Correlation between Residual Stress and hardness response generated by Laser ...
Final Year Dissertation Report
1. 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 alternating
magnetic 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 the
change in G with respect to both the magnetic field strength and frequency. An analytical model has been
built 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 interaction
gave rise to repulsive torques - reducing value of G measured - and that G changes directly with the square of
the magnetic field magnitude. The magnetic torques peak at a frequency of about 40Hz. Our model correctly
predicts this relationship and its order of magnitude, as well as the frequency-dependence measured. In a 1μT
field, the reduction in G is of the order of 1000ppm. The field around the BIPM experiment is about 10nT under
normal operating conditions. This will adjust G by only 0.1-1ppm, which is much smaller than the sensitivity
of the apparatus. The effect we have quantified may have repercussions on the results of other gravitational
experiments. Further study can be undertaken to continue to improve our model, to more accurately detail the
physics that has been approximated, and to extend its application to other G experiments.
3. 1 Introduction
1.1 The Gravitational Constant
Isaac Newton’s gravitational constant, G is the con-
stant of proportionality that governs the interaction
between bodies due to the presence of mass. It is a
fundamental constant of the universe which not only
extends its application to Albert Einstein’s general the-
ory of relativity but is also an essential component of
many different areas of physics. The value of G was
first measured by Henry Cavendish in 1797-98. He de-
termined the density of the Earth to be ρ⊕ = 5.448g
cm–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 by
CODATA (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 uncertainty
of 47 ppm.[1] This differs from Cavendish’s result by
only about 1%. CODATA gives a weighted average,
incorporating the results of many investigations which
use a variety of approaches to measure G. Figure 1.1
shows 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 the
other fundamental forces and this is one of the impor-
tant facts which leads to G being inherently difficult
to measure. It contributes to making G the least well-
defined fundamental constant today. We are sure to
the value of G to hundreds of parts per million. How-
ever, by comparison other similarly universal constants
such as the Planck constant, h and the elementary unit
of charge, e are each known to considerably higher pre-
cision. Their relative standard uncertainties are both
of the order of 0.01ppm. You can increase the gravita-
tional force measured by amplifying the signal, using
greater masses. This, unfortunately, has drawbacks
such as added material expenditure and the reduced
precision of a vast apparatus. Another obstacle faced
when measuring G is that the force of gravity is only
ever attractive. This means it is not possible to shield
against the effects of gravitational perturbations exter-
nal to the experimental apparatus.
Figure 1.1: Spread of G values across different experiments
spanning three decades. The 2001 and 2013 values pub-
lished by BIPM are consistent with one another. (Quinn et
al. 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 then
infer that many of these studies may require more care-
ful consideration to the presence of systematic effects
that could deviate their results from the true value of
G.
In this investigation we have focused on the experiment
undertaken by the collaboration of the Bureau Interna-
tional des Poids et Mesures (BIPM) and the University
of Birmingham. Using their torsion balance (see Fig-
ure 1.2) they have published two values of G, the first
in 2001 and the second in 2013 (Quinn et al. 2001,
Quinn et al. 2013).[13, 14] Compared with the other
values shown in Figure 1.1, the BIPM has measured
relatively 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 value
of G measured. The remainder of this introduction
will summarise how the BIPM experiment functions
and will discuss the motivation behind studying the
effect of magnetic fields (B-fields) in this and similar
experiments.
1.2 BIPM Apparatus
The BIPM experiment (G experiment hereafter) is a
modernised and enhanced successor of the original tor-
sion balance experiments. The apparatus is set up with
a four-mass configuration where there are four pairs of
cylindrical source and test masses of about 11 kg and
1.2 kg, respectively. The diameters of the masses are
given as 118mm and 55mm, respectively, and these val-
ues are also equal to their heights. These masses are
made from a Cu-0.7% Te. Copper was used as it has a
magnetic permeability very close to unity and the in-
clusion of tellurium allows the masses to be machined
to specification so that their masses can be determined
more precisely.
Figure 1.2: The G experiment with the vacuum can and
some of the torsion disk removed. The four pairs of source
and test masses easily identifiable.[15]
The source masses are place on a carousel which
uses an electric motor to rotate them. The test masses
sit in the aluminium torsion disk that is suspended
from a torsion strip made from a Cu-1.8% Be alloy.
Using a strip instead of a wire allows for more mass to
1
4. be supported without having to increase its torsion co-
efficient. This will mean the gravitational torques will
be greater, leading to a larger angle of deflection mea-
sured. An aluminium vacuum can is placed around
the torsion strip and disk assembly so that the disk
can more freely rotate and oscillate without the resis-
tance and the perturbations of the air impacting on
the measurements. The thickness of the can’s walls is
approximately 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 from
which 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 torsion
balance experiments in the way that it incorporates
multiple 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, the
contributions to the uncertainty in the measurements
will be reduced to those which are systematic to the
experiment as a whole.
The first mode follows the same process that Cavendish
would have used, hence the Cavendish method. The
angle of deflection of the test masses/torsion disk due
to the gravitational torque between the masses is mea-
sured by an autocollimator through a series of mirrors.
This deflection angle is reached when the gravitational
torque matches the restoring torque from the torsion
strip. 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 on
the 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 are
rotated to for measurement. The deflection angle is
measured for about 30 minutes at both positive and
negative source mass angle and the difference between
the two is taken, accounting for the initial offset. G
can then be calculated by using the torsion coefficient
of the strip.
The second mode is electrostatic servo-control. The
process is to that of the Cavendish method: an initial
offset is measured and the source masses are placed
at ±18.9◦. However, instead of allowing the disk to
rotate in response to the gravitational torque, a pair
of electrodes placed in the disk are activated to apply
a precisely controlled counter torque, preventing the
disk from rotating. By measuring the voltage across
the electrodes and the change in capacitance between
them with respect to angle, dC/dθ, G can be calcu-
lated.
The third possible mode the G experiment can operate
is called the time-of-swing method. The torsion disk
can be set to oscillate about its equilibrium position
and have its time period measured. When the source
masses are at 0◦ the gravitational torque on the disk
pulls it toward equilibrium, reducing its time period.
When they are rotated to 45◦ the torque will always be
pulling the disk away from its equilibrium, increasing
its time period. We were not able to use this mode
in our investigation because of the strict temperature
stability requirement to achieve desirable uncertainties
which could not be met in the current location of the
G experiment.[12][14][13]
For this study we ran the G experiment, using the
Cavendish and servo-control modes, to discover how
the measurement of G would change in the presence of
an alternating B-field. This ’primary field’ is discussed
in Section 2.1 and the results we obtained from the G
experiment shown in Section 3.1
1.3 Alternating Magnetic Fields
In most laboratory environments there is alternating
mains current flowing in wires and appliances nearby.
From simple electromagnetic theory we know that a
straight wire carrying AC will produce an alternating
B-field radially out from the wire. In the presence of
an alternating B-field, eddy currents will form inside
a conductor which flow to oppose the change in the
field. These eddy currents reduce the magnitude of
the B-field inside the conductor. The skin effect tells
us how the currents inside a conductor are distributed.
At higher frequencies of field, the eddy currents are
larger due to the greater rate of change of field. As the
frequency gets large, the conductor effectively expels
all of the field flowing through it, similar to the effect
of a superconductor. The skin depth, δ of a material
can be given by
δ =
2 ρ
ω μ0 μr
, (1)
where ρ and μr are the material’s resistivity and
relative permeability, ω is the angular frequency of the
field, μ0 is the permeability of free space. The linear
attenuation, a, of a field through a conductor of thick-
ness d can then be given by
a = e– d/δ
. (2)
2
5. The reduction in primary B-field is equivalent to
conductor producing its own opposing, phase-shifted
alternating field. This effect could occur in the masses
of the G experiment in the presence of nearby alternat-
ing B-fields. The fields produced by the masses would
resemble that of magnetic dipoles. The proximity of
the masses means these dipoles will likely introduce
extra forces on the test masses. The aim of this in-
vestigation has been to predict the magnetic effect on
the G experiment and to determine whether the mag-
nitude of this effect was sufficient to impact the value
of 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 the
measurement of G using the G experiment. We ex-
pected there to be additional forces and torques that
would arise due the magnetic interaction of the B-fields
induced in the masses by an external, alternating B-
field. We have aimed to build an analytical model that
can be used to predict the magnitude of the magnetic
torques 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-field
which we placed around the G experiment when it was
running. We have derived and tested what we expect
this primary field to be. We have also derived, using
two approaches, the secondary B-fields we expected to
be induced in the masses. We then evaluated the mag-
netic torque on the test masses. We have tested the
model on the G experiment using different magnitudes
and frequencies of currents, comparing our predictions
with the experimental data. We aimed to correctly pre-
dict the sign and order of magnitude of the magnetic
effect that the G experiment would measure.
The findings in this report do not just apply to an effect
particular to the G experiment. There may be impli-
cations for similar experiments. If they use conducting
masses, pendulums, or other source and measured ob-
jects to which this would be applicable, they would
likely be susceptible to an analogous effect. This could
be relevant to the work done by Harold et al. 2010
(See JILA10 in Figure 1.1) which measured the change
in position of two freely-suspended ”test masses” using
interferometry due to the gravitational interaction of 4
nearby ”source masses”. This will be explored further
in Section 5.
2 The Model
We have constructed an analytical model in MAT-
LAB R to attempt to predict the magnetic torques
experienced by the test masses in the G experiment.
We have used various approximations of the physics in
the model which have been used to make the problem
easier to tackle. These will be discussed in more detail
and 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 around
the 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 the
G experiment, looking down the z-axis. ’S’ and ’T’ refer to
the source and test masses, respectively. The coil’s z-wires
are perpendicular to the page and can be seen to intercept
the page in red. The wires along the y-axis lie in and out
the plane of the page. Not to scale.
The coil has 9 turns of wire encased in plastic tubing
measuring 0.674(1) m wide and 1.170(1) m tall. We re-
quired the calculation of the B-field vector at any point
in a 3-D Cartesian coordinate system. The primary
field, B0 can be determined analytically by integrat-
ing the Biot-Savart law for 4 finite wires, representing
each side of the coil
dB0 =
μ0 I
4π
dl × r
|r|3
, (3)
where I is the current vector, dl is the vector of an
infinitesimal section of wire, and r is the vector of the
wire section to the evaluated point in space. The re-
sultant equations from the integration of the Equation
3 can be found in Appendix A. We also required the
gradients of the field for the calculation of the mag-
netic force in Section 2.3, so each of the primary field
equations was differentiated with respect to each com-
ponent of position.
The field and gradients from each wire were summed
to represent the coil as a whole. Code was written that
would calculate the magnitude of the primary field and
gradients at each point in an assigned volume of space
around the coil. The results of this could be visualised
by using surface plots as shown in Figures 2.2 & 2.3.
3
6. 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
7. To test our prediction of the primary field we used
a magnetometer to measure the field at different posi-
tions in and around the coil. We positioned the meter
at each point in the volume around the coil using a
series of metre rulers. The complete list of test points
we used is shown in Figure 2.4. We limited the num-
ber of points to 27 in order to conserve time required
to complete this process. The field at each point was
measured in x, y and z directions.
Figure 2.4: Diagram of the coil in red, with test positions
in black. The coordinate system is shown in the top-right
corner. m takes the values of -50, 0 and 50. Positions are
given in centimetres relative to the centre of the coil. Not
to scale.
The field at each position was determined by mea-
suring the background field followed by taking half the
difference of the field created using a positive and neg-
ative DC of 1A. The Earth’s static geomagnetic field
was measured to be of the order 10–5T. The result of
this 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 an
uncertainty of approximately 10% (this is because of
the poor choice of resistor used to measure the circuit
voltage, used to calculate the current). The points at
which we expect the field contributions from each wire
to cancel give data that notably deviates from zero. We
recognise that the wires in the coil will not be perfectly
straight 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 to
this deviation. Further measurements could have been
made to map out the primary field in more detail but
this would have taken even more time, and we needed
to focus on the later parts of the model.
The linearity of the magnetometer was tested and the
deviations 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 field
predicted by our equations as a considerable increase
in the number of points would have been required and
would have taken far too long to achieve. We could,
however, visually check gradients by comparing their
surface plots with the fields. It appears that gradient
in Figure 2.3 would match that of the field in Figure
2.2, for example. Also, the divergence of the B-field in
free space should be zero everywhere. The gradients
we have derived follow this rule.
2.2 Secondary Fields
The next step in building our analytical model involved
evaluating the repulsion of the primary field by the cop-
per masses in the G experiment. The resultant field, B
including reduction in the magnitude of uniform alter-
nating B-field, B0 incident on a spherical conductor at
distance r from its centre, outside the sphere’s volume,
is given by Smythe to be
B = 1 +
ˇD
r3
cosθˆr – 1 –
ˇD
2r3
sinθ ˆθ |B0| ˆB0 ,
(4)
where θ is the angle to the direction of the incident
field from the sphere’s centre and ˇD is given by
ˇD =
( 2 μr + 1 ) ν – [ (1 + ν2) + 2 μr ] tanhν
( μr – 1 ) ν + [ (1 + ν2) – μr ] tanhν
a3
c , (5)
where ac is the radius of the sphere and ν is a com-
plex parameter equivalent to
ν =
( 1 + i )
ρ
ac . (6)
Other symbols have their usual meaning.[16] Equa-
tion 4 is given in CGS units. It can be shown that
removing the 1s in each of the brackets of Equation 4
gives an expression that describes the contribution to
the B-field induced in the conductor coil alone. The
conductors field, BS is therefore given by
BS =
ˇD
r3
cosθˆr +
ˇD
2r3
sinθ ˆθ |B0| ˆB0 . (7)
This equation is presented in spherical polar coordi-
nates, so we therefore converted the coordinate system
of Equation 7 to Cartesian coordinates. The gradients
of 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-axis
of the sphere to be aligned with the direction primary
field applied, ˆB0. This does not, however, make the
secondary field equations compatible with the coils co-
ordinate system as the direction of the primary field
5
8. 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 individual
positions used to measure the field.
6
9. 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
10. vector will be different for masses in different positions.
A coordinate system rotation for these equations was
therefore incorporated into the model. Details of this
process and the code used can be found in Appendix
C.
The relative phase of the secondary field to the pri-
mary is incorporated into ˇD. At 60Hz the secondary
field of the source mass lags behind the primary field
by about 155◦. The value and phase of ˇD 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
μ0
ˇD B0 . (8)
By using this method to analytically evaluate the
secondary B-fields we expect to be induced in the
masses, we must make two important assumptions.
Firstly, the direction primary field incident on the
masses in the G experiment is uniform over the vol-
ume of the mass, in the direction we predicted field to
be in at the masses’ centres. The magnitude and direc-
tion of the primary field will not change greatly over
the volume of the source or test masses, providing they
are not positioned near the sides of the coil. The two
source masses which were in closest proximity to the
coil were still about 0.1m from the sides of the coil and
the primary field vector averaged over the volume of the
masses will be approximately equal to the vector that
would pass through their centres. We therefore believe
this approximation is reasonable. Secondly, these sec-
ondary equations are valid for a spherical conductor,
the masses in the G experiment are cylindrical. The
shape of the masses, however, is not too dissimilar from
that of a sphere as their heights and width are equal.
This is the closest they can be to being spherical. We
have included a correction to the value of radius, ac
used for the masses. A multiplicative factor of 3
√
1.5
will give the radius of a sphere with the same volume
as the original cylinder. It is important to match the
volume 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 a
solid phosphor-bronze sphere as the conductor. The
radius of the sphere was measured to be 66.1(2)mm.
The coil was placed on its side, so that its plane was
parallel with the floor, and placed the sphere at its
centre. With the magnetometer positioned just above
the sphere (pointing in the x-direction), measurements
were made of the total field. The magnetometer did
not 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 was
calculated by subtracting the field measured from our
model prediction of the primary field at the chosen
point. The frequency-dependence of the sphere was
tested by changing the frequency of the current and
the radial-dependence was tested by moving the mag-
netometer further from the sphere. The results of these
measurements 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 the
sphere. The current of the field was, again, set to
1 A. The AC B-field measured in the lab reached about
10nN at its maximum and, to avoid mixing of the envi-
ronmental and the primary field, the frequency of 50Hz
was avoided.
The calculated upper and lower bounds shown in
Figure 2.9 reflect the uncertainty in the position of the
magnetometer. The closer to the sphere you measure,
the greater the magnitude of its field. The first point
to note is that the secondary field appears to tend to
the magnitude of the primary field at high frequency,
corresponding to the almost complete expulsion of the
field. The second fact is that it is clear that the slope
of the data points at low frequency do not match up to
the predicted curves. This was a phenomenon we were
unable to resolve. We have avoided placing any other
conductors nearby which could interfere with the mea-
surement. We postulate that some properties of the
sphere, such as ρ or μr, could also be dependent on fre-
quency. There may be additional effects on the sphere
that we do not know about. It was important we ob-
tained the levelling off at high frequency and we made
note of the factor of 3 difference between measured
and 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 Figure
2.8, the reduction of the sphere’s field with distance
has been compared with inverse-square and inverse-
cube functions that have been fit to the start and end
points. This has been implemented in order to examine
the radial-dependence of the field.
Figure 2.8: Plot of secondary field magnitude, measured
at increasing distance from the phosphor bronze sphere.
Inverse-square and inverse-cube relationships have been fit
for comparison.
It was apparent that the field very close to the
sphere’s surface may have a contribution from higher
order terms, which would mean the field would drop
off less with distance. However, as the magnetometer
measured the field further from the pole of the sphere,
the radial-dependence seemed to settle on an inverse-
cube relationship, as we expected.
8
11. Figure 2.9: Plot of secondary field magnitude, measured at 70(5)mm above the phosphor bronze sphere. Sphere is at
the 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 source
mass is positioned 110(5)mm from coil centre. Upper and lower bounds refer to the model prediction of the positional
uncertainty.
9
12. 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 better
analyse this dependence.
When the frequency of the input voltage was increased
the measured current would drop, requiring a greater
input voltage. At 5000Hz the input voltage required to
measure the same voltage in the circuit was between 3
and 4 times larger than that used for 60Hz, to achieve
1 A. A variety of factors could be contributing to this
effect. The power amplifier used to increase the cur-
rent was likely less efficient at high frequencies. The
same problem may have also affected the reading of the
voltage by the multimeter used; these frequencies may
be outside its sensitive limit. Also, at 5000Hz the skin
depth of copper, used in the coil’s wires, is about 1mm
so is of the same size as the wire’s diameter. The skin
effect would produce a counter voltage, lowering the
effective voltage, and hence current, in the coil. Our
results do follow the general trend expected, levelling
off at high frequencies, so we believe by increasing the
input voltage we have counteracted many of these is-
sues.
In addition to the phosphor bronze sphere we were able
to repeat these tests with one of the source masses from
the G experiment. The mass had to be positioned
about 110(5)mm above the centre of the coil, so we
would be expecting a reduced B-field magnitude.
Similarly to the sphere, the field induced on the
source mass approached saturation at high frequency.
The data follows a similar trend that can be seen in
Figure 2.9. This was an indication that the approxi-
mation, whereby the masses are treated as spheres, was
valid. The difference between the data and the predic-
tion at 60Hz is approximately a factor 4, which was
also noted. More points should have been measured at
higher frequencies to more clearly see the tail off of the
curve.
Figure 2.12: Plot of secondary field magnitude, measured at
increasing 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 mostly
comparable to that of the phosphor bronze sphere. The
source mass data gives the impression that higher or-
der terms may continue to be significant to greater radii
before settling on the inverse-cube relationship. This
will likely be a result of the cylindrical shape of the
source mass.
We were able to include secondary fields due to the
source masses as well as the linear attenuation, a, of
the vacuum can to the primary field. An example of
the total resultant field can be seen in Figure 2.11.
Another assumption that has been made is that the
source masses are far enough apart from each other for
the secondary fields to not affect one another. That is
to say, the source masses are made to only experience
the primary field and not the secondary fields from any
of the other source (or test) masses.
Our supervisor, Clive Speake, had also been working
on analytically determining the primary and secondary
fields, parallel to us. The approach Clive had used
10
13. was simpler as no coordinate transformation or rota-
tion was required. This approach from Jackson gives
the field, Bd at radius, r from a magnetic dipole as
Bd =
μ0
4π
3ˆr (m · ˆr) – m
|r|3
, (9)
where ˆr is the unit vector of r and m is the same
magnetic moment which is derived from the previous
approach.[17] The differentiation of Equation 9 is non-
trivial and required the use of Maple TM(which was
also used to check the previous differentiations). This
approach to the secondary field was not utilised until
late into the project for purpose of comparison with
the method outlined earlier in this section. These are
discussed in the upcoming sections.
2.3 Calculation of the Torque
We have been able to calculate the expected magnetic
torques on the test masses about the torsion strip by
evaluating the forces on them. The forces can be de-
rived from the magnetic potential in which the test
masses sit. The potential energy, E of a magnetic
dipole, m in a B-field, B is
E = – m · B . (10)
In the case of the scenario we have modelled, the
field incident on the test masses will be the addition of
the primary field generated by the coil and the sec-
ondary fields induced on the source masses, phase-
shifted relative to the primary. Therefore, B becomes
B0 + BS. The duration of a measurement is substan-
tially longer than the time period of the alternating
B-fields we are using, so we are required to determine
the time averaged energy, Et of a test mass. There-
fore, a factor of 2 being removed and the conjugate of
the field is taken. The result is given as
Et = –
π
μ0
| ˇDt| (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 be
solved by using the appropriate vector identity. The
time-averaged force on a test mass can therefore be
found to be
Ft =
π
μ0
| ˇDt| BT·( BT)∗
+ B∗
T· BT , (13)
where BT ≡ B0 + BS. In this instance the
gradient and the conjugate operators commute so
( BT)∗ ≡ (B∗
T). When including the attenuation
factor, 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 about
the 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 of
torque that will rotate the test masses and the torsion
disk is in the z-direction. The other components of the
torque will not have an effect on the measurement of
G.
The Equations 13 & 14 were included into the model
to so that the torque on all of the test masses due to
the sum of the primary and secondary fields could be
calculated. The code was rewritten so that instead of
evaluating the fields at all points in a given volume of
space, the fields and gradients were evaluated exactly
at the masses. This was done so that the source masses
could be rotated around the torsion disk, akin to the
G experiment, in order to discover how the torque on
the test masses changed with the angle of the source
masses. An overview of the processes in this code is
shown in Appendix D.
The force and torques due to gravity were also included
into the model so that a comparison between magnetic
and gravitational torques could be made. The 2014
CODATA value of G (given in Section 1.1) was used
in these calculations. The magnetic torques in the
z-direction for both secondary field approaches were
compared with the predicted gravitational torques as
a function of source mass angle. The results of these
simulations are shown in the Figures 2.13 & 2.14.
The gravitational torque the model produces has been
checked and we are confident that it closely resembles
what would be measured by the G experiment. The
torque is rotationally symmetrical about 0◦; its magni-
tude is greatest at approximately ±18.4◦, which is very
close to the 18.9◦; and the greatest difference in torque
in G experiment is measured to be about 30nNm, com-
pared with our predicted 36nNm. The magnetic torque
calculated from the original approach in Figure 2.13
predicted an opposite sign to gravity which therefore
made the effect repulsive. This was contrary to what
we had expected the nature of the magnetic interac-
tion to be at the beginning of the investigation. A
200mA current produces a B-field of the order of 1μT
and, at this magnitude of field, Figure 2.13 predicts a
magnetic-to-gravitational torque ratio of about 1%.
11
14. 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
15. 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 alternative
approach. 200mA current.
13
16. Figure 2.17: Difference in torque on test masses when source masses are at ±19◦
as a function of frequency, using original
approach. 200mA current.
Figure 2.18: Difference in torque on test masses when source masses are at ±19◦
as a function of frequency, using
alternative approach. 200mA current.
14
17. Using the second approach yields significantly dif-
ferent results. The order of magnitude of the magnetic
torques is the same but goes the magnetic torque adds
to gravity and the curve appears to be much more off-
set.
The primary field will produce a constant torque on the
test masses as they remain effectively stationary. This
torque will cause a slight offset in the in the equilibrium
angle of the torsion disk which will be accounted for at
the beginning of each G experiment measurement, and
will 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, will
alter 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 be
present in G experiment during its normal operation,
the model was updated again to evaluate the torques
when the source masses were at ±19◦. The difference
in torques between the two orientations could then be
calculated, both of which were done as function of fre-
quency. This way, the effect of different frequencies
and the magnitude of the magnetic torque difference
could be determined.
The original secondary field approach predicts a nega-
tive torque difference - a reduction in the total torque
on the G experiment - which reaches greatest mag-
nitude at about 43Hz. The torque difference decays
toward zero at high frequency as the vacuum can ef-
fectively attenuates the field on the test masses inside
it to zero.
The torque difference calculated using the alternative
approach (shown in Figures 2.16 & 2.18), again, pre-
dicts an attractive magnetic effect instead where the
peak difference in magnitude is at about 40Hz.
Both approaches predict a slight difference in the shape
of the torque with respect to frequency between +19◦
and –19◦. We had expected that both torques would
have the same shape in both configurations but this
may 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. Therefore
the fields and gradients experienced by the test masses
are reduced by a single multiplicative factor, which de-
pends on the skin depth of the can. The effect that has
been employed into the model has been an approxima-
tion of the physics.
In order to discover how the vacuum can would actually
affect the field, we performed numerical finite element
analysis on the B-field using FEMM (Finite Element
Method Magnetics). The coil’s wires were modelled
to generate the primary field and a comparison of the
field with and without the analogue of the aluminium
vacuum 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 greatly
affected too. The effect of the can has clearly been
under-estimated in the code. In the presence of the
can, the field incident at the two source masses nearest
to the wires is increased by about 35% and the other
two source masses experience about a 65% reduction.
This analysis was performed late in the project and
the results have not been included in the model. The
software has the potential to simulate the secondary
fields induced on conductors and to calculate magnetic
potential energy. If more time had been available, fur-
ther finite element analysis could have been performed
to more accurately model the effect of the vacuum can.
15
18. 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
19. 3 Results
The G experiment was run with the coil around the
apparatus to obtain data points of G. [18] These tests
ran with current in the coil changing but the frequency
held constant, and vice versa, to examine how the value
of G would change in the presence of an alternating B-
field. The predictions from the analytical model that
was built were subsequently compared with the G data.
3.1 G Experiment
Firstly, the G experiment was run with a varying a
magnitude of current at 60Hz (including zero current
as a control measurement). 60Hz was used to avoid
mixing with any environmental fields and because this
is the frequency of the mains in the USA, where the G
experiment will be operating soon. For two runs, the
vacuum pump, used to maintain a vacuum inside the
can, was not magnetically shielded, and for the other
runs it was. The pump is continuously active during
measurement, requiring electrical current to function
and will inherently produce an extra source of B-field.
The result of these runs indicates that the effect of the
magnetic torques are in fact repulsive, reducing the
total torque in the system and the value of G mea-
sured. We find this trend in the data strongly follows
a negative-square relationship with the input current
and, hence, primary B-field strength. This correlates
with our model as the multiplication of fields in Equa-
tion 13 would convey a magnetic force proportional
to the square of the current/B-field. The percentage
change in G at 220mA - which produces a μT field -
is of the order of 0.1%. This is a 1000ppm and the
G experiment is sensitive to changes of about 10ppm,
so - with a μT field - the magnetic interaction would
significantly affect the value of G.
Secondly, data points were collected at a con-
stant 220mA at various frequencies to determine the
frequency-dependence of the effect. Few data points
are available, as each one took 2 days to be collected.
The magnitude of the change in G is greatest between
30Hz 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, as
expected.
3.2 Comparison of G Data with Model
Predictions
By finding the ratio between magnetic and gravita-
tional torque differences at ±19◦ we can compare the
G data with the prediction of the model. Both sec-
ondary field approaches were used, the results of which
is shown in Figure 3.2.
We find that our original approach reproduces the same
sign of the effect measured, whereas the alternative
does not. Both, however, do predict the same current-
squared relationship implied by the G results. The
magnitude of the alternative is also more than 3x that
of the original. We are unsure as to the nature of this
ramification. The original approach predicts a change
in 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 would
be 3x-4x lower than the theory anticipated (See Fig-
ure 2.10). If this factor is taken into account, the
model prediction is significantly more comparable with
G data. This also applies to the alternative approach.
If the sign of the effect has been miscalculated in the
code, the prediction will still over-estimate the mag-
netic effect. There could, however, be other factors
that 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 Figure
3.3 match those in Figure 3.2. The model matches the
relationships and predicts an effect that is of the same
order 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
20. Figure 3.2: Percentage change in G measured by the G experiment compared with the model predictions of both
secondary 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 both
secondary field approaches, as a function of frequency; at 220mA.
18
21. 4 Conclusions
We have found that the measurement of G by the G
experiment can indeed be affected by the presence of
alternating B-fields around the apparatus. This mag-
netic forces on the test masses in the experiment are
repulsive, so that it subtracts from the total force the
masses experience during operation. The fractional
change in G measured is proportional to the negative-
square of the B-field, in which the apparatus sits. The
range of field frequencies at which the magnetic effect
is greatest is 30-50Hz. This is close to the frequency
of mains current you would expect to be flowing in a
laboratory environment.
We have built an analytical model in an attempt to
evaluate the total B-field, and gradients thereof, which
will be experienced by the conductive test masses in
the G experiment. The total field is comprised of the
primary field, controlled by a coil of wires to be placed
around the G experiment, and the secondary fields pro-
duced by the conducting source masses. The secondary
fields arise from the masses’ resistance to the change in
B-field through them. This repulsion is equivalent to
the masses having an opposing, phase-shifted B-field
induced 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 expression
that required a coordinate system transformation and
rotation to be compatible with the other model com-
ponents. The second, which was incorporated late into
the project, did not necessitate these actions and could
be directly applied. It would therefore have a lower
chance of having incorrect expressions. The model in-
cluded physical approximations such as: The source
masses were assumed to be spherical, which data in
Section 2.2 showed to be valid; The field through the
masses is uniform, which we estimate to be acceptable;
the aluminium vacuum linearly attenuates the field, to
which 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 the
field due to a source mass by a similar factor. We
are still unsure why the alternative approach used pro-
duces a greater magnitude of this effect, as well as the
opposite 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 project
so little time could be allocated to scrutinise and cor-
rect for minor bugs in the code. It is also possible that
the code may be incorrect for both approaches. Fur-
ther work is required to obtain the same result from
them both.
Finally, the aim of this work has been to acquire an
understanding of the magnetic effects that are present
in gravitational experiments, and we sought to analyt-
ically determine an order of magnitude estimation of
this effect. We have been successful in this endeav-
our. The original approach used to model the effect
correctly predicts the change in G, measured by the
G experiment, due to a local B-field as a function of
magnitude and frequency. The model we have cre-
ated has various physical approximations and exclu-
sions. While, for this project, these have been sufficient
for us to achieve the correct prediction of the trends
and a correct order of magnitude, they can also be
revised and given a more rigorous treatment to more
accurately predict the magnetic effects on G experi-
ments. We had hoped the extra magnetic torques in
the measurement of G would reconcile the different val-
ues published by the various studies. In the case of the
G experiment, the background AC field measured in
the lab was about 10nT. Following the trends in the
data and the model, we predict the change in G to
be of the approximate order of 0.1ppm, which is small
compared to the apparatus’ sensitivity of about 10ppm.
We would not then expect the values measured by the
G experiment to be affected during normal operation.
Other experiments’ values of G may be suppressed by
B-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 the
potential to change the value of G measured, depend-
ing on the magnitude of the local AC B-field. The 780
g test masses and 120 kg source masses that were used
were made of copper and tungsten, respectively. The
resistivity and relative permeability of tungsten are not
dissimilar to copper, so we might expect a similar re-
action to the value of G measured. If the local field is
greater than 100nT the change in G due to magnetic
torques will likely exceed the sensitivity of the mea-
surement, subtracting from its true value.
A solution to counter the effect of B-fields could be
to surround the apparatus with shielding. Therefore,
unless the origin of the field is due an internal device,
the consequential magnetic effects would be effectively
eliminated.
The model in this study has been employed to measure
the magnetic torque present in G experiment when it
is operating in the Cavendish and servo-control modes.
We have not yet considered the change in G using the
time-of-swing mode. In this mode the test masses also
move in the field. Therefore the test masses would ex-
perience a change in both the primary and secondary
fields. The primary field and gradients are approxi-
19
22. mately two orders of magnitude greater than that of
the secondary. This then could result in a magnetic
effect about 10x that of the other modes and a differ-
ence in G of about 1 ppm. Additional modelling of the
this mode of operation would be required to decisively
evaluate the actual effect expected.
6 Acknowledgements
I would firstly like to thank my project partner Peter
Steele for his hard work and dedication throughout this
joint investigation and for his perseverance through the
ups and downs along the way. Secondly I thank my
supervisor, Clive Speake, without whom Peter and I
would not be where we are now. He has worked along-
side us from the beginning and has put aside much of
his time to guide us in the past 6 months. I only hope
we have made a considerable contribution to his work
on the gravitational constant and wish him all the best
in his future endeavours. Finally I wish to mention the
great work John Bryant has put in to make sure Peter
and I have promptly had all the tools and equipment
we could ask for in the lab for our testing. He, along
with Clive, has organised the big G experiment for this
project and they take the credit for the data which we
have been able to use to make our comparisons and
conclusions.
20
23. References
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Physical Constants: 2014”. In: ArXiv e-prints (2015).
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Phys. Rev. Lett. 48 (1982). doi: 10.1103/PhysRevLett.48.121. url:
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url: http://link.aps.org/doi/10.1103/PhysRevLett.91.201101.
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http://dx.doi.org/10.1007/BF02377461.
[5] C.H. Bagley and G.G. Luther. “Preliminary Results of a Determination of the Newtonian Constant of
Gravitation: A Test of the Kuroda Hypothesis”. In: Phys. Rev. Lett. 78 (1997). doi:
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[8] ZK. Hu, JQ. Guo, and J. Luo. “Correction of source mass effects in the HUST-99 measurement of G”.
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http://link.aps.org/doi/10.1103/PhysRevD.71.127505.
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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 physical
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[19] Steele. P.J. 2016.
21
24. 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 + y2
s + z2
–
ye
x2 + y2
e + z2
(–zˆı + x ˆk) , (15)
Bz =
μ0 I
4π
1
(x2 + y2)
zs
x2 + y2 + z2
s
–
ze
x2 + y2 + z2
e
(yˆı – xˆ) (16)
where x, y and z are the distances from a given point in space to the position of the wire, given in Cartesian
coordinates; ys and ye are the positions of the start and end of the y-axis wires that with reference to the
direction 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
ˆr
ˆθ
ˆφ
=
sinθ cosφ sinθ sinφ cosθ
cosθ cosφ cosθ sinφ – sinθ
– sinφ cosφ 0
ˆı
ˆ
ˆk
, (17)
and from this we find the respective conversions for the radial and polar angle unit vectors (ˆr & ˆθ,
respectively) to be
ˆr = sinθ cosφˆı + sinθ sinφˆ + cosθ ˆk , (18)
ˆθ = cosθ cosφˆı + cosθ sinφˆ – sinθ ˆk . (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
ˇD
(x2 + y2 + z2)
5
2
3
2
xzˆı +
3
2
yzˆ + z2
–
x2 + y2
2
ˆk , (24)
where x, y, and z are the distances to a point outside the sphere to its centre, in Cartesian components, and
the other symbols have their usual meanings. The nine components of the gradient of the secondary field are
∂BxS
∂x
=
3
2
B0
ˇD
z ( –4x2 + y2 + z2 )
(x2 + y2 + z2)
7
2
, (25)
∂BxS
∂y
=
3
2
B0
ˇD
– 5 x y z
(x2 + y2 + z2)
7
2
, (26)
22
25. ∂BxS
∂z
=
3
2
B0
ˇD
x ( x2 + y2 – 4z2 )
(x2 + y2 + z2)
7
2
, (27)
∂ByS
∂x
=
3
2
B0
ˇD
– 5 x y z
(x2 + y2 + z2)
7
2
, (28)
∂ByS
∂y
=
3
2
B0
ˇD
z ( x2 – 4y2 + z2 )
(x2 + y2 + z2)
7
2
, (29)
∂ByS
∂z
=
3
2
B0
ˇD
y ( x2 + y2 – 4z2 )
(x2 + y2 + z2)
7
2
, (30)
∂BzS
∂x
=
3
2
B0
ˇD
x ( x2 + y2 – 4z2 )
(x2 + y2 + z2)
7
2
, (31)
∂BzS
∂y
=
3
2
B0
ˇD
y ( x2 + y2 – 4z2 )
(x2 + y2 + z2)
7
2
, (32)
∂BzS
∂z
=
3
2
B0
ˇD
z ( 3x2 + 3y2 – 2z2 )
(x2 + y2 + z2)
7
2
. (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 coordinate
system, 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 correct
magnitude 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. Next
the parameters required to translate components of the spheres coordinate system into the coil’s are
calculated. BvecS1z is the a vector whose first value is the factor that converts the magnitude of
x-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’s
ˆk 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 by
taking 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 is
added to the primary field.
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26. Appendix D Overview of Torque Code with Rotating Source
Masses
Figure D.1: Flow chart detailing the calculations in the model code used to determine the torques on the test masses.
(Steele 2016)[19]
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