The presented circular current loop model reveals that charged fundamental particles such as the electron consist essentially of electric and magnetic energy. The magnetic properties have the same order of magnitude as the electric ones. The electromagnetic field energy is the origin of the inertial mass. The Higgs boson, existing or not, is not needed to “explain” particle mass. The magnetic moment of fundamental particles is not anomalous! The “anomaly” indicates the existence of a small additional amount of kinetic energy. Thus, fundamental particles are not purely field-like such as photons and not (essentially) mass-like such as atoms, they represent a special kind of matter in between. Their kinetic energy is obviously not due to any relativistic effect but is related to an independent physical law that provides, together with the magnetic energy, the angular momentum exactly to be ħ/2. Fundamental particles are (at least) two-dimensional. In the simplest case their core consists of two concentric, nearly identical current loops. Their relative design details, the “anomaly” factor, and the rotational velocity of the uniformly distributed elementary charge follow from the stability condition, i.e. electric and magnetic force balance, and do not depend on the particle’s rest mass! Fundamental particles are objects of classical physics. Their magnetic forces are the true origin of the weak and strong nuclear interactions. For their explanation bosons and gluons are not needed.
Subatomic particles that cannot be subdivided are regarded to be fundamental. This includes that they are stable. Typical fundamental particles in this sense are the electron and the positron. This investigation is provisionally restricted to them. They are characterized by the following four basic entities:
Charge ±e anomaly of magnetic moment a rest mass m0 angular momentum ћ/2.
When no additional entities of fundamental particles are known, these four basic ones are sufficient to describe a fundamental particle. An acceptable model of a fundamental particle must represent its four entities accurately and consistently. This ensures that the model represents also derived entities, e.g.
Bohr magneton μ B = ± e ℏ 2 m 0 (1.1)
Magnetic moment μ = μ B ( 1 + a ) (1.2)
Compton wavelength bar ƛ C = ℏ m 0 c (1.3)
Rest mass energy E 0 = m 0 c 2 (1.4)
Strange but true: As electron and positron do not show any internal structure quantum mechanics regards them to be point particles, i.e. to have no dimensions at all. Such an object would be totally unrealistic and non-physical. It could not have any physical properties: Charge density and mass density would be infinite, angular momentum (“spin”) and magnetic moment could not exist. Fortunately, there is a pioneering exception [
A one-dimensional circular current loop according to
The charge must be distributed homogeneously over its circumference 2πr and it must rotate at a constant velocity vφ. These are the conditions for a constant current
I = e v φ 2 π r = e c 2 π r β φ (2.1)
and a constant magnetic field. This is the only chance that the model can represent a stable, non-radiating particle.
The current I creates a magnetic moment
μ = I π r 2 = e c 2 β φ r (2.2)
where
β φ = v φ c (2.3)
As the presented current loop model is electromagnetic, basically it does not “know” rest mass m0 and rest mass energy m0c2. The model provides an electrostatic field due to its charge ±e and a constant magnetic field due to the rotation of the homogeneously distributed charge at constant velocity. Consequently, the energy inventory is expected to be purely electromagnetic as it is with an electromagnetic wave. The charge ±e does not exist as a concentrated rotating point charge, but as a constant charge density
σ = ± e 2 π r = ± e c 2 π ℏ ƛ C r m 0 (3.1)
where
ƛ C = ℏ m 0 c (3.2)
is the Compton radius of a particle having rest mass m0.
The amount of the elementary charge, | e | , is an entity that is found to be exactly equal with all loaded particles. Additionally, they have a magnetic moment including a strange “anomaly” a. This means that the charge (density) of all these particles is rotating at a constant angular velocity. The circular current loop representing an electromagnetic particle model makes transparent that the rest mass basically is an illusion resulting from the electrostatic (Coulomb) field and from the magnetic (Lorentz) field of the particle. Theoretically the electromagnetic field is spread out infinitely wide. When a particle is accelerated in an electric and/or magnetic field, due to the finite velocity of light the electromagnetic field of the particle is distorted. The distorted field tries to pull its escaping central part back. This restoring force is interpreted as being due to the inertia of mass. The rest mass energy m0c2 of the particle is equivalent to the sum of its electric energy Eel and magnetic energy Emag:
m 0 c 2 = E e l + E m a g (3.3)
The particle does not possess electric energy Eel and magnetic energy Emag and additionally rest mass energy m0c2: The nature of the particle energy m0c2 is electromagnetic, and the rest mass m0 is a fictive entity although it can be measured.
The exact mathematical treatment of the circular current loop is not trivial. But the results of the exact calculation are very useful when the circular current loop is regarded as a model of fundamental particles or at least a basic element for building up such a model. Once done in a proper way the results may serve forever as a universal platform for all respective calculations.
At first the geometric relationship between a circle with radius r, centered at the origin of an orthogonal coordinate system, and a point P(ρ, z) has to be analyzed. In
The distance ζ of a point on the circle having the coordinates r and φ to a point with the coordinates ρ and φ = 0 is given by
ζ 2 = r 2 sin 2 φ + ( ρ − r cos φ ) 2 (4.1)
As
sin 2 φ + cos 2 φ = 1 (4.2)
(4.1) becomes:
ζ 2 = r 2 + ρ 2 − 2 ρ r cos φ (4.3)
The distance D of the point on the circle to a point P(ρ, z) in the ρ-z-plane is then given by
D 2 = ζ 2 + z 2 = r 2 + ρ 2 − 2 ρ r cos φ + z 2 (4.4)
or
( D r ) 2 = 1 + ( ρ r ) 2 + ( z r ) 2 − 2 ρ r cos φ (4.5)
It will turn out to be useful to introduce the abbreviation
k = 2 ρ r 1 + ( ρ r ) 2 + ( z r ) 2 (4.5)
Thus, the distance d becomes
D = r 1 + ( ρ r ) 2 + ( z r ) 2 1 − k cos φ (4.6)
The electrostatic situation is characterized by the continuous and homogenous distribution of the charge q over the circular contour:
d q = q 2 π r r d φ = q 2 π d φ (4.7)
A particular charge element dq generates at the point P(ρ, z) the incremental contribution dV to the electrostatic potential V
d V = 1 4 π ε 0 D q 2 π d φ = q 4 π ε 0 r 1 1 + ( ρ r ) 2 + ( z r ) 2 1 2 π d φ 1 − k cos φ (4.8)
The electrostatic potential V at the point P(ρ, z) is the sum of all these contributions around the complete circle:
V = q 4 π ε 0 r 1 1 + ( ρ r ) 2 + ( z r ) 2 1 2 π ∫ φ = 0 2 π d φ 1 − k cos φ (4.9)
The magnetic situation is described by the vector potential A that is created by the rotation of the uniformly distributed charge at constant velocity vφ, thus according to (2.1) and (2.3) representing a constant current
I = e v φ 2 π r = e c 2 π r β φ (4.10)
According to the vector behavior of the current each element Ids = Irdφ is only effective with its component vertical to the x axis:
I d s e f f = I r cos φ d φ (4.11)
Thus, the increment dA of vector potential is
d A = d A φ = μ 0 4 π I d s e f f D = μ 0 I 2 1 1 + ( ρ r ) 2 + ( z r ) 2 1 2 π cos φ d φ 1 − k cos φ (4.12)
The vector potential A of the current I along the circle at the point P(ρ, z) is
A = A φ = μ 0 I 2 1 1 + ( ρ r ) 2 + ( z r ) 2 1 2 π ∫ φ = 0 2 π cos φ d φ 1 − k cos φ (4.13)
Both the integrals in (4.9) and (4.12) cannot be solved in closed form. It is well known that they can be transformed to combinations of the two standardized elliptical integrals. But this would complicate the situation. The superior method is to develop the identical denominator of the two functions to be integrated into a power series:
1 1 − x = 1 + 1 2 x + 1 ⋅ 3 2 ⋅ 4 x 2 + 1 ⋅ 3 ⋅ 5 2 ⋅ 4 ⋅ 6 x 3 + 1 ⋅ 3 ⋅ 5 ⋅ 7 2 ⋅ 4 ⋅ 6 ⋅ 8 x 4 + ⋯ = ∑ m = 0 ∞ c m x m (4.14)
where
x = k cos φ (4.15)
The coefficients cm can be calculated according to the recursion formula
c 0 = 1 c m = c m − 1 2 m − 1 2 m = c 2 m − 2 ⋅ 2 m − 3 2 m − 2 ⋅ 2 m − 1 2 m m > 0 (4.16)
Thus, the two integrals can be written as
1 2 π ∫ 0 2 π d φ 1 − k cos φ = ∑ m = 0 ∞ [ c m k m 1 2 π ∫ 0 2 π cos m φ d φ ] (4.17)
1 2 π ∫ 0 2 π cos φ d φ 1 − k cos φ = ∑ m = 0 ∞ [ c m k m 1 2 π ∫ 0 2 π cos m + 1 φ d φ ] (4.18)
The following survey shows what happens:
∫ φ = 0 2 π 1 d φ = 2 π (4.19)
∫ φ = 0 2 π x d φ = k ∫ φ = 0 2 π cos φ d φ = k sin φ | 0 2 π = 0 (4.20)
x 2 = k 2 cos 2 φ = 1 2 k 2 ( 1 + cos 2 φ ) (4.21)
∫ φ = 0 2 π x 2 d φ = 1 2 k 2 ( φ + 1 2 sin 2 φ ) | 0 2 π = 1 2 k 2 2 π (4.22)
x 3 = k 3 cos 3 φ = 1 4 k 3 ( 3 cos φ + cos 3 φ ) (4.23)
∫ φ = 0 2 π x 3 d φ = 1 4 k 3 ( 3 sin φ + 1 3 sin 3 φ ) | 0 2 π = 0 (4.24)
Only the terms with even exponents contribute to the solutions. Thus, the integral for the electrostatic potential V selects from the “mother function” (4.14) the even terms and the integral for the vector potential, as each term is multiplied by cosφ, selects the odd terms. This indicates the close relationship between V and A, and it reveals that both are orthogonal to each other.
Generally
∫ φ = 0 2 π cos 2 n + 1 φ d φ = 0 (4.25)
and
∫ 0 2 π cos 2 n φ d φ = 1 2 2 n ( 2 n n ) ⋅ 2 π = 1 2 2 n 2 n ( 2 n − 1 ) ( 2 n − 2 ) ⋯ ( n + 1 ) 1 ⋅ 2 ⋅ 3 ⋯ n ⋅ 2 π (4.26)
From (4.26) follows
∫ 0 2 π cos 2 ( n − 1 ) φ d φ = 1 2 2 n − 2 ( 2 n − 2 n − 1 ) ⋅ 2 π = 1 2 2 n − 2 ( 2 n − 2 ) ( 2 n − 3 ) ( 2 n − 4 ) ⋯ n 1 ⋅ 2 ⋅ 3 ⋯ ( n − 1 ) ⋅ 2 π (4.27)
Consequently, the recursion formula for the integration results is
∫ 0 2 π cos 2 n φ d φ ∫ 0 2 π cos 2 ( n − 1 ) φ d φ = 1 2 2 ⋅ 2 n ( 2 n − 1 ) ( 2 n − 2 ) ⋯ ( n + 1 ) 1 ⋅ 2 ⋅ 3 ⋯ n ⋅ 1 ⋅ 2 ⋅ 3 ⋯ ( n − 1 ) ( 2 n − 2 ) ( 2 n − 3 ) ( 2 n − 4 ) ⋯ n = 1 4 ⋅ 2 n ( 2 n − 1 ) n 2 = 2 n − 1 2 n (4.28)
Accordingly, the first integration results are
1 2 π ∫ 0 2 π cos 2 φ d φ = 1 2 2 n ( 2 n n ) = 1 4 ⋅ 2 1 = 1 2 (4.29)
1 2 π ∫ 0 2 π cos 4 φ d φ = 1 2 4 ( 4 2 ) = 1 16 ⋅ 4 ⋅ 3 2 ⋅ 1 = 3 8 = 1 2 ⋅ 3 4 (4.30)
1 2 π ∫ 0 2 π cos 6 φ d φ = 1 2 6 ( 6 3 ) = 1 64 ⋅ 6 ⋅ 5 ⋅ 4 3 ⋅ 2 ⋅ 1 = 5 16 = 1 2 ⋅ 3 4 ⋅ 5 6 (4.31)
To summarize the even terms of (4.14) it is useful to set m = 2 n . It transforms (4.16) and (4.17) to
c 2 n = c 2 n − 2 ⋅ 4 n − 3 4 n − 2 ⋅ 4 n − 1 4 n (4.32)
and
1 2 π ∫ 0 2 π d φ 1 − k cos φ = 1 + 1 ⋅ 3 2 ⋅ 4 k 2 ⋅ 1 2 + 1 ⋅ 3 ⋅ 5 ⋅ 7 2 ⋅ 4 ⋅ 6 ⋅ 8 k 4 ⋅ 3 8 + ⋯ = ∑ n = 0 ∞ a 2 n k 2 n (4.33)
Consequently, the electrostatic potential according to (4.9) can be written as
V = q 4 π ε 0 r ∑ n = 0 ∞ a 2 n k 2 n 1 + ( ρ r ) 2 + ( z r ) 2 (4.34)
where
a 0 = 1 a 2 n = a 2 n − 2 ⋅ 4 n − 3 4 n − 2 ⋅ 4 n − 1 4 n ⋅ 2 n − 1 2 n = a 2 n − 2 ⋅ 4 n − 1 4 n ⋅ 4 n − 3 4 n (4.35)
or
a 2 n = a 2 n − 2 ( 1 − 1 4 n ) ( 1 − 3 4 n ) (4.36)
The first coefficients are
a 2 = 1 ⋅ 3 4 ⋅ 1 4 = 3 16 a 4 = 3 16 ⋅ 7 8 ⋅ 5 8 = 105 1024 a 6 = 105 1024 ⋅ 11 12 ⋅ 9 12 = 1155 16384 (4.37)
Similarly, the final form of (4.13) is achieved by setting m = 2n + 1. This results in
1 2 π ∫ 0 2 π cos φ d φ 1 − k cos φ = 1 2 k ⋅ 1 2 + 1 ⋅ 3 ⋅ 5 2 ⋅ 4 ⋅ 6 k 3 ⋅ 3 8 + 1 ⋅ 3 ⋅ 5 ⋅ 7 ⋅ 9 2 ⋅ 4 ⋅ 6 ⋅ 8 ⋅ 10 k 5 ⋅ 5 16 + ⋯ = k 4 ∑ n = 0 ∞ b 2 n k 2 n (4.38)
and
A = A φ = μ 0 I 2 1 4 ∑ n = 0 ∞ b 2 n k 2 n + 1 1 + ( ρ r ) 2 + ( z r ) 2 (4.39)
where
b 0 = 1 b 2 n = b 2 n − 2 ⋅ 4 n − 1 4 n ⋅ 4 n + 1 4 n + 2 ⋅ 2 n + 1 2 n + 2 = b 2 n − 2 ⋅ 4 n − 1 4 n ⋅ 4 n + 1 4 n + 4 (4.40)
or
b 2 n = b 2 n − 2 ( 1 − 1 4 n ) ( 1 − 3 4 ( n + 1 ) ) (4.41)
The first coefficients are
b 2 = 1 ⋅ 3 4 ⋅ 5 8 = 15 32 b 4 = 15 32 ⋅ 7 8 ⋅ 3 4 = 315 1024 b 6 = 315 1024 ⋅ 11 12 ⋅ 13 16 = 15015 65536 (4.42)
As a matter of symmetry all points on a second concentric circle having the radius ρ = r2 and axial distance z = z2 from the origin have the same values V1 and A1 generated by a primary circular current loop 1 positioned at the origin.
The electrostatic potential of circle 1 is
V 1 = q 1 4 π ε 0 r 1 ∑ n = 0 ∞ a 2 n k 1 , 2 2 n 1 + ( r 2 r 1 ) 2 + ( z 2 r 1 ) 2 (5.1)
Its vector potential is
A 1 = A φ 1 = μ 0 I 1 2 1 4 ∑ n = 0 ∞ b 2 n k 1 , 2 2 n + 1 1 + ( r 2 r 1 ) 2 + ( z 2 r 1 ) 2 (5.2)
where
k 1 , 2 = 2 r 2 r 1 1 + ( r 2 r 1 ) 2 + ( z 2 r 1 ) 2 (5.3)
When circle 2 with radius r2 and axial distance z2 is loaded with the uniformly distributed charge q2 the electrostatic energy Eel between the two charged loops is
E e l = V 1 ( ρ , z ) ⋅ q 2 (5.4)
or
E e l ( r 2 , z 2 ) = e 2 4 π ε 0 r 1 q 1 e q 2 e ∑ n = 0 ∞ a 2 n k 1 , 2 2 n 1 + ( r 2 r 1 ) 2 + ( z 2 r 1 ) 2 (5.5)
When the charge q2 rotates at velocity v φ 2 = c β φ 2 the magnetic energy Emag is
E m a g = q 2 v φ 2 A 1 φ ( ρ , z ) (5.6)
or
E m a g ( r 2 , z 2 ) = q 2 v φ 2 μ 0 I 1 2 1 4 ∑ n = 0 ∞ b 2 n k 1 , 2 2 n + 1 1 + ( r 2 r 1 ) 2 + ( z 2 r 1 ) 2 (5.7)
where
q 2 v φ 2 μ 0 I 1 2 = q 2 e v φ 2 c ⋅ μ 0 2 q 1 e v φ 1 c 1 2 π r 1 ⋅ e 2 c 2 = μ 0 e 2 c 2 4 π r 1 q 1 e q 2 e β φ 1 β φ 2 (5.8)
To make the electric energy and the magnetic energy comparable and in order to come to a convenient form the following identities are used
e 2 4 π ε 0 = μ 0 c 2 e 2 4 π = α ℏ c = 1.439964445 × 10 − 09 eV ⋅ m (5.9)
α ℏ c = α m 0 c 2 ℏ m 0 c = α m 0 c 2 ƛ C (5.10)
They transform (5.5) and (5.7) to
E e l ( r 2 , z 2 ) = α m 0 c 2 ƛ C r 1 q 1 e q 2 e ∑ n = 0 ∞ a 2 n k 1 , 2 2 n 1 + ( r 2 r 1 ) 2 + ( z 2 r 1 ) 2 (5.11)
and
E m a g ( r 2 , z 2 ) = α m 0 c 2 ƛ C r 1 q 1 e q 2 e β φ 1 β φ 2 1 4 ∑ n = 0 ∞ b 2 n k 1 , 2 2 n + 1 1 + ( r 2 r 1 ) 2 + ( z 2 r 1 ) 2 (5.12)
The standardized energies E e l ( r 2 , z 2 ) α m 0 c 2 ƛ C r 1 q 1 e q 2 e and 1 β φ 1 β φ 2 E m a g ( r 2 , z 2 ) α m 0 c 2 ƛ C r 1 q 1 e q 2 e are presented in
The electric forces Fel z in z direction and Fel ρ in ρ direction are
F e l z ( r 2 , z ) = q 2 ∂ V 1 ( r 2 , z ) ∂ z (6.1)
and
F e l ρ ( r 2 , z ) = q 2 ∂ V 1 ( r 2 , z ) ∂ r 2 (6.2)
where V 1 ( r 2 , z ) is given by (5.1) and (5.3). With applying (5.9) the results are
F e l z ( r 2 , z ) = α ℏ c r 1 2 q 1 e q 2 e ∑ n = 0 ∞ a 2 n ( 4 n + 1 ) k 12 2 n [ 1 + ( r 2 r 1 ) 2 + ( z r 1 ) 2 ] 3 2 z r 1 (6.3)
and
F e l ρ ( r 2 , z ) = α ℏ c r 1 2 q 1 e q 2 e ∑ n = 0 ∞ a 2 n [ 4 n k 12 − ( 4 n + 1 ) r 2 r 1 ] k 12 2 n [ 1 + ( r 2 r 1 ) 2 + ( z r 1 ) 2 ] 3 2 (6.4)
The magnetic forces Fmag z in z direction and Fmag ρ in ρ direction are
F m a g z = q 2 v φ 2 ∂ A 1 φ ( r 2 , z ) ∂ z (6.5)
and
F m a g ρ = q 2 v φ 2 1 r 2 ∂ ( r 2 A 1 φ ) ∂ r 2 (6.6)
where Aφ1 is given by (5.2) and (5.3). By applying (5.8) and 5.9) the results are
F m a g z ( r 2 , z ) = α ℏ c r 1 2 q 1 e q 2 e β φ 1 β φ 2 ∑ n = 0 ∞ b 2 n ( n + 3 4 ) k 12 2 n + 1 [ 1 + ( r 2 r 1 ) 2 + ( z r 1 ) 2 ] 3 2 z r 1 (6.7)
and
F m a g ρ ( r 2 , z ) = α ℏ c r 1 2 q 1 e q 2 e v φ 1 c v φ 2 c ∑ n = 0 ∞ b 2 n [ n + 1 − k 12 ( n + 3 4 ) r 2 r 1 ] k 12 2 n [ 1 + ( r 2 r 1 ) 2 + ( z r 1 ) 2 ] 3 2 (6.8)
According to (5.10) the factor α ℏ c / r 1 2 in (6.3), (6.4), (6.7), and (6.8) may be replaced from
α ℏ c r 1 2 = α m 0 c 2 ƛ C ( ƛ C r 1 ) 2 (6.9)
It will turn out that sometimes it is helpful or necessary to define the average radius
R = r 1 r 2 = r 1 r 2 r 1 (6.10)
A single circular current loop has an electrostatic potential V according to (5.1) and a vector potential A according to (5.2), but no intrinsic electric and magnetic energies. Energies according to (5.11) and (5.12) only exist between (at least) two electric and/or magnetic potentials. Consequently, the simplest electromagnetic model of a charged particle is two-dimensional and consists of two circular current loops. Their total load −e or +e is shared between them. Thus, the radial electrostatic forces of a fundamental particle are always repulsive. To establish the opportunity of a force balance, the radial magnetic forces must be attractive. This requires that both currents have the same direction. As the two current loops of a particle are concentrically positioned in the same plane both axial forces, electric and magnetic, are zero.
The sharing of the indivisible elementary load e to two current loops is no physical sin because one loop is only a mathematic item and no physical object. The simplest model of a stable fundamental particle is a twin-loop where the twins cannot be physically separated. Neutral particles (e.g. the neutron) consist of two fundamental particles of opposite electric polarity similar to the hydrogen atom. In contrary to charged fundamental particles, the range of their electromagnetic field and consequently their size are limited.
The six energy and force equations converge when k < 1. When k is close to unity, the convergence is very poor. For exact calculations in the most interesting range 0.999 ≤ k ≤ 1 more than 108 terms of the series may be needed. When z2 = z1 = 0 and simultaneously r2 = r1 = R, i.e. when the two loops touch each other at their circumference, the maximum value k12 = 1 is reached. In this singular case is v φ 2 = v φ 1 = c , and the derived series for energies and radial forces do not converge. Thus, the respective values cannot be calculated. But the difference E e l − | E m a g | and | F e l ρ | − F m a g ρ converge and can be calculated at nearly unlimited accuracy. Two fascinating results of the calculations with up to 3 × 109 terms are
E e l ( r 1 , 0 ) + 1 β φ 1 β φ 2 E m a g ( r 1 , 0 ) | β φ 1 = β φ 2 = 1 α m 0 c 2 ƛ C r 1 q 1 e q 2 e ≈ 1 2 { ∑ n = 0 3 × 10 9 a 2 n − 1 4 ∑ n = 0 3 × 10 9 b 2 n } = 0 .6366197723 29 ⋯ (7.1)
where
2 π = 0 .636619772368 ⋯ (7.2)
and
F e l ρ ( r 1 , 0 ) + 1 β φ 1 β φ 2 F m a g ρ ( r 1 , 0 ) | β φ 1 = β φ 2 = 1 α m 0 c 2 ƛ C ( ƛ C r 1 ) 2 q 1 e q 2 e ≈ 1 2 2 { ∑ n = 0 3 × 10 9 a 2 n − 1 4 ∑ n = 0 3 × 10 9 b 2 n } = 0 .318309886167 ⋯ (7.3)
where
1 π = 0 .318309886184 ⋅ ⋅ ⋅ (7.4)
It will be shown that the real model slightly differs from this extreme case.
As to positive or negative signs of energies and forces it has to be emphasized that the energy and force equations concern their amounts, i.e. negative signs are omitted. When sums or differences are calculated the correct signs are taken. In the case of (7.3) F e l ρ is negative and F m a g ρ is positive as r 2 / r 1 < 1 . When r 2 / r 1 > 1 it is vice versa. The most important message of (7.1) is: | E m a g | is not essentially smaller than E e l when β φ 1 β φ 2 is close to unity.
Two concentric, identical ( r 2 = r 1 = R ), one-dimensional current loops represent the idealized model of an object that according to (7.1) and (7.2) is characterized by
E e l ( R , 0 ) − | E m a g ( R , 0 ) | v φ 1 = v φ 2 = c = α m 0 c 2 ƛ C R q 1 e q 2 e ⋅ 2 π (7.5)
E e l ( R , 0 ) and E m a g ( R , 0 ) cannot be calculated from the respective power series because they do not converge. But there is a simple way out that was already sketched in Chapter 3: When the twin loop is accelerated the electric field and the magnetic field corresponding to E e l ( R , 0 ) and | E m a g ( R , 0 ) | create a restoring force each. It is traditionally called inertial force and defines the inertial mass. These masses are
m e l = E e l ( R , 0 ) c 2 (7.6)
and
m m a g = | E m a g ( R , 0 ) | c 2 (7.7)
The complete field energy is
E e l ( R , 0 ) + | E m a g ( R , 0 ) | = ( m e l + m m a g ) c 2 = m 0 c 2 (7.8)
When the twin loop is to represent a fundamental particle, its total charge must be
q 1 + q 2 = − e (7.9)
or, in the case of the positron
q 1 + q 2 = + e (7.10)
As the two loops are (idealized) identical is
q 1 = q 2 = ± e 2 (7.11)
and
q 1 q 2 e 2 = 1 4 (7.12)
As already mentioned, the two loops are no physical objects. One loop alone could anyway not possess energy. Electric and magnetic energies arise from the mutual influence of their electromagnetic potentials. Only for mathematical treatment they may carry the non-physical charge e/2. When the two loops are brought together from infinite distance to identical position, energy must be spent to create electric energy against their mutual repulsive force. On the other hand, the magnetic force attracts the loops and thus supports the process of particle creation by contributing negative magnetic energy.
From (7.5) and (7.12) follows
E e l − E m a g = m 0 c 2 ƛ C R α 2 π (7.13)
According to (7.8) is
E e l + E m a g = ( m e l + m m a g ) c 2 = m 0 c 2 (7.14)
Combination of (7.13) and (7.14) results in
E e l = 1 2 m 0 c 2 ( 1 + ƛ C R α 2 π ) (7.15)
and
E m a g = 1 2 m 0 c 2 ( 1 − ƛ C R α 2 π ) (7.16)
The electrical energy is purely static and thus cannot contribute to dynamic effects such as magnetic moment and angular momentum. Thus, these effects are exclusively attributed to the magnetic energy Emag respectively to the “magnetic” mass
m m a g = E m a g c 2 (7.17)
David Bergman formulated this important statement already more than 20 years ago [
m m a g c R ≈ 1 2 m 0 c R ≈ ℏ 2 (7.18)
From (7.16), (7.17), and (7.18) follows
1 2 m 0 ( 1 − ƛ C R α 2 π ) c R ≈ ℏ 2 (7.19)
or
R ≈ ℏ m 0 c ( 1 + α 2 π ) = ƛ C ( 1 + α 2 π ) (7.20)
where
ƛ C = ℏ m 0 c (7.21)
This subject will be investigated exactly in Chapter 9.
With the radius according to (7.20) the energies according to (7.15) and (7.16) become
E e l ≈ 1 2 m 0 c 2 ( 1 + α 2 π 1 + α 2 π ) (7.22)
E m a g ≈ 1 2 m 0 c 2 ( 1 − α 2 π 1 + α 2 π ) (7.23)
Consequently is
E e l − E m a g ≈ m 0 c 2 α 2 π 1 + α 2 π (7.24)
The current in the twin loop is according to (7.20)
I ≈ e c 2 π R ≈ e c 2 π m 0 c ℏ 1 1 + α 2 π (7.25)
From (7.25) follows the magnetic moment
μ = π R 2 I ≈ π R 2 e c 2 π R (7.26)
or
μ ≈ e c 2 R ≈ e c 2 ℏ m 0 c ( 1 + α 2 π ) (7.27)
and finally
μ ≈ e ℏ 2 m 0 ( 1 + α 2 π ) = μ B ( 1 + α 2 π ) (7.28)
where
μ B = e ℏ 2 m 0 (7.29)
is the Bohr magneton and
a e ≈ α 2 π ≈ 1.161 410 × 10 − 3 (7.30)
is the mysterious magnetic moment “anomaly”. All these results and statements and some more are already given in [
β φ 1 β φ 2 | b F m a g ρ = − F e l ρ (8.1)
This means that force balance exists when
β φ 1 β φ 2 | b = − F e l ρ F m a g ρ (8.2)
At the lower value according to (8.2) the force balance is stable. The higher value being extremely close to r 2 / r 1 = 1 represents instable balance. The most interesting region is the range where r2 is close to r1 and
β φ = v φ c = β φ 1 β φ 2 (8.3)
is close to unity. Therefore, it is convenient to consider the differences
δ β b = 1 − β φ 1 β φ 2 | b = 1 − | F e l ρ | F m a g ρ (8.4)
and
δ r = r 1 − r 2 r 1 = 1 − r 2 r 1 (8.5)
The cross at δ r ≈ 2 .4716040 94 × 10 -4 and δ β b ≈ 1 .159246494 × 10 − 3 represents the force balance condition of the electron. Its calculation would have required at least 108 steps. To be sure, the maximum number of steps was set at 3 × 109.
Based on the results of Chapter 9 the resulting force F e l ρ + F m a g ρ according to (6.4) and (6.6) in the vicinity of the electron force balance point is shown in
α ℏ c = 2.307077 × 10 − 28 W ⋅ s ⋅ m = 2.307077 × 10 − 28 N ⋅ m 2 (8.6)
r 1 ≈ R ≈ ƛ C = 386.159268 × 10 − 15 m (8.7)
The absolute forces are
F m a g b = − F e l b ≈ 1 4 α ℏ c ƛ C 2 × 1286.532 = 159.413 × 10 6 N (8.8)
and related to the circumference
F m a g b 2 π R = − F e l b 2 π R ≈ 65.7 × 10 18 N / m (8.9)
The goal of the following considerations is to find the simplest physical, consistent, electromagnetic model of the electron. The energy and force equations will be specialized to the case z = 0, i.e. where the two loops are positioned concentrically in the same plane in order to represent a stable fundamental particle. The radial velocities vφ1 and vφ2 are assumed to be slightly smaller than c:
v φ 1 c = β φ 1 = 1 − δ β φ 1 δ β φ 1 ≪ 1 (9.1)
v φ 2 c = β φ 2 = 1 − δ β φ 2 δ β φ 2 ≪ 1 (9.2)
v φ 1 c v φ 2 c = β φ 1 β φ 2 = ( 1 − δ β φ ) 2 δ β φ ≪ 1 (9.3)
The interesting range of r2 is 0 < r 2 < r 1 where the electric force is negative (repulsive) and the magnetic force is positive (attractive).
According to (7.1) the difference between the electric energy Eel and the magnetic energy is given by
E e l − E m a g β φ 1 β φ 2 = α m 0 c 2 ƛ C r 1 q 1 e q 2 e 2 π (9.4)
As already mentioned in (3.3) the total electromagnetic field energy is
E e l + E m a g = m 0 c 2 (9.5)
Combination of (9.4) and (9.5) results in
E e l m 0 c 2 = 1 1 + ß φ 1 β φ 2 + ß φ 1 β φ 2 1 + ß φ 1 β φ 2 ƛ C r 1 4 q 1 q 2 e 2 α 2 π (9.6)
and
E m a g m 0 c 2 = ß φ 1 β φ 2 1 + ß φ 1 β φ 2 − ß φ 1 β φ 2 1 + ß φ 1 β φ 2 ƛ C r 1 4 q 1 q 2 e 2 α 2 π (9.7)
Relevant for further evaluation is the difference
E e l − E m a g m 0 c 2 = 1 − ß φ 1 β φ 2 1 + ß φ 1 β φ 2 + 2 ß φ 1 β φ 2 1 + ß φ 1 β φ 2 ƛ C r 1 4 q 1 q 2 e 2 α 2 π (9.8)
Now it’s time to eliminate some more misinterpretations and errors from accepted physics:
The magnetic moment arises from the circular current Iφ:
I φ = e v φ 2 π R = e c 2 π R β φ (9.9)
where R and vφ according to (6.10) respectively (8.3) are average values. The magnetic moment is
μ = I φ π R 2 = e c 2 β φ R = 1 2 e v φ R (9.10)
or
μ = e ℏ 2 m 0 β φ R ƛ C (9.11)
The experimental finding is
μ = e ℏ 2 m 0 ( 1 + a ) (9.12)
Of course, this is correct, but the interpretation of a as an “anomaly” is erroneous. As the magnetic moment is due to the current Iφ in φ direction the result must be related to the mass mφ in φ direction rather than to the rest mass m0. What really is measured is
μ = e ℏ 2 m φ ( 1 + a ) (9.13)
The rotational energy component of a rotating object cannot be measured by means of acceleration or gravitational experiments. The result of such measurements is always m0c2. The difference between mφc2 and m0c2 must be interpreted as an additional rotational kinetic energy
E k i n = m 0 c 2 a = 592 .581 eV (9.14)
Consequently is
m φ = m 0 ( 1 + a ) (9.15)
and
μ = e ℏ 2 m φ ( 1 + a ) = e ℏ 2 m 0 ( 1 + a ) ( 1 + a ) (9.16)
Thus, the real situation is as it has to be in terms of classical physics:
μ = e ℏ 2 m 0 = μ B (9.17)
where μ B is the Bohr magneton.
From (9.11) and (9.17) arises the law of conservation of angular momentum:
β φ R = ƛ C = ℏ m 0 c (9.18)
The existence of the kinetic energy Ekin according to (9.14) is important for the angular momentum
J = r m v = ℏ / 2 (9.19)
When (9.19) is applied to the electron it must be observed that only the rotating part
m d y n = E m a g + E k i n c 2 (9.20)
of the mass contributes to the angular momentum. The combination of E m a g and E k i n may appear strange, but it is proven that they are equivalent [
m 0 [ ß φ 1 β φ 2 1 + ß φ 1 β φ 2 − ß φ 1 β φ 2 1 + ß φ 1 β φ 2 ƛ C r 1 4 q 1 q 2 e 2 α 2 π + a ] c β φ R = ℏ 2 (9.21)
where v in (9.19) is substituted from v φ = c β φ and r is substituted according to (6.10). When (9.18) is used in (9.21) the result is
2 ß φ 1 β φ 2 1 + ß φ 1 β φ 2 − 2 ß φ 1 β φ 2 1 + ß φ 1 β φ 2 ƛ C r 1 4 q 1 q 2 e 2 α 2 π + 2 a = 1 (9.22)
and finally
1 − ß φ 1 β φ 2 1 + ß φ 1 β φ 2 + 2 ß φ 1 β φ 2 1 + ß φ 1 β φ 2 ƛ C r 1 4 q 1 q 2 e 2 α 2 π = 2 a (9.23)
If the kinetic energy E k i n = a m 0 c 2 would not have been considered, the right side of (9.23) would have been zero. This would require ß φ 1 β φ 2 > 1 . So, it would have been impossible to present a realistic model of the electron. Thus, the results given in Chapter 7 and in [
Comparison of (9.23) and (9.8) leads to the important result
E e l − E m a g = 2 a m 0 c 2 (9.24)
As the electromagnetic field energy is
E e l + E m a g = m 0 c 2 (9.25)
the following fundamental statements can be formulated:
E e l = ( 1 2 + a ) m 0 c 2 (9.26)
E m a g = ( 1 2 − a ) m 0 c 2 (9.27)
E d y n = E m a g + E k i n = 1 2 m 0 c 2 (9.28)
E t o t a l = E e l + E d y n = ( 1 + a ) m 0 c 2 (9.29)
Now the basic Equations (9.28) and (9.18) explain very simply:
J = m d y n v φ R = 1 2 m 0 c β φ R = 1 2 m 0 c ℏ m 0 c (9.30)
or
J = ℏ 2 (9.31)
Further analysis and evaluation of (9.23) require some reasonable assumptions.
Assumption 1
The charge density on the two circular loops is assumed to be the same:
q 1 r 1 = q 2 r 2 (9.32)
In (7.9) was already stated that the charge condition of the electron is
q 1 + q 2 = e (9.33)
From (9.32) and (9.33) follows
q 1 e = 1 1 + r 2 r 1 (9.34)
and
q 2 e = r 2 r 1 1 + r 2 r 1 (9.35)
The factor
4 q 1 e q 2 e = 4 r 2 r 1 ( 1 + r 2 r 1 ) 2 (9.36)
will turn out to be very close to unity.
Assumption 2
The angular velocities w1 and w2 of the rotating charges q1 and q2 are equal:
ω 2 = ω 1 (9.37)
This means
β φ 2 = β φ 1 r 2 r 1 (9.38)
and
β φ 1 β φ 2 = β φ 1 2 r 2 r 1 (9.39)
When this is applied to (8.3) the average value β φ becomes
β ϕ = β φ 1 β φ 2 = β φ 1 r 2 r 1 (9.40)
Together with (9.18) and (6.10) this leads to
ƛ C r 1 = β φ 1 β φ 2 r 2 r 1 (9.41)
The assumptions resulting in (9.36), (9.40, and (9.41) allow to transform (9.23) to
1 − ß φ 2 1 + ß φ 2 + 2 ß φ 3 1 + ß φ 2 4 ( r 2 r 1 ) 3 2 ( 1 + r 2 r 1 ) 2 α 2 π = 2 a (9.42)
It turns out that a is exclusively given by the ratio r 2 / r 1 because β φ is also determined by r 2 / r 1 via the force balance condition according to (8.4). It allows to calculate for each value of r 2 / r 1 the respective value of β φ = β φ b . Then a can be calculated according to (9.42).
δ β φ = 1 − β φ (9.43)
as a function of
δ r = 1 − r 2 / r 1 . (9.44)
It is meaningful that a > δ β φ as δ r < 2.471 × 10 − 4 and that a < δ β φ as δ r > 2.472 × 10 − 4 .
By iteration the crossover point can be found at extremely high accuracy. At
δ r = 2.471604094 × 10 − 4 (9.45)
respectively
r 2 / r 1 = 0.9997528396 (9.46)
is
δ β φ = a = 1.159246494 × 10 − 3 (9.47)
There is no rigorous proof that this significant point also describes the physical reality, but any other choice would be arbitrary. There is no additional condition that in (9.23) could justify another statement then δ β φ = a . Nevertheless, the following approximation is given as the only reasonable alternative: When in (9.42) according to (9.43) β φ 2 is substituted from ( 1 − δ β φ ) 2 ≈ 1 − 2 δ β φ , then it becomes
δ β φ 1 − δ β φ + 1 − 3 δ β φ 1 − δ β φ 4 ( r 2 r 1 ) 3 2 ( 1 + r 2 r 1 ) 2 α 2 π = 2 a (9.48)
As δ β φ ≪ 1 and r 2 ≈ r 1 it is obvious that
1 − 3 δ β φ 1 − δ β φ 4 ( r 2 r 1 ) 3 2 ( 1 + r 2 r 1 ) 2 α 2 π ≈ α 2 π ≈ a (9.49)
Consequently is
δ β φ 1 − δ β φ ≈ a ≈ α 2 π (9.50)
and
δ β φ ≈ α 2 π 1 + α 2 π = 1 .1600624200 55 × 10 − 3 (9.51)
This corresponds to
r 2 / r 1 = 0.9997526447 (9.52)
and
a = 1.159653926 × 10 − 3 (9.53)
The experimental value of the electron’s magnetic moment “anomaly” is [
a e = 1.159652181 × 10 − 3 (9.54)
Despite of the fact that the values of (9.53) and (9.54) are very close together the solution according to (9.46) and (9.47) is preferred because it looks more natural and more systematic, and it avoids the introduction of an additional parameter. Thus, the charge ±e and the angular momentum ћ/2 are sufficient to derive the new universal constants δ r = 1 − r 2 / r 1 and δ β φ = a . They define the basic structure of fundamental particles and ensure their stability (force balance). The first one is a shape factor while the second one is necessary for the energy inventory. They are dimensionless numbers similar to the fine-structure constant a and are fundamental physical constants because they hold for all respective particles and are invariant against the average radius R = ℏ / ( m 0 v φ ) respectively the rest mass m0. Moreover it means that a fundamental particle may widely increase (not decrease!) its radius and decrease (not increase!) its rotational velocity vφ, e.g. when it joins other charged particles, as far as according to (9.18) the angular momentum conservation condition
R = ƛ C β φ = ƛ C 1 − a = ℏ m 0 v φ (9.55)
is fulfilled. A well-known example is the one-electron atom with the nucleus charge eZ where β φ ≈ α Z and R ≈ a 0 = ƛ C / α . Moreover, it confirms that fundamental particles show mass-like and relativistic behavior, while their internal structure is non-relativistic and essentially field-like.
One important message is that all free fundamental particles having the charge ±e and the angular momentum ћ/2 must possess the same values of r2/r1 and δ β φ = a , regardless of their rest mass m0. There is high evidence that this also holds for the free proton and the free antiproton. It is an important discovery that the magnetic moment “anomaly” a indicates the existence of the kinetic energy E k i n = a m 0 c 2 . It is necessary to make the presented model perfect, and it reveals that fundamental particles internally are not purely field-like such as photons and not (essentially) mass-like such as atoms. They represent a special kind of matter in between. Nevertheless, the fundamental particle is completely an object of classical physics including the non-anomalous magnetic moment μ = e ℏ / ( 2 m 0 ) = e v φ R / 2 . The circular current loop model is two-dimensional. It is the simplest one that is consistent and represents all details of fundamental particles at extremely high accuracy. According to (9.54) the rotational velocity vφe of the electron is given by
β φ e = v φ e c = 1 − a e = 0.998840347819 (9.56)
The respective value of the presented model is
β φ = v ϕ c = 1 − a = 0.998840753506 (9.57)
The maximum and minimum values of the twin loop model are
β φ 1 = β φ r 1 r 2 = 0.998964213363 (9.58)
β φ 2 = β φ r 2 r 1 = 0.99871708907 (9.59)
For practical use it is not necessary to emphasize the twin loop character of a fundamental particle. It will be sufficient to calculate with the average radius according to (9.18)
R = ƛ C β ϕ ≈ ƛ C = ℏ m 0 c (9.60)
The author declares no conflicts of interest regarding the publication of this paper.
Landvogt, G. (2024) The Circular Current Loop as a Model of Fundamental Particles. Journal of High Energy Physics, Gravitation and Cosmology, 10, 303-328. https://doi.org/10.4236/jhepgc.2024.101023