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.

Section 29–3 Sinusoidal waves

(Angular frequency w / Wave number k / w-k relationship)

 

The three interesting points discussed in this section are angular frequency (angular temporal frequency), wave number (angular spatial frequency), and the relationship between angular frequency and wave number.

 

1. Angular frequency:

“The angular frequency ω can be defined as the rate of change of phase with time (radians per second). (Feynman et al., 1963, section 29–3 Sinusoidal waves).”

 

Feynman defines angular frequency (ω) as the rate of change of phase with time (radians per second). For a sinusoidal wave, angular frequency can be simply described as the ratio of an angular displacement to the time taken (e.g., ω = 2p/T in which T is the period of oscillation). Alternatively, we may use the term angular temporal frequency because ω is related to time and it helps to distinguish the concept of wave number that is the rate of change of phase with distance. Similarly, the term acceleration could be defined either as the rate of change of velocity “with time” or “with distance.” To be more specific, we can define angular temporal frequency as the rate of change of an oscillator’s phase (or angular displacement) with respect to time that is applicable to a sinusoidal wave.

 

To recall the concept of phase, we can refer to Chapter 21 in which Feynman suggests: “The constant Δ is sometimes called the phase of the oscillation, but that is a confusion, because other people call ω₀t+Δ the phase, and say the phase changes with time. We might say that Δ is a phase shift from some defined zero. Let us put it differently. Different Δ’s correspond to motions in different phases. That is true, but whether we want to call Δ the phase, or not, is another question...” In short, a phase is a state of an oscillator pertaining to its position and direction of motion. The term phase has been used traditionally to describe the successive stages of various cycles, such as the periodical appearance of the Moon as it orbits the Earth.

 

2. Wave number:

“… we can define a quantity called the wave number, symbolized as k. This is defined as the rate of change of phase with distance (radians per meter). (Feynman et al., 1963, section 29–3 Sinusoidal waves).”

 

Feynman defines wave number (k) as the rate of change of phase with distance (radians per meter). However, this term is a misnomer because it is not strictly a number and k is sometimes used as a complex function. In dispersive media (e.g., water), k is a complex function of frequency that has both real and imaginary parts. Thus, a better term could be angular spatial frequency instead of angular wavenumber or wave number. Specifically, we can define angular spatial frequency as the rate of change of an oscillator’s phase with respect to distance that is applicable to a sinusoidal wave (instead of a wave packet). From an operational perspective, k can be measured as the number of the oscillator’s cycles per unit distance (e.g., k = 2p/l).

 

According to Feynman, the formula for a cosine wave moving in a direction x with a wave number k and an angular frequency ω will be written in general as cos (ωt−kx). Perhaps he could have elaborated that it can be written as a sine function or cosine function, and it can be written as either cos (ωt−kx) or cos (kx−ωt) because cos q = cos (−q). In general, cos (kx−ωt) may refer to a wave moving to the right, whereas cos (kx+ωt) may refer to another wave moving to the left. If we fix t as a specific instant in time (e.g., t = 0), we can have a snapshot graph that shows how a wave’s vertical displacement changes as a function of x. If we fix x as a specific point in space (e.g., x = 0), we can have a history graph that shows how a wave’s vertical displacement changes with time.

 

3. w-k relationship:

“Now in our particular wave there is a definite relationship between the frequency and the wavelength, but the above definitions of k and ω are actually quite general (Feynman et al., 1963, section 29–3 Sinusoidal waves).”

 

Feynman clarifies that there is a definite relationship between the frequency and the wavelength in our particular wave, but the definitions of ω and k provided in this section are quite general. That is, ω and k are not related in the same way in other physical circumstances. Perhaps Feynman could simply explain that the definite relationship shown is applicable only to sinusoidal waves that have only one frequency and one wavelength. In other words, ω and k are related in a complicated manner for a wave packet. This leads to the concept of group velocity and phase velocity that will be covered in chapter 48 when Feynman emphasizes that “[t]he group velocity is the derivative of ω with respect to k, and the phase velocity is ω/k.”

 

In the end of the section, Feynman mentions that equation (29.1) is a legitimate formula because it is applicable to the “wave zone” (the region that is beyond a few wavelengths). However, the term “wave zone” is not commonly used in this context. More interestingly, in section 34-7 The ω, k four-vector, Feynman says, “if ω is thought of as being like t, and k is thought of as being like x divided by c², then the new ω′ will be like t′, and the new k′ will be like x′/c².” That is, the angular frequency ω of a sinusoidal wave and its wave number k transform in the same way as space and time under the Lorentz transformation. Thus, it is worthwhile mentioning that the wave number k and the angular frequency ω are interrelated to the extent they are analogous to the space and time in special relativity.

 

Review Questions:

1. Would you use the term angular frequency and define it as the rate of change of phase with time?

2. Would you use the term wave number and define it as the rate of change of phase with distance?

3. How would you describe the relationship between the angular frequency and wave number of an oscillator?

 

The moral of the lesson: the angular frequency (ω) is the rate of change of phase with time, whereas the wave number (k) is the rate of change of phase with distance; ω = 2p/T = 2pf and k = 2p/l implies ω/k = fl = c (for a sinusoidal wave).

 

Reference:

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.