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the speed c, will move a distance cứo, and will of course have moved over just
one wavelength.
In a physical situation other than that of lght, & is not necessarily related
to œ in this simple way. TỶ we call the distance along an axis #, then the formula
for a cosine wave moving in a direction z with a wave number & and an angular
frequency œ will be written in general as cos (É — &z).
Now that we have introduced the idea of wavelength, we may say something
more about the cireumstances in which (29.1) is a legitimate formula. We recall
that the field is made up of several pieces, one of which varles inversely as r7,
another part which varies inversely as r2, and others which vary even faster. It
would be worth while to know in what circumstances the 1/z part of the field is
the most important part, and the other parts are relatively small. Naturally, the
answer is “if we go “far enoughˆ away,” because terms which vary inversely as
the square ultimately become negligible compared with the 1/z term. How Íar is
“far enough”? The answer is, qualitatively, that the other terms are of order À/r
smaller than the 1/z term. Thus, so long as we are beyond a few wavelengths,
(29.1) is an excellent approximation to the field. Sometimes the region beyond a
few wavelengths is called the “wave zone.”
29-4 Two dipole radiators
Next let us discuss the mathematics involved in combining the efects of two
oscillators to fnd the net fñeld at a given point. This is very easy in the Íew cases
that we considered in the previous chapter. We shall first describe the efects
qualitatively, and then more quantitatively. Let us take the simple case, where
the oscillators are situated with their centers in the same horizontal plane as the
detector, and the line of vibration is vertical.
Figure 29-5(a) represents the top view of 6wo such oscillators, and in this
particular example they are half a wavelength apart in a NÑ-S direction, and are
oscillating together in the same phase, which we call zero phase. NÑow we would
like to know the intensity of the radiation in various directions. By the intensity
we mean the amount of energy that the fñeld carries past us per second, which is
proportional to the square of the fñield, averaged ïn time. So the thing to look at,
when we want to know how bright the light is, is the square of the electric field,
not the electric field itself. (The electric field tells the strength of the force felt
by a stationary charge, but the amount of energy that is going past, in watts per
square meter, is proportional to the square of the electric field. We shall derive
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2 của 2 2 củ ?
4 À/2——4 0 À/2———0
2z |NG z |ÌNG
œ=0 œ=1
(a) (@b)
Fig. 29-5. The intensities In various directions from two dipole oscilla-
tors one-half wavelength apart. Left: in phase (œ = 0). Right: one-half
period out of phase (œ = 7).
the constant of proportionality in Chapter 31.) TÝ we look at the array rom the W
side, both oscillators contribute equally and in phase, so the electric feld is Ewice
as strong as it would be from a single oscillator. Therefore the ?mtensit ¡s [our
times as sfrong ús ?t tuould be tƒ there tuere onÏỤ one oscillator. (The numbers
in Eig. 29-5 represent how strong the intensity would be in this case, compared
with what it would be if there were only a single oscillator of unit strength.) Ñow,
in either the Ñ or S5 direction along the line of the oscillators, since they are half
a wavelength apart, the efect of one oscillator turns out to be out of phase by
exactly half an oscillation from the other, and therefore the fñelds add to zero.
At a certain particular intermediate angle (in fact, at 309) the intensity is 2, and
1t falls of, 4, 2, 0, and so forth. We have to learn how to fnd these numbers at
other angles. It is a question of adding two oscillations with diferent phases.
Let us quickly look at some other cases of interest. Suppose the oscillators
are again one-half a wavelength apart, but the phase œ of one is set half a period
behind the other in its oscillation (Fig. 29-5b). In the W direction the intensity
is now zero, because one oscillator is “pushing” when the other one is “pulling”
But in the N direction the signal from the near one comes at a certain time, and
that of the other comes half a period later. But the latter was originallu half a
period behind in timing, and therefore it is now exactly 7n tưne with the first one,
and so the intensity in this direction is 4 units. The intensity in the direction
at 30” is still 2, as we can prove later.
Now we come to an interesting case which shows up a possibly useful feature.
Let us remark that one of the reasons that phase relations of oscillators are
interesting is for beaming radio transmitters. For instance, if we build an antenna
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system and want to send a radio signal, say, to Hawaii, we set the antennas up
as in Fig. 29-5(a) and we broadcast with our 0wo antennas in phase, because
Hawall is to the west of us. Then we decide that tomorrow we are going %O
broadcast toward Alberta, Canada. Since that is north, not west, all we have
to do 1s to reverse the phase of one of our antennas, and we can broadcast to
the north. 5o we can build antenna systems with various arrangements. Ôurs is
one of the simplest possible ones; we can make them much more complicated,
and by changing the phases in the various antennas we can send the beams In
various directions and send most of the power in the direction in which we wish
to transmit, without ever moving the antennal In both of the preceding cases,
however, while we are broadcasting toward Alberta we are wasting a lot of power
on Easter Island, and it would be interesting to ask whether it is possible to
send it in only øwe direction. At frst sight we might think that with a pair of
antennas of this nature the result is always going to be symmetrical. So let us
consider a case that comes out unsymmetrical, to show the possible variety.
2 Ẫ /4—2
Fig. 29-6. A palr of dipole antennas giving maximum power in one
direction.
Tf the antennas are separated by one-quarter wavelength, and ïf the NÑ one
is one-fourth period behind the S one in time, then what happens (Fig. 29-6)?
In the W direction we get 2, as we will see later. In the SŠ direction we get zero,
because the signal from SŠ comes at a certain time; that from Ñ comes 902 later
in #ữne, but it is already 90° behind in its built-in phase, therefore it arrives,
altogether, 180” out of phase, and there is no efect. On the other hand, in the
NÑ direction, the Ñ signal arrives earlier than the 5 signal by 90” in time, because
1t is a quarter wavelength closer. But its phase is set so, that it is oscillating 90°
behữnd In tìme, which Just compensates the delay diference, and therefore the
two sipnals appear ogether in phase, making the field strength twice as large,
and the energy four tỉmes as great.
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Thus, by using some cÌleverness in spacing and phasing our antennas, we