Lawrence Krauss - The Greatest Story Ever Told--So Far
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on his greatest work, a complete theory of electricity and magnetism.
To do this, Maxwell used Faraday’s discovery as the key to
revealing that the relationship between electricity and magnetism is
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symmetrical. Oersted’s and Faraday’s experiments had shown,
simply, that a current of moving charges produces a magnetic field;
and that a changing magnetic field (produced by moving a magnet
or simply turning on a current to produce a magnet) produces an
electric field.
Maxwell first expressed these results mathematically in 1861, but
soon realized that his equations were incomplete. Magnetism
appeared to be different from electricity. Moving charges create a
magnetic field, but a magnetic field can create an electric field even
without moving—just by changing. As Faraday discovered, turning
on a current, which produces a changing magnetic field as the
current ramps up, produces an electric force that causes a current to
flow in another nearby wire.
Maxwell recognized that to make a complete and consistent set of
equations for electricity and magnetism he had to add an extra term
to the equations, representing what he called a “displacement
current.” He reasoned that moving charges, namely a current,
produce a magnetic field, and moving charges represent one way to
produce a changing electric field (since the field from each charge
changes in space as the charge moves along). So, maybe, a changing
electric field—one that gets stronger or weaker—in a region with no
charges in motion, could produce a magnetic field.
Maxwell envisioned that if he hooked up two parallel plates to
opposite poles of a battery, each plate would get charged with an
opposite charge as current flowed from the battery. This would
produce a growing electric field between the plates and would also
produce a magnetic field around the wires connected to the plates.
For his equations to be completely consistent, Maxwell realized, the
increasing electric field between the plates should also produce a
magnetic field in that empty space between the plates. And that field
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would be the same as any magnetic field produced by a real current
flowing through that space between the plates.
So Maxwell altered his equations by adding a new term
(displacement current) to produce mathematical consistency. This
term effectively behaved like an imaginary current, flowing between
the plates producing a changing electric field identical in magnitude
to the actual changing electric field in the empty space between the
plates. It also was the same as the magnetic field that a real current
would produce if it flowed between the plates. Such a magnetic field
does in fact arise when you perform the experiment with parallel
plates, as undergraduates demonstrate every day in physics
laboratories around the world.
Mathematical consistency and sound physical intuition generally
pay off in physics. This subtle change in the equations may not seem
like much, but its physical impact is profound. Once you remove real
electric charges from the picture, it means that you can describe
everything about electricity and magnetism entirely in terms of the
hypothetical “fields” that Faraday had relied upon purely as a mental
crutch. The connections between electricity and magnetism can thus
be simply stated: A changing electric field produces a magnetic field.
A changing magnetic field produces an electric field.
Suddenly the fields appear in the equations as real physical objects
in their own right and not merely as a way to quantify the force
between charges. Electricity and magnetism became inseparable. It is
impossible to talk about electrical forces alone because, as I will
shortly show, one person’s electric force is another person’s
magnetic force, depending on the circumstances of the observer, and
whether the field is changing in his frame of reference.
We now refer to electromagnetism to describe these phenomena,
for a good reason. After Maxwell, electricity and magnetism were no
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longer viewed as separate forces of nature. They were different
manifestations of one and the same force.
Maxwell published his complete set of equations in 1865 and later
simplified them in his textbook of 1873. These would become
famous as the four Maxwell’s Equations, which (admittedly rewritten
in modern mathematical language) adorn the T-shirts of physics
undergraduates around the world today. We can thus label 1873 as
establishing the second great unification in physics, the first being
Newton’s recognition that the same force governed the motion of
celestial bodies as governed falling apples on Earth. Begun with
Oersted’s and Faraday’s experimental discoveries, this towering
achievement of the human intellect was completed by Maxwell, a
mild-mannered young theoretical physicist from Scotland, exiled to
England by the vicissitudes of academia.
Gaining a new perspective on the cosmos is always—or should be
—immensely satisfying. But science adds an additional and powerful
benefit. New understanding also breeds tangible and testable
consequences, and often immediately.
So it was with Maxwell’s unification, which now made Faraday’s
hypothetical fields literally as real as the nose on your face. Literally,
because it turns out you couldn’t see the nose on your face without
them.
Maxwell’s genius didn’t end just with codifying the principles of
electromagnetism in elegant mathematical form. He used the
mathematics to unravel the hidden nature of that most fundamental
of all physical quantities—which had eluded the great natural
philosophers from Plato to Newton. The most observable thing in
nature: light.
Consider the following thought experiment. Take an electrically
charged object and jiggle it up and down. What happens as you do
this?
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Well, an electric field surrounds the charge, and when you move
the charge, the position of the field lines changes. But, according to
Maxwell, this changing electric field will produce a magnetic field,
which will point in and out of the paper as shown below:
Here the field line pointing into the paper has a cross (the back of
an arrow), and that pointing out of the paper has a dot (the tip of an
arrow). This field will flip direction as the charge changes the
direction of its motion from upward to downward.
But we should not stop there. If I keep jiggling the charged object,
the electric field will keep changing, and so will the induced
magnetic field. But a changing magnetic field will produce an
electric field. Thus there are new induced electric field lines, which
point vertically, changing from up to down as the magnetic field
reverses its sign. I display the electric field line to the right only for
lack of space, but the mirror image will be induced on the left-hand
side.
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/> But that changing electric field will in turn produce a changing
magnetic field, which would exist farther out to the right and left of
the diagram, and so on.
Jiggling a charge produces a succession of disturbances in both
electric and magnetic fields that propagate outward, with the change
in each field acting as a source for the other, due to the rules of
electromagnetism as Maxwell defined them. We can extend the
picture shown above to a 3-D image that captures the full nature of
the changing as shown below:
We see a wave of electric and magnetic disturbances, namely an
electromagnetic wave moving outward, with electric and magnetic
fields oscillating in space, and time, and with the two fields
oscillating in directions that are perpendicular to each other and also
the direction of the wave.
Even before Maxwell had written down the final form of his
equations, he showed that oscillating charges would produce an
electromagnetic wave. But he did something far more significant. He
calculated the speed of that wave, in a beautiful and simple
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calculation that is probably my favorite derivation to show
undergraduates. Here it is:
We can quantify the strength of an electric force by measuring its
magnitude between two charges whose magnitude we already know.
The force is proportional to the product of the charges. Let’s call the
constant of proportionality A.
Similarly we can quantify the strength of the magnetic force
between two electromagnets, each with a current of known
magnitude. This force is proportional to the product of the currents.
Let’s call the constant of proportionality in this case B.
Maxwell showed that the speed of an electromagnetic disturbance
that emanates from an oscillating charge can be rendered precisely
in terms of the measured strength of electricity and the measured
strength of magnetism, which are determined by measuring the
constants A and B in the laboratory. When he used the data then
available for the measured strength of electricity and the measured
strength of magnetism and plugged in the numbers, he derived:
Speed of electromagnetic waves ≈ 311,000,000 meters per second
A famous story claims that when Albert Einstein finished his
General Theory of Relativity and compared its predictions for the
orbit of Mercury to the measured numbers, he had heart
palpitations. One can only imagine, then, the excitement that
Maxwell must have had when he performed his calculation. For this
number, which may seem arbitrary, was well known to him as the
speed of light. In 1849, the French physicist Fizeau had measured the
speed of light, an extremely difficult measurement back then, and
had obtained:
Speed of light ≈ 313,000,000 meters per second
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Given the accuracy available at the time, these two numbers are
identical. (We now know this number far more precisely as
299,792,458 meters per second, which is a key part of the modern
definition of the meter.)
In his typical understated tone, Maxwell noted in 1862, when he
first performed the calculation, “We can scarcely avoid the
conclusion that light consists in the transverse undulations of the
same medium which is the cause of electric and magnetic
phenomena.”
In other words, light is an electromagnetic wave.
Two years later, when he finally wrote his classic paper on
electromagnetism, he added somewhat more confidently, “Light is
an electromagnetic disturbance propagated through the field
according to electromagnetic laws.”
With these words, Maxwell appeared to have finally put to rest
the two-thousand-year-old mystery regarding the nature and origin
of light. His result came, as great insights often do, as an unintended
by-product of other fundamental investigations. In this case, it was a
by-product of one of the most important theoretical advances in
history, the unification of electricity and magnetism into a single
beautiful mathematical theory.
• • •
Before Maxwell, the chief source of wisdom came from a faith in
divinity via Genesis. Even Newton relied upon this source for
understanding the origin of light. After 1862, however, everything
changed.
James Clerk Maxwell was deeply religious, and like Newton
before him, his faith sometimes led him to make strange assertions
about nature. Nevertheless, like the mythical character Prometheus
before him, who stole fire from the gods and gave it to humans to
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use as a tool to forever change their civilization, so too Maxwell stole
fire from the Judeo-Christian God’s first words and forever changed
their meaning. Since 1873, generations of physics students have
proudly proclaimed:
“Maxwell wrote down his four equations and said, Let there be
light!”
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C h a p t e r 4
T H E R E , A N D B A C K A G A I N
He set the earth on its foundations; it can never be moved.
—PSALMS 104:5
When Galileo Galilei was being tried in 1633 for heresy for
“holding as true the false doctrine taught by some that the Sun is the
center of the world,” he allegedly muttered under his breath in front
of his Church inquisitors, “And yet it moves.” With these words, his
revolutionary nature once again sprang forth, in spite of his having
been forced to publicly adhere to the archaic position that the Earth
was fixed.
While the Vatican eventually capitulated on Earth’s motion, the
poor God of the Psalms never got the news. This is somewhat
perplexing since, as Galileo showed a year before the trial, a state of
absolute rest is impossible to verify experimentally. Any experiment
that you perform at rest, such as throwing a ball up in the air and
catching it, will have an identical result if performed while moving at
a constant speed, as, say, might happen while riding on an airplane
in the absence of turbulence. No experiment you can perform on the
plane, if its windows are closed, will tell you whether the plane is
moving or standing still.
While Galileo started the ball rolling, both literally and
metaphorically, in 1632, it took another 273 years to fully lay to rest
this issue (issues, unlike objects, can be laid to rest). It would take
Albert Einstein to do so.
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Einstein was not a revolutionary in the same sense as Galileo, if by
this term one describes those who tear down the dictates of the
authorities who came before, as Galileo had done for Aristotle.
Einstein did just the opposite. He knew that rules that had been
established on the basis of experiment could not easily be tossed
aside, and it was a mark of his genius that he didn’t.
This is so important I want to repeat it for the benefit of those
people who write to me every week or so telling me that they have
discovered a new theory that demonstrates everything we now think
we know about the universe is wrong—and using Einstein as their
exemplar to justify this possibility. Not only is your theory wrong,
but you are doing Einstein a huge disservice: rules that have been
established on the basis of experiment cannot easily be tossed aside.
• • •
Albert Einstein was born in 1879, the same year that James Clerk
Maxwell died. It is tempting to suggest that their combined
brilliance was too much for one simple planet to house at the same
time. But it was just a coincidence, albeit a fortuitous one. If Maxwell
hadn’t preceded him, Einstein couldn’t have been Einstein. He came
from the first generation of young scientists who grew up wrestling
with the new knowledge about light and electromagnetism that
Faraday and Maxwell had generated. This was the true forefront of
physics for young Turks such as Einstein near the end of the
nineteenth century. Light was on everyone’s mind.
Even as a teenager, Einstein was astute enough to realize that
Maxwell’s
beautiful
results
regarding
the
existence
of
electromagnetic waves presented a fundamental problem: they were
inconsistent with the equally beautiful and well-established results of
Galileo regarding the basic properties of motion, produced three
centuries earlier.
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Even before his epic battle with the Catholic Church over the
motion of Earth, Galileo had argued that no experiment exists that
can be performed by anyone to determine whether he or she is
moving uniformly or standing still. But up until Galileo, a state of
absolute rest was considered special. Aristotle had decided that all
objects sought out the state of rest, and the Church decided that rest
was so special that it should be the state of the center of the universe,
namely the planet on which God had placed us.
Like a number of Aristotle’s assertions, although by no means all,
this notion that a state of rest is special is quite intuitive. (For those
who like to quote Aristotle’s wisdom when appealing to his “Prime
Mover” argument for the existence of God, let us remember that he
also claimed that women had a different number of teeth than men,
presumably without bothering to check.)