2026-08-05
Geometric origin of electric and magnetic fields
Why does the same charge at rest create an electric field, but when moving nearby, a magnetic field is also detected? At first glance, these appear to be two different physical entities. However, in the proposed model, there is no gap between them: both fields are derived from a single external branch of the operator \(J\). While the wave center is at rest, this branch manifests entirely as an electric field. As soon as the wave center begins to move, a second, perpendicular projection automatically arises within the same branch. This is what we call the magnetic field.
The purpose of this part of the article is to demonstrate the very principle of the generation of electric and magnetic components from the operator \(J\) and from the geometry of its mapping into three-dimensional space. Here, we are not yet aiming to construct a complete electrodynamics in all possible regimes. It is important to establish the initial relationship: the energy gradient determines the scale of the field, the operator \(J\) determines the ratio of its projections, and spatial geometry imparts directions to these projections.
In previous work, the Coulomb interaction was obtained as an energy gradient between two charges, arising due to the splitting of wave orbits. However, the force characterizes the already completed interaction of the source with another particle. The field must describe the source itself even before a test charge appears nearby. Therefore, the first step is simple: we need to separate the source property from the magnitude of the test charge. We will first do this for a stationary elementary charge, then move on to an arbitrary charge, and finally allow the center of its wave to move.
The main idea is that the one-dimensional space of the model already defines a separate direction of propagation of the force. After mapping into three-dimensional space, this direction becomes a single field line. While the wave center is stationary, the external motion parameter is \(b=0\), and the external branch of the operator has only an electric projection. When the center begins to move along the external coordinate \(\mathfrak e\), the parameter \(b\) becomes greater than zero. Then, the operator \(J\) itself determines the magnitude of the new projection, and the transition to 3D gives it a direction.
Thus, the magnetic field is not introduced here by a separate postulate. We will attempt to obtain it as a geometric consequence of the motion of the existing wave center. It is this transition from \(b=0\) to \(b>0\) that is the main content of the article.
However, this section does not consider a free electromagnetic wave escaping from a source, but a localized wave regime of a particle—a closed standing wave in a certain limited region of space. Its energy is not transferred to infinity, but remains within the system and is redistributed between two mutually perpendicular projections of a single state. It is for this regime that conservation of the full scale of the external branch is further introduced.
Thus, the construction sequence has the form
\[\tag{1} \boxed{ \text{energy gradient} \;\longrightarrow\; E_0(R) \;\longrightarrow\; \mathbf E_0(R) \;\longrightarrow\; J(a,b) \;\longrightarrow\; \mathbf E(R,b),\;\mathbf B(R,b) } \] 1. Electric Field from an Energy Gradient
Let's first consider two elementary charges separated by a distance of \(R\). The interaction energy for them was previously obtained \[\tag{2} U(R)=\frac{\alpha_{\mathrm{fs}}\hbar c}{R}. \]
The sign of the energy is determined by the relative orientation of the charges, but to construct the field scale, we will first consider the absolute value. The force is equal to the negative gradient of the potential energy:
\[\tag{3} F(R)=-\frac{dU}{dR}, \qquad |F(R)|=\frac{\alpha_{\mathrm{fs}}\hbar c}{R^2}. \] The electric field of a source is defined as the force per unit test charge. For a test charge with modulus \(e\), we obtain
\[\tag{4} E_0(R)=\frac{|F(R)|}{e} =\frac{\alpha_{\mathrm{fs}}\hbar c}{eR^2}. \] The index \(0\) here denotes the field of a source at rest, i.e., the state \(b=0\). Using the definition of the fine structure constant,
\[\tag{5} \alpha_{\mathrm{fs}} =\frac{e^2}{4\pi\varepsilon_0\hbar c}, \] Formula (4) can be reduced to standard form:
\[\tag{6} \boxed{ E_0(R) =\frac{\alpha_{\mathrm{fs}}\hbar c}{eR^2} =\frac{e}{4\pi\varepsilon_0R^2} }. \] Therefore, the energy gradient already found contains the electric field strength. The transition from force to field does not require the introduction of a new physical quantity.We: it is sufficient to divide the energy gradient by the test charge.
The meaning of this operation can be expressed quite simply. The force depends on both interacting forces—the source and the test charge. After dividing by the test charge, only the source's characteristic at a given point in space remains. We call this characteristic the electric field.
2. One-Dimensional Direction as a Line of Force
According to the first postulate of the model, the motion of a point always occurs in a single spatial dimension and in time. Therefore, in the original one-dimensional space, the field can only be directed along the coordinate \(R\). The found function \(E_0(R)\) describes the magnitude of the field precisely along this particular direction: \[ E_0(R) = \frac{\alpha_{\mathrm{fs}}\hbar c}{eR^2}. \]
When transitioning to our three-dimensional reality, the one-dimensional coordinate does not disappear. It is mapped to an arbitrarily oriented radial direction, defined by a unit vector.
\[\tag{7} \widehat{\mathbf R}=\frac{\mathbf R}{R}, \qquad |\widehat{\mathbf R}|=1. \] Therefore, the transition to three-dimensional notation is accomplished by simply adding a direction:
\[\tag{8} \boxed{ \mathbf E_0(\mathbf R) =E_0(R)\,\widehat{\mathbf R} }. \] Each choice of \(\widehat{\mathbf R}\) corresponds to one one-dimensional field line. The set of all equivalent radial directions forms the spherically symmetric field of a point charge. Thus, a field line in the three-dimensional description is an image of the original one-dimensional model space, and the field is the set of all such images.
It is important that the formula for the field magnitude does not change with this transition. The law \(1/R^2\) has already been obtained in the one-dimensional calculation. The three-dimensional mapping only tells us which direction the field is directed at each point. Therefore, the factor \(\widehat{\mathbf R}\) is responsible for the direction, but does not create additional energy or change the magnitude of \(E_0(R)\).
3. Field of an Arbitrary Charge
Let the source have charge \(q\), which can be represented as the algebraic sum of elementary charges:
\[\tag{9} q=Ne, \qquad N\in\mathbb R. \] The coefficient \(N=q/e\) contains both the magnitude and sign of the charge. Therefore, the field of an arbitrary point source is
\[\tag{10} \mathbf E_0(\mathbf R) =\frac{q}{e} \frac{\alpha_{\mathrm{fs}}\hbar c}{eR^2} \widehat{\mathbf R}. \] Taking into account formula (5), we finally obtain
\[\tag{11} \boxed{ \mathbf E_0(\mathbf R) =\frac{q}{4\pi\varepsilon_0R^2} \widehat{\mathbf R} }. \] If \(q>0\), the field vector is directed away from the source. If \(q<0\), the formula sign reverses the vector's direction, and the field is directed toward the source. Thus, the sign of the charge determines the orientation of the field line, while the geometric dependence \(1/R^2\) remains unchanged.
In other words, the transition from \(e\) to \(q\) does not change the mechanism by which the field arises. It simply multiplies the number of identical elementary contributions and specifies their overall orientation by the sign of the charge.
4. Electric field and operator J at b=0
The complete state of a particle in split geometry is described by the operator
\[\tag{12} \boxed{ J(a,b) =\jmath^a(-\jmath)^b =\mathfrak e e^{i\pi b} +\bar{\mathfrak e}e^{i\pi a} }, \] where the idempotents satisfy the relations
\[\tag{13} \mathfrak e^2=\mathfrak e, \qquad \bar{\mathfrak e}^{\,2}=\bar{\mathfrak e}, \qquad \mathfrak e\bar{\mathfrak e}=0, \qquad \mathfrak e+\bar{\mathfrak e}=1. \] Parameter \(a\) describes the internal state of the particle, and parameter \(b\) describes its external motion:
\[\tag{14} a=\varpi t, \qquad \pi\varpi=\omega, \qquad b=\frac{\arcsin\beta}{\pi}, \qquad \beta=\frac{v}{c}. \] For a charge at rest, \(v=0\), therefore \(b=0\). The operator takes the form
\[\tag{15} J(a,0) =\mathfrak e +\bar{\mathfrak e}e^{i\omega t}. \] In this notation, the component \(\mathfrak e\) corresponds to the external spatial projection, and \(\bar{\mathfrak e}e^{i\omega t}\) corresponds to the internal periodic state of the particle. Let's associate the dimensional field of a charge at rest with the operator:
\[\tag{16} \mathcal E_0(R,t) =E_0(R) \left( \mathfrak e +\bar{\mathfrak e}e^{i\omega t} \right). \] The observable external part of this operator is equal to \(E_0(R)\mathfrak e\). At \(b=0\), it is directed entirely along the electric field line, and there is no perpendicular component.
Here, it is useful to imagine the internal wave as a periodic process with its own center. The factor \(\bar{\mathfrak e}e^{i\omega t}\)describes the internal rotation of the wave, and the branch \(\mathfrak e\) defines the position of its center in the external one-dimensional space. At \(b=0\), this center is not translated along the external coordinate. Therefore, the external part has no motion capable of generating a second, transverse direction of the field.
5. Splitting of the External Projection at b>0
Now let's impart an external velocity of \(v>0\) to the charge. This means that the center of the internal wave no longer remains at a single point but begins to move along the external coordinate \(\mathfrak e\). According to the definition of the model, such translational motion is described by the parameter \(b>0\). Therefore, the change in \(b\) is not an additional property of the field, but a record of the motion of the wave center itself.
In the operator \(J\), this motion only changes the external branch. The internal frequency \(\omega\) is still in the branch \(\bar{\mathfrak e}\), while the translation of the center is recorded by the factor \(e^{i\pi b}\) at \(\mathfrak e\):
\[\tag{17} J(a,b) =\mathfrak e e^{i\pi b} +\bar{\mathfrak e}e^{i\omega t}. \] Let's decompose the moving outer branch into two ordinary projections:
\[\tag{18} \mathfrak e e^{i\pi b} =\mathfrak e\cos(\pi b) +i\mathfrak e\sin(\pi b). \] This is where the main thing happens. While the wave center is stationary, \(b=0\), therefore \(\sin(\pi b)=0\): the outer branch retains only the original electric projection. But as soon as the wave center begins to move along the coordinate \(\mathfrak e\), \(b>0\), and the sine part is no longer zero. A second term, \(i\mathfrak e\sin(\pi b)\), automatically appears in the outer branch. It does not need to be introduced as a separate postulate or added to \(J\) from the outside—it is initially contained in the moving particle operator.
From the definition of the parameter \(b\) it follows
\[\tag{19} \sin(\pi b)=\beta, \qquad \cos(\pi b) =\sqrt{1-\beta^2} =\frac{1}{\gamma}, \qquad \gamma=\frac{1}{\sqrt{1-\beta^2}}. \] Therefore, the outer branch is represented as the sum of two orthogonal projections:
\[\tag{20} \boxed{ \mathfrak e e^{i\pi b} =\frac{\mathfrak e}{\gamma} +i\mathfrak e\beta }. \] At \(b=0\), the first projection is equal to one, and the second is equal to zero. As the velocity increases, the first projection decreases as \(1/\gamma\), while the perpendicular projection increases as \(\beta\). The absolute value of the outer branch remains unchanged.
In other words, the operator \(J\) itself gives the magnitudes of both outer projections. The cosine part \(\cos(\pi b)=1/\gamma\) shows what fraction of the original field remains electric. The sine part \(\sin(\pi b)=\beta=v/c\) shows the magnitude of the new transverse projection. While we remain in the one-dimensional space of the model, this is not yet the magnetic field vector, but only the magnitude of its future spatial component, already obtained from \(J\).
This can be understood as a rotation of one quantity without changing its length. The electric fraction is equal to the cosine of the angle \(\pi b\), the magnetic fraction is equal to the sine of the same angle. Therefore, the motion does not add a new independent quantity to the field, but rather redistributes the original external projection.
For further derivation, it is not the entire complex notation that is particularly important to us, but rather the sine part of the external branch. Let's denote it by \(\operatorname{Im}J_{\mathrm{ext}}\). Then the operator directly yields
\[\tag{21} \operatorname{Im}J_{\mathrm{ext}} =\mathfrak e\sin(\pi b) =\mathfrak e\beta. \] At \(b=0\), this component disappears. At \(b>0\), it appears automatically and has a relative magnitude of \(\beta=v/c\). Consequently, the operator \(J\) already indicates what fraction of the original electric field should be transferred to the new spatial projection. However, the one-dimensional operator itself cannot yet indicate its direction in three-dimensional space. To do this, we must take into account the geometry of the source's motion and the geometry of the electric field line.
6. Geometric Direction of the Magnetic Field
Result (21) yields the magnitude of the new projection: \(\beta=\sin(\pi b)\). Now let's find its direction. After transitioning from a single field line of the model to a 3D one, the moving source defines two independent spatial vectors:
\[ \boldsymbol{\beta}=\frac{\mathbf v}{c}, \qquad \mathbf E_0(\mathbf R)=E_0(R)\,\widehat{\mathbf R}. \] The first vector defines the direction of motion of the wave center, the second, the direction of the existing electric field line. Together they form an oriented plane. The only direction perpendicular to both vectors is defined by the normal to this plane:
\[ \boldsymbol{\beta}\times\widehat{\mathbf R}. \] The resulting direction is perpendicular to both the velocity and the electric field line. It changes sign at \(\mathbf v\to-\mathbf v\), vanishes at \(b=0\), since then \(\beta=0\), and vanishes at \(\mathbf v\parallel\mathbf R\). All these properties follow from a single vector product.
This direction has all the necessary properties of the magnetic field of a moving charge: it is perpendicular to the motion of the source and the electric field line; changes sign at \(\mathbf v\to-\mathbf v\); vanishes at \(b=0\), since then \(\boldsymbol{\beta}=0\); and vanishes along the motion when \(\mathbf v\parallel\mathbf R\).
Now the two parts of the derivation can be combined without an additional postulate. The operator \(J\) specifies the magnitude of the transverse projection \(\sin(\pi b)=\beta\), and the transition to 3D specifies its direction \(\boldsymbol{\beta}\times\widehat{\mathbf R}\). Therefore, the magnetic field is a transverse spatial mapping of the pre-existing electric field:
Now we can connect magnitude and direction. The original scale of the field is \(E_0(R)\); the operator extracts from it the transverse fraction \(\beta\), and the vector product specifies its orientation. Taking into account the factor \(1/c\), necessary for the dimension of magnetic induction, we obtain
\[\tag{22} \boxed{ \mathbf B(\mathbf R,b) =\frac{1}{c} \boldsymbol{\beta}\times\mathbf E_0(\mathbf R) =\frac{1}{c^2} \mathbf v\times\mathbf E_0(\mathbf R) }. \] Thus, the magnetic field is a transverse reflection of the existing rest field \(\mathbf E_0=E_0\widehat{\mathbf R}\): \(\mathbf B=\boldsymbol{\beta}\times\mathbf E_0/c=\mathbf v\times\mathbf E_0/c^2\). The factor \(1/c\) ensures the correct dimensionality of the magnetic induction. The absolute value of the field is
\[\tag{23} B(R,b,\theta) =\frac{E_0(R)}{c} \beta\sin\theta, \] where \(\theta\) is the angle between \(\mathbf v\) and \(\mathbf R\). For a transverse field line, when \(\theta=\pi/2\), we obtain
\[\tag{24} \boxed{ cB(R,b)=E_0(R)\sin(\pi b)=E_0(R)\beta }. \] For an arbitrary charge \(q\), substituting field (11) leads to the expression
\[\tag{25} \mathbf B(\mathbf R) =\frac{q}{4\pi\varepsilon_0c^2} \frac{\mathbf v\times\widehat{\mathbf R}}{R^2}. \] Since \(\mu_0\varepsilon_0c^2=1\), it takes on the familiar form of the field of a uniformly and slowly moving point charge:
\[\tag{26} \boxed{ \mathbf B(\mathbf R) =\frac{\mu_0q}{4\pi} \frac{\mathbf v\times\widehat{\mathbf R}}{R^2} }. \] Formula (26) shows that the geometric mapping of the sine projection of the operator reproduces the nonrelativistic form of the magnetic field of a moving charge. Subsequently, it is the integration of this elementary result over the distribution of moving charges that should lead to the electric current field.
7. The Electric Component of a Moving Charge
The cosine part of the outer branch preserves the direction \(\widehat{\mathbf R}\). In the model mapping under consideration, the electric component of a moving charge is therefore equal to
\[\tag{27} \boxed{ \mathbf E(\mathbf R,b) =E_0(R)\cos(\pi b)\widehat{\mathbf R} =\frac{E_0(R)}{\gamma}\widehat{\mathbf R} }. \] The complete operator of the electric part, taking into account the constant internal branch, can be written as
\[\tag{28} \boxed{ \mathcal E(R,t;b) =E_0(R) \left[ \frac{\mathfrak e}{\gamma} +\bar{\mathfrak e}e^{i\omega t} \right] }. \] The magnetic projection corresponds to the operator
\[\tag{29} \boxed{ c\mathcal B(R;b) =iE_0(R)\mathfrak e\beta }. \] Formulas (28) and (29) express the main idea of the proposed construction. During motion, a new, independent energy entity does not emerge. Part of the original external projection transforms from the electric component into the perpendicular magnetic component.
It's important not to confuse the two notations here. \(E_0(R)\) is the full initial scale of the field of a charge at rest. The quantity \(E(R,b)=E_0(R)/\gamma\) is its electric projection after the start of motion. Therefore, the magnetic field is directly expressed through \(E_0\). If, however, it is expressed through the now-reduced field \(E\), an additional factor \(\gamma\) appears.
At low speed, this is especially evident. Then \(1/\gamma\approx1\), so the electric part is almost indistinguishable from the field of a charge at rest, and the magnetic part already appears in first order in velocity: \(cB\approx E_0\beta\). This is why, under everyday non-relativistic conditions, the electric field can remain almost unchanged, although a weak magnetic field already exists.
The limit of this result should be emphasized. Formula (27) is a consequence of the adopted split-operator mapping and applies to the transverse geometry of the model under consideration. It is not identical to the standardA dart relativistic transformation of the field of a point charge, in which the spatial distribution of the electric field depends on the reference frame and the observation angle. Here, the model's own projection law is constructed, which must then be separately verified for consistency with full electrodynamics.
8. Preservation of the External Projection in Localized Mode
Now let us clarify the physical meaning of the construction under consideration. The formulas below apply to the localized closed mode of a standing wave. In this mode, the electric and magnetic components are not two independently generated fields: they are quadrature projections of a single state, between which an unchanged full scale can be redistributed. Therefore, conservation should be checked not for each projection separately, but for their combined absolute value.
To calculate the energy value of complex projections, each must be multiplied by its complex conjugate. Literally squaring the magnetic operator would yield a minus sign due to \(i^2=-1\) and therefore would not correspond to the square of its modulus.
For the electric operator, from formula (28), we obtain
\[\tag{30} \begin{aligned} \mathcal E\mathcal E^* &=E_0^2(R) \left( \frac{\mathfrak e}{\gamma} +\bar{\mathfrak e}e^{i\omega t} \right) \left( \frac{\mathfrak e}{\gamma} +\bar{\mathfrak e}e^{-i\omega t} \right) \ &=E_0^2(R) \left( \frac{\mathfrak e}{\gamma^2} +\bar{\mathfrak e} \right). \end{aligned} \] For the magnetic component we have
\[\tag{31} \begin{aligned} c^2\mathcal B\mathcal B^* &=E_0^2(R) \left(i\mathfrak e\beta\right) \left(-i\mathfrak e\beta\right) \ &=E_0^2(R)\mathfrak e\beta^2. \end{aligned} \] The sum of the two contributions is
\[\tag{32} \mathcal E\mathcal E^* +c^2\mathcal B\mathcal B^* =E_0^2(R) \left[ \mathfrak e \left( \frac{1}{\gamma^2}+\beta^2 \right) +\bar{\mathfrak e} \right]. \] But the definition of the Lorentz factor implies the identity
\[\tag{33} \frac{1}{\gamma^2}+\beta^2 =1-\beta^2+\beta^2 =1. \] Taking into account \(\mathfrak e+\bar{\mathfrak e}=1\), we obtain the conservation law for the complete operator field:
\[\tag{34} \boxed{ \mathcal E\mathcal E^* +c^2\mathcal B\mathcal B^* =E_0^2(R) }. \] In the external subspace, the same result has a particularly simple form:
\[\tag{35} \boxed{ E^2(R,b)+c^2B^2(R,b)=E_0^2(R) }. \] This equality should be understood as an internal energy invariant of the proposed model: it expresses the conservation of the modulus of the outer branch when it is rotated by an angle of \(\pi b\). It is not a standard Lorentz invariant of the electromagnetic tensor, which contains the difference \(E^2 - c^2B^2\). Here, the sum of squares arises because the electric and magnetic components are considered as orthogonal projections of a single normalized state.
It is also necessary to distinguish this operator equality from the pointwise description of an arbitrary standing electromagnetic wave. In a conventional resonator, the electric and magnetic components are shifted not only in time but also in space, so their local sum of squares need not be constant at each point. The total energy of the closed volume is constant. In our shorthand notation, \(E\) and \(cB\) play the role of the effective amplitudes of the two projections of the localized regime, and formula (35) expresses the conservation of its overall operator scale.
9. Unified Geometric Notation
For transverse geometry, both external projections can be collected in a single complex formula:
\[\tag{36} \boxed{ E_0(R)e^{i\pi b} =E(R,b)+i\,cB(R,b) }. \] Its real part defines the electric field, and its imaginary part defines the magnetic field:
\[\tag{37} \boxed{ \begin{aligned} E(R,b) &=E_0(R)\cos(\pi b) =\frac{E_0(R)}{\gamma}, \\ cB(R,b) &=E_0(R)\sin(\pi b) =E_0(R)\beta. \end{aligned} } \] Multiplication of formula (36) by its complex conjugate immediately returns the original field scale:
\[\tag{38} \boxed{ \left|E+i\,cB\right|^2 =E^2+c^2B^2 =E_0^2 }. \] Thus, the electric and magnetic components are related in the same way as the cosine and sine projections of a single rotation. The parameter \(b\) does not change the overall field magnitude, but only redistributes it between two mutually perpendicular directions.
Such a quadrature notation is natural precisely for a closed regime, in which the energy is not carried outward, but remains localized and can transition between two forms of manifestation. For a free traveling electromagnetic wave the situation is different: \(\mathbf E\) and \(\mathbf B\) change in phase and are related by the relation \(\mathbf B=\widehat{\mathbf k}\times\mathbf E/c\). Therefore, formulas (36)–(38) inThis article should not be directly applied to the radiation field.
It is important to emphasize that the parameter \(\beta\) does not create any new energy or an independent additional field in the system. As the wave center moves, only the orientation of the external component of the operator \(J\) changes: its constant full scale is redistributed between the longitudinal electric and transverse magnetic projections. Therefore, the magnetic field at \(\beta>0\) should be understood not as new energy added by the motion, but as a spatial manifestation of the existing field in a different direction. In other words, motion reorients \(J\) but does not increase its magnitude: its observed projections change, while the corresponding full magnitude remains unchanged.
Conclusions
This section posed a fundamental, rather than a definitive, goal: to demonstrate how the concepts of \(E\) and \(B\) can arise from a single operator \(J\) and from the geometry of the transition from a one-dimensional direction to three-dimensional space. The electric component is derived from the energy gradient along the field line, the magnitude of the magnetic projection is derived from the sinusoidal component of the outer branch of \(J\), and its spatial direction is derived from the normal to the plane formed by the source's motion and the electric field line.
The electric field in the proposed model is directly derived from the previously determined energy gradient. Dividing the interaction force between two elementary charges by a test charge yields the field strength \(E_0(R)=\alpha_{\mathrm{fs}}\hbar c/(eR^2)\), which, after using the definition of the fine-structure constant, coincides with the Coulomb field of a point charge.
In the original one-dimensional space, this function describes the field along a single spatial direction. When mapped into 3D, a unit vector \(\widehat{\mathbf R}\) is added to it, and a single one-dimensional direction becomes a field line. The set of all its possible orientations forms the three-dimensional source field.
When \(b=0\), the outer branch of the operator completely coincides with the electric projection. At \(b>0\), it decomposes into a real part \(1/\gamma=\cos(\pi b)\) and an imaginary part \(\beta=\sin(\pi b)\). In three-dimensional space, the second part takes the direction \(\widehat{\mathbf v}\times\widehat{\mathbf R}\) and is observed as a magnetic field.
Thus, the magnetic field arises not as an independent complement to the electric field, but as a perpendicular projection of the same external branch of the state of a moving charge. Motion redistributes the original scale of the field between the two components, preserving their full square modulus:
\[\tag{39} \boxed{ b=0 \;\longrightarrow\; \mathbf E_0, \qquad b>0 \;\longrightarrow\; \mathbf E\perp\mathbf B, \qquad E^2+c^2B^2=E_0^2 }. \] Formula (39) in this section pertains to an idealized localized standing wave regime in a closed region. It describes not a free radiation flux, but an internal redistribution of a single scale between the electric and magnetic projections. The parameter \(\beta\) does not create new energy or add a second independent field to the system: it reorients the external branch of \(J\), changing the observed projections while maintaining a constant full modulus.
The obtained result forms the basis for the following sections of the paper. In them, the local construct will be successively expanded: first, from the field of a single moving charge to the field of charge flow and the magnetic force between moving particles, then to the laws of electric current, spatially distributed standing modes, and, finally, to the difference between a closed particle field and a free traveling electromagnetic wave. Thus, the specific geometric conclusion of this article should translate into a more general picture of electrical and magnetic phenomena.