physics 160 laboratory manual - hobart and william smith...

Physics 160 Laboratory Manual

Hobart & William Smith Colleges

Larry CampbellDonald Spector

Ted Allen

Spring 2003

Contents

General Instructions For Physics Lab iii

Recording data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Lab reports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Graphing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Reporting of Experimental Quantities . . . . . . . . . . . . . . . . iv

Collaborative and Individual Work . . . . . . . . . . . . . . . . . . v

General Admonition . . . . . . . . . . . . . . . . . . . . . . . . . . v

Current/Voltage Relationships vi

Procedure I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

Procedure and Analysis II . . . . . . . . . . . . . . . . . . . . . . . ix

Procedure and Analysis III . . . . . . . . . . . . . . . . . . . . . . x

Kirchhoff’s Laws xi

Procedure I: Parallel Circuits . . . . . . . . . . . . . . . . . . . . . xi

Procedure II: Series Circuits . . . . . . . . . . . . . . . . . . . . . . xii

Procedure III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

Procedure IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv

RC Circuits xv

Procedure I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi

Magnetic Field of a Current Loop xix

Procedure I: Dependence of Field on Current . . . . . . . . . . . . xxi

Procedure II: Dependence of Field on Number of Turns . . . . . . xxi

Procedure III: Dependence of Field on Diameter . . . . . . . . . . xxii

i

ii

Prisms and Lenses 1

Procedure I: Refraction . . . . . . . . . . . . . . . . . . . . . . . . 1

Procedure II: Lens Optics . . . . . . . . . . . . . . . . . . . . . . . 4

Procedure III: Lens Optics . . . . . . . . . . . . . . . . . . . . . . . 5

Procedure IV: Lens Optics . . . . . . . . . . . . . . . . . . . . . . . 6

Diffraction Grating 7

Appendix A: Reading the Vernier . . . . . . . . . . . . . . . . . . 10

Notes on Experimental Errors 11

Experimental Error and Uncertainty . . . . . . . . . . . . . . . . . 11

Mean & Standard Deviation . . . . . . . . . . . . . . . . . . . . . . 13

Agreement of Experimental Results . . . . . . . . . . . . . . . . . 17

Notes on Propagation of Errors 19

Sums of Measurements . . . . . . . . . . . . . . . . . . . . . . . . 19

Products and Powers of Measurements . . . . . . . . . . . . . . . 20

Notes on Graphing 23

Graphical Representation and Analysis . . . . . . . . . . . . . . . 23

Linearizing the Data . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Making Good Graphs . . . . . . . . . . . . . . . . . . . . . . . . . 24

Measuring the Slope of a Line . . . . . . . . . . . . . . . . . . . . . 25

Least Squares Method . . . . . . . . . . . . . . . . . . . . . . . . . 27

General Instructions ForPhysics Lab

Come to lab prepared – Bring quadrille-ruled, bound lab notebook, labinstructions, calculator, pen, and fine-lead pencil.

RECORDING DATA

Each student must record all data and observations in ink in his or her ownlab notebook. Mistaken data entries should be crossed out with a singleline, accompanied by a brief explanation of the problem, not thrown away.Do not record data on anything but your lab notebooks.

Put the date, your name, and the name of your lab partner at the top ofyour first data page. Your data and observations should be neat and wellorganized. It will be submitted as part of your laboratory report and mustbe legible.

LAB REPORTS

Specific advice on this matter is given at the end of each lab description,but your report will generally consist of:

Your data and observations, a summary of your experimental andcalculational results, and a discussion of your conclusions. The summaryand discussion may be written out right in the lab book (if done very legiblyand neatly), or you may use word processors and/or spreadsheets if youprefer.

One typed page summarizing what you learned by doing the experi-ment.

Occasionally your instructor may give some additional instructionsin class. It is important that the report be well organized and legible. Ifseveral measurements are to be compared and discussed, they should beentered into a table located near the discussion. Your data and resultsshould always include appropriate units.

iii

General Instructions For Physics Lab iv

Repetitive calculations should not be written out in your report; how-ever, you should include representative examples of any calculations thatare important for your analysis of the experiment.

GRAPHING

Graphs should be drawn with a fine-lead pencil. A complete graph willhave clearly labeled axes, a title, and will include units on the axes.

When you are asked to plot T 2 versus M , it is implied that T 2 will beplotted on the vertical axis and M on the horizontal axis.

Do not choose awkward scales for your graph; e.g., 3 graph divisionsto represent 10 or 100. Always choose 1, 2, or 5 divisions to represent adecimal value. This will make your graphs easier to read and plot. Whenfinding the slope of a best-fit straight line drawn through experimentalpoints:

1. Use a best-fit line to your data. You may do this either by hand or usinga graphing program, such as the one with Microsoft Excel.

2. Do not use differences in data point values for the rise and the run. Youshould use points that are on your best-fit line, not actual data points,which have larger errors than your best-fit line.

3. Do use as large a triangle as possible that has the straight line as itshypotenuse. Then use the lengths of the sides of the triangle as the riseand the run. This will yield the most significant figures for the value ofyour slope.

REPORTING OF EXPERIMENTAL QUANTITIES

Use the rules for significant figures to state your results to the correctnumber of significant figures. Do not quote your calculated results to eightor more digits! Not only is this bad form, it is actually dishonest as itimplies that your errors are very small! Generally you measure quantitiesto about 3 significant figures, so you can usually carry out your calculationsto 4 significant figures and then round off to 3 figures at the final result.After you calculate the uncertainties in your results, you should round offyour calculated result values so that no digits are given beyond the digit(s)occupied by the uncertainty value. The uncertainty value should usually beonly one significant figure, perhaps two in the event that its most significantfigure is 1 or 2. For example: T = 1.54 ± .06 s, or T = 23.8 ± 1.5 s. If youreport a quantity without an explicit error, it is assumed that the error is inthe last decimal place. Thus, T = 2.17 s implies that your error is ±0.005 s.Because the number of significant figures is tied implicitly to the error, it isvery important to take care in the use of significant figures, unlike in puremathematics.

General Instructions For Physics Lab v

COLLABORATIVE AND INDIVIDUAL WORK

You may collaborate with your partner(s) and other students in collectingand analyzing the data, and discussing your results, but for the report eachstudent must write his or her own summary and discussion in his or herown words.

GENERAL ADMONITION

Don’t take your data and then leave the lab early. No one should leave thelab before 4:30 PM without obtaining special permission from the instructor.If you have completed data collection early, stay in the lab and begin youranalysis of the data – you may even be able to complete the analysis beforeleaving. This is good practice because the analysis is easier while thework is still fresh in your mind, and perhaps more importantly, you maydiscover that some of the data is faulty and you will be able to repeat themeasurements immediately.

Current/Voltage Relationships

GOALS AND GENERAL APPROACH

This experiment will give you experience in measuring voltages and cur-rents with multimeters, and enable you to study how the current throughan electrical circuit element depends on the voltage difference across it. Insome cases the current flowing through the element is directly proportionalto the voltage across it (i.e., the electrical potential difference between itstwo ends). Such elements are called resistors and the proportionality con-stant is the resistance of the element in Ohms. (V = IR) However, manycircuit elements do not behave in this simple Ohm’s Law way. You willhave the opportunity to explore both sorts.

In the case of elements which obey Ohm’s Law, a fluid analogy can helpus to understand the voltage/current behavior. Think of the elements andconnecting wires in your circuit as pipes through which water is flowing.The electrical current I corresponds to the rate at which water flows througha section of pipe. In the case of water flowing through a pipe, the rate offlow depends on the difference in water pressure between the two ends ofthe pipe. So if we consider electrical potential to be analogous to waterpressure, we would expect the electrical current through an element to beproportional to the difference in electrical potential between the two endsof the wire (Ohm’s Law).

Pipes which are very narrow, or clogged with scale and obstructionsprovide a high resistance to water flow. A given pressure difference acrosssuch a pipe will result in a very much smaller flow rate than would occurfor a large clean pipe with the same pressure differential. Or, to get the sameflow through the clogged pipe as the clean pipe, one needs a much greaterdifference in pressure across its ends. This analogy helps us to understandwhy thin wires have greater resistance to electrical current flow than thickwires.

Likewise, just as we expect the water to flow out of the pipe at exactlythe same rate it flows into the pipe (unless the pipe has an aneurysm), sowe expect the electric current to be the same at both ends of a wire. (It is acommon misconception that current is getting "used up" as it flows througha circuit.) We will return to this analogy as we move along through theexperiment.

vi

Current/Voltage Relationships vii

APPARATUS

A variable power supply (a source of potential difference and current),two multimeters, a bread board, hookup wire, and some resistive circuitelements (resistor, light bulb, and diode).

Resistors use a system of three colored stripes to indicate their resis-tance values, each color corresponding to a particular integer. The first andsecond stripe give the tens and units values respectively. The third stripegives the power of ten by which the value is to be multiplied. The colorkey is as follows.

Table I Resistor Color Codes

Color Value as Value asdigit multiplier

Black 0 1Brown 1 10Red 2 102

Orange 3 103

Yellow 4 104

Green 5 105

Blue 6 106

Violet 7 107

Gray 8 108

White 9 109

There may be a fourth band on your resistor. It gives the uncertaintyof the color-code resistance value. The following code applies: gold ±5%,silver ±10%, and no band ±20%. Thus, for example, a resistor with stripesbrown, green, orange, and silver would have a nominal resistance value of15 × 103 Ohms with an uncertainty of 10% of that value.

R V Y

27 10 Ω4x

Figure 1 Example of reading the resistor color code. The bands are red,violet, yellow. Thus the value of the resistance is 27× 104Ω.

The circuit that you will use for your measurements is shown in Fig-ure 2.

A is a multimeter wired in series with the power supply and the ele-ment being studied, X. In this configuration it acts as an ammeter (currentmeasuring), and its lead positions and control switches should be set ac-cordingly. V is a multimeter wired in parallel across element X, and in this

Current/Voltage Relationships viii

X+

–

A

V

ammeter

voltm

eter

+ –

+

–

powersupply

Figure 2 The circuit used in this laboratory.

configuration acts as a voltmeter (potential measuring). Its lead positions andcontrol switches should be set accordingly. The power supply is a direct current(dc) device, and so the meters should be set for dc operation.

WARNING! Be sure that your power supply is off while you buildyour circuit. Then have your instructor check your circuit before turningthe power supply on.

Our water pipe analogy will help us here to understand what is hap-pening in this circuit, and some of its limitations. All of the electrical currentcoming out of the power supply passes through the ammeter A. Thus A trulymeasures the current out of the power supply, and we see that A shouldhave very little resistance or it will significantly affect the flow of current.Ideally a meter’s presence should have no effect on a circuit; but clearly nometer can have absolutely no resistance, so the current will be slightly lesswhen the ammeter is in place.

After coming out of A, the current splits into two paths – one throughX and the other through the voltmeter V – and so only part of the currentmeasured by A goes through X. Here we see why it is important that thevoltmeter V have a very large resistance (much larger than that of X). Ifthe resistance of V to current flow is much greater than that of X, themajority of the current will flow through X and only an insignificant tricklewill pass through V. In this case the current measured by A will not differsignificantly from that which actually flows through X. (Since the voltmeteris measuring the loss in energy per unit charge as the current passes fromthe top to the bottom of X, it need only see a small sample of the chargeflowing from top to bottom.)

You might like to think about whether or not one could get aroundthis problem entirely by wiring the circuit in the alternative way shown inFigure 3.

PROCEDURE I

Study the behavior of your light bulb as X in the circuit. Simultaneouslymeasure the current through the light bulb and the voltage across the lightbulb for several different power supply voltages. Be careful not to exceed6 V. After each reading, momentarily remove the voltmeter from the circuitto see whether or not it affects the current reading significantly, and record

Current/Voltage Relationships ix

X

+

–

AV

ammeter

voltm

eter

+

–+

–

powersupply

Figure 3 An alternative circuit.

your results. If it does affect the reading, then you know that your voltmeterresistance is not high enough and your previous result is not be valid. In thiscase, you should use the value measured with the voltmeter disconnectedfrom the circuit as your current value. You should plot your data (I versusV ) in your notebook as you take the data. This will enable you to see whatthe trend is early on, and will make it clear where you may need more datapoints.

ANALYSIS I

1. Make a plot of current (vertical axis in amps) versus voltage (horizontalaxis in volts). The origin (V = 0, I = 0) will be one of your data points.Draw a smooth curve or straight line using your data.

2. Does your light bulb obey Ohm’s Law; i.e., does your data fit a straightline? If it does, find the resistance from your graph (the reciprocal ofthe slope), measure the resistance with your multimeter, and comparethe two values. If your data does not lie on a straight line, you can stillmeasure something called the differential resistance. The differentialresistance is the inverse of the slope at any particular point on the graph,and is different for different points as the slope changes. Measure themaximum and minimum values of the differential resistance for yourlight bulb. Compare them with the value obtained by measuring theresistance of the light bulb with your multimeter. Why do you think thedifferential resistance of the light bulb changes as the current changes?

3. Calculate the power delivered to your light bulb for the highest voltageand current you tried. Note that the power is measured in watts (W) ifthe current is measured in amperes (A) and the voltage is measured involts (V).

PROCEDURE AND ANALYSIS II

1. Carry out the same procedure for your resistor as you did for yourlight bulb, but this time also repeat the measurements with the leadsinto the power supply switched so that the current is running throughthe resistor in the opposite direction. With the leads switched yourvoltage and current readings will be negative. Include both sets ofmeasurements (positive and negative) on a single graph.

Current/Voltage Relationships x

2. Draw a smooth curve or straight line through your data points. Discusswhether Ohm’s Law is obeyed by the current flowing in the forwarddirection and in the backward direction. For data which obey Ohm’sLaw, measure the resistance from the graph.

3. Record the nominal value of the resistor using its color code, and mea-sure its resistance with your multimeter. How do the three resistancevalues compare?

4. Calculate the power delivered to the resistor for the highest voltage andcurrent used.

PROCEDURE AND ANALYSIS III

Carry out the same procedure and analysis for the diode as used for theresistor, for both directions of current flow, but in this case modify yourcircuit, as shown below, by placing your incandescent lamp in series withthe ammeter.

+

–

A

V

ammetervo

ltmet

er

+ –

+

–

powersupply lamp

diod

e

Figure 4 Circuit with diode and current-limiting lamp.

This new arrangement helps to protect your ammeter from overloadsby limiting the current out of the power supply. The diode has very littleresistance in the forward direction. Keep the current in your circuit under120 mA.

For diodes it is particularly important that you test your voltmeter’seffect on the circuit by taking it out momentarily now and again. Forcurrents in one of the two directions, you will find that the current readingchanges significantly when the voltmeter is removed. When that occurs,record the current reading without the voltmeter as your measured value.

You will find very different currents in the forward and backwarddirections. After making a graph which shows both positive and negativecurrents on the same scale, you will have to make a second graph with anew scale for the very low current data (so that it can be better seen). Fromslopes on the graphs, find representative values of the resistance undervarious circumstances. Discuss your findings.

Kirchhoff’s Laws


This experiment provides you with the opportunity to familiarize yourselfwith and test Kirchhoff’s current and potential laws:

1. the algebraic sum of the potential changes around a closed circuit pathmust add up to zero;

2. the algebraic sum of the currents flowing into a junction point must addup to zero. You will also learn a very useful principle governing howvoltage is divided across a set of series resistors.

APPARATUS

Variable power supply, multimeters, resistors, light bulbs, a potentiometer,hook-up wire, and a breadboard.

PROCEDURE I: PARALLEL CIRCUITS

Connect three resistors in parallel across the power supply as shown inFigure 1. Each resistor should have a different value of resistance, but noneshould be less than 200 ohms. Be sure to measure the resistance of eachresistor with your multimeter. Choose a convenient power supply voltage(between 2 V and 7 V) and measure the voltage across every element in thecircuit (including the power supply). Also measure the current coming outof and going into each circuit element.

A convenient way to record your data is to draw a large diagram inyour notebook and record the resistance values, voltage differences, andcurrents directly on the diagram.

ANALYSIS I

1. Show explicitly that your data confirms both the current and potential

xi

Kirchhoff’s Laws xii

R1 R2 R3

+

–

powersupply

Figure 1 Schematic diagram for the circuit used in procedure I.

laws of Kirchhoff.

2. Calculate the equivalent resistance of your three parallel resistors fromthe parallel resistor rule (1/Requiv = 1/R1 + 1/R2 + 1/R3). Then useOhm’s law to calculate how much current such a single resistor woulddraw if it were to replace the three original resistors, assuming thepower supply voltage remains the same. Compare this value with yourmeasured value for the total current drawn from the power supply byyour three parallel resistors.

PROCEDURE II: SERIES CIRCUITS

1. Connect three resistors in series across the power supply as shown infigure 2. Each resistor should have a different resistance, and the sumof the three resistances should be greater than 100 ohms. Be sure tomeasure the resistance of each resistor with your multimeter. Choosea convenient voltage (between 2 V and 7 V) for the power supply andmeasure the voltage across every element in the circuit (including thepower supply). Also measure the current coming out of and going intoeach circuit element.

R1 R2 R3+

–

powersupply

Figure 2 Schematic diagram for the circuit used in procedure II.

ANALYSIS II

1. Show explicitly that your data confirms both the current and potentiallaws of Kirchhoff.

2. Calculate the equivalent resistance of your three series resistors fromthe series resistor rule (Requiv = R1 + R2 + R3). Then use Ohm’s lawto calculate how much current such a single resistor would draw if itwere to replace the three original resistors, assuming the power supplyvoltage remains the same. Compare this value with your measuredvalue for the total current drawn from the power supply by your threeseries resistors.

Kirchhoff’s Laws xiii

3. It is easy to prove from Ohm’s Law and Kirchhoff’s Laws that in sucha set of series resistors, the voltage applied across the ends of the setby the power supply is divided among the individual resistors pro-portionally to their individual resistances. That is, the ratio of thevoltages across any two of the resistors is the same as the ratio oftheir resistances. That is,

R1

R2=V1

V2. (1)

Verify that this holds true for your circuit.

4. Another way of stating this principle is to say that the fraction of thepower supply voltage which appears across any individual resistor,is equal to the fraction of the total series resistance possessed by theresistor in question. That is,

V2

Vpower supply=

R2

(R1 +R2 + · · ·) . (2)

Verify that this holds true for your series circuit.

PROCEDURE III

A potentiometer can be represented schematically in the following way.

+

–

A

B

C

Out

put t

erm

inal

s

Figure 3 Schematic representation of a potentiometer.

The two outer terminals are connected to the ends of a long resistor,and the middle terminal is connected to a sliding contact which can bemoved along the resistor, making contact at any point between the twoends. The resistance between the outer terminals remains constant, but theresistance between the middle terminal and either outer terminal changesas the sliding contact is moved by rotating the control shaft. As the contactis moved to the right, the resistance between B and C decreases while theresistance between A and B increases. The opposite effect is producedwhen the contact is moved to the left.

1. Verify the behavior of the resistance between the terminals of the po-tentiometer by turning the control shaft and measuring the resistancebetween various terminals with your multimeter. Record your obser-vations in your lab book.

Kirchhoff’s Laws xiv

2. Connect the outer terminals of your potentiometer to the power supplyto produce a variable voltage source. Set the power supply to 5 volts.The output voltage between the middle terminal and either outer ter-minal will vary as the potentiometer shaft is turned.

(a) Connect your light bulb between terminals B and C and observe andrecord its behavior as the control shaft is turned.

(b) Measure and record the range of output voltages available betweenB and C when the power supply is set at 5 volts.

(c) Disconnect the light bulb and adjust the potentiometer for an outputbetween B and C of 3 volts, with the power supply still set at 5volts. Disconnect the potentiometer from the circuit, measure theresistance between each pair of terminals, and verify that the voltagedivider relationship holds.

PROCEDURE IV

1. Set the power supply voltage to 5 volts. Connect first one, then two,and then three light bulbs to it in series.

(a) Observe and record the behavior of the light bulbs as more bulbsare added to the circuit.

(b) With three bulbs still connected to the power supply, remove one ofthe bulbs from its socket and observe and record what happens.

2. Repeat the measurements and observations of part 1, but this time wiresuccessive bulbs in parallel to the power supply.

ANALYSIS IV

Explain clearly why the bulbs behave as they do in Procedure IV.

RC Circuits


In this experiment you will study the time behavior of the current andvoltage in a charging and discharging RC circuit. The values of R andC have been chosen so that the times involved can be observed directlyand measured with a stop watch. Later we will return to this circuit anduse an oscilloscope to explore it again for characteristic times much toosmall to be observed directly. The circuit we will use is shown in Figure 1.The switching action shown is achieved by connecting one power leadconnector (banana plug) to the negative terminal of the power and leavingit there, and connecting the other power lead connector alternately to thepositive and then the negative terminals (the banana plugs can be stackedon top of one another when both are connected to the negative terminal).When the switching banana plug is connected to the positive terminal,the capacitor is being charged by the power supply voltage; when it is

R

+

–

powersupply

A

ammeter

V

voltm

eter+

–C

Figure 1 The RC circuit used in this laboratory.

connected to the negative terminal, the power supply is bypassed, thevoltage to the circuit is zero, and the capacitor can discharge back throughthe power supply leads.

The expressions which describe the time dependence of the circuit areas follows.

1. If the capacitor is initially discharged and the clock is started when thepower supply is connected in to the circuit, then;

I(t) = I0 exp(−t/RC) and Vc(t) = V0(1− exp(−t/RC)) , (1)

where I(t) is the current through the resistor and power supply, Vc(t) isthe voltage across the capacitor, I0 = V0/R, and V0 is the power supply

xv

RC Circuits xvi

voltage. Note that ifR is measured in ohms andC is measured in farads(F), then RC will have units of seconds.

2. After the capacitor is completely charged (to the voltage V0), if thepower supply is bypassed by switching the power lead to the negativeterminal on the power supply, and the clock is restarted, then;

I(t) = −I0 exp(−t/RC) and Vc(t) = V0 exp(−t/RC) . (2)

APPARATUS

Power supply, multimeters, resistors, light bulb, stop watch, hook-up wire,bread board.

PROCEDURE I

Build the circuit of Figure 1 using the 0.47 F, NEC capacitor for C and yourlight bulb for R. Be sure to connect the positive side of the circuit to theterminal on the capacitor marked with two lines, and set the output of thepower supply to not more than 5.5 V so that the incandescent lamp won’tbe burned out. Also be sure that each multimeter is connected properlyand set properly for the function it is supposed to serve. Connect thepower supply into the circuit and watch the behavior of the light bulb andthe meters. After the bulb no longer glows, bypass the power supply byswitching the power lead to the negative terminal of the power supply andagain observe the behavior of the light bulb and the meters.

ANALYSIS I

Record your observations along with an explanation for the behavior youobserved.

PROCEDURE II

Rebuild the circuit of Fig. 1, this time using the 1000 µF capacitor for Cand a 0.1 MΩ resistor for R. Measure and record the resistance of theresistor. Connect the power supply and simultaneously start your stopwatch. Record values for the voltage across the capacitor and the currentinto it at 30 second intervals, including t = 0. Your currents will be mostconveniently expressed in µA (10−6 A).

After the capacitor is completely charged, bypass the power supplyand simultaneously start the stop watch. Again, record values for thevoltage and current at 30 second intervals, including t = 0.

RC Circuits xvii

ANALYSIS II

1. For the data taken while charging the capacitor, plot Vc(t), I(t), andln[I(t)] versus time in seconds. By hand, draw a smooth curve throughthe data points.

2. Are your plots of I(t) and Vc(t) consistent with equations (1)?

3. By taking the natural logarithm of both sides of Eq. (1a), one obtains:

ln(I(t)) = ln(I0)− t/RC (3)

Thus the ln(I) plot should be a straight line with a slope equal to−1/RC .If it is, calculate a value for RC from the slope. If it is not, use thoseinitial points which do fall on a straight line to find RC . Compare thisvalue with one calculated from the measured value for the resistanceand C = 1000 µF.

4. For the data taken while discharging the capacitor, plot Vc(t), I(t), andln[Vc(t)] as a function of time in seconds.

5. Are your plots of I(t) and Vc(t) consistent with equations (2)?

6. Is your ln[Vc(t)] plot a straight line? Find the slope using that part ofyour data which lies on a straight line, and from the slope calculate avalue for RC . Compare this value with that obtained by multiplyingthe values for R and C .

7. Plot Vc(t) for the discharging case on semi-log paper. Taking the log (tobase 10) of both sides of equation (2b) we obtain:

log(Vc) = log(V0)−t

RClog(e) . (4)

So this plot should also give a straight line, this time with slope =− log(e)/RC . Calculate a value for RC from your slope. When mea-suring the slope of a line on semi-log paper (a sample of which is at theend of the chapter), choose a triangle whose “rise” is exactly one cycleon the log paper, and set the rise numerically equal to 1.

8. Find the “run” in the usual way from the change in time on the x-axis.

RC Circuits xviii

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1

1

1

2

2

2

2

3

3

3

3

4

4

4

4

5

5

5

5

6

6

6

6

7

7

7

7

8

8

8

8

9

9

9

9

Magnetic Field of a CurrentLoop

GOALS

In this lab you will explore the properties of the magnetic field generated byan electric current flowing through a circular coil of wire. Theory predictsthat the field at the center of such coils is given by:

Bcoil = µ0NI

d. (1)

You will explore how the field at the center of the coil depends on thecurrent, the number of turns, and the diameter of the coil, and comparewhat you find with the theoretical prediction.

APPARATUS

The apparatus you will use is depicted in figure 1 below.

Coil

Power Supply

Compass

Figure 1

It consists of a cylindrical form about which any number of turns ofwire may be wrapped. The resulting coil is connected to a variable currentsupply with a built-in current meter. A magnetic compass is supported atthe center of the coil by a wood shelf. Slots in the cylinder are used to viewthe compass needle from directly overhead and to read the angle by whichit is deflected.

xix

Magnetic Field of a Current Loop xx

EXPERIMENTAL METHOD

The circuit enables us to adjust the current through a wire coil, and measureit in Amps. How do we go about measuring the magnetic field at the centerof that coil? To do this directly would be quite difficult. Fortunately, theearth has a magnetic field, and it is rather easy to measure the ratio betweenthe magnitude of the magnetic field created by the coil and the magnitudeof the earth’s magnetic field. We know from class (or from symmetryconsiderations) that the magnetic field of the coil points along the axis ofthe cylinder about which the coil is wrapped. Suppose we align this axisperpendicular to the earth’s magnetic field. Then the total magnetic field isgiven by the vector sum of these two fields (see Fig. 2a), and the magnitudesof the earth’s and the coil’s magnetic fields are related by

Bcoil = Bearth tan θ . (2)

θ

Bcoil

Bearth

Bnet

wire

Compass

Top View

(a) (b)

Figure 2

Thus if we measure tan(θ), we can find an experimental value forBcoil in terms of the magnetic field of the earth. For most of the analysis(including all of the graphs) we can set Bearth = 1, remembering that all ofour magnetic field values are then being expressed in terms of a unit equalto the magnitude of the earth’s magnetic field. Only when you are asked tofind a value for µ0 will you need to use the value for the earth’s magneticfield, so that your result will be expressed in proper MKS units (Tesla formagnetic fields) and can be compared to the accepted value.

All we need to do is to align these two magnetic fields perpendicularto each other and devise a scheme for measuring the direction of their sum.Both these things are easy to do with a compass, since a compass needlepoints in the direction of the total magnetic field it experiences. Place thecompass at the center of the coil, and with zero current in the coil (and thuswith zero magnetic field from the coil), orient the hoop so that the compassneedle (and hence the earth’s magnetic field) points along the diameter ofthe hoop (see Fig. 2b).

Now when there is a current through the wire coil, the magnetic fieldof the coil will be perpendicular to the magnetic field of the earth, as wedesired. We can now measure the angle θ for whatever configurations ofthe apparatus are of interest to us, and thus determine how the magneticfield of the coil Bcoil varies as we vary the configuration of the apparatus.

Magnetic Field of a Current Loop xxi

PROCEDURE I: DEPENDENCE OF FIELD ON CURRENT

StudyBcoil as a function of the current I through the circuit. Using the next-to-the-smallest cylinder, create a coil of five turns by wrapping the wirestightly around the cylinder, keeping the wires close together and alignedalong the mid-line of the cylinder (use masking tape to keep the wire inplace). After making five complete turns, the wire should be bent to comeaway from the cylinder parallel to the symmetry axis of the cylinder (seeFig. 1). Be sure that the point where the wires come away from the coil isopposite the viewing slots cut into the cylinder. Place the cylinder onto thewooden cradle which will hold it in a stable position.

Place the support shelf in the cylinder and set the compass on it. Makesure the shelf is level and the compass case is oriented parallel to theguidelines drawn on it. Orient the cylinder and cradle until the compassneedle points directly along the diameter of the cylinder. Then tape thecradle to the table to make it easy to return the coil to this orientation. Nowadjust the rotating scale on the compass so that the compass needle pointsto N (0) when it is parallel to the cylinder diameter. This will make it easierto read the angle I directly from the scale of the compass when you makeyour deflection measurements. Once all of these adjustments have beenmade, tape the other chock block to the table to keep the coil in place. Besure that the hoop is still aligned correctly after you have finished tapingthings down.

The wires leading to the coil also generate magnetic fields. We wouldlike to minimize their presence. To do this, tape both wires close togetheronto the table, keeping them parallel to the symmetry axis of the coiluntil they reach the currently supply. Since current is running in oppositedirections in each wire, keeping them close together will result in theirindividual magnetic field contributions being in nearly opposite directionsand thus canceling each other out. Ferrous metals in the lab can also affectyour magnetic field readings, so keep the space under the table free of metalstools, make sure that the drawers in the table don’t contain metal toolsand equipment, and keep your current supply as faraway from the coil asconveniently possible.

At this point you are finally ready to make some measurements. Mea-sure deflection angles (θ) for various values of the current up to a maximumof about 70. For deflections larger than 70 the angular uncertainties willintroduce excessively large uncertainties into your results for Bcoil. Makeone measurement with zero current and at least 5 measurements with non-zero current. Be sure to estimate the uncertainty in I for each measurement.From your deflection measurements calculate values for the magnet field(see Eq. 1).

PROCEDURE II: DEPENDENCE OF FIELD ON NUMBER OF TURNS

Study Bcoil as a function of N , the number of loops of wire on the coil.Using the same cylinder as in Procedure 1, vary the number of loops onthe coil from 5 to 1, keeping the current constant for all the measurements

Magnetic Field of a Current Loop xxii

in this step. Choose a value of the current which gives a deflection ofapproximately 70 with 5 turns of wire. Finally, also take a measurementwith zero turns of wire (but keep the rest of the apparatus the same, so thatthe wire should come up to the hoop and then go away, without wrappingaround it).

PROCEDURE III: DEPENDENCE OF FIELD ON DIAMETER

StudyBcoil as a function of the diameter, d, of the coil. There are cylinders ofdifferent diameters on your table. For each one, wrap a wire coil with threeloops around it and measure the magnetic field produced by a particularvalue of the current. The current should be the same for every measurementin this step, and should be chosen so as to give a deflection of approximately70 for the smallest coil.

ANALYSIS

The field at the center of such coils is given by:

Bcoil = µ0NI

d. (3)

For each procedure, verify this relationship between Bcoil and the pa-rameter being varied (current I , number of turns N , and diameter d, re-spectively). To do this, plotBcoil versus I ,N and 1/d respectively, and lookfor linear behavior.

Include error bars on each of your plots. The errors in deflectionangles will outweigh all other sources of uncertainty. From the uncer-tainties in your values for θ (dθ), you can calculate the corresponding un-certainties in your measured B values (dB) using the following equation:dB = dθ/ cos2 θ. Note: In this equation, dθ must be expressed in radians. Youwill find that the effects of the angular uncertainties on Bcoil will be quitedifferent for small angles than for large angles, so be sure to include errorbars for every point on your graphs.

For each graph draw a straight line which best fits your data points.Measure the slopes of your lines and their uncertainties. An uncertainty inslope may be measured by finding an alternative line which is consistentwith your data and measuring the difference between the two slopes. Sinceyour graphs employ Bearth as the unit of magnetic field, your slopes won’tbe in standard MKS units. To convert slope values to MKS units so as toget the standard value for µ0 , 4π × 10−7 T·m/A, multiply each slope bythe magnetic field of the earth in Tesla (1.6 × 10−5 T). Using the slopes ofthese three lines, calculate three values of µ0, one for each graph (see Eq. 2).Calculate the uncertainties in your µ0 values, from the uncertainties in yourvalues for the slopes. Are your three values consistent with each other?Are they consistent with the value for µ0 found in your textbook?

Prisms and Lenses

REFRACTION

When light passes from one material into another, in general the direction ofthe light ray changes as it crosses the interface between the two materials.This phenomenon is called refraction, and the relationship between theincident and refracted angles (measured with respect to the normal to theinterface) is given by Snell’s Law,

n1 sin θ1 = n2 sin θ2 ,

where the angles are measured from the normal to the interface betweenthe materials, as shown in Fig. 1

θ

θ2

1

nn

1

2

Figure 1 The angles used in Snell’s Law.

All of the imaging produced by lenses and mirrors can be understoodin terms of this law and the law of reflection; together they constitute theprinciples of geometric optics.

PROCEDURE I: REFRACTION

The first exercise is to observe refraction at the two surfaces of a 60o prismand to measure the index of refraction of the prism. This is accomplishedby tracing a line of sight (ray tracing) through the prism, measuring theincident and refracted angles, and using Snell’s Law to calculate the indexof refraction.

Place your 60 prism on the pin-table with a clean sheet of plain paperunder it. Stick pins into the paper and table at positions B and C. Both pinsshould be placed right up against the prism so that they are in contact withthe glass, and both should be placed near the base of the prism so that line

1

Prisms and Lenses 2

BC is as long as possible (see Fig. 2). With your eye at the level of the prism,sight with this eye only until you see the image of pin through the prism.Move your head and eye until pin C and the image of pin B are lined up.Then, continuing to sight through the prism, place pin A about 15 cm awayfrom B so that pin C and the images of A and B are lined up one behind theother. Now add pin D about 15 cm in front of pin C so that pins C and Dare in line with the images of pins A and B. Be careful that the prism doesnot move while all of this is going on.

A

BC

D

Figure 2

Carefully remove the pins and use a fine-lead pencil to trace the outlineof the prism on the paper, being careful not to allow the prism to move whilethis is being done. Then remove the prism from the paper, circle and labelthe holes made by the pins, and use them to draw the rays representing theline of sight through the prism (see the Fig. 2).

Now use a triangle to carefully construct lines perpendicular to theinterfaces at points B and C, and measure the angles of incidence andrefraction at both interfaces. You will have to extend the line BC in bothdirections in order to read these angles off of the protractor. In order toobtain good results, the ray-lines and normals must be constructed withgreat care.

ANALYSIS I

Use Snell’s Law at each interface to calculate the index of refraction of theprism from the measured angles. You will obtain two values of n in thisway, one from each interface. You can take the index of refraction of air tobe 1.00. Compare the two indices of refraction with each other and withtypical values for glass. Are they in good agreement?

Your two n values will probably not be exactly the same. Are theywithin experimental error of each other? To estimate the error in your mea-surement place the paper back on the table without the prism. Stick twopins in it about 15 cm apart. Now with your eye at table level, sight alongthe two pins and place a third pin about 15 cm beyond the farthest pin insuch a way that it appears to line up with the other two. Now remove thepins and use a straight edge to test whether or not the holes lie on a singlestraight line. If they don’t, measure the angle by which the two connectinglines differ. This angle can be taken as the uncertainty introduced intoyour angles by the sighting procedure. In addition, estimate the angularuncertainties introduced by your geometrical constructions and protractormeasurements. Use the quadrature method to add these two angular un-certainties (square, add, and take the square root) to obtain an uncertainty

Prisms and Lenses 3

value for your measured angles. Now repeat one of your Snell’s Law cal-culations using angle values which have been changed by the amount ofyour calculated uncertainty. Increase one of the angles by that amount anddecrease the other angle by the same amount. This gives the worst casescenario. The amount by which this new value for n differs from the oldwill give you an idea of the uncertainty in your n values. Can the differencebetween the n values measured at the two different interfaces be accountedfor by your uncertainty calculation?

LENS OPTICS

As you are learning in class, a refracting material, such as glass, with curvedsurfaces is capable of bending light rays,and when the curved surfaces havethe proper shape, can focus all of the rays that it intercepts from a pointsource, back to another point. Figure 3 illustrates this for a point sourceon the optic axis of the lens, and figure 4 illustrates it for a point source offof the optic axis. The distance of the focus points from the lens dependsonly on the distance of the source points from the lens. Thus a plane objectsource will have all of its points focused onto a plane on the other side ofthe lens, forming an image of the source object.

s s'

ImageObject

Figure 3

s s'

f f

Y

Y'

Image

Object

Figure 4

The distance of the object from the lens,s is related to the distance ofthe image from the lens s′ by the Gaussian lens formula:

1s

+1s′

=1f

; or f =s s′

s+ s′.

where f is a characteristic constant of the lens called the focal length; itdepends on the curvature of the lens surface and the index of refraction of

Prisms and Lenses 4

the lens material. The object height, the image height, and the central raycreate two similar triangles from which we conclude that:

Y ′

Y= − s

′

s= m ,

where m is the magnification. The minus sign in this expression allows anegative value for m to be interpreted as representing an inverted image.

PROCEDURE II: LENS OPTICS

Your experimental equipment includes an optical bench with a built-indistance scale, lens holders, an illuminated arrow to serve as an object, ascreen on which a real image can be projected, a ruler for measuring objectand image sizes, and a lens.

1. Mount the lens with a 10 cm focal length in a lens holder and place it40 cm from the object. Adjust the position of the screen until the sharpestpossible image of the object appears on it. Measure the following.

(a) the object distance s

(b) the image distance s′

(c) the size of the object y

(d) the size of the image y′.

2. Test the precision with which you can measure s′ by moving the screenand then readjusting its position to obtain a sharply focused image.Have your lab partner do it if you did it the first time. Do this three tofour times and record your results. By how much do these values differfrom each other and from your original measurement? This providesyou with an uncertainty value for your s′ measurement. You mayassume that the uncertainty in s can be neglected compared to that ins′.

ANALYSIS II: LENS OPTICS

1. Calculate the quantities

(a) the focal length, f , of the lens and its uncertainty;

(b) the magnification m = y′/y;

(c) the ratio s′/s. (Theory states −s′/s = m).

Note: the uncertainty in f may conveniently be calculated by using

∆f =(f

s′

)2∆s′ . (1)

Prisms and Lenses 5

2. Carefully draw to scale a ray diagram using the measured values forY and s, and the focal length calculated in 1. What does this diagrampredict for Y ′ and s′? Compare these predictions with your measure-ments.

PROCEDURE III: LENS OPTICS

For the next measurement you will use the concept of parallax. This isthe displacement of an object relative to distant objects when viewed fromdifferent directions. Check this out by holding out two fingers (one on eachhand) at different distances from you and approximately lined up. Closeone eye and move your head from side to side and observe how they movewith respect to each other and with respect to the motion of your head.

Now remove the screen from its holder and replace it with the metalpointer. Place your eye (have one eye closed) at the end of the optical bench(the end away from the object) and look towards the object, so that you cansee the image of the object through the lens and the pointer in the same lineof sight. The height of the pointer can be adjusted to facilitate this. Sincethe pointer and the image are at the same location, they should exhibit noparallax; this means that if you move your head from side to side, the imageand the pointer should move together in your field of vision, i.e., they willmaintain a constant distance between them.

To see what happens when the pointer and the image are not at thesame location, move the pointer five or more centimeters closer to the lensand observe what happens when you move your head. Which moves inthe opposite direction from your head relative to the other, the pointer orthe image? Now move the pointer two or more centimeters farther fromthe lens and observe what happens when you move your head? Whichmoves in the opposite direction from your head in this case? By observingwhich of two objects moves opposite your head motion, you can determinewhich is closer to you.

Locate the position of the image again by moving the pointer untilyou have eliminated the parallax entirely. Make this measurement inde-pendently more than once in order to estimate the uncertainty in an s’measured in this way. Be sure to record these values. In later experimentsyou will need to use this method for locating virtual images which cannotbe projected on a screen.

ANALYSIS III: LENS OPTICS

Compare the value for s′ obtained this way with the value obtained usinga real image on the screen.

Prisms and Lenses 6

PROCEDURE IV: LENS OPTICS

Return the screen to the optical bench and use it to measure the imagedistance s′ for four additional object distances. Choose a range of objectdistances which are as large and as small as your optical bench will accom-modate. Try to achieve one case for which s = s′. Measure the image sizefor each case.

ANALYSIS IV

1. Calculate an f value and its uncertainty for each of these four measure-ments. Put all five of your calculated f values (including that from part2.) together in a table for easy comparison. Are they all equal? Makeanother table containingm = y′/y and−s′/s values for every measuredcase. Do the −s′/s values seem to be in reasonable agreement with themagnification values m?

2. From your data make a plot of 1/s′ versus 1/s.

1s

+1s′

=1f⇒ 1

s′= − 1

s+

1f.

Thus plotting 1/s′ versus 1/s should result in a straight line with slope= −1 and y-intercept = 1/f if the lens equation holds. Does your graphconfirm the lens equation? What value for the focal length does it give?Compare this value with those found in part 1.

Diffraction Grating

INTRODUCTION

The spectrometer which you will be using in this experiment (see figurebelow) consists of a collimator tube with an adjustable entrance slit, a ro-tatable table on which the diffraction grating can be mounted, a telescopefor observing the light diffracted from the grating, and an accurate angu-lar scale for measuring the relative directions of the various spectrometercomponents. Your instructor will show you how to use the controls andread the vernier. The collimator has been pre-adjusted so you SHOULDNOT CHANGE THE COLLIMATOR FOCUS CONTROL.

Place your Na lamp in front of the collimator slit, and move the tele-scope until you can observe the illuminated slit through the telescope.Adjust the position of the light source to maximize the brightness of theslit. Adjust the slit width so that it is fairly narrow and adjust both thetelescope focus and the telescope eyepiece so that both the cross hairs andthe illuminated slit are in good focus. Try to eliminate any parallax betweenthe slit image and the cross hairs by the alternate adjustment of the twotelescope controls.

Now mount the grating in a vertical position on the table in the centerof the spectrometer, and adjust it so that its plane is perpendicular to thelight emerging from the collimator and passes approximately through thecenter of the table. Be sure that the vertical lines of the grating run up-and-down and not from side-to-side. Scan the telescope from side to sideto observe the various diffracted lines. The most pronounced lines for Naare a bright yellow doublet, but you will also see fainter lines of otherwavelengths.

The angular position of a line is measured by setting the telescope eye-piece cross hairs on that line and measuring the corresponding angle fromthe scale. The diffraction angle θ for any particular line is the differencebetween the angle measured at zero order (straight on) and the angle mea-sured for that line at some higher order (N = 1, 2, or 3). Each wavelengthwill be diffracted by the same amount to either the right or the left of thezero order maximum, and so diffraction angles measured on either sideshould be equal. For instrumental reasons they often differ slightly, andso a better way to measure θ is to subtract the angular position of the lineon one side of zero order from the angular position of the same line on theother side of zero order, and divide this difference by two.

7


θθ

λn

from collimator

to telescopegr

atin

g

d

Figure 1 Parallel light rays from a collimator diffracted at an angle θinterfere constructively when the path length difference between to adjacentslits, d sin θ, is a multiple of the wavelength, nλ.

E) Telescope

F) focus ring

K) prism table clamp

G,H) telescope clampfine adjust

L) prism table clampfine adjust

I, J) angle scale& vernier

I, J) angle scale& vernier A) grating

B) adjustable slit(twist)

C) Collimator

M) prism tablelevellingscrews

Figure 2 A spectrometer.

Using this method, measure the diffraction angle for both membersof the Na yellow doublet in both first and second order. (Caution: theleft member of the doublet when observing to the left of zero ordercorresponds to the right member of the doublet when observing to theright of zero order.) From these angles, and the known wavelengths of thelines in the doublet (589.0 and 589.6 nm) calculate four separate values ford, the spacing between the slits on your grating. Also calculate the spacingfrom the lines/mm specification stamped on the grating. Make a table ofall of these d values for easy comparison.

When making your measurements, be sure that the spectrometer ta-ble is locked and cannot turn, be sure that the diffraction grating is notdisturbed throughout the course of the measurement, and keep the slitwidth as narrow as possible, consistent with the need for visual clarity. Forweaker (dimmer) lines, you will have to open the slit wider in order to


observe them.

Replace the sodium lamp with a mercury spectral tube, and measurethe wavelengths of three Hg emission lines, using second order diffractionlines. Note the color of each line beside its diffraction angle. Comparethe appearance and intensity of the second order lines with that of thefirst order lines, and record your observations. The yellow doublet will beparticularly revealing in this regard.

Replace the mercury spectral tube with an atomic hydrogen spectraltube, for which you will be able to observe four emission lines from theBalmer series, ranging from red to deep violet. The fourth line, in the deepviolet, will be barely visible in first order, and may not be visible at all insecond order. You may have to open the slit much wider to see this line.Measure the wavelength of each of these lines in second order (except forthe fourth line which will have to be measured in first order if you can’t seeit in second order). Make a table of wavelength values.

n6543

2

∞

1

serieslimit

serieslimit

Lymanseries

Balmerseries

E , eVn0.00

–13.6

–3.40

–1.51–0.85

Energy level diagramfor Hydrogenshowing transitionsfor the Balmer andLyman series.

The Bohr model of the hydrogen atom results in a very simple formulafor the allowed energy levels of the single electron,

En =−13.6 eV

n2 (1)

where n is the principal quantum number of the electron and can take oninteger values from 1 to infinity (it has nothing to do with the diffractionorder N ). The figure to the left shows the energy-level diagram for theelectron in hydrogen which results. When an electron drops from a higherto a lower energy level, it emits a photon of light which carries away theenergy lost by the electron. The energy of the photon is hf = hc/λ, whereh = Planck’s constant, f and λ are the frequency and wavelength of thelight respectively, and c is the velocity of light. So the energy levels resultin the emission of light at discrete wavelengths.

The Balmer series is a series of spectral lines in the visible part of theelectromagnetic spectrum, which arise from photons produced when anelectron falls to the n = 2 level from higher levels (n = 3, 4, 5, . . .). UsingEq. (1) and the relationship c = λf , it can easily be shown that thesewavelengths are given by the following equation:

hc

λ= 13.6 eV(1/22 − 1/n2) , (2)

where n = 3 (red), 4 (blue), 5 (violet) and 6 (deep violet). Putting in theappropriate values for the constants, one obtains:

1/λ = 10.97µm−1(1/4− 1/n2) . (3)

Plot the reciprocal of your wavelengths against 1/n2, and see if theslope and y-intercept support the Bohr model of the atom.


APPENDIX A: READING THE VERNIER

It’s a little tricky to get a reading of the angle from the vernier on theequipment. In this case, you have to read the angle in degrees and minutes,not a decimal degree.

210˚ 15'

10 20 300

210 220

10 20 300

230 240

229˚ 46'

Figure 3 The vernier in the leftmost figure shows the zero mark between210 and 21030′. The 15 line above the scale aligns best with another lineso the reading is 21015′. Similarly, the rightmost figure shows zero markbetween 22930′ and 230. The 16 line above the scale aligns best withanother line, so the angle is 22930′ + 16′ = 22946′.

Notes on Experimental Errors

EXPERIMENTAL ERROR AND UNCERTAINTY

We generally speak of two broad classes of errors – random and systematic.The term random error refers to the random and unpredictable variationsthat occur when a measurement is repeated several times with non-identicalresults, although the results do cluster around the true value. The stop-watch or photogate time measurements that you have performed are verygood examples of this.

xMeasured values of x

i

True value of x

Figure 1 A set of measurements that has a fair amount of random error,but little, if any, systematic error.

Systematic errors are those errors in a measurement that are regularand consistent in the sense that the measurements are consistently too largeor too small, and usually by about the same amount for each measurement.The manual measurement of a time period would be an example of sys-tematic error if the person is consistently late in stopping the clock.

xMeasured values of x

iTrue value of x

Figure 2 A set of measurements that has a fair amount of random error, aswell as significant systematic error.

Other examples of sources of systematic error would be a stop watchthat ran consistently fast or slow, and a meter stick that was mistakenlymanufactured too long or too short.

Any actual measurement will ordinarily be liable to both random and

11


systematic errors. The random errors reveal their presence much morereadily, and can usually be observed by repeating the measurement severaltimes and observing the differences in the values obtained. The measuredvalues will be observed to scatter themselves in some random way aboutan average value.

Systematic errors are usually much more difficult to detect and mea-sure. We are often unaware of their existence. If one is measuring some-thing whose value is already thought to be known, and consistently ob-tains a value that is significantly different, one is certainly led to suspectthe presence of one or more systematic errors. (The alternative is that theexperiments that led to the accepted value contained the systematic error.)However, if one is measuring something that has never been measured be-fore, any systematic error present will not be obvious, and great care mustbe taken to investigate and eliminate every source of systematic error thatone can think of. Experience and ingenuity are the valuable assets in suchan investigation. Obviously sometimes things are overlooked and valuespublished that later turn out to be wrong.

The remainder of these notes will confine themselves to a discussionof random errors. There are several reasons for doing this.

• A discussion of systematic error depends critically on the experimentbeing performed, and cannot easily be generalized.

• Random errors are present and significant in most experiments.

• Random errors are susceptible to a general mathematical treatment bymeans of which their effects can be reduced.

Before going on, it would be well to comment briefly on the use of theword “error.” Random error might better be called random uncertainty.It is the amount by which our measured values uncontrollably fluctuate,and thus provides a measure of our lack of certainty about the value of thequantity we are attempting to measure. The word error connotes mistake,and random error should not be thought of in those terms. The word erroris more appropriate in the term, systematic error, where something morelike a mistake is actually being made. Nevertheless, we will continue to usethe word “error” to denote random uncertainties, because it is so firmlyentrenched in our customary usage.

We will frequently use the term experimental error. Experimental erroris simply total uncertainty in any measured quantity, or any quantity calculatedfrom measured quantities. It is not the amount by which your value differs fromthe generally accepted value. Experimental error will generally have bothsystematic and random components, and one of the most important tasksof the experimenter is to measure, estimate, and assess the experimental“uncertainty” of all experimental and experimentally derived quantities.

How does one go about doing that? Consider first, directly measuredquantities like length of time. One may repeat the experiment several times,taking care to make each measurement independent of the others. Thevariation in these values will enable us to measure the random “error” inthe measurement, using a method to be discussed later in these notes. Manymeasurements will involve reading a scale and interpolating between the


marks on the scale, as well as zeroing the scale at one end or the other. Suchinterpolations cannot be perfect and one is often content to record as theexperimental error simply the uncertainty in one’s mind when making theinterpolation. For example, consider measuring the length of a solid body.If a meter stick is used to measure this length and every measurement yieldsthe same value, we would say that the “error” in the measurement is the“least count” of the meter stick, namely 1 mm. So, we might quote the erroras ± 1 mm. Sometimes the uncertainty will be determined by consultingthe specifications for the equipment being used, where the manufacturerwill state the limits of accuracy of the equipment. All of the possibilitiescannot be covered here, and you will have to use your common sense andingenuity as particular situations arise.

In any event, no experimental measurement is complete until a value of theuncertainty, or experimental error, is determined. Thus every entry in the datapages of your lab book must be accompanied by a value of its uncertainty.This is usually expressed in the following form:

x = 35.7± 0.3 cm

If the experimental uncertainty is the same for a whole series of dataentries, it would be sufficient to record the uncertainty for one entry andindicate that it is the same for all of the others.

One never needs more than two significant figures to specify an exper-imental error, and usually a single significant figure is sufficient. The valueof the measured quantity should be carried out as many decimal places asnecessary so as to include the decimal places used in the experimental error— no more and no less. The following examples illustrate this point.

Correct Incorrect3.4423± 0.0002 s 3.54073± 0.12 m3.54± 0.12 m 3.5± 0.12 kg3.5± 0.1 kg 3± 0.1 m/s

As a final note, NEVER use the term “human error” in discussingyour experimental error. It is meaningless; which is why so many stu-dents are tempted to use it. If you do not know, it is better to simply sayso.

MEAN & STANDARD DEVIATION

Very often an experimentalist will measure a quantity repeatedly, and thenuse an average of the measured values as the final result. Why is thatdone? what is the justification for it? and what is the uncertainty or errorassociated with that average? This is the classic example of random errorand its treatment. If we plot each individual measurement on an axis withappropriate scale and units, it might look something like this.

Such a plot provides a clear and graphic description of the uncertainty,or experimental error, associated with this measurement. If the errors in


xi

Mean

Figure 3 A set of measurements and its mean.

these measurements are truly random, then one would expect that mea-sured values would be larger than the “true value” just as often as theywould be smaller, and by similar amounts. This in turn implies that the“true value” lies somewhere in the center of the distribution of measuredvalues. You might choose a best value by simply using your eye to find thecenter of the distribution. A mathematical procedure that everyone wouldagree upon would of course be better. There is such a procedure, and it iscalled calculating the mean, or average. You already know how to do that.In what sense does that find the center of the distribution?

Let us call the value of the center of the distribution x, and defineit so that if we add up all the differences between it and the measureddata points, we will get zero. (A data point to the left of x will producea negative difference while a data point to the right of x will produce apositive difference.) This can be expressed mathematically in the followingway.

Definition of the Mean

The mean, x, is defined by the relation

N∑i=1

(xi − x) = 0 , (1)

where the xi are individual measurements.

Since the order in which we do the additions and subtractions does notaffect the outcome, this can be rewritten as

N∑i=1

xi −N∑i=1

x = 0, butN∑i=1

x = xN∑i=1

1 = Nx

N∑i=1

xi = Nx , which implies x =1N

N∑i=1

xi .

This last equation is just the ordinary prescription for calculating anaverage. Thus we have shown that the familiar average value of a set ofmeasurements coincides with a reasonable definition of the center of thedistribution of measured values.


Now the question arises, what experimental error should one associatewith each of the measured values in our example? A crude way of doingthis would be to choose the uncertainty value to give a range of values just aswide as that covered by the actual measurements. But this would probablyoverestimate the uncertainty, since a single widely divergent measurementwould make the spread quite large, and would not be truly representativeof most of the values.

Someone might suggest finding, on average, how much each individ-ual value differs from the mean. But we have already seen that the sumof all the deviations from the mean is zero, and therefore such an averagewould be zero — not a very helpful experimental error value. This methodcould be saved by finding the average of the absolute values of the devi-ations from the mean, thus eliminating all of the negative signs from thenegative deviations. Although this method could be used, another methodhas been practically universally accepted; that of the standard deviation.

Definition of the Standard Deviation

The standard deviation of the distribution is calculated in the followingway.

(a) Square each deviation. (This insures that we have only positivequantities to deal with.)

(b) Find the mean of these squared deviations.

(c) Take the square root of this mean. (We want a measure of the devia-tions, not their squares.)

(d) The result of these calculations is the standard deviation, σ, alsocalled the root mean square deviation (r.m.s. deviation) for obviousreasons.

Mathematically we can represent this as follows:

σ =

√√√√ 1N

N∑i=1

(xi − x)2 (2)

We will use the standard deviation as our best estimate of the randomexperimental error associated with each of the individual measurements.That is, each of the individual measurements is uncertain by an amount±σ, whichwe express by writing xi ± σ.

What justification do we have for using the standard deviation in thisway? It has been found that the random errors associated with many typesof measurements follow what is know as a normal probability distribution,which is illustrated in Fig. 1. The height of the curve above any point on thex-axis is proportional to the probability of obtaining an experimental resultvery close to that value of x. This curve meets our crude expectations thatthe likelihood of obtaining any measured value will grow less the more thevalue differs from the average value that lies at the center of the probability


curve. The left-right symmetry of the curve insures that measurements areequally likely to be too small or too large.

More precisely, the area under the probability curve between any twox values, x1 and x2, gives the probability that a measurement will resultin a value lying between x1 and x2. The total area under the curve out to±∞ must equal 1, since the probability that any measured value will liebetween −∞ and +∞ must be 1. But what does this have to do with thestandard deviation?

It can be shown that the area under the normal distribution betweenx− σ and x+ σ equals 0.67. This means that any individual measurementhas a probability of about 67% of lying within one standard deviationfrom the average of many measurements. Similarly it can be shown thatany individual measurement (which obeys the normal distribution) hasa probability of 95% of lying within two standard deviations from theaverage. Thus by using the standard deviation to represent the uncertaintyin our measured values, we are providing some very specific informationabout how much this value is likely to depart from the true average value.

x

− σ + σ

Probability( )x

x x1 2 xFigure 4 A normal probability distribution curve. The total area underthe curve = 1. Two-thirds of the area lies between x − σ and x + σ, whichimplies that 2/3 of the measurements should fall within ±σ of the mean.The probability of obtaining a value between x1 and x2 equals the area underthe curve between x1 and x2.

Now the average of all our measured values is certainly less uncertainthan the individual measurements. What error shall we associate with theaverage? It can easily be shown (see notes on propagation of error) thatthe error to be associated with the average is approximately given by thestandard deviation of the measurements divided by the square root of thenumber of measurements included in the average calculation. We will callthis uncertainty for the average σm. We can summarize our conventions asfollows:


If we have N independent measurements, xi, of a quantity x, then theimportant quantities and their uncertainties are:

• xi ± σ, where xi is any individual measurement,

• and x± σm, where σm = σ/√N .

Two words of caution are in order. First, it is a common mistake to thinkthat the standard deviation σ represents the uncertainty in the average. Itdoes not. Rather, it represent the uncertainties associated with each of theindividual measurements. Second, these statistical methods work wellonly when N is large, say 10 or more, and the larger N is, the better theywork. Occasionally we use them for smaller values of N because we haveno other choice. In such cases we must take our conclusions with a grainof salt.

AGREEMENT OF EXPERIMENTAL RESULTS

Why do we lay such stress on measuring and treating our experimentaluncertainties? This becomes quite clear when we go to compare two ex-perimentally derived numbers. The two numbers might represent twodifferent measurements of the same quantity, taken from the same experi-ment; or they might represent the values arrived at by two different pairsof students in the laboratory; or they might represent your experimentallydetermined value and a value measured and published in the literatureby a professional physicist. The two numbers will almost invariably bedifferent. What does that mean? They don’t agree? Does “agreement”mean identical? You will realize immediately that this cannot be the case.Then what does it mean for values to be in agreement?

For our purposes, the following crude criteria will be sufficient to drawconclusions about the agreement (or lack thereof) of two measured values.

1. Agreement: Two values are in agreement if either value lies within theerror range of the other.

2. Disagreement: Two values disagree if their respective error ranges donot overlap at all.

3. Inconclusive: No clear conclusion can be drawn if the ranges overlap,but neither value lies within the range of the other.

(Technically, what one should check is that the difference between twovalues is consistent with zero. This requires that one use propagationof error to find the error for the difference of two values. If the error isgreater than the difference of the values, then the results are said to be inagreement.)

The following examples and diagrams will help to make this ideaclearer.


Values in agreement:

xA = 5.7± 0.4xB = 6.0± 0.2

Here xB lies within the uncertainty range of xA, though xA does not liewithin the uncertainty range of xB .

Values in disagreement:

xA = 3.5± 0.5xB = 4.3± 0.2

The two uncertainty ranges do not overlap.

Inconclusive:

xA = 8.3± 0.3xB = 8.8± 0.3

Neither value lies within the uncertainty range of the other, although thetwo uncertainty ranges do overlap.

So we see that no experimental measurement can be complete until itsexperimental uncertainty is determined. Otherwise, the measurement is oflittle or no use to us.

Notes on Propagation of Errors

We have learned something about measuring and estimating the uncertain-ties in experimentally measured values. We know that this is importantbecause it allows us to compare two different measurements of the samequantity. It is also important because it allows us to calculate the uncer-tainties in other quantities which are calculated from the directly measuredquantities. In what follows we will determine how to calculate the errorsin a function G(x, y) of the experimentally measured values x, and y.

SUMS OF MEASUREMENTS

For example, suppose thatG(x, y) = x+y, wherex and y are experimentallymeasured quantities with uncertainties dx and dy respectively. What is theuncertainty, dG, in the value ofG that is calculated from the experimentallymeasured values for x and y? The largest and smallest values for G wouldbe:

Gmax = (x+ dx) + (y + dy) = x+ y + (dx+ dy) andGmin = (x− dx) + (y − dy) = x+ y − (dx+ dy) .

In this case we could write: dG = (±)(dx + dy). Graphically theuncertainty in G could be expressed as follows:

dx dy

dGFigure 1

However, in order for the value of G to be this far off, both x and ywould have to be in error by their maximum amounts and in the samesense; i.e., dx and dy both positive or both negative. If the errors in x and yhad opposite signs something like this would happen:

G = (x+ dx) + (y − dy) = x+ y + (dx− dy) .

Here the error in G is smaller than in the previous case, and could be zeroif dx = dy. This case can be represented graphically as follows:

19


dx dy

dGFigure 2

Statistically speaking, it is unlikely that the error in G will be either aslarge as the first assessment or as small as the second. The truth must liesomewhere in between. Our rule for quantities calculated from sums (andalso differences by the way) will be:

dG2 = dx2 + dy2 .

Graphically this can be represented as “error vectors” being added atright angles to each other rather than parallel or anti-parallel:If G(x,y) = ax± by, then the error in G is given by:

dG2 = (a dx)2 + (b dy)2 . (1)

dxdy

dGFigure 3

PRODUCTS AND POWERS OF MEASUREMENTS

Suppose that G(x, y) = Kxpyq, where p and q can be any positive or nega-tive numbers. For example, if we were considering the relation

v = R

√g

2h,

which could be rewritten as v = (√g/2)R1h−1/2, where R and h are the

experimentally measured quantities (corresponding to x and y), then p = 1,q = −1/2, and K =

√g/2.

The rule in this case will be quite similar to that for sums of measure-ments, but in this case it is the fractional uncertainties in G, x, and y (dG/G,


dx/x, and dy/y) that will be important.When G(x,y) = Kxpyq, then the fractional uncertainty in G is given by

(dG

G

)2=(pdx

x

)2+(qdy

y

)2. (2)

An Example Computation

As an example of the application of these rules, we consider finding theuncertainty in the average velocity over an interval when we know theuncertainties in the length and the time.

Suppose that the length of the interval is x = 1.752 ± 0.001 mand the time is t = 0.7759 ± 0.00005 s. The average velocity isv = x/t = 2.258 m/s. The uncertainty is calculated from Eq. (2)above: (

dv

v

)=

√(dx

x

)2+(dt

t

)2

(dv

2.258 m/s

)=

√(0.001 m1.752 m

)2+(

0.00005 s0.7759 s

)2= 0.00057.

Thus, the uncertainty in the velocity is dv = 0.00057 × 2.258 m/s =0.0013 m/s.

An Application: Uncertainty in the Mean

Both rules, for sums and products, are easily generalized to cases with morethan two independent variables, i.e., x, y, z, . . . etc.

The rule for dGwhen G is a sum or difference has a nice application tothe uncertainty in an average value of several measurements. Suppose thatGave = (g1 + g2 + g3 + . . .+ gN )/N . Then by equation (1) the uncertainty inGave is given by:

(dGave)2 =1N2

(dg2

1 + dg22 + dg2

3 + . . .+ dg2N

). (3)

But (1/N)(dg21 + dg2

2 + dg23 + . . . + dg2

N ) is the mean square deviation (σ2)of the g measurements, or the square of the standard deviation of the gmeasurements. Using this identification, we can transform equation (3)to: (dGave)2 = σ2/N , and therefore the uncertainty in the average, dGave,which we sometimes call σm, is

dGave =σ√N. (4)


This confirms our feeling that the uncertainty in an average valueshould be less than the uncertainties in the individual measurements (oth-erwise, why take averages on repeated measurement?), and confirms theformula for the uncertainty in a mean which was given without proof inthe Notes on Experimental Errors.

We summarize these notes with a table showing the propagation oferrors for a few simple cases.

Expression Error in Expression

ax± by√

(a dx)2 + (b dy)2

xy√

(x dy)2 + (y dx)2

x

y

x

y

√(dx

x

)2+(dy

y

)2

√2gx

12

√2gx

∣∣∣∣dxx∣∣∣∣

Notes on Graphing

GRAPHICAL REPRESENTATION AND ANALYSIS

Graphs are used scientifically for two purposes. The first is simply torepresent data in a form that can be understood visually. That is, a graphshows schematically how one variable depends on another. This is theusage that is most familiar to most people. Perhaps the most commongraph of this kind is that of the Dow Jones Industrial Average versustime. One might look at such a graph to see whether stock prices werefalling throughout the day or to see how much volatility there was in theDow. Beyond these general trends, most people do not look further, exceptperhaps to see if the low or high for the day reached or exceeded someparticular level. The size of such a graph does not particularly matter forsuch purposes so newspaper editors generally print the graph small to savespace.

The second and less common use of a graph is to analyze data. Thisis a more sophisticated use of graphing and more stringent criteria applyto making and using such graphs. In a graph of the Dow Jones IndustrialAverage versus time, it is not so important to be able to read the valueof the Dow to great precision. This is not the case with a scientific graphwith which one will analyze data. Just as most students are reluctant toleave out any of the ten digits in a calculated result (even though they mustaccording to the rules of significant figures!), a good scientist ought to bereluctant to lose any precision in analyzing data graphically.

LINEARIZING THE DATA

In analyzing experimental data, we most often will be interested in findinga representation of the data that results in a linear relationship between thequantities. This is because our visual system is acutely sensitive to linesand fairly bad at distinguishing other curves. To the eye, the crest or troughof a sine curve looks identical to a parabola until they are laid on top ofone another. In addition, it is much more difficult to find the curvature ofa parabola, for instance, than it is to find the slope of a line!

The two values that are of interest in a linear graph are the slope andthe y-intercept. For instance, the period of a pendulum, T , and its length L

23


are related theoretically by

T = 2π√L/g

when the amplitude of the motion is very small. If we were to test this rela-tionship experimentally by graphing data, we would square the equationto get

T 2 = (4π2/g)L . (1)

Eq. (1) indicates that if we were to graph T 2 versus L, we should find a linewith slope 4π2/g. In fact, this would be a reasonable way of measuring theacceleration of gravity, g. (Recall that when we say a graph of y versus x,we mean a graph with y along the vertical axis and x along the horizontalaxis.) According to Eq. (1) we should also find an intercept of zero: when Lis zero, T 2 should also be zero. However, we might find that there is a non-zero intercept, which would indicate that perhaps we had not measuredthe length of the pendulum correctly: we may have a non-zero offset.

L

T2

(m)

(sec

)2

1

1

1

4

1

2

3

0.5

5

6

Squared Period vs. Pendulum Length

Figure 1 A well-drawn graph of data for the squared period of a pendulumversus length of the pendulum. The solid line is the best-fit line. The dashedlines are lines of good fit. Note that the line and the data points fill the wholerange of the graph.

MAKING GOOD GRAPHS

The first rule of scientific graphing is to make the graph easy to read ac- MAKE LARGE GRAPHScurately and precisely. This means that graphs must be made as large aspracticable. In practice you should make the graph as large or nearly aslarge as your sheet of graph paper will allow, i.e., the the whole or nearly


the whole sheet. Included in this is the requirement to make the range ofthe axes no larger than is necessary; you want your line to fill up the wholerange. Just as it is more difficult to measure very small objects, it is difficultto read very small graphs accurately. In addition, graphs made very smalllack the precision of larger graphs simply due to the size of the mark madeby even a sharp pencil. In addition to being as large as practicable, a graphmust be labeled in such a way as to be easy to read quickly and accurately.This means that the axes must be labeled in a simple and regular fashion. NUMBER AXES SIMPLYIt is best to use 1, 2, or 5 divisions to represent a decimal value. That waythe graph is easiest to read and to make; one need not guess where a valueof 1.50 lies between marks at 0.85 and 1.70!

In addition to being of good size and having well labeled axes, to TITLE GRAPHS,LABEL AXES,USE UNITS

be complete graphs must have a descriptive title and the quantities beinggraphed must be labeled on the axes with their proper units.

Figure 1 shows a good graph with all the features necessary to a good ATTRIBUTES OF AWELL-DRAWN GRAPHgraph. These good features include

• Large size

• Range of the axes just covers that of data

• Well numbered axes

• Short and Clear Title

• Axes labeled with the appropriate quantity

• Units on the axes

• Data points with uncertainties (error bars)

• A best-fit line

• One or two alternate good-fit lines

MEASURING THE SLOPE OF A LINE

In most of the graphs that we will be using in the introductory laboratorythe most important piece of information will be the slope of the “best-fitline.” There are many ways of obtaining this slope. One way is to usethe formulas at the end of these notes into which one may substitute thecoordinates (xi, yi) and errors σi of the data points. If the errors σi are allof the same size, one might use Microsoft Excel to calculate the slope andintercept. (Excel is not recommended because it is a difficult tool to masterand is not appropriate if there are large error bars on some data points andnot on others.) Last, one may use a hand-drawn graph that is carefullymade. On the hand-drawn graph there should be a best-fit line to the data.It is this line whose slope we wish to calculate. Note that the best-fit linedoes not necessarily go through any of the data points so it is not correct touse the data points to calculate its slope.


Why use widely separated points?

The slope of the line is given by the familiar rule “rise over run.” Now, CHOOSE POINTS FARAPART TO MINIMIZE

ERROR INMEASURING THE

SLOPE

mathematically, it does not matter which two points you choose on a lineto calculate slope; any two points will give the same answer. However, wecannot read the coordinates of the points to arbitrary precision, hencethere is inherent error in calculating the slope from a real graph. The riseis difference in the heights of the two points,

∆y = y2 − y1 ,

so the error in the rise is found from the uncertainty in the y-coordinatesthemselves,

σrise = σ∆y =√σ2y2

+ σ2y1,

and is independent of ∆y. The uncertainty in the rise depends only onthe uncertainty in reading the points on the line, which is minimized bymaking the graph as large as practicable. Similar results hold for the run,∆x. The slope, being the ratio of rise to run, has an error

σbest fit slope

best fit slope=

√(σrise

rise

)2+(σrun

run

)2. (2)

It is useful to pause here to think about Eq. (2). To minimize the errorin the slope, we should minimize the errors in the rise and run, but weshould also maximize the rise and the run themselves. The rise and runare maximized when the points on the line taken to compute the slope areas far apart as possible while still being on the graph. In practice, one willnot choose the points to be way at the ends of the line, but rather one willchoose points quite far apart but lying near the crossing of two lines on thegraph paper so that their coordinates may be read accurately. A graphicalillustration of the Eq. (2) is shown in Fig. 2.

1

1

(a)1

1

(b)

Figure 2 The separation of the points chosen on a line for the computationof the slope has a strong effect on the error in the computed slope. In figure(a) the points, whose widths are exaggerated for clarity, are close together,resulting in a large error in the slope. In figure (b) the points are far apart,resulting in a more precise determination of the slope.


Finding the range of slopes for lines of good fit

It is important to point out that the uncertainty given in Eq. (2) is the ERROR IN REPORTINGTHE SLOPE OF THEBEST-FIT LINE

uncertainty in measuring from your graph the actual slope of the best-fitline. This is logically different from quoting a range of values for linesof good fit! In other words, even if all the points were to lie perfectlyalong a line, there is a limitation in finding the slope of that line due tothe fact that graphing is a physical process and reading the graph has anuncertainty associated with it. In practice, however, the points generallywill not lie along a line and there may be many lines that have a reasonablefit to the data. We’d like to find the range of the slopes for these lines thatfit the data reasonably well. The procedure in this case is to draw twomore lines in addition to the best-fit line. One line will have the largestpossible slope and still fit the data reasonably well and the other will havethe smallest possible slope and still fit the data reasonably well. Examplelines are drawn Figure 1. To find the quoted error in the slope of the best-fitline, take half the difference in the slopes of the two outlying dashed lines.For example, if the slope of the best-fit line in the figure is 4.00 sec2/m,the slope of the steeper dashed line is 4.16 sec2/m, and the slope of theshallower dashed line is 3.90 sec2/m, you would quote the slope as 4.00 ±0.13 sec2/m, because (4.16−3.90)/2 = 0.13. In a pinch, you could draw justone other good-fit line and then the error in the slope is just the differencebetween the best-fit line slope and the slope of the good-fit line.

LEAST SQUARES METHOD

If graphing seems to imprecise to you or you just like algebra better, thereis a method that will give you the slope and the intercept of the best-fit linedirectly without graphing. Of course, you will also forgo the benefits ofseeing the relationship between the data points and the visual estimate ofthe goodness of fit, but you avoid the artistic perils of graphing.

The idea is that the best-fit line ought to minimize the sum of thesquared vertical deviations of the data from the line. That is, you want aline y = mx+ b that has the least value for

∆(m, b) ≡N∑i=1

(yi − (mxi + b))2 .

If the data points have vertical error bars that are±σi for data point (xi, yi),then the correct quantity to minimize is

∆(m, b) ≡N∑i=1

(yi − (mxi + b)

σi

)2

. (3)

The algebra is not difficult to do and the result of minimizing with respectto both m and b yields values for the slope and the y-intercept is

m =S1Sxy − SxSyS1Sx2 − S2

x

, (4)


b =SySx2 − SxSxyS1Sx2 − S2

x

, (5)

where S1, Sx, Sy, Sxy, Sx2 , and Sy2 are defined by

S1 =N∑i=1

σ−2i , Sx =

N∑i=1

σ−2i xi ,

Sy =N∑i=1

σ−2i yi , Sxy =

N∑i=1

σ−2i xiyi ,

Sx2 =N∑i=1

σ−2i x2

i , Sy2 =N∑i=1

σ−2i y2

i .

(6)

The errors in the slope and intercept are given by

dm =

√S1(S1Sx2 + S2

x)

S1Sx2 − S2x

, (7)

db =

√Sx2(S1Sx2 + S2

x)

S1Sx2 − S2x

. (8)

The goodness of fit can be found by evaluating the correlation coefficient

R =S1Sxy − SxSy√

(S1Sx2 − S2x)(S1Sy2 − S2

y), (9)

in the notation of Eq. (6). When the R = 0, there is essentially no linearrelationship present, while a value of R = 1 occurs for perfect correlation,or all points lying on a line.

physics 160 laboratory manual - hobart and william smith...

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