Earlier we looked at obtaining some electron parameters for our mathematical model. In the third part of this work, we need to get a general idea of ​​resonance and the energy generated by it, and again obtain a charge from the time-varying convection current of the electron, which will close our problem.

Let's check the model of an ideal oscillatory circuit, which is an electron. This approach is proposed in this note and has not failed yet. Let's find the wave resistance of the electron [1]: \[ Z_e = \sqrt{L_e \over C_e} \tag{3.1}\] It turns out to be equal to \(30\) Ohm. But it is equal to the reactance of the capacitance and inductance [2]: \[ Z_{C} = {1 \over \omega C_e}, \quad Z_{L} = \omega L_e, \quad Z_e = Z_{C} = Z_{L} \tag{3.2}\] In addition, we know that the current lags behind the voltage by \(\pi/2\) from the formulas (2.4, 2.8). All together this means that there is a resonance in the electron, similar to that which occurs in an oscillatory circuit. And since our circuit is ideal, there are no losses in it, which means that the energy in such a circuit can oscillate practically forever. But the main thing is that in this case the electron does not emit anything outward, which is what needed to be proven.

Let's check how energy is stored in such an ideal oscillatory circuit. We must obtain two energies: from the potential in the capacitor and from the current in the inductance, the sum of which should, in theory, be equal to the Einstein mass-energy of the electron: \[ W_e = m_e c^2 \tag{3.3}\] We can take the data from the previously derived formulas and check that the sum of the energies in the capacitor and inductance of the electron is equal to this energy: \[ {C_e \varphi_0^2 \over 2} + {L_e I_0^2 \over 2} = m_e c^2 \tag{3.4}\] If we take the actual values ​​of current and voltage by their absolute value, the result will be the same. Such energy can also be called the total energy in the oscillatory circuit. Thus, the mass of an electron can be represented as its capacitance and inductance, as well as the potential and current located in it! By the way, the same assumptions lead to the coefficient-free laws of Coulomb and Ampere.

Also, based on this approach, we can assume that any elementary particles can be described using the parameters of capacitance, inductance, potential and current. For each particle, the values ​​of these parameters will, of course, be their own.

In this subsection, we want to close the problem raised in this section, and again, after two differentiations, obtain a charge. This is shown schematically in the following figure, and we will see how it looks strictly from a mathematical point of view further.

And indeed, after the first derivative of charge with respect to time, we get a current (2.4) and the corresponding magnetic field. After the second derivative, a second magnetic field arises, which again generates a charge for us (Fig. 2). To describe mathematically such a charge circulation and its derivatives, we have developed the following formula: \[ q_e = - {\mu\mu_0 \varepsilon\varepsilon_0 \over 4\pi} \iint \limits_S {\partial^2 q_e \over \partial t^2} d S \tag{3.5}\] or the same, but in another form: \[ q_e = - {\mu \varepsilon \over 4\pi c^2} \iint \limits_S {\partial I_e \over \partial t} d S \tag{3.6}\] where \(S\) is the area of ​​the covered surface, \(\mu, \mu_0\) is the relative and absolute magnetic permeability, and \(\varepsilon, \varepsilon_0\) -- relative and absolute permittivity. If we take into account that an electrostatic field has the property of inducing electric charges on surrounding objects, then such a formula can explain their appearance, if alternating or rapidly changing currents flow near them. And the faster the current changes over time, the greater the charge it induces on surrounding objects. Moreover, the author does not focus only on charges with a negative sign :)

Let's check formula (3.6). From the expressions forcurrent (2.4-2.6) it follows: \[ {\partial I_e \over \partial t} = - \omega^2 q_0 \exp(i \omega t) = - {c^2 \over r_e^2} q_0 \exp(i \omega t) \tag{3.7}\] The area of ​​the covered surface for a sphere is found in the classical way: \[ S = 4\pi r_e^2 \tag{3.8}\] Since in this case there are no internal dependencies in the integrands, and the relative permeabilities are equal to unity: \(\mu = \varepsilon = 1\), then we can substitute the previously obtained expressions into the right-hand side of formula (3.6): \[ \left( -{1 \over 4\pi c^2} \right) \left( -{c^2 \over r_e^2} q_0 \exp(i \omega t) \right) 4\pi r_e^2 = q_e \tag{3.9}\] That is, we again obtained the charge in its original definition (2.3).

Formulas (3.5-3.6) explain another fundamental point -- why there is a minus sign before the expression in Faraday's law [3]. You will not find a mathematical justification for the minus sign in this law in any textbook. In our work it is obtained automatically, due to double differentiation \(\exp(i \omega t)\) with respect to time, which is shown, for example, in (2.12). For the same reason, the minus is also in formulas (3.5-3.6), which is needed there to maintain the polarity of the resulting charge.