Download New Experimental Test of Coulomb`s Law: A - Exvacuo

New Experimental Test of
Coulomb’s Law:
A Laboratory Upper Limit on the
Photon Rest Mass
E.R. Williams, J. E. Faller and H.A. Hill (1971)
Abstract
A high-frequency test of Coulomb’s 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
  (1.04  1.2) 10 cm
2
19
2
Abstract
Expressed as a deviation from Coulomb’s
law of the form 1 r 2 q , our experiment
gives q  (2.7  3.1) 1016 . This result
extends the validity of Coulomb’s law by
two orders of magnitude.
Coulomb, Charles (17361806)
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.
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.
Robinson, John (ca. 1725-?)
English doctor who, in 1769, measured
electrical repulsion went as r-2.06 and
attraction as r-c where c < 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.
A deviation from Coulomb’s
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.
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
Coulomb’s law could differ from two by
less than 1/21600.
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 Maxwell’s
limit to 2x10-9.
Bartlett Goldhagen & Phillips (1970)
achieved an upper limit of 1.3x10-13 .
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.
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
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.
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
j  ie 

is the solution for Dirac equation for a
particle interacting with an electromagnetic
field.
  is related to the Dirac operators.
Adding the photon’s mass
If the photon has a mass m0, an additional
term is required •A+ 2A = j
Where    m0c 


(should be h bar).
 h 
The equation show explicitly that the
additional current term is proportional to the
four vector potential A. Therefore they
have a mutual influence.
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, Gauss’s law
becomes
(1)
  E  4   
2
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 > R1 with an
inductor parallel with this spherical capacitor.
To the outer shell is applied a potential V0eiwt.
Developing the necessary equations
forming a spherical Gaussian surface at radius
r between the two shells and then using the
approximation  ( r )  V0 e it
for this
interior region, the integral of Equation(1) over
the volume interior to the Gaussian surface
becomes
(2)
 [  E  4   V e
2
0
it
]d x  0
3
Developing the necessary equations
Therefore E(r) is given by
(3)
E (r )  (qr
2
1 2 it
  V0 e r )r
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 H=0.
Approaching the final equation
Since  H  0 inside,  E  d l  0
t
The voltage appearing across the inductor is
then simply given by
it
V
e
q
2 0
2
2
E

d
l



(
R

R
2
1 )
R
C
6
1
R2
(4)
Approaching the final equation
The differential equation, which describes a
regular LCR equation is
2
d q
dq q
L 2 r
 0
dt
dt C
In the case of a nonzero rest mass
2
it
d q
dq q
2 V0 e
L 2 r
 
dt
dt C
6
(R  R )
2
2
2
1
Final equations describing the
system
  E  4   
2
 m0 c 
 

 h 
2
it
d q
dq q
2 V0 e
L 2 r
 
dt
dt C
6
(R  R )
2
2
2
1
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.
Experimental Setup
Charging a conducting shell
(1.5m in diameter-Large) with
10KVolts peak to peak with
a 4Mhz Sinusoidal voltage.
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.
  c / v  3 *108 / 4 *106  75m
 ( fiber )  1.5 *106 m
Noise
stray electric and magnetic fields
Noise - Solution
Adding 3 shells in order
to prevent stray electric
and magnetic fields
inside the sphere.
Another Noise
Johnson effect = K bTf
gives noise of 10 12V
Adding a Lockin Amplifier
Lock in amplifier
  w'*t
w'   / t  100 / 50sec
cos(t   )
Phase shift
V2 cos( wt )
x
Low pass filter
 V2 cos( )
filter
cos( wt   )V2 cos( wt )  12 V2 [cos( )  cos( 2wt   )]
Lock in Amplifier
signal
reference
Push
pull
signals
RC
vout
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.
Lock In Amplifier Demonstration
3.5
3
2.5
2
Vin+2
1.5
Vref
1
Vout+2
0.5
0
-0.5
-1
-1.5
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.
Lock In Amplifier Demonstration
Lock In Amplifier Demonstration
A full view of the system
We need to check
the System!
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.
Calibration
capacitor
Light beam
Calibration results
The calibration was
done Over 3 cycles .
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
1


Results
The experimental
result is statistically
consistent with the
assumption that the
photon rest mass is
identically zero.
How does the experiment fit in
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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