A Brief Guide to the Great Equations (21 page)

By this time, Maxwell was showing symptoms of what turned out to be abdominal cancer. That November he died. The career of this prodigious, inventive force of nature – whose quiet pursuits transformed the world, Feynman claimed, more profoundly than the Civil War – died at the age of only forty-eight.

Maxwell left unfinished business – exciting ideas suggested by his work that, for one reason or another, he had not pursued. One was the question of producing and detecting electromagnetic waves; another was measuring the ether drift; a third was revamping his series of equations in a concise way for practical use – which was becoming increasingly important with the expansion of telegraphs. All three of these were carried out in the decade following Maxwell’s death.

Heinrich Hertz (1857–1894) was born and raised in Hamburg, and in 1878 began to study in Berlin under Hermann von Helmholtz, who was investigating Maxwell’s electrodynamics. Helmholtz tried
to entice the bright 22-year-old to compete for a prize to be awarded for the person who solved an experimental problem, devised by Helmholtz himself, that would confirm a certain feature of Maxwell’s theory. The youngster declined, afraid the work would absorb several years and not result in a big enough effect to be decisive, and finished his doctoral dissertation instead. In 1885, Hertz moved to Karlsruhe, where he had access to a well-equipped laboratory that he put to use inventively. In 1886, the chance observation that an oscillating current caused sparks to jump across small gaps in a nearby loop of wire set Hertz on a path that led to the publication, in the July 1888 issue of
Annalen der Physik
, of an article entitled ‘On Electromagnetic Waves in Air and Their Reflection.’ Hertz was able to measure the wavelength of these electromagnetic waves, and showed they had the properties of other kinds of waves – including the ability to reflect, refract, interfere, and be polarized, and had a finite speed – in stunning confirmation of Maxwell’s theory.

Meanwhile, a physics professor at Liverpool in England named Oliver Lodge had noted that oscillating currents created waves in wires. In July 1888, Lodge completed a paper on his results and boarded a train to the Alps for a hiking holiday. En route, he pulled out his reading material – that month’s issue of the
Annalen
– to learn of Hertz’s work. Lodge was dismayed; he was planning to attend the annual meeting of the British Association for the Advancement of Science that September in Bath and had expected to be celebrated for his discovery, but now realized that Hertz’s work would overshadow his. Yet Lodge also found himself thrilled by the elegance of Hertz’s experiments, which were much more extensive than his own, for Hertz had detected electromagnetic waves not just in wires but also in air.

The Bath meeting was the first public presentation of Hertz’s discovery to the broader scientific community, and the circumstances were rather dramatic.
¹³
The president of the Mathematics and Physics Section had fallen ill, and his last-minute replacement was Irish physicist George FitzGerald (1851–1901), who had been studying
the possibility of producing electromagnetic waves for almost a decade, and who was therefore well-prepared to state the significance of Hertz’s work. So while this popular meeting featured a new wax phonograph by Thomas Edison, and a speech on ‘Social Democracy’ by George Bernard Shaw, FitzGerald all but stole the show with the news: electromagnetic force does not work through action at a distance, but by waves traveling through the ether. ‘The year 1888’, FitzGerald announced, ‘will be ever memorable as the year in which this great question has been experimentally decided by Hertz in Germany.’ Alerted by FitzGerald’s announcement,
Time
magazine called the news ‘epoch-making.’ Yet confirmation of Maxwell’s ideas also brought to the surface the deep and long-standing dissatisfaction with Maxwell’s impractical formulations: FitzGerald spoke of attempts by meeting participants to ‘murder ψ’ and at least revise the vector potential
A
, and the consensus of the gathering was that some conceptual homicide was necessary.

The dramatic news of the creation and detection of electromagnetic waves – implied by Maxwell’s work but not discussed by him – also provided a classic illustration of the unexpected productivity of equations themselves. As Hertz once said of Maxwell’s equations, ‘One cannot escape the feeling that these mathematical formulae have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than was originally put into them.’
¹⁴

Maxwell’s letter about ether drift, sent to the director of the Nautical Almanac Office, was read to the Royal Society at the beginning of January 1880, 2 months after Maxwell’s death, and then published in
Nature
. One fascinated reader was American physicist Albert A. Michelson (1852–1931). A graduate of the U.S. Naval Academy in Annapolis, Maryland, who remained there to teach science, Michelson was entranced by the challenge of measuring the
speed of light, making attempts in 1878 and 1879, playing hooky from the academy’s traditional July 4 celebration to pursue the work. The 1879 measurement, in which he shot a beam of light down a 2000-foot path and back, had an unprecedented precision, earning the 27-year-old a reputation among U.S. scientists and front-page mention in the August 29 edition of
The New York Times
. Michelson’s fame, however, did not impress the academy enough to release him from a scheduled sea-duty – but he managed to pull strings and secure a leave of absence, allowing him to travel to Europe at the beginning of 1880 to study physics in Helmholtz’s lab. After reading Maxwell’s posthumous letter in
Nature
in January 1880, Michelson invented a device, called an interferential refractometer, that used mirrors to split a beam of light by refracting (bending) it, then sent the two beams along two paths at right angles to each other and back. When the two beams were made to interfere, the difference due to their travels through the ether in different directions would be on the order of a fraction of a wavelength – but this tiny difference would be ‘easily measurable’, he wrote to
Nature
.
¹⁵
Explaining the planned experiment to his children, Michelson asked them to imagine a race between ‘two swimmers, one struggling upstream and back, while the other, covering the same distance, just crosses the river and returns.’ The point, he said, is that ‘The second swimmer will always win, if there is any current in the river.’
¹⁶

A first experiment in 1881 detected no drift, and seemed to have design flaws. Michelson quit active duty, moved to the Case School of Applied Science, in Cleveland, Ohio, and collaborated with Edward Morley (1838–1923), another experimenter, to enlarge and revise the apparatus. This experiment, too, detected no drift, despite an astounding sensitivity of a quarter part per billion. Michelson was baffled and disappointed by the null result, and he and Morley abandoned their plans for further measurements. But other scientists, including George FitzGerald, Dutch physicist Hendrik Lorentz, and French physicist Henri Poincaré, undertook desperate attempts to trust both the Michelson-Morley experiment
and
the existence of the
ether, efforts that set the stage for Albert Einstein’s discovery of special relativity. In 1907, for his role in the magnificent experiment that made it possible – inspired by Maxwell’s letter – Michelson became the tenth person, and the first American citizen, to win the Nobel Prize in Physics.

The standardization was largely due to Oliver Heaviside (1850– 1925), a self-taught electrical engineer, eccentric, and maverick (and discoverer of what was once called the Heaviside layer and now called the ionosphere), who is often called ‘the last amateur of science.’
¹⁷
He left home at sixteen, never had a job in a university, and struggled in poverty, supported by relatives, friends, and a government pension. His one and only job was a 4-year stint as a telegraph operator, and he was avidly interested in the practical issue of improving the flow of energy down telegraph wires. He picked up much of contemporary mathematics on his own, using it in novel ways to improve the state of electromagnetic theory; he introduced imaginary numbers into electricity, for instance. When Heaviside came upon Maxwell’s
Treatise
, his reaction to it was somewhat the same as Maxwell’s own to the then-current state of electrical science: far too complex to be useful to practical folk, for far too many things have to be held in one’s head simultaneously. Maxwell’s formulation of his theory – founded in the vector potential
A
and the electrostatic potential ψ, a relic of the ‘action-at-a-distance’ perspective – was particularly ill-suited to the increasingly urgent concerns of telegraphy, which involved the flow of electromagnetic energy down specific pathways.

The demands of this practical technology, indeed, did much to advance the science of electromagnetism in the 1880s.
¹⁸
Many electromagnetic researchers at the time made clever physical models, involving wheels and connecting bands, to picture to themselves how electrical energy flows from place to place in Maxwell’s theory.
Many were frustrated in particular by Maxwell’s use of the potentials
A
and ψ.

In 1883, in a series of articles in a magazine called the
Electrician
, Heaviside began to examine how Maxwell’s work might be adapted for the practical context of studying the flow of electricity in telegraph wires and circuits. ‘[I]t was only by changing its form of presentation that I was able to see it clearly’, Heaviside wrote later.
¹⁹
His amateur, self-taught condition served him well, for he was not inhibited by current mathematical lore nor impressed by prevailing physical perspectives. His outlook was practical; what was important to him was the energy at each point, and calculating how that energy flowed down a path such as a wire. He was prone to expressing that outlook charmingly, in simple and direct terms, as in the following lead sentence from a paragraph in one of his scientific papers: ‘When energy goes from place to place, it traverses the intermediate space.’
²⁰
He then boldly reworked Maxwell’s sets of equations in terms of
E
and
H
to represent the electric and magnetic forces at each state, and currents
D
and
B
. The result was a sweeping condensation of Maxwell’s work into four equations. These four were pleasingly symmetrical – two electric, two magnetic, and the parallel evident. And they are so thoroughly revamped that they are sometimes called ‘Heaviside’s equations.’
²¹
The equations for free space are the following:

div ε
E
= ρ      curl
H
=
k
E
+ e
Ė

div μ
H
= 0      −curl
E
= μ

and in their more complicated form in the presence of electric charges

div ε
E
= ρ      curl (
H
−
h
₀
−
h
) =
k
E
+ ε
Ė
+
u
ρ

div μ
H
= σ      −curl (
E
−
e
₀
−
e
) =
g
H
+ μ
+
u
σ

Heaviside himself modestly referred to his four equations as ‘Maxwell Redressed’,
²²
though he did promote them, enthusiastically and polemically, as superior to Maxwell’s own equations and
to other revisions thereof. Shortly after the 1888 Bath meeting, for instance, he published a brief note savagely attacking the continued use in propagation equations of the electric potential ψ and the vector potential
A
as ‘metaphysical’ (a term of opprobrium for scientists) and as ‘a mathematical fiction.’
²³
What we measure, after all, are the electric force
E
and the magnetic force
H
, not potentials. These give us real information about the state of the field; these are what propagate when current flows. Keeping ψ and
A
results in ‘an almost impenetrable fog of potentials’ and even inconsistencies, and Heaviside, recalling the Bath conference, advocated their ‘murder.’ Maxwell’s theory works just fine, he concluded, ‘provided that we regard
E
and
H
as the variables.’