Physics 202 Lab Manual Electricity and Magnetism, Sound/Waves, Light

Transcription

1 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 42 E-5a Lenz s Law E-5b Induction - Dropping Magnet E-5c Induction - Fields in Space E-5d Induction - Circuit Elements 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

2 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 93 A. Precision Measurement Devices B. PARALLAX and Notes on using a Telescope NOTE: E=Electricity and Magnetism, S=Sound and Waves, L-Light

3 FORWARD 3 Forward Spring, 2005 This version is only modestly changed from the previous versions. We are gradually revising the manual to improve the clarity and interest of the activities. In particular the dynamic nature of web materials and the change of venue (from Sterling to Chamberlin Hall) has required a number of cosmetic and operational changes. In particular the PASCO computer interface and software have been upgraded from Scientific Workshop to DataStudio. M.J. Winokur In reference to the 1997 edition Much has changed since the implementation of the first edition and a major overhaul was very much in need. In particular, the rapid introduction of the computer into the educational arena has drastically and irreversibly changed the way in which information is acquired, analyzed and recorded. To reflect these changes in the introductory laboratory we have endeavored to create a educational tool which utilizes this technology; hopefully while 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 as a fully integrated web document so that the manual itself has become an interactive tool in the laboratory environment. As always we are indebted to the hard work and efforts by Joe Sylvester to maintain the labortory equipment in excellent working condition. M.J. Winokur M. Thompson From the original edition The experiments in this manual evolved from many years of use at the University of Wisconsin. Past manuals have included cookbooks with directions so complete and detailed that you can perform an experiment without knowing what you are doing or why, and manuals in which theory is so complete that no reference to text or lecture was necessary. This manual avoids the cookbook approach and assumes that correlation between lecture and lab is sufficiently close that explanations (and theory) can be brief: in many cases merely a list of suggestions and precautions. Generally you will need at least an elementary understanding of the material in order to perform the experiment expeditiously and well. We hope that by the time you have completed an experiment, your understanding will have deepened in a manner not achievable by reading books or by working paper problems. 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 have the apparatus out on a table and take a few minutes at the start to name the pieces and give suggestions for use. Also in this way changes in equipment, (sometimes necessary), need not cause confusion.

4 FORWARD 4 Many faculty members have contributed to this manual. Professors Barschall, Blanchard, Camerini, Erwin, Haeberli, Miller, Olsson, Visiting Professor Wickliffe and former Professor Moran have been especially helpful. However, any deficiencies or errors are our responsibility. We welcome suggestions for improvements. Our lab support staff, Joe Sylvester and Harley Nelson (now retired), have made important 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, but also have contributed excellent original circuits and component design for many of the experiments. R. Rollefson H. T. Richards

5 INTRODUCTION 5 Introduction General Instructions and Helpful Hints Physics is an experimental science. In this laboratory, we hope you gain a realistic feeling for the experimental origins, and limitations, of physical concepts; an awareness of experimental errors, of ways to minimize them and how to estimate the reliability of the result in an experiment; and an appreciation of the need for keeping clear and accurate records of experimental investigations. Maintaining a clearly written laboratory notebook is crucial. This lab notebook, at a minimum, should contain the following: 1. Heading of the Experiment: Copy from the manual the number and nameof the experiment. 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 notebook as they are gathered. Original data are the actual readings you have taken. All partners should record all data, so that in case of doubt, the partners lab notebooks can 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 the notebook record after the passage of time. 3. Housekeeping deletions: You may think that a notebook combining all work would soon become quite a mess and have a proliferation of erroneous and superseded material. Indeed it might, but you can improve matters greatly with a little housekeeping work every hour or so. Just draw a box around any erroneous or unnecessary material and hatch three or four parallel diagonal lines across this box. (This way you can come back and rescue the deleted calculations later if you should discover that the first idea was right after all. It occasionally happens.) Append a note to the 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 notebook home to write it up you probably have more important things to do than making a beautiful notebook. (Instructors may permit occasional exceptions if they are satisfied that you have a good enough reason.) 4. Remarks and sketches: When possible, make simple, diagrammatic sketches (rather than pictorial sketches of apparatus. A phrase or sentence introducing each calculation is essential for making sense out of the notebook record after the passage of time. When a useful result occurs at any stage, describe it with at least a word or 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 transparent tape. C. Mark out and plot a simple graph directly in your notebook.

6 INTRODUCTION 6 Show points as dots, circles, or crosses, i.e.,,, or. Instead of connecting points by straight lines, draw a smooth curve which may actually miss most of the points but which shows the functional relationship between the plotted quantities. Fasten directly into the notebook any original data in graphic form (such as the spark tapes of Experiment M4). 6. Units, coordinate labels: Physical quantities always require a number and a dimensional unit to have meaning. Likewise, graphs have abscissas and ordinates which always need labeling. 7. Final data, results and conclusions: At the end of an experiment some written comments and a neat summary of data and results will make your notebook more meaningful to both you and your instructor. The conclusions must be faithful to the data. It is often helpful to formulate conclusion using phrases such as the discrepancy between our measurements and the theoretical prediction was larger than the uncertainty in our measurements. PARTNERS Discussing your work with someone as you go along is often stimulating and of educational value. If possible all partners should perform completely independent calculations. Mistakes in calculation are inevitable, and the more complete the independence of the calculations, the better is the check against these mistakes. Poor results on experiments sometimes arise from computational errors. CHOICE OF NOTEBOOK We recommend a large bound or spiral notebook with paper of good enough quality to stand occasional erasures (needed most commonly in improving pencil sketches or graphs). To correct a wrong number always cross it out instead of erasing: thus ////// since occasionally the correction turns out to be a mistake, and the original number was 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 horizontal lines needed for plain writing. Put everything that you commit to paper right into your notebook. Avoid scribbling notes on loose paper; such scraps often get lost. A good plan is to write initially only on the right-hand pages, leaving the left page for afterthoughts and 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 miscellaneous discussions before you leave the laboratory. Your instructor will check each completed lab report and will usually write down some comments, suggestions or questions in your notebook. Your instructor can help deepen your understanding and feel for the subject. Feel free to talk over your work with him or her.

7 INTRODUCTION 7 Expt. M1 Systematic and Random Errors, Significant Figures, Density of a Solid NAME: Jane.Q. Student Date: 2/29/00 Partner: John Q. Student Purpose: To develop a basic understanding of systematic and random errors in a physical measurement by obtaining the density of metal cylinder. Equiment: Venier caliper, micrometer, precision gauge block, precision balance Theory: ρ= mass/( π r**2 h) ρ= ( m /m)**2.+( h/h)**2.+( 2 r/r)**2. ρ DATA: 1. Calibration of micrometer Reading with jaws fully closed: mm mm mm mm mm mm Micrometer exhibits a systematic zero offset Ave, - + Standard Deviation Measure four calibraton gauge blocks Plot of micrometer error vs. gauge block length Micrometer error (mm) Measure of cylinder diameter: Measure of cylinder height: Measure of cylinder mass CALCULATIONS: Gauge block length (mm) Density=??? Uncertainty from propagation of error. RESULTS and CONCLUSIONS:

8 ERRORS AND UNCERTAINTIES 8 Errors and Uncertainties Reliability estimates of measurements greatly enhance their value. Thus, saying that the average diameter of a cylinder is 10.00±0.02 mm tells much more than the statement that the cylinder is a centimeter in diameter. The reliability of a single measurement (such as the diameter of a cylinder) depends on many factors: FIRST, are actual variations of the quantity being measured, e.g. the diameter of a cylinder may actually be different in different places. You must then specify where the measurement 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 selected places. Then the scatter of the measurements will give a first estimate of the reliability of the average diameter. SECOND, the micrometer caliper used may itself be in error. The errors thus introduced will of course not lie equally on both sides of the true value so that averaging a large number of readings is no help. To eliminate (or at least reduce) such errors, we calibrate the 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 of dimensions approximately equal to those of the cylinder to be measured. We call such errors systematic, and these cause errors in accuracy. THIRD, Another type of systematic error can occur in the measurement of a cylinder: The micrometer will always measure the largest diameter between its jaws; hence if there are small bumps or depressions on the cylinder, the average of a large number of measurements 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 cross section.) 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 may vary because one can t close it with the same force every time. Also the observer s estimate of the fraction of the smallest division varies from trial to trial. Hence the average of a number of these measurements should be closer to the true value than any one measurement. Also the deviations of the individual measurements from the average give an indication of the reliability of that average value. The typical value of this deviation is a measure of the precision. This average deviation has to be calculated from the absolute values of the deviations, since otherwise the fact that there are both positive and negative deviations means that they will cancel. If one finds the average of the absolute values of the deviations, this average deviation from the mean may serve as a measure of reliability. For example, let column 1 represent 10 readings of the diameter of a cylinder taken at one place so that variations in the cylinder do not come into consideration, then column 2 gives the magnitude (absolute) of each reading s deviation from the mean.

9 ERRORS AND UNCERTAINTIES 9 Measurements Deviation from Ave mm Diameter = ±0.001 mm Ave = mm Ave = mm mm Expressed algebraically, the average deviation from the mean is = ( x i x )/n), where x i is the i th measurement of n taken, and x is the mean or arithmetic average of the 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 square deviation). You calculate σ by evaluating σ = 1 n (x i x) n i=1 where x is the mean or arithmetical average of the set of n measurements and x i is the i th measurement. Because of the square, the standard deviation σ weights large deviations more heavily than the average deviation and thus gives a less optimistic estimate of the reliability. In fact, for subtle reasons involving degrees of freedom, σ is really σ = 1 n (x i x) (n 1) 2 σ tells you the typical deviation from the mean you will find for an individual measurement. The mean x itself should be more reliable. That is, if you did several sets of n measurements, the typical means from different sets will be closer to each other than the individual measurements within a set. In other words, the uncertainty in the mean should be less than σ. It turns out to reduce like 1/ n, and is called the standard deviation of the mean σ µ : i=1 σ µ = standard deviation of the mean = σ n = 1 n 1 n 1 n (x i x) 2 For an explanation of the (n 1) factor and a clear discussion of errors, see P.R. Bevington and 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, ɛ have a Gaussian distribution, e ɛ2, about zero), then on average 68% of a large number of measurements will lie closer i=1

10 ERRORS AND UNCERTAINTIES 10 than σ to the true value. While few measurement sets have precisely a normal distribution, the main differences tend to be in the tails of the distributions. If the set of trial measurements are generally bell shaped in the central regions, the normal approximation 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 the reader attempts to reproduce the results in the graph using the procedure described in the report, the reader should expect his or her results to have a 50% chance of falling with 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 ɛ be the error in a measurement whose value is a. Then ( ɛ a ) is the relative error of the measurement, and 100 ( ɛ a )% is the percentage error. These terms are useful in laboratory work.

11 ERRORS AND UNCERTAINTIES 11 UNCERTAINTY ESTIMATE FOR A RESULT INVOLVING MEASUREMENTS OF SEVERAL INDEPENDENT QUANTITIES Let R = f(x, y, z) be a result R which depends on measurements of three different quantities x, y, and z. The uncertainty R in R which results from an uncertainty x in 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 distribution. In this case the uncertainties add in quadrature (the square root of the sum of the squares): ( ) f 2 ( f y + Some examples: R R = x f x f y ) 2 + ( ) f 2 z f z. A.) R = x + y. If errors have a normal or Gaussian distribution and are independent, they combine in quadrature: R = x 2 + y 2. Note that if R = x y, then R/R can become very large if x is nearly equal to y. Hence avoid, if possible, designing an experiment where one measures two large quantities and takes their difference to obtain the desired quantity. B.) R = xy. Again, if the measurement errors are independent and have a Gaussian distribution, 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 πr 2 L where m = mass, r = radius, L = length, are the three measured quantities and ρ = density. Hence ρ m = 1 πr 2 L ρ r = 2m πr 3 L ρ L = Again if the errors have normal distribution, then ( m ) ρ 2 ( ρ = + 2 r ) 2 ( ) L 2 + m r L m πr 2 L 2.

12 ERRORS AND UNCERTAINTIES 12 SIGNIFICANT FIGURES Suppose you have measured the diameter of a circular disc and wish to compute its area A = πd 2 /4 = πr 2. Let the average value of the diameter be ± mm ; dividing d by 2 to get r we obtain ± mm with a relative error r r of = Squaring r (using a calculator) we have r 2 = , with a relative error 2 r/r = , or an absolute error in r 2 of = Thus we can write r 2 = ± 0.04, any additional places in r 2 being unreliable. Hence for this example the first five figures are called significant. Now in computing the area A = πr 2 how many digits of π must be used? A pocket calculator with π = gives A = πr 2 = π ( ± 0.04) = ± 0.11 mm 2 Note that A A = 2 r r = Note also that the same answer results from π = , but that π = gives A = ± 0.11 mm 2 which differs from the correct value by 0.06 mm 2, an amount comparable to the estimated uncertainty. A good rule is to use one more digit in constants than is available in your measurements, and to save one more digit in computations than the number of significant figures in 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 correct number of significant figures. SAMPLE QUESTIONS 1. How many significant figures are there in the following numbers? (a) (b) (c) Round off each of the following numbers to three significant figures. (a) (b) (c) A function has the relationship Z(A, B) = A + B 3 where A and B are found to have uncertainties of ± A and ± B respectively. Find Z in term of A, B and the respective uncertainties assuming the errors are uncorrelated. 4. What happens to σ, the standard deviation, as you make more and more measurements? 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 13 Part I Electricity and Magnetism E-1 Electrostatics OBJECTIVES: To investigate charging by rubbing (triboelectricity); charging by conduction; and charging 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 repel opposite 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 conductor negative charge flows from conductor to ground no 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 the car, you are to some degree protected against injury due to lightning strike. This protection 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 have already 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); test sphere; silk cloth; ebonite rod (black); rabbit fur; acrylic rod (clear). When any two objects rub against each other, electrons move from one object to another. Depending on the nature of the objects, the number of electrons making the move may be very small or very large. One object ends up with too many electrons, and the other with not enough.

14 E-1 ELECTROSTATICS 14 Figure 1: Setup for Expt As shown in the picture, connect the SIGNAL INPUT of the electrometer to the Faraday Cup, and connect the GROUND port of the electrometer to the ground input of the Pasco interface. Make sure the Pasco interface is plugged in. If you insert an object with too many electrons into the Faraday Cup WITHOUT TOUCHING THE SIDES OR TOP, the electrometer will read negative. Insert an object with too few electrons, and it will read positive. Insert an object that is neutral, and the electrometer reading will stay close to zero. 2. Turn ON the electrometer. Set the electrometer to VOLTS FULL SCALE RANGE of 100. Push the ZERO button to remove any charge on the electrometer. The digits display and meter display should both indicate 0. Call over your TA if they don t. Press the ZERO button any time the electrometer reads something other than 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 polish the 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 indicate how 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 in step 5. What did you do that caused the test sphere to become neutral, and what happened to the extra charge?

15 E-1 ELECTROSTATICS Insert the acrylic rod into the Faraday cage. If the electrometer shows charge is present, clean the acrylic rod with soap and water, and verify it is uncharged. If it is still charged, call over your TA. 8. Repeat step 7, using the ebonite rod. 9. Rub the acrylic rod with the silk. What charge does the acrylic rod acquire? (circle one) positive negative 10. Rub the ebonite rod with the rabbit fur. What charge does the ebonite rod acquire? (circle one) positive negative 11. Are your results from steps 9 and 10 in accordance with the Triboelectric Series (see table at right)? 12. You should be able to find at least one item that you brought with you to lab on the Triboelectric Series. Touch the test sphere to the outside of the Faraday Cup, rub the item you brought against the test sphere (which is made of brass), and insert the test sphere into the Faraday Cup to see what type of charge it has acquired. Item you brought: Test sphere charges: positive negative Is this in accordance with the Triboelectric Series? Table I: The Triboelectric Series Various substances ranked from greatest tendency to charge positive when rubbed (hair) to greatest tendency to charge negative when rubbed (teflon) Hair (most positive) Rabbit fur Acrylic (clear plastic) Hands, skin Glass Nylon Wool Quartz Cat Fur Lead Silk Aluminum Paper Cotton (neutral) Steel Wood Amber Ebonite (looks like black plastic) Nickel, copper Brass, silver Gold, platinum Sulfur Rayon Polyester Styrofoam Orlon Saran Polyurethane Polyethylene PVC Teflon (most negative)

16 E-1 ELECTROSTATICS 16 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 does not directly tell you the sign of the charge. The leaves rise when there is charge on the leaves, 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 make sure they are not risen. If they are risen, touch the knob with your finger, and they should 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 the electroscope. The leaves should rise (if they don t, call over your TA). According to the Triboelectric Series table, the acrylic rod should charge positive. Assuming this is true, draw little + signs in the pictures below to indicate where the electric charges are during this process.

17 E-1 ELECTROSTATICS Plug a cable into the ground input of the Pasco interface, and touch the other end of 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 charges on 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 the pictures 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 at the 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?

18 E-1 ELECTROSTATICS 18 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 the electroscope. 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 knob of 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 by looking at the electroscope, the results of steps 1 to 3 allow you to determine that the ebonite rod and the acrylic rod must have charges of opposite sign. Touch the knob of the electroscope with your finger. This should cause all excess charge to depart the electroscope, and the leaves to drop: you have discharged the object. This concludes the first experiment on charging by induction.

19 E-1 ELECTROSTATICS 19 5 Rub the acrylic rod with the silk, and bring the acrylic rod near, but not touching the knob on the electroscope. What happens to the leaves? 6. Assuming the acrylic rod is charged positive, draw little + and - signs on the cartoons to show where the charges are. 7. The charge on the electroscope is defined as the charge on the leaves plus the charge on the knob. Assuming you did not touch the acrylic rod to the knob, what was the charge on the electroscope in the middle cartoon above? 8. Plug a cable into the ground input of the Pasco interface, and repeat step 5, with one difference: while the acrylic rod is near the knob, touch the knob with your end of the cable, and then remove the cable. Only when the cable is gone should you remove the acrylic rod. Sketch what the charges do. 9. What is the sign of the charge that remains on the electroscope? This shows how an object can become charged by induction, and concludes the second experiment on charging by induction.

20 E-1 ELECTROSTATICS 20 For the third experiment on charging by induction we ll return to the apparatus shown in the 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 other objects 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 RANGE of 100. Push the ZERO button to remove any charge on the electrometer. The digits display and meter display should both indicate 0. Call over your TA if they don 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 the instructions 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. The electrometer reads (circle one): positive negative zero 14. TS touches test sphere to outside of Faraday Cup, and then inserts test sphere into Faraday 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 handle of the test sphere. Clean the handle with soap and water and repeat step 14. If the electrometer still doesn?t read zero, call over your TA. 16. ER brings ebonite rod near BUT NOT TOUCHING right side of the big metal sphere, 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 the test 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 big metal 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 as possible 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 the test sphere into the Faraday Cup. The electrometer reads: positive negative zero

21 E-1 ELECTROSTATICS Based on the results of steps 10-23, indicate on the picture below the charge distributions on the ebonite rod and the big metal sphere. This experiment shows how the charge distribution on an object can be measured, one point at a time. It s important that the test sphere be much smaller than the object of which the charge distribution is measured, because every time the test sphere is touched to 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 be measured is not a good measurement. PART III: CONCLUSION QUESTIONS 1. Why does touching the electroscope with your hand ground it? Do you end up with a net charge? If so, how can we see the effects of this? 2. Can the electroscope differentiate between positive and negative charges? Explain based on your observations. 3. Do only insulators acquire charge by rubbing? Explain. 4. Electrostatics experiments often do not work well in humid weather. Explain why this might be so.

22 E-1 ELECTROSTATICS In very dry or cold weather, objects such as clothing, table tops, etc. that somehow acquire a charge tend to hold that charge for a long time, i.e. they are slow to discharge. 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 influence nearby unshielded instruments. If so, ask your instructor for help; e.g. use grounded foil to shield against them. Also remove any clinging loose bits of fiber (e.g. silk or rabbit fur) that may disturb results. 2. The fragile leaves of the electroscope may tear if charged too heavily. Do not disassemble electroscope to attempt repair: see your instructor. 3. To remove charges on the glass windows of the electroscope, lightly rub your hands over the windows while grounding your body. 4. When instructed to touch the hollow sphere with the proofball, avoid rubbing which may result in inaccurate results. (Remember that rubbing is how you created the charge separation on the lucite or rubber rods. Here, we want to charge by conduction.) 5. In humid weather or if there is excess charge on the handle, your proofball may not hold charge. If this happens you can try cleaning or heating the handle (as described in 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 touches the electroscope ground), and then bring parts of the insulating handle close to the electroscope knob. If the leaves move, the handle is charged. To discharge it, hold the handle in a source of ionized air. The charged insulator will attract ions of the opposite sign until it is neutral. An open flame is a simple source of both positive and negative ions. The heated air convects these ions upward such that the handle will attract the correct sign to make it neutral. Hold the insulator at least 10 cm above open flame to avoid heat damage to the handle. 7. Avoid unnecessary handling of insulators because handling may impair their insulating 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 if you move around. Hence, during a given measurement, change position as little as possible and ground yourself. 9. To remove all charge from the cup, press the Zero button. (This connects the electrometer terminals to each other so that any charge flows from/to ground). If the meter does not read within a few percent of zero, notify your instructor.

23 E-1 ELECTROSTATICS Cleaning your paddles before beginning may produce better results. Dirt on the paddles will affect how much charge is transferred (and how actually the same two paddles of the same type are). 11. Always discharge paddles and cup before starting an experiment. To test if an object is charged, put it into the cup and see whether the electrometer deflects. Conductors discharge easily by touching them to a grounded conductor. To discharge an insulator, you must create sufficient ions in the surrounding air. The insulator will then attract ions of the opposite charge until all charge is neutralized. An open flame is a simple source of ionized air; the ions in the flame convect upward with the hot gas. To avoid damage to the insulator, keep it at least 10 cm above the flame! 12. You may measure charge and still avoid spurious effects from charges on the insulating handles if you will touch the charged proofball (or paddle) to the bottom of the cup and then remove it from the cup before reading the electrometer. But remember to discharge the cup (by momentarily grounding) before taking the next reading! However, if the potential of the insulating handle is too large (e.g. way off the least sensitive scale), one can still get spurious effects from leakages.

24 E-2 ELECTRIC FIELDS 24 E-2 Electric Fields OBJECTIVES: To develop an intuitive understanding of relationship between electric fields and equipotential 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 a point charge. What do you expect them to look like for a dipole? A conducting plate? APPARATUS: Power supply (18V, 3A Max); Fluke 115 Digital Multimeter (DMM) & test probes; conductive paper with various conducting electrode configurations; field plotting board; carbon paper; white paper; red and black banana cables Fluke DMM: To use the DMM, choose the DC voltage indicator,, from the dial of measurements. Place one probe on the electrode that is grounded and place the other at a different location on the conductive paper. The DMM will now read the difference in electric potential between those two locations. Pressing harder will not improve your measurement! Note that although the DMM will measure the potential difference between arbitrary points, one probe should always be referenced to ground for the purposes of this lab. EXPERIMENTS: 1. Place white paper on the field plotting board, then a piece of carbon paper. On top of 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 hard pencil or a ball point pen. 3. Using the power supply, place 18 volts across the terminals of the plotting board (as in Fig. 1). Check the quality of the electrodes you are using with the DMM to make sure there are no appreciable potential differences between various locations within the electrodes. If you find any, ask your TA to find you a new one or fix the silver paint quality. 4. Map the equipotential line which corresponds to +3 volts by exploring the region of the grounded electrode with one probe while keeping your other probe on your reference. Make several dark marks where you find a potential of +3V, and make those recorded points are close enough together that you can later draw a smooth line through the points. Note: the points only need to be close together when the direction of the equipotential changes rapidly.

25 E-2 ELECTRIC FIELDS 25 Figure 1: Experimental Setup 5. Map another 8 other equipotential lines. Chose voltages that are spread out evenly to 18V. Remove the white paper when completed. 6. On the white paper, connect the dots representing individual equipotentials with smooth lines. 7. Repeat steps 1-7 for the two configurations shown in Fig. 2. In each case map about ten equipotentials. A B A B 0 Volts 18 Volts 0 Volts 18 Volts (a) (b) Figure 2: Electrode configurations for (a) parallel plates and (b) a lightning rod.

26 E-2 ELECTRIC FIELDS 26 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 electric field strength in the region where the field is largest. This can be done by picking adjacent 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: b V b V a = E d l a and note that V b = V a for any points b and a on the equipotential.) 4. Using your answer in the previous questions, draw about 5 electric field lines on each of 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 a blunt surface. 7. True or false: if a conducting object becomes charged, the charges tend to congregate on the pointy parts of the object. 8. It is clear that lightning rods protect buildings against damage from lightning strikes. 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 readily leak off, thus preventing the building from becoming charged, and attracting lightning; 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 does strike, 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?

27 E-3 CAPACITANCE 27 E-3 Capacitance OBJECTIVES: To examine very carefully the movement of charges in a capacitor; to observe and account for the effects of stray capacitance. EXPERIMENT I: DISCHARGING A CAPACITOR: APPARATUS: Power supply; electrometer; plug-board and capacitors; cables with banana-plug connectors; cable with BNC connectors; BNC female to dual banana-plug adapter (see Fig. 1). Figure 1: BNC female to dual banana-plug adapter. Note the tab on right labelled GND. This indicates which side is electrically connected to the outer conductor of the BNC connector, which is often grounded. 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 these terminals to a pipe driven into the dirt beneath Chamberlin Hall, via the internal circuitry of the electrometer, and the wires in the walls of Chamberlin Hall. These terminals therefore establish a well-defined 0 V, or ground, reference electric potential.

28 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 can be represented schematically as in Fig. 3. Figure 3: Schematic of initial configuration of circuit in Expt. I.

29 E-3 CAPACITANCE In this lab, the only charge carriers are electrons. In order for the charge on an object to change, there must be a conducting path connecting it to some other object. In the above diagram, note that the region in between the two capacitors (it looks like a sideways H ) is not connected to anything. The total charge on this region must therefore 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 in Fig. 4. We have not yet calculated the magnitude of Q 1, but the above two principles require 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 drop across both capacitors, which is the same as the voltage supplied by the power supply. 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 voltage across each capacitor (one at a time). Each voltage should be within a volt of your prediction.

30 E-3 CAPACITANCE The following three steps constitute one attempt to remove charge from the capacitors. Measure the voltage across the capacitor with the electrometer, and record it on the left-hand side of the diagram. Then deduce the charge on each plate of the capacitor (+Q 1, Q 1, +Q 2, Q 2, or 0). Do not make any calculations, but instead use logic alone to make the deduction. Is either capacitor discharged? 11. Restore the original configuration. The following three steps constitute another attempt to remove charge from the capacitors. Is either capacitor discharged? 12. Restore the original configuration. The following four steps constitute one more attempt to remove charge from the capacitors. Is either capacitor discharged?

31 E-3 CAPACITANCE 31 EXPERIMENT II: STRAY CAPACITANCE APPARATUS: Pasco Interface; power supply; electrometer; large parallel-plate capacitor with moveable plate; low-capacitance lead; two brass barrels. (see Fig. 5). Figure 5: Experimental Setup Information about the parallel-plate capacitor: plate diameter D = ± 0.05 cm; an indicated distance of 0 mm corresponds to an actual separation of about 1 mm. Insulating pads are mounted on one of the capacitor plates, so that this is the minimum possible plate separation.

32 E-3 CAPACITANCE 32 THEORY I: Consider a circuit composed of a voltage source and a capacitor, as shown in Fig. 6. Figure 6: Ideal parallel-plate capacitor in circuit. The capacitance of an ideal parallel-plate capacitor is C = ɛ 0 A/d, where A is the plate area and d is the plate spacing. Suppose the plate spacing is set to d, the voltage across the capacitor is set to V, and then all external connections are removed. The capacitor charge Q will henceforth be constant. Since C depends on d, if d is changed, C and therefore V will change: V = Q/C = Qd/ɛ 0 A. The voltage should increase linearly with plate 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 parallelplate capacitor using one of the brass barrels. 4. Connect the center conductor of the electrometer input to the moving plate using the 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 electrometer. Turn on the interface. Your setup should now look like Fig 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 on the moving plate assembly. 9. Move around without touching the capacitor or disturbing the plates. Is there anything you can do that influences the electrometer reading? 10. Charge the capacitor by momentarily touching the positive lead from the power supply to them moveable plate. The electrometer reading should go to 9V and stay there as long as the positive lead is touching the plate.

33 E-3 CAPACITANCE 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; if it did, it would be over 200V at the maximum plate separation. THEORY II Apparently our parallel-plate capacitor is not ideal. The problem is that the charge we put on the moving plate affects the charges on all nearby objects, such as the table. This is an example of stray capacitance, in which charge put on one object changes the potential of all nearby objects (not just the ones we intend), and can make circuits behave in unpredicted ways. Stray capacitance is modeled as a capacitance in parallel with the physical capacitor, as in Fig. 7. Call the stray capacitance C S. Figure 7: Model of real capacitor including the effect of stray capacitance C s. The relation between the charge Q on the parallel-plate capacitor and the voltage V applied to the parallel plate capacitor is then: Q = (C + C s )V, (1) where C = ɛ 0 A/d, the parallel-plate capacitance, changes with plate spacing d, and C s, the stray capacitance, does not. This can be rearranged to solve explicitly for V : V = (Q/C s) d (ɛ 0 A)/C s + d, (2) where d has been separated out because it is changing. This in turn can be rewritten V = A d B + d, (3) where A = Q/C s is the value taken by the voltage when the plate spacing goes to infinity, and B = ɛ 0 A/C s is the value of the plate separation for which the stray capacitance equals the capacitance of the parallel plates. We will now attempt to acquire data to test this model. QUESTIONS: Q1: In step 10 above, when the plate spacing became large, what was the maximum voltage that appeared across the capacitor? Q2: Suppose the stray capacitance C s equals the parallel-plate capacitance C when d = 1 cm. Sketch the capacitor voltage V, as predicted by Eq. 2, as a function of plate separation d, indicating on the graph where d = 1 cm.

34 E-3 CAPACITANCE 34 Q3: Indicate the part of the sketch for which the ideal model of the parallel-plate capacitor is valid (accurately predicts the data). Q4: Indicate the part of the sketch for which the ideal model is invalid (fails to predict the data). Q5: Verify (if you haven t already) that movement of your body can affect the electrometer 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. The data you are about to take should end up looking like the Fig. 8 above. Dexterity is required 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 accurate data 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 this corresponds 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 the capacitor 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 in this position for about five seconds. 7. Return the plate spacing to d = 0.6 cm.

35 E-3 CAPACITANCE Hold the positive lead of the power supply in contact with the moving plate of the capacitor 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 measurements 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 be carried 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 graph that give the clearest evidence for charge leakage. Q7: When trying to verify Eq. 2, is it better to record the voltage assumed immediately upon the plates reaching their final spacing, or to record the voltage some time after the plate have reached their final spacing? 1. From your two datasets, fill in the table in the Pasco setup file of voltage as a function 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 = A d B+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 capacitance. Suppose the stray capacitance in this circuit were cut in half. What would happen to A and B?

36 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 magnetic field; to measure e/m, the electron charge/mass ratio; to estimate the strength of the 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 a thin uniform electron beam. Refer to Figs. 1 & 2 for details. Passing current through a wire filament (F) causes it to become hot. If the filament becomes hot enough, electrons spontaneously leave the surface (a process called thermionic emission). The filament is in close proximity to a higher-potential electrical element (the cylindrical anode C), and so some of the electrons accelerate through the vacuum towards the anode. 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 also contains Mercury (Hg) vapor. When electrons with energies in excess of 10.4 ev collide with Hg atoms, some atoms become ionized or excited, and then quickly recombine and/or de-excite to emit a bluish light. Hence the bluish light marks the path taken by the electron beam (the electrons which collide with atoms are permanently lost from the beam). The potential difference V between the filament and anode accelerates electrons thermionically emitted by the filament. Those electrons traveling toward the slit S emerge with a velocity v given by V e = 1 2 mv2, (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. beam would 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. radius of electron trajectory would get bigger. If the tube is properly oriented, the velocity of the emerging electrons is perpendicular to the magnetic field. Hence the magnetic force vector F B = 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)

37 E-4 ELECTRON CHARGE TO MASS RATIO 37 A C + B F S F C S B A is a hot filament emitting electrons is a cylindrical anode at potential +V with respect to F is a vertical slit thru which electrons emerge is the electron path in the magnetic field is a rod with cross bars at known distances from the filament Electron beam trajectory in presence of Earth s magnetic field Figure 1: The e/m tube viewed along the earth s magnetic field (i.e., B earth is perpendicular to the page). A is a rod with cross bars at known distances from the filament L, L 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 L D C L A filament current out Figure 2: Side view of Figure 1. S F G E G filament current in Eliminating v between (1) and (2) gives e/m = 2V B 2 r 2. (3) The accelerating voltage V is shown on the ANODE Volts display of the power supply. The radius of curvature, r = D/2, is half the distance between the filament F 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 meter m meter m meter m meter m meter m To find e/m we still need to know the magnetic field strength B. Instead of measur-

38 E-4 ELECTRON CHARGE TO MASS RATIO 38 ing B, we calculate it from the dimensions of the Helmholtz coils and the measured coil current, I, needed to bend the beam so that it hits a given cross bar. A Helmholtz coil consist of two identical coaxial coils which are separated by a distance R equal to the radius of either coil, as in Fig. 4. They are useful because near the center (X = R/2) the field is nearly uniform over a large volume. (Many textbooks assign this proof as a problem. Hint: Find how db/dx changes with X for a single coil at X = R/2, etc.) R R X 2 +R 2 X R Figure 3: Helmholtz coil geometry. The field at X = R/2 is in fact: 1 B = 8 ( ) Nµ0 I 125 R (4) as one sees by adding the axial fields B(x) from each of the single coils: Thus when X = R/2, then In Eq. 4, B = 2B(X) = B(X) = Nµ 0 IR 2 2 [R 2 + X 2 ] 3/2. Nµ 0 I 2 R 2 [R 2 + R 2 /4] 3/2 = Nµ 0I R [ ] 5 3/2 = 4 ( 8 Nµ0 I 125 R N = number of turns on each coil (72 for these coils) I = current through each coil in amperes R = mean radius of the coils in meters (approximately 0.33 m but varies slightly from unit to unit) µ 0 = 4π 10 7 tesla meter/ampere. Finally if we substitute (4) into (3) we obtain ). e/m = ( R2 N 2 ) V I 2 r 2 coulombs/kg. (5) CIRCUIT DIAGRAM: Fig. 4 should help you hook up the components. The current has the same direction in both coils of a Helmholtz coil. 1 The actual B at the maximum electron orbit is only 0.5% less. See Price, Electron trajectory in an e/m experiment, Am. J. Phys. 55, 18, (1987)

39 E-4 ELECTRON CHARGE TO MASS RATIO Helmholtz Coils To Power Supply 2 Amps ANODE Volts ANODE milliamps FIELD Amps 44 V. Filament On 22.V Overload Power ANODE Filament E/M POWER SUPPLY GND POS Field GND POS Filament Anode Cylinder with slit To Helmholtz Coils Figure 4: Basic wiring layout. SUGGESTED PROCEDURE: 1. Determine the local direction of the Earth s magnetic field using the compass and dip-needle compass. You should find that it is in the same plane as geographic north but at an angle about 60 degrees from the horizontal. Thus the dip-needle compass should point deep under Canada. Set the axis of the Helmholtz coils along this direction. Rotate the e/m tube about its long axis until the cross bars are facing up. CAUTION: Nearby ferromagnetic material (e.g. steel in the power supply, the table and in the walls) can alter the local direction of B e. 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/I 2, one needs high quality meters for V and I. Our digital meters have an accuracy of ± one digit in the last displayed digit. 4. Start with 22 V between filament and anode. The anode current is displayed on the ANODE milliamps display. With the room dark, gradually turn up the filament control until the beam is visible. The anode current should remain zero until the filament is sufficiently hot, perhaps hot enough that you can see the glowing filament. The filament dial will often be well beyond the halfway position before the anode current departs from zero. You should not expect to see the beam until the anode current is at least a few milliamps. To make the beam more visible, use black cloth and black cardboard to block out stray light: place the black cardboard inside the Helmholtz coils and view from the top. When the beam appears, adjust the 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 should slightly curve the beam. 5. Turn up the Helmholtz coil current until the electron beam is steered into cross bar 1 (the cross bar closest to the filament).

40 E-4 ELECTRON CHARGE TO MASS RATIO Predict what will happen to the beam if the anode voltage is increased to 44 V. Will the 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 necessary to steer the beam into cross bar n. Record I n for each of the five cross bars in the table below. Keep in mind that the electrons are responding to the sum of the Helmholtz coil field and the Earth s field. n r (m) I n(a) I n(a) I n (A) I e (A) e/m B e (T) 9. Rotate the vacuum tube about its long axis by 180. Repeat step 8, using I n to denote the current needed to steer the electrons into the cross bars. CAUTION: Always rotate the tube in the sense to REDUCE the TWIST in the filament and anode leads. Otherwise one can twist the 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 more or less towards the ceiling? 12. Let I n denote the current that would be needed to steer the electrons into post n, if the Earth s field were not present. How is I n calculated from your measurements I n and I n? 13. Let I e denote the current necessary to counteract the Earth s field. How is it calculated from I n and I n? 14. Fill in the columns for I n and I e in the table. 15. Compute the electron s e/m from your data, estimate its uncertainty, and compare it to the accepted value of C/kg. Is the discrepancy between your value and the accepted value larger than or smaller than the uncertainty in your value? 16. Compute the local magnitude of the Earth s magnetic field, B e, from your data, estimate its uncertainty, and compare it to the global average of T. Is the discrepancy between your value and this value larger or smaller than the uncertainty in your value? 17. The local magnitude of the Earth s magnetic field varies from point to point, and at one 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

41 E-4 ELECTRON CHARGE TO MASS RATIO 41 currently accepted value of the strength of the Earth s magnetic field in the vicinity of Madison, WI. One last observation. Note that the electron beam expands in a fan shape after leaving the slit. Interchange the filament leads and note how the fan shape flips to the other side of the beam. Explanation: The fan shape deflection arises from the filament current s magnetic field acting on the beam. Our filament supply purposely uses unfiltered half wave rectification of 60 Hz AC so that for half of the cycle no current flows thru the filament but which is still hot enough to emit sufficient electrons. The electrons emitted in this half cycle constitute the sharp beam edge which we use for measurement purposes. This trick not only avoids deflection effects for this part of the beam, but also avoids the uncertainty in electron energy arising from electrons being emitted from points of different potential along the filament. (During this half cycle of no filament current the filament is an equipotential.) The fan shape deflection relates to electrons emitted during the other half cycle when filament current flows. [See: F.C. Peterson, Am. J. Phys. 51, 320, (1983)]

42 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 a stationary magnet; to explore how the force due to eddy currents depends on the shape of the conductor. YOU NEED TO KNOW: the Lorentz force law; the right-hand rule that determines the direction of the force on a charge moving through a magnetic field. APPARATUS: A permanent magnet; four different pendulums; compass. Z X Figure 1: The magnet and the pendulums. PROCEDURE I: (10 min) 1. Use the compass to determine the direction of the magnetic field between the jaws of the permanent magnet. 2. Set the jaws to be about 1 cm apart, and observe the behavior of each of the four pendulums when they swing through the jaws. Which is affected most? Which least? 3. Reverse the magnet. Observe whether any of the pendulums behave differently when they swing through the jaws. 4. Set pendulum #1 between the jaws of the magnet. Move the magnet rapidly in a direction parallel to the pendulum plate (i.e. without touching jaws). Describe what happens. A Figure 2: Pendulum moves in the +y direction z B y

43 E-5 MAGNETISM 43 QUESTIONS: Refer to Fig. 2. As the leading edge of the pendulum enters the region of the magnetic field: Q1: What is the direction of the force on the electrons in the leading edge? Is it from A to 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 the region 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 bar magnet. INTRODUCTION: In E-5a it was shown that a conductor moving through the magnetic field of a stationary magnet may experience a force. Here we show that a magnet moving in the vicinity of a stationary conductor induces an emf in the conductor. PRECAUTIONS: Figure 3: A moving magnet - magnetic field lines move across a wire YOU NEED TO KNOW: Faraday s Law; Lenz s Law; the right hand rule that determines the direction of the current induced in a conductor by a magnetic field that is changing with time. 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 TOGETHER WITH YOUR HANDS!

44 E-5 MAGNETISM 44 PROCEDURE: 1. Connect the red wire of the voltage sensor to the top of the coil, and the black wire of the voltage sensor to the bottom of the coil. Then connect the other end of the voltage 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 the desktop. 3. Set the distance d 1 15 cm between the top of the plastic tube and the top of the coil. Record this in your lab notebook. 4. The long bar magnet has a narrow cut at one end. This is the North pole of the magnet. 5. Hold the long bar magnet at the top of the tube; the end with the cut should be down, and just one centimeter or so inside the tube. 6. Click on the icon then drop the magnet. 7. Measure the heights V 1 and V 2 of the two peaks using the tool. coordinates 8. Slide the tube upwards, remove the rubber stopper and the magnet, replace the rubber stopper, and slide the tube down again. 9. Sketch in your notebook the graph of voltage vs. time that you see on the screen. QUESTIONS: Figure 4: The setup Q1: The graph you obtained has a fall, a rise, a diminished slope, then a rise and finally a 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 V 1 /V 2? 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 the peaks are what you would expect using the right hand rule (curled fingers = current; thumb = magnetic field). Occasionally a long bar magnet is found to have a reversed polarity, perhaps due in part to repeated mechanical shocks (it s amazing to think that banging a magnet on a table can cause all the atoms to flip their spins). If you think the polarity of your magnet has become reversed, call over your TA.

45 E-5 MAGNETISM Sketch how the graph of voltage vs time ought to look if the magnet were dropped upside-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 predict how the graph would look? 12. Move the coil further down on the tube making the new distance d 2 at least about one-and-a-half times larger than d 1. Record this new distance d 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 V 1/V 2? 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 nearby conductor; to investigate how the induced emf depends on the waveform shape, amplitude, and frequency of the inducing current; to observe how the induced emf depends on the proximity and relative orientation of the two conductors. V 3 c 0 b -3 a Figure 5: A triangular waveform t(ms) APPARATUS: PC and PASCO interface; two large back-to-back coils (N=200 turns; radius r = m, series resistance R 9Ω); a small detecting coil (n=2000 turns); a stand for the small coil. INTRODUCTION: A voltage applied to one conductor (by a power supply, for example) can induce an emf in a nearby conductor. Here we apply a voltage with a triangular waveform to a 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 crosssectional area, and B is the magnetic field created by other nearby objects.

46 E-5 MAGNETISM 46 PROCEDURE: 1. Double click on the Physics202-E5c file in the Pasco/Physics202 folder on the desktop. The monitor should now show a scope window and a window for the control 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 the large coil and analog input B to the small coil, as in Fig 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 voltage applied to the large coil. 5. While recording data, move the small coil around and see how the amplitude of the induced emf changes. 6. Adjust the size of the waves you see in the scope window using the size controls on the side of the scope window, then measure the amplitude V B of the square wave using 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 direction of the magnetic field when the voltage applied to the coil is positive? Draw a sketch showing the current in the large coil, and the direction of the field it produces. Q3: Observe the direction of the winding in the small coil. What is the direction of the induced emf during the rising part of the triangular wave? Draw a sketch to indicate your answer.

47 E-5 MAGNETISM 47 Q4: Orient the small coil to achieve the maximum possible ratio V B / V A. Record this 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 and the 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 amplitude of the induced emf as a function of the distance between the plane of the large coil and 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/r 2? 1/r 3? Or something else? (Hint: consider the field produced by the 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 and Return the frequency to 10 Hz. Rotate the detecting coil 90 so that it is in the horizontal plane. Observe the resulting induced emf. 13. Rotate the coil another 90 so that it is in the vertical plane again. Observe the resulting 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 needful to 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 bar that 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 in another circuit that is not physically connected to it. Here we will observe this same effect, but in addition will see that if the current in a circuit changes, an emf is induced in that same circuit, and will explore the effects of this.

48 E-5 MAGNETISM 48 Figure 7: The schematic setup PROCEDURE: Figure 8: The physical setup 1. Set up the circuit shown in Figs. 7 and 8. Analog input A measures the voltage across the resistor, and is therefore proportional to the current through the outer coil. 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 the desktop. 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 waveform is a square wave. QUESTION: In E-5c, you applied a triangular-wave voltage to one coil, and observed a square wave in a nearby coil. Q1: What is the mathematical relationship between the square wave and the triangular wave?

49 E-5 MAGNETISM 49 Q2: Now we are applying a square wave to one coil. Sketch the waveform we expect to 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, which shows 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/R Q5: 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 = µ 0 N 2 A/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 new inductance of the circuit. Is this closer to your calculation in Q8? Q11: Explain in words why removing the iron bar changed the inductance of the circuit.

50 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 RC Decay. APPARATUS: TDS 2001C or 2002B oscilloscope; plugboard and plugboard kit; multimeter; function generator. INTRODUCTION: The oscilloscope is a high-speed graph-displaying device that displays graphs of electrical 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 in Fig Power on the scope and press the Default Setup button on the second row of buttons at the top of the panel. If you ever get lost, you can press Default Setup and start from scratch.

51 E-6 OSCILLOSCOPES AND RC DECAY Connect the Channel 1 probe to the gold contacts: the probe itself should be clipped to 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 see a very unsteady display that is not scaled properly. 3. To stabilize the graph, adjust the Trigger Level knob. The trigger level is indicated graphically by an arrow on the right side of the display. It is also displayed numerically on the bottom right. You should now see a stable plot that is clipped at the top. 4. Use the Channel 1 Vertical scale knob to display the entire waveform. The numerical 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 the figure. This scale is displayed in µ-seconds on the bottom center of the display above 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 numbers displayed? Q2: What is the relationship between the scales and the grid that the waveform is plotted against? Q3: How does the trigger level need to relate to the amplitude of the waveform to display a stable graph? EXPERIMENT II: BASIC FUNCTIONS: While the display grid will allow you to make crude measurements of amplitude and period, our oscilloscopes offer more sophisticated measurement tools in the form of cursors. 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 cursor position to line up with the maximum amplitude of the square wave. Note that the position of this cursor is displayed on the right side under Cursor Now highlight Cursor 2 and adjust the position to the minimum amplitude of the square wave. The difference between the two cursor positions is displayed at V and should be 5V in this case. 3. Change the Type to Time. Using the same mechanics as above, measure the period and 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 record the result.

52 E-6 OSCILLOSCOPES AND RC DECAY 52 QUESTION: Q1: Why is the measurement made by the multimeter different from the amplitude measured 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 of the function generator using the BNC to micro-grabber adapter. Always use the oscilloscope 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 be displayed on your oscilloscope. The amplitude given by the function generator is the Peak amplitude, or the maximum absolute value of the signal. 3. Measure the Peak-to-Peak amplitude and the Frequency of the waveform. Do they agree with the function generator to within your measurement uncertainties? 4. Now connect the Fluke Multimeter to the function generator output and record the RMS amplitude. Is this measurement consistent with the oscilloscope? 5. Experiment with several different types, amplitudes and frequencies of waveforms created using the function generator. QUESTION: Q1: What is the difference between the 5V@1kHz square wave you first observed and the 5V@1kHz square wave generated by the function generator? Be sure you are confident in your abilities to accurately display several types of waveform on the oscilloscope before moving on. EXPERIMENT IV: RC DECAY: 47 nf 1 KΩ V R V in 47 NF to red 1K to black to ch. 1 to ch. 1 to ch. 2 Figure 2: RC Circuit Schematic Now we will use the scope to characterize an RC Circuit.

53 E-6 OSCILLOSCOPES AND RC DECAY Build the series RC circuit with the plug board kit as shown Fig. 2. There is a raised bump on one leg of the BNC-banana-plug adapter. This bump indicates which leg is ground. Be sure to match the raised bump indicating ground with the figure. 2. Drive the circuit with a 5V@1kHz square wave. 3. Use one probe to monitor the function generator output with the oscilloscope, as you ve done in previous sections. 4. Plug a second probe into Channel 2 of the oscilloscope and use that probe to measure the voltage drop across the resistor. You ll need to press the blue 2 button. You should 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 agree with the calculated value to within your measurement precision? 6. For five different combinations of resistors and capacitors, measure the time constants and compare them to the nominal values τ = RC. You will need to change the driving frequency of the function generator and the horizontal scale of the oscilloscope accordingly. EXPERIMENT V: DIFFERENTIAL AMPLIFIERS: Amplifiers are devices which change the amplitude of the output signal compared to the input signal, without distorting the waveform of the input signal. Most amplifiers increase the amplitude. The customary symbol for an amplifier is a triangular shape:

54 E-6 OSCILLOSCOPES AND RC DECAY 54 V in V out 2 1 (volts) V in 10 5 V out (volts) Figure 4: Amplifier Function In the sine wave example above, the gain is 10 and the input signal is between an input terminal and ground. The internal oscilloscope amplifiers for channels 1 and 2 are of this type and have a common ground. Because of this common ground, one has problems in using a scope to examine simultaneously voltages across individual circuit elements that are in series. Amplifier Differential amplifier V in V out V in V out Figure 5: Amplifier vs. Differential Amplifier We can avoid these problems by interposing differential amplifiers which have two inputs 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 independent of any input ground.

55 E-6 OSCILLOSCOPES AND RC DECAY 55 Figure 6: Inputs to differential amplifiers 1. Connect the resistor and capacitor signal to the differential amplifiers as shown in the figure above. 2. Connect the outputs of the differential amplifiers to the scope inputs, and set both gains to 1. You should see something similar to Fig Describe what is happening to the voltage drop across the resistor when the capacitor is 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

56 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 with impedance 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 across that part and the current though that part. Because impedance is defined as a ratio of voltage/current, impedance is measured in Ohms. For AC circuits the ratio is a complex number: it has an magnitude and a phase. To avoid dealing with complex numbers, resistance and reactance are introduced. For ideal components: The resistance of a resistor is R (the reactance is zero); The reactance of an inductor L is X L = 2πfL (the resistance is zero); The reactance of a capacitor C is X C = 1 2πfC (the resistance is zero). If a resistor, inductor, and capacitor are combined in series (as in Fig. 1), the magnitude of the impedance of the entire circuit is Z = XR 2 + (X L X C ) 2. Note that the impedance Z of a series RLC circuit is a minimum for X L = X C. The frequency for which this occurs is the resonant frequency. At resonance, the current through R is maximum, but the voltage V LC across the LC series combination is zero. The phase angle φ of the series RLC circuit is at resonance this is zero. QUESTION φ = tan 1 ( XL X c R Q1: Derive an expression for the resonant frequency in terms of the inductance and capacitance 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 generator to supply V AC. 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 the circuit (in Ohms). ) ;

57 E-7 SERIES LRC CIRCUITS AND RESONANCE 57 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 V R = IR, V L = IX L, and V C = IX C. 2. Use the digital multimeter (DMM) to measure the RMS voltage drop across the resistor (the voltage between points a and b indicated in the diagram). Check it against 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 drops across 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 function generator 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 with your measurement? 6. While at resonance, use the DMM to measure V bc and V cd, and verify that neither is zero. 7. Now measure V bd. It should be close to zero. Is it? 8. Swap in the 0.47 µf capacitor, and measure the new resonant frequency.

58 E-7 SERIES LRC CIRCUITS AND RESONANCE 58 Q8: Given that the capacitance has increased by a factor of 10, by what factor should the resonant frequency change? Q9: Is this consistent with your measurement? 9. For four different frequencies on either side of resonance, measure the voltage drop across the resistor. Make a table of V R as a function of frequency. 10. Add a third column to the table and calculate the power dissipated in the resistor at each frequency. Use this table to create a plot of power vs frequency. Verify that it looks like a Gaussian. 11. Replace the 1KΩ resistor with a 100Ω resistor and repeat the previous step. What is 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 voltage drop 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 different waveforms. Fig. 2 shows this relationship. 4. Note that at resonance, φ = 0. Determine the resonant frequency using the relative phases 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? V L =I X L V=I Z I V L - V C φ VR =I R =I X LC ωt V C = I X C Figure 2: Phase Relations

59 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 Part C, #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 not discuss, (e.g. a MOSFET: Metal-Oxide Semiconductor-Field-Effect-Transistor). Collector N NPN Collector PNP Collector Emitter Base N P I Base Base Emitter Emitter Figure 1 For an npn transistor the collector is positive relative to the emitter. The baseemitter circuit acts like a diode and is normally conducting (i.e. forward-biased). The base-collector circuit also acts as a diode but is normally non-conducting (reverse biased) if no current flows in the base-emitter circuit. However when current flows in the base-emitter circuit, the high concentration gradient of carriers in the very thin base gives an appreciable diffusion current to the reversed biased collector. The resulting collector current I c depends on the base current I b. We write I c = βi b where β is the current amplification factor. PRECAUTIONS: Although our power transistor is fairly indestructible, its characteristics are temperature sensitive. Hence avoid exceeding the voltages suggested; also leave the power supply off when not making measurements; read currents and voltages to two significant figures only. SUGGESTED EXPERIMENTS: Variable Voltage DC Supply + DMM 2 ma 1. CURRENT Amplification: Measure β for various emitter to collector voltages, V ec. Set up the circuit as in Fig. 2. Figure 2 µa DMM 1

60 E-8: TRANSISTORS 60 Start with the voltage divider completely counter-clockwise so that the emitter-base voltage V eb is a minimum. With the variable power supply set to 3 V, record the I b measured on DMM 1 and the I c on DMM 2 Repeat the readings for ten reasonably spaced (higher) settings of the voltage divider (i.e. higher V eb and hence higher I b. Calculate β(= I c /I b ) and plot it against I b. Over what range of I b is β reasonably constant? Repeat the above but with emitter to collector voltage V ec now at 7 V. 2. (Optional) MEASUREMENT OF I c vs V eb : Remove the multimeter DMM 1 (used as a microammeter) from the circuit of Fig. 2 and use it instead as a voltmeter to measure the emitter to base voltage V eb. With the variable power supply set to give 7 V for V ec, use the voltage divider to vary V eb. Read and record both V eb and I c for 10 reasonably spaced values of V eb. Plot I c vs V eb. The results are very similar to a diode curve (as expected since I c is proportional to I b over the region where β is a constant). 3. VOLTAGE Amplification: By adding a large load resistor in series with the collector, one can convert the current amplification β observed earlier into a voltage amplification. Hook up the circuit plug board as in Fig kω 10 kω Variable Voltage DC Supply +15V kω V out DMM 2 GND V in Figure 3 DMM 1 The input voltage to the transistor V in includes the drop across the 4.7 kω protective resistor. The voltage gain is G = V out / V in. Record input voltages V in and output voltages V out (= V ec ) for ten reasonably spaced values of V in from 0 to 5 volts. Graph V out vs V in and calculate the voltage gain G at the steeply changing part of your graph. The value of V in at the center of this region we call the operating voltage of the amplifier. How would the gain change if the load resistance was 2.2 kω instead of 10 kω? 4. Distortion effects when amplifier is overdriven: Set up the circuit plug board as in Fig. 4. Connect Y 1 and Y 2 to the scope. To connect the signal generator to the board, use the BNC to banana plug adapter and remember that the side with the bump goes to ground.

61 E-8: TRANSISTORS 61 1 kω 10 kω Variable Voltage DC Supply + +15V 4.7 kω GND 0.047µf + V in + Y 2 DMM Y 1 Figure 4 Adjust the voltage divider until the V in is close to the operating voltage found in #3. Vary (and record) the amplitude of the input signal from small values to those which overdrive the amplifier and produce considerable distortion in the output Y 2. Observe and record the effect of changing V in to values outside of the operating range (where β is constant).

62 62 Part II Sound 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 superposing two traveling waves on the string of equal frequency and amplitude, but opposite directions. The distance between nodes (points of minimum motion) is one half wavelength, (λ/2). Since the ends of the string are fixed, the only allowed values of λ are are λ k = 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 certain frequencies are allowed: F/µ f k =. (1) λ k A N N First mode A N N N Second mode λ 2 N N Fourth mode λ Figure 1: The Modes of a String 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 clamp assembly; weight holder & selection of slotted masses; black Dacron string; electronic balance; stroboscope. The set-up consists of an electrically driven speaker which sets up a standing wave in a string stretched between the speaker driver stem and a pulley. Hanging weights on the end of the string past the pulley provides the tension. The computer is configured to generate a digitally synthesized sine wave (in volts versus time) with adjustable frequency and amplitude (max: 10 V). PASCO interface: This transforms the digital signal into a smooth analog signal.

63 S-1 TRANSVERSE STANDING WAVES ON A STRING 63 Precautions: Decrease the amplitude of the signal if the speaker makes a rattling sound. 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 nonlinear effects since the string tension cannot remain constant under the finite amplitude of displacement. [See Elliot, Am. J Phys. 50, 1148, (1982)]. Other oscillatory effects arise from coupling to resonant vibrations of the string between pulley and the weight holder; hence keep this length short. L Speaker Bridge 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 the distance, L, between the bridge and the pin of the speaker be about 1.2 m, and measure 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 the vibration of the string. 5. Double click on the Experiment S-1 file in the Pasco/Physics202 folder on the desktop. The display will appear as shown in Fig You will see that the computer is set to produce a 60 Hz sine wave with an amplitude of 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 of the 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 numeric window and entering a new value with keyboard number entry.

64 S-1 TRANSVERSE STANDING WAVES ON A STRING 64 Figure 4: The Pasco Capstone display. 8. Stop the signal generator, put a 200 g mass on the mass hanger and restart the signal generator. Record the total mass and tension in your lab notebook. In the next few steps you ll keep the tension in the string constant and look for the frequencies of different modes. 9. Adjust the frequency so that the amplitude of the oscillation is at its maximum by changing the frequency in 1 Hz steps. This is best done as follows: First decrease the frequency until the amplitude of the string is very small. Then increase the frequency in 1 Hz steps, and observe that the amplitude first increases and then decreases. Record in your table the frequency f 2 at which the amplitude is maximum. 10. Change the frequency to observe the third mode. Find and record the best frequency f 3 (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 stroboscope. Note that the stroboscope is calibrated in RPM or cycles per minute, NOT Hz (cycles per second). You should find a value close to 70 Hz. 13. Divide the various frequencies f k by k and enter the values in a table. Calculate the average value of f k /k.