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Transcript
Maxwell’s Equations The story goes something like this: In 1819, Hans Christian Oërsted at University of Copenhagen was lecturing to his graduate students on the topic of Electricity, Galvanism and Magnetism. “Electricity” meant more of what we would call static electricity; that is, charged objects. “Galvanism” meant phenomena related to electrical currents. “Magnetism” was the study of compass needles and terrestrial magnetism. At the time, most physicists suspected that there was a connection between all three fields, but there was not a single unifying theory for them. Oërsted was trying to create a demonstration for his students that illustrated the point that galvanism and magnetism were somehow connected. What he came up with was simple: he ran a galvanic (electric) current through a wire and placed a compass needle at right angles to the wire to see if the current changed the magnetic field. Nothing happened. The class filtered in, and, before he could further experiment with his setup, the class began. After class, he tried one more time to see if the compass needle would move. He placed the needle parallel to the wire and turned on the current. The needle moved! It pointed almost at right angles to its original orientation. When Oërsted turned off the current, the needle returned to its original position, and when he turned on the current in the other direction, the needle swung in the opposite direction. Shortly thereafter, André-Marie Ampère, a mathematics professor at the École Polytechnique in Paris, demonstrated that two parallel wires with current would attract each other. By reversing the current in one of the wires, he found that the wires now repelled each other. Another part of the story is: In 1846, a famous English physicist, Sir Charles Wheatstone, was to present a lecture to an audience at the Royal Institution of Great Britain in London. Michael Faraday, a chemist at the Royal Institution, was to introduce Wheatstone to the audience, but Wheatstone had a sudden bout of stage fright, and ran out into the street. Faraday was left to present the lecture. He was known for his experiments: he described a few of them. In one, he wrapped an iron ring with two coils of insulated wire. If he turned on an electrical current in one wire, the other wire, not connected to any current source, would nevertheless exhibit a current. In another experiment, he found that if he simply put a loop of insulated wire around a bar magnet, then moved the wire, the wire itself would exhibit an electrical current. Finally, he described an experiment in which he was able to change the nature (polarization) of a light beam by passing it through a piece of glass that had a magnetic field applied to it. It was this last experiment that caused him to speculate to the audience at the lecture that a moving magnet was surrounded by a force “field” (what we now call an electric field), that changes in that force field generated the forces felt by wires, and, more startlingly, that light was an electromagnetic wave. He later said that, were it not for the sudden opportunity to talk about it, he would never have made his speculations public. As it was at that time, he had no theory to connect his speculations, nor had he any evidence for the “force field” or for the electromagnetic wave nature of light. The final part of the story: In 1854, James Clerk Maxwell, the son of a family who had a baronetcy to its name, had just graduated from the University of Cambridge, earning the title of second wrangler, by having the second-highest score in the arduous (44½ hours over eight days) and difficult (top scores were commonly below 50%) Mathematical Tripos examination. This feat announced to the world at large of his mathematical abilities. However, he was interested in applying his mathematical prowess to problems of physics; at one point, he read one of Faraday’s published papers and started a lively correspondence with him. Faraday, in contrast to Maxwell, did not graduate from college (indeed, all of his degrees, conferred later in his life, were honorary), did not have aristocratic parents (Faraday himself turned down a knighthood on religious grounds) and was dyslexic, which made mathematics impossible forhim. His dyslexia, though, gave him an incredible visualization ability, which he used to describe his electricity and magnetism experiments. Therefore, when Maxwell tried to convince Faraday to convert visualizations into the language of the physics of Newton, Faraday replied: I perceive that I do not use the word 'force' as you define it, 'the tendency of a body to pass from one place to another.' What I mean by the word is the source or sources of all possible actions of the particles or materials of the universe, these being often called the powers of nature when spoken of in respect of the different manners in which their effects are shown. Maxwell nevertheless read Faraday’s papers thoroughly, developing a mathematical version of what Faraday described of his visualizations. Faraday also immersed himself in the works of Oërsted and Ampère, and, over time, realized that, in fact, electricity and galvanism were the same thing, except that electricity, as defined at the time, was static and galvanism was dynamic. Moreover, both of those fields were related to magnetism mathematically. Using calculus, he discovered that he could “re-invent” some previously known laws: The first of Maxwell’s equation is Gauss’s Law, which the University of Gottingen mathematics professor Carl Friedrich Gauss formulated in 1831, describing the existence of electrical charges and the electric fields and forces they create. The second equation is called “Gauss’s Law of Magnetism”, which Gauss did not discover, but is a variation on the original law. Its significance is that it states that there is no such thing as a magnetic charge, unlike an electrical charge which can exist either negatively or positively. The third equation is a version of Faraday’s Law, which states that a time-varying magnetic field induces an electric field, which is the principle underlying all mechanical generators. The fourth equation is Ampère’s Law which states that a magnetic field can be created by an electric current, or, as Maxwell added, by a changing electric field. This latter part caused the most controversy, as Maxwell did not know exactly what an electric field was, so he called it a “displacement current”, and that it existed even in a vacuum. Many critics jumped on that abstraction; William Thomson (later, Lord Kelvin) how such a current could arise out of “nothingness”. It would not be until the acceptance of the abstraction of spacetime (something that Einstein’s theory of relativity would help poularize) that the displacement current would find a fitting visualization. Maxwell’s equations allowed him to predict the speed of the propagation of electromagnetic radiation. Electromagnetic radiation was the result of electric and magnetic fields spreading further from its origin, the electric field inducing the magnetic field, which in turn induced the electric field, and so on. The resulting speed was very close to the measured speed of light. In fact, later work showed that the propagating electric and magnetic fields were electromagnetic radiation; light is an electromagnetic wave, as Faraday had postulated. Epilogue Maxwell published his preliminary ideas in a paper in 1861; this was followed up in two papers and a presentation to the Royal Society of London in 1864 and 1865. He summarized all his work in this field in Treatise on Electricity and Magnetism in 1873. Because vector calculus was still in its infancy, Maxwell originally generated twenty equations and some notation that is no longer used. By the time the mathematical tools were developed, Maxwell had died of stomach cancer at age 48. Oliver Heaviside, an electrical engineer at the Great Northern Telegraph Company, obtained a copy of Maxwell’s Treatise and was determined to study and understand it, at one point, quitting his job and moving back in with his parents. He used the newlydeveloped vector calculus to simplify Maxwell’s twenty equations to the well-known four, and, in doing so, centered the equations around the electric and magnetic fields. Asked why they aren’t named after him, Heaviside replied, in his 1893 publication Electromagnetic Theory, that if we had good reason “to believe that he [Maxwell] would have admitted the necessity of change when pointed out to him, then I think the resulting modified theory may well be called Maxwell’s.”