J. Rogers, SE Ohio, 20 Mar 2025, 0324
Abstract
The definitions of the ampere, coulomb, and electromagnetic constants such as the permittivity of free space () are deeply interconnected through experimental measurements and theoretical consistency. This paper explores how these definitions evolved, starting from the mechanical definition of force and building up to the modern quantum-based definition of the ampere. We highlight the role of force as a foundational concept, showing how it underpins the definitions of charge and electromagnetic laws such as Coulomb’s Law and Gauss’s Law. By tracing this interconnected framework, we demonstrate that the unit system in electromagnetism is not arbitrary but reflects centuries of careful experimentation and theoretical refinement.
1. Introduction
The ampere, coulomb, and electromagnetic constants are fundamental to our understanding of electromagnetism. However, their definitions are often presented in isolation, obscuring the deep connections between them. This paper aims to unify these definitions by tracing their evolution and highlighting the central role of force as a foundational concept. We begin with the original definition of the ampere, show how it leads to the definition of the coulomb, and explore the consistency between Coulomb’s Law and Gauss’s Law. Finally, we discuss the modern quantum-based definition of the ampere and its continuity with the historical force-based approach.
2. The Original Definition of the Ampere
The ampere was originally defined in terms of the force between two parallel current-carrying wires. The setup involves two straight, infinitely long wires placed 1 meter apart in a vacuum. When a current of 1 ampere flows through each wire, the force between them is exactly newtons per meter of wire. This definition connects the ampere to mechanical units: force (newtons), length (meters), and time (seconds).
The force per unit length between the wires is given by:
where is the permeability of free space, and are the currents in the wires, and is the distance between them. For and , this simplifies to:
This equation defines as , establishing a direct link between the ampere and mechanical units.
3. The Definition of the Coulomb
The coulomb is defined as the amount of charge transported by a current of 1 ampere flowing for 1 second:
Thus, 1 coulomb = 1 ampere × 1 second. Because the ampere is defined via force, the coulomb inherits its definition indirectly from mechanical units. This ensures that electromagnetic forces measured in experiments align with the mechanical definition of force.
4. Force Between Charges and Coulomb’s Law
The force between two charges is described by Coulomb’s Law:
where is the permittivity of free space, and are the charges, and is the distance between them. For two charges of 1 coulomb separated by 1 meter, this becomes:
The value of was chosen to make Coulomb’s Law consistent with the definition of the ampere and coulomb. This ensures that the force between charges aligns with the force defined by the ampere.
5. Gauss’s Law and Consistency with Coulomb’s Law
Gauss’s Law, which relates the electric flux through a closed surface to the enclosed charge, is mathematically equivalent to Coulomb’s Law:
The consistency between Coulomb’s Law and Gauss’s Law arises because both depend on how charge is defined via experimental measurements. The force between charges (Coulomb’s Law) aligns perfectly with the force between current-carrying wires (ampere definition), ensuring a unified framework for electromagnetism.
6. The Modern Definition of the Ampere
In 2019, the ampere was redefined based on the elementary charge (), which was fixed at exactly coulombs. Now, 1 ampere corresponds to exactly elementary charges per second. This quantum-based definition preserves continuity with the old force-based definition, ensuring that the numerical value of 1 ampere remains unchanged.
7. Key Insights
Interconnected Definitions: The ampere, coulomb, and electromagnetic constants (, ) are deeply interconnected through experimental measurements and theoretical consistency.
Force as a Foundation: The original mechanical definition of force underpins how charge is defined and how electromagnetic laws like Coulomb’s Law and Gauss’s Law work.
Consistency Across Laws: The force between charges aligns perfectly with the force defined by currents because charge was calibrated to fit these relationships.
8. Conclusion
The definitions of the ampere, coulomb, and electromagnetic constants are not arbitrary but reflect a carefully constructed framework that connects electromagnetism to mechanical units. By tracing the evolution of these definitions, we see how force serves as a foundational concept, ensuring consistency across physical laws. This unified perspective highlights the elegance and interconnectedness of our unit system, demonstrating how centuries of experimentation and theoretical refinement have shaped our understanding of electromagnetism.
References
International Bureau of Weights and Measures (BIPM). (2019). The International System of Units (SI). 9th edition.
Jackson, J. D. (1999). Classical Electrodynamics. 3rd edition. Wiley.
Griffiths, D. J. (2017). Introduction to Electrodynamics. 4th edition. Cambridge University Press.
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