Journal of Modern Physics
Vol.5 No.14(2014), Article ID:49016,4 pages DOI:10.4236/jmp.2014.514125

The Angular Momenta Dipole Moments and Gyromagnetic Ratios of the Electron and the Proton

A. Georgiou

School of Physics Astronomy and Mathematics, University of Hertfordshire, Hatfield, UK

Email: a.georgiou@herts.ac.uk   Received 3 June 2014; revised 1 July 2014; accepted 24 July 2014

ABSTRACT

We had previously obtained analytical formulae for the dipole moments and angular momenta of rotating spherical bodies. The resulting formulae were applied to the Sun, the star 78 Virginis and the Earth. The agreement of the theoretical formulae with the actual real situations is indeed remarkable. In this note we apply the same formulae to the electron and the proton, using the classical values of the radii, so no quantum mechanical treatment is considered.

Keywords:Angular Momenta, Dipole Moments, Electron, Proton 1. Introduction

The purpose of this work is to obtain analytical formulae for the dipole moments and angular momenta of the electron and the proton. For this purpose, exact solutions of the Einstein-Maxwell field equations for volume distributions of rotating charged matter are required. There are difficulties in obtaining such solutions, but these were successfully treated in detail in  . In addition, there are uncertainties regarding the radii of these particles, resulting in some differences between the calculated numerical values and the accepted values of the dipole moments.

2. The Electron

The mass , classical radius , the electric charge of the electron and the vacuum speed of light, are: (1)

The following ratios are required: (2)

If the mass is , then there is an electric charge associated with . Thus, if there is an additional electric charge , the total charge is . The value of is only and so it is negligible compared to the value of in (1). If , we may therefore write for the charge and its square , (3)

In the Reissner-Nordstrom solution  , the coefficient of in the metric, has the form given by (4)

with given by the second of (3).

The coefficient of in the Kerr-Newman metric, is (5)

with . (6)

Equation (5) then, becomes (7)

where is the angular momentum per unit mass; see Equations (18)-(20) of  . On , and so on Equation (7) becomes (8)

which is the same as Equation (2) of  (Note that in reference is used where we use ). It follows that instead of Equations (7)-(8) of  , we have for  (9)

In accordance with the results of  , the dipole moment , total angular momentum and gyromagnetic ratio are: (10)

where (11)

The values in (2) and (3) give for in (9) and  (12)

Equations (10) then give , or (13)

The accepted value is . (14)

From the second of (10), we obtain for  . (15)

The gyromagnetic ratio is therefore, (16)

and this is precisely the value of .

It must be noted that there are various different values for the radius of the classical electron. The value of found is smaller than the accepted value and it depends on which radius we choose. Thus, if we use a radius with numerical value (17)

which is 7 times larger than the classical radius of the electron and repeat the calculations, we find (18)

This is nearer the accepted value in (14).

It was established by Dehmelt  that the upper limit for the electron radius is .

This simply implies that if is the electronic radius, we must have . The values of the radii we used, including the “classical radius of the electron”, are much larger than this.

3. The Proton

The mass, radius, electric charge and mass to radius ratio of the proton, are respectively: (19)

If there is an additional electric charge , the total charge will be . The value of is only and so it is negligible compared to the value of in (1). If , we may therefore write for the total charge and its square , (20)

The same Formulae (12) for and (10) for , and are valid, but with the proton parameters in Equations (19) instead of the electron ones. We find (21)

We note that the gyromagnetic ratio found in the of Equations (21), is precisely the value of .

In the case of the proton the value of found is larger than the accepted value, which is .

4. Conclusion

We have obtained the dipole moments angular momenta and gyromagnetic ratios of the electron and the proton using the analytical formulae developed in  . In the case of the electron, the value of found is smaller than the accepted value, but in the proton case it is larger. If Dehmelt’s  deductions are valid, a complete reevaluation is necessary using our analytical Formulae (10) to find , and . It is not possible to do this, because Dehmelt, does not give any definite values for the radii; he only states that the electron radius should satisfy the inequality . But in any case, the purpose of the calculations here, is to see if the values of the known classical electron and proton radii, give the expected values of , and .

References

1. Georgiou, A. (2012) Journal of Modern Physics, 3, 1301-1310.
2. Blinder, S.M. (2003) Dirac’s Electron via General Relativity. PACS No: 03.50.De, 14.60.C.
3. Dehmelt, H. (1988) Physical Scripta, T22, 102-110. http://dx.doi.org/10.1088/0031-8949/1988/T22/016