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.

Section 29–2 Energy of radiation

(Energy content / Energy flux / Energy loss)

 

The three interesting concepts discussed in this section are the energy content of a wave, energy flux within a conical angle, and energy loss by an oscillating charge.

 

1. Energy content:

“Now we must point out that the energy content of a wave, or the energy effects that such an electric field can have, are proportional to the square of the field… (Feynman et al., 1963, section 29–2 Energy of radiation).”

 

Feynman points out that the energy content of a wave, or the energy effects that an electric field can have, is proportional to the square of the electric field. He explains that it is because the electric field can act on a charge or oscillator and makes it move. However, this does not explain why the energy content of a wave is proportional to the square of the field. In Volume II, section 8–5 Energy in the electrostatic field, Feynman shows that the field energy is equal to U = ϵ₀/2∫E.EdV. In his words, “when an electric field is present, there is located in space an energy whose density (energy per unit volume) is u = ϵ₀(E.E)/2 = ϵ₀E²/2 (Feynman et al., 1964).” Thus, the field energy (potential energy) can be converted into kinetic energy of the charge that is proportional to the square of the electric field.

 

At the beginning of this section, Feynman says that at any particular moment or in any particular place, the strength of the electric field varies inversely as the distance r, as it was mentioned previously. In the last sentence of the previous section, he mentions: “Now, ignoring the angle θ and the constant factors, let us see what that looks like as a function of position or as a function of time (Feynman et al., 1963, section 29-1).” Based on this assumption, the oscillation of a charge clearly causes the electric field to be maximum at the horizontal plane (q = 90°) for any distance r (thus, inversely proportional to the distance r) as shown in Fig 29-1. On the other hand, the electric field is zero at any point on the vertical axis (q = 0°) through the charge. Perhaps the last sentence of section 29-1 on ignoring the angle q should be shifted to the beginning of section 29-2.

 

2. Energy flux:

“So the fact that the amplitude of E varies as 1/r is the same as saying that there is an energy flux which is never lost, an energy which goes on and on, spreading over a greater and greater effective area (Feynman et al., 1963, section 29–2 Energy of radiation).”

 

According to Feynman, the energy that a source can deliver decreases as it is farther away and it varies inversely as the square of the distance. He adds that the energy we can take out of a wave within a conical angle is independent of the distance from where we are. Strictly speaking, the energy flux or energy flow through a cone may vary with the angle θ unless the energy flux through the cone has cylindrical symmetry (through the axis of motion). Thus, the energy flux through any conical angle is not definitely the same and this property allows us to choose the direction of maximum energy flow and help to save the energy needed for an antenna. In section 29-4, Feynman will discuss how multiple antennas will result in a stronger beam (energy flow) in a preferred direction using interference (the title of this chapter).

 

Feynman explains that all the energy we could pick up from a wave in a certain cone at a distance r₁ and at another distance r₂ are the same because the area of the surface intercepted by the cone varies directly as the square of the distance r. It may be worthwhile to mention that the energy flux through the charge is different from its electric flux. More importantly, we may define energy flux using Poynting vector (S) that is mathematically represented by S = (1/m₀)E ´ B in which E is the electric field vector and B is the magnetic field vector. Feynman opines that “[t]here are, in fact, an infinite number of different possibilities for u and S, and so far no one has thought of an experimental way to tell which one is right! (Feynman et al., 1964, section 27-4.)” In other words, energy flux is a calculational device and it is not an observable.

 

3. Energy loss:

“So if we are far enough away that our basic approximation is good enough, the charge cannot recover the energy which has been, as we say, radiated away… We shall study this energy loss further in Chapter 32. (Feynman et al., 1963, section 29–2 Energy of radiation).”

 

Feynman initially says that an oscillating charge would lose some energy which it can never recover. Subsequently, he explains that the charge cannot recover the energy because it has been radiated away. However, he elaborates that the energy is not really lost because it still exists somewhere, and it is available to be picked up by other systems (such as a radio). Perhaps he could have clarified that energy loss does not mean that energy can be destroyed, but it can be transformed into another form of energy. For example, in Volume II, chapter 22, Feynman mentions: “[w]hat about the energy loss when a generator is connected to an arbitrary impedance z? (By ‘loss’ we mean, of course, conversion of electrical energy into thermal energy.)”

 

Feynman ends the section by saying this energy loss will be discussed further in Chapter 32. In Chapter 32, he suggests that “to the driving circuit the antenna acts like a resistance, or a place where energy can be lost (the energy is not really lost, it is really radiated out, but so far as the circuit is concerned, the energy is lost). In an ordinary resistance, the energy which is lost passes into heat; in this case the energy which is lost goes out into space.” In this context, when Feynman uses the phrase energy loss, it may refer to the conversion of electrical energy of a system (antenna) into field energy or the spreading of energy into space. In a sense, the idea of energy loss is dependent on the definition of a system such as an antenna, or the system may include the space surrounding the antenna.

 

Review Questions:

1. Is the strength of an oscillating charge’s electric field really vary inversely as the distance r?

2. Is the energy flux through any conical angle independent of the angle q?

3. Does an oscillating charge lose some energy which it can never recover?

 

The moral of the lesson: the energy content of a wave of an oscillating charge is related to the energy flux through a cone and the energy “loss” by the oscillating charge.

 

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. Feynman, R. P., Leighton, R. B., & Sands, M. (1964). The Feynman Lectures on Physics, Vol II: Mainly electromagnetism and matter. Reading, MA: Addison-Wesley.

Section 29–1 Electromagnetic waves

 (Retarded electric field / Moving electric field / Oscillatory electric field)

 

The three interesting points discussed in this section are retarded electric field, moving electric field, and oscillatory electric field.

 

1. Retarded electric field:

“The electric field E due to a positive charge whose retarded acceleration is a′ (Feynman et al., 1963, section 29–1 Electromagnetic waves).”

 

According to Feynman, if a charge is oscillating with a very small amplitude, the electric field at an angle θ from the axis of the motion is in a direction at right angles to the line of sight and in the plane containing the line of motion and the line of sight. However, it is not straightforward for students to visualize the retarded electric field based on Fig. 29–1. Specifically, Feynman suggests that the understanding of the factor a(t−r/c) is to take the simplest case, θ = 90°, and plot the field graphically. In a sense, the simplest case should be θ = 0° because the formula E(t) = −qa(t−r/c) sin θ/4πϵ₀c²r clearly becomes zero. In short, the retarded electric field at any point on the vertical axis (or line of motion) through the charge is 0.

In Fig. 29-1, Feynman shows that the retarded electric field E due to a positive charge is a′. One may recall the equation (28.6) E_x(t) = (−q4πϵ₀c²/r)a_x(t−r/c) and this leads to the equation (29.1) E(t) = −qa(t−r/c)sin θ/4πϵ₀c²r by simply writing a_x(t−r/c) as a(t−r/c)sin θ. (Note: a(t−r/c) means a_t′=t−r/c) It is worthwhile to elaborate the retarded electric field at θ = 90° refers to any point on the horizontal plane as shown in Fig. 29-1 that is perpendicular to the line of motion. That is, the retarded electric field at any point on any horizontal axis is −qa(t−r/c)/4πϵ₀c²r (maximum value) and it is in the opposite direction to the line of motion. Simply put, the direction of the retarded electric field at any point on the horizontal plane in Fig 29-1 is anti-parallel to the acceleration vector.

 

2. Moving electric field:

“That is, as time goes on the field moves as a wave outward from the source. That is the reason why we sometimes say light is propagated as waves (Feynman et al., 1963, section 29–1 Electromagnetic waves).”

 

According to Feynman, light is sometimes described as a propagation of waves because the retarded electric field moves as a wave outward from the source. It is equivalent to saying its electric field is spreading outward as time goes on. In Volume II, Feynman adds that “[i]f this electric field tries to go away, the changing electric field would create a magnetic field back again. So by a perpetual interplay—by the swishing back and forth from one field to the other—they must go on forever... (Feynman et al., 1964, Chapter 18).” However, we may clarify that the electric field is moving outward in the sense that there is energy in the electric field and its field energy has momentum (See Feynman et al., 1964, Chapter 27).

 

Feynman explains that the retarded electric field at farther points is determined by a charge’s acceleration at earlier times. It can be illustrated by the curve of electric field in Fig. 29–3 that is a “reversed” plot of the charge’s acceleration as a function of time. To be more specific, one may elaborate that the curve of electric field is a “reverse and inverse” plot of the charge’s acceleration graph. That is, the acceleration graph is “reversed” or flipped from left to right because the acceleration of the charge at earlier times affects the electric field at farther points. In addition, the acceleration graph is also “inversed” or flipped from up to down because of the negative sign in E(t) = −qa(t−r/c)sin θ/4πϵ₀c²r.

 

3. Oscillatory electric field:

“An interesting special case is that where the charge q is moving up and down in an oscillatory manner (Feynman et al., 1963, section 29–1 Electromagnetic waves).”

 

Feynman describes a simple example of oscillatory electric field as a result of a charge accelerating up and down along a line having a very small amplitude. In the previous chapter, he has simplified the Heaviside-Feynman expression of electric field by assuming “a still simpler circumstance in which the charges are moving only a small distance at a relatively slow rate. Since they are moving slowly, they do not move an appreciable distance from where they start, so that the delay time is practically constant (Feynman et al., 1963, Section 28-2).” In essence, the simplified electric field formula (29.1) is applicable to any charge that is moving at a relatively slow speed (without relativistic effects) and it is relatively far away from the observer.

 

Feynman expresses the oscillatory electric field of a charge as E = −qsin θa₀cos ω(t−r/c)/4πϵ₀rc² by substituting the charge’s acceleration a =−ω²x₀cos ωt = a₀cos ωt into E(t) = −qa(t−r/c)sin θ/4πϵ₀c²r. However, some may ask whether the expression could be in terms of sine instead of cosine. As an alternative, we may use a =−ω²x₀sin (ωt + a) in which the phase angle a is dependent on the time we choose to start the experiment. Similarly, the expression sin θ in the expression of electric field may be cos (θ + b) instead of sin θ because it depends on the reference line where the angle θ + b is measured from. In summary, sin θ is dependent on the reference line in space, whereas cos ω(t−r/c) is dependent on the clock measured in time.

 

Review Questions:

1. How would you visualize the direction of the retarded electric field using Fig. 29–1?

2. How would you explain the moving electric field as the “reversed” plot of the charge’s acceleration as a function of time?

3. What are the assumptions of the oscillatory electric field?

 

The moral of the lesson: light may be described as a propagation of waves because its electric field is spreading outward from the charge.

 

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. Feynman, R. P., Leighton, R. B., & Sands, M. (1964). The Feynman Lectures on Physics, Vol II: Mainly electromagnetism and matter. Reading, MA: Addison-Wesley.