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An attempt to obtain a new theoretical derivation of the size of the electron microparticle has been done. To this purpose first the Maxwell equation for the electron current has been examined for the case of the one-electron current present in the Bohr model of the hydrogen atom. It has been shown that the equation is satisfied on condition that the microstructure properties of the electron particle are taken into account. In the next step, the quanta of the magnetic field characteristic for the Bohr atom and the electron time periods specific for the electron current along the orbits were substituted in place of parameters entering the classical Oersted equation. This gives an expression for the cross-section radius of the orbits not much different than results for the radius of the electron microparticle obtained in a former electron theory

In order to obtain any classical property of an elementary microparticle, say the electron, an approach combined of both the quantum and classical physical laws seems to be necessary. In the present case―when the classical size parameter of the electron particle is aimed to be deduced―the quantum aspects can be provided by the Bohr model of the hydrogen atom. Here―for any quantum state―we have a definite orbital motion of a single electron in the electrostatic field of a positively charged proton nucleus. The motion―beyond of its orbital track ―has well-defined velocity and energy parameters. However, in order to make use of the equations of classical electrodynamics, especially the Maxwell equations, knowledge of the magnetic field―together with the electric field―in the atom seems to be necessary.

However both of the Maxwell equations―that consider the change of the magnetic induction and that concern the electric line current only (by assuming that the displacement current can be neglected)―take into account the time parameter on different footing [

If we assume the Bohr theory as valid for the hydrogen atom, the electric current given by the one-electron orbital motion is fully stationary for any chosen quantum level n. The velocity of the current composed of a single electron particle is [

since

is the orbit length and

is the time period necessary to travel the distance

The aim of the present paper is, in the first step, to point out that the Maxwell equation concerning the electric line current

where

To this purpose we consider the quanta of the magnetic field

In the next step, in Section 4, the quanta of the magnetic field

Here the path of

It will be found that the indices n in (6) cancel together leaving the formula for the cross-section radius r of the orbit independent of n. This r is expected to approach the radius

The Maxwell equation is written briefly in the form

but it seems to be more convenient to apply an integral form of (8) which is

The magnetic field

The

where

This is an effect of the Lorentz force law in which the wave-vector

The last step in (11a) is due to the fact that the magnetic field is normal to the velocity vector along the orbit. For a full circulation time

identical with (11).

A substitution of

from which we obtain

It is interesting to note that

where

is the absolute electron energy in state n [

is the area occupied by the electron orbit in that state. Here the constants

This gives

which yields the square value of

Usually, when the electron is considered as a charged particle having the radius

In effect

The current (5) is composed, first, of the volume V occupied by the electron particle, so

next the same current should move within a tube having a cross-section area equal approximately to

Since the velocity

The left-hand side of (9) is

A difference between the both sides of (9), or (19) and (20), is represented by the factor of 3/2.

Any current is associated with the magnetic field and the lines of that field circumvent the line of the current. We assume that at the distance r from the center of the current cross-section area the field is

from which we obtain

In effect the cross-section radius of the orbit which approximately can be identified with the radius of the electron microparticle becomes

This result―evidently independent of the index n―is not much different than that given by the well-known formula (16) and the formula derived in [

The Maxwell equations, when applied to electrons, usually neglect the microsize parameters of the electron particle. In Appendix we demonstrate that the Poynting vector

One of aims of the present paper was to indicate that these parameters can be essential in making the Maxwell equations satisfied for a given problem.

The Maxwell equation for the electric current has been examined for the case of the one-electron current present in the Bohr model of the hydrogen atom. It has been shown, for the magnetic field induced by the current, that the equation is satisfied on condition that the microstructure parameter of the electron radius

But the size of the electron microradius can be of importance for itself, especially in quantum electrodynamics, so its calculation becomes a useful task. In the next step of the paper, a substitution of

The result in (25), which can be identified with the size of the radius

Stanisław Olszewski, (2016) Size of the Electron Microparticle Calculated from the Oersted Law. Journal of Modern Physics,07,1297-1303. doi: 10.4236/jmp.2016.711114

The value of the Poynting vector for the energy emission in the hydrogen atom can be easily calculated with the aid of

where

we obtain the absolute value of the Poynting vector

The decrement of

The time period

The emission rate (A4) can be compared with that given by the Joule-Lenz approach [

We see that decrease of (A6) with increase of n is much more rapid than decrease of (A4). Moreover we have

which makes any (A6) much smaller than (A4). The reason of the discrepancy seems to be the choice of S equal to (A2) instead of a much smaller S equal to the toroidal surface enclosing the orbit of the electron circulation about the nucleus.

In any way the Joule-Lenz approximation for the energy emission rate in the hydrogen atom works well as it is indicated by its comparison with the quantum-mechanical theory (see [