## George Green Makes the First Attempt to Formulate a Mathematical Theory of Electricity and Magnetism (1828)

# George Green Makes the First Attempt to Formulate a Mathematical Theory of Electricity and Magnetism (1828)

*Overview*

Over the course of the nineteenth century the science of electricity and magnetism advanced from laboratory curiosity to a fully developed theory that would provide the basis for several major technologies. Essential to this development was the development of a mathematical apparatus to describe the behavior of fields, physical states characterized by a vector or scalar at every point in space. A critical initial step was provided in 1838 by George Green (1793-1841), then a self-taught amateur mathematician. A complete formulation of the behavior of electromagnetic fields was achieved over the next 35 years.

*Background*

While the ancient Greeks were familiar with both static electricity and permanent magnets, the nature of the two phenomena remained a subject of speculation until the beginning of the nineteenth century. In 1800 the Italian physicist Count Alessandro Volta (1745-1827) created the Voltaic "pile," a dependable source of electric current, and vast new experimental possibilities arose. In 1820, the Danish physicist Hans Christian Oersted (1777-1851) reported that a current carrying wire had an effect on compass needles placed around it, a report that quickly brought new investigators into the field. By 1821 Oersted's experiments were being reproduced and expanded upon by two men who would play a major role in the development of the new science of electromagnetism—André Marie Ampère (1775-1836) in France and Michael Faraday (1791-1867) in England.

It was recognized that the electric and magnetic forces had some of the same characteristics as the gravitational force, but were somewhat more complex in character. It was easier, particularly in the case of magnetism, to think of each charged or magnetic object as setting up a disturbance in the space around it which would determine the force that would act on a charged or magnetic object placed at that point. The electric and magnetic fields each assigned a vector quantity to each point in space. As one went from any point to neighboring points, the magnitude and direction of the field would change—the rate of change being determined by the material present.

The analogous problem in fluid flow had been treated by Swiss mathematician Daniel Bernoulli (1700-1782) in a 1789 book on hydrodynamics. In a paper on fluids in 1752, the prolific Swiss mathematician Leonhard Euler (1707-1783) showed that the potential function satisfied a very simple equation involving second partial derivatives, a equation now generally known as Laplace's equation after the French mathematician Pierre Simon Laplace (1749-1827).

In March of 1828, George Green, a self-taught English amateur mathematician, published a work entitled "An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism." In this work Green introduced the notion of potential functions for the electric and magnetic field and showed how to construct the function by adding contributions from each charge. This essay included a very important formula, now known as Green's theorem.

Green's work, distributed to only the 52 individuals who had financially supported the work, might have been lost had it not been for Sir William Thomson (aka Lord Kelvin; 1824-1907), who had it reprinted in a German mathematics journal in 1850. A few years earlier Thomson had noted that the solution to Laplace's equation that took on a defined set of values on the boundary of a region of space would be, of all sufficiently "smooth" functions that satisfied the boundary conditions, that function which minimizes an integral commonly known as the Dirichlet integral after the German Mathematician Lejeune Dirichlet (1805-1859). Green and Thomson's results together make it possible to determine the potential over a region of space, given either its values or that of its derivatives (the field) over the boundary.

Over the course of the nineteenth century, considerable effort was devoted to elucidating the nature of light. This was motivated by the discovery of the polarization of light on reflection by Etienne Louis Malus (1775-1812) in 1808 and the strange "double refraction" of light into polarized beams in crystalline materials such Iceland spar, a form of calcium carbonate. Researcher in optics at the time considered light to be a vibration in a medium, the "luminiferous aether" which filled all space and somehow interacted with matter, but did not slow the motions of bodies moving through it. The Irish mathematician and astronomer Sir William Rowan Hamilton (1805-1865) devoted several years of effort to this problem, not leaving any results of lasting value to optics, but developing the mathematical techniques later to be successfully applied by himself and others, including the German Karl Gustav Jacobi (1804-1851), to problems in mechanics.

The true nature of light became apparent when the English mathematical physicist James Clerk Maxwell (1831-1879) formulated the set of four Maxwell equations describing the behavior of the electric and magnetic fields in space. Applying the mathematical techniques of Green and others to the experimental observations of Ampère and Faraday, Maxwell derived in 1864 a set of four coupled partial differential equations which in empty space could take on the form of wave equations for the components of the electric and magnetic fields. The velocity of the waves was given in the equations in terms of the fundamental force constants of the electric and magnetic force, and turned out to be 300,000 km per second, exactly the measured speed of light in vacuum. There could then be little doubt that light was a form of electromagnetic radiation and that the effects of matter on light, reflection, refraction, and polarization could be calculated from the interactions of the electromagnetic field with the charged particles making up the matter in question.

*Impact*

It is interesting to note the unconventional educational backgrounds of some of the pioneers of electromagnetism. Ampère was born into an upper middle-class family and would most likely have prepared for legal practice or the church were it not for the excesses of the French Revolution, which lead to the execution of his father and the loss of the family fortune. Although he never received a degree, he would be appointed to numerous important university posts in post-revolutionary France. Faraday was born into a working-class family and received an education working in an institution originally founded to provide scientific instruction to the working class. Green was a baker's son who left school early but was nonetheless able to educate himself through independent reading. After making his important contributions Green was admitted to Caius College as a scholar at the age of 40. The science of electromagnetism was thus developed very rapidly during a time of rapid social and economic change by men who would not have been considered educated by traditional standards.

It would be difficult to overstate the impact of the development of electromagnetic theory on the conditions of life in the industrialized world. The principle of electromagnetic induction, discovered by Faraday and incorporated into Maxwell's equations, made possible the design of electric generators and motors, which in turn made it possible to separate the production of electrical energy from its use in industrial production. Maxwell's identification of light as an electromagnetic wave led directly to the discovery of radio waves and the revolution in communications and mass culture that followed. One of the more astonishing conclusions of consequences of Maxwell's theory was that the speed of light would have the same velocity regardless of the relative velocities of the source and the observer. This conclusion led Albert Einstein (1879-1955) to recast the principles of mechanics in the special theory of relativity. It also abolished any need for a "luminiferous aether" in physics. Over the next century, understanding the atomic structure of matter and its interaction with radiation have created a world of lasers and optical fiber communication, none of which would be possible without the mathematical techniques developed by Green and his contemporaries.

**DONALD R. FRANCESCHETTI**

*Further Reading*

Bell, Eric Temple. *The Development of Mathematics.* New York: McGraw-Hill, 1945.

Boyer, Carl B. *A History of Mathematics.* New York: Wiley, 1968.

Crowe, Michael J. *A History of Vector Analysis.* New York: Dover, 1985.

Kline, Morris. *Mathematical Thought from Ancient to Modern Times.* New York: Oxford University Press, 1972.

Whittaker, Sir Edmund T. *A History of the Theories of Aether and Electricity.* Vol. 1. New York: American Institute of Physics, 1987.