TABLE OF CONTENTS. L-2 Find the Focal Length LC-1 Diffraction and Interference LC-4 Photoelectric Effect L-5 The Balmer Series

Transcription

1 Introduction Electricity and Magnetism E-1 Static Electricity TABLE OF CONTENTS E-2 Equipotential Surfaces and Electric Fields E-3 DC Circuits: Resistors and Meters EC-6* Magnetic Fields and Forces EC-5 Magnetic Induction E-4 Magnetic Force on a Charged Particle Light L-2 Find the Focal Length LC-1 Diffraction and Interference LC-4 Photoelectric Effect L-5 The Balmer Series Modern Physics MPC-1 Appendices Radioactive Decay A. Using a D.C. Power Supply B. Using a Digital Multimeter C. Using an Electrometer D. Significant Figures E. Accuracy, Precision and Uncertainty F. Graphing Data by Hand G. Graphing Data and Curve Fitting in Excel * A C in the experiment number means the lab is computerized. TOC - 1

2 RULES OF THE LAB: Please leave the lab in at least as nice a condition as you found it, with equipment put away so that it is ready for the next group to use. When you are done and ready to leave, please help us keep the labs neat: _power off and disconnect all lab equipment _return all lab equipment to its original condition _straighten chairs _keep tables reasonably clean _clean up bits of string, tape, eraser crumbs _put away weights _throw away garbage _take writing utensils with you! During winter: please do not put your feet on bottom shelves of computer carts or on chairs. If you are in the last lab of the day, please help us maximize the computer lifetime by also turning off the computer equipment: _turn off interface _turn off power strip for interface _turn off computer _turn off monitor Thank you for helping preserve and protect the physics lab room and equipment! TOC - 2

3 E-1 STATIC ELECTRICITY What do we do in this lab? This lab has three parts: E-1 Static Electricity A. charging by rubbing (triboelectricity): rub various objects together, and observe whether any of them acquire an electric charge; B. charging by conduction: investigate how charge can be transferred from one object to another by touching the two objects together; C. charging by induction: investigate how the charges on one object can cause the charges on another object to move around, even when the two objects are not touching each other. Why are we doing this lab? The main concept introduced in this lab electric charge is fundamental not only to Physics 104, but also to modern civilization (imagine what life would be like without electricity). Safety concerns: None. You need to know: conductor object through which electric charges can move freely. electric charge a tiny object which can create, and is influenced by, electric fields (example: electrons and protons). If the number of electrons and protons in material object are not equal, the material object is said to carry a net electric charge. electric force force felt by one electric charge when in the vicinity of another (examples: like charges repel, opposite charges attract). grounded connected to the Earth by a conductor (the Earth acts as a reservoir of electric charge that can never be depleted). To the right is a picture of a common symbol used to indicate that an object is grounded. insulator object through which electric charges cannot move. neutral possessing equal amounts of positive and negative electric charge. static electricity any of a number of phenomena related to the sudden passage of electrons from one object to another, often accompanied by a visible spark and an audible snap. E-1-1

4 E-1 STATIC ELECTRICITY Pre-lab 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 negative charge flows no charge flows from ground to conductor from conductor to ground 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 = 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. E-1-2

5 E-1 STATIC ELECTRICITY EQUIPMENT Test Sphere From left to right in the picture above: Ebonite Rod Acrylic Rod Heat gun Matches There is only one of each of these in the lab room. Ethanol lamp Rabbit fur Electroscope Sphere with hole Ebonite rod Acrylic rod Test sphere Silk cloth Two metal spheres Electrometer Faraday cup Every table has one of each of these E-1-3

6 E-1 STATIC ELECTRICITY A. Charging by rubbing (Triboelectricity) 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. A1. 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. Faraday Cup Electrometer Pasco Interface 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. A2. 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. A3. 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). A4. Touch the test sphere against the outside of the Faraday Cup. A5. 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) E-1-4

7 E-1 STATIC ELECTRICITY A6. If all went well, the test sphere should have been charged in step A4, but neutral in step A5. What did you do that caused the test sphere to become neutral, and what happened to the extra charge? A7. 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. A8. Repeat A7, using the ebonite rod. A9. Rub the acrylic rod with the silk. What charge does the acrylic rod acquire? (circle one) positive negative A10. Rub the ebonite rod with the rabbit fur. What charge does the ebonite rod acquire? (circle one) positive negative A11. Are your results from A9 and A10 in accordance with the Triboelectric Series (see table at right)? Hair Table I: The Triboelectric Series 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) Greater tendency to charge positive Greater tendency to charge negative A12. 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 (circle one) Is this in accordance with the Triboelectric Series? E-1-5

8 E-1 STATIC ELECTRICITY B. Charging by Conduction knob leaves 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 at left, the leaves are uncharged). The leaves and the knob of an electroscope are connected by a conductor. B1. 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 they should fall. If you can t get the leaves to fall, call over your TA. B2. 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. B3. 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? E-1-6

9 E-1 STATIC ELECTRICITY B4. Draw little + signs on the diagrams below to indicate what happens to the charges on the oscilloscope during B3. B5. 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. B6. 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? B7. Touch the knob of the electroscope with your finger. What happens to the leaves? E-1-7

10 E-1 STATIC ELECTRICITY C. Charging by Induction In part we ll do three experiments related to charging by induction. C1. 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 step B2, and the leaves of the electroscope should now be risen. C2. 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? C3. Draw + and - signs on the pictures below to show where the charges are: C4. 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 C1 to C3 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. E-1-8

11 E-1 STATIC ELECTRICITY C5. 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? C6. Assuming the acrylic rod is charged positive, draw little + and signs on the cartoons to show where the charges are. C7. 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? C8. Plug a cable into the ground input of the Pasco interface, and repeat step C5, 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. C9. 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. E-1-9

12 E-1 STATIC ELECTRICITY For the third experiment on charging by induction we ll return to the apparatus shown in the picture associated with step A1, 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. C10. 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. C11. 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? C12. 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). C13. ER rubs ebonite rod with rabbit fur, and inserts ebonite rod into Faraday Cup. The electrometer reads (circle one) positive negative zero C14. 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 C15. 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 C14. If the electrometer still doesn t read zero, call over your TA. C16. 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. C17. TS touches test sphere to the left side of the big metal sphere. C18. TS inserts the test sphere into the Faraday Cup. The electrometer reads: positive negative zero C19. 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 C20. 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. C21. 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. C22. TS inserts the test sphere into the Faraday Cup. The electrometer reads: positive negative zero E-1-10

13 E-1 STATIC ELECTRICITY C23. 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 C24. Based on the results of steps C8-C20, 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. Follow-up questions 1. Can a grounded object be charged? Cite one of the steps from this lab which supports your answer. 2. Can a charged object be neutral? 3. Can a neutral object be grounded? 4. An object is isolated from all other objects, so that it never touches any other object. Can the electric charge on this object ever change? 5. Can the distribution of charge on this object ever change? E-1-11

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15 E-2 EQUIPOTENTIAL SURFACES AND ELECTRIC FIELDS E-2 EQUIPOTENTIAL SURFACES AND ELECTRIC FIELDS What do we do in this lab? In this lab you map the equipotential surfaces and electric field lines that surround two sets of conductors. The first set of conductors is called a dipole and the second set is called a parallel plate capacitor. A rough sketch of the equipotential surfaces and electric field lines surrounding the dipole is below. 1 V V V equipotential e V field line V V 0V Why are we doing this lab? In order to get practice visualizing electric equipotential surfaces and fields. Safety concerns: none. You need to know: Electric field line line which is everywhere parallel to the electric field. electric potential property of a point in space, specifically the electric potential energy a unit charge would have if it were located at that point. Units: Volts. electric potential energy property of a charged object, specifically the potential energy possessed by the charged object when placed in an electric field. Units: Joules; or electron Volts, abbreviated ev. equipotential surface collection of points which have the same electric potential. Always and everywhere perpendicular to electric field lines. potential difference difference in electric potential between two points in space. Units: Volts. E-2-1

16 E-2 EQUIPOTENTIAL SURFACES AND ELECTRIC FIELDS Pre-lab Questions 1. Can two equipotential surfaces intersect? 2. Can two electric field lines intersect? 3. When an electric field line intersects an equipotential surface, what is the angle between the electric field line and the equipotential surface? 4. The electric potential caused by a point charge q is V = k e q / r. Is this equation true for all charged objects (for example, for the ebonite rod used in lab E-1), or only for point charges? 5. A test charge q is moved from one point on an equipotential surface to another point on the same equipotential surface. What is the change in electric potential energy of the test charge? 6. What is the difference in electric potential between the two terminals of a 9 Volt battery? EQUIPMENT Notes on equipment The black conductive paper is a conductor, but not a very good conductor. It allows electric charges to flow slowly through it. Your TA may be able to supply you with a special pen that uses silvery-looking conductive ink. This ink is a very good conductor. If you have the special pen, you can use it to draw shapes on the black conductive paper. If you don t have the special pen, you ll have to use the shapes that somebody before you drew. As you may know, the boundary of a good conductor is an equipotential. If you draw a shape on the black conductive paper, and connect the shape to one of the outputs of the power supply, the shape should attain the same potential as the power supply output. E-2-2

17 E-2 EQUIPOTENTIAL SURFACES AND ELECTRIC FIELDS A. Equipotentials And Electric Fields From A Dipole A1. Using an ordinary pen or pencil, draw a positive point charge in the middle of a piece of notebook paper. Make a small dot with your pencil, to the right of the point charge, about 3/4 of the way from point charge to the edge of the paper. A2. Write down the equation for the potential from a point charge: A3. Make a few more dots with your pencil, using the equation to make sure the potential at each dot is the same as the potential at the first dot. When you have enough dots to see the shape you are making, connect the dots to form a line. This line is an equipotential line. What is the shape of this line? A4. Where is the point charge in relation to this shape? A5. Draw another equipotential line, at twice the potential of the first line. Use the equation to tell you where it goes. Now draw a third equipotential line, at four times the potential of the first line. What does your picture look like now? A6. You have just drawn an equipotential map for a single point charge. Would a map like this look any different in any other plane containing the positive point charge? A7. Fold a fresh sheet of paper in half, top meets bottom. Unfold it and turn it sideways so the fold is vertical. In the middle of one half, draw a positive point charge, and in the middle of the other half, an equal magnitude negative point charge. This charge configuration is called a dipole. The fold should bisect the line between the charges, perpendicular to it. A8. What is the potential along the fold? (Hint: use the equation from question A2, plus the principle of superposition.) A9. What is the sign of the potential to the left of the fold? A10. What is the sign of the potential to the right of the fold? A11. (Extra Credit) Draw as many points as you can that are exactly twice as far from the positive charge as they are from the negative charge. These points all fall on a curve. What is the shape of this curve? A12. Make a stack of papers, in order from top to bottom: Conductive paper, with dipole (looks like the figure on page E-2-1) drawn in metallic ink Carbon paper White paper, for copying A13. Place this stack on the wooden board, and move the two metal pieces such that they are holding the stack of papers down, and they are in contact with the metallic ink. The metallic ink will henceforth be referred to as the conductors. A14. Plug in, and turn on the power supply. Connect the + and GND output connections of the power supply to the two terminals on the wooden board using E-2-3

18 E-2 EQUIPOTENTIAL SURFACES AND ELECTRIC FIELDS the banana-plug cables. Set the output voltage to 18 V (see the section in Appendix A entitled Setting a Voltage for details on how to do this). A15. Connect one end of the black probe to the COM terminal of the Fluke multimeter. Connect one end of the red probe to the V Ω terminal of the multimeter. Set the multimeter to measure DC Voltage (see the section in Appendix B entitled Measuring a DC Voltage for details on how to do this). A16. Put the black probe in the connector on the board that is hooked up to the ground of the power supply. This ensures that the multimeter and the power supply agree on what potential to call 0 V. Use the red probe to map out the equipotentials. Hold the probe point at a slant, like a pencil, so you don't punch holes in the paper. Move it lightly across the paper, and don t press too hard, unless you want to make a mark with the carbon paper on the plain paper underneath. Notice that as you get closer to the highpotential side of the dipole, the multimeter measures a higher potential, and as you get closer to the ground side, the multimeter measures a lower potential. A17. Map the 3V equipotential: find five or ten different points that each have a potential of 3 V. At each point, rub the conductive paper with the probe, in order to make a dark mark on the white paper. Then connect the dots on the white paper. The line connecting the dots is the 3 V equipotential. Label it 3 V. A18. Repeat step A17 for 0 V, 6 V, 9 V, 12 V, 15 V, and 18 V. Label each line with the voltage it maps. Make sure everyone on the team gets a chance to map at least one equipotential. You should notice that as you get closer to a conductor, the equipotentials look more like the shape of the conductor. A19. How can you tell from an equipotential map where the electric field will be strongest and weakest? A20. Draw an electric field line: draw a line which crosses all the equipotentials at right angles. A21. Draw five more different electric field lines. Make sure everyone on the team gets a chance to draw at least one field line. A22. Where an electric field line meets the boundary of a conductor, what is the angle between the electric field line and the boundary of the conductor? A23. How can you tell from a map of electric field lines where the electric field will be strongest and weakest? A24. Where is the electric field the strongest? A25. Where is the electric field the weakest? E-2-4

19 E-2 EQUIPOTENTIAL SURFACES AND ELECTRIC FIELDS B. Equipotentials And Electric Fields From Parallel Plates You will be using same equipment as in part A, but this time choose a piece of conductive paper with a painting of two silver tees, the two-dimensional representation of parallel plates. B1. Based on your knowledge of electric potentials, sketch a map of the equipotential lines surrounding two parallel plates, as below. 18 V B2. Fold a piece of notebook paper in half. Unfold it and draw two parallel plates, putting the plates on opposite sides of the fold, equidistant from it, and about 10 cm apart. Assume the plates are at +9 V and -9 V. B3. What is the potential along the fold? B4. Now draw a line that halfway between the fold and the positive plate. Is the potential along this line greater or less than the potential along the fold? B5. Draw another line, halfway between the positive plate and the line you just drew. How does the potential on this new line compare with the potentials on the first line higher or lower? B6. Using conductive ink, draw two parallel plates on a sheet of black conductive paper, or find a sheet that already has two plates drawn on it. On some drawings the conducting ink is very thick and shiny, and in others the conducting ink is very thin, and not very shiny. For which type of drawing is the boundary of the drawn shape most likely to be an equipotential? E-2-5

20 E-2 EQUIPOTENTIAL SURFACES AND ELECTRIC FIELDS B7. Make a stack of papers, in order from top to bottom: Conductive paper, with parallel plates drawn in metallic ink Carbon paper White paper, for copying B8. Map the equipotential surfaces for 0 V, 3 V, 6 V, 9 V, 12 V, 15 V, and 18 V, using the same procedure you used in A12-A18. Be sure you map the lines outside the plates as well as between the plates. Make sure everyone on the team gets a chance to map at least one line. B9. Where is the 0 V equipotential, relative to the boundary of the 0 V plate? B10. Where it the 18 V equipotential, relative to the boundary of the 18 V plate? B 11. Compare the map you made in B8 to the sketch you made in B1. How are they similar, and how are they different? B12. Draw dashed lines that are everywhere perpendicular to the ones on your map. What do these lines represent? B13. Where is the electric field created by the painted tees the strongest? B14. Where is it the weakest? B15. What would happen to the lines in your maps if the tees were the same distance apart, but the power supply voltage was turned up higher? B16. If you are very close to the surface of a positively charged conductor, what is the direction of the electric field relative to that surface does it point directly toward the surface, or directly away? B17. You may have noticed that the shapes of the equipotential lines that are close to the surface of a charged object are very similar in shape to the charged object. Imagine there is a certain amount of positive charge on some object that has a very complicated shape (like, say, a chair). What are the approximate shapes of the equipotential lines that are very far away from the object? E-2-6

21 E-3 DC CIRCUITS AND METERS What do we do in this lab? E-3 DC Circuits: Resistors And Meters This lab has two parts. In the first part of the lab we will measure V and I for a particular resistor, see if Ohm s Law holds, and infer the resistance of the resistor. In the second part of the lab, we will combine resistors in parallel and in series, measure V and I, and infer the resistance R of the combination. Does our inferred value of R agree with what we would predict from the rules we learned in class for adding resistances in parallel and series? Resistors in parallel and in series: which is which? If the resistors are in series all in a line and connected end-to-end--the addition is ordinary addition: R = R 1 + R 2 + R 3 But if they are in parallel side by side with all the leads connected together on both ends the addition rule is different. The total resistance is the inverse of the 1 sum of the inverses: R = R 1 R 2 R 3 Why are we doing this lab? This lab introduces electrical circuits, which are ubiquitous in everyday life. Safety concerns: none. You need to know: parallel combination with a branch point series combination without a branch point A note on prepositions: one speaks of the voltage across a resistor, the current through a resisitor, the resistance of a resistor. E-3-1

22 E-3 DC CIRCUITS AND METERS Pre-lab questions 1. Suppose a 400 Ω, a 200 Ω, and a 100 Ω resistor are all in series with a 9 V power supply. Draw the circuit. 2. What is the equivalent resistance? 3. What is the current flowing out of the power supply? 4. Now suppose the 400 Ω, 200 Ω, and 100 Ω resistors are all in parallel with the power supply. Draw the circuit. 5. Now what is the equivalent resistance? 6. Now what is the current flowing out of the power supply? 7. If current flows through a resistor, power is dissipated. Power is energy per unit time. Where does the energy go? EQUIPMENT Power supply multimeters plugboard resistors short cables E-3-2

23 E-3 DC CIRCUITS AND METERS Multimeter=voltmeter/ammeter/ohmmeter The voltage across a resistor is measured with a voltmeter. The voltmeter is connected in parallel with the resistor. An ideal voltmeter measures the voltage across a resistor without having any effect on it. That is, the voltage is the same with the ideal voltmeter connected or without it. The current through a resistor is measured with an ammeter. The ammeter is connected in series with the resistor. An ideal ammeter measures the current through a resistor without having any effect on the current. The current is the same with the ideal ammeter in the circuit or without it. The resistance of a resistor can be measured with an Ohmmeter. Ohmmeters work best when they are used to measure the resistance of a resistor that is not part of a circuit. In this lab you ll use a pair of multimeters. A multimeter is a device that can function as a voltmeter, ammeter, or ohmmeter. 100 Ω resistor Lines are like wires: each group of nine holes is electrically conected close-up of plugboard resistor A. Ohm s law for a single resistor close-up of plugboard The manufacturer guarantees the resistance of these resistors to an accuracy of 10%. You are going to determine the true resistance of these resistors. Detailed instructions on how to use the power supply are given in Appendix A. The Fluke digital multimeters are identical. Either can be used as an ammeter, a voltmeter, or an ohmmeter. Detailed instructions are given in Appendix B. A1. Set one of the multimeters to act as an ohmmeter. Call it DMM 1. Set up the circuit shown in the figure and photo below, using the 100 Ω resistor. Measure the resistance of the resistor directly using the multimeter. E-3-3

24 E-3 DC CIRCUITS AND METERS Nominal resistance = R nom = 100 Ohms Directly-measured resistance = R meas = Ohms A2. Discuss with you lab partners the uncertainty in your measurement, and record it here: Uncertainty in R meas = ΔR meas = Ohms A3. Construct a table in your lab notebook as below, and fill it in: R nom (Ω) R meas (Ω) uncertainty in R meas ΔR (Ω) Now we ll use two multimeters to measure the voltage across and the current through a resistor simultaneously, and use Ohm s law to calculate the resistance. A4. Set one multimeter to measure current (call it DMM 1 ) and the other to measure voltage (call it DMM 2 ). A5. Set up the circuit shown in the figure and photo below, using the two multimeters, the power supply, the 100 Ω resistor, and the short. Set the power supply to 15 V. E-3-4

25 E-3 DC CIRCUITS AND METERS A6. Based on the readings of the two multimeters in the photo, what is the resistance of the resistor in the photo? deduced resistance = Ohms A7. Which of the following causes charge to flow through the circuit (circle one): Power supply DMM 1 DMM 2 E-3-5

26 E-3 DC CIRCUITS AND METERS A8. Let each lab partner pick a different resistor and fill in the table below for it: set the power supply to five different voltages, and simultaneously measure the current through the resistor and the voltage across the resistor. Discuss with your lab partners how you can estimate the uncertainties in the measurements of the current and voltage. If you can t think of any way to estimate the uncertainties, call over your TA for help. R nom (Ω)= Power Supply Voltage (V) I meas (ma) uncertainty in I meas Δ I meas (ma) V meas (V) uncertainty in V meas ΔV meas (V) A9. Make a nice graph of the results using Excel. Make sure the graph has an informative title (such as the nominal resistance of the resistor), labels on each axis, units on each axis, and vertical and horizontal error bars on each data point Appendix F has some basic information about graphs and error bars. Appendix G has detailed instructions about how to make nice graphs using Excel. Each person in the group should make a graph of a different resistor. A10. Use Excel to add a trendline to the graph (Appendix G has instructions on how to add trendlines to graphs in Excel). Use the trendline formula to infer R calc, the value of the resistance calculated from Ohm s Law: Calculated resistance = R calc = Ohms A11. Does the trendline pass through all of the error bars? A12. Does any of your data lead you to suspect that Ohm s law might not be true for the resistor you chose? A13. Compare R meas and R calc. Does the resistance of any of the resistors appear to change when current flows through it? (circle yes or no for each resistor) 100 Ω: yes/no 200 Ω: yes/no 400 Ω: yes/no A14. Compare R nom, R meas, and R calc. Are R meas and R calc within the manufacturer s stated tolerance of +/- 10% of R nom? (circle yes or no for each resistor) 100 Ω: yes/no 200 Ω: yes/no 400 Ω: yes/no E-3-6

27 E-3 DC CIRCUITS AND METERS B. Resistors in series and parallel B1. Using the equipment you have been given, construct the circuit shown in the diagram below, in which a 400 Ω, a 200 Ω, and a 100 Ω resistor are all in series with the power supply (note that DMM 2 will be moved several times during the next few instructions). B2. Discus with your lab partners what is your best guess of the actual resistance of each of the three resistors you measured in part A, and write it down here: R nom = 100 Ω, R actual = Ω ; R nom = 200 Ω, R actual = Ω ; R nom = 400 Ω, R actual = Ω B3. Use these actual resistances to predict the equivalent resistance of this circuit: R predicted = Ω B4. Set the voltage of the power supply to 9 V. B5. Use DMM 1 to measure the current flowing out of the power supply: Current flowing out of power supply = I series = ma B6. Assuming the voltage shown on the power supply display is exactly correct, use the result of B5 to find the measured value of the equivalent resistance of this circuit: Measured value of equivalent resistance = R measured = Ω E-3-7

28 E-3 DC CIRCUITS AND METERS B7. Use DMM 2 to measure the voltage across each of the three resistors: V R1 = V, V R2 = V, V R3 = V B8. Add you results from B7 together: V B9. What is the percentage difference between your measurement in B8, and the voltage shown on the power supply display? B10. Assume your result from B9 is an estimate of the uncertainty of the voltage shown on the power supply display. Is the discrepancy between the predicted value of the equivalent resistance (B3) and the measured value of the equivalent resistance (B6) larger, smaller, or about the same as, the uncertainty in the voltage shown on the power supply display (circle one): larger about the same as smaller B11. Connect the 400 Ω, 200 Ω, and 100 Ω resistors all in parallel with the power supply (figure out how to do this with the help of your lab partners). B12. Using the actual resistances of these resistors (from B2), predict the equivalent resistance of this circuit: B13. Set the voltage of the power supply to 9 V. R predicted = Ω B14. Measure the voltage across each resistor. Are they the same? B15. Measure the current flowing out of the power supply: Current flowing out of power supply = I parallel = ma B16. Just like you did in B6, calculate the measured value of the equivalent resistance, based on the assumption that the voltage shown on the power supply display is exactly right: Measured value of equivalent resistance = R measured = Ω B17. Is the discrepancy between the predicted and measured values of the equivalent resistance for the parallel circuit larger, smaller, or about the same as the uncertainty in the voltage shown on the power supply display (circle one): larger about the same as smaller B18. Which is larger, R measured for the series circuit, or R measured for the parallel circuit? series B19. Which is larger, I parallel or I series? parallel I parallel I series E-3-8

29 EC-6 MAGNETIC FIELDS AND FORCES What do we do in this lab? EC-6 MAGNETIC FIELDS AND FORCES This lab has three parts: make maps of the magnetic field caused by permanent magnets; measure the strength of the magnetic field caused by permanent magnets; investigate the effect of a magnetic field on a wire carrying electrical current. Why are we doing this lab? Magnetism is a common household phenomenon that remains mysterious to many people, but we hope that doing this lab will lessen the mystery for you. Safety concerns: none. You need to know: Compass long, thin permanent magnet mounted on an axle and free to rotate. Hard magnetic material material that, once magnetized, stays magnetized. Magnetic field invisible, intangible influence permeating the space surrounding a permanent magnet or current-carrying wire that can affect other objects intruding upon that space; cannot be directly perceived by people, but can be accurately measured by electronic devices. Magnetic field lines a way of visualizing magnetic fields; magnetic field lines are drawn so that the direction of a magnetic field line at any point is parallel to the direction of the magnetic field at that point, while the spacing between magnetic field lines in a region is proportional to the magnitude of the magnetic field in that region. Stronger Field Weaker Field Magnetic field strength magnitude of magnetic field; units are Tesla (T). Magnetic force force experienced by a permanent magnet, current-carrying wire, or moving charged particle in a magnetic field. Example: the magnetic force F on a wire of length l, carrying a current I perpendicular to a magnetic field of strength B, has magnitude F= B I l, and direction perpendicular to both the current and the field (as indicated by the righthand rule). Magnetize cause the spins of the atoms in a material to line up. Non-magnetic material material that cannot be magnetized. Permanent magnet magnetized object made of a hard magnetic material. Soft magnetic material material that can be magnetized, but quickly loses its magnetism. EC-6-1

30 EC-6 MAGNETIC FIELDS AND FORCES Pre-lab questions 1. At left is a sketch of the magnetic field near a bar magnet. Also shown are some compasses and compass needles. Assume the Earth s magnetic field is quite similar to what it would be if there were a giant bar magnet at the center of the Earth, and sketch it below. 2. Below is sketch of a bar magnet with a different shape from the one shown above. Sketch the magnetic field lines. S N 3. A permanent magnet can pick up certain paper clips. However, these paper clips can t themselves pick up other paper clips. The paper clips are most likely made of (circle one): a hard magnetic material a soft magnetic material a non-magnetic material 4. A permanent magnet can t pick up an aluminum can, because aluminum is (circle one): a hard magnetic material a soft magnetic material a non-magnetic material 5. Write the equation for the magnetic force on a current-carrying wire, assuming a 90 angle between the magnetic field and the direction of the current: 6. Suppose I = 3A, B=0.08 T, and l=0.02 m. Evaluate F: 7. How much mass has this weight? F = F = N m = g EC-6-2

31 EC-6 MAGNETIC FIELDS AND FORCES A: Mapping Magnetic Field Lines Caused By Permanent Magnets magnetic field sensor magnets compass magnetic field finders Equipment: a white plastic board with two small permanent magnets set into it; compass; two magnetic field finders (clear plastic plates with small iron bars sandwiched between them that act as tiny compasses); clear plastic sheet; transparency and water-based marker pen; magnetic field sensor, which connects to the Pasco interface. The permanent magnets can be rotated so their north-south axes point in any direction in the plane of the board. Red is N and blue is S. Lake Mendota is due north of Chamberlin Hall, and Charter Street (which you can see out the window of the Physics 104 lab room) runs north-south. Make sure you know which way is north. A1. Pick up the compass and walk around with it. Identify two objects that definitely affect the direction the compass points (before you read the compass, pause a few moments to let it stop wobbling). object 1: object 2: A2. Take the compass far away from any objects that seem to affect the direction it points (even if this means going out into the hallway). Which end of the compass points north, the red end or the silver end? (If neither end points north, tell your TA, who may choose to get you another compass). A3. Based on your result from A2, which end of the compass is an N pole, the red end or the silver end? EC-6-3

32 EC-6 MAGNETIC FIELDS AND FORCES A4. Use your compass to verify that both of the magnets in your white board are red N, blue S. If one or both of them are not, call over your TA to confirm. When two permanent magnets are brought close together, the field lines change in one of two ways, depending on the orientation of the magnets (remember that magnetic field lines can never cross). In the first way, the field lines get squashed: S N S N lines can t cross, becomes S N S N In the second way, the field lines reconnect: N S N S lines can t cross, becomes S N S N If the field lines can reconnect, they will; otherwise they just have to get squashed. A5. On the diagram below, make sketches of what you predict the field lines will look like in each of the four possible configurations of the two magnets. Feel free to talk it over with your lab partners before you make the sketches. 1 2 S N S N S N N S Predicted Magnetic line of symmetry line of symmetry Field Line Directions: 3 4 N N N S S S S N line of symmetry line of symmetry EC-6-4

33 EC-6 MAGNETIC FIELDS AND FORCES A6. Set up configuration 1 in the diagram on the previous page. Place both magnetic field finders on top of the white plastic board. Jiggle the finders a little bit to help the small iron bars line up with the magnetic field lines. Place the clear plastic sheet on top of the finders. Place the transparency on top of the clear plastic sheet, and make a sketch of the magnetic field. Reproduce that sketch below: A7. In this configuration, do the field lines get squashed, or do they reconnect? A8. Suppose you reversed the directions of both magnets in orientation, so that N becomes S and S becomes N. The direction of each field line would change, but other than that, would the map look any different? A9. When you placed the field finders on the white plastic board, did some of the small iron bars align themselves quickly? What can you say about the magnitude of the magnetic field in those places? A10. Were there other places where the small bars don't seem to know which way to align? What can you say about the magnitude of the magnetic field in those places? A11. Repeat step A6 for the other 3 configurations, and sketch the measured directions of the field lines below: 1 2 S N S N S N N S Measured Magnetic line of symmetry line of symmetry Field Line Directions: 3 4 N N N S S S S N line of symmetry line of symmetry EC-6-5

34 EC-6 MAGNETIC FIELDS AND FORCES B. Magnetic Field Strength of Permanent Magnets Our magnetic field sensor can measure only one magnetic field component at a time. It has a switch (labeled radial/axial ) to change which component it measures. If you want to measure the magnitude of a magnetic field at a particular position, put the end of the sensor at that position, and then, keeping the tip in place, rotate the sensor. The maximum measurement is the field magnitude. Before making measurements, make sure all lab partners agree which magnetic field component the Hall probe is set up to measure. If uncertain, ask your TA. Calibration information for the Hall probe is at Scroll down to the Magnetic Field of a Magnet section and use the DataStudio set-up file available there. B1. Arrange the magnets in orientation 1 in the figures above (and reprinted below). B2. Each person should do the following: move the magnetic field sensor along the line of symmetry shown in the diagram. Find the maximum field on this line, and write it down (with units). Indicate where along the line of symmetry it occurs, and what its direction is. Try hard not to notice what results your lab partners are getting. 1 Maximum magnetic field on line of symmetry: S N S N line of symmetry B3. Once all your lab team has finished, compare your results. If your results differ from your lab partners, try to figure out why. B4. Compute the average of the maximum magnetic field found by you and your lab partners, and write that down. B5. Compared to the Earth s magnetic field, this is (circle one) maximum B magnet = g Much stronger much weaker about the same B6. (If there s time) Use the magnetic field sensor to try to measure the strength of the Earth s magnetic field (an accurate measurement is made in lab E-4). EC-6-6

35 EC-6 MAGNETIC FIELDS AND FORCES C. Magnetic Force On A Current four- beam balance stand power supply strip magnet Equipment: a permanent magnet with a small gap between the poles, in which the field magnitude is expected to be between 50 and 100 mt; some 1 to 2 cm long currentcarrying copper strips mounted on a small circuit boards (see photo below); a stand suitable for holding a strip in the magnet gap; an adjustable power supply that can supply up to 3 Amps to the strip; a four-beam balance that can measure weights of masses up to 311 g with an accuracy of 0.01 g. Your TA will have a (fancy) magnetic field sensor and may use it to determine the magnetic field in the gap for you, as a check on your calculated value of the field. The readout above the FIELD knob measures the current through the strip (in Amps). Make sure the FIELD knob is set to zero, and the readout shows zero, before you start. A B C D E F Strips EC-6-7

36 EC-6 MAGNETIC FIELDS AND FORCES C1. Pick a strip, and set up the stand, strip, magnet, and balance as shown in the picture on the previous page. Make sure the strip doesn t rub against the magnet. Put the right amount of weight in the other pan of the balance so that the balance reads 0. Caution: Do not let the strip current exceed 3 Amps, and do not leave it turned up any longer than necessary! Too much current will melt the strip! (Why?) C2. Connect the power supply, set the current at 0 Amps, and make sure the balance is still. Then turn up the current briefly to 3 Amps, and bring it back to 0 Amps. What happened to the balance when the current was flowing through the strip? C3. Predict what would happen if you disconnected the cables from the power supply, and then connected them the other way (so the red one goes to GND), and redid C2. C4. Disconnect the cables from the power supply, connect them the other way, and redo C2. Does this match your predictions? C5. Which strip (A, B, C, D, E, or F) do you think will cause the largest deflection to the balance? C6. Using this strip, redo step C2, but this time measure how much weight you have to add or subtract to return the balance to 0 while current is flowing through the strip. Record it here: extra mass = g C7. In pre-lab questions 5-7 you calculated the force that a magnetic field exerted on a current-carrying wire. Which one of Newton s laws says that if object A exerts a force on object B, then object B exerts and equal and opposite force on object A? C8. Draw a free body diagram of the magnet: C9. Draw a free body diagram of the strip. C9. Use your measurement from C6, any other measurements you want to make, and your answers to pre-lab questions 5-7, to estimate the strength B of the magnetic field in the magnet gap: B = T EC-6-8

37 EC-5 MAGNETIC INDUCTION What do we do in this lab? EC-5 MAGNETIC INDUCTION In this lab we ll study three situations in which the magnetic flux changes with time. In part A, a moving aluminum pendulum experiences a changing magnetic flux as it moves through the field of a permanent magnet. In part B, a coil of wire experiences a changing magnetic flux as a permanent magnet falls through it. Finally, in part C one coil experiences a changing magnetic flux due to the changing current in a nearby coil. Why are we doing this lab? Many people find that magnetic flux, and the emf induced by changing magnetic flux, are more easily understood in the laboratory, with reference to actual objects, pictures, or sketches, than through equations. You need to know: magnetic flux the magnetic flux through a surface is the number of magnetic field lines that puncture that surface. Example: if the magnetic field has constant magnitude B throughout space, and the surface is a planar area A whose normal is parallel to the field, then we can write the equation Φ B = B A where Φ B is the symbol for magnetic flux. We won t use that equation in this lab. Instead, whenever possible, we ll draw pictures, and use the definition of magnetic flux given above the magnetic flux through a surface is the number of magnetic field lines that puncture that surface to arrive at a qualitative understanding of how the flux is changing with time. Faraday s Law the emf ε induced in a circuit is the negative of the change of magnetic flux through that circuit with change in time ε = ΔΦ B Δt The right-hand side is the negative of the slope of the graph of the magnetic flux through the circuit as a function of time. In this lab the right-hand side of the equals sign represents the cause (a change in flux with time), and the left-hand side represents the effect (the induced emf). The negative sign is associated with Lenz s Law. Inductor circuit element designed to maximize the importance of induced emf. Inductive time constant pertains to circuits containing inductors; length of time it takes the current to respond when the voltage is changed. EC-5-1

38 EC-5 MAGNETIC INDUCTION Pre-lab Questions A z 1. As shown in the diagram, an object moving to the right is about to pass through a region in which there is a magnetic field directed out of the page. B y Consider the circuit represented by the dashed line. The magnetic flux through this circuit, as a function of time, is shown by the top graph at left. Fill in the bottom two graphs, being careful to make sure that they line up properly with respect to time. 2. The diagram at top shows the object at about which time (circle one): t 1 t 2 3. The object is made of aluminum, and the induced emf causes a current to flow. At time t 1, in which direction is this induced current (circle one): clockwise counter-clockwise 4. At time t 2, in which direction is the induced current (circle one): clockwise counter-clockwise 5. As you know, there is magnetic force on a current-carrying conductor in a magnetic field. In which direction is the magnetic force on this object? (Hint: it is in the same direction at t 1 and t 2.) EC-5-2

39 EC-5 MAGNETIC INDUCTION N 5. The picture at left shows a bar magnet falling though a stationary coil. In which direction is the magnetic flux through the coil (circle one): upwards downwards S 6. Assume the magnet starts from well above the coil and falls downward through the coil, continuing onward until it is well below the coil. Fill out the graphs below. (Hint: The flux starts out very close to zero, when the magnet is far above the coil, and ends up very close to zero, when the magnet is far below the coil.) 7. Now suppose that the coil had the same location and orientation, but the radius was much larger (the hole through the center was really big). What would be different about the emf induced around the coil? 8. Recall that a solenoid a coil of wire besides resistance, also has inductance. If the number of turns of wire is N, the length is l, and the radius is r, what is the inductance L of the solenoid? ε S R L 9. In the circuit at left, what is the approximate time required to start a current flowing through the solenoid after the switch S is closed? (This is called the inductive time constant). EC-5-3

40 EC-5 MAGNETIC INDUCTION A. Magnetic Induction: Stationary Magnet, Moving Object You have four Aluminum pendulums made with differently-shaped ends (see figure below). You have a strong magnet set up so that the pendulums can swing through a gap between the poles. From the pre-lab, you expect that each of the pendulums will experience a magnetic force due to the current induced as it swings through the gap. A B C D A1. Is Aluminum ordinarily magnetic? Fig. 1 Four pendulums A2. Based on Fig. 1, predict which pendulum will slow down the most, and which will slow down the least, due to the magnetic force. predicted to slow down most: predicted to slow down least: A3. Now do the experiment: test all four pendulums, and see which one actually slows down most, and which one actually slows down least. actually slowed down most: actually slowed down least: A4. Take pendulum D in your hand, and move it as quickly as you can through the gap without hitting the sides. Can you feel a force resisting the motion? A5. As far as you can see, does the magnetic force always act in a direction so as to slow the pendulum down, or does it ever act in a direction so as to speed the pendulum up? EC-5-4

41 EC-5 MAGNETIC INDUCTION B. Magnetic Induction: Stationary Object, Moving Magnet Your test apparatus is a 400-turn coil labeled according to its winding direction, mounted on an adjustable stand so that a bar magnet can be dropped through it from different distances above it. There is a rubber stopper to cushion the landing of the bar magnet, and a plastic tube to keep it from falling on the floor. The notched end of the bar magnet is its N pole. 400-turn coil Adjustable stand, showing 400-turn coil You ll use a PASCO voltage sensor to measure the emf induced in the coil by the passage of the magnet through the coil. B1. If you drop the magnet south pole first through the coil, what is the direction of field lines puncturing the surface bounded by the coil, upwards or downwards? B2. As the south pole enters the coil, what is the direction of the field the coil will try to generate in order to oppose the change, upwards or downwards? B3. A little later, when the north pole leaves the coil, what is the direction of the field the coil will try to generate in order to oppose the change, upwards or downwards? EC-5-5

42 EC-5 MAGNETIC INDUCTION B4. When the north pole is passing through the coil, it is moving slightly faster than the south pole was moving when the south pole passed through the coil. How does this fact affect the graph of induced voltage vs. time? B5. How would the graph change if the magnet were reversed and fell north pole first? B6. Turn on the Pasco interface, and connect the voltage sensor to Analog Channel A. B7. Go to and download and launch the Pasco setup file ec5b.ds. The magnet is very brittle, and can break easily. Always hold the bottom of the tube/stopper tightly when dropping the magnet. B8. Use the thumbscrew to raise the coil as high on the stand as it will go. B9. Take the magnet out of the tube. Replace the rubber stopper at the bottom. B10. Click Start. B11. Making sure to hold the bottom of the tube, drop the magnet in the tube, S pole downward. B12. Click Stop. B13. Use the magnifying glass icon (the fourth from left) to blow up the spiky part of the curve you just recorded. The graph you see is the emf induced in the copper coil. B14. Why are the two peaks in different directions? B15. Why is the second peak taller than the first? B16. Move the coil as low on the stand as it will go. What do you expect to see on the graph when you drop the magnet, as compared to the graph from B13 (circle one of each pair): peaks will be taller peaks will be broader peaks will be closer together in time Now drop the magnet and see if you were right. peaks will be not so tall peaks will be narrower peaks will be be further apart in time B17. If you drop the magnet N pole downward, how will the graph be different? Try it and see. EC-5-6

43 EC-5 MAGNETIC INDUCTION B18. Now suppose we mounted the coil from part B on a pendulum like one of the pendulums in part A, and let it swing through the field of a magnet, like in part A (see picture below), and measured the emf induced in the coil with a voltage sensor, like in part B. What would the graph look like would it have two peaks in different directions? EC-5-7

44 EC-5 MAGNETIC INDUCTION C. Faraday's Law And Inductance In part B, you dropped a magnet through a coil, and measured the emf induced in the coil by the changing magnetic flux from the falling magnet. Here, you measure the emf induced in a coil (the detector coil) by the changing magnetic flux from another coil (the driver coil) that has a time-varying current flowing through it. C1. Set up the circuit in the picture below. Driver coil Detector coil Analog Channel A: voltage across 100Ω resistor. This is proportional to the current through the driver coil. Analog Channel B: voltage across detector coil. Analog Channel C: input to Power Amplifier. The driver coil has about 2600 turns of wire, is about 2.8 cm in diameter and 11 cm long, and has a resistance of around 82Ω. The detector coil fits neatly inside the driver coil. There is also an iron rod, which fits neatly inside the detector coil. Putting the iron rod inside the detector coil will increase the flux through the detector coil, and therefore increase the emf induced in the detector coil. C2. Go to and download and launch the Pasco setup file ec5c.ds. C3. Take data with the iron rod removed from the detector coil. C4. Take data with the iron rod inserted into the detector coil. C5. Your data should look similar to the graph on the next page. The purpose of the next few steps is to estimate the inductive time constant of this circuit, with and without the iron rod inserted. You ll need to know the value of e, the base of the natural logarithm: e = EC-5-8

45 EC-5 MAGNETIC INDUCTION C6. All the waveforms in the graph have the same frequency; what is that frequency? C7. Consider the fourth waveform between and seconds. During this time, the voltage exponentially decreases, getting closer and closer to zero. The time constant of this exponential decrease is the time it takes the voltage to drop by a factor of e. What is the time constant of the fourth waveform? τ with = seconds C8. Estimate the time constant of the second waveform. τ without = seconds C9. What is the total resistance of the circuit which includes the Power Amplifer and the driver coil? R = Ohms C10. Using your answers to pre-lab question 9 and C7, estimate the measured inductance of the driver coil with the iron bar in place: Inductance of driver coil with iron bar inserted = Henry C11. Using your answers to pre-lab question 9 C7, estimate the measured inductance of the driver coil by itself, without the iron bar in place: Inductance of driver coil by itself= Henry C12. Using the numbers given on the previous page, and your answer to pre-lab question 8, calculate the theoretical inductance of the driver coil: Theoretical inductance of driver coil by itself = Henry EC-5-9

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47 E-4 MAGNETIC FORCE ON A CHARGED PARTICLE E-4 MAGNETIC FORCE ON A CHARGED PARTICLE What do we do in this lab? In this lab you ll use an apparatus that allows you to see the paths of electrons moving in a magnetic field, and even steer the electrons by turning a knob. The lab has two parts. In the first part, which is qualitative, you see the paths the electrons take for various combinations of initial electron velocity and magnetic field strength. In the second part, you use the same apparatus to make a quantitative measurement of the Earth s magnetic field. Why are we doing this lab? This lab gives you good practice making qualitative predictions of the motions of charged particles in magnetic fields. We hope that you will also find that the lab makes you more comfortable believing that electrons really exist. Finally, many students in the past have said they enjoyed doing this lab. Safety concerns: once the apparatus is set in position, it is secured by a wingnut. If the wingnut isn t tightened properly, the apparatus could suddenly drop onto the table, which could pinch someone s fingers. The apparatus includes a glass vacuum tube, which could break if the apparatus were dropped. You need to know: No new words for this lab E-4-1

48 E-4 MAGNETIC FORCE ON A CHARGED PARTICLE Pre-lab Questions 1. An electron is knocked loose from an atom in the middle of a region of constant magnetic field, and initially moves in the direction shown by the arrow in the diagram. Sketch the subsequent path of the electron. e" 2. Here is the same situation as above, seen from a different viewpoint. Sketch the path of the electron. e" 3. A different electron appears in the same place as the first, and is observed to follow the path shown in the sketch at right. e" However, as seen from the same point of view as question 2, this electron initially has a component of its velocity parallel to the field, as shown in the diagram to the right. Sketch the subsequent path of this electron. 4. What are the one-word names of the paths followed by the two electrons? 5. What is the force F B on a charge q moving with speed v perpendicular to a magnetic field B? F B = 6. Write down the equation for the centripetal force F C acting on an object of mass m and speed v that is observed to move in a circle of radius r: F C = 7. An electron is accelerated from rest by passing through a potential difference ΔV. Write down the equation that relates ΔV to the electron s final speed v: 8. Combine the equations from questions 5, 6, and 7 to come up with a single equation for B r, the magnetic field needed to steer an electron that has been accelerated through a potential difference ΔV into a circle of radius r. v = B r = e" E-4-2

49 E-4 MAGNETIC FORCE ON A CHARGED PARTICLE EQUIPMENT The heart of this experiment is a vacuum tube, from which all the air has been removed, and replaced with a small amount of Mercury vapor. Inside the vacuum tube are a filament, a cylindrical anode with a window in it, and a series of posts, which are 0.065, 0.078, 0.090, and meters from the filament. When current flows through the filament, it heats up. The larger the current, the hotter (and brighter) the filament gets. If enough current flows, the filament gets white hot, and emits electrons. These are accelerated by the anode voltage ΔV towards the anode. Most hit the anode, but a few pass through the window. When they collide with Mercury atoms, blue light is emitted. If there is no magnetic field in the vicinity, the electrons that pass through the window keep going in a straight line, but if there is a magnetic field, the electrons follow a curve. If the field is uniform, the electrons move in a circle or spiral. Helmholtz coils are designed to produce a very uniform magnetic field in the region between the coils. At the center of a pair of Helmholtz coils B = µ 0 NI R, Equation 1 where B is the magnetic field strength in Tesla, N is the number of turns in the coils, R is the coil radius in meters, I is the coil current in Amperes, and µ 0 is the permittivity of free space, 4π x 10-7 Teslameters/ampere. Our particular set of Helmholtz coils have N=72 turns. By varying I, you can steer the electrons into one or the other of the posts. Coil current Helmholtz Coils Tube rotates inside coils Vacuum tube Coil field Coil current Entire unit tilts to align coil field with Earth s field The vacuum tube is mounted inside the Helmholtz coils in such a way that the entire unit can be oriented in space in any direction and at any angle with respect to the horizontal. This allows the magnetic field of the Helmholtz coil to be oriented parallel to the Earth s magnetic field. The vacuum tube can also be rotated 180. Therefore, whether the coil field is aligned parallel to the Earth s field, or anti-parallel to the Earth s field, the tube can be rotated so that electrons can be steered into posts. E-4-3

50 E-4 MAGNETIC FORCE ON A CHARGED PARTICLE Power Supply The current I through the Helmholtz coils is controlled by the Field knob on the power supply, and monitored on the FIELD Amps display. The current through the filament is controlled by the Filament knob, and monitored on the ANODE milliamps display. The switch above the Anode terminal sets the voltage ΔV that accelerates the electrons after they leave the filament. You will also have both an ordinary compass and a dip compass for determining the direction of the earth s magnetic field. A. Observation of Electron Trajectories A1. Use the ordinary compass and dip compass to find the direction of the Earth s magnetic field. A2. Align the Helmholtz coils with the Earth s magnetic field. A3. The glass vacuum tube has 3 terminals. Connect the two that are on opposite sides of the tube to the Filament terminals on the power supply. Connect the remaining terminal to the Anode terminal on the power supply. A4. Turn both the Filament and the Anode knobs to zero. Set the Anode switch to 22 V (the ANODE Volts readout gives a more accurate reading of the anode voltage). A5. Connect the red and black terminals on the Helmholtz coils to the FIELD terminals on the power supply in such a way that the Helmholtz coil field is parallel, not anti-parallel, to the Earth s magnetic field. (Hint: in order to do this, you ll need to know the right-hand rule.) A6. Turn on the power supply. A7. If you want to make the electron beam brighter, which knob do you turn, the filament knob or the field knob? A8. If you want to steer the electron beam into a tighter circle, which knob do you turn, the filament knob or the field knob? A9. Turn the Filament knob up slowly while observing the filament. Make sure the filament current stays below 15 ma. You should see the filament start to glow as it becomes hot (this takes a few seconds). You may need to turn the Filament knob beyond the halfway position before current flows. Increase the current to the anode until you are confident you can clearly see the beam. E-4-4

51 E-4 MAGNETIC FORCE ON A CHARGED PARTICLE A10. Slowly turn up the current in the field coils. Well below 1 ampere, you should be able to observe the electron beam begin to bend into a circle. As the Helmholtz coil current increases, the radius of the circle decreases, so you can make the beam hit any one of the target posts. A11. As alluded to in pre-lab questions 1-4, the path of an electron moving in a magnetic field is in general a helix, a special case of which is the circle (for electrons with velocity perpendicular to the field). Rotate the vacuum tube until you are confident you have seen the electron beam trace out both a helix and a circle. Call over your TA if you find you can t explain why the electron beam traces out the path that it does. B. Measurement of Earth s Magnetic Field The Earth s magnetic field is considerably smaller than the field of the Helmholtz coil, and a vacuum tube large enough to encompass the circles and helices made by electrons spiraling around the Earth s magnetic field lines would be inconveniently large. Nonetheless we can still use our apparatus to measure the Earth s field, in the following way. We already have the Helmholtz coil field aligned with the Earth s field. We ll adjust the Helmholtz coil field as necessary to steer the electron beam into the posts, and record the Helmholtz coil current. Then we ll reverse the current in the Helmholtz coils, so that the Helmholtz coil field is anti-parallel to the Earth s field. We will also have to rotate the tube 180 to compensate. Since the Earth s field is now fighting the Helmholtz coil field, a different Helmholtz coil field, and therefore Helmholtz coil current, will be needed to steer the beam into the posts. The strength of the Earth s field can be found from the difference between the two cases. Once everyone gets to this point (if not before), it is time to darken the room, because this makes it easier to see the blue electron beam. Use the cardboard light shield to block stray light. You may want to have one lab partner stand above the model, looking down, because from that vantage point it is easier to see when the beam hits a post. When it hits a post, it can actually be seen to split in two, with the two parts being deflected to opposite sides of the post. While one lab partner views the model from above, another can adjust the current and a third can record data. Be sure everyone gets to try each position. E-4-5

52 E-4 MAGNETIC FORCE ON A CHARGED PARTICLE B1. Let post 1 be the post closest to the filament. Fill in the first three columns of the table below. For the third column, you ll need to use equation (1) above. Post r (m) I parallel (A) B parallel (T) I anti-parallel (A) B anti-parallel (T) B2. Reverse the direction of the current in the Helmholtz Coils, and spin the vacuum tube 180 around its axis. The Helmholtz coil field should now be antiparallel to the Earth s field, but you should still be able to steer the electron beam into the posts by adjusting the Field knob. B3. Fill in the last four rows of the last two columns. Our power supply can t put out enough current to steer the beam into post 1, when the Helmholtz coil field has to fight the Earth s field. B4. Discuss with your lab partners how you can use the data from the above table to calculate the Earth s magnetic field. Write down the formula here: B Earth = B5. For posts 2-5, solve for B Earth, and fill in the table below. Post B Earth (T) B6. Look up the accepted value of B Earth, and compare it to your result. Further question Have you ever seen an aurora borealis, also known as the northern lights? In many ways this experiment is similar to the aurora. E-4-6

53 L-2 FIND THE FOCAL LENGTH What do we do in this lab? L-2 Find the Focal Length This lab has three parts. The first part doesn t use any equipment, but just gives you practice with sign conventions and ray tracing. In the second part, you measure the focal length of a converging lens, and in the third part, you measure the focal length of a diverging lens. Why are we doing this lab? To gain practical experience with lenses. Safety concerns: none You need to know: Converging lens lens that bends light rays together. Converging lenses are thicker in the middle than at the edge. Diverging lens lens that spreads light rays apart. Diverging lenses are thinner in the middle than at the edge. Converging lens Diverging lens L-2-1

54 L-2 FIND THE FOCAL LENGTH Pre-lab questions 1. Write down the thin lens equation (not the lensmaker s equation!). 2. According to this equation, what two quantities do you need to know in order to be able to calculate the focal length of a lens? 3. What is the equation for the magnification of a thin lens? 4. Is the focal length of a given lens always the same, or does it depend on how far it is from the object? 5. Is the magnification of a given lens always the same, or does it depend on how far it is from the object? 6. Does magnification always mean something looks bigger through the lens? When could it mean otherwise? 7. Suppose you place the light source and the screen on the optical bench, with the source pointing at the screen. Will the source by itself form an image on the screen? Why not? 8. Which kind of lens, converging or diverging, is capable of forming a real image of the source on the screen? 9. Does it make any difference what a lens does, if you turn it around front-toback? A. Sign Conventions and Ray Tracing A1. Using the letter f to stand for focal length, circle the right word ( positive or negative ) under each lens: Converging lens diverging lens f is positive/negative f is positive/negative L-2-2

55 L-2 FIND THE FOCAL LENGTH A2. The light rays all pass through two points called the object and the image. The distance from the object to the lens is called p and is assumed to be positive if the light rays diverge from the object as they travel towards the lens. Another way of saying this is that p is positive if the object is where the light is. Circle the right word below each picture: p is positive/negative p is positive/negative A3. If p is positive we say the object is (circle one) real virtual A4. If p is negative we say the object is (circle one) real virtual. The distance from the image to the lens is called q and is positive if the rays converge after traversing the lens, and negative if they diverge. Another way of saying this is that q is positive if the object is where the light is. Circle the right word below each picture: q is positive/negative q is positive/negative A5. If q is positive we say the image is (circle one): real virtual A6. If q is negative we say the image is (circle one): real virtual. A7. Light rays actually converge at the position of a real image. An image would appear on a screen placed at this position. No image would appear on a screen at the position of a virtual image. Circle the right word below each picture: real/virtual image real/virtual image L-2-3

56 L-2 FIND THE FOCAL LENGTH Here are some drawings for lenses with positive and negative focal lengths. Which is which? (circle the right choice) A8. f is positive negative A9. f is positive negative A10. f is positive negative L-2-4

57 L-2 FIND THE FOCAL LENGTH Equipment You have an optical bench, four lenses of unknown focal lengths that are marked with different colors of tape, a screen for viewing images, and a light source with a set of crosshairs for focusing and a millimeter scale. You also have a ruler for measuring the sizes of images. B. Focal Length of Converging Lens B1. By feeling the lenses, identify which lenses are converging and which are diverging. Write the colors of tape in the appropriate row below: Converging: Diverging: B2. Pick the converging lens with the shortest focal length. Discuss with your lab partners what process you would use to estimate the focal length of this lens if you were marooned on a desert island. If nothing comes to mind, try to use the lens to focus the rays from some nearby object onto any surface you can this might help you get started. Ask your TA for help if you re really stuck. Once you have a process in mind, carry it out, and record the results here: Color of tape on lens: approximate focal length: cm B3. A more accurate method for measuring the focal length of a converging lens: p q Put the light source a distance p from the lens, and then move the screen until an image of the pattern on the front of the light source appears on the screen. The screen is then at a distance q from the lens. Measure p and q, and then use the thin-lens equation (pre-lab question 1) to calculate f: p = cm q = cm f = cm L-2-5

58 L-2 FIND THE FOCAL LENGTH B5. Use the measured p and q and the result from pre-lab question 3 to calculate the predicted magnification of the image: predicted magnification of image: B6. The pattern on the light source is shown at right. The tick marks on the pattern are exactly 1 mm apart, and circles have diameters of 1 cm and 2 cm. Measure the distance between the tick marks on the screen, and use this to calculate the measured magnification of the image: measured magnification of image: C. Focal Length of Diverging Lens C1. Repeat step B3 for the longer-focal-length converging lens: p = cm q = cm f = cm C2. Draw a ray diagram of this setup: C3. If you now added a diverging lens in between the converging lens and the screen, would that move the image closer to, or further from, the light source? closer to further from p 1 q 2 Δ C4. Use this insight to add the diverging lens, and move the screen, so that an image appears (you may need to move the converging lens also). C5. Subscript 1 refers to the converging lens, subscript 2 refers to the diverging lens. The distance Δ between the lenses satisfies the equation q 1 + p 2 = Δ. Measure p 1, q 2, and Δ: p 1 = cm q 2 = cm Δ = cm C4. You now have enough information to solve for p 2, and then for the focal length f 2 of the diverging lens. Record your answers here: p 2 = cm f 2 = cm L-2-6

59 LC-1 DIFFRACTION AND INTERFERENCE LC-1 Interference and Diffraction What do we do in this lab? In the lab we observe single-slit diffraction patterns and double-slit interference patterns. Why are we doing this lab? This lab demonstrates the wave nature of light. Safety concerns: The laser can damage your eyes. Do not look into the laser, put your face near the laser beam, or point the laser near someone else s face. You need to know: Diffraction the bending of light around obstacles, which causes the light to spread out. Interference interaction of one wave with another wave, which results in a third wave which may have considerably larger amplitude (constructive interference) or smaller amplitude (destructive interference) than either of the first two. Slit a rectangular hole. Single-slit diffraction diffraction of light by a single slit (see picture below). There is no light at the angles θ dark at which: asinθ dark = mλ, m = ±1,±2,±3... where a is the slit width and λ is the wavelength of the light. destructive interference m=2 LASER slit width a all in phase m=1 y a L D ( > 1 meter) m= 1 m= 2 m Diffraction minima at =m /a Double-slit interference interference between two single-slit diffraction patterns. The light is bright at d sinθ bright = mλ, m = 0,±1,±2... LASER in phase θ y m=2 m=1 m= 2 m= 1 m=0 a d d a (a << d) m λ θ in phase L ( > 1 meter) Interference maxima at θ= mλ/d LC-1-1

60 LC-1 DIFFRACTION AND INTERFERENCE Pre-lab questions: 1. Below is sketch of two waves interacting. Is this constructive or destructive interference? 2. Below is another sketch is this constructive or destructive interference? 3. Write down the equation for the intensity minima (destructive interference) of the single-slit diffraction pattern: 4. Suppose you knew the distance of the first minimum from the central maximum on a viewing screen. What else would you need to know in order to calculate the wavelength of the light making the pattern? 5. Write down the equation for the intensity maxima (constructive interference) of a double-slit interference pattern. 6. If you knew how far apart the maxima were, what else would you need to know in order to calculate the wavelength of the light making the pattern? The object of this lab is to measure a single-slit diffraction pattern (part A) and a double-slit interference pattern (part B), and use those measurements to infer the wavelength of a laser. LC-1-2

61 LC-1 DIFFRACTION AND INTERFERENCE Equipment light sensor assembly doubleslit set singleslit set laser screen You have an optical bench, sets of single and double slits, a 650 nm red laser, a viewing screen, a light sensor assembly, and the Pasco Interface. The laser has two screws for aiming the beam. One controls vertical aim and the other controls horizontal aim. The single-slit set has a number of different-width slits, and some other shapes. The laser beam must pass through one of the single slits and then enter the light sensor aperture. Likewise for the double-slit set. The light sensor assembly consists of a highsensitivity light sensor, an aperture bracket, and a translation wheel (see photo at right). Use the translation wheel to move the light sensor perpendicular to the laser beam path to quantitatively measure intensity variations of the interfering or diffracted light. The different apertures on the aperture bracket allow different amounts of light into the light sensor, and set the spatial resolution of the light sensor. aperture bracket translation wheel Aperture #4 works well, with a gain setting of 10 (slide switch on light sensor). If this is too sensitive (so that the light sensor saturates, as evidenced by flat tops to the diffraction peaks), use a gain of 1. If it isn t sensitive enough, use 100. LC-1-3

62 LC-1 DIFFRACTION AND INTERFERENCE A. Single-slit Diffraction A1. Put the laser at one end of the optical bench. Connect it to its power supply. A2. Put the screen at the other end of the optical bench. A3. Put the single-slit set in front of the laser. A4. Rotate the single-slit wheel and observe the different diffraction patterns from the different slits. What happens to the diffraction pattern as the slit width decreases? A5. Set the single-slit wheel so the laser illuminates the slit with width a = 0.04 mm. A6. Replace the screen with the light sensor assembly. Connect the light sensor to Analog Channel A of the PASCO interface. Connect cable top side up! A7. Connect the rotary motion sensor: yellow goes in Digital Channel 1, and black in Digital Channel 2. A8. Position aperture #4 on the aperture bracket in front of the input of the light sensor (the 6 o clock position). A9. Align the laser as necessary to make the diffraction pattern appear at the same height as the center of aperture #4. A10. Go to and download and launch file lc1.ds. A11. Move the light sensor off to one side, so that the light isn t hitting it. A12. Click start on DataStudio. Slowly, by hand, move the light sensor across the diffraction pattern. You should see a pattern on your graph. Click stop. A13. If your graph shows just see a small spike with no width, right-click on the graph, and hit Scale. You should now see a diffraction pattern. Call over your TA if you don t. The relevant equations for single-slit diffraction are: asinθ dark = mλ, and the small-angle approximation sin θ ~ tan θ = y/l. m = ±1,±2 A14. Measure L, the distance from the slit (not the laser) to the light sensor: L= m A15. From your graph, determine y, the horizontal distance from the central maximum, for both first-order minima and both second-order minima: first-order minimum on left side of central maximum y 1 left = cm second-order minimum on left side of central maximum y 2 left = cm first-order minimum on right side of central maximum y 1 right = cm second-order minimum on right side of central maximum y 2 right = cm LC-1-4

63 LC-1 DIFFRACTION AND INTERFERENCE A16. Since θ is so small, you can use the small angle approximation sin θ ~ tan θ ~ θ. Use the two equations above to get rid of θ, and solve for λ in terms of y: λ = A17. Use the four measurements from A15 to calculate λ four times. (from y 1 left ) λ = nm (from y 1 right ) λ = nm (from y 2 left ) λ = nm (from y 2 right ) λ = nm A18. On the back of the laser you should see an expected value for the wavelength. Compare your results with this value. B. Double-slit Interference For the double-slit device, aiming and setup are similar. Remember the equations for interference are slightly different than for diffraction: d sinθ bright = mλ, m = 0,±1,±2... sinθ ~ tanθ = y L (d is the distance between the slits, while L is still the distance to the screen). B1. Put the double-slit set in front of the laser and observe the patterns from each of the double slits. B2. Qualitatively, what happens to the interference pattern if the slit width a is kept the same, but the slit spacing d is increased? B3. Qualitatively, what happens to the interference pattern if d is kept the same, but a is increased? B4. Choose the double slit with a = 0.04 mm and d = 0.50 mm. Repeat the procedure from part A, only use y for the first- and second-order maxima instead of minima. Record your measurements in this table: y 1 left y 2 left y 1 right y 2 right = cm (from y 1 left ) λ = nm = cm (from y 2 left ) λ = nm = cm (from y 1 right ) λ = nm = cm (from y 2 right ) λ = nm LC-1-5

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65 LC-4 PHOTOELECTRIC EFFECT What do we do in this lab? LC-4 Photoelectric Effect In this lab we are going to use the photoelectric effect to measure the work function of a particular meta (an alloy of Cesium and Antimony), measure Planck s constant, and investigate whether more intense light causes photoelectrons to be knocked off the Cesium/Antimony alloy faster than less intense light. Why are we doing this lab? We are doing this lab in order to give a balanced account of the quantum nature of light. The existence of the photoelectric effect proves that light consists of particles, which we call photons. Compare lab LC-1 the existence of the interference and diffraction of light proves that light consists of waves. Apparently light has a dual nature, sometimes behaving as if it is made of particles, sometimes behaving as if it is made of waves. Safety concerns: our light source is a Mercury lamp that emits ultraviolet light, which can damage your eyes. Do not look directly at the Mercury lamp. You need to know: Maximum kinetic energy in the context of this lab, Maximum kinetic energy is the maximum kinetic energy an electron ejected from a metal via the photoelectric effect can have, when the metal is illuminated by light of a particular frequency. It therefore depends on the frequency of the light. photon particle of light. Planck s constant fundamental constant of nature, h = 6.63 x J s; ratio of energy to frequency for photons. work function the work function of a metal is the minimum energy a photon needs in order to cause an electron to be ejected from the metal. LC-4-1

66 LC-4 PHOTOELECTRIC EFFECT Pre-Lab Questions 1. What is a photon? 2. What is an ev? 3. Which has more energy, photons of blue light or photons of red light? 4. Write down the equation describing the relationship between the frequency and the wavelength of a photon. 5. Write down the relationship between the energy and the frequency of the photon. 6. Combine the two equations into one that describes the relationship between the energy and the wavelength of a photon. 7. How do you convert between Joules and ev? 8. The wavelength of visible light ranges from about 400 nm to about 700 nm. To what energy range, in ev (electron volts), does this correspond? 9. Write down the equation for the maximum kinetic energy of the electrons in the photoelectric effect in terms of photon frequency. 10. This should be the equation of a straight line, with frequency on the horizontal axis and maximum kinetic energy on the vertical axis. Make a sketch of this line: 11. What is the slope of the line? 12. In terms of the work function, where does the line intersect the vertical (energy) axis? 13. In terms of Planck s constant, where does the line intersect the horizontal (frequency) axis? LC-4-2

67 LC-4 PHOTOELECTRIC EFFECT Equipment photoelectric detector diffraction grating mask mercury lamp multimeter intensity filter color filters The mercury lamp emits light of only five different wavelengths: 578 nm (two overlapping yellow lines); nm (green); nm (blue); nm (violet); and nm (ultraviolet). The ultraviolet line is beyond the range of human vision. These lines are separated by the diffraction grating. The photoelectric detector can be moved from side to side, so that one of the lines falls on the hole in the mask. If you raise the top of the black cylinder behind the mask, you can look inside the photoelectric detector. You should see a white cylinder with an opening on its side. Inside the white cylinder, but not electrically connected to it, is the Cesium/Antimony alloy. When sufficiently energetic photons hit the Cesium/Antimony alloy, they knock off electrons, which then travel to the white cylinder, and collect there. The white cylinder therefore charges negative, and the potential of the white cylinder becomes LC-4-3

68 LC-4 PHOTOELECTRIC EFFECT negative with respect to the Cesium/Antimony alloy. Eventually the electrons knocked off the Cesium/Antimony alloy can t reach the white cylinder. The voltage difference between the Cesium/Antimony alloy and the white cylinder can be monitored from the OUTPUT jacks. This is made possible by a battery-powered differential amplifier inside the detector. With the power switched on, you can check the battery by connecting the Fluke multimeter, set to read volts on the auto scale, between each of the BATTERY TEST jacks in turn and the ground jack (one test jack should give +6V, the other -6V, although higher readings, up to ±8V, are OK; tell your TA if the readings are lower). The PUSH TO ZERO button discharges the white cylinder, so that it is at the same potential as the Cesium/Antimony alloy. The photoelectric detector has to be carefully aligned so that photons can reach the Cesium/Antimony alloy. To align the detector (you may have to do this multiple times during the experiment): raise the black metal cylinder behind the mask, so that you can see the inside of the detector; align the spectral line of your choice so that it falls on the opening of the mask; turn the detector so that the spectral line falls on the opening in the white cylinder, making sure that the line still falls on the opening in the mask; Once you have everything aligned, tighten the thumbscrew on the detector to hold it in place; Be careful not to bump or disturb the housing once you have it aligned. A. Exploration A1. Pull the diffraction grating out from the mercury lamp until it is directly above the hinge. A2. Plug in and turn on the mercury lamp. Don t turn it off until you ve finished the lab it takes a few minutes to get bright. LC-4-4

69 LC-4 PHOTOELECTRIC EFFECT A3. Your TA will now turn the lights off for 5-10 minutes. When the lights are off, move the detector from side to side, and observe the lines that appear on the white mask of the detector. Try to identify: the zeroth-order line (also known as the central maximum ); the first-order lines; the second-order lines; at least one third-order line (but don t worry if you can t see any). If there is any confusion as to which line is which, talk it over with your partners. A4. The filters come in a white plastic box. Hold this up in front of the mask. How many of the first-order lines can you see? A5. Now hold up a piece of white paper in front of the mask. Now how many lines can you see? As noted above, one of the five lines is beyond the range of human vision. However, if all went well, in step A3 above you saw five first-order lines on the mask. If all five lines appear on a surface, that is a good indication that the surface is fluorescent (i.e. glows in the dark ). The mask is fluorescent, the white plastic box is not; some types of paper are and some aren t. A6. Test anything you like to see if it is fluorescent articles of clothing, things you have in your backpack, skin just don t look straight into the light, because the ultraviolet can damage your eyes. Soon your TA will turn the room lights back on. B. Work Function And Planck s Constant B1. Make sure the detector is plugged in (the plug is on the bottom) and turned on. B2. Test the battery. If the battery is not OK, call over your TA. B3. Set the multimeter to measure DC Volts and connect it to the OUTPUT jacks. B4. Align the detector so that the first-order ultraviolet line falls on the Cesium/Antimony alloy. B5. Push the Push to Zero button, and see whether the voltage reading on the multimeter drops momentarily to zero, and then increases, at first rapidly, and then more and more slowly. If you push the button and the voltage does not build up, that may mean that light isn t shining on the target electrode, and the housing needs to be re-aligned (the cables may pull the detector out of alignment). LC-4-5

70 LC-4 PHOTOELECTRIC EFFECT B5. With the detector aligned to the first-order ultraviolet line, measure the maximum voltage measured by the multimeter, after waiting a while for the multimeter reading to stop increasing. Record it in the third column of the table below. Wavelength (nm) Frequency (Hz) Maximum multimeter voltage (V) Maximum kinetic energy (ev) B6. Repeat for the other four first-order lines. Use the yellow and green filters when making measurements with the yellow and green lines. (Why are there no blue, violet and ultraviolet filters? Because measurements on the yellow and green lines are more sensitive to interference from room light and daylight.) B7. Calculate the frequencies for the five lines, and fill in the second column. B8. Discuss with you lab partners the relationship between the maximum multimeter voltage measured for a line, maximum kinetic energy of electrons ejected by light from that line. Fill in the fourth column. B7. Use Excel to make a graph of maximum kinetic energy (vertical axis) vs. frequency (horizontal axis). Make the units of frequency be Hz, and the units maximum kinetic energy be ev. Fit a trendline to this (for tips on how to do this, see the Excel Tutorial appendix in this lab manual). B8. From the graph of maximum kinetic energy vs. frequency, deduce the work function of the metal, and Planck s constant. Compare these to the accepted values (for the Cesium/Antimony alloy we are using, we expect the work function to be about 1.5 ev). Measured value of work function of Cesium/Antimony alloy: ev Measured value of Planck s constant: J s Accepted value of Planck s constant: J s LC-4-6

71 LC-4 PHOTOELECTRIC EFFECT C. Dependence Of Charging Time On Intensity C1. Connect the Pasco Voltage Sensor to the OUTPUT jacks and the Pasco Interface Analog Channel A. C2. Go to and download and launch lc4.ds. C3. Align the detector with the ultraviolet line. C4. Start taking data with DataStudio, then press the PUSH TO ZERO button carefully, so as not to disturb the alignment. Part of your graph should look like the graph below. Call over your TA if it doesn t. C5. Keep taking data until the graph levels off. From the graph, estimate the time it takes for the voltage to reach 90% of its maximum value. Discuss your estimate with your lab partners until you are in agreement. We will call this the charging time. charging time for full-intensity line = τ 100 = s C6. When the PUSH TO ZERO button is released, it can make a big voltage spike on your graph. If, after repeated trials, this prevents you from getting a good estimate of the charging time, call over your TA. C7. The grayscale filter lets you reduce the intensity of a line to 80%, 60%, 40%, or 20% of its initial value. Using the grayscale filter, measure the 90% charging time for each of these four reductions in intensity: τ 80 = s τ 60 = s τ 40 = s τ 20 = s Make sure each team member has a chance to operate the computer, and (carefully!) push the button. LC-4-7

72 LC-4 PHOTOELECTRIC EFFECT C8. If you see the voltage building up fast, what does that imply about the rate at which electrons are being knocked off the alloy? C9. Do you expect the 90% charging time to be proportional to the intensity, or inversely proportional to the intensity? C10. Use τ to denote the 90% charging time. If you think charging time is proportional to intensity, make a graph of τ (on the vertical axis) vs relative intensity (on the horizontal axis). If you think charging time is inversely proportional to intensity, make a graph of 1/τ vs. relative intensity. Make sure to show your graph to your TA. C11. Does your graph support or refute your prediction about whether the 90% charging time is proportional, or inversely proportional, to the intensity of the light? LC-4-8

73 What do we do in this lab? L-5 THE BALMER SERIES L-5 The Balmer Series We measure the wavelengths of three spectral lines emitted by hot Hydrogen gas. Why are we doing this lab? Much of modern physics and chemistry depends on the accurate measurement of the wavelengths spectral lines. If a particular substance is heated it always gives the same set of spectral lines, which clearly follow some sort of pattern. Niels Bohr was the first to (partially) explain such a pattern, with his 1913 theory of the Hydrogen spectrum. This was the first step towards our modern understanding of the atom. This lab is thus of both great conceptual as well as historical interest. Safety concerns: Avoid touching the electrodes holding the glass tube which contains the hydrogen. The glass tube itself gets hot when it is lit don t touch it, and don t let the black cloth touch it. You should know: Balmer Series set of spectral lines emitted by Hydrogen atoms. A Balmer Series line is emitted when an electron in a Hydrogen atom makes a transition to the n=2 level from a higher-energy level. The three longest-wavelength Balmer Series lines are visible to humans: transition color Wavelength (nm) n=3à n=2 red n=4à n=2 blue-green n=5à n=2 violet Blackbody spectrum a spectrum in which all colors are present, such as the spectrum of the sun. Line spectrum a spectrum in which only a few distinct colors (called spectral lines) are present, such as the spectrum of a Hydrogen lamp. Minute of arc 1/60 of a degree; symbol is a single quote mark ( ). An angle of 23 degrees, 34 minutes of arc would be written Example of addition of minutes of arc: = spectrometer device for measuring the wavelengths of spectral lines with great precision E) Telescope M) prism table I, J) angle scale levelling & vernier A) grating screws C) Collimator F) focus ring K) prism table clamp G,H) telescope { clamp fine adjust I, J) angle scale & vernier B) adjustable slit (twist) L) prism table { clamp fine adjust Spectrometer L-5-1

74 L-5 THE BALMER SERIES Pre-lab questions 1. In this experiment, the objective is to measure the wavelength of the visible light emitted by electronic transitions in the atoms of. (What element?) 2. When an electron in transition emits light, it does so because it is moving to a level with higher lower energy. (Circle one) 3. What happens to the electron when the light is absorbed instead of emitted? It moves to a level with higher lower energy. (Circle one) 4. Which level(s) are the electrons in this experiment moving to? 5. Which level(s) are they moving from? 6. Are we measuring the entire Balmer Series in this experiment? 7. Why not? 8. The light from the sun is very nearly identical to a blackbody spectrum, and therefore contains light at all wavelengths, rather than just a few wavelengths like light from a Hydrogen lamp. Name one type of artificial lights that contains all wavelengths, and one that contains just a few: contains all wavelengths: contains only a few wavelengths: 9. What angle is indicated on the following Vernier scale? (If you need to brush up your knowledge of how to read a Vernier scale, please see the Appendix at the end of this write-up.) L-5-2

75 A. Wavelengths of Balmer Series Spectral Lines L-5 The Balmer Series prism table clamp Telescope 1 grating prism table levelling screws Collimator focus ring A1. Adjust the collimator slit to its maximum width by twisting the end furthest from the grating (if you look through the collimator as you do this, you should see the slit getting wider as you twist). A2. Turn on the Hydrogen lamp and place the slit directly in front of it. A4. Loosen the telescope clamp (part G in the figure on the first page). A3. Swing the telescope around so it is directly opposite the collimator. You should see a bright reddish image of the collimator slit (this image should get wider or narrower as you adjust the width of the collimator slit). A4. Twist the collimator, if necessary, so that the image of the slit lines up with the telescope crosshair, and is centered on the crosshair. A5. Close the slit all the way, and then open it 1/8 turn. You should still see a reddish image, although not so wide or bright as before. A6. Swing the telescope left, look through it with your eye, and verify that you can see three slightly separated lines: one violet, one blue-green, and one red. A7. Swing the telescope back through the center and to the right, and verify that you can see three identical lines. A8. The procedure for measuring the wavelength of a line is as follows: with the telescope clamp (part G) loose, swing the telescope so the line is approximately centered on the crosshair. Then tighten the telescope clamp (part G) and use the telescope fine-adjust (part H) to center the line exactly on the crosshair. Record the angles, as indicated in the next step. Then loosen the telescope clamp (part G) and move to the next spectral line. A9. Swing the telescope back again through the center and to the left, and center the violet line in the telescope crosshair. Record the reading of Vernier scale 1 (indicated by 1 in the diagram above) in the column V 1L for row Violet. Record the reading of vernier scale 2 in the column V 2L. Be careful to note that the Vernier reading is degrees and minutes of arc. 2 angle scale & vernier adjustable slit (twist) Violet Blue-green Red V 1L V 2L V 1R V 2R L-5-3

76 L-5 THE BALMER SERIES A10. Swing the telescope back through the center and to the right. Center the other violet line in the telescope. Record the readings of the Vernier scales in the columns V 1R and V 2R. A11. Repeat steps A9 and A10 for the blue-green line, and then for the red line. A12. Convert all your measurements from degrees and minutes of arc to decimal degrees, and enter them in the table below (example: =23.57 ) Violet Blue-green Red V 1L V 2L V 1R V 2R α 1 α 2 A13. Let α represent the angle the telescope moves from one line to the line of the same color on the other side of the center. We have two independent measurements of α, one using Vernier 1 and the other using Vernier 2. For each line, calculate α 1 = V 1L -V 1R and α 2 = V 2L -V 2R and record them in the table above. A14. For each line α 1 and α 2 should not differ by more than 0.1. If they do, go back and recheck your measurements and conversions. A15. The best estimate of α is then α = (α 1 + α 2 )/2, and the best estimate for the diffraction angle of each line is θ = α /2. For each line, enter these in the table below: Violet Bluegreen Red A16. The number of lines per millimeter is recorded on the bottom (or top) of the grating. Record it here: A17. Use this to calculate the distance between lines, d: d = m A18. Use λ = d sin θ to calculate the measured wavelength for each line, and record it in the table above. A19. Compare your measured values to the accepted values, which were given in the you should know section. Record the discrepancy in both nm and % (percentage discrepancy is {λ measured -λ accepted }/λ accepted x 100% ). (accepted values of wavelengths from: L-5-4 α θ λ (measured) nm λ (accepted) nm Discrepancy (nm) Discrepancy (percent)

77 Appendix: How to read a Vernier Scale 0 L-5 The Balmer Series A. Read the degrees: find the zero mark on the upper scale. The degree reading is the first full (as opposed to half ) degree mark to the left of the zero. B. Read the minutes: look for the two lines (one top, one bottom) which match up best. The best-matching lines are indicated by arrows in the picture at right. Read the number of the top line (it ll be between 0 and 30). C. If the zero mark on the upper scale comes to the right of a full degree mark, the number in step B is the minutes. If the zero mark on the upper scale came to the right of a half-degree mark, add 30 to the number you read in step B ' ' Examples L-5-5

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79 MPC-1 RADIOACTIVE DECAY MPC-1 Radioactive Decay What do we do in this lab? In this lab we use a Geiger counter to detect the particles emitted by the radioactive decays of several different substances, and investigate three questions: 1. Is radioactive decay a random process? 2. How do the particles emitted by decaying nuclei spread out in space? 3. What it takes to stop (shield) these particles? Why are we doing this lab? The discovery and understanding of radioactive decay is a major part of modern physics. Safety concerns: do not swallow the radioactive sources You need to know: Atom: Every atom is composed of a nucleus, which is orbited by one or more electrons. Isotope: Radioactive Decay: Alpha particle: Beta particle: Gamma particle: Geiger Counter: Half-life: Activity: Background: Shielding: Species of atom. Every isotope is either stable or unstable. The emission by an unstable nucleus of one (or more) particles. Two protons and two neutrons, moving together as a single particle. An electron or an anti-electron (positron). A high-energy photon. Usually called gamma ray. A device for detecting alpha, beta, or gamma particles. The time one expects to wait before half the atoms of a given isotope undergo radioactive decay. Each isotope has a different half-life. The half-life of a stable isotope is infinity. We ll use the symbol T 1/ 2. The number of radioactive decays that occur per second in a given sample. A sample with a large number of atoms of an isotope with a short half-life will have a high activity. A sample in which 3.7 x atoms decay every second has an activity of 1 Curie (abbreviation = Ci). Particles emitted by radioactive decay of atoms somewhere in the environment, perhaps on the Earth or perhaps in outer space. Material that blocks the passage of one or more types of particle produced by radioactive decay. MPC-1-1

80 MPC-1 RADIOACTIVE DECAY Pre-lab Questions 1. A radioactive source of Cobalt-60 (half-life = 5.27 years) is manufactured in June At the time of its manufacture, it has an activity of 1 µ Ci. What is the activity of this source today? Activity today = µ Ci. 2. Assume our Geiger counter has aperture that is shaped like a circle and has a diameter of 9.14 mm. A radioactive source is placed 10 cm from the aperture. Assume the Geiger counter detects every gamma ray that enters its aperture. What fraction of the decays in the radioactive source will be detected by the Geiger counter? Fraction of decays detected= 3. A radioactive source that emits gamma rays and that has an activity of 1 µ Ci is placed 10 cm from the aperture of the Geiger counter. How many counts per second does the Geiger counter detect? Counts per second = 4. In real life, our Geiger counter only detects about 10% of the gamma rays that enter its aperture. How does that affect your result from question 3 above? 5. Now suppose that shielding composed of a 1/8 thick plate of material X is put between the radioactive source and the Geiger counter. Material X has the property that a 1/8 thickness of it stops 60% of the gamma rays. How does the introduction of this shielding change the Geiger counter count rate from question 4? 6. Will a Geiger counter detect radiation when it is not near any radioactive sources? MPC-1-2

81 MPC-1 RADIOACTIVE DECAY A. Random Nature of Radioactive Decay Perhaps the most surprising thing about radioactive decay is that it is random. If you have a radioactive atom in front of you an atom of Cobalt-60, say you know it will decay at some point, but there s nothing you can do to predict the decay before it happens. In this part of the lab, we ll see what randomness looks like. A1. Set the Cobalt-60 source on the tray in the highest slot it will go. A2. Turn on the Pasco Interface and plug in the Geiger counter to digital channel 1. It should start beeping. A3. The horizontal axis is labeled Geiger Counts (counts/sample). A sample is one second of time. If a second goes by in which the Geiger counter detects 25 counts, the vertical bar above the 25 counts/sample mark on the horizontal axis goes up by one. A4. Click Start, and take data for 10 seconds. You should see the histogram changing as you watch, and it should look something like Histogram 3 in the figure below seconds 100 seconds 10 seconds A5. What is plotted on the vertical axis? MPC-1-3

82 MPC-1 RADIOACTIVE DECAY A6. Now take data for 100 seconds. Watch as the histogram plots itself. A7. Now take data for 1000 seconds. Watch your histogram develop for a while, then go around the room and see other groups histograms. A8. Use Graph to graph one of the runs. Click the Σ button to access the statistics pull-down menu, and tell Pasco to tell you the mean and standard deviation for the run. A9. What does mean mean, in terms of the shape shown on the histogram? A10. What does standard deviation mean, in terms of the shape shown in the histogram? A11. You should expect that 50% of the measurements fall in the range: mean (2/3 x standard deviation) < measurement < mean + (2/3 x standard deviation) Does this appear to be true for the 1000-second dataset? A12. For the three datasets you just took, do the standard deviations get bigger, smaller, or remain the same, the longer you collect data? bigger smaller remain the same A14. What about the accuracy with which you know the mean does that get better, worse, or unchanged, the longer you collect data? gets better gets worse unchanged The accuracy with which you know the mean is called the standard deviation of the mean, and, for random processes like radioactive decay, it does indeed get smaller as more data is collected. MPC-1-4

83 MPC-1 RADIOACTIVE DECAY B. Fall-off of count rate with distance The slots are about / cm apart. The topmost slot that you can put the tray and source in is about / cm below the detector (which you can t see, because it s housed inside the black plastic that you can see). B1. Move the source from the lowest slot to the highest slot, one slot at a time. Take data for 60 seconds for each slot. Calculate the mean, and the standard deviation, of counts per second for each position. Fill in the table below. Slot Distance below detector (cm) 1(lowest) (highest) Counts per second Mean B2. Use Excel to make a plot of count rate as a function of distance (see the Excel Tutorial in the back of the lab manual for tips). B3. Since the gamma rays spread out in all directions just like light, we expect the count rate to be proportional to 1/r 2, where r is the distance from the source to the detector. Use Excel to make a power-law fit to the data (hint: Chart tools>layout>trendline>more trendline options>power>display equation on chart). B5. The gamma rays have to pass through air in order to get from the source to the detector. Based on your graph, what conclusion do you make about the ability of air to stop gamma rays is air good at stopping gamma rays, or not good? B6. Assume the Geiger counter aperture has a diameter of 9.14 mm, and detects 10% of the gamma rays that enter it. Use a calculation similar to that done in prelab questions 2 and 3 to estimate the number of decays per second that are actually occurring in your radioactive source. Activity of source = decays per second MPC-1-5

84 MPC-1 RADIOACTIVE DECAY B7. How much is this in Curie? MPC-1-6 Activity of source = Ci B8. Estimate the age of the source, and write down here the corresponding date (month and year) of manufacture. B9. What is the date stamped on the source? Effect of Shielding C1. You have a box set containing absorbers of various thicknesses (they are labeled A, B, C, D, and so on). Put them in between the source and the detector, and measure count rates. C2. Which absorbers are you certain reduce the count rate? Cite your statistics (the means and standard deviations from runs with and without the absorber). C3. Which absorbers do you suspect, but cannot prove, reduce the count rate. C4. Which absorbers have, as far as you can tell, no effect on the count rate? C5. So far you have only used the Cobalt-60 source, which produces gamma rays (high-energy photons). If there is time, investigate the effect of shielding on the beta particles produced by the Thalium-204 source. Note: the paper label on the top of each Thalium-204 source blocks a significant fraction of the beta particles that are produced, so if you want to measure the largest possible number of counts from this source, place it in the tray label side down. C6. Which absorbers are you certain reduce the count rate from the Thalium-204 source? C7. Are there any absorbers that seem to have no effect? One thing we want you to learn in this lab is that different types of radiation have different abilities to penetrate material substances. Further Questions Look up which are the most common naturally occurring radioactive substances. Which is more radioactive, a typical apple or a typical banana? In some parts of the country, one has to worry about the accumulation of radioactive Radon gas in underground rooms. In other parts of the country, one does not. Which part do you live in?

85 Appendix A Using a DC power supply A DC power supply is a device that provides (as you may have guessed) electrical power in a very regular and specific way. Our supplies are designed to deliver a current at a specified voltage, in accordance with Ohm s Law. The supply you will use in the lab is pictured to the right. It is capable of generating a maximum of 3 Amps at 18 Volts, or 54 Watts of electrical power. For example if you connected the supply set at 10V in series with a 1kΩ resistor, you would draw a current of 10mA and provide 100mW of power to the circuit. Here are the commonly used features of the supply: Power Button: Turns the device on/off Display: Shows the currently provided current or voltage Display Switch: Toggles the display mode between current and voltage. For the purposes of this course, it should always be on Volts. Current Adjust: There are two knobs that adjust the current, fine and coarse. For the purposes of this course, both knobs should be turned fully clockwise. You want the maximum amount of current possible. Voltage Adjust: There are two knobs that control the voltage. These will be the most common adjustments you will make. The Fine adjustment knob has a range of about 2 Volts. The Coarse adjustment knob has a range of about 17 Volts. You will need to turn both of these fully counterclockwise to deliver 0 Volts. Output Connections: You will attach your cables to these. The red connection marked + is the output path for the current. The return path for the current is provided by either the green connection marked GND or the black connection marked -. Notice that these have a thin metal bar connecting the two connections.

86 Setting a voltage: Turn off the supply using the power button. Use only banana cables (see right) to connect the supply to your circuit. Red goes to + and black goes to -. Turn both current adjust knobs fully clockwise. Turn both voltage adjust knobs fully counterclockwise. Press the power button to turn on the supply. The display should read 0V and the green light should be on. Use the coarse voltage adjust knob clockwise to increase the voltage to the desired value. If you are setting 18V, you will also need to turn the fine voltage adjust knob. Troubleshooting: You may hear some clicking noises as the supply adjusts itself to the desired output this is normal. If the displayed voltage does not increase or the red light turns on, try turning the current adjust knobs fully clockwise. If the red light persists, you very likely have a problem with your circuit. Examine your circuit carefully for shorts, direct connections between the terminals of the power supply.

87 Appendix B Using a digital multimeter A digital multimeter (DMM) is a device that measures a variety of electrical properties. In this course, we will use two of these features: measuring a DC voltage and measuring a DC current. Measuring a DC Voltage: At right is the DMM configured to read a DC voltage. Turn the knob to the symbol for DC volts. The red probe connection is plugged into the right-most terminal. The black probe is plugged into the center terminal. Voltage measurements are made by connecting the DMM in parallel with the component that you want to measure. Remember from class that circuit elements connected in parallel have identical voltage drops. In this mode, the DMM acts like a very large resistor so that is does not draw much current away from the circuit. At bottom right is an example of the multimeter connected to a circuit and measuring the voltage drop across a resistor. Note that the sign of the voltage will change if the red and black probes are switched.

88 Measuring a DC Current: At right is the DMM configured to measure a DC current. Turn the knob to the symbol for DC current. The red probe is plugged into the left-most terminal. The black probe is plugged into the center terminal. Current measurements are made by connecting the DMM in series with the component you want to measure. Remember from class that circuit elements connected in series have the same current flowing through them. In this mode, the DMM acts like a bare wire with very little resistance. There is a negligible voltage drop due to the meter. At bottom right is an example of the multimeter connected to a circuit and measuring the current flowing through a resistor. Note that the sign of the current will change if the red and black probes are switched.

89 Troubleshooting: If the DMM turns itself off, just turn the knob to the off position and then back to the correct setting. If any buttons are pressed accidentally or the DMM is acting strangely, turn the DMM to off and then back on again. Double check the leads! Current measurement require the red probe connected to the left-most terminal. Voltage measurements require the red probe connected to the right-most terminal. Double check the knob! Current measurements should use the setting, while voltage measurements should use the setting. Double check your circuit! Current measurement require the DMM to be configured in series. Voltage measurements require the DMM to be in parallel.

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91 Appendix C Using an electrometer An electrometer is a device that measures a DC Voltage. It is distinguished from a digital multimeter by being equivalent to an extremely large resistor (1 GΩ) as compared to the (10 MΩ) equivalence of the DMM. The advantage of this device is that it is capable of measuring voltages across components like capacitors without quickly draining the charge that resides on that component. For example, if we create an RC-Circuit with a 1 mf capacitor and a 1 GΩ resistor, it will have a time constant of over 15 minutes as opposed to the 10 second time constant that connecting the 10 MΩ DMM would provide. Below is a line drawing of the electrometer.

92 Measuring a voltage: At top right is the electrometer configured to measure the voltage drop across a capacitor. Connect the red probe to the electrometer Signal Input port via a BNC cable (shown at middle right). Connect the black probe to the electrometer Ground port via a black banana cable and a BNCbanana adapter (shown at bottom right). Press the ON-OFF button to turn on the electrometer. Set the scale to the smallest number that is larger than the voltage you are supplying by repeatedly pressing the voltage range selector button. The present voltage range is displayed on the voltage range indicators. The electrometer will not change voltage ranges automatically. If the measured voltage is larger than the voltage range setting, the electrometer will not read correctly, and you will need to change the voltage range. Press the Zero button to remove any residual charge on the meter. Do this before connecting to your circuit as it will short your circuit and remove all of the charge you are trying to measure. Make electrical contact in parallel to the component you are measuring by putting the red probe in electrical contact with one side of the component and the black probe in contact with the other side. Record your measurement using either the digits or meter display. Press the ON-OFF button to turn off the electrometer so as not to drain the batteries.

93 APPENDIX D: SIGNIFICANT FIGURES Appendix D: Significant Figures Calculators make it possible to get an answer with a huge number of figures. Unfortunately many of them are meaningless. For instance if you needed to split $1.00 among three people, you could never give them each exactly $ The same is true for measurements. If you use a meter stick with millimeter markings to measure the length of a key, as in figure A-1, you cannot measure more precisely than a quarter or half or a third of a mm. Reporting a number like cm would not only be meaningless, it would be misleading, because it would imply that you really knew the length that precisely. Figure D-1 In your measurement, you can precisely determine the distance down to the nearest millimeter and then improve your precision by estimating the next figure. It is always assumed that the last figure in the number recorded is uncertain. So, you would report the length of the key as 5.81 cm. Since you estimated the 1, it is the uncertain figure. If you don't like estimating, you might be tempted to just give the number that you know best, namely 5.8 cm, but it is clear that 5.81 cm is a better report of the measurement. An estimate is always necessary to report the most precise measurement. When you quote a measurement, the reader will always assume that the last figure is an estimate. Quantifying that estimate is known as estimating uncertainties. Appendix E will illustrate how you might use those estimates to determine the uncertainties in your measurements. APPENDIX D - 1

94 APPENDIX D: SIGNIFICANT FIGURES What are significant figures? The number of significant figures tells the reader the precision of a measurement. Table D-1 gives some examples. Table D-1 Length (centimeters) Number of Significant Figures One of the things that this table illustrates is that not all zeros are significant. For example, the zero in 0.8 is not significant, while the zero in 1.50 is significant. Only the zeros that appear after the first non-zero digit are significant. A good rule is to always express your values in scientific notation. If you say that your friend lives 143 m from you, you are saying that you are sure of that distance to within a few meters (3 significant figures). What if you really only know the distance to a few tens of meters (2 significant figures)? Then you might express the distance as 140 m, or, better, use scientific notation and write 1.4 x 10 2 m. Is it always better to have more figures? Consider the measurement of the length of the key shown in Figure D- 1. If we have a scale with ten etchings to every millimeter, we could use a microscope to measure the spacing to the nearest tenth of a millimeter and guess at the one hundredth millimeter. Our measurement could be cm with the uncertainty in the last figure, four significant figures instead of three. This is because our improved scale allowed our estimate to be more precise. This added precision is shown by more significant figures. The more significant figures a number has, the more precise it is. APPENDIX D - 2

95 APPENDIX D: SIGNIFICANT FIGURES How do I use significant figures in calculations? When using significant figures in calculations, you need to keep track of how the uncertainty propagates. There are mathematical procedures for propagating the uncertainty in the most precise manner, but for this course the simplified uncertainty estimate given below (and also described in Appendix E) will be good enough. Addition and subtraction When adding or subtracting numbers, the number of decimal places must be taken into account. The result should be given to as many decimal places as the term that is given to the smallest number of decimal places. Examples Addition Subtraction The uncertain figures in each number are shown in bold-faced type. Note that when it is necessary to decrease the number of significant figures in the answer, if the last digit is 0, 1, 2, 3, or 4, you round down, and if it is 5, 6, 7, 8, or 9, you round up. Multiplication and division When multiplying or dividing numbers, the number of significant figures must be taken into account. The rule is the same as before: The result should be given to as many significant figures as the term that is given to the smallest number of significant figures. The basis behind this rule is that the least accurately known term in the product will dominate the accuracy of the answer. APPENDIX D - 3

96 APPENDIX D: SIGNIFICANT FIGURES Examples Multiplication x 2.5 x Division = x 10 2 PRACTICE EXERCISES 1. Determine the number of significant figures of the quantities in the following table: Length (centimeters) x x x Number of Significant Figures 2. Add: to 6.7 x 102: [Answer: x 102 = 7.9 x 10 2 ] 3. Multiply: 34.2 and 1.5 x 10 4 [Answer: 34.2 x 1.5 x 104 = 5.1 x 10 5 ] APPENDIX D - 4

97 APPENDIX E: ACCURACY, PRECISION AND UNCERTAINTY Appendix E: Accuracy, Precision and Uncertainty How tall are you? How old are you? When you answered these everyday questions, you probably did it in round numbers such as "five foot, six inches" or "nineteen years, three months." But how true are these answers? Are you exactly 5' 6" tall? Probably not. You estimated your height at 5' 6" and just reported two significant figures. Typically, you round your height to the nearest inch, so that your actual height falls somewhere between 5' 5 1/2" and 5' 6 1/2" tall, or 5' 6" ± 1/2". This ± 1/2" is the uncertainty, and it informs the reader of the precision of the value 5' 6". What is uncertainty? Whenever you measure something, there is always some uncertainty. There are two categories of uncertainty: systematic and random. 1. Systematic uncertainties are those which consistently cause the value to be too large or too small. Systematic uncertainties include such things as reaction time, inaccurate meter sticks, optical parallax and miscalibrated balances. In principle, systematic uncertainties can be eliminated if you know they exist. 2. Random uncertainties are variations in the measurements that occur without a predictable pattern. Random uncertainty can be reduced, but never eliminated. We need a technique to report the contribution of this uncertainty to the measured value. How do I determine the uncertainty? This Appendix will discuss two basic techniques for determining the uncertainty: estimating the uncertainty and measuring the average deviation. Which one you choose will depend on your need for precision. If you need a precise determination of some value, the best technique is to measure that value several times and use the average deviation as the uncertainty. Examples of finding the average deviation are given below. How do I estimate uncertainties? If time or experimental constraints make repeated measurements impossible, then you will need to estimate the uncertainty. When you estimate uncertainties you are trying to account for anything that might cause the measured value to be different if you were to take the measurement again. For example, suppose you were trying to measure the length of a key, as in Figure B-1. APPENDIX E - 1

98 APPENDIX E: ACCURACY, PRECISION AND UNCERTAINTY Figure E-1 If the true value was not as important as the magnitude of the value, you could say that the key s length was 6cm, give or take 1cm. This is a crude estimate, but it may be acceptable. A better estimate of the key s length, as you saw in Appendix D, would be 5.81cm. This tells us that the worst our measurement could be off is a fraction of a mm. To be more precise, we can estimate that fraction to be about a third of a mm, so we can say the length of the key is 5.81 ± 0.3 cm. How do I find the average deviation? If estimating the uncertainty is not good enough for your situation, you can experimentally determine the uncertainty by making several measurements and calculating the average deviation of those measurements. To find the average deviation: 1. Find the average of all your measurements; 2. Find the absolute value of the difference of each measurement from the average (its deviation); 3. Find the average of all the deviations by adding them up and dividing by the number of measurements. Of course you need to take enough measurements to get a distribution for which the average has some meaning. In example 1, a class of six students was asked to find the mass of the same penny using the same balance. In example 2, another class measured a different penny using six different balances. Their results are listed below: Class 1 Penny A, massed by six different students on the same balance. Mass (grams) Average: APPENDIX E - 2

99 APPENDIX E: ACCURACY, PRECISION AND UNCERTAINTY Class 2 The deviations are: 0.011g, 0.004g, 0.001g, 0.005g, 0.001g, 0.001g Sum of deviations: 0.023g Average deviation: (0.023g)/6 = 0.004g Mass of penny A: ± 0.004g Penny B massed by six different students on six different balances Mass (grams) Average: The deviations are: 0.009g, 0.002g, 0.013g, 0.013g, 0.005g, 0.006g Sum of deviations: 0.048g Average deviation: (0.048g)/6= 0.008g Mass of penny B: ± 0.008g However you choose to determine the uncertainty, you should always state your method clearly in your report. For the remainder of this appendix, we will use the results of these two examples. How can I reduce uncertainty? If there are no systematic uncertainties, only random uncertainties, they can be reduced by taking the average of a number of independent measurements of the same quantity. According to statistics, the uncertainty of the average of N 1 independent measurements of the same quantity is times the uncertainty of N one measurement of the quantity. Be careful, this is only true if there are no systematic uncertainties! APPENDIX E - 3

100 APPENDIX E: ACCURACY, PRECISION AND UNCERTAINTY How do I know if two values are the same? If we compare only the average masses of the two pennies we see that they are different. But now include the uncertainty in the masses. For penny A, the most likely mass is somewhere between 3.117g and 3.125g. For penny B, the most likely mass is somewhere between 3.123g and 3.139g. If you compare the ranges of the masses for the two pennies, as shown in Figure B-2, they just overlap. Given the uncertainty in the masses, we are able to conclude that the masses of the two pennies could be the same. If the range of the masses did not overlap, then we ought to conclude that the masses are probably different. Which result is more precise? Figure E-2 Suppose you use a meter stick to measure the length of a table and the width of a hair, each with an uncertainty of 1 mm. Clearly you know more about the length of the table than the width of the hair. Your measurement of the table is very precise but your measurement of the width of the hair is rather crude. To express this sense of precision, you need to calculate the percentage uncertainty. To do this, divide the uncertainty in the measurement by the value of the measurement itself, and then multiply by 100%. For example, we can calculate the precision in the measurements made by class 1 and class 2 as follows: Precision of Class 1's value: (0.004 g g) x 100% = 0.1 % Precision of Class 2's value: (0.008 g g) x 100% = 0.3 % Class 1's results are more precise. This should not be surprising since class 2 introduced more uncertainty in their results by using six different balances instead of only one. Which result is more accurate? Accuracy is a measure of how your measured value compares with the real value. Imagine that class 2 made the measurement again using only one balance. Unfortunately, they chose a balance that was poorly calibrated. They analyzed their results and found the mass of penny B to be ± g. This number is more precise than their previous result since the uncertainty is smaller, but the new measured value of mass is very different from their previous value. We might conclude that this new value for the mass of penny B is different, since the range of the new value does not overlap the range of the previous value. However, that conclusion would be wrong since our uncertainty has not taken into account the APPENDIX E - 4

101 APPENDIX E: ACCURACY, PRECISION AND UNCERTAINTY inaccuracy of the balance, a systematic uncertainty. To determine the accuracy of the measurement, we should check by measuring something that is known. This procedure is called calibration, and it is absolutely necessary for making accurate measurements. Be cautious! It is possible to make measurements that are extremely precise and, at the same time, grossly inaccurate. How can I do calculations with values that have uncertainty? When you do calculations with values that have uncertainties, you will need to estimate (by calculation) the uncertainty in the result. There are mathematical techniques for doing this, which depend on the statistical properties of your measurements. A very simple way to estimate uncertainties is to find the largest possible uncertainty the calculation could yield. This will always overestimate the uncertainty of your calculation, but an overestimate is better than no estimate. The method for performing arithmetic operations on quantities with uncertainties is illustrated in the following examples: Addition: Subtraction: (3.131 ± g) + (3.121 ± g) =? First find the sum of the values: g g = g Next find the largest possible value: g g = g The uncertainty is the difference between the two: g g = g Answer: ± g. Note: This uncertainty can be found by simply adding the individual uncertainties: g g = g (3.131 ± g) (3.121 ± g) =? First find the difference of the values: g g = g Next find the largest possible value: g g = g The uncertainty is the difference between the two: g g = g Answer: 0.010±0.012 g. Note: This uncertainty can be found by simply adding the individual uncertainties: g g = g Notice also, that zero is included in this range, so it is possible that there is no difference in the masses of the pennies, as we saw before. APPENDIX E - 5

102 APPENDIX E: ACCURACY, PRECISION AND UNCERTAINTY Multiplication: Division: (3.131 ± g) x (6.1 ± 0.2 cm) =? First find the product of the values: g x 6.1 cm = 19.1 g-cm Next find the largest possible value: g x 6.3 cm = 19.8 g-cm The uncertainty is the difference between the two: 19.8 g-cm g-cm = 0.7 g-cm Answer: 19.1 ± 0.7g-cm. Note: The percentage uncertainty in the answer is the sum of the individual percentage uncertainties: ( x 100%) + ( x 100%) + ( x 100%) (3.131 ± g) (3.121 ± g) =? First divide the values: g g = Next find the largest possible value: g g = The uncertainty is the difference between the two: = Answer: ± Note: The percentage uncertainty in the answer is the sum of the individual percentage uncertainties: ( x 100%) + ( x 100%) + ( x 100%) Notice also, the largest possible value for the numerator and the smallest possible value for the denominator gives the largest result. The same ideas can be carried out with more complicated calculations. Remember this will always give you an overestimate of your uncertainty. To avoid an overestimate, you can add uncertainties in quadrature. That is, any place above where you would use the sum of uncertainties or the sum of percentage uncertainties, use instead the square root of the sum of the squares of the uncertainties or percentage uncertainties. These techniques help you estimate the random uncertainty that always occurs in measurements. They will not help account for mistakes or poor measurement procedures. There is no substitute for taking data with the utmost of care. A little forethought about the possible sources of uncertainty can go a long way in ensuring precise and accurate data APPENDIX E - 6

103 APPENDIX E: ACCURACY, PRECISION AND UNCERTAINTY PRACTICE EXERCISES: 1. Consider the following results for different experiments. Determine if they agree with the accepted result listed to the right. Also calculate the precision for each result. a) g = 10.4 ± 1.1 m/s 2 g = 9.8 m/s 2 b) T = 1.5 ± 0.1 sec T = 1.1 sec c) k = 1368 ± 45 N/m k = 1300 ± 50 N/m Answers: a) Yes, 11%; b) No, 7%; c) Yes, 3.3% 2. The area of a rectangular metal plate was found by measuring its length and its width. The length was found to be 5.37 ± 0.05 cm. The width was found to be 3.42 ± 0.02 cm. What are the area and the average deviation? Answer: 18.4 ± 0.3 cm 2 3. Each member of your lab group weighs the cart and two mass sets twice. The following table shows this data. Calculate the total mass of the cart with each set of masses and for the two sets of masses combined. Answers: Cart (grams) Mass set 1 (grams) Mass set 2 (grams) Cart and set 1: 299.3±1.6 g. Cart and set 2: 297.0±1.2 g. Cart and both sets: 395.1±1.9 g. APPENDIX E - 7

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105 APPENDIX F: GRAPHING DATA BY HAND Appendix F: Graphing data by hand One of the most powerful tools used for data presentation and analysis is the graph. Used properly, graphs are an important guide to understanding the results of an experiment. They are an easy way to make sense out of your data. Therefore, it is important to graph your data as you take it. The Pasco interface will often provide you with a graph, but on occasion you will have to make your own. Good graphing practice, as outlined below, is a way to save time and effort whilst solving a problem in the laboratory. How do I make a graph? 1. Accurate graphs are drawn on graph paper. Even if you are just making a quick sketch for yourself, it will save you time and effort to use graph paper. That is why every page of your lab journal is a piece of graph paper. Make sure to graph your data as you take it. Never put off drawing a graph until the end. 2. Every graph should have a title to indicate the data it represents. In a large collection of graphs, it is difficult to keep one graph distinct from another without clear, concise titles. 3. The axes of the graph should take up at least half a page. Give yourself plenty of space so that you can see the pattern of the data as it is developing. Both axes should be labeled to show the values being graphed and their appropriate units. 4. The scales on the graphs should be chosen so that the data occupies most of the space of your graph. However, there are times when it is important to include the zero, even if this means the data lies in a small region of the graph. 5. If you have more than one set of data on a set of axes, be sure to label each set to avoid confusion later. How do I plot data and uncertainties? Another technique that makes data analysis easier is to record all your data in a table. A useful data table will always include a title and column headings, so that you will not forget what all the numbers mean. The column headings should include the units of the quantities listed, and usually serve as the labels for the axes of your graph. For example, look at Table F-1 and Graph F-1 on the next page. This is a position-versus-time graph drawn for a hypothetical situation. The uncertainty for each data point is shown on the graph as a line representing a range of possible values with the principal value at the center. The lines are called error bars, and they are useful in determining if your data agrees with your prediction. Any curve (function) that represents your prediction should pass through the error bars of your data. APPENDIX F - 1

106 APPENDIX F: GRAPHING DATA BY HAND Position vs. Time: Exercise 3, Run 1 Time (sec) Position (cm) ± ± ± ± ± ± ± ± ± ± 5 Table F-1 Graph F-1 APPENDIX F - 2

107 APPENDIX F: GRAPHING DATA BY HAND How to fit straight line by eye: If your data points appear to lie on a straight line, then use a clear straight edge and draw a line through all the error bars, making sure the line has as many principal values beneath it as above it. The line does not need to touch any of the principle values. Do not connect the dots. When done correctly, this straight line represents the function that best fits your data. You can read the slope and intercept from your graph. These quantities usually have important physical interpretations. Some computer programs, such as Excel or DataStudio, will determine the best straight line for your data and compute its slope, intercept, and their uncertainties. As an example, look at Graph F-1. The dotted lines are possible linear fits, but the solid line is the best fit. Once you have found the bestfit line, you should determine the slope from the graph and record its value. How do I find the slope of a line? The slope of a line is defined as the ratio of the change in a line's ordinate (vertical axis) to the change in the abscissa (horizontal axis), or the "rise" divided by "run." Your text explains slope. For Graph F-1, the slope of the best-fit line will be the change in the position of the object divided by the time interval for that change in position. To find the value of the slope, look carefully at your best-fit line to find points along the line that have coordinates that you can identify. It is usually not a good idea to use your data points for these values, since your line might not pass through them exactly. For example, the best-fit line on Graph F-1 passes through the points (0.10 sec, 10 cm) and (1.00 sec, 102 cm). This means the slope of the line is: Slope = (102cm 10cm) (1.00sec 0.10sec) = 92cm 0.90sec = 102cm/sec Note that the slope of a position versus time graph has the units of velocity. How do I find the uncertainty in the slope of a line? Look at the dotted lines in Graph F-1. These lines are the largest and smallest values of the slopes that can realistically fit the data. The lines run through the extremes in the uncertainties and they represent the largest and smallest possible slopes for lines that fit the data. You can extend these lines and compute their slopes. These are your uncertainties in the determination of the slope. In this case, it would be the uncertainty in the velocity. APPENDIX F - 3

108 APPENDIX F: GRAPHING DATA BY HAND How do I get the slope of a curve that is not a straight line? The tangent to a point on a smooth curve is just the slope of the curve at that point. If the curve is not a straight line, the slope will change from point to point along the curve. To draw a tangent line at any point on a smooth curve, draw a straight line that only touches the curve at the point of interest, without going inside the curve. Try to get an equal amount of space between the curve and the tangent line on both sides of the point of interest. The tangent line that you draw needs to be long enough to allow you to easily determine its slope. You will also need to determine the uncertainty in the slope of the tangent line by considering all other possible tangent lines and selecting the ones with the largest and smallest slopes. The slopes of these lines will give the uncertainty in the slope of the tangent line. Notice that this is exactly like finding the uncertainty of the slope of a straight line. How do I "linearize" my data? A straight-line graph is the easiest graph to interpret. By seeing if the slope is positive, negative, or zero you can quickly determine the relationship between two measurements. But not all the relationships in nature produce straight-line graphs. However, if we have a theory that predicts how one measured quantity (e.g., position) depends on another (e.g., time) for the experiment, we can make the graph be a straight line. To do this, you make a graph with the appropriate function of one quantity on one axis (e.g., time squared) and the other quantity (e.g., position) on the other axis. This is called "linearizing" the data. For example, if a rolling cart undergoes constant acceleration, the position-versustime graph is curved. In fact, our theory tells us that the curve should be a parabola. To be concrete we will assume that your data starts at a time when the initial velocity of the cart was zero. The theory predicts that the motion is described by x = 1 2 at 2. To linearize this data, you square the time and plot position versus time squared. This graph should be a straight line with a slope of 0.5a. Notice that you can only linearize data if you know, or can guess, the relationship between the measured quantities involved. APPENDIX F - 4

109 APPENDIX F: GRAPHING DATA BY HAND PRACTICE EXERCISES: Is there anything wrong with either of the graphs below: APPENDIX F - 5

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111 Physics Laboratory Excel Tutorial (text adapted from Clemson University Physics and Astronomy Excel Tutorial) Graphing Data and Curve Fitting In this tutorial on graphing, we will examine data taken from an experiment in which the circumferences and radii of several circular objects were measured. The data is displayed on the screen shot to the left. Note that the error in the measurement of the radius is not given; suppose that from your knowledge of radius measurement techniques, you estimate it at +/- 7%. Of course, the equation associated with this data is C=2πr, or the circumference of the circle is equal to two times pi times the circle s radius. In this experiment, the circumferences and radii are measured. We hope to be able to determine the value of π by fitting a straight line to the data. It is our firm belief (although not necessarily the belief of everyone at this university) that beginning laboratory students should learn to plot their data by hand before using a computer application to learn this task. Nonetheless, here we show how to use Excel to plot the data and fit a curve to the data. How to Plot the Data 1. Enter the data onto the worksheet as shown in the above screenshot 2. Click on an empty cell. 3. Click on the Insert tab at the top of the screen. 4. Click on Scatter. 5. Click on Scatter with only Markers. 6. A completely blank chart should now appear. Position the cursor over the blank chart, right- click, and choose Select Data. 7. Click on Add under Legend Entries (Series). This will cause value boxes, like the ones displayed here, to appear. 8. Click on the Collapse Dialog button at the right end of the Series X Values box. This will temporarily shrink the dialog window so you can highlight the x- values that you wish plotted on the horizontal axis. 9. When the dialog box shrinks, you can use the mouse to highlight the x- values that will be plotted along the horizontal axis (in this case the measured radii). Highlight values only, not title.

112 10. When finished click the Expand Dialog button which will return the dialog window to maximum size. 11. Repeat steps 8 and 9 for the Series Y Values, which will tell the computer where along the vertical axis to place the data points. 12. Click OK. A graph should now be visible. 13. Click OK again. The graph should no longer be hidden. However, it lacks titles for the x and y axes, as well as for the entire graph. 14. Double click somewhere in the chart area. The buttons on the top should change. 15. Click on Layout 1 in the Chart Layout area. 16. You should now be able to type titles for the x axis, the y axis, and the entire graph. Select a title so it looks like this. What you type appears in the formula bar at the top, and will appear on the graph when you hit enter. Your graph should now look like this: Making the Graph Look Nice Now that you have plotted the data, take a minute or two to put the finishing touches on the graph. 1. Include units on each axis title: double- click on each axis title and edit. 2. Delete the legend box by right- clicking on it and choosing Delete. 3. Add appropriate symbols. For example, change pi to π by double clicking on the graph title, deleting the i, clicking and dragging so the p is highlighted, placing the mouse over the p and right- clicking, choosing Font, and under Latin text font: choosing symbol. Then click OK.

113 4. Add error bars. a. Select the Layout tab at the top of the screen, click the Error bars pull down menu, and click More error bars options. The Format error bars window should appear. b. Click custom and Specify Value. The Custom Error Bars window should appear. Positive Error Value means how far above each data point the error bar should extend, and Negative Error Value means how far below each data point the error bar should extend. Since (in this example) we measured the error in the circumference, use the Collapse Dialog Button, and highlight the values in column C. Do this for both Positive Error Value and Negative Error Value. Then hit OK. c. Note that Excel has chosen horizontal error bars, which you now have to fix. Position the cursor over a horizontal error bar, and leave it there for a moment till you see Series 1 X Error Bars appear on the screen. Then right- click. Choose Format Error Bars. The Format Error Bars window should appear again, only now it should describe Horizontal Error Bars. In this example, we have estimated the error in measuring the radius at +/- 7%, so click percentage and then enter 7.0. Hit Close. You are finished adding error bars to your graph, which should now look like this: