Physics 160 Laboratory Manual

Transcription

1 Physics 160 Laboratory Manual Hobart & William Smith Colleges Larry Campbell Donald Spector Ted Allen Spring 2003

2 Contents General Instructions For Physics Lab Recording data Lab reports Graphing Reporting of Experimental Quantities Collaborative and Individual Work General Admonition iii iii iii iv iv v v Current/Voltage Relationships vi Procedure I viii Procedure and Analysis II ix Procedure and Analysis III x Kirchhoff s Laws xi Procedure I: Parallel Circuits xi Procedure II: Series Circuits xii Procedure III xiii Procedure IV xiv RC Circuits Procedure I xv xvi Magnetic Field of a Current Loop xix Procedure I: Dependence of Field on Current xxi Procedure II: Dependence of Field on Number of Turns xxi Procedure III: Dependence of Field on Diameter xxii i

3 ii Prisms and Lenses 1 Procedure I: Refraction Procedure II: Lens Optics Procedure III: Lens Optics Procedure IV: Lens Optics Diffraction Grating 7 Appendix A: Reading the Vernier Notes on Experimental Errors 11 Experimental Error and Uncertainty Mean & Standard Deviation Agreement of Experimental Results Notes on Propagation of Errors 19 Sums of Measurements Products and Powers of Measurements Notes on Graphing 23 Graphical Representation and Analysis Linearizing the Data Making Good Graphs Measuring the Slope of a Line Least Squares Method

4 General Instructions For Physics Lab Come to lab prepared Bring quadrille-ruled, bound lab notebook, lab instructions, calculator, pen, and fine-lead pencil. RECORDING DATA Each student must record all data and observations in ink in his or her own lab notebook. Mistaken data entries should be crossed out with a single line, accompanied by a brief explanation of the problem, not thrown away. Do not record data on anything but your lab notebooks. Put the date, your name, and the name of your lab partner at the top of your first data page. Your data and observations should be neat and well organized. It will be submitted as part of your laboratory report and must be legible. LAB REPORTS Specific advice on this matter is given at the end of each lab description, but your report will generally consist of: Your data and observations, a summary of your experimental and calculational results, and a discussion of your conclusions. The summary and discussion may be written out right in the lab book (if done very legibly and neatly), or you may use word processors and/or spreadsheets if you prefer. One typed page summarizing what you learned by doing the experiment. Occasionally your instructor may give some additional instructions in class. It is important that the report be well organized and legible. If several measurements are to be compared and discussed, they should be entered into a table located near the discussion. Your data and results should always include appropriate units. iii

5 General Instructions For Physics Lab iv Repetitive calculations should not be written out in your report; however, you should include representative examples of any calculations that are important for your analysis of the experiment. GRAPHING Graphs should be drawn with a fine-lead pencil. A complete graph will have clearly labeled axes, a title, and will include units on the axes. When you are asked to plot T 2 versus M, it is implied that T 2 will be plotted on the vertical axis and M on the horizontal axis. Do not choose awkward scales for your graph; e.g., 3 graph divisions to represent 10 or 100. Always choose 1, 2, or 5 divisions to represent a decimal value. This will make your graphs easier to read and plot. When finding the slope of a best-fit straight line drawn through experimental points: 1. Use a best-fit line to your data. You may do this either by hand or using a graphing program, such as the one with Microsoft Excel. 2. Do not use differences in data point values for the rise and the run. You should use points that are on your best-fit line, not actual data points, which have larger errors than your best-fit line. 3. Do use as large a triangle as possible that has the straight line as its hypotenuse. Then use the lengths of the sides of the triangle as the rise and the run. This will yield the most significant figures for the value of your slope. REPORTING OF EXPERIMENTAL QUANTITIES Use the rules for significant figures to state your results to the correct number of significant figures. Do not quote your calculated results to eight or more digits! Not only is this bad form, it is actually dishonest as it implies that your errors are very small! Generally you measure quantities to about 3 significant figures, so you can usually carry out your calculations to 4 significant figures and then round off to 3 figures at the final result. After you calculate the uncertainties in your results, you should round off your calculated result values so that no digits are given beyond the digit(s) occupied by the uncertainty value. The uncertainty value should usually be only one significant figure, perhaps two in the event that its most significant figure is 1 or 2. For example: T = 1.54 ±.06 s, or T = 23.8 ± 1.5 s. If you report a quantity without an explicit error, it is assumed that the error is in the last decimal place. Thus, T = 2.17 s implies that your error is ±0.005 s. Because the number of significant figures is tied implicitly to the error, it is very important to take care in the use of significant figures, unlike in pure mathematics.

6 General Instructions For Physics Lab v COLLABORATIVE AND INDIVIDUAL WORK You may collaborate with your partner(s) and other students in collecting and analyzing the data, and discussing your results, but for the report each student must write his or her own summary and discussion in his or her own words. GENERAL ADMONITION Don t take your data and then leave the lab early. No one should leave the lab before 4:30 PM without obtaining special permission from the instructor. If you have completed data collection early, stay in the lab and begin your analysis of the data you may even be able to complete the analysis before leaving. This is good practice because the analysis is easier while the work is still fresh in your mind, and perhaps more importantly, you may discover that some of the data is faulty and you will be able to repeat the measurements immediately.

7 Current/Voltage Relationships GOALS AND GENERAL APPROACH This experiment will give you experience in measuring voltages and currents with multimeters, and enable you to study how the current through an electrical circuit element depends on the voltage difference across it. In some cases the current flowing through the element is directly proportional to the voltage across it (i.e., the electrical potential difference between its two ends). Such elements are called resistors and the proportionality constant is the resistance of the element in Ohms. (V = IR) However, many circuit elements do not behave in this simple Ohm s Law way. You will have the opportunity to explore both sorts. In the case of elements which obey Ohm s Law, a fluid analogy can help us to understand the voltage/current behavior. Think of the elements and connecting wires in your circuit as pipes through which water is flowing. The electrical current I corresponds to the rate at which water flows through a section of pipe. In the case of water flowing through a pipe, the rate of flow depends on the difference in water pressure between the two ends of the pipe. So if we consider electrical potential to be analogous to water pressure, we would expect the electrical current through an element to be proportional to the difference in electrical potential between the two ends of the wire (Ohm s Law). Pipes which are very narrow, or clogged with scale and obstructions provide a high resistance to water flow. A given pressure difference across such a pipe will result in a very much smaller flow rate than would occur for a large clean pipe with the same pressure differential. Or, to get the same flow through the clogged pipe as the clean pipe, one needs a much greater difference in pressure across its ends. This analogy helps us to understand why thin wires have greater resistance to electrical current flow than thick wires. Likewise, just as we expect the water to flow out of the pipe at exactly the same rate it flows into the pipe (unless the pipe has an aneurysm), so we expect the electric current to be the same at both ends of a wire. (It is a common misconception that current is getting "used up" as it flows through a circuit.) We will return to this analogy as we move along through the experiment. vi

8 Current/Voltage Relationships vii APPARATUS A variable power supply (a source of potential difference and current), two multimeters, a bread board, hookup wire, and some resistive circuit elements (resistor, light bulb, and diode). Resistors use a system of three colored stripes to indicate their resistance values, each color corresponding to a particular integer. The first and second stripe give the tens and units values respectively. The third stripe gives the power of ten by which the value is to be multiplied. The color key is as follows. Table I Resistor Color Codes Color Value as Value as digit multiplier Black 0 1 Brown 1 10 Red Orange Yellow Green Blue Violet Gray White There may be a fourth band on your resistor. It gives the uncertainty of the color-code resistance value. The following code applies: gold ±5%, silver ±10%, and no band ±20%. Thus, for example, a resistor with stripes brown, green, orange, and silver would have a nominal resistance value of Ohms with an uncertainty of 10% of that value. R V Y 27 x 10 4 Ω Figure 1 Example of reading the resistor color code. The bands are red, violet, yellow. Thus the value of the resistance is Ω. The circuit that you will use for your measurements is shown in Figure 2. A is a multimeter wired in series with the power supply and the element being studied, X. In this configuration it acts as an ammeter (current measuring), and its lead positions and control switches should be set accordingly. V is a multimeter wired in parallel across element X, and in this

9 Current/Voltage Relationships viii power supply + + A ammeter X + V voltmeter Figure 2 The circuit used in this laboratory. configuration acts as a voltmeter (potential measuring). Its lead positions and control switches should be set accordingly. The power supply is a direct current (dc) device, and so the meters should be set for dc operation. WARNING! Be sure that your power supply is off while you build your circuit. Then have your instructor check your circuit before turning the power supply on. Our water pipe analogy will help us here to understand what is happening in this circuit, and some of its limitations. All of the electrical current coming out of the power supply passes through the ammeter A. Thus A truly measures the current out of the power supply, and we see that A should have very little resistance or it will significantly affect the flow of current. Ideally a meter s presence should have no effect on a circuit; but clearly no meter can have absolutely no resistance, so the current will be slightly less when the ammeter is in place. After coming out of A, the current splits into two paths one through X and the other through the voltmeter V and so only part of the current measured by A goes through X. Here we see why it is important that the voltmeter V have a very large resistance (much larger than that of X). If the resistance of V to current flow is much greater than that of X, the majority of the current will flow through X and only an insignificant trickle will pass through V. In this case the current measured by A will not differ significantly from that which actually flows through X. (Since the voltmeter is measuring the loss in energy per unit charge as the current passes from the top to the bottom of X, it need only see a small sample of the charge flowing from top to bottom.) You might like to think about whether or not one could get around this problem entirely by wiring the circuit in the alternative way shown in Figure 3. PROCEDURE I Study the behavior of your light bulb as X in the circuit. Simultaneously measure the current through the light bulb and the voltage across the light bulb for several different power supply voltages. Be careful not to exceed 6 V. After each reading, momentarily remove the voltmeter from the circuit to see whether or not it affects the current reading significantly, and record

10 Current/Voltage Relationships ix power supply + ammeter + A X + V voltmeter Figure 3 An alternative circuit. your results. If it does affect the reading, then you know that your voltmeter resistance is not high enough and your previous result is not be valid. In this case, you should use the value measured with the voltmeter disconnected from the circuit as your current value. You should plot your data (I versus V ) in your notebook as you take the data. This will enable you to see what the trend is early on, and will make it clear where you may need more data points. ANALYSIS I 1. Make a plot of current (vertical axis in amps) versus voltage (horizontal axis in volts). The origin (V = 0, I = 0) will be one of your data points. Draw a smooth curve or straight line using your data. 2. Does your light bulb obey Ohm s Law; i.e., does your data fit a straight line? If it does, find the resistance from your graph (the reciprocal of the slope), measure the resistance with your multimeter, and compare the two values. If your data does not lie on a straight line, you can still measure something called the differential resistance. The differential resistance is the inverse of the slope at any particular point on the graph, and is different for different points as the slope changes. Measure the maximum and minimum values of the differential resistance for your light bulb. Compare them with the value obtained by measuring the resistance of the light bulb with your multimeter. Why do you think the differential resistance of the light bulb changes as the current changes? 3. Calculate the power delivered to your light bulb for the highest voltage and current you tried. Note that the power is measured in watts (W) if the current is measured in amperes (A) and the voltage is measured in volts (V). PROCEDURE AND ANALYSIS II 1. Carry out the same procedure for your resistor as you did for your light bulb, but this time also repeat the measurements with the leads into the power supply switched so that the current is running through the resistor in the opposite direction. With the leads switched your voltage and current readings will be negative. Include both sets of measurements (positive and negative) on a single graph.

11 Current/Voltage Relationships x 2. Draw a smooth curve or straight line through your data points. Discuss whether Ohm s Law is obeyed by the current flowing in the forward direction and in the backward direction. For data which obey Ohm s Law, measure the resistance from the graph. 3. Record the nominal value of the resistor using its color code, and measure its resistance with your multimeter. How do the three resistance values compare? 4. Calculate the power delivered to the resistor for the highest voltage and current used. PROCEDURE AND ANALYSIS III Carry out the same procedure and analysis for the diode as used for the resistor, for both directions of current flow, but in this case modify your circuit, as shown below, by placing your incandescent lamp in series with the ammeter. power supply + + A ammeter lamp diode + V voltmeter Figure 4 Circuit with diode and current-limiting lamp. This new arrangement helps to protect your ammeter from overloads by limiting the current out of the power supply. The diode has very little resistance in the forward direction. Keep the current in your circuit under 120 ma. For diodes it is particularly important that you test your voltmeter s effect on the circuit by taking it out momentarily now and again. For currents in one of the two directions, you will find that the current reading changes significantly when the voltmeter is removed. When that occurs, record the current reading without the voltmeter as your measured value. You will find very different currents in the forward and backward directions. After making a graph which shows both positive and negative currents on the same scale, you will have to make a second graph with a new scale for the very low current data (so that it can be better seen). From slopes on the graphs, find representative values of the resistance under various circumstances. Discuss your findings.

12 Kirchhoff s Laws GOALS AND GENERAL APPROACH This experiment provides you with the opportunity to familiarize yourself with and test Kirchhoff s current and potential laws: 1. the algebraic sum of the potential changes around a closed circuit path must add up to zero; 2. the algebraic sum of the currents flowing into a junction point must add up to zero. You will also learn a very useful principle governing how voltage is divided across a set of series resistors. APPARATUS Variable power supply, multimeters, resistors, light bulbs, a potentiometer, hook-up wire, and a breadboard. PROCEDURE I: PARALLEL CIRCUITS Connect three resistors in parallel across the power supply as shown in Figure 1. Each resistor should have a different value of resistance, but none should be less than 200 ohms. Be sure to measure the resistance of each resistor with your multimeter. Choose a convenient power supply voltage (between 2 V and 7 V) and measure the voltage across every element in the circuit (including the power supply). Also measure the current coming out of and going into each circuit element. A convenient way to record your data is to draw a large diagram in your notebook and record the resistance values, voltage differences, and currents directly on the diagram. ANALYSIS I 1. Show explicitly that your data confirms both the current and potential xi

13 Kirchhoff s Laws xii power supply + R 1 R 2 R 3 Figure 1 Schematic diagram for the circuit used in procedure I. laws of Kirchhoff. 2. Calculate the equivalent resistance of your three parallel resistors from the parallel resistor rule (1/R equiv = 1/R 1 + 1/R 2 + 1/R 3 ). Then use Ohm s law to calculate how much current such a single resistor would draw if it were to replace the three original resistors, assuming the power supply voltage remains the same. Compare this value with your measured value for the total current drawn from the power supply by your three parallel resistors. PROCEDURE II: SERIES CIRCUITS 1. Connect three resistors in series across the power supply as shown in figure 2. Each resistor should have a different resistance, and the sum of the three resistances should be greater than 100 ohms. Be sure to measure the resistance of each resistor with your multimeter. Choose a convenient voltage (between 2 V and 7 V) for the power supply and measure the voltage across every element in the circuit (including the power supply). Also measure the current coming out of and going into each circuit element. power supply + R 1 R 2 R 3 Figure 2 Schematic diagram for the circuit used in procedure II. ANALYSIS II 1. Show explicitly that your data confirms both the current and potential laws of Kirchhoff. 2. Calculate the equivalent resistance of your three series resistors from the series resistor rule (R equiv = R 1 + R 2 + R 3 ). Then use Ohm s law to calculate how much current such a single resistor would draw if it were to replace the three original resistors, assuming the power supply voltage remains the same. Compare this value with your measured value for the total current drawn from the power supply by your three series resistors.

14 Kirchhoff s Laws xiii 3. It is easy to prove from Ohm s Law and Kirchhoff s Laws that in such a set of series resistors, the voltage applied across the ends of the set by the power supply is divided among the individual resistors proportionally to their individual resistances. That is, the ratio of the voltages across any two of the resistors is the same as the ratio of their resistances. That is, Verify that this holds true for your circuit. R 1 R 2 = V 1 V 2. (1) 4. Another way of stating this principle is to say that the fraction of the power supply voltage which appears across any individual resistor, is equal to the fraction of the total series resistance possessed by the resistor in question. That is, V 2 V power supply = Verify that this holds true for your series circuit. R 2 (R 1 + R 2 + ). (2) PROCEDURE III A potentiometer can be represented schematically in the following way. + A B C Output terminals Figure 3 Schematic representation of a potentiometer. The two outer terminals are connected to the ends of a long resistor, and the middle terminal is connected to a sliding contact which can be moved along the resistor, making contact at any point between the two ends. The resistance between the outer terminals remains constant, but the resistance between the middle terminal and either outer terminal changes as the sliding contact is moved by rotating the control shaft. As the contact is moved to the right, the resistance between B and C decreases while the resistance between A and B increases. The opposite effect is produced when the contact is moved to the left. 1. Verify the behavior of the resistance between the terminals of the potentiometer by turning the control shaft and measuring the resistance between various terminals with your multimeter. Record your observations in your lab book.

15 Kirchhoff s Laws xiv 2. Connect the outer terminals of your potentiometer to the power supply to produce a variable voltage source. Set the power supply to 5 volts. The output voltage between the middle terminal and either outer terminal will vary as the potentiometer shaft is turned. (a) Connect your light bulb between terminals B and C and observe and record its behavior as the control shaft is turned. (b) Measure and record the range of output voltages available between B and C when the power supply is set at 5 volts. (c) Disconnect the light bulb and adjust the potentiometer for an output between B and C of 3 volts, with the power supply still set at 5 volts. Disconnect the potentiometer from the circuit, measure the resistance between each pair of terminals, and verify that the voltage divider relationship holds. PROCEDURE IV 1. Set the power supply voltage to 5 volts. Connect first one, then two, and then three light bulbs to it in series. (a) Observe and record the behavior of the light bulbs as more bulbs are added to the circuit. (b) With three bulbs still connected to the power supply, remove one of the bulbs from its socket and observe and record what happens. 2. Repeat the measurements and observations of part 1, but this time wire successive bulbs in parallel to the power supply. ANALYSIS IV Explain clearly why the bulbs behave as they do in Procedure IV.

16 RC Circuits GOALS AND GENERAL APPROACH In this experiment you will study the time behavior of the current and voltage in a charging and discharging RC circuit. The values of R and C have been chosen so that the times involved can be observed directly and measured with a stop watch. Later we will return to this circuit and use an oscilloscope to explore it again for characteristic times much too small to be observed directly. The circuit we will use is shown in Figure 1. The switching action shown is achieved by connecting one power lead connector (banana plug) to the negative terminal of the power and leaving it there, and connecting the other power lead connector alternately to the positive and then the negative terminals (the banana plugs can be stacked on top of one another when both are connected to the negative terminal). When the switching banana plug is connected to the positive terminal, the capacitor is being charged by the power supply voltage; when it is power supply + ammeter A R C + V voltmeter Figure 1 The RC circuit used in this laboratory. connected to the negative terminal, the power supply is bypassed, the voltage to the circuit is zero, and the capacitor can discharge back through the power supply leads. The expressions which describe the time dependence of the circuit are as follows. 1. If the capacitor is initially discharged and the clock is started when the power supply is connected in to the circuit, then; I(t) = I 0 exp( t/rc) and V c (t) = V 0 (1 exp( t/rc)), (1) where I(t) is the current through the resistor and power supply, V c (t) is the voltage across the capacitor, I 0 = V 0 /R, and V 0 is the power supply xv

17 RC Circuits xvi voltage. Note that if R is measured in ohms and C is measured in farads (F), then RC will have units of seconds. 2. After the capacitor is completely charged (to the voltage V 0 ), if the power supply is bypassed by switching the power lead to the negative terminal on the power supply, and the clock is restarted, then; I(t) = I 0 exp( t/rc) and V c (t) = V 0 exp( t/rc). (2) APPARATUS Power supply, multimeters, resistors, light bulb, stop watch, hook-up wire, bread board. PROCEDURE I Build the circuit of Figure 1 using the 0.47 F, NEC capacitor for C and your light bulb for R. Be sure to connect the positive side of the circuit to the terminal on the capacitor marked with two lines, and set the output of the power supply to not more than 5.5 V so that the incandescent lamp won t be burned out. Also be sure that each multimeter is connected properly and set properly for the function it is supposed to serve. Connect the power supply into the circuit and watch the behavior of the light bulb and the meters. After the bulb no longer glows, bypass the power supply by switching the power lead to the negative terminal of the power supply and again observe the behavior of the light bulb and the meters. ANALYSIS I Record your observations along with an explanation for the behavior you observed. PROCEDURE II Rebuild the circuit of Fig. 1, this time using the 1000 µf capacitor for C and a 0.1 MΩ resistor for R. Measure and record the resistance of the resistor. Connect the power supply and simultaneously start your stop watch. Record values for the voltage across the capacitor and the current into it at 30 second intervals, including t = 0. Your currents will be most conveniently expressed in µa (10 6 A). After the capacitor is completely charged, bypass the power supply and simultaneously start the stop watch. Again, record values for the voltage and current at 30 second intervals, including t = 0.

18 RC Circuits xvii ANALYSIS II 1. For the data taken while charging the capacitor, plot V c (t), I(t), and ln[i(t)] versus time in seconds. By hand, draw a smooth curve through the data points. 2. Are your plots of I(t) and V c (t) consistent with equations (1)? 3. By taking the natural logarithm of both sides of Eq. (1a), one obtains: ln(i(t)) = ln(i 0 ) t/rc (3) Thus the ln(i) plot should be a straight line with a slope equal to 1/RC. If it is, calculate a value for RC from the slope. If it is not, use those initial points which do fall on a straight line to find RC. Compare this value with one calculated from the measured value for the resistance and C = 1000 µf. 4. For the data taken while discharging the capacitor, plot V c (t), I(t), and ln[v c (t)] as a function of time in seconds. 5. Are your plots of I(t) and V c (t) consistent with equations (2)? 6. Is your ln[v c (t)] plot a straight line? Find the slope using that part of your data which lies on a straight line, and from the slope calculate a value for RC. Compare this value with that obtained by multiplying the values for R and C. 7. Plot V c (t) for the discharging case on semi-log paper. Taking the log (to base 10) of both sides of equation (2b) we obtain: log(v c ) = log(v 0 ) t log(e). (4) RC So this plot should also give a straight line, this time with slope = log(e)/rc. Calculate a value for RC from your slope. When measuring the slope of a line on semi-log paper (a sample of which is at the end of the chapter), choose a triangle whose rise is exactly one cycle on the log paper, and set the rise numerically equal to Find the run in the usual way from the change in time on the x-axis.

19 RC Circuits xviii

20 Magnetic Field of a Current Loop GOALS In this lab you will explore the properties of the magnetic field generated by an electric current flowing through a circular coil of wire. Theory predicts that the field at the center of such coils is given by: B coil = µ 0 NI d. (1) You will explore how the field at the center of the coil depends on the current, the number of turns, and the diameter of the coil, and compare what you find with the theoretical prediction. APPARATUS The apparatus you will use is depicted in figure 1 below. Power Supply Compass Coil Figure 1 It consists of a cylindrical form about which any number of turns of wire may be wrapped. The resulting coil is connected to a variable current supply with a built-in current meter. A magnetic compass is supported at the center of the coil by a wood shelf. Slots in the cylinder are used to view the compass needle from directly overhead and to read the angle by which it is deflected. xix

21 Magnetic Field of a Current Loop xx EXPERIMENTAL METHOD The circuit enables us to adjust the current through a wire coil, and measure it in Amps. How do we go about measuring the magnetic field at the center of that coil? To do this directly would be quite difficult. Fortunately, the earth has a magnetic field, and it is rather easy to measure the ratio between the magnitude of the magnetic field created by the coil and the magnitude of the earth s magnetic field. We know from class (or from symmetry considerations) that the magnetic field of the coil points along the axis of the cylinder about which the coil is wrapped. Suppose we align this axis perpendicular to the earth s magnetic field. Then the total magnetic field is given by the vector sum of these two fields (see Fig. 2a), and the magnitudes of the earth s and the coil s magnetic fields are related by B coil = B earth tan θ. (2) B net Top View B coil θ wire Compass B earth (a) (b) Figure 2 Thus if we measure tan(θ), we can find an experimental value for B coil in terms of the magnetic field of the earth. For most of the analysis (including all of the graphs) we can set B earth = 1, remembering that all of our magnetic field values are then being expressed in terms of a unit equal to the magnitude of the earth s magnetic field. Only when you are asked to find a value for µ 0 will you need to use the value for the earth s magnetic field, so that your result will be expressed in proper MKS units (Tesla for magnetic fields) and can be compared to the accepted value. All we need to do is to align these two magnetic fields perpendicular to each other and devise a scheme for measuring the direction of their sum. Both these things are easy to do with a compass, since a compass needle points in the direction of the total magnetic field it experiences. Place the compass at the center of the coil, and with zero current in the coil (and thus with zero magnetic field from the coil), orient the hoop so that the compass needle (and hence the earth s magnetic field) points along the diameter of the hoop (see Fig. 2b). Now when there is a current through the wire coil, the magnetic field of the coil will be perpendicular to the magnetic field of the earth, as we desired. We can now measure the angle θ for whatever configurations of the apparatus are of interest to us, and thus determine how the magnetic field of the coil B coil varies as we vary the configuration of the apparatus.

22 Magnetic Field of a Current Loop xxi PROCEDURE I: DEPENDENCE OF FIELD ON CURRENT Study B coil as a function of the current I through the circuit. Using the nextto-the-smallest cylinder, create a coil of five turns by wrapping the wires tightly around the cylinder, keeping the wires close together and aligned along the mid-line of the cylinder (use masking tape to keep the wire in place). After making five complete turns, the wire should be bent to come away from the cylinder parallel to the symmetry axis of the cylinder (see Fig. 1). Be sure that the point where the wires come away from the coil is opposite the viewing slots cut into the cylinder. Place the cylinder onto the wooden cradle which will hold it in a stable position. Place the support shelf in the cylinder and set the compass on it. Make sure the shelf is level and the compass case is oriented parallel to the guidelines drawn on it. Orient the cylinder and cradle until the compass needle points directly along the diameter of the cylinder. Then tape the cradle to the table to make it easy to return the coil to this orientation. Now adjust the rotating scale on the compass so that the compass needle points to N (0 ) when it is parallel to the cylinder diameter. This will make it easier to read the angle I directly from the scale of the compass when you make your deflection measurements. Once all of these adjustments have been made, tape the other chock block to the table to keep the coil in place. Be sure that the hoop is still aligned correctly after you have finished taping things down. The wires leading to the coil also generate magnetic fields. We would like to minimize their presence. To do this, tape both wires close together onto the table, keeping them parallel to the symmetry axis of the coil until they reach the currently supply. Since current is running in opposite directions in each wire, keeping them close together will result in their individual magnetic field contributions being in nearly opposite directions and thus canceling each other out. Ferrous metals in the lab can also affect your magnetic field readings, so keep the space under the table free of metal stools, make sure that the drawers in the table don t contain metal tools and equipment, and keep your current supply as faraway from the coil as conveniently possible. At this point you are finally ready to make some measurements. Measure deflection angles (θ) for various values of the current up to a maximum of about 70. For deflections larger than 70 the angular uncertainties will introduce excessively large uncertainties into your results for B coil. Make one measurement with zero current and at least 5 measurements with nonzero current. Be sure to estimate the uncertainty in I for each measurement. From your deflection measurements calculate values for the magnet field (see Eq. 1). PROCEDURE II: DEPENDENCE OF FIELD ON NUMBER OF TURNS Study B coil as a function of N, the number of loops of wire on the coil. Using the same cylinder as in Procedure 1, vary the number of loops on the coil from 5 to 1, keeping the current constant for all the measurements

23 Magnetic Field of a Current Loop xxii in this step. Choose a value of the current which gives a deflection of approximately 70 with 5 turns of wire. Finally, also take a measurement with zero turns of wire (but keep the rest of the apparatus the same, so that the wire should come up to the hoop and then go away, without wrapping around it). PROCEDURE III: DEPENDENCE OF FIELD ON DIAMETER Study B coil as a function of the diameter, d, of the coil. There are cylinders of different diameters on your table. For each one, wrap a wire coil with three loops around it and measure the magnetic field produced by a particular value of the current. The current should be the same for every measurement in this step, and should be chosen so as to give a deflection of approximately 70 for the smallest coil. ANALYSIS The field at the center of such coils is given by: B coil = µ 0 NI d. (3) For each procedure, verify this relationship between B coil and the parameter being varied (current I, number of turns N, and diameter d, respectively). To do this, plot B coil versus I, N and 1/d respectively, and look for linear behavior. Include error bars on each of your plots. The errors in deflection angles will outweigh all other sources of uncertainty. From the uncertainties in your values for θ (dθ), you can calculate the corresponding uncertainties in your measured B values (db) using the following equation: db = dθ/ cos 2 θ. Note: In this equation, dθ must be expressed in radians. You will find that the effects of the angular uncertainties on B coil will be quite different for small angles than for large angles, so be sure to include error bars for every point on your graphs. For each graph draw a straight line which best fits your data points. Measure the slopes of your lines and their uncertainties. An uncertainty in slope may be measured by finding an alternative line which is consistent with your data and measuring the difference between the two slopes. Since your graphs employ B earth as the unit of magnetic field, your slopes won t be in standard MKS units. To convert slope values to MKS units so as to get the standard value for µ 0, 4π 10 7 T m/a, multiply each slope by the magnetic field of the earth in Tesla ( T). Using the slopes of these three lines, calculate three values of µ 0, one for each graph (see Eq. 2). Calculate the uncertainties in your µ 0 values, from the uncertainties in your values for the slopes. Are your three values consistent with each other? Are they consistent with the value for µ 0 found in your textbook?

24 Prisms and Lenses REFRACTION When light passes from one material into another, in general the direction of the light ray changes as it crosses the interface between the two materials. This phenomenon is called refraction, and the relationship between the incident and refracted angles (measured with respect to the normal to the interface) is given by Snell s Law, n 1 sin θ 1 = n 2 sin θ 2, where the angles are measured from the normal to the interface between the materials, as shown in Fig. 1 n n 1 2 θ 1 θ 2 Figure 1 The angles used in Snell s Law. All of the imaging produced by lenses and mirrors can be understood in terms of this law and the law of reflection; together they constitute the principles of geometric optics. PROCEDURE I: REFRACTION The first exercise is to observe refraction at the two surfaces of a 60o prism and to measure the index of refraction of the prism. This is accomplished by tracing a line of sight (ray tracing) through the prism, measuring the incident and refracted angles, and using Snell s Law to calculate the index of refraction. Place your 60 prism on the pin-table with a clean sheet of plain paper under it. Stick pins into the paper and table at positions B and C. Both pins should be placed right up against the prism so that they are in contact with the glass, and both should be placed near the base of the prism so that line 1

25 Prisms and Lenses 2 BC is as long as possible (see Fig. 2). With your eye at the level of the prism, sight with this eye only until you see the image of pin through the prism. Move your head and eye until pin C and the image of pin B are lined up. Then, continuing to sight through the prism, place pin A about 15 cm away from B so that pin C and the images of A and B are lined up one behind the other. Now add pin D about 15 cm in front of pin C so that pins C and D are in line with the images of pins A and B. Be careful that the prism does not move while all of this is going on. A B C D Figure 2 Carefully remove the pins and use a fine-lead pencil to trace the outline of the prism on the paper, being careful not to allow the prism to move while this is being done. Then remove the prism from the paper, circle and label the holes made by the pins, and use them to draw the rays representing the line of sight through the prism (see the Fig. 2). Now use a triangle to carefully construct lines perpendicular to the interfaces at points B and C, and measure the angles of incidence and refraction at both interfaces. You will have to extend the line BC in both directions in order to read these angles off of the protractor. In order to obtain good results, the ray-lines and normals must be constructed with great care. ANALYSIS I Use Snell s Law at each interface to calculate the index of refraction of the prism from the measured angles. You will obtain two values of n in this way, one from each interface. You can take the index of refraction of air to be Compare the two indices of refraction with each other and with typical values for glass. Are they in good agreement? Your two n values will probably not be exactly the same. Are they within experimental error of each other? To estimate the error in your measurement place the paper back on the table without the prism. Stick two pins in it about 15 cm apart. Now with your eye at table level, sight along the two pins and place a third pin about 15 cm beyond the farthest pin in such a way that it appears to line up with the other two. Now remove the pins and use a straight edge to test whether or not the holes lie on a single straight line. If they don t, measure the angle by which the two connecting lines differ. This angle can be taken as the uncertainty introduced into your angles by the sighting procedure. In addition, estimate the angular uncertainties introduced by your geometrical constructions and protractor measurements. Use the quadrature method to add these two angular uncertainties (square, add, and take the square root) to obtain an uncertainty

26 Prisms and Lenses 3 value for your measured angles. Now repeat one of your Snell s Law calculations using angle values which have been changed by the amount of your calculated uncertainty. Increase one of the angles by that amount and decrease the other angle by the same amount. This gives the worst case scenario. The amount by which this new value for n differs from the old will give you an idea of the uncertainty in your n values. Can the difference between the n values measured at the two different interfaces be accounted for by your uncertainty calculation? LENS OPTICS As you are learning in class, a refracting material, such as glass, with curved surfaces is capable of bending light rays, and when the curved surfaces have the proper shape, can focus all of the rays that it intercepts from a point source, back to another point. Figure 3 illustrates this for a point source on the optic axis of the lens, and figure 4 illustrates it for a point source off of the optic axis. The distance of the focus points from the lens depends only on the distance of the source points from the lens. Thus a plane object source will have all of its points focused onto a plane on the other side of the lens, forming an image of the source object. s s' Object Image Figure 3 s s' Y Object f f Image Y' Figure 4 The distance of the object from the lens,s is related to the distance of the image from the lens s by the Gaussian lens formula: 1 s + 1 s = 1 f ; or f = s s s + s. where f is a characteristic constant of the lens called the focal length; it depends on the curvature of the lens surface and the index of refraction of

27 Prisms and Lenses 4 the lens material. The object height, the image height, and the central ray create two similar triangles from which we conclude that: Y Y = s s = m, where m is the magnification. The minus sign in this expression allows a negative value for m to be interpreted as representing an inverted image. PROCEDURE II: LENS OPTICS Your experimental equipment includes an optical bench with a built-in distance scale, lens holders, an illuminated arrow to serve as an object, a screen on which a real image can be projected, a ruler for measuring object and image sizes, and a lens. 1. Mount the lens with a 10 cm focal length in a lens holder and place it 40 cm from the object. Adjust the position of the screen until the sharpest possible image of the object appears on it. Measure the following. (a) the object distance s (b) the image distance s (c) the size of the object y (d) the size of the image y. 2. Test the precision with which you can measure s by moving the screen and then readjusting its position to obtain a sharply focused image. Have your lab partner do it if you did it the first time. Do this three to four times and record your results. By how much do these values differ from each other and from your original measurement? This provides you with an uncertainty value for your s measurement. You may assume that the uncertainty in s can be neglected compared to that in s. ANALYSIS II: LENS OPTICS 1. Calculate the quantities (a) the focal length, f, of the lens and its uncertainty; (b) the magnification m = y /y; (c) the ratio s /s. (Theory states s /s = m). Note: the uncertainty in f may conveniently be calculated by using f = ( f s ) 2 s. (1)

28 Prisms and Lenses 5 2. Carefully draw to scale a ray diagram using the measured values for Y and s, and the focal length calculated in 1. What does this diagram predict for Y and s? Compare these predictions with your measurements. PROCEDURE III: LENS OPTICS For the next measurement you will use the concept of parallax. This is the displacement of an object relative to distant objects when viewed from different directions. Check this out by holding out two fingers (one on each hand) at different distances from you and approximately lined up. Close one eye and move your head from side to side and observe how they move with respect to each other and with respect to the motion of your head. Now remove the screen from its holder and replace it with the metal pointer. Place your eye (have one eye closed) at the end of the optical bench (the end away from the object) and look towards the object, so that you can see the image of the object through the lens and the pointer in the same line of sight. The height of the pointer can be adjusted to facilitate this. Since the pointer and the image are at the same location, they should exhibit no parallax; this means that if you move your head from side to side, the image and the pointer should move together in your field of vision, i.e., they will maintain a constant distance between them. To see what happens when the pointer and the image are not at the same location, move the pointer five or more centimeters closer to the lens and observe what happens when you move your head. Which moves in the opposite direction from your head relative to the other, the pointer or the image? Now move the pointer two or more centimeters farther from the lens and observe what happens when you move your head? Which moves in the opposite direction from your head in this case? By observing which of two objects moves opposite your head motion, you can determine which is closer to you. Locate the position of the image again by moving the pointer until you have eliminated the parallax entirely. Make this measurement independently more than once in order to estimate the uncertainty in an s measured in this way. Be sure to record these values. In later experiments you will need to use this method for locating virtual images which cannot be projected on a screen. ANALYSIS III: LENS OPTICS Compare the value for s obtained this way with the value obtained using a real image on the screen.

29 Prisms and Lenses 6 PROCEDURE IV: LENS OPTICS Return the screen to the optical bench and use it to measure the image distance s for four additional object distances. Choose a range of object distances which are as large and as small as your optical bench will accommodate. Try to achieve one case for which s = s. Measure the image size for each case. ANALYSIS IV 1. Calculate an f value and its uncertainty for each of these four measurements. Put all five of your calculated f values (including that from part 2.) together in a table for easy comparison. Are they all equal? Make another table containing m = y /y and s /s values for every measured case. Do the s /s values seem to be in reasonable agreement with the magnification values m? 2. From your data make a plot of 1/s versus 1/s. 1 s + 1 s = 1 f 1 s = 1 s + 1 f. Thus plotting 1/s versus 1/s should result in a straight line with slope = 1 and y-intercept = 1/f if the lens equation holds. Does your graph confirm the lens equation? What value for the focal length does it give? Compare this value with those found in part 1.

30 Diffraction Grating INTRODUCTION The spectrometer which you will be using in this experiment (see figure below) consists of a collimator tube with an adjustable entrance slit, a rotatable table on which the diffraction grating can be mounted, a telescope for observing the light diffracted from the grating, and an accurate angular scale for measuring the relative directions of the various spectrometer components. Your instructor will show you how to use the controls and read the vernier. The collimator has been pre-adjusted so you SHOULD NOT CHANGE THE COLLIMATOR FOCUS CONTROL. Place your Na lamp in front of the collimator slit, and move the telescope until you can observe the illuminated slit through the telescope. Adjust the position of the light source to maximize the brightness of the slit. Adjust the slit width so that it is fairly narrow and adjust both the telescope focus and the telescope eyepiece so that both the cross hairs and the illuminated slit are in good focus. Try to eliminate any parallax between the slit image and the cross hairs by the alternate adjustment of the two telescope controls. Now mount the grating in a vertical position on the table in the center of the spectrometer, and adjust it so that its plane is perpendicular to the light emerging from the collimator and passes approximately through the center of the table. Be sure that the vertical lines of the grating run upand-down and not from side-to-side. Scan the telescope from side to side to observe the various diffracted lines. The most pronounced lines for Na are a bright yellow doublet, but you will also see fainter lines of other wavelengths. The angular position of a line is measured by setting the telescope eyepiece cross hairs on that line and measuring the corresponding angle from the scale. The diffraction angle θ for any particular line is the difference between the angle measured at zero order (straight on) and the angle measured for that line at some higher order (N = 1, 2, or 3). Each wavelength will be diffracted by the same amount to either the right or the left of the zero order maximum, and so diffraction angles measured on either side should be equal. For instrumental reasons they often differ slightly, and so a better way to measure θ is to subtract the angular position of the line on one side of zero order from the angular position of the same line on the other side of zero order, and divide this difference by two. 7