Section 29–4 Two dipole radiators

 (Half wavelength apart / Quarter wavelength apart / 10 wavelengths apart)

 

In this section, Feynman describes the effects of two dipole radiators that are half-wavelength apart, a quarter wavelength apart, and 10 wavelengths apart qualitatively.

 

1. Half wavelength apart:

“Figure 29–5(a) represents the top view of two such oscillators, and in this particular example they are half a wavelength apart in a N–S direction… (Feynman et al., 1963, p. 29–3).”

 

Feynman explains that in either the N or S direction along the line of the two in-phase oscillators that are l/2 apart, the net field becomes zero. It may be worthwhile to introduce a principle of equivalence: if the distance between the two in-phase oscillators is l/2, the effect on any point along the line (except “in-between”) of the two in-phase oscillators is equivalent to two out-of-phase oscillators. Conversely, if the path difference of the two in-phase oscillators to a point is a multiple of wavelength, the effect would be a crest meets a crest and thus, constructive interference occurs. Perhaps Feynman could have used the terms path difference and phase difference to discuss how a path difference of two sources to a point corresponds to the effect of a phase difference (e.g., in phase or out of phase).

 

“By the intensity we mean the amount of energy that the field carries past us per second, which is proportional to the square of the field, averaged in time (Feynman et al., 1963, p. 29–3).”

 

Feynman defines intensity as “the amount of energy that the field carries past us per second, which is proportional to the square of the field, averaged in time.” Some physicists may criticize this definition because of the following reasons. Firstly, they would prefer to specify the context or the object involved, for example, whether it is about sound waves or electromagnetic waves (to be specific, light waves or radio waves). Secondly, some want to state experimental conditions or a physical condition such as the energy flows through a unit area A that is perpendicular to the direction of travel of the waves. Thirdly, we can include the mathematical formula <P>/A that means the average power <P> per unit area A.

 

2. Quarter wavelength apart:

“If the antennas are separated by one-quarter wavelength, and if the N one is one-fourth period behind the S one in time, then what happens? (Feynman et al., 1963, p. 29–4).”

According to Feynman, if the antennas are separated by l/4, in the S direction we get zero intensity (180° out of phase) because the signal from the N antenna is 90° phase behind the S antenna and its signal is further delayed by 90° due to the path difference l/4. In the N direction, both signals appear to be in phase because the delay of 90° phase of the N antenna’s signal is compensated by the extra distance l/4 moved by the signal of S antenna. Specifically, the effect of the two antennas is symmetrical about the N–S direction, but it is unsymmetrical about the E-W direction. However, the phase difference at any point between the two oscillators is not the same as it varies with the path difference that can be between ½l and zero. Feynman ignores this region because his main concern is about how to send signals more efficiently.

 

Feynman says that if we build an antenna system and want to send a radio signal to Hawaii, we can set up the antennas as shown in Fig. 29–5(a) and broadcast with the two antennas in phase. In this section, he uses many terms, such as dipole radiators, dipole oscillators, radio transmitters, antenna system, two antennas in phase, a pair of dipole antennas, and two dipoles that may be confusing. For example, the word dipole suggests that a dipole antenna consists of two poles or two oscillators. In Basic Microwaves, Berkowitz (1966) writes that “[t]he pattern of a half-wave dipole very closely resembles that of the elementary dipole radiator (two isotropic, cophasal radiators spaced l/2 apart) (p. 100).” Thus, the title “two dipole radiators” of the section is potentially misleading because it can be interpreted as four radiators.

 

3. 10 wavelengths apart:

“Let us take a situation in which the separation is ten wavelengths (Fig. 29–7), which is more nearly comparable to the situation in which we experimented in the previous chapter... (Feynman et al., 1963, p. 29–4).”

 

Feynman suggests minimizing the wastage of power by using more antennas: if we draw a line from each antenna to a distant point and the path difference is λ/2, then they will be out of phase. However, it is unclear when he says: “we do indeed have a very sharp beam in the direction we want, because if we just move over a little bit we lose all our intensity.” Perhaps some prefer to understand this using the Double Slit formula or simplifying it as the fringe separation is proportional to the wavelength of light. (Dx = Dl/a where Dx is the fringe separation, D is the distance from the antenna, l is the wavelength, and a is the separation between the antennas.) Furthermore, one may explain using scaling in the sense that an increase in the number of wavelength to 10 is equivalent to a reduction of the wavelength by a factor of 10.

 

“But numbers 3 and 4 are roughly ½ a wavelength out of phase with 1 and 6, and although 1 and 6 push together, 3 and 4 push together too, but in opposite phase (Feynman et al., 1963, p. 29–5).”

 

Feynman provides a rough idea of diffraction grating using 6 antennas to get rid of all the extra maxima to achieve a sharp beam: antenna 3 and 4 are roughly ½ a wavelength out of phase with antenna 1 and 6, and this causes a redistribution of energy. In general, if we have N antennas, antenna 1 and N/2 +1 can interfere destructively; antenna 2 and N/2 +2 can interfere destructively… to increase the intensity of the beam. That is, the presence of more antennas helps the beam to increase its intensity and become narrower. One should realize that the increase in intensity by N² times means that the beam must be narrower in accordance to the law of conservation of energy (or using the formula I = <P>/A).

 

Review Questions:

1. Do you agree with Feynman’s definition of intensity?

2. How would you define a dipole radiator or/and a dipole antenna?

3. How would you explain the effect of 6 antennas that are separated by two wavelengths?

 

The moral of the lesson: the separation of “oscillators” can affect the locations of constructive interference and destructive inference as well as the intensity and width of the maxima(s).

 

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.

2. Berkowitz, B. (1966). Basic Microwaves. New York: Hayden.