Section 30–7 The field of a plane of oscillating charges

 (Radiation field / Infinite plane / Finite charges)

 

In this section, the three interesting concepts involved in a problem are radiation field, an infinite plane of in-phase oscillating charges, and a finite amount of charges on the plane. This section is about an idealization of a plane of in-phase oscillators, but the title could be known as “The far field of an infinite plane of oscillating charges.” This section not only aptly ends with the concept of diffraction that is due to an infinite number of sources, but a formula is derived to explain the origin of refractive index for the next chapter.

 

1. Radiation field:

Suppose that we have a plane full of sources, all oscillating together, with their motion in the plane and all having the same amplitude and phase. What is the field at a finite, but very large, distance away from the plane? … (We cannot get very close, of course, because we do not have the right formulas for the field close to the sources.) (Feynman et al., 1963, p. 30–10).”

 

One may define radiation field as an electromagnetic field that provides the energy radiated by accelerated charges and it is inversely proportional to the distance from the charges. However, the term radiation field could be distinguished into two regions, namely near field and far field, that can be expressed by two different mathematical formulas. To be specific, we can define far field using the equation (28.6) in which the field produced by an accelerating charge is moving non-relativistically and it is located at a very large distance r. This problem also includes the far field of an infinite number of sources that are located at infinity. On the other hand, we do not need the right formulas here for the near field that is close to the sources.

Note: It may be confusing to some why Feynman says “… we do not have the right formulas for the field close to the sources,” but we may use near field to explain the maximum limit of refractive index of a medium (Andreoli et al., 2021).

 

“We know that the radiation field is proportional to the acceleration of the charge, which is −ω²x₀e^iωt…… Using this value for the acceleration as seen from P in our formula for the electric field at large distances from a radiating charge, we get (30.11) (Electric field at P from charge at Q) ≈ q/4πϵ₀c²ω²x₀e^iω(t−r/c)/r (Feynman et al., 1963, p. 30–10).”

 

Feynman clarifies that the formula (30.11) is not quite right because it should not only include the acceleration of the charge but its component perpendicular to the line QP. Theoretically, the component perpendicular to QP of the far field due to the charges would not be canceled out because the oscillations are all in the same direction despite having cylindrical symmetry. From a practical point of view, it is the far field that allows electromagnetic waves to propagate far distances with weak signal strength, but it can still be picked up by receivers or antennas. However, some may prefer saying the acceleration field (instead of radiation field or far field) produced by an accelerating charge varies as 1/r. This is different from the velocity field (Coulomb field) of the charge that is independent of a, but its field varies as 1/r².

 

2. Infinite plane:

“When ρ = 0, we have r = z, so the limits of r are z to infinity… Now e^−i∞ is a mysterious quantity. Its real part, for example, is cos (−∞), which, mathematically speaking, is completely indefinite (although we would expect it to be somewhere—or everywhere (?)—between +1 and −1!). (Feynman et al., 1963, p. 30–11).”

 

Feynman says that e^−i∞ is a mysterious quantity and its real part, cos(−∞), is indefinite. On the contrary, mathematicians prefer to write lim e^−iz and include the condition z ® ∞, but they consider e^−i∞ to be illegal. Furthermore, they would investigate whether a function is holomorphic using Cauchy-Riemann equations u_x = v_y and u_y = –v_x. In general, physicists may define terms involving infinity without rigor or quote a statement that is attributed to Einstein: “Two things are infinite, the universe and human stupidity, and I am not yet completely sure about the universe.” More importantly, the problem based on the infinite plane of constant density of charge is unrealistic and an infinite amount of charges on a “circular ring element” (2pr)Dr divided by the distance r that is infinity could be considered as an undefined quantity.

 

In Fig. 30–11 we have drawn the first five pieces of the sum. Each segment of the curve has the length Δr and is placed at the angle Δθ = −ωΔr/c with respect to the preceding piece (Feynman et al., 1963, p. 30–11).

 

The first term of e^iθ = e^−iωz/c has the most contribution to the total field because z is the shortest distance from the plane to P, but ωz/c is relatively small compared to z and thus the angle of the first arrow (θ) should be small. Similarly, the remaining each segment of the curve has the length Δr and it is placed at the angle Δθ = −ωΔr/c with respect to the preceding piece. One should realize that the real part (x component) starts to decrease in Fig. 30–11 when we add the fourth and fifth term, but this may not really correspond to the physical situation. In other words, not only e^−i∞ is a mysterious quantity, but it is remarkable that e^−iωz/c starts to oscillate right at the beginning. This suggests that the idealized model based on e^−iωz/c has a serious limitation and thus needs tweaking.

 

In Surely You’re Joking, Mr. Feynman!, Feynman (1997) challenged Paul Olum to give him an integral that most people could evaluate with only contour integral, but he could use other methods: “One time I boasted, ‘I can do by other methods any integral anybody else needs contour integration to do.’ So Paul puts up this tremendous damn integral he had obtained by starting out with a complex function that he knew the answer to, taking out the real part of it and leaving only the complex part. He had unwrapped it so it was only possible by contour integration! He was always deflating me like that. He was a very smart fellow (pp. 195-196).” Perhaps Feynman could have explained how the physical situations result in another Cornu spiral or contour integral based on Fermat’s principle of extremum path.

 

3. Finite charges:

“In any real situation the plane of charges cannot be infinite in extent, but must sometime stop... If, however, we let the number of charges in the plane gradually taper off at some large distance from the center (or else stop suddenly but in an irregular shape so for larger ρ the entire ring of width dρ no longer contributes), then the coefficient η in the exact integral would decrease toward zero. (Feynman et al., 1963, p. 30–11).”

 

Feynman suggests tweaking the problem by decreasing the number of charges in the plane that are farther from the center, i.e., the coefficient η in the exact integral would decrease toward zero. We can also include the projection of the acceleration on the plane perpendicular to the line PQ, but we would feel unlucky to even approximate the integral involving an additional 1/r. More importantly, some may argue whether it is mathematically legal to let e^−i∞ equal to zero due to various physical situations. For instance, the coefficient η could be specified as a function that varies with 1/r or e^−r because there is no infinite amount of charges in the real world. Alternatively, Feynman could have discussed whether including the projection of the acceleration on the plane perpendicular to the line PQ would definitely improve the formula, however, he claims that the formula (30.18) or (30.19) is correct at any distance z.

 

It is interesting to note that (iωx₀e^iωt) is just equal to the velocity of the charges, so that we can also write the equation for the field as Total field at P=−ηq²ϵ₀c[velocity of charges] at t−z/c (Feynman et al., 1963, p. 30–11).”

 

Feynman concludes the chapter by saying the formula derived is fortunately rather simple and it is valid for distances far from the plane of oscillating charges… If he assumes that η is a function that varies with 1/r or e^−r, then the integration method has to be changed and it would result in a different formula. The so-called simple formula is a result of idealizations and approximations because we have avoided realistic models such as including the projection of the acceleration on the plane perpendicular to the line PQ (cosine factor). In a sense, this problem beautifully shows the daily life of a physicist in cheating (idealizing models) and tweaking (approximating formulas). However, Feynman could end this chapter by explaining the importance of the formula derived because it relates the delay in phase of the field to refractive index.

Note: It may be worth mentioning this section was delivered at the beginning of the lecture of chapter 31 on refractive index.

 

Review Questions:

1. How would you define the radiation field at a distance far from a plane of oscillators (Idealization)?

2. Should we assume the problem to be based on an infinite plane of constant density of oscillating charges (Limitation)?

3. Is it mathematically legal to let e^−i∞ equal to zero such that we can obtain a reasonable approximate answer (Approximation)?

 

The moral of the lesson: We can understand diffraction further by idealizing (cheating) a plane of in-phase oscillating charges, approximating (tweaking) the density of charges on the plane, and confessing the limitation of the model. In a sense, this is an unfortunate problem, but it comes out—fortunately a rather simple formula by “cheating” and “tweaking” (creating luck).

 

References:

1. Andreoli, F., Gullans, M. J., High, A. A., Browaeys, A., & Chang, D. E. (2021). Maximum Refractive Index of an Atomic Medium. Physical Review X, 11, 011026.

2. Feynman, R. P. (1997). Surely You’re Joking, Mr. Feynman! : Adventures of a Curious Character. New York: Norton.

3. 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 30–6 Diffraction by opaque screens

(Effective light sources / Cornu’s spiral / Geometrical shadow edge)

 

The three interesting concepts discussed in this section are effective light sources, Cornu’s spiral, and geometrical shadow edge.

 

1. Effective light sources:

“Of course, actually there are no sources at the holes, in fact that is the only place that there are certainly no sources… If we use the theorem that we have not yet proved, then we can replace the actual problem by a set of effective sources uniformly distributed over the open space beyond the object (Feynman et al., 1963, p. 30–8).”

 

Some may enjoy how Feynman explained the diffraction of light waves at an opaque sheet with holes in it: We have assumed that there are sources distributed with uniform density across the open holes, but there are actually no sources at the holes. However, there is a gap in his explanation because he has not proved the theorem that justifies why we can replace the problem with a set of effective light sources uniformly distributed over the open space. For example, one may state the theorem as Huygens-Fresnel Principle and then explain that every point on a wavefront is the source of spherical wavelets... In short, the idealized light sources on the wavefront do not exist as physical objects, but they are useful mathematical objects. Perhaps Feynman did not like this principle partly because he wrote that “[a]ctually Huygens’ principle is not correct in optics… (Feynman, 1942, p. 91)” in his PhD thesis.

 

Nevertheless, we get the correct diffraction patterns by considering the holes to be the only places that there are sources; that is a rather peculiar fact. We shall explain later why this is true, but for now let us just suppose that it is (Feynman et al., 1963, p. 30–8).

 

Feynman says that we can get the correct diffraction patterns by considering the holes to be the only places where there are sources, but he has planned to explain later why this is true. Note that he does not provide the explanation for the diffraction patterns in this chapter. In section 31–6 Diffraction of light by a screen, he mentions: “[w]e have the result that the field at P when there are holes in a screen (case b) is the same (except for sign) as the field that is produced by that part of a complete opaque wall which is located where the holes are!... (Feynman et al., 1963).” Furthermore, he remarks that this theory of diffraction is only approximate, and it is valid only if the holes are not too small. In the real world, there are real sources, for example, a beam of radiation incident on an atom causes the electrons in the atom to oscillate and thus the electrons can radiate in various directions (Feynman et al, 1963, Chapter 32).

 

2. Cornu’s spiral:

“To construct that curve involves slightly advanced mathematics, but we can always construct it by actually drawing the arrows and measuring the angles. In any case, we get the marvelous curve (called Cornu’s spiral) shown in Fig. 30–8. (Feynman et al., 1963, p. 30–9).”

 

According to Feynman, Fig. 30–8 shows a marvelous curve that is called Cornu’s spiral. Strictly speaking, the figure does not really show a curve but a series of arrows representing the addition of amplitudes for many in-phase oscillators or antennas. Simply phrased, the arrows having the same length means that the idealized antennas (instead of a continuous line source) have the same electric field strength and are equally spaced. The slightly advanced mathematics is essentially Fresnel integrals that is also known as Euler’s identity: ò₀^u e^i(p/2)u² du = ò₀^u cos (p/2)u² du + iò₀^u sin (p/2)u² du. In other words, Cornu’s spiral is a continuous curve in the complex plane of the points Z = C(z) + iS(z) in which C(z) and S(z) are Fresnel Integrals.

 

A property of Cornu’s spiral is its curvature at any point is linearly proportional to its arc length (distance along the spiral) from the origin. The curve spirals towards a point relatively quickly because the diffraction pattern is mainly due to a small region of effective sources. In essence, the intensity of diffraction pattern is mainly contributed by light rays of the shortest and shorter paths. We can find similar spirals for phenomena including reflection and refraction. Similarly, in his lecture on QED, Feynman (1985) wrote: “[b]elow the graph is the direction of each arrow, and at the bottom is the result of adding all the arrows. It is evident that the major contribution to the final arrow’s length is made by arrows E though I, whose directions are nearly the same because the time of their paths is nearly the same (p. 43).”

 

3. Geometrical shadow edge:

“The intensity near the edge of a shadow. The geometrical shadow edge is at x₀ (Feynman et al., 1963, p. 30–9).”

 

Perhaps Feynman could have clarified the meaning of geometrical shadow edge or geometrical shadow. For instance, the geometrical shadow may be explained as the idealized shadow that would have been seen, assuming there are no diffraction effects. That is, the diffraction of light due to a semi-infinite opaque screen causes the edge of geometrical shadow to be fuzzy and thus physicists define the geometrical shadow edge at x₀. Better still, the geometrical shadow edge could be defined as the boundary between the illuminated region and shadow region. Additionally, the intensity at x₀ is ¼I₀ whereby I₀ is the unobstructed intensity if there is no semi-infinite opaque screen. Perhaps Fig. 30–9 could be modified as shown below. 

Some may ponder whether it is appropriate to say “the intensity is ¼ of the incident light.” This is related to the explanation “actually there are no sources at the holes, in fact that is the only place that there are certainly no sources,” but it is good to stress that I₀ is not the maximum intensity. Theoretically, the intensity at the geometrical shadow edge (x₀) is ¼I₀ implies that I₀ would be the intensity of a plane wave that is parallel to the semi-infinite opaque screen. In other words, the intensity of the incident light is I₀ at all points on the same wavefront and the intensity at the screen or anywhere would remain I₀ if there is no obstacle. Alternatively, we can deduce the intensity at x₀ to be ¼I₀ without using Cornu’s spiral by simply explaining the amplitude is halved because of the semi-infinite opaque screen. However, the plane wave does not exist in the real world and “actually there are no sources at the holes.” (It should also be a plane wave because "we have light coming in from infinity" as mentioned by Feynman.) 

 

Review Questions:

1. Would you explain that there are effective sources or no sources at the holes?

2. Would you say that Fig. 30–8 shows a marvelous curve that is called Cornu’s spiral?

3. How would you define the geometrical shadow edge?

 

The moral of the lesson: The intensity of diffraction pattern is dependent mainly on light rays that traveled by the shortest path to the screen followed by those traveled by slightly shorter paths.

 

References:

1. Feynman, R. P. (1942/2005). Feynman’s thesis: A New Approach to Quantum Theory. Singapore: World Scientific.

2. Feynman, R. P. (1985). QED: The strange theory of light and matter. Princeton: Princeton University Press.

3. 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 30–5 Colored films; crystals

(Single surface reflection / Thin-film interference / Three-dimensional grating)

 

The three interesting concepts discussed in this section are a reflection of a light wave at a surface of a material, thin-film interference (due to front reflection and back reflection), and three-dimensional grating (reflection at atoms of crystals).

 

1. Single surface reflection:

“…when a light wave hits a surface of a material with an index n, let us say at normal incidence, some of the light is reflected. The reason for the reflection we are not in a position to understand right now; we shall discuss it later (Feynman et al., 1963, p. 30–7).”

 

Feynman says that we are not in a position to understand the reflection of a light wave right now. In Volume II, he adds that “the amplitude of a surface reflection is not a property of the material, as is the index of refraction. It is a ‘surface property,’ one that depends precisely on how the surface is made (Feynman et al., 1964, Chapter 33 Reflection from surfaces).” In short, the amount of light reflected by the surface is dependent on the smoothness of the surface or the arrangement of atoms in an object. Essentially, the free electrons of the atoms oscillate in response to the incident light waves and they may cause the reflected light waves to be either strong or weak. In other words, the reflection at the boundary between two media is a process of scattering and interference of electromagnetic waves.

 

“But there are a number of other examples, and even though we do not understand the fundamental mechanism yet, we will someday, and we can understand even now how the interference occurs (Feynman et al., 1963, p. 30–7).”

 

Feynman mentions that we do not understand the fundamental mechanism of reflection and we can understand how the interference occurs. Interestingly, physicists have argued whether two photons can be said to interfere with each other (Glauber, 1995). More important, some may prefer this explanation of reflection: “[w]hen I talk about the partial reflection of light by glass, I am going to pretend that the light is reflected by only the surface of the glass. In reality, a piece of glass is a terrible monster of complexity - huge numbers of electrons are jiggling about. When a photon comes down, it interacts with electrons throughout the glass, not just on the surface. The photon and electrons do some kind of dance, the net result of which is the same as if the photon hit only the surface (Feynman, 1985, p. 16-17).” That is, the fundamental mechanism of reflection can be explained by light waves or photons.

 

2. Thin-film interference:

“Then, if we look at the reflection of a light source in a thin film, we see the sum of two waves; if the thicknesses are small enough, these two waves will produce an interference, either constructive or destructive, depending on the signs of the phases (Feynman et al., 1963, p. 30–7).”

 

According to Feynman, if we look at the reflection of a light source in a thin film, we see an interference pattern that depends on the signs of the phases, provided the thickness of the thin film is small enough. On the other hand, there is also a strong reflection even if the “thin-film” is not small enough (e.g., single crystal x-ray diffraction). To be specific, one may clarify that the interference is observable provided the thickness of the thin film is of the order of about ¼ to 10 wavelengths of visible light. In addition, this is an interference of reflected waves at the front surface and back surface of the thin film. Thus, we may define thin-film interference as an interference of light waves that occurs when light interacts with the front and back surface of a thin film of material.

 

“So we see colors when we look at thin films and the colors change if we look at different angles, because we can appreciate that the timings are different at different angles. Feynman et al., 1963, p. 30–8).”

 

Feynman explains that we can see colors change at thin films because we can appreciate that the timings are different at different angles. Some may be surprised that his explanation is in terms of different timings instead of path difference, and he did not provide a formula. However, the principle can be based on the optical path difference due to the front reflection and back reflection of a thin film (d = nl/4 for constructive inference). In a lecture on QED, Feynman (1985) elaborates that “[t]he ‘front reflection’ arrow is drawn opposite to that of the stopwatch hand when it stops turning… The ‘back reflection’ arrow is drawn in the same direction as the stopwatch hand (pp. 28-29).” One may add that the front reflection is drawn opposite to the stopwatch hand because of a phase shift of 180 degrees when light waves move from a low refractive index medium to a high refractive index medium.

 

3. Three-dimensional grating:

“This principle is used to discover the positions of the atoms in a crystal. The only complication is that a crystal is three-dimensional; it is a repeating three-dimensional array of atoms (Feynman et al., 1963, p. 30–8).”

 

Feynman discusses the principle for determining the positions of atoms: Based on the difference in intensity of the various images, we could find out the shape of the grating scratches, whether the grating was made of wires, sawtooth notches, and so on. Currently, some physicists prefer to use the term condition and state Bragg’s two conditions (or Bragg’s law) and Laue’s condition. Bragg’s first condition is about the regular reflection of x-rays whereby the angle of incidence equals to angle of scattering, whereas the second condition requires the path difference between two scattered waves equals to an integer number of wavelengths (2d sin q = nl). On the other hand, Laue’s condition is a relation of an incident wave and scattering wave from the crystal to the reciprocal lattice vector.

 

According to Feynman, we must use radiation of a very short wavelength, i.e., x-rays, whose wavelength is less than the space between the atoms such that there are diffraction patterns. In a sense, this is not correct because we can use electrons and neutrons instead of x-rays. Furthermore, one should elaborate that glass is an amorphous (non-crystalline) solid in which the atoms are not in regular arrangement (definite lattice pattern). More important, the symmetry (e.g., fourfold symmetry) in the diffraction pattern corresponds to the symmetrical axis or periodicity of atoms. However, the object need not be a crystal, e.g., the “cross shape” diffraction pattern indicates a helical arrangement of DNA.

As a suggestion, you may want to read his lecture on QED: “I can’t resist telling you about a grating that Nature has made: salt crystals are sodium and chlorine atoms packed in a regular pattern. Their alternating pattern, like our grooved surface, acts like a grating when light of the right color (X-rays, in this case) shines on it. By finding the specific locations where a detector picks up a lot of this special reflection (called diffraction), one can determine exactly how far apart the grooves are, and thus how far apart the atoms are (see Fig. 28). It is a beautiful way of determining the structure of all kinds of crystals as well as confirming that X-rays are the same thing as light. Such experiments were first done in 1914. It is very exciting to see, in detail, for the first time how the atoms are packed together in different substances… (Feynman, 1985, pp. 48-49).”

 

Review Questions:

1. How would you explain the reflection of light waves at the boundary between two media?

2. Would you explain that colors change at a thin film if we look at different angles because the timings are different at different angles (or path differences)?

3. How would you state the principle for determining the positions of atoms in a crystal?

 

The moral of the lesson: thin-film interference and the diffraction of light in a crystal are related to the reflection of light waves at the surfaces of a material or atoms in the crystal.

 

References:

1. Feynman, R. P. (1985). QED: The strange theory of light and matter. Princeton: Princeton University Press.

2. 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.

3. Glauber, R. J. (1995). Dirac’s Famous Dictum on Interference: One Photon or Two?. American Journal of Physics, 63(1), 12.