Measuring The Earth’s Magnetic Field
Measuring the Earth’s Magnetic FieldJoshua WebsterPartners: Billy Day & Josh KendrickPHY 3802L12/06/2013
Lab 5: Earth’s Magnetic FieldWebsterAbstractIn this experiment, the Earth’s magnetic field is measured. Using a Helmholtz coil, acalibration constant is determined that is applied to deflection values obtained by a ballisticgalvanometer. The determined values for the vertical, perpendicular, and parallel components ofthe Earth’s magnetic field are, respectively: -38056 732 nT, -28024 521 nT, 26700 607 nT.The horizontal component of the Earth’s magnetic field was determined to be 38708 604 nT,and finally the total magnetic field of the Earth was determined to be 54282 658 nT.1
Lab 5: Earth’s Magnetic FieldWebsterTable of ContentsAbstract . 1Introduction . 3Background . 4Experimental Techniques. 10Diagrams and Images . 10Data . 12Analysis. 17Discussion . 21Conclusion . 23Appendix . 24References . 252
Lab 5: Earth’s Magnetic FieldWebsterIntroductionThe Earth’s magnetic field has been thoroughly studied for hundreds of years. CarlFriedrich Gauss measured its strength in 1835, and from subsequent measurements it has beendetermined that it has a relative decay of 10% over the past 150 years.1 Early studies wereconducted for the purposes of navigation. In fact, compasses have been used since the 11thcentury A.D. Perhaps the most important effect of the Earth’s magnetic field is that it protects theEarth from violent solar winds. Without a strong enough magnetic field to deflect these incomingcharged particles the Earth’s ozone layer would be stripped, leaving Earth to an outcome similarto that of Mars.The experiment described in detail in this report was conducted to measure the intensityof the Earth’s magnetic field at the specific location in which the experiment took place. AnEarth inductor is used to perform the various components of the experiment. It consists of a coilof wire with a specific amount of turns (how many times the coil is wrapped around) and aspecific radius. It is connected to a ballistic galvanometer (a sensitive instrument that measureselectric current), and can be flipped about an axis to induce an EMF (Electromotive Force). Thisinduced EMF is measureable by the deflection in the ballistic galvanometer. The amount ofdeflection is proportional to the component of Earth’s magnetic field that is perpendicular to theEarth inductor. For reliable and comparable data, a calibration is necessary to determine theproportionality constant. In this experiment, calibration was achieved using a Helmholtz coilwith a known magnetic field. A Helmholtz coil consists of two identical coils of wire separatedby a distance that is equal to the radius of the coils. They are commonly used in experiments toproduce a region that has a virtually uniform magnetic field.The following sections of this report will consist of the Background, ExperimentalTechniques, Data, Analysis, Discussion, Conclusion, Appendix, and References. Each sectionhas been carefully composed to provide a depth of knowledge pertaining to the data recorded,calculations, and error analysis.1(Earth's Magnetic Field)3
Lab 5: Earth’s Magnetic FieldWebsterBackgroundWhen the Earth inductor coil is flipped in a magnetic field an EMF is produced that is describedby the equation:Where, in the above equation,coil, and is time.is the EMF,is the number of turns,is the flux through theThe above EMF must equal the potential drop around the circuit, so:Where, in the equation above,is the current in the circuit, andis the resistance.The total charge ( ) that flows into the galvanometer when it is flipped is determined byintegrating over the duration of the flip: For aflip:Where, in the above equation (1.3),is the component of the Earth’s magnetic field that isperpendicular to the coil, and is the area of the coil. Combining equations (1.2) and (1.3) wehave:The deflection of the ballistic galvanometer is proportional to the charge that flows through it:Where, in the above equation (1.5), is the amount of deflection, andis a proportionalityconstant with respect to the galvanometer. Combining equations (1.5) and (1.4) we now have:Equation (1.6) can also be expressed as,4
Lab 5: Earth’s Magnetic FieldWebsterWhere, in the above equation (1.7), the component of the Earth’s magnetic field is related to thedeflection of the galvanometer. Since is unknown to us, we must find it in order to make useof this formula. Sinceis the proportionality constant for the galvanometer, a calibrationprocess must be required. A Helmholtz coil is used to create a known magnetic field, and theeffect of the flip coil is then measured in the total magnetic field (Earth’s field and Helmoltz’sfield). The following algebraic equations describe the relations:Where, in the above equation,is the total magnetic field, andandare the magneticfields of the Earth and the Helmholtz coil, respectively. The deflection of the galvanometerfollows the same additive relationship:Where, in the above equation,is the total deflection, andandare the deflections due tothe Earth and the Helmholtz coil. Equations (2.0) and (2.1) can be combined with (1.7) toproduce a set of relations:()It can also be shown that,The magnetic field of the Helmholtz coil would then be:()Equation (2.4) then leads us to the equation for the Earth’s magnetic field:Let 5
Lab 5: Earth’s Magnetic FieldWebsterWhere, in the above equation (2.6), C is the “calibration” constant that can be determined fromthe calibration measurements using the deflection measured for the Earth’s vertical fieldcomponent. The constant can then be applied to the other components of the Earth’s magneticfield. For each non-zero value of the current, , the measurement of the total deflection,,results in a measurement of the calibration constant, , using the deflection measured for theEarth, , when the current was zero.We need to calculate the uncertainty associated with the value, so using equation A.1 from theappendix: ()( ()()()) Equation (2.6) can also be expressed asThis form makes it obvious that andcan be obtained by plottingversus . The slope ofthe best fit line straight through the points will then be withas the intercept.The magnetic field of the Helmholtz coil is determined by the following formula: Where, in the equation above,is the permeability of free space, is the number of turns ineach coil, is the current flowing through the coils in amperes, and is the radius of the coils inmeters. This equation can be derived by starting with the formula for the on-axis field due to asingle wire loop (which is derived from the Biot-Savart law).2 The uncertainty in the aboveequation (2.9) can be found using the error propagation formula, Appendix A.1:2(Helmholtz Coil)6
Lab 5: Earth’s Magnetic FieldWebster ()()()In this case, the above equation reduces to the following: ( ( ))(( ))We also need to calculate the uncertainty for the calibration constant in (2.6) using the errorpropagation formula A.1 from the Appendix: ()()()For the case of equation (2.6), the above equation becomes: ( ())()()(())From our data, we can also determine the horizontal component of the Earth’s magnetic field: Where, in the above equation,is the horizontal component of Earth’s magnetic field,is thecomponent that is parallel to the meridian, andis the component that is perpendicular to themeridian. The uncertainty for the above equation can be found by using equation A.1 from theAppendix, and adapting it to our needs:7
Lab 5: Earth’s Magnetic FieldWebster ()()() ( )Which just becomes, Now, the total Earth magnetic field,, can be determined by: Where, in the above equation (3.1),is the horizontal component of the Earth’s magnetic field,andis the vertical component. The uncertainty in the total Earth magnetic field can becalculated by using equation A.1 from the appendix, but we just formulated this same type ofequation in (3.01), so similar to before: ( )( ) To find the average C value we can use the formula for the simple arithmetic average (mean):̅ 8
Lab 5: Earth’s Magnetic FieldWebsterThe standard deviation is then: ̅9
Lab 5: Earth’s Magnetic FieldWebsterExperimental TechniquesDiagrams and ImagesImage 1: An Earth inductor (flip coil). This is similar to the type of flip coil used in theexperiments performed in this lab. This is an old Cenco model from the Greenslade Collection.3Diagram 1: This diagram depicts a Helmholtz coil. The radius distance is equal to the spacing ofthe coils. Current flows in through one coil and out at the bottom of the other. The magnetic fieldis uniform in the x-direction.4For the setup, the laboratory table was aligned with the wall, so that it had a knownorientation with respect to the lab. The flip coil was connected to the galvanometer through acurrent limiting series resistor of 700 ohm resistance. The value of the resistance is to be chosen34(The Earth Inductor, or Delzenne's Circle)(Hellwig, 2005)10
Lab 5: Earth’s Magnetic FieldWebstersuch that the deflection of the galvanometer is within range. The components of the Earth’smagnetic field are then to be measured by flipping the coil. At each flip of the coil, the peak ofthe galvanometer deflection is recorded. This is done at least five times for each of the threecomponents of the Earth’s magnetic field vector. The components to be measured are with thecoils axis vertical, parallel to the meridian (North-South), and perpendicular to the meridian(East-West).The second part of the experiment consisted of a calibration process in order to obtain acalibration constant. The calibration process used involved a Helmholtz coil. The Helmholtz coilcreates a known magnetic field, and thus adds to the magnetic field of the Earth. To begin, theflip coil is placed inside the Helmholtz coil with the flip coils axis vertical. The Helmholtz coilwas then connected to the current controlled power supply. Then, 27 deflection values weremeasured at various current values that were chosen to give a range of deflection valuesprimarily near the high ends of the range. The radius of the Helmholtz coil was also measured(0.34 0.005 m), and the number of turns was recorded (72 in each coil).11
Lab 5: Earth’s Magnetic FieldWebsterDataTable 1: This table lists the data for first part of the experiment which consisted of variousorientations of the flip coil in solely the Earth’s magnetic field. dE is the deflection of thegalvanometer due to the Earth’s magnetic field, and (dE) is the associated uncertainty. Theaverages of the values and the standard deviations are provided for each orientation. Theuncertainties are estimated.Vertical OrientationTrialdE (m) (dE) 250.16800.002Avg.0.1673Std.Dev. 3.354E-4Perpendicular Orientation (E-W)TrialdE (m) (dE) 250.12300.002Avg.0.1232Std.Dev.2.236E-4Parallel Orientation (N-S)TrialdE (m) (dE) 0.0025-0.11800.002Avg.-0.1174Std.Dev. 2.837E-4Table 2: This table lists the data for the calibration part of the experiment in which data wasrecorded when using the known magnetic field of a Helmholtz coil. Deflection values listed hereare the total deflection values (corresponding to the sum of the Earth’s and the Helmholtzmagnetic fields). The delta symbol represents the uncertainty in a value, which is an estimate.We must also remember that the deflection data is recorded in centimeters, but must beconverted to meters for use in further calculations. There are 27 values in total.Current (Current) Deflection 2
Lab 5: Earth’s Magnetic 20.20.20.20.20.20.20.20.20.20.20.20.20.2Table 3: The values listed in this table in the order of left to right include: the magnetic field ofthe Helmholtz coil (BH), the uncertainty in the magnetic field of the Helmholtz coil (σBH), thecalibration constant (C), the uncertainty in the calibration constant (σC) using (2.92), theindividual deviations of the calibration constant, the calibration constant divided by theuncertainty squared (C/σC2), and the weight values (1/σC2) for the calibration constant. Alsoincluded at the very bottom is the average calibration constant value, the standard deviation ofthe calibration constant, and the weighted average calibration constant value. The weightedaverage calibration constant is the sum of the C/σC2 values divided by the sum of the individualweights. Note: The (T) units are Tesla.BH 714E-051.904E-052.285E-052.666E-053.047E-05σBH 079.924E-071.010E-061.030E-061.052E-06C 2E-04σC (T/m) Ind. Dev. Of -119.091E-063.922E-1113C/σC2-5.737E 04-1.933E 05-4.320E 05-7.752E 05-9.599E 05-1.661E 04-5.079E 04-1.989E 05-4.402E 05-7.592E 05-9.520E 05-1.174E 06-1.623E 06-2.151E 06-2.725E 06Weight (1/σC2)2.817E 088.554E 081.917E 093.496E 094.296E 098.986E 072.307E 088.957E 081.977E 093.379E 094.239E 095.259E 097.194E 099.544E 091.210E 10
Lab 5: Earth’s Magnetic 15E-134.575E-123.030E-12Std. Dev.2.507E-06-3.268E 06-4.214E 06-4.690E 06-5.291E 06-6.231E 06-7.167E 06-8.026E 06-8.878E 06-9.562E 06-1.031E 07-1.089E 07-1.136E 07-1.177E 07Sum-1.142E 081.433E 101.951E 102.108E 102.359E 102.767E 103.178E 103.542E 103.915E 104.176E 104.504E 104.721E 104.862E 105.049E 10Weighted Avg.-2.277E-04In the above table, the standard deviation from the average C value is roughly on the same orderof magnitude as the individual uncertainties. The standard deviation acts as a good approximateuncertainty for the calibration value.Table 4: The table below shows information for the graph of dT vs. BH given by the programLineFit. The slope is equal to .Slope (Tm-1)-4.3962E 03σSlope (Tm-1)0.0331E 03C (Tm-1)-2.2747E-04σC (Tm-1)3.426E-0614Intercept (dE) (m)1.6678E-01σdE (m)0.0131E-01
Lab 5: Earth’s Magnetic FieldWebsterGraph 1: The graph below shows the total deflection (dT or d T) plotted versus the magneticfield of the Helmholtz coil (BH or B H). From this graph one can find the calibration constantusing the slope, and the deflection caused by the Earth’s magnetic field (dE) is the intercept. Theinformation plotted in this graph is from Table 2 & 3, and the information determined by thegraph is in Table 4.15
Lab 5: Earth’s Magnetic FieldWebsterTable 5 & 6: The data shown in these two tables is the calculated values for the magnetic fieldcomponents as well as the total Earth magnetic field. The magnetic fields are calculated threedifferent ways: with the average C value, with the weighted average C, and with the slopedetermined value for C. BV is the vertical component, B is the parallel component, B is theperpendicular component, B(Earth Total) is the total magnetic field, B(Horizontal) is the horizontalcomponent. The table also shows each components uncertainty to the right of the value. Theunits are Tesla. For comparison and convenience, the values can be multiplied by 109 forconversion to nano-Tesla.Avg. CWtd. Avg. CSlope 1/CBV (T)σBV (T)B (T)σB (T)B (T)σB (T)-3.8720E-05 6.2468E-07 -2.8513E-05 5.5649E-07 2.7166E-05 5.4852E-07-3.8094E-05 6.1916E-07 -2.8053E-05 5.5029E-07 2.6728E-05 5.4223E-07-3.8056E-05 7.3177E-07 -2.8024E-05 6.2058E-07 2.6700E-05 6.0720E-07Avg. CWtd. Avg. CSlope 1/CB(Earth Total) (T) σB(Earth Total) -056.5824E-0716B(Horizontal) (T)3.9383E-053.8747E-053.8708E-05σB(Horizontal) (T)5.5271E-075.3764E-076.0372E-07
Lab 5: Earth’s Magnetic FieldWebsterAnalysisThe largest part of this lab is involved in solving the equations. Understanding how eachintermediate result contributes to the final is of key importance. The following paragraph is myattempt at emphasizing the stages of the calculations.Using the recorded data, the magnetic field of the Helmholtz coil can be determined.With the Helmholtz magnetic field, the calibration constant can be determined. Using thecalibration constant, the vertical, parallel, and perpendicular components of the Earth’s magneticfield can be calculated. Then the horizontal component can be determined by the parallel andperpendicular components. Finally, the total magnetic field can be determined by using thehorizontal and vertical components.We can calculate the magnetic field for the Helmholtz coil by using equation (2.9): The uncertainty in the magnetic field of the Helmholtz coil can now be calculated using equation(2.91): ( ( ) ()( ) )The calibration constant, , can be calculated using equation (2.6) with the magnetic field of theHelmholtz coil determined above. The values forandare the deflection values found inTable 1 and Table 2. They must be converted to meters by dividing by 100. 17
Lab 5: Earth’s Magnetic FieldWebster With the values calculated in this way, we can find a mean (average), and also a standarddeviation. For simplicities sake, I will only calculate the mean and standard deviation of twovalues.̅ (3.2) expressed algebraically is just,̅So in our case,̅The standard deviation of these two values is then, ̅(3.3) expressed algebraically is just, ̅̅̅[ ][]Remember that this mean value and uncertainty is just for the two values shown above,and they simply serve as a means of showing the steps involved in obtaining the actual values(which is a much more lengthy procedure). In Table 2, there is a separate column for the18
Lab 5: Earth’s Magnetic FieldWebsterindividual deviations from the mean. This is just a way to make the calculations easier by doingthe math in parts. It also allows one to see numerically how much a certain data point differsfrom the mean.The weighted average value is found by dividing the sum of the by the sum ofthe values. I will assume that the reader can do simple arithmetic division and omit thesesteps.Now it’s time to calculate the Earth’s magnetic field values. We can do this by using (2.7)with simple arithmetic:For the vertical orientation, using the average vertical deflection and average calibration constantvalue:The uncertainty can be calculated using (2.75): Now with the two previous calculations accomplished, the same goes for the parallel andperpendicular components and their uncertainties.For the horizontal component of the Earth’s magnetic field we must use (3.0), and for itsuncertainty we shall use (3.01). For both calculations I will use the average C magnetic fieldvalues: 19
Lab 5: Earth’s Magnetic FieldWebster With the horizontal component and its uncertainty calculated (along with the vertical componentand its uncertainty), now the total magnetic field of the Earth and its uncertainty can becalculated: This calculated value of the total magnetic field of the Earth and uncertainty is on thesame order of magnitude as the accepted value, though the magnitude differs by a larger degreethan the uncertainty permits.20
Lab 5: Earth’s Magnetic FieldWebsterDiscussionThe determined value for the total magnetic field of the Earth is on the same order as theaccepted value (47655 nT on the 18th of November in Tallahassee, FL), but it is off by around14%. The uncertainty also does not bridge this gap of 7574 nT (our most accurate valuedetermined5 results in a gap of 6627 nT), which hints to something possibly wrong in thecalculations, but I was unable to pinpoint the source of this error. Since the vertical component isclose to its accepted value of 41194 nT that leads me to believe the horizontal component iswhere the problem lies. For the horizontal component value determined previously in theAnalysis section, it is on the same order of magnitude as the accepted value, but much larger (byroughly 39%). The values that go into determining the horizontal component are the parallel andperpendicular components. This means that either the value for the calibration constant is wrong,or there is something wrong with the average deflection value being used. Since the calibrationconstant affects the vertical, parallel, and perpendicular components in the same way I doubt thatis where the major source of error in the horizontal component resides. This has led me tobelieve that one or more of the average deflection values are incorrect. This could be due toincorrect alignment of the flip coil. In addition, the vertical and horizontal components acceptedvalues are both positive, however it was determined in this experiment that the verticalcomponent is negative. Also, examining the accepted values for the vertical and horizontalcomponents shows that the horizontal should have been roughly one-half of the verticalcomponents intensity.Table 6:6 The accepted values for the components of the Earth’s magnetic field for Tallahassee,FL on November 18th, 2013. The “Horizontal Intensity” is the horizontal component, the “NorthComp” is the parallel component, the “East Comp” is the perpendicular component, the “VerticalComp” is the vertical component, and the “Total Field” is the Earth’s total magnetic field. Unitsare in nano-Tesla (nT or 10-9 T).Latitude:Longitude:Elevation:30 27' 0" N84 18' 0" W20.0 MDateDeclination( E -W)Inclination( D -U)HorizontalIntensityNorth Comp( N - S)East Comp( E - W)2013-11-18Change/year-4 23' 39"-6.2'59 49' 3"-5.3'23,958.9 nT-1.4 nT23,888.4 nT-4.8 nT-1,835.6 nT-42.9 nTVerticalComp( D - U)41,194.3 nT-147.0 nTTotal Field47,655.0 nT-127.8 nTThe above accepted values show us that there is a systematic error in our experiment. Theerror arises from the deflection values recorded in the lab. This is more than likely due toimproper alignment of the flip coil axis. If the deflection value is negative, the magnetic fieldwill be positive (since the calibration constant is negative). Therefore, the perpendicular56In which the calibration constant that was determined by the slope of the graph was used.(NOAA)21
Lab 5: Earth’s Magnetic FieldWebstercomponent deflection values should be the only positive deflections. Since ours were recorded asbeing positive, this is a systematic error (a flaw in the experimental procedure).A mis-measurement of the Helmholtz coil radius would also increase the error in thedetermined values for the Earth’s magnetic field. An increase in the radius would make theHelmholtz magnetic field value smaller, which in turn would make the calibration constantlarger. The larger calibration constant would cause the magnetic field component to becomelarger. The overall consequence of a larger radius would decrease the value for the total Earthmagnetic field. For roughly double the radius, the value for the total Earth magnetic fielddecreases by roughly a half. A mis-measurement resulting in a smaller value would just be theopposite of this. I do not believe that there was a mis-measurement of the Helmholtz coils radiusin the case of our experiment. As I am not able to return to the lab currently I will use reasoningto gain a better understanding of our radius. Our radius was determined by measurement to be0.34 0.005 m, which seems like an acceptable radius. Converting 0.34 m to feet (a unit I ammore likely to “eyeball” correctly) gives roughly 1 foot. If memory of the coil size serves mecorrectly, 1 foot definitely seems justifiable.22
Lab 5: Earth’s Magnetic FieldWebsterConclusionThis experiment sought to determine the Earth’s total magnetic field. A calibrationconstant was using the known magnetic field of a Helmholtz coil. This calibration constant wasapplied to deflection values obtained by observing the effects of a flip coil through a ballisticgalvanometer. The determined values for the vertical, perpendicular, and parallel components ofthe Earth’s magnetic field are, respectively: -38056 732 nT, -28024 521 nT, 26700 607 nT.The horizontal component of the Earth’s magnetic field was determined to be 38708 604 nT,and finally the total magnetic field of the Earth was determined to be 54282 658 nT.23
Lab 5: Earth’s Magnetic FieldWebsterAppendixA.1 Formula for the propagation of errors:Given a function, , with variables , , and . The uncertainty in is the square root of the sumof the squares of the partial derivatives of with respect to each variable, and each partialderivative is multiplied by the square of its uncertainty. ()()(24)
Lab 5: Earth’s Magnetic FieldWebsterReferencesEarth's Magnetic Field. (n.d.). Retrieved November 19, 2013, from Wikipedia:http://en.wikipedia.org/wiki/Earth%27s magnetic field#DetectionHellwig, A. (2005, June 20). Helmholtz Coils. Retrieved November 21, 2013, from ltz coils.pngHelmholtz Coil. (n.d.). Retrieved November 21, 2013, from Wikipedia:http://en.wikipedia.org/wiki/Helmholtz coilNOAA. (n.d.). Magnetic Field Calculators. Retrieved November 27, 2103, from NationalGeophysical Data Center: http://www.ngdc.noaa.gov/geomag-web/#igrfwmmThe Earth Inductor, or Delzenne's Circle. (n.d.). Retrieved November 21, 2013, from atus/Electricity/Earth Inductor or Delzennes Circle/Earth Inductor or Delzennes Circle.html25
Dec 06, 2013 · the Earth’s magnetic field are, respectively: -38056 732 nT, -28024 521 nT, 26700 607 nT. The horizontal component of the Earth’s magnetic field was determined to be 38708 604 nT, and finally the total magnetic field of the
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