When developing our electron model, we cannot ignore its mathematical description. Therefore, this part of the work will be devoted exclusively to mathematics, but it will provide an understanding of some unusual patterns that, generally speaking, are not at all obvious. As a result, we will even obtain a formula that explains not only this model, but also any other, more practical one. For example, it will be able to explain where the extra charges on objects come from when an alternating or rapidly changing current flows near them.

First, let's define the terms and some simplifications in the formulas. Many of them are given in the work [1], to which we will constantly refer. Here and below we assume that the electron charge, convection current in the electron and its potential depend on time: \[ q_e = q_e(t), \quad I_e = I_e(t), \quad \varphi_e = \varphi_e(t) \] We define the circular frequency of the electron through the usual standard method \[ \omega = 2\pi \nu_{e} \] from which we immediately obtain the electron wavelength: \[ \lambda_e = {c \over \nu_{e}} = 2\pi r_e \tag{2.1}\] Here \(c\) is the speed of light, and \(r_e\) is the classical radius of the electron. It follows that \[ \omega = {c \over r_e} \tag{2.2}\] These were the initial definitions necessary for further exposition. We will obtain more significant quantities and patterns associated with them further.

We proceed from the fact that the charge is primary, and all electrodynamic and electrostatic fields are formed from it. The charge equation is decisive for all further mathematics in this work. We write it as a wave^(*): \[ q_e = q_0 \exp(i \omega t) \tag{2.3}\] Current, as is known, is a change in charge over time \[ I_e = {d q_e \over dt} = I_0 \exp(i \omega t + \pi/2), \quad I_0 = \omega q_0 \tag{2.4}\] We obtain the first important relationship: \[ I_0 = \omega q_0 \tag{2.5}\] From (2.5), taking into account (2.2), we obtain another expression that will be used in subsequent calculations: \[ I_0 = {q_0 c \over r_e} \tag{2.6}\] The physical and mathematical meaning of (2.6) differs from the model, in in which a point charge rotates around a fixed point, and where this expression would be: \(I = q c/(2\pi r) \). This corresponds to the convection current in the electron obtained in [1], equal to: \(I_0 = 1.7\cdot 10^4\) (A).

The electron capacitance is found in the classical way [3] \[ C_e = 4\pi \varepsilon_0 r_e, \quad \varepsilon=1 \tag{2.7}\] and is \(3.14\cdot 10^{−25}\) (F). We check this value according to [1].

Based on the coefficient of proportionality between potential and charge [3], which is called capacitance, and expression (2.3), we can write the equation for the electron potential: \[ \varphi_e = {q_e \over C_e} = {q_0 \over C_e} \exp(i \omega t) \tag{2.8}\] Here we also select the amplitude value of the electron potential \[ \varphi_0 = {q_0 \over C_e} \tag{2.9}\] which will be \(5.11\cdot 10^{5}\) (V). We will also check this in [1].

Since we consider the electron as an ideal oscillatory circuit, its inductance can be found using the classical Faraday law, as applied to electric circuits: \[ \varphi_e = - L_e {d I_e \over d t} \tag{2.10}\] From where we can derive its value: \[ L_e = - {\varphi_e \over {I_e}^{'}_t} \tag{2.11}\] The derivative of the current is found from (2.4) \[ {I_e}^{'}_t = - I_0 \omega\, \exp(i \omega t) \tag{2.12}\] and the potential is taken from (2.). Then the electron inductance is found as follows: \[ L_e = {q_0 \over C_e I_0 \omega} \tag{2.13}\] If we take into account that \(\varepsilon_0 = 1 / (\mu_0 c^2)\), and formulas (2.2-2.6), we can finally obtain the electron inductance: \[ L_e = {\mu_0 r_e \over 4 \pi} \tag{2.14}\] which is \(2.82\cdot 10^{−22}\) (H). This formula, and the resulting value, also correspond to the work [1].

An ideal oscillatory circuit can give us a ready-made relationship in this case. For a resonant oscillatory circuit in radio engineering, the Thomson formula is used [4]: \[ \nu_e = {1 \over 2\pi \sqrt{L_e C_e}} \tag{2.15}\] We already know all the data in it, and we will immediately get the result. The electron frequency according to Thomson is equal to \(1.69\cdot 10^{22}\) (Hz). We calculate its wavelength according to the classical definition: \[ \lambda_e = {c \over \nu_e} \tag{2.16}\] The wavelength of an electron is \(1.77\cdot 10^{-14}\) (m), which corresponds to [1]. Now we can check all our previous calculations if we compare the wavelength and the circumference of the electron: \[ \lambda_e = 2\pi r_e \tag{2.17}\] This result should have been obtained from the initial formula (2.1). Everything matched, so we are on the right track :)