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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.”