1. The Progressing Electric Field Model

Abstract: Faraday’s law is empirically derived and, as such, may be subject to limitations. Notably, it appears to violate the law of conservation of energy in certain contexts. To establish a more robust formulation, it is necessary to derive the law from first principles. In this article, we theoretically derive Faraday’s law using only Coulomb’s law and special relativity. We present the first stage of this derivation: the construction of the ‘Progressing Electric Field Model.’ This model determines the curl of the electric field produced by moving charges and calculates the electric potential induced in a wire loop within that field

1.     Introduction

Faraday’s law defines the electromotive force (EMF) induced in a wire loop by a varying magnetic field. Although considered a cornerstone of classical electromagnetism, it remains an empirical law; consequently, it may be incomplete regarding phenomena not yet captured by experimental observation. To illustrate this, consider the following experimental setup: suppose two coils, A and B, are positioned side by side, with coil B connected to a resistor R, as shown in Figure 1.

Let the current in coil A, denoted as I_a, vary as follows: I_a increases linearly from zero to I_max, then decreases linearly back to zero. The duration of each phase is Δt. According to Faraday’s law, voltages are induced in coils A and B, which we label V_a and V_b, respectively. Since I_a varies linearly during each phase, V_a and V_b remain constant throughout those intervals. Within resistor R, the voltage V_b generates a current I_b and dissipates electric power equal to |V_bI_b|, both of which are constant in each phase. Consequently, the total work performed in R after both phases is 2|V_bI_b|Dt.

Since I_b is constant, it does not induce a voltage in coil A; therefore, the value of V_a remains unchanged regardless of whether I_b is positive, negative, or zero—just as if coil B were not present. When I_a increases, the voltage in coil A (V_a) is positive, and the electrical work performed in A is given by the integral of V_aI_a . Conversely, when I_a decreases, the voltage in A becomes -V_a, and the work equals the integral of -V_aI_a . Consequently, the total energy consumption of coil A after both phases equals zero.

Since the energy consumption in coil A is zero, A does not transfer any energy to coil B. We therefore encounter a case where B performs work equal to 2|V_bI_b|Dt while receiving no energy from A. This implies that the system consisting of coils A and B performs work without any energy input, which violates the law of conservation of energy.

The cause of this violation is that Faraday’s law predicts zero voltage in A when the current in coil B is constant. By theoretically deriving Faraday’s law from more fundamental laws, we can not only resolve this inconsistency but also uncover new phenomena and achieve a deeper understanding of nature.

An example of such a phenomenon is demonstrated in my experiment, which reveals a tangential electromotive force not predicted by Faraday’s law. You can view the experiment in this video on YouTube: https://www.youtube.com/watch?v=P33Hgj68G9M

To begin the theoretical derivation of Faraday’s law, we must first examine the electric field of a moving electron.

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For more detail, please read « A Derivation of Faraday’s law from Coulomb’s Law and Relativity / 1.The Progressing Electric Field Model » https://pengkuanem.blogspot.com/2026/01/a-derivation-of-faradays-law-from.html https://www.academia.edu/146009113/A_Derivation_of_Faradays_law_from_Coulombs_Law_and_Relativity_1_The_Progressing_Electric_Field_Model

2.     Discussion

This article presents the first stage of a derivation of Faraday’s law based solely on Coulomb’s law and special relativity. We demonstrate that the retarded electric field of a moving electron—which we term the ‘Progressing Electric Field’—is a non-conservative vector field. We have derived the mathematical expression for its curl and have found that several properties of this field are analogous to those of the magnetic field.

We have derived the electric potential induced in a wire loop by a progressing electric field, which shares certain properties with the electromotive force defined by Faraday’s law. However, because this potential is not yet proportional to the rate of change of the magnetic field, it cannot be classified as electromotive force at this stage. Several additional steps of theoretical derivation are required to fully arrive at Faraday’s law.

The essential steps of the derivation presented in this article are as follows:

1. Propagation at the speed of light: In accordance with special relativity, the electric field propagates at the speed of light, c.
2. Iso-intensity circles: The electric field of a moving electron radiates in ‘iso-intensity circles.’ The centers of these circles correspond to the retarded positions of the electron.
3. Application of Coulomb’s law: The intensity of the electric field on an iso-intensity circle is defined by Coulomb’s law relative to the circle’s center.
4. Non-conservative nature: The ‘Progressing Electric Field’ is a deformation of the static electric field and is inherently non-conservative.
5. Instantaneous curl calculation: The Progressing Electric Field is analyzed within a single temporal snapshot, allowing its curl to be computed using instantaneous values.
6. Superposition of charges: The curl of the Progressing Electric Field for a steady current is derived by integrating the fields of individual moving charges.
7. Induction and Lenz’s Law: The Progressing Electric Field induces an electric field within a wire loop; the calculated average value of this field is shown to be consistent with Lenz’s law.

We have constructed our theory upon the Progressing Electric Field, a concept that may initially appear to lack direct experimental verification. One might ask: what if this field is merely a theoretical construct? In reality, experimental evidence for its existence already exists; however, it has historically been overlooked or reinterpreted through the lens of well-established theories.

By applying the ‘Progressing Electric Field Model,’ we will provide a new and comprehensive explanation for this experimental evidence in a forthcoming article.

In summary, the ‘Progressing Electric Field Model’ demonstrates that the induction phenomena traditionally attributed to Faraday’s law can be derived from the relativistic motion of electric charges. By accounting for the propagation delay of the field at speed c, we establish that the resulting electric field is non-conservative and possesses a non-zero curl. While this initial stage of the derivation produces results consistent with Lenz’s law, further refinement is required to achieve full mathematical proportionality with the rate of change of magnetic flux. This theoretical framework not only aligns with the principle of energy conservation but also opens the door to predicting electromagnetic phenomena beyond the reach of classical empirical laws.

For more detail, please read « A Derivation of Faraday’s law from Coulomb’s Law and Relativity / 1.The Progressing Electric Field Model » https://pengkuanem.blogspot.com/2026/01/a-derivation-of-faradays-law-from.html https://www.academia.edu/146009113/A_Derivation_of_Faradays_law_from_Coulombs_Law_and_Relativity_1_The_Progressing_Electric_Field_Model

1.         Introduction       1

2.         The Electric Field of a Moving Electron     2

a)         The Static Case: The Immobile Electron     2

b)         The Dynamic Case: The Moving Electron and the Progressing Field    3

3.         Geometry of the Wire Loop and Iso-intensity Circles 1

a)         Partitioning the Wire Loop into Sectors      4

b)         Determining the Lengths of Segments AB and CD   5

c)         Calculation of the Surface Area of Sector S 6

4.         Potential within the Progressing Electric Field         4

a)         Definition of Potential     7

b)         Calculation of Potential around the Boundary of a Sector      7

c)         Demonstration of Null Potential Variation on Arcs   8

d)         Resultant Potential Variation for a Complete Sector  9

5.         Potential and Field within the Wire Loop    7

a)         Summation of Individual Sector Influences 11

b)         Characterization of the Electric Field in the Wire Loop         12

c)         Application of Stokes’ Theorem to the Induced Field 13

6.         The Influence of Electric Current   13

7.         The Curl of the Progressing Electric Field   13

a)         Calculation of the Curl for a Single Moving Charge  15

b)         Calculation of the Curl due to a Macroscopic Current           16

c)         Connection to the Biot–Savart Law            17

d)         Connection to the Lorentz Force    17

8.         Implications for a Theoretical Faraday’s Law           18

9.         Discussion         19

Letter to the Readers       20 For more detail, please read « A Derivation of Faraday’s law from Coulomb’s Law and Relativity / 1.The Progressing Electric Field Model » https://pengkuanem.blogspot.com/2026/01/a-derivation-of-faradays-law-from.htmlhttps://www.academia.edu/146009113/A_Derivation_of_Faradays_law_from_Coulombs_Law_and_Relativity_1_The_Progressing_Electric_Field_Model