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# Geometrical Calculation of Superluminal Longitudinal Electric Wave Velocity with Toroidal Coordinates

Arvin D Sharma

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## To cite this version:

Arvin D Sharma. Geometrical Calculation of Superluminal Longitudinal Electric Wave Velocity with Toroidal Coordinates. 2025. hal-04988614

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Distributed under a Creative Commons Attribution 4.0 International License
# Geometrical Calculation of Superluminal Longitudinal Electric Wave Velocity with Toroidal Coordinates

Arvin D. Sharma

New Mexico State University

1780 E University Ave, Las Cruces, NM 88003, United States

arvin@nmsu.edu

**Abstract**—The theoretical calculation of superluminal velocities for longitudinal electric waves is presented with a far-field average of $1.52c$ and a maximum value of $1.57077c$, which represents the limiting value as distance tends to infinity. Nikola Tesla's documented result of $1.5708c$ is within .002% of this calculated maximum and agrees with the theoretical result. Both results approximate to $\frac{\pi}{2}c = 1.570796c$, indicating the upper bound for the velocity of longitudinal electric waves as $\frac{\pi}{2}c$. The presented method utilizes geometrical calculations based on the toroidal geometry of the electromagnetic field represented in the bipolar coordinate system of two dimensions and the novel postulate of equal time traveled for both the transverse electromagnetic and longitudinal electromagnetic waves.

## Index Terms—Longitudinal Electric Wave, Superluminal Wireless Transmission

The experimental researches of electricity by Michael Faraday led to the discovery of electromagnetic induction which forms the basis of electromagnetic energy generation and wireless communication systems today. By placing particles of particular types around electrically charged and magnetized materials, the respective induced field can be seen [Fig. 1], [Fig. 2].

To Faraday, these lines were physical parts of reality, meaning, the constituents of space itself polarize and align themselves to form these lines, just like the iron filings and grass seeds. The idea of physical lines of force were later further developed by J.J. Thompson and James Clerk Maxwell [1], [2]. This conception and physics defining the medium is the precursor to Quantum Electrodynamics (QED) and Quantum Field Theory (QFT), which stipulates such electromagnetic field interactions to be mediated by virtual photons, hypothetical particles akin to the photon, which itself is said to have zero mass.

The early developments of electrical theory had the benefit of utilizing the intuition developed from fluid mechanics in order to develop and apply the calculus of flow to electrical problems. This created the vector calculus operators of today, $\nabla F$, $\nabla \cdot F$ and $\nabla \times F$, which are rooted in the theory of flux depicting the flow of fluids. It was after these operators were developed, for the science of electricity, in which they found direct application in the mechanical science of fluids. In electromagnetic theory, this notation is applied to form Maxwell equations, which are defined by Maxwell originally in component form,

PLATE V

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Par. 1931, R. (two thirds scale)

<figure><img src="image_5.png" /><figcaption>Fig. 2. Electric lines of force visualized using grass seeds [7]</figcaption></figure>

in [4], and written using the vector calculus operators by Heaviside in [3]. These four equations, defined by Heaviside, known as Maxwell's equations, are written for free space, in modern notation as

$$ \nabla \times E = -\frac{\partial B}{\partial t} \qquad (1) $$

$$ \nabla \times B = \mu_0 \epsilon_0 \frac{\partial E}{\partial t} \qquad (2) $$

$$ \nabla \cdot E = 0 \qquad (3) $$

$$ \nabla \cdot B = 0 \qquad (4) $$

Where $E$ the electric field intensity, described in free space as $E = \epsilon_0 D$ where $D$ is the electric flux density, $D = \frac{\Psi_e}{A}$, where $\Psi_e$ is the electric flux over the area $A$, and $\epsilon_0$ is the constant of proportionality to the measured
electric field strength based on the medium. $\Psi_e$ is also known as the charge of the medium where $\Psi_e = q$ with units Coulombs [5]. $B$ is the magnetic field intensity where $B = \frac{\Phi_m}{A}$. Equations (1,2) describe the dynamic interaction between the electric and magnetic fields while (3,4) describe the relation between the electric and magnetic fields and the medium.

The displacement current is known as the missing link which allowed for electromagnetic wave propagation to be defined, assumed by Maxwell based on the hypothesized symmetry between the electric and magnetic fields. That is, it has been shown experimentally by Faraday, in his law of induction, a time variation in the magnetic field induces an electromotive force in the form of a circulating electric field, for Maxwell the time variation in the electric field must then also induce a circulating magnetic field. This is described in (3) which allows for the in phase oscillation of energy between the two fields. The displacement current is unique in electrical theory since it depicts what is analogous to the conduction current within the wire, which is modeled from electron flow, in free space, where no charge is present. The displacement current represents energy exchange to and from the electric field, which may result in electromagnetic radiation, or other field interactions. It forms the electrical component of Longitudinal Electric Waves, the subject of which remains in its infancy. At a high level, this describes a means to convey wireless power in a non-electromagnetic manner. A simple proof is to show the Poynting vectors formed by the time varying electric field and circulating magnetic field, which carry energy toward the center and away from the field. If one adopts the dipole radiation model, these Poynting vectors are instead oriented radially outward as a result of the formation of circulating electric fields which close as the polarity switches. In both models, the power is flowing away from the center of the current, transverse to the direction of power flow in the circuit. This displacement current is the flow of electrical energy into and out of the field between the plates of the capacitor in the circuit shown in [Fig. 3]. The wireless transmission system of

<figure><img src="image_6.png" /><figcaption>Fig. 3. RC circuit with constant potential source</figcaption></figure>

Nikola Tesla [Fig. 4] applies the displacement current for the wireless transmission of electrical energy. This system

operates in the same manner as the circuit in [Fig. 3] in which each plate of the capacitor is reshaped into a sphere and series flat-spiral inductors are attached to create circuit of zero impedance, allowing for the lossless transmission of power and the condition of resonance to establish based on the resonant frequency determined by the combination of the respective component's reactance. The propagation speed has been described by Nikola Tesla as faster than light (patent No. 787,412) [8].

<figure><img src="image_7.png" /><figcaption>Fig. 4. Nikola Tesla's wireless transmission system, patent 645576 [8]</figcaption></figure>

The notion of a velocity in electromagnetism, specifically the velocity of light, is derived from the application of Maxwell equations to derive the electric and magnetic field wave equations. A basic sketch is shown here by taking the curl of both sides and substituting (2) into (1):

$$ \nabla \times (\nabla \times E) = \nabla \times (-\frac{\partial B}{\partial t}) = -\frac{\partial}{\partial t}(\nabla \times B) = -\mu_0 \epsilon_0 \frac{\partial^2 E}{\partial t^2} \quad (5) $$

Where the identity $\nabla \times (\nabla \times F) = \nabla(\nabla \cdot F) - \nabla^2 F$ is applied with (3) to produce the following equation

$$ \mu_0 \epsilon_0 \frac{\partial^2 E}{\partial t^2} - \nabla^2 E = 0 \quad (6) $$

Which is of the form of the wave equation, a partial differential equation whose solutions are perturbations propagating with velocity $v$ in the form

$$ \frac{1}{v^2} \frac{\partial^2 u}{\partial t^2} - \frac{\partial^2 u}{\partial x^2} = 0 \quad (7) $$

Where the second order spatial derivative of the function of space and time is equal to the second order time derivative times the reciprocal square of velocity propagation. A sample solution is $y = \sin(x - vt)$ which describes a
wave propagating to the right with velocity v. The velocity, which is now known as the velocity of light, is defined as $\frac{1}{c^2} = \mu_0\epsilon_0$ or equivalently $c = \frac{1}{\sqrt{\mu_0\epsilon_0}}$. However, this wave equation cannot describe the longitudinal form of transmission since the energy flow is in a different direction. The direction of energy transfer in electromagnetism and photon momentum is given by the Poynting vector

$$ \mathcal{P} = \frac{1}{\mu_0} E \times B \qquad (8) $$

As we have shown, and as is evident from [Fig.3] the power is moving longitudinally to the resistor, and the electromagnetism described with the wave equation constitutes radiation energy loss propagating away from the capacitor and circuit. Therefore, the waves described by the wave equation cannot be used to describe the longitudinal power flow, and we must be open to a difference in velocity for this new type of wave. We use the term Longitudinal Electric Wave to describe this new wave based on the work by Eric P. Dollard on this subject [12].

<figure><img src="image_8.png" /><figcaption>Fig. 5. Computationally generated electric (red) and magnetic (blue) field lines using bipolar coordinates</figcaption></figure>

We now perform the task of modeling the velocity of the Longitudinal Electric Wave, specifically the displacement current which constitutes the longitudinal time variation in the electric field. To do so, we utilize a toroidal coordinate system, which in 2 dimensions is described by the bipolar coordinate system [10] and generates the graph shown in [Fig. 5]. The result is the formation of isosurfaces of constant $\sigma$ and $\xi$ which describe the orthogonal lines formed by the electric and magnetic fields respectively in an electromagnetic wave. The isosurface formed with constant $\sigma$ in particular describe the divergent field formed by at the poles of both electric and magnetic dipoles, and is what is used in the calculations which follows, where the field lines of constant $\xi$ describe the corresponding orthogonal field. This field has been generated computationally and shown in [Fig. 5].

The equations used to generate the surfaces are described in [10] and are provided as

$$ x_c = \frac{a}{\tan(\sigma)} \qquad (9) $$

$$ r = \left| \frac{a}{\sin(\sigma)} \right| \qquad (10) $$

Where $x = x_c + r\cos(\theta)$, $y = y_c + r\sin(\theta)$ generates the circle centered at $(x_c, y_c)$ with radius $r$, where $y_c = 0$ for this field (red) and is nonzero for the orthogonal field (blue). In this system, $a$ defines the foci which is placed at $(0, a)$ and $(0, -a)$. The magnetic field lines are generated with

$$ y_c = \frac{a}{\tanh(\xi)} \qquad (11) $$

$$ r = \left| \frac{a}{\sinh(\xi)} \right| \qquad (12) $$

For the purposes of this calculation, these basic equations are sufficient. A rotation about the z-axis of this configuration creates the toroidal coordinate system which accurately describes the three-dimensional field geometry of the actualized system, such as the one shown in [Fig.4].

The presented calculation uses (9,10) by iterating over 256 evenly spaced values of $\sigma$ from $\frac{\pi}{128}$ to $\frac{99\pi}{100}$. Since this field is symmetric, this range is sufficient for a large number of data points for this procedure.

We now assume the following postulate: The time taken for the electromagnetic wave to travel from the source to a point in space is equal to the time for the electric field displacement to propagate along the line of electric flux from the source to that same point. This is the first time to the author's knowledge any such postulate of this kind has been made. If the contrary were true, then either the electromagnetic wave or the displacement are faster. If the displacement was faster, then existing systems will experience superluminal transmission, since the displacement also radiates electromagnetism, and will transmit before the original electromagnetic pulse arrives. If the electromagnetism is faster, this would mean the transverse electromagnetic wave is faster than the longitudinal electric wave. Physical experience says otherwise, for instance, tugging a rope is felt much quicker than a transverse pulse sent down the same rope. It can also be shown, mathematically, assuming the same rope material and length, several orders of magnitude difference in velocity. Therefore, by this postulate, a superluminal velocity is implicit. This is because, the electric flux travels a different path from the straight line path, as shown in [Fig.5], which by the definition of the straight line, is the longer path. For it to arrive at the same time, it must be faster.

The graph of the straight horizontal line distance from the top of the dipole depicted in [Fig.5] to its point on the other side of each circle, $c_l$ against the arc length of the curve subtended by that chord, $s_l$ is given in [Fig.6]. The chord length, $c_l$, is calculated with

$$ c_l = 2\sqrt{r^2 - a^2} \qquad (13) $$
<figure><img src="image_9.png" /><figcaption>Fig. 6. Flux length calculated as the arc length subtended by the chord created by the top of the dipole and its horizontal distance to the other side of the circle.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 7. Chord length to the relative % gain in velocity for the longitudinal electric wave over the arc length to the transverse electromagnetic wave traveling at the velocity of light over the chord length, assuming both arrive at the same time.</figcaption></figure>

Where r is the radius of the circle corresponding to the flux line and a is the distance of each pole or foci from the origin. The arc length of the flux subtended by this chord, $s_l$, is calculated with

$$s_l = 2r(tan^{-1}(\frac{c_l}{2a})) \quad (14)$$

The graph [Fig. 6] shows the chord length is approximately linear with arc length as the radius of the circle increases. We can therefore expect an approximately constant velocity as the arc length grows. We will soon address the nonlinearities at small distances. The longitudinal velocity and percent increase are calculated with

$$t_{cl} = \frac{c_l}{c} \quad (15)$$

$$v_{le} = \frac{s_l}{t_{cl}} \quad (16)$$

$$\%Gain = \frac{(v_{le} - c)}{c} \times 100\% \quad (17)$$

Where $c = 2.9979 \times 10^8$. The velocity tends to c as the chord and arc length tends to zero, due to the distances converging, creating a lower gain at small distances. This is the result of the assumption in which the total time traveled by each wave is the same. The only routine performed on the data prior to determining the averages presented was

the removal of one value of (0,0), representing the flux path in which the dipole is situated directly vertical from its center, causing both the arc length and chord lengths to vanish. Using the equations presented in (15,16,17), we determine an average net gain of approximately 16.1% or 1.161c. If we only consider the far field, chord lengths > 50, then the average gain tends to 52% or 1.52c. We see the results graphed in [Fig.7], which also shows the gain tending to 57% or more specifically, $\frac{\pi}{2}c$ as the distance tends to infinity. The highest calculated value is a gain of 57.077% or 1.57077c. These results assume, the postulate of the time taken for the longitudinal and transverse modes to be equal.

Inspecting our results with others, Nikola Tesla reported a velocity of 471240 km/s [9] which is a percent gain of 57.08% or 1.5708c, which is in line with the calculated theoretical far field average of 1.52c and within .002% to the highest calculated value of 1.57077c. Both of these results approximate $\frac{\pi}{2}c = 1.570796c$ which forms the upper bound of the longitudinal electric wave velocity.

It is common to deny the possibility of superluminal velocities from the notion of their existence resulting in odd interactions with time. One must remember that time is a fictitious concept, meaning, everything we attribute as functions of time are in fact, functions of some other phenomena. In electromagnetics, we speak of time variation when in fact we are dealing with variation in energy, that is $\frac{\partial E}{\partial t}$ represents a variation in time to the variation in electric field intensity, when in fact, the variation in field intensity is the direct consequence of the variation in energy in the electric field at that location. We impose the field to be instead a function of time for convenience, since we can create consistent intervals by which we perform measurement of force or similar physical quantities, and form functions of these results, to describe the phenomena, when in fact, the underlying cause of the variations is distinct from, the choice of reference by which we measure its effects. Ernst Mach describes this further, "Since we only recognize what we call time and space by certain phenomena, spatial and temporal determinations are only determinations by means of other phenomena" [11]. Specifically providing the example, of a moving object whose position is a function of time, is in fact, a function of the rotational position of earth. "The earth's angle of rotation is very ready to our hand, and thus we easily substitute it for other phenomena which are connected with it but less accessible to use..." [11]. Furthermore, the methods presented in this piece require very little new physics, simply making use of apparatus available in the time of Nikola Tesla, therefore, this phenomena of longitudinal electric waves has always been present, and it is of interest now, when it is assumed all is known in classical electrodynamics, to revitalize the ideas of the past, which provide an electrically dynamic perspective of the present.
REFERENCES

[1] Thompson, J.J. (1893). Notes on Recent Researches in Electricity and Magnetism, Intended as a Sequel to Professor Clerk-Maxwell's 'Treatise on Electricity and Magnetism'

[2] Maxwell, J.C. (1861) On Physical Lines of Force.

[3] Heaviside, O. (1892). Electromagnetic Induction and Its Propagation (First Half) (pg. 429-556). *Electrical Papers* (Vol. 1). MacMillan and Co.

[4] Maxwell, J. C. (1865). A Dynamical Theory of the Electromagnetic Field. *Royal Society Transactions*, **155**, 459-512.

[5] Steinmetz, C.P. (1914). Electric Discharges, Waves and Impulses.

[6] Faraday, M. (1851). Faraday's Diary of Experimental Investigations (Vol. VI, entry from December 11, 1851).

[7] Jefimenko, O. (1962). Demonstration of the Electric Fields of Current-Carrying Conductors. American Journal of Physics, 30(1), 19-21. https://doi.org/10.1119/1.1941887

[8] Tesla, N. (1900). Apparatus for Detecting and Measuring the Force of Electricity (U.S. Patent No. 645,576). Retrieved from https://patents.google.com/patent/US645576A/en

[9] Tesla, N. (1905). Electrical transmission of power (U.S. Patent No. 787,412). Retrieved January 6, 2025, from https://patents.google.com/patent/US787412A/en

[10] Lucht, P. (2015). Bipolar Coordinates and the Two-Cylinder Capacitor. Retrieved March 10, 2025.

[11] Mach, E. (1911). History and Root of the Principle of the Conservation of Energy.

[12] Dollard, E.P. (1986). Theory of Wireless Power.