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not take into account the fact that light is goïing at a certain speed, but paints
the world as he sees it. We want to see what his picture would look like. So we
see a dot, representing the charge, moving about in the picture. 'Phe acceleration
of that dot is proportional to the electric field. 'That ¡s all—all we need.
Thus Eq. (28.5) is the complete and correct formula for radiation; even
relativity efects are all contained ín it. However, we often want to apply it to a
still simpler cireumstance in which the charges are moving only a small distanece
at a relatively slow rate. Since they are moving slowly, they do no move an
appreciable distance from where they start, so that the delay time is practically
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constant. 'Then the law ¡s still simpler, because the delay time is fñxed. 'Phus we
imagine that the charge is executing a very tiny motion at an efectively constant
distance. The delay at the distance r is r/c. Then our rule becomes the following:
TÍ the charged object is moving in a very small motion and it is laterally displaced
by the distance #(#), then the angle that the unit vector e¿; is displaced is #/?,
and sinece z is practically constant, the #-component of d2e, /đf? is simply the
acceleration of ø itself at an earlier time divided by z, and so fñnally we get the
law we want, which is
E,0)=— Ta, (: — ^): (28.6)
47coc2r e
Only the component of øx perpendicular to the line of sight is important. Let
us see why that is. Evidently, If the charge is moving in and out straight at us,
the unit vector in that direction does not wiggle at all, and it has no acceleration.
So 1È is only the sidewise motion which is important, only the acceleration that
we see projected on the screen.
28-3 The dipole radiator
As our fundamental “law” of electromagnetic radiation, we are goïing to assume
that (28.6) is true, i.e., that the electric ñeld produced by an accelerating charge
which is moving nonrelativistically at a very large distance ? approaches that
form. 'The electric field varies inversely as r and is proportional to the acceleration
of the charge, projected onto the “plane of sight,” and this acceleration is not
today”s acceleration, but the acceleration that ¡it had at an earlier time, the
amount of delay being a time, r/e. In the remainder of this chapter we shall
discuss this law so that we can understand it better physically, because we are
goïng to use it to understand all of the phenomena of light and radio propagation,
such as refection, refraction, interference, difraction, and scattering. It is the
central law, and is all we need. All the rest of Eq. (28.3) was written down only
to set the stage, so that we could appreciate where (28.6) fts and how i% comes
about.
We shall discuss (28.3) further next year. In the meantime, we shall accept
it as true, but not just on a theoretical basis. We may devise a number of
experiments which illustrate the character of the law. In order to do so, we need
an accelerating charge. It should be a single charge, but if we can make a great
many charges move together, all the same way, we know that the ñeld will be the
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Fig. 28-1. A high-frequency signal generator drives charges up and
down on two wires.
sum oÝ the efects of each of the individual charges; we just add them together.
As an example, consider two pieces oŸ wire connected to a generator, as shown in
Fig. 28-1. The idea is that the generator makes a potential diference, or a field,
which pulls electrons away from piece 4 and pushes them into Ö at one moment,
and then, an infinitesimal time later, it reverses the efect and pulls the electrons
out of and pumps them back into A/ So in these bwo wires charges, leb us say,
are accelerating upward in wire A and upward in wire Ö for one moment, and a
moment later they are accelerating downward in wire 4 and downward in wire Ö.
The fact that we need ÿwo wires and a generator is merely that this is a way of
doïng it. The net result is that we merely have a charge accelerating up and down
as though 4 and were one single wire. A wire that is very short compared
with the distance light travels in one oscillation period is called an elecfric đipolÌe
oscillator. 'hus we have the cireumstanece that we need to apply our law, which
tells us that this charge makes an electric feld, and so we need an instrument to
detect an electric ñeld, and the instrument we use is the same thing—a pair of
wires like A and / If an electric field is applied to such a device, it will produce
a force which will pull the electrons up on both wires or down on both wires.
Thịs signal is detected by means of a rectifier mounted bebween 4 and ?Ø, and
a tiny, ñne wire carries the information into an amplifier, where it is amplified so
we can hear the audiofrequency tone with which the radiofrequency is modulated.
'When this probe feels an electric field, there will be a loud noise coming out of the
loudspeaker, and when there is no electric fñeld driving it, there will be no noise.
Because the room in which the waves we are measuring has other objects in
1%, our electric ñeld will shake electrons in these other objects; the electric field
makes these other charges go up and down, and ín going up and down, these
also produce an efect on our probe. Thus for a successful experiment we must
hold things fairly close together, so that the inuences from the walls and from
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ourselves—the refected waves—are relatively small. 5o the phenomena will not
turn out to appear to be precisely and perfectly in accord with Eq. (28.6), but
will be close enough that we shall be able to appreciate the law.
~ K "° `
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Fig. 28-2. The instantaneous electric field on a sphere centered at a
localized, linearly oscillating charge.
Now we turn the generator on and hear the audio signal. We fnd a strong
fñeld when the detector is parallel to the generator G at point 1 (Eig. 28-2).
We fnd the same amount of fñeld also at any other azimuth angle about the axis
of Œ, because it has no directional efects. On the other hand, when the detector
1s at 3 the field is zero. 'Phat is all right, because our formula said that the field
should be the acceleration of the charge projected perpendicular to the line of
sipht. “Therefore when we look down on Œ, the charge is moving toward and
away from D, and there is no efect. So that checks the frst rule, that there is
no efect when the charge is moving directly toward us. Secondly, the formula
says that the electric fñeld should be perpendicular to z and in the plane of G
and 7; so if we put Ö at 1 but rotate it 90°, we should get no signal. And this is
Just what we fnd, the electric feld is indeed vertical, and not horizontal. When
we move to some intermediate angle, we see that the strongest sipnal occurs
when it is oriented as shown, because although Πis vertical, it does not produce
a fñeld that is simply parallel to itself—it is the projeciton oƒ the acceleration
perpendicular to the line oƒ sight that counts. The signal is weaker at 2 than it is
at 1, because of the proJection efect.