Calculation of the Masses of the Charged Leptons

8/10/2019 Calculation of the Masses of the Charged Leptons

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Calculation of the masses of the charged leptons

A Findlay

E-mail: [email protected]

Abstract. A simple analysis of rotating magnetic fields leads to the derivation of an equationfor the total energy of such a system. Other properties of such systems are derived. Theseinclude; total mass, total charge, angular momentum and magnetic moment. The total energyis scalable. Further analysis suggests two possible systems that might occur naturally. Themasses of these systems can be calculated. Surprisingly, these masses are approximately themass of the Electron and the Tauon. The other properties calculated include the magneticmoment is that of the electron, and a prediction of the as yet unmeasured magnetic moment ofthe Tauon.

1.Introduction

We examine a homogenous magnetic field of field Strength B, which extends a distance r, from the

centre of rotation. We set the south pole of the magnetic field at the centre of rotation and rotate the

North Pole such that the North Pole rotates at a velocity approaching the speed of light. Due to

relativistic considerations, the North Pole changes from a magnetic field into an electric field. Further,

as the velocity of North Pole is not exactly the speed of light, there is a residual magnetic field which

remains at the North Pole. This induces a Lorentz force on the electric field which causes the electric

field to accelerate at right angles to the original magnetic field, inducing a magnetic field at right

angles to the first one. Under the right conditions (see appendix A) this second magnetic field

produces a third magnetic field again at right angles and this third magnetic field cause a replica of the

original magnetic field to occur, creating a stable self- exciting system. Further the electric field

follows a circular motion in a plane. As there are three such electric fields produced, they will repel

and thus arrange themselves equidistance apart. Due to relativistic considerations the three separaterotating electric fields in the plane will have a very small circumference and will appear point like.

The energy associated with this system can then be calculated and hence the mass. The equation for

mass of any such system that we have derived is

Mtotal= 3 (h' c / r + 2 k q2/ r+ 2 k q 2s min/ c r

3) / c

2

Mtotalis the total calculated mass

r isthe distance between the north and south pole = non-relativistic radius of rotation

smin is the relativistic radius of rotation,

h' is the reduced Planck constantc is the speed of light, k is Coulombs constant

q is one third of the charge of the electron


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1.1.

The limiting factor for any such system

As the relativistic mass increases with velocity and any relativistic length will contract with velocity

then

Mtotalmust be proportional to 1/ r.

Reviewing the equation above we notice that this is only the case when s min/r2is a constant. This then

becomes the limiting factor for any such system.

One naturally occurring limiting factor is whensminis the Planck length = 1.6161990 x 10-35

m

Therefore when the constant

smin/ r2= 1.20146942 x10

-11m

It follows thatr= 1.159821235 x10-12m

And the calculated massMtotal= 9.10938284 x10-31

kg

This compares to the measured mass of the electron of 9.10938291 x10-31

kg

The magnetic moment is given asMm= qrc/2 = -9.2847643 x10-24

J/T

This compares to the measured value of the electron of -9.2847643 x10

-24

J/T

A second case of a naturally occurring limit would be when the original Magnetic field strength would

be given as

B= k q /cr2and thusBA= k q/c

Here, A is the area of the rotating magnetic field. Where;

B= qc = qc2/r

Then;

r = k/c3= 3.335640952 x10-16m

And using this radius, the calculated mass isMtotal= 3.16738 x 10-27

kg

This compares to the measured mass of the Tauon of 3.16747 x 10-27

kg

The magnetic moment of the Tauon is then predicted to be

Mm = qrc/2 = -2.6702942 x10-27

J/T


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If this equation would also predict the mass of the Muon, then surely we could rule coincidence or

numerology out?

For the Muon, using

Mm= qrc/2 = -4.49044807 x 10-26J/T

We can calculate a radius r of 5.60931527 x 10-15

m

And using this radius, the calculated mass isMtotal 1.883519 x 10-28

kg

This compares to the measured value of the Muon of 1.883531 x 10-28

kg

2.

Derivation of the mass equation

2.1.1.

Calculation of the energy of the rotating mass.

For the charged Leptons the angular momentum must be h'/2

Where

I= h'/2

and as the mass will appear to be in a thin hoop due to the relativistic effects,

I= mr2/2

then

mrc / 2 = h'/ 2 or mr= h'/c

Therefore the energy of the rotating mass must be

U rotating mass= mc2= h'c/r= h'

2.1.2.Calculation of the Coulomb energy.

Where the electric field is affected by the Lorentz force such that it moves a quarter rotation at right

angles in the time that it would take for a half rotation of the magnetic field and then back to the

original point then it will rotate about a plane. Now as this is rotating at nearly the speed of light, the

circumference will be contracted and the entire electric field will be concentrated in a point like

fashion at a distance r /2 from the center of rotation.

As there are three co-centric magnetic fields, there will be three such point-like electric fields which

will tend to align such that they are at a maximum distance apart.

The sum of the coulomb energy each individual point will feel will then be

U coulomb= 2kqq/r


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2.1.3.Calculation of the energy of the torque produced by the Lorentz force.

The Lorentz force will be given by the equation

F lor= q (E+ v B)

Now where the charge is produced by the rotating magnetic field there will be a very small B field left

Now we can exclude the electrical field of the charge itself, so the Lorentz Force is then given by

F lor= q (v B)

Now B is reduced but by how much?

Assuming that B is reduced by the Lorentz factor (v) then the relativistic force is given by

F lor= q cB / (v)

Where the effective radius of the torque is r /2

The torque Tloris then given by

Tlor=Florr / 2 = q cB r / 2(v)

When

B = q k / c r 2

Tlor= k q2 / 2 (v) r

And the energy associated with the Torque is

U lor= Tlor

As

= 2 = 2 c / r

Then

ULor = 2 c k q2smin/r

3

2.1.4.Calculation of the total system energy and mass.

The energy of a single rotating magnetic field is then

Usingle= h' c /r+ 2 k q2/ r + 2 k q 2smin/ c r

3

The total energy of the system is then


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Usystem = 3(h' c /r+ 2 k q2/ r + 2 k q 2smin/ c r

3)

And the mass of the system is then

M system = 3(h' c /r+ 2 k q2/ r + 2 k q 2smin/ c r

3) / c

2

3. Properties of such systems

Any such system would have then the following properties

1)

All systems have a mass; the ones calculated are approximately the mass of the charged leptons

2)

All systems have a magnetic moment; the ones calculated are approximately the magnetic

moment of the charged leptons.3)All systems have a total charge equal to the charge of the electron

4)All systems can produce an identical system except with an opposite charge by rotating them

about their north-pole instead of the south-pole.

5)

All systems have a single axis angular momentum of h'/2.

6)

All three calculated systems appear "point-like".

7)

All systems have a "Zitterbewegung" due to the non-synchronous Lorentz forces.

8)

Any such system going through a Stern Gerlach apparatus will tend to align one of the magnetic

fields to the apparatus due to the inhomogeneous magnetic field. A random sample of such

systems sent through such an apparatus would have half with rotation in one direction and half

in the opposite direction giving exactly two rotating orientations, one spin "up" and one spin

"down"

9)

Any such stable system, being made out of the same material as photons of light are made out of

should exhibit similar interference or wave like properties as photons exhibit.

10)

All systems are scalable and are different to each other due to the relativistic radius.

11)

This scalability enables such systems to rotate about different radii, for instance rotation due

to a minimum caused by a proton or other atom nuclei.

12)All such systems are "spherical" unless they are affected by other charges near the relativistic

radius.

13)All systems have a fundamental uncertainty as to where they are at any point in time. As the

system takes a minimum time to exist, asking where the system is in a period of less than the

completion time of one system cycle cannot result in a meaningful answer.

14)

Only a limited numbers of such systems can be fitted into scaled up radii.

15)

Only certain scaled up radii are permitted, depending on the ability of the systems to "fit".

4. Conclusion

The masses calculated for all three charged leptons are slightly less than the measured values. This

indicates that there should be a further term in the equation to give the exact mass. We speculate that

this missing term has something to do with the gravitational properties of the mass and/or the mass

density of the vacuum. Further, it seems that the best approach to analyse such a complete system

would be to analyse the wave like properties of the system as a statistical description of extremely fast

periodic dynamics.


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References:

[1] In -- Coulomb (1785b) "Second mmoire sur llectricit et le magntisme," Histoire delAcadmie Royale des Sciences

[2] On Faradays Lines of Force by James Clerk Maxwell 1855

[3] Lorentz, Hendrik Antoon (1892), "La Thorie electromagntique de Maxwell et son application

aux corps mouvants on Internet Archive", Archives nerlandaises des sciences exactes et

naturelles 25: 363552

[4] Albert Einstein (1905) "Zur Elektrodynamik bewegter Krper", Annalen der Physik 17: 891;

[5] Bohr, N.; On the Constitution of Atoms and Molecules. I. Phil. Mag. 26 (1913)

[6] Gerlach, W.; Stern, O. (1922). "Das magnetische Moment des Silberatoms". Zeitschrift fr

Physik 9: 353355.

.

[7] Dirac, P. A. M. (1928). "The Quantum Theory of the Electron". Proceedings of the Royal

Society A: Mathematical, Physical and Engineering Sciences 117 (778): 610

[8] Schrdinger E 1930 Sitz. Preuss. Akad. Wiss. Phys.Math. Kl. 24 41828

[9] Pauli, Wolfgang (1940). "The Connection Between Spin and Statistics". Phys. Rev 58 (8): 716

722.

[10] J. J. Hudson, D. M. Kara, I. J. Smallman, B. E. Sauer, M. R. Tarbutt, E. A. Hinds. Improved

measurement of the shape of the electron. Nature, 2011; 473 (7348): 493 DOI

[11] K.A. Olive et al. (Particle Data Group), Chin. Phys. C, 38, 090001 (2014).

[12] CODATA Value: various constants. Physics.nist.gov. Retrieved on 2014-12-08

AcknowledgmentsI wish to acknowledge the assistance and encouragement from P. Haenggi, Dr. M. Freyer-Mohl, and

my daughters Amanda and Michelle Findlay without whom I would never have been able to complete

this.


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Appendix A: What will the perfect" conditions be?

In order that the charge is travelling at velocity v at exactly the moment that it has moved through an

angle of 90 to the right, the acceleration must be such that the time taken to accelerate due to the

Lorentz force from zero to v must equal the time taken for the charge to move 180 in its original path.

Therefore it follows that these results in the second acceleration to v of the charge moving 90 to the

right will result in the original path of the charge to have travelled 360, and the Lorentz force is then

applied at exactly the correct moment to keep the charge moving at v along the original path.

It follows that there are the following factors

A force accelerates the charge at 90 from zero to v m/s

This acceleration takes place during the time that the original charge moves through 180 at an angularvelocity of

In order for the charge to be accelerated to velocity v at the perfect moment the distance travelled

under this acceleration must be the distance for the original charge to move through an angle of 90

If the charge has the energy Ucoulombthen it has a mass

This is the mass that is accelerated. So if we know the time, the beginning and end velocities we have

the average acceleration, and we can calculate the average force acting on the mass, using the average

distance, and hence find the mass needed to produce the perfect conditions.

S180= 2 rfor 180 and

S90= 2 r for 90

When

v = c

then

t = 2 r/ c

t2= 4 r2/ c 2

1 / t2= c

2/ 4 r2

Average acceleration

Saccel = a t2/ 2 = 2 r

a t2 = 2 2 r

a = 2 2 r/ t2


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a = 2 2 r c 2 / 4 r2

a = 2 c2

/ 2r = 2 c2

/ r

F lor= m a= q (E + v tanBr) = 2 qB c

maccel2 c2/ r = 2 qBc

maccel = 2 qB r/ 2 c = 2 qB r/ c

When

B = q k / c r2

Then

maccel= 2 k q2/ c

2r

This is the mass required for the perfect conditions. This is shown to be the coulomb mass calculatedin the main text.

Appendix B: List of values of constants used in calculations

Planck Length 1.616 199 x 10-35m

Reduced Planck constant 1.054 571 726 x 10-34J s

Speed of light, in vacuum 299 792 458 m s-1

Coulombs constant 8.9875517873681764 x 109Nm

2C

-2

Charge of the electron -1.602 176 565 x 10-19

C1/3 charge of the electron -5.34058855 x 10 -20C2 1.414213562 3.141592654

Calculation of the Masses of the Charged Leptons

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