Laboratory Manual Physics 111: Electricity, Magnetism, Thermodyamics Spring 2014
Laboratory Manual Physics 111: Electricity, Magnetism, Thermodyamics Spring 2014 0 Physics 111 Spring 2014 Laboratory Schedule Week of Experiment 1 February 3 Organizational Meeting (30–45 min) [page 2] 2 February 10 Electrostatics [page 5] 3 February 17 Mapping Electric Potentials and Fields [page 16] Read the Digital Multimeter Primer [page 21] 4 February 24 No lab 5 March 3 DC Circuits I [page 24] Solve preliminary problem before lab[page 37] 6 March 10 DC Circuits II [page 38] 7 March 17 Using an Oscilloscope [page 43] March 24 SPRING BREAK 8 March 31 Solar Cells [page 51] Read the Solar Cell Primer [page 57] 9 April 7 Measurement of Magnetic Fields [page 67] 10 April 14 AC Circuits I [page 72] 11 April 21 AC Circuits II [page 78] 12 April 28 Polarization of Light [page 85] 13 May 5 Calorimetry [page 98] Propagation of Error [page 102] 1 Physics 111 Laboratory GENERAL INFORMATION Dan Styer, Wright 215, x58183, [email protected] (laboratory instructor) General Procedures: Laboratory work will generally be done in Wright 214 by partners following the instructions in the laboratory manual. Each week, before coming to lab, study the instructions and notes for the experiment to be performed, and review relevant material from your textbook and lecture notes. At the beginning of the lab period the instructor will go over special techniques and precautions necessary for the experiment. In general, the instructor will not review material already covered in the notes. For a some of experiments you will be asked to prepare calculations to bring to the lab. The laboratory instructor is there to help you. Feel free to ask questions. The instructor may answer your question with another question to help you think about what you are doing. The emphasis is on careful and objective observation and not on getting preconceived “right answers.” Frequently, in the process of carrying out an experiment, ideas will occur to you about phenomena or processes not mentioned in the instructions that could be investigated with the apparatus at hand. After consultation with the instructor, feel free to pursue your own ideas. No formal written laboratory reports are required. Instead, when you have completed an experiment, consult your instructor to discuss the experiment. After the discussion, your instructor will grade your laboratory work as acceptable (), good (), or exemplary (). If the instructor finds your work unacceptable you must address the concerns raised then return for re-evaluation. Laboratory Notebook: In this laboratory you will keep the same kind of notebook that a research scientist keeps. It is important that your notebook be complete and understandable. It is not necessary to duplicate the laboratory manual in your notebook. Do not force your notes into a rigid “Procedure, Results, Discussion” format. The important point is that anyone should be able to tell from the laboratory notebook what experiment was performed, what data were taken, what the experimental uncertainties were, what calculations were made, and how the final results compared with theory or with other experiments. A notebook with good layout and neatly entered data and text is far more understandable than one with disorganized scribbles. Neatness matters! Your notebook is a journal of your observations. You should record what you observe and interpret what you record, rather than recording something that fits your notion of what should be happening. In research, you often would not know for weeks or months afterwards whether a particular entry is really significant, so you should make the meaning of each entry exact, concise, clear, and complete. Recording “what you observe” means that you do not do mental arithmetic before recording values. For example, suppose a light is placed on an optical bench so that its base is at 10.0 cm on the scale and a lens is placed at 30.0 cm. You 2 should record both of these readings and then subtract to get the separation distance rather than just writing down that the lens is 20.0 cm from the source. That may seem like a trivial point, but having the raw data often helps in tracking down erroneous recordings. Having redundant information often helps eliminate confusion and correct mistakes. You also should record the uncertainty (or estimated accuracy) of any measurement. This can be a specific uncertainty in the measurement as in 10.3 cm 0.1 cm. Or, it can be a global statement recorded early in an experiment, such as “Note that all distance measurements have an uncertainty of 0.1 cm.” Also, when recording numbers pay attention to significant digits. If, for instance, you calculate the value of a capacitance (C) from the slope of a graph, and the slope is uncertain by 10%, don’t give 3 significant digits in stating your calculated value of C. Your laboratory notebook is a permanent record of your work in this laboratory. A research physicist typically uses a bound notebook and tapes in graphs and computer printouts. It is important that your notebook has pre-numbered pages so that you can cross-reference your work and so there is no doubt that no pages have been removed. For this course you should buy one of the (relatively) inexpensive lab notebooks of this sort that can be found in the Chemistry section of the Bookstore’s textbook section. It has a blue cover, is bound, and has numbered pages. It costs about $8. If you have one already – and it is less than half full – that will be fine. Bring your laboratory manual and laboratory notebook to every laboratory session. You will also need to bring a calculator to each laboratory session. One calculator per pair of students is not sufficient. Specific guidelines for keeping your notebook: 1. Leave two or three pages at the beginning for a table of contents, which should be kept up-to-date. Put the date on at least every other page. 2. Write down enough explanation of what you are doing, and why, so that your pages of data can be analyzed later without confusion or ambiguity. Label sections of data and text to correspond to sections of this laboratory manual (e.g., Part A, C4, etc.). 3. Recorded all work legibly in ink in your notebook. Your notebook should be treated as a permanent record. In particular, once any entry has been made on a page in laboratory, do not throw that page away. Experience has shown that pages thrown away often contain critical information! 4. If you make a mistake, draw a line through it (but do not obliterate). Note any incorrect readings or comments and write in corrections. 3 5. Enter directly all data and pertinent information about the equipment and the experimental conditions. Do not record your observations on scratch paper then copy it later into your notebook. Enter directly all raw data and information with which to make corrections. Do not try to make corrections in your head while taking data. Whenever possible, use tables or columns for recording measurements. 6. Most data, after being tabulated, should immediately be graphed, both to increase your immediate understanding of their significance and for easier referral. On graphs give the page number of the data they contain. Graphing will usually be done by computer, then printed and taped into your laboratory notebook. 7. Enter your estimate of the accuracy of each measurement or set of measurements. 8. Complete the notebook immediately after taking the final measurements. Do not delay until the data are cold. Error Analysis: Acquire the habit of estimating experimental uncertainty in every measurement you make. No measurement or final result is complete without an estimate of the precision. The limits of precision can be assessed by such methods as calibrating instruments (or, as an approximation, by noting the manufacturer's specification of accuracy and resolution), checking reproducibility of readings, treating statistically (by error theory) a large number of trial readings under supposedly identical conditions, and by changing the conditions. In discussing experimental results, it is relatively easy to estimate the uncertainty that stems from random measurement errors, but it may be very difficult to evaluate the effects of systematic instrument errors (which would be present, for example, if a meter stick were mis-calibrated). Your instructor will discuss error estimation further in laboratory. In this course error analysis will be somewhat limited. More extensive treatment will occur in Physics 212. Grades and Attendance: Laboratory work is crucial for this course. You must complete every experiment to your instructor’s satisfaction (as confirmed by his signature) to pass this course. Options for making up missed labs are limited – it is important to attend every laboratory class. Your final laboratory grade, based on weekly lab grades, will count 25% towards your final course grade. Should illness or an emergency prevent you from attending your regular laboratory class, contact your instructor immediately to make arrangements to complete the lab at another time. You will find the instructor much more accommodating when such requests are made in advance of, rather than following, an absence. Note to athletes: Your status as an athlete does not give you permission to miss a laboratory. If your practice/game schedule presents conflicts you should meet with the instructor early in the semester and see what accommodation may be made. Usually conflicts can be resolved without major disruption. 4 Physics 111 Laboratory ELECTROSTATICS Introduction All atoms are made up of just three types of particles - protons, neutrons, and electrons. Two of these, protons and electrons, have a property called “electric charge.” Neutrons have no charge and are named for their electrical neutrality. While any everyday object contains a vast amount of charge, most objects contain exactly equal numbers of protons and electrons, and therefore have zero net charge. Anything that has an excess of either type of charge is said to be “charged” and will interact with other charged objects according to this rule: Like charges repel; opposite charges attract. Electrostatics is the study of static charges, i.e., charges that are not moving with respect to one another. Moving charges interact in a more complicated way that will be studied later in the course. In this experiment you will reproduce the classic observations on which the theory of electrostatics is based. Equipment and Discussion Charge Producers There are many combinations of materials that can be used to separate charge simply by rubbing them together. For example, silk can be used to literally rub some of the electrons from a piece of glass, leaving the glass with a net positive charge and the silk with a net negative charge. Such materials are often called “charge producers,” but the name “charge separators” is more exact. It is impossible to create a net charge. You can, however, separate charges to make two equally and oppositely charged objects. In this lab you will use glass, silk, hard rubber, wool, and various types of plastic to separate charges. Charge Detectors After charges are produced we need devices to detect them. One of the simplest is a “pith” ball – In our case a light ball of crushed aluminum foil hung on a string. When a charged object touches a pith ball, charge flows to it. Since the pith ball is so light, it is easy to see small electrical forces attracting or repelling it. A more sophisticated instrument is the electrometer pictured on the next page. This is a very sensitive electronic device that measures the voltage produced by small 5 charges. The actual voltage reading is not important for the purpose of this lab (the idea of voltage will be discussed later). What is important is the polarity of the voltage: + or -. The meter on the electrometer is "zero-centered": it responds to positive charges by swinging to the right; to negative charges by swinging to the left. In this lab, the amount of deflection corresponds to the magnitude of the charge. Some important points for general operation of the electrometer: Never touch the input leads until you have grounded yourself to an earth ground. A person walking across a rug on a cool, dry day can easily acquire a potential of several thousand volts. Similarly, don’t walk around or wave your arms excessively; charge can build up in clothing and affect your results. Between measurements, always press the Zero button to discharge all current from the electrometer. Shorting the test leads together is insufficient. There may still be stray charges within the electrometer circuitry. For good results, it is essential that the electrometer be connected to an earth ground (a water pipe or the ground wire from a 120 V AC socket). Only an earth ground provides a sufficient drain for excess charges that may build up during an experiment. It is also helpful if the experimenter is grounded. Merely touch one hand to a good earth ground just before, or during measurements. The electrometer is used in conjunction with a double cage arrangement often called a Faraday ice pail because Faraday did many of the original studies of 6 electrostatics and he used ice pails in this arrangement for his experiments. The lab apparatus, pictured here, consists of two concentric cylinders that are well insulated from each other. The electrometer is connected to the inner cylinder. When a charge is placed on the inner cylinder, the meter deflects to the left or right, depending on the sign of the charge. In fact, if a charge is even brought near the inner cylinder the meter will deflect, since like charges are repelled from the pail and down the wire toward the electrometer. The outer cylinder serves only to shield the inner cylinder from the effects of stray electric fields produced by fluorescent lights and other sources. In use with the electrometer, the outer pail is connected to ground, as is the ground terminal of the electrometer. Ground is literally the earth, and acts as an infinite reservoir of both positive and negative charges. By connecting the outer pail to ground, the inner pail is shielded from external, unwanted charges such as a wool sweater on a dry winter day. Only the charges that you place inside the inner cylinder should register on the electrometer. Of course, to work perfectly the outer pail would have to completely surround the inner one. However, the outer cylinder you will use is extremely helpful, if not perfect. Procedure The procedure consists of several short experiments. Since these are qualitative experiments, no data or analysis will be done. Instead, what is important to record in your lab notebook is, for each experiment, a description of how you do it (make sketches if it helps), what you observe, and how the observation is explained by the attraction and repulsion of charges. 7 Separation of Charge Your instructor will show you how to make electrical connections and operate the electrometer so you can use it for these experiments. Before you begin, play with the apparatus until you are familiar with it. Always start with the electrometer voltage range in the highest setting (100 V) and adjust down if needed. Analog meters are typically most accurate in the range of 1/3 to 2/3 of full scale. Rub the glass rod with the piece of silk. Then insert the rod into the inner cylinder without allowing it to actually touch the cylinder. When you do this you should get a positive deflection on the electrometer. If you don't, try rubbing the silk on the glass again. If you can't get a reading even after several tries, ask your instructor for help. Remove the glass rod and again note the electrometer reading. If it never touched the pail, the reading should be zero. Why did the electrometer detect charge when the glass rod was inside the inner cylinder? How is the sign of the charge on the glass rod related to the sign indicated by the electrometer? This particular experiment, rubbing glass with a piece of silk, was originally used by Benjamin Franklin to define positive charge. The choice of the charge on the glass as positive was arbitrary, and Franklin didn't know whether the free charges were from the glass or the silk. It turns out that it is the glass that loses free electrons, so electrons are now called "negative." Conservation of Charge The excess electrons on the silk quickly run down your arm and into the ground, so you will probably not be able to detect much charge on the silk. To help you see that there is actually a separation of equal charges, you are provided with two types of plastic mounted on conducting disks that are mounted on rods. The disks are insulated from the rod by white plastic. Do not touch the white insulator with your hands, because oil and dirt will allow charge to leak off. If you have trouble with this experiment, you should try cleaning the insulator with alcohol that is on the front table. (You also have another rod like this that has a metal surface on the conducting disk. This is the "proof plane" which will be used in the next section.) 8 Residual charge may build up in the plastic insulator between the handle and disk of the proof plane and charge producers. Before any experiment make sure to ground these parts by touching them to the shield on the Faraday ice pail. Hold one rod in each hand. Insert the rods one at a time into the inner pail. The rods should be uncharged when you begin, so if you get any reading, ground the disks by touching the back side to the outer pail. Do this until each disk is neutral. Now rub the white and blue surfaces together to separate charges. Light rubbing usually works better than hard pressure. Then insert the rods one at a time into the inner pail. Is the deflection caused by one rod equal and opposite to the deflection caused by the other? Try this with several different levels of charge created by more or less rubbing. Completely remove all charge from the charge producers by grounding them. Do not forget to also remove any stray charge from the necks and handles. Insert both charge producers into the ice pail and rub them together inside the pail. Note the electrometer reading. Do not let the charge producers touch the pail. Remove one charge producer and note the electrometer reading. Replace the first charge producer and remove the other. Note the electrometer readings. Using the magnitude and polarity of the measurements, comment on conservation of charge. Charging by Contact and Induction It is possible, with one charge producer, to deposit either a positive or negative charge on the inner pail! Push the Zero button on the electrometer to remove any residual charge, and make the charge producers neutral. Then rub the charge producers to separate charge, and put one of them aside. Insert the other charge producer into the inner pail and note the sign and magnitude of the charge indicated by the electrometer. Now let the charge producer touch the inside of the ice pail, then remove the charge producer and note the electrometer reading. Why is there now a permanent charge on the inner pail, as indicated by the electrometer? Where did the charge come from? 9 To show that the charge gained by the ice pail was lost by the disk, ground the system by pushing the Zero button, and then insert the charge producer again into the ice pail. Does any charge remain on it? Now, can we use the same charge producer to place an opposite charge on the ice pail? Here is a way to try. Zero everything. Rub the charge producers to separate charge, and then put the same one aside again. Insert the other charge producer into the inner pail and note the sign and magnitude of the charge indicated by the electrometer. Don’t let it touch the pail! With the charge producer still in the pail, ground your hand to the outer shield, and then briefly touch the inner pail with your finger. After that, remove the charge producer. What does the electrometer read now? Why? Where did the charge come from? Could we repeat this process without having to recharge the charge producer? (This is called charging by induction. When the charge producer was first put into the inner pail, it is said to have induced a separation of charge on the pail.) Charge Distribution on a Sphere Since like charges repel, any time you put a charge on a conducting object, the individual charges will spread out until they are as far apart as possible, which means that the charges will be spread out all over the surface of the object. Just exactly how the charges arrange themselves depends on the shape of the conductor. When charges are placed on a sphere, a highly symmetric object, they will distribute themselves evenly over the surface. If another charged object is brought nearby it will polarize the first object, repelling like charges to the far side and attracting unlike charges to the near side. This induced charge separation is an important effect in the behavior of pith balls, but is much easier to observe directly with larger spheres. At your station are two metal spheres on insulating stands. One will be connected to a high voltage power supply and called the charging sphere. The other, called the sampling sphere, will not be connected to anything. To detect the distribution of charge on the spheres, you will use a "'proof plane," which is simply a conducting disk on an insulated rod. When you touch this disk to the sphere, it picks up a small amount of charge from the sphere. By inserting the 10 proof plane into the ice pail, you can determine the size and type of the charge at that location on the sphere. An important aspect of measuring charge distributions is charge conservation. The proof plane removes some charge from the surface it samples. If the proof plane is grounded after each measurement, the surface will be depleted of charge with consecutive measurements. However, by not grounding the proof plane (and by not letting it touch the ice pail), the charge on the surface is not depleted. That charge which the proof plane removed for one measurement is always returned to the surface when the next sampling is made. When the disk of the proof plane touches the surface being sampled, it essentially becomes part of the surface. To minimize distortion of the surface shape when sampling, hold the proof plane flat against the surface, as indicated in the diagram below. Uncharged Sphere Switch the electrometer to the 3-volt scale and zero the meter. Ground the freestanding sphere by touching one hand to the sphere and one hand to the grounded outer cylinder. Also ground the proof plane and check it with the ice pail. Since the electrometer is now on a much more sensitive setting there may be small movements of the meter which are impossible to get rid of. Just ignore them. Now test for charge on the freestanding sphere by touching the proof plane to it at several points and then inserting the plane into the pail. Choose points on all sides to represent an overall sample of the surface. Since the sphere is uncharged, there should be a zero reading. 11 Charged Sphere Set the freestanding sphere aside. Connect the other sphere to the 1000 volt output of the power supply. This power supply is so severely current limited (10 microamps) that it is impossible to get shocked by it even though the voltage is nearly 1000 volts. The voltage source is to be grounded to the same earth ground as the shield and the electrometer. Touch the proof plane to the sphere and then test it in the ice pail. If the reading is off the scale, switch the electrometer to a less sensitive range. By touching the plane to the sphere in various places, verify that the charge is evenly distributed over the sphere. The wire from the supply is attached to the sphere by a binding post that causes a deviation from the spherical shape. Touch the plane to the tip of the binding post. Is more or less charge found on this point compared to the rest of the sphere? Induced Polarity of a Neutral Sphere With the power supply still on, bring the neutral freestanding sphere close to the charged sphere, about an inch away. (Never let the spheres touch!) Now test various places on the freestanding sphere with the proof plane. What has happened? Why? Charging by Induction (again) With the charged sphere still nearby, ground the sampling sphere by touching it briefly with one hand while you touch the grounded outer cylinder with the other hand. Turn off the power supply and move the sphere that is attached to it. Now study the charge on the freestanding sphere: Is there a net charge on it? Why? Is it positive or negative? How is it distributed? Pith Balls-Visual Evidence of the Force Between Charges The electrometer is an extremely sensitive instrument. The pith ball is not. To get a good effect from the pith ball you will use glass and hard rubber rods which produce extremely high voltages that can ruin an electrometer. Therefore turn off your electrometer before beginning this section of the lab. You are also done with the spheres, so you can set them aside as well. Bring the glass rod near to the pith ball. Since both are uncharged, nothing should happen. 12 Charge the rod by rubbing it vigorously with the silk. Once again bring the rod near the pith ball. The ball should first be attracted to the rod, touch it, and then jump away. Why is the uncharged pith ball attracted to the charged rod? Why does the ball jump away after touching the rod? Observe the continued repulsion of the ball by the rod. Leave the charge on the ball. Now charge the hard rubber rod by rubbing it with the fur. Bring it near to the ball, being careful not to let the ball touch it. Why is the ball attracted to the rod? Remove the charge from the ball and the rods by touching them all over. Since the rods are insulators it is necessary to run your hand over the entire rod to discharge it. Now repeat steps 1-3, only this time start using the rubber rod rubbed with fur. Then answer this important question: Why is the uncharged ball attracted to either charged rod? What does this tell you about the force between charges above-andbeyond the fact that opposite charges attract? Completely explain all the observations you have made with the “pith” ball. Capacitance Two conductors, isolated electrically from each other and from their surroundings, are called a capacitor. When a capacitor is charged, the conductors, or plates as they are called, have equal but opposite charges of magnitude Q. A capacitor has the property that the amount of charge on each plate is proportional to the voltage, or potential difference between the plates. (The voltage V, which is actually indicated on the electrometer, has something to do with potential energy of the charges, as you will soon learn in class.) The constant of proportionality is called the capacitance C of the capacitor. This constant depends only on the size and geometry of the conductors of a particular capacitor. Thus, the equation describing the electrical behavior of a capacitor is simply Q = CV. The capacitor to be used in this experiment consists of two metal plates, 20 cm in diameter, which can be adjusted to various separations. The movable plate is mounted on a calibrated slide that gives the plate separation directly in centimeters. Binding posts are provided for electrical connection to each plate. Three plastic spacers are attached to the fixed plate so that when the movable plate is made to touch these spacers, the plate separation is 1 mm. 13 As long as the diameter of the plates is much greater than the distance d between them, and there is nothing between the plates other than air (a vacuum is better), the capacitance of this arrangement is given approximately by C = A/d, where A is the surface area of the plates and is the constant you met in Coulomb’s law. Thus, as you increase the plate separation d the capacitance decreases, and vice-versa. The purpose of the experiments in this part is to qualitatively study the relationship between C, V, and Q for the parallel plate capacitor. Dependence of V on Q Connect the grounded lead of the electrometer to the fixed capacitor plate, and the other lead to the movable plate. Initially set the electrometer scale to 10 V. The electrometer will directly measure the difference in electrical potential between the two plates, which should be about 2-3 mm apart. Zero the electrometer and capacitor. Connect the 1000 V power supply to the charging sphere, which should be placed far enough away from the capacitor that it does not charge the capacitor by induction. Make sure that both the power supply and the electrometer are grounded. Touch the proof plane to the sphere to pick up some charge, and then touch the proof plane to the top of the movable capacitor plate. Observe the voltage between the plates. Do this several times, noting the electrometer reading at each step. If you always touch the sphere and the capacitor plate at the same place, equal amounts of charge will be transferred each time. Do your observations match your expectations? Why is it sufficient to touch only one plate of the capacitor? Of the three quantities V, Q, and C, what is being held constant in this experiment? Repeat this experiment with a doubled plate separation. Compare the readings to the previous case. Dependence of V on C Set the electrometer to the 100 V scale. Put the capacitor plates about 2 mm apart and Zero the system. Important: Change the wire at the power supply from the 1000 V connection to the 30 V connection. 14 Charge the capacitor to 30 V by briefly touching the power supply wire to the movable plate. The electrometer should read 30 V. While watching the electrometer, slide the movable plate gradually so that the plate separation doubles or triples. What happens? Why? Of the three quantities V, Q, and C, what is being held constant in this experiment? Dependence of Q on C Now reconnect the electrometer to the ice pail, and separately connect the 1000 V power supply wire to the movable plate of the capacitor. The fixed plate, the electrometer, and the power supply should all be appropriately grounded. Initially, the plate separation should be about 6 cm. Zero the electrometer. After grounding the proof plane, sample the charge at different locations on one of the capacitor plates, both on the inner and the outer surfaces. Use the ice pail to measure the charge. How is charge distributed on the plate? Check one or two places on the fixed plate. Choose a point near the center of one capacitor plate and measure the charge in this area at different plate separations. How does the charge vary with capacitance? Of the three quantities V, Q, and C, what is being held constant in this experiment? Dependence of Q on V Return the capacitor plate separation to near 6 cm. Measure the charge near the center of one capacitor plate for a power supply voltage of 1000 V, 2000 V, and 3000 V. How does the charge vary with voltage? Of the three quantities V, Q, and C, what is being held constant in this experiment? Clean-Up Be sure that both the power supply and the electrometer are turned off. Straighten up your station before you leave. Take your lab notebook and briefly discuss your results with the instructor. 15 Physics 111 Laboratory MAPPING ELECTRIC FIELDS Objective To find the equipotential lines and from these the electric field that exists in a plane perpendicular to two conductors kept at different electric potentials. At the conclusion of the experiment you should have a clear understanding of the relation between an electric field and its associated potential field. Reference: Halliday, Resnick, & Walker, 9th edition, sections 22-3, 24-4, and 24-5. Introduction An equipotential surface is a surface, every part of which is at the same electric potential. Electric field lines are everywhere perpendicular to equipotential surfaces. So if the equipotential surfaces around a charge distribution can be found, the electric field lines can be mapped by drawing lines perpendicular to the equipotentials. The field lines show the direction of the electric field, and their line density indicates the magnitude of the field. This method has been used many times to determine the electric fields around real irregularly shaped conductors, for which a mathematical solution would be very difficult. Although the field you will investigate is two-dimensional, it is a good approximation to the field found in three dimensions if the length of the objects in the third dimension (into the paper) is very large compared with their sizes in the plane of the paper. Most of the fields which are discussed in the textbook arise from charged conductors assumed to be placed in an evacuated space. While it is technically possible to measure equipotentials in space, the experiment would be quite difficult. Fortunately it can be shown that if any region of space in which an electrostatic field is set up by charged conductors is filled by a homogeneous conducting material, while keeping the potentials of the conductors the same, the electric field and the equipotential surfaces remain the same as they would be in the absence of the conducting material. This is fortunate because it is relatively easy to measure the equipotentials if the electric field exists in a conducting medium. You will paint two conducting electrodes on slightly conducting paper. Then when an electric potential difference is set up between the electrodes, a thin film of current flows through the paper from one electrode to the other, following the electric field lines. This current flow is associated with a potential gradient in the paper which can be directly measured with a voltmeter. You are to use the multimeter to locate and plot the equipotential lines. From these equipotentials you can then find the direction of the electric field at any point, since it is perpendicular to the equipotentials, and also its magnitude, given by the rate of change of potential in that direction. 16 Apparatus multimeter with probe drawing board 1 piece of conducting paper DC power supply conductive ink ruler 2 alligator clips 2 cylindrical head thumbtacks 4 flat head thumbtacks colored pencils Procedure 1. Fasten the black conducting paper to the drawing board with thumbtacks. Read the instructions on the conductive ink dispenser, then carefully draw the set of electrodes shown in the figure below with the conductive ink. 2. Place a metal pushpin in each of the two electrodes. Be sure that the pushpins are stuck in the middle of the silver lines so that good electrical contact is made. You may need to place a dab of ink around the base of each thumbtack where it makes contact with the electrode. 3. Connect the power supply to the electrodes, as shown in the diagram below. Use alligator clips to connect the leads to the pushpins. Make sure the metal grounding strap that connects the negative and ground jacks on the power supply is in place. Connect the COM jack of the multimeter to the power supply ground. Connect a probe to the multimeter V jack. Set the function switch to V. 4. Turn on the power supply. Adjust the voltage output control to provide a 10 volt potential difference between the electrodes. Check the potential difference with the multimeter rather than the meter on the power supply which is not as precise. 5. Use the test lead going to the “V” jack of the multimeter as a probe to verify that all points on each electrode are at the same potential within 5% of the potential difference 17 between the two electrodes. If there is a change of potential along the electrode, the conductive ink is not continuous and you should go over the ink again. 6. Map a set of equipotential lines at regular voltage intervals. You can use the multimeter probe to find points of equal potential and mark their locations with a pencil. Then draw lines connecting the points of equal potential. 7. Draw in the electric field lines using a pencil of a different color. The field lines should be drawn perpendicular to the equipotentials. Make the spacing between the field lines roughly equal to the spacing between equipotential lines in the same region. This way the spacing of the field lines will correctly represents the relative strength of the field in different regions; the field is strong where the lines are close together and weak where they are far apart. 8. Calculate the electric field strength in volts/meter in the neighborhood of four or five different points on the paper, including points where the field is weakest and strongest. 9. Repeat steps 1-8 above for a second electrode geometry chosen from the suggested electrode configurations provided on the last page of the lab instructions. Questions Compare your results with those of other lab groups. Use your observations to support your answers to the following questions: 1. Are there any regions where electric field lines cross each other? Can you exclude this possibility theoretically? Explain. 2. Why are the equipotentials perpendicular to the field lines? 3. Do you notice any distortion of the field due to the finite size of the paper? Can you explain why this occurs? 4. How would you design an electrode configuration to produce a region in space with a uniform electric field? How would you modify your configuration to obtain the maximum field strength for a given potential difference? 5. How would you design an electrode configuration to produce a region with zero electric field? Describe the electric potential within this region. 6. How do the electric fields near sharply pointed parts of electrodes compare to those near more rounded parts? 18 7. There are several items in the back of the room for which the shape of the electrodes and their electric field play a role in the function of the device. Comment on how the shape is related to the function of each. a) Dipole cell membrane bilayer with a dent (injury) on one side. Lightning rod or building beneath a cloud. a) Without dent – a parallel plate capacitor. b) Two parallel nerve fibers with circular cross section, one (the larger one) depolarized by a nerve impulse. c) Two parallel nerve fibers, both normally polarized. d) A conducting lump inside a dipole cell membrane layer. e) A charged linear filament enclosed in a cylindrical sheath. f) A nerve fiber running parallel to a plane conducting membrane surface. If you can find someone who chose electrode configuration b), compare results. The similarity leads to the idea of image charges. 19 g) Electrode structure used to deflect the electron beam in an oscilloscope. h) Illustrates the electrical shielding effect of a conducting enclosure. 20 Introduction to the Digital Multimeter Description of the Multimeter Many of the electrical measurements in this lab are accomplished using a digital multimeter (DMM). A DMM can be used to measure current, voltage, and resistance. It can also be used to test diodes and check an un-powered circuit for continuity. We use several brands and models of DMM’s in the lab, each with slightly different characteristics but with many common features. Each has a function switch that lets you choose the kind of measurement to be made (voltage, current, resistance, etc.). The function switch normally has more than one setting for current measurements – depending on the amount of current to be measured. DMM’s have an AC/DC switch that is used to select alternating (AC) or direct (DC) current or voltage measurements. DMM’s also have a range switch to determine the meter’s sensitivity (i.e., are you measuring 100V or 100 mV?). Typically the range switch is set to “auto-ranging” – meaning that the meter will automatically adjust its sensitivity to give the best reading. Usually there is a way for the user to select manual range, if desired. Connections. All electrical measurements require two wires – conventionally a red and a black lead (for positive and negative), though the color is not important. The DMM usually has three (or four) banana sockets (also called jacks) to choose from in connecting these two leads. One lead (usually black) is always connected to the common socket. Which of the two (or three) remaining sockets is used for the second (red) wire depends on the kind of measurement to be made. For voltage measurements use the socket labeled “V.” For resistance (or continuity) measurements use the socket labeled “.” (This is often the same socket that is labeled V.) Current measurements are more complicated – different sockets are used depending on the magnitude of current to be measured. Usually one socket is used for low currents (say, up to 200 mA) and another is to be used for currents up to 10 A. Sometimes this second socket is also used for very low current measurements (200 A) but requires that you change the function switch setting. Voltage Measurements 1. Set the function switch to V. 2. Connect the red test lead to the V jack and the black lead to the COM jack. 3. The meter is automatically set to dc. To measure ac voltages, press the AC/DC button and “AC” will appear on the display. 4. Connect the test leads to the two points whose potential difference you would like to measure, with the black lead at or near ground potential, if possible. 5. The meter automatically selects the appropriate range, then displays the value of the voltage. 21 Figure 1. Measuring the potential difference between points A and B Current Measurements (Figure 2) To measure current, the meter must be connected in series with the load. When functioning as an ammeter, the multimeter presents a very low resistance. Incorrectly connecting it in parallel may result in large currents and damage the meter and other parts of the circuit. 1. Set the function switch to the desired current range. If this is unknown, start with the highest range. Setting the range too low can damage the meter. 2. For current measurements less than 200 mA, connect the red test lead to the mA jack and the black test lead to the COM jack. Otherwise connect the red lead to the 10 A jack. 3. Press the AC/DC button if an ac current is to be measured. 4. Turn off the power to the circuit to be tested. Open the circuit and connect the meter in series by means of the test leads. 5. Turn on the power and read the value of the current from the display. A single “1” on the display indicates the range is set too low. Figure 2. Measuring the current flowing in the circuit Resistance Measurements For resistance measurements, it is necessary to remove power from the element to be tested. (The DMM supplies a tiny amount of current to the circuit, measures V and I, then takes the 22 ratio to calculate R – it will be very confused if some other power source supplies additional current – it might even damage the DMM.) 1. Set the function switch to the position. 2. Connect the red test lead to the jack and the black test lead to the COM jack. 3. Disconnect from all circuitry the resistance to be measured and connect a test lead to each end. 4. The meter automatically selects the appropriate range and the resistance value is displayed. Important Caveats When students first start using DMMs to measure current it is common for them to make mistakes – mistakes that could damage the DMM. For protection the DMM has a small fuse that may be accessed through the battery compartment. If excessive current flows into the DMM the fuse will “blow.” A blown fuse has no impact on voltage or current measurements, but it does make it impossible to measure current. If you measure zero current when you know there must be non-zero current – you should suspect a blown fuse. The DMM can measure alternating current and voltage, but it is designed only to measure sinusoidal signals, and only those at relatively low frequencies (less than 500 Hz). A DMM is perfect for measuring voltages or currents associated with 60 Hz AC power available at the wall socket. But it is not designed to measure much higher audio frequencies – particularly if they are not pure sine waves. Finally, you should be aware of the limited accuracy of a DMM. Just because it gives you many digits of readout doesn’t mean they are all significant. Consult the DMM’s specification sheet to determine its characteristics. The DMM accuracy depends on the particular type of reading and the scale used. But as a general rule, the reading is only accurate to within 0.5% of the full scale measurement, plus or minus the last digit. That means that if you are on the 200 mV scale, the accuracy is ± 1 mV. So if you are measuring 4.0 mV on this scale – it is uncertain by ± 25%! Don’t be fooled by lots of digits – digits are cheap, calibration is expensive. Credits This document was obtained from W. Bruce Richards Modified 2010-01-23 by John Scofield 23 Physics 111 Laboratory DIRECT CURRENT (DC) CIRCUITS - I I. INTRODUCTION This experiment enables you to design, construct, and understand simple electric circuits. You will become familiar with a power supply, current and voltage meters, various resistors, and capacitors. The basic laws and rules of circuit theory will be studied without attempting to give their full physical justification: theoretical understanding will be attained through lecture, reading, and homework throughout the semester. If you are already an electrical expert, please be patient while the rest of the class catches up. II. BEFORE COMING TO LABORATORY Solve the preliminary problem and bring your solution to laboratory. III. CIRCUIT ELEMENTS A. Power source Technical term: “seat of emf” (electromotive force) Examples: battery, power supply + Circuit symbols: PS - Algebraic symbols: , V Important property: “Potential difference” between terminals Unit: Volt B. Wires Technical term: Conductor Circuit symbol: Important property: Electrically charged particles can move through conductors; in ideal form a wire has the same potential everywhere. C. Current Circuit symbol: Algebraic symbol: i Unit: Ampere 24 Important property: 1 Ampere in a wire corresponds to 6.25 x 1018 electrons/sec (i.e., 1 Coulomb/sec) crossing any plane through the wire. Current can be measured by an ammeter. D. Resistor Circuit symbol: Algebraic symbol: R, r Unit: Ohm, symbol: Ω Important property: For a given power source, resistance determines how large the current is. Examples: Actual resistors take many different physical forms, which you will meet in laboratory. Simple resistors are little plastic-covered cylinders with a wire attached to each end, with colored stripes that denote the numerical value of the resistance according to the following code1: Figure 1 1 Figure taken from The Art of Electronics by P. Horowitz and W. Hill, Cambridge University Press, 1989, p. 1053. 25 E. Complete circuit and definition of resistance of resistor Given the elements introduced above, we may draw a “schematic diagram” of a sample circuit as follows: Figure 2 In this circuit, the symbol represents a current meter (ammeter) which A indicates the magnitude of the current in the circuit. (“A” stands for Ampere, the unit of current. An “ideal” ammeter would have no resistance, but all ammeters have some resistance and therefore affect the amount of current that flows in a circuit.) The symbol represents an ideal voltmeter meter which indicates the V potential difference between its endpoints. (An “ideal” voltmeter would have infinite resistance and therefore no current would pass through it. Real voltmeters have finite resistance and therefore do pass some current.) In the circuit of Figure 2, if we know ahead of time that the power source can maintain a particular potential difference (it may be a 1.5-Volt battery) and if we read a current i through an ideal ammeter, then by definition the resistance of the resistor is R i . For future reference, Ohm’s Law is: “There exist some objects for which R is a constant, independent of the current through them.” IV. APPARATUS A. Digital multimeters (DMM) The meters we shall use in this laboratory are digital multimeters. These meters use solid-state devices (transistors or integrated circuits) to measure voltage and current and display them as numbers on a screen instead of moving pointer. 26 (These meters are much easier to read and not as fragile as older, mechanical meters based on the d’Arsonval movement.) They are called “multimeters” because they can be used for multiple purposes: to measure current, voltage, resistance, and other quantities. Care must be taken in their use. Look over the Introduction to the Digital Multimeter included in this lab manual. Make sure you understand the procedures given there before hooking up the circuits in this lab. When using a multimeter or DMM, it is important to set it to read the property you wish (e.g., current or voltage) before you insert it into the circuit or connect a voltage source to the circuit. Resistance is always measured for components when they are disconnected from a circuit. (If the element is in a circuit, you do not know whether you are measuring its resistance or that of other elements in the circuit.) B. Power supply In these experiments you will use a low-voltage DC power supply that converts the 110 V AC line power to DC, with output voltage adjustable up to a maximum of about 30 volts and maximum output current of about 1 ampere. The power supply can be set so that it will not deliver more than a particular desired value of output current. In this way you can ensure that no more current than the preset value can flow through your circuit. This current-limiting feature may prevent damage to parts of the circuit which are incorrectly wired. The power supply has three output binding posts, labeled -, GND, and +: GN D Figure 3 The function of the power supply is to maintain a fixed potential difference between the and posts (with no restriction on the absolute potential of either post), independent of the current through the supply, provided that the current is less than the set upper limit. The post is held at a higher potential than the post, but it need not be at a positive potential. (Remember that electrical potential at a point is always measured relative to some other point. Usually it is measured relative to ground – the potential of a metal rod driven into the ground. Ground potential is also the potential of one of the three wires in an electrical socket with three holes in it. On these power supplies, the third (green) post is connected to the third prong on the power plug and hence is held at “ground” potential (i.e., the same potential as the building plumbing, steel framework, and so on). This terminal may be connected to any single point in a circuit being studied to prevent a shock hazard which might occur if some of the circuit components were at an elevated voltage with respect to ground. (Careful! As you will learn later, most electrical instruments - oscilloscopes, signal generators, for example - also have 27 one of their terminals connected to ground through the third prong on their power plug. All of these grounds must be connected together in a circuit.) A small, removable metal bracket is provided to connect the GND terminal to either the + or - terminal, which is often a convenient thing to do. In these experiments, the ground terminal will be connected to the negative terminal. The first thing for you to do is to set the power supply so that it will give an output current of no more than 50 mA. Before turning on the power, flip the current range switch to LO, turn the current control knob as far as it will go counterclockwise, and the voltage control knob all the way clockwise. Then plug a wire directly into the + and GND output binding posts. Finally, turn the power on and slowly turn up the current until the current meter reads 0.05 A on the bottom scale. Since the current range switch is set on LO, a full-scale meter reading corresponds to 0.5 A or 500 mA. Hence when the needle points to 0.05 A, a current of 50 mA is passing from the power supply through the external wire. You have now set the maximum current that the power supply can deliver to 50 mA. Turn the voltage control knob all the way back down (counterclockwise). LEAVE THE CURRENT CONTROL KNOB WHERE YOU HAVE SET IT until you want to change the maximum allowed current setting. Always turn the power supply off when you are not using it. Remove the external wire. To operate the power supply, make sure it is plugged in and then: 1. BEFORE TURNING ON THE POWER SUPPLY: If you are assembling a new circuit, turn the COARSE VOLTAGE and FINE VOLTAGE control knobs fully counterclockwise (“all the way down”). Do not change the CURRENT control knob, from the way you pre-set it. 2. After your complete circuit has been assembled and checked by the instructor, turn the POWER switch to ON, and turn the VOLTAGE control knobs clockwise to obtain the desired voltage. 3. The upper meter may be used to read the approximate potential difference between the and posts, and the lower meter reads the approximate current being delivered by the power supply. If the CURRENT range switch is in the HI position, read the upper scale on the current meter; if the range is switched to LO, read the lower scale. In most parts of this experiment you will use a digital multimeter to read currents and voltages more precisely than you can with the meters on the power supply. 4. When you wish to make changes in your circuit or are through experimenting with it, turn the power supply OFF. C. Decade resistance box A decade resistance box is a low wooden or metal box with at least two binding post terminals and several knobs. The box contains an assortment of precision resistors, various combinations of which can be connected to the posts by turning the knobs. The total resistance between the terminals is equal to the sum of the 28 readings of the knobs. Usually these resistance boxes are used only for the most precise work, but we shall use them as convenient sources of adjustable resistance. D. Other resistors Several individual resistors mounted on two-prong plugs are provided. Refer to the color-code table in Figure 1 to determine their alleged resistances. In contrast to the precise values set on resistance boxes, the resistance values of these fixed resistors may differ from their nominal value by 10% or more. Most (commercial) circuits are designed so that these differences are not significant. E. General advice on wiring It will be much easier for you (and your instructor) to inspect your wiring if it “looks” like the circuit diagram. Physically lay out your circuit so that it matches the circuit diagram. Use red wires for positive and black wires for negative potential (to the extent that you can). A messy circuit behaves the same as a neat circuit – bit if there is a wiring mistake, it is much easier to identify in a neat circuit! As you add components to a circuit, there is a tendency for it to become more and more complicated – like tangled strands of spaghetti. Use junction blocks to simplify your wiring. V.Ohm’s Law A. Varying voltage with fixed resistance For many circuit elements the magnitude of the electric current through the device is proportional to the magnitude of the electric potential difference it. When this is the case we say that the device follows Ohm’s Law. THROUGHOUT THIS LABORATORY SESSION, ALWAYS BEGIN YOUR MEASUREMENTS ON A NEW CIRCUIT WITH THE POWER SUPPLY VOLTAGE AT ZERO (FULLY COUNTERCLOCKWISE). Wire up the circuit shown in Figure 4. Use the multimeter to measure the potential difference (voltage) across a resistance box that is adjusted to 2000 ohms. (Set the meter to read DC volts.) Use another multimeter to measure the current through the box. (The highest current you expect to measure is 30 volts/2000 ohms or 15 mA. Therefore set the current meter to the 20-mA scale. If the measured quantity exceeds the scale you choose, the meter will read “1 .” . That is, it will show the digit 1 followed by a several blank spaces and then a period.) Vary the voltage from the power supply over the entire range from 0 to 30 volts. For at least six values of the voltage across the resistor, read and record the voltage V R and corresponding current i . Calculate V R / i . Plot a graph with voltage on the horizontal axis, current on the vertical. Remember that the origin is a data point. 29 General Remark – when you make graphs like this, you typically want to have (roughly) equally-spaced voltages – for this case, 0V, 5V, 10V, etc. Don’t waste time adjusting the power supply to get these exact values – just set the voltage to be in the ballpark and record whatever you measure it to be. If you are making your graph with a computer it is just as easy to plot 5.13V as it is to plot 5.00 V. In all cases, your laboratory notebook should include a schematic diagram of the circuit you are studying. Does your graph show the linear relation expected for a system that obeys Ohm’s Law? Draw a best-fitting straight line and from the slope calculate the resistance to which it corresponds. Is it close to 2000 ohms? (If it is not, something is wrong!) Figure 4 B. Varying the resistance with fixed voltage Now adjust the power supply output so the voltage across the resistor is about 30 volts. Increase the resistance of the decade box in 6 steps up to about 20,000 ohms. Check that the current flowing in the circuit behaves as you expect it to. 30 VI.Resistors in Series and Parallel A. Resistors in Series Connect a second decade box in series with the first one, as shown below. Set the boxes to 2000 and 4000 ohms, respectively. Figure 5 Apply four different voltages across the series combination, measure the current through it for each, and thereby determine the resistance of the combination. Does it agree with what you expect in all cases? For the last voltage you use, predict the voltage across each resistor individually, using Ohm’s Law, and check by measuring them. You have just seen how two resistors can be used as a voltage divider. You connect them in series with each other, then you connect the series combination across the potential difference which you wish to divide into several smaller potential differences. Voltage dividers are used widely in electronic circuits. B. Resistors in Parallel Connect up the parallel combination of resistors as shown in Figure 6. Apply four different voltages across the parallel combination, measure the current for each voltage, calculate the effective resistance for the two resistors in parallel for each, and compare with what you expect on the basis of the formula for parallel resistance. For the last voltage you use, predict the current that should be flowing in each individual resistor and check your predictions. 31 Figure 6 C. A More Complex Circuit Wire the circuit in the figure 7 using the following values: R1 decade resistance box at 400 ohms R2 decade resistance box at 700 ohms R3 fixed resistor of about 470 ohms (on a plug) power supply set at 10 volts Figure 7 Set 10 volts using a digital voltmeter initially and make sure that remains at 10 volts for each measurement. Insert your ammeter so it measures one of the currents, i1, i2, or i3 . After the instructor checks your wiring, measure the currents i1, i2, and i3 with the multimeter and compare the results with your algebraic predictions, evaluated for the given components. While you are measuring i2 and i3 , use another multimeter to record the voltages across R2 and R3 . 32 How good is the agreement between the predicted and measured values of currents? What are the reasons for the discrepancies? Draw the circuit neatly in your laboratory notebook and record your calculations and results. From the currents i2 and i3 and the voltages across R2 and R3 , calculate R2 and R3 . How do they compare with the labeled values? Measure the values of R2 and R3 with one of your DMMs and compare those values to the labeled values and the values you just calculated. (See the DMM instructions.) Explain any significant discrepancies (say, those of more than 5%). VII. Measurement of i vs. V Characteristics of Several Elements A. General Procedure This is the outline of a general procedure to be applied to a study the electrical characteristics of several different devices (or circuit elements) in steps B through E below. You will put the device of interest, called the “device under test” or DUT into the circuit of Figure 8. Your goal is to simultaneously measure the voltage across and the current through the DUT, determining what we call its I-V characteristics. You will use a multimeter to measure voltage, and another multimeter to measure current. The resistor R protects the DUT by limiting the current that can flow through it. You will measure the current through the element as a function of the voltage across it. Figure 8 For each element below, you will start with the applied voltage at a minimum, apply several (e.g., ten) different voltages to the circuit, and thereby gather data with which to make a fairly comprehensive graph of the current through the element as a function of the potential difference applied to it. 33 B. Tungsten Filament (Flashlight Bulb) Reset your power supply for a maximum current of 150 mA. Apply the above procedure to the flashlight bulb. See Figure 9. Do not exceed a voltage of 6.0 volts across the light bulb for any of these measurements. For the protective resistor, use a total of about 150 ohms. (You should have fixed resistors of this size that are capable of dissipating five or ten watts. Do not use a resistance box. They cannot dissipate this much heat!) Figure 9 Plot the current vs. voltage (voltage on the horizontal axis). You may use a lab computer for this or graph paper. In either case, fasten the graph in your notebook. Your current axis should range from 0 to 300 mA. Does the filament obey Ohm’s law? Does the filament have a “resistance”? Explain why the filament behaves differently from the resistor you studied earlier in this experiment. Does it make any difference which direction the charge flows through the filament? Reverse the connections to the bulb for one value of applied voltage and see how the current changes. C. Diode - Forward Current Reset your power supply for a maximum current of 300 mA. Install the small diode (mounted on a double banana plug) into the circuit of Figure 9 with the end with the silver band around it connected to the negative side of the power supply. For the protective resistor, use a fixed resistor of about 100 ohms. Applying the above procedure, determine the current vs. voltage response of the diode; the current should not exceed 300 mA. Most of the variation in current will occur at low voltages - around 0.6 to 0.8 volts. Take at least 10 measurements in this region, and a few at lower voltages. Plot the current vs. voltage (voltage on the horizontal axis). Use the shorter axis for the current. 34 D. Diode - Reverse Current Does it make any difference which direction the charge flows through the diode? Reinstall the diode with its leads reversed and measure the current when the voltage is applied to it in the opposite direction. In particular measure the current when 30 volts is applied to the diode. (The current may be in the microampere range so you may have to adjust the scale on the ammeter.) With this circuit operating and 30 volts applied, unplug the DMM that is measuring the voltage across the diode. Does the current change? What is going on? E. A Forward-Biased Diode Note: This section involves a theoretical calculation. There is no experiment to do. Suppose the diode is used in the circuit shown in Figure 11. The “power supply” puts out a voltage V(t) that varies sinusoidally with time, so its output voltage is V Vo sin(t) . (Don’t worry about the meaning of at this stage. It just determines how many oscillations there are per second.) Let Vo be 6.0 volts and R be 50 ohms. Figure 10 Let’s idealize the diode so that its i vs. V curve looks like that in Figure 11. Consider two cycles of the input sine wave. Predict how you expect the voltage V R (t) to depend on time. Sketch on one graph (with some quantitative accuracy along the voltage axis) the values of both V(t) and V R (t) vs. t. Explain what is going on. 35 Figure 11 36 Direct Current (DC) Circuits Preliminary Problem Please solve this problem and bring your solution to your DC circuits I laboratory session. Suppose three resistors and an emf device (all ideal) are connected as shown below: i1 R1 + R2 E i2 R3 i3 a. Show that i1 E R2 R3 R1 R2 R1 R3 R2 R3 i2 E R3 R1 R2 R1 R3 R2 R3 i3 E R2 R1 R2 R1 R3 R2 R3 b. Are these equations dimensionally correct? c. Do they display the proper symmetry between subscripts 2 and 3? d. Does it make sense that i1 i2 i3 ? e. The current leaving the top of the battery and passing through R1 is i1 . What is the current entering the bottom of the battery? f. If the battery emf E is tripled, what happens to each of the currents, i1 , i2 , i3 ? g. If every resistance value is doubled, what happens to each of the currents? h. If R3 0 , what is i2 ? i. If R3 is much greater than R2 , what is i3 ? Is this what you expect? j. Does current always take “the path of least resistance”? 37 Physics 111 Laboratory DIRECT CURRENT (DC) CIRCUITS - II Here we pick up where we left off in the DC Circuits lab. VIII. Capacitors A. Charging a Capacitor Another device used in electrical circuits is a capacitor. It stores electric charge. It consists of two pieces of electrically conducting material (called electrodes) placed close together. The capacitor you will use in this experiment is contained in a cylinder that has a wire lead sticking out of each end. If you were to cut it open, you would find inside two long strips of metal foil, separated by a thin layer of insulating material (called a dielectric), tightly rolled up to fit into the cylinder. Each of the metal strips is connected to one of the wires. Since the two strips are insulated from each other, electrons cannot flow from one to the other as they do in a conducting wire. However, as you will see, electrons can be made to flow through one wire onto one of the metal strips while at the same time an equal number of electrons leaves the other metal strip and flows out through the other wire. The net effect is to leave a certain quantity of negative charge on the first electrode and an equal quantity of positive charge on the second one. To an outside observer, it is just as if electrons had flowed through the capacitor! Set up the circuit in Figure 12. Let R be 1 megohm. The switch is like a telegraph key: when you press down on the key, it changes the connection. Use your DMM to determine which jacks are connected when the key is up (not pressed) or down. The switch should be inserted so that when the key is up, the DMM measuring voltage is connected to the capacitor, while pressing the switch causes the right end of the 1-megohm resistor to be connected to the capacitor so current can flow to the capacitor. (In the diagram the jack on the switch that is connected to the voltmeter is labeled “NC” to indicate “normally connected.”) Plug the capacitor into a junction box. Connect the end of the capacitor labeled to the switch. Use another DMM to measure current. Procedure: 1. Connect a wire temporarily from A to B. Turn on the power supply and adjust the output voltage until a current of 10.0 A flows. Set this current as well as you can. Remove the wire from A to B. The experiment starts in step 2 below. This is a description of what you are going to do. You will let a current of 10.0 A flow into the capacitor for a period of 30 seconds. (The current will tend to fall during this period, and you will have to adjust the FINE voltage control to maintain this current during the measurement period.) Then you will measure and record the voltage across the 38 capacitor (quickly) and again let current flow onto the capacitor for another 30 seconds. You will carry out this charging/measuring process five times, for a total of 150 seconds. You will make sure that the current remains steady at 10.0 A . Do not discharge the capacitor between these chargings. Figure 12 2. Connect a wire across the capacitor for several seconds to completely discharge it. You are ready to begin a series of capacitor chargings and voltage measurements. 3. Follow the procedure above to charge the capacitor for five 30-second intervals, measuring the voltage across it at the end of each period. Use the key switch to charge (press down) and measure (key up). Use a stop watch to measure the 30second intervals. Keep the voltage-measurement periods as short as possible; otherwise the capacitor will discharge (electrons flowing out of the capacitor) significantly and spoil your results. Remember to keep the current steady! If you have to restart your set of charging measurements, begin with step 2. 4. Make a graph of charge stored on the capacitor vs. voltage across the capacitor. (In order to plot your data, you will need to calculate how many microcoulombs of charge are stored by the capacitor during each 30-second charging period. Recall that a current of one ampere represents one coulomb of charge flowing in a one-second period.) 5. How do the charge stored by the capacitor, Q, and the voltage across it, V, appear to be related? Draw a best-fit straight line that passes through the origin. The observed fact that the ratio Q/V is independent of the magnitude of Q suggests that the ratio be given a name. This is the definition of the capacitance of any two electrodes: C 39 Q . V The unit of capacitance is called the farad, after the 19th-century British scientist Michael Faraday. If two electrodes have a capacitance of 1 farad, then there will be a potential difference between them of 1 volt (named after the Italian scientist Volta) when they have equal and opposite charges on them of 1 coulomb (named after the French scientist Coulomb). This definition is analogous to the definition of resistance. If the ratio V/I for a piece of conducting material is independent of I, then it is useful to define the resistance R V . I Determine the capacitance of the capacitor from the slope of the line on your graph. Write its value, in microfarads, on your graph and in your notebook. B. Discharging a Capacitor 1. Connect the circuit in Figure 13. Change the connections to the switch so that it normally connects the capacitor to the power supply. R represents a 68,000ohm resistor that is normally connected across the voltmeter, not across the capacitor. Turn the coarse control on the power supply down to zero, and then, using its own voltmeter, set the power supply to about 20 volts. (Do not exceed 25 volts, the rating of the capacitor.) You are going to charge the capacitor and then observe its discharge through R, measuring and recording the voltage across it as a function of time. 2. With the power supply at 20 volts (not critical) and the key on the switch up, the capacitor is charged to about 20 volts. If you press the key, the charge on the capacitor will gradually leave by means of current through R (and through the voltmeter, but its resistance of about 10 megohms is a much larger than R). Press the key and see how the voltage across the capacitor decreases. Now it’s time to do this quantitatively. Figure 13 40 3. Charge the capacitor to 20 volts (key up). Press the key and hold it down. Take your first voltage reading as you start the stop watch. 4. Holding the key down, record the value of the voltage at 10-second intervals (keep the stopwatch running) for a period of 140 seconds. This will require that you work closely with your partner. 5. Plot your data on a semi-log graph, with time on the linear axis.2 You should find that it is a straight line. (See below.) Determine the slope of this line. Its 1 units should be s since the logarithm of a quantity must be dimensionless. The voltage across a capacitor C which initially has voltage Vo across it and which discharges through a resistor R is given by V(t) V oe t RC . We divide both sides by V 0 (to make each side dimensionless) and take the natural logarithm and find V (t) t ln . RC V0 Therefore the slope of the curve you have plotted should be 1 . RC Measure R with a DMM and find the value of C from this slope. Does it agree with the value you found from your measurements during the charging process? 6. Explain (in writing) what is going on. Why does the voltage drop more and more slowly? Describe physically what is happening. IX. An Unknown Source of emf A small box labeled STANDARD CELL is alleged to contain a source of emf in series with a resistor. Your task is to perform a series of experiments that will allow you to determine if the unit acts like a source of emf in series with a resistor, and, if it does, to determine the value of the emf and the resistor r . You may wish to connect a circuit like that shown in Figure 14. R is a decade box. The ammeter and voltmeter are DMM’s. Or, you may wish to place the voltmeter and ammeter in other places in the circuit. The point is, you need to find out whether the STANDARD CELL behaves like a source of emf with a resistance in series. Graphing an appropriate voltage against an appropriate current may help. 2 You can make a linear graph of ln(V) versus t or, better yet, you can plot V vs t, but choose to have a logarithmic vertical axis. If you choose the latter, be aware of the conversion of from natural log to base-10 log in calculating the slope. 41 Figure 14 42 Physics 111 Laboratory Using an Oscilloscope A. Introduction: What is an Oscilloscope? The cathode ray oscilloscope (CRO or “scope”) is a very useful and versatile tool. It is used not only by physicists and electrical engineers, but also by chemists, physicians, psychologists, and others. It is to your advantage to learn how to use this important instrument now, because you will be using it throughout the rest of this course and, probably, throughout your scientific career. Fundamentally, an oscilloscope is a voltmeter that displays a graph of voltage against time on a screen. This is an exploration exercise, not a time to rush through these sections. The idea is to play, to experiment, and to learn how this important instrument can be used as a tool. B. Tektronix 2213A Controls: Standard Configuration A high quality oscilloscope like your Tektronix 2213A has a number of special features that adapt it to a variety of applications and hence the control panel is cluttered with a variety of knobs, toggles, and push buttons. Don't panic! Most of the time you will need to adjust only two knobs, one labeled CH1 VOLTS/DIV and the other labeled SEC/DIV. Start your oscilloscope off with the controls set as follows, in what we will call the standard configuration: Trigger system controls (right-most group of controls): VAR HOLDOFF knob: full counterclockwise to NORM SGL SWP button: out P-P AUTO button: depressed NORM button: out SLOPE button: either in or out LEVEL knob: anywhere INT toggle: full up to CH1 SOURCE toggle: full up to INT EXT COUPLING toggle: full up to AC HORIZONTAL and VERTICAL system controls: three POSITION knobs: pointing straight up three red CAL knobs: fully clockwise until they click HORIZONTAL MODE toggle: full left to NO DLY (DLY = delay) (In this mode, the DELAY TIME controls below the SEC/DIV knob have no effect.) SEC/DIV knob: 0.1 msec/div is usually a good place to start. VERTICAL MODE toggle: full left to CH1 43 BW LIMIT button: depressed CH2 INVERT button: out ADD, ALT, CHOP toggle has effect only when VERTICAL MODE is BOTH CH1 (or CH2) VOLTS/DIV knobs: Read the knobs at the 1X position, not the 10X PROBE position. It's usually good to start out at 0.5 volts/div. “Coupling” toggles (below VOLTS/DIV knob - AC, GND, DC): full right to DC Set your scope into standard configuration, plug the power cord in, and depress the POWER button. In a few seconds you will see a single horizontal line on the screen. Adjust the INTENSITY and FOCUS knobs to make this line bright and sharp. Adjust the channel 1 POSITION knob so that the line is immediately behind the x-axis line etched upon the screen. This is a good starting point for every set of measurements. C. The Oscilloscope as a Clock This is just an introduction to this capability, which you will use often throughout the exercise. The horizontal axis of the oscilloscope screen measures time. Convince yourself of this by turning the SEC/DIV knob counterclockwise to 0.5 sec/div. The screen will display a signal dot that traverses two horizontal divisions in one second. When it disappears from the right side of the screen, it pops back up on the left. Turn the knob clockwise to speed up the dot until it blurs into a solid line. D. The Oscilloscope as a DC Voltmeter The vertical axis of the oscilloscope screen measures voltage. Convince yourself of this by connecting the channel 1 inputs of the scope across a 1.5 V battery. (Set the CH1 VOLTS/DIV scale to a larger value if the line jumps off the screen, to a smaller value if the jump is imperceptibly small.) Measure the battery voltage with a conventional voltmeter to check the oscilloscope’s voltage measurement quantitatively. Vary the CH1 VOLTS/DIV scale and observe what happens. Before you go on to the next section, disconnect the wire from one terminal of the battery to the black (ground) input of the oscilloscope. What happens to the signal on the screen? Why? Describe and explain in your notebook. E. The Oscilloscope as a Voltmeter for Time-Varying Signals The power of an oscilloscope is manifest only when it is used to display time-varying voltages. Devices that produce time-varying voltages are called signal generators, and you have one such device before you: an audio generator. 44 Set up the oscilloscope to display the output voltage of your signal generator. As explained in the next section, you must connect the black terminal of the signal generator to the black terminal of the oscilloscope, and similarly for the red terminals. Reversing the connection will not harm the equipment, but it won’t generate a signal either. Experiment with the SINE-SQUARE toggle of the generator, the output level and range controls and the frequency controls. Measure the period of two sinusoidal signals and two square-wave signals and calculate the frequency of each. (To do this, set the time scale so that you may measure the time for several complete periods.) How well do the frequencies agree with the dial on the signal generator? While oscilloscope readings are not highly precise (they can usually be trusted to a few percent), they are considerably better than the readings from the signal generator dial. Before you go on to the next section, disconnect the wire between the black (ground) terminal of the signal generator and the black (ground) input jack of the oscilloscope. What happens to the signal on the screen? Why? Describe and explain in your notebook. F. Grounding This section may explain some of the behavior you observed in section E. If necessary, write a new explanation of what happened when you disconnected the black terminal of the signal generator from the black input jack of the oscilloscope. For reasons of safety, one of the two input terminals on your oscilloscope, the black one, is grounded. This means that the black terminal is connected directly by a wire to the metallic oscilloscope case and to the third prong of the electrical power plug. The third prong is attached by the building’s electrical wiring to the building’s plumbing and allegedly to a large copper plate buried under the building (hence the name “ground”). The same applies to the black terminal on your signal generator, and indeed to one of the two terminals of almost any electronic device that is plugged into a wall socket. This rather elaborate system helps prevent accidental electrocution. Our immediate concern, however, is not why the grounding system exists but rather its implications for electrical measurements. Because of the grounding system, the black terminals of the scope and of the signal generator are directly connected by wires through the third prong of their power plugs. Think about what this means for the following circuit: 45 The voltage across both resistors (VAC) will be twice the voltage across either resistor (VAB or VBC). Set up this circuit and measure the voltages VAC and VBC as follows: Verify that VAC = 2 VBC. Now try to measure VAB with the circuit below: 46 You will measure a voltage that is not equal to the previously measured VBC. This is because the signal generator’s grounded terminal is directly connected to the scope’s grounded terminal via the third prong of the power plug. The true circuit diagram for the above situation is: The bottom resistor is connected in parallel with a piece of wire that runs through the grounding system. Naturally more current flows through the low resistance wire than through the 10k resistor. To summarize, in a proper circuit design the grounded terminals of the oscilloscope and signal generator must be directly connected by wire, because they are connected by wire through the grounding system whether you want the connection or not! G. Transformer Output Plug your 6-volt transformer into an electrical outlet and look at its output with the oscilloscope. Unlike the signal generator, neither output of the transformer is connected to ground. Measure the “peak-to-peak” voltage (maximum to minimum voltage). 47 Note: Since this transformer has no ON-OFF switch, you should unplug it from the wall as soon as you are finished making measurements. In an AC signal the voltage changes with time. Three different numbers are commonly used to characterize the voltage of a sinusoidal signal: the “peak-to-peak” voltage Vpp , V pp V the “amplitude” V o , and the “root-mean-square” voltage V rms o . (The rms 2 2 voltage is the most frequently used, partly because the familiar DC formula for power V2 holds in the AC case if V is the rms voltage.) P R To check these results, measure the output of the 6-volt transformer with your digital multimeter (DMM). Make sure your DMM is set to measure AC voltages rather than DC. Does the DMM reading agree with one of the values ( Vpp , V o , V rms ) you determined with the oscilloscope? Which voltage does the DMM measure? Use the oscilloscope to determine the period of the observed signal. Set the time scale so that you may measure the time for several complete periods. From the period, calculate the frequency. Is it approximately equal to the value you expect? There will be a single set-up on a side table at which you may determine the output voltage of your transformer when significant current is being drawn from it. Take your transformer and DMM to that location, where there is a low-resistance high-power resistor. Measure the resistance of the resistor with your DMM; you may need to switch the DMM to DC to do this. The resistance should be between 1 and 2 ohms. (If not, call the instructor.) Connect the resistor to the transformer, plug in the transformer, and measure its output voltage with your DMM (set to read AC). What voltage does the V meter display? How much current is flowing ( Irms rms )? Why is this transformer R called a 6-volt transformer? H. Studying a Diode with a Dual Trace Oscilloscope Think of a diode (crudely) as a valve that allows current to pass in only one direction. In the experiment on Direct Current (DC) Circuits, you predicted the voltage signals that should appear across the signal generator output and across the resistor in the circuit below. Sketch also a prediction of the voltage across the diode. Connect the circuit shown in Figure 1. With the ground of the oscilloscope connected to point C, observe and record quantitatively the voltage signals between A and C (generator output) and B and C ( V R ). (Note that neither side of the transformer secondary (output) is automatically grounded when the primary is plugged into the wall socket; therefore no point in the circuit is connected to building ground until it is 48 connected to the oscilloscope. Hence any one point in the secondary circuit may be safely connected to the grounded input of the CRO.) Move the oscilloscope ground connection to point B and observe and record the voltage signal between A and B ( V diode ). Compare these observed voltages with your earlier predictions. A diode B t ransformer 1 kohm C Figure 1 Your oscilloscope has a dual trace feature that enables you to observe the diode and resistor voltages simultaneously. Connect the oscilloscope to the circuit as shown in Figure 2, set the VERTICAL MODE toggle to BOTH and the ADD, ALT, CHOP toggle to CHOP. Note that both grounds are, of necessity, connected to the same point in the circuit. (In fact you only need to make one ground connection to the oscilloscope.) In this orientation, channel two is actually displaying the negative of the resistor voltage (because of where the ground connection is). Depress the CH2 INVERT button to show the resistor voltage right-side-up. Note: To make sure that the “zero-level” of voltage is the same for both channels, switch the coupling switches for CH1 and CH2 (below the VOLTS/DIV switches) to GND. That grounds the inputs to these channels so there is no signal going in. Adjust the vertical POSITION knobs so that the two horizontal lines coincide. Now turn the coupling switches back to DC. 49 A CH1 diode t ransf ormer B GND 1 kohm CH2 C Figure 2 Sketch V diode and V R quantitatively on a single graph. Compare these observed voltages with your predictions. Connect a 100 F capacitor in parallel with R, with the +-end of the capacitor connected to point B. Sketch V R . Explain. (Hint: Calculate RC for these components and compare it with the period of the AC signal you are applying.) You have just constructed a simple DC power supply. Remove the capacitor. Usually, as was pointed out at the very beginning of this experiment, the horizontal oscilloscope axis measures time. However, when the SEC/DIV knob is turned fully counterclockwise to X-Y, the oscilloscope displays channel one voltage on the X axis and channel two voltage on the Y axis. In the current set-up channel one displays the voltage across the diode, while the channel two signal is proportional to the current through the diode. Put the oscilloscope into XY mode and see at a glance the I vs. V characteristic curve for your diode! (Recall from the experiment on Direct Current (DC) Circuits how much work it was to produce such a characteristic curve by hand.) I. Side Benches On the side benches are three setups that explore various features of an oscilloscope. Look at and take notes on at least one of them: a. Low frequency signals, triangle signals, sound. b. Musical instruments. c. Music on tape. 50 Physics 111 Laboratory CHARACTERISTICS OF A SOLAR CELL I. Introduction Before coming to laboratory read the essay “Solar Cells and Photovoltaic Power Generation” by John Scofield. Also, go back and review the “unknown EMF” experiment at the end of Experiment 4 as well as your earlier measurements of the I-V characteristics of a diode. You might find this website useful http://inventors.about.com/library/inventors/blsolar3.htm. If you google “how solar cells work” you’ll find many other useful pages. In this experiment you will determine the electrical properties of a silicon solar cell. These properties are: 1. short-circuit current, ISC 2. open-circuit voltage, VOC 3. maximum power, PMAX (under standard test conditions) 4. fill factor, FF 5. energy conversion efficiency, . All of these quantities will be defined later in this write-up. A solar cell is a semiconductor diode. A diode is a two-terminal device which allows current to flow more readily in one direction than in the other. It differs in another fundamental way from resistors and capacitors. These components are said to be “linear” since, when all other conditions are kept fixed, doubling the voltage across them also doubles the current. A diode has a more complicated behavior and is therefore called a nonlinear device. You will recall that you have twice looked at the I-V characteristics of a diode in lab earlier this semester. II. Connecting the Circuit Our goal is to set up a simple circuit with an ammeter and a voltmeter to measure the I-V curve of a solar cell. Ideally this would look like Figure 1 below, where we have this nice power supply for which we can set the voltage anywhere between -600 mV and 600 mV, and it will supply this voltage no matter how much current is required. The B&K Power Supplies you have used, thus far, are not capable of this. (What we want the power supply to do is “source voltage” so that we can measure the resulting current. The B&K power supply is capable of the opposite – controlling current, then you measure the voltage.) Figure 1. Ideal circuit for measuring solar cell I-V characteristics. I and V may be measured with digital multimeters. The power supply has special characteristics that are not obtained with the usual models. 51 Bill Mohler, the electronics expert here at Oberlin College, has designed and built a voltage regulator circuit that works with another DC power supply we have to accomplish this task. You will have to hook up the power supply, voltage regulator circuit, and solar cell as described below. Figure 2. (a) Gilford Instruments DC power supply and (b) Voltage Regulator for the solar cell experiment. The only connections required between yellow, black, and brown banana jacks on both devices (±12 V and ground). The DC power supply and the voltage regulator circuit are to be connected with three banana wires that join their yellow, black and brown connectors. The red/black jacks on the Voltage Regulator labeled Current are to be hooked up to a DMM set to the 200 mA scale for measuring current (i.e., set the DMM to measure DC-A). The other pair of red/black jacks on the Voltage Regulator labeled Volts are to be hooked to a second DMM to measure DC voltage – on the 2 V scale. The long gray cable from the Voltage Regulator will be hooked up directly to your solar cell. The solar cell needs to be mounted on a bracket that will hold it vertically so that light from a slide projector is incident on the cell. To increase the light intensity you want the cell as close to the lens as possible. But you also want to have uniform light intensity over the entire cell, so you will need to keep the cell a few inches from the lens – obviously this is a compromise. A typical arrangement is shown below. Note that it is important to make sure the plane of the solar cell is perpendicular to the light rays. 52 Figure 3. The solar cell is mounted vertically and placed in the light beam of a slide projector as shown. Remember – never set the voltage so high that the current exceeds 200 mA or you will blow the fuse on your DMM ammeter. III. Measurements “In the Dark” First you will measure the I-V characteristics of the solar cell in the dark. To do this place the cell in the dark (i.e., cover it with an opaque cloth). Measure I vs. V, recording your data in a table. Begin with the voltage set to – 600 mV and carefully step up to zero then on into forward bias (i.e., positive voltage across the solar cell). Use the most sensitive current range for your ammeter, but don’t exceed full scale for any range. Be sure to stop increasing the voltage when your current approaches 200 mA. You don’t need to take data for equally spaced voltages. As a rough rule-of-thumb you want to take data so that you are convinced nothing curious is happening in-between adjacent points on your graph. Plot the data using Excel or another graphing program. (Save your file as you will be adding to this graph below.) Choose scales so that voltage is on the horizontal axis (-600 to 600 mV) and current is on the vertical axis (-200 to 200 mA). Did you take sufficient data or do you need to go back and get more detailed data for some range of voltage? At some point take your “opaque shield” off of the cell and see if the measured I or V are affected by background lights. IV. Measurements Under Illumination You are now ready to measure the “light” I-V curve for your solar cell. Ideally you would do this under illumination identical to that given by the sun (same intensity and color characteristics). Of course, the sun’s light depends on where you observe it: Colorado or Ohio, summer or winter, clear or cloudy, etc. Solar scientists have agreed upon a “standard” sunlight which corresponds (approximately) to the noon-day sun at sea level in Arizona on a clear day in June. This has the intensity J 0 1,000W 2 100 mW 2 , m cm 53 Which is called AM1.5 solar illumination.3 Having such a standard allows for ready comparison of the parameters of different solar cells. In the Department we have one AM1.5 solar simulator which you might all use – but it would take too long. Instead, you will use the lamp from a slide projector, and the manufacturer’s rated short-circuit current to calibrate your light source. If you have not already done so, mount your solar cell in the bracket and place it a few inches in front of the slide projector lens. Leave your measurement circuit setup so that you are able to read I and V as you change conditions. Adjust the Voltage Regulator so that the cell voltage is zero (or very close to zero). In the dark this will result in zero current – but not in the light. Note: Once you have used the slide projector bulb, do not turn the projector completely off. Instead, turn the lamp off but leave the switch set so the projector fan runs to cool the bulb. The fan should be run for a few minutes before you turn the projector all the way off. Turn the projector lamp on so that it illuminates your cell. You want to center your cell in the light beam with uniform light intensity over the entire cell so that it produces almost -200 mA of current (say -190 mA to be safe). For the rest of this experiment it is important that your cell remain fixed relative to the projector – a small change in distance will result in a significant change in light intensity. Record the short-circuit current under illumination (i.e., the current when V = 0). Now make measurements of I vs. V with V ranging from -600 mV to 600 mV (or as high as you can go without exceeding the 200 mA current limit). Form a table with three columns, V (mV), I (mA), and P (mW) = I*V. This product is the electrical power dissipated in the solar cell (or power produced when the product is negative). You should now observe significant current even when the cell is reverse-biased. In fact, the light I-V curve should be nearly identical to the dark I-V curve, except that it is shifted “down” by an amount IL, which is the “light-generated current.” (If you turn of the lamp, the curve shifts back up). Add your light I-V curve to the graph you made earlier. Compare the two. Do they have the expected relationship? Graph the power (mW) vs. bias voltage (mV) for your illuminated cell. You should find that the power curve shows an extreme (maximum power generated). Why? Take additional I-V measurements to determine PMAX, and the values VMAX and IMAX corresponding to the maximum power point. Be sure to estimate your uncertainties in these quantities. 3 The AM stands for “air mass.” AM0 has a higher intensity (1380 W/m2) and corresponds to the intensity of sunlight above the Earth’s atmosphere. AM1 light corresponds to sunlight directly overhead on a clear day passing through 1.0 “thickness” of the atmosphere. 54 V. Calculate Device Parameters You are now ready to determine solar cell parameters. Carefully measure the total area of the solar cell with a ruler along with its uncertainty. Next estimate the area of the cell that is covered by the aluminum grid (with its uncertainty). Subtract the grid area from the total cell area to determine the “active” cell area. Here you need to be precise in keeping track of uncertainties as they affect the accuracy with which you can calculate device parameters. From the graph of your data determine VOC (the voltage for which the current is zero), ISC (the current for which voltage is zero), and PMAX. Calculate the fill factor given by the ratio P FF MAX . I SCVOC Refer to your I-V graph and provide (i.e., write out in words) a geometric interpretation of the fill factor. Solar cell efficiency is defined to be the ratio of the maximum electric power produced to the incident light power on the cell. The cell efficiency is given by P MAX , JA where J is the light intensity, A the solar cell area, and PMAX is the maximum power produced. Which area should you use in this expression, the total cell area or the “active area?” (In fact, experts disagree on this. This becomes an issue when different laboratories claim to have fabricated a cell with world-record efficiency. An inventor who has not spent time optimizing the grid pattern will claim that it is correct to use effective area. Another inventor who has invested time and technology to minimize grid area will push for using total cell area. Obviously each is “marketing” their own strengths. Here you should calculate the solar cell efficiency both ways. To do this calculation you must first determine the light intensity for your experiment. Note that ISC is directly proportional to J. The device manufacturer reports that 0 I SC 262 10 mA when illuminated with AM1.5 radiation. If your measured ISC is different, we will ASSUME that your light intensity differs from J0. You can then calculate your ASSUMED light intensity from I J J 0 SC , 0 I SC along with its uncertainty. In other words, we are using the known specs of the solar cell to calibrate your light source. (This is fine so long as the solar cells have exactly the properties given by the manufacturer. In fact, solar cell properties will vary from cell to cell – and if you really have a calibrated light source, then you can determine variations in efficiencies from solar cell to solar cell.) 55 Calculate your light intensity and your corresponding solar cell efficiency. Be sure to include uncertainties in your values. VI. Additional Work Real solar cells, just like real batteries, differ from ideal diodes. One simple correction is that they act like a an ideal diode in parallel with an ideal resistor, RSH, called the “shunt resistance.” When an ideal diode is reverse-biased it would have negligible current. The shunt resistance then offers a parallel path for current to flow that should be noticeable in reverse bias in the dark. Sketch a circuit to represent this “real diode.” Think how the presence of a finite shunt resistor will affect your dark I-V curve. Examine your dark I-V data to estimate the shunt resistance. If you have time, you can actually take your solar cell to the basement lab and measure its short-circuit current under calibrated AM1.5 illumination. This then gives you a precise way to determine what your intensity was with the slide projector, and will allow you to calculate the actual efficiency of your solar cell, rather than assuming it has the manufacture’s specs. See you instructor if you want to try this. Credits Written by John H. Scofield, Oberlin College, 2009. 56 Solar Cells and Photovoltaic Power Generation (by John Scofield) One method for converting the sun’s energy into work would be to use sunlight to heat a reservoir, then use an engine to extract work from this hot reservoir. For instance, sunlight could be used to produce steam from water, and the steam could be used to drive an electric generator. This is a well-understood process, but the conversion efficiency will be limited by the Second Law of Thermodynamics. Moreover, this requires numerous moving parts. An alternate approach is to use the photovoltaic effect to produce electric energy directly from sunlight. Before discussing this we first consider a simpler but related process called the photoelectric effect. 1. The Photoelectric Effect In the late 19th century it was discovered that, when light waves strike a metal, they are capable of ejecting electrons from the metal’s surface. This photoelectric effect was explained by Albert Einstein, work for which he was awarded the 1905 Nobel prize in physics.4 Einstein postulated that electromagnetic waves are composed of ―particles‖ called photons. A photon moves at the speed of light (after all, it is light) and, though it has no mass, has kinetic energy that is determined by its frequency f, namely hf , 34 where h 6.63x10 J s is Planck’s constant, introduced earlier in our discussion of blackbody radiation. Since wavelength and frequency are intimately related ( f c ), the photon energy is simply related to the wavelength of the radiation. Example 1: What is the energy of photons that make up visible light having a wavelength of 600 nm? Solution: E hf 6.63x10 Js3x10 m / s 3.32 x1019 J . 600x109 m 34 E hc 8 The Joule is not a very good unit for discussing energies that are this small. Instead, we use the unit of electron-volt or eV. Recall that voltage is a measure of electric potential energy per unit charge. Thus, if we multiply 1 volt by the fundamental charge e we have an amount of energy, called an eV. The conversion between Joules and eV is 1 eV = 1.602 x 10-19 J . Thus, the answer to the example problem above is usually expressed as 2.07 eV. 4 Surprisingly, Einstein did not earn a Nobel for his more famous theory of relativity. 57 In atoms, negatively charged electrons are bound to positively charged nuclei by the electric force. If a photon strikes an electron with sufficient energy it will knock it loose from the nucleus and eject the electron. If the photon energy is too low the electron cannot be ejected. However, for a certain frequency, called the threshold frequency, the photon will have just enough energy to eject the electron. For higher frequency photons, the extra energy goes into kinetic energy of the ejected electron. If we could collect the ejected electrons in the form of an electric current, this effect could be used to convert electromagnetic wave energy into electrical energy. The amount of energy required to eject the most weakly bound electrons from a metal is called the work function of the metal. The work functions for typical metals are of the order of 10 eV, so wavelengths in the ultraviolet (or shorter) are required to eject electrons from a metal. Since most of the sun’s radiation occurs for longer wavelengths (lower frequencies, thus lower energies), sunlight is not very effective at ejecting electrons from metals. 2. Semiconductors and Metals 2.1 Metals and Energy Bands We know that metals conduct electricity and insulators do not – but on a microscopic level, what causes this incredible difference? Solids are generally composed of atoms that are arranged in regular arrays called crystals. When insulating atoms form a crystal all the electrons of a particular atom remain strongly bound to the nucleus. In contrast, metal atoms, when they form a crystal, give up one (some metal atoms give up more) of their outer electrons and make them available to freely move throughout the entire solid. These conduction electrons, as they are called, are still confined to the metal, but they are free to move easily from atom to atom. If the metal is placed in an electric field the conduction electrons respond to the field to form an electric current. It is these same conduction electrons which get ejected from the metal in the photoelectric effect. Other electrons remain tightly bound to the nuclei of the metal atoms. The conduction electrons are said to be in the conduction band of the metal.5 The number of electrons in the conduction band, n, is a very important quantity. The larger n the better the conductor (i.e., the lower its resistivity). For a metal n is approximately equal to the number of atoms in the crystal, N, since each atom contributes roughly 1 electron to the conduction band. 2.2 Intrinsic Semiconductors A semiconductor is a material that, like its name implies, is not quite a conductor and not quite an insulator. The best known semiconductor is the chemical element silicon (Si). Another elemental semiconductor is germanium (Ge). Other important semiconductors 5 The electronic bands of a solid are analogous to the electronic levels of an isolated atom. Each band can accommodate 2N electrons, where N is the number of atoms in the crystal. In sodium (Na), for example, which which has 11 electrons, the first 10 electrons from each atom fill up the lowest 5 bands. The next band, the th conduction band, is only half-filled by the 11 electron from each atom. 58 include the compounds gallium-arsenide (GaAs), cadmium-telluride (CdTe), and cadmium sulfide (CdS). You will notice that the two elemental semiconductors are both Group-IV elements in the periodic table (see figure below). Moreover, many compound semiconductors are formed by joining one element from column III with another from column V, or, one element from column II with another from column VI. These compound semiconductors are referred to as III-V (pronounced ―three-five‖) or II-VI (pronounced ―two-six‖) semiconductors. I will return to this below when I talk about doped semiconductors. In chemical terms, Group IV elements have four valence electrons per atom. A molecule consisting of one Group III and one Group V (or II and VI) element has, on average, four valence electrons per atom. This is the magic number for a semiconductor. There are numerous ways to achieve this average. 8 1A 3A 4A 5A 6A 7A He B C N O F Ne 2A 1 B 2 B Al Si Cu Zn G a G e A g Cd In S n Au Hg Tl P b P S Cl A r A s S e Br Kr Sb Te I Xe Bi P o A t R n Figure 1. Periodic table showing group IB through VIII elements. III-V semiconductors are formed by binary compounds from the blue columns and II-VI semiconductors are similarly formed from the green columns. Pure semiconductors are insulators at low temperatures because all of the electrons are strongly bound to the nuclei. The most weakly bound electrons are not free to move about the crystal as they are in a metal. These valence electrons, as they are called, nonetheless play an important role because they, if given enough energy, can move into the next higher energy level where they can contribute to conduction. Valence electrons are said to be in the valence band. To be able to move freely throughout the solid, electrons must be elevated to the conduction band. This takes a lot less energy than it does to totally eject the electron via the photoelectric effect. The energy difference between the valence and conduction levels (or bands) is called the band-gap energy, Eg. Bandgap energies for several semiconductors are given in the table below. Material Ge Si CuInSe2 GaAs CdTe CdS Bandgap (eV) 0.7 1.12 1.1 1.4 1.5 2.4 59 transition indirect indirect direct direct direct ? 2.3 Intrinsic Conduction At room temperature there are always thermal fluctuations. Recall from introductory chemistry that it is always possible to thermally excite a system from a lower to a higher energy. The probability that this occurs depends on the temperature and energy difference in the two levels. This process is called thermal activation. As the temperature of a pure semiconductor is raised more and more electrons are thermally excited into the conduction band. The number of electrons in the conduction band is n. When an electron is excited from the valence band to the conduction band, the empty electronic state left behind is called a hole. A hole acts like a positive charge that is free to move in the valence band. The number of holes in the valence band is p. Since each electron in the conduction band leaves behind a companion hole in the valence band, n = p. This balance is destroyed (deliberately) by a process called doping, discussed below. 2.4 Photoconductivity A photon having energy exceeding the bandgap of a semiconductor may excite an electron from the valence to the conduction band. This gives semiconductors the property of photoconductivity – the semiconductor is an insulator in the dark but conducts when exposed to light. Even in bright light there are far fewer conduction electrons in a semiconductor than in a metal. Nevertheless, this property may be exploited to convert the energy of sunlight into electrical energy. The photoconduction process is illustrated in the following figure. Normally all of the allowed states in the valence band are filled with electrons and all of the allowed states in the conduction band are empty. A photon with sufficient energy excites a valence electron into the conduction band and leaves behind a hole (empty electron state) in the valence band. It turns out that both conduction band electrons and valence band holes contribute to the conduction process. As for intrinsic conduction, n = p for this process as well. conduction band light Eg valence band Figure 2. Band diagram for an undoped semiconductor. The vertical direction corresponds to electron energy. The bandgap energy Eg is the minimum energy required to excite an electron from the valence band to the conduction band. A metal is a material which, in the dark, has a completely full valence band and a half-filled conduction band. Thus, metals conduct very well without the assistance of photons. 2.5 Doped (or Extrinsic) semiconductors When a small amount of a phosphorus (P), or another Group-V element with 5 valence electrons, is introduced into Si, four of the valence electrons participate in covalent bonds with neighboring Si atoms, and the extra electron ends up in the conduction band. The 60 number of electrons n in the conduction band is equal to the number Nd of donor impurities introduced. No holes are introduced in the valence band. This process is called doping, and with phosphorus dopant, this results in an n-type semiconductor, that is, a semiconductor which, even in the dark, has an excess number electrons in the conduction band. If, instead, boron (B), or another Group-III element having only 3 valence electrons, is introduced as impurities into the Si,, an electron leaves the Si valence band and is ―borrowed‖ by the B impurity to complete its 4th covalent bond with neighboring Si atoms. This leaves behind a hole in the Si valence band. We call the B impurity an acceptor. The number of holes p in the Si valence band is equal to the number Na of acceptors. Silicon doped with B is said to be a p-type semiconductor. For doped semiconductors, n and p are not in balance as they are for an intrinsic semiconductor. Semiconductor doping is very important for making semiconductor devices such as diodes, transistors, and solar cells. 3. Solar Cell 3.1 Semiconducting Diode A solar cell is a kind of semiconductor device called a diode. A diode is formed by joining an n-type and p-type semiconductor to form a p-n junction (see the figure below). light qVbi Ec p-Si conduction band N P Ef I n+-Si Ev valence band V Figure 3. A diode is formed by joining n-type and p-type semiconductors to form an electrical junction. The sketch on the right shows the potential energy for an electron in the diode. The transition region occurs in the neighborhood of the junction and is called the depletion, or space charge region. A diode acts like a “one-way valve” for current; it allows current to flow from negative to positive, but not in reverse direction. (In constrast, a resistor allows current to flow in either direction with magnitude proportional to the voltage drop. A diode is very definitely a nonlinear electrical device.6 6 So much emphasis is placed on Ohm’s law that it is easy to begin to think that all electrical devices follow Ohm’s law. Not so! Even light bulbs do not follow Ohm’s Law, though we can pretend they do for simple calculations. 61 3.2 Current-Voltage Curves One common method for studying an electrical device is to measure its current-voltage (I-V) characteristics, that is, perform a series of measurements to determine the current through the device for a wide range of applied voltages. The figures below illustrates the I-V characteristics of a resistor (straight line) and a diode (curve). Figure 4. The red curve is the I-V curve for a Si diode. Note that substantial current begins to flow when the forward voltage exceeds 0.7 V. The green line is represents the I-V characteristics of a 10 resistor. When a photon whose energy exceeds the semiconductor bandgap is absorbed, it generates an electron-hole (e-h) pair, as described above in the section on photoconductivity. If the electron and hole are generated near the depletion region, the electron will be swept into the p-side and the hole into the n-side, thereby producing a current. This is the heart of the photovoltaic effect. In designing solar cells, one of the goals is that all absorbed photons will produce e-h pairs that are collected as current. The I-V characteristics for a solar cell are shown on the left below, both in the dark (red) and under direct sunlight (red). Also shown on the right is the power dissipated, P = IV, in the same solar cell. Photons with energies less than the semiconductor bandgap not be absorbed. Photons with energies greater than Eg, however, will be absorbed and will produce photocarriers which are responsible for the light-generated current, IL. It is this light-generated current which shifts the dark I-V curve(red) downward to create the light I-V curve(green) of the solar cell. 62 Figure 5. (a) I-V curves for a solar cell in the dark (red) and under illumination (green). (b) curves showing the power dissipated by the solar cell under similar conditions. The illuminated cell dissiptes negative power in the 4th quadrant – because it is actually producing, not dissipating power. 3.3 Solar Cell Efficiency and More Curves like those shown in Figure 4 are used to determine the efficiency of the solar cell. The I-V curve of an ideal solar cell is shifted downward by exposure to light. This shift is called the light-generated current and is proportional to the light intensity. The maximum power generated by this solar cell occurs when V is about 700 mV. The power is just IV which, at this point on the curve is roughly 50 mW. This may not seem like much, but this is a very small solar cell, having an area of approximately 4 cm2. The intensity of the light used for these measurements was AM1.5 sunlight at 1000 W/m2, or 100 mW/cm2. Thus, the total light energy incident on the solar cell was 400 mW. The efficiency of the solar cell is defined to be maximum electric power , incident light power which, for this solar cell is 12.5%. The light and dark I-V curves for a solar cell are shown again in the figure below. Certain features of the light I-V characteristics are illustrated. The voltage measured across the cell when zero current flows is called the open-circuit voltage, Voc. The current that flows through the cell when there is zero voltage across the cell is called the short-circuit current, Isc. Voc and Isc define a rectangle with area IscVoc. (This area, of course, has units of power.) The maximum power generated by the cell corresponds to a current Ip and a voltage Vp; these values define an ―inscribed‖ area Pmax = IpVp. The fill-factor, FF, is defined as the ratio of the areas of these two rectangles, I pVp P . FF max I scVoc I scVoc 63 The fill factor is always less than unity; the closer it is to unity the better the quality of the solar cell. Voc Vp Ip peak power I sc Figure 6 I-V curves for a solar cell in the dark (red) and under illumination (green). Features of interest include the peak power, short-circuit current Isc, and open-circuit voltage Voc. A solar cell is much like a real battery in that, under illuminated conditions, it consists of an ideal EMF in series with an internal resistance. If we don’t draw significant current from the cell its voltage will be close to the EMF which will be the open-circuit voltage. If, on the other hand we attempt to draw infinite current from the cell we will be unable to do so. Instead, we draw a maximum current equal to the short-circuit current. Of course, maximum power transfer will not be achieved under either of these conditions. Instead, maximum power transfer occurs when we operate the cell at the peak-power point on its I-V curve, which will occur when we connect it to a load resistor that matches its internal resistance. 4. Photovoltaic Energy Conversion 4.1 Role of Bandgap Energy The goal of a solar cell is to have each and every photon which strikes the cell to produce an e-h pair which contributes to the light-generated current. Moreover, it is desirable for the e-h pair to receive all of the energy possible from the photon. Unfortunately, when an e-h pair is generated, any excess energy it receives from the photon over and above the bandgap energy is eventually lost in the form of heat. The solar spectrum contains photons of various energies. No energy is collected from photons whose energies are below the bandgap. At best, only the bandgap energy is obtained from those photons with higher energy. The solar cell is a semiconducting diode. Usually one part of the diode (often the pdoped side) is designed to absorb most of the photons and is called the absorber. If the absorber bandgap is very low, we will absorb more of the sun’s photons, but we will only get a small amount of energy from each photon. Thus, much of the energy of the higher energy photons (visible and UV) is wasted. If, on the other hand we choose a higher bandgap 64 absorber, we will get more energy from each absorbed photons, but will absorb fewer photons since the lower energy ones (IR) will not be absorbed). The figure below shows the maximum possible absorber efficiency as a function of bandgap. This graph illustrates the two competing mechanisms above. Not surprisingly, the best trade-off between these two competing factors is obtained for a bandgap energy which is about where the sun’s spectrum peaks, Eg = 1.5 eV. Theoretically, a solar cell having a GaAs or CdTe absorber layer will have the highest efficiency. Figure 7. Graphs showing the theoretical maximum efficiency, max, of a solar cell versus absorber bandgap energy Eg. The two curves represent two different recombination mechanisms. In both cases, maximum efficiency is obtained with Eg close to 1.5 eV.7 4.2 Direct versus Indirect Transition Another important characteristic of a semiconductor is whether the valence-toconduction-band transition is direct or indirect. The concept is too difficult to discuss in detail here. Suffice it to say that indirect bandgap semiconductors do not absorb light so strongly as direct bandgap semiconductors. If light is incident upon a thick slab of a semiconductor it will not all be absorbed at the surface. Instead, more and more light is absorbed by the semiconductor the farther it travels into the interior. For indirect material, radiation may travel as deep as 20 m into the sample before it is fully absorbed. For direct bandgap semiconductors a typical distance might be only 1 m or less. GaAs and CdTe are both direct bandgap semiconductors. Si and Ge are both indirect. Today’s solar cells are still too expensive to compete in the marketplace. To understand the future viability of a technology it is important to understand what is possible, 7 From A. L. Farhrenbruch and R. H. Bube, Fundamentals of Solar Cells: Photovoltaic Solar Energy Conversion (Academic Press, NY, 1983), p. 14. 65 assuming current problems have been solved. Thermodynamics, for instance, tells us that no matter how hard we work, we will never be able to make an engine with an efficiency that exceeds a certain level. Solar cell efficiencies are not limited by thermodynamics. Instead, the low efficiencies we see today are due to technological problems which will someday be overcome. Silicon is a very well understood material. Crystalline silicon solar cells have been available for many years and are unlikely to see substantial cost reduction. Being an indirect bandgap material, light travels a long distance through Si before being fully absorbed. Moreover, the e-h pairs that are generated by photons absorbed deep in the Si will be collected only if they wander all the way back to the depletion region. This is possible only if the material is very high in quality, free of defects. While Si is an abundant material, high quality single crystal Si is expensive to produce. In contrast, GaAs and CdTe are direct bandgap semiconductors which are much more strong in absorbing light. Thus, an absorber layer from these materials can be, in principle, much thinner than one made from Si. This requires substantially less material and, because the e-h pairs have a smaller distance to diffuse to the depletion region, the material may be of less quality. Solar cells from thin films of these absorber materials deposited on window glass offer much promise for low-cost power generation. 66 Physics 111 Laboratory MEASUREMENT OF MAGNETIC FIELDS I. Introduction This experiment is intended to show some properties of magnetic fields, and how they are measured. To carry out the measurements, you will use a cathode-ray oscilloscope (CRO). The experiment is divided into the sections listed below: Preliminary tasks - before coming to laboratory: a. Consider a plane circular coil of 100 turns of wire with an average radius of 14 cm. Assume that the current in each turn is 1 Amp. If z is the distance along the axis of the coil measured from the center, calculate the numerical value of the magnetic field on the axis of this coil at z = 0, 7, and 14 cm. (See text, Eq. 29-26.) b. Review Ampere’s law. II. Alternating Current Circuits A. What you need to know about AC voltage and current Alternating currents (AC) and voltages are mathematically more complicated than direct currents and voltages. The AC voltage and current are rapidly-changing regular harmonic functions of time which are characterized by three constants -the amplitude, the frequency, and the phase. Expressed as mathematical functions, they are V Vo cos[2t ] Vo cos[t ] I I o cos[2t ] I o cos[t ] where Vo and I o are the amplitudes of the voltage and current, (Greek letter “nu”) is the frequency and 2 is the angular frequency, and and are the phases of the signals relative to some other AC signal (say, with phase angle equal to 0). In this experiment, the phase will not be important. (Many errors are made in calculations by confusing and . The frequency is measured in cycles per second or Hertz. The frequency of the voltage that is available at a wall plug is 60 Hertz. The angular frequency has units of radians per second. It is that appears in most equations for AC impedance and resonance.) A voltage signal is sketched in Figure 1. 67 Parameters other than the simple amplitude are frequently used to indicate the “magnitude” of an AC current or voltage. For example, the easiest AC voltage measurement to make on a CRO is the “peak-to-peak” or “pp” voltage, which is the difference between the maximum and minimum voltages. In Figure 1, the pp voltage amplitude is Vpp = 2 Vo . A second parameter is the root mean square amplitude or VRMS, which is in some sense an “average” voltage. The ordinary long-term time-averaged voltage, V , is zero, of course, since the voltage is negative as often as positive. To get some sort of average that is not zero, it is customary to use instead V RMS V 2 , where the quantity inside the square root is the time-average of V2. To relate VRMS to the parameter Vo, we calculate 2 V RMS Thus, VRMS = Vo / T l T 1 2 2 2 l 2 (V o cos2 t) dt Vo cos 2t dt Vo . T o T o 2 2 . Some basic relations for power dissipated in AC circuits may be written in the same form as the corresponding formulas from dc circuit analysis if they are written in terms of the RMS voltages and currents. The AC voltage available at the wall plugs has a nominal RMS voltage of 120 volts. 68 B. Check of transformer output voltage and frequency (Quick!) Connect the output terminals of a 6-V transformer to the CRO and plug it in. Adjust the CRO for the standard configuration (see Using an Oscilloscope Laboratory instructions). Measure and record in your notebook the peak-to-peak voltage output of the 6-V transformer. Calculate the amplitude Vo and VRMS . Is one of these somewhat near 6 V? Measure and record the period and the frequency of the transformer output voltage. III. Magnetic Field of a Current Loop - Ampere’s Law A. Measurement of the magnetic field at the center of a coil At several stations in the laboratory there is a large single coil wound with 100 turns of wire on an average radius of 14 cm8, plus a mounted resistor of about 1 ohm. (Check the value with your DMM.) With these components assemble the circuit in Figure 2. Also insert a DMM in series with the 1-ohm resistor. A DMM 120 V AC 1 100-turn coil 6.3 V AC Transformer Figure 2 A common way of measuring the current in a circuit with a CRO is to put a known resistor in series with the circuit and then measure the voltage developed across the resistor. In the above circuit, the instantaneous current in the coil may be expressed as a function of time as I Io cos[2 t] where is the “line” frequency you have just measured. Plug in the transformer and use your CRO to measure the voltage across the oneohm resistor. Use this value to determine the current amplitude Io. Use the DMM 8 Confirm this radius by measurement and record its uncertainty. 69 in the circuit to read the RMS current and make sure its values are consistent with what you determine with the CRO. If the current in this coil were constant (i.e., DC), then the magnetic field at a given point near the coil would be proportional to the current. If, for example, the current were 2 amperes, the field at selected points along the axis of the coil would be twice what you calculated before coming to laboratory. If the current, I, in the coil is a sufficiently-slowly varying function of time (60 Hz qualifies), the timevarying magnetic field produced is still proportional to the current at each instant of time. Thus, for our coil with an AC current, the field at different points may be expressed B(r ,t) Bo (r ) cos[2t] , where Bo (r ) is the amplitude of the field at point r that would be produced by a steady current of amplitude Io. If now a second small coil of wire is placed at some point in this varying magnetic field, an EMF given by Faraday’s law will be induced in it. That is, the EMF produced in the “pick-up” coil will be given by N d dt where is the magnetic flux linked by one turn of the coil, and N is the number of turns. The flux is B dA Bn A = Bon A cos[2 t] , where A is the area enclosed by one turn of the pick-up coil, B n is the spaceaverage magnetic field component normal to the plane of the pick-up coil, and Bon is the amplitude of B n . We combine the last three equations - that is, we differentiate with respect to time and multiply by N to get . Thus, for our source coil carrying an AC current, the induced EMF in the pick-up coil will be (t) N Bon A 2 sin 2 t This is the equation of an alternating voltage. What is its frequency and its amplitude? Attached to an arm on one side of the large 100-turn coil is a smaller “pick-up” coil, which has 3000 turns and an average radius of 2.0 cm9. Rotate the pick-up coil so it is in the plane of the large coil near the center of the large coil and use the CRO to observe the voltage induced in it when an AC current of known amplitude is in the large coil. From your knowledge of the current in the larger coil and the 9 You should measure this radius. Keep in mind that that the inner wire loops have smaller radius than do the outer loops. You need to determine the “average” radius which is not the same as the outer radius of the plastic form on which the coil is wound. In the end your measurements should confirm the 2.0 cm figure. 70 observed voltage in the pick-up coil, calculate the amplitude of the magnetic field at the pick-up coil. Compare your measured field value (pick-up coil) with that you calculated for the 100-turn coil before coming to laboratory. How well do they agree? The pick-up coil is not quite at the center of the large coil. Do you expect the field at the pick-up coil to be larger or smaller than the field at the center? B. Ampere’s Law Review with your partner the meaning of Ampere’s law, which may be expressed B ds oI . If the magnetic field and current are slowly-varying functions of time, the lineintegral should be considered to be completely carried out at one instant of time, yielding a value related to the current surrounded by the path of integration at that same instant. For an AC current, a convenient instant is that when I and B have their maximum value. The pick-up coil may be moved on its arm in a closed circular path of radius 18.1 cm10 around the side of the large coil; reference marks are placed on the bearing every 20o around the circle. By observing with your oscilloscope the induced EMF in the pick-up coil, measure every 20o the amplitude of the component of the magnetic field of the large coil that is normal to the plane of the pick-up coil. Make a graph of this field amplitude vs. position around the circle. A careful estimate of the area under this graph for a complete circle should be a good approximation to the line integral B ds for this path. How does its value compare with the prediction of Ampere’s law? What effect does the earth’s magnetic field have on this result? 10 This radius depends on where the coil is mounted on the arm – you should measure this radius and use your measured number – which might be different from the one written here. 71 Physics 111 Laboratory Alternating Current (AC) Circuits I I. Introduction In this experiment you will study the relationships between potential differences and currents in various parts of circuits that include an inductor and a capacitor in addition to resistors. The input voltage will be either a square-wave or a sinusoidal signal, with an adjustable frequency. The voltages will be measured with a cathode-ray oscilloscope. The four parts of the experiment are: Series RC circuit, capacitive impedance, high- and low-pass filters, phase relations between voltage and current. Series RL circuit, inductive impedance, phase relations between voltage and current. Series RLC resonant circuit, frequency dependence of current, phase relations. Lissajous figures and precise frequency comparison. Note: The rate of repetition of an AC signal may be denoted by either of two frequencies. Angular frequency has units of radians/second, (ordinary) frequency or f has units of cycles/second or Hertz. These two frequencies are related by 2 Line voltage (from the sockets in the wall) has a frequency of = 60 Hz. Note: The signal generator has an output impedance of about 600 . That is, it behaves like a pure source of emf with a series resistance of 600 This can seriously affect your results when you have the generator connected directly to a circuit (i.e., not through a transformer), although we have tried to avoid those difficulties by the design of the experiments below. You may also notice an effect near resonance in the series RLC circuit. II. Series RC Circuit – Square-Wave Applied Voltage A. Let us begin by investigating the behavior of “voltage divider” circuits in which one of the elements is a capacitor instead of a resistor. Set your audio generator to produce a square wave output with a frequency of 200 Hz and an amplitude of 2.5 volts (peak-to-peak amplitude of 5.0 V). Observe this waveform with your oscilloscope and make a sketch of it in your notebook. The oscilloscope should be in standard configuration (see Introduction to the Tektronix 2213A Oscilloscope). Wire the circuit shown in Figure 1, a so-called “high-pass filter.” Let R 10,000 (a small box) and C 0.047 F . You can see that the filter is a relative of a voltage divider. 72 Remember that the oscilloscope ground and the signal generator ground are connected through the 110-volt sockets you plug them into. This means that you must connect point B (not point A) in the circuit to the oscilloscope ground. Set the oscilloscope input switch to DC. With your oscilloscope, look at the generator output and the output voltage between points A and B, and sketch what you observe in your notebook. (Pushing in the BW - bandwidth - button on the oscilloscope may reduce high-frequency noise in the signal. Everything in this part of the experiment involves only audio frequencies.) You may understand what you see by thinking of the process as repetitive chargings of the capacitor, first in one direction and then the opposite. The voltage you are observing across R is proportional to the current in the circuit. Compute RC, the time constant for the charging process, and compare it with onehalf the period of the square wave. (Another way to think about what is happening is discussed below, and leads to the name “high-pass filter.”) A C R B Figure 1 B. Replace the high-pass filter by a “low-pass filter,” shown in Figure 2. Observe and record the output. A R C B Figure 2 73 How long should it take the voltage across the capacitor to make 63% or (1 e1) of the change from its initial value to its final value? To see why this is called a low-pass filter, proceed to the next section for a more detailed analysis of these circuits. III. Series RC Circuit - Sinusoidal (AC) Applied Voltage A. Assemble the circuit in figure 3. The circle with part of a sine wave in it represents an audio generator, which produces sinusoidal electrical voltages with frequencies in the audible range. A C 500 ohms B 6-8 ohms R D Figure 3 In this circuit the resistor is now 100 , C is 0.5 F (microfarads), and a transformer has been added. The generator and transformer together form one logical element, namely, the source of the oscillating emf applied across RC. The audio transformer has the effect of reducing the effective output resistance of the voltage source (signal generator and transformer) from about 600 to 6-8 . The side of the transformer labeled 500 should be connected to the signal generator. The transformer secondary (6-8 ohms) is not grounded. Therefore any of points A, B, or D may be put at ground potential when making measurements with the oscilloscope. The audio generator and transformer will be used to apply a sinusoidal voltage with amplitude m to the RC circuit. When a sinusoidal voltage is applied to the RC circuit by the transformer, the voltages across the resistor and capacitor will vary sinusoidally with time. We wish to study the amplitudes of the capacitor and resistor voltages as a function of the frequency of the applied voltage, for a fixed amplitude of applied voltage. 74 Begin your study at a frequency of 200 Hz. Measure the applied voltage (i.e., the voltage between points A and D) and adjust the generator output level so that the amplitude of the applied voltage is 0.3 V. (Check and maintain this level for each frequency you use. The easiest way to measure the amplitude of a sinusoidal voltage with the oscilloscope may be to measure the peak-to-peak amplitude and divide by 2.) The signal for V R , the voltage across R, is small and noisy for low frequencies. Use your oscilloscope to measure the amplitude of the voltage across the capacitor, V C , and across the resistor, V R , at this frequency. (Connecting point B to the ground of the oscilloscope will facilitate this measurement. Using the ADD function on the oscilloscope will also help; see the instructor.) Repeat this process for the frequencies 400 Hz, 800 Hz, 1.2 kHz, 2 kHz, 4 kHz, 8 kHz, 12 kHz and 20 kHz. Using a computer, plot both VC and V R as a function of frequency on a single semi-log graph, with frequency on the logarithmic scale. Notice that there is a frequency 0 near 3,000 Hz where VC = VR. Find and measure 0 with your oscilloscope; don’t rely on the generator dial. Apparently at this frequency the impedances (“impedance” is a generalization of resistance) of these two elements are equal. The impedance of a capacitor is given 1 by . Using the nominal value of your capacitor and the frequency 0 you have C determined, calculate the impedance of the capacitor at 0 and compare the result with 100 , the resistance of R. At frequencies lower than 0 the capacitor has a greater impedance than the resistor, and correspondingly a larger fraction of the voltage applied to the circuit appears across the capacitor. Thus, when observing the voltage across the capacitor, one has a low-frequency-pass filter, in the sense that low-frequency signals pass along the circuit without much attenuation, while less of a signal with frequency higher than 0 survives. If the signal applied to the filter circuit is not sinusoidal but is some other periodic waveform (like a square wave), then by the Fourier theorem it can be regarded as the sum of sinusoidal signals with various frequencies. In going through a low-pass filter, the low-frequency components of the sum would emerge more or less intact, while the high-frequency components would be removed. The pattern you saw when you put a square wave through the low-pass filter may be regarded as a square wave with its high-frequency components removed, which means that the leading and trailing edges of the square pulses were rounded off. Conversely, at frequencies higher than 0 , the capacitor has a smaller impedance than the resistor, and correspondingly a larger fraction of the voltage applied to the circuit appears across the resistor. Thus, when observing the voltage across the resistor, one has a high-frequency-pass filter. The pattern you saw when you put a square wave through the high-pass filter may be regarded as a square wave with its 75 low-frequency components removed, and there is appreciable signal only at times the square wave was changing very rapidly. Under appropriate conditions, high- and low-pass filters may be used to integrate and differentiate electrical signals! B. Look more closely at your data. Note that at every frequency VR VC Vtransf ! Is this a violation of Kirchhoff’s voltage loop theorem? Set your generator at the observed frequency 0 , and measure again VR,VC ,Vtransf . This conundrum is resolved by remembering that the voltages depend upon time. If we consider the actual time-dependent voltages (not peak values) across R, C, and the source, then at any instant of time, Vtransf (t) VR(t) VC (t). But these signals do not attain their peak values at the same time. Therefore it is not surprising that for the voltage amplitudes alone we have VR VC Vtransf . You can easily observe this using the dual trace feature of your oscilloscope. Set the VERTICAL MODE toggle to BOTH, and the ADD/ALT/CHOP toggle to either ALT or CHOP, whichever is easier on your eyes. Display the capacitor voltage on channel 1 and the resistor voltage on channel 2. Because both oscilloscope inputs have the same ground, point B on the diagram must be connected to the grounded (black) terminal. This leads to the mildly distressing fact that although channel 1 is displaying VAB (t) VC (t) , as desired, channel 2 is displaying V DB (t) V R (t) , which is the negative of the voltage across R. You can (and should) eliminate this distress by depressing the CH2 INVERT button. To observe the total voltage drop V AD (t) , you may set the ADD/ALT/CHOP toggle to ADD. Sketch the three curves V C (t) , V R ( t) , and V AD (t) in your notebook with correct relative phase. C. We may obtain a theoretical expression for the current in this RC circuit by setting L = 0 in the equations 31-63 ff of HRW. So doing, we see that if the emf imposed on this circuit is (t) m sin ( t) , then the steady-state current in the circuit should be i (t) i m sin( t ) where im m R (1/ C)2 2 and tan 76 1/ C R . At the frequency 0 do the current amplitude and phase you are observing agree with these predictions? Compare observed and predicted current amplitude and phase at one other frequency. 77 Physics 111 Laboratory Alternating Current (AC) Circuits II IV. Series RL Circuit - Sinusoidal (AC) Applied Voltage A. This section is a repetition of parts of the last, except that the capacitor is replaced by an inductor. Assemble the circuit in Figure 4. Change R to 1,000 ohms. Begin your study at a frequency of 200 Hz. Measure the applied voltage (i.e., the voltage between points A and D) and adjust the generator output level so that the amplitude of the applied voltage is 0.3 V. (Check and maintain this level for each frequency you use.) Use your oscilloscope to measure the amplitude of the voltages across the inductor, V L , and across the resistor, V R , at this frequency. (Connecting point B to the ground of the oscilloscope will facilitate this measurement.) Repeat this process for the frequencies 400 Hz, 800 Hz, 1 kHz, 2 kHz, 4 kHz, 8 kHz, 10 kHz and 20 kHz. Using a computer, plot both V L and V R as a function of frequency on a single semi-log graph, with frequency on the logarithmic scale. A L 500 ohms 6-8 ohms rL B R D Figure 4 78 B. Determine the frequency for which the amplitude of the potential differences across the inductor and resistor are equal. At this frequency, the inductor has an impedance of 1,000 . From the relation L = 1,000 , calculate the inductance of the coil. What have we ignored in the calculation? C. At the frequency for which L = 1,000 , measure the amplitude of the voltage output from the transformer. How does it compare with the sum of the voltage amplitudes across the inductor and resistor? Why don’t they agree? D. Observe the relative phases of the voltages across the resistor and inductor using the dual trace technique described in section IIIB. Sketch VAB (t) , VBD (t) and VAD (t) in your notebook. E. Once again obtaining results from Section 31-9 in HRW, only this time setting C (XC 0) , we find that if the emf imposed on this circuit is (t) m sin ( t) , then the steady-state current in the circuit will be i (t) i m sin( t ) where im m R 2 (L)2 and tan L R . At the frequency for which L = 1,000 , do the current amplitude and phase you are observing agree with these predictions? Check them at one other frequency. 79 V.Series RLC Resonant Circuit A. Assemble the circuit in Figure 5. 1.0 fd B A L 500 ohms 6-8 ohms rL 1 ohm E D Figure 5 The dashed box in Figure 5 represents the coil of wire used earlier: It has inductance L and resistance rL. We will study the current in this circuit as a function of generator frequency . First, quickly sweep through the frequency range while observing VDE (t) on the oscilloscope to determine approximately the frequency at which the current amplitude is a maximum. This quick sweep indicates what frequency ranges are interesting, but it is only approximate because the generator output VAE (t) varies slightly in amplitude as the frequency changes. To correct for this variation you must reset the generator amplitude each time you change the frequency. This is most easily done by displaying VAE (t) on channel 1 of your scope and VDE (t) on channel 2. (Note that point E will then be grounded.) Set the amplitude of the voltage of VAE (t) equal to 0.3 V for every frequency, then record the amplitude of the voltage across the resistor. Carry out this procedure for a range of frequencies, spending little time on the uninteresting frequencies far from resonance. Measure V R with the oscilloscope. Plot the amplitude of the voltage V R across the resistor as a function of frequency, and find the one frequency that leads to maximum current (“resonance”). The 80 resonance (angular) frequency is denoted 0, and theoretically 0 = 1/ LC . How does this check out? B. Investigate the phase relations at resonance using your scope in dual trace mode. There are now four voltages of interest: VAE (t), VAB (t), VBD (t) and VDE (t) , but only two can be displayed at any one time. I recommend that you simultaneously observe VAE (t) and VDE (t), then VDE (t) and VBD (t) (remember to invert the channel 2 display!), then finally VAB (t) and VBD (t) . Sketch all four curves in your notebook so the phases are clearly evident. Note that the capacitor and inductor voltages are quite large, but that they almost precisely cancel each other so that the resistor voltage nearly equals the input voltage. (The inductor coil has many turns of wire and therefore has resistance. Measure this resistance with your DMM.) Verify that at several points in a period, Kirchhoff’s loop rule is satisfied instantaneously. Display both VAE (t) and VDE (t) on your oscilloscope and note how the phase relation varies dramatically as the frequency is swept through resonance. C. The phase relation between the applied voltage and the current in the circuit depends on the frequency of the applied voltage. From Section 31-9 again, the result of applying an emf of of (t) = i(t) m sint to the circuit is to set up a steady-state current m 1 2 R ( L ) C sin ( t ) , 2 L where tan R 1 C . Recall that R in these expressions includes the resistance of the coil and the effective resistance of the voltage source. These expressions should agree with your observations so far. In particular, = 0 at the resonance frequency. At frequencies above resonance, the inductor presents a higher impedance than the capacitor, and the phase difference becomes negative as in an inductive circuit. Below resonance, the impedance is predominantly capacitive, with a positive phase difference. DO NOT DISCONNECT THIS CIRCUIT. YOU WILL NEED IT LATER. D. Several properties of a resonant circuit can be expressed in terms of a quantity called “Q,” which stands for quality. The higher the Q of a resonant circuit, the sharper (i.e., higher quality) the resonance peak is. With Q defined as Q = 0L/R (R being the total resistance in the circuit), you will show when you work Problem 31-49 in HRW that the fractional width of the resonance curve at half-height, 0, is given 81 theoretically exactly by 3 / Q. By measuring the width at half-height of your experimental resonance curve, estimate the Q of your resonant circuit and compare it with the Q calculated directly from the definition. If there is a significant discrepancy, what is the most likely source? E. Q turns up in another context. Let us calculate the amplitude of the voltage across the L or C at resonance. If the amplitude of the emf of the source is m, then at resonance the current amplitude will be im = m/R (see HRW, Eq. 31-63). Thus VL im X L m 0 L Q , R m so at resonance the peak-to-peak voltage across the inductor is greater than the source voltage by a factor of Q. Show that the voltage across the capacitor should be VC Qo , Measure these peak-to-peak voltages discussed and calculate a value of Q. How does it compare with that in part (d)? Which method of finding Q do you expect is more accurate? VI. Lissajous Figures For this part, leave your RLC circuit connected insofar as possible. Just disconnect the signal generator and oscilloscope leads. A “Lissajous figure” is a curve described by a point moving with two sinusoidal motions superimposed at right angles. By observing the form of a given figure, the frequencies of the two sinusoidal motions can be compared if they happen to be related by a ratio of relatively small integers. If the two motions have the same frequency, then measurements of the resulting figures easily yield the difference in their phase, if any. A. Frequency comparison: Suppose a particle is undergoing simple harmonic motion in two dimensions. Its horizontal coordinate varies sinusoidally with time at frequency fh, and its vertical coordinate at frequency fv. If fv = fh, the particle traces out a simple closed curve in a vertical plane as shown in Figure 6a. C D E A B a) f v = f h b) f v = 2f h c) 2f v = 3f h Figure 6 82 This occurs because it takes exactly the same time for the particle to complete one cycle of its horizontal motion as it does for one cycle of its vertical motion. If fv = 2 fh, the particle goes “up-and-down” twice while completing one horizontal cycle, as in Figure 6b. If 2 fv = 3 fh, there will be three vertical cycles for two horizontal cycles and a trace like that in Figure 6c is produced. If we record the particle’s motion and count the number of times nh the particle reaches its extreme horizontal limit (e.g., in Figure 6c, twice; once at A and once at B) and the number of times nv it reaches its extreme vertical limit (three times; at C, D, and E) in one closed orbit, then we can compute the ratio fv / fh as nv / nh. If one of the frequencies is known, the other can be calculated. Exactly the same phenomenon can be examined if we displace the spot on the oscilloscope sinusoidally with time in both horizontal and vertical directions. We shall use line frequency (60 Hz) for controlling displacement in one direction and the variable frequency output of the audio generator for the other. Connect the output of a 6.3-volt transformer to the CH 1 or X amplifier connector. The coupling switch above it should be on AC. Connect the variable-frequency output of the audio generator to the CH 2 or Y amplifier connector, and also switch it to AC coupling. The sweep time SEC/DIV knob should be rotated fully counterclockwise to the XY position. Adjust the CH 1 and CH 2 VOLTS/DIV knobs to get a pattern which fills a large part of the screen but does not overflow it. By observing the Lissajous figures, set the generator to frequencies of 20, 30, 60, 90, 120, 150, and 180 Hz. About what typical percentage error is present in the reading of the generator dial in this range? You now know two things -- the typical accuracy of the frequencies to be used in calculations in the rest of the experiment, and how precise frequencies at regular intervals could be set in relation to one standard known frequency if the precision were required. B. Phase Difference Measurements with a Lissajous Ellipse Suppose that the signal imposed in the horizontal direction is described by the equation x = b sin t, while that in the vertical direction is y = a sin (t +) . In other words, the two signals have the same frequency, but they do not pass through zero at the same time -- there is a phase difference It is asserted that the resulting Lissajous figure would be an ellipse appearing as in Figure 7, and that A/B = |sin |. 83 Figure 7 It may be proved that the figure is an ellipse by eliminating the variable t between the horizontal and vertical equations. The relation between x and y which results is the equation of an ellipse with major axis rotated at some angle to the horizontal. We may derive the equation carrying information about the relative phase as follows. First, we note that x = 0 at times such that t = 0 or . At those times, y has the values [a sin ] and [a sin ( + ) = -a sin ], respectively. Hence the distance A = 2a sin while the distance B is the maximum peak-to-peak displacement in the y-direction, or B = 2a. From these two equalities follows the desired relation. We must enclose the right-hand side of the equation for A/B in absolute value signs, because at all but the lowest frequencies it is impossible to tell whether the spot is rotating clockwise or counter-clockwise. This technique can be used to measure the phase difference, if any, between two sinusoidal voltages of the same frequency. It is important that the Lissajous figure be centered on the oscilloscope screen. Before making any phase measurements and each time you change the sensitivity of either input amplifier, set all oscilloscope inputs to GND and center the beam spot with the horizontal and vertical POSITION controls. Then proceed with the measurements. Return to the RLC circuit you used above. Measure the phase shift between the applied voltage and the resulting current in the RLC circuit at the resonance frequency. (Be sure not to ground two different points in the circuit.) Are the phases of the applied voltage and V R the same? What happens to this phase as you move away from resonance? Measure the phase difference at one other frequency, and compare your result with the theoretical prediction given in Section VC. 84 Physics 111 Laboratory POLARIZATION OF LIGHT Introduction A light beam is characterized by three properties: its intensity, its color, and its polarization. Almost all of visual art is based on varying the intensity and color of light. Polarization is a phenomenon just as important at intensity or color, but our eyes cannot detect polarization so most people don't even know that it exists. Some people are color blind, but all of us are polarization blind. In this laboratory we use our brains and our hands to perceive what our eyes cannot. Polarization is an important concept in physics, chemistry, astronomy, engineering, and other areas. It is also one that can be understood mostly through observation and applied by nonexperts. You have a set of polarizers and other items for observing a number of phenomena. No previous special knowledge or theoretical background is required. Polarized light has many interesting practical applications. Engineers use polarized light to measure stresses, radio waves are polarized, laser beams are often polarized, and even some seismic waves (i.e., from earthquakes) are polarized. The liquid crystal display (LCD) on calculators, watches, cell phones, computers, and televisions work by controlling polarization with electric fields. Bees and other animals navigate using the polarization of skylight. (The eyes of bees, unlike those of humans, can detect polarization.) Artists use polarized light to project a dazzling display of changing color patterns. Other applications include compact disc players, sunglasses, photography, and 3-D movies. Apparatus three linear polarizers scattering solution lamp, table, glass setup LCD display bottle of karo syrup cellophane saran wrap cellophane tape I. Making polarized light The light emitted by a hot solid (incandescent bulb or the sun) is unpolarized – demonstrate this to yourself by looking at the lamp through the polarizer and rotating the polarizer. However, there are many physical processes which modify the light so that afterwards the light is polarized. Such a process may be used either to produce polarized light or to analyze light to find out whether it is polarized. When we say light is "unpolarized" this doesn't mean that the electric field points in no direction. It means that the electric field doesn't point in any particular direction: sometimes it points north, sometimes east, sometimes southwest, etc. If this random 85 direction fluctuates quickly enough, we should call the light "randomly polarized" or "polarized in rapidly fluctuating direction" but in fact we call it "unpolarized." A. Polarization by emission Control of polarization is easiest when you control the process that produces the light. Maxwell's equations imply that when charged particles accelerate, they set up electric and magnetic fields that propagate with finite speed away from the charges. This process is called "radiation" and the fields produced are called "electromagnetic radiation" or "light." Suppose a charge oscillates up and down along the x axis of some coordinate system. It accelerates and thus radiates. If one makes measurements at point P, many wavelengths away from the charge, the radiation is linearly polarized, with E in the wave always within the plane containing the x axis and the line from P to the origin. x θ P q y z This can be seen through the following applets http://physics.usask.ca/~hirose/ep225/radiation.htm Notice that the intensity of the radiation varies with θ and, in particular, that no radiation is emitted along the direction of oscillation (θ = 0º or θ = 180º). Dipole radiation explains why commercial radio emissions are polarized: the radio waves come from huge vertical antennas and hence are vertically polarized. Light emission comes from atoms rather than man-made antennas, and we can't reach in and orient the atoms. Nevertheless this concept will be the key to explaining several polarization processes. B. Polarization by selective absorption If we begin with unpolarized (or "randomly polarized") light and absorb the electric field that points in one direction, we will end up with light polarized in the perpendicular direction. How can we accomplish this? By fixing things so that one component can do work and the other cannot. For example, we could pass the light through a skein of wires all parallel to the y axis (see figure 1). The electric field in the y direction will push electrons in the wires, so this 86 component of the field loses energy (as the wires grow slightly warmer) and eventually dies out altogether. No electric field parallel to y makes it through the skein. The electric field in the x direction also pushes electrons, but the electrons don't move very far because they can't jump out of their wires. This component of the electric field doesn't do work, so it doesn't die out. The light that makes it through the skein is thus polarized in the x direction. Figure 1 The "skein of wires" technique works well for microwaves, but it's not effective at optical frequencies because it's impossible to manufacture wires spaced about a wavelength of light apart. But in 1938 Edwin H. Land figured out how to make the equivalent of such closelyspaced wires. The process starts with a sheet of plastic made up of long hydrocarbon chains. The sheet is stretched in one direction so that most of the chains are parallel. Then the sheet is dipped into a solution containing iodine, which attaches to the chains and makes them conducting. Land called his invention "Polaroid" and founded a company to manufacture it. The company eventually branched into other directions, but it continues to make Polaroid sheets. Mr. Land retired a very wealthy man, but he loved to dabble in his laboratory even in retirement. (Today any kind of polarizer is often called "Polaroid," just as any kind of facial tissue is often called "Kleenex.") A sheet of Polaroid passes only light polarized in one direction, the direction perpendicular to the hydrocarbon chains. This is called the "easy axis" of a sheet of Polaroid. Hold two linear polarizers sandwiched in front of your eye. Slowly rotate one while keeping 87 the other fixed. Record what you see. (The filter near the light source is often called the polarizer, while the one near your eye is called the analyzer.) Hold two polarizers in front of your eye, in such a way that the light intensity is at a minimum. This position is called crossed polarizers. Then, insert the third linear polarizer between the two crossed polarizers and rotate it. Record what you see. C. Polarization by specular reflection - Brewster's angle With your apparatus is a pane of glass. (The back side is painted black to minimize reflection from that surface.) Mount this in the center slot of the wooden block on the angle table, and observe the reflection of the light bulb in the glass. Polaroid 1 1 Figure 2 Test the polarization of the reflected light with a piece of Polaroid. Even though the light from the bulb is unpolarized, there is a particular angle of incidence for which the reflected light is 100% linearly polarized, and hence may be extinguished by the Polaroid. This angle (which is different for different reflecting materials) is called Brewster's angle—measure it. To understand this phenomenon, we investigate what's going on in reflection of light from a glass surface. You know that one light ray comes in (the "incident ray") and two go out (the "reflected ray" and the "refracted ray"). What is the physical source of the reflected and refracted rays? The incident ray causes electrons on the glass surface to oscillate, and (as we've seen) this oscillation causes them to radiate. Figure 3 shows the special case when the reflected ray is perpendicular to the refracted ray. Suppose the incident ray is polarized perpendicular to the plane of the figure (i.e. in and out of the page). Then the surface dipoles oscillate perpendicular to the plane of the figure, and they radiate both in the direction of the reflected ray and in the direction of the refracted ray. But suppose the incident ray is polarized parallel to the plane of the figure. Then the surface dipoles oscillate within the plane of the figure. Remember that oscillating dipoles don't radiate parallel to the direction in which they oscillate. The fact that these dipoles are 88 radiating in the direction of the refracted ray means that they're not radiating in the direction of the reflected ray. On the other hand, suppose the incident ray is unpolarized (that is, sometimes polarized perpendicular and sometimes polarized parallel). The outgoing refracted ray is also unpolarized. But because the reflected ray comes out only when the incident ray perpendicular polarized, the reflected ray is polarized perpendicular. In short, for this particular condition the reflected ray is polarized perpendicular to the plane of the figure. incident ray reflected ray E air 1 E 1 n1 = 1 glass n 2 = n 90 degrees 2 E refracted ray Figure 3 We can calculate a condition for Brewster's angle. The angles of incidence and refraction, namely θ1 and θ2, are related through Snell's law, n1 sin1 n2 sin 2 (1) Brewster's angle is θ1 when θ1 + θ2 = 90. Using nl = 1, n2 = n, and sin θ2 equal to sin (90 - θ1), which is cos θ1, show that tan1 n, 1 Brewster's angle. (2) 3 From your measurement of Brewster's angle for glass, calculate its indexFigure of refraction and the associated uncertainty. The reflection provides a source of light with known polarization. Identify and record the easy axis of transmission for both your Polaroids. Polarizing sunglasses are said to pass most light but eliminate glare, where "glare" means the sunlight reflected from a horizontal surface, such as a puddle. In polarizing sunglasses, should the easy axis be oriented horizontally or vertically? Go out into the hallway and notice that the light from the overhead lamps is unpolarized, but the light reflected from the linoleum floor is polarized. In this way, test your prediction about polarizing sunglasses. D. Polarization by a single scattering The blue color of the sky is due to the fact that the short wavelengths (blue) of light from the 89 sun are scattered more effectively than the longer wavelengths (red). This scattering also means (as we'll see) that skylight is partially polarized, and furthermore is polarized to different extents in different directions. For this reason, if you look at the sky through a polarizer, you can make the blue sky look darker so that the (unpolarized) white clouds stand out beautifully. Photographers use polarizing filters for this purpose. In this experiment we let light from a projector pass into water with a few drops of milk, which scatters the light. If you look at the water through a linear polarizer you can see that the light intensity varies as you turn the polarizer. You can use this to find the direction of polarization. The incident light is unpolarized, but the light scattered out of the liquid is partially polarized. How can we understand this? Light is scattered because electrons in the milk oscillate under the influence of the E-field in the incident light. First, remember that oscillating dipoles don't radiate parallel to the direction of oscillation. Place your eye 90º from in incident beam. If the incident beam is (temporarily) polarized up-down, the electrons oscillate up-down, and they radiate in the direction of your eye. If the incident beam is (temporarily) polarized in-out, the electrons oscillate in-out, and they don't radiate in the direction of your eye. Thus the scattered light is polarized. Why is it only partially polarized? Because some of the light is scattered more than once. In such multiple scattering, the light does not remain in one plane, the line of sight is no longer always perpendicular to the direction of propagation, and multiplyscattered light is less polarized. Make a diagram showing the light source, the tank, and the direction of polarization of the light. Look through a linear polarizer at the blue sky. (For safety's sake, don't look directly at the sun.) The greatest percentage of polarized light lies midway between the solar and antisolar points (Figure 4). (At sunset; i.e., dusk on a clear evening, this midway belt is very prominent if you look directly up while wearing Polaroid sunglasses.) Find the direction of polarization of the skylight. (Hint: Compare the direction of the polarization at different points in the sky with the direction toward the sun.) 90 Figure 4 I said earlier that human eyes cannot detect polarization, and that is almost always correct. But when looking at the polarized portion of the blue sky, you might be able to see a phenomenon called Haidinger’s brushes. This is a faint hour-glass figure subtending an angle of about 3° near the center of your field of vision, oriented perpendicular to the direction of polarization. Some people can see the brushes without using a polarizer, some need a polarizer to see them, and some people can never see them. (When I see them, they pulse. I'm particularly apt to see them if I've just done some intense physical exercise, such as sprinting.) Their probable cause is the selective absorption of light with one direction of polarization by the pigment in the retina. (You might also see the brush when looking through a polarizer at a white sheet of paper illuminated by daylight.) II. Double Refraction A. Physics Take two pieces of Polaroid and cross them so that no light gets through. Now slip a piece of cellophane (say from a candy wrapper or a bread wrapper — almost any kind of clear plastic will do) between the crossed polaroids. Now light gets through! Since the cellophane is perfectly transparent, with none of the "dark" appearance of Polaroid, we know that the cellophane is not absorbing light. Thus this is not the phenomena we noticed before with a Polaroid between two crossed Polaroids. The cellophane is not making polarized light, but changing the character of already-polarized light. Rotate the piece of cellophane between the Polaroids, keeping the Polaroids crossed. You should find two angles at 90 from one another where the cellophane has its greatest effect 91 and two angles at 90 from one another where the cellophane has no effect. Thus cellophane has two special directions, oriented at 90 to one another and lying within the plane of the cellophane, which are related to the rotation of polarization. Not all transparent plastic has this peculiar property. Find a piece of household plastic wrap (such as Saran Wrap) or a piece of the stretchy polyethylene used by dry cleaners to protect clothes. Try this between crossed polaroids. You will find that it has little effect — not much light gets through. The cellophane had two special directions, but for the Saran Wrap all the directions are the same. However, with Saran Wrap you can make special directions. Take a piece of this stretchy plastic and stretch it. Check it between crossed polarizers again. There should now be a huge effect! Furthermore, the greatest transmission comes when the stretched direction is oriented 45º relative to the polarizers. Here is the explanation for the behavior of stretched Saran Wrap. Before the stretching, the long organic molecules of the plastic were coiled in a spaghetti-like mess going in all directions. No direction is special. However, the molecules tend to get straightened out and aligned by the stretching. Now when an E field is applied parallel to the chains, the electrons in the molecules can respond dramatically, whereas when an E field is applied perpendicular to the chains, there is little response. The material is highly polarizable parallel to the chains and weakly polarizable perpendicular to the chains. (Here the word "polarizable" refers to electrostatic polarization, not polarization of light.) The electric susceptibility is different in these two directions, thus the dielectric constant is different in these two directions, thus the index of refraction is different in these two directions. These two directions, the stretch direction and the direction perpendicular to it, are called the optic axes. The optic axis with the larger index of refraction (for E directed along it) is called the slow axis. (Large index of refraction means slow phase velocity.) The other optic axis is called the fast axis. We call the two corresponding indices of refraction ns and nf. (A sheet of cellophane or stretched Saran Wrap or any other material having these properties is called a "retardation plate," and is said to exhibit "double refraction.") So far so good, but how does this anisotropy change the character of polarization? Suppose the incoming light is polarized parallel to the slow axis. Then it emerges polarized parallel to the slow axis, as in the top half of figure 5. If the incoming light is polarized parallel to the fast axis, it emerges polarized parallel to the fast axis, as in the bottom half of figure 5. But what if the incoming light is polarized parallel to some other axis? In that case resolve the incoming light into two components: one component parallel to the slow axis and one component parallel to the fast axis. Each component will emerge with its same polarization, but as seen in figure 5 the two components will no longer be in phase. 92 air cellophane air E parallel to slow axis larger index of refraction shorter wavelength E parallel to fast axis smaller index of refraction longer wavelength Figure 5 At a given point in space, the tip of the total E vector does not simply vibrate up and down, it precesses around in an elliptical path in a plane transverse to the direction of propagation — the light is said to be "elliptically polarized." When such light arrives at the analyzing Polaroid, the E field now has a component parallel to the easy axis of the Polaroid, and that component is transmitted. Thus light passes through the two crossed Polaroids. The phase difference may in fact be calculated. Suppose the cellophane has thickness t. The wavelength λn of the wave in cellophane is related to its vacuum wavelength by n 1 . n In figure 5, there are within the cellophane t/λf = nf t/λ1 wavelengths of the fast component and ns t/λ1 wavelengths of the slow component. Thus there are (ns – nf)t/λ1 more slow wavelengths than fast ones, and since each wavelength occupies 2π radians of phase, the phase difference between the two components leaving the cellophane is t 2 ns n f . 1 If the phase difference δ happens to be an integer multiple of 2π, the light emerging from the cellophane will be linearly polarized just as it went in, and it will be completely blocked by the crossed analyzing Polaroid. B. Colors Equation (3) says that the phase difference depends on the wavelength of the light. The phase difference will be considerably different for violet light (λ = 400 nm) than for red (λ = 650 nm). Suppose that three sheets of cellophane produce a phase change of 2π for yellow light (λ = 590 nm). Then yellow light will not pass through a sandwich of crossed Polaroids with three sheets of cellophane in the middle. But red light will undergo a phase change of less than 2π, so it will pass through. Similarly blue light will undergo a phase change of 93 (3) more than 2π, so it also will pass through. If the system is illuminated with white light, all components but yellow light will pass through. The name of white with yellow removed is "purple". Mount two Polaroids in a crossed position at the two ends of the wooden block, fold the cellophane several times in a random manner, and hold it between the Polaroids. Shine white light through the system, and observe the view from behind the analyzer while slowly rotating the cellophane. If you rotate one of the two polarizers through 90, each color should change into its complementary color. Insert transparent objects between two crossed polarizers and record your observations. Suggested items: plastic rulers, eyeglasses, cellophane tape, flexed plastic protractor. Use the index card or a piece of glass to hold cellophane tape to insert between your polarizers. Try using layers of tape. Produce a work of art. C. Applications The fact that stresses in transparent objects become visible between two polarizers can be used in many different ways, for example: (i) Engineers use plastic models of parts such as those used in airplanes to make visible the stresses under certain loads and to measure the stresses quantitatively. (There is a nice illustration of this technique in the November 1972 issue of Scientific American which shows how to measure the wind stresses on models of Gothic cathedrals.) (ii) Chemists, biologists, geologists, and crime scene investigators: The stresses in chemical compounds, biological samples, and rocks can be viewed using polarized light, e.g., with a microscope equipped with a polarizer and polarization analyzer. This can provide valuable information about the composition and history of the sample. (iii) Glassblowers: When glass cools down after a glassblower removes the torch, stresses develop that can eventually lead to spontaneous breakage. In order to relieve stresses, glassblowers anneal their product by heating the finished glass object and then cooling it very slowly (often overnight). Then glassblowers sometimes examine their products between polarizers to see if any stresses remain. (iv) Opticians: So-called safety eyeglasses can be hardened (tempered) by heating them and then cooling them quickly. This makes glass harder (exactly as rapid cooling makes steel harder) and less likely to break. It also leaves stresses in the glass, which can be seen between polarizers. Note: Some safety glasses are chemically hardened rather than heat-treated and these look like untempered glass when put between polarizers. You may still be able to see some stress near the edges. (Federal regulations require that only safety glasses may be sold. The 94 side windows in many American cars are tempered glass. Windshield glass is laminated glass, not tempered glass. You may notice this difference when you're wearing sunglasses inside a car.) III. Liquid Crystal Displays One important modern application of polarization is the liquid crystal display (LCD), which is used in most digital watches, cell phones, calculators, and flat screen computers. The simplest such displays to understand are those in watches and calculators. The digits on these displays are black on a white background as seen by reflected light. You can convince yourself that polarization matters by putting a polarizer on top of your watch or calculator. Look at the stopwatch LCD available in the lab. You can similarly check out a cell phone or computer display. (When I bought a flat screen TV, I played with it and a sheet of Polaroid for a long time. This was a lot more interesting than the TV shows.) Liquid crystals are special liquids which have many of the properties of solids. The type commonly used for LCDs contains rod-like molecules which tend to align themselves in one direction. In an LCD, the liquid crystal is contained in a thin layer between thin glass plates. Outside these plates are crossed linear polarizers, so if the liquid is removed, no light passes through the display. The inner surfaces of the glass plates are treated so that the molecules in the liquid align along the direction of the polarizer on that plate (see figure 6). Thus the direction of alignment of the molecules gradually changes by 90° in going from one surface to the other. As the polarized light passes through the liquid, the plane of polarization is rotated 90° by the molecules, and the light can pass through the crossed polarizers with little loss. The other important property of liquid crystals used for displays is that the molecules are electric dipoles which tend to align with an electric field. If you look carefully at the dissected display you can faintly see the segments of the display. These are electrodes on the inner surface of one glass plate. If a voltage is applied to one or more segments, the molecules in that area align normally to the plates, and the plane of polarization of the light is no longer rotated. That segment thus appears dark. The electronic circuits in the watch or calculator can turn the individual segments of each digit on separately. With the right combinations of segments, any number can be displayed. 95 Figure 6 IV. Optical Activity Hold a linear polarizer under the beaker of corn syrup and another linear polarizer on top. Let light shine through both filters and through the solution; turn the filter nearest you and record what you see. Without the solution between the polarizers or with plain water, no light would pass through the crossed polarizers. You will notice that this is no longer true with the corn syrup between. The corn syrup rotates the direction of polarization; it is called "optically active." Does the angle of rotation depend on wavelength? How do you know? Does the angle of rotation depend on the thickness of the optically active material? How do you know? 96 This rotation is due to asymmetry in the corn syrup molecules, which can be "right-handed" or "left-handed." (Fruit sugar and turpentine rotate polarized light in opposite directions.) Most such asymmetrical molecules are produced by living organisms — this is one of the mysteries of life. (See Stuart Clark, Stuart. "Polarized starlight and the handedness of life,". American Scientist 87(4) (Jul/Aug 1999) 336-343.) Almost all amino acids produced by living organisms here on Earth have a left-handed optical activity. Amino acids found in certain meteorites are equal mixtures of right- and left-handed molecules. This has been used as evidence of their extraterrestrial origin. Chemists use devices called polarimeters to examine and measure the optical activity of chemical compounds. This can be used to identify compounds, etc. Also, since optical activity changes when a magnetic field is applied (this is called the Faraday effect), optical and radio astronomers study polarized light and radio waves to deduce the magnetic fields in stars and galaxies. Acknowledgements Much of this material was taken from: Electricity and Light Laboratory Manual, Department of Physics, The University of Michigan, Ann Arbor. 97 Physics 111 Laboratory Calorimetry Objectives There are two primary goals of this experiment: to determine the latent heat of vaporization, LV, of liquid nitrogen, and to measure the average specific heat of various metals. You will also use computerized data acquisition in this lab. A useful reference is C. W. Thompson and H. W. White, ―Latent heat and low-temperature heat capacity experiment for the general physics laboratory,‖ Am. J. Phys. 51 (4), 362-364 (1983). A discussion of these quantities can be found in section 18-8 of Fundamental of Physics (ninth edition) by Halliday, Resnick, and Walker. Apparatus stainless steel dewar supply of liquid nitrogen ohmic heater with 30 VDC power supply multimeters (2 per group) thermometer (room temperature) digital balance modified for computerized data-taking computer with interface card standard mass set assortment of various metal samples tongs & safety goggles Experimental Procedure – Latent Heat of Vaporization, Liquid Nitrogen Overview You will place a thermally insulated container (dewar) of liquid nitrogen on a digital balance. Immersed in this will be a resistor attached to a power supply and an ammeter and voltmeter. Initially there will be no current through the resistor. You will measure the ―background‖ mass loss rate of the liquid nitrogen due to thermal contact with room temperature air. Once you have established the background mass loss rate you will turn on the current through the resistor, noting the current through the resistor and the voltage across it. After about a minute of this resistive heating you will turn off the resistor current and monitor the steady mass loss rate for another few minutes. By analyzing a plot of the mass of the system versus time and calculating the energy dissipated in the resistor you will be able to determine the heat of vaporization of the liquid nitrogen. 1. Set up the apparatus as shown in Figure 1. Make sure the heater is completely immersed and does not touch the side of the dewar. You will use two multimeters: one as an ammeter and one as a voltmeter. What is the rated power dissipation of the resistors? If you turn on the power supply to test the circuit, be careful not to exceed the power rating of the resistors. Otherwise you will get some smoke and melt the solder connections! 98 Figure 1: Schematic of the apparatus. The letters refer to the following items: B, balance; C, stainless steel dewar; I, ammeter; V, voltmeter; S, on/off switch; R, resistor. 2. During this experiment you need to coordinate your computerized data-taking with the use of the ohmic heater. Talk over your plan with your partner. Think about what your data plot will look like and discuss which features of the plot you will use to calculate the latent heat. Decide when you are going to turn on the ohmic heater and for how long (about 60 – 80 s is a good value). Do you need to keep track of exactly how long the heater was on? How long are you going to record the mass loss rate after the heater has been turned off? The BALANCE program records a mass reading once a second for as many seconds as you specify when you start data-taking. Set up your data directory on the disk if you have not already done so, and decide on the names you will use to keep track of your data files for this experiment. 3. Using a separate dewar, get a cupful of liquid nitrogen from the instructor. Safely transfer this to your dewar on the balance. Make sure you wear your goggles during this transfer process and for as long as you are working closely with the liquid nitrogen. When the liquid nitrogen appears to have reached a steady boiling rate, begin data-taking. 4. After you have taken enough data to establish the background mass loss rate (2–3 minutes), turn on the heater. Make sure to record the reading on the ammeter and the voltmeter and keep track of exactly how long the heater is on. Continue to take mass data during the heating period. After the heater is turned off, take data for a few more minutes to determine the background loss rate after the heating cycle. 5. Plot your mass vs. time data using the Graphical Analysis program. It is probably a good idea to save your data file and print out this ―overview‖ plot before you continue with your analysis. Clearly label the onset and finish of the heating cycle. Using the method discussed in the Graphical Analysis introduction, break the data up into a pre-heating, heating, and post-heating data series. Analyze them separately to determine 1) background mass loss rate before heating, 2) background mass loss after heating, and 3) the total mass loss due to ohmic heating. Make sure to determine the uncertainties of all these quantities. 6. Using the current and voltage readings for the heater and the length of time that the 99 heater was on, determine the heat input into the system. From this determine the heat of vaporization of liquid nitrogen. Compare this with the accepted value of 47.23 cal/g or 197.7 J/g and compare it with the values obtained by your classmates. Experimental Procedure – Specific Heat of Metals Overview The specific heat of a metal sample is obtained by immersing the sample, initially at room temperature, into a dewar of liquid nitrogen. The mass loss due to this heat input is used to calculate the average specific heat of the metal, where the averaging is over the temperature range from room temperature to the liquid nitrogen boiling point (77 K). 1. Assuming that you will be able to determine the mass loss, M, due to immersing the metal sample in liquid nitrogen, derive an equation for the average molar specific heat of the metal over this temperature range. Discuss this with your classmates and your instructor. 2. Using a setup similar to that shown in Figure 2 and a procedure similar to that used in the first part of the experiment, obtain a plot of mass vs. time for one of your metal samples. Figure 2: Schematic of the apparatus illustrating that the metal sample is transferred from the balance pan into the liquid nitrogen during the specific heat measurement. B is the balance pan and C is the stainless steel dewar. 3. Working with this graph, determine the mass loss of the liquid nitrogen due to the immersion of the room temperature sample in the liquid nitrogen bath. Using the accepted value of LV = 197.7 J/g convert this mass loss into a Q due to the heat content of the metal sample. 4. Working with the data in Table 1, determine the expected average specific heat of the metal you are studying by averaging the data in Table 1 over temperature. You can use the Graphical Analysis program to do the averaging. 100 T [K] 70 80 90 100 120 140 150 160 180 200 220 240 250 260 280 300 Cu 11.0 13.0 14.7 16.2 18.3 19.9 Pb 23.2 23.6 24.0 24.4 24.9 25.1 Al 7.7 9.7 11.4 13.0 C(graphite) 0.9 1.2 1.4 1.7 2.3 2.9 Si 4.3 5.2 6.2 7.2 9.1 10.8 3.6 4.3 5.0 5.7 6.4 12.5 14.1 15.4 16.5 17.4 7.2 7.9 8.6 18.1 18.8 19.5 18.5 21.1 22.0 22.6 23.1 23.6 25.5 25.7 25.9 26.1 26.3 23.9 24.2 24.5 26.5 26.7 27.0 21.5 23.2 24.4 Table 1: Molar specific heat capacities at constant pressure for a range of temperatures for various substances. The units for the molar specific heats are J/mol K. Note the following atomic masses: Cu (63.55 g/mol), Pb (207.2 g/mol), Al (26.98 g/mol), C (12.01 g/mol), Si (28.09 g/mol). 5. Compare your experimental value for the average specific heat of this metal with the predicted value you obtained in the previous step. Think about the sources and sizes of the experimental uncertainties in your measurement. Are your experimental value and the predicted value consistent with each other within the uncertainties of the experiment? 6. If you obtained reasonable agreement in step 5 you may want to try measuring the average specific heat of another metal. Alternatively, you may want to repeat this experiment, trying to improve your procedure. 7. Compare your results with other groups in the class, and discuss your results with the instructor. 101 Propagation of Errors Any measurement contains some uncertainty, even after outright mistakes have been eliminated. It may arise, for example, because of "haziness" in the thing being measured (the height of a tree during a strong wind, or the position of a pendulum bob). Or it may arise from the way in which the measuring instrument must be used (measuring a time interval by starting and stopping a stopwatch, or having to interpolate between the divisions of a ruler). One way to minimize the effect of these errors is to take several repeated measurements. Then, on the assumption that the errors are "random" - that is, as likely to increase a single measurement as to decrease it - the arithmetic average of the several measurements may not be much affected. Be that as it may, it is impossible to eliminate all uncertainty from physical measurements. The question then arises as to how to estimate the uncertainty in some final result obtained by combining arithmetically several individual measurements, each with an associated uncertainty. The analysis of this problem is called "propagation of errors", and there are several procedures that may be used to estimate the error in the final result. First it is necessary to describe two ways of expressing the uncertainty in a measurement. If the result of some measurement is given as a , the number is called the “absolute error”. Most of the time an experimenter can make a good guess or estimate of the absolute error in the measurement; the method of measurement is known, the competence of the experimenter is known, the accuracy with which the instruments have been made is known. As a result he or she can sit back and reflect on the measurement and, after a little practice, write down a good estimated absolute error for the measurement. Notice that a a 1 . a The ratio /a occurring on the right is known as the "relative error", or "fractional error". It may be expressed as a percentage of a. In propagating errors through calculations, both absolute errors and relative errors will be used, and one needs to be able to convert an expression of the error in one form to the other. Now three approaches to error propagation may be described. They are called (a) Brute Force, (b) Follow the Rules, and (c) A General Technique. 102 (a) Brute Force. If one computes the final result of an experiment using several experimental measurements that have associated uncertainties, the best value of the result is obviously obtained by using the best values of the experimental measurements. The uncertainty in the final result may be estimated by repeating the entire calculation many times using assorted values of the experimental measurements that differ from their best values by amounts that fall within the range of their respective uncertainties. Each calculation will yield a different result, and the spread of the different results will give an idea of the uncertainty to be associated with the best estimate of the final result. This method often requires a large amount of calculational effort. (b) Follow the Rules There are several rules that indicate how to estimate the uncertainty in the result of some very simple calculations. The following discussion should make them seem plausible. Addition: Consider the addition of two measurements, each with associated uncertainty. If these are represented by a and b , the sum is a b . What should we put down for ? It is conceivable that both a and b are "off" by their full uncertainty, and hence their sum is incorrect by the sum of the uncertainties. a b a b . However, if the two measurements are totally independent of each other, it is not likely that there is a conspiracy such that both measurements are off to the maximum extent in the same direction. The meaning of the word "likely" can be made more precise by assuming that the uncertainty in the measurements fall in a Gaussian distribution about the value zero. Then it can be shown that the best value to use for is 2 2 in the sense that 68% of the time the uncertainty in the sum will be less than that . As you can see by considering lengths of the sides of triangles, together with the Pythagorean theorem, this is greater than either or , but not as big as ( + ). Thus the rule for addition: 103 The absolute uncertainty in a sum of independent numbers is equal to the square root of the sum of the squares of the absolute uncertainties in each member of the sum. Subtraction: The argument goes the same way as for addition. Conceivably the error in a difference could be as big as the sum of the uncertainties in both terms. But it is not likely. The absolute uncertainty in a difference of two independent numbers is equal to the square root of the sum of the squares of the absolute uncertainties in each term. Multiplication: Here it is more convenient to use relative uncertainties, as shown by the following: a b a 1 b 1 a b ab1 1 a b ab 1 a b a b ab1 . What should we use for ? First, since we can only make estimates of uncertainties anyway, let us choose to ignore the product (/a)(/b), provided both (/a) and (/b) are small numbers much less than one. In that case their product will be much smaller yet, and it will not make an important contribution to the total uncertainty. Then we are faced with the same situation as before; conceivably the could be as big as , but it is not likely. Again we make the same compromise: a b The relative uncertainty in a product of independent numbers is equal to the square root of the sum of the squares of the relative uncertainties in each member of the product (assuming they are small). 104 Division: Assuming that the relative uncertainties are again small numbers, to the same approximation we obtain the same rule as for multiplication. In division, The relative uncertainty in the ratio of independent numbers is equal to the square root of the sum of the squares of the relative uncertainties in the numerator and denominator (assuming they are small). Averages, quantities raised to some power, square or cube roots of measurements, or multiplying or dividing of measurements by arithmetic constants like are other common calculations which have associated rules you should be able to figure out using similar reasoning. If an extensive calculation can be broken down into simple steps, then the above rules may be used to propagate the errors, provided that the uncertainties in the quantities being combined remain independent. (c) A General Technique The techniques of calculus are well adapted to the problem of error propagation, as can be seen by expressing the calculation of the final experimental result as a functional relationship. That is, in f(x, y, z, ...), we may regard the quantities x, y, z, ... as the experimental measurements, and the function f as the computation involving them that yields the desired result. Then the question of error propagation is closely related to the question: “If the quantities x, y, z, ... were slightly different, how different would be the value of f?” Or “How much does f vary if x, y, z, ... are changed by a certain amount?” The answers to these questions will clearly involve derivatives of the function f. Let us first derive the relationship and then adapt the result to error propagation. If you already know about multi-variable Taylor Series expansions, skip to the next paragraph. Otherwise, we may begin with the mean value theorem for a function of one variable, which says that f x x f x df , x dx x x 0 1 In words, on a graph of f(x) there exists some (unspecified) point between x and x + x where the tangent to the curve has the same slope as the straight line through the two points at x and x + x. 105 Thus, if we ask how the function changes as x is changed from x to x + x, the answer is f f x x f x df x. dx x x The result may readily be extended to functions of two variables. Let f f x x, y y f x,y . Subtracting and adding the same intermediate term, f f x x, y y f x, y y f x, y y f x, y f x f x x x , y y y y , 0 1, 0 1. x, y y We do not know the values of or , but if x and y are small enough that the derivatives have essentially the same value throughout the intervals, then we will not be far off if we take . So f f x x, y y f x,y f x f x x,y y y . x ,y With an obvious extension to more variables, the change f that will take place in the value of a function f as its variables are changed from x to x + x, from y to y + y, from z to z + z, …, simultaneously, is approximately f f f f x y x z x ,y,z,... y x,y ,z,... 106 z x,y ,z,... (1) These are the leading terms in a Taylor-series expansion of the multi-variable function f. To use this relationship for error propagation, we make the above-mentioned association of x, y, z, ... as measured parameters and f as the calculation of the results of the experiment. The measured parameters are not exact, but have associated uncertainties. Let x be the estimated absolute error in x, so that its true value might be anywhere in the range x ± x. Equation (1) for f tells about how much the final result will be off if x is in error by x, and similarly for the other variables. Unfortunately, we do not know exactly how far off each measurement is. Conceivably there is a conspiracy among the different measurements so they each have the right size and sign to yield the largest possible f. But if the measurements are independent, such a conspiracy is extremely unlikely. The appropriate compromise, justified by the statistics of Gaussian errors, is to use for error propagation the expression f 2 2 2 2 f f f 2 2 2 x y z x z y Here x, y, z are the estimated absolute uncertainties in the experimental measurements, and f is the corresponding absolute uncertainty in the result calculated via f(x, y, z, …). All of the rules given in part (b) may be derived directly from (2). For example, suppose the calculation is one of division. If f(x, y) = x/y, then f 1 f x , 2 . x y y y Plugging into ( 2 ) we find f 2 x 2 y 2 x2 2 4 y . y Multiplying both sides by (y/x)2, the result is 2 2 2 f x y , f x y 107 (2) which is the rule involving squaring and adding the relative errors in the numerator and denominator to get the square of the relative error in the ratio. In addition, Equation (2) permits you to propagate errors through cosines, logarithms, and other kinds of mathematical functions. Credits W. Bruce Richards, Oberlin College, 2003 108
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