The Angular Momenta, Dipole Moments and Gyromagnetic Ratios of the Neutron and the 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.