(Out of phase / 90° rotation / Unequal retardations)
In this
section, Feynman discusses interferences that are related to 180° out of phase, 90° rotation, and unequal retardations of
two sources.
1. Out of phase:
“Suppose
we make the charges in S1 and S2 both accelerate up and
down, but delay the timing of S2 so that they are 180° out of phase
(Feynman et al., 1963, section 28–4 Interference).”
According to Feynman, if charges in S1 and S2 are 180° out of phase, then the electric
field produced by S1 will be in one direction and the electric field produced by S2 will be in the opposite
direction at any instant. Perhaps it is not obvious
to some students why the fields produced by S1 and S2 are in opposite directions at any instant (except when
the field is zero). Mathematically, if the displacement of a wave at a
phase angle q is +A sin q, then the displacement of the wave at the phase
angle q + 180° is A sin (q + 180°) = A(sin q cos 180° + cos
q sin 180°) = -A sin q. One should realize that a periodic wave can be
divided into two halves whereby the first half (corresponding to first 180°)
can be assigned positive and the second half becomes negative, or vice versa.
Feynman says that we should get no effect at
point 1 if S1 and S2 are 180° out
of phase. In other words, by changing the length of a pipe we can change the
time it takes the signal to arrive at S2 and the two sources can
produce zero if everything is adjusted correctly. Specifically, the resultant displacement of the
two electric fields at point 1 is continuously zero. However, it is incorrect
to conceptualize the destructive
interference of the two electric fields (180° out of phase) results in zero energy and a violation of
conservation of energy. It
is also potentially misleading to use the phrase “no effect” because
destructive interference has the effect of spreading energy so that the total
energy remains constant.
2. 90° rotation:
“First, we restore S1 and S2 to the same phase; that is,
they are again moving together. But now we turn S1 through 90°, as shown in Fig. 28–4
(Feynman et al., 1963, section 28–4 Interference).”
To check interference due to two nonparallel electric
fields, we can first restore the two sources S1 and S2 to the same phase. Next, we
can rotate S1 through 90°, as shown in Fig. 28–4, such that the two electric fields are
perpendicular to each other. Mathematically, we can represent the horizontal
and vertical component of S1 and S2 using A sin wt. Thus, the resultant of the electric fields of S1 and S2 are Ö(A2 sin 2 wt + A2 sin 2 wt) = Ö(2A2 sin 2 wt). In this case, the resultant electric field is
equal to zero only two times in a cycle when wt is equal to 0° and 180°. On the other hand, the
period of the resultant electric field remains unchanged and it is related to
the angular frequency w.
Feynman mentions that it is very interesting to
show the addition of the two fields is in fact a vector addition. However, he
briefly suggests how the resultant electric field can be measured using two
in-phase signals that are perpendicular to each other. In section 29–5 The
mathematics of interference, he explains that “[a]ny cosine function of ωt
can be considered as the horizontal projection of a rotating vector.” Furthermore,
the addition of electric fields can be represented using a complex number for
each vector. Simply put, the result of one plus one in the case of electric
fields is not definitely two, but it is sometimes minus two or a number between
“2 and -2” such as zero.
3. Unequal
retardations:
“Then,
in accordance with the principle that the acceleration should be retarded by an
amount equal to r/c, if the retardations are not equal, the signals are no
longer in phase (Feynman et al., 1963, section 28–4 Interference).”
To demonstrate a signal is
retarded, Feynman explains that it should be possible to find a position at
which the distances of D from S1 and S2 differ by some amount Δ, in such a manner that
there is no net signal. Strictly speaking, this does not happen if the distance
Δ is exactly equal to the distance light goes in one-half of an oscillation of
the generator. This is because the electric field strength at a point is also
dependent on the distance r from the charge. In this example, the
electric field strengths of S1 and S2 at point 2 should be approximately the
same and the net signal is close to zero (instead of exactly equal to zero).
Thus, Feynman clarifies at the end of the chapter that we have not really
checked the 1/r variation of the electric field strength.
Feynman ends the chapter abruptly
without saying the mathematics of interference will be discussed in the next
chapter. It is worth mentioning that he did not use the word interference
in this section except in Fig. 28-3. Interestingly, he explains in the next
chapter that “[i]nterference in ordinary language usually suggests opposition or
hindrance, but in physics we often do not use language the way it was
originally designed!” However, the two terms interference and jamming have been used interchangeably, but
sometimes jamming may mean intentional use of radio waves to disrupt
communications, whereas interference may refer to unintentional forms of
disturbance. Perhaps one may end this chapter by discussing the jamming of
signals in World War II or Star Wars: Episode VI – Return of the
Jedi.
Review Questions:
1. Would you say that there is
no effect at point 1 if S1 and S2 are 180° out of phase?
2.
How would you explain the addition of the two
electric fields is in fact a vector addition?
3. Are the electric field of S1 and S2 exactly the same such that the net signal is equal to zero at point 2?
The moral of the lesson: interferences
are dependent on phase difference (or path difference) and they can be related
to 180° out of
phase, 90°
rotation, and unequal retardations of two sources.
References:
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