Received 31 May 2016; accepted 5 July 2016; published 8 July 2016
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  -  . In the first equation the time action is reduced to the use of a short interval representing the derivative of the magnetic flux with respect to time; in the second equation the time interval enters solely the current velocity, which can be assumed to be a constant term along an arbitrarily long quantity of time. This produces a stationary electric current whose charge density satisfies the equation
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 
is the orbit length and
is the time period necessary to travel the distance about the atomic nucleus.
The aim of the present paper is, in the first step, to point out that the Maxwell equation concerning the electric line current
where is the electron velocity and is the density of the electron particle, can be satisfied only when the microstructure properties of the electron particle are taken into account.
To this purpose we consider the quanta of the magnetic field neglected in the original Bohr model   . These quanta―introduced in Section 2―seem to be of importance (see  ) because they lead to the quanta of the magnetic flux identical with those known experimentally since a long time in superconductors   . Moreover, a combination of the electric and magnetic field present in the atom gives the Poynting vector which approximately provides us with a proper rate of the energy emission due to the process of the electron transition between two quantum levels  . In Section 3 we show that the quanta fulfill the Maxwell equation for the electric current with a satisfactory accuracy.
In the next step, in Section 4, the quanta of the magnetic field―which are due to the electron orbital motion in the atom―are substituted into the equation representing the Oersted law:
Here the path of circumvents the circular cross-section area of the orbit, the area is assumed to have the radius r: it defines the surface of the orbital conductor at which the magnetic field is equal to (see Figure 43 in Ref.  ). Since any orbit can be occupied solely by a single electron particle, r should be independent of the quantum index n. Symbol in (6) represents the current intensity which is coupled with the circulation time period of the electron along the orbit n by the formula
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 of the electron particle moving along the orbit.
2. Maxwell Equation for the Electric Current and the Magnetic Field in the Hydrogen Atom
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 in (9) is a constant term for a given n; see below. The length is given in (3). It should be noted that the integral on the left of (9) does not concern the dot product of and, but is the integral of extended over the line having the length  .
The can be obtained as a result of a constant electric current on the level n if we note that the current is surrounding periodically the nucleus with the frequency
where is given in (4). On the other hand, the is coupled with by the formula  
This is an effect of the Lorentz force law in which the wave-vector of the electron particle satisfies the relation
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 we have. Since we obtain from (11a) the relation
identical with (11).
A substitution of from (4) into (10) gives together with (11) the equation
from which we obtain
It is interesting to note that in (12) can be obtained also from the theory of the cyclotron resonance in metals  . We have the relation 
is the absolute electron energy in state n  , and
is the area occupied by the electron orbit in that state. Here the constants and replace respectively the energy interval and area interval in the real space admitted in course of the change of the quantum state n. A substitution of from (13) and from (14) valid for the hydrogen atom gives together with the formula (11) for the relation:
which yields the square value of in (12).
3. Current Analysis Done with the Aid of a Microstructure Parameter of the Electron Particle
Usually, when the electron is considered as a charged particle having the radius, the potential energy of the charge extended on a spherical surface is assumed to be approximately equal to the rest energy of the electron   :
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 for a given n is a constant [see (2)], we obtain for the right-hand side of (9) the formula
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.
4. The Quanta of the Magnetic Field and Time Periods Entering the Oersted Law Give the Radius of the Electron Microparticle
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 for any orbit n. In this case the formulae (6) and (7) give the relation
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 can be connected with the rest energy of the electron, therefore also with the radius r or.
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 is explicitly taken into account. Here an earlier result can be pointed out that the magnetic field strength entering the Poynting vector constructed for the rate of the emission spectrum in the hydrogen atom cannot be reproduced from the Biot-Savart law unless the electron microstructure radius is applied in the calculations (see    ).
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 and―characte- ristic respectively for the magnetic field quanta and time periods of the electron circulation in the atom―into the Oersted formula gives the expression for the cross-section radius of the electron orbit equal to
The result in (25), which can be identified with the size of the radius of the electron microparticle, does not differ much from the well-known formula (16) as well as the formula quoted in (24).
Appendix: Rest Energy of the Electron Mass Connected with the Poynting Vector of the Hydrogen Atom
The value of the Poynting vector for the energy emission in the hydrogen atom can be easily calculated with the aid of in (12) and the absolute value of the vector of the electric field intensity:
where is the orbit radius applied in (2) and (3). For the spherical surface S having the radius, so
we obtain the absolute value of the Poynting vector equal to
The decrement of due to the change of the quantum state into n for large n is equal to
The time period of the electron circulation in state n of the hydrogen atom [see (4)] is entering the denominator of the last term in (A3) and (A4). This is a characteristic substitution of the transition time between two neighbouring quantum levels and n in the hydrogen atom obtained in a quantum aspect of the Joule-Lenz energy dissipation theory   :
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   ).