2026-08-16
The geometric meaning of the capacitance and inductance of an electron
One of the most obvious consequences of the Wave Electricity theory is the ability to represent an electron as a classical ideal oscillatory circuit, with self-capacitance \(C_e\) and inductance \(L_e\). However, this is not a lumped circuit consisting of a separate capacitor and coil, but a wave system distributed along a closed trajectory. The electric and magnetic components belong to a single internal wave, the wavelength of which coincides with the length of the circuit. In the absence of resistance and radiation, the total energy of this state is not expended but can circulate within the electron indefinitely.
A well-known macroscopic analogy is provided by experiments with closed superconducting rings. After current is induced and the external source is disconnected, the current continues to circulate in the circuit for an extremely long time, since the vanishingly small resistance results in virtually no loss of stored energy. The electron model takes this principle to its ideal limit: a closed wave requires no external source, and its electric and magnetic components form a stable eigenmode. The similarity to a superconductor relates precisely to the undamped circulation of energy; the electron itself is considered not as a material ring, but as a primary closed wave state.
In the main part of the work, the electron's capacitance and inductance were found using well-known electrical engineering relationships. This method is convenient for calculations, but it can create the impression that the electron is an ordinary lumped oscillatory circuit: a separate capacitor connected to a separate coil. In the Wave Electricity model, the meaning of these quantities is different.
The electron is considered a localized closed wave. The electric and magnetic components are not concentrated in two separate elements, but are distributed along the entire closed path. Therefore, it is more correct to speak of a distributed wave contour, and to understand the quantities \(C_e\) and \(L_e\) as its equivalent, or modal, parameters.
This clarification does not invalidate formulas (2.7), (2.14), and (2.15) of the main article. On the contrary, it will be shown below that they follow from the geometry of a closed wave and from the condition that the full period of the electron's natural wave fits within the length of the internal contour.
1. The Electron as a Closed Distributed Contour
Let \(s\) be a one-dimensional coordinate measured along the electron's internal path. In the simplest ring mapping, the length of this path is equal to the circumference of a circle of radius \(r_e\):
\[\tag{A.1} L_e^{(\Gamma)}=2\pi r_e. \] The superscript \((\Gamma)\) here denotes the length of the geometric contour and prevents it from being confused with the inductance \(L_e\). The fundamental motion of the wave remains one-dimensional: at any given moment, it moves only along the coordinate \(s\). The circle appears as a three-dimensional representation of this one-dimensional closed process.
The fundamental eigenmode occurs when the wavelength coincides with the length of the closed path:
\[\tag{A.2} \boxed{ \lambda_e=L_e^{(\Gamma)}=2\pi r_e. } \] This equality has a more fundamental meaning than a simple check of the numerical result. It is a condition for phase closure: after traversing the entire circle, the wave acquires a phase of \(2\pi\) and is consistent with itself. The wave number is therefore equal to
\[\tag{A.3} k_e=\frac{2\pi}{\lambda_e}=\frac{1}{r_e}. \] If an internal wave propagates with velocity \(c\), then its circular and ordinary frequencies are directly obtained from geometry:
\[\tag{A.4} \boxed{ \omega_e=ck_e=\frac{c}{r_e}, \qquad \nu_e=\frac{\omega_e}{2\pi} =\frac{c}{2\pi r_e}. } \] Thus, in the geometric description, the frequency initially arises from the length of the closed path. Thomson's formula then does not introduce a new frequency, but expresses the same eigenmode through its equivalent electric and magnetic parameters.
2. Why does the electron have a spherical capacitance?
At a fundamental level, each field line in the model is a separate one-dimensional direction. For an elementary charge at rest, the field strength along such a direction was previously obtained through the electron's internal energy scale:
\[\tag{A.5} E_0(R) =\frac{\alpha_{\mathrm{fs}}\hbar c}{eR^2} =\frac{e}{4\pi\varepsilon_0R^2}. \] The last equality follows from the definition of the fine structure constant. When the aggregateThe potential of all possible one-dimensional directions is mapped into ordinary three-dimensional space, forming a spherically symmetric field. Therefore, the classical formula for the sphere appears here not because the electron is declared to be a metallic ball, but because its external field has spherical symmetry.
The potential of the boundary of the internal geometry relative to infinity is found by integrating the field strength:
\[\tag{A.6} \varphi_e =\int_{r_e}^{\infty}E_0(R)\,dR =\frac{e}{4\pi\varepsilon_0r_e}. \] Capacitance is defined as the coefficient between a charge and the potential it creates. Therefore,
\[\tag{A.7} \boxed{ C_e=\frac{e}{\varphi_e} =4\pi\varepsilon_0r_e. } \] We have obtained formula (2.7) from the main article, but this time as a consequence of the electron's geometric field. The quantity \(C_e\) characterizes the ability of the entire closed wave structure to create an external electric potential. It does not imply the presence of a separate capacitor inside the electron.
Using the relationship between the classical radius and the electron mass,
\[\tag{A.8} r_e =\frac{e^2}{4\pi\varepsilon_0m_ec^2} =\frac{\alpha_{\mathrm{fs}}\hbar}{m_ec}, \] we can immediately obtain the potential of the electron state:
\[\tag{A.9} \boxed{ \varphi_e=\frac{m_ec^2}{e} \approx 5.11\cdot10^5\, \text{B}. } \] Therefore, the product of the charge and this potential is equal to the total mass-energy of the electron: \(e\varphi_e=m_ec^2\). Below we will see how this energy is divided between the electric and magnetic components of the distributed mode.
3. Inductance as a Magnetic Parameter of the Eigenmode
In a distributed circuit, inductance characterizes the magnetic side of the same wave process, the electric side of which is described by capacitance. For equivalent eigenmode parameters, the resonance relationship is preserved.
\[\tag{A.10} \omega_e^2L_eC_e=1. \] But now both primary geometric quantities are known: the frequency \(\omega_e=c/r_e\) is obtained from the wave closure, and the capacitance \(C_e=4\pi\varepsilon_0r_e\) is obtained from the external spherical field. Therefore, the inductance no longer needs to be introduced as an independent assumption:
\[\tag{A.11} L_e =\frac{1}{\omega_e^2C_e} =\frac{r_e^2}{c^2}\, \frac{1}{4\pi\varepsilon_0r_e}. \] Taking into account the equality \(\varepsilon_0c^2=1/\mu_0\) we obtain
\[\tag{A.12} \boxed{ L_e=\frac{\mu_0r_e}{4\pi}. } \] This is exactly formula (2.14) of the main article. However, its physical meaning has now been clarified: \(L_e\) is not the usual geometric inductance of a thin wire ring, which would also depend on the thickness of the wire, but the effective inductance of the electron's full eigenmode.
From (A.7) and (A.12), we again obtain
\[\tag{A.13} \boxed{ \frac{1}{\sqrt{L_eC_e}}=\frac{c}{r_e}=\omega_e, \qquad \nu_e=\frac{1}{2\pi\sqrt{L_eC_e}}. } \] Thus, the geometric condition \(\lambda_e=2\pi r_e\) and Thomson's electrical formula describe the same process from two perspectives. The first equality shows the spatial closure of the wave, while the second shows the relationship between the electric and magnetic components of its equivalent mode.
4. Two Meanings of Internal Current
When comparing the wave model with a ring, it is important to distinguish between two currents. They relate to the same periodic process but answer different questions. If they are not separated, the factor \(2\pi\) appears unnoticed in the formulas.
Phase current. In the main article, the complex charge is written as \(q_e=q_0\exp(i\omega_et)\). Its derivative characterizes the rate of change of the phase state:
\[\tag{A.14} I_{\omega}(t)=\frac{dq_e}{dt}, \qquad I_{\omega0}=\omega_eq_0. \] When \(q_0=e\), its amplitude is
\[\tag{A.15} \boxed{ I_{\omega0}=\omega_ee =\frac{ec}{r_e} \approx1.70\cdot10^4\, \text{A}. } \] This value is used in the formulas for magnetic energy and equivalent wave impedance in the main article. It is associated with a phase change of one radian and is therefore naturally expressed in terms of the circular frequency \(\omega_e\).
Circulation current. If we choose a stationary section of the ring and ask how much charge passes through it in one complete revolution, the current should be determined in terms of the usual frequency. The period of a complete revolution is
\[\tag{A.16} T_e=\frac{1}{\nu_e}=\frac{2\pi r_e}{c}. \] During this time, one full charge \(e\) passes through the cross section, therefore
\[\tag{A.17} \boxed{ I_{\nu}=\frac{e}{T_e} =e\nu_e =\frac{ec}{2\pi r_e} \approx2.71\cdot10^3\, \text{A}. } \] The two currents are related by a simple relationship:
\[\tag{A.18} \boxed{ I_{\omega0}=2\pi I_{\nu}. } \] Therefore, these are not two independent physical currents inside the electron. \(I_{\omega0}\) measures the rate of phase change of the wave state, while \(I_{\nu}\) measures the total charge transfer through a given cross section in one closed cycle. The first current is needed to describe the magnetic part of the eigenmode, the second is convenient for describing the circulation of energy around the ring.
5. Electric and Magnetic Halves of Energy
The electric part of the energy of a distributed mode is expressed through its equivalent capacitance and potential:
\[\tag{A.19} W_E=\frac{C_e\varphi_e^2}{2} =\frac{e\varphi_e}{2} =\frac{m_ec^2}{2}. \] For the magnetic component, the phase current \(I_{\omega0}=\omega_ee\) is used, since it is the one that enters into the modal relationship with the inductance. Then
\[\tag{A.20} W_H =\frac{L_eI_{\omega0}^2}{2} =\frac{1}{2} \frac{1}{\omega_e^2C_e} \omega_e^2e^2 =\frac{e^2}{2C_e}. \] Since \(e/C_e=\varphi_e\), we obtain the second equal half:
\[\tag{A.21} \boxed{ W_H=\frac{e\varphi_e}{2} =\frac{m_ec^2}{2}. } \] The total energy of the closed wave is therefore equal to
\[\tag{A.22} \boxed{ W_e=W_E+W_H =e\varphi_e =m_ec^2. } \] In a conventional lumped (LC) circuit, the instantaneous electric energy of the capacitor is converted into the magnetic energy of the coil and vice versa. Here, a more general complex waveform is used. The electric and magnetic components are distributed along a closed path and are mutually complementary components of a single normalized mode. Formulas (A.19) and (A.21) refer to their modal amplitudes, not to two separate elements that reach independent maxima at the same time.
The distributed nature of the circuit preserves the physical meaning of sum (A.22): the energy does not switch between two local stores, but circulates continuously along the closed wave. As long as the internal path remains closed, this circulation is not active power and does not imply energy emission outward.
6. Total Energy as Flux per Revolution
The same total energy can be obtained without separately calculating the electric and magnetic halves. For this, the reversal current \(I_{\nu}\) is used. The product of the potential and this current characterizes the internal energy flow through a selected cross-section of the ring:
\[\tag{A.23} P_{\Gamma}=\varphi_eI_{\nu}. \] Here \(P_{\Gamma}\) cannot be understood as the active power consumed by the electron or released into the surrounding space. The potential \(\varphi_e\) is measured relative to infinity and is not the voltage drop across the ring resistance. The quantity \(P_{\Gamma}\) describes the rate of internal transfer of existing energy along a closed circuit.
During one complete period, the charge \(I_{\nu}T_e=e\) passes through the cross-section. Therefore, the energy transferred in one cycle is equal to
\[\tag{A.24} W_e =P_{\Gamma}T_e =\varphi_eI_{\nu}T_e =e\varphi_e. \] Taking into account (A.9), we finally obtain
\[\tag{A.25} \boxed{ W_e =\varphi_eI_{\nu}T_e =\frac{\varphi_eI_{\nu}}{\nu_e} =m_ec^2. } \] This notation clearly demonstrates the wave meaning of the electron's mass-energy: it is equal to the internal energy flux multiplied by the time of one complete geometric closure. The enormous numerical value of the internal power is compensated for by the extremely short orbital period, and no energy escapes.
7. Why split orbits do not disrupt the derivation
A simple circle is an effective representation of the internal contour. In more detailed geometry, the electron wave traverses two closely spaced split orbits lying in the same physical plane, as well as the transitions connecting them. These are not two independent rings or two separate currents: a single wave sequentially traverses the entire two-branch path.
Let us denote the total length of this path by \(L_{\Gamma}\). For the fundamental eigenmode, the general closure condition takes the form
\[\tag{A.26} \boxed{ \lambda_{\Gamma}=L_{\Gamma}, \qquad T_{\Gamma}=\frac{L_{\Gamma}}{c}, \qquad \nu_{\Gamma}=\frac{c}{L_{\Gamma}}. } \] The reversal current changes with the period:
\[\tag{A.27} I_{\Gamma} =\frac{e}{T_{\Gamma}} =\frac{ec}{L_{\Gamma}}. \] However, the product of the current and the period does not depend on the shape and length of the circuit:
\[\tag{A.28} \boxed{ I_{\Gamma}T_{\Gamma}=e. } \] Therefore, splitting can change the transit time, internal frequency, and magnitude of the current through a selected cross-section, but does not change the energy of the full cycle:
\[\tag{A.29} \boxed{ W_e =\varphi_eI_{\Gamma}T_{\Gamma} =e\varphi_e =m_ec^2. } \] At a fundamental level, geometry is replaced by the effectThe active ring is represented by the equality \(\lambda_e=2\pi r_e\). At a more precise level, the full split path \(L_{\Gamma}\) is taken into account. It is precisely the small deviation of the detailed path from the effective circle that can contribute to spin, frequency, and anomalous corrections without destroying the fundamental energy invariant.
The capacitance also retains the fundamental form \(C_e=4\pi\varepsilon_0r_e\), since it is determined by the far spherically symmetric field and the effective boundary of the internal geometry. The fine structure of split orbits can create higher-order corrections but does not change the leading law of the external field.
8. Summary
The capacitance, inductance, potential, and electron current form a single, consistent system. But physically, this is not a set of individual radiotechnical components. All these parameters are different ways of describing a single closed distributed wave.
\[\tag{A.30} \boxed{ \begin{gathered} \lambda_e=2\pi r_e, \qquad \omega_e=\frac{c}{r_e},\\ C_e=4\pi\varepsilon_0r_e, \qquad L_e=\frac{\mu_0r_e}{4\pi},\\ W_E=W_H=\frac{m_ec^2}{2}, \qquad W_e=m_ec^2. \end{gathered} } \] Thus, the formulas of the main article receive a geometric basis. Capacitance describes the electrical manifestation of a closed wave in external space, inductance describes its magnetic component, frequency is determined by the phase closure condition, and mass-energy is equal to the energy circulating through the internal circuit in one complete period.
All of the above considerations apply equally to the positron. Since the masses of the electron and positron are identical, they correspond to identical radius, capacitance, inductance, natural frequency, and total energy of the closed wave. The difference lies not in the structure of the distributed circuit, but in the opposite orientation of its internal wave state, which changes the sign of the charge, potential, and direction of the associated current. Therefore, the positron can be considered the same ideal distributed oscillatory circuit, but with the opposite phase orientation of the circulating wave; in this case, the energy equality \(W=mc^2\) and the relationships between \(C\), \(L\) and the natural frequency are completely preserved.