The Coulomb as a Physicist's Mole: Re-framing Charge as a Dimensionless Count

J. Rogers, SE Ohio

Abstract
For over a century, the system of electromagnetic units has been built upon the premise that electric charge is a fundamental, dimensional substance, measured in Coulombs. This has led to the elementary charge, e, being expressed as a minute fractional value (1.602... × 10⁻¹⁹ C), a source of conceptual difficulty and needless complexity. This paper argues that this formulation is a historical artifact. By drawing a direct analogy to the Mole in chemistry—a unit explicitly defined as a large, dimensionless count of discrete particles—we demonstrate that the Coulomb can and should be re-framed as a similar "counting packet" for elementary charge events. In this view, the elementary charge e is fundamentally the integer 1, and the Coulomb is simply a convenient name for a specific, large number of these events. This re-framing not only simplifies the conceptual foundations of electromagnetism but also reveals the deep, structural similarities in how science bridges the quantum and macroscopic scales across different disciplines.

1. Introduction: The Problem of the "Weird 
The discovery of the quantization of electric charge by Millikan was a landmark achievement of the early 20th century. It revealed a fundamental, indivisible unit of electric interaction. However, the existing 19th-century framework of units was not reformed to reflect this discovery. Instead, the new quantum reality was forced into the old classical system, resulting in the elementary charge e being assigned the arcane value of 1.602... × 10⁻¹⁹ Coulombs.  We are just proposing to invert this value and assign that number directly to the Coulomb as a count.

This "weird e" presents a significant pedagogical and conceptual hurdle. It implies that the most fundamental charge is a tiny fraction of a larger, more "fundamental" unit (the Coulomb), which is the reverse of physical reality. The reality is that charge comes one to a customer.  An electron has a negative charge and a proton has a positive charge. This paper argues that the tiny value for a countable quantum property is a category error and that a solution has existed in a parallel scientific discipline for decades: the chemist's Mole.

2. The Mole: An Established Precedent for Macroscopic Counting
In chemistry and thermodynamics, the problem of bridging the atomic and macroscopic scales is ubiquitous. A single atom is a discrete, countable entity, while a gram of a substance is a convenient, human-scale quantity. The connection between these two scales is not made by defining a new, continuous "substance" of which an atom is a tiny part. Instead, it is made via a dimensionless counting unit.

• Fundamental Unit: The particle (atom, molecule). This is recognized as the indivisible, countable unit.

• Human-Scale Anchor: The gram, a conventional unit of mass.

• The Bridge: Avogadro's Number (N_A ≈ 6.022 × 10²³), a pure, dimensionless number.

• The Definition of the Mole: The Mole is not a fundamental dimension. It is the name given to a count of N_A particles. It is the "chemist's dozen."

This definition serves a clear purpose: it is the exact number of particles required so that the mass of the collection in grams is numerically equal to the mass of a single particle in atomic mass units. It is an elegant, universally understood solution that preserves the quantized, countable nature of the microscopic world while providing a convenient conversion factor to the macroscopic world.

3. The Coulomb: A Misidentified Counting Unit
The problem faced by early 20th-century physicists was identical to the one faced by chemists.

• Fundamental Unit: The elementary charge event, demonstrated by Millikan to be discrete and indivisible.

• Human-Scale Anchor: The Joule and the Volt, conventional units of energy and potential.

• The Problem: How to connect the minuscule energy of a single charge event to the practical scale of Joules and Volts?

Instead of following the chemist's logical path, physics took a historical detour. It retained the Coulomb—a unit defined for continuous, macroscopic currents—as the fundamental unit. This forced the true, indivisible quantum unit to be described as a bizarre fraction of the arbitrary macroscopic unit.

We propose to correct this historical error by applying the same logic as the Mole.

• Fundamental Unit: The elementary charge e, defined as the dimensionless integer 1. It represents one quantum event.

• Human-Scale Anchor: The fundamental identity that Voltage is Energy ([V] = [J]).

• The Bridge: A dimensionless number, N_C = 1 / 1.602... × 10⁻¹⁹ ≈ 6.24 × 10¹⁸.

• The New Definition of the Coulomb: The Coulomb is not a fundamental dimension. It is the name given to a count of N_C elementary charge events. It is the "physicist's dozen."

This definition serves a clear purpose: it is the exact number of charge events required so that the relationship Energy (J) = Potential (V) × Charge (C) is numerically balanced, upholding the deeper identity that Voltage is Energy.

4. Comparing the Conceptual Frameworks
The parallel between the two counting systems is exact and illuminating.

Concept	Chemistry (The Mole)	Electromagnetism (The Coulomb)
Fundamental Unit	1 Particle (indivisible)	1 Charge Event (e=1, indivisible)
Human-Scale Anchor	The Gram	The Joule / Volt
The 'Dozen' Name	The Mole	The Coulomb
The Count (	Avogadro's Number (≈ 6.022 × 10²³)	1/e_trad (≈ 6.24 × 10¹⁸)
Resulting Identity	Mass (g) = Atomic Mass (amu) × N	Energy (J) = Potential (V) × N
Conceptual Status	Explicitly understood as a dimensionless count	Historically mistaken for a fundamental, dimensional substance

5. Discussion: The Benefits of a Counting-Based Framework
Adopting this re-framing has profound benefits:

• Conceptual Clarity: It resolves the paradox of the "weird e". The elementary charge is rightly restored to its status as the fundamental integer 1. The arbitrariness and ugliness of the 1.602... × 10⁻¹⁹ value are correctly relocated to the human-centric definition of the Coulomb.

• Pedagogical Power: Teaching students that the Coulomb is a "big number packet" like the Mole is vastly more intuitive than teaching them that charge is a substance measured in units of C.

• Theoretical Unification: As demonstrated in related work, this single conceptual shift is the key to eliminating Charge as a fundamental dimension, leading to the reformulation of Maxwell's Equations as spacetime mechanics and revealing the deep mechanical nature of electrical units like the Volt (Energy), Ampere (Frequency), and Ohm (Action).

• Philosophical Coherence: It aligns our system of units with physical reality. The world at its smallest scales is discrete and countable. Our units should reflect, not obscure, this fundamental truth.

6. Conclusion
The scientific community already has a successful, well-understood precedent for bridging the quantum and macroscopic scales: the Mole. For over a century, the field of electromagnetism has failed to apply the same clear logic to its own fundamental quantum, the elementary charge. The historical decision to treat the Coulomb as a fundamental unit of substance, rather than as a dimensionless counting packet, has been a long-running source of conceptual confusion.

By explicitly re-defining the elementary charge e as the integer 1 and the Coulomb as the "physicist's dozen"—the specific large count of 6.24 × 10¹⁸ events—we correct this historical error. This re-framing aligns our understanding of charge with the successful methodology of chemistry, simplifies the foundations of electromagnetic theory, and reveals a deeper, more unified geometric structure underlying physical law. The weirdness was never in the electron; it was in our ruler.