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

Gravitational & Magnetic Torque Comparison

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

Current Results

Vacuum can - FEMM

S

S

S

S

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