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Mgiocfrorneigraofic Hồ (cÏfqf6fGrte
28-1 Electromagnetism
'The most dramatic moments in the development of physics are those in which
great syntheses take place, where phenomena which previously had appeared
to be diferent are suddenly discovered to be but diferent aspects of the same
thing. The history of physics is the history of such syntheses, and the basis of
the success of physical science is mainly that we are øble to synthesize.
Perhaps the most dramatie moment in the development of physics during the
19th century occurred to J. C. Maxwell one day in the 1860°s, when he combined
the laws of electricity and magnetism with the laws of the behavior of light. As
a result, the properties of light were partly unravelled—that old and subtle stuf
that is so important and mysterious that it was felt necessary to arrange a special
creation for it when writing Genesis. Maxwell could say, when he was ñnished
with his discovery, “Let there be electricity and magnetism, and there is lightl”
For this culminating moment there was a long preparation in the gradual
discovery and unfolding of the laws of electricity and magnetism. This story we
shall reserve for detailed study next year. However, the story is, briely, as follows.
The gradually discovered properties of electricity and magnetism, of electric Íorces
of attraction and repulsion, and of magnetie forces, showed that although these
forces were rather complex, they all fell off inversely as the square of the distance.
We know, for example, that the simple Coulomb law for stationary charges is
that the electric force field varies inversely as the square of the distance. As a
consequence, for sufficiently great distances there is very little inÑuence of one
system of charges on another. Maxwell noted that the equations or the laws that
had been discovered up to this tìme were mutually inconsistent when he tried to
put them all together, and in order for the whole system to be consistent, he had
to add another term to his equations. With this new term there came an amazing
prediction, which was that a part of the electric and magnetic fields would fall of
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tmmuch more slowly with the distance than the inverse square, namely, inversely as
the first power of the distancel And so he realized that electric currents in one
place can affect other charges far away, and he predicted the basic efects with
which we are familiar today—radio transmission, radar, and so on.
lt seems a miracle that someone talking in Europe can, with mere electrical
inñuences, be heard thousands of miles away in Los Angeles. How is it possible?
lt is because the fields do not vary as the inverse square, but only inversely as
the first power of the distance. Finally, then, even light itself was recognized
to be electric and magnetie inÑuences extending over vast distances, generated
by an almost incredibly rapid oscillation of the electrons in the atoms. All
these phenomena we summarize by the word rad¿øtion or, more specifically,
clectromagnetic radiation, there being one or two other kinds of radiation also.
Almost always, radiation means electromagnetic radiation.
And thus is the universe knit together. The atomic motions of a distant star
siiHl have sufficient inÑuence at this great distance to set the electrons in our eye
in motion, and so we know about the stars. If this law did not exist, we would
all be literally in the dark about the exterior worldl And the electric surgings in
a galaxy fñve billion light years away——which is the farthest object we have found
so far—can still inÑuenee in a signilcant and detectable way the currents in the
great “dish” in front of a radio telescope. And so it is that we see the stars and
the galaxies.
'This remarkable phenomenon is what we shall discuss In the present chapter.
At the beginning of this course in physics we outlined a broad picture of the
world, but we are now better prepared to understand some aspects of it, and
so we shall now go over some parts of it again in greater detail. We begin by
describing the position of physics at the end of the 19%0h century. All that was
then known about the fundamental laws can be summarized as follows.
First, there were laws of forces: one force was the law of gravitation, which
we have written down several times; the force on an object of mass mm, due to
another of mass j, is given by
FPƑ.=GmMe,/rŸ, (28.1)
where e; is a unit vector directed from rn to Mĩ, and r is the distance between
Next, the laws of electricity and magnetism, as known at the end of the
19th century, are these: the electrical forces acting on a charge g can be described
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by two fields, called # and ?Ö, and the velocity ø of the charge g, by the equation
P=q(E+ox Đ). (28.2)
To complete thịs law, we have to say what the formulas for E and Ö are in a
given circumstance: iŸ a number of charges are present, # and the #Ö are each
the sum of contributions, one from each individual charge. So if we can find the
2 and B produced by a single charge, we need only to add all the efects from
all the charges in the universe to get the total # and BI 'This is the principle of
SuperposIfion.
What ¡is the formula for the electric and magnetic field produced by one
individual charge? It turns out that this is very complicated, and it takes a
great deal of study and sophistication to appreciate it. But that is not the
point. We write down the law now only to impress the reader with the beauty
of nature, so to speak, i.e., that it is possible to sunmarize all the fundamental
knowledge on one page, with notations that he is now familiar with. 'This law for
the fields of an individual charge 1s complete and accurate, so far as we know
(except for quantum mechanics) but it looks rather complicated. We shall not
study all the pieces now; we only write it down to give an Impression, to show
that it can be written, and so that we can see ahead of time roughly what ¡it
looks like. As a matter of fact, the most wseƒful way to write the correct laws of
electricity and magnetism is not the way we shall now write them, but involves
what are called field equat¿ons, which we shall learn about next year. But the
mathematical notations for these are different and new, and so we write the law
in an inconvenient form for calculation, but in notations that we now know.
'The electric ñeld, #, is given by
—{ | €Cr: rrd Cự 1 d2
E= 47€o l# + e đdí (#) + c2 đí2 si (28.3)
What do the various terms tell us? Take the frst term, = —qe„:/4meor2.
That, of course, is Coulomb°s law, which we already know: g is the charge that is
produecing the field; ez¿ is the unit vector in the direction from the point where
E2 is measured, z is the distance from ? to g. But, Coulomb's law is wrong. The
discoveries of the 19th century showed that inÑuences cannot travel faster than
a certain fundamental speed c, which we now call the speed of light. I% is not
correct that the first term is Coulomb'°s law, not only because it is not possible to
know where the charge is nøu and at what distance it is œøu, but also because
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