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Physics 124 Lab Manual

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1. 4 4 Experiment 2 Voltmeter Time varying voltages 4 5 Experiment 3 Oscilloscope e 4 6 Experiment 4 Function Generator 0 0 000000 eee 4 7 Experiment 5 A Warning about Grounds 4 8 Experiment 6 Mystery Signals e e e 4 9 Additional Activities 5 Resistors I 5 1 Questions for Lab Preparation e Beds Introductions estos it A it a Ee a li Dior Definitions e eaa ese ees Soe as ee ee hat 6 8G AS Ge eee SA 25 25 25 26 27 27 28 28 29 29 29 30 33 33 38 40 41 41 43 5 4 Schematic Diagrams e 50 Equipment Wi A ia Be eS i aii 5 6 Experiment 1 First Circuit 20020020002 2 eee 5 7 Experiment 2 Kirchhoff s Rules o 2 00000 5 8 Experiment 3 Ohm s LaW 0 0 o 5 9 Experiment 4 Ohmmeter e 5 10 Experiment 5 Correcting Meter Imperfections 5 11 Additional Activities ooa ee Resistors II 6 1 Questions for Lab Preparation 0 000002 ee eee 6 2 Introduction 2 2 4 cau Bako we Bee ee ae Se a ee oA 6 3 Experiment 1 Ohmic vs Dynamic Resistance 6 4 Experiment 2 Impedance 0 0 000 eee ee 6 5 Additional Activities coca eee ow Ea ee a a e 6 6 Formal Lab Report e Capacitors I 7 1 Questions for Lab Preparation 0 0 00000002
2. It would be most helpful to you to try to get through the first four of these items before leaving today since there will not be another opportunity in the laboratory until after the report is due Physics 124 Lab 6 Spring 2012 65 6 3 Experiment 1 Ohmic vs Dynamic Resistance 6 3 1 Finding Ohmic Resistance Q Two of the items you worked with last week are resistors their I V relationship is constant and may be expressed as V IR Ohm s law For the Resistors I lab you should have recorded about twenty data points 1 V and uncertainties 91 9V for both Graph the data using IGOR and fit a line Determine the value of the resistance in each case along with the uncertainty 6 3 2 Finding Dynamic Resistance For the light bulb the dynamic resistance is not constant it does not obey Ohm s law Something more interesting is going on judging from the shape of the I V curve In this section we ll look at a crude calculation of its dynamic resistance The idea here is that dynamic resistance is the slope of the curve of V plotted against I with V on the vertical axis Ideally we would calculate Raynamic dV dI but we can t do that calculation analytically since we don t know V as a function of J On the other hand we can crudely estimate dV dI as AV AI where AV and Al come directly as differences in your data set Q Use IGOR to calculate the dynamic resistance as AV AT from your light bulb data set Plot the dynamic resi
3. C 1 Expressing Experimental Uncertainties C 2 Determining Experimental Uncertainties o C 3 Propagation of Uncertainties C 4 Assessing Uncertainties and Deviations from Expected Results C 5 The User s Guide to Uncertainties D Complex Numbers Bibliography 113 114 115 118 121 123 126 General Instructions The laboratory sessions of Physics 124 are designed to help you become more familiar with fundamental physical concepts by actually carrying out quantitative measurements of physical phenomena The labs are designed to help you develop several basic skills and several higher level skills The basic skills include the following 1 Being able to relate abstract concepts to observable quantities For example know ing how one determines the electrical resistance of a device from easily measured quantities This skill includes the ability to estimate and measure important physical quantities at various levels of precision 2 Knowing how to use common equipment found in professional research laboratories such as voltmeters oscilloscopes signal generators and prototyping breadboards 3 Knowing and applying some generally useful measurement techniques for improving the reliability and precision of measurements such as using repeated measurements and applying comparison methods 4 Being able to estimate the experimental uncertainties in quan
4. New York 2003 p 143 That September Franklin also erected a lightning rod on his own house with an ingenious device to warn of the approaching of a storm The rod which he described in a letter to his friend Collinson was grounded by a wire connected to the pump of a well but he left a six inch gap in the wire as it passed by his bedroom door In the gap were a ball and two bells that would ring when a storm cloud electrified the rod Re create Franklin s alarm system using a suspended pith ball and a pair of plates one of which you connect to each terminal of a Wimshurst machine to represent the Earth and an electrical storm respectively Explain using appropriate diagrams and descriptions how Franklin s storm warning system worked Focus in particular on why the ball alternately strikes one bell and then the other when a charged storm cloud is overhead Physics 124 Lab 3 Spring 2012 28 3 5 2 Spinning Structures If you hook up one of the spinners to a terminal of the Wimshurst machine you will find that it begins to rotate Why Does it matter which terminal you attach it to 3 6 Experiment 5 Van de Graaf Generator CAUTION The Van de Graaf can deliver a momentarily painful shock Does the dome of the Van de Graaf generator become positively or negatively charged See if you can figure out how it works We ll discuss it at the end of lab today 3 7 Experiment 6 Kelvin Water Dropper Pour water i
5. On the other hand your notebook should be reasonably neat and well organized partially so you can find things and partially so that if anyone questions your results not only will they be able to find things but the layout of your notebook will suggest that you investigated the problem carefully and systematically You should use a bound lab notebook that is not a loose leaf notebook So called quadrille Physics 124 Keeping a Lab Notebook Spring 2012 105 notebooks with rectangular grids on each page are particularly handy for making graphs and tables We strongly recommend that you leave every other sheet in your lab notebook free so that you can jot down additional comments and or add graphs onto those blank sheets after the fact If you wish to add a graph done on a computer or a graph done on regular graph paper to the notebook you may simply tape or glue the graph into your notebook When in doubt remember that this is a class in physics not archaeology make sure that you don t put your future self in the position of guessing what you did based on an incomplete historical narrative Next we will discuss some of the information that goes into your lab notebook Introduction You should begin each new experiment on a fresh page in your notebook Leave some room for pre lab lecture notes Start with the date and brief title for the experiment just enough to remind you what that section of your notebook is about Then
6. Several symbols commonly used to mark the ground connection 5 4 9 Power Supply Another device that we ll use from time to time is the power supply Most power supplies are distinguished from batteries in that they are plugged into the wall so their ground terminal is once again the planet that we live on We nevertheless use the battery symbol but we attach the ground symbol to one side or the other depending on whether the power supply is positive or negative Which is which here Your protoboard has three power supplies two of which are variable and one of which is fixed A variable power supply has an arrow through it to indicate that it can be adjusted 2 Figure 5 12 Positive power supply left negative power supply center and variable positive power supply right T T 5 4 10 Function Generator A function generator is depicted by a circle with a waveform within it and two output leads For function generators that plug into the wall such as the CFG 253 one of these Physics 124 Lab 5 Spring 2012 50 leads is attached to ground this is the lead that is the shield of the BNC connection Figure 5 13 The function generator shown generating a sine wave It is unlikely that we can emphasize enough that the ground connections are all attached to one another through some copper spike stuck into the ground in the basement probably You are not required to repeat these connections although it is generally a
7. Sometimes this simple model is too crude and we will need to model light as a wave or a particle However the ray model of light is incredibly useful when discussing the laws of reflection and refraction In this lab you will investigate the fundamental relationship Physics 124 Lab 1 Spring 2012 13 180 Figure 1 1 between incident and refracted light beams This is the foundation for building more com plicated refractive optics such as lenses and optical instruments Thus our treatment of light as a ray will suffice 1 3 Experiment 1 Snell s Law Snell s Law states that for light incident at a normal angle 6 on a smooth interface between two materials of indices of refraction n and na the normal angle 02 of the refracted light is related to 0 by n sin 01 na sin bo 1 1 Q Place the D shaped semicircle of gelatin stiff water within its plastic holder on top of the polar ruled acetate sheet Place the gelatin in such a way that the straight side is aligned with the 90 270 degree line on the paper and so that the midpoint of the straight side sits at the center of the polar grid as shown in Fig 1 1 Set the laser so that the light beam shines through the circular wall of the gelatin and exits at the midpoint of the straight side see Fig 1 1 The light will enter the gelatin normal to its surface You will need to set the height of the laser by propping it up on a book or some such object
8. The rule for simple arithmetic operations on complex numbers is do what comes naturally R Shankar e Addition and Subtraction 21 2 a 3b1 ag jb2 a1 ag j b1 b2 D 2 e Multiplication 2122 a1 jb1 a2 jb2 aiaz jbiaz jaib j bibz aras b1b2 b1a2 a1b2 D 3 Physics 124 Complex Numbers Spring 2012 124 e Norm squared z 32 FZ a jb a jb a jab jba 7 a b D 4 12 VIZ Va b D 5 Note that this always yields a real number In a little while we ll see that this is in some sense the length of 2 e Division 21 ay gb e EE D 6 Za a jbo Of course we d like to write this in the form a jb which means that we have to make the denominator real The easy way to do this is to multiply it by its complex conjugate A 2123 _ a1 jbi a2 jba 1 bib b a9 ajb D 7 gt Fae B 248 a1a2 b1b2 j b1a2 arba D 7 e Argand Diagrams and the Complex Plane e Polar notation Given the representation of Z in the complex plane we can make the following correspon dences between its polar and Cartesian forms a 2 cos b 2 sin0 D 8 Ce T Z Va b tng 2 D 9 3 Nr wa Physics 124 Complex Numbers Spring 2012 125 Figure D 1 The complex number Z can be represented as a point in the complex plane where the hori
9. Zz Finej ees as Ny T CH1 200m CH 500m Wi 500 us CHI 241 rn 25 Jan 11 11 47 1 00000kHz Figure 4 4 A typical oscilloscope display with channel 1 in red and channel 2 in blue You will have only the square wave trace at this point The softmenu selections for channel 1 appear to the right of the voltage graph channels You can also move the trace up and down on the screen by adjusting the vertical knob located above the knob If you press the yellow button marked m you will get a menu of options soft menu on the right side of the oscilloscope display We will discuss some of these options in class If you press the button again the trace will disappear You can make it reappear if you press it again HORIZONTAL SCALE The time interval per division is selected using the knob in the Horizontal control section of the oscilloscope You should try this out now Note what happens to the display when you turn the Scale knob either direction Note also that there is only one Scale knob for the horizontal control each of the two oscilloscope traces is plotted against the same time axis There is also a Horizontal Position knob that moves the entire trace to the right or left Try this adjustment too If you turn this knob too far there is a button that will bring the zero of time back to the center of the display We will define the zero of time and the zero of voltage in a little bit You can also bring up a horizontal contro
10. as measured by the Ohmmeter What sense can you make of what the ohmmeter is doing to make this measurement IMPORTANT Ohmmeters will not give accurate readings for a device that is attached to things like power supplies meters other devices and so forth You need to remove the device from the circuit before attaching it to an ohmmeter IMPORTANT The meter leads red and black are conductors but they also have some small resistance Try touching them together and measuring their resistance prior to measuring each device Technically you should subtract this small resistance from the measured resistance on the meter display Q Can you think of another way to make the ohmmeter leads irrelevant 5 10 Experiment 5 Correcting Meter Imperfections The ammeter and voltmeter are not perfect devices in the sense that the voltmeter does not have an infinite resistance and the ammeter does not have zero resistance These values are practically quite large and quite small respectively compared to the values of the resistances you have measured today but it is possible that these non ideal values will affect your measurement Q Measure the resistances of the ammeter and the voltmeter You will need two fluke multimeters to setup the circuit Think for a minute about the circuit you will need to use These values may change depending on what range setting the voltmeter or ammeter adopts so you will want to override the autorange feature
11. measure electric current we use a device called an ammeter which is represented symbol ically in Fig 5 9 Current must flow through an ammeter in order to be measured The ammeter will display the current that flows into its red lead and out of its black lead If the measured current is negative then the current is actually flowing into the black lead and out of the red lead In some ways this behavior is similar to that of the voltmeter A Figure 5 9 The ammeter 5 4 8 Oscilloscope There isn t any official symbol for an oscilloscope sometimes a little miniature drawing of an oscilloscope is used and other times just the word scope inside a box Physics 124 Lab 5 Spring 2012 49 SCOPE V Figure 5 10 The oscilloscope The funny inverted triangle is the symbol for electrical ground which is an implicit connec tion to the planet we live on All points that are connected to such triangles are actually connected via conductors to one another so it is usually a good idea to put those grounds on the circuit diagram to remind ourselves For the oscilloscope the ground takes the form of the shield of a BNC cable or the little alligator clip on the oscilloscope probe While the grounds are all implicitly connected to one another on a schematic sometimes we will emphasize this point by drawing a line connecting these points together just to make sure that we don t forget t i e de Figure 5 11
12. 1 2 If a screen were placed at s then a bright spot would appear on the screen If instead of a point source we had an extended source a distance s from the lens then a focused image of this extended source would appear on the screen at a position s Indeed if we placed a photographic plate instead of a screen at this position we would have the makings of a camera Physics 124 Lab 1 Spring 2012 15 In this experiment you will test the validity of equation 1 3 There are two or so setups for each minilab so you will need to circulate from station to station Q If an object is very far from the lens s gt f then 1 s lt 1 f In this case we expect from the equation that the distant object will be focused at a distance s f To get a crude idea of the focal length of the lens you are using find some very distant bright object e g a mountain or a tree and measure the distance from the lens to a point where the light is focused Be sure to include with your measurement an estimate of the uncertainty associated with f Q You have been provided with a light source a screen an optical bench and a meter stick Using these set up an experiment to test the validity of Eq 1 3 Plot your data in such a way that if the theory is correct you ought to obtain a straight line graph from which you can obtain the focal length of the lens Again be sure to estimate the uncertainty in your measurement Does your value deduced
13. 106 be recorded and revising your approach So you don t want to get too locked in to one format But you also should avoid writing down data or procedures in the nearest blank space or you ll be cursing yourself when you look for that pieces of information later So the cardinal rule of keeping a lab notebook is this give yourself plenty of space Doing so makes extending tables or descriptions of procedure easy and typically also makes your notebook easier to read If you find that you haven t allotted enough space for a table feel free to start it over on the next page labeling the new table of course and making a note at the old table directing you or someone else to the new one Using only the odd numbered or even numbered if you re left handed pages also works well the blank facing page can be used later to reduce data where you can see both the raw data and the reduced data or for graphs Or you can use all the pages but start out using only the top half of each page Procedure In recording your procedure write in complete sentences and complete paragraphs This is part of the discipline required for keeping a good lab notebook Single words or phrases rapidly become mysterious and only with a sentence or two about what you re measuring such as the period of the pendulum as a function of length will you be able to understand later on what you did Give more details where necessary if for example the lab manual does not
14. I dQ dt There is one more device that finds common use in electronic circuits the inductor The voltage across the inductor is given by V L 9 2 en 9 2 This is the Faraday Lenz law where the constant L is the self inductance of the inductor The SI unit for self inductance is the henry lhenry 1H 1V s A 9 3 Physics 124 Lab 9 Spring 2012 87 As with capacitance the inductance is related to constants of nature and the geometry of the coil The meaning of the minus sign in the Faraday Lenz law is that the sense of the voltage produced by the inductor in response to a changing current is in such a direction as to oppose the change in the current Inductors like there to be no changes in the current and respond by producing a voltage to oppose any changes when they occur The symbol for an inductor is shown in Fig 9 1 As with the capacitor the symbol represents how one could make an inductor by winding a helical coil In fact that is how almost all inductors are manufactured When current flows through the coil a magnetic field is produced within it much as an electric field is produced within a capacitor when it is charged Inductors can store energy like capacitors do although they store it in a magnetic field rather than in an electric field 000 Figure 9 1 In this lab you explore some of the behavior of a series circuit containing an inductor and a resistor 9 3 Experiment 1 RL Circuit S
15. Q Show from Eq 9 5 that during a time interval such as a I t decays exponentially with a time constant given by 7 L R Likewise show from Eq 9 6 that during a time interval such as b I t exponentially approaches a steady non zero value with the same time constant Include this derivation as part of your lab report Show also that with the henry being an abbreviation for volts seconds amperes the combination henries ohms indeed has units of seconds 9 3 2 Experiment Q Let s first look at the response of the RL circuit to a step change in voltage Physics 124 Lab 9 Spring 2012 89 1 We will be using the square wave output of the function generator Begin by setting the amplitude of the function generator to its its largest value You will need to know the output impedance of the square wave function generator on this range it is nominally 50 Q but you should measure it directly using the method in the capacitor lab 2 Record the ACPL number for your coil 3 The coil inductor also has some ordinary resistance which you need to know mea sure it with your digital ohmmeter 4 Set up the following circuit Fig 9 4 using a nominal 100 Q resistor as R V t Ri To Oscilloscope Figure 9 4 As with the capacitor lab the exact frequency of the square wave oscillator is not important but it should be low enough so that each exponential growth or decay is essentially complete before being interrupted by the
16. a measurement depends on the magnitude being measured with higher magnitudes generally yielding measurements with less precision Q What is the precision and dynamic range of the measurement you just made Q Try stacking two D cells together to make a battery and measure the voltage across the battery How does your measurement compare to the individual voltages of the cells you measured earlier What if you reverse the orientation of one of the cells what voltage do you expect and what do you measure With the cells in the reversed orientation your measurement precision is probably about a few mV You can actually do better than this by turning the multifunction knob on your multimeter one click clockwise to mV and then pressing the yellow button a just below the display to make the measurement of constant millivolts This should give you one more digit of precision since your voltage magnitude should be quite small Record this value of the voltage across the two cells Q Keeping the meter on this mV setting measure the voltage across one of the cells What do you measure This is the dynamic range problem in a nutshell If you reverse your leads can you at least tell which lead is at the higher voltage In physics a cell is a single instance of the chemical system that increases the energy of a charge and a battery is a collection of cells In colloquial use cells and batteries in the physics sense a
17. at Eq 7 6 to see why we use this value Ask your instructors to provide the value of the capacitor C if it is not printed in an obvious way somewhere on its side Using this value for C calculate the meter s input impedance in ohms Don t worry about uncertainties rough values are OK here Compare your estimate with what you measured for the voltmeter input impedance you obtained in last week s lab 7 4 Experiment 2 Function Generator Output Impedance Recall that voltage sources typically have a small but finite output impedance sometimes also called the internal resistance We cannot easily measure the output impedance of an emf source by using an ohmmeter By making some clever voltage measurements however we can determine r rather easily We did this last week but as we will need the measurement for this week s lab you will complete the exercise again Get good at doing this you will do this at least one more time before the semester is done Consider the circuit of Fig 7 2 where r denotes the output impedance of the function generator Next consider the circuit of Fig 7 3 Physics 124 Lab 7 Spring 2012 73 1 1 1 l I I I v oscilloscope 1 l 1 1 1 I I I function generator Figure 7 2 A function generator attached to an oscilloscope showing the output impedance r oscilloscope Figure 7 3 When r Rs the output voltage will be half of what it was in the circu
18. been rubbed with silk positive and the charge on a rubber rod after rubbing it with fur negative Armed with this knowledge can you figure out which of your tapes is positive and which is negative Compare with others in the class after making your decision and recording it indelibly in your lab notebook 3 3 Experiment 2 Versorium There are two Versoria you can build a metal one and a wood one Explain how each one works they are slightly different Physics 124 Lab 3 Spring 2012 27 3 4 Experiment 3 Electrophorus Use the electrophorus Does the metal plate become positively or negatively charged 3 5 Experiment 4 Wimshurst Machines BE VERY CAREFUL with the Wimshurst machines they are quite fragile and can deliver a momentarily painful electric shock Turn the crank in the di rection indicated by the arrows on the machine chassis If the belt slips off or the teeth start scraping the wheels contact an instructor Discharge the machine before attaching or removing wires The Wimshurst machine separates charges by induction Can you figure out which terminal is positive and which terminal is negative See if you can figure out how the Wimshurst machine works Keep it as conceptually simple as possible We ll discuss it at the end of lab today 3 5 1 Franklin s Bells Consider the following passage from Walter Isaacson s recent biography Benjamin Franklin An American Life Simon amp Schuster
19. bunch of messy trigonometry Complex numbers consist of real and imaginary parts in general and are themselves a little tricky to use so how on Earth are we supposed to understand this new version of Ohm s law Recall the Euler identity e c0s0 jsin 8 8 12 That is R e7 cos 0 and 3 e79 sind 8 13 Physics 124 Lab 8 Spring 2012 82 are the real and imaginary parts of exp j respectively Suppose now that our synthesizer is generating an output voltage E t Eo coswt This can be written as E t R Epet 8 14 Similarly for I t Io cos wt we have I t R Toet teI 8 15 Note that both t and I t both contain the factor e Let s drop it for now with the understanding that we ll multiply it in before taking the real part Come to think of it let s just drop dealing with the real parts of the equations for our intermediate calculations with the understanding that we ll take the real part at the end after we multiply by e These choices give us a mapping rather than a strict equation E t Eo cos wt E 8 16 I t Ip cos wt gt T Ipet 8 17 Solving the circuit means finding Jp and 6 the two unknowns in this final expression Once you know the current you can figure out the voltage across any device by multiplying the current by that device s complex impedance We now use our generalized Ohm s law V IZ to find lo and 6 for the
20. buttons along the top of the function generator panel fine control is achieved with the ro tary knob on the right The amplitude of the signal is adjusted using a scale button either 0 2V or 0 20V and the nearby rotary knob For now try to dial in a 2V signal at 1 kHz Ask an instructor for help if you are stuck Make sure that the other rotary knobs are all pushed in e g the offset control You should see a signal on channel 2 of the oscilloscope blue It may not be a steady signal i e it may appear to be moving across the screen This is because the trigger is not operating on channel 2 but on channel 1 Q See if you can trigger on the signal from the function generator on channel 2 What happens to the probe comp signal on channel 1 that you were triggering on Q Try adjusting the frequency and the amplitude controls on the function generator Also try the different wave functions Then try the Offset button pull it out and turn it What does it do Physics 124 Lab 4 Spring 2012 40 4 7 Experiment 5 A Warning about Grounds Suppose you were interested in measuring the waveform from the function generator by using the oscilloscope probe rather than the BNC cable There is a subtlety that often confounds the oscilloscope novice the zero of the voltage scale The little clip on the oscilloscope is actually attached to the planet we live on through the oscilloscope ground connection which is the third wire on the plu
21. circuit in which the impedance includes both the resistor and the capacitor Once we have J then we can calculate the voltage across the capacitor by using Ohm s law once again In order to do this we need to have an expression for Z for the resistor and the capacitor For a resistor Z R and the impedance is real For a capacitor however we assert that 5 J Z ah 8 18 where j 1 We use j rather than i here so that we don t end up confusing the symbol with the current We ll derive this expression for the impedance of the capacitor in the future This curious expression for the impedance depends on both the angular frequency w and the capacitance C becoming larger with smaller w Note that it is also purely imaginary Let s see how things go Q Does the fact that the impedance of the capacitor gets big as w gt 0 make sense from the perspective of DC circuits Explain briefly The same rules for series and parallel DC circuits with resistors apply to AC circuits with complex impedances The total impedance of the RC circuit is therefore a Es 8 19 Physics 124 Lab 8 Spring 2012 83 where the first term is due to the resistor and the second term is due to the capacitor and they sum because they are in series Now we can write eV E _ Ew Z R jMW6C wRC j 8 20 We d like to write this expression in the form Ioe so that we can identify the amplitude and phase of the resulting current To
22. eae 1 2 Introduction 4 44 244 ar eee PR EES Ee SS 7 3 Experiment 1 Voltmeter Input Impedance 7 4 Experiment 2 Function Generator Output Impedance 7 5 Experiment 3 Resistor Capacitor Circuit Step change in Voltage 7 6 Additional Activities 000000 ee ee 45 50 53 54 56 58 59 60 64 64 64 65 65 70 70 71 8 Capacitors II 8 1 Questions for Lab Preparation e 8 2 Introductions 2s ape ee oo Aa a aS ee 8 3 Experiment 1 Resistor Capacitor Circuit Sinusoidal Response 8 4 Additional Activities 2 0 002002 a 8 5 Formal Lab Report e 9 Inductors I 9 1 Questions for Lab Preparation e 9 2 Introduction rea dc A ER ee Eo te at 9 3 Experiment 1 RL Circuit Step Response 04 9 4 Experiment 2 RL Circuit Sinusoidal Response 9 5 Experiment 3 RLC Meter 0 0 000 eee ee 9 6 Additional Activities 2 ee 10 Inductors IT 10 1 Questions for Lab Preparation 02000002 eee 10 2 IMTOUCHON cuca eer Se Ace Hd hy ve ee ia Woe poe a i 10 3 Experiment 1 REC Circuit 0 020 02 0 0202020200004 10 4 Formal Lab Report 2 20 20 0200 e A Keeping a Lab Notebook B Formal Lab Report C Uncertainty Analysis 79 79 79 79 85 85 86 86 86 87 90 92 92 94 94 94 99 102 104 108 112
23. err on the side of verbosity and redundancy than to leave out possibly important details Physics 124 General Instructions Spring 2012 8 NEVER ERASE data or calculations from your notebook If you have a good reason to suspect some data for example you forgot to turn on a power supply or a calculation you entered the wrong numbers in your calculator simply draw one line through the data or calculation you wish to ignore and write a comment in the margin It is surprising how often wrong data sets turn out to be useful after all USE PEN to record all of your data so that you will not fall prey to the temptation to erase There will not be any pencils allowed in the lab WRITE ON ONE SIDE of the page only This allows for a blank space to insert graphs or new data and helps with neatness As mentioned above Appendix A gives instructions on how to keep a good lab note book You will be expected to adhere to these guidelines throughout the semester 2 DATA Naturally you will need to pay attention to your data taking technique Throughout the semester you will be learning how to use various types of mea surement equipment sometimes crude and sometimes sophisticated In all cases the quality of your data will depend on your understanding of when and how to use the equipment It is always more important to put care and thought into the setup for a measurement than it is to attain a sometimes deceptively high level of acc
24. field and the direction of propagation So instead of using A for amplitude let s just use E for the electric field In your head you can just replace all the E s with A s Additionally if the manual talks about an electromagnetic wave you can just replace this with the word light Therefore the magnitude of the electric field associated with a beam of light propagating in the x direction may be written as E Fp cos 27 E ft 2 2 This describes a traveling wave with an amplitude Eg and a velocity equal to fA where f is the frequency of the light and A is the light s wavelength Note The intensity of an electromagnetic wave is proportional to the total electric field squared E 2 2 2 Interference Because light is a wave it may exhibit the property of interference Consider what would happen if two electromagnetic waves were traveling in the same direction but with their phases shifted by 180 That is we let one wave have its maximum just as the other is at its minimum The total electric field is just the sum of the electric fields associated with each beam in accordance with the principle of superposition Physics 124 Lab 2 Spring 2012 19 So E E E 2 3 Eo cos 27 5 ft Eo cos 27 G ft 2 4 0 2 5 The field associated with the second beam will exactly cancel the first and there is no net field Since there is no field there is no intensity and hence we would see no li
25. find the norm of a complex number you multiply itself by its complex conjugate and take the square root ee ee 8 21 wRCOY 1 To find the phase shift we need to write I in the form a jb whence the phase shift is given by tan S 1 R 1 Thus 1 t 8 22 an SRC If we were interested in finding I t then we would be all done but in fact we are interested in the voltage across the capacitor We therefore need to use Ohm s law once again just as we would in a DC circuit but this time multiplying the complex current by the complex impedance of the capacitor x VE Pe ee ed j c Vo Zo he 2 o l 8 23 wC V wRC 1 y wRC 1 From this expression we see that E Vo 8 24 e wRC 1 a is the amplitude of the voltage across the capacitor and the overall phase shift is q 9 2 2 The current therefore leads the voltage by 7 2 radians or 90 Recall that tand 1 wRC Having now solved for Veo and dc we must honor our contract for the calculation of Va t Volt RVgel RVepelvttie Voo cos wt dc 8 25 which is the same expression as we had earlier almost Certainly Vco is the same but what about 0 Earlier we found tan c wRC Here we have 6 6 7 2 with tand 1 wRC Are these in fact the same Yes Q Challenging exercise in trigonometry with an elegant geometric solution prove that the two expressions for c are identical Physic
26. give a list of the equipment identifying large pieces of equipment with manufacturer s name and the model For large pieces of equipment record the serial number too With this information you can repeat the experiment with the identical equipment if for some reason you are interrupted and have to the return to the experiment much later Or if you are suspicious of some piece of equipment having this information will let you avoid that particular item Sketch of the Setup It is often a good idea to make a quick sketch of the setup or a schematic diagram for electronics Schematics will be especially helpful in the second semester of physics when you will be connecting various pieces of electronic equipment together in mildly complicated ways Also in optics experiments ray diagrams are useful to keep track of the paths of various light rays Outline of Methods Next give a short paragraph noting the main goal of the experiment and outlining how you expect to carry out your measurements This should not be too detailed since you will probably modify your procedure as you go along But this opening paragraph will help you settle in your own mind what you need to do to get started Particularly as the semester goes on and you develop more and more of the experimental procedure yourself you will find yourself modifying your initial procedure discovering additional variables that should Physics 124 Keeping a Lab Notebook Spring 2012
27. in the resistance The frequency of this current will be the same as the frequency of the oscillatory EMF but one might well expect correctly that the size of the resulting current will be greatest when the frequency of the driving EMF matches the natural resonant frequency of the LC circuit the frequency of natural oscillations that you would predict from Eq 10 8 You might try the analogous experiment with a pendulum constructed from an object tied to a string Hold one end of the string in your hand and swing it at various frequencies Here is another way of coming to the same conclusion If we consider Eq 10 3 and assume Physics 124 Lab 10 Spring 2012 98 that the potential across the capacitor is given by v t V sinwt 10 10 then we see that ilt wCV coswt 10 11 Equation 10 11 tells us two things First the amplitude of the current is related to the amplitude of the oscillating EMF I wCV If we recall Ohm s Law we see that for a capacitor 1 wC plays the role of resistance In the jargon of electronics 1 wC is called the impedance of the capacitor The crucial point is that the impedance of the capacitor varies with frequency For high frequencies for which the current is rapidly oscillating in time the capacitor does not have a chance to charge or discharge very much and its effect impedance in the circuit is small For very low frequencies the capacitor has a very high impedance and prevents very slowly cha
28. level is Taylor Here we will develop a very simple but general method for finding the effects of uncertainties Suppose we want to calculate some result R which depends on the values of several mea sured quantities x y and z R f x y z C 6 Let us also suppose that we know the mean values and standard deviations for each of these quantities Then the uncertainty in R due to the uncertainty in x for example is calculated from GR ZF J 2 da C 7 where the subscript on 6 reminds us that we are calculating the effect due to x alone Note that the partial derivative is evaluated with the mean values of the measured quantities In a similar fashion we may calculate the effects due to dy and 6z N B By calculating each of these contributions to the uncertainty individually we can find out which of the variables has the largest effect on the uncertainty of our final result If we want to improve the experiment we then know how to direct our efforts We now need to combine the individual contributions to get the overall uncertainty in the result The usual argument is the following If we assume that the variables are independent so that variations in one do not affect the variations in the others then we argue that the net uncertainty is calculated as the square root of the sum of the squares of the individual contributions R y eR R 6 R C 8 The formal justification of this statement comes from
29. or the outer shield of the BNC connector Why do you suppose the BNC cable connection is designed this way In this class we are going to make use of the oscilloscope probes which have a BNC plug that goes into a BNC jack You should plug in an oscilloscope probe to channel 1 of the oscilloscope now With the probe installed the voltage difference is measured between the clip at the end of the probe and the alligator clip that dangles down from the middle of the probe This configuration is a little strange but it is standard and has some advantages in some circumstances For these oscilloscopes any time you plug in a probe you should connect it to the probe comp output which generates a square wave signal You must clip the alligator lead to the lower of the two metal connectors and the probe clip to the upper of the two being careful not to short the two metal leads together by accidentally clipping across both leads Once you ve clipped the probe on you should press the button just above it The oscilloscope will then make sure that its internal settings are appropriate to the probe you ve connected DISPLAY Now we re ready to look at the signal on the display You may already be looking at the signal which is fine but follow along with these instructions this first time A quick shorthand method for getting the signal on the screen in a nice way is to press the black AutoSet button in the upper right corner of the fr
30. power supply terminals directly to another of the power supply terminals the price will be a disabled protoboard BNC Connections The two BNC connections on the protoboard can be useful to make connections to external function generators or oscilloscopes You can wire to either the central pin of the BNC cable or to its shield which is also attached internally to ground Variable Resistors There are two variable resistors at the bottom of the protoboard You can wire to either side of the fixed resistor and or to the so called wiper that is used to generate the variable resistance Should you finish these exercises today you can try out these variable resistors and get a sense of how they work see also Sec 5 5 3 Function Generator The protoboard has its own function generator found on the left side The protoboard can generate the same sort of waveforms as can the Tektronix CFG 253 but they are in general inferior in quality We have also found that they tend to fail after a bit of use The controls are fairly self explanatory you can wire either to the signal generator output or to a ground pin For fun you might wire the function generator output to the BNC connection and then use a BNC cable to bring the signal over to the oscilloscope for further examination We will probably not use this function generator so much but it is nice to have on a protoboard if you don t have an external function generator handy 5 6 Experime
31. results This phase of the operation is often mistakenly referred to as Error Analysis In fact the expression for determining the range of uncertainty in a particular measurement is appropriately Uncertainty Analysis Below are some guidelines to help with this process Physics 124 General Instructions Spring 2012 9 2 UNCERTAINTIES The stated results of any measurement are incomplete unless ac companied by the uncertainty in the measured quantity By the uncertainty we mean simply How much greater or smaller than the stated value could the measured quan tity have been before you could tell the difference with your measuring instruments If for instance you measure the distance between two marks as 2 85 cm and judge that you can estimate halves of mm the finest gradations on your meter stick you should report your results as 2 85 0 05 cm More details on uncertainties are given in Appendix C An important if not the most important part of the analysis of an experiment is an assessment of the agreement between the actual results of the experiment and the expected results of the experiment The expected results might be based on theoretical calculations or the results obtained by other experiments If you have correctly determined the experimental uncertainty for your results you should expect your results to agree with the theoretical or previously determined results within the combined uncertainties If your r
32. terference will be halfway between the bright maxima They will also be separated by a Physics 124 Lab 2 Spring 2012 21 distance DA Ay 22 2 11 d If we measure Ay d and D we can in principle measure the wavelength of the incident light In order to perform this experiment a helium neon He Ne laser will be used Lasers create light that propagates only in a particular direction and is monochromatic i e it has only one frequency or wavelength You will determine the wavelength of this laser light from your double slit interference experiment You have been provided with a photographic plate that has a column of double slits For our lab today we will use at least two of these double slits For ease of measurement we recommend the second and third widest double slit pairs on your film note that in the series of double slits the narrowest is actually only a single slit Q For at least these two slit pairs measure the distance d between the slits as follows put the whole photographic plate on an overhead projector to magnify the image Measure the slit separation the image as well as the width of the whole photographic plate and its image at the horizontal line through the slits whose width is being measured Because the slits have a finite width you may want to measure edge to edge and average your results for left edges and right edges to get a value for d Repeat this measurement enough times so that you have a go
33. the inductor You will need to include the following e Include the results from both RL circuit measurements as well as your final RL circuit measurement in which you combine the results from the other two Be sure to include any auxiliary measurements that you made such as the output impedance of the function generator the series resistance of the inductor and the resistances of the two resistors e Include the measurement of L from the RLC meter along with its associated uncer tainty e Include the derivation of the time dependence of the step response for both growth and decay and the time constant 7 e Include and comment briefly on the two RL circuit Bode plots How is the special frequency you find from the Bode plots related to 7 Physics 124 Lab 10 Spring 2012 103 e In the appendix put three phasor plots from the RLC circuit on below and above resonance and two brief comments about the phasor plots e As usual include in the appendix your raw data and fit summaries from IGOR Appendix A Keeping a Lab Notebook Keeping a good lab notebook seems like a simple and obvious task but it requires more care and thought than most people realize It is a skill that requires consistent effort and discipline and is worth the effort to develop Your lab notebook is your written record narrative of everything you did in the lab Hence it includes not only your tables of data but notes on your procedur
34. the potential drop across the resistor Physics 124 Lab 10 Spring 2012 102 IMPORTANT The notation X Y means measure the voltage at point X with CH1 of the oscilloscope and the voltage at point Y with CH2 of the oscilloscope assuming that the MATH channel of the oscilloscope is displaying CH1 CH2 Note also that you will have to measure the phase by hand If you have correctly set the oscilloscope to trigger on the SYNC OUT signal from the function generator you will have little difficulty here From these quantities you can construct a phasor diagram of the voltages across each element of the circuit If Kirchhoff s voltage rule is satisfied at every instant it should be the case that the EMF produced by the function generator is equal to the sum of the voltages across the other three elements Q Repeat these measurements for two more frequencies one above resonance and one below resonance Make sure that the points you pick are not too far away from the reso nance a good spot might be at a frequency at which the voltage drop across the resistor is about one fourth of its value on resonance Q Is Kirchhoff s voltage rule satisfied in every case Q How can the peak amplitude of the voltage across the inductor and or the capacitor be larger than the peak amplitude of the voltage generated by the function generator 10 4 Formal Lab Report The main topic of your formal report is your calculation of the inductance of
35. using the Range button on the multimeter Measure the resistances for the various different range settings Q Why do you think the circuit you used as shown schematically in Figs 5 20 and 5 21 is superior to the circuit shown in Fig 5 23 below Do you think you would have arrived at the same results Q You will want to include a comment that talks about possible systematic effects on your measurement Discuss in particular what might have happened if your test resistance were either really big e g 1 MQ or if it were really small e g 100 Q Can you identify any other systematic effects that might make your measurements less accurate Physics 124 Lab 5 Spring 2012 60 Figure 5 23 Alternate positioning of the ammeter and voltmeter Does this circuit offer any advantages or disadvantages as compared to those in Figs 5 20 and 5 21 5 11 Additional Activities Here are some additional fun things to try if you complete the main laboratory exercises Sec 5 8 above 5 11 1 Body Resistance Q How much resistance does your body have hand to foot Is it different in different places WARNING your skin has a very high resistance but once you prick through the skin your blood is a good conductor and the resistance decreases immensely Don t prick through the skin the battery in the multimeter would produce a current high enough to kill you if your resistance is too low 5 11 2 Circuit Simulator Visit the follow
36. 4 Spring 2012 30 ae VoltAlert E 117 TRUE RMS MULTIMETER O gt Lo Hs ROE imu Rance AUTO V loz OFF y v om A com ep 104 A CATI FUSED C 600v Figure 4 1 The Fluke 117 digital multimeter Input Transducer Voltage Processing Output ADC DAC Experimental physicists make use of both digital and analog electronics in the laboratory We will be focused largely on analog electronics in this class but most modern instruments oscilloscopes signal generators and voltmeters are essentially digital devices For now we can think of voltage or potential as simply a quantity that is measured by a voltmeter Like temperature voltage is a scalar field which means that at every point in space one can assign a value of the voltage in volts Unlike temperature a voltmeter can only tell us the difference in the voltage between two different points This is because voltage is closely related to energy and only differences in energy are meaningful 4 3 Experiment 1 Voltmeter Constant Voltages Our first device is the handheld multimeter which we will be using as a voltmeter today Our multimeter is the Fluke 117 see Fig 4 1 a fairly general purpose device that can measure many electronic quantities Physics 124 Lab 4 Spring 2012 31 The multimeter has three inputs one marked A one marked COM and the third marked VO These ar
37. Physics 124 Lab Manual Amherst College Spring 2012 Contents General Instructions Laboratory Syllabus 1 Optics I Ray Optics 1 1 1 2 1 3 1 4 1 5 Questions for Lab Preparation 2 a Introduction Ba a Bok eh a ee Be A RS Experiment 1 Snell s Law ee Experiment 2 Thin Lenses 0 00002 ee ee eee Additional Activities 2 Optics II Wave Optics 2 1 2 2 2 3 2 4 2 5 2 6 Questions for Lab Preparation 2 00000000 Introduction RG hae Nea PE EG Re ee Yo Experiment 1 Young s Double Slit Experiment 2 Polarization e eee eee Additional Activities 2 ee ee Formal Lab Reports soe ceso io a else Mad Re ee 11 12 12 12 13 14 15 17 17 17 3 Introduction I Electrostatics Sel CITO dut a vale esa ah te an aS ae A a ies Sa SE eh 3 2 Experiment 1 Tape Electrometer 2 2 a eee eee 3 3 Experiment 2 Versorlum e 3 4 Experiment 3 Electrophorus 0 0 0002 ee eee 3 5 Experiment 4 Wimshurst Machines 00000004 3 6 Experiment 5 Van de Graaf Generator 0 00 0000 eae 3 7 Experiment 6 Kelvin Water Dropper 2 004 4 Introduction 11 Equipment 4 1 Questions for Lab Preparation e 42 Introduction 2 2 24 e0use Soke E ee ae Goa a eee aA 4 3 Experiment 1 Voltmeter Constant Voltages
38. Rg _ 1 Re Ri Re 6 1 Q In what sense are these resistors in parallel in these two circuits This is one of the hardest concepts to wrap one s mind around in introductory electronics and may be the hardest question in today s lab After answering this question to the best of your ability in your lab book feel free to draw your instructor into a conversation about it Experimentally one can measure the output impedance in the following way provided the output impedance is much smaller than the voltmeter input impedance this technique will not work for the circuit in Fig 6 2 First measure the output voltage Vin as shown in Fig 6 6 the subscript th is conventional see below Then connect a resistor R in Rin Vin Figure 6 6 Measuring the supply voltage Vin the measured V Vin provided the input impedance of the meter is much larger than Rin parallel to the meter as shown in Fig 6 7 The new measured output voltage V is related Rin Figure 6 7 Measuring the output impedance Rin the measured V can be related to the output impedance of the supply by Eq 6 3 below to Vin R and the output impedance Rip by Vin V IR R Rn R 6 2 Q Show that Eq 6 2 is correct Physics 124 Lab 6 Spring 2012 70 You can easily invert this to find z i R 6 3 As a special example note that if you adjust R until V Vin 2 then Ri R One can neverthele
39. WX Lwav E Ta 9 17 where the sums are over all N measurements i 1 N and the weights wi are the reciprocal squares of the corresponding uncertainties 1 wi 9 18 05 The uncertainty in tway is 1 y 9 19 Vd wi where again the sum runs over all of the measurements i 1 N As an example consider two measurements 11 10 1 and z2 12 2 The uncertainties are 01 1 and o2 2 and the weights are w 1 and wa 0 25 The weighted average is 10 1 12 0 25 wav 10 4 9 20 i 1 0 25 ey and the uncertainty in this weighted average is 1 Cway 0 89 9 21 av TED 921 I would therefore quote the measured value as x 10 4 9 Lab 10 Inductors II 10 1 Questions for Lab Preparation 1 Why is an LC circuit like a harmonic oscillator 2 If an LC circuit is like a harmonic oscillator what is and RLC circuit like 3 What is the resonant frequency of an RLC circuit In this lab we will put together the three linear passive devices with which we have worked in the past few weeks the resistor the capacitor and the inductor Remarkably whenever capacitors and inductors appear together in a circuit Kirchhoff s voltage rule yields the equation for a simple harmonic oscillator One important consequence is that the circuit will display the phenomenon of resonance which you will come to appreciate today 10 2 Introduction 10 2 1 Capacitors Recall that the voltage
40. across a capacitor is proportional to the charge on one of the plates v t salt 10 1 We ll use lower case letters to represent quantities that change as a function of time If we indicate the current t that flows as the capacitor is discharging we have the following situation Fig 10 1 Physics 124 Lab 10 Spring 2012 95 Figure 10 1 Since the current results in the charge q t changing with time we have with the signs used in the figure i t 10 2 C my 10 3 10 2 2 Inductors Now consider the potential difference for an inductor From Lab 9 we know that Oz ap 10 4 Eqs 10 3 and 10 4 suggest that capacitances and inductances are in some sense comple ments of one another e For a capacitor the current is proportional to the time derivative of the voltage e For an inductor the voltage is proportional to the time derivative of the current This situation might remind you of the relationship between velocity and position for the motion of a mass on a spring or the motion of a simple pendulum In some sense it is the interplay of position and velocity that leads to the interesting behavior of springs and pendulums 10 2 3 LC Oscillations a simple case The following circuit is the simplest imaginable one containing both an inductor and a capacitor Fig 10 2 Physics 124 Lab 10 Spring 2012 96 Figure 10 2 Suppose that at t 0 the capacitor is somehow charged as s
41. and voltage across them We will also become familiar with circuit diagrams resistors the use of the handheld multimeter as an ammeter and ohmmeter and the use of the so called breadboards for testing circuits If you finish all of these things there are a number of auxiliary exercises that you can try that will further develop your electronic skills At the end of the two weeks you will be asked to write a lab report on aspects of the laboratory which will require some analysis and the use of IGOR Physics 124 Lab 5 Spring 2012 44 5 3 Definitions The primary quantities that we will work with are current voltage and resistance In this section we ll develop these terms qualitatively to give you a sense of what is happening at the microscopic scale 5 3 1 Current The motion of charge is called an electric current in analogy to the flow of fluid Consider the flow of a fluid like water through a pipe Fig 5 3 1 The current through this pipe P l l l 1 1 1 1 A _ _ __ ___ _ _ _ Figure 5 1 Flow of fluid across a plane P is characterized by the amount of fluid measured by volume flowing past a plane P which is perpendicular to the pipe per unit time Similarly if charge is flowing through a wire the electrical current is characterized by the amount of charge flowing through P per unit time The symbol for electric current is J and its units are units of electric curr
42. any fit with 0 5 lt x2 lt 2 a good fit Physics 124 Uncertainty Analysis Spring 2012 121 C 5 The User s Guide to Uncertainties The rules can be derived using the results of Sec C 3 1 C 5 1 Addition and Subtraction For addition and subtraction one should combine the absolute uncertainties in the measured quantities Typically one calculates the final uncertainty by adding the uncertainties in quadrature which means taking the square root of the sums of the squares For example the quadrature sum of the three uncertainties 621 6x2 and 0x3 is OXtotal y 621 6x2 6x3 C 21 The following simple example shows how to propagate uncertainties for the case of a simple sum Example 1 Alice measures the lengths of the sides of a triangle finding s 2 9 0 2 cm s2 4 2 0 4 cm and s3 4 9 0 1 cm What is the perimeter of the triangle and Alice s uncertainty in its value Answer The equation for the perimeter is P s 52 83 12 0 cm The final uncertainty in its value is found by summing the individual errors in quadrature SP 5s1 682 683 0 5 cm C 22 The value Alice should quote for the perimeter is therefore P 12 0 0 5 cm C 5 2 Multiplication and Division For multiplication and division uncertainties propagate in a slightly different manner One must first calculate the fractional uncertainty of a quantity If some value q has an associated uncertain
43. apacitors I Apr 11 Capacitors II x Apr 18 Inductors I Apr 25 Inductors IT x May 2 No lab Lab 1 Optics I Ray Optics 1 1 Questions for Lab Preparation 1 Explain what the critical angle 0c is 2 There is an unlabeled lens on the table You need to find its focal length Using only the room lights overhead the lens and the table how might you estimate the focal length 3 What does an image distance of infinity mean Is the image really at infinity Almost all microscope objectives have an image distance of infinity why 1 2 Introduction One way to model the behavior of light is to think of it as a simple ray A single ray an arrow just represents the direction the light is propagating Sometimes we can model light as just a single ray for instance a laser pointer traveling through the air and reaching a projection screen might be modeled as a single ray Other times we might model light as two rays or multiple rays A light bulb may be modelled as multiple rays since it sends light out in all directions Modeling light as two or more rays is also useful when modeling a laser beam incident on a lens Even though all the rays are initially propagating in the same direction the curvature of the lens will refract each ray differently Knowing where multiple rays end up not just one ray tells you more about the properties of the lens
44. are general purpose digital oscilloscopes of a kind that has become nearly ubiquitous in research laboratories around the country Unlike the digital multimeter you must plug these oscilloscopes in although there are now battery powered versions that are liberated from the grid 4 5 1 Getting Started TURNING IT ON One you have plugged in the oscilloscope you can turn it on by pressing the power button on the top of the device After a few moments of self testing the main display might look something like in Fig 4 2 The main display graphs the voltage appearing on one or both of its input channels as a function of time This is a major improvement over the voltage averaging provided by the multimeter but with this great power comes great responsibility as attested by the many controls on the front panel In this lab we will spend time on the most essential oscilloscope Technically this is not quite an average but the square root of the average of the squares of the instantaneous signal The reasons for this will be explored further in a future lab Physics 124 Lab 4 Spring 2012 34 l Tektronix TDS ange Save Rocali Measure P g2 93 F gl 5 53 v 5 iS 3 E de Nur uy _ uy a 3 oDODODO 300 v A 300V CAT Sat o 0 5 2722 001 Figure 4 2 The Tektronix 2001c Figure from the Tektronix user manual E KEN 300 V AN 300v CAT II Ext trig Fi
45. as opposed to the charge on the capacitor which is described by Val t QW n cos wt dc 8 10 TH ROP where c is the phase shift for the voltage across the capacitor and is given by Eq 8 6 Note that the amplitude of Va t is given by Eo 1 WRC Veco 8 11 Both the amplitude of this oscillating potential difference and the phase relative to the emf source vary with frequency the product wRC is the controlling factor in both cases It is often convenient to talk about the period T of the oscillating signals T 27 w If T gt RC then dc gt 0 and Vo t and E t have almost the same amplitude and oscillate in phase At the other extreme we have T lt RC then gt 7 2 in this case Vo t has a small amplitude and lags behind E t by 90 Q Using Eq 8 10 draw a graph of Vo t as a function of time for three special cases RC equal to 0 1 100 8 3 2 Theory Il Complex Impedances Let s repeat our analysis of the preceding section by using complex impedances If you have not seen complex numbers before or even if you have there is a brief summary of their properties in Appendix D We are familiar with Ohm s law V IR The complex version of Ohm s law is V IZ where the tilde denotes a complex number The point of using complex numbers is that we can use DC circuit analysis to analyze AC circuits simplifying matters immensely and as a by product avoiding a whole
46. at well abstractly with a first sentence making the most general statement about the manuscript topic followed by a couple more sentences that bring the general to the specific After these introductions you can turn to what it is that you have done which might take a few sentences Then you should proceed to a brief set of concluding statements perhaps moving from specific back to general Writing a good abstract is one of the more difficult things that one does in science It needs to be terse but still convey the excitement of the problem and the contributions that are made in the manuscript Practicing physicists tend to put the writing of the abstract off until the end so that the entire manuscript is laid out and the story line is clear The point of getting into the habit of writing abstracts is not because we want your grade to hinge on how well you can craft them but to get experience with a common form of written scientific expression Indeed abstracts are written for conference proceedings meetings workshops Physics 124 Writing a Formal Lab Report Spring 2012 111 poster sessions and the like not just for manuscripts It can be one of the most common forms of scientific writing Go ahead and give it a try All along you might have been playing with a working title for the manuscript Once you have written the abstract you are ready to make it permanent The title needs also to draw attention to the manuscript since i
47. at your results do not turn out the way you had anticipated and to try to understand what went wrong You should NEVER ERASE data that appear to be wrong In fact it s better for you to use ink for lab notebooks anyway It is perfectly legitimate to state that you are going to ignore some data in your final analysis if you have a justifiable reason to suspect a particular observation or calculation Finally In the midst of these bewildering ground rules it is important to keep in mind that the laboratory exercises can be both challenging and rewarding As with any fundamentally collaborative effort the amount of fun you end up having during these sessions will depend to some degree on the openness humor and good will that you bring with you Enjoy Physics 124 Laboratory Syllabus Spring 2012 11 Laboratory Syllabus We will meet Wednesdays from 2 5pm for the laboratory This semester we will begin and end promptly at the appointed hours Please plan to spend the entire period in class All of the labs must be completed to receive a passing grade For the four sessions marked with a Ye below a formal report will be required Date Lab Reading Jan 25 No Lab Feb 1 Optics I Ray Optics Feb 8 Optics II Wave Optics A Feb 15 Exam Feb 22 Introduction I Electrostatics Feb 29 Introduction Il Equipment Mar 7 Resistors I Mar 14 Resistors II Mar 21 Spring Break Mar 28 Exam Apr 4 C
48. bes the overall output intensity as a function of the angle 0 between the alignment axes of the two polarizers In fact Malus s law says that the output intensity out of the second polarizer and the input intensity out of the first polarizer are related by the factor cos Why does this make sense Q Get a qualitative feel for the effect of crossed polarizers by looking through a stack of two polarizers and rotating one relative to the other You ll probably need to do this in a well lit room What do you notice about the light intensity Do your observations make sense in light of the above discussion 2 5 Additional Activities 2 5 1 Single Slit Experiment In your double slit experiment cover one of your slits so that light is only passing through a single slit What happens to the pattern you observe on the screen Measure the distance to the first minima Does this measurement make sense Use the discussion in section 35 5 of Ohanian s Physics for Scientists and Engineers to help you Physics 124 Lab 2 Spring 2012 23 fixed polarizer second polarizer detector rotates Figure 2 2 2 5 2 Laser Polarization Is the He Ne laser light polarized Using the polarizer make up an experiment to test this question What do you find 2 5 3 Intensity due to Angle between Polarizers On the optical bench set up on the laser a polarizer holder a rotatable polarizer and the detector as shown in Fig 2 2 The first p
49. can be adjusted with the amplitude control on the function generator Note also that this battery has an internal resistance denoted here by r that must be taken into account Q Connect the circuit depicted in Fig 7 6 which is essentially the same as the circuit in Fig 7 5 except that the generic source of emf depicted there is replaced by the function generator Keep in mind that the ground terminal of the function generator must be connected to the ground terminal of the oscilloscope if you do not do this you will see noise with an amplitude of a few millivolts For C use the capacitor you have been provided and record its identification number 1 kQ Scope Ch 1 Scope Ground Figure 7 6 Circuit for measuring the capacitance of a capacitor Set the function generator to produce a large amplitude square wave at about 400 Hz to begin with Observe the waveform then increase and decrease the frequency and see what happens Adjust the time base of the oscilloscope and its vertical sensitivity as desired Then set the function generator at a convenient frequency at which you can see a complete decay of one of the exponential curves but sufficiently spread out so that you can measure it We recommend using the vertical position controls to put the level which the exponential decay is approaching on the bottom line of the screen Measure vc as a function of t for about 10 values of t Be sure to record the uncertainty associate
50. culation and so on may turn out to be correct after all and in any case you want to keep a complete record of your work even the false starts If you believe that a calculation for example is wrong it is better to draw a single line through it and make a note in the margin than to erase the calculation You can always make things neat in your report Sequences of Measurements You will often be performing experiments in which you have two independent variables Usually in such experiments you fix the value of one independent variable and make a series of measurements working through several values of the other variable Then you change the value of the first variable and run through the measurements with the other variable again then you change the first independent variable again make another set of measurements and so on It s usually easier to set up this sort of sequence in your notebook as a series of two column tables or three columns with Remarks rather than a big rectangular grid Title each table with the value of the independent variable that you re holding fixed and keep the format of all of the tables the same Comment on Results Once you have completed the experiment and performed any necessary calculations in the notebook you should look back to the main goal and write down to what extent it was achieved If for example you were making a measurement of g you should include a clear statement of the value of
51. d with each measurement You can use the oscilloscope cursors and you need not record the uncertainty in time which is well controlled by the oscilloscope This is the only measurement you will do by hand Then use the oscilloscope to capture the waveform in its entirety and save it to your memory stick Now change the 1 kQ resistor shown above to a 100 2 resistor and repeat your measurement Do the same for a 10 kQ resistor Physics 124 Lab 7 Spring 2012 78 7 5 3 Analysis Analyze and graph in IGOR at least one of the decays of the sort just described before leaving the lab Show your graph and your results to one of the instructors The theory developed in Section 7 5 1 says that the voltage across the capacitor is of the form ve Ae VOLE 7 7 where A D and E are fit parameters You can obtain the capacitance C from the fit parameter D if you know R Before coming to lab next week plot all three of your data sets and fit them to the functional form of Eq 7 7 Conventionally the plot with the discrete points should have vertical error bars attached to each point For the oscilloscope traces create line plots rather than xy plots and error bars are not typically required Finally find the capacitance and propagate the error correctly 7 6 Additional Activities If you have finished early continue on to the next lab Remember to read the lab and complete the questions for lab preparati
52. differential equation with the initial condition that q 0 at t 0 since the capacitor is initially completely discharged is q WC 1 gaam 7 3 Q Verify that Eq 7 3 provides a solution of the differential equation Eq 8 1 when v Vo by evaluating the derivative of Eq 7 3 and substituting the result and v into the differential equation Note that the product RC determines the time scale required for the capacitor to charge This product is called the time constant for the circuit Te RC Unfortunately we cannot measure the charge stored in the capacitor directly but we can measure the potential difference across the capacitor say with the oscilloscope The electric potential across the capacitor is given by da a W 1 z eFC 7 4 Q Using Eq 7 4 sketch a graph of vc as a function of t Now suppose that the capacitor is completely charged i e the potential difference across it is equal to Vo Then suddenly the potential supplied by the function generator drops back to OV Let s reset our time axis so that this new change occurs at t 0 In this case the charge stored in the capacitor and the potential across the capacitor decrease with time q VoCe RO and 7 5 volt Yye O Note once again that the product RC sets the time scale for the capacitor to discharge Q Using Eq 7 6 sketch a graph of the electrical potential across the capacitor as a function of time 7 5 2 Exper
53. ds what is connected to what using the variable power supplies warning this is a grounded power supply so don t just connect your oscilloscope alligator clip willy nilly in other words don t con nect the alligator clip to anything that isn t directly connected to the ground black terminal on the protoboard at the expense of a new protoboard using the function generator using the BNC connectors to make connections to the Tektronix function generator and the oscilloscope Lab 5 Resistors I 5 1 Questions for Lab Preparation 1 What are devices that obey Ohm s Law called Look up their color codes Do you see a pattern 2 Describe the drawing in Fig 5 19 What does each symbol and letter mean 3 Will the light bulb obey Ohm s Law 5 2 Introduction Last week you learned how to use several instruments the handheld multimeter used as a voltmeter the oscilloscope and the function generator In this lab you will be looking at the relationship between the quantity we worked with last week voltage and the flow of electric charge through a variety of materials current Neither of these concepts will be covered in lecture for several more weeks We will therefore adopt a more phenomenological point of view By the end of this lab which will stretch over two weeks you should be comfortable with the notion that electronic circuit elements are defined by the relationship they implement between the current through them
54. ds of de termining experimental uncertainties As a rule of thumb we usually expect the different methods of determining the uncertainty to agree within a factor of two or three EXAMPLE Suppose that five independent observers measure the distance between two rather fuzzy marks on a piece of paper and obtain the following results d 5 05 cm dy 5 10 cm d3 5 15 cm d4 5 20 cm ds 5 10 cm If the observers were using a scale with 0 1 cm markings method 1 would suggest an uncertainty estimate of about 0 05 cm Method 3 yields a mean value d 5 12 cm and for the standard deviation 0 057 cm 0 06 cm We see that in this case we have reasonable agreement between the two methods of determining the uncertainties We should quote the result of this measurement as 5 12 cm 0 06 cm or 5 12 cm 1 C 3 Propagation of Uncertainties In most measurements some calculation is necessary to link the measured quantities to the desired result The question then naturally arises How do the uncertainties in the measured quantities affect propagate to the results In other words how do we estimate the uncertainty in the desired result from the uncertainties in the measured quantities Physics 124 Uncertainty Analysis Spring 2012 116 C 3 1 General Method The general treatment of the propagation of uncertainties is given in detail in texts on the statistical analysis of experimental data A particularly good reference at this
55. e The units of resistance or dynamic resistance are units of resistance are ohms Q lohm 1V A 19 Devices for which R is a constant are said to be ohmic and in this special case we can write R VJI Ohm s Law AV IR R constant where the constant R is the resistance of the device Devices that obey Ohm s Law are called resistors Keep in mind that most things do not strictly obey Ohm s Law but many of the devices that we will see in the laboratory do 5 4 Schematic Diagrams Since electrical circuits can get enormously complicated an elaborate system of symbols has been developed to help us draw the circuits we study Physics 124 Lab 5 Spring 2012 46 5 4 1 Wires and Nodes The symbol for a wire is a line Wires are conductors for our purposes every point along a wire is at a constant voltage Charges can move freely in a wire so wires support currents Most wires are surrounded by some kind of insulating layer such as plastic or rubber but since air is a good insulator they might well be bare Usually wires have the insulation so that they can be bundled together without fear of having the conductors touching one another accidentally such an unintended connection is often called a short circuit Most devices have conducting metal leads to which wires can be attached No special symbol is used for these kinds of connections the line just connects to the terminal of whatever
56. e and your data analysis as well With practice you will become adept at sharing your time fairly between conducting the experiment and recording relevant information in your notebook as you go along You want all this information in one place for three main reasons and these reasons continue to be valid even after you leave the introductory physics laboratory That is even or rather especially practicing scientists keep lab notebooks First your lab notebook contains the information you will need to write a convincing report on your work whether that report is for a grade in a course or a journal article Second you may need to return to your work months or even years after you have finished an experiment It is surprising how often some early experiment or calculation is important in your later work Hence you need a reasonably complete account of what you have done Third your lab notebook is also the source to which you turn in case someone questions the validity of your results You may have heard of the famous David Baltimore case of alleged scientific fraud in which the lab notebook of one of Baltimore s collaborators has been the subject of careful scrutiny Your notebook therefore serves two purposes that may not be completely compatible with each other On one hand you should write things down pretty much as they occur and before you have a chance to forget them so that you have a complete record of your work in the lab
57. e at the bottom of the multimeter and are a kind of connector known as banana jacks You will often see the corresponding banana plug that fits into the jack It is not entirely clear what the origin of the name is since neither the plug nor the jack looks much like a banana There are two probe leads that fit into the banana jacks one of which is red and the other black The black lead should be plugged into the COM jack and the red lead into the VO jack This is the usual way in which the multimeter is used we will see in a later lab how to use the third multimeter jack Let s turn on the multimeter using the multifunction position knob Turn it clockwise two clicks until the selector points towards V Do not confuse this setting with the one marked V The multimeter is now a voltmeter measuring the voltage difference between the red and black leads Specifically it is measuring Vea Vblack in volts Unless you are moving the leads around or touching them to something in the room it should read a number that is close to zero Keep in mind that positive voltage readings mean that the red lead is at a higher voltage than the black lead and that negative voltage readings mean that the black lead is at a higher voltage than the red lead Q If you want to read the voltage from a battery do you connect to COM or VQ At this point the voltmeter can measure voltages that are reasonably constant i e those that do not change q
58. e result usually an average result of the set of measurements is 5 1 cm but given the conditions under which the measurements were made the fuzziness of the points and the refinement of our distance measuring equipment it is our best judgment that the actual distance might lie between 5 0 cm and 5 2 cm Incidentally an alternative shorthand way of expressing this uncertainty looks like this 5 1 1 cm C 2 where the number in parentheses represents the uncertainty in the last digit Feel free to use this form in your lab work C 1 2 Relative or Percent Uncertainty We might express the same measurement result as 5 1 cm 2 C 3 Here the uncertainty is expressed as a percentage of the measured value Both means of expressing uncertainties are in common use and of course express the same uncertainty An aside on significant figures The number of significant figures quoted for a given result should be consistent with the uncertainty in the measurement In the example above it would be inappropriate to quote the results as 5 cm 0 1 cm too few significant figures in the result or as 5 132 cm 0 1 cm too many significant figures in the result Some scientists prefer to give the best estimate of the next significant figure after the one limited by the uncertainty for example 5 13 cm 0 1 cm The uncertainties since they are estimates are usually quoted with only one significant figure in some cases e g
59. e voltages and currents of Ohm s law and your choice of resistors from your resistor box to revise your circuit to produce such voltages across the devices Your graphs should have the origin somewhere in the middle of the sheet so that an J V graph would look something like that shown in Fig 5 22 Figure 5 22 How your I V graph might look Yours will no doubt properly include error bars however You can do a preliminary analysis of the resistances by hand with a calculator We will do a more thorough analysis next week using IGOR Q Determine the resistances of the Ohmic devices Is the lamp Ohmic If not what can you say about it For example at what voltage and current does the lamp begin to light up What is its dynamic resistance 5 9 Experiment 4 Ohmmeter Y One of the functions of the Fluke 117 multimeter is its ohmmeter function which measures resistance directly between the two leads AFTER you have completed the data collection described above you can measure the resistance of the two mystery devices using the ohmmeter function Write down these numbers in your lab notebook and include these results in your report Consult the multimeter manual for advice on how to determine the uncertainty in these measurements Once we have completed the full analysis it will Physics 124 Lab 5 Spring 2012 59 be interesting to see how these numbers agree with one another What is the resistance of the light bulb
60. electric field aligned with the polymers and re radiating the light that does not We can think of light shining directly on a polarizer as having two components of electric field one aligned with the polymers and one perpendicular to that direction The third dimension perpendicular to the plane of the polarizer doesn t matter and need not be considered here The light that makes it through the polarizer is the perpendicular component We say that the transmitted light is polarized because it comes out having an E field in a single well defined direction that we might mark on the polarizer with an arrow regardless of what its E field orientation was to begin with This is good news for those of us who wear sunglasses since the polarizer acts somewhat as an E field filter and cuts the transmitted light intensity way down Q Why does the third dimension perpendicular to the plane of the polarizer not matter To get polarized light in the first place we can pass unpolarized light light with many random orientations of the E field vector such as the light from our bulb through one polarizer It turns out that in that case half of the incident light intensity makes it through If you shine that polarized light onto a second polarizer the polarization of the output light is determined by the second polarizer as discussed above But the intensity of that light depends on the relative polarization angles of the polarizers Malus s Law descri
61. ely the resonant frequency for the case C 0 01 uF Now find the resonant frequency by any method you like for several i e at least five other C values in the range 0 001 uF to 0 22 uF Don t forget to assign an error bar to your measurement of fo for each C Then measure the values of C with a capacitance meter including uncertainties in that measurement as well Your points will have bidirectional error bars Calculate L Don t forget that 27 1 10 3 4 Phasor Diagram The challenge in this section is to measure the voltage across the individual devices function generator resistor capacitor and inductor without having to reconfigure the circuit You Physics 124 Lab 10 Spring 2012 101 can get a sense of the problem by asking yourself how you could measure the voltage across only the inductor If you hook up your oscilloscope probe to one side of the inductor you measure the voltage across the inductor and the resistor with respect to ground and if you hook up your oscilloscope probe to the other side of the inductor you are measuring the voltage across the capacitor and the function generator with respect to ground If you try to move the ground terminal of the oscilloscope probe to one side of the inductor you end up shorting out components and inadvertently reconfiguring the circuit What to do First of all notice that the voltage across the inductor is actually the difference in the voltages measured on ei
62. ent are amperes amps A 1 ampere 1 C s 1 A 5 3 2 Voltage To get water to flow through a pipe there must be a pressure difference between one end and the other The analogous concept with electrical circuits is potential difference The symbol for electric potential difference is V and its units are units of electric potential are volts V 1 volt 1 J C 1 V Hooking both ends of a hose up to the inlet and outlet of a water pump will cause water to flow through the hose A battery plays a similar role for electric circuits Attaching one end Physics 124 Lab 5 Spring 2012 45 of a wire to the positive end of a battery and the other to the negative end will cause charge to flow through the wire Please take my word for this and refrain from discharging the batteries in this way Batteries are special kinds of electrical pumps because they pretty much always maintain the same potential difference between the positive and negative ends no matter what the current 5 3 3 Resistance The ratio between the voltage difference between two points and the current flowing between those points is called the resistance R where R 5 1 51 We often take the limit where the ratio between AV and J is a derivative i e R dV dl which is the slope of the I V curve this is a bit like defining the velocity v as the slope of the curve of x t curve Strictly speaking this derivative is called the dynamic resistanc
63. er At this point you should be thinking of an overarching narrative structure inhabited by the figures you can then turn from figure to figure in a logical order as you develop that narrative You may find that you need to revisit some of the figure drafts you have created in the previous step in order to make the narrative flow from one figure to the next It is in this step that you can be most creative you need not simply proceed with the tradi tional sections one after the other such as theory apparatus methods data analysis In fact we would prefer that you moved away from this pedantic and rigid ordering scheme whenever possible It often one might even say usually makes sense to mix up these tradi tional categories Your lab report can even be a little bit chatty i e not stuffily formal if that helps you write it Do not however make it into a dialogue between two of your favorite physicists or anything like that The point is to come up with some narrative arc that ties together your work and leads easily from topic to topic Be as creative here as is consistent with this requirement You should also work in the results of the analysis in the main text Often your results themselves can be presented as a figure but other times you need to quote a few numbers and their uncertainties A table can work well for this Technically these are not figures but they serve the same sort of purpose as figures s
64. er than charge what is the gravitational voltage Physics 124 Lab 4 Spring 2012 32 There are two important concepts that one should keep in mind with any measurement device in electronics The first is the concept of precision Roughly speaking precision refers to the number of significant digits that one obtains when measuring some quantity The meters we have generally give four digits with some uncertainty specified in the user s manual in the last digit When you measured the voltage across the cell you could in principle record all four digits which means that your least significant digit was a millivolt mV and your precision is a few mV The second concept is dynamic range Dynamic range refers to level of precision that is possible for a given magnitude of the voltage input When we turn on the multimeter the dynamic range is automatically set to give you the highest precision measurement possible As the magnitude of the input voltage difference increases however the ability of the multimeter to give a highly precise measurement diminishes For instance if you were measuring a potential difference of 8 V the meter could not give you a precision of a few mV instead it would give you a precision of a few tens of mV This is one of many trade offs that occur in electronics We ll observe this effect directly a little later on in the lab today for now it is important just to have a sense that the precision of
65. est effect on the uncertainty of F If we wanted to decrease the uncertainty of our results we ought to work hardest at decreasing the uncertainty in f Finally let s combine the uncertainties using the square root of the sum of the squares method From that computation we find that we ought to give F in the following form F 2 383 0 037 x 10 C 17 or F 2 38 0 04 x 10 C 18 in the appropriate units Note that we have adjusted the number of significant figures to conform to the stated uncertainty As mentioned above for most purposes citing the uncer tainty itself to one significant figure is adequate For certain high precision measurements we might cite the uncertainty to two significant figures C 4 Assessing Uncertainties and Deviations from Expected Results The primary reason for keeping track of measurement uncertainties is that the uncertainties tell us how much confidence we should have in the results of the measurements If the results of our measurements are compared to the results expected on the basis of theoretical calculations or on the basis of previous experiments we expect that if no mistakes have been made the results should agree with each other within the combined uncertainties Note that even a theoretical calculation may have an uncertainty associated with it because there may be uncertainties in some of the numerical quantities used in the calculation or various mathematical approxima
66. esults do not agree with the expected results you must determine why Several common possibilities are the following a You underestimated the experimental uncertainties b There is an undetected systematic error in your measurement c The theoretical calculation is in error d e Some combination of the above The previous measurements are in error or Sometimes these deviations are real and indicate that something interesting has been discovered In most cases unfortunately the explanation of the deviation is rather mundane but nevertheless important Remember that small deviations from expected results have led to several Nobel prizes Analysis After Lab You will prepare four formal lab reports this semester One for each of the major laboratory topics that we will cover optics resistors capacitors and inductors For formal reports you are to prepare a somewhat longer written account of your experimental work These reports should include a complete description of the experiment and its results They should be compiled in a word processor and turned in according to the schedule in the syllabus You should pay special attention to the clarity and conciseness of your writing In fact if we find that your report would benefit from rewriting we may ask you to submit a revised version of the report before a grade is assigned to the report Guidelines for preparation of formal lab reports are
67. for very high precision measurements the uncertainties may be quoted with two significant figures Physics 124 Uncertainty Analysis Spring 2012 114 C 2 Determining Experimental Uncertainties There are several methods for determining experimental uncertainties Here we mention three methods which can be used easily in most of the laboratory measurements in this course C 2 1 Estimate Technique In this method we estimate the precision with which we canmeasure the quantity of interest based on an examination of the measurement equipment scales balances meters etc being used and the quantity being measured which may be fuzzy changing in time etc For example if we were using a scale with 0 1 cm marks to measure the distance between two points on a piece of paper we might estimate the uncertainty in the measured distance to be about 0 05 cm that is we could easily estimate the distance to within 3 of a scale marking C 2 2 Sensitivity Estimate Some measurements are best described as comparison or null measurements in which we balance one or more unknowns against a known quantity For example in a Wheatstone bridge experiment one determines an unknown resistance in terms of a known precision resistance by setting a certain potential difference in the circuit to zero We can estimate the uncertainty in the resulting resistance by slightly varying the precision resistor to see what range of resi
68. g along with its uncertainty Be aware that there are often secondary goals as well to become familiar with a particular physical system or measurement technique for example Comment on your success in attaining these goals as well This serves as a statement of conclusion and gives you the chance to make sure the lab was completed thoroughly and to your satisfaction Appendix B Formal Lab Report In your report we would greatly prefer if you used the style of the American Journal of Physics AJP We will provide a copy of a recent AJP paper that you can use as a guide Note that you should write your report according to your own sense of taste the style that we are referring to is really about the things that should be included in your paper such as title affiliation abstract introduction text figures analysis conclusions and bibliography if required You can include additional technical details e g raw data tables and fits in an appendix at the end if you think they are relevant or illustrate something better than the figures you ve included in your report There are many ways to write a scientific paper The order in which many practicing physicists construct a manuscript is given below 1 Figures and captions 2 Main body of text 3 Analysis including commentary on systematics 4 Conclusions 5 Introduction and 6 Abstract and title front matter Unsurprisingly we will now explore each of these
69. g that goes into the wall Similarly the outer conductor shield of the BNC cable is attached to the planet through the same connection The function generator has the same property Thus you can use the probe in only one way you have to connect the probe clip to the center conductor of the BNC cable and the alligator clip to the BNC shield You should try this out If you reverse the connections you will measure something quite different You can try this out too but be aware that this is a special case if you were to do this to a grounded power supply you might damage it On the other hand if you measure the voltage across something that is not plugged into the wall you needn t worry as much about getting this connection wrong Measuring the voltage across the battery is a good example In general you want to make sure of the following e Make sure that the alligator clips of any oscilloscope probes are connected to the same point in a circuit e Make sure that the shield connection of the function generator is also connected to this point and e Make sure that this point is also the ground connection point of a grounded power supply Note that this problem also doesn t arise with the handheld multimeter which can make differential voltage measurements without worrying about which lead is attached to which part of the circuit Of course this is because it does not plug into the planet through a wall connection If you re w
70. g the frequency back and forth through fo 10 3 2 Bode Plot Q Our first measurement task is to create a Bode plot of the voltage across the resistor Physics 124 Lab 10 Spring 2012 100 as a function of frequency You can use the measurement functions of the oscilloscope to facilitate this experiment you are interested in the relative phase and the relative amplitude of the voltage across the resistor with respect to the voltage generated by the function generator Plot the relative voltage on a log log plot and the relative phase on a semilog plot Please note 1 Use logy not loge 2 For the phase plot indicate the relative phase in units of 7 radians or in degrees 3 Include error bars where appropriate and 4 Indicate on the plot the location of the resonant frequency IMPORTANT If the oscilloscope displays a next to a measurement that means it is not properly measuring that quantity You may have to adjust the horizontal and vertical controls to make the go away 10 3 3 Calculation of L In this part we will calculate the inductance of the inductor L by measuring the resonant frequency fo as a function of the capacitance C We know from the theoretical discussion above that 1 i 10 15 uf which means that we can find L by plotting wo against C fitting to the appropriate func tional form with a parameter that can be related to L with a minimum of fuss Q You will know at least approximat
71. ght This phenomenon is called total destructive interference Alternatively we could imagine two electromagnetic waves having the same amplitude and phase Then the total field would simply double E Fi E2 2 Epcos 27 E ft 2 6 Because the intensity of light is proportional to the square of the electric field the intensity would quadruple This phenomenon is called constructive interference 2 2 3 Diffraction Diffraction and interference seem to go hand and hand in introductory texts Sometimes this classification blurs the definition of diffraction in student s minds Diffraction is just the spreading of a wave with a fixed beam diameter Typically the Huygens Construction is employed to show how this spreading would occur See Section 34 1 in Ohanian s Physics for Engineers and Scientists for more details The divergence angle of the beam 0p mathematically defines how much diffraction or spreading will occur The divergence angle for a particular beam with wavelength A and diameter D is given by 24 This shows that as the beam diameter decreases the amount of diffraction increases Q In the ray optics lab we talked about a lens focusing a beam of light to a point This was referred to as the focal point of the lens Is the focal point real After learning about diffraction do you think we can focus a beam to an infinitesimal point Physics 124 Lab 2 Spring 2012 20 2 3 Experiment 1 Young
72. give a more detailed procedure or if you depart from the procedure in the manual Numerical Data When recording numerical data keep your results in an orderly table You should label the columns and indicate the units in which quantities are measured You should also indicate the uncertainty to be associated with each measurement If the uncertainty is the same for a certain set of data you can simply indicate that uncertainty at the top of the column of that data You will need at least two columns one for the independent variable and one for each dependent variable It s also good to have an additional column usually at the right hand edge of the page labeled Remarks That way if you make a measurement and decide that you didn t quite carry out your procedure correctly you can make a note to that effect in the Remarks column For example suppose that you realize in looking at your pendulum data that one of your measurements must have timed only nine swings instead of ten If you indicate that with say 9 swings you could justify to a suspicious reader your decision to omit that point from your analysis Physics 124 Keeping a Lab Notebook Spring 2012 107 Do not Erase Record your data comments and calculations in ink rather than in pencil That way you avoid the temptation to erase data that you think are incorrect You never should erase calculations data comments etc because the original data cal
73. good idea to do so if only as a reminder that they exist But it is really important not to connect the signal output of a function generator to the ground of an oscilloscope or the output of a grounded power supply to the ground of an oscilloscope or so on these are activities that will generally require us to buy new pieces of equipment even though they are nominally protected against such abuse 5 5 Equipment 5 5 1 Ammeter Your Fluke 117 handheld multimeter can be turned into an ammeter by turning the selector knob to the A position and critically moving the red lead to the leftmost banana input marked A If you fail to do the second step you will never measure any current Unlike a voltmeter which can be attached and detached from a circuit without much penalty the ammeter measures a quantity of charge per unit time that passes through it and therefore requires a reconfiguration of the circuit when it is attached and detached Specifically this means that a segment of wire has to be disconnected from the circuit and the ammeter leads attached to that segment and to the portion of the circuit to which the segment was attached This is a rather disruptive act but it may help to remember that ammeters must be used in this way and if they are removed the circuit must be reconnected where the ammeter leads were attached This will soon become obvious if it is not already see Fig 5 17 5 5 2 Resistors You wil
74. gure 4 3 The three inputs to the oscilloscope Figure from the Tektronix user manual controls As you go on in life you will use more and more of the available features until it becomes your best friend INPUTS There are three inputs to the oscilloscope see Fig 4 3 The two signal inputs are the first two on the left marked and 2 we may talk about the third input marked Ext Trig later These inputs are so called BNC connectors There are special BNC cables that can be plugged into these connectors you should try this out now to get the sense of how these connectors work Note that they push on and then turn to make the solid connection The voltage difference on a BNC cable is measured between the central pin and the outer shield There are also adapters that convert from BNC to banana in which case there is a red connector and a black connector These are mostly similar to the connectors on the handheld multimeter with one important exception that we will discuss in a little while Q Take a minute to look at the connectors Suppose you connected a battery to a BNC cable through the banana BNC connector such that the end of the battery went to the The letters BNC stand for Bayonet Neill Concelman after the style of the connector and its inventors Physics 124 Lab 4 Spring 2012 35 black banana jack and the end went to the red banana jack At the BNC output would the end of the battery be the central pin
75. hanges just as Newton s First Law inhibits sudden velocity changes Similarly the reciprocal of the capacitance plays the role of a spring constant Since you know the solution to Eq 10 9 simple harmonic oscillation with angular frequency w yk m sometimes called for reasons that will become clear in this lab the resonant frequency you can immediately predict the solution to Eq 10 8 Those solutions are oscillatory in time with frequency w 1 V LC 10 2 4 RLC circuit with a sinusoidal EMF What has been omitted from the preceding nearly rigorous theory is the role of the re sistance that is inevitably present Whereas capacitors store energy in electric fields and inductors store energy in the magnetic fields resistors dissipate energy producing heat If a small amount of resistance is present we will observe a damped sinusoidal oscillation a sinusoid of steadily diminishing amplitude If a large amount of resistance is present the overshoot characteristic of oscillation may not even be seen Let us therefore add some resistance to the circuit shown in Figure 10 2 and to compensate for the loss of energy we will add a sinusoidal EMF of adjustable frequency Now our circuit would look like this Fig 10 3 Figure 10 3 The EMF can drive an oscillatory current in this circuit alternately clockwise and coun terclockwise maintaining the amplitude of such a current at a steady value in spite of the energy being dissipated
76. hat type of wavefunction it is Trade back and forth until you feel very comfortable measuring frequency and amplitude of various signals Y Try holding the oscilloscope probe in your hand What signal do you see What is its frequency Does its amplitude depend on where you are or what part of your body the probe clip is touching Where do you think the signal might be coming from Q At various stations around the room will be mystery signals on BNC cables Use your oscilloscope to characterize each of these signals Ask an instructor if you are correct 4 9 Additional Activities 4 9 1 Advanced Oscilloscope and Function Generator Capabilities We will assume in future labs that you have a basic familiarity with the devices you have used today If you have completed everything then you should try to explore some of the more advanced features of the oscilloscope and function generator since there are many more possibilities than we are able to cover in detail in this manual The user manual that comes with the oscilloscope may be of particular use 4 9 2 Protoboard As the final part of this laboratory exercise you will play with the protoboards which include a grounded power supply function generator BNC connections speakers and so forth We will be doing more with the protoboards next week for now these are the interesting things to understand Physics 124 Lab 4 Spring 2012 42 cutting and stripping wires wiring the protoboar
77. here agree with that obtained in Part I While making these measurements observe the characteristics of the image produced Is the image larger or smaller than the original object Is the image inverted Q Repeat the last exercise with a different lens that has a different curvature and hence focal length Which lens has the larger f For which lens are the surfaces more curved i e depart further from being planes Which lens is stronger Q Take one of the negative lenses Estimate its focal length by using the method of parallax or some other method to ascertain the position of virtual images produced by the lens We use parallax all of the time to determine which of two objects is closer than the other You will want to apply the method of parallax to the image of a distant object viewed through the lens and a closer object viewed outside of the lens 1 5 Additional Activities 1 5 1 Intensity of Light Here you will explore the phenomenon that light from a point source propagates outward uniformly in all directions about the source Without any optical interference such as lenses mirrors or interfaces light will propagate outward from a point source in a straight line filling a spherically symmetric volume Because the total energy per unit time produced by the source is constant the intensity energy per unit area per unit time must fall off with distance from the source In fact since the area of a spherical shell s
78. hown As in an RC circuit current will start to flow but unlike the case of the RC circuit q t will not simply decay monotonically to zero it will overshoot The capacitor will become oppositely charged then current will flow clockwise and so on The system continues to oscillate spontaneously much like the undamped oscillations of a mass on a spring Here is the mathematical argument Use Kirchhoff s Voltage Loop Rule and add the voltages around the loop equating the sum to 0 dilt alt L 0 10 5 dt C The sign is correct though the proof of that claim is omitted the sign conventions for i t and q t are those adopted in Fig 10 2 We want to focus our attention on the charge q t so we note that for the discharge of the capacitor i t Eo 10 6 and taking a derivative yields 10 7 dt dt Using Eq 10 7 in Eq 10 5 gives us d q t 1 L 10 8 a al 10 8 But this is an old friend the simple harmonic oscillator equation describing a mass on a spring whose equation of motion is dx t dt Comparing Eqs 10 8 and 10 9 you might well choose to describe L as an inertia mass term for a circuit Just as a mass on a spring overshoots the equilibrium position because m kz t 10 9 Physics 124 Lab 10 Spring 2012 97 of its inertia because of Newton s First Law the charge through a coil tends to keep going Faraday s law inhibits sudden current c
79. ies reversed Then try to measure the resistance of a small resistor lt 10 Q using both meter topologies Compare your results to those obtained with the ohmmeter What happens if you are touching the resistor while you are measuring its resistance For that matter what is the resistance between two points on your body 5 11 6 Temperature The resistance of a resistor can depend on other external quantities such as temperature You can see this effect by measuring the resistance of a resistor and then heating it up by touching it with your fingers In which direction does the resistance change If the resistance gets bigger smaller with increasing temperature then we say that it has a positive negative temperature coefficient One annoying fact with resistors is that their resistances typically do depend on temper ature and to make matters worse they are self heating When current passes through a resistor the resistor gets hotter This can then change the resistance and the properties of the circuit in which it is embedded Part of good electronic design is to make sure that these resistance changes do not upset the circuit properties too much In high end audio systems and other critical analog circuits one often spends a lot of money on special re sistors that have a vanishing temperature coefficient so that the effects of temperature are entirely mitigated These zero tempco resistors can be thousands of times more expensive
80. iment We wish to verify that the equations developed in Sec 7 5 1 describe the response of an RC circuit to step changes in the applied emf We will then use that response to determine Physics 124 Lab 7 Spring 2012 76 the capacitance of the capacitor from measurements of the time constant 7 RC of the circuit For a circuit with a large time constant i e a minute or more it would be possible to study the charge and discharge processes by placing an ordinary high resistance voltmeter across the capacitor and reading the potential difference every few seconds Alternatively the vertical input to an oscilloscope might be connected across the capacitor The time constants of the circuits you are provided with however are only a fraction of a second and these methods would be hopelessly inadequate Instead you can use the trigger controls of the oscilloscope to examine repeated dischargings of the capacitor through the resistor The oscilloscopes are capable of doing some sophisticated averaging of these traces as well although that is not required for our purposes today As with most oscilloscope measurements the goal is to achieve good sensitivity over the interval of time in which you are interested We are interested here in seeing most of a trace from its maximum voltage to close to its minimum voltage with as much vertical resolution as possible Your best results will be obtained by manipulating the horizontal and vertical
81. included in Appendix B Physics 124 General Instructions Spring 2012 10 Grading You must complete all of the labs to pass Physics 124 If you have to miss a lab because of illness family difficulties or other legitimate reasons please let me know in advance whenever possible so we can arrange for a make up time You will receive a grade for each of the writeups in your lab notebook and each formal report according to the grading policies listed on the syllabus Intellectual Responsibility Discussion and cooperation between lab partners is strongly encouraged an indeed often essential during the lab sessions However each student must keep a separate record of the data and must do all calculations independently In addition laboratory partners are expected to share equally in the collection of data The use of any data or calculations other than one s own or the modeling of discussion or analysis after that found in another student s report is considered a violation of the Honor Code We wish to emphasize that intellectual responsibility in lab work extends beyond simply not copying someone else s work to include the notion of scientific integrity i e respect for the data By this we mean you should not alter fudge or make up data just to have your results agree with some predetermined notions Analysis of the data may occasionally cause you to question the validity of those data It is always best to admit th
82. ing web site http phet colorado edu en simulation circuit construction kit dc Construct some of the circuits that you have seen today with the circuit toolkit You will probably have developed some intuition about these circuits does what you see coincide with what you expect 5 11 3 Resistors in Series and Parallel You may have heard that resistors Ri and Ra connected in series add according to the equation R R R2 where R is the effective resistance of the combination For resistors Physics 124 Lab 5 Spring 2012 61 R and R in parallel the effective resistance is given by 1 R 1 R 1 R2 We can use Kirchhoff s rules in order to see why resistances add according to these rules We will often encounter two resistors in series as shown in Fig 5 24 We say that the resistors are in series because the current has no choice of path if it flows through one resistor it must serially flow through the other Later we will show that this new device behaves exactly like a resistor with an effective resistance R R Rg Notice that R is always greater than R or R Because wires typically have a small resistance they can usually be ignored because of this rule WSS Ss gt MM Ri Ra R Figure 5 24 Two resistors in series behave as a single resistance R Ri Ro The other way to hook up two resistors is in parallel as shown in Fig 5 25 In this case the current will divide itself up among
83. ings we probably won t use too much since they are designed for digital electronics We ll now describe some of the aspects of the protoboard that we will use beginning with the big connection panel Wiring The connection panel is the big white breadboard that occupies the center of the protoboard Note that there is a large grid of connections broken into groups of five columns across many rows in height and a number of smaller vertical connections groups of five vertically in two columns One makes a connection to the breadboard by pushing a wire or the lead of a component into the metal connector inside one of the grid elements The breadboard itself has a number of implicit connections between the grid elements which you can then exploit to build up connections between the components and the wires Let s start with the grid elements that consist of five columns across Each group of five is connected together such that a wire or component lead attached to any one of these grid elements is connected to all of the other grid elements and any wires or leads that are attached to them in that group This allows you to make connections between five wires component leads without clipping anything together it s as simple as pushing the wire or lead into the breadboard The grid elements that are in two columns are connected to one another vertically One might think that the grid elements are connected vertically in groups of five anal
84. it of Fig 7 2 Q Build the second circuit Fig 7 3 on your protoboard using the Tektronix function generator as the source of emf Set up the function generator to produce square waves with a frequency of 400 Hz Set the function generator VOLTS OUT in the out position 0 20 V P P The two buttons next to it should also be out The square wave voltage seen on the scope will be smaller in the second circuit than in the first Suppose that you adjust R so that the observed voltage amplitude in the second circuit is half what it is in the first Use that idea to find r For R use a multiturn 1kQ resistor which you can later disconnect from the circuit and measure with an ohmmeter We will need this value of r for the other measurements in this lab 7 5 Experiment 3 Resistor Capacitor Circuit Step change in Voltage Our primary challenge is to understand the charging and discharging a capacitor through a resistor Physics 124 Lab 7 Spring 2012 74 Consider the following circuit 7 4 consisting of a time dependent emf source V t a resistor R and a capacitor C connected in series I t V t C Volt Figure 7 4 Simple time dependent circuit with a resistor and capacitor In what follows we will adopt the engineering convention of using lower case letters to denote time dependence hence v V t i I t and so forth Applying Kirchhoff s Voltage Law to the circuit gives us the following equatio
85. ith your fingers or anything else Physics 124 Lab 3 Spring 2012 26 What do you observe when you bring the tapes near one another Does the effect change if you allow the tapes to touch one another If so how 3 2 2 An Electrometer Prepare the tapes again in the usual way This time stick them carefully at the edge of a table so that they are sticking out horizontally i e parallel to the table surface Try rubbing an insulating that is nonmetal object such as your plastic pen with the fabric of your clothing Bring the object up to the tapes again be careful not to touch unintentionally What is the effect of the object on the tapes What happens if you bring instead the rubbing object e g your clothing near the tapes Repeat this experiment several times with different combinations of rubbing and rubbed objects What patterns emerge from your data If you are careful about handling the tapes you can bring them near charged objects rather than the other way around You can try this in the other experiments described below 3 2 3 Which is Negatively Charged We have called the tapes T and B which is a perfectly fine convention for their charges If we want to communicate with physicists elsewhere however we might want to use a slightly more standard convention and call them positive and negative or and Ben Franklin chose to call the charge on a glass rod after it had
86. ktronix CFG253 see Fig 4 5 generates only three different types of waveforms square sinusoidal and triangle selected by a set of buttons in the upper right corner of the device For now let s work with a sinusoidal waveform The output of the device appears on a BNC connection on the front panel There is also a SYNC connection that s used to drive the oscilloscope trigger if you want For now let s ignore it Go ahead and connect a BNC cable between the function generator output and the oscilloscope input on channel 2 Since we are not using a probe we want to make sure the calibration is correct To do this press the blue button and then cycle through the softmenu selection marked 10x until you get it to read 1x The 1x setting is appropriate for signals that do not come from oscilloscope probes Our oscilloscope probes are all 10x but you can always make sure for any particular probe by doing the probe check steps described earlier in this manual Physics 124 Lab 4 Spring 2012 39 Figure 4 5 The Tektronix CFG253 function generator Unfortunately there is no automatic way to make the oscilloscope choose the 1x setting when you plug in a cable so you will have to remember to do that yourself when you switch between a probe and a BNC cable The two main parts of the function generator besides the type of waveform are the fre quency and the amplitude of the signal You can select the frequency using the row of
87. l Figure 5 14 should help to make the grid element connections explicit A word of advice about making connections try not to have wires and components dangling all over the place when you make circuits Instead shorten component leads as necessary and try to neatly route the wires around Try not to clip alligator clips all over the place either you can make connections using the breadboard Finally you should try color coding your wiring red yellow and blue wires for power supply voltages black for ground and other colors as appropriate Although a color coding scheme may seem silly when you only have a few wires it generally proves extremely useful when you have many many wires to worry about Power Supplies There are three power supplies on your breadboards and one ground Two of the power supplies can be used as variable batteries Although you can connect banana plugs Physics 124 Lab 5 Spring 2012 53 or even wires using the screw connection on these jacks it is generally easier to make connections to the power supply voltages from the four rows of connections just above the connection panel Each of these rows is called a bus and is connected to one of the power supply terminals You can connect these buses to individual connection groups on the panel or you can connect the buses to the vertical bus strips on the panel as you prefer What is most important is that you not connect one of the
88. l have two boxes of resistors that you can use without having to test that they are in fact resistors The resistances are given approximately by their color codes We will make sure that you have a card that will help you decipher the resistor color code but eventually Physics 124 Lab 5 Spring 2012 51 it is probably in your best interest to memorize it since it doesn t take very long to do so and it repays itself in terms of how much time you spend looking at the card Besides how many people do you know who pull out a card of resistor values at a party I thought so 5 5 3 Protoboard Most professional physicists first test a circuit idea on a protoboard which is a multifunction board that allows many impermanent and easily changed connections to be made The main action on a protoboard occurs on the big white connection panel that dominates the center of the protoboard Components can be attached to the protoboard by pushing their leads into the little sockets Many of these sockets are connected to one another directly and with the addition of jumper wires one can make a dizzing array of possible connections There are many other functions scattered about the periphery of the protoboard including power supplies the multicolored banana jacks at the top of the panel a rather crappy function generator BNC connections variable resistors a loudspeaker and several switches and light emitting diode LED arrays Some of these th
89. l probably want to derive the expected voltage across the resistor Vp t using either of the theoretical methods developed in the earlier sections of this manual Note the continuing importance of the product wRC 8 5 Formal Lab Report The formal report should include the step responses of last week analysis of the three curves and the measurement of C Please include as an Appendix to your report the Bode plots for the sinusoidal response and brief answers to the various questions scattered throughout these pages The latter should be written in the manner of a problem set but should be brief and to the point Also remember to include as an Appendix your data tables and the information from your fits Lab 9 Inductors I 9 1 Questions for Lab Preparation 1 Compare the inductor and capacitor How are they similar How are they different 2 Complete the questions from section 9 3 1 You may bring your answers with you to the lab and tape them into your lab notebook 9 2 Introduction We have seen that when a constant current flows through a wire there is a potential difference across the wire proportional to the current That is AV IR For a capacitor we know that the charge Q on the capacitor is related to the potential difference by Q CV where C is the capacitance of the capacitor and depends on its geometry Another way of writing this relation is in terms of the current V a fia 9 1 where we implicitly used
90. l softmenu by pressing the button We won t use that menu so much in this class Physics 124 Lab 4 Spring 2012 37 The controls both horizontal and vertical are used to zoom in on the waveform to get the best signal The same kinds of concerns about dynamic range and precision apply here as do to the multimeter controls 4 5 3 Setting the Trigger The final major oscilloscope control system is the Trigger system The trigger determines when the oscilloscope causes the voltage traces to update on the screen and defines the zero of time If you have a periodic signal then it is important that the zero of time line up from trace to trace otherwise the traces overlap badly and it will be difficult to see what is going on with the signal There are two principal parts to the trigger the trigger level controlled by the knob and the trigger slope control found in the Trigger softmenu after pressing the Trig Menu button The trigger level controls the voltage level that triggers the oscillo scope you can see this on the screen as you move the Level knob around Whenever the input voltage is the same as the trigger level the scope will trigger update the trace Of course for a typical periodic waveform there might be two points in the waveform at which the trigger level equals the voltage at the input one in which the input voltage is decreasing through the trigger level and one in which the input voltage is increasing through the
91. lass today so that we can turn to the important work of analysis of your data next week If you run into trouble with the analysis next week never fear this is why you have two weeks to work on this so that you can get it right for the report Physics 124 Lab 5 Spring 2012 57 As mentioned in the introduction devices are characterized by their I V relationship that is if you apply a voltage across a device a certain current will usually flow through it If that I V relationship is linear then you have a resistor that obeys Ohm s law If it is not then you have some other kind of device Our goal in this set of experiments is to characterize the J V curves for three devices a mystery red device a mystery white device and a lamp At least two of these devices are resistors Ohm s law is not meant to be surprising and you are not proving it here indeed one does not ever prove such a phenomenological relationship one instead determines whether a particular device in the main obeys Ohm s law View your work in this section as an exercise in measurement of resistance with the goal of accurately and precisely determining the resistance of two mystery devices and the dynamic resistance of a third the light bulb Q Build the circuit of Fig 5 20 that will allow you to measure the V relationship for the three devices You can use one multimeter as a voltmeter and the other as an ammeter Use your circuit to obtain abou
92. lds OR y 01 dy 0 12 The traditional rule for handling an additive relationship says that we should add the two absolute uncertainty contributions We see that the traditional method overestimates the uncertainty to some extent Exercise Work out the result for a multiplicative functional relationship R f x y xy Compare our method with the traditional method of adding relative uncertainties Example Suppose we have made some measurements of a mass m a distance r and a frequency f with the following results for the means and standard deviations of the measured quantities m 150 2 0 1 r 5 80 0 02 f 52 3 0 4 Note that we have omitted the units and hence lose 5 points on our lab report From these measured values we want to determine the best value and uncertainty for the following computed quantity F mrf C 13 The best value is computed by simply using the best values of m r and f F 2382875 2 we ll tidy up the number of significant figures later on Physics 124 Uncertainty Analysis Spring 2012 118 Let s use our partial derivative method to find the uncertainty First let s determine the effect to to m F jr Z dm rf 5m 1586 C 14 om Next we look at the effect of r F Or r mf dr 8217 C 15 And finally the effect of f is given by OF F af of 2mr fof 36449 C 16 We see immediately that the measurement of f has the larg
93. ltmeter above Physics 124 Lab 6 Spring 2012 68 Q Why does choosing the ouput impedance of one block to be much smaller than the input impedance of the the following block ensure that the signal is not attenuated It is important therefore that we understand a little about these two types of impedance Theoretically one calculates the output impedance of a circuit by connecting a wire to the output of the device and then imagining raising the potential on that wire by some amount AV In so doing you will be forced to supply some additional current Af such that Kirchhoff s rules are still obeyed The ratio AV AT is the output impedance of the circuit Q What is the output impedance of the power supply attached to the series 10 Mohm resistor Fig 6 2 Q Slightly harder What is the output impedance of the following circuit Ri lt Ra Figure 6 4 What is the output impedance of this circuit Q Similar What is the input impedance of the following circuit Again imagine increas ing the potential on a wire connected to the input and discover what additional current you would need to supply Ri Ra Figure 6 5 What is the input impedance of this circuit Curiously you should have found that the output impedance and the input impedance are both what you would get if you calculate the parallel resistance of the two resistors often Physics 124 Lab 6 Spring 2012 69 written R Ra Ri Re Ri
94. may be written E I t cos wt 9 15 Uero with defined above The output voltage is the voltage across the resistor which is simply ER AA EET cos wt 0 9 16 If this analysis does not seem straightforward it might be of some interest to you to calculate the output voltage across the resistor R using the trigonometric techniques developed in the Capacitor lab Physics 124 Lab 9 Spring 2012 92 9 4 2 Experiment Q Create Bode plots for the circuit with the 1kQ resistor Recall that the first Bode plot is the log log plot of the fractional amplitude Vout Vin vs frequency and the second Bode plot is the semilog plot of relative phase vs frequency Include these two plots in your report 9 5 Experiment 3 RLC Meter Q Measure the inductance of your inductor with the special RLC meter How does this measurement compare to the measurement you made previously 9 6 Additional Activities If you have finished early please move on to the next lab Remember to read the lab and complete the questions for lab preparation first Physics 124 Lab 9 Spring 2012 93 Appendix A Note on Weighted Averages I am quoting this from Taylor An Introduction to Error Analysis 2nd Ed Sausalito University Science Books 1997 p 177 If x1 2 Ty are measurements of a single quantity x with known un certainties 01 09 0n then the best estimate for the true value of x is the weighted average
95. n dq 1 R 7 1 Het GA 7 1 where we have made use of the relationship between the potential difference across the capacitor vc and the charge q stored on one of the capacitor plates Dividing through by R gives a differential equation for the charge q v iR dq v 1 We will be studying two different types of behavior described by this equation the response to step changes in the emf this week and the response to sinusoidal changes in the emf next week 7 5 1 Theory For the first type of behavior we will have a source of emf e g a function generator that jumps very quickly between two voltage values Without loss of generality we ll call one Vo and take the other to be 0 V In fact the function generator may jump quickly between Vo 2 and Vo 2 taking 0 V to be ground Since only differences in potential matter these two pictures are equivalent for our purposes Now let s consider the following scenario Suppose that the voltage provided by the function generator has been 0 V for a long time What long means will become apparent in a moment Then we know that the capacitor will be completely discharged Next let s assume that the function generator output suddenly jumps to the value Vo and stays at that value Let s call the time at which that jump occurs t 0 In the differential equation Physics 124 Lab 7 Spring 2012 75 for q we treat the emf value as a constant and the solution of the
96. n the top of the Kelvin water dropper Try not to get water everywhere Which side becomes positively charged and which side negatively charged Does it change each time you run the experiment Can you force it to go one way rather than the other See if you can figure out how it works Does anything happen to the water streams when the machine becomes charged Lab 4 Introduction 11 Equipment 4 1 Questions for Lab Preparation 1 How are voltage and temperature alike and unalike 2 Do the voltmeter and oscilloscope measure the same thing 3 What is a protoboard 4 2 Introduction The basic idea of electronics in physics is to take some observable quantity pressure temperature position and the like and convert it into a voltage This conversion takes place in a transducer which might be a microphone or a kind of thermometer or strain sensor In analog electronics this voltage is processed by amplification and filtering and then sent to an output device which might be an oscilloscope voltmeter or loudspeaker Input Transducer Voltage 4 Processing Output In digital electronics the voltages are encoded into bits by an analog to digital converter or ADC Once digitized the signal can be processed and stored in a computer or possibly sent to an analog output device after passing through a digital to analog converter DAC Physics 124 Lab
97. nd a current source as opposed to voltage source usually has a large output impedance Note that replacing the 100 Q resistor with a 1 kQ resistor or even a 10 kQ resistor would result in essentially the same current through each Q Show that the current source works as advertised theoretically of course since you won t be able to produce the kinds of voltage required with your protoboard 6 4 2 Output Impedance Obviously if you want to build a voltage supply you don t want to include a 10 MQ resistor in series with the output What is less obvious is that all power supplies do include a small less than a few ohms series resistance For function generators this series resistance can actually be fairly large tens to hundreds of ohms This ubiquitous series resistance is called the output impedance of a circuit The importance of input and output impedances extends beyond signal sources and mea surement devices Electronic devices are usually built out of interconnected blocks where the signal enters one block is processed and then is passed to the next block Input Transducer Amplifier gt Filter H Output In order not to reduce attenuate the signal one requires that the output impedance of a block is small compared to the input impedance of the following block You saw this dramatically in the simple case of the power supply with the 10 MO resistor and the vo
98. next abrupt change of the square wave The oscilloscope measures the voltage across R1 which is proportional to the current thus from what you see on the scope you can measure the desired time constant Measure the waveform for exponential decays when R 100 Q and R 1 kQ Download your data for subsequent analysis 9 3 3 Analysis Q You now have data for two 7 measurements one for each of the two different resistances At this point you will want to find the two values of L with their uncertainties from the time constants you measure Remember that the R in the theoretical expression for T includes not only R but also the output impedance of the function generator r as well as the resistance of the coil itself Rz A more complete circuit diagram taking these impedances into account is shown in Fig 9 5 Be sure to combine your uncertainties appropriately to come up with a final value for L see below Physics 124 Lab 9 Spring 2012 90 E E R Inductor Coil R To Oscilloscope Function Generator Figure 9 5 Figure 9 6 9 4 Experiment 2 RL Circuit Sinusoidal Response Now we ll look at the response of the circuit to sinusoidal inputs of varying frequency Recall that the response may be completely characterized by the Bode plots one for the normalized amplitude as a function of frequency and one for the phase as a function of frequency If you do not remember the details of the Bode plot
99. nging currents from flowing in the circuit Secondly there is a phase difference of 90 between the current and the potential difference for a capacitor Now let s look at an inductor using Eq 10 4 Suppose that the current varies sinusoidally with time i t Isinwt 10 12 Then Eq 10 4 tells us that u t wLI cos wt 10 13 Again we see that there is a 90 phase difference between the current and the potential across the inductor We also see that the product wL gives us the impedance of the inductor This impedance is high at high frequencies because the inductor strongly opposes the rapidly changing current For low frequencies the inductor has almost no effect Look again at Fig 10 3 A capacitor has an impedance of magnitude 1 wC and an inductor has an impedance of wL and there is a 180 phase difference between the potential across the inductor and the potential across the capacitor At some intermediate frequency at which the two impedances are of equal size the maximum amplitude of current can flow This frequency dependent response is called resonance The frequency where the cancela tion occurs is called the resonant frequency denoted fo and the corresponding angular frequency is denoted by wo From our arguments we predict that the resonant frequency satisfies the following condition 1 L 10 14 m0 wol Physics 124 Lab 10 Spring 2012 99 CAUTION wo fo because 27 1 Q Supp
100. ngth from this week Please see the Appendix on formal lab reports for more details Lab 3 Introduction I Electrostatics 3 1 Introduction This lab is really a brief menu of experiments designed to get you thinking about the properties of charged objects and their interactions with other objects both charged and uncharged Since another goal is to become familiar with working in the laboratory this is not meant to be particularly strenuous Have fun Pay close attention though and you may be surprised by some of the things you see 3 2 Experiment 1 Tape Electrometer The equipment for this lab is remarkably simple two roughly 3 cm long pieces of transparent tape Fold a little bit of the ends of the tapes over themselves to make a little handle that won t stick to things This will make it easier to pull the tapes apart and off of surfaces In addition you will need some insulators to rub and some materials to rub them with and a few assorted objects such as magnets and paper clips 1 will provide some of these things others you may be carrying around as everyday items in your backpacks 3 2 1 Preparing the Tapes On one of the two tapes mark the letter B for Bottom and on the other the letter T for Top Stick the two tapes together sticky side of tape T on the non sticky side of tape B Then holding the tapes by their handles pull them apart quickly Try not to touch the tapes excessively w
101. nments related to these preparations in advance of the laboratory meeting Execution During Lab 1 LABORATORY NOTEBOOK A laboratory notebook is one of the most important elements of executing the experiment You will need to keep a step by step record of what you ve done how you ve done it and in what order Obtain an inexpensive permanently bound notebook spiral notebooks and binders are not acceptable for recording your laboratory data your analysis of the data and the conclusions you draw from the measured results along with any other relevant comments BE NEAT The notebook is an informal record of your work but it must be suffi ciently neat and well organized so that both you and the instructor can understand exactly what you have done It is also advantageous for your own professional de velopment that you form the habit of keeping notes on your experimental work notes of sufficient clarity that you can understand them at a later time Developing good lab notebook technique requires consistent effort and discipline skills that will be of great value in any professional career If you become a research scientist you will often while writing reports or planning a new experiment find yourself referring back to work you have done months or even years before it is essential that your notes be sufficiently complete and unambiguous that you can understand exactly what you did then In keeping a laboratory notebook it is better to
102. nt 1 First Circuit Q Measure the voltage between the red and black banana jacks also known as terminals of the power supply You should measure 5 V Now try another measurement between the yellow terminal and the black terminal What happens when you adjust the knob labeled Physics 124 Lab 5 Spring 2012 54 V Try this yet a third time for the blue and and black terminals of the power supply using the V knob Q As a first exercise set the positive supply V to 6 V with respect to the black banana jack which is at ground Plug your light bulb into the connection panel and then route some wires from the power bus to the light bulb If the light goes on you are doing great If not you might want to reassess your connections If you continue to be stuck ask an instructor 5 7 Experiment 2 Kirchhoff s Rules 5 7 1 Theory Current Rule Kirchhoff s name is associated with two rules The first Kirchhoff s current rule ex presses a property of charge namely that it is locally conserved The sum of all currents into a junction is equal to the sum of all currents out of the junction If we have a junction into which the currents J and I gt flow and out of which 13 flows then we know by Kirchhoff s current rule that Ig I Io 5 2 If this weren t the case then charge would build up at the junction After a while we d know if that were happening By way of analogy consider your heart with bl
103. o we ll include them here Physics 124 Writing a Formal Lab Report Spring 2012 110 As you are writing the text you might want to think about a working title for your paper as well B 0 3 Conclusion Summarize your results in this section Sometimes you can indicate future research direc tions but in this introductory course you needn t worry too much about doing that B 0 4 Introduction In the introduction you should provide some context for your manuscript and its results It should answer the question What is important about this research Why should I read this paper What are you going to tell me about Like the abstract see below the introduction should take you from the more general to the more specific until you seamlessly move into the main text of the manuscript Of course the grader will read your paper without having to be convinced by the intro duction but generally speaking the most commonly read parts of a paper after the title and the abstract are the introduction and the conclusion Together these two parts of the paper should provide the takeaway message the remainder of the paper includes the wonderful narrative arc and technical details for the truly interested reader B 0 5 Abstract and Title Finally you should include an abstract The goal of an abstract is to draw a potential reader in by explaining both what it is that you have done and why it is important Abstracts often begin somewh
104. od idea of the uncertainty in d Q Shine your laser beam onto one of the double slits that you have measured Place a screen at large distance D at least a couple of meters away from the double slit You should observe several points of constructive and destructive interference To get maximum sensitivity here it is important that the ambient light be dim Measure the distance between successive minima To get the best precision in this measurement measure the interval over several successive minima and divide by the number of intervals From your measurement determine the wavelength of the He Ne laser Be sure to include an estimate of the uncer tainty in your measurement CAUTION You will see successive brightening and dimming of the maxima as you move away from the center This is an interference effect associated with the finite slit widths Do not confuse it with the double slit interference pattern Repeat the last exercise for your second slit pair Obtain a second value for the He Ne wavelength along with the uncertainty Q Are your two answers the same Explain any differences Which do you have more confidence in Which has greater separation between maxima What do you think you would see for d 2 m Physics 124 Lab 2 Spring 2012 22 2 4 Experiment 2 Polarization Certain materials such as the polymers in our polarizers have the ability to respond to an incident electric field by absorbing all of the light that has an
105. ogous to the horizontal groups of five that we just considered but this is not true In fact the Physics 124 Lab 5 Spring 2012 52 Figure 5 14 The colored lines indicate the connections made by the breadboard The horizontal connec tions are shown in blue the vertical connections in red grid elements are connected to all of the vertical grid elements in its own column Well this is not precisely true the upper half of the grid elements in a vertical column are all connected together and the lower half of the grid elements in a vertical column are all connected together but they are not connected to each other unless you do this explicitly with a wire The purpose of these vertical grid elements is to bring power supply voltages and ground vertically along the breadboard without having to use long wires to do so You can bring other signals along these vertical grid elements too if you wish there s no rule that says you have to make power supply connections to them That s just conventiona
106. olarizer should be aligned so as to pass the inherently polarized laser light beam Direct this light through the second polarizer and adjust its orientation so the transmitted intensity is a maximum The transmission axes of the two polarizers are now parallel to one another Adjust the distance between the second polarizer and the detector so that the measured intensity does not exceed 300 mV on the multimeter for this maximum transmitted intensity Record the measured intensity V every 10 by varying the angle between 90 Because these polarizers are only partially effective you may notice a baseline offset For your data subtract off this baseline and try a plot of V vs cos Beware of software that assumes you ve entered the data in radians What do you find Is it consistent with your expectations 2 5 4 Polarized Sunglasses and Glare Why do polarized sunglasses remove glare To figure this out you will need a polarizer and some glare Typically you can find glare on a waxed floor from the overhead lights so try the hallway Look at the glare without the polarizer Now look at the glare with the polarizer and rotate the polarizer in your hand while still viewing the glare What happens Do you know how polarized sunglasses work now Physics 124 Lab 2 Spring 2012 24 2 6 Formal Lab Report Your lab report should include the measurement of the index of refraction of water from last week and the measurement of the laser wavele
107. on before beginning Lab 8 Capacitors Il 8 1 Questions for Lab Preparation 1 If you have not done so finish analyzing your data from the Capacitors I lab Find the capacitance and propagate the error correctly 2 Complete the questions from section 8 3 1 Bring the answers with you to lab You may tape these answers into your lab notebook 3 Complete the questions from section 8 3 2 Again you may tape these answers into your lab notebook 8 2 Introduction In this lab we will be studying the behavior of an RC circuit with a sinusoidally varying current and voltage 8 3 Experiment 1 Resistor Capacitor Circuit Sinusoidal Response 8 3 1 Theory I Traditional Recall that the governing differential equation is dQ t V t 1 a GR Re on Physics 124 Lab 8 Spring 2012 80 Suppose we let the source of emf vary sinusoidally in time E t Eo cos wt Ep cos 27 ft 8 2 where w 27 f is called the angular frequency Here f is the frequency in hertz Hz although w strictly also has units of Hz to remind ourselves that 27 4 1 we typically write the units of w as inverse seconds sec The differential equation 8 1 describes the behavior of the circuit but V t is replaced by the sinusoidal expression given in Eq 8 2 The general solution to the differential equation in this case is fairly complicated If we perform measurements on this circuit however we find that if we wait a time equal to a few time
108. ondering how to make differential measurements with the oscilloscope given that you can t connect the ground alligator clip arbitrarily in most circumstance you should play with the Math menu which is the red Math button between the 1 and 2 channel buttons on the front of the oscilloscope Keep in mind that a differential measurement is just the difference between what is measured on one channel and the other channel so you want to do math such that you get Vj Vo where V is the voltage measured on channel 1 and V2 is the voltage measured on channel 2 Q Given what we know about voltage does it matter whether we take a differential mea surement in this way Are we losing any information by doing this Physics 124 Lab 4 Spring 2012 41 The voltage associated with the planet is as far as the oscilloscope or function generator or any device plugged into the planet considered to be the zero of the voltage This point is marked with a little arrow to the left of the traces The zero of time is marked similarly with a little arrow at the top of the traces but remember that is related to the trigger signal rather than anything that has to do with the planet 4 8 Experiment 6 Mystery Signals Q Now let s see how adept you are at using the oscilloscope and function generator One partner should set up a secret signal on the function generator and have the other lab partner measure its frequency and amplitude and of course w
109. ont panel In general you can usually start with the AutoSet button if you end up having difficulty getting your signal to display nicely but you should also recognize that it is something of a crutch that you will ultimately want to abandon as you become more comfortable with the oscilloscope After pressing the button you should be looking at a square wave signal some thing like that shown in Fig 4 4 The voltage is given on the vertical axis as a function of time on the horizontal axis The screen will have graticule grid markings on it and at the bottom of the screen will be a guide that tells you what the voltage difference is per division box and the time difference per division box From this information you can characterize the signal pretty well Y Go ahead and characterize the probe comp signal now It will be moving rapidly on the screen and difficult to measure Estimate its amplitude average voltage value and frequency 4 5 2 Setting Scales VERTICAL SCALE You can adjust the sensitivity of the oscilloscope by turning the knob associated with Channel 1 You can try this now Note that the oscilloscope reflects the changes in the vertical scale There are two knobs one for each of the Physics 124 Lab 4 Spring 2012 36 Tek bo Trig d M Pos 0 0005 CH1 m Coupling I I I I O E O O EO O A A A O d ati Ott AA eee ee eee eee eer eee eee SOMHz
110. ood arriving from the two pulmonary arteries and exiting through the aorta If the sum of the blood entering from the pulmonary arteries were not on average the same as the blood leaving through the aorta this would present itself as a serious health issue at the very least One of the most important outcomes of this rule is that the current that passes through several devices serially i e one after the other with no other nodes or current paths introduced along the way must be the same at every point along that path Using the circuit of Fig 5 17 verify Kirchhoff s current rule Use 10 kQ resistors and put the ammeter in only one location at a time You can make your life a lot easier if you use a wire to simulate i e replace an ammeter when you have the actual ammeter located elsewhere in the circuit As always remember to record uncertainties so you have a means of evaluating whether Kirchhoff s current rule is in fact verified Physics 124 Lab 5 Spring 2012 55 L3 node In KE Figure 5 15 Conservation of charge current at a junction of several wires I I I eS Figure 5 16 The same current passes through each of the devices when they are in series Figure 5 17 Circuit for demonstrating Kirchhoff s current rule Voltage Rule The second of Kirchhoff s rules Kirchhoff s voltage rule expresses conservation of en ergy in the circuit The sum of all voltages around any circuit loop mu
111. ope to display the potential produced by the function generator you can use a BNC tee connector to obtain this signal Channel 1 should display the potential across the capacitor Determine the phase differences and whether Vo t leads or lags t for low frequencies high frequencies and for wRC 1 Do the results agree with the predictions of Eq 8 6 4 Log log plots of relative amplitude and semilog plots of phase as a function of fre quency are known as Bode pronounced bow dee plots Include your plots of the relative amplitude and phase of Va t as a function of frequency as an Appendix to your lab report along with a brief comment about how well it matches the theory derived in Sec 8 3 Does anything interesting happen when w RC Physics 124 Lab 8 Spring 2012 85 Notice that what you have produced is a circuit in which the amplitude at low frequencies is basically unchanged but the amplitude at high frequencies is attenuated Q Can you explain more intuitively why the amplitude of the signal becomes attenuated at high frequencies Does this make sense in terms of what you saw with the step changes in emf last week Scope Ch 1 Scope Ground Figure 8 2 Circuit for measuring the amplitude and phase response across the resistor in the RC circuit 8 4 Additional Activities If you have time try switching the capacitor and the resistor and making Bode plots for that case You wil
112. ose L 10 mH 107 H and C 0 01 pF Find the numerical value of the resonant frequency fo This will help you find the resonance in the next section 10 3 Experiment 1 RLC Circuit 10 3 1 Setup Q Construct the circuit shown in Fig 10 4 For R use a 100 resistor from your set of resistors and for L use the inductor you used in Lab 9 For C use a capacitance substitution box set initially at 0 01 uF Using the value of L you found earlier make a rough estimate of the expected resonant frequency You should already have done this above L C Figure 10 4 Set the Function Generator to produce a sinusoidal EMF in the 10 kHz range with its max imum possible amplitude on the 0 20 V scale Connect the SYNC output on the function generator to the EXT TRIG input on the oscilloscope and set up the oscilloscope to trigger on this external signal Now you re ready to find the resonance Measure the voltage across the function generator with one of your oscilloscope probes and measure the voltage across the resistor with the other Pay attention to where the little ground clips attach to your circuit They need to be connected to the ground terminal of the function generator Vary the function generator s frequency until you find the resonance it should be obvious If it is not then probably something in your circuit setup is incorrect Once you have found it admire the resonant character of the circuit s response by varyin
113. owever many of you have probably heard the terms before For the purposes of this lab we need to know that light acts as a transverse wave In a transverse wave the oscillation of the wave occurs perpendicular to the direction of prop agation Transverse waves that you should be familiar with from introductory mechanics Physics 124 Lab 2 Spring 2012 18 include water waves and waves on a string Compare transverse waves to longitudinal waves In a longitudinal wave the oscillation and the propagation of the wave occur in the same dimension A good example of a longitudinal wave is sound A sound wave changes the local density of the medium as it propagates through appearing as a series of compressions and rarefactions If we want to mathematically define a transverse wave we might write down the equation for the amplitude A of the oscillation as a function of both time t and propagation distance x This might look something like A Ag cos 27 E ft 2 1 Here Ap is the maximum amplitude of the wave f is the frequency of the wave and A is the wavelength of the wave Remember from introductory mechanics that if we know f and A we also know the speed of the wave This equation could be used for a water wave a wave on a string or light However in the case of light we usually define the electric field E as the thing that oscillates even though the magnetic field oscillates too in a plane perpendicular to both the electric
114. range controls the horizontal and vertical position controls and the trigger controls You can refer back to earlier chapters of this document to the documentation or help screens for the oscilloscope or ask your instructors if you would like some assistance in the best practices for measuring the trace you want Please make use of the oscilloscope probes unless your signal comes directly from the function generator in which case you can use a BNC cable The periodic charging and discharging of the capacitor could be achieved by using a battery and motor driven switch which made the appropriate connections alternatively at regular intervals The function generator is a more convenient device When set for a square wave output the function generator is designed so that it acts like a battery with an emf that is alternately Vp and 0 switching back and forth between these two values at a rate controlled by the frequency setting on the function generator If for instance the frequency is 500 Hz then for 1 millisecond the function generator acts like the circuit on the left in Fig 7 3 Then for the next millisecond it acts like the circuit on the right Figure 7 5 left For the first half of the cycle the function generator behaves like a battery with output impedance r right In the second half of the cycle the function generator behaves like a wire of impedance resistance r Physics 124 Lab 7 Spring 2012 77 The size of Vo
115. re collectively referred to as batteries Physics 124 Lab 4 Spring 2012 33 4 4 Experiment 2 Voltmeter Time varying voltages Not all voltage sources are constant or nearly so Indeed most interesting transducers generate voltages that can change rapidly with time scales anywhere from hundreds of milliseconds to nanoseconds One such transducer is the simple microphone a device that converts sound waves pressure into an electrical signal voltage Your instructors will have a few microphones when you have a chance try connecting one to your handheld voltmeter Do you see any signal Try talking into the microphone and see if it makes any difference Handheld multimeters can also measure voltages that vary in time up to a certain rate they can then return a sort of average value of the signal If you dial your multimeter to ACV or ACmV you can see what the meter makes of the signal from the microphone when you talk or hum into it Because it is an averaging device the voltmeter is decidedly imperfect Often we would like to know what the voltage is doing as a function of time not what its average value is as a function of time Fortunately there is another device that can do this for us the oscilloscope also known as the physicist s best friend 4 5 Experiment 3 Oscilloscope The oscilloscopes we will use in this class are Tektronix 2001c which can measure signals up to about 50 MHz on two input channels They
116. rises The input impedance of a voltmeter is very large you measured it in last week s lab and you probably found that it is in excess of 10 MQ Even though this is large compared to most of the resistances that one typically uses one should always keep this value in mind when one is performing a measurement For instance consider the following circuit Fig 6 2 1OMQ Figure 6 2 A circuit that should reveal the effect of large but finite input impedance Q Theoretically what is the voltage at the free end of the resistor Build the circuit of Fig 6 2 Measure the voltage between the free end of the resistor and ground using your voltmeter and compare it to the voltage across the power supply terminals What do you observe Can you account for any discrepancy The output impedance for most useful circuits is far less than that of the voltmeter Q Suppose that you wanted to pass 100 mA through a 100 ohm resistor connected between Physics 124 Lab 6 Spring 2012 67 the free end of the 10 MQ resistor and ground using the circuit of Fig 6 2 What is the input impedance of the 100 resistor This is as easy as it looks What battery voltage would be required to generate 100 mA through the 100 Q resistor 10MO V 1009 Figure 6 3 A large output impedance requires a large voltage to supply significant current Such a circuit may more properly be viewed as a current source for small resistances On the other ha
117. s 124 Lab 8 Spring 2012 84 8 3 3 Experiment Q Use the same RC circuit as in Capacitors I with the 1 kQ resistor and the provided capacitor but with the function generator set to produce sine waves Turn the amplitude control to its highest setting and leave it there 1 We will measure the amplitude and phase of the potential across the capacitor as a function of frequency by building the circuit in Fig 8 1 R Scope Ch 1 Scope Ground Figure 8 1 Circuit for measuring the amplitude and phase response of the capacitor in the RC circuit First make a few quick observations at high and low frequencies to see the general trend of the behavior Then judiciously select 10 to 15 frequencies for more careful measurements of the amplitude Use any oscilloscope features that make your job easier 2 Plot your results for the amplitude as a function of frequency Make this plot a log log plot and normalize your amplitude by dividing it by the amplitude of the voltage applied by the function generator Your job can be made easier by noting that in addition to its other amazing talents the oscilloscope can measure frequencies Does your plot conform to the predictions of Eq 8 11 Check these results at a very low frequency at a very high frequency and at the frequency such that wRC 1 3 Now we will look at the phase of Vc t relative to v the function generator signal as a function of frequency Connect channel 2 of the oscillosc
118. s Double Slit Today we would like to observe these wavelike properties of light To do this we will do Young s double slit experiment In this experiment an electromagnetic wave is incident upon two narrow apertures We let the separation between the two apertures be d The electromagnetic wave will exit the two slits with approximately the same amplitude and phase If we place a screen some distance D away from our two slits the electromagnetic waves propagating from the two slits will combine to illuminate the screen see Fig 2 1 AL dsin0 Figure 2 1 The distance traveled by the wave from slit 2 to the screen is longer than that traveled by the wave from slit 1 The difference in the path lengths is approximately equal to AL dsin0 2 8 where 0 is the angle shown in the figure Now if y lt D then 0 is small and sin y D The difference in travel distances is then AL yd D If this path difference is equal to nA where n is an integer then there will be a constructive interference and we will see a bright spot on the screen Thus the positions of the bright spots will be at positions y given by d nDA n m or Yn 2 9 The separation between two successive bright spots will then be given by DA Ay Yn 1 Yn Pa 2 10 Similarly for points where the path difference AL creates a phase shift of 180 there will be destructive interference and no light will be observed These points of destructive in
119. s RC after turning on the circuit then all the voltages in the circuit vary sinusoidally with the same frequency f but may be shifted in phase relative to the emf source and may have different amplitudes Given that information we can guess that the solution to the differential equation 8 1 will look like Q t Qo cos wt 8 3 where is a phase shift If we substitute this form into the differential equation 8 1 we find the following result Qow sin wt p o cos wt amp cos wt 8 4 We have two unknowns Qo the amplitude of the charge oscillation and the phase shift between the driving voltage t and the charge on the capacitor Q t We can find these quantities by judiciously choosing two times to evaluate Eq 8 4 First we use wt 1 2 to find Qo Qouw cos d Ro a 8 5 or equivalently tan wRC 8 6 Note that the product RC appears again Eq 8 6 tells us how the phase difference between Q t and V t depends on the frequency w of the sinusoidally varying emf Next we use the time wt 0 Then we find from Eq 8 4 E Qow sing oo cos 8 7 Using the trigonometric identities 1 cot o sin 6 and cos 8 8 v1 cot d y 1 cot o eo we find after some algebra E Qo a 8 9 W Ry1 1 WRCO Y Physics 124 Lab 8 Spring 2012 81 Once again in practical situations we need to look at the electrical potential across
120. s you should look back at the Capacitors lab 9 4 1 Theory We ll analyze the circuit using complex impedances The first thing to do is to figure out what the complex impedance of the inductor is We ll use a perfect inductor i e an inductor in which the series resistance is vanishingly small in series with a function generator with vanishing output impedance Fig 9 6 From Kirchhoff s voltage rule we have dI E L 0 9 7 with Eo coswt We know that our solution will look something like I t Ip cos wt 6 9 8 Physics 124 Lab 9 Spring 2012 91 where fo is the amplitude of the current and 6 is the phase Using this we find Eo cos wt Lwly sin wt 9 9 Now sin wt sin wt cos sind cos wt and choosing t 0 we find Eo wLIp sin 9 10 Clearly 6 7 2 and 0 lo Zo wL is the magnitude of the impedance Now we can put these pieces together to get the complex impedance Z Zoe jwL 9 11 Now we proceed to analyze the entire RL circuit The total resistance of the circuit is R and the impedance of the inductive part of the inductor is Zz jwL The total impedance is therefore Z R jwL and the current is E E r 5n 9 12 Z R jwL Recall that we are looking for the amplitude of the current Jo and the phase shift 6 The former is E E ee 9 13 0 VUE ars ne and the latter is ST wL tan gt 9 14 ey R 9 14 Thus the current
121. so that you can see the beam as it enters and exits the gelatin Rotate the laser beam until you discover the angle Oc for total internal reflection for water Make sure that the beam is exiting at the midpoint of the straight side Physics 124 Lab 1 Spring 2012 14 Q What is the critical angle Use it to determine the index of refraction of water from the relation sin c 1 ny Q Record the incident and refracted beam angles 01 and 62 respectively starting with 01 0 and increasing 9 in 5 increments for as many data points as you can Make sure to read the 2 values relative to the normal i e relative to the 180 line Determine the index of refraction of the water by using IGOR to fit to the functional form 62 sin Asin 01 1 2 Q Compare the two values that you measure Are they the same 1 4 Experiment 2 Thin Lenses If a point source is a distance s from a thin lens then light diverging from the source will be refracted by the lens and will converge at a distance s from the lens see Fig 1 2 Here s and s are related to the focal length f of the lens by the Gaussian lens equation 1 1 1 ae nee ee 1 3 s S f 1 3 where f is a length characteristic of the particular lens used Its value depends upon the radii of curvature of the spherical surfaces of the lens and on the material of which the lens is made Note in the drawing we assume s gt f Kf f source Figure
122. ss use Eqs 6 2 6 3 with arbitrary R Q Use the oscilloscope to measure the output impedance of the Tektronix CFG253 function generator and the protoboard function generator By the way input impedances are easy to measure experimentally just apply a AV and measure A In some sense this is what an ohmmeter does and what you did in last week s lab You do not have to write up these exercises for your lab report but you need to do them they may appear again in the future in lab and on examinations 6 5 Additional Activities A number of other activities were suggested at the end of last week s lab If you have completed and feel comfortable with the things you have seen this week feel free to try out some of these other exercises Your instructors can provide you with even more fun things to try if you complete these 6 6 Formal Lab Report The formal lab report for this topic should focus on the J V curves for the two resistors and the light bulb Remember to provides some background information that introduces the devices As part of the analysis comment on any systematic uncertainties that you can think of that might be distorting your results Make sure to include the following e diagrams of the circuits used e graphs of the data The graphs should be titled and axes should be labeled e results Lab 7 Capacitors I 7 1 Questions for Lab Preparation 1 How do you measure the output impedance of the function genera
123. st be zero This is really a statement about the conservation of energy and the fact that voltage is defined when the electric field is conservative When we consider the sum of voltages around a loop if we return to our starting point we must return to our starting voltage otherwise voltage would not be well defined and E ds 4 0 A more formal proof of Kirchhoff s voltage rule will have to wait a little while If you re wondering which loop to use the answer is any loop at all although certain loops that follow conductors and cross the devices of a circuit are more interesting than others Physics 124 Lab 5 Spring 2012 56 One of the most important outcomes of this rule is that if you have two or more devices attached at both ends such that the current splits several ways to pass through them the voltage across each of the devices is the same Devices connected in this way are said to be connected in parallel Figure 5 18 The voltage across a set of parallel elements is the same 5 7 2 Experiment Q Verify Kirchhoff s voltage rule by measuring the voltage across each of the devices in the circuit you built to test Kirchhoff s current rule Fig 5 19 Is the sum of the voltages equal to zero Figure 5 19 Circuit for demonstrating Kirchhoff s voltage rule 5 8 Experiment 3 Ohm s Law Your lab report will be based on the work in this section It is recommended that you try to complete this section in c
124. stance on a graph We recommend plotting the dynamic resistance as a function of J and using the average value of J on each interval ATI Q Given that the filament of a light bulb is a material that obeys Ohm s law how might you account for the fact that the dynamic resistance is not constant What else is going on where the dynamic resistance curve is interesting 6 4 Experiment 2 Impedance Impedance is a generalization of resistance for now they can be thought of as identical but when we start considering devices such as capacitors and inductors we will see that impedance includes current voltage relationships other than those given by Ohm s law We shall not prove a remarkable theorem due to Th venin which says that any two terminal combination of signal sources and resistors is equivalent to a single signal source Vi and a single series resistor Rip This is the other reason we care about input and output impedances they can simplify circuit analysis dramatically Physics 124 Lab 6 Spring 2012 66 Figure 6 1 An arbitrary combination of voltage sources and resistors is equivalent to a single voltage source and a single series resistance Th venin s theorem 6 4 1 Input Impedance Devices such as the voltmeter have a finite input impedance which is to say that if you apply a voltage to a voltmeter a small current will flow through it The input impedance is the ratio of the voltage applied to the current that a
125. stance values leads to a balanced bridge within our ability to check for zero potential difference C 2 3 Repeated Measurement Statistical Technique If a measurement is repeated in independent and unbiased ways the results of the measure ments will be slightly different each time A statistical analysis of these results then it is generally agreed gives the best value of the measured quantity and the best estimate of the uncertainty to be associated with that result The usual method of determining the best value for the result is to compute the mean value of the results If x1 22 EN are the N results of the measurement of the quantity x then the mean value of x usually denoted 7 is defined as N __ T tz2 a 1 T N Sna C 4 Physics 124 Uncertainty Analysis Spring 2012 115 The uncertainty in the result is usually expressed as the root mean squared deviation also called the standard deviation usually denoted as dx read delta x Note that here x does not mean the change in x but rather is a measure of the spread in x values in the set of measurements Formally the standard deviation is computed as x E am ER C 5 Although determining the standard deviation may be tedious for a large array of data it is generally accepted as the best estimate of the measurement uncertainty N B In general we cannot expect exact agreement among the various metho
126. such a fuss over measurement uncertainties Indeed in many cases the uncer tainties are so small that for some purposes we needn t worry about them On the other hand there are many situations in which small changes might be very significant A clear statement of measurement uncertainties helps us assess deviations from expected results For example suppose that two scientists report measurements of the speed of light in vac uum Scientist Curie reports 2 99 x 108 m s Scientist Wu reports 2 98 x 108 m s There are several possible conclusions we could draw from these reported results 1 These scientists have discovered that the speed of light is not a universal constant 2 Curie s result is better because it agrees with the accepted value for the speed of light 3 Wu s result is worse because it disagrees with the accepted value for the speed of light 4 Wu made a mistake in measuring the speed of light Without knowing the uncertainties in these measurements however it turns out that we cannot assess the results at all Physics 124 Uncertainty Analysis Spring 2012 113 C 1 Expressing Experimental Uncertainties Suppose that we have measured the distance between two points on a piece of paper There are two common ways of expressing the uncertainty associated with this measurement C 1 1 Absolute Uncertainty We might express the result of the measurement as 5 1 cm 0 1 cm C 1 By this we mean that th
127. symbol without any further ado When wires are attached to other wires however it is customary to use a dot to indicate the junction These dots are also called nodes When wires cross on a schematic but are actually not connected to one another no dot is used on the schematic Figure 5 2 Wires A single wire is shown at left In the center are two wires that cross but are not connected At right are two wires that cross and are connected It is worth mentioning that the wire configuration is a topological configuration and may not actually resemble the schematic when put into place in a physical circuit The connections implied by the wires must be made of course but they may twist and turn in three dimensional space such that a faithful representation of their actual paths would not at all resemble the circuit schematic Circuit schematics usually simplify the connections and make them as short as is consistent with depicting the circuit in question 5 4 2 Battery A battery is represented by the symbols in Fig 5 3 The longer vertical line indicates the position of the positive voltage end of the battery 5 4 3 Resistor A resistor is represented by the drawing in Fig 5 4 We ll return to the details of resistors presently Sec 5 5 2 Physics 124 Lab 5 Spring 2012 47 i i i T Cell Battery Figure 5 3 A simple cell left and battery right MN Resistor Figure 5 4 A resistor 5 4 4 Ligh
128. t 10 1 V values for each device red white lamp and repeat your ten measurements for current going the other way through the device and negative voltages Your laboratory notebook should contain a current voltage graph for each of these devices along with their uncertainties You should refer to the Fluke 117 manual to assess uncertainties for these measurements Figure 5 20 This is the circuit you should use to take data on the mystery devices marked with a and the lamp VERY IMPORTANT Do not apply more than 6 V to or rather across the lamp or it will burn out We recommend starting with the smallest variable voltage the breadboards can generate and then turning up the voltage carefully to something approaching 6 V to avoid this IMPORTANT Reversing the meter leads does NOT reverse the direction of current through the device you are testing but it does multiply your subsequent measurements by the factor 1 Instead you might replace the positive power supply connection with a negative power supply connection and see what happens see Fig 5 21 Make sure you are not applying more than 6V if you are testing the light bulb IMPORTANT You will want to measure the current for voltages that are beyond the reach of the power supplies i e smaller in magnitude than about 1 V Use your knowledge Physics 124 Lab 5 Spring 2012 58 Figure 5 21 Use the negative power supply to test the I V curve for negativ
129. t Bulb A light bulb is often used as a proxy for a resistor in fact as we will see a light bulb is not really like a resistor at all It has the virtue of lighting up when the current through it is sufficiently large Different light bulbs require different amounts of current to turn on 5 4 5 Simple Circuit A circuit which in reality in a cartoon world looks something like that shown in Fig 5 6 is represented symbolically like that shown in Fig 5 7 Current flows in the direction shown from the positive terminal of the battery to the negative terminal through the resistor 5 4 6 Voltmeter As we saw in lab last week we measure the voltage difference between two points with a voltmeter The voltmeter is represented symbolically in Fig 5 8 Voltmeters measure the voltage difference between or across points a and b i e if the potential is Va at point a and V at point b the voltmeter will display Va Vj There is typically a red a and a black lead b the price you pay for confusing these two leads is an extra minus sign in the voltage displayed by the meter N Light Bulb Figure 5 5 A light bulb Physics 124 Lab 5 Spring 2012 48 Figure 5 6 A real circuit with battery and resistor Figure 5 7 A simple circuit diagram a O b Figure 5 8 The voltmeter 5 4 7 Ammeter We have already seen that there is a way to measure the voltage between two points To
130. t bulb is off Is it necessary to subtract the background voltage It it is do this now Q First make a plot of V with background subtracted if necessary vs r Fit the data in IGOR to a line an inverse square law and an exponential decay What is the best fit Does this make sense Can you identify reasons for any discrepancies by looking carefully at your plot Lab 2 Optics 11 Wave Optics 2 1 Questions for Lab Preparation 1 Is light a transverse or longitudinal wave What is sound Light has a polarization and exhibits interference Does sound have these properties Why or why not 2 In Young s time there were not lasers only white light Could you do Young s Double Slit experiment that is described in this manual with white light Why or why not 3 What is the reason the lab manual gives for why polarized sun glasses are useful What is the actual reason Hint It has to do with glare 2 2 Introduction In our last laboratory on ray optics we treated light as if it were composed of rays While this approximation is usually adequate for objects which are large compared to the wavelength of light it is not adequate for describing the interaction of light with small objects In this case it is necessary to consider light as an electromagnetic wave 2 2 1 Light as an Electromagnetic Wave At this point in the semester we have not discussed fields let alone the electric field mag netic field or electromagnetic field H
131. t is the first thing anyone will read If the title doesn t capture the reader s attention then the abstract will never be seen Most scientific papers are not r ead casually however so the most important thing is to represent in some way what the manuscript is about Along with the title one usually includes your lab co workers and their affiliations If you ve declared your major you can write Department of Physics Amherst College Amherst Massachusetts 01002 5000 USA for an affiliation otherwise you should put your Amherst College address The first author should be you followed by your collaborators in alpha betical order When you have put all of these things together then you can include the stylistic conventions of the journal to which you are submitting As mentioned earlier the style we would like looks like that of AJP but we are not going to be looking for things like whether you used the right font etc rather we re interested that all of the pieces are present and presented in a coherent order Appendix C Uncertainty Analysis An intrinsic feature of every measurement is the uncertainty associated with the result of that measurement No measurement is ever exact Being able to determine and assess measurement uncertainties intelligently is an important skill in any type of scientific work The measurement or experimental uncertainty should be considered an essential part of every measurement Why make
132. tep Response 9 3 1 Theory Consider the following circuit Figure 9 2 where V t is a time dependent applied EMF supplied by the CFG2253 function generator In this circuit Kirchhoff s loop rule gives dI vosi y 1R 0 9 4 Physics 124 Lab 9 Spring 2012 88 In our application V t is a square wave switching back and forth from V 0 to its maximum value say Vo V V V V V V Figure 9 3 We are interested in the solution of Eq 9 4 when V t is a constant either 0 or Vo During a time interval such as that denoted by a above we have V 0 Just at the beginning of that time interval V t had been equal to Vo for a long time so the current had the steady value I Vo R So the problem is to solve the equation dI L IR 0 9 5 dt i RB subject to the initial condition that I Vo R at t 0 Similarly during a time interval such as that denoted by b in the sketch V t Vo and just at the beginning of that time interval was 0 So the problem is to solve the equation dI La tIR W 9 6 subject to the initial condition that 0 at t 0 Precisely the same differential equations govern the charging and discharging of a capacitor In our earlier lab on capacitors we saw that in those processes the voltage across a capacitor exponentially approaches its final value with a time constant given by r RC In an exponential of the form e7 7 7 is called the time constant
133. than the resistors you are using in today s lab 5 11 7 Variable Resistors Your instructors can make available a variable resistor for you to try out A variable resistor is a three terminal device with a fixed resistance R 2 between terminals 1 and 2 The third terminal is adjustable and divides the resistor in two such that there is an adjustable resistance between terminals 1 and 3 R13 and a complementary adjustable resistor between Physics 124 Lab 5 Spring 2012 63 terminals 2 and 3 Ra3 such that Ris R23 Rie 5 5 Both Rig and R23 can be adjusted to range from 0 to R 2 provided they also meet the constraint of Eq 5 5 There is also a variable resistor on your protoboard See if you can figure out how to make connections to it Variable resistors used to find frequent use in electronics to adjust things like volume balance tone and so forth Nowadays most of these sorts of controls are digital Lab 6 Resistors II 6 1 Questions for Lab Preparation 1 What is impedance Are there devices without any impedance 2 What is Thevin s theorem Why is it useful 6 2 Introduction Here are the things that you should do today following the introductory comments of your instructor e Finish taking data from last time e Analysis of data using IGOR e Review what is needed for your lab report e Learn about input impedance and output impedance and e If time try out some of the additional activities
134. the theory of statistical distributions and assumes that the distribution of successive measurement values is described by the so called Gaussian or equivalently normal distribution In rough terms we can think of the fluctuations in the results as given by a kind of motion in a space of variables x y and z If the motion is independent in the zx y and z directions then the net speed is given as the square root of the sum of the squares of the velocity components In most cases we simply assume that the fluctuations due to the various variables are independent and use Eq C 8 to calculate the net effect of combining the contributions to the uncertainties Note that our general method applies no matter what the functional relationship between R and the various measured quantities It is not restricted to additive and multiplicative relationships as are the usual simple rules for handling uncertainties Physics 124 Uncertainty Analysis Spring 2012 117 In most cases we do not need extremely precise values for the partial derivatives and we may compute them numerically For example Of d flan 9 2 f T 9 Z Ox dx l C 9 Connection to the traditional simple rules To see where the usual rules for combining uncertainties come from let s look at a simple functional form R x y C 10 Using our procedure developed above we find that d RR x dyR dy C 11 and combining uncertainties yie
135. the various paths two in this case We will show a ane thie two resistors together behave as a single resistor of resistance R where R RI R3 In this case R will always be less than both Ry and Ra AD Figure 5 25 Two resistors in parallel behave as a single resistor of resistance R7 R7 Ry Use the red and white devices and the ohmmeter to test the hypothesis that resistors in series add according to the equation R R Ro 5 3 and that resistors in parallel add according to 1 1 1 5 4 R R R oe 5 11 4 AC Resistance Suppose we hook up a function generator rather than a DC power supply to one of the mystery devices Do you expect Ohm s law to continue to be satisfied See if you can figure out how to use your oscilloscope to measure the voltage across the device and the current through the device Physics 124 Lab 5 Spring 2012 62 IMPORTANT Pay close attention to where you are connecting the oscilloscope grounds If you hook them up to different places in your circuit you will be measuring what you think you are measuring Hint You might want to use at least one additional resistor and possibly the math features of the oscilloscope If what you see makes sense repeat the experiment with the light bulb 5 11 5 Imperfect Voltmeters and Ammeters try measuring the resistance of a really large resistor gt 1 MQ using the I V curve technique Try it again with the meter topolog
136. ther side of the inductor This means that you can use two oscillo scope probes one attached to each terminal of the inductor to measure the voltage across the inductor The key is to subtract the signal measured by one probe from the signal measured by the other This can be done using the MATH button on the front panel of the oscilloscope and setting the MATH mode to CH1 CH2 The oscilloscope will display this trace in red At this point the probe attached to CH1 behaves like the red lead of an ordinary voltmeter and the probe attached to CH2 behaves like the black lead of an ordinary voltmeter Note that you should still attach the little ground clips to the circuit ground point E in Fig 10 5 E C A 3 D v t R E E Figure 10 5 IMPORTANT Make sure that CH1 and CH2 are properly configured for the oscilloscope probes that is they should both be on the 10x setting Pick one of the eight capacitance values you used in Sec 10 3 with a capacitance lt 0 015 uF Set the frequency of the function generator to its resonant value Now measure successively the following quantities e the amplitude of the emf supplied by the function generator A E e the phase of the emf supplied by the function generator should be zero e the amplitude A B and phase of the potential drop across the inductor e the amplitude B D and phase of the potential across the capacitor and e the amplitude D E and phase of
137. tions may have been used in reaching the result There are several ways to assess whether our data support the theory we are trying to test Physics 124 Uncertainty Analysis Spring 2012 119 C 4 1 Rule of Thumb As a rule of thumb if the measured results agree with the expected results within a factor of about two times the combined uncertainties we usually can view the agreement as sat isfactory If the results disagree by more than about two times the combined uncertainties something interesting is going on and further examination is necessary Example Suppose a theorist from Harvard predicts that the value of X in the previous example should be 333 1 N m s Since our result 339 7 N m s overlaps the theoretical pre diction within the combined uncertainties we conclude that there is satisfactory agreement between the measured value and the predicted value given the experimental and theoretical uncertainties However suppose that we refine our measurement technique and get a new result 340 1 0 1 N m s Now the measured result and the theoretical result to not agree Note that our new measured result is perfectly consistent with our previous result with its somewhat larger uncertainty We cannot tell which is right or which is wrong without further investigation and comparison C 4 2 x Statistical Technique The x technique produces a number which tells you how well your data match the theory you are trying
138. tions of the cursors and the values can be read from locations within the softmenu Q Try measuring carefully the amplitude of the square wave signal as well as the period of the waveform How do these values compare to those that you measured more by eye in the earlier part of the lab How do the values compare to the ones listed on the screen Q Try measuring the voltage of the battery in a similar fashion Q Try looking at the signal that comes from a microphone Try various different sounds including a tuning fork and describe the sorts of waveforms that you see 4 5 5 Saving your Data You can save your data on a USB Flash Drive in a number of formats You can use the button to do this Generally you will want to save the actual traces as comma separated value CSV files but you can also save JPG or TIFF files containing the image of the oscilloscope screen instead Later on we ll import CSV files into IGOR so that you can create graphs of the traces and further analyze them To simplify matters see if you can figure out how to program the little printer button so that it will save your data to the Flash drive this will save you some time later Ask your instructors for help if you can t figure it out Go ahead and save a waveform to the flash drive 4 6 Experiment 4 Function Generator The Function Generator generates repeating time varying waveforms The simple function generator we will be using today the Te
139. tities obtained from measurements The higher level skills include the following 1 Planning and preparing for measurements 2 Executing and checking measurements intelligently 3 Analyzing the results of measurements both numerically and graphically This skill includes assessing experimental uncertainties and deviations from expected results 4 Being able to describe talk about and write about physical measurements The laboratory work can be divided into three parts preparation execution and anal ysis The preparation of course must be done before you come to your laboratory Physics 124 General Instructions Spring 2012 7 session The execution and analysis for the most part will be done during the three hour laboratory sessions Some suggestions for performing these three parts successfully are given below Please also refer to Appendix A Keeping a Laboratory Notebook since good note taking will be essential in all phases of the lab Preparation Before Lab 1 Read the laboratory instructions carefully Make sure that you understand what the ultimate goal of the experiment is 2 Review relevant concepts in the text and in your lecture notes 3 Outline the measurements to be made 4 Understand how one goes from the measured quantities to the desired results 5 Organize tables for recording data and the equations needed to relate measured quan tities to the desired results 6 Complete any written assig
140. to test Let s imagine that we have a theory which suggests that two measured variables x and y are related y y x As a test of this theory let s further imagine that we do an experiment which generates x and y 0 We are assuming here that the error in the z is negligibly small The x for this data set is given by N a x y yi s i C 19 i 1 1 where N is the number of data points The reduced x written x is defined by 2 yo C 20 V where v N n and n is the number of free parameters in the fit A straight line fit has two free parameters the slope and the intercept Despite its appearance x is fairly easy to interpret To see this let s consider a large set of data with only a small number of free parameters so that v N If the data fit well to the theory then we would expect on average that y y x 0 Thus we would also expect that x N and therefore x2 1 If on the other hand our theory doesn t fit the data well we would expect y y x gt 0 on average and therefore x2 gt 1 If we have Physics 124 Uncertainty Analysis Spring 2012 120 overestimated our errors or chosen too many free parameters then y y x lt 0 on average and x2 lt 1 A full analysis of this technique is given in Bevington In practice this is a difficult technique to apply with any rigor because estimating errors is so difficult For the purposes of this course we will call
141. topics in slightly more detail and provide a rationale for working on the manuscript in this order Physics 124 Writing a Formal Lab Report Spring 2012 109 B 0 1 Figures We ve talked already about creating figures in the context of presenting and analyzing data see Sec 6 3 above Figures can also include diagrams of apparatus tables and anything else that can be more easily represented by a figure than by a description Each figure requires a caption which should provide a reasonably self contained statement about the meaning of the figure Sometimes it is impossible to provide a succinct summary in which case it is acceptable to give a brief statement followed by see text for details Make sure if you adopt this approach that you actually include the details in the text You are welcome to use figures adapted from or captured from the lab manual i e this document provided you cite such use You can also hand draw your own or use some kind of graphics editing program If you would like to learn how to typeset figures and entire documents in the manner of this lab manual let one of your instructors know and we d be happy to help you Note however that the learning curve for professional typesetting is steep Please number your figures and tables so that you can easily refer to them in the main text B 0 2 Text With the figures completed or at least a draft of them one can proceed to write the text of the pap
142. tor 2 Please answer the questions in section 7 5 1 Bring the answers with you to lab so that you can tape them into your lab notebook 3 What are the units of RC Why is this value useful 7 2 Introduction In this lab we will be performing the following experiments 1 You will measure the input impedance of a voltmeter 2 You will determine the output impedance of the function generator Every real source of emf has some output impedance or internal resistance which in many cases can have an important effect on its operation Here we shall explore a simple but effective method of measuring that internal resistance to illustrate some basic notions about electrical circuits and about measurement philosophy 3 You will observe an RC transient decay of a charged capacitor when a step change in the driving voltage occurs Physics 124 Lab 7 Spring 2012 72 7 3 Experiment 1 Voltmeter Input Impedance There is only one setup for this part You do not have to do this first Q On a bench at the side of the laboratory we have arranged the following circuit Red voltmeter Black Figure 7 1 Circuit for measuring the input impedance of a voltmeter Initially the voltmeter should read about 10 volts call this value Vo Now break the circuit by disconnecting the red wire from the red terminal of the power supply Observe how long it takes the meter reading to decrease to 0 37 Vo Note that e7 0 37 Look
143. trigger level You can select which of these two slopes causes the oscilloscope to trigger through the softmenu There two other aspects of the trigger control that are worth noting both of which are available through the trigger softmenu The first is the trigger type which can be Auto Normal or Single The Auto trigger will display the voltage trace even if the trigger level does not intersect the waveform at all On the other hand the Normal trigger setting will not update the display unless the trigger conditions are met The Single trigger will take one trace and then suspend its operation until the button in the upper right corner of the front panel is pressed again to rearm the oscilloscope The second aspect is the signal that is used for the trigger This can be either of the two input signals a signal that is provided at the external input BNC connector or a variety of other options You should try out the various trigger settings and see if you can figure out what they do 4 5 4 Making a Measurement You can measure voltages and time intervals with the cursor controls reached through the button in the upper part of the front panel You can measure either voltage differences on either Channel 1 or Channel 2 or time differences on both channels by selecting either Voltage or Time cursors The unmarked multifunction knob towards the Physics 124 Lab 4 Spring 2012 38 top of the control panel then controls the posi
144. triggers each photogate t 0 53 0 01 s and t2 1 88 0 02 s What is the speed of the cart and the uncertainty that Cassandra should quote Answer The expression for the speed is of course da d C 25 at C 25 First we compute the numerator and its uncertainty dy is d 141 7 0 4 cm C 26 where we applied the rules for addition and subtraction add absolute uncertainties in quadrature We now do a similar calculation for the denominator to t 1 35 0 02 s C 27 Finally we calculate v using the rules for multiplication and division on the uncertainties in Eqs C 26 and C 27 add fractional uncertainties in quadrature 105 2 cm s C 28 Appendix D Complex Numbers Although most physicists generally use the letter we will adopt instead the engineering convention of choosing j V I Then j 1 and 1 j j A general complex number may be written z a jb where a and b are real numbers Complex numbers include real and imaginary numbers as subsets thus a is a complex number that is also purely real and jb is a complex number that is also purely imaginary We define the real and imaginary parts of Z in the following way R 2 a S Z b Note that both of these quantities are themselves real The complex conjugate of a complex number is denoted 2 and is found by replacing every occurrence of j by 7 2 a jb a jb D 1
145. ty q then fractional uncertainty a C 23 q Once we know the fractional uncertainties for each measured quantity in the product or quotient we can add them in quadrature to get the fractional uncertainty of the result To get the absolute uncertainty of the result simply multiply the fractional uncertainty by the result Example 2 Bob wants to find the area of a triangle He knows the length of the base b 4 2 0 2 cm and the height h 5 8 0 1 cm What is the area of Bob s triangle Physics 124 Uncertainty Analysis Spring 2012 122 Answer The equation for the area is A bh 12 2 cm The final uncertainty in the result is found by summing the fractional errors in quadrature and then multiplying by the sa ay 2Y BY 0s on 024 Bob should quote his total area as A 12 2 0 6 cm result C 5 3 Multiple Operations For combinations of operations the best approach is to break the problem up into pieces that can be solved by using the rules given above and then combine the uncertainties of each of these pieces appropriately The following example should make it clear how one can go about finding the final uncertainty in a more complicated problem Example 3 Cassandra wishes to know the speed of a cart traveling along a level air track She measures the distance of two photogates from the end of the air track dj 18 4 0 2 cm and dz 160 1 0 3 cm and also the times at which the cart
146. uickly This is because the voltmeter samples the voltage at regular intervals usually a few times a second and displays the most recent sampled value If your voltage is varying rapidly more rapidly than the rate at which the display is updated the voltmeter cannot give an accurate voltage reading Q Let s measure the voltage across a D cell Chemical reactions that take place within the cell can add energy to charges that pass through it raising the voltage on one side of the cell with respect to the other As long as the chemical reactions can continue the voltage across the cell should be roughly constant an ideal first test for our voltmeter The D cell is a cylinder with a bump on one of its ends If you bring the red lead into contact with the bump and the black lead into contact with the opposite end you should be able to read the voltage across the battery that is between the red and black leads Q What value do you measure If you reverse the red and black leads such that you connect the bumped end of the cell to the black lead and the flat end to the red lead what voltage do you measure Q Voltage is really a way of talking about the energy of a charge a higher voltage for a positive charge means a higher energy In fact the potential energy U of a charge at a potential V is U qV where q is the charge In some ways this parallels our understanding of the gravitational potential energy U mgh in terms of mass rath
147. uracy from a meter Sure you will want to optimize the accuracy but only when you re sure you re making the right measurement For example if you build a circuit incorrectly it doesn t matter how many digits you get out of a voltage reading If the circuit is wired wrong the results will be wrong too So take care to think first and always critically assess your measurements as you go along to see if they make sense 3 QUICK DATA ASSESSMENT After taking a piece of data ask yourself Does the data make sense Do a quick back of the envelope calculation to see if it does If it doesn t make sure you know why Don t fall into the trap of taking tons of data and analyzing it at the end You need to make sure things are making sense as you go This will prevent you from taking a whole data set with an improperly wired resistor Analysis During Lab 1 GRAPHS AND CALCULATIONS In addition to the ongoing analysis you conduct during data taking you will be expected to perform a more thorough analysis for each experiment Most importantly you will be asked to obtain meaningful physical results from the measurements Often though not always this will be done in the context of graphical analysis That is usually you will fit a curve most often a line to the graph and extract the fit parameters such as the slope Of secondary though not insignificant importance is the need for you to specify some limits of accuracy about your
148. urrounding the source increases as the square of the distance r from the source the intensity I is expected to exhibit inverse square law behavior when measured as a function of r On an optical bench align a light bulb 10 cm in front of a photo sensor light detector Physics 124 Lab 1 Spring 2012 16 filter light detector bulb Figure 1 3 as shown in Fig 1 3 The detector is designed so that it will give an output voltage proportional to the intensity of the light hitting the sensor so long as the voltage is less than 300 mV That is for low intensities the detector responds linearly to the intensity We will want to conduct this experiment in the linear region of the detector To make sure this happens place a holder that contains a collection of light absorbers these are actually just partially darkened overhead transparencies cut into 2 inch squares between the bulb and the detector so that the output reading is close to but not more than 300 mV The absorber filter allows only a fraction of the light to penetrate through it It should be positioned very close to the bulb Q Examine qualitatively what happens to the intensity when the source detector spacing r is gradually increased Take quantitative measurements of the detector output voltage V vs r for at least 10 readings between 10 cm and 80 cm These readings do not need to be evenly spaced Q What is the background voltage or the intensity when the ligh
149. zontal axis is the real axis and the vertical axis is the imaginary axis The length of the complex number is and its phase is 0 Z Z exp 30 e Euler identity e cos sin0 D 10 This can be proven using the Taylor series expansions of e sing and cosg From this identity we can write a complex number in two complementary ways a jb 2 cos0 sin 0 Z e D 11 Note that addition and subtraction are easier using the Cartesian representation and mul tiplication and division are easier using the polar representation e Complex impedances of the passive linear circuit elements Capacitor j wC Inductor jwL Resistor R Bibliography 126

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