Final Year Dissertation Report

University of BirminghamSchool of Physics and Astronomy

The Effects of AC Fields on Gravitational Experiments

Project Report Final

M.C.R.Gilbert

Partner: P.J.Steele

Supervisor: C.C.Speake

Approx. 8641 words

March 24, 2016

Abstract

Newton’s gravitational constant, G has been measured by various experiments, some of which have used con-ductive masses to generate gravitational forces. This project has aimed to evaluate the effect of alternatingmagnetic fields on these experiments by using the BIPM’s ”Big G” apparatus at the University of Birming-ham. A coil of wires was placed around the BIPM experiment to produce a magnetic field and to measure thechange in G with respect to both the magnetic field strength and frequency. An analytical model has beenbuilt that attempts to predict the magnetic torques in this system and the change in G measured that arises,using various physical approximations. With the BIPM experiment, we found that the magnetic interactiongave rise to repulsive torques - reducing value of G measured - and that G changes directly with the square ofthe magnetic field magnitude. The magnetic torques peak at a frequency of about 40Hz. Our model correctlypredicts this relationship and its order of magnitude, as well as the frequency-dependence measured. In a 1μTfield, the reduction in G is of the order of 1000ppm. The field around the BIPM experiment is about 10nT undernormal operating conditions. This will adjust G by only 0.1-1ppm, which is much smaller than the sensitivityof the apparatus. The effect we have quantified may have repercussions on the results of other gravitationalexperiments. Further study can be undertaken to continue to improve our model, to more accurately detail thephysics that has been approximated, and to extend its application to other G experiments.

Contents

1 Introduction 11.1 The Gravitational Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 BIPM Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Alternating Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Outline of this Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 The Model 32.1 The Primary Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Secondary Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Calculation of the Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 Aluminium Vacuum Can . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Results 173.1 G Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Comparison of G Data with Model Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Conclusions 19

5 Discussion and Implications 19

6 Acknowledgements 20

References 21

Appendix A Primary Field Equations 22

Appendix B Coordinate System Transformation of Equation 7 22

Appendix C Rotation of the Secondary Coordinates 23

Appendix D Overview of Torque Code with Rotating Source Masses 24

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1 Introduction

1.1 The Gravitational Constant

Isaac Newton’s gravitational constant, G is the con-stant of proportionality that governs the interactionbetween bodies due to the presence of mass. It is afundamental constant of the universe which not onlyextends its application to Albert Einstein’s general the-ory of relativity but is also an essential component ofmany different areas of physics. The value of G wasfirst measured by Henry Cavendish in 1797-98. He de-termined the density of the Earth to be ρ⊕ = 5.448gcm–3 and, by reverse-engineering Newton’s law of grav-itation, G could be evaluated to be 6.74 x 10–11 m3

kg–1 s–2. The current value of G recommended byCODATA (Committee on Data for Science and Tech-nology) as of their 2014 review is 6.67408(31) x 10–11

m3 kg–1 s–2 which has a relative standard uncertaintyof 47 ppm.[1] This differs from Cavendish’s result byonly about 1%. CODATA gives a weighted average,incorporating the results of many investigations whichuse a variety of approaches to measure G. Figure 1.1shows a distribution of G values from recent experi-ments, some of which will have been utilised by CO-DATA.Gravity is many orders of magnitudes weaker than theother fundamental forces and this is one of the impor-tant facts which leads to G being inherently difficultto measure. It contributes to making G the least well-defined fundamental constant today. We are sure tothe value of G to hundreds of parts per million. How-ever, by comparison other similarly universal constantssuch as the Planck constant, h and the elementary unitof charge, e are each known to considerably higher pre-cision. Their relative standard uncertainties are bothof the order of 0.01ppm. You can increase the gravita-tional force measured by amplifying the signal, usinggreater masses. This, unfortunately, has drawbackssuch as added material expenditure and the reducedprecision of a vast apparatus. Another obstacle facedwhen measuring G is that the force of gravity is onlyever attractive. This means it is not possible to shieldagainst the effects of gravitational perturbations exter-nal to the experimental apparatus.

Figure 1.1: Spread of G values across different experimentsspanning three decades. The 2001 and 2013 values pub-lished by BIPM are consistent with one another. (Quinn etal. 2013) [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]

It is clear from Figure 1.1 that the values of G are

not coherent with one another, even though the indi-vidual uncertainties are relatively small. We may theninfer that many of these studies may require more care-ful consideration to the presence of systematic effectsthat could deviate their results from the true value ofG.In this investigation we have focused on the experimentundertaken by the collaboration of the Bureau Interna-tional des Poids et Mesures (BIPM) and the Universityof Birmingham. Using their torsion balance (see Fig-ure 1.2) they have published two values of G, the firstin 2001 and the second in 2013 (Quinn et al. 2001,Quinn et al. 2013).[13, 14] Compared with the othervalues shown in Figure 1.1, the BIPM has measuredrelatively large values of G. Our investigation has at-tempted to determine whether there is an effect in ad-dition to gravity that can significantly alter the valueof G measured. The remainder of this introductionwill summarise how the BIPM experiment functionsand will discuss the motivation behind studying theeffect of magnetic fields (B-fields) in this and similarexperiments.

1.2 BIPM Apparatus

The BIPM experiment (G experiment hereafter) is amodernised and enhanced successor of the original tor-sion balance experiments. The apparatus is set up witha four-mass configuration where there are four pairs ofcylindrical source and test masses of about 11 kg and1.2 kg, respectively. The diameters of the masses aregiven as 118mm and 55mm, respectively, and these val-ues are also equal to their heights. These masses aremade from a Cu-0.7% Te. Copper was used as it has amagnetic permeability very close to unity and the in-clusion of tellurium allows the masses to be machinedto specification so that their masses can be determinedmore precisely.

Figure 1.2: The G experiment with the vacuum can andsome of the torsion disk removed. The four pairs of sourceand test masses easily identifiable.[15]

The source masses are place on a carousel whichuses an electric motor to rotate them. The test massessit in the aluminium torsion disk that is suspendedfrom a torsion strip made from a Cu-1.8% Be alloy.Using a strip instead of a wire allows for more mass to

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be supported without having to increase its torsion co-efficient. This will mean the gravitational torques willbe greater, leading to a larger angle of deflection mea-sured. An aluminium vacuum can is placed aroundthe torsion strip and disk assembly so that the diskcan more freely rotate and oscillate without the resis-tance and the perturbations of the air impacting onthe measurements. The thickness of the can’s walls isapproximately 5mm.

Figure 1.3: Diagram of the Mk.II version of the BIPM’s”Big G” apparatus used for the measurement 2013 of G: 1)source masses; 2) test mass sitting in the torsion disk; 3)torsion disk; 4) aluminium alloy carousel; 5) gimbal fromwhich the torsion strip and balance are suspended; 6) eddy-current dampers for the gimbal; 7) autocollimator; 8) cen-tral mirror tower on torsion balance; 9) vacuum can po-sition; 10) electrodes for the servo-control. (Quinn et al.2013) [14]

The G experiment is unique amongst other torsionbalance experiments in the way that it incorporatesmultiple modes of operation that can be used to de-termine G independently. By making the values deter-mined by each method consistent with one another, thecontributions to the uncertainty in the measurementswill be reduced to those which are systematic to theexperiment as a whole.The first mode follows the same process that Cavendishwould have used, hence the Cavendish method. Theangle of deflection of the test masses/torsion disk dueto the gravitational torque between the masses is mea-sured by an autocollimator through a series of mirrors.This deflection angle is reached when the gravitationaltorque matches the restoring torque from the torsionstrip. The process begins by measuring the disk’s off-set angle from zero, when the source masses are at 0◦

to their test mass pairs. The gravitational torque onthe torsion disk has been measured to be at its great-est when the source masses are positioned at ±18.9◦,therefore these are the angles the source masses arerotated to for measurement. The deflection angle ismeasured for about 30 minutes at both positive andnegative source mass angle and the difference betweenthe two is taken, accounting for the initial offset. Gcan then be calculated by using the torsion coefficientof the strip.

The second mode is electrostatic servo-control. Theprocess is to that of the Cavendish method: an initialoffset is measured and the source masses are placedat ±18.9◦. However, instead of allowing the disk torotate in response to the gravitational torque, a pairof electrodes placed in the disk are activated to applya precisely controlled counter torque, preventing thedisk from rotating. By measuring the voltage acrossthe electrodes and the change in capacitance betweenthem with respect to angle, dC/dθ, G can be calcu-lated.The third possible mode the G experiment can operateis called the time-of-swing method. The torsion diskcan be set to oscillate about its equilibrium positionand have its time period measured. When the sourcemasses are at 0◦ the gravitational torque on the diskpulls it toward equilibrium, reducing its time period.When they are rotated to 45◦ the torque will always bepulling the disk away from its equilibrium, increasingits time period. We were not able to use this modein our investigation because of the strict temperaturestability requirement to achieve desirable uncertaintieswhich could not be met in the current location of theG experiment.[12][14][13]For this study we ran the G experiment, using theCavendish and servo-control modes, to discover howthe measurement of G would change in the presence ofan alternating B-field. This ’primary field’ is discussedin Section 2.1 and the results we obtained from the Gexperiment shown in Section 3.1

1.3 Alternating Magnetic Fields

In most laboratory environments there is alternatingmains current flowing in wires and appliances nearby.From simple electromagnetic theory we know that astraight wire carrying AC will produce an alternatingB-field radially out from the wire. In the presence ofan alternating B-field, eddy currents will form insidea conductor which flow to oppose the change in thefield. These eddy currents reduce the magnitude ofthe B-field inside the conductor. The skin effect tellsus how the currents inside a conductor are distributed.At higher frequencies of field, the eddy currents arelarger due to the greater rate of change of field. As thefrequency gets large, the conductor effectively expelsall of the field flowing through it, similar to the effectof a superconductor. The skin depth, δ of a materialcan be given by

δ =

√2 ρ

ω μ0 μr, (1)

where ρ and μr are the material’s resistivity andrelative permeability, ω is the angular frequency of thefield, μ0 is the permeability of free space. The linearattenuation, a, of a field through a conductor of thick-ness d can then be given by

a = e–d/δ . (2)

2

The reduction in primary B-field is equivalent toconductor producing its own opposing, phase-shiftedalternating field. This effect could occur in the massesof the G experiment in the presence of nearby alternat-ing B-fields. The fields produced by the masses wouldresemble that of magnetic dipoles. The proximity ofthe masses means these dipoles will likely introduceextra forces on the test masses. The aim of this in-vestigation has been to predict the magnetic effect onthe G experiment and to determine whether the mag-nitude of this effect was sufficient to impact the valueof G measured.

1.4 Outline of this Work

In this project, we set out to investigate whether alter-nating B-fields would have a significant effect on themeasurement of G using the G experiment. We ex-pected there to be additional forces and torques thatwould arise due the magnetic interaction of the B-fieldsinduced in the masses by an external, alternating B-field. We have aimed to build an analytical model thatcan be used to predict the magnitude of the magnetictorques in the case of the G experiment and, therefore,determine how much the measured value of G changes.We have used a coil of wires to generate the B-fieldwhich we placed around the G experiment when it wasrunning. We have derived and tested what we expectthis primary field to be. We have also derived, usingtwo approaches, the secondary B-fields we expected tobe induced in the masses. We then evaluated the mag-netic torque on the test masses. We have tested themodel on the G experiment using different magnitudesand frequencies of currents, comparing our predictionswith the experimental data. We aimed to correctly pre-dict the sign and order of magnitude of the magneticeffect that the G experiment would measure.The findings in this report do not just apply to an effectparticular to the G experiment. There may be impli-cations for similar experiments. If they use conductingmasses, pendulums, or other source and measured ob-jects to which this would be applicable, they wouldlikely be susceptible to an analogous effect. This couldbe relevant to the work done by Harold et al. 2010(See JILA10 in Figure 1.1) which measured the changein position of two freely-suspended ”test masses” usinginterferometry due to the gravitational interaction of 4nearby ”source masses”. This will be explored furtherin Section 5.

2 The Model

We have constructed an analytical model in MAT-LAB R© to attempt to predict the magnetic torquesexperienced by the test masses in the G experiment.We have used various approximations of the physics inthe model which have been used to make the problemeasier to tackle. These will be discussed in more detailand throughout this section.

2.1 The Primary Field

The basis of the model was the primary B-field, gener-ated by the rectangular coil which was placed aroundthe G experiment. Figure 2.1 below illustrates the ge-ometry of the coil around the G apparatus.

Figure 2.1: Illustration of how the coil is placed around theG experiment, looking down the z-axis. ’S’ and ’T’ refer tothe source and test masses, respectively. The coil’s z-wiresare perpendicular to the page and can be seen to interceptthe page in red. The wires along the y-axis lie in and outthe plane of the page. Not to scale.

The coil has 9 turns of wire encased in plastic tubingmeasuring 0.674(1) m wide and 1.170(1) m tall. We re-quired the calculation of the B-field vector at any pointin a 3-D Cartesian coordinate system. The primaryfield, B0 can be determined analytically by integrat-ing the Biot-Savart law for 4 finite wires, representingeach side of the coil

dB0 =μ0 I

4π

dl× r

|r|3, (3)

where I is the current vector, dl is the vector of aninfinitesimal section of wire, and r is the vector of thewire section to the evaluated point in space. The re-sultant equations from the integration of the Equation3 can be found in Appendix A. We also required thegradients of the field for the calculation of the mag-netic force in Section 2.3, so each of the primary fieldequations was differentiated with respect to each com-ponent of position.The field and gradients from each wire were summedto represent the coil as a whole. Code was written thatwould calculate the magnitude of the primary field andgradients at each point in an assigned volume of spacearound the coil. The results of this could be visualisedby using surface plots as shown in Figures 2.2 & 2.3.

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Figure 2.2: Surface plot of modelled primary field (x-component) in the x-y plane through the origin.

Figure 2.3: Surface plot of the primary field gradient, dBx/dx of Figure 2.2.

4

To test our prediction of the primary field we useda magnetometer to measure the field at different posi-tions in and around the coil. We positioned the meterat each point in the volume around the coil using aseries of metre rulers. The complete list of test pointswe used is shown in Figure 2.4. We limited the num-ber of points to 27 in order to conserve time requiredto complete this process. The field at each point wasmeasured in x, y and z directions.

Figure 2.4: Diagram of the coil in red, with test positionsin black. The coordinate system is shown in the top-rightcorner. m takes the values of -50, 0 and 50. Positions aregiven in centimetres relative to the centre of the coil. Notto scale.

The field at each position was determined by mea-suring the background field followed by taking half thedifference of the field created using a positive and neg-ative DC of 1A. The Earth’s static geomagnetic fieldwas measured to be of the order 10–5T. The result ofthis primary field test is compared with the model pre-dictions in Figure 2.5.

The model prediction of the x-component of the pri-mary fits well with the measurement. The data has anuncertainty of approximately 10% (this is because ofthe poor choice of resistor used to measure the circuitvoltage, used to calculate the current). The points atwhich we expect the field contributions from each wireto cancel give data that notably deviates from zero. Werecognise that the wires in the coil will not be perfectlystraight and bunched up in the middle of the tubing,which will result in deviation from the model. The un-certainty of magnetometer’s position would also add tothis deviation. Further measurements could have beenmade to map out the primary field in more detail butthis would have taken even more time, and we neededto focus on the later parts of the model.The linearity of the magnetometer was tested and thedeviations from the average field-current ratio mea-sured by the magnetometer were found to be parts in

103. This performance was satisfactory for the preci-sion we required.We were unable to compare the gradients of the fieldpredicted by our equations as a considerable increasein the number of points would have been required andwould have taken far too long to achieve. We could,however, visually check gradients by comparing theirsurface plots with the fields. It appears that gradientin Figure 2.3 would match that of the field in Figure2.2, for example. Also, the divergence of the B-field infree space should be zero everywhere. The gradientswe have derived follow this rule.

2.2 Secondary Fields

The next step in building our analytical model involvedevaluating the repulsion of the primary field by the cop-per masses in the G experiment. The resultant field, Bincluding reduction in the magnitude of uniform alter-nating B-field, B0 incident on a spherical conductor atdistance r from its centre, outside the sphere’s volume,is given by Smythe to be

B =

[(1 +

D

r3

)cosθ r –

(1 –

D

2r3

)sinθ θ

]|B0| B0 ,

(4)where θ is the angle to the direction of the incident

field from the sphere’s centre and D is given by

D =( 2 μr + 1 ) ν – [ (1 + ν2) + 2 μr ] tanhν

( μr – 1 ) ν + [ (1 + ν2) – μr ] tanhνa3c , (5)

where ac is the radius of the sphere and ν is a com-plex parameter equivalent to

ν =( 1 + i )

ρac . (6)

Other symbols have their usual meaning.[16] Equa-tion 4 is given in CGS units. It can be shown thatremoving the 1s in each of the brackets of Equation 4gives an expression that describes the contribution tothe B-field induced in the conductor coil alone. Theconductors field, BS is therefore given by

BS =

[D

r3cosθ r +

D

2r3sinθ θ

]|B0| B0 . (7)

This equation is presented in spherical polar coordi-nates, so we therefore converted the coordinate systemof Equation 7 to Cartesian coordinates. The gradientsof the secondary field were also determined. The equa-tions obtained for the secondary field and its gradients,as well as the mathematical tools used to derive them,are given in Appendix B. When performing the coor-dinate conversions it was appropriate to set the z-axisof the sphere to be aligned with the direction primaryfield applied, B0. This does not, however, make thesecondary field equations compatible with the coils co-ordinate system as the direction of the primary field

5

Figure 2.5: Comparison of measured primary field with the model. The x-component is at the top followed by y and z.Point 25 in z was stored incorrectly. N.B. The x-axis of these data plots refer to an arbitrary order of the 27 individualpositions used to measure the field.

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Figure 2.6: Surface plot of the modelled secondary field in the x-y plane at z=0.07m (In its own coordinate system)

Figure 2.7: Surface plot of the gradient, dBx/dy of Figure 2.6.

7

vector will be different for masses in different positions.A coordinate system rotation for these equations wastherefore incorporated into the model. Details of thisprocess and the code used can be found in AppendixC.

The relative phase of the secondary field to the pri-mary is incorporated into D. At 60Hz the secondaryfield of the source mass lags behind the primary fieldby about 155◦. The value and phase of D will be dif-ferent between the source and test masses.The magnetic moment vector, m of the spherical con-ductor from the approach used in SI units is given by

m =2π

μ0D B0 . (8)

By using this method to analytically evaluate thesecondary B-fields we expect to be induced in themasses, we must make two important assumptions.Firstly, the direction primary field incident on themasses in the G experiment is uniform over the vol-ume of the mass, in the direction we predicted field tobe in at the masses’ centres. The magnitude and direc-tion of the primary field will not change greatly overthe volume of the source or test masses, providing theyare not positioned near the sides of the coil. The twosource masses which were in closest proximity to thecoil were still about 0.1m from the sides of the coil andthe primary field vector averaged over the volume of themasses will be approximately equal to the vector thatwould pass through their centres. We therefore believethis approximation is reasonable. Secondly, these sec-ondary equations are valid for a spherical conductor,the masses in the G experiment are cylindrical. Theshape of the masses, however, is not too dissimilar fromthat of a sphere as their heights and width are equal.This is the closest they can be to being spherical. Wehave included a correction to the value of radius, acused for the masses. A multiplicative factor of 3

√1.5

will give the radius of a sphere with the same volumeas the original cylinder. It is important to match thevolume of material in which the opposing field is in-duced to obtain the correct value of the field.We tested the secondary field equations by using asolid phosphor-bronze sphere as the conductor. Theradius of the sphere was measured to be 66.1(2)mm.The coil was placed on its side, so that its plane wasparallel with the floor, and placed the sphere at itscentre. With the magnetometer positioned just abovethe sphere (pointing in the x-direction), measurementswere made of the total field. The magnetometer didnot specify the direction of the field, only its mag-nitude. Therefore, because the secondary field is es-sentially anti-phase, its contribution to the field wascalculated by subtracting the field measured from ourmodel prediction of the primary field at the chosenpoint. The frequency-dependence of the sphere wastested by changing the frequency of the current andthe radial-dependence was tested by moving the mag-netometer further from the sphere. The results of thesemeasurements are shown in Figures 2.8 & 2.9.For the frequency-dependence test, the magnetometer

was positioned at 70(5)mm above the centre of thesphere. The current of the field was, again, set to1 A. The AC B-field measured in the lab reached about10nN at its maximum and, to avoid mixing of the envi-ronmental and the primary field, the frequency of 50Hzwas avoided.

The calculated upper and lower bounds shown inFigure 2.9 reflect the uncertainty in the position of themagnetometer. The closer to the sphere you measure,the greater the magnitude of its field. The first pointto note is that the secondary field appears to tend tothe magnitude of the primary field at high frequency,corresponding to the almost complete expulsion of thefield. The second fact is that it is clear that the slopeof the data points at low frequency do not match up tothe predicted curves. This was a phenomenon we wereunable to resolve. We have avoided placing any otherconductors nearby which could interfere with the mea-surement. We postulate that some properties of thesphere, such as ρ or μr, could also be dependent on fre-quency. There may be additional effects on the spherethat we do not know about. It was important we ob-tained the levelling off at high frequency and we madenote of the factor of ' 3 difference between measuredand predicted values at and around 60Hz.The radial-dependence test was carried out using a con-stant current and frequency of 1 A and 55Hz. In Figure2.8, the reduction of the sphere’s field with distancehas been compared with inverse-square and inverse-cube functions that have been fit to the start and endpoints. This has been implemented in order to examinethe radial-dependence of the field.

Figure 2.8: Plot of secondary field magnitude, measuredat increasing distance from the phosphor bronze sphere.Inverse-square and inverse-cube relationships have been fitfor comparison.

It was apparent that the field very close to thesphere’s surface may have a contribution from higherorder terms, which would mean the field would dropoff less with distance. However, as the magnetometermeasured the field further from the pole of the sphere,the radial-dependence seemed to settle on an inverse-cube relationship, as we expected.

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Figure 2.9: Plot of secondary field magnitude, measured at 70(5)mm above the phosphor bronze sphere. Sphere is atthe coil’s centre. Upper and lower bounds refer to the model prediction of the positional uncertainty.

Figure 2.10: Plot of secondary field magnitude, measured at 70(5)mm above the copper source mass. Centre of sourcemass is positioned 110(5)mm from coil centre. Upper and lower bounds refer to the model prediction of the positionaluncertainty.

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Figure 2.11: Total field (x-component) due to coil and 4 source masses in x-y plane (coil coordinates) through the origin.Vacuum can attenuates the field in the centre.

More data points should have been taken to betteranalyse this dependence.When the frequency of the input voltage was increasedthe measured current would drop, requiring a greaterinput voltage. At 5000Hz the input voltage required tomeasure the same voltage in the circuit was between 3and 4 times larger than that used for 60Hz, to achieve1 A. A variety of factors could be contributing to thiseffect. The power amplifier used to increase the cur-rent was likely less efficient at high frequencies. Thesame problem may have also affected the reading of thevoltage by the multimeter used; these frequencies maybe outside its sensitive limit. Also, at 5000Hz the skindepth of copper, used in the coil’s wires, is about 1mmso is of the same size as the wire’s diameter. The skineffect would produce a counter voltage, lowering theeffective voltage, and hence current, in the coil. Ourresults do follow the general trend expected, levellingoff at high frequencies, so we believe by increasing theinput voltage we have counteracted many of these is-sues.In addition to the phosphor bronze sphere we were ableto repeat these tests with one of the source masses fromthe G experiment. The mass had to be positionedabout 110(5)mm above the centre of the coil, so wewould be expecting a reduced B-field magnitude.

Similarly to the sphere, the field induced on thesource mass approached saturation at high frequency.The data follows a similar trend that can be seen inFigure 2.9. This was an indication that the approxi-mation, whereby the masses are treated as spheres, wasvalid. The difference between the data and the predic-tion at 60Hz is approximately a factor 4, which wasalso noted. More points should have been measured athigher frequencies to more clearly see the tail off of thecurve.

Figure 2.12: Plot of secondary field magnitude, measured atincreasing distance from the copper source mass. Inverse-square and inverse-cube relationships have been fit for com-parison.

The radial-dependence of the source mass is mostlycomparable to that of the phosphor bronze sphere. Thesource mass data gives the impression that higher or-der terms may continue to be significant to greater radiibefore settling on the inverse-cube relationship. Thiswill likely be a result of the cylindrical shape of thesource mass.We were able to include secondary fields due to thesource masses as well as the linear attenuation, a, ofthe vacuum can to the primary field. An example ofthe total resultant field can be seen in Figure 2.11.

Another assumption that has been made is that thesource masses are far enough apart from each other forthe secondary fields to not affect one another. That isto say, the source masses are made to only experiencethe primary field and not the secondary fields from anyof the other source (or test) masses.Our supervisor, Clive Speake, had also been workingon analytically determining the primary and secondaryfields, parallel to us. The approach Clive had used

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was simpler as no coordinate transformation or rota-tion was required. This approach from Jackson givesthe field, Bd at radius, r from a magnetic dipole as

Bd =μ0

4π

3 r (m · r) – m

|r|3, (9)

where r is the unit vector of r and m is the samemagnetic moment which is derived from the previousapproach.[17] The differentiation of Equation 9 is non-trivial and required the use of MapleTM(which wasalso used to check the previous differentiations). Thisapproach to the secondary field was not utilised untillate into the project for purpose of comparison withthe method outlined earlier in this section. These arediscussed in the upcoming sections.

2.3 Calculation of the Torque

We have been able to calculate the expected magnetictorques on the test masses about the torsion strip byevaluating the forces on them. The forces can be de-rived from the magnetic potential in which the testmasses sit. The potential energy, E of a magneticdipole, m in a B-field, B is

E = – m ·B . (10)

In the case of the scenario we have modelled, thefield incident on the test masses will be the addition ofthe primary field generated by the coil and the sec-ondary fields induced on the source masses, phase-shifted relative to the primary. Therefore, B becomesB0 + BS. The duration of a measurement is substan-tially longer than the time period of the alternatingB-fields we are using, so we are required to determinethe time averaged energy, 〈Et 〉 of a test mass. There-fore, a factor of 2 being removed and the conjugate ofthe field is taken. The result is given as

〈Et 〉 = –π

μ0|Dt| (B0 + BS) · (B0 + BS)∗ . (11)

The magnetic force, F can be determined by calcu-lating the negative gradient of the energy.

F = –∇E = –∇(m ·B) . (12)

The gradient of the vector dot product can besolved by using the appropriate vector identity. Thetime-averaged force on a test mass can therefore befound to be

〈Ft 〉 =π

μ0|Dt|

[(BT·(∇BT)∗

)+(B∗

T·∇BT

)], (13)

where BT ≡ B0 + BS. In this instance thegradient and the conjugate operators commute so(∇BT)∗ ≡ ∇(B∗

T). When including the attenuationfactor, a, of the vacuum can there is also a factor of|a|2 outside in Equations 11 & 13. Finally, the time-averaged torque, 〈τ 〉 on each of the test masses aboutthe torsion strip can be calculated as

〈τ 〉 = rt × 〈Ft 〉 (14)

where rt is the radial vector of the centre of rota-tion to the position of the test mass. The component oftorque that will rotate the test masses and the torsiondisk is in the z-direction. The other components of thetorque will not have an effect on the measurement ofG.The Equations 13 & 14 were included into the modelto so that the torque on all of the test masses due tothe sum of the primary and secondary fields could becalculated. The code was rewritten so that instead ofevaluating the fields at all points in a given volume ofspace, the fields and gradients were evaluated exactlyat the masses. This was done so that the source massescould be rotated around the torsion disk, akin to theG experiment, in order to discover how the torque onthe test masses changed with the angle of the sourcemasses. An overview of the processes in this code isshown in Appendix D.The force and torques due to gravity were also includedinto the model so that a comparison between magneticand gravitational torques could be made. The 2014CODATA value of G (given in Section 1.1) was usedin these calculations. The magnetic torques in thez-direction for both secondary field approaches werecompared with the predicted gravitational torques asa function of source mass angle. The results of thesesimulations are shown in the Figures 2.13 & 2.14.The gravitational torque the model produces has beenchecked and we are confident that it closely resembleswhat would be measured by the G experiment. Thetorque is rotationally symmetrical about 0◦; its magni-tude is greatest at approximately ±18.4◦, which is veryclose to the 18.9◦; and the greatest difference in torquein G experiment is measured to be about 30nNm, com-pared with our predicted 36nNm. The magnetic torquecalculated from the original approach in Figure 2.13predicted an opposite sign to gravity which thereforemade the effect repulsive. This was contrary to whatwe had expected the nature of the magnetic interac-tion to be at the beginning of the investigation. A200mA current produces a B-field of the order of 1μTand, at this magnitude of field, Figure 2.13 predicts amagnetic-to-gravitational torque ratio of about 1%.

11

Figure 2.13: Model simulation of magnetic and gravitational torques on all test masses due to source mass rotation,using original secondary field approach. 200mA, 60Hz current.

Figure 2.14: Model simulation of magnetic and gravitational torques on all test masses due to source mass rotation,using alternative secondary field approach. 200mA, 60Hz current.

12

Figure 2.15: Torques on test masses when source masses are at ±19◦ as a function of frequency, using original approach.200mA current

Figure 2.16: Torques on test masses when source masses are at ±19◦ as a function of frequency, using alternativeapproach. 200mA current.

13

Figure 2.17: Difference in torque on test masses when source masses are at ±19◦ as a function of frequency, using originalapproach. 200mA current.

Figure 2.18: Difference in torque on test masses when source masses are at ±19◦ as a function of frequency, usingalternative approach. 200mA current.

14

Using the second approach yields significantly dif-ferent results. The order of magnitude of the magnetictorques is the same but goes the magnetic torque addsto gravity and the curve appears to be much more off-set.The primary field will produce a constant torque on thetest masses as they remain effectively stationary. Thistorque will cause a slight offset in the in the equilibriumangle of the torsion disk which will be accounted for atthe beginning of each G experiment measurement, andwill be subtracted when taking the difference in deflec-tion angle. Therefore, the primary field will not di-rectly contribute to a change in the G value measured.The repositioning of the source masses, however, willalter the secondary fields during measurement. Hence,the torques will change and the value of G will differ-ent as result.In order to simulate the magnetic torque expected to bepresent in G experiment during its normal operation,the model was updated again to evaluate the torqueswhen the source masses were at ±19◦. The differencein torques between the two orientations could then becalculated, both of which were done as function of fre-quency. This way, the effect of different frequenciesand the magnitude of the magnetic torque differencecould be determined.The original secondary field approach predicts a nega-tive torque difference - a reduction in the total torqueon the G experiment - which reaches greatest mag-nitude at about 43Hz. The torque difference decaystoward zero at high frequency as the vacuum can ef-fectively attenuates the field on the test masses insideit to zero.The torque difference calculated using the alternativeapproach (shown in Figures 2.16 & 2.18), again, pre-dicts an attractive magnetic effect instead where thepeak difference in magnitude is at about 40Hz.Both approaches predict a slight difference in the shapeof the torque with respect to frequency between +19◦

and –19◦. We had expected that both torques wouldhave the same shape in both configurations but thismay not actually be the case.

2.4 Aluminium Vacuum Can

In the analytical model the attenuation due to the vac-uum can surrounding the test masses has been incorpo-rated by using the linear attenuation factor. Thereforethe fields and gradients experienced by the test massesare reduced by a single multiplicative factor, which de-pends on the skin depth of the can. The effect that hasbeen employed into the model has been an approxima-tion of the physics.In order to discover how the vacuum can would actuallyaffect the field, we performed numerical finite elementanalysis on the B-field using FEMM (Finite ElementMethod Magnetics). The coil’s wires were modelledto generate the primary field and a comparison of thefield with and without the analogue of the aluminiumvacuum can was made.The vacuum can clearly has a considerable effect on the

magnitude and direction field (Figures 2.19 & 2.20).The gradients of the field will, by extension, be greatlyaffected too. The effect of the can has clearly beenunder-estimated in the code. In the presence of thecan, the field incident at the two source masses nearestto the wires is increased by about 35% and the othertwo source masses experience about a 65% reduction.This analysis was performed late in the project andthe results have not been included in the model. Thesoftware has the potential to simulate the secondaryfields induced on conductors and to calculate magneticpotential energy. If more time had been available, fur-ther finite element analysis could have been performedto more accurately model the effect of the vacuum can.

15

Figure 2.19: Fintite element analysis of the primary field in the x-y plane of the coil. Without the vacuum can.

Figure 2.20: Fintite element analysis of the primary field in the x-y plane of the coil. With the vacuum can included.N.B there are mild asymmetries arising from calculation errors, most likely due to the boundary conditions.

16

3 Results

The G experiment was run with the coil around theapparatus to obtain data points of G. [18] These testsran with current in the coil changing but the frequencyheld constant, and vice versa, to examine how the valueof G would change in the presence of an alternating B-field. The predictions from the analytical model thatwas built were subsequently compared with the G data.

3.1 G Experiment

Firstly, the G experiment was run with a varying amagnitude of current at 60Hz (including zero currentas a control measurement). 60Hz was used to avoidmixing with any environmental fields and because thisis the frequency of the mains in the USA, where the Gexperiment will be operating soon. For two runs, thevacuum pump, used to maintain a vacuum inside thecan, was not magnetically shielded, and for the otherruns it was. The pump is continuously active duringmeasurement, requiring electrical current to functionand will inherently produce an extra source of B-field.The result of these runs indicates that the effect of themagnetic torques are in fact repulsive, reducing thetotal torque in the system and the value of G mea-sured. We find this trend in the data strongly followsa negative-square relationship with the input currentand, hence, primary B-field strength. This correlateswith our model as the multiplication of fields in Equa-tion 13 would convey a magnetic force proportionalto the square of the current/B-field. The percentagechange in G at 220mA - which produces a μT field -is of the order of 0.1%. This is a 1000ppm and theG experiment is sensitive to changes of about 10ppm,so - with a μT field - the magnetic interaction wouldsignificantly affect the value of G.Secondly, data points were collected at a con-stant 220mA at various frequencies to determine thefrequency-dependence of the effect. Few data pointsare available, as each one took 2 days to be collected.

The magnitude of the change in G is greatest between30Hz and 50Hz, which does follow the model predic-tion. The change appears to die off due to the at-tenuation of the vacuum can at high frequencies, asexpected.

3.2 Comparison of G Data with ModelPredictions

By finding the ratio between magnetic and gravita-tional torque differences at ±19◦ we can compare theG data with the prediction of the model. Both sec-ondary field approaches were used, the results of whichis shown in Figure 3.2.We find that our original approach reproduces the samesign of the effect measured, whereas the alternativedoes not. Both, however, do predict the same current-squared relationship implied by the G results. Themagnitude of the alternative is also more than 3x thatof the original. We are unsure as to the nature of thisramification. The original approach predicts a changein G measured that is between 3.6x that of the data.However, at 60Hz it was previously found that the sec-ondary field induced in a copper source mass wouldbe 3x-4x lower than the theory anticipated (See Fig-ure 2.10). If this factor is taken into account, themodel prediction is significantly more comparable withG data. This also applies to the alternative approach.If the sign of the effect has been miscalculated in thecode, the prediction will still over-estimate the mag-netic effect. There could, however, be other factorsthat have not been included that would, in reality, re-sult in a lesser difference in G.The frequency-dependence was also compared at a con-stant 220mA.

The trends and differences in magnitude in Figure3.3 match those in Figure 3.2. The model matches therelationships and predicts an effect that is of the sameorder of magnitude that is measured by the G experi-ment.

Figure 3.1: Percentage change in G measured by the G experiment, as a function of current; at 60Hz.

17

Figure 3.2: Percentage change in G measured by the G experiment compared with the model predictions of bothsecondary field approaches, as a function of current; at 60Hz.

Figure 3.3: Percentage change in G measured by the G experiment compared with the model predictions of bothsecondary field approaches, as a function of frequency; at 220mA.

18

4 Conclusions

We have found that the measurement of G by the Gexperiment can indeed be affected by the presence ofalternating B-fields around the apparatus. This mag-netic forces on the test masses in the experiment arerepulsive, so that it subtracts from the total force themasses experience during operation. The fractionalchange in G measured is proportional to the negative-square of the B-field, in which the apparatus sits. Therange of field frequencies at which the magnetic effectis greatest is 30-50Hz. This is close to the frequencyof mains current you would expect to be flowing in alaboratory environment.We have built an analytical model in an attempt toevaluate the total B-field, and gradients thereof, whichwill be experienced by the conductive test masses inthe G experiment. The total field is comprised of theprimary field, controlled by a coil of wires to be placedaround the G experiment, and the secondary fields pro-duced by the conducting source masses. The secondaryfields arise from the masses’ resistance to the change inB-field through them. This repulsion is equivalent tothe masses having an opposing, phase-shifted B-fieldinduced in them; the magnitude of which is depen-dent on the field frequency, given by the skin effect.Two approaches were employed in the model to deter-mine the secondary fields. The first used an expressionthat required a coordinate system transformation androtation to be compatible with the other model com-ponents. The second, which was incorporated late intothe project, did not necessitate these actions and couldbe directly applied. It would therefore have a lowerchance of having incorrect expressions. The model in-cluded physical approximations such as: The sourcemasses were assumed to be spherical, which data inSection 2.2 showed to be valid; The field through themasses is uniform, which we estimate to be acceptable;the aluminium vacuum linearly attenuates the field, towhich analysis in Section 2.4 shows will need revision;and the test masses have no effect on the total field,which is reasonable as nearly all the torques’ magni-tudes arise from nearest source-test mass neighbours,and will therefore not be greatly affected by the rela-tively more distant test masses.The model, using the original secondary field approach,predicted a magnetic effect which followed the same re-pulsive trend seen in the G data (Figure 3.2) but over-estimated the magnitude by a factor of 3-4. This can,however, be somewhat resolved by referring to the mea-surements made of the secondary field (Figure 2.10).The theory used in the model also over-estimated thefield due to a source mass by a similar factor. Weare still unsure why the alternative approach used pro-duces a greater magnitude of this effect, as well as theopposite sign. The underlying physics should be ex-actly the same and therefore produce the same results.It was included into the model late on into the projectso little time could be allocated to scrutinise and cor-rect for minor bugs in the code. It is also possible thatthe code may be incorrect for both approaches. Fur-

ther work is required to obtain the same result fromthem both.Finally, the aim of this work has been to acquire anunderstanding of the magnetic effects that are presentin gravitational experiments, and we sought to analyt-ically determine an order of magnitude estimation ofthis effect. We have been successful in this endeav-our. The original approach used to model the effectcorrectly predicts the change in G, measured by theG experiment, due to a local B-field as a function ofmagnitude and frequency. The model we have cre-ated has various physical approximations and exclu-sions. While, for this project, these have been sufficientfor us to achieve the correct prediction of the trendsand a correct order of magnitude, they can also berevised and given a more rigorous treatment to moreaccurately predict the magnetic effects on G experi-ments. We had hoped the extra magnetic torques inthe measurement of G would reconcile the different val-ues published by the various studies. In the case of theG experiment, the background AC field measured inthe lab was about 10nT. Following the trends in thedata and the model, we predict the change in G tobe of the approximate order of 0.1ppm, which is smallcompared to the apparatus’ sensitivity of about 10ppm.We would not then expect the values measured by theG experiment to be affected during normal operation.Other experiments’ values of G may be suppressed byB-fields but further investigation specific to each ex-periment would be required to determine this.

5 Discussion and Implications

The findings of the effect we have studied can be ap-plied to other experiments used to measure G. In sec-tion 1.4 a similar gravitational experiment was dis-cussed, briefly. An analogous magnetic effect has thepotential to change the value of G measured, depend-ing on the magnitude of the local AC B-field. The 780g test masses and 120 kg source masses that were usedwere made of copper and tungsten, respectively. Theresistivity and relative permeability of tungsten are notdissimilar to copper, so we might expect a similar re-action to the value of G measured. If the local field isgreater than 100nT the change in G due to magnetictorques will likely exceed the sensitivity of the mea-surement, subtracting from its true value.A solution to counter the effect of B-fields could beto surround the apparatus with shielding. Therefore,unless the origin of the field is due an internal device,the consequential magnetic effects would be effectivelyeliminated.The model in this study has been employed to measurethe magnetic torque present in G experiment when itis operating in the Cavendish and servo-control modes.We have not yet considered the change in G using thetime-of-swing mode. In this mode the test masses alsomove in the field. Therefore the test masses would ex-perience a change in both the primary and secondaryfields. The primary field and gradients are approxi-

19

mately two orders of magnitude greater than that ofthe secondary. This then could result in a magneticeffect about 10x that of the other modes and a differ-ence in G of about 1 ppm. Additional modelling of thethis mode of operation would be required to decisivelyevaluate the actual effect expected.

6 Acknowledgements

I would firstly like to thank my project partner PeterSteele for his hard work and dedication throughout thisjoint investigation and for his perseverance through theups and downs along the way. Secondly I thank mysupervisor, Clive Speake, without whom Peter and Iwould not be where we are now. He has worked along-side us from the beginning and has put aside much ofhis time to guide us in the past 6 months. I only hopewe have made a considerable contribution to his workon the gravitational constant and wish him all the bestin his future endeavours. Finally I wish to mention thegreat work John Bryant has put in to make sure Peterand I have promptly had all the tools and equipmentwe could ask for in the lab for our testing. He, alongwith Clive, has organised the big G experiment for thisproject and they take the credit for the data which wehave been able to use to make our comparisons andconclusions.

20

References

[1] P. J. Mohr, D. B. Newell, and B. N. Taylor. “CODATA Recommended Values of the FundamentalPhysical Constants: 2014”. In: ArXiv e-prints (2015).

[2] G.G. Luther and W.R. Towler. “Redetermination of the Newtonian Gravitational Constant G”. In:Phys. Rev. Lett. 48 (1982). doi: 10.1103/PhysRevLett.48.121. url:http://link.aps.org/doi/10.1103/PhysRevLett.48.121.

[3] T. R. Armstrong and M. P. Fitzgerald. “New Measurements of G Using the Measurement StandardsLaboratory Torsion Balance”. In: Phys. Rev. Lett. 91 (2003). doi: 10.1103/PhysRevLett.91.201101.url: http://link.aps.org/doi/10.1103/PhysRevLett.91.201101.

[4] O. V. Karagioz and V. P. Izmailov. “Measurement of the gravitational constant with a torsion balance”.In: Measurement Techniques 39 (). issn: 1573-8906. doi: 10.1007/BF02377461. url:http://dx.doi.org/10.1007/BF02377461.

[5] C.H. Bagley and G.G. Luther. “Preliminary Results of a Determination of the Newtonian Constant ofGravitation: A Test of the Kuroda Hypothesis”. In: Phys. Rev. Lett. 78 (1997). doi:10.1103/PhysRevLett.78.3047. url: http://link.aps.org/doi/10.1103/PhysRevLett.78.3047.

[6] J.H. Gundlach and S.M. Merkowitz. “Measurement of Newton’s Constant Using a Torsion Balance withAngular Acceleration Feedback”. In: Phys. Rev. Lett. 85 (2000). doi: 10.1103/PhysRevLett.85.2869.url: http://link.aps.org/doi/10.1103/PhysRevLett.85.2869.

[7] St. Schlamminger et al. “Measurement of Newton’s gravitational constant”. In: Phys. Rev. D 74 (2006).doi: 10.1103/PhysRevD.74.082001. url: http://link.aps.org/doi/10.1103/PhysRevD.74.082001.

[8] ZK. Hu, JQ. Guo, and J. Luo. “Correction of source mass effects in the HUST-99 measurement of G”.In: Phys. Rev. D 71 (2005). doi: 10.1103/PhysRevD.71.127505. url:http://link.aps.org/doi/10.1103/PhysRevD.71.127505.

[9] J. Luo et al. “Determination of the Newtonian Gravitational Constant G with Time-of-Swing Method”.In: Phys. Rev. Lett. 102 (2009). doi: 10.1103/PhysRevLett.102.240801. url:http://link.aps.org/doi/10.1103/PhysRevLett.102.240801.

[10] H. V. Parks and J. E. Faller. “Simple Pendulum Determination of the Gravitational Constant”. In:Phys. Rev. Lett. 105 (2010). doi: 10.1103/PhysRevLett.105.110801. url:http://link.aps.org/doi/10.1103/PhysRevLett.105.110801.

[11] P. J. Mohr, B. N. Taylor, and D. B. Newell. “CODATA recommended values of the fundamental physicalconstants: 2010”. In: Reviews of Modern Physics 84 (2012). doi: 10.1103/RevModPhys.84.1527.

[12] T. Quinn et al. “The BIPM measurements of the Newtonian constant of gravitation, G”. In:Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and EngineeringSciences 372 (2014). issn: 1364-503X. doi: 10.1098/rsta.2014.0032.

[13] T. J. Quinn et al. “A New Determination of G Using Two Methods”. In: Phys. Rev. Lett. 87 (2001).doi: 10.1103/PhysRevLett.87.111101. url:http://link.aps.org/doi/10.1103/PhysRevLett.87.111101.

[14] T. Quinn et al. “Improved Determination of G Using Two Methods”. In: Phys. Rev. Lett. 111 (2013).doi: 10.1103/PhysRevLett.111.101102.

[15] Neue Zurcher Zeitung. Widerspenstige Gravitationskonstante: Gemeinsam wollen Forscher �BigG� knacken. url: http://physics.nist.gov/cgi-bin/cuu/Value?bg (visited on 03/08/2016).

[16] Smythe W.R. Static and Dynamic Electricity. Second Edition. McGraw-Hill Book Company Inc., 1950.

[17] J.D. Jackson. Classic Electrodynamics. Third Edition. John Wiley & Sons Inc., 1998.

[18] C. Speake and J. Bryant. “Measurements of G using the BIPM apparatus”. Measurements of G withvarious coil field properties.

[19] Steele. P.J. 2016.

21

Appendix A Primary Field Equations

The expression for the B-field for the two wires which run along the y-axis and z-axis are respectively given by

By =μ0 I

4π

1

(x2 + z2)

[ys√

x2 + y2s + z2–

ye√x2 + y2e + z2

](–z ı + x k) , (15)

Bz =μ0 I

4π

1

(x2 + y2)

[zs√

x2 + y2 + z2s–

ze√x2 + y2 + z2e

](y ı – x ) (16)

where x, y and z are the distances from a given point in space to the position of the wire, given in Cartesiancoordinates; ys and ye are the positions of the start and end of the y-axis wires that with reference to thedirection of the current; and zs and ze are the same for the z-axis wires.

Appendix B Coordinate System Transformation of Equation 7

The transformation matrix for converting spherical polar coordinates into Cartesian is given as r

θ

φ

=

sinθ cosφ sinθ sinφ cosθcosθ cosφ cosθ sinφ – sinθ

– sinφ cosφ 0

ı

k

, (17)

and from this we find the respective conversions for the radial and polar angle unit vectors (r & θ,respectively) to be

r = sinθ cosφ ı + sinθ sinφ + cosθ k , (18)

θ = cosθ cosφ ı + cosθ sinφ – sinθ k . (19)

The trigonometric conversion tools used are

sinθ = sin[arccos

(z

r

)]=

√1 –(z

r

)2=

√x2 + y2

x2 + y2 + z2, (20)

cosθ =z

r=

z√x2 + y2 + z2

, (21)

sinφ = sin[arctan

(y

x

)]=

y√x2 + y2

, (22)

cosφ = cos[arctan

(y

x

)]=

x√x2 + y2

. (23)

We then arrive at the secondary field, BS due to a conducting sphere in Cartesian coordinates

BS =B0 D

(x2 + y2 + z2)52

[3

2xz ı +

3

2yz +

(z2 –

x2 + y2

2

)k

], (24)

where x, y, and z are the distances to a point outside the sphere to its centre, in Cartesian components, andthe other symbols have their usual meanings. The nine components of the gradient of the secondary field are

∂BxS

∂x=

3

2B0 D

z ( –4x2 + y2 + z2 )

(x2 + y2 + z2)72

, (25)

∂BxS

∂y=

3

2B0 D

– 5 x y z

(x2 + y2 + z2)72

, (26)

22

∂BxS

∂z=

3

2B0 D

x ( x2 + y2 – 4z2 )

(x2 + y2 + z2)72

, (27)

∂ByS

∂x=

3

2B0 D

– 5 x y z

(x2 + y2 + z2)72

, (28)

∂ByS

∂y=

3

2B0 D

z ( x2 – 4y2 + z2 )

(x2 + y2 + z2)72

, (29)

∂ByS

∂z=

3

2B0 D

y ( x2 + y2 – 4z2 )

(x2 + y2 + z2)72

, (30)

∂BzS

∂x=

3

2B0 D

x ( x2 + y2 – 4z2 )

(x2 + y2 + z2)72

, (31)

∂BzS

∂y=

3

2B0 D

y ( x2 + y2 – 4z2 )

(x2 + y2 + z2)72

, (32)

∂BzS

∂z=

3

2B0 D

z ( 3x2 + 3y2 – 2z2 )

(x2 + y2 + z2)72

. (33)

Evaluating the divergence of the secondary field yield zero for all space outside the surface of the sphere.

Appendix C Rotation of the Secondary Coordinates

Here is a copy of the code used designated for ’Source Mass 1’ which rotates is used to convert its coordinatesystem, used to evaluate the secondary field and gradients, into the coil’ system.

Firstly the Euclidean norm of the primary field is calculated as BS1. This is to calculated the correctmagnitude of the field at the centre of the source mass:

BS1 = norm([BxS1, ByS1, BzS1]);

BxS1 corresponds to the primary field in the x-direction at Source Mass 1. This extends to the y and z. Nextthe parameters required to translate components of the spheres coordinate system into the coil’s arecalculated. BvecS1z is the a vector whose first value is the factor that converts the magnitude ofx-component of the sphere into the z-component of the coil, the second is y-to-z, and so on.

BvecS1z = [BxS1, ByS1, BzS1]/BS1;

The sphere is converted into the coil’s y-direction by calculating the cross product of BvecS1z with the coil’sk vector, denoted by znorm.

BvecS1y = cross(BvecS1z, znorm);

And finally the parameters which translate the sphere into the coil’s x-direction, in BvecS1x can be found bytaking calculating the cross product of BvecS1y with BvecS1z

BvecS1x = cross(BvecS1y, BvecS1z);

These parameters are then multiplied with the appropriate sphere component when the secondary field isadded to the primary field.

23

Appendix D Overview of Torque Code with Rotating SourceMasses

Figure D.1: Flow chart detailing the calculations in the model code used to determine the torques on the test masses.(Steele 2016)[19]

24