**Stable charged particles with cylindrical symmetry**
Probably, the basis of electrons and other leptons is the

**E**,

**H**,

**V** field in the form of closed rings of energy flow and velocity vector. The structure of the particle is not similar to the classical "infinitely thin" circuit with an electric current, where on the elements of the ring the electric and magnetic field differ from zero, due to the fields created by other parts of the circular current.

On the line of "main circuit", the electric and magnetic field is zero, whereas the charge density (~div

**E**) is close to the local maximum.

In computer modeling in a cylindrical coordinate system, the following field values can be a good initial approximation (withs2 = R2 + ρ2 + z2):

vector potential

**A** = Aφ ~ ρ3 / s5

Hρ ~ - ∂Aφ/∂z ~ 5 · ρ3 · z / s7

Hφ = 0

Hz ~ ∂Aφ/∂ρ + Aφ / ρ ~ 4 · ρ2 / s5 - 5 · ρ4 / s7

Jρ = 0

Jφ = ∂Hρ/∂z - ∂Hz/∂ρ ~ - 8 · ρ / s5 + 10 · ρ3 / s7 + 35 · ρ3 · R2 / s9

Jz = 0

scalar potential a ~ ρ4 / s5

Eρ ~ - ∂a/∂ρ ~ - 4 · ρ3 / s5 + 5 · ρ5 / s7

Eφ = 0

Ez ~ - ∂a/∂z ~ 5 · ρ4 · z / s7

div

**E** = ∂Eρ/∂ρ + Eρ / ρ + ∂Ez/∂z

~ - 16 · ρ2 / s5 + 20 · ρ4 / s7 + 35 · ρ4 · R2 / s9

Note that Eρ · Hρ + Ez · Hz = 0,

**E** is perpendicular to

**H** everywhere.

Near the center of the particle is a region where the values div

**E** and rot

**H** are opposite in sign to those found in the rest of space. Meanwhile,

**V** has the same sign everywhere. If the conditional magnetic dipole is directed along the z-axis, with a positive multiplier for

**A** and

**H**, and the total charge of the particle is positive, then near the center there will be a region with a negative divergence

**E** and a negative rotor

**H**, but at the great distance these values are positive.

The velocity is positive everywhere, that is, it is directed counterclockwise with the direction of the z-axis towards us and the x-axis (the start of the counting φ) to the right. div

**E** and rot

**H** must change the sign synchronously so that equality is observed:

**E**′ = 1/ε0 · rot

**H** - grad EV -

**V** · div

**E** = 0

since in a more or less stable particle all fields derivatives in time are zero.

EV = 0,

**V** is perpendicular to

**E** and

**H**, that is, we are talking about a mutually perpendicular triple of vectors in any combination.

A model with a field arrangement closer to the z-axis, for example:

**A** = Aφ ~ ρ / s3

Hρ ~ 3 · ρ · z / s5

Hφ = 0

Hz ~ 2 / s3 - 3 · ρ2 / s5

a ~ ρ2 / s3

Eρ ~ - 2 · ρ / s3 + 3 · ρ3 / s5

Eφ = 0

Ez ~ 3 · ρ2 · z / s5

where on the z-axis there is a pronounced maximum of Hz

and

**J** = Jφ = ∂Hρ/∂z - ∂Hz/∂ρ ~ 15 · R2 · ρ / s7

does not fit, because rot

**H** is positive everywhere, and div

**E** changes sign in the central part. Models with a spherically symmetric scalar potential and electric field are even more inadequate:

a ~ 1 / s

Eρ ~ ρ / s3

Eφ = 0

Ez ~ z / s3

div

**E** ~ 3 ·R2 / s5

there

**E** is not even perpendicular to

**H**, if we take

**A** and

**H** from the previous model.

If

**V**′ ~ (

**D** · V2 - [

**H** ×

**V**]) · div

**E**
then the multiplier V2 prevents the destruction of the particle due to electrical repulsion near the z-axis, although exactly onto it can be also

**D** = 0. With

**V** = Vφ, Vρ = 0 and Vz = 0, there must inevitably be a zone with zero velocity near the z-axis, since the values of the existing quantities in the physical world are finite and only smooth functions with continuous derivatives are permissible.

Precise solutions in degrees of s for real lepton-like particles seem impossible, numerical simulations are required. In the first time after setting the initial approximation, occurs a rapid adaptation of the fields to more accurate values, further stability depends on the adequacy of the model and the accumulation of numerical errors.

Let's try to estimate the magnitudes of what order are the fields at a considerable distance from the z-axis in the perpendicular plane (z = 0, s ≈ r, r2 = ρ2 + z2). From experimental data and works on classical physics it is known that

**E** = Eρ ~ 1 / r2,

**H** = Hz ~ - 1 / r3, u ≈ E2 ~ 1 / r4

**D** · V2 - [

**H** ×

**V**] will tend to zero if

**V** = Vφ ~ 1 / r

The same order of Vφ ~ 1 / r follows from the equation

**W** - u ·

**V** = 0, given that

**W** = Wφ= - Eρ · Hz ~ 1 / r5

**Further directions of research**
All the above mentioned equations are linear in

**E** and

**H**. That is, when these vectors are both multiplied by the same number, they remain true. Since the particles observed in the physical world have strictly defined charges and masses (internal energies), it is logical to assume that the expression

**V**′ may contain nonlinear terms with relatively small factors. Although due to statistical factors, greater resistance to random disturbances, some field formations may be more stable than others.

Author of this work: Mykola (Nicholas) Ivannikov

E-mail:

nikoljob@yahoo.com
Article written 31.07.2022