Received 18 March 2015; accepted 27 June 2015; published 30 June 2015

ABSTRACT

The dipole moments, angular momenta and gyromagnetic ratios of the electron and the proton were obtained earlier. In this note, we derive the corresponding expressions for the neutron and the muon. This work relies on the results obtained earlier for the angular momenta and dipole moments of rotating spherical bodies.

Keywords:

Angular Momenta, Dipole Moments, Neutron, Muon

1. Introduction

The purpose of this note is to derive analytical formulae for the dipole moments, angular momenta and gyromagnetic ratios of the neutron and the muon. The background to this work is fully explained in reference [1] and a parallel paper on the electron and neutron [2] follows the same methods as presented here.

2. The Electromagnetic Field Equations

We shall express the Electromagnetic Field Equations in terms of the 3-vectors representing the electric and magnetic intensities and the corresponding inductions E, H, D, B as follows:

(1)

Here, are the covariant and contravariant forms of the completely anti-

symmetric permutation tensors, is the determinant of the spatial metric tensor

with, and is the Levi-Civita symbol.

For the details of how these expressions are derived, see [1] .

3. The Neutron

The mass of the neutron, its classical radius, the square of the classical radius, and the vacuum speed of light, are

, , and c.

The following quantities are required:

(2)

Associated with there is an electric charge whose numerical value is given by the first of equations (2) above [1] . If there is an additional charge q then the total electric charge will be. We now choose q to be, so that the total charge is zero as required in the case of the neutron. If the total electric charge is zero, the coefficient F of in the Reissner-Nordstrom solution is

(3)

where j is the angular momentum per unit mass [3] . On, and so on and Equation (3) becomes

(4)

which is the same as Equation (2) of [3] . In the case of the neutron, it follows from Equation (78) of [1] , that for we have

(5)

The term is negligible compared with and so Equation (5) becomes

(6)

In accordance with the results of [1] , the dipole moment, total angular momentum and gyromagnetic ratio are

(7)

where is given by Equation (6).

The values in (2) and Equation (6) give for and

(8)

From (6) and the first of (8) we obtain and so

. Since is negligible compared to 1, we may

write. From Equations (79) of [1] , we obtain

Equations (7) give for and

(9)

It follows that the numerical value of is

(10)

We note the important fact that this number, is precisely the value of

(11)

4. The Muon

The mass and classical radius of the muon and its square are

(12)

We then obtain

(13)

For we have

(14)

If then becomes. This is negligible compared to and so Equation (14) becomes

(15)

Equations (7) will then give

(16)

This number for the ratio is precisely the value of and so we have shown that

(17)

as in the case of the neutron.

5. Conclusion

We have obtained the dipole moments angular momenta and gyromagnetic ratios of the neutron and the muon using the analytical formulae developed in [1] . The values found, are consistent with the expected values of these quantities. In particular, the ratio has the value in both the case of the neutron and in the case of the muon. We also note that has the same value as in the case of the electron and the proton [2] and as in the case of other rotating spherical bodies [1] . It is indeed remarkable that the ratio is first developed by us, to deal with rotating spherical bodies of arbitrary masses and radii and we apply it to the case of rotating stars. We now find that it is also valid for elementary particles, which in the classical approximation are assumed to be spherical.

References