final year dissertation seminar slides
TRANSCRIPT
The Effects of AC Fields on Gravitational Experiments
Mark Gilbert
Partner: P.Steele
Supervisor: C.Speake
Outline
Introduction
• Gravitational constant
• BIPM experiment
• Background theory
• Motivation & Aim
Main
• Building of the model
• G experiment results
• Current progress
• Next Steps
Summary
How important is the Gravitational constant, G?
• FUNDAMENTAL constant
• Governs the force of gravity
Quantity Symbol Value
Gravitational constant G 6.674 08(31) x10-11 m3 kg-1 s-1
Planck constant h 6.626 070 040(81) x10-34 J s
Elementary charge e 1.602 176 6208(98) x10-19 C
Least well-defined constant – why?
• Gravity is significantly weak
• Difficult to measure accurately on laboratory scales
• Cannot shield against it
What work is going on at Birmingham?
• BIPM and University of Birmingham collaboration to determine G
• Two values published in 2001 and 2013
• Both 2001 and 2013 are in agreement
• Why might the measured G value be greater than the others?
• Perhaps extra attractive force being measured?
Source: Scientific American, http://www.scientificamerican.com/article/puzzling-measurement-of-big-g-gravitational-constant-ignites-debate-slide-show/
DOI = 10.1103/PhysRevLett.113.039901
How does the BIPM apparatus work?
• 4-Source-4-Test mass torsion balance setup
• Measures torque from gravitational force of Source masses on Test masses
• Multiple independent built-in modes of operation:- Cavendish method - Electrostatic servo control
T
T
TT
S
S
S
S
• Take measurements where torque is greatest at about ± 18.9°
• Time-averaged deflection angle of Test masses measured by an autocollimator
• Torsion strip of known torsion coefficient, k
• Aluminium vacuum can
y
xz
k
Source: Neue Zürcher Zeitung, http://www.nzz.ch/wissenschaft/physik/gemeinsam-wollen-forscher-big-g-knacken-1.18380206
Alternating Fields
• Mains AC wires nearby in laboratory environment
• Alternating current produces radial alternating magnetic field
• AC B-field incident on conductor produces eddy currents, opposing the change
• Eddy currents in conductor produce own repulsive B-field (phase-shifted)
• Conductor now has its own alternating field
• Also skin effect proportional to field frequency
• Source and Test masses will produce their own fields and interact with one another
• Extra magnetic forces & torques!
𝛿 =2𝜌
𝜔𝜇0𝜇𝑟(1) 𝑎 = 𝑒− 𝑑
𝛿 (2)
T
T
TT
S
S
S
S
y
xz
Modelling the Magnetic Torque
• Analytical model
• Primary Field from coil
• Secondary field produced by Source masses
• Attenuation of field due to vacuum can
• Energy at & Force on Test masses
• Torque on Test masses
• Compare with Gravitational torque
z
x
y
I
I
I
The Primary Field
• Rectangular coil of wires to be placed round the BIPM experiment
• Can control input current and frequency
• Calculate B-field due to 4 finite current-carrying wires
Biot-Savart law:
• Required to calculate field gradients – 9 components
• Compare with measured field at some test points using gaussmeter
𝑑𝐵 =𝑢0
𝐼
4𝜋
𝑑 𝑙 × 𝑟
|𝑟|3(3)
The Secondary Field
Magnetic field induced outside a spherical conductor by a polar uniform external field[1]:
where
And
• Convert coordinate system and rotate for application
• Require analytical solutions to field and gradients
𝐵 = − 𝐷
𝑟3 𝑐𝑜𝑠𝜃 𝑟 + 𝐷
2𝑟3 𝑠𝑖𝑛𝜃 𝜃 |𝐵0| 𝐵0 (4)
𝐷 =2𝜇𝑟 + 1 ν − 1 + ν2 + 2𝜇𝑟 tanh(ν)
𝜇𝑟 − 1 ν + 1 + ν2 − 𝜇𝑟 tanh(ν)𝑎𝑐
3 (5)
[1] Smythe, W.R.; Static and Dynamic Electricity; McGraw-Hill Book Company Inc.; 2nd Ed.; 1950; p.398
ν =(1 + 𝑖)
𝛿𝑎𝑐 (6)
zθ
B0
𝑚 =2𝜋
𝜇0
𝐷 𝐵 (7)
Potential Energy and Force
The potential energy for a magnetic dipole moment in an external B-field can be given by:
Then the force on this dipole can be found as:
𝐸 = − 𝑚 . 𝐵 (7)
𝜏 = 𝑟 × 𝐹 (12)
< 𝐸𝑡 >= −2𝜋
𝜇0
|𝐷𝑡|1
2𝑎𝐵𝑠 . 𝑎𝐵𝑠
∗(9)
𝐹 = −𝛻𝐸 (10)
𝐵 𝐵𝑠
< 𝐹𝑡 >=𝜋
𝜇0
|𝐷𝑡| |𝑎|2 𝐵𝑠 . 𝛻𝐵𝑠
∗+ 𝐵𝑠
∗. 𝛻𝐵𝑠 (11)
𝑚 =2𝜋
𝜇0
𝐷 𝐵
Alternative Dipole Approach
Another method used to calculate the induced field, which is equivalent of that due to a magnetic dipole[2]:
𝐵(𝑟) =𝜇0
4𝜋
3 𝑛 𝑝.𝑛 −𝑝
|𝑟|(8)
[2] Jackson, J.D.; Classical Electrodynamics; John Wiley & Sons Inc.; 3rd Ed.; 1999; p.186
Next Steps
• Analyse the effects of the vacuum can using FEMM- Calculate Energy/Forces- Compare FEMM and analytical field values
• Continue to reconcile with possible miscalculations in the original approach of determining the magnetic torque
• Evaluate effect of the dipole fields being attenuated by the nearby test masses- Simplify the fields & forces on test masses to only that from the nearest source massneighbour
Summary
• Experiment to measure G at the university predicts higher value of G than other studies
• Trying to quantify a possible magnetic effect on G measurement due to magnetic fields using analytical modelling with approximations
• G measurement is significantly affected when in an alternating magnetic field
• Field produces overall repulsive force on test masses, reducing measured torque
• Current model predicts a lower magnitude of G disparity
• Further study to be done into incorporating finite element analysis and evaluating existing approximations in order to bring model closer to measured trend
Implications: • Greater attention to presence of local alternating magnetic fields• Effective shielding to reduce unwanted magnetic affects