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28-4 Interference
Next, we may test what happens when we have two sources side by side
sevoral wavelengths apart (Fig. 28-3). The law is that the two sources should
add theïr efects at point 1 when both of the sources are connected to the same
generator and are both moving up and down the same way, so that the total
electric ñeld is the sum of the two and is twice as strong as it was before.
D S2
Fig. 28-3. lllustration of interference of sources.
Now comes an interesting possibility. Suppose we make the charges In S1
and Š› both accelerate up and down, but delay the timing of Š5› so that they are
1802 out of phase. 'Phen the field produced by ŠS¡ will be in one direction and
the field produced by ŠS+ will be in the opposite direction at any instant, and
therefore we should get øoø efect at point 1. The phase of oscillation is neatly
adjustable by means of a pipe which is carrying the signal to S¿. By changing
the length of this pipe we change the time it takes the signal to arrive at 5s and
thus we change the phase of that oscillation. By adjusting this length, we can
indeed fnd a place where there is no more signal left, in spite of the fact that
both 5¡ and 52 are movingl The fact that they are both moving can be checked,
because if we cut one out, we can see the motion of the other. So the bwo of
them together can produce zero iŸ everything is adjusted correctly.
Now, i1 is very interesting to show that the addition of the two fields is in
fact a 0ector addition. We have Jjust checked it for up and down motion, bu§
let us check two nonparallel directions. First, we restore 5 and S2 to the same
phase; that is, they are again moving together. But now we turn 5: through 902,
as shown in Fig. 28-4. Now we should have at point 1 the sum of two efects,
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2. /R
S2
Fig. 28-4. lllustration of the vector character of the combination of
SOUFCeS.
one of which is vertical and the other horizontal. The electric fñeld is the vector
sum oŸ these two in-phase signals—they are both strong at the same time and go
through zero together; the total fñeld should be a signal at 45°. If we turn
to get the maximum noise, it should be at about 45°, and not vertical. And if
we turn i% at right angles to that direction, we should get zero, which is easy to
mmeasure. Indeed, we observe just such behaviorl
Now, how about the retardation? How can we demonstrate that the signal is
retarded? We could, with a great deal of equipment, measure the time at which
1t arrives, but there is another, very simple way. Referring again to Fig. 28-3,
suppose that 5¡ and S52 are in phase. 'Phey are both shaking together, and
they produce equal electric fields at point 1. But suppose we go to a certain
place 2 which is closer to S2 and farther from Š¡. Then, in accordance with the
principle that the acceleration should be retarded by an amount equal to r/e,
1f the retardations are not equal, the signals are no longer in phase. Thus it
should be possible to fnd a position at which the distances of 1 from 5% and S2
difer by some amount A, in such a manner that there is no net signal. 'Phat
is, the distance A is to be the distance light goes in one-half an oscillation of
the generator. We may go still further, and fñnd a poïint where the diferenece is
greater by a whole cycle; that is to say, the signal from the first antenna reaches
point 3 with a delay in time that is greater than that of the second antenna,
by just the length of time it takes for the electric current to oscillate once, and
therefore the two electric fñelds produced at 3 are in phase again. At point 3 the
signal is strong again.
This completes our discussion of the experimental verifcation of some of
the important features of Eq. (28.6). Of course we have not really checked
the 1/r variation of the electric feld strength, or the fact that there is also a
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magnetic fñeld that goes along with the electric ñeld. To do so would require
rather sophisticated techniques and would hardly add to our understanding at
this point. In any case, we have checked those features that are of the greatest
Importance for our later applications, and we shall come back to study some of
the other properties of electromagnetic waves next year.
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Xrfor'for-orte©
29-1 Electromagnetic waves
In this chapter we shall discuss the subject of the preceding chapter more
mathematically. We have qualitatively demonstrated that there are maxima,
and minima in the radiation fñeld from two sources, and our problem now is tO
describe the field in mathematical detail, not just qualitatively.
We have already physically analyzed the meaning of formula (28.6) quite
satisfactorily, but there are a few points to be made about it mathematically. In
the frst place, IŸ a charge is accelerating up and down along a line, in a motion
of very small amplitude, the fñield at some angle Ø from the axis of the motion is
in a direction at right angles to the line of sight and in the plane containing both
the acceleration and the line of sight (Fig. 29-1). Iƒ the distance is called r, then
at time £ the electric fñeld has the magnitude
—qa{(‡ — r/e) sin 8
#(0 = _—_^.. (29.1)
47cgc2r
Fig. 29-1. The electric field E due to a positive charge whose retarded
acceleration is a”.
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Fig. 29-2. The acceleration of a certain charge as a function of time.
Fig. 29-3. The electric field as a function of position at a later time.
(The 1/r variation is ignored.)
where a(# — r/e) is the acceleration at the time (£ — r/c), called the retarded
acceleration.
Now it would be interesting to draw a picture of the fñeld under different
conditions. The thing that is interesting, of course, is the factor ø(£ — r/c),
and to understand it we can take the simplest case, Ø = 90”, and plot the field
graphically. What we had been thinking of before is that we stand in one position
and ask how the fñeld there changes with time. But instead of that, we are now
goïing to see what the field looks like at diferent positions in space at a given
Instant. So what we want is a “snapshot” picture which tells us what the fñeld
1s In diÑerent places. Of course it depends upon the acceleration of the charge.
Suppose that the charge at first had some particular motion: it was Initially
standing still, and ¡it suddenly accelerated in some mamner, as shown in Fig. 29-2,
and then stopped. 'Then, a little bit later, we measure the field at a diferent
place. Then we may assert that the feld will appear as shown in Eig. 29-3. At