Modern physics lab manual

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Published Date:13-07-2017
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PHYSICS 200 LABORATORY MANUAL FOUNDATIONS OFPHYSICSII Spring 20102Contents Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1. Field Superposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2. Equipotentials and Electric Field Lines . . . . . . . . . . . . . . . . . . . . . . 17 3. The Digital Oscilloscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4. Electrical Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 5. RC Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 6. Ohmic & Non-Ohmic Circuit Elements . . . . . . . . . . . . . . . . . . . . . . 57 7. e/m of the Electron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 8. Helmholtz Coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 9. AC Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Appendix A — Least Squares Fits . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Appendix B — Meter Uncertainty Specifications . . . . . . . . . . . . . . . . . . 93 Appendix C — Uncertainty Formulae . . . . . . . . . . . . . . . . . . . . . . . . 97 34Introduction Purpose By a careful and diligent study of natural laws I trust that we shall at least escapethedangersof vagueanddesultory modesofthoughtandacquireahabit of healthy and vigorous thinking which will enable us to recognise error in all the popular forms in which it appears and to seize and hold fast truth whether it be old or new...But I have no reason to believe that the human intellect is able to weave a system of physics out of its own resources without experimental labour. Whenever the attempt has been made it has resulted in an unnatural and self-contradictory mass of rubbush. James Clerk Maxwell Physics and engineering rely on quantitative experiments. Experiments are designed sim- plifications of nature: line drawings rather than color photographs. The hope is that by strippingawaythedetails, theessenceofnatureisrevealed. (Ofcourse, thecriticsofscience wouldarguethattheessenceofnatureislostinsimplification: adissectedfrogisnolongera frog.) While the aim of experiment is appropriate simplification, the design of experiments is anything but simple. Typically it involves days (weeks, months, ...) of “fiddling” before the experiment finally “works”. I wish this sort of creative problem-oriented process could be taught in a scheduled lab period, butlimited time and the many prerequisites make this impossible. Look for more creative labs starting next year Thus this Lab Manual describes experiences (“labs”) that are a caricature of experimental physics. Ourlabs willtypically emphasizethoroughpreparation, anunderlyingmathemati- calmodelofnature,goodexperimentaltechnique,analysisofdata(includingthesignificance of error) ...the basic prerequisites for doing science. But your creativity will be circum- scribed. You will find here “instructions” which are not a part of real experiments (where the methods and/or outcomes are not known in advance). In my real life as a physicist, I have little use for “instructions”, but I’m going to try and force you to follow them in this course. (This year: follow what I say—not what I do.) The goals of these labs are therefore limited. You will: 1. Perform experiments that illustrate the foundations of electricity and magnetism. 2. Become acquainted with some commonly used electronic lab equipment (meters, scopes, sources, etc). 3. Perform basic measurements and recognize the associated limitations (which, when expressed as a number, are called uncertainties or errors). 56 Introduction 4. Practice the methods which allow you to determine how uncertainties in measured quantities propagate to produce uncertainties in calculated quantities. 5. Practicetheprocessofverifyingamathematicalmodel,includingdatacollection, data display, and data analysis (particularly graphical data analysis with curve fitting). 6. Practice the process of keeping an adequate lab notebook. 7. Experience the process of “fiddling” with an experiment until it finally “works”. 8. Develop an appreciation for the highs and lows of lab work. And I hope: learn to learn from the lows. Lab Schedule Thelabschedulecanbefoundinthecoursesyllabus. Youshouldbeenrolledinalabsection for PHYS 200 and you should perform and complete the lab that day/time. Problems meeting the schedule should be addressed—well in advance—to the lab manager. Materials You should bring the following to each lab: notebooks: While one is being graded, the others • Lab notebook. You will need three willbeavailabletouseinthefollowinglabs. Thelabnotebookshouldhavequad-ruled paper (so that it can be used for graphs) and a sewn binding (for example, Ampad 26–251, available in the campus bookstores). The notebooks may be “used” (for example, those used in PHYS 191). • Lab Manual (this one) • TheknowledgeyougainedfromcarefullyreadingtheLabManualbeforeyouattended. • A calculator, preferably scientific. • A straightedge (for example, a 6” ruler). • A pen(we prefer your lab book bewritten in ink, since you’re not supposedto erase). Before Lab: Since you have a limited time to use equipment (other students will need it), it will be to your advantage if you come to the laboratory well prepared. Please read the description of theexperimentcarefully, anddoanypreliminaryworkin your lab notebook beforeyoucome to lab. Note carefully (perhaps by underlining) questions included in the lab description. Typically you will lose points if you fail to answer every question.Introduction 7 During Lab: Note thecondition of your labstation whenyou startsothatyou can returnittothat state when you leave. Check the apparatus assigned to you. Be sure you know the function of each piece of equipment and that all the required pieces are present. If you have questions, ask your instructor. Usually you will want to make a sketch of the setup in your notebook. Prepare your experimental setup and decide on a procedure to follow in collecting data. Keep a running outline in your notebook of the procedure actually used. If the procedure used is identical to that in this Manual, you need only note “see Manual”. Nevertheless, an outline of your procedure can be useful even if you aim to exactly follow the Manual. Prepare tables for recording data (leave room for calculated quantities). Write your data in your notebook as you collect it Check your data table and graph, and make sample calculations, if pertinent, to see if everythinglookssatisfactorybeforegoingontosomethingelse. Mostphysicalquantitieswill appear to vary continuously and thus yield a smooth curve. If your data looks questionable (e.g., a jagged, discontinuous “curve”) you should take some more data near the points in question. Check with the instructor if you have any doubts. Complete the analysis of data in your notebook and indicate your final results clearly. If you makerepeatedcalculations of anyquantity, youneedonlyshow onesamplecalculation. Oftenaspreadsheetwillbeusedtomakerepeatedcalculations. Inthiscaseitisparticularly importanttoreporthoweachcolumnwascalculated. Tapecomputer-generateddatatables, plots and least-squares fit reports into your notebook so that they can be examined easily. Answer all questions that were asked in the Lab Manual. CAUTION: for your protection and for the good of the equipment, please check with the instructor before turning on any electrical devices. Lab Notebook Your lab notebook should represent a detailed record of what you have done in the labora- tory. It should be complete enough so that you could look back on this notebook after a year or two and reconstruct your work. Your notebook should include your preparation for lab, sketches and diagrams to explain the experiment, data collected, initial graphs (done as data is collected), comments on difficulties,samplecalculations, dataanalysis,finalgraphs,results,andanswerstoquestions askedintheLabManual. NEVERdelete, erase, ortearoutsections ofyournotebookthat you want to change. Instead, indicate in the notebook what you want to change and why (such information can bevaluablelater on). Thenlightly drawalinethroughtheunwanted section and proceed with the new work. DO NOT collect data or other information on other sheets of paper and then transfer to your notebook. Your notebook is to be a running record of what you have done, not a formal (all errors eliminated) report. There will be no formal lab reports in this course. When you have finished a particular lab, you turn in your notebook. Ordinarily, your notebook should include the following items for each experiment.8 Introduction NAMES. The title of the experiment, your name, your lab partner’s name, and your lab station number. DATES. The date the experiment was performed. PURPOSE. A brief statement of the objective or purpose of the experiment. THEORY. At least a listing of the relevant equations, and what the symbols represent. Often it is useful to number these equations so you can unambiguously refer to them. Note: These first four items can usually be completed before you come to lab. PROCEDURE. Thissection shouldbean outlineofwhatyou didinlab. Asanabsolute minimum your procedure must clearly describe the data. For example, a column of numbers labeled “voltage” is not sufficient. You must identify how the voltage was measured, the scale settings on the voltmeter, etc. Your diagram of the apparatus (see below) is usually a critical part of this description, as it is usually easier to draw how the data were measured than describe it in words. Sometimes your procedure will be identical to that described in the Lab Manual, in which case the procedure may be abbreviated to something like: “following the procedure in the Lab Manual, we used apparatusZ to measureY as we varied X”. However thereare usually details you can fill in about the procedure. Your procedure may have been different from that described in the Lab Manual. Or points that seem important to you may not have been included. And so on. This section is also a good place to describe any difficulties you encountered in getting the experiment set up and working. DIAGRAMS. A sketch of the apparatus is almost always required. A simple block diagram can often describe the experiment better than a great deal of written expla- nation. DATA. You should record in your notebook (or perhaps a spreadsheet) a concurrent record of your relevant observations (the actual immediately observed data not a recopied version). You should record all the numbers (including every digit displayed by meters) you encounter, including units and uncertainties. If you find it difficult to be neat and organized while the experiment is in progress, you might try using the left-hand pages of your notebook for doodles, raw data, rough calculations, etc., and later transfertheimportantitems totheright-handpages. Thissection oftenincludes computer-generated data tables, graphs and fit reports — just tape them into your lab book (one per page please). You should examine your numbers as they are observed and recorded. Was there an unusual jump? Are intermediate data points required to check a suspicious change? The best way to do this is to graph your data as you acquire it or immediately afterward. CALCULATIONS. Sample calculations should be included to show how results are obtained from the data, and how the uncertainties in the results are related to the uncertainties in the data (see Appendix C). For example, if you calculate the slope of a straight line, you should record your calculations in detail, something like: v −v (4.08−.27) cm/wink 2 1 2 = =0.27214 cm/wink (1) t −t (15−1)wink 2 1Introduction 9 The grader must be able to reproduce your calculated results based on what you have recorded in your notebook. The graders are told to totally disregard answers that appear without an obvious source. It is particularly important to show how each columninaspreadsheethardcopywascalculated (quick &easy via‘self documenting’ equations). RESULTS/CONCLUSIONS. You should end each experiment with a conclusion that summarizesyourresults—whatwereyourresults,howsuccessfulwastheexperiment, and what did you learn from it. This section should begin with a carefully constructed table that collects all of your important numerical results in one place. Numerical values should always include units, an appropriate number of significant digits and the experimental error. You should also compare your results to the theoretical and/or accepted values. Does your experimental range of uncertainty overlap the accepted value? Based on your results, what does the experiment tell you? DISCUSSION/CRITIQUE.Asaservicetousandfuturestudentswewouldappreciate it if you would also include a short critique of the lab in your notebook. Please comment on such things as the clarity of the Lab Manual, performance of equipment, relevance of experiment, timing of the experiment (compared to lecture) and if there is anything you particularly liked or disliked about the lab. This is a good place to blow off a little steam. Don’t worry; you won’t be penalized, and we use constructive criticisms to help improve these experiments. QUICKREPORT. As you leave lab, each lab groupshouldturnin a3”×5”quick report card. You will be told in lab what information belongs on your card. These cards go directly to the instructor who will use them to identify problems. Drop off your lab notebook in your lab instructor’s box. Note: The TA’s cannot accept late labs. Iffor some reason you cannot complete alab ontime, please see the lab manager (Lynn Schultz) in PEngel 139 or call 363–2835. Late labs will only be accepted under exceptional circumstances. If an exception is valid, the lab may still be penalized depending on how responsibly you handled the situation (e.g., did you call BEFORE the lab started?). Grading Each lab in your notebook will be graded separately as follows: 9–10 points: A 8–8.9 points: B 7–7.9 points: C 6–6.9 points: D 0–5.9 points: Unsatisfactory10 Introduction1. Field Superposition Purpose The universe is filled with sources of electric and magnetic fields. In this lab you will test the principle that the field produced by several sources simultaneously is just the vector sum of the fields produced by each source individually. Introduction From Coulomb’s law we know the electric field produced by an isolated charged particle. In a universe of zillions of electrons and protons this result would be without value if we lacked a method of finding the field resulting from multiple sources. Electric fields (and 1 also magnetic fields) combine in the simplest possible way: by vector addition . In this experiment we find it convenient to work with magnetic fields rather than electric fields. We have a readily available source of natural magnetic field from the Earth; we can produce controlled magnetic fields using electromagnets; and we can easily measure the direction of the magnetic field using an ordinary compass. While you have not yet covered magneticfieldsinlecture,allyouwillneedtoknowaboutthemforthislabisthatthesource of magnetic field is electric current and that the magnetic field produced by a current is proportional to that current. Permanent magnets result from orbiting, spinning electrons in iron; Electromagnets result from the easily measured current flowing through copper wires that make up the windings of the coil. (Similar electric currents flowing through the metallic core of the Earth power the Earth’s magnetic field.) Just as an electric field is proportional to the charge that is its source, so the magnetic field of an electromagnet is proportional to its current. Apparatus • 1 power supply • 1 digital multimeter 1 Electricity and magnetism is really just one example, as these two things are, as Einstein showed, really just different aspects of one thing: F . Note that many other force fields (for example that in Einstein’s μν theory of gravity called general relativity) do not satisfy this simple combination rule. 1112 1: Field Superposition I B I Figure 1.1: Magnetic field lines through a finite solenoid of coils carrying current I. • 1 solenoid assembly • 1 compass • 1 set of leads Theory Solenoid Asolenoidisalengthofwirewoundaroundacylinder. Mathematically itiseasiesttothink about a solenoid as a series of circular loops of wire each carrying the same current, but in fact the wire is a tightly spaced helix (spiral). Figure 1.1 shows the magnetic field lines of a solenoid carrying a current I. The result is perhaps a bit surprising: the magnetic field goes through the core of the cylinder as the current winds around the edge of the cylinder. In a month or so you’ll learn how to calculate the magnetic field for such a solenoid, but for this lab it is enough to know that the magnetic field is everywhere proportional to the current. In particular, the magnetic field, B, at he center of our solenoid (where we will be doing our experiment) is given by: B =aI. (1.1) s where I is the electric current flowing around the solenoid (measured in ampere, denoted −3 “A”) and a = 1.14×10 T/A. (The unit of magnetic field is tesla, denoted “T”.) The solenoid will be used as an adjustable source of magnetic field. Earth’s Magnetic Field The magnetic field of the Earth varies with position and time. However, during the course of this lab, at your particular lab table, the Earth’s field may be considered constant in magnitude and direction. (That is the currents producing the Earth’s magnetic field will notchangemuchduringthisexperiment.) GenerallyspeakingtheEarth’sfieldpointsnorth, however here in the northern hemisphere it also points down. The inclination of the field is ◦ quite large (over 70 ) at our latitude.1: Field Superposition 13 Solenoid & Compass Assembly N Power Supply A Figure 1.2: Solenoid circuit. Compass The needle of a compass points in the direction of the magnetic field it experiences. We say that a compass needle shows north, because that is (generally speaking) the direction of the Earth’s magnetic field. The needle is a magnetic dipole made of a small permanent magnet. Just as an electric dipole feels a torque aligning it to the external electric field, so a magnetic dipoletwists until it is aligned with the external magnetic field. We will use the compass to show us the direction of the magnetic field that results from the superposition of the Earth’s field with that of the solenoid. While the compass is graduated in degrees, ◦ we will find it convenient to work in radians, and so the conversion factor 360 =2π radian should be applied to all angles. Setup AsshowninFigure1.2,thecompassisplacedinthecenterofthesolenoid,andtheapparatus is oriented so that with no current flowing the compass needle points perpendicular to the coil-axis in the direction “N”, i.e., θ = 0. When a current is sent through the coil, the needlewill bedeflected topointinthedirection oftheresultingmagnetic field. Thecurrent will be supplied by an adjustable power supply, and accurately measured with a digital multimeter. (For more information on the operation of the multimeter see Figure 4.2 on page 40.) If we denote the magnetic field of the Earth by B and the magnetic field of the e solenoid by B , the resulting total magnetic field should be the vector sum of the two. As s shown in Figure 1.3, we predict B a s tanθ = = I (1.2) B B e e Thus a graph of tanθ vs. I should be a straight line. Note: since the compass needle is only free to rotate in the horizontal plane, only the horizontal components of the magnetic field are detected. Thus B above is actually just e the horizontal component of the Earth’s magnetic field. (Of course, the solenoid has been oriented so that its field is fully horizontal.) At our latitude it turns out that the vertical component of the Earth’s field is more than twice as large as the horizontal component.14 1: Field Superposition B s B e θ Figure 1.3: Adding two magnetic fields: B from the Earth and B from the solenoid. The e s angle of the resulting magnetic field is called θ. Procedure Position the solenoid/compass assembly such that the compass needle is aligned with north (N) on the compass housing. Without moving the assembly, connect the power supply in serieswiththeassemblyandmultimeterasshowninFig. 1.2. Setthemultimeterfunction to measure DC amps, denoted: A, on the 20 mA range. After your instructor checks the circuit, take a series of readings from the compass and multimeter while increasing the current from power supply. ◦ Record about ten well-spaced readings. (Readings should be spaced by about 2 mA or 5 , but it is a waste of time to try to make I or θ a round number. Don’t include the θ = 0, I =0 starting condition as a measurement.) Reverse the leads on the power supplyandtake a similar set of measurements. For this last set of measurements the current, the angle, and the tangent of the angle will be negative. When the measurements have been completed, turn off the power supply and check to see if the compass is still aligned to the north. If the solenoid/compass assembly has moved, the measurements should be redone. Calculations will be easier if you put your measured values into a spreadsheet. The uncer- tainty in the current (δI) can beaccurately determined from Table 4.1 on page 45, however for this lab an error of 0.5% should be accurate enough. The uncertainty in the compass reading (δθ) must be estimated based on your ability to read the compass scale. Typically, for an analog scale, the uncertainty is estimated to be± half of the smallest scale division. You may wish to assign a larger uncertainty if you believe thecompass is unusuallydifficult toread. Because computers generally assume angles are in radians, you will need to convert both θ and δθ to radians Lab Report 1. From your uncertainty in θ, determine an uncertainty for each value of y = tanθ. According to calculus, this is:  2 2 2 δ(tanθ)=δθ/(cosθ) =δθ 1+tan θ =δθ(1+y ) (1.3) combined B1: Field Superposition 15 + (This formula can be entered into WAPP directly.) Alternatively you can estimate errors from the difference tan(θ +δθ)− tanθ ≈ δ(tanθ). Print your spreadsheet and include it in your notebook. (Remember to self-document your spreadsheet or show sample calculations.) Make sure you’ve clearly displayed how each column was calculated. + 2. Use WAPP (Goggle “wapp+” to find it) to determine the line best approximating your tanθ versusI data. (Enter the currentand its uncertainty in amperes, notmA.) Tape the fit report and plot in your notebook. 3. Using the best-fit slope, calculate the Earth’s magnetic field, B , at SJU. (Actually e this is just the horizontal component of of the Earth’s magnetic field.) A “ballpark” −5 value for B is 1.5×10 T, but magnetic materials in the building will affect the e result. Using the uncertainty in slope, calculate the resulting uncertainty in B . e 4. Complete your lab report with a conclusion and lab critique.16 1: Field Superposition2. Equipotentials and Electric Field Lines Anewconceptappearedinphysics,themostimportantinventionsinceNewton’s time: thefield. Itneeded great scientific imagination to realize that it is notthe charges nor the particles but the field in the space between the charges and the particles that is essential for the description of physical phenomena. The field concept proved successful when it led to the formulation of Maxwell’s equations describing the structure of the electromagnetic field. Einstein & Infield The Evolution Of Physics 1938 Purpose To explore the concepts of electrostatic potential and electric field and to investigate the relationships between these quantities. To understandhow equipotential curves are defined and how they can display the behavior of both the potential and field. Introduction Every electric charge in the universe exerts a force on every other electric charge in the universe. Alternatively we can introduce the intermediate concept of an electric field. We then say that the universe’s charge distribution sets up an electric fieldE at every point in space, and then that field produces the electric force on any charged particle that happens to be present. The electric field E is defined in terms of things we can measure if we bring a “test charge” q to the spotwherewewant to measureE. Theelectric field is then defined in terms of the total force, F, experienced by that test charge q: E=F/q (2.1) Note that the field exists independently of any charges that might be used to measure it. 1 For example, fields may be present in a vacuum. At this point is is not obvious that the 1 Star Wars Episode IV: LUKE: You don’t believe in the Force, do you? HAN: Kid, I’ve flown from one side of this galaxy to the other. I’ve seen a lot of strange stuff, but I’ve 1718 2: Equipotentials and Electric Field Lines field concept really explains anything. The utility of the electric field concept lies in the fact that the electric field exists a bit independently of the charges that produce it. For example, there is a time delay between changes in the source charge distribution and the force on distant particles due to propagation delay as the field readjusts to changes in its source. The electric force F does work (F·Δr) on a charge, q, as it is moved from one point to another, say from A to B. This work results in a change in the electrical potential energy ΔU = U −U of the charge. In going from a high potential energy to a low potential B A energy,theforcedoespositivework: W =−ΔU. Consider,foramoment,alineorsurface AB along which the potential energy is constant. No work is done moving along such a line or surface, and therefore the force cannot have a component in this direction: Electric forces (and hence the electric fields) must be perpendicular to surfaces of constant potential energy. This concept is central to our experiment. Example: Gravitational potential energy. Consider a level surface parallel to the surface of the Earth. The potential energy mgh does not change along such a surface. And of course, both the gravitational force mg, and the gravitational field g are perpendicular to this surface, just as we would expect from the above reasoning. Electrical fields and potential energies are less intuitive because less familiar, but they work in much the same way. We can define a new quantity, the potential difference ΔV = V −V between points A B A and B, in the following way: ΔU W AB ΔV = =− (2.2) q q whereW is thework donebytheelectric forceasthetestchargeq is moved fromAtoB. AB Although the potential energy difference ΔU =U −U does depend on q, the ratio ΔU/q B A (andthereforeΔV)isindependentofq. Thatis, inamanneranalogous totheelectric field, potential difference does not depend on the charge q used in its definition, but only on the charge distribution producingit. Also, sinceF is a conservative force, W (and hence the AB potential difference) does not depend on the path taken by q in moving from A to B. Finally, note that only the potential difference has been defined. Specifying the potential itself at any point depends on assigning a value to the potential at some convenient (and arbitrary) reference point. Often we choose the potential to be zero “at infinity”, i.e., far away from the source charges, but this choice is arbitrary. Moreover, if we measure only potential differences, the choice of “ground” (zero volts) doesn’t matter. Apparatus • DC power supply never seen anything to make me believe there’s one all-powerful force controlling everything. There’s no mystical energy field that controls my destiny. Of course, we should replace “force” in the above with “field”.2: Equipotentials and Electric Field Lines 19 – – DC DMM Source + + Probe Figure 2.1: Electric field mapping circuit. • digital multimeter (DMM) • conductive paper with electrode configurations (2) • card board • graph paper • push pins You will be provided two electrode configurations drawn with a special type of conducting ink (graphite suspension) on a special carbon impregnated paper, as sketched in Fig. 2.1. The paper has a very high, but finite, resistance, which allows small currents to flow. Nevertheless, since there is a big difference between the resistance of the conducting ink and the paper, the potential drop within the ink-drawn electrodes is negligible (less than 1%)of that across thepaper. Thepotential differences closely resemblethose understrictly electrostatic (no current) conditions. Theory In this experiment you will use different configurations and orientations of electrodes which have been painted on very high resistance paper with conducting ink. When the electrodes areconnected to a“battery” (actually aDC powersupply: abattery eliminator) an electric field is set up that approximates the electrostatic conditions discussed in lecture. Under truly electrostatic conditions the charges would be fixed, and the battery could be discon- nected and the electric field would continue undiminished. However, in this experiment the battery must continue to make up for the charge that leaks between the electrodes through the paper. These currents are required for the digital multimeter (DMM) to measure the potential difference. Nevertheless, the situation closely approximates electrostatic condi- tions and measurement of the potential differences will be similar to those obtained under strictly electrostatic conditions.20 2: Equipotentials and Electric Field Lines C 5 C B 4 4 B 5 A 4 A 5 B 3 C 3 A 3 A 1 A 2 B 1 B 2 C 2 C 1 Figure 2.2: Some equipotential curves for the circle-line electrodes. Consider Fig. 2.2 where the electrodes are a circle and a line. Suppose we fix one electrode of the digital multimeter (DMM) at some (arbitrarily chosen) point O and then, using the second electrode as a probe, find that set of points A ,A ,A ,... such that the potential 1 2 3 differences between any of these points and O are equal: V =V =V =.... (2.3) A O A O A O 1 2 3 HencethepotentialsatpointsA ,A ,A ,...areequal. Infact, wecanimagineacontinuum 1 2 3 of points tracing out a continuous curve, such that the potential at all the points on this curve is the same. Such a curve is called an equipotential. We label this equipotential V . We could proceed to find a second set of points, B ,B ,B ,... such that the potential A 1 2 3 difference V is the same for n=1,2,3,...; i.e., B O n V =V =V =.... (2.4) B O B O B O 1 2 3 This second set of points defines a second equipotential curve, which we label V . The B actual values of the potentials assigned to these curves, V and V , depend on our choice A B of the reference potential V . However, the potential difference between any two points on O these two equipotentials does not depend on our choice of V ; that is, O (V −V )−(V −V )=V −V (2.5) A O B O A B and so the potential difference V −V can be determined from the measured values V A B AO and V . A mapping of electrostatic equipotentials is analogous to a contour map of BO topographic elevations, since lines of equal elevation are gravitational equipotentials. The elevation contours are closed curves; that is, one could walk in such a way as to remain always at the same height above sea level and eventually return to the starting point. Likewise, electrostatic equipotentials are closed curves. The electric field at a given point is related to the spatial rate of change of the potential at that point, i.e., the gradient, in the potential. Continuing the analogy with the contour map, we find the electric field is

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