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An Essay on the Application of mathematical Analysis to the theories of Electricity and Magnetism
George Green
TL;DR
The Essay addresses the need for a general method to calculate electric and magnetic phenomena beyond solutions tailored to particular geometries. It develops the potential function and related mathematical relations, then applies them to conducting systems and magnetic bodies, deriving general reciprocity, shielding, charge-distribution, and magnetic-force results. The treatment is bounded where atmospheric measurements are too variable for accurate experimental testing and where large separations weaken the plate effect.
Problem
A general method for analyzing electric and magnetic phenomena is needed because existing mathematical treatments are adapted mainly to particular objects.
Method
The Essay relates electric densities and magnetic states to potential functions and develops equations for conducting surfaces, enclosed systems, and magnetic bodies.
Results
The analysis establishes reciprocity of induced potentials, electrical neutrality relations within jars, independence of interior and exterior systems in conducting shells, and determinate magnetic states from the equations.
Takeaways & Limitations
The potential-function framework yields general results for electric charge distributions, conductor shielding, and magnetic action across conducting bodies.
Abstract
from arXiv · showhide
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.
Introductory observations.
The Essay seeks a general mathematical method for analyzing equilibrium in electric and magnetic fluids, centered on relations between force, density, and the potential function. It then applies this framework to conducting bodies, electrical jars, induced charge, and magnetic plates.
- Introductory observations.: The Essay submits equilibrium phenomena of electric and magnetic fluids to mathematical analysis and seeks principles applicable to perfect and imperfect conductors.Green emphasizes that the method produces results with simplicity and generality that ordinary demonstrations would make difficult to obtain.
- Introductory observations.: The potential function represents the combined action of an electrified system through the sum of each electric element divided by its distance from the point considered.Green introduces the term because the function gives forces in a simple form and recurs throughout the Essay.
- Introductory observations.: The proposed method replaces a surface equation lacking a general solution theory with relations between electric density and potential functions.Green presents the ordinary formulation as solvable only when problem-specific considerations make the solution unusually simple.
- Introductory observations.: For an insulated electrical jar, the inner and outer surface densities depend only on the difference between the constant potentials inside the connected conductors.The adjacent surface charges neutralize in total, so the jar’s net charge can be found from the metallic surfaces farthest from the glass.
- Introductory observations.: In a jar connected to a spherical conductor, interior-surface density relates to conductor density as the spherical radius relates to the local glass thickness.For equal similar jars charged by cascade, the total interior electricity equals the charge one jar would receive alone, limiting cascade charging for large accumulation.
- Introductory observations.: The analysis gives induced densities and potentials for spherical surfaces, constructs conducting shapes with rigorously assignable surface densities, and describes rapid magnetic equilibration and plate-mediated forces.For a large soft-iron plate, the force is represented by an infinite aligned sequence of image points multiplied by a small constant, within a stated distance regime.
General preliminary results.
Green establishes the potential function as a general mathematical framework for deriving forces, charge densities, and interior or exterior potentials. He then proves uniqueness, reciprocity, and generalization results for conducting surfaces and systems.
- Definition and force relations: The potential function V is defined as the sum of electric particles divided by their distances from the point under consideration, with forces obtained from its differentials.This formulation replaces separate force calculations with analysis of a single function.
- Conductors and charge density: For a perfectly conducting body, the interior electricity density is zero, while non-perfect conductors require knowledge of the interior potential V to determine density.The result follows from the differential equation relating density to the potential function.
- Uniqueness and construction: Given boundary values of V on a closed surface, only one function satisfies the governing equation and remains nonsingular inside, establishing uniqueness of the potential.Green also constructs the interior and exterior potentials from surface data and induced charge density.
- Reciprocity and multiple bodies: The potential induced by a point charge inside a conducting surface is reciprocal: interchanging the source point and evaluation point leaves its value unchanged.This reciprocity extends to systems of conducting bodies under the corresponding boundary conditions.
- Applications to electrical jars: In electrical jars, charges on corresponding inner coating surfaces neutralize, so the total charge can be calculated from the two exterior surfaces; cascade charging therefore cannot exceed one jar’s grounded charge.Green derives both the local surface-density relation and the system-level charging consequence.
- Shell theorems: A hollow conducting shell isolates its interior system from exterior bodies, while the exterior system behaves as if the interior system were absent and the outer surface carried the combined charge.This separates interior and exterior attractions, repulsions, and surface densities into independent problems.