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Physics 202 Lab Manual

Electricity and Magnetism, Sound/Waves, Light

R. Rollefson, H.T. Richards, M.J. Winokur

August 28, 2015

NOTE: E=Electricity and Magnetism, S=Sound and Waves, L-Light

Contents

Forward 3

Introduction 5

Errors and Uncertainties 8

I Electricity and Magnetism 13

E-1 Electrostatics 13

E-2 Electric Fields 24

E-3 Capacitance 27

E-4 Electron Charge to Mass Ratio 36

E-5 Magnetism 42E-5a Lenz’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42E-5b Induction - Dropping Magnet . . . . . . . . . . . . . . . . . . . . . . 43E-5c Induction - Fields in Space . . . . . . . . . . . . . . . . . . . . . . . . 45E-5d Induction - Circuit Elements . . . . . . . . . . . . . . . . . . . . . . . 47

E-6 Oscilloscopes and RC Decay 50

E-7 Series LRC Circuits and Resonance 56

E-8: Transistors 59

II Sound and Waves 62

S-1 Transverse Standing Waves on a String 62

1

CONTENTS 2

S-2 Velocity of Sound in Air 66

III Light 67

L-1: Diffraction and Interference 67

L-2: Mirrors and Lenses 70

L-3: Optical Instruments 75

L-4: The optics of the eye and resolving power 79

L-5: Spectrometer and the H Balmer Series 82

L-6: Polarization 89

Appendices 93A. Precision Measurement Devices . . . . . . . . . . . . . . . . . . . . . . . . . 93B. PARALLAX and Notes on using a Telescope . . . . . . . . . . . . . . . . . 97

NOTE: E=Electricity and Magnetism, S=Sound and Waves, L-Light

FORWARD 3

Forward

Spring, 2005

This version is only modestly changed from the previous versions. We are gradually re-vising the manual to improve the clarity and interest of the activities. In particular thedynamic nature of web materials and the change of venue (from Sterling to Chamber-lin Hall) has required a number of cosmetic and operational changes. In particular thePASCO computer interface and software have been upgraded from Scientific Workshop toDataStudio.

M.J. Winokur

In reference to the 1997 edition

Much has changed since the implementation of the first edition and a major overhaulwas very much in need. In particular, the rapid introduction of the computer into theeducational arena has drastically and irreversibly changed the way in which information isacquired, analyzed and recorded. To reflect these changes in the introductory laboratorywe have endeavored to create a educational tool which utilizes this technology; hopefullywhile enhancing the learning process and the understanding of physics principles. Thus,when fully deployed, this new edition will be available not only in hard copy but also asa fully integrated web document so that the manual itself has become an interactive toolin the laboratory environment.

As always we are indebted to the hard work and efforts by Joe Sylvester to maintainthe labortory equipment in excellent working condition.

M.J. WinokurM. Thompson

From the original edition

The experiments in this manual evolved from many years of use at the Universityof Wisconsin. Past manuals have included “cookbooks” with directions so complete anddetailed that you can perform an experiment without knowing what you are doing orwhy, and manuals in which theory is so complete that no reference to text or lecture wasnecessary.

This manual avoids the“cookbook” approach and assumes that correlation betweenlecture and lab is sufficiently close that explanations (and theory) can be brief: in manycases merely a list of suggestions and precautions. Generally you will need at least an el-ementary understanding of the material in order to perform the experiment expeditiouslyand well. We hope that by the time you have completed an experiment, your understand-ing will have deepened in a manner not achievable by reading books or by working ”paperproblems”. If the lab should get ahead of the lecture, please read the pertinent material,as recommended by the instructor, before doing the experiment.

The manual does not describe equipment in detail. We find it more efficient to havethe apparatus out on a table and take a few minutes at the start to name the pieces andgive suggestions for use. Also in this way changes in equipment, (sometimes necessary),need not cause confusion.

FORWARD 4

Many faculty members have contributed to this manual. Professors Barschall, Blan-chard, Camerini, Erwin, Haeberli, Miller, Olsson, Visiting Professor Wickliffe and formerProfessor Moran have been especially helpful. However, any deficiencies or errors are ourresponsibility. We welcome suggestions for improvements.

Our lab support staff, Joe Sylvester and Harley Nelson (now retired), have madeimportant contributions not only in maintaining the equipment in good working order,but also in improving the mechanical and aesthetic design of the apparatus.

Likewise our electronic support staff not only maintain the electronic equipment, butalso have contributed excellent original circuits and component design for many of theexperiments.

R. RollefsonH. T. Richards

INTRODUCTION 5

Introduction

General Instructions and Helpful Hints

Physics is an experimental science. In this laboratory, we hope you gain a realisticfeeling for the experimental origins, and limitations, of physical concepts; an awareness ofexperimental errors, of ways to minimize them and how to estimate the reliability of theresult in an experiment; and an appreciation of the need for keeping clear and accuraterecords of experimental investigations.

Maintaining a clearly written laboratory notebook is crucial. This lab notebook, at aminimum, should contain the following:

1. Heading of the Experiment: Copy from the manual the number and nameof theexperiment.Include both the current date and the name(s) of your partner(s).

2. Original data: Original data must always be recorded directly into your notebookas they are gathered. “Original data” are the actual readings you have taken. Allpartners should record all data, so that in case of doubt, the partners’ lab notebookscan be compared to each other. Arrange data in tabular form when appropriate.A phrase or sentence introducing each table is essential for making sense out of thenotebook record after the passage of time.

3. Housekeeping deletions: You may think that a notebook combining all work wouldsoon become quite a mess and have a proliferation of erroneous and supersededmaterial. Indeed it might, but you can improve matters greatly with a little house-keeping work every hour or so. Just draw a box around any erroneous or unnecessarymaterial and hatch three or four parallel diagonal lines across this box. (This wayyou can come back and rescue the deleted calculations later if you should discoverthat the first idea was right after all. It occasionally happens.) Append a note tothe margin of box explaining to yourself what was wrong.

We expect you to keep up your notes as you go along. Don’t take your notebookhome to “write it up” – you probably have more important things to do than makinga beautiful notebook. (Instructors may permit occasional exceptions if they aresatisfied that you have a good enough reason.)

4. Remarks and sketches: When possible, make simple, diagrammatic sketches (ratherthan “pictorial” sketches” of apparatus. A phrase or sentence introducing eachcalculation is essential for making sense out of the notebook record after the passageof time. When a useful result occurs at any stage, describe it with at least a wordor phrase.

5. Graphs: There are three appropriate methods:

A. Affix furnished graph paper in your notebook with transparent tape.

B. Affix a computer generated graph paper in your notebook with transparenttape.

C. Mark out and plot a simple graph directly in your notebook.

INTRODUCTION 6

Show points as dots, circles, or crosses, i.e., ·, , or ×. Instead of connecting pointsby straight lines, draw a smooth curve which may actually miss most of the pointsbut which shows the functional relationship between the plotted quantities. Fastendirectly into the notebook any original data in graphic form (such as the spark tapesof Experiment M4).

6. Units, coordinate labels: Physical quantities always require a number and a dimen-sional unit to have meaning. Likewise, graphs have abscissas and ordinates whichalways need labeling.

7. Final data, results and conclusions: At the end of an experiment some writtencomments and a neat summary of data and results will make your notebook moremeaningful to both you and your instructor. The conclusions must be faithful tothe data. It is often helpful to formulate conclusion using phrases such as “thediscrepancy between our measurements and the theoretical prediction was largerthan the uncertainty in our measurements.”

PARTNERSDiscussing your work with someone as you go along is often stimulating and of educa-

tional value. If possible all partners should perform completely independent calculations.Mistakes in calculation are inevitable, and the more complete the independence of thecalculations, the better is the check against these mistakes. Poor results on experimentssometimes arise from computational errors.CHOICE OF NOTEBOOK

We recommend a large bound or spiral notebook with paper of good enough qualityto stand occasional erasures (needed most commonly in improving pencil sketches orgraphs). To correct a wrong number always cross it out instead of erasing: thus 3.1461//////3.1416 since occasionally the correction turns out to be a mistake, and the original numberwas right. Coarse (1/4 inch) cross-ruled pages are more versatile than blank or line pages.They are useful for tables, crude graphs and sketches while still providing the horizontallines needed for plain writing. Put everything that you commit to paper right into yournotebook. Avoid scribbling notes on loose paper; such scraps often get lost. A good planis to write initially only on the right-hand pages, leaving the left page for afterthoughtsand for the kind of exploratory calculations that you might do on scratch paper.COMPLETION OF WORK

Plan your work so that you can complete calculations, graphing and miscellaneousdiscussions before you leave the laboratory. Your instructor will check each completedlab report and will usually write down some comments, suggestions or questions in yournotebook.

Your instructor can help deepen your understanding and “feel” for the subject. Feelfree to talk over your work with him or her.

INTRODUCTION 7

mass/(

Density of a Solid

Expt. M1 Systematic and Random Errors, Significant Figures,

2. 0.000014 mm

Partner: John Q. Student

CALCULATIONS:

Density= ??? Uncertainty from propagation of error.

Micrometer exhibits a systematic zero offset

Measure four calibraton gauge blocks

Standard DeviationAve,

5. 0.000015 mm

4. 0.000014 mm

3. 0.000012 mm

0.000001 mm1. 0.000013 mm

Reading with jaws fully closed:

1. Calibration of micrometer

DATA:

Theory:

Equiment: Venier caliper, micrometer, precision gauge block, precision balance

physical measurement by obtaining the density of metal cylinder.

Purpose: To develop a basic understanding of systematic and random errors in a

Date: 2/29/00NAME: Jane.Q. Student

ρ

ρ=

error (mm)

Micrometer

Measure of cylinder diameter:

Measure of cylinder mass

Measure of cylinder height:

-+

-+

m∆ρ= (∆

0.002

r/r)**2.∆2h/h)**2.+(∆/m)**2.+(

r**2 h)π∗

0.000

18

Gauge block length (mm)

vs. gauge block length

Plot of micrometer error

2412

-0.002

60

RESULTS and CONCLUSIONS:

ERRORS AND UNCERTAINTIES 8

Errors and Uncertainties

Reliability estimates of measurements greatly enhance their value. Thus, saying that theaverage diameter of a cylinder is 10.00±0.02 mm tells much more than the statement thatthe cylinder is a centimeter in diameter. The reliability of a single measurement (such asthe diameter of a cylinder) depends on many factors:

FIRST, are actual variations of the quantity being measured, e.g. the diameter ofa cylinder may actually be different in different places. You must then specify where themeasurement was made; or if one wants the diameter in order to calculate the volume,first find the average diameter by means of a number of measurements at carefully selectedplaces. Then the scatter of the measurements will give a first estimate of the reliability ofthe average diameter.

SECOND, the micrometer caliper used may itself be in error. The errors thus intro-duced will of course not lie equally on both sides of the true value so that averaging a largenumber of readings is no help. To eliminate (or at least reduce) such errors, we calibratethe measuring instrument: in the case of the micrometer caliper by taking the zero error(the reading when the jaws are closed) and the readings on selected precision gauges ofdimensions approximately equal to those of the cylinder to be measured. We call sucherrors systematic, and these cause errors in accuracy.

THIRD, Another type of systematic error can occur in the measurement of a cylin-der: The micrometer will always measure the largest diameter between its jaws; hence ifthere are small bumps or depressions on the cylinder, the average of a large number ofmeasurements will not give the true average diameter but a quantity somewhat larger.(This error can of course be reduced by making the jaws of the caliper smaller in crosssection.)

FINALLY, if one measures something of definite size with a calibrated instrument,one’s measurements will vary. For example, the reading of the micrometer caliper mayvary because one can’t close it with the same force every time. Also the observer’s esti-mate of the fraction of the smallest division varies from trial to trial. Hence the averageof a number of these measurements should be closer to the true value than any one mea-surement. Also the deviations of the individual measurements from the average give anindication of the reliability of that average value. The typical value of this deviation isa measure of the precision. This average deviation has to be calculated from the absolutevalues of the deviations, since otherwise the fact that there are both positive and negativedeviations means that they will cancel. If one finds the average of the absolute values ofthe deviations, this “average deviation from the mean” may serve as a measure ofreliability. For example, let column 1 represent 10 readings of the diameter of a cylindertaken at one place so that variations in the cylinder do not come into consideration, thencolumn 2 gives the magnitude (absolute) of each reading’s deviation from the mean.


Measurements Deviation from Ave.9.943 mm 0.0009.942 0.0019.944 0.0019.941 0.0029.943 0.0009.943 0.0009.945 0.002 Diameter =9.943 0.0009.941 0.002 9.943±0.001 mm9.942 0.001

Ave = 9.943 mm Ave = 0.0009 mm≈0.001 mmExpressed algebraically, the average deviation from the mean is = (

∑|xi − x|)/n),

where xi is the ith measurement of n taken, and x is the mean or arithmetic average ofthe readings.

Standard Deviation and Normal Distribution:The average deviation shown above is a measure of the spread in a set of measurements.

A more easily calculated version of this is the standard deviation σ (or root mean squaredeviation). You calculate σ by evaluating

σ =

√√√√ 1

n

n∑i=1

(xi − x)2

where x is the mean or arithmetical average of the set of n measurements and xi is theith measurement.

Because of the square, the standard deviation σ weights large deviations more heavilythan the average deviation and thus gives a less optimistic estimate of the reliability. Infact, for subtle reasons involving degrees of freedom, σ is really

σ =

√√√√ 1

(n− 1)

n∑i=1

(xi − x)2

σ tells you the typical deviation from the mean you will find for an individual measure-ment. The mean x itself should be more reliable. That is, if you did several sets of nmeasurements, the typical means from different sets will be closer to each other than theindividual measurements within a set. In other words, the uncertainty in the mean shouldbe less than σ. It turns out to reduce like 1/

√n, and is called the standard deviation

of the mean σµ:

σµ = standard deviation of the mean =σ√n

=1√n

√√√√ 1

n− 1

n∑i=1

(xi − x)2

For an explanation of the (n−1) factor and a clear discussion of errors, see P.R. Bevingtonand D.K Robinson, Data Reduction and Error Analysis for the Physical Sciences, 3rd ed.,McGraw Hill 2003, p. 11.

If the error distribution is “normal” (i.e. the errors, ε have a Gaussian distribution,e−ε

2, about zero), then on average 68% of a large number of measurements will lie closer


than σ to the true value. While few measurement sets have precisely a “normal” distri-bution, the main differences tend to be in the tails of the distributions. If the set of trialmeasurements are generally bell shaped in the central regions, the “normal” approxima-tion generally suffices.

How big should the error bars be?The purpose of the error bars shown on a graph in a technical report is as follows: if thereader attempts to reproduce the results in the graph using the procedure described inthe report, the reader should expect his or her results to have a 50% chance of fallingwith the range indicated by the error bars.

If the error distribution is normal, the error bars should be of length ±0.6745σ.

Relative error and percentage error:Let ε be the error in a measurement whose value is a. Then ( εa) is the relative error of themeasurement, and 100 ( εa)% is the percentage error. These terms are useful in laboratorywork.


UNCERTAINTY ESTIMATE FOR A RESULT INVOLVING MEASUREMENTS OFSEVERAL INDEPENDENT QUANTITIES

Let R = f(x, y, z) be a result R which depends on measurements of three differentquantities x, y, and z. The uncertainty ∆R in R which results from an uncertainty ∆xin the measurement of x is then

∆R =∂f

∂x∆x ,

and the fractional uncertainty in R is

∆R

R=

∂f∂x

f∆x .

In most experimental situations, the errors are uncorrelated and have a normal distribu-tion. In this case the uncertainties add in quadrature (the square root of the sum of thesquares):

∆R

R=

√√√√( ∂f∂x

f∆x

)2

+

(∂f∂y

f∆y

)2

+

(∂f∂z

f∆z

)2

.

Some examples:

A.) R = x + y. If errors have a normal or Gaussian distribution and are indepen-dent, they combine in quadrature:

∆R =√

∆x2 + ∆y2 .

Note that if R = x− y, then ∆R/R can become very large if x is nearly equal to y. Henceavoid, if possible, designing an experiment where one measures two large quantities andtakes their difference to obtain the desired quantity.

B.) R = xy. Again, if the measurement errors are independent and have a Gaussiandistribution, the relative errors will add in quadrature:

∆R

R=

√(∆x

x)2 + (

∆y

y)2 .

Note the same result occurs for R = x/y.C.) Consider the density of a solid (Exp. M1):

ρ =m

πr2L

where m = mass, r = radius, L = length, are the three measured quantities and ρ =density. Hence

∂ρ

∂m=

1

πr2L

∂ρ

∂r=−2m

πr3L

∂ρ

∂L=−mπr2L2

.

Again if the errors have normal distribution, then

∆ρ

ρ=

√(∆m

m

)2

+

(2

∆r

r

)2

+

(∆L

L

)2


SIGNIFICANT FIGURES

Suppose you have measured the diameter of a circular disc and wish to compute itsarea A = πd2/4 = πr2. Let the average value of the diameter be 24.326 ± 0.003 mm; dividing d by 2 to get r we obtain 12.163 ± 0.0015 mm with a relative error ∆r

r of0.0015

12 = 0.00012. Squaring r (using a calculator) we have r2 = 147.938569, with a relativeerror 2∆r/r = 0.00024, or an absolute error in r2 of 0.00024× 147.93 · · · = 0.036 ≈ 0.04.Thus we can write r2 = 147.94± 0.04, any additional places in r2 being unreliable. Hencefor this example the first five figures are called significant.

Now in computing the area A = πr2 how many digits of π must be used? A pocketcalculator with π = 3.141592654 gives

A = πr2 = π × (147.94± 0.04) = 464.77± 0.11 mm2

Note that ∆AA = 2∆r

r = 0.00024. Note also that the same answer results from π = 3.1416,but that π = 3.142 gives A = 464.83 ± 0.11 mm2 which differs from the correct value by0.06 mm2, an amount comparable to the estimated uncertainty.

A good rule is to use one more digit in constants than is available in your measure-ments, and to save one more digit in computations than the number of significant figuresin the data. When you use a calculator you usually get many more digits than you need.Therefore at the end, be sure to round off the final answer to display the correctnumber of significant figures.

SAMPLE QUESTIONS

1. How many significant figures are there in the following numbers?

(a) 976.45

(b) 4.000

(c) 10

2. Round off each of the following numbers to three significant figures.

(a) 4.455

(b) 4.6675

(c) 2.045

3. A function has the relationship Z(A,B) = A+B3 where A and B are found to haveuncertainties of ±∆A and ±∆B respectively. Find ∆Z in term of A, B and therespective uncertainties assuming the errors are uncorrelated.

4. What happens to σ, the standard deviation, as you make more and more measure-ments? what happens to σ, the standard deviation of the mean?

(a) They both remain same

(b) They both decrease

(c) σ increases and σ decreases

(d) σ approachs a constant and σ decreases

13

Part I

Electricity and Magnetism

E-1 Electrostatics

OBJECTIVES:

To investigate charging by rubbing (triboelectricity); charging by conduction; andcharging by induction.

PRELIMINARY QUESTIONS:

1. What is the difference between a “conductor” and an “insulator”?

2. Write down the equation for the electric field caused by a point charge:

3. Circle the statement that is true:opposite charges attract, like charges repelopposite charges repel, like charges attract

4. More precisely, the equation for the force between two point charges is:

5. A conductor is brought into contact with ground. Circle all that are possible:negative charge flows from ground to conductornegative charge flows from conductor to groundno charge flows

6. Write down two occasions on which you have encountered static electricity.

7. You are in a car with the windows rolled up. The frame of the car is made of metal.The car has rubber tires. A thunderstorm develops. Because you are inside thecar, you are to some degree protected against injury due to lightning strike. Thisprotection is due to (circle one):–the fact that you are inside the metal frame, which is a hollow conductor;–the fact that you are isolated from ground by the insulating rubber tires.

8. Record the temperature and relative humidity of the lab room here:temperature = C relative humidity: %

If the air is dry, it is comparatively easy to generate static electricity (you may havealready noticed this). If the air is damp, it is comparatively difficult.

EXPERIMENT I: CHARGING BY RUBBING (TRIBOELECTRICITY):APPARATUS:

Electrometer; Faraday Cup; Pasco Interface (to provide electrical ground); testsphere; silk cloth; ebonite rod (black); rabbit fur; acrylic rod (clear).

When any two objects rub against each other, electrons move from one object toanother. Depending on the nature of the objects, the number of electrons making themove may be very small or very large. One object ends up with “too many” electrons,and the other with “not enough”.

E-1 ELECTROSTATICS 14

Figure 1: Setup for Expt. 1.

1. As shown in the picture, connect the SIGNAL INPUT of the electrometer to theFaraday Cup, and connect the GROUND port of the electrometer to the groundinput of the Pasco interface. Make sure the Pasco interface is plugged in. If you insertan object with too many electrons into the Faraday Cup WITHOUT TOUCHINGTHE SIDES OR TOP, the electrometer will read negative. Insert an object withtoo few electrons, and it will read positive. Insert an object that is neutral, and theelectrometer reading will stay close to zero.

2. Turn ON the electrometer. Set the electrometer to VOLTS FULL SCALE RANGEof 100. Push the “ZERO” button to remove any charge on the electrometer. Thedigits display and meter display should both indicate “0”. Call over your TA if theydon’t. Press the ZERO button any time the electrometer reads something otherthan zero when nothing is inserted in the Faraday Cup.

3. Rub the silk cloth vigorously against the test sphere, as if you are trying to polishthe test sphere. Insert the test sphere into the Faraday Cup (without touching it).Indicate how the electrometer reading changes (circle one):becomes positive doesn’t change becomes negative(if the electrometer reading doesn?t change, call over your TA).

4. Touch the test sphere against the outside of the Faraday Cup.

5. Again insert the test sphere into the Faraday Cup (without touching it), and indicatehow the electrometer reading changes (circle one):becomes positive doesn’t change becomes negative(if the electrometer reading changes a lot, call over your TA)

6. If all went well, the test sphere should have been charged in step 3, but neutral instep 5. What did you do that caused the test sphere to become neutral, and whathappened to the extra charge?


Table I: The Triboelectric Series

Various substances rankedfrom greatest tendency tocharge positive when rubbed (hair)to greatest tendency to chargenegative when rubbed (teflon)

Hair (most positive)Rabbit furAcrylic (clear plastic)Hands, skinGlassNylonWoolQuartzCat FurLeadSilkAluminumPaperCotton (neutral)SteelWoodAmberEbonite (looks like black plastic)Nickel, copperBrass, silverGold, platinumSulfurRayonPolyesterStyrofoamOrlonSaranPolyurethanePolyethylenePVCTeflon (most negative)

7. Insert the acrylic rod into the Faraday cage. Ifthe electrometer shows charge is present, cleanthe acrylic rod with soap and water, and verifyit is uncharged. If it is still charged, call overyour TA.

8. Repeat step 7, using the ebonite rod.

9. Rub the acrylic rod with the silk. What chargedoes the acrylic rod acquire? (circle one)positive negative

10. Rub the ebonite rod with the rabbit fur. Whatcharge does the ebonite rod acquire? (circleone) positive negative

11. Are your results from steps 9 and 10 in accor-dance with the Triboelectric Series (see table atright)?

12. You should be able to find at least one itemthat you brought with you to lab on theTriboelectric Series. Touch the test sphere tothe outside of the Faraday Cup, rub the itemyou brought against the test sphere (which ismade of brass), and insert the test sphere intothe Faraday Cup to see what type of charge ithas acquired.Item you brought:

Test sphere charges: positive negative

Is this in accordance with the TriboelectricSeries?


EXPERIMENT II: CHARGING BY CONDUCTION:APPARATUS: electroscope; plastic rod; silk cloth; ebonite rod; rabbit fur; Pasco interface.

Figure 2: Electroscope

The electroscope is an instrument that detects the presence of electric charge, but doesnot directly tell you the sign of the charge. The leaves rise when there is charge on theleaves, and fall when there is no charge (in the picture above, the leaves are uncharged).The leaves and the knob of an electroscope are connected by a conductor.

1. Inspect the leaves on your electroscope; make sure there are two of them, and makesure they are not risen. If they are risen, touch the knob with your finger, and theyshould fall. If you can’t get the leaves to fall, call over your TA.

2. Rub the acrylic rod with the silk, and touch the acrylic rod to the knob of theelectroscope. The leaves should rise (if they don’t, call over your TA). Accordingto the Triboelectric Series table, the acrylic rod should charge positive. Assumingthis is true, draw little “+” signs in the pictures below to indicate where the electriccharges are during this process.


3. Plug a cable into the ground input of the Pasco interface, and touch the other endof the cable to the knob of the electroscope. What happens to the leaves?

4. Draw little “+” signs on the diagrams below to indicate what happens to the chargeson the oscilloscope during B3.

5. Rub the ebonite rod with the rabbit fur. According to the Triboelectric Series table,the ebonite rod should charge negative. Assuming this is true, draw “-” signs on thepictures below to indicate where the charges are during this process.

6. If the leaves of an electroscope are risen, is there any way to tell just by looking atthe electroscope whether the charge on the leaves is positive or negative?

7. Touch the knob of the electroscope with your finger. What happens to the leaves?


EXPERIMENT III: CHARGING BY INDUCTION:APPARATUS: electroscope; plastic rod; silk cloth; ebonite rod; rabbit fur; electrometer;Faraday cup; Pasco interface (to provide electrical ground).In this part we’ll do three experiments related to charging by induction.

1. Rub the acrylic rod with the silk, and touch the acrylic rod to the knob of theelectroscope. This charges the electroscope by conduction, as in Expt. II step 2,and the leaves of the electroscope should now be risen.

2. Rub the ebonite rod with the rabbit fur, and bring the ebonite rod near the knobof the electroscope without touching it. What happens to the leaves?

3. Draw “”+? and “-” signs on the pictures below to show where the charges are:

4. Even though you can’t tell the sign of the charge on the electroscope leaves just bylooking at the electroscope, the results of steps 1 to 3 allow you to determine thatthe ebonite rod and the acrylic rod must have charges of opposite sign. Touch theknob of the electroscope with your finger. This should cause all excess charge todepart the electroscope, and the leaves to drop: you have “discharged” the object.This concludes the first experiment on charging by induction.


5 Rub the acrylic rod with the silk, and bring the acrylic rod near, but not touchingthe knob on the electroscope. What happens to the leaves?

6. Assuming the acrylic rod is charged positive, draw little + and - signs on the cartoonsto show where the charges are.

7. The charge on the electroscope is defined as the charge on the leaves plus the chargeon the knob. Assuming you did not touch the acrylic rod to the knob, what was thecharge on the electroscope in the middle cartoon above?

8. Plug a cable into the ground input of the Pasco interface, and repeat step 5, withone difference: while the acrylic rod is near the knob, touch the knob with your endof the cable, and then remove the cable. Only when the cable is gone should youremove the acrylic rod. Sketch what the charges do.

9. What is the sign of the charge that remains on the electroscope? This shows howan object can become charged by induction, and concludes the second experimenton charging by induction.


For the third experiment on charging by induction we’ll return to the apparatus shown inthe picture associated with Expt. 1, step 1, and use the test sphere to sample, or “test”,the charge at different points on a conductor. Make sure the rods, fur, silk, and any otherobjects which are likely to be charged are moved far away from the Faraday cup.

10. Turn ON the electrometer. Set the electrometer to VOLTS FULL SCALE RANGEof 100. Push the ZERO button to remove any charge on the electrometer. Thedigits display and meter display should both indicate “0”. Call over your TA if theydon’t.

11. Take one of the big metal spheres, and touch it to the outside of the Faraday Cup.How much charge do you expect to remain on the big metal sphere after this?

12. Choose one person to manipulate the test sphere (we’ll call this person TS in theinstructions below) and one person to manipulate the ebonite rod and rabbit fur(we’ll card this person ER).

13. ER rubs ebonite rod with rabbit fur, and inserts ebonite rod into Faraday Cup. Theelectrometer reads (circle one):positive negative zero

14. TS touches test sphere to outside of Faraday Cup, and then inserts test sphere intoFaraday Cup. The electrometer reads (circle one):positive negative zero

15. If the electrometer doesn’t read zero, it may be because there is charge on the handleof the test sphere. Clean the handle with soap and water and repeat step 14. If theelectrometer still doesn?t read zero, call over your TA.

16. ER brings ebonite rod near BUT NOT TOUCHING right side of the big metalsphere, and holds it there while TS completes the next step.

17. TS touches test sphere to the left side of the big metal sphere.

18. TS inserts the test sphere into the Faraday Cup. The electrometer reads:positive negative zero

19. TS touches the test sphere to the outside of the Faraday Cup, and then inserts thetest sphere into the Faraday Cup. The electrometer reads:positive negative zero

20. ER brings the ebonite rod near BUT NOT TOUCHING the right side of the bigmetal sphere, and holds it there while TS completes the next step.

21. TS touches the test sphere to the right side of the big metal sphere, as close aspossible to the ebonite rod WITHOUT TOUCHING THE EBONITE ROD.

22. TS inserts the test sphere into the Faraday Cup. The electrometer reads:positive negative zero

23. TS touches the test sphere to the outside of the Faraday Cup, and then inserts thetest sphere into the Faraday Cup. The electrometer reads:positive negative zero


24. Based on the results of steps 10-23, indicate on the picture below the charge distri-butions on the ebonite rod and the big metal sphere.

This experiment shows how the charge distribution on an object can be measured, onepoint at a time. It‘s important that the test sphere be much smaller than the object ofwhich the charge distribution is measured, because every time the test sphere is touchedto the object, some charge is taken away from the object. The smaller the test sphere,the less charge is taken away. A “measurement” that significantly changes what is to bemeasured is not a good measurement.

PART III: CONCLUSION QUESTIONS

1. Why does touching the electroscope with your hand ground it? Do you end up witha net charge? If so, how can we see the effects of this?

2. Can the electroscope differentiate between positive and negative charges? Explainbased on your observations.

3. Do only insulators acquire charge by rubbing? Explain.

4. Electrostatics experiments often do not work well in humid weather. Explain whythis might be so.


5. In very dry or cold weather, objects such as clothing, table tops, etc. that somehowacquire a charge tend to hold that charge for a long time, i.e. they are slow todischarge. How might this affect the observations in these experiments?

6. Where might you observe the effects of a net charge in your everyday life?

TROUBLESHOOTING

1. In humid air insulators may adsorb enough moisture that charges leak off rapidly.If so, dry all insulators with a heat gun. Their large surface charges may influencenearby unshielded instruments. If so, ask your instructor for help; e.g. use groundedfoil to shield against them. Also remove any clinging loose bits of fiber (e.g. silk orrabbit fur) that may disturb results.

2. The fragile leaves of the electroscope may tear if charged too heavily. Do notdisassemble electroscope to attempt repair: see your instructor.

3. To remove charges on the glass windows of the electroscope, lightly rub your handsover the windows while grounding your body.

4. When instructed to touch the hollow sphere with the proofball, avoid rubbing whichmay result in inaccurate results. (Remember that rubbing is how you created thecharge separation on the lucite or rubber rods. Here, we want to charge by conduc-tion.)

5. In humid weather or if there is excess charge on the handle, your proofball may nothold charge. If this happens you can try cleaning or heating the handle (as describedin the next item), or putting the lucite rod into the hollow conductor directly.

6. Charges on the insulating handle of a proofball can cause serious measuring errors.Test the handle by grasping the ball with one hand (while the other hand touchesthe electroscope ground), and then bring parts of the insulating handle close to theelectroscope knob. If the leaves move, the handle is charged. To discharge it, holdthe handle in a source of ionized air. The charged insulator will attract ions of theopposite sign until it is neutral. An open flame is a simple source of both positiveand negative ions. The heated air convects these ions upward such that the handlewill attract the correct sign to make it neutral. Hold the insulator at least 10cm above open flame to avoid heat damage to the handle.

7. Avoid unnecessary handling of insulators because handling may impair their insu-lating capability. (Perspiration is a salt solution which is a conductor.)

8. If your clothing or hair has a net charge, the electrometer reading may change ifyou move around. Hence, during a given measurement, change position as little aspossible and ground yourself.

9. To remove all charge from the cup, press the ‘Zero’ button. (This connects theelectrometer terminals to each other so that any charge flows from/to ground). Ifthe meter does not read within a few percent of zero, notify your instructor.


10. Cleaning your paddles before beginning may produce better results. Dirt on thepaddles will affect how much charge is transferred (and how actually the same twopaddles of the same type are).

11. Always discharge paddles and cup before starting an experiment. To test if anobject is charged, put it into the cup and see whether the electrometer deflects.Conductors discharge easily by touching them to a grounded conductor. To dischargean insulator, you must create sufficient ions in the surrounding air. The insulatorwill then attract ions of the opposite charge until all charge is neutralized. An openflame is a simple source of ionized air; the ions in the flame convect upward with thehot gas. To avoid damage to the insulator, keep it at least 10 cm abovethe flame!

12. You may measure charge and still avoid spurious effects from charges on the in-sulating handles if you will touch the charged proofball (or paddle) to the bottomof the cup and then remove it from the cup before reading the electrometer. Butremember to discharge the cup (by momentarily grounding) before taking the nextreading! However, if the potential of the insulating handle is too large (e.g. way offthe least sensitive scale), one can still get spurious effects from leakages.

E-2 ELECTRIC FIELDS 24

E-2 Electric Fields

OBJECTIVES:

To develop an intuitive understanding of relationship between electric fields andequipotential contours; to physically map both in two dimensions.

PRELIMINARY QUESTIONS:

1. Do electric fields extend through a vacuum?

2. Do electric fields extend through the interior of an insulator?

3. Do electric fields extend through the interior of a conductor?

4. What is the relationship between electric field lines and equipotential lines?

5. Draw what you expect the equipotential and electric field lines to look like for apoint charge. What do you expect them to look like for a dipole? A conductingplate?

APPARATUS:

Power supply (18V, 3A Max); Fluke 115 Digital Multimeter (DMM) & test probes;conductive paper with various conducting electrode configurations; field plottingboard; carbon paper; white paper; red and black banana cables

Fluke DMM: To use the DMM, choose the DC voltage indicator, , from the dial ofmeasurements. Place one probe on the electrode that is grounded and place the otherat a different location on the conductive paper. The DMM will now read the differencein electric potential between those two locations. Pressing harder will not improveyour measurement! Note that although the DMM will measure the potential differencebetween arbitrary points, one probe should always be referenced to ground for the purposesof this lab.EXPERIMENTS:

1. Place white paper on the field plotting board, then a piece of carbon paper. On topof those, place the conductive paper with a dipole electrode configuration.

2. Record the shape of the electrodes on the white paper by tracing them with a hardpencil or a ball point pen.

3. Using the power supply, place 18 volts across the terminals of the plotting board (asin Fig. 1). Check the quality of the electrodes you are using with the DMM to makesure there are no appreciable potential differences between various locations withinthe electrodes. If you find any, ask your TA to find you a new one or fix the silverpaint quality.

4. Map the equipotential line which corresponds to +3 volts by exploring the regionof the grounded electrode with one probe while keeping your other probe on yourreference. Make several dark marks where you find a potential of +3V, and makethose recorded points are close enough together that you can later draw a smoothline through the points. Note: the points only need to be close together when thedirection of the equipotential changes rapidly.


Figure 1: Experimental Setup

5. Map another 8 other equipotential lines. Chose voltages that are spread out evenlyto 18V. Remove the white paper when completed.

6. On the white paper, connect the dots representing individual equipotentials withsmooth lines.

7. Repeat steps 1-7 for the two configurations shown in Fig. 2. In each case map aboutten equipotentials.

A B

18 Volts0 Volts

(b)

BA

18 Volts0 Volts

(a)

Figure 2: Electrode configurations for (a) parallel plates and (b) a lightning rod.


QUESTIONS:

1. Explain in a sentence or two how you mapped the equipotential lines.

2. For two different electrode configurations, calculate the mean magnitude of the elec-tric field strength in the region where the field is largest. This can be done by pickingadjacent equipotential contours and measuring the distance between them.

3. What is the relationship between electric potential lines and electric field lines? Hint:Consider the relationship between electric potential and electric field:

Vb − Va = −∫ b

a

~E · d~l

and note that Vb = Va for any points b and a on the equipotential.)

4. Using your answer in the previous questions, draw about 5 electric field lines on eachof the equipotential maps. Do they look like what you might expect?

5. Where in Fig. 2(b) is the electric field strength the highest? Lowest?

6. True or false: electric fields tend to be larger around a sharp point than around ablunt surface.

7. True or false: if a conducting object becomes charged, the charges tend to congregateon the pointy parts of the object.

8. It is clear that lightning rods protect buildings against damage from lightningstrikes. Two theories have been advanced why this might be the case:

a. The lightning rod is a pointy object; charges congregate on the point, and read-ily leak off, thus preventing the building from becoming charged, and attractinglightning;

b. The lightning rod, which must be connected to ground by a thick metal wire,does little to prevent lightning strikes, but ensures that when lightning doesstrike, the current passes harmlessly through the thick wire to the ground,rather than through the building.

One of these theories has a wealth of evidence to back it up, and one does not.Which is which?

E-3 CAPACITANCE 27

E-3 Capacitance

OBJECTIVES:

To examine very carefully the movement of charges in a capacitor; to observe andaccount for the effects of stray capacitance.

EXPERIMENT I: DISCHARGING A CAPACITOR:APPARATUS:

Power supply; electrometer; plug-board and capacitors; cables with banana-plugconnectors; cable with BNC connectors; BNC female to dual banana-plug adapter(see Fig. 1).

Figure 1: BNC female to dualbanana-plug adapter. Notethe tab on right labelled”GND”. This indicates whichside is electrically connectedto the outer conductor of theBNC connector, which is oftengrounded.

1. Turn on the power supply, and set it to 9 V.

2. Turn on the electrometer, and set it to the 10 V scale.

3. Connect the circuit shown in Fig. 2. Note that the power supply “-” and “GND”terminals are shorted together by a metal tab. A conducting path connects theseterminals to a pipe driven into the dirt beneath Chamberlin Hall, via the internalcircuitry of the electrometer, and the wires in the walls of Chamberlin Hall. Theseterminals therefore establish a well-defined 0 V, or “ground”, reference electric po-tential.

E-3 CAPACITANCE 28

Figure 2: Initial configuration of circuit in Expt. I.

4. This symbol is used to indicate an object that is electrically connected to ground:

5. The lines on the plugboard represent electrical connections. Thus this circuit canbe represented schematically as in Fig. 3.

Figure 3: Schematic of initial configuration of circuit in Expt. I.

E-3 CAPACITANCE 29

6. In this lab, the only charge carriers are electrons. In order for the charge on an objectto change, there must be a conducting path connecting it to some other object. Inthe above diagram, note that the region in between the two capacitors (it looks likea sideways “H”) is not connected to anything. The total charge on this region musttherefore remain zero, until it is connected to some other object.

7. Also, the capacitors in this lab have equal and opposite charges on their plates.The distribution of charges on the configuration shown in Fig. 2 is thus as shown inFig. 4. We have not yet calculated the magnitude of Q1, but the above two principlesrequire that all four plates have identical magnitudes of charge.

Figure 4: Distribution of charges of initial configuration of circuit in Expt. I.

8. As configured in Fig. 2, the electrometer is set up to measure the voltage dropacross both capacitors, which is the same as the voltage supplied by the powersupply. Verify that your electrometer reading agrees with the power supply reading.

Q1: Predict the voltage drop across each capacitor.

9. Extract the BNC-to-banana-plug adapter from the plugboard, and measure the volt-age across each capacitor (one at a time). Each voltage should be within a volt ofyour prediction.

E-3 CAPACITANCE 30

10. The following three steps constitute one attempt to remove charge from the capaci-tors. Measure the voltage across the capacitor with the electrometer, and record iton the left-hand side of the diagram. Then deduce the charge on each plate of thecapacitor (+Q1, −Q1, +Q2, −Q2, or 0). Do not make any calculations, but insteaduse logic alone to make the deduction. Is either capacitor discharged?

11. Restore the original configuration. The following three steps constitute anotherattempt to remove charge from the capacitors. Is either capacitor discharged?

12. Restore the original configuration. The following four steps constitute one moreattempt to remove charge from the capacitors. Is either capacitor discharged?

E-3 CAPACITANCE 31

EXPERIMENT II: STRAY CAPACITANCEAPPARATUS:

Pasco Interface; power supply; electrometer; large parallel-plate capacitor withmoveable plate; low-capacitance lead; two brass barrels. (see Fig. 5).

Figure 5: Experimental Setup

Information about the parallel-plate capacitor: plate diameter D = 19.95 ± 0.05 cm; anindicated distance of 0 mm corresponds to an actual separation of about 1 mm. Insulatingpads are mounted on one of the capacitor plates, so that this is the minimum possibleplate separation.

E-3 CAPACITANCE 32

THEORY I: Consider a circuit composed of a voltage source and a capacitor, as shown inFig. 6.

Figure 6: Ideal parallel-plate capacitor in circuit.

The capacitance of an ideal parallel-plate capacitor is C = ε0A/d, where A is the platearea and d is the plate spacing. Suppose the plate spacing is set to d, the voltage acrossthe capacitor is set to V , and then all external connections are removed. The capacitorcharge Q will henceforth be constant. Since C depends on d, if d is changed, C andtherefore V will change: V = Q/C = Qd/ε0A. The voltage should increase linearly withplate separation.

PROCEDURE I: TEST OF THEORY I:

1. Set the power supply voltage to 9 V.

2. Set the electrometer range to 10 V.

3. Connect the GND terminal of the power supply to the stationary plate of the parallel-plate capacitor using one of the brass barrels.

4. Connect the center conductor of the electrometer input to the moving plate usingthe low-capacitance lead (careful-delicate connection), and the other brass barrel.

5. Connect the electrometer to analog input A of the Pasco interface. Zero the elec-trometer. Turn on the interface. Your setup should now look like Fig. 5.

6. Go to Pasco/Physics202 on the desktop, and launch the Physics202-E3 setup file.

7. Start recording data by clicking the “Record” button in the lower left of the graph.

8. Set the relative plate separation to 0.5 cm (0.6 cm actual), using the black line onthe moving plate assembly.

9. Move around without touching the capacitor or disturbing the plates. Is thereanything you can do that influences the electrometer reading?

10. Charge the capacitor by momentarily touching the positive lead from the powersupply to them moveable plate. The electrometer reading should go to 9V and staythere as long as the positive lead is touching the plate.

E-3 CAPACITANCE 33

11. Remove the positive lead, and move the plate while watching the graph on the screen.It is evident that voltage does not continue to rise linearly with plate separation; ifit did, it would be over 200V at the maximum plate separation.

THEORY IIApparently our parallel-plate capacitor is not ideal. The problem is that the charge weput on the moving plate affects the charges on all nearby objects, such as the table. Thisis an example of “stray capacitance”, in which charge put on one object changes thepotential of all nearby objects (not just the ones we intend), and can make circuits behavein unpredicted ways. Stray capacitance is modeled as a capacitance in parallel with thephysical capacitor, as in Fig. 7. Call the stray capacitance CS .

Figure 7: Model of real capacitor including the effect of stray capacitance Cs.

The relation between the charge Q on the parallel-plate capacitor and the voltage Vapplied to the parallel plate capacitor is then:

Q = (C + Cs)V, (1)

where C = ε0A/d, the parallel-plate capacitance, changes with plate spacing d, and Cs,the stray capacitance, does not.

This can be rearranged to solve explicitly for V :

V =(Q/Cs) d

(ε0A)/Cs + d, (2)

where d has been separated out because it is changing. This in turn can be rewritten

V =Ad

B + d, (3)

where A = Q/Cs is the value taken by the voltage when the plate spacing goes to infinity,and B = ε0A/Cs is the value of the plate separation for which the stray capacitance equalsthe capacitance of the parallel plates. We will now attempt to acquire data to test thismodel.QUESTIONS:

Q1: In step 10 above, when the plate spacing became large, what was the maximumvoltage that appeared across the capacitor?

Q2: Suppose the stray capacitance Cs equals the parallel-plate capacitance C when d = 1cm. Sketch the capacitor voltage V , as predicted by Eq. 2, as a function of plateseparation d, indicating on the graph where d = 1 cm.

E-3 CAPACITANCE 34

Q3: Indicate the part of the sketch for which the ideal model of the parallel-plate capac-itor is valid (accurately predicts the data).

Q4: Indicate the part of the sketch for which the ideal model is invalid (fails to predictthe data).

Q5: Verify (if you haven’t already) that movement of your body can affect the electrom-eter reading, even if the plates are motionless. Why might this be?

PROCEDURE II: TEST OF THEORY II

Figure 8: Data for plate separations smaller than 0.6 cm.

The following series of steps allow accurate measurements of V for d < 0.6 cm. Thedata you are about to take should end up looking like the Fig. 8 above. Dexterity isrequired of the experimenter, who must move as little as possible while moving the plate,and bringing the lead intermittently into contact with it. If circumstances prevent accuratedata from being obtained, use the data in the setup file to answer the questions.

1. Set the stop so that the plate separation is limited to d ≤ 0.6 cm (note that thiscorresponds to 0.5 cm on the scale).

2. Move the plate to d = 0.6 cm.

3. Start recording data; then do the following steps without pause.

4. Hold the positive lead of the power supply in contact with the moving plate of thecapacitor for about five seconds.

5. Remove the positive lead.

6. Reduce the plate spacing to d = 0.1 cm (0.0 on the scale). Let the plate remain inthis position for about five seconds.

7. Return the plate spacing to d = 0.6 cm.

E-3 CAPACITANCE 35

8. Hold the positive lead of the power supply in contact with the moving plate of thecapacitor for about five seconds.

9. Repeat the above four steps for d = 0.2 cm, d = 0.3 cm, d = 0.4 cm, d = 0.5 cm.Stop taking data. The data should look like Fig. 8 above.

The following series of steps allow accurate measurement of V for d > 0.6 cm.

1. Set the stop so that the plate separation is limited to d ≥ 0.6 cm.

2. Set the plate separation to d = 0.6 cm.

3. Repeat the above procedure to generate one dataset that includes voltage measure-ments for d = 1.1 cm, d = 2.1 cm, d = 3.1 cm, d = 5.1 cm, d = 7.6 cm, d = 10.1 cm,and d = 12.1 cm.

QUESTIONS:

Q6: The parallel-plate capacitor is surrounded by air, and charge on the plates can becarried off by water molecules in the air. This is one mechanism of charge leakage.Make an accurate sketch of the data from step 12, and circle the parts of the graphthat give the clearest evidence for charge leakage.

Q7: When trying to verify Eq. 2, is it better to record the voltage assumed immediatelyupon the plates reaching their final spacing, or to record the voltage some time afterthe plate have reached their final spacing?

1. From your two datasets, fill in the table in the Pasco setup file of voltage as afunction of plate separation. Do your best to minimize the effect of charge leakage.

2. Make a graph of this data.

3. Figure out how to fit the function V = AdB+d to this data.

4. Record the values of A (in Volts) and B (in cm) here.

QUESTIONS:

Q8: Are the values of A and B from the fit consistent with your graph from step 2?

Q9: A common challenge faced by electrical engineers is minimizing the stray capaci-tance. Suppose the stray capacitance in this circuit were cut in half. What wouldhappen to A and B?

E-4 ELECTRON CHARGE TO MASS RATIO 36

E-4 Electron Charge to Mass Ratio

OBJECTIVES:

To observe magnetic deflection of electrons, at fixed energy, in a uniform magneticfield; to measure e/m, the electron charge/mass ratio; to estimate the strength ofthe Earth’s magnetic field.

APPARATUS:

Vacuum tube; Helmholtz coil; power supply; compass; dip-needle compass.

INTRODUCTION

Your e/m vacuum tube contains a number of features for producing and visualizing athin uniform electron beam. Refer to Figs. 1 & 2 for details. Passing current througha wire filament (F) causes it to become hot. If the filament becomes hot enough,electrons spontaneously leave the surface (a process called thermionic emission). Thefilament is in close proximity to a higher-potential electrical element (the cylindricalanode C), and so some of the electrons accelerate through the vacuum towards theanode. The current between the filament and anode is called the “anode current”.To allow a narrow beam of electrons to leave the vicinity of the anode a thin slit, S,has been cut in the anode cylinder.

Normally the electron beam is invisible to your eye. However the e/m tube alsocontains Mercury (Hg) vapor. When electrons with energies in excess of 10.4 eVcollide with Hg atoms, some atoms become ionized or excited, and then quicklyrecombine and/or de-excite to emit a bluish light. Hence the bluish light marksthe path taken by the electron beam (the electrons which collide with atoms arepermanently lost from the beam).

The potential difference V between the filament and anode accelerates electronsthermionically emitted by the filament. Those electrons traveling toward the slit Semerge with a velocity v given by

V e =1

2mv2 , (1)

provided that the thermal energy at emission is small compared to V e.

Preliminary Questions:

1. What is the purpose of the Helmholtz coil? (Read the appropriate section first.)

2. If you double the potential difference between the filament and the anode: A. beamwould get brighter; B. radius of electron trajectory would get bigger.

3. If you double the current through the filament: A. beam would get brighter; B. radiusof electron trajectory would get bigger.

If the tube is properly oriented, the velocity of the emerging electrons is perpendic-ular to the magnetic field. Hence the magnetic force vector ~FB = e (~v× ~B) supplies

the centripetal force mv2

r for a circular path of radius r. Since ~v is perpendicular to~B,

evB =mv2

r. (2)


F

C

S

B

A

is a hot filament emitting electrons

is the electron path in the magnetic field

is a cylindrical anode at potential +V with respect to F

is a vertical slit thru which electrons emerge

is a rod with cross bars at known distances from the filament+

B

A

CS

F Electron beam trajectory in presence of Earth’smagnetic field

Figure 1: The e/m tube viewed along the earth’s magnetic field (i.e., Bearth is perpendicularto the page).

LD

S F

A

C

L

filamentcurrent out

filamentcurrent in

G GE

A is a rod with cross bars at known distances

from the filament

LL , are insulating plugs in the end of the anode cylinder

D is the distance from the filament to the first cross bar

G , G function as the filament leads and support

E is the anode lead. It supports the anode

and the rod with cross bars

C is the anode cylinder

Figure 2: Side view of Figure 1.

Eliminating v between (1) and (2) gives

e/m =2V

B2r2. (3)

The accelerating voltage V is shown on the “ANODE Volts” display of the powersupply. The radius of curvature, r = D/2, is half the distance between the filamentF and one of the cross bars attached to rod A. The cross bar positions are as follows:

Cross bar No. D=Distance to Filament r=Radius of Beam Path

1 0.065 meter 0.0325 m2 0.078 meter 0.039 m3 0.090 meter 0.045 m4 0.103 meter 0.0515 m5 0.115 meter 0.0575 m

To find e/m we still need to know the magnetic field strength B. Instead of measur-


ing B, we calculate it from the dimensions of the Helmholtz coils and the measuredcoil current, I, needed to bend the beam so that it hits a given cross bar.

A Helmholtz coil consist of two identi-cal coaxial coils which are separated bya distance R equal to the radius of ei-ther coil, as in Fig. 4. They are usefulbecause near the center (X = R/2) thefield is nearly uniform over a large volume.(Many textbooks assign this proof as aproblem. Hint: Find how dB/dX changeswith X for a single coil at X = R/2, etc.)

R

X

R2+X2R

R

Figure 3: Helmholtz coil geometry.

The field at X = R/2 is in fact: 1

B =8√125

(Nµ0I

R

)(4)

as one sees by adding the axial fields B(x) from each of the single coils:

B(X) =Nµ0IR

2

2 [R2 + X2]3/2.

Thus when X = R/2, then

B = 2B(X) =Nµ0I

2R2

[R2 + R2/4]3/2=

Nµ0I

R[

54

]3/2 =8√125

(Nµ0I

R

).

In Eq. 4,

N = number of turns on each coil (72 for these coils)I = current through each coil in amperesR = mean radius of the coils in meters (approximately 0.33 m but varies slightly

from unit to unit)µ0 = 4π × 10−7 tesla meter/ampere.

Finally if we substitute (4) into (3) we obtain

e/m =

(2.47 × 1012 R2

N2

)V

I2r2coulombs/kg . (5)

CIRCUIT DIAGRAM: Fig. 4 should help you hook up the components. The current hasthe same direction in both coils of a Helmholtz coil.

1The actual B at the maximum electron orbit is only ∼ 0.5% less. See Price, “Electron trajectory inan e/m experiment”, Am. J. Phys. 55, 18, (1987)


ANODE milliAmps FIELD AmpsANODE Volts

ANODE

44 V.

22 .V Overload

Filament On

GND POS

Filament

GND POS

FieldPower

2 Amps

E/M POWER SUPPLY

To Helmholtz Coils

Helmholtz

Coils

ToPowerSupply

Anode Cylinderwith slit

Filament

21.3 06.4 2.19

Figure 4: Basic wiring layout.

SUGGESTED PROCEDURE:

1. Determine the local direction of the Earth’s magnetic field using the compass anddip-needle compass. You should find that it is in the same plane as geographic northbut at an angle about 60 degrees from the horizontal. Thus the dip-needle compassshould point deep under Canada. Set the axis of the Helmholtz coils along thisdirection. Rotate the e/m tube about its long axis until the cross bars are facingup.

CAUTION: Nearby ferromagnetic material (e.g. steel in the powersupply, the table and in the walls) can alter the local direction of Be.As long as the field direction does not change during the experiment,there should be no problem.

2. Set the filament supply knob to zero before turning the power on.

3. Since e/m depends on V/I2, one needs high quality meters for V and I. Our digitalmeters have an accuracy of ± one digit in the last displayed digit.

4. Start with ∼ 22 V between filament and anode. The anode current is displayedon the “ANODE milliamps” display. With the room dark, gradually turn up thefilament control until the beam is visible. The anode current should remain zerountil the filament is sufficiently hot, perhaps hot enough that you can see the glowingfilament. The filament dial will often be well beyond the halfway position beforethe anode current departs from zero. You should not expect to see the beam untilthe anode current is at least a few milliamps. To make the beam more visible, useblack cloth and black cardboard to block out stray light: place the black cardboardinside the Helmholtz coils and view from the top. When the beam appears, adjustthe filament control to give 5-10 mA of anode current. Do not exceed 15 mA!With no current through the Helmholtz coils, the earth’s magnetic field shouldslightly curve the beam.

5. Turn up the Helmholtz coil current until the electron beam is steered into cross bar1 (the cross bar closest to the filament).


6. Predict what will happen to the beam if the anode voltage is increased to 44 V. Willthe radius become larger or smaller?

7. Change the anode voltage to 44 V. Does the beam radius become larger or smaller?

8. Return the anode voltage to 22 V. Let I ′n denote the Helmholtz coil current necessaryto steer the beam into cross bar n. Record I ′n for each of the five cross bars in thetable below. Keep in mind that the electrons are responding to the sum of theHelmholtz coil field and the Earth’s field.

n r (m) I ′n(A) I ′′n(A) In(A) Ie(A) e/m Be (T)

9. Rotate the vacuum tube about its long axis by 180. Repeat step 8, using I ′′n todenote the current needed to steer the electrons into the cross bars.

CAUTION: Always rotate the tube in the sense to REDUCE theTWIST in the filament and anode leads. Otherwise one can twistthe leads off.

10. Why is I ′n different from I ′′n?

11. Which direction does the Earth’s field point, more or less towards the floor, or moreor less towards the ceiling?

12. Let In denote the current that would be needed to steer the electrons into post n,if the Earth’s field were not present. How is In calculated from your measurementsI ′n and I ′′n?

13. Let Ie denote the current necessary to counteract the Earth’s field. How is it calcu-lated from I ′n and I ′′n?

14. Fill in the columns for In and Ie in the table.

15. Compute the electron’s e/m from your data, estimate its uncertainty, and compareit to the accepted value of 1.76× 1011 C/kg. Is the discrepancy between your valueand the accepted value larger than or smaller than the uncertainty in your value?

16. Compute the local magnitude of the Earth’s magnetic field, Be, from your data,estimate its uncertainty, and compare it to the global average of 5× 10−5 T. Is thediscrepancy between your value and this value larger or smaller than the uncertaintyin your value?

17. The local magnitude of the Earth’s magnetic field varies from point to point, and atone point at different times. Go to the bulletin board outside room 3320 Chamberlin,find the global and local maps of the Earth’s magnetic field strength, and deduce the


currently accepted value of the strength of the Earth’s magnetic field in the vicinityof Madison, WI.

One last observation. Note that the electron beam expands in a fan shape afterleaving the slit. Interchange the filament leads and note how the fan shape flips tothe other side of the beam.

Explanation: The fan shape deflection arises from the filament current’s magnetic fieldacting on the beam. Our filament supply purposely uses unfiltered half wave recti-fication of 60 Hz AC so that for half of the cycle no current flows thru the filamentbut which is still hot enough to emit sufficient electrons. The electrons emitted inthis half cycle constitute the sharp beam edge which we use for measurement pur-poses. This trick not only avoids deflection effects for this part of the beam, butalso avoids the uncertainty in electron energy arising from electrons being emittedfrom points of different potential along the filament. (During this half cycle of nofilament current the filament is ∼ an equipotential.) The fan shape deflection relatesto electrons emitted during the other half cycle when filament current flows. [See:F.C. Peterson, Am. J. Phys. 51, 320, (1983)]

E-5 MAGNETISM 42

E-5 Magnetism

E-5a Lenz’s Law

OBJECTIVES:To observe the eddy currents induced in a conductor moving through the field of astationary magnet; to explore how the force due to eddy currents depends on theshape of the conductor.

YOU NEED TO KNOW: the Lorentz force law; the right-hand rule that de-termines the direction of the force on a charge moving through a magneticfield.

APPARATUS:

A permanent magnet; four different pendulums; compass.

X

Z

3 4

1 2

Figure 1: The magnet and the pendulums.

PROCEDURE I: (10 min)

1. Use the compass to determine the directionof the magnetic field between the jaws ofthe permanent magnet.

2. Set the jaws to be about 1 cm apart, andobserve the behavior of each of the four pen-dulums when they swing through the jaws.Which is affected most? Which least?

3. Reverse the magnet. Observe whether anyof the pendulums behave differently whenthey swing through the jaws.

4. Set pendulum #1 between the jaws of themagnet. Move the magnet rapidly in adirection parallel to the pendulum plate(i.e. without touching jaws). Describe whathappens.

y

z

B

A

Figure 2: Pendulum moves in the +ydirection

E-5 MAGNETISM 43

QUESTIONS:

Refer to Fig. 2. As the leading edge of the pendulum enters the region of themagnetic field:

Q1: What is the direction of the force on the electrons in the leading edge? Is it from Ato B? Or from B to A?

Q2: What direction is the induced current? (A to B, or B to A)

Q3: In what direction is the force on the pedulum?

Q4: A little later, the pendulum has swung so that the trailing edge is just leaving theregion of the magnetic field. Now what is the direction of the force on the pendulum?

E-5b Induction - Dropping Magnet

OBJECTIVE:

To show that a moving magnet induces an emf within a coil of wire.

APPARATUS:

Pasco interface; voltage sensor; a stand holding a plastic tube and a coil; a long barmagnet.

INTRODUCTION:

In E-5a it was shown that a conductor moving through the magnetic field of astationary magnet may experience a force. Here we show that a magnet moving inthe vicinity of a stationary conductor induces an emf in the conductor.

Figure 3: A moving magnet - magnetic field lines move across a wire

YOU NEED TO KNOW: Faraday’s Law; Lenz’s Law; the right handrule that determines the direction of the current induced in a con-ductor by a magnetic field that is changing with time.

PRECAUTIONS:

• The bar magnet is very fragile. DO NOT DROP IT!

• The bar magnet is easily depolarized. KEEP OTHER MAGNETS AWAY!

• When dropping the magnet, HOLD THE TUBE AND STOPPER TOGETHERWITH YOUR HANDS!

E-5 MAGNETISM 44

PROCEDURE:

1. Connect the red wire of the voltage sensor to the top of the coil, and the black wireof the voltage sensor to the bottom of the coil. Then connect the other end of thevoltage sensor to analog input A of the Pasco interface. Turn on interface.

2. Double click on the “Physics202-E5b” file in the Pasco/Physics202 folder on thedesktop.

3. Set the distance d1 ∼ 15 cm between the top of the plastic tube and the top of thecoil. Record this in your lab notebook.

4. The long bar magnet has a narrow cutat one end. This is the North pole ofthe magnet.

5. Hold the long bar magnet at the top ofthe tube; the end with the cut shouldbe down, and just one centimeter or soinside the tube.

6. Click on the icon then drop the mag-net.

7. Measure the heights V1 and V2 of the

two peaks using the coordinatestool.

8. Slide the tube upwards, remove therubber stopper and the magnet, replacethe rubber stopper, and slide the tubedown again.

9. Sketch in your notebook the graph ofvoltage vs. time that you see on thescreen.

Figure 4: The setup

QUESTIONS:

Q1: The graph you obtained has a fall, a rise, a diminished slope, then a rise and finallya fall. Explain what is happening at these various times.

Q2: Explain why the two voltage peaks you measured in step 7 are not the same size.Can you give an approximately quantitative justification for the ratio V1/V2? HINT:remember Newtonian mechanics formula for free fall v =

√2gh where g is the

acceleration due to gravity .

Q3: Examine the direction of the winding in the coil; verify that the directions of thepeaks are what you would expect using the right hand rule (curled fingers = cur-rent; thumb = magnetic field).

Occasionally a long bar magnet is found to have a reversed polarity, perhaps due inpart to repeated mechanical shocks (it’s amazing to think that banging a magneton a table can cause all the atoms to flip their spins). If you think the polarity ofyour magnet has become reversed, call over your TA.

E-5 MAGNETISM 45

10. Sketch how the graph of voltage vs time ought to look if the magnet were droppedupside-down relative to step 4 (nominal North pole at top).

11. Repeat steps 4-6 with the end with the cut at the top. Did you correctly predicthow the graph would look?

12. Move the coil further down on the tube making the new distance d2 at least aboutone-and-a-half times larger than d1. Record this new distance d2.

13. Repeat steps 4-6. Measure and record the heights V ′1 and V ′2 of the new peaks.

QUESTIONS:

Q4: Is the ratio V ′1/V′

2 larger or smaller than the ratio V1/V2?

Q5: Why has this ratio changed?

E-5c Induction - Fields in Space

OBJECTIVES:

To observe the emf induced in one conductor by a changing current in a nearbyconductor; to investigate how the induced emf depends on the waveform shape,amplitude, and frequency of the inducing current; to observe how the induced emfdepends on the proximity and relative orientation of the two conductors.

a

b

c

100 200

0

-3

3

t(ms)

V

0

Figure 5: A triangular waveform

APPARATUS:

PC and PASCO interface; two large back-to-back coils (N=200 turns; radius r =0.105 m, series resistance R ∼ 9Ω); a small detecting coil (n=2000 turns); a standfor the small coil.

INTRODUCTION:

A voltage applied to one conductor (by a power supply, for example) can induce anemf in a nearby conductor. Here we apply a voltage with a triangular waveform toa large coil, and measure the emf induced in the small coil.

YOU NEED TO KNOW: The emf induced in a (small) coil is E =−nA∆B/∆t, where n is the number of turns in the coil, A its cross-sectional area, and B is the magnetic field created by other nearbyobjects.

E-5 MAGNETISM 46

PROCEDURE:

1. Double click on the “Physics202-E5c” file in the Pasco/Physics202 folder on thedesktop. The monitor should now show a scope window and a window for thecontrol of the signal generator.

2. If necessary adjust the signal generator amplitude to 3 V.

3. Connect the signal generator to the large coil; then connect analog input A to thelarge coil and analog input B to the small coil, as in Fig. 6.

4. Turn on the signal generator and initiate data acquistion by clicking on the icon.You should now see the emf induced in the small coil together with the voltageapplied to the large coil.

5. While recording data, move the small coil around and see how the amplitude of theinduced emf changes.

6. Adjust the size of the waves you see in the scope window using the size controls onthe side of the scope window, then measure the amplitude VB of the square waveusing the crosshair tool. Record this amplitude.

Figure 6: Connections between interface and coils.

QUESTIONS

Q1: Explain why the induced emf is a square wave.

Q2: Observe the direction of the winding in the large coil. What is the directionof the magnetic field when the voltage applied to the coil is positive? Drawa sketch showing the current in the large coil, and the direction of the field itproduces.

Q3: Observe the direction of the winding in the small coil. What is the direction ofthe induced emf during the rising part of the triangular wave? Draw a sketchto indicate your answer.

E-5 MAGNETISM 47

Q4: Orient the small coil to achieve the maximum possible ratio |VB|/|VA|. Recordthis maximum ratio, and make a sketch of this orientation that produced it.

6. Put the small coil on the stand, and move it as close as you can to the large coil.Record the approximate distance (in cm) between the plane of the small coil andthe plane of the large coil, and measure the amplitude of the induced emf.

7. Repeat at distances of 10 and 15 centimeters. Make a simple graph of the amplitudeof the induced emf as a function of the distance between the plane of the large coiland the plane of the small coil, using Excel or any other suitable application.

QUESTION:

Q5: How does the strength of the magnetic field depend on distance? Is it ∼ 1/r ?Or ∼ 1/r2? ∼ 1/r3? Or something else? (Hint: consider the field produced bythe Helmholtz coil in lab E-4.)

8. Replace the detecting coil to its original position at the center of the large coil.Decrease the amplitude of the triangular wave to 2 V.

9. Measure the amplitude of the square wave and record this value.

QUESTIONS:

Q6: Did the slope of the triangular wave change? By how much?Q7: Is the change in amplitude of the square wave what you would expect? Why?

10. Repeat steps 8-9 with a frequency of 15 Hz.

11. Compare the amplitude of the induced emf with the measurements from steps 6 and9.

12. Return the frequency to 10 Hz. Rotate the detecting coil 90 so that it is in thehorizontal plane. Observe the resulting induced emf.

13. Rotate the coil another 90 so that it is in the vertical plane again. Observe theresulting induced emf. Compare with the measurement from step 6.

QUESTION:

Q8: In the “You Need to Know” box that started this section, why was it needfulto specifiy that the coil be “small”?

E-5d Induction - Circuit Elements

OBJECTIVE:

To observe the L/R time constant of an LR circuit.

APPRATUS:

PC and PASCO interface; two voltage sensors; a pair of nested coils; an iron barthat fits inside the inner coil; a plug board; and a 100 Ω resistor.

INTRODUCTION:

In E-5c we showed that a changing current in one circuit can induce an emf inanother circuit that is not physically connected to it. Here we will observe this sameeffect, but in addition will see that if the current in a circuit changes, an emf isinduced in that same circuit, and will explore the effects of this.

E-5 MAGNETISM 48

Figure 7: The schematic setup

Figure 8: The physical setup

PROCEDURE:

1. Set up the circuit shown in Figs. 7 and 8. Analog input A measures the voltageacross the resistor, and is therefore proportional to the current through the outercoil. Analog input B measures the emf induced in the inner coil.

2. Double click on the “Physics202-E5d” file in the Pasco/Physics202 folder on thedesktop. Make sure the iron bar is inserted in the inner coil.

3. Check that the amplitude is ∼ 4 Volts, the frequency is ∼ 10 Hz, and the waveformis a square wave.

QUESTION:

In E-5c, you applied a triangular-wave voltage to one coil, and observed a squarewave in a nearby coil.

Q1: What is the mathematical relationship between the square wave and the trian-gular wave?

E-5 MAGNETISM 49

Q2: Now we are applying a square wave to one coil. Sketch the waveform we expectto observe in the other coil. (Hint: what is the derivative of a square wave?)

5. Click on the icon.

6. The computer monitor should display three graphs:V A, which shows the voltage across the resistor as a function of time, V B, whichshows the emf induced in the inner coil as a function of time, and (at the bottom)the voltage put out by the power supply.

QUESTIONS:

Q3: Does the shape of V B look like your sketch from Q2? (It should.)Q4: The shape of curve A is not quite a ‘square wave’. Why not? HINT: this is an

LR circuit with a time constant τ = L/RQ5: Estimate the time constant of curve A by making measurements on the graph.Q6: Estimate the total R of the circuit from the “asymptotic” value of the current

through the resistor. Why is this different from 100 Ω ?Q7: Estimate the inductance L of the circuit.Q8: The outer coil has N = 2600 turns of fine wire. Measure its radius r and length

l, and calculate its inductance using L = µ0N2A/l . Is this close to the value

you got in Q7?Q9: Remove the iron bar. Take data. Estimate the time constant of curve A.

Q10: Assume the resistance of the circuit hasn’t changed, and estimate the newinductance of the circuit. Is this closer to your calculation in Q8?

Q11: Explain in words why removing the iron bar changed the inductance of thecircuit.

E-6 OSCILLOSCOPES AND RC DECAY 50

E-6 Oscilloscopes and RC Decay

OBJECTIVE:

Learn the basic operation of an oscilloscope; use the oscilloscope to understand RCDecay.

APPARATUS:

TDS 2001C or 2002B oscilloscope; plugboard and plugboard kit; multimeter; func-tion generator.

INTRODUCTION:

The oscilloscope is a high-speed graph-displaying device that displays graphs ofelectrical signals. In most applications the graph shows how signals change over time:the vertical (Y) axis represents voltage and the horizontal (X) axis represents time.This simple graph can tell you many things about a signal, including amplitude,frequency, waveform, and signal-to-noise ratio.

Figure 1: Initial Setup

EXPERIMENT I: INTRODUCING THE OSCILLOSCOPE

In this section, you’ll set up the oscilloscope to display a waveform as shown inFig. 1.

1. Power on the scope and press the “Default Setup” button on the second row ofbuttons at the top of the panel. If you ever get lost, you can press Default Setupand start from scratch.


2. Connect the Channel 1 probe to the gold contacts: the probe itself should be clippedto the top contact and the alligator clip ground should connect to the bottom contact.These contacts output a continuous 5V@1kHz square wave. You should initially seea very unsteady display that is not scaled properly.

3. To stabilize the graph, adjust the “Trigger Level” knob. The trigger level is indi-cated graphically by an arrow on the right side of the display. It is also displayednumerically on the bottom right. You should now see a stable plot that is clippedat the top.

4. Use the Channel 1 “Vertical” scale knob to display the entire waveform. The numer-ical value for this scale is displayed in Volts on the bottom left corner of the display.You may have to readjust the trigger level after scaling.

5. Use the “Horizontal” scale knob to scale the waveform horizontally to match thefigure. This scale is displayed in µ-seconds on the bottom center of the displayabove the date.

6. Play around with all three knobs until you understand what they are doing.

QUESTIONS:

Q1: What is the function of each knob, and how do the knobs relate to the numbersdisplayed?

Q2: What is the relationship between the scales and the grid that the waveform isplotted against?

Q3: How does the trigger level need to relate to the amplitude of the waveform todisplay a stable graph?

EXPERIMENT II: BASIC FUNCTIONS:

While the display grid will allow you to make crude measurements of amplitude andperiod, our oscilloscopes offer more sophisticated measurement tools in the form ofcursors.

1. Press the “Cursor” button on the second row of buttons on the top of the panel.Change the Type to Amplitude. Using the knob at the top left, adjust the cursorposition to line up with the maximum amplitude of the square wave. Note that theposition of this cursor is displayed on the right side under Cursor 1.

2. Now highlight Cursor 2 and adjust the position to the minimum amplitude of thesquare wave. The difference between the two cursor positions is displayed at ∆Vand should be 5V in this case.

3. Change the Type to Time. Using the same mechanics as above, measure the periodand frequency of the displayed waveform.

4. Estimate the error on this measurement and the amplitude measurement above.

5. Probe the gold contacts with the multimeter set to measure AC Volts and recordthe result.


QUESTION:

Q1: Why is the measurement made by the multimeter different from the amplitudemeasured with the cursors? Are the two measurements consistent?

EXPERIMENT III: EXTERNAL SIGNALS:

1. Disconnect the probe from the gold contacts and connect them to the output ofthe function generator using the BNC to micro-grabber adapter. Always use theoscilloscope probes to connect to the inputs. Never use a BNC cable.

2. Turn the function generator on. Its default output 5V@1kHz sine wave should bedisplayed on your oscilloscope. The amplitude given by the function generator isthe Peak amplitude, or the maximum absolute value of the signal.

3. Measure the Peak-to-Peak amplitude and the Frequency of the waveform. Do theyagree with the function generator to within your measurement uncertainties?

4. Now connect the Fluke Multimeter to the function generator output and record theRMS amplitude. Is this measurement consistent with the oscilloscope?

5. Experiment with several different types, amplitudes and frequencies of waveformscreated using the function generator.

QUESTION:

Q1: What is the difference between the 5V@1kHz square wave you first observedand the 5V@1kHz square wave generated by the function generator?

Be sure you are confident in your abilities to accurately display several types of waveformon the oscilloscope before moving on.EXPERIMENT IV: RC DECAY:

47 N

F1K

1 KΩ

47 nf

VR

Vin

to red

to blackto ch. 1 to ch. 1to ch. 2

Figure 2: RC Circuit Schematic

Now we will use the scope to characterize an RC Circuit.


1. Build the series RC circuit with the plug board kit as shown Fig. 2. There is a raisedbump on one leg of the BNC-banana-plug adapter. This bump indicates which legis ground. Be sure to match the raised bump indicating ground with thefigure.

2. Drive the circuit with a 5V@1kHz square wave.

3. Use one probe to monitor the function generator output with the oscilloscope, asyou’ve done in previous sections.

4. Plug a second probe into Channel 2 of the oscilloscope and use that probe to measurethe voltage drop across the resistor. You’ll need to press the blue “2” button. Youshould see something similar to figure 3.

Figure 3: RC Circuit Trace

5. Using the cursor feature, measure the time constant for this circuit. Does it agreewith the calculated value to within your measurement precision?

6. For five different combinations of resistors and capacitors, measure the time con-stants and compare them to the nominal values τ = RC. You will need to changethe driving frequency of the function generator and the horizontal scale of the oscil-loscope accordingly.

EXPERIMENT V: DIFFERENTIAL AMPLIFIERS:

Amplifiers are devices which change the amplitude of the output signal compared tothe input signal, without distorting the waveform of the input signal. Most amplifiersincrease the amplitude. The customary symbol for an amplifier is a triangular shape:


VoutVin

Vout (volts)2

1

−1

−2

(volts)Vin10

5

−5

−10

Figure 4: Amplifier Function

In the sine wave example above, the gain is 10 and the input signal is between aninput terminal and ground. The internal oscilloscope amplifiers for channels 1 and2 are of this type and have a common ground. Because of this common ground, onehas problems in using a scope to examine simultaneously voltages across individualcircuit elements that are in series.

VoutVin

Amplifier

Vin

Vout

Differential amplifier

Figure 5: Amplifier vs. Differential Amplifier

We can avoid these problems by interposing “differential amplifiers” which have twoinputs V ′ and V ′′, (neither at ground), and which amplify only the voltage difference(V ′−V ′′). See Fig. 5. Note now that the ground of the output signal is independentof any input ground.


Figure 6: Inputs to differential amplifiers

1. Connect the resistor and capacitor signal to the differential amplifiers as shown inthe figure above.

2. Connect the outputs of the differential amplifiers to the scope inputs, and set bothgains to 1. You should see something similar to Fig. 7.

3. Describe what is happening to the voltage drop across the resistor when the capacitoris charging and discharging. What is the relative phase of the two components?

4. Now press the pink “Math” button to bring up a selection of mathematical operators.Change the operation to “+”. Describe the significance of what is now displayed.How is it consistent or inconsistent with Kirchoff’s laws?

Figure 7: Differential Amplifier Outputs

E-7 SERIES LRC CIRCUITS AND RESONANCE 56

E-7 Series LRC Circuits and Resonance

OBJECTIVES:

Study voltage and phase relationships in series LRC circuits; gain familiarity withimpedance and reactance; observe resonance.

APPARATUS:

Oscilloscope; plugboard kit; function generator; digital multimeter.

INTRODUCTION:

We define the impedance Z of any part of a circuit as the ratio of the voltage acrossthat part and the current though that part. Because impedance is defined as aratio of voltage/current, impedance is measured in Ohms.

For AC circuits the ratio is a complex number: it has an magnitude and a phase. Toavoid dealing with complex numbers, resistance and reactance are introduced. Forideal components:

• The resistance of a resistor is R (the reactance is zero);

• The reactance of an inductor L is XL = 2πfL (the resistance is zero);

• The reactance of a capacitor C is XC = 12πfC (the resistance is zero).

If a resistor, inductor, and capacitor are combined in series (as in Fig. 1), the mag-nitude of the impedance of the entire circuit is

Z =√X2R + (XL −XC)2 .

Note that the impedance Z of a series RLC circuit is a minimum for XL = XC . Thefrequency for which this occurs is the resonant frequency. At resonance, the currentthrough R is maximum, but the voltage VLC across the LC series combination iszero. The phase angle φ of the series RLC circuit is

φ = tan−1

(XL −Xc

R

);

at resonance this is zero.

QUESTION

Q1: Derive an expression for the resonant frequency in terms of the inductance andcapacitance of the circuit.

EXPERIMENT I: DEPENDENCE OF VOLTAGE ON FREQUENCY

1. Create the circuit in figure 1 using the plug board kit. Use the function generatorto supply VAC . By default it generates a sine wave at a displayed amplitude of 5V(10V Peak-to-Peak) and displayed frequency of 1 kHz.

QUESTIONS:

Q2: Use the formula above to calculate the magnitude of the impedance Z of thecircuit (in Ohms).


Figure 1: A Series LRC Circuit

Q3: Calculate the magnitude of the current I through the circuit (in Amps).

Q4: Calculate the magnitude of the voltage across each component using VR = IR,VL = IXL, and VC = IXC .

2. Use the digital multimeter (DMM) to measure the RMS voltage drop across theresistor (the voltage between points a and b indicated in the diagram). Check itagainst your calculation, and track down any stray factors of

√2, so that your

measurement agrees with your calculation.

3. Use the DMM to measure the RMS voltage drops across the inductor and capacitor.Are these consistent with your calculations?

4. Repeat the last two measurements at 700 Hz and 400 Hz. How do the voltage dropsacross each component change as the frequency changes?

5. Measure the voltage drop across the resistor using channel 1 of the Oscilloscope,and the differential amplifier (as in lab E-6). Vary the frequency of the functiongenerator until you find resonance.

Q5: How do you know when you have found resonance?

Q6: What is the resonant frequency?

Q7: Calculate the expected value of the resonant frequency. Is this consistent withyour measurement?

6. While at resonance, use the DMM to measure Vbc and Vcd, and verify that neitheris zero.

7. Now measure Vbd. It should be close to zero. Is it?

8. Swap in the 0.47 µF capacitor, and measure the new resonant frequency.


Q8: Given that the capacitance has increased by a factor of 10, by what factorshould the resonant frequency change?

Q9: Is this consistent with your measurement?

9. For four different frequencies on either side of resonance, measure the voltage dropacross the resistor. Make a table of VR as a function of frequency.

10. Add a third column to the table and calculate the power dissipated in the resistorat each frequency. Use this table to create a plot of power vs frequency. Verify thatit looks like a Gaussian.

11. Replace the 1KΩ resistor with a 100Ω resistor and repeat the previous step. Whatis the difference?

EXPERIMENT II: DEPENDENCE OF PHASE ON FREQUENCY

1. Switch back to the 1KΩ resistor and use channel 2 of the scope to look at the voltagedrop across the combination of the inductor and capacitor.

2. Use the Math feature to add the two waveforms (channel 1 and channel 2).

3. As you change the frequency, note how the phase changes between the differentwaveforms. Fig. 2 shows this relationship.

4. Note that at resonance, φ = 0. Determine the resonant frequency using the relativephases of the signals on the oscilloscope.

Q10: What is the resonant frequency, as determined by φ = 0?

Q11: What method of measuring resonant frequency appears to be more accurate:looking for maximum I, or looking for φ = 0?

R =I R

C= I X

φ

VL

=I XLC

VL

C

ω

V

t

V=I Z

V

I

=I X

VC

L

-

Figure 2: Phase Relations

E-8: TRANSISTORS 59

E-8: Transistors

OBJECTIVE:To experiment with a transistor and demonstrate its basic operation.

APPARATUS:An npn power transistor; dual trace oscilloscope & manual; signal generator &frequency counter; power supply (±15 V fixed/±9 V variable); circuit plug board &component kit; two digital multimeters (DMM); differential amplifiers.

INTRODUCTION:Our junction transistor (Fig. 1) is like two back to back np diodes (see E-8 PartC, #3). Hence there are two possibilities, an npn transistor and a pnp transistor.Our npn transistor has a central p-type layer (the Base) between two n-type layers(the Emitter and Collector). There are other type transistors which we will notdiscuss, (e.g. a MOSFET: Metal-Oxide Semiconductor-Field-Effect-Transistor).

I

BaseP

Collector

N

Emitter

N

Collector

Base

Emitter

Base

Collector

Emitter

NPN PNP

Figure 1

For an npn transistor the collector is positive relative to the emitter. The base-emitter circuit acts like a diode and is normally conducting (i.e. forward-biased). Thebase-collector circuit also acts as a diode but is normally non-conducting (reversebiased) if no current flows in the base-emitter circuit. However when current flowsin the base-emitter circuit, the high concentration gradient of carriers in the verythin base gives an appreciable diffusion current to the reversed biased collector. Theresulting collector current Ic depends on the base current Ib. We write Ic = βIbwhere β is the current amplification factor.

PRECAUTIONS: Although our power transistor is fairly indestructible, itscharacteristics are temperature sensitive. Hence avoid exceeding the voltagessuggested; also leave the power supply off when not making measurements;read currents and voltages to two significant figures only.

SUGGESTED EXPERIMENTS:

1. CURRENT Amplification:Measure β for various emitterto collector voltages, Vec. Setup the circuit as in Fig. 2.

mA

Variable Voltage

DMM

DC Supply

1

DMM2

Aµ

+

Figure 2

E-8: TRANSISTORS 60

Start with the voltage divider completely counter-clockwise so that the emitter-basevoltage Veb is a minimum. With the variable power supply set to 3 V, record the Ibmeasured on DMM1 and the Ic on DMM2

Repeat the readings for ten reasonably spaced (higher) settings of the voltage divider(i.e. higher Veb and hence higher Ib.

Calculate β(= Ic/Ib) and plot it against Ib. Over what range of Ib is β reasonablyconstant? Repeat the above but with emitter to collector voltage Vec now at 7 V.

2. (Optional) MEASUREMENT OF Ic vs Veb: Remove the multimeter DMM1 (usedas a microammeter) from the circuit of Fig. 2 and use it instead as a voltmeter tomeasure the emitter to base voltage Veb. With the variable power supply set to give7 V for Vec, use the voltage divider to vary Veb. Read and record both Veb and Icfor 10 reasonably spaced values of Veb. Plot Ic vs Veb. The results are very similarto a diode curve (as expected since Ic is proportional to Ib over the region where βis a constant).

3. VOLTAGE Amplification: By adding a large load resistor in series with the col-lector, one can convert the current amplification β observed earlier into a voltageamplification. Hook up the circuit plug board as in Fig. 3.

VoltageDC Supply

Vout

DMM2

GND

Variable

DMM

+15V

1

Vin

4.7 kΩ

10 kΩ

1 kΩ

+

Figure 3

The input voltage to the transistor Vin includes the drop across the 4.7 kΩ protectiveresistor. The voltage gain is G = ∆Vout/∆Vin. Record input voltages Vin and outputvoltages Vout(= Vec) for ten reasonably spaced values of Vin from 0 to 5 volts.

Graph Vout vs Vin and calculate the voltage gain G at the steeply changing partof your graph. The value of Vin at the center of this region we call the “operatingvoltage” of the amplifier. How would the gain change if the load resistance was2.2 kΩ instead of 10 kΩ?

4. Distortion effects when amplifier is overdriven: Set up the circuit plug board as inFig. 4. Connect Y1 and Y2 to the scope. To connect the signal generator to theboard, use the BNC to banana plug adapter and remember that the side with thebump goes to ground.

E-8: TRANSISTORS 61

VoltageDC Supply

+

+15V

GND

1 kΩ

4.7 kΩ

10 kΩ

Vin

DMM

fµ0.047

Variable

+

+

2

Y1

Y

Figure 4

Adjust the voltage divider until the Vin is close to the “operating voltage” found in#3.

Vary (and record) the amplitude of the input signal from small values to those whichoverdrive the amplifier and produce considerable distortion in the output Y2.

Observe and record the effect of changing Vin to values outside of the operatingrange (where β is constant).

62

Part IISound and Waves

S-1 Transverse Standing Waves on a String

OBJECTIVE: To study propagation of transverse waves in a stretched string.

INTRODUCTION:

A standing wave in a string stretched between two points is equivalent to superposingtwo traveling waves on the string of equal frequency and amplitude, but oppositedirections. The distance between nodes (points of minimum motion) is one halfwavelength, (λ/2). Since the ends of the string are fixed, the only allowed values ofλ are are λk = 2L/k, k = 1, 2, 3 . . .

The wave velocity, v, for a stretched string is v =√F/µ where F = tension in

the string and µ = mass per unit length. But v = fλ and hence only certainfrequencies are allowed:

fk =

√F/µ

λk. (1)

N

A

N

N N

A

N

First mode

Second mode

Fourth modeN N

Figure 1: The Modes of a String

λ

2

λ

Figure 2: A close-up

PART A: Waves from a mechanical driver (i.e. a speaker)

APPARATUS:

Basic equipment: Pasco interface; electrically driven speaker; pulley & table clampassembly; weight holder & selection of slotted masses; black Dacron string; electronicbalance; stroboscope.

The set-up consists of an electrically driven speaker which sets up a standing wavein a string stretched between the speaker driver stem and a pulley. Hanging weightson the end of the string past the pulley provides the tension.

The computer is configured to generate a digitally synthesized sine wave (in voltsversus time) with adjustable frequency and amplitude (max: ∼10 V).PASCO interface: This transforms the digital signal into a smooth analog signal.

S-1 TRANSVERSE STANDING WAVES ON A STRING 63

Precautions: Decrease the amplitude of the signal if the speaker makes a rattlingsound. The generator is set to produce sine waves; do not change the waveform.

Note: Although the speaker is intended to excite string vibrations only in a plane,the resultant motion often includes a rotation of this plane. This arises from non-linear effects since the string tension cannot remain constant under the finite ampli-tude of displacement. [See Elliot, Am. J Phys. 50, 1148, (1982)]. Other oscillatoryeffects arise from coupling to resonant vibrations of the string between pulley andthe weight holder; hence keep this length short.

BridgeSpeaker

L

Table

Figure 3: The apparatus

PROCEDURE: CHECKING EQUATION (1)

1. Procure a length of string about 1.5 meters long.

2. Carefully weigh the string, and calculate µ. Note this in your lab notebook.

3. Set up the speaker, bridge, pulley and string as in the above figure. Make thedistance, L, between the bridge and the pin of the speaker be about 1.2 m, andmeasure it accurately using the two meter ruler; record this in your lab notebook.

4. Place the sheet of paper provided on the table; this will make it easier to see thevibration of the string.

5. Double click on the “Experiment S-1” file in the Pasco/Physics202 folder on thedesktop. The display will appear as shown in Fig. 4.

6. You will see that the computer is set to produce a 60 Hz sine wave with an amplitudeof 2 V . To start the string vibrating click the “On” button.

7. Click on the up/down arrow in order to change the amplitude or the frequency ofthe signal.

NOTE: The nominal step sizes for adjusting the amplifier frequency and voltage

may be much too large. To alter the step size use the buttons. To alter the

current or voltage (which of these depends on configuration) use the buttons.You can also change the value directly by clicking the mouse cursor in the numericwindow and entering a new value with keyboard number entry.


Figure 4: The Pasco Capstone display.

8. Stop the signal generator, put a 200 g mass on the mass hanger and restart the signalgenerator. Record the total mass and tension in your lab notebook. In the next fewsteps you’ll keep the tension in the string constant and look for the frequencies ofdifferent modes.

9. Adjust the frequency so that the amplitude of the oscillation is at its maximum bychanging the frequency in 1 Hz steps. This is best done as follows: First decrease thefrequency until the amplitude of the string is very small. Then increase the frequencyin 1 Hz steps, and observe that the amplitude first increases and then decreases.Record in your table the frequency f2 at which the amplitude is maximum.

10. Change the frequency to observe the third mode. Find and record the best frequencyf3(using 10 Hz steps at first may be faster).

11. Find and record the frequency of as many higher modes as you can.

12. OPTIONAL: Check the frequency f of the string in its 2nd mode with the strobo-scope. Note that the stroboscope is calibrated in RPM or cycles per minute, NOTHz (cycles per second). You should find a value close to 70 Hz.

13. Divide the various frequencies fk by k and enter the values in a table. Calculate theaverage value of fk/k.


QUESTION:

Q1: Use this average value of fk/k to calculate the mass per unit length of thestring. How does it compare with your measurement?

14. Choose six masses between 100 g and 1 kg and enter the values in a data table.

15. Determine the resonant frequency of the second mode of the string under thesedifferent tensions and record your results. (Hint: increasing the mass by a factor oftwo increases f2 by nominally a factor of

√2.)

16. Plot a graph of frequency versus mass, m, and include the zero value.

17. Plot a graph of frequency versus√m and again include the zero value.

QUESTIONS:

Q2: Which of the two graphs can be fitted with a straight line? A parabola? Why?

Q3: From the slope of the graph having the linear relationship obtain the mass perunit length of the string.

Q4: Which of the three methods of estimating the mass per unit length of the stringseems most accurate?

FOR FURTHER INVESTIGATION:

Vibrations on a circular membrane are made visible here:

http://www.acs.psu.edu/drussell/Demos/MembraneCircle/Circle.html

S-2 VELOCITY OF SOUND IN AIR 66

S-2 Velocity of Sound in Air

OBJECTIVE:

To calculate the velocity of sound from measurement of the wavelength in air forsound of a certain frequency.

APPARATUS:

Resonance tube with arrangement for varying water level (use only distilled water);rubber tipped hammer; tuning fork; Hg thermometer.

INTRODUCTION:

For a closed tube, resonance occurs at tube lengths of an odd multiple of one-fourthwavelength, i.e. at λ/4, 3λ/4, 5λ/4 etc.

SUGGESTIONS:

1. Find the positions of the water level in the tube for the first three of theseresonances. Use these readings to calculate the speed of sound, v = |~v|. Initiallyhave enough water that you can raise the level above the first resonance position.The tuning fork frequency is on the fork.

Since the effective end of the resonance tube is not at the tube’s end, donot use the position of the tube’s top in your calculations, but rather takedifferences between the other readings.

2. Sound waves in gases have a speed v =√γRT/M . (Recall the formula for the speed

of sound on a string, v =√T/µ (e.g., Lab S-1)). Correct your value of v to 0C

(T = 273.16 K) and compare with that accepted for dry air at 0C: 331.29 ± .07 m/s,[Wong, J. Acoust. Soc. Am., 79, 1559, (1986)]. For humid air see 3. below.

3. We quantify proportions in gas mixtures by the pressure each gas contributes to thetotal pressure. This is called the “partial” or “vapor” pressure. Think of the speedas resulting from an average < γ/M >,

< γ/M > = [(γa/Ma)Pa + (γw/Mw)Pw] /(Pa + Pw) ,

so that

vdry ∼= vhumid√

(γa/Ma)/ < γ/M > ,

where γair = 1.40, γw = 1.33, Pa is the partial pressure of air, Pw is the vaporpressure of water, Ma ∼ 29 kg and Mw = 18 kg.

How should the v.p. of water, Pw, in the tube affect the speed?

OPTIONAL: Humidity changes will affect tuning of what musical instruments?

4. What effect does atmospheric pressure have on the velocity of sound in dry air?(Assume air at these pressures is an ideal gas.)

5. Viscosity and heat conduction in the tube may reduce v by ∼0.1%. See N. Feather,“The Physics of Vibrations and Waves”, Edinburgh Univ. Press, (1961), p. 110-120;this reference also has a delightful historical account (including Newton’s famousgoof).

67

Part III

Light

L-1: Diffraction and Interference

OBJECTIVES:

To observe diffraction; to observe interference; to measure the wavelength, λ, of laserlight.

APPARATUS:

Optical bench; diode laser assembly; a circular wheel with single slits of variouswidths; another circular wheel with various apertures including double slits; screen;Pasco Interface; light sensor; rotary motion sensor; aperture bracket.

INTRODUCTION:

The diode laser provides plane light waves of wavelength 650 (±10) nm. A planewave passing through an opening will spread out (diffract). The spreading is rapidif the opening is narrow, e.g. after passing the narrow slit in Expt. I below. If twoclosely spaced narrow slits (narrow relative to the slit-to-slit spacing) are illuminatedby the same laser beam, as in Expt. II, the two spreading beams will overlap andinterfere.

CAUTION:The laser is a very bright source. Do not allow the laser beam to enter the eyeand do not point the beam at anyone!

EXPERIMENT I: SINGLE SLIT DIFFRACTION:

destructive interference

aλ/m=θ

y

D

m=1

m=2

m=−1

m=−2

all in phase

slit width a

( > 1 meter)

LASER

mλ

θ

a

θ

Diffraction minima at

Figure 1: Schematic of single slit diffraction

L-1: DIFFRACTION AND INTERFERENCE 68

THEORY:

The angular separation in radians of the first minimum from the center of the patternis

θ = λ/a

where a is the width of the slit. (For your derivation you may refer to the text andremember that for small θ, sin θ ∼= θ).


1. Mount the laser and the single-slit wheel on the optical bench as shown in Fig. 1.The wheel should be only a few cm from the laser, and the laser light should fullyilluminate a given slit. Observe the pattern on the supplied white screen andqualitatively explain why in your lab book that the first minimum occurs as shownin Fig. 1 at m = 1.

QUESTIONS:

Q1. Rotate the wheel and illuminate different slits. Qualitatively, how does thepattern vary as the slit width narrows?

Q2. Qualitatively, how do you expect the pattern would vary if the wavelength, λ,were decreased from red to blue-violet?

2. Launch the experiment double-click on the “Physics202-L1” file in thePasco/Physics202 folder on the desktop. Start the data acquisition by clicking the

button. Move the combined light and rotary motion sensor in the lateral direc-tion gently and smoothly by hand. Practice starting on one side of the diffractionpattern and move smoothly towards the other.

3. Turn the PASCO wheel to a single slit width of a=0.08 mm.

4. You now need to set the gain. A larger gain increases the sensor output but canmore easily saturate the output, and may also increase the noise. To set the gain:1) Click on the Hardware Setup icon 2) Select the yellow light bulb icon (or LightSensor) 3) Click the Properties icon and select the desired Gain from the pulldownmenu. Adjust and record the gain setting which give you reasonably good results.

5. Let y be defined as the linear distance between the detector position and the locationof the central maximum. Using the coordinates tool on the PASCO graph display,measure θ = y/D for m= 1 and -1. Use this information, and the known value ofa, to calculate λ. Is your value of λ comparable with the stated value of the diodelaser?

L-1: DIFFRACTION AND INTERFERENCE 69

EXPERIMENT II: TWO-SLIT INTERFERENCE:

For two slit interference, shown in Fig. 2, the distance y to the nth bright fringe fromthe midpoint, y = 0, is

y = D tan θ ∼= Dθ and θ ∼= nλ/d

Henceλ = yd/nD,

(For more information refer to the text and recall that for small θ, θ ∼= sin θ ∼= tan θ.)

dλ/θ= n

n=2n=1

n=0n=−1

n=−2

a

a

(a << d)

d

λm

D

m=1

m=−1

in phase

in phase

y

( > 1 meter)d

θ

LASER

θ

Interference maxima at

Figure 2: Schematic of double (or multiple) slit interference.


1. Rotate the Pasco wheel to the section containing slit pairs and illuminate one of thepairs with the diode laser.

2. Note the difference in pattern and spacing of the resulting interference fringes on thescreen. From Experiment I you know that individually these slit exhibit diffractioneffects. Thus you will observe both (i.e., a combination of single and double slitdiffraction) interference effects in all patterns. Observe the interference pattern forfour slit pairs.

3. Choose one pattern for careful measurement and measure the separations betweena number of adjacent maxima (n= -1, 0, 1, etc.) and calculate λ using the samegeneral procedure as experiment I.

4. Compare your calculated result with the stated value.

L-2: MIRRORS AND LENSES 70

L-2: Mirrors and Lenses

OBJECTIVES:

To study image formation and focal lengths of mirrors and lenses.

APPARATUS:

PASCO optical rail; lenses and mirrors; telescope; illuminated arrow light source,and 12V power supply; sharply pointed rod; desk lamp; white screen.

INTRODUCTION:

We will design, test and then measure mirror and lens assemblies by several tech-niques. In all instances it will be possible to use a simple interactive Java appletto perform a virtual pre-lab exercise. For the physical set-ups it will require testinglocation of image by absence of parallax, and others require focusing a telescope forparallel rays. For both techniques see Appendix B.

EXPERIMENT I: RADIUS OF CURVATURE AND FOCAL LENGTH OF A CONCAVEMIRROR:VIRTUAL PRE-LAB:

1. Point a web browser to http://badger.physics.wisc.edu (an old version of this man-ual) and navigate to lab LC-2. Launch the Concave Mirror Application.

2. After clicking on “Start Me” you should observe a concave mirror, an object and itsimage. On the lower left corner is a cursor position readout.

3. With respect to the figure below and the mirror equation, find the focal length.

4. Adjust the object to get p = q, and obtain the focal length again.

5. Move the object so that p = f . Do you see an image?

6. Move the object so that p < f . Is there an image? Is it real or virtual? Is the imageheight smaller or larger than the object?

The Mirror Equation is given by:

1

p+

1

q=

2

R=

1

f

where f is the focal length andR is the mirror radius curvature.Note that when the object andimage are equally distant fromthe mirror, p = q = R. Youcan use this condition to get anapproximate value for R and f .

illuminatedarrow

screen

p

q

Mirror

Figure 1: Layout of concave lensexperiment.

NOTE: The mirror holder contains a concave and a convex mirroron opposite sides. Be sure you use the concave mirror!

NOTE: You will need to place the white screen off to the side of thetrack and slightly twist the mirror mount to image the arrow.


SUGGESTED PROCEDURE:Follow the steps below to measure the focal length f in three different ways.

1. Obtain f by using object and image distances:

a. Resolve the image of the illuminated arrow formed by the concave mirror on thewhite screen. Vary the object distance, p, and observe how the image distance,q, varies with the object distance. You will need to slightly twist the mirrormount to make the image appear on either side of the light source.

b. Measure the image distance at several different positions of the illuminatedarrow object. From each pair of conjugate object and image positions calculatef for the mirror. Make a table of the results in your lab book, but leave spaceto compare with the results of steps 2 and 3 below.

2. Obtain f = R/2 by imaging at cen-ter of curvature (q = p = R): Replacethe illuminated arrow and screen withthe rod and place the rod at R. If thetip of the rod is at the center of cur-vature, a real inverted image will ap-pear just above it with the tip and itsimage coinciding. An absence of par-allax (see Appendix B.) between thetip and image is the most sensitivetest for coincidence. When the par-allax vanishes, the distance betweenthe tip and the mirror is the radius ofcurvature R. From several determina-tions of R (switching roles with yourlab partners), estimate the reliabilityof your results.

R

Image

Objectside facing mirrorilluminate

eye

Mirror

(quite far back)

Figure 2: Object and image at center ofcurvature, R.

3. Obtain f by placing the object at theprincipal focus (p = f): If the tipof the rod is at f , rays from the tipwill reflect from the mirror as a par-allel bundle and give a sharp imageof the tip in a telescope focused forparallel rays (see Appendix B). Keepthe distance between tip and telescopesmall or the reflected light may missthe telescope. The best test for locat-ing the focal point is absence of par-allax between the tip’s image and thecross hairs in the telescope, but thetelescope must already be focused forparallel rays (e.g., focus on somethingout the window). From several set-tings estimate the reliability of your fmeasurement.

f

Mirror

side facing mirrorilluminate

Object

keep small

telescope focussed forparallel rays

Figure 3: Object at f , image at infinity.


EXPERIMENT II: CONVERGING LENS:VIRTUAL PRE-LAB:

1. Go to http://badger.physics.wisc.edu, lab LC-2, and launch the Converging LensApplication.

2. After clicking on “Start Me” you should observe a convex lens, an object and itsimage. On the lower left corner is a cursor position readout.

3. With respect to the figure below and the lens equation. Find the focal length.

4. Move the object so that its inverted image has the same height. Find the focallength at this point.

5. Move the object so that p = f . Do you see an image?

6. Move the object so that p < f . Is there an image? Is it real or virtual? Is the imageheight smaller or larger than the object? What is the sign of q?


From the thin lens formula

1

object distance+

1

image distance=

1

p+

1

q=

1

f

Measure f using the two methods diagrammed below (by adjustments in the objectdistance). Compare the results from the various methods.a.

arrowilluminated

screenObject

p q

p+q 4fNote: >

lens

b.

Object

p=f

side facing telescopeilluminate

lens

parallel raystelescope focussed for

keep small

EXPERIMENT III: DIVERGING LENS

Since the image is always virtual it is necessary to combine it with a converging lensand space the two lenses so that the final image is real.

VIRTUAL PRE-LAB:

1. Go to http://badger.physics.wisc.edu and launch the Diverging Lens Application.


2. After clicking on “Start Me” you should observe a concave lens, an object and itsvirtual image. On the lower left corner is a cursor position readout.

3. With respect to lens equation and starting positions, find the focal length. Can youfind an object distance for which the image height and object height are the same?

4. Now restart the simulation with a combination of a convex and concave lens byclicking on the “Add 2nd lens” link.

5. This 2nd lens will render a final image which is real. The image with a “1” by it isthat of just the convex lens while the image with a “2” next to it is that of the pair.Move the object and alter its height, if necessary, so that the first image has therelationship q1 ≈ 2p2. (In this simulation |f2| = 3|f1| and qualitatively resemblesthe figure shown below.)

6. Now repeat the last step with |f2| = |f1| by clicking on the “Another 2nd lens” link.Finally reset to the “Another 2nd lens” and then click on the convex lens and dragit past the diverging lens.


1. Use the set-up sketched below. First adjust Lens1 so that with Lens2 removed,q1 ∼ 2p1. Measure q1, insert the diverging Lens2 reasonably close to Lens1 andthen locate the new image distance q2. From p2 = d − q1 (a virtual object) and q2

calculate f2.

screenarrowilluminatedObject

q

pq

p1

1 2

2

1

&

d

1

2

Image1

Object 2

Image2

Lens Lens

2. OPTIONAL: Find f2 by this more sensitive method: with the pointed rod as object,detect the image by a telescope focused for parallel rays (as shown in the followingfigure.)

To align all the objects: tape a piece of paper with a lot of markings on it to thepointed rod. Look through the telescope and move the diverging lens forward andbackward until you can see some markings clearly in the telescope. Then move therod up or down, as necessary, and rotate the telescope from side to side, as necessary,until you can see the tip of the pointed rod. Then move the diverging lens a smallamount until the tip is in focus.

As a last step you will move the pointed rod (i.e. vary p1 and hence q1) until asharp image appears in the telescope with no parallax relative to the cross hairs.(This occurs only if parallel rays leave the diverging lens.) The image from the


converging lens at q1 serves as a virtual object for the concave lens, Lens2. Afterproperly adjusting p1 this virtual object will be at the focal point of the diverginglens, Lens2. Obtain q1 by setting p1 and using the known value of f1. (Make surethat q1 is large enough to accommodate d + |f2|). Now look for the image in thetelescope and adjust p1. Note that since q2 =∞, f2 = p2 and p2 = d− q1.

p q

qfp

illuminate

telescopeside facing

Object

1

d

1

2

222

1

1

Lens

to=

focussed fortelescope

parallel rays

Lens

OPTIONAL: EXPERIMENT IV: FOCAL LENGTH OF A CONVEX MIRROR:

Once again, since convex mirrors give virtually images it is necessary to study themirror in combination with a converging lens.

Measure the focal length of the convex mirror by combining it with a converginglens. Set the pointed rod at twice the focal length of the lens (p1 = 2f1). Nextadjust the mirror position until the inverted image position shows no parallax withthe object. Then R = d− 2f1 and

fmirror = f2 = R/2.

Of course the lens must have f1 > f2, so use a long focal length lens.

eye

(quite far back)illuminate

from this side

mirror

Object

Image

d

lens

=R 2f =2fp1

=q 2f

2 1

1 1

L-3: OPTICAL INSTRUMENTS 75

L-3: Optical Instruments

OBJECTIVE:To construct a number of optical instruments and measure magnification.

APPARATUS:Identical to the of L-2 plus white board, achromatic doublets, Ramsden eyepiece.

SUGGESTION: In constructing an optical instrument catch the real image formedby the first lens on paper. This location can tell where the next lens goes;e.g. for a telescope or microscope place the next lens slightly less than its focallength beyond the first image.

EXPERIMENTS:1. Inverting Telescope

A. First measure the f of a weak converging lens (25 ≤ f ≤ 50 cm) and of astrong converging lens (f ∼ 5 cm). The first lens (f large) typically has thevirtue of intercepting and focusing a fairly large area of light while the secondlens can be positioned to give a large virtual image (and hence magnification.)Notice that if the first image, q1is at a point p2 > f2 with respect to the secondlens you get a real, erect second image. If you are using the web-based labmanual and desire a demonstration, click on the Inverting Telescope Demobutton immediately below.

Use you will use these two lenses to construct an astronomical (inverting) tele-scope. As an object, use the white board and scale while illuminated by abright light. As a procedure for measuring the magnification, vary the spacebetween the objective L1 and the eyepiece L2 until the virtual image I2 of scaleS seen through L2 lies in the plane of S (i.e. p1 + d = |q2|). As a sensitive testof this there should be no parallax between the scale as seen directly by youreye 2 and the scale image seen by eye 1 thru the telescope as shown in Fig. 2.Generally a person will adjust the eyepiece so that the virtual image appearsat the distance of most distinct vision (∼25 cm).

(not to scale)

eyeI2

I1

L2f2L1 1f

p1 p

2q1Scale

(virtual)(real)

Objectd

Objective lensEyepiece

lens

Figure 1: The inverting telescope (not drawn to scale).

L1Scale L2

eye 2

eye 1

Figure 2: View scale with one eye, then the other.


B. Adjust the telescope direction until these imagessuperimpose (as in Fig. 3). The number of divisionson the scale as viewed directly (by eye 2) which fallin one division of the image as seen through thetelescope (by eye 1) is clearly the magnifying powerof the telescope.Compare your measured magnification M with thatcalculated for an astronomical telescope, namelyM = f1/f2 where f1 is the focal length of the ob-jective and f2 that of the eye piece. Why are thetwo different? Calculate the magnification from theactual measured object and image distances. Com-pare with the measured M .

4

23 object

scaleReal

imagescaleVirtual

234

1

5

76

8

Figure 3:

2. Erecting telescope

eye

L1

L3(virtual)

I1

2I

L2

I3

(real)

(real)

Figure 4: Configuration of a three lens erecting telescope.

A. Measure the focal length, f , of a second strong converging lens and use it as theinverting lens L3 (Fig. 4) to form an erecting telescope. L2 must be adjustedto give a real image, I2. Notice that varying the space between I1 and theinverting lens L3 changes the magnification. If you are using the web-basedlab manual and desire a demonstration, click on the Erecting Telescope Demobutton immediately below. In this simulation your are now able to readjustboth the positions and focal lengths!

B. What position of the inverting lens makes the shortest possible telescope, (i.e. itmakes the distance between I1 and I2 a minimum)?

3. Simple microscope or magnifier: Experiment with one of the strong converging lensesas a magnifier until you can make the virtual image appear at any distance youchoose: vary the object distance from zero to the focal length. The parallax testdescribed in Part 1 will enable you to locate the virtual image. When the parallaxbetween the object and the virtual image (seen through the lens) vanishes, the objectand virtual image are in the same plane.

4. Compound microscope: Usethe two strong converginglenses to form a compoundmicroscope.

eye

2I

I1

L2L1

(real)

(virtual)

Figure 5: Setup for compound microscope.


5. Galilean telescope (or opera glass): Make use of the weak converging lens and astrong diverging lens.

OPTIONAL:

1. Ramsden eyepiece

A. Ask the instructor for the holder containing a Ramsden eyepiece, a crosshair,a diaphragm, and an objective lens. See Fig. 6.

d

1

2

S

2 / 3 f

I

f

f

Figure 6: Construction of a Ramsden eyepiece.

The Ramsden eyepiece consists of lenses 1 and 2, plano-convex lenses of f =85 mm. Spherical aberration is less than for a single lens because the raybending is spread over four surfaces. It is a minimum if the curved surfacesface one another as shown. (In addition if the two lenses were their focal lengthapart, chromatic aberration would be a minimum. But, since dirt specks onlens 1 then would be in focus, lenses 1 and 2 are usually set 2f/3 apart). Thetube also contains an objective lens to convert the tube into a telescope ormicroscope.

B. Use the Ramsden eyepiece with d = 2f/3 in the telescope and microscope.Note improvement in image over that of a single eyepiece lens. Adjust eyepiecefor a sharp image of the cross hairs. Then move the objective lens until theimage shows no parallax with respect to the cross hairs.

2. Huygens eyepiece: A common eyepiece in which the image (and hence cross hair)is within the eyepiece. It also employs two plano-convex lenses but both convexsurfaces are toward the incident light. Minimum lateral chromatic aberration existsfor a separation of (f1 + f2)/2. Usually f1 ∼ 3f2.

f1

I2

f2

f2I1

(f1 f2+ ) / 2

S

Figure 7: Construction of a Huygens eyepiece.


3. Eye ring.

A. Point the astronomical telescope at a lamp and explore with a screen the il-lumination back of the eyepiece. [Or remove diaphragm from telescope ormicroscope of optional (1)].

OI

ER

Figure 8: Position of eye ring.

The light coming through the telescope has a minimum cross section at ER.This area ER is the image of the objective formed by the eyepiece. Obviouslyall light which gets through the telescope must go through this image, and thusER is the best spot to place the pupil of the eye. Hence the name “eye ring”.

Note that if one places a diaphragm with aperture appropriately larger thanthe eye ring a little ahead of the eye ring, then an observer looking through theaperture would position the pupil of her/his eye on the eye ring.

B. Using the Ramsden eyepiece, place the diaphragm at the appropriate distancein front of the eye ring. Note the improvement in the ease of observing throughit, especially for the compound microscope.

4. OPTIONAL telescope: Use an achromatic objective lens, of long focal length and acommercial Ramsden eyepiece (from instructor). Note the quality of the image.

5. OPTIONAL microscope: Use a short focal length achromatic objective and a com-mercial Ramsden eyepiece (from instructor). Note the quality of the image.

It is interesting that one does not need an achromatic eyepiece because the virtualimages for red and blue, though at different distances (e.g., red at 2 m, blue at∼ ∞), subtend almost the same angle at eye. As a result the eye doesn’t notice theaberration.

L-4: THE OPTICS OF THE EYE AND RESOLVING POWER 79

L-4: The optics of the eye and resolving power

OBJECTIVES:

To study eye optics and how resolving power depends on aperture.

APPARATUS:

PART I: Eye model, (fill with distilled water to simulate the aqueous & vitreoushumors); light source, lens set, 6“ plastic ruler.

PART II: (2) Optical benches, Na lamps, transparent mm slides, vernier calipers,L-2 telescopes; (1) 15 meter tape.

S1G1

S2G2

RmhR RC L fovea

retina

vitreoushumor

ciliary muscleandsuspensoryligaments

aqueoushumor

lensopticnerve

cornea

iris

pupil

Figure 1: Various schematics for the eye

You may fix the moveable retina (cylindrical) in one of three positions:

R for the normal eye,Rh for the hypermetropic (farsighted) eye,Rm for the myopic (nearsighted) eye.

(An eye retina is actually spherical, but for small images the results are similar.)

The eye model has a fixed cornea C and racks for other lenses.

A lamp box serves as source for most of the experiment.

Set of lenses: (Strength in diopters = 1/f where f is in meters.)

Diopters Focal lengths in air1. Converging spherical + 7. 0.14 m2. Converging spherical +20. 0.05 m3. Converging spherical + 2.00 0.50 m4. Diverging spherical - 1.75 -0.57 m5. Diverging cylindrical - 5.50 -0.182 m6. Converging cylindrical + 1.75 0.57 m7. Diaphragm with small hole . .


PART I - EXPERIMENTS WITH THE EYE MODEL

1. ACCOMMODATION: With the retina in the normal position, point the eye modelat a window or other bright object 4 to 5 meters away. Insert the +7 diopter lensin the water at the inside mount farthest from the cornea. An image of the brightobject should be in focus on the retina.

Next use as object the lamp box and place it about 30 cm from the cornea. Theimage is blurred until one replaces the +7 diopter lens by the +20 diopter lens. Thischange illustrates accommodation. Eye muscles make the lens thicker for close vision.(Under relaxed conditions the ligaments supporting the lens are in radial tension andhence thin the lens. The ciliary muscle contracts against these ligaments allowingthe lens to thicken for viewing near objects. See Fig. 1.)

2. NEAR and FARSIGHTEDNESS (MYOPIA and HYPERMETROPIA): With lampbox at 30 cm and the 20 diopter lens at L, place the retina in the position Rm.Decide which lens placed in slot S1 or S2 will bring the image into focus. Try it.

Make the eye farsighted by moving the retina to Rh, and pick the proper lens toplace in front of the eye to bring the image into focus.

Focus again on a window 4 to 5 meters away when the +7 diopter lens is in thenormal position L and the retina in position R.

Make the eye nearsighted by moving the retina to Rm. Decide which lens will bringthe image into focus. Try it.

Remove the correcting lens; the image again blurs. At S2 place the diaphragm withhole and note image improvement. With sufficient light try a smaller hole (e.g. a holethru masking tape on the diaphragm). Explain the effect of the reduced aperture.

3. ASTIGMATISM: With the +20 diopter lens at L in the eye and the retina in thenormal position, adjust the distance to the lamp box to get a sharp image on theretina. Then produce astigmatism by placing the strong diverging cylindrical lensin mount G1. (In the human eye astigmatism generally results from a cornea withdifferent curvatures in the horizontal and vertical directions.) Turn this lens andnote the effect on the image.

Correct this defect by placingthe converging cylindrical lens infront of the cornea and turningit to the position for which theimage becomes sharp. Note thedirections of the axes of the twocylindrical lenses. (They are notnecessarily aligned with the tab.)

axis

Figure 2: Cylindrical lens.

4. COMPOUND DEFECTS: Combine tests 2 and 3, i.e. make a nearsighted astigmaticeye. Correct by using both a cylindrical and spherical lens in front of the cornea.How is this double correction managed in actual spectacles?

5. CATARACTS and LENS REMOVAL: A cataract results from a cloudy lens. Acataract operation involves removal of the lens. An implanted plastic lens or astrong convergent lens in front of the eye restores vision. Remove the eye lens (L)of the model and place the +7 diopter lens in front of the cornea. A clear imagereappears for nearby objects.


6. Action of a SIMPLE MAGNIFIER: With the lamp box at 35 cm, the +20 diopterlens and the retina in the normal position, measure the size of the image. Now placethe +7 diopter lens in front of the cornea and move the lamp box toward the eyeuntil the image becomes clear. Record the increased size of the image. With thelamp box at 35 cm, the 20 diopter lens and the retina in the normal position, notethe size of the image. Now make the eye nearsighted and bring the lamp box towardthe eye until the image becomes clear. Note the increased size of the image in thenearsighted eye.

NOTE: For a real eye and lens the index of refraction, µ, of the lens varies from 1.42 atits nucleus to 1.36 at its edges, and these values are not so different from those of theadjoining aqueous and vitreous humor (µ = 1.330 and µ = 1.337). Hence the real eye lensis very weak and is primarily for fine focus adjustment. Most of the refraction in the eyeactually occurs at the cornea where µ changes from 1.0 (in air) to 1.37 in the cornea.

PART II - RESOLVING POWERBefore doing this part, read in your text about resolving power.

SUGGESTED EXPERIMENTS:1. Use as a source a slide with mm divisions and illuminated by a sodium discharge

lamp. The wavelength of the sodium yellow light is 589.3 nm.

2. Back away from the slide until the mm lines become indistinct. Then advance farenough so that they are just clearly visible; this procedure insures that eye defectsor the density of rods and cones do not (for this measurement) limit the resolvingpower.

Observing now with only one eye, place a vernier caliper2 in front of the eye pupilwith the caliper jaws parallel to the mm marks and open about 3 mm. Slowly closethe jaws until the mm marks disappear. Note the separation of the jaws, w, andmeasure the distance, D, to the slide from the jaws.

Compare your measurements of the angular separation α of the mm marks at yourobserving distance,

α =1 mm

D (in mm),

with the expression θ = λ/w where θ is the angular limit of resolution, (Rayleigh’scriterion).

3. Mount a telescope so its objective is at the same distance from the slide. Repeat theexperiment with the caliper in front of the objective.3 How do the results compareto your measurement without the telescope?

4. Move the telescope ≥ 4 m from the slide and repeat the experiment.5. Turn the vernier jaws perpendicular to the mm lines and repeat experiment #1.

Comment on the result.

QUESTIONS:1. Would an eye without any defects of vision have better resolving power at dusk or

in bright sunlight? Explain.2. Discuss how a contact lens functions. Would a “soft” contact lens correct well for

astigmatism?

2In all the following be sure you are not looking through the hole at one end of the jaws. It is advisableto cover the hole with masking tape.

3Be sure no light enters the telescope except through the slit jaws. (You may need masking tape onthe caliper jaws to prevent this from happening.)

L-5: SPECTROMETER AND THE H BALMER SERIES 82

L-5: Spectrometer and the H Balmer Series

NOTE: Consult your instructor for assignment of parts suitable for a single lab, e.g. doPart I, II (A or B), Part IV or Part V).OBJECTIVES:

To become familiar with a precision spectrometer and some of its uses.

APPARATUS:

Precision spectrometer; mercury and hydrogen discharge tubes; prism; diffractiongrating; light source for Gaussian eyepiece & its 12V power supply (L-2); ring stand& achromatic lens.

E) Telescope

F) focus ring

K) prism table clamp

G,H) telescope clampfine adjust

L) prism table clampfine adjust

I, J) angle scale& vernier

I, J) angle scale& vernier A) grating

B) adjustable slit(twist)

C) Collimator

M) prism tablelevellingscrews

Figure 1: The spectrometer (side view).

INTRODUCTION and ADJUSTMENTS

PART I - A

1. Become familiar with the clamping and fine adjustment controls for telescope andprism table angles. Never force a motion - you may damage the instrument.If at the end of a fine adjustment, loosen and reclamp! The prism tableclamp sets table elevation: no fine adjustment is necessary.

2. Note that 180 apart are two angle scale reading ports and verniers, Iand J. The verniers have 30 divisions per half degree. Hence each division is(1/30)×(1/2) = (1/60) or one minute (′) of arc.

3. The knurled ring about the collimator, B, controls the slit opening.

4. Staff have already focussed the collimator for parallel light and have set collimatorand telescope ⊥ to spectrometer’s rotation axis.


REMINDER ON READING A VERNIER

1) Position and tighten the telescope.

2) Move the magnifying glass to see the scale marking

3) Look where the zero mark on the upper scale is lo-cated. The degree reading is the first full degree markto the left of the zero (210 in the top figure; 229 inthe bottom figure).

4) Now make the minute reading: if the zero mark onthe upper scale comes after a 1

2 degree mark (theshort lines on the lower scale) then start at 30′, oth-erwise at 0′.

5) Next look for the two lines which match up best be-tween the two scales. Read the number of the appro-priate line on the upper scale and add it to the zeromark is just past a 1

2 degree mark so we begin with30′. The 15′ mark is the one that lines up best so weget 21015′ for the top reading and 229146′ for thebottom one.

6) Now convert to decimal degrees, in this case 15/60 =0.25, so θ = 210.25.

210˚ 15'

10 20 300

210 220

10 20 300

230 240

229˚ 46'

Figure 2: The verniers

PART I - B: FOCUS THE TELESCOPE FOR PARALLEL RAYS

1. i) Slide telescope eyepiece in or out until crosshairs are in sharp

ii) Sight telescope at a distant object (thru an open window); then focus telescopeby rotating its focus ring (Fig. 1) until the object’s clear image falls on the crosshairs. The test for proper focus is absence of parallax between the image andcross hairs (Appendix B). The telescope, now focused for parallel rays, will stayso as long as the focus ring is unmoved; but one may still adjust the eyepieceto suit the observer.

2. An alternative method to focus for parallel light is to use the Gaussian eyepiece +light source as described below in PART II B.

3. With collimator and telescope both properly focused one should get a sharp image ofthe slit and no parallax between slit image and cross hairs. If you still get parallax,recheck your telescope focus and/or consult instructor.

4. Optimizing light thru the collimator: A properly located short focal length lens cangather a large fraction of the light from a source and redirect it

thru the collimator thereby facilitating detection of weak spectral lines. For stronglines it permits narrowing the slit width and thus improving resolution. Use sucha lens (mounted on a ring stand) to find an arrangement which fills the collimatorwith light.


located wellb) Prism is not

reflected beamis not centeredon telescope

a) Prism is in proper location

centered on telescopereflected beam is

B1 B2

A

D

1st position of telescope 2ndposition of telescope

B1B2

Collimating lens

Figure 3: Proper, a), and improper, b) prism locations.

PART II-A: MEASUREMENT OF PRISM ANGLE

1. Mount the prism (in holder) on the dowel pins (in the prism table) so that the prismis far enough from the collimator that beams B1 and B2 are centered with respect tothe spectrometer axis; otherwise much of the reflected light may miss the telescope.Turn the table so that the apex angle A (Fig. 3) splits the beam into nearly equalparts.

2. Using the prism table levelling screws adjust the plane of the prism table so it isclose to horizontal.

3. With the prism table clamped at the proper elevation, set the telescope to receivebeam B1 and form an image of the slit on the intersection of the telescope crosshairs. Record both VERNIER readings (in minutes). Average the two vernierreadings (to eliminate any systematic error from misalignment of the circle scale withrespect to bearing axis), and add the result to one of the angle scale readings. Notethat the angle scale reads in half-degree units and the vernier in minutes (not indecimal degrees!). The zero position on the vernier determines the angle to thenearest half-degree (30 minutes) and the vernier reads the number of minutes pastthe half-degree mark.

4. Then with telescope in the 2nd position, set on the slit image. The change in readingof the same angle scale should be the angle D. (See Fig. 3.) Be sure you don’t get theangle scales mixed up and subtract the first reading of angle scale 1 from a readingof angle scale 2. Also some students incorrectly handle the subtraction when thescale passes through 360 or 0.

5. OPTIONAL: Try to prove that angle D is twice the prism angle A.

PART II-B: ALTERNATE method for prism angle measurement using Gaussian eyepiece

cross hairs objective lens

Lamp

Gaussian eyepiece

B

A

G

reflecting surface

Figure 4: Spectrometer telescope


Introduction: The Gaussian eyepiece (Fig. 4) has a partially reflecting glass plate Gset at 45 to telescope axis so that light from the lamp reflects down the telescopetube, past the cross hairs and out the objective. If in front of the objective you placea reflecting surface perpendicular (⊥) to the telescope axis, the light will reflect backinto the objective and form a real image of the cross hairs. If the cross hairs are inthe focal plane of the objective, their reflected image will form in the same plane.Thus both the cross hairs and the image will be in focus through the eyepiece. Whenone orients the reflecting surface so that the image coincides with the cross hairs,the reflector is accurately ⊥ to telescope axis.

Thus the Gaussian eyepiece permits both focusing the telescope for parallel rays andsetting a reflecting surface ⊥ to the telescope axis.

1. To focus the telescope for parallel light:

a) Rotate the Gaussian eyepiece to open the hole between reflecting plate G andlamp. Adjust eyepiece to give a sharp image of the cross hairs, but don’t turnthe eyepiece to block the light hole. Next, with prism on the table, adjustthe prism table screws to make the table nearly horizontal.

b) Now rotate the prism table until the prism face is approximately ⊥ to telescopeaxis. Clamp the telescope and mount a light on the Gaussian eyepiece. (Checkthat the light hole is still open). With no illumination on the prism exceptthat from the telescope, next rotate the prism table back and forth a fewdegrees until maximum reflected light appears in the eyepiece. Clamp thetable in this position, and then focus the telescope (by turning the focus ring)until the reflected image of the cross hairs appears and shows no parallax withrespect to the cross hairs. The telescope is now focused for parallel rays. Fineadjustments of the prism table leveling screws may help the images coincideand thus set telescope axis accurately ⊥ to the reflecting surface. See Part VI,Sec. 3.

2. To find the prism angle set the telescope ⊥ to first one prism face and then theother. The angle between these two positions is the supplement of the prism angle.See Part IIA for detail about angle readings.

PART III: INDEX OF REFRACTION

Introduction: When the path thru a prism is symmetric, the deviation is a minimum.At this angle of minimum deviation the index of refraction, n, is

n =sin [(A+ d)/δ]

sin (A/2)

where A = angle of prismδ = angle of minimum deviationn = refractive index of the prism.

δ

A

Devise your own methodology and determine the prism’s refractive index for one ormore lines of the Hg spectrum.


PART IV: CALIBRATION OF PRISM SPECTROSCOPE

R for redV for violet

source slit

collimating lenses

A

S

D

R

V

PCB

E

1. Set the prism for minimum deviation for a green line in the Hg spectrum.

2. Determine the angle readings for the yellow, green, blue-violet, and deep-violet Hglines. Repeat this for the red line from a hydrogen discharge.

3. Plot telescope angle vs accepted λ’s for these lines.

4. For the same setting of the prism table, find the angles for the blue-green and twoviolet lines of the hydrogen spectrum.

5. Use your calibration curve to determine λ of the blue-green and violet lines in theatomic hydrogen spectrum. Compare results to accepted values.

OPTIONAL: The calibration curve is very non-linear. Since a more linear plot facilitatesinterpolation of an unknown λ, try plotting deviation vs 1/λ2.

TABLE OF WAVELENGTHS:

Mercury lines Hydrogen lines

λ (in nm) (l/λ2) λ (in nm) (1/λ2)Yellow1 579.0e (unresolved) ×106(nm)−2 ×106(nm)−2

Yellow2 577.0c Avg. = 578 2.993 Red 656.3 2.322Green 546.1 3.353 Blue-green 486.1 4.232Blue 496.0∗ 4.065

49l.6∗ 4.138 Violet 434.0 5.309Blue 435.8 5.265

-violet 434.7∗ Deep Violet 410.2 5.943Deep- 433.9∗

violet 407.8∗ 6.013 NOTE: ∗ implies404.7∗ 6.106 usually faint


PART V: DIFFRACTION GRATING

CAUTION: Do not touch grating surface under any circumstance!

1. Grating constant (number of lines per cm): The lines/cm marked on these replicagratings is only approximate because the plastic often changes dimensions when itis stripped from the master grating. To achieve quantitative results calibration ofthe grating constant is desirable and can be performed using the known λ, 546.1nm, of the Hg green line.

CALIBRATION PROCEDURE:

a) Align telescope with collimator so the slit image falls exactly on the verticalcross hair.

b) Mount grating on the dowel pins in the prism table. Adjust and clamp thetable so the grating is approximately ⊥ to telescope-collimator axis.

c) Then adjust the table so that the front grating face is accurately ⊥ to tele-scope. Use the Gaussian eyepiece (see Part II - B) for the final adjustment: thereflected image of the cross hairs should fall on the cross hairs. (Two reflectedimages can result if the sides of the grating glass are not parallel. If so, testwhich image comes from the grating side by wetting the other side.) Clampprism table in this position and record telescope direction.

d) Next set telescope on the first and second orders of the Hg green line. Useboth ± angles. The ± angles should of course agree. Use them and λgreen =546.1 nm to calculate the grating spacing d from

nλ = d sin θ.

2. Measurement of the wavelengths in the Balmer series in hydrogen:

a) Use the calibrated grating spectroscope to measure λ for three or four atomichydrogen spectral lines.

b) Compare these λ’s to those from the Balmer formula

1

λ= R

[1

22− 1

n2

]where R = 10, 967, 758 m−1

or calculate the frequencies of these lines and compare to

f = cR

[1

22− 1

n2

]where R = 10, 967, 758 m−1.

c) Use Planck’s relation, ∆E = hf , to calculate the energies in eV of the photonsfor each frequency and, therefore, the energy change in the hydrogen atomassociated with each frequency.

d) Assuming that each observed ∆E leaves hydrogen in the n = 2 state, showthe observed transitions on a hydrogen energy level diagram (drawn to scale;consult textbook).


PART VI: ADJUSTMENT OF A SPECTROMETER

NOTE: Not to be performed without permission of the instructor

1. Use Gaussian eyepiece and prism face method to focus telescope for parallel rays.If necessary adjust prism table to make reflected image of the cross hairs coincidewith the cross hairs.

2. Remove prism from the table and align telescope and collimator approximately.Focus the collimator until there is no parallax between the image of the slit and thecross hairs. Adjust the ring on the collimator focusing sleeve so that the collimatoris in focus with the “V” in the slot. Clamp the telescope in line with the collimatorso that the image of the slit (vertical) falls on the vertical cross hairs. Now rotatethe slit through 90 as permitted by the “V” projection and slot. Adjust the levelof the telescope so that the image of the slit falls on the horizontal cross hair. Theaxes of telescope and collimator are now in line but not necessarily perpendicular tothe axis of rotation of the instrument.

3. Replace the prism on the prismtable and clamp it relative to theadjusting screws as shown. (LineAB ⊥ face 1. Line AC ⊥ face2. Thus screw B will adjust face1 without disturbing face 2, andscrew C will adjust face 2 with-out disturbing face 1.

A

B

C

1 2

Bring the telescope to within < 90 of the collimator and use a face of the prismto reflect light from the collimator down the telescope tube. Adjust the prism faceso that the center of the slit image falls on the intersection of the cross hairs. Theprism face is now parallel to the instrument axis.

4. Set the telescope ⊥ to the adjusted prism face (use Gaussian eyepiece method).Return to part 3 setup and adjust the collimator so that the center of the slit imagefalls on the cross hair’s intersection. Telescope and collimator are now properlyfocused and are perp to the instrument axis.

5. To adjust the second prism face ⊥ to the axis of the telescope (again use the Gaussianeyepiece method). Turn only the proper screw on the prism table: otherwise onedisturbs the adjustment of the first face. Recheck the first face and, if necessary,readjust it. Continue this process until both faces are parallel to the axis of theinstrument. Turn collimator slit to vertical position. The spectrometer should nowbe in adjustment.

L-6: POLARIZATION 89

L-6: Polarization

OBJECTIVE: To study polarization and double refraction of light.APPARATUS:

Polarization kit plus a polariscope; Brewster angle assembly (laser + mount on M-2force table, slotted pin & black piece of plastic).

SUGGESTED EXPERIMENTS:

1. Polarization by ABSORPTION: Observe a light source (desk lamp) thru each oftwo polaroids, and then thru both together. Rotate one of the two about the line ofsight. Explain what occurs.

Orient the two polaroids of the polariscope so no light is transmitted. Then introducea third polaroid between the first two and rotate it about the line of sight. Explainwhat happens.HINT: there will be an amplitude component E cos θ in the direction of the trans-mission axis.

2. Polarization by REFLECTION from a dielectric: Observethru the small polaroid (the one with the rim index marks)the light reflected from the black plastic (see figure at right).Rotate the polaroid about the line of sight. Note position ofthe index marks (i.e. the E vector direction) on polaroid whenthe transmitted reflected light is a minimum.

mark

Index

eyepolaroid

θB

air

air

glass

A

B

θiθr

θB

Figure 1: Reflection and transmision of light at Brewster’s angle.

Now find an incident angle θi (and reflection angle θr) for which this minimum iszero. Then θi is Brewster’s angle, θB, and tan θB = n2/n1. Estimate θB roughly.What is the refractive index n2 of the plastic?

Alternatively for greater accuracy use a laser plus polaroid to prepare a beampolarized in the plane containing the incident ray and the normal to the reflect-ing surface. The reflected beam then vanishes at Brewster’s angle. Mount laserplus polaroid on the M-2 force table and point it toward a dielectric (plasticor microscope slide) in the slot of the central pin. Rotate the sample until thereflected beam (on the wall) disappears. A thread from the center pin to thispoint on the wall will help locate the angles on the M-2 table.

Observe any polarization of light reflected from painted or varnished surfaces, floortile, clean metal (e.g. aluminum foil) etc.


3. Polarization by SCATTERING: Try to detect the polarization of skylight: oneneeds a blue sky! Explain (use a diagram) in what direction relative to sunlightshould one look to see this polarization best.

4. Polarization by a DOUBLY REFRACTING CRYSTAL: Uniaxial doubly refractingcrystals (e.g. calcite) are highly anisotropic and, as such, contain an axis of symmetry(the optic axis) for which the velocity of light depends on whether the E-field vector(polarization plane) is ⊥ to this axis (the ordinary “o” ray) or parallel to the opticaxis (the extraordinary “e” ray). In the principal plane (i.e. a plane containing theoptic axis but ⊥ to the surface) the new wave front of the ordinary ray involves theusual spherically expanding wavelets, but to construct the front for the extraordinaryray requires ellipsoidally expanding wavelets where the two axes of the ellipse areproportional to the velocities of the parallel and ⊥ vibrations to the optic axis.

Calcite (CaCO3) has a large velocity differences between the o and e rays, and itsoptic axis is the direction at equal angles with the crystal edges at the obtuse corners.

Observe through the calcite crystal a dot on a piece of paper. Rotate the crystalabout the line of sight. Describe the behavior of the two images of the dot. Introducea polaroid between your eye and the crystal. Note how the appearance of the twoimages changes as you rotate the polaroid about the line of sight. Explain.

NOTE: For the remaining experiments use the POLARISCOPE. The polaroid next tothe light is the “POLARIZER,” the other is the “ANALYZER.” Place the samplesto be studied on top of the polarizer.

5. CIRCULAR POLARIZATION: A thin doubly refracting crystal cut so that the opticaxis is parallel to a surface constitutes a retardation plate. The difference in velocityof the o and e rays then gives a phase difference between the emerging orthogonallypolarized rays. If the phase difference is 90, the retardation is λ/4: hence a “quarterwave plate”.

Crystal plate

Optic axis

Polarizer

Analyzer

45

45

E- vibration

O-vibration

Unpolarizedlight

Figure 2: Obtaining circularly polarized light from unpolarized light

Set the analyzer of the polariscope to transmit no light. Then place the λ/4 platebetween the polarizer and analyzer and rotate the λ/4 plate so that extinction againresults. You will find two such extinction positions 90 apart. Why?HINT: When the optic axis is aligned with the polarization direction of the incomingplane polarized wave, will there be any ordinary ray amplitude? Then when youturn the plate (and thus the optic axis) thru 90, will there be any extraordinaryray amplitude.


Rotate the λ/4 plate to a position halfway between the two extinction positions; thusthe optic axis will make a 45 angle to the incident plane of polarization. Explain whythe plate now looks bright. Leaving the λ/4 plate in this position, rotate the analyzerthrough 2π radians. Why does the brightness stay approximately constant∗?HINT: Recall the conditions for a circular Lissajous figure (see E-8 Part B, Fig. 3).

∗ If it doesn’t, repeat the preceding operations. You may not have set the λ/4plate close enough to the halfway or 45 position. Also remember that a λ/4plate is exactly λ/4 only for a single λ.

6. TWO λ/4 PLATES: Place a λ/4 plate on the polarizer so as to produce circularlypolarized light. Place a second λ/4 plate on top of the first and similarly oriented.Observe and explain what happens when you now rotate the analyzer. Turn the topλ/4 through 90 about the line of sight. Observe and explain what happens nowwhen you rotate the analzyer.

7. COLOR EFFECTS: If the plate is thick enough for retardations of several (but nottoo many) wavelengths, color effects may result. For a constant refractive index:

λred ∼ 660 nm ∼ 3

2(λblue ∼ 440 nm).

Thus a λ/2 plate for red is a 3λ/4 plate for blue. [In general, a n(λ/2) plate forred ∼ 3n(λ/4) plate for blue where n is an odd integer.] Hence for white light sucha plate at 45 between parallel polaroids would completely extinguish the red butpass the circularly polarized blue light. The resultant color is complementary to thatremoved. The intensity (hue and saturation) will of course vary with the relativeorientation of polarizer, the retardation plate, and the analyzer. As n becomes largethe range of λ extinguished narrows. Hence that passed is more nearly white.

a) Place thin mica on the polarizer (with analyzer crossed). Try various micaorientations. Explain.

b) Repeat a) for the mounted specimen of cellophane tape.

8. PHOTO-ELASTIC EFFECTS: Strained isotropic materials become doubly-refracting.

With the polariscope set for extinction:

a) Insert and flex the U-shaped lexan (a polycarbonate resin) sample.

b) Insert a microscope slide vertically. View the long edge and stress the slideby gently flexing it. Note result. (Glass blowers use polarized light to test forresidual glass strains).

9. LCD APPLICATIONS: Watches and calculators often use LIQUID CRYSTAL DIS-PLAYS (LCD) in which nematic liquids (thread like molecules arranged nearly par-allel to each other) can become optically active, rotate the plane of polarizationand thus permit light to pass thru crossed polaroids. A reflector after the analyzerreturns the light through the cell.

To produce the optical rotation: two glass plates, which have been rubbed in onedirection to produce invisible scratches, attach aligned nematic molecules. If a few


glass

thin transparentconductor

polarizer

analyzer

crystalLiquid

glass

glass

Light

dark

conductorthin transparent

bright

Light

glass

Figure 3: Construction of a liquid crystal display

micron thick nematic liquid separates the plates, and if one plate is rotated relativeto the other, then the helical arrangement of the nematic liquid produces a rotationof the plane of polarization.

To extinquish the light one applies a voltage between the transparent (but conduct-ing) tin-oxide coated glass surfaces. A sufficient voltage gradient will align the dipolemoments of the molecules and thus destroy the optical rotatory power. Voltages ap-plied to segments of the seven segment pattern (see figure) permit display of anynumeral 0 to

The power consumption of such an LCD is almost negligible.

OPTIONAL QUESTIONS:

1. How experimentally can one tell whether a light beam is unpolarized? Plane polar-ized? Circularly polarized?

2. How could you tell whether an object is a gray plastic, a polarizing sheet, a λ/4plate or a λ/2 plate?

3. How could you change right circularly polarized light to left circularly polarizedlight?

4. Why does the flexed lexan show colors but not the microscope slide?

APPENDICES 93

Appendices

A. Precision Measurement Devices

Vernier Calipers:

outside calipers

01 2 3 4 5

1 2 3 4 5 6 7 8 9 10 11 12 13

INCH

mm

cenco3

8 9

inside calipers

vernier scale

depth gauge

fixed scale

Figure 1: The vernier caliper

A Vernier consists of a fixed scale and a moving vernier scale. In a metric vernierthe fixed scale is marked in centimeters and millimeters, the vernier scale is nine mil-limeters long, and is divided into ten parts each 0.9 millimeters long. The distancesof each line from the first are therefore 0.9, 1.8, 2.7, . . . ,mm or generally: di = 0.9×i,where di is the distance between the zero line and the ith line of the vernier scale. Ifthe vernier caliper is closed, so that the two jaws touch each other, the zero of thefixed scale should coincide with the zero of the vernier scale. Opening the jaws 0.03cm = 0.3 mm will cause the fourth line (the three line which is a distance of 2.7 mmfrom the zero line of the of the vernier scale) to coincide with the 3 mm line of thefixed scale as shown below.

01

Figure 2: The vernier reads 0.03 cm

Below is another example of vernier reading; the arrow shows which mark on thevernier scale is being used.

9 10

Figure 3: The vernier reads 9.13 cm

APPENDICES 94

EXERCISES:

A. Close the vernier and observe that the first vernier mark coincides with the zero ofthe centimeter scale.

B. Open the jaws of the vernier very slowly and observe how the different vernier markscoincide successively with the millimeter marks on the fixed scale: the first markcoincides with the 1 mm mark on the fixed scale; then the second mark coincideswith the 2 mm mark on the fixed scale; then the third mark coincides with the 3mm mark on the fixed scale and so on.

C. Estimate the dimension of an object using a meter stick and then Use the verniercaliper to measure the dimension precisely.

D. In the four examples of Fig. 4 determine the actual reading.

3 4 4 5

a) b)

6 7 1 2

c) d)

Figure 4: Test cases

Micrometer:

A micrometer can measure distances with more precision than a vernier caliper. Themicrometer has a 0.5 mm pitch screw, this means that you read millimeters and halfmillimeters along the barrel. The sleeve is divided into 50 divisions correspondingto one hundredth of a millimeter (0.01 mm) or 10 µ each. The vernier scale on themicrometer barrel has ten divisions, marked from 2 to 10 in steps of two. The “zero”line is not marked ‘0’, but is longer than the others. The vernier allows you to read tothe nearest thousandth of a millimeter, i.e., to the nearest micron (0.001 mm = 1 µ).

Precaution:

Great care must be taken in using the micrometer caliper; A ratchetknob is provided for closing the caliper on the object being mea-sured without exerting too much force. Treat the micrometer withcare, ALWAYS close the calipers using the ratchet knob, this pre-vents tightening the screw too strongly. Closing the calipers too harddamages the precision screw.

APPENDICES 95

Mitutoyo

0-25 mm 0.001 mm

24

680

0 5

0

5

10

40

45

frame

anvil spindle

sleeveratchet

spindle lock

barrel

Figure 5: The micrometer calipers

Below are two examples of micrometer reading; the arrow shows which mark on thevernier scale is being used.

2

4

680

0 5 10 15 20 25

40

45

0

35

Figure 6: The micrometer reads 20.912 mm

2

4

680

0 5 10 15 20 25

0

5

10

40

45

Figure 7: The micrometer reads 3 µ

In Fig. 7 the zero line on the barrel is barely visible, and the vernier reads 0.003 mm= 3 µ; the zero error is ε0 = 3µ.

APPENDICES 96

A negative zero error, as shown below requires a moment of thought.

2

4

680

0 5 10 15 20 25

0

5

10

40

45

Figure 8: The micrometer reads -4 µ

In Fig. 8 the zero line on the barrel of the micrometer is obscured by the sleeve,(the “zero” line on the sleeve is above the “zero” line on the barrel) this correspondsto a reading of -0.5 mm; the vernier reads 0.496 mm the zero error is then ε0 =−0.5 + 0.496 = −0.004mm = −4 µ.

APPENDICES 97

B. PARALLAX and Notes on using a Telescope

1. PARALLAX:

To do quantitative work in optics one must understand parallax and how it may beeliminated. PARALLAX is defined as apparent motion of an object caused by actualmotion of the observer.

As the observer moves left and right, object 1appears to move to the left and right of object2. The amount by which object 1 appears tomove is proportional to the distance betweenobject 1 and object 2. If object 1 comes infront of object 2, the direction of its apparentmotion reverses.

eye

2

1

Try this with two fingers.

Note that if object 1 is an image and object 2 a cross hair, the absence of parallaxshows that the cross hairs are in the plane of the image.

2. Focusing a Telescope for Parallel Rays:

O = objective lens

E= eye piece. It

may be a single

C=cross-hairs

eye

lens or several

E

lens.

C

TB

O

The eyepiece E slides back and forth in the tube T and one should first adjust theeyepiece to give a clear image of the cross hairs. Then move the tube T back andforth in the barrel B until the image of a distant object, formed by the objectiveO, falls on the plane of the cross hairs. The test for this is the absence of parallaxbetween the cross hairs and image.

The rays from a distant object are nearly parallel. For viewing a distant object, usean open window if the window glass is not accurately plane. Otherwise poor imageformation may result. You can check by trying it both ways. At night use a distantobject in the hallway.

The telescope, now focused for parallel rays, will stay so as long as the distancebetween O and C is unchanged. One may still adjust the eyepiece position to suitthe observer.

3. Finding an Image in a Telescope:

If you have trouble finding an image in a telescope, locate the image first with yourunaided eye, and then pull the telescope in front of your eye, aligning it with yourline of sight. With high magnification it is difficult to find the image in the telescopebecause the alignment must be nearly perfect before the image appears in the fieldof view. The eye has a rather large field of view so that the image will be visibleover a range of positions.

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