[Paper Review] An Essay on the Application of mathematical Analysis to the theories of Electricity and Magnetism

George Green|ArXiv.org|Jul 1, 2008

History and Theory of Mathematics18 citations

TL;DR

This seminal 1828 essay by George Green pioneers the application of mathematical analysis to electricity and magnetism, introducing foundational concepts like potential functions and the mathematical treatment of electric and magnetic fields. It establishes a rigorous framework for deriving electric density distributions on conductors and provides a method to model magnetic wire behavior with high accuracy, validated by comparison to Coulomb’s experimental data using derived constants and differential equations.

ABSTRACT

Green's famous essay (Nottingham, 1828), with which he introduced the potential function, was transcribed from its reprint in Crelle's Journal (1850-54), with several typographical corrections and a reference section added. Green starts with accounts of earlier work, and some introductory remarks motivating his notation and method. Then, he gives a textual summary of later formal calculations, beginning with the general results. Finally, Green applies the results to several problems concerning electricity and magnetism.

Motivation & Objective

• To develop a rigorous mathematical framework for analyzing electric and magnetic phenomena using analysis and potential theory.
• To resolve inconsistencies in earlier treatments of electric repulsion and conduction by introducing a consistent theoretical model.
• To connect the mathematical theory of electricity with that of magnetism through a common law of action.
• To derive accurate expressions for electric density on conducting bodies and magnetic wire behavior, validated by experimental data.

Proposed method

• Introduces the concept of potential functions as a central tool for analyzing electric and magnetic fields.
• Applies hydrostatic equilibrium principles to hypothetical incompressible fluids to complete Cavendish’s proofs, ensuring mathematical rigor.
• Uses differential equations and variable difference equations to model the distribution of electric charge on spherical and ellipsoidal conductors.
• Derives an approximate expression for the integral of the inverse square law over a wire using a constant A, minimized via least squares of error.
• Employs the equation β² = 4(1−g)/(3ga²A) to relate physical constants, with A determined from known logarithmic series.
• Calibrates the model using a single experimental observation to determine the constant K, which scales torsional forces in magnetic wire experiments.

Experimental results

Research questions

• RQ1 How can the mathematical theory of electricity be made rigorous and complete, particularly in light of incomplete proofs by earlier authors like Cavendish?
• RQ2 What is the correct mathematical formulation for the distribution of electric charge on conducting bodies of non-spherical shape?
• RQ3 How can the behavior of magnetic wires under torsional forces be modeled mathematically with high accuracy?
• RQ4 To what extent can theoretical predictions match experimental observations, such as those of Coulomb and Biot, using a unified mathematical framework?

Key findings

• The derived value of g = 0.986636 ensures high accuracy in modeling magnetic wire behavior, with minimal deviation from experimental torsion data.
• For steel wires of radius 1/12 inch, aβ = 0.0548235, a constant that holds across all such wires, enabling consistent modeling.
• The model predicts torsional forces with exceptional accuracy: observed and calculated values differ by less than 0.5° in multiple experiments.
• The formula remains valid for wires longer than 10–15 times their diameter, with high precision even for very fine wires.
• The constant K was determined from a single observation (K ≈ 58.5°), and the model successfully predicts all other torsion values without further calibration.
• Theoretical results for both coarse and fine wires show extremely small residuals between observed and calculated torsions, confirming the model’s robustness.