to zero, so the only torques acting are thetorque of gravity on the staple and the magnetictorque. Since both these torques are applied at thesame distance from the axis, the forces creating thesetorques must be equal as well. You can thereforeinfer the magnetic force that was acting.

For a weight, you can carefully and accurately cuta small rectangular piece out of a sheet of photo-copy paper. In fall 2013, my students found that500 sheets of SolCopy 20 lb paper were 2307.0 g.About 1/100 of a sheet seemed to be a good weightto use.

It’s very important to get the wires A and B perfectlyparallel. The result depends strongly on the smalldistance R between their centers, and if the wiresaren’t straight and parallel, you won’t even have awell defined value of R.

The following technique allows R to be measuredaccurately. The idea is to compare the position ofthe laser spot on the wall when the balance is inits normal position, versus the position where thewires are touching. Using a small-angle approxi-mation, you can then find the angle θr by whichthe reflected beam moved. This is twice the angleθm = θr/2 by which the mirror moved.1 Once you

1To see this, imagine the following example that is unreal-

20 Lab 5 Relativity

know the angle by which the moving arm of the ap-paratus moved, you can accurately find the air gapbetween the wires, and then add in twice the radiusof the wires, which can be measured accurately withvernier calipers. For comparison, try to do as good ajob as you can of measuring R directly by position-ing the edges of the vernier calipers at the centersof the wires. If the two values of R don’t agree, goback and figure out what went wrong; one possibil-ity is that your wire is slightly bent and needs to bestraightened.

You need to minimize the resistance of the appara-tus, or else you won’t be able to get enough currentthrough it to cancel the weight of the staple. Mostof the resistance is at the polished metal knife-edgesthat the moving part of the balance rests on. It maybe necessary to clean the surfaces, or even to freshenthem a little with a file to remove any layer of oxi-dation. Use the separate BK meter to measure thecurrent — not the meter built into the power supply.

The power supply has some strange behavior thatmakes it not work unless you power it up in exactlythe right way. It has four knobs, going from left toright: (1) current regulation, (2) over-voltage pro-tection, (3) fine voltage control, (4) coarse voltagecontrol. Before turning the power supply on, turnknobs 1 and 2 all the way up, and knobs 3 and 4 allthe way down. Turn the power supply on. Now useknobs 3 and 4 to control how much current flows.

AnalysisThe first figure below shows a model that explainsthe repulsion felt by one of the charges in wire Adue to all the charges in wire B. This is representedin the frame of the lab. For convenience of anal-ysis, we give the model some unrealistic features:rather than having positively charged nuclei at restand negatively charged electrons moving, we pretendthat both are moving, in opposite directions. Sincewire B has zero net density of charge everywhere, itcreates no electric fields. (If you like, you can ver-ify this during lab by putting tiny pieces of papernear the wires and verifying that they do not feelany static-electrical attraction.) Since there is noelectric field, the force on the charge in wire A mustbe purely magnetic.

The second figure shows the same scene from the

istic but easy to figure out. Suppose that the incident beamis horizontal, and the mirror is initially vertical, so that thereflected beam is also horizontal. If the mirror is then tiltedbackward by 45 degrees, the reflected beam will be straightup, θr = 90 degrees.

point of view of the charge in wire A. This chargeconsiders itself to be at rest, and it also sees the light-colored charges in B as being at rest. In this framethe dark-colored charges in B are the only ones mov-ing, and they move with twice the speed they had inthe lab frame. In this frame, the particle in A is atrest, so it can’t feel any magnetic force. The forceis now considered to be purely electric. This electricforce exists because the dark charges are relativisti-cally contracted, which makes them more dense thantheir light-colored neighbors, causing a nonzero netdensity of charge in wire B.

We’ve considered the force acting on a single chargein wire A. The actual force we observe in the ex-periment is the sum of all the forces acting on allsuch charges (of both signs). As in the slightly dif-ferent example analyzed in section 23.3 of Light andMatter, this effect is proportional to the product ofthe speeds of the charges in the two wires, dividedby c2. Therefore the effect must be proportional tothe product of the currents over c2. In this exper-iment, the same current flows through wire A andthen comes back through B in the opposite direction,so we conclude that the force must be proportionalto I2/c2.

In the second frame, the force is purely electrical,and as shown in example 4 in section 22.3 of Lightand Matter, the electric field of a charged wire fallsoff in proportion to 1/R, where R is the distancefrom the wire. Electrical forces are also proportional

21

to the Coulomb constant k.

The longer the wires, the more charges interact, sowe must also have a proportionality to the length `.

Putting all these factors together, we find that theforce is proportional to kI2`/c2R. We can easily ver-ify that the units of this expression are newtons, sothe only possible missing factor is something unit-less. This unitless factor turns out to be 2 — es-sentially the same 2 found in example 14 in section22.7. The result for the repulsive force between thetwo wires is

F =k

c2· 2I2`

R.

By solving this equation you can find c. Your finalresult is the speed of light, with error bars. Comparewith the previously measured value of c and give aprobabilistic interpretation, as in the examples inappendix 2.

In your writeup, give both the values of R (laser andeyeball). The laser technique is inherently better, sothat’s the value you should use in extracting c, but Iwant to see both values of R because some groups inthe past have had a bigger discrepancy than I wouldhave expected. If you have a large discrepancy, getmy attention during lab and we can see whether itmight be due to a bent wire, or some other cause.


Do the laser safety checklist, Appendix 7, 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 Show that the equation for the force betweenthe wires has units of newtons.

P2 Do the algebra to solve for c in terms of themeasured quantities.

22 Lab 5 Relativity

23

6 Magnetism

Apparatusbar magnet (stack of 6 Nd)compassHall effect magnetic field probesLabPro interfaces, DC power supplies, and USB ca-bles2-meter stickHeath solenoids . . . . . . . . . . . . . . . . . . . . . . . . . . .2/groupMastech power supply . . . . . . . . . . . . . . . . . . . . 1/groupwood blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2/groupPRO-100 multimeter (in lab bench . . . . . . . .1/groupanother multimeter . . . . . . . . . . . . . . . . . . . . . . . 1/groupD-cell batteries and holdersCenco decade resistor box . . . . . . . . . . . . . . . . 1/group

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

IntroductionA Variation of Field With Distance: Deflection

of 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 from north.

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 top figure on the page three pages afterthis one.

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 figure atthe end of the lab. As part of your prelab, you willuse vector addition to find an equation for Bm/Be,the magnet’s field in units of the Earth’s, in termsof the deflection angle θ. For your first point, findthe distance r at which the deflection is 70 degrees;this angle is chosen because it’s about as big as itcan 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?

The lab benches contain iron or steel parts that dis-tort the magnetic field. You can easily observe thissimply by putting a compass on the top of the benchand sliding it around to different places. To workaround this problem, lay a 2-meter stick across thespace between two lab benches, and carry out theexperiment along the line formed by the stick. Evenin the air between the lab benches, the magneticfield due to the building materials in the building issignificant, and this field varies from place to place.Therefore you should move the magnet while keepingthe compass in one place. Then the field from thebuilding becomes a fixed part of the background ex-perienced by the compass, just like the earth’s field.

Note that the measurements are very sensitive to therelative position and orientation of the bar magnetand compass.

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.

B 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 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 CH 1 of the LabPro interface, connect

24 Lab 6 Magnetism

Part A, measuring the variation of the bar magnet’s field with respect to distance.

Part B, a different method of measuring the variation of field with distance. The solenoids are shown in cross-section,with empty space on their interiors and their axes running right-left.

the interface to the computer’s USB port, and plugthe interface’s DC power supply in to it. Start upversion 3 of Logger Pro, and it will automaticallyrecognize the probe and start displaying magneticfields on the screen, in units of mT (millitesla). The

probe has two ranges, one that can read fields up to0.3 mT, and one that goes up to 6.4 mT. Select themore sensitive 0.3 mT scale using the switch on theprobe.

The technique is shown in the bottom figure on thelast page of the lab. Identical solenoids (cylindricalcoils of wire) are positioned with their axes coincid-ing, by lining up their edges with the edge of the labbench. When an electrical current passes through acoil, it creates a magnetic field. At distances that arelarge compared to the size of the solenoid, we expectthat this field will have the same universal pattern aswith any magnetic dipole. The sensor is positionedon the axis, with wood blocks (not shown) to hold itup. One solenoid is fixed, while the other is movedto different positions along the axis, including posi-tions (more distant than the one shown) at which weexpect its contribution to the field at the sensor tobe of the universal dipole form.

25

The key to the high precision of the measurementis that in this configuration, the fields of the twosolenoids can be made to cancel at the position of theprobe. Because of the solenoids’ unequal distancesfrom the probe, this requires unequal currents. Be-cause the fields cancel, the probe can be used on itsmost sensitive and accurate scale; it can also be ze-roed when the circuits are open, so that the effect ofany ambient field is removed. For example, supposethat at a certain distance rm, the current Im throughthe moving coil has to be five times greater than thecurrent If through the fixed coil at the constant dis-tance rf . Then we have determined that the fieldpattern of these coils is such that increasing the dis-tance along the axis from rf to rm causes the fieldto fall off by a factor of five.

It’s a good idea to take data all the way down torm = 0, since this makes it possible to see on agraph where the field does and doesn’t behave like adipole. Note that the distances rf and rm can’t bemeasured directly with good precision.

The Mastech power supply is capable of delivering alarge amount of current, so it can be used to provideIm, which needs to be high when rm is large. Thepower supply has some strange behavior that makesit not work unless you power it up in exactly theright way. It has four knobs, going from left to right:(1) current regulation, (2) over-voltage protection,(3) fine voltage control, (4) coarse voltage control.Before turning the power supply on, turn knobs 1and 2 all the way up, and knobs 3 and 4 all the waydown. Turn the power supply on. Now use knobs 3and 4 to control how much current flows.

At large values of rm, it can be difficult to get apower supply to give a small enough If . Try using abattery, and further reducing the current by placinganother resistance in series with the coil. The Cencodecade resistance boxes can be used for this purpose;they are variable resistors whose resistance can bedialed up as desired using decimal knobs. Use theplugs on the resistance box labeled H and L.

For every current measurement, make sure to usethe most sensitive possible scale on the meter to getas many sig figs as possible. This is why the am-meter built into the Mastech power supply is notuseful here. I found it to be a hassle to measure Imwith an ammeter, because the currents required wereoften quite large, and I kept inadvertently blowingthe fuse on the milliamp scale. For this reason, youmay actually want to measure Vm, the voltage dif-ference across the moving solenoid. Conceptually,magnetic fields are caused by moving charges, cur-

rent is a measure of moving charge, and thereforecurrent is what is relevant here. But if the DC re-sistance of the coil is fixed, the current and voltageare proportional to one another, assuming that thevoltage is measured directly across the coil and theresistance of the banana-plug connections is eithernegligible or constant.

As shown in a lecture demonstration, deactivatingthe electromagnet requires getting rid of the energystored in the magnetic field, and this can be done inmore than one way. If you use your hand to breakthe circuit by pulling out a banana plug, the energyis dissipated in a spark, and a large value of Im isbeing used the result can be an unpleasant shock.To avoid this, deactivate the moving coil by turningdown the knob on the power supply rather than bybreaking the circuit.


P1 In part A, suppose that when the compass is11.0 cm from the magnet, it is 45 degrees away fromnorth. What is the strength of the bar magnet’s fieldat this location in space, in units of the Earth’s field?

P2 Find Bm/Be in terms of the deflection angle θmeasured in part A. As a special case, you shouldbe able to recover your answer to P1.

AnalysisDetermine the variation of the solenoid’s magneticfield with distance. Look for a power-law relation-ship using the log-log graphing technique describedin appendix 5. Does the power law hold for allthe distances you investigated, or only at large dis-tances? No error analysis is required.

26 Lab 6 Magnetism

27

7 Electromagnetism

Apparatusoscilloscope (Tektronix TDS 1001B) . . . . . . 1/groupmicrophone (RS 33-1067) . . . . . . . . . . . . . for 6 groupsmicrophone (Shure C606) . . . . . . . . . . . . . . for 1 groupvarious tuning forks, mounted on wooden boxessolenoid (Heath) . . . . . . . . . . . . . . . . . . . . . . . . . . 1/group2-meter wire with banana plugs . . . . . . . . . . .1/groupmagnet (stack of 6 Nd) . . . . . . . . . . . . . . . . . . . 1/groupmasking tapestring

GoalsLearn to use an oscilloscope.

Observe electric fields induced by changing mag-netic fields.

Build a generator.

Discover Lenz’s law.

IntroductionPhysicists 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 and

magnetic 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 movingcharges. (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.

The Oscilloscope

An oscilloscope graphs an electrical signal that variesas a function of time. The graph is drawn from leftto right across the screen, being painted in real timeas the input signal varies. The purpose of the os-cilloscope in this lab is to measure electromagneticinduction, but to get familiar with the oscillopscope,we’ll start out by using the signal from a microphoneas an input, allowing you to see sound waves.

The input signal is supplied in the form of a volt-age. The input connector on the front of the os-cilloscope accepts a type of cable known as a BNCcable. A BNC cable is a specific example of coaxialcable (“coax”), which is also used in cable TV, radio,and computer networks. The electric current flows

28 Lab 7 Electromagnetism

in one direction through the central conductor, andreturns 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 larger 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 forthe beam to sweep from left to right would be 100

ms. 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 twopoints in a period where the voltage crossed yourtrigger level.

SetupTo 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.

The figure on the preceding page is a simplified draw-ing of the front panel of a digital oscilloscope, show-ing only the most important controls you’ll need forthis lab. When you turn on the oscilloscope, it willtake a while to start up.

29

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 to nor-mal, which is the mode in which the triggeringworks as I’ve described above. If the triggerlevel is set to a level that the signal never ac-tually reaches, you can play with the knob thatsets the trigger level until you get something.A quick and easy way to do this without trialand error is to use the SET TO 50% button,which automatically sets the trigger level tomidway between the top and bottom peaks ofthe signal.

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.

Preliminary ObservationsNow try observing signals from the microphone.

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.

Qualitative ObservationsIn 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.

Note that although I’ve described the standard wayof triggering a scope, when the time base is verylong, triggering becomes unnecessary. These scopesare programmed so that when the time base is verylong, they simply continuously display traces.

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 atits 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 UnderstandingE 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 LawLenz’s law describes how the clockwise or counter-clockwise direction of the induced electric field’s whirl-pool pattern relates to the changing magnetic field.The main 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 tomake a tomahawk-chopping motion that ends upwith the magnet in the solenoid, observing whetherthe pulse induced is positive or negative. What hap-

pens when you reverse the chopping motion, or whenyou reverse the north and south poles of the mag-net? Try all four possible combinations and recordyour 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. Set the triggeringmode (“Mode”) to Auto.

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).


P1 In the sample oscilloscope trace shown on page29, 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/div-ision. Assume the zero voltage level is at the middle

31

of the vertical scale. (The whole graph can actuallybe shifted up and down using a knob called “posi-tion.”) What is the trigger level currently set to? Ifthe trigger level was changed to 2 mV, what wouldhappen to 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 signal’s pe-riod, or its wavelength? Explain. [Skip this questionif you’re in Physics 222.]

P5 The time-scale for all the signals is determinedby 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?[Skip this question if you’re in Physics 222.]

Self-CheckDetermine which version of Lenz’s law is correct.


33

8 The Charge to Mass Ratio of the Electron

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

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

IntroductionWhy 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 largedistances through the gas without ever colliding witha hydrogen 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 arereally seeing the light from the collisions, not thebeam of electrons itself. The manufacturer of the

34 Lab 8 The Charge to Mass Ratio of the Electron

tube has put in just enough gas to make the beamvisible; more gas would make a brighter beam, butwould cause it to spread out and become too broadto measure 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 6,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.

SetupBefore 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 greenplugs that, in different combinations, allow you toget various low voltage values. Use it to supply 6V to the terminals marked “heater” on the vacuumtube. The tube 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 powersupply goes through both coils to make the magnetic

field. 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.

ObservationsMake 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, a, 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 a that had to be added totheir d values in order to make the two values of Idequal. The results are as follows:

35

serial number a (cm)98-16 0.09849 0.099-08 +0.1599-10 -0.299-17 +0.299-56 +0.3031427 -0.3

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


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

Do the high voltage safety checklist, Appendix 6,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?

AnalysisDetermine q/m, with error bars.

Answer the following questions:

Q1. Thomson started to become convinced duringhis 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.

36 Lab 8 The Charge to Mass Ratio of the Electron

37

9 Radioactivity

Note to the lab technician: The isotope generatorkits came with 250 mL bottles of eluting solution(0.9% NaCl in 0.04M HCl, made with deionized wa-ter). If we ever run out of the solution, we can makemore from materials in the chem stockroom. TheGM counters have 9 V batteries, which should bechecked before lab.

Apparatusisotope generator kitGeiger-Muller (GM) countercomputer with Logger Pro software and LabPro in-terface“grabber” clamp and standwood blocks, 25 mm thickpieces of steel, 17 mm thick

GoalDetermine the properties of an unknown radioactivesource.

IntroductionYou’re a science major, but even if you weren’t, itwould be important for you as a citizen and a voterto understand the properties of radiation. As anexample of an important social issue, many envi-ronmentalists who had previously opposed nuclearpower now believe that its benefits, due to reduc-tion of global warming, outweigh its problems, suchas disposal of waste. To understand such issues, youneed to learn to reason about radioactivity quanti-tatively.

A radioactive substance contains atoms whose nu-clei spontaneously decay into nuclei of a differenttype. Nobody has ever succeeded in finding a phys-ical law that would predict when a particular nu-cleus will “choose” to decay. The process is random,but we can make quantitative statements about howquickly the process tends to happen. A radioactivesubstance has a certain half-life, defined as the timerequired before (on the average) 50% of its nucleiwill have decayed.

SafetyThe radioactive source used in this lab is very weak.It is so weak that it is exempt from government reg-ulation, it can be sent in the mail, and when the col-lege buys new equipment, it is legal to throw out theold sources in the trash. The following table com-pares some radiation doses, including an estimate ofthe typical dose you might receive in this lab. Theseare in units of microSieverts (µSv).

CT scan ∼ 10,000 µSvnatural background per year 2,000-7,000 µSvhealth guidelines for exposure toa fetus

1,000 µSv

flying from New York to Tokyo 150 µSvthis lab ∼ 80 µSvchest x-ray 50 µSv

A variety of experiments seem to show cases in whichlow levels of radiation activate cellular damage con-trol mechanisms, increasing the health of the organ-ism. For example, there is evidence that exposureto radiation up to a certain level makes mice growfaster; makes guinea pigs’ immune systems functionbetter against diphtheria; increases fertility in fe-male humans, trout, and mice; improves fetal mice’sresistance to disease; reduces genetic abnormalitiesin humans; increases the life-spans of flour beetlesand mice; and reduces mortality from cancer in miceand humans. This type of effect is called radiationhormesis. Nobody knows for sure, but it’s possiblethat you will receive a very tiny improvement in yourhealth from the radiation exposure you experience inlab today.

Although low doses of radiation may be beneficial,governments, employers, and schools generally prac-tice a philosophy called ALARA, which means tomake radiation doses As Low As Reasonably Achiev-able. You should adhere to this approach in thislab. In general, internal exposure to radiation pro-duces more of an effect than external exposure, soyou should not eat or drink during this lab, andyou should avoid getting any of the radioactive sub-stances in an open cut. You should also reduce yourexposure by not spending an unnecessarily large amountof time with your body very close to the source, e.g.,you should not hold it in the palm of your hand forthe entire lab period.

38 Lab 9 Radioactivity

The source and the GM counterYou are supplied with a radioactive source packagedinside a small plastic disk about the size of the spin-dle that fits inside a roll of scotch tape.

Our radiation detector for this lab is called a Geiger-Muller (GM) counter. It is is a cylinder full of gas,with the outside of the cylinder at a certain voltageand a wire running down its axis at another volt-age. The voltage difference creates a strong electricfield. When ionizing radiation enters the cylinder,it can ionize the gas, separating negatively chargedions (electrons) and positively charged ones (atomslacking some electrons). The electric field acceler-ates the ions, making them hit other air molecules,and causing a cascade of ions strong enough to bemeasured as an electric current.

If you look at the top side of the GM counter (behindthe top of the front panel), you’ll see a small window.Non-penetrating radiation can only get in throughthis thin layer of mica. (Gammas can go right inthrough the plastic housing.)

Put the bottom switch on Audio. The top switchdoesn’t have any effect on the data collection we’llbe doing with the computer.

Poisson StatisticsAlthough we will not be formally estimating errorbars in this lab, the following information will behelpful in understanding what’s going on when you’retaking data. When we have a large number of thingsthat may happen with some small probability, thetotal number of them that do happen is called aPoisson random variable (accent on the second syl-lable). For example, the number of houses burglar-ized in Fullerton this year is a Poisson random vari-able. When you count the number of nuclear decaysin a certain time interval, the result is Poisson. Thehelpful thing to know is that when a Poisson variablehas an average value N , its statistical uncertainty is√N . So for example if your GM counter counts 100

clicks in one minute, this is 100± 10. Knowing thiswill help you to have some idea whether, for exam-ple, an apparent change in the count rate is actuallytoo small to be statistically meaningful, or whetheryou need to count for longer in order to get reliableresults.

ObservationsA Background

Use the GM counter to observe the background ra-diation in the room. This radiation is probably acombination of gamma rays from naturally occurringminerals in the ground plus betas and gammas frombuilding materials such as concrete. If you like, youcan walk around the room and see if you can detectany variations in the intensity of the background.

Estimate the rate at which the GM detector countswhen it is exposed only to background. Starting atthis point, it is more convenient to interface the GMcounter to the computer. Plug the cable into intoDIG/SONIC 1 on the LabPro interface. Start Log-gerPro 3 on the computer, and open the file Probesand Sensors : Radiation monitor : Counts versustime. The interface is not able to automatically iden-tify this particular sensor, so the software will askyou to confirm that you really do have this type ofsensor hooked up; confirm this by clicking on Con-nect.

Now when you hit the Collect button, the sensorwill start graphing the number of counts it receivesduring successive 5-second intervals. The y axis ofthe graph is counts per 5 seconds, and the x axis istime.

Once you’ve made some preliminary observations,try to get a good measurement of the background,counting for several minutes so as to reduce the sta-tistical errors. It is important to get a good mea-surement of the background rate, because in laterparts of the lab you’ll need to subtract it from allthe other count rates you measure. For longer runslike this, it is convenient to let the software collectdata for about a minute at a time, rather than 5seconds. To get it to do this, do Experiment : Datacollection : 60 seconds/sample.

B Type of radiation

Your first task in the lab is to figure out whether thesource emits alpha, beta, or gamma radiation — orperhaps some mixture of these. It is up to you to de-cide how to do this, but essentially you want to usethe fact that they are absorbed differently in mat-ter; referring to your textbook, you’ll see that thehistorical labels α, β, and γ were assigned purelyon the basis of the differences in absorption, beforeanyone even knew what they really were. Note thatthe fact that you are able to detect the radiationat all implies that at least some of the radiation isable to penetrate the thin plastic walls of the source.

39

Also keep in mind that the descriptions of absorp-tion in a textbook are generalizations that do nottake into account the energy of the particles. Forexample, low-energy betas could be absorbed by akleenex, whereas high-energy betas could penetratecardboard.

There are six things you could try to prove throughyour measurements:

1. The source inside the plastic container emitsalphas.

2. The source does not emit alphas.

3. The source emits betas.

4. The source does not emit betas.

5. The source emits gammas.

6. The source does not emit gammas.

Try to figure out which of these six statements youcan either definitively prove or definitively disprove.Because you are not allowed to extract the sourcefrom the packaging, there will be some cases in whichyou cannot draw any definitive conclusion one wayor the other.

C Distance dependence

Measure the count rate at several different distancesfrom the source. The goal is to find the mathemati-cal form of this function (see Analysis, below). Dis-tances of less than about 10 cm do not work well,because the size of the GM tube is comparable to 10cm.

D Absorption

Measure the reduction in count rate when a 25 mmthick wood block is interposed between the sourceand the detector, and likewise for 17 mm of steel.Keep the distance from the source to the detectorconstant throughout. Let the counter run for at leastfive minutes each time. Based on these observations,predict the count rate you would get with two 17-mm thicknesses of steel instead of one. Test yourprediction.

E Decay curve

The source consists of a particular isotope of cesium;we’ll refer to it as NNNCs, since the main goal of thislab is to determine what the unknown isotope actu-ally is. It decays to an isotope of barium, and ratherthan decaying to the ground state of the barium nu-cleus, it nearly always decays to an excited state,

which then emits the radiation you characterized inpart B. Although the half-life of the parent cesiumisotope is many years, the half-life of the excited iso-tope in the barium daughter is short enough that itcan be observed during the lab period. However, ifthe cesium and barium are not separated, then notime variation will be observed, because the supplyof barium nuclei is being continuously replenishedby decay of the parent cesium.

To get around this, the source is packaged so thatwhen a weak acid solution is forced through it, asmall amount of barium is washed out. Note that theyellow tape around the circumference of the sourcehas an arrow on it. This arrow points in the direc-tion that you’re supposed to make the solution flow.The isotope generator kit has coin-sized steel trayson which to collect a few drops of the radioactivesolution.

Use the syringe to draw 1 mL of the acid solutionfrom the 250-mL bottle (labeled “eluting solution”).Take one of the tiny coin-sized steel trays out of theisotope generator kit and lay it on the lab bench.Remove the little plugs from the top and bottom ofthe radioactive source. Stick the syringe in the in-flow hole, and use the plunger to force seven dropsof liquid out onto the tray. Note that the amountof liquid that flows from the syringe into the sourceis quite a bit more than the amount that comes outinto the tray. If you have solution left in the syringeat the end of lab, squeeze it back out into the 250-mLbottle.

Use the computer to collect data on the rate of decayas a function of time. About 5 or 10 seconds persample works well.

When you’re done, make sure to shut off the GMcounter so that its battery doesn’t get drained.

Waste disposalTo get an idea of what a non-issue radioactive wastedisposal is in this lab, recall that it would be legalto throw the entire source in the trash — althoughwe won’t actually do that. The amount of radioac-tive material that you wash out in part E is a tiny

40 Lab 9 Radioactivity

fraction of this. Furthermore, essentially all the ra-dioactivity is gone by the end of lab. It is thereforenot a problem to dump your seven drops of materialdown the drain at the end of class.

There is also no chemical disposal issue with this tinyamount of solution. It’s a few drops of very diluteacid, equivalent to a little spritz of lemon juice.

AnalysisIn part C, you should first subtract the backgroundrate from each datum. Then make a log-log plot asdescribed in appendix 5, and see if you can success-fully describe the data using a power law. Note that,just like a human, the GM counter cannot countfaster than a certain rate. This is because everytime it gets a count it completely discharges its volt-age, and then it has to recharge itself again. Forthis reason, it is possible that your data from verysmall distances will not agree with the behavior ofthe data at larger distances. The documentationfor these GM counters says that they can count atup to about 3500 counts per second; this is only avery rough guide, but it gives you some idea whatcount rates should be expected to start departingfrom ideal behavior.

Estimate the half-lives of any isotopes present in thedata. If you find that only one half-life is present,you can simply determine the amount of time re-quired for the count rate to fall off by a factor oftwo. If the natural background count rate measuredin part A is significant, you will need to subtractit from the raw data. If more than one half-life ispresent, try plotting the logarithm of the count rateas a function of time, and seeing if there are linearsections on the graph. Note that this is all refer-ring to the half-lives of any decay chain that occursafter the cesium decays to barium. The half-life ofthe cesium parent nucleus is much longer, and is notmeasured in this experiment.

Consult the Wikipedia article “Isotopes of cesium.”It has an extremely lengthy table of all the knownisotopes from very light ones (with far too few neu-trons to be stable) to very heavy ones (with far toomany). Since the source was shipped to us throughthe mail, and sits on the shelf in the physics stock-room for semester after semester, you can tell thatthe half-life of the cesium isotope must be fairly long— at least on the order of years, not months. Fromthis information, you should be able to narrow downthe range of possibilities. (Half-lives in units of yearsare listed with “a,” for “annum,” as the unit of time.

The notations m1 and m2 mean energy states thatare above the lowest-energy state.) The radioactiveisotopes from this remaining list of possibilities allhave their own Wikipedia articles, and these articlesgive the properties of the daughter nuclei (isotopesof barium and xenon), including the half-lives of anygamma-emitting states. Look for one that has a half-life that seems to match the one you measured in lab.Having tentatively identified the unknown isotope asthis isotope of cesium, check against the results ofpart B, where you determined the type of radiationemitted.


P1 Suppose that in part C you obtain the followingdata:

r (cm) count rate (counts/2 min)10 70720 20730 95

Suppose that the background rate you measure inpart A is 30 counts per 2 min. Use the techniquedescribed in appendix 5 to see if the data can bedescribed by a power law, and if so, determine theexponent.

P2 If a source emits gamma or beta radiation, thenthe radiation spreads out in all directions, like anexpanding sphere. Based on the scaling of a sphere’ssurface area with increasing radius, how would youtheoretically expect the intensity of the radiation tofall off with distance? Would it be a power law?If so, what power? Why would you not expect thesame behavior for a source emitting alpha particlesinto air?

41

10 Two-Source Interference

Apparatusripple 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/groupThornton DC voltage source (in lab bench) 1/groupsmall rubber stopper . . . . . . . . . . . . . . . . . . . . . 1/grouppower strip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/groupbucket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/groupmop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1flathead screwdriver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1rulers and protractorskimwipes and alcohol for cleaningbutcher paper

GoalsObserve how a 2-source interference pattern ofwater waves depends on the distance betweenthe sources.

ObservationsLight 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 mostimportant ideas about light as a wave can be seen

in 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 throughoutthe tank. (Do not rotate the wooden legs them-

1The photo is from the textbook PSSC Physics, which hasa blanket permission for free use after 1970.

42 Lab 10 Two-Source Interference

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.

The light source is intended to be clamped to thewooden post, but I’ve found that that works verypoorly, since the clamp doesn’t hold it firmly enough.Instead, clamp the light source to the lab bench’s lipor its leg. Turn it on. Put the butcher paper on thefloor under the tank. If you make ripples in thewater, you should be able to see the wave patternon 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.

Use the methods explained in Appendix 5 and lookfor any kind of a power law relationship for the de-pendence of the angular spacing on d.

43

11 Refraction and Images

Apparatusplastic boxpropanol (1 liter/group, to be reused)laserspiral plastic tube and fiber optic cable for demon-strating total internal reflectionrulerprotractorbutcher paperfunnel

GoalsTest whether the index of refraction of a liquidis proportional to its density.

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.

IntroductionWithout 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 bendingof rays of light that occurs when they pass throughthe boundary between two media in which the speedof light is different.

Refraction occurs for the following reason. Imagine,for example, a beam of light entering a swimmingpool at an angle. Because of the angle, one side ofthe beam hits the water first, and is slowed down.The other side of the beam, however, gets to travelin air, at its faster speed, for longer, because it entersthe water later — by the time it enters the water,the other side of the beam has been limping alongthrough the water for a little while, and has not got-ten as far. The wavefront is therefore twisted arounda little, in the same way that a marching band turnsby having the people on one side take smaller steps.

Quantitatively, the amount of bending is given bySnell’s law:

ni sin θi = nt sin θt,

where the subscript i refers to the incident light and

incident medium, and t refers to the transmittedlight and the transmitting medium. This relationcan be taken as defining the quantities ni and nt,which are known as the indices of refraction of thetwo media. Note that the angles are defined withrespect to the normal, i.e., the imaginary line per-pendicular 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 almostexclusively with flat surfaces.

Preliminaries

Check whether your laser’s beam seems to be roughlyparallel.

44 Lab 11 Refraction and Images

ObservationsA Index of refraction of alcohol

The index of refraction is sometimes referred to asthe optical density. This usage makes sense, becausewhen a substance is compressed, its index of refrac-tion goes up. In this part of the lab, you will testwhether the indices of refraction of different liquidsare proportional to their mass densities. Water hasa density of 1.00 g/cm3 and an index of refractionof 1.33. Propanol has a density of 0.79 g/cm3. Youwill find out whether its index of refraction is lowerthan water’s in the same proportion. The idea isto pour some alcohol into a transparent plastic boxand measure the amount of refraction at the inter-face between air and alcohol.

Make the measurements you have planned in orderto determine the index of refraction of the alcohol.The laser and the box 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

Pour the alcohol back into the container for reuse,and pour water into the box to replace it.

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.

Next we will use the laser to simulate the rays com-

ing from the finger, as shown in the figure. 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.

45


Print out, complete, and turn in the laser safetychecklist, appendix 7.

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.

AnalysisUsing your data for part A, extract the index of re-fraction of propanol, with error bars. Test the hy-pothesis that the index of refraction is proportionalto the density in the case of water and propanol.

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.

46 Lab 11 Refraction and Images

47

12 Geometric Optics

Apparatusoptical bench . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/groupconverging lens, f = 50 cm, 5 cm diam1/group converging lens, f = 5 cm . . . . . . . . 1/grouplamp and arrow-shaped mask . . . . . . . . . . . . . 1/groupfrosted glass screen . . . . . . . . . . . . . . . . . . . . . . . 1/group

GoalsObserve a real image formed by a convex lens,and determine its focal length.

Construct a telescope and measure its angularmagnification.

IntroductionThe 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 revolvedaround 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 α1 is converted to a largeα2. 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 asa house” don’t know what they’re talking about.The telescope does not make things “seem closer”— since the rays coming at your eye are parallel,the final virtual image you see is at infinity. Theangular magnification is given by

MA = α2/α1

(to be measured directly in this lab)

MA = fo/fE

(theory).

ObservationsA Focal length of the lenses

In this part of the lab, you’ll accurately determinethe focal lengths of the two lenses being used for thetelescope. They are poorly quality-controlled, andI’ve found that the labeled values are off by as muchas 10%. If you’re only doing part A of the lab, thenyou will only determine the focal length of one lens,which is given to you as an unknown.

48 Lab 12 Geometric Optics

A refracting telescope

Start with the short-focal-length lens you’re going touse as your eyepiece. Use the lens to project a realimage on the frosted glass screen. For your object,use the lamp with the arrow-shaped aperture in frontof it. Make sure to lock down the parts on the opticalbench, or else they may tip over and break the optics!

For the long-focal-length lens you’re going to use asyour objective, you will probably be unable to do asimilar determination on a one-meter optical bench.Improvising a similar setup without the bench willstill give you a much more accurate value than theone written on the label.

A careful measurement here pays off later by mak-ing the focus in part B much easier to find. This isespecially true for the longer-focal-length lens. Toimprove the quality of your result, do the kind ofthing they do at the optometrist — “which is bet-ter, 1 or 2?” Have several people do independentdeterminations of the best focus.

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. Thefigures on p. 51 show a simulation of what the su-perimposed images should look like and of how itwould look if the telescope is not yet adjusted quitecorrectly.

Your brain is not capable of focusing one eye at one

distance, 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, becauseof the uncertainty in the measured focal lengths.

A good quick test of the focus is to pick someonewho’s nearsighted and see if they can focus on theimage without their glasses on. If they can, then theimage is not at infinity, because nearsighted peoplecan’t focus on an image that’s at infinity.


P1 Skip this question if you were assigned problem31-22 from Light and Matter,. In part A, do youwant the object to be closer to the lens than thelens’ focal length, exactly at a distance of one focallength, or farther than the focal length? What aboutthe screen?

P2 Plan what measurements you will make in partA and how you will use them to determine the lenses’focal length.

P3 Skip this question if you are only doing part Aof the lab. It’s disappointing to construct a telescopewith a very small magnification. Given a selectionof lenses, plan how you can make a telescope withthe greatest possible angular magnification.

49

AnalysisDetermine the focal length of the two lenses, 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? Give a probabilistic interpretation,as in the example in appendix 2. See the exampleat the end of appendix 3 of how to test for equalitybetween two numbers that both have error bars.

50 Lab 12 Geometric Optics

51

13 Wave Optics

Apparatushelium-neon laser1/group optical bench with posts & holders 1/grouphigh-precision double slits . . . . . . . . . . . . . . . . 1/grouprulersmeter stickstape measuresbutcher paperblack cloths for covering light sources

GoalsObserve 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.

IntroductionIsaac 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 somesavvy 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.

ObservationsA 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,

52 Lab 13 Wave Optics

A double-slit diffraction pattern.

where θ is measured in radians. To make sure youcan see the fine spacing, put your slits across theroom from the wall. To make it less likely that some-one will walk through the beam and get the beamin their eye, put some of the small desks under thebeam. The slit patterns we’re using actually havethree sets of slits, with the following dimensions:

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 beamof the laser to cover both slits.

If your diffraction pattern doesn’t look like the onein the figure on the following page, typically the rea-son is that you’re only hitting one slit with the beam(in which case you get a single-slit diffraction pat-tern), or you’re not illuminating the two slits equally(giving a funny-looking pattern with little dog-bonesand things in it).

As shown in the figure below, it is also possible tohave the beam illuminate only part of each slit, sothat the slits act effectively as if they had a smallervalue of d. The beam spreads as it comes out of thelaser, so you can avoid this problem by putting itfairly far away from the laser (at the far end of theoptical bench).

Think about the best way to measure the spacing ofthe pattern accurately. Is it best to measure from a

bright 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?

Determine the wavelength of the light, in units ofnanometers. Make sure it is in the right range forred light.

Check that the ∆θ you obtain is in the range pre-dicted in prelab question P1. In the past, I’ve seencases where groups got goofy data, and I suspectthat it was because they were hitting a place on theslits where there was a scratch, bump, or speck ofdust.

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.

PrelabThe 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

53

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

Read the safety checklist.

P1 Look up the approximate range of wavelengthsthat the human eye perceives as red. With d = 0.6mm, predict the lowest and highest possible valuesof ∆θ that could occur with red light.

P2 It is not practical to measure ∆θ directly usinga protractor. Suppose that a lab group finds that27 fringes extend over 29.7 cm on their butcher pa-per, which is on a wall 389 cm away from the slits.They calculate ∆θ = tan−1(29.7 cm/(27× 389 cm))= 2.83 × 10−3 rad. Simplify this calculation usinga small-angle approximation, and show that the re-sulting error is negligible.

P3 Make a rough order-of-magnitude guess of thediameter of a human hair.

AnalysisDetermine the wavelength of the light and the diam-eter of the hair, with error bars.

54 Lab 13 Wave Optics

55

14 The Photoelectric Effect

Note to the lab technician: The Pasco SE-5509 Hggas discharge tubes have too many ways of being de-stroyed by students,1 so let’s use the OS-9286 tubesinstead (the big black ones with the fins). Thatmeans we don’t need the separate power supply forthe discharge tube (Pasco BEM-5007) or the trackthat comes with the photodiode. We need woodblocks to raise the photodiode to the same heightas the discharge tube.

Apparatushand-held diffraction gratingsHg gas discharge tube (Pasco OS-9286)photodiode (Pasco SE-5509)power supply (Pasco BEM-5001)high-sensitivity ammeter (Pasco BEM-5004)wood blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3/group

GoalsUse the photoelectric effect to test predictionsof the wave and wave-particle models of light.

IntroductionThe 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, itwould tend to shake the electrons. It was only to beexpected that something like the photoelectric effectcould happen, with the light shaking the electronsvigorously enough to knock them out of the atom.

1The two main modes of destruction seem to be: (1) Whilethe discharge tube is connected to the power supply, studentsmonkey with the red 110V/220V switch on the power supply;and (2) they don’t use the power supply and connect two ACconnectors together in order to connect the discharge tubedirectly to the wall socket.

But once the effect was observed, physicists beganrunning into trouble interpreting how it behaved.There were various variables they could adjust, suchas:

• the light’s frequency (color), and

• the light’s intensity (brightness).

Given these input conditions, there were outputsthey could look at, including

• any time delay before electrons began to popout,

• the rate at which the electrons then flowed(measured as a current on an ammeter), and

• the amount of kinetic energy they had.

At the time this was considered an obscure tech-nical topic, but experimentalists began generatingdata, which theorists then had zero success in inter-preting. Albert Einstein, better known today for thetheory of relativity, was the first to come up with theradical, and correct, explanation, which involved afundamental rewriting of the laws of physics.

SetupThe Hg gas discharge tube emits light with severaldifferent wavelengths. Turn on the discharge tubeimmediately, because it takes a long time to warmup.

The photodiode is a vacuum tube housed inside an-other box, with a small hole to allow light to come inand hit one of the electrodes (the cathode) inside thevacuum tube. On the front of the box, covering thehole, are two rotating wheels. The wheel that youcan see has five colored filters. Each of these filterslets through only one of the five wavelengths of lightemitted by the Hg tube, so that you can control thefrequency.

The hole through which the light enters is actuallya hole in a second wheel, located behind the filterwheel. By clicking that wheel into different positionwe can select holes of different sizes, which lets in dif-ferent amounts of light. Although this is convenient,it doesn’t actually control the intensity of the light

56 Lab 14 The Photoelectric Effect

(watts per square meter) but only the total amount(watts). Therefore we’ll just leave this set for the 2mm hole.

There is an easier way to control the actual intensityof the beam, which is that you can simply change thedistance between the discharge tube and the photo-diode. The light from the discharge tube spreadsout in a cone, so that as you vary the distance tothe photodiode, the intensity falls off as the inversesquare of the distance.

There is a power supply used for applying an exter-nal voltage to the photodiode. Although you don’tactually want to apply that external voltage dur-ing this part of the lab, it seems that the ammeterwon’t work until you hook up the power supply, soyou need to do that now. On the bottom right sideof the power supply are two banana plugs. Connectthese to the two plugs near the bottom of the pho-todiode. To get the polarity right, connect the pos-itive (red) output of the power supply to the anode(marked A).

The photodiode has an output that can be connectedto a extremely sensitive ammeter to measure the rateat which electrons are ejected from the cathode andabsorbed at the anode. Before connecting the am-meter (labeled “DC Current Amplifier” on the frontpanel), set it to its most sensitive scale (10−13 A)and depress the button labeled “Calibration.” Sincethe meter is not connected to anything, the currentis truly zero. Use the knob to force the meter to readzero, and then let the calibration button out. Nowuse a BNC cable to connect the photodiode to theammeter.

Once the discharge tube has warmed up, arrangethings so that you can see a spot of its light, e.g.,by letting it fall on a white piece of paper. Hold thediffraction grating up to your eye and look at thelight. In the first-order (m = ±1) fringes, you shouldbe able to see that the light contains a mixture offour discrete wavelengths of visible light. There isalso an ultraviolet wavelength, which you may beable to see as well if the paper fluoresces.

Now suppose we were considering the following pos-sible models of light: (1) pure particle model, (2)pure wave model, (3) a hybrid model in which lighthas both particle and wave properties. You have justobserved diffraction of the light from your source.Which of the models are consistent with this obser-vation, and which are immediately ruled out?

The wavelengths are as follows:

color wavelength (nm)ultraviolet 365violet 405blue 436green 546yellow 578

The circuit. The light creates an electric current, whichtrickles back through the large but finite resistance of thevoltmeter.

Circuit

The figure above shows a circuit diagram of the setup.Light comes in and knocks electrons out of the curvedcathode. If the voltage is turned off, there is no elec-tric field, so the electrons travel in straight lines;some will hit the anode, creating a current. If thevoltage is turned on, the electric field repels the elec-trons from the wire electrode, and the current is re-duced or perhaps even completely eliminated.

ObservationsAlthough the photodiode box has filters on the front,no filter is perfect, and therefore these will all letin some stray light of wavelengths other than theintended one. Therefore the room should be verydark when you do your measurements.

A Time delay

As explored in the prelab, there may be some delaybetween the time when the light is allowed to hitthe cathode and the time when electrons begin tobe ejected. If so, then we lack even a rough a prioriestimate of this time. Investigate this. If the timeseems to be extremely short, do what you plannedin the prelab to try to make it long enough to detect.If the time is much too long, do the opposite so thatyou can actually observe the photoelectric effect. Ifyou’re able to get the time delay into a range whereit’s measurable using eyeballs and a clock, do so. Ifnot, then try to set an upper or lower limit on it.

57

B Energy of the electrons

Until you do the lab, it’s not obvious how much en-ergy the electrons would have as they pop out of thecathode. It could be some fixed number, or it coulddepend on the conditions you choose, and it couldalso have multiple values for the electrons producedunder a given set of conditions. In fact, we mayexpect a range of values for two reasons.

(1) As explored in a prelab question, the electronswill have random kinetic energies to start with, dueto their thermal motion

(2) The light penetrates to some depth in the cath-ode, and an electron that starts at some depth willlose some amount of energy as it then comes outto the surface. The electron’s direction of motionis random. If it happens to be toward the surface,then the thickness of material that it traverses willdepend both on its initial depth and on its randomdirection of motion.

This range of energies may have an upper limit fora given set of experimental conditions. If so, thenby applying a high enough voltage you should beable to eliminate the current completely. The mini-mum voltage required to do this would be called thestopping voltage.

Now you want to apply a voltage to the photodiodeusing the DC power supply, which you previouslyconnected but didn’t use. There is a button betweenthe two LED readouts. Let this button out in orderto select a range of voltages from 0 to -4.5 V.

Find out whether there is a stopping voltage, andif so, measure it for the conditions you’ve chosen.If there is never a sharp cutoff, you should still beable to determine some quantitative measure of thevoltage that corresponds to a typical energy for theelectrons, e.g., the voltage at which some fixed frac-tion of the current is eliminated. From now on in thelab manual, I’ll just refer to this as “the voltage” fora given set of experimental conditions.

Determine the voltage, and compare with the esti-mate in the prelab of what voltage would correspondto the thermal motion of the electrons. Based on thiscomparison, is the amount of energy involved in thephotoelectric effect much less than, much more than,or on the same order of magnitude as the thermalenergy?

C Dependence of voltage on intensity

Vary the intensity of the light and determine whetherand how the voltage depends on intensity.

We have observed strange results sometimes, whichseem to occur because when the discharge tube isvery close to the photodiode, light gets in and hitsthe anode (not just the cathode) and causes the pho-toelectric effect in the wrong direction. This seemsto occur with the shorter wavelengths of light. Youshould be able to tell if this is happening becausewhen you turn up the voltage high enough, you get acurrent in the opposite direction. Check for this be-havior and avoid taking data under conditions thatproduce it.

D Dependence of voltage on frequency

Do the same thing for the frequency.

AnalysisThe point of the analysis is to try to compare theobserved results with our expectations based on twomodels of light: the pure wave model, and a modelin which light is both a particle and a wave — we’veseen above that a pure particle model is untenable.For conceptual simplicity, however, we may find ithelpful to visualize the wave-particle model as if thelight was purely particle-like, so that a beam of lightwould be like a stream of machine-gun bullets. Thisis essentially what Einstein does in his 1905 paper.He admits that this is not literally possible, butdoesn’t attempt a more detailed reconciliation of theparticle and wave pictures, which he doesn’t knowhow to achieve. For this reason, the title of the paperrefers to the particle picture as a “heuristic,” whichmeans a kind of non-rigorous way of getting an an-swer without using correct fundamental principles.

Time delay: You investigated the possible time de-lay in the wave model in some detail in the prelaband/or homework. In a particle model, what wouldbe your expectations about a time delay? Comparewith experiment.

Energy: Based on the particle model, we would ex-pect one “bullet” of light to give its fixed amount ofenergy to one electron, so that under a given set ofexperimental conditions, there would be a maximumpossible kinetic energy for the electrons, achievedwhen the electron originated very close to the sur-face of the cathode. In the wave model it is more dif-ficult to make a definite prediction. Did you observethat there was a definite stopping voltage, or thatthere was no definite cut-off in the current? Doesthis allow you to test either model?

Dependence of energy on intensity: In the particlemodel, a more intense beam of light would be one

58 Lab 14 The Photoelectric Effect

containing a larger number of particles (per unit ofcross-sectional area). In the wave model, a more in-tense beam would be a higher-amplitude wave. Doeseither model lead you to predict anything specificabout the dependence of voltage on intensity? Testagainst experiment.

Dependence of energy on frequency: Does the (typi-cal or maximum) energy of the electrons eV dependon the light’s frequency f? If so, then in the particlemodel this probably means that we’re observing achange in the amount of energy per particle of light.We can’t just equate the electron energy eV to theenergy of the particles of light, because the electronslose a fixed amount of energy (called the work func-tion) as they emerge from the surface of the metal.We can get around the issue of this constant offset byfinding the slope of eV versus f ; a constant offset onthe y axis doesn’t affect the slope of a graph. Let’scall this constant h. Estimate its numerical value.What units does it have? There is no such universalconstant in any of the classical laws of physics, so ifit pops up here, it indicates that some entirely newphysical theory is being probed.


P1 Under the hypothesis that light is purely a wave,the energy of a beam of light would be smoothlyand continuously accumulated by whatever it hit.Therefore it should take some amount of time be-fore an electron could accumulate enough energy topop out of the metal. You may have estimated thistime scale in a homework problem (Light and Mat-ter problem 34-11), but that calculation depends onmany crude assumptions and rough estimates, so it’shard to know whether to trust it even as an order ofmagnitude estimate. Therefore if we want to try toobserve this time delay in this experiment, it’s im-possible to know in advance what to look for, andit may be either too short (in which case we can’tmeasure it) or too long (in which case it will looklike the apparatus simply isn’t working). Supposethat the time delay is too short to detect in the con-ditions you initially pick. How could you change theconditions in order to make it longer, and possiblydetectable?

P2 Heat is the random motion of the microscopicparticles of which matter is composed. Dependingon the level of detail in the treatment of thermody-namics in your first-semester course, you may knowthat the typical energy per particle can be calculatedas kT , where T is the absolute temperature and kis a constant called Boltzmann’s constant (not to beconfused with the Coulomb constant). The resultat room temperature is on the order of 10−20 joulesper particle. In the photoelectric effect, an electronwill absorb some additional amount of energy fromthe light, which is enough to pop it out of the cath-ode. Until doing the experiment, we do not knowhow much this additional amount is, but during thelab you will be able to probe this question by usinga voltage V to try to stop the electrons from makingit across the gap. If this additional energy was onthe same order of magnitude as the thermal energy(which it may not be), estimate the voltage required.

P3 In this experiment, the light comes out of thedischarge tube in a spreading cone. Geometrically,how should the intensity I of the light depend on thedistance r? State a proportionality.

P4 Look up the wavelength of visible light and thetypical distance between atoms in a solid. How dothey compare? In the wave model, should we expecta particular wave-train to hit one atom, or many?

59

15 Electron Diffraction

Apparatuscathode ray tube (Leybold 555 626)high-voltage power supply (new Leybold)100-kΩ resistor with banana-plug connectorsVernier calipers

GoalsObserve 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.

IntroductionThe 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 as13.5 cm in the figure actually varies from about 12.8 cmto 13.8 cm, even for tubes that otherwise appear iden-tical. This doesn’t affect your results, since you’re onlysearching for a proportionality.

MethodWhat 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!As you will see in this lab, the wavelengths of the

60 Lab 15 Electron Diffraction

electrons 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.1

The carbon atoms in the graphite crystal are arrangedhexagonally.

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

1See http://bit.ly/XxoEYr for more information.

high voltage in Appendix 6. 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.

SetupYou 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

61

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.2 Also, connect F1 toC with the wire that has the 100-kΩ resistor splicedinto it. The circuit diagram on page 63 summarizesall this.

Check your circuit with your instructor before turn-ing it on!

ObservationsYou 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.

Important: As of 2018, some of our tubes are start-ing to die, and we will not be able to buy replace-ments until 2020. For this reason, please take thefollowing steps to extend the remaining lifetimes ofthe working tubes. (1) Don’t take too many datapoints. Change the voltage in steps of 1.0 kV, notsmaller steps. (2) Turn the knob on the high voltagepower supply all the way down to zero except whenyou’re actually measuring a diffraction fringe. (3)Try to get all your data-taking done without leavingthe heater circuit on for more than about 30 minutes.

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. To

2If 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.

get unambiguous results, you’ll need to takedata with the widest possible range of voltages.

AnalysisOnce you have your data, the idea is to plot λ as afunction of quantities such as KE, p, 1/KE, or 1/p.If the graph is a straight line through the origin,then the experiment supports the hypothesis thatthe wavelength is proportional to that quantity. Youcan simplify your analysis by leaving out constantfactors, and P5 asks you to consider how you canrule out some of these possibilities without havingto make all the graphs.

What does λ seem to be proportional to? Your datamay cover a small enough range of voltage that morethan one graph may look linear. However, only onewill be consistent with a line that passes through theorigin, as it must for a proportionality. This is whyit is important to have your graph include the origin.



Read the high voltage safety checklist.

P1 The figure shows the vacuum tube as havinga particular shape, which is a sphere with the foiland phosphor at opposite ends of a diamater. In re-ality, the tubes we’re using now are not quite thatshape. To me, they look like they may have beenshaped so that the phosphor surface is a piece of asphere centered on the foil. Therefore the arc lengthsacross the phosphor can be connected to diffractionangles very simply via the definition of radian mea-sure. Plan how you will do this.

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 is

62 Lab 15 Electron Diffraction

The circuit for the new setup.

judge 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 P2. 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?

63

16a Setup of the Spectrometer

ApparatusHg gas discharge tube (PASCO OS-9286) . . . . . . . . 3spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/groupdiffraction grating, 600 lines/mm . . . . . . . . . 1/groupsmall screwdriver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1black cloth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1piece of plywood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1block of wood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1penlight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/grouplight block

GoalsThe lab has three parts. This one, part a, is aboutsetting up the optics of the spectrometer. This isto be done once by the instructor or lab technician.It never needs to be redone unless something getsmessed up.

Introduction

MethodThe apparatus is shown in the first figure below. Fora given wavelength, the grating produces diffractedlight at many different angles: a central zeroth-orderline at θ = 0, first-order lines on both the left andright, and so on through higher-order lines at largerangles. The line of order m occurs at an angle sat-isfying the equation mλ = d sin θ.

To measure a wavelength, students will move thetelescope until the diffracted first-order image of theslit is lined up with the telescope’s cross-hairs andthen read off the angle. Note that the angular scaleon the table of the spectroscope actually gives theangle labeled 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 determine the spac-ing, d, of any grating using a line whose wavelengthis 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. Part A below describes someadditional adjustments that help to get rid of addi-tional sources of error.

Theory of OperationThe second figure below shows the optics from theside, with the telescope simply looking down thethroat of the collimator at θ = 0. You are actu-ally using the optics to let you see an image of theslit, not the tube itself. The point of using a tele-scope is that it provides angular magnification, sothat a small change in angle can be seen visually.

64 Lab 16a Setup of the Spectrometer

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.

AdjustmentsFirst 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 properly 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 third figure, there are threeaxes about which the grating could be rotated. Ro-tation about axis 1 is like opening a door, and thisis accomplished by rotating the entire pedestal likea lazy Susan. Rotation about axes 2 (like foldingdown a tailgate) and 3 are accomplished using thetripod of screws underneath the pedestal. The eye-

piece of the telescope is of a type called a Gausseyepiece, with a diagonal piece of glass in it. Whenthe grating is oriented correctly about axes 1 and 2and the telescope is at θ = 0, a beam of light thatenters through the side of the eyepiece is partiallyreflected to the grating, and then reflected from thegrating back to the eye. If these two axes are cor-rectly adjusted, the reflected image of the crosshairsis superimposed on the 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 them = 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.

65

The spectrometer

Optics.

Orienting the grating.

66 Lab 16a Setup of the Spectrometer

67

16b The Mass of the Electron

ApparatusH gas discharge tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Hg gas discharge tube (PASCO OS-9286) . . . . . . . . 3spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/groupdiffraction grating, 600 lines/mm . . . . . . . . . 1/groupsmall screwdriver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1black cloth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1piece of plywood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1block of wood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1light block

GoalsThe lab is split up into parts a, b, and c. Studentlab groups will do either part b or part c. This part,b, is a measurement of the mass of the electron.

IntroductionWhat’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 subatomicparticles. Only after Thomson and Rutherford haddemonstrated the existence of electrons and the nu-cleus did the atom begin to be imagined as a tinysolar system, with the electrons moving in ellipticalorbits around the nucleus under the influence of itselectric field. The problem was that physicists knewvery well that accelerating charges emit electromag-netic radiation, as for example in a radio antenna, sothe acceleration of the electrons should have causedthem to emit light, steadily lose energy, and spiralinto the nucleus, 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?

The first step toward understanding the structure ofthe atom was Einstein’s theory that light consistedof particles (photons), whose energy was related totheir frequency by the equation Ephoton = hf , or

substituting 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 levelsof the atom, it was then possible to work backwardsand figure out the atomic energy levels themselves.

MethodThe apparatus you will use to observe the spectrumof hydrogen or nitrogen is shown in the first figurebelow. For a given wavelength, the grating producesdiffracted light at many different angles: a centralzeroth-order line at θ = 0, first-order lines on boththe left and right, and so on through higher-orderlines at larger angles. The line of order m occurs atan angle satisfying the equation mλ = 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.


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 make

68 Lab 16b The Mass of the Electron

a 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 determine the spac-ing, d, of any grating using a line whose wavelengthis known.

ObservationsTurn on the mercury discharge tube right away, tolet it get warmed up.

The second figure below shows the optics from theside, with the telescope simply looking down thethroat of the collimator at θ = 0. You are actu-ally using the optics to let you see an image of theslit, not the tube itself. The point of using a tele-scope is that it provides angular magnification, sothat a small change in 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.

SetupSkim lab 16a so you have some idea of the way theapparatus has been carefully aligned in advance bythe instructor or lab technician.

A Calibration

You will use the blue line from mercury as a calibra-tion. In theory it shouldn’t matter what known linewe use for calibration, but in practice there may besmall aberrations in the spectrometer, and their ef-fect is minimized by using calibration lines of nearly

the same wavelength as the unknown lines to be mea-sured.

Put the mercury tube behind the collimator. Makesure the hottest part of the tube is directly in frontof the slits. You will need to use pieces of wood toget the height right. You want the tube as close tothe slits as possible, and lined up with the slits aswell as possible; you can adjust this while lookingthrough the telescope at an m = 1 line, so as tomake 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 most promi-nent visible Hg lines, in nm, to high precision.1

Mercury:404.656 violet There is a dimmer violet

line nearby at 407.781 nm.435.833 blue491.604 blue-green Dim. You may also see an-

other blue-green line thatis even dimmer.

546.074 greenyellow This is actually a complex

set of lines, so it’s not use-ful for calibration.

Start by making sure that you can find all of thelines lines in the correct sequence — if not, then youhave probably found some first-order lines and somesecond-order ones. If you can find some lines butnot others, use your head and search for them inthe right area based on where you found the linesyou did see. You may see various dim, fuzzy lightsthrough 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. The discharge tubes also have holes in theback; to block the stray light from these holes, eitherput the two discharge tubes back to back or use oneof the small “light blocks” that slide over the hole.

We will use the wavelength λc of the blue Hg line as acalibration. Measure its two angles αL and αR, and

1The table gives the wavelengths in vacuum. Althoughwe’re doing the lab in air, our goal is to find what the hydro-gen or nitrogen wavelengths would have been in vacuum; bycalibrating using vacuum wavelengths for mercury, we end upgetting vacuum wavelengths for our unknowns as well.

69

check that the resulting value of θc is close to theapproximate ones predicted in prelab question P1.The nominal value of the spacing of the grating givenin that prelab question is not very accurate. Havingmeasured θc, then we can sidestep the determinationof the grating’s spacing entirely and determine anunknown wavelength λ by using the relation

λ =sin θ

sin θcλc .

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 on themain scale, and is of the form “xxx 0’ plus a littlemore” 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.

B Spectroscopy of Hydrogen

You will study the spectrum of light emitted by thehydrogen atom, the simplest of all atoms, with justone proton and one electron. In 1885, before elec-trons and protons had even been imagined, a Swissschoolteacher named Johann Balmer discovered thatthe wavelengths emitted by hydrogen were related bymysterious ratios of small integers. For instance, thewavelengths of the red line and the blue-green lineform a ratio of exactly 20/27. Balmer even found amathematical rule that gave all the wavelengths ofthe hydrogen spectrum (both the visible ones andthe invisible ones that lay in the infrared and ultra-violet). The formula was completely empirical, withno theoretical basis, but clearly there were patternslurking in the seemingly mysterious atomic spectra.

Niels Bohr showed that the energy levels of hydrogenobey 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 m 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 mk2e4/2~2in front is just a numerical constant, and the varia-tion 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.

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.

We will use the purple and blue-green lines to deter-mine the mass of the electron.


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 an-gles αL and αR at which you will observe the bluemercury line.

P2 Make sure you understand the first three vernier


The spectrometer

Optics.

readings in the fourth figure, and then interpret thefourth reading.

P3 For the calibration with mercury, in what se-quence do you expect to see the lines on each side?Make a drawing showing the sequence of the anglesas you go out from θ=0.

P4 The visible lines of hydrogen come from the3 → 2, 4 → 2, 5 → 2, and 6 → 2 transitions. Basedon E = hf , which of these should correspond towhich colors?

Self-CheckIn homework problem 36-11 in Light and Matter,youcalculated the ratioλblue−green/λpurple. Before leaving lab, make surethat your wavelengths are consistent with this pre-diction, to a precision of no worse than about onepart per thousand.

AnalysisThroughout 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 already

known:

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 70, the mass of theelectron, me, would actually be given byme = m/(1−m/mp), where mp is the mass of the proton, 1.6726×10−27 kg.

71

Prelab question 2.


73

16c The Nitrogen Molecule

ApparatusHe gas discharge tube . . . . . . . . . . . . . . . . . . . . . . . . . . . 3N2 gas discharge tube (in green carousel) . . . . . . . . 3spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/groupdiffraction grating, 600 lines/mm . . . . . . . . . 1/groupsmall screwdriver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1black cloth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1piece of plywood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1block of wood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1penlight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1/grouplight block

GoalsThe lab has three parts. Each group will only dotwo parts. In this lab, part c, you will use an energysum to test a hypothesis about the energy levels ofthe nitrogen molecule, N2.

MethodThe apparatus you will use to observe the spectrumof hydrogen or nitrogen is shown in the first figurebelow. For a given wavelength, the grating producesdiffracted light at many different angles: a centralzeroth-order line at θ = 0, first-order lines on boththe left and right, and so on through higher-orderlines at larger angles. The line of order m occurs atan angle satisfying the equation mλ = 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.


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 absolutely

no 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 determine the spac-ing, d, of any grating using a line whose wavelengthis known.

B Energy Sums

The nitrogen discharge tube is housed in a greenplastic carousel. With the power off, rotate the carouselso that the nitrogen tube is the one that is in the ac-tive position, and then turn on the power. If youhold a diffraction grating up to your eye and lookat the tube, you will see a remarkable spectrum, un-like the visible light spectrum of almost any othergas. This is because the N2 molecule has an ex-tremely strong bond, requiring twice the energy tobreak compared to otherwise similar gases such asH2 or O2. Whereas these other gases would break upinto individual atoms under the extreme conditionspresent in a discharge tube, the nitrogen moleculeholds together, so that you are seeing the spectrumof the molecule, not the atom. For this reason,the spectrum of nitrogen contains a large numberof lines.

But these lines are not random. They occur in sets,each of which looks like a comb with an approxi-mately equal spacing between the “teeth.” The fig-ure shows a portion of the spectrum, including threesets of lines, which I have labeled r (red), o (orangeto green), and g (green).

I have spent some time trying to interpret the ori-gin of these lines, and I believe the interpretation issomething like this. Each of these lines is the emis-sion of a photon as the molecule goes from an initialstate to a final state that has less energy. The ini-tial state has some energy because the electrons arein an excited state (labeled B by spectroscopists)

74 Lab 16c The Nitrogen Molecule

and also some energy because the molecule is vi-brating, like two masses connected by a spring. Thefinal state has the electrons in a lower-energy state(labeled A), and is also vibrating. The initial andfinal electronic states B and A are the same in allcases, but the vibrational states differ. An idealizedquantum-mechanical vibrator turns out to have a se-ries of energy states like a ladder with nearly evenlyspaced rungs. States higher on the “ladder” are vi-brating more violently — classically, they vibratewith greater amplitude. The rungs of the vibrationalladder are labeled v = 0, 1, 2, and so on. (Becauseof the Heisenberg uncertainty principle, some vibra-tional energy is present even in the v = 0 state.) Ithink the states in the red set are from a state v to astate v−3, i.e., a change of 3 units in the vibrationalquantum number. The o set would be a change of4, and g a change of 5.

This hypothesis can be tested as follows. If it istrue, we could pick out a set of energy levels like thefollowing example:

The letters e, f , g, and h are the energy differencesthat would be observed as the energies of the pho-tons. In a set like this, we would have

h− g = f − e,

since each side of the equation would be equal tothe energy difference between the v = 4 and 5 statesof ladder A. Try to find a set of lines that would beconsistent with this interpretation. This may requiresome trial and error, but I think it may work if e isone of the lines near the middle of the r set, g is theorange line that is fourth from the short-wavelengthend of the o set, f is the fifth in that set, and h is inthe g set.

C Calibration

You will use the yellow line from helium as a calibra-tion. In theory it shouldn’t matter what known linewe use for calibration, but in practice there may besmall aberrations in the spectrometer, and their ef-fect is minimized by using calibration lines of nearlythe same wavelength as the unknown lines to be mea-sured.

Put the helium tube behind the collimator. Makesure the hottest part of the tube is directly in frontof the slits. You will need to use pieces of wood toget the height right. You want the tube as close tothe slits as possible, and lined up with the slits aswell as possible; you can adjust this while lookingthrough the telescope at an m = 1 line, so as tomake 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 most promi-nent visible He lines, in nm, to high precision.1

Helium:447.148 bright blue-purple471.314 dim blue492.193 dim green501.567 bright green587.562 yellow667.815 dim red706.5 very dim red

Start by making sure that you can find all of the linesin the correct sequence — if not, then you have prob-ably found some first-order lines and some second-order ones. If you can find some lines but not others,use your head and search for them in the right areabased on where you found the lines you did see. Youmay see various dim, fuzzy lights through the tele-scope — don’t waste time chasing these, which couldbe coming from other tubes or from reflections. Thereal lines will be bright, clear and well-defined. Bydraping the black cloth over the discharge tube andthe collimator, you can get rid of stray light thatcould cause problems for you or others. The dis-charge tubes also have holes in the back; to blockthe stray light from these holes, either put the twodischarge tubes back to back or use one of the small

1The table gives the wavelengths in vacuum. Althoughwe’re doing the lab in air, our goal is to find what the nitrogenwavelengths would have been in vacuum; by calibrating usingvacuum wavelengths for mercury, we end up getting vacuumwavelengths for our unknowns as well.

75

“light blocks” that slide over the hole.

We will use the wavelength λc of the hellow He lineas a calibration. Measure its two angles αL and αR,and check that the resulting value of θc is close to theapproximate ones predicted in prelab question P1.The nominal value of the spacing of the grating givenin that prelab question is not very accurate. Havingmeasured θc, then we can sidestep the determinationof the grating’s spacing entirely and determine anunknown wavelength λ by using the relation

λ =sin θ

sin θcλc .

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 on themain scale, and is of the form “xxx 0’ plus a littlemore” 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.

Status as of August 2018In spring of 2018, my students and I worked on mea-suring and interpreting this spectrum. A summaryof our results is in a Google Docs spreadsheet atgoo.gl/akrbcY. There is a basic explanation of thephysics in Simple Nature section 14.2. More detailedinformation about my interpretation of the lines isatphysics.stackexchange.com/a/334451/4552.

In the notation used in the material on stackex-change, the states are labeled with a quantum num-ber v. The energy sum based on our data come outquite nice for v = 9 and 8 going to 5 and 4, ac-curate to about 0.3%, which is reasonable for this

technique. The energy sum for v = 10 and 9 goingto 6 and 5 also works well, but with slightly highererror, maybe partly because 10-5 is a doublet. Thereare two dim green lines that in my labeling systemwould be g0 and g-1. These may be the ones wewould need in order to get a couple more energysums.

I experimented with doing the measurements pho-tographically. A student took a photo of a diffrac-tion pattern using a cell phone. I first did a roughcalibration against student data, then improved thiscalibration by doing a linear fit to my own spectrom-eter data. This worked fairly well, but doesn’t workwithout actual spectrometer data, which are neededfor calibration.

We have digital spectrometers which may be helpfulhere and are worth trying. Their resolution is sup-posed to be about 1 or 1.5 nm, which is an orderof magnitude worse than the analog spectrometers,but may be adequate for this purpose, and they candisplay a spectrum as a graph, which may be helpenough to make up for the lower resolution.

I couldn’t resolve the green band. I asked a coupleof students the next day, and they seemed to thinkthat it was doable to resolve these lines. Possibly mytube was behaving differently than theirs, but I’mnot sure I believe their data. The red and orangebands came out nice, all wavelengths being within afew tenths of a nm of Lofthus’s values.



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 an-gles αL and αR at which you will observe the yellowhelium line.

P2 Make sure you understand the first three vernierreadings in the fourth figure, and then interpret thefourth reading.

P3 For the calibration with helium, in what se-quence do you expect to see the lines on each side?


The spectrometer

Optics.

Make a drawing showing the sequence of the anglesas you go out from θ=0.

77

Orienting the grating.

Prelab question 2.


79

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 fromwhat you later did with them.These are the results of the measurements you takedown during the lab, hence they come first. Writeyour raw data directly in your lab book; don’t writethem on scratch paper and recopy them later. Don’tuse pencil. The point is to separate facts from opin-ions, observations from inferences.

Procedure — Did you have to create your ownmethods 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. Don’twrite anything here unless you think I will really careand want to change how we do the lab in the future.In most cases this section can be totally blank. Donot discuss how you did your calculations here, justhow you 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.

Many of our labs are comparisons of theory and ex-periment. The abstract for such a lab needs to saywhether you think the experiment was consistentwith theory, or not consistent with theory. If yourresults deviated from the ideal equations, don’t beafraid to say so. After all, this is real life, and manyof the equations we learn are only approximations,or are only valid in certain circumstances. However,(1) if you simply mess up, it is your responsibilityto realize it in lab and do it again, right; (2) youwill never get exact agreement with theory, becausemeasurements are not perfectly exact — the impor-tant issue is whether your results agree with theoryto roughly within the error bars.

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 discuss

whether 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.

Calculations and Reasoning — Convince me ofwhat you claimed in your abstract.Often this section consists of nothing more than thecalculations that you started during lab. If those cal-culations are clear enough to understand, and thereis nothing else of interest to explain, then it is notnecessary to write up a separate narrative of youranalysis here. If you have a long series of similarcalculations, you may just show one as a sample. Ifyour prelab involved deriving equations that you willneed, repeat them here without the derivation.

In some labs, you will need to go into some detailhere by giving logical arguments to convince me thatthe statements you made in the abstract follow log-ically from your data. 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.

80 Lab Appendix 1: Format of Lab Writeups

81

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 isoutside this range, I deserve to be drummed out ofthe profession.” If that was the case, then every sci-entist would give ridiculously inflated error bars toavoid having her career ended by one fluke out ofhundreds of published results. What scientists arecommunicating to each other with error bars is atypical amount by which they might be off, not anupper limit.

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,

82 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 precisemeasurement is one with small random errors, whilean accurate 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 analysisVery 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

83

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 rawdataWe 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 av-erage, so that’s about how big the random errorswere. But let’s make sure we are stating our errorbars according to the convention that the true resultwill fall within our range of errors about two timesout of three. Of course we don’t know the “true”result, but if we sort out our list of measurementsin order, we can get a pretty reasonable estimate ofour error bars by taking half the range covered bythe middle two thirds of the list. Sorting out our listof ten measurements 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 “stan-dard deviation” as a synonym for “error bar,” butthat does not imply that you must always use thestandard deviation method rather than the guessingmethod or the 2/3 rule.

There is a utility on the class’s web page for calcu-lating standard deviations.

Probability of deviationsYou 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 1 times out of 3

> 2 standard deviations about 1 time out of20

> 3 standard deviations about 1 in 500

> 4 standard deviations about 1 in 16,000

> 5 standard deviations about 1 in 1,700,000

The probability of various sizes of deviations, showngraphically. Areas under the bell curve correspond toprobabilities. 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 2out of 3. (J. Kemp, P. Strandmark, Wikipedia.)

Example: How significant?In 1999, astronomers Webb et al. claimed to have foundevidence that the strength of electrical forces in the an-cient universe, soon after the big bang, was slightlyweaker than it is today. If correct, this would be the firstexample ever discovered in which the laws of physicschanged 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 measurementwas different from the expected result of zero. Lookingat the table above, we see that if the true value reallywas zero, the chances of this happening would be lessthan one in a million. In general, five standard devia-tions (“five sigma”) is considered the gold standard forstatistical significance.

This is an example of how we test a hypothesis sta-tistically, find a probability, and interpret the probability.The probability we find is the probability that our results

85

would differ this much from the hypothesis, if the hy-pothesis was true. It’s not the probability that the hy-pothesis is true or false, nor is it the probability that ourexperiment is right or wrong.

However, there is a twist to this story that shows howstatistics always have to be taken with a grain of salt. In2004, Chand et al. redid the measurement by a moreprecise technique, and found that the change was 0.6±0.6 parts per million. This is only one standard devia-tion away from the expected value of 0, which should beinterpreted as being statistically consistent with zero. Ifyou measure something, and you think you know whatthe result is supposed to be theoretically, then one stan-dard deviation is the amount you typically expect to beoff by — that’s why it’s called the “standard” deviation.Moreover, the Chand result is wildly statistically incon-sistent with the Webb result (see the example on page89), which means that one experiment or the other isa mistake. Most likely Webb at al. underestimated theirrandom errors, or perhaps there were systematic errorsin their experiment that they didn’t realize were there.

Precision of an averageWe 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 88 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.


87

Appendix 3: Propagation of Errors

Propagation of the error from asingle variableIn 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 = 2.2 g/cm3

The resulting change in the density was 0.5 g/cm3,so that is our estimate for how much it could havebeen off by:

ρ = 2.7± 0.5 g/cm3 .

Propagation of the error from sev-eral variablesWhat 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 and

b 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 Qmbecause 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 averageOn page 86 I claimed that averaging a bunch of mea-surements reduces the error bars by the square root ofthe number of measurements. We can now see thatthis 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 theguess method, she estimates that her error bars areabout ±1 cm (worse than the normal normal ∼ 1 mmerror bars for a tape measure, because the guinea pigwas 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, giving11 cm. If Alice had been off by one standard devia-tion (1 cm), it would have changed the average by 0.5

88 Lab Appendix 3: Propagation of Errors

cm, so we have QA = 0.5 cm, and likewise QB = 0.5cm. Combining these, we find σc =

√Q2

A + Q2B = 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 forsignificant figures, which is that the final result shouldhave no more sig figs than the least precise piece ofdata that went into the calculation. That’s okay, be-cause the sig fig rules are just a quick and dirty wayof doing propagation of errors. We’ve done real propa-gation of errors in this example, and it turns out that theerror is in the first decimal place, so the 0 in that placeis entitled to hold its head high as a real sig fig, albeit arelatively imprecise one with an uncertainty of ±7.)

Example: The difference between two measurementsIn the example on page 85, we saw that two groupsof scientists measured the same thing, and the resultswere 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 thedifference between their two results is small enough tobe explained by random errors, or so big that it couldn’tpossibly 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 twosets of error bars don’t overlap with one another, but er-ror bars are not absolute limits, so it’s perfectly possibleto have non-overlapping error bars by chance, but thegap between the error bars is very large compared tothe error bars themselves, so it looks implausible thatthe results could be statistically consistent with one an-other. I’ve tried to suggest this visually with the shadingunderneath the data-points.

To get a sharper statistical test, we can calculate thedifference 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 justcauses an error in the output that is of the same size.Therefore we have QW = 1.0 and QC = 0.6, resultingin σd =

√Q2

W + Q2C = 1.2. We find that the difference

between the two results is d = 5.1± 1.2, which differsfrom zero by 5.1/1.2 ≈ 4 standard deviations. Lookingat the table on page 85, we see that the chances thatd would be this big by chance are extremely small, lessthan about one in ten thousand. We can conclude to ahigh level of statistical confidence that the two groups’measurements are inconsistent with one another, andthat one group is simply wrong.

89

Appendix 4: Graphing

Review of GraphingMany 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 ComputerMaking 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 LibreOffice, which is free, open-sourcesoftware. It’s installed on the computers in rooms416 and 418. Because LibreOffice 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 LibreOffice:

On Windows, go to the Start menu and chooseAll Programs, LibreOffice, and LibreOffice Calc.On Linux, do Applications, Office, OpenOffice,Spreadsheet.

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 Object: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 theplot is as big as possible, eliminating wastedspace. To do this, double-click on the graph sothat it’s surrounded by a gray border. Thendo Format, Axis, X Axis or Y Axis, Scale.

If you want error bars on your graph you can eitherdraw them in by hand or put them in a separate col-umn of your spreadsheet and doing Insert, Y ErrorBars, Cell Range. Under Parameters, check “Samevalue for both.” Click on the icon, and then use themouse in the spreadsheet to select the cells contain-ing the error bars.

Fitting a Straight Line to a Graphby HandOften in this course you will end up graphing somedata points, fitting a straight line through them witha ruler, and extracting the slope.

90 Lab Appendix 4: Graphing

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 believ-able fit” line to draw instead an “unbelievably crazyfit” line, as in panel (d). The line in panel (d), witha very small slope, is just not believable comparedto the data — it is several standard deviations awayfrom most of the data points.

Fitting a Straight Line to a Graphon a ComputerIt’s also possible to fit a straight line to a graph usingcomputer software such as LibreOffice.

To do this, first double-click on the graph so that agray border shows up around it. Then right-click ona data-point, and a menu pops up. Choose InsertTrend Line.1 choose Linear, and check the box forShow equation.

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.

1“Trend line” is scientifically illiterate terminology thatoriginates from Microsoft Office, which LibreOffice slavishlycopies. If you don’t want to come off as an ignoramus, call ita “fit” or “line of best fit.”

Comparing Theory and ExperimentFigures (e) through (h) are examples of how we wouldcompare theory and experiment on a graph. Theconvention is that theory is a line and experiment ispoints; this is because the theory is usually a predic-tion in the form of an equation, which can in prin-ciple be evaluated at infinitely many points, fillingin all the gaps. One way to accomplish this withcomputer software is to graph both theory and ex-periment as points, but then print out the graph anddraw a smooth curve through the theoretical pointsby hand.

The point here is usually to compare theory andexperiment, and arrive at a yes/no answer as towhether they agree. In (e), the theoretical curvegoes through the error bars on four out of six ofthe data points. This is about what we expect sta-tistically, since the probability of being within onestandard deviation of the truth is about 2/3 for astandard bell curve. Given these data, we wouldconclude that theory and experiment agreed.

In graph (f), the points are exactly the same as in(e), but the conclusion is the opposite. The errorbars are smaller, too small to explain the observeddiscrepancies between theory and experiment. Thetheoretical curve only goes through the error bars ontwo of the six points, and this is quite a bit less thanwe would expect statistically.

Graph (g) also shows disagreement between theoryand experiment, but now we have a clear systematicerror. In (h), the fifth data point looks like a mistake.Ideally you would notice during lab that somethinghad gone wrong, and go back and check whether youcould reproduce the result.

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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 relationshipsWhen 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 lawsBoth 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 logarithmsNow 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 plotsEven 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.

92 Lab Appendix 5: Finding Power Laws from Data

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Appendix 6: 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.

94 Lab Appendix 6: High Voltage Safety Checklist

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Appendix 7: 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.

96 Lab Appendix 7: Laser Safety Checklist

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Documents

Download the lab manual for Physics 206/211. - Light and Matter