New Experimental Test of Coulomb

1
New Experimental Test of Coulombs Law A
Laboratory Upper Limit on the Photon Rest Mass
A lecture on the Article

• E.R. Williams, J. E. Faller and H.A. Hill (1971)

Porat Amit Oren Zarchin
2
Abstract

• A high-frequency test of Coulombs law is
  described.
• The sensitivity of the experiment is given in
  terms of a finite photon rest mass, using the
  Proca equations.
• The null result of our measurement expressed in
  the form of the photon rest mass squared is

3
Abstract

• Expressed as a deviation from Coulombs law of
  the form , our experiment gives . This
  result extends the validity of Coulombs law by
  two orders of magnitude.

4
Coulomb, Charles (1736-1806)

• French physicist who performed experiments with a
  torsion balance.
• His investigations led him to suggest that there
  were two "fluids" of electricity and magnetism.
• He showed both forces were inverse square, and
  stated that they were unconnected separate
  phenomena.
• The inverse square of electricity has come to be
  known as Coulomb's law.

5
Historical review

• Using a Torsion balance Coulomb demonstrated
  directly that two like charges repel each other
  with a force varies inversely as the square
  distance between them.

6
Robinson, John (ca. 1725-?)

• English doctor who, in 1769, measured electrical
  repulsion went as r-2.06 and attraction as r-c
  where c lt 2. From these results he surmised r-2
  was correct. This determination was made before
  Coulomb proposed just this result, now know as
  Coulomb's law.

7
A deviation from Coulombs law?

• A photon with a finite rest mass will cause a
  deviation, according to Proca equations.
• A deviation from the Euclidian space can cause a
  deviation from the r square law.
• This effect will be neglected when calculating a
  deviation due to the existence of a photon rest
  mass which varies from zero.

8
Historical review

• Cavendish (1773) noted that if the force between
  charges obeys the inverse square law there should
  be no electric forces (I.e. electric fields)
  inside a hollow charge free cavity inside a
  conductor.
• Maxwell has found that the exponent of r in
  Coulombs law could differ from two by less than
  1/21600.

9
Historical review

• Plimpton and Lawton (1936) charged an outer
  sphere with a lowly varying alternating current
  and detected the potential difference between the
  inner and outer spheres. They reduced Maxwells
  limit to 2x10-9.
• Bartlett Goldhagen Phillips (1970) achieved an
  upper limit of 1.3x10-13 .

10
Bartlett Goldhagen Phillips (1970)

• Using five concentric spheres, and applying a
  potential difference of 40 KV at 2500Hz between
  the outer spheres.
• The potential difference between the inner two
  spheres was read using a Lock-in detector.

11
Theory- Proca equations

• In conventional electrodynamics the mass of the
  photon is assumed to vanish. However, a finite
  photon mass may be accommodated in a unique way
  by changing the inhomogeneous Maxwell equations
  to the Proca equations.
• Let us explain the basic concepts which lead to
  these equations

12
Some topics in Quantum Electrodynamics

• The description of the interaction between the
  electromagnetic field and the electron-positron
  field constitutes the main problem of QED.
• We will look on a combination of Maxwell
  equations with the Dirac form of the current
  (comes from the solution of Dirac equation).
• The high-energy experiments test QED in a
  situation where the four-momentum transfer
  characteristic of the experiment, is as large as
  possible. The verdict, as far as the high-energy
  tests are concerned, is that the Maxwell
  equations with the Dirac form of the current for
  the electron and Muon are correct.

13
The electromagnetic field

• We can describe the electromagnetic field by
  means of the equation of retarded potentials
  A?j? (?1,2,3,4)
• A? is the potential of the electric field.
• j? is the current describing the charged
  particles
• is the solution for Dirac equation for a
  particle interacting with an electromagnetic
  field.
• is related to the Dirac operators.

14
Adding the photons mass

• If the photon has a mass m0, an additional term
  is required ? A? ?2A? j?
• Where (should be h bar).
• The equation show explicitly that the additional
  current term is proportional to the four vector
  potential A?. Therefore they have a mutual
  influence.

15
Finally- Proca equation

• Proca equation for a particle of spin 1 and mass
  m0 (such a photon) is
• A? ?2A? (4?/c)j?.
• In a three dimensional notation, Gausss law
  becomes
• (1)

16
Developing the necessary equations

• In order to calculate the sensitivity of the
  system, consider an idealized geometry consisting
  of two concentric, conducting, spherical shells
  of radii R2 gt R1 with an inductor parallel with
  this spherical capacitor.
• To the outer shell is applied a potential V0eiwt.

17
Developing the necessary equations

• forming a spherical Gaussian surface at radius r
  between the two shells and then using the
  approximation for this interior region, the
  integral of Equation(1) over the volume interior
  to the Gaussian surface becomes
• (2)

18
Developing the necessary equations

• Therefore E(r) is given by
• (3)
• Where q is the total charge on the inner shell.
• A complete solution of the fields inside a
  symmetrically charged single sphere will give,
  after neglecting second order terms in the
  electrical filed , equation (3) and H0.

19
Approaching the final equation

• Since inside,
• The voltage appearing across the inductor is then
  simply given by
• (4)

20
Approaching the final equation

• The differential equation, which describes a
  regular LCR equation is
• In the case of a nonzero rest mass

21
Final equations describing the system
22
notes

• Analyzing the signal to noise ratio of the system
  (conventional circuit theory) results that the
  use of
• High frequency
• High Q circuits
• Large apparatus
• High V0
• Will serve to maximize the experimental
  sensitivity.

23
Experimental Setup

• Charging a conducting shell
• (1.5m in diameter-Large) with
• 10KVolts peak to peak with
• a 4Mhz Sinusoidal voltage.

Is it all?
24
Fiber optics

• We would like to transmit data, to and from the
  inner sphere.
• We cannot use Electrical wires since they will
  efffect the measurment.
• So we use Fiber Optics, through a hole in the
  sphere.
• In order to prevent penatration of Outer fields
  through the hole, we use the fiber as a
  Waveguide.
• The waveguide diameter must be smaller than the
  cutoff frequency.

25
Noise

• stray electric and magnetic fields

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Noise - Solution

• Adding 3 shells in order to prevent stray
  electric and magnetic fields inside the sphere.

There is another problem..
27
Another Noise

• Johnson effect
• gives noise of

28
Adding a Lockin Amplifier
Phase shift
Lockin Amplifier
29
Lock in amplifier
Phase shift
x
Low pass filter
filter
30
Lock in Amplifier
Push pull signals
signal
RC
vout
reference
relay

• when the reference signal is positive, the
  signal goes out with no changes.
• when the reference signal is negative, the signal
  goes out up side down.
• The RC integrates over the signal and cancels
  same areas with negative sine.

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Lock In Amplifier Demonstration

• when the reference signal is positive, the
  signal goes out with no changes.
• when the reference signal is negative, the signal
  goes out up side down.
• The RC integrates over the signal and cancels
  same areas with negative sine.

32
Lock In Amplifier Demonstration
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Lock In Amplifier Demonstration
34
A full view of the system

• We need to check the System!

35
Checking calibration

• During a data run, to ensure that our system
  works properly.
• A calibration Voltage is periodically introduced
  into the system on a third light beam while the
  reference beam is working.
• On striking a light sensitive diode induces a
  voltage on the capacitor.

36
Calibration results

• The calibration was done Over 3 cycles .

37
Notes

• As high as possible applied voltage , serves to
  maximize the experimental accuracy .
• In the experiment we use high frequencies in
  order to reduce the skin depth which varies as

38
Results

• The experimental result is statistically
  consistent with the assumption that the photon
  rest mass is identically zero.

39
How does the experiment fit in
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references

• E. R. Williams, J. E. Faller, H. A. Hill. Phys.
  Rev. Let. 26 721 (1971)
• Metrology and Fundamental Constants (oxford 1980)
• D. F. Bartlett, P. E. Goldhagen, E. H. Phillips.
  Phys. Rev. D2 483 (1970)
• Alfred S. Goldhaber, Michael Martin Nieto. Phys.
  Rev. Lett. 21 567 (1968)
• S.J. Plimpton, W. E. Lawton. Phys. Rev. 50 1066
  (1936)
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