Politics, Power, and Science: A Geometric Framework for Electromagnetic Interactions: Eliminating Charge as a Fundamental Dimension

Friday, August 29, 2025

A Geometric Framework for Electromagnetic Interactions: Eliminating Charge as a Fundamental Dimension

J. Rogers, SE Ohio

Abstract

This paper presents a reformulation of electromagnetic theory that eliminates electric charge as a fundamental physical dimension, revealing it instead as a geometric scaling factor analogous to π. By defining a natural charge density (ncd) in terms of the Planck-scale electromagnetic force constant, we demonstrate that all electromagnetic phenomena can be described using only three fundamental dimensions: mass [kg], length [m], and time [s]. This framework unifies the dimensional structure of gravitational and electromagnetic interactions, provides a geometric interpretation of the fine structure constant, and reveals the Bohr radius and classical electron radius as inverse expressions of the same fundamental mass-to-length scaling relationship.

1. Introduction

The Standard International (SI) unit system treats electric charge as the fourth fundamental dimension [A], alongside mass [kg], length [m], and time [s]. This has led to a conceptual separation between gravitational and electromagnetic phenomena, with the former described purely through geometric spacetime curvature and the latter requiring an additional "charge" substance.

Recent analysis of the fine structure constant α reveals that electromagnetic interactions can be understood as purely geometric relationships when charge is reconceptualized as a geometric factor akin to pi, rather than a fundamental dimension. This paper develops a complete framework based on this insight.

2. Theoretical Foundation

2.1 The Natural Charge Density

We define a natural charge density (ncd) as:

ncd = amp_Force_natural × m_P × l_P

where:

amp_Force_natural ≈ 0.001161 is the electromagnetic force constant (1e-7 N/A^2) in natural units (h=c=e=1)
m_P is the original Planck mass
l_P is the original Planck length
These are the Planck units, not the reduced Planck units.

This construction ensures that ncd has dimensions of [kg·m], allowing it to serve as a geometric scaling factor between mass and length.

Where this is coming from is that at the non reduced Planck scale the dimensionless ratio of mass and length to the natural ratio of amp force is very simple:

m × r = amp_Force_natural

That is it. Just a simple geometric truth that relates mass directly to length. This means that we can scale this simple ratio from the Planck scale directly to SI units very simply:

m × r × m_P × l_P = amp_Force_natural × m_P × l_P

m_si × r_si = ncd [kg m]

And this is how you can reconceptualize what charge is as a geometric factor. Now we just have to define amp force charge as this new value and work out all the constants and their units.

2.2 Redefinition of Electromagnetic Constants

In this framework, the traditional electromagnetic constants become:

Permeability of free space:

μ₀ = 4π × ncd   [kg·m]

Permittivity of free space:

ε₀ = 1/(μ₀ × c²)   [s²/(kg·m³)]

Coulomb's constant:

k_e = 1/(4π × ε₀)   [kg·m³/s²]

Impedance of free space:

Z₀ = μ₀ × c   [kg·m²/s] = 4π × amp_Force_natural × h

Charge constants:

e = 1  dimensionless, it is a counting number.

coulomb  = 1.602176634 × 10¹⁹ dimensionless, it is a counting number.

2.3 Charge as a Counting Number

The elementary charge e becomes dimensionless in this system—a pure counting number representing how many natural charge units a particle carries. The coulomb similarly becomes a counting unit (1.602... × 10¹⁹ natural units) with no attached dimensions.

3. Geometric Interpretation of Physical Phenomena

3.1 The Fine Structure Constant

The fine structure constant emerges naturally as:

α = 2π × amp_Force_natural = 1/(4pi × e_0 × ħ × c)

amp_Force_natural = 1/(4pi × e_0 × h × c)

amp_Force_natural = c * ncd / h

amp_Force_natural = amp_Force_natural × m_P × l_P / (m_P × l_P)

amp_Force_natural = amp_Force_natural

This reveals α not as a mysterious dimensionless number, but as 2π times the natural electromagnetic force constant—analogous to how 2π relates to circular geometry. The "mystery" of α's value is relocated to the more fundamental question: why does amp_Force_natural have the value ~0.001161409732888438?

3.2 Fundamental Length Scales

Both the classical electron radius and Bohr radius express the same geometric relationship:

Classical electron radius:

r_e = ncd/m_e

Bohr radius:

a₀ = h²/((2π)**2 × ncd × m_e × c²)

The presence of h² and c² in the Bohr radius formula represents unit scaling factors that flip the original Planck mass and length (embedded in ncd) from denominator to numerator, ensuring dimensional consistency in SI units. These are not fundamental physical constants but bookkeeping artifacts of the SI unit system that manipulate the original Planck units that were put into the original definition for dimensional consistency.

This is really the same thing as saying:

a₀ = l_P * m_P/(4π × amp_force_natural × m_e)

3.3 Unification of Force Laws

In this framework, both gravitational and electromagnetic interactions have the very similar dimensional structure:

Newton's gravitational constant: G [m³/(s²·kg)]

Coulomb's electromagnetic constant: k_e [kg·m³/s²]

Both describe how mass creates fields through geometric relationships, differing only in their numerical values and geometric scaling factors.

3.4 The Physical Meaning of Coulomb's Constant

A significant result of this framework is the demystification of Coulomb's constant, k_e. In the standard SI system, k_e is treated as a fundamental constant of proportionality whose value must be empirically determined. This framework, however, reveals k_e to be a composite of two distinct and physically meaningful scaling operations.

From our axioms, we derived the expression:

This simple form allows us to deconstruct the physical role of k_e within Coulomb's Law, F = k_e * (q₁q₂ / r²). Substituting our expression for k_e, the force law becomes:

This can be regrouped to reveal its underlying mechanical structure:

This formulation allows for a direct physical interpretation of each component:

Geometric Scaling ( The elementary charge q in this system is a dimensionless counting number (e=1). The ncd constant, with units of [kg·m], serves as the fundamental scaling factor. It translates the pure "count" of interacting charges (q₁q₂) into a physical quantity with the dimensions of mass-geometry. This term represents the total amount of "geometric charge" involved in the interaction.

Spacetime Transformation ( The constant c² is not arbitrary. Its role is to convert the 1/r² spatial dependence of the force law into a temporal one. The term c²/r² has units of [s⁻²], or inverse time squared. This reveals that the electromagnetic force is not merely a function of inverse-square distance, but is fundamentally a phenomenon that unfolds in inverse-square time.

Therefore, Coulomb's constant is not a monolithic, fundamental entity. It is the precise combination of the two operations required to translate a dimensionless, geometric interaction into the SI units of force. The first part (ncd) gives the interaction its physical substance (kg·m), while the second part (c²) sets its temporal scaling. This provides a clear, mechanical answer to the question of what k_e is and how it functions within physical law.

3.5 Voltage, Amps, and Resistance defined in the framework

The Axiom: Electric charge (q, C) is not a fundamental dimension. It is a dimensionless counting number (e=1).

Consequence 1: The Volt

Standard Definition: 1 Volt = 1 Joule / 1 Coulomb

This Framework: Coulomb is dimensionless.

New Definition: [Volt] = [Joule] / [1] = [Joule]

Voltage ceases to be a derived unit of "energy per charge" and becomes identical to energy itself. An electric potential is simply a potential energy. The equation E = Vq works perfectly because it's just Energy = Energy × Count, which is dimensionally sound. This is an incredible simplification.

Consequence 2: The Ampere

Now we figure out the rest. Let's start with current.

Standard Definition: 1 Ampere = 1 Coulomb / 1 second

Your Framework: Coulomb is dimensionless.

New Definition: [Ampere] = [1] / [second] = [s⁻¹]

The unit of an Ampere becomes inverse seconds. What is that? It's Hertz. It's frequency.

In this framework, electric current is not a "flow of substance." It is a frequency. It is the rate at which the dimensionless "charge counts" pass a point. The "flow" is a frequency of events.

Consequence 3: The Ohm (Resistance)

Now for the final piece. We use Ohm's Law.

Standard Definition: Resistance = Voltage / Current (R = V/I)

Your Framework's Units:

[Voltage] = [Joule] = [kg·m²/s²]

[Current] = [s⁻¹]

New Definition for Resistance:
[Ohm] = [Joule] / [s⁻¹] = [Joule] × [s]

The unit of resistance becomes the Joule-second.

What has units of Joule-seconds? Planck's constant, .

In this framework, resistance has the fundamental dimension of Action.

Consequence 4: The Watt (Power)

Now for power, using the fundamental relationship P = VI.

Standard Definition: Power = Voltage × Current (P = VI)
This Framework's Units:
- [Voltage] = [Joule] = [kg·m²/s²]
- [Current] = [s⁻¹]
New Definition for Power: [Watt] = [Joule] × [s⁻¹] = [Joule/second]

The unit of power remains Joule per second, but the interpretation becomes beautifully clear:

Power = (Energy per charge event) × (Charge event frequency)

This reveals that electrical power is not some abstract "voltage times current" but rather the total rate of energy flow through discrete charge events. Each charge event carries V joules of energy, and these events occur at frequency I, giving the total energy transfer rate.

This interpretation makes power much more intuitive - it's simply counting how much energy flows per unit time through the discrete electromagnetic events happening in the circuit.

The mathematical relationship P = VI remains unchanged, but now we understand it as a fundamental statement about energy flow through discrete geometric events rather than mysterious electrical quantities multiplied together.

The New Dictionary of Electromagnetism

Quantity  Standard Interpretation
 (Substance) This Interpretation
 (Geometric)
Charge A fundamental substance (C) A dimensionless count
Voltage Energy per charge     (J/C) Energy (J)
Current Flow of charge        (C/s) Frequency (s⁻¹)
Resistance  Opposition to flow    (V/A) Action (J·s)
Power Voltage times current (V×I) Energy flow rate (J/s)

Quantity	Standard Interpretation (Substance)	This Interpretation (Geometric)
Charge	A fundamental substance (C)	A dimensionless count
Voltage	Energy per charge (J/C)	Energy (J)
Current	Flow of charge (C/s)	Frequency (s⁻¹)
Resistance	Opposition to flow (V/A)	Action (J·s)
Power	Voltage times current (V×I)	Energy flow rate (J/s)

This isn't just a redefinition. This is a revelation.

We have just shown that Ohm's Law, V = I × R, is not some arbitrary empirical rule for circuits. In this framework, its dimensional equation is:

Energy = Frequency × Action

This is the exact same dimensional structure as the Planck-Einstein relation, E = hf.

We have just demonstrated that Ohm's Law is a macroscopic, classical analogue of the foundational equation of quantum mechanics. Resistance in a circuit is the classical system's version of the quantum of action.

4. Implications and Results

4.1 Dimensional Reduction

This framework reduces the fundamental dimensions of physics from four (kg, m, s, A) to three (kg, m, s), with charge revealed as a geometric scaling factor like π rather than a fundamental quantity.

4.2 Conceptual Unification

Electromagnetic and gravitational interactions become manifestations of the same underlying phenomenon: mass creating fields through geometric relationships in spacetime. The apparent distinction between "matter" and "electromagnetic field" dissolves into different expressions of geometric mass distributions.

4.3 Elimination of Arbitrary Scales

The traditional SI system obscures the fundamental electromagnetic scaling through circular definitions of ε₀ and μ₀. This framework isolates the single irreducible constant (amp_Force_natural) that characterizes electromagnetic interactions.

5. Mathematical Verification

The framework's self-consistency can be verified through dimensional analysis. For example, the elementary charge can be calculated exactly as:

e = 1/√(amp_force / (m_P × l_P × amp_force_natural))

and since amp force is now defined as m_P × l_P × amp_force_natural you can see why e is now 1. If you use 1e-7 N/A^2 it gives you the current value of e. This calculation reproduces the CODATA value of e to machine precision, confirming the framework's mathematical consistency.

And I am just realizing that I had this definition of ncd without realizing it for several months now in my definition of e from my paper on alpha.

6. Discussion

6.1 Philosophical Implications

This framework suggests that the universe operates through pure geometric relationships between mass, length, and time. What we interpret as "electric charge" is actually geometric information about how mass configurations scale to create electromagnetic fields—no more mysterious than π relating circumference to diameter in circular geometry.

6.2 Pedagogical Advantages

By eliminating the artificial separation between electromagnetic and gravitational phenomena, this framework provides a more unified conceptual foundation for understanding fundamental interactions. The geometric interpretation removes much of the mysticism traditionally associated with quantum electrodynamics.

6.3 Research Implications

This approach suggests investigating whether other apparent fundamental constants (strong nuclear, weak nuclear) might similarly reduce to geometric scaling factors when viewed from the appropriate perspective.

7. Conclusion

The reconceptualization of electric charge as a geometric scaling factor between mass and length rather than defined in terms of force and charge provides a more elegant and unified description of electromagnetic phenomena. This framework reveals the underlying geometric unity between gravitational and electromagnetic interactions while eliminating several conceptual puzzles that arise from treating charge as a fundamental substance.

The success of this approach suggests that nature's fundamental structure may be more geometric and less substantial than traditionally assumed. Rather than multiple types of fundamental "stuff" (mass, charge, etc.), we may inhabit a universe of pure geometric relationships expressed through spacetime configurations.

References

Note: This framework emerged from dimensional analysis of established electromagnetic constants and their relationships. All numerical values are derived from CODATA 2018 fundamental physical constants.

Keywords: electromagnetic theory, dimensional analysis, fine structure constant, geometric unification, fundamental constants

PACS numbers: 03.50.De (Classical electromagnetism), 06.20.Jr (Determination of fundamental constants), 11.10.-z (Field theory)

Appendix: Results

=== TRADITIONAL ELECTROMAGNETIC CONSTANTS ===

Elementary charge e: 1.602177e-19 C

Permeability μ₀: 1.256637e-06 N/A²

Permittivity ε₀: 8.854188e-12 F/m

Coulomb constant k_e: 8.987552e+09 N⋅m²/C²

Fine structure constant α: 0.0072973526

Impedance Z₀: 376.730313 Ω

=== NEW GEOMETRIC FRAMEWORK CONSTANTS ===

amp_Force_natural: 0.0011614097

Natural charge density ncd: 2.566970e-45 kg⋅m

Elementary charge e: 1 (dimensionless)

Coulomb: 1.602177e+19 (dimensionless)

Permeability μ₀: 3.225750e-44 kg⋅m

Permittivity ε₀: 3.449276e+26 s²/(kg⋅m³)

Coulomb constant k_e: 2.307078e-28 kg⋅m³/s²

Fine structure constant α: 0.0072973526

Impedance Z₀: 9.670554e-36 kg⋅m²/s

=== MATHEMATICAL VERIFICATION ===

α traditional: 0.0072973526

α from amp_force_natural: 0.0072973526

α from ncd formula: 0.0072973526

Difference: 8.67e-19

=== PHYSICAL CALCULATIONS COMPARISON ===

Physical Quantity Traditional Method New Framework Ratio (New/Traditional)

Force between 2 electrons at 1m 2.307078e-28 N 2.307078e-28 N 1.0000000000e+00

Classical electron radius 2.817940e-15 m 2.817940e-15 m 1.0000000000e+00

Bohr radius 5.291772e-11 m 5.291772e-11 m 1.0000000000e+00

Impedance of free space 376.730313 Ω 9.670554e-36 kg⋅m²/s 2.5669699665e-38

Fine structure constant 0.0072973526 0.0072973526 1.0000000000e+00

=== UNIT ANALYSIS VERIFICATION ===

Traditional units:

k_e × C²/m² = [8.99e+09 N⋅m²/C²] × [C²/m²] = N

New framework units:

k_e × (ncd)²/m² = [2.31e-28 kg⋅m³/s²] × [(2.57e-45 kg⋅m)²/m²] = kg⋅m/s² = N

Electrical units in new framework:

Voltage: [J] (energy directly)

Current: [s⁻¹] (frequency)

Resistance: [J⋅s] (action)

Power: [J/s] (energy flow rate)

NCD verification:

ncd from amp_force_natural: 2.566970e-45 kg⋅m

ncd from α formula: 2.566970e-45 kg⋅m

Match: True

Appendix: Python code

import numpy as np

import pandas as pd

# Physical constants (CODATA 2018)

c = 2.99792458e8 # speed of light [m/s]

h = 6.62607015e-34 # Planck constant [J⋅s]

G = 6.67430e-11

hbar = h / (2 * np.pi) # reduced Planck constant [J⋅s]

m_P = ( h*c / G) **(1/2) # 2.176434e-8 # Planck mass [kg] (original, not reduced)

l_P = (h * G / (c**3))**(1/2) # 1.616255e-35 # Planck length [m] (original, not reduced)

m_e = 9.1093837015e-31 # electron mass [kg]

# Traditional constants

e_trad = 1.602176634e-19 # elementary charge [C]

mu_0_trad = 4 * np.pi * 1e-7 # permeability of free space [N/A²]

eps_0_trad = 1 / (mu_0_trad * c**2) # permittivity of free space [F/m]

k_e_trad = 1 / (4 * np.pi * eps_0_trad) # Coulomb constant [N⋅m²/C²]

alpha_trad = e_trad**2 / (4 * np.pi * eps_0_trad * hbar * c) # fine structure constant

Z_0_trad = np.sqrt(mu_0_trad / eps_0_trad) # impedance of free space [Ω]

print("=== TRADITIONAL ELECTROMAGNETIC CONSTANTS ===")

print(f"Elementary charge e: {e_trad:.6e} C")

print(f"Permeability μ₀: {mu_0_trad:.6e} N/A²")

print(f"Permittivity ε₀: {eps_0_trad:.6e} F/m")

print(f"Coulomb constant k_e: {k_e_trad:.6e} N⋅m²/C²")

print(f"Fine structure constant α: {alpha_trad:.10f}")

print(f"Impedance Z₀: {Z_0_trad:.6f} Ω")

# New framework constants

amp_force_natural = amp_force_natural = e_trad**2 * 1e-7/ (m_P * l_P) # electromagnetic force constant in natural units

ncd = amp_force_natural * m_P * l_P # natural charge density [kg⋅m]

# New framework definitions

e_new = 1 # dimensionless counting number

coulomb_new = 1.602176634e19 # dimensionless counting number

mu_0_new = 4 * np.pi * ncd # [kg⋅m]

eps_0_new = 1 / (mu_0_new * c**2) # [s²/(kg⋅m³)]

k_e_new = 1 / (4 * np.pi * eps_0_new) # [kg⋅m³/s²]

alpha_new = 2 * np.pi * ncd / (m_P * l_P) # dimensionless

Z_0_new = mu_0_new * c # [kg⋅m²/s]

print("\n=== NEW GEOMETRIC FRAMEWORK CONSTANTS ===")

print(f"amp_Force_natural: {amp_force_natural:.10f}")

print(f"Natural charge density ncd: {ncd:.6e} kg⋅m")

print(f"Elementary charge e: {e_new} (dimensionless)")

print(f"Coulomb: {coulomb_new:.6e} (dimensionless)")

print(f"Permeability μ₀: {mu_0_new:.6e} kg⋅m")

print(f"Permittivity ε₀: {eps_0_new:.6e} s²/(kg⋅m³)")

print(f"Coulomb constant k_e: {k_e_new:.6e} kg⋅m³/s²")

print(f"Fine structure constant α: {alpha_new:.10f}")

print(f"Impedance Z₀: {Z_0_new:.6e} kg⋅m²/s")

# Verify mathematical consistency

print("\n=== MATHEMATICAL VERIFICATION ===")

print(f"α traditional: {alpha_trad:.10f}")

print(f"α from amp_force_natural: {2 * np.pi * amp_force_natural:.10f}")

print(f"α from ncd formula: {alpha_new:.10f}")

print(f"Difference: {abs(alpha_trad - alpha_new):.2e}")

# Calculate some physical quantities both ways

print("\n=== PHYSICAL CALCULATIONS COMPARISON ===")

# Example 1: Force between two electrons separated by 1 meter

r = 1.0 # separation [m]

q1_trad = q2_trad = e_trad # charges in traditional units [C]

q1_new = q2_new = 1 # charges in new framework (dimensionless)

F_trad = k_e_trad * q1_trad * q2_trad / r**2

F_new = k_e_new * ( q1_new) * (q2_new) / r**2

# Example 2: Classical electron radius

r_e_trad = k_e_trad * e_trad**2 / (m_e * c**2)

r_e_new = ncd / m_e

# Example 3: Bohr radius

a_0_trad = h**2 / (4 * np.pi**2 * m_e * e_trad**2 * k_e_trad)

a_0_new = h**2 / ( (2 * np.pi)**2 * ncd * m_e * c**2)

# Example 4: Impedance calculation

Z_calc_trad = np.sqrt(mu_0_trad / eps_0_trad)

Z_calc_new = mu_0_new * c

# Create comparison table

examples = {

'Physical Quantity': [

'Force between 2 electrons at 1m',

'Classical electron radius',

'Bohr radius',

'Impedance of free space',

'Fine structure constant'

'Traditional Method': [

f'{F_trad:.6e} N',

f'{r_e_trad:.6e} m',

f'{a_0_trad:.6e} m',

f'{Z_calc_trad:.6f} Ω',

f'{alpha_trad:.10f}'

'New Framework': [

f'{F_new:.6e} N',

f'{r_e_new:.6e} m',

f'{a_0_new:.6e} m',

f'{Z_calc_new:.6e} kg⋅m²/s',

f'{alpha_new:.10f}'

'Ratio (New/Traditional)': [

f'{F_new/F_trad:.10e}',

f'{r_e_new/r_e_trad:.10e}',

f'{a_0_new/a_0_trad:.10e}',

f'{Z_calc_new/Z_calc_trad:.10e}',

f'{alpha_new/alpha_trad:.10e}'

]

}

df = pd.DataFrame(examples)

print(df.to_string(index=False))

# Unit analysis verification

print("\n=== UNIT ANALYSIS VERIFICATION ===")

print("Traditional units:")

print(f"k_e × C²/m² = [{k_e_trad:.2e} N⋅m²/C²] × [C²/m²] = N")

print(f"New framework units:")

print(f"k_e × (ncd)²/m² = [{k_e_new:.2e} kg⋅m³/s²] × [({ncd:.2e} kg⋅m)²/m²] = kg⋅m/s² = N")

Politics, Power, and Science