Section 28–4 Interference

 (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:

1. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

Section 28–3 The dipole radiator

(Generate electric field / Detect electric fields / Direction of antenna)

 

In this section, Feynman discusses how an antenna generates electric fields, detect electric fields, and how the antenna affects the magnitude of electric fields detected depending on its direction (or orientation).

 

1. Generate electric fields:

“A wire that is very short compared with the distance light travels in one oscillation period is called an electric dipole oscillator (Feynman et al., 1963, section 28–3 The dipole radiator).”

 

According to Feynman, the charges are accelerating upward in wire A and wire B for one moment and later they are accelerating downward in wire A and wire B. This is equivalent to a charge accelerating up and down as if A and B were one single wire. The wire is called an electric dipole oscillator by Feynman, but this setup is sometimes known as the half-wave dipole or Hertz antenna. More importantly, the two horizontal wires in Fig. 28–1 should be shortened or removed because they may cause confusions. The electromagnetic fields generated by the electric currents in the horizontal parts of the two wires are in opposite directions and they almost completely cancel each other.

 

Feynman says that the need of two wires and a generator is merely a way of doing it (or generating electric fields). For example, one may explain the principle of a quarter-wavelength vertical antenna that is also known as the Marconi antenna. It is similar to the Hertz antenna, but it is mounted perpendicular to the Earth and one of its ends is grounded. Perhaps Feynman should have explained whether the electric current in the antenna is flowing in an open circuit or a closed circuit. In a sense, the antenna is an open circuit because there is no continuous path or no movement of electrons at both ends of the antenna. Some physicists may consider the antenna to be a closed circuit if they idealize it as connected to “air” that has high electrical resistance (similar to capacitors).

2. Detect electric fields:

“… we need to apply our law, which tells us that this charge makes an electric field, and so we need an instrument to detect an electric field, and the instrument we use is the same thing—a pair of wires like A and B! (Feynman et al., 1963, section 28–3 The dipole radiator).”

 

Feynman mentions that the signal is detected using a rectifier mounted between A and B, and a tiny wire carries the information into an amplifier, where it is amplified so we can hear the audio frequency tone with which the radiofrequency is modulated. One may emphasize the length of the dipole radiator is related to the wavelength of the electric field and how the signal is detected through a resonance effect. Specifically, the resonant frequency of the antenna (detector) can be varied by adjusting the capacitance of the radiofrequency circuit. The amplification of the signal can be accomplished using a parabolic reflector (instead of an amplifier). In addition, one may elaborate on how the rectifier or diode helps to restrict the direction of electric currents.

 

Feynman explains that when the probe feels an electric field, there will be a loud noise coming out of the loudspeaker, and when there is no electric field driving it, there will be no noise. However, when the electric field (due to the generator G) makes other charges go up and down, and in going up and down, they also produce an induced electric field in the probe (or detector D). Perhaps Feynman could have distinguished the two kinds of electric field: the electric field due to the generator G and the induced electric field in the detector D. It may be worth mentioning that the induced electric field is nonconservative because it does work in moving a charge over a closed path. Alternatively, one may explain that the probe detects changing magnetic field that induces a voltage (induced electromotive force).

 

3. Direction of antenna:

“We find the same amount of field also at any other azimuth angle about the axis of G, because it has no directional effects (Feynman et al., 1963, section 28–3 The dipole radiator).”

 

Feynman says that there is a strong field when the detector D is in parallel to the generator G at point 1 (Fig. 28–2). Thus, we expect a relatively strong induced electric field at point 1. Similarly, the same amount of electric field is induced at other azimuth angle about the axis of G (an axis of symmetry) because the direction of generator G relative to the detector D is still the same (in parallel). In other words, maximum electric field is induced provided the generator G and the detector D are in parallel because the electrons oscillate in the same direction. However, Feynman’s description of Fig 28-2 is potentially misleading. One may clarify that the sphere in Fig 28-2 is not a physical sphere, but an imaginary sphere that compares the relative amount of induced electric field depending on the direction of antenna.

 

Feynman elaborates that the electric field should be perpendicular to r and in the plane of G and r. For example, if the detector D at 1 is rotated by 90°, we should get no signal. Simply put, no electric field is induced in the detector D at 1 because the electric field generated by the generator G is vertical, but the detector D is horizontal. Strictly speaking, there is no induced electric field in the horizontal detector D because we (including Feynman) have idealized the wire of the detector D to have no thickness. In the real world, the wires of the antenna have a thickness and this allows electrons to move a very short distance in the direction of the induced electric field.

 

Review Questions:

1. Is the electric current in the antenna flowing in an open circuit or closed circuit?

2. How would you explain the detection of electric field(s) using an antenna receiver?

3. Should we expect no signal (or close to zero signal) if the detector D at 1 is rotated by 90°?

 

The moral of the lesson: although maximum electric field is induced if the antenna generator and antenna receiver are in parallel, but it can be related to the projection of the acceleration of electrons perpendicular to the line of sight.

 

References:

1. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.