An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism
Encyclopedia
George Green
George Green was a British mathematical physicist who wrote An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism...
, extended the work of Siméon Denis Poisson
Siméon Denis Poisson
Siméon Denis Poisson , was a French mathematician, geometer, and physicist. He however, was the final leading opponent of the wave theory of light as a member of the elite l'Académie française, but was proven wrong by Augustin-Jean Fresnel.-Biography:...
concerning electricity
Electricity
Electricity is a general term encompassing a variety of phenomena resulting from the presence and flow of electric charge. These include many easily recognizable phenomena, such as lightning, static electricity, and the flow of electrical current in an electrical wire...
and magnetism
Magnetism
Magnetism is a property of materials that respond at an atomic or subatomic level to an applied magnetic field. Ferromagnetism is the strongest and most familiar type of magnetism. It is responsible for the behavior of permanent magnets, which produce their own persistent magnetic fields, as well...
. The work's theorem of pure analysis is of the very greatest importance in all branches of physical mathematics. It contains the first exposition of the theory of potential
Potential theory
In mathematics and mathematical physics, potential theory may be defined as the study of harmonic functions.- Definition and comments :The term "potential theory" was coined in 19th-century physics, when it was realized that the fundamental forces of nature could be modeled using potentials which...
. In physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, Green's theorem is mostly used to solve two-dimensional flow integrals
Flux
In the various subfields of physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks.* In the study of transport phenomena , flux is defined as flow per unit area, where flow is the movement of some quantity per time...
, stating that the sum of fluid outflows at any point inside a volume is equal to the total outflow summed about an enclosing area. In plane geometry
Plane geometry
In mathematics, plane geometry may refer to:*Euclidean plane geometry, the geometry of plane figures,*geometry of a plane,or sometimes:*geometry of a projective plane, most commonly the real projective plane but possibly the complex projective plane, Fano plane or others;*geometry of the hyperbolic...
, and in particular, area surveying
Surveying
See Also: Public Land Survey SystemSurveying or land surveying is the technique, profession, and science of accurately determining the terrestrial or three-dimensional position of points and the distances and angles between them...
, Green's theorem can be used to determine the area
Area
Area is a quantity that expresses the extent of a two-dimensional surface or shape in the plane. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat...
and centroid of plane
Centroid
In geometry, the centroid, geometric center, or barycenter of a plane figure or two-dimensional shape X is the intersection of all straight lines that divide X into two parts of equal moment about the line. Informally, it is the "average" of all points of X...
figures solely by integrating over the perimeter
Perimeter
A perimeter is a path that surrounds an area. The word comes from the Greek peri and meter . The term may be used either for the path or its length - it can be thought of as the length of the outline of a shape. The perimeter of a circular area is called circumference.- Practical uses :Calculating...
.
It is in this essay that the term 'potential function' first occurs. Herein also his remarkable theorem
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
in pure mathematics
Pure mathematics
Broadly speaking, pure mathematics is mathematics which studies entirely abstract concepts. From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics, and at variance with the trend towards meeting the needs of...
, since universally known as Green's theorem
Green's theorem
In mathematics, Green's theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C...
, and probably the most important instrument of investigation in the whole range of mathematical physics, made its appearance. We are all now able to understand, in a general way at least, the importance of Green's work, and the progress made since the publication of his essay in 1828. But to fully appreciate his work and subsequent progress one needs to know the outlook for the mathematico-physical sciences as it appeared to Green at this time and to realize his refined sensitiveness in promulgating his discoveries.
The 1828 essay
Poisson's electrical and magnetical investigations were generalized and extended in 1828 by George Green. Green's treatment is based on the properties of the function already used by Lagrange, Laplace, and Poisson, which represents the sum of all the electric or magnetic charges in the field, divided by their respective distances from some given point: to this function Green gave the name potential, by which it has always since been known.In 1828, Green published the paper which is the essay he is most famous for today. When Green published his Essay, it was sold on a subscription basis to 51 people, most of whom were friends and probably could not understand it. The wealthy landowner and mathematician Edward Bromhead bought a copy and encouraged Green to do further work in mathematics. Not believing the offer was sincere, Green did not contact Bromhead for two years.
Upon publishing the work, he first introduced the term 'potential' to denote the result obtained by adding the masses of all the particles of a system, each divided by its distance from a given point; and the properties of this function are first considered and applied to the theories of magnetism and electricity. This was followed by two papers communicated by Sir Bromhead to the Cambridge Philosophical Society
Cambridge Philosophical Society
The Cambridge Philosophical Society is a scientific society at University of Cambridge. It was founded in 1819. The name derives from the medieval use of the word philosophy to denote any research undertaken outside the fields of theology and medicine...
: (1)' On the Laws of the Equilibrium of Fluids analogous to the Electric Fluid ' (12 Nov. 1832); (2)' On the Determination of the Attractions of Ellipsoids of Variable Densities ' (6 May 1833). Both papers display great analytical power, but are rather curious than practically interesting. Green's 1828 essay was neglected by mathematicians till 1846, and before that time most of its important theorems had been rediscovered by Gauss, Chasles, Sturm, and Thomson J. It did influence the work of Lord Kelvin and James Clerk Maxwell
James Clerk Maxwell
James Clerk Maxwell of Glenlair was a Scottish physicist and mathematician. His most prominent achievement was formulating classical electromagnetic theory. This united all previously unrelated observations, experiments and equations of electricity, magnetism and optics into a consistent theory...
.
The self-taught mathematician's essay was one of the greatest advances that were made in the mathematical theory of electricity up to his time. "His researches," as Sir William Thomson
William Thomson, 1st Baron Kelvin
William Thomson, 1st Baron Kelvin OM, GCVO, PC, PRS, PRSE, was a mathematical physicist and engineer. At the University of Glasgow he did important work in the mathematical analysis of electricity and formulation of the first and second laws of thermodynamics, and did much to unify the emerging...
has observed, "have led to the elementary proposition which must constitute the legitimate foundation of every perfect mathematical structure that is to be made from the materials furnished in the experimental laws of Coulomb. Not only do they afford a natural and complete explanation of the beautiful quantitative experiments which havs been so interesting at all times to practical electricians, but they suggest to the mathematician the simplest and most powerful methods of dealing with problems which, if attacked by the mere force of the old analysis, must have remained forever unsolved."
Near the beginning of the memoir is established the celebrated formula connecting surface and volume integrals, which is now generally called Green's Theorem, and of which Poisson's result on the equivalent surface – and volume – distributions of magnetization is a particular application. By using this theorem to investigate the properties of the potential, Green arrived at many results of remarkable beauty and interest. We need only mention, as an example of the power of his method, the following: — Suppose that there is a hollow conducting shell, bounded by two closed surfaces, and that a number of electrified bodies are placed, some within and some without it; and let the inner surface and interior bodies be called the interior system, and the outer surface and exterior bodies be called the exterior system. Then all the electrical phenomena of the interior system, relative to attractions, repulsions, and densities, will be the same as if there were no exterior system, and the inner surface were a perfect conductor, put in communication with the earth; and all those of the exterior system will be the same as if the interior system did not exist, and the outer surface were a perfect conductor, containing a quantity of electricity equal to the whole of that originally contained in the shell itself and in all the interior bodies. It will be evident that electrostatics had by this time attained a state of development in which further progress could be hoped for only in the mathematical superstructure, unless experiment should unexpectedly bring to light phenomena of an entirely new character.
One of the simplest applications of these theorems was to perfect the theory of the Leyden phial
Leyden jar
A Leyden jar, or Leiden jar, is a device that "stores" static electricity between two electrodes on the inside and outside of a jar. It was invented independently by German cleric Ewald Georg von Kleist on 11 October 1745 and by Dutch scientist Pieter van Musschenbroek of Leiden in 1745–1746. The...
, a result which (if we except the peculiar action of the insulating solid medium, since discovered by Faraday
Michael Faraday
Michael Faraday, FRS was an English chemist and physicist who contributed to the fields of electromagnetism and electrochemistry....
) we owe to his genius. He has also shown how an infinite number of forms of conductors may be invented, so that the distribution of electricity in equilibrium on each may be expressible in finite algebraical terms, — an immense stride in the science, when we consider that the distribution of electricity on a single spherical conductor, an uninfluenced ellipsoidal conductor, and two spheres mutually influencing one another, were the only cases solved by Poisson, and indeed the only cases conceived to be solvable by mathematical writers.